Coupling Brownian motions in a useful tool to obtain results in Probability, Geometry and Analysis. In particular, by studying the first meeting time of Brownian motions, we can obtain gradient inequalities for the heat semi group and harmonic functions with explicit constants. In the case of subRiemannian manifolds, this coupling method is particularly interesting as it can deal with some of those problems without the intervention of some geometric or analytic objects that are difficult to define and manipulate.
After an introduction to the subRiemannian structure, we will present and compare several explicit markovian and non markovian coupling methods on some subRiemannian manifolds: the free step 2 Carnot groups (including the Heisenberg group), $SU(2)$ and $SL(2,\mathbb{R})$. We will also present the different inequalities that can be obtained with these methods.
