Weighted stochastic Sobolev spaces and bilinear SPDEs driven by space-time white noise

In this paper we develop basic elements of Malliavin calculus on a weighted L-2(Omega). This class of generalized Wiener functionals is a Hilbert space. It turns out to be substantially smaller than the space of Hida distributions while large enough to accommodate solutions of bilinear stochastic PDEs. As an example, we consider a stochastic advection-diffusion equation driven by space-time white noise in R-d. It is known that for d > 1, this equation has no solutions in L-2(Omega). In contrast, it is shown in the paper that in an appropriately weighted L-2(Omega) there is a unique solution to the stochastic advection-diffusion equation for any d greater than or equal to 1. In addition we present explicit formulas for the Hermite-Fourier coefficients in the Wiener chaos expansion of the solution. (C) 1997 Academic Press.