The extremal process of branching Brownian motion
The extremal process of branching Brownian motion
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DOI：
10.1007/s00440-012-0464-x
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发表时间：
2011-03
影响因子：
2
通讯作者：
L. Arguin;Anton Bovier;N. Kistler
中科院分区：
文献类型：
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作者：
L. Arguin;Anton Bovier;N. Kistler
证明了分支布朗运动的极值过程在大次数极限下弱收敛于簇点过程。极限过程是一个（随机移位的）泊松簇过程，其中簇的位置是一个具有指数密度的强度测度的泊松过程。个别集群的法律特点是分支布朗运动的条件执行“异常大的位移”，并证明了它的存在性。证据结合了三个主要成分。首先，Bramson关于具有一般初始条件的Kolmogorov-Petrovsky-Piscounov方程的解收敛于驻波的结果。第二，这种波的积分表示，首先得到的Lalley和Sellke的情况下，Heaviside初始条件。第三，一个适当的识别与辅助过程的极值过程的尾部（基于Chauvin和Rouault的工作），它完全捕获的极值过程的大时间渐近。通过辅助过程的分析是在平均场自旋玻璃研究中发展起来的腔方法的严格表述。
We prove that the extremal process of branching Brownian motion, in the limit of large times, converges weakly to a cluster point process. The limiting process is a (randomly shifted) Poisson cluster process, where the positions of the clusters is a Poisson process with intensity measure with exponential density. The law of the individual clusters is characterized as branching Brownian motions conditioned to perform “unusually large displacements”, and its existence is proved. The proof combines three main ingredients. First, the results of Bramson on the convergence of solutions of the Kolmogorov–Petrovsky–Piscounov equation with general initial conditions to standing waves. Second, the integral representations of such waves as first obtained by Lalley and Sellke in the case of Heaviside initial conditions. Third, a proper identification of the tail of the extremal process with an auxiliary process (based on the work of Chauvin and Rouault), which fully captures the large time asymptotics of the extremal process. The analysis through the auxiliary process is a rigorous formulation of thecavity methoddeveloped in the study of mean field spin glasses.