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Extinction and coming down from infinity of continuous-state branching processes with competition in a Lévy environment

Part of: Markov processes

Published online by Cambridge University Press:  25 February 2021

H. Leman  and
J. C. Pardo
H. Leman*
Affiliation:
Université de Lyon, Inria, CNRS, ENS de Lyon, UMPA
J. C. Pardo*
Affiliation:
Centro de Investigación en Matemáticas A.C.
*
*Postal address: Université de Lyon, Inria, CNRS, ENS de Lyon, UMPA UMR 5669, 46 allée d’Italie, 69364 Lyon, France.
**Postal address: Centro de Investigación en Matemáticas A.C. Calle Jalisco s/n. 36240 Guanajuato, México. Email address: [email protected]
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Abstract

We are interested in the property of coming down from infinity of continuous-state branching processes with competition in a Lévy environment. We first study the event of extinction for such a family of processes under Grey’s condition. Moreover, if we add an integrability condition on the competition mechanism then the process comes down from infinity regardless of the long-time behaviour of the environment.

Keywords

Continuous-state branching processes in random environmentcompetitionextinctioncoming down from infinity

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60J75: Jump processes
Secondary: 60J85: Applications of branching processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 58 , Issue 1 , March 2021 , pp. 128 - 139
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

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References

Afanasyev, V. I., Böinghoff, C., Kersting, G. and Vatutin, V. A. (2012). Limit theorems for weakly subcritical branching processes in random environment. J. Theoret. Prob., 25, 703–732.CrossRefGoogle Scholar
Ba, M. and Pardoux, E. (2015). Branching processes with interaction and a generalized Ray–Knight theorem. Ann. Inst. H. Poincaré Prob. Statist., 51, 1290–1313.CrossRefGoogle Scholar
Bansaye, V. (2019). Approximation of stochastic processes by non-expansive flows and coming down from infinity. Ann. Appl. Prob., 29, 2374–2438.CrossRefGoogle Scholar
Bansaye, V., Caballero, M. E., and Méléard, S. (2019). Scaling limits of population and evolution processes in random environment. Electron. J. Prob. 24, 19.CrossRefGoogle Scholar
Bansaye, V., Pardo, J. C., and Smadi, C. (2019). Extinction rate of continuous state branching processes in critical Lévy environments. Preprint, arXiv:1903.06058.Google Scholar
Bansaye, V. and Simatos, F. (2015). On the scaling limits of Galton–Watson processes in varying environments. Electron. J. Prob., 20, 75.CrossRefGoogle Scholar
Berestycki, J., Fittipaldi, M. C. and Fontbona, J. (2018). Ray–Knight representation of flows of branching processes with competition by pruning of Lévy trees. Prob. Theory Relat. Fields, 172, 725–788.CrossRefGoogle Scholar
Böinghoff, C., Dyakonova, E. E., Kersting, G. and Vatutin, V. A. (2010). Branching processes in random environment which extinct at a given moment. Markov Process. Relat. Fields, 16, 329–350.Google Scholar
Dawson, D. A. and Li, Z. (2012). Stochastic equations, flows and measure-valued processes. Ann. Prob. 40, 813–857.CrossRefGoogle Scholar
El Karoui, N. and Méléard, S. (1990). Martingale measures and stochastic calculus. Prob. Theory Relat. Fields, 84, 83–101.CrossRefGoogle Scholar
Foucart, C. (2019). Continuous-state branching processes with competition: duality and reflection at infinity. Electron. J. Prob., 24, 33.Google Scholar
González-Casanova, A., Pardo, J. C. and Perez., J. L. (2017). Branching processes with interactions: the subcritical cooperative regime. Preprint, arXiv:1704.04203.Google Scholar
He, H., Li, Z. and Xu, W. (2018). Continuous-state branching processes in Lévy random environments. J. Theoret. Prob., 31, 1952–1974.CrossRefGoogle Scholar
Ikeda, N. and Watanabe, S. (1989). Stochastic Differential Equations and Diffusion Processes. North-Holland, Amsterdam.Google Scholar
Kallenberg, O. (1997). Foundations of Modern Probability. Springer, New York.Google Scholar
Keiding, N. (1975). Extinction and exponential growth in random environments. Theoret. Pop. Biol., 8, 49–63.CrossRefGoogle ScholarPubMed
Kurtz, T. (1978). Diffusion approximations for branching processes. In Advances in Probability and Related Topics, Vol. 5, ed. P. Ney. Dekker, New York, pp. 269–292.Google Scholar
Lambert, A. (2005). The branching process with logistic growth. Ann. Appl. Prob., 15, 1506–1535.CrossRefGoogle Scholar
Le, V. (2014). Processus de branchement avec interaction. Doctoral Thesis, Université Aix-Marseille.Google Scholar
Le, V. and Pardoux, E. (2015). Height and the total mass of the forest of genealogical trees of a large population with general competition. ESAIM: Probab. Stat., 19, 172–193.Google Scholar
Leman, H. and Pardo, J. C. (2019). Extinction time of the logistic branching processes in a Brownian environment. Preprint, arXiv:1906.01395.Google Scholar
Li, P. S. (2019). A continuous-state polynomial branching process. Stoch. Process. Appl., 129, 2941–2967.CrossRefGoogle Scholar
Li, P. S., Yang, Z. and Zhou, X. (2019). A general continuous-state nonlinear branching process. Ann. Appl. Prob. 29, 2523–2555.CrossRefGoogle Scholar
Ma, R. (2015). Lamperti transformation for continuous-state branching processes with competition and applications. Statist. Prob. Lett. 107, 11–17.CrossRefGoogle Scholar
Palau, S. and Pardo, J. C. (2018). Branching processes in a Lévy random environment. Acta Appl. Math. 153, 55–79.CrossRefGoogle Scholar
Pardoux, E. (2016). Probabilistic Models of Population Evolution. Scaling Limits, Genealogies and Interactions. Springer, New York.CrossRefGoogle Scholar
Smith, W. L. and Wilkinson, W. E. (1969). On branching processes in random environments. Ann. Math. Statist. 40, 814–827.CrossRefGoogle Scholar