Optimal transport maps on infinite dimensional spaces

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.

Keywords

Girsanov theorem / entropy / optimal transport map / Wiener space / Lebesgue point

References

[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

[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

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

[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

[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

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

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

[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

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

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

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

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

[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

[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

[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

[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

[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

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

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

[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

[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

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

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

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

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

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

[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

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

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

RIGHTS & PERMISSIONS

Higher Education Press and Springer-Verlag Berlin Heidelberg