Large deviations and averaging for systems of slow--fast stochastic reaction--diffusion equations

TL;DR

This paper establishes a large deviation principle for stochastic reaction-diffusion systems with slow and fast components, using weak convergence methods to analyze the limiting behavior and effective measures in an infinite-dimensional setting.

Contribution

It introduces a novel approach to characterize large deviations in infinite-dimensional slow-fast stochastic systems via viable pairs and decoupling techniques.

Findings

Large deviation principle derived for infinite-dimensional SRDEs

Decoupling of fast process and control simplifies analysis

Variational representation of the large deviation action functional obtained

Abstract

We study a large deviation principle for a system of stochastic reaction--diffusion equations (SRDEs) with a separation of fast and slow components and small noise in the slow component. The derivation of the large deviation principle is based on the weak convergence method in infinite dimensions, which results in studying averaging for controlled SRDEs. By appropriate choice of the parameters, the fast process and the associated control that arises from the weak convergence method decouple from each other. We show that in this decoupling case one can use the weak convergence method to characterize the limiting process via a "viable pair" that captures the limiting controlled dynamics and the effective invariant measure simultaneously. The characterization of the limit of the controlled slow-fast processes in terms of viable pair enables us to obtain a variational representation of the large deviation action functional. Due to the infinite--dimensional nature of our set--up, the proof of tightness as well as the analysis of the limit process and in particular the proof of the large deviations lower bound is considerably more delicate here than in the finite--dimensional situation. Smoothness properties of optimal controls in infinite dimensions (a necessary step for the large deviations lower bound) need to be established. We emphasize that many issues that are present in the infinite dimensional case, are completely absent in finite dimensions.