Stability and exponential convergence of continuous-time Markov chains

Abstract

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.