Functional central limit theorem for super α-stable processes

Transcription

1 874 Science in China Ser. A Mathematics 24 Vol. 47 No Functional central it theorem for super α-stable processes HONG Wenming Department of Mathematics, Beijing Normal University, Beijing 1875, China ( wmhong@bnu.edu.cn) Received November 17, 23; revised February 9, 24 Abstract A functional central it theorem is proved for the centered occupation time process of the super α-stable processes in the finite dimensional distribution sense. For the intermediate dimensions α < d < 2α ( < α 2), the iting process is a Gaussian process, whose covariance is specified; for the critical dimension d = 2α and higher dimensions d > 2α, the iting process is Brownian motion. Keywor: super α-stable processes, occupation time, central it theorem, evolution equation. DOI: 1.136/3ys339 1 Introduction and statement of results he asymptotic behavior of superprocesses has been investigated over the past years [1 6]. A general central it theorem was proved by Iscoe [5] for the (α, d, β)- superprocess in higher dimensions d > α(1 + β), whereas for the intermediate and critical dimensions, the CL was only obtained in the situation that the underlying motion is Brownian motion and the branching is of finite variance, i.e. α = 2, β = 1. Hong et al. [2 4] have considered the iting behavior of the super-brownian motion with super-brownian immigration, and some new and interesting phenomena were revealed for this new model. Recently, Dawson et al. [7] considered the occupation time fluctuations of branching particle system in 3 levels, and complete results were obtained for the situation of finite variance branching and α-stable motion. In this paper, we will focus on the path-valued it behavior, i.e. the functional central it theorem for the occupation time process of super α-stable processes with finite variance branching (β = 1). For the intermediate dimensions α < d < 2α ( < α 2), the norming is ( 3 2 d 2α ) and the iting process is a non-brownian Gaussian process, whose covariance is given explicitly; for the critical dimension d = 2α and higher dimensions d > 2α, the iting process (in the finite dimensional distribution sense) is Brownian motion when the norming is ( log ) and 2 respectively. Similar results appeared for the zero-range process [8] and branching random walk [9]. For the super- Brownian motion, Iscoe [5] has got a result for d = 3, and proved a functional ergodic theorem in ref. [6]; recently, Zhang 1) has considered the path-valued it behavior for 1) Zhang Mei, Functional central it theorem for the super-brownian motion with super-brownian immigration, J. heoret. Probab., to appear. Copyright by Science in China Press 24

2 Functional central it theorem for super α-stable processes 875 the super-brownian motion with super-brownian immigration. We will recall the concept of the superprocess briefly, for the general background we refer the readers to Dawson [1]. Let C(R d ) denote the space of bounded continuous functions on R d. We fix a constant p > d and let φ p (x) := (1+ x 2 ) p/2 for x R d. Let C p (R d ) := {f C(R d ) : f(x) const φ p (x)}. In duality, let M p (R d ) be the space of Radon measures µ on R d such that µ, f := f(x)µ(dx) < for all f C p (R d ). We endow M p (R d ) with the p-vague topology, that is, µ k µ if and only if µ k, f µ, f for all f C p (R d ). hen M p (R d ) is metrizable. hroughout this paper, λ denotes the Lebesgue measure on R d. Suppose that ξ = (ξ t, t ) is an α-stable process ( < α 2) in R d with semigroup (Pt α ) t. Its infinitesimal generator is a fractional power of the Laplacian, α = ( ) α/2. Let p α (t, x, y) be the transition density function of the α-stable process, it is smooth, symmetric and unimodal with the self-similar property (cf. ref. [5]), p α (t, x, y) = p α (t, x y) = t d/α p α (1, t 1/α (x y)). A super α-stable process X = (X t, Q µ ) is an M p (R d )-valued Markov process with X = µ and the transition probabilities given by the following Laplace functional E exp{ X t, f } = exp{ µ, n(t, ) }, f C + p (Rd ), (1.1) where n(, ) is the unique mild solution of the evolution equation ṅ(t) = α n(t) n 2 (t), n() = f. (1.2) Let {g(t, ) : t } be a continuous C p + (Rd )-valued path such that for each a > there is a constant C a > such that g(t) C a φ p for all t [, a]. he weighted occupation time of the super α-stable process may be determined by E exp( t X s, g(s) ) = exp{ µ, m(, t, ) }, (1.3) where m(,, ) is the unique mild solution of ṁ(s) = α m(s) m 2 (s) + g(t s), s t, m() =. See e.g. Iscoe [5]. (1.4) For the path-valued setting, we consider the numerical centered occupation time process Zt (f), t Z t (f) := a 1 [ X s, f E X s, f ], f C p (R d ) +, (1.5) where ( 3 2 d 2α ), d/2 < α < d, a = ( log ) 1 2, d = 2α, (1.6) 1 2, d > 2α.

3 876 Science in China Ser. A Mathematics 24 Vol. 47 No heorem 1.1. For d > α, as, in finite dimensional distribution sense Z t (f) Z t(f), in C([, ), R), and the finite dimensional distribution of (Z t (f)) t is characterized by the Laplace transformation given by { k } E exp θ i Z ti (f) = exp{ˆθaˆθ }, i=1 where t 1 t 2 t k, θ 1, θ 2,, θ k, ˆθ = (θ 1, θ 2,, θ k ), A = (a ij ) k k, (i) When α < d < 2α ( < α 2), a ij = E [Z ti (f)z tj (f)] [ = pα (1, ) λ, f 2 t 3 d/α i + t 3 d/α j 1 C d,α 2 (t i + t j ) 3 d/α 1 ] 2 t i t j 3 d/α, i.e. Z t (f) is a Gaussian process with covariance and E [Z s (f)z t (f)] = 2pα (1, ) λ, f [s 2 3 d/α + t 3 d/α 12 C (s + t)3 d/α 12 ] s t 3 d/α, d,α C d,α = ( d )( α 1 2 d )( 3 d ). α α (ii) When d 2α, a ij = C d min(t i, t j ), i.e. Z t (f) is Brownian motion, with covariance 2C d min(s, t), and p α (1, ) λ, f 2, d = 2α, C d = f(y)pr+r α f(y)dy, d > 2α. Remark 1.1. For d > α, the iting process (Z t (f)) t is something like fractional Brownian motion, but there is some difference in the covariance. (Recall that the fractional Brownian motion with Hurst parameter H has covariance 1{ t 2 1 2H + t 2 2H t 1 t 2 2H } (see for example, ref. [11])). Remark 1.2. We only obtain a it in the finite dimensional distribution sense, the tightness has not been proved yet. Actually, we are uncertain whether tightness hol for this situation, and we leave it as an open problem. As a byproduct, we get a central it theorem in the intermediate and critical dimensions for the super α-stable processes; whereas for the higher dimensions (i.e. d > 2α), the result is well known by Iscoe (heorem 5.4 in ref. [5]). Consider the centered occupation time process Y (f), Y (f) := a 1 Proposition 1.1. As, Copyright by Science in China Press 24 [ X s, f E X s, f ], f C p (R 3 ) +. (1.7) Y (f) Y (f),

4 Functional central it theorem for super α-stable processes 877 where Y (f) is a centered Gaussian random variable and (i) for α < d < 2α ( < α 2), a = ( 3 2 d 2α ), and the variance of Y (f) is E (Y 2 (f)) = 2 C d,α (1 2 1 d/α )p α (1, ) λ, f 2 ; (ii) for d = 2α ( < α 2), a = ( log ) 1/2, and the variance of Y (f) is E (Y 2 (f)) = 1 2 pα (1, ) λ, f 2. he proof is similar as in heorem 1.1, we omit the details. 2 Proofs o simplify the notation, we shall consider k = 2 and let f := a 1 f. From (1.3), (1.4) and (1.5) the Laplace transformation of (Zt 1 (f), Zt 2 (f)), ( t 1 t 2, θ 1, θ 2 ) is given by E exp { θ 1 Z t 1 (f) θ 2 Z t 2 (f) } = E exp { [ X s, θ 1 f 1 [, t1](s) + θ 2 f } ] E ( X s, θ 1 f 1 [, t1](s) + θ 2 f ) { } t2 = exp λ, u 2 (s, ), (2.1) where u (s, x) is the mild solution of the equation u (s) = α u (s) u 2 (s) + θ 1f 1 [, t1]( t 2 s) + θ 2 f, s t 2, u () =, (2.2) i.e. u (s) = α u (s) u 2 (s) + θ 1 f 1 [ (t2 t 1), t 2](s) + θ 2 f, s t 2, u () =, (2.3) and with mild form Let u (s, x) = A (θ 1, θ 2 ; t 1, t 2 ) = P α s r [θ 1f 1 [ (t2 t 1), t 2](r) + θ 2 f ](x) P α s r u (r, ) 2 (x). (2.4) λ, [ P α s r (θ 1f 1 [ (t2 t 1), t 2](r) + θ 2 f )] 2. (2.5)

5 878 Science in China Ser. A Mathematics 24 Vol. 47 No We first note that (let = t 2 t 1 ) where A (1) A (2) A (3) A (4) A (θ 1, θ 2 ; t 1, t 2 ) = a 2 t 2 p α (2s r r, y, z) [θ 1 f(y)1 [, t2](r) + θ 2 f(y)][θ 1 f(z)1 [, t2](r ) + θ 2 f(z)]dydz =: A (1) + A (2) + A (3) + A (4), = θ 2 t 1 a 2 2 = θ 1 θ 2 a 2 = θ 1 θ 2 a 2 = θ 2 2a 2 p α (r + r, y, z)f(y)f(z)dydz, p α (r + r, y, z)f(y)f(z)dydz, p α (r + r, y, z)f(y)f(z)dydz, p α (r + r, y, z)f(y)f(z)dydz. We will consider α < d < 2α ( < α 2) at first by the following two lemmas. Lemma 2.1. Let α < d < 2α ( < α 2), then A (θ 1, θ 2 ; t 1, t 2 ) = ˆθAˆθ, where ˆθ, A are as in heorem 1.1 (with k = 2). Proof. Recall a in (1.6) and the self-similar property of the transition density function of the α-stable process, when α < d < 2α, A (1) = θ 2 1 (r + r ) d α p α (1, ( (r + r )) 1/α (y z))f(y)f(z)dydz, by dominated convergence theorem A(1) = θ 2 1 pα (1, ) λ, f 2 (r + r ) d α = 2 (1 2 1 d/α )p α (1, ) λ, f 2 θ 2 1 C t3 d/α 1. d,α Similarly, we get A(2) = A(3) = λ, f 2 C d,α p α (1, )θ 1 θ 2 [ t 3 d/α 1 +t 3 d/α 2 (t 1+t 2 ) 3 d/α 2 A(4) = 2 (1 2 1 d/α )p α (1, ) λ, f 2 θ C 2t 2 3 d/α 2, d,α where C d,α is given in heorem 1.1. Combining the above, we are done. Copyright by Science in China Press 24 t ] 1 t 2 3 d/α, 2

6 Functional central it theorem for super α-stable processes 879 hen Lemma 2.2. Let α < d < 2α ( < α 2), Proof. β (θ 1, θ 2 ; t 1, t 2 ) := A (θ 1, θ 2 ; t 1, t 2 ) Let G (s) = from (2.4) we know that β (θ 1, θ 2 ; t 1, t 2 ) =. λ, u 2 (s, ). P α s r[θ 1 f 1 [ (t2 t 1), t 2](r) + θ 2 f ], u (s) G (s) C P α s r f, where C is a positive constant (it can take different values in different lines), and hen β (θ 1, θ 2 ; t 1, t 2 ) = G 2 (s) u 2 (s) 2G (s) 2 C λ, G 2 (s) u2 (s) λ, G (s) λ, P α s r f Ca 3 3 d/α r [ P α h f dh] 2 (z)dz Ca 3 5 2d/α P α s ru 2 (r). P α s r u2 (r) r P α s r [ P α r h f dh] 2 p α (2s r r, ) p α (2s r r, ) dh p α (h + h, )dh, which goes to as, because the integral at the right hand side is finite and a 3 5 2d/α = 1/2 d/2α when α < d < 2α. his completes the proof. Remark 2.1. o consider k-dimensional distribution of the process Z t where k > 2, it is enough to replace θ 1 f 1 [, t1]( t 2 s) + θ 2 f by k 1 θ i f 1 [, ti]( t k s) + θ k f i=1 in eq. (2.2), and we can prove the counterpart of Lemma 2.1 and Lemma 2.2 with a bit more complicated calculation. Proof of part (i) of heorem 1.1. Combining Lemma 2.1 and Lemma 2.2 with (2.1), we are done by the discussion in heorem 5.4 of Iscoe [5] on bilateral Laplace transform. Q.E.D.

7 88 Science in China Ser. A Mathematics 24 Vol. 47 No We can prove part (ii) of heorem 1.1 by the same method, so we will only calculate the it of A and the remaining proof is similar as that of part (i). Recall (2.5), and t 1 < t 2, = t 2 t 1. Firstly, note that A (1) = θ 2 1a 2 = θ 2 1a 2 When d > 2α, recall (1.6), A (1) = θ 2 1 (s ) (s ) and by monotone convergence theorem A(1) = θ 2 1 When d = 2α, A(1) = θ 2 1C d t 1. (s ) (s ) = θ2 1 (log ) 1 p α (r + r, y, z)f(y)f(z)dydz θ2 1(log ) 1 = p α (r + r, y, z)f(y)f(z)dydz θ2 1(log ) 1 p α (r + r, y, z)f(y)f(z)dydz p α (r+r, y, z)f(y)f(z)dydz. p α (r + r, y, z)f(y)f(z)dydz, p α (r + r, y, z)f(y)f(z)dydz (s ) (s ) (s ) (s ) a a (s ) a = p α (r, y, z)f(y)f(z)dydz, (s ) +r where a is a positive constant. Breaking up the integral over r at (s ) + a and interchanging the order of integral, we get where But by a simple calculation, A(1) = θ 2 1 a+r [I 1 (, s) + I 2 (, s)], (s ) +a I 1 (, s) = (log ) 1 (r 2a)r 2 2a p α (1, r 1/α (y z))f(y)f(z)dydz, 2(s ) I 2 (, s) = (log ) 1 (2(s ) r)r 2 I 2 (, s) Copyright by Science in China Press 24 (s ) +a p α (1, r 1/α (y z))f(y)f(z)dydz.

8 Functional central it theorem for super α-stable processes 881 and then Similarly, 2(s ) (log ) 1 p α (1, ) λ, f 2 ((s ) a)((s ) + a) 2 (log ) 1 p α (1, ) λ, f 2, (s ) +a I 2 (, s) =. (s ) +a I 1 (, s) (log ) 1 p α (1, ) λ, f 2 r 1 2p α (1, ) λ, f 2, when is large enough. hen by dominated convergence theorem and L Hopsital s rule A(1) = θ 2 1 By the same method, we get where 2a I 1 (, s) = θ 2 1p α (1, ) λ, f 2 t 1. A(2) = A(3) = θ 1 θ 2 C d t 1, A(4) = θ 2 2 C dt 2, p α (1, ) λ, f 2, d = 2α C d = f(y)pr+r α f(y)dy, d > 2α. Acknowledgements he author thanks Prof. D. Dawson for his valuable suggestions and comments, and is grateful for the financial support and the hospitality at Carleton University during his visiting period, where part of the work was done. He also thanks the referee for his/her valuable suggestions. his work was supported by the National Natural Science Foundation of China (Grant Nos.1115 and ), and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry. References 1. Dawson, D. A., he critical measure diffusion process, Z. Wahrsch. Verw. Geb., 1977, 4: Hong, W. M., Longtime behavior for the occupation time of super-brownian motion with random immigration, Stochastic Process, 22, 12: Hong, W. M., Large deviations for super-brownian motion with super-brownian immigration, J. heoret. Probab., 23, 16: Hong, W. M., Li, Z. H., A central it theorem for the super-brownian motion with super-brownian immigration, J. Appl. Probab., 1999, 36: Iscoe, I., A weighted occupation time for a class of measure-valued critical branching Brownian motion, Probab. h. Rel. Fiel, 1986, 71: Iscoe, I., Ergodic theory and a local occupation time for measure-valued critical branching Brownian motion, Stochastics, 1986, 18: Dawson, D. A., Gorostiza, L. G., Wakolbinger, A., Occupation time fluctuations in branching systems, J. heoret. Probab., 21, 14: Quastel, J., Jankowski, H., Sheriff, J., Central it theorem for zero-range processes, Metho Appl. Anal., 22, 9: Zöhle, I., Functional central it theorem for branching random walks, Workshop on Spatially Distributed and Hierarchically Structured Stochastic Systems, Montreal, Dawson, D. A., Measure-valued Markov processes, in Lect. Notes Math., Berlin: Springer-Verlag, 1993, 1541: Samorodnitsky, G., aqqu, M. S., Stable non-gaussian random processes, New York: Chapman and Hall,