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Wasserstein space over the Wiener space
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  • Published: 22 January 2009

Wasserstein space over the Wiener space

  • Shizan Fang1,2,
  • Jinghai Shao2,3 &
  • Karl-Theodor Sturm3 

Probability Theory and Related Fields volume 146, pages 535–565 (2010)Cite this article

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Abstract

The goal of this paper is to study optimal transportation problems and gradient flows of probability measures on the Wiener space, based on and extending fundamental results of Feyel–Üstünel. Carrying out the program of Ambrosio–Gigli–Savaré, we present a complete characterization of the derivative processes for certain class of absolutely continuous curves. We prove existence of the gradient flow curves for the relative entropy w.r.t. the Wiener measure and identify these gradient flow curves with solutions of the Ornstein–Uhlenbeck evolution equation.

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Author information

Authors and Affiliations

  1. I.M.B, Université de Bourgogne, BP 47870, 21078, Dijon, France

    Shizan Fang

  2. School of Mathematics, Beijing Normal University, 100875, Beijing, China

    Shizan Fang & Jinghai Shao

  3. Institut für Angewandte Mathematik, Universität Bonn, 53115, Bonn, Germany

    Jinghai Shao & Karl-Theodor Sturm

Authors
  1. Shizan Fang
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  2. Jinghai Shao
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  3. Karl-Theodor Sturm
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Corresponding author

Correspondence to Karl-Theodor Sturm.

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Fang, S., Shao, J. & Sturm, KT. Wasserstein space over the Wiener space. Probab. Theory Relat. Fields 146, 535–565 (2010). https://doi.org/10.1007/s00440-009-0199-5

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  • Received: 08 January 2008

  • Revised: 17 November 2008

  • Published: 22 January 2009

  • Issue Date: March 2010

  • DOI: https://doi.org/10.1007/s00440-009-0199-5

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Mathematics Subject Classification (2000)

  • Primary: 58B20
  • Secondary: 60J45
  • 60H07
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