A refined factorization of the exponential law

Let $ξ$ be a (possibly killed) subordinator with Laplace exponent $ϕ$ and denote by $I_ϕ=\int_0^{\infty}\mathrm{e}^{-ξ_s}\,\mathrm{d}s$, the so-called exponential functional. Consider the positive random variable $I_{ψ_1}$ whose law, according to Bertoin and Yor [Electron. Comm. Probab. 6 (2001) 95--106], is determined by its negative entire moments as follows: $\mathbb {E}[I_{ψ_1}^{-n}]=\prod_{k=1}^nϕ(k),\qquad n=1,2,...$ In this note, we show that $I_{ψ_1}$ is a positive self-decomposable random variable whenever the Lévy measure of $ξ$ is absolutely continuous with a monotone decreasing density. In fact, $I_{ψ_1}$ is identified as the exponential functional of a spectrally negative (sn, for short) Lévy process. We deduce from Bertoin and Yor [Electron. Comm. Probab. 6 (2001) 95--106] the following factorization of the exponential law ${\mathbf {e}}$: $I_ϕ/I_{ψ_1}\stackrel{\mathrm {(d)}}{=}{\mathbf {e}},$ where $I_{ψ_1}$ is taken to be independent of $I_ϕ$. 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 Lévy 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 $S(α)^α$ is a self-decomposable random variable, where $S(α)$ is a positive stable random variable of index $α\in(0,1)$.

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