A User’s Guide to Optimal Transport

  • Luigi Ambrosio
  • Nicola Gigli
Part of the Lecture Notes in Mathematics book series (LNM, volume 2062)


This text is an expanded version of the lectures given by the first author in the 2009 CIME summer school of Cetraro. It provides a quick and reasonably account of the classical theory of optimal mass transportation and of its more recent developments, including the metric theory of gradient flows, geometric and functional inequalities related to optimal transportation, the first and second order differential calculus in the Wasserstein space and the synthetic theory of metric measure spaces with Ricci curvature bounded from below.



Work partially supported by a MIUR PRIN2008 grant.


  1. 1.
    A. Agrachev, P. Lee, Optimal transportation under nonholonomic constraints. Trans. Am. Math. Soc. 361, 6019–6047 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    G. Alberti, On the structure of singular sets of convex functions. Calc. Var. Partial Differ. Equat. 2, 17–27 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    G. Alberti, L. Ambrosio, A geometrical approach to monotone functions in \({\mathbf{R}}^{n}\). Math. Z. 230, 259–316 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    L. Ambrosio, Lecture notes on optimal transport problem, in Mathematical Aspects of Evolving Interfaces, vol. 1812, ed. by P. Colli, J. Rodrigues. CIME summer school in Madeira (Pt) (Springer, Berlin, 2003), pp. 1–52Google Scholar
  5. 5.
    L. Ambrosio, N. Gigli, Construction of the parallel transport in the Wasserstein space. Meth. Appl. Anal. 15, 1–29 (2008)MathSciNetzbMATHGoogle Scholar
  6. 6.
    L. Ambrosio, S. Rigot, Optimal mass transportation in the Heisenberg group. J. Funct. Anal. 208, 261–301 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    L. Ambrosio, N. Gigli, G. Savaré, in Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2nd edn. Lectures in Mathematics ETH Zürich (Birkhäuser, Basel, 2008)Google Scholar
  8. 8.
    L. Ambrosio, N. Gigli, G. Savaré, Calculus and heat flows in metric measure spaces with ricci curvature bounded below, Comm. Pure and Applied Math. (2011)Google Scholar
  9. 9.
    L. Ambrosio, N. Gigli, G. Savaré, Spaces with riemannian ricci curvature bounded below, Comm. Pure and Applied Math. (2011)Google Scholar
  10. 10.
    L. Ambrosio, B. Kirchheim, A. Pratelli, Existence of optimal transport maps for crystalline norms. Duke Math. J. 125 207–241 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    K. Bacher, K.T. Sturm, Localization and tensorization properties of the curvature-dimension condition for metric measure spaces. J. Funct. Anal. 259, 28–56 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    J.-D. Benamou, Y. Brenier, A numerical method for the optimal time-continuous mass transport problem and related problems, in Monge Ampère Equation: Applications to Geometry and Optimization (Deerfield Beach, FL, 1997). Contemporary Mathematics, vol. 226 (American Mathematical Society, Providence, 1999), pp. 1–11Google Scholar
  13. 13.
    P. Bernard, B. Buffoni, Optimal mass transportation and Mather theory. J. Eur. Math. Soc. (JEMS), 9, 85–127 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    M. Bernot, V. Caselles, J.-M. Morel, The structure of branched transportation networks. Calc. Var. Partial Differ. Equat. 32, 279–317 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    S. Bianchini, A. Brancolini, Estimates on path functionals over Wasserstein spaces. SIAM J. Math. Anal. 42, 1179–1217 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    A. Brancolini, G. Buttazzo, F. Santambrogio, Path functionals over Wasserstein spaces. J. Eur. Math. Soc. (JEMS), 8, 415–434 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    L. Brasco, G. Buttazzo, F. Santambrogio, A Benamou-Brenier approach to branched transport. SIAM J. Math. Anal. 43(2), 1023–1040 (2011). doi:10.1137/10079286XMathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Y. Brenier, Décomposition polaire et réarrangement monotone des champs de vecteurs. C. R. Acad. Sci. Paris I Math. 305, 805–808 (1987)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Y. Brenier, Polar factorization and monotone rearrangement of vector-valued functions. Comm. Pure Appl. Math. 44, 375–417 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    D. Burago, Y. Burago, S. Ivanov, in A Course in Metric Geometry. Graduate Studies in Mathematics, vol. 33 (American Mathematical Society, Providence, 2001)Google Scholar
  21. 21.
    L.A. Caffarelli, Boundary regularity of maps with convex potentials. Comm. Pure Appl. Math. 45, 1141–1151 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    L.A. Caffarelli, The regularity of mappings with a convex potential. J. Am. Math. Soc. 5, 99–104 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    L.A. Caffarelli, Boundary regularity of maps with convex potentials, II. Ann. Math. (2) 144, 453–496 (1996)Google Scholar
  24. 24.
    L.A. Caffarelli, M. Feldman, R.J. McCann, Constructing optimal maps for Monge’s transport problem as a limit of strictly convex costs. J. Am. Math. Soc. 15, 1–26 (2002) (electronic)Google Scholar
  25. 25.
    L. Caravenna, A proof of Sudakov theorem with strictly convex norms. Math. Z. 268(1–2), 371–407 (2011) doi:10.1007/s00209-010-0677-6MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    J.A. Carrillo, S. Lisini, G. Savaré, D. Slepcev, Nonlinear mobility continuity equations and generalized displacement convexity. J. Funct. Anal. 258, 1273–1309 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    T. Champion, L. De Pascale, The Monge problem in \({\mathbb{R}}^{d}\). Duke Math. J. 157(3), 551–572 (2011). doi:10.1215/00127094-1272939MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    T. Champion, L. De Pascale, The Monge problem for strictly convex norms in \({\mathbb{R}}^{d}\). J. Eur. Math. Soc. (JEMS), 12, 1355–1369 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    J. Cheeger, Differentiability of Lipschitz functions on metric measure spaces. Geom. Funct. Anal. 9, 428–517 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    D. Cordero-Erausquin, B. Nazaret, C. Villani, A mass-transportation approach to sharp Sobolev and Gagliardo-Nirenberg inequalities. Adv. Math. 182, 307–332 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    C. Dellacherie, P.-A. Meyer, in Probabilities and Potential. North-Holland Mathematics Studies, vol. 29 (North-Holland, Amsterdam, 1978)Google Scholar
  32. 32.
    Q. Deng, K.-T. Sturm, Localization and tensorization properties of the curvature-dimension condition for metric measure spaces, II. J. Funct. Anal. 260(12), 3718–3725 (2011). doi:10.1016/j.jfa.2011.02.026MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    J. Dolbeault, B. Nazaret, G. Savaré, On the Bakry-Emery criterion for linear diffusions and weighted porous media equations. Comm. Math. Sci 6, 477–494 (2008)zbMATHGoogle Scholar
  34. 34.
    L.C. Evans, W. Gangbo, Differential equations methods for the Monge-Kantorovich mass transfer problem. Mem. Am. Math. Soc. 137, viii+66 (1999)Google Scholar
  35. 35.
    A. Fathi, A. Figalli, Optimal transportation on non-compact manifolds. Isr. J. Math. 175, 1–59 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    D. Feyel, A.S. Üstünel, Monge-Kantorovitch measure transportation and Monge-Ampère equation on Wiener space. Probab. Theor. Relat. Fields 128, 347–385 (2004)zbMATHCrossRefGoogle Scholar
  37. 37.
    A. Figalli, N. Gigli, A new transportation distance between non-negative measures, with applications to gradients flows with Dirichlet boundary conditions. J. Math. Pures Appl. (9), 94(2), 107–130 (2010). doi:10.1016/j.matpur.2009.11.005Google Scholar
  38. 38.
    A. Figalli, F. Maggi, A. Pratelli, A mass transportation approach to quantitative isoperimetric inequalities. Invent. Math. 182, 167–211 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    A. Figalli, L. Rifford, Mass transportation on sub-Riemannian manifolds. Geom. Funct. Anal. 20, 124–159 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    N. Fusco, F. Maggi, A. Pratelli, The sharp quantitative isoperimetric inequality. Ann. Math. (2) 168, 941–980 (2008)Google Scholar
  41. 41.
    W. Gangbo, The Monge mass transfer problem and its applications, in Monge Ampère Equation: Applications to Geometry and Optimization, (Deerfield Beach, FL, 1997). Contemporary Mathematics, vol. 226 (American Mathematical Society, Providence, 1999), pp. 79–104Google Scholar
  42. 42.
    W. Gangbo, R.J. McCann, The geometry of optimal transportation. Acta Math. 177, 113–161 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    N. Gigli, On the geometry of the space of probability measures in R n endowed with the quadratic optimal transport distance, Thesis (Ph.D.)–Scuola Normale Superiore, 2008Google Scholar
  44. 44.
    N. Gigli, Second order analysis on \(({P}_{2}(M),{W}_{2})\). Memoir. Am. Math. Soc. 216(1018), xii+154 (2012). doi:10.1090/S0065-9266-2011-00619-2Google Scholar
  45. 45.
    N. Gigli, On the heat flow on metric measure spaces: existence, uniqueness and stability. Calc. Var. Partial Differential Equations 39(1–2), 101–120 (2010). doi:10.1007/s00526-009-0303-9MathSciNetzbMATHCrossRefGoogle Scholar
  46. 46.
    N. Gigli, On the inverse implication of Brenier-McCann theorems and the structure of \(({P}_{2}(M),{W}_{2})\). Methods Appl. Anal. 18(2), 127–158 (2011)MathSciNetGoogle Scholar
  47. 47.
    R. Jordan, D. Kinderlehrer, F. Otto, The variational formulation of the Fokker-Planck equation. SIAM J. Math. Anal. 29, 1–17 (1998) (electronic)Google Scholar
  48. 48.
    N. Juillet, On displacement interpolation of measures involved in Brenier’s theorem. Proc. Am. Math. Soc. 139(10), 3623–3632 (2011). doi:10.1090/S0002-9939-2011-10891-8MathSciNetzbMATHCrossRefGoogle Scholar
  49. 49.
    L.V. Kantorovich, On an effective method of solving certain classes of extremal problems. Dokl. Akad. Nauk. USSR 28, 212–215 (1940)Google Scholar
  50. 50.
    L.V. Kantorovich, On the translocation of masses. Dokl. Akad. Nauk. USSR 37, 199–201 (1942). English translation in J. Math. Sci. 133(4), 1381–1382 (2006)Google Scholar
  51. 51.
    L.V. Kantorovich, G.S. Rubinshtein, On a space of totally additive functions. Vestn. Leningr. Univ. 7(13), 52–59 (1958)Google Scholar
  52. 52.
    M. Knott, C.S. Smith, On the optimal mapping of distributions. J. Optim. Theor. Appl. 43, 39–49 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
  53. 53.
    K. Kuwada, N. Gigli, S.-I. Ohta, Heat flow on alexandrov spaces, Comm. Pure and Applied Math. (2010)Google Scholar
  54. 54.
    S. Lisini, Characterization of absolutely continuous curves in Wasserstein spaces. Calc. Var. Partial Differ. Equat. 28, 85–120 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  55. 55.
    G. Loeper, On the regularity of solutions of optimal transportation problems. Acta Math. 202, 241–283 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  56. 56.
    J. Lott, Some geometric calculations on Wasserstein space. Comm. Math. Phys. 277, 423–437 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  57. 57.
    J. Lott, C. Villani, Weak curvature conditions and functional inequalities. J. Funct. Anal. 245(1), 311–333 (2007). doi:10.1016/j.jfa.2006.10.018MathSciNetzbMATHCrossRefGoogle Scholar
  58. 58.
    J. Lott, C. Villani, Ricci curvature for metric-measure spaces via optimal transport. Ann. Math. 169(2), 903–991 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  59. 59.
    X.-N. Ma, N.S. Trudinger, and X.-J. Wang, Regularity of potential functions of the optimal transportation problem. Arch. Ration. Mech. Anal. 177, 151–183 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  60. 60.
    F. Maddalena, S. Solimini, Transport distances and irrigation models. J. Convex Anal. 16, 121–152 (2009)MathSciNetzbMATHGoogle Scholar
  61. 61.
    F. Maddalena, S. Solimini, J.-M. Morel, A variational model of irrigation patterns. Interfaces Free Bound. 5, 391–415 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  62. 62.
    R.J. Mccann, A convexity theory for interacting gases and equilibrium crystals. Ph.D. Thesis, Princeton University. ProQuest LLC, Ann Arbor (1994)Google Scholar
  63. 63.
    R.J. McCann, A convexity principle for interacting gases. Adv. Math. 128, 153–179 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  64. 64.
    R.J. McCann, Polar factorization of maps on riemannian manifolds. Geom. Funct. Anal. 11, 589–608 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  65. 65.
    V.D. Milman, G. Schechtman, in Asymptotic Theory of Finite-Dimensional Normed Spaces. Lecture Notes in Mathematics, vol. 1200 (Springer, Berlin, 1986). With an appendix by M. GromovGoogle Scholar
  66. 66.
    G. Monge, Mémoire sur la théorie des d’eblais et des remblais. Histoire de lÕAcadémie Royale des Sciences de Paris (1781), pp. 666–704Google Scholar
  67. 67.
    F. Otto, The geometry of dissipative evolution equations: the porous medium equation. Comm. Partial Differ. Equat. 26, 101–174 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  68. 68.
    A. Pratelli, On the equality between Monge’s infimum and Kantorovich’s minimum in optimal mass transportation. Ann. l’Institut Henri Poincare B Probab. Stat. 43, 1–13 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  69. 69.
    S.T. Rachev, L. Rüschendorf, Mass Transportation Problems, vol. I. Probability and Its Applications (Springer, New York, 1998), pp. xxvi+508 (Theory)Google Scholar
  70. 70.
    R.T. Rockafellar, Convex Analysis (Princeton University Press, Princeton, 1970)zbMATHGoogle Scholar
  71. 71.
    L. Rüschendorf, S.T. Rachev, A characterization of random variables with minimum L 2-distance. J. Multivariate Anal. 32, 48–54 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  72. 72.
    G. Savaré, Gradient flows and diffusion semigroups in metric spaces under lower curvature bounds. C. R. Math. Acad. Sci. Paris 345, 151–154 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  73. 73.
    G. Savaré, Gradient flows and evolution variational inequalities in metric spaces (2010) (in preparation)Google Scholar
  74. 74.
    K.-T. Sturm, On the geometry of metric measure spaces, I. Acta Math. 196, 65–131 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  75. 75.
    K.-T. Sturm, On the geometry of metric measure spaces, II. Acta Math. 196, 133–177 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  76. 76.
    K.-T. Sturm, M.-K. von Renesse, Transport inequalities, gradient estimates, entropy, and Ricci curvature. Comm. Pure Appl. Math. 58, 923–940 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  77. 77.
    V.N. Sudakov, Geometric problems in the theory of infinite-dimensional probability distributions. Proc. Steklov Inst. Math. (2), i–v, 1–178 (1979) (Cover to cover translation of Trudy Mat. Inst. Steklov 141 (1976))Google Scholar
  78. 78.
    N.S. Trudinger, X.-J. Wang, On the Monge mass transfer problem. Calc. Var. Partial Differ. Equat. 13, 19–31 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  79. 79.
    C. Villani, in Topics in Optimal Transportation. Graduate Studies in Mathematics, vol. 58 (American Mathematical Society, Providence, 2003)Google Scholar
  80. 80.
    C. Villani, Optimal Transport, Old and New (Springer, Berlin, 2008)Google Scholar
  81. 81.
    Q. Xia, Optimal paths related to transport problems. Comm. Contemp. Math. 5, 251–279 (2003)zbMATHCrossRefGoogle Scholar
  82. 82.
    Q. Xia, Interior regularity of optimal transport paths. Calc. Var. Partial Differ. Equat. 20, 283–299 (2004)zbMATHCrossRefGoogle Scholar
  83. 83.
    L. Zají ̌cek, On the differentiability of convex functions in finite and infinite dimensional spaces. Czechoslovak Math. J. 29, 340–348 (1979)Google Scholar

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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Scuola Normale SuperiorePisaItaly
  2. 2.Université de Nice, MathématiquesNiceFrance

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