Level 2.5 large deviations for continuous-time Markov chains with time periodic rates

We consider an irreducible continuous-time Markov chain on a finite state space and with time periodic jump rates and prove the joint large deviation principle for the empirical measure and flow and the joint large deviation principle for the empirical measure and current. By contraction, we get the large deviation principle of three types of entropy production flow. We derive some Gallavotti–Cohen duality relations and discuss some applications