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POLAR SETS OF MULTIPARAMETER BIFRACTIONAL BROWNIAN SHEETS

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POLAR SETS OF MULTIPARAMETER BIFRACTIONAL BROWNIAN SHEETSOF,of,POLAR,SETS,Polar,sets

    POLAR SETS OF MULTIPARAMETER BIFRACTIONAL BROWNIAN SHEETS

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    ;ceidr'eccL?tActaMathematicaScientia201030B(3):857m-872蕊一一矾?lhm;,thonnn帮臻秽.::::=::::http://actams.wipm.ac.ca P0LARSETSoFMUIIPARAMETER

    BIFRACTIoNALBRoWNIANSHEETS

    ChenZhenlong(陈振龙)LiHuiqiong(李慧琼)

    CollegeofStatisticsandMathematicsZh~iangGongshangUniversity,Hangzhou310018,C

    hina

    E-mail:zlchenv@163.corn

    AbstractLetB,K={BH,K((),t?)bean(N,d)bifractionalBrowniansheet withHurstindicesH=(Hl,,HN)?(0,1)'?andK=(KI,,KN)?(0,1The

    propertiesofthepolarsetsofBHarediscussed.

    Thesufficientconditionsandnecessary

    conditi0nsforacompactsettobepolarforBHareproved.TheinfimumofHausdorff

dimensionsofitsnonpolarsetsareobtainedbymeansofconstructingaCantortypeset

    toconnectitsHausdorffdimensionandcapacity.

    KevwordsBifractionalBrowniansheet;polarset;Hausdorffdimension;packingdimen

    sion;capacity

    2000MRSubjectClassification60G15:60G17

    1Introduction

    Inrecentyears,therewasofconsiderableinterestinstudyingfractionalBrownianmotion duetoitsapplicationsinvariousscientificareasincludingtelecommunications,turbulence,im

    ageprocessingandfinance.Incontrast,manyauthorsproposedtousemoregeneralselfsimilar

    Gaussianprocessesandrandomfieldsasstochasticmodels(cf.Anheta1.(1999),Benassiet a1.f2000),Addieeta1.f2002),MannersaloandNorros(2002),BonamiandEstrade(2003), Cheriditof2004),Bensoneta1.(2006)).Suchapplicationsraisedmanyinterestingtheoretical

    problemsaboutself-similarGaussianprocessesandrandomfieldsingenera1.However,contrast

    totheextensivestudiesonfractionalBrownianmotion.thereWaslittlesystematicinvestiga- tiononotherself-similarGaussianprocesses.Themainreasonsforthisarethecomplexityof dependencestructuresandthenon.availabilityofconvenientstochasticintegralrepresentations

    forself-similarGaussianprocesseswhichdonothavestationaryincrements. Inpractice,someauthorsstudiedvariouspropertiesofaratherspecialclassofsel~similar Gaussianprocesses.namely,thebifractionalBrownianmotionintroducedbyHoudr6andVilla

    (2003).F0rexample,RussoandTudor(2006)establishedsomepropertiesonthestrongvari

    ations.1ocaltime,andstochasticcalculusofthebifractionalBrownianmotion.Tudorand Xiao(20061developedsomemethodstostudythestronglocallynondeterministic,Chung'S ReceivedOctober8,2007;revisedJanuary27,2008.Researchsupportedbythenationalnatur

alfoundation

    ofChina(70871104),thekeyresearchbaseforhumanitiesandsocialsciencesofZhejiangProvincialhigh

    educationtalents(StatisticsofZh~iangGongshangUniversity)

    858ACTAMATHEMATICASCIENTIAVb1.30Ser.B

    lawoftheiteratedandlocaltimeforthebifractionalBrownianmotion.Themainobjective ofthisarticleistofurtherconsideraspecialclassofsel~similarGaussianfields.namely,the f?,d)bifractionalBrowniansheetintroducedbyTudorandXiaof2006). ForanygivenvectorsH=(H1,,HN)?(0,1)?andK=(K1,-,KN)?(0,1,an

    (N,1)bifractionalBrowniansheetB,={B,K(),t?)withindicesHandKisa

    real-valued,centeredGaussianrandomfieldwithcovariancefunctiongivenby EISO"'K(s)B)=N+t;)I],sR(1.1)

    Itfollowsfrom(1.1)thatS0",isananisotropicGaussianrandomfieldandSO"'(()=0a.s

    foreveryt?a,wheredenotestheboundaryof.

    ConsidertheGaussianrandomfieldBH,K={BH,K(t),t?}indefinedby

    BH,K()=(B'K((),.,B'K(()),(1.2)

    whereBBHd,K

    areindependentcopiesofBO".Then1BHKiscalledanNparameterl

    RdvaluedbifractionalBrowniansheet.

    NotethatifK1==KN=1.thenB,isafractionalBrowniansheetinwith

    HurstindexH?(0,1)?;ifN=1andK=1,thenBH,KisafractionalBrownianmotionin dwithHurstindex1?(0,1);ifN>1,K1==KN=1andHI==HN=1/2,then

    BH,Ki8an(Nd1

    Browniansheet.Hence,BH,Kcanberegardedasanaturalgeneralizationof oneparameterfractionalBrownianmotionin"toGaussianrandomfieldsind.aswellasa generalizationoftheBrowniansheet.Anotherwellknowngeneralizationisthemultiparameter

    fractionalBrownianmotion.

    Letx(t)bearandomfieldonvaluedin.AcompactsetE×FxRdiscalled

    apolarsetfortherandomfield(t,x(t))if

p{(E)nF?)=0

    Thereisalonghistoryofthestudyofpolarsetsofstochasticprocesses.Itiswellknown thatacompactsetisapolarsetforthed-dimensionalBrownianmotion.SeePortandStone (1978)andKahane(1985).TheseresultsareduetoKakutani(1944)andwereextended partiallytomoregeneralprocesseswithstationaryandindependentincrementsbyHawkes (1978)andKahane(1983),tofractionalBrownianmotionbyTestard(1985,1986b)andXiao (1999),andrecentlytotheBrowniansheetbyKhoshnevisan(1997)andChen(1997).Taylor andWatsonf19851alsoprovedsimilarresultsforthepolarsetsfortheheatequation.Inall thesearticle,theindependentincrementpropertyoftheBrowniansheetandthestationary incrementpropertyoffractionalBrownianmotionplayedcrucialroles.Because,ingeneral,the

    bifractionalBrowniansheetBH?Khasneitherthepropertyofindependentincrementsnorthe propertyofstationaryincrementsduetoitscomplexityofdependencestructure,itseemsquite difficulttoinvestigatefinepropertiesofitssamplepaths.Thisaddsmanydifficultiestothe investigationonthepropertiesofitspolarsets.Inthisarticle,wewoulddevelopsystematic methodsandtechnicalestimatestostudythepropertiesofthepolarsetsforthebifractional Browniansheet.Ourmainresultsaresharpandtheirproofsaredifferentfromtheproofsfor No.3Chen&Li:POLARSETSOFMULTIPARAMETERBIFRACTIONALBROW

    NIANSHEETS859

    theBr0?

    rnjansheetandthefractionalBrownianmotion.However,fortheHausdorffdimension ofthesmallestsetsforthe(N,d)bifractionalBrowniansheet,wecouldonlyestablishsome

    inequalities.seeTheorem4.4andTheorem4.5.Itisstillanopenproblemtoshowthebest resuItoftheHausdorffdimensionofthesmallestsetsforthe(N,d)

    bifractionalBrowniansheet.

    Forourpurpose,weshallintroducesomenotationsanddefinitions.Forany0<0c1, wedefineametriconR×by

    p((s,),(t,))=max{Is一(l,lx1),

    where1.1istheusualEuclideannorm.Letbetheclassoffunctions:(0,)_(0,1),which arerightcontinuous,monotoneincreasingwith(0+)=0andsuchthatthereexistsaconstant

clim:

    whereBa(,r)istheopenballinthemetricspace(×,Pa)centeredatwithradiusr.

    WedenotebyB(,r)theopenballintheEuclideanspacecenteredatwithradiusr,andby

    mtheordinaryHausdorffmeasure.

    ItisknownthatmnisametricoutermeasureinthesenseofCarath6odoryandhence everyBorelsetin(xd,Pn)is咖一mQmeasurable(cf.Rogers(1970)).Resultssuchas

    densitytheoremsandFrostman'slemmaanalogoustothoseforordinaryHausdorffmeasure stillh0ld.SeeTaylorandWatson(1985).TheHausdorffdimensiondimaAofAc×is

    definedby

    dimA=inf{'r>0:8m.()=0).(1.6)

    ThesetypeofHausdorffmeasureandHausdorffdimensionwereappliedbyHawkes(1978), TaylorandWatsonf1985),andTestard(1985,1986b)tocharacterizethepolarsetsforsto- chasticprocessesandtheheatequation,andbyTestard(1986a)tostudydoublepolarsets forBrownianmotion.SimilartothedefinitionofpackingmeasuregivenbyTaylorandTricot f1985),onecanalsodefinepackingmeasurePnandpackingdimensionDimnonthemetric

    space(×Rd,pa).Forexample,thepackingdimensionDimaAisdefinedby

    Dim.A=inf{7>0:s7()=O).(1.7)

    Therestofthisarticleisorganizedasfollows.InSection2,weapplytherandommeasure toprovethenecessaryconditionsforacompactsetE×Ftobeapolarsetfor(t,B'K()),

    whichimprovetheresultsoffractionalBrownianmotionandBrowniansheetmentionedabove.

    InSection3.wederivethesufIicientconditionsforasettobepolarbysometechnicalestimates. InSection4.weapplytheaboveresultstocalculatetheHausdorffdimensionofthesmallest

    )whentisrestrictedtoacompactsetEand setsFCthatcanbehitbyBH,K((

    theHausdorffdimensionofthesmallestsetEwhoseimageunderBH,K(()canintersect

    FRd.

    Throughoutthisarticle,theunderlyingparameterspaceisR=0,+?)?,or=

    (0,+?)?.Atypicalparameter,t?iswrittenast=(tl,?-.,(?),oroccasionally,as(c),if

    B

?U

    ACTAMATHEMATICASCIENTIAV01.30Ser.B

    tl==tg=C.Weuse(?,?)and1.1todenotetheordinaryscalarproductandEuclideannorm in,respectively,nomatterthevalueoftheintegerm.WewilluseC,C1,;A1,A2,

    todenoteunspecifiedpositivefiniteconstantswhoseprecisevaluesarenotimportantandmay

    bedifferentineachappearance.

    2NecessaryConditionsofPolarSets

    Inthissection,weprovesomenecessaryconditionsonEandFforEXFtobeapolar setforan(N,d)

    bifractionalBrowniansheet.Becauseofthecomplicatedcovariance,theproof ofnecessityisquiteinvolved.Therefore,wesplittheproofintoseverallemmas,whicharealso

    oftheirowninterest.

    Lemma2.1Letn>0,b>0,0<h1and0<1.If0hk,then, f?+b2h)Ibal.2kaA6)h

    whereaAb=min{a,

    ProofTheinequalityin(2.1)isequivalentto

    [.aVb/~2h(2hk_20

    (2.1)

    whereaVb=max{n,6).For1,let_,()=(h+1)(1).h2.Toprove(2.1),it

    isenoughtoshowthat,(u)?0forall1.Differentiatingf,wehave

    l()=2hk[(u2h+11?u.__

    =

    2~2hk-1[(+)1f"一一

    f1

    Because0<h1,0<kland0hk1,wecandeducethat,(,")0foralll

    Hence,f(u)f(1)=0forallU1.Thisfinishestheproofof(2.1). ForacompactsetE?,wedefinethefunction

    (s,)=E['(s)'(t).[E('(s)B7'(())](s,t?E),(2.2)

    where(s,t)isindependentof=l,,d.

    ThefollowinglemmafollowsfromTudorandXiao(2006).

Lemma2.2LetH(1,,HN)?(0,1)?andK=(1,,KN)?(0,1]?,andlet

    Ebeacompactseton.Then,thereexistpositiveconstantsA1andA2,suchthat,forall

    s=(81,,s?),t=(tl,,tN)?E,wehave ?

    ?

    =

    1

    Lemma2.3LetH

    letEbeacompactseton 8=(81,,SN),t=(tl,

    ls

    N

    

    I(s,()A2?lsl

    =1

    (2.3)

    =

    (Hl,,HN)?(0,1)?andK=(K1,,KN)?(0,1】?,and

    .

    Then,thereexistsapositiveconstantA3,suchthat,forall

    ,tN)?Eanda,b?,wehave

    E'(s)6'K()A3(n_6).(2.4) ,I,

    L

    >

    u

    /f

    1????J

    1??,j一从

    2

h,

    1

    l

    No.3Chen&Li:POLARSETSOFMULTIPARAMETERBIFRACTIONALBROW

    NIANSHEETS861

    where0<1,=1,,?.

    ProofBecauseEisacompactseton,then,thereexistpositiveconstants0<C1<1

    andC2>1,suchthatEcCI,]?.Hence,

    E[.'K(s)'()

    =nE['(s)..E['(t)]2abEEB7'K(s)'(()]

    Obviously

    +;)Ist--ttl].(2.5)

    (s2H~-'}-t;[st-QI.)(s2HlAt;)

    ApplyingLemma2.1withh=Htandk=,wehave 1[(s;+t;)[st-tt~.]s2HtKt八(;

    Itfollowsfromthemeanvaluetheoremand(2.6)that

    1(s2+t(1~t-ttr)

    (2+;)[(s;+;)[st--tt[]

    Ktf,L,2Ht(Kt-1)(2Ht^t;). Ifab0,then,inequalities(2.5)and(2.7)imply

    E.'(s)6'(()]

    ?

    (;H((一2abII(s;HAt2HtKt)

    =1

    N

    ?(s;At2HtKt)(0_6)

    t=l

    N

    ?2HtK~

    (a612

Ifab<0,alsoaccordingtoinequalities(2.5)and(2.8),then,

    E'K(s)6'(t)]. (s2

    N

    T

    gtv2Ht(K~.-1)

    (s;HtKtA8~^t._6)

    fi(

    (2.6)

    (2.7)

    (2.8)

    (2.9)

    (2.10)

    Ch..sing3:nN2_2H ~

    

    (K

    .

    t--I)2H~

    ,by(2.9)and(2.10),wefinishthepr..f.fLemma2.3

    t=l

    ??602一磁

    ??

    +

    ??

    lI

    ??

    +

    ??

    K

    ??嘲

    +

    K

    H

    ?n

    2

    0

    >

    862ACTAMATHEMATICASCIENTIAVb1.30Ser.B Forany8,t??,

    andX,Y?,let

    h(s,t,,Y)=max{(s,(),IxvV)'(2.11)

    where(s,t)isthefunctiondefinedin(2.2). Thefollowingtheoremisthemainresultinthissection.

    Theorem2.4LetH=(H1,,HN)?(0,1)?andK=(,KN)?(0,1】?,and letEandFbecompactsetsonandd\{0),respectively.IfCaph(E×F)>0,then, {(BH,K)(F)nE?0)>0(2.12)

    whereCaph(.)denotesthecapacityon×generatedbythekernelfunctionh(s,t,,Y)

    and0<1,=l,,?.

    ProofIfCaph(E×F)>0,then,thereexistsafinitepositivemeasuresupportedon

    E×F,suchthat

    /EFE~Fh(s)<+?

    Forany(>0,wedefinearandommeasuresonEby ?=()eXp(2(t,e/,

    (2.13)

    (2.14)

    ItfollowsfromKahane(1985)orTestard(1986b)thatifthereexistpositiveconstants

    andC4,suchthat

    E(1luII),E(1Il_)C4,(2.15)

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