Let $f$ be a mapping from an open set in ${\mathbf{R}}^p$ into ${\mathbf{R}}^q$, with $p\>q$. To say that $f$ preserves Brownian motion, up to a random change of clock, means that $f$ is harmonic and that its tangent linear mapping in proportional to a co-isometry. In the case $p=2$, $q=2$, such conditions signify that $f$ corresponds to an analytic function of one complex variable. We study, essentially that case $p=3$, $q=2$, in which we prove in particular that such a mapping cannot be “inner” if it is not trivial. A similar result for $p=4$, $q=2$ would solve...