Fractional pure birth processes

TL;DR

This paper introduces a fractional nonlinear birth process by replacing the standard derivative with a fractional derivative, deriving its probability distribution, and exploring its relation to classical processes through subordination.

Contribution

It develops a fractional version of the nonlinear birth process, providing explicit probability distributions and a subordination representation linking it to classical processes.

Findings

Derived the probability distribution of the fractional birth process.

Established a subordination relation with classical birth processes.

Analyzed the fractional linear birth process in detail.

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.