Smoothness of harmonic maps for hypoelliptic diffusions

Harmonic maps are viewed as maps sending a fixed diffusion to manifold-valued martingales. Under a convexity condition, we prove that the continuity of real-valued harmonic functions implies the continuity of harmonic maps. Then we prove with a probabilistic method that continuous harmonic maps are smooth under Hormander's condition; the proof relies on the study of martingales with values in the tangent bundle.