Branching stable processes and motion by mean curvature flow

TL;DR

This paper establishes a probabilistic connection between solutions of the fractional Allen--Cahn equation and mean curvature flow, using branching stable processes to model long-range dispersal in population dynamics.

Contribution

It introduces a novel probabilistic approach to relate fractional Allen--Cahn solutions to mean curvature flow via branching stable processes, overcoming heavy-tailed distribution challenges.

Findings

Probabilistic representation of fractional Allen--Cahn solutions.

Coupling of stable and Brownian branching motions.

Application to long-range dispersal in populations.

Abstract

We prove a new result relating solutions of the scaled fractional Allen--Cahn equation to motion by mean curvature flow, motivated by the motion of hybrid zones in populations that exhibit long range dispersal. Our proof is purely probabilistic and takes inspiration from Etheridge et al. to describe solutions of the fractional Allen--Cahn equation in terms of ternary branching $\alpha$-stable motions. To overcome technical difficulties arising from the heavy-tailed nature of the stable distribution, we couple ternary branching stable motions to ternary branching Brownian motions subordinated by truncated stable subordinators.