= Logarithm of planar Brownian radius
{title2=$X_t=\log|B_t|$}
For planar Brownian motion started at zero, $\log|B_t|$, considered from time one onward, is a local martingale because $\log|x|$ is harmonic off the polar point zero. Every fixed-time absolute moment is finite, but $\mathbb EX_t=\frac12\log t+\mathbb E\log|B_1|$ increases. It is therefore not a true martingale. Finite moments at individual times do not establish class DL.
// Target: probability-and-statistics.bigb
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