论文标题
关于快速出口的概率和在简单连接的域中的平面布朗运动的长时间停留
On the probability of fast exits and long stays of planar Brownian motion in simply connected domains
论文作者
论文摘要
令$ t^d $表示来自域$ d $的平面布朗尼动议的第一个退出时间。给定了两个简单连接的平面域$ u，w \ neq \ sc \ co $ 0 $，我们调查了我们更有可能退出快速的情况（例如$ {\ bf p}（\ bf p}（t^u <t）> {\ bf p}（\ bf p}（t^w <t）（t^w <t $ for $ t $ for $ u b $ ut $ u b $ w $ w $ w $ w $ w $ w $ w $ w $ w $ w $ w $ w $ w $， p}（t^u> t）> {\ bf p}（t^w> t）$ for $ t $ tame）。我们证明了这些问题的几个结果。特别是，我们表明，快速出口概率的主要因素是边界与原点的接近度，而长期以来一直保持一个重要因素是退出时间的矩。还讨论了激发我们询问的复杂分析理论。
Let $T^D$ denote the first exit time of a planar Brownian motion from a domain $D$. Given two simply connected planar domains $U,W \neq \SC$ containing $0$, we investigate the cases in which we are more likely to have fast exits (meaning for instance ${\bf P}(T^U<t) > {\bf P}(T^W<t)$ for $t$ small) from $U$ than from $W$, or long stays (meaning ${\bf P}(T^U>t) > {\bf P}(T^W>t)$ for $t$ large). We prove several results on these questions. In particular, we show that the primary factor in the probability of fast exits is the proximity of the boundary to the origin, while for long stays an important factor is the moments of the exit time. The complex analytic theory that motivated our inquiry is also discussed.
