Large deviations and mixing for dissipative PDE's with unbounded random kicks

TL;DR

This paper investigates exponential mixing and large deviations for Markov processes driven by dissipative PDEs with unbounded random kicks, extending previous results to more general noise conditions and specific PDE systems.

Contribution

It establishes existence, uniqueness, and exponential stability of stationary measures, and proves a large deviation principle for occupation measures in the context of unbounded noise.

Findings

Existence and uniqueness of stationary measure.

Exponential stability in Kantorovich-Wasserstein metric.

Large deviation principle for occupation measures.

Abstract

We study the problem of exponential mixing and large deviations for discrete-time Markov processes associated with a class of random dynamical systems. Under some dissipativity and regularisation hypotheses for the underlying deterministic dynamics and a non-degeneracy condition for the driving random force, we discuss the existence and uniqueness of a stationary measure and its exponential stability in the Kantorovich-Wasserstein metric. We next turn to the large deviation principle and establish its validity for the occupation measures of the Markov processes in question. The obtained results extend those established earlier for the case of the bounded noise and can be applied to the 2D Navier-Stokes system in a bounded domain and to the complex Ginzburg-Landau equation.