Infinite divisibility of solutions to some self-similar integro-differential equations and exponential functionals of Lévy processes

Abstract

We provide the increasing eigenfunctions associated to spectrally negative self-similar Feller semigroups, which have been introduced by Lamperti. These eigenfunctions are expressed in terms of a new family of power series which includes, for instance, the modified Bessel functions of the first kind and some generalizations of the Mittag-Leffler function. Then, we show that some specific combinations of these functions are Laplace transforms of self-decomposable or infinitely divisible distributions concentrated on the positive line with respect to the main argument, and, more surprisingly, with respect to a parameter of the process. In particular, this generalizes a result of Hartman (1976) obtained for the increasing solution of the Bessel differential equation. Finally, we compute, for some cases, the associated decreasing eigenfunctions and derive the Laplace transform of the exponential functionals of some spectrally negative Lévy processes with a negative first moment.

References in corpus (3)

• Recurrent extensions of self-similar Markov processes and Cramér's condition II
• Weak convergence of positive self-similar Markov processes and overshoots of Lévy processes
• Law of the exponential functional of a new family of one-sided Levy processes via self-similar continuous state branching processes with immigration and the Wright hypergeometric functions

Cited by in corpus (1)

• Law of the exponential functional of a new family of one-sided Levy processes via self-similar continuous state branching processes with immigration and the Wright hypergeometric functions