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Invariant measures for stochastic nonlinear Schrödinger equations : numerical approximations and symplectic structures

Author: Jialin Hong; Xu Wang
Publisher: Singapore : Springer, 2019.
Series: Lecture notes in mathematics (Springer-Verlag), 2251.
Edition/Format:   eBook : Document : EnglishView all editions and formats
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Genre/Form: Electronic books
Material Type: Document, Internet resource
Document Type: Internet Resource, Computer File
All Authors / Contributors: Jialin Hong; Xu Wang
ISBN: 9789813290693 9813290692
OCLC Number: 1118692211
Description: 1 online resource (xiv, 220 pages) : illustrations (some color)
Contents: Intro; Preface; Contents; Notation and Symbols; 1 Invariant Measures and Ergodicity; 1.1 Basic Definitions in Measure Spaces; 1.2 Invariant Measures for Stochastic Processes; 1.3 Ergodicity; 1.4 Strong Feller and Irreducibility Properties; 1.5 Invariant Measures for Hamiltonian Systems; 1.5.1 Stochastic Kubo Oscillator; 1.5.2 Stochastic Dissipative Hamiltonian Systems; 2 Invariant Measures for Stochastic Differential Equations; 2.1 Ergodicity of Solutions to General Stochastic Differential Equations; 2.1.1 Existence of Invariant Measures; 2.1.2 Uniqueness of the Invariant Measure 2.2 Non-degenerate Stochastic Differential Equations and Ergodic Schemes2.3 Stochastic Langevin Equation and Its Discretizations; 2.3.1 Ergodicity for Exact and Numerical Solutions; 2.3.2 Geometric Structure: Conformal Symplecticity; 2.3.3 Schemes of High Weak Convergence Order; 2.4 Approximation of Invariant Measures via Ergodic Schemes; 2.5 Approximation of the Ergodic Limit; 3 Invariant Measures for Stochastic Nonlinear Schrödinger Equations; 3.1 Preliminaries; 3.2 Invariant Measures for Deterministic Nonlinear Schrödinger Equations 3.3 Well-Posedness of Stochastic Nonlinear Schrödinger Equations3.3.1 The Additive Noise Case; 3.3.2 The Multiplicative Noise Case; 3.4 Continuous Dependence of the Solutions on the Initial Data; 3.5 Stochastic Linear Schrödinger Equation with Weak Damping; 3.6 Stochastic Nonlinear Schrödinger Equation with Weak Damping; 3.6.1 One Dimensional Case; 3.6.2 High Dimensional Case; 4 Geometric Structures and Numerical Schemes for Nonlinear Schrödinger Equations; 4.1 Preliminaries; 4.2 Symplectic and Multi-symplectic Methods for Deterministic Schrödinger Equations 4.2.1 Symplectic Temporal Semi-discretizations4.2.2 Multi-symplectic Full Discretizations; 4.3 Stochastic Symplectic Geometric Structure and Numerical Schemes; 4.4 Stochastic Multi-symplectic Geometric Structure and Numerical Schemes; 4.5 Conformal Multi-symplectic Structure for the Damped Case; 5 Numerical Invariant Measures for Damped Stochastic Nonlinear Schrödinger Equations; 5.1 Ergodic Approximation and Numerical Invariant Measures; 5.1.1 Spectral Semi-discretization; 5.1.2 Ergodic Full Discretization; 5.1.3 Weak Error and Error of Invariant Measures; 5.1.4 Numerical Experiments 5.2 Ergodic and Conformal Multi-symplectic Full Approximation5.2.1 Numerical Schemes; 5.2.2 Convergence in Probability; 5.2.3 Numerical Experiments; 6 Approximation of Ergodic Limit for Conservative Stochastic Nonlinear Schrödinger Equations; 6.1 Finite Dimensional Ergodic Approximation; 6.1.1 Finite Dimensional Approximation; 6.1.2 Unique Ergodicity; 6.2 Multi-symplectic Ergodic Fully Discrete Scheme; 6.3 Approximate Error of the Ergodic Limit; 6.4 Numerical Experiments; A Basic Inequalities; B Proof of the Birkhoff-Khinchin Ergodic Theorem; C Proofs of Propositions 5.1, 5.3 and 5.4
Series Title: Lecture notes in mathematics (Springer-Verlag), 2251.
Responsibility: Jialin Hong, Xu Wang.

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