Measure transport on Wiener space and the Girsanov theorem

Denis Feyel; Ali Süleyman Üstünel

doi:10.1016/S1631-073X(02)02326-9

Measure transport on Wiener space and the Girsanov theorem
[Transport de mesure sur l'espace de Wiener et théorème de Girsanov]

Denis Feyel ¹ ; Ali Süleyman Üstünel ²

¹ Département de mathématiques, Université d'Evry-Val-d'Essone, 91025 Evry cedex, France
² ENST, Département Infres, 46, rue Barrault, 75013 Paris, France

Comptes Rendus. Mathématique, Volume 334 (2002) no. 11, pp. 1025-1028.

Résumé
Abstract

Soit (W,H,μ) un espace de Wiener abstrait ; on suppose que ν_i, i=1,2, sont deux probabilités sur $(W, ℬ (W))$ qui sont absolument continues par rapport à μ et que la distance de Wasserstein entre ν₁ et ν₂ est finie. Alors il existe une application T=I_W+ξ de W dans lui-même telle que ξ :W→H soit mesurable, 1-cycliquement monotone et l'image de ν₁ sous T soit égale à ν₂. De plus T est inversible sur le support de ν₂. Nous donnons aussi quelques applications de ce résultat comme l'existence de solutions de l'équation de Monge–Ampère.

Let (W,H,μ) be an abstract Wiener space, and assume that ν_i, i=1,2, are two probability measures on $(W, ℬ (W))$ which are absolutely continuous with respect to μ. Assume that the Wasserstein distance between ν₁ and ν₂ is finite. Then there exists a map T=I_W+ξ of W into itself such that ξ:W→H is measurable and 1-cyclically monotone such that the image of ν₁ under T is ν₂. Moreover T is invertible on the support of ν₂. We give also some applications of this result such as the existence of the solutions of the Monge–Ampère equation in infinite dimensions.

Reçu le : 2002-01-28
Accepté le : 2002-02-19
Publié le : 2002-01-01

DOI : 10.1016/S1631-073X(02)02326-9

Affiliations des auteurs :

Denis Feyel ¹ ; Ali Süleyman Üstünel ²

¹ Département de mathématiques, Université d'Evry-Val-d'Essone, 91025 Evry cedex, France
² ENST, Département Infres, 46, rue Barrault, 75013 Paris, France

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     title = {Measure transport on {Wiener} space and the {Girsanov} theorem},
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     pages = {1025--1028},
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Denis Feyel; Ali Süleyman Üstünel. Measure transport on Wiener space and the Girsanov theorem. Comptes Rendus. Mathématique, Volume 334 (2002) no. 11, pp. 1025-1028. doi : 10.1016/S1631-073X(02)02326-9. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)02326-9/

Bibliographie
Cité par

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[6] A.S. Üstünel Introduction to Analysis on Wiener Space, Lecture Notes in Math., 1610, Springer, 1995

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