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Let $\Xi_n \subset \mathbb R^d$, $n\ge 1$, be a sequence of finite sets andconsider a $\Xi_n$-valued, irreducible, reversible, continuous-time Markovchain $(X^{(n)}_t:t\ge 0)$. Denote by $\mathscr P(\mathbb R^d) $ the set ofprobability measures on $\mathbb R^d$ and by $I_n\colon \mathscr P(\mathbb R^d)\to [0,+\infty)$ the level two large deviations rate functional for $X^{(n)}_t$as $t\to\infty$. We present a general method, based on tools used to prove themetastable behaviour of Markov chains, to derive a full expansion of $I_n$expressing it as $I_n = I^{(0)} \,+\, \sum_{1\le p\le q} (1/\theta^{(p)}_n)\,I^{(p)}$, where $I^{(p)}\colon \mathscr P(\mathbb R^d) \to [0,+\infty]$represent rate functionals independent of $n$ and $\theta^{(p)}_n$ sequencessuch that $\theta^{(1)}_n \to\infty$, $\theta^{(p)}_n / \theta^{(p+1)}_n \to 0$for $1\le p< q$. The speed $\theta^{(p)}_n$ corresponds to the time-scale atwhich the Markov chains $X^{(n)}_t$ exhibits a metastable behavior, and the$I^{(p-1)}$ zero-level sets to the metastable states. To illustrate the theorywe apply the method to random walks in potential fields.

Full $\Gamma$-expansion of reversible Markov chains level two large deviations rate functionals