Géométrie des espaces métriques mesurés : les travaux de Lott, Villani, Sturm

Ledoux, Michel

Séminaire Bourbaki Volume 2007/2008 Exposés 982-996, Astérisque, no. 326 (2009), Exposé no. 990, 23 p.

@incollection{AST_2009__326__257_0,
     author = {Ledoux, Michel},
     title = {G\'eom\'etrie des espaces m\'etriques mesur\'es~: les travaux de {Lott,} {Villani,} {Sturm}},
     booktitle = {S\'eminaire Bourbaki Volume 2007/2008 Expos\'es 982-996},
     series = {Ast\'erisque},
     note = {talk:990},
     pages = {257--279},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {326},
     year = {2009},
     mrnumber = {2605325},
     zbl = {1207.53051},
     language = {fr},
     url = {http://www.numdam.org/item/AST_2009__326__257_0/}
}

TY  - CHAP
AU  - Ledoux, Michel
TI  - Géométrie des espaces métriques mesurés : les travaux de Lott, Villani, Sturm
BT  - Séminaire Bourbaki Volume 2007/2008 Exposés 982-996
AU  - Collectif
T3  - Astérisque
N1  - talk:990
PY  - 2009
SP  - 257
EP  - 279
IS  - 326
PB  - Société mathématique de France
UR  - http://www.numdam.org/item/AST_2009__326__257_0/
LA  - fr
ID  - AST_2009__326__257_0
ER  -

%0 Book Section
%A Ledoux, Michel
%T Géométrie des espaces métriques mesurés : les travaux de Lott, Villani, Sturm
%B Séminaire Bourbaki Volume 2007/2008 Exposés 982-996
%A Collectif
%S Astérisque
%Z talk:990
%D 2009
%P 257-279
%N 326
%I Société mathématique de France
%U http://www.numdam.org/item/AST_2009__326__257_0/
%G fr
%F AST_2009__326__257_0

Ledoux, Michel. Géométrie des espaces métriques mesurés : les travaux de Lott, Villani, Sturm, dans Séminaire Bourbaki Volume 2007/2008 Exposés 982-996, Astérisque, no. 326 (2009), Exposé no. 990, 23 p. http://www.numdam.org/item/AST_2009__326__257_0/

Bibliographie
Cité par

[1] L. Ambrosio, N. Gigli & G. Savaré - Gradient flows in metric spaces and in the space of probability measures, Lectures in Mathematics ETH Zürich, Birkhäuser, 2005. | MR | Zbl

[2] L. Ambrosio & G. Savaré - Gradient flows of probability measures, in Handbook of Differential Equations. Evolutionary equations, vol. 3, Elsevier, 2007. | DOI | MR | Zbl

[3] D. Bakry - Transformations de Riesz pour les semi-groupes symétriques, in Séminaire de probabilités, XIX, 1983/84, Lecture Notes in Math., vol. 1123, Springer, 1985, p. 130-174. | DOI | Numdam | MR | Zbl

[4] D. Bakry -, L'hypercontractivité et son utilisation en théorie des semigroupes, in Lectures on probability theory (Saint-Flour, 1992), Lecture Notes in Math., vol. 1581, Springer, 1994, p. 1-114. | DOI | MR | Zbl

[5] D. Bakry & M. Émery - Diffusions hypercontractives, in Séminaire de probabilités, XIX, 1983/84, Lecture Notes in Math., vol. 1123, Springer, 1985, p. 177-206. | DOI | EuDML | Numdam | MR | Zbl

[6] D. Bakry & M. Ledoux - Sobolev inequalities and Myers's diameter theorem for an abstract Markov generator, Duke Math. J. 85 (1996), p. 253-270. | MR | Zbl

[7] F. Barthe - Autour de l'inégalité de Brunn-Minkowski, Ann. Fac. Sci. Toulouse Math. 12 (2003), p. 127-178. | DOI | EuDML | Numdam | MR | Zbl

[8] S. G. Bobkov & M. Ledoux - From Brunn-Minkowski to Brascamp-Lieb and to logarithmic Sobolev inequalities, Geom. Fund. Anal. 10 (2000), p. 1028-1052. | DOI | MR | Zbl

[9] C. Borell - Diffusion equations and geometric inequalities, Potential Anal. 12 (2000), p. 49-71. | DOI | MR | Zbl

[10] Y. Brenier - Décomposition polaire et réarrangement monotone des champs de vecteurs, C. R. Acad. Sci. Paris Sér. I Math. 305 (1987), p. 805-808. | MR | Zbl

[11] Y. Brenier -, Polar factorization and monotone rearrangement of vector-valued functions, Comm. Pure Appl. Math. 44 (1991), p. 375-417. | DOI | MR | Zbl

[12] M. R. Bridson & A. Haefliger - Metric spaces of non-positive curvature, Grund. Math. Wiss., vol. 319, Springer, 1999. | MR | Zbl

[13] D. Burago, Y. Burago & S. Ivanov - A course in metric geometry, Graduate Studies in Math., vol. 33, Amer. Math. Soc., 2001. | DOI | MR | Zbl

[14] L. A. Caffarelli - The regularity of mappings with a convex potential, J. Amer. Math. Soc. 5 (1992), p. 99-104. | DOI | MR | Zbl

[15] J. Cheeger - Differentiability of Lipschitz functions on metric measure spaces, Geom. Funct. Anal. 9 (1999), p. 428-517. | DOI | MR | Zbl

[16] J. Cheeger & T. H. Colding - On the structure of spaces with Ricci curvature bounded below. I, II, III, J. Differential Geom. 46 (1997), p. 406-480 ; | DOI | MR | Zbl

J. Cheeger & T. H. Colding - On the structure of spaces with Ricci curvature bounded below. I, II, III, J. Differential Geom. 54 (2000), p. 13-35 & 37-74. | DOI | MR | Zbl

[17] D. Cordero-Erausquin - Some applications of mass transport to Gaussiantype inequalities, Arch. Ration. Mech. Anal. 161 (2002), p. 257-269. | DOI | MR | Zbl

[18] D. Cordero-Erausquin -, Quelques exemples d'application du transport de mesure en géométrie euclidienne et riemannienne, in Séminaire de Théorie Spectrale et Géométrie. Vol. 22. Année 2003-2004, Sémin. Théor. Spectr. Géom., vol. 22, Univ. Grenoble I, 2004, p. 125-152. | EuDML | Numdam | MR | Zbl

[19] D. Cordero-Erausquin, R. J. Mccann & M. Schmuckenschläger - A Riemannian interpolation inequality à la Borell, Brascamp and Lieb, Invent. Math. 146 (2001), p. 219-257. | DOI | MR | Zbl

[20] D. Cordero-Erausquin, R. J. Mccann & M. Schmuckenschläger -, Prékopa-Leindler type inequalities on Riemannian manifolds, Jacobi fields, and optimal transport, Ann. Fac. Sci. Toulouse Math. 15 (2006), p. 613-635. | DOI | EuDML | Numdam | MR | Zbl

[21] D. Cordero-Erausquin, B. Nazaret & C. Villani - A mass-transportation approach to sharp Sobolev and Gagliardo-Nirenberg inequalities, Adv. Math. 182 (2004), p. 307-332. | DOI | MR | Zbl

[22] K. Fukaya - Collapsing of Riemannian manifolds and eigenvalues of Laplace operator, Invent. Math. 87 (1987), p. 517-547. | DOI | EuDML | MR | Zbl

[23] S. Gallot, D. Hulin & J. Lafontaine - Riemannian geometry, 3e éd., Universitext, Springer, 2004. | MR

[24] M. Gromov - Metric structures for Riemannian and non-Riemannian spaces, Progress in Math., vol. 152, Birkhäuser, 1999. | MR | Zbl

[25] L. Gross - Logarithmic Sobolev inequalities, Amer. J. Math. 97 (1975), p. 1061-1083. | DOI | MR | Zbl

[26] J. Heinonen - Lectures on analysis on metric spaces, Universitext, Springer, 2001. | DOI | MR | Zbl

[27] R. Jordan, D. Kinderlehrer & F. Otto - The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal. 29 (1998), p. 1-17. | DOI | MR | Zbl

[28] N. Juillet - Geometric inequalities and generalized Ricci bounds in the Heisenberg group, prépublication http://sfb611.iam.uni-bonn.de/uploads/279-komplett.pdf. | DOI | Zbl

[29] K. Kuwae & T. Shioya - Sobolev and Dirichlet spaces over maps between metric spaces, J. reine angew. Math. 555 (2003), p. 39-75. | MR | Zbl

[30] M. Ledoux - The concentration of measure phenomenon, Mathematical Surveys and Monographs, vol. 89, Amer. Math. Soc, 2001. | MR | Zbl

[31] G. Loeper - Continuity of maps solutions of optimal transportation problems, à paraître dans Acta Math.

[32] J. Lott - Some geometric properties of the Bakry-Émery-Ricci tensor, Comment. Math. Helv. 78 (2003), p. 865-883. | DOI | MR | Zbl

[33] J. Lott -, Optimal transport and Ricci curvature for metric-measure spaces, in Surveys in differential geometry. Vol. XI, Surv. Differ. Geom., vol. 11, Int. Press, Somerville, MA, 2007, p. 229-257. | DOI | MR | Zbl

[34] J. Lott & C. Villani - Ricci curvature for metric-measure spaces via optimal transport, à paraître dans Ann. Math. | Zbl

[35] J. Lott & C. Villani - Weak curvature conditions and functional inequalities, J. Funct. Anal. 245 (2007), p. 311-333. | DOI | MR | Zbl

[36] B. Maurey - Inégalité de Brunn-Minkowski-Lusternik, et autres inégalités géométriques et fonctionnelles, Séminaire Bourbaki, vol. 2003/2004, exposé n° 928, Astérisque 299 (2005), p. 95-113. | EuDML | Numdam | MR | Zbl

[37] R. J. Mccann - A convexity principle for interacting gases and equilibrium crystals, Thèse, Princeton University, 1994.

[38] R. J. Mccann -, Existence and uniqueness of monotone measure-preserving maps, Duke Math. J. 80 (1995), p. 309-323. | DOI | MR | Zbl

[39] F. Morgan - Manifolds with density, Notices Amer. Math. Soc. 52 (2005), p. 853-858. | MR | Zbl

[40] S.-I. Ohta - On the measure contraction property of metric measure spaces, Comment. Math. Helv. 82 (2007), p. 805-828. | DOI | MR | Zbl

[41] S.-I. Ohta -, Gradient flows on Wasserstein spaces over compact Alexandrov spaces, à paraître dans Amer. J. Math. | MR | Zbl

[42] F. Otto - The geometry of dissipative evolution équations : the porous medium equation, Comm. Partial Differential Equations 26 (2001), p. 101-174. | DOI | MR | Zbl

[43] F. Otto & C. Villani - Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality, J. Fund. Anal. 173 (2000), p. 361-400. | DOI | MR | Zbl

[44] G. Perelman - The entropy formula for the Ricci flow and its geometric applications, prépublication arXiv:math.DG/0211159. | Zbl

[45] S. T. Rachev & L. Rüschendorf - Mass transportation problems, Springer, 1998. | Zbl

[46] M.-K. Von Renesse - On local Poincaré via transportation, Math. Z. 259 (2008), p. 21-31. | DOI | MR | Zbl

[47] M.-K. Von Renesse & K.-T. Sturm - Transport inequalities, gradient estimates, entropy, and Ricci curvature, Comm. Pure Appl. Math. 58 (2005), p. 923-940. | DOI | MR | Zbl

[48] L. Rüschendorf & S. T. Rachev - A characterization of random variables with minimum $L^{2}$ -distance, J. Multivariate Anal. 32 (1990), p. 48-54. | DOI | MR | Zbl

[49] G. Savaré - Gradient flows and diffusion semigroups in metric spaces under lower curvature bounds, C. R. Math. Acad. Sci. Paris345 (2007), p. 151-154. | DOI | MR | Zbl

[50] A. J. Stam - Some inequalities satisfied by the quantities of information of Fisher and Shannon, Information and Control 2 (1959), p. 101-112. | DOI | MR | Zbl

[51] K.-T. Sturm - Convex functionals of probability measures and nonlinear diffusions on manifolds, J. Math. Pures Appl. 84 (2005), p. 149-168. | DOI | MR | Zbl

[52] K.-T. Sturm -, On the geometry of metric measure spaces. I, Acta Math. 196 (2006), p. 65-131. | DOI | MR | Zbl

[53] K.-T. Sturm -, On the geometry of metric measure spaces. II, Acta Math. 196 (2006), p. 133-177. | DOI | MR | Zbl

[54] C. Villani - Topics in optimal transportation, Graduate Studies in Math., vol. 58, Amer. Math. Soc., 2003. | MR | Zbl

[55] C. Villani -, Transport optimal et courbure de Ricci, in Séminaire : Equations aux Dérivées Partielles. 2005-2006, Sémin. Équ. Dériv. Partielles, exp. 7, École Polytech., 2006. | EuDML | Numdam | MR | Zbl

[56] C. Villani -, Optimal transport, old and new, Grund. Math. Wiss., vol. 338, Springer, 2009. | MR | Zbl