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

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