Logarithm of planar Brownian radius ($X_t=\log |B_t|$)

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 $\mathbb{E}{X}_t=\frac{1}{2}\log t+\mathbb{E}\log |B_1|$ increases. It is therefore not a true martingale. Finite moments at individual times do not establish class DL.

// Target: probability-and-statistics.bigb