Random Attractors for Stochastic Retarded Lattice Dynamical Systems

and Applied Analysis 3 It is worth mentioning that in the absence of the white noise, the existence of a global attractor for 1.3 1.4 was established in 40 . The main contribution of this paper is to extend the method of tail estimates to stochastic retarded LDSs and prove the existence of a random attractor for the infinite dimensional random dynamical system generated by stochastic retarded LDS 1.3 1.4 . It is clear that our method can be used for a variety of other stochastic retarded LDSs, as it was for the nonretarded case. The paper is organized as follows. In the next section, we recall some fundamental results on the existence of a pullback random attractor for random dynamical systems. In Section 3, we establish a necessary and sufficient condition for the relative compactness of sequences in C −ν, 0 ; 2 . In Section 4, we define a continuous random dynamical system for stochastic retarded LDS 1.3 1.4 . The existence of the random attractor for 1.3 1.4 is given in Section 5. 2. Preliminaries In this section, we recall some basic concepts related to random attractors for random dynamical systems. The reader is referred to 18–21, 26, 44, 45 for more details. Let X, ‖ · ‖X be a separable Banach space with Borel σ-algebra B X and Ω,F,P be a probability space. Definition 2.1. Ω,F,P, θt t∈R is called a metric dynamical system if θ : R × Ω → Ω is B R ⊗ F,F -measurable, θ0 is the identity on Ω, θs t θt ◦ θs for all s, t ∈ R, and θtP P for all t ∈ R. Definition 2.2. A set A ⊂ Ω is called invariant with respect to θt t∈R, if for all t ∈ R, it holds θ−1 t A A. 2.1 Definition 2.3. A continuous random dynamical system onX over a metric dynamical system Ω,F,P, θt t∈R is a mapping φ : R ×Ω ×X −→ X, t, ω, x −→ φ t, ω, x , 2.2 which is B R ⊗ F ⊗ B X ,B X -measurable, and for all ω ∈ Ω, i φ t, ω, · : X → X is continuous for all t ∈ R ; ii φ 0, ω, · is the identity on X; iii φ t s,ω, · φ t, θsω, · ◦ φ s,ω, · for all s, t ∈ R . Definition 2.4. A random set D is a multivalued mapping D : Ω → 2 \ ∅ such that for every x ∈ X, the mapping ω → d x,D ω is measurable, where d x, B is the distance between the element x and the set B ⊂ X. It is said that the random set is bounded resp., closed or compact if D ω is bounded resp., closed or compact for P-a.e. ω ∈ Ω. 4 Abstract and Applied Analysis Definition 2.5. A random variable r : Ω → 0,∞ is called tempered with respect to θt t∈R, if for P-a.e. ω ∈ Ω lim t→∞ e−βtr θ−tω 0 ∀β > 0. 2.3 A random set D is called tempered if D ω is contained in a ball with center zero and tempered radius r ω for all ω ∈ Ω. Remark 2.6. If r > 0 is tempered, then for any τ ∈ R, β > 0 and P-a.e. ω ∈ Ω lim t→∞ e−βtr θ−t τω e−βτ · lim t→∞ e−β t−τ r θ−t τω 0. 2.4 Therefore, for any τ ∈ R, r θτ · is also tempered. Moreover, if for P-a.e. ω ∈ Ω, r θtω is continuous in t, then for any ν > 0, supσ∈ −ν,0 r θσ · is measurable and for all β > 0 and P-a.e. ω ∈ Ω lim t→∞ e−βt sup σ∈ −ν,0 r θ−t σω ≤ lim t→∞ e β/2 ν−t · sup s∈ −∞,0 { e β/2 r θsω } 0. 2.5 Hence, for any ν > 0, supσ∈ −ν,0 r θσ · is also tempered. Remark 2.7. If r > 0 is tempered, then for any α > 0 and P-a.e. ω ∈ Ω R ω ∫0 −∞ er θsω ds < ∞. 2.6 Moreover, R is tempered, and if for P-a.e. ω ∈ Ω, r θtω is continuous in t, then R θtω is also continuous in t for such ω. Hereafter, we always assume that φ is a continuous random dynamical system over Ω,F,P, θt t∈R , and D is a collection of random subsets of X. Definition 2.8. A random set K is called a random absorbing set in D if for every B ∈ D and P-a.e. ω ∈ Ω, there exists tB ω > 0 such that φ t, θ−tω, B θ−tω ⊆ K ω ∀t ≥ tB ω . 2.7 Definition 2.9. A random set A is called a D-random attractor D-pullback attractor for φ if the following hold: i A is a random compact set; ii A is strictly invariant, that is, for P-a.e. ω ∈ Ω and all t ≥ 0, φ t, ω,A ω A θtω ; 2.8 Abstract and Applied Analysis 5 iii A attracts all sets in D, that is, for all B ∈ D and P-a.e. ω ∈ Ω,and Applied Analysis 5 iii A attracts all sets in D, that is, for all B ∈ D and P-a.e. ω ∈ Ω, lim t→∞ d ( φ t, θ−tω, B θ−tω ,A ω ) 0, 2.9 where d is the Hausdorff semimetric given by d E, F supx∈Einfy∈F‖x − y‖X for any E ⊆ X and F ⊆ X. Definition 2.10. φ is said to be D-pullback asymptotically compact in X if for all B ∈ D and P-a.e.ω ∈ Ω, {φ tn, θ−tnω, xn }n 1 has a convergent subsequence inX whenever tn → ∞, and xn ∈ B θ−tnω . The following existence result on a random attractor for a continuous random dynamical system can be found in 19, 26 . First, recall that a collection D of random subsets of X is called inclusion closed if whenever E is an arbitrary random set, and F is in D with E ω ⊂ F ω for all ω ∈ Ω, then E must belong to D. Proposition 2.11. LetD be an inclusion-closed collection of random subsets ofX and φ a continuous random dynamical system on X over Ω,F,P, θt t∈R . Suppose that K ∈ D is a closed random absorbing set for φ in D and φ is D-pullback asymptotically compact in X. Then φ has a unique D-random attractorA which is given by A ω ⋂ τ≥0 ⋃ t≥τ φ t, θ−tω,K θ−t . 2.10 In this paper, we will take D as the collection of all tempered random subsets of C and prove the stochastic retarded LDS has a D-random attractor. 3. Compactness Criterion in C −ν, 0 ; 2 In this section, we provide a necessary and sufficient condition for the relative compactness of sequences in C −ν, 0 ; 2 , which will be used to establish the asymptotic compactness of the retarded LDS. Lemma 3.1. Let u ∈ C −ν, 0 ; 2 . Then for every ε > 0, there exists N ε > 0 such that for all k ≥ N ε , sup s∈ −ν,0 ∑ |i|≥k |ui s | < ε. 3.1 Proof. For every ε > 0, by virtue of the uniform continuity of u, there exist −ν s0 < s1 < s2 < · · · < sp 0 such that ∥∥u s − u(sj)∥∥ < √ε 2 , for s ∈ [sj−1, sj], j 1, 2, . . . , p. 3.2 6 Abstract and Applied Analysis Since for each sj , u sj ∈ 2, there exists Nj ε > 0 such that for all k ≥ Nj ε , ∑ |i|≥k ∣∣ui(sj)∣∣2 < ε 4 . 3.3 Take N ε max1≤j≤pNj ε . Then for each s ∈ −ν, 0 , there exists j ∈ {1, 2, . . . , p} such that s ∈ sj−1, sj . Therefore, we get from 3.2 and 3.3 that for all k ≥ N ε , ∑ |i|≥k |ui s | ≤ 2 ∑ |i|≥k ∣∣ui(sj)∣∣2 2∑ |i|≥k ∣∣ui s − ui(sj)∣∣2 ≤ 2 ∑ |i|≥k ∣∣ui(sj)∣∣2 2∥∥u s − u sj ∥∥2 < ε, 3.4 which completes the proof. Theorem 3.2. Let S ⊂ C −ν, 0 ; 2 . Then S is relative compact in C −ν, 0 ; 2 if and only if the following conditions are satisfied: i S is bounded in C −ν, 0 ; 2 ; ii S is equicontinuous; iii limk→∞supu ui i∈Z∈Ssups∈ −ν,0 ∑ |i|≥k |ui s |2 0. Proof. The proof is divided into two steps. We first show the necessity of the conditions and then prove the sufficiency. 1 Assume that S is relative compact in C −ν, 0 ; 2 . Then we want to show conditions i , ii , and iii hold. Clearly, in this case, by the Ascoli-Arzelà theorem, S must be bounded and equicontinuous. So we only need to prove condition iii . Given ε > 0, since S is relative compact, there exists a finite subset E of S such that the balls of radii ε/2 centered at E form a finite covering of S, that is, for each u ∈ S, there exists v ∈ E such that sup s∈ −ν,0 ‖u s − v s ‖ < ε 2 . 3.5 By Lemma 3.1, there exists K∗ ε > 0 such that for all v ∈ E, sup s∈ −ν,0 ∑ |i|≥K∗ ε |vi s | < ε 2 4 . 3.6 By 3.5 and 3.6 , we find that for each u ∈ S, there exists v ∈ E such that sup s∈ −ν,0 ∑ |i|≥K∗ ε |ui s | ≤ 2 sup s∈ −ν,0 ∑ |i|≥K∗ ε |ui s − vi s | 2 sup s∈ −ν,0 ∑ |i|≥K∗ ε |vi s | < ε2. 3.7 Abstract and Applied Analysis 7 Therefore, for all k ≥ K∗ ε , we haveand Applied Analysis 7 Therefore, for all k ≥ K∗ ε , we have sup u ui i∈Z∈S sup s∈ −ν,0 ∑ |i|≥k |ui s | ≤ ε2, 3.8 which implies condition iii . 2 Assume that conditions i , ii , and iii are valid. We want to prove that S is relative compact in C −ν, 0 ; 2 . That is, given ε > 0, we want to show that S has a finite covering of balls of radii ε. By condition iii , we find that there exists K ε > 0 such that for all u ui i∈Z ∈ S, sup s∈ −ν,0 ∑ |i|≥K ε |ui s | < ε 2 4 . 3.9 Consider the set S|K {u|K ui |i|≤K ε : u ui i∈Z ∈ S} in C −ν, 0 ;R2K ε 1 . By conditions i and ii , we know that S|K is bounded and equicontinuous in C −ν, 0 ;R2K ε 1 . Then, by the Ascoli-Arzelà theorem, we obtain that S|K is relative compact in C −ν, 0 ;R2K ε 1 and hence there exists a finite subset H of S|K such that the balls of radii ε/2 centered atH form a finite covering of S|K, that is, for each u|K ∈ S|K, there exists v|K ∈ H such that sup s∈ −ν,0 ∑ |i|≤K ε |ui s − vi s | < ε 2 4 . 3.10 Now for each v|K vi |i|≤K ε ∈ H, we choose ṽ ṽi i∈Z such that ṽi vi for |i| ≤ K ε and ṽi 0 for |i| > K ε . Then by 3.9 and 3.10 , we find that for each u ∈ S, there exists ṽ ∈ H {ṽ : v|K ∈ H} such that sup s∈ −ν,0 ‖u s − ṽ s ‖ ≤ sup s∈ −ν,0 ∑ |i|≤K ε |ui s − vi s | sup s∈ −ν,0 ∑ |i|>K ε |ui s | < ε2, 3.11 which implies that the set S has a finite covering of balls with radii ε. The proof is complete. The next result is a variant of Theorem 3.2 which shows that condition iii in Theorem 3.2 has an equivalent form which is easier to verify for asymptotic compactness of dynamical systems associated with retarded LDSs. Theorem 3.3. Let {un}n 1 { uni i∈Z}n 1 ⊂ C −ν, 0 ; 2 . Then {un}n 1 is relative compact in C −ν, 0 ; 2 if and only if the following conditions are satisfied: i {un}n 1 is bounded in C −ν, 0 ; 2 ; ii {un}n 1 is equicontinuous; iii limk→∞lim supn→∞sups∈ −ν,0 ∑ |i|≥k |ui s |2 0. 8 Abstract and Applied Analysis Proof. If {un}n 1 is relative compact in C −ν, 0 ; 2 , then it follows from Theorem 3.2 that the above conditions i , ii , and iii are satisfied. So, to complete the proof, we only need to show that the above conditions i , ii , and iii imply the conditions in Theorem 3.2. Given ε > 0, it follows from condition iii that there exists K1 ε > 0 such that lim sup n→∞ sup s∈ −ν,0 ∑ |i|≥K1 ε ∣∣uni s ∣∣2 < ε2 2 , 3.12 which implies that there exists N ε > 0 such that sup s∈ −ν,0 ∑ |i|≥K1 ε ∣∣uni s ∣∣2 < ε2, ∀n > N ε . 3.13 By Lemma 3.1, we find that there exists K2 ε > 0 such that sup s∈ −ν,0 ∑ |i|≥K2 ε ∣∣uni s ∣∣2 < ε2, ∀1 ≤ n ≤ N ε . 3.14 Take K ε max{K1 ε , K2 ε }. It follows from 3.13 and 3.14 that sup s∈ −ν,0 ∑ |i|≥K ε ∣∣uni s ∣∣2 < ε2, ∀n ≥ 1, 3.15