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A. Arnold, I. Gamba, M.P. Gualdani, S. Mischler, C. Mouhot, C. Sparber:
"The Wigner-Fokker-Planck equation: stationary states and large time behavior";
in: "ASC Report 25/2010", issued by: Institute for Analysis and Scientific Computing; Vienna University of Technology, W, 2010, ISBN: 978-3-902627-03-2.

We consider the linear Wigner-Fokker-Planck equation subject to
confining potentials which are smooth perturbations of the harmonic oscillator
potential. For a certain class of perturbations we prove that the equation admits
a unique stationary solution in a weighted Sobolev space. A key ingredient
of the proof is a new result on the existence of spectral gaps for Fokker-Planck
type operators in certain weighted L2-spaces. In addition we show that the
steady state corresponds to a positive density matrix operator with unit trace
and that the solutions of the time-dependent problem converge towards the
steady state with an exponential rate.

http://www.asc.tuwien.ac.at/preprint/2010/asc25x2010.pdf