论文标题
在可还原的多元分支布朗运动中散布异常
Anomalous spreading in reducible multitype branching Brownian motion
论文作者
论文摘要
我们考虑了一种两种可还原的布朗尼运动，该运动定义为实际线上的粒子系统，其中两种类型的颗粒根据独立的布朗运动移动，并以恒定的速率产生后代。类型$ 1 $的颗粒可以生下$ 1 $和$ 2 $的类型的颗粒，但是类型$ 2 $的颗粒仅生下$ 2 $的后代。在某些特定条件下，Biggins（2012）表明，此过程表现出异常的传播行为：时间$ t $的最右粒子比仅由$ 1 $或类型$ 2 $组成的分支布朗尼运动中的最右边粒子的预期位置远。 Holzer（2014,2016）也从反应扩散方程的角度研究了这种异常扩散。本文的目的是研究两种可还原分支的布朗尼运动中最远的颗粒位置的渐近行为，特别是获得了最大位移中位数的严格估计。
We consider a two-type reducible branching Brownian motion, defined as a particle system on the real line in which particles of two types move according to independent Brownian motion and create offspring at constant rate. Particles of type $1$ can give birth to particles of types $1$ and $2$, but particles of type $2$ only give birth to descendants of type $2$. Under some specific conditions, Biggins (2012) shows that this process exhibit an anomalous spreading behaviour: the rightmost particle at time $t$ is much further than the expected position for the rightmost particle in a branching Brownian motion consisting only of particles of type $1$ or of type $2$. This anomalous spreading also has been investigated from a reaction-diffusion equation standpoint by Holzer (2014,2016). The aim of this article is to study the asymptotic behaviour of the position of the furthest particles in the two-type reducible branching Brownian motion, obtaining in particular tight estimates for the median of the maximal displacement.
