Approximation of the height process of a continuous state branching process with interaction
Authors:
Ibrahima Dramé and Etienne Pardoux
Journal:
Theor. Probability and Math. Statist. 103 (2020), 3-39
MSC (2020):
Primary 60J80, 60J85; Secondary 60F17
DOI:
https://doi.org/10.1090/tpms/1133
Published electronically:
June 16, 2021
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Additional Information
Abstract: We first show that the properly rescaled height process of the genealogical tree of a continuous time branching process converges to the height process of the genealogy of a (possibly discontinuous) continuous state branching process. We then prove the same type of result for generalized branching processes with interaction.
References
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References
- K. B. Athreya and P. E. Ney, Branching processes, Springer-Verlag, New York, 1972. MR 0373040
- V. Bansaye, M. E. Caballero and S. Méléard, Scaling limits of population and evolution processes in random environment, Electron. J. Probab. 24 (2019), Paper No. 19. MR 3925459
- J. Berestycki, M. C. Fittipaldi and J. Fontbona, Ray–Knight representation of flows of branching processes with competition by pruning of Lévy trees, Probab. Theory Related Fields 172 (2018), no. 3–4, 725–788. MR 3877546
- J. Bertoin, Lévy processes, Cambridge University Press, 1996. MR 1406564
- P. Billingsley, Convergence of Probability Measures, 2nd ed., John Wiley, New York, 1999. MR 1700749
- P. Brémaud, Point processes and queues: martingale dynamics, Springer, 1981.
- E. Çinlar, Probability and Stochastics, Graduate Texts in Mathematics, vol. 261, Springer Science & Business Media, 2011. MR 2767184
- D. A. Dawson and Z. Li, Stochastic equations, flows and measure-valued processes, Ann. Probab. 40 (2012), no. 2, 813–857. MR 2952093
- J.-F. Delmas, Height process for super-critical continuous state branching process, Markov Process. Relat. Fields 14 (2008), 309–326. MR 2437534
- I. Dramé, E. Pardoux and A. B. Sow, Non-binary branching process and non-Markovian exploration process, ESAIM Prob. Stat. 21 (2017), 1–33. MR 3630601
- I. Dramé and E. Pardoux, Approximation of a generalized continuous-state branching process with interaction, Electron. Commun. Probab. 23 (2018). MR 3866046
- T. Duquesne and J.-F. Le Gall, Random trees, Lévy processes and spatial branching processes, Asrérisque, vol. 281, Société mathématique de France, 2002. MR 1954248
- Z. Fu and Z. Li, Stochastic equations of non-negative processes with jumps, Stoch. Proc. Appl. 120 (2010), no. 3, 306–330. MR 2584896
- D. Grey, Asymptotic behaviour of continuous time, continuous state-space branching processes, J. Appl. Probab. 11 (1974), 669–677. MR 408016
- A. Grimvall, On the convergence of sequences of branching processes, Ann. Probab. 2 (1974), 1027–1045. MR 362529
- M. Jiřina, Stochastic branching processes with continuous state space, Czechoslovak Mathematical Journal 8 (1958), no. 2, 292–313.
- A. Joffe and M. Métivier, Weak convergence of sequences of semimartingales with applications to multitype branching processes, Adv. Appl. Prob. 18 (1986), 20–65. MR 827331
- O. Kallenberg, Foundations of Modern Probability, 2nd ed., Springer 2002. MR 4226142
- J. Lamperti, Continuous-state branching processes, Bull. Amer. Math. Soc. 73 (1967), no. 3, 382–386. MR 208685
- J. Lamperti, The limit of a sequence of branching processes, Probab. Theory Related Fields 7 (1967), no. 4, 271–288. MR 217893
- J.-F. Le Gall, Spatial branching processes, random snakes, and partial differential equations, Springer Science & Business Media, 1999. MR 1714707
- J.-F. Le Gall and Y. Le Jan, Branching processes in Lévy processes: the exploration process, Ann. Probab. 26 (1998), 213–252. MR 1617047
- V. Le, E. Pardoux and A. Wakolbinger, “Trees under attack”: a Ray–Knight representation of Feller’s branching diffusion with logistic growth, Probab. Theory and Rel. Fields 155 (2013), no. 3–4, 583–619. MR 3034788
- P. S. Li, A continuous-state polynomial branching process, Stoch. Proc. Appl. 129 (2019), 2941–2967. MR 3980150
- P. S. Li, X. Yang and X. Zhou, A general continuous-state nonlinear branching process, Ann. Appl. Probab. 29 (2019), 2523–2555. MR 3983343
- Z. Li, Measure-valued branching Markov processes, Springer, Berlin, 2011. MR 2760602
- Z. Li, Continuous-state branching processes, arXiv preprint arXiv:1202.3223, 2012.
- Z. Li, E. Pardoux and A. Wakolbinger, The Height process of a general CSBP with interaction, J. Theor. Probab. (2020), https://doi.org/10.1007/s10959-020-01054-5.
- E. Pardoux, Probabilistic models of population evolution scaling limits and interactions, Springer, 2015. MR 3496029
- E. Pardoux and A. Rascanu, Stochastic Differential Equations, Backward SDEs, Partial Differential Equations, Springer, 2014. MR 3308895
- P. Protter, Stochastic integration and differential equations, Springer, 2004 MR 2020294
- D. Revuz and M. Yor, Continuous martingales and Brownian motion, 3rd ed., Springer Verlag, New York, 1999. MR 1725357
- M. L. Silverstein, A new approach to local times (local time existence and property determination using real valued strong Markov process basis), Journal of Mathematics and Mechanics 17 (1968), 1023–1054. MR 0226734
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Additional Information
Ibrahima Dramé
Affiliation:
Université Cheikh Anta Diop de Dakar, FST, LMA, 16180 Dakar-Fann, Sénégal
Email:
iboudrame87@gmail.com
Etienne Pardoux
Affiliation:
Aix-Marseille Univ, CNRS, Centrale Marseille, I2M, Marseille, France
Email:
etienne.pardoux@univ-amu.fr
Keywords:
Continuous-state branching processes,
scaling limit,
Galton-Watson processes,
Lévy processes,
local time,
height process
Received by editor(s):
December 1, 2019
Published electronically:
June 16, 2021
Article copyright:
© Copyright 2020
Taras Shevchenko National University of Kyiv
