Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 201 / 5

The function $u(x)=ϵ/|x|$ is harmonic on $\{|x|>ϵ\}$, equals one on the target sphere, and tends to zero at infinity. Optional stopping at the first hit of radius $ϵ$ and the first exit from a ball of radius $R$ givesLetting $R\to \infty$ and taking $|x|=1$ yields ${\mathbb{P}}_x({\tau }_{ϵ}<\infty )=ϵ$.

Project Brownian motion modulo ${\mathbb{Z}}^3$ to the compact three-dimensional torus. The projected process has normalized volume as invariant probability measure and is irreducible, so it visits every nonempty open set infinitely often almost surely. The image of $S_3$ contains the radius-$ϵ$ ball about the origin. Hence the original Brownian motion hits $S_3$ at an unbounded set of times.

Yes. Quotient only the first coordinate modulo $\mathbb{Z}$. The resulting process lives on $\mathbb{T}\times {\mathbb{R}}^2$ and the image of $S_1$ is the radius-$ϵ$ ball about $(0,0,0)$. The two noncompact coordinates form planar Brownian motion, which is recurrent; during its infinitely many returns to a smaller disc, the independent circle coordinate has a fixed positive chance of lying in the required interval. The Strong Markov property then shows that the target ball is visited infinitely often. Lifting back proves that $S_1$ is hit at unbounded times.