Metastable Γ-expansion of finite state Markov chains level two large deviations rate functions

Overview & Synopsis

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 (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 time-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∈Spωjπ(p)j for some probability measure ω in Sp.