An optimal regularity result for Kolmogorov equations and weak uniqueness for some critical SPDEs

Abstract

We show uniqueness in law for the critical SPDE dXt=AXtdt+(−A)1/2F(X(t))dt+dWt,X0=x∈H, where A:dom(A)⊂H→H is a negative definite self-adjoint operator on a separable Hilbert space H having A−1 of trace class and W is a cylindrical Wiener process on H. Here, F:H→H can be continuous with, at most, linear growth (some functions F which grow more than linearly can also be considered). This leads to new uniqueness results for generalized stochastic Burgers equations and for three-dimensional stochastic Cahn–Hilliard-type equations which have interesting applications. To get weak uniqueness, we use an infinite dimensional localization principle and also establish a new optimal regularity result for the Kolmogorov equation λu−Lu=f associated to the SPDE when F=0 (λ>0, f:H→R Borel and bounded). In particular, we prove that the first derivative Du(x) belongs to dom((−A)1/2), for any x∈H, and supx∈H|(−A)1/2Du(x)|H=‖(−A)1/2Du‖0≤C‖f‖0.