Metastable Distributions of Semi-Markov Processes
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866913573685428224 |
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| author | Koralov, Leonid Imtiyas, Ishfaaq Mohammed |
| author_facet | Koralov, Leonid Imtiyas, Ishfaaq Mohammed |
| contents | In this paper, we consider semi-Markov processes whose transition times and transition probabilities depend on a small parameter $\varepsilon$. Understanding the asymptotic behavior of such processes is needed in order to study the asymptotics of various randomly perturbed dynamical and stochastic systems. The long-time behavior of a semi-Markov process $X^\varepsilon_t$ depends on how the point $(1/\varepsilon, t(\varepsilon))$ approaches infinity. We introduce the notion of complete asymptotic regularity (a certain asymptotic condition on transition probabilities and transition times), originally developed for parameter-dependent Markov chains, which ensures the existence of the metastable distribution for each initial point and a given time scale $t(\varepsilon)$. The result may be viewed as a generalization of the ergodic theorem to the case of parameter-dependent semi-Markov processes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_04795 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Metastable Distributions of Semi-Markov Processes Koralov, Leonid Imtiyas, Ishfaaq Mohammed Probability 60J27, 60K15 In this paper, we consider semi-Markov processes whose transition times and transition probabilities depend on a small parameter $\varepsilon$. Understanding the asymptotic behavior of such processes is needed in order to study the asymptotics of various randomly perturbed dynamical and stochastic systems. The long-time behavior of a semi-Markov process $X^\varepsilon_t$ depends on how the point $(1/\varepsilon, t(\varepsilon))$ approaches infinity. We introduce the notion of complete asymptotic regularity (a certain asymptotic condition on transition probabilities and transition times), originally developed for parameter-dependent Markov chains, which ensures the existence of the metastable distribution for each initial point and a given time scale $t(\varepsilon)$. The result may be viewed as a generalization of the ergodic theorem to the case of parameter-dependent semi-Markov processes. |
| title | Metastable Distributions of Semi-Markov Processes |
| topic | Probability 60J27, 60K15 |
| url | https://arxiv.org/abs/2411.04795 |