摘要翻译：
我们解析地计算了连续时间布朗运动（有漂移和无漂移）在第一次通过原点之前达到最大值的时间$t_m$的概率密度$P(t_m)$。我们还计算了最大值$M$和$t_m$的联合概率密度$p(M,t_m)$。在无漂移情况下，我们发现$P(t_m)$具有幂律尾:$P(t_m)\sim t_m^{-3/2}$对于大$t_m$和$P(t_m)\sim t_m^{-1/2}$对于小$t_m$。当存在向原点的漂移时，对于大的$t_m$，$P(t_m)$呈指数衰减。数值模拟的结果与我们的解析预测非常一致。
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英文标题：
《Distribution of the time at which the deviation of a Brownian motion is
  maximum before its first-passage time》
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作者：
Julien Randon-Furling (LPTMS), Satya N. Majumdar (LPTMS)
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最新提交年份：
2008
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分类信息：

一级分类：Physics        物理学
二级分类：Statistical Mechanics        统计力学
分类描述：Phase transitions, thermodynamics, field theory, non-equilibrium phenomena, renormalization group and scaling, integrable models, turbulence
相变，热力学，场论，非平衡现象，重整化群和标度，可积模型，湍流
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一级分类：Mathematics        数学
二级分类：Probability        概率
分类描述：Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory
概率论与随机过程的理论与应用：例如中心极限定理，大偏差，随机微分方程，统计力学模型，排队论
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英文摘要：
  We calculate analytically the probability density $P(t_m)$ of the time $t_m$ at which a continuous-time Brownian motion (with and without drift) attains its maximum before passing through the origin for the first time. We also compute the joint probability density $P(M,t_m)$ of the maximum $M$ and $t_m$. In the driftless case, we find that $P(t_m)$ has power-law tails: $P(t_m)\sim t_m^{-3/2}$ for large $t_m$ and $P(t_m)\sim t_m^{-1/2}$ for small $t_m$. In presence of a drift towards the origin, $P(t_m)$ decays exponentially for large $t_m$. The results from numerical simulations are in excellent agreement with our analytical predictions.
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PDF链接：
https://arxiv.org/pdf/708.2101