Poisson-type processes governed by fractional and higher-order recursive differential equations

By Luisa Beghin, Enzo Orsingher. 2009

107 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Modeling and Simulation, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/18a4f20bf6e0e86f2b3fdd29bb74da4a9bbfde63

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In Applied Mathematics

In Modeling and Simulation

In Numerical Analysis

Citations in the corpus

A fractional counting process and its connection with the Poisson process (2015, methodology): “Moreover, Beghin and Orsingher (2010) study the properties of Poisson-type fractional processes, governed by fractional recursive differential equations, obtained substituting regular derivatives with fractional derivatives.”
The Fractional Poisson Process and the Inverse Stable Subordinator (2010, background): “Theorem 2.2 extends a result in Behgin and Orsingher [5].”