Randomly Stopped Nonlinear Fractional Birth Processes

We present and analyse the nonlinear classical pure birth process $\text{mathpzc}N(t)$, $t>0$, and the fractional pure birth process $\text{mathpzc}{N}^{\nu }(t)$, $t>0$, subordinated to various random times, namely the first-passage time $T_t$ of the standard Brownian motion $B(t)$, $t>0$, the $\alpha$-stable subordinator $\text{mathpzc}{S}^{\alpha }(t)$, $\alpha \in (0,1)$, and others. For all of them we derive the state probability distribution $\hat{p}k (t)$,$k \geq 1$ and, in some cases, we also present the corresponding governing differential equation. We also highlight interesting interpretations for both the subordinated classical birth process $\widehat{\text{mathpzc}N}(t)$, $t>0$, and its fractional counterpart ${\widehat{\text{mathpzc}N}}^{\nu }(t)$, $t>0$ 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 $\text{mathpzc}{N}^{\nu }(t)$ have been examined in the last part of the paper. In particular, the processes $\text{mathpzc}{N}^{\nu }(T_t)$, $\text{mathpzc}{N}^{\nu }(\text{mathpzc}{S}^{\alpha }(t))$, $\text{mathpzc}{N}^{\nu }(T_{2\nu }(t))$, have been analysed, where $T_{2\nu }(t)$, $t>0$, is a process related to fractional diffusion equations. Also the related process $\text{mathpzc}N(\text{mathpzc}{S}^{\alpha }(T_{2\nu }(t)))$ is investigated and compared with $\mathpzc{N}(T{2\nu}(\mathpzc{S}^\alpha(t))) = \mathpzc{N}^\nu (\mathpzc{S}^\alpha(t))$. As a byproduct of our analysis, some formulae relating Mittag–Leffler functions are obtained.