论文标题
布朗蜜蜂群的持续波动
Persistent fluctuations of the swarm size of Brownian bees
论文作者
论文摘要
“布朗蜜蜂”模型描述了$ n $独立分支布朗尼颗粒的系统。在每个分支事件中，距离原点最远的粒子被去除，因此颗粒的数量始终保持恒定。 Berestycki等。 （2020）证明，在$ n \至\ infty $时，该粒子系统的粗粒空间密度生活在球形对称域中，并通过自由边界问题的解决方案来描述确定性反应 - 扩散方程。此外，他们表明，长期以来，该解决方案接近具有紧凑支持的独特球形对称稳态：半径$ \ ell_0 $的球体取决于空间尺寸$ d $。在这里，由于分支布朗尼运动的随机特征，我们研究了该系统中的波动，我们专注于群体大小的持续波动。我们评估了概率密度$ \ MATHCAL {p}（\ ell，n，t）$，即粒子与原点的最大距离保持小于指定的值$ \ ell <\ ell_0 $，或大于指定的值$ \ ell> \ ell> \ ell_0 $，在时间间隔$ 0 <t <t $中，$ 0 <t $ t $，其中$ t $非常大。我们认为$ \ MATHCAL {p}（\ ell，n，t）$展示了大差异表格$ - \ ln \ Mathcal {p} \ simeq n t r_d（\ ell）$。对于所有$ d $，我们获得了速率函数$ r_d（\ ell）$的渐近学，$ \ ell \ ll \ ll \ ell_0 $，$ \ ell \ gg \ gg \ ell_0 $和$ | \ ell-\ ell-\ ell_0 | \ ell_0 | \ ll \ ll \ ell_0 $。对于$ d = 1 $，可以通过分析计算整个费率函数。我们通过确定基于指定的$ \ ell $条件的群的最佳（最可能的）密度曲线来获得这些结果，并认为此密度曲线与其中心位于原点上是球形对称的。
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 state with compact support: a sphere which radius $\ell_0$ depends on the spatial dimension $d$. Here we study fluctuations in this system in the limit of large $N$ due to the stochastic character of the branching Brownian motion, and we focus on persistent fluctuations of the swarm size. We evaluate the probability density $\mathcal{P}(\ell,N,T)$ that the maximum distance of a particle from the origin remains smaller than a specified value $\ell<\ell_0$, or larger than a specified value $\ell>\ell_0$, on a time interval $0<t<T$, where $T$ is very large. We argue that $\mathcal{P}(\ell,N,T)$ exhibits the large-deviation form $-\ln \mathcal{P} \simeq N T R_d(\ell)$. For all $d$ we obtain asymptotics of the rate function $R_d(\ell)$ in the regimes $\ell \ll \ell_0$, $\ell\gg \ell_0$ and $|\ell-\ell_0|\ll \ell_0$. For $d=1$ the whole rate function can be calculated analytically. We obtain these results by determining the optimal (most probable) density profile of the swarm, conditioned on the specified $\ell$, and by arguing that this density profile is spherically symmetric with its center at the origin.
