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Infinite divisibility of solutions to some self-similar integro-differential equations and exponential functionals of Lévy processes. (English) Zbl 1180.31010
This paper is concerned with analytic properties of monotone eigenfunctions associated to a family of integro-differential operators with parameters. First, the authors obtain that the increasing eigenfunctions are uniquely determined by the distribution of a spectrally negative, infinitely divisible random variable, with characteristic exponent \(\psi\). The 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 generalized Mittag-Leffler functions. In the second part of the paper the authors prove that some specific combinations of these increasing eigenfunctions are in fact Laplace transforms of self-decomposable or infinitely divisible distributions concentrated on the positive line with respect to the main argument. Finally, associated decreasing eigenfunctions are considered and their properties.

MSC:
31C05 Harmonic, subharmonic, superharmonic functions on other spaces
60G18 Self-similar stochastic processes
33E12 Mittag-Leffler functions and generalizations
20C20 Modular representations and characters
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