Comptes Rendus
Measure transport on Wiener space and the Girsanov theorem
Comptes Rendus. Mathématique, Volume 334 (2002) no. 11, pp. 1025-1028.

Let (W,H,μ) be an abstract Wiener space, and assume that νi, i=1,2, are two probability measures on (W,(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:WH 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.

Soit (W,H,μ) un espace de Wiener abstrait ; on suppose que νi, i=1,2, sont deux probabilités sur (W,(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 :WH 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.

Received:
Accepted:
Published online:
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
@article{CRMATH_2002__334_11_1025_0,
     author = {Denis Feyel and Ali S\"uleyman \"Ust\"unel},
     title = {Measure transport on {Wiener} space and the {Girsanov} theorem},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1025--1028},
     publisher = {Elsevier},
     volume = {334},
     number = {11},
     year = {2002},
     doi = {10.1016/S1631-073X(02)02326-9},
     language = {en},
}
TY  - JOUR
AU  - Denis Feyel
AU  - Ali Süleyman Üstünel
TI  - Measure transport on Wiener space and the Girsanov theorem
JO  - Comptes Rendus. Mathématique
PY  - 2002
SP  - 1025
EP  - 1028
VL  - 334
IS  - 11
PB  - Elsevier
DO  - 10.1016/S1631-073X(02)02326-9
LA  - en
ID  - CRMATH_2002__334_11_1025_0
ER  - 
%0 Journal Article
%A Denis Feyel
%A Ali Süleyman Üstünel
%T Measure transport on Wiener space and the Girsanov theorem
%J Comptes Rendus. Mathématique
%D 2002
%P 1025-1028
%V 334
%N 11
%I Elsevier
%R 10.1016/S1631-073X(02)02326-9
%G en
%F CRMATH_2002__334_11_1025_0
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

Cited by Sources:

Comments - Policy


Articles of potential interest

Some remarks about the positivity of random variables on a Gaussian probability space

Denis Feyel; A. Suleyman Üstünel

C. R. Math (2004)


The strong solution of the Monge–Ampère equation on the Wiener space for log-concave densities

Denis Feyel; Ali Suleyman Üstünel

C. R. Math (2004)


A necessary and sufficient condition for invertibility of adapted perturbations of identity on Wiener space

A. Suleyman Üstünel

C. R. Math (2008)