Linear Parabolic Stochastic PDE and Wiener Chaos

Abstract

We study Cauchy's problem for a second-order linear parabolic stochastic partial differential equation (SPDE) driven by a cylindrical Brownian motion. Existence and uniqueness of a generalized (soft) solution is established in Sobolev, Hölder, and Lipschitz classes. We make only minimal assumptions, virtually identical to those common to similar deterministic problems. A stochastic Feynman--Kac formula for the soft solution is also derived. It is shown that the soft solution allows a Wiener chaos expansion and that the coefficients of this expansion can be computed recursively by solving a simple system of parabolic PDEs.