Global Analysis in Mathematical Physics: Geometric and Stochastic Methods

'I. Finite-Dimensional Differential Geometry and Mechanics.- 1 Some Geometric Constructions in Calculus on Manifolds.- 1. Complete Riemannian Metrics and the Completeness of Vector Fields.- 1.A A Necessary and Sufficient Condition for the Completeness of a Vector Field.- 1.B A Way to Construct Complete Riemannian Metrics.- 2. Riemannian Manifolds Possessing a Uniform Riemannian Atlas.- 3. Integral Operators with Parallel Translation.- 3.A The Operator S.- 3.B The Operator ?.- 3.C Integral Operators.- 2 Geometric Formalism of Newtonian Mechanics.- 4. Geometric Mechanics: Introduction and Review of Standard Examples.- 4.A Basic Notions.- 4.B Some Special Classes of Force Fields.- 4.C Mechanical Systems on Groups.- 5. Geometric Mechanics with Linear Constraints.- 5.A Linear Mechanical Constraints.- 5.B Reduced Connections.- 5.CLength Minimizing and Least-Constrained Nonholonomic Geodesics.- 6. Mechanical Systems with Discontinuous Forces and Systems with Control: Differential Inclusions.- 7. Integral Equations of Geometric Mechanics: The Velocity Hodograph.- 7.A General Constructions.- 7.B Integral Formalism of Geometric Mechanics with Constraints.- 8. Mechanical Interpretation of Parallel Translation and Systems with Delayed Control Force.- 3 Accessible Points of Mechanical Systems.- 9. Examples of Points that Cannot Be Connected by a Trajectory.- 10. The Main Result on Accessible Points.- 11. Generalizations to Systems with Constraints.- II. Stochastic Differential Geometry and its Applications to Physics.- 4 Stochastic Differential Equations on Riemannian Manifolds.- 12. Review of the Theory of Stochastic Equations and Integrals on Finite-Dimensional Linear Spaces.- 12.A Wiener Processes.- 12.B The Itô Integral.- 12.C The Backward Integral and the Stratonovich Integral.- 12.D The Itô and Stratonovich Stochastic Differential Equations.- 12.E Solutions of SDEs.- 12.F Approximation by Solutions of Ordinary Differential Equations.- 12.G A Relationship Between SDEs and PDEs.- 13. Stochastic Differential Equations on Manifolds.- 14. Stochastic Parallel Translation and the Integral Formalism for the Itô Equations.- 15. Wiener Processes on Riemannian Manifolds and Related Stochastic Differential Equations.- 15.A Wiener Processes on Riemannian Manifolds.- 15.B Stochastic Equations.- 15.C Equations with Identity as the Diffusion Coefficient.- 16. Stochastic Differential Equations with Constraints.- 5 The Langevin Equation.- 17. The Langevin Equation of Geometric Mechanics.- 18. Strong Solutions of the Langevin Equation, Ornstein-Uhlenbeck Processes.- 6 Mean Derivatives, Nelson's Stochastic Mechanics, and Quantization.- 19. More on Stochastic Equations and Stochastic Mechanics in ?n.- 19.A Preliminaries.- 19.B Forward Mean Derivatives.- 19.C Backward Mean Derivatives and Backward Equations.- 19.D Symmetric and Antisymmetric Derivatives.- 19.E The Derivatives of a Vector Field Along ?(t) and the Acceleration of ?(t).- 19.F Stochastic Mechanics.- 20. Mean Derivatives and Stochastic Mechanics on Riemannian Manifolds.- 20.A Mean Derivatives on Manifolds and Related Equations.- 20.B Geometric Stochastic Mechanics.- 20.C The Existence of Solutions in Stochastic Mechanics.- 21. Relativistic Stochastic Mechanics.- III. Infinite-Dimensional Differential Geometry and Hydrodynamics.- 7 Geometry of Manifolds of Diffeomorphisms.- 22. Manifolds of Mappings and Groups of Diffeomorphisms.- 22.A Manifolds of Mappings.- 22.B The Group of H8-Diffeomorphisms.- 22.C Diffeomorphisms of a Manifold with Boundary.- 22.D Some Smooth Operators and Vector Bundles over Ds(M).- 23. Weak Riemannian Metrics and Connections on Manifolds of Diffeomorphisms.- 23.A The Case of a Closed Manifold.- 23.B The Case of a Manifold with Boundary.- 23.C The Strong Riemannian Metric.- 24. Lagrangian Formalism of Hydrodynamics of an Ideal Barotropic Fluid.- 24.A Diffuse Matter.- 24.B A