Full Γ-expansion of reversible Markov chains level two large deviations rate functionals

Abstract

Let ${\mathrm{\Xi }}_{\mathit{n}}\subset {\mathbb{R}}^{\mathit{d}}$, $\mathit{n}\ge 1$, be a sequence of finite sets and consider a ${\mathrm{\Xi }}_{\mathit{n}}$-valued, irreducible, reversible, continuous-time Markov chain $({\mathit{X}}_{\mathit{t}}^{(\mathit{n})}:\mathit{t}\ge 0)$. Denote by $\mathcal{P}({\mathbb{R}}^{\mathit{d}})$ the set of probability measures on ${\mathbb{R}}^{\mathit{d}}$ and by ${\mathcal{I}}_{\mathit{n}}:\mathcal{P}({\mathbb{R}}^{\mathit{d}})\to [0,\mathbf{+}\infty )$ the level two large deviations rate functional for ${\mathit{X}}_{\mathit{t}}^{(\mathit{n})}$ as $\mathit{t}\to \infty$. We present a general method, based on tools used to prove the metastable behaviour of Markov chains, to derive a full expansion of ${\mathcal{I}}_{\mathit{n}}$ expressing it as ${\mathcal{I}}_{\mathit{n}}={\mathcal{I}}^{(0)}\mathbf{+}{\sum }_{1\le \mathit{p}\le \mathfrak{q}}(1/{\mathit{\theta }}_{\mathit{n}}^{(\mathit{p})}){\mathcal{I}}^{(\mathit{p})}$, where ${\mathcal{I}}^{(\mathit{p})}:\mathcal{P}({\mathbb{R}}^{\mathit{d}})\to [0,\mathbf{+}\infty ]$ represent rate functionals independent of n and ${\mathit{\theta }}_{\mathit{n}}^{(\mathit{p})}$ sequences such that ${\mathit{\theta }}_{\mathit{n}}^{(1)}\to \infty$, ${\mathit{\theta }}_{\mathit{n}}^{(\mathit{p})}/{\mathit{\theta }}_{\mathit{n}}^{(\mathit{p}\mathbf{+}1)}\to 0$ for $1\le \mathit{p}<\mathfrak{q}$. The speed ${\mathit{\theta }}_{\mathit{n}}^{(\mathit{p})}$ corresponds to the time-scale at which the Markov chains ${\mathit{X}}_{\mathit{t}}^{(\mathit{n})}$ exhibits a metastable behaviour, and the ${\mathcal{I}}^{(\mathit{p}-1)}$ zero-level sets to the metastable states. To illustrate the theory we apply the method to random walks in potential fields.