Wick Calculus For Nonlinear Gaussian Functionals

TL;DR

This paper reviews key results on Wick product and Wick renormalization within the framework of abstract Wiener space, highlighting their properties, conditions for integrability, and connections with Malliavin calculus and second quantization.

Contribution

It provides a comprehensive survey of Wick calculus, including classical results, conditions for integrability, and relations with advanced stochastic analysis tools.

Findings

Conditions for integrability of Wick products

Relations between Wick calculus and Malliavin calculus

Description of the $S$-transform as a useful tool

Abstract

This paper surveys some results on Wick product and Wick renormalization. The framework is the abstract Wiener space. Some known results on Wick product and Wick renormalization in the white noise analysis framework are presented for classical random variables. Some conditions are described for random variables whose Wick product or whose renormalization are integrable random variables. Relevant results on multiple Wiener integrals, second quantization operator, Malliavin calculus and their relations with the Wick product and Wick renormalization are also briefly presented. A useful tool for Wick product is the $S$-transform which is also described without the introduction of generalized random variables.