Analytic Theory of Itô-Stochastic Differential Equations with Non-smooth Coefficients

139 Pages·2022·2.078 MB·English

by  Haesung Lee, Wilhelm Stannat, Gerald Trutnau

SpringerBriefs in
Probability and Mathematical Statistics
Haesung Lee · Wilhelm Stannat · Gerald Trutnau
Analytic Theory of Itô-
Stochastic Differential
Equations with Non-
smooth Coefficients
SpringerBriefs in Probability and Mathematical
Statistics
Editor-in-Chief
GesineReinert,UniversityofOxford,Oxford,UK
SeriesEditors
NinaGantert,TechnischeUniversitätMünchen,Munich,Germany
TailenHsing,UniversityofMichigan,AnnArbor,MI,USA
RichardNickl,UniversityofCambridge,Cambridge,UK
SandrinePéché,UniversitéParisDiderot,Paris,France
YosefRinott,HebrewUniversityofJerusalem,Jerusalem,Israel
AlmutE.D.Veraart,ImperialCollegeLondon,London,UK
SpringerBriefs present concise summaries of cutting-edge research and practical
applications across a wide spectrum of fields. Featuring compact volumes of 50
to 125 pages, the series covers a range of content from professionalto academic.
Briefsarecharacterizedbyfast,globalelectronicdissemination,standardpublishing
contracts, standardized manuscript preparation and formatting guidelines, and
expeditedproductionschedules.
Typicaltopicsmightinclude:
(cid:129) Atimelyreportofstate-of-thearttechniques
(cid:129) A bridge between new research results, as published in journal articles, and a
contextualliteraturereview
(cid:129) Asnapshotofahotoremergingtopic
(cid:129) Lectureofseminarnotesmakinga specialisttopicaccessible fornon-specialist
readers
(cid:129) SpringerBriefs in Probability and Mathematical Statistics showcase topics of
currentrelevanceinthefieldofprobabilityandmathematicalstatistics
Manuscriptspresentingnewresultsinaclassicalfield,newfield,oranemerging
topic,orbridgesbetweennewresultsandalreadypublishedworks,areencouraged.
This series is intended for mathematicians and other scientists with interest in
probabilityandmathematicalstatistics.Allvolumespublishedinthisseriesundergo
athoroughrefereeingprocess.
TheSBPMS seriesispublishedundertheauspicesoftheBernoulliSocietyfor
MathematicalStatisticsandProbability.
All titles in this seriesare peer-reviewedto the usualstandardsof mathematics
anditsapplications.
Haesung Lee (cid:129) Wilhelm Stannat (cid:129) Gerald Trutnau
Analytic Theory
of Itô-Stochastic Differential
Equations with Non-smooth
Coefficients
HaesungLee WilhelmStannat
DepartmentofMathematicsandComputer InstitutfürMathematik
Science TechnischeUniversitätBerlin
KoreaScienceAcademyofKAIST Berlin,Germany
Busan,Korea(Republicof)
GeraldTrutnau
DepartmentofMathematicalSciences
andResearchInstituteofMathematics
SeoulNationalUniversity
Seoul,Korea(Republicof)
ISSN2365-4333 ISSN2365-4341 (electronic)
SpringerBriefsinProbabilityandMathematicalStatistics
ISBN978-981-19-3830-6 ISBN978-981-19-3831-3 (eBook)
https://doi.org/10.1007/978-981-19-3831-3
©TheAuthor(s),underexclusivelicensetoSpringerNatureSingaporePteLtd.2022
Thisworkissubjecttocopyright.AllrightsaresolelyandexclusivelylicensedbythePublisher,whether
thewhole orpart ofthematerial isconcerned, specifically therights oftranslation, reprinting, reuse
ofillustrations, recitation, broadcasting, reproductiononmicrofilmsorinanyotherphysicalway,and
transmissionorinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilar
ordissimilarmethodologynowknownorhereafterdeveloped.
Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication
doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant
protectivelawsandregulationsandthereforefreeforgeneraluse.
Thepublisher,theauthors,andtheeditorsaresafetoassumethattheadviceandinformationinthisbook
arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor
theeditorsgiveawarranty,expressedorimplied,withrespecttothematerialcontainedhereinorforany
errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional
claimsinpublishedmapsandinstitutionalaffiliations.
ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSingaporePteLtd.
The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721,
Singapore
Acknowledgments
TheresearchofHaesungLeewassupportedbytheBasicScienceResearchProgram
throughtheNationalResearchFoundationofKorea(NRF)fundedbytheMinistry
of Education(2020R1A6A3A01096151)and by the DFG throughthe IRTG 2235
“Searching for the regular in the irregular: Analysis of singular and random
systems.” The research of Wilhelm Stannat was partially supported by the DFG
throughthe Research Unit FOR 2402“Rough paths, stochastic partial differential
equations and related topics.” The research of Gerald Trutnau was supported by
theBasicScienceResearchProgramthroughtheNationalResearchFoundationof
Korea(NRF)fundedbytheMinistryofEducation(2017R1D1A1B03035632).
v
Contents
1 Introduction .................................................................. 1
1.1 MethodsandResults.................................................... 3
1.2 OrganizationoftheBook............................................... 7
2 TheAbstractCauchyProbleminLr-SpaceswithWeights............. 9
2.1 TheAbstractSetting,ExistenceandUniqueness...................... 9
2.1.1 FrameworkandBasicNotations............................... 12
2.1.2 ExistenceofMaximalExtensionsonRd ...................... 15
2.1.3 UniquenessofMaximalExtensionsonRd .................... 24
2.2 ExistenceandRegularityofDensitiestoInfinitesimally
InvariantMeasures...................................................... 37
2.2.1 ClassofAdmissibleCoefficientsandtheMainTheorem .... 37
2.2.2 Proofs............................................................ 40
2.2.3 Discussion....................................................... 43
2.3 RegularSolutionstotheAbstractCauchyProblem................... 44
2.4 IrreducibilityofSolutionstotheAbstractCauchyProblem.......... 52
2.5 CommentsandReferencestoRelatedLiterature...................... 56
3 StochasticDifferentialEquations .......................................... 59
3.1 Existence ................................................................ 59
3.1.1 RegularSolutionstotheAbstractCauchyProblem
asTransitionFunctions......................................... 59
3.1.2 ConstructionofaHuntProcess ................................ 62
3.1.3 Krylov-typeEstimate........................................... 70
3.1.4 IdentificationoftheStochasticDifferentialEquation......... 73
3.2 GlobalProperties........................................................ 79
3.2.1 Non-explosionResultsandMomentInequalities............. 80
3.2.2 TransienceandRecurrence..................................... 91
3.2.3 Long Time Behavior: Ergodicity, Existence
and Uniqueness of Invariant Measures,
Examples/Counterexamples.................................... 98
vii
viii Contents
3.3 Uniqueness.............................................................. 103
3.3.1 PathwiseUniquenessandStrongSolutions ................... 104
3.3.2 UniquenessinLaw(ViaL1-Uniqueness) ..................... 111
3.4 CommentsandReferencestoRelatedLiterature...................... 115
4 ConclusionandOutlook..................................................... 117
References......................................................................... 121
Index............................................................................... 125
Notations and Conventions
VectorSpacesandNorms
(cid:2)·(cid:2) theEuclideannormonthed-dimensionalEuclideanspaceRd
(cid:3)·,·(cid:4) theEuclideaninnerproductinRd
|·| theabsolutevalueinR
(cid:2)·(cid:2)B thenormassociatedwithaBanachspaceB
B(cid:5) thedualspaceofaBanachspaceB
SetsandSetOperations
Rd thed-dimensionalEuclideanspace
Rd the one-pointcompactificationof Rd with the point at infinity
(cid:2)
“(cid:2)”
(Rd)S setofallfunctionsfromS toRd,whereS ⊂[0,∞)
(cid:2) (cid:2)
V theclosureofV ⊂Rd
B (x) forx ∈Rd,r >0,definedas{y ∈Rd :(cid:2)x−y(cid:2)<r}
r
B (x) definedas{y ∈Rd :(cid:2)x−y(cid:2)≤r}
r
B shortforB (0)
r r
R (r) theopencubeinRd withcenterx ∈Rd andedgelengthr >0
x
R (r) theclosureofR (r)
x x
A+B definedas{a+b:a ∈A,b∈B},forsetsA,Bwithanaddition
operation
Measuresandσ-Algebras
In this book, any measure is always non-zero, and positive and if a measure is
defined on a subset of Rd, then it is a Borel measure, i.e., defined on the Borel
subsets.
μ=ρdx denotestheinfinitesimallyinvariantmeasure(see(2.14),Theo-
rem2.24andRemark2.28)
dx theLebesguemeasureonB(Rd)
dt theLebesguemeasureonB(R)
ix
x NotationsandConventions
B(Rd) the Borel subsets of Rd or the space of Borel measurable
functionsf :Rd →R
B(Rd) defined as {A ⊂ Rd : A ∈ B(Rd)orA = A ∪{(cid:2)}, A ∈
(cid:2) (cid:2) 0 0
B(Rd)}
B(X) smallest σ-algebra containing the open sets of a topological
spaceX
a.e. almosteverywhere
supp(ν) thesupportofameasureν onRd
supp(u) forameasurablefunctionu:Rd →Rdefinedassupp(|u|dx)
δ Diracmeasureatx ∈Rd
x (cid:2)
P (x,dy) the sub-probability measure defined by P (x,A) = P 1 (x),
t t t A
A∈B(Rd),(x,t)∈Rd ×(0,∞)(seeProposition3.1)
DerivativesofFunctions,VectorFields
∂ f (weak)partialderivativeinthetimevariablet
t
∂ f (weak)partialderivativeinthei-thspatialcoordinate
i
∇f (weak)spatialgradient,∇f :=(∂ f,...,∂ f)
1 d
∂ f second-order(weak)partialderivatives,∂ f :=∂ ∂ f
ij ij i j
∇2f (weak)Hessianmatrix,(∇(cid:2)2f)=(∂ijf)1≤i,j≤d
(cid:2)f (weak)Laplacian,(cid:2)f = di=1∂iif
divF (we(cid:2)ak)divergenceofthevectorfieldF=(f1,...,fd),defined
as di=1∂ifi
(∇B)i for 1 ≤ i ≤ d and a matrix B = (bij)1≤i,j≤d of functions,
(cid:2)(∇B)i is the divergence of the i-th row of B, i.e., defined as
d ∂ b
j=1 j ij
∇B definedas((∇B) ,...,(∇B) )
1 d
BT foramatrixB,thetransposedmatrixisdenotedbyBT
trace(B) t(cid:2)race of a matrix of functions B = (bij)1≤i,j≤d, trace(B) =
d b
i=1 ii
A diffusionmatrixA=(aij)1≤i,j≤d
G in Sect. 2.1 the drift G satisfies G = (g ,...,g ) ∈
1 d
L2 (Rd,Rd,μ) (cf. (2.13)). From Sect. 2.2.1 on the drift
loc
satisfiesG=(g ,...,g )= 1∇(A+CT)+H(seeassumption
1 d 2
(a)inSect.2.2.1and(2.55),butalsoRemark2.23)
βρ,B logarithmicderivative(ofρ associatedwithB =(cid:3)(cid:2)(bij)1≤i,j≤d),
βρ,B =(cid:4) (β1ρ,B,...,βdρ,B), where βiρ,B = 12 dj=1∂jbij+
b ∂jρ , i.e., βρ,B = 1∇B + 1 B∇ρ (see (2.19) and
ij ρ 2 2ρ
Remark2.28)
B B = G−βρ,A, divergencezerovectorfield with respectto μ
(see(2.22),(2.23))