Logarithmic radius of planar Brownian motion ($M_t=\log |B_t|$)

Started at radius one, planar Brownian motion avoids the origin, and its logarithmic radius is a continuous local martingale because the logarithmic potential is harmonic away from zero. It is a strict local martingale: at deterministic $t>0$, the angular average of a logarithmic potential gives $\mathbb{E}\log |B_t|={\int }_1^{\infty }{e}^{-r^2/(2t)}dr/r>0$, whereas the initial value is zero. Absolute integrability follows from integrability of $r|\log r|$ near zero and the Gaussian tail. Its positive expectation does not contradict nonnegative-local-martingale bounds, since this process takes both signs.

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.