We study a continuous-time branching random walk (BRW) on the lattice ℤd, d ∈ 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 spatially homogeneous, symmetric and irreducible but, in contrast to the majority of previous investigations, the random walk transition intensities a(x, y) decrease as |y − x|−(d+α) for |y − x| → ∞, where α ∈ (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 Markov 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 βc > 0 is found for every d ≥ 1. In particular, if β > βc 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 > βc called supercritical. Classification of the BRW treated as subcritical (β < βc) or critical (β = βc) 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 ∈ ℤd and of the particle population on ℤd according to the ratio d/α.

中文翻译：

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多维格上的重尾分支随机游走。片刻接近

我们研究了格子上的连续时间分支随机游走 (BRW) ℤd,d∈ ℕ，具有单一分支源，即可以发生粒子出生和死亡的格点。随机游走被假定为空间均匀、对称且不可约，但与之前的大多数研究相比，随机游走转变强度一种(X,是的) 减少为 |是的-X|-(d+α)为|是的-X| → ∞，其中α∈ (0, 2)，这导致随机游走跳跃的无限方差。粒子在源头的诞生和死亡的机制是由一个连续时间的马尔可夫分支过程控制的。源强度由某个参数表征β. 我们计算每个格点处的粒子数和总种群大小的所有整数矩的长期渐近行为。关于参数β, 一个重要的临界点βC> 0 为每个找到d≥ 1. 特别是，如果β>βC进化算子为具有正特征值的粒子数产生一阶矩的行为。正特征值的存在产生指数增长吨案例中的粒子数β>βC叫超临界. BRW 的分类被视为亚临界(β<βC） 要么危急(β=βC) 对于重尾随机游走跳跃比对于具有有限跳跃方差的随机游走更复杂。我们研究了任意点多个粒子的所有整数矩的渐近行为是的∈ ℤd和 ℤ 上的粒子群d根据比例d/α.