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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.

MSC:
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
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