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.
The research of this project is supported by the National Key R&D Program of China (No. 2020YFA0712900). The research of Yan-Xia Ren is supported in part by NSFC (Grant Nos. 11731009, 12071011 and 12231002) and The Fundamental Research Funds for Central Universities, Peking University LMEQF. The research of Renming Song is supported in part by a grant from the Simons Foundation (#960480, Renming Song).
We thank the referees for very helpful comments and suggestions. Part of the research for this paper was done while the second-named author was visiting Jiangsu Normal University, where he was partially supported by a grant from the National Natural Science Foundation of China (11931004, Yingchao Xie).
"Branching Brownian motion in a periodic environment and existence of pulsating traveling waves." Electron. J. Probab. 28 1 - 50, 2023. https://doi.org/10.1214/23-EJP960