双曲空间中有限速度运动粒子的级联
Cascades of Particles Moving at Finite Velocity in Hyperbolic Spaces
摘要：
研究了在双曲空间（Poincaré半平面和Poincaré圆盘）测地线上以有限速度运动的粒子的分支过程.每个粒子可以分裂成两个粒子只有一次在泊松起搏时间和偏离正交分裂时。在时间$t$，在$N(t)$泊松事件之后，有$N(t)+1$粒子沿着不同的测地线移动。我们能够得到粒子云质心的平均双曲距离的精确表达式。我们从两个不同的和独立的方式得到这样的平均双曲线距离，我们研究的行为的相关表达式作为$t$的增加和不同的值的参数$c$（双曲线运动速度）和$λ$（再生率）。每个移动粒子的平均双曲线距离也检查和一个有用的表示，作为一个随机停止的粒子移动的主测地线的距离，提出。
英文摘要：
A branching process of particles moving at finite velocity over the geodesic lines of the hyperbolic space (Poincaré half-plane and Poincaré disk) is examined. Each particle can split into two particles only once at Poisson paced times and deviates orthogonally when splitted. At time $t$, after $N(t)$ Poisson events, there are $N(t)+1$ particles moving along different geodesic lines. We are able to obtain the exact expression of the mean hyperbolic distance of the center of mass of the cloud of particles. We derive such mean hyperbolic distance from two different and independent ways and we study the behavior of the relevant expression as $t$ increases and for different values of the parameters $c$ (hyperbolic velocity of motion) and $λ$ (rate of reproduction). The mean hyperbolic distance of each moving particle is also examined and a useful representation, as the distance of a randomly stopped particle moving over the main geodesic line, is presented.
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