The strong Feller property for singular stochastic PDEs

TL;DR

This paper proves that the Markov semigroups of various singular stochastic PDEs, including KPZ and $ ext{Phi}^4_3$, satisfy the strong Feller property, ensuring uniqueness of invariant measures in some cases.

Contribution

It establishes the strong Feller property for a broad class of singular stochastic PDEs, advancing understanding of their ergodic behavior.

Findings

Strong Feller property holds for KPZ and $ ext{Phi}^4_3$ models.

Uniqueness of the invariant measure for KPZ with periodic boundary conditions.

Implications for ergodicity and long-term behavior of these SPDEs.

Abstract

We show that the Markov semigroups generated by a large class of singular stochastic PDEs satisfy the strong Feller property. These include for example the KPZ equation and the dynamical $\Phi^4_3$ model. As a corollary, we prove that the Brownian bridge measure is the unique invariant measure for the KPZ equation with periodic boundary conditions.