Journal of Applied Probability

State-dependent fractional point processes

R. Garra, E. Orsingher, and F. Polito

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


In this paper we analyse the fractional Poisson process where the state probabilities pkνk(t), t ≥ 0, are governed by time-fractional equations of order 0 < νk ≤ 1 depending on the number k of events that have occurred up to time t. We are able to obtain explicitly the Laplace transform of pkνk(t) and various representations of state probabilities. We show that the Poisson process with intermediate waiting times depending on νk differs from that constructed from the fractional state equations (in the case of νk = ν, for all k, they coincide with the time-fractional Poisson process). We also introduce a different form of fractional state-dependent Poisson process as a weighted sum of homogeneous Poisson processes. Finally, we consider the fractional birth process governed by equations with state-dependent fractionality.

Article information

J. Appl. Probab., Volume 52, Number 1 (2015), 18-36.

First available in Project Euclid: 17 April 2015

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G55: Point processes 26A33: Fractional derivatives and integrals
Secondary: 34A08: Fractional differential equations 60G22: Fractional processes, including fractional Brownian motion

Dzhrbashyan-Caputo fractional derivative Poisson process stable process Mittag-Leffler function pure birth process


Garra, R.; Orsingher, E.; Polito, F. State-dependent fractional point processes. J. Appl. Probab. 52 (2015), no. 1, 18--36. doi:10.1239/jap/1429282604.

Export citation


  • Balakrishnan, N. and Kozubowski, T. J. (2008). A class of weighted Poisson processes. Statist. Prob. Lett. 78, 2346–2352.
  • Beghin, L. and Macci, C. (2013). Large deviations for fractional Poisson processes. Statist. Prob. Lett. 83, 1193–1202.
  • Beghin, L. and Orsingher, E. (2009). Fractional Poisson processes and related planar random motions. Electron. J. Prob. 14, 1790–1827.
  • D'Ovidio, M., Orsingher, E. and Toaldo, B. (2014). Fractional telegraph-type equations and hyperbolic Brownian motion. Statist. Prob. Lett. 89, 131–137.
  • Fedotov, S., Ivanov, A. O. and Zubarev, A. Y. (2013). Non-homogeneous random walks, subdiffusive migration of cells and anomolous chemotaxis. Math. Model. Nat. Phenom. 8, 28–43.
  • Garra, R. and Polito, F. (2011). A note on fractional linear pure birth and pure death processes in epidemic models. Physica A 390, 3704–3709.
  • Hilfer, R. and Anton, L. (1995). Fractional master equation and fractal time random walks. Phys. Rev. E 51, R848–R851.
  • Laskin, N. (2003). Fractional Poisson process. Commun. Nonlinear Sci. Numerical Simul. 8, 201–213.
  • Laskin, N. (2009). Some applications of the fractional Poisson probability distribution. J. Math. Phys. 50, 113513.
  • Mainardi, F., Gorenflo, R. and Scalas, E. (2004). A fractional generalization of the Poisson process. Vietnam J. Math. 32, 53–64.
  • Mathai, A. M. and Haubold, H. J. (2008). Special Functions for Applied Scientists. Springer, New York.
  • Meerschaert, M. M., Nane, E. and Vellaisamy, P. (2011). The fractional poisson process and the inverse stable subordinator. Electron. J. Prob. 16, 1600–1620.
  • Olver, F. W. J., Lozier, D. W., Boisvert, R. F. and Clark, C. W. (eds) (2010). NIST Handbook of Mathematical Functions. Cambridge University Press.
  • Orsingher, E. and Polito, F. (2010). Fractional pure birth processes. Bernoulli 16, 858–881.
  • Orsingher, E. and Polito, F. (2012). The space-fractional Poisson process. Statist. Prob. Lett. 82, 852–858.
  • Podlubny, I. (1999). Fractional Differential Equations. Academic Press, San Diego, CA.
  • Prabhakar, T. R. (1971). A singular integral equation with a generalized Mittag–Leffler function in the kernel. Yokohama Math. J. 19, 7–15.
  • Repin, O. N. and Saichev, A. I. (2000). Fractional Poisson law. Radiophysics Quantum Electron. 43, 738–741.
  • Saxena, R. K., Mathai, A. M. and Haubold, H. J. (2006). Solutions of fractional reaction-diffusion equations in terms of Mittag–Leffler functions. Internat. J. Scientific Res. 15, 1–17.
  • Sixdeniers, J.-M., Penson, K. A. and Solomon, A. I. (1999). Mittag–Leffler coherent states. J. Phys. A 32, 7543–7563.