We first discuss a class of degenerate martingales (which we will call rank-n martingales) that arises naturally as the diffusion associated with minimal submanifolds, mean curvature flow, and some sub-Riemannian structures. This provides a unified approach to "coarse" properties, such as transience, of such structures. We then specialize to minimal surfaces in R^3, in which case the associated rank-2 martingale (which is just Brownian motion on the surface, viewed as a process in R^3) has the additional property that the tangent plane also evolves as a martingale. Taking advantage of this extra structure, we develop an extrinsic analogue of the mirror coupling of two Brownian motions. This allows us to study finer geometric and analytic properties of minimal surfaces, such as intersection results (strong halfspace-type theorems) and Liouville properties.