Logarithmic Sobolev inequalities for inhomogeneous Markov semigroups
ESAIM: Probability and Statistics, Tome 12 (2008), pp. 492-504.

We investigate the dissipativity properties of a class of scalar second order parabolic partial differential equations with time-dependent coefficients. We provide explicit condition on the drift term which ensure that the relative entropy of one particular orbit with respect to some other one decreases to zero. The decay rate is obtained explicitly by the use of a Sobolev logarithmic inequality for the associated semigroup, which is derived by an adaptation of Bakry’s $\Gamma$-calculus. As a byproduct, the systematic method for constructing entropies which 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.

DOI : https://doi.org/10.1051/ps:2007042
Classification : 60J60,  47D07
Mots clés : inhomogeneous Markov process, logarithmic Sobolev inequality, relative entropy
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author = {Collet, Jean-Fran\c{c}ois and Malrieu, Florent},
title = {Logarithmic {Sobolev} inequalities for inhomogeneous {Markov} semigroups},
journal = {ESAIM: Probability and Statistics},
pages = {492--504},
publisher = {EDP-Sciences},
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Collet, Jean-François; Malrieu, Florent. Logarithmic Sobolev inequalities for inhomogeneous Markov semigroups. ESAIM: Probability and Statistics, Tome 12 (2008), pp. 492-504. doi : 10.1051/ps:2007042. http://www.numdam.org/articles/10.1051/ps:2007042/

[1] 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. | Zbl 0982.46026

[2] D. Bakry, M. Émery, Hypercontractivité de semi-groupes de diffusion. CRAS Ser. I 299 (1984) 775-778. | MR 772092 | Zbl 0563.60068

[3] D. Bakry, L'hypercontractivité et son utilisation en théorie des semigroupes. Lect. Notes Math. 1581 (1994) 1-114. | MR 1307413 | Zbl 0856.47026

[4] 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. | MR 1654503

[5] G.I. Barenblatt, Scaling, self-similarity, and intermediate asymptotics. Cambridge Texts in Applied Mathematics 14 Cambridge University Press (1996). | MR 1426127 | Zbl 0907.76002

[6] J. Bricmont, A. Kupiainen and G. Lin, Renormalization group and asymptotics of solutions of nonlinear parabolic equations. Comm. Pure Appl. Math. 47 (1994) 893-922. | MR 1280993 | Zbl 0806.35067

[7] D. Chafaï, Entropies, Convexity en Functional Inequalities. Journal of Mathematics of Kyoto University 44 (2004) 325-363. | MR 2081075 | Zbl 1079.26009

[8] J.F. Collet, Extensive Lyapounov functionals for moment-preserving evolution equations. C.R.A.S. Ser. I 334 (2002) 429-434. | MR 1892947 | Zbl 1090.82026

[9] P. Del Moral, M. Ledoux and L. Miclo, On contraction properties of Markov kernels. Probab. Theory Related Fields 126 (2003) 395-420. | MR 1992499 | Zbl 1030.60060

[10] W.J. Ewens, Mathematical population genetics. I, Interdisciplinary Applied Mathematics, Vol 27. Springer-Verlag (2004). | MR 2026891 | Zbl 1060.92046

[11] R. Kubo, H-Theorems for Markoffian Processes, in Perspectives in Statistical Physics, H.J. Raveché Ed., North Holland Publishing (1981). | MR 626365

[12] S. Kullback and R.A. Leibler, On Information and Sufficiency. Ann. Math. Stat. 22 (1951) 79-86. | MR 39968 | Zbl 0042.38403

[13] F. Otto and C. Villani, Generalization of an inequality by Talagrand, and links with the Sobolev Logarihmic Inequality. J. Func. Anal. 173 (2000) 361-400. | MR 1760620 | Zbl 0985.58019

[14] M.S. Pinsker, Information and Information Stability of Random Variables and Processes. Holden-Day Inc. (1964). | MR 213190 | Zbl 0125.09202

[15] G. Toscani, Remarks on entropy and equilibrium states. Appl. Math. Lett. 12 (1999) 19-25. | MR 1750055 | Zbl 0940.35168

[16] G. Toscani and C. Villani, On the trend to equilibrium for some dissipative systems with slowly increasing a priori bounds. J. Stat. Phys. 98 (2000) 1279-1309. | MR 1751701 | Zbl 1034.82032

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