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

Abstract

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.