Representations of Urbanik's classes and multiparameter Ornstein-Uhlenbeck processes

Abstract

A class of integrals with respect to homogeneous Lévy bases on $\mathbb{R}^k$ is considered. In the one-dimensional case $k=1$ this class corresponds to the selfdecomposable distributions. Necessary and sufficient conditions for existence as well as some representations of the integrals are given. Generalizing the one-dimensional case it is shown that the class of integrals corresponds to Urbanik's class $ L_{k-1}(R)$. Finally, multiparameter Ornstein-Uhlenbeck processes are defined and studied.