Estimates on the amplitude of the first Dirichlet eigenvector in discrete frameworks  

作　　者：DIACONIS Persi MICLO Laurent 

机构地区：[1]Department of Mathematics, Stanford University, California 94305, USA [2]Institut de Matheatiques de Toulouse, Universite de Toulouse and CNRS, UMR 5219, France

出　　处：《Science China Mathematics》2016年第2期205-226,共22页中国科学：数学（英文版）

基　　金：supported by Agence Nationale de la Recherche(Grant Nos.ANR-11-LABX-0040-CIMI;ANR-11-IDEX-0002-02 and ANR-12-BS01-0019)

摘　　要：Consider a finite absorbing Markov generator, irreducible on the non-absorbing states. PerronFrobenius theory ensures the existence of a corresponding positive eigenvector ψ. The goal of the paper is to give bounds on the amplitude max ψ/ min ψ. Two approaches are proposed: One using a path method and the other one, restricted to the reversible situation, based on spectral estimates. The latter approach is extended to denumerable birth and death processes absorbing at 0 for which infinity is an entrance boundary. The interest of estimating the ratio is the reduction of the quantitative study of convergence to quasi-stationarity to the convergence to equilibrium of related ergodic processes, as seen by Diaconis and Miclo(2014).Consider a finite absorbing Markov generator, irreducible on the non-absorbing states. PerronFrobenius theory ensures the existence of a corresponding positive eigenvector ψ. The goal of the paper is to give bounds on the amplitude max ψ/ min ψ. Two approaches are proposed： One using a path method and the other one, restricted to the reversible situation, based on spectral estimates. The latter approach is extended to denumerable birth and death processes absorbing at 0 for which infinity is an entrance boundary. The interest of estimating the ratio is the reduction of the quantitative study of convergence to quasi-stationarity to the convergence to equilibrium of related ergodic processes, as seen by Diaconis and Miclo（2014）.

关 键 词：finite absorbing Markov process first Dirichlet eigenvector path method spectral estimates denumerable absorbing birth and death process entrance boundary 

分 类 号：O211.62[理学—概率论与数理统计]