Two explicit Martin boundary constructions英文文献资料

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1、-296-TwoexplicitMartinboundaz~constructions.F.Spitzer.A.Theboundaryforaclassofrandomwalks(jointworkwithP.Ney,Trans.Amer.Math.SoT.1966).ConsiderrandomwalkontheN-dimensionalintegersZNwithtransitionfunctionP(x,y)=P(o,y-x),meanvector/J=~"xP(o,x)~0andsuchthatP(o,x)~Of

2、oronlyfinitelymanyxandforsomexineachhalfspace.ThentheGreenfunctionOOG(x,y)=~Pn(x,y)n=oisfiniteandstrictlypositive.Thefunctiona,x~(a)=~P(o,x)e,ae~N'accordingtoHennequinhasthepropertythatD=EaI~(a)g13isacompactconvexsubsetofRN,thatgrad~iscontinuousandnon-zeroonthebo

3、undarybDofD.Finallythesphere~-1=fxIXe~N,Ixl=1]ismappedhomeomorphicallyonto~DbythemapSN_1~~BD~rad~(u)=v~u=~(v)Igrad~(u)lItisprovedthatZNuSN_1isacompactmetricspacewiththe~etricII~-~I,x,y~z~,(x,y)I~-yl,x~ZN,y~SN_I,Ix-yl,x,yESN_1Xwhere~-,andthatinthistopologythefunct

4、ionsfy(.),definedforeach1+Ixly~ZNby[~xeZNG(o,x)'fy(x)=l~(x).yLe,x~SN_1-297-arecontinuousonZNu~_I.Corollax-j:TheusualPoisson-Martin-Doobrepresentationforthenon-negativeharmonicfunctionshasS%(x).yh(y)=ep(dx)~=probabilitymeasureonSN_i.SN-IB.Theboundaryforaclassofbra

5、nchingprocesses(jointworkwithH.Kesten,toappear)ConsideraGalton-Watsonprocesswithgeneratingfunctionf(s)=I-m+ms,Oj~>0andtheGreenfunctionooG(x,j)=~Pn(k,j)<~,wherePn(k,j)=coefficientofsj

6、inn=o[fn(S)]k,,ithfn=°f°""°f,ntimes.ItisshownthatifN=setofpositiveintegers,T=[o,1),thenNoTbecomesacom-pactmetricspaceifoneintroducesthemetricIIi9,(x,y)Ix-yl,x,yeTwhere~isthefractionalpartoflogx/log(~),andthatinthistopologythefunc-tionsfj(.),definedforeachjeNbyG(x

7、,j),:~"eNfj(x)=xeT"V=-cSOarecontinuousonNuT.Corollar~(proveddirectlybyH.Dinges):Theinvariantmeasures,i.e.soiutions~0of(SOk=S-298-havethePoissonrepresentation/u(j)=JZj(x)v(d=),j~~,Tfordifferentprobabilitymeasures~onT.AnalogousresultsobtainforallGaltonWatsonprocess

8、eswithm=~kpk