Convergence of spatial branching processes to $α$-stable CSBPs: Genealogy of semi-pushed fronts

Abstract

We consider an inhomogeneous branching diffusion on an unbounded domain of $\mathbb{R}^d$ and propose a simple condition under which we expect the size process (i.e., the number of particles) and the genealogy of the system to converge to those of an $α$-stable continuous-state branching process, with $α\in(1,2)$. This condition can be seen as the spatial analogue of the classical assumption that the tail of the offspring distribution of a Galton--Watson process is regularly varying. We make a first step towards establishing this result by providing a set of sufficient conditions under which the branching diffusion, seen as a random marked metric measure space that captures both the positions and the genealogical structure of the population, converges to an $α$-stable genealogy. These conditions are based on the convergence of the moments of the process, which can be efficiently computed via recursive formulas. We apply this framework to a one-dimensional branching Brownian motion with inhomogeneous branching rate and negative drift. This model was introduced by Tourniaire as a toy model to investigate the internal dynamics of fluctuating pushed fronts. By using our general set of conditions we prove convergence of the genealogy of the process in the semipushed regime, which was conjectured to hold by Birzu, Hallatschek, and Korolev.