The oscillation of the occupation time process of super-Brownian motion on Sierpinski gasket

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盟一β=:A + BE-f"28EO. (..1xl≤2whereUn/3fYdenotes the largest integral part of n/3.Byf" 2雨熬插IxI≤2°f@ 2 BE-we have1256SCIENCE IN CHINAE" "Series AEOVol.43B e°1xl≤@212rx6号)门中≤f μE iFal( 35招)0≤Lip"E-f"292@where ρ is a positive constant less than 1 .Remember that φ has bounded supportE-so by stting &~ dyz@ $ 2"yL@"A" dyE@gainf the sup-ports of pn contract tf" 0£θE@ Thus for the first summand if" 282@ve haveA≤」,x| >{f(drfi" rExEyECof~dy2@≤JlyI≤y;1x> ghnfr"dxtL。df "r&rEy3_γ-xI18"1-rf"dy中. pnwx中drLIr0p{- 1.()”}. |yI≤41x1>221-718p&"dy2$drLar=) ”gE- f"30E0lyI≤lyrwhere L3 is some positive constant. The last inequality is due to the following estimation for the transitiondensity function of linear Brownian motion29"y_ x2pr 1ErEyEO-√2πcp{-E2j≤LsIγ-x1-3E-~312@and equation| y-x18"1-+21-183cr{-12(,r0("2L20/}"320Inf" 31£@x£y∈R'£rand Ls is a positive constant. If' °328@x£y∈ GCR2. Now for large n we continuef"30ECwithA≤L6|lyI≤4j'fp*"dyE$1x1> OnsErYf"dxf@ y-x 1-3121->28-32“"1->200≤L6]lyI≤4,'1f%~"dy申1x1> Ron3sef"dx≤4[x|> 2Inf"dx£G x 1-32*"1->2888≤L7k= 0*°<1x1≤2',.rx.x1-323/2"1->20D4ipS0n/3E$h .! 51090550= Lsi"E-f33⑥where 0< λ < 1. Summarizing the above we geti味1ErE⑥viK< Lm SVffM中国煤化工measure by 0. .f[~34E@This shows that t-1 X, converges in distribution to 8More preciselyE- we can also getYHCNMH G .PRT5-"Xs°£pi4> e2@f~1- e-°f@发~1 - expf0i^5- "xsnEp i&⑥王1- e-°01 - expf0-i”μEn%~ 1ExE@yO≤Ljma& λEpZ@.By Borel-Cantelli lemma&ir'5- Xs"£φ ip >0fu. s. f and becausei^t-IX,fpilK $'5-"XsnEpiA-5"≤t≤5"+1No.12OSCILLATION OF CCUPATION TIME PROCESS1257we have for all t&p' t-'X{f∞it°→0fa.s.The above deduction and computation yieldTheorem 3.1. For almost all sample paths the measure t - 1 X, converges vaguely to the zero mea-sure as t-➢∞.Considering l"1+08%,[ra= -log32r = 5"Ereq.f" 242@becomesβ~ log5'5"uG" sExE0-| dr|f" rExEyE$ "2"yE@"&i" dyE⑥. |"drf"s- rExEyO5"'u&^ r2yEQt Bi"dy£OBy a method similar to that in sec. 2E-we get 5"u”s£rfO' h" sExfO-6° sEx fQsatisfyingf" s£x20=|. drf" rExEy20 dr|fp"s- rExEyfQl+2" r&yEQ dyEOf~35E@Consequently a weak result that i2 1+战, converges vaguely to zero measure almost surely can be de-rived.Acknowledgements This work was supported by the National Natural Science Foundation of Chin&~ Grant No. 1980101920References1. BarlowE-M.T. f- PerkinsEE. A. f-Brownian motion on the Sierpinsk gasketEProbab. Th. Rel. Fields£- 1988E 792543.2. BarlowE-M. T. E-Bass£-R. F . E-Construction of Browmnian motion on the Siepinski capetE-Amm. Inst. H. PoincareL- 1989E-25E8225.3. Dawson£-D. A. ffleischmannf-K . E7A super Brownian motion with a single point catalyst£ -Stoch. Proc. Appl . E-1994E-49E9.4. Dawson£-D. A. E-FleischmannE-K . f Critical branching in a highly fluctuating randon medium£E-Probab. Th. Rel. Fields£-1991f- 90£241.5. Dawson£-D. A. f-FleischmannfK. fDiffusion and reaction caused by point catalysts£ 6IAM J. Appl. Math. £H1992E 529163.6. Fleischmann£-K . f-CGirtnerE-J. E Occupation time pssesses a critical pointE-Math. Nachr . E1986E-1252275.7. IscoeEH. E-A weighted occupation time for a class of measu中国煤化工hab. Th. Rel. FielsL-1986-712985 .8. Guo Junyid -Wu Rongf 6uper- Brownian motion on the SierpMYHC N M H Gdium£-in Trends in Pobabilityand Related Analysif" eds. KonoE-N . f -6hich2 N. R. Singapore29World Scientific£ 1997E-159.9. Guo Junyi. The local extinction of super- Brownian motion on Sierpinski gaskelE- fcience in China£ feries AE-1998E-41£9260.0. Kumagai£-T. f Construction and some properties of a class of nonsymmetric difusion processes on the Sierpinski gaskefE-inAsymptotic Problems in Probablity Theory£ Stochastic Models and Difusions on Fractalf "eds. ElworthyfK. D. E-HhedaE+N.2国-New York&9John Wiley and Sons Ine. E-1993E-19i4247.11. Lindst万行数掘ownian motion on nested fractalsL-Mem. Am. Math. Soc . E-19902-420.