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April 2015 Limit theorems for Smoluchowski dynamics associated with critical continuous-state branching processes
Gautam Iyer, Nicholas Leger, Robert L. Pego
Ann. Appl. Probab. 25(2): 675-713 (April 2015). DOI: 10.1214/14-AAP1008


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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Gautam Iyer. Nicholas Leger. Robert L. Pego. "Limit theorems for Smoluchowski dynamics associated with critical continuous-state branching processes." Ann. Appl. Probab. 25 (2) 675 - 713, April 2015.


Published: April 2015
First available in Project Euclid: 19 February 2015

zbMATH: 1312.60100
MathSciNet: MR3313753
Digital Object Identifier: 10.1214/14-AAP1008

Primary: 60J80
Secondary: 35Q70 , 60G18 , 82C28

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

Rights: Copyright © 2015 Institute of Mathematical Statistics


Vol.25 • No. 2 • April 2015
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