THE WIGNER–FOKKER–PLANCK EQUATION: STATIONARY STATES AND LARGE TIME BEHAVIOR

ARNOLD, ANTON; GAMBA, IRENE M.; GUALDANI, MARIA PIA; MISCHLER, STÉPHANE; MOUHOT, CLEMENT; SPARBER, CHRISTOF

Type

Article

Authors

ARNOLD, ANTON
GAMBA, IRENE M.
GUALDANI, MARIA PIA
MISCHLER, STÉPHANE
MOUHOT, CLEMENT
SPARBER, CHRISTOF

Date

2012-09-10

Online Publication Date

2012-09-10

Print Publication Date

2012-11

Permanent link to this record

http://hdl.handle.net/10754/599977

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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 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 FokkerPlanck 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. © 2012 World Scientific Publishing Company.

Citation

ARNOLD A, GAMBA IM, GUALDANI MP, MISCHLER S, MOUHOT C, et al. (2012) THE WIGNER–FOKKER–PLANCK EQUATION: STATIONARY STATES AND LARGE TIME BEHAVIOR. Mathematical Models and Methods in Applied Sciences 22: 1250034. Available: http://dx.doi.org/10.1142/S0218202512500340.

Sponsors

A.A. acknowledges partial support from the FWF (project "Quantum Transport Equations: Kinetic, Relativistic, and Diffusive Phenomena" and Wissenschaftskolleg "Differentialgleichungen"), the OAD (Amadeus project). I.M.G. is supported by NSF-DMS 0807712. M.P.G. is supported by NSF-DMS-1109682. C.M. would like to thank Cambridge University for providing repeated hospitality in 2009, thanks to the Award No. KUK-I1-007-43, funded by the King Abdullah University of Science and Technology (KAUST). Support from the Institute of Computational Engineering and Sciences at the University of Texas at Austin is also gratefully acknowledged.