Logarithmic escape rate of three-dimensional Brownian motion (${\lim }_{t\to \infty }\frac{\log |W_t|}{\log t}=\frac{1}{2}\text{almost surely}$)

For $W$ started at a point of radius one, its radial process has the law of the inverse-clock process $e^{B_{J_t}+J_t/2}$, where $J$ inverts $H_s={\int }_0^s{e}^{2B_r+r}dr$. The logarithmic growth of an exponential Brownian clock with positive drift gives $\log H_s/s\to 1$, whereas $\log (e^{B_s+s/2})/s\to 1/2$. Substituting $s=J_t$ proves the displayed almost sure escape rate and divergence of the radius. Independently, Brownian scaling gives the same rate in probability; if an almost sure logarithmic rate is already known to exist, that identifies its value.