AI 中文总结
本文针对平面布朗运动凸包的内切半径与逆内切半径，通过推导一维布朗运动极差期望闭式、沙漏域首出时间矩，改进了其期望与方差的界。
AI 中文摘要
我们研究标准平面布朗运动凸包的内切半径（inradius）与逆内切半径（inverse inradius），主要目标是改进这些量期望的现有界并建立方差界。为计算上述界，本文得到的核心贡献是一系列中间结果：我们求出两个独立一维布朗运动的极差最小值与最大值期望的闭式表达式，该结果不仅改进了内切半径期望的上界，还给出了文献中多次研究、此前仅知数值评估的某量的显式表达式；此外，我们研究平面布朗运动从沙漏形无界区域的首出时间，并计算其一阶与二阶矩，该结果使逆内切半径期望的上界较之前改进近5倍，且对建立对应方差上界起关键作用。
英文摘要
We study the inradius and inverse inradius of the convex hull of standard planar Brownian motion. Our main goals are to improve the existing bounds on the expected values of these quantities and to establish variance bounds. The main contributions of the paper, however, are the following intermediate results we obtain in order to compute our bounds. We find closed-form expressions for the expected values of the minimum and maximum of the ranges of two independent one-dimensional Brownian motions. Besides improving the upper bound on the expected inradius, this result also gives an explicit expression for a quantity that has been investigated repeatedly in the literature and for which only a numerical evaluation has been known so far. Furthermore, we study the exit time of planar Brownian motion from an unbounded region that we call the hourglass domain and compute its first two moments. This result leads to a nearly fivefold improvement of the previous upper bound on the expected inverse inradius and plays an essential role in establishing the corresponding upper bound on the variance.
Comments27 pages, 2 figures, 1 table
