Gross–Sobolev spaces on path manifolds: uniqueness and intertwining by Itô maps

K.David Elworthy; Xue-Mei Li

doi:10.1016/j.crma.2003.10.004

Probability Theory

Gross–Sobolev spaces on path manifolds: uniqueness and intertwining by Itô maps
[Espaces de Gross–Sobolev sur les espaces des chemins : unicité et entrelacement par les applications d'Itô]

Présenté par : Jean-Michel Bismut

K.David Elworthy ¹ ; Xue-Mei Li ²

¹ Mathematics Institute, University of Warwick, Coventry CV4 7AL,UK
² The Department of Computing and Mathematics, The Nottingham Trent University, Nottingham NG7 1AS, UK

Comptes Rendus. Mathématique, Volume 337 (2003) no. 11, pp. 741-744.

Résumé
Abstract

Nous donnons des conditions sous lesquelles les applications d'Itô $ℐ$ donnant la solution d'une équation différentielle stochastique sur une variété Riemannienne M entrelace l'opérateur de dérivation d sur l'espace de chemins de M, ainsi que celui de l'espace de Wiener canonique, de $d_{Ω} ℐ^{*} = ℐ^{*} d_{C_{x_{0}} M}$ . Nous en déduisons une propriété d'unicité de d sur l'espace de chemins. Des résultats sur les dérivées d'ordre supérieur ainsi que sur les dérivées covariantes sont également donnés.

Conditions are given under which the solution map $ℐ$ of a stochastic differential equation on a Riemannian manifolds M intertwines the differentiation operator d on the path space of M and that of the canonical Wiener space, $d_{Ω} ℐ^{*} = ℐ^{*} d_{C_{x_{0}} M}$ . A uniqueness property of d on the path space follows. Results are also given for higher derivatives and covariant derivatives.

Reçu le : 2003-10-01
Accepté le : 2003-10-06
Publié le : 2003-11-14

DOI : 10.1016/j.crma.2003.10.004

Affiliations des auteurs :

K.David Elworthy ¹ ; Xue-Mei Li ²

¹ Mathematics Institute, University of Warwick, Coventry CV4 7AL,UK
² The Department of Computing and Mathematics, The Nottingham Trent University, Nottingham NG7 1AS, UK

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     author = {K.David Elworthy and Xue-Mei Li},
     title = {Gross{\textendash}Sobolev spaces on path manifolds: uniqueness and intertwining by {It\^o} maps},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {741--744},
     publisher = {Elsevier},
     volume = {337},
     number = {11},
     year = {2003},
     doi = {10.1016/j.crma.2003.10.004},
     language = {en},
}

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K.David Elworthy; Xue-Mei Li. Gross–Sobolev spaces on path manifolds: uniqueness and intertwining by Itô maps. Comptes Rendus. Mathématique, Volume 337 (2003) no. 11, pp. 741-744. doi : 10.1016/j.crma.2003.10.004. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2003.10.004/

Bibliographie
Cité par

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[3] A.B. Cruzeiro; S.-Z. Fang An L² estimate for Riemannian anticipative stochastic integrals, J. Funct. Anal., Volume 143 (1997) no. 2, pp. 400-414

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[5] B.K. Driver A Cameron–Martin type quasi-invariance theorem for Brownian motion on a compact Riemannian manifold, J. Funct. Anal., Volume 100 (1992), pp. 272-377

[6] B.K. Driver The non-equivalence of Dirichlet forms on path spaces, Stochastic Analysis on Infinite-Dimensional Spaces, Longman, 1994, pp. 75-87

[7] K.D. Elworthy; Y. Le Jan; X.-M. Li Concerning the geometry of stochastic differential equations and stochastic flows, New Trends in Stochastic Analysis, World Scientific, 1997

[8] K.D. Elworthy; Y. LeJan; X.-M. Li On the Geometry of Diffusion Operators and Stochastic Flows, Lecture Notes in Math., 1720, Springer, 1999

[9] K.D. Elworthy, X.-M. Li, Itô map and the chain rule in Malliavin calculus, in preparation

[10] K.D. Elworthy; X.-M. Li Special Itô maps and an L² Hodge theory for one forms on path spaces, Stochastic Processes, Physics and Geometry: New Interplays, I, American Mathematical Society, 2000, pp. 145-162

[11] X.-D. Li Sobolev spaces and capacities theory on path spaces over a compact Riemannian manifold, Probab. Theory Related Fields, Volume 125 (2003), pp. 96-134

[12] D. Nualart The Malliavin Calculus and Related Topics, Springer-Verlag, 1995

[13] I. Shigekawa A quasihomeomorphism on the Wiener space, Proc. Sympos. Pure Math., Volume 57 (1995), pp. 473-486

[14] H. Sugita On a characterization of the Sobolev spaces over an abstract wiener space, J. Math. Kyoto Univ., Volume 25 (1985) no. 4, pp. 717-725

Cité par Sources :

^☆ Research partially supported by NSF grant DMS 0072387 and EPSRC GR/NOO 845.

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