The Annals of Applied Probability

Limit theorems for Smoluchowski dynamics associated with critical continuous-state branching processes

Gautam Iyer, Nicholas Leger, and Robert L. Pego

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We investigate the well-posedness and asymptotic self-similarity of solutions to a generalized Smoluchowski coagulation equation recently introduced by Bertoin and Le Gall in the context of continuous-state branching theory. In particular, this equation governs the evolution of the Lévy measure of a critical continuous-state branching process which becomes extinct (i.e., is absorbed at zero) almost surely. We show that a nondegenerate scaling limit of the Lévy measure (and the process) exists if and only if the branching mechanism is regularly varying at 0. When the branching mechanism is regularly varying, we characterize nondegenerate scaling limits of arbitrary finite-measure solutions in terms of generalized Mittag–Leffler series.

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Ann. Appl. Probab., Volume 25, Number 2 (2015), 675-713.

First available in Project Euclid: 19 February 2015

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Zentralblatt MATH identifier

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60G18: Self-similar processes 35Q70: PDEs in connection with mechanics of particles and systems 82C28: Dynamic renormalization group methods [See also 81T17]

Continuous-state branching process critical branching limit theorem scaling limit Smoluchowski equation coagulation self-similar solution Mittag–Leffler series regular variation Bernstein function


Iyer, Gautam; Leger, Nicholas; Pego, Robert L. Limit theorems for Smoluchowski dynamics associated with critical continuous-state branching processes. Ann. Appl. Probab. 25 (2015), no. 2, 675--713. doi:10.1214/14-AAP1008.

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