Immigration process in catalytic medium

中國科學(xué)A000108中國科學(xué)AR資源系統SCIENCE IN CHINA(SERIESA)數字化期刊WANFANG DATA ( CHINAINFO)DIGITIZED PERIODICAL2000V ol.43No.1 P.59-64Immigration process in catalytic mediumHONG Wenming (洪文明)( Institute of Mathematics, Fudan University, Shanghai 200433, China )WANG Zikun (王梓坤)( Department of Mathematics, Beijing Normal University, Beijing 100875, China )Abstract : The longtime behavior of the immigration process associated with a catalytic super -Brownian motion is studied. A large number law is proved in dimension d≤3 and a central limittheorem is proved for dimension d=3.Keywords : immigration process, branching rate functional, Brownian collision local time, catalyticsuper - Brownian motion.▲It is well known that the measure-valued branching process, or superprocess, describes theevolution of a population that evolves according to the law of chance. If we consider a situationwhere there are some additional source of population from which immigration occurs during theevolution, we need to consider a measure-valued branching process with immigration, or simplyimmigration process [ 1,2] . Some limit theorem for the immigration process were obtained in refs.[ 3,4 ] . Recently, much attention is focused on the superprocess in random environment.Randomizing the branching rate functional, Dawson and Fleischmann [5] constructed a super-Brownian motion in catalytic medium, the so-called catalytic super- Brownian motion in dimensiond≤3, whose branching rate functional is random and is given by the Brownian collision local time(BCLT). The BCLT is determined by another super- Brownian motion ρ , which is called a catalyticmedium (referref. [ 5 ] for details). A central limit theorem for the occupation time of the catalyticsuper- Brownian motion is proved inref. [ 6 ] .The situation is also interesting for the immigration process. In this paper, we consider theimmigration process associated with catalytic super-Brownian motion (ICSBM) XP . And we obtainthe weak large number law (d≤3) and the central limit theorem (d=3) for the ICSBM XP and itsoccupation time process.1 Main resultsLet W=[ w,∩s.as,t≥0,a∈Rd ] denote a standard Brownian motion in Rd with semigroup{P,t≥0}. Let C(Rd) denote the Banach space of continuous bound中國煤化Iequipped withCNMHG,the supreme norm. Let φ p(a) : =(1+|al2)-P/2 for a∈Rd, and let Cp(ky.t{ieC(K",|f(x)|≤C(φ p，fle///E/ Vqk/zgkx- exzgx000/0001000108.htm(第1/ 8頁(yè)) 2010-3-23 15:53:26中國科學(xué)A000108(x) for some constant Cp}. Let M,(Rd) : = { Radon measuresμ on Rd such that」(1+|x|P)-lμ (dx)<∞}. Suppose that Mp(Rd) is endowed with the p-vague topology. Note <μ ,f> :=J f(x)μ (dx).Let λ denote the Lebesgue measure. We shall take p> d, so thatλ∈Mp(Rd).Suppose that we are given an ordinary Mp(R)-valued critical branching super- Brownianmotionρ :=[ρ ,Q1,Ps.. ,t2s≥0μ∈Mp(R4)] . (We writePp for Po.. .) For d≤3 Dawsonand Fleischmann[5]proved the existence of the Brownian collision local time (BCLT) L[w,ρ ](dr) of ρ , which is an additive function of W. And for f∈Cp(Rd)+o,(db)p(r-s， a, b)(b).(1.1)Furthermore, it is the branching rate functional. We refer to ref. [5] for details.For Px -a.s. ρ , the ICSBM starting from μ with the immigration rate V is denoted by XP : =[XP ,Q2,PP μ、,t≥0,μ ,V∈M,(Rd) ] . The Laplace functional of its transition probabilities isP..exp(- →<λ ,f> in probabilityfle///E/ Vqk/zgkx- exzgEx000001000108.htm (第2/ 8頁(yè)) 2010-3-23 15:53:26中國科學(xué)A000108Let S(R4) be the space of rapidly decreasing, infinitely differentiable functions of Rd whose allpartial derivatives are also rapidly decreasing, and let S' (Rd) be the dual space of S(Rd). Let你，f>:= r-'2[-儀，f>-(λ，f)], f∈S(R).(1.4)Theorem2. Let d=3. Then we have' γP→y∞in distribution, where' y∞is a centeredGaussian variable in S' (R3). Its covariance isCor( 3/4), a2(t)=tβ ,a3(t)=tY ,β ,Y >0.(i)d=3, a∈R3Proof.We prove (i) only. By the same method, (ii) can be obtained. Consider the I aplacetransition functional of the occupation time of pPerp[-.。 0, uniformly in S there isp.{a(o)-1f6C(f(D>(P.1(b)P -~dr[(P_(b)Pdb1≥e}<-20(1)-. Vas,"r^,.(db)(.(b))≤0(rcd)→0(as 1→∞),where c(1)=2a -3/2>0, c(2)=2β -η (β >η > 0), c(3)=2y . This completes the proof.Q.E.D.Proof of Theorem 1. It suffices to provelimQexp(- r-'+(r.1 =0.(2.7)Then from (2.4) together with (2.6) and (2.7), (2.3) is obtained. This completes the proof.Q.E.D.Proof of Theorem2. Letft : =t-1/2f. From (1.2) and (1.3), with respect to Q, the Laplacefunctionalof yP tis中國煤化工MHCNM HG，fle///E/ Vqk/zgkx- exzgEx000001000108.htm (第5/ 8頁(yè)) 2010-3-23 15:53:26中國科學(xué)A000108Qexp(-不，0))= P.exp{(2.8)where v(,t;) is the solution of (1.3) with f being replaced by f. Becausej6r(-P-f6(6)2 ≤川°(1λ -32).<2.1→0(2.9)ast→∞. By Lemma 1 and the dominated convergence theorem, under probability Px , we have。+∞J0≤lirm" drlp,(db)P-A(b)2 =0，1+∞J0(2.10)1 xJ0=lin1 *∞。0= limt"-1'asi'rf(P._,)f(b))2db: | dr|(P,( b))2dbJ0= <入，fGf),(2.11)where G is the potential operator of Brownian motion. From (1.3)(P.S(b)2-(u(r.1,b)2≤2(Pf(b)).| dh|p%',(dx)o(h,l,x)p(h- r,b.x).Using Lemma 1 and H. lder inequality, we get中國煤化工MHCNM HGfle///E/ Vqk/zgkx- exzgx000/0001000108.htm(第6/ 8頁(yè)) 2010-3-23 15:53:26中國科學(xué)A0001080≤limds)p,(db)[(P,_f(b))2- (o(r,l,b))2]→∞J0 J= lim _ds|dr|d6[(P._. f(b)) - (o(r,l,b))]1∞J 0≤2 lim| ds| dr|db(P,f(b)).dh|pn(dx)o(h,l,x)Pp(h- r,b.x)1*∞。(≤2 limds| dr}db( P,_f(b))]2≤C●lin≤C●lim[dh(1 A (I- h)-3/2)J0=C●1-1/2dh(1 A h-32)as t→∞. Combining (2.11) and (2.12), we havelim"ds dr|p,(db)(o(r.t,b) = →<λ ,f> in probability中國煤化工MHCNM HG，fle///E Vqk/zgkx- exzgEx000001000108.htm (第7/ 8頁(yè)) 2010-3-23 15:53:26中國科學(xué)A000108w/ := 1-s4[(v,/-x.f-.f]. r∈s(Rt).(2.14)Theorem4. Letd=3. TheninS' (R3),了P→γ∞in distribution under Q,where y∞is a centered Gaussian variable in S' (R3). Its covariance isCor