Fractional Pure Birth Processes
نویسندگان
چکیده
We consider a fractional version of the classical non-linear birth process of which the YuleFurry 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 relationship Nν (t) = N (T2ν (t)) where N (t) is the classical generalised 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.
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