On the trend to global equilibrium in spatially inhomogeneous entropy-dissipating systems: The linear Fokker-Planck equation

We study the long-time behavior of kinetic equations in which transport and spatial confinement (in an exterior potential or in a box) are associated with a (degenerate) collision operator acting only in the velocity variable. We expose a general method, based on logarithmic Sobolev inequalities and the entropy, to overcome the well-known problem, due to the degeneracy in the position variable, of the existence of infinitely many local equilibria. This method requires that the solution be somewhat smooth. In this paper, we apply it to the linear Fokker-Planck equation and prove decay to equilibrium faster than O(t−1/ϵ) for all ϵ > 0. © 2001 John Wiley & Sons, Inc.