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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 & 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, ().   Google Scholar

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J. Math. Pures Appl. (9), 88 (2007), 219-229. doi: 10.1016/j.matpur.2007.06.003.  Google Scholar

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Comm. Partial Differential Equations, 26 (2001), 101-174. doi: 10.1081/PDE-100002243.  Google Scholar

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J. Funct. Anal., 173 (2000), 361-400. doi: 10.1006/jfan.1999.3557.  Google Scholar

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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, ().   Google Scholar

[2]

Preprint (2011), arXiv:1106.2090. Google Scholar

[3]

Preprint (2011), arXiv:1111.3730 . Google Scholar

[4]

J. Math. Pures Appl. (9), 80 (2001), 669-696. doi: 10.1016/S0021-7824(01)01208-9.  Google Scholar

[5]

Geom. Funct. Anal., 9 (1999), 428-517. doi: 10.1007/s000390050094.  Google Scholar

[6]

Accepted paper at CPAM (2011), arXiv:1008.1319. Google Scholar

[7]

Ann. Probab., 37 (2009), 2480-2498. doi: 10.1214/09-AOP470.  Google Scholar

[8]

Markov Process. Rel. Fields, 16 (2010), 635-736.  Google Scholar

[9]

Preprint 2011. Google Scholar

[10]

Bull. Amer. Math. Soc. (N.S.), 44 (2007), 163-232. doi: 10.1090/S0273-0979-07-01140-8.  Google Scholar

[11]

J. Math. Pures Appl. (9), 88 (2007), 219-229. doi: 10.1016/j.matpur.2007.06.003.  Google Scholar

[12]

Comm. Partial Differential Equations, 26 (2001), 101-174. doi: 10.1081/PDE-100002243.  Google Scholar

[13]

J. Funct. Anal., 173 (2000), 361-400. doi: 10.1006/jfan.1999.3557.  Google Scholar

[14]

Geom. Funct. Anal., 6 (1996), 587-600. doi: 10.1007/BF02249265.  Google Scholar

[15]

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.  Google Scholar

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