A user's guide to optimal transport

Lecture Notes in Mathematics

Volumn 2062, Issue , 2013, Pages 1-155

  Ambrosio, Luigi ^a   Gigli, Nicola ^b  

^a SCUOLA NORMALE SUPERIORE   (Italy)

^b UNIVERSITÉ CÔTE D'AZUR   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84879327546     PISSN: 00758434     EISSN: None     Source Type: Book Series    
DOI: 10.1007/978-3-642-32160-3_1     Document Type: Article

References (83)

1. A. Agrachev, P. Lee, Optimal transportation under nonholonomic constraints. Trans. Am. Math. Soc. 361, 6019-6047 (2009)
2. G. Alberti, On the structure of singular sets of convex functions. Calc. Var. Partial Differ. Equat. 2, 17-27 (1994)
3. G. Alberti, L. Ambrosio, A geometrical approach to monotone functions in Rn. Math. Z. 230, 259-316 (1999)
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. 1-52
5. L. Ambrosio, N. Gigli, Construction of the parallel transport in the Wasserstein space. Meth. Appl. Anal. 15, 1-29 (2008)
6. L. Ambrosio, S. Rigot, Optimal mass transportation in the Heisenberg group. J. Funct. Anal. 208, 261-301 (2004)
7. L. Ambrosio, N. Gigli, G. Savaŕe, in Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2nd edn. Lectures in Mathematics ETH Z̈urich (Birkḧauser, Basel, 2008)
8. L. Ambrosio, N. Gigli, G. Savaŕe, Calculus and heat flows in metric measure spaces with ricci curvature bounded below, Comm. Pure and Applied Math. (2011)
9. L. Ambrosio, N. Gigli, G. Savaŕe, Spaces with riemannian ricci curvature bounded below, Comm. Pure and Applied Math. (2011)
10. L. Ambrosio, B. Kirchheim, A. Pratelli, Existence of optimal transport maps for crystalline norms. Duke Math. J. 125 207-241 (2004)
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)
12. J.-D. Benamou, Y. Brenier, A numerical method for the optimal time-continuous mass transport problem and related problems, in Monge Amp̀ere Equation: Applications to Geometry and Optimization (Deerfield Beach, FL, 1997). Contemporary Mathematics, vol. 226 (American Mathematical Society, Providence, 1999), pp. 1-11
13. P. Bernard, B. Buffoni, Optimal mass transportation and Mather theory. J. Eur. Math. Soc. (JEMS), 9, 85-127 (2007)
14. M. Bernot, V. Caselles, J.-M. Morel, The structure of branched transportation networks. Calc. Var. Partial Differ. Equat. 32, 279-317 (2008)
15. S. Bianchini, A. Brancolini, Estimates on path functionals over Wasserstein spaces. SIAM J. Math. Anal. 42, 1179-1217 (2010)
16. A. Brancolini, G. Buttazzo, F. Santambrogio, Path functionals over Wasserstein spaces. J. Eur. Math. Soc. (JEMS), 8, 415-434 (2006)
17. L. Brasco, G. Buttazzo, F. Santambrogio, A Benamou-Brenier approach to branched transport. SIAM J. Math. Anal. 43(2), 1023-1040 (2011). doi:10.1137/10079286X
18. Y. Brenier, D́ecomposition polaire et ŕearrangement monotone des champs de vecteurs. C. R. Acad. Sci. Paris I Math. 305, 805-808 (1987)
19. Y. Brenier, Polar factorization and monotone rearrangement of vector-valued functions. Comm. Pure Appl. Math. 44, 375-417 (1991)
20. D. Burago, Y. Burago, S. Ivanov, in A Course in Metric Geometry. Graduate Studies in Mathematics, vol. 33 (American Mathematical Society, Providence, 2001)
21. L.A. Caffarelli, Boundary regularity of maps with convex potentials. Comm. Pure Appl. Math. 45, 1141-1151 (1992)
22. L.A. Caffarelli, The regularity of mappings with a convex potential. J. Am. Math. Soc. 5, 99-104 (1992)
23. L.A. Caffarelli, Boundary regularity of maps with convex potentials, II. Ann. Math. (2) 144, 453-496 (1996)
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)
25. L. Caravenna, A proof of Sudakov theorem with strictly convex norms. Math. Z. 268(1-2), 371-407 (2011) doi:10.1007/s00209-010-0677-6
26. J.A. Carrillo, S. Lisini, G. Savaŕe, D. Slepcev, Nonlinear mobility continuity equations and generalized displacement convexity. J. Funct. Anal. 258, 1273-1309 (2010)
27. T. Champion, L. De Pascale, The Monge problem in Rd . Duke Math. J. 157(3), 551-572 (2011). doi:10.1215/00127094-1272939
28. T. Champion, L. De Pascale, TheMonge problem for strictly convex norms in Rd . J. Eur.Math. Soc. (JEMS), 12, 1355-1369 (2010)
29. J. Cheeger, Differentiability of Lipschitz functions on metric measure spaces. Geom. Funct. Anal. 9, 428-517 (1999)
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)
31. C. Dellacherie, P.-A. Meyer, in Probabilities and Potential. North-Holland Mathematics Studies, vol. 29 (North-Holland, Amsterdam, 1978)
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.2011.02.026
33. J. Dolbeault, B. Nazaret, G. Savaŕe, On the Bakry-Emery criterion for linear diffusions and weighted porous media equations. Comm. Math. Sci 6, 477-494 (2008)
34. L.C. Evans, W. Gangbo, Differential equations methods for the Monge-Kantorovich mass transfer problem. Mem. Am. Math. Soc. 137, viii+66 (1999)
35. A. Fathi, A. Figalli, Optimal transportation on non-compact manifolds. Isr. J. Math. 175, 1-59 (2010)
36. D. Feyel, A.S. Ü stünel, Monge-Kantorovitch measure transportation and Monge-Ampère equation on Wiener space. Probab. Theor. Relat. Fields 128, 347-385 (2004)
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.2009.11.005
38. A. Figalli, F. Maggi, A. Pratelli, A mass transportation approach to quantitative isoperimetric inequalities. Invent. Math. 182, 167-211 (2010)
39. A. Figalli, L. Rifford, Mass transportation on sub-Riemannian manifolds. Geom. Funct. Anal. 20, 124-159 (2010)
40. N. Fusco, F. Maggi, A. Pratelli, The sharp quantitative isoperimetric inequality. Ann. Math. (2) 168, 941-980 (2008)
41. W. Gangbo, The Monge mass transfer problem and its applications, in Monge Amp̀ere Equation: Applications to Geometry and Optimization, (Deerfield Beach, FL, 1997). Contemporary Mathematics, vol. 226 (American Mathematical Society, Providence, 1999), pp. 79-104
42. W. Gangbo, R.J. McCann, The geometry of optimal transportation. Acta Math. 177, 113-161 (1996)
43. N. Gigli, On the geometry of the space of probability measures in Rn endowed with the quadratic optimal transport distance, Thesis (Ph.D.)-Scuola Normale Superiore, 2008
44. N. Gigli, Second order analysis on .P2.M/;W2/.Memoir. Am.Math. Soc. 216(1018), xii+154 (2012). doi:10.1090/S0065-9266-2011-00619-2
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/s00526-009-0303-9
46. N. Gigli, On the inverse implication of Brenier-McCann theorems and the structure of .P2.M/;W2/. Methods Appl. Anal. 18(2), 127-158 (2011)
47. R. Jordan, D. Kinderlehrer, F. Otto, The variational formulation of the Fokker-Planck equation. SIAM J. Math. Anal. 29, 1-17 (1998) (electronic)
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/S0002-9939-2011-10891-8
49. L.V. Kantorovich, On an effective method of solving certain classes of extremal problems. Dokl. Akad. Nauk. USSR 28, 212-215 (1940)
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)
51. L.V. Kantorovich, G.S. Rubinshtein, On a space of totally additive functions. Vestn. Leningr. Univ. 7(13), 52-59 (1958)
52. M. Knott, C.S. Smith, On the optimal mapping of distributions. J. Optim. Theor. Appl. 43, 39-49 (1984)
53. K. Kuwada, N. Gigli, S.-I. Ohta, Heat flow on alexandrov spaces, Comm. Pure and Applied Math. (2010)
54. S. Lisini, Characterization of absolutely continuous curves in Wasserstein spaces. Calc. Var. Partial Differ. Equat. 28, 85-120 (2007)
55. G. Loeper, On the regularity of solutions of optimal transportation problems. Acta Math. 202, 241-283 (2009)
56. J. Lott, Some geometric calculations on Wasserstein space. Comm. Math. Phys. 277, 423-437 (2008)
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. 018
58. J. Lott, C. Villani, Ricci curvature for metric-measure spaces via optimal transport. Ann. Math. 169(2), 903-991 (2009)
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)
60. F.Maddalena, S. Solimini, Transport distances and irrigation models. J. Convex Anal. 16, 121-152 (2009)
61. F. Maddalena, S. Solimini, J.-M. Morel, A variational model of irrigation patterns. Interfaces Free Bound. 5, 391-415 (2003)
62. R.J. Mccann, A convexity theory for interacting gases and equilibrium crystals. Ph.D. Thesis, Princeton University. ProQuest LLC, Ann Arbor (1994)
63. R.J. McCann, A convexity principle for interacting gases. Adv. Math. 128, 153-179 (1997)
64. R.J. McCann, Polar factorization of maps on riemannian manifolds. Geom. Funct. Anal. 11, 589-608 (2001)
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. Gromov
66. G.Monge,Ḿemoire sur la th́eorie des d'eblais et des remblais. Histoire de l̃OAcad́emie Royale des Sciences de Paris (1781), pp. 666-704
67. F. Otto, The geometry of dissipative evolution equations: the porous medium equation. Comm. Partial Differ. Equat. 26, 101-174 (2001)
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)
69. S.T. Rachev, L. R̈uschendorf, Mass Transportation Problems, vol. I. Probability and Its Applications (Springer, New York, 1998), pp. xxvi+508 (Theory)
70. R.T. Rockafellar, Convex Analysis (Princeton University Press, Princeton, 1970)
71. L. R̈uschendorf, S.T. Rachev, A characterization of random variables with minimum L2-distance. J. Multivariate Anal. 32, 48-54 (1990)
72. G. Savaŕe, Gradient flows and diffusion semigroups in metric spaces under lower curvature bounds. C. R. Math. Acad. Sci. Paris 345, 151-154 (2007)
73. G. Savaŕe, Gradient flows and evolution variational inequalities in metric spaces (2010) (in preparation)
74. K.-T. Sturm, On the geometry of metric measure spaces, I. Acta Math. 196, 65-131 (2006)
75. K.-T. Sturm, On the geometry of metric measure spaces, II. Acta Math. 196, 133-177 (2006)
76. K.-T. Sturm, M.-K. von Renesse, Transport inequalities, gradient estimates, entropy, and Ricci curvature. Comm. Pure Appl. Math. 58, 923-940 (2005)
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))
78. N.S. Trudinger, X.-J. Wang, On the Monge mass transfer problem. Calc. Var. Partial Differ. Equat. 13, 19-31 (2001)
79. C. Villani, in Topics in Optimal Transportation. Graduate Studies in Mathematics, vol. 58 (American Mathematical Society, Providence, 2003)
80. C. Villani, Optimal Transport, Old and New (Springer, Berlin, 2008)
81. Q. Xia, Optimal paths related to transport problems. Comm. Contemp. Math. 5, 251-279 (2003)
82. Q. Xia, Interior regularity of optimal transport paths. Calc. Var. Partial Differ. Equat. 20, 283-299 (2004)
83. L. Zaj́ cek, On the differentiability of convex functions in finite and infinite dimensional spaces. Czechoslovak Math. J. 29, 340-348 (1979)