2014/12/22 by Volker Betz, Betz, Volker, Stéphane Le Roux +1 · 2 citations
Mathematics · Physics and Astronomy · #60J10 #60J22 #Complex Network Analysis Techniques #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.1412.6979
openalex publication_date 2014/12/22 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28
We consider a simple but important class of metastable discrete time Markov\nchains, which we call perturbed Markov chains. Basically, we assume that the\ntransition matrices depend on a parameter \ε, and converge as\n\ε. We further assume that the chain is irreducible for\n\ε but may have several essential communicating classes when\n\ε. This leads to metastable behavior, possibly on multiple time\nscales. For each of the relevant time scales, we derive two effective chains.\nThe first one describes the (possibly irreversible) metastable dynamics, while\nthe second one is reversible and describes metastable escape probabilities.\nClosed probabilistic expressions are given for the asymptotic transition\nprobabilities of these chains, but we also show how to compute them in a fast\nand numerically stable way. As a consequence, we obtain efficient algorithms\nfor computing the committor function and the limiting stationary distribution.\n