Existence and Uniqueness of Invariant Measures for Stochastic Evolution Equations with Weakly Dissipative Drifts

TL;DR

This paper establishes new decay estimates for stochastic evolution equations with weakly dissipative drifts, proving the existence, uniqueness, and convergence rates of invariant measures, and applies these results to specific singular stochastic PDEs.

Contribution

It introduces a novel decay estimate that ensures the uniqueness and existence of invariant measures for a class of stochastic evolution equations with weakly dissipative drifts.

Findings

Proves the existence of invariant measures for the studied equations.

Establishes the uniqueness of invariant measures using decay estimates.

Determines the convergence rate of the transition semigroup to the invariant measure.

Abstract

In this paper, a new decay estimate for a class of stochastic evolution equations with weakly dissipative drifts is established, which directly implies the uniqueness of invariant measures for the corresponding transition semigroups. Moreover, the existence of invariant measures and the convergence rate of corresponding transition semigroup to the invariant measure are also investigated. As applications, the main results are applied to singular stochastic $p$-Laplace equations and stochastic fast diffusion equations, which solves an open problem raised by Barbu and Da Prato in [Stoc. Proc. Appl. 120(2010), 1247-1266].