Support Theorem for Diffusion Processes on Hilbert Spaces

dX(t) = a(X(t))-dw(t) + b(X(t))dt X(t) = x where oeCl(Rn -> Rn (x) Rm\ beCl(Rn^>Rn\ 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 Px of X(t) which is a probability measure on C x([0, T], Rn) 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)