Hitting probability of semiball

Question

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

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.

Answer 1

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

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