The vanishing of L2 harmonic one-forms on based path spaces

TL;DR

This paper demonstrates the triviality of the first L2 cohomology class on based path spaces of Riemannian manifolds with Brownian measure, showing that all L2 harmonic one-forms vanish, and provides explicit formulas for these forms.

Contribution

It introduces explicit Clark-Ocone type formulas for closed and co-closed one-forms on path spaces, and relates the exterior derivative to the path space structure, proving spectral gap existence.

Findings

First L2 cohomology class is trivial on path spaces

Explicit formulas for harmonic one-forms are derived

Laplacian on one-forms has a spectral gap

Abstract

We prove the triviality of the first L2 cohomology class of based path spaces of Riemannian manifolds furnished with Brownian motion measure, and the consequent vanishing of L2 harmonic one-forms. We give explicit formulae for closed and co-closed one-forms expressed as differentials of functions and co-differentials of L2 two-forms, respectively; these are considered as extended Clark-Ocone formulae. A feature of the proof is the use of the temporal structure of path spaces to relate a rough exterior derivative operator on one-forms to the exterior differentiation operator used to construct the de Rham complex and the self-adjoint Laplacian on L2 one-forms. This Laplacian is shown to have a spectral gap.