The shape of the front of multidimensional branching Brownian motion

TL;DR

This work analyzes the frontier of multidimensional branching Brownian motion in $\mathbb{R}^d$ ($d\ge2$) by first detailing the extremal landscape around near-maximal particles and then establishing a sharp 3/2-scaling limit for the front as a random surface generated from a $\text{Bessel}(3)$ process. The authors introduce the extremal landscape $\mathcal{E}_{t,\ell}^{\mathrm{land}}$, proving its convergence to a decorated PPP $\mathcal{E}_{\infty}^{\mathrm{land}}$ with law $\nu$, which encodes the full local geometry around extremal particles, including transversal directions. They then translate this landscape into a precise description of the front around the maximal particle: the front converges to a deterministic surface $\rho(s,\theta)$ given by a Fenchel-Legendre transform of a $\text{Bessel}(3)$-driven process, leading to a surface of the form revolved about the radial axis. The analysis combines spine decomposition, last-exit-time controls, and KPP/Gartner-type arguments to relate the geometric front to a variational problem, thereby extending the one-dimensional decoding of extremes to the richer multidimensional setting. Overall, the results deepen the understanding of how extremal particles shape the frontier and illuminate the transverse structure invisible to the standard extremal process alone.

Abstract

We study the shape of the outer envelope of a branching Brownian motion (BBM) in $\mathbb{R}^d$, $d\geq 2$. We focus on the extremal particles: those whose norm is within $O(1)$ of the maximal norm amongst the particles alive at time $t$. Our main result is a scaling limit, with exponent $3/2$, for the outer-envelope of the BBM around each extremal particle (the "front"); the scaling limit is a continuous random surface given explicitly in terms of a Bessel(3) process. Towards this end, we introduce a point process that captures the full landscape around each extremal particle and show convergence in distribution to an explicit point process. This complements the global description of the extremal process given in Berestycki et. al. (Ann. Probab. 52 (2024), no. 3, 955-982), where the local behavior at directions transversal to the radial component of the extremal particles is not addressed.

Figures (4)

• Figure 1: (Top-left) A simulation of 2D BBM run until time $t= 8.5$, with trajectories of particles colored according to norm, $u^*$ marked in red, and particles closest to $u^*$ circled in red. (Top-right) Roughly, the front around $u^*$ in the top-left simulation. (Bottom) A simulation of the front run until $t=225$. In blue, the particles constituting the point process $\mathcal{C}_t$. In solid orange, the deterministic curves $y=\pm 2|x|^{3/2}$. In red, the front $\mathfrak{h}_{\mathcal{C}_t}^{\epsilon}$ defined with intervals of size $0.3$ (i.e., $p_i^{(1)} \in (-s, -s+0.3]$ in \ref{['def:front']}) for a better rendition.
• Figure 2: An artistic rendition of the extremal cluster and the front in $d=2$. Drawn are only the particles up to distance $L$ (in the direction $\mathbf{e}_1$) behind the extremal particle (at the origin). The front for $\theta=+1$ is drawn in red, for $\theta=-1$ in blue. The dashed lines correspond to $y=\pm x^{3/2}$. The particles in the strip $x\in (-L,-L+1]$ used to define the front at $t=1$ are shaded darker than other particles. Note the vertical lines used to define the height of the front at $t=1$.
• Figure 3: (Left) A simulation of $\rho(s)$ for $s\in [0,200]$. (Right) The paraboloid formed by revolving $\rho(s)$ around an axis (the axis was not taken to be $\mathbf{e}_1$ for better resolution): this is a simulation of the scaling limit of the front $(\rho(s,\theta))_{s\in[0,200], \theta\in\mathbb{S}^{1}}$ of 3D BBM, rotated.
• Figure 4: A simulation of a point process approximating the extremal cluster $\mathcal{C}$ in dimension $2$. The trajectory of the spine is approximated by $S_s := (-\sqrt2s-R_s, Y_s)$, where $R_s$ is a Bessel(3) process and $Y_s$ an independent standard Brownian motion (black). See \ref{['rk:spine-bessel']} for an explanation of why $A_s \approx -\sqrt2s-R_s$. The branching times are approximated by a Poisson point process of rate $2$, the location of the spine at each branching time is marked with a uniquely-colored "$\times$", and the associated BBM point cloud is plotted in the same color. The simulation contains 25 BBM point clouds.

Theorems & Definitions (75)

• Definition 1.1: Front of a point process
• Definition 1.2: Front of the BBM, extremal cluster
• Theorem 1
• Remark 1.3
• Remark 1.4
• Remark 1.5
• Definition 1.6
• Theorem 2
• Corollary 3
• Remark 1.7