The class of distributions on \mathbb{R} generated by convolutions of Γ-distributions and the class generated by convolutions of mixtures of exponential distributions are generalized to higher dimensions and denoted by T(\mathbb{R}^d) and B(\mathbb{R}^d) . From the Lévy process \{X_t^{(\mu)}\} on \mathbb{R}^d with distribution μ at t=1, Υ(μ) is defined as the distribution of the stochastic integral \int_0^1 \log(1/t)\d X_t^{(\mu)} . This mapping is a generalization of the mapping Υ introduced by Barndorff-Nielsen and Thorbjørnsen in one dimension. It is proved that \Upsilon(ID(\mathbb{R}^d))=B(\mathbb{R}^d) and \Upsilon(L(\mathbb{R}^d))=T(\mathbb{R}^d) , where ID(\mathbb{R}^d) and L(\mathbb{R}^d) are the classes of infinitely divisible distributions and of self-decomposable distributions on \mathbb{R}^d , respectively. The relations with the mapping Φ from μ to the distribution at each time of the stationary process of Ornstein-Uhlenbeck type with background driving Lévy process \{X_t^{(\mu)}\} are studied. Developments of these results in the context of the nested sequence L_m(\mathbb{R}^d), m=0,1,\ldots,\infty , [math] , are presented. Other applications and examples are given.