= Transience of Brownian motion in dimension at least three
Standard <Brownian motion> $B$ in $\mathbb R^d$ is transient for $d\geq3$: $|B_t|\to\infty$ <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 $1/|B_t|\to0$ 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.
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