Open Access
February 2011 On a fractional linear birth–death process
Enzo Orsingher, Federico Polito
Bernoulli 17(1): 114-137 (February 2011). DOI: 10.3150/10-BEJ263


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 $p_0^ν(t)$ 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.


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Enzo Orsingher. Federico Polito. "On a fractional linear birth–death process." Bernoulli 17 (1) 114 - 137, February 2011.


Published: February 2011
First available in Project Euclid: 8 February 2011

zbMATH: 1284.60157
MathSciNet: MR2797984
Digital Object Identifier: 10.3150/10-BEJ263

Keywords: extinction probabilities , fractional derivatives , fractional diffusion equations , generalized birth–death process , iterated Brownian motion , Mittag–Leffler functions

Rights: Copyright © 2011 Bernoulli Society for Mathematical Statistics and Probability

Vol.17 • No. 1 • February 2011
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