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 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 $T_{2\nu }(t)$ 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.

Published in at http://dx.doi.org/10.3150/09-BEJ235 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

References in corpus (2)

• Fractional diffusion equations and processes with randomly varying time
• Fractional processes: from Poisson to branching one

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