Electronic Journal of Probability

Poisson-Type Processes Governed by Fractional and Higher-Order Recursive Differential Equations

Luisa Beghin and Enzo Orsingher

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We consider some fractional extensions of the recursive differential equation governing the Poisson process, i.e. $\partial_tp_k(t)=-\lambda(p_k(t)-p_{k-1}(t))$, $k\geq0$, $t>0$ by introducing fractional time-derivatives of order $\nu,2\nu,\ldots,n\nu$. We show that the so-called "Generalized Mittag-Leffler functions" $E_{\alpha,\beta^k}(x)$, $x\in\mathbb{R}$ (introduced by Prabhakar [24] )arise as solutions of these equations. The corresponding processes are proved to be renewal, with density of the intearrival times (represented by Mittag-Leffler functions) possessing power, instead of exponential, decay, for $t\to\infty$. On the other hand, near the origin the behavior of the law of the interarrival times drastically changes for the parameter $\nu$ varying in $(0,1]$. For integer values of $\nu$, these models can be viewed as a higher-order Poisson processes, connected with the standard case by simple and explict relationships.

Article information

Electron. J. Probab., Volume 15 (2010), paper no. 22, 684-709.

Accepted: 20 May 2010
First available in Project Euclid: 1 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K05: Renewal theory
Secondary: 33E12: Mittag-Leffler functions and generalizations 26A33: Fractional derivatives and integrals

Fractional difference-differential equations Generalized Mittag-Leffler functions Fractional Poisson processes Processes with random time Renewal function Cox process

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Beghin, Luisa; Orsingher, Enzo. Poisson-Type Processes Governed by Fractional and Higher-Order Recursive Differential Equations. Electron. J. Probab. 15 (2010), paper no. 22, 684--709. doi:10.1214/EJP.v15-762. https://projecteuclid.org/euclid.ejp/1464819807

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