Fractal Dimension of a Random Invariant Set and Applications

Abstract

We prove an abstract result on random invariant sets of finite fractal dimension. Then this result is applied to a stochastic semilinear degenerate parabolic equation and an upper bound is obtained for the random attractors of fractal dimension.

1. Introduction

The notion of random attractors is a generalization of the classical concept of global attractors for deterministic dynamical systems (see, e.g., [1–3]). Random attractors are compact invariant random sets attracting all the orbits. The asymptotic behavior of a random dynamical system (RDS) is captured by random attractors, which were first introduced in [4]. The existence of random attractors associated with stochastic partial differential equations has been extensively studied by many authors [4–9].

As in the deterministic case, finite dimensionality is an important property of random attractors which can be established for several random dynamical systems. In [10], Crauel and Flandoli developed a method to obtain finite Hausdorff dimension of a random invariant set. However, their assumptions are very restrictive. They proposed certain bounds on the derivative of the RDS as well as on the rate of approximation of the RDS by its derivative to hold uniformly in 𝜔∈Ω (Ω denotes certain probability space). This drawback was overcome by using a “random squeezing property” in [11]. In [12], Debussche used the method involving the Lyapunov exponents (see [3]) to obtain an upper bound on the Hausdorff dimension for a random invariant set, and this method was developed in a recent paper [13] for bounding the fractal dimension of random invariant sets. Motivated by [14], we give a new criterion for the upper bound on fractal dimension of random invariant sets. This result does not require 𝐶¹-smoothness of the RDS. Therefore, it can be applied to more stochastic models. However, as mentioned in [14], the estimate based on our theorems usually turns out to be conservative.

In the next section, we formulate and prove our main abstract results. In Section 3, we apply our abstract results to the random attractor for the RDS generated by a stochastic semilinear degenerate parabolic equation and obtain an upper bound of fractal dimension of the random attractor. Throughout this paper, we denote by ‖⋅‖_𝑋 the norm of Banach space 𝑋. The inner product and norm of 𝐿²(Ω) are written as (⋅,⋅) and ‖⋅‖, respectively. The letter 𝑐 denotes any positive constant which may be different from line to line even in the same line.

2. Preliminaries and Main Results

In this section, we give the main abstract results for the finite fractal dimension of a random invariant set. For that matter, we need some basic concepts.

Definition 1. Let 𝑀 be a compact set in a metric space 𝑋. The fractal (box-counting) dimension dim_𝑓𝑀 of 𝑀 is defined by dim_𝑓𝑀=limsup_𝜀→0ln𝑛(𝑀,𝜀)(),ln1/𝜀 (1) where 𝑛(𝑀,𝜀) is the minimal number of closed balls of the radius 𝜀 which cover the set 𝑀.

For other alternative formulations of the definition of the box-counting dimension, see Definition 3.1 in Falconer's book [15].

Definition 2. Let 𝑋 be a complete metric space endowed with the metric 𝑑 and let 𝑀 be a bounded closed set in 𝑋. Assume that 𝜚 is a pseudometric defined on 𝑀. Let 𝐵⊂𝑀 and 𝜀>0.(i)A subset 𝒰 in 𝐵 is said to be (𝜀,𝜚)-distinguishable if 𝜚(𝑥,𝑥^′)>𝜀 for any 𝑥,𝑥^′∈𝒰, 𝑥≠𝑥^′. We denote by 𝑚_𝜚(𝐵,𝜀) the maximal cardinality of an (𝜀,𝜚)-distinguishable subset of 𝐵.(ii)The pseudometric 𝜚 is said to be compact on 𝑀 if and only if 𝑚_𝜚(𝑀,𝜀) is finite for every 𝜀>0.(iii)For any 𝑟>0 we define a local (𝑟,𝜀,𝜚)-capacity of the set 𝑀 by the formula 𝒞_𝜚(𝑀;𝑟,𝜀)=supln𝑚_𝜚(𝐵,𝜀)∶𝐵⊂𝑀,diam𝐵≤2𝑟. (2)

We next recall some notions related to RDS. The reader is referred to [4–8, 16] for more details. Let (Ω,ℱ,ℙ) be a probability space and let 𝑋 be a Banach space.

Definition 3. (1) (Ω,ℱ,ℙ,(𝜃_𝑡)_𝑡∈ℝ) is called a metric dynamical system (MDS) if 𝜃∶ℝ×Ω→Ω is (ℬ(ℝ)×ℱ,ℱ)-measurable, 𝜃₀ is the identity on Ω, 𝜃_𝑠+𝑡=𝜃_𝑠∘𝜃_𝑡 for all 𝑠, 𝑡∈ℝ, and 𝜃_𝑡ℙ=ℙ for all 𝑡∈ℝ.
(2) An RDS on 𝑋 over an MDS (Ω,ℱ,ℙ,(𝜃_𝑡)_𝑡∈ℝ) is a mapping 𝜙∶ℝ⁺×Ω×𝑋→𝑋, (𝑡,𝜔,𝑥)↦𝜙(𝑡,𝜔,𝑥) which is (ℬ(ℝ⁺)×ℱ×ℬ(𝑋),ℬ(𝑋))-measurable and satisfies, for ℙ-a.s. 𝜔∈Ω,(a)𝜙(0,𝜔,⋅)=id on 𝑋,(b)𝜙(𝑡+𝑠,𝜔,⋅)=𝜙(𝑡,𝜃_𝑠𝜔,⋅)∘𝜙(𝑠,𝜔,⋅) (cocycle property) on 𝑋 for all 𝑠, 𝑡∈ℝ⁺.
An RDS is said to be continuous on 𝑋 if 𝜙(𝑡,𝜔)∶𝑋→𝑋 is continuous for all 𝑡∈ℝ⁺ and ℙ-a.s. 𝜔∈Ω.

Let 𝜃 be a measure-preserving ergodic transformation on (Ω,ℱ,ℙ) and let 𝑆(𝜔) be a family of maps from 𝑋 to 𝑋. We assume that 𝒜(𝜔), 𝜔∈Ω, is a compact measurable set satisfying, for ℙ-a.s. 𝜔∈Ω, 𝑆(𝜔)𝒜(𝜔)=𝒜(𝜃𝜔). (3)

Our aim is to study the fractal dimension of the sets 𝒜(𝜔), 𝜔∈Ω. We define a discrete RDS {𝑆^𝑛,𝑛∈ℤ⁺} by 𝑆^𝑛(𝜔)∶=𝑆(𝜃_𝑛−1𝜔)𝑆(𝜃_𝑛−2𝜔)⋯𝑆(𝜃𝜔)𝑆(𝜔). In the proof of the following theorem we keep track of the “𝜀-approximate 𝜈 volume”: 𝑉_𝜈(𝑋,𝜀)∶=𝜀^𝜈𝑛(𝑋,𝜀). (4) Our main results read as follows.

Theorem 4. Let 𝑋 be a Banach space, and 𝑆(𝜔) satisfies, for ℙ-a.s. 𝜔∈Ω, the following:(i)𝑆(𝜔) is Lipschitz on 𝒜(𝜔); that is, there exists 𝐿>0 independent of 𝜔 such that 𝑆(𝜔)𝑣₁−𝑆(𝜔)𝑣_2𝑋≤𝐿𝑣₁−𝑣_2𝑋,𝑣₁,𝑣₂∈𝒜(𝜔), (5)(ii)there exist compact seminorms 𝑛₁(𝑥), 𝑛₂(𝑥) (independent of 𝜔) on 𝑋 such that 𝑆(𝜔)𝑣₁−𝑆(𝜔)𝑣_2𝑋≤𝜂𝑣₁−𝑣_2𝑋+𝐾𝑛₁𝑣₁−𝑣₂+𝑛₂×𝑆(𝜔)𝑣₁−𝑆(𝜔)𝑣₂, (6)for any 𝑣₁,𝑣₂∈𝒜(𝜔), where 0≤𝜂<1/2𝑒 and 𝐾>0 are constants independent of 𝜔 (a seminorm 𝑛(𝑥) on 𝑋 is said to be compact if and only if for any bounded set 𝐵⊂𝑋 there exists a sequence {𝑥_𝑛}⊂𝐵 such that 𝑛(𝑥_𝑚−𝑥_𝑛)→0 as 𝑚,𝑛→∞). Then 𝒜(𝜔) has finite fractal dimension in 𝑋; that is, for ℙ-a.s. 𝜔∈Ω, 𝑑_𝑓𝒜(𝜔)≤ln𝑚₀8𝑒𝐾1+𝐿^21/2,1−2𝑒𝜂 (7) where 𝑚₀(𝑅) is the maximal number of pairs (𝑥_𝑖,𝑦_𝑖) in 𝑋×𝑋 possessing the properties 𝑥_𝑖²_𝑋+𝑦_𝑖²_𝑋≤𝑅²,𝑛₁𝑥_𝑖−𝑥_𝑗+𝑛₂𝑦_𝑖−𝑦_𝑗>1,𝑖≠𝑗. (8)

Proof. We set 𝜚_𝜔(𝑥,𝑦)=𝐾(𝑛₁(𝑥−𝑦)+𝑛₂(𝑆(𝜔)𝑥−𝑆(𝜔)𝑦)); then, for every 𝜔∈Ω, 𝜚_𝜔 is compact on 𝒜(𝜔) in the sense of Definition 2. From [14], we see that the local (𝑟,𝜀,𝜚_𝜔)-capacity of the set 𝒜(𝜔) admits the estimate 𝒞_𝜚𝜔(𝒜(𝜔);𝑟,𝜀)≤ln𝑚₀2𝐾(1+𝐿²)^1/2𝑟𝜀,ℙ-a.s.𝜔, (9) where 𝑚₀(𝑅) is the maximal number of pairs (𝑥_𝑖,𝑦_𝑖) in 𝑋×𝑋 possessing the properties 𝑥_𝑖²_𝑋+𝑦_𝑖²_𝑋≤𝑅²,𝑛₁𝑥_𝑖−𝑥_𝑗+𝑛₂𝑦_𝑖−𝑦_𝑗>1,𝑖≠𝑗. (10)
For any fixed 𝜀₀>0, we assume that {𝐵_𝑖∶𝑖=1,…,𝑛(𝒜(𝜔),𝜀₀)} is the minimal covering of 𝒜(𝜔) by closed balls of the radius 𝜀₀. Set 𝐹_𝑖=𝐵_𝑖∩𝒜(𝜔),𝑖=1,…,𝑛(𝒜(𝜔),𝜀₀). Let 𝛿=(1/4𝑒)−(𝜂/2) and let {𝑥^𝑖_𝑗;𝑗=1,…,𝑛_𝑖}⊂𝐹_𝑖 be a maximal (𝛿𝜀₀,𝜚_𝜔)-distinguishable subset of 𝐹_𝑖. Since 𝜚_𝜔 is compact, this finite set exists, and then we have 𝑛_𝑖=𝑚_𝜚𝜔𝐹_𝑖,𝛿𝜀₀≤exp𝐶_𝜚𝜔𝒜(𝜔);𝜀₀,𝛿𝜀₀≤𝑚₀2𝐾1+𝐿^21/2𝛿=∶𝑒^𝜈,𝒜(𝜔)⊂^{𝑛(𝒜(𝜔),𝜀0)𝑛}_𝑖=1^𝑖_𝑗=1𝐵_𝑖,𝑗,𝐵_𝑖,𝑗=𝑣∈𝐹_𝑖∶𝜚_𝜔𝑣,𝑥^𝑖_𝑗≤𝛿𝜀₀. (11) Therefore, 𝑆(𝜔)𝒜(𝜔)⊂^{𝑛(𝒜(𝜔),𝜀0)𝑛}_𝑖=1^𝑖_𝑗=1𝑆(𝜔)𝐵_𝑖,𝑗. (12) For any 𝑦₁,𝑦₂∈𝐵_𝑖,𝑗(⊂𝒜(𝜔)), we get from (6) that 𝑑𝑆(𝜔)𝑦₁,𝑆(𝜔)𝑦₂≤𝜂𝑑𝑦₁,𝑦₂+𝜚_𝜔𝑦₁,𝑥^𝑖_𝑗+𝜚_𝜔𝑦₂,𝑥^𝑖_𝑗≤2𝜂+𝛿𝜀₀. (13) Thus diam 𝑆(𝜔)𝐵_𝑖,𝑗≤2(𝜂+𝛿)𝜀₀ for any 𝑖,𝑗. Therefore, 𝑛𝑆(𝜔)𝒜(𝜔),2𝜂+𝛿𝜀₀≤𝑛_𝑖𝑛𝒜(𝜔),𝜀₀≤𝑒^𝜈𝑛𝒜(𝜔),𝜀₀. (14)
For general 𝑛∈ℕ, replacing 𝜚_𝜔, 𝑆(𝜔), and 𝒜(𝜔) by 𝜚_𝜃𝑛𝜔, 𝑆(𝜃_𝑛𝜔), and 𝑆^𝑛(𝜔)𝒜(𝜔), respectively, in the above procedure and noting that 𝑆^𝑛(𝜔)𝒜(𝜔)=𝒜(𝜃_𝑛𝜔), we can get that 𝑆𝜃_𝑛𝜔𝑆^𝑛(𝜔)𝒜(𝜔)⊂^{𝑛(𝒜(𝜃𝑛𝜔),𝜀0)𝑛}_𝑖=1^′𝑖_𝑗=1𝑆𝜃_𝑛𝜔𝐵^′_𝑖,𝑗, (15) where 𝑛^′_𝑖≤𝑒^𝜈, and 𝑛𝑆𝜃_𝑛𝜔𝑆^𝑛(𝜔)𝒜(𝜔),2𝜂+𝛿𝜀₀≤𝑛^′_𝑖𝑛𝒜𝜃_𝑛𝜔,𝜀₀≤𝑒^𝜈𝑛𝒜𝜃_𝑛𝜔,𝜀₀. (16) Thus, 𝑛𝑆^𝑛+1(𝜔)𝒜(𝜔),2𝜂+𝛿𝜀₀≤𝑒^𝜈𝑛𝑆^𝑛(𝜔)𝒜(𝜔),𝜀₀. (17) Setting 𝑞=2(𝜂+𝛿), then by a standard induction procedure we deduce that 𝑛𝑆^𝑛(𝜔)𝒜(𝜔),𝑞^𝑛𝜀₀≤𝑒^𝑛𝜈𝑛𝒜(𝜔),𝜀₀. (18)
Multiplying (18) by 𝑞^𝑛𝜈𝜀^𝜈₀ we get 𝑉_𝜈𝑆^𝑛(𝜔)𝒜(𝜔),𝑞^𝑛𝜀₀≤𝑞^𝑛𝜈𝑒^𝑛𝜈𝜀^𝜈₀𝑛𝒜(𝜔),𝜀₀=𝜅^𝑛𝜈𝑉_𝜈𝒜(𝜔),𝜀₀, (19) where 𝜅=𝑞𝑒=2(𝜂+𝛿)𝑒<1. Then we can get from the above inequality that 𝑉_𝜈𝒜𝜃_𝑛𝜔,𝑞^𝑛𝜀₀≤𝜅^𝑛𝜈𝑉_𝜈𝒜(𝜔),𝜀₀. (20) Therefore, 𝑉_𝜈𝒜(𝜔),𝑞^𝑛𝜀₀≤𝜅^𝑛𝜈𝑉_𝜈𝒜𝜃_−𝑛𝜔,𝜀₀. (21) On one hand, for the above 𝑞, setting 𝛽=ln𝑞⁻¹, then we have −𝛽+𝛿≤ln𝑞≤−𝛽−𝛿,forany𝛿>0. (22) That is, −𝛽+𝛿𝑘≤ln𝑞^𝑘≤−𝛽−𝛿𝑘,forany𝑘∈ℕ,any𝛿>0. (23) This implies that 𝜀₀𝑒^{−(𝛽+𝛿)𝑘}≤𝑞^𝑘𝜀₀≤𝑒^{−(𝛽−𝛿)𝑘}𝜀₀,forany𝑘∈ℕ,any𝛿>0. (24) On the other hand, for any 𝑀>0, we consider the following set: Ω_𝑀=𝜔∈Ω∶𝑉_𝜈𝒜(𝜔),𝜀₀≤𝑀. (25) Then, Ω_𝑀1⊂Ω_𝑀2 for any 𝑀₁≤𝑀₂ and Ω=⋃_𝑀>0Ω_𝑀. We can choose 𝑀₀ large enough such that ℙ(Ω_𝑀)>0 for all 𝑀≥𝑀₀. It follows from the Poincaré recurrence theorem (see [13]) that, for every element 𝜔∈Ω_𝑀 (𝑀≥𝑀₀), there exists a sequence 𝑘_𝑗=𝑘_𝑗(𝜔)→∞ such that 𝜃_−𝑘𝑗𝜔∈Ω_𝑀 (𝑀≥𝑀₀). Therefore, from (21), for all 𝜔∈Ω_𝑀 (𝑀≥𝑀₀), 𝑉_𝜈𝒜(𝜔),𝑞^𝑘𝑗𝜀₀≤𝜅^𝑘𝑗𝜈𝑉_𝜈𝒜𝜃_−𝑘𝑗𝜔,𝜀₀≤𝑀𝜅^𝑘𝑗𝜈. (26) From (24) we see that 𝑞^𝑘𝑗𝜀₀ satisfies the assumptions of Lemma 2.2 in [13]. Then, (26) and the related result in [13] yield that dim_𝑓𝒜(𝜔)≤𝜈,𝜔∈Ω_𝑀𝑀≥𝑀₀. (27) Since 𝑃(Ω_𝑀)→1 as 𝑀→∞, this yields that the above inequality holds for ℙ-a.s. 𝜔∈Ω. The proof is complete.

As in the deterministic case [14], the following result can be easily deduced by Theorem 4.

Theorem 5. Let 𝑋 and 𝑌 be Banach spaces such that 𝑌 is compactly embedded in 𝑋. Let 𝒜(𝜔) be a compact measurable set invariant under 𝑆(𝜔). Assume that, for ℙ-a.s. 𝜔∈Ω, 𝑆(𝜔)𝑣₁−𝑆(𝜔)𝑣_2𝑌≤𝐿𝑣₁−𝑣_2𝑋,𝑣₁,𝑣₂∈𝒜(𝜔), (28) where 𝐿 is a constant independent of 𝜔. Then 𝒜(𝜔) has finite fractal dimension in 𝑋 and admits the estimate dim_𝑓𝒜(𝜔)≤ln𝑚_𝑌,𝑋(8𝑒𝐿),ℙ-𝑎.𝑠.𝜔, (29) where 𝑚_𝑌,𝑋(𝑅) is the maximal number of points 𝑥_𝑖 in the ball of the radius 𝑅 in 𝑌 possessing the properties ‖𝑥_𝑖−𝑥_𝑗‖_𝑋>1, 𝑖≠𝑗.

Remark 6. Recalling that we have defined 𝑛(𝑀,𝜀) in Definition 1, we call ℋ_𝜀(𝑀)∶=log₂𝑛(𝑀,𝜀) the Kolmogorov 𝜀-entropy of 𝑀. Then the number 𝑚_𝑌,𝑋(𝑅) can be bounded by the Kolmogorov entropy. To show this we assume that 𝑖 is the compact embedding of 𝑌 into 𝑋 in Theorem 5 and denote by 𝐵_𝑌(𝑅) the ball of the radius 𝑅 in 𝑌. By the definition of 𝑚_𝑌,𝑋(𝑅), one can easily show that 𝑚_𝑌,𝑋(𝑅)≤𝑛𝑖𝐵_𝑌(𝑅),12. (30) That is, 𝑚_𝑌,𝑋(𝑅)≤2^{ℋ1/2(𝑖𝐵𝑌(𝑅))}=2^{ℋ1/2(𝑅𝑖𝐵𝑌)}=2^{ℋ1/2R(𝑖𝐵𝑌)}, (31) where 𝐵_𝑌=𝐵_𝑌(1). Moreover, the Kolmogorov entropy is closely related to the entropy numbers 𝑒_𝑘(𝑖), 𝑘∈ℕ (see definition in [17]). Thus, one can estimate 𝑚_𝑌,𝑋(𝑅) by using the entropy numbers, and here we omit the details. We refer the readers to [17] for more details about the entropy numbers 𝑒_𝑘(𝑖).

In the concrete application of Theorems 4 and 5, one can define 𝑆(𝜔)=𝜙𝑇^∗,𝜔forsome𝑇^∗>0, (32) where 𝑇^∗ is independent of 𝜔 and 𝜙(𝑡,𝜔) is an RDS on 𝑋 over an MDS (Ω,ℱ,ℙ,(𝜃_𝑡)_𝑡∈ℝ). Then from the cocycle property we have 𝑆^𝑛(𝜔)∶=𝜙𝑛𝑇^∗𝑇,𝜔=𝜙^∗,𝜃_{(𝑛−1)𝑇}^∗𝜔∘𝜙𝑇^∗,𝜃_{(𝑛−2)𝑇}^∗𝜔∘⋯∘𝜙𝑇^∗𝜃,𝜔=𝑆_{(𝑛−1)𝑇}^∗𝜔∘𝑆𝜃_{(𝑛−2)𝑇}^∗𝜔(𝜔)Θ∘⋯∘𝑆=𝑆_𝑛−1𝜔∘𝑆Θ_𝑛−2𝜔∘⋯∘𝑆(𝜔). (33) This implies that {𝑆^𝑛(𝜔)}_𝑛∈ℕ is a discrete RDS over the MDS (Ω,ℱ,ℙ,(Θ_𝑛)_𝑛∈ℤ) on 𝑋, where Θ_𝑛(𝜔)=𝜃_𝑛𝑇^∗(𝜔). (34)

3. Applications

Our abstract results can be applied to many stochastic models. In this section, we consider the following stochastic semilinear degenerate parabolic equation with variable, nonnegative coefficients defined on an arbitrary domain (bounded or unbounded) 𝐷⊂ℝ^𝑁 with 𝑁≥2 (we refer the reader to [9] for more details): 𝑑𝑢+−div(𝜎(𝑥)∇𝑢)+𝜆𝑢+𝑓(𝑢)𝑑𝑡=^𝑚_𝑗=1ℎ_𝑗𝑑𝑤_𝑗in𝐷×ℝ⁺;𝑢(𝑥,𝑡)=0on𝜕𝐷×ℝ⁺;𝑢(𝑥,0)=𝑢₀(𝑥),in𝐷, (35) where 𝜆>0 and the nonlinear term 𝑓∈𝐶¹(ℝ,ℝ) satisfies the following assumptions: 𝑓(0)=0,𝑓^′(𝑠)≥−𝑙,∀𝑠∈ℝ, (36) with positive constant 𝑙.(1)The case when 𝐷 is bounded is as follows: 𝛼₁|𝑠|^𝑝−𝛽₁≤𝑓(𝑠)𝑠≤𝛼₂|𝑠|^𝑝−𝛽₂,∀𝑠∈ℝ, (37) with positive constants 𝛼₁, 𝛼₂, 𝛽₁, and 𝛽₂.(2)The case when 𝐷 is unbounded is as follows: 𝑓(𝑥,𝑠)𝑠≥𝛼₁|𝑠|^𝑝−𝑘₁(𝑥),𝑓(𝑥,𝑠)≤𝛼₂|𝑠|^𝑝−1+𝑘₂(𝑥), (38)with positive constants 𝛼₁ and 𝛼₂, and 𝑘₁∈𝐿¹(𝐷)∩𝐿^∞(𝐷), and 𝑘₂∈𝐿²(𝐷)∩𝐿^𝑞(𝐷), where (1/𝑝)+(1/𝑞)=1.

The degeneracy of problem (35) is considered in the sense that the measurable, nonnegative diffusion coefficient 𝜎(𝑥) is allowed to have at most a finite number of essential zeros. We assume that the function 𝜎∶𝐷→ℝ⁺∪{0} satisfies the following assumptions:(ℋ_𝛼)𝜎∈𝐿¹_loc(𝐷) and, for some 𝛼∈(0,2), liminf_𝑥→𝑧|𝑥−𝑧|^−𝛼𝜎(𝑥)>0 for every 𝑧∈𝐷, when the domain 𝐷 is bounded;(ℋ^∞_𝛼,𝛽)𝜎 satisfies condition ℋ_𝛼 and liminf_|𝑥|→∞|𝑥|^−𝛽𝜎(𝑥)>0 for some 𝛽>2, when the domain 𝐷 is unbounded.

We use the natural energy space 𝒟₀^1,2(𝐷,𝜎) defined as the closure of 𝐶^∞₀ with respect to the norm: ‖𝑢‖_𝒟0^1,2_(𝐷,𝜎)∶=_𝐷𝜎(𝑥)|∇𝑢|²𝑑𝑥^1/2. (39) The space 𝒟₀^1,2(𝐷,𝜎) is a Hilbert space with respect to the scalar product: (𝑢,𝑣)_𝜎∶=_𝐷𝜎(𝑥)∇𝑢∇𝑣𝑑𝑥. (40) Moreover, 𝒟₀^1,2(𝐷,𝜎)↪𝐿²(𝐷) compactly for both bounded (when assumption ℋ_𝛼 holds true) and unbounded (when assumption ℋ^∞_𝛼,𝛽 holds true) domain 𝐷.

We consider the following parameterized evolution equation: 𝜕𝑣𝜃𝜕𝑡+𝐴𝑣+𝜆𝑣+𝑓𝑣+𝑧_𝑡𝜔=−𝐴𝑧𝜃_𝑡𝜔, (41) where 𝑣(𝑡)=𝑢(𝑡)−𝑧(𝜃_𝑡𝜔) and 𝑢(𝑡) is a solution of (35). Also 𝐴𝑣∶=−div(𝜎(𝑥)∇𝑣) and 𝑧 is an Ornstein-Uhlenbeck process.

We denote by 𝜙(𝑡,𝜔,𝑢₀)=𝑢(𝑡,𝜔,𝑢₀) the RDS generated by (35) and 𝒜(𝜔) the random attractor in 𝐿²(𝐷) for the RDS 𝜙. We now verify the compact Lipschitz condition (28) as follows.

Lemma 7. Under the assumptions (36), (37), and ℋ_𝛼 for bounded domain ((36), (38), and ℋ^∞_𝛼,𝛽 hold for unbounded domain, resp.), one has that, for any 𝑡≥1 and ℙ-a.s. 𝜔∈Ω, 𝜙𝑡,𝜔,𝑢_0,1−𝜙𝑡,𝜔,𝑢_{0,2𝒟01,2(𝐷,𝜎)}≤𝑐𝑒^𝑐𝑡𝑢_0,1−𝑢_0,2,∀𝑢_0,1,𝑢_0,2∈𝐿²(𝐷), (42) where 𝑐 is independent of 𝜔.

Proof. Setting 𝑣_0,1=𝑢_0,1−𝑧(𝜔) and 𝑣_0,2=𝑢_0,2−𝑧(𝜔), we assume that 𝑣₁(𝑡) and 𝑣₂(𝑡) are two solutions of (41) with the initial functions 𝑣_0,1 and 𝑣_0,2, respectively. We consider the difference 𝑤(𝑡)=𝑣₁(𝑡)−𝑣₂(𝑡), and then 𝑤(𝑡) satisfies 𝑤_𝑡(𝑡)+𝐴𝑤(𝑡)+𝜆𝑤(𝑡)+𝑙(𝑡,𝜔)𝑤(𝑡)=0, (43) where 𝑙(𝑡,𝜔)=∫¹₀𝑓^′[𝑠(𝑣₁(𝑡,𝜔)+𝑧(𝜃_𝑡𝜔))+(1−𝑠)(𝑣₂(𝑡,𝜔)+𝑧(𝜃_𝑡𝜔))]𝑑𝑠.
We first take the inner product of (43) with 𝑤(𝑡) in 𝐿²(𝐷) to get 12𝑑‖𝑤‖𝑑𝑡²+‖𝑤‖²_{𝒟01,2(𝐷,𝜎)}+𝜆‖𝑤‖²+(𝑙(𝑡,𝜔)𝑤,𝑤)=0. (44) Also 𝑓^′(𝑢)≥−𝑙 implies that 𝑙(𝑡,𝜔)≥−𝑙, and we can get from the above equation that 𝑑‖𝑤‖𝑑𝑡²+2‖𝑤‖²_{𝒟01,2(𝐷,𝜎)}+2𝜆‖𝑤‖²≤𝑐‖𝑤‖². (45) It is easy to deduce from (45) that ‖𝑤(𝑡)‖²≤𝑒^𝑐𝑡‖𝑤(0)‖²,𝑡≥0. (46) For any 𝑡≥0, we integrate (45) into (𝑡,𝑡+1) to get _𝑡^𝑡+1‖𝑤(𝑠)‖²_{𝒟01,2()𝐷,𝜎}𝑑𝑠≤𝑐‖𝑤(𝑡)‖²+𝑐_𝑡^𝑡+1‖𝑤(𝑠)‖²𝑑𝑠. (47) Putting (46) into the above inequality we obtain that _𝑡^𝑡+1‖𝑤(𝑠)‖²_{𝒟01,2()𝐷,𝜎}𝑑𝑠≤𝑐𝑒^𝑐𝑡‖𝑤(0)‖². (48)
Next, we multiply (43) by 𝐴𝑤, and we have 12𝑑‖𝑤‖𝑑𝑡^{2𝒟01,2()𝐷,𝜎}+‖𝐴𝑤‖²+𝜆‖𝑤‖²_{𝒟01,2()𝐷,𝜎}+(𝑙(𝑡,𝜔)𝑤,𝐴𝑤)=0. (49) This implies that 𝑑‖𝑤(𝑡𝑑𝑡)‖²_{𝒟01,2()𝐷,𝜎}≤𝑐‖𝑤(𝑡)‖²_{𝒟01,2()𝐷,𝜎}. (50) Applying the uniform Gronwall lemma (noting that the uniform Gronwall inequality also holds true when the right-hand side of (48) is dependent on 𝑡!), it yields that ‖𝑤(𝑡)‖²_{𝒟01,2()𝐷,𝜎}≤𝑐𝑒^𝑐𝑡‖𝑤(0)‖². (51)
Finally, using the relationship 𝑢(𝑡,𝜔,𝑢₀)=𝑣(𝑡,𝜔,𝑣₀)+𝑧(𝜃_𝑡𝜔), one can easily deduce the result. This completes the proof.

Choosing 𝑡=1 in Lemma 7 and setting 𝑆(𝜔)=𝜙(1,𝜔), we see that 𝑆(𝜔) satisfies the assumptions in Theorem 5 with 𝑋=𝐿²(𝐷) and 𝑌=𝒟₀^1,2(𝐷,𝜎). Therefore, we have the following.

Theorem 8. Let the assumptions of Lemma 7 hold. Then the random attractor 𝒜(𝜔) for the RDS  𝜙 has finite fractal dimension in 𝐿²(𝐷); that is, for ℙ-a.s. 𝜔∈Ω, dim_𝑓𝒜(𝜔)≤𝑐, (52) where 𝑐 is a constant independent of 𝜔.

Acknowledgments

The authors are grateful to the anonymous referees for helpful comments and suggestions that greatly improved the presentation of this paper. The paper was supported by NSF of China 10871078 and FRF for the Central Universities of China 2012QN034 and 2013ZZGH027.