= Last visit to a bounded ball for three-dimensional Brownian motion
{title2=$\mathbb P(\exists t\ge\sigma_R:|B_t|\le r)=r/R$}
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.
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