Self-similar growth-fragmentations as scaling limits of Markov branching processes

TL;DR

This paper establishes explicit conditions under which Markov branching processes converge, when scaled, to self-similar growth-fragmentation processes with negative index, including their genealogical structures as real trees.

Contribution

It provides the first explicit criteria linking the transition kernel of a Markov branching process to its scaling limit as a self-similar growth-fragmentation.

Findings

Derived explicit conditions for convergence to growth-fragmentation

Established scaling limits for genealogical structures as real trees

Extended results to processes with negative index

Abstract

We provide explicit conditions, in terms of the transition kernel of its driving particle, for a Markov branching process to admit a scaling limit toward a self-similar growth-fragmentation with negative index. We also derive a scaling limit for the genealogical embedding considered as a compact real tree.