Comptes Rendus
Probability Theory
Geometry of foliations on the Wiener space and stochastic calculus of variations
Comptes Rendus. Mathématique, Volume 339 (2004) no. 9, pp. 637-642.

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.

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.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2004.09.009
Hélène Airault 1; Paul Malliavin 2; Jiagang Ren 3

1 Laboratoire CNRS UMR 6140, LAMFA (Amiens), INSSET, université de Picardie Jules Verne, 48, rue Raspail, 02100 Saint-Quentin, France
2 10, rue Saint-Louis en l'Isle, 75004 Paris, France
3 Department of Mathematics, Zhongshan University, Guangzhou, Guangdong 510275, China
@article{CRMATH_2004__339_9_637_0,
     author = {H\'el\`ene Airault and Paul Malliavin and Jiagang Ren},
     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},
}
TY  - JOUR
AU  - Hélène Airault
AU  - Paul Malliavin
AU  - Jiagang Ren
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
DO  - 10.1016/j.crma.2004.09.009
LA  - en
ID  - CRMATH_2004__339_9_637_0
ER  - 
%0 Journal Article
%A Hélène Airault
%A Paul Malliavin
%A Jiagang Ren
%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
%R 10.1016/j.crma.2004.09.009
%G en
%F CRMATH_2004__339_9_637_0
Hélène Airault; Paul Malliavin; Jiagang Ren. Geometry of foliations on the Wiener space and stochastic calculus of variations. Comptes Rendus. Mathématique, Volume 339 (2004) no. 9, pp. 637-642. doi : 10.1016/j.crma.2004.09.009. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.09.009/

[1] H. Airault; P. Malliavin Intégration géométrique sur l'espace de Wiener, Bull. Sci. Math. (2), Volume 112 (1988), pp. 3-52

[2] H. Airault; J. Van Biesen 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] H. Airault; P. Malliavin Functorial analysis in geometric probability theory, Stochastic Analysis and Mathematical Physics, Progr. Probab., vol. 50, Birkhäuser, Boston, MA, 2001, pp. 1-37

[4] P. Malliavin Smooth sigma-fields (Mayer-Wolf; Merzbach; Schwartz, eds.), Stochastic Analysis, Academic Press, 1991

[5] P. Malliavin Stochastic Analysis, Grundlehren Math. Wiss., vol. 313, Springer-Verlag, 1997

[6] A.S. Ustunel; M. Zakai On the structure of independence on Wiener space, J. Funct. Anal., Volume 90 (1990) no. 1, pp. 113-137

Cited by Sources:

Comments - Policy


Articles of potential interest

Fellerian pants

Zdzisław Brzeźniak; Remi Léandre

C. R. Math (2006)


Support of Virasoro unitarizing measures

Hélène Airault; Paul Malliavin; Anton Thalmaier

C. R. Math (2002)


Riemannian connections and curvatures on the universal Teichmuller space

Hélène Airault

C. R. Math (2005)