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

Elworthy, K.David; Li, Xue-Mei

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

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 ${\mathrm{d}}_{\Omega }{ℐ}^{*}={ℐ}^{*}{\mathrm{d}}_{C_{x_0}M}$. 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, ${\mathrm{d}}_{\Omega }{ℐ}^{*}={ℐ}^{*}{\mathrm{d}}_{C_{x_0}M}$. A uniqueness property of d on the path space follows. Results are also given for higher derivatives and covariant derivatives.

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