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Estimation of spectral gap for Markov chains

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Abstract

The study of the convergent rate (spectral gap) in theL 2-sense is motivated from several different fields: probability, statistics, mathematical physics, computer science and so on and it is now an active research topic. Based on a new approach (the coupling technique) introduced in [7] for the estimate of the convergent rate and as a continuation of [4], [5], [7–9], [23] and [24], this paper studies the estimate of the rate for time-continuous Markov chains. Two variational formulas for the rate are presented here for the first time for birth-death processes. For diffusions, similar results are presented in an accompany paper [10]. The new formulas enable us to recover or improve the main known results. The connection between the sharp estimate and the corresponding eigenfunction is explored and illustrated by various examples. A previous result on optimal Markovian couplings[4] is also extended in the paper.

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Authors and Affiliations

  1. Department of Mathematics, Beijing Normal University, 100875, Beijing, China

    Chen Mufa

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  1. Chen Mufa
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Research supported in part by NSFC, Qin Shi Sci & Tech. Foundation and the State Education Commission of China.

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Mufa, C. Estimation of spectral gap for Markov chains. Acta Mathematica Sinica 12, 337–360 (1996). https://doi.org/10.1007/BF02106789

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  • DOI: https://doi.org/10.1007/BF02106789

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