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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.

Put $R_t=|W_t|$. First verify that the origin is not reached, so that differentiating the radius is legitimate. On ${\mathbb{R}}^3\setminus \{0\}$ the function $F(w)=|w{|}^{-1}$ is harmonic. For $T_{\epsilon }=\inf \{t:R_t=\epsilon \}$ with $0<\epsilon <1$, the Itô formula makes $F(W_{t\wedge T_{\epsilon }})$ a martingale on each finite horizon: its stochastic integrand has norm at most ${\epsilon }^{-2}$. Since $F(W_0)=1$,Let $t\to \infty$, then $\epsilon \downarrow 0$. Any path hitting zero must first hit every positive $\epsilon$, so $\mathbb{P}(\exists t:R_t=0)=0$.

For $r(w)=|w|$, ${\partial }_{i}r=w_i/r$ and $\Delta r=2/r$ in three dimensions. Itô's formula, initially localized away from zero, yieldsThe continuous local martingale ${\gamma }_t={\sum }_i{\int }_0^t(W_s^i/R_s)d{W}_s^i$ has bracket $[\gamma {]}_t={\int }_0^t{\sum }_i(W_s^i/R_s{)}^{2}ds=t$ and starts at zero. By the Lévy characterization of Brownian motion, $\gamma$ is Brownian motion. ThusThis proves that the radial process is a three-dimensional Bessel process solving the same equation as $X$ in part (b). The permitted uniqueness in distribution identifies their laws as processes.

The strong law in part (a) gives $Y_s\to \infty$, and the inverse clock satisfies $J_t\to \infty$. Hence $X_t=Y_{J_t}\to \infty$. Equality of process laws transfers this almost sure path property to $R$.

For the further logarithmic rate, fix $0<\eta <1/2$. Almost surely, for all sufficiently large $s$, $|B_s|\le \eta s$. With a finite random lower cutoff $S$, for $u>S$,Taking logarithms and dividing by $u$ gives lower and upper asymptotic bounds $1-2\eta$ and $1+2\eta$. Letting $\eta \downarrow 0$ along a countable sequence proves the logarithmic growth of an exponential Brownian clock with positive drift, $\log H_u/u\to 1$. Also $\log Y_u/u=B_u/u+1/2\to 1/2$. Since $t=H_{J_t}$,Again transfer the full path event by uniqueness in distribution. We obtain the logarithmic escape rate of three-dimensional Brownian motion

For an independent way to identify the value once existence of the almost sure limit is granted, use Brownian scaling. At each fixed $t$, $W_t/\sqrt{t}$ has the law of $G+(1/\sqrt{t},0,0)$, where $G$ is standard three-dimensional Gaussian. Its norm converges in distribution to the strictly positive finite variable $|G|$, so $\log (|W_t|/\sqrt{t})$ is tight. ThereforeAn existing almost sure limit must agree with this probability limit. This guesses and identifies the value through scaling, while the inverse-clock argument proves that the almost sure limit actually exists.