Moderate deviations for systems of slow-fast stochastic reaction-diffusion equations

TL;DR

This paper establishes the Moderate Deviation Principle for slow-fast stochastic reaction-diffusion systems, providing explicit rate functions and addressing challenges unique to infinite-dimensional settings.

Contribution

It derives the exact form of the moderate deviations rate function for infinite-dimensional reaction-diffusion systems with multiple scales, using weak convergence and stochastic control methods.

Findings

Explicit rate function for moderate deviations in reaction-diffusion systems

Handling of infinite-dimensional issues absent in finite-dimensional cases

Comparison of moderate deviations with large deviation principles

Abstract

The goal of this paper is to study the Moderate Deviation Principle (MDP) for a system of stochastic reaction-diffusion equations with a time-scale separation in slow and fast components and small noise in the slow component. Based on weak convergence methods in infinite dimensions and related stochastic control arguments, we obtain an exact form for the moderate deviations rate function in different regimes as the small noise and time-scale separation parameters vanish. Many issues that appear due to the infinite dimensionality of the problem are completely absent in their finite-dimensional counterpart. In comparison to corresponding Large Deviation Principles, the moderate deviation scaling necessitates a more delicate approach to establishing tightness and properly identifying the limiting behavior of the underlying controlled problem. The latter involves regularity properties of a solution of an associated elliptic Kolmogorov equation on Hilbert space along with a finite-dimensional approximation argument.