We consider a fractional version of the classical non-linear 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 $\mathcal{N}_\nu \left( t \right)$ 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 relationship $\mathcal{N}_\nu \left( t \right) = \mathcal{N} \left( T_{2 \nu} \left( t \right) \right)$ where $\mathcal{N} \left( t \right)$ is the classical generalised birth process and $T_{2 \nu} \left( t \right)$ is a random time whose distribution is related to the fractional diffusion equation. The fractional linear birth process is examined in detail in section \ref{sec-exp} and various forms of its distribution are given and discussed.

### Fractional Pure birth processes

#### Abstract

We consider a fractional version of the classical non-linear 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 $\mathcal{N}_\nu \left( t \right)$ 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 relationship $\mathcal{N}_\nu \left( t \right) = \mathcal{N} \left( T_{2 \nu} \left( t \right) \right)$ where $\mathcal{N} \left( t \right)$ is the classical generalised birth process and $T_{2 \nu} \left( t \right)$ is a random time whose distribution is related to the fractional diffusion equation. The fractional linear birth process is examined in detail in section \ref{sec-exp} and various forms of its distribution are given and discussed.
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2010
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http://arxiv.org/pdf/1008.2145.pdf
http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.bj/1281099887
Airy Functions; Branching Processes; Dzherbashyan-Caputo Fractional Derivative; Iterated Brownian Motion; Mittag-Leﬄer Functions; Non-Linear Birth Process; Stable Processes; Vandermonde Determinants; Yule-Furry Process
E. ORSINGHER;F. POLITO
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2318/93102