Harnack Inequalities and Ergodicity of Stochastic Reaction-Diffusion Equation in $L^p$

TL;DR

This paper establishes Harnack inequalities for stochastic reaction-diffusion equations in $L^p$, leading to results on the ergodicity and uniqueness of invariant measures for the associated Markov semigroup.

Contribution

It introduces a novel coupling by change of measure approach to derive Harnack inequalities for these equations in $L^p$, enabling ergodicity analysis.

Findings

Harnack inequalities hold for stochastic reaction-diffusion equations in $L^p$.

The Markov semigroup has a unique ergodic invariant measure in $L^p$.

Results are robust to super-linear growth drifts with negative leading coefficients.

Abstract

We derive Harnack inequalities for a stochastic reaction-diffusion equation with dissipative drift driven by additive irregular noise in the $L^p$-space for any $p \ge 2$. These inequalities are utilized to investigate the ergodicity of the corresponding Markov semigroup $(P_t)$. The main ingredient of our method is a coupling by the change of measure. Applying our results to the stochastic reaction-diffusion equation with a super-linear growth drift having a negative leading coefficient, perturbed by a Lipschitz term, indicates that $(P_t)$ possesses a unique and thus ergodic invariant measure in $L^p$ for all $p \ge 2$, which is independent of the Lipschitz term.