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Ordinary and Stochastic Differential Geometry as aTool for Mathematical Physics
Mathematics and Its Applications
Managing Editor :
M. HAZEWINKEL
Centre for Mathematics and Computer Science , Amsterdam, The Netherlands
Volume 374
Ordinary and StochasticDifferential Geometryas a Tool forMathematical Physics
by
Yuri E. GliklikhMathematics Faculty,Voronezh State University,Voronezh, Russia
Springer-Science+Business Media, B.Y
A c.I.P. Catalogue record for this book is available from the Library of Congress.
Printedon acid-freepaper
All Rights Reserved
© 1996 Springer Science+Business Media DordrechtOriginally published by Kluwer Academic Publishers in 1996.Softcover reprint of the hardcover 1st edition 1996
No part of the material protected by this copyright notice may be reproduced orutilized in any form or by any means , electronic or mechanical,including photocopying, recording or by any information storage andretrieval system, without written permission from the copyright owner.
ISBN 978-90-481-4731-1 ISBN 978-94-015-8634-4 (eBook)DOI 10.1007/978-94-015-8634-4
TABLE OF CONTENTS
ACKNOWLEDGEMENTS
INTRODUCTION
Chapter 1. ELEMENTS OF COORDINATE-FREE DIFFERENTIALGEOMETRY
1. Elementary introduction to manifolds and fibre bundles
1.A. Manifolds
LB. Fibre bundles
I.e. Tangent, cotangent and frame bundles
1.0. Vector and covector fields on manifolds
I .E. Riemannian metrics
1.P. Lie groups and Lie algebras
2. Connections and related objects
2.A. The structure of the tangent bundle
2.B. Connection and connector on the tangent bundle
2.C. Covariant derivative, parallel translation and geodesics
2.0. The case of Riemannian and semi -Riemannian manifolds.
Levi-Civita connection.
3. General construction of connections
3.A. Connections on principal and associated bundles
3.B. The case of frame bundles and the tangent bundle
3.e. Geodesics. exponential map and normal charts
4. Cartan development and integral operators with parallel
translation
4.A. Cartan development and operator S
4.B. Properties of the operator rt.s
4.e. Integral operators with parallel translation
5. Geometric formalism for classical physics
5.A. Newtonian mechanical systems
5.B. Mechanical systems with constraints
5.e. Mechanical systems on groups
ix
xi
1
1
3
5
7
9
10
11
11
14
16
19
22
22
25
26
28
28
31
32
34
34
36
38
vi Table of Contents
S.D. Integral form of Newton's law and the velocity hodograph
equation 39
5.E. Elements of Relativity Theory 41
Chapter II. INTRODUCTION TO STOCHASTIC ANALYSIS IN Rn 45
6. Some preliminary notions from probability theory 45
7. Stochastic integrals, It6 processes and stochastic
differential equations 50
7.A. Stochastic integrals with respect to a Wiener process 50
7.B. It6 processes 54
7.C. Stochastic differential equations 56
8. Mean derivatives of stochastic processes and their calculation 60
8.A. General definitions and results 60
8.B. Calculation of mean derivatives for a Wiener process
and for solutions of It6 equations 66
8.C. Calculation of mean derivatives for It6 processes 69
Chapter III. STOCHASTIC DIFFERENfIAL EQUATIONS ONMANIFOLDS 75
9. It6 stochastic differential equations on manifolds 75
9.A. It6 bundles and It6 equations 75
9.B. It6 vector fields. Belopolskaya-Dalecky approach 77
9.C. Mean derivatives 82
10. Stochastic integrals with parallel translation and It6
processes on finite-dimensional Riemannian manifolds 86
1O.A. General construction 87
10.8. Stochastically complete Riemannian manifolds 92
IO.C. Mean Derivatives 95
Chapter IV. LANGEVIN'S EQUATION IN GEOMETRIC FORM 99
11. Langevin's equation on Riemannian manifolds
and its weak solutions 99
12. Strong solutions of Langevin's equation and
Ornstein - Uhlenbeck processes 103
Table of Contents
Chapter V. NELSON'S STOCHASTIC MECHANICS
13. Stochastic Mechanics in Rn and the basic existence theorem
13.A. Principal ideas of Nelson's stochastic mechanics
13.B. Basic existence theorem
14. Geometrically-invariant form of Stochastic Mechanics and
the existence theorem on Riemannian manifolds
l4.A. Some comments on Stochastic Mechanics on Riemannian
manifolds
14.B. Existence theorem
15. Relativistic Stochastic Mechanics
l5.A. Stochastic Mechanics in Minkowski space
l5.B. Stochastic mechanics in the space-times
of General Relativity
Chapter VI. THE LAGRANGIAN APPROACH TOHYDRODYNAMICS
16. Geometry of manifolds of Sobolev diffeomorphisms
16.A. Brief account of Sobolev spaces
16.B. Manifolds of maps and groups of Sobolev diffeomorphisms
16.C. Weak Riemannian metric and related objects
16.D. A strong Riemannian metric
17. Lagrangian hydrodynamical systems of perfect barotropic
and incompressible fluids
17.A. The diffuse matter
17.B. Perfect barotropic fluid
17.C. Perfect incompressible fluid
18. Stochastic differential geometry of groups of
diffeomorphisms of a flat n-dimensional torus
19. Viscous incompressible fluid
19.A. Main construction
19.B. The case of viscous fluid in a domain in R"
with a frictionless boundary
vii
107
107
107
112
117
117
120
125
125
132
137
137
137
138
142
145
146
146
147
149
152
157
158
162
viii Table of Content s
APPENDIX. Solution of the Newton - Nelson equation withrandom initial data (Yu.E.G/ik/ikh and TJ.zastawniak)
REFERENCES
INDEX
166
175
183
ACKNOWLEDGEMENTS
The research presented in this book was made possible in part by Grant NZBOOO
from the International Science Foundation, by Grant NZB300 from the International
Science Foundation and Russian Government, and by Grant 94-378 from INTAS.
1 would like also to express my thanks to K.D. Elworthy for his hospitality at
the University of Warwick and very useful discussions. All these let me ventilate the
problems and the results with many people whom I could not have met otherwise. I am
grateful to TJ. Zastawniak who drew my attention to some confusions and mistakes,
which , in particular, led to our mutual development of the subject in the Appendix to
this book . 1 am also indebted to A. Truman for his interest in my work and for fruitful
discussions .
Mathematics Faculty
Voronezh State University
394693 Voronezh Russia
Yuri Gliklikh
December 1995
INTRODUCTION
The geometrical methods in modem mathematical physics and the developments
in Geometry and Global Analysis motivated by physical problems are being
intensively worked out in contemporary mathematics. In particular, during the last
decades a new branch of Global Analysis, Stochastic Differential Geometry, was
formed to meet the needs of Mathematical Physics. It deals with a lot of various
second order differential equations on finite and infinite-dimensional manifolds
arising in Physics, and its validity is based on the deep inter-relation between modem
Differential Geometry and certain parts of the Theory of Stochastic Processes,
discovered not so long ago.
The foundation of our topic is presented in the contemporary mathematical
literature by a lot of publications devoted to certain parts of the above-mentioned
themes and connected with the scope of material of this book. There exist some
monographs on Stochastic Differential Equations on Manifolds (e.g. [9,36,38,87])
based on the Stratonovich approach. In [7] there is a detailed description of It6
equations on manifolds in Belopolskaya-Dalecky form. Nelson's book [94] deals with
Stochastic Mechanics and mean derivatives on Riemannian Manifolds. The books and
survey papers on the Lagrange approach to Hydrodynamics [2,31,73 ,88], etc., give
good presentations of the use of infinite-dimensional ordinary differential geometry in
ideal hydrodynamics. We should also refer here to [89,102], to the previous books by
the author [53,64], and to many others .
As compared with the above-mentioned works, this book is devoted to a unified
geometrical approach to several directions of Mathematical Physics centred on the
applications of Stochastic Differential Geometry and has the following characteristic
points:
A) The content of this book and the manner of presentation are determined by the
applications to the Langevin equation of Statistical Mechanics, to Nelson's Stochastic
Mechanics (a version of Quantum Mechanics), and to the Hydrodynamics of Viscous
Incompressible Fluid treated with the modem Lagrange formalism (see the survey of
the contents below). This unification is very natural, since it is based on the fact that
the equations of motion in the above three theories appear to be related to different
stochastic generalizations of the well-known geometric form of Newton's second law
of motion. So it becomes possible to use common geometric machinery for their
investigation.
xi
xii Introduction
B) The book unifies three independently developed approaches to Stochastic
Differential Equations on Manifolds, namely the Theory of Ito equations in the form
of Belopolskaya-Dalecky, Nelson's construction of the so-called mean derivatives of
stochastic processes and the author's constructions of stochastic line integrals with
Riemannian parallel translation . It is shown that these approaches have a natural
geometric interconnection and their mutual application allows one to obtain a lot of
new and deep results.
C) The book contains preliminary and background material from coordinate-free
Differential Geometry and from the Theory of Stochastic Differential Equations
sufficient to make the book self-contained and convenient for specialists in ordinary
Differential Geometry and Global Analysis not familiar with Stochastics, for
specialists in Stochastic Analysis not familiar with Geometry and (maybe) for
specialists in Mathematical Physics not familiar with both these branches. Since the
above two branches of Mathematics are traditionally (and mistakenly) considered as
being far apart, the author hopes that this point, together with the reasonable size of
the book, will widen the realm of scientists interested in the further development of
Stochastic Differential Geometry, and in its applications to the Mathematical Physics.
The previous book by the author [53] and its revised and enlarged English edition
[64] deal with a broader set of problems from Mathematical Physics than this book. So
they could only touch upon the material of this book without going into details . They
also lack the background material and do not include a lot of new methods and deep
new results obtained since 1989. In the text we make many references to [64] in order
to indicate some directions of possible further developments or alternative approaches
to the subject. The main difference from [64] is that here 'we dig deeper, but in a
smaller area, and have made all necessary preparations '. Generally speaking, it is a
good idea to read both books to be familiar with the scope of the subject.
The book consists of 6 chapters.
The first chapter is devoted to the preliminaries from coordinate-free Differential
Geometry. The main topics are Connections on Tangent and Principal Bundles, the
Covariant Derivative, Parallel Translation, the Exponential Map, the notion of a
Mechanical System (in particular, with constraints), Newton's law, Relativity Theory ,
etc.. The exposition is brief but quite complete . We also describe the author's basic
constructions of integral operators with parallel translation and of the velocity
hodograph equation, which is an ordinary integral equation in a single tangent space to
the manifold.
Introduction xiii
The second chapter deals with some basic notions from Probability Theory,
Martingales, Wiener processes, Stochastic integrals, Stochastic Differential Equations,
Ito processes etc . in R". Apart from the description of classical notions and results it
contains new material connected with the notions of Nelson's mean derivatives and
their calculation for some types of processes in R".
In the third chapter we consider Stochastic Differential Equations on manifolds.
Recall that the right hand side of an Ito equation is a field of non-tensorial character
with respect to changes of coordinates. We introduce a certain special fibre bundle
whose sections are Ito equations, and then direct our attention to the Belopolskaya
Dalecky approach based on the consideration of special tensor-type fields
corresponding to Ito equations with respect to a given connection on the manifold. We
define Nelson's mean derivatives for processes on manifolds and show that the
I3elopolskaya - Dalecky approach is properly compatible with them. In particular, we
calculate some mean derivatives for solutions to Ito equations. Then we describe the
author's construction of the Ito line integral with parallel translation, and define the
notions of Ito development (a certain generalization of the classical Cartan
development), of Ito and Wiener processes on manifolds (as Ito developments of the
corresponding processes in tangent spaces), of the stochastic completeness of a
Riemannian manifold, and so on. This chapter provides the machinery for the
applications below.
The rest of the book is devoted to applications. Recall that Newton's law of
motion , as a second order differential equation on a Riemannian manifold M, can be
described as a first order differential equation on the tangent bundle TM . There are
three obvious possibilities for inserting a (perturbed) white noise in the latter equation
in order to obtain a well-defined stochastic differential equation on TM. Namely, we
can use the horizontal lift of the noise , the vertical lift and, lastly, we can use both lifts
simultaneously. We should point out that, roughly speaking, all these possibilities are
realized in mathematical physics (with suitable modifications, of course).
The fourth chapter deals with the Langevin equation in geometrically invariant
form, which describes the motion of a material point on a curved space under the
action of a force with both deterministic and stochastic components. This equation
corresponds to the case where the white noise is inserted into Newton 's law via its
vertical lift (see above), and therefore this equation is the simplest for investigation.
We describe it in a mathematically correct form in terms of the integrals with parallel
translation. Its velocity hodograph equation appears to be an Ito equation of a well-
xiv Introduction
known type in a single tangent (i.e. vector) space, and hence we are able to use a lot of
classical methods and to make a quite deep investigation of it. In particular, we
consider the Ornstein - Uhlenbeck processes, i.e., strong solutions of the Langevin
equation with some special forces.
In the fifth chapter we study Nelson's Stochastic Mechanics, a theory based on
the ideas of classical physics but giving the same predictions as Quantum Mechanics.
Sometimes it is considered as the third method of quantization, differing from the
well-known Hamiltonian and Lagrangian (based on the path integrals) methods. The
deep analysis of its equation of motion, called the Newton - Nelson equation, shows
that this case corresponds to white noise inserted in Newton's law by both vertical and
horizontal lifts (see above). We include the necessary facts from the foundations of
Stochastic Mechanics as well as new results. For the sake of the readers' convenience
we start with Stochastic Mechanics in the Euclidean space R'' and then consider the
general cases of Riemannian manifolds and of relativistic Stochastic Mechanics (in
particular, General Relativity). The main difference between our approach and the
ones previously used is that we assume the trajectory (i.e., a solution of the Newton
Nelson equation) to be an It6 process, not necessarily a diffusion process, and so it
may not be a Markovian one. Nevertheless, we show that the correspondence with
ordinary Quantum Mechanics remains the same as was described for Markovian
diffusions .
Recall that for mechanical systems in Rn with potential forces the existence of a
solution of the Newton - Nelson equation in the class of Markovian diffusions was
proved earlier by the methods of the Theory of Partial Differential Equations under
very broad conditions on the force field. Our existence theorem is essentially more
general. First, we consider forces of a much more general type, in particular non
potential and non-gyroscopic ones, i.e., we are able to quantize the systems where
other methods of quantization are not applicable . Second, we consider systems on
Riemannian manifolds, assuming the latter to have only bounded Ricci curvature and
its covariant derivative. For this case we show the existence of a trajectory of a
stochastic-mechanical system in the class of It6 processes under conditions on the
force field which are in fact very close to those on potential forces of the case in R"
mentioned above. The result is generalized also to space-times in General Relativity
Theory. Our constructions are based on the methods of Stochastic Differential
Geometry developed in the previous chapters. In particular we find a solution of a
certain equation in a single tangent space (a sort of velocity hodograph equation) and
Introduction xv
then obtain a trajectory on the manifold as the Ito development of that solution. The
main construction deals with deterministic initial data for the solution and this leads to
a singularity at the initial time instant. In the case of General Relativity the singularity
gives a certain model of the Big Bang.
In the Appendix we describe a modification of the construction covering random
initial data. There is no singularity at the zero instant in this case.
In the sixth chapter we consider the modem Lagrangian approach to
hydrodynamics, where we focus our attention on viscous incompressible fluid. This
approach was suggested by V.1. Arnold and then developed by Ebin and Marsden,
mainly for perfect fluids. As it was shown by the above authors, the hydrodynamics of
perfect fluids has a very natural interpretation in terms of infinite-dimensional
differential geometry. Namely, under the absence of external forces the flow of a
perfect incompressible fluid on a compact orientable Riemannian manifold is a
geodesic curve of the Levi-Civita connection for the natural weakly Riemannian
metric on the Hilbert manifold (group) of the volume preserving diffeomorphisms of
the initial finite-dimensional manifold. In the case of non-zero external forces the flow
is described in terms of the corresponding Newton's law on the manifold of
diffeomorphisms . Only after the transition to the tangent space at the identical
diffeomorphism (which plays the role of 'algebra' for the group of diffeomorphisms)
one obtains the classical Euler equation of hydrodynamics, losing the derivatives. We
give a brief description of this material as well as of the perfect barotropic fluid, and of
the necessary facts from the geometry of groups of diffeomorphisms.
Viscous incompressible fluid was considered by Ebin and Marsden in terms of an
additional force field constructed from the Laplace operator. That is why the
corresponding equation on the manifold of diffeomorphisms lost derivatives from the
outset and did not have natural geometric properties.
We derive a new approach to viscous incompressible fluid based on the
Stochastic Differential Geometry of the manifold of diffeomorphisms. We show that
via this approach viscous incompressible fluid is described as naturally as perfect fluid
was described in terms of ordinary differential geometry. Namely, under the absence
of external forces a flow of the fluid is the mathematical expectation of a certain
stochastic analogue of a geodesic curve on the manifold of diffeomorphisms, and is
governed by an analogue of Newton's law when a force is present. Here, after the
transition to the 'algebra ' we obtain Navier-Stokes equation, losing the derivatives .
The machinery of mean derivatives and Ito equations in Belopolskaya-Dalecky form
xvi Introduction
is involved in the construction. Note that the analogue of Newton's law is formulated
in terms of covariant backward mean derivatives, and analysis shows that it can
be considered as an ordinary Newton's law perturbed by the horizontal lift of a
'backward white noise' (see above).
We study the model (but the most important) examples of a fluid moving on a flat
n-dimensional torus or in a bounded domain in R'' . Note that even for a flat finite
dimensional manifold (such as the torus and R") the manifold of volume-preserving
diffeomorphisms is not flat , and so we have to use the entire theory developed in the
previous chapters.
Recommended