Generalised Clark-Ocone formulae for differential forms

Details Generalised Clark-Ocone formulae for differential forms Generalised Clark-Ocone formulae for differential forms

2011-11-04

arxiv.org/abs/1111.1194v3

Mathematics

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Scientific paper

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.

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