Bernoulli

  • Bernoulli
  • Volume 16, Number 3 (2010), 858-881.

Fractional pure birth processes

Enzo Orsingher and Federico Polito

Full-text: Open access

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 difference-differential equations which govern the probability law of the process with the Dzherbashyan–Caputo fractional derivative. We derive the probability distribution of the number $\mathcal{N}_{\nu}(t)$ 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 $\mathcal{N}_{\nu}(t)=\mathcal{N}(T_{2\nu}(t))$, where $\mathcal{N}(t)$ is the classical generalized birth process and T2ν(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.

Article information

Source
Bernoulli Volume 16, Number 3 (2010), 858-881.

Dates
First available in Project Euclid: 6 August 2010

Permanent link to this document
http://projecteuclid.org/euclid.bj/1281099887

Digital Object Identifier
doi:10.3150/09-BEJ235

Mathematical Reviews number (MathSciNet)
MR2730651

Zentralblatt MATH identifier
05945288

Keywords
Airy functions branching processes Dzherbashyan–Caputo fractional derivative iterated Brownian motion Mittag–Leffler functions nonlinear birth process stable processes Vandermonde determinants Yule–Furry process

Citation

Orsingher, Enzo; Polito, Federico. Fractional pure birth processes. Bernoulli 16 (2010), no. 3, 858--881. doi:10.3150/09-BEJ235. http://projecteuclid.org/euclid.bj/1281099887.


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