Analysis of generalized Levy white noise functionals

摘要

Employing the Segal-Bargmann transform (S-transform for abbreviation) of regular Levy white noise functionals, we define and study the generalized Levy 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 Levy white noise functionals obtained in our previous paper, we are able to define and study the renormalization of products of Levy white noises, multiplication operator by Levy white noises, and the differential operators with respect to a Levy white noise and their adjoint operators. In the courses of our investigation we also obtain a formula for the products of multiple Levy-It (o) over cap stochastic integrals. As applications, we discuss the existence of Hitsuda-Skorokhod integral for Levy processes, Kubo-Takenaka formula for Levy processes, and It (o) over cap formula for generalized Levy white noise functionals. (C) 2003 Elsevier Inc. All rights reserved.