x := T x + T x and T The subdiffusive case

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

1 m t, x + T 2 m t, x, 29 where T 1 m

t, x and T

2 m t, x, are defined in the case 1 and the case 2 with m 1 = m 2 = m and κ 1 = κ 2 = κ.

4.2 The subdiffusive case

We extend the above results to the subdiffusive case, meant for which the time-fractional derivative is ∂ β ∂ t β , β ∈ 0, 1, see Section 1 in the system 2, that is, ∂ β ∂ t β ‚ u v Œ = −µI − ∆ γ 2 −∆ α 2 ‚ u v Œ + B ‚ u v Œ , µ, α, γ 0, 30 The time-fractional derivative ∂ β ∂ t β , for any β 0, can be seen, for example, the book of Djrbashian [8]. For 0 β 1, it is d β f d t β t = 1 Γ1 − β Z t f ′ τ t − τ β d τ, 31 where f t is causal i.e., f vanishes for t 0. 971 The solution of 30 can be obtained via the fractional both in time and in space procedure see, for example, [20, 21]; under the Condition DM, we express the solution as the convolution of the fractional both in time and in space Green function and the initial data, as follows. wt, x; w · = Z R n P ‚ G β t, x − y; d 1 G β t, x − y; d 2 Œ P −1 ‚ u y v y Œ d y 32 with the fractional Green function G β t, x; d j is defined via the transformation E β −µ|λ| α 1 + |λ| 2 γ 2 t β + d j t β = Z R n e i x,λ G β t, x; d j d x, j ∈ {1, 2}, where E β · is the Mittag-Leffler function defined by see, for example, [2] or [8, Chapter 1] E β z = ∞ X p=0 z p Γβ p + 1 , z ∈ C. 33 For the properties of the Mittag-Leffler function, we refer the classic book by Erdélyi et.al. [10] pp. 206-212, in particular p. 206 7 and p. 210 21. The following results are time-fractional versions of those in the above; however, the sub-diffusivity brings some new feature. For the large-scalings of homogenization of the system, we need to take an additional scaling on the matrix B in the system 30 in order to compromise the effect of this sub-diffusivity upon the interaction between u and v; while the sub-diffusivity has no influence on the small-scalings and thus it is the same as that in Section 3. To make the situation clear, in the following we denote the vector solution by wt, x; w ·, B. We only state some partial assertions in the below. Proposition 4. Let {wt, x; w ·, B, t 0, x ∈ R n } be the solution-vector of the initial value prob- lem 30 and 3, satisfying Conditions DM, SGRID, and LD. Denote again the long-range-dependence indices κ j and the Hermite ranks m j , j ∈ {1, 2}. Assume that m 1 κ 1 m 2 κ 2 . Large-scalings: under m 1 κ 1 , m 2 κ 2 min{2α, n} the finite-dimensional distributions of the rescaled random field T 1 ǫ t, x := ǫ m1κ1 α L m 1 ǫ − β α − 1 2 n w ǫ −1 t, ǫ − β α x; w ·, ǫ β B − Cǫ −1 t; ǫ β B o , t 0, x ∈ R n , converge weakly, as ǫ → 0, to the finite-dimensional distributions of the random field T 1 t, x = ‚ p 11 p 22 T 1 t, x; d 1 − p 12 p 21 T 1 t, x; d 2 p 21 p 22 T 1 t, x; d 1 − p 21 p 22 T 1 t, x; d 2 Œ , t 0, x ∈ R n , where C ǫ −1 t; ǫ β B = Ct; B = P ‚ E β d 1 t β E β d 2 t ⠌ P −1 ‚ C 1 C 2 Œ , and for j ∈ {1, 2} T 1 t, x; d j is expressed by the following multiple-Wiener integral: C 1 m 1 p m 1 Kn, κ 1 m1 2 Z ′ R n ×m1 e i x,λ 1 +···+λ m1 E β −µ|λ 1 + · · · + λ m 1 | α t β + d j t β |λ 1 | · · · |λ m 1 | n −κ1 2 m 1 Y l=1 W 1 dλ l . 34 972 Small-scalings: under m 1 κ 1 , m 2 κ 2 min{2α + γ, n} the finite-dimensional distributions of the rescaled random field M 1 ǫ := [ ǫ m 1 κ 1 χ L m 1 ǫ −χ ] − 1 2 n w ǫt, ǫ β α+γ x; w ǫ − β α+γ −χ ·, B − ‚ 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 M 1 t, x = ‚ M 1 t, x Œ , t 0, x ∈ R n , where M 1 = C 1 m 1 p m 1 Kn, κ 1 m1 2 Z ′ R n ×m1 e i x,λ 1 +···+λ m1 E β −µ|λ 1 + · · · + λ m 1 | α+γ t β |λ 1 | · · · |λ m 1 | n −κ1 2 m 1 Y l=1 W 1 dλ l .

4.3 Two remarks for future study