Electronic Communications in Probability

Approximation of a generalized continuous-state branching process with interaction

Ibrahima Dramé and Étienne Pardoux

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In this work, we consider a continuous–time branching process with interaction where the birth and death rates are non linear functions of the population size. We prove that after a proper renormalization our model converges to a generalized continuous state branching process solution of the SDE \[\begin{aligned} Z_t^x=& x + \int _{0}^{t} f(Z_r^x) dr + \sqrt{2c} \int _{0}^{t} \int _{0}^{Z_{r}^x }W(dr,du) + \int _{0}^{t}\int _{0}^{1}\int _{0}^{Z_{r^-}^x}z \ \overline{M} (ds, dz, du)\\ &+ \int _{0}^{t}\int _{1}^{\infty }\int _{0}^{Z_{r^-}^x}z \ M(ds, dz, du), \end{aligned} \] where $W$ is a space-time white noise on $(0,\infty )^2$ and $\overline{M} (ds, dz, du)= M(ds, dz, du)- ds \mu (dz) du$, with $M$ being a Poisson random measure on $(0,\infty )^3$ independent of $W,$ with mean measure $ds\mu (dz)du$, where $(1\wedge z^2)\mu (dz)$ is a finite measure on $(0, \infty )$.

Article information

Electron. Commun. Probab., Volume 23 (2018), paper no. 73, 14 pp.

Received: 25 May 2018
Accepted: 2 October 2018
First available in Project Euclid: 17 October 2018

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Zentralblatt MATH identifier

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60F17: Functional limit theorems; invariance principles 92D25: Population dynamics (general)

continuous-state branching processes interaction Galton-Watson processes tightness

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Dramé, Ibrahima; Pardoux, Étienne. Approximation of a generalized continuous-state branching process with interaction. Electron. Commun. Probab. 23 (2018), paper no. 73, 14 pp. doi:10.1214/18-ECP176. https://projecteuclid.org/euclid.ecp/1539763346

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