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Generalised Clark-Ocone formulae for differential forms. (English) Zbl 1331.60106
Summary: 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 I. Shigekawa’s approach [J. Math. Kyoto Univ. 26, 191–202 (1986; Zbl 0611.58006)] and the chaos-theoretic version [Y. Yang, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 16, No. 1, Article ID 1350008, 14 p. (2013; Zbl 1277.58002)]. This new approach has the potential of carrying over to curved path spaces, as indicated by the vanishing result for harmonic one-forms in [K. D. Elworthy and Y. Yang, J. Funct. Anal. 264, No. 5, 1168–1196 (2013; Zbl 1264.58001)]. For the flat path group, the generalised Clark-Ocone formulae can be derived using the Itō map.

60H07 Stochastic calculus of variations and the Malliavin calculus
60H05 Stochastic integrals
60H30 Applications of stochastic analysis (to PDEs, etc.)
58A14 Hodge theory in global analysis
60J65 Brownian motion