Full Γ-expansion of Reversible Markov Chains Level Two Large Deviations Rate Functionals

Let Ξ_n ⊂ℝ^d, n≥ 1, be a sequence of finite sets and consider a Ξ_n-valued, irreducible, reversible, continuous-time Markov chain (X^(n)_t:t≥ 0). Denote by 𝒫(ℝ^d) the set of probability measures on ℝ^d and by I_n𝒫(ℝ^d) → [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 I_n expressing it as I_n = I^(0) + ∑_1≤ p≤ q (1/θ^(p)_n) I^(p), where I^(p)𝒫(ℝ^d) → [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 behavior, 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.