Analytic and Geometric Logarithmic Sobolev Inequalities
Journées équations aux dérivées partielles (2011), article no. 7, 15 p.

We survey analytic and geometric proofs of classical logarithmic Sobolev inequalities for Gaussian and more general strictly log-concave probability measures. Developments of the last decade link the two approaches through heat kernel and Hamilton-Jacobi equations, inequalities in convex geometry and mass transportation.

DOI : 10.5802/jedp.79
Classification : 60H, 35K, 58J
Mots clés : Logarithmic Sobolev inequality, heat kernel, Brunn-Minkowski inequality
Ledoux, Michel 1

1 Institut de Mathématiques de Toulouse, Université de Toulouse, F-31062 Toulouse, France, and Institut Universitaire de France
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Ledoux, Michel. Analytic and Geometric Logarithmic Sobolev Inequalities. Journées équations aux dérivées partielles (2011), article  no. 7, 15 p. doi : 10.5802/jedp.79. http://archive.numdam.org/articles/10.5802/jedp.79/

[1] Ané, C. et al. Sur les inégalités de Sobolev logarithmiques. Panoramas et Synthèses, vol. 10. Soc. Math. de France (2000). | MR | Zbl

[2] Bakry, D. L’hypercontractivité et son utilisation en théorie des semigroupes. Ecole d’Eté de Probabilités de St-Flour. Springer Lecture Notes in Math. 1581, 1-114 (1994). | MR | Zbl

[3] Bakry, D. Functional inequalities for Markov semigroups. Probability Measures on Groups: Recent Directions and Trends. Proceedings of the CIMPA-TIFR School (2002). Tata Institute of Fundamental Research, New Delhi, 91-147 (2006). | MR | Zbl

[4] Bakry, D. and Émery, M. Diffusions hypercontractives. Séminaire de Probabilités, XIX. Springer Lecture Notes in Math. 1123, 177-206 (1985). | Numdam | MR | Zbl

[5] Bakry, D., Gentil, I. and Ledoux, M. Forthcoming monograph (2012).

[6] Bakry, D. and Ledoux, M. A logarithmic Sobolev form of the Li-Yau parabolic inequality. Revista Mat. Iberoamericana 22, 683-702 (2006). | MR | Zbl

[7] Barthe, F. Autour de l’inégalité de Brunn-Minkowski. Ann. Fac. Sci. Toulouse Math. 12, 127-178 (2003). | Numdam | MR | Zbl

[8] Bobkov, S. and Ledoux, M. From Brunn-Minkowski to Brascamp-Lieb and to logarithmic Sobolev inequalities. Geom. Funct. Anal. 10, 1028-1052 (2000). | MR | Zbl

[9] Bobkov, S. and Ledoux, M. From Brunn-Minkowski to sharp Sobolev inequalities. Annali di Matematica Pura ed Applicata 187, 369-384 (2008). | MR

[10] Bobkov, S., Gentil, I. and Ledoux, M. Hypercontractivity of Hamilton-Jacobi equations. J. Math. Pures Appl. 80, 669-696 (2001). | MR | Zbl

[11] Cordero-Erausquin, D. Some applications of mass transport to Gaussian type inequalities (2000). Arch. Rational Mech. Anal. 161, 257-269 (2002). | MR | Zbl

[12] Cordero-Erausquin, D., Nazaret, B. and Villani, C. A mass-transportation approach to sharp Sobolev and Gagliardo-Nirenberg inequalities. Adv. Math. 182, 307-332 (2004). | MR | Zbl

[13] Davies, E. B. Heat kernel and spectral theory. Cambridge Univ. Press (1989). | MR

[14] Demange, J. Porous media equation and Sobolev inequalities under negative curvature. Bull. Sci. Math. 129, 804-830 (2005). | MR | Zbl

[15] Evans, L. C. Partial differential equations. Graduate Studies in Math. 19. Amer. Math. Soc. (1997). | MR | Zbl

[16] Federbush, P. A partially alternate derivation of a result of Nelson. J. Math. Phys. 10, 50-52 (1969). | Zbl

[17] Gardner, R. J. The Brunn-Minkowski inequality. Bull. Amer. Math. Soc. 39, 355-405 (2002). | MR | Zbl

[18] Gross, L. Logarithmic Sobolev inequalities. Amer. J. Math. 97, 1061-1083 (1975). | MR | Zbl

[19] Das Gupta, S. Brunn-Minkowski inequality and its aftermath. J. Multivariate Anal. 10, 296-318 (1980). | MR | Zbl

[20] Ledoux, M. The geometry of Markov diffusion generators. Ann. Fac. Sci. Toulouse IX, 305-366 (2000). | EuDML | Numdam | MR | Zbl

[21] Ledoux, M. Géométrie des espaces métriques mesurés : les travaux de Lott, Villani, Sturm. Séminaire Bourbaki, Astérisque 326, 257-280 (2009). | Numdam | MR | Zbl

[22] Leindler, L. On a certain converse of Hölder’s inequality II. Acta Sci. Math. Szeged 33, 217-223 (1972). | MR | Zbl

[23] Li, P. and Yau, S.-T. On the parabolic kernel of the Schrödinger operator. Acta Math. 156, 153-201 (1986). | MR | Zbl

[24] Otto, F. and Villani, C. Generalization of an inequality by Talagrand, and links with the logarithmic Sobolev inequality. J. Funct. Anal. 173, 361-400 (2000). | MR | Zbl

[25] Prékopa, A. On logarithmic concave measures and functions. Acta Sci. Math. Szeged 34, 335-343 (1973). | MR | Zbl

[26] Royer, G. An initiation to logarithmic Sobolev inequalities. Translated from the 1999 French original. SMF/AMS Texts and Monographs 14. Amer. Math. Soc. / Soc. Math. de France (2007). | MR | Zbl

[27] Stam, A. Some inequalities satisfied by the quantities of information of Fisher and Shannon. Inform. Control 2, 101-112 (1959). | MR | Zbl

[28] Villani, C. Topics in optimal transportation. Graduate Studies in Mathematics 58. Amer. Math. Soc. (2003). | MR | Zbl

[29] Villani, C. Optimal transport, old and new. Grundlehren der Mathematischen Wissenschaften, 338. Springer (2009). | MR | Zbl

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