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We investigate the well-posedness and asymptotic self-similarity of solutionsto a generalized Smoluchowski coagulation equation recently introduced byBertoin and Le Gall in the context of continuous-state branching theory. Inparticular, this equation governs the evolution of the L\'{e}vy measure of acritical continuous-state branching process which becomes extinct (i.e., isabsorbed at zero) almost surely. We show that a nondegenerate scaling limit ofthe L\'{e}vy measure (and the process) exists if and only if the branchingmechanism is regularly varying at 0. When the branching mechanism is regularlyvarying, we characterize nondegenerate scaling limits of arbitraryfinite-measure solutions in terms of generalized Mittag-Leffler series.

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