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Random perturbations of dynamical systems / M. I. Freĭdlin
Titre : Random perturbations of dynamical systems Type de document : texte imprimé Auteurs : M. I. Freĭdlin ; Alexander D. Wentzell ; Joseph Szücs Mention d'édition : 3rd ed. Importance : 1 online resource (xxviii, 458 pages) Présentation : illustrations ISBN/ISSN/EAN : 978-3-642-25847-3 Langues : Anglais (eng) Mots-clés : Mathematics. Distribution (Probability theory) Probability Theory and Stochastic Processes. waarschijnlijkheidstheorie probability theory stochastische processen stochastic processes wiskunde Mathematics (General) Wiskunde (algemeen) Index. décimale : 519 Résumé : 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. Random perturbations of dynamical systems [texte imprimé] / M. I. Freĭdlin ; Alexander D. Wentzell ; Joseph Szücs . - 3rd ed. . - [s.d.] . - 1 online resource (xxviii, 458 pages) : illustrations.
ISBN : 978-3-642-25847-3
Langues : Anglais (eng)
Mots-clés : Mathematics. Distribution (Probability theory) Probability Theory and Stochastic Processes. waarschijnlijkheidstheorie probability theory stochastische processen stochastic processes wiskunde Mathematics (General) Wiskunde (algemeen) Index. décimale : 519 Résumé : 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. Réservation
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