Metastable Distributions of Markov Chains with Rare Transitions

In this paper we consider Markov chains $$X^\varepsilon _t$$Xtε with transition rates that depend on a small parameter $$\varepsilon $$ε. We are interested in the long time behavior of $$X^\varepsilon _t$$Xtε at various $$\varepsilon $$ε-dependent time scales $$t = t(\varepsilon )$$t=t(ε). The asymptotic behavior depends on how the point $$(1/\varepsilon , t(\varepsilon ))$$(1/ε,t(ε)) 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 $$t(\varepsilon )$$t(ε). 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.

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