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 :
then .
References