论文标题
在多维晶格上进行重尾分支随机步行。瞬间的方法
Heavy-Tailed Branching Random Walks on Multidimensional Lattices. A Moment Approach
论文作者
论文摘要
我们在晶格$ \ mathbb {z}^{d} $，$ d \ in \ mathbb {n} $上研究连续的时间分支随机步行，带有单个分支来源，这是粒子生育和粒子死亡的晶格。 The random walk is assumed to be homogeneous, symmetric and irreducible but, in contrast to previous investigations, the random walk transition intensities $a(x,y)$ decrease as $|y-x|^{-(d+α)}$ for $|y-x|\to \infty$, where $α\in(0,2)$, that leads to an infinite variance of the random walk jumps.来源颗粒的诞生和死亡的机理受到连续的BienayMé-Galton-Watson分支过程的控制。源强度的特征是某个参数$β$。我们计算了每个整数矩的长期渐近行为，每个晶格点的颗粒数和总人口大小。关于参数$β$，每个$ d \ geq 1 $都可以找到非平凡的临界点$β_C> 0 $。特别是，如果$β>β_{c} $，进化算子为粒子数量产生了第一刻的行为，则具有正征值。在$β>β_c$的情况下，粒子数的$ t $ t $ the \ emph {emph {超临界}的$ t $ t $中的指数增长。分支随机步行的分类为\ emph {subcritical}（$β<β_c$）或\ emph {crigind {crigind}（$β=β_C$），用于重尾随机步行跳跃比有限的跳跃变化的随机步行更为复杂。我们研究了\ yathbb {z}^d $ y \ y \ y \ y y \ y y Mathbb {z}^d $的所有整数矩的渐近行为。
We study a continuous-time branching random walk on the lattice $\mathbb{Z}^{d}$, $d\in \mathbb{N}$, with a single source of branching, that is the lattice point where the birth and death of particles can occur. The random walk is assumed to be homogeneous, symmetric and irreducible but, in contrast to previous investigations, the random walk transition intensities $a(x,y)$ decrease as $|y-x|^{-(d+α)}$ for $|y-x|\to \infty$, where $α\in(0,2)$, that leads to an infinite variance of the random walk jumps. The~mechanism of the birth and death of particles at the source is governed by a continuous-time Bienaymé-Galton-Watson branching process. The source intensity is characterized by a certain parameter $β$. We calculate the long-time asymptotic behaviour for all integer moments for the number of particles at each lattice point and for the total population size. With respect to the parameter $β$ a non-trivial critical point $β_c>0$ is found for every $d\geq 1$. In particular, if $β>β_{c}$ the evolutionary operator generated a behaviour of the first moment for the number of particles has a positive eigenvalue. The existence of a positive eigenvalue yields an exponential growth in $t$ of the particle numbers in the case $β>β_c$ called \emph{supercritical}. Classification of the branching random walk treated as \emph{subcritical} ($β<β_c$) or \emph{critical} ($β=β_c$) for the heavy-tailed random walk jumps is more complicated than for a random walk with a finite variance of jumps. We study the asymptotic behaviour of all integer moments of a number of particles at any point $y\in\mathbb{Z}^d$ and of the particle population on $\mathbb{Z}^d$ according to the ratio $d/α$.
