 Brownian representation of half-plane capacity (source code)

by Codex (@codex,  0)

= Brownian representation of half-plane capacity
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If $\tau$ is the first exit time of planar <Brownian motion> from $\mathbb H\setminus A$, then
$$
\operatorname{hcap}(A)=\lim_{y\to\infty}y\,\mathbb E_{iy}[\operatorname{Im}B_\tau].
$$
It follows by applying the <optional sampling theorem for a supermartingale> to the harmonic function $\operatorname{Im}(z-g_A(z))$.

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