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

TL;DR

This work addresses Harnack inequalities and ergodicity for a stochastic reaction-diffusion equation in the Banach space $L^p$, driven by additive irregular noise and featuring a dissipative, polynomial-type drift. The authors develop a coupling by change of measure, enabling log- and power-Harnack inequalities that yield strong Feller properties and, under suitable dimension conditions, uniqueness of the invariant measure. They prove existence (and, with extra hypotheses, uniqueness and ergodicity) of an invariant measure with full support in $L^p$, and provide density estimates relative to this measure via the Harnack framework. The results extend Harnack-inequality techniques to SPDEs in Banach spaces and provide quantitative ergodic guarantees for stochastic Allen–Cahn-type equations with unbounded noise operators, with implications for long-time behavior and regularity of solutions.

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.

Theorems & Definitions (23)

• Remark 2.1
• Remark 2.2
• Example 2.1
• Example 2.2
• Remark 2.3
• Theorem 2.1
• Theorem 2.2
• Lemma 3.1
• Remark 3.1
• Remark 3.2