A shape theorem for periodic branching diffusion

Abstract

We prove a shape theorem for a multidimensional periodic branching diffusion, which is a branching particle system where particle trajectories evolve according to a diffusion, drift, branching rate and offspring distribution which all depend periodically on space. More precisely, we show that almost surely as $t\to\infty$, the particle cloud at time $t$, normalized by $t$, converges in Hausdorff distance to a deterministic convex set $\mathcal{W}$. We further establish that the boundary of the asymptotic shape $\mathcal{W}$ is of class $C^2$, strictly convex, and has positive Gaussian curvature bounded away from zero. This work generalizes the approach of arXiv:2507.10515 for $g$-BBM and furthers our understanding of the shape $\mathcal{W}$ in that setting.