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For $0<x<M$, a three-dimensional Bessel process stopped at $M$ has the law of Brownian motion conditioned to hit $M$ before zero and stopped at $M$. On $(\epsilon ,M)$, the Girsanov theorem and the Itô formula give terminal density $X_{\tau }/x$. Its value is $M/x$ at the upper exit and $\epsilon /x$ at the lower exit. Letting $\epsilon \downarrow 0$ in bounded stopped-path functionals gives density $(M/x)1_{\{T_M<T_0\}}$ relative to the corresponding stopped Wiener measure. The exit probability is $x/M$ by the optional stopping theorem, so the density is normalized. On the unstopped full path space the same density defines the conditioned Brownian law; pushing it forward by the stopping map gives the stopped Bessel law. The already-stopped law is not absolutely continuous relative to unstopped Wiener measure, since eventually constant paths have Wiener probability zero.

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.

For $w\ne 0$ in ${\mathbb{R}}^3$, the radial function $r(w)=\Vert w\Vert$ hasThe multidimensional Itô formula, initially stopped away from zero, givesThe martingale term has quadratic variation $dt$, since ${\sum }_i(W_t^i/R_t{)}^2=1$. The Lévy characterization of Brownian motion says that a continuous local martingale starting at zero with quadratic variation $t$ is a standard Brownian motion. Consequently the radial equation isOn $[\epsilon ,M]$ the drift $1/r$ is Lipschitz continuous. Extend it to a bounded globally Lipschitz function on the real line. The standard existence and pathwise uniqueness theorem for stochastic differential equations with globally Lipschitz coefficients applies to this extension, and gives uniqueness in law up to the first exit from $[\epsilon ,M]$. Both this radial process and the process in part (a) solve the same equation from $x$. Thuswhere each exit time is computed from its own coordinate path.

For completeness, the radial equation is valid globally: the three-dimensional Bessel process does not hit zero. Apply the Itô formula to $1/R$ on the annulus. Its drift is zero, and the stopped process is a bounded martingale. At the first exit,soThe exit time is finite because one coordinate of the underlying Brownian motion reaches $M$ in finite time. Letting $\epsilon \downarrow 0$ and then considering arbitrarily large $M$ proves that zero is never reached. The radial martingale therefore has quadratic variation $t$ on the whole time axis, completing the Lévy characterization of Brownian motion argument.