长程色散的空间生长过程：微观、中观和扩散率的差异
Spatial growth processes with long range dispersion: microscopics, mesoscopics, and discrepancy in spread rate
摘要：
本文考虑了{连续时间-连续空间}分支随机游动的传播速度，并附加了一个限制条件，即在任意空间点的出生率不能超过$1$ .色散核被认为具有多项式衰减为$|x|^{- 2α}$，$x \to \infty$的密度。我们表明，如果$α> 2$，则系统以线性速度传播，{而对于$α\in (\frac 12 ,2]$，传播比线性速度快}。我们还考虑了对应于微观随机系统的介观方程。我们表明，在对比的微观过程中，介观方程的解以指数速度传播每$α> \frac 12$。
英文摘要：
We consider the speed of propagation of a {continuous-time continuous-space} branching random walk with the additional restriction that the birth rate at any spatial point cannot exceed $1$. The dispersion kernel is taken to have density that decays polynomially as $|x|^{- 2α}$, $x \to \infty$. We show that if $α> 2$, then the system spreads at a linear speed, {while for $α\in (\frac 12 ,2]$ the spread is faster than linear}. We also consider the mesoscopic equation corresponding to the microscopic stochastic system. We show that in contrast to the microscopic process, the solution to the mesoscopic equation spreads exponentially fast for every $α> \frac 12$.
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