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.
Reviewer: Marius Iosifescu (Bucureşti)
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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