Mark I. Freidlin - Böcker

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.

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.

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.

CONTENTS: M.I. Freidlin: Semi-linear PDE's and limit theorems for large deviations.- J.F. Le Gall: Some properties of planar Brownian motion.

The usefulness of a joint consideration of classical theory of deterministic perturbations together with stochastic perturbations is illustrated in this section.

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.

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.