= Cauchy winding law at the first unit-disc exit
{c}
{title2=$\theta_{T_{\mathbb D}}\sim\operatorname{Cauchy}(0,-\log|Z_0|)$}
For a start $Z_0=r_0e^{i\alpha_0}$ in the punctured <unit disc>, the logarithmic lift $-i\log Z$ starts at $\alpha_0-i\log r_0$ in the <upper half-plane>. Local <conformal invariance of planar Brownian motion> changes its clock by $\int|Z_s|^{-2}ds$. That clock is finite at the <disc> exit because the stopped path is <compact> and avoids zero. Its terminal real coordinate is therefore the usual upper-half-plane <Cauchy exit law from a Brownian half-plane>. Subtracting $\alpha_0$ gives the displayed winding law. Its first absolute moment diverges logarithmically, even though the final point modulo angle lies on a <compact> circle.
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