Hyperbolic branching Brownian motion: the empirical limit measure

Abstract

We study branching Brownian motion in hyperbolic space. As hyperbolic Brownian motion is transient, the normalised empirical measure of branching Brownian motion converges to a random measure ${\mu }_{\infty }$ on the boundary. We show that the Hausdorff dimension of ${\mu }_{\infty }$ is $(2\beta )\wedge 1$ where $\beta$ is the branching rate, and that ${\mu }_{\infty }$ admits a Lebesgue density for $\beta >1/2$. This is very different to the behaviour of the set of accumulation points on the boundary where ${\beta }_c=1/8$ which has been shown by Lalley and Sellke \cite{lalley_hyperbolic_1997}. This answers several questions posed by Woess \cite{woess_notes_2024} and similar questions posed by Candellero and Hutchcroft \cite{candellero_boundary_2023}. We believe that our methods also apply to branching random walks on non-elementary hyperbolic groups.