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

We consider some fractional extensions of the recursive differential equation governing the Poisson process, i.e. ∂ₜpₖ(t)=-λ(pₖ(t)-pₖ₋₁(t)), k≥0, $t>0$ by introducing fractional time-derivatives of order ν,2ν,…,nν. We show that the so-called "Generalized Mittag-Leffler functions" Eα,βᵏ(x), x (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→∞. On the other hand, near the origin the behavior of the law of the interarrival times drastically changes for the parameter ν varying in $(0,1]$. For integer values of ν, these models can be viewed as a higher-order Poisson processes, connected with the standard case by simple and explict relationships.