Comptes Rendus
no. 11
Measure transport on Wiener space and the Girsanov theorem
[Transport de mesure sur l'espace de Wiener et théorème de Girsanov]
Denis Feyel1 ; Ali Süleyman Üstünel2
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.

Soit (W,H,μ) un espace de Wiener abstrait ; on suppose que νi, 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=IW+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 νi, 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=IW+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.

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

[1] Y. Brenier Polar factorization and monotone rearrangement of vector valued functions, Comm. Pure Appl. Math., Volume 44 (1991), pp. 375-417

[2] D. Feyel; A. de La Pradelle Capacités gaussiennes, Ann. Inst. Fourier, Volume 41 (1991) no. 1, pp. 49-76

[3] D. Feyel; A.S. Üstünel The notion of convexity and concavity on Wiener space, J. Funct. Anal., Volume 176 (2000), pp. 400-428

[4] R.J. McCann Existence and uniqueness of monotone measure-preserving maps, Duke Math. J., Volume 80 (1995), pp. 309-323

[5] T. Rockafellar Convex Analysis, Princeton University Press, Princeton, 1972

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

[7] A.S. Üstünel; M. Zakai Transformation of Measure on Wiener Space, Springer, 1999

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