= Solution
The <half-plane capacity> is the Laurent coefficient
$$
\boxed{\operatorname{hcap}(K)=a_K=\lim_{y\to\infty}iy\bigl(g_K(iy)-iy\bigr).}
$$
It is real and nonnegative. For <planar Brownian motion> starting at $z\in D$, let $\tau_D$ be its first exit from $D$. The <Brownian representation of half-plane capacity> is
$$
\boxed{\operatorname{hcap}(K)=\lim_{y\to\infty}y\,\mathbb E_{iy}\bigl[\operatorname{Im}B_{\tau_D}\bigr].}
$$
One may also state the underlying harmonic identity, valid throughout $D$,
$$
\operatorname{Im}g_K(z)=\operatorname{Im}z-\mathbb E_z[\operatorname{Im}B_{\tau_D}].
$$
The process is killed at the hull or real boundary; real-boundary exits contribute zero. These are the Brownian characterizations, with the normalization that a <Chordal Loewner equation> at speed $2$ produces capacity $2t$.
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