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

Handle URI:
http://hdl.handle.net/10754/599977
Title:
THE WIGNER–FOKKER–PLANCK EQUATION: STATIONARY STATES AND LARGE TIME BEHAVIOR
Authors:
ARNOLD, ANTON; GAMBA, IRENE M.; GUALDANI, MARIA PIA; MISCHLER, STÉPHANE; MOUHOT, CLEMENT; SPARBER, CHRISTOF
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.
Publisher:
World Scientific Pub Co Pte Lt
Journal:
Mathematical Models and Methods in Applied Sciences
Issue Date:
Nov-2012
DOI:
10.1142/S0218202512500340
Type:
Article
ISSN:
0218-2025; 1793-6314
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.
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Full metadata record

DC FieldValue Language
dc.contributor.authorARNOLD, ANTONen
dc.contributor.authorGAMBA, IRENE M.en
dc.contributor.authorGUALDANI, MARIA PIAen
dc.contributor.authorMISCHLER, STÉPHANEen
dc.contributor.authorMOUHOT, CLEMENTen
dc.contributor.authorSPARBER, CHRISTOFen
dc.date.accessioned2016-02-28T06:33:40Zen
dc.date.available2016-02-28T06:33:40Zen
dc.date.issued2012-11en
dc.identifier.citationARNOLD 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.en
dc.identifier.issn0218-2025en
dc.identifier.issn1793-6314en
dc.identifier.doi10.1142/S0218202512500340en
dc.identifier.urihttp://hdl.handle.net/10754/599977en
dc.description.abstractWe 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.en
dc.description.sponsorshipA.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.en
dc.publisherWorld Scientific Pub Co Pte Lten
dc.subjectFokkerPlanck operatoren
dc.subjectlarge-time behavioren
dc.subjectspectral gapen
dc.subjectstationary solutionen
dc.subjectWigner transformen
dc.titleTHE WIGNER–FOKKER–PLANCK EQUATION: STATIONARY STATES AND LARGE TIME BEHAVIORen
dc.typeArticleen
dc.identifier.journalMathematical Models and Methods in Applied Sciencesen
dc.contributor.institutionTechnische Universitat Wien, Vienna, Austriaen
dc.contributor.institutionUniversity of Texas at Austin, Austin, United Statesen
dc.contributor.institutionCentre de Recherche en Mathematiques de la Decision, Paris, Franceen
dc.contributor.institutionUniversity of Cambridge, Cambridge, United Kingdomen
dc.contributor.institutionUniversity of Illinois at Chicago, Chicago, United Statesen
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