Abstract
A criterion for algebraic convergence of the entropy is presented and an algebraic convergence result for the entropy of an exclusion process is improved. A weak entropy inequality is considered and its relationship to entropic convergence is discussed.
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References
Liggett T M. L 2 rates of convergence of attractive reversible nearst paticle systems: The critical case. Ann Probab, 19: 935–959 (1991)
Deuschel J D. Algebraic L 2 decay of attractive critical processes on the lattics. Ann Probab, 15: 780–799 (1987)
Chen M F, Wang Y Z. Algebraic convergence of Markov chains. Ann Appl Prob, 13: 604–627 (2003)
Mao Y H. Algebraic convergence for discrete-time Markov chains. Sci China Ser A-Math, 46: 621–630 (2003)
Janvresse E, Landim C, Quastel J, Yau H T. Relaxation to equilibrium of conservative dynamics I: Zero range processes. Ann Probab, 27: 325–360 (1999)
Landim C, Yau H T. Convergence to equilibrium of conservative paticle systems on Z d. Ann Probab, 31: 115–147 (2003)
Röckner M, Wang F Y. Weak Poincaré inequalities and L 2-convergence rates of markov semigroup. J Funct Anal, 185: 564–603 (2001)
Bertini L, Zegarlinski B. Coercive inequalities for Kawasaki dynamics: The product case. Markov Process Related Fields, 5: 125–162 (1999)
Cattiaux P, Gentil I, Guillin A. Weak logarithmic Sobolev inequalities and entropic convergence. Probability Theory and Related Fields, 139: 563–603 (2007)
Bakry D. L’hypercontactivite et son utilisation en theorie des semi-groupes, Ecole d’eté de probabilités de Saint-Flour XXII, 1992. Lecture Notes in Math, Vol 1581, 1994, 1–114
Chen M F. Eigenvalues, Inequalities and Ergodic Theory. New York: Springer, 2004
Saloff-Coste L. Lectures on finite Markov chains. In: Lecture Notes in Math, Vol 1665, 1997, 301–413
Wang F Y. Functional Inequalities, Markov Semigroups and Spectral Theory. Beijing-New York: Science Press, 2005
Fukushima M, Oshima Y, Takeda M. Dirichlet Forms and Symmetric Markov Processes. Berlin: Walter de Gruyter, 1994
Ma Z M, Röckner M. Introduction to Theory of (Non-symmetric) Dirichlet Forms. New York: Springer-Verlag, 1992
Gross L. Logarithmic Sobolev inequalities. Amer J Math, 97: 1061–1083 (1976)
Chen M F. Nash inequalities for general symmetric forms. Acta Math Sin Engl Ser, 15: 353–370 (1999)
Zhang S Y, Mao Y H. Exponential convergence rate in Boltzman-Shannon entropy. Sci China Ser A-Math, 44: 280–285 (2000)
Wang F Y, Zhang Q Z. Weak Poincaré ineualities, Decay of Markov semigroups and concentration of measures. Acta Math Sin Engl Ser, 21(4): 937–942 (2005)
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This work was supported by the National Natural Science Foundation of China (Grant No. 10571139)
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Gao, F., Li, L. Weak entropy inequalities and entropic convergence. Sci. China Ser. A-Math. 51, 1798–1806 (2008). https://doi.org/10.1007/s11425-008-0058-3
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DOI: https://doi.org/10.1007/s11425-008-0058-3