Flows, currents, and cycles for Markov Chains: large deviation asymptotics

We consider a continuous time Markov chain on a countable state space. We prove a joint large deviation principle (LDP) of the e m- pirical measure and current in the limit of large time interval. The proo f is based on results on the joint large deviations of the empirical meas ure and flow obtained in [5]. By improving such results we also show, under additional assumptions, that the LDP holds with the strong L 1 topology on the space of currents. We deduce a general version of the Galla votti– Cohen (GC) symmetry for the current field and show that it implies th e so–called fluctuation theorem for the GC functional. We also analyze the large deviation properties of generalized empirical currents assoc iated to a fundamental basis in the cycle space, which, as we show, are given by the first class homological coefficients in the graph underlying the Ma rkov chain. Finally, we discuss in detail some examples.