发表机构
CMAP, École polytechnique; ICJ, Université Lyon 1（巴黎综合理工学院 CMAP; 里昂第一大学 ICJ）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对后代分布为临界且吸引α-稳定律的分支随机游走，研究其范围线性增长率，扩展分支容量至α-稳定情形并估计球的对应容量。
AI 中文摘要
我们研究了在后代分布μ为临界且吸引α-稳定律时，尺寸条件分支随机游走（BRW）的范围线性增长率。这通过Le Gall与Lin引入的无限不变BRW，以及一种将范围的增长率与基础树的维度概念普遍关联的新准则来实现。随后，在暂态情形（即范围确实呈线性增长时），我们将分支容量的概念扩展至该α-稳定情形。我们证明其仍与BRW（或其无限版本）到达ℤᵈ中远处集合的渐近概率相关，并估计了球的α-稳定分支容量。
英文摘要
We study the linear growth rate of the range of size-conditioned Branching Random Walks (BRW) when the offspring distribution $μ$ is critical and attracted to an $α$-stable law. This is done via the infinite invariant BRW introduced by Le Gall & Lin and a new criterion which relates this growth rate of the range to a notion of dimension of the underlying tree in a general way. Then, in the transient case (that is, when the range does grow linearly), we extend the notion of branching capacity to this $α$-stable case. We show that it is still related to the asymptotic probability that a BRW (or its infinite version) reaches a distant set in $\mathbb Z^d$, and we estimate the $α$-stable branching capacity of balls.
Comments36 pages, 6 figures
