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.

History

The link between semi-elliptic operators and stochastic processes, followed by their use to solve boundary value problems, is repeatedly and independently rediscovered in the early-mid-20th century.

The connection that Kakutani makes between stochastic differential equations and the Itō process is effectively the same as Kolmogorov's forward equation, made in 1931, which is only later recognized as the Fokker–Planck equation, first presented in 1914-1917. The solution of a boundary value problem by means of expectation values over stochastic processes is now more commonly known not under Kakutani's name, but as the Feynman–Kac formula, developed in 1947.

These results are founded on the use of the Itō integral, required to integrate a stochastic process. But this is also independently rediscovered as the Stratonovich integral; the two forms can be translated into one-another by an offset.

Introduction: Kakutani's solution to the classical Dirichlet problem

be a domain (an open and connected set be the Laplace operator be a bounded function on the boundary, and consider the problem:

is the expected value for a canonical Brownian motion. See theorem 3 in Kakutani 1944, p.710.

The Dirichlet–Poisson problem

be a semi-elliptic differential operator 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 on compactly-supported can be taken to be the solution to the stochastic differential equation:

is n-dimensional Brownian motion, has components as above, and the matrix field is chosen so that:

. Let denote the first exit time.

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

is a bounded function and that:

It turns out that one further condition is required:

almost surely in finite time. Under this assumption, the candidate solution above reduces to:

functions), then:

:

.

References