α-TRANSIENCE AND α-RECURRENCE FOR RANDOM WALKS AND LEVY PROCESSES

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【出版日期】2005-01-25

【摘要】<正>The authors investigate the α-transience and α-recurrence for random walks and Levy processes by means of the associated moment generating function, give a dichotomy theorem for not one-sided processes and prove that the process X is quasi-symmetric if and only if X is not α-recurrent for all α< 0 which gives a probabilistic explanation of quasi-symmetry, a concept originated from C. J. Stone.

【刊名】Chinese Annals of Mathematics

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5 1 .Introduetion Random walks andL乙vy proeesses on渺are very interesting and important(see【1,3, 4,6一8」).The statespaeeofanyL巨vyproeessisaloeallyeompaetAbeliangroupG.IfGis eompaetly generated,then G 15 the direet sum of a eompaet grouPD,a veetor groupV,and a lattiee groupL(see{2,p.go}).There are two integers a and b such that V 15 isomorphie to渺and五15 isomorphie to zb.Thus the properties ofL‘vy proeesses on俨m盯be generalized toL乙vy Proeesses on a eomPaetly generated loeally eomPaet Abelian group with little effort.Let X be aL乙vy proeess on俨with eonvolution semigroup{二,}or a random walk with transition probability拼.W七know thatL乙vy proeesses(or random walks)ean be divided into two elasses:reeurrenee and transienee.If the genuine dimension of X 15 greater thanZ,then X 15 transient.Thus the question that how to elassify the transient L乙vy Proeess or randorn walks more finely 15 very interesting. In the present PaPer,we shall diseussa一transienee anda一reeurrenee witha兰0 ofL乙vy proeesses(resp.random walk).W七shall show that for anya<0,X isa一reeurrent if and only ifE(e一“LN)=00 for all open neighborhoods ofo:while X isa一transient if and only ifE(e一“LN)<、for all bounded open neighborhoods ofo·We eall a probability measure 夕quasi一symmetrie if lim Sup。‘(兀)令=一for some eompaet set万.The property of quasi- Zes十O〔 symmetrie measure has been studied in【9}.W七eall X quasi一symmetrie if二l(or if拼)15 quasi一symmetrie. for alla<O,that W七shall show that X 15 quasi一symmetrie if and only if X isa一reeurrent 15 to say,tlie speed at which X ese即es from bounded oPen set Can not be any exPonential.This gives a Probabilistie exPlanation ally oounQea oPer of quasi一symmetry. If X 15 【28 万H人N〔了,H 2..ZHAO,朋.2.&YING,J.召 rz()t(。:ze一sided,tllen a diehotomy theorenz liolds.、Ve may also use the nx(〕I,ient ge,l(·rati了19 fl;n(·ti(〕,It()(、士lara(沈erize thea一trarlsienee anda一reeurrenee.Ono;;r an(,ther paper!10}、‘() ‘·la、、i称’tlle(l1,asi一symmotrieL映vy pro(、esses more finely,we stlldy the polyn()nlial ree、;rrerl。。、 andp()lynomial transienee. Throughout this paper,for any finite positive measure户on皿九,define the eliara(、teri、ti(、 fi,n。‘10,1 of赵on渺as户(x):=报。e,(‘,,)拼(d。)and define‘he mor-nent generatingfu,,(、‘i‘),1 ()f‘户。、£l,(二):=报。。戈了‘)户(d,).If乞isapositiveinteger,weuse拼‘todenoteth。:乞一t’()101 (·oilvolution of补.Let月o:=凡.FOr any Borel set且,let LA denote the last exit ti:工xef全on, 几,lotG(A)denote the elosed group generated by几and let A denote the elos,:re of几.f,()r 、11y,一任吧.1(:tx土denote the spaee{。:(。,x)=O}and占二de,lote tlle Dira〔mea、1,re at t蛋lel)()ixzt x.Let杖denote the eoll(:etion of all setsN二R?‘sueh that N 15 relative(、()niPa、、t ()Ioerl、、n(a()〔N.凡r any人全。,let了、:={(二‘,二2二,二”)任渺:}x‘1三夸f()rl三Z兰,;} 号2.。一肠ansienee and。一Reeurrenee of Random丫Valks I·、·t尤二{X‘;P£}l〕。a ra,idonl walk on渺 l,i、genllinelyn一dirnensional,that 15,there with trarlsition Probability 1llea、,lre赵,wh、、z、、 15 no proPer linear 51一bsPa(沦of渺 、一l一)I)户.城、1、e p to de,、ote po for eonvenienee.A point劣‘渺15 ealledp()ssible (二ontairls iff()r eacll rzeiglzb()rllo()(1 N of 0 there 15 an integer乞 、et()f找11 po、、iblep〔)intsl)y E.ThenE二 0 suell thatP(X,任x+N)>0.We denol、th,、 supvl,乞whieh 15 the smallest elosed semigro,z一) >一一加U 2=0 t 11、、t(、ontains()and suppl‘.The elosed group generated by supp拼‘一s,ipp户‘15 ilzdep(·l、(lelzt ()f tlze positive integer,and we denote it by Gl.Let G be the gro一lp generated by suppl,. r}lezl GI生G.Let斤:。、be the Haar measure on G and let护denote the eolleetioxl()f:、11 b‘),lr,ded Borel sets A on G sueh that。。(A)>0 and me(先(A))=0.Here口〔、(且)15 tll,、 bourldary of且relative to G. 、Ves盯拜15 normalized(see!5,pp,64一75〕,or see}71)if there 15 an integer7*10三,*:三,;. 二,nd there are real numbers ai,,二,a二1 sueh that 户(2二nl, 2万n跳,O O)=exp(2二乞(n,a:+…+nol aol)) 丘)l、alli:lte罗rs,‘l 才乙几i, R)r arzv‘丫 〔皿and any and{户(口)}<1 for all other values of夕. BOrel subset A on皿几、define Va(A) =又。一“,l、‘(A)·TI‘e“ 之=0 15 a mea、:zre with supPort艺.In this seetion,we assume that。兰0 whi〔、h 15 tl:e onl、 irlterestin吕case. Definition 2.1.Gz.en an,x,岁,叨e£a夕that习ean be reaehed 扩介:a1心N〔杖, 己。汀z了n祝几zea忿已,and P了(X;任,+N)>0 for、ooe乞nt印er乞全 加。二,and二二比:二夕. 0 .Wosa珍that x and刀 。:,tox什军,扩x and岁ean 6e reaehed加二eaeh oth。:. Lemnia 2.1.(l) S叩Pose that Ar,且2,且 2)for any,nt印e:、乞,j. al产℃ Bo代1 set,and Al+且:侄且. T/tf,,z l,‘+了(互)全l,‘(A,)拜J(且 (2)The relat,on料 zsa几已q牡乞uale几t relat饭on on渺 Proof.(1)Sinee几飞+AZ旦A,A:互A一for allz任A2.For any integer、饭,7 。认一(‘,一关。。名‘A一,。了(“·,:人2 02(“一,。了(由,:户!(“1)。了(AZ, (2) there rzeed onlv to show that了 任杖sueh thatNI+Nz and以杆£imPly thatx什之,FOr anyN任状. .Sineex二军and夕二;子,‘(军+万.一,1)二 yN 什C一 a一TRANSIENCE ANDa一RECURRENCE 129 px(X,〔夕+N1)>0 and拼7(z+N1一,)=p,(凡〔z+Nl)>0 for some乞,j.Then by(1), px(X,+,任:+N)=拼‘+了(:+N一x)全科‘(y+Nl一x)户,(:+Nl一,)>0 .Heneex。:. Similarly,之几x.Thereforex什岑. Then渺15 divided into disjoint equivalent elasses ealled eommunieating elasses.Clearly, the set that ean be reaehed from x isx+艺and the set that ean eommunieate with x 15 x+£n(一黝. A vector 0 15 said to be a sided veetor for拼if。并0 and拼{x:(。,二)>O}=O,strietly sided veetor for拼if赵{x:(

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