FRACTAL DIMENSION OF RANDOM ATTRACTOR FOR STOCHASTIC NON-AUTONOMOUS DAMPED WAVE EQUATION WITH LINEAR MULTIPLICATIVE WHITE NOISE

. In this paper, we ﬁrst present some conditions for bounding the fractal dimension of a random invariant set of a non-autonomous random dy- namical system on a separable Banach space. Then we apply these conditions to prove the ﬁniteness of fractal dimension of random attractor for stochastic damped wave equation with linear multiplicative white noise.


1.
Introduction. It is well known that the finite dimensionality of attractor is one of most important topics in studying the asymptotic behavior of infinite-dimensional dynamical systems. Recently, the random attractors for autonomous and nonautonomous random dynamical systems have been studied widely since Crauel and Flandoli [8] established a criterion for the existence of a random attractor for an autonomous random dynamical systems, see [3-5, 7, 9, 10, 16, 24-27] and the references wherein. As we know, there are several approaches to estimate the upper bound of Hausdorff and fractal dimension of random attractors, see [9,11,12,19,20,29]. Of those works, Crauel and Flandoli [9], Debussche [11,12] developed some technique for bounding the Hausdorff dimension of random attractors for autonomous random dynamical systems; and Langa [20] generalized the method of [12] to the fractal dimension but requiring differentiability of random dynamical system. Wang and Tang [29] gave a method to bound the fractal dimension of random attractor similar to [20], but requiring some "strong" conditions that the Lipshitz constant of system and the "contraction" coefficient of the infinite-dimensional part of system are deterministic constants independent of the sample points which are only satisfied by some special random systems with uniform bounded derivative of the nonlinearity. In fact, the difficulty to a random system is that a random attractor is not uniformly bounded along the sample path of the sample points.
The finite fractal dimension of attractor plays a very important role in the finitedimensional reduction theory of infinite dimensional dynamical systems basing on the fact that a compact set A in a metric space with fractal dimension dim f (A) less than m/2 for some m ∈ N can be placed in the graph of Hölder continuous mapping which maps a compact subset of R m onto A. But no such a finite parametrization is available for a set if just knowing the boundedness of its Hausdorff dimension (see [20,22]).
An example of such nonlinearity is f (u, x) = |u| p−1 u arising in the relativistic quantum mechanics equation [23]. The attractor for damped wave equation of type (1) has been studied widely. For the global attractors, pullback attractors (or kernel sections) and the bounds of their Hausdorff and fractal dimensions for the deterministic autonomous and non-autonomous wave equations without noise (i.e., a = 0), we can see [2,6,17,18,23]. For the random attractor and the bounds of its Hausdorff dimensions for the stochastic wave equations (1) with additive noise (i.e., the random term is "adW (t)" independent of u), we can see [9,10,11,13,21,27,30,32]. For the stochastic system (1) with linear multiplicative noise "au • dW (t)" and sufficient small coefficient a, when the nonlinear function f and its derivative f u are both uniformly bounded, Fan [14] proved the existence of random attractor and obtained an upper bound of the Hausdorff and fractal dimension of the random attractor by using the method of [12]. Later, Wang [25] proved the existence of random attractor under condition (H). We notice that in [14], the obtained upper bound of the dimension of random attractor is independent of the random term and also that of the dimension of attractor for the corresponding deterministic system without noise. However, in general, the random attractor should depend on the noise term for nonlinear stochastic equations.
Notice that equation (1) is a non-autonomous stochastic equation that the external term g is time-dependent. For this case, Wang have established an efficacious theory about the existence and upper semi-continuity of random attractors for nonautonomous stochastic systems by introducing two parametric spaces, see [24][25][26][27].
Here we first prove that when f satisfies (H) and the coefficient a is small enough, the system (1) has a random attractor in iteration", then we apply our criteria to obtain an upper bound of fractal dimension of random attractor which implies that the random attractor of (1) can be embedded in a finite dimensional Euclidean space. We will see that our obtained upper bound is influenced by the noise term. It is worth mentioning that we need to prove the higher regularity of random attractor by a recurrence method, for this aim, we establish some new argument basing on the idea of [31].
This paper is organized as follows. In Section 2, we first present some concepts related to non-autonomous random dynamical system and random attractor, then we give some sufficient conditions to obtain an upper bound of fractal dimension of a random invariant set. In section 3, we apply our method to get an upper bound of fractal dimension of random attractor of system (1).

2.
Fractal dimension of random invariant sets. In this section, we give some sufficient conditions to bound the fractal dimension of a random invariant set for a non-autonomous random dynamical system. Let (Ω, F, P) be a probability space and X be a separable Banach space with Borel σ-algebra B(X). First we present some basic concepts related with nonautonomous random dynamical system and random attractor (see [1,26] for details).
Let D = D(X) be the collection of the tempered families of nonempty subsets of X.
Definition 2.5. A family K = {K(τ, ω) : τ ∈ R, ω ∈ Ω} ∈ D is called a measurable D-pullback attracting (or absorbing) set for Φ if (i) K is measurable with respect to the P-completion of F in Ω; (ii) for all τ ∈ R, ω ∈ Ω and for every B ∈ D, it holds: where "d H (·, ·)" denotes the Hausdorff semi-distance between two subsets of X.
Theorem 2.7. Let Φ be a continuous RDS on X over R and (Ω, F, P, {θ t } t∈R ). If Φ has a compact measurable (w.r.t. F) D-pullback attracting set K in D, then Φ has a unique D-pullback random attractor A in D given by: for each τ ∈ R and ω ∈ Ω, For the finiteness of fractal dimension of a random invariant set for a nonautonomous random dynamical system, we have the following result originated from the idea of Theorem 1.4 of [11].
In the following, we first prove the existence of random attractor A(τ, ω) of Φ in E based on Theorem 2.7 and study the boundness of A(τ, ω) in E 1 by "iteration", then we prove the finiteness of the fractal dimension of A(τ, ω) by Theorem 2.8.

SHENGFAN ZHOU AND MIN ZHAO
where .

3.3.
Existence of random attractor. In this subsection, we assume that (H), (69) and (97) hold. We will prove the existence of a random attractor A(τ, ω) for Φ by Theorem 2.7 and prove that A(τ, ω) is included in the bounded ball of E 1 basing on Lemma 3.6 and the "iteration" method similar to [31]. To construct a non-empty compact measurable tempered attracting set for Φ in E 1 , we need the following result which generalizes Lemma 1.3 of [31] to random dynamical system.
To this end, combining Lemmas 3.1, 3.9 and Theorem 2.7, our main result about the existence of a random attractor for the RDS Φ is as follows.