AI 中文总结
本文研究平面Ornstein-Uhlenbeck过程在环形扇区和主轴矩形上的首次退出，通过本征函数展开获得退出时间与退出位置的联合定律，并用蒙特卡洛模拟验证，应用于光镊捕获胶体粒子。
AI 中文摘要
我们研究各向同性平面Ornstein-Uhlenbeck过程从环形扇区（由两个同心圆弧和两条径向线段围成的区域）的首次退出，并获得了相关退出泛函的显式本征函数展开。在通过基态变换使生成元自伴之后，极坐标分离将问题简化为径向合流超几何方程，其两个一般非整数阶的独立（Whittaker）解通过一个双半径行列式结合，从而确定谱。由此我们获得了生存概率、退出时间的密度和矩，以及主要贡献——退出时间与退出边界的联合定律，该定律同时确定了退出时刻和退出发生的边界片段。作为伴随的可分离情形，我们还处理了主轴矩形上真正相关的可逆平面Ornstein-Uhlenbeck过程，其中径向函数被抛物柱面函数取代：生存概率分解为一维问题，而退出时间与退出边的联合定律则不能分解。这些展开仅需一维求根和求积，并通过蒙特卡洛模拟验证。我们还讨论了对光镊捕获胶体粒子的应用，在该场景中环形几何自然出现。
英文摘要
We study the first exit of an isotropic planar Ornstein-Uhlenbeck process from an annular sector, the region bounded by two concentric circular arcs and two radial segments, and obtain explicit eigenfunction expansions for the associated exit functionals. After the ground-state transformation that renders the generator self-adjoint, polar separation reduces the problem to a radial confluent hypergeometric equation whose two independent (Whittaker) solutions of generally non-integral order are combined through a two-radius determinant that fixes the spectrum. In this way we obtain the survival probability, the density and moments of the exit time, and -the principal contribution -the joint law of the exit time and the exit boundary, which resolves both the instant of exit and the boundary piece through which it occurs. As a companion separable case, we also treat a genuinely correlated, reversible planar Ornstein-Uhlenbeck process on a principal-axis rectangle, where the radial functions are replaced by parabolic cylinder functions: the survival probability factorizes into one-dimensional problems, while the joint law of the exit time and the exit side does not. The expansions require only one-dimensional root-finding and quadrature and are validated against Monte Carlo simulation. An application to an optically trapped colloidal particle is discussed, a setting in which the annular geometry arises naturally.
