Probability Theory and Related Fields

, Volume 128, Issue 3, pp 347–385 | Cite as

Monge-Kantorovitch Measure Transportation and Monge-Ampère Equation on Wiener Space

  • D. Feyel
  • A. S. ÜstünelEmail author


Let (W,μ,H) be an abstract Wiener space assume two ν i ,i=1,2 probabilities on (W,ℬ(W)). We give some conditions for the Wasserstein distance between ν1 and ν2 with respect to the Cameron-Martin space to be finite, where the infimum is taken on the set of probability measures β on W×W whose first and second marginals are ν1 and ν2. In this case we prove the existence of a unique (cyclically monotone) map T=I W +ξ, with ξ:WH, such that T maps ν1 to ν2. Moreover, if ν2≪μ, then T is stochastically invertible, i.e., there exists S:WW such that ST=I W ν1 a.s. and TS=I W ν2 a.s. If, in addition, ν1=μ, then there exists a 1-convex function φ in the Gaussian Sobolev space such that ξ=∇φ. These results imply that the quasi-invariant transformations of the Wiener space with finite Wasserstein distance from μ can be written as the composition of a transport map T and a rotation, i.e., a measure preserving map. We give also 1-convex sub-solutions and Ito-type solutions of the Monge-Ampère equation on W.


Monge-Kantorovitch problem Measure transportation Monge-Ampère equation Wiener space Girsanov theorem Transport inequalities 


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  1. 1.
    Abdellaoui, T., Heinich, H.: Sur la distance de deux lois dans le cas vectoriel. CRAS, Paris Série I, Math. 319, 397–400 (1994)Google Scholar
  2. 2.
    Anderson, R.D., Klee, V.L.Jr.: Convex functions and upper semicontinuous collections. Duke Math. J. 19, 349–357 (1952)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Appell, P.: Mémoire sur déblais et les remblais des systèmes continus ou discontinus. Mémoires présentées par divers savants à l’Académie des Sciences de l’Institut de France. Paris, I. N. 29, 1–208 (1887)Google Scholar
  4. 4.
    Appell, P.: Le problème géométrique des déblais et des remblais. Mémorial des Sciences Mathématiques, fasc. XXVII, Paris, (1928)Google Scholar
  5. 5.
    Bickel, P.J., Freedman, D.A.: Some asymptotic theory for the bootstrap. Ann. Statis. 9(6), 1196–1217 (1981)zbMATHGoogle Scholar
  6. 6.
    Brenier, Y.: Polar factorization and monotone rearrangement of vector valued functions. Comm. pure Appl. Math. 44, 375-417 (1991)zbMATHGoogle Scholar
  7. 7.
    Caffarelli, L.A.: The regularity of mappings with a convex potential. J. Am. Math. Soc. 5, 99–104 (1992)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Dacarogna, B., Moser, J.: On a partial differential equation involving the Jacobian determinant. Ann. Inst. Henri Poincaré, Analyse non-linéaire. 7, 1–26 (1990)Google Scholar
  9. 9.
    Dellacherie, C., Meyer, P.A.: Probabilités et Potentiel, Ch. I à IV. Paris, Hermann, 1975Google Scholar
  10. 10.
    Djellout, H., Guillin, A., Wu, L.: Transportation cost-information inequalities for random dynamical systems and diffusions. Preprint, December 2002Google Scholar
  11. 11.
    Dunford, N., Schwartz, J.T.: Linear Operators. 2, Interscience, 1963Google Scholar
  12. 12.
    Fernique, X.: Extension du théorème de Cameron-Martin aux translations aléatoires. Comptes Rendus Mathématiques 335(1), 65–68 (2002)CrossRefzbMATHGoogle Scholar
  13. 13.
    Fernique, X.: Comparaison aux mesures gaussiennes, espaces autoreproduisants. Une application des propriétés isopérimétriques. PreprintGoogle Scholar
  14. 14.
    Feyel, D., de La Pradelle, A.: Capacités gaussiennes. Annales de l’Institut Fourier t.41, f.1, 49–76 (1991)Google Scholar
  15. 15.
    Feyel, D., Üstünel, A.S.: The notion of convexity and concavity on Wiener space. J. Funct. Anal. 176, 400–428 (2000)CrossRefMathSciNetzbMATHGoogle Scholar
  16. 16.
    Feyel, D., Üstünel, A.S.: Transport of measures on Wiener space and the Girsanov theorem. Comptes Rendus Mathématiques. 334(1), 1025–1028 (2002)CrossRefzbMATHGoogle Scholar
  17. 17.
    Gangbo, W., McCann, R.J.: The geometry of optimal transportation. Acta Mathematica 177, 113–161 (1996)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Ito, K., Nisio, M.: On the convergence of sums of independent Banach space valued random variables. Osaka J. Math. 5, 35–48 (1968)zbMATHGoogle Scholar
  19. 19.
    Kantorovitch, L.V.: On the transfer of masses. Dokl. Acad. Nauk. SSSR 37, 227–229 (1942)Google Scholar
  20. 20.
    Marton, K.: Bounding -distance by informational divergence: a method to prove measure concentration. Ann. Probab. 24(2), 857–866 (1996)Google Scholar
  21. 21.
    McCann, R.J.: Existence and uniqueness of monotone measure-preserving maps. Duke Math. J. 80, 309–323 (1995)MathSciNetzbMATHGoogle Scholar
  22. 22.
    McCann, R.J.: A convexity principle for interacting gases. Adv. Math. 128, 153–179 (1997)CrossRefMathSciNetzbMATHGoogle Scholar
  23. 23.
    Monge, G.: Mémoire sur la théorie des déblais et des remblais. Histoire de l’Académie Royale des Sciences, Paris, 1781Google Scholar
  24. 24.
    Rachev, S.T.; The Monge-Kantorovitch transference problem. Th. prob. Appl. 49, 647–676 (1985)Google Scholar
  25. 25.
    Rockafellar, T.: Convex Analysis. Princeton University Press, Princeton, 1972Google Scholar
  26. 26.
    Sudakov, V.N.: Geometric problems in the theory of infinite dimensional probability distributions. Proc. Steklov Inst. Math. 141, 1–178 (1979)MathSciNetGoogle Scholar
  27. 27.
    Talagrand, M.: Transportation cost for Gaussian and other product measures. Geom. Funct. Anal. 6, 587–600 (1996)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Thomas, E.: The Lebesgue-Nikodym theorem for vector valued Radon measures. Memoirs of A.M.S. 139, (1974)Google Scholar
  29. 29.
    Üstünel, A.S.: Representation of distributions on Wiener space and Stochastic Calculus of Variations. J. Funct. Anal. 70, 126–139 (1987)MathSciNetGoogle Scholar
  30. 30.
    Üstünel, A.S.: Introduction to Analysis on Wiener Space. Lecture Notes in Math. 1610, Springer, 1995Google Scholar
  31. 31.
    Üstünel, A.S., Zakai, M.: Transformation of Measure on Wiener Space. Springer Monographs in Mathematics. Springer Verlag, 1999Google Scholar

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© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  1. 1.Université d’Evry-Val-d’EssoneEvry CedexFrance
  2. 2.ENST, Dépt., InfresParis Cedex 13France

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