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

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)