= Complex exponential construction of planar Brownian motion
{title2=$Z_t=e^{\beta_t+i\theta_t},\quad A_t=\int_0^t e^{2\beta_s}ds$}
For independent one-dimensional <Brownian motions> $\beta,\theta$, the real and imaginary parts of $Z$ are continuous local martingales with equal bracket $A$ and zero cross bracket. The <Divergence of a driftless Brownian exponential clock> makes $A_\infty=\infty$, and the <common-clock time change of orthogonal local martingales> makes $Z_{A^{-1}(r)}$ planar Brownian motion started at $1$. The exponential never vanishes, proving that this Brownian motion never hits zero. Recurrent visits of $\beta$ to arbitrarily negative levels prove visits arbitrarily near zero at arbitrarily large Brownian times.
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