The extremal point process of branching Brownian motion in \(\mathbb{R}^d\)

We consider a branching Brownian motion in \(\mathbb{R}^d\) with \(d \geq 1\) in which the position \(X_t^{(u)}\in \mathbb{R}^d\) of a particle \(u\) at time \(t\) can be encoded by its direction \(\theta^{(u)}_t \in \mathbb{S}^{d-1}\) and its distance \(R^{(u)}_t\) to 0. We prove that the {\it extremal point process} \(\sum \delta_{\theta^{(u)}_t, R^{(u)}_t - m_t^{(d)}}\) (where the sum is over all particles alive at time \(t\) and \(m^{(d)}_t\) is an explicit centring term) converges in distribution to a randomly shifted decorated Poisson point process on \(\mathbb{S}^{d-1} \times \mathbb{R}\). More precisely, the so-called {\it clan-leaders} form a Cox process with intensity proportional to \(D_\infty(\theta) e^{-\sqrt{2}r} ~\mathrm{d} r ~\mathrm{d} \theta \), where \(D_\infty(\theta)\) is the limit of the derivative martingale in direction \(\theta\) and the decorations are i.i.d. copies of the decoration process of the standard one-dimensional branching Brownian motion. This proves a conjecture of Stasiński, Berestycki and Mallein (Ann. Inst. H. Poincar\'{e} 57:1786--1810, 2021), and builds on that paper and on Kim, Lubetzky and Zeitouni (arXiv:2104.07698).