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)

• Kohn's Proof of the Hypoellipticity of the Hörmander Operators.- Compactness Criteria for the Resolvent of Schrödinger Operators.- Global Pseudo-differential Calculus.- Analysis of some Fokker-Planck Operator.- Return to Equillibrium for the Fokker-Planck Operator.- Hypoellipticity and Nilpotent Groups.- Maximal Hypoellipticity for Polynomial of Vector Fields and Spectral Byproducts.- On Fokker-Planck Operators and Nilpotent Techniques.- Maximal Microhypoellipticity for Systems and Applications to Witten Laplacians.- Spectral Properties of the Witten-Laplacians in Connection with Poincaré Inequalities for Laplace Integrals.- Semi-classical Analysis for the Schrödinger Operator: Harmonic Approximation.- Decay of Eigenfunctions and Application to the Splitting.- Semi-classical Analysis and Witten Laplacians: Morse Inequalities.- Semi-classical Analysis and Witten Laplacians: Tunneling Effects.- Accurate Asymptotics for the Exponentially Small Eigenvalues of the Witten Laplacian.- Application to the Fokker-Planck Equation.- Epilogue.- Index.

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)

• Kohn's Proof of the Hypoellipticity of the Hörmander Operators.- Compactness Criteria for the Resolvent of Schrödinger Operators.- Global Pseudo-differential Calculus.- Analysis of some Fokker-Planck Operator.- Return to Equillibrium for the Fokker-Planck Operator.- Hypoellipticity and Nilpotent Groups.- Maximal Hypoellipticity for Polynomial of Vector Fields and Spectral Byproducts.- On Fokker-Planck Operators and Nilpotent Techniques.- Maximal Microhypoellipticity for Systems and Applications to Witten Laplacians.- Spectral Properties of the Witten-Laplacians in Connection with Poincaré Inequalities for Laplace Integrals.- Semi-classical Analysis for the Schrödinger Operator: Harmonic Approximation.- Decay of Eigenfunctions and Application to the Splitting.- Semi-classical Analysis and Witten Laplacians: Morse Inequalities.- Semi-classical Analysis and Witten Laplacians: Tunneling Effects.- Accurate Asymptotics for the Exponentially Small Eigenvalues of the Witten Laplacian.- Application to the Fokker-Planck Equation.- Epilogue.- Index.