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

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.