Weighted Stochastic Sobolev Spaces and Bilinear SPDEs Driven by Space–Time White Noise

✍ Scribed by David Nualart; Boris Rozovskii

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 426 KB
• Volume: 149
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.1996.3091

✦ Synopsis

In this paper we develop basic elements of Malliavin calculus on a weighted L 2 (0). 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 (0). In contrast, it is shown in the paper that in an appropriately weighted L 2 (0) there is a unique solution to the stochastic advection-diffusion equation for any d 1. In addition we present explicit formulas for the Hermite Fourier coefficients in the Wiener chaos expansion of the solution.

1997 Academic Press where W is a white noise on [0, T ]_R d , and L is a uniformly elliptic second order differential operator.