= Solution
For $w\ne0$ in $\mathbb R^3$, the radial function $r(w)=\lVert w\rVert$ has
$$
\partial_i r=\frac{w_i}{r},\qquad
\partial_{ij}r=\frac{\delta_{ij}}{r}-\frac{w_iw_j}{r^3},\qquad
\Delta r=\frac2r.
$$
The multidimensional <Itô formula>, initially stopped away from zero, gives
$$
dR_t=\sum_{i=1}^3\frac{W_t^i}{R_t}\,dW_t^i+\frac{dt}{R_t}.
$$
The 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 is
$$
\boxed{dR_t=dB_t+\frac{dt}{R_t}.}
$$
On $[\varepsilon,M]$ 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 $[\varepsilon,M]$. Both this radial process and the process in part (a) solve the same equation from $x$. Thus
$$
\boxed{\mathcal L(R_{\cdot\wedge\tau_R})
=\mathcal L_{\mathbb Q^{M,\varepsilon}}(X_{\cdot\wedge\tau_X}),}
$$
where 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,
$$
\frac1x=\frac1\varepsilon\mathbb P(T_\varepsilon<T_M)
+\frac1M\mathbb P(T_M<T_\varepsilon),
$$
so
$$
\mathbb P(T_\varepsilon<T_M)=\frac{\varepsilon(M-x)}{x(M-\varepsilon)}\longrightarrow0.
$$
The exit time is finite because one coordinate of the underlying <Brownian motion> reaches $M$ in finite time. Letting $\varepsilon\downarrow0$ 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.
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