Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 201 / 5 / a / Solution

Planar Brownian motion is recurrent. More explicitly, the planar Brownian annulus hitting probability giveswhen $|x|>1$. Thus the unit disc is hit almost surely from every starting point. Applying the Strong Markov property after each departure and return shows that such returns occur after arbitrarily large times. Therefore