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A Note on 2-Level Superprocesses

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A Note on 2-Level Superprocesses Journal of Applied Probability.Vol.41(1), 202-210. A Note On 2-Level Superprocesses Wenming HONG1 Department of Mathematics, Beijing Normal University,Beijing 100875, P.R.China Abstract. We prove some central limit theorems for a 2-level super-Brownian motion with random immigration, which lead to limiting Gaussian random fields. The covariance of those Gaussian fields are explicitly characterized. Key words: super-Brownian motion, 2-level superprocess, random immigra- tion, central limit theorem. AMS 1991 Subject Classifications: Primary 60J80; Secondary 60F05. 1. Introduction and statement of main results Measure-valued processes whose dynamics is given by a random reproduction of mass (producing mass fluctuations) together with a mass flow (partly smoothing out the fluc- tuation) are an interesting object of study. In this paper, as a mass reproduction, we will consider 2-level branching, with a critical reproduction on the individual as well as on the family level; the mass flow is that of Brownian motion, leading to so called 2-level super-Brownian motion. Those 2-level superprocesses have been studied by many authors including Dawson, Gorostiza, Hochberg, Wakobinger and Wu ([5], [7], [12], [13]) etc. Wu ([13]) confirmed Dawson’s conjecture about extinction in lower dimensions d · 4; whereas Gorostiza et al ([7]) proved the persistent property in higher dimensions d > 4 and thus established that d = 4 is the critical dimension. For 1-level superprocesses, as we known, this important property was proved by Dawson in the famous paper [1] in which the critical dimension is d = 2. The aim of this note is to consider the central limit behavior of the 2-level super- Brownian motion. The long-time fluctuations of the occupation time of 1-level super- Brownian motion were studied by Iscoe ([11]). Those of (aggregated) 2-level branching particle systems (with a more general class of particle motion) by Dawson et al ([4]). In [4] it was indicated which terms in the covariance functional of the occupation time 1Supported by the National Natural Science Foundation of China (Grant No.10101005 and No. 10121101) . E-mail: [email protected] 1 fluctuations drop out in the superprocess limit. We shall actually establish the central limit theorem for the occupation time process of the 2-level super-Brownian motion for d > 4 (Theorem 1), which is parallel to the result of [4]. Our interest is focused on the 2-level super-Brownian motion with 2-level super- Brownian immigration, where the random path of a 2-level super-Brownian motion acts as a source of “family” level immigration for another 2-level super-Brownian motion. In this model, the immigration “families” involve two kind of evolution a) the variability inherent in the immigration source, b) the variability produced by the family branching after the immigration. The fluctuation limit theorems for itself (Theorem 2) and its occupation time (Theorem 3) are proved (d > 4), we will see that in lower dimensions a) makes contribution to the fluctuation limits, and in higher dimensions b) makes contribution. Interestingly, in the critical dimension both of them make contributions. A similar phenomenon has been observed for the 1-level super-Brownian motion by Hong & Li ([10]) and Hong ([9]). Moreover, it reveals that the random immigration “smooths” the critical dimension in the sense that there is no “ log ” term in the normalization for the critical dimension. We first recall some background material and notations. Let C(IRd) denote the space d of continuous bounded functions on IR . We fix a constant p > d and let Áp(x) := 2 ¡p=2 d d d (1 + jxj ) for x 2 IR . Let Cp(IR ) := ff 2 C(IR ): jf(x)j · const¢Áp(x)g. In duality, d d R let Mp(IR ) be the space of Radon measures ¹ on IR such that h¹; fi := f(x)¹(dx) < 1 d d for all f 2 Cp(IR ). Throughout this paper, ¸ denotes the Lebesgue measure on IR . d Suppose that W = (wt; t ¸ 0) is a standard Brownian motion in IR with semigroup (Pt)t¸0. Let (Tt)t¸0 be the semigroup of the (one-level) super-Brownian motion which was determined by + d E expf¡hXt; fig = expf¡h¹; n(t; ¢)ig; f 2 Cp (IR ); (1) where n(¢; ¢) is the unique mild solution of the evolution equation ( n˙ (t) = ∆n(t) ¡ n2(t) (2) n(0) = f: From this it is easy to obtain the following Proposition 1. Let (Tt)t¸0 be the semigroup of the super-Brownian motion. Then for + d f 2 Cp (IR ) we have Tt(h¢; fi)(¹) = h¹; Ptfi (3) 2 and 2 2 R t 2 Tt(h¢; fi )(¹) = h¹; Ptfi + 2h¹; 0 Ps(Pt¡sf) dsi: (4) 2 Let (Tt)t¸0 be the semigroup of the super-Brownian motion. Let p > d and let ½ ZZ ¾ 2 d d Mp (IR ) := º 2 M(Mp(IR ))) : Áp(x)¹(dx)º(d¹) < 1 : 2 d The 2-level superprocess with branching rate ® > 0 is an Mp (IR )-valued continuous homogenous Markov process X ´ fX (t); t ¸ 0g such that E [exp(¡hhX (t);F iijX (0) = º] = exp(¡hhº; vF (t)ii) (5) d + d for F of the form F (¹) = h(h¹; fi) with f 2 Cc(IR ) and h 2 Cb(IR ), where vF (t; ¹) is the solution of the non-linear integral equation Z t 2 v(t; ¹) = Ttv(0; ¢)(¹) ¡ ® Tt¡s(v(s; ¢)) (¹)ds; v(0; ¹) = F (¹); (6) 0 R where hhº; F ii := F (¹)º(d¹). See [7], [13]; and [3] or [12]. As pointed out in [7], the 2-level superprocess X can be thought of as a “super- superprocess”. We will consider the occupation time process Y ´ fY(t); t ¸ 0g of the R t 2-level superprocess X , where Y(t) := 0 X (s)ds is determined by the Laplace transition functional R t E [exp(¡ 0 hhX (s);F iidsjX (0) = º] = exp(¡hhº; uF (t)ii); (7) where uF (t; ¹) is the solution of the non-linear integral equation Z t Z t 2 u(t; ¹) = TsF (¹)ds ¡ ® Tt¡s(u(s; ¢)) (¹)ds; u(0; ¹) = 0: (8) 0 0 Let X (0) = R1, which is the canonical equilibrium measure of the 1-level superprocess non-trivial equilibrium state X(1) for d ¸ 3 (see Dawson & Perkins [6], Gorostiza et al [7]). Consider the centered process Z(t), ¡1 hhZ(t);F ii := ad(t) [hhY(t);F ii ¡ E hhY(t);F ii]; (9) where the normalizing factor ad(t) is given by 8 > 3 <> t 4 ; d = 5; 1 a (t) = 2 (10) d > (t log t) ; d = 6; :> 1 t 2 ; d ¸ 7: 3 The Green potential operator G of the Brownian motion is defined by Z 1 d + Gf = Ptfdt; f 2 Cc(IR ) ; 0 and that the (operator) powers of G are given by Z 1 k 1 k¡1 G f = t Ptfdt; k ¸ 2: (k ¡ 1)! 0 Then we obtained the following central limit theorem Theorem 1 Let d ¸ 5 and X (0) = R1. Then as t ! 1, Z(t) converges in distribution to a centered Gaussian field Z(1) with covariance ( ®C h¸; fih¸; hi; d = 5; 6 Cov(hZ(1);F i; hZ(1);Hi) = d ®hf; G3hi; d ¸ 7: p 16 ¡5=2 ¡3 d where C5 = 9 (2 ¡ 2)(2¼) , C6 = (2¼) =4, ¸ is the Lebesgue measure on IR , d + d + F (¹) = h¹; fi and H(¹) = h¹; hi for f 2 Cc(IR ) and h 2 Cc(IR ) . 2 This result was indicated by Dawson et al (see 2.6 and Theorem 3.2.1 of [4]), we omit the details of the proof. Now we consider the 2-level superprocess with random immigration. Hong and Li ([10]) studied this model in the 1-level situation, where some interesting phenomenon in asymptotic behavior is revealed. Let # ´ (#t; t ¸ 0) be a critical 2-level superprocess with branching rate ¯ > 0. Taking the path of # as the immigration rate, we get another 2-level superprocess X # with immigration, which is determined by · ¯ ¸ n o ¯ # # ¯ E exp ¡hhXt ;F ii = E E expf¡hhXt ;F iig¯fσ(#s; s · t)g n o R t (11) = E exp ¡hhX0; v(t; ¢)ii ¡ o hh#s; v(t ¡ s; ¢)iids = exp f¡hhX0; v(t; ¢)ii ¡ hh#0; v(t; ¢)iig : where v(¢; ¢) is the solution of the non-linear integral equation R t R t 2 v(t; ¹) = 0 Tt¡sv(s; ¹)ds ¡ ¯ 0 Tt¡s(v(s; ¢)) (¹)ds; v(0; ¹) = 0 (12) and v(¢; ¢) is the solution of the equation (6). Consider the centered process X #(t), # ¡1 # # hhX (t);F ii := bd(t) [hhX (t);F ii ¡ E hhX (t);F ii] ; (13) where the normalizing factor bd(t) is given by ( 3 t 4 ; d = 5 bd(t) = 1 (14) t 2 ; d ¸ 6: 4 We shall prove the following central limit theorem: Theorem 2 Let d ¸ 5, X (0) = a1R1, #(0) = a2R1 and a1; a2 > 0. Then as t ! 1, X #(t) converges in distribution to a centered Gaussian field X #(1) with covariance Cov(hhX #(1);F ii; hhX #(1);Hii) 8 <> a2¯Cdh¸; fih¸; hi; d = 5; 1 2 = a2¯Cdh¸; fih¸; hi + 2 a2®hf; G hi; d = 6; :> 1 2 2 a2®hf; G hi; d ¸ 7: d + d + for F (¹) = h¹; fi and H(¹) = h¹; hi with f 2 Cc(IR ) and h 2 Cc(IR ) , where Z 1 Z 1 ¡d=2 2 ¡d=2 Cd = (4¼) s ds (s + r) dr; 0 0 for d = 5; 6.
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