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

Consider a sequence $(\eta^N(t) :t\ge 0)$ of continuous-time, irreducible Markov chains evolving on a fixed finite set $E$, indexed by a parameter $N$. Denote by $R_N(\eta,\xi)$ the jump rates of the Markov chain $\eta^N_t$, and assume that for any pair of bonds $(\eta,\xi)$, $(\eta',\xi')$ $\arctan \{R_N(\eta,\xi)/R_N(\eta',\xi')\}$ 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 $\theta^1_N, \dots, \theta^{\mf p}_N$, $\theta^j_N \ll \theta^{j+1}_N$, and of coarsening partitions $\{\ms E^j_1, \dots, \ms E^j_{\mf n_j}, \Delta^j\}$, $1\le j\le \mf p$, of the set $E$. Let $\phi_j: E \to \{0,1, \dots, \mf n_j\}$ be the projection defined by $\phi_j(\eta) = \sum_{x=1}^{\mf n_j} x \, \mb 1\{\eta \in \ms E^j_x\}$. For each $1\le j\le \mf p$, we prove that the hidden Markov chain $X^j_N(t) = \phi_j(\eta^N(t\theta^j_N))$ converges to a Markov chain on $\{1, \dots, \mf n_j\}$.