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 (X (n) t : 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 X (n) t 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) + Σ 1≤p≤q(1/θ (p) n )I(p), where I(p) : P(Rd ) → [0,+ ∞] represent rate functionals independent of n and θ (p) n sequences such that θ (1) n →∞, θ (p) n /θ (p+1) n → 0 for 1 ≤ p < q. The speed θ (p) n corresponds to the time-scale at which the Markov chains X (n) t 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.