Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 203 / 1 / b / Solution

Put $F(z)=z+z^{-1}$. On the unit semicircle, $F(e^{i\theta })=2\cos \theta$, andThe conformal invariance of planar Brownian motion and the Poisson kernel for the upper half-plane therefore give the exit density with respect to $d\theta$:As $z\to \infty$ in $\mathbb{H}$,uniformly in $\theta \in [0,\pi ]$. Hence