Global Analysis in Mathematical Physics : Geometric and Stochastic Methods

The first edition of this book entitled Analysis on Riemannian Manifolds and Some Problems of Mathematical Physics was published by Voronezh Univer­ sity Press in 1989. For its English edition, the book has been substantially revised and expanded. In particular, new material has been added to Sections 19 and 20. I am grateful to Viktor L. Ginzburg for his hard work on the transla­ tion and for writing Appendix F, and to Tomasz Zastawniak for his numerous suggestions. My special thanks go to the referee for his valuable remarks on the theory of stochastic processes. Finally, I would like to acknowledge the support of the AMS fSU Aid Fund and the International Science Foundation (Grant NZBOOO), which made possible my work on some of the new results included in the English edition of the book. Voronezh, Russia Yuri Gliklikh September, 1995 Preface to the Russian Edition The present book is apparently the first in monographic literature in which a common treatment is given to three areas of global analysis previously consid­ ered quite distant from each other, namely, differential geometry and classical mechanics, stochastic differential geometry and statistical and quantum me­ chanics, and infinite-dimensional differential geometry of groups of diffeomor­ phisms and hydrodynamics. The unification of these topics under the cover of one book appears, however, quite natural, since the exposition is based on a geometrically invariant form of the Newton equation and its analogs taken as a fundamental law of motion.

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 Barotropic Fluid.- 8 Lagrangian Formalism of Hydrodynamics of an Ideal Incompressible Fluid.- 25. Geometry of the Manifold of Volume-Preserving Diffeomorphisms and LHSs of an Ideal Incompressible Fluid.- 25.A Volume-Preserving Diffeomorphisms of a Closed Manifold.- 25.B Volume-Preserving Diffeomorphisms of a Manifold with Boundary.- 25.C LHS’s of an Ideal Incompressible Fluid.- 26. The Flow of an Ideal Incompressible Fluid on a Manifold with Boundary as an LHS with an Infinite-Dimensional Constraint on the Group of Diffeomorphisms of a Closed Manifold.- 27. The Regularity Theorem and a Review of Results on the Existence of Solutions.- 9 Hydrodynamics of a Viscous Incompressible Fluid and Stochastic Differential Geometry of Groups of Diffeomorphisms.- 28. Stochastic Differential Geometry on the Groups of Diffeomorphisms of the n-Dimensional Torus.- 29. A Viscous Incompressible Fluid.- Appendices.- A. Introduction to the Theory of Connections.- Connections on Principal Bundles.- Connections on the Tangent Bundle.- Covariant Derivatives.- Connection Coefficients and Christoffel Symbols.- Second-Order Differential Equations and the Spray.- The Exponential Map and Normal Charts.- B. Introduction to the Theory of Set-Valued Maps.- C. Basic Definitions of Probability Theory and the Theory of Stochastic Processes.- Stochastic Processes and Cylinder Sets.- The Conditional Expectation.- Markovian Processes.- Martingales and Semimartingales.- D. The Itô Group and the Principal Itô Bundle.- E. Sobolev Spaces.- F. Accessible Points and Closed Trajectories of Mechanical Systems (by Viktor L. Ginzburg).- Growth of the Force Field and Accessible Points.- Accessible Points in Systems with Constraints.- Closed Trajectories of Mechanical Systems.- References.