论文标题
临界分支随机步行的不变度量
Invariant measures of critical branching random walks in high dimension
论文作者
论文摘要
在这项工作中，我们表征了在运动定律为$α$稳定或具有有限的离散范围时，对于所有足够大d的关键分支空间过程的群集不变点过程。更准确地说，当运动为$α$稳定时，$α$ $ \ le $ 2 $ $ 2，而后代定律$μ$的分支过程具有沉重的尾巴，使得$ $ $ $（k）$ \ sim $ k -2- $β$，那么我们需要Dimension D比关键的Dimension D比关键的Dimension $ $ $ $ $ $α$/$β$。特别是，当运动是布朗尼人，而后代定律$μ$的第二刻时，这个临界维度为2。与[BCG97]中Bramson，Cox和Greven的先前工作相反，其证明使用PDE技术，我们的证明仅使用概率工具。
In this work, we characterize cluster-invariant point processes for critical branching spatial processes on R d for all large enough d when the motion law is $α$-stable or has a finite discrete range. More precisely, when the motion is $α$-stable with $α$ $\le$ 2 and the offspring law $μ$ of the branching process has an heavy tail such that $μ$(k) $\sim$ k --2--$β$ , then we need the dimension d to be strictly larger than the critical dimension $α$/$β$. In particular, when the motion is Brownian and the offspring law $μ$ has a second moment, this critical dimension is 2. Contrary to the previous work of Bramson, Cox and Greven in [BCG97] whose proof used PDE techniques, our proof uses probabilistic tools only.
