Large deviations from a stationary measure for a class of dissipative PDE's with random kicks

TL;DR

This paper establishes a large deviation principle for occupation measures of dissipative PDEs with random kicks, demonstrating how solutions deviate from their stationary measure over time.

Contribution

It introduces a large deviation framework for dissipative PDEs with non-degenerate random forcing, extending understanding of their long-term probabilistic behavior.

Findings

Proves a large deviation principle for occupation measures

Uses Kifer's criterion and Lyapunov-Schmidt reduction

Shows uniqueness of stationary measure under non-degenerate forcing

Abstract

We study a class of dissipative PDE's perturbed by a bounded random kick force. It is assumed that the random force is non-degenerate, so that the Markov process obtained by the restriction of solutions to integer times has a unique stationary measure. The main result of the paper is a large deviation principle for occupation measures of the Markov process in question. The proof is based on Kifer's large deviation criterion, a Lyapunov-Schmidt type reduction, and an abstract result on large-time asymptotic for generalised Markov semigroups.