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 sequence of Markov chains. 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 (Xt(n):t≥0) evolving on a fixed finite state space V. Under a hypothesis on the jump rates, we prove the existence of time-scales θn(p) and probability measures with disjoint supports πj(p), j∈Sp, 1≤p≤q, such that (a) θn(1)→∞, θn(k+1)/θn(k)→∞, (b) for all p, x∈V, t>0, starting from x, the distribution of Xtθn(p)(n) converges, as n→∞, to a convex combination of the probability measures πj(p). The weights of the convex combination naturally depend on x and t. Let In be the level two large deviations rate functional for Xt(n), 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/θn(p))I(p) for some rate functionals I(p) which take finite values only at convex combinations of the measures πj(p): I(p)(μ)<∞ if, and only if, μ=∑j∈S pωjπj(p) for some probability measure ω in Sp.