Also appears in collection : Sub-Riemannian manifolds : from geodesics to hypoelliptic diffusion / Géométrie sous-riemannienne : des géodésiques aux diffusions hypoelliptiques

We discuss hypoelliptic and subelliptic diffusions; the lectures include the following topics: Malliavin calculus; Hormander's theorem; smoothness of transition probabilities under Hormander's brackets condition; control theory and Stroock-Varadhan's support theorems; hypoelliptic heat kernel estimates; gradient estimates and Harnack type inequalities for subelliptic diffusion semi-groups; notions of curvature related to sub-Riemannian diffusions.