= Solution
In the <half-plane-capacity parameterization>, let $W$ be standard real <Brownian motion> and put $\xi_t=\sqrt\kappa\,W_t$. Solve the <Chordal Loewner equation>
$$
\partial_tg_t(z)=\frac2{g_t(z)-\xi_t},\qquad g_0(z)=z.
$$
The points for which this flow ceases to exist form the growing <compact H-hulls> $K_t$, with <half-plane capacity> $2t$. The process is an <SLE> trace when its initial segments generate these hulls and
$$
\boxed{\gamma_t=\lim_{y\downarrow0}g_t^{-1}(\xi_t+iy),
\qquad \xi_t=\sqrt\kappa\,W_t.}
$$
The limit is the continuous <trace of a Loewner chain> started at zero. If the curve disconnects a region from infinity, that region is included in $K_t$; the hull is not generally just the visited curve. For $\kappa=0$ the driver is identically zero and the trace is the deterministic vertical slit $\gamma_t=2i\sqrt t$.
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