= Logarithmic radius of planar Brownian motion
{title2=$M_t=\log|B_t|$}
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.
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