Invariant measures for stochastic nonlinear Schrödinger equations : numerical approximations and symplectic structures / Jialin Hong, Xu Wang.

Includes bibliographical references and index.

Online resource; title from PDF title page (SpringerLink, viewed September 11, 2019).

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

This book provides some recent advance in the study of stochastic nonlinear Schrödinger equations and their numerical approximations, including the well-posedness, ergodicity, symplecticity and multi-symplecticity. It gives an accessible overview of the existence and uniqueness of invariant measures for stochastic differential equations, introduces geometric structures including symplecticity and (conformal) multi-symplecticity for nonlinear Schrödinger equations and their numerical approximations, and studies the properties and convergence errors of numerical methods for stochastic nonlinear Schrödinger equations. This book will appeal to researchers who are interested in numerical analysis, stochastic analysis, ergodic theory, partial differential equation theory, etc.