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© 2012

Random Perturbations of Dynamical Systems

  • Third revised and enlarged edition

  • New chapters and enlarged bibliographic references

  • A very detailed and deep mathematical treatment of the long term behavior of randomly perturbed dynamical systems

Textbook

Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 260)

Table of contents

  1. Front Matter
    Pages I-XXVIII
  2. Mark I. Freidlin, Alexander D. Wentzell
    Pages 1-28
  3. Mark I. Freidlin, Alexander D. Wentzell
    Pages 29-53
  4. Mark I. Freidlin, Alexander D. Wentzell
    Pages 54-84
  5. Mark I. Freidlin, Alexander D. Wentzell
    Pages 85-116
  6. Mark I. Freidlin, Alexander D. Wentzell
    Pages 117-141
  7. Mark I. Freidlin, Alexander D. Wentzell
    Pages 142-191
  8. Mark I. Freidlin, Alexander D. Wentzell
    Pages 192-257
  9. Mark I. Freidlin, Alexander D. Wentzell
    Pages 258-354
  10. Mark I. Freidlin, Alexander D. Wentzell
    Pages 355-389
  11. Mark I. Freidlin, Alexander D. Wentzell
    Pages 390-404
  12. Mark I. Freidlin, Alexander D. Wentzell
    Pages 405-440
  13. Back Matter
    Pages 441-458

About this book

Introduction

Many notions and results presented in the previous editions of this volume have since become quite popular in applications, and many of them have been “rediscovered” in applied papers.
 
In the present 3rd edition small changes were made to the chapters in which long-time behavior of the perturbed system is determined by large deviations. Most of these changes concern terminology. In particular, it is explained that the notion of sub-limiting distribution for a given initial point and a time scale is identical to the idea of metastability, that the stochastic resonance is a manifestation of metastability, and that the theory of this effect is a part of the large deviation theory. The reader will also find new comments on the notion of quasi-potential that the authors introduced more than forty years ago, and new references to recent papers in which the proofs of some conjectures included in previous editions have been obtained.
 
Apart from the above mentioned changes the main innovations in the 3rd edition concern the averaging principle. A new Section on deterministic perturbations of one-degree-of-freedom systems was added in Chapter 8. It is shown there that pure deterministic perturbations of an oscillator may lead to a stochastic, in a certain sense, long-time behavior of the system, if the corresponding Hamiltonian has saddle points. The usefulness of a joint consideration of classical theory of deterministic perturbations together with stochastic perturbations is illustrated in this section. Also a new Chapter 9 has been inserted in which deterministic and stochastic perturbations of systems with many degrees of freedom are considered. Because of the resonances, stochastic regularization in this case is even more important.

Keywords

60F10, 34E10, 60H10, 60J60 averaging principle exit problems large deviations metastability perturbations of Hamiltonian systems

Authors and affiliations

  1. 1.Department of MathematicsUniversity of MarylandCollege ParkUSA
  2. 2.Department of MathematicsTulane UniversityNew OrleansUSA

Bibliographic information

  • Book Title Random Perturbations of Dynamical Systems
  • Authors Mark I. Freidlin
    Alexander D. Wentzell
  • Series Title Grundlehren der mathematischen Wissenschaften
  • DOI https://doi.org/10.1007/978-3-642-25847-3
  • Copyright Information Springer-Verlag Berlin Heidelberg 2012
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
  • Hardcover ISBN 978-3-642-25846-6
  • Softcover ISBN 978-3-642-44687-0
  • eBook ISBN 978-3-642-25847-3
  • Series ISSN 0072-7830
  • Edition Number 3
  • Number of Pages XXVIII, 460
  • Number of Illustrations 0 b/w illustrations, 0 illustrations in colour
  • Topics Probability Theory and Stochastic Processes
  • Buy this book on publisher's site
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Reviews

From the reviews of the third edition:

“The celebrated work of Ventsel and Freidlin has proved to be a major contribution in this development, with their phenomenal text Random Perturbations of Dynamical Systems, now in its third edition, playing a unique role. … The book under review has evolved since its first English edition was published in 1984, a translation from the Russian original of 1979. … it will attract an ever growing population of applied mathematicians to the fascinating new frontier of stochastic dynamics.” (Hong Qian and Hao Ge, SIAM Review, Vol. 55 (3), 2013)