Grundlehren der mathematischen Wissenschaften

Random Perturbations of Dynamical Systems

Authors: Freidlin, Mark I., Wentzell, Alexander D.

  • 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
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About this Textbook

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.

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)


Table of contents (11 chapters)

  • Random Perturbations

    Freidlin, Mark I. (et al.)

    Pages 1-28

  • Small Random Perturbations on a Finite Time Interval

    Freidlin, Mark I. (et al.)

    Pages 29-53

  • Action Functional

    Freidlin, Mark I. (et al.)

    Pages 54-84

  • Gaussian Perturbations of Dynamical Systems. Neighborhood of an Equilibrium Point

    Freidlin, Mark I. (et al.)

    Pages 85-116

  • Perturbations Leading to Markov Processes

    Freidlin, Mark I. (et al.)

    Pages 117-141

Buy this book

eBook $109.00
price for USA (gross)
  • ISBN 978-3-642-25847-3
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover $149.00
price for USA
  • ISBN 978-3-642-25846-6
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Softcover $149.00
price for USA
  • ISBN 978-3-642-44687-0
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Rent the ebook  
  • Rental duration: 1 or 6 month
  • low-cost access
  • online reader with highlighting and note-making option
  • can be used across all devices
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Bibliographic Information

Bibliographic Information
Book Title
Random Perturbations of Dynamical Systems
Authors
Translated by
Szücs, J.
Series Title
Grundlehren der mathematischen Wissenschaften
Series Volume
260
Copyright
2012
Publisher
Springer-Verlag Berlin Heidelberg
Copyright Holder
Springer-Verlag Berlin Heidelberg
eBook ISBN
978-3-642-25847-3
DOI
10.1007/978-3-642-25847-3
Hardcover ISBN
978-3-642-25846-6
Softcover ISBN
978-3-642-44687-0
Series ISSN
0072-7830
Edition Number
3
Number of Pages
XXVIII, 460
Topics