## Publication 130

*Coagulation and universal scaling limits for critical Galton-Watson
processes*

##### Authors:

##### 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 Lévy 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 Lévy 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 Lévy triples and a continuity theorem for Bernstein transforms (Laplace exponents). We develop these in a self-contained appendix.

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