  We prove a large-deviation principle (LDP) for the sample paths of jump
Markov processes in the small noise limit when, possibly, all the jump rates
vanish uniformly, but slowly enough, in a region of the state space. We further
discuss the optimality of our assumptions on the decay of the jump rates. As a
direct application of this work we relax the assumptions needed for the
application of LDPs to, e.g., Chemical Reaction Network dynamics, where
vanishing reaction rates arise naturally particularly the context of mass
action kinetics.
