Stability and exponential convergence of continuous-time Markov chains

For finite, homogeneous, continuous-time Markov chains having a unique stationary distribution, we derive perturbation bounds which demonstrate the connection between the sensitivity to perturbations and the rate of exponential convergence to stationarity. Our perturbation bounds substantially improve upon the known results. We also discuss convergence bounds for chains with diagonalizable generators and investigate the relationship between the rate of convergence and the sensitivity of the eigenvalues of the generator; special attention is given to reversible chains.

[1]  Alexander I. Zeifman.  Upper and lower bounds on the rate of convergence for nonhomogeneous birth and death processes , 1995 .

[2]  P. Diaconis,et al.  Geometric Bounds for Eigenvalues of Markov Chains , 1991 .

[3]  O. Thoumine,et al.  A probabilistic model for ligand-cytoskeleton transmembrane adhesion: predicting the behavior of microspheres on the surface of migrating cells. , 2000, Journal of theoretical biology.

[4]  P. Diaconis,et al.  Nash inequalities for finite Markov chains , 1996 .

[5]  N. Kampen,et al.  Stochastic processes in physics and chemistry , 1981 .

[6]  Investigation of stochastic fluctuations in the signal formation of microbiosensors , 1998 .

[7]  A. Zeifman.  Stability of birth-and-death processes , 1998 .

[8]  Nan-Fu Peng.  Spectral representations of the transition probability matrices for continuous time finite Markov chains , 1996 .

[9]  B. Granovsky,et al.  Nonstationary Markovian queues , 2000 .

[10]  Alexander Zeifman,et al.  On strong ergodicity for nonhomogeneous continuous-time Markov chains , 1994 .

[11]  E. Seneta.  Coefficients of ergodicity: structure and applications , 1979, Advances in Applied Probability.

[12]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[13]  N. Kartashov.  Inequalities in Theorems of Ergodicity and Stability for Markov Chains with Common Phase Space. I , 1986 .

[14]  F G Ball,et al.  Stochastic models for systems of interacting ion channels. , 2000, IMA journal of mathematics applied in medicine and biology.

[15]  Q Zheng,et al.  Note on the non-homogeneous Prendiville process. , 1998, Mathematical biosciences.

[16]  J. A. Fill.  Eigenvalue bounds on convergence to stationarity for nonreversible markov chains , 1991 .

[17]  Stability for continuous-time nonhomogeneous Markov chains , 1985 .

[18]  P. Diaconis,et al.  LOGARITHMIC SOBOLEV INEQUALITIES FOR FINITE MARKOV CHAINS , 1996 .

[19]  Alexander I. Zeifman,et al.  The decay function of nonhomogeneous birth-death processes, with application to mean-field models , 1997 .

[20]  A. Zeifman,et al.  On Ergodicity and Stability Estimates for Some Nonhomogeneous Markov Chains , 2002 .

[21]  Alexander I. Zeifman,et al.  The N-limit of spectral gap of a class of birth-death Markov chains , 2000 .