Full Γ-expansion of reversible Markov chains level two large deviations rate functionals

Let Ξn⊂Rd, n≥1, be a sequence of finite sets and consider a Ξn-valued, irreducible, reversible, continuous-time Markov chain (Xt(n):t≥0). Denote by P(Rd) the set of probability measures on Rd and by In:P(Rd)→[0,+∞) the level two large deviations rate functional for Xt(n) as t→∞. We present a general method, based on tools used to prove the metastable behaviour of Markov chains, to derive a full expansion of In expressing it as In=I(0)+∑ 1≤p≤q(1/θn(p))I(p), where I(p):P(Rd)→[0,+∞] represent rate functionals independent of n and θn(p) sequences such that θn(1)→∞, θn(p)/θn(p+1)→0 for 1≤p<q. The speed θn(p) corresponds to the time-scale at which the Markov chains Xt(n) exhibits a metastable behaviour, and the I(p−1) zero-level sets to the metastable states. To illustrate the theory we apply the method to random walks in potential fields.