Asymptotic Analysis of Multiscale Markov Chain

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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 O(1ε) comparing to the inter-transition rates among them, where 0 < ε≪ 1. Several asymptotic results are obtained as ε→ 0 concerning the convergence of Kolmogorov backward equation, Poincaré constant, (modified) logarithmic Sobolev constant to their counterparts of certain reduced Markov chain. Both reversible and irreversible Markov chains are considered.