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

Nous donnons des conditions sous lesquelles les applications d'Itô 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 * = * 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 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 * = * 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 :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2003.10.004
Elworthy, K.David 1 ; Li, Xue-Mei 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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Elworthy, K.David; Li, Xue-Mei. Gross–Sobolev spaces on path manifolds: uniqueness and intertwining by Itô maps. Comptes Rendus. Mathématique, Tome 337 (2003) no. 11, pp. 741-744. doi : 10.1016/j.crma.2003.10.004. http://www.numdam.org/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.