Metastable $\Gamma$-expansion of finite state Markov chains level two large deviations rate functions

We examine two analytical characterisation of the metastable behavior of a Markov chain. The first one expressed in terms of its transition probabilities, and the second one in terms of its large deviations rate functional. Consider a sequence of continuous-time Markov chains $(X^{(n)}_t:t\ge 0)$ evolving on a fixed finite state space $V$. Under a hypothesis on the jump rates, we prove the existence of times-scales $\theta^{(p)}_n$ and probability measures with disjoint supports $\pi^{(p)}_j$, $j\in S_p$, $1\le p \le q$, such that (a) $\theta^{(1)}_n \to \infty$, $\theta^{(k+1)}_n/\theta^{(k)}_n \to \infty$, (b) for all $p$, $x\in V$, $t>0$, starting from $x$, the distribution of $X^{(n)}_{t \theta^{(p)}_n}$ converges, as $n\to\infty$, to a convex combination of the probability measures $\pi^{(p)}_j$. The weights of the convex combination naturally depend on $x$ and $t$. Let $I_n$ be the level two large deviations rate functional for $X^{(n)}_t$, as $t\to\infty$. Under the same hypothesis on the jump rates and assuming, furthermore, that the process is reversible, 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 $I^{(p)}$ which take finite values only at convex combinations of the measures $\pi^{(p)}_j$: $I^{(p)}(\mu) < \infty$ if, and only if, $\mu = \sum_{j\in S_p} \omega_j\, \pi^{(p)}_j$ for some probability measure $\omega$ in $S_p$.