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We construct differential forms of all orders and a covariant derivative together with its adjoint on the probability space of a standard Poisson process, using derivation operators. In this framewok we derive a de Rham-Hodge-Kodaira decomposition as well as Weitzenböck and Clark-Ocone formulas for random differential forms. As in the Wiener space setting, this construction provides two distinct approaches to the vanishing of harmonic differential forms. [ABSTRACT FROM AUTHOR]

A law of iterated logarithm (LIL) in small time and an asymptotic estimate of modulus of continuity are proved for Brownian motion on the loop group ℒ( G ) over a compact connected Lie group G . Upper bounds are obtained via infinite-dimensional deviation inequalities for functionals on the path space ℙ(ℒ( G )) on ℒ( G ), such as the supremum of Brownian motion on ℒ( G ), which are proved from the Clark–Ocone formula on ℙ(ℒ( G )). The lower bounds rely on analog finite-dimensional results that are proved separately on Riemannian path space. [ABSTRACT FROM AUTHOR]