Coercivity, hypocoercivity, exponential time decay and simulations for discrete Fokker- Planck equations

International audienceIn this article, we propose and study several discrete versions of homogeneous and inhomogeneous one-dimensional Fokker-Planck equations. In particular, for these dis-cretizations of velocity and space, we prove the exponential convergence to the equilibrium of the solutions, for time-continuous equations as well as for time-discrete equations. Our method uses new types of discrete Poincaré inequalities for a " two-direction " discretization of the derivative in velocity. For the inhomogeneous problem, we adapt hypocoercive methods to the discrete cases