REMAINING-LIFETIME AGE-STRUCTURED BRANCHING PROCESSES

doi:10.1007/s10473-025-0319-z

Abstract

Abstract: We study age-structured branching models with reproduction law depending on the remaining lifetime of the parent. The lifespan of an individual is determined at its birth and its remaining lifetime decreases at the unit speed. The models, without or with immigration, are constructed as measure-valued processes by pathwise unique solutions of stochastic equations driven by time-space Poisson random measures. In the subcritical branching case, we give a sufficient condition for the ergodicity of the process with immigration. Two large number laws and a central limit theorem of the occupation times are proved.

Key words: branching process, remaining lifetime, immigration, stochastic equation, ergodicity, occupation time, large number law, central limit theorem

CLC Number:

60J80

Ziling CHENG, Zenghu LI. REMAINING-LIFETIME AGE-STRUCTURED BRANCHING PROCESSES[J].Acta mathematica scientia,Series B, 2025, 45(3): 1107-1136.

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References

[1] Bellman R, Harris T. On age-dependent binary branching processes. Ann Math, 1952, 55: 280-295
[2] Bienaymé I J. De la loi de multiplication et de la durée des families. Soc Philomat Paris Extraits, 1845, 5: 37-39
[3] Bose A. A law of large numbers for the scaled age distribution of linear birth-and-death processes. Canad J Statist, 1986, 14: 233-244
[4] Bose A, Kaj I. Diffusion approximation for an age-structured population. Ann Appl Probab, 1995, 5: 140-157
[5] Bose A, Kaj I. A scaling limit process for the age-reproduction structure in a Markov population. Markov Process Relat Fields, 2000, 6: 397-428
[6] Champagnat N, Ferrière R, Méléard S.Individual-based probabilistic models of adpatative evolution and various scaling approximations//Dalang R C, Russo F, Dozzi M. Seminar on Stochastic Analysis, Random Fields and Applications V. Basel: Birkhäuser, 2005: 75-113
[7] Crump K, Mode C J. A general age-dependent branching process, I. J Math Anal Appl, 1968, 24: 494-508
[8] Crump K, Mode C J. A general age-dependent branching process, II. J Math Anal Appl, 1969, 25: 8-17
[9] Dawson D A, Gorostiza L G, Wakolbinger A. Occupation time fluctuations in branching systems. J Theoret Probab, 2001, 14(3): 729-796
[10] Dawson D A, Gorostiza L G, Li Z. Nonlocal branching superprocesses and some related models. Acta Appl Math, 2002, 74: 93-112
[11] Dawson D A, Li Z. Skew convolution semigroups and affine Markov processes. Ann Probab, 2006, 34: 1103-1142
[12] Dawson D A, Li Z. Stochastic equations, flows and measure-valued processes. Ann Probab, 2012, 40: 813-857
[13] Doney R A. Age-dependent birth and death processes. Z Wahrscheinlichkeitsth, 1972, 22: 69-90
[14] Doney R A. A limit theorem for a class of supercritical branching processes. J Appl Probab, 1972, 9: 707-724
[15] Fan J Y, Hamza K, Jagers P, Klebaner F C. Convergence of the age structure of general schemes of population processes. Bernoulli, 2020, 26(2): 893-926
[16] Fournier N, Méléard S.A microscopic probabilistic description of a locally regulated population and macroscopic approximations. Ann Appl Probab, 2004, 14: 1880-1919
[17] Hamza K, Jagers P, Klebaner F C. The age structure of population-dependent general branching processes in environments with a high carrying capacity. Proc Steklov Inst Math, 2013, 282(1): 90-105
[18] He H, Li Z, Yang X. Stochastic equations of super-Lévy processes with general branching mechanism. Stochastic Process Appl, 2014, 124: 1519-1565
[19] Hong W M. Longtime behavior for the occupation time processes of a super-Brownian motion with random immigration. Stoch Process Appl, 2002, 102(1): 43-62
[20] Iscoe I. A weighted occupation time for a class of measure-valued branching processes. Probab Th Rel Fields, 1986, 71: 85-116
[21] Iscoe I. Ergodic theory and a local occupation time for measure-valued critical branching Brownian motion. Stochastics, 1986, 18: 197-243
[22] Jagers P. A general stochastic model for population development. Skand Aktuar Tidskr, 1969, 52: 84-103
[23] Jagers P.Branching Processes with Biological Application. New York: Wiley, 1975
[24] Jagers P. General branching processes as Markov fields. Stoch Process Appl, 1989, 32: 183-212
[25] Jagers P, Klebaner F C. Population-size-dependent and age-dependent branching processes. Stoch Process Appl, 2000, 87: 235-254
[26] Jagers P, Klebaner F C. Population-size-dependent, age-structured branching processes linger around their carrying capacity. J Appl Probab, 2011, 48A: 249-260
[27] Ji L, Li Z. Construction of age-structured branching processes by stochastic equations. J Appl Probab, 2022, 59: 670-684
[28] Kaj I, Sagitov S. Limit processes for age-dependent branching particle systems. J Theoret Probab, 1998, 11: 225-257
[29] Kendall D G. Stochastic processes and population growth. J R Statist Soc B, 1949, 11: 230-264
[30] Li Z. Some central limit theorems for super Brownian motion. Acta Math Sci, 1999, 19B: 121-126
[31] Li Z.Continuous-state branching processes with immigration//Jiao Y, (eds). From Probability to Finance. Singapore: Springer, 2020: 1-69
[32] Li Z.Measure-Valued Branching Markov Processes. Heidelberg: Springer, 2022
[33] Metz J A J, Tran V C. Daphnias: From the individual based model to the large population equation. J Math Biol, 2013, 66: 915-933
[34] Oelschläger K. Limit theorems for age-structured populations. Ann Probab, 1990, 18: 290-318
[35] Pardoux E.Probabilistic Models of Population Evolution: Scaling Limits, Genealogies and Interactions. Switzerland: Springer, 2016
[36] Situ R.Theory of Stochastic Differential Equations with Jumps and Applications. Berlin: Springer, 2005
[37] Tang J. Limit theorems for the weighted occupation time for super-Brownian motions on $\mathrm{H}^d$. Progress in Natural Science, 2006, 16(8): 803-807
[38] Tran V C. Large population limit and time behaviour of a stochastic particle model describing an age-structured population. ESAIM Probab Stat, 2008, 12: 345-386
[39] Watson H W, Galton F. On the probability of the extinction of families. J Anthropol Inst Great Britain and Ireland, 1874, 4: 138-144
[40] Xiong J. Super-Brownian motion as the unique strong solution to an SPDE. Ann Probab, 2013, 41: 1030-1054