Large fluctuations of the area under a constrained Brownian excursion
Large fluctuations of the area under a constrained Brownian excursion
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约束布朗偏移下面积的大波动
DOI：
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发表时间：
2018
期刊：
影响因子：
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通讯作者：
B. Meerson
中科院分区：
文献类型：
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作者：
B. Meerson
我们研究在时间间隔 上布朗偏移下面积的大波动，限制为远离移动壁 x0(t)，使得 和 x0(|t| < T) > 0。我们关注由广义抛物线族描述的壁函数，其中 。使用最优波动法（OFM），我们计算了长时间区域的大偏差函数（LDF）。 OFM 以布朗运动的最佳路径或射线的形式提供了面积波动的简单描述。我们证明，LDF 在 的临界值处的三阶导数有跳跃。这种奇点是最优路径发生质变的结果，可以解释为三阶动态相变。尽管 OFM 不适用于典型的（小）面积波动，但我们认为它正确地捕获了 T 的幂律缩放，其指数连续依赖于 和 k。我们还考虑余弦墙来说明最佳路径和典型波动缩放的不同可能行为。对于某些壁函数，由于多个 OFM 解决方案共存而产生的附加相变应该是可能的。
We study large fluctuations of the area under a Brownian excursion on the time interval , constrained to stay away from a moving wall x0(t) such that and x0(|t| < T) > 0. We focus on wall functions described by a family of generalized parabolas , where . Using the optimal fluctuation method (OFM), we calculate the large deviation function (LDF) of the area at long times. The OFM provides a simple description of the area fluctuations in terms of optimal paths, or rays, of the Brownian motion. We show that the LDF has a jump in the third derivative with respect to at a critical value of . This singularity results from a qualitative change of the optimal path, and it can be interpreted as a third-order dynamical phase transition. Although the OFM is not applicable for typical (small) area fluctuations, we argue that it correctly captures their power-law scaling of with T, with an exponent that depends continuously on and on k. We also consider the cosine wall to illustrate a different possible behavior of the optimal path and of the scaling of typical fluctuations. For some wall functions additional phase transitions, which result from a coexistence of multiple OFM solutions, should be possible.