Invariant measures for stochastic nonlinear Schrödinger equations: numerical approximations and symplectic structures
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[2019]
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| Series: | Lecture notes in mathematics
volume 2251 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | xiv, 220 Seiten Diagramme |
| ISBN: | 9789813290686 |
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| adam_text | Jialin Hong • Xu Wang
Invariant Measures
for Stochastic Nonlinear
Schrödinger Equations
Numerical Approximations and Symplectic
Structures
^ Springer
Universitäts- und
Landesbibliothek
Darmstadt
Jialin Hong
LSEC, ICMSEC, Academy of Mathematics
and Systems Science
Chinese Academy of Sciences
Beijing, China
School of Mathematical Sciences
University of Chinese Academy of Sciences
Beijing, China
Xu Wang
LSEC, ICMSEC, Academy of Mathematics
and Systems Science
Chinese Academy of Sciences
Beijing, China
School of Mathematical Sciences
University of Chinese Academy of Sciences
Beijing, China
ISSN 0075-8434 ISSN 1617-9692 (electronic)
Lecture Notes in Mathematics
ISBN 978-981-32-9068-6 ISBN 978-981-32-9069-3 (eBook)
https://doi org/10 1007/978-981-32-9069-3
© Springer Nature Singapore Pie Ltd 2019
This work is subject to copyright All rights are reserved by the Publisher, whether the whole or part
of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,
recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission
or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar
methodology now known or hereafter developed
The use of general descriptive names, registered names, trademarks, service marks, etc in this
publication does not imply, even in the absence of a specific statement, that such names are exempt from
the relevant protective laws and regulations and therefore free for general use
The publisher, the authors and the editors are safe to assume that the advice and information in this
book are believed to be true and accurate at the dale of publication Neither the publisher nor the
authors or the editors give a warranty, expressed or implied, with respect to the material contained
herein or for any errors or omissions that may have been made The publisher remains neutral with regard
to jurisdictional claims in published maps and institutional affiliations
This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd
The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721,
Singapore
/^ 0=H amp;8
Contents
1 Invariant Measures and Ergodicity 1
1 1 Basic Definitions in Measure Spaces 1
1 2 Invariant Measures for Stochastic Processes 5
1 3 Ergodicity 9
1 4 Strong Feller and Irreducibility Properties 15
1 5 Invariant Measures for Hamiltonian Systems 17
151 Stochastic Kubo Oscillator 19
152 Stochastic Dissipative Hamiltonian Systems 24
Summary 29
2 Invariant Measures for Stochastic Differential Equations 31
2 1 Ergodicity of Solutions to General Stochastic Differential
Equations 32
211 Existence of Invariant Measures 32
212 Uniqueness of the Invariant Measure 34
2 2 Non-degenerate Stochastic Differential Equations
and Ergodic Schemes 36
2 3 Stochastic Langevin Equation and Its Discretizations 39
231 Ergodicity for Exact and Numerical Solutions 40
232 Geometric Structure: Conformal Symplecticity 42
233 Schemes of High Weak Convergence Order 46
2 4 Approximation of Invariant Measures via Ergodic Schemes 51
2 5 Approximation of the Ergodic Limit 56
Summary 60
3 Invariant Measures for Stochastic Nonlinear Schrödinger
Equations 63
3 1 Preliminaries 63
3 2 Invariant Measures for Deterministic Nonlinear Schrödinger
Equations 65
ix
Contents
3 3 Well-Posedness of Stochastic Nonlinear Schrödinger
Equations 68
331 The Additive Noise Case 68
332 The Multiplicative Noise Case 70
3 4 Continuous Dependence of the Solutions on the Initial Data 73
3 5 Stochastic Linear Schrödinger Equation with Weak Damping 75
3 6 Stochastic Nonlinear Schrödinger Equation with Weak
Damping 77
361 One Dimensional Case 77
362 High Dimensional Case 78
Summary 79
4 Geometric Structures and Numerical Schemes for Nonlinear
Schrödinger Equations 81
4 1 Preliminaries 81
4 2 Symplectic and Multi-symplectic Methods for Deterministic
Schrödinger Equations 85
421 Symplectic Temporal Semi-discretizations 85
422 Multi-symplectic Full Discretizations 87
4 3 Stochastic Symplectic Geometric Structure and Numerical
Schemes 92
4 4 Stochastic Multi-symplectic Geometric Structure and Numerical
Schemes 100
4 5 Conformal Multi-symplectic Structure for the Damped Case 103
Summary 108
5 Numerical Invariant Measures for Damped Stochastic Nonlinear
Schrödinger Equations 109
5 1 Ergodic Approximation and Numerical Invariant Measures 109
511 Spectral Semi-discretization 110
512 Ergodic Full Discretization 114
513 Weak Error and Error of Invariant Measures 120
514 Numerical Experiments 132
5 2 Ergodic and Conformal Multi-symplectic Full Approximation 135
521 Numerical Schemes 135
522 Convergence in Probability 142
523 Numerical Experiments 149
Summary 152
6 Approximation of Ergodic Limit for Conservative Stochastic
Nonlinear Schrödinger Equations 153
6 1 Finite Dimensional Ergodic Approximation 153
611 Finite Dimensional Approximation 154
612 Unique Ergodicity 156
Contents xi
6 2 Multi-symplectic Ergodic Fully Discrete Scheme 159
6 3 Approximate Error of the Ergodic Limit 165
6 4 Numerical Experiments 177
Summary 180
Appendix A: Basic Inequalities 181
Appendix B: Proof of the Birkhoff-Khinchin Ergodic Theorem 191
Appendix C: Proofs of Propositions 5 1, 5 3 and 5 4 195
References 209
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| id | DE-604.BV046164223 |
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| isbn | 9789813290686 |
| language | English |
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| series2 | Lecture notes in mathematics |
| spelling | Hong, Jialin Verfasser (DE-588)1200141504 aut Invariant measures for stochastic nonlinear Schrödinger equations numerical approximations and symplectic structures Jialin Hong, Xu Wang Singapore Springer [2019] © 2019 xiv, 220 Seiten Diagramme txt rdacontent n rdamedia nc rdacarrier Lecture notes in mathematics volume 2251 Nichtlineare Schrödinger-Gleichung (DE-588)4278277-6 gnd rswk-swf Stochastische partielle Differentialgleichung (DE-588)4135969-0 gnd rswk-swf Nichtlineare Schrödinger-Gleichung (DE-588)4278277-6 s Stochastische partielle Differentialgleichung (DE-588)4135969-0 s DE-604 Wang, Xu Verfasser (DE-588)1176035479 aut Erscheint auch als Online-Ausgabe 978-981-32-9069-3 Lecture notes in mathematics volume 2251 (DE-604)BV000676446 2251 HEBIS Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=031544096&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Hong, Jialin Wang, Xu Invariant measures for stochastic nonlinear Schrödinger equations numerical approximations and symplectic structures Lecture notes in mathematics Nichtlineare Schrödinger-Gleichung (DE-588)4278277-6 gnd Stochastische partielle Differentialgleichung (DE-588)4135969-0 gnd |
| subject_GND | (DE-588)4278277-6 (DE-588)4135969-0 |
| title | Invariant measures for stochastic nonlinear Schrödinger equations numerical approximations and symplectic structures |
| title_auth | Invariant measures for stochastic nonlinear Schrödinger equations numerical approximations and symplectic structures |
| title_exact_search | Invariant measures for stochastic nonlinear Schrödinger equations numerical approximations and symplectic structures |
| title_full | Invariant measures for stochastic nonlinear Schrödinger equations numerical approximations and symplectic structures Jialin Hong, Xu Wang |
| title_fullStr | Invariant measures for stochastic nonlinear Schrödinger equations numerical approximations and symplectic structures Jialin Hong, Xu Wang |
| title_full_unstemmed | Invariant measures for stochastic nonlinear Schrödinger equations numerical approximations and symplectic structures Jialin Hong, Xu Wang |
| title_short | Invariant measures for stochastic nonlinear Schrödinger equations |
| title_sort | invariant measures for stochastic nonlinear schrodinger equations numerical approximations and symplectic structures |
| title_sub | numerical approximations and symplectic structures |
| topic | Nichtlineare Schrödinger-Gleichung (DE-588)4278277-6 gnd Stochastische partielle Differentialgleichung (DE-588)4135969-0 gnd |
| topic_facet | Nichtlineare Schrödinger-Gleichung Stochastische partielle Differentialgleichung |
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