Transience of Brownian motion in dimension at least three

Brownian motion in ${\mathbb{R}}^d$ is transient for $d\ge 3$: 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.

For standard three-dimensional Brownian motion started at $x$ with $r<|x|<R$,The harmonic function $x\mapsto 1/|x|$ gives a bounded stopped martingale in the annulus, so the optional stopping theorem gives the formula. Taking $R\to \infty$ yields ${\mathbb{P}}_x({\tau }_r<\infty )=r/|x|$ for $|x|>r$. This limit uses path continuity on the finite interval before a hit, rather than assuming transience to prove transience.

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.