论文标题
在半平面和马丁边界上倾斜反射的布朗运动的占用密度的渐近行为
Asymptotic behavior of the occupancy density for obliquely reflected Brownian motion in a half-plane and Martin boundary
论文作者
论文摘要
令$π$为在半平面上倾斜反射的布朗运动的占用密度，让（$ρ$，$α$）是上半平面上一个点的极性坐标。这项工作确定了$π$（$ρ$，$α$）的确切渐近行为为$ρ$ $ \ rightarrow $ $ $ $ \ infty $，$α$ $ \ in $（0，$π$）。我们发现明确的功能A，B，C，使得$π$（$ρ$，$α$）$ \ sim $ $ $ $ $ $ $ $ $ \ rightarrow $$ \ infty $ \ infty $ a（$α$）$ρ$ b（$α$）e -c（$α$）$ρ$。这关闭了J. Michael Harrison教授于2013年8月首先提出的一个开放问题。我们还计算了边界占用度量的尾巴分布的确切渐近学，并且我们获得了$π$的显式积分表达式。最后，我们通过找到过程的马丁边界并提供满足斜诺伊曼边界问题的所有相应谐波函数。
Let $π$ be the occupancy density of an obliquely reflected Brownian motion in the half plane and let ($ρ$, $α$) be the polar coordinates of a point in the upper half plane. This work determines the exact asymptotic behavior of $π$($ρ$, $α$) as $ρ$ $\rightarrow$ $\infty$ with $α$ $\in$ (0, $π$). We find explicit functions a, b, c such that $π$($ρ$, $α$) $\sim$ $ρ$$\rightarrow$$\infty$ a($α$)$ρ$ b($α$) e --c($α$)$ρ$. This closes an open problem first stated by Professor J. Michael Harrison in August 2013. We also compute the exact asymptotics for the tail distribution of the boundary occupancy measure and we obtain an explicit integral expression for $π$. We conclude by finding the Martin boundary of the process and giving all of the corresponding harmonic functions satisfying an oblique Neumann boundary problem.
