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.
Keywords: Logarithmic Sobolev inequality, heat kernel, Brunn-Minkowski inequality
@article{JEDP_2011____A7_0,
author = {Michel Ledoux},
title = {Analytic and {Geometric} {Logarithmic} {Sobolev} {Inequalities}},
journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
eid = {7},
publisher = {Groupement de recherche 2434 du CNRS},
year = {2011},
doi = {10.5802/jedp.79},
language = {en},
url = {https://jedp.centre-mersenne.org/articles/10.5802/jedp.79/}
}
TY - JOUR AU - Michel Ledoux TI - Analytic and Geometric Logarithmic Sobolev Inequalities JO - Journées équations aux dérivées partielles PY - 2011 PB - Groupement de recherche 2434 du CNRS UR - https://jedp.centre-mersenne.org/articles/10.5802/jedp.79/ UR - https://doi.org/10.5802/jedp.79 DO - 10.5802/jedp.79 LA - en ID - JEDP_2011____A7_0 ER -
Michel Ledoux. Analytic and Geometric Logarithmic Sobolev Inequalities. Journées équations aux dérivées partielles (2011), article no. 7, 15 p. doi : 10.5802/jedp.79. https://jedp.centre-mersenne.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). | 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
Cited by Sources: