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Iyer, Gautam; Leger, Nicholas; Pego, Robert L.

Limit theorems for Smoluchowski dynamics associated with critical continuous-state branching processes. (English) Zbl 1312.60100

Ann. Appl. Probab. 25, No. 2, 675-713 (2015).
Summary: 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.

Cited in 6 Documents

MSC:

60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60G18 Self-similar stochastic processes
35Q70 PDEs in connection with mechanics of particles and systems of particles
82C28 Dynamic renormalization group methods applied to problems in time-dependent statistical mechanics

Keywords:

continuous-state branching process; critical branching; scaling limit; Smoluchowski equation; coagulation; self-similar solution; Mittag-Leffler series; regular variation; Bernstein function
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\textit{G. Iyer} et al., Ann. Appl. Probab. 25, No. 2, 675--713 (2015; Zbl 1312.60100)
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