Three-dimensional Bessel process ($d{R}_t=d{B}_t+R_t^{-1}dt$)

The radial part of three-dimensional Brownian motion started away from the origin is a Bessel process of dimension three. The radial Laplacian is $\Delta |x|=2/|x|$, so the Itô formula gives the displayed equation. The radial martingale has unit quadratic variation rate and is Brownian motion by the Lévy characterization of Brownian motion. The reciprocal $1/R$ is a local martingale. Stopping it on $(\epsilon ,M)$ gives ${\mathbb{P}}_x(T_{\epsilon }<T_M)=\epsilon (M-x)/(x(M-\epsilon ))$. Letting $\epsilon \downarrow 0$ shows that zero is not hit.

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.