A Note on Transience Versus Recurrence for a Branching Random Walk in Random Environment
A Note on Transience Versus Recurrence for a Branching Random Walk in Random Environment
复制标题
关于随机环境中分支随机游走的瞬态与递归的注释
DOI：
10.1023/a:1004539225064
复制
发表时间：
1999
影响因子：
1.6
通讯作者：
S. Popov
中科院分区：
文献类型：
--
作者：
F. den Hollander;M. Menshikov;S. Popov
我们考虑了ℤd上随机环境中的分枝随机游动，其中粒子按照给定的子代分布进行独立的简单随机游动，并在密度在无穷远趋于零的位置的随机子集上进行分支。假定最初有一个粒子从原点开始，我们确定了分离瞬变和重现的分支点密度的临界衰减率，即子体以概率<；1命中原点。=1.我们证明了对于d≥3，临界衰减率存在二分性，这取决于分枝点处的平均子代是高于还是低于与简单随机游动的返回概率相关的某个值。这种二分法标志着命中原点的后代的行为从局部行为向全局行为的转变。我们还考虑了分支位置出现在两种或两种以上类型中、具有不同后代分布的情况，并表明由于类型之间可能存在相互作用，因此分类更加微妙。本笔记是第二作者和多位合著者研究随机介质中随机运动的瞬变与重现问题的一系列论文的一部分。
We consider a branching random walk in random environment on ℤd where particles perform independent simple random walks and branch, according to a given offspring distribution, at a random subset of sites whose density tends to zero at infinity. Given that initially one particle starts at the origin, we identify the critical rate of decay of the density of the branching sites separating transience from recurrence, i.e., the progeny hits the origin with probability <1 resp. =1. We show that for d≥3 there is a dichotomy in the critical rate of decay, depending on whether the mean offspring at a branching site is above or below a certain value related to the return probability of the simple random walk. The dichotomy marks a transition from local to global behavior in the progeny that hits the origin. We also consider the situation where the branching sites occur in two or more types, with different offspring distributions, and show that the classification is more subtle due to a possible interplay between the types. This note is part of a series of papers by the second author and various co-authors investigating the problem of transience versus recurrence for random motions in random media.