= Logarithmic escape rate of three-dimensional Brownian motion
{title2=$\lim_{t\to\infty}\frac{\log|W_t|}{\log t}=\frac12\quad\text{almost surely}$}
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 $H_s=\int_0^se^{2B_r+r}dr$. The <logarithmic growth of an exponential Brownian clock with positive drift> gives $\log H_s/s\to1$, whereas $\log(e^{B_s+s/2})/s\to1/2$. 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.
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