On a fractional linear birth--death process

TL;DR

This paper introduces a fractional linear birth-death process by replacing the derivative with a fractional derivative, providing explicit formulas for probabilities, and connecting it to classical processes through subordination, with analysis of mean and variance.

Contribution

The paper develops a fractional birth-death process with explicit formulas and subordination relationships, extending classical models with fractional calculus techniques.

Findings

Explicit formulas for extinction and state probabilities are derived.

The process's behavior is analyzed for different birth and death rate scenarios.

Connections to fractional pure birth processes are established.

Abstract

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.