ON SUBORDINATORS, SELF-SIMILAR MARKOV PROCESSES AND SOME FACTORIZATIONS OF THE EXPONENTIAL VARIABLE

Let be a subordinator with Laplace exponent Phi, I = integral(infinity)(0)exp(-xi(s))ds the so-called exponential functional, and X (respectively, (X) over cap) the self-similar Markov process obtained from xi (respectively, from (xi) over cap = -xi) by Lamperti's transformation. We establish the existence of a unique probability measure rho on]0, infinity[ with k-th moment given for every k is an element of N by the product Phi(1)...Phi(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 p then IR is a standard exponential variable, that the function t -> E(1/X-t) coincides with the Laplace transform of rho, and that rho is the 1-invariant distribution of the sub-markovian process (X) over cap. A number of known factorizations of an exponential variable are shown to be of the preceding form IR for various subordinators xi.