Large Deviations for Slow-Fast Mean-Field Diffusions

TL;DR

This paper studies large deviations in slow-fast mean-field diffusions, extending existing results to include the influence of the fast process laws on the slow component, and provides an explicit rate function formula.

Contribution

It introduces a novel approach using functional occupation measures and feedback controls to establish large deviation principles in complex mean-field systems.

Findings

Derived explicit rate function for large deviations.

Extended large deviation results to systems with law-dependent fast processes.

Established upper and lower bounds for the Laplace principle.

Abstract

The aim of this paper is to investigate the large deviations for a class of slow-fast mean-field diffusions, which extends some existing results to the case where the laws of fast process are also involved in the slow component. Due to the perturbations of fast process and its time marginal law, one cannot prove the large deviations based on verifying the powerful weak convergence criterion directly. To overcome this problem, we employ the functional occupation measure, which combined with the notion of the viable pair and the controls of feedback form to characterize the limits of controlled sequences and justify the upper and lower bounds of Laplace principle. As a consequence, the explicit representation formula of the rate function for large deviations is also presented.