2010/08/01 by Enzo Orsingher, Federico Polito · 65 citations
Mathematics · Physics and Astronomy · #Advanced Mathematical Theories and Applications #Applied mathematics #Distribution (mathematics) #Fractional Differential Equations Solutions #Fractional calculus #Mathematical analysis #Mathematics #Pure mathematics #Statistical Mechanics and Entropy #Statistics #Stochastic process #math.PR #math.ST #stat.TH
paper · pdf · doi:10.3150/09-bej235
published in Bernoulli 16(3) (Chapman and Hall London) · 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)
openalex publication_date 2010/08/01 · arxiv created 2011/02/14 · arxiv updated 2014/03/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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 Nν(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 Nν(t)=N(T2ν(t)), where N(t) is the classical generalized birth process and T2ν(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.