Improved moment estimates for invariant measures of semilinear diffusions in Hilbert spaces and applications

TL;DR

This paper establishes new regularity and integrability estimates for invariant measures of semilinear stochastic PDEs in Hilbert spaces, and applies these results to prove uniqueness of associated operators, with examples including Burgers and thin-film models.

Contribution

It introduces a novel approach combining pathwise estimates and a new technique to prove $L^1$-uniqueness of the Kolmogorov operator for semilinear diffusions.

Findings

A priori estimates for invariant measures.

Proof of $L^1$-uniqueness of the Kolmogorov operator.

Applications to stochastic Burgers and thin-film equations.

Abstract

We study regularity properties for invariant measures of semilinear diffusions in a separable Hilbert space. Based on a pathwise estimate for the underlying stochastic convolution, we prove a priori estimates on such invariant measures. As an application, we combine such estimates with a new technique to prove the $L^1$-uniqueness of the induced Kolmogorov operator, defined on a space of cylindrical functions. Finally, examples of stochastic Burgers equations and thin-film growth models are given to illustrate our abstract result.