Large deviations for interacting particle systems: joint mean-field and small-noise limit

TL;DR

This paper establishes a large deviations principle for a system of interacting particles considering both the mean-field limit and the small-noise limit simultaneously, providing a rigorous probabilistic analysis of rare events.

Contribution

It introduces a novel joint large deviations framework for empirical measures and currents in interacting particle systems under combined mean-field and small-noise limits.

Findings

Proves a large deviations principle for empirical measures and currents

Uses tilting and entropy methods for the proof

Employs deterministic control problems for analysis

Abstract

We consider a system of stochastic interacting particles in $\mathbb{R}^d$ and we describe large deviations asymptotics in a joint mean-field and small-noise limit. Precisely, a large deviations principle (LDP) is established for the empirical measure and the stochastic current, as the number of particles tends to infinity and the noise vanishes, simultaneously. We give a direct proof of the LDP using tilting and subsequently exploiting the link between entropy and large deviations. To this aim we employ consistency of suitable deterministic control problems associated to the stochastic dynamics.