An L2 theory for differential forms on path spaces I

Abstract

An L 2 theory of differential forms is proposed for the Banach manifold of continuous paths on a Riemannian manifold M furnished with its Brownian motion measure. Differentiation must be restricted to certain Hilbert space directions, the H -tangent vectors. To obtain a closed exterior differential operator the relevant spaces of differential forms, the H -forms, are perturbed by the curvature of M. A Hodge decomposition is given for L 2 H -one-forms, and the structure of H -two-forms is described. The dual operator d * is analysed in terms of a natural connection on the H -tangent spaces. Malliavin calculus is a basic tool.