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An analytic version of Wiener-Itô decomposition on abstract Wiener spaces. (English) Zbl 1427.60010

Summary: In this paper, we first establish an analogue of Wiener-Itô theorem on finite-dimensional Gaussian spaces through the inverse \(S\)-transform, that is, the Gauss transform on Segal-Bargmann spaces. Based on this point of view, on infinite-dimensional abstract Wiener space \((H,B)\), we apply the analyticity of the \(S\)-transform, which is an isometry from the \(L^2\)-space onto the Bargmann-Segal-Dwyer space, to study the regularity. Then, by defining the Gauss transform on Bargmann-Segal-Dwyer space and showing the relationship with the \(S\)-transform, an analytic version of Wiener-Itô decomposition will be obtained.

MSC:

60B11 Probability theory on linear topological spaces
46E20 Hilbert spaces of continuous, differentiable or analytic functions
46E50 Spaces of differentiable or holomorphic functions on infinite-dimensional spaces
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