Propagation of Branching Random Walk on Periodic Graphs

Pub Date : 2024-07-11 DOI:10.1134/s0081543824010073

E. Vl. Bulinskaya

引用次数: 0

Abstract

A model of a branching random walk on \(\mathbb Z^d\) with a periodic set of sources of reproduction and death of particles is studied. For this model, an asymptotic description of the propagation of the population of particles with time is obtained for the first time. The intensities of the sources may be different. The branching regime is assumed to be supercritical, and the tails of the jump distribution of the walk are assumed to be “light.” The main theorem establishes the Hausdorff-metric convergence of the properly normalized random cloud of particles that exist in the branching random walk at time \(t\) to the limit set, as \(t\) tends to infinity. This convergence takes place for almost all elementary outcomes of the event meaning nondegeneracy of the population of particles under study. The limit set in \(\mathbb R^d\), called the asymptotic shape of the population, is found in an explicit form.

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周期图上分支随机漫步的传播

摘要研究了一个在 \(\mathbb Z^d\) 上的分支随机行走模型，该模型具有一组周期性的粒子繁殖和死亡源。对于这个模型，首次得到了粒子群随时间传播的渐近描述。源的强度可能不同。假设分支机制是超临界的，并假设行走的跳跃分布的尾部是 "轻的"。当 \(t\) 趋于无穷大时，主定理确定了分支随机行走中存在的粒子的适当归一化随机云在 \(t\) 时间到极限集的豪斯多夫计量收敛性。这种收敛发生在事件的几乎所有基本结果上，这意味着所研究的粒子群的非整数性。在 \(\mathbb R^d\) 中的极限集被称为粒子群的渐近线形状，它是以明确的形式找到的。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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