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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.
Onishchik, A. A. Kirillov, and E. B. Vinberg, who obtained their
first results on Lie groups in Dynkin's seminar. At a later stage,
the work of the seminar was greatly enriched by the active
participation of 1. 1. Pyatetskii Shapiro. As already noted, Dynkin
started to work in probability as far back as his undergraduate
studies. In fact, his first published paper deals with a problem
arising in Markov chain theory. The most significant among his
earliest probabilistic results concern sufficient statistics. In
[15] and [17], Dynkin described all families of one-dimensional
probability distributions admitting non-trivial sufficient
statistics. These papers have considerably influenced the
subsequent research in this field. But Dynkin's most famous results
in probability concern the theory of Markov processes. Following
Kolmogorov, Feller, Doob and Ito, Dynkin opened a new chapter in
the theory of Markov processes. He created the fundamental concept
of a Markov process as a family of measures corresponding to var
ious initial times and states and he defined time homogeneous
processes in terms of the shift operators ()t. In a joint paper
with his student A.
Contents: Azencott, R. : Large deviations and applications.-
Freidlin, Mark I. Semi-linear PDE's and limit theorems for large
deviations- Varadhan, Srinivasa R.S.: Large deviations and
applications.
Surveys in Applied Mathematics is a series of volumes, each of
which contains expo of several topics in mathematics and their
applications. They are written at a sitions level accessible to
advanced graduate students and interested nonspecialists, but they
also contain the results of recent research. Volume I consists of
three articles. The first is the classic paper of J. B. Keller and
R. M. Lewis, "Asymptotic Methods for Partial Differential
Equations: The Reduced Wave Equation and Maxwell's Equations." The
second is by D. W. McLaughlin and E. A. Overman on "Whiskered Tori
for Integrable Pde's: Chaotic Behavior in Near Integrable Pde's."
This is a systematic analytical and numerical study of near
integrable wave equations, including the sine-Gordon equations and
the perturbed nonlinear SchrOdinger equation. The third article is
by G. Papanicolaou on "Diffusion in Random Media." It is an
introductory survey of homogenization methods for the diffusion
equation with random diffusivity."
Probabilistic methods can be applied very successfully to a number
of asymptotic problems for second-order linear and non-linear
partial differential equations. Due to the close connection between
the second order differential operators with a non-negative
characteristic form on the one hand and Markov processes on the
other, many problems in PDE's can be reformulated as problems for
corresponding stochastic processes and vice versa. In the present
book four classes of problems are considered: - the Dirichlet
problem with a small parameter in higher derivatives for
differential equations and systems - the averaging principle for
stochastic processes and PDE's - homogenization in PDE's and in
stochastic processes - wave front propagation for semilinear
differential equations and systems. From the probabilistic point of
view, the first two topics concern random perturbations of
dynamical systems. The third topic, homog- enization, is a natural
problem for stochastic processes as well as for PDE's. Wave fronts
in semilinear PDE's are interesting examples of pattern formation
in reaction-diffusion equations. The text presents new results in
probability theory and their applica- tion to the above problems.
Various examples help the reader to understand the effects.
Prerequisites are knowledge in probability theory and in partial
differential equations.
CONTENTS: M.I. Freidlin: Semi-linear PDE's and limit theorems for
large deviations.- J.F. Le Gall: Some properties of planar Brownian
motion.
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."
Eugene B. Dynkin published his first paper, on Markov chain theory,
whilst still an undergraduate student at Moscow State University.
He went on to make fundamental contributions to the theory of
Markov processes and to Lie groups, generating entire schools in
these areas. This volume features original mathematical papers,
written to honour E.B. Dynkin's 70th birthday. It contains papers
dealing with problems in stochastic analysis, probability theory
and mathematical physics.
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