 Solution

Use the half-plane-capacity parameterization in which $g_t(z)=z+2t/z+O(z^{-2})$ at infinity. The Chordal Loewner equation isIt 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. Thusis 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.