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

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.

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