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For fixed x and hemisphere H of radius r and centered at the origin, I wonder what is $P_{x}(T_{H}<\infty)$.

enter image description here

Attempt

Firstly, I wonder if there is any relation between $P_{x}(T_{H}<N)$ and $\frac{1}{N}\int_{0}^{N}\int_{H}p(x,t,y)dydt$ where $p(x,t,y)$ is the transition density of Brownian motion.

Secondly, in Landkoff, we have $Cap(H)=\frac{2r}{\pi}(1-\frac{1}{\sqrt{3}})$. Thus, we have

$\frac{2r}{\pi}(1-\frac{1}{\sqrt{3}})=lim_{t\to \infty}\frac{1}{t}\int_{\mathbb{R}^{3}/H}P_{x}(T_{H}<t)dx$.

Any ideas or references? I will post as I find things.

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1 Answer 1

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As explained here, you need to solve the electrostatic problem of a unit point charge outside a grounded hemisphere. The hitting probability then equals minus the induced charge on the hemisphere, or equivalently, the surface integral of the normal component of the electric field.

It is unlikely that a simple closed-form solution exists for this electrostatic problem. Such a solution does exist if the hemisphere is on a grounded plane, see for example these lecture notes. So this will answer the following modification of your problem:

What is the probability $P$ for Brownian motion to hit the hemisphere $x^2+y^2+z^2=r^2$, $x>0$, starting from the point $(x_0,0,0)$, $x_0>r$, before passing through the plane $x=0$? The answer is

$$P=1-(1-r/x_0)\sqrt{1+r/x_0}.$$

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  • $\begingroup$ thanks for the link; any other sources that helped you understand it (books, lecture notes etc)? A bit more mathematical. $\endgroup$
    – TKM
    Jan 6, 2015 at 1:48
  • $\begingroup$ Also, do you know if anyone has computed the hitting probabilities for many kinds of sets using method-of-images? $\endgroup$
    – TKM
    Feb 11, 2015 at 19:32
  • $\begingroup$ no, the method of images only works for high-symmetry configurations; for a general solution I would just use some numerical Poisson solver, that's really the easiest way to go. $\endgroup$ Feb 11, 2015 at 19:36

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