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

Liu, Zhihui

Published 2020-08-04Version 1

We derive Harnack inequalities for a stochastic reaction-diffusion equation with dissipative drift driven by additive rough noise in the $L^p(\OOO)$-space, for any $p \ge 2$, where $\OOO$ is a bounded, open subset of $\rr^d$. These inequalities are used to study the ergodicity and contractivity of the corresponding Markov semigroup $(P_t)_{t \ge 0}$. The main ingredients of our method are a coupling by the change of measure and a uniform exponential moments' estimation $\sup_{t \ge 0} \ee \exp(\epsilon \|\cdot\|_p^p)$ with some positive constant $\epsilon$ for the solution. Applying our results to the stochastic reaction-diffusion equation with a super-linear growth drift having negative leading coefficient, perturbed by a Lipschitz term, indicates that $(P_t)_{t \ge 0}$ possesses a unique and thus ergodic invariant measure and is supercontractive in $L^p$, which is independent of the Lipschitz term.