Distribution of particles near the front in supercritical branching Brownian motion with compactly supported branching

TL;DR

This paper studies the distribution of particles near the front in supercritical branching Brownian motion with compact support, showing convergence of normalized particle counts and dependence on domain location.

Contribution

It introduces sharp asymptotics for PDE solutions to analyze particle distribution near the front, revealing new dependence on domain location.

Findings

Normalized particle counts converge in distribution and moments near the front.

Limiting distribution depends on the asymptotic position of the domain.

Results extend understanding of particle behavior in supercritical branching processes.

Abstract

We investigate the long-time behavior of a $d-$dimensional supercritical branching Brownian motion with a compactly supported branching potential. It is known that, for $\mathbf{v}\in \mathbb{R}^d$, all the moments of the normalized number of particles in a bounded domain centered at $\mathbf{v} t$ converge, as $t \rightarrow \infty$, provided that $\|\mathbf{v}\|$ is strictly less than the asymptotic speed of the front. The limiting distribution does not depend on $\mathbf{v}$. Using sharp asymptotics for the solutions of parabolic PDEs with compact potential, we prove that the normalized number of particles in a bounded time-dependent domain located near the front converges in distribution and with all the moments. The limit, however, now depends on the asymptotic location of the domain.