Large deviations for interacting diffusions with path-dependent McKean-Vlasov limit

TL;DR

This paper establishes a large deviation principle for a system of path-dependent interacting diffusions in random media, characterizing the limiting behavior via a path-dependent McKean-Vlasov diffusion and applying it to models like delayed Kuramoto and Galves-L"ocherbach.

Contribution

It introduces a large deviation principle for path-dependent mean-field diffusions with explicit rate function and characterizes the limit as a McKean-Vlasov diffusion.

Findings

Large deviation principle with explicit rate function established.

Characterization of the limit as a path-dependent McKean-Vlasov diffusion.

Application to delayed stochastic Kuramoto and Galves-L"ocherbach models.

Abstract

We consider a mean-field system of path-dependent stochastic interacting diffusions in random media over a finite time window. The interaction term is given as a function of the empirical measure and is allowed to be non-linear and path dependent. We prove that the sequence of empirical measures of the full trajectories satisfies a large deviation principle with explicit rate function. The minimizer of the rate function is characterized as the path-dependent McKean-Vlasov diffusion associated to the system. As corollary, we obtain a strong law of large numbers for the sequence of empirical measures. The proof is based on a decoupling technique by associating to the system a convenient family of product measures. To illustrate, we apply our results for the delayed stochastic Kuramoto model and for a SDE version of Galves-L\"ocherbach model.