Metastable Γ-expansion of finite state Markov chains level two large deviations rate functions

Abstract

We examine two analytical characterisation of the metastable behavior of a sequence of Markov chains. The first one expressed in terms of its transition probabilities, and the second one in terms of its large deviations rate functional.

Consider a sequence of continuous-time Markov chains $({\mathit{X}}_{\mathit{t}}^{(\mathit{n})}:\mathit{t}\ge 0)$ evolving on a fixed finite state space V. Under a hypothesis on the jump rates, we prove the existence of time-scales ${\mathit{\theta }}_{\mathit{n}}^{(\mathit{p})}$ and probability measures with disjoint supports ${\mathit{\pi }}_{\mathit{j}}^{(\mathit{p})}$, $\mathit{j}\in {\mathit{S}}_{\mathit{p}}$, $1\le \mathit{p}\le \mathfrak{q}$, such that (a) ${\mathit{\theta }}_{\mathit{n}}^{(1)}\to \infty$, ${\mathit{\theta }}_{\mathit{n}}^{(\mathit{k}\mathbf{+}1)}/{\mathit{\theta }}_{\mathit{n}}^{(\mathit{k})}\to \infty$, (b) for all p, $\mathit{x}\in \mathit{V}$, $\mathit{t}>0$, starting from x, the distribution of ${\mathit{X}}_{\mathit{t}{\mathit{\theta }}_{\mathit{n}}^{(\mathit{p})}}^{(\mathit{n})}$ converges, as $\mathit{n}\to \infty$, to a convex combination of the probability measures ${\mathit{\pi }}_{\mathit{j}}^{(\mathit{p})}$. The weights of the convex combination naturally depend on x and t.

Let ${\mathcal{I}}_{\mathit{n}}$ be the level two large deviations rate functional for ${\mathit{X}}_{\mathit{t}}^{(\mathit{n})}$, as $\mathit{t}\to \infty$. Under the same hypothesis on the jump rates and assuming, furthermore, that the process is reversible, we prove that ${\mathcal{I}}_{\mathit{n}}$ can be written as ${\mathcal{I}}_{\mathit{n}}={\mathcal{I}}^{(0)}\mathbf{+}{\sum }_{1\le \mathit{p}\le \mathfrak{q}}(1/{\mathit{\theta }}_{\mathit{n}}^{(\mathit{p})}){\mathcal{I}}^{(\mathit{p})}$ for some rate functionals ${\mathcal{I}}^{(\mathit{p})}$ which take finite values only at convex combinations of the measures ${\mathit{\pi }}_{\mathit{j}}^{(\mathit{p})}$: ${\mathcal{I}}^{(\mathit{p})}(\mathit{\mu })<\infty$ if, and only if, $\mathit{\mu }={\sum }_{\mathit{j}\in {\mathit{S}}_{\mathit{p}}}{\mathit{\omega }}_{\mathit{j}}{\mathit{\pi }}_{\mathit{j}}^{(\mathit{p})}$ for some probability measure ω in ${\mathit{S}}_{\mathit{p}}$.