Table Of ContentLecture 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-MichelMorel,CMLA,ÉcoleNormaleSupérieuredeCachan,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
AlessioFigalli,DepartmentofMathematics,SwissFederalInstituteofTechnology,
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
AngelaKunoth,MathematicalInstitute,UniversityofCologne,Cologne,Germany
Ariane Mézard, Institut Mathématique de Jussieu-Paris Rive Gauche, Sorbonne
University, PARIS, France
MarkPodolskij,DepartmentofMathematics,AarhusUniversity,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
(cid:129)
Invariant Measures
for Stochastic Nonlinear
ö
Schr dinger Equations
Numerical Approximations and Symplectic
Structures
123
JialinHong XuWang
LSEC,ICMSEC, Academy of Mathematics LSEC,ICMSEC, Academy of Mathematics
andSystems Science andSystems Science
ChineseAcademy of Sciences ChineseAcademy of Sciences
Beijing,China Beijing,China
Schoolof Mathematical Sciences Schoolof Mathematical Sciences
UniversityofChineseAcademyofSciences UniversityofChineseAcademyofSciences
Beijing,China Beijing,China
ISSN 0075-8434 ISSN 1617-9692 (electronic)
Lecture Notesin Mathematics
ISBN978-981-32-9068-6 ISBN978-981-32-9069-3 (eBook)
https://doi.org/10.1007/978-981-32-9069-3
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Preface
Stochasticdifferentialequations,originatedfromtheworkofA.Einsteinandputon
solidermathematicalfootingbyK.ItôandR.L.Stratonovich,arenowwidelyused
to model various phenomena caused by random medium or stochastic external
sources,suchaspropagationofnonlineardispersivewavesinrandommediumand
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.
Themainobjectofthismonographistheanalysisandnumericalapproximations
of invariant distributions for stochastic nonlinear Schrödinger equations. The con-
sidered model shows good performances over longtime due to its dynamical
behaviorandgeometricstructure.Amongtheseproperties,ergodicityandstochastic
(conformal) multi-symplecticity play fundamental roles in the study of the dynam-
icalbehaviorandgeometricstructureforstochasticnonlinearSchrödingerequations.
The stochastic multi-symplecticity is an extension of the multi-symplectic conser-
vationlawinthedeterministiccase(see[34,117]andreferencestherein).Itimplies
that the spatio-temporal geometric structure of the system is preserved over long-
time, and isan essential part ofsymplectic geometric theory.
Ergodic theory, as an important branch of mathematics, studies the asymptotic
behaviorofmeasure-preservingtransformations.Itoriginatesfromthestudyofthe
“ergodic hypothesis” which is the basic hypothesis in statistical mechanics and
informationtheory,andisnowdevelopedasapowerfulamalgamofmethodsused
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
stochasticHamiltoniandissipativesystemsareintroducedastwoexamplestostudy
their invariant measures, geometric structures, and numerical approximations.
Tohaveabetterunderstandingofthelongtimebehaviorofstochasticdifferential
equations (SDEs) with ergodic invariant measures, we then investigate the fol-
lowing fundamental problems:
(cid:129) Whatkindofsystemsareergodic,andwhatistheconvergenceratebetweenthe
R R
temporal average T1 0TuðPsxÞds and the spatial average Mudl?
(cid:129) 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?
(cid:129) Is the temporal average oRf a numerical approximation (possibly not ergodic) a
proper approximation of udl?
M
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 dif-
ferential 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 mea-
sures,geometricstructures,andthelargedeviationprinciple.Werecallsomeresults
about invariant measures for both deterministic and stochastic Schrödinger equa-
tions 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 H1ð0;LÞwithL[0.Forthestochasticcasewithweak damping, the
solution is shown to be Gaussian for the linear case. As a result, the distribution
ofthesolutionconvergestoauniqueC-valuedGaussiandistribution.Forthecubic
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.
InChap.4,weconcentrateonthegeometricstructuresforbothdeterministicand
stochastic nonlinear Schrödinger equations, as well as their numerical approxima-
tions.Severaltemporalsemi-discretizationsandfulldiscretizationsarereviewedfor
the deterministic nonlinear Schrödinger equation. The stochastic symplectic and
multi-symplecticconservationlawsareprovedforstochasticnonlinearSchrödinger
equations with a linear multiplicative noise in Stratonovich sense. The numerical
approximationsforthedeterministiccasearegeneralizedtothestochasticcase.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.
Chapter5isdevotedtostudyingthenumericalapproximationsofthestochastic
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
thatthesystempossessesaconformalmulti-symplecticgeometricstructure instead
of the multi-symplectic structure in conservative case. We are interested in the
investigationofthegeometricstructureforthisdampedequation,andaimtogivea
computable approximation to the original invariant measure, as well as the
approximate error defined in the standard way.
WeconsiderinChap.6thestochasticnonlinearSchrödingerequationwithlinear
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.
Ourpurposesaretofindanergodicfinitedimensionalapproximationoftheoriginal
system, and to approximate its ergodic limit through an ergodic, multi-symplectic
full discretization. In addition, the temporal average of the fully discrete schemeis
shown to converge to the spatial average of the finite dimensional approximation
withaspecificrate.Theseresultsmayhelptofindanewwaytoshowtheergodicity
of conservative equations as an afterthought.
viii Preface
Last butnotleast, wefindit anefficient waytogive a visible descriptionofthe
ergodic limit via numerical experiments, and several numerical experiments are
giveninChaps.5and6tomakeitclearertothereaderwhatthelongtimebehaviors
of ergodic processes are.
Beijing, China Jialin Hong
May 2017 Xu Wang
Acknowledgements Itisourpleasuretothanktheauthorsofourreferences fortheirimportant
contributionsandmotivatingustogetintothesewonderfulmathematicalfields.Wewouldliketo
thank all the referees, whose comments and suggestions are of great help in improving this
monograph.WewouldalsoliketothankDr.ChuchuChen,YulanLu,JianboCui,LiyingSun,
ChuyingHuang,DiancongJinandDeruiShengfortheircarefulreadingandpointingoutalotof
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)foritsfinancialsupport.
Contents
1 Invariant Measures and Ergodicity. . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Basic Definitions in Measure Spaces . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Invariant Measures for Stochastic Processes. . . . . . . . . . . . . . . . . 5
1.3 Ergodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.4 Strong Feller and Irreducibility Properties . . . . . . . . . . . . . . . . . . 15
1.5 Invariant Measures for Hamiltonian Systems . . . . . . . . . . . . . . . . 17
1.5.1 Stochastic Kubo Oscillator . . . . . . . . . . . . . . . . . . . . . . . . 19
1.5.2 Stochastic Dissipative Hamiltonian Systems . . . . . . . . . . . 24
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2 Invariant Measures for Stochastic Differential Equations. . . . . . . . . 31
2.1 Ergodicity of Solutions to General Stochastic Differential
Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.1.1 Existence of Invariant Measures . . . . . . . . . . . . . . . . . . . . 32
2.1.2 Uniqueness of the Invariant Measure . . . . . . . . . . . . . . . . 34
2.2 Non-degenerate Stochastic Differential Equations
and Ergodic Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.3 Stochastic Langevin Equation and Its Discretizations . . . . . . . . . . 39
2.3.1 Ergodicity for Exact and Numerical Solutions. . . . . . . . . . 40
2.3.2 Geometric Structure: Conformal Symplecticity . . . . . . . . . 42
2.3.3 Schemes of High Weak Convergence Order . . . . . . . . . . . 46
2.4 Approximation of Invariant Measures via Ergodic Schemes . . . . . 51
2.5 Approximation of the Ergodic Limit . . . . . . . . . . . . . . . . . . . . . . 56
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3 Invariant Measures for Stochastic Nonlinear Schrödinger
Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.2 Invariant Measures for Deterministic Nonlinear Schrödinger
Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
ix
