= Winding at logarithmic radial passage levels
{title2=$\mathbb E e^{i\lambda\theta_{T_r}}=e^{-r|\lambda|},\quad T_r=\inf\{t:|B_t|=e^{-r}\}$}
For <planar Brownian motion> started at $1$, continuous winding angles sampled at the displayed radii have independent stationary increments. At $T_r$, apply the <Strong Markov property>, rotate by the current angle, and rescale space by $e^r$ and time by $e^{-2r}$. The future radial-level winding increment has the original law and is independent of the past. The logarithmic radius and winding angle are independent Brownian coordinates on the common clock $\int|B_s|^{-2}ds$. At radial passage they are therefore $U_{\eta_r}=-r$ and $V_{\eta_r}$, where $\eta_r$ is a <Brownian first-passage time>. Its <Laplace transform> gives the displayed characteristic function. The level process is a symmetric <Cauchy process>; replacing the passage clock by its right-continuous version supplies a <càdlàg> modification without changing any fixed-level law.
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