= Solution
{c}
For a radial function $f(r)$ in three dimensions, the <Laplacian> is $f''+2f'/r$, while the imposed radial <drift> contributes $-\alpha f'/r$. Thus the radial <generator of an SDE diffusion> is
$$
L_rf=f''+\frac{2-\alpha}{r}f'.
$$
The <hitting probability> for the inner <sphere> is <harmonic> for this <diffusion generator> in the shell, with value one on the inner <sphere> and zero on the outer <sphere>. At $\alpha=1$, the <boundary value problem> is therefore
$$
p''(r)+\frac1rp'(r)=0,
\qquad p(R_1)=1,\qquad p(R_2)=0.
$$
Integrating $(rp')'=0$ gives $p(r)=A\log r+B$. The two <boundary conditions> determine
$$
\boxed{p(r_0)=\frac{\log(R_2/r_0)}{\log(R_2/R_1)}.}
$$
Both arguments of the <logarithms> are dimensionless. The result decreases from one to zero across the shell and lies strictly between zero and one for $R_1<r_0<R_2$.
For a direct <probability> interpretation, stop at the first contact $T$ with either <sphere>. The stopped process $p(r(t\wedge T))$ is a bounded <martingale> by <Itô's formula>. The shell has positive inner <radius>, so its <drift> is bounded and its <diffusion> is nondegenerate; exit occurs almost surely. <Optional stopping> and <dominated convergence> therefore give
$$
p(r_0)=\mathbb E p(r(T))=\mathbb P\bigl(r(T)=R_1\bigr).
$$
The <logarithm> appears because the inward <drift> cancels one of the two geometric $f'/r$ terms in the three-dimensional radial <Laplacian>, leaving the same radial <diffusion generator> as a planar <Brownian motion> with <diffusion coefficient> one.
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