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

TL;DR

This paper investigates branching Brownian motion in a periodic environment, establishing the existence of pulsating travelling waves in certain cases and linking these solutions to martingale limits.

Contribution

It introduces a probabilistic approach to prove the existence of pulsating travelling waves for the F-KPP equation in a periodic setting, using martingale techniques.

Findings

Existence of pulsating travelling waves in supercritical and critical cases.

Non-existence of pulsating travelling waves in subcritical case.

Martingale methods effectively characterize wave solutions.

Abstract

We study the limits of the additive and derivative martingales of one-dimensional branching Brownian motion in a periodic environment. Then we prove the existence of pulsating travelling wave solutions of the corresponding F-KPP equation in the supercritical and critical cases by representing the solutions probabilistically in terms of the limits of the additive and derivative martingales. We also prove that there is no pulsating travelling wave solution in the subcritical case. Our main tools are the spine decomposition and martingale change of measures.