XXVI Workshop on Geometric Methods in Physics 1-7.07.2007

David Elworthy

An L2 differential form theory on path spaces.

A self-adjoint Hodge Laplacian is constructed on spaces of one- and two- forms over the manifold of continuous paths on a compact Riemannian manifold. The measure used is Brownian motion measure. The q-vectors involved, dual to the forms, are perturbations of the usual exterior powers of"finite energy" tangent vectors by a curvature term. The "damped Markovian connection" on path space plays an important role, and will be simply described. A proposal for extending this to forms of all degrees will be given. The talk is based on: "An L2 theory for differential forms on path spaces I", K. D. Elworthy & Xue-Mei Li, available on http://www.xuemei.org/bib.html