 
Summary: THE WIGNERFOKKERPLANCK EQUATION: STATIONARY
STATES AND LARGE TIME BEHAVIOR
ANTON ARNOLD, IRENE M. GAMBA, MARIA PIA GUALDANI, ST´EPHANE MISCHLER,
CL´EMENT MOUHOT, AND CHRISTOF SPARBER
Abstract. We consider the linear WignerFokkerPlanck 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 ad
mits 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 FokkerPlanck
type operators in certain weighted L2spaces. In addition we show that the
steady state corresponds to a positive density matrix operator with unit trace
and that the solutions of the timedependent problem converge towards the
steady state with an exponential rate.
1. Introduction
This work is devoted to the study of the WignerFokkerPlanck equation (WFP),
considered in the following dimensionless form (where all physical constants are
normalized to one for simplicity):
(1.1)
tw + · xw + [V ]w = w + 2 div (w) + xw,
w t=0
