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

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.