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

Consider a sequence $(η^N(t) :t\ge 0)$ of continuous-time, irreducible Markov chains evolving on a fixed finite set $E$, indexed by a parameter $N$. Denote by $R_N(η,ξ)$ the jump rates of the Markov chain $η^N_t$, and assume that for any pair of bonds $(η,ξ)$, $(η',ξ')$ $\arctan \{R_N(η,ξ)/R_N(η',ξ')\}$ 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 $θ^1_N, \dots, θ^{\mf p}_N$, $θ^j_N \ll θ^{j+1}_N$, 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 $X^j_N(t) = ϕ_j(η^N(tθ^j_N))$ converges to a Markov chain on $\{1, \dots, \mf n_j\}$.