Coagulation and universal scaling limits for critical Galton-Watson processes

Abstract

The basis of this paper is the elementary observation that the $n$-step descendant distribution of any Galton-Watson process satisfies a discrete Smoluchowski coagulation equation with multiple coalescence. Using this we obtain necessary and sufficient criteria for the convergence of scaling limits of Galton-Watson processes that are simpler (and equivalent) to the classical criteria obtained by Grimvall in 1974. Our results provide a clear and natural interpretation, and an alternate proof, of the fact that the L\'evy jump measure of certain CSBPs satisfies a generalized Smoluchowski equation. (This result was previously proved by Bertoin and Le Gall in 2006.) Moreover, our analysis shows that the nonlinear scaling dynamics of CSBPs becomes linear and purely dilatational when expressed in terms of the L\'evy triple associated with the branching mechanism. We use this to prove existence of universal critical Galton-Watson and CSBPs analogous to W. Doeblin's "universal laws". Namely, these universal processes generate all possible critical and subcritical CSBPs as subsequential scaling limits. Our convergence results rely on a natural topology for L\'evy triples and a continuity theorem for Bernstein transforms (Laplace exponents). We develop these in a self-contained appendix.