[Géométrie des foliations sur l'espace de Wiener et calcul des variations stochastiques.]
Le Calcul Stochastique des variations considère classiquement des applications de l'espace de Wiener dans un espace de dimension finie ; dans ce contexte s'inscrit la théorie des applications non dégénérées pour lesquelles on peut établir la régularité des lois ainsi que l'existence de désintégrations continues. L'Analyse stochastique en dimension infinie et singulièrement la théorie des SPDE, pose la question naturelle de l'étude des applications de l'espace de Wiener dans un espace de dimension infinie. Nous approchons ce problème de manière intrinsèque, privilégiant l'étude géomètrique des sous tribus à travers leurs foliations associées.
Stochastic Calculus of variations deals with maps defined on the Wiener space, with finite dimensional range; within this context appears the notion of non-degenerate map, which corresponds roughly speaking to some kind of infinite dimensional ellipticity; a non-degenerate map has a smooth law; by conditioning, it generates a continuous desintegration of the Wiener measure. Infinite dimensional Stochastic Analysis and particularly SPDE theory raise the natural question of what can be done for maps with an infinite dimensional range; our approach to this problem emphasizes an intrinsic geometric aspect, replacing range by generated σ-field and its associated foliation of the Wiener space.
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@article{CRMATH_2004__339_9_637_0,
author = {Airault, H\'el\`ene and Malliavin, Paul and Ren, Jiagang},
title = {Geometry of foliations on the {Wiener} space and stochastic calculus of variations},
journal = {Comptes Rendus. Math\'ematique},
pages = {637--642},
publisher = {Elsevier},
volume = {339},
number = {9},
year = {2004},
doi = {10.1016/j.crma.2004.09.009},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2004.09.009/}
}
TY - JOUR AU - Airault, Hélène AU - Malliavin, Paul AU - Ren, Jiagang TI - Geometry of foliations on the Wiener space and stochastic calculus of variations JO - Comptes Rendus. Mathématique PY - 2004 SP - 637 EP - 642 VL - 339 IS - 9 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2004.09.009/ DO - 10.1016/j.crma.2004.09.009 LA - en ID - CRMATH_2004__339_9_637_0 ER -
%0 Journal Article %A Airault, Hélène %A Malliavin, Paul %A Ren, Jiagang %T Geometry of foliations on the Wiener space and stochastic calculus of variations %J Comptes Rendus. Mathématique %D 2004 %P 637-642 %V 339 %N 9 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2004.09.009/ %R 10.1016/j.crma.2004.09.009 %G en %F CRMATH_2004__339_9_637_0
Airault, Hélène; Malliavin, Paul; Ren, Jiagang. Geometry of foliations on the Wiener space and stochastic calculus of variations. Comptes Rendus. Mathématique, Tome 339 (2004) no. 9, pp. 637-642. doi : 10.1016/j.crma.2004.09.009. http://www.numdam.org/articles/10.1016/j.crma.2004.09.009/
[1] Intégration géométrique sur l'espace de Wiener, Bull. Sci. Math. (2), Volume 112 (1988), pp. 3-52
[2] Le processus d' Ornstein–Uhlenbeck sur une sous-variété de l'espace de Wiener, Bull. Sci. Math. (2), Volume 115 (1991), pp. 185-210
[3] Functorial analysis in geometric probability theory, Stochastic Analysis and Mathematical Physics, Progr. Probab., vol. 50, Birkhäuser, Boston, MA, 2001, pp. 1-37
[4] Smooth sigma-fields (Mayer-Wolf; Merzbach; Schwartz, eds.), Stochastic Analysis, Academic Press, 1991
[5] Stochastic Analysis, Grundlehren Math. Wiss., vol. 313, Springer-Verlag, 1997
[6] On the structure of independence on Wiener space, J. Funct. Anal., Volume 90 (1990) no. 1, pp. 113-137
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