Randomly Stopped Nonlinear Fractional Birth Processes

الأصل · EN

We present and analyse the nonlinear classical pure birth process N (t), t>0, and the fractional pure birth process Nν(t), t>0, subordinated to various random times, namely the first-passage time Tₜ of the standard Brownian motion B(t), t>0, the α-stable subordinator Sα(t), α∈ (0,1), and others. For all of them we derive the state probability distribution pₖ (t), k ≥ 1 and, in some cases, we also present the corresponding governing differential equation. We also highlight interesting interpretations for both the subordinated classical birth process N (t), t>0, and its fractional counterpart Nν(t), t>0 in terms of classical birth processes with random rates evaluated on a stretched or squashed time scale. Various types of compositions of the fractional pure birth process Nν(t) have been examined in the last part of the paper. In particular, the processes Nν(Tₜ), Nν(Sα(t)), Nν(T₂ν(t)), have been analysed, where T₂ν(t), t>0, is a process related to fractional diffusion equations. Also the related process N(Sα(T₂ν(t))) is investigated and compared with N(T₂ν(Sα(t))) = Nν(Sα(t)). As a byproduct of our analysis, some formulae relating Mittag--Leffler functions are obtained.