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

Using one-dimensional branching Brownian motion in a periodic environment, we give probabilistic proofs of the asymptotics and uniqueness of pulsating traveling waves of the Fisher-Kolmogorov-Petrovskii-Piskounov (F-KPP) equation in a periodic environment. This paper is a sequel to 'Branching Brownian motion in a periodic environment and existence of pulsating travelling waves' (Ren et al., 2022), in which we proved the existence of the pulsating traveling waves in the supercritical and critical cases, using the limits of the additive and derivative martingales of branching Brownian motion in a periodic environment.