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

Consider a sequence of continuous-time Markov chains (X-t((n)) : t = 0) evolving on a fixed finite state space V. Let I-n be the level two large deviations rate functional for X-t((n)), t, as t -> infinity. Under a hypothesis on the jump rates, we prove that I-n can be written as I-n = I-(0) + Sigma(1 <= p <= q)(1/theta((p))(n))I-(p) for some rate functionals I-(p). The weights theta((p))(n) correspond to the time-scales at which the sequence of Markov chains X-t((n)) exhibit a metastable behavior, and the zero level sets of the rate functionals I-(p) identify the metastable states. (c) 2023 Elsevier B.V. All rights reserved.