• Bernoulli
  • Volume 17, Number 1 (2011), 114-137.

On a fractional linear birth–death process

Enzo Orsingher and Federico Polito

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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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Bernoulli, Volume 17, Number 1 (2011), 114-137.

First available in Project Euclid: 8 February 2011

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


Orsingher, Enzo; Polito, Federico. On a fractional linear birth–death process. Bernoulli 17 (2011), no. 1, 114--137. doi:10.3150/10-BEJ263.

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  • [1] Bailey, N.T.J. (1964). The Elements of Stochastic Processes with Applications to the Natural Sciences. New York: Wiley.
  • [2] Beghin, L. and Orsingher, E. (2009). Fractional Poisson processes and related planar random motions. Electron. J. Probab. 14 1790–1826.
  • [3] Cahoy, D.O. (2007). Fractional Poisson processes in terms of alpha-stable densities. Ph.D. thesis.
  • [4] Feller, W. (1968). An Introduction to Probability Theory and Its Applications, Volume 1, 3rd ed. New York: Wiley.
  • [5] Laskin, N. (2003). Fractional Poisson process. Commun. Nonlinear Sci. Numer. Simul. 8 201–213.
  • [6] Orsingher, E. and Beghin, L. (2004). Time-fractional telegraph equations and telegraph processes with Brownian time. Probab. Theory Related Fields 128 141–160.
  • [7] Orsingher, E. and Beghin, L. (2009). Fractional diffusion equations and processes with randomly-varying time. Ann. Probab. 37 206–249.
  • [8] Orsingher, E. and Polito, F. (2010). Fractional pure birth processes. Bernoulli 16 858–881.
  • [9] Podlubny, I. (1999). Fractional Differential Equations. San Diego: Academic Press.
  • [10] Uchaikin, V.V., Cahoy, D.O. and Sibatov, R.T. (2008). Fractional processes: From Poisson to branching one. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 18 2717–2725.
  • [11] Uchaikin, V.V. and Sibatov, R.T. (2008). A fractional Poisson process in a model of dispersive charge transport in semiconductors. Russian J. Numer. Anal. Math. Modelling 23 283–297.