Fractional pure birth processes

Abstract

We consider a fractional version of the classical nonlinear birth process of which the Yule Furry model is a particular case. Fractionality is obtained by replacing the first order time derivative in the differencedifferential equations which govern the probability law of the process with the Dzherbashyan Caputo fractional derivative. We derive the probability distribution of the number N(v)(t) of individuals at an arbitrary time t. We also present an interesting representation for the number of individuals at time t, in the form of the subordination relation N(v)(t) = N(T(2v) (t)), where N(t) is the classical generalized birth process and T(2v) (t) is a random time whose distribution is related to the fractional diffusion equation. The fractional linear birth process is examined in detail in Section 3 and various forms of its distribution are given and discussed

Composite fractional time evolutions of cumulants in fractional generalized birth processes

“…Fractional generalizations of models with long memory in complex systems have been studied extensively [1][2][3][4][5][6][7][8][9][10][11]. There are various approaches for the fractional generalizations:…”

“…(1) A naive method is that the normal time derivative is replaced by a fractional derivative: in this respect (a) fractional Master equation (Saichev and Zaslavsky [5]; Laskin [6], Orsingher and Polito, [7][8], Konno [9]), (b) fractional Fokker-Planck equation (Barkai [10]) and (c) fractional Langevin equation (Franosch et al [11]) were studied. (2) When a normal spatial derivative is replaced by a spatial fractional derivative, one obtains a fractional diffusion equation (Lévy diffusion) [3] (3) When normal spatial and temporal derivatives are replaced by fractional derivatives, one obtains fractional spatial-temporal derivative diffusion equation (Zaslavsky [5]) and various fractional transport equations.…”

Randomly Stopped Nonlinear Fractional Birth Processes

“…is the Mittag-Leffler function and λ k , k ≥ 1, are the birth rates (see Orsingher and Polito [15]). For ν = 1, we retrieve from (1.1) and (1.3) the classical distributions of nonlinear and linear pure birth process, by taking into account that E 1,1 (x) = e x .…”

“…. , T n t , are independent first-passage times and 15) where B j (t), t > 0, 1 ≤ j ≤ n, are independent Brownian motions. In particular, for ν = 1 we show that the state probabilities 16) satisfy the 2 n th order equations…”

Discussion on the paper “On simulation and properties of the stable law” by L. Devroye and L. James

“…One can think for example at simulation of stochastic processes, generation of pseudo random numbers, cryptography, Monte Carlo and MCMC techniques, and so forth. A simple example in which generation of random deviates that are functions of stable random variables is needed, and which can make evident the important of the topic, regards the simulation of trajectories of time-fractional point processes such as the fractional Poisson process [2,10,11] or the fractional Yule process (fractional pure birth process) [13] (or in general of renewal processes with inter-arrival times distribution related to the stable law). The fractional Poisson process for example can be indeed constructed by exploiting its renewal structure.…”