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

Wei Liu,Jonas Toelle

doi:10.1214/ecp.v16-1643

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].

Highlights

In recent years, the variational approach has been used intensively by many authors to analyze semilinear and quasilinear stochastic partial differential equations
The existence of an invariant measure has been established by Wang and the first named author in [18, 15], and recently by Barbu and Da Prato in [2] for stochastic fast diffusion equations under some weaker assumptions
In this work we prove the uniqueness of invariant measures in a more general setting and do not assume any non-degeneracy of the noise

Summary

Introduction

The variational approach has been used intensively by many authors to analyze semilinear and quasilinear stochastic partial differential equations. The existence of an invariant measure has been established by Wang and the first named author in [18, 15], and recently by Barbu and Da Prato in [2] for stochastic fast diffusion equations under some weaker assumptions. It is more difficult, to derive the uniqueness of invariant measures for this type of stochastic equations due to the lack of strong dissipativity for the drift. (A1) Hemicontinuity of A: The map λ → V ∗ A(v1 + λv2), v V is continuous on R

H v1 β V

Proof of the main results

Applications