I have a disk of radius R removed from the plane. Brownian excursions are emanated from infinity. I am tasked to find a probability that the brownian motion impacts an arc of the disk boundary. I am also asked to do this for a point outside the disk and for replacing the disk with an ellipse; which part of the ellipse receives more particles?
My original idea was to take the common example for a particle inside to hit an arc of the disk and invert the argument somehow. Any hints would be appreciated.
Cheers!
In all three cases the conformal invariance of harmonic measure / Brownian motion helps a lot.
The distribution of exit position is invariant under rotation of the circle. There is only one rotation-invariant probability measure on the circle.
Reduce the problem to 1 by mapping the point to $\infty$ by a fractional linear transformation such as $z\mapsto \frac{1}{z-a}$
Reduce the problem to 2 using a conformal map from the exterior of the disk to the exterior of ellipse. Namely, $z\mapsto c(z+z^{-1})$ maps the exterior of disk of radius $R>1$ onto the exterior of the ellipse with semi-axes $c(R+R^{-1})$ and $c(R-R^{-1})$.