Functional Integrals: Approximate Evaluation and Applications

420 Pages·1993·3.25 MB·English

by  A. D. Egorov, P. I. Sobolevsky, L. A. Yanovich (auth.)

Functional Integrals:
Approximate Evaluation and Applications
Mathematics and Its Applications
Managing Editor:
M. HAZEWINKEL
Centre for Mathematics and Computer Science. Amsterdam. The Netherlands
Volume 249
Functional Integrals:
Approximate Evaluation
and Applications
by
A. D. Egorov,
P.1. Sobolevsky
and
L. A. Yanovich
Institute ofM athematics,
Be/arus Academy of Sciences,
Minsk, Byelo-Russia
SPRINGER SCIENCE+BUSINESS MEDIA, B.V.
Library of Congress Cataloging-in-Publication Data
Egorov, A. D. (A~eksandr D~itr1ev1ch)
[Pribl1zhennye metody vych1slen1 fa kont1nual 'nykh integralov.
Engl1shl
Functional integrals : approximate evaluat10n and applications
by A.D. Egorov, P.I. Sobolevsky, and L.A. Yanov1ch.
p. cm. -- (Mathemat1cs and its appl1cations ; v. 249)
Includes bibliographical references and index.
ISBN 978-94-010-4773-9 ISBN 978-94-011-1761-6 (eBook)
DOI 10.1007/978-94-011-1761-6
1. Linear topological spaces. 2. Integration, Functional.
1. Sobolevskil, P. 1. (Pavel Iosifov1chl II. fAnov1ch, L. A.
(Leonid Aleksandrovichl III. T1tle. IV. Ser1es: Mathematics and
its appl1cat1ons (Kluwer Academic Publishersl ; v. 249.
QA322.E3813 1993
515' .73--dc20 93-9599
ISBN 978-94-010-4773-9
Printed on acid-free paper
This is an updated and revised translation of the original work
Approximate Evaluation of Continuallntegrals
Nauka and Tekhnika, Minsk © 1985, 1987
All Rights Reserved
© 1993 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 1993
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
including photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner.
Contents
Preface IX
1 Backgrounds from Analysis on Linear Topological Spaces 1
1.1 Cylindric Functions, Functional Polynomials, Derivatives 1
1.2 Definition of Functional Integrals with Respect to Measure, Quasi-
measure and Pseudomeasure, Relations with Random Process
Theory 5
1.3 Characteristic Functionals of Measures 7
1.4 Moments, Semi-invariants, Integrals of Cylindric Functions 11
2 Integrals with Respect to Gaussian Measures and Some
Quasimeasures: Exact Formulae, Wick Polynomials, Diagrams 15
2.1 Some Properties of Spaces with Gaussian Measure. Formulae
for Change of Integration Variables 15
2.2 Exact Formulae for Integrals of Special Functionals. Infinitesimal
Change of Measure 20
2.3 Integrals of Variations and of Derivatives of Functionals. Wick
Ordering. Diagrams 26
2.4 Integration with Respect to Gaussian Measure in Particular Spaces 34
3 Integration in Linear Topological Spaces of Some Special Classes 47
3.1 Inductive Limits of Linear Topological Spaces 47
3.2 Projective Limits of Linear Topological Spaces 48
3.3 Generalized Function Spaces 52
3.4 Integrals in Product Spaces 55
4 Approximate Interpolation-Type Formulae 65
4.1 Interpolation of Functionals 65
4.2 Repeated Interpolation. Taylor's Formula 67
4.3 Construction Rules for Divided Difference Operators 68
4.4 Approximate Interpolation Formulae 77
5 Formulae Based on Characteristic Functional Approximations,
which Preserve a Given Number of Moments 81
5.1 Approximations of Characteristic Functionals 81
5.2 Reducing the Number of Terms in Approximations 89
5.3 Approximate Formulae 101
6 Integrals with Respect to Gaussian Measures 109
6.1 Formulae of Given Accuracy in Linear Topological Spaces 109
6.2 Formulae Based on Approximations of the Correlation Functional 119
vi
6.3 Stationary Gaussian Measures 128
6.4 Error Estimates for Approximate Formulae Based on Approxi-
mations of the Argument 130
6.5 Formulae which are Exact for Special Kinds of Functionals 1:34
6.6 Convergence of Functional Quadrature Processes 139
7 Integrals with Respect to Conditional Wiener Measure 147
7.1 Approximations of Conditional Wiener Process which Preserve a
Given Number of Moments 147
7.2 Formulae of First Accuracy Degree 155
7.3 Third Accuracy Degree 158
7.4 Arbitrary Accuracy Degree 161
8 Integrals with Respect to Measures which Correspond to
Uniform Processes with Independent Increments 167
8.1 Formulae of First, Third and Fifth Accuracy Degrees 168
8.2 Arbitrary Accuracy Degree 176
8.:3 Integrals with Respect to Measures Generated by Multidimen-
sional Processes 189
8.4 Convergence of Composite Formulae 193
8.5 Cubature Formulae for Multiple Probabilistic Integrals 200
9 Approximations which Agree with Diagram Approaches 211
9.1 Formulae which are Exact for Polynomials of Wick Powers 211
9.2 Approximate Integration of Functionals of Wick Exponents 215
9.3 Formulae which are Exact for Diagrams of a Given Type 219
9.4 Approximate Formulae for Integrals with Respect to Quasimeasures 226
9.5 Some Extensions. Composite Formulae 229
10 Approximations of Integrals Based on Interpolation of Measure 235
10.1 Approximations of Integrals with Respect to Ornstein-Uhlenbeck
Measure 235
10.2 Integrals with Respect to Wiener Measure, Conditional Wiener
Measure, and Modular Measure 241
10.3 Formulae Based on Measure Interpolation for Integrals of
Non-Differentiable Functionals 245
11 Integrals with Respect to Measures Generated by Solutions of
Stochastic Equations. Integrals Over Manifolds 249
11.1 Approximate Formulae for Integrals with Respect to Measures
Generated by Solutions of Stochastic Equations 249
11.2 Approximations of Integrals with Respect to Measures Generated
by Stochastic Differential Equations over Martingales 25:3
vii
11.3 Formula of Infinitesimal Change of Measure in Integrals with
Respect to Measures Generated by Solutions of Ito Equations 260
11.4 Approximate Formulae for Integrals over Manifolds 266
12 Quadrature Formulae for Integrals of Special Form 277
12.1 Formulae Based on Algebraic Interpolation 277
12.2 Formulae Based on Trigonometric Interpolation 282
12.:3 Quadrature Formulae with Equal Coefficients 292
12.4 Tables of Nodes and Coefficients of Quadrature Formula of Highest
Accuracy Degree for Some Integrals :300
12.5 Formulae with the Minimal Residual Estimate :319
13 Evaluation of Integrals by Monte-Carlo Method :327
1:3.1 Definitions and Facts Related to Monte-Carlo Method :327
1:3.2 Estimates for Integrals with Respect to Wiener Measure :3:31
13.:3 Estimation of Integrals with Respect to Arbitrary Gaussian Measure
in Space of Continuous Functions :3:34
13.4 A Sharper Monte-Carlo Estimate of Functional Integrals :338
14 Approximate Formulae for Multiple Integrals with Respect to
Gaussian Measure 34:3
14.1 Formulae of Third Accuracy Degree :344
14.2 Formulae of Fifth Accuracy Degree :350
14.:3 Formulae of Seventh Accuracy Degree :357
14.4 Cubature Formulae for Multiple Integrals of a Certain Kind :3.59
15 Some Special Problems of Functional Integration :367
15.1 Application of Functional Integrals to Solution of Certain Kinds
of Equations :367
15.2 Application of Approximations Based on Measure Interpolation
to Evaluation of Ground-State Energy for Certain Quantum
Systems :375
15.3 Mean-Square Approximation of Some Classes of Linear Functionals 378
15.4 Exact Formulae for Integrals with Respect to Gaussian and
Conditional Gaussian Measures of Special Types of Functionals 391
Bibliography 401
Index 417
Preface
Functional integration is a relatively new and sufficiently broad area of scientific
research. In addition to the ongoing development of the mathematical theory, ex-
tensive research is being carried out on applications to a wide spectrum of applied
problems.
Quantum statistical physics, field theory, solid-state theory, nuclear physics, optics,
quantum optics, statistical radiotechnics, radiation physics of high-energy particles,
probability theory, stochastic differential equations are some of the areas in which
applications are found [1]-[10], and this list steadily grows.
An important condition for the applicability of functional integrals is the existence
of efficient evaluation methods. The development of these methods, however, has en-
countered serious problems due to the fact that the elaboration of many issues from
analysis on infinite-dimensional spaces is far from being finished. This is also true
in the case of the theory of functional integration and, in particular, the theory of
integrals w.r.t. quasimeasures including Feynman integrals. At present, the most e-
laborated theory deals with functional integration w.r.t. count ably additive measures
[11]-[17].
This monograph is mainly devoted to methods of evaluation of functional integrals
w.r.t. count ably additive measures and certain quasimeasures on general and concrete
spaces and, in particular, of integrals w.r.t. measures generated by random processes
and quasimeasures which correspond to fundamental solutions of partial differential
equations.
An approximate evaluation of functional integrals was initiated in the papers of
Cameron [18], Vladimirov [19], Gelfand and Chentsov [20], devoted to the evaluation
of Wiener integrals. More recently, the ideas of these authors have been extended in
[21]-[33].
An evaluation of functional integrals is also considered in more physics-oriented
papers (see [34]-[39] and the bibliography therein).
Research on some issues of approximate evaluation of integrals w.r.t. Gaussian
measures is given in the papers [40]-[58].
Recently, the authors have developed methods of approximate evaluation of inte-
grals w.r.t. measures which correspond to various random processes including pro-
cesses with independent increments, of integrals w.r.t. quasimeasures. A number of
new results have also been obtained concerning the approximate evaluation of inte-
grals w.r.t. Gaussian measures. In particular a method has been developed which
agrees with the Feynman diagram method; formulae have been constructed which
employ various ways for the specification of Gaussian measures; approximations have
been constructed for integrals w.r.t. measures on spaces of functions defined on
infinite intervals; interpolation formulae have been derived for integrals w.r.t. non-
Gaussian measures. Formulae have also been obtained for integrals w.r.t. measures
generated by the solutions of stochastic differential equations w.r.t. martingales, and
w.r.t. measures generated by Gaussian processes on Riemann manifolds. An approx-
ix
x
imate method has been developed for the evaluation of integrals which is based on
the formula of infinitesimal change of measure. All these issues comprise the contents
of this book.
Most of the approximate formulae considered in here are based on the require-
ment that they are exact for functional polynomials of a given degree and that they
converge to the exact value of the integral. For the construction of these formulae,
we use various approximations for the argument of the integrated functional in the
general case, and in the case of the measure defined by a random process, we use
approximations of the process.
Attention is paid to the construction of approximate formulae for concrete mea-
sures. In particular, formulae are given for integrals w.r.t. measures which correspond
to Wiener, conditional Wiener and other Gaussian processes, the Gamma-process,
and Laplace, Poisson and telegraph processes. Integrals w.r.t. measures defined by
multidimensional processes and random fields are also considered.
For integrals w.r.t. the Gaussian measure of functionals of special kinds, approx-
imate formulae in the form of quadrature sums are investigated. An evaluation of
integrals w.r.t. Gaussian measure by the Monte-Carlo method is considered.
Approximation expressions for most of the approximate formulae considered con-
tain multiple integrals; therefore cubature formulae for the evaluation of certain class-
es of such integrals are obtained. They are constructed based on the formulae of a
given degree of accuracy for the corresponding functional integrals, and therefore
multiplicity is of no principal importance for their construction.
This monograph considers applications of the constructed approximate formulae
to the solution of applied problems, in particular, to the solution of certain integral
equations and partial differential equations, to the determination of the energy for the
ground state of model quantum systems and, to the evaluation of the expectations
for functionals of random processes. Certain extremal problems of approximation
theory are solved, and exact formulae are given for the evaluation of integrals w.r.t.
conditional and unconditional Gaussian measures of special kinds of functionals most
commonly occurring in applications.
This book also sketches the necessary background from analysis on infinite-dimen-
sional spaces.
We would like to thank our colleagues from the Institute of Mathematics of the
Byelorussian Academy of Sciences for fruitful discussions on the scope and the main
results of the book, and Dr. N. Korneenko for the translation and TEX setting of
the manuscript.
We also wish to express our gratitude to Kluwer Academic Publishers, whose
proposal stimulated us to prepare this book.
Chapter 1
Backgrounds from Analysis on
Linear Topological Spaces
The book is devoted to functional integrals defined on separable locally convex linear
topological spaces (or, briefly, on linear topological spaces). The accepted degree of
the generality of the exposition allows to embed into a general scheme the issues of
evaluation of functional integrals which are most commonly encountered in literature.
1.1 Cylindric Functions, Functional Polynomials,
Derivatives
Let X be a linear topological space; X' is the dual space of linear continuous func-
tionals on X. For 1 E X' and x E X, the value of 1 on x will be denoted by (1, XI or
byl(x).
We would like to mention two classes of functionals on X which are of special
importance in functional integration: cylindric functionals and functional polyno-
mials. A functional F( x) is called cylindric, if it may be represented in the form of
F( x) = f( (h, x I,' .. ,( In' X I), where f (u) is a function defined on the n-dimensional Eu-
clidean space Rn, u = (Ul,'" ,un), lj E X', j = 1,2,···,n (n = 1,2,·· .). In general,
this representation is not unique. Cylindric functionals are closely related to the defi-
nition of functional integrals (as we shall see, functional integrals of cylindric function-
als may be written in the explicit form), and moreover, a wide class of functionals may
be approximated by the cylindric ones. Let us consider the simplest example. Let X
be a linear topological space with basis {ej}, j = 1,2, ... , i.e., X :1 x = I:i=l (lj, x lej,
where the series converges under topology of X, {lj}"j = 1,2,···, is the dual basis in
X'. Let further F(x) be a continuous functional on X. Then F(x) = liIDn-+oo Fn(x),
where Fn(x) = F(I:i=l(lj,x)ej) is a cylindric functional.
1