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We consider a class of martingales on Cartan-Hadamard manifolds that includesBrownian motion on a minimal submanifold. We give sufficient conditions forsuch martingales to be transient, extending previous results on the transienceof minimal submanifolds. We also give conditions for the almost sureconvergence of the angular component (in polar coordinates) of a martingale inthis class, including both the negatively pinched case (using earlier resultson martingales of bounded dilation), and the radially symmetric case withquadratic decay of the upper curvature bound. Applied to minimal submanifolds,this gives curvature conditions on the ambient Cartan-Hadamard manifold underwhich any minimal submanifold admits a non-constant, bounded, harmonicfunction. Though our discussion is primarily motivated by minimal submanifolds,this class of martingales includes diffusions naturally associated to ancientsolutions of mean curvature flow and to certain sub-Riemannian structures, andwe briefly discuss these contexts as well. Our techniques are elementary,consisting mainly of comparison geometry and Ito's rule.

Martingales arising from minimal submanifolds and other geometric contexts