2024/12/05 by Mercer, Jacob · 1 citation
Economics, Econometrics and Finance · Mathematics · #FOS: Mathematics #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2412.04527
openalex publication_date 2024/12/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
N-Brownian bees is a branching-selection particle system in ℝd in which N particles behave as independent binary branching Brownian motions, and where at each branching event, we remove the particle furthest from the origin. We study a variant in which d=1 and particles have an additional drift μ∈ℝ. We show that there is a critical value, μcN, and three distinct regimes (sub-critical, critical, and super-critical) and we describe the behaviour of the system in each case. In the sub-critical regime, the system is positive Harris recurrent and has an invariant distribution; in the super-critical regime, the system is transient; and in the critical case, after rescaling, the system behaves like a single reflected Brownian motion. We also show that the critical drift μcN is in fact the speed of the well-studied N-BBM process, and give a rigorous proof for the speed of N-BBM, which was missing in the literature.