= Solution
Use the <half-plane-capacity parameterization> in which $g_t(z)=z+2t/z+O(z^{-2})$ at infinity. The <Chordal Loewner equation> is
$$
\boxed{\partial_tg_t(z)=\frac{2}{g_t(z)-W_t},\qquad g_0(z)=z,\qquad W_t=\sqrt\kappa\,\beta_t.}
$$
It holds up to the <Loewner swallowing time> of $z$. Although $\beta$ is nowhere differentiable, it is continuous: the equation for $g$ is an ordinary integral equation with a continuous time-dependent coefficient away from its pole.
For $x\in\mathbb R\setminus\{0\}$, the same real-valued equation has a unique solution until $g_t(x)-W_t$ first reaches zero. Its solution agrees with the boundary value of the <conformal map> from the unswallowed side. One can also obtain it by <Schwarz reflection> across an unswallowed real interval. Thus
$$
\boxed{T(x)=\inf\{t:g_t(x)=W_t\}}
$$
is the <boundary-point swallowing time for a Loewner chain>; it can be infinite. The collision definition agrees with membership in the closed hull. The initial point has $T(0)=0$. Swallowing a real point is not the same as visiting it: a curve can cut off an entire real interval in one step.
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