On a fractional linear birth–death process

In this paper we introduce and examine a fractional linear birth-death process $N_\nu \left( t \right)$, $t>0$, whose fractionality is obtained by replacing the time-derivative with a fractional derivative in the system of difference-differential equations governing the state probabilities $p_k^\nu \left( t \right)$, $t>0$, $k \geq 0$. We present a subordination relationship connecting $N_\nu \left( t \right)$, $t>0$ with the classical birth-death process $N \left( t \right)$, $t>0$ by means of the time process $T_{2 \nu} \left( t \right)$, $t>0$ whose distribution is related to a time-fractional diffusion equation. We obtain explicit formulas for the extinction probability $p_0^\nu \left( t \right)$, and the state probabilities $p_k^\nu \left( t \right)$, $t>0$, $k \geq 1$ in the three relevant cases $\lambda > \mu$, $\lambda < \mu$, $\lambda = \mu$ (where $\lambda$ and $\mu$ are respectively the birth and the 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 $\mathbb{E} N_\nu \left( t \right)$ and $\mathbb{V}\!\text{ar}N_\nu \left( t \right)$ are derived and analysed.