The Wigner-Fokker-Planck equation: Stationary states and large time behaviorReport as inadecuate




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1 Institut für Analysis und Scientific Computing 2 Departement of Mathematics Austin 3 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision 4 DPMMS-CMS 5 UIC - Department of Mathematics, Statistics and Computer Science Chicago

Abstract : 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 $L^2$-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.

Keywords : Wigner transform Fokker Planck operator Spectral gap Stationary solution Large time behavior





Author: Anton Arnold - Irene Gamba - Maria Pia Gualdani - Stéphane Mischler - Clément Mouhot - Christof Sparber -

Source: https://hal.archives-ouvertes.fr/



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