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


 
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1. Title Title of document Poisson-Type Processes Governed by Fractional and Higher-Order Recursive Differential Equations
 
2. Creator Author's name, affiliation, country Luisa Beghin; Sapienza University of Rome; Italy
 
2. Creator Author's name, affiliation, country Enzo Orsingher; Sapienza University of Rome; Italy
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) Fractional difference-differential equations; Generalized Mittag-Leffler functions; Fractional Poisson processes; Processes with random time; Renewal function; Cox process.
 
3. Subject Subject classification 60K05; 33E12; 26A33
 
4. Description Abstract 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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7. Date (YYYY-MM-DD) 2010-05-20
 
8. Type Status & genre Peer-reviewed Article
 
8. Type Type
 
9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ejp.ejpecp.org/article/view/762
 
10. Identifier Digital Object Identifier 10.1214/EJP.v15-762
 
11. Source Journal/conference title; vol., no. (year) Electronic Journal of Probability; Vol 15
 
12. Language English=en
 
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