Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-23T12:31:31.606Z Has data issue: false hasContentIssue false
  • English
  • Français

GLOBAL SUBELLIPTIC ESTIMATES FOR KRAMERS–FOKKER–PLANCK OPERATORS WITH SOME CLASS OF POLYNOMIALS

Part of: Spectral theory and eigenvalue problems Close-to-elliptic equations and systems Equations of mathematical physics and other areas of application General theory of linear operators

Published online by Cambridge University Press:  22 June 2020

Mona Ben Said Open the ORCID record for Mona Ben Said [Opens in a new window]
Mona Ben Said*
Affiliation:
Laboratoire Analyse, Géométrie et Applications, Université Paris 13, 99 Avenue, Jean Baptiste Clément, 93430Villetaneuse, France ([email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

In this article, we study some Kramers–Fokker–Planck operators with a polynomial potential $V(q)$ of degree greater than two having quadratic limiting behaviour. This work provides an accurate global subelliptic estimate for Kramers–Fokker–Planck operators under some conditions imposed on the potential.

Keywords

subelliptic estimatescompact resolventKramers–Fokker–Planck operator

MSC classification

Primary: 35Q84: Fokker-Planck equations 35H20: Subelliptic equations 47A10: Spectrum, resolvent
Secondary: 35P05: General topics in linear spectral theory 14P10: Semialgebraic sets and related spaces
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 21 , Issue 2 , March 2022 , pp. 675 - 711
Check for updates
Copyright
© The Author(s) 2020. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aleman, A. and Viola, J., On weak and strong solution operators for evolution equations coming from quadratic operators, J. Spectr. Theory 8(1) (2018), 33–121.CrossRefGoogle Scholar
Ben Said, M., Nier, F. and Viola, J., Quaternionic structure and analysis of some Kramers–Fokker–Planck, preprint, 2018, arXiv:1807.01881.Google Scholar
Bony, J.-M. and Lerner, N., Quantification asymptotique et microlocalisations d’ordre supérieur, Ann. Sci. Éc. Norm. Supér. (4) 22 (1989), 377–433.CrossRefGoogle Scholar
Helffer, B. and Nier, F., Hypoelliptic Estimates and Spectral Theory for Fokker–Planck Operators and Witten Laplacians, Lecture Notes in Mathematics, Volume 1862, x + 209 pp (Springer-Verlag, Berlin, 2005).CrossRefGoogle Scholar
Helffer, B. and Nourrigat, J., Hypoellipticité maximale pour des opérateurs polynômes de champs de vecteurs, Progress in Mathematics, Volume 58 (Birkhäuser Boston Inc., 1985).Google Scholar
Hérau, F. and Nier, F., Isotropic hypoellipticity and trend to equilibrium for the Fokker–Planck equation with a high-degree potential, Arch. Ration. Mech. Anal. 171(2) (2004), 151–218.CrossRefGoogle Scholar
Hitrik, M. and Pravda-Starov, K., Spectra and semigroup smoothing for non-elliptic quadratic operators, Math. Ann. 344(4) (2009), 801–846.CrossRefGoogle Scholar
Hörmander, L., Symplectic classification of quadratic forms, and general Mehler formulas, Math. Z. 219 (1995), 413–449.CrossRefGoogle Scholar
Hörmander, L., The Analysis of Linear Partial Differential Operators. I. Distribution Theory and Fourier Analysis. Reprint of the second (1990) edition [Springer, Berlin; MR1065993], Classics in Mathematics, x + 440 pp (Springer-Verlag, Berlin, 2003).Google Scholar
Hörmander, L., The Analysis of Linear Partial Differential Operators. II. Differential Operators with Constant Coefficients. Reprint of the 1983 original, Classics in Mathematics, viii + 392 pp (Springer-Verlag, Berlin, 2005).Google Scholar
Li, W.-X., Global hypoellipticity and compactness of resolvent for Fokker–Planck operator, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 11(4) (2012), 789–815.Google Scholar
Li, W.-X., Compactness criteria for the resolvent of Fokker–Planck operator.prepublication, preprint, 2015, arXiv:1510.01567.Google Scholar
Nier, F., Hypoellipticity for Fokker–Planck operators and Witten Laplacians, in Lectures on the Analysis of Nonlinear Partial Differential Equations, Morningside Lecture in Mathematics, Volume 1, pp. 31–84 (Int. Press, Somerville, MA, 2012). Part 1.Google Scholar
Nourrigat, J., Subelliptic estimates for systems of pseudo-differential operators. Course in Recife. University of Recife, 1982.Google Scholar
Reed, M. and Simon, B., Fourier Analysis, Self-adjointness, xv + 361 pp (Academic Press, Harcourt Brace Jovanovich, New York, London, 1975).Google Scholar
Simon, B., Convexity: An Analytic Viewpoint, Cambridge Tracts in Mathematics, Volume 187 (Cambridge University Press, Cambridge, 2011).CrossRefGoogle Scholar
Villani, C., Hypocoercivity, Mem. Amer. Math. Soc. 202(950) (2009), iv + 141.Google Scholar
Viola, J., Spectral projections and resolvent bounds for partially elliptic quadratic differential operators, J. Pseudo-Differ. Oper. Appl. 4(2) (2013), 145–221.CrossRefGoogle Scholar
Viola, J., The elliptic evolution of non-self-adjoint degree-2 Hamiltonians, preprint, 2017, arXiv:1701.00801.Google Scholar