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

Abstract

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 \(H^1\)-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 \(L^2\)-distance between a Fokker–Planck-solution and the steady state has always a sharp decay estimate of the order \(\mathcal O\big ( (1+t)e^{-t\nu /2}\big )\), with \(\nu \) the friction parameter. The presented method also gives new hypoelliptic regularization results for kinetic Fokker–Planck equations (from a weighted \(L^2\)-space to a weighted \(H^1\)-space).