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Metastable Distributions of Semi-Markov Processes

Leonid Koralov, Ishfaaq Mohammed Imtiyas

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.

Metastable Distributions of Semi-Markov Processes

Abstract

In this paper, we consider semi-Markov processes whose transition times and transition probabilities depend on a small parameter . 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 depends on how the point 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 . The result may be viewed as a generalization of the ergodic theorem to the case of parameter-dependent semi-Markov processes.

Paper Structure

This paper contains 11 sections, 20 theorems, 183 equations.

Table of Contents

  1. Introduction.
  2. Assumptions on the dynamics of the system and formulation of the main result.
  3. Examples of systems leading to parameter-dependent semi-Markov processes.
  4. Random perturbations of dynamical systems with finitely many stable equilibrium points
  5. Random perturbations of diffusions with invariant submanifolds
  6. Random perturbations of flows with heteroclinic networks
  7. Reduction to a process with nearly independent transition times and transition probabilities.
  8. Asymptotic results for discrete-time Markov chains
  9. Hierarchy of clusters
  10. Ergodic Theorems for Semi-Markov Processes
  11. Proof of Theorem \ref{['prelimt2']}

Key Result

Lemma 2.2

Suppose $f_1(\varepsilon), \dots f_m(\varepsilon)$ and $g_1(\varepsilon), \dots , g_{n}(\varepsilon)$ are positive functions which satisfy for $1 \le i \le m$ and $1 \le j \le n.$ Then,

Theorems & Definitions (28)

  • Remark 2.1
  • Lemma 2.2
  • Theorem 2.3
  • Remark 2.4
  • Lemma 4.1
  • Lemma 4.2
  • Corollary 4.3
  • Theorem 4.4
  • Remark 4.5
  • Proposition 4.6
  • ...and 18 more