Moderate deviations for fully coupled multiscale weakly interacting particle systems

TL;DR

This paper establishes a moderate deviations principle for the empirical distribution of fully coupled weakly interacting diffusions in a two-scale environment, using weak convergence methods and control problem representations.

Contribution

It introduces a novel approach to analyze moderate deviations in multiscale interacting particle systems via effective control problems and Sobolev representations.

Findings

Derived a moderate deviations rate function in a Sobolev form.

Established ergodic theorems related to the system.

Analyzed regularity of Poisson equations for McKean-Vlasov problems.

Abstract

We consider a collection of fully coupled weakly interacting diffusion processes moving in a two-scale environment. We study the moderate deviations principle of the empirical distribution of the particles' positions in the combined limit as the number of particles grow to infinity and the time-scale separation parameter goes to zero simultaneously. We make use of weak convergence methods, which provide a convenient representation for the moderate deviations rate function in terms of an effective mean field control problem. We rigorously obtain equivalent representations for the moderate deviations rate function in an appropriate "negative Sobolev" form, which is reminiscent of the large deviations rate function for the empirical measure of weakly interacting diffusions obtained in the 1987 seminal paper by Dawson-G\"{a}rtner. In the course of the proof we obtain related ergodic theorems and we rigorously study the regularity of Poisson type of equations associated to McKean-Vlasov problems, both of which are topics of independent interest.