White Noise on Bialgebras
Stochastic processes with independent increments on a group are generalized to the concept of "white noise" on a Hopf algebra or bialgebra. The main purpose of the book is the characterization of these processes as solutions of quantum stochastic differen
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Lecture Notes in Mathematics Editors: A. Dold, Heidelberg B. Eckmann, Zurich F. Takens, Groningen
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Michael Schiirmann
White Noise
on Bialgebras
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest
Author Michael Schurmann Institut fur Angewandte Mathematik Universitat Heidelberg 1m Neuenheimer Feld 294 W-6900 Heidelberg, Germany
Mathematics Subject Classification (1991): 81S25, 81R50, 60130, 81S05, 60B15 ISBN 3-540-56627-9 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-56627-9 Springer-Verlag New York Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1993 Printed in Germany Typesetting: Camera-ready by author/editor 46/3140-543210 - Printed on acid-free paper
To Jutta, Antje and Matthias
CONTENTS
Introduction 1. Basic concepts and first results 1.1. Preliminaries 1.2. Quantum probabilistic notions 1.3. Independence 1.4. Commutation factors 1.5. Invariance of states 1.6. Additive and multiplicative white noise 1.7. Involutive bialgebras 1.8. Examples 1.9. White noise on involutive bialgebras 2. Symmetric white noise on Bose Fock space 2.1. Bose Fock space over L2 (R.+, H) 2.2. Kernels and operators 2.3. The basic formula 2.4. Quantum stochastic integrals and quantum Ito's formula 2.5. Coalgebra stochastic integral equations 3. Symmetrization 3.1. Symmetrization of bialgebras 3.2. Schoenberg correspondence 3.3. Symmetrization of white noise 4. White noise on Bose Fock space 4.1. Group-like elements and realization of white noise 4.2. Primitive elements and additive white noise 4.3. Az ema noise and quantum Wiener and Poisson processes 4.4. Multiplicative and unitary white noise 4.5. Co commutative white noise and infinitely divisible representations of groups and Lie algebras 5. Quadratic components of conditionally positive linear funetionals 5.1. Maximal quadratic components 5.2. Infinitely divisible states on the Weyl algebra 6. Limit theorems 6.1. A coalgebra limit theorem 6.2. The underlying additive noise as a limit 6.3. Invariance principles REFERENCES SUBJECT INDEX
1 12 12 15 17 19 21 22 26 30 35 41
41 45 52 57 65 69 69 74 79 81 81 85 90
94 103 114 114 122 128 128 130 132 138 143
Introduction These notes are a contribution to the field of quantum (or non-commutative] probability theory. Quantum probability can be regarded as an attempt of a unified approach to classical probability and the quantum theory of irreversible processes. Our special inter
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