Exponential mixing property for absorbing Markov processes

Abstract

In this paper, we study the speed of mixing of absorbing Markov processes with state space E, that is, the rate of the convergence in the following limits: limt→∞Ex[f(Xpt)g(Xt)|T>t]=∫Ef(y)β(dy)∫Eg(y)α(dy),limt→∞Ex[f(Xpt)g(Xqt)|T>t]=∫Ef(y)β(dy)∫Eg(y)β(dy),where f,g are bounded and measurable functions defined on E, x∈E, 0<p<q<1, T is the absorption time of the process, α is a quasi-stationary distribution and β is a quasi-ergodic distribution. Under general conditions, we show that the convergence rates of the limits are exponential. As an application, we apply our result to the logistic Feller diffusion process and the birth–death process with catastrophes.