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 Hörmander’s condition; the proof relies on the study of martingales with values in the tangent bundle.
"Smoothness of harmonic maps for hypoelliptic diffusions." Ann. Probab. 28 (2) 643 - 666, April 2000. https://doi.org/10.1214/aop/1019160255