An introduction to the study of the stochastic properties of dynamical systems: Stochastic properties of dynamical systems
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| Main Author: | |
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| Format: | Book |
| Language: | English |
| Published: |
Paris
Société Mathématique de France
2022
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| Series: | Cours spécialisés
30 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | xxii, 249 Seiten |
| ISBN: | 9782856299678 |
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Record in the Search Index
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CONTENTS Preamble . xiii Introduction . xv List of Symbols . xxi 1. Probability preserving dynamical systems, recurrence, ergodicity, mixing. 1.1. Dynamical system, invariant measure . 1.1.1. Definition of probability preserving dynamical systems . 1.1.2. First examples . 1.2. Recurrence . 1.3. Ergodicity . 1.3.1. Definition of ergodicity by visits to sets of positive measure . 1.3.2. Link between ergodicity and density of orbits . 1.3.3. Characterization of ergodicity by (sub)-invariant sets and invariant functions . 8 1.3.4. Characterization of ergodicity by almost surely (sub)-invariant sets and almost surely invariant functions . 10 1.3.5. ★ Ergodicity and Birkhoff sums
. 1.4. Mixing . 1.4.1. Definition and properties . 1.4.2. Examples of mixing and non mixing dynamical systems . 1.5. Exercises . 1.6. Solutions of the exercises . 14 15 16 18 21 23 2. Factors, extensions, isomorphisms . 2.1. Definitions and first examples . 2.2. Properties preserved by factorization . 2.3. Natural extension . 2.4. Exercises . 2.5. Solutions of the exercises . 33 33 35 36 42 43 1 1 1 2 5 7 7 7 3. Stationary processes, Markov chains, Transfer operator . 49 3.1. Stationary processes and dynamical systems . 49 3.2. Markov chains, stationary measures
. 50 SOCIÉTÉ MATHÉMATIQUE DE FRANCE 2022
CONTENTS 3.2.1. Markov operator and Markov properties . 3.2.2. Stationary measures of Markov chains . 3.3. Transfer operator of a dynamical system . 3.3.1. Definitions and first properties . 3.3.2. Invariant measures and transfer operator. 3.4. Exercises on dynamical systems isomorphic to Markov chains . 3.5. Solutions of the exercises . 51 53 55 55 60 70 73 4. Ergodic theorems, asymptotic variance . 83 4.1. The von Neumann ergodic Theorem . 84 4.2. The Birkhoff ergodic theorem . 87 4.3. Asymptotic variance . 91 4.4. Exercises . 94 4.5. Solutions of the exercises . 97 5. Martingale approximation method .105 5.1. Reversed martingales
. 106 5.2. Central Limit Theorem for reversed martingales and applications . 109 5.3. Martingale-coboundary decomposition . 114 5.4. Non-degeneracy of the Gaussian limit. 122 5.5. Other limit theorems . 123 5.6. Exercises . 125 5.7. Solutions of the exercises . 126 6. Quasi-compactness of transfer operators . 131 6.1. Review of spectral theory . 133 6.2. A variant of the Ionescu Tulcea and Marinescu theorem . 136 6.2.1. Application . 138 6.2.2. Proof of Theorem 6.7 . 139 6.3. ★ Further spectral theory: Essential spectrum and quasi-compact operators . 145 6.3.1. ★ Essential spectrum and decomposition of quasi-compact operators 145 6.3.2. ★ Other
characterizations of the essential spectral radius . 151 6.3.3. ★ Quasi-compactness and Doeblin Fortet type Condition . 155 6.4. Application to the study of dynamical systems . 157 6.5. Exercise . 160 6.6. Solution of the exercise . 162 7. The Nagaev-Guivarc’h operator perturbation method . 169 7.1. A perturbation theorem based on the implicit function theorem . 171 7.2. Application to the Central Limit Theorem . 175 7.3. Exercises . 180 7.4. if The Keller and Liverani perturbation theorem . 181 7.4.1. ★ Continuity of the résolvant . 183 COURS SPÉCIALISÉS 30
CONTENTS xi 7.4.2. ★ Continuity of the dimension of the generalized eigenspaces. 188 7.4.3. ★ Proof of Theorem 7.8 .190 7.5. ★ Exercises . 191 7.6. Solutions of the exercises . 194 8. ★ Central Limit Theorem via decorrelation . 209 8.1. ★ Central Limit Theorem under a general decorrelation assumption . 211 8.2. ★ A general strategy to prove the decorrelation assumptions using conditioning . 219 8.3. ★ Central Limit Theorem in a general (partially) hyperbolic context . . 221 8.4. ★ Application to ergodic toral automorphisms . 224 8.4.1. ★ Exponential rate of decorrelation for Hôlder observables . 228 8.4.2. ★ Proof of the exponential convergence of conditional expectations 232 8.5. ★ Solutions of the exercises . 237 Bibliography . 243 Index . 251 SOCIÉTÉ MATHÉMATIQUE DE FRANCE 2022 |
| adam_txt |
CONTENTS Preamble . xiii Introduction . xv List of Symbols . xxi 1. Probability preserving dynamical systems, recurrence, ergodicity, mixing. 1.1. Dynamical system, invariant measure . 1.1.1. Definition of probability preserving dynamical systems . 1.1.2. First examples . 1.2. Recurrence . 1.3. Ergodicity . 1.3.1. Definition of ergodicity by visits to sets of positive measure . 1.3.2. Link between ergodicity and density of orbits . 1.3.3. Characterization of ergodicity by (sub)-invariant sets and invariant functions . 8 1.3.4. Characterization of ergodicity by almost surely (sub)-invariant sets and almost surely invariant functions . 10 1.3.5. ★ Ergodicity and Birkhoff sums
. 1.4. Mixing . 1.4.1. Definition and properties . 1.4.2. Examples of mixing and non mixing dynamical systems . 1.5. Exercises . 1.6. Solutions of the exercises . 14 15 16 18 21 23 2. Factors, extensions, isomorphisms . 2.1. Definitions and first examples . 2.2. Properties preserved by factorization . 2.3. Natural extension . 2.4. Exercises . 2.5. Solutions of the exercises . 33 33 35 36 42 43 1 1 1 2 5 7 7 7 3. Stationary processes, Markov chains, Transfer operator . 49 3.1. Stationary processes and dynamical systems . 49 3.2. Markov chains, stationary measures
. 50 SOCIÉTÉ MATHÉMATIQUE DE FRANCE 2022
CONTENTS 3.2.1. Markov operator and Markov properties . 3.2.2. Stationary measures of Markov chains . 3.3. Transfer operator of a dynamical system . 3.3.1. Definitions and first properties . 3.3.2. Invariant measures and transfer operator. 3.4. Exercises on dynamical systems isomorphic to Markov chains . 3.5. Solutions of the exercises . 51 53 55 55 60 70 73 4. Ergodic theorems, asymptotic variance . 83 4.1. The von Neumann ergodic Theorem . 84 4.2. The Birkhoff ergodic theorem . 87 4.3. Asymptotic variance . 91 4.4. Exercises . 94 4.5. Solutions of the exercises . 97 5. Martingale approximation method .105 5.1. Reversed martingales
. 106 5.2. Central Limit Theorem for reversed martingales and applications . 109 5.3. Martingale-coboundary decomposition . 114 5.4. Non-degeneracy of the Gaussian limit. 122 5.5. Other limit theorems . 123 5.6. Exercises . 125 5.7. Solutions of the exercises . 126 6. Quasi-compactness of transfer operators . 131 6.1. Review of spectral theory . 133 6.2. A variant of the Ionescu Tulcea and Marinescu theorem . 136 6.2.1. Application . 138 6.2.2. Proof of Theorem 6.7 . 139 6.3. ★ Further spectral theory: Essential spectrum and quasi-compact operators . 145 6.3.1. ★ Essential spectrum and decomposition of quasi-compact operators 145 6.3.2. ★ Other
characterizations of the essential spectral radius . 151 6.3.3. ★ Quasi-compactness and Doeblin Fortet type Condition . 155 6.4. Application to the study of dynamical systems . 157 6.5. Exercise . 160 6.6. Solution of the exercise . 162 7. The Nagaev-Guivarc’h operator perturbation method . 169 7.1. A perturbation theorem based on the implicit function theorem . 171 7.2. Application to the Central Limit Theorem . 175 7.3. Exercises . 180 7.4. if The Keller and Liverani perturbation theorem . 181 7.4.1. ★ Continuity of the résolvant . 183 COURS SPÉCIALISÉS 30
CONTENTS xi 7.4.2. ★ Continuity of the dimension of the generalized eigenspaces. 188 7.4.3. ★ Proof of Theorem 7.8 .190 7.5. ★ Exercises . 191 7.6. Solutions of the exercises . 194 8. ★ Central Limit Theorem via decorrelation . 209 8.1. ★ Central Limit Theorem under a general decorrelation assumption . 211 8.2. ★ A general strategy to prove the decorrelation assumptions using conditioning . 219 8.3. ★ Central Limit Theorem in a general (partially) hyperbolic context . . 221 8.4. ★ Application to ergodic toral automorphisms . 224 8.4.1. ★ Exponential rate of decorrelation for Hôlder observables . 228 8.4.2. ★ Proof of the exponential convergence of conditional expectations 232 8.5. ★ Solutions of the exercises . 237 Bibliography . 243 Index . 251 SOCIÉTÉ MATHÉMATIQUE DE FRANCE 2022 |
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| spelling | Pène, Françoise Verfasser (DE-588)127739914X aut An introduction to the study of the stochastic properties of dynamical systems Stochastic properties of dynamical systems Françoise Pène Paris Société Mathématique de France 2022 xxii, 249 Seiten txt rdacontent n rdamedia nc rdacarrier Cours spécialisés 30 Stochastik (DE-588)4121729-9 gnd rswk-swf Dynamisches System (DE-588)4013396-5 gnd rswk-swf Dynamisches System (DE-588)4013396-5 s Stochastik (DE-588)4121729-9 s DE-604 Cours spécialisés 30 (DE-604)BV012699276 30 Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=034129174&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Pène, Françoise An introduction to the study of the stochastic properties of dynamical systems Stochastic properties of dynamical systems Cours spécialisés Stochastik (DE-588)4121729-9 gnd Dynamisches System (DE-588)4013396-5 gnd |
| subject_GND | (DE-588)4121729-9 (DE-588)4013396-5 |
| title | An introduction to the study of the stochastic properties of dynamical systems Stochastic properties of dynamical systems |
| title_auth | An introduction to the study of the stochastic properties of dynamical systems Stochastic properties of dynamical systems |
| title_exact_search | An introduction to the study of the stochastic properties of dynamical systems Stochastic properties of dynamical systems |
| title_exact_search_txtP | An introduction to the study of the stochastic properties of dynamical systems Stochastic properties of dynamical systems |
| title_full | An introduction to the study of the stochastic properties of dynamical systems Stochastic properties of dynamical systems Françoise Pène |
| title_fullStr | An introduction to the study of the stochastic properties of dynamical systems Stochastic properties of dynamical systems Françoise Pène |
| title_full_unstemmed | An introduction to the study of the stochastic properties of dynamical systems Stochastic properties of dynamical systems Françoise Pène |
| title_short | An introduction to the study of the stochastic properties of dynamical systems |
| title_sort | an introduction to the study of the stochastic properties of dynamical systems stochastic properties of dynamical systems |
| title_sub | Stochastic properties of dynamical systems |
| topic | Stochastik (DE-588)4121729-9 gnd Dynamisches System (DE-588)4013396-5 gnd |
| topic_facet | Stochastik Dynamisches System |
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