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."
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