Transience of Brownian motion in dimension at least three
= Transience of Brownian motion in dimension at least three
Brownian motion in $\mathbb R^d$ is transient for $d\geq3$: its distance from the origin tends to infinity almost surely. In dimension three, the positive local martingale $|B_t|^{-1}$ and <Brownian scaling> give a short proof.
It is a dimension-dependent long-time property of <Brownian motion>.