Support Theorem for Diffusion Processes on Hilbert Spaces

Abstract

Introduction In this paper we will prove a support theorem for infinite dimensional diffusion processes on Hilbert spaces. In finite dimensional cases we have a celebrated Stroock-Varadhan's theory ([1], [2]). We briefly review their theory. (See §2 for the below notation.) Let X(t) (0 ^ t ^ T) be the diffusion process governed by the following stochastic differential equation (abbreviation, SDE). (1) dX(t) = a(X(t))-dw(t) + b(X(t))dt X(t) = x where oeCl(R n-> R n (x) R m \ beCl(R n ^>R n \ and w(t) is an m-dimensional Brownian motion. The notation-dw(t) denotes the Stratonovich stochastic differential. The problem is to determine the topological support of the diffusion measure P x of X(t) which is a probability measure on C x ([0, T], R n) endowed with the uniform convergence topology. To prove the support theorem they first used the approximation theorem in the following. Let £(•, h) be the solution of the following ordinary differential equation (ODE), (2) {(t, fc) = *(£(*, h))h(t) + b($(t, h)) £(0, x) = x where ft is a piecewise smooth function from [0, T] to R m with ft(0) = 0 and let T aT Then £(•, w k) converges to X(t, w) uniformly as fc->oo a.s., which yields Communicated by H. Araki, April 25, 1989.