Global and Stochastic Analysis with Applications to Mathematical Physics (eBook)

Yuri E. Gliklikh (Autor)

eBook Download: PDF

2010 | 2011
XXIV, 436 Seiten
Springer London (Verlag)
978-0-85729-163-9 (ISBN)

Infos

<p>Methods of global analysis and stochastic analysis are most often applied in mathematical physics as separate entities, thus forming important directions in the field. However, while combination of the two subject areas is rare, it is fundamental for the consideration of a broader class of problems.</p>

<p>This book develops methods of Global Analysis and Stochastic Analysis such that their combination allows one to have a more or less common treatment for areas of mathematical physics that traditionally are considered as divergent and requiring different methods of investigation.</p>

<p>Global and Stochastic Analysis with Applications to Mathematical Physics covers branches of mathematics that are currently absent in monograph form. Through the demonstration of new topics of investigation and results, both in traditional and more recent problems, this book offers a fresh perspective on ordinary and stochastic differential equations and inclusions (in particular, given in terms of Nelson's mean derivatives) on linear spaces and manifolds. Topics covered include classical mechanics on non-linear configuration spaces, problems of statistical and quantum physics, and hydrodynamics.</p>

<p>A self-contained book that provides a large amount of preliminary material and recent results which will serve to be a useful introduction to the subject and a valuable resource for further research. It will appeal to researchers, graduate and PhD students working in global analysis, stochastic analysis and mathematical physics.</p>

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The main aim of this book is to develop the methods of Global Analysis and of Stochastic Analysis such that their combination allows one to have a more or less common treatment for areas of mathematical physics that traditionally are considered as quite distant from each other and requiring different methods of investigation. Among those areas we mention classical mechanics on non-linear configuration spaces, some problems of statistical and quantum physics, hydrodynamics, etc. The idea, yielding the unification of these topics, is based on the use of a geometrically invariant form of Newton's second law and its analogs (stochastic, set-valued, infinite-dimensional, etc.) as a fundamental equation of motion.

Inhalt

Preface 7
Contents 9
Introduction 14
Global Analysis 23
Manifolds and Related Objects 24
Manifolds, Vectors and Covectors. A Glossary 24
Lie Groups and Lie Algebras 32
Fiber Bundles 35
Riemannian and Semi-Riemannian Metrics 39
Tensors 42
Differential Forms and Polyvectors 46
The Lie Derivative 53
Connections 56
The Structure of a Tangent Bundle to a Vector Bundle 56
Connections on Vector Bundles 61
Connections on Manifolds 71
Geodesics 74
Curvature and Torsion Tensors 76
Riemannian Connections. The Levi-Civitá Connection 77
Connections on Principal Bundles 81
A Connection on the Total Space of a Vector Bundle 85
Second Order Tangent Vectors and Connections 86
Ordinary Differential Equations 88
Global in Time Existence of Solutions of Ordinary Differential Equations 88
A necessary and sufficient condition for completeness of a vector field of one-sided type 88
A generalization to the infinite-dimensional case 91
A necessary and sufficient condition for completeness of a vector field of two-sided type 103
Some sufficient conditions 105
Integral Operators with Parallel Translation 107
The operator S 107
The operator 110
Integral operators 111
Second Order Differential Equations (Special Vector Fields) 113
Elements of the Theory of Set-Valued Mappings 118
Set-Valued Mappings and Differential Inclusions 118
Special Approximations 121
Analysis on Groups of Diffeomorphisms 126
General Concepts 126
The Group of Diffeomorphisms of a Flat Torus 132
Stochastic Analysis 134
Essentials from Stochastic Analysis in Linear Spaces 135
Some Definitions from Probability Theory and the Theory of Stochastic Processes 135
Stochastic processes. Cylinder sets 135
Conditional expectation 137
Markov processes 138
Martingales and semi-martingales 138
Weak convergence of probability measures 139
A Survey on Stochastic Integrals and Equations 140
White noise and Wiener processes 140
Stochastic integrals 143
Stochastic differential equations 148
Stochastic Flows and their Generators 155
Stochastic Analysis on Manifolds 159
Stochastic Differential Equations in Stratonovich Form on a Manifold 159
General construction 159
Riemannian uniform atlases 163
The Itô Bundle and Itô Equations on a Manifold 166
Itô Equations in Belopolskaya-Daletskii Form 171
Completeness of Stochastic Flows 178
Setting up the problem and a necessary condition for completeness 178
A necessary and sufficient condition for completeness of flows continuous at infinity 179
Remarks on L1-complete stochastic flows 183
A Condition for Weak Compactness of Measures Corresponding to Solutions of Stochastic Differential Equations 184
Stochastic Development and Parallel Translation 188
The Eells-Elworthy and Itô developments 188
Wiener processes on Riemannian manifolds. Stochastic completeness 192
Parallel translation along a stochastic process. Itô processes on manifolds 196
The Integral Approach to Stochastic Differential Equations on Manifolds 197
General constructions 197
Stochastic differential equations in terms of Wiener processes in tangent spaces 202
Equations with unit diffusion coefficients 204
Mean Derivatives in Linear Spaces 207
General Definitions and Results 207
Calculation of Mean Derivatives for a Wiener Process and for Diffusion Processes 219
Calculation of Mean Derivatives for Itô Processes 223
First Order Differential Equations and Inclusions with Mean Derivatives 229
The Case of P-mean Derivatives 240
Mean Derivatives on Manifolds 245
Forward and Backward Mean Derivatives 245
Current and Osmotic Velocities 250
Mean Derivatives of Vector Fields Along Stochastic Processes 252
The Quadratic Mean Derivative 254
Mean Derivatives of Itô Processes on Manifolds 257
Equations and Inclusions with Mean Derivatives 259
Stochastic Differential Inclusions in Terms of Infinitesimal Generators 263
Stochastic Analysis on Groups of Diffeomorphisms 267
The General Case 267
The Case of a Flat Torus 269
Applications to Mathematical Physics 273
Newtonian Mechanics 274
A Geometric Language for Newtonian Mechanics 274
Mechanical Systems on Lie Groups 276
Conservative Mechanical Systems 277
Hamilton's Principle of Least Action 279
Noether's Theorem 282
Geometric Mechanics with Linear Constraints 285
The notion of a linear mechanical constraint 286
Reduced connections 287
Length minimizing and least constrained non-holonomic geodesics 288
Mechanical Systems with Discontinuous Forces and Systems with Control. Differential Inclusions 290
Integral Equations of Geometric Mechanics. The Velocity Hodograph 294
General constructions 294
An integral formalism of geometric mechanics with constraints 296
Mechanical Interpretation of Parallel Translation and Systems with Delayed Control Forces 297
Accessible Points and Sub-Manifolds of Mechanical Systems. Controllability 302
Discussion of the Problem 302
Examples of Points that Cannot be Connected by a Trajectory 304
Existence of Solutions 307
Generalizations to Systems with Constraints 316
Some Problems on Lorentz Manifolds 318
Introduction to Relativity Theory 318
Space-times 318
World lines. The light cone. Proper time 322
Reference frames and 3-dimensional notions 324
Some consequences 327
The electromagnetic field 331
Gravitational fields 333
A Two-Point Boundary Value Problem on a Lorentz Manifold Arising in A. Poltorak's Concept of Reference Frame 335
Discussion of the problem 335
The reference frame with flat connection 337
The reference frame with Riemannian connection 340
A Classical Particle in a Classical Gauge Field 343
A brief introduction to gauge fields and some preliminary constructions 344
The equation of motion 347
Mechanical Systems with Random Perturbations 350
Setting Up the Problem 350
The Langevin Equation and Ornstein-Uhlenbeck Processes on Manifolds 352
Set-Valued Forces. Langevin Type Inclusions 361
Systems with Random Perturbation of Velocity 367
The Newton-Nelson Equation 374
Stochastic Mechanics in Rn 375
Principal ideas of Nelson's stochastic mechanics 375
Existence theorems 379
The Geometric Form of Stochastic Mechanics 386
Some comments on stochastic mechanics on Riemannian manifolds 386
Existence theorems 388
Relativistic Stochastic Mechanics 395
Stochastic mechanics in Minkowski space 395
Stochastic mechanics in the space-times of general relativity 402
Hydrodynamics 406
The Lagrangian Formalism of the Hydrodynamics of an Ideal Barotropic Fluid 406
Diffuse matter 406
A barotropic fluid 408
Lagrangian Hydrodynamical Systems of an Ideal Incompressible Fluid 411
The Regularity Theorem and a Review of Results on the Existence of Solutions 414
Description of Deterministic Viscous Hydrodynamics Via a Stochastic Version of Newton's Law on Groups of Diffeomorphisms 421
General construction 421
Solutions of Burgers, Reynolds and Navier-Stokes equations via stochastic perturbations of inviscid flows 427
References 434
Index 445

Produkt-Details

• Erscheint lt. Verlag: 7.12.2010
• Reihe/Serie: Theoretical and Mathematical Physics
• Zusatzinfo: XXIV, 436 p.
• Verlagsort: London
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Mathematik / Informatik ► Mathematik ► Statistik
• Themenwelt: Naturwissenschaften ► Physik / Astronomie
• Themenwelt: Technik
• Schlagworte: Global Analysis • Manifolds • Nelson's mean derivatives • Set-valued mappings • stochastic analysis
• ISBN-10: 0-85729-163-7 / 0857291637
• ISBN-13: 978-0-85729-163-9 / 9780857291639