Maximum of branching Brownian motion in a periodic environment

Eyal Lubetzky, Chris Thornett, Ofer Zeitouni

Research output: Contribution to journalArticlepeer-review


We study the maximum of Branching Brownian motion (BBM) with branching rates that vary in space, via a periodic function of a particle's location. This corresponds to a variant of the F-KPP equation in a periodic medium, extensively studied in the last 15 years, admitting pulsating fronts as solutions. Recent progress on this PDE due to Hamel, Nolen, Roquejoffre and Ryzhik ('16) implies tightness for the centered maximum of BBM in a periodic environment. Here we establish the convergence in distribution of specific subsequences of this centered maximum, and identify the limiting distribution. Consequently, we find the asymptotic shift between the solution to the corresponding F-KPP equation with Heavyside initial data and the pulsating wave, thereby answering a question of Hamel et al. Analogous results are given for the cases where the Brownian motion is replaced by an Ito diffusion with periodic coefficients, as well as for nearest-neighbor branching random walks.

Original languageEnglish (US)
Pages (from-to)2065-2093
Number of pages29
JournalAnnales de l'institut Henri Poincare (B) Probability and Statistics
Issue number4
StatePublished - Nov 2022


  • Branching Brownian motion
  • F-KPP in periodic medium

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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