A continuum-tree-valued Markov process
A continuum-tree-valued Markov process
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DOI：
10.1214/11-aop644
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发表时间：
2009-04
影响因子：
2.3
通讯作者：
R. Abraham;Jean-François Delmas
中科院分区：
文献类型：
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作者：
R. Abraham;Jean-François Delmas
本文利用探索过程和Girsanov定理构造了与超临界连续状态分支过程相关的Levy连续统随机树。我们还将剪枝程序扩展到这种超临界情况。让$\psi$成为关键分支机制。我们设置$\psi_\theta(\cdot)=\psi(\cdot+\theta)-\psi(\theta)$。设$\Theta=(\theta_\infty,+\infty)$或$\Theta=[\theta_\infty,+\infty)$为$\theta$的一组值，其中$\psi_\theta$是一个分支机制。剪枝过程允许构造一个递减的levy - crt值马尔可夫过程$(\ct_\theta,\theta\in\Theta)$，使得$\mathcal{T}_\theta$具有分支机制$\psi_\theta$。如果是次临界$\theta>0$，如果是超临界$\theta<0$。然后我们考虑CRT的爆炸时间$A$：较小的（负）时间$\theta$，其中$\mathcal{T}_\theta$具有有限的质量。我们描述了$A$的规律以及在这个爆炸时间之后阴极射线管的分布。爆炸后的阴极射线管可以看作是一个条件不灭的阴极射线管，它被修剪成与$A$相关的独立强度。我们还研究了爆炸时间后crt值过程的演变。这扩展了Aldous和Pitman关于高尔顿-沃森树的结果。对于二次分支机构的特殊情况，我们证明了爆炸后CRT的总质量表现为指数为1/2的稳定次级体的倒数。这一结果与Aldous' CRT碎片标记片段的大小有关。
We present a construction of a Levy continuum random tree (CRT) associated with a super-critical continuous state branching process using the so-called exploration process and a Girsanov's theorem. We also extend the pruning procedure to this super-critical case. Let $\psi$ be a critical branching mechanism. We set $\psi_\theta(\cdot)=\psi(\cdot+\theta)-\psi(\theta)$. Let $\Theta=(\theta_\infty,+\infty)$ or $\Theta=[\theta_\infty,+\infty)$ be the set of values of $\theta$ for which $\psi_\theta$ is a branching mechanism. The pruning procedure allows to construct a decreasing Levy-CRT-valued Markov process $(\ct_\theta,\theta\in\Theta)$, such that $\mathcal{T}_\theta$ has branching mechanism $\psi_\theta$. It is sub-critical if $\theta>0$ and super-critical if $\theta<0$. We then consider the explosion time $A$ of the CRT: the smaller (negative) time $\theta$ for which $\mathcal{T}_\theta$ has finite mass. We describe the law of $A$ as well as the distribution of the CRT just after this explosion time. The CRT just after explosion can be seen as a CRT conditioned not to be extinct which is pruned with an independent intensity related to $A$. We also study the evolution of the CRT-valued process after the explosion time. This extends results from Aldous and Pitman on Galton-Watson trees. For the particular case of the quadratic branching mechanism, we show that after explosion the total mass of the CRT behaves like the inverse of a stable subordinator with index 1/2. This result is related to the size of the tagged fragment for the fragmentation of Aldous' CRT.