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 weightedL2(Ω). 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 Rd. It is known that ford>1, this equation has no solutions inL2(Ω). In contrast, it is shown in the paper that in an appropriately weightedL2(Ω) there is a unique solution to the stochastic advection-diffusion equation for anyd⩾1. In addition we present explicit formulas for the Hermite–Fourier coefficients in the Wiener chaos expansion of the solution.