Large deviations of mean-field interacting particle systems in a fast varying environment

TL;DR

This paper establishes a large deviation principle for the joint behavior of mean-field particle systems and fast-varying environments, extending existing diffusion results to jump processes with a stochastic exponential approach.

Contribution

It introduces a path-space large deviation principle for coupled mean-field systems with jumps in a fast environment, broadening the scope of previous diffusion-based models.

Findings

Path-space large deviation principle derived

Characterization of the rate function via variational problem

Extension from diffusions to jump processes

Abstract

This paper studies large deviations of a ``fully coupled" finite state mean-field interacting particle system in a fast varying environment. The empirical measure of the particles evolves in the slow time scale and the random environment evolves in the fast time scale. Our main result is the path-space large deviation principle for the joint law of the empirical measure process of the particles and the occupation measure process of the fast environment. This extends previous results known for two time scale diffusions to two time scale mean-field models with jumps. Our proof is based on the method of stochastic exponentials. We characterise the rate function by studying a certain variational problem associated with an exponential martingale.