Fractional pure birth processes

Details Fractional pure birth processes Fractional pure birth processes

2010-08-12

arxiv.org/abs/1008.2145v2

Bernoulli 2010, Vol. 16, No. 3, 858-881

Mathematics

Probability

Published in at http://dx.doi.org/10.3150/09-BEJ235 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statisti

Scientific paper

10.3150/09-BEJ235

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.

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