# Fractal Dimension of a Random Invariant Set and Applications

Authors: Gang Wang; Yanbin Tang
Journal: Journal of Applied Mathematics
 (2013)

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## 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.


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## Body

## 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 Section3, 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.
LetM be a compact set in a metric space X. The fractal (box-counting) dimension dimfM of M is defined by
(1)dimfM=limsupε→0lnn(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 [15].Definition 2.
LetX 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 subset𝒰 in B is said to be (ε,ϱ)-distinguishable if ϱ(x,x′)>ε for any x,x′∈𝒰, x≠x′. We denote by mϱ(B,ε) the maximal cardinality of an (ε,ϱ)-distinguishable subset of B.(ii)
The pseudometricϱ is said to be compact on M if and only if mϱ(M,ε) is finite for every ε>0.(iii)
For anyr>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):B⊂M,diamB≤2r}.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,(θt)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, θs+t=θs∘θt for all s, t∈ℝ, and θtℙ=ℙ for all t∈ℝ.
(2) An RDS onX over an MDS (Ω,ℱ,ℙ,(θt)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,θsω,·)∘ϕ(s,ω,·) (cocycle property) on X for all s, t∈ℝ+.
An RDS is said to be continuous onX if ϕ(t,ω):X→X is continuous for all t∈ℝ+ and ℙ-a.s. ω∈Ω.Letθ 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,
(3)S(ω)𝒜(ω)=𝒜(θω).Our aim is to study the fractal dimension of the sets𝒜(ω), ω∈Ω. We define a discrete RDS {Sn,n∈ℤ+} 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.
LetX 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
(5)∥S(ω)v1-S(ω)v2∥X≤L∥v1-v2∥X,v1,v2∈𝒜(ω),(ii)
there exist compact seminormsn1(x), n2(x) (independent of ω) on X such that
(6)∥S(ω)v1-S(ω)v2∥X≤η∥v1-v2∥X+K[n1(v1-v2)+n2+Kc×(S(ω)v1-S(ω)v2)],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 {xn}⊂B such that n(xm-xn)→0 as m,n→∞). Then 𝒜(ω) has finite fractal dimension in X; that is, for ℙ-a.s. ω∈Ω,
(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)∥xi∥X2+∥yi∥X2≤R2,n1(xi-xj)+n2(yi-yj)>1,i≠j.Proof.
We setϱω(x,y)=K(n1(x-y)+n2(S(ω)x-S(ω)y)); then, for every ω∈Ω, ϱω is compact on 𝒜(ω) in the sense of Definition 2. From [14], 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),ℙ-a.s.ω,
where m0(R) is the maximal number of pairs (xi,yi) in X×X possessing the properties
(10)∥xi∥X2+∥yi∥X2≤R2,n1(xi-xj)+n2(yi-yj)>1,i≠j.
For any fixedε0>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=Bi∩𝒜(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={v∈Fi:ϱω(v,xji)≤δε0}.
Therefore,
(12)S(ω)𝒜(ω)⊂⋃i=1n(𝒜(ω),ε0)‍⋃j=1ni‍S(ω)Bi,j.
For any y1,y2∈Bi,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,j≤2(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 generaln∈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=1ni′‍S(θnω)Bi,j′,
where ni′≤eν, and
(16)n(S(θnω)Sn(ω)𝒜(ω),2(η+δ)ε0)≤ni′n(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),foranyδ>0.
That is,
(23)-(β+δ)k≤lnqk≤-(β-δ)k,(β+δ)cforanyk∈ℕ,anyδ>0.
This implies that
(24)ε0e-(β+δ)k≤qkε0≤e-(β-δ)kε0,ccccforanyk∈ℕ,anyδ>0.
On the other hand, for any M>0, we consider the following set:
(25)ΩM={ω∈Ω:Vν(𝒜(ω),ε0)≤M}.
Then, ΩM1⊂ΩM2 for any M1≤M2 and Ω=⋃M>0‍ΩM. We can choose M0 large enough such that ℙ(ΩM)>0 for all M≥M0. It follows from the Poincaré recurrence theorem (see [13]) that, for every element ω∈ΩM (M≥M0), there exists a sequence kj=kj(ω)→∞ such that θ-kjω∈ΩM (M≥M0). Therefore, from (21), for all ω∈ΩM (M≥M0),
(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 [13]. Then, (26) and the related result in [13] yield that
(27)dimf𝒜(ω)≤ν,ω∈ΩM(M≥M0).
Since P(ΩM)→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.
LetX 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,
(28)∥S(ω)v1-S(ω)v2∥Y≤L∥v1-v2∥X,v1,v2∈𝒜(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),ℙ-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-xj∥X>1, i≠j.Remark 6.
Recalling that we have definedn(M,ε) in Definition 1, we call ℋε(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)≤2ℋ1/2(iBY(R))=2ℋ1/2(RiBY)=2ℋ1/2R(iBY),
where BY=BY(1). Moreover, the Kolmogorov entropy is closely related to the entropy numbers ek(i), k∈ℕ (see definition in [17]). Thus, one can estimate mY,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 ek(i).In the concrete application of Theorems4 and 5, one can define
(32)S(ω)=ϕ(T*,ω)forsomeT*>0,
where T* is independent of ω and ϕ(t,ω) is an RDS on X over an MDS (Ω,ℱ,ℙ,(θt)t∈ℝ). 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∈ℕ is a discrete RDS over the MDS (Ω,ℱ,ℙ,(Θn)n∈ℤ) 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)D⊂ℝN with N≥2 (we refer the reader to [9] for more details):
(35)du+[-div(σ(x)∇u)+λu+f(u)]dt=∑j=1mhjdwjinD×ℝ+;u(x,t)=0on∂D×ℝ+;u(x,0)=u0(x),inD,
where λ>0 and the nonlinear term f∈C1(ℝ,ℝ) satisfies the following assumptions:
(36)f(0)=0,f′(s)≥-l,∀s∈ℝ,
with positive constant l.(1)
The case whenD is bounded is as follows:
(37)α1|s|p-β1≤f(s)s≤α2|s|p-β2,∀s∈ℝ,
with positive constants α1, α2, β1, and β2.(2)
The case whenD is unbounded is as follows:
(38)f(x,s)s≥α1|s|p-k1(x),|f(x,s)|≤α2|s|p-1+k2(x),with positive constants α1 and α2, and k1∈L1(D)∩L∞(D), and k2∈L2(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→ℝ+∪{0} satisfies the following assumptions:(ℋα)
σ∈Lloc1(D) and, for some α∈(0,2), liminfx→z|x-z|-ασ(x)>0 for every z∈D-, when the domain D is bounded;(ℋα,β∞)
σ satisfies condition ℋα and liminf|x|→∞|x|-βσ(x)>0 for some β>2, when the domain D is unbounded.We use the natural energy space𝒟01,2(D,σ) defined as the closure of C0∞ with respect to the norm:
(39)∥u∥𝒟01,2(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)∇u∇vdx.
Moreover, 𝒟01,2(D,σ)↪L2(D) compactly for both bounded (when assumption ℋα holds true) and unbounded (when assumption ℋα,β∞ holds true) domain D.We consider the following parameterized evolution equation:(41)∂v∂t+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 byϕ(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 ℋα for bounded domain ((36), (38), and ℋα,β∞ hold for unbounded domain, resp.), one has that, for any t≥1 and ℙ-a.s. ω∈Ω,
(42)∥ϕ(t,ω,u0,1)-ϕ(t,ω,u0,2)∥𝒟01,2(D,σ)≤cect∥u0,1-u0,2∥,∥ϕ(t,ω,u0,1)-(t,ω,u0,2)∥𝒟01,2(D,σ)c∀u0,1,u0,2∈L2(D),
where c is independent of ω.Proof.
Settingv0,1=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)12ddt∥w∥2+∥w∥𝒟01,2(D,σ)2+λ∥w∥2+(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)ddt∥w∥2+2∥w∥𝒟01,2(D,σ)2+2λ∥w∥2≤c∥w∥2.
It is easy to deduce from (45) that
(46)∥w(t)∥2≤ect∥w(0)∥2,t≥0.
For any t≥0, we integrate (45) into (t,t+1) to get
(47)∫tt+1‍∥w(s)∥𝒟01,2(D,σ)2ds≤c∥w(t)∥2+c∫tt+1‍∥w(s)∥2ds.
Putting (46) into the above inequality we obtain that
(48)∫tt+1‍∥w(s)∥𝒟01,2(D,σ)2ds≤cect∥w(0)∥2.
Next, we multiply (43) by Aw, and we have
(49)12ddt∥w∥𝒟01,2(D,σ)2+∥Aw∥2+λ∥w∥𝒟01,2(D,σ)2+(l(t,ω)w,Aw)=0.
This implies that
(50)ddt∥w(t)∥𝒟01,2(D,σ)2≤c∥w(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)2≤cect∥w(0)∥2.
Finally, using the relationshipu(t,ω,u0)=v(t,ω,v0)+z(θtω), one can easily deduce the result. This completes the proof.Choosingt=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 Lemma7 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,
(52)dimf𝒜(ω)≤c,
where c is a constant independent of ω.



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