Generalised Clark-Ocone formulae for differential forms

TL;DR

This paper extends the Clark-Ocone formula to differential forms on Wiener space, providing explicit representations and a new proof of the triviality of L^2 de Rham cohomology groups, with potential applications to curved path spaces.

Contribution

It introduces generalized Clark-Ocone formulae for differential forms on Wiener space, offering explicit expressions and an alternative proof of cohomology triviality.

Findings

Explicit formulas for closed and co-closed differential forms

New proof of L^2 de Rham cohomology triviality on Wiener space

Potential extension to curved path spaces

Abstract

We generalise the Clark-Ocone formula for functions to give analogous representations for differential forms on the classical Wiener space. Such formulae provide explicit expressions for closed and co-closed differential forms and, as a by-product, a new proof of the triviality of the L^2 de Rham cohomology groups on the Wiener space, alternative to Shigekawa's approach [16] and the chaos-theoretic version [18]. This new approach has the potential of carrying over to curved path spaces, as indicated by the vanishing result for harmonic one-forms in [6]. For the flat path group, the generalised Clark-Ocone formulae can be proved directly using the It\^o map.