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

Let ${\mathrm{\Xi }}_n\subset {\mathbb{R}}^d$, $n\ge 1$, be a sequence of finite sets and consider a ${\mathrm{\Xi }}_n$-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 $I_n\text{\hspace{0.17em}⁣}:\mathcal{P}({\mathbb{R}}^d)\to [0,+\mathrm{\infty })$ the level two large deviations rate functional for $X_t^{(n)}$ as $t\to \mathrm{\infty }$. 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 $I_n=I^{(0)}+{\sum }_{1\le p\le q}(1\mathrm{/}{\theta }_n^{(p)})I^{(p)}$, where $I^{(p)}\text{\hspace{0.17em}⁣}:\mathcal{P}({\mathbb{R}}^d)\to [0,+\mathrm{\infty }]$ represent rate functionals independent of $n$ and ${\theta }_n^{(p)}$ sequences such that ${\theta }_n^{(1)}\to \mathrm{\infty }$, ${\theta }_n^{(p)}\mathrm{/}{\theta }_n^{(p+1)}\to 0$ 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.