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

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

doi:10.1142/S0218202512500340

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

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

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17 Scopus citations

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.

Original language	English (US)
Pages (from-to)	1250034
Journal	Mathematical Models and Methods in Applied Sciences
Volume	22
Issue number	11
DOIs	https://doi.org/10.1142/S0218202512500340
State	Published - Sep 10 2012
Externally published	Yes

Bibliographical note

KAUST Repository Item: Exported on 2020-10-01
Acknowledgements: 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.
This publication acknowledges KAUST support, but has no KAUST affiliated authors.

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title = "THE WIGNER–FOKKER–PLANCK EQUATION: STATIONARY STATES AND LARGE TIME BEHAVIOR",

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. {\textcopyright} 2012 World Scientific Publishing Company.",

author = "ANTON ARNOLD and GAMBA, {IRENE M.} and GUALDANI, {MARIA PIA} and ST{\'E}PHANE MISCHLER and CLEMENT MOUHOT and CHRISTOF SPARBER",

note = "KAUST Repository Item: Exported on 2020-10-01 Acknowledgements: 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. This publication acknowledges KAUST support, but has no KAUST affiliated authors.",

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AU - ARNOLD, ANTON

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AU - MISCHLER, STÉPHANE

AU - MOUHOT, CLEMENT

AU - SPARBER, CHRISTOF

N1 - KAUST Repository Item: Exported on 2020-10-01 Acknowledgements: 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. This publication acknowledges KAUST support, but has no KAUST affiliated authors.

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N2 - 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.

AB - 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.

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JO - Mathematical Models and Methods in Applied Sciences

JF - Mathematical Models and Methods in Applied Sciences

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ER -

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

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