标题： 关于平面布朗运动第一象限和第三象限的占用时间 显示英文标题

标题： On occupation times of the first and third quadrants for planar Brownian motion

摘要： 一个感兴趣的开放问题，最初在Bingham和Doney于1988年的工作中引入应用概率领域（见\cite{Bingham}），其表述如下：求平面布朗运动的象限占用时间的分布。 在本文的简要通信中，我们研究了这个长期开放问题的一种替代表述：设$X(t), Y(t), t \geq 0$分别是从$x,y$出发的标准布朗运动。 求总时间$T=Leb\{t \in [0,1]: X(t) \times Y(t) >0\}$的分布，当$x=y=0$，即第一象限和第三象限并集的占用时间。 如果使用两个相邻的象限，问题会变得简单得多，$T$的分布遵循反正弦定律。 显示英文摘要

摘要： An open problem of interest, first infused into the applied probability community in the work of Bingham and Doney in 1988, (see \cite{Bingham}) is stated as follows: find the distribution of the quadrant occupation time of planar Brownian motion. In this short communication, we study an alternate formulation of this longstanding open problem: let $X(t), Y(t), t \geq 0$ be standard Brownian motions starting at $x,y$ respectively. Find the distribution of the total time $T=Leb\{t \in [0,1]: X(t) \times Y(t) >0\}$, when $x=y=0$, i.e., the occupation time of the union of the first and third quadrants. If two adjacent quadrants are used, the problem becomes much easier and the distribution of $T$ follows the arcsine law.