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Coagulation and universal scaling limits for critical Galton–Watson processes

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

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.
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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 

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