- Volume 16, Number 3 (2010), 858-881.
Fractional pure birth processes
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 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 , where 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.
Bernoulli Volume 16, Number 3 (2010), 858-881.
First available in Project Euclid: 6 August 2010
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Airy functions branching processes Dzherbashyan–Caputo fractional derivative iterated Brownian motion Mittag–Leffler functions nonlinear birth process stable processes Vandermonde determinants Yule–Furry process
Orsingher, Enzo; Polito, Federico. Fractional pure birth processes. Bernoulli 16 (2010), no. 3, 858--881. doi:10.3150/09-BEJ235. https://projecteuclid.org/euclid.bj/1281099887.