We consider a simple but important class of metastable discrete time Markovchains, which we call perturbed Markov chains. Basically, we assume that thetransition matrices depend on a parameter $\varepsilon$, and converge as$\varepsilon$. We further assume that the chain is irreducible for$\varepsilon$ but may have several essential communicating classes when$\varepsilon$. This leads to metastable behavior, possibly on multiple timescales. For each of the relevant time scales, we derive two effective chains.The first one describes the (possibly irreversible) metastable dynamics, whilethe second one is reversible and describes metastable escape probabilities.Closed probabilistic expressions are given for the asymptotic transitionprobabilities of these chains, but we also show how to compute them in a fastand numerically stable way. As a consequence, we obtain efficient algorithmsfor computing the committor function and the limiting stationary distribution.
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