On a fractional linear birth--death process

In this paper, we introduce and examine a fractional linear birth--death process $N_{\nu}(t)$, $t>0$, whose fractionality is obtained by replacing the time derivative with a fractional derivative in the system of difference-differential equations governing the state probabilities $p_k^{\nu}(t)$, $t>0$, $k\geq0$. We present a subordination relationship connecting $N_{\nu}(t)$, $t>0$, with the classical birth--death process $N(t)$, $t>0$, by means of the time process $T_{2\nu}(t)$, $t>0$, whose distribution is related to a time-fractional diffusion equation. We obtain explicit formulas for the extinction probability $p_0^{\nu}(t)$ and the state probabilities $p_k^{\nu}(t)$, $t>0$, $k\geq1$, in the three relevant cases $\lambda>\mu$, $\lambda<\mu$, $\lambda=\mu$ (where $\lambda$ and $\mu$ are, respectively, the birth and death rates) and discuss their behaviour in specific situations. We highlight the connection of the fractional linear birth--death process with the fractional pure birth process. Finally, the mean values $\mathbb{E}N_{\nu}(t)$ and $\operatorname {\mathbb{V}ar}N_{\nu}(t)$ are derived and analyzed.