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摘要翻译：
我们导出了一类约束布朗运动的最大值M的联合概率密度P(M,t_m)和达到最大值的时间t_m。特别地，我们给出了与布朗运动相关的漂移、弯曲和反射桥的显式结果。然后在M上积分，在每种情况下都以双无穷级数的形式得到边缘密度P(t_m)。对于漂移和弯曲，我们详细分析了P(t_m)的矩和渐近极限，表明理论结果与数值模拟结果非常吻合。我们的主要推导方法是基于路径积分技术；然而，另一种方法也被概述，它建立在布朗运动过程的概率研究中更普遍地遇到的某些“一致公式”上。
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英文标题：
《On the time to reach maximum for a variety of constrained Brownian
  motions》
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作者：
Satya. N. Majumdar (LPTMS), Julien Randon-Furling (LPTMS), Michael J.
  Kearney, Marc Yor (PMA)
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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 derive P(M,t_m), the joint probability density of the maximum M and the time t_m at which this maximum is achieved for a class of constrained Brownian motions. In particular, we provide explicit results for excursions, meanders and reflected bridges associated with Brownian motion. By subsequently integrating over M, the marginal density P(t_m) is obtained in each case in the form of a doubly infinite series. For the excursion and meander, we analyse the moments and asymptotic limits of P(t_m) in some detail and show that the theoretical results are in excellent accord with numerical simulations. Our primary method of derivation is based on a path integral technique; however, an alternative approach is also outlined which is founded on certain "agreement formulae" that are encountered more generally in probabilistic studies of Brownian motion processes.
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PDF链接：
https://arxiv.org/pdf/802.2619

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