LINEAR PARABOLIC STOCHASTIC PDES AND WIENER CHAOS

We study Cauchy's problem for a second-order linear parabolic stochast ic partial differential equation (SPDE) driven by a cylindrical Browni an motion. Existence and uniqueness of a generalized (soft) solution i s established in Sobolev, Holder, and Lipschitz classes. We make only minimal assumptions, virtually identical to those common to similar de terministic problems. A stochastic Feynman-Kac formula for the soft so lution is also derived. It is shown that the soft solution allows a Wi ener chaos expansion and that the coefficients of this expansion can b e computed recursively by solving a simple system of parabolic PDEs.