We show uniqueness in law for the critical SPDE
where is a negative definite self-adjoint operator on a separable Hilbert space H having of trace class and W is a cylindrical Wiener process on H. Here, 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 associated to the SPDE when (, Borel and bounded). In particular, we prove that the first derivative belongs to , for any , and .
"An optimal regularity result for Kolmogorov equations and weak uniqueness for some critical SPDEs." Ann. Probab. 49 (3) 1310 - 1346, May 2021. https://doi.org/10.1214/20-AOP1482