Metastability from the large deviations point of view: A $\mathrm{\Gamma }$-expansion of the level two large deviations rate functional of non-reversible finite-state Markov chains

Consider a sequence of continuous-time Markov chains $(X_t^{(n)}:t\ge 0)$ evolving on a fixed finite state space $V$. Let $I_n$ be the level two large deviations rate functional for $X_t^{(n)}$, as $t\to \mathrm{\infty }$. Under a hypothesis on the jump rates, we prove that $I_n$ can be written as $I_n=I^{(0)}+{\sum }_{1\le p\le q}(1\mathrm{/}{\theta }_n^{(p)})I^{(p)}$ for some rate functionals $I^{(p)}$. The weights ${\theta }_n^{(p)}$ 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.