3 Partially Reflected Brownian Motion - Laboratoire de Physique de ...

Partially Reflected Brownian Motion:A Stochastic Approach to TransportPhenomena ∗arXiv:math.PR/0610080 v1 2 Oct 2006Denis S. Grebenkov †Laboratoire de Physique de la Matière Condensée,C.N.R.S. – Ecole Polytechnique 91128 Palaiseau Cedex, FranceReceived: August 2004; Published: September 2006AbstractTransport phenomena are ubiquitous in nature and known to be importantfor various scientific domains. Examples can be found in physics, electrochemistry,heterogeneous catalysis, physiology, etc. To obtain new informationabout diffusive or Laplacian transport towards a semi-permeable orresistive interface, one can study the random trajectories of diffusing particlesmodeled, in a first approximation, by the partially reflected Brownianmotion. This stochastic process turns out to be a convenient mathematicalfoundation for discrete, semi-continuous and continuous theoretical descriptionsof diffusive transport.This paper presents an overview of these topics with a special emphasison the close relation between stochastic processes with partial reflectionsand Laplacian transport phenomena. We give selected examples of thesephenomena followed by a brief introduction to the partially reflected Brownianmotion and related probabilistic topics (e.g., local time process andspread harmonic measure). A particular attention is paid to the use of theDirichlet-to-Neumann operator. Some practical consequences and furtherperspectives are discussed.Keywords: Diffusion with Reflections; Mixed Boundary Value Problems; LaplacianTransport Phenomena.∗ This article partially reproduces the chapter which has been written by the author for thevolume “Focus on Probability Theory”, and it should be referenced as D. S. Grebenkov, in Focuson Probability Theory, Ed. L. R. Velle, pp. 135-169 (Nova Science Publishers, 2006). Thebibligraphic reference (ISBN) is 1-59454-474-3. Further information about this volume can befound on https://www.novapublishers.com/catalog/† E-mail address: denis.grebenkov@polytechnique.edu1

Partially Reflected Brownian Motion... 3of potential theory, variational analysis and probability theory. The second sectionis devoted to remind some basic definitions of stochastic process theory: stoppingtimes, reflected Brownian motion, local time process, harmonic measure,etc. In the third section, we introduce the partially reflected Brownian motion andshow its properties for a planar surface. An important relation to the Dirichletto-Neumannoperator is revealed and then illustrated by several examples. Thelast section presents different stochastic descriptions of Laplacian transport phenomena:a recently developed continuous approach and two other methods. In theconclusion, we summarize the essential issues of the paper.1 Laplacian Transport PhenomenaThe transport of species between two distinct “regions” separated by an interfaceoccurs in various biological systems: water and minerals are pumped by rootsfrom the earth, ions and biological species penetrate through cellular membranes,oxygen molecules diffuse towards and pass through alveolar ducts, and so on.Transport processes are relevant for many other scientific domains, for example,heterogeneous catalysis and electrochemistry. In this section, we shall give three 1important examples of the particular transport process, called Laplacian or diffusivetransport.1.1 Stationary Diffusion across Semi-permeable MembranesLet us begin by considering the respiration process of mammals. Inbreathing afresh air, one makes it flow from the mouth to the dichotomic bronchial tree of thelungs (Fig. 1). For humans, first fifteen generations of this tree serve for convectionaltransport of the air towards pulmonary acini, terminal gas exchange units[1]. A gradual increase of the total cross-section of bronchiae leads to a decreaseof air velocity. At the entrance of the acinus, it becomes lower than the characteristicdiffusion velocity [2]. As a consequence, one can describe the gas exchangeinside the acinus as stationary diffusion of oxygen molecules in air from the entrance(“source” with constant concentration C 0 during one cycle of respiration)to the alveolar membranes [3]. In the bulk, the flux density is proportional to thegradient of concentration (Fick’s law), J = −D∇C, where D is the diffusioncoefficient. The mass conservation law, written locally as div J = 0, leads to theLaplace equation ∆C = 0 in the bulk. The flux density towards the interface issimply J n = D ∂C/∂n, where the normal derivative ∂/∂n is directed to the bulk.1 Diffusive NMR phenomena present another important example when the transport propertiesare considerably affected by irregular geometry. In this paper, we do not discuss this case since ithas been recently reviewed in a separate paper [4].

Partially Reflected Brownian Motion... 4Arrived to the alveolar membrane, oxygen molecules can penetrate across theboundary for further absorption in blood, or to be “bounced” on it and to continuethe motion. The “proportion” of absorbed and reflected molecules can be characterizedby permeability W varying from 0 (perfectly reflecting boundary) to infinity(perfectly absorbing boundary). In this description, the flux density acrossthe alveolar membrane is proportional to the concentration, J n = WC. Equatingthese two densities on the alveolar membrane, one gets a mixed boundarycondition, D(∂C/∂n) = WC, called also Fourier or Robin boundary condition.Resuming these relations, one provides the following mathematical descriptionfor the diffusion regime of human or, in general, mammalian respiration:∆C = 0 in the bulk (1)C = C 0 on the source (2)[I − Λ ∂ ]C = 0 on the alveolar membrane (3)∂nwhere the underlying physics and physiology are characterized by a single parameterΛ = D/W , which is homogeneous to a length (I stands for the identityoperator). Note also that the dependence on constant C 0 is irrelevant. In whatfollows, we address to this “classical” boundary value problem. The essentialcomplication resides in a very irregular geometry of the pulmonary acinus, whichpresents a branched structure of eight generations (for humans), “sticked” by alveolarducts (Fig. 1). For small Λ, only a minor part of the boundary is involved tothe transport process (so-called Dirichlet active zone), whereas the flux acrossthe rest of the boundary is almost zero (this effect is called diffusional screening[3, 5, 6, 7, 8, 9]). With an increase of Λ, larger and larger part of the boundary becomesactive. As a result, the efficiency of human lungs depends on the parameterΛ in a nontrivial manner that implies different physiological consequences [10].The trajectory of a chosen oxygen molecule can be seen as Brownian motion fromthe source towards the alveolar membrane, with multiple bounces on the boundaryand final absorption. This is in fact what we call the partially reflected Brownianmotion (Section 3). A profound study of the interplay between the irregular geometryof the acinus and the erratic random motion of oxygen molecules inside itshould help to better understand physiological functioning of human lungs.1.2 Heterogeneous CatalysisA similar description can be brought to the molecular regime of heterogeneouscatalysis omnipresent in petrochemistry. One considers reactive molecules A injectedinto a solvent and then diffusing towards a catalyst. Hitting the catalyticsurface, they can be transformed into other molecules A ∗ (with a finite reaction

Partially Reflected Brownian Motion... 5Figure 1: On the left, a cast of human lungs; on the right, a cast of pulmonaryacinus [1, 11] (by E. Weibel).rate K), or to be bounced for further diffusion in the bulk. The new moleculesA ∗ , collected by appropriate physical or chemical technique, do not further contributeto the transport process. Assuming the presence of a remote source ofreactive molecules A, one can model, in a first approximation 2 , the heterogeneouscatalysis by the mixed boundary value problem (1–3) with a characteristic lengthΛ = D/K [15, 16, 17, 18]. The keynote of this similitude is related to the factthat each reactive molecule arrived onto the boundary terminates its motion aftera number of successive reflections. The mechanism leading to its terminationis different: for the oxygen diffusion, the molecules are absorbed by the alveolarmembrane and transferred to the blood, while for the heterogeneous catalysis,the reactive molecules are transformed by chemical reaction into other moleculeswhich do not further participate to the process. Since the overall production ofnew molecules A ∗ depends on the total surface area of the catalytic surface, onetries to design catalysts with the largest possible surface (for given volume), realizingporous and very irregular boundaries (Fig. 2). As a consequence, the diffusionalscreening becomes important to understand numerous industrial processesin petrochemistry. Since random trajectories of reactive molecules correspond tothe partially reflected Brownian motion, its study may allow a design of more2 This description is probably too simplified in order to model the heterogeneous catalysis quantitatively.First, the presence of molecules A ∗ near the catalyst may “obstruct” the access to thecatalytic surface. Second, parasite reactions happen on the boundary that implies a progressivedeactivation of the catalyst. Consequently, the reactivity K becomes dependent on the spatialposition on the catalytic surface, leading to an inhomogeneous boundary condition. Finally, themolecular diffusion can be applied only if the mean free path of reactive molecules is much lowerthan the geometrical features of the catalyst (in the opposite case, one deals with Knudsen diffusion[12, 13, 14, 17]). Nevertheless, the simple description (1–3) permits to take into accountmany important features related to the catalytic process.

Partially Reflected Brownian Motion... 6Figure 2: On the left, an example of an irregular catalytic surface (by J. S. Andradejr.); at the center, photo of a rough metallic surface of nickel electrode (byE. Chassaing); on the right, photo of an irregular metallic electrode used to studythe Laplacian transport phenomena experimentally (by B. Sapoval).efficient catalysts.1.3 Electric Transport in ElectrochemistryThe other example of Laplacian transport phenomena can be found in electrochemistry:the electric current between two metallic electrodes into an electrolyteis described by the same boundary value problem. Indeed, the electric potential Vobeys the Laplace equation in the bulk since the electrolyte is locally neutral. Takingone electrode of very low resistance (counter-electrode), one writes the correspondingboundary condition as V = V 0 , where V 0 is the applied tension. For theother electrode of surface resistance r (working electrode), one obtains the mixedboundary condition by equating the volume current density −ρ −1 ∇V (ρ is theelectrolyte resistivity) and the surface current density V/r: Λ ∂V/∂n = V , whereΛ = r/ρ is again the physical length of the problem. The similar description canbe brought even in the case of an alternative tension [19, 20].For electric transport, one cannot associate directly the mixed boundary valueproblem with the partially reflected Brownian motion since there is no diffusingparticle. From this point of view, the electrochemical problem has only aformal analogy with two previous examples. At the same time, the electrochemistryis an appropriate domain to study experimentally the influence of the irregulargeometry on the (average) transport properties. Taking metallic electrodesof different shapes with micro- or macro-roughness (e.g., see Fig. 2), onecan directly measure the spectroscopic impedance or admittance (see below).These characteristics are equivalent to the total flux across the boundary for diffusionalproblems [21, 22]. The observation of anomalous impedance behavior[23] had provoked numerous theoretical, numerical and experimental stud-

Partially Reflected Brownian Motion... 7ies of the role of a geometrical irregularity in Laplacian transport phenomena[24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38] (for more information,see [39] and references therein).1.4 Discrete and Semi-continuous ApproachesAmong different theoretical approaches developed to study Laplacian transportphenomena, we have to mention the double layer theory of Halsey and Leibig[26, 27, 28, 29] and the formalism of the Brownian self-transport operator proposedby Filoche and Sapoval [21]. In Section 4.3, we shall show how the originalGreen function description by Halsey and Leibig can be related to the Brownianmotion reflected with jump (this stochastic reformulation will be referred toas “semi-continuous” approach). In turn, Filoche and Sapoval considered latticerandom walks with partial reflections to derive a spectral representation for themacroscopic response of an irregular interface (see Section 4.4, where this formalismis referred to as “discrete” approach). Although both methods accuratelydescribe Laplacian transport phenomena (e.g., they give an explicit formula forthe total flux across the boundary), their major inconvenience resides in the dependenceon an artificial length scale: jump distance a for the semi-continuousapproach and lattice parameter a for the discrete approach. A physical intuitionsuggests that, if these descriptions are correct, there should exist a well definedcontinuous limit as a tends to 0. Certain substantial arguments to justify the existenceof this limit were brought in [39] (and they will be strengthened in thispaper), but a rigorous mathematical proof is still required. To overcome this difficulty,a new theoretical approach has been recently developed in [40]. We shallcall it “continuous” since it is tightly related to a continuous stochastic process,namely, the partially reflected Brownian motion. This approach will integratethe advantages of the previous ones, being a mathematical foundation for understandingLaplacian transport phenomena. We shall return to these questions inSection 4.2 Basic DefinitionsIn this section, we recall the basic definitions related to the Brownian motion andreflected Brownian motion that can be found in extensive literature, e.g., [41, 42,43, 44, 45, 46, 47]. The familiar reader may pass over this section.

Partially Reflected Brownian Motion... 82.1 Brownian Motion and Dirichlet Boundary Value ProblemThe Brownian motion can be defined in different ways [41]. Throughout thispaper, we use the following definition.Definition 2.1 A stochastic process W t (t ≥ 0) defined on the chosen probabilisticspace is called one-dimensional Brownian motion (or Wiener process) startedfrom the origin, if• its trajectories are continuous almost surely (with probability 1);• it starts from the origin almost surely, P{W 0 = 0} = 1;• its joint distribution isP{W t1 ∈ Γ 1 , ..., W tn ∈ Γ n } =∫ ∫dx 1 ... dx n g(0, x 1 ; t 1 ) g(x 1 , x 2 ; t 2 − t 1 ) ... g(x n−1 , x n ; t n − t n−1 )Γ 1 Γ nfor any integer n, any real numbers 0 < t 1 < ... < t n and arbitrary intervalsΓ 1 , ..., Γ n , where g(x, x ′ ; t) is the Gaussian densityg(x, x ′ ; t) = √ 1 exp[− (x − ]x′ ) 2x, x ′ ∈ R, t ∈ R + (4)2πt 2tBy definition, g(0, x ; t)dx is the probability to find the Brownian motion indx vicinity of point x at time t:P{W t ∈ (x, x + dx)} = g(0, x ; t)dxThe collection W t = (Wt 1,..., W t d ) of d independent one-dimensional Brownianmotions Wt k is called d-dimensional Brownian motion started from the origin(in the following, we shall omit the pointing on the dimension). The translatedstochastic process, x + W t , is called Brownian motion started from the pointx ∈ R d .Various properties of the Brownian motion and its relation to other mathematicalfields (like partially differential equations or potential theory) are well knownand can be found in [41, 42, 43, 44, 45, 46, 47].As one can see, the Brownian motion W t is defined for the whole space R d ,without any binding to a particular domain. However, physical processes are usuallyconfined into a certain domain Ω ⊂ R d . The “presence” of its boundary∂Ω can be introduced by a specific condition for a quantity we are looking for.To illustrate this notion, let us introduce the harmonic measure ω x defined as theprobability measure to hit different subsets of the boundary ∂Ω for the first time.

Partially Reflected Brownian Motion... 9Definition 2.2 Let Ω ⊂ R d be a domain with boundary ∂Ω. For any x ∈ R d , arandom variable T x = inf{t > 0 : (x + W t ) ∈ ∂Ω} is called stopping timeon the boundary ∂Ω (it gives the first moment when the Brownian motion startedfrom x hits the boundary). For any subset A from the Borel σ-algebra B(∂Ω), onedefines its harmonic measure ω x {A} (hitting probability) as:ω x {A} = P{W T x ∈ A, T x < ∞}(we remind that the Borel σ-algebra B(∂Ω) is generated by all open subsets of∂Ω).We gave this classical definition of the harmonic measure in order to outlinethat the boundary ∂Ω is present in the problem only through the stopping time T x .In other words, its introduction does not change the definition of the Brownianmotion itself. This feature considerably simplifies the following analysis.Up to this moment, we did not specify the domain Ω and its boundary ∂Ω,since the harmonic measure can be well defined for very irregular domains [48,49, 50]. However, the following definitions will need some restrictions on theboundary. Throughout this paper, we shall consider a domain Ω ⊂ R d (d ≥ 2)with bounded smooth boundary ∂Ω (twice continuous differentiable manifold).One the one hand, this condition can be weakened in different ways, but it wouldrequire more sophisticated analysis overflowing the frames of this paper (e.g., see[51, 52]). One the other hand, our primary aim is to describe Laplacian transportphenomena listened in Section 1. Dealing with physical problems, one can alwayssmooth a given boundary ∂Ω whatever its original irregularity. Indeed, thephysics naturally provides a minimal cut-off δ (e.g., mean free path of diffusingor reacting molecules) which determines the “admissible” scales of the boundary.All geometrical features of the boundary smaller than δ should be irrelevant(otherwise, the proposed physical description would be incorrect). Smoothingthese geometrical elements, one can obtain a boundary that may be (very) irregularon length scales larger than δ, but smooth on length scales lower than δ. Fora smooth boundary ∂Ω, one can introduce the harmonic measure density ω x (s)such that ω x (s)ds is the probability that the Brownian motion started at x hits theboundary in ds vicinity of the boundary point s.The harmonic measure, generated by the Brownian motion, gives a generalsolution of the Dirichlet boundary value problem with a given function f on ∂Ω:∆u = 0 (x ∈ Ω), u = f (x ∈ ∂Ω) (5)Indeed, the harmonic measure density is equal to the normal derivative of theGreen function for the Dirichlet problem, so that one writes the solution u(x)

Partially Reflected Brownian Motion... 10explicitly [53]:or as following expectation∫u(x) =∂Ωf(s) ω x (s)dsu(x) = E { f(W T x) } (6)One can give a physical interpretation to this mathematical relation. In order tocalculate the expectation, one considers all possible trajectories of the Brownianmotion started from the point x ∈ Ω. For each trajectory terminated at boundarypoints s = W T x, one assigns the weight f(s) and then averages over all thesetrajectories. Giving this interpretation, we do not discuss the mathematical realizationof such average over all possible trajectories. To do this operation properly,one can introduce the Wiener measure on the space of continuous functions andthen define the corresponding functional integrals [41]. It is interesting to remarkthat this reasoning traced to the Feynman’s description of quantum mechanics bypath integrals [54]. Note also that the relation (6) is the mathematical foundationto Monte Carlo numerical tools for solving the Dirichlet problem (5). In fact,launching a large number of random walkers from the point x, one determines,for each trajectory, its hitting point s and assigns the corresponding weight f(s).The average over all random walkers gives an approximate value of the solutionu(x) at point x.One can see that the Brownian motion is an efficient mathematical tool tostudy Dirichlet boundary value problems. However, it becomes useless for othertypes of boundary conditions like, e.g., the Neumann condition. The simple physicalreason is the following. As we have mentioned above, the Dirichlet boundarycondition is introduced through the stopping time T x . It means that we are interestedonly in the Brownian motion W t for times t between 0 and T x . Since themotion with t > T x is irrelevant for this problem, one may think that the Brownianmotion is absorbed on the boundary ∂Ω at the first hit. In other words, the Dirichletcondition corresponds to a purely absorbing interface ∂Ω. For the Neumanncondition, the situation changes drastically. The normal derivative representinga flux leads to the notion of reflection on the boundary: if one would like to fixthe flux density across the boundary, certain particles should be reflected. Theprobabilistic description of the Neumann boundary condition necessitates thus anintroduction of the other stochastic process called reflected Brownian motion.2.2 Reflected Brownian MotionThe fact of reflection on the boundary implies three essential distinctions withrespect to the (simple) Brownian motion:

Partially Reflected Brownian Motion... 11• the definition of the reflected Brownian motion will depend on the domain Ω(as a consequence, it will be necessarily more sophisticated than the abovedefinition of the Brownian motion);• the type and direction of each reflection should be prescribed (e.g., right oroblique);• some restrictions on the boundary ∂Ω should be introduced, for instance,the normal vector should be well defined at each point (as a consequence,the boundary cannot be very irregular).It is not thus surprising that the definition of the reflected Brownian motion requiresstochastic differential equations. We do not intend to reproduce the wholeanalysis leading to the reflected Brownian motion since one can find it in correspondingliterature (e.g., see [41, 55, 56, 57]). In the case of smooth boundaries,the following definition is quite classical. The situation becomes essentially moredifficult when one tries to extend it for nonsmooth domains.Definition 2.3 Let Ω ⊂ R d be a domain with boundary ∂Ω, and n(s) is a vectorvaluedfunction on ∂Ω. For a given point x ∈ Ω, one considers the stochasticequation in the following form:dŴt = dW t + n(Ŵt)I ∂Ω (Ŵt)dl t Ŵ 0 = x, l 0 = 0 (7)where W t is d-dimensional Brownian motion and I ∂Ω is the indicator of the boundary∂Ω. By a solution of this equation, we mean a pair of almost surely continuousprocesses Ŵt and l t , satisfying (7), adapted to the underlying family of σ-fieldsand satisfying, with probability 1, the following conditions:• Ŵt belongs to Ω ∪ ∂Ω;• l t is a nondecreasing process which increases only for t ∈ T , T = {t >0 : Ŵ t ∈ ∂Ω} having Lebesgue measure zero almost surely.The process Ŵt is called Brownian motion normally reflected on the boundary (orreflected Brownian motion), the process l t is called local time on the boundary(or local time process).The following theorem ensures the existence and uniqueness of these stochasticprocesses in the case of smooth boundaries.Theorem 2.4 Let Ω ⊂ R d be a bounded domain with twice continuous differentiableboundary ∂Ω, n(s) is the vector of the inward unit normal at boundarypoint s (orthogonal to the boundary at s and oriented towards the domain). For

Partially Reflected Brownian Motion... 12a given point x ∈ Ω ∪ ∂Ω, the stochastic equation (7) possesses a unique solution,i.e., there exist the reflected Brownian motion Ŵt and the local time on theboundary l t satisfying the above conditions, and they are unique.Proof can be found in [41, 55].We should note that this theorem can be extended in different ways. For example,one can consider the Brownian motion, reflected on the boundary in thedirection given by another vector-valued field than the field n(s) of the inwardunit normals. The assumption that the domain is bounded can be replaced by amore subtle hypothesis that allows to extend the definition of the reflected Brownianmotion for some classes of unbounded domains. At last, one may definethis motion for a general case of second order elliptic differential operators (withcertain restrictions on their coefficients). The interested reader may consult thecorresponding literature, e.g., [41, 45].Although the rigorous mathematical definition of stochastic differential equationsis more difficult than in the case of ordinary differential equations, an intuitivemeaning of its elements remains qualitatively the same. For example, thestochastic equation (7) states that an infinitesimal variation dŴt of the reflectedBrownian motion Ŵt in the domain Ω (bulk) is governed only by the variationdW t of the (simple) Brownian motion W t (the second term vanishes due to the indicatorI ∂Ω ). When the motion hits the boundary, the second term does not allowto leave the domain leading to a variation directed along the inward unit normaln(s) towards the interior of the domain. On the other hand, each hit of the boundaryincreases the local time l t . Consequently, the single stochastic equation (7)defines simultaneously two random processes, Ŵ t and l t , strongly dependent eachof other.As an example, one can consider one-dimensional Brownian motion reflectedat zero which can be written as mirror reflection of the (simple) Brownian motion:Ŵ t = |x + W t |. Applying Itô’s formula to this function, one obtains:Ŵ t = |x + W t | = x +∫ t∫ tsign(x + W t ′)dW t ′ + 1 δ(x + W t ′)dt ′200One can show that the second term is equivalent to a Brownian motion W t, ′ whereasthe third term, denoted as l t , is a continuous, nondecreasing random process whichincreases only on the set T = {t > 0 : x + W t = 0} of the Lebesgue measurezero. The previous expression can thus be written as Ŵt = x + W t ′ + l t or, indifferential form, as dŴt = dW t ′ + dl t which is the particular case of the stochasticequation (7). For the local time l t , Lévy proved the following representation

Partially Reflected Brownian Motion... 13[43, 44]:∫1tl t = lim I [0,a] (Ŵt a→0 2a′)dt′ (8)0This relation makes explicit the meaning of the local time l t : it shows how “manytimes” the reflected Brownian motion passed in an infinitesimal vicinity of zeroup to the moment t. Lévy also gave another useful representation for the localtime:l t = lim aN t (a) (9)a→0where N t (a) is the number of passages of the reflected Brownian motion throughthe interval [0, a] up to the moment t. If one introduces a sequence of stoppingtimes at points 0 and a,τ (0)nτ (0)0 = inf{t > 0 : Ŵ t = 0} τ (a)0 = inf{t > τ (a)0 : Ŵ t = a}= inf{t > τ (a)n−1 : Ŵ t = 0}the number of passages can be defined asN t (a) = sup{n > 0 : τ (0)n < t}τ (a)n = inf{t > τ (0)n−1 : Ŵ t = a}Note that the representations (8) and (9) can be extended for a general case ofd-dimensional reflected Brownian motion.3 Partially Reflected Brownian Motion3.1 Definition and Certain PropertiesBearing in mind the description of Laplacian transport phenomena, we would liketo extend the concept of the reflected Brownian motion in order to deal with themixed boundary condition (3).Definition 3.1 For a given domain Ω ⊂ R d with smooth bounded boundary ∂Ω,let Ŵ t be the reflected Brownian motion started from x ∈ Ω ∪ ∂Ω, and l t be therelated local time process. Let χ be a random variable, independent of Ŵt and l tand distributed according to the exponential law with a positive parameter Λ:The stopping timeP{χ ≥ λ} = exp[−λ/Λ] (λ ≥ 0) (10)T x Λ = inf{t > 0 : l t ≥ χ}gives the first moment when the local time process l t exceeds the random variableχ. The process Ŵt conditioned to stop at random moment t = T x Λ is calledpartially reflected Brownian motion (PRBM).

Partially Reflected Brownian Motion... 14First of all, we stress that the partially reflected Brownian motion is not a newstochastic process: it reproduces completely the reflected Brownian motion Ŵt upto the moment T x Λ . The only difference between them resides in the fact that weare not interested in what happens after this moment. Consequently, the conditionto stop at t = T x Λ may be thought as an absorption on the boundary ∂Ω. It explainsthe term “partially reflected”: after multiple reflections, the process will beabsorbed on the boundary (see Section 4 for further comments). Roughly speaking,the whole term “partially reflected Brownian motion” is a shorter version ofthe phrase “reflected Brownian motion conditioned to stop at random momentT x Λ ”.In the particular case Λ = 0, the exponential distribution (10) is degenerated:P{χ = 0} = 1 and P{χ > 0} = 0. Consequently, the stopping time becomes:T x 0 = inf{t > 0 : l t > 0}. Since the first moment of an increase of the localtime process l t corresponds to the first hit of the boundary ∂Ω, one obtains thestopping time of the (simple) Brownian motion: T x 0 = Tx . One concludes that,for Λ = 0, the partially reflected Brownian motion becomes the Brownian motionconditioned to stop at the first hit of the boundary.To study the partially reflected Brownian motion, one can introduce a measurequantifying absorptions on different subsets of the boundary ∂Ω.Definition 3.2 For any subset A from the Borel σ-algebra B(∂Ω), one defines itsspread harmonic measure ω x,Λ {A} as:ω x,Λ {A} = P{ŴT x Λ ∈ A, Tx Λ < ∞}As the harmonic measure itself, ω x,Λ {A} satisfies the properties of a probabilisticmeasure, in particular, ω x,Λ {∂Ω} = 1. When Λ goes to 0, the spreadharmonic measure tends to the harmonic measure: ω x,Λ {A} → ω x {A}. Sincethe present definition of the PRBM requires the smoothness of the boundary, thespread harmonic measure can be characterized by its density ω x,Λ (s).Dealing with the Brownian motion, one could formally take the starting pointx on the boundary ∂Ω, but it would lead to trivial results: the stopping time T xbecomes 0 and the harmonic measure ω x is degenerated to the Dirac point measure:ω x {A} = I A (x) (if x ∈ ∂Ω). In the case of the partially reflected Brownianmotion, the starting point x can belong to the domain Ω or to its boundary ∂Ω: inboth cases the spread harmonic measure has nontrivial properties.It is convenient to separate each random trajectory of the PRBM in two parts,before and after the first hit of the boundary. The first part, Ŵ 0≤t≤T x, coincideswith the (simple) Brownian motion started from x and conditioned to stop on theboundary, while the second part, Ŵ T x ≤t≤T x , coincides with the reflected Brownianmotion started on the boundary (at the first hitting point) andΛconditioned

Partially Reflected Brownian Motion... 15to stop on the same boundary at random moment T x Λ . Since these two parts areindependent, one can write the spread harmonic measure density as∫ω x,Λ (s) = ds ′ ω x (s ′ ) T Λ (s ′ , s) T Λ (s ′ , s) ≡ ω s ′ ,Λ(s) (11)∂ΩThe integral kernel T Λ (s ′ , s) represents the probability density that the PRBMstarted from the boundary point s ′ is stopped (absorbed) in an infinitesimal vicinityof the boundary point s. Consequently, it is sufficient to determine the probabilitiesof displacements between two boundary points in order to reconstruct thewhole spread harmonic measure density.Lemma 3.3 For any subset A from B(∂Ω) and fixed positive Λ, the spread harmonicmeasure ω x,Λ {A}, considered as a function of x, solves the mixed boundaryvalue problem:[∆ω x,Λ {A} = 0 (x ∈ Ω), I − Λ ∂ ]ω x,Λ {A} = I A (x) (x ∈ ∂Ω)∂n(12)This lemma generalizes the Kakutani theorem for the harmonic measure (whenΛ = 0) [58]. We do not reproduce the proof of this lemma since it would requiremany technical details. It can be also reformulated for the spread harmonic measuredensity:Lemma 3.4 For any boundary point s ∈ ∂Ω and fixed positive Λ, the spreadharmonic measure density ω x,Λ (s), considered as a function of x, satisfies thefollowing conditions:[∆ω x,Λ (s) = 0 (x ∈ Ω), I − Λ ∂ ]ω x,Λ (s) = δ(s − x) (x ∈ ∂Ω)∂n(13)where δ(s − x) is the Dirac function (distribution) on the boundary.According to this lemma, the solution of a general mixed boundary value problem[∆u = 0 (x ∈ Ω), I − Λ ∂ ]u = f (x ∈ ∂Ω)∂nwith a given function f on ∂Ω and fixed positive Λ can be written in two equivalentforms:∫u(x) = f(s) ω x,Λ (s)ds = E { f(ŴT x)}Λ∂Ω

Partially Reflected Brownian Motion... 16Again, one can give a physical interpretation of this relation: one averages thefunction f over all possible trajectories of the partially reflected Brownian motionstarted from the point x. Each trajectory is weighted by f(s) according to theboundary point s of its final absorption.3.2 Planar SurfaceWe remind that the physical motivation of this work is a possibility to describe diffusingparticles near semi-permeable interfaces by the partially reflected Brownianmotion. Indeed, the mixed boundary value problem (1–3) is an averaged descriptionfor the concentration of particles, while the stochastic description permits to“follow” the trajectory of one individual particle. This analysis may provide anew information: typical or average distance between the first hitting point andthe final absorption point; proportion of “flatten” trajectories, going near the interface,with respect to remote trajectories, moving away from the interface and thenreturning to it, etc. In this subsection, we briefly consider the particular case of theplanar surface (boundary of a half space), when the partially reflected Brownianmotion can be constructed in a simple way, without stochastic equations. Consequently,many related characteristics can be obtained explicitly. In addition, thisconstruction for the half space brings an example of the PRBM for an unboundeddomain.Let Ω be the upper half space, Ω = {x ∈ R d : x d > 0}, with smoothboundary ∂Ω = {x ∈ R d : x d = 0}. Let {Wt k } are d independent Brownianmotions started from the origin. Then, the Brownian motion, started from a givenpoint x ∈ Ω and reflected on the boundary ∂Ω, can be written in a simple way as(x 1 + Wt 1,..., x d−1 + Wtd−1 , |x d + Wt d |). The particular simplification is broughtby the fact that reflections happen in a single direction, being involved through theone-dimensional reflected Brownian motion |x d +Wt d |. Without loss of generality,we can consider the reflected Brownian motion started from the origin (x = 0):the translational invariance along the hyperplane ∂Ω permits to move the startingpoint in ∂Ω, whereas the convolution property (11) allows displacements in orthogonaldirection. The local time process l t can be introduced either through thestochastic equations (7), or with the help of Lévy’s formulae (8) or (9).Lemma 3.5 Let Ω be the upper half space, Ω = {x ∈ R d : x d > 0}. For anypositive Λ, the stopping time T 0 Λ , defined in 3.1, is distributed according toP{T 0 Λ ∈ (t, t + dt)} = ρ Λ(t)dt, ρ Λ (t) =∫ ∞0z e −z2 /2t e −z/ΛΛ √ 2π t 3/2 dz (14)

Partially Reflected Brownian Motion... 17Proof. Since the local time l t and the random variable χ are independent, one canwrite the probability P{T 0 Λ ∈ (t, t + dt)} as∫∞P{ T 0 Λ ∈ (t, t+dt) } =0{}P inf{τ > 0 : l τ = z} ∈ (t, t+dt) P{ χ ∈ (z, z+dz) }Then, the first factor is the well known density of the inverse local time process[42],{}P inf{τ > 0 : l τ = z} ∈ (t, t + dt) = dt z /2t√ e−z2 2π t3/2while the second factor is given by the exponential law density (10) that implies(14). □Note that the integral in (14) can be represented with the help of the Gaussianerror function[ρ Λ (t) = 1( ) ] 1 1 √t/2Λ√ √2Λ − K 2, K(z) = 2 ∫∞√ e z2 −x 2 dx2 π t/2Λ2 πOne finds the asymptotic behavior of the density ρ Λ (t):ρ Λ (t) ∼ (√ 2πΛ ) −1 t −1/2 (t → 0), ρ Λ (t) ∼ (√ 2π/Λ ) −1 t −3/2 (t → ∞)Once the distribution of stopping time T 0 Λ is determined, one can calculate thespread harmonic measure density ω x,Λ (s).Lemma 3.6 Let Ω be the upper half space, Ω = {x ∈ R dpositive Λ, the spread harmonic measure density ω x,Λ (s) isω x,Λ (s 1 , ..., s d−1 ) =where |k| =∫ ∞−∞...∫ ∞−∞√k 2 1 + ... + k 2 d−1 .dk 1 ...dk d−1(2π) d−1[exp −i∑d−1j=1z: x d > 0}. For any] e−x d |k|k j (x j − s j )1 + Λ|k|(15)Proof. First, the probability kernel T Λ (s, s ′ ), defined for two boundary pointss, s ′ ∈ ∂Ω, is translationally invariant in the hyperplane ∂Ω, T Λ (s, s ′ ) = t Λ (s−s ′ ),wheret Λ (s 1 , ..., s d−1 )ds 1 ...ds d−1 ≡{}P W 1 T∈ (s 0 1 , s 1 + ds 1 ) , ... , W d−1 ∈ (sΛT 0 d−1 , s d−1 + ds d−1 )Λ

Partially Reflected Brownian Motion... 18The stopping time T 0 Λ is related to the orthogonal motion and, consequently, independentof lateral motions Wt 1 d−1, ..., Wt. Therefore, the above probability can bewritten ast Λ (s 1 , ..., s d−1 )ds 1 ...ds d−1 =∫ ∞ {}P Wt 1 ∈ (s 1 , s 1 + ds 1 ), ... , Wt d−1 ∈ (s d−1 , s d−1 + ds d−1 ) ρ Λ (t)dt0Since the lateral motions are independent between themselves, the first factor isequal to the product of Gaussian densities (4):t Λ (s 1 , ..., s d−1 ) =∫ ∞0dt ρ Λ (t)Using the integral representation (14), one findst Λ (s 1 , ..., s d−1 ) = Γ(d/2)π d/2 Λ∫ ∞0dz∏d−1j=1e −s2 j /2t√2πtz e −z/Λ[s21 + ... + s 2 d−1 + z2] d/2where Γ(z) stands for Euler gamma function (see [39] for details).Substituting the well known harmonic measure density ω x (s) for the upperhalf space (generalized Cauchy distribution),ω x (s 1 , ..., s d−1 ) = Γ(d/2)π d/2x d[(x1 − s 1 ) 2 + ... + (x d−1 − s d−1 ) 2 + (x d ) 2] d/2into convolution (11), one finally obtains the expression (15) for the spread harmonicmeasure density. □One can easily verify that the spread harmonic measure density ω x,Λ (s) andthe probability kernel T Λ (s, s ′ ) satisfy the following conditions:1. Normalization condition:∫ds ω x,Λ (s) = 1∂Ω∫∂Ωds ′ T Λ (s, s ′ ) = 12. Dirichlet limit (Λ → 0):ω x,Λ (s) −→ ω x (s) T Λ (s, s ′ ) −→ δ(s − s ′ )

Partially Reflected Brownian Motion... 193. Translational invariance:ω x,Λ (s) = ω x−s,Λ (0) T Λ (s, s ′ ) = T Λ (s − s ′ , 0) ≡ t Λ (s − s ′ )One can also deduce the asymptotic behavior of the function t Λ (s) as |s| → 0 or|s| → ∞. For this purpose, it is convenient to define the new function η d (z) byrelationt Λ (s) = η d(|s|/Λ)ω(0,...,0,Λ) (s)where the second factor is the harmonic measure density for the Brownian motionstarted from the point (0, ..., 0, Λ). Using the explicit formulae for t} {{ }Λ (s) andd−1ω (0,...,0,Λ) (s), one obtains:η d (z) = ( 1 + z 2) d/2∫ ∞Its asymptotic behavior for z going to infinity iswhereas for z going to 0, one has0t e −t dt(t 2 + z 2 ) d/2 (16)η d (z) = 1 − 5d 2 z−2 + O(z −4 ) (17)η d (z) ∼ Γ(d/2)π d/2 z 2−d (d > 2), η d (z) ∼ 1 πln z (d = 2)These relations can be used for qualitative study of the partially reflectedBrownian motion. For instance, one identifies the parameter Λ as a characteristiclength scale of the problem: the magnitude of any distance (e.g., |s|) has tobe compared with Λ. Interestingly, the asymptotic behavior (17) for large z meansthat the function t Λ (s) is close to the harmonic measure density ω (0,...,0,Λ) (s).Roughly speaking, for large |s|/Λ, the partially reflected Brownian motion startedfrom the origin is qualitatively equivalent to the (simple) Brownian motion startedfrom the point (0, ..., 0, Λ). In other words, the partial reflections on the boundarylead to a spreading of the harmonic measure with characteristic scale Λ (see alsorelation (11)). The explicit analytical results can also be derived in presence of anabsorbing barrier at a given height [59].The knowledge of the probability kernel T Λ (s, s ′ ) brings an important informationabout the partially reflected Brownian motion in the (upper) half space. Asan example, we calculate the probability P Λ (r) that the PRBM started from the

Partially Reflected Brownian Motion... 20.. Λ .. . ..... . ... .. . . .. . . .. .. . . ...... . . . . ... ........ . . ... ... ..... ... . . .. .. . .. .. . .. ..... . ... . .. . ....... ... . .... .... . ........ .... .... . ..... . . .... .... . . ..... . ...... ... ..... .... . . . . . .. . . .... ..... .... .∂C.. Λ . ∂n = C Λ C = 0.... . .. .. . . .... .. . .. ... .. . Figure 3: Land Surveyor Approximation: the total flux across the boundary canbe approximately calculated when the mixed boundary condition ∂C/∂n = C/Λon a given irregular curve (on the left) is replaced by the Dirichlet condition C = 0on the coarse-grained boundary (on the right). The last one is obtained by replacingcurvilinear intervals of length Λ by corresponding linear chords.origin is finally absorbed on the disk B d−1r = { (x 1 , ..., x d ) ∈ R d : x 2 1 + ... +x 2 d−1 ≤ r2 , x d = 0 } of radius r centered at the origin:P Λ (r) =∫B d−1rds t Λ (s) = 2 Γ(d 2 )Γ( d−12 )√ π∫ ∞0te −t dt∫r/Λ0x d−2 dx[x 2 + t 2 ] d/2 (18)This probability shows how far the partially reflected Brownian motion can goaway after the first hit of the boundary. One sees that this function depends onlyon the ratio r/Λ, going to 0 for small radii and to 1 for large radii. Again, theparameter Λ is the characteristic length scale of the problem. In two-dimensionalcase, P Λ (Λ/2) is the probability that the PRBM is absorbed on the linear segmentof length Λ, centered at the origin (the first hitting point). The numericalcalculation of the integral in (18) gives P Λ (Λ/2) ≃ 0.4521, i.e., about half of theparticles is absorbed on this region. In other words, the length of the characteristicabsorption region (where half of the particles is absorbed) is approximatelyequal to Λ. It has been shown recently that this result is qualitatively valid for alarge class of irregular boundaries [60]. Roughly speaking, if the one-dimensionalboundary (curve) has no deep pores (fjords) and its perimeter is large with respectto the scale Λ, then the curvilinear interval of length Λ, centered on the first hittingpoint, absorbs approximately half of the diffusing particles.This result can be considered as a first mathematical justification of the LandSurveyor Approximation (LSA) developed by Sapoval [19]. According to thisapproximation, a given one-dimensional interface (curve) can be coarse-grainedwith physical scale Λ in order to replace the mixed boundary condition [I −Λ∂/∂n]C = 0 by the Dirichlet condition C = 0 (see Fig. 3). Some heuristicphysical arguments allowed to state that the total flux across the irregular semipermeableinterface was approximately equal to the total flux across this coarse-

Partially Reflected Brownian Motion... 21grained boundary with Dirichlet condition. This statement provided a simple butpowerful tool to investigate Laplacian transport phenomena. The land surveyorapproximation had been checked numerically [33, 34], but not mathematically.The study of the partially reflected Brownian motion brings its justification andfurther understanding. Actually, the coarse-graining procedure generates the regionsof length Λ, where about half of the particles is absorbed. The LSA isbased on two simplifications which can be clearly explained in terms of diffusingparticles:1. The Dirichlet boundary condition on the coarse-grained boundary meansthat all particles arrived to the characteristic absorption region are absorbed.This approximation does not take into account half of the particles whichescaped this region.2. The linear chords, generated by coarse-graining, are deterministic regions.This approximation neglects the fact that the characteristic absorption regionsshould be centered at the random position of the first hit of the boundary.Although these simplifications seem to be rough, the numerical simulations showthat the LSA reproduces the transport properties with good accuracy. However,this approximation has no any kind of small parameter which would allow tocontrol its applicability. More accurate theoretical approaches will be discussedin Section 4.One can go further by extending the Land Surveyor Approximation to thethree-dimensional case, which still remains poorly understood. Indeed, the numericalcalculation in 3D leads to P Λ (Λ) ≃ 0.4611, i.e., about half of the particlesis absorbed on the disk of radius Λ centered at the first hitting point. Consequently,if one finds a convenient cover of a semi-permeable irregular interface by disk-likesets of characteristic radius Λ, the LSA may be still valid, i.e., the total flux acrossa given interface would be approximated by the total flux across the perfectlyabsorbing coarse-grained interface (with Dirichlet condition). An accurate mathematicalformulation of this extension and its numerical verification present openinteresting problems.3.3 Relation to the Dirichlet-to-Neumann OperatorThe construction of the partially reflected Brownian motion for a given domain Ωrequires the resolution of the stochastic differential equation (7), a quite difficultproblem. Fortunately, many characteristics of this process, e.g., the spread harmonicmeasure density ω x,Λ (s), can be obtained in another way. This subsection

Partially Reflected Brownian Motion... 22is devoted to the Dirichlet-to-Neumann operator and its relation to the partiallyreflected Brownian motion.Definition 3.7 For a given domain Ω ⊂ R d (d ≥ 2) with smooth bounded boundary∂Ω, let u : Ω ∪ ∂Ω → R be a harmonic function with Dirichlet conditionu = f, a function f being from the Sobolev space H 1 (∂Ω) (in other words, u isthe solution of the boundary value problem (5)). Applying the normal derivativeto u, one obtains a new function g = ∂u/∂n belonging to the space L 2 (∂Ω) ofmeasurable and square integrable functions. Then the operator M, acting fromH 1 (∂Ω) to L 2 (∂Ω), which associates the new function g with a given f, is calledDirichlet-to-Neumann operator.It is known that the Dirichlet-to-Neumann operator M is self-adjoint pseudodifferentialoperator of the first order, with discrete positive spectrum {µ α } andsmooth eigenfunctions forming a complete basis in L 2 (∂Ω) [51, 52, 61, 62, 63,64, 65, 66, 67, 68]. One can also define its resolvent operator T Λ = [I + ΛM] −1 ,called spreading operator. This is an analytic operator function in the whole complexplane, except a denumerable set of points, C\{−µ −1α }. In particular, T Λ iswell defined for any positive Λ.Lemma 3.8 For any strictly positive Λ, the spreading operator T Λ acts fromL 2 (∂Ω) to L 2 (∂Ω) as a compact integral operator,∫[T Λ f](s) = ds ′ f(s ′ ) T Λ (s ′ , s)∂Ωwhere the kernel T Λ (s, s ′ ) is given by (11).Proof. The probability kernel T Λ (s, s ′ ) is a positive function satisfying the normalization:∫T Λ (s, s ′ )ds ′ = 1∂Ωsince the partially reflected Brownian motion is conditioned to be finally absorbedon the boundary. Therefore, one obtains:∫ ∫ds ds ′ |T Λ (s, s ′ )| 2 = S tot < ∞∂Ω ∂Ωwhere S tot is the total surface area of the boundary ∂Ω. The integral operator T Λdefined by the kernel T Λ (s, s ′ ) is a Hilbert-Schmidt operator and, consequently, acompact operator.

Partially Reflected Brownian Motion... 23The boundary condition in lemma 3.4 can be written with the help of theDirichlet-to-Neumann operator:[I + ΛM]TΛ (s, s ′ ) = δ(s − s ′ )that implies that the integral operator T Λ coincides with the resolvent [I +ΛM] −1of the Dirichlet-to-Neumann operator M. □This simple lemma creates a “bridge” between the partially reflected Brownianmotion and the Dirichlet-to-Neumann operator. In particular, the relation(11) for the spread harmonic measure density ω x,Λ (s) can now be understood asapplication of the spreading operator T Λ to the harmonic measure density ω x (s).Consequently, once the Dirichlet-to-Neumann operator is constructed for a givendomain, one can calculate the density ω x,Λ (s) without solving the stochastic differentialequations (7).The self-adjointness of the Dirichlet-to-Neumann operator allows one to applyefficient tools of the spectral theory. For example, one can rewrite the relation (11)as spectral decomposition of the harmonic measure density on eigenfunctions V αof the operator M:ω x,Λ (s) = ∑ (ωx · V ∗ ∫α)(L 2V α (s) ωx · V1 + Λµα)∗ = ωL 2 x (s ′ ) Vα ∗ (s′ ) ds ′αα∂Ω(19)where ( · ) L 2 denotes the scalar product in L 2 (∂Ω) space. The advantage of thisrelation is an explicit dependence on the physical parameter Λ.3.4 ExamplesIn order to illustrate the underlying concepts, we consider several examples.3.4.1 Two-Dimensional DiskWe are going to study the partially reflecting Brownian motion in a unit disk, Ω ={ x ∈ R 2 : |x| < 1 } (its boundary is a unit circle, ∂Ω = { x ∈ R 2 : |x| = 1 }).In this case, the harmonic measure density ω x (s) ≡ ω(r, θ) is a function oftwo real variables: the distance 0 ≤ r < 1 between the starting point x ∈ Ω andthe origin, and the angle 0 ≤ θ < 2π between directions onto points x and s ∈ ∂Ωfrom the origin. The harmonic measure density is known as Poisson kernel:ω(r, θ) =1 − r 22π(1 − 2r cosθ + r 2 )(20)

Partially Reflected Brownian Motion... 24The rotational invariance of the domain Ω implies that the eigenbasis of theDirichlet-to-Neumann operator M is the Fourier basis,V α (θ) = eiαθ√2π(α ∈ Z)Taking Fourier harmonic as boundary condition, u(r = 1, θ) = e iαθ , one findsa regular solution of the corresponding Dirichlet problem: u(r, θ) = r |α| e iαθ .Since the normal derivative coincides with the radius derivative, one obtains theeigenvalues of the Dirichlet-to-Neumann operator:µ α = |α| (α ∈ Z)These eigenvalues are doubly degenerated (expect µ 0 = 0).The spread harmonic measure density is given by relation (19):ω x,Λ (s) ≡ ω Λ (r, θ) = 1 ∞∑ r |α| e iαθ2π 1 + Λ|α|α=−∞(the scalar product of the harmonic measure density ω x (s) and eigenfunctionsVα ∗(θ) is shown to be equal to r|α| , with r = |x|). In the case Λ = 0, one retrievesthe Poisson representation for the harmonic measure density (20) just as required.The kernel of the resolvent operator T Λ isT Λ (θ, θ ′ ) = 12π∞∑α=−∞e iα(θ−θ′ )1 + Λ|α|For the exterior problem, when Ω = {x ∈ R 2 : |x| > 1}, one obtains exactlythe same results.3.4.2 Three-Dimensional BallThe similar arguments can be applied for higher dimensions. For example, in thethree-dimensional case, one considers the unit ball Ω = { x ∈ R 3 : |x| < 1 }.The harmonic measure density is known to be1 − r 2ω x (s) ≡ ω(r, θ) ≡4π ( 1 − 2r cosθ + r 2) 3/2(s ∈ ∂Ω)where r = |x| < 1 is the distance between the starting point x ∈ Ω and the origin,and θ is the angle between directions onto points x and s ∈ ∂Ω from the origin.This function can be expanded on the basis of spherical harmonics asω x (s) =∞∑l=0 m=−ll∑r l Y l,m (s) Y l,m (x/r)

Partially Reflected Brownian Motion... 25The rotational symmetry of the problem implies that the eigenbasis of theDirichlet-to-Neumann operator is formed by spherical harmonics Y l,m . A regularsolution of the Dirichlet problem (5) in the unit ball can be written in sphericalcoordinates r, θ and ϕ asu(r, θ, ϕ) =∞∑ l∑f l,m r l Y l,m (θ, ϕ)l=0 m=−lwhere f l,m are coefficients of the expansion of a given boundary function f (Dirichletcondition) on the complete basis of spherical harmonics. Since the normalderivative coincides with the radius derivative, one obtains[Mf](θ, ϕ) =( ) ∂u=∂n∂Ω( ) ∂u=∂rr=1∞∑l∑l=0 m=−li.e., the eigenvalues of the Dirichlet-to-Neumann operator M aref l,m l Y l,m (θ, ϕ)µ l = l (l ∈ {0, 1, 2, ...}) (21)Note that the l-th eigenvalue is degenerated n l = (2l + 1) times.Interestingly 3 , the Dirichlet-to-Neumann operator M for the unit ball Ω ={x ∈ R 3 : |x| < 1} coincides with an operator introduced by Dirac in quantummechanics [69]. It is known that the hydrogen atom is described by threequantum numbers: the main quantum number n, the orbital quantum number land magnetic quantum number m. Two last numbers are associated with indicesof spherical harmonics. Thus, the Dirichlet-to-Neumann operator in the ball isapparently the orbital quantum number operator for the hydrogen atom. In particular,the degeneracy of eigenvalues of this operator can be understood from thepoint of view of spin degeneracy.The spread harmonic measure density ω x,Λ (s) and the spreading operator kernelT Λ (s, s ′ ) can be written explicitly as spectral decompositions on the eigenbasisof the Dirichlet-to-Neumann operator M as in the two-dimensional case.The eigenvalues of the Dirichlet-to-Neumann operator for d-dimensional unitball are still given by (21) with degeneracyn l =(2l + d − 2)(d − 2)(l + d − 3)!(d − 3)! l!The exterior problem for Ω = {x ∈ R 3 : |x| > 1} can be considered in thesame manner. The harmonic measure density is ˜ω x (s) = (1/r) ω(1/r, θ), with3 The author thanks Dr. S. Shadchin for valuable discussions on this relation.

Partially Reflected Brownian Motion... 26r = |x| > 1. Using the same expansion on spherical harmonics, one obtainsµ l = l + 1 with l ∈ {0, 1, 2, ...}. In particular, the lowest eigenvalue µ 0 = 1 isstrictly positive. This difference with respect to the spectrum µ l for the interiorproblem has a simple probabilistic origin: the Brownian motion in three dimensionsis transient, i.e., there is a positive probability (equal to 1 − 1/r) to neverreturn to the ball. Another explication follows from the theory of boundary valueproblems for elliptic differential operators: the exterior Neumann problem has aunique solution, while the solution of the interior Neumann problem is definedup to a constant. Consequently, the Dirichlet-to-Neumann M operator should beinvertible for the exterior problem that implies a simple condition for its eigenvalues:µ α ≠ 0. On the contrary, M is not invertible for the interior problemproviding the condition that at least one eigenvalue is zero.4 Stochastic Approaches to Laplacian Transport PhenomenaIn this section, we return to the Laplacian transport phenomena, discussed at thebeginning. First, we are going to introduce the notion of source, diffusing particlesstarted from. The definition of the partially reflected Brownian motion will requireonly a minor modification. After that, a recently developed continuous approachwill be presented with a special emphasis on its physical significance. Finally,we shall mention two other physical descriptions which can now be considered asuseful approximations to the continuous approach.4.1 Notion of SourceThe description of the partially reflected Brownian motion given in the previoussection does not involve a source, an important element for Laplacian transportphenomena. In this subsection, we are going to discuss the extension of previousdefinitions in order to introduce the source. As one will see, a minor modificationwill be sufficient.Throughout this subsection, we consider a bounded domain Ω with twice continuousdifferentiable boundary composed of two disjoint parts, ∂Ω and ∂Ω 0 , referredto as working interface and source respectively 4 . In practice, the workinginterface and the source are well separated in space, therefore one may think abouta circular ring as generic domain.4 Previously, the whole boundary of the domain had been considered as the working interface.For this reason, we preserve the same notation ∂Ω for this object and hope that it will not lead toambiguities.

Partially Reflected Brownian Motion... 27As previously, one considers the reflected Brownian motion Ŵt, started fromany point x ∈ Ω ∪ ∂Ω ∪ ∂Ω 0 and reflected on the whole boundary ∂Ω ∪ ∂Ω 0 , thecorresponding local time process l t , and the stopping time T x Λ defined in 3.1. Letus introduce a new stopping time τ as the first moment when the process Ŵt hitsthe source ∂Ω 0 :τ = inf{t > 0 : Ŵ t ∈ ∂Ω 0 }Then, the spread harmonic measure can be introduced for any subset A from Borelσ-algebra B(∂Ω) (defined on the working interface alone!) asω x,Λ {A} = P{ŴT x ∈ A, Tx Λ Λ < τ < ∞} (22)We outline two distinctions with respect to the previous definition 3.2:• The measure is considered on Borel subsets of the working interface ∂Ωonly, whereas the reflected Brownian motion Ŵt and the local time processl t are defined on the whole boundary ∂Ω ∪ ∂Ω 0 .• There is a supplementary condition T x Λ < τ providing that the process Ŵtshould be stopped (absorbed) on the working interface before hitting thesource.Note that 1 − ω x,Λ {∂Ω} is the probability that the process Ŵt started from agiven point x ∈ Ω hits the source ∂Ω 0 before its final absorption on the workinginterface ∂Ω.One can easily extend the lemma 3.3 to this spread harmonic measure:Lemma 4.1 For any subset A from B(∂Ω) and fixed positive Λ, the spread harmonicmeasure ω x,Λ {A}, considered as a function of x, solves the boundary valueproblem:[I − Λ ∂ ]ω∆ω x,Λ {A} = 0 (x ∈ Ω),x,Λ {A} = I A (x) (x ∈ ∂Ω)∂nω x,Λ {A} = 0 (x ∈ ∂Ω 0 )(23)Proof is similar to that of the lemma 3.3. The last condition holds since x ∈ ∂Ω 0implies τ = 0. □Corollary 4.2 Function C Λ (x) = C 0 (1 − ω x,Λ {∂Ω}) solves the boundary valueproblem (1–3).

Partially Reflected Brownian Motion... 28Proof is a direct verification. □Consequently, a simple introduction of the source allows one to apply theprevious description of the partially reflected Brownian motion to study Laplaciantransport phenomena. Due to reversibility of the Brownian motion, one may thinkthat (1 − ω x,Λ {∂Ω})dx gives also the probability to find the partially reflectedBrownian motion, started from the absorbing source, in dx vicinity of the pointx ∈ Ω, under partially absorbing condition on the working interface ∂Ω. Note thatsuch way of reasoning, being intuitive and useful, is quite formal. In particular,the (simple) Brownian motion started from the source returns to it infinitely manytimes with probability 1. If one really needs to define such a process, the startingpoint should be taken slightly above the source.Since the boundary ∂Ω is supposed to be smooth, one can introduce the spreadharmonic measure density ω x,Λ (s). In turn, the kernel of the spreading operator isdefined as previously, T Λ (s, s ′ ) ≡ ω s,Λ (s ′ ). In particular, one retrieves the relation(11):∫ω x,Λ (s) = ds ′ ω x,0 (s ′ ) T Λ (s ′ , s)∂Ωwhere the harmonic measure density ω x,0 (s) is defined by relation (22) with Λ =0.The definition of the Dirichlet-to-Neumann operator can also be extended todomains with a source. For a given function f ∈ H 1 (∂Ω), one solves the Dirichletproblem in the domain Ω:∆u = 0 (x ∈ Ω),u = f (x ∈ ∂Ω)u = 0 (x ∈ ∂Ω 0 )(in principle, one could consider another function on the source). For a givenfunction f on ∂Ω, the Dirichlet-to-Neumann operator M, acting from H 1 (∂Ω)to L 2 (∂Ω), associates the new function g = ∂u/∂n on the working interface ∂Ω.One can prove general properties of this operator and its relation to the partiallyreflected Brownian motion in a straight way. In particular, the spreading operatorT Λ , defined by its kernel T Λ (s, s ′ ), coincides with the resolvent operator [I +ΛM] −1 . However, some normalization properties may be changed. In particular,for the probability kernel T Λ (s, s ′ ), one has∫T Λ (s, s ′ )ds ′ < 1∂Ωsince the PRBM started from the working interface ∂Ω can now be absorbed onthe source.

Partially Reflected Brownian Motion... 294.2 Continuous ApproachThe stochastic treatment by means of the partially reflected Brownian motionbrings the solution to the problem (1–3) describing Laplacian transport phenomena:C Λ (x) = C 0 (1 − ω x,Λ {∂Ω}) (see corollary 4.2). One can go further usingthe close relation to the Dirichlet-to-Neumann operator [39, 40]. According tothe lemma 4.1, the density ω x,Λ {∂Ω}, considered as a function of x, solves theboundary value problem:[I − Λ ∂ ]ω∆ω x,Λ {∂Ω} = 0 (x ∈ Ω),x,Λ {∂Ω} = 1 (x ∈ ∂Ω)∂nω x,Λ {∂Ω} = 0 (x ∈ ∂Ω 0 )The restriction of the function ω x,Λ {∂Ω} on ∂Ω can be written with the help ofthe Dirichlet-to-Neumann operator M asω s,Λ {∂Ω} = [ (I + ΛM) −1 1 ] (s) = [T Λ 1](s)where 1 stands for a constant (unit) function on the working interface.One defines then the flux density φ Λ (s) across the working interface ∂Ω:φ Λ (s) = D ∂C Λ∂n (s) = −DC 0∂ω x,Λ {∂Ω}(s)∂nSince the normal derivative of a harmonic function can be represented as the applicationof the Dirichlet-to-Neumann operator to the restriction of this functionon the boundary, one writesφ Λ (s) = DC 0 [Mω s,Λ {∂Ω}](s) = DC 0 [MT Λ 1](s)(the sign is changed due to particular orientation of the normal derivative). TakingΛ = 0, one finds φ 0 (s) = DC 0 [M1](s) and finallyφ Λ (s) = [T Λ φ 0 ](s)The transport properties of the working interface can be characterized by aphysical quantity called spectroscopic impedance. We remind that the impedanceof an electric scheme is defined as the tension applied between two external poles,divided by the total electric current passing through. The formal analogy betweenthe electric problem and the diffusive transport, discussed in Section 1, leads toa natural definition of the impedance in our case as the concentration C 0 on thesource ∂Ω 0 divided by the total flux across the working interface ∂Ω:Z cell (Λ) =C 0∫ds φ Λ (s)∂Ω

Partially Reflected Brownian Motion... 30Taking Λ = 0, one deals with a purely absorbing interface ∂Ω: any particlearrived to ∂Ω is immediately absorbed (without reflections). In other words, suchinterface has no resistance for passage across it. Consequently, the impedanceZ cell (0) represents the “access resistance” by the bulk: the possibility that theBrownian motion can return to the source without hitting the working interface.The resistance of the working interface alone can thus be characterized by thedifference between Z cell (Λ) and Z cell (0), called spectroscopic impedance:Z sp (Λ) = Z cell (Λ) − Z cell (0)Using the simple identityC 0((φ0 − φ Λ ) · 1 ) L 2 = Λ D(φΛ · φ 0)L 2 (24)one writes the spectroscopic impedance as( )φΛ · φ 0Z sp (Λ) = Λ D(L 2φΛ · 1 ) (φ0 · 1 ) L 2 L 2Applying again the identity (24), one finds a more convenient form:The new function1Z sp (Λ) =1Z(Λ) − 1Z cell (0)Z(Λ) =D( Λ TΛ φ h 0 · )φh 0can be called effective impedance, whereφ h 0 (s) =φ 0(s)(φ0 · 1 ) L 2L 2is the normalized flux density towards the perfectly absorbing working interface∂Ω. Finally, the spectral decomposition of the spreading operator T Λ on the basisof eigenfunctions V α of the Dirichlet-to-Neumann operator M leads to theimportant relation for the effective impedance:Z(Λ) = Λ D∑αF αF α = ( φ h 01 + Λµ · V ) ( )α L φh 2 0 · Vα∗ (25)L 2 αThis relation presents the central result of the continuous approach developedin [40]. Let us briefly discuss its physical meaning. The spectroscopic impedance

Partially Reflected Brownian Motion... 31Z sp (Λ) or, equivalently, the effective impedance Z(Λ), is a physical quantity thatcharacterizes the transport properties of the whole working interface. More importantly,this quantity can be measured directly in experiment (e.g., in electrochemistry).On the other hand, the local transport properties of the working interfaceare described by the single physical parameter Λ, being related to the membranepermeability W , the electrode resistance r or the catalyst reactivity K (see Section1). Varying the parameter Λ, one changes the local transport properties ateach boundary point and, consequently, the whole linear response of the workinginterface. At first sight, one may think that an increase of the local boundary resistancewould imply a proportional increase of the whole boundary resistance,i.e., Z(Λ) ∼ Λ. This reasoning, being true for a planar surface, becomes invalidin a general case due to geometrical irregularities and related screening effects. Infact, an irregular geometry modifies considerably the linear response of the workinginterface [24, 25, 26, 27, 28, 29, 30, 31, 32]. The boundary value problem(1–3), describing Laplacian transport phenomena on the average, allows formallyto study such geometrical influence. Practically, however, this is a very difficultproblem. In contrast, the continuous approach provides an efficient tool to carryout these studies both in theoretical and numerical ways. In particular, the relation(25) makes explicit the impedance dependence on the local transport properties(parameter Λ) and allows one to identify contributions due to the physics and dueto the geometry, originally involved in the problem in a complex manner. In otherwords, whatever the physical problem (diffusion across semi-permeable membranes,heterogeneous catalysis or electric transport), the geometry enters onlythrough the spectral characteristics of the Dirichlet-to-Neumann operator M: itseigenvalues µ α and the spectral components F α of the normalized flux densityφ h 0 (s) on the basis of its eigenfunctions V α(s).The other important meaning of the relation (25) can be outlined if one considersthe inverse problem [39]: what is the most available information that onecan retrieve from a measurement of the spectroscopic impedance of an unknownworking interface? The mathematical response can be given immediately if onerewrites (25) as Laplace transform of the new function ζ(λ):Z(Λ) = 1 D∫ ∞dλ e −λ/Λ ζ(λ),ζ(λ) ≡ ∑ αF α e −λµα0Under assumption to be able to measure the impedance with an absolute precision,one can reconstruct the function ζ(λ) and, consequently, the set of characteristics{µ α , F α } which may thus be called harmonic geometrical spectrum ofthe working interface. The hierarchical structure of this spectrum for self-similarboundaries has been recently investigated [70]. Many interesting properties of thefunction ζ(λ) remain poorly understood.

Partially Reflected Brownian Motion... 32In the two following subsections, we are going to discuss some aspects of thesemi-continuous and the discrete descriptions of Laplacian transport phenomena.These approaches are based on more intuitive notion of partial reflections on theboundary. Since these descriptions turn out to be approximations to the continuousapproach, we do not present the circumstantial details.4.3 Semi-continuous ApproachHalsey and Leibig gave the first theoretical description of the electrolytic doublelayer response with emphasis on electrochemical applications [29]. This description,involving the Green function of the electrolytic cell, can be reformulated inthe following stochastic language (for details, see [39]). For a given domain Ωwith smooth bounded boundary ∂Ω, one considers the Brownian motion startedfrom a point x ∈ Ω. When the diffusing particle hits the boundary at some points, two complementary events may happen:• with probability ε, the Brownian motion is reflected to the interior bulk points + an(s), slightly above the boundary (here n(s) is the unit normal vectorto the boundary at point s, a is a small positive parameter); the Brownianmotion continues from this point;• or, with probability 1 − ε, the Brownian motion is terminated at this point s(absorbed on the boundary).This stochastic process is continued until the absorption on the boundary and canbe called Brownian motion reflected with jump. Two new parameters, the jumpdistance a and the reflection probability ε, are related to the given physical lengthΛ [22]:1ε =(26)1 + (a/Λ)Now, one can calculate the probability ω (a)x,Λ(s)ds that this process is finallyabsorbed in ds vicinity of the boundary point s. Since the motions before andafter each reflection are independent, this probability can be obtained as the sumof probabilities to be absorbed after 0, 1, 2, ... reflections:ω (a)x,Λ (s)ds = [ ω x (s)ds ] (1 − ε) +∫+∫∂Ω ∂Ω∫∂Ω[ωx (s 1 )ds 1]ε[ωs1 +an(s 1 )(s)ds ] (1 − ε)+[ωx (s 1 )ds 1]ε[ωs1 +an(s 1 )(s 2 )ds 2]ε[ωs2 +an(s 2 )(s)ds ] (1 − ε) + ...For example, the third term represents the probability to hit the boundary in ds 1vicinity of the point s 1 , to be reflected to the neighboring point s 1 + an(s 1 ), to hit

Partially Reflected Brownian Motion... 33again the boundary in ds 2 vicinity of the point s 2 , to be reflected to the neighboringpoint s 2 + an(s 2 ), to hit the boundary for the last time in ds vicinity of the points, and to be finally absorbed. Introducing the integral operator Q (a) , acting fromL 2 (∂Ω) to L 2 (∂Ω) as∫[Q (a) f](s) = ds ′ f(s ′ ) ω s ′ +an(s ′ )(s)∂Ωone rewrites the previous sum as the application of the new integral operator T (a)Λto the harmonic measure density ω x (s):ω (a) (a)x,Λ(s) = [TΛ ω x](s) with T (a)Λ∞ = (1 − ε) ∑( ) εQ(a) kk=0(27)What happens when the jump distance a goes to 0? Hitting the boundary, theBrownian motion will be reflected to interior points lying closer and closer to theboundary, i.e., displacements of the Brownian motion between two serial hits aregetting smaller and smaller. At the same time, the reflection probability ε tendsto 1 according to relation (26), i.e., the average number of reflections increases.Indeed, the distribution of the random number N of reflections until the finalabsorption is simplyP{N = n} = (1 − ε) ε n (28)implying that the average number E{N } = ε(1 − ε) −1 goes to infinity. Does alimiting process exist? The situation is complicated by the local choice betweenreflection and absorption: at each hitting point, the motion can be absorbed withvanishing probability 1 − ε. In order to overcome this difficulty, one can considerthis process from a slightly different point of view. Actually, one can replace thelocal condition of the absorption (with probability 1 − ε) by its global analog: theprocess is absorbed on the boundary when the number of reflections exceeds arandom variable N distributed according to the geometrical law (28). Evidently,this modification does not change at all the properties of the process. At the sametime, we gain that the condition of the absorption becomes independent of theBrownian motion between serial hits. As a consequence, one can consider thecorresponding limits (as a → 0) separately. So, the Brownian motion reflectedwith jump should tend to the reflected Brownian motion as the jump distancea vanishes. This motion, however, is conditioned to stop when the number ofreflections exceeds the random variable N . Since the average number E{N } goesto infinity in the limit a → 0, it is convenient to consider a normalized variableχ = aN obeying the following distribution:∞∑P{χ ≥ λ} = P{N ≥ λ/a} = P{N = n} ≃ ε [λ/a] ≃ exp[−λ/Λ] (29)[λ/a]

Partially Reflected Brownian Motion... 34(the last equality is written with the help of (26) for a going to 0). Since thenumber of reflections on jump distance a, multiplied by a, tends to the local timeprocess according to Lévy’s formula (9), the previous condition of absorption canbe reformulated: the motion is absorbed when its local time process exceeds arandom variable distributed according to the exponential law (29). One thus concludesthat the Brownian motion reflected with jump should tend to the partiallyreflected Brownian motion defined in 3.1.The above analysis, presented as a sketch (without proofs), does not pretendto a mathematical rigour. It may be considered rather as a possible justificationwhich can be brought for the semi-continuous approach if necessary. In particular,one can demonstrate that the density ω (a)x,Λ(s), given by relation (27), tends to thespread harmonic measure density as the jump distance a goes to 0:ω x,Λ (s) = lim ω (a)a→0x,Λ (s)This relation may be useful for numerical computations (in particular, it was appliedin [60]). Similarly, the integral operator T (a)Λshould converge to the spreadingoperator T Λ as a → 0. Calculating the geometrical series in (27) and representingT (a)Λas T (a)Λ = (1 − ε)( I − εQ (a)) −1 =(I + Λ I − ) −1Q(a)aone obtains the following approximation for the Dirichlet-to-Neumann operator:M = lima→0I − Q (a)aAgain, this relation may be useful for the numerical computation of this operator.The advantages of the semi-continuous approach are based on an apparentintuitive meaning of partial reflections on the boundary. Moreover, this approachprovides even a more realistic description of physico-chemical processes at microscopiclevel. For example, if one considers the partially reflected Brownian motionstarted from a boundary point, the number of hits of the boundary is infinite forany moment t > 0 that sounds impossible for real physical species. The keypointis that, for diffusion across a semi-permeable membrane or heterogeneous reactionon a catalytic surface, the description by the boundary value problem (1–3)cannot be justified on length scales less than the mean free path of diffusing particles.Since the continuous limit a → 0 requires such non-physical scales, it is notsurprising that the limiting process (the PRBM) presents some irrealistic propertiesfrom the physical point of view. The similar limitation happens for the electric

Partially Reflected Brownian Motion... 35transport problem for which the smallest physical scale is given by the thicknessof the double layer, being close to the Debye-Hückel length [26, 29]. Evidently,this remark does not devaluate the efficiency of the continuous approach basedon the partially reflected Brownian motion. On the contrary, the mathematicalrigour of this approach justifies the semi-continuous description and simplifies itsstudy by introducing the Dirichlet-to-Neumann operator. However, when dealingwith a mathematical description of a physical problem, one should take care thatdeduced consequences do not go beyond the ranges of the model.The capabilities of the semi-continuous approach are essentially limited by thefact that the governing operator Q (a) is not self-adjoint (the function ω s+an(s) (s ′ )is not symmetric with respect to the permutation of s and s ′ except specific cases).As a consequence, one cannot develop the spectral decomposition (25) of theimpedance. In particular, there is no possibility to distinguish contributions fromdifferent eigenmodes. Although the operator Q (a) is defined naturally by the harmonicmeasure density, it does not provide a proper description of the problem asit was done with the Dirichlet-to-Neumann operator.4.4 Discrete ApproachAnother stochastic approach to Laplacian transport phenomena was developedby Filoche and Sapoval [21]. The main idea is to model the partially reflectedBrownian motion by lattice random walks with partial reflections on the boundary.Actually, one discretizes a given domain Ω by d-dimensional hypercubiclattice of mesh a and considers the following stochastic process: started from aremote source, a random walker jumps to a neighboring site at each step withprobability (2d) −1 . When the walker arrives to a boundary site, it can be reflectedto its neighboring site (belonging to the bulk) with probability ε (and the motioncontinues), or it can be absorbed with probability (1 − ε). The motion continuesuntil the final absorption on the boundary, or the return to the source. One canshow [22] that the discrete parameters a and ε are related by the expression (26)involving the continuous physical parameter Λ.In the discrete description, the harmonic measure density is replaced by thedistribution of hitting probabilities (P 0 ) j on boundary sites j, (simple) randomwalks being started from a remote source. Let Q (a)j,kdenote the probability toarrive to the boundary site k starting from the boundary sites by a random walkin the bulk without hitting the boundary or the source during the walk 5 . One canthus calculate the distribution of probabilities (P Λ ) j to be finally absorbed on the5 We use the same notation Q (a) for the integral operator in semi-continuous approach andfor the matrix of these probabilities since they have the same meaning and even may be used toapproximate each other.

Partially Reflected Brownian Motion... 36boundary sites j, when random walks with partial reflections are started from aremote source. Indeed, the Markov property of this process allows to calculate(P Λ ) j as the sum of contributions provided by random trajectories with 0, 1, 2, ...reflections before the final absorption:(P Λ ) j = (P 0 ) j (1−ε)+ ∑ (P 0 ) k1 εQ (a)k 1 ,j (1−ε)+∑ ∑(P 0 ) k1 εQ (a)k 1 ,k 2εQ (a)k 2 ,j (1−ε)+...k 1 k 1 k 2(we remind that ε = (1 + a/Λ) −1 ). For example, the second term represents theproduct of the following probabilities: to hit a boundary site k 1 , to be reflected toits neighboring site, to arrive to the boundary site j, and to be finally absorbed onit. If one considers P 0 and P Λ as vectors and Q (a) as matrix, the summation overintermediate sites k 1 , k 2 , ... can be understood as matrix product:[∞∑ (P Λ = (1 − ε)) ]εQ(a) nn=0i.e., the distribution of absorption probabilities (P Λ ) j is obtained as the applicationof a linear operator, depending on Q (a) and Λ (or ε), to the distribution ofhitting probabilities (P 0 ) j . The symmetric matrix Q (a) represents a self-adjointoperator, called Brownian self-transport operator. Using the normalization property|Q (a) | ≤ 1 and relation (26) between Λ and ε, one obtains:P Λ = T (a)Λ P 0P 0[T (a)Λ = I + Λ I − ] −1Q(a)aThe operator T (a)Λ, depending on the lattice parameter a, is called (discrete) spreadingoperator. The previous relation, written explicitly as(P Λ ) j = ∑ k((a))(P 0 ) k TΛ k,jallows one to separate random trajectories in two independent parts:• the random walker started from a remote source arrives to the boundary sitek (first factor);• it continues the motion with partial reflections until the final absorption onthe boundary site j (second factor).One concludes that the absorption probabilities (P Λ ) j provide a discrete analogof the spread harmonic measure density, while the matrix ( T (a) )Λis a discreteanalog of the kernel T Λ (s, s ′ ) of the spreading operator T Λ = [I + ΛM] −1k,j.

Partially Reflected Brownian Motion... 37In particular, the bounded operators (I − Q (a) )/a can be understood as discreteapproximations of the Dirichlet-to-Neumann operator M (in resolvent sense). Asfor the semi-continuous approach, we do not furnish the corresponding proofs (see[39] for more details).The advantage of the discrete description with respect to the semi-continuousapproach is based on the fact that the Brownian self-transport operator Q (a) and,consequently, the (discrete) spreading operator T (a)Λare self-adjoint. This propertyallows to employ all the machinery of the spectral theory in order to express thephysical characteristics of Laplacian transport through eigenmodes of this operatorin an explicit way. For example, the spectral decomposition (25) can be writtenin the discrete case. Such decompositions have been used to study Laplaciantransport towards irregular geometries [22, 39]. Moreover, the discrete descriptionsuggests at least two different ways to study the problem numerically: directMonte Carlo simulations and discrete boundary elements method.The discrete description, being intuitively the most simple and useful, maylead to mathematical difficulties when one tries to proceed the continuous limita going to 0. Although the partially reflected Brownian motion is the naturallimit of random walks with partial reflections, its rigorous demonstration, in ourknowledge, is not yet realized in details. The interested reader can find moreinformation on this topic in [71, 72, 73, 74, 75, 76].5 ConclusionThe application of stochastic processes to represent the solution of boundary valueproblems is well known and wide used. In particular, Monte Carlo simulations aregenerally based on this concept. In this paper, we gave a brief overview of Laplaciantransport phenomena in different scientific domains (e.g., physics, electrochemistry,chemistry, physiology) and related stochastic approaches to describethem. The most attention has been paid to the recently developed continuous approachbased on the partially reflected Brownian motion. This stochastic processcan be thought as rigorous mathematical description for random trajectories ofdiffusing particles hitting a semi-permeable interface, in comparison with moreintuitive physical descriptions by semi-continuous and discrete approaches. Thepartially reflected Brownian motion turns out to be the natural limit of the Brownianmotion reflected with jump (semi-continuous approach) and of the latticerandom walks with partial reflections (discrete approach).The profound relation between the partially reflected Brownian motion andthe spectral properties of the Dirichlet-to-Neumann operator M are shown to beuseful for practical purposes. In particular, the kernel of the resolvent operator

Partially Reflected Brownian Motion... 38T Λ = [I + ΛM] −1 gives the probability density T Λ (s, s ′ ) allowing to reconstructthe spread harmonic measure ω x,Λ . Moreover, the spectral decomposition on thecomplete basis of the Dirichlet-to-Neumann operator eigenfunctions leads to theexplicit analytical formula for its density. Consequently, the use of the operatorM is an efficient way to study different probability distributions related to thepartially reflected Brownian motion.The spectral decomposition of the spectroscopic impedance, characterizingthe linear response of the whole working interface, leads to an explicit analyticaldependence on the physical parameter Λ allowing to identify physical and geometricalcontributions which were involved in a complex manner. The harmonicgeometrical spectrum of the working interface contains the complete informationabout its transport properties. The combined use of stochastic characteristics ofthe partially reflected Brownian motion and spectral properties of the Dirichletto-Neumannoperator opens encouraging possibilities for further understandingvarious physical and chemical transport processes in nature. In this light, a moreprofound mathematical analysis of these objects seems to be an important perspectivefor the present study.AcknowledgementThe author thanks Professor B. Sapoval and Professor M. Filoche for valuablediscussions and fruitful collective work on physical aspects of Laplacian transportphenomena.References[1] E. R. Weibel, The Pathway for oxygen. Structure and function in the mammalianrespiratory system (Harvard University Press, Cambridge, Massachusettsand London, England, 1984).[2] B. Mauroy, M. Filoche, E. R. Weibel, and B. Sapoval, “An OptimalBronchial Tree May Be Dangerous, Nature 427, 633 (2004).[3] B. Sapoval, M. Filoche, and E. R. Weibel, “Branched Structures, AcinusMorphology and Optimal Design of Mammalian Lungs”, in Branching innature, Eds. by V. Fleury, J.-F. Gouyet, and M. Leonetti, pp. 225-242 (EDPSciences/Springer Verlag, 2001).[4] D. S. Grebenkov, “NMR survey of the Reflected Brownian Motion”,Rev. Mod. Phys. (submitted).

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