Metastability from the large deviations point of view: A Γ-expansion of the level two large deviations rate functional of non-reversible finite-state Markov chains

Consider a sequence of continuous-time Markov chains (Xt(n):t≥0) evolving on a fixed finite state space V. Let In be the level two large deviations rate functional for Xt(n), as t→∞. Under a hypothesis on the jump rates, we prove that In can be written as In=I(0)+∑1≤p≤q(1/θn(p))I(p) for some rate functionals I(p). The weights θn(p) correspond to the time-scales at which the sequence of Markov chains Xt(n) exhibit a metastable behavior, and the zero level sets of the rate functionals I(p) identify the metastable states.