SURVEY ARTICLE

Optimal transport maps on infinite dimensional spaces

  • Shizan FANG ,
  • Vincent NOLOT
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  • I.M.B., BP 47870, Université de Bourgogne, Dijon, France In memory of Denis Feyel

Received date: 02 Feb 2015

Accepted date: 11 Feb 2015

Published date: 05 Jun 2015

Copyright

2014 Higher Education Press and Springer-Verlag Berlin Heidelberg

Abstract

We will give a survey on results concerning Girsanov transformations, transportation cost inequalities, convexity of entropy, and optimal transport maps on some infinite dimensional spaces. Some open Problems will be arisen.

Cite this article

Shizan FANG , Vincent NOLOT . Optimal transport maps on infinite dimensional spaces[J]. Frontiers of Mathematics in China, 2015 , 10(4) : 715 -732 . DOI: 10.1007/s11464-015-0474-y

1
Ambrosio L, Gigli N, Savaré G. Gradient Flows in Metric Spaces and in the Space of Probability Measures. Lect in Math, ETH Zürich. Basel: Birkhäuser, 2005

2
Ambrosio L, Kirchheim B, Pratelli A. Existence of optimal transport maps for crystalline norms. Duke Math J, 2004, 125: 207-241

DOI

3
Ambrosio L, Pratelli A. Existence and stability results in the L1 theory of optimal transportation. In: Morel J-M, Takens F, Teissier B, eds. Optimal Transportation and Applications. Lecture Notes in Math, Vol 1813. Berlin: Springer, 2003, 123-160

DOI

4
Bao J, Wang F-Y, Yuan C G. Transportation cost inequalities for neutral functional stochastic equations. Z Anal Anwend, 2013, 32: 457-475

DOI

5
Bogachev V I, Kolesnikov A V. Sobolev regularity for Monge-Ampère equation in the Wiener space. arXiv: 1110.1822v1, 2011

6
Brenier Y. Polar factorization and monotone rearrangement of vector valued functions. Comm Pure Appl Math, 1991, 44: 375-417

DOI

7
Champion T, De Pascale L. The Monge problem in ℝd. Duke Math J, 2010, 157: 551-572

8
Djellout H, Guilin A, Wu L. Transportation cost-information inequalities for random dynamical systems and diffusions. Ann Probab, 2004, 32: 2702-2732

DOI

9
Driver B. Integration by parts and quasi-invariance for heat measures on loop groups. J Funct Anal, 1997, 149: 470-547

DOI

10
Driver B, Lohrentz T. Logarithmic Sobolev inequalities for pinned loop groups. J Funct Anal, 1996, 140: 381-448

DOI

11
Driver B, Srimurthy V K. Absolute continuity of heat kernel measure with pinned Wiener measure on loop groups. Ann Probab, 2001, 29: 691-723

12
Fang S. Introduction to Malliavin Calculus. Math Ser for Graduate Students. Beijing/Berlin: Tsinghua University Press/Springer, 2005

13
Fang S, Malliavin P. Stochastic analysis on the path space of a Riemannian manifold. J Funct Anal, 1993, 131: 249-274

DOI

14
Fang S, Nolot V. Sobolev estimates for optimal transport maps on Gaussian spaces. J Funct Anal, 2014, 266: 5045-5084

DOI

15
Fang S, Shao J. Transportation cost inequalities on path and loop groups. J Funct Anal, 2005, 218: 293-317

DOI

16
Fang S, Shao J. Optimal transport maps for Monge-Kantorovich problem on loop groups. J Funct Anal, 2007, 248: 225-257

DOI

17
Fang S, Shao J, Sturm K T. Wasserstein space over the Wiener space. Probab Theory Related Fields, 2010, 146: 535-565

DOI

18
Fang S, Wang F Y, Wu B. Transportation-cost inequality on path spaces with uniform distance. Stochastic Process Appl, 2008, 118(12): 2181-2197

DOI

19
Feyel D, Üstünel A S. Monge-Kantorovich measure transportation and Monge-Ampère equation on Wiener space. Probab Theory Related Fields, 2004, 128: 347-385

DOI

20
Feyel D, Üstünel A S. Solution of the Monge-Ampère equation on Wiener space for general log-concave measures. J Funct Anal, 2006, 232: 29-55

DOI

21
Gross L. Logarithmic Sobolev inequalities on Lie groups. Illinois J Math, 1992, 36: 447-490

22
Knott M, Smith C S. On the optimal mapping of distributions. J Optim Theory Appl, 1984, 43(1): 39-49

DOI

23
Kolesnikov A V. On Sobolev regularity of mass transport and transportation inequalities. arXiv: 1007.1103v3, 2011

24
Lassalle R. Invertibility of adapted perturbations of the identity on abstract Wiener space. J Funct Anal, 2012, 262: 2734-2776

DOI

25
Lott J, Villani C. Ricci curvature for metric-measure spaces via optimal transport. Ann Math, 2009, 169: 903-991

DOI

26
Ma Y. Transportation inequalities for stochastic differential equations with jumps. Stochastic Process Appl, 2010, 120: 2-21

DOI

27
Malliavin P. Stochastic Analysis. Grundlehren Math Wiss, Vol 313. Berlin: Springer, 1997

28
Malliavin P. Hypoellipticity in infinite dimension. In: Pinsky M, ed. Diffusion Processes and Related Problems in Analysis, Vol I: Diffusions in Analysis and Geometry. Progress in Probability. Boston: Birkhäuser, 2012, 17-33

29
McCann R. Polar factorization of maps on Riemannian manifolds. Geom Funct Anal, 2001, 11: 589-608

DOI

30
Nolot V. Convexity and problems of optimal transport maps on the Wiener space. Ph D Thesis, University of Bourgogne, 2013

31
Rachev S T, Rüschendorf L. Mass Transportation Problems. Probab Appl. New York: Springer-Verlag, 1998

32
Shao J. From the heat measure to the pinned Wiener measure on loop groups. Bull Sci Math, 2011, 135: 601-612

DOI

33
Sturm K T. On the geometry of metric measure spaces. Acta Math, 2006, 196: 65-131

DOI

34
Sturm K T, Von Renesse M K. Transport inequalities, gradient estimates, entropy and Ricci curvature. Comm Pure Appl Math, 2005, 58: 923-940

DOI

35
Tiser J. Differentiation theorem for Gaussian measures on Hilbert space. Trans Amer Math Soc, 1988, 308: 655-666

DOI

36
Üstünel A S. Entropy, invertibility and variational calculus of adapted shifts on Wiener space. J Funct Anal, 2009, 257: 3655-3689

DOI

37
Üstünel A S. Variational calculation of Laplace transforms via entropy on Wiener space and applications. J Funct Anal, 2014, 267: 3058-3083

DOI

38
Villani C. Optimal Transport, Old and New. Grundlehren Math Wiss, Vol 338. Berlin: Springer-Verlag, 2009

39
Villani C. Regularity of optimal transport and cut-locus: from non smooth analysis to geometry to smooth analysis. Discrete Contin Dyn Syst, 2011, 30: 559-571

DOI

40
Wang F-Y. Probability distance inequalities on Riemannian manifolds and path spaces. J Funct Anal, 2004, 206(1): 167-190

DOI

41
Wang F-Y. Transportation-cost inequalities on path space over manifolds with boundary. Doc Math, 2013, 18: 297-322

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