= Solution
For the <Bessel process>
$$
dX_t=dB_t+\frac{d-1}{2X_t}dt,
$$
the <scale function of a one-dimensional diffusion> is $s(x)=x^{2-d}$ when $d\ne2$. If $0<\epsilon<x<R$, the <boundary hitting probability from a diffusion scale function> gives
$$
\mathbb P_x(T_R<T_\epsilon)
=\frac{s(x)-s(\epsilon)}{s(R)-s(\epsilon)}.
$$
If $d<2$, then $s(0)=0$ and $s(R)\to\infty$. Letting $\epsilon\downarrow0$ and then $R\to\infty$ shows that the process cannot escape to infinity before reaching zero. The exit time from each bounded interval is finite almost surely, so $T_0<\infty$ almost surely.
If $d>2$, then $s(\epsilon)\to\infty$ in absolute value as $\epsilon\downarrow0$. Equivalently,
$$
\mathbb P_x(T_\epsilon<T_R)
=\frac{s(R)-s(x)}{s(R)-s(\epsilon)}
\longrightarrow0.
$$
Thus the process does not hit zero. The borderline case $d=2$ has scale function $\log x$ and also does not hit zero. This is the <Hitting-zero classification for a Bessel process>.
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