Abstract
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 L\'{e}vy 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 L\'{e}vy 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.
Highlights
Bertoin and Le Gall [3] observed a connection between the Smoluchowski coagulation equation and any critical continuousstate branching process that becomes extinct with probability one
Our general goal in this paper is to establish criteria for the existence of dynamic scaling limits in such branching processes, by extending methods that were recently used to analyze coagulation dynamics in the classically important ‘solvable’ cases
As shown in [3], the Levy measure of a critical CSBP which becomes extinct almost surely satisfies a generalized type of Smoluchowski coagulation equation
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 continuousstate branching theory. This equation governs the evolution of the Levy 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 Levy 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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