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.
be a domain (an open and connected set) in
be the Laplace operator, let
be a bounded function on the boundary
, and consider the problem: It can be shown that if a solution
at the (random) first exit point from
for a canonical Brownian motion starting at
See theorem 3 in Kakutani 1944, p. 710.
be a semi-elliptic differential operator on
of the form: where the coefficients
are continuous functions and all the eigenvalues of the matrix
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
can be taken to be the solution to the stochastic differential equation: where
is n-dimensional Brownian motion,
as above, and the matrix field
is chosen so that: For a point
denote the law of
given initial datum
denote expectation with respect to
denote the first exit time of
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
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
satisfies (P2) and there exists a constant