Infinite divisibility of solutions to some self-similar integro-differential equations and exponential functionals of L\'evy processes

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 L\'evy processes with a negative first moment.

[1]  J. Kent.  Some Probabilistic Properties of Bessel Functions , 1978 .

[2]  R. Blumenthal.  On construction of Markov processes , 1983 .

[3]  P. Levy.  Wiener's Random Function, and Other Laplacian Random Functions , 1951 .

[4]  M. Caballero,et al.  Weak convergence of positive self-similar Markov processes and overshoots of Lévy processes , 2004, math/0406015.

[5]  J. Pitman,et al.  Bessel processes and infinitely divisible laws , 1981 .

[6]  F. Steutel,et al.  Infinite Divisibility of Probability Distributions on the Real Line , 2003 .

[7]  M. Yor,et al.  Sur les fonctionnelles exponentielles de certains processus de lévy , 1994 .

[8]  Marc Yor,et al.  The Entrance Laws of Self-Similar Markov Processes and Exponential Functionals of Lévy Processes , 2002 .

[9]  J. Pitman,et al.  Self-similar processes with independent increments associated with Lévy and Bessel processes , 2002 .

[10]  J. Trujillo,et al.  Differential equations of fractional order:methods results and problem —I , 2001 .

[11]  S. Wolfe,et al.  On the Unimodality of $L$ Functions , 1971 .

[12]  Bert Zwart,et al.  Tail asymptotics for exponential function-als of L evy processes , 2006 .

[13]  佐藤 健一.  Lévy processes and infinitely divisible distributions , 2013 .

[14]  Geoffrey S. Watson,et al.  "Normal" Distribution Functions on Spheres and the Modified Bessel Functions , 1974 .

[15]  M. Yor,et al.  Loi de l'indice du lacet Brownien, et distribution de Hartman-Watson , 1980 .

[16]  Marc Yor,et al.  On the entire moments of self-similar Markov processes and exponential functionals of Lévy processes , 2002 .

[17]  P. Hartman.  Completely monotone families of solutions of $n$-th order linear differential equations and infinitely divisible distributions , 1976 .

[18]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1967 .

[19]  M. Yor,et al.  Exponential functionals of Levy processes , 2005, math/0511265.

[20]  K. Chung.  Review: William Feller, An Introduction to Probability Theory and its Applications 2 , 1973 .

[21]  R. Wolpert.  Lévy Processes , 2000 .

[22]  S. Wolfe.  On a continuous analogue of the stochastic difference equation Xn = ρ X n–1 + Bn , 1980, Advances in Applied Probability.

[23]  Victor Rivero,et al.  Recurrent extensions of self-similar Markov processes and Cramér’s condition II , 2005 .

[24]  T. MacRobert.  Higher Transcendental Functions , 1955, Nature.

[25]  P. Patie.  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 , 2007, 0712.1115.

[26]  R. A. Silverman,et al.  Special functions and their applications , 1966 .

[27]  Victor Manuel Rivero Mercado,et al.  Recurrent extensions of self-similar Markov processes and Cramer's condition , 2007 .

[28]  J. Lamperti.  Semi-stable Markov processes. I , 1972 .

[29]  A. Erdélyi,et al.  Higher Transcendental Functions , 1954 .

[30]  M. Yor.  Exponential Functionals of Brownian Motion and Related Processes , 2001 .

[31]  Anatoly A. Kilbas,et al.  On solution of integral equation of Abel-Volterra type , 1995, Differential and Integral Equations.

[32]  M. Yor,et al.  Continuous martingales and Brownian motion , 1990 .

[33]  S. Wolfe.  On a continuous analogue of the stochastic difference equation Xn=ρXn-1+Bn , 1982 .

[34]  M. Yor,et al.  Variations sur une formule de Paul Lévy , 1987 .

[35]  Z. Ciesielski,et al.  First passage times and sojourn times for Brownian motion in space and the exact Hausdorff measure of the sample path , 1962 .