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

Abstract

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