Full $\Gamma$-expansion of reversible Markov chains level two large deviations rate functionals

Abstract

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 $(X_t^{(n)}:t\ge 0)$. Denote by $\mathcal{P}({\mathbb{R}}^d)$ the set of probability measures on ${\mathbb{R}}^d$ and by $I_n:\mathcal{P}({\mathbb{R}}^d)\to [0,+\infty )$ the level two large deviations rate functional for $X_t^{(n)}$ 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 $I_n=I^{(0)}+{\sum }_{1\le p\le q}(1/{\theta }_n^{(p)})I^{(p)}$, where $I^{(p)}:\mathcal{P}({\mathbb{R}}^d)\to [0,+\infty ]$ represent rate functionals independent of $n$ and ${\theta }_n^{(p)}$ sequences such that ${\theta }_n^{(1)}\to \infty$, ${\theta }_n^{(p)}/{\theta }_n^{(p+1)}\to 0$ for $1\le p<q$. The speed ${\theta }_n^{(p)}$ corresponds to the time-scale at which the Markov chains $X_t^{(n)}$ 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.

arXiv comment39 pages. More explanations added in various parts of the text + introduction extended. Journal version

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