= Harmonic limit along an infinite strip arm
{title2=$u(x,y)\to(a+b)/2+(a-b)x/2$}
Suppose an unbounded domain contains the upper strip $(-1,1)\times(1,\infty)$ with its two vertical sides on the boundary. A bounded <harmonic function> with constant values $a$ on the right side and $b$ on the left side, and with valid <bounded harmonic representation with almost sure Brownian exit>, satisfies $u(x,y)\to a(1+x)/2+b(1-x)/2$ as $y\to\infty$. The horizontal <Brownian exit time> is finite; on that finite random interval the vertical Brownian path has a finite minimum. Consequently reaching the bottom before a side has probability tending to zero. The remaining two exit probabilities are the one-dimensional interval probabilities.
Back to article page
