Research Article | Open Access
Fractal Dimension of a Random Invariant Set and Applications
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.
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 . 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 , 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 ( denotes certain probability space). This drawback was overcome by using a “random squeezing property” in . In , Debussche used the method involving the Lyapunov exponents (see ) to obtain an upper bound on the Hausdorff dimension for a random invariant set, and this method was developed in a recent paper  for bounding the fractal dimension of random invariant sets. Motivated by , we give a new criterion for the upper bound on fractal dimension of random invariant sets. This result does not require -smoothness of the RDS. Therefore, it can be applied to more stochastic models. However, as mentioned in , 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 are written as and , respectively. The letter 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 be a compact set in a metric space . The fractal (box-counting) dimension 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 .
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 .(i)A subset in is said to be -distinguishable if for any , . We denote by the maximal cardinality of an -distinguishable subset of .(ii)The pseudometric is said to be compact on if and only if is finite for every .(iii)For any we define a local -capacity of the set by the formula
Definition 3. (1) is called a metric dynamical system (MDS) if is -measurable, is the identity on , for all , , and for all .
(2) An RDS on over an MDS is a mapping , which is -measurable and satisfies, for -a.s. ,(a) on ,(b) (cocycle property) on for all , .
An RDS is said to be continuous on if is continuous for all and -a.s. .
Let be a measure-preserving ergodic transformation on and let be a family of maps from to . We assume that , , is a compact measurable set satisfying, for -a.s. ,
Our aim is to study the fractal dimension of the sets , . We define a discrete RDS by . In the proof of the following theorem we keep track of the “-approximate volume”: Our main results read as follows.
Theorem 4. Let be a Banach space, and satisfies, for -a.s. , the following:(i) is Lipschitz on ; that is, there exists independent of such that (ii)there exist compact seminorms , (independent of ) on such that for any , where and 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 as ). Then has finite fractal dimension in ; that is, for -a.s. , where is the maximal number of pairs in possessing the properties
Proof. We set ; then, for every , is compact on in the sense of Definition 2. From , we see that the local -capacity of the set admits the estimate
where is the maximal number of pairs in possessing the properties
For any fixed , we assume that is the minimal covering of by closed balls of the radius . Set . Let and let be a maximal -distinguishable subset of . Since is compact, this finite set exists, and then we have Therefore, For any , we get from (6) that Thus diam for any . Therefore,
For general , replacing , , and by , , and , respectively, in the above procedure and noting that , we can get that where , and Thus, Setting , then by a standard induction procedure we deduce that
Multiplying (18) by we get where . Then we can get from the above inequality that Therefore, On one hand, for the above , setting , then we have That is, This implies that On the other hand, for any , we consider the following set: Then, for any and . We can choose large enough such that for all . It follows from the Poincaré recurrence theorem (see ) that, for every element (), there exists a sequence such that (). Therefore, from (21), for all (), From (24) we see that satisfies the assumptions of Lemma 2.2 in . Then, (26) and the related result in  yield that Since as , this yields that the above inequality holds for -a.s. . The proof is complete.
Theorem 5. Let and be Banach spaces such that is compactly embedded in . Let be a compact measurable set invariant under . Assume that, for -a.s. , where is a constant independent of . Then has finite fractal dimension in and admits the estimate where is the maximal number of points in the ball of the radius in possessing the properties , .
Remark 6. Recalling that we have defined in Definition 1, we call 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 by the ball of the radius in . By the definition of , one can easily show that That is, where . Moreover, the Kolmogorov entropy is closely related to the entropy numbers , (see definition in ). Thus, one can estimate by using the entropy numbers, and here we omit the details. We refer the readers to  for more details about the entropy numbers .
In the concrete application of Theorems 4 and 5, one can define where is independent of and is an RDS on over an MDS . Then from the cocycle property we have This implies that is a discrete RDS over the MDS on , where
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) with (we refer the reader to  for more details): where and the nonlinear term satisfies the following assumptions: with positive constant .(1)The case when is bounded is as follows: with positive constants , , , and .(2)The case when is unbounded is as follows: with positive constants and , and , and , where .
The degeneracy of problem (35) is considered in the sense that the measurable, nonnegative diffusion coefficient is allowed to have at most a finite number of essential zeros. We assume that the function satisfies the following assumptions:() and, for some , for every , when the domain is bounded;() satisfies condition and for some , when the domain is unbounded.
We use the natural energy space defined as the closure of with respect to the norm: The space is a Hilbert space with respect to the scalar product: Moreover, compactly for both bounded (when assumption holds true) and unbounded (when assumption holds true) domain .
We consider the following parameterized evolution equation: where and is a solution of (35). Also and is an Ornstein-Uhlenbeck process.
Proof. Setting and , we assume that and are two solutions of (41) with the initial functions and , respectively. We consider the difference , and then satisfies
We first take the inner product of (43) with in to get Also implies that , and we can get from the above equation that It is easy to deduce from (45) that For any , we integrate (45) into to get Putting (46) into the above inequality we obtain that
Next, we multiply (43) by , and we have This implies that Applying the uniform Gronwall lemma (noting that the uniform Gronwall inequality also holds true when the right-hand side of (48) is dependent on !), it yields that
Finally, using the relationship , one can easily deduce the result. This completes the proof.
Theorem 8. Let the assumptions of Lemma 7 hold. Then the random attractor for the RDS has finite fractal dimension in ; that is, for -a.s. , where is a constant independent of .
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.
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Copyright © 2013 Gang Wang and Yanbin Tang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.