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

TL;DR

This paper proves weak uniqueness for certain critical stochastic partial differential equations (SPDEs) with locally H"older continuous drift, using an infinite dimensional localization principle and optimal regularity results for associated Kolmogorov equations.

Contribution

It establishes new weak uniqueness results for critical SPDEs with non-globally Lipschitz drift, extending previous work to more general nonlinearities.

Findings

Proves weak uniqueness for critical SPDEs with locally H"older continuous drift.

Develops an infinite dimensional localization principle for proving uniqueness.

Derives an optimal regularity result for Kolmogorov equations in infinite dimensions.

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)].