# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [1] R. P. Agarwal. A propos d’une note de M. Pierre Humbert. C. R. Math. Acad. Sci. Paris 236 (1953) 2031-2032. · Zbl 0051.30801 [2] V. Bally and L. Stoica. A class of Markov processes which admit a local time. Ann. Probab. 15 (1987) 241-262. · Zbl 0615.60069 [3] J. Bertoin. Lévy Processes . Cambridge Univ. Press, Cambridge, 1996. · Zbl 0861.60003 [4] J. Bertoin and M. Yor. The entrance laws of self-similar Markov processes and exponential functionals of Lévy processes. Potential Anal. 17 (2002) 389-400. · Zbl 1004.60046 [5] J. Bertoin and M. Yor. On the entire moments of self-similar Markov processes and exponential functionals of Lévy processes. Ann. Fac. Sci. Toulouse Math. 11 (2002) 19-32. · Zbl 1031.60038 [6] J. Bertoin and M. Yor. Exponential functionals of Lévy processes. Probab. Surv. 2 (2005) 191-212. · Zbl 1189.60096 [7] Ph. Biane and M. Yor. Variations sur une formule de Paul Lévy. Ann. Inst. H. Poincaré Probab. Statist. 23 (1987) 359-377. · Zbl 0623.60099 [8] R. M. Blumenthal. On construction of Markov processes. Z. Wahrsch. Verw. Gebiete 63 (1983) 433-444. · Zbl 0494.60071 [9] Ph. Carmona, F. Petit and M. Yor. Sur les fonctionnelles exponentielles de certains processus de Lévy. Stoch. Stoch. Rep. 47 (1994) 71-101. (English version in [38], p. 139-171.) · Zbl 0830.60072 [10] M. E. Caballero and L. Chaumont. Weak convergence of positive self-similar Markov processes and overshoots of Lévy processes. Ann. Probab. 34 (2006) 1012-1034. · Zbl 1098.60038 [11] Z. Ciesielski and S. J. Taylor. First passage times and sojourn times for Brownian motion in space and the exact Hausdorff measure of the sample path. Trans. Amer. Math. Soc. 103 (1962) 434-450. JSTOR: · Zbl 0121.13003 [12] E. B. Dynkin. Markov Processes I , II . Die Grundlehren der Mathematischen Wissenschaften, Bände 121 122 . Academic Press, New York, 1965. [13] A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi. Higher Transcendental Functions 3 . McGraw-Hill, New York, 1955. · Zbl 0064.06302 [14] W. E. Feller. An Introduction to Probability Theory and Its Applications 2 , 2nd edition. Wiley, New York, 1971. · Zbl 0219.60003 [15] I. I. Gikhman and A. V. Skorokhod. The Theory of Stochastic Processes II . Springer, Berlin, 1975. · Zbl 0348.60042 [16] P. Hartman. Completely monotone families of solutions of n -th order linear differential equations and infinitely divisible distributions. Ann. Sc. Norm. Super. Pisa Cl. Sci. IV-III (1976) 267-287. · Zbl 0386.34016 [17] P. Hartman and G. S. Watson. “Normal” distribution functions on spheres and the modified Bessel functions. Ann. Probab. 2 (1974) 593-607. · Zbl 0305.60033 [18] P. Humbert. Quelques résultats relatifs à la fonction de Mittag-Leffler. C. R. Math. Acad. Sci. Paris 236 (1953) 1467-1468. · Zbl 0050.10404 [19] M. Jeanblanc, J. Pitman and M. Yor. Self-similar processes with independent increments associated with Lévy and Bessel processes. Stochastic Process. Appl. 100 (2002) 223-232. · Zbl 1059.60052 [20] J. Kent. Some probabilistic properties of Bessel functions. Ann. Probab. 6 (1978) 760-770. · Zbl 0402.60080 [21] A. A. Kilbas and J. J. Trujillo. Differential equations of fractional orders: Methods, results and problems. Appl. Anal. 78 (2001) 153-192. · Zbl 1031.34002 [22] A. A. Kilbas and M. Saigo. On solution of integral equations of Abel-Volterra type. Differential Integral Equations 8 (1995) 993-1011. · Zbl 0823.45002 [23] J. Lamperti. Semi-stable Markov processes. Z. Wahrsch. Verw. Gebiete 22 (1972) 205-225. · Zbl 0274.60052 [24] N. N. Lebedev. Special Functions and Their Applications . Dover, New York, 1972. · Zbl 0271.33001 [25] P. Lévy. Wiener’s random functions, and other Laplacian random functions. In Proc. Sec. Berkeley Symp. Math. Statist. Probab., 1950 II . 171-187. California Univ. Press, Berkeley, 1951. · Zbl 0044.13802 [26] K. Maulik and B. Zwart. Tail asymptotics for exponential functionals of Lévy processes. Stochastic Process. Appl. 116 (2006) 156-177. · Zbl 1090.60046 [27] P. A. Meyer. Processus à accroissements indépendants et positifs. Séminaire de probabilités de Strasbourg 3 (1969) 175-189. · Zbl 0181.44901 [28] G. Mittag-Leffler. Sur la nouvelle function E \alpha ( x ). C. R. Math. Acad. Sci. Paris 137 (1903) 554-558. · JFM 34.0435.01 [29] P. Patie. Exponential functional of one-sided Lévy processes and self-similar continuous state branching processes with immigration. Bull. Sci. Math. · Zbl 1171.60009 [30] J. Pitman and M. Yor. Bessel processes and infinitely divisible laws. In Stochastic Integrals ( In Proc. Sympos. Univ. Durham, Durham, 1980 ) 285-370. D. Williams (ed.). Lecture Notes in Math. 851 . Springer, Berlin, 1981. · Zbl 0469.60076 [31] V. Rivero. Recurrent extensions of self-similar Markov processes and Cramér’s condition. Bernoulli 11 (2005) 471-509. · Zbl 1077.60055 [32] K. Sato. Lévy Processes and Infinitely Divisible Distributions . Cambridge Univ. Press, Cambridge, 1999. · Zbl 0973.60001 [33] F. W. Steutel and K. van Harn. Infinite divisibility of probability distributions on the real line. Monographs and Textbooks in Pure and Applied Mathematics 259 . Marcel Dekker Inc., New York, 2004. · Zbl 1063.60001 [34] J. Vuolle-Apiala. Itô excursion theory for self-similar Markov processes. Ann. Probab. 22 (1994) 546-565. · Zbl 0810.60067 [35] S. J. Wolfe. On the unimodality of L functions. Ann. Math. Stat. 42 (1971) 912-918. · Zbl 0219.60026 [36] S. J. Wolfe. On a continuous analogue of the stochastic difference equation X n = \rho X n - 1 + B n . Stochastic Process. Appl. 12 (1982) 301-312. · Zbl 0482.60062 [37] M. Yor. Loi de l’indice du lacet brownien et distribution de Hartman-Watson. Z. Wahrsch. Verw. Gebiete 53 (1980) 71-95. · Zbl 0436.60057 [38] M. Yor. Exponential Functionals of Brownian Motion and Related Processes . Springer, Berlin, 2001. · Zbl 0999.60004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.