Stochastic Processes and Boundary Value Problems - The Dirichlet-Poisson Problem

The Dirichlet-Poisson Problem

Let D be a domain in Rn and let L be a semi-elliptic differential operator on C2(Rn; R) of the form

where the coefficients b_i and a_ij are continuous functions and all the eigenvalues of the matrix a(x) = (a_ij(x)) are non-negative. Let f ∈ C(D; R) and g ∈ C(∂D; R). Consider the Poisson problem

The idea of the stochastic method for solving this problem is as follows. First, one finds an Itō diffusion X whose infinitesimal generator A coincides with L on compactly-supported C2 functions f : Rn → R. For example, X can be taken to be the solution to the stochastic differential equation

where B is n-dimensional Brownian motion, b has components b_i as above, and the matrix field σ is chosen so that

For a point x ∈ Rn, let Px denote the law of X given initial datum X₀ = x, and let Ex denote expectation with respect to Px. Let τ_D denote the first exit time of X from D.

In this notation, the candidate solution for (P1) is

provided that g is a bounded function and that

It turns out that one further condition is required:

i.e., for all x, the process X starting at x almost surely leaves D in finite time. Under this assumption, the candidate solution above reduces to

and solves (P1) in the sense that if denotes the characteristic operator for X (which agrees with A on C2 functions), then

Moreover, if v ∈ C2(D; R) satisfies (P2) and there exists a constant C such that, for all x ∈ D,

then v = u.