Poisson-type processes governed by fractional and higher-order recursive differential equations
L Beghin, E Orsingher - 2010 - projecteuclid.org
2010•projecteuclid.org
We consider some fractional extensions of the recursive differential equation governing the
Poisson process, ie \partial_tp_k(t)=-λ(p_k(t)-p_k-1(t)), k\geq0, t>0 by introducing fractional
time-derivatives of order ν,2ν,...,nν. We show that the so-called" Generalized Mittag-Leffler
functions" E_α,β^k(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 …
Poisson process, ie \partial_tp_k(t)=-λ(p_k(t)-p_k-1(t)), k\geq0, t>0 by introducing fractional
time-derivatives of order ν,2ν,...,nν. We show that the so-called" Generalized Mittag-Leffler
functions" E_α,β^k(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 …
Abstract
We consider some fractional extensions of the recursive differential equation governing the Poisson process, i.e. , , by introducing fractional time-derivatives of order . We show that the so-called "Generalized Mittag-Leffler functions" , (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 . On the other hand, near the origin the behavior of the law of the interarrival times drastically changes for the parameter varying in . For integer values of , these models can be viewed as a higher-order Poisson processes, connected with the standard case by simple and explict relationships.