vol.43 no. 12science in chinae" "series ae@december 2000the oscillation of the occupation time process ofsuper-brownian motion on sierpinski gasketguo junye""1u%u0a£odepartment of mathematicse- nankai universitye-tianjin 300071£-chinae. " emaile9yguo@ public. tpt. tj. cnf⑥received october 22e-1999abstractthe occupation time process of super-brownian motion on the sierpinski gasket is stud-ied. it is shown that this process does not possess stable property in the long run£ but oscillates peri-odically in some sense. other convergence properties are also studied.keywordsl? occupation time process£ superprocess£ catalytic point .let g denote the sierpinski gasket which is a fractal subset of r . the brownian motionb”t£@on g was constructed by barlow and perkins'2 in 1988. later on£ different kind of diffu-sion were constructed in different kinds of fractal structure5'2i4ey. up to nowf -the properties ofthis kinds of diffusions are comparatively clear . the corresponding superprocess and the superpro-cess with spatial motion in euclidian space 4252 behave differently. as will be proved in this pa-pere-the occupation time process of this super- brownian motion oscillates regularly in the long runwhich differs from the known fact thatf in euclidian spaces£- the occupation time process of super-brownian motion has a tendency to stabilize in the critical case .let x; be the super- brownian motion with spatial motion b” t f@and branching mechanismzl β where0< β≤1. then the laplace functionals for x, satisfye,expf0-i”xeφiry= expf0-i μff~ i£a&-where f" tfgs the solution to the equationf" tex2o= f" texeye$~ y2qi dy2@__dr|f"t- rexeyf@l e" reyf@" "dyfo ["220in the above equation a special measure μ on g is used as the initial measure of e”tf@and actu-ally serves as the reference measure of the transition density function f texeye@f b" td ref.£ulfxg briefly speakingf-it is a hausdorff measure normalized to be 1 on the sierpinski gasketwith unit side. for the propertiesof μ ,and f texeye@one can refer to ref.£u1fy we will alsoadopt some other notations in ref .£u1£yand use l2-e" i = 1£22 - fco denote the generic con-stants. in most papers£- the bounded continuous functions are treated as the test functions£ -buthere we take a special set of test functions fe-f =eqpeφ∈c$" gf国h f~xe中国煤化工1-where k1f- 72 are positive constants that may deplmh.cn m h grxund c" ceqs the setof all bounded continuous functions on g. the corresponding dual space ism = 20军vffikk ∞forallf∈ f 李-where f denotes the set of nonnegative functions in f. xi and the following occupation timeprocesses月窍数据m .no.12oscillation of 0ccupation time process1251if a point catalyst c∈g works£ the relevant superprocess£- called catalytic super brownianmotion with point catalyst at c and denoted by卢中te国-has laplace functionals satisfyinge,expfci° °理ψiay= expfci μff leo&x用-f320where 6 teoexegs the solution to the equation6" lzoexe@=| fr lexeye$" yfq" dy20 drf"t - rexeof⑥l 2° reoee£⑤→f"420where ψ∈f £-0< β≤1. consider the corresponding occupation time processes x2-ricswithx=jox,drfef"ec0=. r,"car. their laplace functionals ful6l respectivelye,expfuli 'x2φiy= expfq-i ' μch~ lf甲a-f5eoe,expfi f°腔ψiθy= expfci μf° tlσfp吠→f~6e@where h" texf国fl teoexe@are solutions of the following relevant equations£2js.,"x2@r-|. dr|p"i- rexeyf@l 2" reyx@" dy2@-f720e" teoexeo_| s,ydr -| drf~t - rexeoi⑥l e" rzoeee◎f"8eoas usualeσeψ∈f e-s, is the transition semigroup of b" tf⑥for the superprocesses in catalyt-ic medium one can refer to refs £u3 i @6ey for the occupation time process£ see ref £u7ey in thesequelf- the behaviour of random measures t-1x, and t-1yrcc-as t tends to infinityf- will bediscussed.1 oscillations of the occupation time processlet ψ be a function in f that has a bounded closed support. we then have the followingtheorem.theorem1.1. ifβf~2/d,f@ 1e-t = 5"le-l > 0f then the random measures 1-1 fic20converge in distribution to some nondegenerate random measure f~ leqe where ifh" lfgs a ran-dom variable that satisfies m 5le@ ra~ le国d, = log9/ log5.the proof of the theorem will be given in the next section. theorem1 shows that 1-f,cc2isnot convergent as t- ∞fhbut converges when t goes along a discrete time series 5"l. that is tosay£- it oscillates regularly when t becomes largef and the period of oscillation increases accordingto a geometric series 5"l. this must result in some scaling regularity of the limit measuresfas isshown in the theorem. we first prove some lemmas .lemma 1.1. there exists an f∈f such that for any fixed t > 0 and any bounded sub-set d of the sierpinski gasket gdrp" rexey2@≤f" :xa国y∈de-t ∈euoeteyproof. by theorem 7.9 of ref .£u1-o1中国煤化工0f we have:mhcnmh g{fβ°reπ&γyq≤l1r-:/2xp[-2(r)hence for a fixed ε > 0fr≤t£ we have「iy-x12~1->国‘tr1252science in chinae" "series aeovol.43≤mexp0l n; 1y-sx1871->ey .≤[m2exnf0- n21 x 191-日-x≠2q∈g2咒”zede@≤ε-l qf-x∈fq∈ger”zzd2@≤ε&y[m2exrfql n21 x 1|9*1-r8rx ≠2q∈g2眾" z&de@≤e-s lm3expfq_ n31 x |19^1-xe22-x ∈fq∈g&a" zfdf@≤e£y≤mexpf0_ ni x |18"1-空号-where m f-m2e-ne- n2e-m3f-n3f q are positive constants£-m = maxf0m2-m3fj-n =mirfuv2ean3ey the existence of m3e 7n3 is guaranteed by the boundedness of£c∈gea”zedeo≤ε£y thusf"x20= mexfol n| x 191->@y .is what we need and the proof is complete.for any t > 0f-tlefine fioerito be the set of all continuous maps 9~* texegfuoffeyinto fsuch that | f." terxeo|≤gf" xeo∈f . it is a banach space with norm ii f ii =sup |i" s£x2@ x. we havelemma 1.2. if β=l、1f-hen for any fixed 8e-l > 0e the integral equationf" sex2o= | drf" lr2xeoec_ 1|" df"t"s - r2oxzθ2@) 2" r2ol@~920has a unique positive solution in fe0oefey it is obvious that the solution depends on l. if we writeit as asexf-then the scaling invariant property 3f&" s£2xe@ f sex f@holds.proof.uniqueness. let us give a simple inequality which will be used in the proof of thislemma and the following onese>that ise=|al β- bl bi≤2"a b2@a- b12-a£b≥0.ifjf安s&x8@f"￥sfxeg feiozreyare two positive solutions of eq.2 *9i国then for s≤τf-| f"^sexf@_ f"2 s£x2@\≤l|_ drf"2"s - re⑥xeθx@2j@f 空r2θl@ f中rfof@so .i叫jf"壁r2θeo_ f"2e rfθ2@≤l|° drf"t"s- r20xeβφa2l 8i0( 公df" lr£o2ox申°~j叫jf" rfθeo_ f" rfθ2@≤max| f"贮rf_ f"e r£@|。jcf" τ2国0≤r≤τwhere lim&^" r&o= lnm2 890(. °f°lrloeolqr= 0. hencemaxf"*s2@_jf"e*s2g|。≤f"renax||f“*s2@_ jf"*sl@| ..0≤8≤tfor a suficiently small τewe have jf" s£@中国煤化工uing this procedure atthe beginning times rf2tfi-frwe have the dehcnmh gexistence. byf^ 6f6: 8e国the laplace funcuonaustot~ i utiie,expfqi°t1j°eψiay= expf0-i ' μe话teoep哩哪阳号一where ψ∈f has bounded support and f~ s£rfgsatisfiessloexeo=|" tr-'s;ψdr-| drf" s - rexeoq@! e" reoee2◎f£"1020oscillation of 0ccupation time process1253let t= l5"e-f~ s£o£xe@ 0f~ l5"s£eoe2"x$9b". then 城" seofx fgs the solution to the fol-lowing equation29%~ s&oxrfo_」drjf^ 1r&xexyso"$~2"yf@"dy&q drp(i"s- rsoxδ )明/ros号).f["11£@if we take c王0£0£6∈ofthen bf"11e0| o%seθlx2@_ % seθfxfcjdr|f" lr&xey2@f"dyf@_」 'dr),f" lrexey2q&" 'dy炯 l| drfi"""s - rf⑤x£0e@ o1/ 2" r£ol@£c_ vh e" r£oe@eo0≤|u& "dye中drp~ lrexey2@- pf~ dye@ drfi" lr2xeyf@l drfx~s - r⑤xfθ風2~i 0需rl0e@f@ 1 ue~ rlθeθeo 80iq uf%"rlθfθz@_ cf~ r2σl@2@ .f" 12e0here p& dyf@ 3"$^ "2"xf@~ dy£@-ψ has bounded closed support andi ' ψepi| u&~ gec δ.obviously the suppot of n≥1egs contained in the bounded support of ~ y£@" *dy2◎ it is .easily known that μn converges weakly to δi 48~ dye国where δo is a dirac measure on g that as-signs mass 1 orf" 0eθeo by lemma 1.1 and the weak convergence of μn£ we havelim|p&~ dy2中drf" lrexeye0-|" drf" lrexzθfof" 13£⑥joandf~ s£o2x&@ |f 'dyxq"drfi" lrex&ye@≤f" x2@∈f .nowl" 12eccan be rewritten as嘴s20exeq_ i slθsx伸≤| |u% "dy2@ drf" lrexey2o |r% dy£(drf" lrexeyedj0. 4lif|b| dr"t"s- re⑤x2θ2∈ v% : rloe8ec_ uf~ rloeeeq .£" 14e0combiningf~ 1286 132@andf~ 14e@eildssup)|| o%~ sfθexz@_ uf~ sioexz@| ∞≤sup| r%" dy中drf" lrexeys@ & "dy中drf" lrexeyeg8≤ti∞o 4l|f|&ljsup|, d&" ir204./2 supll %~ rfθexeo_ u&" rfθexeq|。["1520for any τ>0. take τ sufficiently small so that中国煤化工4llfl|& l|.myhcnmhgthen it fllws fronf" 15e@hat 0- h" sexegs a cauchy sequence in feooererl thus there exists f" s£r2o∈feor-ef andlima~ s£oex£o= f" sexfdy≤t.2" 16eorepeating互貉据procedure we see that for any t > 0f there exists g" ser&c feioerersuch thaf" 16e01254science in chinae" "series aeovol.43holds for s≤t. consequently 9" sexegs the solution to eq £ °9eo the existence proof is completed.2 proof of main theoremwe prove theorem 1.1 in this section as follows.proof. taking eq .l" 1 1ecinto consideration£ we can first obtaini' t 1eoex£oxiμ=i° 01 l5"eoexf⑥yx iμwhat we want to know is the limiting behaviour ofi μe功° 1exe@μ let δ=° ψepx iuirf °9e@ then| f s£xfc_ f" s£xf@["drlp(q"s- r28xf)- fr"t"s- r27686p2 0 ro&号) 1jd"s- r&r862:1 0( r2o2ql)-f' e (r&f) lj]df"s- r2oxz02φri a (号)-f r26q=:i ii ii ive-f1720where忧”yf@ 3"$^ 2"y£⑥'the four terms will be considered separately. by lemma 1.1 and the bound-edness of the support of ψf drfi" lrexeyfq≤ f∈f holds for all y∈sup$~ ψ2◎this2 together withthe weak convergence of e yfq~ dy£qoi ' ψejx i腾~ dyfo-gives lim i| i|l 。=0. on the other hande-t s£ofx2@≤| _dr|f" lrexeyfqe~ 'yt@i "dy2o<|°" xq&" y国" dy2o= j"xoψxniμf"18e@the estimation on p" texey ae"theorem 5 in ref .2u12zqande" 18e@ensure thatlimll ii .≤liml|f" x$⑥ψ&mil↓β drllp(i"s- r国πε品)_ fr"tt"s- ri0xe6e@| 。∞=0.f" 19l0eq.£" °9e@can be used to derive lim ii iv|i . =0.the item ii must be dealt with carefully. byf" 9egs" 1128e: 182国-or any s euoefeywe haveil≤2l max ii 说~ reofcf" req|∞0≤r≤8"dfi"t"s- rf0r2o电r l ψsni胸 (0。df irlx06zφ' )2≤6°ψnillfl& (anf;- 120.r)口max|i %"r2o20_ f"r2q| .i叫|, d"y"s- re604/20≤r≤$中国煤化工≤lj口max ii g reoqc_ g" rfgf"20200≤r≤5.yhcnmh g .choose a small τ >0 such that lt1-2 < 1. then byf" 20e0lim. sup |i i||∞≤lτ1-之 lim. sup |i i|i .8-which mey丽月敝掘p_ ii i。= 0e-because lr'-2 <1.no.12oscillation of 0ccupation time process1255the above deduction shows that infu0erf2vn converges in feoerxyto f. using the above method re-patedlyefinally dn converges tof in feoezrfor any fixed t>0.with the help of" 10fcs" 11e8s" 16e@and lemma 1. 12- we conclude that when t= l5"e-e,expfqli^ 1-1f°理ψiay= exp0-i "p&oi" 1eofx£ayexf0li ”puff" 1exeay= expf0-i ' μe3f#" 1ezxeay= expfuli "μff" 1eπz@9-f"21£@and that the solution f" sex£@o eq上*9f2@satisfies f°~ sex 840e>otherwise we can derive fronf^" 9e@hat0=i" pf$" s£@l= δs- 1 drfl r" rf02o_ δs ≠0£-f"22@which is a contradiction. these show that t-1 产c converges in distribution to some nondegenerate ran-dom measure h leq and f *5le@ rf~ l2⑥3 convergence property of occupation time process x,we turm to the study of occupation time process without the participation of catalytic points. let t =5”f 7analogous t&~ 11e@we havee,expfq-i'xs e5- "piay= expf0li "μeof 1exq-"232@where φ∈f and uf sexeqsatisfies1g sex20="rjf" reπeys￥° 2yx@"f"dyq@_f~2420ande,expf0-i”xxs.e5- "piay= expf0-i ”pμ2m% sexf@ayf25e⑥we start witlf" 23eqe" 252@o prove that t - 1 x, converges vaguely to zero measurl t is not necessarily e-qual to 5"£@ this means that it has local extinction behaviour .by mean value theorem2-let ξn be a point in2u1e2fysuch thati “ ul p ξ2⑥p iμ=fiut"8μ ir. integrating the two sides of" 24e@yieldsir uf s£πεoxiμ= i μζφil()“了山山89xi10.f" 26e@hence we getie ,52mik后u师2 r&@xihr≤21 "p2qi(号)”.["2720fronfe" 25e@t fllows thai”uf rf国px iμus an increasing function of r≥0. therefore by holderis inequali-ty and eq 224e国-we havei' f 1exepix un eexeqp iμxf s£xf@~ dah2dxe⑥中国煤化工myhcn mh grzo