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The exact statement of the problem is:

For each z in the slit open unit disk, find the probability that a Brownian particle will exit the region through the circular part of the boundary.

So this is a complex analysis class I'm in and the professor has taken a bit of a detour away from our textbook, thus I feel a bit adrift.

Basically I'm looking for the relevant theorems and formulas required to solve this problem. I think there is this important integral which describes the probability of exiting a region starting from a certain point, I'm not exactly sure. Hopefully someone can help me out here, thanks.

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Hint: If you call this probability $h(z)$, then $h$ is a harmonic function on the interior of the slit disk. And you know its values on the boundary... –  Nate Eldredge Apr 10 '12 at 1:00
    
Ya I remember my professor talking about that, but there really isn't anything in the book, I'm just looking for the relevant equations not the solution. I couldn't find a wikipedia page on it beyond a short excerpt about the mean value property on the Harmonic Function page. Is there a site that explains what I'm after? I don't take notes so I don't have any of it. –  heat death Apr 10 '12 at 1:08
    
Sorry I just realized after posting my answer: when you say slit unit disk, do you mean the origin removed or are you removing a line segment, say $[0,1]$? –  Alex R. Apr 10 '12 at 1:15
    
The line segment –  heat death Apr 10 '12 at 1:25
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1 Answer

I'm not sure how much probability you have seen so I'll just mention the technique briefly in hopes that your lecture notes cover this in some form. Feel free to ask for clarification. I refrain from using any probability jargon simply because you probably aren't expected to know it. One way of solving this is to assume you start within a slit unit disk and ask "What is the chance that the particle will exit the region from the circular boundary, BEFORE hitting $[0,1)$." Perhaps your professor covered this but, this is equivalent to solving for a harmonic function with the value $1$ on the circular boundary and 0 on $[0,1)$. It will help to use the laplacian in spherical coordinates for this problem.

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Ok so I need to find $f$ such that $\frac{{\partial}^2f}{\partial x^2} + i\frac{{\partial}^2f}{\partial y^2} = 0$ and $f(r_2)=1$ and $f((-1,0])=0$. Is this correct? I don't really understand the Laplacian or its connection. –  heat death Apr 10 '12 at 1:22
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