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

We study the long-rime 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 equili brium faster than O(t(-1/epsilon)) for all epsilon > 0. (C) 2001 John Wiley & Sons, Inc.