Local Lyapunov Exponents Sublimiting Growth Rates of Linear Random D
Establishing a new concept of local Lyapunov exponents the author brings together two separate theories, namely Lyapunov exponents and the theory of large deviations. Specifically, a linear differential system is considered which is controlled by a stocha
- PDF / 2,523,592 Bytes
- 263 Pages / 439.37 x 666.142 pts Page_size
- 84 Downloads / 200 Views
1963
Wolfgang Siegert
Local Lyapunov Exponents Sublimiting Growth Rates of Linear Random Differential Equations
123
Wolfgang Siegert Allianz Lebensversicherungs - AG Reinsburgstrasse 19 70178 Stuttgart Germany [email protected]
ISBN 978-3-540-85963-5 e-ISBN 978-3-540-85964-2 DOI: 10.1007/978-3-540-85964-2 Lecture Notes in Mathematics ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 Library of Congress Control Number: 2008934460 Mathematics Subject Classification (2000): 60F10, 60H10, 37H15, 34F04, 34C11, 58J35, 91B28, 37N10, 92D15, 92D25 c 2009 Springer-Verlag 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, 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. Violations are liable to prosecution under the German Copyright Law. 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: SPi Publishing Services Printed on acid-free paper 987654321 springer.com
Preface
Establishing a new concept of local Lyapunov exponents, two separate theories are brought together, namely Lyapunov exponents and the theory of large deviations. Specifically, for the stochastic differential system dZtε = A (Xtε ) Ztε dt √ dXtε = b (Xtε ) dt + ε σ (Xtε ) dWt the new concept is introduced. Due to stationarity, the Lyapunov exponents of Ztε (which by Oseledets’ Multiplicative Ergodic Theorem describe the exponential growth rates of Ztε ) do not depend on the initial position x of X ε . Now the goal of this work is to provide a Lyapunov-type number for each regime of the drift b. As this characteristic number shall depend on the domain in which X ε , a dynamical system perturbed by additive white noise, is starting, it yields a concept of locality for the Lyapunov exponents of Ztε . Furthermore, the locality of such local Lyapunov exponents is to be understood as reflecting the quasi-deterministic behavior of X ε which asserts that in the limit of small noise, ε → 0, the process X ε has metastable states depending on its initial value as well as on the time scale chosen (Freidlin-Wentzell theory). Up to now local Lyapunov exponents have been defined as finite time versions of Lyapunov exponents by several authors, but here we target at investigating the large time asymptotics t → ∞. So the goal is to connect the large parameters t and ε−1 in the customary definition of the Lyapunov exponents in order to approach the sublimiting distributions (Freidlin) which are supporte
Data Loading...
