Metastable Distributions for Semi-Markov Processes

Abstract

In this work, we consider semi-Markov processes whose transition times and transitionprobabilities depend on a small parameter ε. 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 Xε t depends on how the point (1/ε, t(ε)) 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(ε). The result may be viewed as a generalization of the ergodic theorem to the case of parameter-dependent semi-Markov processes.