We prove that the distance between two reflected Brownian motions, driven by the same white noise, outside a sphere in a $3$-dimensional flat torus does not converge to $0$, a.s., if the radius of the sphere is sufficiently small, relative to the size of the torus.
"Brownian earthworm." Ann. Probab. 41 (6) 4002 - 4049, November 2013. https://doi.org/10.1214/12-AOP831