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REAL AND STOCHASTIC ANALYSIS Current Trends 8940_9789814551274_tp.indd 1 17/10/13 11:37 AM October 24, 2013 10:1 9in x 6in Real and Stochastic Analysis: Current Trends b1644-fm TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk R E A L A N D S T O C H A S T I C A N A LY S I S Current Trends Edited by: M. M. Rao University of California at Riverside, USA World Scientific NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI 8940_9789814551274_tp.indd 2 17/10/13 11:37 AM Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE Library of Congress Cataloging-in-Publication Data Real and stochastic analysis (World Scientific (Firm)) Real and stochastic analysis : current trends / Malempati Madhusudana Rao, University of California, Riverside, USA. pages cm Includes bibliographical references. ISBN 978-9814551274 (hard cover : alk. paper) 1. Stochastic analysis. I. Rao, M. M. (Malempati Madhusudana), 1929– editor of compilation. II. Title. QA274.2.R424 2014 519.2'2--dc23 2013027573 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Copyright © 2014 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. In-house Editor: Angeline Fong Printed in Singapore October 24, 2013 10:54 9in x 6in Real and Stochastic Analysis: Current Trends b1644-fm CONTENTS Preface ix Introduction and Overview xi 1. Gaussian Measures on Infinite-Dimensional Spaces 1 V. I. Bogachev 0 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 0.1 Notation and terminology . . . . . . . . . . . . . . . . 2 1 Gaussian Measures on Rd . . . . . . . . . . . . . . . . . . . 3 2 Infinite-Dimensional Gaussian Distributions . . . . . . . . . 6 3 The Wiener Measure . . . . . . . . . . . . . . . . . . . . . . 12 4 Radon Gaussian Measures . . . . . . . . . . . . . . . . . . . 17 5 The Cameron–Martin Space and Measurable Linear Operators . . . . . . . . . . . . . . . . . . . . . . . . 19 6 Zero-one Laws and Dichotomies . . . . . . . . . . . . . . . . 32 7 The Ornstein–Uhlenbeck Semigroup . . . . . . . . . . . . . 32 8 The Hermite–Chebyshev Polynomials. . . . . . . . . . . . . 34 9 Sobolev Classes over Gaussian Measures . . . . . . . . . . . 42 10 Transformations of Gaussian Measures . . . . . . . . . . . . 51 11 Convexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 12 Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . 73 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 2. Random Fields and Hypergroups 85 Herbert Heyer 0 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 1 Commutative Hypergroups . . . . . . . . . . . . . . . . . . 86 1.1 Definition and first examples. . . . . . . . . . . . . . . 86 1.2 Some harmonic analysis . . . . . . . . . . . . . . . . . 91 1.3 Basic constructions of hypergroups . . . . . . . . . . . 98 2 Random Fields over Hypergroups . . . . . . . . . . . . . . . 110 v October 24, 2013 10:1 9in x 6in Real and Stochastic Analysis: Current Trends b1644-fm vi Real and Stochastic Analysis 2.1 Second order random fields . . . . . . . . . . . . . . . 110 2.2 Translation and decomposition . . . . . . . . . . . . . 118 2.3 Harmonizability . . . . . . . . . . . . . . . . . . . . . . 129 3 Generalized Random Fields over Hypergroups . . . . . . . . 142 3.1 Segal algebras . . . . . . . . . . . . . . . . . . . . . . . 142 3.2 The extended Feichtinger algebra . . . . . . . . . . . . 148 3.3 Covariance and duality . . . . . . . . . . . . . . . . . . 161 3.4 Suggestions for further research . . . . . . . . . . . . . 176 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 3. A Concise Exposition of Large Deviations 183 F. Hiai 0 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 1 Definitions and Generalities . . . . . . . . . . . . . . . . . . 185 2 The Cramér Theorem . . . . . . . . . . . . . . . . . . . . . 191 3 The Ga¨rtner-Ellis Theorem . . . . . . . . . . . . . . . . . . 199 4 Varadhan’s Integral Lemma . . . . . . . . . . . . . . . . . . 211 5 The Sanov Theorem . . . . . . . . . . . . . . . . . . . . . . 218 6 Large Deviations for Random Matrices . . . . . . . . . . . . 230 7 Quantum Large Deviations in Spin Chains . . . . . . . . . . 245 8 Applications of Large Deviations . . . . . . . . . . . . . . . 252 8.1 Boltzmann-Gibbs entropy and mutual information . . . . . . . . . . . . . . . . . . . . . . . . 252 8.2 Free entropy and orbital free entropy . . . . . . . . . . 257 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 4. Quantum White Noise Calculus and Applications 269 Un Cig Ji and Nobuaki Obata 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 2 Elements of Gaussian Analysis . . . . . . . . . . . . . . . . 273 2.1 Standard construction of countable Hilbert spaces . . . 273 2.2 Gaussian space . . . . . . . . . . . . . . . . . . . . . . 276 2.3 Fock spaces and the Wiener–Itô decomposition . . . . 279 2.4 Underlying spaces . . . . . . . . . . . . . . . . . . . . . 282 3 White Noise Distributions . . . . . . . . . . . . . . . . . . . 285 3.1 Standard CKS-space . . . . . . . . . . . . . . . . . . . 285 3.2 Brownian motion . . . . . . . . . . . . . . . . . . . . . 291 3.3 The S-transform . . . . . . . . . . . . . . . . . . . . . 292 October 24, 2013 10:1 9in x 6in Real and Stochastic Analysis: Current Trends b1644-fm Contents vii 3.4 Infinite dimensional holomorphic functions . . . . . . . 297 4 White Noise Operators . . . . . . . . . . . . . . . . . . . . . 300 4.1 White noise operators and their symbols . . . . . . . . 300 4.2 Quantum white noise . . . . . . . . . . . . . . . . . . . 302 4.3 Integral kernel operators and Fock expansion . . . . . 306 4.4 Characterizationof operator symbols . . . . . . . . . . 310 4.5 Wick product and wick multiplication operators . . . . 311 4.6 Multiplication operators . . . . . . . . . . . . . . . . . 314 4.7 Convolution operators . . . . . . . . . . . . . . . . . . 315 5 Quantum Stochastic Gradients . . . . . . . . . . . . . . . . 321 5.1 Annihilation, creation and conservation processes . . . 321 5.2 Classical stochastic gradient . . . . . . . . . . . . . . . 322 5.3 Creation gradient . . . . . . . . . . . . . . . . . . . . . 324 5.4 Annihilation gradient . . . . . . . . . . . . . . . . . . . 327 5.5 Conservation gradient . . . . . . . . . . . . . . . . . . 329 6 Quantum Stochastic Integrals . . . . . . . . . . . . . . . . . 331 6.1 The Hitsuda–Skorohodintegral . . . . . . . . . . . . . 331 6.2 Creation integral . . . . . . . . . . . . . . . . . . . . . 331 6.3 Annihilation integral . . . . . . . . . . . . . . . . . . . 334 6.4 Conservation integral . . . . . . . . . . . . . . . . . . . 335 7 Quantum White Noise Derivatives . . . . . . . . . . . . . . 336 7.1 Quadratic functions of quantum white noise . . . . . . 336 7.2 Quantum white noise derivatives . . . . . . . . . . . . 337 7.3 Wick derivations . . . . . . . . . . . . . . . . . . . . . 340 7.4 Quantum white noise differential equations of Wick type. . . . . . . . . . . . . . . . . . . . . . . . 342 7.5 The implementation problem . . . . . . . . . . . . . . 343 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348 5. Weak Radon-Nikody´m Derivatives, Dunford-Schwartz Type Integration, and Cramér and Karhunen Processes 355 Yuîchiroˆ Kakihara 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 2 Hilbert Space Valued Measures . . . . . . . . . . . . . . . . 357 2.1 Radon-Nikody´m property . . . . . . . . . . . . . . . . 357 2.2 Weak Radon-Nikody´m derivatives . . . . . . . . . . . . 360 2.3 Existence and uniqueness . . . . . . . . . . . . . . . . 364 2.4 Orthogonally scattered measures and dilation . . . . . 374 October 24, 2013 10:1 9in x 6in Real and Stochastic Analysis: Current Trends b1644-fm viii Real and Stochastic Analysis 2.5 Dunford-Schwartz type integration . . . . . . . . . . . 377 2.6 Bimeasure integration . . . . . . . . . . . . . . . . . . 382 3 Hilbert-Schmidt Class Operator Valued Measures . . . . . . 384 3.1 The space of Hilbert-Schmidt class operators as a normal Hilbert module . . . . . . . . . . . . . . . 384 3.2 The space L1(ξ) . . . . . . . . . . . . . . . . . . . . . . 386 3.3 Weak Radon-Nikody´m derivatives . . . . . . . . . . . . 390 3.4 The spaces L1 (ξ) and L1(ξ) . . . . . . . . . . . . . . 396 DS ∗ 3.5 The spaces L1DS(η) and L2(Fη) . . . . . . . . . . . . . 404 3.6 The spaces L1∗(ξ) and L2∗(Mξ) . . . . . . . . . . . . . . 406 4 Cramér and Karhunen Processes . . . . . . . . . . . . . . . 414 4.1 Infinite dimensional second order stochastic processes . . . . . . . . . . . . . . . . . . . . 414 4.2 Cramér processes . . . . . . . . . . . . . . . . . . . . . 416 4.3 Karhunen processes . . . . . . . . . . . . . . . . . . . . 420 4.4 Operator representation . . . . . . . . . . . . . . . . . 423 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428 6. Entropy, SDE-LDP and Fenchel-Legendre-Orlicz Classes 431 M. M. Rao 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 431 2 Error Estimation Problems from Probabilty Limit Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 434 3 Higher Order SDE and Related Classes . . . . . . . . . . . 450 4 Entropy, Action/Rate Functionals and LDP . . . . . . . . . 461 5 Vector Valued Processes and Multiparameter FLO Classes. . . . . . . . . . . . . . . . . . . . . . . . . . . 483 6 Evaluations and Representations of Conditional Means . . . 493 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498 7. Bispectral Density Estimation in Harmonizable Processes 503 H. Soedjak 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 503 2 Assumptions and a Resampling Procedure . . . . . . . . . . 505 3 The Limit Distribution of the Estimator . . . . . . . . . . . 515 4 Final Remarks and Suggestions . . . . . . . . . . . . . . . . 558 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 560 Contributors 561 October 24, 2013 10:1 9in x 6in Real and Stochastic Analysis: Current Trends b1644-fm PREFACE Just as in the case of the earlier volumes published in 1986, 1997 and 2004underthe generaltitle ofRealandStochasticAnalysis,exhibiting the usefulness of the real (also called functional analytic) methods in advanc- ing stochastic analysis, the current volume again aims to exemplify these methods and concentrate on several (related) areas of stochastic processes andfields thataredeemedto be ofconsiderableinterest.The purposeis to present some interesting parts of stochastic analysis by active researchers which crucially employ such functional analytic methods. Thus each chap- ter highlights the current state of the subject considered, and presents, by an active researcher, emphasizing what is completed and what are the currenttrendsinthesubject.Thematerialisnotonlyasurveybutcontains considerable amount of new material as well. The seven chapters deal with different classes of the subject by the invited authors. The special role and motivation of the Brownian Motion and some serious extensions to infinite dimensional spaces have been focussed. Also included are reasons for the need of infinite dimensional extensions and use of abstract methods, by including some concrete illus- trations. In this setting applications to quantum stochastic analysis is dis- cussed.AlsotheLDPandrandomfieldsongeneralstructures(hypergroups) aswellasrepresentationofcertainclassesrelatedtoCramérandKarhunen processesaswellasstatisticalestimationproblemsforfamiliesofharmoniz- able (nonstationary) classes are also treated. Some aspects of free random analysisextendingtheLDPresultsisconsideredaswell.Asusualallarticles are reviewed. I hope that the work stimulates both the young and seasoned researchers.Forpresentingthearticlesinthedesiredformat,Iwouldliketo thank the authors in doing some revisions, and for meeting the deadlines. Thanks are also due to my collegues Dr. L. O. Ferguson and particularly Dr. Y. Kakihara for advice and help on the arrangements, and the UCR ix