arXiv 1102.18882011-02math.PR

分支布朗运动中稳定点过程的一个注记

A note on stable point processes occurring in branching Brownian motion

Pascal Maillard

发表信息：: Electronic Communications in Probability 18, 5 (2013) 1-9
DOI：: 10.1214/ECP.v18-2390

摘要：

我们称点过程$Z$在$\mathbb R$上为\emph{exp-1-stable}，如果对于每个具有$e^α+e^β=1$的$α,β\in\mathbb R$，$Z$在规律上等于$T_αZ+T_βZ'$，其中$Z'$是$Z$的独立拷贝，$T_x$是$x$的翻译。这种过程出现在研究的极端粒子的分支布朗运动和分支随机行走和几个作者已经证明，设置存在一个点过程$D$对$\mathbb R$，使$Z$是等于在法律上的$\sum_{i=1}^\infty T_{ξ_i} D_i$，其中$(ξ_i)_{i\ge1}$是原子的泊松过程的强度$e^{-x}\,\mathrm d x$对$\mathbb R$和$(D_i)_{i\ge 1}$是独立的副本$D$和独立的$(ξ_i)_{i\ge1}$。在这篇文章中，我们展示了这种分解是如何从（并）稳定点过程的经典\emph{LePage decomposition}得到的。此外，在$\mathbb R$上随机测度的一般情况下，给出了它的一个简短的证明.

英文摘要：

We call a point process $Z$ on $\mathbb R$ \emph{exp-1-stable} if for every $α,β\in\mathbb R$ with $e^α+e^β=1$, $Z$ is equal in law to $T_αZ+T_βZ'$, where $Z'$ is an independent copy of $Z$ and $T_x$ is the translation by $x$. Such processes appear in the study of the extremal particles of branching Brownian motion and branching random walk and several authors have proven in that setting the existence of a point process $D$ on $\mathbb R$ such that $Z$ is equal in law to $\sum_{i=1}^\infty T_{ξ_i} D_i$, where $(ξ_i)_{i\ge1}$ are the atoms of a Poisson process of intensity $e^{-x}\,\mathrm d x$ on $\mathbb R$ and $(D_i)_{i\ge 1}$ are independent copies of $D$ and independent of $(ξ_i)_{i\ge1}$. In this note, we show how this decomposition follows from the classic \emph{LePage decomposition} of a (union)-stable point process. Moreover, we give a short proof of it in the general case of random measures on $\mathbb R$.

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英文摘要：

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