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

Abstract

We consider some fractional extensions of the recursive differential equation governing the Poisson process, i.e. d dt p k (t) = −λ(p k (t) − p k−1 (t)), k ≥ 1, t > 0 by introducing fractional time-derivatives of order ν, 2ν, ..., nν . We show that the so-called "Generalized Mittag-Leffler functions" E k α,β (x) (introduced by Prabhakar [20]) 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.analytical form, in terms of infinite series or successive derivatives of Mittag-Leffler functions. We recall the definition of the (two-parameters) Mittag-Leffler function:A different definition of Poisson fractional process has been proposed by Wang and Wen [30] and successively studied in [31]-[32]: in analogy to the well-known fractional Brownian motion, the new process is defined as a stochastic integral with respect to the Poisson measure. It displays properties similar to fractional Brownian motion, such as self-similarity (in the wide-sense) and long-range dependence.Another approach was followed by Repin and Saichev [23]: they start by generalizing, in a fractional sense, the distribution of the interarrival times U j between two Poisson events. This is expressed, in terms of Mittag-Leffler functions, as follows, for ν ∈ (0, 1] :and coincides with the solution to the fractional equationwhere δ(·) denotes the Dirac delta function and again the fractional derivative is intended in the Riemann-Liouville sense. For ν = 1 formula (1.2) reduces to the well-known density appearing in the case of a homogeneous Poisson process, N (t), t > 0, with intensity λ = 1, i.e. f (t) = e −t . The same approach is followed by Mainardi et al.[14]-[15]- [16], where a deep analysis of the related process is performed: it turns out to be a true renewal process, while it looses the Markovian property. Their first step is the study of the following fractional equation (instead of (1.3)) 4) with initial condition ψ(0 + ) = 1 and with fractional derivative defined in the Caputo sense. The solution ψ(t) = E ν,1 (−t ν ) to (1.4) represents the survival probability of the fractional Poisson process. As a consequence its probability distribution is expressed in terms of derivatives of Mittag-Leffler functions, while the density of the k-th event waiting time is a fractional generalization of the Erlang distribution and coincides with the k-fold convolution of (1.2). The analysis carried out by Beghin and Orsingher [2] starts, as in [11], from the generalization of the equation governing the Poisson process, where the time-derivative is substituted by the fractional derivat...

Fractional Relaxation Equations and Brownian Crossing Probabilities of a Random Boundary

“…By comparing (2.39) with the results in[4], we can deduce that the crossing probability ψ k 1/2 (t) can be written in terms of the fractional Poisson process of order ν = 1 .…”

“…The first form represents the probability of no events up to time t (or survival probability) for the so-called fractional Poisson process N ν (t), t ≥ 0 (see, amongst others, [2], [4], [14], [18], and [32]). The first form represents the probability of no events up to time t (or survival probability) for the so-called fractional Poisson process N ν (t), t ≥ 0 (see, amongst others, [2], [4], [14], [18], and [32]).…”

Poisson process with different Brownian clocks

“…where, in the last step, we have applied formula (2.31) of Beghin and Orsingher [4]. We note that, for u ¼ 1, formula (2.34) reduces to one, while for u ¼ 0 it gives p el 0 ðtÞ, since it is …”

“…We check that the two expressions of (2.21) coincide, by applying the relation holding for generalized Mittag-Leffler functions proved in Beghin and Orsingher [4] (see formula (3.6), for n ¼ 0, m ¼ 2, z ¼ 1, and n ¼ 1 2 ): …”

“…In order to obtain the recursive differential equation governing the distribution of the process N el , we note that p el k ðtÞ (in the form given in (2.26)) can be rewritten in terms of the probability distribution p 1=2 k ðtÞ, k $ 1, of the fractional Poisson process N n , with parameters n ¼ 1 2 and l ffiffi 2 p (see Beghin and Orsingher [4]). We recall that…”