Asymptotic Analysis of Multiscale Markov Chain

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 $\mathcal{O}(\frac{1}{ϵ})$ comparing to the inter-transition rates among them, where $0<ϵ\ll 1$. Several asymptotic results are obtained as $ϵ\to 0$ 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.