Large-scalings getdoccb7b. 234KB Jun 04 2011 12:04:59 AM

To understand the stochastic structure of the limiting fields, we state, for instance, the following covariance result of Y ∗∗∗ 1 t, x Y ∗∗∗ 2 t, x. Proposition 2. For each fixed t 0, the limiting vector field € Y ∗∗∗ 1 t, x Y ∗∗∗ 2 t, x Š in the case 3 of Theorem 2 is spatial-homogeneous and its covariance matrix has the following spectral representation E ‚ Y ∗∗∗ 1 t, x Y ∗∗∗ 2 t, x Œ € Y ∗∗∗ 1 t ′ , x ′ Y ∗∗∗ 2 t ′ , x ′ Š = Z R n e i x−x ′ , λ S λ; t, t ′ , α, γdλ, where S λ; t, t ′ , α, γ = Kn, mκ e −µt+t ′ |λ|α +γ |λ| n −mκ d ia gC 1 m 2 , C 2 m 2 . Remark. In view of the singularity of the diagonal spectral matrix near the origin, we may con- clude that, for limiting vector field in the case 3, the long-range-dependence only exists within each individual component. In Proposition 3 below, we will find that there is also long-range-dependence between the different components of the large-scale limiting fields. 4 Extensions

4.1 Large-scalings

In this subsection, we state the large-scale or say the macro limits of the system, in which only the Riesz parameter α plays its role in the scaling scheme. The result is compared to the single-equation case in And and Leonenko [2, Theorems 2.2 and 2.3]; it shows various limit fields may happen because of the different relations of the various parameters. The proof can be proceeded by the decoupling method. Proposition 3. Let wt, x; w · := ut, x; u ·, vt, x; v ·, t 0, x ∈ R n , be the solution- vector of the initial value problem 2 and 3, satisfying the Conditions MD, SGRID, and LD. In the following, Qt; d 1 , d 2 is the matrix defined in 7, p i j is the entry in 6, and the two Gaussian noise fields W j , j ∈ {1, 2}, are totally independent. Again let m 1 , m 2 , κ 1 and κ 2 denote the parameters in the Conditions SGRID and LD for u and v . 1 If m 2 κ 2 m 1 κ 1 and d 1 d 2 , then the finite-dimensional distributions of the rescaled random field [ǫ m1κ1 α L m 1 ǫ − 1 α ] − 1 2 e −d 1 t ǫ n w t ǫ , x ǫ 1 α ; w · − Q t ǫ ; d 1 , d 2 ‚ C 1 C 2 Œ o , t 0, x ∈ R n , converge weakly, as ǫ → 0, to the finite-dimensional distributions of the random field T 1 m 1 t, x := ‚ p 11 p 22 X 1 m 1 t, x p 21 p 22 X 1 m 1 t, x Œ , t 0, x ∈ R n , where X 1 m 1 t, x := C 1 m 1 p m 1 Kn, κ 1 m1 2 Z ′ R n ×m1 e i x,z 1 +...+z m1 −µt|z 1 +...+z m1 | α |z 1 | · · · |z m 1 | n −κ1 2 m 1 Y l=1 W 1 dz l , 27 970 with W 1 · is a complex Gaussian white noise on R n . 2 If m 1 κ 1 m 2 κ 2 and d 1 d 2 , then the finite-dimensional distributions of the rescaled random field [ǫ m2κ2 α L m 2 ǫ − 1 α ] − 1 2 e −d 1 t ǫ n w t ǫ , x ǫ 1 α ; w · − Q t ǫ ; d 1 , d 2 ‚ C 1 C 2 Œ o , t 0, x ∈ R n , converge weakly, as ǫ → 0, to the finite-dimensional distributions of the random field T 2 m 2 t, x := ‚ −p 11 p 12 X 2 m 2 t, x −p 21 p 12 X 2 m 2 t, x Œ , t 0, x ∈ R n , where X 2 m 2 t, x := C 2 m 2 p m 2 Kn, κ 2 m2 2 Z ′ R n ×m2 e i x,z 1 +...+z m2 −µt|z 1 +...+z m2 | α |z 1 | · · · |z m 2 | n −κ2 2 m 2 Y l=1 W 2 dz l , 28 and W 2 · is a complex Gaussian white noise on R n . 3 If m 1 = m 2 := m, κ 1 = κ 2 := κ, and d 1 d 2 , then the finite-dimensional distributions of the rescaled random field [ǫ m κ α L m ǫ − 1 α ] − 1 2 e −d 1 t ǫ n w t ǫ , x ǫ 1 α ; w · − Q t ǫ ; d 1 , d 2 ‚ C 1 C 2 Œ o , t 0, x ∈ R n , converge weakly, as ǫ → 0, to the finite-dimensional distributions of the random field T 3 m

t, x := T