JAM Journal of Applied Mathematics 1687-0042 1110-757X Hindawi Publishing Corporation 415764 10.1155/2013/415764 415764 Research Article Fractal Dimension of a Random Invariant Set and Applications http://orcid.org/0000-0001-6287-5273 Wang Gang Tang Yanbin Van Vleck Erik 1 School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan, Hubei 430074 China hust.edu.cn 2013 7 11 2013 2013 29 04 2013 26 09 2013 2013 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.

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., ). 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 .

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 ωΩ (Ω 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 C1-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 ·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 dimfM of M is defined by (1)dimfM=limsupε0lnn(M,ε)ln(1/ε), 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 .

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 BM and ε>0.

A subset 𝒰 in B is said to be (ε,ϱ)-distinguishable if ϱ(x,x)>ε for any x,x𝒰, xx. We denote by mϱ(B,ε) the maximal cardinality of an (ε,ϱ)-distinguishable subset of B.

The pseudometric ϱ is said to be compact on M if and only if mϱ(M,ε) is finite for every ε>0.

For any r>0 we define a local (r,ε,ϱ)-capacity of the set M by the formula (2)𝒞ϱ(M;r,ε)=sup{lnmϱ(B,ε):BM,diamB2r}.

We next recall some notions related to RDS. The reader is referred to [48, 16] for more details. Let (Ω,,) be a probability space and let X be a Banach space.

Definition 3.

(1) (Ω,,,(θt)t) is called a metric dynamical system (MDS) if θ:×ΩΩ is (()×,)-measurable, θ0 is the identity on Ω, θs+t=θsθt for all s, t, and θt= for all t.

(2) An RDS on X over an MDS (Ω,,,(θt)t) is a mapping ϕ:+×Ω×XX, (t,ω,x)ϕ(t,ω,x) which is ((+)××(X),(X))-measurable and satisfies, for -a.s. ωΩ,

ϕ(0,ω,·)=id on X,

ϕ(t+s,ω,·)=ϕ(t,θsω,·)ϕ(s,ω,·) (cocycle property) on X for all s, t+.

An RDS is said to be continuous on X if ϕ(t,ω):XX is continuous for all t+ and -a.s. ωΩ.

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

Our aim is to study the fractal dimension of the sets 𝒜(ω), ωΩ. We define a discrete RDS {Sn,n+} by Sn(ω):=S(θn-1ω)S(θn-2ω)S(θω)S(ω). In the proof of the following theorem we keep track of the “ε-approximate ν volume”: (4)Vν(X,ε)=ενn(X,ε). Our main results read as follows.

Theorem 4.

Let X be a Banach space, and S(ω) satisfies, for -a.s. ωΩ, the following:

S(ω) is Lipschitz on 𝒜(ω); that is, there exists L>0 independent of ω such that (5)S(ω)v1-S(ω)v2XLv1-v2X,v1,v2𝒜(ω),

there exist compact seminorms n1(x), n2(x) (independent of ω) on X such that (6)S(ω)v1-S(ω)v2Xηv1-v2X+K[n1(v1-v2)+n2+Kc×(S(ω)v1-S(ω)v2)],

for any v1,v2𝒜(ω), where 0η<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 BX there exists a sequence {xn}B such that n(xm-xn)0 as m,n). Then 𝒜(ω) has finite fractal dimension in X; that is, for -a.s. ωΩ, (7)df𝒜(ω)lnm0(8eK(1+L2)1/21-2eη), where m0(R) is the maximal number of pairs (xi,yi) in X×X possessing the properties (8)xiX2+yiX2R2,n1(xi-xj)+n2(yi-yj)>1,ij.

Proof.

We set ϱω(x,y)=K(n1(x-y)+n2(S(ω)x-S(ω)y)); then, for every ωΩ, ϱω is compact on 𝒜(ω) in the sense of Definition 2. From , we see that the local (r,ε,ϱω)-capacity of the set 𝒜(ω) admits the estimate (9)𝒞ϱω(𝒜(ω);r,ε)lnm0(2K(1+L2)1/2rε),-a.s.  ω, where m0(R) is the maximal number of pairs (xi,yi) in X×X possessing the properties (10)xiX2+yiX2R2,n1(xi-xj)+n2(yi-yj)>1,ij.

For any fixed ε0>0, we assume that {Bi:i=1,,n(𝒜(ω),ε0)} is the minimal covering of 𝒜(ω) by closed balls of the radius ε0. Set Fi=Bi𝒜(ω),i=1,,n(𝒜(ω),ε0). Let δ=(1/4e)-(η/2) and let {xji;j=1,,ni}Fi be a maximal (δε0,ϱω)-distinguishable subset of Fi. Since ϱω is compact, this finite set exists, and then we have (11)ni=mϱω(Fi,δε0)exp{Cϱω(𝒜(ω);ε0,δε0)}m0(2K(1+L2)1/2δ)=:eν,𝒜(ω)i=1n(𝒜(ω),ε0)j=1niBi,j,Bi,j={vFi:ϱω(v,xji)δε0}. Therefore, (12)S(ω)𝒜(ω)i=1n(𝒜(ω),ε0)j=1niS(ω)Bi,j. For any y1,y2Bi,j(𝒜(ω)), we get from (6) that (13)d(S(ω)y1,S(ω)y2)ηd(y1,y2)+ϱω(y1,xji)+ϱω(y2,xji)2(η+δ)ε0. Thus diam S(ω)Bi,j2(η+δ)ε0 for any i,j. Therefore, (14)n(S(ω)𝒜(ω),2(η+δ)ε0)nin(𝒜(ω),ε0)eνn(𝒜(ω),ε0).

For general n, replacing ϱω, S(ω), and 𝒜(ω) by ϱθnω, S(θnω), and Sn(ω)𝒜(ω), respectively, in the above procedure and noting that Sn(ω)𝒜(ω)=𝒜(θnω), we can get that (15)S(θnω)Sn(ω)𝒜(ω)i=1n(𝒜(θnω),ε0)j=1niS(θnω)Bi,j, where nieν, and (16)n(S(θnω)Sn(ω)𝒜(ω),2(η+δ)ε0)nin(𝒜(θnω),ε0)eνn(𝒜(θnω),ε0). Thus, (17)n(Sn+1(ω)𝒜(ω),2(η+δ)ε0)eνn(Sn(ω)𝒜(ω),ε0). Setting q=2(η+δ), then by a standard induction procedure we deduce that (18)n(Sn(ω)𝒜(ω),qnε0)enνn(𝒜(ω),ε0).

Multiplying (18) by qnνε0ν we get (19)Vν(Sn(ω)𝒜(ω),qnε0)qnνenνε0νn(𝒜(ω),ε0)=κnνVν(𝒜(ω),ε0), where κ=qe=2(η+δ)e<1. Then we can get from the above inequality that (20)Vν(𝒜(θnω),qnε0)κnνVν(𝒜(ω),ε0). Therefore, (21)Vν(𝒜(ω),qnε0)κnνVν(𝒜(θ-nω),ε0). On one hand, for the above q, setting β=lnq-1, then we have (22)-(β+δ)lnq-(β-δ),forany  δ>0. That is, (23)-(β+δ)klnqk-(β-δ)k,(β+δ)cfor  any  k,any  δ>0. This implies that (24)ε0e-(β+δ)kqkε0e-(β-δ)kε0,ccccfor  any  k,any  δ>0. On the other hand, for any M>0, we consider the following set: (25)ΩM={ωΩ:Vν(𝒜(ω),ε0)M}. Then, ΩM1ΩM2 for any M1M2 and Ω=M>0ΩM. We can choose M0 large enough such that (ΩM)>0 for all MM0. It follows from the Poincaré recurrence theorem (see ) that, for every element ωΩM (MM0), there exists a sequence kj=kj(ω) such that θ-kjωΩM (MM0). Therefore, from (21), for all ωΩM (MM0), (26)Vν(𝒜(ω),qkjε0)κkjνVν(𝒜(θ-kjω),ε0)Mκkjν. From (24) we see that qkjε0 satisfies the assumptions of Lemma 2.2 in . Then, (26) and the related result in  yield that (27)dimf𝒜(ω)ν,ωΩM(MM0). Since P(ΩM)1 as M, this yields that the above inequality holds for -a.s. ωΩ. The proof is complete.

As in the deterministic case , 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 -a.s. ωΩ, (28)S(ω)v1-S(ω)v2YLv1-v2X,v1,v2𝒜(ω), where L is a constant independent of ω. Then 𝒜(ω) has finite fractal dimension in X and admits the estimate (29)dimf𝒜(ω)lnmY,X(8eL),-a.s.ω, where mY,X(R) is the maximal number of points xi in the ball of the radius R in Y possessing the properties xi-xjX>1, ij.

Remark 6.

Recalling that we have defined n(M,ε) in Definition 1, we call ε(M):=log2n(M,ε) the Kolmogorov ε-entropy of M. Then the number mY,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 BY(R) the ball of the radius R in Y. By the definition of mY,X(R), one can easily show that (30)mY,X(R)n(iBY(R),12). That is, (31)mY,X(R)21/2(iBY(R))=21/2(RiBY)=21/2R(iBY), where BY=BY(1). Moreover, the Kolmogorov entropy is closely related to the entropy numbers ek(i), k (see definition in ). Thus, one can estimate mY,X(R) by using the entropy numbers, and here we omit the details. We refer the readers to  for more details about the entropy numbers ek(i).

In the concrete application of Theorems 4 and 5, one can define (32)S(ω)=ϕ(T*,ω)for  some  T*>0, where T* is independent of ω and ϕ(t,ω) is an RDS on X over an MDS (Ω,,,(θt)t). Then from the cocycle property we have (33)Sn(ω)=ϕ(nT*,ω)=ϕ(T*,θ(n-1)T*ω)ϕ(T*,θ(n-2)T*ω)ϕ(T*,ω)=S(θ(n-1)T*ω)S(θ(n-2)T*ω)S(ω)=S(Θn-1ω)S(Θn-2ω)S(ω). This implies that {Sn(ω)}n is a discrete RDS over the MDS (Ω,,,(Θn)n) on X, where (34)Θn(ω)=θnT*(ω).

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) DN with N2 (we refer the reader to  for more details): (35)du+[-  div(σ(x)u)+λu+f(u)]dt=j=1mhjdwjin  D×+;u(x,t)=0onD×+;u(x,0)=u0(x),in  D, where λ>0 and the nonlinear term fC1(,) satisfies the following assumptions: (36)f(0)=0,f(s)-l,s, with positive constant l.

The case when D is bounded is as follows: (37)α1|s|p-β1f(s)sα2|s|p-β2,s, with positive constants α1, α2, β1, and β2.

The case when D is unbounded is as follows: (38)f(x,s)sα1|s|p-k1(x),|f(x,s)|α2|s|p-1+k2(x),

with positive constants α1 and α2, and k1L1(D)L(D), and k2L2(D)Lq(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+{0} satisfies the following assumptions:

σLloc1(D) and, for some α(0,2), liminfxz|x-z|-ασ(x)>0 for every zD-, when the domain D is bounded;

σ satisfies condition α and liminf|x||x|-βσ(x)>0 for some β>2, when the domain D is unbounded.

We use the natural energy space 𝒟01,2(D,σ) defined as the closure of C0 with respect to the norm: (39)u𝒟01,2(D,σ)=(Dσ(x)|u|2dx)1/2. The space 𝒟01,2(D,σ) is a Hilbert space with respect to the scalar product: (40)(u,v)σ=Dσ(x)uvdx. Moreover, 𝒟01,2(D,σ)L2(D) compactly for both bounded (when assumption α holds true) and unbounded (when assumption α,β holds true) domain D.

We consider the following parameterized evolution equation: (41)vt+Av+λv+f(v+z(θtω))=-Az(θtω), where v(t)=u(t)-z(θtω) 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 α for bounded domain ((36), (38), and α,β hold for unbounded domain, resp.), one has that, for any t1 and -a.s. ωΩ, (42)ϕ(t,ω,u0,1)-ϕ(t,ω,u0,2)𝒟01,2(D,σ)cectu0,1-u0,2,ϕ(t,ω,u0,1)-(t,ω,u0,2)𝒟01,2(D,σ)cu0,1,u0,2L2(D), where c is independent of ω.

Proof.

Setting v0,1=u0,1-z(ω) and v0,2=u0,2-z(ω), we assume that v1(t) and v2(t) are two solutions of (41) with the initial functions v0,1 and v0,2, respectively. We consider the difference w(t)=v1(t)-v2(t), and then w(t) satisfies (43)wt(t)+Aw(t)+λw(t)+l(t,ω)w(t)=0, where l(t,ω)=01f[s(v1(t,ω)+z(θtω))+(1-s)(v2(t,ω)+z(θtω))]ds.

We first take the inner product of (43) with w(t) in L2(D) to get (44)12ddtw2+w𝒟01,2(D,σ)2+λw2+(l(t,ω)w,w)=0. Also f(u)-l implies that l(t,ω)-l, and we can get from the above equation that (45)ddtw2+2w𝒟01,2(D,σ)2+2λw2cw2. It is easy to deduce from (45) that (46)w(t)2ectw(0)2,t0. For any t0, we integrate (45) into (t,t+1) to get (47)tt+1w(s)𝒟01,2(D,σ)2dscw(t)2+ctt+1w(s)2ds. Putting (46) into the above inequality we obtain that (48)tt+1w(s)𝒟01,2(D,σ)2dscectw(0)2.

Next, we multiply (43) by Aw, and we have (49)12ddtw𝒟01,2(D,σ)2+Aw2+λw𝒟01,2(D,σ)2+(l(t,ω)w,Aw)=0. This implies that (50)ddtw(t)𝒟01,2(D,σ)2cw(t)𝒟01,2(D,σ)2. 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 (51)w(t)𝒟01,2(D,σ)2cectw(0)2.

Finally, using the relationship u(t,ω,u0)=v(t,ω,v0)+z(θtω), 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=𝒟01,2(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 -a.s. ωΩ, (52)dimf𝒜(ω)c, 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.

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