= Solution
For chordal <SLE> with parameter $\kappa$, the <Loewner trace> $\gamma$ starts at zero and is parametrized by <half-plane capacity> $2t$. Let $D_t$ be the unbounded component of $\mathbb H\setminus\gamma[0,t]$, and let $K_t=\mathbb H\setminus D_t$ be its filled <compact H-hull>. When the trace is not simple, the hull includes the regions it disconnects from infinity. The associated <mapping-out functions of compact H-hulls> have expansion
$$
g_t(z)=z+\frac{2t}{z}+O(z^{-2})
$$
and satisfy the <Chordal Loewner equation>
$$
\boxed{\partial_tg_t(z)=\frac2{g_t(z)-\xi_t},\qquad
g_0(z)=z,\qquad\xi_t=\sqrt\kappa\,B_t,}
$$
where $B$ is standard real <Brownian motion>. The equation is solved until $z$ is swallowed by the hull. The <Loewner transform> $\xi$ records the real boundary point at which the mapped hull grows; its relation to the trace is
$$
\gamma_t=\lim_{y\downarrow0}g_t^{-1}(\xi_t+iy).
$$
Equivalently the tip maps to $\xi_t$ in the appropriate boundary prime-end sense. Thus the flow encodes the surviving domains, the transform drives that flow, and the trace generates its hulls. For $\kappa=0$, the driver is identically zero, $g_t(z)=\sqrt{z^2+4t}$ with the branch asymptotic to $z$, and $\gamma_t=2i\sqrt t$.
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