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

In this paper, we first 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 traveling 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 traveling wave solution in the subcritical case. Our main tools are the spine decomposition and martingale change of measures.

Automated summary

In this paper, we investigate the limits of the additive and derivative martingales of one-dimensional branching Brownian motion in a periodic environment. We then demonstrate the existence of pulsating traveling wave solutions of the corresponding F-KPP equation in the supercritical and critical cases by probabilistically representing the solutions in terms of the limits of the additive and derivative martingales. Additionally, we prove the absence of pulsating traveling wave solutions in the subcritical case. The main techniques employed are spine decomposition and martingale change of measures.