Random perturbations of dynamical systems:
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 asympto...
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | German English Russian |
| Published: |
Berlin [u.a.]
Springer
2012
|
| Edition: | 3. ed. |
| Series: | Die Grundlehren der mathematischen Wissenschaften
260 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Summary: | 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 centred 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 |
| Item Description: | Aus dem Russ. übers. |
| Physical Description: | XXVIII, 458 S. graph. Darst. |
| ISBN: | 9783642258466 |
Staff View
MARC
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| 240 | 1 | 0 | |a Fluktuacii v dinamičeskich sistemach pod dejstviem malych slučajnych vozmuščenij |
| 245 | 1 | 0 | |a Random perturbations of dynamical systems |c Mark I. Freidlin ; Alexander D. Wentzell |
| 250 | |a 3. ed. | ||
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2012 | |
| 300 | |a XXVIII, 458 S. |b graph. Darst. | ||
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| 490 | 1 | |a Die Grundlehren der mathematischen Wissenschaften |v 260 | |
| 500 | |a Aus dem Russ. übers. | ||
| 520 | 3 | |a 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 | |
| 520 | |a The authors' main tools are the large deviation theory the centred 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 | ||
| 520 | |a 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 | ||
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Record in the Search Index
| _version_ | 1804148572476669952 |
|---|---|
| adam_text | Contents
Preface
to the Third Edition
....................................
v
Preface to the Second Edition
...................................
vii
Preface
...................................................... ix
Contents
..................................................... xi
Introduction
.................................................. xv
CHAPTER
1
Random Perturbations
......................................... 1
1
Probabilities and Random Variables
............................ 1
2
Random Processes. General Properties
.......................... 3
3
Wiener Process. Stochastic Integral
............................. 9
4
Markov Processes and Semigroups
............................. 15
5
Diffusion Processes and Differential Equations
................... 19
CHAPTER
2
Small Random Perturbations on a Finite Time Interval
.............. 29
1
Zeroth Order Approximation
.................................. 29
2
Expansion in Powers of a Small Parameter
....................... 36
3
Elliptic and Parabolic Differential Equations with a Small Parameter
. 44
CHAPTER
3
Action Functional
............................................. 54
1
Laplace s Method in a Function Space
.......................... 54
2
Exponential Estimates
........................................ 57
3
Action Functional. General Properties
........................... 63
4
Action Functional for Gaussian Random Processes and Fields
...... 75
xii Contents
CHAPTER
4
Gaussian Perturbations of Dynamical Systems. Neighborhood of an
Equilibrium Point
............................................. 85
1
Action Functional
........................................... 85
2
The Problem of Exit from a Domain
............................ 89
3
Properties of the
Quasipotential.
Examples
...................... 100
4
Asymptotics of the Mean Exit Time and Invariant Measure
......... 105
5
Gaussian Perturbations of General Form
......................... 114
CHAPTER
5
Perturbations Leading to Markov Processes
....................... 117
1
Legendre Transformation
..................................... 117
2
Locally Infinitely Divisible Processes
........................... 124
3
Special Cases. Generalizations
................................. 134
4
Consequences. Generalization of Results of Chap.
4............... 137
CHAPTER
6
Markov Perturbations on Large Time Intervals
.................... 142
1
Auxiliary Results. Equivalence Relation
......................... 142
2
Markov Chains Connected with the Process
(Χξ,
P%)
............. 150
3
Lemmas on Markov Chains
................................... 157
4
The Problem of the Invariant Measure
.......................... 165
5
The Problem of Exit from a Domain
............................ 172
6
Decomposition into Cycles. Metastability
....................... 178
7
Eigenvalue Problems
......................................... 184
CHAPTER
7
The Averaging Principle. Fluctuations in Dynamical Systems with
Averaging
.................................................... 192
1
The Averaging Principle in the Theory of Ordinary Differential
Equations
.................................................. 192
2
The Averaging Principle when the Fast Motion is a Random Process
. 196
3
Normal Deviations from an Averaged System
.................... 198
4
Large Deviations from an Averaged System
...................... 212
5
Large Deviations Continued
................................... 219
6
The Behavior of the System on Large Time Intervals
.............. 226
7
Not Very Large Deviations
.................................... 230
8
Examples
.................................................. 235
9
The Averaging Principle for Stochastic Differential Equations
...... 244
CHAPTER
8
Random Perturbations of Hamiltonian Systems
.................... 258
1
Introduction
................................................ 258
2
Main Results
............................................... 269
3
Proof of Theorem
2.2 ........................................ 275
Contents
хш
4
Proof of Lemmas
3.1
to
3.4 ................................... 285
5
Proof of Lemma
3.5.......................................... 300
6
Proof of Lemma
3.6.......................................... 311
7
Remarks and Generalizations
.................................. 316
8
Deterministic Perturbations of Hamiltonian Systems. One Degree of
Freedom
................................................... 332
CHAPTER
9
The Multidimensional Case
.....................................355
1
Slow Component Lives on an Open Book Space
.................. 355
2
The Results Outside the Singularities
........................... 360
3
Weakly Coupled Oscillators. Formulation of the Results
........... 367
4
The Markov Process
(Y(t),
Py) on
Г:
Existence and Uniqueness.
.. 372
5
Proof of Theorem
3.2 ........................................ 376
6
Deterministic Coupling
....................................... 384
CHAPTER
10
Stability Under Random Perturbations
...........................390
1
Formulation of the Problem
................................... 390
2
The Problem of Optimal Stabilization
........................... 396
3
Examples
.................................................. 401
CHAPTER
11
Sharpenings and Generalizations
................................405
1
Local Theorems and Sharp Asymptotics
......................... 405
2
Large Deviations for Random Measures
......................... 412
3
Processes with Small Diffusion with Reflection at the Boundary
..... 419
4
Wave Fronts in
Semilinear
PDEs and Large Deviations
............ 423
5
Random Perturbations of Infinite-Dimensional Systems
............ 433
References
...................................................441
Index
........................................................457
|
| any_adam_object | 1 |
| author | Frejdlin, Mark I. Wentzell, Alexander D. 1937- |
| author_GND | (DE-588)109061047 |
| author_facet | Frejdlin, Mark I. Wentzell, Alexander D. 1937- |
| author_role | aut aut |
| author_sort | Frejdlin, Mark I. |
| author_variant | m i f mi mif a d w ad adw |
| building | Verbundindex |
| bvnumber | BV039699718 |
| callnumber-first | Q - Science |
| callnumber-label | QA274 |
| callnumber-raw | QA274 |
| callnumber-search | QA274 |
| callnumber-sort | QA 3274 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 810 SK 820 |
| ctrlnum | (OCoLC)767777549 (DE-599)BVBBV039699718 |
| dewey-full | 519.2 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 519 - Probabilities and applied mathematics |
| dewey-raw | 519.2 |
| dewey-search | 519.2 |
| dewey-sort | 3519.2 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 3. ed. |
| format | Book |
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| id | DE-604.BV039699718 |
| illustrated | Illustrated |
| indexdate | 2024-07-10T00:09:16Z |
| institution | BVB |
| isbn | 9783642258466 |
| language | German English Russian |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-024548295 |
| oclc_num | 767777549 |
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| owner_facet | DE-739 DE-824 DE-19 DE-BY-UBM DE-11 DE-384 DE-188 DE-83 |
| physical | XXVIII, 458 S. graph. Darst. |
| publishDate | 2012 |
| publishDateSearch | 2012 |
| publishDateSort | 2012 |
| publisher | Springer |
| record_format | marc |
| series | Die Grundlehren der mathematischen Wissenschaften |
| series2 | Die Grundlehren der mathematischen Wissenschaften |
| spelling | Frejdlin, Mark I. Verfasser aut Fluktuacii v dinamičeskich sistemach pod dejstviem malych slučajnych vozmuščenij Random perturbations of dynamical systems Mark I. Freidlin ; Alexander D. Wentzell 3. ed. Berlin [u.a.] Springer 2012 XXVIII, 458 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Die Grundlehren der mathematischen Wissenschaften 260 Aus dem Russ. übers. 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 centred 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 Dynamische systemen gtt Perturbation (Mathématiques) Perturbation (Mathématiques) ram Processos estocasticos larpcal Processus stochastiques Processus stochastiques ram Stochastische processen gtt Perturbation (Mathematics) Stochastic processes Dynamisches System (DE-588)4013396-5 gnd rswk-swf Störungstheorie (DE-588)4128420-3 gnd rswk-swf Stochastischer Prozess (DE-588)4057630-9 gnd rswk-swf Dynamisches System (DE-588)4013396-5 s Störungstheorie (DE-588)4128420-3 s Stochastischer Prozess (DE-588)4057630-9 s DE-604 Wentzell, Alexander D. 1937- Verfasser (DE-588)109061047 aut Die Grundlehren der mathematischen Wissenschaften 260 (DE-604)BV000000395 260 Digitalisierung UB Passau application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024548295&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Frejdlin, Mark I. Wentzell, Alexander D. 1937- Random perturbations of dynamical systems Die Grundlehren der mathematischen Wissenschaften Dynamische systemen gtt Perturbation (Mathématiques) Perturbation (Mathématiques) ram Processos estocasticos larpcal Processus stochastiques Processus stochastiques ram Stochastische processen gtt Perturbation (Mathematics) Stochastic processes Dynamisches System (DE-588)4013396-5 gnd Störungstheorie (DE-588)4128420-3 gnd Stochastischer Prozess (DE-588)4057630-9 gnd |
| subject_GND | (DE-588)4013396-5 (DE-588)4128420-3 (DE-588)4057630-9 |
| title | Random perturbations of dynamical systems |
| title_alt | Fluktuacii v dinamičeskich sistemach pod dejstviem malych slučajnych vozmuščenij |
| title_auth | Random perturbations of dynamical systems |
| title_exact_search | Random perturbations of dynamical systems |
| title_full | Random perturbations of dynamical systems Mark I. Freidlin ; Alexander D. Wentzell |
| title_fullStr | Random perturbations of dynamical systems Mark I. Freidlin ; Alexander D. Wentzell |
| title_full_unstemmed | Random perturbations of dynamical systems Mark I. Freidlin ; Alexander D. Wentzell |
| title_short | Random perturbations of dynamical systems |
| title_sort | random perturbations of dynamical systems |
| topic | Dynamische systemen gtt Perturbation (Mathématiques) Perturbation (Mathématiques) ram Processos estocasticos larpcal Processus stochastiques Processus stochastiques ram Stochastische processen gtt Perturbation (Mathematics) Stochastic processes Dynamisches System (DE-588)4013396-5 gnd Störungstheorie (DE-588)4128420-3 gnd Stochastischer Prozess (DE-588)4057630-9 gnd |
| topic_facet | Dynamische systemen Perturbation (Mathématiques) Processos estocasticos Processus stochastiques Stochastische processen Perturbation (Mathematics) Stochastic processes Dynamisches System Störungstheorie Stochastischer Prozess |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024548295&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000395 |
| work_keys_str_mv | AT frejdlinmarki fluktuaciivdinamiceskichsistemachpoddejstviemmalychslucajnychvozmuscenij AT wentzellalexanderd fluktuaciivdinamiceskichsistemachpoddejstviemmalychslucajnychvozmuscenij AT frejdlinmarki randomperturbationsofdynamicalsystems AT wentzellalexanderd randomperturbationsofdynamicalsystems |