A shape theorem for BBM in a periodic environment

TL;DR

This paper proves that binary branching Brownian motion in a periodic environment converges to a deterministic convex shape over time, with the shape's speed and form characterized by probabilistic tail bounds and reaction-diffusion equations.

Contribution

It establishes a shape theorem for BBM in periodic environments, linking probabilistic tail bounds with reaction-diffusion front speeds.

Findings

Almost sure convergence to a convex shape

Asymptotic propagation speed in all directions

Connection between BBM and Fisher-KPP front speeds

Abstract

We consider the long-time behaviour of binary branching Brownian motion (BBM) where the branching rate depends on a periodic spatial heterogeneity. We prove that almost surely as $t\to\infty$, the heterogeneous BBM at time $t$, normalized by $t$, approaches a deterministic convex shape with respect to Hausdorff distance. Our approach relies on establishing tail bounds on the probability of existence of BBM particles lying in half-spaces, which in particular yields the asymptotic speed of propagation of projections of the BBM in every direction. Our arguments are primarily probabilistic in nature, but additionally exploit the existence of a "front speed" (or minimal speed of a pulsating traveling front solution) for the Fisher-KPP reaction-diffusion equation naturally associated to the BBM.