Multidimensional branching Brownian motion extremal process conjecture

Let $d\geq 1$, let $\mathcal{N}_t$ be the particles at time $t$, and let $X_t(u)\in\mathbb{R}^d$ be the position of particle $u$. Define

$$r_t=\sqrt{2}t+\frac{d-4}{2\sqrt{2}}\log t$$

and, for $u\in\mathcal{N}_t$, define its direction by $D_t(u)=X_t(u)/\\|X_t(u)\\|$. Let $Z_\infty(\theta)$ be the limiting derivative-martingale measure density on $\mathbb{S}^{d-1}$, and let $\sigma$ denote surface measure on $\mathbb{S}^{d-1}$. Multidimensional extremal process conjecture. There exists $c_d^\star>0$ such that

$$\lim_{t\to\infty}\sum_{u\in\mathcal{N}_t}\delta_{(D_t(u),\\|X_t(u)\\|-r_t)}=\mathcal{L}(\mathrm{d}\theta,\mathrm{d}x)\quad\text{in law},$$

where $\mathcal{L}$ is a decorated Poisson point process: if $(\theta_j,\xi_j)_{j\geq 1}$ are the atoms of a Poisson point process with intensity

$$c_d^\star Z_\infty(\theta)\\,\sigma(\mathrm{d}\theta)e^{-\sqrt{2}x}\\,\mathrm{d}x,$$

and $(D_j)_{j\geq 1}$ are independent identically distributed point processes on $\mathbb{R}$ with common distribution $\mathcal{D}$, then

$$\mathcal{L}=\sum_{j\geq 1}\sum_{x\in D_j}\delta_{(\theta_j,\xi_j+x)}.$$

This conjecture predicts the full extremal point process of multidimensional branching Brownian motion and, in particular, the asymptotic behaviour of the derivative martingale. It is a multidimensional analogue of known descriptions for one-dimensional branching Brownian motion, while the convergence and the decorated Poisson structure in the multidimensional setting remain to be proved.