Ultrametric pseudodifferential equations and applications

255 Pages·2018·1.134 MB·English

by  Khrennikov, A., Kozyrev, Sergei V., Zúñiga-Galindo, W. A

ULTRAMETRIC PSEUDODIFFERENTIAL
EQUATIONS AND APPLICATIONS
Startingfromphysicalmotivationsandleadingtopracticalapplications,thisbook
providesaninterdisciplinaryperspectiveonthecuttingedgeofultrametric
pseudodifferentialequations.Itshowsthewaysinwhichtheseequationslink
differentfields,includingmathematics,engineering,andgeophysics.Inparticular,
theauthorsprovideadetailedexplanationofthegeophysicalapplicationsof p-adic
diffusionequationsusefulwhenmodelingtheflowsofliquidsthroughporousrock.
p-Adicwaveletstheoryand p-adicpseudodifferentialequationsarealsopresented,
alongwiththeirconnectionstomathematicalphysics,representationtheory,the
physicsofdisorderedsystems,probability,numbertheory,and p-adicdynamical
systems.
Materialthatwaspreviouslyspreadacrossmanyarticlesinjournalsofmany
differentfieldsisbroughttogetherhere,includingrecentworkonthevanderPut
seriestechnique.Thisbookprovidesanexcellentsnapshotofthefascinatingfieldof
ultrametricpseudodifferentialequations,includingtheiremergingapplicationsand
currentlyunsolvedproblems.
EncyclopediaofMathematicsandItsApplications
Thisseriesisdevotedtosignificanttopicsorthemesthathavewideapplicationin
mathematicsormathematicalscienceandforwhichadetaileddevelopmentofthe
abstracttheoryislessimportantthanathoroughandconcreteexplorationofthe
implicationsandapplications.
BooksintheEncyclopediaofMathematicsandItsApplicationscovertheir
subjectscomprehensively.Lessimportantresultsmaybesummarizedasexercises
attheendsofchapters.Fortechnicalities,readerscanbereferredtothe
bibliography,whichisexpectedtobecomprehensive.Asaresult,volumesare
encyclopedicreferencesormanageableguidestomajorsubjects.
Encyclopedia of Mathematics and its Applications
AllthetitleslistedbelowcanbeobtainedfromgoodbooksellersorfromCambridge
UniversityPress.Foracompleteserieslistingvisit
www.cambridge.org/mathematics.
119 M.DezaandM.DutourSikiric´GeometryofChemicalGraphs
120 T.NishiuraAbsoluteMeasurableSpaces
121 M.PrestPurity,SpectraandLocalisation
122 S.KhrushchevOrthogonalPolynomialsandContinuedFractions
123 H.NagamochiandT.IbarakiAlgorithmicAspectsofGraphConnectivity
124 F.W.KingHilbertTransformsI
125 F.W.KingHilbertTransformsII
126 O.CalinandD.-C.ChangSub-RiemannianGeometry
127 M.Grabischetal.AggregationFunctions
128 L.W.BeinekeandR.J.Wilson(eds.)withJ.L.GrossandT.W.TuckerTopicsinTopologicalGraph
Theory
129 J.Berstel,D.PerrinandC.ReutenauerCodesandAutomata
130 T.G.FaticoniModulesoverEndomorphismRings
131 H.MorimotoStochasticControlandMathematicalModeling
132 G.SchmidtRelationalMathematics
133 P.KornerupandD.W.MatulaFinitePrecisionNumberSystemsandArithmetic
134 Y.CramaandP.L.Hammer(eds.)BooleanModelsandMethodsinMathematics,ComputerScience,and
Engineering
135 V.BerthéandM.Rigo(eds.)Combinatorics,AutomataandNumberTheory
136 A.Kristály,V.D.RadulescuandC.VargaVariationalPrinciplesinMathematicalPhysics,Geometry,and
Economics
137 J.BerstelandC.ReutenauerNoncommutativeRationalSerieswithApplications
138 B.CourcelleandJ.EngelfrietGraphStructureandMonadicSecond-OrderLogic
139 M.FiedlerMatricesandGraphsinGeometry
140 N.VakilRealAnalysisthroughModernInfinitesimals
141 R.B.ParisHadamardExpansionsandHyperasymptoticEvaluation
142 Y.CramaandP.L.HammerBooleanFunctions
143 A.Arapostathis,V.S.BorkarandM.K.GhoshErgodicControlofDiffusionProcesses
144 N.Caspard,B.LeclercandB.MonjardetFiniteOrderedSets
145 D.Z.ArovandH.DymBitangentialDirectandInverseProblemsforSystemsofIntegralandDifferential
Equations
146 G.DassiosEllipsoidalHarmonics
147 L.W.BeinekeandR.J.Wilson(eds.)withO.R.OellermannTopicsinStructuralGraphTheory
148 L.Berlyand,A.G.KolpakovandA.NovikovIntroductiontotheNetworkApproximationMethodfor
MaterialsModeling
149 M.BaakeandU.GrimmAperiodicOrderI:AMathematicalInvitation
150 J.Borweinetal.LatticeSumsThenandNow
151 R.SchneiderConvexBodies:TheBrunn–MinkowskiTheory(SecondEdition)
152 G.DaPratoandJ.ZabczykStochasticEquationsinInfiniteDimensions(SecondEdition)
153 D.Hofmann,G.J.SealandW.Tholen(eds.)MonoidalTopology
154 M.CabreraGarcíaandÁ.RodríguezPalaciosNon-AssociativeNormedAlgebrasI:TheVidav–Palmerand
Gelfand–NaimarkTheorems
155 C.F.DunklandY.XuOrthogonalPolynomialsofSeveralVariables(SecondEdition)
156 L.W.BeinekeandR.J.Wilson(eds.)withB.ToftTopicsinChromaticGraphTheory
157 T.MoraSolvingPolynomialEquationSystemsIII:AlgebraicSolving
158 T.MoraSolvingPolynomialEquationSystemsIV:BuchbergerTheoryandBeyond
159 V.BerthéandM.Rigo(eds.)Combinatorics,WordsandSymbolicDynamics
160 B.RubinIntroductiontoRadonTransforms:WithElementsofFractionalCalculusandHarmonicAnalysis
161 M.GherguandS.D.TaliaferroIsolatedSingularitiesinPartialDifferentialInequalities
162 G.MolicaBisci,V.D.RadulescuandR.ServadeiVariationalMethodsforNonlocalFractionalProblems
163 S.WagonTheBanach–TarskiParadox(SecondEdition)
164 K.BroughanEquivalentsoftheRiemannHypothesisI:ArithmeticEquivalents
165 K.BroughanEquivalentsoftheRiemannHypothesisII:AnalyticEquivalents
166 M.BaakeandU.Grimm(eds.)AperiodicOrderII:CrystallographyandAlmostPeriodicity
167 M.CabreraGarcíaandÁ.RodríguezPalaciosNon-AssociativeNormedAlgebrasII:Representation
TheoryandtheZel’manovApproach
168 A.Yu.Khrennikov,S.V.KozyrevandW.A.Zúñiga-GalindoUltrametricPseudodifferentialEquations
andApplications
Encyclopedia of Mathematics and its Applications
Ultrametric Pseudodifferential
Equations and Applications
ANDREI YU. KHRENNIKOV
Linnéuniversitetet,Sweden
SERGEI V. KOZYREV
SteklovInstituteofMathematics,Moscow
W. A. ZÚÑIGA-GALINDO
CentrodeInvestigaciónydeEstudiosAvanzados
delInstitutoPolitécnicoNacional,Mexico
UniversityPrintingHouse,CambridgeCB28BS,UnitedKingdom
OneLibertyPlaza,20thFloor,NewYork,NY10006,USA
477WilliamstownRoad,PortMelbourne,VIC3207,Australia
314–321,3rdFloor,Plot3,SplendorForum,JasolaDistrictCentre,NewDelhi-110025,India
79AnsonRoad,#06-04/06,Singapore079906
CambridgeUniversityPressispartoftheUniversityofCambridge.
ItfurtherstheUniversity’smissionbydisseminatingknowledgeinthepursuitof
education,learningandresearchatthehighestinternationallevelsofexcellence.
www.cambridge.org
Informationonthistitle:www.cambridge.org/9781107188822
DOI:10.1017/9781316986707
©CambridgeUniversityPress2018
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithoutthewritten
permissionofCambridgeUniversityPress.
Firstpublished2018
PrintedintheUnitedKingdombyClays,StIvesplc
AcataloguerecordforthispublicationisavailablefromtheBritishLibrary
ISBN978-1-107-18882-2Hardback
CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracy
ofURLsforexternalorthird-partyinternetwebsitesreferredtointhispublication,
anddoesnotguaranteethatanycontentonsuchwebsitesis,orwillremain,
accurateorappropriate.
DedicatedtoVasiliiSergeevichVladimirov

Contents
Preface pagexi
1 p-AdicAnalysis:EssentialIdeasandResults 1
1.1 TheFieldof p-AdicNumbers 1
1.2 TopologyofQN 2
p
1.3 TheBruhat–SchwartzSpaceandtheFourierTransform 3
1.4 Distributions 4
1.5 SomeFunctionSpaces 6
2 UltrametricGeometry:ClusterNetworksandBuildings 8
2.1 Introduction 8
2.2 Clustering,Trees,andUltrametricSpaces 9
2.3 FamilyofMetricsandMulticlustering 12
2.4 AffineBruhat–TitsBuildingsandClusterNetworks 13
2.5 GroupsActingonTreesandtheVladimirovOperator 17
3 p-AdicWavelets 20
3.1 Introduction 20
3.2 Basisof p-AdicWavelets 26
3.3 CoherentStates 28
3.4 OrbitsofMean-ZeroTestFunctionsasWaveletFrames 30
3.5 MultidimensionalWaveletsandRepresentationTheory 33
3.6 WaveletswithMatrixDilations 34
3.7 WaveletTransformofDistributions 40
3.8 RelationtotheHaarBasisontheRealLine 41
3.9 p-AdicMultiresolutionAnalysis 43
3.10 p-AdicOne-DimensionalHaarWaveletBases 45
3.11 p-AdicScalingFunctions 48
3.12 MultiresolutionFramesofWavelets 49
3.13 MultidimensionalMultiresolutionWaveletBases 51
vii
viii Contents
3.14 p-AdicShannon–KotelnikovTheorem 53
3.15 SpectralTheoryof p-AdicPseudodifferentialOperators 54
3.16 WaveletsandOperatorsforGeneralUltrametricSpaces 59
4 UltrametricityintheTheoryofComplexSystems 63
4.1 Introduction 63
4.2 p-AdicParametrizationoftheParisiMatrix 65
4.3 DynamicsonComplexEnergyLandscapes 67
4.4 ActomyosinMolecularMotor 70
4.5 2-AdicModeloftheGeneticCode 73
5 SomeApplicationsofWaveletsandIntegralOperators 76
5.1 PseudodifferentialEquations 76
5.2 Non-linearEquationsandtheCascadeModelofTurbulence 78
5.3 p-AdicBrownianMotion 81
6 p-AdicandUltrametricModelsinGeophysics 83
6.1 Tree-likeStructuresinNature 84
6.2 p-AdicConfigurationSpaceforNetworksofCapillariesand
BalanceEquationsforDensitiesofFluids 85
6.3 Non-linear p-AdicDynamics 89
7 RecentDevelopmentoftheTheoryof p-AdicDynamical
Systems 94
7.1 VanderPutSeriesandCoordinateRepresentationsof
DynamicalMaps 96
7.2 RecentResultsaboutMeasure-PreservingFunctionsand
ErgodicDynamics 99
7.3 ErgodicDynamicalSystemsBasedon1-LipschitzFunctions 105
8 Parabolic-TypeEquations,MarkovProcesses,andModelsof
ComplexHierarchicalSystems 114
8.1 Introduction 114
8.2 OperatorsW,Parabolic-TypeEquations,andMarkov
Processes 115
8.3 EllipticPseudodifferentialOperators,Parabolic-Type
EquationsandMarkovProcesses 121
8.4 Non-ArchimedeanReaction–UltradiffusionEquationsand
ComplexHierarchicSystems 123
9 StochasticHeatEquationDrivenbyGaussianNoise 133
9.1 Introduction 133
9.2 p-AdicParabolic-TypePseudodifferentialEquations 134
9.3 Positive-DefiniteDistributionsandtheBochner–Schwartz
Theorem 136
9.4 StochasticIntegralsandGaussianNoise 138