Metastability of finite state Markov chains: a recursive procedure to identify slow variables for model reduction

Abstract

Consider a sequence $(\hat{I}{\cdot }^N(t):t\ge 0)$ of continuous-time, irreducible Markov chains evolving on a fixed finite set $E$, indexed by a parameter $N$. Denote by $R_N(\hat{I}\cdot ,\hat{I}¾)$ the jump rates of the Markov chain $\hat{I}{\cdot }_t^N$, and assume that for any pair of bonds $(\hat{I}\cdot ,\hat{I}¾)$, $(\hat{I}{\cdot }^{\prime },\hat{I}{¾}^{\prime })$ $\arctan \{R_N(\hat{I}\cdot ,\hat{I}¾)/R_N(\hat{I}{\cdot }^{\prime },\hat{I}{¾}^{\prime })\}$ converges as $N\uparrow \infty$. Under a hypothesis slightly more restrictive (cf. \eqref{mhyp} below), we present a recursive procedure which provides a sequence of increasing time-scales $θ^1_N, \dots, θ^{\mf p}_N$, $\hat{I}{¸}_N^j\ll \hat{I}{¸}_N^{j+1}$, and of coarsening partitions $\{\ms E^j_1, \dots, \ms E^j_{\mf n_j}, Δ^j\}$, $1\le j\le \mf p$, of the set $E$. Let $ϕ_j: E \to \{0,1, \dots, \mf n_j\}$ be the projection defined by $ ϕ_j(η) = \sum_{x=1}^{\mf n_j} x \, \mb 1\{η\in \ms E^j_x\}$. For each $1\le j\le \mf p$, we prove that the hidden Markov chain $X_N^j(t)=\ddot{I}{•}_j(\hat{I}{\cdot }^N(t\hat{I}{¸}_N^j))$ converges to a Markov chain on $\{1, \dots, \mf n_j\}$.