Hostname: page-component-8448b6f56d-wq2xx Total loading time: 0 Render date: 2024-04-19T23:18:33.073Z Has data issue: false hasContentIssue false
  • English
  • Français

Coagulation and universal scaling limits for critical Galton–Watson processes

Part of: Low-dimensional dynamical systems Markov processes Equations of mathematical physics and other areas of application

Published online by Cambridge University Press:  26 July 2018

Gautam Iyer ,
Nicholas Leger  and
Robert L. Pego
Gautam Iyer*
Affiliation:
Carnegie Mellon University
Nicholas Leger*
Affiliation:
University of Houston
Robert L. Pego*
Affiliation:
Carnegie Mellon University
*
* Postal address: Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, USA.
*** Postal address: Department of Mathematics, University of Houston, 4800 Calhoun Rd., Houston, TX 77004, USA. Email address: nleger@math.uh.edu
* Postal address: Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

The basis of this paper is the elementary observation that the n-step descendant distribution of any Galton–Watson process satisfies a discrete Smoluchowski coagulation equation with multiple coalescence. Using this we obtain simple necessary and sufficient criteria for the convergence of scaling limits of critical Galton–Watson processes in terms of scaled family-size distributions and a natural notion of convergence of Lévy triples. Our results provide a clear and natural interpretation, and an alternate proof, of the fact that the Lévy jump measure of certain continuous-state branching processes (CSBPs) satisfies a generalized Smoluchowski equation. (This result was previously proved by Bertoin and Le Gall (2006).) Our analysis shows that the nonlinear scaling dynamics of CSBPs become linear and purely dilatational when expressed in terms of the Lévy triple associated with the branching mechanism. We prove a continuity theorem for CSBPs in terms of the associated Lévy triples, and use our scaling analysis to prove the existence of universal critical Galton–Watson processes and CSBPs analogous to Doeblin's `universal laws'. Namely, these universal processes generate all possible critical and subcritical CSBPs as subsequential scaling limits. Our convergence results rely on a natural topology for Lévy triples and a continuity theorem for Bernstein transforms (Laplace exponents) which we develop in a self-contained appendix.

Keywords

Smoluchowski equationmultiple coalescenceGalton–Watson processCSBPuniversal lawBernstein transform

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 37E20: Universality, renormalization 35Q70: PDEs in connection with mechanics of particles and systems
Type
Original Article
Information
Advances in Applied Probability , Volume 50 , Issue 2 , June 2018 , pp. 504 - 542
Copyright
Copyright © Applied Probability Trust 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Aldous, D. J. (1999). Deterministic and stochastic models for coalescence (aggregation and coagulation): a review of the mean-field theory for probabilists. Bernoulli 5, 3–48. Google Scholar
[2]Athreya, K. B. and Ney, P. E. (2004). Branching Processes. Dover, Mineola, NY. Google Scholar
[3]Bacaër, N. (2011). A Short History of Mathematical Population Dynamics. Springer, London. Google Scholar
[4]Bansaye, V. and Simatos, F. (2015). On the scaling limits of Galton-Watson processes in varying environments. Electron. J. Prob. 20, 75. Google Scholar
[5]Berestycki, N. (2009). Recent Progress in Coalescent Theory (Math. Surveys 16). Sociedade Brasileira de Matemática, Rio de Janeiro. Google Scholar
[6]Berestycki, J., Berestycki, N. and Limic, V. (2014). A small-time coupling between λ-coalescents and branching processes. Ann. Appl. Prob. 24, 449–475. Google Scholar
[7]Berestycki, J., Berestycki, N. and Schweinsberg, J. (2008). Small-time behavior of beta coalescents. Ann. Inst. H. Poincaré Prob. Statist. 44, 214–238. Google Scholar
[8]Bertoin, J. (2000). Subordinators, Lévy processes with no negative jumps, and branching processes. Preprint. Université Pierre et Marie Curie. Google Scholar
[9]Bertoin, J. (2006). Random Fragmentation and Coagulation Processes (Camb. Studies Adv. Math. 102). Cambridge University Press. Google Scholar
[10]Bertoin, J. and Le Gall, J.-F. (2000). The Bolthausen-Sznitman coalescent and the genealogy of continuous-state branching processes. Prob. Theory Relat. Fields 117, 249–266. Google Scholar
[11]Bertoin, J. and Le Gall, J.-F. (2003). Stochastic flows associated to coalescent processes. Prob. Theory Relat. Fields 126, 261–288. Google Scholar
[12]Bertoin, J. and Le Gall, J.-F. (2005). Stochastic flows associated to coalescent processes. II. Stochastic differential equations. Ann. Inst. H. Poincaré Prob. Statist. 41, 307–333. Google Scholar
[13]Bertoin, J. and Le Gall, J.-F. (2006). Stochastic flows associated to coalescent processes. III. Limit theorems. Illinois J. Math. 50, 147–181. Google Scholar
[14]Birkner, M.et al. (2005). Alpha-stable branching and beta-coalescents. Electron. J. Prob. 10, 303–325. Google Scholar
[15]Caballero, M. E., Lambert, A. and Uribe Bravo, G. (2009). Proof(s) of the Lamperti representation of continuous-state branching processes. Prob. Surveys 6, 62–89. Google Scholar
[16]Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. II, 2nd edn. John Wiley, New York. Google Scholar
[17]Gnedin, A., Iksanov, A. and Marynych, A. (2014). λ-coalescents: a survey. In Celebrating 50 Years of The Applied Probability Trust (J. Appl. Prob. 51A), pp. 23–40. Google Scholar
[18]Grey, D. R. (1974). Asymptotic behaviour of continuous time, continuous state-space branching processes. J. Appl. Prob. 11, 669–677. Google Scholar
[19]Grimvall, A. (1974). On the convergence of sequences of branching processes. Ann. Prob. 2, 1027–1045. Google Scholar
[20]Grosjean, N. and Huillet, T. (2016). On a coalescence process and its branching genealogy. J. Appl. Prob. 53, 1156–1165. Google Scholar
[21]Heyde, C. C. and Seneta, E. (1977). I. J. Bienaymé: Statistical Theory Anticipated. Springer, New York. Google Scholar
[22]Iyer, G., Leger, N. and Pego, R. L. (2015). Limit theorems for Smoluchowski dynamics associated with critical continuous-state branching processes. Ann. Appl. Prob. 25, 675–713. Google Scholar
[23]Kingman, J. F. C. (1982). The coalescent. Stoch. Process. Appl. 13, 235–248. Google Scholar
[24]Kingman, J. F. C. (1982). On the genealogy of large populations. In Essays in Statistical Science (J. Appl. Prob. 19A), pp. 27–43. Google Scholar
[25]Kyprianou, A. E. (2014). Fluctuations of Lévy Processes with Applications, 2nd edn. Springer, Heidelberg. Google Scholar
[26]Lambert, A. (2003). Coalescence times for the branching process. Adv. Appl. Prob. 35, 1071–1089. Google Scholar
[27]Lambert, A. (2007). Quasi-stationary distributions and the continuous-state branching process conditioned to be never extinct. Electron. J. Prob. 12, 420–446. Google Scholar
[28]Lamperti, J. (1967). Continuous state branching processes. Bull. Amer. Math. Soc. 73, 382–386. Google Scholar
[29]Lamperti, J. (1967). The limit of a sequence of branching processes. Z. Wahrscheinlichkeitsth. 7, 271–288. Google Scholar
[30]Laurençot, P. and van Roessel, H. (2015). Absence of gelation and self-similar behavior for a coagulation-fragmentation equation. SIAM J. Math. Anal. 47, 2355–2374. Google Scholar
[31]Li, Z.-H. (2000). Asymptotic behaviour of continuous time and state branching processes. J. Austral. Math. Soc. Ser. A 68, 68–84. Google Scholar
[32]Menon, G. and Pego, R. L. (2004). Approach to self-similarity in Smoluchowski's coagulation equations. Commun. Pure Appl. Math. 57, 1197–1232. Google Scholar
[33]Menon, G. and Pego, R. L. (2008). The scaling attractor and ultimate dynamics for Smoluchowski's coagulation equations. J. Nonlinear Sci. 18, 143–190. Google Scholar
[34]Pakes, A. G. (2008). Conditional limit theorems for continuous time and state branching process. In Records and Branching Processes, Nova, New York, pp. 63–103. Google Scholar
[35]Pakes, A. G. (2010). Critical Markov branching process limit theorems allowing infinite variance. Adv. Appl. Prob. 42, 460–488. Google Scholar
[36]Pitman, J. (1999). Coalescents with multiple collisions. Ann. Prob. 27, 1870–1902. Google Scholar
[37]Sagitov, S. (1999). The general coalescent with asynchronous mergers of ancestral lines. J. Appl. Prob. 36, 1116–1125. Google Scholar
[38]Schilling, R. L., Song, R. and Vondraček, Z. (2010). Bernstein Functions (De Gruyter Studies Math. 37). De Gruyter, Berlin. Google Scholar
[39]Schweinsberg, J. (2003). Coalescent processes obtained from supercritical Galton-Watson processes. Stoch. Process. Appl. 106, 107–139. Google Scholar
[40]Vatutin, V. A. and Zubkov, A. M. (1985). Branching processes. I. In Probability Theory: Mathematical Statistics: Theoretical Cybernetics, Akad. Nauk SSSR, Moscow, pp. 3–67. Google Scholar
[41]Vatutin, V. A. and Zubkov, A. M. (1993). Branching processes. II. J. Soviet Math. 67, 3407–3485. Google Scholar
[42]Von Smoluchowski, M. (1916). Drei Vorträge über Diffusion, Brownsche Bewegung und Koagulation von Kolloidteilchen. Physik. Z. 17, 557–585. Google Scholar
[43]Von Smoluchowski, M. (1918). Experiments on a mathematical theory of kinetic coagulation of coloid solutions. Z. Physikalische Chem. Stoch. Verwandtschaftslehre 92, 129–168. Google Scholar
[44]Watson, H. W. and Galton, F. (1875). On the probability of the extinction of families. J. Anthropological Inst. Great Britain Ireland 4, 138–144. Google Scholar