Random Perturbations of Dynamical Systems (Grundlehren der mathematischen Wissenschaften Book 260)

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.

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 shastic 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 shastic, 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 shastic perturbations is illustrated in this section. Also a new Chapter 9 has been inserted in which deterministic and shastic perturbations of systems with many degrees of freedom are considered. Because of the resonances, shastic regularization in this case is even more important.

This volume is concerned with various kinds of limit theorems for stochastic processes defined as a result of random perturbations of dynamical systems, especially with the long-time behavior of the perturbed system. In particular, exit problems, metastable states, optimal stabilization, and asymptotics of stationary distributions are also carefully considered. The authors' main tools are the large deviation theory the centered limit theorem for stochastic processes, and the averaging principle - all presented in great detail. The results allow for explicit calculations of the asymptotics of many interesting characteristics of the perturbed system. Most of the results are closely connected with PDEs, and the authors' approach presents a powerful method for studying the asymptotic behavior of the solutions of initial-boundary value problems for corresponding PDEs. Main innovations in this edition concern the averaging principle. A new section on deterministic perturbations of one-degree-of-freedom systems was added in Chap. 8. We show 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. To give a rigorous meaning to this statement, one should, first, regularize the system by the addition of small random perturbations. It turns out that the stochasticity of long-time behavior is independent of the regularization. The stochasticity is an intrinsic property of the original system related to the instability of saddle points. This shows usefulness of a joint consideration of classical theory of deterministic perturbations together with stochastic perturbations This volume is concerned with various kinds of limit theorems for stochastic processes defined as a result of random perturbations of dynamical systems; especially with the long-time behavior of the perturbed system. In particular, exit problems, metastable states, optimal stabilization, and asymptotics of stationary distributions are also carefully considered. The authors' main tools are the large deviation theory, the central limit theorem for stochastic processes, and the averaging principle--all presented in great detail. The results allow for explicit calculations of the asymptotics of many interesting characteristics of the perturbed system. Most of the results are closely connected with PDE's and the author's approach presents a powerful method for studying the asymptotic behavior of the solutions of initial-boundary value problems for corresponding PDE's. The most essential additions and changes in this new edition concern the averaging principle. A new chapter on random perturbations of Hamiltonian systems has been added along with new results on fast oscillating perturbations of systems with conservation laws. New sections on wave front propagation in semilinear PDE's and on random perturbations of certain infinite-dimensional dynamical systems have been incorporated into the chapter on Sharpenings and Generalizations. Front Matter....Pages I-XXVIII Random Perturbations....Pages 1-28 Small Random Perturbations on a Finite Time Interval....Pages 29-53 Action Functional....Pages 54-84 Gaussian Perturbations of Dynamical Systems. Neighborhood of an Equilibrium Point....Pages 85-116 Perturbations Leading to Markov Processes....Pages 117-141 Markov Perturbations on Large Time Intervals....Pages 142-191 The Averaging Principle. Fluctuations in Dynamical Systems with Averaging....Pages 192-257 Random Perturbations of Hamiltonian Systems....Pages 258-354 The Multidimensional Case....Pages 355-389 Stability Under Random Perturbations....Pages 390-404 Sharpenings and Generalizations....Pages 405-440 Back Matter....Pages 441-458