Gliklikh Y.E. Global and Stochastic Analysis with Applications to Mathematical Physics
Подождите немного. Документ загружается.
x Contents
3.2 Integral Operators with Parallel Translation . . . . ........... 86
3.2.1 The operator S .................................. 86
3.2.2 The operator Γ .................................. 89
3.2.3 Integral operators . . .............................. 90
3.3 Second Order Differential Equations (Special Vector Fields) . . 92
4 Elements of the Theory of Set-Valued Mappings .......... 97
4.1 Set-ValuedMappings and DifferentialInclusions............ 97
4.2 Special Approximations ................................. 100
5 Analysis on Groups of Diffeomorphisms .................. 105
5.1 GeneralConcepts....................................... 105
5.2 TheGroup ofDiffeomorphisms of aFlat Torus............. 111
Part II Stochastic Analysis
6 Essentials from Stochastic Analysis in Linear Spaces ...... 115
6.1 Some Definitions from Probability Theory and the Theory
ofStochasticProcesses.................................. 115
6.1.1 Stochastic processes. Cylinder sets . . . . . . . ........... 115
6.1.2 Conditional expectation . .......................... 117
6.1.3 Markov processes . . . .............................. 118
6.1.4 Martingales and semi-martingales .................. 118
6.1.5 Weak convergence of probability measures ........... 119
6.2 ASurveyon StochasticIntegrals and Equations ............ 120
6.2.1 White noise and Wiener processes .................. 120
6.2.2 Stochastic integrals . .............................. 123
6.2.3 Stochastic differential equations . . .................. 128
6.3 Stochastic Flowsand theirGenerators .................... 135
7 Stochastic Analysis on Manifolds ......................... 139
7.1 Stochastic Differential Equations in Stratonovich Form on a
Manifold .............................................. 139
7.1.1 General construction . . . . .......................... 139
7.1.2 Riemannian uniform atlases . ...................... 143
7.2 The Itˆo Bundle and ItˆoEquationson aManifold ........... 146
7.3 ItˆoEquationsin Belopolskaya-DaletskiiForm .............. 151
7.4 Completenessof StochasticFlows......................... 158
7.4.1 Setting up the problem and a necessary condition for
completeness..................................... 158
7.4.2 A necessary and sufficient condition for completeness
of flows continuous at infinity ...................... 159
7.4.3 Remarks on L
1
-complete stochastic flows............ 163
7.5 A Condition for Weak Compactness of Measures
Corresponding to Solutions of Stochastic Differential
Equations ............................................. 164
Contents xi
7.6 Stochastic Development and Parallel Translation ........... 168
7.6.1 The Eells-Elworthy and Itˆodevelopments............ 168
7.6.2 Wiener processes on Riemannian manifolds.
Stochasticcompleteness ........................... 172
7.6.3 Parallel translation along a stochastic process.
Itˆoprocesseson manifolds......................... 176
7.7 The Integral Approach to Stochastic Differential Equations
onManifolds........................................... 177
7.7.1 General constructions . . . .......................... 177
7.7.2 Stochastic differential equations in terms of Wiener
processes intangentspaces ........................ 182
7.7.3 Equations with unit diffusion coefficients . ........... 184
8 Mean Derivatives in Linear Spaces........................ 187
8.1 GeneralDefinitions and Results .......................... 187
8.2 Calculation of Mean Derivatives for a Wiener Process and
forDiffusion Processes .................................. 199
8.3 Calculation of Mean Derivatives for ItˆoProcesses........... 203
8.4 First Order Differential Equations and Inclusions with Mean
Derivatives ............................................ 209
8.5 The Case of P-mean Derivatives.......................... 220
9 Mean Derivatives on Manifolds ........................... 225
9.1 Forwardand Backward MeanDerivatives.................. 225
9.2 Currentand Osmotic Velocities .......................... 230
9.3 Mean Derivatives of Vector Fields Along Stochastic Processes 232
9.4 TheQuadratic Mean Derivative .......................... 234
9.5 Mean Derivatives of ItˆoProcesseson Manifolds ............ 237
9.6 Equationsand Inclusions withMean Derivatives............ 239
9.7 Stochastic Differential Inclusions in Terms of Infinitesimal
Generators ............................................ 243
10 Stochastic Analysis on Groups of Diffeomorphisms ....... 247
10.1 The General Case . ..................................... 247
10.2 The Case of a Flat Torus . . .............................. 249
Part III Applications to Mathematical Physics
11 Newtonian Mechanics .................................... 255
11.1 A Geometric Language for Newtonian Mechanics ........... 255
11.2 Mechanical Systems on Lie Groups . ...................... 257
11.3 Conservative Mechanical Systems . . . ...................... 258
11.4 Hamilton’s Principle of Least Action ...................... 260
11.5 Noether’s Theorem ..................................... 263
11.6 Geometric Mechanics with Linear Constraints . . . ........... 266
11.6.1 The notion of a linear mechanical constraint . . ....... 267
xii Contents
11.6.2 Reduced connections . . . . .......................... 268
11.6.3 Length minimizing and least constrained
non-holonomic geodesics .......................... 269
11.7 Mechanical Systems with Discontinuous Forces and Systems
withControl. Differential Inclusions....................... 271
11.8 Integral Equations of Geometric Mechanics. The Velocity
Hodograph ............................................ 275
11.8.1 General constructions . . . .......................... 275
11.8.2 An integral formalism of geometric mechanics with
constraints ...................................... 277
11.9 Mechanical Interpretation of Parallel Translation and
Systems withDelayed ControlForces ..................... 278
12 Accessible Points and Sub-Manifolds of Mechanical
Systems. Controllability .................................. 283
12.1 Discussion of the Problem . .............................. 283
12.2 Examples of Points that Cannot be Connected
bya Trajectory ........................................ 285
12.3 Existence of Solutions . . . . .............................. 288
12.4 Generalizations to Systems with Constraints . . . . ........... 297
13 Some Problems on Lorentz Manifolds ..................... 299
13.1 Introduction to Relativity Theory . . ...................... 299
13.1.1 Space-times. ..................................... 299
13.1.2 World lines. The light cone. Proper time . ........... 303
13.1.3 Reference frames and 3-dimensional notions . . ....... 305
13.1.4 Some consequences . .............................. 308
13.1.5 The electromagnetic field . . . . ...................... 312
13.1.6 Gravitational fields . .............................. 314
13.2 A Two-Point Boundary Value Problem on a Lorentz
Manifold Arising in A. Poltorak’s Concept of Reference
Frame................................................. 316
13.2.1 Discussion of the problem . . . ...................... 316
13.2.2 The reference frame with flat connection . ........... 318
13.2.3 The reference frame with Riemannian connection . . . . . 321
13.3 A Classical Particle in a Classical Gauge Field . . ........... 324
13.3.1 A brief introduction to gauge fields and some
preliminary constructions.......................... 325
13.3.2 The equation of motion . .......................... 328
14 Mechanical Systems with Random Perturbations ......... 331
14.1 Setting Up the Problem . . . .............................. 331
14.2 The Langevin Equation and Ornstein-Uhlenbeck Processes
onManifolds........................................... 333
14.3 Set-Valued Forces. Langevin Type Inclusions . . . . ........... 342
14.4 Systems with Random Perturbation of Velocity . ........... 348
Contents xiii
15 The Newton-Nelson Equation............................. 355
15.1 Stochastic Mechanics in R
n
.............................. 356
15.1.1 Principal ideas of Nelson’s stochastic mechanics . . . . . . 356
15.1.2 Existence theorems . .............................. 360
15.2 The Geometric Form of Stochastic Mechanics . . . ........... 367
15.2.1 Some comments on stochastic mechanics on
Riemannian manifolds ............................ 367
15.2.2 Existence theorems . .............................. 369
15.3 Relativistic Stochastic Mechanics . . . ...................... 376
15.3.1 Stochastic mechanics in Minkowski space . ........... 376
15.3.2 Stochastic mechanics in the space-times of general
relativity ........................................ 383
16 Hydrodynamics ........................................... 387
16.1 The Lagrangian Formalism of the Hydrodynamics of an
IdealBarotropic Fluid .................................. 387
16.1.1 Diffuse matter . . . . . .............................. 387
16.1.2 A barotropic fluid . . .............................. 389
16.2 Lagrangian Hydrodynamical Systems of an Ideal
Incompressible Fluid.................................... 392
16.3 The Regularity Theorem and a Review of Results on the
Existence ofSolutions................................... 395
16.4 Description of Deterministic Viscous Hydrodynamics
Via a Stochastic Version of Newton’s Law on Groups of
Diffeomorphisms ....................................... 402
16.4.1 General construction . . . . .......................... 402
16.4.2 Solutions of Burgers, Reynolds and Navier-Stokes
equations via stochastic perturbations of inviscid flows 408
References .................................................... 415
Index ......................................................... 427
Introduction
The main aim of this book is to develop and combine the methods of Global
Analysis and Stochastic Analysis allowing a more or less common treatment
of areas of mathematical physics that are traditionally considered distant
from one another and which have formerly required different methods of
investigation. Among the areas we mention are classical mechanics on non-
linear configuration spaces and some problems of statistical physics, quantum
physics and hydrodynamics. The idea, yielding the unification of these top-
ics, 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. The realization of this idea allows one to
elaborate general approaches to the investigation of the above-mentioned the-
ories whose modification in each concrete case permits us to create effective
methods of investigation and to obtain new important results.
The principal project of the book incorporates a huge amount of math-
ematical machinery including, among other things, some branches of global
analysis, stochastic analysis, set-valued analysis and analysis on infinite di-
mensional manifolds. The large amount of space devoted to preliminary ma-
terial and recent results in the above-mentioned branches of mathematics is
a result of the author’s desire to make the book as self-contained as possible.
Some of this preliminary material can be used as a first introduction to the
subject. In those cases where a detailed description of the material would be
too lengthy to include, we simply give a survey of notions and constructions
without detailed proofs. Generally we limit ourselves to the material that is
necessary for the applications to mathematical physics which appear in the
last part of the book. Nevertheless, for the sake of completeness, we often
describe notions, results and constructions that are not used in the book but
that are close to its main theme.
The book consists of three parts (subdivided into chapters, sections and
sometimes subsections), devoted, respectively, to global analysis, stochastic
analysis and applications to mathematical physics. The contents of these
parts are interrelated, however each part can be read independently. The
xv
xvi Introduction
first part contains an introduction to the geometry of manifolds from the very
beginning to branches that are often absent in textbooks and monographs.
In particular, it includes the machinery that is used in stochastic analysis
on manifolds. Thus the first two parts together present a rather complete
introduction to the latter. Some chapters of Part 3 are related only to Part 1
and can be read without reference to the material of Part 2. Other chapters
use results and constructions from both Part 1 and Part 2.
The material of Part 1 is devoted to global analysis and forms the basis
for the subsequent exposition. We begin with a glossary of commonly used
definitions and formulae from the theory of manifolds. Following this we look
at Lie groups and algebras, fiber bundles and related topics, and introduce
Riemannian metrics, tensors, differential forms and Lie derivatives.
Most important for the forthcoming material is Chapter 2, devoted to the
theory of connections. For vector bundles we use the approach to connections
based on the so-called connectors (connection maps) first introduced by P.
Dombrovski in the 1960s (see [56]). This is easily generalized to, say, man-
ifolds of maps. The connections on manifolds are introduced as connections
on their tangent bundles. We look at curvature and torsion tensors and Rie-
mannian connections (in particular, the Levi-Civit´a connection), connections
on principal bundles and a connection on the total space of a vector bundle
generated by a connection on the bundle and by a connection on the base. We
conclude the chapter with the notion of second order tangent vectors that are
transformed into ordinary (i.e., first order) tangent vectors by a connection.
Chapter 3 is devoted to ordinary differential equations on manifolds. The
first topic addresses the necessary and sufficient conditions for completeness
of flows of vector fields. We show how to modify the sufficient conditions
for completeness in both one-sided and two-sided cases in order to obtain
necessary and sufficient conditions. In both cases the necessary and sufficient
conditions involve a transition to the extended phase space. One-sided con-
ditions are formulated in terms of the existence of a proper function on the
extended phase space such that its derivative in the direction of the natural
extension of the right-hand side of the equation onto the extended phase space
is uniformly bounded. For this type of necessary and sufficient condition we
also find a natural infinite-dimensional generalization that is applicable also
to some cases where the right-hand side is given only on an everywhere dense
subset of the phase space.
Two-sided conditions are formulated in terms of the existence of a complete
Riemannian metric on the extended phase space such that the norm of the
natural extension of the right-hand side of the equation is uniformly bounded
with respect to this metric.
We then describe the basic construction and properties of integral opera-
tors with parallel translation, elaborated by the author. These operators and
their stochastic generalizations are applied to the investigation of various
equations below.
Introduction xvii
In the remaining part of the chapter we describe second order differential
equations on manifolds as special vector fields on tangent bundles and as given
in terms of covariant derivatives. The latter, in particular, involve geodesic
sprays of connections. For completeness we conclude the chapter with a brief
account of Hamiltonian systems.
Chapter 4 is devoted to elements of set-valued analysis and forms the
basis for all following set-valued problems. Besides the standard notions of
upper and lower semi-continuous set-valued maps, differential inclusions, etc.,
here we also present two new results establishing the existence, in the finite-
dimensional case, of a sequence of special ε-approximations for an upper semi-
continuous set-valued map with convex values that point-wise converges to
a Borel measurable selector of the set-valued map as ε → 0. These results
are important in the later study of various stochastic differential inclusion
problems.
In Chapter 5 we describe analysis on groups of Sobolev diffeomorphisms
of a compact manifold. These groups are natural configuration spaces for sys-
tems of hydrodynamics in the framework suggested by Arnold-Ebin-Marsden.
The case of the group of diffeomorphisms of a flat torus is of special interest
since a fluid motion on the torus is often considered in classical hydrodynam-
ics.
The second part is devoted to the constructions and results from con-
temporary stochastic analysis with the main focus on stochastic analysis on
manifolds. In Chapter 6 we recall some material that is not generally in-
cluded in a standard university course. We describe, among other topics,
conditional expectations, martingales, weak convergence of probability mea-
sures, stochastic integrals and stochastic differential equations in Euclidean
spaces (both in Itˆo and in Stratonovich forms) and stochastic flows and their
generators (forward and backward).
In the next chapter we pass to manifolds. It should be pointed out that
topological and geometric constructions are used considerably in stochastic
analysis on manifolds. Thus the material of this chapter is based on the ma-
terial of Part 1 (first of all on the theory of connections) and, of course, on
the material of the previous chapter. First we describe stochastic differential
equations on manifolds in Stratonovich form. This formalism is widely used
since the right-hand sides of such equations are transformed under coordinate
changes like ordinary tangent vectors. Then we show that on each manifold
there exists a Riemannian metric having the so-called uniform Riemannian
atlas. If the coefficients of a Stratonovich equation are C
1
-smooth and uni-
formly C
1
bounded with respect to such a metric, the flow is complete.
We then turn to Itˆo stochastic differential equations on manifolds. We de-
scribe an approach treating such equations as cross-sections of a special fiber
bundle (the Itˆo bundle) with an interesting structure group, interrelated ap-
proaches elaborated by Belopolskaya and Daletskii and by Baxendale (based
on the theory of connections), an approach based on integral operators with
xviii Introduction
parallel translation along stochastic processes (due to the author), and some
other contemporary constructions.
We should mention a necessary and sufficient condition for completeness of
a stochastic flow, continuous at infinity. In some sense this is a generalization
of the one-sided necessary and sufficient condition for completeness of the
flow of a vector field described in section 3.1. There is also a result on criteria
of weak compactness of measures on path spaces corresponding to solutions
of stochastic differential equations on manifolds.
Besides the well-known Eells-Elworthy development we introduce another
one, the so-called Itˆo development whose construction is based on Itˆoequa-
tions. On this basis we introduce the notion of Itˆo processes on manifolds (as
Itˆo developments of Itˆo processes in tangent spaces), along which the parallel
translation is well-defined. The use of Itˆo processes makes the construction
of parallel translation simple and clear.
In addition, some general existence of solution theorems are proved, in par-
ticular for the so-called equations with unit diffusion coefficient on stochas-
tically complete Riemannian manifolds.
The chapter concludes with the notion of a martingale with respect to a
connection.
We then consider (in Chapter 8) a version of differential calculus for
stochastic processes, Nelson’s theory of mean derivatives. Later some equa-
tions of mathematical physics (in mechanics with random perturbation of
forces or velocities, in quantum theory and in hydrodynamics) are given in
terms of these derivatives.
The classical Nelson mean derivatives give information only about the drift
of a stochastic process. By a slight modification of a certain idea of Nelson we
introduce a new sort of mean derivative (called the quadratic mean deriva-
tive) that is responsible for the diffusion term. Considering the quadratic
derivative together with Nelson’s classical derivatives, we investigate first or-
der differential equations and inclusions with mean derivatives: with forward
mean derivatives, with backward mean derivatives and with the so-called
current velocity, having a physical interpretation as a stochastic analog of
ordinary physical velocity. In particular we prove some existence of solution
theorems. We create a list of the first and second mean derivatives for a
Wiener process, for solutions of Itˆo equations, for Itˆo diffusion type processes
with unit diffusion coefficient, etc., that allow us in forthcoming chapters to
prove the existence of solutions of higher order equations in mean derivatives.
The features of mean derivatives on manifolds are of special interest. The
construction of forward and backward mean derivatives on a manifold in-
volves a connection. It turns out that the forward and backward derivatives,
determined with respect to a certain connection, are naturally related to Itˆo
equations in Belopolskaya-Daletskii form, determined by the same connec-
tion. We show that forward and backward mean derivatives are related to for-
ward and backward generators of a flow that is governed by the Itˆo equation
(cross-section of the Itˆo bundle) corresponding to the Belopolskaya-Daletskii
Introduction xix
equation with respect to the given connection. This relationship is described
in terms of the same connection as a fiber-wise linear mapping from the sec-
ond order tangent bundle to the ordinary (i.e., first order) tangent bundle to
the manifold.
Note that the quadratic mean derivative and current velocity are defined
without using connections and have the form of a (2, 0)-tensor field and a
vector field, respectively.
Some existence of solutions theorems for equations and inclusions in mean
derivatives on manifolds are proved.
We conclude Part 2 with elements of stochastic analysis on groups of
diffeomorphisms. We consider right-invariant Itˆo stochastic differential equa-
tions in Belopolskaya-Daletskii form on general groups of diffeomorphisms
and on volume preserving groups. The Wiener process, used in the construc-
tion of the equations, is finite-dimensional. In the general case it is taken
from a Euclidean space in which the finite dimensional manifold is embedded
by Nash’s theorem. For the particular case of the group of diffeomorphisms
of a flat n-dimensional torus a special n-dimensional Wiener process is con-
structed that allows one to apply the corresponding equations to the investi-
gation of viscous hydrodynamics described below. Some existence of solution
theorems are obtained.
Making use of the material of Parts 1 and 2, Part 3 is devoted to the de-
scription and investigation of various mechanical and physical systems. The
exposition begins with a description of classical Newtonian mechanics in the
language of invariant geometry and topology. Newton’s second law is intro-
duced in terms of the covariant derivative of the Levi-Civit´a connection of a
Riemannian metric that determines the kinetic energy on the configuration
space. After introducing such mechanical systems in a very general form, we
consider the special case of conservative systems, including Hamilton’s prin-
ciple of least action and Noether’s theorem. We also consider systems with
group structure, systems with discontinuous forces (where Newton’s law is
given in terms of differential inclusions), systems with delayed forces (de-
scribed in terms of parallel translation), systems with constraints given in
geometric form due to Vershik and Faddeev (including non-holonomic me-
chanics and the so-called vakonomic systems, i.e., variational problems with
constraints), integral equations of geometric mechanics (involving parallel
translations), velocity hodographs, and so on.
In Chapter 12 we apply the machinery developed above to the qualitative
behavior of trajectories of mechanical systems. We consider the two-point
boundary value problem for trajectories, i.e., whether it is possible to join
two points of configuration space by a trajectory. It should be noted that
on non-linear configuration spaces (i.e., on Riemannian manifolds), even for
smooth bounded forces independent of velocities, this problem may not have a
solution at all, unlike the case of linear configuration spaces. This may happen
if the points are conjugate along all geodesics of the Levi-Civit´a connection
joining them (this is true for all types of forces, e.g., for forces depending on