Geometrical optics of constrained Brownian motion: three short stories
Geometrical optics of constrained Brownian motion: three short stories
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约束布朗运动的几何光学：三个短篇故事
DOI：
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发表时间：
2019
期刊：
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通讯作者：
Naftali R. Smith
中科院分区：
文献类型：
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作者：
B. Meerson;Naftali R. Smith
几何光学的最优涨落方法使人们对布朗运动的大偏差有了更深刻的认识。在这里，我们通过讲述三个关于布朗运动的小故事来说明这一点，布朗运动被约束“推”到了一个大偏差区域。在故事1中，我们计算了布朗粒子在平面内绕反射盘游荡时的缠绕角的短时大偏差函数。故事2讨论了平面上吸收障碍物上方的拉伸布朗运动。我们计算了幸存的布朗粒子在中间点的位置的短时LDF。故事3涉及布朗粒子在1 + 1维中抵抗壁吸收的生存，壁根据幂律前进，其中。我们还计算了LDF的粒子位置在较早的时间，条件是生存的较晚的时间。在这三个故事中，我们发现的LDFs有一个简单的几何起源，可以解释为动力学相变的奇异性。我们还利用几何光学的小偏差极限来重建典型涨落的分布。我们认为，在故事2和故事3中，这是Ferrari-Spohn分布。
The optimal fluctuation method—essentially geometrical optics—gives a deep insight into large deviations of Brownian motion. Here we illustrate this point by telling three short stories about Brownian motions, ‘pushed’ into a large-deviation regime by constraints. In story 1 we compute the short-time large deviation function (LDF) of the winding angle of a Brownian particle wandering around a reflecting disk in the plane. Story 2 addresses a stretched Brownian motion above absorbing obstacles in the plane. We compute the short-time LDF of the position of the surviving Brownian particle at an intermediate point. Story 3 deals with survival of a Brownian particle in 1 + 1 dimension against absorption by a wall which advances according to a power law , where . We also calculate the LDF of the particle position at an earlier time, conditional on the survival by a later time. In all three stories we uncover singularities of the LDFs which have a simple geometric origin and can be interpreted as dynamical phase transitions. We also use the small-deviation limit of the geometrical optics to reconstruct the distribution of typical fluctuations. We argue that, in stories 2 and 3, this is the Ferrari–Spohn distribution.