Asymptotic Analysis of Multiscale Markov Chain

TL;DR

This paper analyzes the asymptotic behavior of multiscale continuous-time Markov chains with fast intra-cluster and slow inter-cluster transitions, providing convergence results for key constants as the scale parameter approaches zero.

Contribution

It offers new asymptotic results on the convergence of Kolmogorov backward equations and functional inequalities for multiscale Markov chains, including both reversible and irreversible cases.

Findings

Convergence of Kolmogorov backward equation as ε→0

Convergence of Poincaré and logarithmic Sobolev constants

Applicable to both reversible and irreversible Markov chains

Abstract

We consider continuous-time Markov chain on a finite state space X. We assume X can be clustered into several subsets such that the intra-transition rates within these subsets are of order $\mathcal{O}(\frac{1}{\epsilon})$ comparing to the inter-transition rates among them, where $0 < \epsilon \ll 1$. Several asymptotic results are obtained as $\epsilon \rightarrow 0$ concerning the convergence of Kolmogorov backward equation, Poincar\'e constant, (modified) logarithmic Sobolev constant to their counterparts of certain reduced Markov chain. Both reversible and irreversible Markov chains are considered.