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

TL;DR

This paper establishes large deviation principles for empirical measures and currents in continuous-time Markov chains, proving a Gallavotti-Cohen symmetry and analyzing cycle-based currents with applications to fluctuation theorems.

Contribution

It extends large deviation results to joint measures and currents, introduces strong topology LDP, and explores homological cycle currents in Markov chains.

Findings

Proved joint large deviation principle for empirical measure and current.

Established Gallavotti-Cohen symmetry for the current field.

Analyzed large deviations of cycle-based empirical currents.

Abstract

We consider a continuous time Markov chain on a countable state space. We prove a joint large deviation principle (LDP) of the empirical measure and current in the limit of large time interval. The proof is based on results on the joint large deviations of the empirical measure and flow obtained in \cite{BFG}. 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 Gallavotti-Cohen (GC) symmetry for the current field and show that it implies the so-called fluctuation theorem for the GC functional. We also analyze the large deviation properties of generalized empirical currents associated 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 Markov chain. Finally, we discuss in detail some examples.