Let $\Xi_n \subset \mathbb R^d$, $n\ge 1$, be a sequence of finite sets and consider a $\Xi_n$-valued, irreducible, reversible, continuous-time Markov chain $(X^{ (n)}_t:t\ge 0)$. Denote by $\mathscr P(\mathbb R^d) $ the set of probability measures on $\mathbb R^d$ and by $I_n\colon \mathscr P(\mathbb R^d) \to [0,+\infty)$ the level two large deviations rate functional for $X^{ (n)}_t$ as $t\to\infty$. 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 $I_n = I^{ (0)} \,+\, \sum_{ 1\le p\le q} (1/\theta^{ (p)}_n)\, I^{ (p)}$, where $I^{ (p)}\colon \mathscr P(\mathbb R^d) \to [0,+\infty]$ represent rate functionals independent of $n$ and $\theta^{ (p)}_n$ sequences such that $\theta^{ (1)}_n \to\infty$, $\theta^{ (p)}_n / \theta^{ (p+1)}_n \to 0$ for $1\le p< q$. The speed $\theta^{ (p)}_n$ 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.
Categories: math.PR cond-mat.stat-mech math.AP