Probabilities on the Heisenberg Group Limit Theorems and Brownian Mo
The Heisenberg group comes from quantum mechanics and is the simplest non-commutative Lie group. While it belongs to the class of simply connected nilpotent Lie groups, it turns out that its special structure yields many results which (up to now) have not
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Daniel Neuenschwander
Probabilities on the Heisenberg Group Limit Theorems and Brownian Motion
Springer
Author Daniel Neuenschwander Universite de Lausanne Ecole des Hautes Etudes Commerciales Institut de Sciences Actuarielles CH-1015 Lausanne, Switzerland and Universitat Bern Institut fur mathematische Statistik und Versicherungslehre Sidlerstrasse 5 CH-3012 Bern, Switzerland
Cataloging-Data applied for Die Deutsche Bibliothek - CIP-Einheitsaufnahme Neuenschwander, Daniel: Probabilities on the Heisenberg group: limit theorems and Brownian motion. - Berlin; Heidelberg; New York; Barcelona; Budapest; Hong Kong; London; Milan; Paris ; Tokyo: Springer, 1996 (Lecture notes in mathematics; 1630) ISBN 3-540-61453-2
NE:GT Mathematics Subject Classification (1991): 60-02, 60B 15, 22E25, 47D06, 60F05, 60F15, 60F17, 60F25, 60G 10, 60G 17, 60G 18, 60G42, 60H05, 60125, 60135, 60J45, 60J55,60J60,60J65,62F35
ISSN 0075-8434 ISBN 3-540-61453-2 Springer-Verlag Berlin Heidelberg New York 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 1996 Printed in Germany Typesetting: Camera-ready TEX output by the author SPIN: 10479780 46/3142-543210 - Printed on acid-free paper
Preface
Probability theory on algebraic and geometric structures such as e.g. topological groups has attracted much interest in the literature during the past decades and is a subject of growing importance. Stimuli which can not be overestimated for the research work which has and is currently been done in the field of probability theory on groups and related structures are the regular Oberwolfach conferences organized by L. Schmetterer, H. Heyer, and A. Mukherjea as well as the recent foundation of the" Journal of Theoretical Probability" also by A. Mukherjea. In this work we will have, from the probabilistic point of view, a closer look at the so-called Heisenberg group. Its structure reflects the Heisenberg uncertainty principle as non-commutativity of the location and the momentum operator. In a certain sense, it is the simplest non-commutative Lie group, so it is clear that in generalizing classical results of probability theory to the non-commutative situation, one naturally passes by this group. Our aim will be to survey, under the limit theoretic aspect and its relation to Brownian motion, certain results which turned out to be valid on the Heisenberg group but w
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