Random attractors for stochastic evolution equations driven by fractional brownian motion

Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. SIAM J. MATH. ANAL.c 2014 Society for Industrial and Applied Mathematics Vol. 46, No. 4, pp. 2281–2309 RANDOM ATTRACTORS FOR STOCHASTIC EVOLUTION EQUATIONS DRIVEN BY FRACTIONAL BROWNIAN MOTION∗ HONGJUN GAO†,MAR ´ IA J. GARRIDO-ATIENZA‡,AND BJ ¨ ORN SCHMALFUSS§ Abstract. The main goal of this article is to prove the existence of a random attractor for a stochastic evolution equation driven by a fractional Brownian motion with Hurst parameter H∈ (1/2,1). We would like to emphasize that we do not use the usual cohomology method, consisting of transforming the stochastic equation into a random one, but we deal directly with the stochastic equation. In particular, in order to get adequate a priori estimates of the solution needed for the existence of an absorbing ball, we will introduce stopping times to control the size of the noise. In the first part of this article we shall obtain the existence of a pullback attractor for the nonautonomous dynamical system generated by the pathwise mild solution of an nonlinear infinitedimensional evolution equation with a nontrivial H¨older continuous driving function. In the second part, we shall consider the random setup: stochastic equations having as a driving process a fractional Brownian motion with H∈(1/2,1). Under a smallness condition for that noise we will show the existence and uniqueness of a random attractor for the stochastic evolution equation. Key words. fractional derivatives, pathwise mild solutions, nonautonomous and random dynamical systems, fractional Brownian motion, pullback and random attractors AMS subject classifications. 37L55, 60H15, 37L25, 35R60, 58B99 DOI. 10.1137/130930662 1. Introduction. This article shows the existence of random attractors for a new class of stochastic evolution equations. These equations contain a nontrivial fractional noise with a Hurst parameter H>1/2. In particular, it generalizes the results of the conference proceedings by Garrido-Atienza, Maslowski, and Schmalfuss [12], where random attractors are studied for ordinary stochastic equations containing a fractional noise. The idea of this article is based on the modern theory of stochastic integration for fractional Brownian motions (fBm) with H>1/2; see, for instance, Z¨ahle [27]. These integrals are related to the Young integration [26] and one of its presentations is based on a sort of generalized integration by parts formula with respect to fractional derivatives. The main advantage of this integration with respect to the classical Itˆo integration theory, where the integrator is a white noise (or equivalently a fractional Brownian motion with Hurst parameter H=1/2), is that integrals can be defined pathwise. However, Itˆo integrals are only defined almost surely and exceptional sets may depend on the integrand. But this fact contradicts the definition of a random dynamical system, where initial state dependent exceptional sets are not allowed. By ∗Received by the editors July 25, 2013; accepted for publication (in revised form) April 4, 2014; published electronically July 1, 2014. http://www.siam.org/journals/sima/46-4/93066.html †Institute of Mathematics, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210046, China ([email protected]). This author was supported by China NSF grant 11171158, National Basic Research Program of China (973 Program) 2013CB834100, the Natural Science Foundation of Jiangsu Province (BK2011777), Qing Lan and “333” Project of Jiangsu Province, and the NSF of the Jiangsu Higher Education Committee of China (11KJA110001) and the Jiangsu Collaborative Innovation Center for Climate Change. ‡Dpto. Ecuaciones Diferenciales y An´alisis Num´erico, Universidad de Sevilla, 41080 Sevilla, Spain ([email protected]). This author was partially supported by the European Funds for Regional Development and Ministerio de Economia y Competitividad (Spain) under grant MTM2011-22411. §Institut f¨ur Mathematik, 07737 Jena, Germany ([email protected]). This author was partially supported by the European Funds for Regional Development and Ministerio de Economia y Competitividad (Spain) under grant MTM2011-22411. 2281 Downloaded 06/13/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. 2282 H. GAO, M. J. GARRIDO-ATIENZA, AND B. SCHMALFUSS using fractional derivatives to define stochastic integrals we are able to avoid this dependence on exceptional sets. We remark that the results presented in this article do not cover the white noise case. For the existence of pathwise solutions for this case (and more generally for the cases in which H∈(1/3,1/2]) we refer to [10], [11] and the forthcoming paper [9]. We want to emphasize here that the techniques to obtain such a pathwise solution are qualitatively different from the methods presented in this article. This article is divided into two parts. In the first part, sections 2 and 3, we mention that nonlinear infinite-dimensional evolution equations driven by a H¨older continuous function with H¨older index greater than 1/2 have a pathwise mild solution which generates a nonautonomous dynamical system, and we obtain the existence of a unique pullback attractor associated to it if we restrict this dynamical system to a discrete time set. Existence results for this kind of equation have been studied in Maslowski and Nualart [20] and in Garrido-Atienza, Lu, and Schmalfuss [8] when the driving function is an fBm with H>1/2, and very recently by Chen, Gao, Garrido-Atienza, and Schmalfuss [5]. In this last paper, the authors were able to overcome the lack of the regularity of the semigroup generated by the linear part of the evolution equation by considering suitable modifications of the space of β-H¨older continuous functions as phase space, where β>1/2 is related to the H¨older index of the driving function, and they established the existence and uniqueness of a pathwise mild solution. These results are thus briefly included as a preliminary step, where the construction of the integral for H¨older continuous integrators is the main tool. Then, by standard arguments we can consider the pathwise mild solution to be a nonautonomous dynamical system. According to Flandoli and Schmalfuss [7] or Schmalfuss [23], in order to prove the existence of a nonautonomous or pullback attractor, appropriate a priori estimates of the solution are necessary. In particular (see Theorem 2.2 below), we should construct a compact pullback absorbing set belonging to a certain family. However, by the particular structure of the estimates of the pathwise integrals, the standard Gronwall lemma is not available and, therefore, we have to formulate a special Gronwall lemma for discrete time, which will be shown to work only when the estimates of these integrals are not too large. To ensure that this smallness condition holds true we will introduce (non-Markovian) stopping times with the property that they themselves satisfy the cocycle property, and assuming that these stopping times have special asymptotic properties, the existence of a unique pullback attractor is obtained. Then, in the second part of the article (section 4), we show how to adapt the previous results to obtain the random attractor associated to stochastic evolution equations driven by an fBm with H>1/2. In fact, by choosing the canonical version of the fBm, we show that the nonautonomous dynamical system becomes a random dynamical system and that the stopping times are measurable. In addition, it suffices to choose fractional noises with covariance operators having small trace, since this condition ensures that the required asymptotical properties of the stopping times hold with probability one. All the previous study finally concludes with the existence and uniqueness of a random attractor. To finish the introduction, we would like to stress that there are a few papers whose main concern is the study of the random attractor for stochastic differential equations and stochastic partial differential equations (SPDEs) driven by an fBm with H>1/2; see [13], [15], [16], and [17]. But in all these papers the noise has a special structure (more precisely, either pure additive noise or a very simple multiplicative noise) making it possible to use the cohomology method, and hence the random attractor is obtained by transforming the stochastic equation into a random one. Our Downloaded 06/13/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. RANDOM ATTRACTORS FOR SPDEs DRIVEN BY AN FBM 2283 aim here is to consider more general noisy terms with the challenge of getting the random attractor with no transformation. We also should mention that in [12] the existence of the random attractor for a stochastic differential equation driven by such an fBm was successfully obtained without using the cohomology method. Indeed the technique we are going to follow here has some similarities to the one used in [12], in the sense that there the authors also considered stopping times and the discrete nonautonomous dynamical system as a preliminary step to get the random attractor. However, since the nature of infinite dynamical systems is different from that of their finite counterparts, the construction of the absorbing set in this paper in much more involved, mainly due to the lack of regularity of the semigroup and the way of getting the compactness of the absorbing ball. 2. Preliminaries. In this section we collect definitions and properties of dynamical systems and present the construction of the integral with H¨older continuous integrator and H¨older exponent greater than 1/2. Although this construction has already been done in the recent paper [5], we present it here for the sake of completeness, since a good understanding of this integral is the starting point to obtain the cocycle property and afterward the random attractor associated to the corresponding stochastic evolution equation. 2.1. Dynamical systems. Let (V,|·|) be a Banach space and let T+=R+or Z+. A mapping ϕ:T+×V→Vhaving the semigroup property ϕ(t, ·)◦ϕ(τ,u0)=ϕ(t+τ,u0),ϕ(0,u 0)=u0t, τ ∈T+,u 0∈V, is called an (autonomous) dynamical system. A dynamical system ϕhas a global attractor A⊂Vwith respect to the set system D, consisting of the bounded sets of V,ifAis nonempty, compact, and invariant, that is, ϕ(t, A)=Afor all t∈T+ and lim T+t→∞ dist(ϕ(t, D),A) = 0 for every D∈D. dist(X,Y ) denotes the Hausdorff semidistance defined by dist(X,Y )=sup x∈X inf y∈Y|x−y|. A comprehensive presentation of the concept of attractors can be found in the work by Babin and Vishik [2], Hale [18], or Temam [25]. We want to consider a generalization of the concept of global attractors to nonautonomous and random dynamical systems given by the pullback attractors.Asa first ingredient, for the time set T=Ror Z, we introduce the flow (θt)t∈Ton the set Ωofnonautonomous perturbations by θ:T×Ω→Ω, θt◦θτ=θt+τ,θ 0ω=ωfor t, τ ∈T, ω ∈Ω. The easiest example for such a flow is given by Ω = Tand θji=i+jfor j∈T. Downloaded 06/13/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. 2284 H. GAO, M. J. GARRIDO-ATIENZA, AND B. SCHMALFUSS As a generalization of the semigroup property we consider a cocycle to be a mapping ϕ:T+×Ω×V→Vsuch that ϕ(t+τ,ω,u0)=ϕ(t, θτω,·)◦ϕ(τ,ω,u0),(2.1) ϕ(0,ω,u 0)=u0 for all t, τ ∈T+,u0∈V,andω∈Ω. ϕis also called a nonautonomous dynamical system. Let us now equip (Ω,θ) with a measurable structure. Consider the probability space (Ω,F,P)whereFis a σ-algebra on Ω and Pis a measure, which is supposed to be invariant and ergodic with respect to θ.Then(Ω,F,P,θ) is called an ergodic metric dynamical system.AB(T+)⊗F⊗B(V),B(V) measurable mapping ϕhaving the cocycle property (2.1) is called a random dynamical system with respect to the metric dynamical system (Ω,F,P,θ). Before we give the notion of an attractor for a nonautonomous dynamical system we introduce set systems that will be attracted by that attractor. Let Dbe a set of families (D(ω))ω∈Ωsuch that ∅ =D(ω)⊂Vand satisfies a general property P,which will be determined later. In addition we assume the following completeness condition for D:letD=(D(ω))ω∈Ωsuch that ∅ =D(ω)⊂Vand satisfies the property P; if there exists a D∈Dsuch that D(ω)⊂D(ω)forω∈Ω, then D∈D.Fora given ν>0, an example of such a system is the backward ν-exponentially growing sets Dν V:D=(D(ω))ω∈Ωis an element of this set system if there exists a mapping Ωω→ r(ω)∈R+such that ∅ =D(ω)⊂BV(0,r(ω)) and lim sup Tt→−∞ log+r(θtω) |t|<ν.(2.2) BV(0,r(ω)) denotes the ball in Vwith center 0 and radius r(ω). We also stress that sometimes we write Dν Z,V or Dν R,V to emphasize the corresponding time set. We note that it is enough to consider these properties along an orbit t∈T{θtω}. However, we define families for (D(ω))ω∈Ω. We now consider these families of sets in a random setup. Assuming that Vis aseparable Banach space, we consider D=(D(ω))ω∈Ωsuch that ∅ =D(ω)⊂Vis closed and verifies the following property P: the mapping ω→ dist({u},D(ω)) is a random variable for u∈V.ThenDis a random set; see Castaing and Valadier [4, Chapter III]. Indeed, we are interested in considering the set ˆ Dof random sets having backward and forward subexponential growth (ν= 0), which means that for a random set Dthere exists a positive random variable rsuch that D(ω)⊂BV(0,r(ω)) and lim Tt→±∞ log+r(θtω) |t|=0 forω∈Ω.(2.3) These sets are also called the random tempered sets. In this case we are only interested in the time set T=R. By the following lemma it is only interesting to consider the case ν=0. Lemma 2.1. Let (Ω,F,P)be an ergodic metric dynamical system. If Dis a νexponentially growing random set (ν<∞), then Dis just subexponentially growing. Downloaded 06/13/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. RANDOM ATTRACTORS FOR SPDEs DRIVEN BY AN FBM 2285 The result follows directly, since by Arnold [1, Proposition 4.1.3] we have that lim t→+∞ log+r(θtω) t= lim t→−∞ log+r(θtω) |t|∈{0,+∞}.(2.4) A family A=(A(ω))ω∈Ωis a pullback attractor for the nonautonomous dynamical system ϕwith respect to Dif A∈Dis nonempty, compact, and invariant in the sense that ϕ(t, ω, A(ω)) = A(θtω) for all t∈T+,ω∈Ω, and lim Tt→+∞dist(ϕ(t, θ−tω,D(θ−tω)),A(ω)) = 0 for all ω∈Ω,D∈D. In the case that ϕis a random dynamical system and ˆ Dconsists of the random tempered sets introduced above, we call Aa random attractor. This notion has been introduced in Schmalfuss [23]. Finally, a family B=(B(ω))ω∈Ωis called pullback absorbing for Dif ϕ(t, θ−tω,D(θ−tω)) ⊂B(ω)fort≥˜ T(D,ω)∈T+ for any D∈Dand ω∈Ω. ˜ T(D,ω) is the so-called absorption time. We have the following main theorem about the existence of a pullback/random attractor (see Flandoli and Schmalfuss [7] or Schmalfuss [24]). Theorem 2.2. Let ϕbe a continuous nonautonomous dynamical system. Suppose that ϕhas a compact pullback absorbing set Bin D.Thenϕhas a unique pullback attractor with respect to D.Ifϕis a random dynamical system and D=ˆ Dconsists of the tempered sets, then the associated pullback attractor is a random attractor. In both cases A(ω)=  τ≥T0(B,ω) t≥τ ϕ(t, θ−tω,B(θ−tω)), where T0(B,ω)is the absorption time corresponding to B. 2.2. Integrals in Hilbert spaces for H¨older continuous integrators with H¨older exponent >1/2. We begin this subsection by introducing some function spaces. Assume (V,(·,·),|·|) is a separable Hilbert space with the complete orthonormal base (ei)i∈N.LetCβ([T1,T 2]; V) be the Banach space of H¨older continuous functions with exponent β>0 having values in V. A norm on this space is given by uβ=uβ,T1,T2=sup s∈[T1,T2] |u(s)|+|||u|||β,T1,T2 with |||u|||β,T1,T2=sup T1≤s<t≤T2 |u(t)−u(s)| |t−s|β. C([T1,T 2]; V) denotes the space of continuous functions on [T1,T 2] with values in V with finite supremum norm, and let Cβ,∼([T1,T 2]; V)⊂C([T1,T 2]; V) be equipped with the norm uβ,∼=uβ,∼,T1,T2=sup s∈[T1,T2] |u(s)|+sup T1<s<t≤T2 (s−T1)β|u(t)−u(s)| |t−s|β. Downloaded 06/13/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. 2286 H. GAO, M. J. GARRIDO-ATIENZA, AND B. SCHMALFUSS For every ρ>0 we can consider the equivalent norm uβ,ρ,∼,T1,T2=sup s∈[T1,T2] e−ρ(s−T1)|u(s)|(2.5) +sup T1<s<t≤T2 (s−T1)βe−ρ(t−T1)|u(t)−u(s)| |t−s|β. When no confusion is possible, we will write uβ,ρ,∼instead of uβ,ρ,∼,T1,T2. Lemma 2.3 (see [5]). Cβ,∼([T1,T 2]; V)is a Banach space. Now we introduce integrals where the integrator is a H¨older continuous function. The definition of these integrals shall have as a foundation the definition and properties of the pathwise integrals given by Z¨ahle [27]. Let us then assume that ˜ V, ˆ Vare separable Hilbert spaces; then for 0 <α<1 and general measurable functions K: [T1,T 2]→ ˆ Vand ω:[T1,T 2]→ ˜ V, we define the following Weyl fractional derivatives: Dα T1+K[r]= 1 Γ(1 −α)K(r) (r−T1)α+αr T1 K(r)−K(q) (r−q)1+αdq∈ˆ V, D1−α T2−ωT2−[r]=(−1)1−α Γ(α)ω(r)−ω(T2−) (T2−r)1−α+(1−α)T2 r ω(r)−ω(q) (q−r)2−αdq∈˜ V, where ωT2−(r)=ω(r)−ω(T2−), ω(T2−) is the left side limit of ωat T2,andΓ(·)is the Gamma function. Let f∈L1((T1,T 2); R)andα>0. Following Samko, Kilbas, and Marichev [22], the left-sided and right-sided fractional Riemann–Liouville integrals of fof order α are defined for almost all x∈(T1,T 2)by Iα T1+f(x)= 1 Γ(α)x T1 (x−r)α−1f(r)dr, Iα T2−f(x)=(−1)−α Γ(α)T2 x (r−x)α−1f(r)dr, respectively. Denote by Iα T1+(Lp((T1,T 2); R)) (resp., Iα T2−(Lp((T1,T 2); R))) the image of Lp((T1,T 2); R)(resp.,Lp((T1,T 2); R)) by the operator Iα T1+(resp., Iα T2−). Suppose now that k∈Iα T1+(Lp((T1,T 2); R)),ζ T2−∈I1−α T2−(Lp((T1,T 2); R)) with 1/p +1/p≤1, where αp < 1, and that k(T1+), the right-side limit of kat T1, exists. Then following Z¨ahle [27] we define T2 T1 kdζ =(−1)αT2 T1 Dα T1+k[r]D1−α T2−ζT2−[r]dr,(2.6) where ζT2−(r)=ζ(r)−ζ(T2−), for r∈(T1,T 2). In fact, under the above assumptions, the appearing fractional derivatives in the integrals are well defined taking ˜ V=ˆ V=R. Consider now the separable Hilbert space L2(V) of Hilbert–Schmidt operators on Vwith the usual norm · L2(V)and inner product (·,·)L2(V). A base in this space is given by Eijek=0:j=k, ei:j=k. (2.7) Let us consider mappings K:[T1,T 2]→L2(V)andω:[T1,T 2]→Vsuch that kji = (K, Eji)L2(V)∈Iα T1+(Lp((T1,T 2); R)) and kji(T1+) exists and αp < 1. Moreover, ζiT2−=(ωT2−(t),e i)∈I1−α T2−(Lp((T1,T 2); R)), where 1/p +1/p≤1. In addition, Downloaded 06/13/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. RANDOM ATTRACTORS FOR SPDEs DRIVEN BY AN FBM 2287 [T1,T 2]r→Dα T1+K[r]L2(V)|D1−α T2−ωT2−[r]|∈L1((T1,T 2); R). We then introduce T2 T1 Kdω := (−1)αT2 T1 Dα T1+K[r]D1−α T2−ωT2−[r]dr.(2.8) Due to Pettis’s theorem and the separability of V, the integrand is weakly measurable and hence measurable. In addition, we can present this integral by T2 T1 Kdω = j iT2 T1 Dα T1+kji[r]D1−α T2−ζiT2−[r]drej with norm given by T2 T1 Kdω = j iT2 T1 Dα T1+kji[r]D1−α T2−ζiT2−[r]dr 21 2 ≤T2 T1 Dα T1+K[r]L2(V)|D1−α T2−ωT2−[r]|dr. For H>1/2, in what follows we fix parameters 1/2<β<β <β  <H. Under this choice, let Ω be the (θt)t∈R-invariant set of paths ω:R→Vwhich are β-H¨older continuous on any compact subinterval of Rand are zero at zero. Later we will need to formulate asymptotical conditions for the set of these paths. As the flow (θt)t∈Ron Ω of nonautonomous perturbations we consider θ:R×Ω→Ω,θ tω(·)=ω(t+·)−ω(t).(2.9) Lemma 2.4 (see [5]). Suppose that K:[T1,T 2]→ L2(V)is β-H¨older continuous where 1−β<α<β. Then the integral T2 T1 K(r)dω(r)∈V is well defined in the sense of (2.8).Iffor0≤τ≤T1the mapping K:[T1−τ,T2]→ L2(V)is β-H¨older continuous, then T2 T1 K(r)dω(r)=T2−τ T1−τ K(r+τ)dθτω(r). 3. Evolution equations driven by a H¨older continuous function. We now consider the following evolution equation on [0,T]: du =Au dt +F(u)dt +G(u)dω, u(0) = u0∈V,(3.1) driven by a H¨older continuous path ω. The integral with respect to dω is interpreted in the sense of the previous section. In what follows we describe the different terms on the right-hand side of (3.1). Let −Abe a strictly positive and symmetric operator with a compact inverse which is the generator of an analytic exponential decreasing semigroup Son V.Forδ≥0we introduce the spaces Vδ:= D((−A)δ) with norm |·| Vδ=|(−A)δ·|such that V=V0. Downloaded 06/13/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. 2288 H. GAO, M. J. GARRIDO-ATIENZA, AND B. SCHMALFUSS The spaces Vδ,δ>0 are supposed to be compactly embedded in V.Fromnowon, assume that (ei)i∈Nis the complete orthonormal base generated by the eigenelements of −Awith associated eigenvalues (λi)i∈N. Let L(Vδ,V γ) denote the space of continuous linear operators from Vδinto Vγ. Then there exists a constant c>0 such that we have the estimates |S(t)|L(Vα,Vγ)=|(−A)γS(t)|L(Vα,V )≤c tγ−αe−λ1tfor γ≥α, t ∈(0,T],(3.2) |S(t)−id|L(Vσ+μ,Vθ+μ)≤ctσ−θ,for θ≥0,σ∈(θ,1+θ],μ∈R;(3.3) see [6, Chapter 2]. In particular, formula (1.14) on p. 83 of that book becomes (3.2) above in bounded intervals, where we shall use it. From these inequalities, for μ, η ∈(0,1] and 0 ≤δ<γ+μthere exists a c>0 such that for 0 ≤q≤r≤s≤t, we can derive that |S(t−r)−S(t−q)|L(Vδ,Vγ)≤c(r−q)μ(t−r)−μ−γ+δ, |S(t−r)−S(s−r)−S(t−q)+S(s−q)|L(V)≤c(t−s)μ(r−q)η(s−r)−(μ+η). As usual, L(V) denotes L(V,V ). Coming back to (3.1), the mapping F:V→Vis supposed to be Lipschitz continuous, but we are going to assume that F≡0, simplification that we make for thesakeofbrevitysincethedt-nonlinearity is not the interesting term in system (3.1), and of course that we would achieve the same existence results as we obtain below assuming that Fwere Lipschitz. However, for the existence of an attractor we also would need to assume that Fhas a sufficiently small Lipschitz constant. The nonlinear operator G:V→L2(V) is assumed to be a twice continuously Fr´echet-differentiable operator with bounded first and second derivatives. Let us denote, respectively, by cDG,c D2Gthe bounds for these derivatives and set cG= G(0)L2(V). Then, for u1,u 2,v 1,v 2∈V,wehave G(u1)−G(v1)L2(V)≤cDG|u1−v1|, G(u1)L2(V)≤cG+cDG|u1|, G(u1)−G(v1)−(G(u2)−G(v2))L2(V) ≤cDG|u1−v1−(u2−v2)|+cD2G|u1−u2|(|u1−v1|+|u2−v2|). (The proof of the last inequality can be found in [20].) Finally, the driving function ω∈Cβ([0,T]; V)for1/2<β<β <Hand 1−β<α<β. We are then interested in solving (3.1) in a mild sense, which means that for t∈[0,T]wehavetosolve u(t)=S(t)u0+t 0 S(t−r)G(u(r))dω.(3.4) According to the definition of the integral given in the previous section, we need to estimate the fractional derivative of the term S(t−·)G(u(·)). Lemma 3.1 (see [5]). Let Sbe the semigroup generated by −A, assume that Gis Lipschitz and u∈Cβ,∼([T1,T 2]; V). Then for the orthonormal base (ei)i∈Nof V,for any i, j ∈N, (ei,S(t−·)G(u(·))ej)∈Iα T1+(Lp((T1,T 2); R)) Downloaded 06/13/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. RANDOM ATTRACTORS FOR SPDEs DRIVEN BY AN FBM 2289 if αp < 1. In addition, the mapping r→ Dα T1+S(t−·)G(u(·))[r]is measurable on [T1,t]for t≤T2and satisfies the estimate ||Dα T1+S(t−·)G(u(·))[r]||L2(V)≤c(1 + uβ,∼)(r−T1)−α1+(r−T1)β (t−r)β. Similarly, assuming that u∈Cβ([T1,T 2]; V),then ||Dα T1+S(t−·)G(u(·))[r]||L2(V)≤c(1 + uβ)(r−T1)−α1+(r−T1)β+(r−T1)β (t−r)β. In addition, (ωT2−(t),e i)∈I1−α T2−(Lp((T1,T 2); R)) for any p>1and ω∈Cβ([T1,T 2]; V) for β>1−α,and |D1−α T2−ωT2−[r]|≤c|||ω|||β,T1,T2(T2−r)β+α−1. Existence of solutions of (3.4) has been investigated in [20] and [8] when considering as integrators of the integrals a fractional Brownian motion with Hurst parameter in (1/2,1) and certain phase spaces, which are not as natural as the space of H¨older continuous functions. In this article we present the existence theory in the space of H¨older continuous solutions for appropriate exponents. More precisely, we have the following existence result, whose proof can be found in [5]. Theorem 3.2 (see [5]). If u0∈V, then for every T>0,(3.4) has a unique solution uin Cβ,∼([0,T]; V). If u0∈Vβ, then for every T>0,(3.4) has a unique solution uin Cβ([0,T]; V). The main reason to consider the space Cβ,∼([0,T]; V)isthatt→ S(t)u0is not a β-H¨older continuous function but an element of that space. However, if u0∈Vβ,then t→ S(t)u0is an element of Cβ([0,T]; V). Considering the equivalent norm from (2.5) on Cβ,∼([0,T]; V), we are able to adapt ρto the data of our problem (in particular to T) such that the right-hand side of (3.4) forms an operator which satisfies the conditions of the Banach fixed point theorem if u0∈V. In particular (see [5]), we obtain estimates of the integral like    · 0 S(·−r)G(u(r))dω   β,∼,0,T ≤cT|||ω|||β,0,T (1 + uβ,∼,0,T ),(3.5)    · 0 S(·−r)G(u(r))dω   β,0,T ≤cT|||ω|||β,0,T (1 + uβ,0,T ).(3.6) Assuming that u0∈V, the corresponding solution uof (3.4) satisfies u(T)∈Vβfor every T>0, due to |u(T)|Vβ≤cT−β|u0|+cT|||ω|||β,0,T (1 + uβ,∼,0,T )<∞.(3.7) The constant cTin the above formulas depends on the operators S, G and is bounded when Tis bounded. cTcan be chosen independently of ω. In addition to the previous estimates we also have the following one. Lemma 3.3. For 1−β<α<βand u∈Cβ([0,T]; V), T 0 S(T−r)G(u(r))dωVβ ≤cT|||ω|||β,0,T (1 + uβ,0,T ). Downloaded 06/13/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. 2296 H. GAO, M. J. GARRIDO-ATIENZA, AND B. SCHMALFUSS The first term on the right-hand side is backward ν-exponentially growing, which follows from the assumption that (|T(θTi(ω)ω)|−β)i∈Zis subexponentially growing. Indeed it is a product of two terms where one factor satisfies (2.2) while the other is ν-exponentially growing. Therefore, for any D∈D ν Z,V the set F(j, ω)∈D ν Z,Vβand then there is a sequence (BVβ(0,ρ(i, ω)))i∈Zbackward ν-exponentially growing such that these balls in Vβwith center zero and radius ρ(i, ω) contain the sets F(i, ω). Moreover, from Corollary 3.8 we immediately have that sup u0∈F(j−i,ω) |Φ(i, j −i, ω, u0)|≤(1 + k1)i−1e−λ1 2Ti−1(θTj−i(ω)ω)(k0ρ(j−i, ω)+k2) + i−1  m=1 2k2(1 + k1)i−1−me−λ1 2(Ti−1(θTj−i(ω)ω)−Tm(θTj−i(ω)ω)). Taking into account that the stopping times satisfy the cocycle property, it holds that Ti−1(θTj−i(ω)ω)−Tm(θTj−i(ω)ω)=Ti−1−m(θTj+m−i(ω)ω)=−T−i+m+1(θTj−1(ω)ω) and thus |Φ(i, j −i, ω, u0)|≤(1 + k1)i−1e λ1 2T−i+1(θTj−1(ω)ω)(k0ρ(j−i, ω)+k2)(3.24) + i−1  m=1 2k2(1 + k1)i−1−me λ1 2T−i+m+1(θTj−1(ω)ω) =(1+k1)i−1e λ1 2T−i+1(θTj−1(ω)ω)(k0ρ(j−i, ω)+k2) + 0  m=2−i 2k2(1 + k1)−me λ1 2Tm(θTj−1(ω)ω). Note that for every >0thereisanm>0 such that for m<0,|m|>m ,then |Tm(θTj−1(ω)ω)|>(d−)|m|,orequivalently,λ1 2Tm(θTj−1(ω)ω)<m λ1 2(d−), which is a consequence of the second inequality in (3.18). Therefore, (1 + k1)−me λ1 2Tm(θTj−1(ω)ω)=e−mlog(1+k1)+ λ1 2Tm(θTj−1(ω)ω) ≤em(−log(1+k1)+ λ1 2(d−)) ≤eνm, where the last estimate is true since λ1 2d−log(1 + k1)−ν−λ1 2>0 for small >0, which follows from the last inequality of (3.18). Hence, the sum on the right-hand side of (3.24) converges to R(j) 2when i→∞. On the other hand, the first term in (3.24) converges to zero for i→∞due to (3.18). To obtain the absorbing property it remains to mention that, thanks to the cocycle property for Φ, for sufficient large i∈Z+, |Φ(i+1,j−i−1,ω,v 0)|=|Φ(i, j −i, ω, Φ(1,j−i−1,ω,v 0))| ≤(1 + k1)ie −λ1 2Ti(θTj−i−1(ω)ω) ×(k0|Φ(1,j−i−1,ω,v 0)|Vβ+k2)+R(j)/2, Downloaded 06/13/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. RANDOM ATTRACTORS FOR SPDEs DRIVEN BY AN FBM 2297 hence sup v0∈D(j−i−1) |Φ(i+1,j−i−1,ω,v 0)|≤ sup u0∈F(j−i,ω) |Φ(i, j −i, ω, u0)|<R(j) for sufficient large i∈Z+. Lemma 3.10. Under the conditions of Lemma 3.9, the absorbing set B(ω)given by (3.21) is contained in Dν Z,V . Proof. We show that (R(i))i∈Zis ν-exponentially growing, for which we will use (3.18). Since Tm(θTi−1(ω)ω)=Tm+i−1(ω)−Ti−1(ω)=Tm+i−1(ω)+T−i+1(θTi−1(ω)ω), we obtain that R(i)=2 0  m=−∞ 2k2(1 + k1)−me λ1 2Tm(θTi−1(ω)ω) =2 0  m=−∞ 2k2(1 + k1)−meλ1 2(Tm+i−1(ω)+T−i+1(θTi−1(ω)ω)). Furthermore, due to (3.18), for any sufficiently small δthere exists j0(δ)∈Z−such that for j≤j0(δ)wehave Tj(ω)≤dj +δ 2|j|, which together with T−i+1(θTi−1(ω)ω)≤−i+ 1 implies that R(i)≤2 0  m=−∞ 2k2(1 + k1)−meλ1 2((Tm+i−1(ω)−d(i+m−1))+d(i+m−1)−(i−1)) ≤2 0  m=−∞ 2k2(1 + k1)−meλ1 2(δ|i+m−1|+d(i+m−1)−(i−1)) ≤2eλ1 2δ|i−1|e(1−d)|i−1| 0  m=−∞ 2k2(1 + k1)−meλ1 2(δ|m|+dm). For sufficiently small δthe sum on the right-hand side is finite. Moreover, for ν> (1 −d) (see (3.19)) and δ>0small lim i→−∞ eνiR(i)≤2eλ1 2δlim i→−∞ eλ1 2δ|i|+νie(1−d)|i−1| 0  m=−∞ 2k2(1 + k1)−meλ1 2(δ|m|+dm)=0 such that B⊂D ν Z,V . The conditions of the last two lemmas are always grantable if the nonautonomous perturbation is small in the following sense: μcould be chosen small. Then, if ν>0 is also small, dshould satisfy essentially the next three conditions: d<1,d+ν>1,d>2ν/λ1. And we can always find an appropriate dsolving these three inequalities. Note that the worst case happens when the first eigenvalue λ1is small, which forces νto be chosen small enough and dto be close to one. But dclose to one means that the stopping times Tiare close to one in the average, in the sense of the second inequality in (3.18), or in other words, that the contribution of |||ω|||β,0,τ for the construction of the stopping time is small in the average; see (3.9). Downloaded 06/13/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. 2298 H. GAO, M. J. GARRIDO-ATIENZA, AND B. SCHMALFUSS Theorem 3.11. Consider ω∈Ωsuch that the assumptions of Lemma 3.9 hold. Then the discrete nonautonomous dynamical system Φ(·,ω)has a pullback attractor {A(i, ω)}i∈Zwith respect to the system of ν-exponentially growing sets Dν Z,V . Proof.NotethatΦ(i, j, ω, ·) is continuous on V.LetB(ω)⊂D ν Z,V be the absorbing set from Lemma 3.9. For any j∈Z,let ˜ T(B,j,ω) be the absorption time of B(j, ω)∈D ν Z,V by itself. Set T∗=T∗(j)= ˜ T(B,j,ω). Notice that Φ(T∗,−T∗−1+j, ω, B(−T∗−1+j, ω)) is absorbing, and set C(j, ω):=Φ(1,−1+j, ω, Φ(T∗,−T∗−1+j, ω, B(−T∗−1+j, ω))) ⊂B(j, ω). Hence, the definition of T∗ensures that C(ω)=(C(j, ω))j∈Zis Dν Z,V -pullback absorbing and contained in Dν Z,V . In addition, by (3.7) and the compact embedding of Vβ⊂Vthe sets C(j, ω) are also compact. Now we can apply Theorem 2.2, giving the existence of a pullback attractor {A(i, ω)}i∈Zfor Φ. The conclusion of the last theorem will be used to study ϕas a nonautonomous dynamical system. In particular we show that this dynamical system has a pullback attractor. 3.3. Attracting sets for the nonautonomous dynamical systems associated to (3.4). We now study the nonautonomous dynamical system ϕgiven by (3.4). For this purpose it would be enough to consider this mapping ϕalong one orbit t∈R{θtω}for a fixed arbitrary ω∈Ω. However, we are going to consider measurable mappings ϕon the entire set Ω. In this sense the following definition is given. Consider the family of closed nonempty sets (D(ω))ω∈Ω⊂D 0 R,V with D(ω)= BV(0,r(ω)) such that lim sup Rt→−∞ log+r(θtω) |t|=0. For such a Ddefine the sets (G(i, ω))i∈Zby (3.25) G(i, ω)=  τ∈(Ti−1(ω),Ti(ω)] D(θτω),i∈Z, which are elements of D0 Z,V . In the contrary case, we would find an ω, a subsequence (i(i, ω))i∈Z−,i(i, ω):=i∈Z−,ti∈[Ti−1(ω),T i(ω)], and ui∈D(θtiω) such that 0<lim sup i→−∞ log+|ui| |i|. But we can estimate this expression by lim sup i→−∞ log+|ui| |Ti−1(ω)|lim sup i→−∞ |Ti−1(ω)| |i|≤lim sup i→−∞ log+|ui| |ti|lim sup i→−∞ |Ti−1(ω)| |i|=0, due to the fact that ui∈D(θtiω). Let us emphasize that, by the definition of the stopping times, the last factor in the last term is finite. We then can conclude the following. Lemma 3.12. For D∈D 0 R,V we have that ED(i, ω)=  −τ∈[ˆ T(θTi(ω)ω),0]  u0∈D(θ−τθTi(ω)ω) {ϕ(τ,θ−τθTi(ω)ω,u0)}(3.26) Downloaded 06/13/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. RANDOM ATTRACTORS FOR SPDEs DRIVEN BY AN FBM 2299 defines a set in D0 Z,V . In addition, HD(i, ω)=  τ∈[0,T(θTi(ω)ω)]  u0∈D(i,ω) {ϕ(τ,θTi(ω)ω,u0)} is in Dν Z,V provided D∈D ν Z,V . Proof. First, similar to the proof of Lemma 3.9 we obtain that (1 −cμ)uβ,∼,0,T (θTj−1(ω)ω)≤c(|u0|+μ). On the other hand, for −s∈[ˆ T(θTi(ω)ω),0], due to (3.5) we have |ϕ(s, θ−sθTi(ω)ω,u0)|≤c|u0|+s 0 S(s−r)G(u(r))dθ−sθTi(ω)ω ≤c|u0|+c(1 + uβ,∼,0,T (θTi(ω)ω))|||θ−sθTi(ω)ω|||β,0,s, and because |||θ−sθTi(ω)ω|||β,0,s ≤ |||θTi(ω)ω|||β,ˆ T(θTi(ω)ω),0≤μ, we obtain that (3.27) |ϕ(s, θ−sθTi(ω)ω,u0)|≤c|u0|+cμ1+ c 1−cμ|u0|+cμ 1−cμ. Hence the norm of any element in E(i, ω) is bounded by sup u0∈G(i,ω) c|u0|1+ cμ 1−cμ+cμ1+ cμ 1−cμ for G(i, ω) introduced in (3.25), which gives the desired property. The second property follows similarly. Lemma 3.13. Under the conditions of Theorem 3.11, define the family of sets Aby A(ω):=A(0,ω),whereA(i, ω)has been defined in Theorem 3.11.ThenAis invariant and attracting for elements from D0 R,V with respect to ϕ. Proof.Fort>0 there exists a unique i∗=i∗(ω)∈Z−such that −t∈ (Ti∗(ω)−1(ω),T i∗(ω)(ω)]. Because of T−i∗(θTi∗ω)=−Ti∗(ω), due also to the cocycle property and the relationship between ϕand Φ, we can conclude that for D∈D 0 R,V ϕ(t, θ−tω,D(θ−tω)) =ϕ(T−i∗(θTi∗ω),θ Ti∗(ω)ω,ϕ(t−T−i∗(θTi∗ω),θ −t−Ti∗θTi∗ω,D(θ−t−Ti∗θTi∗ω))) =Φ(−i∗,i ∗,ω,ϕ(t−T−i∗(θTi∗ω),θ −t−Ti∗θTi∗ω,D(θ−t−Ti∗θTi∗ω))) ⊂Φ(−i∗,i ∗,ω,E D(i∗,ω)), where ED∈D 0 Z,V is defined by (3.26) and which can be also written as ED(i, ω)=  τ∈[Ti−1(ω),Ti(ω)]  u0∈D(θτω) {ϕ(Ti(ω)−τ,θτ−Ti(ω)θTi(ω)ω,u0)}. Hence Aattracts Dwith respect to ϕsince 0≤dist(ϕ(t, θ−tω,D(θ−tω)),A(ω)) ≤dist(Φ(−i, i, ω, ED(i, ω)),A(0,ω)), and lim i→−∞ dist(Φ(−i, i, ω, ED(i, ω)),A(0,ω)) = 0. Downloaded 06/13/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. 2300 H. GAO, M. J. GARRIDO-ATIENZA, AND B. SCHMALFUSS NowweprovetheinvarianceofA.Fort>0wedenotebyi(ω) the largest isuch that Ti(ω)<tfor any ω∈Ω. Hence by (3.11) for t2=t, t1=Ti(ω) ···≤Ti−1(ω)<T −1(θtω)+t≤Ti(ω)<t ≤Ti+1(ω)<T 1(θtω)+t≤Ti+2(ω)<T 2(θtω)+t≤··· . Since (A(i+j, ω))j∈Zis pullback attracting with respect to Φ, by (3.23) the same property is true for the following compact set: ˜ A(j, θtω):=ϕ(t−Ti+j(ω)+Tj(θtω),θ Ti+j(ω)ω,A(i+j, ω)). Because of |t−Ti+j(ω)+Tj(θtω)|≤1and{A(j, ω)}j∈Z∈D ν Z,V , applying the second part of Lemma 3.12 we have that ˜ A(j, θtω)∈D ν Z,V .Furthermore Φ(1,j,θ tω, ˜ A(j, θtω)) =ϕ(T(θTj(θtω)θtω),θ Tj(θtω)θtω,ϕ(t+Tj(θtω)−Ti+j(ω),θ Ti+j(ω)ω,A(i+j, ω))) =ϕ(t+Tj+1(θtω)−Ti+j(ω),θ Ti+j(ω)ω,A(i+j, ω)) =ϕ(t+Tj+1(θtω)−Ti+j+1(ω),θ Ti+j+1(ω)ω,A(i+j+1,ω)) = ˜ A(j+1,θ tω), which shows that ˜ A(·,θ tω)isacompactinvariant set in Dν Z,V for the cocycle Φ(·,·,θ tω,·). Since ˜ A(·,θ tω)∈D ν Z,V , it is attracted by A(·,θ tω), and due to the invariance of the former ˜ A(j, θtω)⊂A(j, θtω).(3.28) Now we define ˆ A(i+j+1,ω):=ϕ(Ti+j+1(ω)−t−Tj(θtω),θ Tj(θtω)θtω,A(j, θtω)), which implies that ˆ Ahas qualitatively the same properties as ˜ A.Inparticular, ˆ A(j, ω)⊂A(j, ω). But we have A(θtω)=A(0,θ tω)(3.29) =Φ(1,−1,θ tω,A(−1,θ tω)) = ϕ(t−Ti(ω),θ Ti(ω)ω, ˆ A(i(ω),ω)) ⊂ϕ(t−Ti(ω),θ Ti(ω)ω,A(i(ω),ω)) = ˜ A(0,θ tω) such that, by (3.28) and (3.29), ˜ A(0,θ tω)=A(θtω)and ϕ(t, ω, A(ω)) = ϕ(t−Ti(ω),θ Ti(ω)ω,Φ(i,0,ω,A(0,ω)) = A(θtω), and hence Ais invariant. Lemma 3.14. Suppose that (3.30) 1 ≥ˆ d:= lim sup i→−∞ |Ti(ω)| |i|,ˇ d:= lim inf i→−∞ |Ti(ω)| |i|≥d>0. (i) We have that A∈D ν/ˇ d R,V . (ii) Suppose that ˆ d> ˇ dand ν>0. Then there exists an ω∈Ωand D∈D ν/ˇ d R,V such that the restriction of t→D(θtω)to (Ti(ω))i∈Zis in Dνˆ d/ ˇ d Z,V but not in Dν Z,V along an orbit t∈R{θtω}. Downloaded 06/13/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. RANDOM ATTRACTORS FOR SPDEs DRIVEN BY AN FBM 2301 (iii) If D∈D 0 R,V then the restriction of t→D(θtω)to (Ti(ω))i∈Nis in D0 Z,V . Proof.(i)AsA(θTi(ω)ω)=A(i, ω), we obtain lim sup i→−∞ log+(supu∈A(θTi(ω)ω)|u|) |Ti(ω)|≤lim sup i→−∞ log+(supu∈A(i,ω)|u|) |i|lim sup i→−∞ |i| |Ti(ω)|≤ν ˇ d. In addition, we can show a similar estimate for |ϕ(t, θTi(ω)ω,u0)|as in (3.27). By using that estimate we obtain the desired result simply taking t∈(Ti−1(ω),T i(ω)] and applying the above convergence result. (ii) Let ibe a sequence such that lim sup i→−∞ |Ti(ω)| |i|= lim i→−∞ |Ti(ω)| |i|. On some orbit t∈R{θtω}we define an element D∈D ν/ˇ d R,V by D(θtω)={0}for t=Ti(ω). In addition, for a sufficiently small >0 we choose a sequence (ui)isuch that lim i→−∞ log+|ui| |Ti(ω)|=ν ˇ d− and set D(θTi(ω)ω)={ui}.Wehavethat lim sup i→−∞ log+(supu∈D(θTi(ω)ω)|u|) |i|= lim sup i→−∞ log+(supu∈D(θTi(ω)ω)|u|) |Ti(ω)| |Ti(ω)| |i| = lim i→−∞ log+(supu∈D(θTi(ω)ω)|u|) |Ti(ω)| |Ti(ω)| |i| =ν ˇ d−ˆ d<νˆ d ˇ d and it is bigger than νwhen >0 is chosen sufficiently small. (iii) This follows similar to the proof of (ii). We would like to emphasize the following: by part (ii) in the above result we know that the restriction of particular elements D∈D ν/ˇ d R,V to the discrete time sequence (Ti(ω))i∈Zis only in Dˆ dν/ ˇ d Z,V but not in Dν Z,V in general. Therefore, that restriction is not attracted by (A(i, ω))i∈Zif ˆ d> ˇ d, and consequently the set Ais not an attractor for ϕ, since in that case it should attract all sets in Dν/ˇ d R,V . By these reasons Ais only called a partially attracting set, since it is contained in Dν/ˇ d R,V but we cannot prove that Ais attracting for the entire system Dν/ˇ d R,V . However, in Theorem 4.6 below and in a measurable setting, we will prove stronger growth conditions for A, which will allow us to conclude that Ais a (random) attractor. 4. Attractors for random dynamical systems of SPDEs with fractional Brownian motion. In this section we study the nonautonomous dynamical system under measurability assumptions. In particular, we need to introduce a metric dynamical system. It will be crucial that the integrals with H¨older continuous integrators are defined pathwise. This is a qualitative difference to the definition of the classical stochastic integral where the integrand is a white noise. We recall that the pathwise Downloaded 06/13/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. 2302 H. GAO, M. J. GARRIDO-ATIENZA, AND B. SCHMALFUSS definition of the former integral just gave us the nonautonomous dynamical system ϕ from section 3.1 defined by (3.8). A one-dimensional fBm is a centered Gauss process on Rwith Hurst parameter H∈(0,1) having the covariance R(s, t)=1 2(|t|2H+|s|2H−|t−s|2H) defined on an appropriate probability space. Similarly we can define an fBm with values in Vand covariance Q,whereQis a positive symmetric operator on Vof trace class. It is known that Qhas a discrete spectrum (qi)i∈Nrelated to the complete orthonormal system in Vgiven by (fi)i∈N. This process has a version ωin C0(R, V ), the space of continuous functions which are zero at zero, and q−1 2 iπiωare independently and identically distributed one-dimensional fBm, where πiis the projection on the ith mode of the base (fi)i∈N, such that we can define the integrals as in subsection 2.2. For simplicity we identify in the following the base (fi)i∈Nwith the base (ei)i∈N. In what follows we consider a canonical version of this process given by the probability space (C0(R, V ),B(C0(R, V )),P), where Pis the Gauss measure generated by the fBm. On this probability space we can also introduce the shift θtω(·)= ω(·+t)−ω(t). Lemma 4.1. (C0(R, V ),B(C0(R, V )),P,θ)is an ergodic metric dynamical system. The proof of this lemma can be found in [21] and with a deeper analysis in [14]. This (canonical) process has a version which is β-H¨older continuous on any interval [−k,k]forβ <H. Let us denote by Ωβ ⊂C0(R, V )thesetofelementswhichareβ-H¨older continuous on any interval [−k,k],k∈N, and are zero at 0. Lemma 4.2 (see [5]). We have Ωβ ∈B(C0(R, V )). In addition, Ωβ is (θt)t∈R- invariant. Choose μ, ν such that (3.18) holds. Then by the ergodic theory there exists a (θt)t∈R-invariant set Ω1∈B(C0(R, V )) of full measure such that lim t→±∞ 1 tt 0supr∈[0,1] |||θr+qω|||β ,−1,0+μ μ1 β −β dq(4.1) =EPsupr∈[0,1] |||θrω|||β ,−1,0+μ μ1 β −β =: d−1<∞. For the construction of the invariant set Ω1we refer to Arnold [1, p. 538, Ergodic Theorem (iv)]. The above expectation can be controlled by choosing a Qwith small trace such that dis close to 1; see, for instance, the construction of the random variable Kin Kunita [19, Theorem 1.4.1]. Let us also choose tr Qsmall enough such that (3.19) holds true, i.e., ν+d>1. In addition, there exists a (θt)t∈R-invariant set Ω2∈B(C0(R, V )) of full measure such that Rt→ supr∈[0,1] |||θt+rω|||β ,−1,0has a sublinear growth, which follows by Arnold [1, Proposition 4.1.3]. Indeed, supr∈[0,1] |||θrω|||β ,−1,0≤ |||ω|||β ,−1,1,where the right-hand side of this inequality has finite moments of any order, by Kunita [19, Theorem 1.4.1]. Setting Ω = Ω1∩Ω2∩Ωβ ,letFbe the trace-σ-algebra of B(C0(R, V )) with respect to Ω, and let us consider the restriction of the measure Pto this σ-algebra, Downloaded 06/13/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. RANDOM ATTRACTORS FOR SPDEs DRIVEN BY AN FBM 2303 which we denote again by P.SinceΩis(θt)t∈R-invariant it is not hard to see that the restriction of θto R×Ω is a random flow and that (Ω,F,P,θ) is an ergodic metric dynamical system; see [3] for a general proof of these properties. We are now in a position to derive from the non–autonomous dynamical system ϕa random dynamical system. Lemma 4.3. (i) Let ϕbe the nonautonomous dynamical system given by (3.8). Then this mapping is B(R+)⊗F⊗B(V),B(V)measurable, and hence ϕis a random dynamical system. (ii) The stopping times T, ˆ Tare measurable. Proof. First note that (r, ω)→ ω(r) is measurable such that (r, ω)→ D1−α T−ωT−[r] is measurable. Assuming in addition that (r, ω)→ k(r, ω)∈L2(V) is measurable and that for any ωthe product Dα 0+k(·,ω)[r]L2(V)|D1−α T−ωT−[r]|is integrable, then the stochastic integral ω→ T 0 S(T−r)G(u(r, ω))dω is measurable, since k(r, ω)=G(u(r, ω)) has the above properties. Since the solution uof (3.4) is given by the Banach fixed point theorem, it can be constructed by successive iterations in the space Cβ,∼([0,T]; V), starting this procedure with the constant function u0. Although these approximations are given on different time intervals [Ti(ω),T i+1(ω)] they can be finitely concatenated to one approximation on [0,T] converging to a solution of (3.4) in the separable Banach space C([0,T]; V) which is measurable. If we pick initial conditions in a sufficiently small neighborhood of u0, the contraction constants for the corresponding mapping in the Banach fixed point theorem can be chosen to be the same, and therefore u0→ ϕ(t, ω, u0) is continuous. Then, by Castaing and Valadier [4], the mapping ϕis measurable on B([0,T])⊗F⊗B(V),B(V). Hence, the B(R+)⊗F⊗B(V),B(V) measurability of this expression follows immediately. (ii) Since ω→ |||ω|||β,0,τ is measurable, the measurability of ω→ T(ω) follows. In the same manner we can argue for ˆ T. In order to establish the growth of the stopping times we need the following lemma. Lemma 4.4. Let N(ω)∈Nbe the random number of stopping times in [−1,0] defined by (3.9) and (3.10). Then we have for ω∈Ωβ N(ω)≤|||ω|||β ,−1,0+μ μ1 β −β . Proof. The following estimates follow easily on Ωβ : μ=sup ˆ T(ω)≤s<t≤0 |ω(t)−ω(s)| |t−s|β+μ(−ˆ T(ω))1−β ≤sup ˆ T(ω)≤s<t≤0 |ω(t)−ω(s)| |t−s|β +μ(−ˆ T(ω))1−β (−ˆ T(ω))β −β ≤(|||ω|||β ,−1,0+μ)(−ˆ T(ω))β −β such that |ˆ T(ω)|≥μ |||ω|||β ,−1,0+μ1 β −β . Downloaded 06/13/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. 2304 H. GAO, M. J. GARRIDO-ATIENZA, AND B. SCHMALFUSS The same estimate holds for ˆ T(θT−i(ω)ω)aslongas ˆ T(θT−i(ω)ω)+T−i(ω)=T−i−1(ω)≥ −1.For the smallest number ihaving this property we set N(ω)=i+ 1. Then due to the definition of the stopping times (3.10) we conclude that 1≥|T−N(ω)(ω)|= N(ω)−1  i=0 |ˆ T(θT−i(ω)ω)|≥N(ω)μ |||ω|||β ,−1,0+μ1 β −β , and then the result follows. Lemma 4.5. On Ωwe have lim inf k→−∞ |Tk(ω)| |k|= lim inf k→−∞ Tk(ω) k≥d>0, where dhas been defined in (4.1).Onthisset|T(θTi(ω)ω)|−βis subexponentially growing for i→−∞. Proof.LetMj(ω) be the number of stopping times Tk(ω)in(j−1,j]. For j∈−Z+let kj(ω) be the biggest integer such that Tkj(ω)≤j. Then choosing t2=jand t1=Tkj(ω)(ω) from (3.11) we know ···<T kj(ω)−2(ω)≤j+ˆ T(θjω)+ ˆ T(θˆ T(θjω)+jω)=j+T−2(θjω) <T kj(ω)−1(ω)≤j+ˆ T(θjω)=j+T−1(θjω)<T kj(ω)(ω)≤j such that Mj(ω)=N(θjω)orMj(ω)=N(θjω)−1, where N(ω) has been introduced in Lemma 4.4. Hence lim sup n→∞,n>1n−1 j=0 M−j(ω) n−1≤lim sup n→∞,n>1n−1 j=0 N(θ−jω) n−1 ≤lim sup n→∞,n>1n−1 j=0 |||θ−jω|||β,−1,0+μ μ1 β −β n−1, which follows from Lemma 4.4 for ω∈Ωβ . The terms under the sum of the numerator on the right-hand side can be estimated by supr∈[0,1] |||θr+q−jω|||β ,−1,0+μ μ1 β −β for any q∈[−1,0] and hence N(θ−jω)≤0 −1supr∈[0,1] |||θr+q−jω|||β ,−1,0+μ μ1 β −β dq. Then we obtain the estimate for n>1 n−1 j=0 N(θ−jω) n−1≤1 n−10 −nsupr∈[0,1] |||θr+qω|||β ,−1,0+μ μ1 β −β dq, where the right-hand side converges to 1/d for ω∈Ω1and n→∞by (4.1). Then we have lim inf n→∞  T−n−1 j=0 M−j(ω)(ω) n−1 j=0 M−j(ω)≥lim inf n→∞ (n−1) n−1 j=0 Mj−1(ω)≥d. Downloaded 06/13/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. RANDOM ATTRACTORS FOR SPDEs DRIVEN BY AN FBM 2305 To see that finally the conclusion holds true, choose for every k∈−Nan n=n(k,ω) such that − n  j=0 M−j(ω)<k≤− n−1  j=0 M−j(ω). Thus lim inf k→∞ |T−k(ω)| |k|≥lim inf n→−∞  T−n−1 j=0 M−j(ω)(ω) n j=0 M−j(ω)≥lim inf n→−∞  T−n−1 j=0 M−j(ω)(ω) n−1 j=0 M−j(ω)+N(θ−nω) = lim inf n→−∞ n−1 n−1 j=0 M−j(ω)≥d by the sublinear convergence of n→ N(θnω) (thanks to the ergodic theorem) and the at least asymptotical linear growth of n→ n−1 j=0 M−j(ω)+N(θ−nω). To prove the second part of the statement, note that similar to the estimate of ˆ T in the proof of Lemma 4.4, for ω∈Ω, |T(θTi(ω)ω)|−β≤|||θTi(ω)ω|||β ,0,1+μ μβ β −β for large |i|, which follows from the fact that the mapping i→ |||θTi(ω)ω|||β ,0,1grows sublinearly in Ω ⊂Ω2, and therefore, the previous inequality shows the subpolynomial growth of |T(θTi(ω)ω)|−β. We can finally prove the main result of this section. Theorem 4.6. The pullback attractor stated in Lemma 3.13 is a random attractor attracting the random tempered sets ˆ D,where ˆ Dhas been defined in (2.3). Proof. Since the stopping times Tiare random variables we obtain that the radii of the absorbing balls R(i, ω) are random variables, and therefore the balls B(i, ω) are random sets. Note that A(ω)=  j∈Z+ i∈Z+,i≥j Φ(i, −i, ω, B(−i, ω)). Φ inherits the measurable properties of ϕand in particular u→ Φ(i, −i, ω, u)is continuous. Hence Φ(i, −i, ω, B(−i, ω)) is a random set. Then by [7] we have that A is a random set. On the other hand, we have that A(θTi(ω)ω)⊂B(i, ω)andA∈D ν ˇ d R,V for ˇ dgiven by (3.30). Furthermore, Lemma 2.1 shows that Ais a set in ˆ D, and since ˆ D⊂D 0 R,V the conclusion follows from Lemma 3.13. 5. Appendix. We present here the proof of Lemma 3.7. Proof. We have to estimate uβ,Ti−1(θTjω),Ti(θTjω)=sup Ti−1(θTjω)≤t≤Ti(θTjω) |u(t)|+|||u|||β,Ti−1(θTjω),Ti(θTjω). First, considering u0∈Vβ, from (3.2) and (3.3) we obtain S(·)u0β,Ti−1(θTjω),Ti(θTjω)≤ce−λ1Ti−1(θTjω)|u0|Vβ. Thanks to the additivity of the above integral (see [5]), for t∈[Ti−1(θTjω),T i(θTjω)] the following splitting holds true: Downloaded 06/13/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php