Donsker-Varadhan asymptotics for degenerate jump Markov processes

We consider a class of continuous time Markov chains on a compact metric space that admit an invariant measure strictly positive on open sets together with absorbing states. We prove the joint large deviation principle for the empirical measure and flow. Due to the lack of uniform ergodicity, the zero level set of the rate function is not a singleton. As corollaries, we obtain the Donsker-Varadhan rate function for the empirical measure and a variational expression of the rate function for the empirical flow.