The logarithmic correction for the order of the maximum for two-speed branching Brownian motion changes discontinuously when approaching slopes $\sigma_1^2=\sigma_2^2=1$ which corresponds to standard branching Brownian motion. In this article we study this transition more closely by choosing $\sigma_1^2=1\pm t^{-\alpha}$ and $\sigma_2^2=1\pm t^{-\alpha}$. 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 $0<\alpha<\frac{1}{2}$. This is due to the localisation of extremal particles at the time of speed change which depends on $\alpha$ 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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从1到6：微扰分支布朗运动的精细分析

当接近斜率 $\sigma_1^2=\sigma_2^2=1$ 时，两速分支布朗运动的最大值阶数的对数校正不连续变化，这对应于标准分支布朗运动。在本文中，我们通过选择 $\sigma_1^2=1\pm t^{-\alpha}$ 和 $\sigma_2^2=1\pm t^{-\alpha}$ 来更仔细地研究这种转变。我们表明，最大值阶次的对数校正现在在 iid 情况下的校正之间平滑地插值 $\frac{1}{2\sqrt 2}\ln(t),\;\frac{3}{2\ sqrt 2}\ln(t)$ 和 $\frac{6}{2\sqrt 2}\ln(t)$ 当 $0<\alpha<\frac{1}{2}$。这是由于极值粒子在速度变化时的定位取决于 $\alpha$ 并且不同于标准分支布朗运动中的定位。