Electronic Communications in Probability

Approximation of a generalized continuous-state branching process with interaction

Ibrahima Dramé and Étienne Pardoux

Full-text: Open access

Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1539763346

Digital Object Identifier
doi:10.1214/18-ECP176

Mathematical Reviews number (MathSciNet)
MR3866046

Zentralblatt MATH identifier
1401.60159

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

Keywords
continuous-state branching processes interaction Galton-Watson processes tightness

Rights
Creative Commons Attribution 4.0 International License.

Citation

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


Export citation

References

  • [1] Aldous, D.: Stopping times and tightness. Ann. Probability, 6, (1978), 335–340.
  • [2] Athreya, K. B. and Ney, P. E.: Branching processes. Grundlehren der Mathematischen Wissenschaften, 196, Springer-Verlag, New York, 1972. xi+287 pp.
  • [3] Ba, M. and Pardoux, E.: Branching process with competition and a generalized Ray Knight Theorem. Ann. Inst. Henri Poincaré Probab. Stat., 51, (2015), 1290–1313.
  • [4] Billingsley, P.: Convergence of Probability Measures, 2d ed. John Wiley, New York, 1999. x+277 pp.
  • [5] Dawson, D. A. and Li, Z.: Stochastic equations, flows and measure-valued processes. Ann. Probability, 40, (2012), 813–857.
  • [6] Dramé, I., Pardoux, E., and Sow, A. B.: Non-binary branching process and non-markovian exploration process. ESAIM Probab. Stat. 21, (2017), 1–33.
  • [7] Duquesne T. and Le Gall J.-F.: Random trees, Lévy processes and spatial branching processes, Astérisque 281, Société mathématique de France, 2002. vi+147 pp.
  • [8] Grey, D.: Asymptotic behaviour of continuous time, continuous state-space branching processes. J. Appl. Probability 11, (1974), 669–677.
  • [9] Grimvall, A.: On the convergence of sequences of branching processes. Ann. Probability 2, (1974), 1027–1045.
  • [10] Jacod, J. and Shiryaev, A. N.: Limit theorems for stochastic processes, 2d ed. Grundlehren der Mathematischen Wissenschaften, 288, Springer-Verlag, 2003. xx+661 pp.
  • [11] Li, Z.: Measure-valued branching Markov processes. Springer, Berlin, 2011. xii+350 pp.
  • [12] Pardoux, E.: Probabilistic models of population evolution. Scaling limits and interactions. Mathematical Biosciences Institute Lecture Notes series, Stochastics in Biological Systems, 1.6. Springer, 2016. viii+125 pp.
  • [13] Silverstein, M. L.: A new approach to local times (local time existence and property determination using real valued strong markov process basis). J. Math. Mech., 17, (1968), 1023–1054.