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Logarithmic Sobolev inequalities for inhomogeneous Markov Semigroups

Published online by Cambridge University Press:  01 November 2008

Jean-François Collet  and
Florent Malrieu
Jean-François Collet
Affiliation:
Laboratoire J.A. Dieudonné, Université de Nice Sophia Antipolis, Parc Valrose, 06108 Nice Cedex 02, France.
Florent Malrieu
Affiliation:
IRMAR, Université Rennes 1, Campus de Beaulieu, 35042 Rennes Cedex, France.
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Abstract

We investigate the dissipativity properties of a class of scalar secondorder parabolic partial differential equations with time-dependentcoefficients. We provide explicit condition on the drift term which ensurethat the relative entropy of one particular orbit with respect to some otherone decreases to zero. The decay rate is obtained explicitly by the use ofa Sobolev logarithmic inequality for the associated semigroup, which is derived by an adaptation of Bakry's Γ-calculus.As a byproduct, the systematic method for constructing entropieswhich we propose here also yields the well-known intermediate asymptotics for the heat equation in a very quick way, and without having to rescale the original equation.

Keywords

Inhomogeneous Markov processlogarithmic Sobolev inequalityrelative entropy
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 12 , 2008 , pp. 492 - 504
Copyright
© EDP Sciences, SMAI, 2008

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References

C. Ané, S. Blachère, D. Chafaï, P. Fougères, I. Gentil, F. Malrieu, C. Roberto and G. Scheffer, Sur les inégalités de Sobolev logarithmiques. Collection “Panoramas et Synthèses”, SMF(2000) No. 10.
D. Bakry, M. Émery, Hypercontractivité de semi-groupes de diffusion. CRAS Ser. I 299 (1984) 775–778.
Bakry, D., L'hypercontractivité et son utilisation en théorie des semigroupes. Lect. Notes Math. 1581 (1994) 1–114. CrossRef
D. Bakry, On Sobolev and logarithmic Sobolev inequalities for Markov semigroups, in New trends in stochastic analysis (Charingworth, 1994), River Edge, Taniguchi symposium, World Sci. Publishing, NJ (1997) 43–75.
G.I. Barenblatt, Scaling, self-similarity, and intermediate asymptotics. Cambridge Texts in Applied Mathematics 14 Cambridge University Press (1996).
Bricmont, J., Kupiainen, A. and Lin, G., Renormalization group and asymptotics of solutions of nonlinear parabolic equations. Comm. Pure Appl. Math. 47 (1994) 893–922. CrossRef
Chafaï, D., Entropies, Convexity en Functional Inequalities. Journal of Mathematics of Kyoto University 44 (2004) 325–363. CrossRef
Collet, J.F., Extensive Lyapounov functionals for moment-preserving evolution equations. C.R.A.S. Ser. I 334 (2002) 429–434.
Del Moral, P., Ledoux, M. and Miclo, L., On contraction properties of Markov kernels. Probab. Theory Related Fields 126 (2003) 395–420. CrossRef
W.J. Ewens, Mathematical population genetics. I, Interdisciplinary Applied Mathematics, Vol 27. Springer-Verlag (2004).
R. Kubo, H-Theorems for Markoffian Processes, in Perspectives in Statistical Physics, H.J. Raveché Ed., North Holland Publishing (1981).
Kullback, S. and Leibler, R.A., Information, On and Sufficiency. Ann. Math. Stat. 22 (1951) 79–86. CrossRef
Otto, F. and Villani, C., Generalization of an inequality by Talagrand, and links with the Sobolev Logarihmic Inequality. J. Func. Anal. 173 (2000) 361–400. CrossRef
M.S. Pinsker, Information and Information Stability of Random Variables and Processes. Holden-Day Inc. (1964).
Toscani, G., Remarks on entropy and equilibrium states. Appl. Math. Lett. 12 (1999) 19–25. CrossRef
Toscani, G. and Villani, C., On the trend to equilibrium for some dissipative systems with slowly increasing a priori bounds. J. Stat. Phys. 98 (2000) 1279–1309. CrossRef