  The classical Bakry-\'Emery calculus is extended to study, for degenerated
(non-elliptic, non-reversible, or non-diffusive) Markov processes, questions
such as hypoellipticity, hypocoercivity, functional inequalities or Wasserstein
contraction. In particular we obtain the optimal speed of convergence to
equilibrium for any ergodic Ornstein-Uhlenbeck process, which is given by the
spectral gap of the drift matrix and the size of the corresponding Jordan
blocks. We also study chains of $N$ interacting overdamped particles and
establish for their invariant measures log-Sobolev inequalities with constants
of order $N^2$, which is optimal.
