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1、第22卷第4期广西师范大学学报(自然科学版DVol.22No.42004年12月JOURNALOFGUANGXINORMALUNIVERSITYDecember2004=================================================================ONT~E~AUSDORFFDIMENSIONOFALGEBRAICSUMcOFT~EIMAGESETSFORGOUPN12DENGGuo-he,YANGXiang-gun(1.CollegeofMathematicsandComputerScience,GuangxiNormalUn
2、iversity,Guilin541004,China;2.CollegeofMathematicsandComputerScience,~unanNormalUniversity,Changsha410081,ChinaDNccNAbstract,LetX(tD=R+-RbeGOUPN.ForanycompactsetsE,FCR+{0},the~ausdorffdimensionofthealgebraicsumoftheimagesetsofX(ED-X(FDforXisconsideredinthispaper.Thisresultcon-tainsthecorres
3、pondingoneofBroWniansheetandgeneraliZedBroWniansheet.Keywords,generaliZedOUP;algebraicsumoftheimagesets;~ausdorffmeasureCLCnumber,O211.6Documentcode,AArticleID,1001-6600(2004D04-0031-050IntroductionandpreliminariesNLetW={W(tD=(W1(tD,W2(tD,,Wc(tDD,tER+}beGaussianrandomfieldsWithmeanZeroincNR,
4、andWI={WI(tD,tER+}(1{I{cDbei.i.d.real-valueprocessesWithcovariancefunctionE{WI(tDNNWI(SD}=IFZ(0,SZ/tZ,WhereS=(S1,S2,,SND,t=(t1,t2,,tNDER+,andFZ(1{Z{NDisabsolutelyZ=1NNcontinuousLebesgue-stieliemeasureW.r.t.LebesguemeasureonR+.Itisobviousthat{WI(tD,tER+}1iscalledN-parameter1-dimensionalgenera
5、liZedBroWniansheet(denotedbyGBSND.LetX={X(tD=N(I1(tD,I2(tD,,Ic(tDD,tER+},thenXbecalledN-parameterc-dimensionalgeneraliZedOrenstein-Uh-Nlenbeckprocesses,ifII={II(tD,tER+}(1{I{cDaremutualindependentreal-valueprocessesinR,andII(tD=I(tDI(0D+WI(tD,(1{I{cD,WhereI(0Darerandomvariablesthatareindepen
6、denttoWI,NI(tD,(1{I{cDareboundedcontinuousfunctionintWithouttakingZerovalueanddefinedonR+.ForthesakeofconvenientWriting,WedenoteN-parameterc-dimentionalgeneraliZedOrenstein-UhlenbeckcprocessbyGOUPN.DenoteX(ED-X(FD={X(tD-X(SD=tEE,SEF},thenX(ED-X(FDbecalledthealgebraicsumNoftheimagesetsofX,Whe
7、reE,FCR+.The~ausdorffdimensionoftheimagesetsforGaussianrandomfieldsWereWidelystudied,forex-1ample,Mountfordhasconsideredthe~ausdorffmeasureofimagesetsforBroWniansheet.Recently,thealgebraicsumoftheN-parameterc-dimensionalGaussianfieldshavebeenobtain
