Large deviation properties of weakly interacting processes via weak convergence methods

TL;DR

This paper establishes a large deviation principle for systems of weakly interacting particles modeled by SDEs, using a weak convergence approach that applies to various models including those with delay.

Contribution

It introduces a novel weak convergence method to derive large deviation principles for weakly interacting particle systems, extending to non-diffusion SDEs with delays.

Findings

Large deviation principles are proven for particle systems via weak convergence.

The approach applies to SDEs with delay and non-diffusion types.

Method avoids discretization, simplifying analysis.

Abstract

We study large deviation properties of systems of weakly interacting particles modeled by It\^{o} stochastic differential equations (SDEs). It is known under certain conditions that the corresponding sequence of empirical measures converges, as the number of particles tends to infinity, to the weak solution of an associated McKean-Vlasov equation. We derive a large deviation principle via the weak convergence approach. The proof, which avoids discretization arguments, is based on a representation theorem, weak convergence and ideas from stochastic optimal control. The method works under rather mild assumptions and also for models described by SDEs not of diffusion type. To illustrate this, we treat the case of SDEs with delay.