Complex Wiener-It\^o Chaos Decomposition Revisited

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.