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Metastable distributions of Markov chains with rare transitions. (English) Zbl 1379.37006
Authors’ abstract: In this paper we consider Markov chains \(X_{t}^{\varepsilon }\) with transition rates that depend on a small parameter \(\varepsilon \). We are interested in the long time behavior of \( X_{t}^{\varepsilon }\) at various \(\varepsilon \)-dependent time scales \( t=t(\varepsilon )\). The asymptotic behavior depends on how the point \( (1/\varepsilon ,t(\varepsilon ))\) approaches infinity. We introduce a general notion of complete asymptotic regularity (a certain asymptotic relation between the ratios of transition rates), which ensures the existence of the metastable distribution for each initial point and a given time scale \(t(\varepsilon )\). The technique of i-graphs allows one to describe the metastable distribution explicitly. The result may be viewed as a generalization of the ergodic theorem to the case of parameter-dependent Markov chains.

MSC:
37A25 Ergodicity, mixing, rates of mixing
37A30 Ergodic theorems, spectral theory, Markov operators
60J27 Continuous-time Markov processes on discrete state spaces
60J28 Applications of continuous-time Markov processes on discrete state spaces
60F10 Large deviations
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