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 ${\eta }_t^N$, and assume that for any pair of bonds $(\eta ,\xi )$, $({\eta }^{\mathrm{\prime }},{\xi }^{\mathrm{\prime }})$ $\arctan {R_N(\eta,\xi)/R_N(\eta’,\xi’)}$convergesas$N\uparrow\infty$. Under a hypothesis slightly more restrictive (cf. \eqref{mhyp} below), we present a recursive procedure which provides a sequence of increasing time-scales ${\theta }_N^1,\dots ,{\theta }_N^{\text{mf}p}$, ${\theta }_N^j\ll {\theta }_N^{j+1}$, and of coarsening partitions $\{\text{ms}{E}_1^j,\dots ,\text{ms}{E}_{\text{mf}{n}_j}^j,{\mathrm{\Delta }}^j\}$, $1\le j\le \text{mf}p$, of the set $E$. Let ${\varphi }_j:E\to \{0,1,\dots ,\text{mf}{n}_j\}$ be the projection defined by $ \phi_j(\eta) = \sum_{x=1}^{\mf n_j} x , \mb 1{\eta \in \ms E^j_x}$\mathrm{.}Foreach$1\le j\le \mf p$, we prove that the hidden Markov chain $X_N^j(t)={\varphi }_j({\eta }^N(t{\theta }_N^j))$ converges to a Markov chain on $\{1,\dots ,\text{mf}{n}_j\}$.