Ergodicity of a nonlinear stochastic reaction-diffusion equation with memory

TL;DR

This paper investigates the long-term statistical behavior of a class of nonlinear stochastic reaction-diffusion equations with memory, establishing conditions for ergodicity and the existence of unique invariant measures.

Contribution

It extends previous ergodicity results to a broader class of semi-linear Volterra equations with memory and stochastic forcing, using the generalized coupling method.

Findings

Existence of statistically steady states with regularity properties

Uniqueness of invariant probability measure under sufficient stochastic forcing

Exponential attraction to the invariant measure

Abstract

We consider a class of semi-linear differential Volterra equations with memory terms, polynomial nonlinearities and random perturbation. For a broad class of nonlinearities, we study statistically steady states of the system and find that they possess regularities compatible with those of the weak solutions. Moreover, if sufficiently many directions in the phase space are stochastically forced, we employ the \emph{generalized coupling} approach to establish the existence and uniqueness of the invariant probability measure to which the system is exponentially attractive. This extends ergodicity results previously established in [Bonaccorsi et al., SIAM J. Math. Anal., 44 (2012)].