Weak uniqueness by noise for singular stochastic PDEs

Federico Bertacco, Carlo Orrieri, Luca Scarpa

Published 2023-08-03Version 1

We prove weak uniqueness for a general class of SPDEs on a Hilbert space. The main novelty is that the drift is only defined on a Sobolev-type subspace and no H\"older-continuity assumptions are required. This allows us to cover examples such as equations with divergence-form drift and Cahn-Hilliard-type equations with possibly singular perturbations. The main idea is to consider a suitable coloured Wiener noise so that both the solvability of the SPDE and the regularising effect of the Kolmogorov operator are preserved via stochastic maximal regularity results. As a by-product, this also allows us to generalise the available results of uniqueness by noise for perturbations of the heat equation to higher dimensions.