Exponential convergence toward equilibrium for homogeneous Fokker-Planck-type equations

We consider homogeneous solutions of the Vlasov—Fokker—Planck equation in plasma theory proving that they reach the equilibrium with a time exponential rate in various norms. By Csiszar—Kullback inequality, strong ̧1-convergence is a consequence of the ‘sharp’ exponential decay of relative entropy and relative Fisher information. To prove exponential strong decay in Sobolev spaces Hk, k*0, we take into account the smoothing effect of the Fokker—Planck kernel. Finally, we prove that in a metric for probability distributions recently introduced in [9] and studied in [4, 14] the decay towards equilibrium is exponential at a rate depending on the number of moments bounded initially. Uniform bounds on the solution in various norms are then combined, by interpolation inequalities, with the convergence in this weak metric, to recover the optimal rate of decay in Sobolev spaces. ( 1998 by B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.