Functional Integrals: Approximate Evaluation and Applications

Transcription

1 Functional Integrals: Approximate Evaluation and Applications

2 Mathematics and Its Applications Managing Editor: M. HAZEWINKEL Centre for Mathematics and Computer Science. Amsterdam. The Netherlands Volume 249

3 Functional Integrals: Approximate Evaluation and Applications by A. D. Egorov, P.1. Sobolevsky and L. A. Yanovich Institute of Mathematics, Be/arus Academy of Sciences, Minsk, Byelo-Russia SPRINGER SCIENCE+BUSINESS MEDIA, B.V.

4 Library of Congress Cataloging-in-Publication Data Egorov, A. D. ( A ~ e k sd a~ nid tr r 1 e v 1 c h ) [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 ISBN (ebook) DOI / 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 QA322.E '.73--dc ISBN 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.

5 Contents Preface IX 1 Backgrounds from Analysis on Linear Topological Spaces Cylindric Functions, Functional Polynomials, Derivatives Definition of Functional Integrals with Respect to Measure, Quasimeasure and Pseudomeasure, Relations with Random Process Theory Characteristic Functionals of Measures Moments, Semi-invariants, Integrals of Cylindric Functions 11 2 Integrals with Respect to Gaussian Measures and Some Quasimeasures: Exact Formulae, Wick Polynomials, Diagrams Some Properties of Spaces with Gaussian Measure. Formulae for Change of Integration Variables Exact Formulae for Integrals of Special Functionals. Infinitesimal Change of Measure Integrals of Variations and of Derivatives of Functionals. Wick Ordering. Diagrams Integration with Respect to Gaussian Measure in Particular Spaces 34 3 Integration in Linear Topological Spaces of Some Special Classes Inductive Limits of Linear Topological Spaces Projective Limits of Linear Topological Spaces Generalized Function Spaces Integrals in Product Spaces 55 4 Approximate Interpolation-Type Formulae Interpolation of Functionals Repeated Interpolation. Taylor's Formula Construction Rules for Divided Difference Operators Approximate Interpolation Formulae 77 5 Formulae Based on Characteristic Functional Approximations, which Preserve a Given Number of Moments Approximations of Characteristic Functionals Reducing the Number of Terms in Approximations Approximate Formulae Integrals with Respect to Gaussian Measures Formulae of Given Accuracy in Linear Topological Spaces Formulae Based on Approximations of the Correlation Functional 119

6 vi 6.3 Stationary Gaussian Measures Error Estimates for Approximate Formulae Based on Approximations of the Argument Formulae which are Exact for Special Kinds of Functionals 1: Convergence of Functional Quadrature Processes Integrals with Respect to Conditional Wiener Measure Approximations of Conditional Wiener Process which Preserve a Given Number of Moments Formulae of First Accuracy Degree Third Accuracy Degree Arbitrary Accuracy Degree Integrals with Respect to Measures which Correspond to Uniform Processes with Independent Increments Formulae of First, Third and Fifth Accuracy Degrees Arbitrary Accuracy Degree :3 Integrals with Respect to Measures Generated by Multidimensional Processes Convergence of Composite Formulae Cubature Formulae for Multiple Probabilistic Integrals Approximations which Agree with Diagram Approaches Formulae which are Exact for Polynomials of Wick Powers Approximate Integration of Functionals of Wick Exponents Formulae which are Exact for Diagrams of a Given Type Approximate Formulae for Integrals with Respect to Quasimeasures Some Extensions. Composite Formulae Approximations of Integrals Based on Interpolation of Measure Approximations of Integrals with Respect to Ornstein-Uhlenbeck Measure Integrals with Respect to Wiener Measure, Conditional Wiener Measure, and Modular Measure Formulae Based on Measure Interpolation for Integrals of Non-Differentiable Functionals Integrals with Respect to Measures Generated by Solutions of Stochastic Equations. Integrals Over Manifolds Approximate Formulae for Integrals with Respect to Measures Generated by Solutions of Stochastic Equations Approximations of Integrals with Respect to Measures Generated by Stochastic Differential Equations over Martingales 25:3

7 11.3 Formula of Infinitesimal Change of Measure in Integrals with Respect to Measures Generated by Solutions of Ito Equations Approximate Formulae for Integrals over Manifolds Quadrature Formulae for Integrals of Special Form Formulae Based on Algebraic Interpolation Formulae Based on Trigonometric Interpolation :3 Quadrature Formulae with Equal Coefficients Tables of Nodes and Coefficients of Quadrature Formula of Highest Accuracy Degree for Some Integrals : Formulae with the Minimal Residual Estimate : 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: A Sharper Monte-Carlo Estimate of Functional Integrals : Approximate Formulae for Multiple Integrals with Respect to Gaussian Measure 34: Formulae of Third Accuracy Degree : Formulae of Fifth Accuracy Degree : :3 Formulae of Seventh Accuracy Degree : Cubature Formulae for Multiple Integrals of a Certain Kind : Some Special Problems of Functional Integration : Application of Functional Integrals to Solution of Certain Kinds of Equations : Application of Approximations Based on Measure Interpolation to Evaluation of Ground-State Energy for Certain Quantum Systems : Mean-Square Approximation of Some Classes of Linear Functionals Exact Formulae for Integrals with Respect to Gaussian and Conditional Gaussian Measures of Special Types of Functionals 391 Bibliography 401 Index 417 vii

8 Preface Functional integration is a relatively new and sufficiently broad area of scientific research. In addition to the ongoing development of the mathematical theory, extensive 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 encountered 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 integrals w.r.t. measures which correspond to various random processes including processes with independent increments, of integrals w.r.t. quasimeasures. A number of new results have also been obtained concerning the approximate evaluation of integrals 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 approxix

9 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 requirement 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 measures. 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, approximate 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 contain multiple integrals; therefore cubature formulae for the evaluation of certain classes 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-dimensional 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.