Ito-Wiener chaos and the Hodge decomposition on an abstract Wiener space

TL;DR

This paper provides a new proof of the triviality of $L^2$ cohomology groups on an abstract Wiener space using Boson-Fermion Fock space structure and symmetric group representation theory, offering an alternative to previous methods.

Contribution

It introduces a novel proof technique for $L^2$ cohomology triviality on Wiener spaces utilizing Boson-Fermion Fock space and symmetric group representations.

Findings

Established triviality of $L^2$ cohomology groups on abstract Wiener spaces.

Characterized spaces of exact and co-exact forms via representation theory.

Provided an alternative proof to Shigekawa's result.

Abstract

Using the structure of the Boson-Fermion Fock space and an argument taken from [2], we give a new proof of the triviality of the $L^2$ cohomology groups on an abstract Wiener space, alternative to that given by Shigekawa [9]. We apply some representation theory of the symmetric group to characterise the spaces of exact and co-exact forms in their Boson-Fermion Fock space representation.