Spectral analysis and $k$-spine decomposition of inhomogeneous branching Brownian motions. Genealogies in fully pushed fronts

TL;DR

This paper analyzes a space-dependent branching Brownian motion model for invasive populations, establishing genealogical convergence and spectral properties in the fully pushed front regime.

Contribution

It introduces a spectral and genealogical analysis of inhomogeneous branching Brownian motions, focusing on the fully pushed regime and proving convergence to a Brownian Coalescent Point Process.

Findings

Genealogies converge to a Brownian Coalescent Point Process.

Spectral decomposition determines the invariant measure and mixing time.

Yaglom law established for the critical branching process.

Abstract

We consider a system of particles performing a one-dimensional dyadic branching Brownian motion with space-dependent branching rate, negative drift $-\mu$ and killed upon reaching $0$. More precisely, the particles branch at rate $r(x)=(1+W(x))/2,$ where $W$ is a compactly supported and non-negative smooth function and the drift $\mu$ is chosen in such a way that the system is critical in some sense. This particle system can be seen as an analytically tractable model for fluctuating fronts, describing the internal mechanisms driving the invasion of a habitat by a cooperating population. Recent studies from Birzu, Hallatschek and Korolev suggest the existence of three classes of fluctuating fronts: pulled, semi pushed and fully pushed fronts. Here, we focus on the fully pushed regime. We establish a Yaglom law for this branching process and prove that the genealogy of the particles converges to a Brownian Coalescent Point Process using a method of moments. In practice, the genealogy of the BBM is seen as a random marked metric measure space and we use spinal decomposition to prove its convergence in the Gromov-weak topology. We also carry the spectral decomposition of a differential operator related to the BBM to determine the invariant measure of the spine as well as its mixing time.