Levy stable distributions via associated integral transform

Author(s)

Gorska, K. ^{[1, 2, 3]} ; Penson, K. A. ^[3]

Affiliation

^[1] Univ Sao Paulo, Inst Fis, BR-05315970 Sao Paulo - Brazil

^[2] Polish Acad Sci, H Niewodniczanski Inst Nucl Phys, PL-31342 Krakow - Poland

^[3] Univ Paris 06, CNRS, LPTMC, UMR 7600, F-75252 Paris 05 - France

• Document type: Journal article
• Source: Journal of Mathematical Physics; v. 53, n. 5 MAY 2012.
• Web of Science Citations: 14

Abstract

We present a method of generation of exact and explicit forms of one-sided, heavy-tailed Levy stable probability distributions g(alpha)(x), 0 <= x < infinity, 0 < alpha < 1. We demonstrate that the knowledge of one such a distribution g a ( x) suffices to obtain exactly g(alpha)p ( x), p = 2, 3, .... Similarly, from known g(alpha)(x) and g(beta)(x), 0 < alpha, beta < 1, we obtain g(alpha beta)( x). The method is based on the construction of the integral operator, called Levy transform, which implements the above operations. For a rational, alpha = l/k with l < k, we reproduce in this manner many of the recently obtained exact results for g(l/k)(x). This approach can be also recast as an application of the Efros theorem for generalized Laplace convolutions. It relies solely on efficient definite integration. (C) 2012 American Institute of Physics. {[}http://dx.doi.org/10.1063/1.4709443] (AU)

• FAPESP's process: 10/15698-5 - Coherent states and semiclassical description of quantum dissipative systems, spinning systems, and quantum relativistic particles.
• Grantee: Katarzyna Górska
• Support Opportunities: Scholarships in Brazil - Post-Doctoral