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Poisson-type processes governed by fractional and higher-order recursive differential equations. (English) Zbl 1228.60093
Summary: We consider some fractional extensions of the recursive differential equation governing the Poisson process, i.e.,
\[ \frac {d}{dt}p_k(t)= -\lambda \big(p_k(t)-p_{k-1}(t)\big), \quad k\geq 0,\;t>0, \] by introducing fractional time-derivatives of order \(\nu,2\nu,\dots,n\nu\). We show that the so-called “generalized Mittag-Leffler functions” \(E_{\alpha,\beta}^k(x)\), \(x\in\mathbb R\) (introduced by T. R. Prabhakar [Yokohama Math. J. 19, 7–15 (1971; Zbl 0221.45003)]) arise as solutions of these equations. The corresponding processes are proved to be renewal, with density of the interarrival times (represented by Mittag-Leffler functions) possessing power, instead of exponential, decay, for \(t\rightarrow\infty\). 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 explicit relationships.

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