David Elworthy (University of Warwick)
kde@maths.warwick.ac.uk
The most natural measure on path or loop spaces of Riemannian manifolds appear to be those derived from Brownian motions on the manifolds. For such measures, and more generally, any integration by parts theory or construction of operators such as d* require differentiation only in some specified Hilbert spaces of directions. This leads to difficulties in even defining a DeRham complex involving Hilbert spaces of square integrable forms, and although there is a good substitute for a Laplacian on functions on these spaces there is as yet no firm theory of Laplacians on forms. In this talk I will describe the background and review recent work, presenting a candidate for a Hodge Theory on path spaces obtained in joint work with Xue-Mei Li.