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arXiv 2406.04526math.PR
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分支布朗运动一致最大位移的相变

Phase transition of the consistent maximal displacement of branching Brownian motion

Julien Berestycki, Jiaqi Liu, Bastien Mallein, Jason Schweinsberg

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AI总结：

本文研究分支布朗运动粒子轨迹保持在曲线$(\sqrt{2}+\varepsilon)s$附近的问题，发现当时间$t$达到$\varepsilon^{-3/2}$量级时发生相变，并据此确定了带左漂移且原点吸收的分支布朗运动为避免灭绝所需的初始位置。

摘要：

考虑分支布朗运动：初始时在原点有一个粒子，粒子独立地按照布朗运动移动，并以速率1分裂为两个粒子。众所周知，在时间$t$时最右侧的粒子将位于$\sqrt{2} t$附近。Roberts研究了所谓的一致最大位移，并证明以高概率，在时间$t$会存在一个粒子，其祖先在所有$s \in [0, t]$时间内都保持在曲线$s \mapsto \sqrt{2} s$的距离$ct^{1/3}$以内，其中$c = (3 \pi^2)^{1/3}/\sqrt{2}$。我们考虑的问题是：对于所有$s \in [0, t]$，一个粒子的轨迹能够多接近地保持在曲线$s \mapsto (\sqrt{2} + \varepsilon) s$附近，其中$\varepsilon > 0$很小。我们发现存在一个相变，当$t$的量级为$\varepsilon^{-3/2}$时行为发生变化。这一结果使我们能够确定，对于粒子具有向左漂移$\sqrt{2} + \varepsilon$且在原点被杀死（吸收）的分支布朗运动，粒子在零时刻需要从什么位置出发，才能以高概率使该过程直到时间$t$都避免灭绝。

英文摘要：

Consider branching Brownian motion in which we begin with one particle at the origin, particles independently move according to Brownian motion, and particles split into two at rate one. It is well-known that the right-most particle at time $t$ will be near $\sqrt{2} t$. Roberts considered the so-called consistent maximal displacement and showed that with high probability, there will be a particle at time $t$ whose ancestors stayed within a distance $ct^{1/3}$ of the curve $s \mapsto \sqrt{2} s$ for all $s \in [0, t]$, where $c = (3 π^2)^{1/3}/\sqrt{2}$. We consider the question of how close the trajectory of a particle can stay to the curve $s \mapsto (\sqrt{2} + \varepsilon) s$ for all $s \in [0, t]$, where $\varepsilon> 0$ is small. We find that there is a phase transition, with the behavior changing when $t$ is of the order $\varepsilon^{-3/2}$. This result allows us to determine, for branching Brownian motion in which particles have a drift to the left of $\sqrt{2} + \varepsilon$ and are killed at the origin, the position at which a particle needs to begin at time zero for there to be a high probability that the process avoids extinction until time $t$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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