Random attractor for second-order stochastic delay lattice sine-Gordon equation

In this paper, we prove the existence of random D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{D}$\end{document}-attractor for the second-order stochastic delay sine-Gordon equation on infinite lattice with certain dissipative conditions, and then establish the upper bound of Kolmogorov ε-entropy for the random D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{D}$\end{document}-attractor.


Second-order stochastic delay lattice sine-Gordon equation
Denote l p (p ≥ 1) defined by with the norm In particular, l 2 is a Hilbert space with the inner product (·, ·) and norm · given by for any u = (u i ) i∈Z k , v = (v i ) i∈Z k ∈ l 2 . Define a linear operator in the following way: (Au i ) i∈Z k = 2ku (i 1 ,...,i j ,...,i k )u (i 1 -1,...,i j ,...,i k ) -· · ·u (i 1 ,...,i j -1,...,i k ) -· · ·u (i 1 ,...,i j ,...,i k -1) u (i 1 +1,...,i j ,...,i k ) -· · ·u (i 1 ,...,i j +1,...,i k ) -· · ·u (i 1 ,...,i j ,...,i k +1) , Then B * j is the adjoint operator of B j , and By using the above equalities, we have For any u = (u i ) i∈Z k , v = (v i ) i∈Z k ∈ l 2 , we define a new inner product and norm on l 2 by It is obvious that Denote Then the norms · and · λ are equivalent. Let H = l 2 λ × l 2 be endowed with the inner product and norm In the following, we consider the probability space ( , F, P), where = ω ∈ C R, l 2 : ω(0) = 0 , F is the Borel σ -algebra induced by the compact-open topology of , and P the corresponding Wiener measure on ( , F). We will identify ω with Define the time shift by Then ( , F, P, (θ t ) t∈R ) is a metric dynamical system with the filtration For convenience, we rewrite Eq. (1.1) as where δ is a positive constant and satisfies then Eq. (3.2) can be rewritten as Also, we make the following assumptions:

By (H 1 ) and (H 3 ), we know that
where d 1 = max{2δ 2 + 54k 2 λ + 3λ + 6δ 2 (δ-α) 2 λ , 6(δα) 2 + 2(4k + λ)}. Thus, F and D map the bounded sets into bounded sets. In this way, by the standard theory of differential equations, we find that there exists a unique local solution. Then calculations in blow shows that this local solution is actually global. Indeed, suppose the solutions ϕ (1) Applying the Gronwall inequality, we have The proof is complete. Proof The proof is similar to that of Theorem 3.2 in [9], so here it is omitted.
For convenience, we denote ψ 2 where i ∈ Z k and i = max 1≤j≤k |i j |.
The proof is complete.

An upper bound of the Kolmogorov ε-entropy
In this section, we study the upper bound of the Kolmogorov ε-entropy of the global random D-attractor A(ω) given by Theorem 4.1.