We provide a simple overview of some hypoellipticity results with sharp indices for a class of kinetic equations and we outline a general strategy based on some geometrical properties.

Harmonic maps are viewed as maps sending a xed diiusion 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 HH ormander's condition; the proof relies on the study of martingales with values in th...