T H`ESE - ENS Cachan

N ◦ d’Ordre : 3264T H È S EPrésentéeDEVANT L’UNIVERSITÉ DE RENNES Ipour obtenirle grade de DOCTEUR DE L’UNIVERSITÉ DE RENNES IMention Mathématiques et ApplicationsparEric GAUTIERInstitut de Recherche Mathématique de RennesEcole Normale Supérieure de Cachan, Antenne de Bretagne&Centre de Recherche en Economie et Statistique - INSEEÉcole Doctorale MATISSEU.F.R. de MathématiquesTITRE DE LA THÈSE :Grandes déviations pour des équations de Schrödinger non linéairesstochastiques et applicationsSoutenue le 09 décembre 2005 devant la Commission d’ExamenCOMPOSITION DU JURY :Mme Anne DE BOUARD ExaminatriceM François COQUET Examinateur, membre invitéM Arnaud DEBUSSCHE Directeur de thèseM Josselin GARNIER RapporteurM Ying HU ExaminateurMme Sylvie MÉLÉARD ExaminatriceMme Marta SANZ-SOLÉ Rapportrice

RemerciementsDans un premier temps, je souhaiterais exprimer ma profonde gratitudeà mon directeur de thèse Arnaud Debussche. Merci de m’avoir fait confianceet de m’avoir fourni un sujet que j’ai trouvé tout à fait passionnant, à lafrontière entre les probabilités, les équations aux dérivées partielles et laphysique. J’ai le sentiment d’avoir appris beaucoup de belles mathématiques.Je le remercie aussi pour les précieuses qualités scientifiques et pédagogiquesde son encadrement et pour sa grande disponibilité.Je remercie très sincèrement Marta Sanz-Solé et Josselin Garnier d’avoiraccepté de rapporter cette thèse, pour leur lecture attentive de ce manuscritet pour leurs commentaires.Je tiens aussi à exprimer ma gratitude à Sylvie Méléard, François Coquetet Ying Hu, qui par ailleurs dirige l’équipe Processus Stochastiques del’IRMAR où je suis rattaché, de s’être intéressés à ce travail et d’avoir acceptéd’être membres de mon jury. Je suis aussi très reconnaissant à Anne deBouard des discussions que nous avons eues et de m’avoir invité au congrèsde la SMAI qui m’a permis de discuter des applications physiques. Je laremercie également de faire partie de ce jury.J’adresse de très sincères remerciements à Michel Pierre pour m’avoiraccueilli au plateau de mathématiques de l’antenne de Bretagne de l’ENSCachan où il est toujours très agréable de venir travailler. Je le remercieaussi pour sa gentillesse et ses encouragements.Ma gratitude va aussi à Christian Robert qui a bien voulu m’accueillirdans son équipe de Statistique au CREST et pour plusieurs discussionsfructueuses sur des travaux encore en cours lors de la rédaction de cettethèse.Je tiens à remercier les personnes de l’INSEE qui m’ont permis de menercette thèse conjointement à mon travail quotidien à l’Unité de MéthodologieStatistique et tout particulièrement Pascal Ardilly, Daniel Verger et MarcChristine. Je salue au passage mes collègues et amis de l’INSEE.Toute ma reconnaissance va également à mes différents professeurs demathématiques à l’antenne de Bretagne de l’ENS Cachan, aux universitésde Rennes 1, d’Orsay et d’avant. Ils ont su me transmettre leur goût pour

les mathématiques et leurs applications qui a suscité mon engagement danscette thèse.Je tiens également à exprimer ma grande sympathie envers mes collègueset amis de Rennes et Paris : Pierre Alquier, Patrice Bertail, Patrice Boissolles,Virginie Bonnaillie-Noël, Gabriel Caloz, Michel Delecroix, Paul Doukhan,Christophe Dupont, Emmanuelle Gautherat, Ghislaine Gayraud, HugoHarari-Kermadec, Cyril Joutard, Nicolas Landais, Jimy Lamboley, BenoîtMerlet, Mounir Rihani, Judith Rousseau, Mathieu Rosenbaum, Cyril Odasso,Karel Pravda-Starov, Rozenn Texier-Picard, Sandrine Toldo, Grégory Vial,Julien Vovelle, Jean-Michel Zakoian, mais aussi les membres du laboratoirede Processus Stochastiques.Je souhaite par ailleurs remercier les équipes administratives toujourstrès disponibles.J’ai aussi une pensée pour les baillacois, les voileux, Matthieu, Guéno,Florian et Julie, Amaury, Yassine et Natacha, Céline, Joan et Louba, Eric,Solen et Damien, Etienne, Karine et Joaquín, Vincent et Catherine, Romain,Alexandre M, Vincent et Teresa, Didier et Fabiana, Alexandre P, Emilie etAntonin et tous mes autres amis.Enfin, j’ai une pensée toute particulière pour mes parents, Alain etBérénice et les autres membres de ma famille qui m’ont beaucoup soutenu.Je suis aussi très reconnaissant à Anne d’avoir partagé ce travail à mes côtéset de m’avoir toujours encouragé. Je remercie également Chantal, Claude etDamien pour leur soutien.

Table des matières1 Introduction 71.1 Éléments sur les équations déterministes . . . . . . . . . . . . 71.1.1 Propriétés du groupe linéaire . . . . . . . . . . . . . . 81.1.2 L’équation non linéaire, caractère localement bien posédu problème de Cauchy . . . . . . . . . . . . . . . . . 101.1.3 Invariants de l’équation, existence globale et explosionen temps fini . . . . . . . . . . . . . . . . . . . . . . . 111.1.4 Les ondes solitaires . . . . . . . . . . . . . . . . . . . . 131.2 Les perturbations aléatoires . . . . . . . . . . . . . . . . . . . 161.2.1 Des motivations . . . . . . . . . . . . . . . . . . . . . 161.2.2 Le bruit, le processus de Wiener et les mesures Gaussiennes. . . . . . . . . . . . . . . . . . . . . . . . . . . 201.2.3 Les équations de Schrödinger non linéaires stochastiques 251.3 Les grandes déviations . . . . . . . . . . . . . . . . . . . . . . 271.3.1 Présentation . . . . . . . . . . . . . . . . . . . . . . . 271.3.2 Des résultats généraux . . . . . . . . . . . . . . . . . . 291.3.3 Transport par image directe de PGDs . . . . . . . . . 302 Présentation des résultats 332.1 Grandes déviations et support pour un bruit additif . . . . . 332.2 Grandes déviations uniformes pour un bruit multiplicatif . . . 392.3 Application à la transmission par solitons . . . . . . . . . . . 422.4 Application à la sortie d’un domaine d’attraction . . . . . . . 472.5 Le cas d’un bruit additif fractionnaire . . . . . . . . . . . . . 513 Conclusion et perspectives 57A Large deviations in the additive case and applications 61A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613

A.2 Notations and preliminary results . . . . . . . . . . . . . . . . 64A.2.1 Properties of the group . . . . . . . . . . . . . . . . . 66A.2.2 Topology and trajectory spaces . . . . . . . . . . . . . 66A.2.3 Statistical properties of the noise . . . . . . . . . . . . 69A.2.4 The random perturbation . . . . . . . . . . . . . . . . 70A.2.5 Continuity with respect to the perturbation . . . . . . 73A.3 Sample path large deviations . . . . . . . . . . . . . . . . . . 75A.4 The support of the law of the solution . . . . . . . . . . . . . 78A.5 Applications to the blow-up times . . . . . . . . . . . . . . . 79A.5.1 Probability of blow-up after time T . . . . . . . . . . . 80A.5.2 Probability of blow-up before time T . . . . . . . . . . 82A.5.3 Bounds for the approximate blow-up time . . . . . . . 82A.6 Applications to nonlinear optics . . . . . . . . . . . . . . . . . 84A.6.1 Upper bounds . . . . . . . . . . . . . . . . . . . . . . . 88A.6.2 Lower bounds . . . . . . . . . . . . . . . . . . . . . . . 89B Large deviations in the multiplicative case 95B.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95B.2 Preliminaries and statement of the result . . . . . . . . . . . 97B.3 Continuity of the skeleton with respect to the control on thesets C a and exponential tail estimates . . . . . . . . . . . . . 101B.4 Almost continuity of the Itô map . . . . . . . . . . . . . . . . 109B.5 End of the proof of the uniform LDP . . . . . . . . . . . . . . 114B.6 Applications to the blow-up times . . . . . . . . . . . . . . . 116C Application to the timing jitter 119C.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119C.2 Notations and preliminaries . . . . . . . . . . . . . . . . . . . 124C.3 Tails of the the mass and center with additive noise . . . . . 132C.4 Tails of the center in the multiplicative case . . . . . . . . . . 142C.5 Annex - proof of Theorem C.2.1 . . . . . . . . . . . . . . . . 146D Exit from a domain for weakly damped equations 149D.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149D.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 152D.3 Exit from a domain of attraction in L 2 . . . . . . . . . . . . . 156D.4 Exit from a domain of attraction in H 1 . . . . . . . . . . . . . 170D.5 Annex - proof of Theorem D.2.1 . . . . . . . . . . . . . . . . 180

E The case of a fractional additive noise 185E.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185E.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 187E.3 Local well posedness and necessary results . . . . . . . . . . . 192E.4 The main results . . . . . . . . . . . . . . . . . . . . . . . . . 197

Chapitre 1Introduction1.1 Éléments sur les équations déterministesLes équations de Schrödinger non linéaires (NLS) apparaissent dans denombreux modèles de la physique. Elles interviennent pour décrire l’évolutiond’enveloppes de trains d’ondes monochromatiques de la formeɛΨ exp (ik · x − ωt) ,où k est le vecteur d’onde, dans un milieu faiblement non linéaire et dispersif.L’amplitude est supposée petite, i.e. ɛ

Introductionpeuvent s’obtenir à partir d’équations des ondes non linéaires en utilisantnotamment un développement autour de l’onde monochromatique et uneanalyse multi-échelle. On trouve aussi ces équations en mécanique quantiqueen localisant le potentiel d’intéraction V de l’équation de Hartreei ∂u∂t + 1 2 ∆u + ( V ∗ |u| 2) u = 0ou en physique des solides pour certaines structures moléculaires comme approximationcontinue de systèmes de molécules ou d’atomes en intéractiondisposés sur un réseau (chaînes anharmoniques d’atomes, agrégats moléculairesde Scheibe...).D’une manière générale, ces équations s’écrivent sous la formei ∂u∂t = div (D∇u) + f(x, |u|2 )u.Nous nous restreindrons au cas où la matrice D vaut l’identité, l’opérateurnon borné est alors le Laplacien, et au cas où la non linéarité est de la formef(x, |u| 2 )u = λ|u| 2σ u où λ = ±1. Nous considérons l’équation dans toutl’espace R d .1.1.1 Propriétés du groupe linéaireConsidérons ici l’équation linéaireappelée aussi équation libre.∂u∂t = −i∆uIntroduisons les espaces de Hilbert de fonctions à valeurs complexes L 2 ouH s , s ≥ 0. L’espace de Lebesgue L 2 des fonctions de carré intégrable est munidu produit scalaire défini pour u et v dans L 2 par (u, v) L 2 = Re ∫ R d u(x)v(x).L’espace de Sobolev H s est l’espace des distributions tempérées u dont latransformée de Fourier û satisfait (1 + |ξ| 2 ) s/2 û ∈ L 2 . L’opérateur −i∆u estun opérateur non borné sur les espaces H s . Il est défini sur un domaine, parexemple D(−i∆u) = H 2 dans le cas d’un opérateur dans L 2 . Les espaces L psont les espaces de Lebesgue standard et les espaces W k,p sont les espacesde Sobolev de fonctions de L p ayant des dérivées partielles jusque l’ordre kdans L p .L’opérateur étant anti auto-adjoint, il est, d’après le théorème de Stone,le générateur infinitésimal d’un groupe unitaire (U(t)) t∈Rfortement continu.8

IntroductionLes opérateurs U(t) sur l’espace considéré, L 2 ou H 1 , sont des isométries. Cegroupe a une forme explicite pour des fonctions de la classe de Schwartz, ils’agit de la convolution par le noyau)1K(t, x) = exp(− i|x|2 ,(4iπt) d 2 4tc’est à dire∀t ≠ 0, ∀u 0 ∈ S(R d ), U(t)u 0 = K(t, ·) ∗ u 0Le groupe ne possède pas de propriété de régularisation globale. Il admetpar contre des propriétés de régularisation locale.Le groupe possède des propriétés d’intégrabilité que nous utiliserons trèsfréquemment. Tout d’abord, nous avons des estimées de décroissance :∀p ≥ 2, ∀t ≠ 0, ∀u 0 ∈ L p′ (R d ), ‖U(t)u 0 ‖ L p (R d ) ≤ (4π|t|) −d 12 − 1 pNous observons qu’en particulier nous avons, pour p > 2,lim ‖U(t)u 0‖ Lt→±∞ p (R d ) = 0.‖u 0 ‖ L p ′ (R d ) .Ainsi, bien que le groupe sur L 2 soit une isométrie, nous avons la décroissancevers 0 de la norme L ∞ et de toutes les normes L p entre 2 et ∞. Une solutionmême localisée au départ ”s’étale” au cours du temps. Il s’agit d’une propriétéde dispersion. Nous pouvons aussi exprimer cette propriété autrement.Considérons une onde plane monochromatique u(t, x) = exp (ik · x − iωt) etinjectons cette fonction dans l’équation linéaire. Nous obtenons la relationde dispersion réelle ω = |k| 2 et nous avons detD k ω ≠ 0. Dans ce cas, lesondes de vecteurs d’ondes différents se propagent à des vitesses différentes.Il en résulte un étalement du paquet d’ondes.Rappelons désormais les inégalités de Strichartz.Une paire (r(p), p) est dite admissible si 2 ≤ p < 2dd−2lorsque d > 2 (2 ≤ p

Introduction(i)il existe C positif tel que pour u 0 dans L 2 , T positif et (r(p), p) paireadmissible,‖U(·)u 0 ‖ L r(p) (0,T ;L p ) ≤ C ‖u 0‖ L 2.La seconde inégalité de Strichartz donne des propriétés d’intégrabilité de la”convolution” par le groupe linéaire. Cette convolution apparaît lorsque nousconsidérons l’équation non linéaire, l’équation non linéaire stochastique, ouune équation non homogène, comme perturbations de l’équation linéaire ;c’est ce que nous faisons lorsque nous étudions des solutions mild. Ces solutionsd’équations aux dérivées partielles (EDP) s’expriment sous une formeanalogue à celle donnée par la méthode de la variation de la constante pourles équations différentielles ordinaires (EDO), dans ce cas on parle aussi deforme de Duhamel. La deuxième inégalité de Strichartz s’écrit(ii)Pour tout T positif, (r(p), p) et (r(q), q) des paires admissibles,s et ρ tels que 1 s + 1r(q) = 1 et 1 ρ + 1 q= 1, il existe C positif tel quepour f dans L s (0, T ; L ρ ),∥ ∫ ·0 U(· − s)f(s)ds∥ ∥L r(p) (0,T ;L p ) ≤ C‖f‖ L s (0,T ;L ρ ).Notons que les exposants p et q peuvent être choisis de manière complètementindépendante, ceci résultant d’un argument de dualité.1.1.2 L’équation non linéaire, caractère localement bien posédu problème de CauchyL’équation non linéaire est traitée comme une perturbation de l’équationlinéaire. Nous nous intéressons à des solutions faibles ou de manière équivalenteà des solutions mild. Celles-ci satisfont pour t positifu(t) = U(t)u 0 − iλ∫ t0U(t − s)|u(s)| 2σ u(s)dsoù u 0 est la donnée initiale que nous prenons dans L 2 ou dans H 1 pourdes valeurs de σ convenables. Dans le cas d’une équation dans H 1 avecdonnée initiale dans H 1 et une dimension d’espace, on peut vérifier, grâceaux inégalités de Hölder et de Gagliardo-Nirenberg, que la non linéarité estLipschitzienne sur les bornés de H 1 quelque soit σ. Sinon, la non linéaritén’est pas localement Lipschitzienne et l’inégalité de Strichartz (ii) nous permetde donner un sens à la ”convolution” avec le groupe. On procède dans10

Introductionle moment angulaireM (u(t)) = 2Re( ∫ () )i x ∧ u(t, x)∇u(t, x) dx .R dLe flot de l’EDP est Hamiltonien, on peut réécrire l’équation sous la formeoù dH(u) · h =∂u∂t(δH(u)δu , h ).= iδH(u)δu=δH (u)δuCitons également les deux relations qui suivent.Le centre de masse, ou position, défini par la relation Y (u(t)) = ∫ Rx|u(t, x)| 2 dxdsatisfait l’équationdY (u(t))= P (u(t)) .dtLe centre de masse est aussi souvent défini en divisant la quantité précédentepar la masse totale en t, rappelons que celle-ci est ici constante.La variance, définie par V (u(t)) = ∫ R|x| 2 |u(t, x)| 2 dx, satisfait la relationdappelée identité de la variance, ou théorème du viriel,d 2 V (u(t))dt 2 = 8H (u(t)) − 4 dσ − 2 ∫σ + 1 q |u(t, x)| 2σ+2 dx.R dL’invariance de la masse entraîne que les solutions existent globalement dansL 2 . Dans le cas H 1 , on obtient, grâce à l’invariance de la masse et de l’Hamiltonien,que le problème de Cauchy est bien globalement posé dès que σ < 2 d ,cas d’une non linéarité sous-critique. Lorsque la non linéarité est critique ousur-critique ,σ = 2 d et 2 d < σ < 2d−2, le problème est globalement bien posélorsque λ = −1, cas de la non linéarité répulsive ou défocalisante. Si λ = 1 et2d ≤ σ < 2d−2 , si d ≥ 3, ou si λ = 1 et 2 d≤ σ, si d = 1 ou d = 2, les solutionspeuvent exploser en temps fini. Cette propriété correspond en physique àun transfert énergétique violent des grandes aux petites échelles. Le premierrésultat rigoureux est dû à Glassey exploite l’identité de la variance, voir[87]. Plusieurs résultats existent, citons en particulierTheorem 1.1.1 Pour une donnée initiale u 0 ∈ H 1 (R d ) telle que V (u 0 )

Introductionsi l’une des conditions suivantes est satisfaite :(i) H (u 0 ) < 0,(ii) H (u 0 ) = 0 et Im ∫ R d x.u 0 (x)∇u 0 (x)dx < 0,(iii) H (u 0 ) > 0 et Im ∫ R d x.u 0 (x)∇u 0 (x)dx ≤ −4 √ H (u 0 )‖xu 0 ‖ L 2 (R d ).Theorem 1.1.2 Si d = 1 et σ = 2, toute condition initiale de H 1 (R d ) telleque H (u 0 ) < 0 explose en temps fini.Si d ≥ 2 et 2 d ≤ σ ≤ 2d−2 ∧ 2, toute condition initiale de H1 (R d ) à symétrieradiale, i.e. ne dépendant que de |x|, telle que H (u 0 ) < 0, explose en tempsfini.Ce deuxième résultat dû à Ogawa et Tsutsumi présente l’intérêt de s’affranchirde la restriction à des données initiales de variance finie. Il existeégalement un certains nombre de résultats fins, voir [136], sur l’explosion :taux d’explosion, auto-similarité et concentration de la masse en dimensioncritique...1.1.4 Les ondes solitairesUn équilibre entre effet non linéaire et dispersion peut entraîner l’existencede solutions que l’on appelle ondes solitaires ou, dans certains cas,solitons. Ce phénomène a été découvert empiriquement par l’ingénieur JohnScott Russel en 1834. Se promenant sur le bord d’un canal, il vit se déplacerune onde qui s’était formée à la proue d’un bateau tiré par des chevaux.Cette onde se mit à cheminer seule, alors que le bateau s’était arrêté, surune longue distance, sans que sa forme ou sa vitesse ne s’altère. L’EDPdécrivant la propagation de vagues de grande longueur d’onde et de petiteamplitude A à la surface d’un canal de faible profondeur h 0 est l’équation deKorteweg-de Vries. Elle peut s’obtenir à partir des équations d’Euler. Ellepossède des solutions de type ondes progressives de la forme ϕ(x − ct). Dansle cas d’une dimension en espace, la fonction ϕ s’écrit(√ )ϕ(z) = Asech 2 3Az ,la vitesse est donnée par c = √ gh 0(1 + A2h 0). Rappelons que la fonctionsécante hyperbolique s’exprime sech(z) = 1cosh(z). La solution est donc localisée.Les ondes solitaires interviennent désormais dans plusieurs branches dela physique : en optique, en physique des plasmas, en astrophysique, en4h 3 013

Introductionhydrodynamique... La vague de mascaret qui remonte le long de certainsfleuves est en général suivie d’un train de solitons. Des ondes solitaires sepropagent aussi sur la couche thermocline de l’océan et sont engendréespar la topographie du sol. Des ondes solitaires atmosphériques existent et semanifestent sous la forme du nuage ”Morning Glory” en Australie. Celui-ci sedéveloppe en présence d’une inversion de température et peut être engendrépar une activité orageuse ou une collision entre des fronts de brise océaniqueopposés. Les solitons ont aussi été introduits en physique des solides (pourcertaines chaînes d’atomes, certains des cristaux) et en biologie (cinétiquedes réactions biologiques, dénaturation thermique de l’ADN).Des ondes solitaires existent pour l’équation de Schrödinger non linéairelorsque la non linéarité est attractive ou focalisante, i.e. le cas où λ = 1.Ce ne sont plus cette fois des solutions onde progressive mais plutôt desétats stationnaires, c’est à dire des solutions de la forme exp(iωt)ϕ(x) où ωest strictement positif. La terminologie stationnaire vient de la mécaniquequantique. En effet, l’intensité de l’onde est invariante par translation temporellece qui correspond à la probabilité de trouver la particule à un endroitparticulier.Le profil ϕ satisfait l’EDP∆ϕ − ωϕ + |ϕ| 2σ ϕ = 0, (1.1.1)la condition ω > 0 assure que lim |x|→∞ ϕ(x) = 0 c’est à dire que les solutionssont localisées. Cette équation admet une unique solution positive lorsquela dimension d’espace vaut d = 1, elle est donnée parϕ(z) = (ω(σ + 1)) 12σ sech1σ(√ ωσz).Les fonctions ϕ sont aussi les points critiques dans H 1 de la fonctionnelle deLyapunov appelée aussi fonctionnelle d’actionEn effet, l’EDP peut se réécrireS(u) = 1 {H(u) + ωN(u)} .2δS(u)δu = 0.Nous verrons, c.f. annexe D, que le problème d’optimisation dual intervientlorsque l’on cherche à caractériser le point de sortie d’un niveau d’énergiepour des équations de Schrödinger non linéaires avec un petit bruit additifet un amortissement faible.14

IntroductionDu point de vue de l’optique, Hasegawa et Tappert ont proposé en1973 d’utiliser les solitons de l’équation de Schrödinger non linéaire pourcoder des bits et transmettre sur de très longues distances des données àtrès haut débit. Les solitons de l’équation de Schrödinger non linéaires permettentaussi de rendre compte des transferts d’énergie dans les structuresmoléculaires, par exemple les protéines, les molécules d’ADN ou la photosynthèse.En effet, dans le cas d’un potentiel d’intéraction entre plus prochesvoisins avec un terme harmonique et un terme d’ordre 4, en postulant desoscillations internes et des solutions solitons, les physiciens obtiennent quela limite continue de l’équation de l’enveloppe du soliton est l’équation deSchrödinger non linéaire. Enfin, il semblerait également que les solitons del’équation de Schrödinger non linéaire modélisent les vagues scélérates à lasurface de l’océan.1.2 Les perturbations aléatoiresNous allons nous intéresser à des perturbations aléatoires des équationsprécédentes. Un terme aléatoire supplémentaire est ajouté, il peut s’agird’une force, terme supplémentaire de type ˙η(t, x), ou d’un potentiel, termede type u ˙η(t, x). Nous parlons de bruit additif ou multiplicatif. Il est paramétrépar le temps et les variables d’espaces. Les équations bruitées quenous considérons sont motivées par la physique. En optique ou en physiquedes solides (cristaux, chaînes d’atomes de type Fermi Pasta Ulam, agrégatsmoléculaires de Scheibe) le bruit peut être additif ou multiplicatif.1.2.1 Des motivationsDans de nombreuses modélisations le bruit est introduit afin de rendrecompte du fait que l’on ne peut pas décrire parfaitement le système à partirdes grandeurs à disposition. Ceci est le cas par exemple en économieoù une partie des grandeurs est inobservable. Les systèmes évoluant defaçon déterministe au court du temps, et pour un temps ”continu”, sontsouvent décrits par une EDO ou une EDP si l’évolution temporelle estliée à des fluctuations spatiales. On peut alors rendre compte de l’incertitudesur l’état du système, par exemple l’évolution d’une particule enprésence de fluctuation de la température ou la valeur d’un produit financier,par un terme de bruit dans l’équation. Nous sommes alors en présenced’équations différentielles stochastiques (EDS) ou aux dérivées partielles stochastiques(EDPS). Prenons le cas des équations aux dérivées partielles detype équations d’évolutions. Ces équations sont souvent valables sur une16

Introductioncertaine plage de valeurs des paramètres. Elles décrivent l’évolution dansdes milieux idéalisés et par exemple ne tiennent pas compte des impuretéset des fluctuations des propriétés du milieu notamment en fonction des fluctuationsde la température. Elles correspondent aussi à des approximationsde modèles plus complexes et négligent des termes d’ordres plus élevés. Leterme aléatoire peut alors rendre compte des termes que l’on a négligés et/oudes fluctuations du milieu. De ce point de vue, il est intéressant d’évaluerla robustesse des résultats qualitatifs obtenus pour ces modèles idéalisés enprésence d’une petite perturbation. Nous allons voir par la suite que le bruitpeut aussi être tout à fait intrinsèque.Le système peut aussi être excité par des forces extérieures que l’on n’arrivepas à décrire. Il peut s’agir, par exemple, d’une impulsion électrique surun neurone, voir [139]. Celle ci arrive à des intervalles de temps irréguliers età divers endroits du neurone. En optique des fibres l’équation de Schrödingernon linéaire apparaît comme modèle. Un signal permet de coder un 1,l’absence de signal un 0. Un tel signal est en fait amorti, ceci empêche la propagationsur de longues distances, typiquement plus de 1000 km. Plusieurstypes d’amplifications permettent de palier à ce problème. Considérons dansun premier temps les amplificateurs par dopage à l’Erbium. Dans ce cas, deschaînes d’amplificateurs régulièrement espacés le long de la fibre permettentde compenser l’amortissement. Le bruit est alors tout à fait intrinsèque auphénomène d’amplification. Du point de vue de la mécanique quantique,la mesure d’une quantité physique se fait nécessairement avec une certaineimprécision et il ne peut pas exister de mesure optique de l’intensité lumineusesans incertitude. De ce fait montrons qu’un amplificateur qui neprésenterait pas d’émission spontanée de bruit violerait le principe d’incertituded’Heisenberg. Celui-ci stipule qu’il existe une limite fondamentale àla mesure simultanée du moment p et de la position x d’une particule. Lesincertitudes étant notées ∆p et ∆x satisfont∆p∆x ≥ 2où = h2πest la constante de Planck. Dans notre cas, les particules sontdes photons. En supposant qu’ils se propagent le long de l’axe des x, nousobtenons alors la borne suivante sur l’incertitude de la mesure conjointe del’énergie E du photon et de son temps d’arrivée t∆E∆t ≥ 2 .17

IntroductionEn effet, nous avons p = k = ω c= E c, k étant le nombre d’onde et cla célérité de la lumière, et ∆t = ∆xc. Supposons alors que l’incertitudesur l’énergie soit liée à l’incertitude sur le nombre de photons, la fréquenceν étant elle parfaitement déterminée. L’incertitude sur l’énergie vaut alors∆E = hν∆n et celle sur la phase ∆ϕ = 2πν∆t. Nous obtenons alors laborne d’incertitude nombre-phase∆n∆ϕ ≥ 1 2 .Supposons qu’un amplificateur sans émission spontanée de bruit existe etqu’il soit de gain G. Soit n 0 ± ∆n 0 le nombre de photons incidents, ϕ 0 ±∆ϕ 0 la phase du signal d’entrée, n ± ∆n = G (n 0 ± ∆n 0 ) le nombre dephotons à la sortie, et ϕ ± ∆ϕ = ϕ 0 ± ∆ϕ 0 + θ la phase du signal de sortie ;la translation de phase θ rend compte de la propagation du signal dansl’amplificateur. Supposons que nous mesurions le signal après l’amplificateuravec un détecteur pour lequel l’incertitude est minimale alors nous avons larelation ∆n∆ϕ = 1 2 . Cette incertitude correspond à l’incertitude ∆n 0∆ϕ 0 =12G < 1 2sur le signal incident avant amplification. Ceci contredit le principed’incertitude d’Heisenberg. On peut aussi montrer, voir [52], que l’amplitudeminimale P N du bruit qui permette de respecter le principe d’incertituded’Heisenberg à l’entrée et à la sortie de l’amplificateur est donnée parP N =hν(G − 1)2T= hνB(G − 1),T représente l’intervalle de temps sur lequel le détecteur mesure le signal etB la largeur de bande de l’amplificateur. Une description quantique du bruitest possible mais devient beaucoup plus complexe, voir en particulier [52]. Ilest par contre remarquable que, même si le bruit a des origines quantiques, iln’est pas nécessaire de rentrer précisément dans les principes de la mécaniquequantique afin de décrire l’effet du bruit sur la transmission d’un signal. Engénéral, on modélise pour ce type d’amplification le bruit comme un bruitadditif, voir par exemple [50, 51, 62, 64, 65]. Ceci est aussi le cas dans lepremier exemple du neurone.Il existe aussi d’autres types d’amplification, c.f. [52]. Citons des amplificateursparamétriques.L’amplification de Raman dans une fibre monomode a été suggérée en 1989-1992. Le processus physique qui rend possible l’amplification de Raman estl’intéraction de la lumière avec les phonons optiques. Dans ce cas, l’évolution18

Introductiondu nombre de photons n est régie par l’équation différentielleP p± dndx = g R (n + 1) − αnA effP pA effoù g R est le gain de Raman, correspond à l’intensité injectée et α à laperte de la fibre.Le mélange paramétrique de quatre ondes ou mélange stimulé de 4 photonsest étudié pour la première fois en 1974-1975. Il est appliqué en 1980-1985pour l’amplification d’ondes progressives. Ces amplificateurs sont un autretype d’amplificateurs paramétriques.Ces deux types d’amplification sont aussi accompagnés d’émission spontanéede bruit. Dans les deux cas, le bruit apparaît sous forme multiplicative, voir[59, 98] pour l’amplification de Raman et [101] pour le processus de mélangede quatre ondes. L’amplification de Raman contribue aussi à la nonlinéaritéde Kerr ; il alors figure également dans l’équation un terme de réponse deRaman supplémentaire, c.f. [98].Dans le cas des systèmes de communication optiques, prendre en comptele bruit doit permettre de quantifier la dégradation du signal depuis la sourced’émission et le récepteur à l’extrémité de la fibre. Plus particulièrement, àl’extrémité de la fibre, un récepteur converti la lumière en courant électrique.Les photons incidents sont capturés et un électron est émis, simultanémentune transition entre états liés a lieu. Le signal électrique est préamplifié puisintégré. Un circuit électronique mesure alors l’énergie sur une fenêtre quis’exprime en fonction de la période inter émission des bits et la compare àun niveau de référence. Ceci permet de décider si le signal est un 1 ou un 0.Le taux d’erreur sur les bits (BER pour bit-error rate) s’exprimeBER = P(1|0)p(0) + P(0|1)p(1)où P(1|0) est la probabilité conditionnelle que le circuit électronique décidede manière erronée qu’un 1 a été émis alors qu’un 0 était émis (on définitde manière analogue P(0|1)) ; p(0) et p(1) sont les probabilités de transmettreles symboles 0 et 1. Lorsque le nombre de bits du message est élevéon peut supposer que p(0) = p(1) = 1 2. L’obtention du BER nécessite deconnaître les probabilités conditionnelles et donc la loi de la quantité mesuréeen l’extrémité de la fibre en fonction de la donnée initiale. L’approximationGaussienne est souvent utilisée en pratique et cette pratique est critiquéepar plusieurs physiciens. Mais aussi, nous sommes intéressés par l’influencede la longueur de la fibre et des paramètres liés à l’émission du signal (amplitude,période inter émission...) sur le BER. Il est donc souhaitable que,19

Introductionc’est une martingale et un processus de Markov) et tel que pour t > s,β √t−s t−β ssuit la loi N (0, 1). La dernière propriété est une propriété de stationnarité.De manière équivalente, il s’agit d’un processus Gaussien (les marginalesfini-dimensionnelles (β t1 , ..., β tn ) sont des vecteurs Gaussiens), centré,de fonction de covariance E(B t B s ) = t ∧ s. Le mouvement Brownien est lui,contrairement au bruit blanc, corrélé en temps ; comme cela est précisé plushaut, il s’agit d’un processus de Markov. La mesure de Wiener est la mesureimage sur C([0, T ]) de la probabilité P sur l’espace probabilisé sous-jacent Ωpar l’application qui à ω ∈ Ω associe la trajectoire t ↦→ B t (ω).Considérons le bruit blanc à valeurs dans R, notons S n = ∑ ni=1 X ila marche aléatoire (S 0 = 0) et définissons la trajectoire interpolée S t =S ⌊t⌋ + (t − ⌊t⌋)X ⌊t⌋+1 . Le théorème de Donsker indique qu’à changementd’échelle et, à la renormalisation correspondant à celle du théorème de lalimite centrale près, la marche aléatoire interpolée converge en loi vers unetrajectoire ( Brownienne. ) Il donne plus précisément que la loi du processusSt n = 1σ √ n S nt , où t≥0 σ2 = E(Xi 2 ), converge étroitement vers la mesure deWiener.Par ailleurs, le mouvement Brownien définit une variable aléatoire à valeursdans l’espace de Banach C ([0, T ]) ou dans L 2 (0, T ). L’opérateur decovariance Q dans L 2 (0, T ), muni du produit scalaire usuel (·, ⋆) L 2 s’exprimeen fonction de la fonction de covariance par, étant donné ϕ et ψ dansL 2 (0, T ),(Qϕ, ψ) L 2 =∫ T0(t ∧ s)ϕ(s)ds.L’opérateur est auto-adjoint compact et nous avons, en corollaire du théorèmeusuel de diagonalisation des opérateurs auto-adjoints compacts, la décompositionde Karhunen-Loeve :il existe une famille (ξ j ) j∈Nde variables aléatoires de loi normale standardN (0, 1) indépendantes et (e j ) j∈Nune base Hilbertienne de L 2 (0, T )constituée de fonctions propres de Q associées aux valeurs propres λ j tellesqueβ(t) = ∑ j∈N√λj ξ j e j (t).Dans le cas où T = 1, nous avons)πt)β(t) = √ 2 ∑ sin (( n + 1ξ j ( )2j∈N n +1.2 π21

IntroductionBien qu’à priori non sommable, la dérivée formelle de la série ci-dessus est lasomme d’une série à coefficients aléatoires où chaque composante d’une secondebase de L 2 (0, T ) est indépendante et de loi N (0, σ 2 ). Plus précisément,le bruit Gaussien blanc en temps continu est la dérivée au sens des distributionsdu mouvement Brownien. Il s’agit ainsi, même en dimension infinie,d’un vecteur dont toutes les composantes sont indépendantes et deloi N (0, σ 2 ). L’intégrale stochastique permet de donner un sens sous formeintégrale à des EDOs perturbées par un bruit blanc.Nous considérerons dans ce qui suit des mouvements Brownien W àtrajectoires dans un espace de Hilbert ou plus généralement de Banach,appelés processus de Wiener. Dans la première définition du mouvementBrownien, il suffit de remplacer βt−βs √ t−sloi une mesure Gaussienne centrée µ.suit la loi N (0, 1) par Wt−Ws √ t−sa pourRappelons qu’une mesure de probabilité µ sur un espace de Banach réelséparable E, muni de sa tribu Borélienne, est une mesure Gaussienne sipour tout élément e ∗ du dual topologique E ∗ , la forme linéaire continue< e ∗ , · > E ∗ ,E sur E, où le crochet est celui de la dualité E ∗ − E, définit unevariable aléatoire réelle centrée. La mesure de Wiener sur C([0, T ]) est unemesure Gaussienne. Le théorème de Fernique donne en particulier l’existencede moments de tout ordre.Introduisons la notion d’espace de Hilbert noyau auto-reproduisant, voirpar exemple [23] pour une introduction. Il s’agit d’un espace d’énergie liéà la covariance, il caractérise la mesure Gaussienne centrée. Nous verronsque cet espace intervient dans la fonction de taux d’un principe de grandesdéviations pour des familles de mesures Gaussiennes et pour caractériserleurs supports. Définissons par R l’applicationR : E ∗ → Ee ∗ ↦→ R(e ∗ ) = ∫ E < e∗ , e > E ∗ ,E e µ(de)l’intégrale est une intégrale de Bochner, elle est bien définie comme unélément de E, d’après le théorème de Fernique. L’espace de Hilbert, noyauauto-reproduisant H µ de la mesure Gaussienne centrée µ sur E, est lecomplété de R(E ∗ ) pour le produit scalaire∫〈R(e ∗ ), R(f ∗ )〉 Hµ= < e ∗ , e > E ∗ ,E< f ∗ , e > E ∗ ,E µ(de)ERappelons aussi que H µ s’injecte continûment dans E (l’injection sera notéei) et que cette image est de mesure nulle pour la mesure µ. Le support d’une22

IntroductionEtelle mesure Gaussienne µ est donné par( )supp µ = H µ , c.f. [8]. Signalonsaussi que pour toute base Hilbertienne e ∗ j de j∈N H∗ µ, base duale de (e j ) j∈N,et e dans E, nous avons, d’après le théorème de convergence des martingales,limn→∞j=0n∑< e ∗ j, e > E ∗ ,E e j = e.Une telle décomposition en série permet, par exemple, de définir l’intégralestochastique par rapport à un processus de Wiener sur E à partir de l’intégralestochastique usuelle par rapport au mouvement Brownien. Enfin, nous avonsla représentation en diagrammeE ∗i∗↩→ H ∗ µR←→ H µi↩→ E.L’application R est en fait l’identification de Riesz entre le dual topologiqueH ∗ µ et H µ . Ce formalisme est celui des espaces de Wiener abstraits.Considérons le cas de la loi d’un processus Gaussien à trajectoires continuessur [0, 1]. Le dual topologique de C([0, 1]) étant l’espace des mesuressignées sur [0, 1], nous avons, d’après le théorème de Fubini, pour t appartenantà [0, 1]Re ∗ (t) = δ t (Re ∗ )= ∫ E < e∗ , e > E ∗ ,E e(t)µ(de)= ∫ ( ∫ )1E 0 e(s)e∗ (ds) e(t)µ(de)= ∫ 1(∫0 E e(s)e(t)µ(de)) e ∗ (ds)= ∫ 10 γ(s, t)e∗ (ds)où γ(s, t) est la fonction de covariance. Dans le cas particulier de la mesurede Wiener nous obtenonsetRe ∗ (t) =〈Re ∗ , Rf ∗ 〉 Hµ=∫ 10∫ t0e ∗ ([s, 1])ds(Re ∗ ) ′ (t) (Rf ∗ ) ′ (t)dt.L’espace de Hilbert noyau auto-reproduisant est l’espace de Sobolev H 1 0 (0, 1).Il est constitué de fonctions absolument continues, nulles en 0. On l’appelleaussi espace de Cameron-Martin.23

IntroductionLorsque l’espace E est un espace de Hilbert réel séparable H muni duproduit scalaire (·, ⋆) H , la mesure µ admet un opérateur de covariance Qsur H satisfaisant pour tout h et k dans H∫(Qh, k) H= (h, l) H (k, l) H µ(dl) = E [(h, ·) H (k, ·) H ] .HDans ce cas on peut montrer que l’espace de Hilbert noyau auto-reproduisantest isométrique à ImQ 1 2 muni de la structure image.Etant donné un espace de Hilbert réel séparable H, considérons la famille(µ e1 ,...,e n) des mesures Gaussiennes centrées de dimension finie et de covariancel’identité indexée par les familles orthonormales de H sur les tribuscylindriquesσ {h ∈ H : ((e 1 , h) H , ..., (e n , h) H ) ∈ B(R n )} ,où B(R n ) est la tribu Borélienne de R n . Elle définit sur la réunion des tribuscylindriques une fonction d’ensembles µ. Cette fonction est appelée mesurecylindrique. Néanmoins, la réunion des tribus cylindriques n’est pas unetribu et µ n’est pas σ−additive. Si H et H ′ sont des espaces de Hilbert et siΦ est un opérateur linéaire continu de H dans H ′ et µ une mesure cylindriquesur H alors l’image directe Φ ∗ µ est σ−additive (et donc peut être étendueen une mesure de Radon) si et seulement si Φ est Hilbert-Schmidt. LorsqueH ′ est un espace de Banach, nous pouvons introduire la notion d’opérateurγ−radonifiant, voir en particulier [17, 18].Un processus de Wiener cylindrique W c sur un espace de Hilbert réelséparable H est tel que, quelle que soit (e j ) j∈Nune base Hilbertienne de H,il existe une suite (β j ) j∈Nde mouvements Browniens indépendants telle queW c = ∑ j∈N β je j . Ce processus n’est pas bien défini, en particulier ses marginales,W c (t) pour t positif, n’admettent pas de moment d’ordre 2. Par contreson image directe par un opérateur Φ Hilbert-Schmidt ou γ−radonifiantest bien définie. Si Φ est à valeurs un espace de Hilbert, la covariance duprocessus image vaut Q = ΦΦ ∗ . D’après ce qui précède, un processus deWiener W de covariance Q sur un espace de Hilbert réel séparable H estl’image directe, via l’injection Hilbert-Schmidt de ImQ 1 2 dans H, d’un processusde Wiener cylindrique sur le noyau auto-reproduisant de la loi deW (1). Comme mentionné plus haut, l’intégrale stochastique par rapport àun processus de Wiener est définie via l’intégrale stochastique usuelle dansR grâce à la décomposition en série, voir [34] pour un cadre Hilbert. Lebruit considéré dans nos équations aux dérivées partielles stochastiques est24

Introductionla dérivée en temps, au sens des distributions, d’un processus de Wiener.La décomposition formelle de la dérivée en temps d’un processus de Wienercylindrique est la somme d’un développement en série sur le produit tensorieldes bases de L 2 (0, T ) et de H dont les coefficients sont une collectiondénombrable de variables aléatoires N (0, σ 2 ) indépendantes. Nous appelonsce bruit un bruit blanc espace-temps. Dans le cas d’un processus de Wienerusuel, le bruit est blanc en temps et coloré en espace.Il existe également une seconde approche tout à fait équivalente du calculstochastique permettant de donner un sens aux EDP stochastiques c.f. [79,126]. Cette approche s’appelle parfois approche variationnelle. Elle est aussiassez répandue, voir par exemple [22, 29, 139]. Sous cette approche, dansle cas d’un domaine borné, nous prenons par exemple [0, 1], et d’un tempsdans [0, T ], un bruit blanc W d’intensité λ, la mesure de Lebesgue, est unefonction sur les éléments de la tribu B des Boréliens de [0, 1] × [0, T ] telleque :(i)(ii)W (A) est une variable aléatoire N (0, λ(A))si A ∩ B = ∅ alors W (A) et W (B) sont indépendantset W (A ∪ B) = W (A) + W (B) p.s.La condition (ii) signifie que W est une fonction additive d’ensembles.Le champ Gaussien W x,t = W ([0, x] × [0, t]) est appelé drap Brownien ; ilcorrespond aussi à ∫ x0 W c(t, x)dx où W c est un processus de Wiener cylindriquesur L 2 (0, 1). L’intégrale stochastique dans ce cadre consiste à définirune intégrale par rapport à des mesures martingales.1.2.3 Les équations de Schrödinger non linéaires stochastiquesNous étudions dans cette thèse deux types d’équations de Schrödingernon linéaires stochastiques. Une avec bruit additif et une avec un bruitmultiplicatif particulier.L’équation avec bruit additif s’écrit sous forme d’Itôidu − ( ∆u + λ|u| 2σ u ) dt = dWoù W est un processus de Wiener sur L 2 ou H 1 selon que nous étudions dessolutions dans L 2 ou dans H 1 .L’équation avec bruit multiplicatif s’écrit quant à elleidu − ( ∆u + λ|u| 2σ u ) dt = u ◦ dW25

Introductionoù ◦ désigne le produit Stratonovich et W est un processus de Wiener surL 2 R ∩ Lα ou H 1 R ∩ W1,α selon que nous étudions des solutions dans L 2 oudans H 1 . L’indice R signifie que nous considérons des espaces de fonctions àvaleurs dans R. La condition sur α est α > 2d. Lorsqu’on définit W commeΦW c avec Φ γ−radonifiant de H (par exemple L 2 ) dans L 2 R ∩Lα ou H 1 R ∩W1,p ,nous pouvons réécrire le produit Stratonovich en fonction du produit Itô, viaun terme de tendance supplémentaire qui correspond au terme de crochet.Nous obtenons(idu − ∆u + λ|u| 2σ u − i )2 uF Φ dt = udW,où F Φ (x) = ∑ j∈N (Φe j(x)) 2 pour x dans R d et (e j ) j∈Nest une base Hilbertiennede H.Notons que pour certaines EDPS particulières, en particulier pour l’équationde la chaleur en dimension 1, il est possible d’étudier le bruit blanc. Ce sontdes équations où le semi-groupe linéaire a des propriétés de régularisationglobale. Dans ce cas, dans la forme mild, le semi-groupe S qui apparaîtdans la convolution stochastique, par exemple ∫ t0S(t − s)dW (s) dans le casd’un bruit additif, possède lui-même la propriété d’être Hilbert-Schmidt dèsque t > s. Ici le groupe est une isométrie sur les espaces de Hilbert baséssur L 2 , il ne possède pas de telles propriétés. Notons qu’en physique les auteursétudient généralement le cas de perturbations aléatoires d’équations deSchrödinger nonlinéaires de type bruit blanc. Nous n’arrivons pas à donnerde sens mathématique au bruit blanc dans ce cas. Une limite bruit blanc,dans un sens qui sera précisé, est par contre considérée dans les sections 2.1et 2.3.Il est prouvé dans [36, 37] que le problème de Cauchy est localementbien posé dans L 2 ou dans H 1 pour des exposants σ suffisamment petits.Les solutions sont des solutions faibles au sens de l’analyse des équationsaux dérivées partielles. Ce sont de manière équivalente des solutions mild.Dans le cas du bruit additif, la condition sur σ est la même que dans le casdéterministe. Dans le cas du bruit multiplicatif, la condition sur σ est lamême que dans le cas déterministe si d = 1 ou d = 2. Sinon, dans le cas L 2 ,la condition est σ < 2 d ∧ 1d−1 et, dans le cas H1 , σ < 2 si d = 3 et 1 2 ≤ σ < 2d−2ou σ < 1d−1si d ≥ 4. Par ailleurs, le bruit étant réel et le produit étant unproduit Stratonovich, la masse est conservée. Cette propriété est motivée parla physique. Le cadre H 1 est alors adapté à l’étude de l’explosion en tempsfinie. En effet, on ne peut pas observer le phénomène d’explosion dans L 2 ,26

Introductionen tout cas pour un bruit multiplicatif, car la masse est conservée. Dans cecas, lorsque le problème est localement bien posé, il est aussi globalementbien posé. Pour les équations stochastiques dans H 1 , avec bruit additif oumultiplicatif, les solutions sont globales pour des non linéarités sous-critiquesou dans le cas défocalisant dès que le problème est localement bien posé, c.f.[37]. Ce résultat coincide avec celui pour les équations déterministes.Les équations ont aussi été étudiées d’un point de vue numérique dans[13, 43]. Dans [43], le cas de la dimension 1 est considéré. L’influence des deuxtypes de bruit sur la propagation des solitons et sur l’explosion en tempsfini est étudiée. Il est obtenu qu’un bruit additif a tendance à accélérerl’explosion en temps fini alors que le bruit multiplicatif a tendance à laretarder. Mais aussi, il est obtenu que toute donnée initiale explose en tempsfini dans les cas critiques et sur-critiques. Le cas du bruit blanc est aussiétudié. Il apparaît que le bruit blanc multiplicatif empêche l’explosion entemps fini. Dans [13], le cas de la dimension 2 est étudié. L’effet du bruit surl’explosion en temps fini est étudié d’un point de vue théorique dans [38, 39,40] pour des bruits additifs complexes et réels et des bruits multiplicatifs.Sous des hypothèses légèrement plus fortes sur la régularité du processus deWiener, il est montré que, pour des non linéarités sur-critiques et des donnéesinitiales non nulles, quel que soit t strictement positif, la probabilité que lasolution explose avant t est strictement positive.1.3 Les grandes déviations1.3.1 PrésentationLes résultats de grandes déviations permettent de quantifier une loi faibledes grands nombres. Supposons qu’une famille de mesures de probabilités(µ ɛ ) ɛ>0, sur un espace de Banach muni de sa tribu Borélienne, convergeétroitement vers une mesure de Dirac en un point x, alors nous savons que,pour tout Borélien A ne contenant pas x dans son intérieur, lim ɛ→0 µ ɛ (A) =0. Dans le langage des probabilités un tel évènement A est un évènementde grandes déviations. La déviation est grande car l’ensemble A ne dépendpas de ɛ. Un résultat de grandes déviations quantifie la convergence vers0 à vitesse ɛ. L’espace de Banach dans cette thèse sera généralement unespace de trajectoires ou la droite réelle. Cela pourrait aussi être un espacede mesures, par exemple dans le cas de mesures empiriques, nous parlonsdans ce cas de grandes déviations de niveau 2.27

IntroductionLe cas le plus élémentaire est celui où (X j ) j≥1est une famille de variablesaléatoires réelles, centrées, indépendantes, et de même loi, et où ons’intéresse à quantifier la convergence en loi vers 0 de la variable aléatoireS n = 1 n∑ nj=1 X j. Si la loi des variables aléatoires est N (0, 1), la loi de S nest encore Gaussienne et nous pouvons vérifier aisément que1limn→∞ n log P ( |S n | ≥ δ ) = − δ22 .Le théorème de Cramer donne que ce résultat est encore valable dans le casnon Gaussien. La limite peut ne plus être définie, le résultat donne malgrétout un encadrement des limites inférieures et supérieures. Cet encadrementdépend de la loi des variables aléatoires X j via une fonctionnelle quenous appelons fonction de taux. Nous appelons cela un principe de grandesdéviations. Le théorème de Cramer s’énonce de la façon qui suit.Theorem 1.3.1 La famille des lois µ n des variables aléatoires S n satisfaitun principe de grandes déviations (PGD) de vitesse 1 net de fonction de tauxΛ ∗ , la transformée de Fenchel-Legendre, définie parΛ ∗ (x) = sup {λx − Λ(λ)}λ∈Roù Λ(λ) est le logarithme de la transformée de Laplace de la loi de X 1 . End’autres termes, pour tout Borélien A de R, nous avons la suite d’inégalités− inf Λ ∗ 1(x) ≤ lim n→∞x∈Ån log µ 1n(A) ≤ lim n→∞n log µ n(A) ≤ − inf Λ ∗ (x)x∈ACet énoncé nous montre qu’un principe de grandes déviations transformeun problème de calcul des probabilités en un problème de minimisation.Dans cette thèse nous serons confrontés à des problèmes de contrôle optimalvoire de calcul des variations. Mais aussi, un problème de minimisation peuttrouver une réponse probabiliste. Par ailleurs, nous voyons qu’un résultat degrandes déviations fait intervenir la topologie et que les bornes supérieureset inférieures sont d’autant meilleures que la topologie est fine. Enfin, lafonction de taux est une fonctionnelle semi-continue inférieurement. Unebonne fonction de taux I est une fonction de taux telle que l’ensemble desniveaux inférieurs à un réel strictement positif c, i.e. I −1 ([0, c]), est compact.Cette propriété assure que, sur de tels ensembles, la fonction de taux atteintson minimum.Dans un espace polonais E (nous notons par B ses boules), nous pouvonsréécrire de façon équivalente les bornes supérieures et inférieures d’un PGD28

Introductionpour une famille de mesures (µ ɛ ) ɛ>0, de vitesse ɛ et de bonne fonction detaux I.La borne supérieure prend la forme :quels que soient c et δ strictement positifs, K(c) δ un δ−voisinage de I −1 ([0, c]),nous avonslim ɛ→0 ɛ log µ ɛ (( K(c) δ) c)≤ −c.La borne inférieure quant à elle se réécrit sous la forme :quels que soient e dans E et δ strictement positif, nous avonslim ɛ→0 ɛ log µ ɛ (B(e, δ)) ≥ −I(e).Nous appelons cette écriture écriture à la Freidlin-Wentzell.1.3.2 Des résultats générauxLe théorème de Cramer s’étend à R d et à la dimension infinie. Desrésultats de grandes déviations existent aussi pour des suites de variables noni.i.d., c’est le cas du théorème de Gärtner-Ellis ou des résultats de grandesdéviations pour les chaînes de Markov, c.f. [48]. Nous présentons désormaisdans cette section deux résultats de grandes déviations qui nous serons utilespar la suite.Du théorème de Cramer en dimension infinie nous pouvons déduire unrésultat général de grandes déviations pour les mesures Gaussiennes sur lesespaces de Banach réels séparables. Considérons un tel espace de Banach E,une mesure Gaussienne µ sur E, d’espace de Hilbert noyau auto-reproduisantH µ , et la famille de mesures (µ ɛ ) ɛ>0images directes de µ par la transformationx ↦→ √ ɛx sur E. Nous avons le résultat qui suit, c.f. [53].Theorem 1.3.2 La famille (µ ɛ ) ɛ>0satisfait un PGD sur E, de vitesse ɛet de bonne fonction de taux la transformée de Fenchel-Legendre Λ ∗ µ. Parailleurs la transformée de Fenchel-Legendre Λ ∗ µ se réécritΛ ∗ µ(e) ={ 12 ‖e‖2 H µsi e ∈ H µ∞ si e ∈ E \ H µ.Nous pouvons déduire de ce résultat le théorème de Schilder qui énonce unPGD pour les trajectoires du mouvement Brownien.29

(NEBRASKA DEPARTMENT OF CORRECTIDNAL SERVICESTECUMSEH STATE CORRECTIONAL INSTITUTIONOPERATIONAL MEMORANDUM 205.02.01, VISITING PROCEDURESPage 31 of 37c. Inmates are not permitted near the vending machines at any time for anyreason.d. Visitors are not permitted to loiter near the vending machines.e. Inmates in a non-contact visit are not allowed vending machine items;their visitors may purchase vending machine items for their ownconsumption, but not for the inmate.10. Use of Restrooma. Visitors will use the restrooms located on the east wall of the VisitingRoom designated for visitor use.A Visitor needing to use the restroom will notify the Visiting RoomOfficer. The Visiting Room Officer will escort the visitor to one of thesearch rooms located on the east wall of the Visiting room. The Officerwill pat search the visitor and escort him/her to the restroom. The Officerwill wait until the visitor has finished using the restroom and will escorthim/her back to search room to be re-pat searched before being allowedto return to the Visiting Room. After the Officer has completed searchingthe Visitor, the Visitor will be allowed to resume the visit.b. Inmates will use the restroom located on the secure side of the searchrooms of the Visiting Room.(1) An inmate needing to use the restroom will notify the VisitingRoom Officer. The Visiting Room Officer will escort the inmateto one of the search rooms. The officer will strip search theinmate and escort him to the restroom. The officer will wait untilthe inmate has finished using the restroom and will escort himback to the Visiting Room. The inmate will be under the officer'sdirect supervision at all times and will not be allowed to havecontact with any other inmate.(2) Under no circumstances will the assigned Visiting Room Officerleave the Visiting Room for assisting in search duties. Theassigned officer(s) will perform search duties for those inmateson approved visit who need to use the restroom during their visit.c. Visitors are not permitted to loiter in the restrooms or near the vendingmachines.XXIII.INMATE DRESS CODE/POSSESSIONS DURING VISITINGA. Inmates in general population will be required to be in possession of their 1.0. card. Thefollowing items of clothing will be authorized for inmates on pass to visit in theAdministration Building. (Soiled, torn, or otherwise inappropriate clothing shall not beworn in the visiting rooms/areas.):1. Pants (institutional issue or approved civilian-blue jeans only).(a. All pants will be kept neat and properly buttoned/zipped at all times.

IntroductionIl est cependant possible d’imposer une propriété plus faible que la continuitépour f. Nous aurons besoin de résultats de ce type pour montrer unPGD pour les lois des trajectoires des équations de Schrödinger non linéairesstochastiques lorsque le bruit est multiplicatif. En effet, la démarche généralepour montrer un PGD au niveau des trajectoires des équations de Schrödingernon linéaires stochastiques est de transférer par image directe un PGDpour le processus de Wiener, résultat du type du théorème de Schilder pourle mouvement Brownien. Voici une première approche, cela ne sera pas celleadoptée dans le présent papier. Cette méthode peut permettre de montrerun PGD pour les trajectoires d’une EDS (voir par exemple [48, 66]).Theorem 1.3.6 Soit (µ ɛ ) ɛ>0une famille de mesures de probabilité satisfaisantun PGD de bonne fonction de taux I sur un espace topologique séparéX . Soit également une suite de fonctions continues (f j ) j∈Nde X dans unespace métrique (Y, d). Supposons qu’il existe une application f de X dansY telle que∀c > 0, lim j→∞supx∈X : y=f(x)d (f m (x), f(x)) = 0.Alors, toute famille de mesures de probabilité (˜µ ɛ ) ɛ>0telle quelim lim ɛ→0ɛ log P ɛ,j (Γ δ ) = −∞, (1.3.1)j→∞où P ɛ,j est le produit tensoriel de (f j ) ∗µ ɛ et de ˜µ ɛ etΓ δ = { (y, z) ∈ Y 2 : d(y, z) > δ } ,satisfait un PGD sur Y de vitesse ɛ et de bonne fonction de tauxI ′ (y) =inf I(x).y∈Y: y=f(x)La relation (1.3.1) signifie que les mesures ( (f j ) ∗µ ɛ) sont des approximationsexponentiellement bonnes des mesures (˜µ ɛ ) ɛ>0ɛ>0.Nous utiliserons une autre extension du principe de contraction de Varadhan,celle-ci utilise le lemme d’Azencott appelé aussi inégalité de Freidlin-Wentzell. Elle sera présentée dans le chapitre qui suit. Notons aussi que, dansle cas des PGDs pour les trajectoires de nos EDPS, nos familles de mesuressont paramètrées par la donnée initiale. Nous nous intéresserons aussi àdes PGDs uniformes, en un sens qui sera précisé, en la donnée initiale. Onpeut aussi trouver dans [131] une extension du principe de contraction deVaradhan permettant de transporter des PGDs uniformes pour une formulationà la Freidlin-Wentzell.31

Chapitre 2Présentation des résultatsNous présentons dans cette partie les résultats des différents articlesque rassemble cette thèse. Les articles figurent en annexe. Nous étudionsles équations de Schrödinger non linéaires stochastiques et considérons lalimite lorsque le bruit tend vers 0. Nous prouvons donc des principes degrandes déviations au niveau des lois des trajectoires des solutions. Ils sonténoncés dans des espaces de trajectoires explosives et pour des topologiesrelativement fines rendant compte des propriétés d’intégrabilité du groupede Schrödinger. Nous donnons des applications à l’asymptotique des tempsd’explosion et aux fluctuations de la masse et de la position d’un signalde type soliton. Ce dernier problème trouve par exemple des applications àl’évaluation des erreurs de transmission par solitons dans les fibres optiques.Nous donnons aussi une application au temps moyen de sortie d’un domaineet aux points de sortie dans le cas d’équations faiblement amorties. Nousétudions aussi le cas de bruits additifs plus généraux de type bruits fractionnairesen dimension infinie. Nous prouvons aussi pour des bruits additifs desthéorèmes de support dans les espaces considérés.2.1 Grandes déviations et théorèmes de supportpour des équations de Schrödinger non linéairesstochastiques avec bruit additifNous présentons dans ce paragraphe un principe de grande déviations auniveau des trajectoires et un théorème de support ainsi que leurs applicationsdans le cas d’une équation de type NLS perturbée par un bruit additif. Le33

Présentation des résultatslecteur trouvera plus de détails ainsi que des preuves dans l’annexe A quireprend l’article [81] publié dans ESAIM : Probability and Statististics.Nous nous intéressons aux équationsidu ɛ − (∆u ɛ + λ|u ɛ | 2σ u ɛ )dt = √ ɛdW,λ = ±1, x ∈ R ddans H 1 . L’exposant σ est tel que σ > 0 si d = 1, 2 et σ < 2d−2sinon.Nous posons W = ΦW c , où Φ est Hilbert-Schmidt de L 2 dans H 1 et W c estun processus de Wiener cylindrique sur L 2 . Le paramètre ɛ est l’amplitudedu bruit. Nous cherchons à quantifier la convergence faible des lois des trajectoiresvers la masse de Dirac en la solution déterministe, lorsque ɛ tendvers zéro. Nous nous intéressons donc à un principe de grandes déviationstrajectoriel.Nous souhaitons dans un premier temps pouvoir traiter des non-linéaritéscritiques et sur-critiques où les solutions peuvent exploser en temps fini.Par ailleurs, nous savons que, un PGD s’énonçant pour A borélien del’espace des trajectoires− inf I(u) ≤ lim ɛ→0 ɛ log P (u ɛ ∈ A) ≤ lim ɛ→0 ɛ log P (u ɛ ∈ A) ≤ − inf I(u),u∈Åu∈Aplus la topologie est forte plus les bornes sont précises. En outre un PGDse transporte en un PGD pour les mesures images directes par applicationscontinues entre espaces topologiques séparés. Ainsi nous pouvons déduired’un PGD pour une topologie forte un PGD pour une topologie plus faibleà condition que celle-ci reste séparée.Nous introduisons donc un espace de trajectoires explosives, noté E ∞ ,muni d’une topologie relativement fine exploitant les propriétés d’intégrabilitédu groupe linéaire de Schrödinger. L’espace E ∞ est une partie de l’espaceE(H 1 ) des fonctions f continues à valeurs H 1 ∪ {∆} muni de la convergenceuniforme sur les intervalles [0, T ] où T est inférieur strictement au tempsd’explosionT (f) = inf{t > 0 : f(t) = ∆}et ∆ est un point cimetière. L’espace H 1 ∪ {∆} est muni de la topologie quiest engendrée par les ouverts de H 1 et les complémentaires dans H 1 ∪{∆} desfermés bornés de H 1 . Les fonctions f de E ∞ vérifient en outre des propriétésd’intégrabilité sur les intervalles [0, T ] où T < T (f). Nous posons en effet si34

Présentation des résultatsd > 2E ∞ ={[ )f ∈ E(H 1 2d) : ∀ p ∈ 2, , ∀ T ∈ [0, T (f)), f ∈ L r(p) ( 0, T ; W 1,p)} .d − 2Lorsque d = 1 ou d = 2 nous écrivons p ∈ [2, ∞). L’espace est muni de latopologie définie pour ϕ 1 dans E ∞ par la base de voisinages suivanteW T,p,ɛ (ϕ 1 ) = {ϕ ∈ E ∞ : T (ϕ) ≥ T, ‖ϕ 1 − ϕ‖ X (T,p) ≤ ɛ} .où T < T (ϕ 1 ), p est comme défini plus haut et ɛ est strictement positif.Il s’agit d’un espace topologique séparé. Si nous notons toujours parT : E ∞ → [0, ∞] le temps d’explosion, l’application est mesurable et semicontinueinférieurement.Nous vérifions la continuité de l’application qui envoie la convolutionstochastique Z définie par Z(t) = ∫ t0U(t − s)dW (s) en la solution. Maisaussi nous vérifions que la convolution stochastiques définit bien une variablealéatoire à valeurs dans l’espace topologique considéré et a pour loi unemesure Gaussienne centrée. Nous énonçons pour les lois de √ ɛZ un PGDqui découle du théorème de Dawson-Gartner pour les limites projectives etdu PGD abstrait pour des familles de mesures Gaussiennes. Le PGD pour leslois des trajectoires de l’équation de Schrödinger non linéaire stochastiqueµ uɛ est alors déduit par contraction. Quantifier la probabilité d’un évènementde grande déviation revient à résoudre un problème de contrôle optimal, lafonctionnelle à minimiser est la fonction de tauxI(u) = 1 {}inf ‖h‖ 22 h∈L 2 (0,∞;L 2 L): S(h)=u2 (0,∞;L 2 ),où inf ∅ = ∞ et S(h), appelé squelette de l’équation stochastique, est l’uniquesolution mild de {idudt = ∆u + λ|u|2σ u + Φh,u(0) = u 0 ∈ H 1 ,c’est à dire qu’elle s’écrit sous la formeu(t) = U(t)u 0 − iλ∫ t0U(t − s)|u(s)| 2σ u(s) − iLe PGD s’énonce de la façon suivante.∫ t0U(t − s)Φh(s)ds.Théorème 2.1.1 (PGD) La famille de mesures de probabilité ( µ uɛ )ɛ≥0 surE ∞ satisfait un PGD de vitesse ɛ et de bonne fonction de taux I.35

Présentation des résultatsNous prouvons ensuite dans ce cadre le théorème de support.Théorème 2.1.2 (Théorème de support) Le support de la loi des solutionsest caractérisé parsupp µ u1 = ImS E∞ .Nous commençons par appliquer les deux résultats ci-dessus aux tempsd’explosion et obtenons les résultats qui suivent. Dans ce qui suit nous notonsu d la solution de l’équation déterministe.Proposition 2.1.1 Si u 0 ∈ H 3 et si l’image de Φ est dense alors pour toutT > 0,P(T (u 1 ) > T ) > 0.Proposition 2.1.2 Si u 0 ∈ H 3 , si l’image de Φ est dense et si T ≥ T (u d ),où u d est la solution de l’équation déterministe avec donnée initiale u 0 , ilexiste c ∈ [0, ∞) tel queProposition 2.1.3 Si T < T (u d ),lim ɛ→0 ɛ log P (T (u ɛ ) > T ) ≥ −c.lim ɛ→0 ɛ log P (T (u ɛ ) ≤ T ) ≤ −U [0,T ] < 0.où U [0,T ] = 1 2 inf h∈L 2 (0,∞;L 2 ):T (S(h))≤T{‖h‖ 2 L 2 (0,∞;L 2 )}.A chaque fois nous n’avons une inégalité que dans un sens, l’autre inégalitédonne un résultat trivial. Des résultats pour des approximations des tempsd’explosions définis parT R (f) = {t ∈ [0, ∞) : ‖f(t)‖ H 1 ≥ R}sont donnés en annexe A, nous obtenons en corollaire des bornes inférieureset supérieures non triviales de P (S < T R (u ɛ ) ≤ T ) pour S < T < T R (u d )ou T R (u d ) < S < T .Enfin, nous étudions dans cette annexe l’erreur de transmission par solitonsdans les fibres optiques. Cette deuxième étude est prolongée dans lasection 2.3 et l’annexe C. Il a été suggéré en 1973 par Hasegawa et Tappert[93], de tirer profit de la dispersion et de la non-linéarité, apparemmentfacteurs limitants pour transmettre des données codées par des solitons.Nous considérons le cas de l’équation cubique avec non linéarité focalisante36

Présentation des résultatsetCD {h 1 = ∈ L 2 (0, T ; L 2 ) : il existe η ∈ HD1h(t, x) = i η′ (t)η(t) Ψ η(t, x) − i √ (2η ′ (t) exp −iNous obtenons∫ t0)η 2 (s)ds η(t)x sinh }cosh 2 (η(t)x) .Proposition 2.1.5 Pour tout T > 0, γ ∈ (0, 1) et D dense dans (0, 1),pour toute suite d’opérateurs Hilbert-Schmidt (Φ n ) n∈Nde L 2 dans L 2 telleque pour tout h ∈ C D , Φ n h converge vers h dans L 1 ( 0, T ; L 2) . Alors nousavons la borne supérieure, l’exposant n rappelant que Φ est remplacé par Φ n ,lim n→∞,ɛ→0 ɛ log P (N (u ɛ,n (T )) ≥ 4(1 − γ)) ≥ − 2(1 − γ)(12 + π2 ).9TPour une donnée initiale profil de soliton nous avons la borne inférieuresuivante.Proposition 2.1.6 Pour tout T > 0, γ dans (0, 1), et tout opérateur Φdans L 2 (L 2 , H 1 ), l’inégalité suivante est satisfaitelim ɛ→0 ɛ log P (N (u ɛ (T )) < 4(1 − γ)) ≤ −2T ‖Φ‖ 2 c(1 + γ) 2En considérant les mêmes solitons modulés pour des paramètres dans{HD 2 = η : [0, T ] → R, il existe ˜γ ∈ D tel que η(t) = η˜γ,T (t)=(2 − ˜γ − 2 √ ) ( t1 − ˜γTγ 2) 2 (+ 2 −1 + √ ) } t1 − ˜γT + 1et les contrôles, correspondant au cas où Φ = I, dans CD 2montrons la borne supérieure suivante.associés, nousProposition 2.1.7 Pour tout T > 0, γ ∈ (0, 1) et D dense dans (0, 1),pour toute suite d’opérateurs Hilbert-Schmidt (Φ n ) n∈Nde L 2 dans L 2 telleque pour tout h ∈ C D , Φ n h converge vers h dans L 1 ( 0, T ; L 2) . Alors nousavons la borne supérieure, l’exposant n rappelant que Φ est remplacé par Φ n ,lim n→∞,ɛ→0 ɛ log P ɛ (N (u ɛ,n (T )) < 4(1 − γ)) ≥ − 2(2 − γ − 2√ 1 − γ)(12 + π 2 ).9T38

Présentation des résultatsLes bornes que nous obtenons sont du même ordre en T . Elles sont à chaquefois en 1 Tce qui correspond à ce qui est obtenu en physique. Dans le cas d’unedonnée initiale nulle, elles sont aussi du même ordre en γ. Nous obtenonsque les queues de la masse sont, sur une échelle logarithmique, les mêmesque celles d’une loi exponentielle. Il s’agit d’un résultat également obtenupar les physiciens, voir le résultat de [63] sur l’amplitude. Dans le cas d’unedonnée initiale profil de soliton, les bornes ne sont plus du même ordre en γ.Mais la borne inférieure nous permet en tout cas de conclure que les queuesne sont pas Gaussiennes. Les supposer Gaussiennes entraîne une évaluationtrompeuse de l’erreur.2.2 Grandes déviations uniformes pour des équationsde Schrödinger non linéaires stochastiques avecbruit multiplicatifNous décrivons dans ce paragraphe un principe de grandes déviationsuniforme au niveau des trajectoires des solutions d’une équation de typeNLS perturbée par un bruit multiplicatif. Nous donnons des applications àl’asymptotique des temps d’explosion. Les détails ainsi que les preuves sontdonnées dans l’annexe B qui correspond à l’article [82] publié dans StochasticProcesses and their Applications.Dans cet article nous considérons des équations de Schrödinger non linéairesstochastiques avec bruit multiplicatifidu ɛ,u 0− (∆u ɛ,u 0+ λ|u ɛ,u 0| 2σ u ɛ,u 0)dt = √ ɛu ɛ,u 0◦ dW, λ = ±1, x ∈ R d .L’exposant σ satisfait σ > 0 si d = 1, 2 et σ < 2d−2sinon et la donnée initialeu 0 est dans H 1 . Nous posons W = ΦW c , où Φ est Hilbert-Schmidt de L 2dans H s R , espace de fonctions à valeurs réelles avec s > d 2 + 1 et W c est unprocessus de Wiener cylindrique sur L 2 . Le symbole ◦ correspond au produitStratonovich. Le paramètre ɛ, intensité du bruit, tend vers zéro.Nous prouvons dans cet article un PGD trajectoriel uniforme (en desdonnées initiales dans des compacts de H 1 ). Il est lui aussi énoncé dansun espace de trajectoires explosives muni d’une topologie analogue à unetopologie limite projective rendant compte des propriétés d’intégrabilité dugroupe de Schrödinger. Nous notons à nouveau cet espace E ∞ . Il est cette foisdéfini pour d dans N ∗ par l’ensemble des fonctions f de E(H 1 ) tel que pourtout p dans A(d) et tout T dans [0, T (f)), f appartienne à L r(p) ( 0, T ; W 1,p) .39

Présentation des résultatsL’ensemble d’exposants [ A(d) ) est [ l’ensemble ) [2, ∞) lorsque d = 1 ou d = 2 etrespectivement 2, 2(3d−1)2d3(d−1)et 2,d−1lorsque d ≥ 3. L’espace E ∞ est munide la topologie définie pour ϕ 1 dans E ∞ par la base de voisinagesW T,p,r (ϕ 1 ) = {ϕ ∈ E ∞ : T (ϕ) > T, ‖ϕ 1 − ϕ‖ X (T,p) ≤ r}où T < T (ϕ 1 ), p appartient à A(d) et r est strictement positif. On peutvérifier que cet espace topologique est bien séparé. Si on note toujours parT : E ∞ → [0, ∞] le temps d’explosion, l’application T est mesurable etsemi-continue inférieurement.Le squelette de l’équation stochastique est l’unique solution mild du problèmede contrôle {idudt = ∆u + λ|u|2σ u + uΦh,u(0) = u 0 ∈ H 1 , h ∈ L 2 ( 0, ∞; L 2) .Le résultat est énoncé ci-après, nous notons K ⊂⊂ H 1 lorsque K est uncompact de H 1 , Int(A) l’intérieur de A et B (E ∞ ) les boréliens de E ∞ .Theorem 2.2.1 La famille des lois des trajectoires ( µ ) uɛ,u 0ɛ>0PGD uniforme de vitesse ɛ et de bonne fonction de tauxsatisfait unI u 0(w) = 1 2 inf h∈L 2 (0,∞;L 2 ):w=S c (u 0 ,h) ‖h‖ 2 L 2 (0,∞;L 2 ) ,i.e. quelque soit K ⊂⊂ H 1 , et A dans B (E ∞ ), nous avons la borne inférieure− supu 0 ∈Ket la borne supérieureinfw∈Int(A) Iu 0(w) ≤ lim ɛ→0 ɛ log inf Pu 0 ∈K (uɛ,u 0∈ A)lim ɛ→0 ɛ log supu 0 ∈KP (u ɛ,u 0∈ A) ≤ −infw∈A,u 0 ∈KI u 0(w).Ce résultat se déduit cette fois d’un PGD pour le processus de Wiener (etnon la convolution stochastique). La preuve est ici plus complexe que dansle cas d’un bruit additif. En effet, les trajectoires des solutions ne sont pasimages directes par une application continue des trajectoires du processusde Wiener.Il serait aussi envisageable d’utiliser un principe de contraction et deprouver un PGD pour les trajectoires rugueuses au dessus des solutions àpartir d’un PGD pour les trajectoires rugueuses au dessus du processus deWiener. Cela nécessiterait par contre de montrer un résultat de continuité à40

Présentation des résultatsla T. Lyons, c.f. [107, 108], mais pour l’EDP stochastique. Nous pourrionsnotamment nous inspirer de [105, 103].Nous adoptons ici une preuve plus classique basée essentiellement surle lemme d’Azencott aussi appelé inégalité de Freidlin-Wentzell. Ce lemmecorrespond à ce qui remplace ici la continuité au niveau des trajectoires.Il s’énonce de la manière suivante, C a désigne le ensembles des niveauxinférieurs à a de la fonction de taux d’un PGD pour le processus de Wiener.Le squelette S(u 0 , f) est cette fois ci celui où l’on remplace Φh par ∂f∂t .Proposition 2.2.2 Pour tout a, R et ρ strictement positifs, u 0 dans H 1 ,f dans C a , T < T (S(u 0 , f)), p dans A(d), il existe ɛ 0 , γ et r strictementpositifs tels que pout tout ɛ dans (0, ɛ 0 ] et ũ 0 dans B H 1(u 0 , r),( ∥∥ɛ log P u ɛ,ũ 0− S(u 0 , f) ∥ X (T,p) ≥ ρ; ∥ √ ɛW − f ∥ )C([0,T ];H sR ) < γ ≤ −R.La preuve de ce lemme nécessite les estimées de décroissance exponentielledes queues dans des espaces de Banach suivantes.Proposition 2.2.3 Si Z, défini par Z(t) = ∫ t0U(t − s)ξ(s)dW (s), est telqu’il existe η positif tel que ‖ξ‖ 2 C([0,T ];H 1 )≤ η p.s., alors pour tout p dansÃ(d), T et δ positifs,P ( ‖Z‖ C([0,T ];H 1 ) ≥ δ ) ( )≤ 3 exp − δ2)κ(1 (η))P(‖Z‖ L r(p) (0,T ;W 1,p ) ≥ δ ≤ c exp − δ2κ 2 (η)où c = 2e + exp((2ek 0 !) 1k 0), k 0 = max(2, min{k ∈ N : 2k ≥ r(p)})c8cκ 2 (η) =κ 1 (η) = T 4c (∞) 2 ‖Φ‖ 2 η,L 0,s2( ) 2 r(p)d 1−2 T 2r(p)(d + 1)(d + p)‖Φ‖ 2 L 0,s1 − 4r(p)( )r(p)d2et c(∞) sont les normes des injections continues H s RH s R ⊂ W1,∞ R .2η,⊂ r(p)dW1, 2RNous utilisons également la continuité du squelette modifié par rapport auxcontrôles sur les ensembles C a des niveaux inférieurs à a strictement positifs.Le résultat de continuité s’énonce comme suit.41et

Présentation des résultatsProposition 2.2.4 Pour tout u 0 dans H 1 , a positif et f dans C a , S(u 0 , f)existe et est défini de manière unique. L’application est continue de H 1 × C adans E ∞ , où C a a la topologie induite par celle de C ([0, ∞); H s R ).Nous donnons dans cet article une première application du PGD. Ils’agit d’une application aux temps d’explosion. Si on note u u 0dla solution del’équation déterministe, nous prouvonsProposition 2.2.5 Si T < inf u0 ∈K T ( u u )0d , où K ⊂⊂ H 1 , il existe c strictementpositif tel quelim ɛ→0 ɛ log sup P (T (u ɛ,u 0) ≤ T ) ≤ −c.u 0 ∈KProposition 2.2.6 Soit U u 0la solution de l’équation de Schrödinger libreavec une donnée initiale u 0 dans H s et supposons que l’espace vectoriel engendrépar { |U u 0(t)| 2 , t ∈ [0, 2T ] } appartienne à l’image de Φ pour T >T ( u u )0d . Il existe alors c strictement positif tel quelim ɛ→0 ɛ log P (T (u ɛ,u 0) > T ) ≥ −c.2.3 Asymptotique de petits bruits pour la fluctuationdes temps d’arrivée dans la transmissionpar solitonsNous décrivons dans ce paragraphe une application des principes degrandes déviations trajectoriels pour les solutions d’équations de type NLSperturbée par un bruit additif ou multiplicatif. Nous étudions l’asymptotiquedes queues du temps d’arrivée d’une donnée initiale profil de solitonlorsque l’amplitude ɛ du bruit tend vers 0. L’article [44] correspondant figureen annexe C. La fluctuation du temps d’arrivée est une source d’erreur detransmission. Nous avons commencé à traiter le sujet des erreurs de transmissiondans la section 2.1. Nous considérons toujours le cas de l’équationavec non-linéarité cubique focalisante en dimension 1.Nous étudions dans un premier temps le cas d’un petit bruit complexeadditif. Il correspond à l’émission spontanée de bruit par des amplificateursrégulièrement espacés le long de la fibre afin de palier à la perte par amortissement.Nous considérons aussi le cas de petits bruits multiplicatifs réels.Ceux ci correspondent à d’autres types d’amplification : l’amplification de42

Présentation des résultats(T est très grand) longueur de la fibre et en R (de l’ordre de l’unité). L’ordrede grandeur en l’amplitude A de la donnée initiale est celui obtenu par lesphysiciens pour le bruit additif.Pour l’étude de la fluctuation de la position nous avons besoin de PGDslégèrement différents des PGDs donnés précédemment. En effet, nous souhaitonsnous placer dans un espace de trajectoires où la position est biendéfinie. Nous introduisons donc les espacesetmunis des normesNous prouvons alorsΣ = { f ∈ H 1 : x ↦→ xf(x) ∈ L 2} ,Σ 1 2 ={f ∈ H 1 : x ↦→ √ |x|f(x) ∈ L 2}‖f‖ 2 Σ = ‖f‖ 2 H+ ‖x ↦→ xf(x)‖ 2 1 L 2 ,‖f‖ 2 = ‖f‖ 2 ∥Σ 1 2 H+ ∥x ↦→ √ |x|f(x)1 ∥ 2 .L 2Théorème 2.3.1 Supposons que Φ (tel que W = ΦW c soit le processus deWiener dirigeant l’équation stochastique) soit Hilbert-Schmidt de L 2 dans Σdans le cas additif et de L 2 dans H s (R, R)) avec s > 3/2 dans le cas multiplicatif.Supposons que la donnée initiale u 0 appartienne à Σ. Alors les solutionsdes équations stochastiques sont presque sûrement dans C([0, T ]; Σ 1 2 ).De plus elles définissent de variables aléatoires à valeurs dans C([0, T ]; Σ 1 2 )et leurs lois ( µ ) uɛ,u 0satisfont des PGDs de vitesse ɛ et de bonnes fonctionsdeɛ>0tauxI u 0(w) = 1 2infh∈L 2 (0,T ;L 2 ): w=S(u 0 ,h) ‖h‖2 L 2 (0,T ;L 2 ) ,où S(u 0 , ·) = S a,u 0(·) dans le cas additif et S(u 0 , ·) = S m,u 0(·) dans le casmultiplicatif, avec la convention inf ∅ = ∞.Le squelette S a,u 0de l’énoncé ci-dessus est la solution mild du problème decontrôle {idudt = ∆u + |u|2 u + Φh,u(0) = u 0 ∈ Σ and h ∈ L 2 ( 0, T ; L 2) .Le squelette S m,u 0est la solution mild du problème de contrôle où l’équationest remplacée pari dudt = ∆u + |u|2 u + uΦh.44

Présentation des résultatstelle que pour tout h dans HT,A D , Φ nh converge vers h dans L 1 (0, T ; Σ). Alorsl’inégalité suivante est vérifiée, l’exposant n rappelle que Φ est remplacé parΦ n ,( ( ) )lim n→∞,ɛ→0 ɛ log P Y u ɛ,Ψ0 A ,n (T ) ≥ R ≥ − π2 R 2128T 3 A 3 .Les deux bornes sont en − 1 lorsque T est grand. Cela confirme que la fluctuationdu centre est prépondérante sur la fluctuation de la masse en ce quiT 3concerne l’erreur de transmission. En effet, T étant très grand, les queues dela position sont plus épaisses que celles du centre. En outre sur une échellelogarithmique les queues sont indistingables de queues Gaussiennes. Si la loiétait effectivement Gaussienne, le facteur T 3 correspondrait exactement àla variance proportionnelle à T 3 de l’effet Gordon-Haus. Cette variance enT 3 est supérieure à celle du mouvement Brownien en T et on appelle aussicette fluctuation une super diffusion. Une partie de la littérature physiqueest consacrée à la loi de la position. Nous retrouvons ici le fait qu’au premierordre on peut bien considérer que la loi de la position est Gaussienne.Si nous considérons également la suite d’opérateurs (Φ n ) n∈Ndans la bornesupérieure, le comportement en A grand n’est pas contradictoire. Nous obtenonsque l’ordre de grandeur en A du logarithme de la queue est supérieurà − 1 . Il s’agit aussi de l’ordre de grandeur obtenu en physique.A 3Dans le cas des bruits multiplicatifs nous ne sommes pas arrivés à obtenirune borne inférieure car les contrôles suggérés par l’approche calcul desvariations ne sont ni dans L 2 ni dans l’image de Φ (H s (R, R), s > 3 2). Nousobtenons par contre une borne supérieure. Nous considérons ensuite que lebruit est à valeurs dans H s (R, R) ⊕ xL 1 (0, T ; R). Alors, après avoir donnéun sens aux équations stochastiques et au squelette associé nous prouvonsla borne inférieure suivante.Proposition 2.3.3 Quels que soient T , A et R positifs, l’inégalité suivanteest satisfaite() )lim ɛ→0 ɛ log P Y(u ɛ,Ψ0 A (T ) ≥ R ≥ − 3R2128A 2 T 3Pour cette borne nous exploitons l’identité d’énergie qui suit.) (Y(S m,Ψ0 A (h)(t) = 2Re i∫ t0∫RS m,Ψ0 A(h)(s, x)∂ x S m,Ψ0 A (h)(s, x)dxds),Nous obtenons également la borne supérieure correspondante.46

Présentation des résultatsProposition 2.3.4 Quels que soient T , A et R positifs, l’inégalité suivanteest satisfaite() ) ( 3 2Rlim ɛ→0 ɛ log P Y(u ɛ,Ψ0 A (T ) ≥ R ≤ −(16) 2).A 2 T 3 ‖Φ‖ 2 L c(L 2 ,W 1,∞ (R,R)) ∨ 1Nous obtenons, comme dans le cas additif, le facteur − 1 et que sur uneT 3échelle logarithmique les queues sont bien indistingables des queues Gaussiennes.Il s’agit bien du résultat auquel on s’attend d’après la référence [59].Les ordres de grandeur en A sont en − 1 . Nous nous attendrions pourtantA 2dans ce cas, au vu de [59] à un ordre de grandeur en − 1 .A 42.4 Application à la sortie d’un domaine d’attractionpour des équations de Schrödinger nonlinéaires stochastiques faiblement amortiesNous présentons dans cette section une autre application des principesde grandes déviations trajectoriels pour les solutions d’équations de typeNLS faiblement amorties perturbée par un bruit additif ou multiplicatif.Nous étudions ici l’asymptotique lorsque l’amplitude du bruit tend vers 0du temps moyen et du point de sortie d’un voisinage de 0. L’article [84]correspondant figure en annexe D.Dans cet article nous étudions des équations faiblement amorties. Dansle cas du bruit additif nous avonsidu ɛ,u 0= (∆u ɛ,u 0+ λ|u ɛ,u 0| 2σ u ɛ,u 0− iαu ɛ,u 0)dt + √ ɛdW,λ = ±1, x ∈ R doù α et ɛ sont strictement positifs et où la donnée initiale u 0 est dans L 2ou H 1 . Le bruit est toujours coloré en espace et le processus de Wienerest à valeur L 2 ou H 1 . Quand le bruit est multiplicatif réel avec produitStratonovich, l’équation s’écritidu ɛ,u 0= (∆u ɛ,u 0+λ|u ɛ,u 0| 2σ u ɛ,u 0−iαu ɛ,u 0)dt+ √ ɛu ɛ,u 0◦dW, λ = ±1, x ∈ R d .Le processus de Wiener est alors à valeurs dans H s R avec s > d 2+ 1, espacede Sobolev basé sur L 2 de fonctions à valeurs réelles. Les données initialessont dans ce cas dans H 1 . Dans L 2 le phénomène de sortie n’existe pas car lamasse décroît. On peut trouver des résultats sur les équations déterministespar exemple dans [86, 90, 138].47

Présentation des résultatsEn l’absence de bruit les solutions tendent vers zéro dans L 2 (respectivementdans H 1 ). En présence de bruit par contre, les trajectoires aléatoiressortent de voisinages de zéro invariants par le flot de l’équation déterministe.Le comportement est donc en cela complètement différent de celui du systèmedéterministe. Le temps de sortie est exponentiel et plus le bruit est faibleplus le temps est grand. Mais aussi, les trajectoires qui sortent du domaineou les points de sortie les plus probables minimisent une certaine énergie(énergie ”effective” en physique).En présence d’un unique extremum le comportement le plus probableest le comportement qui minimise l’énergie et l’évolution est essentiellementdéterministe. Sinon, à chaque trajectoire ou point de sortie est associé unpoids. Notons que l’étude menée dans [67] correspond au cas où plusieurséquilibres coexistent.L’étude de ce problème trouve par exemple des applications en physique(mécanique quantique et statistique, optique...). Elle intervient égalementen économie et en particulier en macro économie financière. Voici donc unexemple d’application à l’optique ou l’hydrodynamique et plus généralementtous les domaines où intervient notre EDPS.Nous menons dans ce papier une étude analogue à celle initiée par Freidlinet Wentzell pour les équations différentielles perturbées par un petit bruitadditif, c.f. [73] Chapitre 4. Cette étude a été généralisée au cas de bruitsmultiplicatifs mais la dimension finie est utilisée très régulièrement dans lapreuve afin d’obtenir de la compacité c.f. par exemple [48] Chapitre 5. Le casde la dimension infinie est plus complexe, les propriétés de compacité disparaissenten général et le semi-groupe d’évolution n’est pas uniformémentcontinu mais seulement fortement continu. Ce problème a déjà été étudiépour certaines EDPS, c.f. par exemple [29, 34, 68]. Le problème que nousétudions pose certaines difficultés particulières : le groupe n’a pas de propriétésrégularisantes, les variables d’espace vivent dans tout l’espace R d , lanon linéarité n’est jamais localement Lipschitzienne sauf lorsque d = 1 pourdes solutions dans H 1 .Nous montrons les deux théorèmes qui suivent valables dans L 2 (pour unbruit additif) et dans H 1 (pour un bruit additif ou multiplicatif). Le domaineD est un borélien de L 2 (respectivement H 1 ) borné dans L 2 (respectivementH 1 ) contenant zéro dans son intérieur et invariant par le flot de l’équationdéterministe. Le temps de sortie est défini parτ ɛ,u 0= inf {t ≥ 0 : u ɛ,u 0(t) ∈ D c } .48

Présentation des résultatsNous utiliserons pour l’étude de ce problème des PGDs et notons doncS(u 0 , h) le squelette de l’équation stochastique, i.e. la solution mild del’équation contrôlée{i( dudt + αu) = ∆u + λ|u| 2σ u + Φh,u(0) = u 0où u 0 appartient à L 2 ou H 1 dans le cas additif ou{i( dudt + αu) = ∆u + λ|u| 2σ u + uΦh,u(0) = u 0où u 0 appartient à H 1 dans le cas multiplicatif.La fonction de taux des PGDs trajectoriels est alors toujours définiecommeI u 0T (w) = 1 ∫ Tinf‖h(s)‖ 22 h∈L 2 (0,T ;L 2 Lds.): S(u 0 ,h)=w2 0Nous notons dans ce qui suit par N (B, ρ) pour ρ positif le ρ−voisinage dansL 2 de l’ensemble B.Nous définissonse = inf { I 0 T (w) : w(T ) ∈ D c , T > 0 } .Si ρ est positif et suffisamment petit nous posonse ρ = inf { I u 0T (w) : ‖u 0‖ L 2 ≤ ρ, w(T ) ∈ (D −ρ ) c , T > 0 } ,où D −ρ = D \ N (∂D, ρ) et ∂D désigne le bord de D dans L 2 .Enfin nous définissonse = limρ→0e ρ .Dans le cas H 1 , il convient de remplacer dans ce qui précède L 2 par H 1 .Dans tous les cas 0 < e ≤ e. Nous ne montrons pas dans ce papier quee = e, c’est un problème de contrôle qui semble difficile et qui sera étudiéultérieurement.Le résultat sur le temps de sortie s’énonce dans L 2 et dans H 1 pour desdomaines dans L 2 ou dans H 1 comme suit.49

Présentation des résultatsTheorem 2.4.1 Quel que soit u 0 dans D et δ strictement positif, il existeL strictement positif tel que( ( ( ) ( )))lim ɛ→0 ɛ log P τ ɛ,u 0 e − δ e + δ/∈ exp , exp≤ −L,ɛɛet quelque soit u 0 dans D,e ≤ lim ɛ→0 ɛ log E (τ ɛ,u 0) ≤ lim ɛ→0 ɛ log E (τ ɛ,u 0) ≤ e.De plus, quel que soit δ strictement positif, il existe L strictement positif telque( ( ))lim ɛ→0 ɛ log sup P τ ɛ,u 0 e + δ≥ exp≤ −L,u 0 ∈Dɛetlim ɛ→0 ɛ log sup E (τ ɛ,u 0) ≤ e.u 0 ∈DLe deuxième théorème caractérise formellement les points de sortie. Nousdéfinissons pour ρ strictement positif et suffisamment petit, N un fermé de∂D (pour la topologie de L 2 , respectivement H 1 ) dans L 2 (respectivementH 1 ),e N,ρ = inf { I u 0T (w) : ‖u 0‖ L 2 ≤ ρ, w(T ) ∈ (D \ N (N, ρ)) c , T > 0 } ,dans le cas H 1 remplacer ‖u 0 ‖ L 2 par ‖u 0 ‖ H 1 et N (N, ρ) par le ρ−voisinagedans H 1 . Nous posons ensuitee N = limρ→0e N,ρ .Comme e ρ ≤ e N,ρ nous avons bien e ≤ e N . Nous prouvons doncThéorème 2.4.1 Si e N > e, alors quel que soit u 0 dans D, il existe Lstrictement positif tel quelim ɛ→0 ɛ log P (u ɛ,u 0(τ ɛ,u 0) ∈ N) ≤ −L.Nous pouvons en déduire le corollaireCorollaire 2.4.2 Supposons que v ∗ appartenant à ∂D soit tel que quel quesoit δ strictement positif et N = {v ∈ ∂D : ‖v − v ∗ ‖ L 2 ≥ δ} (dans le cas H 1noter ‖v − v ∗ ‖ H 1) nous avons e N > e alors∀δ > 0, ∀u 0 ∈ D, ∃L > 0 : lim ɛ→0 ɛ log P (‖u ɛ,u 0(τ ɛ,u 0) − v ∗ ‖ L 2 ≥ δ) ≤ −L,(dans le cas H 1 noter ‖ · ‖ H 1).50

Présentation des résultatsCes résultats sont démontrés grâce à des PGDs uniformes en les donnéesinitiales avec une formulation des bornes supérieures et inférieures du typeFreidlin-Wentzell. Nous exploitons en particulier les propriétés d’intégrabilitédu groupe libre de Schrödinger, l’évolution de la masse et de l’Hamiltonien,des estimées exponentielles de queues d’intégrales stochastiques et la propriétéde Markov de la solution. Nous verrons que le cas d’un bruit multiplicatifse traite comme celui d’un bruit additif, l’argument de troncature estseulement légèrement différent.Nous concluons en remarquant que si nous étions capables de prouvere = e pour la sortie d’un domaine de H 1 avec un bruit additif blanc enespace et en temps, la caractérisation du point de sortie serait reliée auxondes solitaires. Remarquons qu’il a été obtenu numériquement dans [46]pour des équations de Korteweg-de Vries que l’énergie injectée par le bruitorganise le système et crée des ondes solitaires. Cela serait peut être uneconfirmation de ce fait.2.5 Grandes déviations et théorèmes de support,le cas d’une équation en dimension 1 avec bruitfractionnaire additifDans cette section nous nous intéressons à des équations de Schrödingernon linéaires stochastiques avec un bruit additif fractionnaire en dimension1. Il s’agit donc d’une extension des résultats de l’article [81] à des bruitsGaussiens plus généraux et colorés en temps. Les preuves sont données dansl’article qui figure en annexe E.L’équation stochastique s’écritidu − (∆u + λ|u| 2σ u)dt = dW H , λ = ±1,la donnée initiale u 0 est une fonction de H 1 . Nous considérons à nouveau dessolutions faibles au sens de l’analyse des équations aux dérivées partielles,ou de manière équivalente à des solutions mildsu(t) = U(t)u 0 − iλ∫ t0U(t − s)(|u(s)| 2σ u(s))ds − i∫ t0U(t − s)dW H (s),où (U(t)) t∈Rest le groupe de Schrödinger sur H 1 . Le processus W H est unprocessus de Wiener fractionnaire. Le paramètre H, paramètre de Hurst,appartient à (0, 1). Nous supposons que W H est l’image directe par Φ51

Présentation des résultatsHilbert-Schmidt d’un processus de Wiener cylindrique fractionnaire sur L 2 .Le processus cylindrique est tel que pour toute base Hilbertienne (e j ) j∈Nde)L 2 , il existe des mouvements Browniens fractionnaires(β j H(t) tels queW c (t) = ∑ j∈N βH j (t)e j. Nous faisons l’hypothèse (A) suivanteΦ est Hilbert-Schmidt de L 2 dans H 1+γavec (1 − 2H) < γ < 1 si H < 1 2 et 0 ≤ γ < 1 si H > 1 2 .Un mouvement Brownien fractionnaire est un processus Gaussien centré àaccroissements stationnairesde covarianceE( ∣∣β H (t) − β H (s) ∣ ∣ 2) = |t − s| 2H , t, s > 0,E ( β H (t)β H (s) ) = 1 2(s 2H + t 2H − |s − t| 2H) .La covariance des accroissements passés et futurs est négative si H < 1 2 etpositive si H > 1 2 . Le cas H = 1 2où les accroissements sont indépendantsest celui du mouvement Brownien. Le processus admet une modification àtrajectoires α−Höldériennes pour α < H. Il peut s’écrire, quitte à élargirl’espace de probabilités, sous la formeβ H (t) =∫ t0K H (t, s)dβ(s),où K H est le noyau de carré intégrable triangulaire, i.e. K H (t, s) = 0 sis > t,( ) ∫K H (t, s) = c H (t − s) H− 1 1 t( (2 + cH2 − H (u − s) H− 3 s 22 1 −u)1−H) du,(2.5.1)pour une certaine constante c H . Ce n’est pas une semi-martingale et nousconsidérons l’intégrale stochastique comme une intégrale de Skohorod, commecela a été défini dans [3].Cette intégrale se réécrit comme une intégrale de Skohorod pour le mouvementBrownien. Cela revient à définir dans un premier temps un espacede Hilbert noyau auto reproduisant au niveau du bruit, noté H. On peutreprésenter cet espace au moyen de l’opérateur linéaire KT∗ de l’ensembledes fonctions en escalier E dans L 2 (0, T ) défini pour ϕ dans E par(K ∗ T ϕ) (s) = ϕ(s)K(T, s) +52∫ Tss(ϕ(t) − ϕ(s))K(dt, s).t≥0

Présentation des résultatsAlors H est l’adhérence de E pour la norme ‖ϕ‖ H = ‖K ∗ T ϕ‖ L 2 (0,T ). Pour desfonctions ϕ de E et h de L 2 (0, T ) nous avons∫ T0(K ∗ T ϕ) (t)h(t)dt =∫ T0ϕ(t)(Kh)(dt).Cette dualité permet d’étendre l’intégrale par rapport à Kh(dt), définiepour les fonctions en escalier, à des intégrandes de H. Mais aussi enfin dedéfinir une intégrale stochastique de type Skohorod en définissant, pour desintégrandes ϕ dans H,δ X (ϕ) =∫ T0(K ∗ T ϕ) (t)δβ(t) =∫ T0(K ∗ T ϕ) (t)dβ(t).Pour des intégrandes adaptées l’intégrale de Skohorod s’écrit donc commeune intégrale stochastique usuelle pour le mouvement Brownien.Nos intégrandes seront déterministes et nous envisagerons des intégrandesà valeurs dans un espace de Hilbert comme cela est fait dans [137]. Ellerevient à définir KT ∗ comme un opérateur sur des fonctions à valeurs un espacede Hilbert avec l’expression ci-dessus. Dans ce cas, l’intégrale est uneintégrale au sens de Bochner.La relation de dualité est encore vraie pour des fonctions en escalier àvaleurs dans un espace de Hilbert. L’opérateur KT ∗ commute avec le produitscalaire sur l’espace de Hilbert. Ainsi le produit scalaire de l’intégrale stochastiqueest l’intégrale stochastique dans R du produit scalaire. Lorsquel’intégrande est une famille d’opérateurs (Λ(t)) t∈[0,T ]d’un espace de HilbertE dans un espace de Hilbert E ′ satisfaisant∑j∈N∫ Tl’intégrale est définie pour t positif par∫ t0Λ(s)dW H (s) = ∑ j∈N0∫ t0‖ (KT ∗ Λ(·)Φe j ) (t)‖ 2 E ′dt < ∞,Λ(s)Φe j dβ H j (s) = ∑ j∈N∫ t0(K ∗ t Λ(·)Φe j ) (s)dβ j (s).Nous commençons alors par montrer le résultat suivant sur la convolutionstochastique.53

Présentation des résultatsLemme 2.5.1 La convolution stochastique Z : t ↦→ ∫ t0 U(t − s)dW H (s) estbien définie. De plus, sous l’hypothèse (A), elle admet une modification àtrajectoires α−Hölderiennes pour α < γ 2∧H si γ > 0 et seulement continuessi γ = 0 (cela n’est possible que si H > 1 2 ).Elle définit une variable aléatoire à valeurs dans C ( [0, ∞); H 1) . De plus,les mesures images directes µ Z,T de µ Z par la restriction sur C ( [0, T ]; H 1)pour T positif sont des mesures Gaussiennes centrées.Nous considérons par la suite une telle modification. Nous notons v u 0(z) lasolution mild de{idvdt = ∆v + λ|v − iz|2σ (v − iz)u(0) = u 0 ∈ H 1 ,où z appartient à C ( [0, ∞); H 1) . Le problème est localement bien posé grâceà un argument de point fixe dans C ( [0, T ]; H 1) pour T suffisamment petit.Celui-ci utilise le fait que la non linéarité est Lipschitzienne sur les bornésde H 1 .Nous définissons alors les solutions comme des solutions dans l’espace detrajectoires explosives E(H 1 ). Rappelons que nous commençons par ajouterun point ∆ à l’espace H 1 et que l’on munit l’ensemble de la topologie telle queles ouverts sont ceux de H 1 et les complémentaires dans H 1 ∪{∆} des fermésbornés. L’ensemble C([0, ∞); H 1 ∪ {∆}) est alors bien défini comme l’intersectiondes espaces C([0, T ]; H 1 ∪ {∆}) pour T positif. Le temps d’explosionde f dans C([0, ∞); H 1 ∪ {∆}) est alors T (f) = inf{t ∈ [0, ∞) : f(t) = ∆},avec la convention que inf ∅ = ∞. Nous pouvons alors définirE(H 1 ) = { f ∈ C([0, ∞); H 1 ∪ {∆}) : f(t 0 ) = ∆ ⇒ ∀t ≥ t 0 , f(t) = ∆ } ,muni de la topologie définie par la base de voisinagesV T,r (ϕ 1 ) = { ϕ ∈ E(H 1 ) : T (ϕ) > T, ‖ϕ 1 − ϕ‖ C([0,T ];H 1 ) ≤ r }de ϕ 1 dans E(H 1 ) pour T < T (ϕ 1 ) et r positif. C’est un espace séparé.Si nous posons G u 0l’applicationG u 0: z ↦→ v u 0(z) − iz,nous avons u ɛ,u 0= G u 0( √ ɛZ) où Z est la convolution stochastique.Nous notons (F t ) t≥0la filtration engendrée par le processus de Wienerfractionnaire.Nous avons alors les deux résultats qui suivent.54

Présentation des résultatsLemme 2.5.2 L’applicationest continue.C ( [0, ∞); H 1) → E ( H 1)z ↦→ G u 0(z)Théorème 2.5.1 Sous l’hypothèse (A) et pour des données initiales u 0 F 0mesurables à valeurs dans H 1 , il existe une unique solution mild au problèmede Cauchy continue à valeurs dans H 1 . Elle est définie sur un intervalle detemps aléatoire [0, τ ∗ (u 0 , ω)) où τ ∗ (u 0 , ω) est un temps d’arrêt tel queτ ∗ (u 0 , ω) = ∞ orlim ‖u(t)‖t→τ ∗ H(u 0 ,ω) 1 = ∞.En outre, τ ∗ est presque sûrement semi continue par rapport à u 0 . La solutionu définit une variable aléatoire à valeurs dans E ( H 1) .Lemme 2.5.3 L’opérateur de covariance de Z sur L 2 ( 0, T ; L 2) est donnépour h dans L 2 ( 0, T ; L 2) parQh(t) = ∑ ∫ T ∫ t∧uj∈N 0 0(K∗T1l [0,t] (·)U(t − ·)Φe j)(s)((K∗T1l [0,u] (·)U(u − ·)Φe j)(s), h(u))L 2 dsdu,lorsque H > 1 2nous pouvons écrire Qh(t) comme(c 2 H H −2) 1 2 (B 2 − 2H, H − 1 ) ∫ T ∫ t ∫ s|u−v| 2H−2 U(t−v)ΦΦ ∗ U(u−s)h(s)dudvds,2 0 0 0où B est la fonction Beta.L’espace de Hilbert noyau auto reproduisant de µ Z,T est Im Q 1 2 avec lanorme de la structure image. Il vaut également ImL où L est défini pour hdans L 2 ( 0, T ; L 2) par∫ t(K∗T 1l [0,t] (·)U(t − ·)Φe j)(s)(h(s), ej ) L 2ds.Lh(t) = ∑ j∈N0De plus les mesures images directes pour ɛ positif de x ↦→ √ ɛx sur C ( [0, ∞); H 1)satisfont un PGD de vitesse ɛ et de bonne fonction de tauxI Z (f) = 1 {}inf ‖h‖ 22 h∈L 2 (0,∞;L 2 L):L(h)=f2 (0,∞;L 2 ).55

Présentation des résultatsNous en déduisons les résultats suivants.Soit µ uɛ,u0 les lois sur E(H 1 ) des solutions mild u ɛ,u 0de{ idu − (∆u + λ|u| 2σ u)dt = √ ɛdW H ,u(0) = u 0 ∈ H 1 .(2.5.2)Nous avons le PGD suivant.Théorème 2.5.2 Les lois µ uɛ,u0 sur E(H 1 ) satisfont un PGD de vitesse ɛet de bonne fonction de tauxI u 0(w) = 1 2{}inf ‖h‖ 2h∈L 2 (0,∞;L 2 L):S(u 0 ,h)=w2 (0,∞;L 2 ),où S(u 0 , h), le squelette, est la solution mild dans E(H 1 ) du problème decontrôle⎧⎨ i ∂u∂t − (∆u + λ|u|2σ u) = Φ ˙Kh,u(0) = u⎩0 ∈ H 1h ∈ L 2 ( 0, ∞; L 2) ;Seule l’intégrale, ou celle de la forme mild, du membre de droite est définieà partir de la relation de dualité.Nous obtenons alors avec les mêmes arguments que dans [81] le théorème desupport.Théorème 2.5.3 Le support de la loi µ u1,u0sur E(H 1 ) est donné parsupp µ u1,u0 = ImL E(H1 ).56

Chapitre 3Conclusion et perspectivesNous avons étudié plusieurs aspects des grandes déviations lorsque lebruit tend vers 0 pour des équations de Schrödinger non linéaires stochastiques.Nous nous sommes intéressés à des principes de grandes déviationstrajectoriels. Nous avons traité des non linéarités sous critiques et surcritiqueset déduit des résultats sur l’explosion en temps fini. Nous noussommes aussi intéressés à des applications en physique et avons appliquénos résultats à l’étude de l’erreur de transmission par solitons dans les fibresoptiques. Nous avons obtenu de manière rigoureuse plusieurs résultats dephysique dont les preuves semblent difficiles à justifier mathématiquement.Certains résultats sont nouveaux. Nous nous sommes intéressés à l’étude destemps et des points de sortie d’un domaine d’équilibre pour des équationsfaiblement amorties. Enfin nous avons commencé à étudier le cas de bruitsGaussiens plus généraux.Plusieurs approfondissements seraient possibles.Tout d’abord, nous avons vu que l’étude de l’asymptotique des queuespour des bruits tendant vers 0 est reliée à un problème de contrôle optimal.Les queues de la position sont plus épaisses que celles de la masse pourde longues fibres. Le risque de voir la position excéder un seuil (relativementpetit par rapport à la longueur T de la fibre) est supérieur à celuique la masse excède un seuil. Néanmoins il est possible de concevoir desfibres avec éléments de contrôle tels que les queues, en particulier de la position,soient moins épaisses. Dans [128] il est suggéré que du point de vuede l’ingénierie il faille optimiser sur de tels champs externes tout en satisfaisantune contrainte de coût. Le nouveau problème de contrôle optimalnécessite une double optimisation. Nous pourrions alors diminuer exponen-57

Conclusion et perspectivestiellement les fluctuations non désirées. Dans [63], différents éléments decontrôle réduisant la fluctuation du centre sont suggérés. Par contre il n’estpas proposé d’optimiser sur ces éléments afin d’en définir avec des propriétésoptimales.Cette même idée peut permettre de réduire ou d’augmenter le tempsmoyen de sortie d’un domaine d’attraction ou d’agir sur le point de sortie.Lorsque plusieurs points d’équilibre existent cette technique peut permettrede rendre plus ou moins fréquentes certaines transitions mais aussi de définira priori la forme des transitions.Il est possible également de trancher la question que se posent dansbeaucoup de papiers les physiciens sur le caractère Gaussien ou non de la loidu centre. Nous avons obtenu que sur une échelle logarithmique les queuessont les mêmes que celle d’une loi exponentielle. La différence se situeraitau niveau des facteurs pré-exponentiels. Ceux-ci peuvent être obtenus pardes grandes déviations précises. On pourrait donc chercher à montrer desgrandes déviations précises pour les équations de equations de Schrödingernon linéaires stochastiques et s’inspirer notamment de [6, 7, 14, 121, 122,123, 124].Il serait aussi très intéressant de regarder de plus près le résultat sur lasortie d’un domaine dans H 1 . En effet, nous avons vu que pour des bruitsadditifs celui-ci semble intimement lié aux ondes solitaires. Il conviendraitégalement de chercher à résoudre les problèmes de contrôle qui subsistent.Mais aussi, pour certains systèmes purement Hamiltoniens, ce qui est lecas pour les équations sans amortissement, l’asymptotique de l’espérance dupremier temps de sortie de domaines délimités par des ensembles de niveaude l’Hamiltonien ou de toute autre quantité invariante est souvent obtenuaprès changement de temps, application de la formule d’Itô aux quantitésinvariantes évaluées en la solution aléatoire et la technique de ”stochasticaveraging” (voir par exemple [69, 70, 71, 72, 73, 74, 133, 134, 135]). Le changementde temps est tel que le mouvement lent entre les différents niveauxdes quantités invariantes du flot déterministe soit de vitesse 1. Les fluctuationsrapides s’effectuant le long des ensembles de niveau. En général l’échellede temps n’est plus exponentielle. Il serait intéressant de regarder ce typede problème pour des équations de Schrödinger non-linéaires stochastiques.Par ailleurs plusieurs résultats de stabilité orbitale (résultats de stabilitétenant compte des groupes de symétrie pour l’équation), c.f. par58

Conclusion et perspectivesexemple [26, 91, 127, 140], ou de stabilité asymptotique, c.f. par exemple[19, 20, 31, 32, 75, 130], existent pour les équations de Schrödinger nonlinéaires. Il serait très intéressant d’étudier l’influence d’un petit bruit surun résultat de stabilité asymptotique, mais ces résultats sont partiels et parexemple valables pour certaines données initiales seulement, ils reposent surdes hypothèses sur l’opérateur linéarisé au voisinage de l’onde solitaire...On pourrait chercher à prouver des grandes déviations pour les mesuresinvariantes lorsque le bruit tend vers 0 comme cela est fait dans [28, 73, 132].Des résultats sur les mesures invariantes pour une équation de Schrödingernon linéaire stochastique avec non-linéarité cubique et focalisante sur undomaine borné et avec un bruit additif ont été prouvés dans [45].Il serait aussi possible d’étudier des bruits multiplicatifs relativementgénéraux en adoptant une approche basée sur les trajectoires rugueuses.Il conviendrait alors de montrer un résultat de continuité par rapport auxtrajectoires rugueuses du processus dirigeant l’équation analogue à celui de[107] pour les EDS. Nous pourrions en déduire des principes de grandesdéviations et des théorèmes de support en procédant comme dans [103].Enfin, nous pourrions aussi nous intéresser à des grandes déviations pourla famille de mesures d’occupation( 1T∫ T0)1l B (u u 0(s))dsT >0où B est un borélien de L 2 ou H 1 et plus généralement pour la famille demesures empiriques( ∫ 1 T)δT u u 0 (s) ds0T >0où u u 0est la solution de l’équation stochastique issue de u 0 . Les principes degrandes déviations sont alors des principes de grandes déviations de niveau 2car les lois des mesures aléatoires ci-dessus sont des mesures sur des espacesde mesures. Ils quantifient un résultat de convergence faible vers la mesurede Dirac en la mesure invariante. Pour cela il est possible de se référer à [48]pour le cas des chaînes de Markov, à [53] pour des résultats plus généraux età [30, 96, 115, 120] pour des diffusions . On peut aussi se référer aux articlesde Donsker et Varadhan [54, 55, 56, 57] et à l’application [58] au problèmedu Polaron.59

Appendix ALarge deviations and supportfor nonlinear Schrödingerequations with additive noiseand applicationsAbstract: Sample path large deviations for the laws of the solutions ofstochastic nonlinear Schrödinger equations when the noise converges to zeroare presented. The noise is a complex additive Gaussian noise. It is white intime and colored in space. The solutions may be global or blow-up in finitetime, the two cases are distinguished. The results are stated in trajectoryspaces endowed with projective limit topologies. In this setting, the supportof the law of the solution is also characterized. As a consequence, results onthe law of the blow-up time and asymptotics when the noise converges tozero are obtained. An application to the transmission of solitary waves infiber optics is also given.A.1 IntroductionIn this article, the stochastic nonlinear Schrödinger (NLS) equation with apower law nonlinearity and an additive noise is studied. The deterministicequation occurs as a basic model in many areas of physics: hydrodynamics,plasma physics, nonlinear optics, molecular biology. It describes the propagationof waves in media with both nonlinear and dispersive responses. It isan idealized model and does not take into account many aspects such as in-

Appendix A.Large deviations in the additive case and applicationshomogeneities, high order terms, thermal fluctuations, external forces whichmay be modeled as a random excitation (see [50, 59, 62, 63, ?, 114]). Propagationin random media may also be considered. The resulting re-scaledequation is a random perturbation of the dynamical system of the followingform:i ∂ ∂t ψ − (∆ψ + λ|ψ|2σ ψ) = ξ, x ∈ R d , t ≥ 0, λ = ±1, (A.1.1)where ξ is a complex valued space-time white noise with correlation function,following the notation used in [62],E [ ξ(t 1 , x 1 )¯ξ(t 2 , x 2 ) ] = Dδ t1 −t 2⊗ δ x1 −x 2D is the noise amplitude and δ denotes the Dirac mass. When λ = 1 thenonlinearity is called focusing, otherwise it is defocusing.With the notations of the next section, the unbounded operator −i∆on L 2 with domain H 2 is skew-adjoint. Stone’s theorem gives thus thatit generates a unitary group (S(t) = e −it∆ ) t∈R . The Fourier transformgives that this group is also unitary on every Sobolev space based on L 2 .Consequently, there is no smoothing effect in the Sobolev spaces. We arethus unable to treat the space-time white noise and will consider a complexvalued centered Gaussian noise, white in time and colored in space.In the present article, the formalism of stochastic evolution equationsin Banach spaces as presented in [34] is adopted. This point of view ispreferred to the field and martingale measure stochastic integral approach,see [139], in order to use a particular property of the group, namely hypercontractivity.The Strichartz inequalities, presented in the next section,show that some integrability property is gained through time integrationand ”convolution” with the group. In this setting, the Gaussian noise isdefined as the time derivative in the sense of distributions of a Q-Wienerprocess (W (t)) t∈[0,∞) on H 1 . Here Q is the covariance operator of the lawof the H 1 −random variable W (1), which is a centered Gaussian measure.With the Itô notations, the stochastic evolution equation is writtenidu − (∆u + λ|u| 2σ u)dt = dW.(A.1.2)The initial datum u 0 is a function of H 1 . We will consider solutions of NLSthat are weak solutions in the sense used in the analysis of partial differentialequations or equivalently mild solutions which satisfyu(t) = S(t)u 0 −iλ∫ t0S(t−s)(|u(s)| 2σ u(s))ds−i62∫ t0S(t−s)dW (s). (A.1.3)

Appendix A.Large deviations in the additive case and applicationsThe well posedness of the Cauchy problem associated to (A.1.1) in the deterministiccase depends on the size of σ. If σ < 2 d, the nonlinearity issubcritical and the Cauchy problem is globally well posed in L 2 or H 1 . Ifσ = 2 d , critical nonlinearity, or 2 d < σ < 2d−2 when d ≥ 3 or simply σ > 2 d otherwise,supercritical nonlinearity, the Cauchy problem is locally well posedin H 1 ; see [99]. In this latter case, if the nonlinearity is defocusing, the solutionis global. In the focusing case some initial data yield global solutionswhile it is known that other initial data yield solutions which blow up infinite time; see [25, 136].In [37], the H 1 results have been generalized to the stochastic case and existenceand uniqueness results are obtained for the stochastic equation underthe same conditions on σ. Continuity with respect to the initial data andthe perturbation is proved. It is shown that the proof of global existencefor a defocusing nonlinearity or for a focusing nonlinearity with a subcriticalexponent, could be adapted in the stochastic case even if the massand HamiltonianH (u(t)) = 1 2∫N (u(t)) = ‖u(t)‖ 2 L 2R d |∇u(t, x)| 2 dx −λ ∫|u(t, x)| 2σ+2 dx2σ + 2 R dare no longer conserved. For a focusing nonlinearity and critical or supercriticalexponents, the solution may blow-up in finite time. The blow-up timeis denoted by τ(ω). It satisfies either lim t→τ(ω) ‖u(t)‖ H 1 = ∞ or τ(ω) = ∞,even if the solution is obtained by a fixed point argument in a ball of a spaceof more regular functions than C([0, T ]; H 1 ).In this article, we are interested in the law of the paths of the randomsolution. When the noise converges to zero, continuity with respect to theperturbation gives that the law converges to the Dirac mass on the deterministicsolution. In the following, a large deviation result is shown. It givesthe rate of convergence to zero, on a logarithmic scale, of the probabilitythat paths are in sets that do not contain the deterministic solution. A generalresult is stated for the case where blow-up in finite time is possible anda second one for the particular case where the solutions are global. Also, thestronger the topology, the sharper are the estimates. We will therefore takeadvantage of the variety of spaces that can be considered for the fixed pointargument, due to the integrability property, and present the large deviationprinciples in trajectory spaces endowed with projective limit topologies. Acharacterization of the support of the law of the solution in these trajectory63

Appendix A.Large deviations in the additive case and applicationsspaces is proved. The two results can be transferred to weaker topologiesor more generally by any continuous mapping. The first application is aproof that, for certain noises, with positive probability some solutions blowup after any time T . Some estimates on the law of the blow-up time whenthe noise converges to zero are also obtained. This study is yet anothercontribution to the study of the influence of a noise on the blow-up of thesolutions of the focusing supercritical NLS; see in the case of an additivenoise [38, 40]. A second application is given. It consists in obtaining similarresults as in [62] with an approach based on large deviations. The aim isto compute estimates of error probability in signal transmission in opticalfibers when the medium is random and nonlinear, for small noises. Uniformlarge deviations for small noise asymptotics when the noise enters linearlyas a random potential the NLS equation are studied in [82]. In that casewe had to use a more elaborate proof based on the Freidlin and Wentzellinequality and the continuity of the skeleton with respect to the control onthe sets levels of the rate function of the initial Wiener process less or equalto a positive constant since in that case the Itô map fails to be continuousat the level of paths for the topologies we consider.Section A.2 is devoted to notations and properties of the group, of thenoise and of the stochastic convolution. An extension of the result of continuitywith respect to the stochastic convolution presented in [37] is alsogiven. In Section A.3, the large deviation principles (LDP) is presented.Section A.4 is devoted to the support result and the two last sections to theapplications.A.2 Notations and preliminary resultsThroughout the paper the following notations will be used.The set of positive integers and positive real numbers are denoted respectivelyby N ∗ and R ∗ +, while the set of real numbers different from 0 isdenoted by R ∗ .For p in N ∗ , L p is the classical Lebesgue space of complex valued functionsand W 1,p is the associated Sobolev space of L p functions with first orderderivatives, in the sense of distributions, in L p . When p = 2, H s denotesthe fractional Sobolev space of tempered distributions v ∈ S ′ such that theFourier transform ˆv satisfies (1 + |ξ| 2 ) s/2ˆv ∈ L 2 . The space L 2 is endowedwith the inner product defined by (u, v) L 2 = Re ∫ R d u(x)v(x)dx. Also, whenit is clear that µ is a Borel measure on a specified Banach space, we simplywrite L 2 (µ) and do not specify the Banach space and Borel σ−field.64

Appendix A.Large deviations in the additive case and applicationsIf I is an interval of R, (E, ‖·‖ E ) a Banach space and r belongs to [1, ∞],then L r (I; E) is the space of strongly Lebesgue measurable functions f fromI into E such that t → ‖f(t)‖ E is in L r (I). Let L r loc(0, ∞; E) be the respectivespaces of locally integrable functions on (0, ∞). They are endowed withtopologies of Fréchet space. The spaces L r (Ω; E) are defined similarly.We recall that a pair (r, p) of positive numbers is called an admissiblepair if p satisfies 2 ≤ p 2 (2 ≤ p < ∞ when d = 2 and( )2 ≤ p ≤ ∞ when d = 1) and r is such that 2 r = d 12 − 1 p. For example(∞, 2) is an admissible pair.When E is a Banach space, we will denote by E ∗ its topological dual space.For x ∗ ∈ E ∗ and x ∈ E, the duality will be denoted < x ∗ , x > E ∗ ,E.We recall that Φ is a Hilbert Schmidt operator from a Hilbert space H intoa Hilbert space ˜H if it is a linear continuous operator such that, given acomplete orthonormal system (e H j ) j∈N of H, ∑ j∈N ‖ΦeH j ‖2˜H < ∞. We willdenote by L 2 (H, ˜H) the space of Hilbert Schmidt operators from H into ˜Hendowed with the norm‖Φ‖ L2 (H, ˜H) = tr (ΦΦ∗ ) = ∑ j∈N‖Φe H j ‖ 2˜H,where Φ ∗ denotes the adjoint of Φ and tr the trace. We denote by L s,r2 thecorresponding space for H = H s and ˜H = H r . In the introduction Φ hasbeen taken in L 0,12 .When A and B are two Banach spaces, A ∩ B, where the norm of anelement is defined as the maximum of the norm in A and in B, is a Banachspace. The following Banach spaces defined for the admissible pair (r(p), p)and positive T byX (T,p) = C ( [0, T ]; H 1) ∩ L r(p) ( 0, T ; W 1,p)will be of particular interest.The probability space will be denoted by (Ω, F, P). Also, x ∧ y standsfor the minimum of the two real numbers x and y and x ∨ y for the maximum.We recall that a rate function I is a lower semicontinuous functionand that a good rate function I is a rate function such that for every c > 0,{x : I(x) ≤ c} is a compact set. Finally, we will denote by supp µ the supportof a probability measure µ on a topological vector space. It is thecomplement of the largest open set of null measure.65

Appendix A.Large deviations in the additive case and applicationsA.2.1Properties of the groupWhen the group acts on the Schwartz space S, the Fourier transform givesthe following analytic expression∀u 0 ∈ S, ∀t ≠ 0, S(t)u 0 =1(4iπt) d 2∫R d e−i|x−y|24t u 0 (y)dy.The Fourier transform also gives that the adjoint of S(t) in L 2 and in everySobolev space on L 2 is S(−t), the same bounded operator with time reversal.The Strichartz inequalities, see [99], are the following(i)There exists C positive such that for u 0 in H 1 , T positive and(r(p), p) admissible pair,‖U(·)u 0 ‖ X (T,p) ≤ C ‖u 0 ‖ L 2 ,(ii)For every T positive, (r(p), p) and (r(q), q) admissible pairs, s and ρsuch that 1 s + 1r(q) = 1 and 1 ρ + 1 q= 1, there exists C positive suchthat for f in L s ( 0, T ; W 1,ρ) ,∥ ∫ ·0 U(· − s)f(s)ds∥ ∥X (T,p) ≤ C‖f‖ L s (0,T ;W 1,ρ ).Remark A.2.1 The first estimate gives the integrability property of thegroup, the second gives the integrability of the convolution that allows totreat the nonlinearity.A.2.2Topology and trajectory spacesLet us introduce a topological space that allows us to treat the subcriticalcase or the defocusing case. When d > 2, we set⋂X ∞ =X (T,p) ,T ∈R ∗ 2d+ , 2≤p< d−2it is endowed with the projective limit topology; see [15] and [48]. Whend = 2 and d = 1 we write ( p ∈ [[2, ∞). ) )The set of indices R ∗ 2+ × 2,d−2, ≺ when d > 2 or ( R ∗ + × [2, ∞) , ≺ )when d = 2 or d = 1, where (T, p) ≺ (S, q) if T ≤ S and p ≤ q, is a partiallyordered right-filtering set.66

Appendix A.Large deviations in the additive case and applicationsIf (T, p) ≺ (S, q) and u ∈ X (S,q) , Hölder’s inequality gives that for α suchthat 1 p = α q + 1−α2 , ‖u(t)‖ L p ≤ ‖u(t)‖ 1−αL 2 ‖u(t)‖ α L q.Consequently,‖u(t)‖ W 1,p ≤ (d + 1)‖u(t)‖ 1−αH 1 ‖u(t)‖ α W 1,q .By time integration, along with Hölder’s inequality, the fact that r(q) =αr(p) and that T ≤ S, we obtain that u is a function of X (T,p) and‖u‖ X (T,p) ≤ (d + 1)‖u‖ X (S,q).(A.2.1)If we denote by p (T,p)(S,q) the dense and continuous embeddings from X(S,q) intoX (T,p) , they satisfy the consistency conditions∀ (T, p) ≺ (S, q) ≺ (R, r), p (T,p)(R,r) = p(S,q) (R,r) ◦ p(T,p) (S,q) .Consequently, the projective limit topology is well defined by the followingneighborhood basis, given for ϕ 1 in X ∞ by⎧⎫⎨U(ϕ 1 ; (T, p); ɛ) =⎩ ϕ ∈⋂⎬X (T ′ ,p ′) : ‖ϕ − ϕ 1 ‖ X (T,p) < ɛ⎭ .(T ′ ,p ′ )∈JIt is the weakest topology on the intersection such that for every (T, p) ∈ J,the injection p (T,p) : X ∞ → X (T,p) is continuous. It is a standard fact, see[15], that X ∞ is a Hausdorff topological space.Following from (A.2.1), a countable neighborhood basis of ϕ 1 is given))by ( U ( ϕ 1 ; (n, p(l)); 1 kd > 2. If d = 2 and d = 1, we take p(l) = l.Also it is convenient, for measurability issues, to note that X ∞ can beturned into a complete separable metric space, i.e. a Polish space, setting∀(x, y) ∈ X 2 ∞, d(x, y) =(n,k,l)∈(N ∗ ) 3 , where p(l) = 2 + 4d−2 − 1 land l > d−24if∑n> d−2412 n (‖x − y‖ X (n,p(n)) ∧ 1) .It can be checked that it is also a locally convex Fréchet space.The following spaces are introduced for the case where blow-up mayoccur. Adding a point ∆ to the space H 1 and adapting slightly the proof ofAlexandroff’s compactification, it can be seen that the open sets of H 1 and67

Appendix A.Large deviations in the additive case and applicationsthe complement in H 1 ∪{∆} of the closed bounded sets of H 1 define the opensets of a topology on H 1 ∪ {∆}. This topology induces on H 1 the topologyof H 1 . Also, with such a topology H 1 ∪{∆} is a Hausdorff topological space.Note that in [5], where diffusions are studied, the compactification of R d isconsidered. Nonetheless, compactness is not an important feature and theabove construction is enough for the following.The space C([0, ∞); H 1 ∪ {∆}) is the space of continuous functions withvalue in H 1 ∪ {∆}. Also, if f belongs to C([0, ∞); H 1 ∪ {∆}) we denote theblow-up time byT (f) = inf{t ∈ [0, ∞) : f(t) = ∆}.As in [5], a space of exploding paths, where ∆ acts as a cemetery, is introduced.We setE(H 1 ) = { f ∈ C([0, ∞); H 1 ∪ {∆}) : f(t 0 ) = ∆ ⇒ ∀t ≥ t 0 , f(t) = ∆ } .It is endowed with the topology defined by the following neighborhood basisgiven for ϕ 1 in E(H 1 ) byV T,ɛ (ϕ 1 ) = { ϕ ∈ E(H 1 ) : T (ϕ) ≥ T, ‖ϕ 1 − ϕ‖ L ∞ ([0,T ];H 1 ) ≤ ɛ } ,where T < T (ϕ 1 ) and ɛ > 0.As a consequence of the topology of E(H 1 ), the function T : E(H 1 ) →[0, ∞] is sequentially lower semicontinuous, this is to say that if a sequence offunctions (f n ) n∈N converges to f then lim n→∞ T (f n ) ≥ T (f). Following from(A.2.1), the topology of E(H 1 ) is also defined by the countable neighborhoodbasis given for ϕ 1 ∈ E(H 1 ) by(V T (ϕ1 )− 1 n , 1 k(ϕ 1 ))(n,k)∈(N ∗ ) 2.Therefore T is alower semicontinuous mapping.Note that, as topological spaces, the two following spaces satisfy theidentity{f ∈ E(H 1 ) : T (f) = ∞ } = C([0, ∞); H 1 ).Finally, the analogue of the intersection in the subcritical case endowed withprojective limit topology is defined, when d > 2, by{[ )E ∞ = f ∈ E(H 1 2d) : ∀ p ∈ 2, , ∀ T ∈ [0, T (f)), f ∈ L r(p) ( 0, T ; W 1,p)} .d − 2When d = 2 and d = 1 we write p ∈ [2, ∞). It is endowed with the topologydefined for ϕ 1 in E ∞ by the following neighborhood basisW T,p,ɛ (ϕ 1 ) = {ϕ ∈ E ∞ : T (ϕ) ≥ T, ‖ϕ 1 − ϕ‖ X (T,p) ≤ ɛ} .68

Appendix A.Large deviations in the additive case and applicationswhere T < T (ϕ 1 ), p is as above and ɛ > 0. From the same arguments asfor the space X ∞ , E ∞ is a Hausdorff topological space. Also, as previously,(A.2.1) gives that the topology can be defined for ϕ 1 in E ∞ by the countableneighborhood basis(W T (ϕ1 )− 1 n ,p(n), 1 k(ϕ 1 ))(n,k)∈(N ∗ ) 2 : n> d−24If we denote again by T : E ∞ → [0, ∞] the blow-up time, since E ∞is continuously embedded into E(H 1 ), T is lower semicontinuous. Thus,since {[0, t], t ∈ [0, ∞]} is a π−system that generates the Borel σ−algebra of[0, ∞], T is measurable. Note also that, as topological spaces, the followingspaces are identicalA.2.3{f ∈ E ∞ : T (f) = ∞} = X ∞ .Statistical properties of the noiseThe Q-Wiener process W is such that its trajectories are in C([0, ∞); H 1 ).We assume in the following that Q = ΦΦ ∗ where Φ is a Hilbert-Schmidtoperator from L 2 into H 1 . The Wiener process can therefore be written asW = ΦW c where W c is a cylindrical Wiener process.We recall that for any orthonormal basis (e j ) j∈N of L 2 , there exists asequence∑of real independent Brownian motions (β j ) j∈N such that W c =j∈N β je j . The sum W c = ∑ j∈N β je j is well defined in every Hilbert spaceH such that L 2 is embedded into H with a Hilbert Schmidt embedding. Wesay that it is cylindrical because it is such that the decomposition of W c (1)on cylinder sets (e 1 , ..., e N ) are the finite dimensional centered Gaussian variables(β 1 (1), ..., β N (1)) with a covariance equal to the identity. The law ofW (1) is thus the direct image measure by the Hilbert-Schmidt mapping Φof the natural extension of the corresponding sequence of centered Gaussianmeasures in finite dimensions, with a covariance equal to identity. In otherwords it is the bona-fide σ−additive direct image measure of a Gaussiancylindrical measure. Also, formally, for T positive the coefficients of theseries expansion of the derivative of W c on the tensor product of the completeorthonormal systems of L 2 and of L 2 (0, T ), given for example by thetime derivative of the eigenvectors of the correlation operator of the law onC([0, T ]) of the Brownian motions, is a sequence of independent real-valuedstandard normal random variables. It is thus a Gaussian white noise.In reference [62] the authors define the correlation function by the quantity[ ∂E∂t W c(t + s, x + z) ∂ ]∂t W c(t, x) ,.69

Appendix A.Large deviations in the additive case and applicationswriting down formally the series expansion we obtain in the case of the whitenoise 2δ 0 (s) ⊗ δ 0 (z). In the case of our space-colored noise, we obtain themultiplication of δ 0 (s) by the L 2 function ∑ j∈N Φe j(x + z)Φe j (x), where(e j ) j∈N is a complete orthonormal system of L 2 . Also as the operator Φ belongsto L 0,12 and thus to L 0,02 it may be defined through the kernel K(x, y) =12∑j∈N Φe j(x)e j (y) of L 2 (R d × R d ), considering that (e j ) j∈Nconsists of(f j ) j∈N a complete orthonormal system of L 2 (R d , R) and of (if j ) j∈N . Thismeans that for any square integrable function u, Φu(x) = ∫ RK(x, y)u(y)dy.dIn that case we could write the correlation function(2∫)K(x + z, u)K(x, u)du δ 0 (s).R dIn the following we assume that the probability space is endowed withthe filtration F t = N ∪ σ{W s , 0 ≤ s ≤ t} where N denotes the P−null sets.A.2.4The random perturbationWe define the stochastic convolution by Z(t) = ∫ t0S(t − s)dW (s) and theoperator L on L 2 (0, T ; L 2 ) byLh(t) =∫ twhere I is the injection of H 1 into L 2 .0I ◦ S(t − s)Φh(s)ds, h ∈ L 2 (0, T ; L 2 ),Proposition A.2.2 The stochastic convolution defines a measurable mappingfrom (Ω, F) into ( X ∞ , B X) , where B X stands for the Borel σ−field. Itslaw is denoted by µ Z .The direct images µ Z;(T,p) = p (T,p)∗ µ Z on the real Banach spaces X (T,p)are centered Gaussian measures of reproducing kernel Hilbert space (RKHS)H µ Z;(T,p) = ImL with the norm of the image structure.Proof. Setting F (t) = ∫ t0 S(−u)dW (u), for t ∈ R+ , Z(t) = S(t)F (t) follows.Indeed, if (f j ) j∈N is a complete orthonormal system of H 1 , a straightforwardcalculation gives that (Z(t), f j ) H 1 = (S(t)F (t), f j ) H 1 for every j in N. Thecontinuity of the paths in H 1 follows from the construction of the stochasticintegral with respect to the Wiener process since the deterministic operatorintegrand satisfies ∫ T0 ‖S(−u)Φ‖2 L 0,12< ∞ and from the strong continuity ofthe group.Step 1: We claim that the mapping Z is measurable from (Ω, F) into70

Appendix A.Large deviations in the additive case and applications(X (T,p) , B (T,p)) , where B (T,p) denotes the associated Borel σ−field.Since X (T,p) is a Polish space, every open set is a countable union ofopen balls and consequently B (T,p) is generated by open balls. Note that theevent {ω ∈ Ω : ‖Z(ω) − x‖ X (T,p) ≤ r} is equal to( ⋂s∈Q∩[0,T ] {ω ∈ Ω : ‖Z(s)(ω) − x‖ H 1 ≤ r} )⋂ { ω ∈ Ω : ‖Z(ω) − x‖ L r(p) (0,T ;W 1,p ) ≤ r }.Also, note that, since (Z(t)) t∈R + is a collection of H 1 random variables,the first part is a countable intersection of elements of F. Consequently, itsuffices to show that : ω ↦→ (t ↦→ Z(t)) defines a L r(p) (0, T ; W 1,p ) randomvariable.Consider (Φ n ) n∈N a sequence of operators of L 0,22 converging to Φ forthe topology of L 0,12 and Z n the associated stochastic convolutions. TheSobolev injections along with Hölder’s inequality give that when d > 2 and2 ≤ p ≤ 2dd−2 , H1 is continuously embedded in L p . It also gives that, whend = 2, H 1 is continuously embedded in every L p for every p ∈ [2, ∞) andfor every p ∈ [2, ∞] when d = 1. Consequently, for every n in N, Z n definesa C([0, T ]; H 2 ) random variable and therefore a L r(p) ( 0, T ; W 1,p) randomvariable for the corresponding values of p.Revisiting the proof of Proposition 3.1 in reference [37] and letting 2σ+2be replaced by any of the previous values of p besides p = ∞ when d = 1, thenecessary measurability issues to apply the Fubini’s theorem are satisfied.Also, one gets the same estimates and that there exists a constant C(d, p)such that for every n and m in N,[]E ‖Z n+m (ω) − Z n (ω)‖ r ≤ C(d, p)T r L r(p) (0,T ;W 1,p 2 −1 ‖Φ)n+m − Φ n ‖ r .L 0,12The sequence (Z n ) n∈N is thus a Cauchy sequence of the Banach spaceL r ( Ω; L r ( 0, T ; W 1,p)) and converges to ˜Z. The previous calculation alsogives that[]E ‖Z n (ω) − Z(ω)‖ r ≤ C(d, p)T r L r(p) (0,T ;L p 2 −1 ‖Φ)n − Φ‖ r .L 0,12Therefore ˜Z = Z, Z belongs to L r(p) ( 0, T ; W 1,p) and it defines a measurablemapping as expected.Note that in X ∞ , to simplify the notations, we did not write the casesp = ∞ when d = 1 or p =2dd−2when d > 2. We are indeed interestedin results on the laws of the solutions of stochastic NLS and not really on71

Appendix A.Large deviations in the additive case and applicationsthe stochastic convolution. Also, the result of continuity in the next sectionshows that we necessarily lose on p in order to interpolate with 2 < p < p ′and have a nonzero exponent on the L 2 −norm. Therefore, even if it seemsat first glance that we lose on the Sobolev’s injections, it is not a restriction.Step 2: We show that the mapping Z is measurable with values in X ∞with the Borel σ−field B X∞ .From step 1, given x ∈ X ∞ , for every n in N ∗ such that n > d−24the mapping ω ↦→ ‖Z(ω) − x‖ X (n,p(n)) from (Ω, F) into (R + , B(R + )), whereB(R + ) stands for the Borel σ−field of R + , is measurable. Thusω ↦→ d(Z(ω), x) = limN∑N→∞n=112 n (‖Z(ω) − x‖ X (n,p(n)) ∧ 1)is measurable. Consequently, for every r in R + , {ω ∈ Ω : d(Z(ω), x) < r}belongs to F.Note that the law µ Z;X∞ of Z on the metric space X ∞ , which is a positiveBorel measure, is therefore also regular and consequently it is a Radonmeasure.Step 3 (Statements on the measures µ Z;(T,p) ): For (T, p) in R ∗ +×[ )2,2d−2when d > 2 or R ∗ + × [2, ∞) when d = 2 or d = 1, let i (T,p) denote the continuousinjections from X (T,p) into L 2 (0, T ; L 2 ) and µ Z;L = (i (T,p) ) ∗ µ Z;(T,p) .The σ−field on L 2 (0, T ; L 2 ) is the Borel σ−field. Let h ∈ L 2 (0, T ; L 2 ), then(h, i(T,p) (Z) ) L 2 (0,T ;L 2 ) = ∫ T0∞∑i,j=1∫ t0(e j , S(t − s)Φe i ) L 2dβ i (s)(h(t), e j ) L 2and from classical computation it is the almost sure limit of a sum of independentcentered Gaussian random variables, thus µ Z;L is a centered Gaussianmeasure.Every linear continuous functional on L 2 (0, T ; L 2 ) defines by restriction alinear continuous functional on X (T,p) . Thus, L 2 (0, T ; L 2 ) ∗ could be thoughtof as a subset of ( X (T,p)) ∗. Since i(T,p) is a continuous injection, L 2 (0, T ; L 2 ) ∗is dense in ( X (T,p)) (∗ (Xfor the weak ∗ topology σ(T,p) ) )∗, X(T,p). Thismeans that, given x ∗ ∈ ( X (T,p)) ∗, there exists a sequence (hn ) n∈Nof elementsof L 2 (0, T ; L 2 ) such that for every x ∈ X (T,p) ,(hn , i (T,p) (x) ) L 2 (0,T ;L 2 ) =< x∗ , x > (X (T,p)) ∗ .,X (T,p)limn→∞In other words, the random variable < x ∗ , · > (X (T,p)) ∗ ,X (T,p)is a pointwiselimit of ( h n , i (T,p) (·) ) which are, from the above, centered GaussianL 2 (0,T ;L 2 )72

Appendix A.Large deviations in the additive case and applicationsrandom variables. As a consequence, µ Z;(T,p) is a centered Gaussian measure.Recall that the RKHS H µ Z;L of µ Z;L is ImR L where R L is the mappingfrom H ∗ µ Z;L = L 2 (0, T ; L 2 ) ∗L2 (µ Z;L )with the inner product derived from theone in L 2 (µ Z;L ) into L 2 (0, T ; L 2 ) defined for ϕ in H ∗ µ Z,L∫R L (ϕ) =L 2 (0,T ;L 2 )xϕ(x)µ Z;L (dx).The same is true for H µ Z;(T,p) replacing L 2 (0, T ; L 2 ) by X (T,p) and µ Z;L byµ Z;(T,p) .Since µ Z;L is the image of µ Z;(T,p) , taking x ∗ ∈ L 2 (0, T ; L 2 ) ∗ , we obtainthatby‖x ∗ ‖ L 2 (µ Z;L )= ∫ L 2 (0,T ;L 2 ) < x∗ , x > 2 L 2 (0,T ;L 2 ) ∗ ,L 2 (0,T ;L 2 ) µZ;L (dx)= ∫ X (T,p) < x ∗ , x > 2 L 2 (0,T ;L 2 ) ∗ ,L 2 (0,T ;L 2 ) µZ;(T,p) (dx)= ∫ X (T,p) < x ∗ , x > 2 (X (T,p) ) ∗ ,X (T,p) µZ;(T,p) (dx) = ‖x ∗ ‖ L 2 (µ Z;(T,p) ) .Therefore, from Lebesgue’s dominated convergence theorem, we obtain that(X (T,p)) ∗= L 2 (0, T ; L 2 ) ∗σ((X(T,p) ) ∗ ,X (T,p) ) ⊂ L 2 (0, T ; L 2 ) ∗L2 (µ Z;(T,p) )= H∗µ Z;L .It follows that H ∗ µ Z;(T,p) ⊂ H ∗ µ Z;L .The reverse inclusion follows from the fact that L 2 (0, T ; L 2 ) ∗ ⊂ ( X (T,p)) ∗.The conclusion follows from the quite standard fact that the RKHS ofµ Z;L , which is a centered Gaussian measure on a Hilbert space, is equalto ImQ 1 2 , with the norm of the image structure. Q denotes the covarianceoperator of the centered Gaussian measure, it is given, see [34], for h ∈L 2 (0, T ; L 2 ), byQh(v) =∫ T ∫ u∧v00IS(v − s)ΦΦ ∗ S(s − u)I ∗ h(u)dsdu.Corollary B.5 of reference [34] finally gives that ImL = ImQ 1 2 .□A.2.5Continuity with respect to the perturbationRecall that the mild solution of stochastic NLS (A.1.3) could be written asa function of the perturbation.73

Appendix A.Large deviations in the additive case and applicationsLet v(x) denotes the solution of{idvdt − (∆v + |v − ix|2σ (v − ix)) = 0,v(0) = u 0 ,(A.2.2)or equivalently a fixed point of the functionalF z (v)(t) = S(t)u 0 − iλ∫ t0S(t − s)(|(v − iz)(s)| 2σ (v − iz)(s))ds,where z is an element of [ X (T,p) ), p is such that p ≥ 2σ + 2 and (T, p) is anarbitrary pair in R ∗ 2+ × 2,d−2when d > 2 or R ∗ + × [2, ∞) when d = 2 ord = 1.If u is such that u = v(Z) − iZ where Z is the stochastic convolution,note that its regularity is given in the previous section, then u is asolution of (A.1.3). Consequently, if G denotes the mapping that satisfiesG(z) = v(z) − iz we obtain that u = G(Z).The local existence follows from the fact that for R > 0 and r > 0fixed, taking ‖z‖ X (T,2σ+2) ≤ R and ‖u 0 ‖ H 1 ≤ r, there exists a sufficientlysmall T2σ+2∗ such that the closed ball centered at 0 of radius 2r is invariantand F z is a contraction for the topology of L ∞ ([0, T2σ+2 ∗ ]; L2 ) ∩L r (0, T2σ+2 ∗ ; Lp ). Note that a closed ball of X (T 2σ+2 ∗ ,2σ+2) is complete forthe topology of L ∞ ([0, T2σ+2 ∗ ]; L2 ) ∩ L r (0, T2σ+2 ∗ ; Lp ). The proof uses extensivelythe Strichartz’ estimates; see [37] for a detailed proof. The same fixedpoint argument can be used for ‖z‖ X (T,p) ≤ R in a closed ball of radius 2r inX (T p ∗,p) for every Tp ∗ sufficiently small and p ≥ 2σ + 2 such that (Tp ∗ , p) ∈ J.From (A.2.1), there exists a unique maximal solution v(z) that belongs toE ∞ .It could be deduced from Proposition 3.5 of [37], that the mapping Gfrom X ∞ into E ∞ is a continuous mapping from ⋂ T ∈R ∗ X(T,2σ+2) with the+projective limit topology into E(H 1 ).follows.The result can be strengthened asProposition A.2.3 The mapping G from X ∞ into E ∞ is continuous.Proof. Let ˜z be a function of X ∞ and T < T ( ˜Z). Revisiting the proof ofProposition 3.5 of [37] and taking ɛ > 0, p ′ ≥ 2σ + 2, R = 1 + ‖˜z‖ X (T,p ′ ),r = 1 + ‖v(˜z)‖ C([0,T ];H 1 ), and 2 < p < p ′ , there exists η > 0 satisfyingη

Appendix A.Large deviations in the additive case and applicationsThe constant α is the one that appears in the application of Hölder’s inequalitybefore (A.2.1). Consequently, since v(z) and v(˜z) are functions of theclosed ball centered at 0 and of radius 2r in X (T,p) , the triangle inequalitygives that‖v(z) − v(˜z)‖ X (T,p ′ ) ≤ 4r.The application of both Hölder’s inequality and the triangle inequality allowto conclude that∀z ∈ X ∞ : ‖z − ˜z‖ X (T,p ′ ) ≤ η, ‖G(z) − ‡(˜x)‖ X (T,p) ≤ ɛwhich, from the definition of the neighborhood basis of E ∞ , gives the continuity.□The following corollary is a consequence of the last statement of SectionA.2.2.Corollary A.2.4 In the focusing subcritical case or in the defocusing case,G is a continuous mapping from X ∞ into X ∞The continuity allows us to define the law of the solutions of the stochasticNLS equations on E ∞ and in the cases of global existence in X ∞ as the directimage µ u = G ∗ µ Z , the same notation will be used in both cases.Let consider the solutions ofidu ɛ − (∆u ɛ + λ|u ɛ | 2σ u ɛ )dt = √ ɛdW,(A.2.3)where ɛ ≥ 0. The laws of the solutions u ɛ in the corresponding trajectoryspaces are denoted by µ uɛ , or equivalently G ∗ µ Zɛ where µ Zɛ is the directimage of µ Z under the transformation x ↦→ √ ɛx on X ∞ . The continuity alsogives that the family converges weakly to the Dirac mass on the deterministicsolution u d as ɛ converges to zero. Next section is devoted to the study ofthe convergence towards 0 of rare events or tail events of the law of thesolution u ɛ , namely large deviations. It allows to describe more preciselythe convergence towards the deterministic measure.A.3 Sample path large deviationsTheorem A.3.1 The family of probability measures ( µ)ɛ≥0 uɛ on E ∞ satisfiesa LDP of speed ɛ and good rate functionI(u) = 1 {}inf ‖h‖ 22 h∈L 2 (0,∞;L 2 L):S(h)=u2 (0,∞;L 2 ),75

Appendix A.Large deviations in the additive case and applicationswhere inf ∅ = ∞ and S(h), called the skeleton, is the unique mild solutionof the following control problem:{idudt = ∆u + λ|u|2σ u + Φh,u(0) = u 0 ∈ H 1 .This is to say that for every Borel set A of E ∞ ,− inf I(u) ≤ lim ɛ→0 ɛ log µ uɛ (A) ≤ lim ɛ→0 ɛ log µ uɛ (A) ≤ − inf I(u).u∈Åu∈AThe same result holds in X ∞ for the family of laws of the solutions in thecases of global existence.Proof. The general LDP for centered Gaussian measures [ on)real Banachspaces, see [53], gives that for a given pair (T, p) in R ∗ 2+× 2,d−2when d > 2or R ∗ + × [2, ∞) when d = 2 or d = 1, the family ( p (T,p)∗ µ Zɛ) ɛ≥0 satisfies aLDP on X (T,p) of speed ɛ and good rate function defined for z ∈ X (T,p) by,{ 1I Z;(T,p) 2(z) =‖z‖2 H , if z ∈ H µ Z;(T,p) µ Z;(T,p),∞, otherwise,which, using Proposition A.2.2, is equal toI Z;(T,p) (z) ={ 12 ‖z|2 ImL, if z ∈ ImL,∞, otherwise,.Dawson-Gärtner’s theorem, see [48], along with the monotone convergencetheorem, allows us to deduce that the family ( µ Zɛ) satisfies the LDP withɛ≥0the good rate function defined for z ∈ X ∞ byI Z {(z) = sup (T,p)∈J I Z;(T,p) (z) }⎧ { ) }⎨−11(sup= (T,p)∈J 2(Φ ‖ dz|KerΦ ⊥dt + i∆z) ‖ 2 L 2 (0,T ;L 2 )⎩∞ if dzdt+ i∆z /∈ ImΦ{}= 1 2 inf h∈L 2 (0,∞;L 2 ):L(h)=z ‖h‖ 2 L 2 (0,∞;L 2 ).It has been shown in Section A.2.2 and A.2.5 that G is a continuous functionfrom a Hausdorff topological space into another Hausdorff topologicalspace. Consequently, both results follow from Varadhan’s contraction principlealong with the fact that if G ◦ L(h) = u then u is the unique mildsolution of the control problem (i.e. u = S(h)).□76

Appendix A.Large deviations in the additive case and applicationsRemark A.3.2 The rate function can be written⎧∫1 T (u)(−1 (i⎪⎨du2 0 ∥ Φ |KerΦ ⊥)dt − ∆u − λ|u|2σ u ) (s)∥ dsI(u) =L 2if i ⎪⎩dudt − ∆u − λ|u|2σ u ∈ ImΦ,∞ otherwise.Remark A.3.3 In the cases where blow-up may occur, the argument thatwill[follow)allows us to prove the weaker result that, given an (T, p) in R ∗ + ×22,d−2when d > 2 or R ∗ + × [2, ∞) when d = 2 or d = 1 andI (T,p) (u) = 1 2{}inf ‖h‖ 2h∈L 2 (0,T ;L 2 L):S(h)=u2 (0,T ;L 2 ),then for every bounded Borel set A of X (T,p)− inf I (T,p) (u) ≤ lim ɛ→0 ɛ log P (u ɛ ∈ A) ≤ lim ɛ→0 ɛ log P (u ɛ ∈ A) ≤ − inf I (T,p) (u).u∈Åu∈A(A.3.1)Indeed, if u ɛ belongs to A, there exists a constant R such that ‖u ɛ ‖ X (T,p) ≤ R.Denoting by u ɛ,R the solution of the following fixed point problemu ɛ,R (t) = S(t)u 0 −iλ∫ t0S(t−s)(|(u ɛ,R −i √ ɛZ)(s)| 2σ (u ɛ,R −i √ ɛZ)(s))1l ‖u ɛ,R ‖ X (s,p) ≤Rds,the arguments used previously allow to show that √ ɛZ → u ɛ,R is a continuousmapping from every X (T,p′) into X (T,p) for p ′ > p. The result (A.3.1)with u ɛ,R follows from Varadhan’s contraction principle replacing S(h) byS R (h) with the truncation in front of the nonlinearity. Finally, the statementfollows from the fact that ‖u ɛ ‖ X (T,p) ≤ R implies that u ɛ,R = u ɛ andthatinf {‖h‖ 2h∈L 2 (0,T ;L 2 ):S R L(h)∈A2 (0,T ;L 2 ) } = inf {‖h‖ 2h∈L 2 (0,T ;L 2 L):S(h)∈A2 (0,T ;L 2 ) }.Note that writing ∂ ∂th instead of h in the optimal control problem leads toa rate function consisting in the minimisation of 1 2 ‖h‖2 . This spaceH0 1(0,∞;L2 )is somehow the equivalent of the Cameron-Martin space for the Brownianmotion. Specifying only the law µ of W (1) on H 1 and dropping Φ in thecontrol problem would lead to a rate function consisting in the minimisationof 1 2 ‖h‖2 H0 1(0,∞;Hµ), where H µ stands for the RKHS of µ.77

Appendix A.Large deviations in the additive case and applicationsThe formalism of a LDP stated in the intersection space with a projectivelimit topology allows, for example, to deduce by contraction, when there isno blow-up in finite time, a variety of sample path LDP on every X (T,p) .The rate function could be interpreted as the minimal energy to implementcontrol.LDP for the family of laws of u ɛ (T ), for a fixed T , could be deduced bycontraction in the cases of global existence. The rate function is then theminimal energy needed to transfer u 0 to x from 0 to T .Next section gives a characterization of the support of the law of thesolution in our setting. Section A.5 is devoted to some consequences ofthese results on the blow-up time.A.4 The support of the law of the solutionTheorem A.4.1 (The support theorem) The support of the law of thesolution is characterized bysupp µ u = ImS E∞and in the cases of global existence bysupp µ u = ImS X∞[ )Proof. Step 1: From Proposition A.2.3, given (T, p) in R ∗ 2+× 2,d−2whend > 2 or R ∗ + × [2, ∞) when d = 2 or d = 1, µ Z;(T,p) is a Gaussian measureon a Banach space and its RKHS is ImL. Consequently, see [8] Theorem(IX,2;1), its support is ImL X(T,p) . Also, from the definition of the imagemeasure we have that()) )µ Z p −1(T,p)(ImL X(T,p) = µ(ImL Z;(T,p) X(T,p) = 1.As a consequence the first inclusion followssupp µ Z ⊂ ⋂)(ImL X(T,p) = ImL X∞ .(T,p)p −1(T,p)It then suffices to show that ImL ⊂ supp µ Z . Suppose that x /∈ supp µ Z ,then there exists a neighborhood V of x in X ∞ , satisfying V = ⋂ ni=1 V (T i,p i )where V (T i,p i ) is a neighborhood of x in X (T i,p i ) , n is a finite integer and78

Appendix A.Large deviations in the additive case and applications[ )(T i , p i ) a finite sequence of elements of R ∗ 2+ × 2,d−2when d > 2 orR⋂ ∗ + ×[2, ∞) when d = 2 or d = 1, such that µ Z (V ) = 0. It can be shown thatni=1 X(T i,p i ) is still a separable Banach space. It is such that X ∞ is continuouslyembedded into it, and such that the Borel direct image probabilitymeasure is a Gaussian measure of RKHS ImL. The support of this measureis then the closure of ImL for the topology defined by the maximum of thenorms on each factor. Thus, V ∩ ImL = ∅ and x /∈ ImL.Step 2: We conclude using the continuity of G.Indeed since G(ImL) ⊂ G(ImL) E∞ , ImL ⊂ G −1 (G(ImL) E∞) . Since G iscontinuous, the right side is a closed set of X ∞ and from step 1,supp µ Z ⊂ G −1 ( Im (G ◦ L) E∞) ,andµ Z ( G −1 ( ImS E∞ ))= 1,thussupp µ u ⊂ ImS E∞ .Suppose that x /∈ supp µ u , there exists a neighborhood V of x in E ∞ suchthat µ u (V ) = µ Z ( G −1 (V ) ) = 0, consequently G −1 (V ) ⋂ ImL = ∅ and x /∈ImS. This gives the reverse inclusion.The same arguments hold replacing E ∞ by X ∞ .□Note that the result of step 2 is general and gives that the support of thedirect images µ E of the law µ u by any continuous mapping f from eitherE ∞ or X ∞ into a topological vector space E is Im (f ◦ S) E . For example, inthe cases of global existence, given a positive T , the support of the law inH 1 of u(T ) is ImS(T ) H1 .Remark A.4.2 Remark that the LDP and support theorem may be provedfor more general driving noises provided that the stochastic convolution remainsa Gaussian process. The case of a noise derived from a fractionalWiener process which is a one parameter generalization of the usual Wienerprocess has been studied. The results will appear elsewhere.A.5 Applications to the blow-up timesIn this section the equation with a focusing nonlinearity, i.e. λ = 1, is considered.In this case, it is known that some solutions of the deterministic79

Appendix A.Large deviations in the additive case and applicationsequation blow up in finite time for a critical or subcritical nonlinearity. Ithas been proved in Section A.2.2 that T is a measurable mapping fromE ∞ to [0, ∞], both spaces are equipped with their Borel σ−fields. Incidentally,T (u) is a F t −stopping time. Also, if B is a Borel set of [0, ∞],P (T (u) ∈ B) = µ u ( T −1 (B) ) .The support theorem allows us to determine whether an open or a closedset of the form T −1 (B) is such that µ u ( T −1 (B) ) > 0 or µ u ( T −1 (B) ) < 1respectively. An application of this fact is given in Proposition A.5.1.For a Borel set B such that { Int ( T −1 (B) )} ∩ ImS E∞is nonempty, whereInt ( T −1 (B) ) stands for the interior set of T −1 (B), P(T (u) ∈ B) > 0 holds.Also, T is not continuous and Varadhan’s contraction principle does notallow to obtain a LDP for the law of the blow-up time. Nonetheless, theLDP for the family ( µ)ɛ>0 uɛ gives the interesting result that{− infu∈Int(T −1 (B)) I(u) ≤ lim ɛ→0 ɛ log P (T (u ɛ ) ∈ B)lim ɛ→0 ɛ log P (T (u ɛ ) ∈ B) ≤ − inf u∈T −1 (B) I(u).Note also that the interior or the closure of sets in E ∞ are difficult to characterize.In that respect, the semicontinuity of T makes the sets (T, ∞] and[0, T ] particularly interesting.A.5.1Probability of blow-up after time TProposition A.5.1 If u 0 ∈ H 3 and the range of Φ is dense then for everypositive T ,P(T (u) > T ) > 0.Proof. Since T is lower semicontinuous, T −1 ((T, ∞]) is an open set.Consider H = −∆u 0 − |u 0 | 2σ u 0 which satisfies G ◦ Λ(H) = u 0 , where Λhas been defined in Section A.2.1, then T (S(H)) = ∞. Also, using Φ onedefines, in a natural way, an operator from L 2 loc (0, ∞; L2 ) into L 2 loc (0, ∞; H1 )and it can be shown, that it still has a dense range. Consequently, thereexists a sequence (h n ) n∈N of L 2 loc (0, ∞; L2 ) functions such that (Φ(h n )) n∈Nconverges to H in L 2 loc (0, ∞; H1 ).Using the semicontinuity of T , the continuity of G, the fact that S =G◦Λ◦Φ, the following lemma and the fact that L 2 loc (0, ∞; H1 ) is continuouslyembedded in L 1 loc (0, ∞; H1 ), lim n→∞ T (S(h n )) ≥ ∞, i.e. lim n→∞ T (S(h n )) =∞, follows. Therefore T (S(h n )) > T for n large enough and T −1 ((T, ∞]) ∩(ImS) is nonempty.The conclusion follows then from the support theorem.□80

Appendix A.Large deviations in the additive case and applicationsAs a corollary, taking the complement of T −1 ((T, ∞]), P(T (u) ≤ T ) < 1follows. This is related to the results of [38] where it is proved that for everypositive T , P(T (u) < T ) > 0 and to the graphs in Section A.4 of [43].Lemma A.5.2 The operator Λ, defined in Section A.2.1, is continuousfrom L 1 loc (0, ∞; H1 ) into X ∞ .Proof. The result follows from (ii) of the Strichartz inequalities when s = 1and ρ = 2, the fact that the partial derivatives with respect to one spacevariable commutes with both the integral and the group and the definitionof the projective limit topology.□The following result holds when the amplitude of the noise converges tozero.Proposition A.5.3 If u 0 ∈ H 3 , the range of Φ is dense and T ≥ T (u d ),where u d is the solution of the deterministic NLS equation with initial datumu 0 , there exists c in [0, ∞) such thatProof. DefineL (T,∞] = 1 2The result follows then fromlim ɛ→0 ɛ log P (T (u ɛ ) > T ) ≥ −c.{}inf ‖h‖ 2h∈L 2 (0,∞;L 2 L):T (S(h))>T2 (0,∞;L 2 ).−L (T,∞] ≤ lim ɛ→0 ɛ log P (T (u ɛ ) > T )and that, from the arguments of the proof of Proposition A.5.1, for every Tsuch that T ≥ T (u d ) the set {h ∈ L 2 (0, ∞; L 2 ) : T (S(h)) > T } is nonempty.□Remark A.5.4 The assumption that u 0 ∈ H 3 could be dropped using similararguments as in Proposition 3.3 of [38].Note that the LDP does not give interesting information on the upper boundeven if the bounds have been sharpened using the rather strong projectivelimit topology. It is zero since h = 0 belongs to T −1 ((T, ∞]) as for everyT > 0, T −1 ((T, ∞]) = E ∞ . Indeed, if a function f of E ∞ is given and blowsup at a particular time T (f) such that T > T (f), it is possible to build asequence (f n ) n∈N of functions of E ∞ equal to f on [ 0, T (f) − 1 n]and suchthat T (f n ) > T . The same problem will appear in the next section where81

Appendix A.Large deviations in the additive case and applicationsthe LDP gives a lower bound equal to −∞. Indeed, Int ( T −1 ([0, T ]) ) is thecomplement of the above and thus an empty set. To overcome this problemthe approximate blow-up time is introduced. Note also that it is possiblethat L (T,∞] = 0.Also, the case T < T (u d ) has not been treated. Indeed, the associatedevent is not a large deviation event and the LDP only gives thatA.5.2lim ɛ log P (Tɛ→0 (uɛ ) > T ) = 0.Probability of blow-up before time TIn that case we obtain−∞ ≤ lim ɛ→0 ɛ log P (T (u ɛ ) ≤ T ) ≤ lim ɛ→0 ɛ log P (T (u ɛ ) ≤ T ) ≤ −U [0,T ]where U [0,T ] = 1 2 inf h∈L 2 (0,∞;L 2 ):T (S(h))≤T{‖h‖ 2 L 2 (0,∞;L 2 )}.Proposition A.5.5 If T < T (u d ),lim ɛ→0 ɛ log P (T (u ɛ ) ≤ T ) ≤ −U [0,T ] < 0.Proof. Let (h n ) n∈N be a sequence of L 2 (0, ∞; L 2 ) functions convergingto zero. It follows from Lemma 5.2 and the fact that L 2 (0, ∞; H 1 ) is continuouslyembedded into L 1 loc (0, ∞; H1 ) that S = G ◦Λ◦Φ is continuous fromL 2 (0, ∞; L 2 ) into E ∞ . Also, from the semicontinuity of T , lim n→∞ T (S(h n )) ≥T (u d ). Thus there exists N large enough such that for every n ≥ N,T (S(h n )) > T . As a consequence we obtain that necessarily U [0,T ] > 0.□When T ≥ T (u d ), the probability is not supposed to tend to zero. Also,as h = 0 is a solution, the upper bound is zero and none of the bounds areinteresting.A.5.3Bounds for the approximate blow-up timeTo overcome the limitation that T −1 ((T, ∞]) = E ∞ , which does not allow tohave two interesting bounds simultaneously, we introduce for every positiveR the mappings T R defined for f ∈ E ∞ byT R (f) = inf{t ∈ [0, ∞) : ‖f(t)‖ H 1 ≥ R}.It corresponds to the approximation of the blow-up time used in [43]. Weobtain the following bounds.82

Appendix A.Large deviations in the additive case and applicationsProposition A.5.6 When T ≥ T R (u d ), the following inequality holds−c < −L (T,∞]R≤ lim ɛ→0 ɛ log P (T R (u ɛ ) > T )≤ lim ɛ→0 ɛ log P (T R (u ɛ ) > T ) ≤ − sup α>0 L (T,∞]R+α .Also, when T < T R (u d ), we have that− inf α>0 U [0,T ]R+α ≤ lim ɛ→0ɛ log P (T R (u ɛ ) ≤ T )≤ lim ɛ→0 ɛ log P (T R (u ɛ ) ≤ T ) ≤ −U [0,T ]R< 0.In the above c is nonnegative and the numbers L (T,∞]Rand U [0,T ]Ras L (T,∞] and U [0,T ] replacing T by T R .are definedProof. The result follows from the facts that T R , which is not continuous,is lower semicontinuous, that for every α > 0, T −1−1R((T, ∞]) ⊂ TR+α ((T, ∞]),thus T −1R+α ([0, T ]) ⊂ Int ( T −1R([0, T ])) and from the arguments used in theproofs of the last two propositions.□We also obtain the following estimates of other large deviation events.Corollary A.5.7 If S, T < T R (u d ), for every c > 0, there exists ɛ 0 > 0such that if ɛ ≤ ɛ 0 ,) ())exp(− inf α>0 U [0,T ]R+α +cɛ1 − exp(− U [0,S]R −inf α>0 U [0,T ]R+αɛ≤ P (S(< T R (u ɛ ) ≤ T )≤ exp − U [0,T ]R −cɛ) (1 − exp(− inf α>0 U [0,S]R+αɛ−U[0,T ]R)).If S, T > T R (u d ), for every positive c, there exists a positive ɛ 0 such that ifɛ ≤ ɛ 0 ,( ) ())exp − L(S,∞] R +cɛ1 − exp(− sup α>0 L(T,∞] R+α −L(T,∞] Rɛ≤ P (S < T R (u ɛ ) ≤ T )≤ exp(− sup α>0 L(S,∞]R+α −cɛ) (1 − exp(− L(T,∞] R−sup α>0 L (S,∞]R+αɛ)).Proof. When S, T < T R (u d ), the result follows from the inequalities andfrom the fact thatP (S < T R (u ɛ ) ≤ T )= P ({T R (u ɛ ) ≤ T }(\ {T R (u ɛ ) ≤ S}))= P (T R (u ɛ ) ≤ T ) 1 − P(T R(u ɛ )≤S)P(T R (u ɛ )≤T ).83

Appendix A.Large deviations in the additive case and applicationsWhen S, T > T R (u d ), we useP (S < T R (u ɛ ) ≤ T ) = P ({T R (u ɛ ) > S}(\ {T R (u ɛ ) > T }))= P (T R (u ɛ ) > S) 1 − P(T R(u ɛ )>T )P(T R (u ɛ )>S).□A.6 Applications to nonlinear opticsThe NLS equation when d = λ = σ = 1 is called the noisy cubic focusingnonlinear Schrödinger equation. It is a model used in nonlinear optics.Recall that for the above values of the parameters the solutions are global.The variable t stands for the one dimensional space coordinate and x for thetime. The deterministic equation is such that there exists a particular classof solutions, which are localized in space (here time), that propagate at afinite constant velocity and keep the same shape. These solutions are calledsolitons or solitary waves. The functionsΨ η (t, x) = √ 2η exp ( −iη 2 t ) sech(ηx), η > 0,form a family of solitons. They are used in optical fibers as informationcarriers to transmit the datum 0 or 1 at high bit rates over long distances.The noise stands for the noise produced by in-line amplifiers.Let u ɛ denotes the solution with u 0 (·) = Ψ 1 (0, ·) as initial datum andɛ as noise amplitude like in Section A.3 and u ɛ,n denotes the solution withnull initial datum and the same noise amplitude. The mass of u 0 is 4.At a particular coordinate T of the fiber, when a window [−l, l] is given,the square of the L 2 (−l, l)−norm, or measured mass, is recorded. It is closeto the mass in the deterministic case for sufficiently high l since the wave islocalized. A decision criterium is to accept that we have 1 if the measuredmass is above a certain threshold and 0 otherwise. We set a threshold of theform 4(1 − γ), where γ is a real number in [0, 1].As the soliton is progressively distorted by the noise, it is possible eitherto wrongly decide that the source has emitted a 1, or to wrongly discard a1. The two error probabilities consist of(∫ l)P |0ɛ = P |u ɛ,n (T, x)| 2 dx ≥ 4(1 − γ)−landP |1ɛ(∫ l)= P |u ɛ (T, x)| 2 dx < 4(1 − γ) .−l84

Appendix A.Large deviations in the additive case and applicationsIn modern communication systems the error rate is less than 10 −9 whichis beyond the scope of statistics, moreover due to the nonlinearity of thesystem the measured mass does not have a gaussian law. This justifies thatwe use theoretical arguments to characterize these error probabilities. Weshow in this section how the LDP applies to this problem.We obtain similar results, in the case of an unbounded window, as inreference [62] for the first error probability and as [63] for the second errorprobability. In these articles the heuristic argument of the collective coordinatesis used. This is a physical argument which unables to reduce theproblem to a finite dimensional system involving modulated parameters. In[62], the authors explain what the leading parameters are and reduce theproblem to a three dimensional problem, that the fluctuations of the parametersare described by SDEs where the noises are some spatial integralsof the initial noise and that the averaging over the initial noise is equivalentto averaging over new noises with zero cross correlations. They explainthat a decrease in the soliton power Q ɛ = N(uɛ (T )) 24and a timing jitterR +∞−∞ x|uɛ (T,x)| 2 dxT ɛ =, which characterizes the shifts in the arrival time ofN(u ɛ (T )) 2the pulse, are mainly responsible for the loss of the pulse. Thus they writedown the probability density function of the joint law of the two processes,using a formalism called the instanton formalism, as a quantity which is theaveraging over the noise of the path integral over arbitrary functions for themodulated parameters, taking into account the finite dimensional evolution,of the exponential of an integral over t in [0, T ] of the so called effectiveLagrangian. Finally they compute the path integral using a saddle-pointapproximation with boundary conditions and obtain an expression of theprobability density function in the small noise asymptotic. Details on thecalculation are given in [63]. The overall argument seems very difficult tojustify rigorously. In particular, the reduction to a three dimensional problemis obtained by minimizing the Lagrangian on a small space of curveswhereas NLS is obtained by minimizing over all paths. Note that they recoveranalytically the empirical Gordon-Haus effect that the dispersion intiming is much larger than that of the mass. The authors also explain that,for the first error probability, the optimal way to create a large signal is togrow a soliton and obtain a small noise asymptotic expression of the probabilitydensity function of the amplitude at the coordinate T of the solutionwith null initial datum.In the following we make the assumption that ImΦ has a dense range.Indeed, from the arguments used in the proof of Proposition A.5.3, it isneeded for controllability issues to guarantee that the infima are not taken85

Appendix A.Large deviations in the additive case and applicationsover empty sets. Also T is fixed, γ 0 ∈ ( 0, 2) 1 is fixed and the size l of thewindow is such that∫ l−l|u d (T, x)| 2 dx ∧∫ l−l(|Ψ 1 (0, x)| 2 dx > 41 − γ 02We finally stress that for the events we study in dimension d = 1 we couldconsider a L 2 setting instead of a H 1 setting. However we chose to work inH 1 to keep the coherence of the article. The H 1 setting is necessary whenwe consider higher dimensions or larger powers of nonlinearities and studyevents like {∫ l}|u ɛ,n (T, x)| 2 dx ≥ 4 − γ, T (u ɛ,n ) > T−lor{∫ l}|u ɛ (T, x)| 2 dx < 4 − γ, T (u ɛ ) > T .−lThe LDP proved herein allows to state the following proposition.Proposition A.6.1 For every γ in [γ 0 , 1 − γ 0 ] besides an at most countableset of points, the following equivalents for the probabilities of error hold{}lim ɛ→0 ɛ log P |0ɛ = − 1 2 inf h∈L 2 (0,∞;L 2 ): R l−l |˜S(h)(T,x)| ‖h‖ 2 2 dx≥4(1−γ) L 2 (0,∞;L 2 )}lim ɛ→0 ɛ log P |1ɛ= − 1 2 inf h∈L 2 (0,∞;L 2 ): R l−l |S(h)(T,x)|2 dx

Appendix A.Large deviations in the additive case and applicationsFor every δ > 0, U 0 (γ) ≤ L 0 (γ) ≤ U 0 (γ −δ) and U 1 (γ) ≤ L 1 (γ) ≤ U 1 (γ +δ)hold.The function γ ↦→ U 0 (γ) is positive and decreasing. Also, since the rangeof Φ is dense, there exists a sequence (h 0 n) n∈N of functions of L 2 (0, ∞; L 2 ) sothat Φh n converges towhereH 0 (t) = i du0dt − ∆u0 − λ|u 0 | 2σ u 0u 0 t(t) = 1l t≤TT Ψ 1(0, ·)and by the continuity proved in Section A.5.1(ϕ ◦ ˜S(h)0n) convergesn∈Nto ϕ ◦ ˜S(H 0 ) > 4 ( 1 − γ )02> 4(1 − γ0 ). Consequently, h 0 n belongs to theminimizing set for n large enough. Thus, U 0 (γ 0 ) < ∞ follows. Consequently,the function γ ↦→ U 0 (γ) possesses an at most countable set of points ofdiscontinuity.Similarly, the function γ ↦→ U 1 (γ) is a bounded increasing function.Also, if (h 1 n) n∈N and H 1 (t) are defined as previously replacing u 0 (t) byu 1 (t) = 1l t≤T(1 −(1 −√γ02) tT)Ψ 1 (0, ·),the sequence (ϕ◦S(h 1 n)) n∈N converges to ϕ◦S(H 1 ) ≤ 2γ 0 = 4 ( 1 − ( 1 − γ ))02.Thus, for n large enough h 1 n belongs to the minimizing set. Consequently,the function γ ↦→ U 1 (γ) has an at most countable set of points of discontinuity.Thus, for a well chosen γ, letting δ converge to zero, we obtain fori ∈ {0, 1} that L i (γ) = U i (γ) and the equivalents follow.From the arguments used in the proof of Proposition A.5.5, ˜S is a continuousmapping from L 2 (0, ∞; H 1 ) into X ∞ . Since ϕ is continuous, if((H n ) n∈N is a sequence of functions converging to zero in L 2 (0, ∞; H 1 ) thenϕ ◦ ˜S(H)n ) converges to ϕ ◦ ˜S(0) = 0. Similarly (ϕ ◦ S(H n )) n∈Nconvergesto ϕ ◦ S(0) > 4 ( 1 − γ )n∈N02. We have now proved the last point of ourresult that is both infima are positive.□In the two following sections we concentrate on the mass, we take l = ∞as if the window were not bounded. Somehow, if we forget the coefficient,we concentrate on the tails of the marginal law of the soliton power when theinitial datum is a soliton or the tails of the amplitude of the solution withnull initial datum as ɛ converges to zero. We recall, it has been pointed outin the introduction, that the mass is no longer preserved in the stochasticcase and is such that its expected value increases.87

Appendix A.Large deviations in the additive case and applicationsA.6.1Upper boundsThe norm of the linear continuous operator Φ of L 2 is thereafter denoted by‖Φ‖ c .Proposition A.6.2 For every positive T , γ in (0, 1), and every operator Φin L 2 (L 2 , H 1 ), the inequalitieslim ɛ→0 ɛ log P |0ɛ≤ − 1 − γ2T ‖Φ‖ 2 candhold.lim ɛ→0 ɛ log P |1ɛγ 2≤ −2T ‖Φ‖ 2 c(1 + γ) .Proof. Multiplying by −iu the equationi dudt − ∆u − λ|u|2σ u = Φh,integrating over time and space and taking the real part gives that( ∫ T ∫ )‖u(T )‖ 2 L− ‖u 2 0 ‖ 2 L= 2Re −i Φhu dxdt .20 RFirst bound: The boundary conditions ‖u(T )‖ 2 L 2 (R)≥ 4(1 − γ) andu 0 = 0 along with Cauchy Schwarz inequality imply both that4(1 − γ) ≤ 2‖Φ‖ c ‖h‖ L 2 (0,T ;L 2 )‖u‖ L 2 (0,T ;L 2 ),and that∫ T0 ‖u(t)‖2 L 2 dt = 2 ∫ T0 Re (−i ∫ t0 Φhu dxds )dt≤ 2T ‖Φ‖ c ‖h‖ L 2 (0,T ;L 2 )‖u‖ L 2 (0,T ;L 2 ),thus,‖h‖ 2 L 2 (0,∞;L 2 ) ≥ 1 − γT ‖Φ‖ 2 .cSecond bound: The boundary conditions ‖u(T )‖ 2 L< 4(1 − γ) and2‖u 0 ‖ 2 L= 4 give both that along with the Cauchy-Schwarz inequality24γ < 2‖Φ‖ c ‖h‖ L 2 (0,∞;L 2 )‖u‖ L 2 (0,T ;L 2 )88

Appendix A.Large deviations in the additive case and applicationsand also along with Cauchy Schwarz and integration over time‖u‖ 2 L 2 (0,T ;L 2 ) − 4T ≤ 2T ‖Φ‖ c‖h‖ L 2 (0,∞;L 2 )‖u‖ L 2 (0,T ;L 2 ).Consequently, it follows that( √)4‖u‖ L 2 (0,T ;L 2 ) ≤ T ‖Φ‖ c ‖h‖ L 2 (0,T ;L 2 ) 1 + 1 +T ‖Φ‖ 2 c‖h‖ 2 .L 2 (0,T ;L 2 )Thus, we obtain( √)2γT ‖Φ‖ 2 < ‖h‖ 2 4L 2 (0,T ;L 2 )1 + 1 +cT ‖Φ‖ 2 c‖h‖ 2 L 2 (0,T ;L 2 )( √ )and since the function x → x 1 + 1 + 4 xis increasing on R ∗ +,The upper bound follows.‖h‖ 2 L 2 (0,∞;L 2 ) > γ 2T ‖Φ‖ 2 c(1 + γ) .Remark A.6.3 The estimates used in the proof of the above result onlyuse the fact that the nonlinearity acts as a potential. Indeed the same resultholds for any nonlinearity of this type.A.6.2Lower boundsWe prove the following lower bounds.Proposition A.6.4 For every positive T , γ ∈ (0, 1) and ˜γ in to a densesubset of (0, 1), for every sequence of operators (Φ n ) n∈Nin L 0,12 such thatfor every h in L 2 ( 0, T ; L 2) of the formh(t, x) = i η′ (t)η(t) Ψ η(t, x) − i √ ( ∫ t)2η ′ (t) exp −i η 2 (s)ds η(t)x sinhcosh 2 (η(t)x)whereΨ η (t, x) = √ (2η(t) exp −i∫ t00)η 2 (s)ds sech(η(t)x)and η is of one of the following parameterized form( ) t 2η˜γ,T (t) = (1 − ˜γ)T□(A.6.1)89

Appendix A.Large deviations in the additive case and applicationsorη˜γ,T (t) =(2 − ˜γ − 2 √ ) ( )t 2 (1 − ˜γ + 2 −1 + √ t1 − ˜γ)TT + 1,Φ n h converges to h in L 1 (0, T ; L 2 ), we obtain the following inequalities wherethe n in the error probabilities is there to recall that Φ is replaced by Φ nandlim n→∞,ɛ→0 ɛ log P |0ɛ,n ≥ − 2(1 − γ)(12 + π2 )9Tlim n→∞,ɛ→0 ɛ log P |1ɛ,n ≥ − 2(1 − √ 1 − γ) 2 (12 + π 2 ).9TProof. Consider first that ”Φ = I”, denote the corresponding skeletons by˜S W N when the initial datum is 0 and by S W N when it is Ψ(x) = √ 2sech(x),they are defined from the (ii) of the Strichartz inequalities on L 1 (loc 0, ∞; H1 ) ,suppose also that η is any function of C([0, T ]). Since the initial data0 or Ψ belong to H 2 , for h ∈ L 2 , S W N (h) and ˜S W N (h) are functions ofC([0, T ]; H 2 ) ∩ C 1 ([0, T ]; L 2 ), consequently t → η(t) = 1 4 ‖Ψ η(t, ·)‖ 2 Lis necessarilya function in C 1 ([0, T ]). Also, for controls h η parameterized as in the2above assumptions, η is in C 1 ([0, T ]), the controls belong to L 1 (loc 0, ∞; H1 ) ,the skeletons are the prescribed paths Ψ η and we obtain}inf η∈ C 1 ([0,T ]):‖˜S W N (h η)(T,·)‖{‖h 2L 2 ≥4(1−γ) η ‖ 2 L 2 (0,∞;L 2 )∫ T= inf η∈ C 1 ([0,T ]),b.c. 0 F (η(t), η′ (t)) dt,where the Lagrangian F isF (z, p) = 1 9 (12 + π2 ) p2z ,and b.c. stands for the boundary conditions η(0) = 0 and η(T ) ≥ 1 − γ.Indeed, since ˜S W N (h)(T ) is a function of (h(t)) t∈[0,T ] , the infimum could betaken on functions set to zero almost everywhere after T , thus ‖h‖ 2 L 2 (0,∞;L 2 )in the left hand side could be replaced by ‖h‖ 2 L 2 (0,T ;L 2 ). A scaling argumentgives that the terminal boundary condition is necessarily saturated.Similarly, for the second error probability, ˜S W N is replaced by S W N andb.c. is η(0) = 1 and η(T ) = 1 − γ.The usual results of the indirect method do not apply to the problem90

Appendix A.Large deviations in the additive case and applicationsof the calculus of variations, nonetheless solutions of the boundary valueproblem associated to the Euler-Lagrange equation(2 η′′ η′η = ηprovide upper bounds when we compute the integral of the Lagrangian. Ifwe suppose that η is in C 3 ([0, T ]) and that it is positive on (0, T ), we obtainby derivation of the ODE that on (0, T ),η ′′′ = 0.Also, looking for solutions of the form at 2 +bt+c, we obtain that necessarilyb 2 = 4ac. Thus C 3 ([0, T ]) positive solutions are necessarily of the forma ( t −2a) b 2. From the boundary conditions, we obtain that for the firsterror probability the function defined by( ) t 2η 0 (t) = (1 − γ)Tis a solution of the boundary value problem. For the second error probability,the boundary conditions imply that the two following functions defined by(η 1,1 (t) = 1 + √ ) ( ) 2 t 2 (1 − γ + 2 −1 − √ t1 − γ)TT + 1) 2andη 1,2 (t) =(1 − √ ) ( ) 2 t 2 (1 − γ + 2 −1 + √ t1 − γ)TT + 1are solutions of the boundary value problem. The second function gives thesmallest value when we compute the integral of the Lagrangian.From the assumptions on the operators Φ n and Lemma A.5.2, for functionsh of the assumptions of the proposition, (G ◦ Λ ◦ Φ n )h converges to(G ◦ Λ)h = ˜S W N (h) in C ( [0, T ]; L 2) . In the above G is the mapping definedin Section A.2.5 with null initial datum. Thus ‖˜S n (h)(T, .)‖ 2 Lconverges to2‖˜S W N (h)(T, .)‖ 2 L. As a consequence, for h in the particular parameterized2set of controls where ˜γ = γ − δ and δ > 0, there exists N 0 large enoughsuch that for any n ≥ N 0 , ‖˜S W N (h)(T, .)‖ 2 L≥ 4(1 − γ + δ) implies that2‖˜S n (h)(T, .)‖ 2 L> 4(1 − γ). Thus the infimum in the rate for a fixed γ is2smaller than the infimum on the smallest particular set of controls h, whichis itself smaller than the square of the L 2 −norm of the control h correspondingto η γ,T . Indeed, h is such that ‖˜S W N (h)(T, .)‖ 2 L≥ 4(1 − γ + δ), for n291

Appendix A.Large deviations in the additive case and applicationslarge enough, which implies the expected boundary condition. We concludefrom the upper bound obtained in the previous study of the problem of calculusof variations by guessing the likeliest path, by taking the limit inf inn of the opposite and from the fact that δ is then arbitrary. The end of theproof of the lower bound for the second error probability is the same. □We have finally obtained upper and lower bounds that agree up to constantsin their behavior in large T for the two error probabilities. In thecase of the first error probability they also agree in their behavior in γ forγ near 1, it is not quite the case for the second error probability and γ nearzero. The bounds for the first error probability are of the same order asthe one we could obtain from the results of [62]. Indeed, with a slightlydifferent normalization on the NLS equation and when the noise is the idealwhite noise and thus ‖Φ‖ c = 1, a result of [62] is that the probability densityfunction of the mass of the pulse at the coordinate T of the fiber whenthe initial datum is null is asymptotically that of an exponential law ofparameter ɛT 2. Integrating the density over [2(1 − γ), ∞), we obtain thatlim ɛ→0 ɛ log P |0ɛ = − 4(1−γ) which is indeed in between the two bounds andTvery close to the lower bound though obtained with a more general parametrization.Also in Section 6 of [63], the authors study numerically thesecond error probability by integrating the joint probability density functionover a domain for the soliton power and timing jitter which depends onthe size of the window and the threshold. They also consider the unrealisticcase where the size of the window is large and of the order of the coordinateT of the location of the receiver in the fiber. The effect of the timingjitter could then be neglected and they obtain an error probability given= − c(γ)T, with a constant c(γ) which is a function of thethreshold. It indeed exhibits the same behavior in T as the one obtained inour previous calculations.by lim ɛ→0 ɛ log P |1ɛRemark A.6.5 In the proposition we would { like to impose that the operator }1Φ is acting as the identity map on spancosh(ax) , x sinh (ax); a ∈ [0, 1] butcosh 2it seems too strong to be compatible with the Hilbert-Schmidt assumption.On the other hand, we may check that the assumptions made here can easilybe fulfilled. Also, under these assumptions, the noise is as close as possibleto the space-time white noise considered in [62, 63] that we are not able totreat mathematically.Remark A.6.6 Note that it is natural to obtain that the opposite of theerror probabilities are decreasing functions of T . Indeed, the higher is T ,92

Appendix A.Large deviations in the additive case and applicationsthe less energy is needed to form a signal which mass gets above a fixedthreshold at the coordinate T . Replacing above by under, we obtain thesame result in the case of a soliton as initial datum. Consequently, thehigher is T the higher the error probabilities get. Also, in the case of thefirst error probability, both upper and lower bounds in Proposition A.6.2 andProposition A.6.4 are increasing functions of γ. Similarly, in the case ofthe second error probability, the bounds are decreasing functions of γ. Thiscould be interpreted as the higher is the threshold, the more energy is neededto form a signal which mass gets above the threshold at the coordinate T andconversely in the case of a soliton as initial datum.Remark A.6.7 The results obtained numerically for a parametrization bythe amplitude solely without modulating the phase and other shape parametersor of the phase η 2 (t)t instead of ∫ t0 η2 (s)ds in (A.6.1), gave lessinteresting lower bounds that did not exhibit the desired properties in T .Remark A.6.8 In [62] the following parametrization of the solution√2η(t) exp (iβ(t)x + iα(t) + iτ(t)) [sech(η(t)(x − y(t))) + v(t, x)] ,where τ ′ (t) = η 2 (t), is studied. The authors give a physical justification ofthe fact that for large T the field v could be neglected. This could be comparedwith the results on asymptotic stability for the deterministic nonlinearSchrödinger equation.93

Appendix BUniform large deviations forthe nonlinear Schrödingerequation with multiplicativenoiseAbstract: Uniform large deviations at the level of the paths for the stochasticnonlinear Schrödinger equation are presented. The noise is a realmultiplicative Gaussian noise, white in time and colored in space. The trajectoryspace allows blow-up. It is endowed with a topology analogous to aprojective limit topology. Asymptotics of the tails of the blow-up time areobtained as corollaries.B.1 IntroductionThe nonlinear Schrödinger (NLS) equation with a power law nonlinearity isa generic model in many areas of physics among which solid state physicsand optics. It describes the propagation of slowly varying envelopes of awave packet in media with both nonlinear and dispersive responses. In somecases, spatial and temporal fluctuations of the parameters of the mediumhave to be considered to account for example for thermal fluctuations orinhomogeneities in the medium; see for example [10, 11, 59]. Sometimesthe only source of fluctuation that has significant effect on the dynamicsenters linearly as a random potential. In fiber optics it accounts for Ramancoupling to the thermal phonon; see [59] for details.95

Appendix B.Large deviations in the multiplicative caseIn the following, the noise is the time derivative in the sense of distributionsof a Wiener process (W (t)) t∈[0,∞) defined on some real separableHilbert space of real valued functions. As the unitary group (U(t)) t∈Rgeneratedby −i∆ on H 1 , space of complex valued functions, is an isometry,there is no smoothing effect in the Sobolev spaces based on L 2 . We are thusunable to treat the space-time white noise often considered in physics. Thenoise is thus white in time and colored in space. With the Itô notations wewriteidu − (∆u + λ|u| 2σ u)dt = u ◦ dW.(B.1.1)The symbol ◦ stands for the Stratonovich product. It corresponds, in termsof the Itô product, to(idu − ∆u + λ|u| 2σ u − i )2 uF Φ dt = udW, (B.1.2)where F Φ (x) = ∑ j∈N (Φe j(x)) 2 for x in R d . When λ = 1 the nonlinearity iscalled focusing, otherwise it is defocusing.The well posedness of the Cauchy problem associated to the deterministic,see [25, 136], and stochastic, see [37], NLS equations depends on thesize of σ. If σ < 2 d, the nonlinearity is subcritical and the Cauchy problemis globally well posed in L 2 or H 1 . If σ = 2 d, critical nonlinearity, or2d < σ < 2d−2 when d ≥ 3 (σ > 2 dwhen d=1,2), supercritical nonlinearity,the Cauchy problem is locally well posed in H 1 . In this latter case, if thenonlinearity is defocusing, solutions are global. In the focusing case andfor the deterministic equation some initial data yield global solutions whileother yield solutions which blow up in finite time. Results on the influenceof a multiplicative noise on the blow-up appeared in a series of numericaland theoretical papers; see for example [13, 43] and [39].In this article we prove a sample path large deviation principle (LDP) forequation (B.1.2). We give an application to the blow-up time. This type ofstudy has been done for a noise of additive type in [81] where applications toerror in fiber optic soliton transmission are also given. Applications to thestudy of the diffusion in position of a soliton pulse are derived in [44]. Notethat our approach allows to prove rigorously results obtained by heuristicarguments in the physics literature.A first type of proof for a LDP when the noise is multiplicative usesan extension of Varadhan’s contraction principle; see [53], [48][Proposition4.2.3] and [66]. It requires a sequence of approximations of the measurableItô map by continuous maps uniformly converging on the sets of levels ofthe initial good rate function less or equal to a positive constant and that96

Appendix B.Large deviations in the multiplicative casethe resulting sequence of family of laws are exponentially good approximationsof the laws of the solutions. A second type is based on the estimateof Proposition B.4.1. In [5], LDP for diffusions that may blow up in finitetime are proved this way. This second type of proof is generally adoptedin the case of SPDEs; see [22, 27, 29, 116]. It is that of the present paper.Note that in [22, 29] the approach to the stochastic calculus is based onmartingale measures whereas in [27, 116] it is infinite dimensional. A thirdtype of proof is based on the continuity theorem of T. Lyons for rough pathsin the p−variation topology; see [104] in the case of diffusions.Uniformity with respect to initial data in compact sets allows to studythe first exit time and the most probable exit points from a neighborhoodof an asymptotically stable equilibrium point. We have studied the case ofweakly damped stochastic nonlinear Schrödinger equations when the noise isof additive or multiplicative type. It will appear elsewhere. Uniform LDPsalso yield LDPs for the family of invariant measures of Markov transitionsemi-groups defined by SPDEs, when the noise goes to zero and when themeasures weakly converge to a Dirac mass on 0. Results on invariant measuresfor some stochastic NLS equations are given in [45]. Uniform LDPsare proved in [5, 48] for diffusions and in [27, 29, 116, 131] for particularSPDEs.B.2 Preliminaries and statement of the resultThroughout the paper the following notations will be used. The constant Cis generic and may take different values, even within the same line.The set of positive integers and positive real numbers are denoted by N ∗and R ∗ +, while the set of real numbers different from 0 is denoted by R ∗ .For p ∈ N ∗ , L p is the Lebesgue space of complex valued functions. For kin N ∗ , W k,p is the Sobolev space of L p functions with partial derivatives upto level k, in the sense of distributions, in L p . For p = 2 and s in R ∗ +, H s isthe Sobolev space of tempered distributions v of Fourier transform ˆv suchthat (1 + |ξ| 2 ) s/2ˆv belongs to L 2 . We denote the spaces by L p R , Wk,p Rand Hs Rwhen the functions are real-valued. The space L 2 is endowed with the innerproduct (u, v) L 2 = Re ∫ Ru(x)v(x)dx. If I is an interval of R, (E, ‖ · ‖ d E ) aBanach space and r belongs to [1, ∞], then L r (I; E) is the space of stronglyLebesgue measurable functions f from I into E such that t → ‖f(t)‖ E is inL r (I).The space of linear continuous operators from B into ˜B, where B and˜B are Banach spaces is L c(B, ˜B). When B = H and ˜B = ˜H are Hilbert97

Appendix B.Large deviations in the multiplicative casespaces, such an operator is Hilbert-Schmidt when ∑ j∈N ‖ΦeH j ‖2˜H < ∞. Theset of such operators is denoted by L 2 (H, ˜H), or L s,r2 when H = H s R˜H and= H r R . The previous sum is the square of a norm that we denote by‖Φ‖ L2 (H, ˜H) .Recall that when A and B are two Banach spaces, A∩B, where the normof an element is the maximum of the norms in A and in B, is a Banach space.We recall that a pair (r(p), p) of positive numbers is called an admissiblepair if p satisfies 2 ≤ p 2 (2 ≤ p < ∞ when d = 2 and( )22 ≤ p ≤ ∞ when d = 1) and r(p) is such thatr(p) = d 12 − 1 p. Given anadmissible pair (r(p), p) and S and T such that 0 ≤ S < T , the spaceX (S,T,p) = C ( [S, T ]; H 1) ∩ L r(p) ( S, T ; W 1,p) ,denoted by X (T,p) when S = 0, is of particular interest for the NLS equation.Now, let us present the space of exploding paths, see [81] for the mainproperties. We add a point ∆ to the space H 1 and embed the space with thetopology such that its open sets are the open sets of H 1 and the complementin H 1 ∪{∆} of the closed bounded sets of H 1 . The set C([0, ∞); H 1 ∪{∆}) isthen well defined. We denote the blow-up time of f in C([0, ∞); H 1 ∪ {∆})by T (f) = inf{t ∈ [0, ∞) : f(t) = ∆}, with the convention that inf ∅ = ∞.We may now define the setE(H 1 ) = { f ∈ C([0, ∞); H 1 ∪ {∆}) : f(t 0 ) = ∆ ⇒ ∀t ≥ t 0 , f(t) = ∆ } ,it is endowed with the topology defined by the neighborhood basisV T,r (ϕ 1 ) = { ϕ ∈ E(H 1 ) : T (ϕ) > T, ‖ϕ 1 − ϕ‖ C([0,T ];H 1 ) ≤ r } ,of ϕ 1 in E(H 1 ) given T < T (ϕ 1 ) and r positive.We define[A(d) and)Ã(d) [ by the)sets [2, ∞) when d = 1 or d = 2 andrespectively 2, 2(3d−1)2d3(d−1)and 2,d−1when d ≥ 3. The space E ∞ is nowdefined for any d in N ∗ by the set of functions f in E(H 1 ) such that for everyp ∈ A(d) and all T ∈ [0, T (f)), f belongs to L r(p) ( 0, T ; W 1,p) . It is endowedwith the topology defined for ϕ 1 in E ∞ by the neighborhood basisW T,p,r (ϕ 1 ) = {ϕ ∈ E ∞ : T (ϕ) > T, ‖ϕ 1 − ϕ‖ X (T,p) ≤ r}where T < T (ϕ 1 ), p belongs to A(d) and r is positive. The space is a Hausdorfftopological space and thus we may consider applying generalizationsof Varadhan’s contraction principle. If we denote again by T : E ∞ → [0, ∞]98

Appendix B.Large deviations in the multiplicative casethe blow-up time, the mapping is measurable and lower semicontinuous.We denote by x∧y and by x∨y respectively the minimum and maximumof x and y. Recall that a rate function I is a lower semicontinuous functionand that it is good if for every c positive, {x : I(x) ≤ c} is a compact set.Let us recall the well known properties on the linear group (U(t)) t∈Rgenerated by the deterministic linear equation. First for every p ≥ 2, t ≠ 0and u 0 in L p′ ,‖U(t)u 0 ‖ W 1,p ≤ (4π|t|) −d 12 − 1 p‖u 0 ‖ W 1,p ′ , 1p + 1 p ′ = 1.(B.2.1)We also have the Strichartz inequalities, see [136],(i)There exists C positive such that for u 0 in H 1 , T positive and(r(p), p) admissible pair,(ii)‖U(·)u 0 ‖ X (T,p) ≤ C ‖u 0 ‖ L 2 ,For every T positive, (r(p), p) and (r(q), q) admissible pairs, s and ρsuch that 1 s + 1r(q) = 1 and 1 ρ + 1 q= 1, there exists C positive suchthat for f in L s ( 0, T ; W 1,ρ) ,∥ ∫ ·0 U(· − s)f(s)ds∥ ∥X (T,p) ≤ C‖f‖ L s (0,T ;W 1,ρ ).We consider W originated from a cylindrical Wiener process on L 2 , i.e.W = ΦW c where Φ is Hilbert-Schmidt. It is known that for any orthonormalbasis (e j ) j∈N of L 2 Rthere exists a sequence of real independent Brownianmotions (β j ) j∈N such that W c (t, x, ω) = ∑ j∈N β j(t, ω)e j (x). In [37], theassumption on Φ is that it is γ−radonifying from L 2 R into W1,α ∩ L 0,12 whereα > 2d, this ensures that the noise lives in the Banach space W 1,α ∩ H 1 R .In the proof of the continuity with respect to the potential on the sets oflevels less or equal to a positive constant of the rate function of the samplepath LDP for the Wiener process we use the continuous embedding of H s R inW 1,∞R. In that respect, it would be enough to impose that Φ is γ−radonifyingfrom L 2 into H 1 R ∩ W1+s,β for any sβ > d. This embedding is again invokedin the proof of the exponential tail estimates, where in addition we use anexpansion on a complete orthonormal system and thus that the image is aHilbert space. Thus, we make the stronger assumption (A) which is givenin terms of Hilbert spacesfor some s > d 2+ 1, Φ ∈ L0,s 2 . (A)99

Appendix B.Large deviations in the multiplicative caseWe finally assume that the probability space is endowed with the filtrationF t = N ∪ σ{W s , 0 ≤ s ≤ t} where N denotes the P−null sets.In the following we restrict ourselves to the case where 1 2≤ σ ifd = 1, 2 or 1 2 ≤ σ < 2d−2if d ≥ 3. We are interested, for ɛ positive, in themild solutions(idu ɛ,u 0= ∆u ɛ,u 0+ λ|u ɛ,u 0| 2σ u ɛ,u 0− iɛ )2 F Φu ɛ,u 0dt + √ ɛu ɛ,u 0dW,with initial datum u 0 ∈ H 1 , i.e. solutions of the integral equationu ɛ,u 0(t) =U(t)u 0 − ∫ t0 U(t − s) ( iλ|u ɛ,u 0(s)| 2σ u ɛ,u 0(s) + ɛ )2 uɛ,u 0(s)F Φ ds−i √ ɛ ∫ t0 U(t − s)uɛ,u 0(s)dW (s).Recall that F Φ (x) = ∑ j∈N (Φe j(x)) 2 . Note that from the assumptions onΦ, F Φ belongs to W 1,∞R . The product with uɛ,u 0in H 1 is thus well defined.For the weaker assumptions on Φ in [37], the ”convolution” of the productis meaningful in the space considered for the fixed point, proving the localwell posedness, thanks to the (ii) of the Strichartz inequalities.Since the stronger the topology, the sharper the estimates, we take advantageof the variety of spaces where the fixed point can be conducted, asit has been done in [81], due to the integrability property and state the LDPin E ∞ . Note that we may check quite easily from the fixed point argumentin [37] that the solutions belong to E ∞ . Also, using similar arguments as in[81], it can be shown that the solutions define random variables with valuesin E ∞ . The laws are denoted by µ uɛ,u0 . The rate function of the LDP forthese family of measures is the infimum of the L 2 -norm of the controls, ofthe control problem{idudt = ∆u + λ|u|2σ u + uΦh, h ∈ L 2 ( 0, ∞; L 2) ,u(0) = u 0 ∈ H 1 ,producing the prescribed path. We may write the mild solution, or skeleton,S c (u 0 , h) = U(t)u 0 −i∫ t0U(t−s) [ S c (u 0 , h)(s) ( λ|S c (u 0 , h)(s)| 2σ + Φh(s) )] ds.If we replace Φh by ∂f∂twhere f belongs to H1 0 (0, ∞; Hs R) which is the subspaceof C ([0, ∞); H s R) of locally square integrable in time and with locallysquare integrable in time time derivative functions, with null initial datum,we write∫ t[ (S(u 0 , f) = U(t)u 0 − i U(t − s) S(u 0 , f)(s) λ|S(u 0 , f)(s)| 2σ + ∂f )]ds.∂s0100

Appendix B.Large deviations in the multiplicative caseThe topology on C ([0, ∞); H s R) is that of the uniform convergence on thecompact subsets of [0, ∞). In the following we write K ⊂⊂ H 1 when K is acompact subset of H 1 and by Int(A) the interior of A.Theorem B.2.1 The family ( µ ) uɛ,u 0ɛ>0and good rate functionsatisfies a uniform LDP of speed ɛI u 0(w)= inf f∈H 10 (0,∞;H s R):w=S(u 0 ,f) IW (f)= 1 2 inf h∈L 2 (0,∞;L 2 ):w=S c (u 0 ,h) ‖h‖ 2 L 2 (0,∞;L 2 ) ,where I W is the rate function of the sample path LDP for the Wiener process,i.e. ∀K ⊂⊂ H 1 , ∀A ∈ B (E ∞ ) ,− sup u0 ∈K inf w∈Int(A) I u 0(w) ≤ lim ɛ→0 ɛ log inf u0 ∈K P (u ɛ,u 0∈ A)≤ lim ɛ→0 ɛ log sup u0 ∈K P (u ɛ,u 0∈ A) ≤ − inf w∈A,u0 ∈K Iu 0(w).This result is proved in Section B.3 and B.4. The aim is to push forwardthe following sample path LDP for the Wiener process. It follows from thegeneral LDP for Gaussian measures on a real separable Banach space, see[53], the fact that the laws of the restrictions of √ ɛW on C ([0, T ]; H s R ) andL 2 (0, T ; H s R) have same RKHS and Dawson-Gärtner’s theorem for projectivelimits.Proposition B.2.2 The family of laws of ( √ ɛW ) ɛ>0on C([0, ∞); H s R ) satisfiesa LDP of speed ɛ and good rate functionI W (f) = 1 2infh∈L 2 (0,∞;L 2 ):f= R ·0 Φh(s)ds ‖h‖ 2 L 2 (0,∞;L 2 ) .We denote by C a the set { f ∈ C ([0, ∞); H s R ) : IW (f) ≤ a } also equal to{∥ f ∈ H 1 0 (0, ∞; H s R ) : f(0) = 0, ∂f ∥∥∥∂t ∈ ImΦ, Φ −1 ∂f|(KerΦ) ⊥ ∂t ∥ ≤ √ }2a .L 2 (0,∞;L 2 )B.3 Continuity of the skeleton with respect to thecontrol on the sets C a and exponential tail estimatesWe start by a result on the continuity of the skeleton with respect to thecontrol on the sets C a . It is used to prove the lower bound of the LDP andthat the rate function is good; see Section B.5. In that respect, our proof is101

Appendix B.Large deviations in the multiplicative casecloser to the proofs in [22, 29]. The authors of [27, 116] use some slightlydifferent arguments and prove separately the compactness of the set of levelsof the rate function less or equal to a positive constant, the lower and upperbounds of the LDP using their characterizations in metric spaces. Note thatin the applications, B.6, we need to compute infima of a quantity involvingthe rate function and the continuity proves to be very useful. We prove thecontinuity with respect to the control and initial datum as suggested in [29]though the continuity with respect to the initial data is not used in the proofof the uniform LDP.Proposition B.3.1 For every u 0 ∈ H 1 , a positive and f in C a , S(u 0 , f)exists and is uniquely defined. It is a continuous mapping from H 1 ×C a intoE ∞ , where C a has the topology induced by that of C ([0, ∞); H s R ).Proof. Let F denote the mapping such that∫ t[ (F(u, u 0 , f) = U(t)u 0 − i U(t − s) u(s) λ|u(s)| 2σ + ∂f )]ds.∂s0Let a and r be positive, f in C a and u 0 such that ‖u 0 ‖ H 1 ≤ r, set R = 2crwhere c is the norm of the continuous mapping of the (i) of the Strichartz inequalities.From (i) and (ii) of the Strichartz inequalities along with Hölder’sinequality, the Sobolev injections and the continuity of Φ, (for any T positive, p in A(d), u and v in X (T,p) and any ν init is a well defined interval since σ < 2d−2‖F(u, u 0 , f)‖ X (T,p)≤ c‖u 0 ‖ H 1 + CT ν ‖u‖ 2σ+1 + CT 1 X (T,p) 2 − 1∥r(p)‖u‖ C([0,T ];H 1 )≤ c‖u 0 ‖ H 1 + CT ν ‖u‖ 2σ+1X (T,p) + C √ aT 1 2 − 1r(p)‖u‖ X (T,p).Similarly we obtain),0, 1 − σ(d−2)2, there exists C positive such that∥∥ ∂f ∂s∥ r(p)dL 21,0,T ;W 2‖F(u, u 0 , f) − F(v, u 0 , f)‖ X (T,p)()≤ C[T ν ‖u‖ 2σ + ‖v‖ 2σ + T 1 X (T,p) X (T,p) 2 − 1 √ ]r(p)a ‖u − v‖ X (T,p).Thus, for T = T ∗ r,a,p small enough depending on r, a and p, the ball centeredat 0 of radius R is invariant and the mapping F(·, u 0 , f) is a 3 4 −contraction.We denote by S 0 (u 0 , f) the unique fixed point of F(·, u 0 , f) in X (T ∗ r,a,p ,p) .Also when T is positive, we can solve the fixed point problem on any102

Appendix B.Large deviations in the multiplicative caseinterval [ ⌊ ⌋kTr,a,p, ∗ (k + 1)Tr,a,p] ∗ with 1 ≤ k ≤ TT. The fixed point is denotedby S k (u k , f) where u k = S k−1 (u k−1 , f)(kTr,a,p), ∗ as long as ‖u k ‖ H 1 ≤ r.r,a,p∗Existence and uniqueness of a maximal solution S(u 0 , f) follows. It coincideswith S k (u k , f) on the above intervals when it is defined. We may alsoshow that the blow-up time corresponds to the blow-up of the H 1 −norm,thus S(u 0 , f) is an element of E ∞ .We shall now prove the continuity. Take u 0 in H 1 , a positive and f inC a . From the definition of the neighborhood basis of the topology of E ∞ ,it is enough to see that for ɛ positive, T < T (S(u 0 , f)), and p ∈ A(d),there exists η positive such that for every ũ 0 in H 1 and g in C a satisfying‖u 0 − ũ 0 ‖ H 1 + ‖f − g‖ C([0,T ];H sR ) ≤ η then ‖S(u 0, f) − S(ũ 0 , g)‖ X (T,p) ≤ ɛ.⌊ ⌋We set r = ‖S(u 0 , f)‖ X (T,p) + 1, N = TT, δr,a,p∗ N+1 = ɛN+1∧ 1, and definefor k ∈ {0, ..., N} , δ k and η k such that 0 < δ k < δ k+1 , 0 < η k < η k+1 < 1 and∥∥S k+1 (u k , f) − S k+1 (ũ k , g) ∥ ≤ δ X(kT k+1,r,a,p ∗ ,(k+1)T r,a,p ∗ ,p)if‖u k − ũ k ‖ H 1 + d C([0,∞);H sR ) (f, g) ≤ η k+1.The distance d C([0,∞);H sR )(·, ·) is one of the classical distances of the topologyof uniform convergence on compact subsets of [0, ∞). It is possible to choose(δ k ) k=0,...,Nas long as we prove the continuity for k = 0. The maximal solutionS(ũ 0 , g) then necessarily satisfies T (S(ũ 0 , g)) > T . We conclude settingη = η 1 and using the triangle inequality.We now prove the continuity for k = 0. First note that ‖u 0 ‖ H 1 ≤ r and‖ũ 0 ‖ H 1 ≤ r since η < 1, thus if Υ 1 denotes ∥ ∥S 0 (u 0 , f) − S 0 (ũ 0 , g) ∥ ∥X (T r,a,p ∗ ,p) ,Υ 1 = ∥ ∥F(S 0 (u 0 , f), u 0 , f) − F(S 0 (ũ 0 , g), ũ 0 , g) ∥ ∥X (T ∗ r,a,p ,p)≤ ∥ ∥ F(S 0 (u 0 , f), u 0 , f) − F(S 0 (u 0 , f), ũ 0 , g) ∥ ∥X (T ∗ r,a,p ,p)+ ∥ ∥F(S 0 (u 0 , f), ũ 0 , g) − F(S 0 (ũ 0 , g), ũ 0 , g) ∥ ∥X (T ∗ r,a,p ,p) ,and since F(·, ũ 0 , g) is a 3 4 −contraction,Υ 1 ≤ 4 ∥ ∥F(S 0 (u 0 , f), u 0 , f) − F(S 0 (u 0 , f), ũ 0 , g) ∥ ∥X (T ∗ r,a,p ,p)≤ 4 (c‖u 0 − ũ 0 ‖ H 1 + Υ 2 )where, using Hölder’s inequality and taking p < ˜p,Υ 2 = ∥ ∫ ·0 U(· − s)S0 (u 0 , f) ∂(f−g)∂s(s)ds∥ X(Tr,a,p ∗ ,p) ≤ Υ 3 ∨ Υ 4103

Appendix B.Large deviations in the multiplicative caseand( ∥∥∥ ∫ ·Υ 3 =0 U(· − s)S0 (u 0 , f) ∂(f−g)∂s(s)ds∥ θ C([0,Tr,a,p];H ∗ 1 )× ∥ ∫ ·0 U(· − s)S0 (u 0 , f) ∂(f−g)∂s(s)ds∥ 1−θL r(˜p) (0,T ∗ r,a,p;W 1,˜p )),with θ = ˜p−p˜p−2 ,Υ 4 = ∥ ∫ ·0 U(· − s)S0 (u 0 , f) ∂(f−g)∂s(s)ds∥ C([0,T ∗ r,a,p ];H 1 ) .From the (ii) of the Strichartz inequalities we obtainΥ 4≤ C( √a (T∗r,a,p) 12 − 1r(˜p)R) 1−θ.It is now enough to show that when g is close enough to f, Υ 3 can be madearbitrarily small. Take n in N and set for i in {0, ..., n} , t i = iT r,a,p∗nandS 0,n (u 0 , f) = U(t − t i ) (S(u 0 , f)(t i )) , for t i ≤ t < t i+1 .As in the previous calculations we obtain∥ ∫ ·0 U(· − s) ( S 0 (u 0 , f) − S 0,n (u 0 , f) ) ∂(f−g)∂s(s)ds∥ C([0,T ∗ r,a,p ];H 1 )≤ C √ a ( Tr,a,p∗ ) 12 − 1 ∥r(p) ∥S 0 (u 0 , f) − S 0,n (u 0 , f) ∥ C([0,T ∗ r,a,p ];H 1 )) 12 − 1≤ C √ a ( Tr,a,p∗ r(p)∥ ∥∥ ∫ [ (·sup i∈{0,...,n−1} t iU(· − s) S 0 (u 0 , f) λ ∣ ∣S 0 (u 0 , f) ∣ )]2σ + ∂f ∥∂s(s)ds≤ C √ a ( [Tr,a,p∗ ) 1(2 − 1r(p)R 2σ+1 T ∗) ν (r,a,p T ∗) 1n + r,a,p 2 − 1r(p)nR √ ]a .(B.3.1)∥C([ti ,t i+1 ];H 1 )It can be made arbitrarily small for large n. Now it remains to bound theC ([ 0, T ∗ r,a,p]; H1 ) −norm of∫ t0 U(t − s)S0,n (u 0 , f) ∂(f−g)∂s(s)ds= ∑ n−1∫ (ti+1 ∧ti=0 t i ∧tU(t − t i ∧ t)S 0 (u 0 , f)(t i ∧ t) ∂(f−g)∂s)(s) ds= ∑ n−1i=0 U(t − t i ∧ t) [ S 0 (u 0 , f)(t i ∧ t) ((f − g)(t i+1 ∧ t) − (f − g)(t i ∧ t)) ]since, from the Sobolev injections and the fact that U(t − t i ∧ t) is a groupon H 1 , for any i in {0, ..., n − 1} and v in H s R ,∥∥U(t − t i ∧ t) ( S 0 (u 0 , f)(t i ∧ t)v )∥ ∥H 1 ≤ C ∥ ∥S 0 (u 0 , f)(t i ∧ t) ∥ ∥H 1 ‖v‖ H sR104

Appendix B.Large deviations in the multiplicative caseand as ∂(f−g)∂sis Bochner integrable, we obtain an upper bound of the formrCn‖f − g‖ C([0,T ];H sR ). For fixed n, it can be made arbitrarily small takingg sufficiently close to f.□We now give the exponential tail estimates.Lemma B.3.2 Assume that ξ is a point-predictable process, that p belongsto Ã(d), T is positive and that there exists η positive such that ‖ξ‖2 C([0,T ];H 1 ) ≤η a.s., then for every t in [0, T ] and δ positive,(∫ t0))P sup∥ U(t − s)ξ(s)dW (s)∥ ≥ δ ≤ exp(1 − δ2,W 1,p κ(η)whereand ct 0 ∈[0,T ]04cκ(η) =( ) 2 r(p)d 1−2 T 4r(p)(d + 1)(d + p)‖Φ‖ 2 L 0,s1 − 4r(p)( )r(p)d2is the norm of the continuous embedding H s R2η,r(p)d2⊂ W1,R.Proof. Let us denote by g a (f) = ( 1 + a‖f‖ p ) 1pthe real-valued functionW 1,pparameterized∫by a positive and by M the martingale defined by M(t 0 ) =t00 U(t − s)ξ(s)dW (s). The function g a is twice Fréchet differentiable, thefirst and second derivatives at point M(t) in the direction h are denoted byDg a (M(t)).h and D 2 g a (M(t), M(t)).(h, h), they are continuous. Also, thesecond derivative is uniformly continuous on the bounded sets. By the Itôformula the following decomposition holdswhere E a (t 0 ) is equal to∫ t00g a (M(t 0 )) = 1 + E a (t 0 ) + 1 2 R a(t 0 )Dg a (M(s)).U(t−s)ξ(s)dW (s)− 1 2∫ t0‖Dg a (M(s)).U(t−s)ξ(s)‖ 2 L 2 (L 2 ,R) dsand R a (t 0 ) to∫ t00 ‖Dg a(M(s)).U(t − s)ξ(s)‖ 2 L 2 (L 2 ,R) ds+ ∫ t 0∑0 j∈N D2 g a (M(s), M(s)).(U(t − s)ξ(s)Φe j , U(t − s)ξ(s)Φe j )ds,where (e j ) j∈N is a complete orthonormal system of L 2 . We denote by[ ∫q(u, h) = ReRu|u| p−2 hdx + ∑ d∫]d k=1 R∂ d xk u|∂ xk u| p−2 ∂ xk hdx ,105

Appendix B.Large deviations in the multiplicative casethen Dg a (u).h = a ( 1 + a‖u‖ p W 1,p ) 1p −1 q(u, h), andD 2 g a (u, u).(h, g) = a 2 (1 − p) ( 1 + a‖u‖ p ) 1W 1,p p −2 q(u, h)q(u, g)+a ( 1 + a‖u‖ p ) [( 1)W 1,p p −1 1 + p−22Re ∫ (R|u| p−2 gh + ∑ )dd k=1 |∂ x ku| p−2 ∂ xk g∂ xk h dx+ p−22 Re ∫ (Ru 2 |u| p−4 gh + ∑ ) ]dd k=1 ∂ x ku 2 |∂ xk u| p−4 ∂ xk g∂ xk h dx .From the series expansion of the Hilbert-Schmidt norm along with Hölder’sinequality, we obtain that R a (t 0 ) is less than⎡⎛⎞a(d + 1) ∫ 2(p−1)t 0 ⎢0 ⎣a(d + 1) ∑ j∈N ‖U(t − s)ξ(s)Φe j‖ 2 ⎝ ‖M(s)‖ W 1,p W 1,p ⎠1+a‖M(s)‖ p 1pW 1,p−(p − 1)a(d + 1) ( 1 + a‖M(s)‖ p ) 1W 1,p p −2 ∑j∈N q (M(s), U(t − s)ξ(s)Φe j) 2(p − 1)‖M(s)‖ p−2 (W 1 + a‖M(s)‖p) 11,p W 1,p p −1 ∑]j∈N ‖U(t − s)ξ(s)Φe j‖ 2 Wds.1,pSince the term in parenthesis in the first part is an increasing function of‖M(s)‖ W 1,p, the second term is non positive and‖M(s)‖ p−2W 1,p (1 + a‖M(s)‖pW 1,p ) 1p −1= a 2 p −1 ( a‖M(s)‖ p W 1,p ) 1−2p ( 1 + a‖M(s)‖ p W 1,p ) 1p −1≤ a 2 p −1 ( 1 + a‖M(s)‖ p W 1,p ) 1−2p ( 1 + a‖M(s)‖ p W 1,p ) 1p −1 ≤ a 2 p −1 ,we obtain thatR a (t 0 ) ≤ (d + 1)(d + p)a 2 p∫ t00∑‖U(t − s)ξ(s)Φe j ‖ 2 Wds. 1,pFinally, from (B.2.1), Hölder’s inequality and the Sobolev injections we obtainthat for any t 0 in [0, T ],R a (t 0 ) ≤ 2(d + 1)(d + p)a 2 p2j∈N( r(p)dc2) 2 ∫ T‖Φ‖ 2 η |t − s| − 4L 0,sr(p)ds,2 0the integral is finite since p

Appendix B.Large deviations in the multiplicative case()≤ exp −(1 + aδ p ) 1 p + 1 + κ(η)a p24()≤ e exp −a 1 p δ + κ(η)a p24.The last inequality holds for arbitrary a positive.finally obtains the desired estimate.Minimizing on a, one□∫Proposition B.3.3 (Exponential tail estimates) If Z, defined by Z(t) =t0 U(t−s)ξ(s)dW (s), is such that there exists η positive such that ‖ξ‖2 C([0,T ];H 1 ) ≤η a.s., then for any p in Ã(d), T and δ positive,( )P ( ‖Z‖ C([0,T ];H 1 ) ≥ δ ) ≤ 3 exp)P(‖Z‖ L r(p) (0,T ;W 1,p ) ≥ δ ≤ c exp− δ2κ 1 (η)(− δ2κ 2 (η)where c = 2e + exp((2ek 0 !) 1k 0), k 0 = 2 ∨ min{k ∈ N : 2k ≥ r(p)}c8cκ 2 (η) =κ 1 (η) = T 4c (∞) 2 ‖Φ‖ 2 η,L 0,s2( ) 2 r(p)d 1−2 T 2r(p)(d + 1)(d + p)‖Φ‖ 2 L 0,s1 − 4r(p)( )r(p)d2and c(∞) are the norms of the continuous embeddings H s Rand H s R ⊂ W1,∞ R .)2η,⊂ r(p)dW1, 2RProof. The first estimate. We recall that in H 1 we may write, using theseries expansion of the Wiener process and that (U t ) t∈Ris a unitary group,see [81], Z(t) = U(t) ∫ t0U(−s)ξ(s)dW (s). Since U(−s) is an isometry, oneobtains that for every s in [0, T ],‖U(−s)ξ(s)Φ‖ L2 (L 2 ,H 1 ) ≤ ‖L s ‖ Lc(H s R ,H1 ) ‖Φ‖ L 0,s2≤ c (∞) ‖L s ‖ Lc(W 1,∞R ,H 1 ) ‖Φ‖ L 0,s2≤ c (∞) ‖ξ(s)‖ H 1‖Φ‖ L0,s2where L is such that L s u = ξ(s)u. Consequently, we obtain that∫ T0‖U(−s)ξ(s)Φ‖ 2 L 0,12ds ≤ c (∞) 2 ‖Φ‖ 2 T ‖ξ(s)‖ 2 L 0,s C([0,T ],H 1 ) ds2107

Appendix B.Large deviations in the multiplicative caseWe conclude using [117][Theorem 2.1]. The result still holds when the operatortakes its value in another Hilbert space.The second estimate. From Markov’s inequality it is enough to show that[ (∫1·)]2E expκ 2 (η) ∥ U(· − s)ξ(s)dW (s)∥≤ c.0L r(p) (0,T ;W 1,p )For k ≥ k 0 , Jensen’s inequality along with Fubini’s theorem, Lemma B.3.2,the change of variables and the integration by parts formulae give that[ ( 1E √ ∥ ∫ ) ] 2k·κ2 (η) 0 U(· − s)ξ(s)dW (s)∥ ∥L r(p) (0,T ;W 1,p )≤ 1 T≤ 1 T≤ 1 T∫ T0 E [ (T∫ T0∫ T0(κ 2 (η)) r(p)2(∫ ∥∥∥∞ ∫ t0P(∫ ∞≤ e ∫ ∞0exp0e exp(−2u 1 k∥ ∫ ∥ ) 2kt∥∥0 U(t − s)ξ(s)dW (s) r(p) r(p)W 1,p0 U(t − s)ξ(s)dW (s) ∥ ∥∥W1,p ≥ (− κ 2(η)2T r(p))u 1 kκ(η))dudt](κ 2 (η)) r(p)2Tdt) )1r(p)1u 2k dudtdu = e ∫ ∞0kv k−1 exp(−2v)dv = 2e ∫ ∞0v k exp(−2v)dv.Thus, using Fubini’s theorem one obtains[ ( 1E √ ∥ ∫ ) ] 2k0·κ2 (η) 0 U(· − s)ξ(s)dW (s)∥ ∥L r(p) (0,T ;W 1,p )≤ k 0 ! ∑ [ (1k≥k 0 k! E √ 1 ∥ ∫ ·κ2 (η) 0 U(· − s)ξ(s)dW (s)∥ ∥L r(p) (0,T ;W 1,p )≤ k 0 ! ∑ 1k≥k 0 k! 2e ∫ ∞0v k exp(−2v)dv≤ k 0 ! ∑ k∈N 1 k! 2e ∫ ∞0v k exp(−2v)dv = 2ek 0 !,hence from Hölder’s inequality∑ k0 −1k=0= ∑ k 0 −1k=0≤ ∑ k 0 −1k=0≤ ∑ k 0 −1k=0[ (1k! E √ 1 ∥ ∫ ·κ2 (η) 0 U(· − s)ξ(s)dW (s)∥ ∥L r(p) (0,T ;W 1,p )) 2k]) 2k][ {(∥∥1κ 2 (η) k k! E ∫ ) } k]·0 U(· − s)ξ(s)dW (s)∥ 2k0 k∥0L r(p) (0,T ;W 1,p )[ (∥∥1κ 2 (η) k k! E ∫ ·1 k(2ek 0 !) k 0k!≤ exp0 U(· − s)ξ(s)dW (s)∥ ∥L r(p) (0,T ;W 1,p )((2ek 0 !) 1k 0).108) 2k0] kk 0

Appendix B.Large deviations in the multiplicative caseThe end of the proof is now straightforward.□B.4 Almost continuity of the Itô mapProposition B.4.1 For every positive a, R and ρ, u 0 in H 1 , f in C a ,T < T (S(u 0 , f)), p in A(d), there exists positive ɛ 0 , γ and r such that forevery ɛ in (0, ɛ 0 ] and ũ 0 in B H 1(u 0 , r),( ∥∥ɛ log P u ɛ,ũ 0− S(u 0 , f) ∥ X (T,p) ≥ ρ; ∥ √ ɛW − f ∥ )C([0,T ];H sR ) < γ ≤ −R.Proof. Take u 0 in H 1 , f in C a , a, R and ρ positive, T < T (S(u 0 , f)) andp in A(d).Step 1: Change of measure to center the Wiener process aroundf. The function f in C a is such that there exists h in L 2 ( 0, T ; L 2) such thatf(·) = ∫ ·0 Φh(s)ds and 1 2 ‖h‖2 L 2 (0,T ;L 2 ) ≤ a. We denote by W ɛ the processW (t)−√ 1 ∫ t∂fɛ ∂s ds = W (t)− √ 1 ∫ tɛ[ (Since E exp 1∫ T200Φh(s)ds = Φ(W c (t) − √ 1 ∫ t)h(s)ds .ɛ 0) ]0 ‖h(s)‖2 Lds < ∞, the Novikov condition is satisfied2and the Girsanov theorem gives that W ɛ is a µ−Wiener process on C ([0, T ]; H s R )under the probability P ɛ defined bydP ɛdP( ∫ 1√ɛ t∣ = exp (h, dW c (s)) L 2 − 1Ft2ɛ(Set u ɛ (t) = exp − √ 1 ∫ )tɛ 0 (h, dW c(s)) L 2A =0∫ T0)‖h(s)‖ 2 Lds 2and λ such that a − λ < −R and{ ∥∥u ɛ,ũ 0− S(u 0 , f) ∥ ∥X (T,p) ≥ ρ; ∥ ∥ √ ɛW − f ∥ ∥C([0,T ];H sR ) < γ },is a uni-( (then since exp − √ 1 ∫ tɛ 0 (h, dW c(s)) L 2 − 1 ∫ ))t2ɛ 0 ‖h(s)‖2 Lds 2formly integrable martingale{ }P(A) ≤ E dP Pɛ dP ɛ1l A∩{u ɛ (T )≤exp( λ ɛ )}+ P ( u ɛ (T ) > exp ( λ(ɛ≤ E Pɛ{1l A exp λɛ + 1 ∫ )}T2ɛ 0 ‖h(s)‖2 Lds 2≤ exp ( λ+aɛ)Pɛ (A) + exp ( a−λɛ).t∈[0,T ]))+ exp ( − λ ɛ)E (u ɛ (T ))109

Appendix B.Large deviations in the multiplicative caseFinally it is enough to prove that there exists positive ɛ 0 , γ and r such thatfor every ɛ in (0, ɛ 0 ] and ũ 0 in B H 1(u 0 , r), ɛ log P ɛ (A) ≤ −R − λ − a, orequivalently thatɛ log P ɛ( ∥∥vɛ,ũ 0− S(u 0 , f) ∥ ∥X (T,p) ≥ ρ; ∥ ∥ √ ɛW ɛ∥∥C([0,T];H sR ) < γ )≤ −R − λ − a,where v ɛ,ũ 0satisfies v ɛ,ũ 0(0) = ũ 0 and(idv ɛ,ũ 0= ∆v ɛ,ũ 0+ λ|v ɛ,ũ 0| 2σ v ɛ,ũ 0+ ∂f∂t vɛ,ũ 0− iɛ )2 F Φv ɛ,ũ 0dt + √ ɛv ɛ,ũ 0dW ɛ .Step 2: Reduction to estimates for the stochastic convolution.Note, this is standard fact, that the unboundedness of the drift and coefficientof the Wiener process is not a limitation since the result of PropositionB.4.1 is local. A localisation argument is therefore used to overcomethe apparent difficulty. We replace T byτ ρ = inf { t : ‖v ɛ,ũ 0− S(u 0 , f)‖ X (t,p) ≥ ρ } ∧ T.Since T < T (S(u 0 , f)), v ɛ,ũ 0satisfies‖v ɛ,ũ 0‖ X (τρ,p) ≤ ρ + ‖S(u 0 , f)‖ X (τρ,p) = D.With computations similar to that of the proofs of Proposition B.3.1 hereinand of Theorem 4.1 in [37] with a cutt-off function in front of the nonlinearity( ‖S(u0) ( ),f)‖of the form θX (s,p)‖vDand θɛ,ũ 0 ‖ X (s,p)D, we obtain for t positive()and ν ∈ 0, 1 − σ(d−2)2,∥ vɛ,ũ 0− S(u 0 , f) ∥ X (t∧τρ,p) ≤ C‖ũ 0 − u 0 ‖ H 1 + √ ɛ ∥ ∫ ·0 U(· − s)vɛ,ũ 0(s)dW ɛ (s) ∥ X (t∧τρ,p)+C[(t ∧ τ ρ ) ν (D 2σ )(1 + D) + (t ∧ τ ρ ) 1 2 − 1 √ ]r(p)a + ɛ(t ∧ τ ρ ) 1− 2 ∥∥r(p)v ɛ,ũ 0− S(u 0 , f) ∥ X (t∧τρ,p)+Cɛ(t ∧ τ ρ ) 1− 2r(p)‖S(u 0 , f)‖ X (t∧τρ,p).Set ɛ ≤ 1, then for t = t ∗ small enough we obtain∥∥v ɛ,ũ 0− S(u 0 , f) ∥ X (t ∗ ∧τρ,p) ≤ 2 (C‖ũ 0 − u 0 ‖ H 1 + CɛD+ √ ɛ ∥ ∫ ·0 U(· − s)vɛ,ũ 0(s)dW ɛ (s) ∥ X (τρ,p)).(B.4.1)Set N =⌊τρt ∗ ∧τ ρ⌋, and for i in {0, ..., N}, T i = it ∗ and T N+1 = T . Inequality(B.4.1) also holds for ∥ ∥v ɛ,ũ 0− S(u 0 , f) ∥ ∥X(T i ,T i+1 ,p)for every i in {0, ..., N},110

Appendix B.Large deviations in the multiplicative casereplacing ‖ũ 0 − u 0 ‖ H 1 by ∥ ∥v ɛ,ũ 0(T i ) − S(u 0 , f)(T i ) ∥ H 1.As for i in {1, ..., N}, ∥ vɛ,ũ 0(T i ) − S(u 0 , f)(T i ) ∥ H 1 ≤ ∥ vɛ,ũ 0− S(u 0 , f) ∥ X(T i−1 ,T i ,p),we obtain using the triangle inequality that∥ vɛ,ũ 0− S(u 0 , f) ∥ X (τρ,p)≤ 2(N + 1) (√ ɛ ∥ ∫ ·0 U(· − s)vɛ,ũ 0(s)dW ɛ (s) ∥ X (τρ,p) + CɛD )+2C ∑ ∥N−1i=1∥v ɛ,ũ 0− S(u 0 , f) ∥ X(T i−1 ,T i ,p) + 2C‖u 0 − ũ 0 ‖ H 1( ∑N−1) (√ɛ ∥∫ ≤ 2(N + 1)·i=0 (2C)i 0 U(· − s)vɛ,ũ 0(s)dW ɛ (s) ∥ X (τρ,p) + CɛD )+(2C) N ‖u 0 − ũ 0 ‖ H 1.We may choose 2C > 1, then it is enough to show that there exists positiveɛ 0 , γ and r such that (2C) N r < ρ and for every ɛ in (0, ɛ 0 ] and ũ 0 inB H 1(u 0 , r),ɛ log P ɛ( √ɛ ∥ ∥∫ ·0 U(· − s)vɛ,ũ 0(s)dW ɛ (s) ∥ ∥X (τρ,p) + CɛD ≥ (2C−1)(ρ−(2C)N r)2(N+1)((2C) N −1) ;‖ √ ɛW ɛ ‖ C([0,T ];H sR ) < γ )≤ −R − λ − a.Step 3: The case of the stochastic convolution. We now need thatfor u 0 in H 1 , f in C a , a, R and ρ positive, T < T (S(u 0 , f)) and p in A(d),there exists ɛ 0 , γ and r positive such that for every ɛ in (0, ɛ 0 ] and ũ 0 inB H 1(u 0 , r),( √ɛ ∥∫ ɛ log P ɛ ·0 U(· − s)vɛ,ũ 0(s)dW ɛ (s) ∥ X (τρ,p) ≥ ρ; ‖ √ )ɛW ɛ ‖ C([0,τρ];H s R) < γ≤ −R.For n in N and i in {0, ..., n}, we set t i = iτρnas (B.3.1)and the same approximationv ɛ,ũ 0,n (t) = U(t − t i ) ( v ɛ,ũ 0(t i ) ) , for t i ≤ t < t i+1 .For any δ positive we may write( √ɛ ∥∫ P ɛ ·0 U(· − s)vɛ,ũ 0(s)dW ɛ (s) ∥ X (τρ,p) ≥ ρ; ‖ √ )ɛW ɛ ‖ C([0,τρ];H s < γR≤ P 1 + P 2 + P 3 ,where(√ ∥∫ P 1 = P ɛ ɛ ·0 U(· − s) ( v ɛ,ũ 0(s) − v ɛ,ũ0,n (s) ) dW ɛ (s) ∥ X (τρ,p) ≥ ρ∥2 ;∥v ɛ,ũ 0− v ∥ )ɛ,ũ 0,n C([0,τρ];H 1 ) < δ ,P 2 = P ɛ( ∥∥v ɛ,ũ 0− v ɛ,ũ 0,n ∥ ∥C([0,τρ];H 1 ) ≥ δ ),111

Appendix B.Large deviations in the multiplicative case( √ɛ ∥∫ P 3 = P ɛ ·0 U(· − s)vɛ,ũ0,n (s)dW ɛ (s) ∥ X (τρ,p) ≥ ρ 2 ; ‖√ ɛW ɛ ‖ C([0,τρ];H∥s R) < γ;∥v ɛ,ũ 0− v ∥ )ɛ,ũ 0,n C([0,τρ];H 1 ) < δ .From Proposition B.3.3,(ρ 2 )P 1 ≤ C exp −4ɛ (κ 1 (δ 2 ) ∨ κ 2 (δ 2 ,))thus for any ɛ < 1 and δ small enough, P 1 < −R − 1.Also P 2 ≤ P 21 + P 22 where(∥∫ √ ∥∥∥ ·P 21 = P ɛ ɛsupi∈{0,...,n−1})U(t − s)v ɛ,ũ 0(s)dW ɛ (s)∥ ≥ δt i C([ti ,t i+12];H 1 )and P 22 equals(∫ ·[P ɛ sup∥ U(t − s) λ ∣ ∣v ɛ,ũ 0(s) ∣ 2σ + ∂fi∈{0,...,n−1} t i∂s (s) − iɛ ])2 F Φ v ɛ,ũ 0(s)ds∥ ≥ δ .C([ti ,t i+12];H 1 )( )From Proposition B.3.3, P 21 ≤ 3n exp − Cδ2 nτ ρDand P 2 22 = 0 for n largeenough. Indeed, with calculations similar to that of the proof of Theorem4.1 in [37] and of Proposition B.3.1, we obtain for ɛ < 1,∥ ∥∥ ∫ [·sup i∈{0,...,n−1} t iU(t − s) λ ∣ ∣v ɛ,ũ 0(s) ∣ ]2σ + ∂f∂s (s) − iɛ 2 F Φ v ɛ,ũ 0 (s)ds∥ [ C([ti ,t i+1 ];H 1 )(τρ) ν≤ Cn D 2σ+1 + ( τ ρ) 12 − 1r(p)nD √ a + 1 ( τρ) 1−2]r(p)2 nD ,then, for every δ positive and 0 < ɛ < 1, for n large enough ɛ log P 2 < −R−1.So far we have obtained that for δ small enough and n large enough, forany 0 < ɛ < 12 log(2) , ɛ log (P 1 + P 2 ) ≤ −R − 1 2 .Now fix δ as above. Note that the condition ∥ ∥ vɛ,ũ 0− v ɛ,ũ 0,n ∥ ∥C([0,τρ];H 1 )

Appendix B.Large deviations in the multiplicative casewhere( √ɛ∥ ∥∥ ∫ ·P 31 = P ɛ · U(· − s)vɛ,ũ0,n (s)dW ɛ (s) ∥ ≥ n 1+ 2r(˜p) +η 12;L r(˜p) (0,τ ρ;W 1,˜p )∥∥v ɛ,ũ 0− v ∥ )ɛ,ũ 0,n C([0,τρ];H 1 ) < δ ,( √ɛ∥ ∥∥ ∫ ·P 32 = P ɛ · U(· − s)vɛ,ũ0,n (s)dW ɛ (s) ∥ θ n 1+ 2r(˜ρ) +η 1−θ2≥ ρC([0,τ ρ];H 1 )8 ;∥ vɛ,ũ 0− v ∥ )ɛ,ũ 0,n C([0,τρ];H 1 ) < δ ,( √ɛ∥ ∥∥ ∫ ·P 33 = P ɛ · U(· − s)vɛ,ũ0,n (s)dW ɛ (s) ∥ ≥ ρ C([0,τρ];H 1 )8 ;∥∥v ɛ,ũ 0− v ∥ )ɛ,ũ 0,n C([0,τρ];H 1 ) < δ ,( ∥∫ √ɛ ∥∥∥ ·P 34 = P ɛ U(· − s)v ɛ,ũ0,n (s)dW ɛ (s) ∥0( ∥∫ √ɛ ∥∥∥ ·P 35 = P ɛ U(· − s)v ɛ,ũ0,n (s)dW ɛ (s) ∥0∥L r(p) (0,τ ρ;W 1,p )∥C([0,τρ];H 1 )≥ ρ 8 ; E )≥ ρ 8 ; E )The probability P 31 is less than(∑ n−1 √ɛ∥ ∥∥i=0 P ∫ ti+1ɛ t iU(· − s)v ɛ,ũ0,n (s)dW ɛ (s) ∥ ≥ n −1+ 2r(˜p) +η 12;L r(˜p) (t i ,t i+1 ;W 1,˜p )∥∥v ɛ,ũ 0− v ∥ )ɛ,ũ 0,n C([0,τρ];H 1 ) < δwhich is itself, from Proposition B.3.3, less than⎛⎞nnC exp ⎝−−1+ 2r(˜p) +ηC ( ⎠τ ρ) 1−2.r(˜p)nɛ(δ + D) 2Thus, for n large enough, ɛ ≤ 1 and δ positive, ɛ log P 31 < −R − 1.The first exponential tail estimate of Proposition B.3.3 gives that⎛⎞P 32 ≤ nC exp ⎝−Cnρ 21+ 2r(˜p) +η (1−θ) ( τρn)ɛ(δ + D) 2and, from the choice of ˜p, for n large enough, for any ɛ and δ positive,ɛ log P 32 < −R − 1.The same holds for P 33 even more clearly.⎠ ,.,113

Appendix B.Large deviations in the multiplicative caseThe decay estimate (B.2.1) along with Hölder’s inequality give that themapping w ↦→ U(t − t j )v ɛ,ũ 0(t j )w from H s R into W1,p is continuous. Thus,we may write∥ ∫ ·0 U(· − s)vɛ,ũ0,n (s)dW ɛ (s) ∥ L r(p) (0,T ;W 1,p )= ∥ ∑ n−1i=1 1l ∑ i−1∫ tj+1t i ≤t

Appendix B.Large deviations in the multiplicative caseSuppose that I u 0is the rate function of the LDP, then from PropositionB.3.1, since the sets (I u 0) −1 ([0, a]) are the direct image by S(u 0 , ·) of thesets C a which are compact, it is a good rate function.Now the set A is a Borel set of E ∞ and u 0 is some initial datum in H 1 .An upper bound. In the case where inf w∈AI u 0(w) = 0 there is nothingto prove. Otherwise, take 0 < a < inf w∈AI u 0(w) and R > a. Suppose thatf is such that I W (f) ≤ a, thenI u 0(S(u 0 , f)) ≤ a < inf I u 0(w),w∈Athus S(u 0 , f) /∈ A and there exists a neighborhood of S(u 0 , f)V u0 ,f = {v ∈ E ∞ : T (v) > T and ‖v − S(u 0 , f)‖ X (T,p) < ρ u0 ,f }such that V u0 ,f ⊂ A c . Also, from Proposition B.4.1, there exists ɛ u0 ,f , γ u0 ,fand r u0 ,f positive such that for every ɛ ≤ ɛ u0 ,f and ũ 0 in B H 1(u 0 , r u0 ,f ),( ∥∥ɛ log P u ɛ,ũ 0− S(u 0 , f) ∥ X (T,p) ≥ ρ u0 ,f ; ∥ √ ɛW − f ∥ )C([0,T ];H sR ) < γ u 0 ,f ≤ −R.Let denote by O u0 ,f the set O u0 ,f = B C([0,T ];H sR ) (f, γ u 0 ,f ). The family(O u0 ,f ) f∈Cais a covering by open sets of the compact set C a , thus thereexists a finite sub-covering of the form ⋃ Ni=1 O u 0 ,f i. We can now writeP ( u ɛ,ũ 0∈ A ) ( {u≤ Pɛ,ũ 0∈ A } { √ɛW ⋃ })∩ ∈ Ni=1 O u 0 ,f( i√ɛW ⋃ )+P /∈ Ni=1 O u 0 ,f i∑ N∑ N≤i=1 P ({ u ɛ,ũ 0∈ A } ∩ { √ ɛW ∈ O u0 ,f i} ) + P ( √ ɛW /∈ C a )≤i=1 P ({ u ɛ,ũ }0√/∈ V u0 ,f i ∩ { ɛW ∈ Ou0 ,f i} ) + exp ( − a )ɛ ,( ∧N)for ɛ ≤ ɛ 0 for some ɛ 0 positive. Thus, for ɛ ≤ ɛ 0 ∧i=1 ɛ u 0 ,f iwe obtainfor ũ 0 in B H 1(u 0 , r u0 ) where r u0 = ∧ Ni=1 r u 0 ,f i,P ( u ɛ,ũ 0∈ A ) (≤ N exp − R ɛand)+ exp(− a ),ɛɛ log P ( u ɛ,ũ 0∈ A ) ≤ ɛ log 2 + (ɛ log N − R) ∨ (−a).Finally, there exists r u0 such that for any ũ 0 in B H 1(u 0 , r u0 ),lim ɛ→0 ɛ log P ( u ɛ,ũ 0∈ A ) ≤ −a.115

Appendix B.Large deviations in the multiplicative caseSince a is arbitrary, we obtain,lim ɛ→0,ũ0 →u 0ɛ log P ( u ɛ,ũ 0∈ A ) ≤ − inf I u 0(w).w∈AA lower bound. Suppose that inf w∈Int(A) I u 0(w) < ∞, otherwise there isnothing to prove, and take w in Int(A) such that I u 0(w) < ∞.The continuity of S(u 0 , ·) along with the compactness of the set C I u 0 (w)+1give that there exists f such that w = S(u 0 , f) and I u 0(w) = I W (f). TakeV u0 ,f an elementary neighborhood of S(u 0 , f) included in A and O u0 ,f definedas previously, η positive and R > I u 0(w) + η. We obtain(exp− R−ηɛ)(≤ exp− IW (f)ɛ)≤ P ( √ ɛW ∈ O u0 ,f )≤ P ({ u ɛ,ũ 0/∈ V u0 ,f}∩ {√ ɛW ∈ Ou0 ,f } ) + P ( u ɛ,ũ 0∈ A ) .Thus, there exists r u0 and ɛ 0 positive such that for every ũ 0 in B H 1(u 0 , r u0 )and ɛ ≤ ɛ 0 ,−R + η ≤ ɛ log 2 + ( ɛ log P ( u ɛ,ũ 0∈ A )) ∨ (−R)and there exists ɛ 1 ≤ ɛ 0 such that for every ɛ ≤ ɛ 1 ,−I u 0(w) ≤ ɛ log 2 + ɛ log P ( u ɛ,ũ 0∈ A ) .As a consequence, we obtain that for every u 0 in H 1 , there exists r u0 positivesuch that for every ũ 0 in B H 1(u 0 , r u0 ),andlim ɛ→0 ɛ log P ( u ɛ,ũ 0∈ A ) ≥ −I u 0(w)lim ɛ→0 ɛ log P ( u ɛ,ũ 0∈ A ) ≥ − infw∈Int(A) Iu 0(w)since w in Int(A) is arbitrary.The uniform LDP follows from the two bounds and Corollary 5.6.15 in[48].B.6 Applications to the blow-up timesIn this section, the equation with a focusing nonlinearity is considered.Then, some solutions of the deterministic equation blow up in finite timefor critical or supercritical nonlinearities. If B is a Borel set of [0, ∞],116

Appendix B.Large deviations in the multiplicative caseP (T (u ɛ,u 0) ∈ B) = µ ( uɛ,u 0T −1 (B) ) .for K ⊂⊂ H 1 ,Thus, the uniform LDP gives that− supu 0 ∈Kinfu∈Int(T −1 (B)) Iu 0(u) ≤ lim ɛ→0 ɛ log inf P (Tu 0 ∈K (uɛ,u 0) ∈ B)andlim ɛ→0 ɛ log supu 0 ∈KP (T (u ɛ,u 0) ∈ B) ≤ −infu∈T −1 (B),u 0 ∈KI u 0(u).Since T is lower semicontinuous, the sets (T, ∞] and [0, T ] are particularlyinteresting. We recall, see [81] for more details, that for every T positive,T −1 ((T, ∞]) = E ∞ and Int ( T −1 ([0, T ]) ) = ∅. Thus, for the two types ofsets, at least one bound is trivial. Considering approximate blow-up times,see [19], allows us to obtain two interesting bounds and to treat intervals ofthe form (S, T ] where 0 ≤ S < T . We do not consider this latter questionin this chapter. We finally recall that when T < T (u u 0d) the LDP gives thatlim ɛ→0 ɛ log P (T (u ɛ,u 0) > T ) = 0, indeed this is not a large deviation event.We obtain similarly, when T > T (u u 0d ), lim ɛ→0 ɛ log P (T (u ɛ,u 0) ≤ T ) = 0.Proposition B.6.1 If T < TK i = inf u 0 ∈K T ( u u )0d , where K ⊂⊂ H 1 , thenthere exists c positive such thatlim ɛ→0 ɛ log sup P (T (u ɛ,u 0) ≤ T ) ≤ −c.u 0 ∈KProof. Since T is lower semicontinuous, T −1 ([0, T ]) is a closed set. Supposethat there exists a sequence (u n , h n ) in K × L 2 (0, ∞; L 2 ) such thatT (S c (u n , h n )) ≤ T and lim n→∞ h n = 0. Since K is a compact set we mayextract a subsequence u ϕ(n) such that u ϕ(n) converges to some ũ. Also, ifwe denote by f n (·) = ∫ ·0 Φh n(s)ds, f n converges to zero in C ([0, ∞); H s R )and satisfies T (S(u n , f n )) ≤ T . Moreover, there exists a positive such thatfor every n in N, f n in C a . The semicontinuity of T along with PropositionB.3.1 give thatT ≥ lim n→∞ T ( S(u ϕ(n) , f ϕ(n) ) ) ≥ T (S(ũ, 0)) ≥ T i K > T,which is contradictory.□In the following we consider the case d = 2 or d = 3 and a cubic nonlinearity,i.e. σ = 1. In that case blow-up may occur.117

Appendix B.Large deviations in the multiplicative caseProposition B.6.2 Let U u 0be the solution of the free Schrödinger equationwith initial datum u 0 in H s and assume that span { |U u 0(t)| 2 , t ∈ [0, 2T ] }belongs to the range of Φ for T > T ( u u )0d . There exists c positive such thatlim ɛ→0 ɛ log P (T (u ɛ,u 0) > T ) ≥ −c.Remark B.6.3 It is known that for some Gaussian initial data u 0 , see [13],the solutions of NLS blow up in finite time. Also, the solutions of the freeequation are smooth and strongly decreasing at infinity, thus we may checkthat it is possible to define a Hilbert-Schmidt operator Φ satisfying the lastassumption.Proof. Define F u 0by F u 0(t) = − ∫ t∧2T0|U u 0(s)| 2 ds. The control is suchthat S(u 0 , F u 0) = U u 0on [0, 2T ] thus T (S(u 0 , F u 0)) ≥ 2T since S(u 0 , F u 0)does not blow up. Also, F u 0belongs to C ([0, ∞), H s R ) since, for s > d 2 , Hsis an algebra and U u 0belongs to C([0, ∞), H s ). Finally, from the assumptionon Φ, there exists h in L 2 (0, ∞; L 2 ) setting h = 0 after 2T such thatΦh(s) = |U u 0(s)| 2 1l s≤2T and F u 0belongs to C a for some a positive. We thusobtain that F u 0belongs to {f ∈ C ([0, ∞), H s R ) : T (S(u 0, f)) > T } and thatI W (F u 0) ≤ a < ∞.□Remark B.6.4 A result on compact sets K in H s for T > sup u0 ∈K T ( u u )0dholds provided that span { |U u 0(t)| 2 , t ∈ [0, 2T ], u 0 ∈ K } belongs to the rangeof Φ restricted to a ball of L 2 ( 0, 2T ; L 2) .118

Appendix CSmall noise asymptotic ofthe timing jitter in solitontransmissionAbstract: We consider random perturbations of the focusing cubic onedimensional nonlinear Schrödinger equation. The noises, either additive ormultiplicative, are white in time and colored in space. In the additive case a”white noise limit” is considered. We study the small noise asymptotic of thetails of the center and mass of a pulse at a fixed coordinate when the initialdatum is null or a soliton profile. Our main tools are large deviation resultsat the level of paths. Upper and lower bounds are obtained from boundsfor the optimal control problems derived from the rate function of the largedeviation principles. Our results are in perfect agreement with several resultsfrom physics. These results had been obtained with arguments which seemsdifficult to fully justify mathematically. Some results are new.C.1 IntroductionThe nonlinear Schrödinger (NLS) equation occurs as a generic model inmany areas of physics and describes the propagation of slowly varying envelopesof a wave packet in media with both nonlinear and dispersive responses.The one-dimensional equation with a cubic focusing nonlinearityis for example a model in the context of long-haul transmission lines in fiberoptics; see for example [88] for a derivation of the equation in that context.The variable t stands for the space coordinate and x for some retarded time.119

Appendix C.Application to the timing jitterResulting from a balance between the focusing nonlinearity and the dispersivelinear part, localized (here in time) waves propagate, they are calledsolitons or solitary waves. The functions√2Asech(A(x−x0 )+2AΩt) exp ( −i(A 2 − Ω 2 )t + iΩ(x − x 0 ) + iθ 0)(C.1.1)where A > 0 is the amplitude, Ω is the group velocity or angular carrierfrequency, x 0 and θ 0 are respectively the initial position and phase, aresolitons. In soliton based amplitude-shifted-keyed systems (ASK) communicationsystems, solitons are used as information carriers to transmit thedatum 0 or 1. A 1 corresponds to the emission of a soliton at time 0 withnull velocity Ψ 0 A (x) = √ 2Asech(Ax). It is produced by a laser beam. Atthe far end T of the fiber a receiver records1l∫ l2− l 2|u u 0(T, x)| 2 dx, u 0 = 0 or u 0 = Ψ 0 A,[−l2 , l 2]is a window in time; l may be chosen small since the wave uu 0,solution of the NLS equation, is localized and remains centered. When theabove quantity is above a threshold I d it is decided that a 1 has been emitted,otherwise it is decided that a 0 has been emitted.However, it is physically more relevant to consider random perturbationsand then error in transmission may occur. Phenomena such as a fluctuatingdielectric permittivity, a deviating fiber radius or a random initial shapemaybe taken into account in a perturbation term. Moreover noise is somehowintrinsic to such systems.To counterbalance for loss in the fiber, regularly spaced amplifiers areplaced along the line and the distance between amplifiers is small comparedto the length of the line. If we suppose that the gain is adjusted to counterbalanceexactly for loss, there remains a spontaneous emission noise. Thiscould be justified theoretically thanks to Heisenberg’s uncertainty principle.This noise could be modeled as a random external force; see for example[51, 63, 110]. We could formally write the equation asi ∂uɛ,u 0∂t= ∆u ɛ,u 0+ |u ɛ,u 0| 2 u ɛ,u 0+ √ ɛξ, (C.1.2)where ɛ stands for the small noise amplitude, ξ is a complex Gaussian spacetimenoise and u 0 is the initial datum. The functions are complex valued.Note that this equation also appears in the context of anharmonic atomicchains in the presence of thermal fluctuation; see for example [16].Other types of amplification among which Raman coupling to thermal120

Appendix C.Application to the timing jitterphonon, see [52, 59, 98], and four-wave-mixing, see [52, 101], also lead tospontaneaous emission of noise. However in this case the noise enters as areal multiplicative noise. Note that in the case of the Raman amplificationa Raman nonlinear response also appears in the equation and the Ramaneffect also contributes to the Kerr effect, i.e. the power law nonlinearity. Itis assumed that the extra Raman nonlinear response may be neglected to afirst approximation in a treatment of the noise effect on the frequency andthus, by dynamical coupling, on the position of the pulse since it producesessentially a deterministic shift in frequency. The evolution equation maybe written formally asi ∂uɛ,u 0∂t= ∆u ɛ,u 0+ |u ɛ,u 0| 2 u ɛ,u 0+ √ ɛu ɛ,u 0ξ, (C.1.3)in that case the noise ξ is a real Gaussian noise. Note that this model isalso introduced in the context of crystals; see for example [10, 11, 12].In the presence of noise, the soliton is progressively distorted by thenoise, even though it is small, and with small probability an error in transmissionmay occur in the sense that 1 is discarded. Also, when the noise isadditive, it may create from nothing a structure that might be mistaken asa 1.When a 1 is emitted, it is assumed that two processes are mainly responsiblefor the loss of the signal: a decrease of the mass)∥∥N(u ɛ,Ψ0 A (T ) = ∥u ɛ,Ψ0 A ∥∥2(T )and a diffusion in position, characterized by the center of the pulse) ∫∣∣Y(u ɛ,Ψ0 A (T ) = x ∣u ɛ,Ψ0 A ∣∣2(T, x) dx.RThe fluctuation of the center results in a shift in the arrival time. It is calledtiming jitter. The event that for null initial datum a 1 is detected onlyresults from a large fluctuation of the mass.When the noise is of multiplicative type the mass is invariant and weshall only focus on the timing jitter.Considering that the probability of sending a 1 is 1 2, the bit error rate isdefined as( ∫ ) (l12 P 1∣2∣∣u ɛ,Ψ0 A ∣∣2(T, x) dx ≤ Id + 1 ∫ ll − l 2 P 1 2 ∣∣u ɛ,0 (T, x) ∣ )2 dx > I d ,l − l 22L 2121

Appendix C.Application to the timing jitterthe probabilities that the measured quantities are below or above the thresholdare conditional probabilities. Again, in the case of a multiplicative noisethe second conditionnal probability is null. In practical applications, thisbit error rate might be less than 10 −9 . Moreover it is widely admitted thatthe statistics are not Gaussian. Thus a statistical treatment for inference ofthe bit error rate requires a theoretical evaluation.In the physics literature the amplitude of the noise is assumed to besmall. Physical techniques often rely on an adiabatic perturbation theorywhere the pulse is approximated by a soliton ansatz with finite fluctuatingcollective variables; it requires that the noise is small.Some articles from physics study the variance of the center; see for exampleof [16, 59, 88]. In the seminal paper [88] of Gordon and Haus it isobtained that the variance of the center is of the order of T 3 (superdiffusion,i.e. stronger than that of the Brownian motion which is linear) and thatthe fluctuation of the center is connected with a shift in the soliton carrierfrequency. It is assumed that the timing jitter is the most troublesome andupper limit of the information rate is derived based on a Gaussian assumption.In [59], the only paper from physics we found on noise induced timingjitter when the noise is multiplicative, a Raman-modified NLS equation isconsidered; independent complex additive and real multiplicative noises appearboth in the equation. The contribution of each noise to the variance ofthe center is of the order T 3 . They however exhibit a different behavior inthe initial amplitude A.Other articles study the deviation from the Gaussian assumption. Againusing the perturbation theory of solitons, see for example [89, 94], physicistshave obtained that the statistics of the center may be non Gaussian whenthere is soliton interaction or filtering, see for example [51, 64, 65, 85, 111].Otherwise it could be considered as Gaussian in the first order only; see forexample [2, 51, 97]. In [114] as in [88] the model is a juxtaposition of deterministicevolutions with randomly perturbed initial initial data in betweenamplifiers. The log of the tails of the amplitude and center are evaluatednumerically via an importance sampled Monte Carlo estimator. Simulationsare obtained from a distribution where the small probability event is a centralevent; they are weighted by a likelihood ratio weight. It is obtainedthat the log of tails of the amplitude only differs significantly from that ofGaussian tails. Note that we may expect to use the numerical methodologybased on a genealogical particle analysis developed in [47]. In this referencethe importance sampling and Monte Carlo methodologies are compared toa particle system approach and it is applied to the estimation of probabilityof rare events due to polarization-mode dispersion in optical fibers.122

Appendix C.Application to the timing jitterIn [51, 63, 110], probability density functions (PDF) are examinated. In[63] the PDF of the joint law of the mass and center at coordinate T , whenthe initial datum is a soliton profile, are approximated from a PDF of therandom parameters of a solution described as a soliton with a finite set offluctuating parameters. The parameters are assumed to evolve according todynamically coupled SDEs. This latter PDF is obtained via a saddle pointapproximation of a corresponding finite dimensional Martin-Siggia-Rose effectiveaction. The complete infinite dimensional effective action, see forexample [95] is not treated. The PDF of the amplitude (a multiple of themass with the parametrization) is obtained when the initial datum is null.The probability of loosing a 1 is numerically evaluated under the assumptionof a very large window. In [51] the Fokker-Planck equation is used to obtainthe PDF of the mass at T . In [110] a similar result is obtained. Howeverthe PDF of the marginal law of the center has not been evaluated.Note that infinite dimensional effective actions in physics are intimatelyrelated to the rate function of a sample path large deviation principle (LDP).Paths minimizing the action for certain configurations of the system arecalled optimal fluctuations or instantons, see also for example [9, 129]. Notethat in [67], where the large deviations approach is adopted, the problemof transitions between stable equilibrium configurations (tunnelling) of unforcednonlinear heat equations in the limit of small noise is studied. Themost likely transitions are the instantons from quantum mechanics; they aresaddle points of the equilibrium action functional related to the rate functionof the sample path LDP. Exit from neighborhoods of zero for weaklydamped stochastic NLS equations is studied in the article [84].In the present article we apply sample path LDPs to the study of thetails of the law of the mass and center of the pulse at the end of the fiber.We thus study cumulative distribution functions (CDFs) instead of PDFsbut do not study the bulk of the distribution. As we will see, we are notable to treat mathematically the case of the space-time white noise whichis mainly used in the physical models. We thus restrict ourselves to noisesthat are colored in space. In the case of a noise of additive type we willconsider sequences of noises that mimic the white noise in the limit. Thelog of the tails in the limit of small noise are of the order of the oppositeof the infima of a functional derived from the rate functions of the LDPsdivided by the noise amplitude. The infima are optimal control problems.We give upper and lower bounds using energy inequalities and modulatedsolitons. The two bounds mostly differ up to multiplicative constants andthe orders in T and A are compared to that of the physicists.123

Appendix C.Application to the timing jitterC.2 Notations and preliminariesFor p ≥ 1, L p is the classical Lebesgue space of complex valued functionson R and W 1,p is the associated Sobolev space of L p functions with firstorder derivatives, in the sense of distributions, in L p . If I is an interval ofR, (E, ‖ · ‖ E ) a Banach space and r belongs to [1, ∞], then L r (I; E) is thespace of strongly Lebesgue measurable functions f from I into E such thatt → ‖f(t)‖ E is in L r (I). The space L 2 with the inner product defined by(u, v) L 2 = Re ∫ R u(x)v(x)dx is a Hilbert space. The Sobolev spaces Hs arethe Hilbert spaces of functions of L 2 with partial derivatives up to order s inL 2 . When s is fractional it is defined classically via the Fourier transform.When the functions are real valued we specify it, for example we writeH s (R, R). The following Hilbert spaces of spatially localized functionsΣ = { f ∈ H 1 : x ↦→ xf(x) ∈ L 2} ,Σ 1 2 ={f ∈ H 1 : x ↦→ √ |x|f(x) ∈ L 2}are also introduced and endowed with the norms‖f‖ 2 Σ = ‖f‖ 2 H 1 + ‖x ↦→ xf(x)‖ 2 L 2 ,‖f‖ 2 Σ 1 2= ‖f‖ 2 ∥H+ ∥x ↦→ √ |x|f(x)1 ∥ 2 .L 2We denote by ‖Φ‖ Lc(A,B) the norm of Φ as a linear continuous operatorfrom A to B, where A and B are normed vector spaces. We recall that Φ is aHilbert-Schmidt operator from H to ˜H, where H and ˜H are Hilbert spaces,if it is a linear continuous operator such that, given a complete orthonormalsystem (e H j )∞ j=1 of H, ∑ ∞j=1 ‖ΦeH j ‖2˜H < ∞. We will denote by L 2 (H, ˜H) thespace of Hilbert-Schmidt operators from H to ˜H endowed with the norm‖Φ‖ L2 (H, ˜H) = tr (ΦΦ∗ ) =∞∑‖Φe H j ‖ 2˜H.We also recall that a cylindrical Wiener process W c in a Hilbert spaceH is such that for any complete orthonormal system (e j ) ∞ j=1 of H, thereexists∑a sequence of independent Brownian motions (β j ) ∞ j=1 such that W c =∞j=1 β je j . This sum does not converge in H but in any Hilbert space Usuch that the embedding H ⊂ U is Hilbert-Schmidt. The image of theprocess W c by a linear mapping Φ on H is a well defined process in H whenthe mapping is Hilbert-Schmidt on H, i.e. Φ ∈ L 2 (H) = L 2 (H, H). Then,124j=1

Appendix C.Application to the timing jitterW = ΦW c is such that W (1) is well defined with a covariance operator ΦΦ ∗ .We recall that a rate function I is a lower semicontinuous function andthat a good rate function I is a rate function such that for every positive c,{x : I(x) ≤ c} is a compact set.Let us now recall some mathematical aspects of the stochastic NLS equations.The equations, written as SPDEs in the Itô form, are in the additivecaseidu ɛ,u 0− (∆u ɛ,u 0+ |u ɛ,u 0| 2 u ɛ,u 0)dt = √ ɛdW, (C.2.1)and in the multiplicative caseidu ɛ,u 0− (∆u ɛ,u 0+ |u ɛ,u 0| 2 u ɛ,u 0)dt = √ ɛu ɛ,u 0◦ dW.(C.2.2)The symbol ◦ stands for the Stratonovich product. In the case of equation(C.2.2), see [37], the massN (u ɛ,u 0(t)) = ‖u ɛ,u 0(t)‖ 2 L 2 , t > 0is a conserved quantity. Precise assumptions on Φ such that W = ΦW c aremade below. These equations are supplemented with an initial datumu ɛ,u 0(0) = u 0 .In this paper, we consider initial data in Σ ⊂ H 1 and work with the solutionconstructed in [37]. Since we work with a subcritical non linearity, we couldalso consider solutions in L 2 with initial data in L 2 . However, the H 1 −settingis preferred in order to be able to consider the spaces Σ and Σ 1 2 and studythe center of the pulse∫Y(u ɛ,u 0(t)) = x|u ɛ,u 0(t, x)| 2 dx, t > 0,Rdefined when u ɛ,u 0(t) belongs to Σ 1 2 .We are concerned by weak solutions or equivalently by mild solutionswhich, in the additive case, satisfyu ɛ,u 0(t) =U(t)u 0 − i−i √ ɛ∫ t0∫ t0U(t − s)(|u ɛ,u 0(s)| 2 u ɛ,u 0(s))dsU(t − s)dW (s)(C.2.3)125

Appendix C.Application to the timing jitterwhere (U(t)) t∈Rstands for the Schrödinger group, U(t) = e −it∆ , t ∈ R. Thelast term is called the stochastic convolution. In the multiplicative case, themild equation isu ɛ,u 0(t) =U(t)u 0 − i∫ t∫ t0U(t − s)(|u ɛ,u 0(s)| 2 u ɛ,u 0(s))ds−i √ ɛ U(t − s)u ɛ,u 0(s)dW (s) − iɛ U(t − s)F Φ u ɛ,u 0(s)ds02 0(C.2.4)where the stochastic integral is a Itô integral and, given (e j ) ∞ j=1 an orthonormalbasis of L 2 , F Φ (x) = ∑ ∞j=1 (Φe j) 2 (x). The term ɛ 2 F Φ(x) is the Itôcorrection.The noise is the time derivative in the sense of distributions of the Wienerprocess W . It corresponds to a white noise in time. A space-time white noisewould correspond to Φ equal to the identity. We cannot handle such roughnoises and make the assumption that the two noises are colored in space.The basic limitation is that, unlike semi-groups like the Heat semi-group,the Schrödinger group is an isometry and does not allow smoothing in theSobolev spaces based on L 2 . For instance, in the additive case, it can beseen that the stochastic convolution is a well defined process with paths inL 2 if and only if Φ is a Hilbert-Schmidt operator on L 2 .In fact, we make even stronger assumptions. In the additive case weassume that W is a Wiener process on Σ. In the multiplicative case, it isimposed that W is a Wiener process on H s (R, R) where s satisfies s > 3 2 .We know that the Cauchy problem is globally well posed in H 1 ; see [37]for a general discussion on the local well posedness and the global existencefor more general nonlinearities and dimensions. Note that the present deterministicNLS equation is integrable thanks to the inverse scattering method.We will not use these techniques in the article. Results on the influence ofthe noise on the blow-up time, for more general nonlinearities and dimensionsare given in [39, 40]. In [13, 43] the ideal white noise and results onthe influence of a noise on the blow-up are studied numerically.Sample path LDPs for stochastic NLS equations are proved in [81, 82].These LDPs do not(allow to treat ) the center of the solution and we shallconsider LDPs in C [0, T ]; Σ 1 2 where T is positive (the length of the fiberline). The rate function of the LDP in the additive case is defined in termsof the mild solution of the control problem{idudt = ∆u + |u|2 u + Φh,u(0) = u 0 ∈ Σ and h ∈ L 2 ( 0, T ; L 2) (C.2.5).126∫ t

Appendix C.Application to the timing jitterWe denote the solution by u = S a,u 0(h). The mapping h → S a,u 0(h) is calledthe skeleton and (C.2.5) the skeleton equation.In the multiplicative case, the controlled equation isi dudt = ∆u + |u|2 u + uΦh,(C.2.6)whose mild solution is denoted by u = S m,u 0(h). The mapping S m,u 0isagain called the skeleton and (C.2.6) the skeleton equation.In this article, when describing properties which hold both in the additiveand multiplicative cases, we use the symbol S(u 0 , h) to denote either S a,u 0(h)or S m,u 0(h).Let us now state the sample path LDPs. The proof is given in the annex.Theorem C.2.1 Assume that Φ belongs to L 2 (L 2 , Σ) in the additive caseand Φ ∈ L 2 (L 2 , H s (R, R)) with s > 3/2 in the multiplicative case. Assumealso that the initial datum u 0 is in Σ. Then the solutions of the stochasticnonlinear Schrödinger equations (C.2.3) and (C.2.4) are almost surelyin C([0, T ]; Σ 1 2 ). Moreover, they define C([0, T ]; Σ 1 2 ) random variables andtheir laws ( µ ) uɛ,u 0satisfy a LDP of speed ɛ and good rate functionɛ>0I u 0(w) = 1 2infh∈L 2 (0,T ;L 2 ): w=S(u 0 ,h) ‖h‖2 L 2 (0,T ;L 2 ) ,where S(u 0 , ·) = S a,u 0(·) in the additive case and S(u 0 , ·) = S m,u 0(·) in themultiplicative case, and with ( the convention ) that inf ∅ = ∞. It means thatfor every Borel set B of C [0, T ]; Σ 1 2 , we have the lower boundand the upper bound− infw∈Int(B) Iu 0(w) ≤ lim ɛ→0 ɛ log P (u ɛ,u 0∈ B)lim ɛ→0 ɛ log P (u ɛ,u 0∈ B) ≤ − inf I u 0(w).w∈BThese sample path LDPs allow for example to evaluate the probability that,originated from a soliton profileΨ 0 A(x) = √ 2Asech (Ax) ,the random solution be significantly different from the deterministic solitonsolutionΨ A (t, x) = Ψ 0 A(x) exp ( −iA 2 t ) .127

Appendix C.Application to the timing jitterIndeed, for T , δ and η positive and ɛ small enough, the LDP implies that(exp − C ) ( ∥∥∥u ∥ ) (1ɛ,Ψ≤ P 0 ∥∥C([0,TA − ΨA 1 > δ ≤ exp − C )2,ɛ];Σ 2 ) ɛwhereandC 1 =C 2 =infw: ‖w−Ψ A ‖C([0,T ];Σ 1 >δ IΨ0 A (w) + η2 )infw: ‖w−Ψ A ‖C([0,T ];Σ 1 ≥δ IΨ0 A (w) − η.2 )Recall that, since the rate function is a good rate function, if B is a closed setand inf w∈B I Ψ0 A(w) < ∞, then there is an f in B, optimal fluctuation, suchthat I Ψ0 A(f) = inf w∈B I Ψ0 A(w). Thus if B does not contain the deterministicsolution then necessarily inf w∈B I Ψ0 A(w) > 0. Consequently η may be chosensuch that C 2 is positive and the above probability of a deviation from thedeterministic path is exponentially small in the small ɛ limit.In this article we are interested in estimating the probability of particulardeviations from the deterministic paths. Namely, we wish to study how themass and the center of a solution at coordinate T deviate from their valuein the ”frozen” deterministic system (i.e. when ɛ = 0). In the absence ofnoise, the mass is a conserved quantity and for initial data being either 0 orΨ 0 Athe center remains equal to zero.We know from [81] that we may push forward by continuity the LDP forthe paths to a LDP for the mass at T and obtain a LDP with speed ɛ andgood rate function for an initial datum u 0I u 0N (m) = 1 {}inf‖h‖ 22 h∈L 2 (0,T ;L 2 ): N(S a,u L0 (h)(T ))=m2 (0,T ;L 2 ).In the case of a multiplicative noise, the mass is a conserved quantity. Thus,in this case, the mass cannot deviate from the deterministic behavior.Similarly, the mapping Y is continuous from Σ 1 2 into R. We may thusdefine by direct image the measures ( µ Y(uɛ,u 0 (T )) ) for an initial datumɛ>0u 0 in Σ. We obtain by contraction that they satisfy a LDP of speed ɛ andgood rate functionI u 0Y (y) = 1 {}inf‖h‖ 22 h∈L 2 (0,T ;L 2 L): Y(S(u 0 ,h)(T ))=y2 (0,T ;L 2 ),the skeleton S is either that of the additive or multiplicative case.128

Appendix C.Application to the timing jitterLet us briefly explain our strategy to estimate the probability of some event.Let us consider for instance the event D ɛ = { Y ( u ɛ,0 (T ) ) ∈ [a, b] } where [a, b]is an interval which does not contain 0. We use the LDP to obtain− infy∈(a,b) I0 Y (y) ≤ lim ɛ→0 ɛ log P (D ɛ ) ≤ lim ɛ→0 ɛ log P (D ɛ ) ≤ − infy∈[a,b] I0 Y (y).(C.2.7)To estimate the upper bound, we use energy type inequalities. These giveestimates of the minimum L 2 norm of the control h required to change thedeterministic behavior and have the center in [a, b] at time T . Namely, weobtain a constant c such thatThis clearly impliesif Y (S(u 0 , h)(T )) ∈ [a, b] then 1 2 ‖h‖2 L 2 (0,T ;L 2 ) ≥ c.lim ɛ→0 ɛ log P (D ɛ ) ≤ −c.The second step is to find a particular function h such that Y(S(u 0 , h)(T )) ∈(a, b) and ˜c = 1 2 ‖h J‖ 2 L 2 (0,T ;L 2 )is as small as possible. Then−˜c ≤ lim ɛ→0 ɛ log P (D ɛ ) .In this second step, we are led to solve a control problem.The difficulty is to have sufficiently sharp energy estimates and to find agood solution to the control problem so that c and ˜c are as close as possible.We see below that we are able to do so in some interesting situations andderive good estimates on such probabilities.Note also that proceeding as in [81] for the mass, we may prove in theadditive case that inf y∈J I u 0Y(y) < ∞ for every nonempty interval J andany u 0 provided the range of Φ is dense. Indeed, for every real numberaT π2a, a solution of the form u(t, x) = (1 + atx)u 0 satisfies Y (u(T )) =3.Plugging this solution into equation (C.2.5), we find a control such thatthe solution reaches any interval at time T . Using the continuity of h ↦→Y (S a,u 0(h)(T )) from L 2 (0, T ; L 2 ) into R and the density of the range of Φ,we obtain inf y∈J I u 0Y(y) < ∞. This shows that in this case the two extremebounds in (C.2.7) are finite implying that P(D ɛ ) goes to zero exponentiallyfast when ɛ goes to 0.Remark C.2.2 Also, using similar arguments as in [81], we can prove thatfor every positive R besides an at most countable set of points, we can replacelim and lim by lim in the LDP and obtainlim ɛ→0 ɛ log P (Y (u ɛ,u 0(T )) ≥ R)= − 1 2ɛ inf h∈L 2 (0,T ;L 2 ): Y(S(u 0 ,h)(T ))≥R{}‖h‖ 2 L 2 (0,T ;L 2 )129

Appendix C.Application to the timing jitterlim ɛ→0 ɛ log P (Y (u ɛ,u 0(T )) ≤ −R)= − 1 2ɛ inf h∈L 2 (0,T ;L 2 ): Y(S(u 0 ,h)(T ))≤−R{}‖h‖ 2 L 2 (0,T ;L 2 ).This uses the fact that a monotone and bounded function is continuous almosteverywhere.We end this section with some remarks which will be useful in the developmentof our method when we consider the center of the solution. Let usconsider an initial datum is Ψ 0 A. The probability of tail events of the centerare related to the behavior of Y ( S(Ψ 0 A , h)) . If h ≠ 0, S(Ψ 0 A , h)(t) ≠ Ψ Aand the center may move. An equation for the motion of the center is givenin [141] in the case of an external potential. The first step consists in multiplyingthe controlled PDE by −ixu, taking the real part, and integratingby part the term involving the Laplace operator. We then obtain for thecontrolled PDE associated to the multiplicative case[)] ( ∫)′Y(S m,Ψ0 A (h)(t) = 2Re i S m,Ψ0 A(h)(t, x)∂ x S m,Ψ0 A (h)(t, x)dx ,R(C.2.8)while in the additive case we obtain[ ( )] ′ (Y S a,Ψ0 A(h)(t) = 2Re i ∫ )RA(h)(t, Sa,Ψ0 x)∂ x S a,Ψ0 A(h)(t, x)dx(−2Re i ∫ )RA(h)(t, xSa,Ψ0 x) (Φh) (t, x)dx .The quantity( ∫)P(u) = 2Re i u(x)∂ x u(x)dx , u ∈ H 1 .R(C.2.9)on the right hand side of (C.2.8) and (C.2.9) is usually called the momentum.As a consequence of (C.2.8) we see that in the multiplicative case, thecenter of the solution of the control problem cannot move unless its phasedepends on the space variable. For instance, if the control is chosen so thatthe solution S a,Ψ0 A(h)(t) is a modulated soliton of type (C.1.1) with varyingamplitude and group velocity,S a,Ψ0 A(h)(t) =√ 2A(t)sech(A(t)(x − x 0 ) + 2A(t)Ω(t)t)exp ( −i(A(t) 2 − Ω(t) 2 )t + iΩ(t)(x − x 0 ) + iθ 0)we have the well known identity[ ()] ′Y S m,Ψ0 A (h)(t) = −2Ω(t)N(Sm,Ψ 0 A (h)(t)) = −8Ω(t)A(t).130

Appendix C.Application to the timing jitterIt will be convenient to choose controlled solutions of the form above. Sincethe initial datum is Ψ 0 A, we necessarily have Ω(0) = 0, hence Ω cannot bechosen constant. We will see that it is sufficient to have a constant amplitudeA in order to get sharp bounds. Thus we will use modulated solitons assolutions of the controlled problem with constant amplitude when studyingthe motion of the center.The first idea to find a control giving a solution whose center or massverify some desired property is to take the above modulated soliton and plugit into the skeleton equation. This gives an explicit form of the control interms of the various parameters. Then, we compute the space-time L 2 normof this control. We obtain a function of the parameters which we can tryto minimize thanks to the calculus of variations. This approach is not easyto perform, the function to minimize has a complicated form and is oftensingular. Thus, we also have chosen a simpler approach which consists infinding directly controls giving solutions with the desired properties. Notehowever that the calculus of variations approach has allowed us to guess theform of the modulated soliton we should choose.Let us consider the following controlled nonlinear Schrödinger equationi dudt = ∆u + |u|2 u + λ(t)xu(C.2.10)with initial datum Ψ 0 A . The function λ is taken in L1 (0, T ; R). This correspondsto the multiplicative skeleton equation with Φh = λ(t)x or tothe additive one with Φh = λ(t)xu. We use well known transformationto compute explicitly the solution of (C.2.10) which we denote by Ψ A,λ .We ( first ( may check that the functions v 1 and v 2 defined by v 1 (t, x) =∫ ) )(texp i0 λ(s)ds x u(t, x) and v 2 (t, x) = exp −i ∫ t(∫ )s0 0 λ(τ)dτ) 2 ds v 1 (t, x)(gauge transform) satisfy the PDEsandi ∂v (∫1∂t = ∂2 v t2 (∫ t)1∂x 2 + |v 1| 2 ∂v1v 1 − λ(s)ds)v 1 − 2i λ(s)ds00 ∂x( (∫ t∂v2i∂t + 20) ) ∂v2λ(s)ds∂x= ∂2 v 2∂x 2 + |v 2| 2 v 2with initial datum Ψ 0 A. We conclude using the methods of characteristicsthat v 3 defined by∫ t ∫ s)v 3 (t, x) = v 2(t, x + 2 λ(u)duds13100

Appendix C.Application to the timing jitteris a solution of the usual NLS equation with initial datum Ψ 0 A. Thus weobtain that v 3 (t, x) = Ψ A (t, x) and that the solution of the Cauchy problemassociated to (C.2.10) isΨ A,λ (t, x) = √ 2Asech[exp −iA 2 t + i ∫ t0(A(x − 2 ∫ t0∫ s0 λ(τ)dτds ))(∫ s0 λ(τ)dτ) 2 ∫ (t∫ ) (ds − ix0 λ(s)ds + 2i t ∫0 λ(s)ds t0We obtain a modulated soliton with group velocity given by Ω(t) = ∫ t0 λ(s)ds.In the additive case, it is possible to obtain a control such that the solutionhas same center and group velocity and such that the space-time L 2 norm ofthe control is simpler to compute. It is obtained thanks to the observationthat using the gauge transform the solution of the Cauchy problem{(i dvdt = ∆v + |v|2 v + λ(t)v(0) = Ψ 0 A ,is given by(˜Ψ A,λ (t, x) = exp 2i∫ t0λ(s)x − 2 ∫ t0∫ s ∫ τ00∫ s0 λ(τ)dτds )v)λ(σ)dσdτds Ψ A,λ (t, x).∫ s0 λ(τ)dτds )].(C.2.11)Remark C.2.3 Note that, for the controls chosen above, relation (C.2.8)holds also in the additive case. Thus the second term in (C.2.9) which, atfirst glance, could be useful to act on the center is in fact useless.Also, it could be thought that the choice of more complicated group velocitiescould be useful. We have tried to consider a space dependent group velocitybut the calculus of variations approach shows that optimality is reachedwhen it does not depend on space.C.3 Tails of the the mass and center with additivenoiseIn the case of an additive noise, both the mass and center may deviate fromthe deterministic behavior and result in error in transmission.We shall study tails and thus the probability of a deviation from themean. The constant R will quantify this deviation. We are not really interestedin large R. In practice R may be assumed to be in (0, 4). But, sinceɛ goes to zero and the factor in the exponential should be multiplied by 1 ɛwhile R is of order 1. It results in very unlikely events. These significant132

Appendix C.Application to the timing jitterexcursions of the mass and position are exactly large deviation events.Moreover another parameter is particularly interesting. It is T the lengthof the fiber optical line. It is assumed to be large. For example we couldthink of a fiber optical line between Europe and America.We first recall the results obtained in [81] for the tails of mass of thepulse at the end of the line. The initial datum may be u 0 = 0 or u 0 = Ψwhere Ψ(x) = √ 2sech(x). We could consider a soliton profile with anyamplitude A as well but for simplicity, we consider the case A = 1. Howeverwe consider the parameter A for the timing jitter in order to compare withresults from physics.Let us begin with upper bounds of the tails. As already mentioned, theyare obtained thanks to energy estimates. For the second bound we considerthe case of the emission of a signal. In that case only a decrease of the massis troublesome and causes in error in transmission. Thus the bound givenonly accounts for a significant decrease of the mass.Proposition C.3.1 For every positive T and R (R in (0, 4) for the secondinequality) and every operator Φ in L 2 (L 2 , H 1 ), the following inequalitiesholdlim ɛ→0 ɛ log P ( N ( u ɛ,0 (T ) ) ≥ R ) R≤ −8T ‖Φ‖ 2 ,L c(L 2 ,L 2 )lim ɛ→0 ɛ log P ( N ( u ɛ,Ψ (T ) ) − 4 < −R ) ≤ −8T ‖Φ‖ 2 L c(L(4 + R).2 ,L 2 )Proof. We only give a sketch of the proof. Details can be found in [81].We treat the first inequality. The proof for the second inequality is similar.Multiplying by −iu the equationi dudt − ∆u − λ|u|2 u = Φh,integrating over time and space and taking the real part gives, for t ∈ [0, T ],( ∫ t ∫ () )‖S a,0 (h)(t)‖ 2 L− ‖u 2 0 ‖ 2 L= 2Re −i (Φh)(s, x)S a,0 (h)(s, x) dxds .20(C.3.1)We first integrate once more with respect to t ∈ [0, T ] and use the Cauchy-Schwarz inequality to obtain(∫ T1/2 (∫ T1/2‖S a,0 (h)(s)‖ 2 Lds)≤ 2T ‖Φ‖ 2 Lc(L 2 ,L 2 ) ‖h(s)‖ 2 Lds).20133R0R 2

Appendix C.Application to the timing jitterThen, taking t = T in (C.3.1), using again the Cauchy-Schwarz inequalityand the above bound, we deduceIt follows‖S a,0 (h)(T )‖ 2 L 2 ≤ 4T ‖Φ‖ 2 L c(L 2 ,L 2 )IN 0 (m) = 1 inf2x≥8T ‖Φ‖ 2 .L c(L 2 ,L 2 )∫ Th∈L 2 (0,T ;L 2 ): N(S a,0 (h)(T ))=m0‖h(s)‖ 2 L 2 ds.{}‖h‖ 2 L 2 (0,T ;L 2 )Now, by the LDP on the mass, we havelim ɛ→0 ɛ log P ( N ( u ɛ,0 (T ) ) ≥ R ) ≤ −infx∈[R,∞] Iu 0N (m)and the result follows.□Let us now consider lower bounds. We use modulated solitons as solutionsof the controlled equation. We have found that it is sufficient that onlythe amplitude is varying. We take the solution of (C.2.5) of the form√2A(t) exp(−i∫ t0)A 2 (s)ds sech(A(t)x).(C.3.2)The singular Euler-Lagrange equation given by the calculus of variationswhen minimizing the energy of the controls giving such solutions has allowedto guess a good parametrization when the initial datum is either 0 or Ψ.Define the following sets of time dependent functions, depending on a set ofparameters D,A 1 D={( )A : [0, T ] → R, there exists ˜R ∈ D such that A(t) = ˜R t 2 }2TandA 2 D = {A : [0, T ] → R, there exists ˜R ∈ D such thatA(t) =(8 − ˜R√− 4 4 − ˜R) ( t2T) 2 ( √+ −4 + 2 4 − ˜R) } t2T + 1 .134

Appendix C.Application to the timing jitterModulated amplitude taken in A 1 D or A2 Dthe setset are associated to controls inCD {h i = ∈ L 2 (0, T ; L 2 ), there exists A ∈ A i Dh(t, x) = i A′ (t)A(t) Ψ A(t, x) − i √ (2A ′ (t) exp −i∫ t0)A 2 (s)ds A(t)x sinh }cosh 2 (A(t)x)where i = 1 or i = 2.We have the following proposition whose proof follows from the lowerbound of the LDP for the mass. The proof is given in [81]. It uses that theinfimum of the rate function is smaller than the infimum on the smaller setsof controls CD 1 and C2 Dcorresponding to well-chosen modulated amplitudes.The assumptions can easily be fulfilled. They are made to be as close aspossible to the space-time white noise considered in physics that we are notable to treat mathematically.Proposition C.3.2 Let T and R be positive numbers (R in (0, 4) for thesecond inequality), take D dense in [R, R + 1] and a sequence of operators(Φ n ) n∈Nin L 2(L 2 , L 2) such that for every h ∈ C 1 D we have Φ nh convergesto h in L 1 ( 0, T ; L 2) . Then we obtainlim n→∞,ɛ→0 ɛ log P ( N ( u ɛ,0,n (T ) ) ≥ R ) ≥ − R(12 + π2 ).18TReplacing in the above C 1 D by C2 Dwe obtainlim n→∞,ɛ→0 ɛ log P ( N ( u ɛ,Ψ,n (T ) ) − 4 < −R ) ≥ − 2(8 − R − 4√ 4 − R)(12 + π 2 ).36TThe exponent n is there to recall that Φ is replaced by Φ n ,Note that the result in Proposition C.3.1 depends on Φ only through itsnorm as a bounded operator in L 2 . It is not difficult to see that there existssequences of operators (Φ n ) n∈Nsatisfying the assumptions of PropositionC.3.2, i.e. which are Hilbert-Schmidt from L 2 to L 2 and Φ n approximatesthe identity on the good set of controls, and are uniformly bounded asoperators on L 2 by a constant independent on T . For such sequences ofoperators, the upper and lower bounds given above agree up to constants intheir behavior in large T .It is obtained in [63], for the ideal white noise and using the heuristicarguments recalled in the introduction, that the probability density function135

Appendix C.Application to the timing jitterof the amplitude of the pulse at coordinate T when the initial datum isnull is asymptotically that of an exponential law of parameter ɛT 2 . Theamplitude is a constant times the mass for the modulated soliton solutionsconsidered [63]. Integrating this density over [ R 2, ∞) and taking into accountthe different normalisation, we obtain lim ɛ→0 ɛ log P ( N ( u ɛ,0 (T ) ) ≥ R ) =− R T. It is in between our two bounds and very close to our lower bound.A surprising fact is that, we obtain our result by parameterizing only theamplitude whereas in [63] a much more general parametrization is used.Both bounds exhibit the right behavior in R and T . Moreover, the order inR confirms physical and numerical results that the law is not Gaussian. On alog scale the order in R is that of tails of an exponential law. In such a casethe Gaussian approximation leads to incorrect tails and error estimates.Let us now comment on our results in the case of a soliton as initialdatum. In [63], the error probability when the size of the measurementwindow is of the order of the coordinate T is obtained. It is given bylim ɛ→0 ɛ log P ( N ( u ɛ,Ψ (T ) ) − 4 < −R ) = − c(R)T, with a constant c(R). Itexhibits the same behavior in T as in our calculations. The discussion onthe behavior with respect to R is less clear. Our bounds are not of the sameorder. In [51, 110] the PDF of the mass at coordinate T for a soliton profileas initial datum is not Gaussian. The numerical simulations in [114] alsoexhibit a significant difference between the log of the tails of the amplitudeand that of a Gaussian law. Our lower bound indicates that again the tailsare larger than Gaussian tails. Thus we give a rigorous proof of the factthat a Gaussian approximation is incorrect.Finally, it is natural to obtain that the tails of the mass are increasingfunctions of T since the higher is T , the less energy is needed to form asignal whose mass gets above a fixed threshold at T . Replacing above byunder, the same holds in the case of a soliton as initial datum.Remark C.3.3 The H 1 setting is not required here. We could as well workwith L 2 solutions and a LDP in L 2 . However, it is required to work in H 1for the study of the center below.We now estimate the tails of the center. As for the mass, the rate is hardto handle since it involves an optimal control problem for controlled NLSequations. We again deduce the asymptotic of the tails from the LDP lookingat upper and lower bounds. We consider that the initial datum is Ψ 0 Asince only in this case the timing jitter might be troublesome.Let us begin with an upper bound. It is deduced from the equation ofmotion of the center in the controlled NLS equation (C.2.9).136

Appendix C.Application to the timing jitterProposition C.3.4 For every positive T , A and R and every operator Φin L 2(L 2 , Σ ) , the following inequality holds() )lim ɛ→0 ɛ log P Y(u ɛ,Ψ0 A (T ) ≥ R≤ −R 2).8T (2T + 1)(4A 2 + R2T +1‖Φ‖ 2 L c(L 2 ,Σ)Proof. Differentiating the momentum of the solution with respect to timeand replacing the time derivative of the solution with the correspondingterms of the equation we obtain[)] ∫′ (P(S a,Ψ0 A (h)(t) = 4Re S a,Ψ0 A (h)(t, x) ∂x Φh ) (t, x)dx.RIndeed by successive integration by parts all terms cancel besides the oneinvolving the forcing term. Since Y ( Ψ 0 ( )A)= 0 and P Ψ0A= 0, thanks to(C.2.9), we obtain the identityY(S a,Ψ0 A(h)(t)) =( ∫ t4Re(0−2Re∫ s ∫0∫i ∫ t0)RA(h)(σ, Sa,Ψ0 x) (∂ x Φh) (σ, x)dxdσds)RA(h)(s, xSa,Ψ0 x) (Φh) (s, x)dxds .From this identity it follows that the controls h in the minimizing set of theLDP applied to the event we consider necessarily satisfy()R ≤ Y S a,Ψ0 A(h)(T ) ≤ 4T ‖Φ‖ Lc(L 2 ,H 1 )‖h‖ L 2 (0,T ;L 2 )‖S a,Ψ0 A(h)‖ L 2 (0,T ;L 2 )+2‖Φ‖ Lc(L 2 ,Σ)‖h‖ L 2 (0,T ;L 2 )‖S a,Ψ0 A(h)‖ L 2 (0,T ;L 2 ).Moreover, arguing as in the proof of Proposition C.3.1, see also [81],‖S a,Ψ0 A(h)‖ L 2 (0,T ;L 2 ) ≤(T ‖Φ‖ Lc(L 2 ,L 2 )‖h‖ L 2 (0,T ;L 2 ) )√4A1 + 1 + .T ‖Φ‖ 2 Lc(L 2 ,L 2 ) ‖h‖2 L 2 (0,T ;L 2 )A lower bound on 1 2 ‖h‖2 L 2 (0,T ;L 2 )follows easily since the function x ↦→( √ )x 1 + 1 + 4 xis increasing on R ∗ +. The result follows.□A lower bound is obtained considering controls suggested at the endof Section 2 and minimizing on the smaller set of controls. We define thefollowing set of control for A, T positive and D a subset of (0, ∞)H D A,T =({h ∈ L2 (0, T ; L 2 ), h(t, x) = λ(t)137x − 2 ∫ t0∫ s0 λ(τ)dτds )˜ΨA,λ (t, x),with λ(t) = 3 ˜R(T −t)8AT 3 , ˜R ∈ D}

Appendix C.Application to the timing jitterProposition C.3.5 Let T , A and R be positive. Assume that, for a denseset D of [R, R + 1], (Φ n ) n∈Nis a sequence of operators in L 2(L 2 , Σ ) suchthat for any h in H D T,A , Φ nh converges to h in L 1 (0, T ; Σ). Then we havethe following inequality where the n in the exponent recalls that Φ is replacedby Φ n ,( ( ) )lim n→∞,ɛ→0 ɛ log P Y u ɛ,Ψ0 A ,n (T ) ≥ R ≥ − π2 R 2128T 3 A 3 .Proof. By the LDP for the center Y, we know that for a fixed n a lowerbound is given bywhereI Ψ0 AY,n (y)= 1 2− infy>R IΨ0 AY,n (y)infh∈L 2 (0,T ;L 2 ): Y S a,Ψ0 A ,n (h)(T ) =y{}‖h‖ 2 L 2 (0,T ;L 2 ).Again, the n is there to recall that in the skeleton equation, Φ is replaced byΦ n . To minorize this quantity, we first treat the case Φ = I. Note that thestochastic equation has no meaning in this case but the skeleton equationhas a well defined solution provided h ∈ L 2 (0, T ; L 2 ). We denote by S a,Ψ0 AW Nthe skeleton when Φ = I. It is not difficult to see that S a,Ψ0 AW N(h) belongs toL ∞ ([0, T ]; Σ) when h belong to L 1 (0, T ; Σ). A standard argument to provethis is to compute the second derivative with respect to time of the varianceV(u) = ∫ R x2 |u(t, x)| 2 dx when u = S a,Ψ0 AW N(h). It is also standard to provethat, for each t, the mapping h → S a,Ψ0 AW N(h)(t) is weakly continuous fromL 1 (0, T ; Σ) to Σ and strongly continuous from L 1 (0, T ; Σ) to H 1 . Therefore,since Y is weakly continuous on Σ, thanks to our assumptions, we knowthat for h ∈ HT,AD() () ()Y S a,Ψ0 A ,n (h)(T ) = Y S a,Ψ0 AW N (Φ nh)(T ) → Y S a,Ψ0 AW N (h)(T ) when n → ∞.(C.3.3)Let ˜H T,A be the same set of controls as above but where λ is only assumedto belong to L 2 (0, T ; R)˜H T,A(= {h ∈ L 2 (0, T ; L 2 ), h(t, x) = λ(t)x − 2 ∫ t0∫ s0 λ(τ)dτds )˜ΨA,λ (t, x),λ ∈ L 2 (0, T ; R)}.138

Appendix C.Application to the timing jitterClearly,infh∈L 2 (0,T ;L 2 ): Y S a,Ψ0 AW N≥ (h)(T ) ˜R≤infh∈ ˜H T,A : Y S a,Ψ0 AW N≥ (h)(T ) ˜R= infλ∈L 2 (0,T ;R), R T0∫ tR t0λ(s)dsdt≥ ˜R8A‖h‖ 2 L 2 (0,T ;L 2 )‖h‖ 2 L 2 (0,T ;L 2 )π 2 ∫ Tλ 2 (t)dt3A 0Note that the contraint ∫ T˜R0 0λ(s)dsdt ≥8A, is not a boundary conditionas in the usual calculus of variations. To solve this minimization problem,we use the quantity L T,A, ˜R(λ) defined by∫ T∫ TL T,A, ˜R(λ) = π2λ 2 (t)dt − γ3A 00∫ t0λ(s)dsdt,where γ belongs to R. We then impose that our guess λ ∗ is a criticalT,A, ˜Rpoint of L T,A, ˜R(λ) and that it satisfies the constraint ∫ T ∫ t˜R0 0λ(s)dsdt =8A .We obtainλ ∗ T,A, ˜R (t) = 3 ˜R(T − t)8AT 3 .We do not claim that the minimization problem is solved, we simply writeLet us setinfλ∈L 1 (0,T ;R), R T R t0 0∫ T≤ π23Ah ∗˜R(t, x) = λ ∗ T,A, ˜R (t) (x − 20λ(s)dsdt≥ ˜R8Aλ ∗ (t)dt =π2 T,A, ˜R∫ t ∫ s00π 2 ∫ Tλ 2 (t)dt3A 0˜R264A 3 T 3λ ∗ T,A, ˜R (τ)dτds )˜Ψ A,λ ∗ (t, x).T,A, ˜RBy (C.3.3), we have for ˜R ∈ D,(Y S a,Ψ0 A ,n (h ∗˜R)(T ) ()) → Y S a,Ψ0 AW N (h∗˜R)(T ) when n → ∞.Therefore, for n large enough,(Y S a,Ψ0 A ,n (h ∗˜R)(T )) > R.139

Appendix C.Application to the timing jitterWe deduceinfx>R IΨ0 AY,n (x) ≤π2 ˜R264A 3 T 3 .Since this is true for ˜R in a dense set of [R, R + 1] we deduce the result. □The upper and lower bounds given in Proposition C.3.4 and C.3.5 are inperfect agreement in their behavior with respect to R and to T when T islarge. Indeed, for T large, the upper bound in Proposition C.3.4 is close toR 2. However, we have to be careful before doing such a comparison.Indeed, the bounds can be compared only if we are able to consider128T 3 A‖Φ‖ Lc(L 2 ,Σ)a sequence of operators (Φ n ) n∈N satisfying the assumptions of PropositionC.3.5 and such that ‖Φ n ‖ Lc(L 2 ,Σ) is bounded uniformly in n.It seems possible to construct such a sequence. For instance we maychoose ˜Φ in L 2 (L 2 , Σ) such that ˜Φk = k for k in K A , the closure in L 2 ofthe vector space spanned by {(x − a)sech (A(x − b)) , a ∈ [0, 1], b ∈ [0, 1]}.We believe that K A is embedded in Σ in a Hilbert-Schmidt way. For T andA sufficiently large and D ⊂ [R, R + 1], each h in the set HA,T D is such thath(t) ∈ K A for t ∈ [0, T ], thus ˜Φh = h and we can take Φ n = ˜Φ in PropositionC.3.5. In this case, the two bounds are comparable and are of the sameorder in R and T . Note that ‖Φ n ‖ Lc(L 2 ,Σ) is independent on R and T .In fact, many such sequences probably exist. Therefore, it seems thatthe bounds can be compared in many circumstances. Roughly speaking,the fact that this can be done means that we are treating noises which aresufficiently localized around the soliton Ψ 0 A .If the sequence (Φ n ) n∈N converges pointwise to the identity, i.e. if wewish to understand what happens in the white noise limit, then this localizationassumption does not hold. In this case, the lower bound is meaningfulwhereas the upper bound converges to zero and provides no information.The comparison of the behavior of the bounds with respect to A is lessclear. The two bounds seem contradictory for large A. This is due to thefact that it is not possible to choose a sequence of operators (Φ n ) n∈N satisfyingthe assumptions of Proposition C.3.5 and such that ‖Φ n ‖ Lc(L 2 ,Σ) isuniformly bounded with respect to A. Indeed such a sequence necessarilysatisfies‖h‖ L 1 (0,T ;Σ) ≤ lim n→∞ ‖Φ n ‖ Lc(L 2 ,Σ)‖h‖ L 1 (0,T ;L 2 )for any h ∈ HT,A D . It is easily seen that for A and T sufficiently large, theratio of ‖h‖ L 1 (0,T ;Σ) and ‖h‖ L 1 (0,T ;L 2 ) is of the order A.In fact this shows that the upper bound in Proposition C.3.4 is alwaysRlarger than a constant times 2for a sequence satisfying the assumptionsT 3 A 3140

Appendix C.Application to the timing jitterof Proposition C.3.5. Thus there is no contradiction.We can probably go further. Indeed, there may exist sequences ofoperators satisfying the assumptions of Proposition C.3.5 and such that‖Φ n ‖ Lc(L 2 ,Σ) ≤ cA for some constant c. In this case the bounds are of thesame order with respect to A, R and T . An example could be constructedin the same way as above. It suffices to take Φ n equal to the identity onK A and zero on a complementary space. Indeed, it can be shown that‖h‖ Σ ≤ cA‖h‖ L 2 for some constant c.Therefore, the two bounds are also comparable in their behavior withrespect to A under a localization assumption on the noise.Let us now compare our result with the results obtained in the physicsliterature. First, we note that we obtain that on a log scale the tails areequivalent to Gaussian tails. This is indeed the kind of result obtained byarguments from the physical theory of perturbation of solitons.Remark C.3.6 We are missing the pre exponential factors to concludewhether or not the tails are Gaussian. We could think of using sharp Laplaceasymptotics to obtain these factors.Now, suppose the law were indeed Gaussian, then the asymptotic of thetails may be written in terms of the variance. By doing so, we find that thevariance of the timing jitter is of the order T 3 . It agrees perfectly with theinitial results of [88]. Also the order in both A and T seems to agree perfectlywith the orders of the contribution of the additive noise to the variance ofthe timing jitter in equation (3.18) in [59]. Note however that in [88, 97],where the model is instead a juxtaposition of deterministic evolutions withrandom initial data in between amplifiers, the order in A seems to be − c A .We end this section noticing that our result confirms the fact that, inthe presence of additive noise, the timing jitter is more troublesome thanthe fluctuation of the mass when we consider the problem of losing a signal.Indeed we have ( found)that the error probability due to timing jitter is of theorder of exp − c 1(R)and an error probability due to the fluctuation of theɛT 3(mass is of the order of exp− c 2(R)ɛT)which is clearly negligible compared tothe first for large T . Recall that T represents the length of a fiber opticalline and is thus assumed to be very large.Remark C.3.7 From an engineering point of view it is possible to exponentiallyreduce the probability of undesired deviations of the center by introducinginline control elements; see for example [63]. We could also useideas given in [128] and optimize on such external fields for a limited cost141

Appendix C.Application to the timing jitteror penalty functional. The new optimal control problem requires then doubleoptimization.Remark C.3.8 Note that the methodology developed herein could also beapplied to the determination of the small noise asymptotic of the tails ofthe position of an isolated vortex, defined by ∮ ∇ arg u(t, x) · dl, in a Bosecondensates or superfluid Helium as in [125]. There the physical perturbationapproach along with the Fokker-Planck equation are used. The small noiseacts as the small temperature.C.4 Tails of the center in the multiplicative caseIn the case of the multiplicative noise, the mass is a conserved quantity andwe restrict our attention to the case of the law of the center of the pulsewhen the initial datum is the soliton profile Ψ 0 A .Again, let us begin with upper bounds obtained from an equation forthe motion of the center in the controlled NLS equation.From relation (C.2.8) and integration by parts, we obtain the equationin [141],] ∫ ′′ [Y(S m,Ψ0 A (h)(t)) = 2 |S m,Ψ0 A (h)(t, x)| 2 (∂ x Φh) (t, x)dx. (C.4.1)We may thus deduce the next proposition.RProposition(C.4.1 For every positive T , A and R and every operator Φin L 2 L 2 , H s (R, R) ) , where s > 3 2the following inequality holds() ) ( ) 3 2Rlim ɛ→0 ɛ log P Y(u 2ɛ,Ψ0 A (T ) ≥ R ≤ −16 2A 2 T 3 ‖Φ‖ 2 .L c(L 2 ,W 1,∞ (R,R))(′Proof. From equation (C.4.1), the fact that Y S A(h)) m,Ψ0 (0) = P(Ψ0A) =0, that for such values of s the injection of H s (R, R) into W 1,∞ (R, R) iscontinuous and that the mass is conserved and thus remains equal to 4 weobtain that(′Y S A(h)(t)) m,Ψ0 ≤ 8A‖Φ‖ Lc(L 2 ,W 1,∞ (R,R))‖h‖ L 1 (0,t;L 2 )≤ 8A √ t‖Φ‖ Lc(L 2 ,W 1,∞ (R,R))‖h‖ L 2 (0,T ;L 2 )142

Appendix C.Application to the timing jitterThen, since Y ( Ψ 0 A)= 0, we obtain integrating the above inequality that)R ≤ Y(S m,Ψ0 A (h)(T ) ≤ 16AT 3 2‖Φ‖3 Lc(L 2 ,W 1,∞ (R,R))‖h‖ L 2 (0,T ;L 2 )and the conclusion follows.□Let us consider now lower bounds. We need to find controls which havethe desired effect on the center. We have seen that in the additive case, goodcontrols are given by functions in HA,T D . Recalling the transformations onthe equation made at the end of Section 2, we can equivalently take controlsof the form λ(t)xΨ A,λ which correspond to the solution Ψ A,λ . Thus, in themultiplicative case, a good control is given by h(t, x) = λ(t)x. Unfortunatelythese controls do not belong to the range of Φ nor to L 2 ( 0, T ; L 2) and arenot admissible.We have tried to approximate these controls by admissible ones. Sincethe control is multiplied by Ψ A,λ in the equation, it seems that it has no effectoutside a set centered around the center of Ψ A,λ and that we could replaceλ(t)x by a truncation. We have not been able to get any information by sucharguments. We have tried several other choices of control corresponding tovarious modulated solitons especially with a phase nonlinear in x. Theynever yielded the right order of the lower bound with respect to A or T . Wetherefore impose a new assumption that Φ takes its values in H s (R, R) ⊕xL 1 (0, T ; R). In other words we consider the slightly different equationidũ ɛ,u 0= ( ∆ũ ɛ,u 0+ |ũ ɛ,u 0| 2 ũ ɛ,u 0 ) dt + ũ ɛ,u 0◦ √ ɛdW (t) + √ ɛxũ ɛ,u 0◦ dβ(t)(C.4.2)where β is a standard Brownian motion independant of W and the correspondingcontrolled PDEi d dt ˜S u 0(h 1 , h 2 ) = ∆ ˜S u 0(h 1 , h 2 ) + | ˜S u 0(h 1 , h 2 )| 2 ˜Su 0(h 1 , h 2 )+ ˜S u 0(h 1 , h 2 )Φh 1 + x ˜S u 0(h 1 , h 2 )h 2where h 1 belongs to L 2 ( 0, T ; L 2) and h 2 belongs to L 2 (0, T ; R), the initialdatum is u 0 and in the sequel u 0 = Ψ 0 A. We may guess by successiveapplications of the Itô formula, multiplying ũ ɛ,u 0by the random phase termexp (ix √ ɛβ(t)), and similar transformations as in Section 2 (stochastic gaugetransform, stochastic methods of characteristics...) that we should considerthe function(exp ix √ ∫ t) (ɛβ(t) − iɛ β 2 (s)ds ũ ɛ,u 0t, x + 2 √ ∫ t)ɛ β(s)ds .01430

Appendix C.Application to the timing jitterIt indeed satisfies equation (C.2.2) with same initial datum. We deduce thatũ ɛ,u 0(t, x) =exp(−ix √ ɛβ(t) + iɛ ∫ t0 β2 (s)ds + 2iɛβ(t) ∫ t0 β(s)ds )u ɛ,u 0(t, x − 2 √ ɛ ∫ t0 β(s)ds ).A similar computation shows that(˜S u 0(h 1 , h 2 )(t, x) = exp −ix √ ɛ ∫ t0 h 2(s)ds + i ∫ t(∫ s0 0 h 2(u)du ) 2 ds+2i ∫ t0 h 2(s)ds ∫ t ∫ ) (s0 0 h 2(u)duds S m,u 0(h 1 ) t, x − 2 ∫ t ∫ )s0 0 h 2(u)du .The functions ũ ɛ,u 0and ˜S u 0(h 1 , h 2 ) are well defined functions of L 2 (0, T ; Σ)and we may compute their centers. We obtain a lower bound of the asymptoticof the tails of the center of the new solutions.Proposition C.4.2 For every positive T , A and R and every operator Φin(L 2 L 2 , H s (R, R) ) where s > 3 2the following inequality holds() )lim ɛ→0 ɛ log P Y(u ɛ,Ψ0 A (T ) ≥ R ≥ − 3R2128A 2 T 3 .( )Proof. Consider the mapping F from C [0, T ]; Σ 1 2 × C ([0, T ]; R) into Rsuch that∫F (u, b) =(Take u and u ′ in C [0, T ]; Σ 1 2R∫ T)∣ ∣∣∣2|x|∣(T, u x − 2 b(s)ds dx.0)and b and b ′ in C ([0, T ]; R), then by thetriangle and inverse triangle inequalities and the change of variables weobtain|F (u, b) − F (u ′ , b ′ )|≤ ∫ ∣R∣∣x + 2 ∫ ∣ ∣T ∣∣ ∣∣x0 b(s)ds ∫ T− + 20 b′ (s)ds∣∣ |u(T, x)| 2 dx+ ∣ ∫ ∣R∣x + 2 ∫ T0 b′ (s)ds∣ ( |u(T, x)| 2 − |u ′ (T, x)| 2) dx∣≤ 2 ∣ ∫ T0 b(s)ds − ∫ T0 b′ (s)ds∣ ∫ R |u(T, x)|2 dx+ ∫ R |x| ||u(T, x)| − |u′ (T, x)|| (|u(T, x)| + |u ′ (T, x)|) dx+2 ∣ ∫ T0 b′ (s)ds∣ ∫ R ||u(T, x)| − |u′ (T, x)|| (|u(T, x)| + |u ′ (T, x)|) dxwe conclude from the inverse triangle and Hölder inequalities that F iscontinuous. We may then push forward the LDP for the paths of u ɛ,Ψ0 A and144

Appendix C.Application to the timing jitterof the Brownian motion by the mapping F using a slight modification ( of the )result of exercise 4.2.7 of [48] and obtain a LDP for the laws of Y ũ ɛ,Ψ0 A(T )(which is that of F u ɛ,Ψ0 A, √ )ɛβ of speed ɛ and good rate function definedas a function of the rate function of the original solutions and of the ratefunction I β of the sample path LDP for the Brownian motionĨ Ψ0 AY (x) = inf (u,b): F (u,b)=x (I u 0(u) + I β (b))≤ 1 2 inf (h 1 ,h 2 ): FS m,Ψ0 A (h 1 ), R ·≤0 h 2(s)ds12 inf (h 1 ,h 2 ): Y˜SΨ 0 A (h 1 ,h 2 )(T )=x{}‖h 1 ‖ 2 =x L 2 (0,T ;L 2 ) + ‖h 2‖ 2 L 2 (0,T ;R){}‖h 1 ‖ 2 L 2 (0,T ;L 2 ) + ‖h 2‖ 2 L 2 (0,T ;R).Thus considering solely controls of the from (0, h 2 ), we minimize in h 2 for γin R,∫ T∫ T ∫ th 2 2(t)dt − γ h 2 (s)ds,00 0where we impose that∫ TY (Ψ A,h2 (T )) = 8A0∫ t0h 2 (s)ds = ˜R > R.The conclusion follows.□( ) ( )Remark C.4.3 We may check that Y u ɛ,Ψ0 A = Y ũ ɛ,Ψ0 A −8 √ ɛ ∫ T0 β(s)dsand that ∫ TT 30β(s)ds is a centered Gaussian random variable with variance3 .The corresponding upper bound for this modified stochastic NLS equationis() ) ( 3 2Rlim ɛ→0 ɛ log P Y(u ɛ,Ψ0 A (T ) ≥ R ≤ −(16) 2).A 2 T 3 ‖Φ‖ 2 L c(L 2 ,W 1,∞ (R,R)) ∨ 1Note that the lower bound do not require to consider a sequence of operators(Φ n ) n∈Nand we may indeed compare the upper and lower bounds. They areof the same order in T and in A. Note also that, as in the additive case, weobtain that on a log scale the tails are equivalently that of Gaussian tails.Also, our tails are of the order in T that we expect from the contributionof the multiplicative noise to the variance of the timing jitter in equation(3.18) in [59].However, concerning the amplitude, it is not of the order of − c asA 4145

Appendix C.Application to the timing jitterwe would expect from [59]. This is probably due to the fact that we haveconsidered a colored noise with a term x d dtβ that grows linearly in time(the x variable). We have explained that, otherwise, we fail to obtain alower bound. We have obtained, that for large A, and thus for even morelocalized in time solitons, the tails of the center in the additive noise arelarger than that in the multiplicative noise. Note that it is predicted in[59] that the quantum Raman noise is a dominant source of fluctuations inphase and arrival time for sub-picosecond solitons and that on the otherhand for longer solitons, Raman effects are reduced compared to the usualGordon-Haus jitter. It seems at first glance to be in contradiction with ourresults but their result is obtained for A = 1 and time corresponds to thetypical pulse duration considered for scaling purposes in order to obtain theNLS equation; also our order in A differs from theirs.C.5 Annex - proof of Theorem C.2.1We denote herein by V(f) = ∫ R |x|2 |f(x)| 2 dx the variance defined for f inΣ.Let us start with the additive case. We denote by v u 0(z) the solution of{idvdt = ∆v + λ|v − iz|2σ (v − iz),u(0) = u 0 ∈ Σwhere z belongs to X (T,2σ+2,Σ) = C([0, T ]; Σ) ∩ L r ( 0, T ; W 1,2σ+2) and r issuch that 2 r = 1 2 − 12σ+2 . We also denote by Gu 0the mappingz ↦→ v u 0(z) − iz,it is such that u ɛ,u 0= G u 0( √ ɛZ) where Z is the stochastic convolutiondefined by Z(t) = ∫ t0U(t − s)dW (s).We can check from similar arguments as those of the proof of Proposition1 in [81] that the stochastic convolution is a X (T,2σ+2,Σ) random variablewhose law µ Z is a centered Gaussian measure. Let z belong to X (T,2σ+2,Σ) ,take s < t < T , the triangle along with the Hölder inequalities then allowto compute∣ ∫ R |x| ( |G u 0(z)(t, x)| 2 − |G u 0(z)(s, x)| 2) dx ∣ ≤ ∫ R |x|(|Gu 0(z)(t, x)| + |G u 0(z)(s, x)|)|(|G u 0(z)(t, x)| − |G u 0(z)(s, x)|)|dx≤ ‖G u 0(z)(t) − G u 0(z)(s)‖ L 2√V(|G u 0(z)(t)| + |G u 0(z)(s)|)≤ 2 √ 2‖G u 0(z)(t) − G u 0(z)(s)‖ L 2( √V(v √ √ √ )× u 0(z)(t)) + V(v u 0(z)(s)) + V(z(t)) + V(z(s)) .146

Appendix C.Application to the timing jitterThe application of the Gronwall inequality in the proof of Proposition 3.5 in[39], along with the Sobolev injection allow to prove that G u 0(z) belongs toC([0, T ]; Σ 1 2 ). The computation above also shows that the mapping G u 0iscontinuous from X (T,2σ+2,Σ) to C([0, T ]; Σ 1 2 ). The general result on LDP forGaussian measures gives the LDP for the measures µ Zɛ , the direct imagesof µ Z under the transformation x ↦→ √ ɛx on X (T,2σ+2,Σ) . We conclude withthe contraction principle.In the multiplicative case, it is also required to revisit the proof of theLDP in [82]. Note that in the following when Φh is replaced by ∂f∂t where fbelongs to H 1 0 (0, T ; Hs (R, R)) which is the subspace of C ([0, T ]; H s (R, R)) offunctions null at time 0, square integrable in time and with square integrablein time time derivative. The skeleton is then denoted by ˜S m,u 0(f).We may check using the above calculation and the fact that for every t ∈[0, T ], ˜S m,u 0(f)(t) belongs to Σ that(˜Sm,u) ()V 0(f)(t) ≤ 4‖˜S m,u 0(f)(t)‖ 2 C([0,T ];H 1 ) + V(u 0) e T ,see the arguments of the proof of Proposition 3.2 in [40] used for the skeleton,that the skeleton is continuous from the sets of levels of the rate function ofthe Wiener process less or equal to a positive constant, with the topologyinduced by that of C ([0, T ]; H s (R, R)), to C([0, T ]; Σ 1 2). The only differencein the proof of Proposition 4.1 in [82], the Azencott lemma (also calledFreidlin-Wentzell inequality or almost continuity of the Itô map) is in step2. It is the reduction to estimates on the stochastic convolution. We useV ( v ɛ,ũ 0(t) ) ()≤ 4‖v ɛ,ũ 0(t)‖ 2 C([0,T ];H 1 ) + V(ũ 0) e T ,see the proof of Proposition 3.2 in [40], where v ɛ,ũ 0satisfies v ɛ,ũ 0(0) = ũ 0and(idv ɛ,ũ 0= ∆v ɛ,ũ 0+ λ|v ɛ,ũ 0| 2σ v ɛ,ũ 0+ ∂f∂t vɛ,ũ 0− iɛ )2 F Φv ɛ,ũ 0dt + √ ɛv ɛ,ũ 0dW ɛ ,f(·) = ∫ ·0 Φh(s)ds, W ɛ(t) = W (t) − √ 1 ∫ t ∂fɛ 0 ∂s ds = W (t) − √ 1 ∫ tɛ 0 Φh(s)ds,F Φ (x) = ∑ ∞j=1 (Φe j(x)) 2 and (e j ) ∞ j=1 is any complete orthonormal systemof L 2 . The bound remains the same as in [40] because of the cancelationof the extra term in the application of the Itô formula and the cancelationof the Itô-Stratonovich correction with the second order Itô correctionterm when the Itô formula is applied to the truncated varianceV r (v) = ∫ R exp(−r|x|2 )|x| 2 |v(x)| 2 dx.□147

Appendix C.Application to the timing jitterRemark C.5.1 Uniform LDPs hold (uniform with respect to initial datain balls) in the Freidlin-Wentzell formulation or compact sets in the presentformulation with lim and lim. More general nonlinearities and dimensionsand the case where blow-up may occur could be considered. It is still possibleto state the result in spaces of exploding paths with a projective limit topologyaccounting for the various integrability. Uniformity could be useful since inoptical experiments the initial pulse is a laser output and it is known up toa certain level of uncertainty.148

Appendix DExit from a neighborhood ofzero for weakly dampedstochastic nonlinearSchrödinger equationsAbstract: Exit from a neighborhood of zero for weakly damped stochasticnonlinear Schrödinger equations is studied. The small noise is either complexand of additive type or real and of multiplicative type. It is white in time andcolored in space. The neighborhood is either in L 2 or in H 1 . The small noiseasymptotic of both the first exit times and the exit points are characterized.D.1 IntroductionThe study of the first exit time from a neighborhood of an asymptoticallystable equilibrium point, the exit place determination or the transition betweentwo equilibrium points in randomly perturbed dynamical systems isimportant in several areas of physics among which statistical and quantummechanics, the natural sciences, financial macro economics... The problemis relevant in nonlinear optics; see for example [100]. We shall consider thecase of weakly damped nonlinear Schrödinger equations. It is a model innonlinear optics, hydrodynamics, biology, field theory, Fermi-Pasta-Ulamchains of atoms...For a fixed noise amplitude and for diffusions, the first exit time andthe distribution of the exit points on the boundary of the domain can be149

Appendix D.Exit from a domain for weakly damped equationscharacterized respectively by the Dirichlet and Poisson equations. However,when the dimension is larger than one, we may seldom solve explicitly theseequations and large deviation techniques are precious tools when the noiseis assumed to be small; see for example [48, 73]. The techniques used in thephysics literature is often called optimal fluctuations or instanton formalismand are closely related to the large deviations.An energy then characterizes the transition between two states and theexit from a neighborhood of an asymptotically stable equilibrium point. Theenergy is derived from the rate function of the sample path large deviationprinciple (LDP). The paths that minimize this energy are the most likelyexiting paths or transitions and when the infimum is unique the systemshows an almost deterministic behavior. Note that the first order of theprobability are that of the Boltzman theory and the amplitude of the smallnoise acts as the temperature. The deterministic dynamics is sometimes interpretedas the evolution at temperature 0 and the small noise as the smalltemperature nonequilibrium case. In the pioneering article [67], a nonlinearheat equation perturbed by a small noise of additive type is considered.Transitions in that case are the instantons of quantum mechanics. Also in[106], predictions for a noisy exit problem are confirmed both numericallyand experimentally.We will consider weakly damped nonlinear Schrödinger equations in R d .Equations are perturbed by a small noise. The noise is white in time andof additive or multiplicative type. We define it as the time derivative in thesense of distributions of a Hilbert space-valued Wiener process W . The twotypes of noises are physically relevant; see for example [44]. When the noiseis of additive type, the Hilbert space is L 2 or H 1 , spaces of complex valuedfunctions. The evolution equation is thenidu ɛ,u 0= (∆u ɛ,u 0+ λ|u ɛ,u 0| 2σ u ɛ,u 0− iαu ɛ,u 0)dt + √ ɛdW,(D.1.1)where α and ɛ are positive and u 0 is an initial datum in L 2 (respectively H 1 ).When the noise is of multiplicative type, the Hilbert space is the Sobolevspace based on L 2 of real valued functions H s R for s > d 2+ 1 and the productis a Stratonovich product. In that case the equation may be writtenidu ɛ,u 0= (∆u ɛ,u 0+ λ|u ɛ,u 0| 2σ u ɛ,u 0− iαu ɛ,u 0)dt + √ ɛu ɛ,u 0◦ dW.(D.1.2)The Wiener process W is always assumed to be colored in space since thelinear group does not have global regularizing properties and is an isometryon the Sobolev spaces based on L 2 . The power σ in the nonlinearity satisfiesσ < 2 dand thus solutions do not exhibit blow-up.150

Appendix D.Exit from a domain for weakly damped equationsIn [81] and [82] we have proved sample paths LDPs for the two types ofnoises but without damping and deduced the asymptotic of the tails of theblow-up times. In [81] we also deduced the tails of the mass, defined later,of the pulse at the end of a fiber optical line. We have thus evaluated theerror probabilities in optical soliton transmission when the receiver recordsthe signal on an infinite time interval. In [44] we have applied the LDP tothe problem of the diffusion in position of the soliton and studied the tailsof the random position. Our results are in perfect agreement with resultsfrom physics obtained via heuristic arguments. The damping term in thedrift here is often physically relevant but small and neglected in the models.For example in [44], in the case of an additive noise, we have considered thatthe gain of the amplifiers is adjusted to compensate exactly for loss and thatthere remains only a spontaneous emission noise.The flow in the equations above has Hamiltonian, gradient and randomcomponents. The mass∫N (u) = |u| 2 dxR dcharacterizes the gradient component. The Hamiltonian denoted by H(u),defined for functions in H 1 , has a kinetic and a potential term, it may bewrittenH (u) = 1 ∫|∇u| 2 dx −λ ∫|u| 2σ+2 dx.2 R d 2σ + 2 R dNote that the vector fields associated to the mass and Hamiltonian areorthogonal. Recall that the mass and Hamiltonian are invariant quantitiesof the equation without noise and damping. Other quantities like the linearor angular momentum are also invariant for nonlinear Schrödinger equations.We could rewrite, for example equation (D.1.1) as(du ɛ,u 0 δH (uɛ,u 0)=− α δN (u ɛ,u 0))2 δu ɛ,u dt − i √ ɛdW.0δu ɛ,u 0Without noise solutions are uniformly attracted to zero in L 2 and in H 1 .We will prove that because of noise, the behavior is completely different.Though for finite times the probabilities of large excursions off neighborhoodsof zero go to zero exponentially fast with ɛ, if we wait long enough,the time scale is exponential, such large fluctuations occur and exit from aneighborhood of zero takes place. In the L 2 case, we only consider noises ofadditive type where, because of noise, mass is injected or pumped randomlyin the system. It would also be possible to treat rather general multiplicativenoises as long as noise allows injection of mass. In H 1 we consider the151

Appendix D.Exit from a domain for weakly damped equationstwo types of noises.We use large deviation techniques to prove the corresponding result inour infinite dimensional setting. In [68], the case of a space variable in aunidimensional torus is treated for a particular SPDE and the regularizingproperty of the Heat semi-group is a central tool. Let us stress that theSchrödinger linear group is an isometry on every Sobolev space based onL 2 . In [29], the neighborhood is defined for a strong topology of β-Hölderfunctions and is relatively compact for a weaker topology, the space variableis again in a bounded subset of R d . Note also that one particular difficultyin infinite dimensions, along with compactness, is that the linear group isstrongly and not uniformly continuous. In this article the neighborhood isnot relatively compact, we work on the all space R d , the nonlinearity is locallyLipschitz only in H 1 for d = 1.However, there remain difficult problems from the control of nonlinearPDEs to prove for example that the upper and lower bounds on the exittime are equal. Also, it seems formally that, in the case of a noise of additivetype which is white in time and in space, the escape off levels of theHamiltonian less than a constant is intimately related to the solitary waves.We will not address these last issues in the present article.D.2 PreliminariesThroughout the paper the following notations will be used.The set of positive integers and positive real numbers are denoted by N ∗and R ∗ +. For p ∈ N ∗ , L p is the Lebesgue space of complex valued functions.For k in N ∗ , W k,p is the Sobolev space of L p functions with partial derivativesup to level k, in the sense of distributions, in L p . For p = 2 and s in R ∗ +, H sis the Sobolev space of tempered distributions v of Fourier transform ˆv suchthat (1 + |ξ| 2 ) s/2ˆv belongs to L 2 . We denote the spaces by L p R , Wk,p Rand Hs Rwhen the functions are real-valued. The space L 2 is endowed with the innerproduct (u, v) L2 = Re ∫ Ru(x)v(x)dx. If I is an interval of R, (E, ‖ · ‖ d E ) aBanach space and r belongs to [1, ∞], then L r (I; E) is the space of stronglyLebesgue measurable functions f from I into E such that t → ‖f(t)‖ E is inL r (I).The space of linear continuous operators from B into ˜B, where B and˜B are Banach spaces is L c(B, ˜B). When B = H and ˜B = ˜H are Hilbertspaces, such an operator is Hilbert-Schmidt when ∑ j∈N ‖ΦeH j ‖2˜H < ∞ forevery (e j ) j∈Ncomplete orthonormal system of H. The set of such operatorsis denoted by L 2 (H, ˜H), or L s,r2 when H = H s and ˜H = H r . When H = H s R152

Appendix D.Exit from a domain for weakly damped equationsand ˜H = H r R, we denote it by Ls,r2,R. When s = 0 or r = 0 the Hilbert spaceis L 2 or L 2 R .We also denote by Bρ 0 and Sρ 0 respectively the open ball and the spherecentered at 0 of radius ρ in L 2 . We denote these by Bρ 1 and Sρ 1 in H 1 . We willdenote by N 0 (A, ρ) the ρ−neighborhood of a set A in L 2 and N 1 (A, ρ) theneighborhood in H 1 . In the following we impose that compact sets satisfythe Hausdorff property.We will use in Lemma D.3.5 the integrability of the Schrödinger lineargroup which is related to the dispersive property. Recall that (r(p), p) is anadmissible pair if p is such that 2 ≤ p < 2dd−2when d > 2 (2 ≤ p < ∞ when( )2d = 2 and 2 ≤ p ≤ ∞ when d = 1) and r(p) satisfiesr(p) = d 12 − 1 p.For every (r(p), p) admissible pair and T positive, we define the BanachspacesY (T,p) = C ( [0, T ]; L 2) ∩ L r(p) (0, T ; L p ) ,andX (T,p) = C ( [0, T ]; H 1) ∩ L r(p) ( 0, T ; W 1,p) ,where the norms are the maximum of the norms in the two intersectedBanach spaces. The Schrödinger linear group is denoted by (U(t)) t≥0; it isdefined on L 2 or on H 1 . Let us recall the Strichartz inequalities, see [25],(i)There exists C positive such that for u 0 in L 2 , T positive and(r(p), p) admissible pair,(ii)‖U(t)u 0 ‖ Y (T,p) ≤ C ‖u 0 ‖ L 2 ,For every T positive, (r(p), p) and (r(q), q) admissible pairs, s and ρsuch that 1 s + 1r(q) = 1 and 1 ρ + 1 q= 1, there exists C positive suchthat for f in L s (0, T ; L ρ ),∥ ∫ ·0 U(· − s)f(s)ds∥ ∥Y (T,p) ≤ C‖f‖ L s (0,T ;L ρ ).Similar inequalities hold when the group is acting on H 1 , replacing L 2 byH 1 , Y (T,p) by X (T,p) and L s (0, T ; L ρ ) by L s ( 0, T ; W 1,ρ) .It is known that, in the Hilbert space setting, only direct images ofuncorrelated space wise Wiener processes by Hilbert-Schmidt operators arewell defined. However, when the semi-group has regularizing properties,the semi-group may act as a Hilbert-Schmidt operator and a white in spacenoise may be considered. It is not possible here since the Schrödinger groupis an isometry on the Sobolev spaces based on L 2 . The Wiener process W153

Appendix D.Exit from a domain for weakly damped equationsis thus defined as ΦW c , where W c is a cylindrical Wiener process on L 2 andΦ is Hilbert-Schmidt. Then ΦΦ ∗ is the correlation operator of W (1), it hasfinite trace.We consider the following Cauchy problems{ iduɛ,u 0= (∆u ɛ,u 0+ λ|u ɛ,u 0| 2σ u ɛ,u 0− iαu ɛ,u 0)dt + √ ɛdW,u ɛ,u 0(0) = u 0(D.2.1)with u 0 in L 2 and Φ in L 0,02 or u 0 in H 1 and Φ in L 0,12 , and{ iduɛ,u 0= (∆u ɛ,u 0+ λ|u ɛ,u 0| 2σ u ɛ,u 0− iαu ɛ,u 0)dt + √ ɛu ɛ,u 0◦ dW,u ɛ,u 0(0) = u 0(D.2.2)with u 0 in H 1 and Φ in L 0,s2,R where s > d 2+ 1. When the noise is of multiplicativetype, we may write the equation in terms of a Itô product,idu ɛ,u 0= (∆u ɛ,u 0+ λ|u ɛ,u 0| 2σ u ɛ,u 0− iαu ɛ,u 0− iɛ 2 uɛ,u 0F Φ )dt + √ ɛu ɛ,u 0dW,where F Φ (x) = ∑ j∈N (Φe j(x)) 2 for x in R d and (e j ) j∈Na complete orthonormalsystem of L 2 . We consider mild solutions; for example the mildsolutions of (D.2.1) satisfiesu ɛ,u 0(t) =U(t)u 0 − iλ ∫ t0 U(t − s)(|uɛ,u 0(s)| 2σ u ɛ,u 0(s) − iαu ɛ,u 0(s))ds−i √ ɛ ∫ t0U(t − s)dW (s), t > 0.The Cauchy problems are globally well posed in L 2 and H 1 with the samearguments as in [37].The main tools in this article are the sample paths LDPs for the solutionsof the three Cauchy problems. They are uniform in the initial data. Unlikein [44, 81, 82] we use a Freidlin-Wentzell type formulation of the upper andlower bounds of the LDPs. Indeed it seems that the restriction that initialdata be in compact sets in [82] is a real limitation in particular for stochasticNLS equations. Indeed the linear Schrödinger group is not compact due tothe lack of smoothing effect and to the fact that we work on the whole spaceR d . This limitation disappears when we work with the Freidlin-Wentzelltype formulation; we may now obtain bounds for initial data in balls ofL 2 (respectively H 1 ) for ɛ small enough. Note that it is well known that inmetric spaces and for non uniform LDPs the two formulations are equivalent.A proof will be given and we will stress, in the multiplicative case, on theslight differences with the proof of the result in [82].154

Appendix D.Exit from a domain for weakly damped equationsWe denote by S(u 0 , h) the skeleton of equation (D.2.1) or (D.2.2), i.e.the mild solution of the controlled equation{i( dudt + αu) = ∆u + λ|u| 2σ u + Φh,u(0) = u 0where u 0 belongs to L 2 or H 1 in the additive case and the mild solution of{i( dudt + αu) = ∆u + λ|u| 2σ u + uΦh,u(0) = u 0where u 0 belongs to H 1 in the multiplicative case.The rate functions of the LDPs are always defined asI u 0T (w) = 1 ∫ Tinf‖h(s)‖ 22 h∈L 2 (0,T ;L 2 Lds.): S(u 0 ,h)=w2 0We denote for T and a positive by K u 0T (a) = ( I u )0 −1T ([0, a]) the levels of therate function less or equal to aK u 0T (a) = {w ∈ C ( [0, T ]; L 2) : w = S(u 0 , h),∫1 T}‖h(s)‖ 2 Lds ≤ a .22 0We also denote by d C([0,T ];L 2 ) the usual distance between sets of C ( [0, T ]; L 2)and by d C([0,T ];H 1 ) the distance between sets of C ( [0, T ]; H 1) .We also denote by ˜S(u 0 , f) the skeleton of equation (D.2.2) where we replaceΦh by ∂f∂t where f belongs to H1 0 (0, T ; Hs R ), the subspace of C ([0, T ]; Hs R )of square integrable in time and with square integrable in time time derivativefunctions, null at t = 0. Also C a denotes the set{f ∈ H 1 0 (0, T ; H s R ) : ∂f∂t ∈ ImΦ, IW T (f) = 1 2∥∂t ∥∂f∥ Φ−1 |(KerΦ) ⊥[and A(d) the set [2, ∞) when d = 1 or d = 2 and2, 2(3d−1)3(d−1)2L 2 (0,T ;L 2 )≤ a})when d ≥ 3.The above ITW is the good rate function of the LDP for the Wiener process.The uniform LDP with the Freidlin-Wentzell formulation that we will needin the remaining is then as follows. In the additive case we consider the L 2and H 1 case while in the multiplicative case we only consider the H 1 casebecause we will not need a L 2 result. Indeed in the case of the multiplicativenoise the L 2 norm remains invariant.155

Appendix D.Exit from a domain for weakly damped equationsTheorem D.2.1 In the additive case and in L 2 we have:for every a, ρ, T , δ and γ positive,(i) there exists ɛ 0 positive such that for every ɛ in (0, ɛ 0 ), u 0 such that‖u 0 ‖ L 2 ≤ ρ and ã in (0, a],P ( (d C([0,T ];L 2 ) uɛ,u 0, K u 0T (ã)) ≥ δ ) (< exp −ã − γ ),ɛ(ii)there exists ɛ 0 positive such that for every ɛ in (0, ɛ 0 ), u 0 such that‖u 0 ‖ L 2 ≤ ρ and w in K u 0T (a),)P(‖u ɛ,u 0− w‖ C([0,T ];L 2 ) < δ > exp(− Iu 0T (w) + γ ).ɛIn H 1 , the result holds for additive and multiplicative noises replacing in theabove ‖u 0 ‖ L 2 by ‖u 0 ‖ H 1 and C ( [0, T ]; L 2) by C ( [0, T ]; H 1) .Note that the extra conditionFor every a positive and K compact in L 2 , the set KT K(a) = ⋃ u 0 ∈K Ku 0T (a)is a compact subset of C ( [0, T ]; L 2)often appears to be part of a uniform LDP. It will not be used in the following.The proof of this result is given in the annex.D.3 Exit from a domain of attraction in L 2In this section we only consider the case of an additive noise. Recall thatfor the real multiplicative noise the mass is decreasing and thus exit is impossible.We may easily check that the mass N (S(u 0 , 0)) of the solution of thedeterministic equation satisfiesN (S(u 0 , 0)(t)) = N (u 0 ) exp (−2αt) .(D.3.1)With noise though, the mass fluctuates around the deterministic decay. Recallhow the Itô formula applies to the fluctuation of the mass, see [37] fora proof,N (u ɛ,u 0(t)) − N (u 0 ) =−2 √ ɛIm ∫ ∫ tR d 0 uɛ,u 0dW dx−2α ‖u ɛ,u 0‖ 2 L 2 (0,t;L 2 ) + ɛt‖Φ‖2 .L 0,02(D.3.2)Assume that D is a bounded measurable subset of L 2 containing 0 in itsinterior and invariant by the deterministic flow, i.e.∀u 0 ∈ D, ∀t ≥ 0, S(u 0 , 0)(t) ∈ D;156

Appendix D.Exit from a domain for weakly damped equationsit may be an open ball. There exists R positive such that D ⊂ B R .We define byτ ɛ,u 0= inf {t ≥ 0 : u ɛ,u 0(t) ∈ D c }the first exit time of the process u ɛ,u 0off the domain D.An easy information on the exit time is obtained as follows. The expectationof an integration via the Duhamel formula of the Itô decomposition, theprocess u ɛ,u 0being stopped at the first exit time, gives E [exp (−2ατ ɛ,u 0)] =1 − 2αRɛ‖Φ‖ 2 L 0,02. Without damping we obtain E [τ ɛ,u 0] =Rɛ‖Φ‖ 2 L 0,02precise information for small noises we use LDP techniques.Let us introducee = inf { I 0 T (w) : w(T ) ∈ D c , T > 0 } .When ρ is positive and small enough, we sete ρ = inf { I u 0T (w) : ‖u 0‖ L 2 ≤ ρ, w(T ) ∈ (D −ρ ) c , T > 0 } ,. To get morewhere D −ρ = D \ N 0 (∂D, ρ) and ∂D is the the boundary of ∂D in L 2 . Wedefine thene = limρ→0e ρ .We shall denote in this section by ‖Φ‖ c the norm of Φ as a bounded operatoron L 2 . Let us start with the following lemma.Lemma D.3.1 0 < e ≤ e.Proof. It is clear that e ≤ e. Let us check that e > 0. Let d denote thepositive distance between 0 and ∂D. Take ρ small such that the distancebetween Bρ 0 and (D −ρ ) c is larger than d 2. Multiplying the evolution equationby −iS(u 0 , h), taking the real part, integrating over space and using theDuhamel formula we obtainN (S(u 0 , h)(T )) − exp (−2αT ) N (u 0 )= 2 ∫ (T∫ )0 exp (−2α(T − s)) Im RS (u d 0 , h)Φhdxds .If S(u 0 , h)(T ) ∈ (D −ρ ) c and correspond to the first escape off D then∫dT2≤ 2‖Φ‖ c 0 exp (−2α(T − s)) ‖S(u 0, h)(s)‖ L 2 ‖h(s)‖ L 2ds( ∫ ) 1T≤ 2R‖Φ‖ c 0 exp (−4α(T − s)) ds 2‖h‖ L 2 (0,T ;L 2 ),157

Appendix D.Exit from a domain for weakly damped equationsthusand the result follows.αd 28R 2 ‖Φ‖ 2 c≤ 1 2 ‖h‖2 L 2 (0,T ;L 2 ) ,□Note that we would expect e and e to be equal. We should prove that,for a fixed level of energy, we may find ρ arbitrarily small and a controlof energy less than the fixed level such that the controlled solution goesfrom 0 to u 0 in Bρ 0 in finite time. We should also find a second control ofenergy smaller than the fixed level such that the controlled solution goesfrom ∂D −ρ to D c in finite time. Note that control arguments for nonlinearSchrödinger equations where the control enters the equation as an externalforce or potential are used in [38, 39] in the study of the blow-up time forstochastic nonlinear Schrödinger equations. Here it seems more intricateand the arguments of [38, 39] do not seem to apply. If these two boundswere indeed equal, they would also correspond to{}E(D) = 1 2 inf ‖h‖ 2 L 2 (0,∞;L 2 ) : ∃T > 0 : S(0, h)(T ) ∈ ∂D= inf v∈∂D V (0, v)where the quasi-potential is defined asV (u 0 , u f ) = inf { I 0 T (w) : w ∈ C ( R + ; L 2) , w(0) = u 0 , w(T ) = u f , T > 0 } .We shall prove in this section the two following results. The first theoremcharacterizes the first exit time from the domain.Theorem D.3.2 For every u 0 in D and δ positive, there exists L positivesuch that( ( ( ) ( )))lim ɛ→0 ɛ log P τ ɛ,u 0 e − δ e + δ/∈ exp , exp≤ −L, (D.3.3)ɛɛand for every u 0 in D,e ≤ lim ɛ→0 ɛ log E (τ ɛ,u 0) ≤ lim ɛ→0 ɛ log E (τ ɛ,u 0) ≤ e.Moreover, for every δ positive, there exists L positive such that( ( ))lim ɛ→0 ɛ log sup P τ ɛ,u 0 e + δ≥ exp≤ −L,u 0 ∈Dɛandlim ɛ→0 ɛ log sup E (τ ɛ,u 0) ≤ e.u 0 ∈D(D.3.4)(D.3.5)(D.3.6)158

Appendix D.Exit from a domain for weakly damped equationsThe second theorem characterizes formally the exit points. We shall definefor ρ positive small enough, N a closed subset of ∂De N,ρ = inf { I u 0T (w) : ‖u 0‖ L 2 ≤ ρ, w(T ) ∈ ( D \ N 0 (N, ρ) ) c, T > 0}.We then defineNote that e ρ ≤ e N,ρ and thus e ≤ e N .e N = limρ→0e N,ρ .Theorem D.3.3 If e N > e, then for every u 0 in D, there exists L positivesuch thatlim ɛ→0 ɛ log P (u ɛ,u 0(τ ɛ,u 0) ∈ N) ≤ −L.Thus the probability of an escape off D via points of N such that e ρ ≤ e N,ρgoes to zero exponentially fast with ɛ. Suppose that we were able to solve theprevious control problem, then as noise goes to zero, the probability of anexit via closed subsets of ∂D where the quasi-potential is not minimal goesto zero. As the expected exit time is finite, an exit occurs almost surely. Itis exponentially more likely that it occurs via infima of the quasi-potential.When there are several infima the exit measure is a probability measureon ∂D. When there exists only one infimum we may state the followingcorollary.Corollary D.3.4 Assume that v ∗ in ∂D is such that for every δ positiveand N = {v ∈ ∂D : ‖v − v ∗ ‖ L 2 ≥ δ} we have e N > e then∀δ > 0, ∀u 0 ∈ D, ∃L > 0 : lim ɛ→0 ɛ log P (‖u ɛ,u 0(τ ɛ,u 0) − v ∗ ‖ L 2 ≥ δ) ≤ −L.We need to prove a few lemmas before proving the two theorems.Let us defineσ ɛ,u 0ρ = inf { t ≥ 0 : u ɛ,u 0(t) ∈ B 0 ρ ∪ D c} ,where B 0 ρ ⊂ D.Lemma D.3.5 For every ρ and L positive with Bρ 0 ⊂ D, there exists T andɛ 0 positive such that for every u 0 in D and ɛ in (0, ɛ 0 ),P ( σ ɛ,u 0ρ > T ) (≤ exp − L ).ɛ159

Appendix D.Exit from a domain for weakly damped equationsProof. The result is straightforward if u 0 belongs to Bρ. 0 Suppose now thatu 0 belongs to D \ Bρ. 0 From equation (D.3.1), the bounded subsets of L 2 areuniformly attracted to zero by the flow of the deterministic equation. Thusthere exists a positive time T 1 such that for every u 1 in the ρ 8 −neighborhoodof D \ Bρ 0 and t ≥ T 1 , S (u 1 , 0) (t) ∈ B 0 ρ . We shall choose ρ < 8 and follow8three steps.Step 1: Let us first recall why there exists M ′ = M ′ (T 1 , R, σ, α) suchthatsupu 1 ∈N 0 (D\Bρ, 0 ρ 8)‖S(u 1 , 0)‖ Y (T 1 ,2σ+2) ≤ M ′ . (D.3.7)From the Strichartz inequalities, there exists C positive such that∥‖S(u 1 , 0)‖ Y (t,2σ+2) ≤ C ‖u 1 ‖ L 2 + C ∥|S(u 1 , 0)| 2σ+1∥ ∥∥Lγ ′ (0,t;L s′ )+Cα ‖S (u 1 , 0)‖ L 1 (0,t;L 2 )where γ ′ and s ′ are such that 1 γ+ 1 ′ r(˜p) = 1 and 1 s+1˜p= 1 and (r(˜p), ˜p) is′an admissible pair. Note that the first term is smaller than C(R + 1). Fromthe Hölder inequality, setting2σ2σ + 2 + 12σ + 2 = 1 s ′ ,2σω + 1r(2σ + 2) = 1 γ ′ ,we can write∥∥|S(u 1 , 0)| 2σ+1∥ ∥∥L ≤ C ‖S(u 1, 0)‖γ ′ L(0,t;L s′ ) r(2σ+2) (0,t;L 2σ+2 ) ‖S(u 1, 0)‖ 2σL ω (0,t;L 2σ+2 ) .It is easy to check that since σ < 2 d, we have ω < r(2σ + 2). Thus it followsthat‖S(u 1 , 0)‖ Y (t,2σ+2) ≤ C(R+1)+Ct ωr(2σ+2)r(2σ+2)−ω‖S(u 1 , 0)‖ 2σ+1Y (t,2σ+2) +Cα √ t ‖S(u 1 , 0)‖ Y (t,2σ+2) .The function x ↦→ C(R + 1) + Ct ωr(2σ+2)r(2σ+2)−ωx 2σ+1 + Cα √ tx − x is positive on aneighborhood of zero. For t 0 = t 0 (R, σ, α) small enough, the function has atleast one zero. Also, the function goes to ∞ as x goes to ∞. Thus, denotingby M(R, σ) the first zero of the above function, we obtain by a classicalargument that ‖S(u 1 , 0)‖ Y (t 0 ,2σ+2) ≤ M(R, σ) for every u 1 in N 0 ( D \ B 0 ρ, ρ 8).Also, as for every t in [0, T ], S(u 1 , 0)(t) belongs to N 0 ( D \ B 0 ρ, ρ 8), repeatingthe previous argument, u 1 is replaced by S(u 1 , 0)(t 0 ) and so on, we obtainsup ‖S(u 1 , 0)‖ Y (T 1 ,p) ≤ M ′ ,u 1 ∈N 0 (D\Bρ 0, ρ 8)160

Appendix D.Exit from a domain for weakly damped equationswhere M ′ =⌈T1t 0⌉M proving (D.3.7).Step2: Let us now prove that for T large enough, to be defined later,and larger than T 1 , we have{T ρ = w ∈ C ( [0, T ]; L 2) ( : ∀t ∈ [0, T ], w(t) ∈ N 0 D \ Bρ, 0 ρ )}⊂ K u 0T8(2L)c .(D.3.8)Since K u 0T (2L) is included in the image of S(u 0, ·) it suffices to consider win T ρ such that w = S(u 0 , h) for some h in L 2 (0, T ; L 2 ). Take h such thatS(u 0 , h) belongs to T ρ we have‖S(u 0 , h) − S(u 0 , 0)‖ C([0,T1 ];L 2 ) ≥ ‖S(u 0, h)(T 1 ) − S(u 0 , 0)(T 1 )‖ L 2 ≥ 3ρ 4 ,but also, necessarily, for the admissible pair (r(2σ + 2), 2σ + 2),‖S(u 0 , h) − S(u 0 , 0)‖ Y (T 1 ,2σ+2) ≥ 3ρ 4 .(D.3.9)Denote by S M ′ +1 the skeleton corresponding to the following control problem{i ( (dudt + αu) ‖u‖Y)(t,2σ+2)= ∆u + λθM ′ +1|u| 2σ u + Φh,u(0) = u 1where θ is a C ∞ function with compact support, such that θ(x) = 0 if x ≥ 2and θ(x) = 1 if 0 ≤ x ≤ 1. Then (D.3.9) implies that∥∥S M ′ +1 (u 0 , h) − S M ′ +1 (u 0 , 0) ∥ ≥ 3ρ Y (T 1 ,2σ+2)4 .We shall now split the interval [0, T 1 ] in many parts. We shall denote hereby Y s,t,2σ+2 for s < t the space Y t,2σ+2 on the interval [s, t]. Applying theStrichartz inequalities on a small interval [0, t] with the computations in theproof of Lemma 3.3 in [36], we obtain∥∥S M ′ +1 (u 0 , h) − S M ′ +1 (u 0 , 0) ∥ ≤ Cα√ ∥t ∥S M ′ +1 (uY (t,2σ+2) 0 , h) − S M ′ +1 (u 0 , 0) ∥ Y (t,2σ+2)+C M ′ +1t 1− dσ ∥2 ∥S M ′ +1 (u 0 , h) − S M ′ +1 (u 0 , 0) ∥ + C√ t‖Φ‖Y (t,2σ+2) c ‖h‖ L 2 (0,t;L 2 )where C M ′ +1 is a constant which depends on M ′ + 1. Take t 1 small enough1−σdsuch that C M ′ 2+1t1 + Cα √ t 1 ≤ 1 2. We obtain then∥∥S M ′ +1 (u 0 , h) − S M ′ +1 (u 0 , 0) ∥ ≤ 2C√ tY (t 1 ,2σ+2) 1 ‖Φ‖ c ‖h‖ L 2 (0,t 1 ;L 2 ).161

Appendix D.Exit from a domain for weakly damped equationsIn the case where 2t 1 < T 1 , let us see how such inequality propagateson [t 1 , 2t 1 ]. We now have two different initial data S M ′ +1 (u 0 , h) (t 1 ) andS M ′ +1 (u 0 , 0) (t 1 ). We obtain similarly∥∥S M ′ +1 (u 0 , h) − S M ′ +1 (u 0 , 0) ∥Y (t 1 ,2t 1 ,2σ+2)≤ 2C √ ∥t 1 ‖Φ‖ c ‖h‖ L 2 (0,t 1 ;L 2 ) + 2 ∥S M ′ +1 (u 0 , h) (t 1 ) − S M ′ +1 (u 0 , 0) (t 1 )H 1≤ 2C √ ∥t 1 ‖Φ‖ c ‖h‖ L 2 (0,T 1 ;L 2 ) + 2 ∥S M ′ +1 (u 0 , h) (t 1 ) − S M ′ +1 (u 0 , 0) (t 1 ) ∥ .Y (0,t 1 ,2σ+2){ ⌊ ⌋}Then iterating on each interval of the form [kt 1 , (k+1)t 1 ] for k in 1, ..., T1t 1− 1 ,the remaining term can be treated similarly, and using the triangle inequalitywe obtain thatl m∥∥S M ′ +1 (u 0 , h) − S M ′ +1 T1(u 0 , 0) ∥ ≤ 2 t1 +1 √Y (T 1 ,2σ+2)C t1 ‖Φ‖ c ‖h‖ L 2 (0,t 1 ;L 2 ).We may then conclude thatwhere M ′′ =ρ 28C(t 1 ,T 1 )‖Φ‖ 2 c12 ‖h‖2 L 2 (0,T 1 ;L 2 ) ≥ M ′′and C (t 1 , T 1 ) is a constant which depends onlyon t 1 and T 1 . Note that we have used for later purposes that 3ρ 2 > ρ 2 .Similarly replacing [0, T 1 ] by [T 1 , 2T 1 ] and u 0 respectively by S (u 0 , h) (T 1 )and S (u 0 , 0) (T 1 ) in (D.3.9), the inequality still holds true. Thus thanks tothe inverse triangle inequality we obtain on [T 1 , 2T 1 ]∥∥∥S M ′ +1 (u 0 , h) − S M ′ +1 (u 0 , 0)()∥= S M ′ +1 (u 0 , h)(T 1 ), h − S M ′ +1≥ 3ρ 4∥S M ′ +1∥Y (T 1 ,2T 1 ,2σ+2)(S M ′ +1 (u 0 , 0)(T 1 ), 0)∥ ∥∥Y(0,T 1 ,2σ+2)Thus from the inverse triangle inequality along with the fact that for bothS M ′ +1 (u 0 , h)(T 1 ) and S M ′ +1 (u 0 , 0)(T 1 ) as initial data the deterministic solutionsbelong to the ball B 0 ρ , we obtain8∥∥S M ′ +1 ( S M ′ +1 (u 0 , h)(T 1 ), hWe finally obtain the same lower bound)( )∥− S M ′ +1S M ′ +1 ∥∥Y(u 0 , h)(T 1 ), 0(0,T 1 ,2σ+2)12 ‖h‖2 L 2 (T 1 ,2T 1 ;L 2 ) ≥ M ′′162≥ ρ 2 .

Appendix D.Exit from a domain for weakly damped equationsas before.Iterating the argument we obtain if T > 2T 1 ,12 ‖h‖2 L 2 (0,2T 1 ;L 2 ) = 1 2 ‖h‖2 L 2 (0,T 1 ;L 2 ) + 1 2 ‖h‖2 L 2 (T 1 ,2T 1 ;L 2 ) ≥ 2M ′′ .Thus for j positive and T > jT 1 , we obtain, iterating the above argument,that12 ‖h‖2 L 2 (0,jT 1 ;L 2 ) ≥ jM ′′ .The result (D.3.8) is obtained for T = jT 1 where j is such that jM ′′ > 2L.Step 3: We may now conclude from the (i) of Theorem D.2.1 since,P (σ ɛ,u 0ρ > T ) = P ( ∀t ∈ [0, T ], u ɛ,u 0(t) ∈ D \ Bρ0 )= P ( (d C([0,T ];L 2 ) uɛ,u 0, Tρc )>ρ8),≤ P ( (d C([0,T ];L 2 ) uɛ,u 0, K u 0T (2L)) ≥ ρ 8),taking a = 2L, ρ = R where D ⊂ B R , δ = ρ 8and γ = L.Note that if ρ ≥ 8, we should replace R + 1 by R + ρ 8 and M ′ + 1 byM ′ + ρ 8. Anyway, we will use the lemma for small ρ.□Lemma D.3.6 For every ρ positive such that Bρ 0 ⊂ D and u 0 in D, thereexists L positive such thatlim ɛ→0 ɛ log P ( u ɛ,u ( 0σ ɛ,u ) )0ρ ∈ ∂D ≤ −LProof. Take ρ positive satisfying the assumptions of the lemma and takeu 0 in D. When u 0 belongs to Bρ 0 the result is straightforward. Suppose nowthat u 0 belongs to D \ Bρ. 0 Let T be defined as{T = inft ≥ 0 : S (u 0 , 0) (t) ∈ B 0 ρ2then since S (u 0 , 0) ([0, T ]) is a compact subset of D, the distance d betweenS (u 0 , 0) ([0, T ]) and D c is well defined and positive. The conclusion followsthen from the fact thatP ( u ɛ,u 0 ( σ ɛ,u 0ρ},)∈ ∂D)≤ P(‖u ɛ,u 0− S (u 0 , 0)‖ C([0,T ];L 2 ) ≥ ρ ∧ d2the LDP and the fact that, from the compactness of the sets K u 0T(a) for apositive, we haveinf‖h‖ 2h∈L 2 (0,T ;L 2 ): ‖S(u 0 ,h)−S(u 0 ,0)‖ C([0,T ];L 2 ) ≥ ρ∧d L 2 (0,T ;L 2 ) > 0.2163),

Appendix D.Exit from a domain for weakly damped equationsWe have used the fact that the upper bound of the LDP in the Freidlin-Wentzell formulation implies the classical upper bound. Note that this is awell known result for non uniform LDPs. Indeed we do not need a uniformLDP in this proof.□The following lemma replaces Lemma 5.7.23 in [48]. Indeed, the case of astochastic PDE is more intricate than that of a SDE since the linear groupis only strongly and not uniformly continuous. However, it is possible toprove that the group on L 2 when acting on bounded sets of H 1 is uniformlycontinuous. We shall proceed in a different manner and thus we will notloose in regularity. Indeed, the Schrödinger group does not have regularizingproperties and we would obtain a weaker result with extra assumptions onΦ and the initial data.Lemma D.3.7 For every ρ and L positive such that B2ρ 0 ⊂ D, there existsT (L, ρ) < ∞ such that(‖Φ‖ 2 L 0,02lim ɛ→0 ɛ log sup Pu 0 ∈Sρ0)sup (N (u ɛ,u 0(t)) − N (u 0 )) ≥ 3ρ 2 ≤ −Lt∈[0,T (L,ρ)]Proof. Take L and ρ positive. Note that for every ɛ in (0, ɛ 0 ) where ɛ 0 =ρ 2, for T (L, ρ) ≤ 1 we have ɛT (L, ρ)‖Φ‖ 2 < ρ 2 . Thus from equationL 0,02(D.3.2), we know that it is enough to prove that there exists T (L, ρ) ≤ 1such that for ɛ 1 small enough, ɛ 1 < ɛ 0 , and all ɛ < ɛ 0 ,( (ɛ log sup P sup −2 √ ∫ t) )ɛIm uu 0 ∈Sρ0 t∈[0,T (L,ρ)]∫R ɛ,u0,τ dW dx ≥ 2ρ 2 ≤ −L,d 0where u ɛ,u0,τ is the process u ɛ,u 0stopped at τ ɛ,u 0, the first time when u ɛ,u S2ρ0 0hits S2ρ 0 . Setting Z(t) = Im ∫ ∫ tR d 0 uɛ,u0,τ dW dx, it is enough to show that(ɛ log sup Pu 0 ∈Sρ0)sup |Z(t)| ≥ √ ρ2≤ −L,t∈[0,T (L,ρ)] ɛand thus to show exponential tail estimates for the process Z(t). Our proofnow follows closely that of [117][Theorem 2.1]. We introduce the functionf l (x) = √ 1 + lx 2 , where l is a positive parameter. We now apply the Itôformula to f l (Z(t)) and the process decomposes into 1 + E l (t) + R l (t) whereE l (t) =∫ t02lZ(t)√1 + lZ(t) 2 dZ(t) − 1 2164∫ t0(2lZ(t)√1 + lZ(t) 2) 2d < Z > t ,

Appendix D.Exit from a domain for weakly damped equationsandR l (t) = 1 2∫ t0() 22lZ(t)√ d < Z > t +1 + lZ(t) 2∫ t0ld < Z >(1 + lZ(t) 2 ) 3 t .2Moreover, given (e j ) j∈Na complete orthonormal system of L 2 ,< Z(t) >=∫ t0∑(u ɛ,u0,τ , −iΦe j ) 2 L(s)ds,2j∈Nwe prove with the Hölder inequality that |R l (t)| ≤ 12lρ 2 ‖Φ‖ 2 t, for everyL 0,02u 0 in D. We may thus write(P sup t∈[0,T (L,ρ)] |Z(t)| ≥ ρ2(= P sup t∈[0,T (L,ρ)] exp (f l (Z(t))) ≥ exp(≤ P sup t∈[0,T (L,ρ)] exp (E l (t)) ≥ exp√ ɛ)(f l(ρ 2√ ɛ)))(f l(ρ 2√ ɛ)− 1 − 12lρ 2 ‖Φ‖ 2 L 0,02))T (L, ρ) .The Novikov condition is also satisfied and E l (t) is such that (exp (E l (t))) t∈R +is a uniformly integrable martingale. The exponential tail estimates followfrom the Doob inequality optimizing on the parameter l. We may then write(⎛⎞sup Pu 0 ∈Sρ0)sup |Z(t)| ≥ √ ρ2≤ 3 expt∈[0,T (L,ρ)] ɛ⎝−ρ 248ɛ‖Φ‖ 2 L 0,02⎠ .T (L, ρ)We now conclude setting T (L, ρ) =enough.ρ 250‖Φ‖ 2 L 0,02L and choosing ɛ 1 < ɛ 0 small□Proof of Theorem D.3.2. Let us first prove (D.3.6) and deduce (D.3.5).Fix δ positive and choose h and T 1 such that S(0, h)(T 1 ) ∈ D c andI 0 T 1(S(0, h)) = 1 2 ‖h‖2 L 2 (0,T ;L 2 ) ≤ e + δ 5 .Let d 0 denote the positive distance between S(0, h) (T 1 ) and D. With similararguments as in [37] or with a truncation argument we may prove that theskeleton is continuous with respect to the initial datum for the L 2 topology.Thus there exists ρ positive, a function of h which has been fixed, such thatif u 0 belongs to B 0 ρ then‖S (u 0 , h) − S(0, h)‖ C([0,T1 ];L 2 ) < d 02 .165

Appendix D.Exit from a domain for weakly damped equationsWe may assume that ρ is such that B 0 ρ ⊂ D. From the triangle inequalityand the (ii) of Theorem D.2.1, there exists ɛ 1 positive such that for all ɛ in(0, ɛ 1 ) and u 0 in B 0 ρ,P (τ ɛ,u 0< T 1 )()≥ P ‖u ɛ,u 0− S(0, h)‖ C([0,T1 ];L 2 ) < d 0(≥ P ‖u ɛ,u 0− S (u 0 , h)‖ C([0,T1 ];L 2 ) < d 0()≥ exp − Iu 0T (S(u 0 ,h))+ δ 1 5ɛ.From Lemma D.3.5, there exists T 2 and ɛ 2 positive such that for all ɛ in(0, ɛ 2 ),inf P ( σ ɛ,u ) 10ρ ≤ T 2 ≥u 0 ∈D 2 .Thus, for T = T 1 + T 2 , from the strong Markov property we obtain that forall ɛ < ɛ 3 < ɛ 1 ∧ ɛ 2 .q = inf u0 ∈D P (τ ɛ,u 0≤ T ) ≥ inf u0 ∈D P (σ ɛ,u 0ρ ≤ T 2 ) inf u0 ∈Bρ 0 P (τ ɛ,u 0≤ T 1 )()≥ 1 2 exp − Iu 0T (S(u 0 ,h))+ δ 1 5ɛ()≥ exp − Iu 0T (S(u 0 ,h))+ 2δ1 5ɛ.Thus, for any k ≥ 1, we haveP (τ ɛ,u 0> (k + 1)T ) = [1 − P (τ ɛ,u 0≤ (k + 1)T |τ ɛ,u 0> kT )] P (τ ɛ,u 0> kT )≤ (1 − q)P (τ ɛ,u 0> kT )≤ (1 − q) k .We may now compute, since I u 0T 1(S (u 0 , h)) = I 0 T 1(S (0, h)) = 1 2 ‖h‖2 L 2 (0,T ;L 2 )2)sup u0 ∈D E (τ ɛ,u 0)∫ ∞= sup u0 ∈D 0P (τ ɛ,u 0> t) dt≤ T [1 + ∑ ∞k=1 sup x∈D P (τ ɛ,u 0> kT )]≤ T q≤ T exp( )e+ 3δ5ɛ.It implies that there exists ɛ 4 small enough such that for ɛ in (0, ɛ 4 ),( )sup E (τ ɛ,u 0 e +4δ5) ≤ exp . (D.3.10)u 0 ∈Dɛ166

Appendix D.Exit from a domain for weakly damped equationsThus the Chebychev inequality gives that( ( ))sup P τ ɛ,u 0 e + δ≥ exp≤ expu 0 ∈Dɛin other words( ( ))sup P τ ɛ,u 0 e + δ≥ expu 0 ∈Dɛ(− e + δ )ɛsup E (τ ɛ,u 0) ,u 0 ∈D(≤ exp − δ ). (D.3.11)5ɛRelations (D.3.10) and (D.3.11) imply (D.3.6) and (D.3.5).Let us now prove the lower bound on τ ɛ,u 0. Take δ positive. Remindthat we have proved that e > 0. Take ρ positive small enough such thate − δ 4 ≤ e ρ and B2ρ 0 ⊂ D. We define the following sequences of stoppingtimes, θ 0 = 0 and for k in N,τ k = inf { t ≥ θ k : u ɛ,u 0(t) ∈ B 0 ρ ∪ D c} ,θ k+1 = inf { t > τ k : u ɛ,u 0(t) ∈ S 0 2ρ},where θ k+1 = ∞ if u ɛ,u 0(τ k ) ∈ ∂D. Fix T 1 = T ( e − 3δ4 , ρ) given in LemmaD.3.7. We know that there exists ɛ 1 positive such that for all ɛ in (0, ɛ 1 ), forall k ≥ 1 and u 0 in D,P (θ k − τ k−1 ≤ T 1 ) ≤ expFor u 0 in D and an m in N ∗ , we have(− e − 3δ4ɛP (τ ɛ,u 0≤ mT 1 ) ≤ P (τ ɛ,u 0= τ 0 ) + ∑ mk=1 P (τ ɛ,u 0= τ k )+P (∃k ∈ {1, ..., m} : θ k − τ k−1 ≤ T 1 )= P (τ ɛ,u 0= τ 0 ) + ∑ mk=1 P (τ ɛ,u 0= τ k )+ ∑ mk=1 P (θ k − τ k−1 ≤ T 1 ) .).(D.3.12)In other words the escape before mT 1 can occur either as an escape withoutpassing in the small ball Bρ 0 (if u 0 belongs to D \ Bρ) 0 or as an escape withk in {1, ...m} significant fluctuations off Bρ, 0 i.e. crossing S2ρ 0 , or at leastone of the m first transitions between Sρ 0 and S2ρ 0 happens in less than T 1.The latter is known to be arbitrarily small. Let us prove that the remainingprobabilities are small enough for small ɛ.For every k ≥ 1 and T 2 positive, we may writeP (τ ɛ,u 0= τ k ) ≤ P (τ ɛ,u 0≤ T 2 ; τ ɛ,u 0= τ k ) + P ( σ ɛ,u 0ρ > T 2).167

Appendix D.Exit from a domain for weakly damped equationsFix T 2 as in Lemma D.3.5 with L = e − 3δ4 . Thus there exists ɛ 2 smallenough such that for ɛ in (0, ɛ 2 ),P ( (σ ɛ,u )0ρ > T 2 ≤ exp − e − )3δ4.ɛAlso, from the (i) of Theorem D.2.1, we obtain that there exists ɛ 3 positivesuch that for every u 1 in B 0 ρ and ɛ in (0, ɛ 3 ),P (τ ɛ,u 1≤ T 2 )(≤ P(≤ exp≤ expd C([0,T2 ];L 2 )− eρ− δ 2ɛ(−e−3δ4ɛ)).(u ɛ,u 1, K u (1T eρ 2− 4) ) )δ ≥ ρThus the above bound holds for P (τ ɛ,u 0≤ T 2 ; τ ɛ,u 0= τ k ) replacing u 1 byu ɛ,u 0(τ k−1 ) since as k ≥ 1, u ɛ,u 0(τ k−1 ) belongs to Bρ 0 and τ k − τ k−1 ≤ T 2and using the Markov property. The inequality (D.3.12) gives that for all ɛin (0, ɛ 0 ) where ɛ 0 = ɛ 1 ∧ ɛ 2 ∧ ɛ 3 ,P (τ ɛ,u 0≤ mT 1 ) ≤ P ( u ɛ,u ( 0σ ɛ,u ) )0ρ ∈ ∂D + 3m exp(− e − 3δ4ɛ⌈ ( )⌉Fix m = 1T 1exp e−δɛ, then for all ɛ in (0, ɛ 0 ),( ( ))P τ ɛ,u 0≤ exp e−δɛ≤ P (τ ɛ,u 0≤ mT 1 )≤ P (u ɛ,u 0(σ ɛ,u 0ρ ) ∈ ∂D) + 3 T 1exp ( − δ 4ɛ).We may now conclude with Lemma D.3.6 and obtain the expected lowerbound on E (τ ɛ,u 0) from the Chebychev inequality.□Proof of Theorem D.3.3. Let N be closed subset of ∂D. When e N = ∞we shall replace in the proof that follows e N by an increasing sequence ofpositive numbers. Take δ such that 0 < δ < e N −e3, ρ positive such thate N − δ 3 ≤ e N,ρ and B2ρ 0 ⊂ D. Define the same sequences of stopping times(τ k ) k∈Nand (θ k ) k∈Nas in the proof of Theorem D.3.2.Take L = e N − δ and T 1 and T 2 = T (L, ρ) as in Lemma D.3.5 and D.3.7.Thanks to Lemma D.3.5 and the uniform LDP, with a computation similarto the one following inequality (D.3.12), we obtain that for ɛ 0 small enough).168

Appendix D.Exit from a domain for weakly damped equationsand ɛ ≤ ɛ 0 ,sup u0 ∈S2ρ 0 P (uɛ,u 0(σ ɛ,u 0ρ ) ∈ N)≤ sup u0 ∈S2ρ 0 P (uɛ,u 0(σ ɛ,u 0ρ ) ∈ N, σ ɛ,u 0ρ ≤ T 1 ) + sup u0 ∈S2ρ 0 P (σɛ,u 0ρ > T 1 )(≤ sup u0 ∈B(d2ρ 0 P C([0,T1 ];L 2 ) u ɛ,u 0, K u (0T eN,ρ 1− 3) ) )δ ≥ ρ+ sup u0 ∈D P (σ ɛ,u 0( ) ρ > T 1 )≤ 2 exp .− e N −δɛPossibly choosing ɛ 0 smaller, we may assume that for every positive integerl and every ɛ ≤ ɛ 0 ,)sup u0 ∈D P (τ l ≤ lT 2 ) ≤ l sup u0 ∈S(supρ 0 P t∈[0,T2 ] (N (u ɛ,u 0(t)) − N (u 0 )) ≥ ρ( )≤ l exp .− e N −δɛThus if u 0 belongs to B 0 ρP (u ɛ,u 0(τ ɛ,u 0) ∈ N) ≤ P (τ ɛ,u 0> τ l ) + ∑ lk=1 P (uɛ,u 0(τ ɛ,u 0) ∈ N, τ ɛ,u 0= τ k )≤ P (τ ɛ,u 0> lT 2 ) + P (τ l ≤ lT 2 )+l sup u0 ∈S2ρ 0 P (uɛ,u 0(σ ɛ,u 0ρ ) ∈ N)( )≤ P (τ ɛ,u 0> lT 2 ) + 3l exp .− e N −δɛ⌈Take now l = 1T 2exp ( ) ⌉ e+δɛand use the upper bound (D.3.11), possiblychoosing ɛ 0 smaller, we obtain that for ɛ ≤ ɛ 0sup u0 ∈Bρ 0 P (uɛ,u 0(τ ɛ,u 0) ∈ N) ≤ exp ( ( )−5ɛ) δ +4T 2exp − e N −e+2δɛ≤ exp ( −5ɛ) δ +4T 2exp ( − δ )ɛ .Finally, when u 0 is any function in D, we conclude thanks toP (u ɛ,u 0(τ ɛ,u 0) ∈ N) ≤ P ( u ɛ,u ( 0σ ɛ,u ) )0ρ ∈ ∂D + sup P (u ɛ,u 0(τ ɛ,u 0) ∈ N)u 0 ∈Bρ0and to Lemma D.3.6.□Remark D.3.8 Note that it has been proposed in [128] to introduce controlelements in order to reduce or enhance exponentially the expected exit timeor to act on the exiting points, for a limited cost. We may then think ofoptimizing on such external fields. However the problem is computationallyinvolved since the optimal control problem requires double optimisation.169

Appendix D.Exit from a domain for weakly damped equationsD.4 Exit from a domain of attraction in H 1We now consider a measurable bounded subset D of H 1 invariant by theflow of the deterministic equation; D and R are such that D ⊂ B 1 R . Weconsider both (D.2.1) and (D.2.2) where the noise is either of additive or ofmultiplicative type. In this section we are interested in both the fluctuationof the L 2 norm and that of the L 2 norm of the gradient. The Hamiltonianand a modified Hamiltonian will thus be of particular interest. We shall firstdistinguish the case where the nonlinearity is defocusing (λ = −1) where theHamiltonian takes non negative values from the case where the nonlinearityis focusing (λ = 1) where the Hamiltonian may take negative values.We may prove, see for example [86], thatddt H (S(u 0, 0)(t)) + 2αΨ (S(u 0 , 0)) = 0,where S(u 0 , 0) is the solution of the deterministic weakly damped nonlinearSchrödinger equation with initial datum u 0 in H 1 andΨ (S(u 0 , 0)) = 1 2 ‖∇S(u 0, 0)‖ 2 L 2 − λ ∫|S(u 0 , 0)(x)| 2σ+2 dx.2 R dThus, when the nonlinearity is defocusing we have0 ≤ H (S(u 0 , 0)(t)) ≤ H (u 0 ) exp (−2αt) . (D.4.1)As it is done in [45], we shall consider in the focusing case a modifiedHamiltonian denoted by ˜H(u) defined for u in H 1 by˜H(u) = H(u) + β(σ, d)C ‖u‖ 2+4σ2−σdL 2where the constant C is that of the third inequality in the following sequenceof inequalities where we use the Gagliardo-Nirenberg inequality12σ + 2 ‖u‖2σ+2 ≤ C‖u‖ 2σ+2−σd ‖∇u‖ σdL 2σ+2 L 2L≤ 1 2 4 ‖∇u‖2 L+ C‖u‖ 2+22σ(2−σd)4σ2−σd,L 2and β(σ, d) =(σ+2)(2−σd)+2σ(4σ+3)∨ 2. When evaluated at the deterministicsolution, the modified Hamiltonian satisfies()0 ≤ ˜H (S(u 0 , 0)(t)) ≤ ˜H 3(σ + 1)(u 0 ) exp −2α4σ + 3 t . (D.4.2)170

Appendix D.Exit from a domain for weakly damped equationsAlso, when the nonlinearity is defocusing we now have, for every β positive,0 ≤ ˜H (S(u 0 , 0)(t)) ≤ ˜H (u 0 ) exp (−2αt) . (D.4.3)From the Sobolev inequalities, for ρ positive, the sets}˜H ρ ={u ∈ H 1 : ˜H(u) = ρ = ˜H −1 ({ρ}) , ρ > 0are closed subsets of H 1 and}˜H 0are open subsets of H 1 .Also, ˜H is such that4σ12 ‖∇u‖2 2+L+βC‖u‖ 2 2−σd≤ ˜H(u) ≤ 3 L 2 4 ‖∇u‖2 L+(β +1)C‖u‖ 2when the nonlinearity is defocusing and4σ14 ‖∇u‖2 2+L+ C‖u‖ 2 2−σd≤ ˜H(u) ≤ 1 L 2 2 ‖∇u‖2 L+ β(σ, d)C‖u‖ 24σ2+ 2−σdL 24σ2+ 2−σdL 2(D.4.4)(D.4.5)when it is focusing. Thus the sets ˜H

Appendix D.Exit from a domain for weakly damped equationsProof. The result follows from the Itô formula. The main difficulty is injustifying the computations. We may proceed as in [37].□Also, when the noise is of multiplicative type we obtain the followingproposition.Proposition D.4.2 When u denotes the solution of equation (D.2.1), (e j ) j∈Na complete orthonormal system of L 2 , the following decomposition holds˜H (u(t)) = ˜H (u 0 )−2α ∫ (t0 Ψ (u(s)) ds − 2βC 1 + 2σ2−σd+ √ ɛIm ∫ ∫ tR d 0u(s)∇u(s)∇dW (s)dx+ ɛ ∑ ∫ t ∫2 j∈N 0 R|u(s)| 2 |∇Φe d j | 2 dxds.)α ∫ t02−σdL 2‖u(s)‖2+4σThe first exit time τ ɛ,u 0from the domain D in H 1 is defined as in SectionD.2. Note that the domain D may be a domain of attraction of the form˜H 0 } ,and for ρ positive small enoughe ρ = inf{I u 0T (w) : }˜H (u0 ) ≤ ρ, w(T ) ∈ (D −ρ ) c , T > 0 ,where D −ρ = D \ N 1 (∂D, ρ). Then we sete = limρ→0e ρ .Also, for ρ positive small enough, N a closed subset of the boundary of D,we define{e N,ρ = inf I u 0T (w) : ˜H (u0 ) ≤ ρ, w(T ) ∈ ( D \ N 1 (N, ρ) ) }c, T > 0andWe finally also introduceσ ɛ,u 0ρ= infe N = limρ→0e N,ρ .{t ≥ 0 : u ɛ,u 0(t) ∈ ˜H

Appendix D.Exit from a domain for weakly damped equationsLemma D.4.3 0 < e ≤ e.Proof. We only have to prove the first inequality. Integrating the equationdescribing the evolution of ˜H (S (u 0 , h) (t)) via the Duhamel formula wherethe skeleton is that of the equation with an additive noise we obtain()˜H (S(u 0 , h)(T )) − exp −2α 3(σ+1)4σ+3 T ˜H (u0 )≤ ∫ ()T[Im0 exp −2α 3(σ+1)4σ+3 (T − s) ( ∫R d ∇S(u0 , h)∇Φh ) (s, x)dx−λIm ∫ (R d |S(u0 , h)| 2σ S(u 0 , h)Φh ) ( )(s, x)dx−2Cβ 1 + 2σ2−σdIm ∫ (R S d (u0 , h) Φh ) ](s, x)dx ds,with a focusing or defocusing nonlinearity. Let d denote the positive distancebetween 0 and ∂D. Take ρ such that the distance between Bρ 1 and (D −ρ ) c islarger than d 2. We then have, from the fact that the Sobolev injection fromH 1 into L 2σ+2 ,d2 ≤ ∫ ()T0 exp −2α 3(σ+1)4σ+3[R‖Φ‖Lc(L (T − s) 2 ,H 1 )‖h‖ L 2+CR 2σ+1 ‖Φ‖ Lc(L 2 ,H 1 )‖h‖ L 2( )]+2Cβ 1 + 2σ2−σdR‖Φ‖ Lc(L 2 ,L 2 )‖h‖ L 2 ds,We conclude as in Lemma D.3.1 and use that from the choice of β thecomplementary of a ball is included in the complementary of a set ˜H d 2 + 1, in particular Φ belongs to L (c L 2 , W 1,∞) . □The theorems of Section D.2 still hold for a domain of attraction in H 1and a noise of additive and multiplicative type.Theorem D.4.4 For every u 0 in D and δ positive, there exists L positivesuch that( ( ( ) ( )))lim ɛ→0 ɛ log P τ ɛ,u 0 e − δ e + δ/∈ exp , exp≤ −L, (D.4.6)ɛɛand for every u 0 in D,e ≤ lim ɛ→0 ɛ log E (τ ɛ,u 0) ≤ lim ɛ→0 ɛ log E (τ ɛ,u 0) ≤ e.(D.4.7)173

Appendix D.Exit from a domain for weakly damped equationsMoreover, for every δ positive, there exists L positive such that( ( ))lim ɛ→0 ɛ log sup P τ ɛ,u 0 e + δ≥ exp≤ −L,u 0 ∈Dɛ(D.4.8)andlim ɛ→0 ɛ log sup E (τ ɛ,u 0) ≤ e.u 0 ∈D(D.4.9)Theorem D.4.5 If e N > e, then for every u 0 in D, there exists L positivesuch thatlim ɛ→0 ɛ log P (u ɛ,u 0(τ ɛ,u 0) ∈ N) ≤ −L.Again we may deduce the corollaryCorollary D.4.6 Assume that v ∗ in ∂D is such that for every δ positiveand N = {v ∈ ∂D : ‖v − v ∗ ‖ L 2 ≥ δ} we have e N > e then∀δ > 0, ∀u 0 ∈ D, ∃L > 0 : lim ɛ→0 ɛ log P (‖u ɛ,u 0(τ ɛ,u 0) − v ∗ ‖ L 2 ≥ δ) ≤ −L.The proof of these results still relies on three lemmas and the uniform LDP.Let us now state the lemmas for both a noise of additive and of multiplicativetype.Lemma D.4.7 For every ρ and L positive with ˜H T ) (≤ exp − L ).ɛProof. We proceed as in the proof of Lemma D.3.5.Let d denote the positive distance between 0 and D \ ˜H

Appendix D.Exit from a domain for weakly damped equationsin [37] derived from the Strichartz inequalities on smaller intervals; we shallalso replace in the proof of Lemma D.3.5 ρ αρ8by8 .In Step 2 for the multiplicative case, we also introduce the truncation infront of the term uΦh in the controlled PDE.The end of the proof is identical to that of Lemma D.3.5, the LDP is theLDP in C ( [0, T ]; H 1) , for additive or multiplicative noises.□Lemma D.4.8 For every ρ positive such that ˜H ρ ⊂ D and u 0 in D, thereexists L positive such thatlim ɛ→0 ɛ log P ( u ɛ,u ( 0σ ɛ,u ) )0ρ ∈ ∂D ≤ −LProof. It is the same proof as for Lemma D.3.6. We only have to replaceby any ball in H 1 centered at 0 and included in ˜H

Appendix D.Exit from a domain for weakly damped equationsThen, setting c(σ) = 3(σ+1)2σ4σ+3and m(σ, d) = 1+2−σd, dropping the exponentsɛ and u 0 to have more concise formulas, we obtain in the additive case˜H (u(t)) − exp (−2αc(σ)t) ˜H (u 0 )≤ √ (ɛ Im ∫ ∫ tR d 0exp (−2αc(σ)(t − s)) ∇u(s)∇dW (s)dx−λIm ∫ ∫ tR d 0 exp (−2αc(σ)(t − s)) |u(s)|2σ u(s)dW (s)dx+2βCm(σ, d)Im ∫ )∫ 4σtR d 0 exp (−2αc(σ)(t − s)) ‖u(s)‖ 2−σdu(s)dW (s)dx− λɛ2+ ɛ∑j∈N∫ t0 exp (−2αc(σ)(t − s)) ∫ R d [|u(s)| 2σ |Φe j | 24αc(σ) (1 − exp (−2αc(σ)t)) ‖∇Φ‖2 L 0,0∫ t+ɛβCm(σ, d)‖Φ‖ 2 L 0,02+ɛβC 4σ2−σd m(σ, d) ∑ j∈N2L 20 exp (−2αc(σ)(t − s)) ‖u(s)‖ 4σ∫ t0+ 2σ |u(s)| 2σ−2 (Re(u(s)Φe j )) 2 ]dxds2−σdL 2dsexp (−2αc(σ)(t − s)) ‖u(s)‖2(2σ2−σd −1)L 2 (Re∫R d u(s)Φe j dx ) 2 ds.We again use a localization argument and replace the process u by theprocess u τ stopped at the first exit time off ˜H

Appendix D.Exit from a domain for weakly damped equationsWe then obtaind < Z 1 > t ≤ exp (4αc(σ)t) ∑ j∈N (∇uτ (t), −i∇Φe j ) 2 L 2 dtd < Z 2 > t ≤ exp (4αc(σ)t) ∑ j∈N(|u τ (t)| 2σ u τ (t), −iΦe j) 2L 2 dtd < Z 3 > t ≤ 4β 2 C 2 m(σ, d) 2 exp (4αc(σ)t) ‖u τ 2−σd(t)‖L∑j∈N 2 (uτ (t), −iΦe j ) 2 Ldt. 2Using the Hölder inequality and, for Z 2 , the continuous Sobolev injection ofH 1 into L 2σ+2 we obtaind < Z 1 > t ≤ exp (4αc(σ)t) ‖Φ‖ 2 b(ρ, σ, d)dtL 0,12d < Z 2 > t ≤ exp (4αc(σ)t) c(1, 2σ + 2) 2(2σ+2) ‖Φ‖ 2 L 0,12d < Z 3 > t ≤ 4β 2 C 2 m(σ, d) 2 exp (4αc(σ)t) b(ρ, σ, d) (1+8σb(ρ, σ, d) 2σ+1 dt4σ2−σd) ‖Φ‖2We can then bound each of the three remainders ( Rl i(t)) i=1,2,3that of Lemma D.3.7 using the inequality Rl i(t) ≤ 3l ∫ t0 d < Z i > t .We conclude that it is possible to choose T (L, ρ) equal to⎛14αc(σ) logL 0,12dt.similar to⎝ αc(σ)ρ 290b(ρ,σ,d)‖Φ‖ 2 L 0,1 max 1,c(1,2σ+2) 2(2σ+1) b(ρ,σ,d) 2σ ,4β 2 C 2 m(σ,d) 2 b(ρ,σ,d)4σ 2−σd2When the noise is of multiplicative type we obtain˜H (u(t)) − exp (−2αc(σ)t) ˜H (u 0 )≤ √ ɛIm ∫ ∫ tR d 0exp (−2αc(σ)(t − s)) u(s)∇u(s)∇dW (s)dx+ ɛ ∑ ∫ t2 j∈N 0 exp (−2αc(σ)(t − s)) ∫ R|u(s)| 2 |∇Φe d j | 2 dxds.Again we use a localization argument and consider the process u stoppedat the exit off ˜H 2ρ . As Φ is Hilbert-Schmidt from L 2 into H s R, the secondɛterm of the right hand side is less than4αc(σ) ‖Φ‖2 b(ρ, σ, d) and for ɛ smallL 0,s2enough, it is enough to prove the result for the stochastic integral replacingρ by ρ 2. We know that it is enough to obtain an upper bound of the bracketof∫ tZ(t) = Im exp (2αc(σ)s) u∫R τ (s)∇u τ (s)∇dW (s)dx.d 0We obtaind < Z > t ≤ exp (4αc(σ)t) ∑ j∈N(∇u τ (t), −iu τ (t)∇Φe j ) 2 L 2 dt.⎞⎠ .Denoting by c(s, ∞) the norm of the Sobolev injection of H s Rdeduce thatd < Z > t ≤ exp (4αc(σ)t) c(s, ∞) 2 ‖Φ‖ 2 b(ρ, σ, d) 2 dt.L 0,s2into W1,∞Rwe177

Appendix D.Exit from a domain for weakly damped equationsFinally, we conclude that we may choose⎛⎞1T (L, ρ) =4αc(σ) log ⎝αc(σ)ρ 2⎠10b(ρ, σ, d) 2 c(s, ∞) 2 ‖Φ‖ 2 .LL 0,s2□We may now prove Theorem D.4.6 and D.4.7.Here are some of the specific aspects of the proof of Theorem D.4.6.Proof of Theorem D.4.6. There is no difference in the proof of the upperbound on τ ɛ,u 0. Let us thus focus on the lower bound. Take δ positive.Since e > 0, we now choose ρ positive such that e − δ 4 ≤ e ρ, ˜H2ρ ⊂ D and˜H 2ρ ⊂ D−ρ. c We define the sequences of stopping times θ 0 = 0 and for k inN,τ kθ k+1= inf{t ≥ θ k : u ɛ,u 0(t) ∈ ˜H} τ k : u ɛ,u 0(t) ∈ ˜H}2ρ ,where θ k+1 = ∞ if u ɛ,u 0(τ k ) ∈ ∂D. Let us fix T 1 = T ( e − 3δ4 , ρ) given byLemma D.4.9. We now use that for u 0 in D and m a positive integer,P (τ ɛ,u 0≤ mT 1 ) ≤P (τ ɛ,u 0= τ 0 ) + ∑ mk=1 P (τ ɛ,u 0= τ k )+ ∑ mk=1 P (θ k − τ k−1 ≤ T 1 )(D.4.11)and conclude as in the proof of Theorem D.3.2.□We may also check that the proof of Theorem D.3.3 also applies to proveTheorem D.4.5, the LDPs are those in H 1 and the sequences of stoppingtimes are those defined above.Again the control argument to prove that e = e seems difficult. We mayhowever apply in the H 1 case the Sobolev injection in order to treat thenonlinearity.Let us now make an interesting comment. Assume that we are ableto prove Theorem D.4.4 with e = e at least for an additive noise. Theexit points are then characterized by the infimum of the quasi potential onthe boundary of the domain of attraction. Under assumptions such that Φcommutes with the Laplacian and that Φ does not change the phase, wehave an explicit expression of the quasi potential since the vector-field inthe drift is the sum of a gradient vector-field and a vector-field which isorthogonal to the first one, see for example [68, 73]. These assumptions onΦ are such that we can mimick the computations for the ideal white noise.178

Appendix D.Exit from a domain for weakly damped equations( ) ∥−1 ∥∥∥2The quasipotential is proportional to N H (u) =∥ Φ |KerΦ ⊥ u . Indeed,Lthe rate function of the LDP applied to u, for T positive, may be 2 writtenfor γ in R,I u 0T (u)∫ T( ) −1 (i∥∂u0 ∥(Φ |KerΦ ⊥∂t + iα(1 − γ)u + iαγu − ∆u − λ|u|2σ u )) 2(s)∥ dsL 2∫ T( ) −1 (i∥∂u0 ∥(Φ |KerΦ ⊥∂t + iα(1 − γ)u − ∆u − λ|u|2σ u )) 2(s)∥ ds( ) L 2+ αγ2 [N H(u(T )) − N H (u 0 )] + α 2 γ 2∫2 + (1 − γ)γ T0 N H(u(s))ds.= 1 2= 1 2The last term is equal to zero if and only if γ = 2 or γ = 0. When γ = 2 weobtain{∫1 T( ) −1 (V (0, u f ) = inf ∥2 0 ∥(Φ ∂u|KerΦ ⊥∂t − αu + i∆u + iλ|u|2σ u )) 2(s)∥ dsL 2+αN H (u f ) : u(0) = 0, u(T ) = u f , T > 0}≥ αN H (u f ).In order to prove the converse inequality, we should prove that there exists asequence of functions satisfying the boundary conditions such that the firstterm is arbitrarily small; it is another control problem. Assume that we areable to solve it, then the quasi potential is indeed proportional to the mass.Suppose now that the domain of attraction is a set of the form ˜H

Appendix D.Exit from a domain for weakly damped equationsdynamical behavior of the solutions of the nonlinear equation under theinfluence of a noise. Indeed, it would give an indication that the energyinjected by the noise organizes and creates solitary waves. Note that suchbehavior has been observed numerically in [46] on the Korteweg-de Vriesequation.D.5 Annex - proof of Theorem D.2.1The following lemma proves to be at the core of the proof of the uniformLDPs. It is often called Azencott lemma or Freidlin-Wentzell inequality.The differences with the result of [82] are that here the initial data are thesame for the random process and the skeleton and that the ”for every ρpositive” stands before ”there exists ɛ 0 and γ positive”. We shall only stresson the differences in the proof.Lemma D.5.1 For every a, L, T , δ and ρ positive, f in C a , p in A(d),there exists ɛ 0 and γ positive such that for every ɛ in (0, ɛ 0 ), ‖u 0 ‖ H 1 ≤ ρ,(∥ ∥∥u ɛ,uɛ log P 0− ˜S(u ) 0 , f) ∥ ≥ δ; ‖√ ɛW − f‖X (T,p) C([0,T ];H sR ) < γ ≤ −L.Proof. There are still three steps in the proof of this result. The first stepis a change of measure to center the process around f. It uses the Girsanovtheorem and is the same as in [82].The second step is a reduction to estimates for the stochastic convolution.It strongly involves the Strichartz inequalities but it is slightly different thanin [82]. The truncation argument has to hold for all ‖u 0 ‖ H 1 ≤ ρ. Thus weuse the fact that there exists M = M(T, ρ, σ) positive such that∥sup ∥˜S(u 1 , f) ∥ ≤ M.u 1 ∈Bρ1 X (T,p)The proof of this fact follows from the computations in [37], we will recallthe arguments in L 2 in the proof of Lemma D.3.5. The result in H 1 willagain be used in the proof of Lemma D.4.7. As the initial data are the samefor the random process and the skeleton, the remaining of the argumentdoes not require restrictions on ρ.The third step corresponds to estimates for the stochastic convolution. It isthe same as in [82].Note that the extra damping term in the drift is treated easily thanks tothe Strichartz inequalities.□180

Appendix D.Exit from a domain for weakly damped equationsWe shall now prove Theorem D.2.1.Proof of Theorem D.2.1. Let us start with the case of an additive noise.Recall that, in that case, the mild solution of the stochastic equation couldbe written as a function of the perturbation in the convolution form. Letv u 0(Z) denote the solution of{i∂v∂t − ( ∆v + |v − iZ| 2σ (v − iZ) − iα(v − iZ) ) = 0,v(0) = u 0 ,or equivalently a fixed point of the functional F Z such thatF Z (v)(t) =U(t)u 0 − iλ ∫ t0 U(t − s) ( |(v − iZ)(s)| 2σ (v − iZ)(s) ) ds−α ∫ t0U(t − s)(v − iZ)(s)ds,where Z belongs to C ( [0, T ]; L 2) (respectively C ( [0, T ]; H 1) ). If u ɛ,u 0isdefined as u ɛ,u 0= v u 0(Z ɛ ) − iZ ɛ where Z ɛ is the stochastic convolutionZ ɛ (t) = √ ɛ ∫ t0 U(t − s)dW (s) then uɛ,u 0is a solution of the stochastic equation.Consequently, if G (·, u 0 ) denotes the mapping from C ( [0, T ]; L 2) (respectivelyC ( [0, T ]; H 1) ) to C ( [0, T ]; L 2) (respectively C ( [0, T ]; H 1) ) definedby G (Z, u 0 ) = v u 0(Z) − iZ, we obtain u ɛ,u 0= G (Z ɛ , u 0 ). We may also checkwith arguments similar to that of [37, 81], involving the Strichartz inequalitiesthat the mapping G is equicontinuous in its first arguments for secondarguments in bounded sets of L 2 (respectively H 1 ). The result now followsfrom Proposition 5 in [131].Let us now consider the case of a multiplicative noise. Initial data belongto H 1 and we consider paths in H 1 . The proof is very close to that in [82].The main tool is again the Azencott lemma or almost continuity of theItô map. We need the slightly different result from that in [82].Let us see how the above lemma implies (i) and (ii).We start with the upper bound (i). Take a, ρ, T and δ positive. TakeL > a. For ã in (0, a], we denote byA u 0ã = { v ∈ C ( [0, T ]; H 1) : d C([0,T ];H 1 )(v, Ku 0T (ã)) ≥ δ } .Note that we have A u 0a ⊂ A u 0ãand C ã ⊂ C a . Take ã ∈ (0, a] and f such thatIT W (f) < ã.We shall now apply the Azencott lemma and choose p = 2. We obtainɛ ρ,f,δ and γ ρ,f,δ positive such that for every ɛ ≤ ɛ ρ,f,δ and u 0 such that‖u 0 ‖ H 1 ≤ ρ,(∥ ∥∥u ɛ,uɛ log P 0− ˜S(u 0 , f) ∥ ≥ δ; ∥ √ ɛW − f ∥ )X (T,p) C([0,T ];H sR ) < γ ρ,f,δ ≤ −L.181

Appendix D.Exit from a domain for weakly damped equationsLet us denote by O ρ,f,δ the set O ρ,f,δ = B C([0,T ];H sR ) (f, γ ρ,f,δ). The family(O ρ,f,δ ) f∈Cais a covering by open sets of the compact set C a , thus thereexists a finite sub-covering of the form ⋃ Ni=1 O ρ,f i ,δ. We can now writeP ( u ɛ,u 0∈ A u 0ã) ( {u≤ Pɛ,u 0∈ A u } {0 √ɛW ⋃ã ∩ ∈ Ni=1 O ρ,f i ,δ}))≤≤( √ɛW ⋃+P /∈ N ∑ i=1 O ρ,f i ,δNi=1 P ({ u ɛ,u 0∈ A u } √0ã ∩ { ɛW ∈ Oρ,fi ,δ} )+P ( √ ∑ ɛW({∥ /∈ C a )N ∥∥ui=1 P ɛ,u 0− ˜S(u } 0 , f) ∥ ≥ δ ∩ { √ )ɛW ∈ O ρ,fi ,δ}X (T,p)+ exp ( − a )ɛ ,for ɛ ≤ ɛ 0 for some ɛ 0 positive. We used thatd C([0,T ];H 1 )(˜S(u0 , f), A u 0ã)≥ δ,which is a consequence of the definition of the sets A u 0ã .As a consequence, for ɛ ≤ ɛ 0 ∧ (min i=1,..,N ɛ u0 ,f i) we obtain for u 0 in B 1 ρ,P ( (u ɛ,u 0∈ A u )0ã ≤ N exp − L ) (+ exp − a ),ɛɛand for ɛ 1 small enough, for every ɛ ∈ (0, ɛ 1 ),ɛ log P ( u ɛ,u 0∈ A u )0ã ≤ ɛ log 2 + (ɛ log N − L) ∨ (−a).If ɛ 1 is also chosen such that ɛ 1 Iu 0(w). Chooseɛ ρ,f,δ positive and O ρ,f,δ , the ball centered at f of radius γ ρ,f,δ defined aspreviously, such that for every ɛ ≤ ɛ ρ,f,δ and u 0 such that ‖u 0 ‖ H 1 ≤ ρ,(∥ ∥∥u ɛ,uɛ log P 0− ˜S(u 0 , f) ∥ ≥ δ; ∥ √ ɛW − f ∥ )X (T,p) C([0,T ];H sR ) < γ ρ,f,δ ≤ −L.182

Appendix D.Exit from a domain for weakly damped equationsWe obtain(exp − IW T(f)ɛ)≤ P ( √ ɛW ∈ O ρ,f,δ )({∥ ∥∥u≤ Pɛ,u 0− ˜S(u } 0 , f)≥ δ ∩ { √ )ɛW ∈ OX (T,p) ρ,f,δ }(∥ ∥∥u+Pɛ,u 0− ˜S(u )0 , f) ∥ < δ .X (T,p)Thus, for ɛ ≤ ɛ ρ,f,δ , for every u 0 such that ‖u 0 ‖ H 1 ≤ ρ,( (∥−I u 0 ∥∥u ɛ,u(w) ≤ ɛ log 2 + ɛ log P 0− ˜S(u )) 0 , f) ∥ < δ ∨ (−L)X (T,p)and for ɛ 1 small enough and such that ɛ 1 log(2) < γ, for every ɛ positivesuch that ɛ < ɛ 1 , for every u 0 such that ‖u 0 ‖ H 1 ≤ ρ,(∥−I u 0 ∥∥u ɛ,u(w) − γ ≤ ɛ log P 0− ˜S(u )0 , f) ∥ < δ .X (T,p)It ends the proof of (i) and (ii).□183

Appendix ELarge deviations and supportfor one-dimensional NLSequations with a fractionaladditive noiseAbstract: In this article we consider one-dimensional stochastic nonlinearSchrödinger equations driven by a fractional additive noise. We provethe local well posedness of the Cauchy problem in H 1 , sample path largedeviations and a support result. The latter results are stated in a space ofexploding paths.E.1 IntroductionWe shall consider in this article stochastic nonlinear Schrödinger (NLS)equations, with a Kerr nonlinearity, driven by fractional noises of additivetype. With the Itô notations, the stochastic evolution equation is writtenidu − (∆u + λ|u| 2σ u)dt = dW H , λ = ±1, (E.1.1)u is a complex valued function of time and space. The initial datum u 0 is afunction of H 1 . We consider weak solutions in the sense used in the analysisof partial differential equations or equivalently mild solutions which satisfyu(t) = U(t)u 0 − iλ∫ t0U(t − s)(|u(s)| 2σ u(s))ds − i185∫ t0U(t − s)dW H (s),(E.1.2)

Appendix E.The case of a fractional additive noisewhere (U(t)) t∈Ris the Schrödinger linear group on H 1 generated by theskew adjoint unbounded operator −i∆. Recall that the group is a groupof isometries on the Sobolev spaces based on L 2 . We consider the onedimensionalequation in H 1 . Under such assumptions the nonlinearity isLipschitz on the bounded sets. Otherwise the Strichartz inequalities areneeded to treat the nonlinearity and it is required that the stochastic convolution∫ t0 U(t − s)dW H (s) satisfies certain integrability properties whichseems much more difficult to prove than in the case of the standard Brownianmotion treated in [37]. The well posedness of the Cauchy problem associatedto (E.1.1) in the deterministic case depends on the size of σ. If σ < 2, thenonlinearity is subcritical and the Cauchy problem is globally well posed. Ifσ = 2, critical nonlinearity, or σ > 2, supercritical nonlinearity, the Cauchyproblem is locally well posed in H 1 and solutions may blow up in finite time;see [99]. The local well posedness and results on the the global existencewhen the noise is derived from a Wiener process are proved in [37].Here we have denoted by W H a fractional Wiener process, the fractionalnoise is then the time derivative in the sense of distributions of the continuousin time Hilbert space valued Gaussian process. The parameter H iscalled the Hurst parameter; it belongs to (0, 1). When H ≠ 1 2, the noise iscolored in both time and space. The coloration in space assumption is necessarysince the group does not have regularizing properties in the Sobolevspaces based on L 2 . Note that in optics the time variable corresponds tospace and the space variable to some retarded time. Therefore in such situationswe introduce extra correlations in space. The white in space, thusin time, noise can be approximated considering the limit of a sequence ofcolored noises mimicking the white in space noise in the limit; see [44, 81].Fractional noises have been introduced in hydrology, economics and telecommunications.Though mostly white noises are considered for perturbationsof the NLS equations in the physics literature, fractional noises could possiblyalso be relevant; however it is of interest from a mathematical point ofview.We extend the results of [81] by considering more general driving noises.Note that in [82] we have studied large deviations for a noise of multiplicativetype. In [44] we have applied our results to the problem of error insoliton transmission while in [84] we have studied the problem of exit froma domain of attraction.In this article we prove that the Cauchy problem is locally well posed andwe give a sample path large deviation principle (LDP) in a space of explodingpaths, we also prove a support theorem. Note that the proofs hold forquite general SPDEs with a locally Lipschitz nonlinearity and a fractional186

Appendix E.The case of a fractional additive noiseadditive noise as long as we may prove that the stochastic convolution is acontinuous in time process.In the article we do not investigate global well posedness. This wouldrequire the application of the Itô formula, see [3], for the Hamiltonian andmass to a certain power as in [37]. The integrands of the stochastic integralsare anticipating and integrals remain in a Skohorod sense. It seems to bemuch more complicated than in the standard case. It is also the reason whywe did not consider the case of multiplicative noises. This will be the objectof future investigations.E.2 PreliminariesThe space of Lebesgue square integrable functions L 2 with the inner productdefined by (u, v) L 2 = Re ∫ Ru(x)v(x)dx is a Hilbert space. The Sobolevspaces H s are the Hilbert spaces of functions of L 2 with partial derivativesup to order s in L 2 . When s is fractional it is defined classically via theFourier transform. If I is an interval of R, (E, ‖ · ‖ E ) a Banach space and rbelongs to [1, ∞], then L r (I; E) is the space of strongly Lebesgue measurablefunctions f from I into E such that t → ‖f(t)‖ E is in L r (I). The integralis the Bochner integral. The space of bounded operators from B to B ′ ,two Banach spaces, is denoted by L c (B, B ′ ). The space of Hilbert-Schmidtoperators Φ from E to E ′ , two Hilbert spaces, is denoted by L 2 (E, E ′ ),when endowed with the norm ‖Φ‖ 2 L 2 (E,E ′ ) = trΦΦ∗ = ∑ j∈N ‖Φe j‖ 2 Ewhere′(e j ) j∈Nis a complete orthonormal system of E it is a is a Hilbert space. Wedenote by L 0,s2 the above space when E = L 2 and E ′ = H s .We denote by x∧y and by x∨y respectively the minimum and maximumof x and y. Recall that a rate function I is a lower semicontinuous functionand that it is good if for every c positive, {x : I(x) ≤ c} is a compact set.Let us now recall, for the sake of completeness, the main properties ofthe fractional Brownian motion.A fBm is a centered Gaussian process with stationary incrementsThe covariance is given byE( ∣∣β H (t) − β H (s) ∣ ∣ 2) = |t − s| 2H , t, s > 0.E ( β H (t)β H (s) ) = 1 2(s 2H + t 2H − |s − t| 2H) .The increments are no longer independent for H ≠ 1 2. The covariance offuture and past increments is negative if H < 1 2 and positive if H > 1 2 .187

Appendix E.The case of a fractional additive noiseFBms present long range dependence for H > 1 2as the covariance betweenincrements at a distance u decays as u 2H−2 . These processes are also selfsimilar,i.e. the law of the paths t ↦→ β H (at) where a is positive are thatof t ↦→ a H β H (t). Note that the solution of the NLS equation displays aself similar behavior near blow-up for supercritical nonlinearities but in thespace variable, see [136]. The case where H = 1 2is that of the standardBrownian motion.Enlarging if necessary the probability space (Ω, F, P), the fBm may bedefined in terms of a standard Brownian motion (β(t)) t≥0via a square integrabletriangular kernel K H , i.e. K H (t, s) = 0 if s > t,whereβ H (t) =∫ t0K H (t, s)dβ(s),( ) ∫K H (t, s) = c H (t − s) H− 1 1 t2 + cH2 − H (u − s) H− 3 2andFrom (E.2.1) we may obtain(2HΓ ( 32c H =− H)) 12Γ ( H + 1 ) .2 Γ (2 − 2H)∂K H ( ) 1(t, s) = c H∂t2 − H (t − s) H− 3 2s( ( s 21 −u)1−H) du,(E.2.1)( st) 12 −H . (E.2.2)We shall now denote by (F t ) t≥0the filtration generated by the above fBm.The fBm admits a modification with α−Hölder continuous sample pathswhere α < H; see for example [42]. The paths also have 1 H−finite variation.For H ≠ 1 2, the fBm is neither Markovian nor a semi martingale. As a consequenceof the latter stochastic integration with respect to fBM requires adifferent approach from that of integration with respect to semi-martingales.Several approaches to it exist and we will adopt that of [3] where it is definedas a Skohorod integral. A rough paths approach may also be considered,see for example [33, 113]. FBms are particular Volterra processes, seefor example [41], defined for T positive asX(t) =∫ t0K(t, s)dβ(s), K ∈ L 2 ([0, T ] × [0, T ]) , T > 0, K(t, s) = 0 if s > t.188

Appendix E.The case of a fractional additive noiseThe covariance of such processes isR(t, s) =∫ t∧s0K(t, r)K(s, r)dr,the covariance operator when the process is considered as a L 2 (0, T )−randomvariable is nuclear, i.e. it has finite trace, and is defined through the kernelR(t, s), i.e. for h in L 2 (0, T ), Rh(t) = ∫ T0R(t, s)h(s)ds. The operator R issuch that R = KK ∗ where K is the Hilbert-Schmidt operator that satisfiesfor h ∈ L 2 (0, T ), Kh(t) = ∫ T0 K(t, s)h(s)ds = ∫ t0 K(t, s)h(s)ds and K∗ isits adjoint. These processes admit modifications with continuous samplepaths; they are Gaussian processes. Thus it is classical that the reproducingkernel Hilbert space (RKHS) of the Gaussian measure in L 2 (0, T ), equal tothe range of R 1 2 denoted by Im R 1 2 with the norm of the image structure, isalso equal to Im K; it is also the RKHS of the measure in C([0, T ]) whichis a Banach space continuously embedded in L 2 (0, T ). When we considerdirectly the RKHS of the measure on C([0, T ]) we also know that it is isometricto the closure in L 2 (µ), where µ is the Gaussian measure, of thedual of the Banach space defined by means of the evaluation at points t in[0, T ] and thus to the first Wiener chaos. After such characterizations ofthe RKHS of the fBm it is then standard fact, see for example [8, 53], thatthe rate function of a LDP for the family of Gaussian measures defined asdirect images of µ via the mapping x ↦→ √ ɛx is given by 1 2 ‖ · ‖2 Hand thatthe support of the law of the Gaussian measure is the closure of the RKHSfor the norm of the Banach space. We aim to transport such results to thelaw of the solution of the stochastic NLS equation driven by such noises.In order to define a stochastic integration we consider another RKHSwhich is at the level of the noise and not of the process, we denote it byH. It may be seen as generated by step functions on [0, T ]; the stochasticintegral of a step function 1l [0,T ] should coincide with the evaluation at pointt of the fBm. The set of such step functions is denoted by E. The innerproduct is then defined asR(t, s) = 〈 〉1l [0,t] , 1l [0,s] H = ( )K(t, ·)1l [0,t] , K(s, ·)1l [0,s] L 2 (0,T ) .Also a representation of H may be obtained considering the linear operatorK ∗ T from E into L2 (0, T ) defined for ϕ in E by(K ∗ T ϕ) (s) = ϕ(s)K(T, s) +∫ Ts(ϕ(t) − ϕ(s))K(dt, s).(E.2.3)189

Appendix E.The case of a fractional additive noiseIt is such that for any ϕ in E and h in L 2 (0, T ) we have∫ T0(K ∗ T ϕ) (t)h(t)dt =∫ T0ϕ(t)(Kh)(dt).The space H may then be represented as the closure of E with respect to thenorm ‖ϕ‖ H = ‖KT ∗ ϕ‖ L 2 (0,T ). The operator KT ∗ is then an isometry betweenH and a closed subspace of L 2 (0, T ); we write H = (KT ∗ ( )−1 L 2 (0, T ) ) . Theabove duality relation allows to extend integration with respect to Kh(dt)to integrands in H; it extends that of step functions. It also allows to definea stochastic integration and for integrands ϕ in H, the Skohorod integralsatisfiesδ X (ϕ) =∫ T0(K ∗ T ϕ) (t)δβ(t) =∫ T0(K ∗ T ϕ) (t)dβ(t).From now on we shall restrict our attention to the particular case of thefBm and denote the kernel and the operator by K instead of K H .We shall use several time the following necessary property that we mayeasily check for the fBm thanks to (E.2.1) and (E.2.3) that for 0 < t < T ,(K∗T 1l [0,t] ϕ ) (s) = (K ∗ t ϕ) (s)1l [0,t] (s). (E.2.4)Recall that for the smoother kernels such that H > 1 2, relation (E.2.3)has the simpler and more natural form(K ∗ T ϕ) (s) =∫ Tsϕ(r)K(dr, s).(E.2.5)The formulation in (E.2.3) allows to extend this definition to singular kernels,i.e. when H < 1 2 . For H such that H > 1 2, the inner product in H ofϕ and ψ is given by∫ T0〈ϕ, ψ〉 H== c 2 H∫ T0 ϕ(u)ψ(v) ∫ u∧v ∂K ∂K0 ∂u(u, s)∂v(v, s)dsdudv( )H −1 2 ( ) ∫2 B 2 − 2H, H −1 T2 0∫ T0 ϕ(u)ψ(v)|u − v|2H−2 dudv,from a computation given in [4]; B denotes the Beta function. It correspondsto the covariance of the stochastic integrals with respect to the fBm[∫ T∫ T]E ϕ(u)dβ H (u) ψ(v)dβ H (v) ;00the space H is thus what would be a RKHS at the level of the noise inL 2 (0, T ) which covariance is ∫ u∧v ∂K ∂K0 ∂u(u, s)∂v(v, s)ds.190

Appendix E.The case of a fractional additive noiseIn infinite dimensions, we assume that W H is the direct image by aHilbert-Schmidt operator Φ of a cylindrical fractional Wiener process onL 2 , i.e. W H = ΦWc H .We shall assume thatΦ belongs to L 2 (L 2 , H 1+γ ) with 0 ≤ γ < 1 for H > 1 2and (1 − 2H) < γ < 1 for H < 1 2 .(A)This assumption will be used along with the fact that for any γ ∈ (0, 1) andt a real number‖U(t) − I‖ Lc(H 1+γ ,H 1 ) ≤ 2 1−γ γ2 |t| 2 ; (E.2.6)it could be proved using the Fourier transform.A cylindrical fractional Wiener process on a Hilbert space is such that forevery orthonormal basis (e j ) j∈Nthere exists independent fractional Brownianmotions (fBm)) (β j H(t) such that W c(t) = ∑ j∈Nt≥0 βH j (t)e j. Note thatin what follows it is also possible to assume that the parameter H takesdifferent values H j on each coordinate. The Hilbert-Schmidt assumption isknown to be the exact assumption needed in Hilbert space so that the marginalsW H (t) are well defined Gaussian measures. Stochastic integrationwith respect to fractional Wiener processes in a Hilbert space E ′ , see forexample [137], when integrands are deterministic is defined as above but forstep functions multiplied by elements of the Hilbert space. It is such that ascalar product by an element of E ′ is the one dimensional stochastic integralof the scalar product of the integrand. Operators KT ∗ are still well definedwhen the RKHS H is made of functions with values in E ′ . Integrals of deterministicbounded operator valued integrands Λ from the Hilbert space Eto E ′ are defined for t positive as∫ t0Λ(s)dW H (s) = ∑ j∈N∫ twhen (Λ(t)) t∈[0,T ]is such that∑j∈N0∫ T0Λ(s)Φe j dβ H j (s) = ∑ j∈N∫ t‖ (KT ∗ Λ(·)Φe j ) (t)‖ 2 E ′dt < ∞.0(K ∗ t Λ(·)Φe j ) (s)dβ j (s),Note that the duality relation (E.1.2) still holds, the integral is a Bochnerintegral, and that KT ∗ commutes with the scalar product with an element ofE ′ . We assume now that E = L 2 , E ′ = H 1 and that (e j ) j∈Nis an orthonormalbasis of L 2 .191

Appendix E.The case of a fractional additive noiseWe may also check from (E.1.2) that the linear group (U(t)) t∈Rcommuteswith K ∗ T .Let us now present the space of exploding paths. Indeed, since the nonlinearityis only Lipschitz on the bounded sets of H 1 , solutions may blowup in finite time. We shall proceed as in [81]; see the reference for moredetails. We add a point ∆ to the space H 1 and embed the space with thetopology such that its open sets are the open sets of H 1 and the complementin H 1 ∪{∆} of the closed bounded sets of H 1 . The set C([0, ∞); H 1 ∪{∆}) isthen well defined. We denote the blow-up time of f in C([0, ∞); H 1 ∪ {∆})by T (f) = inf{t ∈ [0, ∞) : f(t) = ∆}, with the convention that inf ∅ = ∞.We may now define the setE(H 1 ) = { f ∈ C([0, ∞); H 1 ∪ {∆}) : f(t 0 ) = ∆ ⇒ ∀t ≥ t 0 , f(t) = ∆ } ,it is endowed with the topology defined by the neighborhood basisV T,r (ϕ 1 ) = { ϕ ∈ E(H 1 ) : T (ϕ) > T, ‖ϕ 1 − ϕ‖ C([0,T ];H 1 ) ≤ r } ,of ϕ 1 in E(H 1 ) given T < T (ϕ 1 ) and r positive. The space is a Hausdorfftopological space and thus we may consider applying the Varadhan contractionprinciple.E.3 Local well posedness and necessary resultsWe shall proceed as in [37] though we do not have to check the integrabilityproperty. Indeed in dimension one we do not need the Strichartz inequalities.Let us first check the following lemma.Lemma E.3.1 The stochastic convolution Z : t ↦→ ∫ t0 U(t − s)dW H (s) iswell defined. Moreover, under assumption (A) it has a modification withα−Hölder continuous sample paths where α < γ 2 ∧ H. If H ≥ 1 2 and γ =0 the modification is only continuous. It defines a C ( [0, ∞); H 1) randomvariable. Moreover, the direct images µ Z,T of its law µ Z by the restrictionon C ( [0, T ]; H 1) for T positive are centered Gaussian measures.Proof. The stochastic convolution is well defined since for t positive∑j∈N∫ t0 ‖(K∗ t U(t − ·)Φe j ) (u)‖ 2 Hdu 1= ∑ ∫ t ∥j∈N 0∥U(−u)Φe j K(t, u) + ∫ tu (U(−r) − U(−u)) Φe jK(dr, u) ∥ 2 duH 1≤ 2‖Φ‖ 2 ∫ tL 0,1 0 K(t, u)2 du + 2 ∑ ∫ tj∈N 0∥ ∫ tu (U(−r) − U(−u)) Φe jK(dr, u) ∥ 2 du2H 1≤ T 1 + T 2192

Appendix E.The case of a fractional additive noiseThe integral in T 1 is equal to E[(β H (t)) 2 ] = t 2H . To obtain an upper boundfor the second term we have to distinguish the cases H < 1 2 and H > 1 2 .When H < 1 2, we apply (E.2.6) and obtain() 2 ∫T 2 ≤ 2 2−γ ‖Φ‖ 2 1t(∫ t(( )L 0,1+γ2 c H H −1(r − u) H− 3 2 + γ r H−1 222 dr)du2 0 uu)thusT 2 ≤ 2 2−γ ‖Φ‖ 2 L 0,1+γ2() 2 ∫1t(∫ t) 2( )c H H −1(r − u) H− 3 2 + γ 2 dr du,2 0 uthe integral is well defined since H − 3 2 + γ 2> −1. We finally obtain2 2−γ ‖Φ‖ 2 (2LT 2 ≤0,1+γ12( ) ( ))2H − 1 + γ c H H −12 H −12 + γ t 2H+γ .2When H > 1 2we do not need to use (A), the kernel is null on the diagonaland its derivative is integrable. We obtainT 2 ≤ 2 4−γ ‖Φ‖ 2 L 0,1+γ2∫ t0K(t, u) 2 du = 2 4−γ ‖Φ‖ 2 t 2H .L 0,1+γ2The fact that µ Z,T are Gaussian measures follows from the fact that Z isdefined as∑∫ t ( )K∗T 1l [0,t] (·)U(t − ·)Φe j (s)dβj (s).j∈N0The law is Gaussian since the law of the action of an element of the dualis a pointwise limit of Gaussian random variables; see for example [81].Note that as we are dealing with a centered Gaussian process, we maycontrol the higher moments controlling the second order moments; see [34]for a proof in the infinite dimensional setting. It is therefore enough to showthat for every positive T there exists positive C and α such that for every(t, s) ∈ [0, T ] 2 .E [ ‖Z(t) − Z(s)‖ 2 H 1 ]≤ C|t − s| α ,and then control higher moments and conclude with the Kolmogorov criterion.When 0 < s < t, we haveZ(t) − Z(s) = U(s) (U(t − s) − I) ∑ ∫ T( )j∈N 0 K∗T1l [0,t] (·)U(−·)Φe j (w)dβj (w)+U(s) ∑ ∫ Tj∈N 0 ((K∗ t U(−·)Φe j ) (w) − (Ks ∗ U(−·)Φe j ) (w)) dβ j (w)= ˜T 1 (t, s) + ˜T 2 (t, s).193

Appendix E.The case of a fractional additive noiseLet us begin with the case where γ > 0. We may write] [ ∥∥∥ ] [ ∥∥∥E[‖Z(t) − Z(s)‖ 2 H 1 ≤ 2E ˜T1 (t, s) ∥ 2 ]+ 2E ˜T2 (t, s) ∥ 2 .H 1 H 1From the above when (A) is such that γ > 0 there exists a constant C(T )such that[ ∥∥∥ ]E ˜T1 (t, s) ∥ 2 H 1≤ ‖U(t − s) − I‖ 2 ∑ ∫ TL c(H 1+γ ,H 1 ) j∈N 0∥ ( KT ∗ 1l ) ∥2 ∥∥2[0,t](·)U(−·)Φe j (w) dwH 1≤ 2 1−γ C(T )‖Φ‖ 2 |t − s| γ .L 0,12Also,[ ∥∥∥ ]E ˜T2 (t, s) ∥ 2 H 1= ∑ j∈N≤ ∑ j∈N∫ T0 ‖U(−u)Φe jK(t, u) + ∫ tu (U(−r) − U(−u)) Φe jK(dr, u)−ϕ(u)K(s, u) − ∫ s( u (U(−r) − U(−u)) Φe jK(dr, u) ‖ 2 Hdu 1˜Tj21 + ˜T j 22 + ˜T)j23 ,where, using the fact that the kernel is triangular,˜T j 21 = ∫ s ∥0∥U(−u) (K(t, u) − K(s, u)) + ∫ ts (U(−r) − U(−u)) Φe jK(dr, u) ∥ 2 du,H˜T j 1 22 = 2 ∫ ts ‖U(−u)Φe jK(t, u)‖ 2 Hdu 1˜T j 23 = 2 ∫ ts∥ ∫ tu (U(−r) − U(−u)) Φe jK(dr, u) ∥ 2 du.H 1We havethusand˜T j 21˜T j 21˜T j 22= ∫ s0∥ ∫ ts U(−r)Φe jK(dr, u) ∥ 2( duH ∫ ) 1t2s |K(dr, u)| du= ‖Φe j ‖ 2 H 1 ∫ s0= ‖Φe j ‖ 2 H 1 ∫ s0 (K(t, u) − K(s, u))2 du≤ ‖Φe j ‖ 2 ∫ tH 1 0 [ (K(t, u) − K(s, u))2 du(β≤ ‖Φe j ‖ 2 HEH (t) − β H (s) ) ] 2 1≤ ‖Φe j ‖ 2 H 1 |t − s| 2H ,= 2 ‖Φe j ‖ 2 H 1 ∫ ts K(t, u)2 du= 2 ‖Φe j ‖ 2 H 1 ∫ ts (K(t, u) − K(s, u))2 du194

Appendix E.The case of a fractional additive noisethus˜T j 22≤ 2 2−γ ‖Φ‖ 2 L 0,1+γ2≤ 2 4−γ ‖Φ‖ 2 L 0,1+γ2≤ 2 ‖Φe j ‖ 2 H 1 ∫ t0 (K(t, u) − K(s, u)2 du≤ 2 ‖Φe j ‖ 2 H 1 |t − s| 2H ,finally the same computations as above shows that when H < 1 2 , sinceH − 3 2 + γ 2> 0 we have( ) 2˜T j ∫ (t ∫ t23s u (r − 3 u)H− 2 + γ (2 r H−1 22u)dr)du1c H(H− 1 2)|t − s| 2H+γ .and when H > 1 2the kernel is null on the diagonal and its derivative isnegative and integrable thus we have˜T j 23≤ 4 ∫ (t ∫ )s ‖Φe j‖ 2 t 2H 1 u |K(dr, u)| du≤ 4 ‖Φe j ‖ 2 ∫ tH 1 s K(t, u)2 du≤ 4 ‖Φe j ‖ 2 HE [ |β H (t) − β H (s)| 2]1≤ 4 ‖Φe j ‖ 2 H|t − s| 2H .1We may conduct similar computations when 0 < t < s < T and we are nowable to conclude that Z admits a modification with α-Hölder continuoussample paths with α < γ 2 ∧ H.Let us now consider the case where H > 1 2and γ = 0. Since the groupis an isometry we have‖Z(t) − Z(s)‖ H 1 ≤∥∥(U(t − s) − I) ∑ j∈N+ ∥ ˜T 2 (t, s) ∥ .H 1∫ T0(K∗T1l [0,t] (·)U(−·)Φe j)(w)dβj (w)∥∥H 1Since the group is strongly continuous and since, from the above,∑∫ T(K∗T 1l [0,t] (·)U(−·)Φe j)(w)dβj (w)j∈N0belongs to H 1 the first term of the right hand side goes to zero as s convergesto t. Also we may write∥ ˜T 2 (t, s) ∥ ≤ ‖Y (t) − Y (s)‖ H 1 H 1where (Y (t)) t∈[0,T ]defined for t ∈ [0, T ] byY (t) = ∑ j∈N∫ T0(K ∗ t U(−·)Φe j ) (w)dβ j (w)195

Appendix E.The case of a fractional additive noiseis a Gaussian process. We again conclude from the Kolmogorov criterionthat Y (t) admits a modification with continuous sample paths. Thus forsuch a modification of Y , Z has continuous sample paths. We shall nowconsider this modification.It is now a standard fact to prove that the process defines a C ( [0, ∞); H 1)random variable, see for example [81].□In the following we consider such a modification. Let us now denote byv u 0(z) the solution of{idvdt = ∆v + λ|v − iz|2σ (v − iz)u(0) = u 0 ∈ H 1 ,where z belongs to C ( [0, ∞); H 1) . The local well posedness is obtained by afixed point argument on C ( [0, T ]; H 1) for a sufficiently small time interval;it uses the fact that the nonlinearity is Lipschitz on the bounded sets of H 1 .We then define G u 0the mappingG u 0: z ↦→ v u 0(z) − iz,it is such that u ɛ,u 0= G u 0( √ ɛZ) where Z is the stochastic convolutiondefined by Z(t) = ∫ t0 U(t − s)dW H (s).We may now check from similar arguments as in [37, 81] the two nextresults.Lemma E.3.2 The mappingis continuous.C ( [0, ∞); H 1) → E ( H 1)z ↦→ G u 0(z)Theorem E.3.3 Assume (A) and that the initial datum u 0 is F 0 measurablewith values in H 1 ; then there exists a unique solution to (E.1.2) withcontinuous H 1 valued paths such that u(0) = u 0 . The solution is defined ona random interval [0, τ ∗ (u 0 , ω)) where τ ∗ (u 0 , ω) is a stopping time such thatτ ∗ (u 0 , ω) = ∞ orlim ‖u(t)‖t→τ ∗ H(u 0 ,ω) 1 = ∞.Furthermore, τ ∗ is almost surely lower semi continuous with respect to u 0 .The solution u defines a E ( H 1) random variable.196

Appendix E.The case of a fractional additive noiseE.4 The main resultsLet us first study large deviations for the laws µ uɛ,u0solutions u ɛ,u 0ofon E(H 1 ) of the mild{ idu − (∆u + λ|u| 2σ u)dt = √ ɛdW H ,u(0) = u 0 ∈ H 1 .(E.4.1)Lemma E.4.1 The covariance operator of Z on L 2 ( 0, T ; L 2) is given for hin L 2 ( 0, T ; L 2) byQh(t) = ∑ ∫ T ∫ t∧uj∈N 0 0(K∗T1l [0,t] (·)U(t − ·)Φe j)(s)((K∗T1l [0,u] (·)U(u − ·)Φe j)(s), h(u))L 2 dsdu,when H > 1 2we may write Qh(t) as(c 2 H H −2) 1 2 (β 2 − 2H, H − 1 ) ∫ T2 0∫ t ∫ s00|u−v| 2H−2 U(t−v)ΦΦ ∗ U(u−s)h(s)dudvds.Also, the RKHS of µ Z,T is Im Q 1 2 with the norm of the image structure. Itis also ImL where L is defined for h in L 2 ( 0, T ; L 2) by∫ t(K∗T 1l [0,t] (·)U(t − ·)Φe j)(s)(h(s), ej ) L 2ds.Lh(t) = ∑ j∈N0Moreover the direct image measures for ɛ positive of x ↦→ √ ɛx on C ( [0, ∞); H 1)satisfy a LDP of speed ɛ and good rate functionI Z (f) = 1 2{}inf ‖h‖ 2h∈L 2 (0,∞;L 2 L):L(h)=f2 (0,∞;L 2 ).Proof. We may first check with the same computations as those used inLemma E.3.1 that L is well defined and that for h in L 2 (0, T ; L 2 ), Lh belongsto L 2 (0, T ; L 2 ); it also holds replacing L 2 by H 1 . Take h and k in L 2 (0, T ; L 2 ),we have[ ∫ TE0 (Z(u), h(u)) L 2 du ∫ ]T0 (Z(t), k(t)) L 2 dt= ∑ [ ∫j∈N E T ∫ (T ∫ T( )0 0 0 K∗T1l [0,u] (·)U(u − ·)Φe j (s)dβj (s), h(u))( ∫L 2T( ) ]0 K∗T1l [0,t] (·)U(t − ·)Φe j (v)dβj (v), k(t))L 2= ∫ T0 (Qh(t), k(t)) L 2 dt 197

Appendix E.The case of a fractional additive noisewhere Q is defined in the lemma. The result for H > 1 2is obtained with theparticular form of the inner product in H for such values of H.Checking that for k in L 2 (0, T ; L 2 ),L ∗ k(s) = ∑ j∈N∫ Ts((K∗T 1l [0,t] (·)U(t − ·)Φe j)(s), k(t))L 2 e j dt,we obtain that Q = LL ∗ .We may thus deduce, see for example [81], that the RKHS of µ Z,T is alsoImL with the norm of the image structure. It is indeed the RKHS of thedirect image of µ Z,T on L 2 (0, T ; L 2 ) but it is standard fact, see for example[34, 81], that the two measures have same RKHS.We may now deduce from the general LDP for Gaussian measures onBanach spaces, see [53], that the direct images of µ Z,T by the mappingx ↦→ √ ɛx satisfy a LDP of speed ɛ and good rate functionI Z,T (f) = 1 2 inf { ‖h‖ 2 ImL : f = Lh }with the convention that inf ∅ = ∞.We conclude letting T go to infinity using Dawson-Gartner’s theorem forprojective limits and Lebesgue’s dominated convergence theorem. □We may now deduce from Lemma E.3.2 and E.4.1, the fact that(G u 0◦ L) (·) = S(u 0 , ·),and the Varadhan contraction principle the following theorem. Recall thatthe Varadhan contraction principle requires that C ( [0, ∞); H 1) and E(H 1 )be Hausdorff topological spaces which is true.Theorem E.4.2 The laws µ uɛ,u0 on E(H 1 ) satisfy a LDP of speed ɛ andgood rate functionI u 0(w) = 1 {}inf ‖h‖ 22 h∈L 2 (0,∞;L 2 L):S(u 0 ,h)=w2 (0,∞;L 2 ),where S(u 0 , h) denotes the mild solution in E(H 1 ) of the following controlproblem⎧⎨ i ∂u∂t − (∆u + λ|u|2σ u) = Φ ˙Kh,u(0) = u⎩0 ∈ H 1h ∈ L 2 ( 0, ∞; L 2) ;it is called the skeleton. Only the integral, or the integral in the mild formulation,of the right hand side is defined; it is by means of the dualityrelation.198

Appendix E.The case of a fractional additive noiseRemark E.4.3 We could also prove a uniform LDP.The characterization of the support follows with the same arguments as in[81]. However we shall recall the proof for the sake of completeness.Theorem E.4.4 The support of the law µ u1,u0supp µ u1,u0 = ImL E(H1 ).on E(H 1 ) is given byProof. Step 1: We have shown that µ Z;(T,p) is a Gaussian measure on aBanach space and that its RKHS is ImL. Consequently, see [8] Theorem(IX,2;1), its support is ImL C(0,T ;H1 ). Also, from the definition of the imagemeasure we have that(µ Z p −1T(ImL C([0,T ];H1 ) )) (= µ Z,T ImL C([0,T ];H1 ) ) = 1,where p T denotes the projection of C ( [0, ∞); H 1) into C ( [0, T ]; H 1) . As aconsequence it follows that(ImL C([0,T ];H1 ) ) = ImL C([0,∞);H1 ) .supp µ Z ⊂ ⋂ Tp −1TIt then suffices to show that ImL ⊂ supp µ Z . Suppose that x /∈ supp µ Z ,then there exists a neighborhood V of x in C ( [0, ∞); H 1) which is a neighborhoodof x in C ( [0, T ]; H 1) for T large such that µ Z (V ) = 0. Since thesupport of µ Z,T is the closure of ImL for the topology of C ( [0, T ]; H 1) ,V ∩ ImL = ∅ and x /∈ ImL.Step 2: We conclude using the continuity of G. ()Since G u 0(ImL) ⊂ G u E(H0 (ImL) 1 ), ImL ⊂ (Gu 0) −1 G u E(H0 (ImL) 1 ).Because G u 0is continuous, the right side is a closed set of C ( [0, ∞); H 1)and from step 1,()supp µ Z ⊂ (G u 0) −1 Im (G u E(H0 ◦ L) 1 ),and( ())µ Z (G u 0) −1 ImS(u 0 ) E(H1 )= 1,thussupp µ u ⊂ ImS(u 0 ) E(H1 ).Suppose that x /∈ supp µ u1,u0 , there exists ) a neighborhood V of x in E(H 1 )such that µ u1,u0 (V ) = µ((G Z u 0) −1 (V ) = 0, consequently (G u 0) −1 (V ) ⋂ ImLis empty and x /∈ ImS(u 0 ). This gives the reverse inclusion.□199

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Grandes déviations pour des équations de Schrödinger nonlinéaires stochastiques et applicationsRésumé: Dans cette thèse nous étudions l’asymptotique de petits bruitspour des perturbations aléatoires d’équations de Schrödinger non linéaires.Les bruits sont Gaussiens, la plupart du temps blancs en temps et toujourscolorés en espace, additifs ou multiplicatifs. Un évènement de grandesdéviations est un évènement où le système diffère significativement du systèmedéterministe. Lorsque le bruit tend vers zéro, la probabilité d’un tel évènementrare tend vers zéro sur une échelle logarithmique avec pour vitesse l’amplitudedu bruit. Nous prouvons des principes de grandes déviations trajectoriels.Dans ce cas le facteur multiplicatif de la vitesse, le taux, est relié à unproblème de contrôle optimal. Les résultats sont appliqués aux temps d’explosion.Nous étudions ensuite l’asymptotique de petits bruits des queuesde la masse et de la position du signal dans une ”limite bruit blanc”. Lesfluctuations de ces quantités sont les causes principales d’erreur de transmissionpar solitons dans les fibres optiques. Nous considérons également leproblème des temps moyens et des points de sortie d’un voisinage de zéropour des équations faiblement amorties. Enfin, nous présentons un principede grandes déviations et un théorème de support pour des bruits Gaussiensfractionnaires additifs.Mots clés: Equation de Schrödinger non linéaire, équations aux dérivéespartielles stochastiques, grandes déviations, ondes solitaires, explosion entemps fini, mouvement Brownien fractionaire, théorèmes de support.Large deviations for stochastic nonlinear Schrödinger equationsand applicationsAbstract: This thesis is dedicated to the study of the small noise asymptoticin random perturbations of nonlinear Schrödinger equations. The noisesare Gaussian, mostly white in time and always colored in space, of additiveand multiplicative types. Large deviations are such that the behaviorof the stochastic system differs significantly from the deterministic one. Asthe noise goes to zero the probability of such rare events goes to zero on alogarithmic scale with speed given by the noise amplitude. We prove largedeviation principles at the level of paths. The rate of convergence to zero ofthe logarithm of the probabilities is related to an optimal control problem.Our first application is to the blow-up times. We then apply our results tothe study of the small noise asymptotic of the tails of the mass and positionof the soliton-like pulse in a ”white noise limit”. The fluctuations of thesequantities are the main causes of error in optical soliton transmission. Wealso consider the problem of the mean exit times and the exit points from aneighborhood of zero for weakly damped equations. Finally we present largedeviations and a support theorem for fractional additive Gaussian noises.Keywords: Nonlinear Schrödinger equation, large deviations, stochasticpartial differential equations, solitary waves, blow-up, fractional Brownianmotion, support theorems.2000 Mathematics Subject Classification: 35Q55, 35Q51,60H15,60F10, 78A60.