In mathematics, some boundary value problems can be solved using the methods of stochastic analysis. Perhaps the most celebrated example is Shizuo Kakutani's 1944 solution of the Dirichlet problem for the Laplace operator using Brownian motion. However, it turns out that for a large class of semi-elliptic second-order partial differential equations the associated Dirichlet boundary value problem can be solved using an Itō process that solves an associated stochastic differential equation.
Introduction: Kakutani's solution to the classical Dirichlet problem
Let be a domain (an open and connected set) in . Let be the Laplace operator, let be a bounded function on the boundary , and consider the problem:
It can be shown that if a solution exists, then is the expected value of at the (random) first exit point from for a canonical Brownian motion starting at . See theorem 3 in Kakutani 1944, p. 710.
The Dirichlet–Poisson problem
Let be a domain in and let be a semi-elliptic differential operator on of the form:
where the coefficients and are continuous functions and all the eigenvalues of the matrix are non-negative. Let and . Consider the Poisson problem:
The idea of the stochastic method for solving this problem is as follows. First, one finds an Itō diffusion whose infinitesimal generator coincides with on compactly-supported functions . For example, can be taken to be the solution to the stochastic differential equation:
where is n-dimensional Brownian motion, has components as above, and the matrix field is chosen so that:
For a point , let denote the law of given initial datum , and let denote expectation with respect to . Let denote the first exit time of from .
In this notation, the candidate solution for (P1) is:
provided that is a bounded function and that:
It turns out that one further condition is required:
For all , the process starting at almost surely leaves 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 (which agrees with on functions), then:
Moreover, if satisfies (P2) and there exists a constant such that, for all :