Random Dynamical Systems
This book is the first systematic presentation of the theory of random dynamical systems, i.e. of dynamical systems under the influence of some kind of randomness. The theory comprises products of random mappings as well as random and stochastic different
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Springer-Verlag Berlin Heidelberg GmbH
Ludwig Arnold
Random Dynamical Systems With 40 Figures
Springer
Ludwig Arnold University of Bremen Institute for Dynamical Systems Postfach 33 04 40 28334 Bremen Germany e-mail: [email protected] Library of Congress Cataloging-in-Publication Data Arnold, L. (Ludwig), 1937Random dynamical systems I Ludwig Arnold. p.cm.-- (Springer monographs in mathematics) Includes bibliographical references and index. ISBN 3-540-63758 (hardcover: alk. paper) 1. Stochastic differential equations. 2. Differentiable dynamical systems. 3. Ergodic theory. I. Title. II. Series QA274.23.A75 1998 519.2--dc21 98-27207 CIP
Corrected 2nd printing 2003
Mathematics Subject Classification (2000): 34F05, 37Hxx, 60HIO, 93E03
ISSN 1439-7382 ISBN 978-3-642-08355-6 ISBN 978-3-662-12878-7 (eBook) DOI 10.1007/978-3-662-12878-7 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, reuse of illustrations, recitation, broadcasting, reproduction on microfilm 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 Berlin Heidelberg GmbH. Violations are liable for prosecution under the German Copyright Law.
http://www.springer.de ©Springer-Verlag Berlin Heidelberg 1998 Originally published by Springer-Verlag Berlin Heidelberg New York in 1998
Softcover reprint of the hardcover 1st edition 1998 The use of general descriptive names, registered names, trademarks etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: Erich Kirchner, Heidelberg Typesetting: The Author's I!'fEC input files have been edited and reformatted by Le-TeX Jelonek, Schmidt & Vockler GbR, Leipzig using a Springer Jli.TEC macro package 41/3142ck-5 4 3 21 0 Printed on acid-free paper SPIN 10785408
Preface
Background and Scope of the Book This book continues, extends, and unites various developments in the intersection of probability theory and dynamical systems. I will briefly outline the background of the book, thus placing it in a systematic and historical context and tradition. Roughly speaking, a random dynamical system is a combination of a measure-preserving dynamical system in the sense of ergodic theory, (D,F,lP', (B(t))tE'lf), 'II'= JR+, IR, z+, Z, with a smooth (or topological) dynamical system, typically generated by a differential or difference equation :i: = f(x) or Xn+l = tp(x.,), to a random differential equation :i: = f(B(t)w,x) or random difference equation Xn+l = tp(B(n)w, Xn)· Both components have been very well investigated separately. However, a symbiosis of them leads to a new research program which has only partly been car
