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Optimal Transportation and Functional Inequalities

  • Dominique Bakry
  • Ivan Gentil
  • Michel Ledoux
Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 348)

Abstract

This chapter is a brief investigation of the links between optimal transportation methods and functional inequalities in the Markov operator framework of this monograph. After a brief introduction to the basic material on optimal transportation, the main topic of transportation cost inequalities and first examples for Gaussian measures are presented. Interpolation along the geodesics of optimal transport is used towards logarithmic Sobolev inequalities and transportation cost inequalities comparing relative entropy and Wasserstein distances between probability measures. An alternate approach to sharp Sobolev or Gagliardo–Nirenberg inequalities in Euclidean space is provided next along these lines. Non-linear Hamilton–Jacobi equations and hypercontractivity properties of their solutions, analogous to the ones for linear heat equations, are investigated in the further sections towards the relationships between (quadratic) transportation cost inequalities and logarithmic Sobolev inequalities. Contraction properties in Wasserstein space along with the heat semigroup are investigated in the Markov operator setting. The last section is a very brief overview of recent developments towards a notion of Ricci curvature lower bounds based on optimal transportation and the connection with the Γ-calculus developed in this work.

Keywords

Sobolev Inequality Logarithmic Sobolev Inequality Markov Operator Functional Inequality Optimal Transportation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Bibliography

  1. 3.
    M. Agueh, N. Ghoussoub, X. Kang, Geometric inequalities via a general comparison principle for interacting gases. Geom. Funct. Anal. 14(1), 215–244 (2004) CrossRefzbMATHMathSciNetGoogle Scholar
  2. 9.
    L. Ambrosio, N. Gigli, G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2nd edn. Lectures in Mathematics ETH Zürich (Birkhäuser, Basel, 2008) zbMATHGoogle Scholar
  3. 10.
    L. Ambrosio, N. Gigli, G. Savaré, Bakry-Emery curvature-dimension condition and Riemannian Ricci curvature bounds. Preprint, 2012 Google Scholar
  4. 11.
    L. Ambrosio, N. Gigli, G. Savaré, Metric measure spaces with Riemannian Ricci curvature bounded from below. Preprint, 2012 Google Scholar
  5. 12.
    L. Ambrosio, N. Gigli, G. Savaré, Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below. Invent. Math. (2013). doi: 10.1007/s00222-013-0456-1 Google Scholar
  6. 31.
    D. Bakry, F. Bolley, I. Gentil, Dimension dependent hypercontractivity for Gaussian kernels. Probab. Theory Relat. Fields 154(3), 845–874 (2012) CrossRefzbMATHMathSciNetGoogle Scholar
  7. 37.
    D. Bakry, I. Gentil, M. Ledoux, On Harnack inequalities and optimal transportation. Ann. Sc. Norm. Sup. Pisa (2012). doi: 10.2422/2036-2145.201210_007 Google Scholar
  8. 43.
    Z.M. Balogh, A. Engulatov, L. Hunziker, O.E. Maasalo, Functional inequalities and Hamilton–Jacobi equations in geodesic spaces. Potential Anal. 36(2), 317–337 (2012) CrossRefzbMATHMathSciNetGoogle Scholar
  9. 44.
    G. Barles, Solutions de Viscosité des Équations de Hamilton-Jacobi. Mathématiques & Applications (Berlin) [Mathematics & Applications], vol. 17 (Springer, Paris, 1994) zbMATHGoogle Scholar
  10. 56.
    W. Beckner, Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality. Ann. Math. (2) 138(1), 213–242 (1993) CrossRefzbMATHMathSciNetGoogle Scholar
  11. 71.
    G. Blower, The Gaussian isoperimetric inequality and transportation. Positivity 7(3), 203–224 (2003) CrossRefzbMATHMathSciNetGoogle Scholar
  12. 76.
    S.G. Bobkov, I. Gentil, M. Ledoux, Hypercontractivity of Hamilton-Jacobi equations. J. Math. Pures Appl. 80(7), 669–696 (2001) zbMATHMathSciNetGoogle Scholar
  13. 77.
    S.G. Bobkov, F. Götze, Exponential integrability and transportation cost related to logarithmic Sobolev inequalities. J. Funct. Anal. 163(1), 1–28 (1999) CrossRefzbMATHMathSciNetGoogle Scholar
  14. 85.
    F. Bolley, I. Gentil, A. Guillin, Dimensional contraction via Markov transportation distance. Preprint, 2013 Google Scholar
  15. 94.
    Y. Brenier, Polar factorization and monotone rearrangement of vector-valued functions. Commun. Pure Appl. Math. 44(4), 375–417 (1991) CrossRefzbMATHMathSciNetGoogle Scholar
  16. 101.
    L.A. Caffarelli, Boundary regularity of maps with convex potentials. II. Ann. Math. (2) 144(3), 453–496 (1996) CrossRefzbMATHMathSciNetGoogle Scholar
  17. 102.
    L.A. Caffarelli, A priori estimates and the geometry of the Monge Ampère equation, in Nonlinear Partial Differential Equations in Differential Geometry, Park City, UT, 1992. IAS/Park City Math. Ser., vol. 2. (Amer. Math. Soc., Providence, 1996), pp. 5–63 Google Scholar
  18. 103.
    L.A. Caffarelli, Monotonicity properties of optimal transportation and the FKG and related inequalities. Commun. Math. Phys. 214(3), 547–563 (2000) CrossRefzbMATHMathSciNetGoogle Scholar
  19. 104.
    P. Cannarsa, C. Sinestrari, Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control. Progress in Nonlinear Differential Equations and Their Applications, vol. 58 (Birkhäuser, Boston, 2004) zbMATHGoogle Scholar
  20. 112.
    J.A. Carrillo, R.J. McCann, C. Villani, Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates. Rev. Mat. Iberoam. 19(3), 971–1018 (2003) CrossRefzbMATHMathSciNetGoogle Scholar
  21. 113.
    J.A. Carrillo, R.J. McCann, C. Villani, Contractions in the 2-Wasserstein length space and thermalization of granular media. Arch. Ration. Mech. Anal. 179, 217–263 (2006) CrossRefzbMATHMathSciNetGoogle Scholar
  22. 117.
    P. Cattiaux, A. Guillin, On quadratic transportation cost inequalities. J. Math. Pures Appl. 86(4), 341–361 (2006) MathSciNetGoogle Scholar
  23. 128.
    M.-F. Chen, Trilogy of couplings and general formulas for lower bound of spectral gap, in Probability Towards 2000, New York, 1995. Lecture Notes in Statist., vol. 128 (Springer, New York, 1998), pp. 123–136 CrossRefGoogle Scholar
  24. 129.
    M.-F. Chen, F.-Y. Wang, Application of coupling method to the first eigenvalue on manifold. Prog. Nat. Sci. 5(2), 227–229 (1995) Google Scholar
  25. 133.
    D. Cordero-Erausquin, Some applications of mass transport to Gaussian-type inequalities. Arch. Ration. Mech. Anal. 161(3), 257–269 (2002) CrossRefzbMATHMathSciNetGoogle Scholar
  26. 134.
    D. Cordero-Erausquin, W. Gangbo, C. Houdré, Inequalities for generalized entropy and optimal transportation, in Recent Advances in the Theory and Applications of Mass Transport. Contemp. Math., vol. 353 (Am. Math. Soc., Providence, 2004), pp. 73–94 CrossRefGoogle Scholar
  27. 135.
    D. Cordero-Erausquin, R.J. McCann, M. Schmuckenschläger, A Riemannian interpolation inequality à la Borell, Brascamp and Lieb. Invent. Math. 146(2), 219–257 (2001) CrossRefzbMATHMathSciNetGoogle Scholar
  28. 136.
    D. Cordero-Erausquin, R.J. McCann, M. Schmuckenschläger, Prékopa-Leindler type inequalities on Riemannian manifolds, Jacobi fields, and optimal transport. Ann. Fac. Sci. Toulouse 15(4), 613–635 (2006) CrossRefzbMATHMathSciNetGoogle Scholar
  29. 137.
    D. Cordero-Erausquin, B. Nazaret, C. Villani, A mass-transportation approach to sharp Sobolev and Gagliardo-Nirenberg inequalities. Adv. Math. 182(2), 307–332 (2004) CrossRefzbMATHMathSciNetGoogle Scholar
  30. 148.
    M. Del Pino, J. Dolbeault, The optimal Euclidean L p-Sobolev logarithmic inequality. J. Funct. Anal. 197(1), 151–161 (2003) CrossRefzbMATHMathSciNetGoogle Scholar
  31. 175.
    M. Erbar, The heat equation on manifolds as a gradient flow in the Wasserstein space. Ann. Inst. Henri Poincaré Probab. Stat. 46(1), 1–23 (2010) CrossRefzbMATHMathSciNetGoogle Scholar
  32. 176.
    M. Erbar, K. Kuwada, K.-T. Sturm, On the equivalence of the entropy curvature-dimension condition and Bochner’s inequality on metric measure spaces. Preprint, 2013 Google Scholar
  33. 177.
    J.F. Escobar, Sharp constant in a Sobolev trace inequality. Indiana Univ. Math. J. 37(3), 687–698 (1988) CrossRefzbMATHMathSciNetGoogle Scholar
  34. 179.
    L.C. Evans, Partial Differential Equations, 2nd edn. Graduate Studies in Mathematics, vol. 19 (American Mathematical Society, Providence, 2010) zbMATHGoogle Scholar
  35. 180.
    L.C. Evans, R.F. Gariepy, Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics (CRC Press, Boca Raton, 1992) zbMATHGoogle Scholar
  36. 188.
    Y. Fujita, An optimal logarithmic Sobolev inequality with Lipschitz constants. J. Funct. Anal. 261(5), 1133–1144 (2011) CrossRefzbMATHMathSciNetGoogle Scholar
  37. 197.
    I. Gentil, Ultracontractive bounds on Hamilton-Jacobi solutions. Bull. Sci. Math. 126(6), 507–524 (2002) CrossRefzbMATHMathSciNetGoogle Scholar
  38. 198.
    I. Gentil, The general optimal L p-Euclidean logarithmic Sobolev inequality by Hamilton-Jacobi equations. J. Funct. Anal. 202(2), 591–599 (2003) CrossRefzbMATHMathSciNetGoogle Scholar
  39. 206.
    N. Gozlan, A characterization of dimension free concentration in terms of transportation inequalities. Ann. Probab. 37(6), 2480–2498 (2009) CrossRefzbMATHMathSciNetGoogle Scholar
  40. 208.
    N. Gozlan, Transport-entropy inequalities on the line. Electron. J. Probab. 17(49), 1–18 (2012) MathSciNetGoogle Scholar
  41. 209.
    N. Gozlan, C. Léonard, A large deviation approach to some transportation cost inequalities. Probab. Theory Relat. Fields 139(1–2), 235–283 (2007) CrossRefzbMATHGoogle Scholar
  42. 210.
    N. Gozlan, C. Léonard, Transport inequalities. A survey. Markov Process. Relat. Fields 16, 635–736 (2010) zbMATHGoogle Scholar
  43. 211.
    N. Gozlan, C. Roberto, P.-M. Samson, From concentration to logarithmic Sobolev and Poincaré inequalities. J. Funct. Anal. 260(5), 1491–1522 (2011) CrossRefzbMATHMathSciNetGoogle Scholar
  44. 212.
    N. Gozlan, C. Roberto, P.-M. Samson, Hamilton Jacobi equations on metric spaces and transport entropy inequalities. Rev. Mat. Iberoam. (2013, to appear) Google Scholar
  45. 213.
    N. Gozlan, C. Roberto, P.-M. Samson, Characterization of Talagrand’s transport-entropy inequality in metric spaces. Ann. Probab. 41(5), 3112–3139 (2013) CrossRefzbMATHMathSciNetGoogle Scholar
  46. 259.
    R. Jordan, D. Kinderlehrer, F. Otto, The variational formulation of the Fokker-Planck equation. SIAM J. Math. Anal. 29(1), 1–17 (1998) CrossRefzbMATHMathSciNetGoogle Scholar
  47. 266.
    H. Knothe, Contributions to the theory of convex bodies. Mich. Math. J. 4, 39–52 (1957) CrossRefzbMATHMathSciNetGoogle Scholar
  48. 267.
    M. Knott, C.S. Smith, On the optimal mapping of distributions. J. Optim. Theory Appl. 43(1), 39–49 (1984) CrossRefzbMATHMathSciNetGoogle Scholar
  49. 270.
    K. Kuwada, Duality on gradient estimates and Wasserstein controls. J. Funct. Anal. 258(11), 3758–3774 (2010) CrossRefzbMATHMathSciNetGoogle Scholar
  50. 271.
    K. Kuwada, Space-time Wasserstein controls and Bakry-Ledoux type gradient estimates. Preprint, 2013 Google Scholar
  51. 278.
    M. Ledoux, The Concentration of Measure Phenomenon. Mathematical Surveys and Monographs, vol. 89 (American Mathematical Society, Providence, 2001) zbMATHGoogle Scholar
  52. 291.
    J. Lott, C. Villani, Hamilton-Jacobi semigroup on length spaces and applications. J. Math. Pures Appl. 88(3), 219–229 (2007) zbMATHMathSciNetGoogle Scholar
  53. 292.
    J. Lott, C. Villani, Ricci curvature for metric-measure spaces via optimal transport. Ann. Math. (2) 169(3), 903–991 (2009) CrossRefzbMATHMathSciNetGoogle Scholar
  54. 296.
    F. Maggi, C. Villani, Balls have the worst best Sobolev inequalities. II. Variants and extensions. Calc. Var. Partial Differ. Equ. 31(1), 47–74 (2008) CrossRefzbMATHMathSciNetGoogle Scholar
  55. 298.
    K. Marton, Bounding \(\overline{d}\)-distance by informational divergence: a method to prove measure concentration. Ann. Probab. 24(2), 857–866 (1996) CrossRefzbMATHMathSciNetGoogle Scholar
  56. 304.
    R.J. McCann, Existence and uniqueness of monotone measure-preserving maps. Duke Math. J. 80(2), 309–323 (1995) CrossRefzbMATHMathSciNetGoogle Scholar
  57. 305.
    R.J. McCann, A convexity principle for interacting gases. Adv. Math. 128(1), 153–179 (1997) CrossRefzbMATHMathSciNetGoogle Scholar
  58. 316.
    V.D. Milman, G. Schechtman, Asymptotic Theory of Finite-Dimensional Normed Spaces. Lecture Notes in Mathematics, vol. 1200 (Springer, Berlin, 1986). With an appendix by M. Gromov zbMATHGoogle Scholar
  59. 324.
    B. Nazaret, Best constant in Sobolev trace inequalities on the half-space. Nonlinear Anal. 65(10), 1977–1985 (2006) CrossRefzbMATHMathSciNetGoogle Scholar
  60. 339.
    F. Otto, The geometry of dissipative evolution equations: the porous medium equation. Commun. Partial Differ. Equ. 26(1–2), 101–174 (2001) CrossRefzbMATHMathSciNetGoogle Scholar
  61. 340.
    F. Otto, C. Villani, Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality. J. Funct. Anal. 173(2), 361–400 (2000) CrossRefzbMATHMathSciNetGoogle Scholar
  62. 341.
    F. Otto, M. Westdickenberg, Eulerian calculus for the contraction in the Wasserstein distance. SIAM J. Math. Anal. 37(4), 1227–1255 (2005) (electronic) CrossRefMathSciNetGoogle Scholar
  63. 351.
    S.T. Rachev, Probability Metrics and the Stability of Stochastic Models. Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics (Wiley, Chichester, 1991) zbMATHGoogle Scholar
  64. 352.
    S.T. Rachev, L. Rüschendorf, Mass Transportation Problems. Vol. I. Probability and Its Applications (New York) (Springer, New York, 1998). Theory Google Scholar
  65. 353.
    S.T. Rachev, L. Rüschendorf, Mass Transportation Problems. Vol. II. Probability and Its Applications (New York) (Springer, New York, 1998). Applications Google Scholar
  66. 373.
    L. Rüschendorf, S.T. Rachev, A characterization of random variables with minimum L 2-distance. J. Multivar. Anal. 32(1), 48–54 (1990) CrossRefzbMATHGoogle Scholar
  67. 399.
    K.-T. Sturm, On the geometry of metric measure spaces. I. Acta Math. 196(1), 65–131 (2006) CrossRefzbMATHMathSciNetGoogle Scholar
  68. 400.
    K.-T. Sturm, On the geometry of metric measure spaces. II. Acta Math. 196(1), 133–177 (2006) CrossRefzbMATHMathSciNetGoogle Scholar
  69. 404.
    M. Talagrand, Transportation cost for Gaussian and other product measures. Geom. Funct. Anal. 6(3), 587–600 (1996) CrossRefzbMATHMathSciNetGoogle Scholar
  70. 407.
    H. Tanaka, An inequality for a functional of probability distributions and its application to Kac’s one-dimensional model of a Maxwellian gas. Z. Wahrscheinlichkeitstheor. Verw. Geb. 27, 47–52 (1973) CrossRefzbMATHGoogle Scholar
  71. 408.
    H. Tanaka, Probabilistic treatment of the Boltzmann equation of Maxwellian molecules. Z. Wahrscheinlichkeitstheor. Verw. Geb. 46(1), 67–105 (1978/79) CrossRefGoogle Scholar
  72. 424.
    C. Villani, Topics in Optimal Transportation. Graduate Studies in Mathematics, vol. 58 (American Mathematical Society, Providence, 2003) zbMATHGoogle Scholar
  73. 426.
    C. Villani, Optimal Transport, old and new. Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 338 (Springer, Berlin, 2009) CrossRefzbMATHGoogle Scholar
  74. 427.
    M.-K. von Renesse, K.-T. Sturm, Transport inequalities, gradient estimates, entropy, and Ricci curvature. Commun. Pure Appl. Math. 58(7), 923–940 (2005) CrossRefzbMATHGoogle Scholar
  75. 431.
    F.-Y. Wang, Functional Inequalities, Markov Processes, and Spectral Theory (Science Press, Beijing, 2004) Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Dominique Bakry
    • 1
  • Ivan Gentil
    • 2
  • Michel Ledoux
    • 1
  1. 1.Institut de Mathématiques de ToulouseUniversité de Toulouse and Institut Universitaire de FranceToulouseFrance
  2. 2.Institut Camille JordanUniversité Claude Bernard Lyon 1VilleurbanneFrance

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