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

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.