Averaging principle for stochastic real Ginzburg-Landau equation driven by $\alpha$-stable process

In this paper, we study a system of stochastic partial differential equations with slow and fast time-scales, where the slow component is a stochastic real Ginzburg-Landau equation and the fast component is a stochastic reaction-diffusion equation, the system is driven by $\alpha$-stable process with $\alpha\in (1,2)$. Using the classical Khasminskii approach based on time discretization and the techniques of stopping times, we show that the slow component strong converges to the solution of the corresponding averaged equation under some suitable conditions.


Introduction
In this paper, we are interested in studying the averaging principle for stochastic real Ginzburg-Landau equation driven by α-stable process, i.e., considering the following stochastic slow-fast system on torus T = R/Z: where ε > 0 is a small parameter describing the ratio of time scales between the slow component X ε t and fast component Y ε t . The coefficients f and g satisfy some suitable conditions. {L t } t 0 and {Z t } t 0 are mutually independent cylindrical α-stable process, α ∈ (1, 2). Under some assumptions, we aim to prove X ε convergent toX in the strong sense, i.e., holds for any 0 < κ < 1, whereX is the solution of the corresponding averaged equation (see Eq. (2.13) below). The theory of averaging principle has a long and rich history. Bogoliubov and Mitropolsky [2] first studied the averaging principle for the deterministic systems. Later on, the theory of averaging principle for stochastic differential equations was firstly proved by Khasminskii [21], see, e.g., [14,15,16,18,19,28] for further generalization. Recently, averaging principle for stochastic reaction-diffusion systems has become an active research area which attracted much attention (see for instance [3,4,5,6,10,11,12,22,24,25]).
To the best of our knowledge, there are rarely results on the averaging principle for stochastic partial differential equations with nonlinear term on this topic. The averaging principle for one dimensional stochastic Burgers equation and two dimensional stochastic Navier-Stokes equation have been studied in [7] and [17] repsectively. Averaging principle for stochastic Kuramoto-Sivashinsky equation with a fast oscillation was studied by Gao in [13]. But the noise considered in above references are Winer noise. This paper focus on studying the averaging principle for stochastic real Ginzburg-Landau equation driven by α-stable process. Bao et al. [1] study the averaging principle for two-time scale stochastic partial differential equations driven by α-stable noise without the nonlinear term. Xu et al. [26] study the strong convergence of the averaging principle for slowfast SPDEs driven by Poisson random measures. However, α-stable noise does not have second moment, so we can't take p-th moment of the solution (p 2), hence some methods developed in [7,17,26] do not work in this situation. So, the most challenge here is how to deal with the nonlinear term and α-stable noise. To overcome these difficulties, we shall deal with the nonlinear term and the stable noise more carefully. The techniques of stopping times will be used frequently.
The proof of our main result is divided into several steps. Firstly, we follow the skills in [27] to give a priori estimate of the solution (X ε t , Y ε t ), which is very important to constrict some stopping times later. Meanwhile, we prove an estimate of |X ε t − X ε s | when s, t before the stopping time. Secondly, based on the Khasminskii discretization introduced in [21], we split the interval [0, T ] into some subintervals of size δ > 0 which depends on ε, and on each interval [kδ, (k + 1)δ)], k 0, we construct an auxiliary process (X ε t ,Ŷ ε t ) which associate with the system (1.1). Finally, by controlling the difference processes X ε t −X ε t andX ε t −X t respectively, we obtain (1.2) when time before the stopping time. Moreover, we use the priori estimates of the X ε t andX t to get a control the term of time after stopping time. The paper is organized as follows. In the next section, we introduce some notation and assumptions that we use throughout the paper, and give out the main result. The section 3 is devoted to prove the strong convergence. The final section is the appendix, where we show the detailed proof of existence and uniqueness of solution, and the corresponding Galerkin approximation.
Along the paper, C, C p , C T and C p,R,T denote positive constants which may change from line to line, where C p depends on p, C T depends on T , and C p,R,T depends on p, R, T .

Notations and main results
For p 1, let L p (T) be the space of p-th power integrable R-valued functions on torus T and | · | L p be the usual norm. For k ∈ N, W k,2 (T) is the Sobolev space of all functions in L 2 (T) whose differentials belong to L 2 (T) up to the order k. Let be a separable real Hilbert space with inner product and norm

Denote Laplacian operator ∆ by
Ax := ∆x := ∂ 2 ∂ξ 2 x, x ∈ W k,2 (T) ∩ H. The eigenfunctions of ∆ are given by which is a orthonormal basis of H. For any k ∈ Z * := Z \ {0}, For any s ∈ R, we define and with the associated norm It is easy to see H 0 = H and H −s be the dual space of H s . Notice that the dual action is also denoted by ·, · without confusion. Denote V := H 1 , which is densely and compactly embedded in H. It is well known that A is the infinitesimal generator of a strongly continuous semigroup {e tA } t 0 . Define nonliear operator, We shall often use the following inequalities: The proof of (2.6)-(2.8) can be founded in [27,Appendix] and we will show (2.5) in the Appendix.
With the above notations, the system (1.1) can be rewritten as: where {L t } t 0 and {Z t } t 0 are mutually independent cylindrical α-stable process given by where α ∈ (1, 2), {β k } k∈Z * and {γ k } k∈Z * are two given sequence of positive numbers and {L k t } k∈Z * and {Z k t } k∈Z * are independent one dimensional α-stable processes satisfying for any k ∈ Z * and t 0, We impose the conditions on the functions f, g : H × H → H.
A1. f and g are Lipschitz continuous, i.e., there exist constants C > 0 and L f , L g > 0 such that for any

A2.
There exist constants C 1 , C 2 > 0 such that A3. f is uniformly bounded, i.e.,there exists C > 0 such that A4. The smallest eigenvalue λ 1 of −A and the Lipschitz constant L g satisfy Remark 2.1. Under the condition A1-A3, for any given initial value x, y ∈ H, ε > 0, system (2.9) exists a unique mild solution X ε ∈ D([0, ∞); H) ∩D((0, ∞); V ), Y ε t ∈ H (see Appendix below). By [20], in general, (2.10) in condition A2 does not imply that Y ε ∈ D([0, ∞); H), only imply Y ε t ∈ H, but it is enough for us to prove our main result. Condition A4 is called the dissipative condition, which used to give the uniform estimate of Y ε with respect to ε and the exponential ergodicity of frozen equation (see Proposition 3.7 below).
Based on the Banach fixed point theorem, we have the following existence and uniqueness of the mild solution of system (2.9), whose proof is given in the appendix. Theorem 2.3. Assume the conditions A1-A3 hold. Then for every ε > 0, x ∈ H, y ∈ H, system (2.9) admits a unique mild solution The main result of this paper is the following theorem.
Theorem 2.4. Assume the conditions A1-A4 hold. Then for any x ∈ H θ with θ ∈ (1/2, 1], y ∈ H, T > 0 and 0 < κ < 1, whereX t is the solution of the corresponding averaged equation: Here µ x is the unique invariant measure of the frozen equation Z t is a version of Z t and independent of L t and Z t . Remark 2.5. Since we only have the priori estimate E sup 0 t T X ε t < ∞ ( see Lemma 3.1 below), the main result (2.12) holds only for 0 < κ < 1.

Proof of Theorem 2.4
In this section, we are devoted to proving Theorem 2.4. The proof consists of the following several steps. In the first step, we give a priori estimate of the solution (X ε t , Y ε t ) in Lemma 3.1, which is used to construct a stopping time τ ε R . Then Lemma 3.2 gives a uniform estimate of X ε t θ when t T ∧τ ε R for θ ∈ (1/2, 1], which is used to obtain an estimate of the expectation of X ε t − X ε s when 0 s t T ∧ τ ε R in Lemma 3.3. In the second step, following the idea inspired by Khasminskii in [21], we introduce an auxiliary process (X ε t ,Ŷ ε t ) ∈ H × H and also give the uniform bounds, see Lemma 3.4. Meanwhile, we introduce a new stopping timẽ τ ε R τ ε R . Then Lemma 3.5 is used to deduce an estimate of the difference process X ε t −X ε t when t T ∧τ ε R , which will be stated in Lemma 3.6. In the third step, we study the frozen equation and average equation. After defining another stopping timeτ ε R τ ε R , we give a control ofX ε t −X t when t T ∧τ ε R in Lemma 3.10. Finally, in order to prove the main result, it is sufficient to control the term of time after the stoppingτ ε R , which will be done by the priori estimates of the X ε t ,X t (see Lemma 3.9).
3.1. Some priori estimates of (X ε t , Y ε t ). We first prove the uniform bounds for p-moment of the solutions X ε t and Y ε t for the system (2.9), with respect to ε ∈ (0, 1) and t ∈ [0, T ]. The main proof follows the techniques in [8], [9] and [27], where the authors deal with the 2D stochastic Navier-Stokes equation, 1D stochastic Burgers' equation and stochastic real Ginzburg-Landau equation driven by α-stable noise, respectively. Inspired by the above references, we first have a fast review about the purely jump Lévy process as following. L k t are independent one dimensional α-stable processes, so they are purely jump Lévy processes and have the same characteristic function, i.e., is a complex valued function called Lévy symbol given by and the corresponding compensated Poisson measure is given by By Lévy-Itô's decomposition, one has

Lemma 3.1.
Under conditions A1-A4, for any x, y ∈ H and T > 0, there exists a constant C T > 0 such that for all ε ∈ (0, 1), Proof. For m ∈ N * , put H m = span{e k , |k| m} and let π m be the projection from H to H m . Consider the Galerkin approximation of system (2.9): Then for any x, y ∈ H m , For I m 1 (t), By (2.8) and condition A3, there exists C > 0 such that For I m 2 (t), by the Burkholder-Davis-Gundy inequality and (3.4), we have For I m 3 (t), the Taylor's expansion follows Notice that condition A2 implies k∈Z * β k < ∞. Then combining (3.5)-(3.9), we have where the constant C T depends on T . By Theorem 4.2 in the appendix below, for any t > 0, Hence by Fatou's Lemma in (3.10) implies Notice that Then for any t 0, by property (2.3), we have which is also a cylindrical α-stable process. Then by [23, (4.12)], This and (3.11), we have for any t T , Hence (3.2) holds due to L g < λ 1 in condition A4. The proof is complete.
In order to study the high regularity of the slow component X ε t , we need to construct the following stopping time, i.e., for any ε ∈ (0, 1), R > 0, Proof. Recall that According to properties (2.4) and (2.5), for any θ ∈ (1/2, 1], we have Using the interpolation inequality, Then Then by the Gronwall's lemma, we obtain By Remark 2.2, for any 1 p < α, we have The proof is complete.
Because that we will use the approach based on time discretization later, we first give an estimate of X ε t+h − X ε t when 0 t t + h T ∧ τ ε R .

3.2.
Estimates of the auxiliary process (X ε t ,Ŷ ε t ). Following the idea inspired by Khasminskii [21], we introduce an auxiliary process (X ε t ,Ŷ ε t ) ∈ H × H. Specifically, we split the interval [0, T ] into some subintervals of size δ > 0, where δ is a positive number depends on ε and will be chosen later. With the initial valueŶ ε 0 = Y ε 0 = y, we construct the processŶ ε t as follows: where t(δ) = [ t δ ]δ is the nearest breakpoint proceeding t. Then we construct the processX ε t as follows: The following Lemma gives a control of the auxiliary process (X ε t ,Ŷ ε t ). Since the proof almost follows the steps in the proof of Lemma 3.1, we omit the proof here.

Lemma 3.4. Under conditions A1-A4, for any x, y ∈ H and T > 0, there exists a constant
Here τ ε R comes from Lemma 3.2.
Proof. By the construction of Y ε t andŶ ε t , we have Then for any t > 0, By Fubini's theorem, By Lemma 3.3 and L g < λ 1 , we have The proof is complete.
In the next lemma, we shall deal with the difference process X ε t −X ε t . To this end, we construct another stopping time, i.e., for any ε ∈ (0, 1), R > 0, Lemma 3.6. Under the conditions A1-A4, for any x ∈ H θ with θ ∈ (1/2, 1], y ∈ H, 1 p < α, T > 0 and R > 0 there exists a constant C p,R,T > 0 such that Proof. In view of (3.19) and (2.11), we write Using properties (2.1) and (2.6), condition A1, we get Then by the definition ofτ ε R , we have sup The Gronwall's inequality implies Notice thatτ ε R τ ε R , then it follows from Lemmas 3.3 and 3.5, we have The proof is complete.

The frozen and averaged equation.
For any fixed x ∈ H, we first consider the following frozen equation associated with the fast component: whereZ t is a version of Z t and independent of {L t } t 0 and {Z t } t 0 . Since g(x, ·) is Lipshcitz continuous, it is easy to prove that for any fixed x, y ∈ H, the Eq. (3.21) has a unique mild solution denoted by Y x,y t . For any x ∈ H, let P x t be the transition semigroup of Y x,y t , that is, for any bounded measurable function ϕ on H and t 0, , y ∈ H. The asymptotic behavior of P x t has been studied in many literatures, the following result shows the existence and uniqueness of the invariant measure and gives the exponential convergence to the equilibrium (see [ The following lemma is used to prove the existence and uniqueness of the solution of corresponding averaged equation, we state it ahead.
By the Lipschitz continuous of g, we get The Gronwall's inequality implies The proof is complete. Now, we introduce the averaged equation, which satisfies: (3.22) wheref The existence and uniqueness of the solution and its priori estimates of Eq. (3.22) is the following lemma.
Moreover, for any x ∈ H and T > 0, there exists a constant C T > 0 such that Proof. It is sufficient to check that thef is Lipschitz continuous and bounded, then the results can be easily obtained by following the procedures in Theorem 2.3 and Lemma 3.1.
Obviously,f is bounded by the boundness of f . It remain to showf is Lipschitz. In fact, for any x 1 , x 2 , y ∈ H and t > 0, by Proposition 3.7 and Lemma 3.8, we have Hence, the proof is completed by letting t → ∞. Now, we intend to estimate the difference process X ε t −X ε t . Similar as the argument in Lemma 3.6, we further construct a new stopping time, i.e., for any ε ∈ (0, 1), R > 0, Proof. From (3.19) and (3.23), it is easy to seê For J 1 (t), according to (2.6), we have For J 3 (t) and J 4 (t), by the Lipschitz continuity off , we obtain Then by (3.25) to (3.28), we have The Gronwall's inequality and the definition ofτ ε R imply Notice thatτ ε R τ ε R , then by Lemma 3.6, we obtain Now, it is remain to estimate J 2 (t). Set n t = [ t δ ], we write J 2 (t) = J 2,1 (t) + J 2,2 (t) + J 2,3 (t), For J 2,2 (t), we have For J 2,3 (t), it follows from the boundness of f , For J 2,1 (t), from the construction ofŶ ε t , we obtain that, for any k ∈ N * and s ∈ [0, δ), where Z kδ (t) := Z t+kδ − Z kδ is the shift version of Z t , which is also a cylindrical α-stable process.
Recall thatZ t be a cylindrical α-stable process which is independent of (X ε kδ ,Ŷ ε kδ ). We construct a process Y whereẐ t := ε 1/αZ t ε is again a cylindrical α-stable process by self-similar property of stable Lévy processes. Then the uniqueness of the solution to Eq. (3.32) and Eq. (3.33) implies that the distribution of (X ε kδ ,Ŷ ε s+kδ ) 0 s δ coincides with the distribution of (X ε kδ , Y Then we try to control J 2,1 (t) : Now, let's estimate Ψ k (s, r). Define F s := σ{Y x,y u , u s} Then for s > r, by the Markov property, Proposition 3.7 and condition A3, In the last inequality, Lemmas 3.1 and 3.4 have been used. Hence This, together with (3.30), (3.31), (3.34) and Lemma 3.3, we get According to the estimates (3.29) and (3.35), we obtain which complete the proof.

4.2.
The existence and uniqueness of solution of system (2.9). Fix ε > 0, for all ω ∈ Ω, define (4.1) For each T > 0, define By Remark 2.2, for every k ∈ N, there exists some set N ε k such that P(N ε k ) = 0 and K ε k (ω) < ∞, ω ∈ N ε k . Define N ε = ∪ k 1 N ε k , it is easy to see P(N ε ) = 0 and that for all where C is some constant depending on x , σ and K ε 1 (ω). 1 + x 2σ .
Proof. We shall apply the Banach fixed point theorem. Since these two statements will be proved by the same method, we only prove statement (i).
Let 0 < T 1 and B > 0 be some constants to be determined later. For σ = 1 6 , define Given any u = (u 1 , Then (4.2) We shall prove that there exist T 0 > 0 and B 0 > 0 such that whenever T ∈ (0, T 0 ] and B > B 0 , the following two statements hold: It is obvious that F (u)(0) = (x, y). By the condition A3, properties (2.4) and (2.5), we have It is easy to see the continuity of F (u) 1 (t) and F (u) 2 (t). As T > 0 is sufficiently small and B is large enough, statement (a) follows from (4.3)-(4.5). Now, let's prove statement (b). Given any u = (u 1 , Notice that u 1 (s) 2σ s −σ B and v 1 (s) 2σ s −σ B, we have This implies  By (4.6) to (4.8) and choosing T small enough, it is easy to see statement (b) holds. Finally, system (4.1) has a unique solution in S by the Banach fixed point theorem. Let (W · , V · ) ∈ S be the solution obtained by the above, for every σ ∈ [ 1 6 , 1 2 ], where C is some constant depending on x , σ and K ε 1 (ω). The proof is complete.