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

   
@article{EJP762,
	author = {Luisa Beghin and Enzo Orsingher},
	title = {Poisson-Type Processes Governed by Fractional and Higher-Order Recursive Differential Equations},
	journal = {Electron. J. Probab.},
	fjournal = {Electronic Journal of Probability},
	volume = {15},
	year = {2010},
	keywords = {Fractional difference-differential equations; Generalized Mittag-Leffler functions; Fractional Poisson processes; Processes with random time; Renewal function; Cox process.},
	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.},
	pages = {no. 22, 684-709},
	issn = {1083-6489},
	doi = {10.1214/EJP.v15-762},    
        url = {http://ejp.ejpecp.org/article/view/762}}