The Wigner-Fokker-Planck equation: Stationary states and large time behavior

Gualdani, Maria Pia; Gamba, Irene M.; Mischler, Stéphane; Sparber, Christof; Arnold, Anton; Mouhot, Clément

Gualdani, Maria Pia; Gamba, Irene M.; Mischler, Stéphane; Sparber, Christof; Arnold, Anton; Mouhot, Clément (2012), The Wigner-Fokker-Planck equation: Stationary states and large time behavior, Mathematical Models and Methods in Applied Sciences, 22, 11, p. n°1250034. http://dx.doi.org/10.1142/S0218202512500340

Type

Article accepté pour publication ou publié

Date

2012

Journal name

Mathematical Models and Methods in Applied Sciences

Volume

Number

Publisher

World Scientific

Pages

n°1250034

Abstract (EN)

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.

Subjects / Keywords

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