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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.

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          Time-fractional telegraph equations and telegraph processes with brownian time

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            Fractional diffusion equations and processes with randomly varying time

            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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              Poisson fractional processes

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                Author and article information

                Journal
                10.3150/09-BEJ235
                1008.2145

                Probability,Statistics theory
                Probability, Statistics theory

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