Gliklikh Global and Stochastic Analysis with Applications to Mathematical Physics

2011
ISBN: 978-0-85729-163-9
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 436 Seiten

Reihe: Theoretical and Mathematical Physics

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.

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.

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.

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.

Autoren/Hrsg.

Gliklikh, Yuri E.

Inhaltsverzeichnis

1;Preface;7
2;Contents;9
3;Introduction;14
4;Global Analysis;23
4.1;Manifolds and Related Objects;24
4.1.1;Manifolds, Vectors and Covectors. A Glossary;24
4.1.2;Lie Groups and Lie Algebras;32
4.1.3;Fiber Bundles;35
4.1.4;Riemannian and Semi-Riemannian Metrics;39
4.1.5;Tensors;42
4.1.6;Differential Forms and Polyvectors;46
4.1.7;The Lie Derivative;53
4.2;Connections;56
4.2.1;The Structure of a Tangent Bundle to a Vector Bundle ;56
4.2.2;Connections on Vector Bundles;61
4.2.3;Connections on Manifolds;71
4.2.4;Geodesics;74
4.2.5;Curvature and Torsion Tensors;76
4.2.6;Riemannian Connections. The Levi-Civitá Connection;77
4.2.7;Connections on Principal Bundles;81
4.2.8;A Connection on the Total Space of a Vector Bundle;85
4.2.9;Second Order Tangent Vectors and Connections;86
4.3;Ordinary Differential Equations;88
4.3.1;Global in Time Existence of Solutions of Ordinary Differential Equations;88
4.3.1.1;A necessary and sufficient condition for completeness of a vector field of one-sided type;88
4.3.1.2;A generalization to the infinite-dimensional case;91
4.3.1.3;A necessary and sufficient condition for completeness of a vector field of two-sided type;103
4.3.1.4;Some sufficient conditions;105
4.3.2;Integral Operators with Parallel Translation;107
4.3.2.1;The operator S;107
4.3.2.2;The operator ;110
4.3.2.3;Integral operators;111
4.3.3;Second Order Differential Equations (Special Vector Fields);113
4.4;Elements of the Theory of Set-Valued Mappings;118
4.4.1;Set-Valued Mappings and Differential Inclusions;118
4.4.2;Special Approximations;121
4.5;Analysis on Groups of Diffeomorphisms;126
4.5.1;General Concepts;126
4.5.2;The Group of Diffeomorphisms of a Flat Torus;132
5;Stochastic Analysis;134
5.1;Essentials from Stochastic Analysis in Linear Spaces;135
5.1.1;Some Definitions from Probability Theory and the Theory of Stochastic Processes;135
5.1.1.1;Stochastic processes. Cylinder sets;135
5.1.1.2;Conditional expectation;137
5.1.1.3;Markov processes;138
5.1.1.4;Martingales and semi-martingales;138
5.1.1.5;Weak convergence of probability measures;139
5.1.2;A Survey on Stochastic Integrals and Equations;140
5.1.2.1;White noise and Wiener processes;140
5.1.2.2;Stochastic integrals;143
5.1.2.3;Stochastic differential equations;148
5.1.3;Stochastic Flows and their Generators;155
5.2;Stochastic Analysis on Manifolds;159
5.2.1;Stochastic Differential Equations in Stratonovich Form on a Manifold;159
5.2.1.1;General construction;159
5.2.1.2;Riemannian uniform atlases;163
5.2.2; The Itô Bundle and Itô Equations on a Manifold;166
5.2.3;Itô Equations in Belopolskaya-Daletskii Form;171
5.2.4;Completeness of Stochastic Flows;178
5.2.4.1;Setting up the problem and a necessary condition for completeness;178
5.2.4.2;A necessary and sufficient condition for completeness of flows continuous at infinity;179
5.2.4.3;Remarks on L1-complete stochastic flows;183
5.2.5;A Condition for Weak Compactness of Measures Corresponding to Solutions of Stochastic Differential Equations;184
5.2.6;Stochastic Development and Parallel Translation;188
5.2.6.1;The Eells-Elworthy and Itô developments;188
5.2.6.2;Wiener processes on Riemannian manifolds. Stochastic completeness;192
5.2.6.3;Parallel translation along a stochastic process. Itô processes on manifolds;196
5.2.7;The Integral Approach to Stochastic Differential Equations on Manifolds;197
5.2.7.1;General constructions;197
5.2.7.2;Stochastic differential equations in terms of Wiener processes in tangent spaces;202
5.2.7.3;Equations with unit diffusion coefficients;204
5.3;Mean Derivatives in Linear Spaces;207
5.3.1;General Definitions and Results;207
5.3.2;Calculation of Mean Derivatives for a Wiener Process and for Diffusion Processes;219
5.3.3;Calculation of Mean Derivatives for Itô Processes;223
5.3.4;First Order Differential Equations and Inclusions with Mean Derivatives;229
5.3.5;The Case of P-mean Derivatives;240
5.4;Mean Derivatives on Manifolds;245
5.4.1;Forward and Backward Mean Derivatives;245
5.4.2;Current and Osmotic Velocities;250
5.4.3;Mean Derivatives of Vector Fields Along Stochastic Processes;252
5.4.4;The Quadratic Mean Derivative;254
5.4.5;Mean Derivatives of Itô Processes on Manifolds;257
5.4.6;Equations and Inclusions with Mean Derivatives;259
5.4.7;Stochastic Differential Inclusions in Terms of Infinitesimal Generators;263
5.5;Stochastic Analysis on Groups of Diffeomorphisms;267
5.5.1;The General Case;267
5.5.2;The Case of a Flat Torus;269
6;Applications to Mathematical Physics;273
6.1;Newtonian Mechanics;274
6.1.1;A Geometric Language for Newtonian Mechanics;274
6.1.2;Mechanical Systems on Lie Groups;276
6.1.3;Conservative Mechanical Systems;277
6.1.4;Hamilton's Principle of Least Action;279
6.1.5;Noether's Theorem;282
6.1.6;Geometric Mechanics with Linear Constraints;285
6.1.6.1;The notion of a linear mechanical constraint;286
6.1.6.2;Reduced connections;287
6.1.6.3;Length minimizing and least constrained non-holonomic geodesics;288
6.1.7;Mechanical Systems with Discontinuous Forces and Systems with Control. Differential Inclusions;290
6.1.8;Integral Equations of Geometric Mechanics. The Velocity Hodograph;294
6.1.8.1;General constructions;294
6.1.8.2;An integral formalism of geometric mechanics with constraints;296
6.1.9;Mechanical Interpretation of Parallel Translation and Systems with Delayed Control Forces;297
6.2;Accessible Points and Sub-Manifolds of Mechanical Systems. Controllability;302
6.2.1;Discussion of the Problem;302
6.2.2;Examples of Points that Cannot be Connected by a Trajectory;304
6.2.3;Existence of Solutions ;307
6.2.4;Generalizations to Systems with Constraints;316
6.3;Some Problems on Lorentz Manifolds;318
6.3.1;Introduction to Relativity Theory;318
6.3.1.1;Space-times;318
6.3.1.2;World lines. The light cone. Proper time;322
6.3.1.3;Reference frames and 3-dimensional notions;324
6.3.1.4;Some consequences;327
6.3.1.5;The electromagnetic field;331
6.3.1.6;Gravitational fields;333
6.3.2;A Two-Point Boundary Value Problem on a Lorentz Manifold Arising in A. Poltorak's Concept of Reference Frame;335
6.3.2.1;Discussion of the problem;335
6.3.2.2;The reference frame with flat connection;337
6.3.2.3;The reference frame with Riemannian connection;340
6.3.3;A Classical Particle in a Classical Gauge Field;343
6.3.3.1;A brief introduction to gauge fields and some preliminary constructions;344
6.3.3.2;The equation of motion;347
6.4;Mechanical Systems with Random Perturbations;350
6.4.1;Setting Up the Problem;350
6.4.2;The Langevin Equation and Ornstein-Uhlenbeck Processes on Manifolds;352
6.4.3;Set-Valued Forces. Langevin Type Inclusions;361
6.4.4;Systems with Random Perturbation of Velocity;367
6.5;The Newton-Nelson Equation;374
6.5.1;Stochastic Mechanics in Rn;375
6.5.1.1;Principal ideas of Nelson's stochastic mechanics;375
6.5.1.2;Existence theorems;379
6.5.2;The Geometric Form of Stochastic Mechanics;386
6.5.2.1;Some comments on stochastic mechanics on Riemannian manifolds;386
6.5.2.2; Existence theorems;388
6.5.3;Relativistic Stochastic Mechanics;395
6.5.3.1;Stochastic mechanics in Minkowski space;395
6.5.3.2;Stochastic mechanics in the space-times of general relativity;402
6.6;Hydrodynamics;406
6.6.1;The Lagrangian Formalism of the Hydrodynamics of an Ideal Barotropic Fluid;406
6.6.1.1;Diffuse matter;406
6.6.1.2;A barotropic fluid;408
6.6.2;Lagrangian Hydrodynamical Systems of an Ideal Incompressible Fluid;411
6.6.3;The Regularity Theorem and a Review of Results on the Existence of Solutions;414
6.6.4;Description of Deterministic Viscous Hydrodynamics Via a Stochastic Version of Newton's Law on Groups of Diffeomorphisms;421
6.6.4.1;General construction;421
6.6.4.2;Solutions of Burgers, Reynolds and Navier-Stokes equations via stochastic perturbations of inviscid flows;427
7;References;434
8;Index;445