Quasi-Ergodicity of Transient Patterns in Stochastic Reaction-Diffusion Equations

TL;DR

This paper investigates the transient patterns in stochastic reaction-diffusion equations using quasi-stationary measures, establishing existence, uniqueness, and convergence properties without small noise restrictions.

Contribution

It introduces a spectral gap-based framework for analyzing quasi-ergodic measures in SPDEs, extending understanding of transient dynamics in stochastic systems.

Findings

Proves existence and uniqueness of quasi-stationary measures.

Establishes exponential convergence rates in $L^2$ norm.

Characterizes system behavior near invariant manifolds conditioned on staying in neighborhoods.

Abstract

We study transient patterns appearing in a class of SPDE using the framework of quasi-stationary and quasi-ergodic measures. In particular, we prove the existence and uniqueness of quasi-stationary and quasi-ergodic measures for a class of reaction-diffusion systems perturbed by additive cylindrical noise. We obtain convergence results in $L^2$ and almost surely, and demonstrate an exponential rate of convergence to the quasi-stationary measure in an $L^2$ norm. These results allow us to qualitatively characterize the behaviour of these systems in neighbourhoods of an invariant manifold of the corresponding deterministic systems at some large time $t>0$, conditioned on remaining in the neighbourhood up to time $t$. The approach we take here is based on spectral gap conditions, and is not restricted to the small noise regime.