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2010 Poisson-Type Processes Governed by Fractional and Higher-Order Recursive Differential Equations
Luisa Beghin, Enzo Orsingher
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Electron. J. Probab. 15: 684-709 (2010). DOI: 10.1214/EJP.v15-762


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.


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Luisa Beghin. Enzo Orsingher. "Poisson-Type Processes Governed by Fractional and Higher-Order Recursive Differential Equations." Electron. J. Probab. 15 684 - 709, 2010.


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

zbMATH: 1228.60093
MathSciNet: MR2650778
Digital Object Identifier: 10.1214/EJP.v15-762

Primary: 60K05
Secondary: 26A33, 33E12


Vol.15 • 2010
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