Path Spaces and Their Tangent Spaces

Abstract

Stochastic calculus of variation on a Lie group: Reduced variation and adjoint representation — Path groups: left infinitesimal quasi-invariance of Wiener measure — Path group on a compact Lie group — Orthonormal frame bundles over a Riemannian manifold: Levi-Civita parallelism, structure equations — Lifting to the frame bundle of the Riemannian diffusion: the stochastic parallel displacement — The horizontal stochastic flow, determination of its Jacobian on a Riemannian diffusion — Invariance of Brownian motion by orthogonal transformations — Tangent processes — Stochastic analysis on the path space of a Riemannian manifold: Twisted differential, Clark-Bismut-Ocone representation — Harnack estimates via the reduced variation — Loop spaces.