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Flows, currents, and cycles for Markov chains: Large deviation asymptotics

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  • Bertini, Lorenzo
  • Faggionato, Alessandra
  • Gabrielli, Davide

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 Bertini et al. (in press). By improving such results we also show, under additional assumptions, that the LDP holds with the strong L1 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.

Suggested Citation

  • Bertini, Lorenzo & Faggionato, Alessandra & Gabrielli, Davide, 2015. "Flows, currents, and cycles for Markov chains: Large deviation asymptotics," Stochastic Processes and their Applications, Elsevier, vol. 125(7), pages 2786-2819.
    • Handle: RePEc:eee:spapps:v:125:y:2015:i:7:p:2786-2819
    DOI: 10.1016/j.spa.2015.02.001
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    References listed on IDEAS

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    1. Maes, Christian & Netočný, Karel & Wynants, Bram, 2008. "Steady state statistics of driven diffusions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(12), pages 2675-2689.
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