 Last visit to a bounded ball for three-dimensional Brownian motion

For Brownian motion started at zero in dimension three, let ${\sigma }_R$ be its first hit of the sphere of radius $R>r$. The optional stopping theorem applied to $|B_t{|}^2-3t$ proves that ${\sigma }_R$ is finite almost surely. The Strong Markov property and the Brownian sphere-hitting probability in dimension three then give the displayed return probability. Choose radii $R_m=2^{m}r$. The return events after ${\sigma }_{R_m}$ decrease and their probabilities tend to zero. Almost surely one of these exit times is followed by no visit to the closed ball of radius $r$. Apply this to all positive integer $r$ to obtain $|B_t|\to \infty$ almost surely. This gives full transience of Brownian motion in dimension at least three, rather than only a positive escape probability.