Generalised Clark-ocone Formulae for Differential Forms

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 de Rham cohomology groups on the Wiener space, alternative to Shigekawa’s approach [15] and the chaos-theoretic version [17]. This new approach has the potential of carrying over to curved path spaces, as indicated by the vanishing result for harmonic one-forms in [5]. For the flat path group, the generalised Clark-Ocone formulae can be derived using the Itô map.