Fractal Dimension of a Random Invariant Set and Applications

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 Csmoothness 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 L(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.


Introduction
The notion of random attractors is a generalization of the classical concept of global attractors for deterministic dynamical systems (see, e.g., [1][2][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][5][6][7][8][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 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  1smoothness 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 ‖ ⋅ ‖  the norm of Banach space .The inner product and norm of  2 (Ω) are written as (⋅, ⋅) and ‖ ⋅ ‖, respectively.The letter  denotes any positive constant which may be different from line to line even in the same line.

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  be a compact set in a metric space .The fractal (box-counting) dimension dim   of  is defined by where (, ) is the minimal number of closed balls of the radius  which cover the set .
For other alternative formulations of the definition of the box-counting dimension, see Definition 3.1 in Falconer's book [15].Definition 2. Let  be a complete metric space endowed with the metric  and let  be a bounded closed set in .Assume that  is a pseudometric defined on .Let  ⊂  and  > 0.
(ii) The pseudometric  is said to be compact on  if and only if   (, ) is finite for every  > 0.
(iii) For any  > 0 we define a local (, , )-capacity of the set  by the formula We next recall some notions related to RDS.The reader is referred to [4][5][6][7][8]16] for more details.Let (Ω, F, P) be a probability space and let  be a Banach space.
measurable,  0 is the identity on Ω,  + =   ∘   for all ,  ∈ R, and   P = P for all  ∈ R.
An RDS is said to be continuous on  if (, ) :  →  is continuous for all  ∈ R + and P-a.s. ∈ Ω.
Let  be a measure-preserving ergodic transformation on (Ω, F, P) and let () be a family of maps from  to .We assume that A(),  ∈ Ω, is a compact measurable set satisfying, for P-a.s. ∈ Ω,  () A () = A () . ( Our aim is to study the fractal dimension of the sets A(),  ∈ Ω.We define a discrete RDS {  ,  ∈ Z + } by   () := ( −1 )( −2 ) ⋅ ⋅ ⋅ ()().In the proof of the following theorem we keep track of the "-approximate ] volume": Our main results read as follows.
As in the deterministic case [14], the following result can be easily deduced by Theorem 4. Theorem 5. Let  and  be Banach spaces such that  is compactly embedded in .Let A() be a compact measurable set invariant under ().Assume that, for P-a.s. ∈ Ω, where  is a constant independent of .
where 0 ≤  < 1/2 and  > 0 are constants independent of  (a seminorm () on  is said to be compact if and only if for any bounded set  ⊂  there exists a sequence {  } ⊂  such that (  −  ) → 0 as ,  → ∞).
Remark 6. Recalling that we have defined (, ) in Definition 1, we call H  () := log 2 (, ) the Kolmogorov -entropy of .Then the number  , () can be bounded by the Kolmogorov entropy.To show this we assume that  is the compact embedding of  into  in Theorem 5 and denote