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 ≥ 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 ∈ Sp, 1 ≤ p ≤ q, such that (a) θ (1) n → ∞, θ (k+1) n /θ (k) n → ∞, (b) for all p, x ∈ V , t > 0, starting from x, the distribution of X (n) tθ (p) n converges, as n → ∞, to a convex combination of the probability measures π (p) j . The weights of the convex combination naturally depend on x and t. Let In be the level two large deviations rate functional for X (n) t , as t → ∞. Under the same hypothesis on the jump rates and assuming, furthermore, that the process is reversible, we prove that In can be written as In = I(0) + ∑ 1≤p≤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) < ∞ if, and only if, μ = ∑ j∈Sp ωj π (p) j for some probability measure ω in Sp.