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

Abstract

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

Stochastic Partial Differential Equations Driven by Purely Spatial Noise

“…[11,12,25]. For historical remarks regarding other types of generalized solutions and applications to SPDEs see the review paper [18] and the references therein.…”

A note on stochastic elliptic models

“…(1) for a log-normal coefficient a(x, x) is square integrable in the probability space, the strong ellipticity still fails since a log-normal random field is not strictly positive from below, which implies that the Lax-Milgram lemma cannot be applied directly. Then we have to study the solution in a space larger than L 2 ðFÞ, where we usually associate a weight either to the probability measure or to each term of the orthonormal basis of the L 2 ðFÞ space [7,12,9,13], see Section 3.2, where F indicates a complete probability space.…”

“…Let aðx; xÞ ¼ a2J a a ðxÞH a ðnÞ and a }ðÀ1Þ ðx; xÞ ¼ P a2Jã b ðxÞ-H b ðnÞ. Substituting them into Eq (12). and comparing the coefficients of H a (n), we obtaiñ…”

Lyapunov exponent estimates of a class of higher-order stochastic Anderson models

“…There have been extensive studies on this model (see e.g. [6,7,8,9,10,11]). In those articles, the authors proved the existence and uniqueness of the solution and studied at greater length the asymptotic behaviors of the solution to (1.1).…”