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Strong ergodicity for Markov processes by coupling methods

Part of: Spectral theory and eigenvalue problems Markov processes

Published online by Cambridge University Press:  14 July 2016

Yong-Hua Mao
Yong-Hua Mao*
Affiliation:
Beijing Normal University
*
∗ Postal address: Department of Mathematics, Beijing Normal University, Beijing 100875, People's Republic of China. Email address: maoyh@bnu.edu.cn
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Abstract

In this paper, we apply coupling methods to study strong ergodicity for Markov processes, and sufficient conditions are presented in terms of the expectations of coupling times. In particular, explicit criteria are obtained for one-dimensional diffusions and birth-death processes to be strongly ergodic. As a by-product, strong ergodicity implies that the essential spectra of the generators for these processes are empty.

Keywords

Markov semigroupsstrong ergodicitycouplingessential spectrum

MSC classification

Primary: 60J60: Diffusion processes 60J75: Jump processes 35P15: Estimation of eigenvalues, upper and lower bounds

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 39 , Issue 4 , December 2002 , pp. 839 - 852
Copyright
Copyright © Applied Probability Trust 2002 

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Footnotes

Research supported in part by RFDP (No. 20010027007), 973 Project, NSFC (10121101) and NSFC for Distinguished Young Scholars (No. 10025105).

References

[1]. Aldous, D., and Diaconis, P. (1987). Strong uniform times and finite random walks. Adv. Appl. Math. 8, 69–97.Google Scholar
[2]. Anderson, W. (1991). Continuous-Time Markov Chains. Springer, New York.Google Scholar
[3]. Chen, M.-F. (1992). From Markov Chains to Non-Equilibrium Particle Systems. World Scientific, Singapore.Google Scholar
[4]. Chen, M.-F. (1994). Optimal couplings and applications to Riemannian geometry. In Probability Theory and Mathematical Statistics, Vol. 1, eds Grigelionis, B. et al., TEV, Vilnius, pp. 121–142.Google Scholar
[5]. Chen, M.-F. (1996). Estimation of spectral gap for Markov chains. Acta Math. Sinica 12, 337–360.Google Scholar
[6]. Chen, M.-F. (1998). Trilogy of coupling and general formulas for lower bound of spectral gap. In Probability Towards 2000 (Lecture Notes Statist. 128), eds Accardi, L. and Heyde, C. C., Springer, New York, pp. 123–136.Google Scholar
[7]. Chen, M.-F., and Li, S.-F. (1989). Coupling methods for multidimensional diffusion processes. Ann. Prob. 17, 151–177.Google Scholar
[8]. Chen, M.-F., and Wang, F.-Y. (1994). Application of coupling methods to the first eigenvalue on manifolds. Sci. China A 37, 1–14.Google Scholar
[9]. Chen, M.-F., and Wang, F.-Y. (1995). Estimation of the first eigenvalue of second order elliptic operators. J. Funct. Anal. 131, 345–363.Google Scholar
[10]. Chen, M.-F., and Wang, F.-Y. (1997). General formula for lower bound of the first eigenvalue on Riemannian manifolds. Sci. China A 40, 384–394.CrossRefGoogle Scholar
[11]. Cranston, M. (1991). Gradient estimates on manifolds using coupling. J. Funct. Anal. 99, 110–124.CrossRefGoogle Scholar
[12]. Diaconis, P. (1988). Group Representations in Probability and Statistics (IMS Lecture Notes—Monogr. Ser. 11). Institute of Mathematical Statistics, Hawyard, CA.Google Scholar
[13]. Diaconis, P., and Saloff-Coste, L. (1996). Nash's inequality for finite Markov chains. J. Theoret. Prob. 9, 459–510.Google Scholar
[14]. Karlin, S., and Taylor, H. (1981). A Second Course in Stochastic Processes. Academic Press, Boston.Google Scholar
[15]. Kendall, W. (1986). Nonnegative Ricci curvature and the Brownian coupling property. Stochastics 19, 111–129.Google Scholar
[16]. Mao, Y.-H. (2000). On empty essential spectrum for Markov process in dimension one. Preprint.Google Scholar
[17]. Martínez, S., and Ycart, B. (2001). Decay rates and cutoff for convergence and hitting times of Markov chains with countably infinite state space. Adv. Appl. Prob. 33, 185–205.Google Scholar
[18]. Meyn, S., and Tweedie, R. L. (1993). Markov Chains and Stochastic Stability. Springer, London.CrossRefGoogle Scholar
[19]. Roberts, G. O., and Tweedie, R. L. (2000). Rates of convergence of stochastically monotone and continuous time Markov models. J. Appl. Prob. 37, 359–373.Google Scholar
[20]. Saloff-Coste, L. (1994). Precise estimates on the rate at which certain diffusions tend to equilibrium. Math. Z. 217, 641–677.Google Scholar
[21]. Wang, F.-Y. (1994). Application of coupling methods to the Neumann eigenvalue problem. Prob. Theory Relat. Fields 98, 299–306.CrossRefGoogle Scholar
[22]. Zhang, S.-Y. (1999). Existence of the optimal measurable coupling and ergodicity for Markov processes. Sci. China A 42, 58–67.CrossRefGoogle Scholar
[23]. Zhang, Y.-H. (2001). Strong ergodicity for single birth processes. J. Appl. Prob. 38, 270–277.CrossRefGoogle Scholar