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

Abstract

We show uniqueness in law for the critical SPDE \begin{eqnarray} \label{qq1} dX_t = AX_t dt + (-A)^{1/2}F(X(t))dt + dW_t,\;\; X_0 =x \in H, \end{eqnarray} where $A$ $ : \text{dom}(A) \subset H \to 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 \to H $ can be locally H\"older 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. We do not know if uniqueness holds under the sole assumption of continuity of $F$ plus growth condition as stated in [Priola, Ann. of Prob. 49 (2021)]. To get weak uniqueness we use an infinite dimensional localization principle and an optimal regularity result for the Kolmogorov equation $ \lambda u - L u = f$ associated to the SPDE when $F = z \in H$ is constant and $\lambda >0$. This optimal result is similar to a theorem of [Da Prato, J. Evol. Eq. 3 (2003)].