Branching Brownian motion in a periodic environment and uniqueness of pulsating travelling waves

TL;DR

This paper provides probabilistic proofs for the asymptotics and uniqueness of pulsating travelling waves in the F-KPP equation within a periodic environment, building on previous work establishing their existence.

Contribution

It introduces probabilistic methods to prove asymptotic behavior and uniqueness of pulsating travelling waves in a periodic setting, complementing earlier existence results.

Findings

Proves asymptotics of pulsating travelling waves

Establishes uniqueness of these waves in a periodic environment

Builds on martingale limits from branching Brownian motion

Abstract

Using one-dimensional branching Brownian motion in a periodic environment, we give probabilistic proofs of the asymptotics and uniqueness of pulsating travelling waves of the F-KPP equation in a periodic environment. This paper is a sequel to [Ren et al. Branching Brownian motion in a periodic environment and existence of pulsating travelling waves], in which we proved the existence of the pulsating travelling waves in the supercritical and critical cases using the limits of the additive and derivative martingales of branching Brownian motion in a periodic environment.