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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

Nicola Gigli 1, and  Michel Ledoux 2,

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.
Keywords: log-Sobolev inequality, Talagrand inequality., Metric measure spaces.
Mathematics Subject Classification: Primary: 51F99; Secondary: 39B6.
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.

show all references

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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