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Characterization of Talagrand’s transport-entropy inequalities in metric spaces. (English) Zbl 1283.60029
The authors establish that a modified log-Sobolev inequality implies the corresponding transport-entropy inequality, thus extending a classical result of Otto and Villani. As main result they obtain the equivalence of the transport-entropy inequality with a modified log-Sobolev inequality called $$(\tau)$$-log Sobolev inequality. As an application it is deduced that these inequalities remain stable under perturbations (Holley-Stroock perturbation result).

##### MSC:
 60E15 Inequalities; stochastic orderings 26D10 Inequalities involving derivatives and differential and integral operators
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##### References:
 [1] Ané, C., Blachère, S., Chafaï, D., Fougères, P., Gentil, I., Malrieu, F., Roberto, C. and Scheffer, G. (2000). Sur les Inégalités de Sobolev Logarithmiques. Panoramas et Synthèses [ Panoramas and Syntheses ] 10 . Société Mathématique de France, Paris. · Zbl 0982.46026 [2] Barthe, F. and Roberto, C. (2008). Modified logarithmic Sobolev inequalities on $$\mathbb{R}$$. Potential Anal. 29 167-193. · Zbl 1170.26010 [3] Bobkov, S. G., Gentil, I. and Ledoux, M. (2001). Hypercontractivity of Hamilton-Jacobi equations. J. Math. Pures Appl. (9) 80 669-696. · Zbl 1038.35020 [4] Bobkov, S. G. and Götze, F. (1999). Exponential integrability and transportation cost related to logarithmic Sobolev inequalities. J. Funct. Anal. 163 1-28. · Zbl 0924.46027 [5] Bobkov, S. G. and Ledoux, M. (1997). Poincaré’s inequalities and Talagrand’s concentration phenomenon for the exponential distribution. Probab. Theory Related Fields 107 383-400. · Zbl 0878.60014 [6] Cattiaux, P. and Guillin, A. (2006). On quadratic transportation cost inequalities. J. Math. Pures Appl. (9) 86 341-361. · Zbl 1118.58017 [7] Gentil, I. (2008). From the Prékopa-Leindler inequality to modified logarithmic Sobolev inequality. Ann. Fac. Sci. Toulouse Math. (6) 17 291-308. · Zbl 1175.26036 [8] Gentil, I., Guillin, A. and Miclo, L. (2005). Modified logarithmic Sobolev inequalities and transportation inequalities. Probab. Theory Related Fields 133 409-436. · Zbl 1080.26010 [9] Gentil, I., Guillin, A. and Miclo, L. (2007). Modified logarithmic Sobolev inequalities in null curvature. Rev. Mat. Iberoam. 23 235-258. · Zbl 1123.26022 [10] Gozlan, N. (2007). Characterization of Talagrand’s like transportation-cost inequalities on the real line. J. Funct. Anal. 250 400-425. · Zbl 1135.46022 [11] Gozlan, N. (2009). A characterization of dimension free concentration in terms of transportation inequalities. Ann. Probab. 37 2480-2498. · Zbl 1201.60016 [12] Gozlan, N. and Léonard, C. (2007). A large deviation approach to some transportation cost inequalities. Probab. Theory Related Fields 139 235-283. · Zbl 1126.60022 [13] Gozlan, N. and Léonard, C. (2010). Transport inequalities. A survey. Markov Process. Related Fields 16 635-736. · Zbl 1229.26029 [14] Gozlan, N., Roberto, C. and Samson, P. M. (2010). From concentration to logarithmic Sobolev and Poincaré inequalities. J. Funct. Anal. 260 1491-1522. · Zbl 1226.60024 [15] Gozlan, N., Roberto, C. and Samson, P. M. (2011). A new characterization of Talagrand’s transport-entropy inequalities and applications. Ann. Probab. 39 857-880. · Zbl 1233.60007 [16] Gross, L. (1975). Logarithmic Sobolev inequalities. Amer. J. Math. 97 1061-1083. · Zbl 0318.46049 [17] Helffer, B. (2002). Semiclassical Analysis , Witten Laplacians , and Statistical Mechanics. Series in Partial Differential Equations and Applications 1 . World Scientific, River Edge, NJ. · Zbl 1046.82001 [18] Holley, R. and Stroock, D. (1987). Logarithmic Sobolev inequalities and stochastic Ising models. J. Stat. Phys. 46 1159-1194. · Zbl 0682.60109 [19] Ledoux, M. (2001). The Concentration of Measure Phenomenon. Mathematical Surveys and Monographs 89 . Amer. Math. Soc., Providence, RI. · Zbl 0995.60002 [20] Lott, J. and Villani, C. (2007). Hamilton-Jacobi semigroup on length spaces and applications. J. Math. Pures Appl. (9) 88 219-229. · Zbl 1210.53047 [21] Marton, K. (1986). A simple proof of the blowing-up lemma. IEEE Trans. Inform. Theory 32 445-446. · Zbl 0594.94003 [22] Maurey, B. (1991). Some deviation inequalities. Geom. Funct. Anal. 1 188-197. · Zbl 0756.60018 [23] Monge, G. (1781). Mémoire sur la théorie des déblais et des remblais. Histoire de L’Académie Royale des Sciences de Paris 666-704. [24] Otto, F. and Villani, C. (2000). Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality. J. Funct. Anal. 173 361-400. · Zbl 0985.58019 [25] Papageorgiou, I. (2011). A note on the modified log-Sobolev inequality. Potential Anal. 35 275-286. · Zbl 1234.60022 [26] Royer, G. (1999). Une Initiation aux Inégalités de Sobolev Logarithmiques . Société Mathématique de France, Paris. · Zbl 0927.60006 [27] Samson, P. M. (2000). Concentration of measure inequalities for Markov chains and $$\Phi$$-mixing processes. Ann. Probab. 28 416-461. · Zbl 1044.60061 [28] Stam, A. J. (1959). Some inequalities satisfied by the quantities of information of Fisher and Shannon. Information and Control 2 101-112. · Zbl 0085.34701 [29] Talagrand, M. (1991). A new isoperimetric inequality and the concentration of measure phenomenon. In Geometric Aspects of Functional Analysis (1989 - 90). Lecture Notes in Math. 1469 94-124. Springer, Berlin. · Zbl 0818.46047 [30] Talagrand, M. (1996). Transportation cost for Gaussian and other product measures. Geom. Funct. Anal. 6 587-600. · Zbl 0859.46030 [31] Villani, C. (2009). Optimal Transport : Old and New. Grundlehren der Mathematischen Wissenschaften [ Fundamental Principles of Mathematical Sciences ] 338 . Springer, Berlin. · Zbl 1156.53003
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