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Metastable $Γ$-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 $θ^{(p)}_n$ and probability measures with disjoint supports $π^{(p)}_j$, $j\in S_p$, $1\le p \le q$, such that (a) $θ^{(1)}_n \to \infty$, $θ^{(k+1)}_n/θ^{(k)}_n \to \infty$, (b) for all $p$, $x\in V$, $t>0$, starting from $x$, the distribution of $X^{(n)}_{t θ^{(p)}_n}$ converges, as $n\to\infty$, to a convex combination of the probability measures $π^{(p)}_j$. 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 $I_n = I^{(0)} \,+\, \sum_{1\le p\le q} (1/θ^{(p)}_n) \, I^{(p)}$ for some rate functionals $I^{(p)}$ which take finite values only at convex combinations of the measures $π^{(p)}_j$: $I^{(p)}(μ) < \infty$ if, and only if, $μ= \sum_{j\in S_p} ω_j\, π^{(p)}_j$ for some probability measure $ω$ in $S_p$.