 Solution

The second coordinate ${\zeta }^2$ is one-dimensional Brownian motion, whose zero set is unbounded almost surely by recurrence. The clock $\tau$ and its inverse are continuous, increasing, and unbounded, so the unwrapped angle ${\Theta }_t={\zeta }_{\tau (t)}^2$ has zeros at arbitrarily large original times.

To ensure the times are stopping times for the original Brownian filtration, define directlyThe logarithmic lift makes $\Theta$ a continuous adapted process. Hitting a closed set after a stopping time is a stopping time under the usual filtration conditions, and the unbounded zero set proves inductively that each $T_n$ is finite almost surely. Since $T_n\ge T_{n-1}+1$, the sequence tends to infinity, and continuity givesGeometrically, the Brownian path returns to the positive real ray at arbitrarily late times with zero net unwrapped winding angle about the origin. The assertion is stronger than principal argument zero, which also allows any integer number of completed turns.