  We consider a class of martingales on Cartan-Hadamard manifolds that includes
Brownian motion on a minimal submanifold. We give sufficient conditions for
such martingales to be transient, extending previous results on the transience
of minimal submanifolds. We also give conditions for the almost sure
convergence of the angular component (in polar coordinates) of a martingale in
this class, including both the negatively pinched case (using earlier results
on martingales of bounded dilation), and the radially symmetric case with
quadratic decay of the upper curvature bound. Applied to minimal submanifolds,
this gives curvature conditions on the ambient Cartan-Hadamard manifold under
which any minimal submanifold admits a non-constant, bounded, harmonic
function. Though our discussion is primarily motivated by minimal submanifolds,
this class of martingales includes diffusions naturally associated to ancient
solutions of mean curvature flow and to certain sub-Riemannian structures, and
we briefly discuss these contexts as well. Our techniques are elementary,
consisting mainly of comparison geometry and Ito's rule.
