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Monge-Kantorovitch Measure Transportation and Monge-Ampère Equation on Wiener Space
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  • Published: 26 November 2003

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

  • D. Feyel1 &
  • A. S. Üstünel2 

Probability Theory and Related Fields volume 128, pages 347–385 (2004)Cite this article

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Abstract

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:W→H, such that T maps ν1 to ν2. Moreover, if ν2≪μ, then T is stochastically invertible, i.e., there exists S:W→W such that S○T=I W ν1 a.s. and T○S=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.

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Authors and Affiliations

  1. Université d’Evry-Val-d’Essone, 91025, Evry Cedex, France

    D. Feyel

  2. ENST, Dépt., Infres, 46 ,rue Barrault, 75634, Paris Cedex 13, France

    A. S. Üstünel

Authors
  1. D. Feyel
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  2. A. S. Üstünel
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Corresponding author

Correspondence to A. S. Üstünel.

Additional information

cf. Theorem 6.1 for the precise hypothesis about ν1 and ν2.

In fact this hypothesis is too strong, cf. Theorem 6.1.

Mathematics Subject Classification (2000): 60H07, 60H05,60H25, 60G15, 60G30, 60G35, 46G12, 47H05, 47H1, 35J60, 35B65, 35A30, 46N10, 49Q20, 58E12, 26A16, 28C20

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Feyel, D., Üstünel, A. Monge-Kantorovitch Measure Transportation and Monge-Ampère Equation on Wiener Space. Probab. Theory Relat. Fields 128, 347–385 (2004). https://doi.org/10.1007/s00440-003-0307-x

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  • Received: 17 November 2002

  • Revised: 20 September 2003

  • Published: 26 November 2003

  • Issue Date: March 2004

  • DOI: https://doi.org/10.1007/s00440-003-0307-x

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Keywords

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