Table Of ContentLecture Notes in Mathematics 2314
Jialin Hong
Liying Sun
Symplectic
Integration
of Stochastic
Hamiltonian
Systems
Lecture Notes in Mathematics
Volume 2314
Editors-in-Chief
Jean-MichelMorel,CMLA,ENS,Cachan,France
BernardTeissier,IMJ-PRG,Paris,France
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MarkPolicott,MathematicsInstitute,UniversityofWarwick,Coventry,UK
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Jialin Hong (cid:129) Liying Sun
Symplectic Integration of
Stochastic Hamiltonian
Systems
JialinHong LiyingSun
AcademyofMathematicsandSystems AcademyofMathematicsandSystems
Science Science
ChineseAcademyofSciences ChineseAcademyofSciences
Beijing,Beijing,China Beijing,Beijing,China
ThisworkwassupportedbyNationalNaturalScienceFoundationofChinaNo.91530118No.91630312
No. 11711530017 No. 11926417 No. 11971470 No. 11871068 No. 12031020 No. 12026428 No.
12101596No.12171047
ISSN0075-8434 ISSN1617-9692 (electronic)
LectureNotesinMathematics
ISBN978-981-19-7669-8 ISBN978-981-19-7670-4 (eBook)
https://doi.org/10.1007/978-981-19-7670-4
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Preface
As numerous modern challenges in scientific questions, industrial needs, and
societal requirements emerge, the demand for designing numerical methods to
solvetremendouslycomplexphenomenaconcerningstochasticity,imprecision,and
vagueness also grows. Even though the computational power increases rapidly,
stochastic numerical methods still cannot compute sophisticated stochastic prob-
lems well over longer time intervals. The qualitative properties of stochastic
numerical methods for such problems are of vital importance to the accuracy of
the numerical simulation and the reliability of long-term predictions. Recently,
stochastic structure-preserving numerical methods, which promise considerable
benefits such as good numerical stability and probabilistic superiority, have been
ofgreatconcern.
Thismonographisdevotedtostochasticstructure-preservingnumericalmethods
for stochastic differential equations. The emphasis is placed on the systematic
constructionandprobabilisticsuperiorityofstochasticsymplecticmethods,which
preserve the symplectic geometric structure of the stochastic flow of finite and
infinite dimensional stochastic Hamiltonian systems. The objects considered here
arerelatedtoseveralfascinatingmathematicalfields:numericalanalysis,stochastic
analysis, symplectic geometry, rough path theory, partial differential equation,
probabilitytheory,etc.
Chapter1introducesthestochasticflowassociatedwithStratonovichstochastic
differentialequations,whichisacornerstoneinthestudyofthestochasticgeometric
structure. It also collects long-time behaviors of stochastic (rough) Hamiltonian
systems, stochastic forced Hamiltonian systems, and stochastic Poisson systems
together. Compared to existing textbooks on stochastic Hamiltonian systems, this
monographpresentssomeup-to-datemathematicaltools,suchasthestochasticθ-
.
generatingfunctionwithparameterθ ∈ [0,1],stochasticHamilton–Jacobitheory,
.
which can be exploited to characterize intrinsic properties of the original systems
andtoconstructeffectivestochasticsymplecticmethods.
The philosophy behind the preservation of the structure is to identify intrinsic
propertiesofthecontinuoussystemsandthendesignnumericalmethodsinheriting
the original properties at discrete time points. At this stage, the convergence has
v
vi Preface
been considered as a basic and crucial aspect for deciding the utility of stochastic
numerical methods. The convergence of existing stochastic numerical methods is
mainly studied in two different aspects: strong and weak convergence. In Chap.2,
we focus on the systematical construction of strongly and weakly convergent
stochastic structure-preserving numerical methods proposed recently, including
stochasticsymplecticmethods,stochasticpseudo-symplecticmethods,andstochas-
tic numerical methods preserving invariants, etc., as well as the construction of
rough symplectic Runge–Kutta methods. Chapter 3 provides the methodology
of constructing higher weak order and ergodic numerical methods based on the
modifiedstochasticdifferentialequationandstochasticgeneratingfunction.Chap-
ter 4 deals with several novel strongly convergent stochastic structure-preserving
numerical methods for stochastic Maxwell equations, stochastic wave equations,
and stochastic Schrödinger equations, which are infinite-dimensional stochastic
Hamiltoniansystemsofsignificantvalue.
Different from the deterministic case, the solution of the stochastic numerical
methods can be regarded as a stochastic process, namely a collection of random
variables.Animportantshiftofkeyideascomesaboutbytreatingtheprobabilistic
behaviorsofstochasticnumericalmethodsviathelargedeviationprinciple,which
is concerned with the exponential decay of probabilities of very rare events. It
allowsabetterunderstandingonthesuperiorityofstochasticsymplecticmethodsfor
thestochasticHamiltoniansystem.Oneofthepeculiarfeaturesinthismonograph
consistsinthenumericalmeritsofstochasticsymplecticmethodsforthestochastic
linearoscillatorandstochasticlinearSchrödingerequation,whicharepresentedin
Chaps.2and4.Thegeometricpropertiesofstochasticnumericalmethodscouldalso
be translated into the preservation of structure on the level of stochastic modified
equationsbytheasymptoticseriesexpansion.Thischaracteristicisdemonstratedin
Chap.3 for stochastic differential equations, especially for stochastic Hamiltonian
systems.
We hope that Master or PhD students and researchers who have a prior
understandingofthebasictheoryofstochasticprocessesandstochasticdifferential
equationsandareinterestedinthemathematicalornumericalaspectsofstochastic
simulations will find this monograph useful. The monograph could also be used
as a textbook for a semester-long graduate-level course. The topics discussed in
thismonographwillprovideusefultoolsforthereaderstohaveagoodknowledge
of stochastic structure-preserving integration, especially for stochastic symplectic
methodsofstochasticHamiltoniansystems.
Beijing,China JialinHong
Beijing,China LiyingSun
April2022
Acknowledgments
Thistextwouldnothavebeenpossiblewithouttheinvaluablesupportandguidance
ofsomanypeople.WewarmlythankDrs.JianboCui,ChuchuChen,ZihengChen,
Tonghe Dang, Baohui Hou, Diancong Jin, Derui Sheng, Tau Zhou, and Xiaojing
Zhangforhelpingustotrackdownanuncountablenumberoftypos.Withouttheir
help,themanuscriptwouldbemuchworsethanitisnow.Wealsoacknowledgethe
NationalNaturalScienceFoundationofChina(No.91530118,No.91630312,No.
11711530017,No.11926417,No.11971470,No.11871068,No.12031020andNo.
12026428,No.12101596,No.12171047)foritsfinancialsupport.
vii
Contents
1 StochasticHamiltonianSystems ........................................... 1
1.1 StratonovichStochasticDifferentialEquations....................... 1
1.2 StochasticHamiltonianSystems....................................... 9
1.2.1 StochasticSymplecticStructure............................... 10
1.2.2 StochasticVariationalPrinciple ............................... 15
1.2.3 Stochasticθ-GeneratingFunction............................. 23
.
1.3 Non-canonicalStochasticHamiltonianSystems...................... 32
1.3.1 StochasticForcedHamiltonianSystems...................... 32
1.3.2 StochasticPoissonSystems.................................... 38
1.4 RoughHamiltonianSystems........................................... 44
1.5 EndNotes............................................................... 50
2 StochasticStructure-PreservingNumericalMethods.................... 53
2.1 StochasticNumericalMethods ........................................ 53
2.1.1 StrongandWeakConvergence................................ 55
2.1.2 NumericalMethodsviaTaylorExpansions................... 60
2.1.3 StochasticRunge–KuttaMethods............................. 65
2.2 StochasticSymplecticMethods........................................ 68
2.2.1 SymplecticMethodsviaPadéApproximation................ 71
2.2.2 SymplecticMethodsviaGeneratingFunction................ 80
2.2.3 StochasticGalerkinVariationalIntegrators................... 86
2.2.4 Stochastic(Rough)SymplecticRunge–KuttaMethods...... 91
2.3 ApplicationofSymplecticMethodstoStochasticPoisson
Systems ................................................................. 105
2.4 SuperiorityofStochasticSymplecticMethodsviaLarge
DeviationPrinciple(LDP) ............................................. 113
2.4.1 LDPforNumericalMethods .................................. 117
2.4.2 AsymptoticalPreservationforLDP........................... 124
ix
x Contents
2.5 StochasticPseudo-SymplecticMethodsandStochastic
MethodsPreservingInvariants......................................... 130
2.5.1 StochasticPseudo-SymplecticMethods ...................... 130
2.5.2 StochasticNumericalMethodsPreservingInvariants........ 135
2.6 EndNotes............................................................... 142
3 StochasticModifiedEquationsandApplications......................... 145
3.1 StochasticModifiedEquations ........................................ 145
3.1.1 ModifiedEquationviaBackwardKolmogorovEquation.... 148
3.1.2 ModifiedEquationviaStochasticGeneratingFunction...... 153
3.1.3 ModifiedEquationforRoughSymplecticMethods.......... 167
3.2 StochasticNumericalMethodsBasedonModifiedEquation........ 174
3.2.1 HighOrderMethodsviaBackwardKolmogorov
Equation ........................................................ 175
3.2.2 HighOrderSymplecticMethodsviaGenerating
Function ........................................................ 179
3.3 ConformalSymplecticandErgodicMethodsforStochastic
LangevinEquation ..................................................... 184
3.4 EndNotes............................................................... 202
4 Infinite-DimensionalStochasticHamiltonianSystems .................. 205
4.1 Infinite-DimensionalStochasticHamiltonianSystems............... 205
4.2 StochasticMaxwellEquation.......................................... 211
4.2.1 StochasticExponentialIntegrator............................. 219
4.2.2 StochasticSymplecticRunge–KuttaDiscretizations......... 230
4.3 StochasticSchrödingerEquation ...................................... 234
4.3.1 StochasticSymplecticRunge–KuttaDiscretizations......... 234
4.3.2 ApproximatingLargeDeviationRateFunctionsvia
SpectralGalerkinMethod ..................................... 237
4.4 StochasticWaveEquationwithCubicNonlinearity.................. 250
4.4.1 SpectralGalerkinMethodandExponentialIntegrability .... 252
4.4.2 ExponentiallyIntegrableFullDiscretization ................. 262
4.5 EndNotes............................................................... 268
AppendixA ....................................................................... 271
A.1 StochasticStratonovichIntegral....................................... 271
A.2 LargeDeviationPrinciple.............................................. 273
A.3 ExponentialIntegrability............................................... 276
A.4 StochasticMulti-SymplecticStructure................................ 277
References......................................................................... 287
