[Transport de mesure sur l'espace de Wiener et théorème de Girsanov]
Soit (W,H,μ) un espace de Wiener abstrait ; on suppose que νi, i=1,2, sont deux probabilités sur 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 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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@article{CRMATH_2002__334_11_1025_0,
author = {Feyel, Denis and \"Ust\"unel, Ali S\"uleyman},
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},
url = {http://www.numdam.org/articles/10.1016/S1631-073X(02)02326-9/}
}
TY - JOUR AU - Feyel, Denis AU - Üstünel, Ali Süleyman 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 UR - http://www.numdam.org/articles/10.1016/S1631-073X(02)02326-9/ DO - 10.1016/S1631-073X(02)02326-9 LA - en ID - CRMATH_2002__334_11_1025_0 ER -
%0 Journal Article %A Feyel, Denis %A Üstünel, Ali Süleyman %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 %U http://www.numdam.org/articles/10.1016/S1631-073X(02)02326-9/ %R 10.1016/S1631-073X(02)02326-9 %G en %F CRMATH_2002__334_11_1025_0
Feyel, Denis; Üstünel, Ali Süleyman. Measure transport on Wiener space and the Girsanov theorem. Comptes Rendus. Mathématique, Tome 334 (2002) no. 11, pp. 1025-1028. doi : 10.1016/S1631-073X(02)02326-9. http://www.numdam.org/articles/10.1016/S1631-073X(02)02326-9/
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