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

Part of: Representations of solutions Markov processes

Published online by Cambridge University Press:  09 November 2022

Yan-Xia Ren ,
Renming Song  and
Fan Yang
Yan-Xia Ren*
Affiliation:
Peking University
Renming Song*
Affiliation:
University of Illinois at Urbana-Champaign
Fan Yang*
Affiliation:
Peking University
*
*Postal address: LMAM School of Mathematical Sciences and Center for Statistical Science, Peking University, Beijing, 100871, P. R. China.
***Postal address: Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA. Email address: rsong@illinois.edu
*Postal address: LMAM School of Mathematical Sciences and Center for Statistical Science, Peking University, Beijing, 100871, P. R. China.
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Abstract

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.

Keywords

F-KPP equationasymptotic behaviorBessel-3 processmartingale change of measures

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 35C07: Traveling wave solutions
Type
Original Article
Information
Advances in Applied Probability , Volume 55 , Issue 2 , June 2023 , pp. 510 - 548
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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