周期分支扩散的形状定理
A shape theorem for periodic branching diffusion
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中文总结 AI 辅助
该研究证明了多维周期分支扩散的形状定理，确定粒子云归一化后收敛的渐近凸集性质，并推广了arXiv:2507.10515的方法以深化对该形状的理解。
中文摘要 AI 辅助
我们证明了多维周期分支扩散的形状定理，该模型是一种分支粒子系统，其中粒子轨迹根据扩散、漂移、分支率和后代分布演化，所有这些均随空间呈周期性变化。更确切地说，我们证明几乎必然当t→∞时，时刻t的粒子云经t归一化后，在豪斯多夫距离下收敛到确定性凸集𝒲。我们进一步证明渐近形状𝒲的边界属于C²类、严格凸，且具有远离零的正高斯曲率。本工作推广了arXiv:2507.10515中针对g-BBM的方法，深化了我们对该情形下形状𝒲的理解。
英文摘要
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.
