Abstract
This note is devoted to study the exponential convergence rate in the total variation for reversible Markov processes by comparing it with the spectral gap. It is proved that in a quite general setup, with a suitable restriction on the initial distributions, the rate is bounded from below by the spectral gap. Furthermore, in the compact case or for birth-death processes or half-line diffusions, the rate is shown to be equal to the spectral gap.
Similar content being viewed by others
References
Liggett T M. ExponentialL 2 convergence of attractive reversible nearest particle systems. Ann Probab, 1989, 17: 403–432
Chen M F. ExponentialL 2-convergence andL 2-spectral gap for Markov processes. Acta Math Sin New Ser, 1991, 7(1): 19–37
Chen M F. From Markov Chains to Non-Equilibrium Particle Systems. Singapore: World Scientific, 1992
Chen M F. Estimation’ of spectral gap for Markov chains. Acta Math Sin New Ser, 1996, 12(4): 337–360
Chen M F, Wang F Y. Estimation of spectral gap for elliptic operators. Trans Amer Math Soc, 1997, 349: 1239–1267
Chen M F, Wang F Y. General formula for lower bound of the first eigenvalue on Riemannian manifolds. Sci Sin 1997, 27(1): 34–42 (Chinese Edition); 1997, 40(4): 384–394 (English Edition)
Chen M F, Wang F Y. Estimation of the first eigenvalue of second order elliptic operators. J Funct Anal, 1995, 131(2): 345–363
Diaconis, Stroock. Geometric bounds for eigenvalues of Markov chains. Ann Appl Prob, 1991, 1(1): 36–61
Diaconis P, Saloff-coste L. Nash inequality for finite Markov chains. J Theor Prob, 1996, 9(2): 459–510
Rosenthal J S. Markov chain convergence: From finite to infinite. Stoch Proc Appl, 1996, 62(1): 55–72
Saloff-Coste L. Convergence to equilibrium and Logarithmic Sobolev constant on manifolds with Ricci curvature bounded below. Coll Math, 1994, 109–121
Sinclair A J, Jerrum M R. Approximate counting, uniform generation, and rapidly mixing Markov chains. Inform and Comput, 1989, 82: 93–133
Wang Y Z. Convergennce rate in total variation for diffusion processes. Preprint 1996
Author information
Authors and Affiliations
Additional information
Research supported in part by NSFC, Qiu Shi Sci. & Tech. Found. DPFIHE, MCSEC and Univ., of Rome I, Italy
Rights and permissions
About this article
Cite this article
Mufa, C. Estimate of exponential convergence rate in total variation by spectral gap. Acta Mathematica Sinica 14, 9–16 (1998). https://doi.org/10.1007/BF02563878
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF02563878