Branching Brownian Motion, mean curvature flow and the motion of hybrid zones

TL;DR

This paper establishes a probabilistic link between Allen-Cahn equation, mean curvature flow, and the evolution of hybrid zones in population genetics, using duality with branching and coalescing random walkers.

Contribution

It provides a new probabilistic proof connecting Allen-Cahn equation and mean curvature flow, and extends this to the spatial Lambda-Fleming-Viot process with hybrid zone modeling.

Findings

Probabilistic proof of the Allen-Cahn and mean curvature flow connection.

Extension of the result to population genetics hybrid zones.

Duality with branching and coalescing random walkers.

Abstract

We provide a probabilistic proof of a well known connection between a special case of the Allen-Cahn equation and mean curvature flow. We then prove a corresponding result for scaling limits of the spatial $\Lambda$-Fleming-Viot process with selection, in which the selection mechanism is chosen to model what are known in population genetics as hybrid zones. Our proofs will exploit a duality with a system of branching (and coalescing) random walkers which is of some interest in its own right.