Invariant Measures for Stochastic Nonlinear Schrödinger Equations: Numerical Approximations and Symplectic Structures [1st ed. 2019] 978-981-32-9068-6, 978-981-32-9069-3

Table of contents :
Front Matter ....Pages i-xiv
Invariant Measures and Ergodicity (Jialin Hong, Xu Wang)....Pages 1-29
Invariant Measures for Stochastic Differential Equations (Jialin Hong, Xu Wang)....Pages 31-61
Invariant Measures for Stochastic Nonlinear Schrödinger Equations (Jialin Hong, Xu Wang)....Pages 63-79
Geometric Structures and Numerical Schemes for Nonlinear Schrödinger Equations (Jialin Hong, Xu Wang)....Pages 81-108
Numerical Invariant Measures for Damped Stochastic Nonlinear Schrödinger Equations (Jialin Hong, Xu Wang)....Pages 109-152
Approximation of Ergodic Limit for Conservative Stochastic Nonlinear Schrödinger Equations (Jialin Hong, Xu Wang)....Pages 153-180
Back Matter ....Pages 181-220

Citation preview

Lecture Notes in Mathematics 2251

Jialin Hong Xu Wang

Invariant Measures for Stochastic Nonlinear Schrödinger Equations Numerical Approximations and Symplectic Structures

Lecture Notes in Mathematics Volume 2251

Editors-in-Chief Jean-Michel Morel, CMLA, École Normale Supérieure de Cachan, Cachan, France Bernard Teissier, Equipe Géométrie et Dynamique, Institut Mathématique de Jussieu-Paris Rive Gauche, Paris, France Advisory Editors Karin Baur, School of Mathematics, University of Leeds, Leeds, UK Michel Brion, Institut Fourier, Université Grenoble Alpes, Grenoble, France Camillo De Lellis, Institute for Advanced Study, Princeton, NJ, USA Alessio Figalli, Department of Mathematics, Swiss Federal Institute of Technology, Zurich, Switzerland Annette Huber, Mathematical Institute, Albert-Ludwigs-Universität, Freiburg, Germany Davar Khoshnevisan, Department of Mathematics, The University of Utah, Salt Lake City, UT, USA Ioannis Kontoyiannis, Department of Engineering, University of Cambridge, Cambridge, UK Angela Kunoth, Mathematical Institute, University of Cologne, Cologne, Germany Ariane Mézard, Institut Mathématique de Jussieu-Paris Rive Gauche, Sorbonne University, PARIS, France Mark Podolskij, Department of Mathematics, Aarhus University, Aarhus, Denmark Sylvia Serfaty, Courant Institute of Mathematics, New York University, New York, USA Gabriele Vezzosi, Dipartimento Matematica Informatica, University of Florence, FIRENZE, Italy Anna Wienhard, Mathematical Institute, Heidelberg University, Heidelberg, Germany

More information about this series at http://www.springer.com/series/304

Jialin Hong Xu Wang •

Invariant Measures for Stochastic Nonlinear Schrödinger Equations Numerical Approximations and Symplectic Structures

123

Jialin Hong LSEC, ICMSEC, Academy of Mathematics and Systems Science 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

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 Pte 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 date 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

Preface

Stochastic differential equations, originated from the work of A. Einstein and put on solider mathematical footing by K. Itô and R. L. Stratonovich, are now widely used to model various phenomena caused by random medium or stochastic external sources, such as propagation of nonlinear dispersive waves in random medium and unstable stock prices in financial market. There are fruitful results on the study of geometric structures, dynamical behaviors, statistical properties, and some other important internal properties for stochastic differential equations. In this monograph, we take both the geometric structure and dynamical behavior, more precisely, the (conformal) multi-symplecticity and ergodicity, of stochastic nonlinear Schrödinger equations into consideration when constructing numerical approximations. Thus, the problems considered in this monograph are related to several fascinating research hotspots: numerical analysis, stochastic analysis, ergodic theory, partial differential equation theory, and so on. The main object of this monograph is the analysis and numerical approximations of invariant distributions for stochastic nonlinear Schrödinger equations. The considered model shows good performances over longtime due to its dynamical behavior and geometric structure. Among these properties, ergodicity and stochastic (conformal) multi-symplecticity play fundamental roles in the study of the dynamical behavior and geometric structure for stochastic nonlinear Schrödinger equations. The stochastic multi-symplecticity is an extension of the multi-symplectic conservation law in the deterministic case (see [34, 117] and references therein). It implies that the spatio-temporal geometric structure of the system is preserved over longtime, and is an essential part of symplectic geometric theory. Ergodic theory, as an important branch of mathematics, studies the asymptotic behavior of measure-preserving transformations. It originates from the study of the “ergodic hypothesis” which is the basic hypothesis in statistical mechanics and information theory, and is now developed as a powerful amalgam of methods used for the analysis of statistical properties of dynamical systems. Thus, it is widely

v

vi

Preface

investigated by researchers in dynamical systems, probability theory, physics, biology, chemistry, etc. (see [39, 53, 85, 129, 170, 189] and references therein). We first give a systematic summary of the definitions and sufficient conditions of the existence and uniqueness of invariant measures as well as the ergodicity for stochastic Markov processes in Chap. 1. Moreover, stochastic Kubo oscillator and stochastic Hamiltonian dissipative systems are introduced as two examples to study their invariant measures, geometric structures, and numerical approximations. To have a better understanding of the longtime behavior of stochastic differential equations (SDEs) with ergodic invariant measures, we then investigate the following fundamental problems: • What kind of systems are ergodic, and what is the convergence rate between the R RT temporal average T1 0 uðPs xÞds and the spatial average M udl? • Are there any numerical approximations which could inherit the ergodicity of the original system, and how to estimate the error between the numerical invariant measure and the original one? • Is the temporal average of R a numerical approximation (possibly not ergodic) a proper approximation of M udl? Chapter 2 answers these questions with respect to finite dimensional stochastic differential equations, and take the stochastic Langevin equation as an example to construct its ergodic numerical schemes. Some more specific conditions which ensure the ergodicity of the solution of stochastic differential equations under proper assumptions are also given in Chap. 2. The first two chapters will help the readers who are not familiar with invariant measures and ergodicity to get a better understanding on these concepts and their properties. However, the answers to above questions with respect to stochastic partial differential equations (SPDEs) are far from complete. The main contribution of this monograph is the investigation of ergodic numerical approximations for stochastic nonlinear Schrödinger equations, which gives answers to the last two questions mentioned above for stochastic nonlinear Schrödinger equations. In Chap. 3, the local and global existence and uniqueness of the solution to stochastic nonlinear Schrödinger equations with both additive and multiplicative noises are briefly introduced. The continuous dependence of the solution on initial data is also given to ensure that the semigroup generated by the solution is Feller, which is a fundamental condition when studying the existence of invariant measures, geometric structures, and the large deviation principle. We recall some results about invariant measures for both deterministic and stochastic Schrödinger equations with weak damping. For the deterministic case, its invariant measure is constructed based on a finite approximation of the considered equation in the energy space H1 ð0; LÞ with L [ 0. For the stochastic case with weak damping, the solution is shown to be Gaussian for the linear case. As a result, the distribution of the solution converges to a unique C-valued Gaussian distribution. For the cubic

Preface

vii

nonlinear case with weak damping, the considered model is shown to posses the mixing property with a unique invariant measure, and the rate of convergence to equilibrium is at least polynomial of any power due to the damping term. The existence result for ergodic invariant measures is extended to the high dimension and unbounded domain case with certain conditions on the relation between the nonlinear term and the spatial dimension. In Chap. 4, we concentrate on the geometric structures for both deterministic and stochastic nonlinear Schrödinger equations, as well as their numerical approximations. Several temporal semi-discretizations and full discretizations are reviewed for the deterministic nonlinear Schrödinger equation. The stochastic symplectic and multi-symplectic conservation laws are proved for stochastic nonlinear Schrödinger equations with a linear multiplicative noise in Stratonovich sense. The numerical approximations for the deterministic case are generalized to the stochastic case. The temporal semi-discretization based on the Runge–Kutta methods is studied for stochastic nonlinear Schrödinger equations. The symplectic condition is also established, under which the numerical solution preserves the discrete stochastic symplectic structure. A full discretization based on the midpoint scheme in both spatial and temporal directions is given afterward to inherit the stochastic multi-symplectic structure of the stochastic nonlinear Schrödinger equation. While for stochastic nonlinear Schrödinger equation with weak damping, its geometric structure is shown to satisfy a conformal stochastic multi-symplectic principle instead of the multi-symplectic conservation law. Chapter 5 is devoted to studying the numerical approximations of the stochastic nonlinear Schrödinger equation with additional damped term and an infinite dimensional additive noise. This model is shown to be ergodic with a unique invariant measure [70], but is apparently not multi-symplectic anymore. We show that the system possesses a conformal multi-symplectic geometric structure instead of the multi-symplectic structure in conservative case. We are interested in the investigation of the geometric structure for this damped equation, and aim to give a computable approximation to the original invariant measure, as well as the approximate error defined in the standard way. We consider in Chap. 6 the stochastic nonlinear Schrödinger equation with linear multiplicative noise, which also possesses the charge conservation law almost surely. For this conservative equation, it has been shown to possess the multi-symplectic structure [119], while its ergodicity turns to be an open problem. Our purposes are to find an ergodic finite dimensional approximation of the original system, and to approximate its ergodic limit through an ergodic, multi-symplectic full discretization. In addition, the temporal average of the fully discrete scheme is shown to converge to the spatial average of the finite dimensional approximation with a specific rate. These results may help to find a new way to show the ergodicity of conservative equations as an afterthought.

viii

Preface

Last but not least, we find it an efficient way to give a visible description of the ergodic limit via numerical experiments, and several numerical experiments are given in Chaps. 5 and 6 to make it clearer to the reader what the longtime behaviors of ergodic processes are. Beijing, China May 2017

Jialin Hong Xu Wang

Acknowledgements It is our pleasure to thank the authors of our references for their important contributions and motivating us to get into these wonderful mathematical fields. We would like to thank all the referees, whose comments and suggestions are of great help in improving this monograph. We would also like to thank Dr. Chuchu Chen, Yulan Lu, Jianbo Cui, Liying Sun, Chuying Huang, Diancong Jin and Derui Sheng for their careful reading and pointing out a lot of typos in the draft of this monograph. We also acknowledge the National Natural Science Foundation of China (No. 11021101, No. 11290142, No. 91130003, No. 91530118 and No. 91630312) for its financial support.

Contents

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 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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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 Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 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 . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1 1 5 9 15 17 19 24 29

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3 Invariant Measures for Stochastic Nonlinear Schrödinger Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Invariant Measures for Deterministic Nonlinear Schrödinger Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63 63 65

ix

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Contents

3.3 Well-Posedness of Stochastic Nonlinear Schrödinger Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 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 . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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77 77 78 79

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85 85 87

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92

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-discretizations . . . . . . . . . . . 4.2.2 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 . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . 100 . . 103 . . 108

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 Approximation . . 5.2.1 Numerical Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Convergence in Probability . . . . . . . . . . . . . . . . . . . . . 5.2.3 Numerical Experiments . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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109 109 110 114 120 132 135 135 142 149 152

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 . . . . . . . . . . . . . . . . . . . . . . . . .

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153 153 154 156

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Contents

6.2 Multi-symplectic Ergodic Fully Discrete Scheme 6.3 Approximate Error of the Ergodic Limit . . . . . . 6.4 Numerical Experiments . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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159 165 177 180

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 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

Notation and Symbols

H S N (resp. N þ ) R (resp. R þ ) Rd (resp. Td ) BðHÞ PðHÞ L ðH; KÞ Lp ðH; KÞ Lp ðH; lÞ Bb ðHÞ Cb ðHÞ C1 b ðHÞ C1 p ðHÞ Wk;p ðHÞ Wk;1 ðHÞ Hp ðRd Þ ðX; F ; PÞ ðM; G ; lÞ pt ðx; GÞ

Hilbert space Unit sphere in the corresponding space Set of all nonnegative (resp. positive) integers Set of all real numbers (resp. set of all positive real numbers) d-dimensional real space (resp. d-dimensional torus) Borel r-algebra on H Space of all probability measures on ðH; BðHÞÞ Space of all linear bounded operators from space H to K, denoted by L ðHÞ if K ¼ H Space of all K-valued functions defined on H which are pth integrable Space of all functions defined on H which are pth integrable with respect to measure l, also denoted by Lp ðHÞ for simplicity Banach space of all Borel bounded functions on H Banach space of all uniformly continuous and bounded functions on H Space of all smooth and bounded functions with bounded derivatives of any order Space of all smooth functions with polynomial growth Sobolev space consisting of all R-valued functions on H whose first k weak derivatives are functions in Lp Sobolev spaces of all functions on H whose derivatives up to order k have finite L1 norm Sobolev spaces of all functions on Rd whose derivatives up to order p are square integrable Probability space Measure space Transition probability for a stochastic process starting from x to hit set G at time t

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xiv

Nð0; 1Þ jj k k k kF hMi h; iH ð; Þ Ec M U s ^ t (resp. s _ t) RðuÞ (resp. JðuÞ) DomðAÞ

Notation and Symbols

Standard normal distribution Absolute value in R or C Euclidean norm in the corresponding finite dimensional real space Frobenius norm of matrices in the corresponding space Quardratic variation process of stochastic process M Inner product in some Hilbert space H; in particular, if H is a finite dimensional real-valued space, denote h; i for simplicity Inner product in a finite dimentional complex-valued space Complementary set of set E Closure of M under the corresponding Euclidean norm Conjugate of complex number U Minimum (resp. maximum) of s; t 2 R Real (resp. imaginary) part of u Domain of operator A

Chapter 1

Invariant Measures and Ergodicity

1.1 Basic Definitions in Measure Spaces In classical ergodic theory, the behaviors of automorphisms, endomorphisms, flows, and semiflows are studied in measure spaces. Let (M, G , μ) be a measure space with a normalized measure μ. This section introduces some basic concepts concerning invariant measures and ergodicity for endomorphisms and semiflows. We refer to [53] for more details about the ergodic theory of dynamical systems in measure spaces. Definition 1.1 An endomorphism of M is a surjection P : M → M such that for any A ∈ G , P −1 A ∈ G and μ(P −1 A) = μ(A) with P −1 A denoting the inverse image of A. The measure μ is said to be invariant under the endomorphism P. For continuous dynamical systems, a family of endomorphisms will be taken into consideration with measure μ still being invariant under these endomorphisms. Definition 1.2 Let {Pt }t≥0 be a one-parameter semigroup of endomorphisms on (M, G , μ) satisfying Pt+s x = Pt ◦ Ps x for all t, s ≥ 0 and x ∈ M. Then {Pt }t≥0 is called a semiflow if ϕ(Pt x) is measurable on M × R+ for any measurable function ϕ on M. Remark 1.1 It shows immediately that μ is an invariant measure of {Pt }t≥0 , i.e., 

 M

ϕ(Pt x)μ(d x) =

M

ϕ(x)μ(d x)

© Springer Nature Singapore Pte Ltd. 2019 J. Hong and X. Wang, Invariant Measures for Stochastic Nonlinear Schrödinger Equations, Lecture Notes in Mathematics 2251, https://doi.org/10.1007/978-981-32-9069-3_1

1

2

1 Invariant Measures and Ergodicity

for any measurable and bounded function ϕ on M, according to the approximation by simple functions. In fact, for indicator function 1 A of measurable set A ∈ G , we have   1 A (Pt x)μ(d x) = 1 Pt−1 A (x)μ(d x) = μ(Pt−1 A). M

M

Note that μ(Pt−1 A) = μ(A) since Pt is an endomorphism, which leads to 

 M

1 A (Pt x)μ(d x) = μ(A) =

M

1 A (x)μ(d x).

  In some circumstances, we also use the notation M ϕdμ := M ϕ(x)μ(d x) for simplicity. The expression “dynamical system” could stand for either endomorphisms or semiflows, or even for automorphisms and flows whose definitions are omitted to make the content simple and clear. In this case, the measure space (M, G , μ) is known as the phase space. Definition 1.3 A set G ∈ G is said to be invariant with respect to {Pt }t≥0 if x ∈ G is equivalent to Pt x ∈ G for any t ≥ 0, i.e., 1G (Pt x) = 1G (x) for any t ≥ 0 and for all x ∈ M. In addition, the dynamical system {Pt }t≥0 is said to be ergodic if all the invariant set G are trivial, i.e., μ(G) = 0 or μ(G) = 1. The measure μ is called an ergodic invariant measure for {Pt }t≥0 . When considering the asymptotic behavior of endomorphism P or semiflow {Pt }t≥0 , the Birkhoff–Khinchin ergodic theorem (see e.g. [22, 53]) shows that the limit lim

N →∞

N −1 1  ϕ(P n x) or N n=0

lim

T →∞

1 T



T

ϕ(Ps x)ds

0

always exists for μ-almost every x ∈ M. Theorem 1.1 (Birkhoff–Khinchin ergodic theorem) For ϕ ∈ L1 (M, μ), the limits lim

N →∞

N −1 1  ϕ(P n x) N n=0

1.1 Basic Definitions in Measure Spaces

3

and lim

T →∞

1 T



T

ϕ(Ps x)ds

0

exist for μ-almost every x ∈ M when P is an endomorphism and {Pt }t≥0 is a semiflow. The proof of above theorem can also be used to prove the Von Neumann theorem introduced in the next section and is given in Appendix B for the reader’s convenience. Denote by ϕ(x) ˜ := lim

T →∞

1 T



T

ϕ(Ps x)ds.

(1.1)

0

Based on the results above, we claim that Aa := {ϕ(x) ˜ < a} is an invariant set for any a ∈ R, that is, for any x ∈ Aa , Pt x ∈ Aa for any t ≥ 0, and vice versa. In fact,   1 T 1 T ϕ(Ps ◦ Pt x)ds = lim ϕ(Ps+t x)ds T →∞ T 0 T →∞ T 0  1 T ϕ(Ps x)ds = ϕ(x), ˜ = lim T →∞ T 0

ϕ(P ˜ t x) = lim

which verifies the claim. It then leads to the following corollary when the semiflow {Pt }t≥0 is ergodic. Corollary 1.1 If {Pt }t≥0 is ergodic, then ϕ(x) ˜ equals a constant μ-almost surely, and the constant is exactly the spatial average of ϕ, i.e.,  ϕ(x) ˜ =

M

 ϕdμ ˜ =

M

ϕdμ.

Proof Since ϕ(x) ˜ exists for μ-almost every x ∈ M, we derive according to above claim and the ergodicity of semiflow {Pt }t≥0 that there must exist a0 = b0 ∈ R such that μ(Aa0 ) = 0 and μ(Ab0 ) = 1. As a result, ˜ ≤ b0 , μ-a.s. a0 ≤ ϕ(x) Choose any c ∈ (a0 , b0 ), then μ(Ac ) = 0 or 1 since the ergodicity of the semiflow. If μ(Ac ) = 0, we denote a1 = c and b1 = b0 , otherwise we denote a1 = a0 and b1 = c. It then ensures that ˜ ≤ b1 , μ-a.s. a1 ≤ ϕ(x) Following this procedure, we get a family of closed intervals {[an , bn ]}n∈N . According to the principle of nested intervals, one derives lim an = lim bn ,

n→∞

n→∞

4

1 Invariant Measures and Ergodicity

which ensures that ϕ˜ is a constant μ-almost surely. Moreover, integrating (1.1) with respect to μ, we obtain 

  1 T ϕdμ ˜ = lim ϕ(Ps x)μ(d x)ds ϕ(x) ˜ = T →∞ T 0 M M    1 T ϕ(x)μ(d x)ds = ϕdμ, = lim T →∞ T 0 M M 

where we have used Remark 1.1 in the third step.

Remark 1.2 The converse of above corollary is also true: assume that the following limit is a constant μ-almost surely lim

T →∞

1 T



T

 ϕ(Pt x)dt =

0

M

ϕdμ.

Then for any invariant set G, we choose ϕ = 1G in above equation and get  μ(G) =

M

1G dμ = lim

T →∞

1 T



T

1G (Pt x)dt = 1G (x),

0

which indicates that G is trivial and thus {Pt }t≥0 is ergodic. It then gives an equivalent definition of ergodicity: 1 lim T →∞ T



T

 ϕ(Pt x)dt =

0

M

ϕdμ

for ϕ ∈ L1 (M, μ). For the initial value problem of a specific differential equation defined on M = Rd d X (t) = f (X (t)), dt

X (0) = x, t ≥ 0

one can easily define a Markov semigroup Pt x := X x (t), where X x (·) denotes the solution starting from the initial value x. In the following, for some test function ϕ, we use the notation Pt ϕ(x) instead of ϕ(Pt x) to emphasize the linearity of the operator Pt for any t ≥ 0. For Markov semigroup {Pt }t≥0 generated by the solution of differential equations or stochastic differential equations, its invariant measures and ergodicity are also widely studied to investigate the longtime behavior of its ensemble average. Different from the results given above for endomorphisms, the existence of invariant probability measures for Markov semigroup {Pt }t≥0 is not trivial. In the following sections, we give definitions of invariant measures and ergodicity in view of stochastic differential equations by introducing the Markov semigroup generated by the solution of a stochastic differential equation. Some sufficient conditions

1.1 Basic Definitions in Measure Spaces

5

on existence and uniqueness of the invariant measure and ergodicity for stochastic processes or Markov chains are also incorporated. We refer to [58, 60, 120, 143, 182] and references therein for more details.

1.2 Invariant Measures for Stochastic Processes Let (Ω, F , P) be a complete probability space with filtration {Ft }t≥0 and H be a separable Hilbert space. For an initial value problem of the following stochastic differential equation 

d X (t) = b(X (t))dt + σ (X (t))dW (t), X (0) = x ∈ H

(1.2)

with W being the standard Wiener process associated to filtration {Ft }t≥0 , there exists a unique solution X : Ω × [0, T ] → H which is an H-valued stochastic Markov process under certain conditions. We denote by X x (t) the solution at time t starting from x when it is necessary to point out the dependence of the initial value x, and eliminate the variable ω ∈ Ω. Assumptions on the drift coefficient b and the diffusion coefficient σ will be given in the next subsection to ensure the well-posedness of (1.2). For ϕ ∈ Bb (H) with Bb (H) denoting the space of all measurable and bounded functions defined on H, we define the transition semigroup {Pt }t≥0 as Pt ϕ(x) = E[ϕ(X x (t))],

(1.3)

which is a Markov semigroup on Bb (H). Here E[·] denotes the expectation of a random variable. In addition, we denote by πt (x, G) = P(X x (t) ∈ G) the transition probability that X x (t) hits G for any G ∈ B(H), where B(H) is the Borel σ -algebra. Then πt (x, ·) is a probability measure (also called a probability kernel, see [58, 116]) and satisfies   Pt ϕ(x) = ϕ(X x (t))dP = ϕ(y)πt (x, dy). (1.4) Ω

H

In addition, choosing ϕ = 1G for some G ∈ B(H), we have πt (x, G) = Pt 1G (x) = E[1G (X x (t))] = P(X x (t) ∈ G).

6

1 Invariant Measures and Ergodicity

We define the discrete version of πt similarly: for a discrete H-valued Markov chain {X n }n∈N , for example, the solution of some numerical scheme for (1.2), we denote by πn (x, G) = P(X n ∈ G|X 0 = x) the discrete transition kernel for x ∈ H and G ∈ B(H). Under certain assumptions which ensure that the solution X x (t) to (1.2) is uniformly continuous with respect to the initial value x, the Markov semigroup {Pt }t≥0 defined in (1.3) is Feller: for any ϕ ∈ Cb (H), Pt ϕ ∈ Cb (H) (see e.g. Chap. 3) with Cb (H) being the space of all continuous and bounded functions. We also call X a Feller process if the Markov semigroup {Pt }t≥0 associated to X is Feller. Next, we introduce some useful conditions which ensure the existence of invariant measures for Markov Feller processes. We denote by H a separate Hilbert space and {Pt }t≥0 the Markov transition semigroup associated to the solution X . Definition 1.4 ([58]) A probability measure μ ∈ P(H) is said to be invariant for X if   Pt ϕ(x)μ(d x) = ϕdμ (1.5) H

H

for all ϕ ∈ Bb (H) and t ≥ 0, where Pt ϕ(x) := E[ϕ(X x (t))] and P(H) denotes the space of all the probability measures on H. In the sequel, by saying “invariant measure”, we always mean “invariant probability measure”. Remark 1.3 If Markov semigroup {Pt }t≥0 possesses an invariant measure, it can be uniquely extended to a family of linear bounded operators on L p (H, μ), p ≥ 1, such that Pt L (L p (H,μ)) ≤ 1, t ≥ 0. One could find immediately that if μ is an invariant measure of X , then  μ(G) =

H

πt (x, G)μ(d x), ∀ t ≥ 0

(1.6)

by choosing ϕ = 1G , and in addition, the integration of E[ϕ(X x (t))] with respect to μ keeps a constant for any t ≥ 0. An intuitive description of invariant measures is that if the probability distribution πt (x, ·) of X x (t) converges to a probability measure μ, then μ is an invariant measure of X x (t). The following theorem gives a sufficient condition for the existence of invariant measures, which is less restrictive than the description above. Definition 1.5 ([58]) If a subset Λ ⊂ P(H) satisfies that, for any ε > 0, there exists a compact set K ε such that μ(K ε ) ≥ 1 − ε for all μ ∈ Λ, then Λ is said to be tight.

1.2 Invariant Measures for Stochastic Processes

7

The following theorem indicates the existence of invariant measures under the tightness assumption taking advantage of the Prokhorov theorem (see Theorem 6.7, [58]). Theorem 1.2 (Krylov–Bogoliubov theorem) If X is a Markov Feller process and for some initial value x0 ∈ H, the set {μT }T >0 with definition μT (G) :=



1 T

T

E[1G (X x0 (t))]dt =

0

1 T



T

πt (x0 , G)dt, ∀ G ∈ B(H)

0

is tight, then there exists an invariant measure μ ∈ P(H) for X . Proof Since {μT }T >0 ⊂ P(H) is tight, according to the Prokhorov theorem, there exist {μTk }k∈N ⊂ {μT }T >0 and μ ∈ P(H) such that {μTk }k∈N is weakly convergent to μ. That is, for ψ ∈ Cb (H),   ψdμTk = ψdμ. lim k→∞ H

H

On the other hand,  lim

k→∞ H

ψdμTk = lim

k→∞

1 Tk

 0

Tk

 H

ψ(x)πt (x0 , d x)dt = lim

k→∞

1 Tk



Tk

Pt ψ(x0 )dt.

0

For any ϕ ∈ Cb (H), utilizing the Feller property of {Pt }t≥0 , we choose ψ = Ps ϕ ∈ Cb (H) in above equations for s ≥ 0 and derive from above equations that  H

 Tk 1 Pt+s ϕ(x0 )dt k→∞ Tk 0  Tk   Tk +s  s 1 Pt ϕ(x0 )dt + Pt ϕ(x0 )dt − Pt ϕ(x0 )dt = lim k→∞ Tk 0 Tk 0  = ϕdμ,

Ps ϕdμ = lim

H

which indicates that μ is an invariant measure for X .



Remark 1.4 Note that the tightness of transition probabilities {πt (x0 , ·)}t>0 ensures the tightness of {μT }T >0 defined in the theorem above, which can be obtained through the Definition 1.5. Thus, it also ensures the existence of invariant measures for X . As a consequence of the theorem above, another frequently used sufficient condition for the existence of invariant measures, or even the existence of ergodic invariant measures, is stated below utilizing Lyapunov functionals. Theorem 1.3 (Proposition 7.10, [58]) If there exist some x0 ∈ H and a constant C = C(x0 ) > 0 such that

8

1 Invariant Measures and Ergodicity

E[V (X x0 (t))] ≤ C(x0 ), ∀ t ≥ 0, then there exists an invariant measure for X , where V : H → [1, +∞] is a Borel function (Lyapunov functional) whose level sets K a := {x ∈ H : V (x) ≤ a} are compact for any a > 0. Proof For any ε > 0, let a(ε) = c of K a(ε) satisfies tary set K a(ε)

C(x0 ) . ε

We obtain for any t > 0 that the complemen-

 c πt (x0 , K a(ε) )

=

πt (x0 , dy)    V (y) πt (x0 , dy) ≤ H a(ε) 1 E[V (X x0 (t))] ≤ ε, = a(ε) {V (x)>a(ε)}

which yields apparently πt (x0 , K a(ε) ) > 1 − ε. Hence {πt (x0 , ·)}t>0 is tight, which ensures the existence of invariant measures according to Remark 1.4.  The assumption in Theorem 1.3 is usually called the Lyapunov condition, which is often characterized by some other sufficient conditions. For example, a dissipative condition x · b(x) ≤ −β|x|2 + C for some β > 0 and C > 0 is used in [179]. The details of other conditions which ensure the existence of invariant measures will be discussed in Chap. 2. Furthermore, the theorem above could also be applied to a discrete Markov chain {X n }n∈N to gain the existence of invariant measures if the condition E[V (X n )] ≤ C(X 0 ) is satisfied for some constant C(X 0 ) depending on initial data X 0 . Example 1.1 Consider the stochastic differential equation d X (t) = −X (t)dt +

√ 2d B(t),

X (0) = ξ ∈ R, t ≥ 0,

(1.7)

where B is a one-dimensional standard Brownian motion. Applying Itô’s formula to |X (t)|2 with | · | denoting the absolute value, we have E|X (t)|2 = e−2t E|ξ |2 + 1 − e−2t ≤ E|ξ |2 + 1,

1.2 Invariant Measures for Stochastic Processes

9

which implies the existence of invariant measures based on Theorem 1.3 by choosing the Lyapunov functional V (·) = | · |2 + 1. It is well known that (1.7) can be solved explicitly, whose solution is given by the Ornstein–Uhlenbeck process  t√ 2e−(t−s) d B(s). X (t) = e ξ + −t

0

If the initial value ξ is a constant, then the probability kernel of X (t) is a Gaussian distribution satisfying πt (ξ, ·) = N (e−t ξ, 1 − e−2t ) → N (0, 1) as t → ∞. If the initial value ξ is an N (0, 1)-distributed random variable, then X (t) is also an N (0, 1)-distributed random variable for any t ≥ 0, which implies the invariance of the measure μ = N (0, 1) (see also [58]). More precisely, we claim that μ satisfies the identity in Definition 1.4. In fact, by choosing ϕ in Definition 1.4 as ϕ(x) = eihx for any h ∈ R which can be chosen as proper approximations of functions in Cb (R), we have 

 R

1   t √ −(t−s) x2 −t d B(s)) E eih (e x+ 0 2e √ e− 2 d x 2π R  2 h2 1 1 −t e 2 (e h+ix ) − 2 d x =: f (t). =√ 2π R

Pt ϕ(x)μ(d x) =

Since f (t) ≡ 0, we have  R

1 Pt ϕ(x)μ(d x) = f (t) = f (0) = √ 2π

 R

x2

eihx− 2 d x =

 R

ϕ(x)μ(d x),

which completes the proof of the claim. Next, we give a brief introduction of ergodicity for both stochastic processes and invariant measures, based on which we will show that the invariant measure of (1.7) is also ergodic, strongly mixing and unique.

1.3 Ergodicity As an important asymptotic behavior, ergodicity is studied in a number of different research areas, while we concentrate on the ergodicity of the stochastic flows generated by stochastic differential equations. In this section, we introduce definitions of ergodicity and mixing property for the solution X to (1.2) (see also [58,

10

1 Invariant Measures and Ergodicity

74, 143]), which characterize different convergence rates of the temporal average of Pt ϕ(x) = E[ϕ(X x (t))]. For Markov semigroup {Pt }t≥0 , similar to the results concerned on semiflows shown in the Birkhoff–Khinchin ergodic theorem, the limit 1 lim T →∞ T exists with ϕ(x) ˜ satisfying



T

Pt ϕ(x)dt =: ϕ(x) ˜

0



 H

ϕdμ ˜ =

H

ϕdμ

based on the Von Neumann theorem (see e.g. [58, 60]). Following are some definitions related to this limit. Definition 1.6 (see e.g. [60]) Let μ be an invariant measure of X . (i) X is said to be ergodic on H if 1 lim T →∞ T



T

 E[ϕ(X x (t))]dt =

0

H

ϕdμ in L2 (H, μ)

(1.8)

for all ϕ ∈ L2 (H, μ). (ii) X is said to be strongly mixing on H if  lim E[ϕ(X x (t))] =

t→∞

H

ϕdμ in L2 (H, μ)

for all ϕ ∈ L2 (H, μ). (iii) X is said to be exponentially mixing on H if there exist ρ > 0 and a positive function C(·) such that for any bounded Lipschitz continuous function ϕ on H, all t > 0 and all x ∈ H,







≤ C(x)L ϕ e−ρt ,

E[ϕ(X x (t))] − ϕdμ



H

where L ϕ denotes the Lipschitz constant of ϕ. The convergence in L2 (H, μ) here and below is interpreted with respect to the initial value x ∈ H. The ergodicity of a stochastic process is usually described as the case its temporal average equals its spatial average. The spatial average is also known as the ergodic limit, which is approximated in Chap. 6. A stochastic differential equation is usually called ergodic (resp. strongly mixing, exponentially mixing) if its solution is ergodic (resp. strongly mixing, exponentially mixing). Different definitions above show that Pt ϕ(x) converges to the spatial average  H ϕdμ in different senses or with different rates.

1.3 Ergodicity

11

Example 1.2 We still consider the equation in Example 1.1 with a deterministic initial value ξ . The probability kernel of the solution X ξ (t) is πt (ξ, d x) = 

−t

1 2π(1 − e−2t )

Hence, we can calculate  E[ϕ(X ξ (t))] = ϕ(x)πt (ξ, d x) =  R

1 t→∞ −−−→ √ 2π

e

2

ξ) − (x−e 2(1−e−2t )

1



2π(1 − e−2t )   2 − x2 ϕ(x)e d x = ϕdμ R

d x.

R

ϕ(x)e

−t

2

ξ) − (x−e 2(1−e−2t )

dx

R

with μ denoting the Gaussian distribution. As a result, the solution X ξ (t) to (1.7) is strongly mixing, and thus is ergodic. For a discrete H-valued Markov chain {X n }n∈N , we define its ergodicity if it possesses an invariant measure μ and in addition it satisfies that lim

N →∞

 N 1  E[ϕ(X n )|X 0 = x] = ϕdμ in L2 (H, μ). N n=1 H

We would like to mention that the following is an equivalent definition about ergodicity with respect to invariant measure μ according to the Birkhoff–Khinchin ergodic theorem introduced in Sect. 1.1. Definition 1.7 Let μ be an invariant measure of X . Then μ is said to be ergodic if any invariant set is trivial. More precisely, if G ∈ B(H) satisfies P(X x (t) ∈ G) = 1G (x), μ-a.s. for any t ≥ 0, then μ(G) = 0 or μ(G) = 1. It is worth noticing that the definition above coincides with those in Definition 1.6 (i) according to Corollary 1.1 and Remark 1.2. The set G is invariant since its characteristic function 1G is invariant under {Pt }t≥0 : recall that P(X x (t) ∈ G) = E[1G (X x (t))] = Pt 1G (x). Thus, the condition in Definition 1.7 turns to be Pt 1G (x) = 1G (x), μ-a.s. for any t ≥ 0.

12

1 Invariant Measures and Ergodicity

The relationship between invariant measures and ergodic measures is stated in the following proposition, which can also be regard as a sufficient condition of the ergodicity of an invariant measure. Proposition 1.1 (Theorem 5.16, [58]) Assume that the invariant measure μ for X is unique, then μ is ergodic. Proof Assume by contradiction that μ is not ergodic. Then there exists a nontrivial invariant set G such that 0 < μ(G) < 1 based on Definition 1.7. Define a measure μG based on the invariant set G μG (A) :=

μ(G ∩ A) , μ(G)

A ∈ B(H).

We claim that μG = μ is also an invariant measure for X , which gives rise to a contradiction that μ is the unique invariant measure. To prove the claim, we only need to verify that (1.5) holds when ϕ = 1 A for any A ∈ B(H), i.e.,   Pt 1 A dμG = 1 A dμG . H

H

Note that the left hand side of above equation shows  H

    1 1 Pt 1 A dμG = Pt 1 A dμ = Pt 1 A∩G dμ + Pt 1 A∩G c dμ μ(G) G μ(G) G G   1 1 μ(A ∩ G) = Pt 1 A∩G dμ = 1 A∩G dμ = , μ(G) H μ(G) H μ(G)

while the right hand side shows  H

1 A dμG = μG (A) =

μ(A ∩ G) . μ(G)

Here, we have used the fact that 0 ≤ Pt 1 A∩G c ≤ Pt 1G c = 1G c μ-a.s. since G is an invariant set. Thus, Pt 1 A∩G c = 0 μ-almost surely on G.



It follows directly from Example 1.2 that the invariant measure obtained in Example 1.1 is unique. Taking advantage of above proposition, we derive a stronger result than that in Theorem 1.3 if the Lyapunov condition holds uniformly. Theorem 1.4 If there exists a positive constant C0 such that E[V (X x (t))] ≤ C0 , ∀ x ∈ H, t ≥ 0,

1.3 Ergodicity

13

then the set of all the invariant measures Λ ⊂ P(H) is tight and there exists an ergodic invariant measure for X . Proof Based on Theorem 1.3, Λ is not empty. Note that there is a natural embedding of P(H) into C∗b (H) by setting  Fμ (ϕ) := which satisfies

H

ϕdμ, ∀ ϕ ∈ Cb (H),







Fμ := sup ϕdμ

≤ 1. ϕ =1

H

Then Λ is a convex subset of C∗b (H). Step 1. We first show that Λ is tight. For any δ > 0, t > 0 and μ ∈ Λ, by choosing V and utilizing the fact that μ is an invariant measure, we the test function ϕ = 1+δV have

  V (x) V (x) μ(d x) = μ(d x) Pt 1 + δV (x) H 1 + δV (x) H    V (X x (t)) μ(d x) = E 1 + δV (X x (t)) H  ≤ E[V (X x (t))]μ(d x) ≤ C0 . H

Let δ → 0, we derive

 sup

μ∈Λ H

For any ε > 0, denoting a(ε) = we have  c )= μ(K a(ε)

V dμ ≤ C0 .

C0 ε

and K a(ε) as the definition in Theorem 1.3,  V (x) μ(d x) ≤ ε, μ(d x) ≤ {V (x)>a(ε)} H a(ε)

which indicates μ(K a(ε) ) > 1 − ε, ∀ μ ∈ Λ. Step 2. Now we show the existence of ergodic invariant measures for X . The Krein–Milman theorem (see e.g. [190]) says that a non-empty compact convex subset of a locally convex linear topological space has at least one extremal point. It then suffices to show that the extremal points of Λ coincide with the ergodic invariant measures for X . To show that all the extremal points of Λ are ergodic invariant measures for X , we denote by μ an extremal point of Λ, that is, if there exist μ1 , μ2 ∈ Λ and α ∈ (0, 1) such that

14

1 Invariant Measures and Ergodicity

μ = αμ1 + (1 − α)μ2 , then μ1 = μ2 = μ. Assume by contradiction that μ is not ergodic. Then following the same procedure in the proof of Proposition 1.1, there exists a nontrivial invariant set G such that μG , μG c ∈ Λ with the definitions μG (A) =

μ(G ∩ A) μ(G c ∩ A) , μG c (A) = , ∀ A ∈ B(H) μ(G) μ(G c )

and α := μ(G) ∈ (0, 1). It then give rise to a contradiction since μ = αμG + (1 − α)μG c and μG = μG c . We finally show that if μ is ergodic, then it is an extremal point in Λ. Let μ be an ergodic invariant measure for X . Assume by contradiction that it is not an extremal point in Λ, i.e., there exist μ1 , μ2 ∈ Λ with μ1 = μ2 and α ∈ (0, 1) such that μ = αμ1 + (1 − α)μ2 . It is clear that μ1 and μ2 are absolutely continuous with respect to μ, that is, if A ∈ B(H) satisfies μ(A) = 0, then μ1 (A) = μ2 (A) = 0, denoted by μ1  μ and μ2  μ. Then for any A ∈ B(H), according to the Birkhoff–Khinchin ergodic theorem in Sect. 1.1 and that μ is ergodic, we have lim

T →∞

1 T



T 0

 Pt 1 A (x)dt =

H

1 A dμ = μ(A), μ-a.s.,

which also holds μ1 -almost surely since μ1  μ. Hence integrating above equation with respect to μ1 yields   1 T μ(A) = lim Pt 1 A (x)dμ1 dt T →∞ T 0 H   1 T 1 A (x)dμ1 dt = μ1 (A), = lim T →∞ T 0 H where we used the fact that μ1 ∈ Λ in the second step. It then indicates μ1 = μ, and  also μ2 = μ in the same procedure, which is a contradiction to μ1 = μ2 .

1.4 Strong Feller and Irreducibility Properties

15

1.4 Strong Feller and Irreducibility Properties Strong Feller and irreducibility properties are frequently utilized to ensure the uniqueness of the invariant measure, as well as ergodicity. More details about the sufficient conditions of strong Feller and irreducibility properties will be given in the following chapter. Definition 1.8 (see [58, 60]) Let X (t) be the solution to (1.2) with the probability kernel πt (·, ·).  (i) X (t) is strong Feller if H ϕ(y)πt (·, dy) ∈ Cb (H) for any ϕ ∈ Bb (H). (ii) X (t) is irreducible if πt (x, B(x0 , r )) > 0 for all x0 , x ∈ H, r > 0 and any t > 0, where B(x0 , r ) denotes the ball centred at x0 with radius r . In other words, the strong Feller property implies that πt (x, ·) is continuous in x ∈ H, while the irreducibility property means that any open sets are reachable with positive probability by the solution X x (t) started at any x ∈ H. These two properties yield the following properties of μ, whose proof is based on [58, 60]. Theorem 1.5 If X is strong Feller and irreducible, and possesses an invariant measure μ, then (i) μ is equivalent to the probability kernel πt (x, ·) for any t > 0 and all x ∈ H. (ii) μ is an ergodic measure of X (t). (iii) μ is the unique invariant measure of X (t). Here, two measures μ and ν are called equivalent if each is absolutely continuous with respect to the other, i.e., μ  ν and ν  μ. More precisely, for every G ∈ B(H), μ(G) = (>)0 ⇔ ν(G) = (>)0. Proof (i) Step 1. We first prove that for t > 0, πt (x, ·) are equivalent for all x ∈ H. To this end, it suffices to show that for any fixed x ∈ H, if G ∈ B(H) such that πt (x, G) > 0, then πt (y, G) > 0 for all y ∈ H. In fact, as  πt (x, G) =

H

πs (y, G)πt−s (x, dy) > 0

for any s ∈ (0, t) based on the Chapman–Kolmogorov equation of Markov processes, there exists some y0 ∈ H such that πs (y0 , G) > 0. It follows from the strong Feller property that πs (y, ·) is continuous in y. Thus, there exists r > 0 such that πs (w, G) > 0 for all w ∈ B(y0 , r ), and also, πt (y, B(y0 , r )) > 0 for any t > 0 and y ∈ H based on the irreducibility property. Hence, it shows  πt (y, G) =

 H

πs (w, G)πt−s (y, dw) ≥

B(y0 ,r )

πs (w, G)πt−s (y, dw) > 0.

16

1 Invariant Measures and Ergodicity

Step 2. μ is equivalent to πt (x, ·) for any t > 0 and x ∈ H. Firstly, if G ∈ B(H) such that πt (x, G) = 0 for some x ∈ H, then we have πt (y, G) = 0 for all y ∈ H based on Step 1. It yields μ  πt (x, ·) as  μ(G) =

H

πt (y, G)μ(dy) = 0

based on (1.6). On the other hand, if μ(G) = 0, then there must exist some v0 ∈ H such that πt (v0 , G) = 0. We hence get πt (x, ·)  μ for all x ∈ H based on Step 1. (ii) μ is ergodic. Let G ∈ B(H) be an invariant set of X (t) with μ(G) > 0, whose definition is given in Definition 1.7. Thus, P(X x (t) ∈ G) = πt (x, G) > 0, ∀ x ∈ H according to the fact πt (x, G) is equivalent to μ for all x ∈ H proved in (i). Since G is an invariant set, we derive 1G (x) = P(X x (t) ∈ G) > 0, μ-a.e., which implies μ(G) = 1. It indicates that any invariant set is trivial, so μ is ergodic. (iii) μ is unique. Assume by contradiction that there is another invariant measure ν for X (t) which also satisfies the assumptions in this theorem, then ν is equivalent to μ and is also ergodic. Let G ∈ B(H) be such that μ(G) = ν(G). We set   

1 T

E[1G (X x (t))]dt = μ(G) , E := x ∈ H lim T →∞ T 0   

1 T

E[1G (X x (t))]dt = ν(G) . F := x ∈ H lim T →∞ T 0 Then based on the ergodicity of μ and ν with ϕ = 1G in Definition 1.6, we have μ(E) = 1, ν(F) = 1 and E ∩ F = ∅. It then yields that F ⊂ E c , which leads to μ(F) = μ(E c ) = 0, which is in contradiction to the fact that μ and ν are equivalent.  Above conditions and properties can also be defined for a discrete Markov chain {X n }n∈N :  (i) {X n }n∈N is strong Feller if H ϕ(y)πn (·, dy) ∈ Cb (H) for any ϕ ∈ Bb (H). (ii) {X n }n∈N is irreducible if πn (x, B(x0 , r )) > 0 for all x, x0 ∈ H, r > 0 and any n ∈ N+ . In particular, if H is a finite dimensional Banach space with norm · H , another kind of sufficient condition for ergodicity is stated as follows. Corollary 1.2 (Theorem 5.5, [182]) Assume that the homogeneous Markov chain {X n }n∈N is strong Feller and irreducible. In addition, there exists a compact set G 0 ∈ B(H) with positive Legesgue measure m(G 0 ) > 0 such that

1.4 Strong Feller and Irreducibility Properties



17





E X n+1 H − X n H | X n = x ≤

− C1 , ∀ x ∈ / G0, C2 < ∞, ∀ x ∈ G 0

(1.9)

for some positive constants C1 and C2 . Then {X n }n∈N is ergodic. It is easy to check that condition (1.9) is a sufficient condition for the Lyapunov condition in Theorem 1.3, and thus this corollary also leads to the results given in / G 0 , we have E[ X n+1 H ] ≤ E[ X n H ], while Theorem 1.5. In fact, for X n = x ∈ for X n = x ∈ G 0 , we get E[ X n+1 H ] ≤ αE[ X n H ] + (1 − α)E[ X n H ] + C2 ≤ αE[ X n H ] + (1 − α)C + C2

for some α ∈ (0, 1), since G 0 is bounded. The desired result is then obtained by recurrence.

1.5 Invariant Measures for Hamiltonian Systems A class of specific stochastic dynamical systems—stochastic Hamiltonian systems— plays an important role in stochastic dynamics, whose geometric structure has been investigated by many authors. For instance, stochastic Hamiltonian systems with multiplicative noises and additive noises are considered in [145] and [146], respectively. The symplectic structure of the systems, as well as numerical methods with the same property, are studied in these papers. The author in [149] studies symplectic methods obtained by composition of stochastic flows of simpler Hamiltonian systems. The author in [134] shows that an averaging principle holds for a completely integrable stochastic Hamiltonian system. Authors in [119] propose the stochastic multi-symplectic conservation law for stochastic Hamiltonian partial differential equations, and develop a stochastic multi-symplectic method for the stochastic nonlinear Schrödinger equation driven by a linear multiplicative noise. The ergodic theory of Hamiltonian systems could be tracked back to the study of statistical ensembles, i.e., a collection of stationary distributions on the phase space for a given deterministic Hamiltonian system, which is also called ‘monode’ by Boltzmann (see e.g. [23, 85]). For classical mechanical systems, the Lagrangian is denoted by L(q, q) ˙ = T (q, q) ˙ + U (q) with q = (q1 , · · · , qd ) ∈ Rd being the generalized coordinate, T and U being the kinetic energy and potential energy, respectively. According to the Hamilton’s principle, one seeks the extremal q such that the action functional 

T 0

L(q, q)dt ˙

18

1 Invariant Measures and Ergodicity

is stationary under variation with δq(0) = δq(T ) = 0. Hence, L and q obey the Lagrange equation ∂L d ∂L = . dt ∂ q˙ ∂q By introducing the generalized momentum, which is also known as the Poisson’s variables, given by the Legendre transform p= and the Hamiltonian

∂L (q, q) ˙ ∂ q˙

(1.10)

˙ H ( p, q) := p  q˙ − L(q, q),

one get the following equations p˙ = −

∂H ∂H ( p, q), q˙ = ( p, q), ∂q ∂p

which constitute the Hamiltonian systems. Equivalently, we have X˙ = J −1 ∇ H (X ) by denoting X = ( p  , q  ) ∈ R2d and the 2d-dimensional standard symplectic matrix

0 Id . J= −Id 0 2d×2d Here, Id denotes the d-dimensional identity matrix, and the following condition is required that (1.10) defines a continuously differentiable bijection p ↔ q˙ for any q, which is called the Legendre transform. For an autonomous Hamiltonian system with M being the phase space, its phase flow φt : x → X x (t) preserves phase volume 

 M

f (φt (x))d S =

f d S, M

where d S denotes the phase volume element of M. This identity is also referred to as Liouville’s theorem and φt is called an equi-measure transformation (see e.g. [190]). Then the ergodic hypothesis of Boltzmann that the temporal average of a physical quantity should be equal to the spatial average of this physical quantity can be expressed by   f dS 1 T f (φt (x))dt = M lim T →∞ T 0 M dS

1.5 Invariant Measures for Hamiltonian Systems

19

 with the assumption that the total volume of the system is finite M d S < ∞. A generalization of above argument to the Markov process is the condition of the existence of invariant measures for the considered systems. Hence, the existence and ergodicity of invariant measures for stochastic Hamiltonian systems (see e.g. [179]), as well as its numerical approximations, are also of vital importance and remain to be further investigated. It motivates us to consider both geometric structure and dynamical behavior of stochastic nonlinear Schrödinger equations over longtime numerically introduced in Chaps. 5 and 6. In this section, we investigate the invariant measures, geometric structures and symplectic integrators for two simple Hamiltonian systems–stochastic Kubo oscillator and stochastic dissipative Hamiltonian systems.

1.5.1 Stochastic Kubo Oscillator In this subsection, we denote H = R2d , and investigate the geometric structure as well as existence of invariant measures for the Kubo oscillator. Consider the following R2d -valued stochastic differential equation in the Stratonovich sense 

dp = − qdt − q ◦ d B(t), p(0) = p0 , dq = pdt + p ◦ d B(t), q(0) = q0

(1.11)

with deterministic initial value ( p0 , q0 ) ∈ R2d and B being a one-dimensional R-valued standard Brownian motion. Denoting X = ( p  , q  ) = ( p 1 , · · · , p d , q 1 , · · · , q d ) ∈ R2d and H (X ) = H ( p, q) :=

1 ( p 2 + q 2 ) 2

with · denoting the Euclidean norm in the corresponding Euclidean spaces, we can rewrite (1.11) into a compact form d X = J −1 ∇ H (X )dt + J −1 ∇ H (X ) ◦ d B(t)

(1.12)

with x := X (0) = ( p0 , q0 ) . Here J denotes the 2d-dimensional standard symplectic matrix. Energy conservation law. Equations of the form (1.12) are called stochastic Hamiltonian equations. We can find out that the Hamiltonian H ( p, q) of this system is a conserved quantity (or invariant or first integral). Actually, the solution of (1.11) can be expressed explicitly 

p(t) = − sin(t + B(t))q0 + cos(t + B(t)) p0 , q(t) = cos(t + B(t))q0 + sin(t + B(t)) p0 .

20

1 Invariant Measures and Ergodicity

Fig. 1.1 a Hamiltonian H ( p, q) and b phase trajectories starting from different initial values (d = 1, T = 104 )

Its Hamiltonian, as well as its phase trajectory, can be simulated directly as in Fig. 1.1. Moreover, if the drift and diffusion coefficients involve the same Hamiltonian function, it always holds that the Hamiltonian function is a first integral. More precisely, for any t ≥ 0, the solution X x (t) of (1.12) with H (x) = c0 lies on the isoenergetic surface Σc0 := {X ∈ H | H (X ) = c0 } almost surely, i.e., H ( p(t), q(t)) = H ( p0 , q0 ) P-a.s. In fact, based on the Stratonovich chain rule, we get d H (X (t)) =∇ H (X (t)) ◦ d X (t) =∇ H (X (t)) J −1 ∇ H (X (t))dt + ∇ H (X (t)) J −1 ∇ H (X (t)) ◦ d B(t) =0 since J −1 is skew symmetric. In the sequel, we omit the notation ‘P-a.s.’ or ‘almost surely’ unless it is necessary. Symplectic conservation law. The phase flow φt : x → X of Eq. (1.12) is linear, and possesses stochastic symplectic structure

or equivalently,

∂X ∂x



J

∂X ∂x

= J,

1.5 Invariant Measures for Hamiltonian Systems

d p(t) ∧ dq(t) :=

d 

21

d pi (t) ∧ dq i (t) = d p0 ∧ dq0

i=1

with ‘d’ denoting the exterior derivative and d p ∧ dq being a differential 2-form. We introduce here the geometric description of symplectic transformations for Hamiltonian systems. The reader is referred to [10, 82, 95] and references therein for more details. The 2-form d p(t) ∧ dq(t) on R2d are expressed as the sum of d 2-forms d pi (t) ∧ dq i (t), i = 1, · · · , d, on R2 . For the two dimensional case, i.e. d = 1, the preservation of the symplectic structure is equivalent to the preservation of the phase area. In fact, we set two vectors ξ = (ξ1 , ξ2 ) , η = (η1 , η2 ) ∈ R2 , which span a parallelogram with oriented area or.ar ea(ξ, η) := det

ξ1 η1 ξ2 η2

= ξ1 η2 − ξ2 η1 = ξ  J η.

The area of the parallelogram under the projection of the symplectic mapping A satisfying A J A = J turns to be or.ar ea(Aa, Ab) = (Aa) J (Ab) = a  (A J A)b = or.ar ea(a, b), which indicates the area preservation property of A. One can find from Fig. 1.2 that the area keeps invariant under the linear flow φt of (1.12). Symplectic integrators. A natural question in application is to construct numerical integrators which could also preserve the stochastic symplectic structure. We next introduce two equivalent ways of proving the symplecticity of numerical integrators.

(a)

(b)

3 3 2

1

1

0

0

q

q

2

-1

-1

-2

-2

-3 -3 -3

-2

-1

0

p

1

2

3

4

-3

-2

-1

0

1

2

3

4

p

Fig. 1.2 Images under the phase flow with initial image being (a) a square determined by (1, 1), (1, 2), (2, 2) and (2, 1) or (b) a triangle determined by (1, 1.5), (2, 2) and (3, 1) (d = 1, T = 10)

22

1 Invariant Measures and Ergodicity

Definition 1.9 ([95]) A numerical one-step method is called symplectic if the onestep map Y1 = φn (Y0 ) is symplectic whenever the method is applied to a smooth Hamiltonian system. As an example, we consider the midpoint scheme applied to (1.12) X n+1 = X n + J

−1

∇H

X n+1 + X n 2

(τ + n+1 B) ,

where τ denotes the uniform time step-size and n+1 B = B(tn+1 ) − B(tn ) is the increment of the Brownian motion B with tn = nτ and n ∈ N. Denote f = J −1 ∇ H , and Yn+1 := By noting that

and

X n+1 + X n 1 = X n + f (Yn+1 ) (τ + n+1 B) . 2 2

∂ X n+1 ∂ f (Yn+1 ) = I2d + (τ + n+1 B) ∂ Xn ∂ Xn ∂Yn+1 1 ∂ f (Yn+1 ) = I2d + (τ + n+1 B) , ∂ Xn 2 ∂ Xn

we conclude the symplecticity of the midpoint scheme:

∂ X n+1  ∂ X n+1 J ∂ Xn ∂ Xn

∂ f (Yn+1 ) ∂ f (Yn+1 )  =J +J (τ + n+1 B) + J (τ + n+1 B) ∂ Xn ∂ Xn

∂ f (Yn+1 )  ∂ f (Yn+1 ) (τ + n+1 B)2 + J ∂ Xn ∂ Xn 



 ∂Yn+1 1 ∂ f (Yn+1 ) ∂ f (Yn+1 ) =J + − J (τ + n+1 B) (τ + n+1 B) ∂ Xn 2 ∂ Xn ∂ Xn 



∂ f (Yn+1 )  ∂Yn+1 1 ∂ f (Yn+1 ) + J − (τ + n+1 B) (τ + n+1 B) ∂ Xn ∂ Xn 2 ∂ Xn

∂ f (Yn+1 )  ∂ f (Yn+1 ) (τ + n+1 B)2 + J ∂ Xn ∂ Xn

1.5 Invariant Measures for Hamiltonian Systems



23

 ∂Yn+1 + =J + J J (τ + n+1 B) ∂ Xn 

  ∂Yn+1 ∂Yn+1   2 2 − =J + ∇ H (Yn+1 ) + ∇ H (Yn+1 )J J (τ + n+1 B) ∂ Xn ∂ Xn ∂Yn+1 ∂ Xn





∂ f (Yn+1 ) ∂ Xn



∂ f (Yn+1 ) ∂ Xn





= J, where in the last step we used the fact J − J = −I2d . Equivalently, the symplecticity of integrators can also be proved utilizing differential 2-forms by rewriting the midpoint scheme above as 

˜ n+1 , pn+1 = pn − Hq  ˜ n+1 , qn+1 = qn + H p 

˜ n+1 := τ + n+1 B, and H p := ∂ p H and Hq := ∂q H are evaluated at where  ( pn+12+ pn , qn+12+qn ). Then the total differentials of pn+1 and qn+1 read

⎧ d pn+1 + d pn dqn+1 + dqn ˜ ⎪ ⎪ d p = d p − H + H n+1 , n qp qq ⎨ n+1 2 2

⎪ dp + d pn dqn+1 + dqn ˜ ⎪ ⎩ dqn+1 = dqn + H pp n+1 + H pq n+1 . 2 2 Note that H pp = Hqq = 1 and H pq = Hq p = 0 for the stochastic Kubo oscillator. It then shows immediately 

   ˜ n+1 ˜ n+1   d pn+1 + dqn+1 ∧ d pn+1 + dqn+1 − 2 2     ˜ n+1 ˜ n+1   d pn + dqn ∧ d pn − dqn , = 2 2 which indicates that d pn+1 ∧ dqn+1 = d pn ∧ dqn , ∀ n ∈ N ˜2 

> 0 for any n ∈ N. since 1 + n+1 4 Invariant measures. The existence of invariant measures for (1.11) can be proved based on the Krylov–Bogoliubov Theorem in Sect. 1.2. The invariant measure for (1.11) is not unique in the whole space R2d since solutions starting from initial values with different norms lie in different spheres. Theorem 1.6 The Kubo oscillator (1.11) possesses invariant measures.

24

1 Invariant Measures and Ergodicity

Proof Fix some initial value x ∈ R2d which satisfies that x = c0 . We consider the transition probability πt (x, ·) ∈ P(R2d ). We claim that the subset {μT }T >0 ⊂ P(R2d ) defined through 1 μT (G) = T



T

πt (x, G)dt

0

is tight if the set {πt (x, ·)}t>0 is tight. In fact, if {πt (x, ·)}t>0 is tight, then for any ε > 0, there exists a compact set K ε such that 1 μT (K ε ) = T



T 0

1 πt (x, K ε )dt ≥ T



T

(1 − ε)dt = 1 − ε,

0

which verifies the claim. The tightness of {μT }T >0 ensures that (1.11) possesses an invariant measure according to Theorem 1.2. It then suffices to show the tightness of {πt (x, ·)}t>0 . Note that the isoenergetic surface Σc0 ⊂ R2d is tight, then for any ε > 0, πt (x, Σc0 ) = P(X x (t) ∈ Σc0 ) = 1 > 1 − ε, 

which completes the proof.

1.5.2 Stochastic Dissipative Hamiltonian Systems In this section, we aim to introduce an fundamental procedure of showing the existence and uniqueness of invariant measures for stochastic differential equations with non-global Lipschitz drift coefficients and degenerate noises. Hence, we turn to consider stochastic Hamiltonian systems with dissipative terms of the type 

dp = − Hq ( p, q)dt − F( p, q)H p ( p, q)dt + dW (t),

p(0) = p0 ,

dq = H p ( p, q)dt, q(0) = q0 with p, q ∈ Rd and W being a d-dimensional standard Brownian motion. Here we only focus on the one-dimensional case with F( p, q) ≡ α > 0 and H ( p, q) = 21 p 2 + 41 q 4 , i.e., 

dp =(−q 3 − αp)dt + dW (t), dq = pdt, q(0) = q0 ∈ R

p(0) = p0 ∈ R,

(1.13)

to make everything clear. We refer to [179] for the general case with certain assumptions on F and H .

1.5 Invariant Measures for Hamiltonian Systems

25

Geometric structure. For the system (1.13) with dissipative term, the symplectic conservation law is not satisfied anymore. Instead, its differential 2-form d p(t) ∧ dq(t) decays exponentially. Theorem 1.7 The phase flow of (1.13) satisfies the conformal symplectic conservation law, that is, d p(t) ∧ dq(t) = e−αt d p0 ∧ dq0 , ∀ t ≥ 0. Proof By calculating d (d p(t) ∧ dq(t)) = d[dp(t)] ∧ dq(t) + d p(t) ∧ d[dq(t)] = d[(−Hq ( p, q) − αp)dt + d W (t)] ∧ dq(t) + d p(t) ∧ d[ p(t)dt] = [(−Hqq ( p, q)dq(t) − αd p(t)) ∧ dq(t) + d p(t) ∧ d p(t)]dt = − α[d p(t) ∧ dq(t)]dt,

we get the result immediately, where we have used the fact that Ad p ∧ d p = 0 for any symmetric matrices A ∈ Rd×d and p ∈ Rd .



Invariant measures. To show the existence of invariant measures for a Markov process utilizing the Krylov–Bogoliubov theorem, the process need to be Feller in addition. That is, the solution of the considered model continuously depends on the initial data. This property is ensured by the local Lipschitz coefficients and the uniform boundedness of the solution stated below following the idea of Talay [179]. Lemma 1.1 If the initial value ( p0 , q0 ) ∈ R2 has finite moments of all order. Then the moments of all order for the solution of (1.13) are bounded uniformly in time, i.e., for any k ∈ N, there exists a constant C = C(k, α, p0 , q0 ) such that sup E

 k < C. | p(t)|2 + |q(t)|2

t≥0

Proof We first set the Lyapunov functional V as 2

2 α +1 α V ( p, q) := H ( p, q) + pq + 2 4

2   α 2 1 1 2 1 2 α2 + 1 p + q2 + p+ q + q − = 4 2 2 2 4  1 2 p + q2 . ≥ 4 Then Itô’s formula applied to V ( p(t), q(t)) yields that

26

1 Invariant Measures and Ergodicity

1 d V ( p(t), q(t)) = V p ( p(t), q(t))dp(t) + Vq ( p(t), q(t))dq(t) + V pp dt 2

2 1 α α α p(t)q(t) + dt = − p(t)2 − q(t)4 − 2 2 2 2   α + p(t) + q(t) dW (t). 2 Taking expectation to both sides of above equation, we obtain 

α dE[V ( p(t), q(t))] = −αE[V ( p(t), q(t))] − q(t)4 + α 4



α2 + 1 4

2

 1 + dt 2

≤ − αE[V ( p(t), q(t))]dt + C(α)dt, which indicates that E[V ( p(t), q(t))] ≤ e

−αt



t

E[V ( p0 , q0 )] + C(α)

e−α(t−s) ds

0

≤ E[V ( p0 , q0 )] + C(α). This give the result for k = 1. For higher moments, assume that   sup E V k ( p(t), q(t)) ≤ C. t≥0

Then for k + 1, we have   dE V k+1 ( p(t), q(t))

  1 α α α2 p(t)q(t) + dt = (k + 1)E V k ( p(t), q(t)) − p(t)2 − q(t)4 − 2 2 2 2  2   1 α dt + (k + 1)kE V k−1 ( p(t), q(t)) p(t) + q(t) 2 2   ≤ − α(k + 1)E V k+1 ( p(t), q(t)) dt + Cdt, where we have used the fact 

α 2 α2 2 α2 2 p + q ≤ 2p + q ≤ 4 2 ∨ V ( p, q) 2 2 2

in the last step. The proof is complete by induction.



Moreover, the lemma above also ensures the existence of invariant measures for (1.13) according to Theorem 1.3. To gain the uniqueness of the invariant measure, we next show that all the invariant measures for the solution of (1.13) admit strictly positive densities with respect to the Lebesgue measure.

1.5 Invariant Measures for Hamiltonian Systems

27

Lemma 1.2 (Lemma 2.2, [179]) For any stochastic initial value X 0 := ( p0 , q0 ) ∈ R2 and t > 0, the distribution of the solution X (t) := ( p(t), q(t)) has a strictly positive density ρt with respect to the Lebesgue measure. Proof For any stochastic initial value X 0 with initial probability distribution ν0 , the distribution of the solution X (t) reads  πt (x, A)ν0 (d x), ∀ A ∈ B(R2 ). πtν0 (A) = R2

As a result, we only need to show that for any deterministic initial value x := ( p0 , q0 ), the transition probability πt (x, ·) admits a positive density ρ˜t (x, ·) ∈ L1 (R2 ) for any t > 0 which ensures that πtν0 (·) also has a positive density  ρt (y) =

R2

ρ˜t (x, y)ν0 (d x)

since πtν0 (A) =





ρ˜t (x, y)dyν0 (d x)

  ρ˜t (x, y)ν0 (d x) dy, ∀ A ∈ B(R2 ). = R2

A

A

R2

One can check that the span of the vector fields and their Lie brackets of system (1.13) is equal to the whole space, which is known as the Hörmander condition. It ensures the existence of the jointly continuous density ρ˜t (x, ·) for the law πt (x, ·) according to Theorem 2.2. This result will be discussed in detail in Chap. 2, so we omit it here. We only show the positivity of ρ˜t (x, ·). Following the argument used in [139], we consider the associated control problem 

d p˜ =(−q˜ 3 − α p)dt ˜ + du(t), d q˜ = pdt ˜

with a smooth control function u ∈ C 1 (0, T ), which has the following equivalent form d q˜ du d 2 q˜ + q˜ 3 = . +α 2 dt dt dt For any fixed T > 0 and any vectors x0 = ( p0 , q0 ), x + = ( p + , q + ) ∈ R2 , we construct q˜ ∈ C∞ (0, T ) such that

d q˜ d q˜ − − q(0), ˜ q(T ˜ ), (0) = (q , p ) and (T ) = (q + , p + ) dt dt

28

1 Invariant Measures and Ergodicity

using polynomial interpolation. Hence we can also get the control function u ∈ C∞ (0, T ) with u(0) = 0. Denoting X := ( p, q), X˜ := ( p, ˜ q), ˜ F(X ) :=

p −q 3 − αp

and Γ :=

00 . 01

(1.14)

We achieve that X (t) − X˜ (t) =



t

F(X (s)) − F( X˜ (s))ds + Γ (W (t) − u(t)), ∀ t ∈ [0, T ].

0

Note that the following property holds for Brownian motions P



sup W (t) − u(t) < ε > 0, ∀ ε > 0. 0≤t≤T



If the event {ω ∈ Ω : sup0≤t≤T W (t) − u(t) < ε} happens, then the Gronwall inequality in a small time interval and its continuation yield that for some δ(ε),



sup X (t) − X˜ (t) < δ(ε) 0≤t≤T

according to the facts that F is continuously differentiable and thus locally Lipschitz, and that the ranges of X (t) and X˜ (t) (t ∈ [0, T ]) are both compact sets. As a result, for any δ > 0, by choosing ε > 0 small enough, we finally obtain   P |X (T ) − x + | < δ = πT (x0 , B(x + , δ)) > 0, where X˜ (T ) = x + and B(x + , δ) denotes the open ball centered at x + with radius δ.  The proof of Lemma 1.2 also shows that X is strong Feller and irreducible, based on which one can get the uniqueness of the invariant measure. Theorem 1.8 There exists a unique invariant measure for the solution of (1.13), which admits a strictly positive density with respect to the Lebesgue measure. Proof For an invariant measure μ, choosing initial values which satisfy the distribution μ, then the solution X at any time t ≥ 0 admits the same law since  μ(A) = =



 R2 R2

1 A (x)μ(d x) =

R2

Pt 1 A (x)μ(d x)

1 A (X x (t))μ(d x) = P(X (t) ∈ A), ∀ A ∈ B(R2 ), t ≥ 0.

1.5 Invariant Measures for Hamiltonian Systems

29

It is then deduced from the lemma above that μ has a strictly positive density with respect to the Lebesgue measure. Based on the proof of Lemma 1.2, X is strong Feller and irreducible which ensures that μ is the unique invariant measure and is ergodic according to Theorem 1.5. 

Summary This chapter gives the definitions of invariant measures and ergodicity for stochastic processes, as well as several sufficient conditions for the existence and uniqueness of invariant measures, which provide fundamental tools of studying the ergodicity of differential equations and their numerical approximations. For a deterministic Hamiltonian system, it is well known that its Hamiltonian and symplectic structure are conserved by the flow of the system. As a result, the solution lies in the isoenergetic surface, which indicates that the invariant measure of the system will not be unique. It is also the case for the stochastic Kubo oscillator driven by standard Brownian motions introduced in Sect. 1.5.2. Moreover, even though symplectic schemes possess the discrete symplectic conservation law and perform better than non-symplectic schemes over long time, they could not preserve the Hamiltonian in general. Hence, the ergodicity for Hamiltonian systems as well as their numerical approximations remains unclear. By bringing in a dissipative term, [179] considers a kind of stochastic dissipative Hamiltonian systems driven by degenerate noises, whose unique global solution is an ergodic process. The implicit Euler scheme applied to the considered model is shown to converge to the equilibrium exponentially. As a specific example of stochastic dissipative Hamiltonian systems, the stochastic Langevin equation will be studied in the following chapter, and high order schemes possessing ergodicity and the conformal symplectic structure will be constructed. If the stochastic differential equations are driven by rough paths instead of standard Brownian motions, the solution is not a Markov process any more. The invariant measures and ergodicity for stochastic differential equations driven by rough paths are studied in [42, 97, 98] and references therein. It is still unclear and worth considering whether it is possible to approximate the equilibrium of rough differential equations through a proper numerical approximation.

Chapter 2

Invariant Measures for Stochastic Differential Equations

For ergodic SDEs, when solving them numerically, it is extremely important to choose proper schemes which could possess the properties under consideration and be applicable to practice. To the best of our knowledge, the numerical analysis of ergodic SDEs usually follows two directions. One is to construct numerical schemes which could inherit the ergodicity of the original system, and then to give the approximate error between the numerical invariant measure and the original one. We refer to [139] for the study of ergodic numerical schemes of SDEs, to [178] and [179] for the approximations of invariant measures of general SDEs and stochastic Hamiltonian systems, respectively, to [2] for high order approximations of invariant measures of ergodic SDEs, to [31] for approximation of the invariant measure for parabolic SPDEs, and to [50] for stochastic nonlinear Schrödinger equation case. The other one is to investigate the convergence rate of the temporal average for the numerical solution to the ergodic limit for the underlying SDEs, see [148] for the research about the Langevin equation, [140] for general SDEs, and [32] for parabolic SPDEs. This chapter encompasses a brief introduction on above results concerning ergodicity of both SDEs and numerical schemes as well as approximate error of invariant measures and the ergodic limit, mainly based on [139, 140, 178] and references therein. In Sect. 2.1, several sufficient conditions related to the drift and diffusion terms are introduced to ensure the existence and uniqueness of invariant measures for SDEs. More specific SDEs, i.e., non-degenerate SDEs and stochastic Langevin equations, as well as their numerical approximations, are studied in Sects. 2.2 and 2.3 to illustrate the analysis in Sect. 2.1. For an ergodic numerical approximation, the error between its invariant measure and the one of the original equation is studied in Sect. 2.4. For general (probably not ergodic) numerical approximations, the temporal average of the numerical solution may also be a proper approximation of the invariant measure of the original equation, which is stated in Sect. 2.5. © Springer Nature Singapore Pte Ltd. 2019 J. Hong and X. Wang, Invariant Measures for Stochastic Nonlinear Schrödinger Equations, Lecture Notes in Mathematics 2251, https://doi.org/10.1007/978-981-32-9069-3_2

31

32

2 Invariant Measures for Stochastic Differential Equations

Let H = Rd for some d ∈ N throughout this chapter. We still denote by X x (t) the solution of the following initial value problem 

d X (t) = b(X (t))dt + σ (X (t))dW (t), t ≥ 0 X (0) = x ∈ H

(2.1)

with b : H → H, σ : H → Rd×m and W an m-dimensional Wiener process. In the following sections, precise assumptions on coefficients b and σ will be given to get certain properties (see e.g. Theorems 2.2 and 2.3). We denote by {X n }n∈N a homogeneous Markov chain with deterministic initial value X 0 = x, which may be the solution of a numerical approximation for (2.1).

2.1 Ergodicity of Solutions to General Stochastic Differential Equations It has been shown in Chap. 1 that the Lyapunov condition ensures the existence of invariant measures for (2.1) while the uniqueness of the invariant measure is obtained if the strong Feller and irreducibility properties are satisfied. Now we introduce some conditions which concentrate on a compact set instead of the whole state space.

2.1.1 Existence of Invariant Measures We denote by L the infinitesimal generator of the process X in (2.1): L :=

d  i=1

bi

d ∂ ∂2 1  + (σ σ  )i j . ∂ xi 2 i, j=1 ∂ xi ∂ x j

(2.2)

The following assumption on b and σ is a sufficient condition of the Lyapunov condition given in Theorem 1.3. We denote by  ·  the Euclidean norm of vectors in H. Assumption 2.1 ([139]) There exist real numbers a1 , a2 ∈ (0, ∞) and a function V : H → [1, ∞) with lim V (y) = ∞ y→∞

such that L V (y) ≤ −a1 V (y) + a2 , ∀ y ∈ H.

2.1 Ergodicity of Solutions to General Stochastic Differential Equations

33

Proposition 2.1 ([139]) If the infinitesimal generator L of X in (2.1) satisfies Assumption 2.1, we have E[V (X n )] ≤αE[V (X n−1 )] + β

(2.3)

with X n := X (nT ) for some real numbers T ∈ (0, +∞), α ∈ (0, 1) and β ∈ [0, +∞), which leads to the Lyapunov condition in Theorem 1.3. Proof Itô’s formula applied to V (X (t)) yields that d V (X (t)) = L V (X (t))dt + ∇V (X (t)) σ (X (t))dW (t) ≤ − a1 V (X (t))dt + a2 dt + ∇V (X (t)) σ (X (t))dW (t). Then taking expectation to both sides of above formula and using the Gronwall inequality, we obtain E[V (X (t))] ≤ e−a1 (t−s) E[V (X (s))] +

 a2  1 − e−a1 (t−s) , a1

which implies (2.3) with t = nT , s = (n − 1)T , α = e−a1 T and β = this recursive formula, it indicates

a2 . a1

Based on

E[V (X n )] ≤ αE[V (X n−1 )] + β ≤ α n E[V (X 0 )] + β(1 + α + · · · + α n−1 ) 1 − α n n→∞ β , = α n EV (X 0 ) + β −−−→ 1−α 1−α which implies the uniform boundedness of E[V (X n )] and coincides with the Lyapunov condition given in Theorem 1.3.  In particular, if the Markov chain {X n }n∈N is generated through a numerical scheme X n+1 = F(X n , τ, n+1 W ) of (2.1) with time step-size τ , increment n+1 W = W (tn+1 ) − W (tn ) and X n being an approximation of X (tn ) at tn = nτ , then the following proposition is frequently used to test whether the scheme could inherit the Lyapunov structure. The idea of its proof comes from that of Theorem 7.2 in [139]. Proposition 2.2 If F ∈ C∞ (Rd × Rm , H) and Assumption 2.1 holds for an essentially quadratic function V : C1 (1 + y2 ) ≤ V (y) ≤ C2 (1 + y2 ), ∇V (y) ≤ C3 (1 + y), ∀ y ∈ H for some constants C1 , C2 , C3 > 0. Then (2.3) holds for the proposed scheme when applied to Eq. (2.1) if (i) there exist c1 , ε > 0, independent of τ > 0, such that EX (t1 ) − X 1 2 ≤ c1 (1 + x2 )τ ε+2 for any initial value x ∈ H,

34

2 Invariant Measures for Stochastic Differential Equations

(ii) there exists c2 = c2 (r ) > 0, independent of τ > 0, such that EX 1 r ≤ c2 (1 + xr ) for all r ≥ 1 and x ∈ H. Proof Noting that C1 (1 + EX (t)2 ) ≤ E[V (X (t))] ≤ e−a1 t V (x) +

 a2  1 − e−a1 t , a1

we have EX (t)2 ≤ C(1 + x2 ), where C denotes generic constants which may be different from one to another. For some fixed numerical solution X n−1 , we denote by X X n−1 (τ ) the exact solution of (2.1) at time τ starting from X (0) = X n−1 . Thus, based on conditions above, we conclude E[V (X n )] ≤ E[V (X X n−1 (τ ))] + E|V (X X n−1 (τ )) − V (X n )|  1    ≤ e−a1 τ E[V (X n−1 )] + C + E  ∇V (θ X n + (1 − θ)X X n−1 (τ )), X X n−1 (τ ) − X n dθ  

0

1 2 ≤ e−a1 τ E[V (X n−1 )] + C E[1 + X n 2 + X X n−1 (τ )2 ]E[X X n−1 (τ ) − X n 2 ] + C  ε ≤ e−a1 τ E[V (X n−1 )] + C 1 + EX n−1 2 τ 2 +1 ≤ e−a˜ 1 τ E[V (X n−1 )] + C, ε

where we have used the fact that e−a1 τ + Cτ 2 +1 ≤ e−a˜ 1 τ for some a˜ 1 > 0 and sufficiently small τ in the last step.  For examples which satisfy conditions in Proposition 2.2, we refer to Corollaries 7.4 and 7.5 in [139].

2.1.2 Uniqueness of the Invariant Measure Based on the probability kernel πt (x, ·) (resp. πn (x, ·)) for Markov process {X x (t)}t≥0 (resp. Markov chain {X n }n∈N ) defined in Sect. 1.2, the following assumption given in a compact set is used as a necessary condition for strong Feller and irreducibility properties. Assumption 2.2 ([139]) Suppose for some fixed compact set G 0 ∈ B(H) that (i) for some x0 ∈ int(G 0 ) and any r > 0, there exists t∗ = t∗ (r ) such that πt∗ (x, B(x0 , r )) > 0, ∀ x ∈ G 0 ; (ii) πt (x, ·) possesses a density pt (x, y), i.e.,  πt (x, G) =

pt (x, y)dy, ∀ x ∈ G 0 , ∀ G ∈ B(H) ∩ B(G 0 ) G

2.1 Ergodicity of Solutions to General Stochastic Differential Equations

35

with pt (x, y) jointly continuous in (x, y) ∈ G 0 × G 0 . Assumption 2.2, together with Assumption 2.1, gives an alternative way to Theorems 1.3 and 1.5 of showing the existence and uniqueness of the invariant measure, which is stated in the following theorem. Theorem 2.1 (Theorem 2.5, [139]) If X satisfies Assumptions 2.1 and 2.2, or alternatively, {X n }n∈N defined in Proposition 2.1 satisfies (2.3) and Assumption 2.2 by replacing t with n, then {X (t)}t≥0 (resp. {X n }n∈N ) possesses a unique invariant measure with 

 2β  G 0 = y V (y) ≤ ρ−α 1

for some ρ ∈ (α 2 , 1). Here, α and β are the same as those in Proposition 2.1. In general, Assumption 2.2 is not easy to verify. Thus, a sufficient condition stated in the following proposition is frequently used to verify Assumption 2.2 (ii). That is, a Lie bracket condition, which is also known as the Hörmander condition (see [111, 168] and references therein), implies the hypoelliptic setting of the generator L and that πt has a continuous density (Theorem 38.16, [168]). Theorem 2.2 (Hörmander’s theorem) Assumption 2.2 (ii) is satisfied by the transition probability of (2.1) if (i) b, σi ∈ C∞ (H), i = 1, · · · , m with σ = (σ1 , · · · , σm ); (ii) the Hörmander condition is satisfied, more precisely, Λn (y) = H for some n ∈ N and for all y ∈ H, where Λ0 (y) = span{b(y), σi (y), i = 1, · · · , m}, Λn+1 (y) = span{ f (y), [g, f ](y) : f ∈ Λn , g ∈ Λ0 } and [ f, g](y) = (∇g(y)) f (y) − (∇ f (y))g(y) denotes the Lie bracket. Remark 2.1 Operator L is called hypoelliptic if L f ∈ C∞ (H) implies that f ∈ C∞ (H), but this condition is not easy to verify. Thus, one usually uses the Hörmander condition, which actually is a sufficient condition of hypoelliptic operators (Theorem 1.1, [111]), as the hypoelliptic setting. As applications of the above theory, the following two sections concern two kinds of SDEs: non-degenerate SDEs with dissipative conditions and the stochastic Langevin equation with degenerate noises. We give their ergodicity results under specific conditions.

36

2 Invariant Measures for Stochastic Differential Equations

2.2 Non-degenerate Stochastic Differential Equations and Ergodic Schemes For general SDEs (2.1) or Markov chains on H, we give some other sufficient conditions (see also [2, 140, 178–180] and references therein), which are easy to verify for a specific equation, to show its ergodicity. Theorem 2.3 Assume that (i) functions b and σ are smooth and have bounded derivatives to any order, and furthermore, σ is also bounded itself; (ii) there exist a constant ζ > 0 and a compact set G 0 such that y, b(y) ≤ −ζ y2 , ∀ y ∈ H − G 0 ; (iii) the associated operator L is uniformly elliptic, i.e., for some η > 0, y  (σ (z)σ (z) )y ≥ ηy2 , ∀ y, z ∈ H. Then (2.1) is ergodic with a unique invariant measure μ, and furthermore, μ possesses a smooth density. Proof To verify the Lyapunov condition, we denote V (y) = y2 . Then V (y) is bounded for y ∈ G 0 . For y ∈ H − G 0 , we have L V (y) =2 y, b(y) +

d 

(σ (y)σ (y) )i j

i, j=1

≤ − 2ζ V (y) + σ (y)2F , where  ·  F denotes the Frobenius norm of matrices. It immediately implies that Assumption 2.1 holds as σ is uniformly bounded, and ensures the existence of invariant measures. In addition, the elliptic setting ensures that the Dirichlet problem for following equation L v = 0 admits a unique solution (see e.g. Lemma 3.4 and Remark 3.10, [120]). This fact, together with the Existence Theorem for weak solutions (see e.g. Theorem 4, Chap. 6.2, [77]) in classical theory of differential equations, indicates that L ∗ρ = 0 possesses a unique nontrivial solution ρ∞ ∈ C∞ (H) satisfying L ∗ denotes the L2 (H)-adjoint operator of L .

H

ρ∞ d x = 1, where

2.2 Non-degenerate Stochastic Differential Equations and Ergodic Schemes

37

We then construct a probability measure μ satisfying dμ(x) = ρ∞ (x)d x. Denote u(x, t) = E[ϕ(X x (t))], then Itô’s formula yields that  u(x, t) = u(x, 0) +

t

L u(x, s)ds.

0

Taking the L2 (H)-inner product between both sides of above equation and ρ∞ , and denoting u(t) = u(x, t) for short, we have  u(t), ρ∞ L2 (H) = u(0), ρ∞ L2 (H) +

0

 = u(0), ρ∞ L2 (H) +

t

0

t

L u(s), ρ∞ L2 (H) ds u(s), L ∗ ρ∞ L2 (H) ds = u(0), ρ∞ L2 (H) ,

which implies that 

 H

E[ϕ(X x (t))]dμ(x) =

H

ϕdμ(x)

and thus μ is the unique invariant measure of (2.1) as a result of the uniqueness of  ρ∞ . Furthermore, it holds that if (2.1) possesses an invariant measure with a smooth density ρ∞ with respect to the Lebesgue measure, then ρ∞ is a solution of the steady state Fokker–Planck equation L ∗ ρ∞ = 0 (see e.g. Theorem 4, Chap. IX, [174]). The assumptions in Theorem 2.3 are extremely strong for a stochastic process to be ergodic. Actually, it can be weakened onto some compact set. Theorem 2.4 Assume that there exists a bounded domain U ⊂ H with regular boundary such that (i) in U and some neighborhood thereof, denoted by U˜ , y  (σ (z)σ (z) )y ≥ ηy2 , ∀ y ∈ H, z ∈ U˜ ; (ii) defining a stopping time τ x = inf{t : X x (t) ∈ U }, one has E[τ x ] < ∞ for any x ∈ H − U , and supx∈K E[τ x ] < ∞ for any compact set K ⊂ H. Then (2.1) is also ergodic with a unique invariant measure μ. We refer to [120] for the proof of this theorem: the existence of invariant measures is shown in Theorem 4.1, the ergodicity of the solution is proved in Theorem 4.2 and Corollary 4.3, while the uniqueness of the invariant measure is given in Corollary 4.4. For ergodic SDEs (2.1), a natural question is whether one can construct numerical schemes which could inherit this dynamical property. We can find through Theorem 2.1 that the Lyapunov structure and the strong Feller and irreducibility properties are

38

2 Invariant Measures for Stochastic Differential Equations

crucial and easy to verify when proving the ergodicity of (2.1). Numerical schemes are also constructed in order to inherit these properties and gain a unique numerical invariant measure. d×m ) and W = (W1 , · · · , Wm ) ∈ Rm . We Let σ = (σ1 , · · · , σm ) ∈ C∞ b (H, R now show the ergodicity of the Euler–Maruyama scheme with sufficiently small time step-size τ : X n+1 = X n + b(X n )τ + σ (X n )n+1 W,

(2.4)

where n+1 W = W (tn+1 ) − W (tn ). It is apparent that the numerical solution {X n }n∈N of (2.4) exists and is an Ftn -adapted homogenous Markov chain. Theorem 2.5 The numerical solution of (2.4) applied to (2.1), under the assumptions in Theorem 2.3, is ergodic for sufficiently small time step-size τ . Proof We can find out that the matrix σ (x) is of full rank for any initial value x ∈ Rd according to the assumption (iii) in Theorem 2.3. It also implies that, for any open ball B o ⊂ G 0 , the transition probability π1 (x, B o ) = P(x + b(x)τ + σ (x)1 W ∈ B o ) > 0 since 1 W is Gaussian. Furthermore, since the distribution of n+1 W has a smooth density with respect to the Lebesgue measure for any n ∈ N, the transition kernel πn (x, ·) = [π1 (x, ·)]n has a smooth density. We then conclude that (2.4) satisfies Assumption 2.2. Based on the assumptions (i) and (ii) in Theorem 2.3, we have that (2.3) holds: / G0 for X n ∈ EX n+1 2 =EX n 2 + 2τ E X n , b(X n ) + τ 2 Eb(X n )2 + Eσ (X n )n+1 W 2 ≤(1 − 2ζ τ )EX n 2 + τ 2 Eb(0) + ∇b(θ X n )X n 2 + Cτ ≤(1 − Cτ )EX n 2 + Cτ for sufficiently small τ independent of n, and θ ∈ [0, 1]. If X n ∈ G 0 , it can be deduced that EX n+1 2 ≤ (1 − Cτ )EX n 2 + Cτ EX n 2 + Cτ, where EX n 2 is bounded as G 0 is compact. We complete the proof taking advantage of Theorem 2.1.



Example 2.1 Recall the one-dimensional Ornstein–Uhlenbeck process given through (1.7) in Example 1.1. It is apparent that this equation satisfies the non-degenerate

2.2 Non-degenerate Stochastic Differential Equations and Ergodic Schemes

39

assumptions in Theorem 2.3 and thus possesses a unique invariant measure μ = N (0, 1). Scheme (2.4) applied to (1.7) reads X n+1 = X n − X n τ +

√

2n+1 B

(2.5)

with time step-size τ ∈ (0, 1), deterministic initial value X 0 = ξ ∈ R and the increments n+1 B = B(tn+1 ) − B(tn ). Thus, the numerical solution {X n }n∈N obeys the following distribution X 1 ∼ N ((1 − τ )ξ, 2τ ) =: π1 (ξ, ·),   X 2 ∼ N (1 − τ )2 ξ, 2τ (1 − τ )2 + 2τ =: π2 (ξ, ·),    X 3 ∼ N (1 − τ )3 ξ, 2τ (1 − τ )4 + (1 − τ )2 + 1 =: π3 (ξ, ·), ···

X n ∼ N (1 − τ )n ξ,

  2  1 − (1 − τ )2n =: πn (ξ, ·), 2−τ

···   2 We finally obtain that the distributions of X n converge to μτ := N 0, 2−τ as n →   2 τ , ∞. We claim that μ is an invariant measure for {X n }n∈N . In fact, if X n ∼ N 0, 2−τ   2 . In addition, {X n }n∈N then one can calculate through (2.5) that X n+1 ∼ N 0, 2−τ can also be shown to be strong Feller and irreducible, taking advantage of Gaussian distributions πn , which then yields that μτ is the unique invariant measure of (2.5). It is worth noticing that the conditions given in Theorems 2.3 and 2.4 are still very strong, and are not satisfied by a wide class of models. In some cases where the conditions in Theorem 2.3 are not satisfied, the Lyapunov structure may be easily destroyed by numerical schemes, especially explicit ones, since it is a global structure instead of concentrating on some compact set. We refer to the one-dimensional stochastic differential equation d X = −X 3 dt + dW as a counterexample given in [139], whose drift coefficient has polynomial growth. The Euler–Maruyama scheme applied to this equation fails to be ergodic (Lemma 6.3, [139]), and even not converge in strong sense ([113, 114]). As a result, numerical schemes need to be specially constructed when conditions in Theorems 2.3 and 2.4 are not satisfied. In the following, we show the ergodicity, as well as its geometric structure, for a type of SDEs with degenerate noise.

2.3 Stochastic Langevin Equation and Its Discretizations In this section, we study a stochastic dissipative Hamiltonian system similar to that studied in Sect. 1.5.2 utilizing the argument introduced in the former two sections.

40

2 Invariant Measures for Stochastic Differential Equations

We still denote H = Rd . Consider a stochastic Langevin equation with the hypoelliptic setting (see Theorem 2.2 and Remark 2.1): ⎧ m  ⎪ ⎨ dp = − ∇ F(q)dt − γ pdt − σi dWi , ⎪ ⎩

i=1

(2.6)

dq = pdt

with p, q ∈ H denoting the position and momentum of a particle, respectively. It is a damped Hamiltonian system with degenerate additive noises and satisfies the stochastic conformal symplectic structure, which will be introduced in the following. Here F ∈ C∞ (H, R+ ), γ > 0, σ = (σ1 , · · · , σm ) ∈ Rd×m and W = (W1 , · · · , Wm ) is a standard m-dimensional Brownian motion. Equation (2.6) admits a separable Hamiltonian function H ( p, q) = 21  p2 + F(q). In particular, we assume that m ≥ d and rank{σi ∈ H, i = 1, · · · , m} = d.

2.3.1 Ergodicity for Exact and Numerical Solutions The ergodicity of (2.6) has been considered by several authors (see e.g. [43, 120, 139, 167]). We introduce one of these proofs through the following three lemmas according to Theorem 1.5. Lemma 2.1 There exists a Lyapunov functional V : R2d → [1, +∞] which satisfies Assumption 2.1 with infinitesimal generator L associated to (2.6) under condition F(q) +

3γ 2 q2 − ∇ F(q), q ≤ C 8

(2.7)

for any q ∈ H and some constant C > 0, and thus ensures the existence of invariant measures for (2.6). Proof Define V ( p, q) = 2F(q) + c1 q2 + c2  p2 + c3 p, q + 1 with coefficients ci , i = 1, 2, 3, chosen such that Assumption 2.1 is satisfied. Then we have L V ( p, q) = p, 2∇ F(q) + 2c1 q + c3 p − ∇ F(q) + γ p, 2c2 p + c3 q + c2 σ 2F = (2c1 − γ c3 ) p, q + (c3 − 2γ ) p2 − c3 ∇ F(q), q + σ 2F by choosing c2 = 1 in the last step. We then choose c3 = γ and 2c1 = γ c3 for simplicity, and obtain

2.3 Stochastic Langevin Equation and Its Discretizations

41

L V ( p, q) = − γ  p2 − γ ∇ F(q), q + σ 2F  3  γ 21  3  γ 2 2 γ  + γ F(q) + 3γ q2 = − V ( p, q) −  q − p   2 2 2 8 γ 2 − γ ∇ F(q), q + + σ  F 2

 3γ 2 γ γ q2 − ∇ F(q), q + + σ 2F . ≤ − V ( p, q) + γ F(q) + 2 8 2 According to condition (2.7), Assumption 2.1 holds for V ( p, q) = 2F(q) + γ2 q2 +  p2 + γ p, q + 1.  2

The condition (2.7) can of course be modified by choosing other constants c1 , c2 and c3 in V ( p, q). We refer to [139] for a general form of condition (2.7). In the following lemma, we show the strong Feller property via the Hörmander condition under the assumption m ≥ d. We can also find out through its proof that if m < d, there are not enough noises to drive the solution to any points in the state space. Lemma 2.2 Equation (2.6) satisfies the hypoelliptic setting and thus its solution is a strong Feller process. Proof This proof is a part of that of Theorem 3.2, [139]. Rewrite (2.6) into d



  m  σi p −∇ F(q) − γ p dWi = dt − 0 q p i=1

= : Y0 ( p, q)dt +

m 

Yi dWi .

(2.8)

i=1

For m ≥ d and rank{Yi , i = 1, · · · , m} = d, we can deduce that

[Yi , Y0 ] = (∇Y0 )Yi =

−γ Id −∇ 2 F(q) 0 Id



−σi 0



=

γ σi −σi



are independent of Yi , i = 1, · · · , m, and hence rank{Yi , [Yi , Y0 ], i = 1, · · · , m} = 2d, which completes the proof.



Lemma 2.3 (Lemma 3.4, [139]) The solution to (2.6) is irreducible. In addition, one can also show that the invariant measure μ of (2.6) possesses a smooth Boltzmann–Gibbs density (see e.g. [139]) 

 2 p + F(q) , ρ( p, q) ∝ exp −γ 2

42

2 Invariant Measures for Stochastic Differential Equations

taking advantage of the Fokker–Planck equation. One can then calculate the ergodic limit based on the density ρ, precisely, 

 R2d

ϕ( p, q)dμ( p, q) =

Rd ×Rd

ϕ( p, q)ρ( p, q)dpdq,

which can be used as a reference value to see whether a numerical scheme could approximate the ergodic limit properly (see Sect. 2.5). We refer to [2, 25, 167] and references therein for more details. For this kind of SDEs with degenerate additive noise, one can also construct ergodic numerical schemes benefiting especially from the additive noise. As is shown in (2.8), the Langevin equation (2.6) can be rewritten as d X = Y0 (X )dt + Y dW by denoting X = ( p  , q  ) , Y0 (X ) := Y0 ( p, q), Y = (Y1 , · · · , Ym ) ∈ R2d×m and W = (W1 , · · · , Wm ) . The authors in [139] study three schemes: the Euler– Maruyama scheme X n+1 = X n + Y0 (X n )h + Y n+1 W,

(2.9)

the backward Euler method X n+1 = X n + Y0 (X n+1 )h + Y n+1 W,

(2.10)

the split-step backward Euler method 

X ∗ = X n + Y0 (X ∗ )h, X n+1 = X ∗ + Y n+1 W,

(2.11)

and obtain the ergodicity for the numerical solution {X n }n∈N as stated in the following theorem. Theorem 2.6 (Corollary 7.4, [139]) Consider the Langevin equation (2.6) with hypoelliptic setting. Assume that essentially quadratic function F ∈ C∞ (Rd , R+ ) satisfies condition (2.7) and that ∇ F is globally Lipschitz. Solutions {X n }n∈N of schemes (2.9), (2.10) and (2.11) are all ergodic, and each of them possesses a unique invariant measure.

2.3.2 Geometric Structure: Conformal Symplecticity It is well known that Hamiltonian systems in both deterministic and stochastic cases involve symplectic structures (see [10, 47, 51, 56, 82, 95, 100, 103, 110, 119, 169,

2.3 Stochastic Langevin Equation and Its Discretizations

43

187, 188, 195] and references therein). For the stochastic case, as an example, in the following form d X = J −1 ∇ H0 (X )dt + J −1 ∇ H1 (X ) ◦ dW (t)

(2.12)

with initial value X (0) = ( p0 , q0 ) , p0 , q0 ∈ H. Here, W (t) is an m-dimensional standard Brownian motion and 

0 Id J= −Id 0 is the standard symplectic matrix. If we decompose X as X = ( p  , q  ) with p, q ∈ H and p(0) = p0 , q(0) = q0 , then the phase flow ( p0 , q0 ) → ( p, q) of (2.12) preserves the symplectic structure d p ∧ dq = d p0 ∧ dq0 almost surely with ‘d’ denoting the exterior derivative and d p ∧ dq being a differential 2-form. For the stochastic Langevin equation (2.6), it can be described as a damped Hamiltonian system (see e.g. [179]), and degenerates to a Hamiltonian system when the absorption coefficient γ = 0. Thus, it possesses another important geometric structure, that is, conformal symplectic structure. Actually, rewrite (2.6) as   d X = J −1 ∇ H0 (X ) + ΛX dt + J −1 ∇ H1 (X ) ◦ dW

(2.13)

with X = ( p  , q  ) , x0 := ( p0 , q0 ) , H0 =

1  p2 + F(q), 2

H1 = σ q and Λ =

 −γ Id 0 . 0 0

Hence similar to [95] and based on (2.13), the phase flow φt : x0 → X (t) of (2.13) satisfies ⎡

∂φt d⎣ ∂ x0



 J

 = J −1 ∇ 2 H0 (φt ) 

⎤         ∂φt ∂φt ⎦ ∂φt ∂φt ∂φt = d + J J d ∂ x0 ∂ x0 ∂ x0 ∂ x0 ∂ x0

∂φt ∂φt +Λ ∂ x0 ∂ x0



 J

∂φt ∂ x0



 dt + J −1 ∇ 2 H1 (φt )

∂φt ◦ dW ∂ x0



 J

∂φt ∂ x0



       ∂φt ∂φt ∂φt ∂φt ∂φt J J −1 ∇ 2 H0 (φt ) +Λ J J −1 ∇ 2 H1 (φt ) ◦ dW dt + ∂ x0 ∂ x0 ∂ x0 ∂ x0 ∂ x0 ⎡   ⎤ ∂φt ∂φt ⎦ =−γ ⎣ J dt, ∂ x0 ∂ x0 +

44

2 Invariant Measures for Stochastic Differential Equations

where in the last step we have used the fact that J − J = −I2d and ΛJ + J Λ = −γ J . Thus, we obtain

∂φt ∂ x0



J

∂φt ∂ x0



= e−γ t J,

which implies immediately the following stochastic conformal symplectic structure d p(t) ∧ dq(t) = e−γ t d p0 ∧ dq0 . It indicates that the phase flow of (2.6) is not area preserving any more, but possesses the exponentially decay area instead. Namely, denote by Vol(t) the phase volume on the domain Dt ⊂ R2d at time t, then  ∂( p(t), q(t)) dp0 dq0 , dp(t)dq(t) = det ∂( p0 , q0 ) D0 

 Vol(t) = Dt

where the determinant of the Jacobian matrix

∂( p(t), q(t)) det ∂( p0 , q0 )

∂( p(t),q(t)) ∂( p0 ,q0 )



satisfies

= e−γ td .

Recall that schemes (2.9), (2.10) and (2.11) for (2.6) are all ergodic, but they fail to preserve the conformal symplectic structure of the original system. To construct numerical schemes which could inherit both the ergodicity and the conformal symplecticity, we introduce the following systems based on the splitting technique (see [147] and references therein) d p˜ = − ∇ F(q)dt ˜ − σ dW, d q˜ = pdt, ˜ q(0) ˜ = q˜0 ; d pˆ = − γ pdt, ˆ p(0) ˆ = pˆ 0 , d qˆ = 0, q(0) ˆ = qˆ0 .

p(0) ˜ = p˜ 0 , (2.14) (2.15)

We would like to mention that other kinds of splitting, for example, Lie–Trotter splitting method (see e.g. [3, 25]), may be also available to construct conformal symplectic schemes. Note that the system (2.15) can be solved exactly and explicitly by p(t) ˆ = e−γ t pˆ 0 and q(t) ˆ = qˆ0 . Furthermore, we apply the midpoint scheme to the system (2.14), which together with the exact solution of (2.15) yields the scheme

2.3 Stochastic Langevin Equation and Its Discretizations

⎧ ⎪ ⎪ p˜ n+1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ q˜n+1 ⎪ ⎪ ⎪ ⎪ pn+1 ⎪ ⎪ ⎪ ⎩ qn+1

= pn − τ ∇ F

q˜n+1 + qn 2

45

 − σ n+1 W,

τ ( p˜ n+1 + pn ), 2 −γ τ =e p˜ n+1 , = qn + = q˜n+1

with time step-size τ , numerical solutions { pn , qn }n∈N and n+1 W = W (tn+1 ) − W (tn ). The above scheme can be rewritten into a compact form by eliminating intermediate variables p˜ n+1 and q˜n+1 :

 ⎧ qn+1 + qn ⎪ ⎨ pn+1 = e−γ τ pn − τ e−γ τ ∇ F − e−γ τ σ n+1 W, 2 ⎪ τ γτ ⎩q n+1 = qn + (e pn+1 + pn ). 2

(2.16)

Theorem 2.7 Let F be essentially quadratic. The solution { pn , qn }n∈N of the proposed scheme (2.16) for (2.6) is ergodic with a unique invariant measure when sampled at even n, i.e., { p2n , q2n }n∈N is ergodic. In addition, it also preserves the discrete stochastic conformal symplectic structure d pn+1 ∧ dqn+1 = e−γ τ d pn ∧ dqn . Proof We show that the proposed scheme could inherit the Lyapunov structure according to Proposition 2.2, which is sufficient to show the existence of the invariant measure for scheme (2.16). Actually, it is not hard to verify that the condition (ii) in Proposition 2.2 holds for the proposed scheme. Based on Lemma 2.1 and the above fact, we can also get condition (i) in Proposition 2.2 with ε = 1. To get the uniqueness of the invariant measure, we only need to verify that Assumption 2.2 holds for the Markov chain { p2n , q2n }n∈N . We first show that the transition probability π2 (x0 , ·) has a smooth density with x0 = ( p0 , q0 ) . We define a function K : R2d × R2d → R2d as    2 1 u 2 − e−γ τ u 1 + τ e−γ τ ∇ F v +v + e−γ τ σ n+1 W 1 1 2 2 2 K (u , v , u , v ) = , v2 − v1 − τ2 (eγ τ u 2 + u 1 ) whose Jacobian matrix ∂ K (u 1 , v1 , u 2 , v2 ) = ∂(u 2 , v2 )

Id − τ2 eγ τ

2 1 τ −γ τ 2 e ∇ F( v +v ) 2 2 Id



is positive definite for any (u 1 , v1 ) and sufficiently small τ . Hence, based on the implicit function theorem, there exists a continuous differentiable function k such that ( pn+1 , qn+1 ) = k( pn , qn ). This fact ensures that the transition probability of

46

2 Invariant Measures for Stochastic Differential Equations

( pn+1 , qn+1 ) possesses a continuous density, i.e., Assumption 2.2 (ii) holds, as n+1 W possesses a smooth density. Next we show that Assumption 2.2 (i) holds, i.e, π2 (x0 , B(y, r )) > 0 for any x0 , y ∈ R2d and r > 0. For any fixed x0 and x2 ∈ B(y, r ), we consider the first two steps in the Markov chain { pn , qn }n∈N p2 = e

−γ τ

p1 − τ e

−γ τ

∇F

q2 + q1 2



− e−γ τ σ 2 W ;

τ γτ (e p2 + p1 ), 2 

q1 + q0 − e−γ τ , p1 = e−γ τ p0 − τ e−γ τ ∇ F 2 τ q1 = q0 + (eγ τ p1 + p0 )σ 1 W 2

q2 = q1 +

(2.17) (2.18) (2.19) (2.20)

with ( p0 , q0 ) = x0 . From (2.18) and (2.20), p1 and q1 can be uniquely determined to ensure that ( p2 , q2 ) = x2 . Thus, 2 W and 1 W can also be determined according to (2.17) and (2.19), respectively. As the Brownian motion hits any open ball with positive probability, we finally get that π2 (x0 , B(y, r )) > 0. We now conclude from all the above that the numerical solution possesses a unique invariant measure based on Theorem 2.1. On the other hand, it is well known that the midpoint scheme preserves the discrete symplectic structure for stochastic Hamiltonian systems. Thus, we have d pn+1 ∧ dqn+1 = e−γ τ d p˜ n+1 ∧ dq˜n+1 = e−γ τ d pn ∧ dqn , which shows the conformal symplecticity of scheme (2.16).



This approach is also available if we apply the symplectic Euler method to system (2.14), and an explicit scheme will be obtained in that circumstance. The midpoint scheme and symplectic Euler scheme applied to (2.6) are both of order one in weak convergence sense. Schemes of higher weak convergence orders are constructed based on generating functions and modified equations in the following subsection.

2.3.3 Schemes of High Weak Convergence Order In this subsection, an approach of designing conformal symplectic schemes with a higher weak convergence order is introduced by transforming (2.6) to a homogenous Hamiltonian system. The generating function is then available to design symplectic schemes for Hamiltonian systems (see e.g. [7, 8, 185, 186, 188]). Denote Xˆ (t) = eγ t p(t) and Yˆ (t) = q(t) with components X l and Yl , l = 1, · · · , d, respectively. Then Itô’s formula applied to Xˆ and Yˆ yields

2.3 Stochastic Langevin Equation and Its Discretizations

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

d Xˆ = − eγ t ∇ F(Yˆ )dt − eγ t

47 m 

σi dWi (t),

i=1

d Yˆ =e−γ t Xˆ dt,

which forms a non-autonomous stochastic Hamiltonian system ⎧ m  ∂ Hˆ 0 ∂ Hˆ i ⎪ ⎪ ˆ dt − ◦ dWi , ⎪dX = − ⎪ ⎨ ˆ ∂ Yˆ i=1 ∂ Y m ⎪  ∂ Hˆ 0 ∂ Hˆ i ⎪ ⎪ ˆ ⎪ d Y = dt + ◦ dWi ⎩ ˆ ∂ Xˆ i=1 ∂ X

with Hamiltonians 1 Hˆ 0 = eγ t F(Yˆ ) + e−γ t | Xˆ |2 , 2

Hˆ i = eγ t σi · Yˆ

and ‘◦’ indicating that equations above hold in Stratonovich integral sense. Bringing in two new variables X d+1 and Yd+1 such that d X d+1 = −

m  ∂ Hˆ i ∂ Hˆ 0 dt − ◦ dWi (t), ∂t ∂t i=1

 1 | p0 |2 + σi · q0 , 2 m

X d+1 (0) = F(q0 ) +

i=1

dYd+1 =dt, Yd+1 (0) = 0,

and denoting X = ( Xˆ  , X d+1 ) = (X 1 , · · · , X d , X d+1 ) and Y = (Yˆ  , Yd+1 ) = (Y1 , · · · , Yd , Yd+1 ) , we get a (2d + 2)-dimensional autonomous stochastic Hamiltonian system ⎧ m  ∂ H0 ∂ Hi ⎪ ⎪ d X = − dt − ◦ dWi , ⎪ ⎪ ⎨ ∂Y ∂Y i=1 (2.21) m ⎪  ∂ H0 ∂ Hi ⎪ ⎪ ⎪ dt + ◦ dWi , ⎩ dY = ∂X ∂X i=1 where 1 H0 (X, Y ) = eγ Yd+1 F(Yˆ ) + e−γ Yd+1 | Xˆ |2 + X d+1 , 2 Hi (X, Y ) = eγ Yd+1 σi · Yˆ . Denote X (0) = x and Y (0) = y for convenience. It is revealed in [185] that the generating function S(X, y, t) related to (2.21) is the solution of the following stochastic Hamilton–Jacobi partial differential equation

48

2 Invariant Measures for Stochastic Differential Equations

dt S(X, y, t) = H0

∂S X, y + ∂X

 dt +

m  i=1

Hi

∂S X, y + ∂X

 ◦ dWi .

(2.22)

Moreover, the mapping (x, y) → (X (t), Y (t)) defined by X (t) = x −

∂ S(X (t), y, t) ∂ S(X (t), y, t) , Y (t) = y + ∂y ∂X

(2.23)

is the stochastic flow of (2.21). Based on the Itô representation theorem and stochastic Taylor-Stratonovich expansion, S(X, y, t) has a series expansion (see e.g. [7]) S(X, y, t) =

 α

G α (X, y)Jαt ,

(2.24)

where Jαt =

 t 0

sl



s2

···

0

◦dW j1 (s1 ) ◦ dW j2 (s2 ) ◦ · · · ◦ dW jl (sl )

0

with multi-index α = ( j1 , j2 , · · · , jl ) ∈ {0, 1, · · · , m}⊗l , l ≥ 1 and dW0 (s) := ds. Before calculating coefficients G α (X, y) in (2.24), we first specify some notations. Let l(α) denote the length of α, and α− be the multi-index resulting from discarding the last index of α. Define α ∗ α  = ( j1 , · · · , jl , j1 , · · · , jl ) where α = ( j1 , · · · , jl ) and α  = ( j1 , · · · , jl ). The concatenation ‘∗’ between a set of multi-indices Λ and α is Λ ∗ α = {β ∗ α|β ∈ Λ}. Furthermore, define

Λα,α =

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

{( j1 , j1 ), ( j1 , j1 )}, if l = l  = 1, {Λ( j1 ),α − ∗ ( jl ), α  ∗ ( j1 )}, if l = 1, l  = 1, {Λα−,( j1 ) ∗ ( jl ), α ∗ ( j1 )}, if l = 1, l  = 1, {Λα−,α ∗ ( jl ), Λα,α − ∗ ( jl )}, if l = 1, l  = 1.

For k > 2, let Λα1 ,··· ,αk = {Λβ,αk : β ∈ Λα1 ,··· ,αk−1 }. Substituting (2.24) into (2.22) and taking Taylor expansions to Hi (i = 0, 1, · · · , m) at (X, y), we obtain G α = Hi with α = (i) being a single index, and G α (X, y) =

l(α )−1  r =1

1 r!

d+1 

∂ r H jl (X, y)

k1 ,··· ,kr =1

∂ yk1 · · · ∂ ykr

 l(α 1 ) + · · · + l(αr ) = l(α) − 1 α− ∈ Λα 1 ,··· ,αr

∂G α 1 ∂ X k1

···

∂G αr ∂ X kr

for multi-indices α = ( j1 , j2 , · · · , jl ) with l ≥ 2. We refer to [7, 146, 185] for more details about generating functions and weakly convergent symplectic numerical schemes obtained by truncating the generating

2.3 Stochastic Langevin Equation and Its Discretizations

49

function. In this case, simulation of multiple integrals is involved to obtain schemes with high weak convergence orders. To reduce the simulation of multiple integrals, it is more convenient to consider the modified stochastic Hamiltonian system ⎧ m  ∂ HiM (X M , Y M ) ∂ H0M (X M , Y M ) ⎪ M ⎪ d X = − dt − ◦ dWi , ⎪ ⎪ ⎨ ∂Y M ∂Y M

X M (0) = x,

i=1

m ⎪  ∂ HiM (X M , Y M ) ∂ H0M (X M , Y M ) ⎪ M ⎪ ⎪ dt + ◦ dWi , Y M (0) = y. ⎩ dY = M ∂XM ∂ X i=1 (2.25) Here,

HiM (X M , Y M ) = Hi (X M , Y M ) + Hi(1) (X M , Y M )τ + · · · + Hi() (X M , Y M )τ  (2.26) ( j) with some fixed step-size τ ∈ (0, 1) and undetermined functions Hi for j = 1, · · · ,  and i = 0, · · · , m. Rewrite the multiple Stratonovich integrals Jαt as Jαt =

⎧ ⎪ Cαβ Iβt , l(α) ≥ 2, ⎨ β

⎪ ⎩ I t , l(α) = 1, α β

where β = (i 1 , i 2 , · · · , il ) ∈ {0, 1, · · · , m}⊗l are indices of length l ≥ 1, Cα are certain constants given in [122], and Iβt :=

 t 0

sl



s2

···

0

dWi1 (s1 )dWi2 (s2 ) · · · dWil (sl )

0

are multiply integrals in the Itô sense. Denote by ˆ S(X, y, t) =



Gˆ α (X, y)

α



Cαβ Iβt ,

(2.27)

l(β)≤k

the truncated modified generating function (see e.g. [7, 122]), where Gˆ α (X, y) =

l(α )−1  r =1

1 r!

d+1 

∂ r H jM (X, y)

k1 ,··· ,kr =1

∂ yk1 · · · ∂ ykr

l

 l(α 1 ) + · · · + l(αr ) = l(α) − 1 α− ∈ Λα 1 ,··· ,αr

∂ Gˆ α 1 ∂ X k1

···

∂ Gˆ αr ∂ X kr

for l(α) ≥ 2, and Gˆ (i) = HiM for i = 0, 1, · · · , m. Then (2.27) determines a one-step approximation

50

2 Invariant Measures for Stochastic Differential Equations

X1 = x −

ˆ 1 , y, τ ) ˆ 1 , y, τ ) ∂ S(X ∂ S(X . , Y1 = y + ∂y ∂ X1

(2.28)

( j)

By choosing proper functions Hi in (2.26), we will be able to construct symplectic schemes approximating (2.21) with weak order k + k  , that is, the local weak error is of order k + k  + 1 

|Eφ(X (τ ), Y (τ )) − Eφ(X 1 , Y1 )| = O(τ k+k +1 ),

(2.29)

even though the scheme is only of weak order k when approximating the modified Eq. (2.25). We take the case k = k  = 1 as an example in which Hamiltonians Hi , i = 0, · · · , m, should be modified as in (2.26) with  = 1. Let φ ∈ C6p (R2d+2 , R). It then suffices to determine functions Hi(1) , i = 0, · · · , m. Utilizing the Taylor expansion to φ(X (τ ), Y (τ )) and φ(X 1 , Y1 ) at (x  , y  ) = (x1 , · · · , xd+1 , y1 , · · · , yd+1 ) and comparing the terms on both sides of (2.29), we choose Hi(1) , i = 0, · · · , m such that ∂ Hi(1) = 0, i = 0, · · · , m, ∂ xd+1 ∂ Hi(1) ∂ Hi(1) 1 1 = γ eγ yd+1 σi , = σi , i = 1, · · · , m, ∂(y1 , · · · , yd ) 2 ∂(x1 , · · · , xd ) 2 d (1) 2 ∂ F(y) 1  ∂ F(y) 1 ∂ H0 = x j + γ eγ yd+1 , r = 1, · · · , d, ∂ yr 2 j=1 ∂ yr ∂ y j 2 ∂ yr ∂ H0(1) 1 ∂ F(y) 1 −γ yd+1 = − γe xr , r = 1, · · · , d. ∂ xr 2 ∂ yr 2 Then Sˆ is determined, based on which one can get the expression of the one-step approximation (2.28) as well as the numerical solution {(X n , Yn )}n∈N . To deduce an approximation of the original system (2.6) based on scheme (2.28), we denote by X n− and Yn− the first d components of X n and Yn respectively, and denote pn = e−γ tn X n− , qn = Yn− . We finally get the numerical scheme for (2.6) ⎧  τ2 γ τ −γ τ ⎪ ⎪ ⎪ pn+1 =e−γ τ pn − ∇ 2 F(qn ) pn+1 − τ 1 + e ∇ F(qn ) ⎪ ⎪ 2 2 ⎪ ⎨  γ τ −γ τ − 1+ e σ n+1 W, ⎪ 2 ⎪ ⎪  ⎪ ⎪ γ τ γτ τ2 τ ⎪ ⎩ qn+1 =qn + τ 1 − e pn+1 + ∇ F(qn ) + σ n+1 W 2 2 2

(2.30)

2.3 Stochastic Langevin Equation and Its Discretizations

51

with n+1 W = W (tn+1 ) − W (tn ) and tn = nτ , n ∈ N, which admits the discrete conformal symplectic structure and is of weak order two. Theorem 2.8 Scheme (2.30) possesses the discrete conformal symplectic conservation law, i.e, d pn+1 ∧ dqn+1 = e−γ τ d pn ∧ dqn . Proof Note that   τ2 d pn+1 ∧ dqn+1 =d pn+1 ∧ dqn + d pn+1 ∧ ∇ 2 F(qn )dqn 2 

τ2 2 = Id + ∇ F(qn ) d pn+1 ∧ dqn 2   γ τ −γ τ e ∇ F(qn ) ∧ dqn =d e−γ τ pn − τ 1 + 2 =e−γ τ d pn ∧ dqn . 

It then completes the proof.

Theorem 2.9 Assume that ∇ F is globally Lipschitz continuous with linear growth. Then scheme (2.30) admits an invariant measure and has convergence order two in weak sense. More precisely, for a fixed T > 0, |Eϕ ( p(tn ), q(tn )) − Eϕ ( pn , qn )| = O(τ 2 ) for all ϕ ∈ C6p (R2d , R) and tn ≤ T, n ∈ N, where C6p (R2d , R) denotes the space of continuous functions whose derivatives are of polynomial growth up to the 6th order. The readers are referred to [108] for the proof of above theorem. Moreover, the weak error plays an important role in approximating the invariant measure of the original system, which will be introduced in the following two sections.

2.4 Approximation of Invariant Measures via Ergodic Schemes The error estimate for ergodic SDEs usually contains two aspects: the error between invariant measures and the error between the temporal average of the numerical solution and the ergodic limit. In this section, we concentrate on the first aspect. If the numerical solution is proved or assumed to be ergodic with a numerical invariant measure μτ , the error between the original invariant measure μ and the numerical one is defined as      ϕdμτ  e(ϕ) :=  ϕdμ − H

H

52

2 Invariant Measures for Stochastic Differential Equations

for some kind of test functions ϕ. One can investigate the convergence order of e(ϕ) to determine whether μτ is a proper approximation of μ (see [3, 140, 178, 182] and references therein). Note that the ergodicity of the exact solution X and the numerical one {X n }n∈N ensures the following equations 1 lim T →∞ T

 0

T

 E[ϕ(X (t))]dt =

H

ϕdμ,

 N −1 1  lim E[ϕ(X n )] = ϕdμτ . N →∞ N H n=0

For a fixed time step-size τ , the difference between above equations shows that   N −1    1  tn+1   e(ϕ) =  lim (E[ϕ(X (t)] − E[ϕ(X n )]) dt  . T =N τ →∞ N τ  t n=0 n It indicates that the error between invariant measures is the same as the weak error if the weak error is shown to be independent of the time interval. However, the time-independent weak error is usually a difficult criterion to verify. The following theorem gives an error estimate of invariant measures via the solution of Kolmogorov equation avoiding proving that the weak error is time independent. In Sect. 2.2, we show that both Eq. (2.1) and the Euler–Maruyama scheme (2.4) applied to it are ergodic under the assumptions in Theorem 2.3. Based on that result, the following theorem gives the error between invariant measures, whose proof is similar to that of Theorem 3.3 in [178]. We also refer to [178] for higher order approximations. Theorem 2.10 Under the same assumptions as in Theorem 2.3, the Euler– Maruyama scheme (2.4) possesses a numerical invariant measure μτ . The error between invariant measures μ and μτ is of order one, i.e., e(ϕ) = O(τ ), ∀ ϕ ∈ C∞ p (H), where C∞ p (H) denotes the space of smooth functions on H with all the derivatives having polynomial growth. Proof We first claim that the numerical solution of (2.4) is uniformly bounded EX n  p ≤ K p (1 + x p exp(−α p tn ))

(2.31)

with deterministic initial value X 0 = x and positive constants K p and α p depending on p. In fact, we have known from the proof of Theorem 2.5 that EX n 2 ≤ (1 − Cτ )EX n−1 2 + Cτ ≤ e−α2 tn x2 + C

2.4 Approximation of Invariant Measures via Ergodic Schemes

53

for some positive constant α2 , which verifies (2.31) with p = 2. For p = 4, note that EX n+1 4 = E X n 2 + 2τ X n , b(X n ) + τ 2 b(X n )2 + σ (X n )n+1 W 2 !2 + 2 X n + b(X n )τ, n+1 W !2 !2 = E X n 2 + 2τ X n , b(X n ) + τ 2 b(X n )2 + E σ (X n )n+1 W 2 !  + 2E X n 2 + 2τ X n , b(X n ) + τ 2 b(X n )2 σ (X n )n+1 W 2 ! + 4E σ (X n )n+1 W 2 X n + b(X n )τ, n+1 W !2 + 4E X n + b(X n )τ, n+1 W ≤ (1 − Cτ )EX n 4 + Cτ 2 + Cτ (1 − Cτ )EX n 2 ,

which, together with the estimate for EX n 2 , verifies (2.31). Following this procedure, we then complete the claim by recursion. In addition, (2.4) has been shown to be of weak order one |E[ϕ(X (tn ))] − E[ϕ(X n )]| ≤ Cn τ with tn = nτ and some constant Cn depending on n (see also [122, 144, 177]). Recall in Sect. 2.2 that u(x, t) = E[ϕ(X x (t))] is the solution of the following Kolmogorov equation ⎧ ⎨ ∂ u(x, t) = L u(x, t), ∂t ⎩ u(x, 0) = ϕ(x), and the derivatives of u(x, t) also have the boundedness similar to X n (Theorem 3.4, [178]) ∂ι u(x, t) ≤ Cι (1 + x pι ) exp(−αι t)

(2.32)

with multi-index ι and positive constants pι , αι depending on ι. The Taylor expansion performed on u(tk , X n ) at X n−1 shows that E[u(tk , X n )] = E[u(tk , X n−1 )] + E[∇u(tk , X n−1 )(X n − X n−1 )]  1  + E ∇ 2 u(tk , X n−1 )(X n − X n−1 )2 2  1  + E ∇ 3 u(tk , X n−1 )(X n − X n−1 )3 3!

54

2 Invariant Measures for Stochastic Differential Equations

 1  4 E ∇ u(tk , θ (X n − X n−1 ) + X n−1 )(X n − X n−1 )4 4!   (2.33) = E[u(tk , X n−1 )] + τ E L u(tk , X n−1 ) + R1k,n τ 2 , +

where θ ∈ (0, 1) and ∇ p u(t, x)y p := ∇ p u(t, x)(y, · · · , y) is the Fréchet derivative for some p ∈ N. The remainder R1k,n can be expressed by  τ   1  R1k,n = E ∇ 2 u(tk , X n−1 )b(X n−1 )2 + E ∇ 3 u(tk , X n−1 )b(X n−1 )3 2 6   1 + E ∇ 4 u(tk , θ (X n − X n−1 ) + X n−1 )(X n − X n−1 )4 . 24τ 2 According to (2.31) and (2.32), we have |R1k,n | ≤ C(1 + EX n  p + EX n−1  p ) exp(−ξ1 tk ) ≤ C(1 + x p ) exp(−ξ1 tk ) for some p and ξ1 . In the same procedure as above, based on Itô’s formula, we derive   E[u(tk+1 , X n−1 )] = E[u(tk , X n−1 )] + τ E L u(tk , X n−1 ) + R2k,n τ 2

(2.34)

with remainder |R2k,n | ≤ C(1 + xq ) exp(−ξ2 tk ) for some positive q and ξ2 . The difference between (2.33) and (2.34) leads to E[u(tk , X n )] = E[u(tk+1 , X n−1 )] + (R1k,n − R2k,n )τ 2 = E[u(tk+n , x)] +

k+n 

j,n+k− j

(R1

j,n+k− j

− R2

)τ 2 .

j=k

Choosing k = 0 and taking the average of above terms for n = 1, · · · , N , we obtain  N −1  N −1 N −1 ∞ 1    1    j,n− j 1    j,n− j  2 E[u(0, X n )] − E[u(tn , x)] ≤ − R2  R1 τ N  N N n=0 n=0 n=0 j=0 ≤ C(1 + x p + xq )τ 2

∞ 

exp(−(ξ1 ∧ ξ2 ) jτ ) ≤ C(1 + x p + xq )τ,

j=0

" since the series τ ∞ j=0 exp(−(ξ1 ∧ ξ2 ) jτ ) converges and is uniformly bounded with respect to τ . Let N → ∞ in above inequality. The ergodicity of both X n and X (tn ) indicates that  N −1 N −1 1  1  lim E[u(0, X n )] = lim E[ϕ(X n )] = ϕdμτ N →∞ N N →∞ N H n=0 n=0

2.4 Approximation of Invariant Measures via Ergodic Schemes

55

and N −1 N −1 1  1  E[u(tn , x)] = lim E[ϕ(X (tn ))] N →∞ N N →∞ N n=0 n=0 N −1  1  tn+1 = lim E[ϕ(X (tn ))]ds N →∞ N τ n=0 tn  N −1  tn+1 1  r 2 E[ϕ(X (s))]ds + O((1 + x )τ ) = lim N →∞ N τ tn n=0  = ϕdμ + O((1 + xr )τ ),

lim

H

where we have used the numerical integration and the polynomial growth of ∇ϕ, more precisely, ∇ϕ(θ (X n − X (s)) + X (s)) ≤ C(1 + xr ) for any θ ∈ (0, 1) and some r ∈ N. It then completes the proof.  For ergodic processes or chains, their invariant measures are usually unknown. The error between invariant measures depends heavily on the uniform boundedness of the exact and numerical solutions. We next give an example, in which the invariant measures for both the exact solution and the numerical one have explicit forms. In this circumstance, the error between invariant measures is immediately obtained through the error between their densities. Example 2.2 We still take the Ornstein–Uhlenbeck process with invariant measure μ = N (0, 1) in Example 1.1 as an instance. The Euler–Maruyama scheme is shown 2 to be ergodic with a unique invariant measure μτ = N (0, 2−τ ) in Example 2.1. Thus,     − 21

  (2−τ )x 2 1 2 x2   e(ϕ) =  ϕ(x) √ e− 4 e− 2 − dx  R  2 − τ 2π     1

 (2−τ )x 2 1  2−τ 2 τ x2  = √  ϕ(x)e− 4 e− 4 − dx ,  2 2π  R where  



  τ x 2 2 − τ  21   τ x2 1  τ   − 4   − exp ln 1 − − e  = exp −   2 4 2 2   2  τx 1  τ  . ≤  + ln 1 − 4 2 2  Hence, we finally get e(ϕ) = O(τ ) for any ϕ ∈ C∞ p (R) based on the fact that p − (2−τ )x 2 4 d x < ∞ for p ∈ N. Rx e

56

2 Invariant Measures for Stochastic Differential Equations

For SDEs with degenerate additive noise, such as the Langevin equation (2.6), and general numerical schemes in the form X n+1 = F(X n , n+1 W ), the following theorem gives the error between invariant measures. Theorem 2.11 (Theorem 7.3, [139]) For SDEs with degenerate additive noise and satisfying Assumptions 2.1 and 2.2, if the numerical solution of scheme X n+1 = F(X n , n+1 W ) satisfies assumptions in Proposition 2.2 and Assumption 2.2, then both the exact solution and the numerical one possess unique invariant measures μ and μτ , respectively. If, in addition, EX (tn ) − X n 2 ≤ C(1 + x2 )τ s for any n ∈ N with initial value X (0) = x, then there exists some constant ρ ∈ (0, 21 ) such that e(ϕ) ≤ Cτ sρ , ∀ ϕ ∈ Cb (H). We refer to [179] and references therein for the study of degenerate SDEs with multiplicative noises and the implicit Euler scheme. The author shows the ergodicity of the exact solution and the numerical one, and gives the error between invariant measures.

2.5 Approximation of the Ergodic Limit In some other circumstances, the average of a broad class of empirical functions ϕ with respect to the invariant measure μ, i.e., the ergodic limit ϕ¯ := H ϕdμ, may be of more importance than the behavior of the solution itself. To approximate the such that ϕdμ ergodic limit, one usually construct a sequence of measures μ N N → H ϕdμ in some sense as N → ∞, while the numerical solution need not to be ergodic H anymore. Let X denote the exact solution of (2.1), and {X n }n∈N be a numerical solution of some proper numerical scheme. Denote the auxiliary measures μ N , N ∈ N, such that  H

ϕdμ N :=

 N   N 1  1  τn ϕdδ X n = [τn ϕ(X n )] t N n=1 t N n=1 H

" with tn = nk=1 τk , τn = tn − tn−1 and δ X n being the Dirac measure centered on the point X n . When the step-size τn is decreasing, i.e., limn→∞ τn = 0 and limn→∞ tn = ∞, it has been shown that   ϕdμ N → ϕdμ =: ϕ, ¯ a.s. (2.35) H

H

2.5 Approximation of the Ergodic Limit

57

We refer to [130, 131, 142, 159] and references therein for more details. However, if τn ≡ τ , we have tn = nτ and  H

ϕdμ N =

N 1  ϕ(X n ), N n=1

which together with the local weak error, also yields (2.35) (see e.g. [140, 148]) taking advantage of the Poisson equation (see e.g. [160–162, 175]). This approach relies especially on the local error of the numerical schemes, and is also applicable to the case with decreasing step-size τn . This section contains a brief introduction of the approach mentioned above for (2.1) possessing a unique invariant measure μ under either uniform elliptic setting (see Theorem 2.3) or hypoelliptic setting (see Theorem 2.2). We first give the Poisson equation on H associated to (2.1) L Φ = ϕ − ϕ, ¯

(2.36)

where L is defined in (2.2) and the function ϕ − ϕ¯ is centered in the sense H (ϕ − ϕ)dμ ¯ = 0. The well-posedness of (2.36) has been studied in [160] for the elliptic setting and in [162] for the" degenerate case. However, to gain the convergence rate N ϕ(X n ) when approximate the ergodic limit ϕ, ¯ it is of the temporal average N1 n=1 essential for Φ to be regular enough. Thus, in the following, we assume that the state space is compact, namely, H = Td for simplicity, where Td denotes the torus in Rd . In this case, the regularity of Φ is stated in the following theorem. Theorem 2.12 (Theorem 4.1, [140]) Assume that (2.1) possesses a unique invariant measure under the elliptic setting (resp. hypoelliptic setting) with smooth coefficients b and σ . For any ϕ ∈ W p,∞ (H) (resp. ϕ ∈ W p+2,∞ (H)) with p ∈ N, there exists a unique solution Φ ∈ W p+2,∞ (H) to (2.36). Assume that a numerical scheme could be solved explicitly by X n+1 = X n + τ F(X n , τ ) + G(X n , τ )n+1 W

(2.37)

with increments n+1 W = W (tn+1 ) − W (tn ) and, in addition, F and G satisfy that b(x) − F(x, τ ) + σ (x) − G(x, τ ) F ≤ Cτ

(2.38)

for all x ∈ H, sufficiently small τ and some constant C. Note that there is an equivalent condition to (2.38), which reads that the local weak error between the exact solution and the numerical one is of order two, and is stated in the following proposition. Proposition 2.3 Consider (2.1) and (2.37) with smooth coefficients b and σ . For X (0) = X 0 and ϕ ∈ W4,∞ (H), the following two conditions are equivalent:

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2 Invariant Measures for Stochastic Differential Equations

(i) b(x) − F(x, τ ) + σ (x) − G(x, τ ) F ≤ Cτ , ∀ x ∈ H, (ii) |E[ϕ(X (τ ))] − E[ϕ(X 1 )]| = O(τ 2 ). Proof Based on Itô’s formula and the uniform boundedness of b, σ and ∇ p ϕ, p = 1, · · · , 4, on a compact set, we have 

τ

E[L ϕ(X (s))]ds,  τ s   = ϕ(x) + τ L ϕ(x) + E L 2 ϕ(X (r )) dr ds

E[ϕ(X (τ ))] = ϕ(x) +

0

0

0

= ϕ(x) + τ L ϕ(x) + O(τ 2 ).

(2.39)

According to the Taylor expansion, it shows E[ϕ(X 1 )]

! ! 1 = ϕ(x) + E ∇ϕ(x) (τ F + G1 W ) + E ∇ 2 ϕ(x) (τ F + G1 W )2 2 ! ! 1 3 1 4 3 +E ∇ ϕ(x) (τ F + G1 W ) + E ∇ ϕ(θ x + (1 − θ )X 1 ) (τ F + G1 W )4 3! 4!   τ2  2 1 2 2 ∇ ϕ(x)F 2 + ∇ 3 ϕ(x)F G 2 = ϕ(x) + τ ∇ϕ(x)F + ∇ ϕ(x)G + 2 2 ! τ3 3 1 ∇ ϕ(x)F 3 + E ∇ 4 ϕ(θ x + (1 − θ )X 1 ) (τ F + G1 W )4 + (2.40) 6 24

with F and G taking values at (x, τ ) and θ ∈ (0, 1). Here, we write ∇ p ϕ(x)(y1 , · · · , y p ) for the pth derivative evaluated in directions y j , j = 1, · · · , p, and ∇ p ϕ(x)y p for short if all the directions are the same. The difference between (2.39) and (2.40) yields E[ϕ(X (τ ))] − E[ϕ(X 1 )]    τ 2  1 =τ ∇ϕ(x)(b(x) − F) + ∇ 2 ϕ(x) σ (x)2 − G 2 − ∇ 2 ϕ(x)F 2 + ∇ 3 ϕ(x)F G 2 2 2 ! 1 τ3 3 ∇ ϕ(x)F 3 − E ∇ 4 ϕ(θ x + (1 − θ )X 1 ) (τ F + G1 W )4 + O(τ 2 ), − 6 24

from which one can get the equivalence of conditions (i) and (ii), since (i) could also imply the boundedness of F and G.  It is worth mentioning that most numerical schemes can not be expressed as (2.37), but as X n+1 = X n + K (X n , τ, n+1 W )

(2.41)

for some function K : H × (0, 1) × Rm → H, according to the well-posedness of the scheme. The condition (2.38) is not available anymore in this circumstance. Thus,

2.5 Approximation of the Ergodic Limit

59

for general schemes, condition (ii) in Proposition 2.3 is frequently used to obtain the approximate error of the ergodic limit. Theorem 2.13 For any ϕ ∈ W4,∞ (H), assume that the assumptions in Theorem 2.12 and condition (ii) in Proposition 2.3 hold for (2.1) and (2.41). Then  N −1 

 1   1   E[ϕ(X n )] − ϕ¯  ≤ C τ +  N  T n=0 for T = N τ and X 0 = X (0) = x. We refer to [140] for a similar version of this theorem, which is proved under the equivalent condition (2.38), and for a general version when the numerical scheme has local weak order p + 1 which is also introduced at the end of this section. The proof of Theorem 2.13 is given here for the readers’ convenience. Proof As (2.36) admits a unique solution Φ ∈ W4,∞ (H) according to Theorem 2.12, we derive that N −1 N −1 1  1  E[ϕ(X n )] − ϕ¯ = E[L Φ(X n )]. N n=0 N n=0

(2.42)

If we take X n as an initial value and denote the solution at time τ as X X n (τ ), then 

τ

E[Φ(X X n (τ ))] = E[Φ(X n )] + E



L Φ(X (s))ds  = E[Φ(X n )] + τ E[L Φ(X n )] + E 0

0 2

τ



s

 L Φ(X (r ))dr ds 2

0

= E[Φ(X n )] + τ E[L Φ(X n )] + O(τ ) as Φ ∈ W4,∞ (H). On the other hand, since E[ϕ(X (τ ))] − E[ϕ(X 1 )] = O(τ 2 ) for any initial value x ∈ H and any ϕ ∈ W4,∞ (H), it leads to E[Φ(X n+1 )] − E[Φ(X X n (τ ))] = O(τ 2 ). Thus, we have N −1 N −1 1  1  E[Φ(X X n (τ )) − Φ(X n )] + O(τ ) E[L Φ(X n )] = N n=0 N n=0 τ

=

N −1 1  E[Φ(X n+1 ) − Φ(X n )] 1 + O(τ ) = (E[Φ(X N )] − Φ(x)) + O(τ ), N n=0 τ T

60

2 Invariant Measures for Stochastic Differential Equations

which, together with the uniform boundedness of Φ and (2.42), completes the proof.  Based on the results above, one can get the approximate error of the ergodic limit, which contains two parts intuitively: the weak error between the numerical solution and the exact one, and the error between the temporal average and the spatial one. More precisely, it can be separated into the following two parts N −1 N −1  1  tn+1  1  E[ϕ(X n )] − E[ϕ(X (t))] dt E[ϕ(X n )] − ϕ¯ = N n=0 T n=0 tn   1 T + E[ϕ(X (t))]dt − ϕdμ. T 0 H

This approach is also applicable for high order cases, stated as follows. Assumption 2.3 For all ϕ ∈ W2( p+2),∞ (H) and sufficiently small τ |E[ϕ(X (τ ))] − E[ϕ(X 1 )]| ≤ Cτ p+1 , where the constant C depends on the initial value X (0) = X 0 = x, and is uniform over all ϕ with ϕW2( p+2),∞ (H) ≤ 1. Theorem 2.14 (Theorem 5.6, [140]) Assume that (2.1) and the numerical solution to (2.41) satisfy the assumptions in Theorem 2.12 and Assumption 2.3. Then for any ϕ ∈ W2( p+1),∞ (H) with ϕW2( p+1),∞ (H) ≤ 1 and X 0 = X (0) = x,  N −1 

 1   1   p E[ϕ(X n )] − ϕ¯  ≤ C τ +  N  T n=0 with T = N τ . We refer to [140] for the proof of above theorem, which can also be found in Chap. 6 for a finite dimensional approximation of stochastic Schrödinger equations.

Summary This chapter mainly focuses on the existence and uniqueness of invariant measures for stochastic ordinary differential equations and numerical approximations of invariant measures. Non-degenerate SDEs and an important kind of degenerate SDEs– stochastic Langevin equations—are taken as the keystone to illustrate the procedure of approximating invariant measures via numerical schemes. The stochastic Langevin equation introduced in Sect. 2.3 admits a unique invariant Boltzmann–Gibbs measure, which gives the probability that a system will be in a

2.5 Approximation of the Ergodic Limit

61

certain state and shows that states with lower energy will always have a higher probability of being occupied than the states with higher energy. We also refer to [27, 156, 166] for the study of Boltzmann–Gibbs measures for deterministic Korteweg– de Vries equations, and refer to [27, 28, 157, 181] for the study of deterministic nonlinear Schrödinger equations, which is also briefly introduced in Sect. 3.2. Moreover, as a stochastic dissipative Hamiltonian system, the stochastic Langevin equation can be transformed to an equivalent autonomous stochastic Hamiltonian system. The generating function is then employed to construct conformal symplectic schemes for the stochastic Langevin equation, and the modified equation technique (inspired by the backward error analysis) is adopted to improve the accuracy of the proposed schemes and to release the simulation of the multiple integrals. There is plenty of work related to the construction of high order schemes for both deterministic and stochastic systems. We refer to [41, 122, 141, 164, 176, 195] for high order symplectic Runge–Kutta methods, to [7, 8, 83, 154, 185, 186, 188] for symplectic schemes constructed through generating functions, and to [1, 2, 46, 69, 126, 127, 163, 171, 180] for high order integrators based on modified equations.

Chapter 3

Invariant Measures for Stochastic Nonlinear Schrödinger Equations

This chapter focuses on the existence and uniqueness of invariant measures for stochastic nonlinear Schrödinger equations, and recalls several results related to the well-posedness and continuous dependence on the initial value as the starting point of this chapter. Section 3.1 introduces some notations used throughout the following chapters. In addition, we recall some fundamental conserved quantities of deterministic nonlinear Schrödinger equations, which are essential in proving the well-posedness of Schrödinger equations. Section 3.2 gives the expression of an invariant measure for the deterministic nonlinear Schrödinger equation under periodic setting, utilizing a Fourier finite dimensional truncation of the considered model. Section 3.3 introduces the existence and uniqueness of solutions for stochastic nonlinear Schrödinger equations driven by additive or multiplicative noises. Section 3.4 shows the pathwise continuous dependence on the initial data, which indicates that the transition semigroup generated by the solution is Markovian and Feller. These results form the basis of the study on invariant measures for stochastic Schrödinger equations. Sections 3.5 and 3.6 are devoted to the study of invariant measures and ergodicity for stochastic Schrödinger equations with weak damping in different dimensions.

3.1 Preliminaries We first introduce some notations used throughout the following chapters, and recall some classical results for deterministic nonlinear Schrödinger equations. • Let (Ω, F , P, {Ft }t≥0 ) be a filtered probability space. © Springer Nature Singapore Pte Ltd. 2019 J. Hong and X. Wang, Invariant Measures for Stochastic Nonlinear Schrödinger Equations, Lecture Notes in Mathematics 2251, https://doi.org/10.1007/978-981-32-9069-3_3

63

64

3 Invariant Measures for Stochastic Nonlinear Schrödinger Equations

• Let {βk }k∈N be a family of mutually independent R-valued standard Brownian motions on (Ω, F , P, {Ft }t≥0 ). • Denote by L p (O) the classical Lebesgue space of C-valued functions for p ≥ 1. When p = 2, L2 (O) denotes the space of C-valued square integrable functions with inner product  (u, v) =

u(x)v(x)d x O

for u, v ∈ L2 (O). In particular, L2 (O; R) denotes the space of R-valued square integrable functions. Domain O ⊂ Rd , d ≥ 1, may be different in the following sections, and will be pointed out at the beginning of each section. • Denote by Hs (O) := Ws,2 (O) the classical Sobolev space for s ≥ 0, which is a Hilbert space endowed with inner product (u, v)Hs (O ) =

   O |α|≤s

 D α u D α v d x, u, v ∈ Hs (O).

• Denote by L2s := L2 (L2 (O); Hs (O)), s ≥ 0, the space of Hilbert–Schmidt operators from L2 (O) to Hs (O) equipped with the norm L 2s =

 

 21 ek 2Hs (O )

,  ∈ L2s .

k∈N

• For a Banach space B, the space of γ -radonifying operators from L2 (O) to B is denoted by R(L2 (O); B) with norm

 R(L2 (O );B)

⎛

2 ⎞ 21





= ⎝E˜

γk ek ⎠ ,  ∈ R(L2 (O); B),



k∈N

B

where {γk }k∈N is any sequence of mutually independent R-valued normal random ˜ and the norm is independent of ˜ F˜ , P), variables on another probability space (Ω, {γk }k∈N and {ek }k∈N . In particular, if B is a Hilbert space, then R(L2 (O); B) = L2 (L2 (O); B). The Schrödinger equation, originating in quantum mechanics, was first presented by E. Schrödinger in 1926. Physicists investigate the molecular structure and atomic structure of actual substance, and even subatomic and macroscopic systems, via solving time dependent evolution problems for the Schrödinger equation. It is central to the applications of quantum mechanics, statistical mechanics, plasma physics, as well as nonlinear optics. Recall that the classical nonlinear Schrödinger equation

3.1 Preliminaries

65

du = iΔudt + iλ|u|2σ udt, u(0, x) = u 0 (x), x ∈ O ⊂ Rd ,

(3.1)

has an infinite number of invariant quantities. Among them, the following conservation laws are frequently considered in practical problems (see [29, 34, 81, 117] and references therein). Charge conservation law. The charge of (3.1) is defined as  M(u(t)) := u(t)2L2 (O ) =

O

|u(t, x)|2 d x,

in which |u(x, t)|2 represents the density for the particle to appear in state x at time t in quantum mechanics. The law of conservation of the charge reads M(u(t)) = M(u 0 ) for (3.1) under Dirichlet boundary condition. Energy conservation law. The Hamiltonian 1 H (u(t)) := 2



λ |∇u(t, x)| d x − 2σ + 2



2

O

O

|u(t, x)|2σ +2 d x

is shown to be preserved for any time when the solution is well-posed and sufficiently smooth, i.e., H (u(t)) = H (u 0 ).

3.2 Invariant Measures for Deterministic Nonlinear Schrödinger Equations In this section, we consider the invariant distributions of the following one dimensional nonlinear Schrödinger equation under periodic setting: ⎧ 2σ ⎪ ⎨ u˙ = iΔu + iλ|u| u, u(t, 0) = u(t, L), t ∈ R, ⎪ ⎩ u(0, x) = u 0 (x), x ∈ (0, L).

(3.2)

Denote L p := L p (0, L). We obtain from above that its Hamiltonian reads H (u) =

1 λ ∇u2L2 − uL2σ2σ+2 +2 , 2 2σ + 2

based on which the authors in [132] define a formal and unnormalized Gibbs measure

66

3 Invariant Measures for Stochastic Nonlinear Schrödinger Equations

e−β H (φ)



dφ(x)

(3.3)

x∈[0,L]

⎤ ⎡     λβ β p = exp φL p ⎣exp − ∇φ2L2 dφ(x)⎦ 2σ + 2 2 x∈[0,L] 

with dφ denoting the Lebesgue measure in the complex plane. The measure above is ill-defined since there is no analogue of Lebesgue measure on infinite dimensional Banach spaces. Even though it is possible to define Gaussian measure on infinite dimensional spaces, it can not be defined through a density in that case. The authors in [132] then modify the definition of this formal measure as  exp

 λβ p φL p 1φ φ2 ≤K  dμβ (φ) 2σ + 2 L2

with μβ a Wiener measure on an appropriate statistical ensemble, such that it is well-defined and normalizable under the global well-posedness assumption of the equation. Since the Hamiltonian is not bounded below, the identity function based on the truncation of the L2 -norm of φ is added to ensure that the measure is normalizable. Based on these investigations, the author in [27] proves further that the measure is invariant under the flow of (3.2) via a finite dimensional approximation of equation. More precisely, one can consider a finite dimensional approximation

  u˙ N = iΔu N + iλπ N |u N |2σ u N , u N (0) = π N u 0

(3.4)

with π N being the projection operator such that π N u(x, t) =



an (t)e2πinx .

|n|≤N

For the finite dimensional equation (3.4), it possesses the discrete charge and energy conservation laws: ⎛ M N (a(t)) := ⎝



⎞ 21 |an (t)|2 ⎠ = M N (a(0)),

|n|≤N



λ n 2 |an (t)|2 − HN (a(t)) := 2π 2 2σ + 2 |n|≤N



L 0

 2σ +2     2πinx   an e d x,   |n|≤N 

where a(t) := (an )|n|≤N ∈ C2N +1 . Furthermore, (3.4) is also a Hamiltonian system

3.2 Invariant Measures for Deterministic Nonlinear Schrödinger Equations

a˙ = −i

67

∂ HN ∂a

with a denoting the conjugate of the complex value a, or equivalently, p˙ N =

∂ Hˆ N ∂ Hˆ N , q˙ N = − , ∂q N ∂ pN

  4N +2 where ( p  , a = p N + iq N and Hˆ N ( p N , q N ) = HN (a). For this finite N , qN ) ∈ R dimensional case, we can also define the measure similar to (3.3):

dμ N =e−HN (a) da  2σ +2 ⎞ ⎛   L   λ 2πinx   = exp ⎝ a e dx⎠ · n   2σ + 2 0 |n|≤N  ⎞ ⎤ ⎛ ⎡  ⎣da0 ⊗ exp ⎝−2π 2 n 2 |an (t)|2 ⎠ d(an )|n|≤N ,n =0 ⎦ |n|≤N ,n =0

 2σ +2 ⎞   L   λ 2πinx   ⎝ =: exp an e d x ⎠ [da0 ⊗ dρ N ] ,   2σ + 2 0 |n|≤N  ⎛

where ρ N is a measure on C2N . associated to a 2N -dimensional Remark 3.1 The measure ρ N is a Gaussian   1 measure ξn (ω) |n|≤N ,n =0 and of course can be norGaussian random variable X (ω) := 2πn malizable, where {ξn }|n|≤N ,n =0 is a family of independent normal random variables. In fact, for any Borel set A ∈ B(C2N ), 

 |an |2 P(X (ω) ∈ A) = exp −  2 d(an )|n|≤N ,n =0 1 A |n|≤N ,n =0 2 2πn ⎞ ⎛   = exp ⎝−2π 2 n 2 |an (t)|2 ⎠ d(an )|n|≤N ,n =0 = ρ N (A). 



A

|n|≤N ,n =0

The measure μ N is well-defined (see [27, 132]) when choosing the space     Ω N ,K := a = (an )|n|≤N a ≤ K as the statistical ensemble, where  ·  denotes the Euclidean norm in the corresponding space. When considering the limits ρ and μ of ρ N and μ N , respectively, one can derive from Lemma 3.10 in [27] that ρ and μ are well-defined for σ ≤ 2. Moreover, Eq. (3.2)

68

3 Invariant Measures for Stochastic Nonlinear Schrödinger Equations

is globally well-posed μ-almost everywhere and admits an invariant measure with proper σ . Theorem 3.1 ([27]) For σ ≤ 2, the initial value problem (3.2) is globally wellposed. If 1 ≤ σ ≤ 2, the measure μ is invariant under the flow of (3.2). For the study of invariant measures for deterministic nonlinear Schrödinger equation in high dimensions, we refer to [26, 27, 30, 132] and references therein for more details. In the following, we will turn our attention to the study of stochastic nonlinear Schrödinger equations, including their well-posedness, invariant measures, geometric structures (see Chap. 4) and numerical approximations (see Chaps. 4, 5 and 6).

3.3 Well-Posedness of Stochastic Nonlinear Schrödinger Equations In this section, the local and global existence and uniqueness for the solution of stochastic nonlinear Schrödinger equations on O = Rd are introduced, based on the evolution of charge and energy of the solution. Higher regularity of the solution is obtained for the one dimensional case O = (0, 1). Let the noise in the considered stochastic nonlinear Schrödinger equation be a Cvalued Wiener process for the additive noise case, and be a R-valued Wiener process for the multiplicative noise case. The precise assumption on the noise will be given separately for each case.

3.3.1 The Additive Noise Case We first consider the nonlinear Schrödinger equation perturbed by an additive noise

  du = i Δu + λ|u|2σ u dt + dW (t), u(0, x) = u 0 (x), x ∈ Rd ,

(3.5)

where W is a Q-Wiener process on L2 (Rd ) with symmetric and positive definite covariance operator Q. More precisely, W (t) has the following Karhunen–Loève expansion ∞  1 W (t) = Q 2 ek βk (t) k=1

with {ek }k∈N being an orthonormal basis for L2 (Rd ), and {βk }k∈N being a sequence of R-valued mutually independent and identically distributed Brownian motions.

3.3 Well-Posedness of Stochastic Nonlinear Schrödinger Equations

69

We give the well-posedness of the mild solution of the considered equation in this section. Process u is called a mild solution to (3.5) on [0, T ] if for any t ∈ [0, T ] it satisfies  t  t S(t − s)|u(s)|2σ u(s)ds + S(t − s)dW (s) (3.6) u(t) = S(t)u 0 + i 0

0

almost surely and each integral in the right hand side of above equation is welldefined, where S(t) = eitΔ is the unitary group on Hs (Rd ) generated by the linear equation u˙ = iΔu. The local existence of the solution of (3.6) is given first by defining a stopping time τ ∗ (u 0 ) such that τ ∗ (u 0 ) = +∞ or

lim u(t)H1 (Rd ) = +∞

(3.7)

t→τ ∗ (u 0 )

holds almost surely. Theorem 3.2 (Theorem 3.1, [64]) Assume that u 0 ∈ H1 (Rd ) is F0 measurable, 1 Q 2 ∈ L21 and that ⎧ ⎪ ⎨ [0, +∞) if d = 1, 2,   d σ ∈ I := 2 ⎪ if d ≥ 3. ⎩ 0, d −2 Then there exists a unique solution u ∈ H1 (Rd ) on [0, τ ∗ (u 0 )) such that u(0) = u 0 . To derive the global existence of the solution, that is τ ∗ (u 0 ) = +∞ almost surely, a priori estimates for the charge and energy of (3.5) are required. Different from the deterministic case (3.1), the charge and energy of (3.5) are not conserved any more, but satisfy the following evolution principles. Proposition 3.1 ([64]) Under the assumptions in Theorem 3.2, the charge and the energy of the solution of (3.5) satisfy  M(u(τ )) = M(u 0 ) + 2

 k∈N

0

τ

 1 2

1

(u(s), Q ek )dβk (s) + τ Q 2 2L 0 2

and  H (u(τ )) =H (u 0 ) −  −

 λ τ 2 0 k∈N

  1 1 τ (Δu + λ|u|2σ u, d W (t)) + ∇(Q 2 ek )2L2 (Rd ) ds 2 0 k∈N 0 



2

2 1 1

σ 2



|u| Q ek 2 d + 2σ |u|σ −1 [u Q 2 ek ] 2 d ds. τ

L (R )

L (R )

Moreover, for any p ∈ N, the uniform estimate for the pth moment of M(u(t)) holds: there exists a constant C p > 0 such that

70

3 Invariant Measures for Stochastic Nonlinear Schrödinger Equations

 E

 ! sup M p (u(t)) ≤ C p E M p (u 0 ) .

t∈[0,τ ]

Based on the a priori estimate above, the global well-posedness of the solution is derived by showing the uniform boundedness of the second moment of u(t)H1 (Rd ) , i.e.,   E sup u(t)2H1 (Rd ) ≤ C (T, Q, u 0 ) t≤τ

with any given T > 0 and stopping time τ < inf{T, τ ∗ (u 0 )}. Theorem 3.3 (Theorem 3.4, [64]) Assume that the assumptions in Theorem 3.2 hold, and that σ < d2 or λ = −1. Then the solution given in Theorem 3.2 is global, i.e., τ ∗ (u 0 ) = +∞ almost surely.

3.3.2 The Multiplicative Noise Case For nonlinear Schrödinger equation perturbed by the linear multiplicative noise

  du = i Δu + λ|u|2σ u dt + iu ◦ dW (t), u(0, x) = u 0 (x), x ∈ Rd ,

the driven noise W (t) =

∞ "

(3.8)

1

Q 2 ek βk (t) is a Q-Wiener process on L2 (Rd ; R) with a

k=0

symmetric and positive definite covariance operator Q. Here, {ek }k∈N is an orthonormal basis in L2 (Rd ; R), and {βk }k∈N as above. The stochastic integral arising on the right hand side of (3.8) holds in the Stratonovich sense, and the solution possesses the charge conservation law almost surely. In this subsection, the global existence and uniqueness of the mild solution of (3.8) in L2 (Rd ), H1 (Rd ) and Hs (0, 1) for s ≥ 2 are introduced. 1

Theorem 3.4 (Theorem 2.1, [61]) Assume that Q 2 ∈ L20 (Rd ) ∩ R(L2 (Rd ), L2+δ (Rd )) for some δ > 2(d − 1). Let parameters p, r and ρ satisfy $ #  ⎧ 2 ⎪ ⎪ + 1 ≤ p if d = 1, max 2σ + 2, 2 ⎨ δ $ #  ⎪ 2d 2 ⎪ ⎩ max 2σ + 2, 2 +1 ≤p≤ if d ≥ 2, δ d −1 2 =d r



1 1 − 2 p



#  and ρ ≥ max r, (2σ + 2)

$ 4σ +1 . 2 − dσ

3.3 Well-Posedness of Stochastic Nonlinear Schrödinger Equations

71

Then for any F0 -measurable initial datum u 0 ∈ Lρ (Ω; L2 (Rd )) and T > 0, there is a unique solution u ∈ Lρ (Ω; C([0, T ]; L2 (Rd ))) ∩ L1 (Ω; Lr ([0, T ]; L p (Rd ))) of (3.8), which possesses the charge conservation law almost surely M(u(t)) = M(u 0 ), ∀ t ≥ 0. To study the well-posedness of (3.8) in energy space H1 (Rd ), similar to the additive noise case, local existence and uniqueness is given first utilizing the stopping time τ ∗ (u 0 ) defined through (3.7). Theorem 3.5 (Theorem 4.1, [64]) Assume that u 0 ∈ H1 (Rd ) is F0 -measurable, 1 Q 2 ∈ L21 ∩ R(L2 (Rd ); W1,α (Rd )) with α > 2d and that ⎧ (0, +∞) if d = 1, 2, ⎪ ⎪ ⎪ ⎨ (0, 2) if d = 3, σ ∈ I d :=     ⎪ 1 1 2 ⎪ ⎪ ⎩ ∪ 0, if d ≥ 4. , 2 d −2 d −1 Then there exists a unique solution u ∈ C([0, τ ]; H1 (Rd )) ∩ Lr ([0, τ ]; W1, p (Rd )) almost surely for any τ < τ ∗ (u 0 ) and some r ≥ 2 and p satisfying where stopping time τ ∗ (u 0 ) is defined through (3.7).

2 r

=d

%

1 2

−

1 p

& ,

The energy for the multiplicative noise case is not preserved, but satisfies the following evolution principle. Proposition 3.2 (Proposition 4.5, [64]) Assume that the assumptions in Theorem 3.5 hold. Then the solution u given in Theorem 3.5 satisfies  H (u(τ )) =H (u 0 ) −  +

1 2

∞  τ  k=0

0

∞   k=0

τ

% & 1 ∇u, u∇(Q 2 ek ) dβk (s)



0 1

u∇(Q 2 ek )2L2 (Rd ) ds,

for any stopping time τ such that τ < τ ∗ (u 0 ) almost surely, where H (u) is the same as that in Sect. 3.1. The global well-posedness of (3.8) in H1 (Rd ) is obtained based on the uniform estimate of the energy H (u(t))

72

3 Invariant Measures for Stochastic Nonlinear Schrödinger Equations

  E sup u(t)2H1 (Rd ) ≤ C(T, Q, u 0 ) t≤τ

as stated in the additive noise case. Theorem 3.6 (Theorem 4.6, [64]) Assume that the assumptions in Theorem 3.5 hold and that either σ < d2 or λ = −1. Then the solution given in Theorem 3.5 is global, i.e., τ ∗ (u 0 ) = +∞ almost surely. The global well-posedness result above is valid for the equation with a subcritical or defocusing nonlinearity. Here, the subcritical condition is given as σ < d2 , which 2 proposed in is named as a comparison with the Hs (Rd )-critical condition σ < d−2s the deterministic case (see e.g. [27, 44, 91]). We refer the reader to [15, 16] for the existence and uniqueness of strong solutions in L2 (Rd ) and H1 (Rd ) of stochastic nonlinear Schrödinger equations with multiplicative noises, including both conservative and nonconservative cases. For stochastic nonlinear Schrödinger equation (3.8) on bounded domain O with homogenous Dirichlet boundary condition, i.e., u(t, x) = 0 on (0, T ] × ∂O for some T > 0, the uniform boundedness of the solution in the energy space is also obtained for the defocusing case. Proposition 3.3 (Corollary 1, [49]) Let p ≥ 1, λ = −1, σ = 1, O ⊂ Rd be a bounded Lipschitz domain. The initial value u 0 is F0 measurable with homogenous Dirichlet boundary condition and E[H p (u 0 )] < ∞. Assume that u is the solution of (3.8) in domain (0, T ] × O with homogenous Dirichlet boundary condition. Then there exists a constant C = C(Q, T, u 0 , p) such that  E

% & 2p 4p sup ∇u(t)L2 (O ) + u(t)L4 (O ) ≤ C.

0≤t≤T

In particular, for the one dimensional case O = (0, 1), uniform boundedness of the solution in H1 (0, 1) and H2 (0, 1) is derived for both focusing and defocusing cases. Proposition 3.4 ([49, 55]) Let O = (0, 1), σ = 1 and p ≥ 1. Assume that the initial value u 0 ∈ H2 (0, 1) is F0 measurable with homogenous Dirichlet boundary 1 condition and Q 2 ∈ L22 . Then there exists a constant C = C(Q, T, u 0 , p) such that  E

 2p sup u(t)H2 (0,1) ≤ C.

0≤t≤T

The regularity of the solution can be further improved for the one dimensional case O = (0, 1) utilizing the Lyapunov functional   f (u) := ∇ s u2L2 − λ (−Δ)s−1 u, |u|2 u , u ∈ Hs (0, 1) for s ≥ 2. We refer to [55] for more details.

3.4 Continuous Dependence of the Solutions on the Initial Data

73

3.4 Continuous Dependence of the Solutions on the Initial Data The continuous dependence of the solutions on the initial data plays an important role in proving the existence of invariant measures. In fact, the Krylov–Bogoliubov theorem (see Theorem 1.2) is frequently used in proving the existence of invariant measures for time homogenous Markov processes (see e.g. [31, 76]). It requires the Feller property of the semigroup generated through the solution, which is a consequence of the continuous dependence on the initial data (see e.g. [60, 76]). The continuous ' t dependence on the initial data and noise is established in [64] by setting z(t) = 0 S(t − s)dW (s) and v(t) = u(t) − z(t). If v(t) satisfies  v(t) = S(t)u 0 + iλ

t

S(t − s)|v(s) + z(s)|2σ (v(s) + z(s))ds,

(3.9)

0

then u(t) is the mild solution of (3.5). We denote by v(z, u 0 , ·) the solution of (3.9) if exists. Theorem 3.7 (Proposition 3.5, [64]) Assume that u ∗0 ∈ H1 (Rd ) z ∗ ∈ C([0, T ]; H1 (Rd )) ∩ Lr ([0, T ]; W1,2σ +2 (Rd )) with r = 4(σdσ+1) and solution v(z ∗ , u ∗0 , ·) to (3.9) exists in C([0, T ]; H1 (Rd )). Then there exist neighborhoods Bz ∗ ⊂ C([0, T ]; H1 (Rd )) ∩ Lr ([0, T ]; W1,2σ +2 (Rd )) of z ∗ and Bu ∗0 ⊂ C([0, T ]; H1 (Rd )) of u ∗0 such that for arbitrary (z, u 0 ) ∈ Bz ∗ × Bu ∗0 there exists a unique solution v(z, u 0 , ·) ∈ C([0, T ]; H1 (Rd )) ∩ Lr ([0, T ]; W1,2σ +2 (Rd )) to (3.9). In addition, the mapping f : Bz ∗ × Bu ∗0 → C([0, T ]; H1 (Rd )) (z, u 0 ) → v(z, u 0 , ·) is continuous. Based on above results, one can get the large deviation results: for ε > 0, the solution u ε of the following equation   du ε = i Δu ε + λ|u ε |2σ u ε dt + εdW (t) converges to the solution u 0 of the deterministic equation (ε = 0) in C([0, T ]; H1 (Rd )) almost surely as ε → 0. More details can be found in [63, 64].

74

3 Invariant Measures for Stochastic Nonlinear Schrödinger Equations

For the multiplicative noise case with O = (0, 1) and σ = 1, the continuous dependence of solutions to (3.8) on initial data and noises is derived by showing the exponential integrability of the solution. 1

Lemma 3.1 ([54, 55]) Let O = (0, 1), σ = 1, q ≥ 1 and Q 2 ∈ L22 . Assume that the initial value u 0 is F0 measurable and satisfies 

!



E exp(H (u 0 )) + E exp

u 0 6L2 (0,1) 2

 + 4q T 2

2

u 0 2L2 (0,1)

exp(C(u 0 ))

0, where 1

C(u 0 ) = 2L 2 T Q 2 2L 2 u 0 2L2 (0,1) 2

and L is a constant such that  f L∞ (0,1) ≤ L f H1 (0,1) for any f ∈ H1 (0, 1). Then there exists a constant C = C(Q, T, u 0 , q) such that 

  E exp q

T

0

 u 0 L2 (0,1) ∇u(t)2L2 (0,1) dt

≤ C.

Exponential integrability of exact and numerical solutions is important in the numerical analysis to obtain the strong convergence rate of numerical schemes especially when the equation has a non-globally Lipschitz continuous nonlinearity (see e.g. [54, 55, 112]). As an application of the exponential integrability, the continuous dependence on both initial value and noises are given in [55]. Theorem 3.8 (Corollary 3.1, [55]) Assume that the assumptions in Lemma 3.1 hold for two different initial values u 0 and v0 with q = 4 p for p = 2 or p ≥ 4. Then there exists a constant C = C(T, Q, p, u 0 , v0 ) such that  1p

  E

sup u(t) −

t∈[0,T ]

p v(t)L2 (0,1)

)& 21p % ( 2p ≤ C E u 0 − v0 L2 (0,1) ,

where u and v denote the solution of (3.8) with initial values u 0 and v0 respectively. Furthermore, the large deviation result also holds for the multiplicative noise case. For ε > 0, the solution u ε of the following equation du ε = i(Δu ε + λ|u ε |2 u ε )dt + iεu ε ◦ dW (t) satisfies

 1p

  E

sup u ε (t) −

t∈[0,T ]

p u 0 (t)L2 (0,1)

≤ Cε

under the assumptions in Theorem 3.8, which is proved in [55].

3.4 Continuous Dependence of the Solutions on the Initial Data

75

When studying the microscopic dynamical trajectory of an individual macroscopic system, it is more convenient to study appropriate ensembles in the state space instead. It then turns our focus on the study of invariant measures for the considered system. The following sections are devoted to showing the invariant distributions of stochastic Schrödinger equations with damping (see [70, 76, 109]).

3.5 Stochastic Linear Schrödinger Equation with Weak Damping This section is devoted to studying the distribution of the following linear equation 1

du = (iΔu − αu + iλu)dt + Q 2 dW

(3.10)

with α > 0, λ ∈ R, and operator Q commuting with Δ. Moreover, W is a cylindrical Wiener process on L2 (O) with O = (0, 1) such that W =

∞ 

em βm

m=1

with {em }m∈N being an orthogonal basis for L2 (O; R) and {βm }m∈N being a sequence of mutually independent and identically distributed C-valued Brownian motions. Rewriting above equation through its components u m := u, em , we obtain du m = (−λαm + iλ)u m dt +

∞  1 Q 2 ei , em dβi , m ∈ N. i=1

Its solution is given by an Ornstein–Uhlenbeck process u (t) = e m

(−λαm +iλ)t m

u (0) +

∞   i=1

t

α

e(−λm +iλ)(t−s) Q 2 ei , em dβi (s), 1

0

where u m (0) = u 0 , em  and λαm are the eigenvalues for the linear operator −iΔ + α I d with I d being the identity operator. Note that {u m (t)}t≥0 satisfies a complex Gaussian distribution N (m, C, R) defined by its mean m, covariance C and relation R:  ! !  α m u m (t) :=E u m (t) = e(−λm +iλ)t m u m (0) ,     2   1 − e−2αt 1 C u m (t) :=E u m (t) − m u m (t)  = e−2αt C u m (0) + Q 2 em 2 , α    2    α R u m (t) :=E u m (t) − m u m (t) = e2(−λm +iλ)t R u m (0) .

76

3 Invariant Measures for Stochastic Nonlinear Schrödinger Equations

m m m We use the notation μm t := N (m(u (t)), C(u (t)), R(u (t))) for simplicity.

Remark 3.2 We consider a one-dimensional C-valued Gaussian random variable Z = a + ib with a and b being two R-valued Gaussian random variables. If its relation vanishes, i.e., R(Z ) = E|a − Ea|2 − E|b − Eb|2 + 2i(E[ab] − EaEb) = 0, it implies E|a − Ea|2 = E|b − Eb|2 and E[ab] = EaEb. Since a and b are both Gaussian, we obtain equivalently that a and b are independent with the same covariance. Remark 3.3 The characteristic function of a one-dimensional complex Gaussian variable Z with distribution ν = N (m, C, R) reads (see e.g. [6])  exp{i(cz)}ν(dz) νˆ (c) :=E[exp{i(cZ )}] = C $ # 1 = exp i(cm) − (cCc + (cRc)) , c ∈ C. 4 It can be generalized for the infinite dimensional case utilizing inner product in H : $ # 1 ν(w) ˆ := exp iw, m − (Cw, w + Rw, w) , w ∈ H. 4 Hence, we get that the unique invariant measure of (3.10) is a complex Gaussian distribution, which is stated in the following theorem. We refer to [70, 76] and references therein for the existence of invariant measures for the nonlinear case, and refer to [32, 60] and references therein for other types of SPDEs. Theorem 3.9 Assume that Q is a nonnegative and symmetric trace operator. The solution u in (3.10) possesses a unique invariant measure  μ∞ = N

 1 0, Q, 0 . α

Proof Based on Remark 3.2, we define 1

Q 2 em  (ξm + irm ) √ 2α

um ∞ =

with {ξm , rm }m∈N being independent standard R-valued normal random variables, i.e., ξm , rm ∼ N (0, 1). Apparently,  1 Q 2 em 2 , 0 =: μm 0, ∞. α

 um ∞ ∼N

3.5 Stochastic Linear Schrödinger Equation with Weak Damping

77

We claim that the following random variable has the distribution μ∞ : ∞ 

u ∞ :=

um ∞ em =

m=1

1 ∞  Q 2 em  (ξm + irm )em . √ 2α m=1

" m Compared with u(t) = ∞ m=1 u (t)em , it then suffices to show that the distribution m m m μt of u (t) converges to μ∞ . As a result of Remark 3.3, the characteristic function of μm t is # !    1  α α (c) = exp i(ce(−λm +iλ)t E u m (0) ) −  e2(−λm +iλ)t R u m (0) c2 μˆ m t 4 $  −2αt 1 −2αt  m  1 − e 1 2 2 2 − e Q em  |c| C u (0) + 4 α 1 2

Q em  |c|2 } = μˆ m and μˆ m t (c) → exp{− ∞ (c). 4α 2



3.6 Stochastic Nonlinear Schrödinger Equation with Weak Damping In this section, we consider the stochastic damped nonlinear Schrödinger equation in the following form

du =(iΔu − αu + iλ|u|2σ u)dt + dW u(0) =u 0 , x ∈ O

(3.11)

with α > 0 and W being a Q-Wiener process on L2 (O) with a symmetric and positive definite covariance operator Q such that 1

Q 2 ek =

√

ηk ek , ηk > 0, k ∈ N.

We still denote the linear operator Aα := −iΔ + α I d.

3.6.1 One Dimensional Case Let O = (0, 1). Note that the real parts of eigenvalues of the operator −Aα are negative. This model is called a weakly damped stochastic nonlinear Schrödinger equation in [70]. This establishes the ergodicity and even polynomial mixing for (3.11) with a more general assumption: there exists some N∗ ∈ N+ such that ηk > 0

78

3 Invariant Measures for Stochastic Nonlinear Schrödinger Equations

for all k ≤ N∗ . The geometric structure and numerical approximations for this model are considered in Sect. 4.5 and Chap. 5, respectively. Let {Pt }t≥0 be the Markov semigroup generated by the solution u of the considered model. 1

Theorem 3.10 (Theorem 1.1, [70]) Assume that Q 2 L 23 < ∞. There exists a ˙ 1 (O) with unique stationary probability measure μ of {Pt }t≥0 on H  C p :=

2p

˙ 1 (O ) H

y1 dμ(y) < ∞

for any p ∈ N+ . Moreover, the measures converge to equilibrium with polynomial speed at any order, i.e., Pt ν − μW ≤ C˜ p (1 + t)− p (1 + C1 ) ˙ 1 (O)). for some C˜ p > 0 and any ν ∈ P(H Here,  · W is a Wasserstein type norm, and we refer to [70] for more details.

3.6.2 High Dimensional Case Let O = Rd . The existence of invariant measures and ergodic measures is established in H1 (O) by [76]. 1

Theorem 3.11 Assume that u 0 ∈ H1 (O) is F0 -measurable, Q 2 ∈ L21 and that

σ ∈ Iα,d

⎧ 2 ⎪ ⎪ [0, ) if λ = 1, ⎪ ⎪ d ⎨ := [0, +∞) if λ = −1, d = 1, 2, ⎪ ⎪ ⎪ ⎪ ⎩ [0, 2 ) if λ = −1, d ≥ 3. d −2

Then Eq. (3.11) possesses an invariant measure. We also refer to [27, 28, 30, 70, 76, 121, 155] for more details on invariant measures and ergodicity for Schrödinger equations in both the deterministic and the stochastic case. It is not difficult to find out that the ergodicity of (3.11) can be obtained based on the above results. Thus, the temporal average converge to the ergodic limit almost surely as a result of the mixing property:  lim E[ f (u(t, u 0 ))] =

t→∞

˙ 1 (O ) H

f (y)dμ(y)

3.6 Stochastic Nonlinear Schrödinger Equation with Weak Damping

79

with f ∈ Cb and u(t, u 0 ) denoting the solution starting from u 0 . It also leads to the ergodicity   1 T lim E[ f (u(t, u 0 ))]dt = f (y)dμ(y), (3.12) T →∞ T 0 ˙ 1 (O ) H which will be considered in Chaps. 5 and 6.

Summary In this chapter, the invariant measures for nonlinear Schrödinger equations, together with the well-posedness and continuous dependence on the initial value as the preliminaries, are introduced. The existence of an invariant Gibbs measure for deterministic nonlinear Schrödinger equations, which are also Hamiltonian partial differential equations, has been investigated by many authors taking advantage of the Hamiltonian, see [20, 27, 28, 40, 132, 157, 173, 183, 184, 192–194] and references therein. For stochastic Schrödinger equations, the Hamiltonian is not conserved anymore in general. Hence, dissipative terms, which have applications in forced and damped quantum, are usually involved to ensure the existence and uniqueness of an invariant measure for stochastic nonlinear Schrödinger equations. We refer to [70, 76, 121, 150] for more details. Numerical approximation of invariant measures for stochastic nonlinear Schrödinger equations will be studied in Chaps. 5 and 6. There are lots of profound researches on invariant measures for other kinds of SPDEs, see e.g. [17, 37, 60, 89, 128, 137, 152, 158, 165], and we refer to [31, 32] for numerical approximations of invariant measures. In particular, the study of invariant measures for stochastic Navier–Stokes equations can be found in [5, 33, 38, 59, 73, 84, 96] and references therein. It still remains to be investigated whether stochastic Hamiltonian partial differential equations admit a unique invariant measure.

Chapter 4

Geometric Structures and Numerical Schemes for Nonlinear Schrödinger Equations

The stochastic (conformal) symplectic structure in the finite dimension case has been introduced briefly in Chap. 1 for the stochastic Kubo oscillator and stochastic dissipative Hamiltonian systems. In this chapter, we turn to considering the geometric structures—stochastic symplectic and multi-symplectic structures—in the infinite dimension case for stochastic nonlinear Schrödinger equations as well as their numerical approximations. Section 4.1 shows the symplectic and multi-symplectic structures for deterministic nonlinear Schrödinger equations, based on which Sect. 4.2 gives several numerical approximations which could inherit the symplecticity or multi-symplecticity. Sections 4.3 and 4.4 generalize these definitions to SPDEs, and show that stochastic nonlinear Schrödinger equations with either additive or linear multiplicative noise preserve the symplectic and multi-symplectic conservation laws. Numerical approximations which could inherit these properties are also given in these sections, which are also used in Chap. 6. Section 4.5 focuses on stochastic nonlinear Schrödinger equations with weak damping. The conformal multi-symplectic conservation law for the considered equation is introduced, which is used in the numerical analysis of invariant measures in Chap. 5.

4.1 Preliminaries It is introduced for the finite dimensional case in Sect. 4.1 that the Lagrange equation and the Hamiltonian system are equivalent based on the unique Legendre transform. In infinite dimensional case, the Lagrange–Hamiltonian duality is no longer uniquely defined. One can define the symplectic structure for infinite dimensional Hamiltonian systems or the multi-symplectic structure when regarding the considered model as a Hamiltonian partial differential equation. © Springer Nature Singapore Pte Ltd. 2019 J. Hong and X. Wang, Invariant Measures for Stochastic Nonlinear Schrödinger Equations, Lecture Notes in Mathematics 2251, https://doi.org/10.1007/978-981-32-9069-3_4

81

82

4 Geometric Structures and Numerical Schemes for Nonlinear Schrödinger Equations

We now recall the geometric structures for the deterministic nonlinear Schrödinger equation ⎧ 2σ ⎪ ⎨ du = iΔudt + iλ|u| udt, u(0, x) = u 0 (x), x ∈ O ⊂ Rd ,

⎪ ⎩

(4.1)

u(t, x) = 0, x ∈ ∂O, t ≥ 0.

It is known that (4.1) can be regarded as an infinite dimensional Hamiltonian system. Therefore its geometric structures could be investigated in a similar way to the finite dimensional case by taking average with respect to the spatial variable. Denoting by p and q the real and imaginary parts of solution u, which satisfy 

 σ p˙ = − Δq − λ p 2 + q 2 q, σ  q˙ =Δp + λ p 2 + q 2 p,

(4.2)

we can transform (4.1) into p˙ =

δ Hˆ δ Hˆ ( p, q), q˙ = − ( p, q), δq δp

where δ denotes the variation and    2 σ +1 1 λ Hˆ ( p, q) := H (u) = |∇ p|2 + |∇q|2 d x − p + q2 d x. 2 O 2σ + 2 O Theorem 4.1 The symplectic conservation law holds for (4.2): the averaged differential 2-form O d p(t) ∧ dq(t)d x is invariant at any time, i.e.,

O

d p(t) ∧ dq(t)d x =

O

d p0 ∧ dq0 d x, ∀ t ≥ 0

with p(0) = p0 and q(0) = q0 . Proof Denoting  (t) :=

d p(t) ∧ dq(t)d x  ∂ p ∂q ∂ p ∂q d p0 ∧ dq0 d x, − = ∂ p0 ∂q0 ∂q0 ∂ p0 O O

it then suffices to show that  ∂ p ∂q d ∂ p ∂q d d p0 ∧ dq0 d x = 0.  (t) = − dt ∂ p0 ∂q0 ∂q0 ∂ p0 O dt

4.1 Preliminaries

83

In fact, by denoting F( p, q) :=

σ +1 λ  2 p + q2 2σ + 2

and calculating d dt

d dt d dt d dt

   2   ∂ ∂F ∂ 2 σ = −Δq − −Δq − λ p + q q = ∂ p0 ∂ p0 ∂q    2 2 ∂ F ∂ F ∂q ∂p ∂q − − , =−Δ ∂ p0 ∂ p∂q ∂ p0 ∂q 2 ∂ p0     ∂q ∂2 F ∂ 2 F ∂q ∂p ∂p =−Δ − − , ∂q0 ∂q0 ∂ p∂q ∂q0 ∂q 2 ∂q0     ∂p ∂q ∂q ∂2 F ∂ p ∂2 F =Δ + + , ∂ p0 ∂ p0 ∂ p 2 ∂ p0 ∂q∂ p ∂ p0     ∂p ∂2 F ∂ p ∂2 F ∂q ∂q =Δ + + , ∂q0 ∂q0 ∂ p 2 ∂q0 ∂q∂ p ∂q0 

∂p ∂ p0



we finally get    ∂q ∂q ∂q ∂q −Δ +Δ ∂ p0 ∂q0 ∂q0 ∂ p0 O      ∂p ∂p ∂p ∂p −Δ d p0 ∧ dq0 d x +Δ ∂q0 ∂ p0 ∂ p0 ∂q0 =− d(Δq) ∧ dq + d(Δp) ∧ d pd x O =− ∇ [d(∇q) ∧ dq + d(∇ p) ∧ d p] d x = 0

d  (t) = dt





O

due to the homogenous boundary condition.



Next we turn to investigating the geometric structures for (4.1) in the view of the Hamiltonian partial differential equation. In this case, the derivatives of u with respect to the spatial variable should also be taken as a conjugate variable when defining the Legendre transform introduced in Sect. 1.5. Definition 4.1 A deterministic partial differential equation is called a Hamiltonian partial differential equation if it can be written in the form M z t + K z x = ∇ S(z), z ∈ Rn , n ≥ 3, where M and K are skew-symmetric matrices and S is a smooth function of the state variable z.

84

4 Geometric Structures and Numerical Schemes for Nonlinear Schrödinger Equations

For simplicity, let d = 1 and O = (0, 1). Rewriting (4.1) utilizing the real part p and imaginary part q of u, we obtain 

pt = − qx x − λ( p 2 + q 2 )σ q, qt = px x + λ( p 2 + q 2 )σ p.

Denoting auxiliary variables v = px , w = qx , we transform above system into the following form ⎧ σ  ⎪ pt + wx = − λ p 2 + q 2 q, ⎪ ⎪ ⎪ σ  ⎨ qt − vx =λ p 2 + q 2 p, (4.3) ⎪ px =v, ⎪ ⎪ ⎪ ⎩ qx =w with Hamiltonian function  σ +1 λ  2 1 2 v + w2 − p + q2 . H˜ ( p, q, v, w) := 2 2σ + 2 Denoting further z = ( p, q, v, w) , two skew symmetric matrices ⎛

0 ⎜−1 M4 = ⎜ ⎝ 0 0

1 0 0 0

0 0 0 0

⎞ 0 0⎟ ⎟, 0⎠ 0

⎛

0 ⎜0 K4 = ⎜ ⎝1 0

⎞ 0 −1 0 0 0 −1⎟ ⎟ 0 0 0⎠ 1 0 0

and S(z) = 2σλ+2 ( p 2 + q 2 )σ +1 + 21 (v2 + w2 ), we obtain its equivalent Hamiltonian partial differential equation form M4 z t + K 4 z x = ∇ S(z). Theorem 4.2 The nonlinear Schrödinger equation (4.1) possesses the multisymplectic conservation law: ∂ ∂ ω(t, x) + κ(t, x) = 0 ∂t ∂x with ω = 21 dz ∧ M4 dz and κ = 21 dz ∧ K 4 dz. More precisely,   ∂  ∂ d p ∧ dq − d p ∧ dv + dq ∧ dw = 0. ∂t ∂x Proof Performing partial derivative to ω with respect to t yields

4.1 Preliminaries

85

 1 ∂ ω = dz t ∧ M4 dz + dz ∧ M4 dz t ∂t 2 = dz ∧ d(M4 z t ), where we used the fact that M4 = −M4 and dz t ∧ M4 dz = M4 dz t ∧ dz = dz ∧ M4 dz t . Similarly, we have

∂ κ = dz ∧ d(K 4 z x ). ∂x

As a result, ∂ ∂ ω+ κ = dz ∧ d(M4 z t + K 4 z x ) = dz ∧ d∇ S(z) = dz ∧ ∇ 2 S(z)dz = 0 ∂t ∂x 

due to the symmetry of ∇ 2 S(z).

4.2 Symplectic and Multi-symplectic Methods for Deterministic Schrödinger Equations This section is devoted to introducing several symplectic temporal semidiscretizations or multi-symplectic full discretizations for the deterministic nonlinear Schrödinger equation (4.1).

4.2.1 Symplectic Temporal Semi-discretizations Since the midpoint scheme possesses the discrete symplectic conservation law when applied to finite dimensional Hamiltonian systems (see e.g. [95]), we consider the midpoint scheme as the temporal semi-discretization for the nonlinear Schrödinger equation. We apply the midpoint scheme to (4.2) and get  ⎧ 2  2 σ 1 n+1 n n+ 21 n+ 21 n+ 21 ⎪ ⎪ = p − τ Δq − τλ p + q q n+ 2 , ⎨p    σ  ⎪ 1 1 2 1 2 1 ⎪ ⎩ q n+1 = q n + τ Δp n+ 2 + τ λ p n+ 2 + q n+ 2 p n+ 2 , 1

1

(4.4)

where τ is the uniform time step-size, p n+ 2 := 21 ( p n+1 + p n ) and q n+ 2 := 21 (q n+1 + q n ). Scheme (4.4) is equivalent to the midpoint scheme applied to (4.1) directly, that

86

4 Geometric Structures and Numerical Schemes for Nonlinear Schrödinger Equations

is, it can be written in the following form 1

1

1

u n+1 = u n + iΔu n+ 2 τ + iλ|u n+ 2 |2σ u n+ 2 τ 1

with u n+ 2 := 21 (u n+1 + u n ) and u n = p n + iq n for n ∈ N. Theorem 4.3 The temporal semi-discretization based on the midpoint scheme for nonlinear Schrödinger equation (4.1) possesses the discrete symplectic conservation law, i.e., n+1 n+1 dp ∧ dq d x = d p n ∧ dq n d x. O

O

Proof We use the same notation F( p, q) :=

λ ( p 2 + q 2 )σ +1 2σ + 2

as that in the proof of Theorem 4.1. Then according to scheme (4.4), we have ⎧   ∂2 F ∂2 F 1 1 1 n+1 ⎪ ⎪ d p n+ 2 − τ 2 dq n+ 2 , − d p n = − τ Δ dq n+ 2 − τ ⎨dp ∂ p∂q ∂q   2 2 ⎪ ⎪ ⎩ dq n+1 − dq n = τ Δ d p n+ 21 + τ ∂ F d p n+ 21 + τ ∂ F dq n+ 21 . ∂ p2 ∂q∂ p 1

Performing the wedge product between d p n+1 − d p n and dq n+ 2 and taking integral with respect to x, we get

 n+1  1 dp − d p n ∧ dq n+ 2 d x O    ∂2 F ∂ 2 F n+ 1 1 n+ 21 n+ 21 2 = −τ Δ dq ∧ dq n+ 2 d x dp −τ − τ 2 dq ∂ p∂q ∂q O ∂2 F 1 1 d p n+ 2 ∧ dq n+ 2 d x =−τ O ∂ p∂q and similarly dp O

n+ 21



∧ dq

n+1

− dq

n



dx = τ

O

1

d p n+ 2 ∧

∂2 F 1 dq n+ 2 d x. ∂q∂ p

Adding the above two equations, we finally complete the proof.



More generally, we apply the s-stage Runge–Kutta method to (4.2) in the temporal direction and get

4.2 Symplectic and Multi-symplectic Methods for Deterministic Schrödinger Equations

87

⎧ s    ⎪ n ⎪ ⎪ P = p − τ ai j ΔQ j + λ(P j2 + Q 2j )σ Q j , i ⎪ ⎪ ⎪ ⎪ j=1 ⎪ ⎪ ⎪ s ⎪  ⎪   ⎪ n ⎪ Q = q + τ ai j ΔP j + λ(P j2 + Q 2j )σ P j , ⎪ i ⎪ ⎨ j=1

s ⎪  ⎪   ⎪ n+1 n ⎪ p = p −τ bi ΔQ i + λ(Pi2 + Q i2 )σ Q i , ⎪ ⎪ ⎪ ⎪ i=1 ⎪ ⎪ ⎪ s ⎪  ⎪   ⎪ n+1 n ⎪ ⎪ q = q + τ bi ΔPi + λ(Pi2 + Q i2 )σ Pi ⎩

(4.5)

i=1

with parameters ai j and bi for i, j = 1, · · · , s. Theorem 4.4 If the coefficients ai j and bi satisfy bi b j = bi ai j + b j a ji , ∀ i, j = 1, · · · , s,

(4.6)

then the temporal semi-discretization (4.5) has a solution, and it possesses the discrete symplectic conservation law

O

d p n+1 ∧ dq n+1 d x =

O

d p n ∧ dq n d x.

Condition (4.6) is named the symplectic condition for Runge–Kutta methods. The proof of Theorem 4.4 is a special case of that of Theorem 4.7 for the stochastic case, and thus is omitted here. It is not hard to find out that the midpoint scheme (4.4) is a 1-stage Runge–Kutta method with coefficients a11 = 21 and b1 = 1, which satisfy the symplectic condition apparently.

4.2.2 Multi-symplectic Full Discretizations When regarding (4.1) as a partial differential equation instead of an evolution equation in time, discretizations in the spatial direction will be taken into consideration. We discretize (4.3) utilizing the midpoint scheme in both temporal and spatial directions. Based on the notations v = px and w = qx , the full discretization reads

88

4 Geometric Structures and Numerical Schemes for Nonlinear Schrödinger Equations

⎧ n+1 n+ 1 n+ 1 p j+ 1 − p nj+ 1 ⎪ w j+12 − w j 2 ⎪ ⎪ 2 2 ⎪ + ⎪ ⎪ τ h ⎪ ⎪ ⎪ ⎪ n+1 ⎪ n n+ 21 n+ 1 ⎪ q j+ 1 − q j+ 1 ⎪ v j+1 − v j 2 ⎪ 2 2 ⎪ − ⎨ τ h 1 ⎪ n+ n+ 1 ⎪ ⎪ p j+12 − p j 2 ⎪ ⎪ ⎪ ⎪ ⎪ h ⎪ ⎪ ⎪ ⎪ n+ 21 n+ 1 ⎪ ⎪ q j+1 − q j 2 ⎪ ⎩ h

=−λ =λ





n+ 1 p j+ 12 2

2

 n+ 1 2 σ n+ 1 + q j+ 12 q j+ 12 , 2

2

  n+ 1 2 σ n+ 1 n+ 1 2 p j+ 12 + q j+ 12 p j+ 12 , 2

2

2

(4.7)

n+ 1

=v j+ 12 , 2

n+ 1

=w j+ 12 , 2

which is known as the Preissman scheme (see e.g. [35]). Here, n+ 21

pj

:=

 1  n+1 p j + p nj , 2

 1 n p j+1 + p nj , 2

p nj+ 1 := 2

and τ and h denote the uniform time and space step-sizes, respectively. Theorem 4.5 The Preissman scheme (4.7) preserves the discrete multi-symplectic structure, that is,  1  n+1 n n d p j+ 1 ∧ dq n+1 1 − d p j+ 1 ∧ dq j+ 1 j+ 2 2 2 2 τ   1 1 n+ 2 n+ 21 n+ 21 n+ 1 d p j+1 ∧ dv j+1 − d p j ∧ dv j 2 − h  1  n+ 21 n+ 1 n+ 1 n+ 1 dq j+1 ∧ dw j+12 − dq j 2 ∧ dw j 2 = 0. − h Proof Utilizing the Hamiltonian function H˜ ( p, q, v, w) introduced for (4.3) in Sect. 4.1, scheme (4.7) can be expressed as ⎧ n+1 n+ 21 n+ 21 ⎪ p j+ 1 − p nj+ 1 w − w ⎪ j+1 j ⎪ 2 2 ⎪ + ⎪ ⎪ ⎪ τ h ⎪ ⎪ ⎪ n+1 ⎪ n n+ 21 n+ 1 ⎪ − q q ⎪ v j+1 − v j 2 j+ 21 j+ 21 ⎪ ⎪ − ⎨ τ h ⎪ n+ 21 n+ 21 ⎪ ⎪ p − p ⎪ j+1 j ⎪ ⎪ ⎪ ⎪ h ⎪ ⎪ ⎪ n+ 1 n+ 1 ⎪ ⎪ ⎪ q j+12 − q j 2 ⎪ ⎩ h where

=

∂ ˜ n+ 21 H 1, ∂q j+ 2

=−

∂ ˜ n+ 21 H 1, ∂ p j+ 2

=

∂ ˜ H ∂v

=

∂ ˜ n+ 21 H 1, ∂w j+ 2

n+ 21 j+ 21

 n+ 1 n+ 1 n+ 1 n+ 1  n+ 1 H˜ j+ 12 := H˜ p j+ 12 , q j+ 12 , v j+ 12 , w j+ 12 . 2

2

2

2

2

,

(4.8)

4.2 Symplectic and Multi-symplectic Methods for Deterministic Schrödinger Equations

89

Taking the exterior derivative of the first equation in (4.8) and performing the n+ 1

wedge product between the obtained equation and dq j+ 12 , we obtain 2

      1 1 n+ 1 n+ 1 n+ 1 n+ 1 d p n+11 − p nj+ 1 ∧ dq 12 − dq 12 ∧ d w j+12 − w j 2 j+ 2 j+ 2 j+ 2 τ h 2   1 d p n+11 ∧ dq n+11 + d p n+11 ∧ dq nj+ 1 − d p nj+ 1 ∧ dq n+11 − d p nj+ 1 ∧ dq nj+ 1 = j+ 2 j+ 2 j+ 2 j+ 2 2τ 2 2 2 2   1 1 1 1 1 1 n+ 2 n+ 2 n+ 2 n+ 2 n+ 2 n+ 21 n+ 21 n+ 1 dq j+1 ∧ dw j+1 − dq j+1 ∧ dw j + dq j ∧ dw j+1 − dq j ∧ dw j 2 − 2h  2  2 ∂ ˜ n+ 21 ∂ n+ 1 n+ 1 n+ 1 n+ 1 n+ 1 H˜ 12 d p 12 ∧ dq 12 + H = dq 12 ∧ dq 12 1 2 j+ 2 j+ 2 j+ 2 j+ 2 j+ 2 ∂ p∂q j+ 2 ∂q  2  2 ∂ ∂ n+ 21 n+ 21 n+ 21 n+ 21 n+ 21 n+ 1 H˜ H˜ + dv 1 ∧ dq 1 + dw 1 ∧ dq 12 j+ 2 j+ 2 j+ 2 j+ 2 ∂v∂q j+ 21 ∂w∂q j+ 21  2 1 1 1 ∂ n+ n+ n+ H˜ 2 d p 12 ∧ dq 12 . = j+ 2 j+ 2 ∂ p∂q j+ 21

Similarly, for the second equation in (4.8), we take the exterior derivative, perform n+ 1

the wedge product with d p j+ 12 and obtain 2

 1  n+1 n+1 n+1 n n n n d p j+ 1 ∧ dq n+1 − d p ∧ dq + d p ∧ dq − d p ∧ dq 1 1 1 1 j+ 2 j+ 2 j+ 2 j+ 2 j+ 21 j+ 21 j+ 21 2 2τ   1 1 1 1 1 1 1 1 n+ n+ n+ n+ n+ n+ n+ n+ 1 d p j+12 ∧ dv j+12 − d p j+12 ∧ dv j 2 + d p j 2 ∧ dv j+12 − d p j 2 ∧ dv j 2 − 2h  2 ∂ n+ 21 n+ 21 n+ 1 ˜ = − d p j+ 1 ∧ H j+ 1 dq j+ 12 . 2 2 2 ∂q∂ p Adding the above two equations and utilizing the fact that 

∂ 2 ˜ n+ 21 H 1 ∂ p∂q j+ 2



n+ 1 d p j+ 12 2

∧

n+ 1 dq j+ 12 2

=

n+ 1 d p j+ 12 2

 ∧

∂ 2 ˜ n+ 21 H 1 ∂q∂ p j+ 2



n+ 1

dq j+ 12 , 2

we derive  1  n+1 n n d p j+ 1 ∧ dq n+1 1 − d p j+ 1 ∧ dq j+ 1 j+ 2 2 2 2 τ  1  n+ 21 n+ 21 n+ 21 n+ 1 n+ 1 n+ 1 n+ 1 n+ 1 dq j+1 ∧ dw j+1 − dq j+1 ∧ dw j 2 + dq j 2 ∧ dw j+12 − dq j 2 ∧ dw j 2 − 2h  1  n+ 21 n+ 1 n+ 1 n+ 1 n+ 1 n+ 1 n+ 1 n+ 1 d p j+1 ∧ dv j+12 − d p j+12 ∧ dv j 2 + d p j 2 ∧ dv j+12 − d p j 2 ∧ dv j 2 − 2h = 0. We use a similar argument for the last two equations in (4.8). Taking their exterior n+ 1

n+ 1

2

2

derivatives, performing the wedge product with dv j+ 12 and dw j+ 12 respectively and

90

4 Geometric Structures and Numerical Schemes for Nonlinear Schrödinger Equations

adding the two derived equations together, we get  1  n+ 21 n+ 1 n+ 1 n+ 1 n+ 1 n+ 1 n+ 1 n+ 1 d p j+1 ∧ dv j+12 + d p j+12 ∧ dv j 2 − d p j 2 ∧ dv j+12 − d p j 2 ∧ dv j 2 2h  1  n+ 21 n+ 1 n+ 1 n+ 1 n+ 1 n+ 1 n+ 1 n+ 1 dq j+1 ∧ dw j+12 + dq j+12 ∧ dw j 2 − dq j 2 ∧ dw j+12 − dq j 2 ∧ dw j 2 + 2h = 0. We finally conclude the required result in the theorem based on above two equations.  Note that there is an alternative way to show the multi-symplecticity of scheme (4.7) since it can be rewritten into the following form n+ 21

M4 δt z nj+ 1 + K 4 δx z j 2

 n+ 1  = ∇ S z j+ 12 2

with the notation z nj = ( p nj , q nj , vnj , wnj ) , δt z nj :=

− z nj z n+1 j τ

and δx z nj :=

z nj+1 − z nj h

.

Some more details will be given in the proof of the stochastic Preissman scheme in Sect. 4.4. This scheme is also called the centered cell scheme (see e.g. [117]) or the central box scheme (see e.g. [107]). As for the nonlinear Schrödinger equations, we concern mainly on the value of u, equivalently p and q. As a result, it is more convenient to eliminate auxiliary variables v and w and get an equivalent numerical scheme for (4.1). Corollary 4.1 The Preissman scheme (4.7) has the following equivalent form 

   n+1 n n n u n+1 + u n+1 j+1 + 2u j j−1 − u j+1 + 2u j + u j−1

2τ     1  n+ 21 2σ n+ 21  n+ 21 2σ n+ 21 = iλ u j+ 1  u j+ 1 + u j− 1  u j− 1 2 2 2 2 2

n+ 1

−i

n+ 21

u j+12 − 2u j

n+ 1

+ u j−12

h2

as a numerical scheme for (4.1). Proof For simplicity, we consider the equivalent form (4.8) of (4.7) based on the Hamiltonian function H˜ . According to the first equation in scheme (4.8), we have p n+1 − p nj+ 1 j+ 1 2

2

τ

n+ 1

+

n+ 21

w j+12 − w j h

=

∂ ˜ n+ 21 H 1, ∂q j+ 2

4.2 Symplectic and Multi-symplectic Methods for Deterministic Schrödinger Equations

p n+1 − p nj− 1 j− 1 2

2

τ

n+ 21

+

wj

n+ 1

− w j−12 h

∂ ˜ n+ 21 H 1, ∂q j− 2

=

whose summation shows     n+1 n+1 n n n p n+1 − p + 2 p + p + 2 p + p j+1 j j−1 j+1 j j−1 ∂ ˜ n+ 21 ∂ ˜ n+ 21 = H j+ 1 + H 1. 2 ∂q ∂q j− 2

91

2τ

n+ 1

+

n+ 1

w j+12 − w j−12 h

Furthermore, the last equation in (4.8) leads to n+ 1

n+ 21

n+ 21

q j+12 − q j

−

h

n+ 1

− q j−12

qj

h

=

 ∂ ˜ n+ 21 1  n+ 21 ∂ ˜ n+ 21 n+ 1 w j+1 − w j−12 . H j+ 1 − H j− 1 = 2 2 ∂w ∂w 2

Combining the above two equations together, we derive 

=

   n+1 n n n p n+1 + p n+1 j+1 + 2 p j j−1 − p j+1 + 2 p j + p j−1

∂ ˜ H ∂q

n+ 21 j+ 21

+

∂ ˜ H ∂q

2τ n+ 21 j− 21

+

 n+ 1  n+ 1 n+ 1 2 q j+12 − 2q j 2 + q j−12 h2

.

The same procedure applied to the second and third equations in (4.8) yields     n+1 n n n q n+1 + q n+1 j+1 + 2q j j−1 − q j+1 + 2q j + q j−1  =−

∂ ˜ H ∂p

n+ 21 j+ 21

+

∂ ˜ H ∂p

2τ

n+ 21 j− 21

−

 n+ 1  n+ 1 n+ 1 2 p j+12 − 2 p j 2 + p j−12 h2

.

Then utilizing the notation u nj := p nj + iq nj , we finally get     n+1 n+1 n n n u n+1 − u + 2u + u + 2u + u j+1 j j−1 j+1 j j−1 2τ     1 2σ 1 n+  n+   n+ 1 2σ n+ 1 = iλ u j+ 12  u j+ 12 + u j− 12  u j− 12 , 2

2

which completes the proof.

2

n+ 1

− 2i

n+ 21

u j+12 − 2u j

n+ 1

+ u j−12

h2

2



92

4 Geometric Structures and Numerical Schemes for Nonlinear Schrödinger Equations

4.3 Stochastic Symplectic Geometric Structure and Numerical Schemes The symplecticity and multi-symplecticity are preserved by stochastic nonlinear Schrödinger equations with additive or linear multiplicative noises, similar to the deterministic case. We show these properties in this section and the following one based on the variational principle with a stochastic forcing. The classical Hamilton’s principle seeks the extremal q(t) such that the action

T ˙ is stationary with δq(0) = δq(T ) = 0 under variation, functional 0 L(t, q, q)dt where L is the Lagrangian of the deterministic Hamiltonian system. Based on the , the author in [92] Legendre transform H ( p, q, t) = pq˙ − L(t, q, q) ˙ with p = δL δ q˙ gives the modified Hamilton’s principle

T

δ

pq˙ − H ( p, q, t)dt ≡ 0.

0

However, for Hamiltonian systems influenced by an additional nonconservative force, i.e., an external noise, its Hamiltonian energy turns out to be H0 ( p, q, t) + H1 ( p, q, t) ◦ χ˙ instead of H ( p, q, t). Here, H1 ( p, q, t) ◦ χ˙ denotes the work done by a spatio-temporal noise χ˙ , and χ˙ = dWdt(t) is a formal time derivative of an Rvalued Wiener process W . In this case, especially when p, q depending on the spatial variable x ∈ Rd , by introducing the generalized action functional

T

G( p, q) := 0

 Rd

 pqd ˙ x − H0 ( p, q, t) − H1 ( p, q, t) ◦ χ˙ dt,

the generalized modified Hamilton’s principle reads δG = δ = 0

≡0



T

0 T Rd

Rd

 pqd ˙ x − H0 ( p, q, t) − H1 ( p, q, t) ◦ χ˙ dt



pδ q˙ + qδp ˙ −

 δ H0 δ H0 δ H1 δ H1 δp − δq − ◦ χ˙ δp − ◦ χ˙ δq d xdt δp δq δp δq (4.9)

under the condition of fixed endpoints, that is, the increments at time 0 and T are zeros: δq(0) = δq(T ) = 0. The principle above, together with the fact that

T

T ˙ = − 0 pδqdt, ˙ yields the generalized Hamiltonian equations of the mo0 pδ qdt tion with noise: ⎧ δ H0 δ H1 ⎪ ⎪ ⎨ p˙ = − δq − δq ◦ χ˙ , δH δ H1 ⎪ ⎪ ⎩ q˙ = 0 + ◦ χ˙ , δp δp

4.3 Stochastic Symplectic Geometric Structure and Numerical Schemes

93

which is rigorously interpreted by the following SPDE ⎧ δ H0 δ H1 ⎪ ⎪ ⎨ dp = − δq dt − δq ◦ dW, δ H0 δ H1 ⎪ ⎪ ⎩ dq = dt + ◦ dW, δp δp

(4.10)

where the variation of H0 is defined by  d  δ H0 ( p, q) = H0 ( p + εδp, q + εδq), dε ε=0 and similarly for H1 . For (4.10) with a fixed initial value ( p(0), q(0)) = ( p0 , q0 ) , we refer to [90, 120] for the differentiability of the solution with respect to initial data. Theorem 4.6 If the solution of (4.10) is differentiable with respect to the initial value. Then the phase flow of system (4.10) preserves the symplectic structure almost surely, that is,  (t) :=

O

d p(t) ∧ dq(t)d x =

O

d p0 ∧ dq0 d x =  (0).

Next, we focus on the symplectic structure and numerical approximations for the stochastic nonlinear Schrödinger equation (3.8) with an R-valued linear multiplicative noise. Denoting by p and q the real and imaginary parts of solution u, we have that (3.8) is equivalent to 

  dp = − Δq + λ( p 2 + q 2 )σ q dt − q ◦ dW,   dq = Δp + λ( p 2 + q 2 )σ p dt + p ◦ dW.

(4.11)

It is not hard to check that (4.11) can be rewritten in the form (4.10) with functionals (see also [47])  2 σ +1 1 λ H0 ( p, q) = − p + q2 |∇ p|2 + |∇q|2 d x + dx 2 O 2σ + 2 O and H1 ( p, q) =

1 2



 O

 p 2 + q 2 d x.

To clear up ambiguities, we give an explanation about the notation to calculate the variation of H0 :

δ H0 . δp

It is easy

94

4 Geometric Structures and Numerical Schemes for Nonlinear Schrödinger Equations

 d  δ H0 ( p, q) =  H0 ( p + εδp, q + εδq) dε ε=0  2 σ =− p + q 2 ( pδp + qδq)d x [∇ p∇(δp) + ∇q∇(δq)] d x + λ O O

   2   σ   2 σ = p δp + Δq + λ p 2 + q 2 q δq d x Δp + λ p + q O

for any increments δp and δq. Compared with (4.9), we just denote  σ δ H0 = Δp + λ p 2 + q 2 p. δp Similarly, we get

 σ δ H0 = Δq + λ p 2 + q 2 q, δq

δ H1 δp

H1 = p and δδq = q. σ  Denote V := λ p 2 + q 2 . Then the variations of

δ H0 δp

and

δ H0 δq

are

  δ H0 = Δ + V p p + V δp + Vq pδq, δp    δ H0 = V p qδp + Δ + Vq q + V δq δ δq 

δ

with V p :=

∂V ∂p

and Vq :=

∂V ∂q

, such that

δ 2 H0 δ 2 H0 δ 2 H0 δ 2 H0 = V = V = Δ + V p + V, p, q, = Δ + Vq q + V, p q p δp 2 δpδq δqδp δq 2 and similarly,

δ 2 H1 δ 2 H1 δ 2 H1 δ 2 H1 = = 0, = 1, = 1. 2 δp δpδq δqδp δq 2

Hence, we derive the following equalities related to the directional derivatives of p and q in the directions p0 and q0 :  d

∂p ∂ p0



 =−

δ 2 H0 ∂ p δ 2 H0 ∂q + δqδp ∂ p0 δq 2 ∂ p0



 dt −

δ 2 H1 ∂ p δ 2 H1 ∂q + δqδp ∂ p0 δq 2 ∂ p0

◦ dW

4.3 Stochastic Symplectic Geometric Structure and Numerical Schemes

=−  d

∂p ∂q0

 d 

and d

    ∂q ∂q ∂p dt − Δ + Vq q + V + Vpq ◦ dW, ∂ p0 ∂ p0 ∂ p0





   ∂q ∂p ∂q + Vpq ◦ dW, = − Δ + Vq q + V dt − ∂q0 ∂q0 ∂q0



    ∂p ∂q ∂p + Vq p ◦ dW = Δ + Vp p + V dt + ∂ p0 ∂ p0 ∂ p0



    ∂p ∂p ∂q = Δ + Vp p + V dt + + Vq p ◦ dW. ∂q0 ∂q0 ∂q0

∂q ∂ p0 ∂q ∂q0

95

Based on above equalities, one can check that  d

∂ p ∂q ∂ p ∂q − ∂ p0 ∂q0 ∂q0 ∂ p0







 ∂q ∂q ∂p ∂p = −Δ + Δ ∂ p0 ∂q0 ∂ p0 ∂q0    ∂q ∂q ∂p ∂p dt, +Δ − Δ ∂q0 ∂ p0 ∂q0 ∂ p0

(4.12)

where we have used the fact that V p q = Vq p. Now we define the averaged differential 2-form  (t) := O d p(t) ∧ dq(t)d x similar to the deterministic case. Then, based on (4.12), the derivative with respect to time gives the stochastic symplectic conservation law (see also [47]) d (t) := d =− =−





O O O

d p ∧ dqd x =

d O

∂ p ∂q ∂ p ∂q − ∂ p0 ∂q0 ∂q0 ∂ p0

d p0 ∧ dq0 d x

[d(Δp) ∧ d p + d(Δq) ∧ dq] d xdt ∇ [d(∇ p) ∧ d p + d(∇q) ∧ dq] d xdt

= 0, where the homogenous boundary condition is used in the last step. It then shows the symplecticity of (4.11). For stochastic nonlinear Schrödinger equations, their temporal semidiscretizations based on the s-stage Runge–Kutta methods with certain conditions could inherit the symplecticity of the original systems as in the case of deterministic nonlinear Schrödinger equations (Theorems 4.3 and 4.4). The s-stage Runge–Kutta method applied to (4.11) in temporal direction reads

96

4 Geometric Structures and Numerical Schemes for Nonlinear Schrödinger Equations

⎧ s s    ⎪ (0)  n 2 2 σ ⎪ ⎪ P ΔQ − = p − τ a + λ(P + Q ) Q ai(1) i j j ⎪ j j ij j Q j δn+1 W, ⎪ ⎪ ⎪ j=1 j=1 ⎪ ⎪ ⎪ s s ⎪  ⎪   ⎪ (0)  n 2 2 σ ⎪ Q ΔP + = q + τ a + λ(P + Q ) P ai(1) ⎪ i j j j j i j j P j δn+1 W, ⎪ ⎨ j=1

j=1

s s ⎪  ⎪   ⎪ (0)  n+1 n 2 2 σ ⎪ p = p −τ bi ΔQ i + λ(Pi + Q i ) Q i − bi(1) Q i δn+1 W, ⎪ ⎪ ⎪ ⎪ i=1 i=1 ⎪ ⎪ ⎪ s s ⎪   ⎪   ⎪ n+1 ⎪ ⎪ = qn + τ bi(0) ΔPi + λ(Pi2 + Q i2 )σ Pi + bi(1) Pi δn+1 W, ⎩q i=1

(4.13)

i=1

where tn = nτ and δn+1 W := W (tn+1 ) − W (tn ) denotes the increment of the Wiener (0) (1) (1) process. Two classes of parameters {ai(0) j , bi }i, j=1,··· ,s and {ai j , bi }i, j=1,··· ,s may be different. Scheme (4.13) can be expressed as an approximation for the original system (3.8) by denoting Ui := Pi + iQ i and u n := p n + iq n : ⎧ s s   ⎪  ⎪ (0)  n 2σ ⎪ U ΔU + i = u + iτ a + λ|U | U ai(1) ⎪ i j j j ij j U j δn+1 W, ⎪ ⎨ j=1

j=1

s s ⎪    ⎪ (0)  ⎪ n+1 n 2σ ⎪ u ΔU + i = u + iτ b + λ|U | U bi(1) Ui δn+1 W. ⎪ i i i i ⎩ i=1

(4.14)

i=1

Theorem 4.7 ([47]) Assume that the coefficients satisfy (0) (0) (0) (0) bi(0) b(0) j = bi ai j + b j a ji , (0) (1) (1) (0) bi(0) b(1) j = bi ai j + b j a ji , (1) (1) (1) (1) bi(1) b(1) j = bi ai j + b j a ji .

The temporal semi-discretization (4.13) possesses a solution in L2 (O), which preserves both the discrete charge conservation law u n+1 2L2 (O ) = u n 2L2 (O ) and the discrete symplectic conservation law

dp

O

n+1

∧ dq

n+1

dx =

O

d p n ∧ dq n d x, ∀ n ∈ N

almost surely. Proof Step 1. We first show the existence of the solution, which follows from a standard Galerkin method and Brouwer’s theorem. In fact, we first consider the

4.3 Stochastic Symplectic Geometric Structure and Numerical Schemes

97

following finite dimensional approximation  2σ  ˜ u˜ dt + iπ M (u˜ ◦ dW (t)) , d u˜ = iΔudt ˜ + iλπ M |u|

(4.15)

which is obtained based on the spectral Galerkin approximation. Here, π M is the projection operator from L2 (O) to an M-dimensional space VM . The well-posedness for (4.15) follows from the argument of the well-posedness for (3.8). Scheme (4.14) applied to (4.15) yields ⎧ s s   ⎪ ⎪ (0) ˜ ⎪ ˜ i = u˜ n + iτ ˜ U a + i ai(1) A ⎪ j ij j Bj, ⎪ ⎨ j=1

j=1

s s ⎪   ⎪ (0) ˜ ⎪ n+1 n ⎪ u ˜ = u ˜ + iτ b + i bi(1) B˜ i , A ⎪ i i ⎩ i=1

i=1

where we use the notations     and B˜ i := π M U˜ i δn+1 W . A˜ i := ΔU˜ i + λπ M |U˜ i |2σ U˜ i Based on the scheme above, we have 1

u˜ n+ 2 = u˜ n +

s s iτ  (0) ˜ i  (1) ˜ bi Ai + b Bi 2 i=1 2 i=1 i

and 1

u˜ n+1 − u˜ n + u˜ n 2 s s s s   iτ  (0) ˜ i  (1) ˜ ˜ ˜ = bi Ai + bi Bi + U˜ i − iτ ai(0) ai(1) j Aj − i j Bj. 2 i=1 2 i=1 j=1 j=1

u˜ n+ 2 =

Defining the continuous function 1

1

g(u˜ n+ 2 ) :=u˜ n+ 2 − u˜ n −

s s iτ  (0) ˜ i  (1) ˜ bi Ai − b Bi , 2 i=1 2 i=1 i

we obtain based on the assumption on the coefficients that

98

4 Geometric Structures and Numerical Schemes for Nonlinear Schrödinger Equations

    1 1 1 1 g(u˜ n+ 2 ), u˜ n+ 2 = u˜ n+ 2 − u˜ n , u˜ n+ 2 +

s  τ 2   (0) (0) (0) (0) bi b j − bi(0) ai(0) ( A˜ i , A˜ j ) j − b j a ji 4 i, j=1

−

s  τ   (0) (1) (1) (0) bi b j − bi(0) ai(1) ( A˜ i , B˜ j ) − b a j j ji 2 i, j=1

s  1   (1) (1) (1) (1) bi b j − bi(1) ai(1) ( B˜ i , B˜ j ) − b a j j ji 4 i, j=1  s s iτ  (0) ˜ i  (1) ˜ ˜ − bi Ai + b Bi , Ui 2 i=1 2 i=1 i ! !   1 !2 1 ! = !u˜ n+ 2 ! 2 − u˜ n , u˜ n+ 2 L (O ) ! ! ! ! ! ! 1 ! ! ! n+ 21 ! ≥ !u˜ n+ 2 ! 2 ! 2 − !u˜ n !L2 (O ) . !u˜

+

L (O )

L (O )

Then utilizing the C-valued version of Brouwer’s fixed-point result (see e.g. [4]), we get the existence for the solution of the finite dimensional approximation. The existence of (4.13) (or equivalently (4.14)) then follows from the convergence analysis of the spectral Galerkin approximation in [57]. Step 2. Now we show that the numerical solution is charge-preserved. We consider the second equation in scheme (4.14) u n+1 − u n = iτ

s  i=1

bi(0) Ai + i

s 

bi(1) Bi

i=1

with notations Ai := ΔUi + λ|Ui |2σ Ui and Bi := Ui δn+1 W. Then we can conclude    ! ! ! 1 ! n+1 n n+ 21 !u n+1 !2 2 − !u n !2 2 = u =0 − u , u L (O ) L (O ) 2 following the same procedure in Step 1. Step 3. We finally prove the symplecticity of the numerical scheme. By denoting

4.3 Stochastic Symplectic Geometric Structure and Numerical Schemes

99

AiQ :=ΔQ i + λ(Pi2 + Q i2 )σ Q i , AiP :=ΔPi + λ(Pi2 + Q i2 )σ Pi , BiQ :=Q i δn+1 W, BiP :=Pi δn+1 W, we rewrite scheme (4.13) as ⎧ s s   ⎪ (0) Q Q n ⎪ ⎪ P = p − τ a A − ai(1) i ⎪ ij j j Bj , ⎪ ⎪ ⎪ j=1 j=1 ⎪ ⎪ ⎪ s s ⎪  (0)  ⎪ ⎪ n P P ⎪ Q = q + τ a A + ai(1) ⎪ i j ij j Bj , ⎪ ⎨ j=1

j=1

s s ⎪   ⎪ ⎪ (0) Q n+1 n ⎪ p = p − τ b A − bi(1) BiQ , ⎪ i i ⎪ ⎪ ⎪ i=1 i=1 ⎪ ⎪ ⎪ s s ⎪   ⎪ ⎪ (0) P ⎪ q n+1 = q n + τ ⎪ b A + bi(1) BiP . i ⎩ i i=1

i=1

Performing the wedge product between the last two equations above and noting that ⎛ ⎞ ⎧ s s   ⎪ ⎪ Q Q⎠ ⎪ ⎪ d p n =d Pi + d ⎝τ ai(0) ai(1) , ⎪ j Aj + j Bj ⎪ ⎨ j=1 j=1 ⎛ ⎞ ⎪ s s ⎪   ⎪ ⎪ dq n =dQ − d ⎝τ P P⎠ ⎪ ai(0) ai(1) , ⎪ i j Aj + j Bj ⎩ j=1

j=1

we obtain  dp

n+1

∧ dq

n+1

=d p ∧ dq + d Pi ∧ d τ n

n

 −d τ

bi(0) AiP +

i=1 s  i=1

− τ2

s 

bi(0) AiQ +

s 

bi(1) BiQ

s 

bi(1) BiP

i=1

∧ dQ i

i=1

s    (0) (0) (0) (0) bi(0) b(0) d AiQ ∧ d A Pj j − bi ai j − b j a ji i, j=1

−τ

s    (0) (1) (1) (0) bi(0) b(1) d AiQ ∧ dB jP − b a − b a j i ij j ji i, j=1

100

4 Geometric Structures and Numerical Schemes for Nonlinear Schrödinger Equations

−τ

s    (1) (0) (0) (1) bi(1) b(0) dBiQ ∧ d A Pj − b a − b a j i ij j ji i, j=1

−

s  

 (1) (1) (1) (1) bi(1) b(1) dBiQ ∧ dB jP . j − bi ai j − b j a ji

i, j=1

Note that d Pi ∧ dBiP = 0, dBiQ ∧ dQ i = 0 and d Pi ∧ d AiP − d AiQ ∧ dQ i d x  2  ∂ F d Pi = − d(∇ Pi ) ∧ d(∇ Pi ) + d(∇ Q i ) ∧ d(∇ Q i ) + d Pi ∧ ∂ p2 O   2 ∂2 F ∂ F ∂2 F dx dQ d P − dQ ∧ dQ + d P ∧ − ∧ dQ i i i i i i ∂q 2 ∂q∂ p ∂ p∂q =0 O

with F( p, q) :=

σ +1 λ  2 p + q2 2σ + 2

and the partial derivatives of F taking value at (Pi , Q i ). We then conclude that

O

d p n+1 ∧ dq n+1 d x =

O

d p n ∧ dq n d x

based on the assumption on the coefficients.



As a special case of the 1-stage stochastic Runge–Kutta methods, the midpoint (0) (1) scheme with a11 = a11 = 21 and b1(0) = b1(1) = 1 also possesses the discrete symplectic conservation law.

4.4 Stochastic Multi-symplectic Geometric Structure and Numerical Schemes The stochastic symplectic structure investigated above is obtained when regarding (3.8) as an infinite dimensional evolution equation in time. However, when the spatial variation is of interest for a Hamiltonian partial differential equation, such as spatially periodic waves, the multi-symplectic structure is usually involved, which is first investigated in [34] for the deterministic case. For simplicity, we show the stochastic

4.4 Stochastic Multi-symplectic Geometric Structure and Numerical Schemes

101

multi-symplecticity for (3.8) on the domain [0, 1] ⊂ R with homogeneous boundary condition u(t, 0) = u(t, 1) = 0. ∂L , the LaIn this point of view, in addition of the conjugate momenta p = ∂q t grangian L = L(t, x, q, qt ( p, q), qx , px ) also depends on space derivatives qx and ∂L , the generalized Hamilpx . Defining the conjugate coordinates v = ∂∂pLx and w = ∂q x ton’s principle shows

T

δG( p, q, v, w) = δ 1

1

pqt + v px + wqx − S0 ( p, q, v, w) − S1 ( p, q, v, w) ◦ χd ˙ xdt

0

   ∂ S0 ∂ S0 ∂ S1 ∂ S1 − ◦ χ˙ δp + − pt − wx − − ◦ χ˙ δq ∂p ∂p ∂q ∂q 0 0     ∂ S0 ∂ S0 ∂ S1 ∂ S1 + px − − ◦ χ˙ δv + qx − − ◦ χ˙ δwd xdt ≡ 0. ∂v ∂v ∂w ∂w

=

0



T



qt − v x −

 σ +1 1  2  By introducing S0 = 2σλ+2 p 2 + q 2 + 2 v + w2 and S1 = equation above shows

1 2



 p 2 + q 2 , the

σ  qt − vx =λ p 2 + q 2 p + p ◦ χ˙ , σ  − pt − wx =λ p 2 + q 2 q + q ◦ χ˙ , px =v, qx =w. It gives the equivalent multi-symplectic form of (4.11): M4 dz + K 4 z x dt = ∇ S0 dt + ∇ S1 ◦ dW with z = ( p, q, v, w) and skew-symmetric matrices M4 , K 4 being the same as those in Sect. 4.1. For the numerical approximation of the above equation, we introduce the stochastic Preissman scheme similar to the deterministic case ⎛ n+1 ⎛ ⎞ ⎞ n+ 21 n+ 21 z j+ 1 − z nj+ 1 z − z j+1 j 2 2 ⎠ ⎠ M4 ⎝ + K4 ⎝ τ h n+ 1

n+ 1

2

2

= ∇ S0 (z j+ 12 ) + ∇ S1 (z j+ 12 )

δn+1 W j+ 21 τ

,

(4.16)

where δn+1 W j+ 21 = W (tn+1 , x j+ 21 ) − W (tn , x j+ 21 ) denotes the increments of W at x j+ 21 = 21 (x j+1 + x j ). Scheme (4.16) has the following equivalent form

102

4 Geometric Structures and Numerical Schemes for Nonlinear Schrödinger Equations

⎧ n+1 n+ 1 n+ 1 p 1 − pn 1 ⎪ w j+12 − w j 2 ⎪ j+ 2 j+ 2 ⎪ ⎪ + ⎪ ⎪ ⎪ τ h ⎪ ⎪ ⎪ ⎪ n+1 n n+ 21 n+ 1 ⎪ − q q ⎪ v j+1 − v j 2 ⎪ j+ 21 j+ 21 ⎪ ⎪ − ⎨ τ h 1 ⎪ n+ n+ 1 ⎪ ⎪ p j+12 − p j 2 ⎪ ⎪ ⎪ ⎪ ⎪ h ⎪ ⎪ ⎪ ⎪ n+ 21 n+ 1 ⎪ ⎪ q j+1 − q j 2 ⎪ ⎪ ⎩ h

σ n+ 1  n+ 1 n+ 1 n+ 1 δn+1 W j+ 21 = − λ ( p 12 )2 + (q 12 )2 q 12 − q 12 , j+ 2 j+ 2 j+ 2 j+ 2 τ σ n+ 1  n+ 1 n+ 1 n+ 1 δn+1 W j+ 21 =λ ( p 12 )2 + (q 12 )2 p 12 + p 12 , j+ 2 j+ 2 j+ 2 j+ 2 τ =v

n+ 21 j+ 21

=w

,

n+ 21 j+ 21

.

Now we show that the stochastic Preissman scheme preserves the stochastic multisymplectic structure. Theorem 4.8 ([119]) The full discretization (4.16) preserves the stochastic multisymplectic conservation law, i.e., n+ 1

ωn+1 − ωnj+ 1 j+ 1 2

+

2

τ

n+ 21

2

=0

h

with ωnj+ 1 := 21 dz nj+ 1 ∧ M4 dz nj+ 1 and κ j 2

n+ 21

κ j+12 − κ j

2

n+ 21

:= 21 dz j

n+ 21

∧ K 4 dz j

.

Proof Taking the exterior derivative in the phase space on both sides of (4.16), we get ⎛ M4 ⎝

dz n+1 − dz nj+ 1 j+ 1 2

2

τ

⎛

⎞

⎠ + K4 ⎝

n+ 1

n+ 21

dz j+12 − dz j h

⎞ ⎠

 n+ 1 n+ 1 δn+1 W j+ 21 n+ 1 dz j+ 12 . = ∇ 2 S0 (z j+ 12 ) + ∇ 2 S1 (z j+ 12 ) 2 2 2 τ Then performing the wedge product between n+ 1 dz j+ 12 2

=

dz n+1 + dz nj+ 1 j+ 1 2

2

2

n+ 1

=

n+ 21

dz j+12 + dz j 2

and (4.17) yields ⎛ ⎝

dz n+1 + dz nj+ 1 j+ 1 2

2

2 ⎛

+⎝

n+ 1

⎠ ∧ M4 ⎝

n+ 21

dz j+12 + dz j 2

⎛

⎞

⎞

dz n+1 − dz nj+ 1 j+ 1 2

2

τ ⎛

⎠ ∧ K4 ⎝

n+ 1

⎞ ⎠

n+ 21

dz j+12 − dz j h

⎞ ⎠

(4.17)

4.4 Stochastic Multi-symplectic Geometric Structure and Numerical Schemes n+ 1 = dz j+ 12 2

 ∧ ∇ S0 + ∇ S1 2

2

δn+1 W j+ 21 τ

n+ 1

dz j+ 12 = 0

103

(4.18)

2

due to the symmetry of ∇ 2 S0 and ∇ 2 S1 . Moreover, according to the fact that matrices M4 and K 4 are skew-symmetric, we get − dz n+1 ∧ M4 dz nj+ 1 = 0, dz nj+ 1 ∧ M4 dz n+1 j+ 1 j+ 1 2

2

2

n+ 1 dz j 2

n+ 1 K 4 dz j+12

n+ 1 dz j+12

∧

−

2

∧

n+ 1 K 4 dz j 2

= 0, 

which, together with (4.18), completes the proof.

4.5 Conformal Multi-symplectic Structure for the Damped Case Some modifications to the canonical Schrödinger equation arise as mathematical models to suppress the loss of the soliton signal caused by the Gordon–Haus effect (see e.g. [93]). Several types of modifications with damped terms and noises are investigated in [70, 76, 79, 80] and references therein. As one of the modifications above, the following one-dimensional damped stochastic nonlinear Schrödinger equation (see also [50, 70]) ⎧   2 ⎪ ⎨ du = iΔu − αu + iλ|u| u dt + εdW (4.19) u(t, 0) = u(t, 1) = 0, t ≥ 0 ⎪ ⎩ u(0, x) = u 0 (x), x ∈ O := (0, 1) with the absorption coefficient α > 0 will be mainly studied in the following sections. This section is devoted to introducing some internal properties of (4.19), including geometric structures and the charge evolution law, with α > 0, λ = ±1 and the noise intensity ε. Note that the solution u is a C-valued random field on a probability space (, F , P). We consider W = W1 + iW2 as a Q-Wiener process on L2 (O) with W1 and W2 being its real and imaginary parts, respectively. The covariance operator Q is assumed to be a symmetric, positive definite and trace operator such that W has the following Karhunen–Loève expansion W =

∞  k=1

1

Q 2 ek βk =

∞  √ k=1

ηk ek βk , ηk > 0 and η :=

∞ 

ηk < ∞,

k=1

where {ek }k∈N is an orthogonal basis for L2 (O; R), {βk = βk1 + iβk2 }k∈N is associated to a filtration {Ft }t≥0 , and {βki }i=1,2 k∈N is a family of independent and identically

104

4 Geometric Structures and Numerical Schemes for Nonlinear Schrödinger Equations

distributed R-valued Wiener processes. In addition, we assume that Q commutes with the Laplace operator, and hence {ek }k∈N is also the eigenbasis of the Dirichlet Laplacian. Actually, the assumption on Q can be generalized to the degenerate case, which will be explained in Remark 5.1. We set the linear operator Aα := −iΔ + α I d, and the semigroup S(t) := e−t Aα = t (iΔ−α I d) is generated by −Aα . The mild solution of (4.19) exists globally and can e be written as T T S(t − s)|u(s)|2 u(s)ds + ε S(t − s)dW (s). u(t) = S(t)u 0 + iλ 0

0

Let {λαk }k∈N+ be an increasing sequence of eigenvalues of Aα with 1 ≤ |λαk | → +∞. Definition 4.2 For all s ∈ N, we define the normed linear space ∞ ∞ "  #   s   (u, ek )2 |λα |s < ∞ , ˙ s := Dom(Aα2 ) = u u = H (u, ek )ek ∈ L2 (O) s.t. k k=1

k=1

endowed with the s-norm ∞      (u, ek ) 2 λα s us := k

1 2

.

k=1

˙ 0. In particular, u0 = uL2 (O ) for any u ∈ H In the sequel, we use the notation L2 := L2 (O) and Hs := Hs (O). It is easy to check that the above norms satisfy ur ≤ us for any 0 ≤ r ≤ s, and us ∼ = ˙ s , where uHs denotes the natural Sobolev norm. uHs (s = 0, 1, 2) for any u ∈ H The operator norm is defined as EL (H˙ s ,H˙ r ) = sup

˙s u∈H

Eur , ∀ r, s ∈ N us

for any operator E. Hence, for 0 ≤ r ≤ s, we have

S(t)L

˙ s ,H ˙ r) (H

= sup

$   2  r  21 ∞   t (iΔ−α I d) u, ek  λαk  k=1 e us

˙s u∈H

= sup

˙s u∈H

e−αt ur ≤ e−αt . us

1

In the following, operator Q 2 is assumed to be a Hilbert–Schmidt operator from L2 ˙ s with norm to H 1

Q 2 2L 2s :=

∞  k=1

1

Q 2 ek 2s =

∞  k=1

|λαk |s ηk < ∞.

4.5 Conformal Multi-symplectic Structure for the Damped Case

105

We define the spatio-temporal noise χ˙ = dW , set u = p + iq, χ˙ = χ˙ 1 + iχ˙ 2 with dt dW1 dW2 p, q, χ˙ 1 = dt and χ˙ 2 = dt , and rewrite (4.19) as ⎧ ⎨

pt + qx x + αp + λ( p 2 + q 2 )q = εχ˙ 1 ,

⎩ −q + p − αq + λ( p 2 + q 2 ) p = −εχ˙ . t xx 2

(4.20)

Denoting v = px , w = qx , z = ( p, q, v, w) , Eq. (4.20) can be transformed into a compact form M4 dt z + K 4 ∂x zdt = −α M4 zdt + ∇ S0 (z)dt + ∇ S1 (z) ◦ dW1 + ∇ S2 (z) ◦ dW2 , (4.21) where M4 , K 4 are the same as those in Sect. 4.1 and S0 (z) =

λ 2 1 ( p + q 2 )2 + (v2 + w2 ), S1 (z) = −εq, S2 (z) = εp. 4 2

Without pointing it out, the equations below hold in the sense P-a.s. We now prove that (4.19) possesses the stochastic conformal multi-symplectic conservation law, whose definition is also given in the following theorem. Theorem 4.9 Equation (4.19) is a stochastic conformal multi-symplectic Hamiltonian system, and preserves the stochastic conformal multi-symplectic conservation law dt ω(t, x) + ∂x κ(t, x)dt = −αω(t, x)dt, which means x1 ω(t1 , x)d x − x0

x1 x0



t1

ω(t0 , x)d x + t0

=−



κ(t, x1 )dt − x1 x0

t1

κ(t, x0 )dt

t0



t1

(4.22)

αω(t, x)dtd x,

t0

where ω = 21 dz ∧ M4 dz and κ = 21 dz ∧ K 4 dz are two differential 2-forms associated with two skew-symmetric matrices M4 and K 4 . Proof To simplify the proof, we denote (z 1 , z 2 , z 3 , z 4 ) := ( p, q, v, w) = z  and (zl )tx := zl (t, x) for l = 1, 2, 3, 4. Noting that ω = dz 2 ∧ dz 1 and κ = dz 1 ∧ dz 3 + dz 2 ∧ dz 4 , we have

106

4 Geometric Structures and Numerical Schemes for Nonlinear Schrödinger Equations



=

x1

x 0x1 x0



x1

ω(t1 , x)d x −

ω(t0 , x)d x

x0

  d(z 2 )tx1 ∧ d(z 1 )tx1 − d(z 2 )tx0 ∧ d(z 1 )tx0 dx

 4  ∂(z 1 )tx x0 1 = ∧ x0 d(z i )t0 ∂(z ) i x0 t0 l=1 i=1  4  4  x x  ∂(z 2 )t  ∂(z 1 )t x0 x0 0 0 dx − d(z ) d(z ) ∧ l i t0 t0 ∂(zl )tx00 ∂(z i )tx00 l=1 i=1  4  x1  4   ∂(z 2 )tx1 ∂(z 1 )tx1 ∂(z 2 )tx0 ∂(z 1 )tx0 = d x d(zl )tx00 ∧ d(z i )tx00 x0 x0 − x0 x0 ∂(z ) ∂(z ) ∂(z ) ∂(z ) l i l i x t t t t 0 0 0 0 0 l=1 i=1

=:

x1

  4

4  4 

∂(z 2 )tx1 d(zl )tx00 ∂(zl )tx00

Hl,i (t1 , x1 )d(zl )tx00 ∧ d(z i )tx00

(4.23)

l=1 i=1

with Hl,i (t1 , x1 ) =

t1

x1  ∂(z2 )tx1

x ∂(zl )t00

x0



4  4  

∂(z 2 )tx1 ∂(zl )tx00

4 4  

=:

t1

t1

−

∂(z 2 )tx0 ∂(z 1 )tx0

x ∂(zl )t00

 d x. Similarly,

x ∂(z i )t00

κ(t, x0 )dt

t0

 −

t0

l=1 i=1

−

x ∂(z i )t00

κ(t, x1 )dt −

t0

=

∂(z 1 )tx1

∂(z 4 )tx1 ∂(z i )tx00

∂(z 1 )tx1 ∂(z 3 )tx1 ∂(z 1 )tx0 ∂(z 3 )tx0 + ∂(z i )tx00 ∂(zl )tx00 ∂(z i )tx00 ∂(zl )tx00  ∂(z 2 )tx0 ∂(z 4 )tx0 dt d(zl )tx00 ∧ d(z i )tx00 + ∂(zl )tx00 ∂(z i )tx00

Ml,i (t1 , x1 )d(zl )tx00 ∧ d(z i )tx00

(4.24)

l=1 i=1

and

x1 x0

= 2α



t1

αω(t, x)dtd x

t0

4  4   l=1 i=1

= : 2α

4  4 

x1 x0

t0

t1



∂(z 2 )tx ∂(z 1 )tx ∂(zl )tx00 ∂(z i )tx00



 dtd x d(zl )tx00 ∧ d(z i )tx00

Nl,i (t1 , x1 )d(zl )tx00 ∧ d(z i )tx00 .

l=1 i=1

Adding (4.23), (4.24) and (4.25) together, we get that (4.22) holds if

(4.25)

4.5 Conformal Multi-symplectic Structure for the Damped Case

107

Hl,i (t1 , x1 ) + Ml,i (t1 , x1 ) + 2αNl,i (t1 , x1 ) = 0

(4.26)

for any l, i ∈ {1, 2, 3, 4}, t1 ∈ R+ and x1 ∈ R. In fact, rewritting (4.20) as ⎧ ∂ S0 (z) ⎪ ⎪ dt + εdW1 , ⎨ dt z 1 = − ∂x (z 4 )dt − αz 1 dt + ∂z 2 ∂ S (z) ⎪ ⎪ ⎩ dt z 2 =∂x (z 3 )dt − αz 2 dt − 0 dt + εdW2 , ∂z 1 and taking partial derivatives with respect to (z i )tx00 and (zl )tx00 respectively, we have ⎧ 4  ⎪ ∂(z 1 )tx ∂ S1 (z) ∂(zl )tx ∂ ∂(z 4 )tx ∂(z 1 )tx ⎪ ⎪ d = − dt − α dt + dt, ⎪ t ⎪ ⎨ ∂(z i )tx00 ∂ x ∂(z i )tx00 ∂(z i )tx00 ∂(z 2 )tx ∂(zl )tx ∂(z i )tx00 l=1 4 ⎪  ⎪ ∂(z 2 )tx ∂ S1 (z) ∂(z i )tx ∂ ∂(z 3 )tx ∂(z 2 )tx ⎪ ⎪ = dt − α dt − dt. d ⎪ t x x x ⎩ ∂(zl ) 0 ∂ x ∂(zl ) 0 ∂(zl )t00 ∂(z 1 )tx ∂(z i )tx ∂(zl )tx00 t0 t0 i=1

Furthermore,   ∂(z 2 )tx1 ∂(z 2 )tx1 ∂(z 1 )tx1 ∂(z 1 )tx1 + dx d d t x0 t1 ∂(zl )tx00 ∂(zl )tx00 1 ∂(z i )tx00 x0 ∂(z i )t0  x1 ∂(z 1 )tx1 ∂ ∂(z 3 )tx1 ∂(z 2 )tx1 = dt1 d x − α x0 ∂ x ∂(zl )tx00 ∂(zl )tx00 x0 ∂(z i )t0  x1 ∂(z 2 )tx1 ∂ ∂(z 4 )tx1 ∂(z 1 )tx1 dt1 d x − +α x0 ∂ x ∂(z i )tx00 ∂(z i )tx00 x0 ∂(z l )t0

dt1 Hl,i (t1 , x1 ) =

=

x1

∂(z 1 )tx10 ∂(z 3 )tx10 ∂(z 1 )tx11 ∂(z 3 )tx11 − − ∂(z i )tx00 ∂(zl )tx00 ∂(z i )tx00 ∂(zl )tx00 x1 ∂(z 1 )tx1 ∂(z 2 )tx10 ∂(z 4 )tx10 + − 2α x0 ∂(zl )tx00 ∂(z i )tx00 x0 ∂(z i )t0

∂(z 2 )tx11 ∂(z 4 )tx11 ∂(zl )tx00 ∂(z i )tx00 ∂(z 2 )tx1 d xdt1 ∂(zl )tx00

= − dt1 Ml,i (t1 , x1 ) − 2αdt1 Nl,i (t1 , x1 ), which together with the fact that Hl,i (t0 , x1 ) + Ml,i (t0 , x1 ) + 2αNl,i (t0 , x1 ) = 0 yields (4.26).  Next, we show that the charge of the solution u(t), although is not conserved anymore, satisfies an exponential type evolution law. Proposition 4.1 Assume that M(u 0 ) < ∞, then the solution of (4.19) is uniformly bounded with EM(u(t)) = e−2αt EM(u 0 ) + where M(u) = u2L2 .

ε2 η (1 − e−2αt ), α

(4.27)

108

4 Geometric Structures and Numerical Schemes for Nonlinear Schrödinger Equations

Proof Itô’s formula applied to M(u(t)) yields 

1

d M(u(t)) = −2α M(u(t))dt + 2ε

 ud xdW + 2ε2 ηdt,

0

where [·] denotes the real part of a complex value. Taking the expectation on both sides of the above equation and solving the ordinary differential equation, we derive EM(u(t)) = e

−2αt



t

2ε ηe 2

2αs

ds + EM(u 0 )

0

= e−2αt EM(u 0 ) +

ε2 η (1 − e−2αt ). α 

We hence conclude that the damped stochastic nonlinear Schrödinger equation (4.19) possesses the stochastic conformal multi-symplectic conservation law (4.22) with its solution being uniformly bounded with (4.27).

Summary The phase flow of Schrödinger equations in both deterministic and stochastic cases preserves the symplectic structure when the equations are regarded as infinite dimensional Hamiltonian systems, and possesses the multi-symplectic conservation law when the equations are interpreted as Hamiltonian partial differential equations. Symplectic temporal semi-discretization and multi-symplectic full discretizations are proposed to inherit the geometric structures of the original system in this chapter. We refer to [11, 117, 124] and [11, 34, 105, 117] for the study of symplectic and multi-symplectic schemes of deterministic Schrödinger equations, respectively. Symplectic and multi-symplectic schemes for stochastic Schrödinger equations can be found in [47, 48, 56] and [56, 107, 110, 119], respectively. Moreover, Maxwell equations and wave equations are also Hamiltonian partial differential equations with symplectic and multi-symplectic structures. We refer to [51, 101, 103, 125] for the design of structure-preserving methods for both deterministic and stochastic Maxwell equations, and refer to [104, 135, 164] for those of wave equations.

Chapter 5

Numerical Invariant Measures for Damped Stochastic Nonlinear Schrödinger Equations

Stochastic nonlinear Schrödinger equations (NLSEs) arise in many applications, for instance, to model the nonlinear dispersive waves with perturbation due to random media or thermal fluctuations. In this chapter and Chap. 6, we investigate the construction of ergodic numerical approximations and the analysis of the error between invariant measures or error to the ergodic limit for stochastic NLSEs. This chapter mainly focuses on the stochastic NLSE with additive noise introduced in Sects. 3.3.1 and 4.5.

5.1 Ergodic Approximation and Numerical Invariant Measures This section is devoted to constructing a full discretization of ⎧   2 ⎪ ⎨ du = iΔu − αu + iλ|u| u dt + εdW, u(t, 0) = u(t, 1) = 0, t ≥ 0, ⎪ ⎩ u(0, x) = u 0 (x), x ∈ O := (0, 1)

(5.1)

with W the same as that in (4.19), and to solving the first problem mentioned at the beginning of Chap. 2. To this end, an ergodic semi-discretization and an ergodic full discretization are given in Sects. 5.1.1 and 5.1.2, respectively. In addition, the weak error, as well as the error between invariant measures, is given in Sect. 5.1.3. In particular, we choose the noise intensity ε = 1 throughout this section. Throughout this chapter, the s-norm  · s , s ∈ N, defined in Sect. 4.5 will be used.

© Springer Nature Singapore Pte Ltd. 2019 J. Hong and X. Wang, Invariant Measures for Stochastic Nonlinear Schrödinger Equations, Lecture Notes in Mathematics 2251, https://doi.org/10.1007/978-981-32-9069-3_5

109

110

5 Numerical Invariant Measures for Damped Stochastic …

5.1.1 Spectral Semi-discretization The M-dimensional spectral space with finite M ∈ N+ is defined as M . VM := span{ek }k=1

˙ 0 → VM be a projection operator, which is defined as Let π M : H πM u =

M ∞   ˙ 0. (u, ek )ek , ∀ u = (u, ek )ek ∈ H k=1

k=1

We use u M as an approximation to the original solution u, and the spatial semidiscrete scheme is expressed as ⎧

⎨ du M = iΔu M − αu M + iλπ M |u M |2 u M  dt + π M dW ⎩ u (0, x) = π u (x), M

where π M dW =

(5.2)

M 0

M √ ηk ek dβk , and the projection operator π M is bounded k=1 π M L (H˙ s ,L2 ) ≤ 1, ∀ s ∈ N.

Theorem 5.1 Let u M be the solution of Eq. (5.2), then u M possesses a unique invariant measure, denoted by μ M . Thus, u M is ergodic. Proof Following from Theorems 1.3 and 1.5, we need to show three properties of u M ,“strong Feller”, “irreducibility” and the “Lyapunov condition”, in order to show the ergodicity of u M . Thus the proof is divided into three parts as follows. Part 1. Strong Feller.  We transform (5.2) into an equivalent finite-dimensional SDE. Denoting ak (t) = u M (t, x), ek (x) , we have u M (t, x) =

M 

ak (t)ek (x).

k=1

Itô’s formula applied to ak (t) leads to     √ dak (t) = − λαk ak (t) + iλπ M |u M |2 u M , ek dt + ηk dβk (t)

(5.3)

for 1 ≤ k ≤ M. We decompose the above equation into its real and imaginary parts by denoting ak = ak1 + iak2 , λαk = λ1k + iλ2k and βk = βk1 + iβk2 , where {βki }i=1,2 1≤k≤M is a family of independent R-valued Wiener processes and the superscripts 1 and 2 mean the real and imaginary parts of a complex number, respectively, and obtain

5.1 Ergodic Approximation and Numerical Invariant Measures

111

⎧     √ ⎪ ⎨ dak1 = − λ1k ak1 + λ2k ak2 + iλπ M |u M |2 u M , ek dt + ηk dβk1 (t),     √ ⎪ ⎩ dak2 = − λ2k ak1 − λ1k ak2 +  iλπ M |u M |2 u M , ek dt + ηk dβk2 (t). With notations X (t) = (a11 (t), a12 (t), · · · , a 1M (t), a 2M (t)) , 1 2 , βM ) ∈ R2M , F = diag{Λ1 , · · · , Λ M }, βM

β = (β11 , β12 , · · · ,

   ⎞

 iλπ M  |u M |2 u M  , e1 ⎜  iλπ M |u M |2 u M , e1 ⎟  1 2  ⎜ ⎟ −λi λi ⎜ ⎟ .. Λi = ⎟ 2 1 , G(X (t)) = ⎜ . −λi −λi ⎜  ⎟    2 ⎝ iλπ M |u M | u M , e M ⎠      iλπ M |u M |2 u M , e M ⎛

and

⎛√ η1 √ ⎜ η1 ⎜ ⎜ .. Z =⎜ . ⎜ √ ⎝ ηM

⎞

√ ηM

⎟ ⎟ ⎟ ⎟ := (Z 11 , Z 12 · · · , Z 1M , Z 2M ), ⎟ ⎠

we get an equivalent form of (5.2) 2 2 M  M      Z ki dβki := Y (X (t)) dt + Z ki dβki . d X (t) = F X (t) + G X (t) dt + k=1 i=1

k=1 i=1

It is obvious that span{Z 11 , Z 12 , · · · , Z 1M , Z 2M } = R2M , which means that the Hörmander condition holds. According to Theorem 2.2 (see also [111]), X (t) is a strong Feller process. Part 2. Irreducibility. By using the same notations as above, we have d X = Y (X )dt + Z dβ,

(5.4)

with X = X (t) ∈ R2M , X (0) = y and Z being invertible. Using a similar technique as [139], we consider the associated control problem d X = Y (X )dt + Z dU,

(5.5)

with X = X (t) and a smooth control function U ∈ C1 (0, T ). For any fixed T > 0, polynomial interpolation, we derive a continuous y ∈ R2M and y + ∈ R2M , using  function X (t), t ∈ [0, T ] such that X (0) = y and X (T ) = y + . Hence,

112

5 Numerical Invariant Measures for Damped Stochastic …

  dU = Z −1 d X − Y (X )dt , and the control function U satisfies (5.5) with X (0) = y, X (T ) = y + and U (0) = 0. We subtract the resulting Eqs. (5.4) and (5.5), and achieve 

t

X (t) − X (t) =

Y (X (s)) − Y (X (s))ds + Z (β(t) − U (t)), t ∈ [0, T ].

0

Note that Y is locally Lipschitz continuous because of its continuous differentiability, and the ranges of X (t) and X (t) (t ∈ [0, T ]) are both compact sets. According to the following property of Brownian motion  P

     ≤ ε > 0, ∀ ε > 0, sup β(t) − U (t) 0≤t≤T

the Gronwall inequality in a small time interval and its continuation yield that   sup  X (t) − X (t) → 0 as ε → 0 0≤t≤T

holds with positive probability (see also [139]). For any δ > 0, by choosing ε > 0 small enough, we finally obtain

P |X (T ) − y + | < δ > 0. In other words, X (T ) hits B(y + , δ) with positive probability. The irreducibility has been proved. The above two conditions ensure the uniqueness of the invariant measure of X (t). It suffices to show the existence of invariant measures in the following. Part 3. Lyapunov condition. A useful tool for proving the existence of invariant measures is provided by Lyapunov functionals, which is introduced in Theorem 1.3. Itô’s formula applied to u M (t)20 implies that  du M (t)20 = − 2αu M (t)20 dt + 2

1 0

u M (t)π M dW (t)d x + 2

M 

ηk dt, (5.6)

k=1

where we have used the fact that      

1 1 2 4 2 |u M | − (I d − π M )(|u M | u M )u M d x

iλ π M (|u M | u M )u M d x = iλ 0



= −λ (I d − π M )(|u M |2 u M ), u M = 0

0

5.1 Ergodic Approximation and Numerical Invariant Measures

113

with I d being the identity operator. Taking the expectation on both sides of (5.6), we obtain d Eu M (t)20 = −2αEu M (t)20 + C M , dt where C M = 2

M k=1

Eu M (t)20 = e−2αt

ηk ≤ 2η. It is solved as

 t 0

C M e2αs ds + Eu M (0)20 ≤ e−2αt Eu M (0)20 + C, ∀ t > 0.

On the other hand,  u M (t)20

1

= 0

M  2   ak (t)ek (x) d x = X (t)l22 (R2M ) .  k=1

Define V =  · l 2 (R2M ) : R2M → [0, +∞]. The level sets of V are tight by the Heine– Borel theorem. Therefore, X (t) is ergodic according to Theorem 1.5. We mention that the ergodicity of X (t) is equivalent to the existence of a ran1 2 , ξM ) such that the following convergence holds dom variable ξ = (ξ11 , ξ12 , · · · , ξ M in distribution lim X (t) = ξ, i.e., lim aki (t) = ξki , ∀ k = 1, · · · , M, i = 1, 2.

t→∞

t→∞

It then leads to lim u M (t) =

t→∞

M   1  ξk + iξk2 ek , k=1

which shows the ergodicity of u M (t).



Remark 5.1 The above proof can be easily generalized for the degenerate case as mentioned in Sect. 3.6: there exists some N∗ ∈ N+ such that ηk > 0 for any k ≤ N∗ , in which case we need in addition that M ≥ N∗ . That is, the covariance operator Q may have zero eigenvalues. According to the proof of the Lyapunov condition, we have the following uniform boundedness for the 0-norm. Moreover, the 1-norm and 2-norm are also uniformly bounded, which is stated in the following proposition. In sequel, all the constants C are independent of the end point T of the time interval and may be different from line to line. ˙ 1 , Q 21 L 1 < ∞ and p ≥ 1. There exist posProposition 5.1 Assume that u 0 ∈ H 2 itive constants c0 and C = C(α, p, u 0 , c0 , Q), such that for any t > 0, (i) Eu M (t)0 ≤ e−2αpt Eu M (0)0 + C ≤ C, 2p

2p

(ii) E[H (u M (t)) p ] ≤ e−αpt E[H (u M (0)) p ] + C ≤ C,

114

5 Numerical Invariant Measures for Damped Stochastic …

where H (u M (t)) = 21 ∇u M (t)20 − λ4 u M (t)4L4 + c0 u M (t)60 . In addition, if as˙ 2 and Q 21 L 2 < ∞, we also have sume further u 0 ∈ H 2 (iii) Eu M (t)22 ≤ C. The proof of above proposition is given in Appendix C.1 for the convenience of readers. Remark 5.2 The uniform boundedness of the original solution u can also be obtained ˙ 2 -regularity for both the by the same procedure as Proposition 5.1 or [70]. As the H original solution and numerical solutions is essential to obtain the time-independent ˙ 2 and Q 21 L 2 < ∞ in the error analysis. weak error, we need the assumption u 0 ∈ H 2

5.1.2 Ergodic Full Discretization In this subsection, a modified implicit Euler scheme is applied in the temporal direction to obtain a full discretization. The well-posedness and uniform boundedness in  · 0 -norm of the numerical solution is first given such that the Lyapunov condition is satisfied, which implies the existence of invariant measures. Furthermore, the uniqueness of the invariant measure is proved utilizing the uniqueness of the numerical solution, which shows immediately that the numerical solution is ergodic. In addition, the higher regularity of the numerical solution, that is, the uniform boundedness of the numerical solution in  · 1 - and  · 2 -norms is also obtained, which is essential to get the order of convergence in weak sense. We apply a modified implicit Euler scheme to approximate (5.2), and obtain the following scheme    ⎧ n 2 −ατ n−1 2 |u | + |e u | ⎪ n−1 M M n −ατ n n ⎨u − e u uM τ + π M δn W M M = iΔu M + iλπ M 2 ⎪ ⎩ 0 u M = π M u 0 (x), (5.7) where u nM is an approximation of u M (tn ), τ represents the uniform time step-size, tn = nτ , and δn W = W (tn ) − W (tn−1 ). The well-posedness of scheme (5.7), together with the uniform boundedness of the numerical solution, is stated in the following proposition. ˙ 0 . For sufficiently small τ , there exists a unique Proposition 5.2 Assume that u 0 ∈ H VM -valued and {Ftn }n∈N -adapted solution {u nM }n∈N of (5.7), which satisfies that for any integer p ≥ 2, there exists a constant C = C( p, α, u 0M ) > 0, such that p

Eu nM 0 ≤ C, ∀ n ∈ N.

5.1 Ergodic Approximation and Numerical Invariant Measures

115

Proof Step 1. Existence and uniqueness of the solution. Similar to [65], we fix a family of deterministic functions gn ∈ VM , n ∈ N. We ˜ nM ∈ VM of also fix u˜ n−1 M ∈ VM , the existence of solution u  u˜ nM

−

e−ατ u˜ n−1 M

=

iτ Δu˜ nM

+ iλτ π M

2 |u˜ nM |2 + |e−ατ u˜ n−1 M | u˜ nM 2

 +

√

τ gn

(5.8)

can be proved by using Brouwer’s fixed point theorem. Indeed, multiplying (5.8) by n u˜ M , integrating with respect to x and taking the real part, we obtain 2 −2ατ 2 u˜ n−1 u˜ nM 20 + u˜ nM − e−ατ u˜ n−1 M 0 − e M 0  1   1 √ n −ατ n−1 −ατ n−1 =2 τ

(u˜ M − e u˜ M )gn d x + (e u˜ M )gn d x 0

0

2 −2ατ 2 2 u˜ n−1 ≤u˜ nM − e−ατ u˜ n−1 M 0 + e M 0 + 2τ gn 0 ,

i.e., 2 2 u˜ nM 20 ≤ 2e−2ατ u˜ n−1 M 0 + 2τ gn 0 .

(5.9)

Define Λ : VM × VM → P(L2 ), ˜ nM |u˜ nM are solutions of (5.8)}, (u˜ n−1 M , gn )  → {u where P(L2 ) is the power set of L2 . The inequality (5.9) implies that Λ is bounded, and its graph is closed by the closed graph theorem. When the spaces are endowed with their Borel σ -algebras, there is a measurable continuous function κ : VM × VM → L2 such that κ(u, g) ∈ Λ(u, g), ∀ (u, g) ∈ VM × VM . n Assume that u n−1 M ∈ VM is a Ftn−1 -measurable random variable, then u M = n−1 π M√δn W ) is an L2 -valued solution of (5.7). Moreover, κ(u M , τ

 (1 −

iΔτ )u nM

=

e−ατ u n−1 M

+ iλτ π M

2 |u nM |2 + |e−ατ u n−1 M | u nM 2

 + π M δn W ∈ V M .

Hence, u nM is actually a VM -valued solution of (5.7). and sufficiently small time step-size τ , the solution u nM is For any given u n−1 M uniquely defined, which can be proved in a similar procedure as [4]. This fact will be used in proving the ergodicity of the numerical solution {u nM }n∈N .

116

5 Numerical Invariant Measures for Damped Stochastic …

In fact, suppose that U and W are two solutions of the scheme, then it follows 

   τ 2 | (U − W ) . U − W = iτ Δ U − W + iλ π M |U |2 U − |W |2 W + |e−ατ u n−1 M 2 Multiplying the equation above by U − W , integrating in space and taking the real and imaginary part respectively, we have τ g(U ) − g(W )L 43 U − W L4 , 2 1 λ 2 2 ∇(U − W )20 ≤ g(U ) − g(W )L 43 U − W L4 + e−ατ u n−1 M L4 U − W L4 , 2 2

U − W 20 ≤

where g(U ) := |U |2 U and g(U ) − g(W )

4

L3

 3 4 4 1 3  2 2 = |U | U − |W | W  d x 0

 3 4 4 1 3  2 2 = |U | (U − W ) + |W | (U − W ) + U W (U − W ) d x 0

  1  1 2 2 4 1 1   2 2 4 ≤ |U − W | d x |U | + |W | + |U W | d x 0

0

2  ≤|U | + |W |L4 U − W L4 .

Since U − W 4L4 ≤ U − W 30 ∇(U − W )0

23 1 τ ≤ g(U ) − g(W )L 43 U − W L4 g(U ) − g(W )L 43 U − W L4 2 2 1

2 |λ| −ατ n−1 2 e u M L4 U − W 2L4 + 2

21 2 3  1 3 2 ≤ τ 2 |U | + |W |L4 |U | + |W |L4 + |λ|u n−1  U − W 4L4 4 M L 4

4 3  1 3  4 U − W 4L4 , ≤ τ 2 |U | + |W |L4 + |λ||U | + |W |L4 u n−1  L M 4 if U = W , then 

   |U | + |W |4 4 + |λ||U | + |W |3 4 u n−1 L4 M L L 

4 6  3 ≤C0 τ 2 |U | + |W | 4 + |λ||U | + |W | 4 + |λ|u n−1 2 4 .

1 3 1≤ τ2 4

L

L

M

L

5.1 Ergodic Approximation and Numerical Invariant Measures

117

For cases λ = 0 or −1, the L4 -norm of the solution is uniformly bounded, which leads to a contradiction when τ is sufficiently small. For case λ = 1, according to the fact that 3  9      |U | + |W |6 4 ≤ |U | + |W | 2 ∇(|U | + |W |) 2 ≤ M 29 |U | + |W |6 , L 0 0 0 9

3

we have C0 M 2 τ 2 > 1, which is also a contradiction when τ is sufficiently small. Thus, the numerical solution for (5.7) is unique. Step 2. Boundedness of the p-moments. We use the notation C to denote a generic constant, which does not depend on time and may be different from line to line. (i) p = 2. To show the boundedness, we multiply (5.7) by u nM , integrate in [0,1] with respect to the space variable, take expectation and take the real part,  1 n−1 2 2 −2ατ  − e Eu  = 2 E u nM π M δn W d x Eu nM 20 + Eu nM − e−ατ u n−1 0 0 M M 0  1  n  −ατ n−1 n −ατ n−1 2 u M − e u M π M δn W d x ≤ Eu M − e u M 0 + Eπ M δn W 20 . =2 E 0

We can derive 2 −2ατ n Eu 0M 20 Eu nM 20 ≤e−2ατ Eu n−1 M 0 + Cτ ≤ e   + Cτ 1 + e−2ατ + · · · + e−2ατ (n−1) Cτ ≤ Eu 0M 20 + C, ≤e−2αtn Eu 0M 20 + 1 − e−2ατ

where we have used e−2ατ < 1 − 2ατ e−2α for τ ∈ (0, 1). (ii) p = 4. In the case p = 2, without taking the expectation, we have 2 n −ατ n−1 2 u M 0 = 2

u nM 20 − e−2ατ u n−1 M 0 + u M − e



1 0

u nM π M δn W d x.

Multiplying both sides by u nM 20 , taking the expectation and taking the real part, we obtain

2 n 2 n −ατ n−1 2 u M 0 u nM 20 (L H S) =Eu nM 40 − e−2ατ Eu n−1 M 0 u M 0 + E u M − e

1

2 1 4 n 2 −2ατ 2 = Eu nM 40 − e−4ατ Eu n−1 u n−1 M 0 + E u M 0 − e M 0 2

2 n −2ατ n−1 2 n 2 + E u M − e u M 0 u M 0

118

5 Numerical Invariant Measures for Damped Stochastic …

and 

1

(R H S) =2 E 

0 1

=2 E 0

u nM 20 u nM π M δn W d x 

  u nM 20 u nM − e−ατ u n−1 π M δn W d x M

 1

 n 2  2 −ατ n−1 u M 0 − e−2ατ u n−1 u M π M δn W d x + 2 E M 0 e 0



n 2 n 2 n 2 2 ≤E u M − e−ατ u n−1 M 0 u M 0 + E u M 0 π M δn W 0

2 1 2 2 + E u nM 20 − e−2ατ u n−1 + 4e−2ατ Eu n−1 M 0 M π M δn W 0 4

1

2 2 n 2 n 2 −2ατ 2 ≤E u nM − e−ατ u n−1 u n−1 + Cτ. M 0 u M 0 + E u M 0 − e M 0 2 Comparing (L H S) with (R H S), we obtain 4 Eu nM 40 ≤ e−4ατ Eu n−1 M 0 + Cτ ≤ C.

(iii) p = 3. Using (i) and (ii), it is easy to check that the following holds true Eu nM 30 ≤ E

u nM 20 + u nM 40 ≤ C. 2

(iv) p > 4. By repeating the above procedure, we complete the proof.



Before showing the weak error between u M (t) and u nM , we need some a priori estimates on u nM 1 and u nM 2 given in Propositions 5.3 and 5.4, whose proofs are given in Appendices C.2 and C.3. ˙ 1 , u 0 = π M u 0 and Q 2 L 1 < Proposition 5.3 Assume that λ = 0 or −1, u 0 ∈ H M 2 ∞. Then for any p ≥ 1, there exists a constant C = C(α, u 0 , p) independent of M and tn , such that E[Hnp ] ≤ C, ∀ n ∈ N, 1

where Hn := ∇u nM 20 − λ2 u nM 4L4 . Corollary 5.1 Under the assumptions in Proposition 5.3, p Eu nM − e−ατ u n−1 M 0 ≤ Cτ , 2p

where constant C is independent of M and tn . Proof It is easy to check this by multiplying both sides of (C.9) by u nM − e−ατ u n−1 M , integrating with respect to x and taking the expectation,

5.1 Ergodic Approximation and Numerical Invariant Measures

Eu nM − e−ατ u n−1 M 0   1  =E τ  ∇u nM ∇(u nM − e−ατ u n−1 )d x +

M

119

2p

0



1 0

  dx π M δn W u nM − e−ατ u n−1 M

 n 2  n  n  τ 2 +  |u M | + |e−ατ u n−1 u M + e−ατ u n−1 dx u M − e−ατ u n−1 M | M M 4 0    p p 0 ≤Cτ p E ∇u nM 0 ∇ u nM − e−ατ u n−1 M



2p 2p n−1 2 p n 2p + Cτ p E u nM 1 + u n−1 u   + u  M 0 1 0 M M 1

p

1 2p 2p + CEπ M δn W 0 + Eu nM − e−ατ u n−1 M 0 2 1 2p p ≤ Eu nM − e−ατ u n−1 M 0 + Cτ . 2 

Then we complete the proof by Proposition 5.3.

˙ 2 and Q 2 L 2 < ∞, Proposition 5.4 Under the assumptions λ = 0 or −1, u 0 ∈ H 2 the uniform boundedness of the 2-norm holds, i.e., 1

Eu nM 22 ≤ C, ∀ n ∈ N, where C is also independent of M and tn . The numerical solution {u nM }n∈N is ergodic possessing a unique invariant measure utilizing the uniform boundedness given above. Theorem 5.2 For all τ sufficiently small, the solution {u nM }n∈N of scheme (5.7) has a unique invariant measure μτM . Thus, it is ergodic. Proof (i) Lyapunov condition. Based on Proposition 5.2, we can take essentially quadratic function V (·) = 1 +  · 20 as the Lyapunov functional, and the Lyapunov condition holds. (ii) Minorization condition (see Assumption 2.1 of [139]). In scheme (5.7), it gives PMn =e−ατ PMn−1 − τ ΔQ nM   n τλ 2 π M |PMn |2 + |Q nM |2 + |e−ατ PMn−1 |2 + |e−ατ Q n−1 − M | QM 2 M  √ + (5.10) ηk ek δn βk1 , Q nM

k=1 −ατ n−1 =e QM

+ τ ΔPMn   n τλ 2 π M |PMn |2 + |Q nM |2 + |e−ατ PMn−1 |2 + |e−ατ Q n−1 PM + M | 2

120

5 Numerical Invariant Measures for Damped Stochastic …

+

M  √ ηk ek δn βk2 ,

(5.11)

k=1

where PMn and Q nM denote the real and imaginary part of u nM respectively, that is   M √ ηk ek δn βk1 + iδn βk2 , where δn βk1 and u nM = PMn + iQ nM . Also, π M δn W = k=1 δn βk2 are the real and imaginary part of δn W respectively. For any y1 = a1 + ib1 , y2 = a2 + ib2 ∈ VM with ai and bi denoting the real M is a basis of VM , and imaginary part of yi (i = 1, 2) respectively, as {ek }k=1 1 2 M {δn βk , δn βk }k=1 can be uniquely determined to ensure that (PMn−1 , Q n−1 M ) = (a1 , b1 ) and (PMn , Q nM ) = (a2 , b2 ), which implies the irreducibility of u nM . As stated in Proposition 5.2, the Ftn -measurable solution {u nM }n∈N is uniquely π M√δn W ), where δn W has a C∞ density. Thus, the transition defined by u nM = κ(u n−1 M , τ kernel P1 (x, G) with G ∈ B(VM ) possesses a jointly continuous density p1 (x, y). Furthermore, densities pn (x, y) are achieved by the time-homogeneous property of Markov chain {u nM }n∈N . In conclusion, the minorization condition together with the Lyapunov condition leads to the existence and uniqueness of the invariant measure according to Theorem 2.5 in [139]. 

5.1.3 Weak Error and Error of Invariant Measures In this section, the error analysis in weak sense is studied for the considered model (5.1) in the linear sense (i.e., λ = 0). This result can be extended to the globally Lipschitz case as is investigated in [67]. For non-globally Lipschitz case, the exponential integrability of both the exact and numerical solutions are need in addition to get the order of convergence in both mean-square and weak sense, see for instance [55, 57] for the study of stochastic NLSEs and [112–115, 118] for other kinds of SDEs or SPDEs. We first establish the weak convergence error for the spatial semi-discretization (5.2) utilizing a transformation of u M (t) and the corresponding Kolmogorov equation. ˙ 2 and Q 2 L 2 < ∞. For any f ∈ Theorem 5.3 Let λ = 0. Assume that u 0 ∈ H 2 C2b (L2 ), there exists a constant C = C(u 0 , f, Q) independent of T such that 1

        E f u M (T ) − E f u(T )  ≤ C M −2 , ∀ T > 0.   Before proving Theorem 5.3, the error between semigroups of the original system and the spatial semi-discretization is given in the following lemma.

5.1 Ergodic Approximation and Numerical Invariant Measures

121

Lemma 5.1 Assume that S(t) and π M are defined as before. We have the following estimation S(t) − S(t)π M L (H˙ s ,L2 ) ≤ Ce−αt M −s . ˙ s , we have Proof For any u ∈ H  S(t)u − S(t)π M u0 = e  ≤e

−αt

|λ M |

− 2s

∞ 

−αt

u − π M u0 = e

−αt

 21

∞ 

|(u, ek )|

2

k=M+1

 21 |λαk |s |(u, ek )|2

≤ Ce−αt M −s us .

k=M+1

 Proof (of Theorem 5.3) We split the proof into three steps. Step 1. Calculation of E [ f (u(T ))]. For the case λ = 0, the mild solution of the considered Eq. (5.1) is 

T

u(T ) = S(T )u 0 +

S(T − t)dW (t),

0

which leads to   E[ f (u(T ))] = E f S(T )u 0 +

T



S(T − t)dW (t)

.

(5.12)

0

Denoting v(T − t, y) := E[ f (u(T ))|S(T − t)u(t) = y], it follows

∂v(T − t, y) 1 1 1 = − T r (S(T − t)Q 2 )∗ D 2 v(T − t, y)S(T − t)Q 2 . ∂t 2 In addition, according to the mild solution of (5.1) given above, we have    v(T − t, y) = E f y +

T

 S(T − s)dW (s)

.

t

For any h ∈ L2 , similar to [67, Lemma 5.13], we have  (Dv(T − t, y), h) = E

 Df

 y+

T

 S(T − s)dW , h

 ,

t

which satisfies |(Dv(T − t, y), h)| ≤ C f C1b h0 .

(5.13)

122

5 Numerical Invariant Measures for Damped Stochastic …

Similarly, we have       D 2 v(T − t, y), h , h  ≤ C f C2b h20 .

(5.14)

Step 2. Calculation of E [ f (u M (T ))]. The mild solution of the spatial semi-discretization (5.2) is 

T

u M (t) = S(t)π M u 0 +

S(t − s)π M dW (s).

0

We consider an auxiliary stochastic process: Y M (t) = S(T − t)u M (t), which satisfies dY M (t) = S(T − t)π M dW (t). Itô’s formula applied to t → v(T − t, Y M (t)) yields ∂v (T − t, Y M (t))dt ∂t + (Dv(T − t, Y M (t)), S(T − t)π M d W (t))

1 1 1 + T r (S(T − t)π M Q 2 )∗ D 2 v(T − t, Y M (t))S(T − t)π M Q 2 dt 2 = (Dv(T − t, Y M (t)), S(T − t)π M d W (t))

1 1 1 + T r (S(T − t)π M Q 2 )∗ D 2 v(T − t, Y M (t))S(T − t)π M Q 2 dt 2

1 1 1 − T r (S(T − t)Q 2 )∗ D 2 v(T − t, Y M (t))S(T − t)Q 2 dt. 2

dv(T − t, Y M (t)) =

Therefore, v(0, Y M (T )) =v(T, Y M (0)) +

 T 0

(Dv(T − s, Y M (s)), S(T − s)π M d W (s))



1 1 1 T T r (S(T − t)π M Q 2 )∗ D 2 v(T − t, Y M (t))S(T − t)π M Q 2 dt + 2 0 

1 1 1 T − T r (S(T − t)Q 2 )∗ D 2 v(T − t, Y M (t))S(T − t)Q 2 dt. 2 0

(5.15) According to the definition of Y M , we have Y M (0) = S(T )π M u 0 and Y M (T ) = u M (T ).

5.1 Ergodic Approximation and Numerical Invariant Measures

123

According to the representation of v, we have v(0, Y M (T )) = E [ f (u(T ))|u(T ) = Y M (T )] = E [ f (u M (T ))|u(T ) = u M (T )] and v(T, Y M (0)) =E [ f (u(T ))|S(T )u(0) = Y M (0)]    T

  S(T − t)d W (t) S(T )u(0) = S(T )π M u 0 . =E f S(T )π M u 0 + 0

Taking the expectation of the two sides of (5.15) and we obtian   E [ f (u M (T ))] =E f S(T )π M u 0 +

T

S(T − t)d W (t)



0

 T!

1 1 1 + E T r (S(T − t)π M Q 2 )∗ D 2 v(T − t, Y M (t))S(T − t)π M Q 2 2 0

" 1 1 − T r (S(T − t)Q 2 )∗ D 2 v(T − t, Y M (t))S(T − t)Q 2 dt. (5.16)

Step 3. Weak error of the solutions. Subtracting the resulting Eqs. (5.12) and (5.16) leads to E [ f (u M (T ))] − E [ f (u(T ))]   T 

1 S(T − t)Q 2 d W (t) − f S(T )u 0 + =E f S(T )π M u 0 + 0

T

1

S(T − t)Q 2 d W (t)



0

 T!

1 1 1 + E T r (S(T − t)π M Q 2 )∗ D 2 v(T − t, Y M (t))S(T − t)π M Q 2 2 0

" 1 1 − T r (S(T − t)Q 2 )∗ D 2 v(T − t, Y M (t))S(T − t)Q 2 dt = : I + I I.

(5.17)

Due to Lemma 5.1, term I is estimated as |I | ≤C  f C1b E S(T )u 0 − S(T )π M u 0 0 ≤Ce−αT  f C1b Eu 0 2 M −2 ≤Ce−αT M −2 .

(5.18) 1

Let us now estimate term I I . As (S(T − t)π M − S(T − t))Q 2 is a bounded linear operator and so is D 2 v as shown in (5.14), we have 

 T r (S(T − t)π M Q 21 )∗ D 2 v(T − t, Y M (t))S(T − t)π M Q 21 

124

5 Numerical Invariant Measures for Damped Stochastic …

 1 1  − T r (S(T − t)Q 2 )∗ D 2 v(T − t, Y M (t))S(T − t)Q 2  

 1 1   = T r ((S(T − t)π M − S(T − t))Q 2 )∗ D 2 v(T − t, Y M (t))(S(T − t)π M + S(T − t))Q 2  ≤CS(T − t)π M − S(T − t)L ≤Ce

−α(T −t)

M

−2

˙ 2 ,L2 ) Q (H

1 2

L 2  f C2 S(T − t)L 2

b

(L2 ,L2 ) Q

1 2

L

0 2

.

Hence, integrating the above equation leads to |I I | ≤

C −2 M . α

(5.19)

Plugging (5.18) and (5.19) into (5.17), we conclude           E f u M (T ) − E f u(T )  ≤ C e−αT + 1 M −2 ≤ C M −2 ,   α

(5.20) 

in which C is independent of time T .

We still use modified processes to calculate the weak error of the fully discrete scheme in temporal direction. Denote Sτ = (I d − iτ Δ)−1 e−ατ , then scheme (5.7) is rewritten as   n 2 −ατ n−1 2 | + |e u | |u n−1 M M u nM + eατ Sτ π M δn W u nM =Sτ u M + iλτ eατ Sτ π M 2   n 2  |u lM |2 + |e−ατ u l−1 M | n 0 ατ n+1−l l uM =Sτ u M + iλτ e Sτ πM 2 l=1 + eατ

n 

Sτn+1−l π M δl W.

(5.21)

l=1

Lemma 5.2 For any k ∈ N and sufficiently small τ , we have the following estimates, (i) Sτn − S(t)L (H˙ 2 ,L2 ) ≤ C(t + τ ) 2 e−αt τ 2 , t ∈ [tn−1 , tn+1 ], 1

(ii) Sτn − S(t)L (H˙ 1 ,H˙ 1 ) ≤ Ce−αt ,

1

t ∈ [tn−1 , tn+1 ],

where the constant C = C(α) is independent of n and τ . Proof Step 1. Let t = tn . As S(t) is the operator semigroup generated by the linear operator of the following equation ˙ 2, du(t) = (iΔ − α I d)u(t)dt, u(0) = u 0 ∈ H and Sτ = (I d − iτ Δ)−1 e−ατ is the discrete operator semigroup, we have

(5.22)

5.1 Ergodic Approximation and Numerical Invariant Measures

Sτn u(0) = u n = e−ατ u n−1 + iτ Δu n , S(tn )u(0) = u(tn ) = e−ατ u(tn−1 ) +



125

(5.23) tn

ie−α(tn −s) Δu(s)ds,

(5.24)

tn−1

where u n is the numerical solution of the proposed scheme applied to (5.22) n and  n u(tn ) is the exact solution of (5.22) at time tn . Denote E n = u − u(tn ) = Sτ − S(tn ) u(0) with E 0 = 0, then E n = e−ατ E n−1 + iτ ΔE n + i



tn



Δu(tn ) − e−α(tn −s) Δu(s) ds.

tn−1

Multiplying the above formula by E n , integrating with respect to x, and taking the real part, we derive 1 E n 20 + E n − e−ατ E n−1 20 − e−2ατ E n−1 20 2    1  tn  tn −α(tn −r ) ΔE n ie Δu(r )dr dsd x = i  ≤C ≤Ce

0 tn

tn−1 tn



s

Δu n − Δu(tk )0 Δu(r )0 dr ds

tn−1 s −2αtn

Δu(0)20 τ 2 ,

where we have used the fact that Δu n 20 ≤ e−2αtn Δu 0 20 , Δu(t)0 ≤ Ce−αt Δu(0)0 . In fact, multiplying (5.23) by Δu n − e−ατ Δu n−1 , integrating in space and taking the imaginary part, we obtain Δu n 20 ≤ e−2ατ Δu n−1 20 ≤ e−2αtn Δu 0 20 . Then it is easy to check that the fact E n 20 ≤ e−2ατ E n−1 20 + Ce−2αtn Δu(0)20 τ 2 leads to

E n 20 ≤ Ctn e−2αtn Δu(0)20 τ, 1

which finally yields Sτn − S(tn )L (H˙ 2 ,L2 ) ≤ Ctn2 e−αtn τ 2 in (i). 1

(5.25)

126

5 Numerical Invariant Measures for Damped Stochastic …

For (ii), by noticing that Δek = −k 2 π 2 ek , we have ∞  2

  n    α  −αtn 2 2 −n −ik 2 π 2 tn  S − S(tn ) u(0)2 = e (1 + iτ k (u(0), e π ) − e )  k  |λk | τ 1 k=1

≤4e−2αtn

∞ 

|(u(0), ek )|2 |λαk | = 4e−2αtn u(0)21 .

k=1

In the following two steps, we only give the proof of (i), and (ii) can be proved in a same procedure. Step 2. If t ∈ [tn−1 , tn ], then Sτn − S(t)L (H˙ 2 ,L2 ) ≤Sτn − S(tn )L (H˙ 2 ,L2 ) + S(tn ) − S(t)L (H˙ 2 ,L2 ) 1

1

1

1

1

1

≤Ctn2 e−αtn τ 2 + e−αt |e−α(tn −t) − 1| ≤Ctn2 e−αtn τ 2 + e−αt

∞  1 (ατ )n n! n=1

≤Ctn2 e−αtn τ 2 + e−αt ατ

eατ − 1 ατ

≤C(t + τ ) 2 e−αt τ 2 . 1

1

ατ

We have used the fact that e ατ−1 is uniformly bounded for ατ ∈ [0, 1]. Step 3. If t ∈ [tn , tn+1 ], we have Sτn − S(t)L (H˙ 2 ,L2 ) ≤Sτn − S(tn )L (H˙ 2 ,L2 ) + S(tn ) − S(t)L (H˙ 2 ,L2 ) 1

≤Ctn2 e−αtn τ 2 + e−αt |e−α(tn −t) − 1| 1

1

≤Ctn2 e−αt eα(t−tn ) τ 2 + e−αt ατ 1

eατ −1 ατ

≤C(t + τ ) 2 e−αt τ 2 , 1

1

where the fact eα(t−tn ) ≤ eατ ≤ e has been used.



Remark 5.3 From (5.23), we can also prove that Sτn L (L2 ,L2 ) ≤ Ce−αt , where n and t satisfy t ∈ [tn−1 , tn+1 ]. The next theorem gives the time-independent weak error of the solutions for different cases.

5.1 Ergodic Approximation and Numerical Invariant Measures

127

˙ 2 , u 0 = u M (0) = π M u 0 and Theorem 5.4 Let λ = 0. Assume that u 0 ∈ H M 1 Q 2 2L 2 < ∞. The weak error between (5.2) and (5.7) is independent of time and of 2

order 21 , i.e., for any f ∈ C2b (L2 ), there exists a constant C = C(u 0 , f ) independent of M, T and N such that   1  N  )] ≤ Cτ 2 , ∀ T = N τ. E[ f (u M (T ))] − E[ f (u M

Corollary 5.2 Under above assumptions,   1  N  )] ≤ Cτ 2 E[ f (u M (t))] − E[ f (u M holds for any t ∈ [(N − 1)τ, (N + 1)τ ]. Proof As    N  )] E[ f (u M (t))] − E[ f (u M        N  =E[ f (u M (T ))] − E[ f (u M (t))] + E[ f (u M (T ))] − E[ f (u M )] and     E[ f (u M (T ))] − E[ f (u M (t))] ≤  f C1b Eu M (T ) − u M (t)0   2 ≤ f C1b (T − t) sup Eu M (t)2 + Eu M (t)0 + Eu M (t)1 u M (t)0 t≥0

  1 +  f C1b Eπ M W (T ) − W (t) 0 ≤ Cτ 2 , we then complete the proof according to Theorem 5.4.



Proof (of Theorem 5.4) We split it into several steps. Step 1. Calculation of E[ f (u M (T ))]. Recall the process we constructed in the proof of Theorem 5.3, dY M (t) = S(T − t)π M dW (t). Denote v M (T − t, y) = E[ f (Y M (T ))|Y M (t) = y]. Then v M (0, Y M (T ))



=v M (T, Y M (0)) + 0

T



Dv M (T − t, Y M (t)), S(T − t)π M dW (t) ,

(5.26)

128

5 Numerical Invariant Measures for Damped Stochastic …

where v M (0, Y M (T )) = E[ f (u M (T ))|Y M (T ) = u M (T )] and v M (T, Y M (0)) =E[ f (Y M (T ))|Y M (0) = S(T )u M (0)]    T

  S(T − s)π M dW Y M (0) = S(T )u M (0) . =E f S(T )u M (0) + 0

The expectation of (5.26) implies,    E[ f (u M (T ))] = E f S(T )u M (0) +

T

 S(T − s)π M dW

.

(5.27)

0 N )]. Step 2. Calculation of E[ f (u M Similar to [71], we define a discrete modified process

Y Mn :=SτN −n u nM =SτN u 0M + eατ

n 

SτN +1−l π M δl W.

l=1

Consider the following time continuous interpolation of Y Mn , which is also VM -valued and {Ft }t≥0 -adapted, Y˜ M (t) :=SτN u 0M + eατ

 t N 0

SτN +1−l π M 1l (s)dW (s),

l=1

where 1l (s) = 1 if s ∈ [tl−1 , tl ], otherwise, 1l (s) = 0. In particular for t ∈ [tl−1 , tl ],

l−1 + eατ SτN +1−l π M W (t) − W (tl−1 ) , Y˜ M (t) =Y M

(5.28)



l + eατ SτN +1−l π M W (t) − W (tl ) . Y˜ M (t) =Y M

(5.29)

or equivalently,

Apply Itô’s formula to t → v M (T − t, Y˜ M (t)), dv M (T − t, Y˜ M (t)) =

N

 ∂v M (T − t, Y˜ M (t))dt + Dv M , eατ SτN +1−l π M 1l (t)dW (t) ∂t l=1

5.1 Ergodic Approximation and Numerical Invariant Measures

1 + Tr 2 

 e

ατ

N 

∗ 1 SτN +1−l π M Q 2 1l (t)

l=1

= Dv M , e

ατ

N 

 2

D vM e

ατ

129 N 

 1 SτN +1−l π M Q 2 1l (t)

dt

l=1



SτN +1−l π M 1l (t)dW (t)

l=1

+

1 2

N 

Tr

eατ SτN +1−l π M Q 2

1

∗



1 D 2 v M eατ SτN +1−l π M Q 2 1l (t)dt

l=1



1 1 ∗ 1 T r S(T − t)π M Q 2 D 2 v M S(T − t)π M Q 2 1l (t)dt, 2 l=1 N

−

where Dv M and D 2 v M are evaluated at (T − t, Y˜ M (t)). The same as before, integrating the formula above from 0 to T , and taking the expectation based on the fact that N N )|Y M (T ) = u M ] v M (0, Y˜ M (T )) =E[ f (Y M (T ))|Y M (T ) = Y˜ M (T )] = E[ f (u M

and v M (T, Y˜ M (0)) =E[ f (Y M (T ))|Y M (0) = Y˜ M (0)]      T  N N  S(T − s)π M dW Y M (0) = Sτ u M (0) , =E f Sτ u M (0) + 0

we obtain    N )] =E f SτN u M (0) + E[ f (u M +

1 2

N 

 S(T − s)π M dW

0



T

E 0

l=1

T





1 ∗ 1 T r eατ SτN +1−l π M Q 2 D 2 v M eατ SτN +1−l π M Q 2

− S(T − t)π M Q

1 2

∗

2

D vM



 S(T − t)π M Q 1l (t)dt. 1 2

(5.30)

Step 3. Weak convergence order. Subtracting (5.27) from (5.30), we derive N E[ f (u M )] − E[ f (u M (T ))]      T  S(T − s)π M d W − f S(T )u M (0) + =E f SτN u M (0) + 0

0

T

 S(T − s)π M d W

  T N



1 ∗ 1 1 E T r eατ SτN +1−l π M Q 2 D 2 v M eατ SτN +1−l π M Q 2 + 2 0 l=1

130

5 Numerical Invariant Measures for Damped Stochastic …



 1 ∗ 1 − S(T − t)π M Q 2 D 2 v M S(T − t)π M Q 2 1l (t)dt.

= : I + I I.

Now we estimate terms I and I I separately. The constants C below may be different and are all independent of T and τ .      T  N  S(T − s)π M dW |I | =E f Sτ u M (0) + 0      T   S(T − s)π M dW − E f S(T )u M (0) +  0

≤C f C1b SτN u M (0) − S(T )u M (0)0 ≤C f C1b SτN − S(T )L (H˙ 2 ,L2 ) u M (0)2 ≤C(T + τ ) 2 e−αT τ 2 , 1

1

(5.31)

˙ 2. where we have used Lemma 5.2 and u M (0) = π M u 0 ∈ H For term I I , similar to the same part in the proof of Theorem 5.3, we have II =

  T N



1 1 ∗ 1 E T r eατ SτN +1−l π M Q 2 D 2 v M eατ SτN +1−l π M Q 2 2 l=1 0



 1 ∗ 1 2 − S(T − t)π M Q 2 D v M S(T − t)π M Q 2 1l (t)dt

  T N

  ατ N +1−l 1 1 ∗ = E Tr e Sτ − S(T − t) π M Q 2 D 2 v M 2 l=1 0

   1 ατ N +1−l 2 · e Sτ − S(T − t) π M Q 



   ατ N +1−l 1 ∗ 1 2 2 2 1l (t)dt − S(T − t) π M Q D v M S(T − t)π M Q + 2T r e Sτ   T N 

 1 1 ∗ = E T r e2ατ SτN +1−l − S(T − t) π M Q 2 D 2 v M 2 l=1 0

  1 · SτN +1−l − S(T − t) π M Q 2 



 1 ∗ 1 + 2e2ατ SτN +1−l − S(T − t) π M Q 2 D 2 v M S(T − t)π M Q 2



 1 ∗ 1 2ατ 2 + (e − 1) S(T − t)π M Q 2 D v M S(T − t)π M Q 2 1l (t)dt 1 E 2 l=1 N

=:



T 0

(Al + 2Bl + Cl )1l (t)dt,

5.1 Ergodic Approximation and Numerical Invariant Measures

131

where Al , Bl and Cl satisfy 1 2 2 ˙ 2 ,L2 ) π M Q L (L2 ,H ˙ 2 )  f C2b L (H

E|Al | ≤CSτN +1−l − S(T − t))2 ≤C(T − t + τ )e−2α(T −t) τ,

1

E|Bl | ≤CSτN +1−l − S(T − t))L (H˙ 2 ,L2 ) π M Q 2 2

˙ 2) L (L2 ,H

≤C(T − t +

 f C2 S(T − t)L (L2 ,L2 ) b

1 1 τ ) 2 e−2α(T −t) τ 2

and E|Cl | ≤ Cτ π M Q 2 2L (L2 ,L2 )  f C2b S(T − t)2L (L2 ,L2 ) ≤ Ce−2α(T −t) τ. 1

It follows 1

|I I | ≤ Cτ 2 .

(5.32)

We then conclude from (5.31) and (5.32) that    1   N )  ≤ Cτ 2 , E [ f (u M (T ))] − E f (u M where C is independent of M, T and N .



Remark 5.4 For the linear case (λ = 0), the weak convergence order depends heavily on the regularity of the solution, which depends only on the regularity of the initial value and noise. We can achieve higher order by increasing the regularity of the initial value and the noise. For example, the weak order turns out to be 1 if we ˙ 4 and Q 21 L 4 < ∞. assume u 0 ∈ H 2 Based on the ergodicity of stochastic processes u and u M , for any deterministic ˙ 2 , we have the following two equations u0 ∈ H     1 T E f u(t) dt = f (y)dμ(y), T →∞ T 0 L2     1 T E f u M (t) dt = f (y)dμ M (y) lim T →∞ T 0 VM lim

for any f ∈ C2b (L2 ). Due to the time-independence of the weak error in Theorem 5.3, it turns out for any fixed α and M,               1 T    f (y)dμ M (y) =  lim E f u(t) − E f u M (t) dt   2 f (y)dμ(y) − T →∞ T 0 L VM     1 T   E f u(t) − E f u M (t)  dt ≤ lim T →∞ T 0

132

5 Numerical Invariant Measures for Damped Stochastic …

1 ≤ lim T →∞ T C ≤ M −2 , α



T

 C e

−αt

0

1 + α



M −2 dt

which implies that μ M is a proper approximation of μ. Thus, we give the following theorem. ˙ 2 , Q 21 L 3 < ∞ and f ∈ C2 (L2 ). Theorem 5.5 Let λ = 0. Assume that u 0 ∈ H b 2 The error between the invariant measures μ and μ M is of order 2, i.e.,    



L2

f (y)dμ(y) − VM

  C f (y)dμ M (y) < M −2 . α

Remark 5.5 Although the time-independent weak error between u and u M is ob1 tained under the assumption Q 2 L 22 < ∞, it is necessary to assume in addition 1 that Q 2 L 23 < ∞ in order to get the unique ergodicity of u given in Theorem 3.10. The error between the invariant measures μ M and μτM is derived in the same way. ˙ 2 , Q 21 L 2 < ∞ and f ∈ C2 (L2 ), the Theorem 5.6 Let λ = 0. Assume that u 0 ∈ H b 2 error between invariant measures μ M and μτM is of order 21 , i.e.,    

 

 f (y)dμ M (y) − VM

VM

f (y)dμτM (y)

1

< Cτ 2 .

5.1.4 Numerical Experiments This section provides numerical experiments to test the longtime behavior of scheme (5.7) for the case λ = 0. Based on the spatial semi-discretization (5.3) with λαk = i(kπ )2 + α: dak (t) = −i(kπ )2 ak (t)dt − αak (t)dt +

√ ηk dβk (t), 1 ≤ k ≤ M,

we derive an equivalent form of the full discretization (5.7) as ⎛ ⎜ an − e−ατ an−1 = − iτ π 2 ⎝

1

⎞

⎞ η1 δn β1 ⎟ n ⎜ ⎟ .. .. ⎠a + ⎝ ⎠, . . √ 2 M η M δn β M ⎛ √

where an := (a1n , · · · , a nM ) is an approximation of a(t) := (a1 (t), · · · , a M (t)) and δn βk = βk (tn ) − βk (tn−1 ) for 1 ≤ k ≤ M. In the sequel, we take α = 1, M = 100. Furthermore, we choose 1000 realizations to approximate the expectations.

5.1 Ergodic Approximation and Numerical Invariant Measures =2

-4

10 -4

8 initial(1) initial(2) initial(3) initial(4) initial(5)

6 5 4 3 2 1

Temporal Average

Temporal Average

10 -4

133 =2

-4

initial(1) initial(2) initial(3) initial(4) initial(5)

6

4

2

0

0 0

20

40

60

80

0

100

20

40

Fig. 5.1 The temporal averages 2−4 , T = 100)

60

80

100

t

t

1 N

 N −1 n=0

E[ f (an )] started from different initial values (τ =

 N −1 Ergodic limit. Figure 5.1 shows the temporal averages N1 n=0 E[ f (an )] of the fully discrete scheme starting from five different initial values initial(1) = (1, 0, · · · , 0) , initial(2) = (0.0003i, 0, · · · , 0) ,        2 100 1 π , sin π , · · · , sin π initial(3) = sin , 101 101 101   2+i (1, 2, · · · , 100) , initial(4) = 20        2i 100i i , exp − , · · · , exp − . initial(5) = exp − 50 50 50 They converge to the same value with error τ 2 before time T , where τ = 2−4 and T = 100. We choose ηk = k −3 in this experiment. This result verifies that the temporal averages converge to the spatial average, which is a constant, for almost every initial values in the whole space. √ Weak error. In Figs. 5.2 and 5.3, we fix the initial value u 0 (x) as 2 sin(π x), such that ak (0) = (u 0 , ek ) and a0 = a(0) = (1, 0, · · · , 0) . Figure 5.2 displays the weak error E[ f (a(tn )) − f (an )] over long time T = 103 for different time step-sizes and test functions: 1

(a) τ = 2−2 , f (a) = a2 exp(−a2 ), (b) τ = 2−4 , f (a) = a2 exp(−a2 ), (c) τ = 2−2 , f (a) = sin(a), (d) τ = 2−4 , f (a) = sin(a).

134

5 Numerical Invariant Measures for Damped Stochastic … 10 -4

8

=2

-2

6

=2

-4

10

Weak Error

Weak Error

10 -4

15

4 2

5

0

0 -2

-5 0

200

400

600

800

1000

0

200

400

t

10 -4

8

600

800

1000

600

800

1000

t

=2

-2

10 -4

15

=2

-4

Weak Error

Weak Error

6 4 2

10

5

0 0 -2 0

200

400

600

800

1000

0

200

400

t

t

Fig. 5.2 The weak error E[ f (a(tn )) − f (an )] for different f and step-size τ with tn = nτ ∈ [0, T ] and T = 103 Weak Order

Weak order

Strong Order 100

100

100

10-1

10-1

10-1

Order 0.4 Order 0.6 Order 1.0

10-2

k k

10-3 10-3

k

10-2

Order 0.4 Order 0.6 Order 1.0

10-2

=k -1 =k

k

-3 k

=k -5

10

-1

10-3 10-3

k

10-2

=k

Order 0.4 Order 0.6 Order 1.0

10-2

-1 k

=k

-3

=k

-5

k

10

-1

10-3 10-3

k

10-2

=k -1 =k

-3

=k -5

10-1

˙ 2 and H ˙ 4 , i.e., ηk = k −1 , k −3 , k −5 (T = 1, Fig. 5.3 The strong and weak orders for noise in L2 , H −i τ ∈ {2 , 5 ≤ i ≤ 9})

The reference values are generated for the time step-size τ = 2−8 , and the noise is ˙ 2 , i.e., ηk = k −3 . Figure 5.2 shows that the weak error is independent of chosen in H 1 time interval and can be controlled by Cτ 2 .

5.1 Ergodic Approximation and Numerical Invariant Measures

135

Convergence order. Figure 5.3 displays (a) the strong convergence order, (b) the rate of weak convergence for f (a) = a2 exp(−a2 ), (c) the rate of weak convergence for f (a) = sin(a). −12 The reference √ values are generated for the time step-size τ = 2 . As the initial value u 0 (x) = 2 sin(π x) is regular enough, both the strong and weak convergence order depend heavily on the regularity of the noise for the linear case. It shows in Fig. 5.3 ˙ 2 to H ˙ 4 (i.e., ηk from k −1 that the orders slightly increase as the noise from L2 via H −3 −5 via k to k ), which verifies Remark 5.4. Note that the orders are a little bit better than the theoretical results, because the truncation of the noise makes the noise more regular than it should be. Numerical tests also show that the weak convergence order is almost the same as the strong convergence order, which is similar to the statement in Remark 5.11, [67].

5.2 Ergodic and Conformal Multi-symplectic Full Approximation In this section, we further take the geometric structure into consideration, and aim to propose a fully discrete scheme to inherit both the ergodicity and the conformal multi-symplecticity of the original system (5.1).

5.2.1 Numerical Schemes We apply the central difference scheme to (5.1) in the spatial direction and obtain ⎧   K  ⎪ u j+1 − 2u j + u j−1 √ ⎪ 2 ⎪ − αu j + iλ|u j | u j dt + ε ηk ek (x j )dβk (t), ⎪ ⎨ du j = i h2 ⎪ ⎪ u 0 (t) = u M+1 (t) = 0, ⎪ ⎪ ⎩ u j (0) = u 0 (x j ),

k=1

(5.33) where h is the uniform spatial step-size and u j := u j (t) is an approximation of u(x j , t) with x j = j h, j = 1, 2, · · · , M and (M + 1)h = 1. With the notation U = (u 1 , · · · , u M ) ∈ C M , β = (β1 , · · · , β K ) ∈ C K , √ √ F(U ) = diag{|u 1 |2 , · · · , |u M |2 }, Λ = diag{ η1 , · · · , η K },

136

5 Numerical Invariant Measures for Damped Stochastic …

⎞ ⎞ ⎛ −2 1 e1 (x1 ) · · · e K (x1 ) ⎟ ⎜ 1 −2 1 ⎟ ⎜ ⎟ ⎜ .. M×K ∈ R M×M and σ = ⎝ ... A=⎜ , ⎠∈R .. .. .. ⎟ . ⎝ . . .⎠ e1 (x M ) · · · e K (x M ) 1 −2 ⎛

we rewrite (5.33) into a finite dimensional SDE ⎧   ⎪ ⎨ dU = i 1 AU − αU + iλF(U )U dt + εσ Λdβ, h2 ⎪ ⎩ U (0) = (u (x ), · · · , u (x )) . 0

1

0

(5.34)

M

In the sequel, we still denote by  ·  the Euclidean norm for vectors or matri 1/2  M 2 ces, i.e., v = for a vector v = (v1 , · · · , v M ) ∈ C M and A = j=1 |v j | √ max{ μ : μ is an eigenvalue of A A}. Lemma 5.3 Matrix A is uniformly bounded for any dimension M, more precisely, A ≤ 4. Proof Based on the definition of A, we only need to show that the maximum absolute value λ∗ := max{|λ| : det(λI d − A) = 0} of eigenvalues of A ∈ R M×M is uniformly bounded with respect to dimension M, and no larger than 4, then the lemma holds. In fact, according to the Gerschgorin theorem and the properties of # MA, all the G i (A) eigenvalues λi , i = 1, · · · , M, of A lie in the Gerschgorin area G M = i=1 with G i (A) := {z ∈ R : |z + 2| ≤ Ri }, i = 1, · · · , M.  Here, Ri := M j=1, j=i |ai j | = 2 for any i = 1, · · · , M with ai j being the components of matrix A = (ai j ) M×M . As a result, we have λi ∈ G M = {z ∈ R : −4 ≤ z ≤ 0} for all i = 1, · · · , M, which completes the proof.



The solution of (5.34) is uniformly bounded, which is stated in the following proposition. Proposition 5.5 Assume that Eu 0 2L2 < ∞, then the solution U of (5.34) is uniformly bounded with hEU (t)2 ≤ e−2αt hEU (0)2 + where η(K ) :=

K k=1

ηk .

2ε2 η(K ) (1 − e−2αt ), α

(5.35)

5.2 Ergodic and Conformal Multi-symplectic Full Approximation

137

Proof Similar to the proof of Proposition 4.1, we apply Itô’s formula to U (s)2 and obtain

dU (s)2 = 2 [U dU ] + (εσ Λdβ) (εσ Λdβ)

= −2αU (s)2 ds + 2 [U εσ Λdβ] ⎡   K ⎤ K M   √ √ ⎣ + ε2 ηk ek (x j )dβk ηk ek (x j )dβk ⎦ . j=1

k=1

(5.36)

k=1

Taking the expectation on both sides of (5.36) leads to dEU (s)2 = − 2αEU (s)2 ds + 2ε2

M  K 

ηk ek2 (x j )ds.

j=1 k=1

Thus, multiplying the above equation by he2αs and integrating from 0 to t leads to  t 0

he2αs dEU (s)2 +

Based on the fact that

 t 0

2αhe2αs EU (s)2 ds = 2ε2 h

M  K 

ηk ek2 (x j )

j=1 k=1

M

2 j=1 en (x j )

 t

e2αs ds.

0

≤ 2M ≤ 2h −1 , we have  ε2 h 2αt (e − 1) ηk ek2 (x j ) α j=1 k=1 M

e2αt hEU (t)2 − hEU (0)2 = ≤

K

2ε2 η(K ) 2αt (e − 1) α

(5.37)

which completes the proof. In addition, noting that for h = 1/(M + 1) and x j = j h, j = 1, · · · , M, Eu 0 2L2

=E

M   j=1

=E

xj

|u 0 (x)|2 d x

x j−1

M 

|u 0 (x j )|2 h + O(h) = hEU (0)2 + O(h),

j=1

the uniform boundedness under the assumption Eu 0 2L2 < ∞ is obtained.



Remark 5.6 Scheme (5.33) is equivalent to the symplectic Euler scheme applied to (4.21), i.e.,

138

5 Numerical Invariant Measures for Damped Stochastic …

⎧ p j+1 − p j = hv j+1 , ⎪ ⎪ ⎪ ⎪ ⎪ q j+1 − q j = hw j+1 , ⎪ ⎪ ⎪ ⎪ K ⎪  ⎨ dβ 2 (t) √ v j+1 − v j = h(q j )t + αhq j − h(( p j )2 + (q j )2 ) p j − ε ηk ek (x j ) k , dt ⎪ ⎪ k=1 ⎪ ⎪ ⎪ K ⎪  ⎪ dβk1 (t) √ ⎪ 2 2 ⎪ w . − w = −h( p ) − αhp − h(( p ) + (q ) )q + ε η e (x ) ⎪ j+1 j j t j j j j k k j ⎩ dt k=1

For the construction of fully discrete schemes, the splitting technique is applied such that the proposed scheme could inherit the properties of (5.1). We drop the linear terms and the stochastic term for the moment and consider the following equation dU (t) − iλF(U (t))U (t)dt = 0

(5.38)

first. Multiplying F(U (t)) to both sides of (5.38) and taking the imaginary part, we obtain U (t)2 = U (0)2 , which implies that F(U (t)) = F(U (0)). Thus, (5.38) is shown to possess a unique solution U (t) = eiλF(U (0))t U (0). For the linear equation  1 AU (t) + iαU (t) dt = εσ Λdβ, dU (t) − i h2 

a modified midpoint scheme is applied to obtain its full discretization. Now we define the following splitting scheme initialized with U 0 = U (0): U n+1 + e− 2 ατ U˜ n iτ U n+1 + e− 2 ατ U˜ n 1 U n+1 = e− 2 ατ U˜ n + 2 A − ατ + εσ Λδn+1 β, h 2 4 (5.39a) n n iλF(U )τ n U˜ = e U , (5.39b) 1

1

where U n = (u n1 , · · · , u nM ) ∈ C M , τ denotes the uniform time step-size, δn+1 β = β(tn+1 ) − β(tn ) and tn = nτ , n ∈ N. Note that scheme (5.39) can be rewritten as   τ n A U n+1 + e f (U ) U n 2 2h  1  n − ατ U n+1 + e f (U ) U n + εσ Λδn+1 β, 4

U n+1 − e f (U ) U n =i n

which can also be expressed in the following explicit form

(5.40)

5.2 Ergodic and Conformal Multi-symplectic Full Approximation

139

−1   iτ iτ 1 1 n I d + 2 A − ατ I d e f (U ) U n = I d − 2 A + ατ I d 2h 4 2h 4 (5.41)  −1 iτ 1 + I d − 2 A + ατ I d εσ Λδn+1 β 2h 4 

U

n+1

  with f (U n ) = − 21 α I d + iλF(U n ) τ . Thus, there uniquely exists an adapted solution {U n }n∈N+ of (5.40) for sufficiently small τ . As for the proposed splitting scheme (5.39), the solution of the one-step approximation (5.39b) coincides with the exact solution of the Hamiltonian system dU (t) − iλF(U (t))U (t)dt = 0. That is, the phase flow U n → U˜ n preserves the symplectic structure. It then suffices to show that (5.39a) possesses the conformal multi-symplectic conservation law, which is stated in the following theorem. Theorem 5.7 The one-step approximation defined through (5.39a) possesses the discrete conformal multi-symplectic conservation law

e

−ατ

dz n+1 ∧ M4 dz n+1 − dz nj ∧ M4 dz nj j j τ

n+ 21

+

dz j

n+ 1

n+ 1

∧ (K 41 dz j+12 − K 42 dz j−12 ) h

1 n+ 1 n+ 1 = − αdz j 2 ∧ M4 dz j 2 , 2 n+ 1

where z nj = ( p nj , q nj , vnj , wnj ) , z j 2 = 21 (z n+1 + e− 2 ατ z nj ), vnj+1 := ( p nj+1 − p nj )h −1 j and wnj+1 := (q nj+1 − q nj )h −1 with p nj and q nj being the real and imaginary parts of u nj , respectively. Moreover, ⎛

0 ⎜ 0 K 41 = ⎜ ⎝0 0

1

⎛ ⎞ 0 −1 0 0 ⎜ 0 0 −1 ⎟ ⎟, K2 = ⎜0 4 ⎝1 0 0 0⎠ 0 0 0 0

0 0 0 1

⎞ 0 0⎟ ⎟ 0⎠ 0

0 0 0 0

such that K 4 := K 41 + K 42 and M4 are the same as those in Sect. 4.1. Proof We denote U˜ n still by U n for convenience in this proof, and we have U n+1 + e− 2 ατ U n iτ U n+1 + e− 2 ατ U n − ατ + εσ Λδn+1 β U + 2A h 2 4 1

U

n+1

=e

− 21 ατ

1

n

  with U n = u n1 , · · · , u nM ∈ C M . Denote by δn+1 β 1 and δn+1 β 2 the real and imaginary parts of δn+1 β, respectively. Noticing that the jth component of h −2 AU n can be expressed as h −1 (vnj+1 − vnj ) + ih −1 (wnj+1 − wnj ), we decompose (5.39a) with its real and imaginary parts respectively and derive

140

5 Numerical Invariant Measures for Damped Stochastic …

⎧ n+1 1 n+1 p j − e− 2 ατ p nj wn+1 wn − wnj ⎪ j+1 − w j ⎪ − 21 ατ j+1 ⎪ + + e ⎪ ⎪ ⎪ τ 2h 2h ⎪ ⎪ ⎪ 1 1 ⎪ n+1 − ατ n ⎪ = − α( p j + e 2 p j ) + εσ Λδn+1 β 1 , ⎨ 4 n+1 n+1 − 21 ατ n ⎪ q j v j+1 − vn+1 vn − vnj ⎪ qj − e j ⎪ − 21 ατ j+1 ⎪ − − e ⎪ ⎪ ⎪ τ 2h 2h ⎪ ⎪ ⎪ 1 1 ⎪ ⎩ = − α(q n+1 + e− 2 ατ q nj ) + εσ Λδn+1 β 2 . j 4 Combining formula vnj+1 = ( p nj+1 − p nj )h −1 , wnj+1 = (q nj+1 − q nj )h −1 with above equations, we have z n+1 − e− 2 ατ z nj j

n+ 1

1

M4

τ

+

K 41

n+ 21

z j+12 − z j h

n+ 1

k+ 21

+

K 42

zj

n+ 1

− z j−12 h

n+ 1

1 n+ 1 n+ 1 = − α M4 z j 2 + ξ j 2 , 2

n+ 1

where ξ j 2 := (εσ Λδn+1 β 2 , −εσ Λδn+1 β 1 , v j 2 , w j 2 ) . Taking differential in phase space on both sides of the above equation, and performing the wedge product n+ 21

with dz j

respectively, we show the discrete conformal multi-symplectic conservan+ 21

tion law based on the symmetry of matrix −K 41 + K 42 and the fact dz j n+ 21

K 42 )dz j

n+ 21

= 0, dz j

n+ 21

∧ dξ j

∧ (−K 41 +

= 0.



Remark 5.7 It is also feasible to show that scheme (5.39) are conformal symplectic in time, which together with Remark 5.6, yields the conformal multi-symplecticity of the fully discrete scheme (5.40). 1

Proposition 5.6 Assume that Eu 0 2L2 < ∞, Q 2 ∈ L22 and K ≤ C∗ (M + 1) for some constant C∗ ≥ 1, then the solution {U n }n∈N+ of (5.40) is uniformly bounded, i.e., hEU n 2 ≤ e−αtn hEU 0 2 + C

(5.42)

with tn = nτ and the constant C depending on α, ε, Q and C∗ .   Proof We multiply U n+1 + e f (U n ) U n to (5.40), take the real part and expectation, and obtain EU n+1 2 − e−ατ EU n 2

     1 n) n+1 f (U n ) n 2 n+1 f (U n = − ατ EU +e U  +E U −e U εσ Λδn+1 β 4    τ  1 n = − ατ EU n+1 + e f (U ) U n 2 + E − i 2 A U n+1 + e f (U n ) U n 4 2h

5.2 Ergodic and Conformal Multi-symplectic Full Approximation

141

   1  n+1 n) f (U n − ατ U +e U + εσ Λδn+1 β εσ Λδn+1 β 4 1 1 n n ≤ − ατ EU n+1 + e f (U ) U n 2 + ατ EU n+1 + e f (U ) U n 2 4 8 1 n −2 2 + Cτ Eh Aεσ Λδn+1 β + ατ EU n+1 + e f (U ) U n 2 8 + Cτ Eεσ Λδn+1 β2 + Eεσ Λδn+1 β2 .

(5.43)

For the smooth functions ek (x), k = 1, · · · , K , we have     Δek (x j ) − ek (x j+1 ) − 2ek (x j ) + ek (x j−1 )  ≤ Ck 4 h 2 ≤ Ck 2 , k ≥ 1   2 h based on the fact kh ≤ K (M + 1)−1 ≤ C∗ . Thus, 2  K M    e (x ) − 2e (x ) + e (x ) √ k j+1 k j k j−1   Eh −2 Aεσ Λδn+1 β2 = ε2 E ηk δ β  n+1 k   h2 j=1

≤ 2ε2

K M  

k=1

K   2 ηk |Δek (x j )| + Ck 2 τ ≤ C Mτ k 4 ηk ≤ Ch −1 τ.

j=1 k=1

(5.44)

k=1

In the last step, we have used the fact

K  k=1

1

k 4 ηk ≤ CQ 2 L 22 ≤ C. Similarly,

2  K M    √   E ηk ek (x j )δn+1 βk  ≤ C Mητ ≤ Ch −1 τ. (5.45) Eεσ Λδn+1 β = ε   2

2

j=1

k=1

Substituting (5.44) and (5.45) into (5.43), we obtain hEU n+1 2 ≤ e−ατ hEU n 2 + Cτ ≤ e−αtn+1 hEU 0 2 + Cτ which, together with the fact

1−e−αtn 1−e−ατ

≤

1 , ατ

1 − e−αtn , 1 − e−ατ

completes the proof.



Theorem 5.8 Under the assumptions in Proposition 5.6 and ηk > 0 for k = 1, · · · , K , the solution {U n }n∈N+ of (5.40) is uniquely ergodic with a unique invariant measure, denoted by μτh , satisfying  N −1 1  n lim E f (U ) = f dμτh , ∀ f ∈ Cb (C M ). N →∞ N M C n=0

(5.46)

142

5 Numerical Invariant Measures for Damped Stochastic …

Proof For any fixed h > 0, we choose V (·) := h · 2 as the Lyapunov functional, which satisfies that the level sets K c := {u ∈ C M : V (u) ≤ c} are compact for any c > 0 and E[V (U n )] ≤ V (U 0 ) + C for any n ∈ N. Thus, the Markov chain {U n }n∈N possesses an invariant measure according to Theorem 1.3 (see also [58, Proposition 7.10]). Next we show that {U n }n∈N is irreducible and strong Feller (also known as the minorization condition in Assumption 2.1 of [139]), which yields the uniqueness of the invariant measure. In fact, for any u, v ∈ C M , we can derive from (5.40) that δ1 β can be chosen as εσ Λδ1 β = v − e f (u) u − i

  1   τ A v + e f (u) u + ατ v + e f (u) u 2 2h 4

such that U 0 = u, U 1 = v, where we have used the fact that σ is full rank and Λ is invertible. Thus, we can conclude based on the homogenous property of the Markov chain {U n }n∈N that the transition kernel Pn (u, A) := P(U n ∈ A|U 0 = u) > 0, which implies the irreducibility of the chain. On the other hand, as δ1 β has a C∞ density, it follows from (5.41) that U 1 also has a C∞ density for any deterministic initial value U 0 = u. Then explicit construction shows that {U n }n∈N possesses a family of C∞ densities and is strong Feller.  The conformal multi-symplecticity, uniform boundedness of the charge and ergodicity for scheme (5.40) are clearly consistent with the continuous results (4.22), (4.27) and (3.12), respectively. The next result concerns the error estimation of the proposed scheme, where the truncation technique will be used to deal with the nonglobally Lipschitz nonlinearity.

5.2.2 Convergence in Probability In this section, we focus on the approximate error for the proposed scheme in temporal direction. As the nonlinear term is not global Lipschitz, we consider the following truncated function first   1 (5.47) dU R = i 2 AU R − αU R + iλFR (U R )U R dt + εσ Λdβ, h with U R := U R (t) = (u R,1 (t), · · · , u R,M (t)) and the initial value U R (0) = U (0).

F(v) for any vector v ∈ C M and a cut-off function θ ∈ Here FR (v) := θ v R ∞ C (R) satisfying θ (x) = 1 for x ∈ [0, 1] and θ (x) = 0 for x ≥ 2 (see also [67, 136]). In addition, we have  FR (U R ) = θ

U R  R



 max |u R, j |2 ≤ θ

1≤ j≤M

 U R  U R 2 ≤ 4R 2 . R

5.2 Ergodic and Conformal Multi-symplectic Full Approximation

143

As a result, the nonlinear term FR (U R )U R is globally Lipschitz with respect to the norm  · . The proposed scheme (5.41) applied to the truncated Eq. (5.47) yields the following scheme −1    iτ iτ 1 1 n I d + 2 A − ατ I d e f R (U R ) U Rn U Rn+1 = I d − 2 A + ατ I d 2h 4 2h 4 (5.48) −1  iτ 1 + I d − 2 A + ατ I d εσ Λδn+1 β, 2h 4   where f R (U Rn ) := − 21 α I d + iλFR (U Rn ) τ and U Rn = (u nR,1 , · · · , u nR,M ) . 1

Theorem 5.9 For Eq. (5.47) and scheme (5.48), assume that Eu 0 2L2 < ∞, Q 2 ∈ L22 , α ≥ 21 and τ = O(h 4 ). For T = N τ , there exists a constant C R which depends on α, ε, R, Q, u 0 and is independent of T and N such that hEU R (T ) − U RN 2 ≤ C R τ 2 . Proof Denote the semigroup operator S(t) := e Et which is generated by the linear operator E := i h12 A − α2 I d, then the mild solution of (5.47) is  U R (tn+1 ) =S(τ )U R (tn ) +  − tn

tn+1

S(tn+1 − s)iλFR (U R (s))U R (s)ds

tn tn+1

α S(tn+1 − s) U R (s)ds + 2



tn+1

(5.49)

S(tn+1 − s)εσ Λdβ(s).

tn

Subtracting (5.48) from (5.49), we obtain U R (tn+1 ) − U Rn+1 −1    1 1 n I d + Eτ e f R (U R ) U Rn =S(τ )U R (tn ) − I d − Eτ 2 2  tn+1  tn+1 α + S(tn+1 − s)iλFR (U R (s))U R (s)ds − S(tn+1 − s) U R (s)ds 2 tn tn −1    tn+1  1 + S(tn+1 − s) − I d − Eτ εσ Λdβ(s) 2 tn  −1      1 1 n I d + Eτ =S(τ ) U R (tn ) − U R + S(τ ) − I d − Eτ U Rn 2 2 −1      1 1 n I d + Eτ U Rn − e f R (U R ) U Rn + I d − Eτ 2 2  tn+1  tn+1 1 + S(tn+1 − s)iλFR (U R (s))U R (s)ds − S(tn+1 − s) αU R (s)ds 2 tn tn

144

5 Numerical Invariant Measures for Damped Stochastic …

 + tn

tn+1





1 S(tn+1 − s) − I d − Eτ 2

−1 

εσ Λdβ(s)

= : I + I I + I I I + I V + V + V I. To show the strong convergence order of (5.48), we give the estimates of above terms, respectively. For terms I and I I , we have  1 α 2   EI 2 = E e(i h2 A− 2 I d)τ (U R (tn ) − U Rn ) = e−ατ EU R (tn ) − U Rn 2 and

EI I 2 ≤CE(Eτ )3 U Rn 2 ≤ Cτ 6 E 3 2 EU Rn 2 ≤Ch −13 τ 6 A6 ≤ Ch −13 τ 6

(5.50)

(5.51)

  based on e x − (1 − x2 )−1 (1 + x2 ) = O(x 3 ) as x → 0, Lemma 5.3 and Proposition 5.6. For term V I , the Taylor expansion yields that   EV I  ≤2E   2

2  (S(tn+1 − s) − S(τ )) εσ Λdβ(s)  tn  2  −1    1   + 2E  S(τ ) − I d − Eτ εσ Λδn+1 β    2 tn+1

(5.52)

≤Cτ 2 EEεσ Λδn+1 β2 ≤ Ch −1 τ 3 . It then remains to estimate terms I I I , I V and V . We obtain the following equation in the same way as that of (5.43) U Rn+1 2 − e−ατ U Rn 2 ≤ Cτ h −2 Aεσ Λδn+1 β2 + Cεσ Λδn+1 β2 . Multiplying the above equation by U Rn+1 2 , we derive 2  U Rn+1 4 + U Rn+1 2 − e−ατ U Rn 2 − e−2ατ U Rn 4   ≤Cτ U Rn+1 2 − e−ατ U Rn 2 h −2 Aεσ Λδn+1 β2 + Cτ e−ατ U Rn 2 h −2 Aεσ Λδn+1 β2   + C U Rn+1 2 − e−ατ U Rn 2 εσ Λδn+1 β2 + Ce−ατ U Rn 2 εσ Λδn+1 β2 2  ≤ U Rn+1 2 − e−ατ U Rn 2 + τ e−2ατ U Rn 4 + Cτ h −2 Aεσ Λδn+1 β4 C + εσ Λδn+1 β4 . τ

5.2 Ergodic and Conformal Multi-symplectic Full Approximation

145

Based on (5.44) and (5.45), we take the expectation of the above equation and derive EU Rn+1 4 ≤ (1 + τ )e−2ατ EU Rn 4 + Ch −2 τ ≤ (1 + τ )n+1 e−2ατ (n+1) EU R0 4 + Ch −2 ≤ Ch −2 , where we have used the fact that EU R0 4 ≤ (EU R0 2 )2 ≤ Ch −2 and (1 + τ )e−2ατ < 1 for α ≥ 21 . Similarly, we derive EU Rn 8 ≤ Ch −4 , which implies that ⎛

⎞2 M    u n 4 ⎠ ≤ EU n 8 ≤ Ch −4 , ∀ n ∈ N. EFR (U Rn )4 = E ⎝ R, j R j=1

Thus, by the Taylor expansion, we have III + IV + V (5.53)  −1  

1 1 = I d − Eτ I d + Eτ − f R (U Rn ) + O( f (U Rn )2 ) U Rn 2 2  tn+1  tn+1 1 + S(tn+1 − s)iλFR (U R (s))U R (s)ds − S(tn+1 − s) αU R (s)ds 2 tn tn   −1    tn+1  1 1 n n I d + Eτ FR (U R )U R ds =iλ S(tn+1 − s)FR (U R (s))U R (s) − I d − Eτ 2 2 tn   −1     tn+1 1 1 1 S(tn+1 − s)U R (s) − I d − Eτ − α I d + Eτ U Rn ds 2 tn 2 2  −1   1 1 + I d − Eτ I d + Eτ O( f R (U Rn )2 )U Rn 2 2 = : I I˜ I + I ˜V + V˜ . (5.54)

Now we estimate the above terms one by one. For I I˜ I , we have 

−1   1 1 I d + Eτ I I˜ I =iλ S(tn+1 − s) − I d − Eτ FR (U Rn )U Rn ds 2 2 tn  tn+1

+ iλ S(tn+1 − s) FR (U R (s))FR (U R (s)) − FR (U Rn )U Rn ds 

tn+1



tn

= : I I˜ I 1 + I I˜ I 2 , which satisfies  2  −1     1 1 E FR (U Rn )U Rn  I d + Eτ  S(tn+1 − s) − I d − 2 Eτ  ds 2 tn ⎡ ⎤ M     n 6 ⎦ ≤Ch −12 τ 8 EFR (U Rn )U Rn 2 ≤ Ch −12 τ 8 E ⎣ u R, j  ≤ Ch −15 τ 8

E I I˜ I 1 2 ≤τ



tn+1

j=1

146

5 Numerical Invariant Measures for Damped Stochastic …

and E I I˜ I 2 2 ≤τ



tn+1

tn





2   ES(tn+1 − s) FR (U R (s))FR (U R (s)) − FR (U Rn )U Rn  ds

tn+1

≤Cτ tn

EU R (s) − U R (tn )2 ds + Cτ 2 e−ατ EU R (tn ) − U Rn 2 .

Note that  s  s  α EU R (s) − U R (tn )2 =E S(s − r )iλF (U (r ))U (r )dr − S(s − r ) U R (r )dr R R R  2 tn tn 2  s  −3 2 + S(s − r )εσ Λdβ(r )  ≤h τ . tn

We then get E I I˜ I 2 ≤ Ch −15 τ 8 + Ch −3 τ 4 + Cτ 2 e−ατ EU R (tn ) − U Rn 2 .

(5.55)

Term I ˜V can be estimated in the same way as the estimation of I I˜ I . Term V˜ turns to be    4  1    2 n 2 n 2 4 n 2 n 2   ˜ EV  ≤CE f R (U R ) U R ≤ Cτ E sup − α + iλ|u R, j |  U R  2 1≤ j≤M ⎛ ⎛

⎞8 ⎞ 21 M     1 u n 2 ⎠ ⎠ EU n 4 2 ≤ Ch −5 τ 4 . ≤Cτ 4 ⎝E ⎝ R, j R

(5.56)

j=1

From (5.50)–(5.56), we conclude hEU R (tn+1 ) − U Rn+1 2 ≤h(1 + Cτ 2 )e−ατ EU R (tn ) − U Rn 2 + Cτ 3 + Ch −4 τ 4 + Ch −12 τ 6 + Ch −14 τ 8 ≤Cτ 2 + Ch −4 τ 3 + Ch −12 τ 5 + Ch −14 τ 7 ≤ Cτ 2 , where in the last two steps we have used the facts that τ = O(h 4 ) and (1 +  Cτ 2 )e−ατ < 1 for sufficiently small τ . Based on the estimates on the truncated equation and its numerical scheme, we are now in the position to give the approximate error between U (t) and U n . The proof of the following theorem is motivated by [67, 136] and holds for any fixed T > 0 without other restrictions. 1

Theorem 5.10 For Eq. (5.34) and scheme (5.41), assume that Eu 0 2L2 < ∞, Q 2 ∈ L22 , α ≥ 21 and τ = O(h 4 ). For any T > 0, scheme (5.41) converges with order one in probability, i.e.,

5.2 Ergodic and Conformal Multi-symplectic Full Approximation

 lim P

C→∞

√

sup

1≤n≤[T /τ ]

147

 hU (tn ) − U  ≥ Cτ n

= 0.

(5.57)

Proof For any γ ∈ (0, 1), we define n γ := inf{1 ≤ n ≤ [T /τ ] : U (tn ) − U n  ≥ γ } and then deduce that )

( sup

1≤n≤[T /τ ]

U (tn ) − U  ≥ γ n

)

 ( ⊂

(

sup U (tn ) ≥ R − 1 ∩ 0≤n≤n γ

( ∪

)

) sup

1≤n≤[T /τ ]

(

sup U (tn ) < R − 1 ∩ 0≤n≤n γ

( ⊂

U (tn ) − U n  ≥ γ

sup

U (tn ) − U  ≥ γ

sup

U (tn ) − U  ≥ γ

1≤n≤[T /τ ]

)

)  n

sup U (tn ) ≥ R − 1 0≤n≤n γ

( ∪

)

sup U (tn ) < R − 1 ∩ 0≤n≤n γ

(

)  1≤n≤[T /τ ]

.

n

* + If sup0≤n≤n γ U (tn ) < R − 1 happens, it is easy to show that U n  ≤ U (tn ) − U n  + U (tn ) < R − 1 + γ < R, FR (U Rk ) = F(U Rk ), U Rk = U n for k = 0, 1, · · · , n γ − 1 and U R (tn ) = U (tn ) for 0 ≤ n ≤ n γ . Furthermore, comparing scheme (5.48) with (5.41) and noting that n −1 f R (U Rγ )



 1 n γ −1 = − α I d + iλFR (U R ) τ 2   1 n γ −1 ) τ = f (U n γ −1 ), = − α I d + iλF(U 2

n

we have U Rγ = U n γ , which implies n

U R (tn γ ) − U Rγ  = U (tn γ ) − U n γ  ≥ γ . We conclude that for any γ ∈ (0, 1), there exists n γ ∈ N such that )

(

sup U (tn ) < R − 1 ∩ 0≤n≤n γ n

⊂{U R (tn γ ) − U Rγ  ≥ γ }.

(

) sup

1≤n≤[T /τ ]

U (tn ) − U n  ≥ γ

148

5 Numerical Invariant Measures for Damped Stochastic …

Thus, for some constants C, C1 > 0, choosing γ = deduce   √ n hU (tn ) − U  ≥ Cτ P sup 

1≤n≤[T /τ ]

≤P

√

sup



√

hU (tn ) ≥ C1 + P

0≤n≤n γ

hE sup U (tn )2 0≤n≤n γ

≤

h −1 Cτ and R =

√

h −1 C1 , we

n

hU R (tn γ ) − U Rγ  ≥ Cτ



n

hEU R (tn γ ) − U Rγ 2 . C 2τ 2

+

C12

√

(5.58)

We claim that e2αt U (t)2 is a submartingale, which ensures that  hE

 sup U (tn )

2

0≤n≤n γ

 ≤ hE

 sup e

2αtn

U (tn )

2

0≤n≤n γ

 ≤ e2αT hE U (tn γ )2 ≤ Ce2αT

based on a martingale inequality and Proposition 5.5. In fact, denoting C M,K := M  K 2 2αt U (t)2 similar to (5.36), j=1 k=1 ηk ek (x j ) and applying Itô’s formula to e we derive  T   C M,K  2αt e −1 , e2αs U (s)εσ Λdβ(s) + e2αt U (t)2 = U (0)2 + 2 α 0 where 2

,T 0

 e2αs U (s)εσ Λdβ(s) is a martingale. Apparently, we have

 r



 C M,K 2αt e E e2αt U (t)2 |Fr = U (0)2 + 2 e2αs U (s)εσ Λdβ(s) + −1 α 0  r

 C M,K e2αr − 1 e2αs U (s)εσ Λdβ(s) + ≥ U (0)2 + 2 α 0 = e2αr U (r )2

for r ≤ t, which completes the claim. Hence, based on the above claim and Theorem 5.9, inequality (5.58) turns to be  P

sup

1≤n≤[T /τ ]

√

 hU (tn ) − U  ≥ Cτ n

≤

Ce2αT CR + 2, 2 C C1

which approaches to 0 as C1 , C → +∞ for any T > 0.



5.2 Ergodic and Conformal Multi-symplectic Full Approximation 25

149

25 Numerical Limit

20

20

15

15

Charge

Charge

Numerical Exact

10

5

0 0

10

5

5

10

15

20

25

30

35

0

0

Time t

5

10

15

20

25

30

35

Time t

Fig. 5.4 Evolution of the discrete charge hEU n 2 with t = nτ for a ε = 0 and b ε = 1 (h = 0.1, τ = 2−6 , T = 32)

5.2.3 Numerical Experiments In this section, we provide several numerical experiments to illustrate the accuracy and capability of the fully discrete scheme (5.40), which can be calculated explicitly. We investigate the good performance in longtime simulation of the proposed scheme and check the temporal accuracy by fixing the space step-size. In the sequel, we take λ = 1, α = 0.5, truncate the infinite series of Wiener process to K = 100 terms and choose 500 realizations to approximate the expectation. Charge evolution. For the semi-discretization, the charge of the solution satisfies the evolution formula (5.37). To investigate the recurrence relation for the discrete charge of the fully discrete scheme, Fig. 5.4 plots the discrete charge for different values of ε with initial value u 0 (x) = sin(π x), ηk = k −6 , h = 1/(M + 1) = 0.1, τ = 2−6 and T = 32. We can observe that the discrete charge inherits the charge dissipation law without the noise term, i.e., ε = 0, and preserves the charge dissipation K 2  2 law approximately with a limit εαh M j=1 k=1 ηk ek (x j ) calculated through (5.37) for ε = 1. Ergodic limit. Based on the definition of ergodicity, if numerical solution U n 1  N −1 is ergodic, its temporal averages N n=1, E[ f (U n )] starting from different initial values will converge to the spatial average C M f dμτh . To verify this property, Fig. 5.5 shows the temporal averages of the fully discrete scheme starting from five different initial values initial(1) =(1, 0, · · · , 0) , initial(2) =(0.0003i, 0, · · · , 0) ,

150

5 Numerical Invariant Measures for Damped Stochastic … 3.5

10 -3

2.5 2 1.5 1 0.5

initial(1) initial(2) initial(3) initial(4) initial(5)

1.5

Temporal average

Temporal average

3

0 0

10 -3

2 initial(1) initial(2) initial(3 initial(4) initial(5)

1 0.5 0 -0.5 -1

50

100

150

200

Time t

250

300

350

-1.5

0

50

100

150

200

250

300

350

Time t

 N −1 Fig. 5.5 The temporal averages N1 n=1 E[ f (U n )] starting from different initial values for 2 bounded functions a f = exp(−U  ) and b f = sin(U 2 ) (h = 0.1, ε = 1, τ = 2−6 , T = 350)

1

2

100

π , sin π , · · · , sin π ) , 101 101 101 2+i initial(4) = (1, 2, · · · , 100) , 20 i 2i 100i initial(5) =(exp(− ), exp(− ), · · · , exp(− )) . 50 50 50 initial(3) =(sin

These temporal averages tend to the same value with test functions: (a) f (U ) = exp(−U 2 ), (b) f (U ) = sin(U 2 ). Weak error. As stated in Theorems 5.9 and 5.10, the mean-square convergence  1 error hEU R (T ) − U RN 2 2 with respect to the truncated Eq. (5.47) is independent of time T , and convergence in probability sense with respect to the original equation is also independent of time T . To clarify this property, we define the mean-square convergence error as

21 Eh,τ := hEU (T ) − U N 2 , T = N τ. Figure 5.6 displays the error Eh,τ till time T = 103 for different time step-sizes: (a) τ = 2−8 and (b) τ = 2−10 with h = 0.25, and shows that the mean-square convergence error is independent of time interval, which coincides with our theoretical results. Convergence order. We investigate the mean-square convergence order in temporal direction of the proposed method (5.40) in this experiment. Let h = 0.1, T = 1 and the initial value u 0 (x) = sin(π x). We plot Eh,τ against τ on a log-log scale with

5.2 Ergodic and Conformal Multi-symplectic Full Approximation 10 -3

20

Strong Error

Strong Error

15

10

5

10

0

-5

10 -3

20

15

151

5

0

0

100

200

300

400

500

600

700

800

-5

900 1000

0

100

200

300

400

500

600

700

800

900 1000

Time t

Time t



1 2 Fig. 5.6 The mean-square convergence error hEU (T ) − U N 2 for step-sizes a τ = 2−8 and b τ = 2−10 (h = 0.25, ε = 1, T = 103 )

10

-9

10 -1 Order 1.0 Numerical

Order 2.0 Numerical

Mean-square Error L2

Mean-square Error L2

10 -8

10 -10 10 -11 10 -12 10 -13 10 -14 -4 10

10 -3

10 -2

10 -2

10 -3

10 -4 -4 10

10 -3

10 -2

Fig. 5.7 Rates of convergence of (5.40) for a ε = 0 and b ε = 1, respectively (h = 0.1, T = 1, τ = 2−l , 11 ≤ l ≤ 14)

various combinations of (α, ε) and take method (5.40) with small time step-size τ = 2−16 as the reference solution. We then compare it with method (5.40) evaluated with time step-sizes (22 τ, 23 τ, 24 τ, 25 τ ) in order to show the rate of convergence. Figure 5.7 presents the mean-square convergence order for the error Eh,τ with various sizes of ε. Figure 5.7 shows that the proposed scheme (5.40) is of order 2 for the deterministic case, i.e., ε = 0, and of order 1 for the stochastic case with ε = 1, which coincides with the theoretical analysis.

152

5 Numerical Invariant Measures for Damped Stochastic …

Summary This chapter focuses on the numerical approximations of the invariant measure for the damped stochastic NLSE. An ergodic full discretization combining the spectral Galerkin method in space and modified implicit Euler scheme in time is proposed in Sect. 5.1. The error for invariant measures is given based on the weak error between the exact solution and the numerical one. As for the damped stochastic NLSE, it possesses the conformal symplectic conservation law, an ergodic scheme with the discrete conformal symplectic structure is then constructed in Sect. 5.2 to inherit the internal properties of the original system as more as possible. We would like to mention that the error between invariant measures, for ergodic systems and schemes, can also be derived directly utilizing the expansion of the solution of the Kolmogorov equation without requiring that the weak error is independent of time interval. We refer to [178, 180] for the analysis of the Euler and Milstein schemes. This idea could be modified to analyze the approximation error of the ergodic limit in Chap. 6. Except for the approximation of invariant measures, there is also a lot of work related to the density for solutions of SDEs and error analysis between the density for exact solution and that for the numerical one. We refer to [12, 24, 42, 151, 153] for the study of densities for both finite and infinite dimensional stochastic systems, and to [13, 14] for the convergence of the density for the Euler scheme applied to SDEs under the Hörmander condition. The existence, regularity and convergence of the density for numerical schemes of SPDEs are also interesting and important topics, and are needed further investigations.

Chapter 6

Approximation of Ergodic Limit for Conservative Stochastic Nonlinear Schrödinger Equations

This chapter is mainly developed to consider stochastic NLSEs possessing the stochastic multi-symplectic conservation law and the charge conservation law (see e.g. [64, 119]). For this kind of conservative equations, it then suffices to consider its dynamic behavior on the unit sphere without loss of generality. We show in Sect. 6.1 that the finite dimensional approximation (FDA) based on the midpoint scheme is ergodic with a unique invariant measure. Also, the ergodic limit of this FDA can be approximated via the temporal average of an ergodic fully discrete scheme (FDS), see Sect. 6.3. We also prove in Sect. 6.2 that the FDS could inherit the internal properties, i.e., charge conservation law and multi-symplecticity, of the original system.

6.1 Finite Dimensional Ergodic Approximation In Sects. 6.1 and 6.2, we concentrate on the stochastic NLSE on a bounded interval O = (0, 1) with σ = 1: ⎧   2 ⎪ ⎨ du = i Δu + λ|u| u dt + iu ◦ dW, (6.1) u(t, 0) = u(t, 1) = 0, t > 0, ⎪ ⎩ u(0, x) = u 0 (x), x ∈ (0, 1). Here, λ = ±1, and W is a Q-Wiener process on L2 (O; R) under the Dirichlet boundary condition. The covariance operator Q is symmetric and positive definite such that the Karhunen–Loève expansion of W is as follows W =

∞  k=1

1

Q 2 ek βk =

∞  √ ηk ek βk , ηk > 0, k ∈ N, k=1

© Springer Nature Singapore Pte Ltd. 2019 J. Hong and X. Wang, Invariant Measures for Stochastic Nonlinear Schrödinger Equations, Lecture Notes in Mathematics 2251, https://doi.org/10.1007/978-981-32-9069-3_6

153

154

6 Approximation of Ergodic Limit for Conservative Stochastic Nonlinear …

√ where {ek := 2 sin(kπ ·)}k∈N is an orthogonal basis for L2 (O; R) with homogeneous boundary condition and {βk }k∈N is a sequence of R-valued mutually independent and identically distributed Brownian motions. In addition, Q is assumed to commute with the Laplacian and hence {ek }k∈N is also an eigenbasis of the Dirichlet Laplacian Δ in L2 (O; R). The well-posedness, as well as the corresponding assumptions on u 0 and Q, is given in Sect. 3.3.2. Now we turn our attention to the numerical approximation of (6.1). To investigate longtime behaviors of (6.1) numerically, we first apply the central finite difference scheme to (6.1) in the spatial variable to obtain an FDA, which is also a Hamiltonian system, in a finite dimensional space R M for some M ∈ N+ . It is worth noticing that the charge conservation law for the FDA also holds, which allows us to consider properties of the FDA restricted onto the unit sphere S ⊂ R M . When showing the ergodicity of this conservative system, we construct an invariant control set M0 ⊂ S with respect to a control function introduced in Sect. 6.1.2. The FDA is proved to be ergodic in M0 based on the Krylov–Bogoliubov theorem and the Hörmander condition.

6.1.1 Finite Dimensional Approximation Based on the central finite difference scheme and the notation u j := u j (t), j = 1, · · · , M, we consider the following spatial semi-discretization du j = i

K  u j+1 − 2u j + u j−1 √ 2 dt + iu + λ|u | u ηk ek (x j ) ◦ dβk (t) j j j h2 k=1

K √ with a truncated noise k=1 ηk ek (x)βk (t), K ∈ N+ , a given uniform spatial step1 for some M ≤ K and x j = j h, j = 1, · · · , M. The condition M ≤ K size h = M+1 here ensures the existence of the solution for the control function. Denoting vectors U := U (t) = (u 1 , · · · , u M ) ∈ C M , β(t) = (β1 (t), · · · , β K (t)) ∈ R K , and matrices F(U ) = diag{|u 1 |2 , · · · , |u M |2 }, E k = diag{ek (x1 ), · · · , ek (x M )}, Λ = √ √ diag{ η1 , · · · , η K }, Z (U ) = diag{u 1 , · · · , u M }E M K Λ, ⎛

−2 ⎜ 1 ⎜ A=⎜ ⎝

⎞ ⎞ ⎛ 1 e1 (x1 ) · · · e K (x1 ) ⎟ −2 1 ⎟ ⎜ .. ⎟ ∈ R M×M , E M K = ⎝ ... , .. .. .. ⎟ . ⎠ ⎠ . . . e1 (x M ) · · · e K (x M ) M×K 1 −2

then the FDA is in the following form

6.1 Finite Dimensional Ergodic Approximation

155

⎧

⎪ ⎨ dU = i 1 AU + λF(U )U dt + iZ (U ) ◦ dβ(t), h2 ⎪ ⎩ U (0) = c (u (x ), · · · , u (x )) , ∗ 0 1 0 M

(6.2)

where c∗ is a normalized constant such that U (0) ∈ S. The noise term in (6.2) has an equivalent Itô form iZ (U ) ◦ dβ(t) =i

K K K   1 √ √ ηk E k U ◦ dβk (t) = − ηk E k2 U dt + i ηk E k U dβk (t) 2

k=1

k=1

ˆ dt + i =: − EU

K  √

k=1

ηk E k U dβk (t)

(6.3)

k=1

K with Eˆ = 21 k=1 ηk E k2 . In the sequel, denote by · the Euclidean norm for both matrices and vectors, which satisfies BV ≤ B

V for any matrices B ∈ Cm×n and vectors V ∈ Cn , m, n ∈ N, and denote by · F the Frobenius norm for matrices. Matrix A is bounded in · -norm independent of M as stated in Lemma 5.3. Proposition 6.1 FDA (6.2) possesses the charge conservation law, i.e.,

U (t) 2 = U (0) 2 , ∀ t ≥ 0, P-a.s.,  M 1 1 2 2 2 where U (t) = ( P(t) 2 + Q(t) 2 ) 2 = m=1 (| pm (t)| + |qm (t)| ) , P(t) = ( p1 (t), · · · , p M (t)) and Q(t) = (q1 (t), · · · , q M (t)) are the real and imaginary parts of U (t) respectively. Proof Noticing that matrices A and F(U ) are symmetric and the linear function Z (U ) satisfies ⎛  ⎜ U Z (U ) = (u 1 , · · · , u M ) ⎝

u1

⎞ ..

⎟ ⎠ EMK

.

uM ⎛√ = (|u 1 | , · · · , |u M | )E M K 2

2

⎛√ η1 ⎜ .. ⎝ .

⎜ ⎝

η1

⎞ ..

.

√ ηK

⎞ √

⎟ ⎠ ηK

⎟ K ⎠∈R ,

(6.4)



where U denotes the conjugate of U , we multiply (6.2) by U , take the real part, and then get the charge conservation law for U .  In the sequel, without pointing it out explicitly, all equations hold in the sense P-a.s.

156

6 Approximation of Ergodic Limit for Conservative Stochastic Nonlinear …

Remark 6.1 Equation (6.1) can be rewritten into an infinite dimensional Hamiltonian system (see [119]). It is easy to verify that the central finite difference scheme (6.2) applied to (6.1) is equivalent to the symplectic Euler scheme applied to the infinite dimensional Hamiltonian form of (6.1), which implies the symplecticity of (6.2).

6.1.2 Unique Ergodicity As the charge of (6.2) is conserved shown in Proposition 6.1, for any fixed initial value U (0), the solution U (t) of (6.2) is trapped in the equipotential surface, and there will not be a unique invariant measure on the whole domain. Without loss of generality, we assume that U (0) ∈ S and only investigate the unique ergodicity of (6.2) on S. The most challenging thing is that the diffusion term Z (U ) depends linearly on the solution U . As a result, the Hörmander condition, which is frequently used to show the ergodicity of SDEs with multiplicative noises, does not uniformly hold for U ∈ S. It depends on the number of zero entries of the vector U . Instead, we need to construct an invariant control set, which is the union of several disjoint subsets, such that the Hörmander condition holds in each subset. Definition 6.1 (see e.g. [9]) A subset M = ∅ of S is called an invariant control set for the control system

1 dφ = i 2 Aφ + λF(φ)φ dt + iZ (φ)dΨ (t) h

(6.5)

of (6.2) with a differentiable deterministic function Ψ , if O + (x) = M for any x ∈ M, and M is maximal with respect to inclusion, where O + (x) denotes the set of points reachable from x (i.e., connected with x) in any finite time and M denotes the closure of M. The uniqueness of the invariant measure on an invariant set is given through the following theorem, which will be used to show the uniqueness of the invariant measure for FDA (6.2). Theorem 6.1 (Theorem 5.1, [9]) Let M be an invariant control set. Assume that there exists a point x0 ∈ M such that the Hörmander condition given in Theorem 2.2 holds for x0 . Then there is at most one invariant probability measure μ with suppμ = M and μ(M) = 1. We state one of our main results in the following theorem. Theorem 6.2 FDA (6.2) possesses a unique invariant probability measure μh on an invariant control set M0 with supp(μh ) = S and μh (M0 ) = 1,

6.1 Finite Dimensional Ergodic Approximation

157

which implies the ergodicity of (6.2). Proof Step 1. Existence of Invariant Measures From Proposition 6.1, we find πt (U (0), S) = 1 for all t ≥ 0, where πt (U (0), ·) denotes the transition probability (probability kernel) of U (t). As the finite dimensional unit sphere S is tight, the family of measures {πt (U (0), ·)}t≥0 is tight, which implies the existence of invariant measures by the Krylov–Bogoliubov theorem [58]. Step 2. Invariant Control Set Denote U = P + iQ with P = (P1 , · · · , PM ) , Q = (Q 1 , · · · , Q M ) ∈ R M being the real and imaginary parts of U , respectively. Note that the subset S0 := {U ∈ S : ∃ 1 ≤ i ≤ M s.t. Pi = 0 or Q i = 0} is a union of finite number of lower dimensional unit spheres. We have m(S0 ) = 0 with m(·) being the Lebesgue measure in R M . We denote further X i := Pi and X M+i = Q i for 1 ≤ i ≤ M for convenience. We first consider the following subset of S as an example: S(1,··· ,M) := {U ∈ S\S0 : X i > 0, 1 ≤ i ≤ M}. For any t > 0, y, z ∈ S(1,··· ,M) , there exists a differentiable function φ satisfying φ(s) = (φ1 (s), · · · , φ M (s)) ∈ S(1,··· ,M) , s ∈ [0, t], φ(0) = y and φ(t) = z by polynomial interpolation argument. As rank(Z (φ(s))) = M for φ(s) ∈ S(1,··· ,M) and M ≤ K , the linear system

1 Z (φ(s))X = −iφ (s) − 2 Aφ(s) + λF(φ(s))φ(s) h 



possesses a solution X ∈ C M . As in addition Z (φ(s)) = diag{φ1 (s), · · · , φ M (s)} E M K Λ, where diag{φ1 (s), · · · , φ M (s)} is invertible for φ(s) ∈ S(1,··· ,M) , the solution X depends continuously on s and is denoted by X (s). Thus, there exists a · differentiable function Ψ (·) := 0 X (s)ds which, together with φ defined above, satisfies the control system (6.5) with the initial datum Ψ (0) = 0. That is, for any y, z ∈ S(1,··· ,M) , y and z are connected, denoted by y ↔ z. We also call S(1,··· ,M) a self-connected set. The above argument also holds for the following subsets S+ σ :={U ∈ S\S0 : σ = (σ1 , · · · , σ M ), X σk > 0, k = 1, · · · , M}, S− σ :={U ∈ S\S0 : σ = (σ1 , · · · , σ M ), X σk < 0, k = 1, · · · , M}, where σ ∈ Σ := {ν = (ν1 , · · · , ν M ) : 1 ≤ ν1 < ν2 < · · · < ν M ≤ 2M}. − That is, each element in the collection {S+ σ , Sσ }σ ∈Σ is self-connected.

158

6 Approximation of Ergodic Limit for Conservative Stochastic Nonlinear …

− Moreover, for any two different elements S1 , S2 ∈ {S+ σ , Sσ }σ ∈Σ and any y ∈ + − S1 , z ∈ S2 , there exist S3 ∈ {Sσ , Sσ }σ ∈Σ , r1 and r2 , satisfying r1 ∈ S1 ∩ S3 = ∅, r2 ∈ S2 ∩ S3 = ∅ and y ↔ r1 ↔ r2 ↔ z. Thus,

M0 :=

  − S+ σ ∪ Sσ = S\S0 σ ∈Σ

is an invariant control set for (6.5) with M0 = S since it is the maximal set in which any two points are connected. Step 3. Uniqueness of the Invariant Measure We rewrite (6.2) with P and Q according to its equivalent form in the Itô sense and obtain      − Eˆ − h12 A − λF(P, Q) P P d = 1 dt ˆ Q Q A + λF(P, Q) − E h2    K  √ 0 −E k P ηk + dβk (t) Ek 0 Q k=1

=:X 0 (P, Q)dt +

K 

X k (P, Q)dβk (t).

(6.6)

k=1

To derive the uniqueness of the invariant measure, we consider the Lie algebra generated by the diffusions of (6.6)     L(X 0 , X 1 , · · · , X K ) = span X l , [X i , X j ], X l , [X i , X j ] , · · · , 0 ≤ l, i, j ≤ K . Choosing p∗ = 0 and q∗ = √−1M (1, · · · , 1) such that z ∗ := p∗ + iq∗ ∈ S− (M+1,··· ,2M) ⊂ M0 , we derive that the following vectors ⎛

⎞ ek (x1 ) . ⎜ . ⎟ ⎜ ⎟  ⎜ . ⎟  ⎟ ηk ⎜ ηk (x ) e k M ⎜ ⎟ X k ( p∗ , q∗ ) = ⎟ , [X 0 , X k ]( p∗ , q∗ ) = M 0 M⎜ ⎜ ⎟ ⎜ . ⎟ ⎝ .. ⎠

⎛

⎞ ek (x1 ) ⎜ ⎟ − Eˆ ⎝ ... ⎠

⎛

⎞

⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ (x ) e k ⎛M ⎜ ⎟ ⎞ ⎜ ⎟ (x ) e 1 k ⎜ ⎟ ⎜ 1 ⎟ ⎜ ⎟ . 1 I d) ⎝( 2 A + M ⎝ .. ⎠⎠ h

0

ek (x M )

are independent of each other for k = 1, · · · , M, which hence implies the following Hörmander condition dim L(X 0 , X 1 , · · · , X K )(z ∗ ) = 2M.

6.1 Finite Dimensional Ergodic Approximation

159

Then there is at most one invariant measure with supp(μh ) = S according to Theorem 6.1. Actually, according to the above procedure, we obtain that Hörmander condition holds uniformly for any z ∈ M0 . Combining the three steps above, we conclude that there exists a unique invariant  measure μh on M0 for the FDA, with μh (M0 ) = 1. For some other nonlinearities such that the equation still possesses the charge conservation law, e.g., iF(x, |u|)u with F being some real valued potential function, we can still get the ergodicity of the finite dimensional approximation of the original equation through the procedure used in Theorem 6.2. The procedure could also applied to higher dimensional Schrödinger equations with proper well-posed assumptions, but it may be more technical to verify the Hörmander condition. Remark 6.2 According to the ergodicity of (6.2) and noticing that 1 = μh (M0 ) ≤ μh (S) ≤ 1, we have   1 T E f (U (t))dt = f dμh , ∀ f ∈ Bb (S), in L2 (S, μh ), lim T →∞ T 0 S  where S f dμh is known as the ergodic limit with respect to the invariant measure μh .

6.2 Multi-symplectic Ergodic Fully Discrete Scheme An FDS with the discrete multi-symplectic structure and the discrete charge conservation law is constructed in this section, which also inherits the unique ergodicity of the FDA. More precisely, we apply the midpoint scheme to (6.2), and obtain the following FDS ⎧ ⎨ U n+1 − U n = i τ AU n+ 21 + iλτ F(U n+ 21 )U n+ 21 + iZ (U n+ 21 )δn+1 β, h2 ⎩ U 0 = U (0) ∈ S,

(6.7)

where τ denotes the uniform time step-size, tn = nτ , U n = (u n1 , · · · , u nM ) ∈ C M , 1 U n+ 2 = 21 (U n+1 + U n ) and δn+1 β = β(tn+1 ) − β(tn ). For FDS (6.7), which is implicit in both drift and diffusion terms, its well-posedness is stated in the following proposition, and it converges to FDA (6.2) in Stratonovich sense since it is consistent with the Stratonovich rule. Proposition 6.2 For any initial value U 0 = U (0) ∈ S, there exists a unique solution {U n }n∈N of (6.7), and it possesses the discrete charge conservation law, i.e.,

U n+1 2 = U n 2 = 1, ∀ n ∈ N.

160

6 Approximation of Ergodic Limit for Conservative Stochastic Nonlinear … 1

Proof We multiply both sides of (6.7) by U n+ 2 , take the real part, and obtain the existence of the numerical solution by Brouwer’s fixed point theorem as well as the discrete charge conservation law. For the uniqueness, we assume that X = (X 1 , · · · , X M ) and Y = (Y1 , · · · , Y M ) are two solutions of (6.7) with U n = z = (z 1 , · · · , z M ) ∈ S. It follows that X, Y ∈ S and   τ X −Y iλτ X −Y δn+1 β, (6.8) X −Y =i 2A + H (X, Y, z) + iZ h 2 8 2 where ⎛ ⎜ H (X, Y, z) = ⎝

|X 1 + z 1 |2 (X 1 + z 1 ) − |Y1 + z 1 |2 (Y1 + z 1 ) .. .

⎞ ⎟ ⎠.

|X M + z M |2 (X M + z M ) − |Y M + z M |2 (Y M + z M ) Based on the fact that |a|2 a − |b|2 b = |a|2 (a − b) + |b|2 (a − b) + ab(a − b) for any a, b ∈ C, we have 





 (X − Y ) H (X, Y, z) = 



M 

 (X m + z m )(Ym + z m )(X m − Ym )

2

m=1

with [V ] denoting the imaginary part of V . Multiplying (6.8) by (X − Y ) and taking the real part, we obtain  λτ 

X − Y 2 = −  (X − Y ) H (X, Y, z) 8   τ τ max |X m + z m ||Ym + z m | X − Y 2 ≤ X − Y 2 , ≤ 8 1≤m≤M 2 where we have used the fact X, Y, z ∈ S and (6.4). For τ < 1, we get X = Y and complete the proof.  The proposition above shows that (6.7) possesses the discrete charge conservation law. Furthermore, (6.7) also inherits the unique ergodicity of the FDA and the stochastic multi-symplecticity of the original equation, which are stated in the following two theorems. Theorem 6.3 FDS (6.7) is also ergodic with a unique invariant measure μτh on the control set M0 , such that μτh (M0 ) = 1. Also, lim

N →∞

 N −1 1  f (U n ) = f dμτh , ∀ f ∈ Bb (S), in L2 (S, μτh ). N n=0 S

6.2 Multi-symplectic Ergodic Fully Discrete Scheme

161

Proof Based on the charge conservation law for {U n }n∈N , we obtain the existence of the invariant measure similar to the proof of Theorem 6.2. To obtain the uniqueness of the invariant measure, we show that the Markov chain {U 3n }n∈N satisfies Assumption 2.2. Firstly, Proposition 6.2 implies that for a given U n ∈ S, solution U n+1 can be defined through a continuous function U n+1 = κ(U n , δn+1 β). As δn+1 β has a C∞ density, we derive a jointly continuous density for U n+1 . Secondly, similar to Theorem 6.2, for any given y, z ∈ M0 , there must exist σ1 , σ2 , σ3 ∈ Σ and r1 , r2 ∈ M0 , such that y ∈ Sσ1 , z ∈ Sσ2 , r1 ∈ Sσ1 ∩ Sσ3 and 1 1 ∈ Sσ1 and Z ( y+r ) is invertible, δ3n+1 β can be chosen to r2 ∈ Sσ2 ∩ Sσ3 . As y+r 2 2 ensure that     y + r1 y + r1 y + r1 τ y + r1 + iλτ F + iZ δ3n+1 β r1 − y = i 2 A h 2 2 2 2 2 holds, i.e., r1 = κ(y, δ3n+1 β). Similarly, based on the fact r1 +r ∈ Sσk and r22+z ∈ Sσk , 2 we have r2 = κ(r1 , δ3n+2 β) and z = κ(r2 , δ3n+3 β). That is, for any given y, z ∈ M0 , δ3n+1 β, δ3n+2 β, δ3n+3 β can be chosen to ensure that U 3n = y and U 3(n+1) = z. Finally we obtain that, for any δ > 0,

   P3 (y, B(z, δ)) := P U 3 ∈ B(z, δ)U 0 = y > 0, where B(z, δ) denotes the open ball of radius δ centered at z.



The infinite dimensional system (6.1) has been shown to preserve the stochastic multi-symplectic conservation law locally (see e.g. [119]) d(d p ∧ dq) − ∂x (d p ∧ dv + dq ∧ dw)dt = 0 with p, q denoting the real and imaginary parts of the solution u, respectively, and v = px , w = qx being the derivatives of p and q with respect to x. We now show that this ergodic FDS (6.7) not only possesses the discrete charge conservation law as shown in Proposition 6.2 but also preserves the discrete stochastic multi-symplectic structure. Theorem 6.4 The implicit FDS (6.7) preserves the discrete multi-symplectic structure 1 1 n+ 1 n+ 1 n+ 1 n+ 1 ∧ dq n+1 − d p nj ∧ dq nj ) − (d p j 2 ∧ dv j+12 − d p j−12 ∧ dv j 2 ) (d p n+1 j j τ h 1 n+ 21 n+ 21 n+ 21 n+ 1 − (dq j ∧ dw j+1 − dq j−1 ∧ dw j 2 ) = 0, h where p nj , q nj denote the real and imaginary parts of u nj , respectively, v j = h1 ( p nj − p nj−1 ) and w j = h1 (q nj − q nj−1 ) with j = 1, · · · , M − 1 and n ∈ N.

162

6 Approximation of Ergodic Limit for Conservative Stochastic Nonlinear …

Proof Rewriting (6.7) with the real and imaginary parts of the components u nj of U n , we have   ⎧ 1 1 1 1 1 1 1 n+1 n ) − 1 (v n+ 2 − v n+ 2 ) = ( p n+ 2 )2 + (q n+ 2 )2 p n+ 2 + p n+ 2 ζ K , ⎪ ⎪ (q − q ⎪ j j j j+1 j j j j j ⎪ τ h ⎪ ⎪ ⎪   ⎪ ⎪ 1 1 n+ 1 n+ 1 n+ 1 n+ 1 n+ 1 n+ 1 ⎪ ⎪ − pnj ) − (w j+12 − w j 2 ) = ( p j 2 )2 + (q j 2 )2 q j 2 + q j 2 ζ jK , ⎨ − ( pn+1 τ j h ⎪ 1 n+ 21 ⎪ n+ 1 n+ 1 ⎪ ⎪ − p j−12 ) = v j 2 , (pj ⎪ ⎪ h ⎪ ⎪ ⎪ ⎪ 1 n+ 21 n+ 1 n+ 1 ⎪ ⎩ (q j − q j−12 ) = w j 2 , h

where ζ jK =

(6.9) K √ n+ 21 n+ 21 n+ 21 n+ 21 η e (x )δ β (t). Denoting z = ( p , q , v , k k j n+1 k j j j j k=1

n+ 21 

wj

) and taking differential in the phase space on both sides of (6.9), we obtain ⎛

⎛

⎞ q n+1 − q nj j 1 ⎜ −( p n+1 − p nj )⎟ j ⎟ d⎜

τ ⎝

n+ 21

n+ 21

)

⎞

⎜ ⎟ n+ 1 n+ 1 ⎟ 1 ⎜ ⎜−(w j+12 − w j 2 )⎟ ⎟ 1 1 ⎠+ hd⎜ ⎜ p n+ 2 − p n+ 2 ⎟ j j−1 ⎠ ⎝ n+ 1 n+ 1 q j 2 − q j−12

0 0

=∇ 2 S1 (z j

n+ 1

−(v j+12 − v j

n+ 21

)dz j

n+ 21

+ ∇ 2 S2 (z j

n+ 21

)dz j

ζ jK ,

(6.10)

where n+ 21

S1 (z j

)=

1  n+ 21 2 n+ 1 ( p j ) + (q j 2 )2 4

and n+ 21

S2 (z j

)=

1  n+ 21 pj 2 n+ 21

Then the wedge product between dz j symmetry of ∇ 2 S1 and ∇ 2 S2 .

2

2

+

1  n+ 21 v 2 j

+

1  n+ 21 q 2 j

2

2

+

1  n+ 21 wj 2

2

.

and (6.10) concludes the proof based on the 

Before giving the approximate error of the ergodic limit, we give some essential a priori estimates about the stability of FDS (6.7) and FDA (6.2). In the following, C denotes a generic constant independent of T , N , τ and h while C h denotes a generic constant depending on h. Lemma 6.1 For any initial value U 0 ∈ S and γ ≥ 1, if the covariance operator 1

1

+ε

Q 2 ∈ L22 for some ε > 0, then there exists a constant C = C(γ , ε) such that the solution {U n }n∈N of (6.7) satisfies ! !2γ E !U n+1 − U n ! ≤ C(τ 2γ h −4γ + τ γ ), ∀ n ∈ N.

6.2 Multi-symplectic Ergodic Fully Discrete Scheme

163

Proof As proved in Proposition 6.2 that U n = 1 for any n ∈ N, for the nonlinear term, we have  M   M  !2γ !   n+ 1 6γ   n+ 1 2 ! n+ 21 n+ 21 ! 6γ −2 2 2 )U ≤E ≤1 E !F(U ! =E u m  u m  · 1 m=1

m=1 n+ 1

1

by the convexity of S, i.e., U n+ 2 ≤ 1 and |u m 2 | ≤ 1, a.s. The noise term can be estimated as ⎛  K 2 ⎞γ M   ! !  1 2γ 1 √ n+   ! ! u m 2 ek (xm ) ηk δn+1 βk  ⎠ E ! Z (U n+ 2 )δn+1 β ! =E ⎝    m=1 k=1 ⎛ ⎡ $ K %2 ⎤⎞γ M     1 2 √ 2 ⎣u n+ =E ⎝ ek (xm ) ηk δn+1 βk ⎦⎠ m  $

m=1

k=1

M    K   n+ 21 2  √ ≤E 2 ηk |δn+1 βk | u m  m=1

$ γ

≤2 E

K 

2

%γ

k=1

%2γ

1 2

ηk |δn+1 βk |

k=1

 K   1 1+2ε ηk2 k 2γ ≤C

2γ 2γ −1



2γ −1

k=1

E

K 

 k −(1+2ε) |δn+1 βk |2γ

k=1

≤Cτ γ ,

(6.11)

where we have used the fact K   k=1

1 2

ηk k

1+2ε 2γ

2γ 2γ −1



K  γ  γ 1−γ 2γ −1 (1+2ε) (1+2ε) 2γ −1 2γ −1 ηk k k = k=1

 K   2γγ−1  2γγ−1  K  γ  γ (1+2ε) (1−γ )(1+2ε) 2γ −1 ηk k 2γ −1 k 2γ −1 ≤ k=1

k=1

 2γγ−1  K  K  2γγ −1 −1    1+2ε −(1+2ε) ηk k = k ≤C k=1

k=1

2γ −1 γ −1

 2γγ −1 −1

164

6 Approximation of Ergodic Limit for Conservative Stochastic Nonlinear … 1

1

according to the assumption Q 2 ∈ L22

+ε

. In conclusion,

! !2γ E !U n+1 − U n !  ! !2γ !2γ ! ! !2γ  !τ ! ! ! n+ 21 ! n+ 21 n+ 21 ! n+ 21 ≤C E ! 2 AU )U )δn+1 β ! ! + E !λτ F(U ! + E !Z (U h !   Cτ 2γ ! 1 !2γ ! ≤ 4γ E !U n+ 2 ! + Cτ 2γ + Cτ γ ≤ C τ 2γ h −4γ + τ γ , h where we have used the fact that A ≤ 4 in Lemma 5.3.



Lemma 6.2 For any initial value U (0) ∈ S and γ ≥ 1, there exists a constant C such that the solution U (t) of (6.2) satisfies E U (tn+1 ) − U (tn ) 2γ ≤ C(τ 2γ h −4γ + τ γ ), ∀ n ∈ N. Proof From (6.2) and (6.3), based on Hölder’s inequality, we obtain E U (tn+1 ) − U (tn ) 2γ ! tn+1 !2γ

 tn+1 ! ! 1 ! ˆ i 2 AU + iλF(U )U − EU dt + =E ! iZ (U )dβ(t)! ! h tn tn !2γ  tn+1   tn+1 ! 2γ −1 ! 1 ! 2γ ˆ ! ≤C E! 1 2γ −1 dt !i h 2 AU + iλF(U )U − EU ! dt tn tn ! tn+1 !2γ  ! ! + E! iZ (U )dβ(t)! ! ! tn ! !2γ  tn+1 !1 ! ! E U 2γ dt + Cτ 2γ + Cτ γ ≤Cτ 2γ −1 ! ! h 2 A! tn ≤C(τ 2γ h −4γ + τ γ ), where we have used the boundedness of F(U )U in S similar to that in Lemma 6.1. In the third step of the equation above, we also used ⎛

 K 2 ⎞γ M      ˆ 2γ ≤ CE ⎝ ηk ek2 (xm )u m  ⎠ E EU    m=1 k=1 ⎛ $ K %2 ⎞γ M   ≤CE ⎝ |u m |2 ηk ⎠ ≤ Cη2γ E U 2γ ≤ C m=1

k=1

6.2 Multi-symplectic Ergodic Fully Discrete Scheme

165

and ! ! E! !

tn+1

tn

!2γ  ! ! iZ (U )dβ(t)! ≤ C

⎛  ⎝ ≤C

tn+1 tn

 ≤C

tn+1

$ $

tn+1 tn

 2γ E Z (U ) F

M  K    u m ek (xm )√ηk 2 E m=1 k=1

  γ  1 E 2η U 2 γ dt

γ

%γ % γ1

1 γ

γ dt

⎞γ dt ⎠

≤ Cτ γ

tn

according to the Burkholder–Davis–Gundy inequality, where · F denotes the Frobenius norm. 

6.3 Approximate Error of the Ergodic Limit To approximate the ergodic limit of (6.2) and get the approximate error, we give an estimate of the local weak convergence error between U (τ ) and U 1 based on the Poisson equation associated to (6.2) (see also [140]). Recall that SDE (6.2) in the Stratonovich sense has an equivalent Itô form

1 ˆ dt + iZ (U )dβ(t) dU = i 2 AU + iλF(U )U − EU h = : b(U )dt + σ (U )dβ(t)

(6.12)

 based on (6.3). For any fixed f ∈ W4,∞ (S), let fˆ := S f dμh and ϕ be the unique solution of the Poisson equation L ϕ = f − fˆ, where L denotes the infinitesimal generator defined in (2.2). It is easy to find out that (6.12) satisfies the hypoelliptic setting (see e.g. [140]) according to the Hörmander condition in Theorem 6.2. Thus, ϕ ∈ W4,∞ (S) according to Theorem 4.1 in [140]. Note that ϕ has a unique extension to a function in W4,∞ (B(0, 1)) with B(0, 1) being the unit open ball centered at 0. In fact, the harmonic equation (

Δg =0 in B(0, 1) g =ϕ on S

has a smooth solution in W4,∞ (B(0, 1)) such that g|S = ϕ. For simplicity, we still use the notation ϕ to denote the extension on B(0, 1). Based on the existence and uniqueness of the numerical solution {U n }n∈N , (6.7) can be rewritten in the following explicit form

166

6 Approximation of Ergodic Limit for Conservative Stochastic Nonlinear …

U n+1 = U n + τ Φ(U n , τ, h, δn+1 β)

(6.13)

for some function Φ. Denoting by D k ϕ(u)(Φ1 , · · · , Φk ) the kth order Fréchet derivative evaluated in the directions Φ j , j = 1, · · · , k with D k ϕ(u)(Φ)k for short if all the directions are the same, we have

1 1 ϕ(U n+1 ) =ϕ(U n ) + τ Dϕ(U n )Φ n + τ D 2 ϕ(U n )(Φ n )2 + D 3 ϕ(U n )(τ Φ n )3 + RnΦ 2 6 1 = : ϕ(U n ) + τ L Φ ϕ(U n ) + D 3 ϕ(U n )(τ Φ n )3 + RnΦ , (6.14) 6

where Φ n := Φ(U n , τ, h, δn+1 β), 1 L Φ ϕ(U n ) := Dϕ(U n )Φ n + τ D 2 ϕ(U n )(Φ n )2 , 2 and RnΦ :=

1 6



1

s 3 D 4 ϕ(sU n + (1 − s)U n+1 )(τ Φ n )4 ds.

0

Here, sU n + (1 − s)U n+1 ∈ B(0, 1). Adding (6.14) together from n = 0 to n = N − 1 for some fixed N ∈ N, then dividing the result by T = N τ , and noticing that L ϕ(U n ) = f (U n ) − fˆ, we obtain ϕ(U N ) − ϕ(U 0 ) 1 = Nτ N + =

 N −1

N −1  Φ   L ϕ(U n ) − L ϕ(U n ) + L ϕ(U n )

n=0

1 τ

N −1  n=0

n=0

1 3 1 D ϕ(U n )(τ Φ n )3 + 6 τ

N −1 

RnΦ



n=0

N −1 

 Φ  1 3 D ϕ(U n )(τ Φ n )3 L ϕ(U n ) − L ϕ(U n ) + 6τ n=0 $ N −1 % N −1 1  1  Φ + f (U n ) − fˆ + R , N n=0 N τ n=0 n

1 N

which shows   N −1      N −1   1       1  1     E ϕ(U N ) − ϕ(U 0 )  +  f (U n ) − fˆ  ≤  ERnΦ  E     N n=0 Nτ N τ n=0  N −1 

 1  1 3  Φ n n n n 3  + D ϕ(U )(τ Φ )  =: I + I I + I I I. E L ϕ(U ) − L ϕ(U ) + N  6τ n=0 (6.15)

6.3 Approximate Error of the Ergodic Limit

167

 N −1 The average N1 n=0 f (U n ) is regarded as an approximation of fˆ. We next begin to investigate the approximate error by estimating I , I I and I I I respectively. According to the fact that ϕ ∈ W4,∞ (B(0, 1)) and Lemma 6.1, we have I ≤

C 2 ϕ 0,∞ ≤ Nτ T

(6.16)

and II ≤ ≤

N −1 N −1 * !4 !4 * 1  )! C  )! E !τ Φ n ! D 4 ϕ L ∞ ≤ E !U n+1 − U n ! N τ n=0 N τ n=0 N −1    C   4 −8 τ h + τ 2 ≤ C τ 3 h −8 + τ , N τ n=0

(6.17)

where ϕ γ ,∞ := sup|α|≤γ ,u∈B(0,1) |D α ϕ(u)|, γ ∈ N. It then remains to estimate the term I I I . To this end, we need the estimate about the local weak convergence, which is stated in the following theorem. 1

1

+ε

Theorem 6.5 Assume that Q 2 ∈ L22 for some ε > 0. On one hand, for a fixed h ∈ (0, 1), the local weak error satisfies    E ϕ(U (τ )) − ϕ(U 1 )  ≤ C h τ 2 with some constant C h = C(ϕ, η, h). On the other hand, for any h, τ ∈ (0, 1) satis1 fying ρ := τ 2 h −2 < C, it holds    E ϕ(U (τ )) − ϕ(U 1 )  ≤ Cρ τ 23 with some constant Cρ = C(ϕ, η, ρ). Proof Based on the Taylor expansion, Lemmas 6.1 and 6.2, we obtain        E ϕ(U (τ )) − ϕ(U 1 ) = E Dϕ(U 1 ) U (τ ) − U 1 + O U (τ ) − U 1 2      =E Dϕ(U 0 ) U (τ ) − U 1 + E D 2 ϕ(U 0 )(U 1 − U 0 , U (τ ) − U 1 )    + O E U 1 − U 0 2 U (τ ) − U 1 + E U (τ ) − U 1 2 =:A + B + C . We give the mild solution and the discrete mild solution of (6.2) and (6.7) respectively, U (τ ) =e

i h12 Aτ

 +

 U + 0

 1 ˆ (s) ds ei h2 A(τ −s) iλF(U (s))U (s) − EU

0

τ

e 0

τ

i h12 A(τ −s)

iZ (U (s))dβ(s),

168

6 Approximation of Ergodic Limit for Conservative Stochastic Nonlinear …

 1 iτ iτ iτ 1 −1 0 −1 A) (I d + A)U + (I d − A) iλτ F U2 U2 2 2 2 2h 2h 2h  1 iτ −1 + (I d − 2 A) iZ U 2 δ1 β. 2h

U 1 =(I d −

Estimation of A . Considering the difference between above equations, we have   iτ iτ i h12 Aτ −1 − (I d − 2 A) (I d + 2 A) U 0 U (τ ) − U = e 2h 2h

 τ iτ i h12 A(τ −s) −1 e λF(U (s))U (s)ds +i − (I d − 2 A) 2h 0 τ * )  1 iτ 1 +i (I d − 2 A)−1 λ F(U (s))U (s) − F U 2 U 2 ds 2h 0  −1   τ iτ i h12 A(τ −s) +i − Id − 2 A e Z (U (s))dβ(s) 2h 0 −1  τ iτ Id − 2 A Z (U (s) − U 0 )dβ(s) +i 2h 0    −1  τ i iτ i h12 A(τ −s) ˆ 1 0 Id − 2 A Z (U − U )δ1 β + e − EU (s)ds 2 2h 0 1

=:a + b + c + d + e + f, which, together with the fact that E[Dϕ(U 0 )d] = E[Dϕ(U 0 )e] = 0, yields that         A =E Dϕ(U 0 )a + E Dϕ(U 0 )b + E Dϕ(U 0 )c + E Dϕ(U 0 )f = : A1 + A2 + A3 + A4 . Based on the estimates e x − (1 − x2 )−1 (1 + x2 ) = O(x 3 ) for x < 1, and ! ! τ ! ! i 1 A(τ −s) iτ −1 ! !e h2

A ≤ Cτ h −2 , ∀ s ∈ [0, τ ], (6.18) ≤ C − (I d − A) ! ! 2h 2 h2 we have |A1 | ≤ C ϕ 1,∞ τ h −2 A 3 E U 0 ≤ Cτ 3 h −6

(6.19)

and  |A2 | ≤ C ϕ 1,∞

τ

τ h −2 A

F(U (s))U (s) ds ≤ Cτ 2 h −2 .

0

Term A3 can be estimated based on Lemmas 6.1 and 6.2. In fact,

(6.20)

6.3 Approximate Error of the Ergodic Limit

169

  τ    iτ 0 −1  F(U (s))U (s) − F(U 0 )U 0 |A3 | =E Dϕ(U ) (I d − 2 A) 2h 0

   1  1 0 0 − F U 2 U 2 − F(U )U ds . Denote g(V ) := F(V )V, V ∈ B(0, 1), which is a continuous differentiable function satisfying |D k g(V )| ≤ C for V ≤ 1 and k ∈ N, such that F(U (s))U (s) − F(U 0 )U 0 = g(U (s)) − g(U 0 )  1 =Dg(U 0 )(U (s) − U 0 ) + r D 2 g(rU 0 + (1 − r )U (s))(U (s) − U 0 )2 dr 0  s   s i ˆ (r )dr + AU (r ) + iλF(U (r ))U (r ) − EU Z (U (r ))dβ(r ) =Dg(U 0 ) 2 0 h 0  1 r D 2 g(rU 0 + (1 − r )U (s))(U (s) − U 0 )2 dr + 0

 1 1 for s ∈ [0, τ ], and the same for the term F U 2 U 2 − F(U 0 )U 0 . Based on s   Lemma 6.2 with γ = 1 and the fact that E Dg(U 0 ) 0 Z (U (r ))dβ(r ) = 0, we hence get |A3 | ≤ C(τ 2 h −2 + τ 3 h −4 + τ 2 )

(6.21)

similar to the proof of Lemma 6.2. Rewrite ⎛ 1 u 1 − u 01 ⎜ 1 0 .. Z (U − U )δ1 β = ⎝ . u 1M − u 0M √ k=1 ek (x 1 ) ηk δ1 βk

⎞ ⎟ ⎠ E M K Λδ1 β

⎛ K ⎜ =⎝

⎞ ..

.

K

k=1 ek (x M )

√ ηk δ1 βk

⎟ 1 0 ⎠ (U − U )

= : G(U 1 − U 0 ), 1

where G satisfies that E[GU 0 ] = 0. Utilizing that E[G F(U 0 )U 0 ] = 0 and U 2 = 1 (U 1 − U 0 ) + U 0 , we rewrite term A4 as 2 $  %

−1  τ 1 i iτ i 2 A(τ −s) ˆ 0 1 0 h Id − 2 A A4 = − E Dϕ(U ) G(U − U ) + e EU (s)ds 2 2h 0 

 −1  1 1 1 1 iτ τ i G i 2 AU 2 + iλτ F(U 2 )U 2 + iGU 2 = − E Dϕ(U 0 ) I d − 2 A 2 2h h

170

6 Approximation of Ergodic Limit for Conservative Stochastic Nonlinear …



 τ 1 i A(τ −s) ˆ − E Dϕ(U 0 ) e h2 EU (s)ds 

=

0

  −1 τ 0 ) I d − iτ A 1 − U 0) E Dϕ(U G A(U 4h 2 2h 2    −1  1 1 1 iτ 0 0 0 + λτ E Dϕ(U ) I d − 2 A G F(U 2 )U 2 − F(U )U 2 2h    −1 iτ 1 0 2 1 0 G (U − U ) + E Dϕ(U ) I d − 2 A 4 2h  $ % −1  τ 1 iτ 1 2 0 i 2 A(τ −s) ˆ 0 h Id − 2 A + E Dϕ(U ) G U − e EU (s)ds 2 2h 0

=:A4,1 + A4,2 + A4,3 + A4,4 ,

where  5  1  1 |A4,1 | ≤ Cτ h −2 E G 2 2 E U 1 − U 0 2 2 ≤ C τ 2 h −4 + τ 2 h −2 and  ! !2  21 1 1 ! 0 0! 2 2 E !F(U )U − F(U )U !  5 1 3  ≤Cτ 2 E U 1 − U 0 2 2 ≤ C τ 2 h −2 + τ 2 .

 1 |A4,2 | ≤Cτ E G 2 2

According to E[G 3 U 0 ] = 0, A4,3 can be expressed as    −1  1 1 1 τ 1 iτ i 0 2 1 0 G i 2 AU 2 + iτ λF(U 2 )U 2 + G(U − U ) . E Dϕ(U ) I d − 2 A 4 2 2h h

For any U ∈ C M , we have $

K  √ E GU = E Z (U )δ1 β ≤ CE U

ηk |δ1 βk | 2

2

% 21

1 1  ≤ Cτ 2 E U 2 2 .

k=1 1

1

Hence, E G 3 (U 1 − U 0 ) ≤ Cτ 2 (E G 2 (U 1 − U 0 ) 2 ) 2 can be further estimated based on Lemma 6.1 with γ = 2 and (6.11) with γ = 4, that is, 2 ⎤ $ %2  M   K    √ 1 0  ⎦  E G 2 (U 1 − U 0 ) 2 =E ⎣ e (x ) η δ β (u − u ) k m k 1 k m m    m=1  k=1 ⎡

6.3 Approximate Error of the Ergodic Limit

171

⎡

$

≤E ⎣ U 1 − U 0 2

K 

%4 ⎤ 1 2

ηk |δ1 βk |

⎦

k=1

⎛ $

  ≤ E U 1 − U 0 4 ⎝E 1 2

K 

%8 ⎞ 21 1 2

ηk |δ1 βk |

⎠

k=1 4 −4

≤C(τ h

+ τ ). 3

1

Combining with Lemma 6.1 and U 2 ≤ 1, we get |A4,1 + A4,2 + A4,3 | ≤ C(τ 2 h −4 + τ 2 h −2 + τ 2 ). 5

For the term A4,4 , we have 

 −1 iτ 2 0 E Dϕ(U ) I d − 2 A G U 2h ⎡ ⎛ K 2 ⎞⎤ 2 0  −1 k=1 ek (x 1 )ηk (δ1 βk ) u 1 iτ ⎢ ⎜ ⎟⎥ .. =E ⎣ Dϕ(U 0 ) I d − 2 A ⎝ ⎠⎦ . 2h K 2 2 0 k=1 ek (x M )ηk (δ1 βk ) u M 

0

and ⎛ K

2 k=1 ek (x 1 )ηk u 1 (s)

ˆ (s) = 1 ⎜ EU ⎝ 2 K

.. .

⎞ ⎟ ⎠.

2 k=1 ek (x M )ηk u M (s)

Thus, we obtain ⎡



1 ⎢ iτ Dϕ(U 0 ) I d − 2 A A4,4 = E ⎢ 2 ⎣ 2h

−1

⎛ K

2 2 0 k=1 ek (x 1 )ηk (δ1 βk ) u 1

⎜ ⎜ ⎝ K

⎞⎤

⎟⎥ .. ⎟⎥ ⎠⎦ . 2 (x )η (δ β )2 u 0 e k=1 k M k 1 k M ⎡ ⎛ K ⎞ ⎤ 2  τ 1 k=1 ek (x 1 )ηk u 1 (s) ⎟ ⎥ 1 ⎢ i 2 A(τ −s) ⎜ .. 0 ⎜ ⎟ ds ⎥ h − E⎢ Dϕ(U ) e ⎣ ⎝ ⎠ ⎦ . 2 0 K 2 e (x )η u (s) M k M k=1 k ⎡ ⎛ K ⎞⎤ 2 0 2  −1 k=1 ek (x 1 )ηk ((δ1 βk ) − τ )u 1 ⎜ ⎟⎥ 1 ⎢ iτ .. ⎜ ⎟⎥ Dϕ(U 0 ) I d − 2 A = E⎢ ⎝ ⎠⎦ . 2 ⎣ 2h K 2 0 2 k=1 ek (x M )ηk ((δ1 βk ) − τ )u M

172

6 Approximation of Ergodic Limit for Conservative Stochastic Nonlinear …

⎡ +

1 ⎢ E ⎢ Dϕ(U 0 ) 2 ⎣ ⎡

1 ⎢ Dϕ(U 0 ) − E⎢ 2 ⎣



$

Id −

0



−1

⎛ K

%

2 0 k=1 ek (x 1 )ηk u 1

⎞

⎤

⎜ ⎜ ⎝ K

⎟ ⎥ .. ⎟ ds ⎥ ⎠ ⎦ . 2 (x )η u 0 e k=1 k M k M ⎛ K   ⎞ ⎤ 2 (x )η u (s) − u 0 e 1 k=1 k 1 k 1 ⎟ ⎥ i 1 A(τ −s) ⎜ .. ⎜ ⎟ ds ⎥ . e h2 (6.22) ⎝ ⎠ ⎦ .   K 2 0 k=1 ek (x M )ηk u M (s) − u M

τ

τ 0

iτ A 2h 2

−e

i

1 h2

A(τ −s)

Note that the first term in (6.22) vanishes as E(δ1 βk )2 = τ . Replacing U (s) − U 0 by the integral type of (6.2), we obtain |A4,4 | ≤ C(τ 2 h −2 + τ 2 ) based on (6.18) and the technique used in (6.21). It then leads to  5 |A4 | ≤ C τ 2 h −4 + τ 2 h −2 + τ 2 .

(6.23)

We then conclude from (6.19)–(6.23) that  5 |A | ≤ C τ 3 h −6 + τ 2 h −4 + τ 2 h −2 + τ 2 .

(6.24)

Estimation of C . Estimations of A1 and A2 show that   E a + b 2 ≤ C τ 6 h −12 + τ 4 h −4 .

(6.25)

Based on Hölder’s inequality, Itô isometry, Lemmas 6.1 and 6.2, together with the expression Z (U 1 − U 0 )δ1 β = G(U 1 − U 0 ), we have  E c + d 2 ≤Cτ

τ



1

E U (s) − U 2 2 ds +

0

τ

Cτ 2 h −4 ds

0

≤C(τ 3 h −4 + τ 3 ), ⎡

(6.26)

! !2 ⎤ −1 ! !   iτ ! ! E e 2 ≤CE ⎣ Z U (s) − U 0 ! ds ⎦ ! Id − 2 A ! 2h 0 ! F    τ M K       u m (s) − u 0 ek (xm )√ηk 2 ds ≤C E 

0

τ

m

m=1 k=1  3 −4 2

 ≤C τ h

+τ

6.3 Approximate Error of the Ergodic Limit

173

and  E f 2 ≤CE Z (U 1 − U 0 )δ1 β 2 + Cτ

τ

ˆ (s) 2 ds E EU

0

 1  1 ≤C E G 4 2 E U 1 − U 0 4 2 + Cτ 2 ≤C(τ 3 h −4 + τ 2 ). We then conclude that   E U (τ ) − U 1 2 ≤ C τ 6 h −12 + τ 3 h −4 + τ 2 ,

(6.27)

which yields  1  1 E U 1 − U 0 4 2 E U (τ ) − U 1 2 2 + E U (τ ) − U 1 2   ≤C τ 6 h −12 + τ 4 h −8 + τ 3 h −4 + τ 2 . (6.28)

|C | =O

   Estimation of B. As for B = E D 2 ϕ(U 0 ) U 1 − U 0 , a + b + c + d + e + f , according to Hölder’s inequality, (6.25) and (6.26), we have   2   E D ϕ(U 0 ) U 1 − U 0 , a + b + c + d   1  1 ≤C E U 1 − U 0 2 2 E a + b + c + d 2 2  7 5 ≤C τ 4 h −8 + τ 2 h −6 + τ 2 h −4 + τ 2 h −2 + τ 2 . Note that    E D 2 ϕ(U 0 ) U 1 − U 0 , e + f $   

% −1 iτ 1 Id − 2 A Z (U (s) − U )dβ(s) =E D ϕ(U ) U − U , i 2h 0  $ %  −1 1 iτ 2 0 1 0 1 0 + E D ϕ(U ) U − U , i I d − 2 A Z (U − U )δ1 β 2 2h  

 τ i h12 A(τ −s) ˆ 2 0 1 0 − E D ϕ(U ) U − U , e EU (s)ds 2

0

1

0

τ

0

= : B1 + B2 + B3 , where

  |B1 | ≤ C τ 6 h −12 + τ 3 h −4 + τ 2

according to (6.27) and Lemma 6.1. Furthermore,

174

6 Approximation of Ergodic Limit for Conservative Stochastic Nonlinear …

 −1

 τ 1 1 1 1 iτ B2 = E D 2 ϕ(U 0 ) i 2 AU 2 + iτ λF(U 2 )U 2 , i I d − 2 A Z (U 1 − U 0 )δ1 β 2 h 2h $ %  −1

 1 0 U −U iτ 1 2 0 1 0 Z (U − U )δ1 β δ1 β, i I d − 2 A + E D ϕ(U ) iZ 2 2 2h  −1

 iτ 1 Z (U 1 − U 0 )δ1 β + E D 2 ϕ(U 0 ) iZ (U 0 )δ1 β, i I d − 2 A 2 2h = : B2,1 + B2,2 + B2,3

 5 |B2,1 + B2,2 | ≤ C τ 2 h −4 + τ 2 h −2 + τ 2

with

similar to the estimate of E f 2 . Replacing U 1 − U 0 again by (6.7) and using the notation Z (U )δ1 β = GU , we obtain   1   |B2,3 | ≤  E D 2 ϕ(U 0 ) GU 0 , I d − 2   1   =  E D 2 ϕ(U 0 ) GU 0 , I d − 4

 −1 1 iτ  2 A G U 2  + C(τ 2 h −2 + τ 2 )  2h 2 −1

 iτ  2 1 0 A G (U − U )  + C(τ 2 h −2 + τ 2 )  2h 2

≤C(τ 2 h −2 + τ 2 ),

where we used the fact E[G 3 U 0 ] = 0 since U 0 is F0 -adapted. For term B3 ,      τ 1   1 1 1 τ i A(τ −s) ˆ |B3 | ≤ E D 2 ϕ(U 0 ) i 2 AU 2 + iτ λF(U 2 )U 2 , e h2 EU (s)ds  h 0      τ 1    1 i A(τ −s) 2 0 0 2  ˆ h 2 + E D ϕ(U ) iZ (U )δ1 β, e E U (s) − U ds  0      τ 1   1 i A(τ −s) ˆ 0 2 0  + E D ϕ(U ) iZ (U 2 )δ1 β, e h2 EU ds  0      τ 1  1  i A(τ −s) ˆ 0 2 −2 2 2 0 ≤C(τ h + τ ) + E D ϕ(U ) iZ (U 1 − U 0 )δ1 β, e h2 EU ds  2 0 ≤C(τ 2 h −2 + τ 2 ).

We finally obtain  5 |B| ≤ C τ 6 h −12 + τ 2 h −4 + τ 2 h −2 + τ 2 , which, together with (6.24) and (6.28), leads to     E ϕ(U (τ )) − ϕ(U 1 )  ≤ C τ 6 h −12 + τ 4 h −8 + τ 3 h −6 + τ 25 h −4 + τ 2 h −2 + τ 2 . That is, for a fixed step-size h ∈ (0, 1),

6.3 Approximate Error of the Ergodic Limit

175

   E ϕ(U (τ )) − ϕ(U 1 )  ≤ C h τ 2 . However, for any h, τ ∈ (0, 1) satisfying ρ := τ 2 h −2 ≤ C, we have 1

    E ϕ(U (τ )) − ϕ(U 1 )  ≤ C ρ 6 τ 3 + ρ 4 τ 2 + (ρ 3 + ρ 2 + ρ)τ 23 + τ 2 ≤ Cρ τ 23 , with some constant Cρ depending on ρ instead of h.



We would like to mention that the dependence on h or ρ may be released if higher regularity of both {U (t)}t≥0 and {U n }n∈N can be derived uniformly. More precisely, 1 AU (t) and h12 AU n are uniformly bounded in strong sense. h2 Now we are in the position to show the approximate error between the time average of FDS and the ergodic limit of FDA. Theorem 6.6 Under the assumptions in Theorem 6.5 on any time interval [0, T ] and for any f ∈ W4,∞ (S), for a fixed h ∈ (0, 1), there exists a positive constant C h = C( f, η, h) such that   N −1      1  1   n +τ . f (U ) − fˆ  ≤ C h E   N T n=0

Proof Based on (6.15)–(6.17), it suffices to estimate term I I I . For any f ∈ W4,∞ (S), we know from the statement above that the solution to the Poisson equation L ϕ = f − fˆ can be extended to ϕ ∈ W4,∞ (B(0, 1)). Based on (6.14), Lemma 6.1 and the condition τ = O(h 4 ), we have    1  Eϕ(U 1 ) =Eϕ(U 0 ) + τ E L Φ ϕ(U 0 ) + E D 3 ϕ(U 0 )(U 1 − U 0 )3 + O(τ 2 ) 6   =Eϕ(U 0 ) + τ E L Φ ϕ(U 0 ) + O(τ 2 ), (6.29) where in the last step we used the fact that )

*



E D 3 ϕ(U 0 )(U 1 − U 0 )3 =E



1 1 1 1 τ D 3 ϕ(U 0 ) i 2 AU 2 + iλτ F(U 2 )U 2 + iZ (U 2 )δ1 β h

3 

 3 1 + O(τ 2 h −2 + τ 2 ) =E D 3 ϕ(U 0 ) iZ (U 2 )δ1 β   3  i 3 0 1 0 0 Z (U − U )δ1 β + iZ (U )δ1 β =E D ϕ(U ) 2 + O(τ 2 h −2 + τ 2 ) =O(τ 2 h −2 + τ 2 )

(6.30)

3  based on the linearity of Z , Lemma 6.1 and the fact that E iZ (U 0 )δ1 β = 0. We can also get the following expression similar to (6.29) based on the Taylor expansion

176

6 Approximation of Ergodic Limit for Conservative Stochastic Nonlinear …

and Lemma 6.2

 τ 1 E Dϕ(U 0 )b(U (t)) + D 2 ϕ(U 0 ) (σ (U (t)))2 dt Eϕ(U (τ )) =Eϕ(U 0 ) + 2 0  τ

+E Dϕ(U 0 )σ (U (t))dβ(t) 0

 1  + E D 3 ϕ(U 0 )(U (τ ) − U 0 )3 + O(τ 2 ) 6 

τ

=Eϕ(U 0 ) + E

L˜ t ϕ(U 0 )dt + O(τ 2 ),

(6.31)

0

where

1 L˜ t ϕ(U 0 ) := Dϕ(U 0 )b(U (t)) + D 2 ϕ(U 0 ) (σ (U (t)))2 2  τ  and E 0 Dϕ(U 0 )σ (U (t))dβ(t) = 0. Thus, subtracting (6.29) from (6.31), we derive 

  τ      E τ L Φ ϕ(U 0 ) − ˜ t ϕ(U 0 )dt  ≤ E ϕ(U (τ )) − ϕ(U 1 )  + Cτ 2 . (6.32) L   0

Note that   τ ) *   τ      0 0 0 0    E L˜ t ϕ(U ) − L ϕ(U ) dt  ≤  E Dϕ(U ) b(U (t)) − b(U ) dt   0 0   τ  1  2    0 0 0  (6.33) + E D ϕ(U ) σ (U (t)) − σ (U ), σ (U (t)) + σ (U ) dt  . 2 0 For the first term in (6.33), we have        )  E Dϕ(U 0 ) b(U (t)) − b(U 0 )  = E Dϕ(U 0 ) i 1 A U (t) − U 0 2 h*  0 0 0 ˆ (t) − U )  ≤ C(th −2 + t), + iλ F(U (t))U (t) − F(U )U − E(U where we used the fact that g(V ) = F(V )V, V ∈ S, is a continuous differentiable function and replaced U (t) − U 0 by the integral form of (6.2) similar to the estimate in Theorem 6.5. The second term in (6.33) can be estimated in the same way. Thus, we have  τ ) *    ˜ t ϕ(U 0 ) − L ϕ(U 0 ) dt  ≤ C(τ 2 h −2 + τ 2 ). E L (6.34)   0

We hence conclude based on (6.30), (6.32), (6.34) and Theorem 6.5 that

6.3 Approximate Error of the Ergodic Limit

177

 N −1

 1  1 3  Φ n n n n+1 n 3  III = D ϕ(U )(U E L ϕ(U ) − L ϕ(U ) + −U )  N  6τ n=0 

  τ )  τ *     1 ≤ sup E τ L Φ ϕ(U 0 ) − L˜ t ϕ(U 0 )dt  +  E L˜ t ϕ(U 0 ) − L ϕ(U 0 ) dt  τ U 0 ∈S 0 0 + C(τ h −2 + τ ) ≤ C h τ.

(6.35)

From (6.16), (6.17) and (6.35), we finally obtain   N −1      1  1   n ˆ ≤ C E + τ . f (U ) − f   h   N T n=0

 Remark 6.3 Based on the theorem above and the ergodicity of (6.2), for a fixed h, we obtain   N −1   T   1  1   f (U n ) − f (U (t))dt  ≤ C h (B(T ) + τ ), E   N T 0 n=0

which implies that the global weak error is of order one, i.e.,  ) *   ˜ + τ ), t ∈ [nτ, (n + 1)τ ], E f (U n ) − f (U (t))  ≤ C h ( B(t) ˜ ) → 0 as T → ∞. On the other hand, a time independent where B(T ) → 0 and B(T weak error in turn leads to the result stated in Theorem 6.6.

6.4 Numerical Experiments In this section, numerical experiments are given to test several properties of the fully discretization (6.7) of the stochastic NLSE (Eq. 3.8) in focusing case, i.e., λ = 1. Since these properties are considered in mean-square sense, we simulate noises δn β √ by identically distributed random variables τ ξn with ξn being independent K dimensional N (0, 1)-random variables, and approximate the expectation by taking averaged value over 1000 paths. In addition, the proposed scheme, which is implicit, is numerically solved utilizing the fixed point iteration. 1

1

+ε

2 2 ∈ L in Theorem 6.5, the eigenvalues {ηk }k∈N According to the assumption 2 ∞ Q(1+ε) of Q should satisfy that k=1 k ηk < ∞. We choose ηk = k −4 throughout the numerical experiments, and the first K terms will be used in the scheme. As we omit the boundary nodes in the simulation, we may choose the normalized initial value U 0 = c∗ (U 0 (1), · · · , U 0 (M)) based on function u 0 (x) satisfying U 0 (m) = u 0 (mh), m = 1, · · · , M, in which u 0 (x) need not to satisfy the boundary condition

6 Approximation of Ergodic Limit for Conservative Stochastic Nonlinear … 1

10 -11 =2 -4 =2 -5 =2 -7

0

-1 0 10 20 30 40 50 60 70 80 90 100

E||Un||2 -1

E||Un||2 -1

=2 -6

1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1

=2 -4 =2 -5 =2

-6

E||Un||2 -1

178

=2 -7

0

0.5

t

1

1.5

2

2.5

3

1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1

=2 -10 =2 -11 =2 -12 =2 -13

0 0.005 0.01 0.015 0.02 0.025 0.03

t

t

Fig. 6.1 Charge evolution E U n 2 − 1 for a the proposed scheme with T = 100 under step-sizes τ = 2−i (i = 4, 5, 6, 7), b IME scheme with T = 3 under step-sizes τ = 2−i (i = 4, 5, 6, 7), and c EM scheme with T = 2−5 under step-sizes τ = 2−i (i = 10, 11, 12, 13) (h = 0.05, K = 30)

in (6.1). Let u 0 (x) = 1. Then the normalized initial value U 0 satisfies U 0 = 1, which is used in Figs. 6.1, 6.4 and 6.3. Charge evolution. To verify Proposition 6.2 and show the superiority of the proposed scheme, we first simulate the discrete charge for the proposed scheme compared with Euler–Maruyama (EM) scheme and implicit Euler (IE) scheme, respectively. Figure 6.1 shows that the proposed scheme possesses the discrete charge conservation law E U n 2 = 1, while both the EM scheme and the IE scheme deviate from the initial charge gradually. Since the EM scheme is not stable and will blow up in a short time, we choose the time step-size τ small enough for the EM scheme in the experiments. M γ Ergodic limit. We use the notation U γ := m=1 (| pm |γ + |qm |γ ) for U ∈ C M and  γ = 3, 4 with P = ( p1 , · · · , p M ) , Q = (q1 , · · · , p M ) being the real and imaginary parts of U . To verify the ergodicity shown in Theorem 6.3, which states that the  temporal averages will converge to the ergodic limit S f dμh for almost every initial  N −1 E[ f (U n )] for the provalue U 0 ∈ S, we simulate the temporal averages N1 n=0 posed scheme stating from different initial values. In Fig. 6.2, we choose test functions 4 f ∈ W4,∞ as (a) f (U ) = U 33 , (b) f (U ) = sin( U 44 ) and (c) f (U ) = e− U 4 , and choose five different initial values Ul0 = c∗ (Ul0 (1), · · · , Ul0 (M)) , l = 1, · · · , 5 based on the following five functions 1 i u 0,1 (x) = √ + √ , u 0,2 (x) = 1, u 0,3 (x) = 2x, 2 2 % $  1 π (exp − 1) (1 − exp (x(1 − x))), u 0,4 (x) = 1 − 2 4 x x u 0,5 (x) =c∗ sech( √ ) exp (i ) 2 2

6.4 Numerical Experiments

179

f(U)=||U||3

f(U)=sin(||U||4)

3 U 02

U 05

0.25 0.2

Time average

0.1 0.08 0.06

U

0 1 0 2

U0 3

U0 4 0

0.95

U5

0.9

0.04 0.15

U 33 , 10

0.85 0

0.02 0 2 4 6 8 10 12 14 16 18 20

0 2 4 6 8 10 12 14 16 18 20

t

t

Fig. 6.2 The temporal averages b f (U ) = 10

sin( U 44 )

1 N

 N −1 n=0

40

60

80 100 120 140

t

E[ f (U n )] for the proposed scheme with a f (U ) = 4

-3

10

10

-3

10

Weak error

8

6 4 2

6 4

-3

2

6 4 2

0

0

-2

-2

-2

0 100 200 300 400 500 600 700 800 900 1000

10

8

0

t

20

and c f (U ) = e− U 4 (τ = 2−6 , h = 0.05, K = 30)

8

Weak error

0.12

U

Weak error

Time average

U 04

f(U)=e -||U||4

1 0 U1 0 U 2 U0 3 0 U4 U05

0.14

U 03

0.3

4

4

0.16

U 01

Time average

0.35

0 100 200 300 400 500 600 700 800 900 1000

t

0 100 200 300 400 500 600 700 800 900 1000

t

Fig. 6.3 The weak error |E[ f (U n ) − f (U (T ))]| for a f (U ) = U 33 , b f (U ) = sin( U 44 ) and 4 c f (U ) = e− U 4 (τ = 2−12 , h = 0.05, T = 103 , K = 30)

with Ul0 (m) = u 0,l (hm), 1 ≤ m ≤ M and c∗ being normalized constants. Figure 6.2 shows that the proposed scheme starting from different initial values converges to the same value with error no more than O(τ ) with h = 0.05 and τ = 2−6 , which also coincides with the error analysis in Theorem 6.6. Weak error and convergence order. For a fixed h, Figs. 6.3 and 6.4 show that the weak error over long time and weak convergence order in temporal direction, respectively, with test functions f being the same as those given above. Based on the ergodicity for both FDS and FDA, the weak error is supposed to be independent of time interval when time is large enough. To verify this property, we simulate the weak error over long time in Fig. 6.3. It shows that the weak error for the proposed scheme would not increase before T = 1000. Furthermore, Fig. 6.4 shows that the proposed scheme is of order one in the weak sense which coincides with the statement in Remark 6.3.

180

6 Approximation of Ergodic Limit for Conservative Stochastic Nonlinear … Global Weak Order

10 -1

Order 0.5 Order 1.0

Global Weak Order

10 -1

10 -1

10 -2

10 -2

10 -2

10 -3

10 -3

10 -3

10 -4 10 -4

10 -3

(a)

f (U) = U33

Global Weak Order Order 0.5 Order 1.0

Order 0.5 Order 1.0

10 -4 -4 10

(b)

10 -3

f (U) = sin(U44 )

10 -4 10 -4

10 -3

(c)

4 f (U) = e−U4

Fig. 6.4 The weak convergence order of |E[ f (U n ) − f (U (T ))]| with a f (U ) = U 33 , b f (U ) = 4 sin( U 44 ) and c f (U ) = e− U 4 (τ = 2−i , 10 ≤ i ≤ 13, h = 0.05, T = 2−1 , K = 30)

Summary In this chapter, stochastic NLSE with a linear multiplicative noise is studied. Similar to the deterministic case, this model possesses both the symplectic and multisymplectic conservation laws and is charge-conserved almost surely. A finite dimensional approximation based on the central difference scheme is shown to be chargeconserved and ergodic on the unit sphere with charge one. A full discretization is then proposed possessing the discrete charge conservation law and the multi-symplectic conservation law. The numerical solution is also ergodic on the unit sphere, whose temporal approximation converges to the ergodic limit of the finite dimensional approximation. The ergodicity for the exact solution of stochastic systems without dissipative conditions is still unknown up to the knowledge of the authors. It is introduced that the deterministic NLSE possesses an invariant Gibbs measure in Chap. 3. For stochastic cases, the behavior of the solution will be definitely influenced by the noise. There is a series of papers considering the blow-up phenomenon for stochastic NLSEs theoretically [62, 63, 66] and numerically [18, 68], and also the large deviation principle to give the asymptotics of the tails of the blow-up time [86–88]. We refer to [36, 45, 138] for large deviations for invariant measures, to [52, 72, 75] for more details about the theory and applications of large deviations. The large deviations related to numerical approximations for SDEs, as well as their invariant measures, are also worth considering as a way to investigate the behavior of the exact solution.

Appendix A

Basic Inequalities

This chapter is devoted to giving some basic inequalities (see e.g. [99]) which are frequently used in the study of the well-posedness and numerical approximations for differential equations. Lemma A.1 (ε-Young inequality) Let ε > 0, p, q > 1 such that

1 p

+

1 q

= 1. Then

q

q ε− p |b|q ε|a| p + ≤ ε|a| p + ε− p |b|q |a||b| ≤ p q

for any a, b ∈ R. For the case ε = 1, the inequality above is known as the Young inequality for products, which is also called Cauchy inequality if in particular p = q = 2. In what follows, we introduce the Young inequality for integral operators.  Lemma A.2 Let X and Y be measurable spaces. Denote T f (x) := Y K (x, y) f (y)dy with K : X × Y → R satisfying sup K (x, ·)Lr (Y ) ≤ C0 x∈X

and sup K (·, y)Lr (X ) ≤ C0 y∈Y

for r ≥ 1. Then for 1 ≤ p, q ≤ ∞ satisfying 1 +

1 q

=

1 r

+ 1p , it holds

T f Lq (X ) ≤ C0  f L p (Y ) . Let O ⊂ Rd , d ≥ 1, be a measurable set.

© Springer Nature Singapore Pte Ltd. 2019 J. Hong and X. Wang, Invariant Measures for Stochastic Nonlinear Schrödinger Equations, Lecture Notes in Mathematics 2251, https://doi.org/10.1007/978-981-32-9069-3

181

182

Appendix A: Basic Inequalities

Lemma A.3 (Hölder’s inequality) Let 1 ≤ p, q ≤ ∞ such that f ∈ L p (O), g ∈ Lq (O), it holds

1 p

+

1 q

= 1. For any

 f gL1 (O ) ≤  f L p (O ) gLq (O ) . For the case p = q = 2, the inequality above is also called the Schwarz inequality. Lemma A.4 (Minkowski’s inequality) Let 1 ≤ p ≤ ∞. For any f, g ∈ L p (O), the following inequality holds  f + gL p (O ) ≤  f L p (O ) + gL p (O ) . Sobolev Type Inequalities Sobolev inequalities are frequently used to show the Sobolev embedding between different Sobolev spaces. We refer to [77] for more details. Let k be a non-negative integer and 1 ≤ p ≤ ∞. Denote by Wk, p (O) the space of equivalence classes of functions u ∈ L p (O) such that D α u ∈ L p (O) for all derivations of length |α| ≤ k, where Dα u =

∂ α1 ∂ αd u α1 · · · ∂ x1 ∂ xdαd

for each multi-index α = (α1 , · · · , αd ). The Gagliardo–Nirenberg interpolation inequality introduced below gives the estimates on weak derivatives of functions on Sobolev spaces. Lemma A.5 (Gagliardo–Nirenberg inequality) Let 1 ≤ p, q, r ≤ ∞ and j, m be two integers such that 0 ≤ j < m and 1 j = +a p d



1 m − r d

 +

1−a , q

where a ∈ [ mj , 1]. If r > 1 and m − j − dr = 0, choose a < 1. Then there exists d C = C(d, m, j, a, q, r ) such that for any u ∈ C∞ 0 (R ), one has  |α|= j

⎛ ∂ uL p (Rd ) ≤ C ⎝ α

 |α|=m

⎞a ∂ uLr (Rd ) ⎠ u1−a . Lq (Rd ) α

Lemma A.6 (Gagliardo–Nirenberg–Sobolev inequality) For any 1 ≤ p < d, there d exists a constant C depending only on d and p such that for any u ∈ C∞ 0 (R ), one has uL p∗ (Rd ) ≤ CDuL p (Rd ) with

1 p∗

=

1 p

− d1 .

Appendix A: Basic Inequalities

183 ∗

The result above implies the Sobolev embedding W1, p (Rd ) → L p (Rd ). The general case is stated in the Sobolev embedding theorem. Theorem A.1 (Sobolev embedding theorem) Let O = Rd , d ≥ 1. (i) If k > l and 1 ≤ p < q < ∞ satisfy (k − l) p < d and 1 k 1 l − = − , p d q d then one has the embedding Wk, p (Rd ) → Wl,q (Rd ) and the embedding is continuous. (ii) If d < p and k r +α 1 − =− p d d with α ∈ (0, 1], then one has the embedding Wk, p (Rd ) → Cr,α (Rd ). The second part of the Sobolev embedding theorem is a direct consequence of the Morrey inequality. Lemma A.7 (Morrey inequality) Let d < p ≤ ∞ and γ = 1 − dp . Then there exists C = C(d, p) such that uC0,γ (Rd ) ≤ CuW1, p (Rd ) for all u ∈ C1 (Rd ) ∩ L p (Rd ). The proof of the Sobolev embedding theorem relies on the following inequality, which is also known as the Hardy–Littlewood–Sobolev fractional integration theorem. Lemma A.8 (Hardy–Littlewood–Sobolev inequality) Assume that r > 1, 1 < p < q < ∞ and 1 1 1 1+ = + . q r p Then there exists C = C( p, q) such that Ir f Lq (Rd ) ≤ C f L p (Rd ) , where Ir f (x) =



|x − y|− r f (y)dy denotes the Riesz potential. d

Rd

184

Appendix A: Basic Inequalities

Gronwall Type Inequalities The classical Gronwall inequality is established by Gronwall [94]. It is then modified by Bellman [19], Bihari [21] and other researchers to deal with stability and uniqueness problems of differential equations. Lemma A.9 (Gronwall–Bellman inequality) Let T > 0 and u, F : [0, T ] → R be continuous and non-negative functions. If there exists K > 0 such that

t

u(t) ≤ K +

F(s)u(s)ds, ∀ t ∈ [0, T ],

0



then u(t) ≤ K exp

t

 F(s)ds , ∀ t ∈ [0, T ].

0

Lemma A.9 has a discrete version, which is frequently used in the numerical analysis of differential equations. Lemma A.10 Let {u n }n∈N and {Fn }n∈N be non-negative sequences. If there exists K > 0 such that N −1  Fn u n , N ∈ N, uN ≤ K + n=1

then uN ≤ K

N −1 

(1 + Fn ) ≤ K exp

n=1

N −1 

Fn , N ∈ N.

n=1

The Gronwall–Bellman inequality, together with its discrete version, is widely used to show the stability and uniqueness of the solutions of differential equations and their numerical discretizations. This inequality is then generalized by Bihari to establish a bound for the difference of solutions of differential equations with perturbations. The proof is given below for the readers’ convenience. Lemma A.11 (Bihari inequality) Let u, F be non-negative and continuous functions in [a, b] ⊂ R, and K > 0 be a constant. Assume that  is a non-negative nondecreasing continuous function in [0, ∞) such that

u(t) ≤ K +

t

F(s) (u(s))ds, ∀ t ∈ [a, b],

a

then one has u(t) ≤ Θ

−1



Θ(K ) + a

t

 F(s)ds , ∀ t ∈ [a, T ]

Appendix A: Basic Inequalities

185

for T ≤ b such that Θ(K ) + [a, T ]. Here,

t a

F(s)ds is within the domain of Θ −1 for any t ∈

v

Θ(v) =

v0

1 dx  (x)

for v0 > 0, v ≥ 0 and Θ −1 denotes the inverse function of Θ, which exists due to the monotony of  . Proof Denoting



t

y(t) := K +

F(s) (u(s))ds > 0,

a

then we have u(t) ≤ y(t) for any t ∈ [a, b]. Furthermore,  (u(t)) ≤  (y(t)) and  (y(t)) > 0 since  is non-decreasing. It then yields that y (t) F(t) (u(t)) = ≤ F(t).  (y(t))  (y(t)) Integrating with respect to t, we have

t

Θ(y(t)) − Θ(y(a)) = a

1 dy(s) ≤  (y(s))



t

F(s)ds a

with Θ(y(a)) = Θ(K ). As a result, u(t) ≤ y(t) ≤ Θ

−1



Θ(K ) +



t

F(s)ds a

since Θ −1 is also non-decreasing. Furthermore, we claim that the result above is independent of the choice of v0 in the definition of function Θ. In fact, we define another function

v 1 ¯ d x. Θ(v) :=  (x) v1 By denoting

δ :=

v1 v0

we get

and hence

1 d x,  (x)

¯ Θ(v) = Θ(v) − δ Θ¯ −1 (w) = Θ −1 (w + δ).

186

Appendix A: Basic Inequalities

Finally, we derive 

−1 ¯ ¯ Θ(K ) + Θ



t

F(s)ds

a

 

t ¯ Θ(K ) + =Θ F(s)ds + δ a  

t −1 Θ(K ) + F(s)ds =Θ −1

a



and complete the proof. Example A.1 As an example, we choose  (x) := x. Then Θ(v) = ln(v) − ln(v0 ) for v, v0 > 0 and

Θ −1 (w) = exp(w + ln(v0 )) for w ∈ R.

Assume that the condition in Lemma A.9 holds, that is,

t

u(t) ≤ K +

F(s)u(s)ds. 0

Then according to the Bihari inequality, we derive  

t F(s)ds u(t) ≤Θ −1 Θ(K ) + 0  

t F(s)ds + ln(v0 ) = exp ln(K ) − ln(v0 ) + 0   t F(s)ds , =K exp 0

which verifies the result of the Gronwall–Bellman inequality. Example A.2 If a continuous function u : [a, b] → R satisfies

u (t) ≤ 2

u 20

+

t

F(s)|u(s)|ds a

with u 0 ∈ R and F being non-negative and continuous in [a, b], we claim that it holds

1 t F(s)ds. |u(t)| ≤ |u 0 | + 2 a In fact, by denoting y0 := u 20 , y(t) := u 2 (t) for t ∈ [a, b] and choosing  (x) = x for x ∈ R+ such that 1 2

1

1

Θ(v) = 2v 2 − 2v02

Appendix A: Basic Inequalities

187

and

w

Θ −1 (w) =

we have

2 t

y(t) ≤ y0 +

1

+ v02

2

,

F(s) (y(s))ds.

a

Then according to the Bihari inequality, it leads to 

y(t) ≤Θ −1 Θ(y0 ) + 

1 = y0 + 2 1 2





t

F(s)ds

a

2

t

,

F(s)ds a

which verifies the claim. More generally, one can modify the Gronwall–Bellman inequality by replacing the constant K by a function depending on t, which is stated below. Theorem A.2 Assume that continuous functions u, Φ and F satisfy that F is nonnegative in [a, b] and

u(t) ≤ Φ(t) +

t

F(s)u(s)ds, ∀ t ∈ [a, b].

a

Then

u(t) ≤ Φ(t) +



t

t

F(s)Φ(s) exp a

 F(r )dr ds, ∀ t ∈ [a, b].

s

Proof Denoting

y(t) :=

t

F(s)u(s)ds a

with y(a) = 0, we get 

y (t) =F(t)u(t) ≤ F(t) Φ(t) +



t

F(s)u(s)ds

a

=F(t)Φ(t) + F(t)y(t). t Solving this ordinary differential equation by multiplying exp(− a F(s)ds), we derive   t   t  d y(t) exp − F(s)ds ≤ F(t)Φ(t) exp − F(s)ds . dt a a

188

Appendix A: Basic Inequalities



It then leads to



t

y(t) ≤

t

F(s)Φ(s) exp a

 F(r )dr ds,

s

which completes the proof by using u(t) ≤ Φ(t) + y(t).



The theorem above also has a discrete version. Corollary A.1 Let {u n }n∈N , {K n }n∈N and {Fn }n∈N be non-negative sequences which satisfy  Fl u l , ∀ n ≥ 0. un ≤ Kn + 0≤l