 Transience of Brownian motion in dimension at least three

Standard Brownian motion $B$ in ${\mathbb{R}}^d$ is transient for $d\ge 3$: $|B_t|\to \infty$ almost surely. In dimension three, start away from zero and use the positive local martingale $1/|B_t|$. It is a supermartingale, hence converges to a finite limit. Brownian scaling and the normal distribution of $B_t/\sqrt{t}$ show that $1/|B_t|\to 0$ in probability, so its almost-sure limit must be zero. Thus $|B_t|\to \infty$.

For a start at zero, apply the same argument from any positive deterministic time; the position there is nonzero almost surely. In higher dimensions, project onto three coordinates: the full radius dominates a radius already tending to infinity. This differs from recurrence of planar Brownian motion, where every neighbourhood is visited repeatedly.