Quasi-invariance of the Wiener measure on the path space over a compact Riemannian manifold
The quasi-invariance property of the Wiener measure in the flat Euclidean space is the well known Cameron-Martin theorem. For Brownian motion on a compact Riemannian manifold significant results on the quasi-invariance of the flow on the path space are due to \textit{B. K. Driver} [J. Funct. Anal. 110, No. 2, 272-376 (1992; Zbl 0765.60064)]. This property is utilized to show that the directional derivative and gradient operators are closable, together with the associated Dirichlet form. The integration by parts formula is proved and used to compute explicitly the Ornstein-Uhlen-Beck operator in the path space, and to show that the set of cylindrical functions lies in its domain of definition.
scientific article; zbMATH DE number 850628