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  • © 2005

Hypoelliptic Estimates and Spectral Theory for Fokker-Planck Operators and Witten Laplacians

Part of the book series: Lecture Notes in Mathematics (LNM, volume 1862)

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Table of contents (19 chapters)

  1. Front Matter

    Pages I-X
  2. 1. Introduction

    • Bernard Helffer, Francis Nier
    Pages 1-9
  3. 4. Global Pseudo-differential Calculus

    • Bernard Helffer, Francis Nier
    Pages 27-42
  4. 5. Analysis of Some Fokker-Planck Operator

    • Bernard Helffer, Francis Nier
    Pages 43-64
  5. 6. Return to Equilibrium for the Fokker-Planck Operator

    • Bernard Helffer, Francis Nier
    Pages 65-72
  6. 7. Hypoellipticity and Nilpotent Groups

    • Bernard Helffer, Francis Nier
    Pages 73-78
  7. 9. On Fokker-Planck Operators and Nilpotent Techniques

    • Bernard Helffer, Francis Nier
    Pages 89-95
  8. 13. Decay of Eigenfunctions and Application to the Splitting

    • Bernard Helffer, Francis Nier
    Pages 147-161
  9. 14. Semi-classical Analysis and Witten Laplacians: Morse Inequalities

    • Bernard Helffer, Francis Nier
    Pages 163-172
  10. 15. Semi-classical Analysis and Witten Laplacians: Tunneling Effects

    • Bernard Helffer, Francis Nier
    Pages 173-180
  11. 17. Application to the Fokker-Planck Equation

    • Bernard Helffer, Francis Nier
    Pages 189-191
  12. 18. Epilogue

    • Bernard Helffer, Francis Nier
    Pages 193-193
  13. References and Index

    • Bernard Helffer, Francis Nier
    Pages 195-209

About this book

There has recently been a renewal of interest in Fokker-Planck operators, motivated by problems in statistical physics, in kinetic equations, and differential geometry. Compared to more standard problems in the spectral theory of partial differential operators, those operators are not self-adjoint and only hypoelliptic. The aim of the analysis is to give, as generally as possible, an accurate qualitative and quantitative description of the exponential return to the thermodynamical equilibrium. While exploring and improving recent results in this direction, this volume proposes a review of known techniques on: the hypoellipticity of polynomial of vector fields and its global counterpart, the global Weyl-Hörmander pseudo-differential calculus, the spectral theory of non-self-adjoint operators, the semi-classical analysis of Schrödinger-type operators, the Witten complexes, and the Morse inequalities.

Keywords

  • Eigenvalue
  • Fokker-Planck operators
  • Hypoellipticity
  • Maximum
  • Witten Laplacians
  • calculus
  • compactness
  • compactness criteria
  • return to equilibrium
  • partial differential equations

Reviews

From the reviews of the first edition:

"The aim of this text is to give an account of how the known techniques from partial differential equations and spectral theory can be applied for the analysis of Fokker-Plank operators or Witten Laplacians … . This synthetic text is very challenging and useful for researchers in partial differential equations, probability theory and mathematical physics." (Viorel Iftimie, Zentralblatt MATH, Vol. 1072, 2005)

Bibliographic Information

Buying options

eBook USD 39.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book USD 54.99
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions