Hypoelliptic Estimates and Spectral Theory for Fokker-Planck Operators and Witten Laplacians (Lecture Notes in Mathematics Book 1862)

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. 1. Introduction....Pages 1-9 2. Kohn’s Proof of the Hypoellipticity of the Hörmander Operators....Pages 11-18 3. Compactness Criteria for the Resolvent of Schrödinger Operators....Pages 19-26 4. Global Pseudo-differential Calculus....Pages 27-42 5. Analysis of Some Fokker-Planck Operator....Pages 43-64 6. Return to Equilibrium for the Fokker-Planck Operator....Pages 65-72 7. Hypoellipticity and Nilpotent Groups....Pages 73-78 8. Maximal Hypoellipticity for Polynomial of Vector Fields and Spectral Byproducts....Pages 79-87 9. On Fokker-Planck Operators and Nilpotent Techniques....Pages 89-95 10. Maximal Microhypoellipticity for Systems and Applications to Witten Laplacians....Pages 97-112 11. Spectral Properties of the Witten-Laplacians in Connection with Poincaré Inequalities for Laplace Integrals....Pages 113-131 12. Semi-classical Analysis for the Schrödinger Operator: Harmonic Approximation....Pages 133-145 13. Decay of Eigenfunctions and Application to the Splitting....Pages 147-161 14. Semi-classical Analysis and Witten Laplacians: Morse Inequalities....Pages 163-172 15. Semi-classical Analysis and Witten Laplacians: Tunneling Effects....Pages 173-180 16. Accurate Asymptotics for the Exponentially Small Eigenvalues of $\Delta_{f,h}^{(0)}$ ....Pages 181-188 17. Application to the Fokker-Planck Equation....Pages 189-191 18. Epilogue....Pages 193-193 References and Index....Pages 195-209 There has been recently 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 thisvolume 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