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

Presented by: 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.

Abstract
Résumé

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.

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.

Received: 2003-10-01
Accepted: 2003-10-06
Published online: 2003-11-14

DOI: 10.1016/j.crma.2003.10.004

Author's affiliations:

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},
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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/

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

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