Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Continuous-state branching processes, extremal processes and super-individuals

Clément Foucart and Chunhua Ma

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


The long-term behavior of flows of continuous-state branching processes are characterized through subordinators and extremal processes. The extremal processes arise in the case of supercritical processes with infinite mean and of subcritical processes with infinite variation. The jumps of these extremal processes are interpreted as specific initial individuals whose progenies overwhelm the population. These individuals, which correspond to the records of a certain Poisson point process embedded in the flow, are called super-individuals. They radically increase the growth rate to $+\infty$ in the supercritical case, and slow down the rate of extinction in the subcritical one.


Les comportements en temps long des flots de processus de branchement en temps et espace continus sont caractérisés par des subordinateurs et des processus extrémaux. Les processus extrémaux apparaissent dans le cas des processus sur-critiques de moyenne infinie et des processus sous-critiques à variation infinie. Les sauts de ces processus extrémaux sont interprétés comme des individus initiaux spécifiques dont les descendances envahissent la population. Ces individus, qui correspondent aux instants de records d’un certain processus ponctuel de Poisson, sont appelés super-individus. Ils augmentent de façon radicale la vitesse de divergence dans le cas sur-critique et diminuent celle d’extinction dans le cas sous-critique.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 55, Number 2 (2019), 1061-1086.

Received: 21 November 2016
Revised: 5 December 2017
Accepted: 16 April 2018
First available in Project Euclid: 14 May 2019

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60G70: Extreme value theory; extremal processes 60G55: Point processes

Continuous-state branching process Subordinator Extremal process Infinite mean Infinite variation Super-exponential growth Grey martingale Non-linear renormalisation


Foucart, Clément; Ma, Chunhua. Continuous-state branching processes, extremal processes and super-individuals. Ann. Inst. H. Poincaré Probab. Statist. 55 (2019), no. 2, 1061--1086. doi:10.1214/18-AIHP909.

Export citation


  • [1] A. D. Barbour and H.-J. Schuh. Functional normalization for the branching process with infinite mean. J. Appl. Probab. 16 (3) (1979) 513–525.
  • [2] J. Berestycki, A. E. Kyprianou and A. Murillo-Salas. The prolific backbone for supercritical superprocesses. Stochastic Process. Appl. 121 (6) (2011) 1315–1331.
  • [3] J. Bertoin, J. Fontbona and S. Martínez. On prolific individuals in a supercritical continuous-state branching process. J. Appl. Probab. 45 (3) (2008) 714–726.
  • [4] J. Bertoin and J.-F. Le Gall. The Bolthausen–Sznitman coalescent and the genealogy of continuous-state branching processes. Probab. Theory Related Fields 117 (2) (2000) 249–266.
  • [5] N. H. Bingham. Continuous branching processes and spectral positivity. Stochastic Process. Appl. 4 (3) (1976) 217–242.
  • [6] A. Bovier and I. Kurkova. Gibbs measures of Derrida’s generalised random energy models and genealogies of Neveu’s continuous state branching process. Available at Accessed: 2016-06-30.
  • [7] H. Cohn and A. G. Pakes. A representation for the limiting random variable of a branching process with infinite mean and some related problems. J. Appl. Probab. 15 (2) (1978) 225–234.
  • [8] D. A. Dawson and Z. Li. Stochastic equations, flows and measure-valued processes. Ann. Probab. 40 (2) (2012) 813–857.
  • [9] T. Duquesne and C. Labbé. On the Eve property for CSBP. Electron. J. Probab. 19 (2014) paper no. 6, 31 pp.
  • [10] T. Duquesne and M. Winkel. Growth of Lévy trees. Probab. Theory Related Fields 139 (3) (2007) 313–371.
  • [11] M. Dwass. Extremal processes, II. Illinois J. Math. 10 (3) (1966) 381–391.
  • [12] K. Fleischmann and A. Sturm. A super-stable motion with infinite mean branching. Ann. Inst. Henri Poincaré Probab. Stat. 40 (5) (2004) 513–537.
  • [13] K. Fleischmann and V. Wachtel. Large scale localization of a spatial version of Neveu’s branching process. Stochastic Process. Appl. 116 (7) (2006) 983–1011.
  • [14] D. R. Grey. Asymptotic behaviour of continuous time, continuous state-space branching processes. J. Appl. Probab. 11 (1974) 669–677.
  • [15] D. R. Grey. Almost-sure convergence in Markov branching process with infinite mean. J. Appl. Probab. 14 (1977) 702–716.
  • [16] T. Huillet. Energy cascades as branching processes with emphasis on Neveu’s approach to Derrida’s random energy model. Adv. in Appl. Probab. 35 (2) (2003) 477–503.
  • [17] M. Jiřina. Stochastic branching processes with continuous state space. Czechoslovak Math. J. 8 (83) (1958) 292–313.
  • [18] A. E. Kyprianou. Fluctuations of Lévy Processes with Applications, 2nd edition. Universitext. Springer, Heidelberg, 2014.
  • [19] A. E. Kyprianou, J.-L. Pérez and Y.-X. Ren. The backbone decomposition for spatially dependent supercritical superprocesses. In Séminaire de Probabilités XLVI 33–59. Lecture Notes in Math. 2123. Springer, Cham, 2014.
  • [20] A. E. Kyprianou and Y.-X. Ren. Backbone decomposition for continuous-state branching processes with immigration. Statist. Probab. Lett. 82 (1) (2012) 139–144.
  • [21] C. Labbé. Genealogy of flows of continuous-state branching processes via flows of partitions and the Eve property. Ann. Inst. Henri Poincaré Probab. Stat. 50 (3) (2014) 732–769.
  • [22] J. Lamperti. Continuous state branching processes. Bull. Amer. Math. Soc. 73 (1967) 382–386.
  • [23] J. Lamperti. The limit of a sequence of branching processes. Z. Wahrsch. Verw. Gebiete 7 (1967) 271–288.
  • [24] Z. H. Li. Measure-Valued Branching Markov Processes. Springer, Heidelberg, 2011.
  • [25] Z. H. Li. Path-valued branching processes and nonlocal branching superprocesses. Ann. Probab. 42 (1) (2014) 41–79.
  • [26] J. Neveu. A continuous state branching process in relation with the GREM model of spin glass theory. Rapport interne (non imprimé) 267, Ecole Polytechnique, 1992.
  • [27] S. I. Resnick. Extreme Values, Regular Variation and Point Processes. Springer Series in Operations Research and Financial Engineering. Applied Probability 4. Springer, New York, 2008.
  • [28] S. I. Resnick and M. Rubinovitch. The structure of extremal processes. Adv. in Appl. Probab. 5 (1973) 287–307.