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Consider a sequence of continuous-time Markov chains $(X^{(n)}_t:t\ge 0)$evolving on a fixed finite state space $V$. Let $I_n$ be the level two largedeviations rate functional for $X^{(n)}_t$, as $t\to\infty$. Under a hypothesison the jump rates, we prove that $I_n$ can be written as $I_n = I^{(0)} \,+\,\sum_{1\le p\le q} (1/\theta^{(p)}_n) \, I^{(p)}$ for some rate functionals$I^{(p)}$. The weights $\theta^{(p)}_n$ correspond to the time-scales at whichthe sequence of Markov chains $X^{(n)}_t$ exhibit a metastable behavior, andthe zero level sets of the rate functionals $I^{(p)}$ identify the metastablestates.

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