Exponential Stability and Hypoelliptic Regularization for the Kinetic Fokker-Planck Equation with Confining Potential.

This paper is concerned with a modified entropy method to establish the large-time convergence towards the (unique) steady state, for kinetic Fokker-Planck equations with non-quadratic confinement potentials in whole space. We extend previous approaches by analyzing Lyapunov functionals with non-constant weight matrices in the dissipation functional (a generalized Fisher information). We establish exponential convergence in a weighted H1-norm with rates that become sharp in the case of quadratic potentials. In the defective case for quadratic potentials, i.e. when the drift matrix has non-trivial Jordan blocks, the weighted L2-distance between a Fokker-Planck-solution and the steady state has always a sharp decay estimate of the order O((1+t)e-tν/2), with ν the friction parameter. The presented method also gives new hypoelliptic regularization results for kinetic Fokker-Planck equations (from a weighted L2-space to a weighted H1-space).