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# Fractional pure birth processes

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### 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.

### Most cited references7

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### Time-fractional telegraph equations and telegraph processes with brownian time

(2004)
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### Fractional diffusion equations and processes with randomly varying time

(2011)
In this paper the solutions $$u_{\nu}=u_{\nu}(x,t)$$ to fractional diffusion equations of order $$0<\nu \leq 2$$ are analyzed and interpreted as densities of the composition of various types of stochastic processes. For the fractional equations of order $$\nu =\frac{1}{2^n}$$, $$n\geq 1,$$ we show that the solutions $$u_{{1/2^n}}$$ correspond to the distribution of the $$n$$-times iterated Brownian motion. For these processes the distributions of the maximum and of the sojourn time are explicitly given. The case of fractional equations of order $$\nu =\frac{2}{3^n}$$, $$n\geq 1,$$ is also investigated and related to Brownian motion and processes with densities expressed in terms of Airy functions. In the general case we show that $$u_{\nu}$$ coincides with the distribution of Brownian motion with random time or of different processes with a Brownian time. The interplay between the solutions $$u_{\nu}$$ and stable distributions is also explored. Interesting cases involving the bilateral exponential distribution are obtained in the limit.
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### Fractional processes: from Poisson to branching one

,  ,   (2010)
Fractional generalizations of the Poisson process and branching Furry process are considered. The link between characteristics of the processes, fractional differential equations and Levy stable densities are discussed and used for construction of the Monte Carlo algorithm for simulation of random waiting times in fractional processes. Numerical calculations are performed and limit distributions of the normalized variable Z=N/ are found for both processes.
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### Author and article information

###### Journal
12 August 2010
2011-02-14
1008.2145
10.3150/09-BEJ235

IMS-BEJ-BEJ235
Bernoulli 2010, Vol. 16, No. 3, 858-881
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)
math.PR math.ST stat.TH
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