On a fractional linear birth–death process

Orsingher, Enzo ; Polito, Federico

Bernoulli, Tome 17 (2011) no. 1, p. 114-137 / Harvested from Project Euclid

Résumé

In this paper, we introduce and examine a fractional linear birth–death process N_ν(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^ν(t), t>0, k≥0. We present a subordination relationship connecting N_ν(t), t>0, with the classical birth–death process N(t), t>0, by means of the time process T_2ν(t), t>0, whose distribution is related to a time-fractional diffusion equation. ¶ We obtain explicit formulas for the extinction probability p₀^ν(t) and the state probabilities p_k^ν(t), t>0, k≥1, in the three relevant cases λ>μ, λ<μ, λ=μ (where λ and μ 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.

Détail

Publié le : 2011-02-15
Classification:  extinction probabilities,  fractional derivatives,  fractional diffusion equations,  generalized birth–death process,  iterated Brownian motion,  Mittag–Leffler functions