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. d d t pk(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” Ek α,β(x), x ∈ 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 →∞. 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.