Harnack Inequality and Applications for Stochastic Evolution Equations with Monotone Drifts

TL;DR

This paper proves a dimension-free Harnack inequality for stochastic evolution equations with monotone drifts and explores its implications for ergodicity, contractivity, and spectral properties of the associated transition semigroups.

Contribution

It establishes a general Harnack inequality for a broad class of stochastic evolution equations with monotone drifts, extending previous results and applications.

Findings

Proved dimension-free Harnack inequality for stochastic evolution equations.

Established ergodicity, hyper-contractivity, and compactness of transition semigroups.

Demonstrated exponential convergence to invariant measures and spectral gap existence.

Abstract

In this paper, the dimension-free Harnack inequality is proved for the associated transition semigroups to a large class of stochastic evolution equations with monotone drifts. As applications, the ergodicity, hyper-(or ultra-)contractivity and compactness are established for the corresponding transition semigroups. Moreover, the exponential convergence of the transition semigroups to invariant measure and the existence of a spectral gap are also derived. The main results are applied to many concrete stochastic evolution equations such as stochastic reaction-diffusion equations, stochastic porous media equations and the stochastic p-Laplace equation in Hilbert space.