Barycentric Brownian Bees

带漂移的布朗蜂：寻找临界性

Brownian Bees with Drift: Finding the Criticality

本文研究带漂移 Brownian Bees 的临界行为，确定区分常返与暂态的漂移阈值，证明小漂移下收敛至唯一不变分布，并讨论 N-BBM。

Brownian蜜蜂的临界漂移与反射布朗运动不变性原理

Critical Drift for Brownian Bees and a Reflected Brownian Motion Invariance Principle

本文研究一维带漂移的$N$-Brownian蜜蜂分支选择系统，确定临界漂移及三种相态，并证明该临界值等于$N$-BBM速度。

Brownian bees in the infinite swarm limit

The Brownian bees model is a branching particle system with spatial selection. It is a system of $N$ particles which move as independent Brownian motions in $\mathbb{R}^d$ and independently branch at rate 1, and, crucially, at each branching event, the particle which is the furthest away from the origin is removed to keep the population size constant. In the present work we prove that as $N \to \infty$ the behaviour of the particle system is well approximated by the solution of a free boundary problem (which is the subject of a companion paper), the hydrodynamic limit of the system. We then sh…

Fluctuations of a swarm of Brownian bees

The ``Brownian bees" model describes an ensemble of $N$ independent branching Brownian particles. When a particle branches into two particles, the particle farthest from the origin is eliminated so as to keep a constant number of particles. In the limit of $N\to \infty$, the spatial density of the particles is governed by the solution of a free boundary problem for a reaction-diffusion equation. At long times the particle density approaches a spherically symmetric steady state solution with a compact support. Here we study fluctuations of the ``swarm of bees" due to the random character of the…

Persistent fluctuations of the swarm size of Brownian bees

The "Brownian bees" model describes a system of $N$ independent branching Brownian particles. At each branching event the particle farthest from the origin is removed, so that the number of particles remains constant at all times. Berestycki et al. (2020) proved that, at $N\to \infty$, the coarse-grained spatial density of this particle system lives in a spherically symmetric domain and is described by the solution of a free boundary problem for a deterministic reaction-diffusion equation. Further, they showed that, at long times, this solution approaches a unique spherically symmetric steady …

Probabilities of moderately atypical fluctuations of the size of a swarm of Brownian Bees

The ``Brownian bees'' model describes an ensemble of $N=$~const independent branching Brownian particles. The conservation of $N$ is provided by a modified branching process. When a particle branches into two particles, the particle which is farthest from the origin is eliminated simultaneously. The spatial density of the particles is governed by the solution of a free boundary problem for a reaction-diffusion equation in the limit of $N \gg 1$. At long times, the particle density approaches a spherically symmetric steady state solution with a compact support of radius $\bar{\ell}_0$. However,…