Metastable Distributions of Semi-Markov Processes

TL;DR

This paper investigates the asymptotic behavior of semi-Markov processes with small parameter dependence, introducing conditions that ensure metastable distributions and generalize ergodic theorems for such systems.

Contribution

It introduces the concept of complete asymptotic regularity for semi-Markov processes, enabling the analysis of metastable distributions and extending ergodic results to parameter-dependent cases.

Findings

Existence of metastable distributions under asymptotic regularity.

Generalization of ergodic theorem for parameter-dependent semi-Markov processes.

Framework for analyzing long-time behavior of perturbed stochastic systems.

Abstract

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.