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Construction of age-structured branching processes by stochastic equations

Part of: Stochastic analysis Markov processes

Published online by Cambridge University Press:  31 May 2022

Lina Ji Open the ORCID record for Lina Ji [Opens in a new window]  and
Zenghu Li
Lina Ji*
Affiliation:
Shenzhen MSU-BIT University
Zenghu Li*
Affiliation:
Beijing Normal University
*
*Postal address: Faculty of Computational Mathematics and Cybernetics, Shenzhen MSU-BIT University, Shenzhen 518172, People’s Republic of China. Email: jiln@smbu.edu.cn
**Postal address: Laboratory of Mathematics and Complex Systems, School of Mathematical Sciences, Beijing Normal University, Beijing 100875, People’s Republic of China. Email: lizh@bnu.edu.cn
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Abstract

We provide constructions of age-structured branching processes without or with immigration as pathwise-unique solutions to stochastic integral equations. A necessary and sufficient condition for the ergodicity of the model with immigration is also given.

Keywords

Distribution-function-valued processergodicitymeasure-valuednon-localimmigration

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60J85: Applications of branching processes 60H15: Stochastic partial differential equations
Type
Original Article
Information
Journal of Applied Probability , Volume 59 , Issue 3 , September 2022 , pp. 670 - 684
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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References

Athreya, K. and Ney, P. (1972). Branching Processes. Springer, Berlin.10.1007/978-3-642-65371-1CrossRefGoogle Scholar
Bansaye, V. and Méléard, S. (2015). Stochastic Models for Structured Populations: Scaling Limits and Long Time Behavior. Springer, Cham.Google Scholar
Bellman, R. and Harris, T. (1952). On age-dependent binary branching processes. Ann. Math. 55, 280–295.10.2307/1969779CrossRefGoogle Scholar
Bose, A. (1986). A law of large numbers for the scaled age distribution of linear birth-and-death processes. Canad. J. Statist. 14, 233–244.10.2307/3314800CrossRefGoogle Scholar
Bose, A. and Kaj, I. (1995). Diffusion approximation for an age-structured population. Ann. Appl. Prob. 5, 140–157.10.1214/aoap/1177004833CrossRefGoogle Scholar
Bose, A. and Kaj, I. (2000). A scaling limit process for the age-reproduction structure in a Markov population. Markov Process. Relat. Fields 6, 397–428.Google Scholar
Champagnat, N., Ferrière, R. and Méléard, S. (2006). Individual-based probabilistic models of adpatative evolution and various scaling approximations. In Seminar on Stochastic Analysis, Random Fields and Applications V (Progress in Prob. 59, eds R. C. Dalang, F. Russo and M. Dozzi. Birkhäuser, Basel.Google Scholar
Crump, K. S. and Mode, C. J. (1968). A general age-dependent branching process I. J. Math. Anal. Appl. 24, 494–508.10.1016/0022-247X(68)90005-XCrossRefGoogle Scholar
Crump, K. S. and Mode, C. J. (1968). A general age-dependent branching process II. J. Math. Anal. Appl. 25, 8–17.10.1016/0022-247X(69)90210-8CrossRefGoogle Scholar
Dawson, D. A., Gorostiza, L. G. and Li, Z. (2002). Nonlocal branching superprocesses and some related models. Acta Appl. Math. 74, 93–112.10.1023/A:1020507922973CrossRefGoogle Scholar
Doney, R. A. (1972). Age-dependent birth and death processes. Z. Wahrscheinlichkeitsth. 22, 69–90.10.1007/BF00538906CrossRefGoogle Scholar
Etheridge, A. M. (2000). An Introduction to Superprocesses. American Mathematical Society, Providence, RI.10.1090/ulect/020CrossRefGoogle Scholar
Fournier, N. and Méléard, S. (2004). A microscopic probabilistic description of a locally regulated population and macroscopic approximations. Ann. Appl. Prob. 14, 1880–1919.10.1214/105051604000000882CrossRefGoogle Scholar
Harris, T. E. (1963). The Theory of Branching Processes. Springer, Berlin.10.1007/978-3-642-51866-9CrossRefGoogle Scholar
Jagers, P. (1969). A general stochastic model for population development. Skand. Aktuar. Tidskr. 52, 84–103.Google Scholar
Jagers, P. (1975). Branching Processes with Biological Applications. Wiley, New York.Google Scholar
Jagers, P. and Klebaner, F. C. (2000). Population-size-dependent and age-dependent branching processes. Stoch. Process. Appl. 87, 235–254.10.1016/S0304-4149(99)00111-8CrossRefGoogle Scholar
Jagers, P. and Klebaner, F. C. (2011). Population-size-dependent, age-structured branching processes linger around their carrying capacity. J. Appl. Prob. 48A, 249–260.10.1239/jap/1318940469CrossRefGoogle Scholar
Kaj, I. and Sagitov, S. (1998). Limit processes for age-dependent branching particle systems. J. Theoret. Prob. 11, 225–257.10.1023/A:1021607311191CrossRefGoogle Scholar
Kendall, D. G. (1949). Stochastic processes and population growth. J. R. Statist. Soc. B 11, 230–264.Google Scholar
Li, Z. (2011). Measure-Valued Branching Markov Processes. Springer, Heidelberg.10.1007/978-3-642-15004-3CrossRefGoogle Scholar
Li, Z. (2020). Continuous-state branching processes with immigration. In From Probability to Finance, ed. Y. Jiao. Springer, Singapore.10.1007/978-981-15-1576-7_1CrossRefGoogle Scholar
Li, Z. (2021): Ergodicities and exponential ergodicities of Dawson–Watanabe type processes. Teor. Veroyatnost. i Primenen. 66, 342–368.10.4213/tvp5341CrossRefGoogle Scholar
Metz, J. A. J. and Tran, V. C. (2013). Daphnias: From the individual based model to the large population equation. J. Math. Biol. 66, 915–933.10.1007/s00285-012-0619-5CrossRefGoogle Scholar
Oelschläger, K. (1990). Limit theorems for age-structured populations. Ann. Prob. 18, 290–318.10.1214/aop/1176990950CrossRefGoogle Scholar
Pardoux, E. (2016). Probabilistic Models of Population Evolution: Scaling Limits, Genealogies and Interactions. Springer, Switzerland.10.1007/978-3-319-30328-4CrossRefGoogle Scholar
Pazy, A. (1983). Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York.10.1007/978-1-4612-5561-1CrossRefGoogle Scholar
Tran, V. C. (2008). Large population limit and time behaviour of a stochastic particle model describing an age-structured population. ESAIM Prob. Stat. 12, 345–386.10.1051/ps:2007052CrossRefGoogle Scholar