The class of distributions on $\mathbb{R}$ generated by convolutions of $\Gamma$-distributions and the one 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 L\'evy process $\{X_{t}^{(\mu)}\}$ on $\mathbb{R}^{d}$ with distribution $\mu$ at $t=1$, $\Upsilon(\mu)$ is defined as the distribution of the stochastic integral $\int_{0}^{1} \log(1/t)dX_{t}^{(\mu)}$. This mapping is a generalization of the mapping $\Upsilon$ 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 selfdecomposable distributions on $\mathbb{R}^{d}$, respectively. The relations with the mapping $\Phi$ from $\mu$ 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 $L_{m}(\mathbb{R}^{d})$, $m=0,1,\ldots ,\infty$, are presented. Other applications and examples are given.