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.