travelling waves solutions to the k-p-p equation at the critical wave speed continuing simo

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1、TravellingwavessolutionstotheK-P-Pequationatthecriticalwavespeed:continuingSimonHarris'probabilisticanalysisA.E.KyprianouApril18,2000AbstractRecentlyHarris(1999),usingprobabilisticargumentsalone,hasgivennewproofsofthe(known)existence,asymptoticsandunique-nessoftravellingwavesolutionstothe

2、K-P-Pequation.ThispaperisasequeltoKyprianou(2000b)whichprovidesalternativeprobabilisticargumentsforsupercriticalwavespeeds.Wecompleteourprobabilis-ticanalysishereforthemoredicultcaseofcriticalwavespeeds.Theanalysisiscenteredaroundthestudyofadditiveandmultiplicativemartingalesandtheconstru

3、ctionofsize-biasedmeasuresonaspaceofnon-homogenousmarkedtreesgeneratedbyatruncatedbranchingBrownianmotion.Aspartofourresults,wealsoobtainamarti-naleconvergencetheoremforthederivativeoftheadditivemartin-gale.SomeofthemainideasareinspiredbythetechniquesfoundinKyprianouandBiggins(2000)andLyon

4、s(1997).Thevalueofthesenewprobabilisticproofsistheirgenericnaturewhichinprinciplecanbegeneralizedtostudyothertypesofspatialbranchingdiusionsandassociatedtravellingwaves.Keywordsandphrases.BranchingBrownianMotion,K-P-Pequa-tion,Travellingwavesolutions,AdditiveMartingales,DerivativeMar-ting

5、ales,MultiplicativeMartingales,ConditionedBrownianMotion,Bessel-3Processes.AMS1991subjectclassication.Primary60J80.DepartmentofMathematics,TheUniversityofUtrecht,Budapestlaan6,3584CDUtrecht,TheNetherlands.Email:kyprianou@math.uu.nl11IntroductionAbranchingBrownianmotionisconstructedasfoll

6、ows.Aninitialancestorbeginsitsexistenceattheoriginofone-dimensionalEuclideanspaceandtime.ThisindividualisimmortalandmovesaccordingtoanindependentcopyofstandardBrownianmotionB.Theinitialancestorproducesaran-domnumberofospring,X;attimeswhichformaPoissonprocess,n;withrate>0.Weshallassumethat

7、Xhasdistribution(p:k0)suchkPthatm:=kp<1.Startingfromtheirpointofcreationonthekk0pathoftheirparent,eachofthesechildrenmovesandreproducesaccord-ingtoanindependentcopyofthetriple(B;n;X).LetZbethepointtprocessdescribingthenumberandpositionsofindividualsaliveatti