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 problemedit
Let be a domain in and let be a semi-elliptic differential operator on of the form:
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:
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 .
Referencesedit
Kakutani, Shizuo (1944). "Two-dimensional Brownian motion and harmonic functions". Proc. Imp. Acad. Tokyo. 20 (10): 706–714. doi:10.3792/pia/1195572706.
Kakutani, Shizuo (1944). "On Brownian motions in n-space". Proc. Imp. Acad. Tokyo. 20 (9): 648–652. doi:10.3792/pia/1195572742.
Øksendal, Bernt K. (2003). Stochastic Differential Equations: An Introduction with Applications (Sixth ed.). Berlin: Springer. ISBN 3-540-04758-1. (See Section 9)