Area fluctuations of a Brownian excursion are described by the Airy distribution, which found applications in different areas of physics, mathematics and computer science. Here we generalize this distribution to describe the area fluctuations on a \emph{subinterval} of a Brownian excursion. In the first version of the problem (Model 1) no additional conditions are imposed. In the second version (Model 2) we study the distribution of the area fluctuations on a subinterval given the excursion area on the whole interval. Both versions admit convenient path-integral formulations. In Model 1 we obtain an explicit expression for the Laplace transform of the area distribution on the subinterval. In both models we focus on large deviations of the area by evaluating the tails of the area distributions, sometimes with account of pre-exponential factors. When conditioning on very large areas in Model 2, we uncover two singularities in the rate function of the subinterval area fraction. They can be interpreted as dynamical phase transitions of second and third order.

中文翻译：

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布朗偏移子区间的面积波动

布朗偏移的面积波动由艾里分布描述，该分布在物理、数学和计算机科学的不同领域都有应用。在这里，我们概括这个分布来描述布朗偏移的 \emph {subinterval} 上的面积波动。在问题的第一个版本（模型 1）中没有附加条件。在第二个版本（模型 2）中，我们研究了给定整个区间的偏移区域的子区间上的区域波动分布。两个版本都承认方便的路径积分公式。在模型 1 中，我们获得了子区间上面积分布的拉普拉斯变换的显式表达式。在这两种模型中，我们通过评估面积分布的尾部来关注面积的大偏差，有时会考虑指数前因素。在模型 2 中对非常大的区域进行调节时，我们发现了子区间面积分数的速率函数中的两个奇点。它们可以解释为二阶和三阶动态相变。