On subordinators, self-similar Markov processes and some factorizations of the exponential variable

Abstract

Let ξ be a subordinator with Laplace exponent Φ, I=∫∞0exp(−ξs)ds the so-called exponential functional, and X (respectively, X^) the self-similar Markov process obtained from ξ (respectively, from ξ^=−ξ) by Lamperti's transformation. We establish the existence of a unique probability measure ρ on ]0,∞[ with k-th moment given for every k∈N by the product Φ(1)⋯Φ(k), and which bears some remarkable connections with the preceding variables. In particular we show that if R is an independent random variable with law ρ then IR is a standard exponential variable, that the function t→E(1/Xt) coincides with the Laplace transform of ρ, and that ρ is the 1-invariant distribution of the sub-markovian process X^. A number of known factorizations of an exponential variable are shown to be of the preceding form IR for various subordinators ξ.