arXiv 1808.054452018-08math.PR

从$1$到$6$：扰动分支布朗运动的精细分析

From $1$ to $6$: a finer analysis of perturbed branching Brownian motion

Anton Bovier, Lisa Hartung

备注：: 28 pages, 2 figures, introduction rewritten and misprints corrected

摘要：

双速分支布朗运动的最大值阶数的对数修正在接近标准分支布朗运动的斜率$σ_1^2=σ_2^2=1$时不连续变化。在本文中，我们通过选择$σ_1^2=1\pm t^{-α}$和$σ_2^2=1\pm t^{-α}$来更仔细地研究这种转变。我们表明，对数校正的顺序的最大值，现在顺利地插入之间的校正在iid的情况下$\frac{1}{2\sqrt 2}\ln(t),\;\frac{3}{2\sqrt 2}\ln(t)$和$\frac{6}{2\sqrt 2}\ln(t)$时，$0<α<\frac{1}{2}$。这是由于在速度变化时的极值粒子的局部化，这取决于$α$，并且不同于标准分支布朗运动中的极值粒子。我们还建立了在所有情况下的最大值和极值过程的渐近律，这与标准分支布朗运动的渐近律基本一致。

英文摘要：

The logarithmic correction for the order of the maximum for two-speed branching Brownian motion changes discontinuously when approaching slopes $σ_1^2=σ_2^2=1$ which corresponds to standard branching Brownian motion. In this article we study this transition more closely by choosing $σ_1^2=1\pm t^{-α}$ and $σ_2^2=1\pm t^{-α}$. We show that the logarithmic correction for the order of the maximum now smoothly interpolates between the correction in the iid case $\frac{1}{2\sqrt 2}\ln(t),\;\frac{3}{2\sqrt 2}\ln(t)$ and $\frac{6}{2\sqrt 2}\ln(t)$ when $0<α<\frac{1}{2}$. This is due to the localisation of extremal particles at the time of speed change which depends on $α$ and differs from the one in standard branching Brownian motion. We also establish in all cases the asymptotic law of the maximum and characterise the extremal process, which turns out to coincide essentially with that of standard branching Brownian motion.

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