# Complex Wiener-It\^o Chaos Decomposition Revisited

## TL;DR

This paper revisits complex Wiener-Itô chaos, exploring properties of complex multiple integrals and Ornstein-Uhlenbeck operators, providing formulas and moment expansions relevant for advanced stochastic analysis.

## Contribution

It introduces new properties and formulas for complex Wiener-Itô integrals and Ornstein-Uhlenbeck operators, enhancing theoretical understanding and computational tools.

## Key findings

- Derived Stroock's, Hu-Meyer, and Clark-Ocone formulas for complex integrals.
- Established hypercontractivity of complex Ornstein-Uhlenbeck semigroups.
- Provided new expansions for the fourth moments of complex Wiener-Itô integrals.

## Abstract

In this article, some properties of complex Wiener-It\^o multiple integrals and complex Ornstein-Uhlenbeck operators and semigroups are obtained. Those include Stroock's formula, Hu-Meyer formula, Clark-Ocone formula and the hypercontractivity of complex Ornstein-Uhlenbeck semigroups. As an application, several expansions of the fourth moments of complex Wiener-It\^o multiple integrals are given.