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 differencedifferential equations which govern the probability law of the process with the Dzherbashyan Caputo fractional derivative. We derive the probability distribution of the number N(v)(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 N(v)(t) = N(T(2v) (t)), where N(t) is the classical generalized birth process and T(2v) (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.

Fractional pure birth processes / Orsingher, Enzo; Polito, Federico. - In: BERNOULLI. - ISSN 1350-7265. - 16:3(2010), pp. 858-881. [10.3150/09-bej235]

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 differencedifferential equations which govern the probability law of the process with the Dzherbashyan Caputo fractional derivative. We derive the probability distribution of the number N(v)(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 N(v)(t) = N(T(2v) (t)), where N(t) is the classical generalized birth process and T(2v) (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.
Scheda breve Scheda completa
2010
nonlinear birth process; vandermonde determinants; stable processes; iterated brownian motion; dzherbashyan-caputo fractional derivative; branching processes; yule-furry process; airy functions; yule furry process; mittag-leffler functions
01 Pubblicazione su rivista::01a Articolo in rivista
Fractional pure birth processes / Orsingher, Enzo; Polito, Federico. - In: BERNOULLI. - ISSN 1350-7265. - 16:3(2010), pp. 858-881. [10.3150/09-bej235]
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11573/360206`
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