Donsker-Varadhan asymptotics for degenerate jump Markov processes

TL;DR

This paper establishes large deviation principles for certain degenerate jump Markov processes, extending Donsker-Varadhan asymptotics to non-uniformly ergodic cases with absorbing states.

Contribution

It provides the first joint large deviation results for empirical measure and flow in degenerate jump Markov processes without uniform ergodicity.

Findings

Derived the joint large deviation principle for empirical measure and flow.

Obtained the Donsker-Varadhan rate function for the empirical measure.

Presented a variational expression for the rate function of the empirical flow.

Abstract

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.