Taiwanese Journal of Mathematics

An Analytic Version of Wiener-Itô Decomposition on Abstract Wiener Spaces

Yuh-Jia Lee and Hsin-Hung Shih

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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.

Article information

Taiwanese J. Math., Volume 23, Number 2 (2019), 453-471.

Received: 8 May 2018
Accepted: 10 December 2018
First available in Project Euclid: 28 December 2018

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Mathematical Reviews number (MathSciNet)

Primary: 60B11: Probability theory on linear topological spaces [See also 28C20] 46E20: Hilbert spaces of continuous, differentiable or analytic functions 46E50: Spaces of differentiable or holomorphic functions on infinite- dimensional spaces [See also 46G20, 46G25, 47H60]
Secondary: 28C20: Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.) [See also 46G12, 58C35, 58D20, 60B11]

abstract Wiener space abstract Wiener measure Wiener-Itô decomposition Segal-Bargmann transform Gauss transform


Lee, Yuh-Jia; Shih, Hsin-Hung. An Analytic Version of Wiener-Itô Decomposition on Abstract Wiener Spaces. Taiwanese J. Math. 23 (2019), no. 2, 453--471. doi:10.11650/tjm/181207. https://projecteuclid.org/euclid.twjm/1545966024

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