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二维布朗在楔形中先碰哪条射线

Hit the ray

专题
Probability / 概率
难度
L4

题目详情

Let OaOa and ObOb be two rays in the two dimensional plane such that ∠aOb\angle aOb contains the point (1,0)(1,0) as shown in the picture below. What is the probability that the two dimensional Brownian motion starting at (1,0)(1,0) hits the ray OaOa before hitting the ray ObOb ?

解析

二维布朗运动的命中概率满足调和方程（Dirichlet 问题）。在楔形区域内，命中两条边界射线的概率只与角度有关，并且是角度的线性函数。

令楔形开口角为

θ=∠aOb∈(0,2π),\theta=\angle aOb\in(0,2\pi),

令起点 (1,0)(1,0) 相对射线 OaOa 的极角为

φ=∠aO(1,0)∈(0,θ).\varphi=\angle aO(1,0)\in(0,\theta).

则

P(先碰 Oa)=1−φθ=θ−φθ,\boxed{\mathbb{P}(\text{先碰 }Oa)=1-\frac{\varphi}{\theta}=\frac{\theta-\varphi}{\theta}}, P(先碰 Ob)=φθ.\boxed{\mathbb{P}(\text{先碰 }Ob)=\frac{\varphi}{\theta}}.