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 empirical measure and flux pair weakly interacting Markov jump processes. As corollary, we obtain such deviation principle in context processes time-periodic rates which period-length converges to 0.