Quasi-continuity method for mean-field diffusions: large deviations and central limit theorem

TL;DR

This paper establishes large deviation principles and a central limit theorem for empirical measures of mean-field diffusions with path-dependent coefficients, including degenerate cases, extending classical methods to a mean-field context.

Contribution

It introduces a quasi-continuity method for mean-field diffusions, enabling analysis of large deviations and CLT in complex, path-dependent, and degenerate settings.

Findings

Proved pathwise large deviation principle in Wasserstein topology.

Established a pathwise central limit theorem for empirical measures.

Derived uniform-in-time-step fluctuation and large deviation estimates.

Abstract

A pathwise large deviation principle in the Wasserstein topology and a pathwise central limit theorem are proved for the empirical measure of a mean-field system of interacting diffusions. The coefficients are path-dependent. The framework allows for degenerate diffusion matrices, which may depend on the empirical measure, including mean-field kinetic processes. The main tool is an extension of Tanaka's pathwise construction to non-constant diffusion matrices. This can be seen as a mean-field analogous of Azencott's quasi-continuity method for the Freidlin-Wentzell theory. As a by-product, uniform-in-time-step fluctuation and large deviation estimates are proved for a discrete-time version of the meanfield system. Uniform-in-time-step convergence is also proved for the value function of some mean-field control problems with quadratic cost.