The Bakry-Emery condition applied to the functional inequality and contraction in Wasserstein distance

 Abstract : 

During the lessons we would like to introduce the Bakry-Emery condition in the Euclidean space R^n and also in a Riemannian manifold. First we study the two classical examples: the Euclidean Heat  semigroup and the Ornstein-Uhlenbeck semigroup. Then we define the general setting: a Markov semigroup associated to  the Gamma and the Gamma-2 operators. Under the Bakry-Emery condition we obtain coercive inequalities which implies for instance functional inequality such as Poincaré or logarithmic-Sobolev inequalities.  The last part of the course is devoted to the contraction in wasserstein distance under the Bakry-Emery condition.