Well-posedness, mean attractors and invariant measures of stochastic discrete long-wave-short-wave resonance equations driven by locally Lipschitz nonlinear noise

W ell-p osedness, mean attractors and in v arian t measures of sto c hastic discrete long-w a v e-short-w a v e resonance equations driv en b y lo cally Lipsc hitz nonlinear noise ∗ Xia P an a , Jianh ua Huang a , Jun tao W u b , Jiangw ei Zhang c † a College of Science, National Universit y of Defense T echnology , Changsha, Hunan 410073, P .R. China b Sc ho ol of Mathematics and Statistics, W uhan Univ ersity , W uhan, Hub ei 430072, P .R. China c Institute of Applied Physics and Computational Mathematics, Beijing, 100088, P .R. China Abstract This pap er is dev oted to in vestigating the random dynamics of stochastic discrete long-w av e-short- w av e resonance equation, whic h are characterized b y the following features: (1) the equation con tains lo cally Lipsc hitz nonlinear coupling terms u m v m and ( B ( | u ( t ) | 2 )) m for m ∈ Z ; (2) the nonlinear co eﬃcien ts of noises satisfy lo cal Lipsc hitz conditions; and (3) the system couples real and complex equations and is inﬁnite-dimensional. These inheren t structural properties preven t the analysis from b eing carried out in a standard pro duct space of the same order and make it diﬃcult to directly v erify the tightness of the distribution family of solutions. T o address these challenges, we adopt a high-order product space L 4 (Ω , C ([ τ , τ + T ] , ℓ 2 c )) × L 2 (Ω , C ([ τ , τ + T ] , ℓ 2 )) as the phase space and emplo y the tec hnique of uniform tail-end estimates. The main results include: establishing the global w ell-p osedness of the nonautonomous stochastic discrete long-wa ve-short-w av e resonance equations driv en by nonlinear noise in L 4 (Ω , C ([ τ , τ + T ] , ℓ 2 c )) × L 2 (Ω , C ([ τ , τ + T ] , ℓ 2 )); based on this, deﬁning the mean random dynamical system and proving the existence and uniqueness of weak D -pullbac k mean random attractors. When the external forcing terms are indep endent of time and sample, we in vestigate the existence of inv ariant measures for the corresponding autonomous system and examine the limiting b eha vior of the inv ariant measure as the noise intensit y tends to zero. Keyw ords: W eak mean attractor; Long-wa ve-short-w a ve resonance equation; Tightness; Inv ariant measure; Limiting b eha vior. MSC 2010: 35B40, 35B41, 37L55, 60H10. 1 In tro duction In this paper, we are concerned with the follo wing nonautonomous sto c hastic discrete lattice long- w av e-short-wa ve resonance equation driven by nonlinear noise deﬁned on the integer set Z :                  idu m ( t ) = (2 u m ( t ) − u m +1 ( t ) − u m − 1 ( t )) dt − iαu m ( t ) dt + u m ( t ) v m ( t ) dt + f m ( t ) dt + ε ∞ P k =1 ( b k,m ( t ) + h k,m ( u m ( t ))) dW k ( t ) , m ∈ Z , t > τ , τ ∈ R , dv m ( t ) = − β v m ( t ) dt − λ ( B ( | u ( t ) | 2 )) m dt + g m ( t ) dt + ε ∞ P k =1 ( γ k,m ( t ) + σ k,m ( v m ( t ))) dW k ( t ) , m ∈ Z , t > τ , τ ∈ R , (1.1) ∗ The researc h is supp orted b y National Natural Science F oundation of China (No. 12371198, 12031020) † Corresponding author: jwzhang0202@yeah.net 1 with initial data u m ( τ ) = u m,τ , v m ( τ ) = v m,τ , m ∈ Z , τ ∈ R , (1.2) where u m ( t ) ∈ C and v m ( t ) ∈ R , for m ∈ Z , denote the unknown functions, and here R and C represen t the real n umbers and complex num b ers, resp ectiv ely; the symbol i denotes the imaginary unit such that i 2 = − 1, α, β , λ > 0 are the positive constants, | u | 2 =  | u m | 2  m ∈ Z , ε ∈ [0 , ε 0 ] with ε 0 > 0 is the intensit y of noise, f = ( f m ) m ∈ Z , g = ( g m ) m ∈ Z , b = ( b k,m ) k ∈ N ,m ∈ Z and γ = ( γ k,m ) k ∈ N ,m ∈ Z are progressiv ely measurable time-dep endent sequences, h k,m and σ k,m are tw o sequences of lo cally Lipschitz contin uous functions, { W k } k ∈ N is a sequence of independent tw o-side real-v alued Wiener pro cesses deﬁned on a complete ﬁltered probability space (Ω , F , { F t } t ∈ R , P ) satisfying the usual conditions. The equation (1.1), as a lattice mo del, can b e regarded as a discretization in the one-dimensional real line R with resp ect to the spatial v ariable x of the follo wing non-autonomous long-wa ve–short-w av e resonance equation:        idu ( t ) + u xx ( t ) dt + iαu ( t ) dt = u ( t ) v ( t ) dt + f ( t ) dt + ε ∞ P k =1 ( b k ( t ) + h k ( x, u ( t ))) dW k ( t ) , dv ( t ) + β v ( t ) dt + λ ( | u ( t ) | 2 ) x dt = g ( t ) dt + ε ∞ P k =1 ( γ k ( t ) + σ k ( x, v ( t ))) dW k ( t ) , (1.3) The long-w av e-short-w av e resonance equation (1.3) of the ab ov e t yp e originate from ﬂuid dynamics and plasma ph ysics [2]. They describ e a physical mec hanism in which resonan t interactions occur b et ween long and short wa ves. Under such resonance, energy can b e transferred b et ween the tw o, leading to the formation of coupled w av e structures. There is an extensiv e b o dy of literature in vestigating this class of problems, see, e.g., [3 – 7] and the references therein. In this w ork, our main ob jective is to in vestigate the discrete version of (1.3). Lattice systems arise in a wide range of applications, including biology , c hemical reactions, pattern formation, nerve-pulse propagation, electrical circuits, and physics, see, e.g., [8 – 12, 24, 31], and the references therein. Ov er the past few decades, substantial research has b een devoted to v arious classes of deterministic and sto c hastic (dela y) lattice systems, with ma jor eﬀorts directed tow ard w ell-p osedness and the long-time b ehavior of solutions, including global and random attractors [13, 22, 30, 32, 42 – 44], chaotic prop erties of solutions [41], tra veling w av es [26 – 28], and inv arian t measures [18, 21, 23, 35 – 38]. In recent y ears, there has b een growing atten tion to and researc h on stochastic PDEs driv en b y state-dep endent noise, also referred to as nonlinear noise. Unlike sto chastic PDEs with additive or linear m ultiplicative noise, such systems generally cannot b e transformed via the con ven tional Ornstein-Uhlenbeck approach in to pathwise deterministic equations with random co eﬃcien ts. This has motiv ated the developmen t of sev eral frameworks for studying this class of problems. Regarding the attractors for sto chastic partial diﬀeren tial equations driven b y nonlinear noise, B. W ang developed a more general framework based on the results in [25], enabling the study of w eak mean random attractors for the corresp onding sto chastic models [39, 40]. Recently , several theoretical approac hes ha ve b een developed to analyze the statistical prop erties of stochastic PDEs on un b ounded domains and sto chastic lattice systems deﬁned on the in teger set Z (or Z n ), including measure attractors, in v ariant measures, ergo dicity , and large deviation principles, providing to ols to describe the asymptotic b eha vior of their solutions, see e.g., [14 – 17, 19, 21, 35 – 38, 45 – 47, 47 – 50]. The previously published literature has inv estigated v arious types of lattice systems, addressing issues such as weak mean random attractors, in v ariant measures, and ergodicity , including (fractional) reaction-diﬀusion lattice systems, p -Laplace lat- tice systems, rev ersible Selk o v lattice systems, Sc hr¨ odinger lattice systems, and Klein-Gordon-Schr¨ odinger lattice systems. How ev er, for the target equation (1.1) to be studied in this paper, no results are av ailable. Therefore, this pap er inv estigates the nonautonomous stochastic discrete long-w av e-short-w av e resonance equation driven by locally Lipschitz nonlinear noise. It should b e noted that the analysis of this equation is not a straigh tforward extension of existing mo del results, mainly due to the follo wing reasons: • Structural Complexity . The sp ecial structure of the equation itself giv es rise to some estimation diﬃculties. Sp eciﬁcally , the nonlinear coupling terms betw een the complex and real equations prev ent the existence and asymptotic behavior of solutions from b eing considered in the phase space L 4 (Ω , C ([ τ , τ + T ] , ℓ 2 c )) × L 2 (Ω , C ([ τ , τ + T ] , ℓ 2 )). • Lo cal Lipschitz Contin uit y . Since the ﬁrst equation in the coupled system (1.1) is a complex equation and the second is a real equation, the lo cal Lipschitz contin uity of the nonlinear terms 2 mak es it imp ossible to directly use classical truncation in [34] to globally appro ximate the nonlinear terms in the process of pro ving the global existence of the solution. • Tightness. The proof of the tightness of the solution’s probability distribution requires the use of uniform tail estimates. How ever, this result cannot b e directly obtained by applying Itˆ o formula to ∥ ρ n u ε ∥ 4 and ∥ ρ n v ε ∥ 2 , as such an approac h w ould not lead to the desired estimate. • High-Order Phase Space. Since the phase space is no longer the classical L 2 (Ω , ℓ 2 c ) × L 2 (Ω , ℓ 2 ), but rather the high-order Bo chner space L 4 (Ω , ℓ 2 c ) × L 2 (Ω , ℓ 2 ), proving the tigh tness of the distri- bution family of the solution and the F eller property of the semigroup is no longer trivial. • Contin uous Dep endence and Con vergence. When pro ving the contin uous dependence of the solution and the con v ergence of the in v ariant measure with resp ect to the noise intensit y in the high- order Bo chner space, there will b e some technical diﬃculties in obtaining the necessary stabilit y estimates for the solution. The arguments mentioned ab o ve, which diﬀer from those used in the study of global w ell-p osedness and long-time b ehavior of solutions to random lattice systems driven b y nonlinear noise of other types, lead to the main diﬃculties of this pap er. W e will fo cus on addressing these distinct c hallenges, with three main inno v ations as follo ws: ( i ) F or the system of sto c hastic equations driven by locally Lipschitz nonlinear noise and coupled by complex and real equations, w e prop ose a new truncation metho d, whic h includes the technique of approximating with global Lipsc hitz functions, as w ell as truncation function estimation techniques when p erforming uniform tail estimates for the solution. ( ii ) F or the sto chastic discrete long-wa ve-short-w a ve resonance equation driven by lo cally Lipschitz nonlinear noise, we hav e explored the use of high-order pro duct space L 4 (Ω , C ([ τ , τ + T ] , ℓ 2 c )) × L 2 (Ω , C ([ τ , τ + T ] , ℓ 2 )) as the phase space to study the global well-posedness and long-time dynamics of the solution. ( iii ) F or the sto c hastic discrete long-w av e-short-wa v e resonance equation, regarding the con tin uous dep endence of the solution in L 4 (Ω , C ([ τ , τ + T ] , ℓ 2 c )) × L 2 (Ω , C ([ τ , τ + T ] , ℓ 2 )) and the estimation of solution conv ergence with respect to the noise intensit y , both fourth-order moment and second-order moment estimates m ust be emplo yed to o vercome the diﬃculties introduced b y the nonlinear terms. Inspired b y the aforementioned methods, the main results of the current article are the global well- p osedness, the existence and uniqueness of weak D -pullbac k mean random attractors, the existence of in v ariant measures, and the limiting b ehavior of the inv ariant measures as the noise in tensity tends to zero for the nonautonomous sto chastic discrete long-wa ve-short-w a ve resonance equation (1.1)-(1.2) driven by nonlinear noise in high-order Bo chner space. The article is organized as follows. In Section 2, we introduce some notations and transform the original equation into an abstract non-autonomous stochastic diﬀerential equation. In Section 3, w e pro ve the existence and uniqueness of solutions to equation (1.1)-(1.2). In Section 4, we pro ve the existence and uniqueness of weak pullback mean random attractors in high-order Bochner space. In Section 5, w e establish the existence of in v ariant measures, and in the ﬁnal section, w e study the limiting b ehavior of in v ariant measures as the noise in tensity ε → 0. 2 Preliminaries In this section, w e ﬁrst deﬁne some Hilb ert spaces consisting of real-v alued and complex-v alued summable bi-inﬁnite sequences and then in tro duce sev eral op erators. F urthermore, w e make some basic assumptions with resp ect to time-dependent external force terms and nonlinear noise co eﬃcients. Finally , w e reform ulate the system (1.1)– (1.2) as an abstract equation. Throughout this pap er, w e denote by c a generic positive constant, whic h is allo wed to v ary in diﬀeren t line, and let R τ := [ τ , ∞ ) and R + := [0 , ∞ ). Since the problem (1.1)–(1.2) is studied in the phase space L 4 (Ω , ℓ 2 c ) × L 2 (Ω , ℓ 2 ), it is necessary to deﬁne the spaces ℓ 2 and ℓ 2 c . They are explicitly described as follo ws: ℓ 2 c = ( u = ( u m ) m ∈ Z   u m ∈ C and X m ∈ Z | u m | 2 < + ∞ ) , ℓ 2 = ( v = ( v m ) m ∈ Z   v m ∈ R and X m ∈ Z | v m | 2 < + ∞ ) , 3 where both ℓ 2 c and ℓ 2 are equipped with the follo wing inner products and norms: ( u 1 , u 2 ) = X m ∈ Z u 1 m u 2 m , ∥ u 1 ∥ = p ( u 1 , u 1 ) , ∀ u 1 =  u 1 m  m ∈ Z , u 2 =  u 2 m  m ∈ Z ∈ ℓ 2 c , ( v 1 , v 2 ) = X m ∈ Z v 1 m v 2 m , ∥ v 1 ∥ = p ( v 1 , v 1 ) , ∀ v 1 =  v 1 m  m ∈ Z , v 2 =  v 2 m  m ∈ Z ∈ ℓ 2 , where u 2 m denotes the conjugate of u 2 m . F or any u = ( u m ) m ∈ Z , the linear operators A and B are deﬁned by ( Au ) m = − u m − 1 + 2 u m − u m +1 and ( B u ) m = u m +1 − u m , ∀ u = ( u m ) m ∈ Z ∈ ℓ 2 or ℓ 2 c . W e deﬁne the adjoin t operator B ∗ of B by ( B ∗ u ) m = u m − 1 − u m , ∀ u = ( u m ) m ∈ Z ∈ ℓ 2 or ℓ 2 c . By simple calculation, w e can verify the follo wing facts: ( Au, v ) = ( B ∗ B u, v ) = ( B u, B v ) , ( B u, v ) = ( u, B ∗ v ) , ∀ v = ( v m ) m ∈ Z , u = ( u m ) m ∈ Z ∈ ℓ 2 or ℓ 2 c . Ob viously , w e can deduce that ( Au, u ) = ∥ B u ∥ 2 ≥ 0 , ∀ u = ( u m ) m ∈ Z ∈ ℓ 2 or ℓ 2 c . In particular, for u = ( u m ) m ∈ Z ∈ ℓ 2 or ℓ 2 c , w e can get ∥ B u ∥ 2 ≤ 4 ∥ u ∥ 2 . T o discuss the existence, uniqueness and long-time uniform estimates of solutions, we imp ose the follo wing assumptions. First, for the time-dependent external force terms, the following condition m ust b e satisﬁed. ( H 1 ) Assume that f = ( f m ) m ∈ Z and b k = ( b k,m ) m ∈ Z b elong to L 4 loc ( R , L 4 (Ω , ℓ 2 c )), and g = ( g m ) m ∈ Z , γ k = ( γ k,m ) m ∈ Z b elong to L 4 loc ( R , L 4 (Ω , ℓ 2 )). Namely , for any τ ∈ R and T > 0, Z τ + T τ E   ∥ f ( t ) ∥ 4 + ∥ g ( t ) ∥ 4 + X k ∈ N ∥ b k ( t ) ∥ 2 ! 2 + X k ∈ N ∥ γ k ( t ) ∥ 2 ! 2   dt < ∞ , where E [ · ] denotes the mathematical exp ectation of a random v ariable. F or the sake of simplicity , we also denote ∥ b ( t ) ∥ 2 = X k ∈ N ∥ b k ( t ) ∥ 2 and ∥ γ ( t ) ∥ 2 = X k ∈ N ∥ γ k ( t ) ∥ 2 , ∀ t ∈ R . Moreo ver, for the nonlinear noise coeﬃcients, we give the follo wing assumptions. ( H 2 ) Assume that h k,m : C → C and σ k,m : R → R are lo cally Lipsc hitz contin uous uniformly with resp ect to k ∈ N and m ∈ Z . More precisely , for an y compact subsets I 1 ⊂ C and I 2 ⊂ R , there exist the constan ts L j k,m, I j > 0 ( j = 1 , 2) suc h that for all k ∈ N and m ∈ Z ,   h k,m  s 1  − h k,m  s 2    ≤ L 1 k,m, I 1   s 1 − s 2   , ∀ s 1 , s 2 ∈ I 1 , and   σ k,m  s 1  − σ k,m  s 2    ≤ L 2 k,m, I 2   s 1 − s 2   , ∀ s 1 , s 2 ∈ I 2 , where w e set L j =  L j k,m, I j  k ∈ N ,m ∈ Z ∈ ℓ 2 with   L j   2 = P k ∈ N P m ∈ Z |L j k,m, I j | 2 < ∞ for eac h j = 1 , 2. ( H 3 ) F or an y k ∈ N , m ∈ Z , there exists a positive sequence δ k,m suc h that for s ∈ C or R , | h k,m ( s ) | ∨ | σ k,m ( s ) | ≤ δ k,m (1 + | s | ) , where w e set δ = ( δ k,m ) k ∈ N ,m ∈ Z ∈ ℓ 2 with ∥ δ ∥ 2 = P k ∈ N P m ∈ Z | δ k,m | 2 < ∞ . F or each k ∈ N , w e deﬁne the op erators h k : ℓ 2 c → ℓ 2 c and σ k : ℓ 2 → ℓ 2 b y h k ( u ) = ( h k,m ( u m )) m ∈ Z , ∀ u = ( u m ) m ∈ Z ∈ ℓ 2 c , and σ k ( v ) = ( σ k,m ( v m )) m ∈ Z , ∀ v = ( v m ) m ∈ Z ∈ ℓ 2 . 4 By ( H 3 ) w e can derive that for every u = ( u m ) m ∈ Z ∈ ℓ 2 c and v = ( v m ) m ∈ Z ∈ ℓ 2 , X k ∈ N ∥ h k ( u ) ∥ 2 ≤ 2 ∥ δ ∥ 2  1 + ∥ u ∥ 2  and X k ∈ N ∥ σ k ( v ) ∥ 2 ≤ 2 ∥ δ ∥ 2  1 + ∥ v ∥ 2  , (2.1) whic h implies that h k : ℓ 2 c → ℓ 2 c and σ k : ℓ 2 → ℓ 2 are well-deﬁned. F urthermore, it is easy to verify that h k : ℓ 2 c → ℓ 2 c and σ k : ℓ 2 → ℓ 2 are lo cally Lipschitz con tinuous, that is, for every n > 0, there exist c 1 = c 1 ( n ) > 0 and c 2 = c 2 ( n ) > 0 suc h that for all u 1 , u 2 ∈ ℓ 2 c and v 1 , v 2 ∈ ℓ 2 with ∥ u 1 ∥ , ∥ u 2 ∥ ≤ n and ∥ v 1 ∥ , ∥ v 2 ∥ ≤ n , it holds X k ∈ N ∥ h k ( u 1 ) − h k ( u 2 ) ∥ 2 ≤ c 1 ∥ u 1 − u 2 ∥ 2 and X k ∈ N ∥ σ k ( v 1 ) − σ k ( v 2 ) ∥ 2 ≤ c 2 ∥ v 1 − v 2 ∥ 2 . (2.2) Remark 2.1. In assumption ( H 1 ) , we c an also supp ose that f = ( f m ) m ∈ Z and b k = ( b k,m ) m ∈ Z b elong to L 4 loc ( R , L 4 (Ω , ℓ 2 c )) , and g = ( g m ) m ∈ Z , γ k = ( γ k,m ) m ∈ Z b elong to L 2 loc ( R , L 2 (Ω , ℓ 2 )) . It should b e emphasize d that the assumptions r e gar ding the time-dep endent external for c e se quenc es do not aﬀe ct the subse quent c onclusions on the existenc e and uniqueness of solutions or the r andom dynamics. • W e deﬁne tw o op erators F : ℓ 2 c × ℓ 2 → ℓ 2 c and G : ℓ 2 c → ℓ 2 b y F ( u, v ) = ( u m v m ) m ∈ Z and G ( u ) = λ  B ( | u | 2 )  m  m ∈ Z for any u = ( u m ) m ∈ Z ∈ ℓ 2 c and v = ( v m ) m ∈ Z ∈ ℓ 2 . It is easy to obtain that for an y u 1 , u 2 ∈ ℓ 2 c and v 1 , v 2 ∈ ℓ 2 , ∥ F ( u 1 , v 1 ) − F ( u 2 , v 2 ) ∥ 2 = X m ∈ Z | u 1 m ( v 1 m − v 2 m ) + v 2 m ( u 1 m − u 2 m ) | 2 ≤ 2 X m ∈ Z | u 1 m | 2 | v 1 m − v 2 m | 2 + 2 X m ∈ Z | v 2 m | 2 | u 1 m − u 2 m | 2 ≤ 2  1 + ∥ u 1 ∥ 2 + ∥ v 2 ∥ 2   ∥ u 1 − u 2 ∥ 2 + ∥ v 1 − v 2 ∥ 2  . (2.3) Moreo ver, for an y u 1 , u 2 ∈ ℓ 2 c , it holds ∥ G ( u 1 ) − G ( u 2 ) ∥ 2 ≤ 4 λ 2 ∥| u 1 | 2 − | u 2 | 2 ∥ 2 ≤ 8 λ 2  ∥ u 1 ∥ 2 + ∥ u 2 ∥ 2  ∥ u 1 − u 2 ∥ 2 . (2.4) It follows from (2.3)-(2.4) that F ( u, v ) and G ( u ) satisfy lo cally Lisp chiz conditions. In other w ords, for ev ery n ∈ N , there exist c 3 ( n ) > 0 and c 4 ( n ) > 0 suc h that for any u 1 , u 2 ∈ ℓ 2 c , v 1 , v 2 ∈ ℓ 2 and ∥ u 1 ∥ ≤ n , ∥ u 2 ∥ ≤ n , ∥ v 1 ∥ ≤ n , ∥ v 2 ∥ ≤ n , ∥ F ( u 1 , v 1 ) − F ( u 2 , v 2 ) ∥ 2 ≤ c 3 ( n )  ∥ u 1 − u 2 ∥ 2 + ∥ v 1 − v 2 ∥ 2  , ∥ G ( u 1 ) − G ( u 2 ) ∥ 2 ≤ c 4 ( n ) ∥ u 1 − u 2 ∥ 2 . (2.5) With the help of the aforemen tioned notations, we can rewrite the system (1.1)-(1.2) in ℓ 2 c × ℓ 2 as follo ws:            idu ( t ) = ( Au ( t ) − iαu ( t ) + F ( u ( t ) , v ( t )) + f ( t )) dt + ε ∞ P k =1 ( h k ( u ( t )) + b k ( t )) dW k ( t ) , t > τ , dv ( t ) = ( − β v ( t ) − G ( u ( t )) + g ( t )) dt + ε ∞ P k =1 ( σ k ( v ( t )) + γ k ( t )) dW k ( t ) , t > τ , ( u ( τ ) , v ( τ )) = ( u τ , v τ ) , τ ∈ R . (2.6) In order to represen t the abov e system (2.6) in an abstract form, we introduce some addition notations. Let φ = ( u, v ) T , G ( φ, t ) = ( − iF ( u ( t ) , v ( t )) − if ( t ) , − G ( u ( t )) + g ( t )) T , and A =  iA + αI 0 0 β I  , H k ( φ, t ) =  − ih k ( u ( t )) − ib k ( t ) σ k ( v ( t )) + γ k ( t )  . Then, w e ha ve the follo wing abstract non-autonomous sto c hastic diﬀerential equations:    dφ + A φdt = G ( φ, t ) dt + ε ∞ P k =1 H k ( φ, t ) dW k ( t ) , t > τ , τ ∈ R , φ ( τ ) = φ τ , τ ∈ R , (2.7) where φ τ ∈ ℓ 2 c × ℓ 2 . 5 3 Existence and uniqueness of solutions This section mainly studies the existence and uniqueness of solutions to problem (2.7). T o this end, based on the preparatory w ork in Section 2, we consider the solutions of problem (2.7) in the follo wing sense: Deﬁnition 3.1. F or any τ ∈ R and F τ -me asur able φ τ ∈ L 4 (Ω , ℓ 2 c ) × L 2 (Ω , ℓ 2 ) , an ℓ 2 c × ℓ 2 -value d F t - adapte d sto chastic pr o c ess φ ( t ) = ( u ( t ) , v ( t )) is c al le d a solution of system (2.7) if φ ( t ) ∈ L 2 (Ω , C ([ τ , τ + T ] , ℓ 2 c )) × L 2 (Ω , C ([ τ , τ + T ] , ℓ 2 )) for any T > 0 , and for almost sur ely ω ∈ Ω , φ ( t ) + Z t τ A φ ( s ) ds = φ τ + Z t τ G ( φ ( s ) , s ) ds + ε ∞ X k =1 Z t τ H k ( φ ( s ) , s ) dW k ( s ) (3.1) in ℓ 2 c × ℓ 2 for al l t ≥ τ . By assumption ( H 3 ), w e know that the nonlinear drift terms F , G and the nonlinear diﬀusion co eﬃcien ts h k , σ k are lo cally Lipsc hitz contin uous. T o establish the global existence of a solution to system (2.6) in the sense of Deﬁnition 3.1, it is necessary to approximate the lo cally Lipschitz nonlinearities by globally Lipschitz con tinuous ones. F or this purpose, for ev ery n ∈ N , w e introduce t wo cut-oﬀ functions ρ C n : C → C and ρ R n : R → R deﬁned b y ρ C n ( z ) = ( z , if | z | ≤ n, n z | z | , if | z | > n, and ρ R n ( s ) = ( s, if | s | ≤ n, n s | s | , if | s | > n. (3.2) W e observe that ρ C n : C → C and ρ R n : R → R are both increasing and satisfy the global Lipschitz condition: | ρ C n ( z 1 ) − ρ C n ( z 2 ) | ≤ 2 | z 1 − z 2 | , ∀ z 1 , z 2 ∈ C , (3.3) | ρ R n ( s 1 ) − ρ R n ( s 2 ) | ≤ | s 1 − s 2 | , ∀ s 1 , s 2 ∈ R . (3.4) In addition, we hav e ρ C n (0) = 0 , | ρ C n ( z ) | ≤ n, ∀ z ∈ C and ρ R n (0) = 0 , | ρ R n ( s ) | ≤ n, ∀ s ∈ R . (3.5) P articularly , let arg ( ρ C n ( z )) = arg ( z ) for z  = 0. F or every k , n ∈ N and any u = ( u m ) m ∈ Z ∈ ℓ 2 c and v = ( v m ) m ∈ Z ∈ ℓ 2 , let F n ( u, v ) =  ( ρ C n u m )( ρ R n v m )  m ∈ Z , G n ( u ) = λ  B  | ρ C n u | 2  m  m ∈ Z , (3.6) and h n k ( u ) = h k ( ρ C n ( u )) = ( h k,m ( ρ C n ( u m ))) m ∈ Z , σ n k ( v ) = σ k ( ρ R n ( v )) = ( σ k,m ( ρ R n ( v m ))) m ∈ Z . (3.7) By (2.2), (2.5) and (3.3)-(3.6) we know that F n : ℓ 2 c × ℓ 2 → ℓ 2 c , G n : ℓ 2 c → ℓ 2 , h n k : ℓ 2 c → ℓ 2 c and σ n k : ℓ 2 → ℓ 2 are globally Lipsc hitz con tinuous; that is, for ev ery k , n ∈ N , there exist Q 1 ( n ) > 0, Q 2 ( n ) > 0, Q 3 ( n ) > 0, Q 4 ( n ) > 0 depending only on n suc h that for all u, u 1 , u 2 ∈ ℓ 2 c and v , v 1 , v 2 ∈ ℓ 2 , it holds ∥ F n ( u 1 , v 1 ) − F n ( u 2 , v 2 ) ∥ 2 ≤ Q 1 ( n )  ∥ u 1 − u 2 ∥ 2 + ∥ v 1 − v 2 ∥ 2  , ∥ G n ( u 1 ) − G n ( u 2 ) ∥ 2 ≤ Q 2 ( n ) ∥ u 1 − u 2 ∥ 2 , (3.8) and X k ∈ N ∥ h n k ( u 1 ) − h n k ( u 2 ) ∥ 2 ≤ Q 3 ( n ) ∥ u 1 − u 2 ∥ 2 , X k ∈ N ∥ σ n k ( v 1 ) − σ n k ( v 2 ) ∥ 2 ≤ Q 4 ( n ) ∥ v 1 − v 2 ∥ 2 . (3.9) Moreo ver, we hav e F n (0 , 0) = 0, G n (0) = 0 and X k ∈ N ∥ h n k ( u ) ∥ 2 ≤ 2 ∥ δ ∥ 2 (1 + ∥ u ∥ 2 ) , X k ∈ N ∥ σ n k ( v ) ∥ 2 ≤ 2 ∥ δ ∥ 2 (1 + ∥ v ∥ 2 ) . (3.10) 6 F or every n ∈ N , w e consider the following appro ximate sto chastic system in ℓ 2 c × ℓ 2 ,    dφ n ( t ) + A φ n ( t ) dt = G n ( φ n ( t ) , t ) dt + ε ∞ P k =1 H n k ( φ n ( t ) , t ) dW k ( t ) , t > τ , τ ∈ R , φ n ( τ ) = φ τ , τ ∈ R , (3.11) where φ n ( t ) = ( u n ( t ) , v n ( t )) T , A φ n ( t ) =  iAu n ( t ) + αu n ( t ) β v n ( t )  , G n ( φ n ( t ) , t ) =  − iF n ( u n ( t ) , v n ( t )) − if ( t ) − G n ( u n ( t )) + g ( t )  , and H n k ( φ n ( t ) , t ) =  − ih n k ( u n ( t )) − ib k ( t ) σ n k ( v n ( t )) + γ k ( t )  . F ollo wing the standard theory of the existence of solutions for sto chastic diﬀeren tial equations on the en tire space R n (see e.g., [1]), it follo ws from (3.8)-(3.10) that for all n ∈ N , τ ∈ R , and any F τ -measurable φ τ ∈ L 2 (Ω , ℓ 2 c × ℓ 2 ), the approximate system (3.11) admits a unique solution φ n ∈ L 2 (Ω , C ([ τ , ∞ ) , ℓ 2 c × ℓ 2 )) in the sense of Deﬁnition 3.1, with F , G and H k replaced b y F n , G n and H n k , respectively . W e now establish the existence and uniqueness of solutions to system (2.7) in the sense of Deﬁnition 3.1 via a limiting pro cedure. Sp eciﬁcally , w e will examine the limiting b ehavior of the solution sequence { φ n } ∞ n =1 of the approximate system (3.11) as n → ∞ . As a k ey step, for ev ery n ∈ N , t ≥ τ , and T > 0, w e introduce the stopping time τ n deﬁned b y τ n = inf { t ≥ τ : ∥ u n ( t ) ∥ + ∥ v n ( t ) ∥ > n } , (3.12) where, as usual, τ n = + ∞ if { t ≥ τ : ∥ u n ( t ) ∥ + ∥ v n ( t ) ∥ > n } = ∅ . F or con v enience, we let φ n τ n = φ n ( t ∧ τ n ), then the sequence { φ n τ n } ∞ n =1 is consisten t in the follo wing sense. Lemma 3.2. Supp ose that ( H 1 ) − ( H 3 ) hold, and let φ n ( t ) = ( u n ( t ) , v n ( t )) T b e the solution of system (3.11) . Then, we have the fol lowing c onclusions φ n +1 τ n = φ n τ n and τ n +1 ≥ τ n , a.s., ∀ t ≥ τ , n ∈ N , (3.13) wher e τ n is the stopping time given by (3.12) . Pr o of. Applying Ito’s formula and com bining with (3.11), by taking the real part we hav e that a.s., ∥ u n +1 ( t ∧ τ n ) − u n ( t ∧ τ n ) ∥ 2 + 2 α Z t ∧ τ n τ ∥ u n +1 ( r ) − u n ( r ) ∥ 2 dr = 2 Z t ∧ τ n τ Im  F n +1  u n +1 ( r ) , v n +1 ( r )  − F n ( u n ( r ) , v n ( r )) , u n +1 ( r ) − u n ( r )  dr + ε 2 ∞ X k =1 Z t ∧ τ n τ ∥ h n +1 k  u n +1 ( r )  − h n k ( u n ( r )) ∥ 2 dr + 2 ε ∞ X k =1 Z t ∧ τ n τ Im  u n +1 ( r ) − u n ( r )   h n +1 k  u n +1 ( r )  − h n k ( u n ( r ))  dW k ( r ) . (3.14) By (3.11) and applying Ito’s formula again, w e can derive that a.s., ∥ v n +1 ( t ∧ τ n ) − v n ( t ∧ τ n ) ∥ 2 + 2 β Z t ∧ τ n τ ∥ v n +1 ( r ) − v n ( r ) ∥ 2 dr + 2 Z t ∧ τ n τ  G n +1  u n +1 ( r )  − G n ( u n ( r )) , v n +1 ( r ) − v n ( r )  dr = ε 2 ∞ X k =1 Z t ∧ τ n τ ∥ σ n +1 k  v n +1 ( r )  − σ n k  v n ( r )  ∥ 2 dr + 2 ε ∞ X k =1 Z t ∧ τ n τ  v n +1 ( r ) − v n ( r )   σ n +1 k  v n +1 ( r )  − σ n k  v n ( r )  dW k ( r ) . (3.15) 7 According to the Riesz representation theorem, w e know that the last terms u n +1 ( r ) − u n ( r ) and v n +1 ( r ) − v n ( r ) in (3.14) and (3.15) are iden tiﬁed with the elemen ts in the dual space of ℓ 2 c and ℓ 2 , respectively . By (3.12) we ﬁnd that for any r ∈ [ τ , τ n ), it holds ∥ u n ( r ) ∥ ≤ n and ∥ v n ( r ) ∥ ≤ n . Hence, for all n ∈ N and r ∈ [ τ , τ n ), w e ha ve F n +1 ( u n ( r ) , v n ( r )) = F n ( u n ( r ) , v n ( r )) , G n +1 ( u n ( r )) = G n ( u n ( r )) , (3.16) and h n +1 k ( u n ( r )) = h n k ( u n ( r )) , σ n +1 k ( v n ( r )) = σ n k ( v n ( r )) . (3.17) W e now deal with the ﬁrst term on the right-hand side of (3.14) and the third term on the left-hand side of (3.15). By (3.6), (3.8), (3.16) and Y oung’s inequalit y we can deriv e that for all r ∈ [ τ , τ n ), 2     Z t ∧ τ n τ Im  F n +1  u n +1 ( r ) , v n +1 ( r )  − F n ( u n ( r ) , v n ( r )) , u n +1 ( r ) − u n ( r )  dr     = 2     Z t ∧ τ n τ Im  F n +1  u n +1 ( r ) , v n +1 ( r )  − F n +1 ( u n ( r ) , v n ( r )) , u n +1 ( r ) − u n ( r )  dr     ≤ Z t ∧ τ n τ  ∥ F n +1  u n +1 ( r ) , v n +1 ( r )  − F n +1  u n ( r ) , v n ( r ) ∥ 2  + ∥ u n +1 ( r ) − u n ( r ) ∥ 2  dr ≤ 2 ( Q 1 ( n + 1) + 1) Z t ∧ τ n τ  ∥ u n +1 ( r ) − u n ( r ) ∥ 2 + ∥ v n +1 ( r ) − v n ( r ) ∥ 2  dr , (3.18) and     2 Z t ∧ τ n τ  G n +1  u n +1 ( r )  − G n ( u n ( r )) , v n +1 ( r ) − v n ( r )  dr     ≤ ( Q 2 ( n + 1) + 1) Z t ∧ τ n τ  ∥ u n +1 ( r ) − u n ( r ) ∥ 2 + ∥ v n +1 ( r ) − v n ( r ) ∥ 2  dr . (3.19) F urthermore, in light of (3.7) and (3.9), it can b e deduced from (3.17) that the second term on the righ t-hand side of (3.14) and the ﬁrst term on the right-hand side of (3.15) satisfy ε 2 ∞ X k =1 Z t ∧ τ n τ ∥ h n +1 k  u n +1 ( r )  − h n k ( u n ( r )) ∥ 2 dr ≤ ε 2 0 Q 3 ( n + 1) Z t ∧ τ n τ ∥ u n +1 ( r ) − u n ( r ) ∥ 2 dr ≤ Q 3 ( n + 1) Z t ∧ τ n τ ∥ u n +1 ( r ) − u n ( r ) ∥ 2 dr , (3.20) and ε 2 ∞ X k =1 Z t ∧ τ n τ ∥ σ n +1 k  v n +1 ( r )  − σ n k  v n ( r )  ∥ 2 dr ≤ Q 4 ( n + 1) Z t ∧ τ n τ ∥ v n +1 ( r ) − v n ( r ) ∥ 2 dr , (3.21) resp ectiv ely , where we set ε 0 ≤ 1 for con venience. T ogether with (3.14)-(3.15) and (3.18)-(3.21), w e can infer that there exists Q 5 ( n ) > 0 such that ∥ u n +1 ( t ∧ τ n ) − u n ( t ∧ τ n ) ∥ 2 + ∥ v n +1 ( t ∧ τ n ) − v n ( t ∧ τ n ) ∥ 2 ≤ Q 5 ( n ) Z t ∧ τ n τ  ∥ u n +1 ( r ) − u n ( r ) ∥ 2 + ∥ v n +1 ( r ) − v n ( r ) ∥ 2  dr + 2 ε ∞ X k =1 Z t ∧ τ n τ Im  u n +1 ( r ) − u n ( r )   h n +1 k  u n +1 ( r )  − h n k ( u n ( r ))  dW k ( r ) + 2 ε ∞ X k =1 Z t ∧ τ n τ  v n +1 ( r ) − v n ( r )   σ n +1 k  v n +1 ( r )  − σ n k  v n ( r )  dW k ( r ) . (3.22) T aking the supremum of b oth sides of inequalit y (3.22) ov er the in terv al [ τ , t ] and then computing the 8 exp ectation of the resulting inequality , b y (3.17) w e obtain E  sup τ ≤ r ≤ t  ∥ u n +1 ( r ∧ τ n ) − u n ( r ∧ τ n ) ∥ 2 + ∥ v n +1 ( r ∧ τ n ) − v n ( r ∧ τ n ) ∥ 2   ≤ Q 5 ( n ) Z t τ E  sup τ ≤ r ≤ s  ∥ u n +1 ( r ∧ τ n ) − u n ( r ∧ τ n ) ∥ 2 + ∥ v n +1 ( r ∧ τ n ) − v n ( r ∧ τ n ) ∥ 2   ds + 2 ε 0 E " sup τ ≤ r ≤ t ∧ τ n      ∞ X k =1 Z r τ Im  u n +1 ( s ) − u n ( s )   h n +1 k  u n +1 ( s )  − h n +1 k ( u n ( s ))  dW k ( s )      # + 2 ε 0 E " sup τ ≤ r ≤ t ∧ τ n      ∞ X k =1 Z r τ  v n +1 ( s ) − v n ( s )   σ n +1 k  v n +1 ( s )  − σ n +1 k  v n ( s )  dW k ( s )      # . (3.23) W e handle the second and third terms on the right-hand side of (3.23). By (3.9) and the Burkholder- Da vis-Gundy (BDG) inequalit y w e obtain there exist Q 6 ( n ) > 0 and Q 7 ( n ) > 0 such that 2 ε 0 E " sup τ ≤ r ≤ t ∧ τ n      ∞ X k =1 Z r τ Im  u n +1 ( s ) − u n ( s )   h n +1 k  u n +1 ( s )  − h n +1 k ( u n ( s ))  dW k ( s )      # ≤ 8 √ 2 ε 0 E   Z t ∧ τ n τ ∥ u n +1 ( s ) − u n ( s ) ∥ 2 ∞ X k =1 ∥ h n +1 k  u n +1 ( s )  − h n +1 k ( u n ( s )) ∥ 2 ! ds ! 1 2   ≤ 8 √ 2 ε 0 E    sup τ ≤ s ≤ t ∥ u n +1 ( s ∧ τ n ) − u n ( s ∧ τ n ) ∥  Z t ∧ τ n τ ∞ X k =1 ∥ h n +1 k  u n +1 ( s )  − h n +1 k ( u n ( s )) ∥ 2 ds ! 1 2   ≤ 16 ε 0 p Q 3 ( n + 1) E "  sup τ ≤ r ≤ t ∥ u n +1 ( r ∧ τ n ) − u n ( r ∧ τ n ) ∥   Z t ∧ τ n τ ∥ u n +1 ( r ) − u n ( r ) ∥ 2 dr  1 2 # ≤ 1 2 E  sup τ ≤ r ≤ t ∥ u n +1 ( r ∧ τ n ) − u n ( r ∧ τ n ) ∥ 2  + Q 6 ( n ) Z t τ E  sup τ ≤ r ≤ s ∥ u n +1 ( r ∧ τ n ) − u n ( r ∧ τ n ) ∥ 2  ds, (3.24) and 2 ε 0 E " sup τ ≤ r ≤ t ∧ τ n      ∞ X k =1 Z r τ  v n +1 ( s ) − v n ( s )   σ n +1 k  v n +1 ( s )  − σ n +1 k  v n ( s )  dW k ( s )      # ≤ 1 2 E  sup τ ≤ r ≤ t ∥ v n +1 ( r ∧ τ n ) − v n ( r ∧ τ n ) ∥ 2  + Q 7 ( n ) Z t τ E  sup τ ≤ r ≤ s ∥ v n +1 ( r ∧ τ n ) − v n ( r ∧ τ n ) ∥ 2  ds. (3.25) Substituting (3.24) and (3.25) in to (3.23), we hav e that there exists Q 8 ( n ) > 0 such that E  sup τ ≤ r ≤ t  ∥ u n +1 ( r ∧ τ n ) − u n ( r ∧ τ n ) ∥ 2 + ∥ v n +1 ( r ∧ τ n ) − v n ( r ∧ τ n ) ∥ 2   ≤ Q 8 ( n ) Z t τ E  sup τ ≤ r ≤ s  ∥ u n +1 ( r ∧ τ n ) − u n ( r ∧ τ n ) ∥ 2 + ∥ v n +1 ( r ∧ τ n ) − v n ( r ∧ τ n ) ∥ 2   ds, whic h implies that E  sup τ ≤ r ≤ t  ∥ u n +1 τ n ( r ) − u n τ n ( r ) ∥ 2 + ∥ v n +1 τ n ( r ) − v n τ n ( r ) ∥ 2   ≤ Q 8 ( n ) Z t τ E  sup τ ≤ r ≤ s  ∥ u n +1 τ n ( r ) − u n τ n ( r ) ∥ 2 + ∥ v n +1 τ n ( r ) − v n τ n ( r ) ∥ 2   ds. (3.26) Applying Gron wall’s inequalit y to (3.26), we can obtain E  sup τ ≤ r ≤ t  ∥ φ n +1 τ n ( r ) − φ n τ n ( r ) ∥ 2   = 0 , ∀ t ≥ τ , from which w e ha ve φ n +1 τ n ( t ) = φ n τ n ( t ) for an y t ≥ τ almost surely , i.e., u n +1 τ n ( t ) = u n τ n ( t ) and v n +1 τ n ( t ) = v n τ n ( t ) for an y t ≥ τ almost surely . Therefore, by (3.12) we ha ve τ n +1 ≥ τ n almost surely , which sho ws the desired result (3.13). This pro of is ﬁnished. 9 Remark 3.3. As a pr eliminary, we notic e that system (3.11) is known to have a unique solution in L 2 (Ω , C ([ τ , ∞ ) , ℓ 2 c × ℓ 2 )) . The estimates pr ovide d in L emma 3.4 subse quently ensur e the existenc e and uniqueness of this solution in the mor e r e gular sp ac e L 4 (Ω , C ([ τ , ∞ ) , ℓ 2 c )) × L 2 (Ω , C ([ τ , ∞ ) , ℓ 2 )) . This foundation al lows us to establish the uniqueness of the solution for system 2.7 within this high-or der setting. Next, we ﬁrst establish uniform estimates for solutions φ n of the appro ximate system (3.11). Based on these estimates, w e then prov e that the asso ciated stopping times τ n tend to inﬁnity as n → ∞ . Lemma 3.4. Supp ose that ( H 1 ) − ( H 3 ) hold, and let φ n = ( u n ( t ) , v n ( t )) T b e the solution of system (3.11) . Then, for e ach T > 0 , φ n satisﬁes the fol lowing estimates: E  sup τ ≤ r ≤ τ + T  ∥ u n ( r ) ∥ 4 + ∥ u n ( r ) ∥ 2   ≤ ρ 0 , (3.27) wher e ρ 0 = c e c (1+ T ) 2  E  ∥ u τ ∥ 4 + ∥ v τ ∥ 2  + (1 + T ) Z t τ E  ∥ f ( r ) ∥ 4 + ∥ g ( r ) ∥ 4 + ∥ b ( r ) ∥ 4 + ∥ γ ( r ) ∥ 4  dr + (1 + T ) 2  , (3.28) with c > 0 b eing a c onstant indep endent of τ , φ τ , n and T . In p articular, let τ n b e the stopping time given by (3.12) and τ b e deﬁne d by τ = lim n →∞ τ n = sup n ∈ N τ n , then we have τ = ∞ , a.s. (3.29) Pr o of. Step 1. W e ﬁrst prov e (3.27). Applying Ito’s form ula for the pro cess ∥ u ( t ) ∥ p , by (3.11) and taking the real part we obtain that a.s. ∥ u n ( t ) ∥ p + αp Z t τ ∥ u n ( r ) ∥ p dr = ∥ u τ ∥ p + p Z t τ ∥ u n ( r ) ∥ p − 2 · ( f ( r ) , u n ( r )) dr + p Z t τ ∥ u n ( r ) ∥ p − 2 · Im ( F n ( u n ( r ) , v n ( r )) , u n ( r )) dr | {z } =0 + pε ∞ X k =1 Z t τ ∥ u n ( r ) ∥ p − 2 Im ( u n ( r ) , ( h n k ( u n ( r )) + b k ( r )) dW k ( r )) + p 2 ε 2 Z t τ ∥ u n ( r ) ∥ p − 2 ∞ X k =1 ∥ h n k ( u n ( r )) + b k ( r ) ∥ 2 dr + p ( p − 2) 2 ε 2 Z t τ ∥ u n ( r ) ∥ p − 4 ∞ X k =1 ( u n ( r ) , h n k ( u n ( r )) + b k ( r )) 2 dr . (3.30) Using Ito’s formula for the pro cess ∥ v ( t ) ∥ 2 , b y (3.11) w e get that a.s. ∥ v n ( t ) ∥ 2 + 2 β Z t τ ∥ v n ( r ) ∥ 2 dr = ∥ v τ ∥ 2 + 2 Z t τ ( g ( r ) , v n ( r )) dr − 2 Z t τ ( G n ( u n ( r )) , v n ( r )) dr + ε 2 ∞ X k =1 Z t τ ∥ σ n k  v n ( r )  + γ k ( r ) ∥ 2 dr + 2 ε ∞ X k =1 Z t τ  v n ( r ) ,  σ n k  v n ( r )  + γ k ( r )  dW k ( r )  . (3.31) Com bining with (3.30) and (3.31) yields ∥ u n ( t ) ∥ 4 + ∥ v n ( t ) ∥ 2 + 4 α Z t τ ∥ u n ( r ) ∥ 4 dr + 2 β Z t τ ∥ v n ( r ) ∥ 2 dr ≤ ∥ u τ ∥ 4 + ∥ v τ ∥ 2 + 4 Z t τ ∥ u n ( r ) ∥ 2 | ( f ( r ) , u n ( r )) | dr + 2 Z t τ | ( g ( r ) , v n ( r )) | dr 10 + 2 Z t τ | ( G n ( u n ( r )) , v n ( r )) | dr + 6 ε 2 Z t τ ∥ u n ( r ) ∥ 2 ∞ X k =1  ∥ h n k ( u n ( r )) + b k ( r ) ∥ 2  dr + ε 2 ∞ X k =1 Z t τ ∥ σ n k  v n ( r )  + γ k ( r ) ∥ 2 dr + 2 ε      ∞ X k =1 Z t τ  v n ( r ) ,  σ n k  v n ( r )  + γ k ( r )  dW k ( r )       + 4 ε      ∞ X k =1 Z t τ ∥ u n ( r ) ∥ 2 ( u n ( r ) , ( h n k ( u n ( r )) + b k ( r )) dW k ( r ))      . (3.32) W e deal with the third to seven th terms on the right-hand side of (3.32), b y H¨ older’s inequality and Y oung’s inequality we can infer that a.s. 4 Z t τ ∥ u n ( r ) ∥ 2 | ( f ( r ) , u n ( r )) | dr ≤ α Z t τ ∥ u n ( r ) ∥ 4 dr + c Z t τ ∥ f ( r ) ∥ 4 dr , 2 Z t τ | ( g ( r ) , v n ( r )) | dr ≤ β 2 Z t τ ∥ v n ( r ) ∥ 2 dr + c Z t τ ∥ g ( r ) ∥ 2 dr , 2 Z t τ | ( G n ( u n ( r )) , v n ( r )) | dr ≤ c Z t τ ∥ u n ( r ) ∥ 4 dr + β 2 Z t τ ∥ v n ( r ) ∥ 2 dr , (3.33) and b y (3.10) w e ha ve 6 ε 2 Z t τ ∥ u n ( r ) ∥ 2 ∞ X k =1  ∥ h n k ( u n ( r )) + b k ( r ) ∥ 2  dr + 2 ε 2 ∞ X k =1 Z t τ ∥ σ n k  v n ( r )  + γ k ( r ) ∥ 2 dr ≤ 12 ε 2 0 Z t τ ∞ X k =1  ∥ u n ( r ) ∥ 2 ∥ b k ( r ) ∥ 2 + ∥ γ k ( r ) ∥ 2  dr + 2 ε 2 0 Z t τ ∞ X k =1  ∥ u n ( r ) ∥ 2 ∥ h n k ( u n ( r )) ∥ 2 + ∥ σ n k  v n ( r )  ∥ 2  dr ≤ α Z t τ ∥ u n ( r ) ∥ 4 dr + c Z t τ ∞ X k =1 ∥ b k ( r ) ∥ 2 ! 2 dr + 12 ε 2 0 Z t τ ∞ X k =1 ∥ γ k ( r ) ∥ 2 dr + 6 ε 2 0 ∥ δ ∥ 2 ( t − τ ) + 6 ε 2 0 ∥ δ ∥ 2 Z t τ ∥ u n ( r ) ∥ 4 dr + 4 ε 2 0 ∥ δ ∥ 2 Z t τ ∥ v n ( r ) ∥ 2 dr ≤ α Z t τ ∥ u n ( r ) ∥ 4 dr + c Z t τ ∞ X k =1 ∥ b k ( r ) ∥ 2 ! 2 + ∞ X k =1 ∥ γ k ( r ) ∥ 2 ! 2 dr + c ( t − τ ) + c Z t τ ∥ u n ( r ) ∥ 4 dr + c Z t τ ∥ v n ( r ) ∥ 2 dr . (3.34) According to (3.32)-(3.34), w e conclude that for all t ∈ [ τ , τ + T ], E  sup τ ≤ r ≤ t  ∥ u n ( r ) ∥ 4 + ∥ v n ( r ) ∥ 2   ≤ E  ∥ u τ ∥ 4 + ∥ v τ ∥ 2  + c Z t τ  ∥ f ( r ) ∥ 4 + ∥ g ( r ) ∥ 4 + ∥ b ( r ) ∥ 4 + ∥ γ ( r ) ∥ 4  dr + c ( t − τ ) + c Z t τ E  sup τ ≤ r ≤ s  ∥ u n ( r ) ∥ 4 + ∥ v n ( r ) ∥ 2   ds + 2 ε E " sup τ ≤ r ≤ t      ∞ X k =1 Z r τ  v n ( s ) ,  σ n k  v n ( s )  + γ k ( s )  dW k ( s )       # + 4 ε E " sup τ ≤ r ≤ t      ∞ X k =1 Z r τ ∥ u n ( s ) ∥ 2 ( u n ( s ) , ( h n k ( u n ( s )) + b k ( s )) dW k ( s ))      # . (3.35) F or the last t wo terms on the righ t-hand side of (3.35), b y the BDG inequality and (3.10), we can get 2 ε E " sup τ ≤ r ≤ t      ∞ X k =1 Z r τ  v n ( s ) ,  σ n k  v n ( s )  + γ k ( s )  dW k ( s )       # 11 ≤ 2 ε 0 E   Z t τ ∞ X k =1 ∥ v n ( r ) ∥ 2 ∥ σ n k  v n ( r )  + γ k ( r ) ∥ 2 dr ! 1 2   ≤ 1 2 Z t τ E  sup τ ≤ r ≤ s ∥ v n ( r ) ∥ 2  ds + c E " ∞ X k =1 Z t τ ∥ σ n k  v n ( r )  + γ k ( r ) ∥ 2 dr # ≤ 1 2 Z t τ E  sup τ ≤ r ≤ s ∥ v n ( r ) ∥ 2  ds + c Z t τ E  ∥ γ ( r ) ∥ 4  dr + c ( t − τ ) + c Z t τ E  ∥ v n ( r ) ∥ 2  dr , (3.36) and 4 ε E " sup τ ≤ r ≤ t      ∞ X k =1 Z s τ ∥ u n ( s ) ∥ 2 ( u n ( s ) , ( h n k ( u n ( s )) + b k ( s )) dW k ( r ))      # ≤ 4 ε 0 E   Z t τ ∞ X k =1 ∥ u n ( r ) ∥ 6 ∥ h n k  v n ( r )  + b k ( r ) ∥ 2 dr ! 1 2   ≤ 4 ε 0 E    sup τ ≤ r ≤ s ∥ u n ( r ) ∥ 3  Z t τ ∞ X k =1 ∥ h n k  v n ( s )  + b k ( s ) ∥ 2 ds ! 1 2   ≤ 4 ε 0 E    sup τ ≤ r ≤ s ∥ u n ( r ) ∥ 3  Z t τ ∞ X k =1 ∥ h n k  v n ( s )  + b k ( s ) ∥ 2 ds ! 1 2   ≤ 1 2 Z t τ E  sup τ ≤ r ≤ s ∥ u n ( r ) ∥ 4  dr + c ( t − τ ) E   Z t τ ∞ X k =1 ∥ h n k  u n ( r )  + b k ( r ) ∥ 2 dr ! 2   ≤ 1 2 Z t τ E  sup τ ≤ r ≤ s ∥ u n ( r ) ∥ 4  dr + c ( t − τ ) E  Z t τ ∥ u n ( r ) ∥ 4 dr  + c ( t − τ ) 2 + c ( t − τ ) Z t τ E  ∥ b ( r ) ∥ 4  dr . (3.37) Note that the ab ov e all c is indep endent of τ , φ τ , n and T . Thus, by (3.35)-(3.37) w e obtain for all t ∈ [ τ , τ + T ], E  ∥ u ( r ) ∥ 4 + ∥ u ( r ) ∥ 2 + ∥ v ( r ) ∥ 2  ≤ E  ∥ u τ ∥ 4 + ∥ u τ ∥ 2 + ∥ v τ ∥ 2  + c Z t τ E  ∥ f ( r ) ∥ 4 + ∥ g ( r ) ∥ 4 + ∥ b ( r ) ∥ 4 + ∥ γ ( r ) ∥ 4  dr + c + c Z t τ E  ∥ u ( r ) ∥ 4 + ∥ u ( r ) ∥ 2 + ∥ v ( r ) ∥ 2  ds. (3.38) Applying Gron wall’s inequalit y to (3.38), we hav e E  sup τ ≤ r ≤ t  ∥ u n ( r ) ∥ 4 + ∥ v n ( r ) ∥ 2   ≤ e c e c (1+ T ) 2 , (3.39) where e c = 2 E  ∥ u τ ∥ 4 + ∥ v τ ∥ 2  + c (1 + T ) Z t τ E  ∥ f ( r ) ∥ 4 + ∥ g ( r ) ∥ 4 + ∥ b ( r ) ∥ 4 + ∥ γ ( r ) ∥ 4  dr + c (1 + T ) 2 . It follo ws from (3.39) that (3.27) holds. Step 2. W e no w pro ve (3.29). F or an arbitrary T ∈ N , b y (3.12) w e ha ve { τ n < τ + T } ⊆  sup τ ≤ r ≤ τ + T ( ∥ u n ( r ) ∥ + ∥ v n ( r ) ∥ ) ≥ n  . Then, b y Cheb ychev’s inequality , Y oung’s inequalit y and (3.27) we get P { τ n < τ + T } ≤ P  sup τ ≤ r ≤ τ + T ( ∥ u n ( r ) ∥ + ∥ v n ( r ) ∥ ) ≥ n  ≤ 2 n 2 E  sup τ ≤ s ≤ τ + T  ∥ u n ( s ) ∥ 4 + ∥ v n ( s ) ∥ 2   + 1 2 n 2 ≤ 4 ρ 0 + 1 2 n 2 , (3.40) 12 where ρ 0 , independent of n , is the same num b er as in (3.27). F rom (3.40), we obtain ∞ X n =1 P { τ n < τ + T } ≤ 4 ρ 0 + 1 2 ∞ X n =1 1 n 2 < ∞ . (3.41) Setting Ω T = ∞ T l =1 ∞ T n = l { τ n < τ + T } . Combining (3.41) with the Borel-Cantelli lemma yields P (Ω T ) = P ∞ \ l =1 ∞ \ n = l { τ n < τ + T } ! = 0 , whic h shows that there exists a subset Ω T of Ω such that P (Ω T ) = 0 and for each ω ∈ Ω \ Ω T , there exists n 0 = n 0 ( ω ) > 0 suc h that τ n ( ω ) ≥ τ + T for all n ≥ n 0 . Th us, we get τ ( ω ) ≥ τ + T , ∀ ω ∈ Ω \ Ω T . Let Ω 0 = ∞ S T =1 Ω T . Then P (Ω 0 ) = 0, and for all ω ∈ Ω \ Ω 0 and all T ∈ N , w e hav e τ ( ω ) ≥ τ + T . Hence, we deduce that τ ( ω ) = ∞ for all ω ∈ Ω \ Ω 0 , from whic h w e get that (3.29) holds. This completes the proof. In what follows, we address the existence and uniqueness of solutions to system (2.7). Theorem 3.5. Supp ose that ( H 1 ) − ( H 3 ) hold, and let φ τ ∈ L 4 (Ω , ℓ 2 c ) × L 2 (Ω , ℓ 2 ) b e F τ -me asur able for every τ ∈ R . Then, for e ach T > 0 , the non-autonomous sto chastic system (2.7) p ossesses a unique solution φ ( t, τ , φ τ ) = ( u ( t, τ , u τ ) , v ( t, τ , v τ )) T in the sense of Deﬁnition 3.1. Mor e over, the solution φ satisﬁes E h ∥ u ∥ 4 C ([ τ ,τ + T ] ,ℓ 2 c ) + ∥ v ∥ 2 C ([ τ ,τ + T ] ,ℓ 2 ) i (3.42) ≤ ρ 1 e ρ 1 T T + E  ∥ u τ ∥ 4 + ∥ v τ ∥ 2  + Z τ + T τ E  ∥ g ( t ) ∥ 4 + ∥ f ( t ) ∥ 4 + ∥ b ( t ) ∥ 4 + ∥ γ ( t ) ∥ 4  dt ! , wher e ρ 1 > 0 is a c onstant indep endent of u τ , τ and T . Pr o of. By Lemmas 3.2 and 3.4, w e ﬁnd that there exists a subset Ω 1 of Ω with P (Ω 1 ) = 0 such that, for all n ∈ N , ω ∈ Ω \ Ω 1 , and n ≥ n 0 , it holds τ ( ω ) = lim n →∞ τ n ( ω ) = ∞ and φ n +1 τ n ( t, ω ) = φ n τ n ( t, ω ). It follows that for each ω ∈ Ω 1 and t ≥ τ , there exists n 0 = n 0 ( t, ω ) ≥ 1 suc h that τ n ( ω ) > t and φ n ( t, ω ) = φ n 0 ( t, ω ) , ∀ n ≥ n 0 . (3.43) Deﬁne a mapping φ : R τ × Ω → ℓ 2 c × ℓ 2 b y φ ( t, ω ) = ( φ n ( t, ω ) , ω ∈ Ω 1 , t ∈ [ τ , τ n ( ω )] , φ τ ( ω ) , ω ∈ Ω \ Ω 1 , t ∈ R τ . (3.44) Then, it is easy to c heck that the mapping φ deﬁned in (3.44) is w ell-deﬁned. Moreo ver, since φ n is a contin uous ℓ 2 c × ℓ 2 -v alued pro cess, it follows from (3.44) that φ is also an almost surely contin uous ℓ 2 c × ℓ 2 -v alued pro cess in time t . By (3.44), we obtain that for every ﬁxed t ≥ τ , lim n →∞ φ n ( t, ω ) = φ ( t, ω ) , ∀ ω ∈ Ω 1 . (3.45) Observ e that the F t -adaptedness of φ inherits directly from that of φ n through (3.45). F urthermore, applying F atou’s lemma in conjunction with Lemma 3.2 and (3.45), w e obtain the for any T > 0, E h ∥ u ∥ 4 C ([ τ ,τ + T ] ,ℓ 2 c ) + ∥ v ∥ 2 C ([ τ ,τ + T ] ,ℓ 2 ) i ≤ ρ 0 , (3.46) where ρ 0 is from Lemma 3.4. 13 Next, we prov e that φ = ( u, v ) T is a solution of system (2.7). Recall that φ n = ( u n , v n ) T is the solution of system (3.11) and hence it satisﬁes φ n ( t ∧ τ n ) = φ τ + Z t ∧ τ n τ ( − A φ n ( r ) + G n ( φ n ( r ) , r )) dr + ε ∞ X k =1 Z t ∧ τ n τ H n k ( φ n ( r ) , r ) dW k ( r ) . (3.47) By (3.44) w e know that φ n ( t ∧ τ n ) = φ ( t ∧ τ n ) almost surely . Thus, w e can obtain from the deﬁnitions of G n and H n k that a.s. G n ( φ n ( r ) , r ) = G ( φ ( r ) , r ) and H n k ( φ n ( r ) , r ) = H k ( φ ( r ) , r ) for r ∈ [ τ , τ n ] , whic h, together with (3.47), can deduce that a.s. φ ( t ∧ τ n ) = φ τ + Z t ∧ τ n τ ( − A φ ( r ) + G ( φ ( r ) , r )) dr + ε ∞ X k =1 Z t ∧ τ n τ H k ( φ ( r ) , r ) dW k ( r ) . (3.48) Thanks to lim n →∞ τ n = ∞ almost surely , it follows from (3.48) that for all t ≥ τ , φ ( t ) = φ τ + Z t τ ( − A φ ( r ) + G ( φ ( r ) , r )) dr + ε ∞ X k =1 Z t τ H k ( φ ( r ) , r ) dW k ( r ) P -a.s. , whic h, along with (3.48), implies that φ is a solution of system (2.7) in the sense of Deﬁnition 3.1. Finally , w e turn to the uniqueness of solutions of system (2.7). Let φ 1 = ( u 1 , v 1 ) T and φ 2 = ( u 2 , v 2 ) T b e tw o solutions of system (2.7) in the sense of Deﬁnition 3.1. W e are committed to proving the following assertion, i.e., for an y T > 0, it holds P  ∥ φ 1 ( t ) − φ 2 ( t ) ∥ ℓ 2 c × ℓ 2 = 0 for all t ∈ [ τ , τ + T ]  = P  ∥ u 1 ( t ) − u 2 ( t ) ∥ + ∥ v 1 ( t ) − v 2 ( t ) ∥ = 0 for all t ∈ [ τ , τ + T ]  = 0 . (3.49) F or each n ∈ N , τ ∈ R and T > 0, we deﬁne the following stopping time: T n = ( τ + T ) ∧ inf { t ≥ τ : ∥ φ 1 ( t ) ∥ ≥ n or ∥ φ 2 ( t ) ∥ ≥ n } . (3.50) By (2.7) we obtain that a.s., u 1 ( t ∧ T n ) − u 2 ( t ∧ T n ) + α Z t ∧ T n τ  u 1 ( r ) − u 2 ( r )  dr + i Z t ∧ T n τ  Au 1 ( r ) − Au 2 ( r )  dr + i Z t ∧ T n τ  F  u 1 ( r ) , v 1 ( r )  − F  u 2 ( r ) , v 2 ( r )  dr = u 1 ( τ ) − u 2 ( τ ) − iε ∞ X k =1 Z t ∧ T n τ  h k  u 1 ( r )  − h k  u 2 ( r )  dW k ( r ) , (3.51) and v 1 ( t ∧ T n ) − v 2 ( t ∧ T n ) + β Z t ∧ T n τ  v 1 ( r ) − v 2 ( r )  dr + Z t ∧ T n τ  G  u 1 ( r )  − G  u 2 ( r )  dr = v 1 ( τ ) − v 2 ( τ ) + ε ∞ X k =1 Z t ∧ T n τ  σ k  v 1 ( r )  − σ k  v 2 ( r )  dW k ( r ) . (3.52) Using (3.51) and applying Ito’s form ula to the pro cess ∥ u 1 ( t ∧ T n ) − u 2 ( t ∧ T n ) ∥ 2 + ∥ u 1 ( t ∧ T n ) − u 2 ( t ∧ T n ) ∥ 4 , b y taking the real part we obtain ∥ u 1 ( t ∧ T n ) − u 2 ( t ∧ T n ) ∥ 2 + ∥ u 1 ( t ∧ T n ) − u 2 ( t ∧ T n ) ∥ 4 + 2 α Z t ∧ T n τ ∥ u 1 ( r ) − u 2 ( r ) ∥ 2 dr + 4 α Z t ∧ T n τ ∥ u 1 ( r ) − u 2 ( r ) ∥ 4 dr ≤ ∥ u 1 ( τ ) − u 2 ( τ ) ∥ 2 + ∥ u 1 ( τ ) − u 2 ( τ ) ∥ 4 + ε 2 ∞ X k =1 Z t ∧ T n τ ∥ h k  u 1 ( r )  − h k  u 2 ( r )  ∥ 2 dr 14 + 6 ε 2 ∞ X k =1 Z t ∧ T n τ ∥ u 1 ( r ) − u 2 ( r ) ∥ 2 ∥ h k  u 1 ( r )  − h k  u 2 ( r )  ∥ 2 dr + 2 Z t ∧ T n τ    F  u 1 ( r ) , v 1 ( r )  − F  u 2 ( r ) , v 2 ( r )  , u 1 ( r ) − u 2 ( r )    dr + 4 Z t ∧ T n τ ∥ u 1 ( r ) − u 2 ( r ) ∥ 2    F  u 1 ( r ) , v 1 ( r )  − F  u 2 ( r ) , v 2 ( r )  , u 1 ( r ) − u 2 ( r )    dr + 2 ε ∞ X k =1 Z t ∧ T n τ Im  u 1 ( r ) − u 2 ( r )   h k  u 1 ( r )  − h k  u 2 ( r )  dW k ( r ) + 4 ε ∞ X k =1 Z t ∧ T n τ ∥ u 1 ( r ) − u 2 ( r ) ∥ 2 Im  u 1 ( r ) − u 2 ( r )   h k  u 1 ( r )  − h k  u 2 ( r )  dW k ( r ) . (3.53) By (3.52) and Ito’s form ula w e can infer ∥ v 1 ( t ∧ T n ) − v 2 ( t ∧ T n ) ∥ 2 + 2 β Z t ∧ T n τ ∥ v 1 ( r ) − v 2 ( r ) ∥ 2 dr + 2 Z t ∧ T n τ  G  u 1 ( r )  − G  u 2 ( r )  , v 1 ( r ) − v 2 ( r )  dr = ∥ v 1 ( τ ) − v 2 ( τ ) ∥ 2 + ε 2 ∞ X k =1 Z t ∧ T n τ ∥ σ k  v 1 ( r )  − σ k  v 2 ( r )  ∥ 2 dr + 2 ε ∞ X k =1 Z t ∧ T n τ  v 1 ( r ) − v 2 ( r )   σ k  v 1 ( r )  − σ n  v 2 ( r )  dW k ( r ) . (3.54) By H¨ older’s inequalit y , Y oung’s inequalit y and (2.5), it is easy to derive that there exists c 5 = c 5 ( n ) > 0 suc h that 2 Z t ∧ T n τ    F  u 1 ( r ) , v 1 ( r )  − F  u 2 ( r ) , v 2 ( r )  , u 1 ( r ) − u 2 ( r )    dr + 4 Z t ∧ T n τ ∥ u 1 ( r ) − u 2 ( r ) ∥ 2    F  u 1 ( r ) , v 1 ( r )  − F  u 2 ( r ) , v 2 ( r )  , u 1 ( r ) − u 2 ( r )    dr ≤ Z t ∧ T n τ  ∥ F  u 1 ( r ) , v 1 ( r )  − F  u 2 ( r ) , v 2 ( r )  ∥ 2 + ∥ u 1 ( r ) − u 2 ( r ) ∥ 2  dr + 2 Z t ∧ T n τ ∥ u 1 ( r ) − u 2 ( r ) ∥ 2  ∥ F  u 1 ( r ) , v 1 ( r )  − F  u 2 ( r ) , v 2 ( r )  ∥ 2 + ∥ u 1 ( r ) − u 2 ( r ) ∥ 2  dr ≤ ( c 3 ( n ) + 1) Z t ∧ T n τ ∥ u 1 ( r ) − u 2 ( r ) ∥ 2 dr + c 3 ( n ) (4 c 5 + 1) Z t ∧ T n τ ∥ v 1 ( r ) − v 2 ( r ) ∥ 2 dr + 2( c 3 ( n ) + 1) Z t ∧ T n τ ∥ u 1 ( r ) − u 2 ( r ) ∥ 4 dr , (3.55) and      2 Z t ∧ T n τ  G  u 1 ( r )  − G  u 2 ( r )  , v 1 ( r ) − v 2 ( r )  dr      ≤ Z t ∧ T n τ ∥ G  u 1 ( r )  − G  u 2 ( r )  ∥ 2 dr + Z t ∧ T n τ ∥ v 1 ( r ) − v 2 ( r ) ∥ 2 dr ≤ c 4 ( n ) Z t ∧ T n τ ∥ u 1 ( r ) − u 2 ( r ) ∥ 2 dr + Z t ∧ T n τ ∥ v 1 ( r ) − v 2 ( r ) ∥ 2 dr . (3.56) By (2.2), (2.5) and (3.53)-(3.56), we obtain that there exists c 6 = c 6 ( n ) = max { c 1 ε 2 0 + c 3 ( n ) + 1 + c 4 ( n ) , 6 c 1 ε 2 0 + 2( c 3 ( n ) + 1) , c 3 ( n ) (4 c 5 + 1) + c 2 ε 2 0 + 1 } suc h that the following inequality holds ∥ u 1 ( t ∧ T n ) − u 2 ( t ∧ T n ) ∥ 2 + ∥ u 1 ( t ∧ T n ) − u 2 ( t ∧ T n ) ∥ 4 + ∥ v 1 ( t ∧ T n ) − v 2 ( t ∧ T n ) ∥ 2 15 ≤ ∥ u 1 ( τ ) − u 2 ( τ ) ∥ 2 + ∥ u 1 ( τ ) − u 2 ( τ ) ∥ 4 + ∥ v 1 ( τ ) − v 2 ( τ ) ∥ 2 + c 6 Z t ∧ T n τ  ∥ u 1 ( r ) − u 2 ( r ) ∥ 2 + ∥ u 1 ( r ) − u 2 ( r ) ∥ 4 + ∥ v 1 ( r ) − v 2 ( r ) ∥ 2  dr + 2 ε ∞ X k =1 Z t ∧ T n τ Im  u 1 ( r ) − u 2 ( r )   h k  u 1 ( r )  − h k  u 2 ( r )  dW k ( r ) + 4 ε ∞ X k =1 Z t ∧ T n τ ∥ u 1 ( r ) − u 2 ( r ) ∥ 2 Im  u 1 ( r ) − u 2 ( r )   h k  u 1 ( r )  − h k  u 2 ( r )  dW k ( r ) + 2 ε ∞ X k =1 Z t ∧ T n τ  v 1 ( r ) − v 2 ( r )   σ k  v 1 ( r )  − σ n  v 2 ( r )  dW k ( r ) , from whic h w e ha ve E  sup τ ≤ r ≤ t  ∥ u 1 T n ( r ) − u 2 T n ( r ) ∥ 2 + ∥ u 1 T n ( r ) − u 2 T n ( r ) ∥ 4 + ∥ v 1 T n ( r ) − v 2 T n ( r ) ∥ 2   ≤ E  ∥ u 1 ( τ ) − u 2 ( τ ) ∥ 2 + ∥ u 1 ( τ ) − u 2 ( τ ) ∥ 4 + ∥ v 1 ( τ ) − v 2 ( τ ) ∥ 2  + c 6 Z t τ E  sup τ ≤ r ≤ s  ∥ u 1 T n ( r ) − u 2 T n ( r ) ∥ 2 + ∥ u 1 T n ( r ) − u 2 T n ( r ) ∥ 4 + ∥ v 1 T n ( r ) − v 2 T n ( r ) ∥ 2   ds + 2 ε E " sup τ ≤ s ≤ t ∧ T n      ∞ X k =1 Z s τ Im  u 1 ( r ) − u 2 ( r )   h k  u 1 ( r )  − h k  u 2 ( r )  dW k ( r )      # + 4 ε E " sup τ ≤ s ≤ t ∧ T n      ∞ X k =1 Z s τ ∥ u 1 ( r ) − u 2 ( r ) ∥ 2 Im  u 1 ( r ) − u 2 ( r )   h k  u 1 ( r )  − h k  u 2 ( r )  dW k ( r )      # + 2 ε E " sup τ ≤ s ≤ t ∧ T n      ∞ X k =1 Z s τ  v 1 ( r ) − v 2 ( r )   σ k  v 1 ( r )  − σ n  v 2 ( r )  dW k ( r )      # . (3.57) W e deal with the last three terms on the righ t-hand side of (3.57). Using (2.2) and the BDG inequalit y can deriv e that for all t ∈ [ τ , τ + T ] with T > 0, it holds 2 ε E " sup τ ≤ s ≤ t ∧ T n      ∞ X k =1 Z s τ Im  u 1 ( r ) − u 2 ( r )   h k  u 1 ( r )  − h k  u 2 ( r )  dW k ( r )      # + 4 ε E " sup τ ≤ s ≤ t ∧ T n      ∞ X k =1 Z s τ ∥ u 1 ( r ) − u 2 ( r ) ∥ 2 Im  u 1 ( r ) − u 2 ( r )   h k  u 1 ( r )  − h k  u 2 ( r )  dW k ( r )      # ≤ 8 √ 2 ε 0 E   Z t ∧ T n τ ∥ u 1 ( r ) − u 2 ( r ) ∥ 2 ∞ X k =1 ∥ h k  u 1 ( r )  − h k  u 2 ( r )  ∥ 2 dr ! 1 2   + 16 √ 2 ε 0 E   Z t ∧ T n τ ∥ u 1 ( r ) − u 2 ( r ) ∥ 6 ∞ X k =1 ∥ h k  u 1 ( r )  − h k  u 2 ( r )  ∥ 2 dr ! 1 2   ≤ 8 √ 2 ε 0 E   sup τ ≤ r ≤ t ∥ u 1 T n ( r ) − u 2 T n ( r ) ∥ Z t ∧ T n τ ∞ X k =1 ∥ h k  u 1 ( r )  − h k  u 2 ( r )  ∥ 2 dr ! 1 2   + 16 √ 2 ε 0 E   sup τ ≤ r ≤ t ∥ u 1 T n ( r ) − u 2 T n ( r ) ∥ 3 Z t ∧ T n τ ∞ X k =1 ∥ h k  u 1 ( r )  − h k  u 2 ( r )  ∥ 2 dr ! 1 2   ≤ 1 2 E  sup τ ≤ r ≤ t  ∥ u 1 T n ( r ) − u 2 T n ( r ) ∥ 2 + ∥ u 1 T n ( r ) − u 2 T n ( r ) ∥ 4   + 64 ε 2 0 c 1 Z t τ E  sup τ ≤ r ≤ s ∥ u 1 T n ( r ) − u 2 T n ( r ) ∥ 2  ds +  2 3  − 3 (16 ε 0 ) 4 c 2 1 T Z t τ E  sup τ ≤ r ≤ s ∥ u 1 T n ( r ) − u 2 T n ( r ) ∥ 4  ds, and 2 ε E " sup τ ≤ s ≤ t ∧ T n      ∞ X k =1 Z s τ  v 1 ( r ) − v 2 ( r )   σ k  v 1 ( r )  − σ n  v 2 ( r )  dW k ( r )      # ≤ 8 √ 2 ε 0 E   sup τ ≤ r ≤ t ∥ v 1 T n ( r ) − v 2 T n ( r ) ∥ Z t ∧ T n τ ∞ X k =1 ∥ σ k  v 1 ( r )  − σ k  v 2 ( r )  ∥ 2 dr ! 1 2   16 ≤ 1 2 E  sup τ ≤ r ≤ t  ∥ v 1 T n ( r ) − v 2 T n ( r ) ∥ 2   + 64 ε 2 0 c 2 Z t τ E  sup τ ≤ r ≤ s ∥ v 1 T n ( r ) − v 2 T n ( r ) ∥ 2  ds. Therefore, w e can obtain that there exists c 7 = c 6 + 64 ε 2 0 ( c 1 + c 2 ) +  2 3  − 3 (16 ε 0 ) 4 c 2 1 suc h that for all t ∈ [ τ , τ + T ], E  sup τ ≤ r ≤ t  ∥ u 1 T n ( r ) − u 2 T n ( r ) ∥ 2 + ∥ u 1 T n ( r ) − u 2 T n ( r ) ∥ 4 + ∥ v 1 T n ( r ) − v 2 T n ( r ) ∥ 2   ≤ 2 E  ∥ u 1 ( τ ) − u 2 ( τ ) ∥ 2 + ∥ u 1 ( τ ) − u 2 ( τ ) ∥ 4 + ∥ v 1 ( τ ) − v 2 ( τ ) ∥ 2  + 2 c 7 (1 + T ) · Z t τ E  sup τ ≤ r ≤ s  ∥ u 1 T n ( r ) − u 2 T n ( r ) ∥ 2 + ∥ u 1 T n ( r ) − u 2 T n ( r ) ∥ 4 + ∥ v 1 T n ( r ) − v 2 T n ( r ) ∥ 2   ds. (3.58) By applying Gronw all’s lemma to (3.58), we hav e E  sup τ ≤ r ≤ τ + T  ∥ u 1 T n ( r ) − u 2 T n ( r ) ∥ 2 + ∥ u 1 T n ( r ) − u 2 T n ( r ) ∥ 4 + ∥ v 1 T n ( r ) − v 2 T n ( r ) ∥ 2   ≤ 2 e 2 c 7 (1+ T ) T E  ∥ u 1 ( τ ) − u 2 ( τ ) ∥ 2 + ∥ u 1 ( τ ) − u 2 ( τ ) ∥ 4 + ∥ v 1 ( τ ) − v 2 ( τ ) ∥ 2  . (3.59) Owing to φ 1 ( τ ) = φ 2 ( τ ) in L 4 (Ω , ℓ 2 c ) × L 2 (Ω , ℓ 2 ), w e can derive from (3.59) that E  sup τ ≤ r ≤ τ + T  ∥ u 1 T n ( r ) − u 2 T n ( r ) ∥ 4 + ∥ v 1 T n ( r ) − v 2 T n ( r ) ∥ 2   = 0 , from which we ha ve ∥ u 1 T n ( r ) − u 2 T n ( r ) ∥ + ∥ v 1 T n ( r ) − v 2 T n ( r ) ∥ = 0 for all t ∈ [ τ , τ + T ] almost surely . W e ﬁnd T n = τ + T for large enough n by the con tinuit y of φ 1 and φ 2 in t . Consequen tly , we conclude that ∥ u 1 ( t ) − u 2 ( t ) ∥ + ∥ v 1 ( t ) − v 2 ( t ) ∥ = 0 for all t ∈ [ τ , τ + T ] almost surely; that is, ∥ φ 1 ( t ) − φ 2 ( t ) ∥ = 0 for all t ∈ [ τ , τ + T ] almost surely . This establishes (3.49). By the arbitrariness of T , it follo ws from (3.59) that P ( ∥ φ 1 ( t ) − φ 2 ( t ) ∥ = 0 for all t ≥ τ ) = 1 , whic h implies the uniqueness of solutions. In particular, simlar to the argument (3.27), we can also verify (3.42). This proof is ﬁnished. 4 Existence of w eak pullback attractors In this section, w e focus on studying the existence and uniqueness of w eak pullbac k mean random attractors of system (2.7) in L 4 (Ω , ℓ 2 c ) × L 2 (Ω , ℓ 2 ) o ver (Ω , F , {F t } t ∈ R , P ). T o this end, w e supp ose that α − 18 λ 2 β > 0 . (4.1) F rom now on, w e will b egin b y deﬁning a mean random dynamical system for system (2.7), whic h will serve as the basis for inv estigating the existence and uniqueness of w eak D -pullback mean random attractors. F or every τ ∈ R and t ∈ R + , we deﬁne the mapping Φ( t, τ ) : L 4 (Ω , ℓ 2 c ) × L 2 (Ω , ℓ 2 ) → L 4 (Ω , ℓ 2 c ) × L 2 (Ω , ℓ 2 ) b y Φ( t, τ , φ τ ) = φ ( t + τ , τ , φ τ ) , ∀ φ τ ∈ L 4 (Ω , ℓ 2 c ) × L 2 (Ω , ℓ 2 ) , (4.2) where φ ( t + τ , τ , φ τ ) denotes the solution of system (2.7) with initial data φ τ . It is easy to see that Φ(0 , τ ) is the iden tity operator on L 4 (Ω , ℓ 2 c ) × L 2 (Ω , ℓ 2 ). Additionally , by the uniqueness of solution of (2.7) w e ﬁnd that for an y τ ∈ R and s, t ∈ R + , Φ( t + s, τ , φ τ ) = Φ( t, s + τ , Φ( s, τ , φ τ )) , i.e., Φ( t + s, τ ) = Φ( t, s + τ ) ◦ Φ( s, τ ). Therefore, Φ is called a mean random dynamical system asso ciated with (2.7) on L 4 (Ω , ℓ 2 c ) × L 2 (Ω , ℓ 2 ) o v er (Ω , F , {F t } t ∈ R , P ) b y [39, Deﬁnition 2 . 9]. In order to a void confusion with symbols, w e write Φ( t + s, τ , φ τ ) = Φ( t + s, τ ) φ τ . 17 Let B = { B ( τ ) ⊆ L 4 (Ω , ℓ 2 c ) × L 2 (Ω , ℓ 2 ) : τ ∈ R } denote a family of all nonempt y b ounded sets of L 4 (Ω , ℓ 2 c ) × L 2 (Ω , ℓ 2 ) suc h that lim τ →−∞ e κτ ∥ B ( τ ) ∥ 2 L 4 (Ω ,ℓ 2 c ) × L 2 (Ω ,ℓ 2 ) = 0 , (4.3) where ∥ B ( τ ) ∥ L 4 (Ω ,ℓ 2 c ) × L 2 (Ω ,ℓ 2 ) = sup φ ∈ B ( τ ) ∥ φ ∥ L 4 (Ω ,ℓ 2 c ) × L 2 (Ω ,ℓ 2 ) . W e denote by D the collection of all families of nonempty b ounded sets of L 4 (Ω , ℓ 2 c ) × L 2 (Ω , ℓ 2 ), it is deﬁned b y D = { B = { B ( τ ) ⊆ L 4 (Ω , ℓ 2 c ) × L 2 (Ω , ℓ 2 ) : B ( τ )  = ∅ b ounded , τ ∈ R } : B satisﬁes (4 . 3) } . (4.4) T o obtain the existence of weak D -pullbac k mean random attractors of problem (2.7), we further suppose Z τ −∞ e κr E  ∥ f ( r ) ∥ 4 + ∥ g ( r ) ∥ 4 + ∥ b ( r ) ∥ 4 + ∥ γ ( r ) ∥ 4  dr < ∞ , ∀ τ ∈ R . (4.5) W e derive the follo wing uniform estimates of solution to (2.7) in L 4 (Ω , ℓ 2 c ) × L 2 (Ω , ℓ 2 ). Lemma 4.1. Supp ose that ( H 1 ) − ( H 3 ) , (4.1) and (4.5) hold. Then ther e exists ε 0 = max ( r α 24 ∥ δ ∥ 2 , s β 48 ∥ δ ∥ 2 ) such that for any ε ∈ [0 , ε 0 ] , τ ∈ R and B = { B ( t ) } t ∈ R ∈ D , ther e exists T = T ( τ , B ) > 0 such that for al l t ≥ T , the solution φ ( t ) = ( u ( t ) , v ( t )) T of system (2.7) satisﬁes E h ∥ u ( τ , τ − t, u τ − t ) ∥ 4 + ∥ v ( τ , τ − t, v τ − t ) ∥ 2 i ≤ M 0 + M 0 Z τ −∞ e κ ( r − τ ) E  ∥ f ( t ) ∥ 4 + ∥ g ( t ) ∥ 4 + ∥ b ( r ) ∥ 4 + ∥ γ ( t ) ∥ 4  dr , (4.6) wher e φ τ − t = ( u τ − t , v τ − t ) T ∈ B ( τ − t ) , and M 0 > 0 is a c onstant indep endent of τ and B . Pr o of. Applying Ito’s form ula to the process ∥ u ( t ) ∥ p , by (2.7) and taking the real part w e obtain that a.s. d ∥ u ( t ) ∥ p + αp ∥ u ( t ) ∥ p dt = p ∥ u ( t ) ∥ p − 2 Im ( f ( t ) , u ( t )) dt + pε ∞ X k =1 ∥ u ( t ) ∥ p − 2 Im ( u ( t ) , h k ( u ( t )) + b k ( t )) dW k ( r ) + p 2 ε 2 ∥ u ( t ) ∥ p − 2 ∞ X k =1 ∥ h k ( u ( t )) + b k ( t ) ∥ 2 dt + p ( p − 2) 2 ε 2 ∥ u ( t ) ∥ p − 4 ∞ X k =1 ( u ( t ) , h k ( u ( t )) + b k ( t )) 2 dt. (4.7) Using Ito’s formula to the pro cess ∥ v ( t ) ∥ 2 , b y (2.7) w e get that a.s. ∥ v ( t ) ∥ 2 + 2 β ∥ v ( t ) ∥ 2 dt = 2 ( g ( t ) , v ( t )) dt − 2 ( G ( u ( t )) , v ( t )) dt + ε 2 ∞ X k =1 ∥ σ k  v ( t )  + γ k ( t ) ∥ 2 dt + 2 ε ∞ X k =1  v ( t ) , σ k  v ( t )  + γ k ( t )  dW k ( t ) . (4.8) Com bining with (4.7) and (4.8), and by taking expectation w e ha ve E  ∥ u ( t ) ∥ 4 + ∥ v ( t ) ∥ 2  + 4 α E  ∥ u ( t ) ∥ 4  dt + 2 β E  ∥ v ( t ) ∥ 2  dt ≤ 4 E  ∥ u ( t ) ∥ 2 | ( f ( t ) , u ( t )) |  dt + 2 E [ | ( g ( t ) , v ( t )) | ] dt + 2 E [ | ( G ( u ( t )) , v ( t )) | ] dt + 6 ε 2 E " ∥ u ( t ) ∥ 2 ∞ X k =1  ∥ h k ( u ( t )) + b k ( t ) ∥ 2  # dt + ε 2 ∞ X k =1 E  ∥ σ k  v ( t )  + γ k ( t ) ∥ 2  dt. (4.9) 18 W e no w estimate the second, third, and fourth terms on the right-hand side of (4.9). By Y oung’s inequalit y and H¨ older’s inequality , w e obtain 4 E  ∥ u ( t ) ∥ 2 | ( f ( t ) , u ( t )) |  ≤ α E  ∥ u ( t ) ∥ 4  + 27 α E  ∥ f ( t ) ∥ 4  , 2 E [ | ( g ( t ) , v ( t )) | ] ≤ β 2 E  ∥ v ( t ) ∥ 2  + 2 β E  ∥ g ( t ) ∥ 2  , 2 E [ | ( G ( u ( t )) , v ( t )) | ] ≤ β 2 E  ∥ v ( t ) ∥ 2  dr + 16 λ 2 β E  ∥ u ( t ) ∥ 4  . (4.10) F or the ﬁfth and sixth terms on the right-hand side of (4.9). By virtue of assumption ( H 3 ), Y oung’s inequalit y and H¨ older’s inequality , one gets 6 ε 2 E " ∥ u ( t ) ∥ 2 ∞ X k =1  ∥ h k ( u ( t )) + b k ( t ) ∥ 2  # + ε 2 ∞ X k =1 E  ∥ σ k  v ( t )  + γ k ( t ) ∥ 2  ≤ 12 ε 2 E " ∞ X k =1  ∥ u ( t ) ∥ 2 ∥ b k ( t ) ∥ 2 + ∥ γ k ( t ) ∥ 2  # + 12 ε 2 E " ∞ X k =1  ∥ u ( t ) ∥ 2 ∥ h k ( u ( t )) ∥ 2 + ∥ σ k  v ( t )  ∥ 2  # ≤  α + 24 ε 2 0 ∥ δ ∥ 2  E  ∥ u ( t ) ∥ 4  + 24 ε 2 0 ∥ δ ∥ 2 E  ∥ v ( t ) ∥ 2  + 36 ε 4 0 α E  ∥ b ( t ) ∥ 4  + 12 ε 2 0 E  ∥ γ ( t ) ∥ 2  + 48 ε 2 0 ∥ δ ∥ 2 ≤  α + 24 ε 2 0 ∥ δ ∥ 2  E  ∥ u ( t ) ∥ 4  + 24 ε 2 0 ∥ δ ∥ 2 E  ∥ v ( t ) ∥ 2  + 36 ε 4 0 α E  ∥ b ( t ) ∥ 4  + 12 ε 2 0 E  ∥ γ ( t ) ∥ 4  + 6 ε 2 0 (1 + 8 ∥ δ ∥ 2 ) . (4.11) Com bining with (4.9)-(4.11), let κ = min n α − 18 λ 2 β , β 2 o , w e see that d dt E  ∥ u ( t ) ∥ 4 + ∥ v ( t ) ∥ 2  + κ E  ∥ u ( t ) ∥ 4 + ∥ v ( t ) ∥ 2  ≤ e κ E  ∥ f ( t ) ∥ 4 + ∥ g ( t ) ∥ 4 + ∥ b ( r ) ∥ 4 + ∥ γ ( t ) ∥ 4  + e κ, (4.12) where e κ = max n 27 α , 2 β , 9 β 2 576 α ∥ δ ∥ 4 , β 4 ∥ δ ∥ 2 , β (1+8 ∥ δ ∥ 2 ) 8 ∥ δ ∥ 2 o . Multiplying (4.12) by e κt , and then in tegrating from τ − t ( t > 0) to τ , w e deduce that E h ∥ u ( τ , τ − t, u τ − t ) ∥ 4 + ∥ v ( τ , τ − t, v τ − t ) ∥ 2 i ≤ e − κt E h ∥ u τ − t ∥ 4 + ∥ v τ − t ∥ 2 i + e κe − κτ Z τ τ − t e κr E  ∥ f ( t ) ∥ 4 + ∥ g ( t ) ∥ 4 + ∥ b ( r ) ∥ 4 + ∥ γ ( t ) ∥ 4  dr + e κ κ . (4.13) Thanks to φ τ − t ∈ B ( τ − t ) and B = { B ( t ) } t ∈ R ∈ D , w e kno w lim t →∞ e − κt  E h ∥ u τ − t ∥ 4 + ∥ v τ − t ∥ 2 i ≤ lim t →∞ e − κt  E h ∥ B ( τ − t ) ∥ L 4 (Ω ,ℓ 2 c ) × L 2 (Ω ,ℓ 2 ) i = 0 . Th us, there exists T = T ( τ , B ) > 0 suc h that e − κt  E h ∥ u τ − t ∥ 4 + ∥ v τ − t ∥ 2 i ≤ e κ κ , ∀ t ≥ T , (4.14) whic h, together with (4.13) and (4.14), conclude the desired result (4.6). This pro of is ﬁnished. According to Lemma 4.1, we shall directly give the existence of w eakly compact D -pullback bounded absoring set. Lemma 4.2. Supp ose that ( H 1 ) − ( H 3 ) , (4.1) and (4.5) hold. Then the me an r andom dynamic al system Φ gener ate d by system (2.7) has a unique we akly c omp act D -pul lb ack b ounde d absorbing set K = {K ( τ ) : τ ∈ R } ∈ D in L 4 (Ω , ℓ 2 c ) × L 2 (Ω , ℓ 2 ) , her e for any given τ ∈ R , K ( τ ) is denote d as fol lows K ( τ ) = { φ ∈ L 4 (Ω , ℓ 2 c ) × L 2 (Ω , ℓ 2 ) : E  ∥ u ( t ) ∥ 4 + ∥ v ( t ) ∥ 2  ≤ R 0 ( τ ) } , (4.15) wher e R 0 ( τ ) = M 0 + M 0 R τ −∞ e κ ( r − τ ) E  ∥ f ( t ) ∥ 4 + ∥ g ( t ) ∥ 4 + ∥ b ( r ) ∥ 4 + ∥ γ ( t ) ∥ 4  dr . 19 Pr o of. Since K ( τ ) is a b ounded closed conv ex subset of the reﬂexive pro duct Banach space L 4 (Ω , ℓ 2 c ) × L 2 (Ω , ℓ 2 ), then it is obvious that K ( τ ) is weakly compact in L 4 (Ω , ℓ 2 c ) × L 2 (Ω , ℓ 2 ). In particular, for an y τ ∈ R and B = { B ( t ) } t ∈ R ∈ D , by Lemma 4.1 there exists T = T ( τ , B ) > 0 suc h that Φ( τ , τ − t, B ( τ − t )) = φ ( τ , τ − t, B ( τ − t )) ⊆ K ( τ ) , ∀ t ≥ T . W e ﬁnally v erify K ∈ D . F rom (4.6), we can get 0 ≤ lim τ →−∞ e κτ ∥K ( τ ) ∥ 2 L 4 (Ω , F τ ,ℓ 2 c ) × L 2 (Ω , F τ ,ℓ 2 ) ≤ lim τ →−∞ e κτ R 0 ( τ ) = 0 , whic h implies that K satisﬁes (4.3). This pro of is ﬁnished. Ultimately , we presen t a main theorem establishing the existence and uniqueness of a w eak D - pullbac k mean random attractor for the mean random dynamical system Φ. Theorem 4.3. Supp ose that ( H 1 ) − ( H 3 ) , (4.1) and (4.5) hold. Then the me an r andom dynamic al system Φ gener ate d by the system (2.7) p ossesses a unique D -pul lb ack me an r andom attr actor A = {A ( τ ) : τ ∈ R } ∈ D in L 4 (Ω , ℓ 2 c ) × L 2 (Ω , ℓ 2 ) over (Ω , F , {F t } t ∈ R , P ) . In p articular, for every τ ∈ R , A ( τ ) c an b e r epr esente d as : A ( τ ) = \ s ≥ 0 [ t ≥ s Φ( t, τ − t ) K ( τ − t ) w , wher e the closur e is taken with r esp e ct to the we ak top olo gy of L 4 (Ω , ℓ 2 c ) × L 2 (Ω , ℓ 2 ) . Pr o of. Combining with Lemmas 4.1 and 4.2, b y [39, Theorem 2 . 7] we can immediately get the existence and uniqueness of w eak D -pullback mean random attractors A ∈ D of Φ. This completes the pro of. 5 Existence of in v arian t measures In this section, we shall establish the existence of in v ariant measures of autonomous stochastic discrete long-w av e–short-wa ve resonance equation (2.7) driven by nonlinear noise. T o this end, we shall pro ceed with the argument in three stages: ( i ) deriving uniform estimates for solutions to problem (2.7) with initial time τ = 0; ( ii ) establishing the tightness of the family of probabilit y distributions of the solutions in ℓ 2 c × ℓ 2 ; ( iii ) applying the Krylov-Bogolyubov metho d to conclude the existence of an inv arian t measure. W e assume that the external forcing terms f , g , b k , γ k are independent of time t and the sample ω ∈ Ω in this section. Under this time-indep endent and sample-indep endent assumptions, w e present the follo wing assumption: ∥ f ∥ 2 + ∥ g ∥ 2 + ∞ X k =1 ∥ b k ∥ 2 + ∞ X k =1 ∥ γ k ∥ 2 < ∞ . (5.1) In order to clearly represen t the dependence of solutions on the noise in tensity ε ∈ [0 , ε 0 ], we denote b y φ ε ( t, 0 , φ 0 ) = ( u ε ( t, 0 , u 0 ) , v ε ( t, 0 , v 0 )) T the solution of system (2.7) with initial data φ 0 = ( u 0 , v 0 ) T at initial time 0. Next, w e shall ﬁrst give the uniform estimates of solution φ ε ( t, 0 , φ 0 ) of problem (2.7) in L 4 (Ω , ℓ 2 c ) × L 2 (Ω , ℓ 2 ) for all t ≥ 0 for c hecking the tigh tness of probabilit y distributions of solutions and the existence of in v ariant measures. Lemma 5.1. Supp ose that the assumptions ( H 2 ) , ( H 3 ) , (4.1) and (5.1) hold. Then for al l ε ∈ [0 , ε 0 ] with ε 0 = max  q α 24 ∥ δ ∥ 2 , q β 48 ∥ δ ∥ 2  , the solution φ ε ( t, 0 , φ 0 ) of system (2.7) with initial data φ 0 at inital time τ = 0 satisﬁes sup t ≥ 0 sup ε ∈ [0 ,ε 0 ] E  ∥ u ε ( t, 0 , u 0 ) ∥ 4 + ∥ v ε ( t, 0 , u 0 ) ∥ 2  ≤ E [ ∥ u 0 ∥ 4 + ∥ v 0 ∥ 2 ] + M 1  1 + ∥ f ∥ 4 + ∥ g ∥ 4 + ∥ b ∥ 4 + ∥ γ ∥ 4  , (5.2) wher e M 1 > 0 is a c onstant indep endent of ε , φ 0 , t . 20 Pr o of. Since f , g , b k , γ k are time-indep endent deterministic functions. Then by Gronw all’s inequality w e can obtain from (4.12) that for all t ≥ 0, E  ∥ u ε ( t ) ∥ 4 + ∥ v ε ( t ) ∥ 2  ≤ e − κt E  ∥ u 0 ∥ 4 + ∥ v 0 ∥ 2  + e κ κ  ∥ f ∥ 4 + ∥ g ∥ 4 + ∥ b ∥ 4 + ∥ γ ∥ 4 + 1  , from whic h w e can obtain the desired conclusion. This proof is ﬁnished. Next, we shall pro ve the tightness of the family of probability distributions of solutions on ℓ 2 c × ℓ 2 , whic h can be ac hieved by deriving the following uniform tail-estimates of solutions for system (2.7). Lemma 5.2. Supp ose that the assumptions ( H 2 ) , ( H 3 ) , (4.1) and (5.1) hold. Then for every c omp act subset K (:= e K × b K ) of ℓ 2 c × ℓ 2 ∗ , ε ∈ [0 , ε 0 ] with ε 0 = max  q α 24 ∥ δ ∥ 2 , q β 48 ∥ δ ∥ 2  and ζ > 0 , ther e exists a p ositive numb er N = N ( K , ζ ) ∈ N such that the solution φ ε ( t ; 0 , φ 0 ) of system (2.7) with φ 0 ∈ K satisﬁes sup n ≥ N sup t ≥ 0 sup ε ∈ [0 ,ε 0 ] sup φ 0 ∈ K      X | m |≥ n E  | u ε m ( t, 0 , u 0 ) | 2    2 + X | m |≥ n E  | v ε m ( t, 0 , v 0 ) | 2     < ζ , (5.3) for al l n ≥ N and t ≥ 0 . Pr o of. Let ρ : R → R be a smo oth function suc h that 0 ≤ ρ ( s ) ≤ 1 for all s ∈ R , and ρ ( s ) = ( 0 , for | s | ≤ 1 , 1 , for | s | ≥ 2 . (5.4) It is easy to v erify from (5.4) that there is a constant C 0 > 0 suc h that | ρ ′ ( s ) | ≤ C 0 for every s ∈ R . F or eac h n ∈ N and φ = ( φ m ) m ∈ Z , w e set ρ n =  ρ  m n  m ∈ Z and ρ n φ =  ρ  m n  φ m  m ∈ Z . W e also apply analogous notation to the other terms. By (2.7) w e get d ( ρ n φ ε ( t )) + A ( ρ n φ ε ( t )) dt = ρ n G ( φ ε ( t ) , t ) dt + ε ∞ X k =1 ρ n H k ( φ ε ( t ) , t ) dW k ( t ) . (5.5) Similar to (4.7), combining with Ito’s formula and (5.5), and by taking the real part of the resulting expression w e ha ve d ∥ ρ n u ε ( t ) ∥ p ≤ − pα ∥ ρ n u ε ( t ) ∥ p dt + p Im ( ∥ ρ n u ε ( t ) ∥ p − 2 ( f ( t ) , ρ 2 n u ε ( t ))) dt + pε ∞ X k =1 ∥ ρ n u ε ( t ) ∥ p − 2 Im  h k ( u ε ( t )) + b k , ρ 2 n z ε ( t )  dW k ( t ) + p ( p − 1) 2 ε 2 ∥ ρ n u ε ( t ) ∥ p − 2 ∞ X k =1 ∥ ρ n h k ( u ε ( t )) + ρ n b k ( t ) ∥ 2 dt, (5.6) and d ∥ ρ 2 n v ε ( t ) ∥ 2 ≤ − 2 β ∥ ρ 2 n v ε ( t ) ∥ 2 dt + 2 | ( G ( u ε ( t )) , ρ 4 n v ε ( t )) | dt + 2( g ( t ) , ρ 4 n v ε ( t )) dt + 2 ε ∞ X k =1  σ k ( v ε ( t )) + γ k , ρ 4 n v ε ( t )  dW k ( t ) + ε 2 ∞ X k =1   ρ 2 n σ k ( v ε ( t )) + ρ 2 n γ k   2 dt. (5.7) Com bining with (5.6)-(5.7) and then taking exp ectation, w e obtain d dt E  ∥ ρ n u ε ( t ) ∥ 4 + ∥ ρ 2 n v ε ( t ) ∥ 2  ≤ − 4 α E [ ∥ ρ n u ε ( t ) ∥ 4 ] − 2 β E [ ∥ ρ 2 n v ε ( t ) ∥ 2 ] + 2 E    G ( u ε ( t ) , ρ 4 n v ε ( t ))    + 4 E [ ∥ ρ n u ε ( t ) ∥ 2 Im ( f ( t ) , ρ 2 n u ε ( t ))] + 2 E [( g, ρ 4 n v ε ( t ))] + 6 ε 2 ∞ X k =1 E h ∥ ρ n u ε ( t ) ∥ 2 ∥ ρ n h k ( u ε ( t )) + ρ n b k ( t ) ∥ 2 i + ε 2 ∞ X k =1 E h   ρ 2 n σ k ( v ε ( t )) + ρ 2 n γ k   2 i . (5.8) ∗ e K and b K are compact subsets of ℓ 2 c and ℓ 2 , resp ectively . 21 F or the third term on the righ t-hand side of (5.8), we use Y oung’s and H¨ older’s inequalities to obtain that 2 E    ( G ( u ε ( t ) , ρ 4 n v ε ( t ))    ≤ 2 λ 2 β E h   ρ 2 n B ( | u ε ( t ) | 2 )   2 i + β 2 E h   ρ 2 n v ε ( t )   2 i = 2 λ 2 β E " X m ∈ Z  ρ 2 ( m n )( | u ε m +1 ( t ) | 2 − | u ε m ( t ) | 2 )  2 # + β 2 E h   ρ 2 n v ε ( t )   2 i ≤ 4 λ 2 β E " X m ∈ Z ρ 4 ( m n )( | u ε m +1 ( t ) | 4 + | u ε m ( t ) | 4 ) # + β 2 E h   ρ 2 n v ε ( t )   2 i ≤ 4 λ 2 β E " X m ∈ Z ρ 4 ( m n ) | u ε m +1 ( t ) | 4 # + 4 λ 2 β E " X m ∈ Z ρ 4 ( m n ) | u ε m ( t ) | 4 # + β 2 E h   ρ 2 n v ε ( t )   2 i ≤ 4 λ 2 β E   X m ∈ Z ρ 2 ( m n ) | u ε m +1 ( t ) | 2 ! 2   + 4 λ 2 β E  ∥ ρ n u ε ∥ 4  + β 2 E h   ρ 2 n v ε ( t )   2 i ≤ 4 λ 2 β E   2 X m ∈ Z ρ 2 ( m + 1 n ) | u ε m +1 ( t ) | 2 + 2 X m ∈ Z  ρ ( m n ) − ρ ( m + 1 n )  2 | u ε m +1 ( t ) | 2 ! 2   + 4 λ 2 β E  ∥ ρ n u ε ∥ 4  + β 2 E h   ρ 2 n v ε ( t )   2 i ≤ 8 λ 2 β E   2 X m ∈ Z  ρ ( m n ) − ρ ( m + 1 n )  2 | u ε m +1 ( t ) | 2 ! 2   + 36 λ 2 β E  ∥ ρ n u ε ∥ 4  + β 2 E h   ρ 2 n v ε ( t )   2 i ≤ 32 λ 2 C 4 0 n 4 β E h ∥ u ε ( t ) ∥ 2 i + 12 λ 2 β E  ∥ ρ n u ε ∥ 4  + β 2 E h   ρ 2 n v ε ( t )   2 i . (5.9) F or the fourth and ﬁfth terms on the righ t-hand side of (5.8), we hav e 4 E [ ∥ ρ n u ε ( t ) ∥ 2 Im ( f ( t ) , ρ 2 n u ε ( t ))] ≤ α E h ∥ ρ n u ε ( t ) ∥ 4 i + c E [ ∥ ρ n f ∥ 4 ] , (5.10) 2 E [ | ( g , ρ 4 n v ε ( t )) | ] ≤ β 2 E [   ρ 2 n v ε ( t )   2 ] + c E [ ∥ ρ 2 n g ∥ 2 ] . (5.11) F or the sixth term on the righ t-hand side of (5.8), by Y oung’s inequality , H¨ older’s inequality as well as the prop erties of smo oth function ρ deﬁned in (5.4), w e can deriv e that 6 ε 2 ∞ X k =1 E h ∥ ρ n u ε ( t ) ∥ 2 ∥ ρ n h k ( u ε ( t )) + ρ n b k ( t ) ∥ 2 i =6 ε 2 E " ∥ ρ n u ε ( t ) ∥ 2 ∞ X k =1 X m ∈ Z ρ 2 ( m n )( h k,m ( u ε m ( t )) + b k,m ) 2 # ≤ 12 ε 2 E " ∥ ρ n u ε ( t ) ∥ 2 ∞ X k =1 X m ∈ Z ρ 2 ( m n )(( h k,m ( u ε m ( t ))) 2 + | b k,m | 2 ) # ≤ 12 ε 2 E " ∥ ρ n u ε ( t ) ∥ 2 ∞ X k =1 X m ∈ Z ρ 2 ( m n )( h k,m ( u ε m ( t ))) 2 # + 12 ε 2 E " ∥ ρ n u ε ( t ) ∥ 2 ∞ X k =1 X m ∈ Z ρ 2 ( m n ) | b k,m | 2 # ≤ 12 ε 2 E " ∥ ρ n u ε ( t ) ∥ 2 ∞ X k =1 X m ∈ Z ρ 2 ( m n )( δ k,m (1 + | u ε m ( t ) | )) 2 # + 12 ε 2 E  ∥ ρ n u ε ( t ) ∥ 2 ∥ ρ n b ∥ 2  ≤ 24 ε 2 0 E " ∥ ρ n u ε ( t ) ∥ 2 ∞ X k =1 X m ∈ Z ρ 2 ( m n ) | δ k,m | 2 # + 24 ε 2 0 E " ∥ ρ n u ε ( t ) ∥ 2 ∞ X k =1 X m ∈ Z ρ 2 ( m n ) | δ k,m | 2 | u ε m ( t ) | 2 # + 12 ε 2 0 E  ∥ ρ n u ε ( t ) ∥ 2 ∥ ρ n b ∥ 2  ≤ α E  ∥ ρ n u ε ∥ 4  + c E  ∥ ρ n δ ∥ 4  + c E  ∥ ρ n b ∥ 4  . (5.12) 22 A similar argument applied to the sev enth term on the right-hand side of (5.12) yields ε 2 ∞ X k =1 E h   ρ 2 n σ k ( v ε ( t )) + ρ 2 n γ k   2 i ≤ β 2 E  ∥ ρ 2 n v ε ∥ 2  + c E  ∥ ρ 2 n δ ∥ 4  + c E  ∥ ρ 2 n γ ∥ 4  . (5.13) Substituting (5.9)-(5.13) in to (5.8), b ecause f , g , δ k , b k , γ k are indep endent of t and ω , then we can get that d dt E  ∥ ρ n u ε ( t ) ∥ 4 + ∥ ρ 2 n v ε ( t ) ∥ 2  + 2( α − 18 λ 2 β ) E  ∥ ρ n u ε ( t ) ∥ 4  + β E  ∥ ρ n v ε ( t ) ∥ 2  ≤ 16 λ 2 C 4 0 n 4 β E h ∥ u ε ( t ) ∥ 2 i + c  ∥ ρ n f ∥ 4 +   ρ 2 n g   2 + ∥ ρ n δ ∥ 4 + ∥ ρ n b ∥ 4 + ∥ ρ 2 n γ ∥ 4  ≤ 16 λ 2 C 4 0 n 4 β E h ∥ u ε ( t ) ∥ 2 i + c  ∥ ρ n f ∥ 4 + ∥ ρ n g ∥ 2 + ∥ ρ n δ ∥ 4 + ∥ ρ n b ∥ 4 + ∥ ρ n γ ∥ 4  . (5.14) W e know from Lemma 4.1 that there exists C 1 = C 1 ( K ) > 0 indep endent of n such that for all φ 0 ∈ K and t ≥ 0, it holds E h ∥ u ε ( t ) ∥ 2 i ≤ C 1 . Therefore, for an y ζ > 0, there exists N 1 := N 1 ( ζ ) > 0 suc h that for any n ≥ N 1 , 16 λ 2 C 4 0 n 4 β E h ∥ u ε ( t ) ∥ 2 i ≤ κζ 12 , (5.15) Moreo ver, by ( H 3 ) and (5.1), w e can deduce that for any ζ > 0 there exists N 2 := N 2 ( ζ ) suc h that for an y n ≥ N 2 , c ∥ ρ n f ∥ 4 = c   X | m |≥ n | f m | 2   2 ≤ κζ 12 , c ∥ ρ n g ∥ 4 = c   X | m |≥ n | g m | 2   2 ≤ κζ 12 , (5.16) and c  ∥ ρ n δ ∥ 4 + ∥ ρ n b ∥ 4 + ∥ ρ n γ ∥ 4  = c   X | m |≥ n ∞ X k =1 | δ k,m | 2   2 + c   X | m |≥ n ∞ X k =1 | b k,m | 2   2 + c   X | m |≥ n ∞ X k =1 | γ k,m | 2   2 ≤ κζ 4 . (5.17) By (5.14)-(5.17) and applying the Gron wall’s inequality , w e can get that for any t ≥ 0 E  ∥ ρ n u ε ( t, 0 , u 0 ) ∥ 4 + ∥ ρ 2 n v ε ( t, 0 , v 0 ) ∥ 2  ≤ e − κt E  ∥ ρ n u (0) ∥ 4 + ∥ ρ 2 n v (0) ∥ 2  + ζ 2 ≤ E  ∥ ρ n u (0) ∥ 4 + ∥ ρ 2 n v (0) ∥ 2  + ζ 2 . (5.18) Since K is a compact subset of ℓ 2 c × ℓ 2 and φ 0 = ( u 0 , v 0 ) ∈ K , th us w e can get that as n → ∞ , sup φ 0 ∈ K sup t ≥ 0 E  ∥ ρ n u 0 ∥ 4 + ∥ ρ 2 n v 0 ∥ 2  ≤ sup φ 0 ∈ K sup t ≥ 0 E      X | m |≥ n | u 0 ,m | 2   2 + X | m |≥ n | v 0 ,m | 2    → 0 , where the last step follows from the uniform tail estimate and the fact that  P | m |≥ n | u 0 ,m | 2  2 dominates P | m |≥ n | u 0 ,m | 2 for large n when the tail is small. That is to say , there exists N = N 3 ( K, ζ ) > 0 suc h that for any n ≥ N 3 and φ 0 ∈ K , E  ∥ ρ n u (0) ∥ 4 + ∥ ρ 2 n v (0) ∥ 2  ≤ ζ 2 , 23 whic h, together with (5.18), conclude that there exists N = max { N 1 , N 2 , N 3 } such that for all t ≥ 0, n ≥ N and φ 0 ∈ K , E      X | m |≥ 2 n | u ε m ( t, 0 , u 0 ) | 2   2 + X | m |≥ 2 n | v ε k ( t, 0 , v 0 ) | 2    ≤ E  ∥ ρ n u ε ( t, 0 , u 0 ) ∥ 4 + ∥ ρ 2 n v ε ( t, 0 , v 0 ) ∥ 2  ≤ ζ . (5.19) Th us, the conclusion (5.3) holds. This pro of is ﬁnished. W e denote the probability distribution of solution φ ( t, 0 , u 0 ) of the sto chastic discrete long-w av e- short-w av e resonance system (2.7) b y L ( φ ( t, 0 , u 0 )). It follo ws from Lemmas 5.1 and 5.2 that the family of distributions for the solutions to system (2.7) (with τ = 0) is tight. Lemma 5.3. Supp ose that the assumptions ( H 2 ) , ( H 3 ) , (4.1) and (5.1) hold. Then for every c om- p act subset K (:= e K × b K ) in ℓ 2 c × ℓ 2 and ε ∈ [0 , ε 0 ] with ε 0 = max  q α 24 ∥ δ ∥ 2 , q β 48 ∥ δ ∥ 2  , the family { L ( φ ( t, 0 , φ 0 )) : t ≥ 0 , φ 0 ∈ K } of the distributions of the solutions of system (2.7) is tight in ℓ 2 c × ℓ 2 . Mor e pr e cisely, for every ζ > 0 , ther e exists a c omp act subset K ζ ε of ℓ 2 c × ℓ 2 such that for al l φ 0 ∈ K , P  { ω ∈ Ω : φ ( t, 0 , φ 0 ) / ∈ K ζ ε }  < ζ , ∀ t ≥ 0 . (5.20) Pr o of. F or given n ∈ N , let I [ − n,n ] a the characteristic function of [ − n, n ]. If φ ( t, 0 , φ 0 ) is the solution of sto c hastic discrete long-w av e-short-w av e resonance system (2.7) with τ = 0, then we write e φ ε n ( t, φ 0 ) = ( I [ − n,n ] ( m ) φ ε m ( t, 0 , u 0 )) m ∈ Z and b φ ε n ( t, φ 0 ) = ((1 − I [ − n,n ] ( m )) φ ε m ( t, 0 , φ 0 )) m ∈ Z . It follo ws that for all n ∈ N and t ≥ 0, φ ε ( t, 0 , φ 0 ) = e φ ε n ( t, u 0 ) + b φ ε n ( t, u 0 ) . (5.21) By Lemma 5.2, for every ζ > 0 and k ∈ N , there exists an integer n k dep ending on K , ζ and k such that for all t ≥ 0 and φ 0 ∈ K , E      X | m |≥ n k | u m ( t, 0 , u 0 ) | 2   2    < ζ 2 2 8 k +4 and E   X | m |≥ n k | v m ( t, 0 , v 0 ) | 2   < ζ 2 4 k +2 , (5.22) whic h means that for all t ≥ 0 and φ 0 ∈ K , E  ∥ b u ε n k ( t, u 0 ) ∥ 4  < ζ 2 2 8 k +4 and E  ∥ b v ε n k ( t, v 0 ) ∥ 2  < ζ 2 4 k +2 . (5.23) Since φ 0 b elongs to the compact subset K in ℓ 2 c × ℓ 2 , whic h along with Lemma 5.1 can deduce that there exists c = c ( K ) > 0 suc h that for all t ≥ 0 and φ 0 ∈ K , E  ∥ u ( t, 0 , u 0 ) ∥ 4 + ∥ v ( t, 0 , v 0 ) ∥ 2  ≤ c 2 . (5.24) F or every m ∈ N , w e deﬁne Y ε m =  ϖ = ( ϖ m ) m ∈ Z ∈ ℓ 2 c × ℓ 2 : ϖ m = 0 for | m | > n k and ∥ ϖ ∥ ℓ 2 c × ℓ 2 ≤ 2 k +1 (1 + c ) √ ζ  , (5.25) and K ε m =  ϖ ∈ ℓ 2 c × ℓ 2 : ∥ ϖ − w ∥ ℓ 2 c × ℓ 2 ≤ 1 2 k for some w ∈ Y ε m  , (5.26) where c is the same constan t as in (5.24). Then b y (5.21), (5.25) and (5.26), we obtain that { ω ∈ Ω : φ ε ( t, 0 , φ 0 ) / ∈ K ε m } ⊆  ω ∈ Ω : e φ ε n k ( t, φ 0 ) / ∈ Y ε m  ∪  ω ∈ Ω : φ ( t, 0 , φ 0 ) / ∈ K ε m and e φ ε n k ( t, u 0 ) ∈ Y ε m  24 ⊆  ω ∈ Ω : e φ ε n k ( t, φ 0 ) / ∈ Y ε m  ∪  ω ∈ Ω : ∥ b φ ε n k ( t, φ 0 ) ∥ ℓ 2 c × ℓ 2 > 1 2 k  . (5.27) By (5.24), (5.25) and Cheb ychev’s inequality we get that for all t ≥ 0 and u 0 ∈ K , P ( { ω ∈ Ω : e φ ε n k ( t, φ 0 ) / ∈ Y ε m } ) ≤ P  ω ∈ Ω : ∥ e φ ε n k ( t, φ 0 ) ∥ ℓ 2 c × ℓ 2 > 2 k +1 (1 + c ) √ ζ  ≤ ζ 2 2( k +1) (1 + c ) 2 E h ∥ e φ ε n k ( t, φ 0 ) ∥ 2 ℓ 2 c × ℓ 2 i ≤ ζ 2 2 k +1 (1 + c ) 2 E h ∥ φ ε ( t, 0 , φ 0 ) ∥ 2 ℓ 2 c × ℓ 2 i ≤ ζ 2 2 k (1 + c ) 2 E  1 + ∥ u ε ( t, 0 , u 0 ) ∥ 4 + ∥ v ε ( t, 0 , v 0 ) ∥ 2  ≤ ζ 2 2 k , (5.28) here ∥ φ ε ( t, 0 , φ 0 ) ∥ = ∥ u ε ( t, 0 , u 0 ) ∥ + ∥ v ε ( t, 0 , v 0 ) ∥ , Combining with (5.23) and Chebyc hev’s inequality . we obtain P  ω ∈ Ω : ∥ b φ ε n k ( t, φ 0 ) ∥ ℓ 2 c × ℓ 2 > 1 2 k  ≤ 2 2 k E h ∥ b φ ε n k ( t, φ 0 ) ∥ 2 ℓ 2 c × ℓ 2 i ≤ 2 2 k +1 q E  ∥ b u ε n k ( t, u 0 ) ∥ 4  + 2 2 k +1 E  ∥ b v ε n k ( t, v 0 ) ∥ 2  ≤ ζ 2 2 k . (5.29) By (5.27)-(5.29) we hav e that for any t ≥ 0 and φ 0 ∈ K , P { ω ∈ Ω : φ ε ( t, 0 , φ 0 ) / ∈ K ε m } ≤ ζ 2 2 k − 1 . (5.30) Let K ε ζ = ∩ ∞ m =1 K ε m . Then K ε ζ is compact in ℓ 2 c × ℓ 2 , since it is closed and totally b ounded. Therefore, b y (5.30) w e know that for any t ≥ 0 and φ 0 ∈ K , P { ω ∈ Ω : φ ε ( t, 0 , φ 0 ) / ∈ K ε ζ } ≤ ∞ X k =1 ζ 2 2 k − 1 < ζ . (5.31) This proof is ﬁnished. In the sequel, w e shall contin ue to study the existence of inv arian t measures of sto chastic discrete long-w av e-short-wa ve resonance equation (2.7) with nonlinear noise on ℓ 2 c × ℓ 2 . Let Ξ : ℓ 2 c × ℓ 2 → R b e a b ounded Borel function, w e deﬁne  P ε s,t Ξ  ( ϕ ) = E [Ξ ( φ ε ( t, s, ϕ ))] , ∀ 0 ≤ s ≤ t, ϕ ∈ ℓ 2 c × ℓ 2 . (5.32) The family  P ε s,t  0 ≤ s ≤ t with parameter ε is called the transition semigroup (or op erator) associated with system (2.7). In particular, let B  ℓ 2 c × ℓ 2  denote the Borel σ -algebra of ℓ 2 c × ℓ 2 , then for an y Γ ∈ B  ℓ 2 c × ℓ 2  and 0 ≤ s ≤ t , we can deﬁne P ε ( s, ϕ ; t, Γ) =  P ε s,t χ Γ  ( ϕ ) = P ( { ω ∈ Ω : φ ε ( t, s, ϕ ) ∈ Γ } ) , ϕ ∈ ℓ 2 c × ℓ 2 , (5.33) where χ Γ is a c haracteristic function of Γ. Note that P ε ( s, ϕ, t, · ) is the probabilit y distribution of the solution φ ε ( t, s, ϕ ), namely , P ε ( s, ϕ, t, · ) = L ( φ ε ( t, s, ϕ )). F urthermore, we give the following deﬁnition of in v ariant measures, see [29] for more details. A probability measure µ on ( ℓ 2 c × ℓ 2 , B ( ℓ 2 c × ℓ 2 )) is called an in v ariant measure if Z ℓ 2 c × ℓ 2 ( P ε t Λ ) ( ϕ ) µ ( dϕ ) = Z ℓ 2 c × ℓ 2 Λ ( ϕ ) µ ( dϕ ) , ∀ t ≥ 0 , Λ ∈ C b ( X ) , (5.34) where C b ( X ) is the collection of all bounded contin uous functions, and the Mark ov semigroup P ε t is the abbreviation of P ε 0 ,t , it is deﬁned as follo ws: P ε t Λ ( ϕ ) = Z H Λ ( ξ ) P ( t, ϕ, dξ ) . W e now demonstrate the F eller prop erty of P ε s,t for 0 ≤ s ≤ t , whic h is needed for proving the existence of inv arian t measures. 25 Lemma 5.4. Supp ose that the assumptions ( H 2 ) , ( H 3 ) , (4.1) and (5.1) hold. If Ξ : ℓ 2 c × ℓ 2 → R is b ounde d and c ontinuous, then for every 0 ≤ s ≤ t , the function P s,t Ξ : ℓ 2 c × ℓ 2 → R is also b ounde d and c ontinuous. Pr o of. Let { z n = ( e z n , b z n ) } ∞ n =1 b e a sequence of ℓ 2 c × ℓ 2 † , and z = ( e z, b z ) ∈ ℓ 2 c × ℓ 2 suc h that z n → z in ℓ 2 c × ℓ 2 (i.e., e z n → e z and b z n → b z in ℓ 2 c and ℓ 2 , resp ectively) as n → ∞ . W e now pro ceed to prov e the follo wing limit lim n →∞ E [Ξ( φ ( t 0 , s 0 , z n ))] = E [Ξ( φ ( t 0 , s 0 , z ))] , ∀ t 0 ≥ s 0 ≥ 0 . (5.35) Since the set { z , z n } ∞ n =1 is compact in ℓ 2 c × ℓ 2 , then b y using Lemma 5.3 we can obtain that the family  L ( φ ( t 0 , s 0 , z )) , L ( φ ( t 0 , s 0 , z n ))  ∞ n =1 of the distributions of φ ( t 0 , s 0 , z ) and φ ( t 0 , s 0 , z n ) is tigh t in ℓ 2 c × ℓ 2 . Th us, for ev ery ζ > 0, there exists a compact subset K ε ζ of ℓ 2 c × ℓ 2 suc h that for all n ∈ N , P ε ( s 0 , z ; t 0 , K ε ζ ) > 1 − ζ 4 and P ε ( s 0 , z n ; t 0 , K ε ζ ) > 1 − ζ 4 . (5.36) Accoridng to the contin uity of Ξ, it is easy to ﬁnd that Ξ is uniformly contin uous in K ε ζ . Then there exists b δ > 0 suc h that for all φ 1 , φ 2 ∈ K ε ζ with ∥ φ 1 − φ 2 ∥ < b δ , there holds | Ξ( φ 1 ) − Ξ( φ 2 ) | < ζ . (5.37) Thanks to z n → z in ℓ 2 c × ℓ 2 , by Theorem 3.5 we obtain that there exists b c 1 = b c 1 ( s 0 , t 0 ) > 0 indep enden t of n and ε such that E " sup t ∈ [ s 0 ,t 0 ]  ∥ u ( t, s 0 , e z n ) ∥ 4 + ∥ v ( t, s 0 , b z n ) ∥ 2  # + E " sup t ∈ [ s 0 ,t 0 ]  ∥ u ( t, s 0 , e z ) ∥ 4 + ∥ v ( t, s 0 , b z ) ∥ 2  # ≤ b c 1 . This together with Cheb yshev’s inequalit y can derive that for ev ery R > 0, P ( ω ∈ Ω : sup t ∈ [ s 0 ,t 0 ] ∥ φ ( t, s 0 , z n ) ∥ ℓ 2 c × ℓ 2 > R )! ≤ 2  √ b c 1 + b c 1  R 2 , P ( ω ∈ Ω : sup t ∈ [ s 0 ,t 0 ] ∥ φ ( t, s 0 , z ) ∥ ℓ 2 c × ℓ 2 > R )! ≤ 2  √ b c 1 + b c 1  R 2 . (5.38) F rom (5.38), we ha ve that there exists b R = b R ( ζ ) > 0 suc h that for all n ∈ N , P ( ω ∈ Ω : sup t ∈ [ s 0 ,t 0 ] ∥ φ ( t, s 0 , z n ) ∥ ℓ 2 c × ℓ 2 > b R )! < ζ 4 , P ( ω ∈ Ω : sup t ∈ [ s 0 ,t 0 ] ∥ φ ( t, s 0 , z ) ∥ ℓ 2 c × ℓ 2 > b R )! < ζ 4 . (5.39) F or n ∈ N , let us deﬁne the following stopping time: τ n = inf n t ≥ s 0 : ∥ φ ( t, s 0 , z n ) ∥ ℓ 2 c × ℓ 2 > b R o and τ = inf n t ≥ s 0 : ∥ φ ( t, s 0 , z ) ∥ ℓ 2 c × ℓ 2 > b R o . It follows from (2.2), (2.3), (2.4), and the argument in (3.59) that there exists b c 2 = b c 2 ( s 0 , t 0 , b R ) > 0 such that for all n ∈ N , E  sup t ∈ [ s 0 ,t 0 ]  ∥ u ( t ∧ τ n ∧ τ , s 0 , e z n ) − u ( t ∧ τ n ∧ τ , s 0 , e z ) ∥ 4 (5.40) + ∥ v ( t ∧ τ n ∧ τ , s 0 , b z n ) − v ( t ∧ τ n ∧ τ , s 0 , b z ) ∥ 2   ≤ b c 2  ∥ e z n − e z ∥ 2 + ∥ e z n − e z ∥ 4 + ∥ b z n − b z ∥ 2  , whic h implies that P  ω ∈ Ω : ∥ φ ( t 0 ∧ τ n ∧ τ , s 0 , z n ) − φ ( t 0 ∧ τ n ∧ τ , s 0 , z ) ∥ ℓ 2 c × ℓ 2 ≥ ι  (5.41) † { e z n } ∞ n =1 and { b z n } ∞ n =1 are sequences of ℓ 2 c and ℓ 2 , resp ectively . 26 ≤ 2 ι 2  p b c 2  ∥ e z n − e z ∥ 2 + ∥ e z n − e z ∥ 4 + ∥ b z n − b z ∥ 2  1 / 2 + b c 2  ∥ e z n − e z ∥ 2 + ∥ e z n − e z ∥ 4 + ∥ b z n − b z ∥ 2   . Giv en n ∈ N , we deﬁne the following set Ω ε, 1 ζ ,n = ( ω ∈ Ω : φ ( t 0 , s 0 , z n ) ∈ K ε ζ and sup t ∈ [ s 0 ,t 0 ] ∥ φ ( t 0 , s 0 , z n ) ∥ ℓ 2 c × ℓ 2 ≤ b R ) , Ω ε, 2 ζ = ( ω ∈ Ω : φ ( t 0 , s 0 , z ) ∈ K ε ζ and sup t ∈ [ s 0 ,t 0 ] ∥ φ ( t 0 , s 0 , z ) ∥ ℓ 2 c × ℓ 2 ≤ b R ) . (5.42) Let Ω ε ζ ,n = Ω ε, 1 ζ ,n ∩ Ω ε, 2 ζ , then by (5.36) and (5.39) we can deduce P  Ω \ Ω ε ζ ,n  ≤ P  Ω \ Ω ε, 1 ζ ,n  + P  Ω \ Ω ε, 2 ζ  < ζ , from whic h w e ha ve P  Ω ε ζ ,n  > 1 − ζ . (5.43) Owing to τ n ( ω ) , τ n ( ω ) ≥ t 0 for all ω ∈ Ω ε ζ ,n . Then w e can get that for for all ω ∈ Ω ε ζ ,n , φ ( t 0 ∧ τ n ∧ τ , s 0 , z n ) = φ ( t 0 , s 0 , z n ) and φ ( t 0 ∧ τ n ∧ τ , s 0 , z ) = φ ( t 0 ∧ τ n ∧ τ , s 0 , z ) . (5.44) By (5.44) we hav e P  ω ∈ Ω ε ζ ,n : ∥ φ ( t 0 , s 0 , z n ) − φ ( t 0 , s 0 , z ) ∥ ℓ 2 c × ℓ 2 ≥ ι  ≤ P  ω ∈ Ω : ∥ φ ( t 0 ∧ τ n ∧ τ , s 0 , z n ) − φ ( t 0 ∧ τ n ∧ τ , s 0 , z ) ∥ ℓ 2 c × ℓ 2 ≥ ι  , (5.45) Com bining with (5.41) and (5.45), we infer that for all n ∈ N , P  ω ∈ Ω ε ζ ,n : ∥ φ ( t 0 , s 0 , z n ) − φ ( t 0 , s 0 , z ) ∥ ℓ 2 c × ℓ 2 ≥ ι  (5.46) ≤ 2 ι 2  p b c 2  ∥ e z n − e z ∥ 2 + ∥ e z n − e z ∥ 4 + ∥ b z n − b z ∥ 2  1 / 2 + b c 2  ∥ e z n − e z ∥ 2 + ∥ e z n − e z ∥ 4 + ∥ b z n − b z ∥ 2   . Since Ξ is bounded, then there exists b c 3 = b c 3 ( φ ) > 0 such that | Ξ( φ ) | ≤ b c 3 , ∀ φ ∈ ℓ 2 c × ℓ 2 . (5.47) By (5.46) and (5.47) and (5.37) w e ha ve Z Ω ε ζ ,n | Ξ( φ ( t 0 , s 0 , z n )) − Ξ( φ ( t 0 , s 0 , z )) | d P ≤ Z Ω ε ζ ,n ∩{ ω ∈ Ω: ∥ φ ( t 0 ,s 0 ,z n ) − φ ( t 0 ,s 0 ,z ) ∥ ℓ 2 c × ℓ 2 ≥ ι } | Ξ( φ ( t 0 , s 0 , z n )) − Ξ( φ ( t 0 , s 0 , z )) | d P + Z Ω ε ζ ,n ∩{ ω ∈ Ω: ∥ φ ( t 0 ,s 0 ,z n ) − φ ( t 0 ,s 0 ,z ) ∥ ℓ 2 c × ℓ 2 <ι } | Ξ( φ ( t 0 , s 0 , z n )) − Ξ( φ ( t 0 , s 0 , z )) | d P ≤ 4 b c 3 ι 2  p b c 2 + b c 2    ∥ e z n − e z ∥ 2 + ∥ e z n − e z ∥ 4 + ∥ b z n − b z ∥ 2  1 / 2 + ∥ e z n − e z ∥ 2 + ∥ e z n − e z ∥ 4 + ∥ b z n − b z ∥ 2  + ζ . (5.48) Since as n → ∞ , e z n → e z in ℓ 2 c and b z n → b z in ℓ 2 , respectively . Then b y (5.48) w e get lim sup n →∞ Z Ω ε ζ ,n | Ξ( φ ( t 0 , s 0 , z n )) − Ξ( φ ( t 0 , s 0 , z )) | d P ≤ ζ . F urthermore, it follo ws from (5.43) and (5.47) that Z Ω \ Ω ε ζ ,n | Ξ( φ ( t 0 , s 0 , z n )) − Ξ( φ ( t 0 , s 0 , z )) | d P ≤ 2 b c 3 ζ . 27 Therefore, we hav e lim sup n →∞ R Ω | Ξ( φ ( t 0 , s 0 , z n )) − Ξ( φ ( t 0 , s 0 , z )) | d P ≤ (1 + 2 b c 3 ) ζ , which along with the arbitrariness of ζ can obtain lim sup n →∞ Z Ω | Ξ( φ ( t 0 , s 0 , z n )) − Ξ( φ ( t 0 , s 0 , z )) | d P = 0 . This completes the proof. A t the end of this section, w e giv e the existence theorem of in v ariant measures. This can b e pro ved b y using Krylo v-Bogolyub ov metho d. Theorem 5.5. Supp ose that the assumptions ( H 2 ) , ( H 3 ) , (4.1) and (5.1) hold. Then for every ε ∈ [0 , ε 0 ] with ε 0 = max  q α 24 ∥ δ ∥ 2 , q β 48 ∥ δ ∥ 2  , the sto chastic discr ete long-wave-short-wave r esonanc e e quation (2.7) p ossesses an invariant me asur e on ℓ 2 c × ℓ 2 . Pr o of. W e organize the proof in to four steps, following a standard metho dology . Step 1. F eller prop erty . It follows from Lemma 5.4 that the transition semigroup P ε s,t is F eller for an y 0 ≤ s ≤ t . Step 2. Marko v prop erty . By the uniqueness of solutions to system (2.7) and the reasoning in Section 9 of [29], we can obtain that for every b ounded and con tinuous Ξ : ℓ 2 c × ℓ 2 → R , the solution φ ( t, s, φ 0 ) of (2.7) is a ℓ 2 c × ℓ 2 -v alued Marko v pro cess, that is, for any 0 ≤ s ≤ r ≤ t and φ 0 ∈ L 4 (Ω , ℓ 2 c ) × L 2 (Ω , ℓ 2 ), E [Ξ( φ ( t, r, φ 0 )) |F r ] = E [Ξ( φ ( t, r , z ))] | z = φ 0 P -a.s., i.e., E [Ξ( φ ( t, r , φ 0 )) |F r ] = ( P ε r,t Ξ)( z ) | z = φ 0 , P − a.s., from whic h w e ha ve E [Ξ( φ ( t, s, φ 0 )) |F r ] = ( P ε r,t Ξ)( z ) | z = φ ( r,s,φ 0 ) , P − a.s. This implies ℓ 2 c × ℓ 2 -v alued Marko v prop erty of solution φ ( t, s, φ 0 ) of system (2.7). Step 3. Chapman-Kolmogorov equation. As an immediate consequence of Step 2 , we can get that for any b ounded Borel function Ξ ∈ ℓ 2 c × ℓ 2 → R and 0 ≤ s ≤ r ≤ t , ( P ε s,t Ξ)( φ 0 ) = ( P ε s,r ( P ε r,t Ξ))( φ 0 ) , P − a.s., (5.49) where φ 0 ∈ L 4 (Ω , ℓ 2 c ) × L 2 (Ω , ℓ 2 ). In particular, and for every Γ ∈ B  ℓ 2 c × ℓ 2  , the follo wing Chapman- Kolmogoro v equation P ( s, φ 0 ; t, Γ) = Z ℓ 2 c × ℓ 2 P ε ( s, φ 0 ; r, dy ) P ε ( r , y ; t, Γ) , ∀ 0 ≤ s ≤ r ≤ t (5.50) holds for any φ 0 ∈ L 4 (Ω , ℓ 2 c ) × L 2 (Ω , ℓ 2 ). Step 4. T ranslat ion prop ert y . According to the argumen t of [29], the op erator P ε s,t satisﬁes the follo wing translation property P ε ( s, φ 0 ; t, · ) = P ε (0 , φ 0 ; t − s, · ) (5.51) holds for any φ 0 ∈ L 4 (Ω , ℓ 2 c ) × L 2 (Ω , ℓ 2 ) and 0 ≤ s ≤ r ≤ t . Step 5. Existence of inv ariant measures. W e consider the solution φ ( t, 0 , φ 0 ) of system (2.7) with initial data φ 0 ∈ L 4 (Ω , ℓ 2 c ) × L 2 (Ω , ℓ 2 ), and let µ n ( · ) = 1 n Z n 0 P ε (0 , φ 0 ; t, · ) dt, ∀ n ∈ N . (5.52) By virtue of Lemma 5.3 we get that the family { µ n } ∞ n =1 is tigh t on ℓ 2 c × ℓ 2 . Thus, there exists a probability measure µ ∈ ℓ 2 c × ℓ 2 suc h that, up to a subsequence, µ n k → ˜ µ as k → ∞ . This along with (5.52) can get that for any s, t ≥ 0 and Λ ∈ C b ( H ), Z ℓ 2 c × ℓ 2 Λ ( x ) dµ ( x ) = lim n →∞ 1 n Z n 0 Z ℓ 2 c × ℓ 2 Λ ( x ) P ε (0 , φ 0 ; s, dx ) ! ds 28 = lim n →∞ 1 n Z n − s − s Z ℓ 2 c × ℓ 2 Λ ( x ) P ε (0 , φ 0 ; s + t, dx ) ! ds = lim n →∞ 1 n Z 0 − s Z ℓ 2 c × ℓ 2 Λ ( x ) P ε (0 , φ 0 ; s + t, dx ) ! ds + lim n →∞ 1 n Z n 0 Z ℓ 2 c × ℓ 2 Λ ( x ) P ε (0 , φ 0 ; s + t, dx ) ! ds + lim n →∞ 1 n Z n − s n Z ℓ 2 c × ℓ 2 Λ ( x ) P ε (0 , φ 0 ; s + t, dx ) ! ds, whic h, together with (5.51), can deduce Z ℓ 2 c × ℓ 2 Λ ( x ) dµ ( x ) = lim n →∞ 1 n Z n 0 Z ℓ 2 c × ℓ 2 Λ ( x ) P ε (0 , φ 0 ; s + t, dx ) ! ds = lim n →∞ 1 n Z n 0 Z ℓ 2 c × ℓ 2 Λ ( x ) Z ℓ 2 c × ℓ 2 P ε ( s, y ; s + t, dx ) P ε (0 , φ 0 ; s, dy ) ! ds = lim n →∞ 1 n Z n 0 Z ℓ 2 c × ℓ 2 Z ℓ 2 c × ℓ 2 Λ ( x ) P ε (0 , y ; t, dx ) P ε (0 , φ 0 ; s, dy ) ! ds = Z ℓ 2 c × ℓ 2 Z ℓ 2 c × ℓ 2 Λ ( x ) P ε (0 , y ; t, dx ) ! dµ ( y ) = Z ℓ 2 c × ℓ 2 Z ℓ 2 c × ℓ 2 Λ ( y ) P ε (0 , x ; t, dy ) ! dµ ( x ) . This completes the proof. 6 Limiting b eha vior of inv arian t measures with resp ect to the noise intensit y This section in v estigates the limiting behavior b et ween the inv ariant measures of the stochastic discrete long-wa v e-short-wa ve resonance equation (2.7) and its deterministic coun terpart with ε = 0, under the assumption that the external forcing terms f , g , b k , and γ k are indep endent of time t and the sample ω ∈ Ω. T o this end, we ﬁrst establish the following the con vergence result in probability of the solutions, whic h will b e used in the proof of our main theorem. Lemma 6.1. Supp ose that the assumptions ( H 2 ) , ( H 3 ) , (4.1) and (5.1) hold. Then for any b ounde d subset B (:= e B × b B ) of ℓ 2 c × ℓ 2 ‡ , T > 0 , η > 0 , and ε 1 , ε 2 ∈ [0 , ε 0 ] with ε 0 = max  q α 24 ∥ δ ∥ 2 , q β 48 ∥ δ ∥ 2  , ther e holds lim ε 1 → ε 2 sup φ 0 ∈ B P  ω ∈ Ω : sup 0 ≤ t ≤ T ∥ φ ε 1 ( t, 0 , φ 0 ) − φ ε 2 ( t, 0 , φ 0 ) ∥ ℓ 2 c × ℓ 2 ≥ η  = 0 , (6.1) wher e φ ε 1 ( t, 0 , φ 0 ) = ( u ε 1 ( t, 0 , u 0 ) , v ε 1 ( t, 0 , v 0 )) T and φ ε 2 ( t, 0 , φ 0 ) = ( u ε 2 ( t, 0 , u 0 ) , v ε 2 ( t, 0 , v 0 )) T ar e two solutions of system (2.7) with initial value φ 0 = ( u 0 , v 0 ) T . Pr o of. F or simplicity’s sake, we use φ ε 1 ( t ) = ( u ε 1 ( t ) , v ε 1 ( t )) T and φ ε 2 ( t ) = ( u ε 2 ( t ) , v ε 2 ( t )) T to denote φ ε 1 ( t, 0 , φ 0 ) = ( u ε 1 ( t, 0 , u 0 ) , v ε 1 ( t, 0 , v 0 )) T and φ ε 2 ( t, 0 , φ 0 ) = ( u ε 2 ( t, 0 , u 0 ) , v ε 2 ( t, 0 , v 0 )) T , resp ectively . F or any n ∈ N and T > 0, w e deﬁne the following stopping time T n : T n = inf { t ≥ 0 : ∥ φ ε 1 ( t ) ∥ ≥ n } ∧ inf { t ≥ 0 : ∥ φ ε 2 ( t ) ∥ ≥ n } ∧ T . (6.2) W e can get from the equation (2 . 7) the follo wing inden ties u ε 1 ( t ∧ T n ) − u ε 2 ( t ∧ T n ) + α Z t ∧ T n 0 ( u ε 1 ( r ) − u ε 2 ( r )) dr + i Z t ∧ T n 0 ( Au ε 1 ( r ) − Au ε 2 ( r )) dr (6.3) ‡ e B and b B are b ounded subsets of ℓ 2 c and ℓ 2 , resp ectively . 29 + i Z t ∧ T n 0 ( F ( u ε 1 ( r ) , v ε 1 ( r )) − F ( u ε 2 ( r ) , v ε 2 ( r ))) dr = − iε 2 ∞ X k =1 Z t ∧ T n 0 ( h k ( u ε 1 ( r )) − h k ( u ε 2 ( r ))) dW k ( r ) − i ( ε 1 − ε 2 ) ∞ X k =1 Z t ∧ T n 0 ( h k ( u ε 1 ( r )) + b k ) dW k ( r ) , and v ε 1 ( t ∧ T n ) − v ε 2 ( t ∧ T n ) + β Z t ∧ T n 0 ( v ε 1 ( r ) − v ε 2 ( r )) dr + Z t ∧ T n 0 ( G ( u ε 1 ( r )) − G ( u ε 2 ( r ))) dr (6.4) = ε 2 ∞ X k =1 Z t ∧ T n 0  σ k  v ε 1 ( r )  − σ k  v ε 2 ( r )  dW k ( r ) + ( ε 1 − ε 2 ) ∞ X k =1 Z t ∧ T n 0  σ k  v ε 1 ( r )  + γ k  dW k ( r ) . Applying Ito’s formula and com bining with (6.3), b y taking the real part we hav e that a.s., ∥ u ε 1 ( t ∧ T n ) − u ε 2 ( t ∧ T n ) ∥ 2 + ∥ u ε 1 ( t ∧ T n ) − u ε 2 ( t ∧ T n ) ∥ 4 + 2 α Z t ∧ T n 0 ∥ u ε 1 ( r ) − u ε 2 ( r ) ∥ 2 dr + 4 α Z t ∧ T n 0 ∥ u ε 1 ( r ) − u ε 2 ( r ) ∥ 2 dr ≤ 2 Z t ∧ T n 0 | ( F ( u ε 1 ( r ) , v ε 1 ( r )) − F ( u ε 2 ( r ) , v ε 2 ( r )) , u ε 1 ( r ) − u ε 2 ( r )) | dr + 4 Z t ∧ T n 0 ∥ u ε 1 ( r ) − u ε 2 ( r ) ∥ 2 | ( F ( u ε 1 ( r ) , v ε 1 ( r )) − F ( u ε 2 ( r ) , v ε 2 ( r )) , u ε 1 ( r ) − u ε 2 ( r )) | dr + ∞ X k =1 Z t ∧ T n 0 ∥ ε 2 ( h k ( u ε 1 ( r )) − h k ( u ε 2 ( r )) + ( ε 1 − ε 2 ) ( h k ( u ε 1 ( r )) + b k )) ∥ 2 dr + 6 ∞ X k =1 Z t ∧ T n 0 ∥ u ε 1 ( r ) − u ε 2 ( r ) ∥ 2 ∥ ε 2 ( h k ( u ε 1 ( r )) − h k ( u ε 2 ( r )) + ( ε 1 − ε 2 ) ( h k ( u ε 1 ( r )) + b k )) ∥ 2 dr + 2 ∞ X k =1 Z t ∧ T n 0 Im  u ε 1 ( r ) − u ε 2 ( r ) , ε 2 ( h k ( u ε 1 ( r )) − h k ( u ε 2 ( r ))) + ( ε 1 − ε 2 ) ( h k ( u ε 1 ( r )) + b k )  dW k ( r ) + 4 ∞ X k =1 Z t ∧ T n 0 ∥ u ε 1 ( r ) − u ε 2 ( r ) ∥ 2 Im  u ε 1 ( r ) − u ε 2 ( r ) , ε 2 ( h k ( u ε 1 ( r )) − h k ( u ε 2 ( r ))) + ( ε 1 − ε 2 ) ( h k ( u ε 1 ( r )) + b k )  dW k ( r ) . (6.5) By (6.4) and Ito’s form ula w e can also derive that a.s., ∥ v ε 1 ( t ∧ T n ) − v ε 2 ( t ∧ τ n ) ∥ 2 + 2 β Z t ∧ T n 0 ∥ v ε 1 ( r ) − v ε 2 ( r ) ∥ 2 dr ≤ 2 Z t ∧ T n 0 | ( G ( u ε 1 ( r )) − G ( u n ( r )) , v ε 1 ( r ) − v n ( r )) | dr + ∞ X k =1 Z t ∧ T n 0 ∥ ε 2  σ k  v ε 1 ( r )  − σ k  v ε 2 ( r )  + ( ε 1 − ε 2 )  σ k  v ε 1 ( r )  + γ k  ∥ 2 dr + 2 ∞ X k =1 Z t ∧ T n 0  v ε 1 ( r ) − v ε 2 ( r ) , ε 2  σ k  v ε 1 ( r )  − σ k  v ε 2 ( r )  + ( ε 1 − ε 2 ) ( σ k ( v ε 1 ( r )) + γ k )  dW k ( r ) . (6.6) By summing (6.5)-(6.6) w e deduce that ∥ u ε 1 ( t ∧ T n ) − u ε 2 ( t ∧ T n ) ∥ 2 + ∥ u ε 1 ( t ∧ T n ) − u ε 2 ( t ∧ T n ) ∥ 4 + ∥ v ε 1 ( t ∧ T n ) − v ε 2 ( t ∧ T n ) ∥ 2 ≤ 2 Z t ∧ T n 0 | ( F ( u ε 1 ( r ) , v ε 1 ( r )) − F ( u ε 2 ( r ) , v ε 2 ( r )) , u ε 1 ( r ) − u ε 2 ( r )) | dr + 4 Z t ∧ T n 0 ∥ u ε 1 ( r ) − u ε 2 ( r ) ∥ 2 | ( F ( u ε 1 ( r ) , v ε 1 ( r )) − F ( u ε 2 ( r ) , v ε 2 ( r )) , u ε 1 ( r ) − u ε 2 ( r )) | dr 30 + 2 Z t ∧ T n 0 | ( G ( u ε 1 ( r )) − G ( u n ( r )) , v ε 1 ( r ) − v n ( r )) | dr + ∞ X k =1 Z t ∧ T n 0 ∥ ε 2 ( h k ( u ε 1 ( r )) − h k ( u ε 2 ( r )) + ( ε 1 − ε 2 ) ( h k ( u ε 1 ( r )) + b k )) ∥ 2 dr + 6 ∞ X k =1 Z t ∧ T n 0 ∥ u ε 1 ( r ) − u ε 2 ( r ) ∥ 2 ∥ ε 2 ( h k ( u ε 1 ( r )) − h k ( u ε 2 ( r )) + ( ε 1 − ε 2 ) ( h k ( u ε 1 ( r )) + b k )) ∥ 2 dr + ∞ X k =1 Z t ∧ T n 0 ∥ ε 2  σ k  v ε 1 ( r )  − σ k  v ε 2 ( r )  + ( ε 1 − ε 2 )  σ k  v ε 1 ( r )  + γ k  ∥ 2 dr + 2 ∞ X k =1 Z t ∧ T n 0 Im  u ε 1 ( r ) − u ε 2 ( r ) , ε 2 ( h k ( u ε 1 ( r )) − h k ( u ε 2 ( r ))) + ( ε 1 − ε 2 ) ( h k ( u ε 1 ( r )) + b k )  dW k ( r ) + 4 ∞ X k =1 Z t ∧ T n 0 ∥ u ε 1 ( r ) − u ε 2 ( r ) ∥ 2 Im  u ε 1 ( r ) − u ε 2 ( r ) , ε 2 ( h k ( u ε 1 ( r )) − h k ( u ε 2 ( r ))) + ( ε 1 − ε 2 ) ( h k ( u ε 1 ( r )) + b k )  dW k ( r ) + 2 ∞ X k =1 Z t ∧ T n 0  v ε 1 ( r ) − v ε 2 ( r ) , ε 2  σ k  v ε 1 ( r )  − σ k  v ε 2 ( r )  + ( ε 1 − ε 2 ) ( σ k ( v ε 1 ( r )) + γ k )  dW k ( r ) := 9 X l =1 J l ( t ∧ T n ) . (6.7) Analogous to the pro ofs of (3.55) and (3.56), the ﬁrst and third terms on the right-hand side of (6.7) can b e handled. Namely , by (2.3) and (2.4) w e can derive that for an y b ounded subsetset B of ℓ 2 c × ℓ 2 , there exists a constan t c 8 = c 8 ( B ) > 0 indep endent of ε 1 and ε 2 suc h that for all t ∈ [0 , T ] with T > 0, 3 X l =1 J l ( t ∧ T n ) ≤ c 8 Z t ∧ T n 0  ∥ u ε 1 ( r ) − u ε 2 ( r ) ∥ 2 + ∥ u ε 1 ( r ) − u ε 2 ( r ) ∥ 4 + ∥ v ε 1 ( r ) − v ε 2 ( r ) ∥ 2  dr . (6.8) F or the fourth and sixth terms on the right-hand side of (6.7), by (2.2) we can infer that there exists a constan t c 9 = c 9 ( B ) > 0 indep endent of ε 1 and ε 2 suc h that for all t ∈ [0 , T ] with T > 0, 6 X l =4 J l ( t ∧ T n ) ≤ 2 ε 2 2 Z t ∧ T n 0 ∞ X k =1  ∥ h k ( u ε 1 ( r )) − h k ( u ε 2 ( r )) ∥ 2 + ∥ σ k  v ε 1 ( r )  − σ k  v ε 2 ( r )  ∥ 2  dr + 2( ε 1 − ε 2 ) 2 Z t ∧ T n 0 ∞ X k =1  ∥ h k ( u ε 1 ( r )) + b k ∥ 2 + ∥ σ k  v ε 1 ( r )  + γ k ∥ 2  dr + 12 ε 2 2 Z t ∧ T n 0 ∥ u ε 1 ( r ) − u ε 2 ( r ) ∥ 2 ∞ X k =1 ∥ h k ( u ε 1 ( r )) − h k ( u ε 2 ( r )) ∥ 2 dr + 12( ε 1 − ε 2 ) 2 Z t ∧ T n 0 ∥ u ε 1 ( r ) − u ε 2 ( r ) ∥ 2 ∞ X k =1 ∥ h k ( u ε 1 ( r )) + b k ∥ 2 dr ≤ c 9 Z t ∧ T n 0  ∥ u ε 1 ( r ) − u ε 2 ( r ) ∥ 2 + ∥ u ε 1 ( r ) − u ε 2 ( r ) ∥ 4 + ∥ v ε 1 ( r ) − v ε 2 ( r ) ∥ 2  dr + | ε 1 − ε 2 | 2 c 9 T  1 + ∥ b ∥ 4 + ∥ γ ∥ 4  . (6.9) By (6.7)-(6.9), we obtain that there exists c 10 = c 8 + c 9 suc h that for all t ∈ [0 , T ] with T > 0, E  sup 0 ≤ r ≤ t ∥ u ε 1 ( r ∧ T n ) − u ε 2 ( r ∧ T n ) ∥ 2 + ∥ v ε 1 ( r ∧ T n ) − v ε 2 ( r ∧ T n ) ∥ 2  ≤ c 10 Z t ∧ T n 0  ∥ u ε 1 ( r ) − u ε 2 ( r ) ∥ 2 + ∥ u ε 1 ( r ) − u ε 2 ( r ) ∥ 4 + ∥ v ε 1 ( r ) − v ε 2 ( r ) ∥ 2  dr + | ε 1 − ε 2 | 2 c 10 T  1 + ∥ b ∥ 4 + ∥ γ ∥ 4  + E " sup τ ≤ s ≤ t ∧ T n      9 X l =7 J l ( s )      # . (6.10) 31 W e handle the last tw o terms on the righ t-hand side of (6.10). By (2.2) and the BDG inequality we obtain that there exists c 11 = c 11 ( B ) > 0 indep enden t of ε 1 and ε 2 suc h that for all t ∈ [0 , T ] with T > 0, E  sup τ ≤ s ≤ t ∧ T n | J 7 ( s ) + J 8 ( s ) |  ≤ 8 √ 2 E  Z t ∧ T n 0 ∥ u ε 1 ( r ) − u ε 2 ( r ) ∥ 2 ∞ X k =1 ∥ ε 2 ( h k ( u ε 1 ( r )) − h k ( u ε 2 ( r ))) + ( ε 1 − ε 2 ) ( h k ( u ε 1 ( r )) + b k ) ∥ 2 dr  1 2  + 16 √ 2 E  Z t ∧ T n 0 ∥ u ε 1 ( r ) − u ε 2 ( r ) ∥ 4 ∞ X k =1 ∥ ε 2 ( h k ( u ε 1 ( r )) − h k ( u ε 2 ( r ))) + ( ε 1 − ε 2 ) ( h k ( u ε 1 ( r )) + b k ) ∥ 2 dr  1 2  ≤ 16 ε 2 2 E   Z t ∧ T n 0 ∥ u ε 1 ( r ) − u ε 2 ( r ) ∥ 2 ∞ X k =1 ∥ h k ( u ε 1 ( r )) − h k ( u ε 2 ( r )) ∥ 2 dr ! 1 2   + 16 | ε 1 − ε 2 | 2 E   Z t ∧ T n 0 ∥ u ε 1 ( r ) − u ε 2 ( r ) ∥ 2 ∞ X k =1 ∥ h k ( u ε 1 ( r )) + b k ∥ 2 dr ! 1 2   + 32 ε 2 2 E    sup 0 ≤ r ≤ t ∥ u ε 1 ( r ∧ T n ) − u ε 2 ( r ∧ T n ) ∥ 3  Z t ∧ T n 0 ∞ X k =1 ∥ h k ( u ε 1 ( r )) − h k ( u ε 2 ( r )) ∥ 2 dr ! 1 2   + 32 | ε 1 − ε 2 | 2 E    sup 0 ≤ r ≤ t ∥ u ε 1 ( r ∧ T n ) − u ε 2 ( r ∧ T n ) ∥ 3  Z t ∧ T n 0 ∞ X k =1 ∥ h k ( u ε 1 ( r )) + b k ∥ 2 dr ! 1 2   ≤ c E   sup 0 ≤ r ≤ t ∥ u ε 1 ( r ∧ T n ) − u ε 2 ( r ∧ T n ) ∥ + sup 0 ≤ r ≤ t ∥ u ε 1 ( r ∧ T n ) − u ε 2 ( r ∧ T n ) ∥ 3  · Z t ∧ T n 0 ∞ X k =1 ∥ h k ( u ε 1 ( r )) − h k ( u ε 2 ( r )) ∥ 2 dr ! 1 2  + c | ε 1 − ε 2 | 2 E   sup 0 ≤ r ≤ t ∥ u ε 1 ( r ∧ T n ) − u ε 2 ( r ∧ T n ) ∥ + sup 0 ≤ r ≤ t ∥ u ε 1 ( r ∧ T n ) − u ε 2 ( r ∧ T n ) ∥ 3  · Z t ∧ T n 0 ∞ X k =1 ∥ h k ( u ε 1 ( r )) + b k ∥ 2 dr ! 1 2  ≤ 1 2 E  sup 0 ≤ r ≤ t ∥ u ε 1 ( r ∧ T n ) − u ε 2 ( r ∧ T n ) ∥ 2  + 1 2 E  sup 0 ≤ r ≤ t ∥ u ε 1 ( r ∧ T n ) − u ε 2 ( r ∧ T n ) ∥ 4  + c 11 Z t 0 E  sup τ ≤ r ≤ s  ∥ u ε 1 ( r ∧ T n ) − u ε 2 ( r ∧ T n ) ∥ 2 + ∥ u ε 1 ( r ∧ T n ) − u ε 2 ( r ∧ T n ) ∥ 4   ds + | ε 1 − ε 2 | 2 c T (1 + ∥ b ∥ 4 ) , (6.11) and simlarly , we deriv e that there exists c 12 = c 12 ( B ) > 0 indep endent of ε 1 and ε 2 suc h that for all t ∈ [0 , T ] with T > 0, E  sup τ ≤ s ≤ t ∧ T n | J 9 ( s ) |  ≤ 1 2 E  sup 0 ≤ r ≤ t ∥ v ε 1 ( r ∧ T n ) − v ε 2 ( r ∧ T n ) ∥ 2  + c 12 Z t 0 E  sup τ ≤ r ≤ s ∥ v ε 1 ( r ∧ T n ) − v ε 2 ( r ∧ T n ) ∥ 2  ds + | ε 1 − ε 2 | 2 c 12 T (1 + ∥ γ ∥ 4 ) . (6.12) Substituting (6.11) and (6.12) in to (6.10), we get that there exists c 13 = c 10 + c 11 + c 12 suc h that for all t ∈ [0 , T ] with T > 0, E  sup 0 ≤ r ≤ t  ∥ u ε 1 ( r ∧ T n ) − u ε 2 ( r ∧ T n ) ∥ 4 + ∥ φ ε 1 ( r ∧ T n ) − φ ε 2 ( r ∧ T n ) ∥ 2 ℓ 2 c × ℓ 2   32 ≤ c 13 Z t 0 E  sup τ ≤ r ≤ s  ∥ u ε 1 ( r ∧ T n ) − u ε 2 ( r ∧ T n ) ∥ 4 + ∥ φ ε 1 ( r ∧ T n ) − φ ε 2 ( r ∧ T n ) ∥ 2 ℓ 2 c × ℓ 2   ds + | ε 1 − ε 2 | 2 c 13 T (1 + ∥ b ∥ 4 + ∥ γ ∥ 4 ) . (6.13) By Gron wall’s lemma, w e obtain from (6.13) that for all t ∈ [0 , T ], E  sup 0 ≤ r ≤ t  ∥ u ε 1 ( r ∧ T n ) − u ε 2 ( r ∧ T n ) ∥ 4 + ∥ v ε 1 ( r ∧ T n ) − v ε 2 ( r ∧ T n ) ∥ 2   ≤ | ε 1 − ε 2 | 2 c 13 T (1 + ∥ b ∥ 4 + ∥ γ ∥ 4 ) e c 13 T . (6.14) Note that T n = T for large enough n since φ ε 1 and φ ε 2 is contin uous in t . Then, taking the limit in (6.14) as n → ∞ , w e obtain E  sup 0 ≤ t ≤ T  ∥ u ε 1 ( t ) − u ε 2 ( t ) ∥ 4 + ∥ v ε 1 ( t ) − v ε 2 ( t ) ∥ 2   ≤ | ε 1 − ε 2 | 2 c 13 T (1 + ∥ b ∥ 4 + ∥ γ ∥ 4 ) e c 13 T . (6.15) Therefore, b y (6.15) and Chebyc hev’s inequalit y w e can obtain sup φ 0 ∈ B P  ω ∈ Ω : sup 0 ≤ t ≤ T ∥ φ ε 1 ( t, 0 , φ 0 ) − φ ε 2 ( t, 0 , φ 0 ) ∥ ℓ 2 c × ℓ 2 ≥ η  ≤ 2 η 2 E  sup 0 ≤ t ≤ T  ∥ u ε 1 ( t, 0 , u 0 ) − u ε 2 ( t, 0 , u 0 ) ∥ 2 + ∥ v ε 1 ( t, 0 , v 0 ) − v ε 2 ( t, 0 , v 0 ) ∥ 2   ≤ 2 η 2  E  sup 0 ≤ t ≤ T ∥ u ε 1 ( t, 0 , u 0 ) − u ε 2 ( t, 0 , u 0 ) ∥ 4  1 / 2 + E  sup 0 ≤ t ≤ T ∥ v ε 1 ( t, 0 , v 0 ) − v ε 2 ( t, 0 , v 0 ) ∥ 2  ! ≤ 2 η 2 | ε 1 − ε 2 |  e c 13 T 2 p c 13 T (1 + ∥ b ∥ 4 + ∥ γ ∥ 4 ) + | ε 1 − ε 2 | c 13 T (1 + ∥ b ∥ 4 + ∥ γ ∥ 4 ) e c 13 T  → 0 , as ε 1 → ε 2 . This completes the proof. T o inv estigate the limiting b ehavior of inv ariant measures of system (2.7) as ε v aries, we ﬁrst establish the following key result. Note that Theorem 5.5 guaran tees that the set ℑ ε men tioned below is non-empt y . Theorem 6.2. Supp ose that the assumptions ( H 2 ) , ( H 3 ) , (4.1) and (5.1) hold. Then we have the fol lowing c onclusions: ( i ) The union ∪ ε ∈ [0 ,ε 0 ] ℑ ε is tight on ℓ 2 c × ℓ 2 , wher e ℑ ε denotes the c ol le ction of al l invariant me asur es of system (2.7) for every ε ∈ [0 , ε 0 ] with ε 0 = max  q α 24 ∥ δ ∥ 2 , q β 48 ∥ δ ∥ 2  . ( ii ) If ε n → 0 and µ ε n ∈ ℑ ε n , then ther e exist a subse quenc e { ε n k } ∞ k =1 of { ε n } ∞ n =1 and an invariant me asur e µ 0 ∈ ℑ 0 such that µ ε n k → µ 0 we akly ℓ 2 c × ℓ 2 . Pr o of. Pro of of ( i ) . First, we observ e that all estimates in Lemmas 5.1 and 5.2 are uniform in ε ∈ [0 , ε 0 ]. Therefore, one can verify that the union ∪ ε ∈ [0 ,ε 0 ] ℑ ε is tight. Since the proof is analogous to that of Lemma 5.3, w e omit the details. Pro of of ( ii ) . F rom part ( i ), we know that the set { µ ε : ε ∈ [0 , ε 0 ] } is tigh t. Consequently , there exist a subsequence { ε n k } ∞ k =1 of { ε n } ∞ n =1 and a probabilit y measure µ 0 suc h that µ ε n k → µ 0 w eakly . Since ε n k → 0, it follo ws from Lemma 6.1 and [20] that µ is an in v arian t measure of system (2.7) with ε = 0, yielding µ 0 ∈ ℑ 0 . This completes the proof. Theorem 6.3. Supp ose that the assumptions ( H 2 ) , ( H 3 ) , (4.1) and (5.1) hold. If µ ε ar e the invariant me asur es of the systems (2.7) with ε ∈ [0 , ε 0 ] , her e ε 0 = max  q α 24 ∥ δ ∥ 2 , q β 48 ∥ δ ∥ 2  , then µ ε → µ 0 we akly as ε → 0 . Pr o of. By Theorem 5.5, w e kno w that for ev ery ε ∈ [0 , ε 0 ] with ε 0 = max  q α 24 ∥ δ ∥ 2 , q β 48 ∥ δ ∥ 2  , system (2.7) admits an inv ariant measure. Com bining this fact with Theorem 6.2 yields the desired conclusion. 33 7 Conclusions and remarks This work inv estigates the existence and uniqueness of solutions to the stochastic discrete long- w av e-short-wa ve resonance equation, the existence of a weak pullback mean random attractor, and the existence of in v ariant measures together with their limiting b ehavior with resp ect to the noise inten- sit y . The sp eciﬁc structure of the equation, particularly its coupled nonlinear terms, mak es the classical L 2 (Ω , C ([ τ , τ + T ] ℓ 2 c × ℓ 2 )) space unsuitable as the phase space, th us requiring analysis in the high-order space L 4 (Ω , C ([ τ , τ + T ] , ℓ 2 c )) × L 2 (Ω , C ([ τ , τ + T ] , ℓ 2 )). This approach presen ts signiﬁcant c hallenges in obtaining long-time uniform estimates of solutions, tail estimates, and con vergence analysis. T o address these issues, we develop several nov el analytical techniques. F urthermore, the high-order phase space re- quires us to reestablish the tightness of the solution distributions and the F eller prop erty of the transition semigroup for the stochastic discrete long-w av e-short-wa ve resonance equation. A limitation of this study is the lac k of established mixing prop erty and ergodicity for the in v ariant measures. In principle, demonstrating exp onential conv ergence of solutions for the sto chastic discrete long-w av e-short-wa ve resonance equation o ver extended time scales would lead to exp onential mixing of the in v ariant measures, and consequently establish their uniqueness and ergo dicity . How ev er, proving suc h exp onential con vergence in the L 4 (Ω , ℓ 2 c ) × L 2 (Ω , ℓ 2 ) setting remains b eyond the reach of curren t analytical tec hniques, thereb y prev enting us from obtaining these stronger qualitativ e results at the presen t stage. It is noteworth y that under the lo cal Lipsc hitz condition on the noise co eﬃcients of the sto chastic discrete long-w av e-short-w av e resonance equation, the results established in this paper can be fully ex- tended to the more general high-order Bo chner space L 2 p (Ω , ℓ 2 c ) × L p (Ω , ℓ 2 ) ( p ≥ 2). The main challenges in suc h an extension lie in establishing the p th-order Itˆ o energy equalities and deriving more sophisticated and reﬁned high-order momen t estimates, which will b e addressed in our future w ork. Declarations Author Con tributions All authors contributed equally to this w ork. Sp eciﬁcally , X. P an, J. Huang, J. W u, and J. Zhang w ere in volv ed in: conceptualization and metho dology design; formal analysis and theoretical pro ofs; v alidation and in vestigation; writing-original draft preparation; and writing-review and editing. All authors hav e read and agreed to the published v ersion of the man uscript. Conﬂict of interest The authors hav e no conﬂicts to disclose. Av ailabilit y of date and materials Not applicable. References [1] L. Arnold, Sto chastic diﬀerential Equations: theory and applications. John Wiley and Sons Inc, New Y ork, 1974. [2] D.J. Benney , A general theory for interactions b etw een short and long w av es. Stud. Appl. Math. , 56(1977), 81–94. [3] Y. Li, Long time b ehavior for the w eakly damp ed driv en long-w av e-short-wa ve resonance equations. J. Dif. Equ. , 223(2006), 261–289. [4] C. Zhao, S. Zhou, Compact kernel sections of long-wa ve-short-w a ve resonance equations on inﬁnite lattices. Nonline ar Anal. , 68(2008), 652–670. [5] C.Z. W ang, X. Gang, C. Zhao, In v arian t Borel probability measures for discrete long-w av e-short- w av e resonance equation. Applie d Mathematics and Computation , 339(2018), 853-865. [6] B. Guo, L. Chen, Orbital stabilit y of solitary wa v es of the long wa ve-short w av e resonance equations. Math. Metho ds Appl. Sci. , 21(1998), 883–894. 34 [7] R. Liu, H. Liu, J. Xin, A ttractor for the non-autonomous long w av e-short w av e resonance interaction equation with damping. J. Appl. Anal. Comput. , 10(2020), 1149–1169. [8] J. Bell, C. Cosner, Threshold b ehaviour and propagation for nonlinear diﬀeren tial-diﬀerence systems motiv ated by mo deling m yelinated axons. Quart. Appl. Math. , 42(1984), 1–14. [9] S.N. Chow, J. Mallet-P aret, E.S. V an Vleck, P attern formation and spatial chaos in spatially discrete ev olution equations. R andom Comput. Dyn. , 4(1996), 109–178. [10] R. Kap v al, Discrete mo dels for chemically reacting systems. J. Math. Chem. , 6(1991), 113–163. [11] J.P . Keener, Propagation and its failure in coupled systems of discrete excitable cells. SIAM J. Appl. Math. , 47(1987), 556–572. [12] S.N. Cho w, Lattice dynamical systems, Dynamical Systems. L e ctur e Notes in Math. , Springer, Berlin, 1822. [13] B. W ang, Dynamics of systems on inﬁnite lattices. J. Diﬀer ential Equations , 221(2006), 224-245. [14] B. W ang, W eak pullback attractors for mean random dynamical systems in bo chner spaces. J. Dynam. Diﬀer ential Equations , 31(2019), 2177-2204. [15] Z. Chen, X. Li, B. W ang, Inv ariant measures of sto chastic dela y lattice systems. Discr ete Contin. Dyn. Syst. Ser. B , 26(2021), 3235-3269. [16] Z. Chen, B. W ang, W eak mean attractors and inv arian t measures for sto chastic Schr¨ odinger dela y lattice systems. J. Dynam. Diﬀer ential Equations , 35(2023), 3201-3240. [17] D. Li, B. W ang, X. W ang, Limiting b ehavior of in v ariant measures of stochastic dela y lattice systems. J. Dynam. Diﬀer ential Equations , 34(2022), 1453-1487. [18] B. W ang, Dynamics of sto chastic reaction-diﬀusion lattice systems driven by nonlinear noise. J. Math. Anal. Appl. , 477(2019), 104-132. [19] J. Zhang, X. W ang, D. Huang. Pullbac k measure attractors for nonautonomous stochastic lattice rev ersible Selko v systems. Sto ch. Dyn. , 2025.8, doi.org/10.1142/S0219493725400039. [20] L. Chen, Z. Dong, J. Jiang, J. Zhai, On limiting b ehavior of stationary measures for sto chastic ev olution systems with small noisy intensit y . Sci. China Math , 63(2020), 1463-1504. [21] P . Chen, M. F reitas, X. Zhang, Random Attractor, Inv ariant measures and ergo dicity of lattice p -laplacian equations driven by sup erlinear noise. J. Ge om. Anal. , 33(2023), 33-98. [22] X. W ang, K. Lu, B. W ang, Exp onential stability of non-autonomous sto chastic delay lattice systems with m ultiplicative noise. J. Dynam. Diﬀer ential Equations , 28(2016), 1309-1335. [23] Z. Chen, B. W ang, Asymptotic b eha vior of sto chastic complex lattice systems driven b y sup erlinear noise. J. The or. Pr ob ab. , 36(2023), 1487-1519. [24] J.P . Keener, Propagation and its failure in coupled systems of discrete excitable cells. SIAM J. Appl. Math. , 47(1987), 556-572. [25] P .E. Kloeden, T. Lorenz, Mean-square random dynamical systems. J. Diﬀer. Equ. , 253(2012), 1422-1438. [26] P .W. Bates, X. Chen, A. Chma j, T rav eling wa ves of bistable dynamics on a lattice. SIAM J. Math. A nal. , 35(2003), 520–546. [27] C.E. Elmer, E.S. V an Vlec k, Analysis and computation of trav eling wa v e solutions of bistable diﬀeren tial-diﬀerence equations. Nonline arity , 12(1999), 771–798. [28] C.E. Elmer, E.S. V an Vlec k, T rav eling wa ves solutions for bistable diﬀeren tial-diﬀerence equations with perio dic diﬀusion. SIAM J. Appl. Math. , 61(2001), 1648–1679. [29] D. Prato, G. Zab czyk, Ergo dicit y for Inﬁnite Dimensional Systems, Cambridge Univ ersity Press, Cam bridge, 1996. [30] P .W. Bates, H. Lisei, K. Lu, Attractors for sto chastic lattice dynamical systems. Sto ch. Dyn. , 6(2006), 1-21. [31] S.R.S. V aradhan, Asymptotic probabilities and diﬀeren tial equations. Comm. Pur e Appl. Math. , 19(1966), 261-286. [32] T. Caraballo, K. Lu, A ttractors for sto c hastic lattice dynamical systems with a multiplicativ e noise. F r ont. Math. China , 3(2008), 317-335. [33] B. W ang, Attractors for reaction-diﬀusion equations in unbounded domains. Physic a D , 128(1999), 41-52. [34] B. W ang, Dynamics of sto chastic reaction–diﬀusion lattice systems driven b y nonlinear noise. J. Math. Anal. Appl. , 477(2019), 104-132. [35] Z. Chen, B. W ang, W eak mean attractors and inv arian t measures for sto chastic Schr¨ odinger dela y lattice systems. J. Dynam. Diﬀer ential Equations , 35(2023), 3201-3240. [36] D. Li, B. W ang, X. W ang, P erio dic measures of sto chastic delay lattice systems. J. Diﬀer ential Equations , 272(2021), 74-104. [37] B. W ang, R. W ang, Asymptotic behavior of sto chastic Sc hr¨ odinger lattice systems driven by non- linear noise. Sto ch. Anal. Appl. , 38(2020), 213-237. 35 [38] R. W ang, B. W ang, Random dynamics of p-Laplacian lattice systems driv en b y inﬁnite-dimensional nonlinear noise. Sto ch. Pr o c ess. Appl. , 130(2020), 7431-7462. [39] B. W ang, W eak pullbac k attractors for mean random dynamical systems in Bo c hner spaces. J. Dyn. Diﬀer. Equ. , 31(2019), 2177-2204. [40] B. W ang, W eak pullback attractors for stochastic Na vier-Stokes equations with nonlinear diﬀusion terms. Pr o c. Amer. Math. So c. , 147(2019), 1627-1638. [41] S.N. Cho w, J. Mallet-P aret, Pattern formation and spatial c haos in lattice dynamical systems. IEEE T r ans. Cir cuits Systems I F und. The ory Appl. , 42(1995), 746-751. [42] J.M. Pereira, Global attractor for a generalized discrete nonlinear Sc hr¨ odinger equation. A cta Appl. Math. , 134(2014), 173-183. [43] S. Zhou, Attractors and approximations for lattice dynamical systems. J. Diﬀer ential Equations , 200(2004), 342-368. [44] T. Caraballo, F. Morillas, J. V alero, Attractors of sto c hastic lattice dynamical systems with a m ultiplicative noise and non-Lipschitz nonlinearities. J. Diﬀer ential Equations , 253(2012), 667-693. [45] B. W ang, Large deviations of fractional sto chastic equations with non-Lipschitz drift and multi- plicativ e noise on unbounded domains. J. Diﬀer ential Equations , 376(2023), 1–38. [46] B. W ang, Large deviation principles of stochastic reaction–diﬀusion lattice systems. Discr ete Contin. Dyn. Syst. Ser. B , 29(2024), 1319–1343. [47] D. Li and B. W ang, Pullbac k measure attractors for nonautonomous stochastic reaction-diﬀusion equations on thin domains. J. Diﬀer. Equ. , 397(2024), 232–261. [48] J. Zhang, K. Liu, J. Huang, Uniform measure attractors for nonautonomous sto c hastic tamed 3D Na vier–Stokes equations with almost p erio dic forcing. Appl. Math. L ett. , 160(2025), 109345. [49] S. Mi, D. Li and T. Zeng, Pullbac k measure attractors for nonautonomous stochastic lattice systems. Pr o c. R. So c. Edinb., Se ct. A Math. , (2024), 1-20. [50] Y. W u, J. Zhang, J. Huang, W eak mean attractors, ergo dicity and large deviation principle of sto c hastic Klein-Gordon-Schr¨ odinger lattice systems. Sto ch. Dyn. , 25(2025), 2550018. 36

Well-posedness, mean attractors and invariant measures of stochastic discrete long-wave-short-wave resonance equations driven by locally Lipschitz nonlinear noise

Original Paper

Comments & Academic Discussion

Leave a Comment

Original Paper

Related Papers

Comments & Academic Discussion

Leave a Comment