Central limit theorems for supercritical superprocesses with immigration
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Central limit theorems for supercritical superprocesses withimmigration
Li Wang the date of receipt and acceptance should be inserted later
Abstract
In this paper, we establish a central limit theorem for a large class of general supercriticalsuperprocesses with immigration with spatially dependent branching mechanisms satisfying a secondmoment condition. This central limit theorem extends and generalizes the results obtained by Ren, Songand Zhang [21]. We first give law of large numbers for supercritical superprocesses with immigration sincethere is few convergence result on immigration superprocesses, then based on these results, we establishthe central limit theorem.
Keywords
Central limit theorem ¨ Supercritical superprocess with immigration ¨ Excursion measure
Mathematics Subject Classification (2010)
Primary 60J68 ¨ Secondary 60F05
In recent years, there have been many papers on law of large numbers type convergence theorems andon central limit theorem types convergence results for branching Markov processes and superprocess, seefor instance, [1,2,3,4,9,10,11,17,18,19,20,21,22] and the references therein. Especially in [21], Ren, Songand Zhang proved one central limit theorem which generalizes and unifies all the central limit theoremsof [18,19] and the advantage of this central limit theorem is that it allows us to characterize the limitingGaussian field and reveals more independent structures of the limiting Gaussian field.The focus of this paper is, to take the next step, on central limit theorems for superprocesses withimmigration. First, we will give law of large numbers on supercritical superprocesses with immigrationsince there is few convergence result on immigration superprocesses, then based on the these results, wewill give the central limit theorems. The main tool of this paper is the excursion measure of superprocessand immigration superprocess. As a continuation of [21], the underlying spatial process in this paper isthe same as in [21], we will state it in the next subsection for reader’s convenience.1.1 Spatial processE is a locally compact separable metric space and m is a σ -finite Borel measure on E with full support. B is a point not contained in E and will be interpreted as the cemetery point. Every function f on E isautomatically extended to E B : “ E Y tBu by setting f pBq “
0. We will assume that ξ “ t ξ t , Π x u is an Supported by NSFC 11301020 grants.Li WangSchool of Sciences, Beijing University of Chemical Technology, Beijing 100029, P.R. China.E-mail: [email protected] L Wang m -symmetric Hunt process on E and ζ : “ inf t t ą ξ t “ Bu is the lifetime of ξ . The semigroup of ξ will be denoted by t P t : t ě u . We will always assume that there exists a family of continuous strictlypositive symmetric functions t p t p x, y q , t ą u on E ˆ E such that P t f p x q “ ż E p t p x, y q f p y q m p dy q . It is well known that, for p ě t P t : t ě u is a strongly continuous contraction semigroup on L p p E, m q .Define r a t p x q : “ p t p x, x q . We will always assume that r a t p x q satisfies the following two conditions:(a) For any t ą
0, we have ż E r a t p x q m p dx q ă 8 . (b) There exists t ą t ě t , r a t p x q P L p E, m q .These two conditions are satisfied by a lot of Markov processes, see the examples in [20].1.2 Superprocess with immigrationLet B b p E q ( B ` b p E q ) be the set of (positive) bounded Borel measurable functions on E . Let X “ t X t : t ě u be a superprocess determined by the following parameters: a spatial motion ξ “ t ξ t , Π x u satisfyingthe assumptions of the previous subsection, a branching rate function β p x q on E which is a non-negativebounded measurable function and a branching mechanism ψ of the form ψ p x, λ q “ ´ a p x q λ ` b p x q λ ` ż p , `8q p e ´ λy ´ ` λy q n p x, dy q , x P E, λ ą , where a P B b p E q , b P B ` b p E q and n is a kernel from E to p , satisfyingsup x P E ż y n p x, dy q ă 8 . Let M p E q be the space of finite measures on E equipped with the topology of weak convergence. Theexistence of such superprocess is well known, see, for instance, [5] or [16]. X is a c´adl´ag Markov processtaking values in M p E q . For any µ P M p E q , we denote the law of X with initial configuration µ by P µ .Then X has transition semigroup p Q t q t ě defined by ż M p E q e ´ ν p f q Q t p µ, dν q “ exp t´ µ p V t p f qqu , where V t p f q is the unique positive solution to the equation V t p f qp x q ` Π x ż t ψ p ξ s , V t ´ s f p ξ s qq β p ξ s q ds “ Π x f p ξ t q , (1.1)where ψ pB , λ q “ , λ ą . Let M p E q ˝ : “ M p E qzt u , where 0 is the null measure. Define ϕ p f q “ η p f q ` ż M p E q ˝ p ´ e ´ ν p f q q H p dν q , f P B ` b p E q , (1.2)where η P M p E q and ν p q H p dν q is a finite measure on M p E q ˝ .Suppose that tp Y t , G t q : t ě u is a Markov process in M p E q with transition semigroup p Q Nt q t ě givenby ż M p E q e ´ ν p f q Q Nt p µ, dν q “ exp " ´ µ p V t p f qq ´ ż t ϕ p V s p f qq ds * , (1.3) LT for superprocesses with immigration 3 where V t p f q is the unique positive solution to the equation (1.1). We call tp Y t , G t q : t ě u a superprocesswith immigration associated with p Q t q t ě or t X t : t ě u . For the exsitence and properties of suchprocesses, we refer the reader to [12,13,14]. For any µ P M p E q , we still use P µ to denote the law of Y with initial configuration µ when there is no confusion.Define the random set Z : “ t t ě } Y t } “ u . In this paper, we assume that the process X , or equivalently Y , is supercritical, then clearly P µ tD t ą } Y t } “ u “ Z is bounded P µ -a.s. for µ P M p E q .Let α p x q : “ β p x q a p x q and A p x q : “ β p x q ˆ b p x q ` ż y n p x, dy q ˙ . Then, by our assumptions, α p x q P B b p E q , A p x q P B ` b p E q . Thus there exists M ą x P E p| α p x q| ` A p x qq ď M. For f P B b p E q and p t, x q P p ,
8q ˆ E , define T t f p x q “ Π x ” e ş t α p ξ s q ds f p ξ t q ı . (1.4)It is well known that T t f p x q “ E δ x rx f, X t ys for every x P E . Then (1.1) can be written into V t p f qp x q ` ż t ż E ψ p y, V t ´ s f p y qq β p y q T s p x, dy q ds “ T t f p x q , (1.5)where ψ p x, λ q “ ψ p x, λ q ` a p x q λ , see Theorem 2.23 in [16].It is shown in [20] that there exists a family of continuous strictly positive symmetric functions t q t p x, y q , t ą u on E ˆ E such that q t p x, y q ď e Mt p t p x, y q and for any f P B b p E q , T t f p x q “ ż E q t p x, y q f p y q m p dy q . It follows immediately that, for any p ě t T t : t ě u is a strongly continuous semigroup on L p p E, m q and } T t f } pp ď e Mpt } f } pp . Define a t p x q : “ q t p x, x q . It follows from the assumptions (a) and (b) in the previous subsection that a t enjoys the following properties:(i) For any t ą
0, we have ż E a t p x q m p dx q ă 8 . (ii) There exists t ą t ě t , a t p x q P L p E, m q . L Wang
It follows from (i) above that, for any t ą T t is a compact operator. The infinitesimal generator of t T t : t ě u in L p E, m q has purely discrete spectrum with eigenvalues ´ λ ą ´ λ ą ´ λ ą ¨ ¨ ¨ . Thefirst eigenvalue ´ λ is simple and the eigenfunction φ associated with ´ λ can be chosen to be strictlypositive everywhere and continuous. We will assume that } φ } “ φ is sometimes denoted as φ p q . For k ą
1, let t φ p k q j , j “ , , ¨ ¨ ¨ n k u be an orthogonal basis of the eigenspace (which is finite dimensional)associated with ´ λ k . It is well-known that t φ p k q j , j “ , , ¨ ¨ ¨ n k ; k “ , , . . . u forms a complete orthogonalbasis of L p E, m q and all the eigenfunctions are continuous. For any k ě j “ , . . . , n k and t ą
0, wehave T t φ p k q j p x q “ e ´ λ k t φ p k q j p x q and e ´ λ k t { | φ p k q j p x q| ď a t p x q { , x P E. It follows from the relation above that all the eigenfunctions φ p k q j belong to L p E, m q . For any x, y P E and t ą
0, we have q t p x, y q “ ÿ k “ e ´ λ k t n k ÿ j “ φ p k q j p x q φ p k q j p y q , where the series is locally uniformly convergent on E ˆ E . The basic facts recalled in this paragraphare well known, for instance, one can refer to ([8], Section 2). In this paper, since we assume that thesuperprocess X is supercritical, λ ă We will use x¨ , ¨y m to denote the inner product in L p E, m q . Any f P L p E, m q admits the followingexpansion: f p x q “ ÿ k “ n k ÿ j “ a kj φ p k q j p x q , (2.1)where a kj “ x f, φ p k q j y m and the series converges in L p E, m q . a will sometimes be written as a . For f P L p E, m q , define γ p f q : “ inf t k ě j with 1 ď j ď n k such that a kj ‰ u , where we use the usual convention inf H “ 8 . Define f p x q : “ n γ p f q ÿ j “ a γ p f q j φ p γ p f qq j p x q . We note that if f P L p E, m q is nonnegative and m p x : f p x q ą q ą
0, then x f, φ y m ą γ p f q “
1. Define C l : “ g p x q “ ÿ λ k ă λ n k ÿ j “ a kj φ p k q j p x q : a kj P R and g ‰ + , C c : “ g p x q “ n k ÿ j “ a kj φ p k q j p x q : 2 λ k “ λ , a kj P R and g ‰ + and C s : “ g p x q P L p E, m q X L p E, m q : g ‰ λ ă λ γ p g q ( . LT for superprocesses with immigration 5
Note that C l consists of these functions in L p E, m q X L p E, m q that only have nontrivial projection ontothe eigen-spaces corresponding to those “large” eigenvalues ´ λ k satisfying λ ą λ k . The space C l isof finite dimension. The space C c is the (finite dimensional) eigenspace corresponding to the “critical”eigenvalue ´ λ k with λ “ λ k . Note that there may not be a critical eigenvalue and in this case, C c is empty. The space C s consists of these functions in L p E, m q X L p E, m q that only have nontrivialprojections onto the eigen-spaces corresponding to those ”small” eigenvalues ´ λ k satisfying λ ă λ k .The space C s is of infinite dimensional in general.We use x f, ν y : “ ş R f p x q ν p dx q . And whenever we deal with an initial configuration µ P M p E q , we areimplicitly assuming that it has compact support.2.1 Excursion measures of t Y t , t ě u Let D denote the space of all paths t w t : t ě u from r , to M p E q that are right continuous in M p E q having zero as a trap. Let p A , A t q denote the natural σ -algebras on D generated by the coordinate process.It is known from [16, Chapter 8] that one can associate with t P δ x : x P E u a family of σ -finite measures t N x : x P E u , defined on p D , A q such that N x pt uq “ N x p ´ e ´x f,w t y q “ ´ log P δ x p e ´x f,X t y q , f P B ` b p E q , t ą , (2.2)and for every 0 ă t ă ¨ ¨ ¨ ă t n ă 8 , and nonzero µ , ¨ ¨ ¨ , µ n P M p E q , N x p w t P dµ , ¨ ¨ ¨ , w t n P dµ n q “ N x p w t P dµ q P µ p X t ´ t P dµ q ¨ ¨ ¨ P µ n ´ p X t n ´ t n ´ P dµ n q . For earlier work on excursion measures of superprocesses, see [6,7,15]. Next we list some properties of N x which will be used later. Proposition 2.1 If P δ x |x f, X t y| ă 8 , then ż D x f, w t y N x p dw q “ P δ x x f, X t y . If P δ x x f, X t y ă 8 , then ż D x f, w t y N x p dw q “ V ar δ x x f, X t y . Assume that Condition 8.5 in Li [16] holds: there is a spatially constant local branching mechanism z ÞÑ ψ ˚ p z q so that ψ p z q Ñ 8 as z Ñ 8 and ψ is bounded below by ψ ˚ p z q in the sense ψ p x, f q ě ψ ˚ p f p x qq , x P E, f P B ` b p E q , then it is sufficient for the cumulant semigroup p V t q t ě to admit the following representation for all x P E : V t f p x q “ λ t p x, f q ` ż M p E q ˝ p ´ e ´ ν p f q q L t p x, dν q , where λ t p x, dy q is a bounded kernel on E and p ^ ν p qq L t p x, dν q is a bounded kernel from E to M p E q ˝ .Let γ t “ ż t ż E η p dx q λ s p x, ¨q ds, H t “ ż E η p dx q L t p x, ¨q ` HQ ˝ t and G t “ ż t H s ds, t ě , where p Q ˝ t q t ě denotes the restriction of p Q t q t ě to M p E q ˝ . By Li [16, Theorems 9.5-9.6], p G t q t ě is anentrance rule for p Q ˝ t q t ě and has the decomposition G t “ ż t G st ´ s ζ p ds q , t ě , L Wang where ζ p ds q is a diffuse Radon measure on r , and tp G st q t ą : s ě u is a family of entrance laws for p Q ˝ t q t ě . Thus by Li [16, Theorem A.40], there corresponds a σ -finite measure Q s p dw q on p D , A q suchthat for every 0 ă t ă ¨ ¨ ¨ ă t n ă 8 , and nonzero µ , ¨ ¨ ¨ , µ n P M p E q , Q s p w t P dµ , ¨ ¨ ¨ , w t n P dµ n q “ G st p dµ q Q ˝ t ´ t p µ , dµ q ¨ ¨ ¨ Q ˝ t n ´ t n ´ p µ n ´ , dµ n q . Let F t “ σ p X s : s P r , t sq . Suppose that N p ds, dw q is a Poisson random measure on p , D independentof tp X t , F t q : t ě u with intensity ζ p ds q Q s p dw q , in a probability space p r Ω, r F , P q . Define another process t Λ t : t ě u by Λ t : “ γ t ` ż p ,t s ż D w t ´ s N p ds, dw q , t ě . The Markov property of p Λ t q t ě follows from a similar proof as that of Li [16, Theorem 9.29]. Moreover, P r exp t´ Λ t p f qus “ exp ´ γ t p f q ´ ż t ζ p ds q ż M p E q ˝ p ´ e ´x f,µ y q G st ´ s p dµ q + “ exp ´ ż t η p V s f q ds ´ ż t ż M p E q ˝ P µ r ´ exp t´x f, X t ´ s yus H p dµ q ds + “ exp ´ ż t η p V s f q ds ´ ż t ż M p E q ˝ p ´ exp t´x µ, V s f yuq H p dµ q ds + “ exp " ´ ż t ϕ p V s f q ds * . Let r F t be the σ -algebra generated by random variables t N H p A q : A P B pr , t sq ˆ A t u and Y t “ X t ` Λ t , X “ µ and H t “ σ p F t Y r F t q . Then tp Y t , H t q : t ě u is an superprocess with immigration whose transition semigroup is determinedby p Q Nt q t ě . That is, t Y , p H t q t ě , P µ u has the same law as t Y, p G t q t ě , P µ u .Notice that, for f P L p E, m q , N x px| f | , ω t yq “ T t | f |p x q ă 8 , which implies that N x px| f | , ω t y “ 8q “ f P L p E, m q , P µ ´ e iθ x f,Y t y ¯ “ exp "ż E ż D p e iθ x f,w t y ´ q N x p dw q µ p dx q * ` exp iθγ t p f q ` ż t ζ p ds q ż M p E q ˝ p exp t iθ x f, µ yu ´ q G st ´ s p dµ q + . Thus, by the Markov property of the process, we have P µ r exp t iθ x f, Y t ` s yu| Y t s“ exp "ż E ż D p e iθ x f,w s y ´ q N x p dw q Y t p dx q * ` exp iθγ s p f q ` ż t ` st ζ p du q ż M p E q ˝ p exp t iθ x f, µ yu ´ q G ut ` s ´ u p dµ q + “ exp "ż E ż D p e iθ x f,w s y ´ q N x p dw q Y t p dx q * ` exp s η p V u p iθf qq du ` ż s ż M p E q ˝ P µ p exp t iθ x f, X s ´ u yu ´ q H p dµ q du + . (2.3)The intuitive meaning of (2.3) is clear, given Y t , the population at time t ` s is made up of two parts;the native part generated by the mass Y t and the immigration in the time interval p t, t ` s s . LT for superprocesses with immigration 7 t Y t , t ě u For reader’s convenience, we first recall some results about the semigroup p T t q , the proofs of which canbe found in [20]. For two positive functions f and g on E , by f p x q À g p x q , we will denote the fact thatthere exists a constant c ą f p x q ď cg p x q for any x P E (whose exact value is not relevant tofollowing calculations). Lemma 2.1 [20, Lemma 2.4] For any f P L p E, m q , x P E and t ą , we have T t f p x q “ ÿ k “ γ p f q e ´ λ k t n k ÿ j “ a kj φ p k q j p x q (2.4) and lim t Ñ8 e λ γ p f q t T t f p x q “ n γ p f q ÿ j “ a γ p f q j φ p γ p f qq j p x q , (2.5) where the series in (2.4) converges absolutely and uniformly in any compact subset of E . Moreover, forany t ą , sup t ą t e λ γ p f q t | T t f p x q| ď e λ γ p f q t } f } ˆż E a t { p x q m p dx q ˙ p a t p x qq { , (2.6)sup t ą t e p λ γ p f q` ´ λ γ p f q q t | e λ γ p f q t T t f p x q ´ f p x q|ď e λ γ p f q` t } f } ˆż E a t { p x q m p dx q ˙ p a t p x qq { . Lemma 2.2 [16, Proposition 9.14] Suppose that ş M p E q ˝ ν p q H p dν q ă 8 . Then for t ě , µ P M p E q and f P B b p E q we have P µ x Y t , f y “ µ p T t f q ` ż t Γ p T s f q ds (2.7) and P µ x Y t , f y “ ˆ µ p T t f q ` ż t Γ p T s f q ds ˙ ` ż E ż t T s r A p T t ´ s f q sp x q dsµ p dx q` ż t ż E ż u T s r A p T u ´ s f q sp x q dsΓ p dx q du ` ż t ż M p E q ˝ ν p T s f q H p dν q ds, (2.8) where t Ñ T t f is defined by (1.4) and Γ p f q “ η p f q ` ş M p E q ˝ ν p f q H p dν q . Let m p t q “ ż t Γ p a { s q ds and n p t q “ ż t ż M p E q ˝ ” ν ´ a { s ¯ı H p dν q ds. Assumption 2.1
We assume throughout this paper that there exists t ą ż t Γ p a { s q ds ă 8 and ż t ż M p E q ˝ r ν p a { s qs H p dν q ds ă 8 . L Wang
Remark 2.1
It follows from Assumption 2.1 and the fact t Ñ a t is decreasing that, for any t ą Γ p a { t q ă 8 . Furthermore, for any t ě t , m p t q ă 8 , n p t q ă 8 . Since e ´ λ k t { | φ p k q j p x q| ď a t p x q { , we have | Γ p φ p k q j q| ď e λ k t { | Γ p a { t q| ă 8 for any k ě j “ , . . . , n k .For f P L p E, m q X L p E, m q , and x P E , it follows from H¨older’s inequality that p T t ´ s f q p x q ď e M p t ´ s q T t ´ s p f qp x q . (2.9)Thus, using a routine limit argument, one can easily check that (2.7) and (2.8) also hold for f P L p E, m qX L p E, m q under Assumption 2.1. Lemma 2.3
Under Assumption 2.1, for any f P L p E, m q X L p E, m q ,(1) If λ ă λ γ p f q , then for any x P E , lim t Ñ8 e λ t { P δ x x f, Y t y “ and lim t Ñ8 e λ t P δ x x f, Y t y “ ż e λ s x A p T s f q , φ y m dsφ p x q` ż ż l e λ l x A p T u ´ l f q , φ y m dudlΓ p φ q . (2.11) (2) If λ “ λ γ p f q , then for any p t, x q P p t ,
8q ˆ E , lim t Ñ8 t ´ e λ t V ar δ x x f, Y t y “ ˆ φ p x q ` Γ p φ q´ λ ˙ x Af , φ y m . (2.12) (3) If λ ą λ γ p f q , then for any x P E , lim t Ñ8 e λ γ p f q t V ar δ x x f, Y t y “ ż e λ γ p f q s T s p Af qp x q ds ` ż ż E ż l e λ γ p f q l T u ´ l r A p f q sp x q duΓ p dx q dl ` ż ż M p E q ˝ e λ γ p f q s ν p f qp x q H p dν q ds. (2.13) Proof. (1) If λ ă λ γ p f q , by the moment formula (2.7), e λ t { | P δ x x f, Y t y| ď e p λ ´ λ γ p f q q t { p e λ γ p f q t | T t f p x q|q ` e λ t { ż t | Γ p T s f q| ds. Using H¨older’s inequality, we can get, | T s f p x q| ď ż E q s p x, y q| f |p y q m p dy q ď } f } ˆż E q s p x, y q m p dy q ˙ { “ } f } a s p x q { . (2.14) LT for superprocesses with immigration 9
Then for any t ą t ą
0, we have by (2.6), e λ t { ż t Γ p T s f q ds “ e λ t { ż t Γ p T s f q ds ` e λ t { ż tt Γ p T s f q ds ď e λ t { } f } ż t Γ p a { s q ds ` ż tt e λ p t ´ s q{ e p λ ´ λ γ p f q q s { Γ p e λ γ p f q s T s f q ds À e λ t { } f } m p t q ` r e p λ ´ λ γ p f q q t { ` e λ t { s Γ p a { t q . (2.15)Combining the estimates above, we get lim t Ñ8 e λ t { P δ x x f, Y t y “ . For the second moment, we have already known from the proof of Lemma 2.3 in [20] thatlim t Ñ8 e λ t ż t T s r A p T t ´ s f q sp x q ds “ ż e λ s x A p T s f q , φ y m dsφ p x q and for s ą t , e λ s ż s T u r A p T s ´ u f q sp x q du À a t p x q { . (2.16)For any s ď t , using (2.9) and (2.14), we get ż E ż s T u r A p T s ´ u f q sp x q duΓ p dx q ď ż E ż s M e M p s ´ u q T s p f qp x q duΓ p dx qď e Mt } f } Γ p a { s q . Combining with (2.16), we get e λ t ż t ż E ż s T u r A p T s ´ u f q sp x q duΓ p dx q ds “ ż t t e λ p t ´ s q ż E e λ s ż s T u r A p T s ´ u f q sp x q duΓ p dx q ds ` e λ t ż t ż E ż s T u r A p T s ´ u f q sp x q duΓ p dx q ds À Γ p a { t q ` e λ t e Mt } f } m p t q . (2.17)Thus, by the dominated convergence theorem and let l “ t ´ s , we have, e λ t ż t ż E ż s T s ´ u r A p T u f q sp x q duΓ p dx q ds “ ż t e λ p t ´ s q ż E e λ s ż s T s ´ u r A p T u f q sp x q duΓ p dx q ds “ ż t e λ l ż E e λ p t ´ l q ż t ´ l T t ´ l ´ u r A p T u f q sp x q duΓ p dx q dl “ ż t ż E e λ l e λ p t ´ l q ż tl T t ´ u r A p T u ´ l f q sp x q duΓ p dx q dl Ñ ż ż l e λ l x A p T u ´ l f q , φ y m Γ p φ q dudl, as t Ñ 8 . For the last term in (2.8), for t ą t , we have by (2.14), e λ t ż t ż M p E q ˝ ν p T s f q H p dν q ds “ ż tt e λ p t ´ s q ż M p E q ˝ e p λ ´ λ γ p f q q s ν p e λ γ p f q s T s f q H p dν q ds ` e λ t ż t ż M p E q ˝ ν p T s f q H p dν q ds À p e p λ ´ λ γ p f q q t ` e λ t q ż M p E q ˝ ν ´ a { t ¯ H p dν q ` e λ t } f } n p t q , (2.18)so we have lim t Ñ8 e λ t ż t ż M p E q ˝ ν p T s f q H p dν q ds “ . (2) If λ “ λ γ p f q , t ´ e λ t V ar δ x x f, Y t y “ t ´ e λ t ż t T s r A p T t ´ s f q sp x q ds ` t ´ e λ t ż t ż E ż u T s r A p T u ´ s f q sp x q dsΓ p dx q du ` t ´ e λ t ż t ż M p E q ˝ ν p T s f q H p dν q ds : “ A p t, x q ` A p t q ` A p t q . We have already known from the proof of Lemma 2.3 in [20] that for t ą t , | A p t, x q ´ x Af , φ y m φ p x q| À t ´ a t p x q { . Thus, by similar estimate to (2.17), we have A p t q “ ż t t ´ e λ p t ´ s q ż E e λ s ż s T u r A p T s ´ u f q sp x q duΓ p dx q ds “ ż t t st e λ p t ´ s q s ´ e λ s ż E ż s T u r A p T s ´ u f q sp x q duΓ p dx q ds ` t ´ e λ t ż t ż E ż s T u r A p T s ´ u f q sp x q duΓ p dx q ds À t ´ Γ p a { t q ` x Af , φ y m Γ p φ q ` t ´ e λ t } f } m p t q . Let l “ t ´ s , by the dominated convergence theorem, we have as t Ñ 8 , A p t q “ ż t e λ l t ´ lt ż E p t ´ l q ´ e λ p t ´ l q ż tl T t ´ u r A p T u ´ l f q sp x q duΓ p dx q dl Ñ ż e λ l Γ p φ qx Af , φ y m dl and similarly to (2.18), we can prove that A p t q Ñ t Ñ 8 . (3) If λ ą λ γ p f q , we have already known from the proof of Lemma 2.3 in [20] thatlim t Ñ8 e λ γ p f q t ż t T s r A p T t ´ s f q sp x q ds “ ż e λ γ p f q s T s p Af qp x q ds. LT for superprocesses with immigration 11
For the second term in (2.8), by (2.37) in [20], for t ą t , e λ γ p f q t ż t ż E ż s T u r A p T s ´ u f q sp x q duΓ p dx q ds “ ż t t e λ γ p f q p t ´ s q ż E e λ γ p f q s ż s T u r A p T s ´ u f q sp x q duΓ p dx q ds ` ż t e λ γ p f q t ż E ż s T u r A p T s ´ u f q sp x q duΓ p dx q ds À Γ p a { t q ` e λ γ p f q t e Mt } f } m p t q . (2.19)Consequently, we can use the dominated convergence theorem to get e λ γ p f q t ż t ż E ż s T u r A p T s ´ u f q sp x q duΓ p dx q ds “ ż t ż E e λ γ p f q l e λ γ p f qp t ´ l q ż tl T u ´ l r A p T t ´ u f q sp x q duΓ p dx q dl Ñ ż ż E ż l e λ γ p f q l T u ´ l r A p f q sp x q duΓ p dx q dl. For the last term in (2.8), e λ γ p f q t ż t ż M p E q ˝ ν p T s f q H p dν q ds “ ż tt e λ γ p f q p t ´ s q ż M p E q ˝ ν p e λ γ p f q s T s f q H p dν q ds ` e λ γ p f q t ż t ż M p E q ˝ ν p T s f q H p dν q ds À ż M p E q ˝ ν ´ a { t ¯ H p dν q ` e λ γ p f q t } f } n p t q . (2.20)Then it follows by the dominated convergence theorem that as t Ñ 8 , e λ γ p f q t ż t ż M p E q ˝ ν p T s f q H p dν q ds “ ż t e λ γ p f q p t ´ s q ż M p E q ˝ ν p e λ γ p f q s T s f q H p dν q ds Ñ ż ż M p E q ˝ e λ γ p f q s ν p f q H p dν q ds. Combining the above, we getlim t Ñ8 e λ γ p f q t V ar δ x x f, Y t y “ ż e λ γ p f q s T s p Af qp x q ds ` ż ż E ż l e λ γ p f q l T u ´ l r A p f q sp x q duΓ p dx q dl ` ż ż M p E q ˝ e λ γ p f q s ν p f q H p dν q ds. Thus the proof of (3) is now complete. l t Y t , t ě u Define W k,jt : “ e λ k t x φ p k q j , Y t y ,H k,jt : “ e λ k t x φ p k q j , Y t y ´ λ ´ k p e λ k t ´ q Γ p φ p k q j q , t ě . Lemma 3.1 t H k,jt : t ě u is a martingale under P µ . Moreover, if λ ą λ k , then sup t ą t P µ p H k,jt q ă8 . Thus the limit H k,j : “ lim t Ñ8 H k,jt exists P µ -a.s. and in L p P µ q .Proof. By the moment formula and the fact that T t φ p k q j p x q “ e ´ λ k t φ p k q j p x q , P µ r H k,jt ` s | G t s “ e λ k t x φ p k q j , Y t y ` e λ k p t ` s q ż t ` st Γ p T t ` s ´ u φ p k q j q du ´ λ ´ k p e λ k p t ` s q ´ q Γ p φ p k q j q“ e λ k t x φ p k q j , Y t y ´ λ ´ k p e λ k t ´ q Γ p φ p k q j q“ H k,jt . By (2.19) and (2.20), we have thatsup t ą t P µ r H k,jt s À sup t ą t e λ k t V ar µ x φ p k q j , Y t y ` µ p φ p k q j q À x a { t , µ y ` Γ p a { t q ` ż M p E q ˝ ν ´ a { t ¯ H p dν q ` m p t q ` n p t q ` µ p φ p k q j q from which the convergence asserted in the lemma follows easily. l Remark 3.1
We know from Remark 2.1 that Γ p φ p k q j q ă 8 . If λ ą λ k , then W k,j : “ lim t Ñ8 e λ k t x φ p k q j , Y t y “ H k,j ´ λ ´ k p e λ k t ´ q Γ p φ p k q j q “ H k,j ` λ ´ k Γ p φ p k q j q exists P µ -a.s. and in L p P µ q for any k ě j “ , . . . , n k . Theorem 3.1 If f P L p E, m q X L p E, m q with λ ą λ γ p f q and Γ p φ p γ p f qq j q ă 8 , then as t Ñ 8 , e λ γ p f q x f, Y t y Ñ n γ p f q ÿ j “ a γ p f q j W γ p f q ,j , in L p P µ q . Proof.
By using the moment estimates of Y , the proof is similar to that of [20, Theorem 1.6], we omitthe details here. l Remark 3.2
When γ p f q “
1, let Ă W be short for W , . Therefore as t Ñ 8 , e λ t x f, Y t y Ñ x f, φ y m Ă W , in L p P µ q . In particular, the convergence also holds in P µ -probability. LT for superprocesses with immigration 13 t Y t , t ě u For f P C s and h P C c , we define σ f : “ ż e λ s x A p T s f q , φ y m ds and ρ h : “ x Ah , φ y m . For g p x q “ ř k :2 λ k ă λ ř n k j “ a kj φ p k q j p x q P C l , we define I s g p x q “ ÿ k :2 λ k ă λ n k ÿ j “ e λ k s a kj φ p k q j p x q and β g : “ ż e ´ λ s @ A p I s g q , φ D m ds. Theorem 3.2 If f P C s , h P C c and g p x q “ ř k :2 λ k ă λ ř n k j “ a kj φ p k q j p x q P C l , then σ f ă 8 , ρ h ă 8 and β g ă 8 . Furthermore, it holds that, as t Ñ 8 , ˜ e λ t x φ , Y t y , x g, Y t y ´ ř k :2 λ k ă λ ř n k j “ e ´ λ k t a kj W k,j a x φ , Y t y , x h, Y t y a t x φ , Y t y , x f, Y t y a x φ , Y t y ¸ d Ñ p Ă W , G p g q , G p h q , G p f qq , where G p g q „ N p , β g q , G p h q „ N p , ρ h q and G p f q „ N p , σ f q . Moreover, Ă W , G p g q , G p h q and G p f q are independent. Remark 3.3
In [21], Corollaries 1.5, 1.6 and 1.7 are excellent complementary of Theorem 1.4. They allhold in our situation only with X t replaced by Y t . We will not restate them here.3.3 Proof of the central limit theorem of Y The general methodology is similar to that of [20] and [21]. We recall some facts about weak convergencewhich will be used later. For f : R n Ñ R , let } f } L : “ sup x ‰ y | f p x q ´ f p y q|{} x ´ y } and } f } BL : “} f } ` } f } L . For any distributions ν and ν on R n , define d p ν , ν q : “ sup "ˇˇˇˇż f dν ´ ż f dν ˇˇˇˇ : } f } BL ď * . Then d is a metric. It follows from [9, Theorem 11.3.3] that the topology generated by d is equivalent to theweak convergence topology. From the definition, we can easily see that, if ν and ν are the distributionsof two R n -valued random variables X and Y respectively, then d p ν , ν q ď E } X ´ Y } ď a E } X ´ Y } . Before the proof of Theorem 3.2, we prove several lemmas first. The first lemma below was proved in[21], we state it here for reader’s convenience. Recall the excursion measure N x defined by (2.2) on theprobability space p D , A q , define r H k,jt p w q : “ e λ k t x φ p k q j , w t y , t ě , w P D . Lemma 3.2 ([21, Lemma 3.1]) If λ ą λ k , then the limit r H k,j : “ lim t Ñ8 r H k,jt exists N x -a.e., in L p N x q and in L p N x q . Lemma 3.3 If f P C s , then σ f ă 8 and for any nonzero µ P M p E q , it holds under P µ that ´ e λ t x φ , Y t y , e λ t { x f, Y t y ¯ Ñ ˆĂ W , G p f q bĂ W ˙ , t Ñ 8 , where G p f q „ N p , σ f q . Moreover, Ă W and G p f q are independent.Proof. We need to consider the limit of the R -valued random variable defined by U p t q : “ ´ e λ t x φ , Y t y , e λ t { x f, Y t y ¯ , or equivalently, we need to consider the limit of U p s ` t q as t Ñ 8 for any s ą
0. For s, t ą t , U p s ` t q “ ´ e λ p t ` s q x φ , Y t ` s y , e λ p t ` s q{ x f, Y t ` s y ´ e λ p t ` s q{ x T s f, Y t y ¯ ` ´ , e λ p t ` s q{ x T s f, Y t y ¯ . We will prove that the second term on the right hand has no contribution to the double limit, first as t Ñ 8 and then s Ñ 8 . The double limit of the first term is equal to another R -valued random variable U p s, t q where U p s, t q : “ ´ e λ t x φ , Y t y , e λ p t ` s q{ x f, Y t ` s y ´ e λ p t ` s q{ x T s f, Y t y ¯ . We claim that U p s, t q d Ñ p Ă W , bĂ W G p s qq , as t Ñ 8 , where G p s q „ N p , σ f p s qq with σ f p s q to be given later. Denote the characteristic function of U p s, t q under P µ by κ p θ , θ , s, t q : κ p θ , θ , s, t q “ P µ ´ exp ! iθ e λ t x φ , Y t y ` iθ e λ p t ` s q{ x f, Y t ` s y ´ iθ e λ p t ` s q{ x T s f, Y t y )¯ “ P µ ` exp iθ e λ t x φ , Y t y` ż E ż D ´ exp t iθ e λ p t ` s q{ x f, w s yu ´ ´ iθ e λ p t ` s q{ x f, w s y ¯ N x p dw q Y t p dx q` ż s ż M p E q ˝ P µ ´ exp t iθ e λ p t ` s q{ x f, X s ´ u yu ´ ¯ H p dµ q du ` ż s η p V u p iθ e λ p t ` s q{ f qq du *˙ . (3.1)Let P µ ˆ exp "ż E ż D ´ exp t iθ e λ p t ` s q{ x f, w s yu ´ ´ iθ e λ p t ` s q{ x f, w s y ¯ N x p dw q Y t p dx q *˙ “ P µ ˆ exp " ´ θ e λ t x V s , Y t y ` x R s p e λ p t ` s q{ θ , ¨q , Y t y *˙ , where V s p x q “ e λ s V ar δ x x f, X s y and R s p θ, x q “ ż D ˆ exp x iθf, w s y ´ ´ iθ x f, w s y ` θ x f, w s y ˙ N x p dw q . By Remark 3.2 and the fact that V s p x q À a t p x q { P L p E, m q X L p E, m q , we havelim t Ñ8 e λ t x V s , Y t y “ x V s , φ y m Ă W , in probability . LT for superprocesses with immigration 15
Let Z s : “ e λ s { x f, w s y and h p x, s, t q : “ N x ˆ Z s ˆ θ e λ t { Z s ^ ˙˙ . We know from (2.16) in [21] that h p x, s, t q Ó t Ò 8 and for t ą t , h p x, s, t q ď N x p Z s q “ e λ s V ar δ x px f, X s yq À a t p x q { P L p E, m q . By (3.8) in [21], | R s p e λ p t ` s q{ θ , x q| ď θ e λ t h p x, s, t q ď θ e λ t a t p x q { P L p E, m q . We claim that, as t Ñ 8 , P µ |x R s p e λ p t ` s q{ θ , ¨q , Y t y| ď θ e λ t „ T t p h p¨ , s, t qq ` ż t Γ p T v h q dv Ñ . (3.2)For the first term on the right hand side of (3.2). By the results in [21], for any u ă t ,lim sup t Ñ8 e λ t r T t p h p¨ , s, t qqs ď lim sup t Ñ8 e λ t r T t p h p¨ , s, u qqs “ x h p¨ , s, u q , φ y m φ p x q . Letting u Ñ 8 , we get lim t Ñ8 e λ t r T t p h p¨ , s, t qqs “
0. For the second term, we havelim sup t Ñ8 e λ t ż t Γ p T v p h p¨ , s, t qqq dv ď lim sup t Ñ8 ż t e λ p t ´ v q Γ p e λ v T v p h p¨ , s, u qqq dv ď ż e λ v x h p¨ , s, u q , φ y m dvΓ p φ qÀ Γ p φ qxp a t q { , φ y m . By the dominated convergence theorem,lim t Ñ8 e λ t ż t Γ p T v p h p¨ , s, t qqq dv ď lim u Ñ8 lim t Ñ8 ż t e λ p t ´ v q Γ p e λ v T v p h p¨ , s, u qqq dv “ . For the last two terms on the right hand of (3.1), we know from (1.5) that V t | f |p x q ď T t | f |p x q for all t ě x P E. Thus by (2.15), we get ˇˇˇˇˇ exp s η p V u p iθ e λ p t ` s q{ f qq du ` ż M p E q ˝ ż s P µ ´ exp t iθ e λ p t ` s q{ x f, X u yu ´ ¯ duH p dµ q +ˇˇˇˇˇ À exp θ e λ p t ` s q{ ż s η p T u | f |qq du ` θ e λ p t ` s q{ ż M p E q ˝ ż s P µ rx| f | , X u ys duH p dµ q + À exp ! θ e λ t { ´ e λ s { } f } m p t q ` r e p λ ´ λ γ p f q q s { ` e λ s { s Γ p a { t q ¯) Ñ , as t Ñ 8 . (3.3)Hence by the dominated convergence theorem, we getlim t Ñ8 κ p θ , θ , s, t q “ P µ ˆ exp t iθ Ă W u exp " ´ θ x V s , φ y m Ă W *˙ . Since e λ p t ` s q x φ , Y t ` s y ´ e λ t x φ , Y t y Ñ t Ñ 8 , we easily get that under P µ , U p s, t q : “ ´ e λ p t ` s q x φ , Y t ` s y , e λ p t ` s q{ x f, Y t ` s y ´ e λ p t ` s q{ x T s f, Y t y ¯ d Ñ p Ă W , bĂ W G p s qq , as t Ñ 8 . By (2.15) in [21], lim s Ñ8 V s p x q “ lim s Ñ8 e λ s V ar δ x x f, X s y “ σ f φ p x q . Thus lim s Ñ8 σ f p s q “ σ f . So lim s Ñ8 d p G p s q , G p f qq “ σ f .Let D p s ` t q and r D p s, t q be the distribution of U p s ` t q and U p s, t q respectively, and let D p s q and D bethe distributions of p Ă W , bĂ W G p s qq and p Ă W , bĂ W G p f qq respectively. Then, by a similar argumentas in [20] and using the definition of lim sup t Ñ8 ,lim sup t Ñ8 d p D p t q , D q ď lim sup t Ñ8 d p D p s ` t q , D qď lim sup t Ñ8 d p D p s ` t q , r D p s, t qq ` d p r D p s, t q , D p s qq ` d p D p s q , D qď lim sup t Ñ8 p P µ p e λ p t ` s q{ x T s f, Y t yq q { ` ` d p D p s q , D q . Letting s Ñ 8 , we getlim sup t Ñ8 d p D p t q , D q ď lim sup s Ñ8 lim sup t Ñ8 p P µ p e λ p t ` s q{ x T s f, Y t yq q { . Therefore, we are left to prove thatlim sup s Ñ8 lim sup t Ñ8 e λ p t ` s q P µ px T s f, Y t yq “ . (3.4)By (2.8), we have that P µ px T s f, Y t yq “ ˆ µ p T s ` t f q ` ż t Γ p T u ` s f q du ˙ ` ż E ż t T t ´ u r A p T u ` s f q sp x q duµ p dx q` ż t ż E ż v T v ´ u r A p T u ` s f q sp x q duΓ p dx q dv ` ż t ż M p E q ˝ ν p T u ` s f q H p dν q du : “ B ` B ` B ` B . By (2.10), we getlim t Ñ8 e λ p t ` s q B “ lim t Ñ8 ˆ e λ p t ` s q{ ˆ µ p T s ` t f q ` ż t Γ p T u ` s f q du ˙˙ “ t Ñ8 e λ p t ` s q B “ . For B , by a similar proof as of Lemma 3.2 in [21], we get e λ p t ` s q ż t ż E ż v T v ´ u r A p T u ` s f q sp x q duΓ p dx q dv “ ż t e λ p t ´ v q e λ p v ` s q ż E ż v T v ´ u r A p T u ` s f q sp x q duΓ p dx q dv À ż t e λ p t ´ v q e p λ ´ λ γ p f q q s dvΓ p a { t q . LT for superprocesses with immigration 17
As for the last term B , we have by (2.6) that, for s ą t , e λ p t ` s q ż t ż M p E q ˝ ν p T u ` s f q H p dν q du ď e p λ ´ λ γ p f q q s ż t e p λ ´ λ γ p f q q u e λ p t ´ u q ż M p E q ˝ ν p e λ γ p f q p s ` u q T s ` u f q duH p dν qÀ e p λ ´ λ γ p f q q s ż M p E q ˝ ν p a { t q H p dν q . Combining the above estimates, we will get the desired result by first letting t Ñ 8 and then letting s Ñ 8 . l Lemma 3.4 If f P C s and h P C c . Then ´ e λ t x φ , Y t y , t ´ { e λ t { x h, Y t y , e λ t { x f, Y t y ¯ d Ñ ˆĂ W , G p h q bĂ W , G p f q bĂ W ˙ , where G p f q „ N p , σ f q and G p h q „ N p , ρ h q . Moreover, Ă W , G p f q and G p h q are independent.Proof. We will use the idea suggested in [21] with some modifications to prove the result. In the proof,we always assume t ą t . We define an R -valued random variable by U p t q : “ ´ e λ t x φ , Y t y , t ´ { e λ t { x h, Y t y , e λ t { x f, Y t y ¯ . Let n ą U p nt q “ ´ e λ nt x φ , Y nt y , p nt q ´ { e λ nt { x h, Y nt y , e λ nt { x f, Y nt y ¯ . To consider the limit of U p t q as t Ñ 8 , it is equivalent to consider the limit of U p nt q for any n ą t ą t , n ą U p nt q “ ˜ e λ nt x φ , Y nt y , e λ nt { px h, Y nt y ´ x T p n ´ q t h, Y t yqp nt q { ,e λ nt { px f, Y nt y ´ x T p n ´ q t f, Y t yq ¯ ` ´ , p nt q ´ { e λ nt { x T p n ´ q t h, Y t y , e λ nt { x T p n ´ q t f, Y t y ¯ . (3.5)We will prove that the second term on the right hand has no contribution to the double limit, first as t Ñ 8 and then n Ñ 8 . The double limit of the first term is equal to another R -valued random variable U p n, t q where U p n, t q : “ ˜ e λ t x φ , Y t y , e λ nt { px h, Y nt y ´ x T p n ´ q t h, Y t yqpp n ´ q t q { ,e λ nt { px f, Y nt y ´ x T p n ´ q t f, Y t yq ¯ . We claim that U p n, t q d Ñ p Ă W , bĂ W G p h q , bĂ W G p f qq , as t Ñ 8 . Denote the characteristic function of U p n, t q under P µ by κ p θ , θ , θ , n, t q . Define Z p t, θ q : “ θ t ´ { e λ t { x h, X t y , Z p t, θ q : “ θ e λ t { x f, X t y , t ą , and Z t p θ , θ q : “ Z p t, θ q ` Z p t, θ q . We define the corresponding random variables on D by r Z p t, θ q , r Z p t, θ q and r Z t p θ , θ q . Using an argu-ment similar to that leading to (3.1), we get κ p θ , θ , θ , n, t q “ P µ ˆ exp " iθ e λ t x φ , Y t y ` ż E ż D ´ exp t ie λ t { r Z p n ´ q t p θ , θ qp w qu´ ´ ie λ t { r Z p n ´ q t p θ , θ qp w q ¯ N x p dw q Y t p dx q` ż ntt ż M p E q ˝ P µ ´ exp t ie λ u { Z nt ´ u p θ , θ qu ´ ¯ H p dµ q du ` ż p n ´ q t η p V u p iθ rp n ´ q t s ´ { e λ nt { h ` iθ e λ nt { f qq du +¸ , (3.6)where Z nt ´ u p θ , θ qp w q “ p nt ´ u p n ´ q t q { Z p nt ´ u, θ qp w q ` Z p nt ´ u, θ qp w q . Define R t p θ, x q : “ ż D ´ exp t iθ r Z t p θ , θ qp w qu ´ ´ iθ r Z t p θ , θ qp w q` θ p r Z t p θ , θ qp w qq ˙ N x p dw q and J p n, t, x q : “ ż E ż D ´ exp t ie λ t { r Z p n ´ q t p θ , θ qp w qu´ ´ ie λ t { r Z p n ´ q t p θ , θ qp w q ¯ N x p dw q Y t p dx q . Then J p n, t, x q “ ´ e λ t N x p r Z p n ´ q t p θ , θ qq ` R n ´ q t p e λ t { , x q and κ p θ , θ , θ , n, t q “ P µ p exp t iθ e λ t x φ , Y t y ` x J p n, t, ¨q , Y t yuq . Let V nt p x q : “ N x p r Z p n ´ q t p θ , θ qq . Then x J p n, t, ¨q , Y t y “ ´ e λ t x V nt , Y t y ` x R n ´ q t p e λ t , ¨q , Y t y : “ J p n, t q ` J p n, t q . We first consider J p n, t q . It follows from the estimate (3.26) in [21] thatlim t Ñ8 J p n, t q “ lim t Ñ8 ´ e λ t p θ ρ h ` θ σ f qx φ , Y t y “ ´ p θ ρ h ` θ σ f q Ă W . For J p n, t q , by an similar argument as in (3.2), we have J p n, t q Ñ t Ñ 8 . We can use the same method as in the proof of (3.3) that the last two terms on the right hand side of(3.6) tend to 1 as t goes to infinity. Hence, combining the above calculations, we get thatlim t Ñ8 κ p θ , θ , θ , n, t q “ P µ „ exp t iθ Ă W u exp " ´ p θ ρ h ` θ σ f q Ă W * . LT for superprocesses with immigration 19
Similarly as in the proof of Lemma 3.3, in order to get the desired result, we now need only to show thatlim n Ñ8 lim sup t Ñ8 p nt q ´ e λ nt P µ px T p n ´ q t h, Y t yq “ , lim n Ñ8 lim sup t Ñ8 e λ nt P µ px T p n ´ q t f, Y t yq “ . In fact, using (2.12) and the fact that x T t h, µ y “ e ´ λ t { x h, µ y , we have p nt q ´ e λ nt P µ px T p n ´ q t h, Y t yq “ p nt q ´ e λ t V ar µ x h, Y t y ` p nt q ´ e λ t p P µ x h, Y t yq À n ´ p ` t ´ q . Using the same method as the proof of (3.4) with s “ p n ´ q t , and then letting t Ñ 8 , we get e λ nt P µ px T p n ´ q t f, Y t yq Ñ . The proof is now complete. l Recall that g p x q “ ÿ k :2 λ k ă λ n k ÿ j “ a kj φ p k q j p x q and I u g p x q “ ÿ k :2 λ k ă λ n k ÿ j “ e λ k u a kj φ p k q j p x q . Note that the sum over k is a sum over a finite number of elements. Define H p w q : “ ÿ k :2 λ k ă λ n k ÿ j “ a kj r H k,j p w q , w P D . By Lemma 3.2, we have, as u Ñ 8x I u g, w u y Ñ H , N x -a.e., in L p N x q and in L p N x q . Since N x x I u g, w u y “ P δ x x I u g, X u y “ g p x q , we get N x p H q “ g p x q and by (3.37) and (3.38) in [21], N x p H q “ ż T s »– A ˜ ÿ k :2 λ k ă λ n k ÿ j “ e λ k s a kj φ p k q j ¸ fifl p x q ds À p a t p x qq { P L p E, m q X L p E, m q . (3.7) Proof of Theorem 3.2.
Consider an R -valued random variable U p t q defined by: U p t q : “ ˜ e λ t x φ , Y t y , e λ t { ˜ x g, Y t y ´ ÿ k :2 λ k ă λ n k ÿ j “ e ´ λ k t a kj W k,j ¸ ,t ´ { e λ t { x h, Y t y , e λ t { x f, Y t y ¯ . To get the conclusion of Theorem 3.2, it suffice to show that, under P µ , U p t q d Ñ ˆĂ W , bĂ W G p g q , bĂ W G p h q , bĂ W G p f q ˙ , where Ă W , G p g q , G p h q and G p f q are independent. Note that, by Lemma 3.1,lim u Ñ8 x I u g, Y t ` u y “ ÿ k :2 λ k ă λ n k ÿ j “ a kj W k,j p w q , P µ -a.s.Denote the characteristic function of U p t q under P µ by κ p θ , θ , θ , θ , t q . Then we have κ p θ , θ , θ , θ , t q “ lim u Ñ8 P µ ´ exp ! iθ e λ t x φ , Y t y ` iθ e λ t { px g, Y t y ´ x I u g, Y t ` u yq` iθ t ´ { e λ t { x h, Y t y ` iθ e λ t { x f, Y t y )¯ “ lim u Ñ8 P µ ´ exp ! iθ e λ t x φ , Y t y ` iθ t ´ { e λ t { x h, Y t y` iθ e λ t { x f, Y t y ` x J u p t, ¨q , Y t y` ż u ż M p E q ˝ Q µ ´ exp t´ iθ e λ t { x I u g, X u ´ s yu ´ ¯ dsH p dµ q` ż u η p V s p´ iθ e λ t { I u g qq ds *˙ , (3.8)where J u p t, x q “ ż D ´ exp t´ iθ e λ t { x I u g, w u yu ´ ` iθ e λ t { x I u g, w u y ¯ N x p dw q . By (3.42) in [21], we havelim u Ñ8 J u p t, x q “ N x ´ exp t´ iθ e λ t { H u ´ ` iθ e λ t { H ¯ : “ J p t, x q and lim u Ñ8 x J u p t q , Y t y “ x J p t, ¨q , Y t y , P µ -a.s.For the last two terms on the right hand side of (3.8), we have ˇˇˇˇˇ exp u η p V s p´ iθ e λ t { I u g qq ds ` ż M p E q ˝ ż u P µ ´ exp t´ iθ e λ t { x I u g, X s yu ´ ¯ dsH p dµ q +ˇˇˇˇˇ À exp θ e λ t { ż u η p T s | I u g |qq ds ` θ e λ t { ż M p E q ˝ ż u P µ rx| I u g | , X s ys dsH p dµ q + “ exp " θ e λ t { ż u Γ p T s | I u g |q ds * ď exp θ e λ t { ÿ k :2 λ k ă λ n k ÿ j “ | a kj | e λ k u ż u Γ p T s | φ p k q j |q ds + Ñ u Ñ 8 , then letting t Ñ 8 and using (2.15). Consequently, by the dominated convergencetheorem, we obtain κ p θ , θ , θ , θ , t q “ P µ ´ exp ! iθ e λ t x φ , Y t y ` iθ t ´ { e λ t { x h, Y t y` iθ e λ t { x f, Y t y ` x J p t, ¨q , Y t y )¯ . LT for superprocesses with immigration 21
Let R s p θ, x q “ N x ˆ exp t iθH u ´ ´ iθH ` θ H ˙ . Then, x J p t q , Y t y “ ´ θ e λ t x V, Y t y ` x R p´ e λ t { θ , ¨q , Y t y , where V p x q : “ N x p H q . By the results in [21] and similar arguments as (3.2), we havelim t Ñ8 x R p´ e λ t { θ , ¨q , Y t y “ V P L p E, m q X L p E, m q , we have by Remark 3.2 thatlim t Ñ8 e λ t x V, Y t y “ x V, φ y m Ă W in probability. (3.10)Therefore, combing (3.9) and (3.10), we getlim t Ñ8 exp tx J p t, ¨q , Y t yu “ exp " ´ θ x V, φ y m Ă W * in probability. (3.11)Recall that lim t Ñ8 e λ t x φ , Y t y “ Ă W , P µ -a.s. Thus by (3.11) and the fact that | exp tx J p t, ¨q , Y t yu| ď t Ñ 8 , ˇˇˇˇ P µ ˆ exp "ˆ iθ ´ θ x V, φ y m ˙ e λ t x φ , Y t y ` iθ t ´ { e λ t { x h, Y t y` iθ e λ t { x f, Y t y )¯ ´ κ p θ , θ , θ , θ , t q ˇˇˇ ď ˇˇˇˇ P µ exp tx J p t, ¨q , Y t yu ´ exp " ´ θ x V, φ y m e λ t x φ , Y t y *ˇˇˇˇ Ñ . Consequently, by Lemma 3.4, we havelim t Ñ8 P µ ˆ exp "ˆ iθ ´ θ x V, φ y m ˙ e λ t x φ , Y t y ` iθ t ´ { e λ t { x h, Y t y` iθ e λ t { x f, Y t y )¯ “ P µ „ exp t iθ Ă W u exp " ´ p θ x V, φ y m ` θ ρ h ` θ σ f q Ă W * . By (3.7), we get x V, φ y m “ ż e λ s x A p I s g q , φ y m ds. The proof is now complete. l Acknowledgements
The parts of this paper were written while the author visited Concordia. The author would like togive sincere thanks to Professor Xiaowen Zhou for his encouragement and helpful discussions and hospitality at Concordia.The author also thank an anonymous referee for useful comments.2 L Wang
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