Journal of Statistical Physics

, Volume 141, Issue 1, pp 68–93

Fractional Non-Linear, Linear and Sublinear Death Processes



This paper is devoted to the study of a fractional version of non-linear Open image in new window, t>0, linear Mν(t), t>0 and sublinear \(\mathfrak{M}^{\nu}(t)\), t>0 death processes. Fractionality is introduced by replacing the usual integer-order derivative in the difference-differential equations governing the state probabilities, with the fractional derivative understood in the sense of Dzhrbashyan–Caputo. We derive explicitly the state probabilities of the three death processes and examine the related probability generating functions and mean values. A useful subordination relation is also proved, allowing us to express the death processes as compositions of their classical counterparts with the random time process T(t), t>0. This random time has one-dimensional distribution which is the folded solution to a Cauchy problem of the fractional diffusion equation.


Fractional diffusion Dzhrbashyan–Caputo fractional derivative Mittag-Leffler functions Linear death process Non-linear death process Sublinear death process Subordinated processes 


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Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Enzo Orsingher
    • 1
  • Federico Polito
    • 1
  • Ludmila Sakhno
    • 2
  1. 1.Dipartimento di Statistica, Probabilità e Statistiche Applicate“Sapienza” Università di RomaRomeItaly
  2. 2.Department of Probability Theory, Statistics and Actuarial MathematicsTaras Shevchenko National University of KyivKyivUkraine

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