= Solution
Use the following deterministic boundary fact from the <Chordal Loewner equation>. For a continuous <trace of a Loewner chain> started at zero and generating its hulls, the real flow at $b>0$ has a <boundary swallowing time> $T_b$. Until that time it is the reflected boundary value of $g_t$, and
$$
X_t=g_t(b)-\xi_t>0.
$$
A finite $T_b$ is its first collision with the driver, $X_t\to0$ as $t\uparrow T_b$. Topologically, the boundary point $b$ is then visited or separated from infinity precisely when the trace has reached some point in $[b,\infty)$. A boundary crosscut can swallow an interval, so this is not a claim that the trace visits the particular point $b$.
Consequently the <SLE boundary swallowing criterion> is
$$
\boxed{\{\gamma\text{ hits }[b,\infty)\}=\{T_b<\infty\}.}
$$
For the <SLE> driver this real flow obeys
$$
dX_t=\frac2{X_t}\,dt-\sqrt\kappa\,dW_t,\qquad X_0=b,
$$
up to its first hit of zero. This is the <Boundary-point Bessel flow for SLE>.
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