中国科学a000108中国科学ar资源系统science in china(seriesa)数字化期刊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中国煤化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页) 2010-3-23 15:53:26中国科学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(-
