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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
##### Keywords:
Markov chain; metastable distribution; large deviations
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##### References:
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