We study stochastic partly dissipative lattice systems with random coupled coefficients and multiplicative/additive white noise in a weighted space of infinite sequences. We first show that these stochastic partly dissipative lattice differential equations generate a random dynamical system. We then establish the existence of a tempered random bounded absorbing set and a global compact random attractor for the associated random dynamical system.

1. Introduction

Stochastic lattice differential equations (SLDE’s) arise naturally in a wide variety of applications where the spatial structure has a discrete character and random spatiotemporal forcing, called noise, is taken into account. These random perturbations are not only introduced to compensate for the defects in some deterministic models, but are also rather intrinsic phenomena. SLDE’s may also arise as spatial discretization of stochastic partial differential equations (SPDE’s); however, this need not to be the case, and many of the most interesting models are those which are far away from any SPDE’s.

The long term behavior of SLDE’s is usually studied via global random attractors. For SLDE’s on regular spaces of infinite sequences, Bates et al. initiated the study on existence of a global random attractor for a certain type of first-order SLDE’s with additive white noise on 1D lattice ℤ [1]. Continuing studies have been made on various types of SLDS’s with multiplicative or additive noise, see [2–7].

Note that regular spaces of infinite sequences may exclude many important and interesting solutions whose components are just bounded, considering that a weighted space of infinite sequences can make the study of stochastic LDE’s more intensive. More importantly, all existing works on SLDE’s consider either a noncoupled additive noise or a multiplicative white noise term at each individual node whereas in a realistic system randomness appears at each node as well as the coupling mode between two nodes. Han et al. initiated the asymptotic study of such SLDE’s in a weighted space of infinite sequences, with not only additive/multiplicative noise but also coefficients which are randomly coupled [8].

In this work, following the idea of [8], we will investigate the existence of a global random attractor for the following stochastic partly dissipative lattice systems with random coupled coefficients and multiplicative/additive white noise in weighted spaces:

()

where u_i, h_i, g_i, a_i, b_i ∈ ℝ f_i ∈ C¹(ℝ, ℝ); (i ∈ ℤ), λ, α, σ, μ > 0 are positive constants; A is the coupling operator, η_i,−q(ω), …, η_i,0(ω), … , η_i,+q(ω), i ∈ ℤ, q ∈ ℕ, are random variables, and w(t), {w_i(t) : i ∈ ℤ} are two-sided Brownian motions on proper probability spaces.

For deterministic partly dissipative lattice systems without noise, the existence of the global attractor has been studied in [9–13]. For stochastic lattice system (1.2) with additive noises, when q = 1, η_i,±1(ω) ≡ 1, η_i,0(ω)≡−2 for all i ∈ ℤ, Huang [4] and Wang et al. [14] proved the existence of a global random attractor for the associated RDS in the regular phase space l² × l². In this work we will consider the existence of a compact global random attractor in the weighted space , which attracts random tempered bounded sets in pullback sense, for stochastic lattice systems (1.1) and (1.2). Here we choose a positive weight function ρ : ℤ → (0, M₀] such that . If ∑_i∈ℤρ(i) < ∞, then contains any infinite sequences whose components are just bounded and l² ⊂ l^∞. Note that when ρ(i) ≡ 1, our results recover the results obtained in [4, 14] while is reduced to the standard l². Moreover, the required conditions in this work for the existence of a random attractor for system (1.2)-(1.1) in weighted space are weaker than those in l² × l².

The rest of this paper is organized as follows. In Section 2, we present some preliminary results for global random attractors of continuous random dynamical systems in weighted spaces of infinite sequences. We then discuss the existence of random attractors for stochastic lattice systems (1.1) and (1.2) in Sections 3 and 4, respectively.

2. Preliminaries

In this section, we present some concepts related to random dynamical systems (RDSs) and random attractors [1, 8, 15] on weighted space of infinite sequences.

Let ρ be a positive function from ℤ to (0, M₀] ⊂ ℝ⁺, where M₀ is a finite positive constant. Define for any i ∈ ℤ, ρ_i = ρ(i) and

()

then

is a separable Hilbert space with the inner product 〈u,v〉_ρ = ∑_i∈ℤρ_iu_iv_i and norm

for u = (u_i) _i∈ℤ,

. Moreover, define

()

with inner product

()

and norm

()

then H is also a separable Hilbert space.

Let (Ω, ℱ, ℙ) be a probability space and {θ_t : Ω → Ω, t ∈ ℝ} be a family of measure-preserving transformations such that (t, Ω) ↦ θ_tΩ is (ℬ(ℝ) × ℱ, ℱ)-measurable, θ₀ = Id_Ω and θ_t+s = θ_tθ_s for all s, t ∈ ℝ. The space (Ω, ℱ, ℙ, (θ_t) _t∈ℝ) is called a metric dynamical system. In the following, “property (P) holds for a.e. ω ∈ Ω with respect to (θ_t) _t∈ℝ" means that there is with and such that (P) holds for all .

Recall the following definitions from existing literature.

(i)
A stochastic process {S(t, ω)} _{t≥0,ω∈Ω} is said to be a continuous RDS over (Ω, ℱ, ℙ, (θ_t) _t∈ℝ) with state space H, if S : ℝ⁺ × Ω × H → H is (ℬ(ℝ⁺) × ℱ × ℬ(H), ℬ(H))-measurable, and for each ω ∈ Ω, the mapping S(t, ω) : H → H, u ↦ S(t, ω)u is continuous for t ≥ 0, S(0, ω)u = u and S(t + s, ω) = S(t, θ_sω)S(s, ω) for all u ∈ H and s, t ≥ 0.
(ii)
A set-valued mapping ω ↦ D(ω) ⊂ H (may be written as D(ω) for short) is said to be a random set if the mapping ω ↦ dist_H(u, D(ω)) is measurable for any u ∈ H.
(iii)
A random set D(ω) is called a closed (compact) random set if D(ω) is closed (compact) for each ω ∈ Ω.
(iv)
A random set D(ω) is said to be bounded if there exist u₀ ∈ H and a random variable r(ω) > 0 such that D(ω)⊂{u ∈ H : ∥u − u₀∥_H ≤ r(ω)} for all ω ∈ Ω.
(v)
A random bounded set D(ω) is said to be tempered if for a.e. ω ∈ Ω,
()
Denote by 𝒟(H) the set of all tempered random sets of H.
(vi)
A random set B(ω) is said to be a random absorbing set in 𝒟(H) if for any D(ω) ∈ 𝒟(H) and a.e. ω ∈ Ω, there exists T_D(ω) such that S(t, θ_−tω)D(θ_−tω) ⊂ B(ω) for all t ≥ T_D(ω).
(vii)
A random set A(ω) is said to be a random attracting set if for any D(ω) ∈ 𝒟(H), we have
()
in which dist_H is the Hausdorff semidistance defined via dist_H(E, F) = sup _u∈Einf _v∈F∥u−v∥_ρ for any .
(viii)
A random compact set A(ω) is said to be a random global 𝒟 attractor if it is a compact random attracting set and S(t, ω)A(ω) = A(θ_tω) for a.e. ω ∈ Ω and t ≥ 0.

Definition 2.1 (see [8].){S(t, ω)} _{t≥0,ω∈Ω} is said to be random asymptotically null in 𝒟(H), if for any D(ω) ∈ 𝒟(H), a.e. ω ∈ Ω, and any ɛ > 0, there exist T(ɛ, ω, D(ω)) > 0 and I(ɛ, ω, D(ω)) ∈ ℕ such that

()

Theorem 2.2 (see [8].)Let {S(t, ω)} _{t≥0,ω∈Ω} be a continuous RDS over (Ω, ℱ, ℙ, (θ_t) _t∈ℝ) with state space H and suppose that

(a)
there exists a random bounded closed absorbing set B(ω) ∈ 𝒟(H) such that for a.e. ω ∈ Ω and any D(ω) ∈ 𝒟(H), there exists T_D(ω) > 0 yielding S(t, θ_−tω)D(θ_−tω) ⊂ B(ω) for all t ≥ T_D(ω);
(b)
{S(t, ω)} _{t≥0,ω∈Ω} is random asymptotically null on B(ω); that is, for a.e. ω ∈ Ω and for any ɛ > 0, there exist T(ɛ, ω, B(ω)) > 0 and I(ɛ, ω, B(ω)) ∈ ℕ such that

()

Then the RDS {S(t, ω)} _{t≥0,ω∈Ω} possesses a unique global random 𝒟 attractor A(ω) given by

()

3. Stochastic Partly Dissipative Lattice Systems with Multiplicative Noise in Weighted Spaces

This section is devoted to the study of asymptotic behavior for system (1.1) in weighted space . We first transform the stochastic lattice system (1.1) to random lattice system in Section 3.1. We then show in Section 3.2 that (1.1) generates random dynamical system in H. Finally we prove in Section 3.3 the existence of a global random attractor for system (1.1).

Throughout the rest of this paper, a positive weight function ρ : ℤ → ℝ⁺ is chosen to satisfy

(P0)
0 < ρ(i) ≤ M₀ and ρ(i) ≤ c · ρ(i ± 1), for all i ∈ ℤ for some positive constants M₀ and c.

(e.g., ρ(x) = 1/(1 + ϵ²x²) ^q, q > 1/2 [16, 17] and ρ(x) = e^−ϵ|x|, x ∈ ℤ where ϵ > 0).

3.1. Mathematical Setting

Define Ω₁ = {ω ∈ C(ℝ, ℝ) : ω(0) = 0} = C₀(ℝ, ℝ), and denote by ℱ₁ the Borel σ-algebra on Ω₁ generated by the compact open topology (see [2, 15]) and ℙ₁ the corresponding Wiener measure on ℱ₁. Defining (θ_t) _t∈ℝ on Ω₁ via θ_tω(·) = ω(·+t) − ω(t) for t ∈ ℝ, then (Ω₁, ℱ₁, ℙ₁, (θ_t) _t∈ℝ) is a metric dynamical system.

Consider the stochastic lattice system (1.1) with random coupled coefficients and multiplicative white noise:

()

where u = (u_i) _i∈ℤ, v = (v_i) _i∈ℤ; f(u) = (f_i(u_i)) _i∈ℤ with f_i ∈ C¹(ℝ, ℝ) (i ∈ ℤ), g = (g_i) _i∈ℤ, h = (h_i) _i∈ℤ; λ, α, σ, μ are positive constants; η_i,−q(ω), …, η_i,0(ω), …, η_i,+q(ω) (q ∈ ℕ) are random variables on the probability space (Ω₁, ℱ₁, ℙ₁); A(·) is a linear operator on

defined by

()

w(t) is a Brownian motion (Wiener process) on the probability space (Ω₁, ℱ₁, ℙ₁); ∘ denotes the Stratonovich sense of the stochastic term.

For convenience, we first transform (3.1) into a random differential equation without white noise. Let

()

then δ(θ_tω) is an Ornstein-Uhlenbeck process on (Ω₁, ℱ₁, ℙ₁, (θ_t) _t∈ℝ) that solves the following Ornstein-Uhlenbeck equation (see [2, 15] for details)

()

where w(t)(ω) = w(t, ω) = ω(t) for ω ∈ Ω₁, t ∈ ℝ, and possesses the following properties.

Lemma 3.1 (see [2], [15].)There exists a θ_t-invariant set of Ω₁ of full ℙ₁ measure such that for , one has

(i)
the random variable |δ(ω)| is tempered;
(ii)
the mapping δ(θ_tω)
()
is a stationary solution of Ornstein-Uhlenbeck equation (3.4) with continuous trajectories;

(iii)

()

The mapping of θ on possesses same properties as the original one if we choose the trace σ-algebra with respect to to be denoted also by ℱ₁. Therefore we can change our metric dynamical system with respect to , still denoted by the symbols (Ω₁, ℱ₁, ℙ₁, (θ_t) _t∈ℝ).

Let

()

where (u(t, ω), v(t, ω)) is a solution of (3.1), then (u(t, ω), v(t, ω))↦(x(t, ω), y(t, ω)) is a homomorphism in H. System (3.1) can then be transformed to the following random system with random coefficients but without white noise:

()

Letting z = (x, y), (3.8) are equivalent to

()

where

()

We now make the following standing assumptions on f_i, g_i, h_i, and η_i,j, (j = −q, …, q) i ∈ ℤ and study in the following subsections asymptotic behavior of system (3.9).

(H1)
g = (g_i) _i∈ℤ, .
(H2)
Let
()
η(θ_tω) (<∞) belongs to with respect to t ∈ ℝ for each ω ∈ Ω₁.
()
and η(ω) is tempered, that is, there exists a θ_t-invariant set Ω₁₀ ∈ ℱ₁ of full ℙ₁ measure such that for ω ∈ Ω₁₀,
()
In the following, we will consider and still write as Ω₁.
(H3)
, where .
(H4)
There exists a function R ∈ C(ℝ⁺, ℝ⁺) such that

()

(H5)
f_i ∈ C¹(ℝ, ℝ), f_i(0) = 0, , , and there exists a constant a ≥ 0 such that , for all s ∈ ℝ, i ∈ ℤ.

3.2. Random Dynamical System Generated by Random Lattice System

In this subsection, we show that the random lattice system (3.9) generates a random dynamical system on H.

Definition 3.2. We call z : [0, T) → H a solution of the following random differential equation

()

where ω ∈ Ω₀, if z ∈ C([0, T), H) satisfies

()

Theorem 3.3. Let T > 0 and (P0), (H1), (H2), (H4), and (H5) hold. Then for any ω ∈ Ω₁ and any initial data z₀ = (x(0), y(0)) ∈ H, (3.9) admits a unique solution z(·; ω, z₀) = (x(·; ω, z₀), y(·; ω, z₀)) ∈ C([0, T), H) with z(0; ω, z₀) = z₀.

Proof. (1) Denote E = l² × l², we first show that if z₀ ∈ E and (h, g) ∈ E, then (3.9) admits a unique solution z(t; ω, z₀, h, g) ∈ E on [0, T) with z(0; ω, z₀, g, h) = z₀. Given z ∈ E, ω ∈ Ω₁, and (h, g) ∈ E, note that F(z, ω) is continuous in z and measurable in ω from E × Ω₁ to E.

By (3.2) and (H2),

()

By (H4),

()

and therefore

()

For any z⁽¹⁾ = (x⁽¹⁾, y⁽¹⁾), z⁽²⁾ = (x⁽²⁾, y⁽²⁾) ∈ E, and for some ϑ ∈ (0,1)

()

Also

()

It then follows that

()

For any compact set D ⊂ E with sup _z∈D∥z∥≤r, defining random variable ζ_D(ω) ≥ 0 via

()

we have

()

and for any z, z⁽¹⁾, z⁽²⁾ ∈ D,

()

According to [15, 19, 20], problem (3.9) possesses a unique local solution z(·; ω, z₀, g, h) ∈ C([0, T_max), E) (0 < T_max ≤ T) satisfying the integral equation

()

We will next show that T_max = T. Since the set C₀(ℝ) of continuous random process in t is dense in the set L¹(ℝ) (see [18, 21]), for each ω ∈ Ω₁, there exists a sequence of continuous random process in t ∈ ℝ such that

()

Consider the random differential equation with initial data z₀ ∈ E:

()

where

()

Follow the same procedure as above, (3.28) has a unique solution

, that is,

()

and by the continuity of A_m(s, ω) in s, there holds

()

Note that

()

multiplying (3.31) by

and sum over i ∈ ℤ results in

()

Applying Gronwall’s inequality to (3.33) we obtain that

()

where κ(t) ∈ C([0, T), ℝ) is independent of m, which implies that the interval of existence of z^(m)(t) is [0, T), and z^(m)(·; ω, z₀, g, h) ∈ C¹([0, T), E).

By (3.34),

()

Since

for some K(T, ω) > 0 and t ∈ [0, T), then for any t, τ ∈ [0, T), m ∈ ℕ,

()

By the Arzela-Acoli Theorem, there exists a convergent subsequence

such that

()

and

is continuous on t ∈ [0, T). Moreover,

for t ∈ [0, T). By (3.27), (3.35), assumption (H2), and the Lebesgue Dominated Convergence Theorem we have

()

Thus replacing m by m_k in (3.31) and letting k → ∞ give

()

By the uniquness of the solutions of (3.9), we have

for t ∈ [0, T_max). By (3.34),

for t ∈ [0, T_max), which implies that the solution z(t) of (3.9) exists globally on t ∈ [0, T).

(2) Next we prove that for any z₀ ∈ H and (h, g) ∈ H, (3.9) has a solution z(t; ω, z₀, h, g) on [0, T) with z(0; ω, z₀, h, g) = z₀. Let z_1,0, z_2,0 ∈ E and h₁ = (h_1,i) _i∈ℤ, h₂ = (h_2,i) _i∈ℤ, g₁ = (g_1,i) _i∈ℤ,g₂ = (g_2,i) _i∈ℤ ∈ l². Let be two solutions of (3.28) with initial data z_1,0, z_2,0 and h, g replaced by h₁, h₂, g₁, g₂, respectively. Set . Take inner product 〈·, ·〉 _H of (d/dt)d^(m) with d^(m) and evaluate each term as follows. By (P0), (H1), (H2), and (H4),

()

It then follows that

()

For T > 0, applying Gronwall’s inequality to (3.41) on [0, T] implies that

()

for some constant C_T depending on T, and thus

()

where

is a constant depending on T. Denote by

, where

with the norm ∥·∥_ρ. By (3.43), there exists a mapping

such that Φ^(m)(z₀, g, h) = z^(m)(t; ω, z₀, g, h), where z^(m)(t; ω, z₀, g, h) is the solution of (3.28) on [0, T) with z^(m)(0; ω, z₀, g, h) = z₀. Since

is dense in

, the mapping Φ^(m) can be extended uniquely to a continuous mapping

For given z₀ ∈ H and (g, h) ∈ H, for T > 0. There exist sequences , such that

()

Let

be the solution of (3.28), then it satisfies the integral equation

()

By the continuity of

, we have for t ∈ [0, T),

()

Thus for each i ∈ ℤ,

()

Moreover,

is bounded in n. Let n → ∞, then we have

()

and

satisfies the differential equation (3.31).

Multiply equation (3.31) by and sum over i ∈ ℤ, we obtain

()

Similar to the process (3.35)–(3.39) in part (1), we obtain the existence of a unique solution z(t; ω, z₀, g, h) ∈ H of (3.9) with initial data z₀ ∈ H, which is the limit function of a subsequence of {z^(m)(t; ω, z₀, g, h)} in H for t ∈ [0, T). In the latter part of this paper, we may write z(t; ω, z₀, h, g) as z(t; ω, z₀) for simplicity.

Theorem 3.4. Assume that (P0), (H1), (H2), (H4), and (H5) hold. Then (3.9) generates a continuous RDS over (Ω₁, ℱ₁, ℙ₁, (θ_t) _t∈ℝ) with state space H:

()

Moreover,

()

defines a continuous RDS

over (Ω₁, ℱ₁, ℙ₁, (θ_t) _t∈ℝ) associated with (3.1).

Proof. By Theorem 3.3, the solution z(t; ω, z₀) of (3.9) with z(0; ω, z₀) = z₀ exists globally on [0, ∞). It is then left to show that z(t; ω, z₀) = z(t; ω, z₀, h, g) is measurable in (t, ω, z₀).

In fact, for z₀ ∈ E and (h, g) ∈ E, the solution of (3.9) z(t; ω, z₀, h, g) ∈ E for t ∈ [0, ∞). In this case, function F(z, t, ω, h, g) = F(z, t, ω) is continuous in z,h, g and measurable in t, ω, which implies that z : [0, ∞) × Ω₁ × E × E → E,(t; ω, z₀, h, g) ↦ z(t; ω, z₀, h, g) is (ℬ([0, ∞) × ℱ₁ × ℬ(E) × ℬ(E), ℬ(E))-measurable.

For z₀ ∈ H and (h, g) ∈ H, the solution z(t; ω, z₀, h, g)∈H for t ∈ [0, ∞). For any given N > 0, define T_N : H → E, (u, v) = ((u_i), (v_i)) _i∈ℤ → T_N(u, v) = ((T_N(u, v)) _i) _i∈ℤ by

()

and write

()

Then T_N is continuous and for any z₀ ∈ H, (h, g) ∈ H, and

()

Thus z : [0, ∞) × Ω₁ × E × E → H is (ℬ([0, ∞)) × ℱ₀ × ℬ(E) × ℬ(E), ℬ(H))-measurable. Observe also that (Id, Id, T_N, T_N):[0, ∞) × Ω₁ × H × H → [0, ∞) × Ω₀ × E × E is (ℬ([0, ∞)) × ℱ₀ × ℬ(H) × ℬ(H), ℬ([0, ∞)) × ℱ₀ × ℬ(E) × ℬ(E))-measurable. Hence z_N = z∘(Id, Id, T_N, T_N):[0, ∞) × Ω₁ × H × H → H is (ℬ([0, ∞)) × ℱ₀ × ℬ(H) × ℬ(H), ℬ(H))-measurable. It then follows from (3.54) that z : [0, ∞) × Ω₁ × H × H → H is (ℬ([0, ∞)) × ℱ₁ × ℬ(H) × ℬ(H), ℬ(H))-measurable. Therefore, fix (h, g) ∈ H we have that z(t; ω, z₀) = z(t; ω, z₀, h, g) is measurable in (t, ω, z₀). The other statements then follow directly.

Remark 3.5. If (h, g) ∈ E, system (3.1) defines a continuous RDS {φ(t)} _t≥0 over (Ω₁, ℱ₁, ℙ₁, (θ_t) _t∈ℝ) in both state spaces E and H.

3.3. Existence of Tempered Random Bounded Absorbing Sets and Global Random Attractors in Weighted Space

In this subsection, we study the existence of a tempered random bounded absorbing set and a global random attractor for the random dynamical system generated by (3.1) in weighted space H.

Theorem 3.6. Assume that (P0), (H1)–(H5) hold, then there exists a closed tempered random bounded absorbing set B_1ρ(ω) ∈ 𝒟(H) of such that for any D(ω) ∈ 𝒟(H) and each ω ∈ Ω₁, there exists T_D(ω) > 0 yielding φ(t, θ_−tω)D(θ_−tω)⊂B_1ρ(ω) for all t ≥ T_D(ω). In particular, there exists T_1ρ(ω) > 0 such that φ(t, θ_−tω)B_1ρ(θ_−tω) ⊂ B_1ρ(ω) for all t ≥ T_1ρ(ω).

Proof. (1) For initial condition z₀ ∈ E and (h, g) ∈ E, let z^(m)(t, ω) = z^(m)(t; ω, z₀(ω), h, g) be a solution of (3.28) with z₀(ω) = e^−δ(ω)z₀ ∈ E, where ω ∈ Ω₁, then z^(m)(t, ω) ∈ E for all t ≥ 0. Let ϵ₁ > 0 be such that

()

By (H4) and (H5), we have

()

Applying Gronwall’s inequality to (3.57), we obtain that for t > 0,

()

(2) For any z₀ ∈ H and (h, g) ∈ H, let {z_0n} ⊂ E and {(h_n, g_n)} ⊂ E be sequences such that

()

Then z^(m)(t; ω, z_0n, h_n, g_n) → z^(m)(t; ω, z₀, h, g) as n → ∞ in H, and (3.58) holds for z₀ ∈ H. Therefore,

()

where

()

For any β > 0, since

()

then r_1ρ(ω) is tempered.

Let z(t, ω) = ψ(t, ω)z₀(ω) = z(t; ω, z₀(ω), h, g) be a solution of equation (3.9) with z₀(ω) = e^−δ(ω)(u₀, v₀) ∈ H, where ω ∈ Ω₁ and (h, g) ∈ H, then z(t, ω) ∈ H, and there exists a subsequence converging to z(t, ω) as m_k → ∞ for all t ≥ 0. Inequality (3.60) still holds after replacing z^(m)(t, ω) by z(t, ω) since the right hand of (3.60) is independent of m. Thus for (u₀, v₀) ∈ D(θ_−tω),

()

Let

. By properties of η(θ_±tω) and D ∈ 𝒟(H), we have

()

and hence

()

Denote by

, it follows that

()

is a tempered closed random absorbing set for

Theorem 3.7. Assume that (P0), (H1)–(H5) hold, then the RDS generated by (3.1) possesses a unique global random 𝒟 attractor given by

()

Proof. According to Theorem 2.2, it remains to prove the asymptotically nullness of ; that is, for any ɛ > 0, there exists T(ɛ, ω, B_1ρ) > T_1ρ(ω) and I(ɛ, ω) ∈ ℕ such that when t ≥ T(ɛ, ω, B_1ρ), the solution φ(t, ω)(u₀, v₀) = ((u_i, v_i)(t; ω, u₀, v₀))_i∈ℤ ∈ H of (3.1) with (u₀, v₀) ∈ B_1ρ(θ_−tω) satisfies

()

Choose a smooth increasing function ξ ∈ C¹(ℝ⁺, [0,1]) such that

()

Let (u, v)(t; ω, u₀, v₀, h, g) = ((u_i, v_i)(t; ω, u₀, v₀, h, g)) _i∈ℤ be a solution of (3.1), then

()

is a solution of (3.9) with z₀(ω) = e^−δ(ω)(u₀, v₀) ∈ H.

Let z_0n = T_nz₀, (h_n, g_n) = T_n(h, g), where T_n is as in (3.52). Then z_0n ∈ E, (h_n, g_n) ∈ E and z(t; ω, z_0n, h_n, g_n) → z(t; ω, z₀, h, g) in H. For any n ≥ 1, let z^(m)(t) = z^(m)(t; ω, z_0n(ω), h_n, g_n) be the solution of (3.28), where z^(m)(0) = z_0n(ω). By Theorem 3.4, z^(m)(·) ∈ C([0, ∞), E)∩C¹((0, ∞), E). Let M be a suitable large integer (will be specified later); multiply (3.31) by and sum over i ∈ ℤ, we obtain

()

Applying Gronwall’s inequality to (3.71) from

to t gives

()

Therefore for (u₀, v₀) ∈ B_1ρ(θ_−tω)∩H,

()

We next estimate terms (i), (ii), (iii) on the right-hand side of (3.73). By (3.61),

()

which implies that for all ɛ > 0, there exists T₁(ɛ, ω, B_1ρ) ≥ T_1ρ such that for t ≥ T₁(ɛ, ω, B_1ρ),

()

and

()

there exists I₁(ɛ, ω) ∈ ℕ such that for M > I₁(ɛ, ω),

()

Note that , then by (H2), η(ω) is tempered and it follows that there exists a such that

()

Therefore

()

For t ≫ T_1ρ, by (H2) and (3.6), there exists

such that

()

Let

, and t > T₂ + T_1ρ, write

()

of which

()

Equation (3.79) together with (3.82) implies that there exist T₃(ɛ, ω, B_1ρ) ≥ T₂ and I₂(ɛ, ω) ∈ ℕ such that for M > I₂(ɛ, ω), t ≥ T₃(ɛ, ω, B_1ρ),

()

In summary, let

()

Then for t > T(ɛ, ω, B_1ρ) and M ≥ I(ɛ, ω), we have

()

Since

as m_k → ∞, by (3.85),

()

Letting n → ∞ in (3.86), we obtain

()

That is,

is asymptotically null on B_1ρ(ω), which completes the proof.

4. Stochastic Partly Dissipative Lattice Systems with Additive White Noise in Weighted Spaces

This section is devoted to the study of asymptotic behavior for system (1.2) in weighted space . The structure and the idea of proofs are similar to that of Section 3, and we will present our major results without elaborting the details of proofs in this section.

4.1. Mathematical Setting

Define Ω₂ = {ω ∈ C(ℝ, l²) : ω(0) = 0}, and denote by ℱ₂ the Borel σ-algebra on Ω₂ generated by the compact open topology [1] and ℙ₂ is the corresponding Wiener measure on ℱ₂, then (Ω₂, ℱ₂, ℙ₂, (θ_t) _t∈ℝ) is a metric dynamical system. Let the infinite sequence eⁱ (i ∈ ℤ) denote the element having 1 at position i and 0 for all other components. Write

()

where {w_i(t) : i ∈ ℤ} are independent two-sided Brownian motions on probability space (Ω₂, ℱ₂, ℙ₂); then W¹(t, ω) and W²(t, ω) are Wiener processes with values in l² defined on the probability space (Ω₂, ℱ₂, ℙ₂).

Consider stochastic lattice system (1.2) with random coupled coefficients and additive independent white noises:

()

where u_i, v_i, h_i, g_i, a_i, b_i ∈ ℝ; u = (u_i) _i∈ℤ, v = (v_i) _i∈ℤ, f(u) = (f_i(u_i)) _i∈ℤ, g = (g_i) _i∈ℤ, h = (h_i) _i∈ℤ, a = (a_i) _i∈ℤ ∈ l², b = (b_i) _i∈ℤ ∈ l², f_i ∈ C¹(ℝ, ℝ) (i ∈ ℤ); λ, α, σ, μ are positive constants; η_i,−q(ω),…, η_i,0(ω),…,η_i,+q(ω), i ∈ ℤ, q ∈ ℕ are random variables; A is defined as in (3.2).

To transform (4.2) into a random equation without white noise, let

()

Then δ¹(θ_tω), δ²(θ_tω) are both Ornstein-Uhlenbeck processes on (Ω₂, ℱ₂, ℙ₂) and solve the following Ornstein-Uhlenbeck equations (see [1]), respectively,

()

Lemma 4.1 (see [1].)There exists a θ_t-invariant set of Ω₂ of full ℙ measure such that for ,

(i)
lim _t→±∞∥ω(t)∥/t = 0;
(ii)
the random variables ∥δ^j(ω)∥ are tempered and the mappings
()
are stationary solutions of Ornstein-Uhlenbeck equations (4.4) in l² with continuous trajectories;
(iii)

()

In the following, we consider the completion of the probability space , still denoted by (Ω₂, ℱ₂, ℙ₂),

Let

()

then system (4.2) becomes the following random system with random coefficients but without white noise:

()

In addition, we make the following assumptions on functions f_i, i ∈ ℤ:

(H6)
f_i ∈ C¹(ℝ, ℝ) satisfy
()
where μ, d_i, d_f are positive constants, p ∈ ℕ, and d = (d_i) _i∈ℤ ∈ l².

4.2. Random Dynamical System Generated by Random Lattice System

Denote by , we have the following.

Theorem 4.2. Let T > 0 and assume that (P0), (H1), (H2), (H4), and (H6) hold. Then for every ω ∈ Ω₂ and any initial data , problem (4.8) admits a unique solution with .

Proof. Similar to the proof of Theorem 3.3.

Theorem 4.3. Assume that (P0), (H1), (H2), (H4), and (H6) hold. Then system (4.8) generates a continuous RDS over (Ω₂, ℱ₂, ℙ₂, (θ_t) _t∈ℝ) with state space H:

()

Moreover,

()

where (u₀, v₀) ∈ H,t ≥ 0, ω ∈ Ω₂, defines a continuous RDS

over (Ω₂, ℱ₂, ℙ₂, (θ_t) _t∈ℝ) associated with (4.2).

Proof. It follows immediately from similar arguments to the proof of Theorem 3.4.

4.3. Existence of Tempered Bounded Random Absorbing Set and Random Attractor in Weighted Space

In this subsection, we study the existence of a tempered random bounded absorbing set and a global random attractor for the random dynamical system generated by (4.2) in weighted space H.

Theorem 4.4. Assume that (P0), (H1)–(H4), and (H6) hold. Then

(a)
there exists a closed tempered bounded random absorbing set B_2ρ(ω) ∈ 𝒟(H) of RDS such that for any D ∈ 𝒟(H) and each ω ∈ Ω₂, there exists yielding , . In particular, there exists T_2ρ(ω) > 0 such that , for all t ≥ T_2ρ(ω);
(b)
the RDS generated by equations (4.2) possesses a unique global random 𝒟 attractor given by

()

Proof. Similar to the proofs of Theorems 3.6 and 3.7.

References

1 Bates P. W., Lisei H., and Lu K., Attractors for stochastic lattice dynamical systems, Stochastics and Dynamics. (2006) 6, no. 1, 1–21, 2210679, https://doi.org/10.1142/S0219493706001621, ZBL1105.60041.
10.1142/S0219493706001621
Web of Science® Google Scholar
2 Caraballo T. and Lu K., Attractors for stochastic lattice dynamical systems with a multiplicative noise, Frontiers of Mathematics in China. (2008) 3, no. 3, 317–335, 2425157, https://doi.org/10.1007/s11464-008-0028-7, ZBL1155.60324.
10.1007/s11464-008-0028-7
Web of Science® Google Scholar
3 Han X., Random attractors for stochastic sine-Gordon lattice systems with multiplicative white noise, Journal of Mathematical Analysis and Applications. (2011) 376, no. 2, 481–493, 2747772, https://doi.org/10.1016/j.jmaa.2010.11.032, ZBL1209.60038.
10.1016/j.jmaa.2010.11.032
Web of Science® Google Scholar
4 Huang J., The random attractor of stochastic FitzHugh-Nagumo equations in an infinite lattice with white noises, Physica D. (2007) 233, no. 2, 83–94, 2375097, https://doi.org/10.1016/j.physd.2007.06.008, ZBL1126.37048.
10.1016/j.physd.2007.06.008
Web of Science® Google Scholar
5 Lv Y. and Sun J., Dynamical behavior for stochastic lattice systems, Chaos, Solitons and Fractals. (2006) 27, no. 4, 1080–1090, 2166690, https://doi.org/10.1016/j.chaos.2005.04.089, ZBL1134.37350.
10.1016/j.chaos.2005.04.089
Web of Science® Google Scholar
6 Wang X., Li S., and Xu D., Random attractors for second-order stochastic lattice dynamical systems, Nonlinear Analysis: Theory, Methods & Applications. (2010) 72, no. 1, 483–494, https://doi.org/10.1016/j.na.2009.06.094, 2574958, ZBL1181.60103.
10.1016/j.na.2009.06.094
Web of Science® Google Scholar
7 Zhao C. and Zhou S., Sufficient conditions for the existence of global random attractors for stochastic lattice dynamical systems and applications, Journal of Mathematical Analysis and Applications. (2009) 354, no. 1, 78–95, 2510419, https://doi.org/10.1016/j.jmaa.2008.12.036, ZBL1192.37106.
10.1016/j.jmaa.2008.12.036
Web of Science® Google Scholar
8 Han X., Shen W., and Zhou S., Random attractors for stochastic lattice dynamical systems in weighted spaces, Journal of Differential Equations. (2011) 250, no. 3, 1235–1266, 2737203, https://doi.org/10.1016/j.jde.2010.10.018, ZBL1208.60063.
10.1016/j.jde.2010.10.018
Web of Science® Google Scholar
9 Abdallah A., Upper semicontinuity of the attractor for lattice dynamical systems of partly dissipative reaction diffusion systems, Journal of Applied Mathematics. (2005) 2005, no. 3, 273–288, 2-s2.0-33750122870, https://doi.org/10.1155/JAM.2005.273.
10.1155/JAM.2005.273
Google Scholar
10 Li X. and Lv H., Uniform attractor for the partly dissipative nonautonomous lattice systems, Advances in Difference Equations. (2009) 2009, 21, 916316, https://doi.org/10.1155/2009/916316, 2540304, ZBL1185.37163.
10.1155/2009/916316
Google Scholar
11 Li X.-J. and Wang D.-B., Attractors for partly dissipative lattice dynamic systems in weighted spaces, Journal of Mathematical Analysis and Applications. (2007) 325, no. 1, 141–156, 2273034, https://doi.org/10.1016/j.jmaa.2006.01.054, ZBL1104.37047.
10.1016/j.jmaa.2006.01.054
Web of Science® Google Scholar
12 Li X. and Zhong C., Attractors for partly dissipative lattice dynamic systems in l² × l², Journal of Computational and Applied Mathematics. (2005) 177, no. 1, 159–174, 2118665, https://doi.org/10.1016/j.cam.2004.09.014, ZBL1061.37060.
10.1016/j.cam.2004.09.014
Web of Science® Google Scholar
13 Van Vleck E. and Wang B., Attractors for lattice FitzHugh-Nagumo systems, Physica D. (2005) 212, no. 3-4, 317–336, 2187515, https://doi.org/10.1016/j.physd.2005.10.006, ZBL1086.34047.
10.1016/j.physd.2005.10.006
Web of Science® Google Scholar
14 Wang Y., Liu Y., and Wang Z., Random attractors for partly dissipative stochastic lattice dynamical systems, Journal of Difference Equations and Applications. (2008) 14, no. 8, 799–817, 2433921, https://doi.org/10.1080/10236190701859542, ZBL1143.37050.
10.1080/10236190701859542
Web of Science® Google Scholar
15 Arnold L., Random Dynamical Systems, 1998, Springer, Berlin, Germany, Springer Monographs in Mathematics, 1723992.
10.1007/978-3-662-12878-7
Google Scholar
16 Beyn W.-J. and Pilyugin S. Yu., Attractors of reaction diffusion systems on infinite lattices, Journal of Dynamics and Differential Equations. (2003) 15, no. 2-3, 485–515, 2046728, https://doi.org/10.1023/B:JODY.0000009745.41889.30, ZBL1041.37040.
10.1023/B:JODY.0000009745.41889.30
Google Scholar
17 Wang B., Dynamics of systems on infinite lattices, Journal of Differential Equations. (2006) 221, no. 1, 224–245, 2193849, https://doi.org/10.1016/j.jde.2005.01.003, ZBL1085.37056.
10.1016/j.jde.2005.01.003
Web of Science® Google Scholar
18 Adams R. A. and Fournier J. J. F., Sobolev Spaces, 2003, 140, 2nd edition, Elsevier, Amsterdam, The Netherlands, Pure and Applied Mathematics, 2424078.
Web of Science® Google Scholar
19 Pazy A., Semigroups of Linear Operator and Applications to Partial Differential Equations, 2007, Springer, Berlin, Germany.
Google Scholar
20 Chueshov I., Monotone Random Systems Theory and Applications, 2002, 1779, Springer, Berlin, Germany, Lecture Notes in Mathematics, 1902500.
10.1007/b83277
Google Scholar
21 Robinson J. C., Infinite-Dimensional Dynamical Systems, 2001, Cambridge University Press, Cambridge, UK, Cambridge Texts in Applied Mathematics, 1881888.
10.1007/978-94-010-0732-0
Google Scholar

All articles

Asymptotic Behavior of Stochastic Partly Dissipative Lattice Systems in Weighted Spaces

Abstract

1. Introduction

2. Preliminaries

3. Stochastic Partly Dissipative Lattice Systems with Multiplicative Noise in Weighted Spaces

3.1. Mathematical Setting

3.2. Random Dynamical System Generated by Random Lattice System

3.3. Existence of Tempered Random Bounded Absorbing Sets and Global Random Attractors in Weighted Space

4. Stochastic Partly Dissipative Lattice Systems with Additive White Noise in Weighted Spaces

4.1. Mathematical Setting

4.2. Random Dynamical System Generated by Random Lattice System

4.3. Existence of Tempered Bounded Random Absorbing Set and Random Attractor in Weighted Space

References

References

Information

About Wiley Online Library

Help & Support

Opportunities

Connect with Wiley

Asymptotic Behavior of Stochastic Partly Dissipative Lattice Systems in Weighted Spaces

Abstract

1. Introduction

2. Preliminaries

3. Stochastic Partly Dissipative Lattice Systems with Multiplicative Noise in Weighted Spaces

3.1. Mathematical Setting

3.2. Random Dynamical System Generated by Random Lattice System

3.3. Existence of Tempered Random Bounded Absorbing Sets and Global Random Attractors in Weighted Space

4. Stochastic Partly Dissipative Lattice Systems with Additive White Noise in Weighted Spaces

4.1. Mathematical Setting

4.2. Random Dynamical System Generated by Random Lattice System

4.3. Existence of Tempered Bounded Random Absorbing Set and Random Attractor in Weighted Space

References

References

Related

Information