Flux large deviations of weakly interacting jump processes via well-posedness of an associated Hamilton-Jacobi equation

TL;DR

This paper proves a large deviation principle for weakly interacting Markov jump processes by establishing well-posedness of an associated Hamilton-Jacobi equation, including time-periodic cases with vanishing period length.

Contribution

It introduces a novel approach linking Hamilton-Jacobi equation well-posedness to large deviations in jump processes, extending to time-periodic jump rates.

Findings

Uniqueness of solutions to the Hamilton-Jacobi equations established.

Large deviation principle derived for empirical measure and flux trajectories.

Results extended to time-periodic jump rates with diminishing period length.

Abstract

We establish uniqueness for a class of first-order Hamilton-Jacobi equations with Hamiltonians that arise from the large deviations of the empirical measure and empirical flux pair of weakly interacting Markov jump processes. As a corollary we obtain a large deviation principle for the trajectory of the empirical measure and empirical flux pair of such processes. As a second corollary we get same result in the setting where the jump-rates are time-periodic, with period-length that decreases to 0.