ERGODICITY OF THE STOCHASTIC COUPLED FRACTIONAL GINZBURG-LANDAU EQUATIONS DRIVEN BY α -STABLE NOISE

. The current paper is devoted to the ergodicity of stochastic coupled fractional Ginzburg-Landau equations driven by α -stable noise on the Torus T . By the maximal inequality for stochastic α -stable convolution and commutator estimates, the well-posedness of the mild solution for stochas- tic coupled fractional Ginzburg-Landau equations is established. Due to the discontinuous trajectories and non-Lipschitz nonlinear term, the existence and uniqueness of the invariant measures are obtained by the strong Feller property and the accessibility to zero.


1.
Introduction. In recent years, the stochastic partial differential equations driven by Lévy noise have attracted a lot of attention, see [1,8,16,17,24]) and references therein. But most of them required the square integrability for Lévy noise, which clearly rules out the interesting α-stable noise. It is worthy to weak this restriction since α-stable noises have been deeply studied and widely applied to physics, queueing theory, mathematical finances and others. There have been some study of the stochastic equations driven by α-stable noises (see for instance [6,17,18,19,23,26,27]). The authors in [17] investigated the structural properties of solutions for the stochastic nonlinear equations with bounded and Lipschitz nonlinearities driven by cylindrical stable processes. While [23] studied the ergodicity of the stochastic equation with unbounded and non-Lipschitz dissipative function driven by α-stable noises with α ∈ (1, 2). The exponential mixing of the SPDEs driven by α-stable noises has been established in [18,19,23]. Dong in [6] proved the exponential ergodicity and strong Feller of the stochastic Burgers equations driven by α 2 -subordinated cylindrical Brownian motions with α ∈ (1, 2). The existence of the invariant measure has been shown for 2D stochastic Navier-stokes equation forced by α-stable noises with α ∈ (1, 2) in [8]. Specially, Xu in [26] studied the ergodicity of the stochastic real Ginzburg-Landau equation driven by α-stable noises with α ∈ ( 3 2 , 2) and established a maximal inequality for the stochastic α-stable convolution which is useful for studying other SPDEs forced by α-stable noise.
Ginzburg-Landau(GL) equations are usually applied to describe a class of optical fiber materials. There have been extensive study of the GL equations(see [3,4,9,10,11,25] and reference therein). The exact homoclinic wave and soliton solution of the GL equations have been studied in [4]. Guo et al in [11] proved the existence of a global attractor for the GL equation. The coupled Ginzburg-Landau(CGL) equations have attracted considerable attention in modeling a class of nonlinear optical fiber materials with active and passive coupled cores. There are also many papers concerning the CGL equations(see [5,15,20] and reference therein). The existence of the stable solutions and exponential attractors for the CGL systems has been proved in [5] and [20] respectively. Recently, the well-posedness and dynamics of the stochastic coupled fractional Ginzburg-Landau equation with multiplicative noise have been studied in [21].
Recalling that the fractional Laplace operator is the infinitesimal generator of the Feller semigroup associated with symmetric α-stable Lévy noise. Hence, there is a natural question, how about the ergodicity of the stochastic fractional Ginzburg-Landau(CFGL) equations driven by α-stable noise? vMotivated by the work in [26] and [21], in the present paper, we consider the following coupled fractional Ginzburg-Landau(CFGL) equations driven by α-stable noise on the Torus T: with the initial conditions and the periodic boundary conditions: where δ ∈ ( 1 2 , 1), u and v denote the amplitude of the electromagnetic wave in a dual-core system, t denotes the time, x is the horizontal axis of the plane wave, γ 2 > 0, µ 2 > 0, µ 3 > 0 and µ 4 > 0 are dissipation coefficients , γ 1 > 0, µ 1 > 3 , γ 3 , σ 1 , σ 2 are real numbers and L t is cylindrical α-stable noise (be specialized later). The fractional Laplacian (−∆) δ can be regarded as a pseudo differential operator with |ξ| 2δ and can be realized through the Fourier transform [22]: where u is the Fourier transform of u.
In the current paper, we consider the ergodicity of the stochastic CFGL equations driven by α-stable noise with α ∈ ( 1+2δ 3δ−δ 2 , 2). The main contribution of the current paper is to apply the commutator estimates developed by Kato and Ponce in [13] to overcome the difficulty in the convergence since higher order estimates cannot be obtained caused by the nonlocal fractional diffusion. We apply the maximal inequality for the stochastic α-stable convolution developed by Xu in [26] and Banach Fixed point theorem to show the existence and uniqueness of the global mild solution. By a priori estimates for the Galerkin approximation equations and classical Bogoliubov-Krylov theorem, the existence of invariant measure is proved. We truncate the nonlinear term to prove the strong Feller property by a gradient estimate developed by Priola and Zabcyzk in [17]. Due to the discontinuous trajectories of α-stable noise, we prove that the system (1) is accessible to zero instead of the irreducibility, and present the ergodicity by the criterion in Hairer in [12].
The rest of the paper is organized as follows. In section 2 is devoted to the introduction of the functional setting, the maximal inequality for the stochastic αstable convolution and commutator estimates. In section 3, we apply the Banach fixed point theorem and the commutator estimates to show the well-posedness of the mild solution for the stochastic FCGL equation. In section 4, some a priori estimates for the Galerkin approximation equation are presented and the existence of the invariant measures for equation (1) is obtained. Finally, the strong Feller property and the accessibility of the system (1) is shown to obtain the uniqueness of the invariant measure for the stochastic FCGL equations.
In this section, we first introduce some notations for the working function space, and then present the maximal inequality for the stochastic α-stable convolution and commutator estimates, which are the key tools to show the well-posedness of the solution for stochastic FCGL equations on the torus T.
Firstly, we introduce some notations as follows We recall some nations to the fractional derivative and fractional Sobolev space.
Since u is a periodic function, it can be expressed by a Fourier series where u(ξ) := T e −iξy u(y)dy.
Then for δ ∈ R, we can write Let A = − , the operator A δ with δ ≥ 0 can be defined by where e k is an orthonormal basis of H and Z * = Z\0 .
Let C σ > 0 be some constants depending on σ, then it follows that

TIANLONG SHEN AND JIANHUA HUANG
Denote Equation (1) can be rewritten as where L t = Σ k∈Z * β k l k (t)e k is a cylindrical α-stable processes on H with {l k (t)} k∈Z * being 1 dimensional symmetric α-stable process sequence with α > 1. In addition, we assume that there exist C 1 , C 2 > 0 such that where β > δ 2 + 1 2αδ can get that the following convolution Z 1t and Z 2t are in V . Consider the following Ornstein-Uhlenbeck process where Here is an important lemma from [26] that plays a crucial role in proving the well-posedness, strong Feller and accessibility for the solution of equation (1).
The following two lemmas which describe commutator estimates developed by Kato and Ponce in [13] and Kening et al in [14] are the key technique tools to show the well-posedness of the mild solution and the estimates on the nonlinearity N (φ).
where Λ denotes the square root of the Laplacian (−∆) Suppose that Λ s f ∈ L q , then f ∈ L p and there is a constant C ≥ 0 such that and if f = Λ −s g for g ∈ L q , then 2. Well-posedness of the mild solution. In this section, we will apply the commutator estimates and the Banach fixed point theorem to show the well-posedness of the mild solution for stochastic FCGL equation.
], g has a left limit and its right continuous at t. Let Then the equations (1) can be changed into the following equations For each T > 0, define Lemma 1.1 yields that for every k ∈ N, there exists some set N k ∈ Ω such that P(N k ) = 0 and Let N = k≥1 N k , it is easy to see P(N ) = 0 and that for all T > 0 We introduce the working space S by where ζ 1 = 1 6δ and ζ 2 ∈ (0, 1 2 ). For any ψ = (b, c), ϕ = (d, e) ∈ S, we endow S the metric D(·, ·) by then (S, D) is a closed metric space.  (7) holds, then we have where C is some constant depending on u 0 H , v 0 H and K 1 (ω).
Proof. For the sake of simplicity, we will omit the variable ω. Let 0 < T ≤ 1 and B > 0 be some constants to be determined later. For any ψ ∈ S, define a map F : We claim that there exist T 0 > 0 and B 0 > 0 such that the following (a) and (b) hold for t ∈ (0, T 0 ] and B ≥ B 0 : (a) Fψ ∈ S for any ψ ∈ S. (b) D(Fψ, Fϕ) ≤ 1 2 D(ψ, ϕ) for any ψ, ϕ ∈ S. If fact, it follows from (4), Lemma 1.3 and Young's inequality that Lemma 1.3 and Young's inequality implies that Similarly, which gives and Choosing T > 0 be small enough and B be large enough, then we derive the claim (a) holds from (21)- (26). Moreover, (23) and (26) imply the continuity of Fψ in H × H.
Next, we prove the claim-(b) holds. For any ψ, ϕ ∈ S, it follows from (4), Lemma 1.3 and Young's inequality that

TIANLONG SHEN AND JIANHUA HUANG
Similarly, and Combining (27)-(29) gives Choosing T be small enough, then we obtain claim-(b). By the Banach fixed point theorem and the claims (a) and (b), we obtain that equation (1) admits a unique solution in S.

It follows from Gronwall's inequality that
Thus, the proof of Lemma 2.3 is completed.
Theorem 2.4. Under the conditions (7), the following statements hold.
(A1) For u 0 , v 0 ∈ H, and ω ∈ Ω, equation (1) possesses a unique mild solution has the following form: where C is some constant depending on T , α, β and ω. 3. The existence of the invariant measure. In this section, we follow the method in [7] to prove the existence of invariant measures. Let e k{k∈Z * } be an orthogonal basis of H and define H m := span{e k ; |k| ≤ m}.
It is known that H m is a finite dimensional Hilbert space equipped with the norm adopted from H. For any m > 0, let π m : H → H m be the projection from H to H m .
The Galerkin approximation of (1) has the following form where C is some constant depending on u 0 W , v 0 W , T and K T (ω). (D2) For u 0 , v 0 ∈ W with W = H, V and ω ∈ Ω a.s., it follows that It follows that Let m → ∞, then Similarly to (21)-(26), we have Direct calculation shows where K = sup 0≤s≤t,m ( u V + u m V ) ≤ 2Ĉ 1 and 1 p + 1 q = 1 with 1 ≤ p ≤ 2.
which implies lim Thus, the proof of Lemma 3.1 is completed. Proof. We follow the method in [7]. Define , v m . It follows from Young's inequality that Similar to the argument in [7], we obtain Hence, we have It follows from Young's inequality that which with the classical Bogoliubov-Krylov's argument implies the existence of invariant measures and that the support of invariant measure is V × V .

4.
Uniqueness of the invariant measure. In this section, we will follow the idea of [26] to show the uniqueness of the invariant measure. Due to the discontinuous trajectories and non-Lipschitz nonlinear term, the existence and uniqueness of the invariant measures are obtained by the strong Feller property and the accessibility to zero instead of the irreducibility. The noise (L t ) t≥0 under the norm · V needs to get a gradient estimate for the OU semigroup corresponding to (Z 1t ) t≥0 and (Z 2t ) t≥0 to show the strong Feller property of the semigroup (P t ) t≥0 on B b (V × V ).
Since the nonlinear term N (·) is not bounded, we consider the equation with truncated nonlinearity as follows: where with the estimates It follows from Lemma 1.2, Lemma 1.3 and δ > 1 For any φ, ϕ ∈ V × V , it follows that (57).

Lemma 1.3 gives
: It follows from Riesz representation theorem that there exists some Df (x, y) ∈ V × V such that

TIANLONG SHEN AND JIANHUA HUANG
Define Noticing that N ρ is a bounded Lipschitz function. We deduce from Lemma 5.9 of [17] and β < 3 2 − 1 α that following proposition holds.
and C > 0 depends on ρ, α and θ. Proof. Assume f ∞ = 1. For T 0 > 0, it suffices to show that for all t ∈ (0, T 0 ] Lemma 1.1 and the Markov inequality gives Let ρ be large enough such that u 0 V < √ ρ 2 , v 0 V < √ ρ 2 and A := {K T0 ≤ ρ 4 }. It follows from Lemma 1.1 that there exists some 0 < t 0 ≤ T 0 depending on ρ such that for all ω ∈ A, Hence, we deduce that It follows from ω ∈ A and (60) that, Hence, we have and P( sup Define the stopping time by Then for any t ∈ [0, t 0 ], Since for t ∈ [0, τ ), equation (1) and equation (57) both has a unique mild solution, It follows from (61) that Proposition 1 and (61) give Thus, we deduce that For all ε > 0, let ρ ≥ max{ 12C ε , 2 u 0 For t 0 < t ≤ T 0 , we derive from Markov property and the strong Feller property that Proof. For any T 0 > 0, it suffices to show that for all t ∈ (0, Let Ω N := max{{sup 0≤t0≤T0 Z 1t V }, {sup 0≤t0≤T0 Z 2t V }} ≤ N and it follows from Lemma 1.1 and Chebyshev's inequality that Observe that u − u = I 1 + I 2 , where It follows from (4) that Due to Hölder inequality, we obtain Theorem 2.4 implies that Similarly, u V ≤ C and v V ≤ C. Hence, For r ∈ (0, t 0 ], define Φ r = sup Choosing r small enough such that Cr 1 2 ≤ 1 4 , we get Φ r + Ψ r ≤ C( u 0 − u 0 H + v 0 − v 0 H ), which implies that For all 0 < t ≤ T 0 , by the Markov property, we obtain where s = t 2 ∧ r and 1 Ω N |. We derive from (62) that G 1 ≤ 2c f ∞ N . Combining (65), Theorem 4.1 with the dominated convergence theorem, we obtain that Thus, the proof of Theorem 4.2 is completed.

4.2.
The accessibility. Because we can't get the irreducibility, we fail to apply the classical Doob's Theorem to get the ergodicity. Alternatively, we apply a criterion in [12]. Finally, let's introduce the conception of accessibility.
Definition 4.3. (Accessibility) Let (X t ) t≥0 be a stochastic process valued on a metric space E and let (P t (x, .)) x∈E be the transition probability family. (X t ) t≥0 is said to be accessible to x 0 ∈ E if the resolvent R λ satisfies for all x ∈ E and all neighborhoods U of x 0 , where λ > 0 is arbitrary.
The following theorem developed by M.Hairer in [12] is the key tool to show the uniqueness of the invariant measure.