  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.
