Cauchy winding law at the first unit-disc exit (${\theta }_{T_{\mathbb{D}}}\sim \mathrm{Cauchy}(0,-\log |Z_0|)$)

For a start $Z_0=r_0{e}^{i{\alpha }_0}$ 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.