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May  2013, 33(5): 1927-1935. doi: 10.3934/dcds.2013.33.1927

From log Sobolev to Talagrand: A quick proof

 1 Laboratoire J. A. Dieudonné, Université de Nice, Parc Valrose, 06108 Nice, France 2 Institut de Mathématiques de Toulouse, Université de Toulouse, 31062 Toulouse, France

Received  December 2011 Revised  February 2012 Published  December 2012

We provide yet another proof of the Otto-Villani theorem from the log Sobolev inequality to the Talagrand transportation cost inequality valid in arbitrary metric measure spaces. The argument relies on the recent development [2] identifying gradient flows in Hilbert space and in Wassertein space, emphasizing one key step as precisely the root of the Otto-Villani theorem. The approach does not require the doubling property or the validity of the local Poincaré inequality.
Citation: Nicola Gigli, Michel Ledoux. From log Sobolev to Talagrand: A quick proof. Discrete and Continuous Dynamical Systems, 2013, 33 (5) : 1927-1935. doi: 10.3934/dcds.2013.33.1927
References:
 [1] L. Ambrosio, N. Gigli and G. Savaré, Heat Flow and calculus over spaces with Ricci curvature bounded from below - the compact case, To appear in Rend. Acc. Naz. Lnice, in memory of E. Magenes. [2] _______, Calculus and heat flows in metric measure spaces with Ricci curvature bounded from below, Preprint (2011), arXiv:1106.2090. [3] _______, Density of lipschitz functions and equivalence of weak gradients in metric measure spaces, Preprint (2011), arXiv:1111.3730 . [4] S. Bobkov, I. Gentil and M. Ledoux, Hypercontractivity of Hamilton-Jacobi equations, J. Math. Pures Appl. (9), 80 (2001), 669-696. doi: 10.1016/S0021-7824(01)01208-9. [5] J. Cheeger, Differentiability of Lipschitz functions on metric measure spaces, Geom. Funct. Anal., 9 (1999), 428-517. doi: 10.1007/s000390050094. [6] N. Gigli, K. Kuwada and S. Ohta, Heat flow on Alexandrov spaces, Accepted paper at CPAM (2011), arXiv:1008.1319. [7] N. Gozlan, A characterization of dimension free concentration in terms of transportation inequalities, Ann. Probab., 37 (2009), 2480-2498. doi: 10.1214/09-AOP470. [8] N. Gozlan and C. Léonard, Transport inequalities. A survey, Markov Process. Rel. Fields, 16 (2010), 635-736. [9] N. Gozlan, C. Roberto and P.-M. Samson, Characterization of Talagrand's transport-entropy inequalities in metric spaces, Preprint 2011. [10] J. Heinonen, Nonsmooth calculus, Bull. Amer. Math. Soc. (N.S.), 44 (2007), 163-232. doi: 10.1090/S0273-0979-07-01140-8. [11] J. Lott and C. Villani, Hamilton-Jacobi semigroup on length spaces and applications, J. Math. Pures Appl. (9), 88 (2007), 219-229. doi: 10.1016/j.matpur.2007.06.003. [12] F. Otto, The geometry of dissipative evolution equations: the porous medium equation, Comm. Partial Differential Equations, 26 (2001), 101-174. doi: 10.1081/PDE-100002243. [13] F. Otto and C. Villani, Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality, J. Funct. Anal., 173 (2000), 361-400. doi: 10.1006/jfan.1999.3557. [14] M. Talagrand, Transportation cost for Gaussian and other product measures, Geom. Funct. Anal., 6 (1996), 587-600. doi: 10.1007/BF02249265. [15] C. Villani, "Optimal Transport," Grundlehren der Mathematischen Wissenschaften. [Fundamental Principles of Mathematical Sciences], 338, Springer-Verlag, Berlin, 2009, Old and new. doi: 10.1007/978-3-540-71050-9.

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References:
 [1] L. Ambrosio, N. Gigli and G. Savaré, Heat Flow and calculus over spaces with Ricci curvature bounded from below - the compact case, To appear in Rend. Acc. Naz. Lnice, in memory of E. Magenes. [2] _______, Calculus and heat flows in metric measure spaces with Ricci curvature bounded from below, Preprint (2011), arXiv:1106.2090. [3] _______, Density of lipschitz functions and equivalence of weak gradients in metric measure spaces, Preprint (2011), arXiv:1111.3730 . [4] S. Bobkov, I. Gentil and M. Ledoux, Hypercontractivity of Hamilton-Jacobi equations, J. Math. Pures Appl. (9), 80 (2001), 669-696. doi: 10.1016/S0021-7824(01)01208-9. [5] J. Cheeger, Differentiability of Lipschitz functions on metric measure spaces, Geom. Funct. Anal., 9 (1999), 428-517. doi: 10.1007/s000390050094. [6] N. Gigli, K. Kuwada and S. Ohta, Heat flow on Alexandrov spaces, Accepted paper at CPAM (2011), arXiv:1008.1319. [7] N. Gozlan, A characterization of dimension free concentration in terms of transportation inequalities, Ann. Probab., 37 (2009), 2480-2498. doi: 10.1214/09-AOP470. [8] N. Gozlan and C. Léonard, Transport inequalities. A survey, Markov Process. Rel. Fields, 16 (2010), 635-736. [9] N. Gozlan, C. Roberto and P.-M. Samson, Characterization of Talagrand's transport-entropy inequalities in metric spaces, Preprint 2011. [10] J. Heinonen, Nonsmooth calculus, Bull. Amer. Math. Soc. (N.S.), 44 (2007), 163-232. doi: 10.1090/S0273-0979-07-01140-8. [11] J. Lott and C. Villani, Hamilton-Jacobi semigroup on length spaces and applications, J. Math. Pures Appl. (9), 88 (2007), 219-229. doi: 10.1016/j.matpur.2007.06.003. [12] F. Otto, The geometry of dissipative evolution equations: the porous medium equation, Comm. Partial Differential Equations, 26 (2001), 101-174. doi: 10.1081/PDE-100002243. [13] F. Otto and C. Villani, Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality, J. Funct. Anal., 173 (2000), 361-400. doi: 10.1006/jfan.1999.3557. [14] M. Talagrand, Transportation cost for Gaussian and other product measures, Geom. Funct. Anal., 6 (1996), 587-600. doi: 10.1007/BF02249265. [15] C. Villani, "Optimal Transport," Grundlehren der Mathematischen Wissenschaften. [Fundamental Principles of Mathematical Sciences], 338, Springer-Verlag, Berlin, 2009, Old and new. doi: 10.1007/978-3-540-71050-9.
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