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

TL;DR

This paper studies the long-term behavior of solutions to a generalized Smoluchowski coagulation equation linked to critical continuous-state branching processes, establishing conditions for their asymptotic self-similarity and describing their scaling limits.

Contribution

It characterizes the existence of nondegenerate scaling limits of solutions based on the regular variation of the branching mechanism, introducing generalized Mittag-Leffler series for description.

Findings

Scaling limits exist if and only if the branching mechanism is regularly varying at zero.

Nondegenerate limits are characterized by generalized Mittag-Leffler series.

The results connect coagulation dynamics with branching process theory.

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.