Large deviations for Markov jump processes with mean-field interaction via the comparison principle for an associated Hamilton-Jacobi equation

TL;DR

This paper establishes a large deviation principle for mean-field interacting Markov jump processes using viscosity solutions and Hamilton-Jacobi equations, with applications to models like Ehrenfest and Curie-Weiss dynamics.

Contribution

It introduces a novel analytic approach based on the comparison principle for Hamilton-Jacobi equations to prove large deviations for a broad class of processes.

Findings

Large deviation principle proven for mean-field Markov jump processes.

Comparison principle for Hamilton-Jacobi equations established.

Method to identify Lyapunov functions for McKean-Vlasov equations demonstrated.

Abstract

We prove the large deviation principle for the trajectory of a broad class of mean field interacting Markov jump processes via a general analytic approach based on viscosity solutions. Examples include generalized Ehrenfest models as well as Curie-Weiss spin flip dynamics with singular jump rates. The main step in the proof of the large deviation principle, which is of independent interest, is the proof of the comparison principle for an associated collection of Hamilton-Jacobi equations. Additionally, we show that the large deviation principle provides a general method to identify a Lyapunov function for the associated McKean-Vlasov equation.