Analysis of generalized Lévy white noise functionals

Abstract

Employing the Segal–Bargmann transform ( S-transform for abbreviation) of regular Lévy white noise functionals, we define and study the generalized Lévy white noise functionals by means of their functional representations acting on test functionals. The main results generalize (Gaussian) white noise analysis initiated by T. Hida to non-Gaussian cases. Thanks to the closed form of the S-transform of Lévy white noise functionals obtained in our previous paper, we are able to define and study the renormalization of products of Lévy white noises, multiplication operator by Lévy white noises, and the differential operators with respect to a Lévy white noise and their adjoint operators. In the courses of our investigation we also obtain a formula for the products of multiple Lévy–Itô stochastic integrals. As applications, we discuss the existence of Hitsuda–Skorokhod integral for Lévy processes, Kubo–Takenaka formula for Lévy processes, and Itô formula for generalized Lévy white noise functionals.