A refined factorization of the exponential law

Let $\xi$ be a (possibly killed) subordinator with Laplace exponent $\phi$ and denote by $I_{\phi}=\int_0^{\infty}\mathrm{e}^{-\xi_s}\,\mathrm{d}s$, the so-called exponential functional. Consider the positive random variable $I_{\psi_1}$ whose law, according to Bertoin and Yor [Electron. Comm. Probab. 6 (2001) 95--106], is determined by its negative entire moments as follows: $$\mathbb {E}[I_{\psi_1}^{-n}]=\prod_{k=1}^n\phi(k),\qquad n=1,2,...$$ In this note, we show that $I_{\psi_1}$ is a positive self-decomposable random variable whenever the L\'{e}vy measure of $\xi$ is absolutely continuous with a monotone decreasing density. In fact, $I_{\psi_1}$ is identified as the exponential functional of a spectrally negative (sn, for short) L\'{e}vy process. We deduce from Bertoin and Yor [Electron. Comm. Probab. 6 (2001) 95--106] the following factorization of the exponential law ${\mathbf {e}}$: $$I_{\phi}/I_{\psi_1}\stackrel{\mathrm {(d)}}{=}{\mathbf {e}},$$ where $I_{\psi_1}$ is taken to be independent of $I_{\phi}$. We proceed by showing an identity in distribution between the entrance law of an sn self-similar positive Feller process and the reciprocal of the exponential functional of sn L\'{e}vy processes. As a by-product, we obtain some new examples of the law of the exponential functionals, a new factorization of the exponential law and some interesting distributional properties of some random variables. For instance, we obtain that $S(\alpha)^{\alpha}$ is a self-decomposable random variable, where $S(\alpha)$ is a positive stable random variable of index $\alpha\in(0,1)$.