We investigate the statistics of encounters of a diffusing particle with different subsets of the boundary of a confining domain. The encounters with each subset are characterized by the boundary local time on that subset. We extend a recently proposed approach to express the joint probability density of the particle position and of its multiple boundary local times via a multi-dimensional Laplace transform of the conventional propagator satisfying the diffusion equation with mixed Robin boundary conditions. In the particular cases of an interval, a circular annulus and a spherical shell, this representation can be explicitly inverted to access the statistics of two boundary local times. We provide the exact solutions and their probabilistic interpretation for the case of an interval and sketch their derivation for two other cases. We also obtain the distributions of various associated first-passage times and discuss their applications.

中文翻译：

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多目标多边界本地时间联合分布及相关首航时间问题

我们研究了扩散粒子与限制域边界的不同子集相遇的统计数据。与每个子集的相遇以该子集上的边界本地时间为特征。我们扩展了最近提出的方法，通过满足具有混合罗宾边界条件的扩散方程的传统传播器的多维拉普拉斯变换来表达粒子位置及其多个边界局部时间的联合概率密度。在间隔、圆环和球壳的特殊情况下，可以显式反转该表示以访问两个边界本地时间的统计数据。我们为区间的情况提供了精确的解决方案及其概率解释，并概述了其他两种情况的推导。