Pathwise solutions and attractors for retarded SPDEs with time smooth diffusion coefficients

arXiv:1302.2400v1 [math.DS] 11 Feb 2013 PATHWISE SOLUTIONS AND ATTRACTORS FOR RETARDED SPDES WITH TIME SMOOTH DIFFUSION COEFFICIENTS HAKIMA BESSAIH, MAR´ IA J. GARRIDO-ATIENZA, AND BJ¨ ORN SCHMALFUSS Abstract. In this paper we study the long–time dynamics of mild solutions to retarded stochastic evolution systems driven by a Hilbert-valued Brownian motion. As a preparation for this purpose we have to show the existence and uniqueness of a cocycle solution of such an equation. We do not assume that the noise is given in additive form or that it is a very simple multiplicative noise. However, we need some smoothing property for the coefficient in front of the noise. The main idea of this paper consists of expressing the stochastic integral in terms of non-stochastic integrals and the noisy path by using an integration by parts. This latter term causes that in a first moment only a local mild solution can be obtained, since in order to apply the Banach fixed point theorem it is crucial to have the H¨older norm of the noisy path to be sufficiently small. Later, by using appropriate stopping times, we shall derive the existence and uniqueness of a global mild solution. Furthermore, the asymptotic behavior is investigated by using the Random Dynamical Systems theory. In particular, we shall show that the global mild solution generates a random dynamical system that, under an appropriate smallness condition for the time lag, have associated a random attractor. 1. Introduction The purpose of this paper is to show the existence of a random dynamical system generated by the solution of stochastic partial differential equations with delay of the following form (1.1) (du = (Au(t) + F(ut))dt +G(ut)dW(t),for t≥0 u(t) = ξ(t),for t∈[−µ, 0] in a separable Hilbert space H, where Ais the infinitesimal generator of an analytic semigroup on H,Fand Gare appropriate nonlinear terms, and Wis a two-sided Wiener process with values in a separable Hilbert space U. The term utis given by ut(s) = u(t+s) with s∈[−µ, 0], where µ > 0 is given and the initial condition is a continuous function on [−µ, 0]. Retarded differential systems arise naturally in several situations in the area of applied mathematics due to biological motivations like species growth or incubation time in delayed transmission of disease, see for instance [23] and [31], or due to physical reasons with non–instant transmission phenomena such as high velocity fields in wind tunnel experiments, see [21]. Further examples can be found in biochemical reactions in the field of gene regulation where lengthy transcription has been modeled with delayed dynamics, see [28]. The asymptotic behavior of such models has meaningful interpretations like permanence, instability and chaotic developments. From the mathematical point of view, there is a huge literature concerning the study of retarded stochastic differential systems, we refer here to the monographs by Mao [26, 27], and to the papers [30], [32], [9], [4] and [20], to mention a few of them. In this paper we are interested in analyzing the long-time behavior of the (mild) solution to (1.1) by obtaining the random attractor associated to the random dynamical system generated by the mild solution. However, even when dealing with non-retarded equations, a fundamental problem in the study of the dynamics of a stochastic partial differential equation is to show that it generates a random dynamical system. Nevertheless, it is well-known that a large class of partial differential equations with stationary random coefficients and Ito stochastic ordinary differential equations generate random dynamical systems, see the monograph by Arnold [1]. However, for the stochastic partial differential equations driven by Brownian motion the problem is much more difficult, and the reason is twofold: on the one hand, the stochastic integral is only defined almost surely where the exceptional set may depend on the initial state, which contradicts the definition of the cocycle property, and on the other, Kolmogorov’s theorem in an appropriate form is only true for finite dimensional random fields, see Kunita [24] Theorem 1.4.1. In spite of that there are some partial results for additive as well as simple multiplicative Brownian noises, see for instance the papers [18], [16, 17] and [8], to mention 1 2 HAKIMA BESSAIH, MAR´ IA J. GARRIDO-ATIENZA, AND BJ ¨ ORN SCHMALFUSS only a few of them. In the case of having retarded stochastic systems there are also positive results, as shown in the papers [7], [5] and [10]. In all the mentioned articles the main ingredient consists of transforming the stochastic equation into a random one, being possible to deal with the latter by using deterministic techniques. This transformation is known as cohomology, which consists of a stationary coordinate change by means of which flows of stochastic differential equations may be viewed as ordinary differential equations with a random parameter. This method presents the restriction that it is not always possible to find this appropriate change of variable, since it is applicable only when considering an additive noise or very particular cases of multiplicative noise. Our technique is not based on the comohology, rather on considering diffusion terms Gwith the smoothness property that the corresponding stochastic integral can be expressed, by means of the integration by parts formula, in terms of two non-stochastic integrals and the noise path as well (see formula (2.9) below), which in particular also means that our delayed system can be reduced to a deterministic delayed system with a random parameter. This idea of removing the stochastic Itˆo integral is partially borrowed from Bensoussan and Frehse [3]. Nevertheless, the main novelty in this article is the fact that we are able to consider non-trivial delayed diffusion terms, which is not at all a trivial problem as stressed by Mohammed [29]. As pointed out before, we shall investigate the existence of mild solutions to (1.1) generating a random dynamical system. Due to the mentioned transformation of the stochastic integral, the mild solution will be given in terms of the noise path. This fact allows the establishment of the existence and uniqueness of a local mild solution. As it will be shown in Section 2, see condition (2.11), the Banach fixed point argument will ensure the existence and uniqueness of a mild solution provided that an appropriate H¨older–norm of the noise path is sufficiently small. As a consequence, we shall consider stopping times {Ti}i∈Nwith the property that in every interval [Ti, Ti+1] we can find a unique local mild solution, and thus, we finally can build a global mild solution for the problem (1.1) by glueing all these local solutions. Note that the idea of considering a smoothing diffusion term was also pointed out by Mohammed and Scheutzow [30]. In that paper they construct the infinite-dimensional stochastic semiflow generated by the solution of stochastic functional differential equations, but when having a p-dimensional Brownian motion and not a Hilbert-valued Brownian motion. There are more differences with respect to our paper, since they consider a different phase space than in our setting and do not cover the existence of the global attractor associated to the flow. Once the existence of a mild solution is already established, we want to investigate its longtime behavior by analyzing the existence of random attractors associated to the random dynamical systems generated by the solution of (1.1). For an overview about the theory of random attractors we refer to [1], [6], [11], [19], [22], [34], amongst many others. In particular we will obtain that, under an appropriate smallness condition for the time lag, there exists a tempered absorbing ball which will ensure the existence of a random attractor for our retarded system. The content of the paper is as follows. In Section 2 we first establish the framework in which our analysis is carried out, introducing the basic notations and assumptions, and defining the mild solution as a sum of different terms in which there are no stochastic integrals. We also prove the existence and uniqueness of local solutions in adequate time intervals, that with the help of stoping times, will be sufficient to establish the existence and uniqueness of a global mild solution. This global solution generates a random dynamical system in the space C([−µ, 0]; H). We also exhibit an example to illustrate the different regularity conditions for the non-linear terms appearing in (1.1). Section 3 is devoted to the study of the random attractor associated to the random dynamical system obtained in the previous section. 2. Pathwise solutions We start this section by introducing the abstract definition of a random dynamical system. Definition 2.1. Let Vbe a Banach space. A mapping ϕ:R+×V→Vhaving the semigroup property ϕ(t, ·)◦ϕ(τ, u0) = ϕ(t+τ, u0), ϕ(0, u0) = u0for t, τ ∈R+and u0∈V is called an autonomous dynamical system. PATHWISE SOLUTIONS AND ATTRACTORS FOR RETARDED SPDES 3 We want to consider a generalization of the concept of an autonomous dynamical system to non-autonomous and random dynamical systems. As first we introduce as a model for a noise a metric dynamical system (Ω,F,P, θ) where (Ω,F,P) is a probability space and θis a B(R)⊗ F,Fmeasurable flow θ= (θt)t∈R, i.e. θt◦θτ=θt+τ, θ0= idΩfor t, τ ∈R, ω ∈Ω such that Pis ergodic with respect to θ. In the following we consider the Brownian motion metric dynamical system: let Ube a separable Hilbert space and let C0(R;U) be the set of continuous functions on Rwith values in Uwhich are zero at zero equipped with the compact open topology. We consider the Wiener measure Pon B(C0(R;U)) having a trace–class covariance operator Qon U. Then Kolmogorov’s fundamental theorem and Kolmogorov’s theorem about a (H¨older-)continuous version give the canonical probability space (C0(R;U),B(C0(R;U)),P), which becomes an ergodic metric dynamical system if we add the Wiener shift (2.1) θtω(·) = ω(·+t)−ω(t), ω ∈Ω. Let us consider for some fixed β∈(0,1/2) the set of paths Ω in C0(R;U) which have a finite β-H¨older-seminorm on any interval [−k, k], k ∈N. Denote by k · kβ,a,b (and very often simply by k · kβ) the β-H¨older-seminorm on an interval [a, b]. Again by Komogorov’s theorem about a H¨older-continuous version, this set contained in B(C0(R;U)) has measure one, and in addition it is invariant with respect to θ= (θt)t∈Rdefined by (2.1). We choose Fthe trace-σ-algebra of B(C0(R;U)) with respect to Ω, and for the restriction of Pto this new σ-algebra we use again the symbol P. In the following we will work with this metric dynamical system (Ω,F,P, θ). We also note that from the above canonical Brownian motion it is not hard to derive a filtered Brownian motion (Ω,F,(Ft)t≥0,P) or its corresponding P-completion (Ω,¯ F,(¯ Ft)t≥0,¯ P), where the filtration ( ¯ Ft)t≥0satisfies the usual conditions. As a generalization of the semigroup property introduced in Definition 2.1, we consider a random dynamical system, RDS for shorten, for some metric dynamical system (Ω,F,P, θ), which is given by a B(R+)⊗ F ⊗ B(V),B(V)-measurable mapping ϕ:R+×Ω×V→V such that ϕ(0, ω, u0) = u0, ϕ(t+τ, ω, u0) = ϕ(t, θτω, ·)◦ϕ(τ, ω, u0),for all t, τ ∈R+, u0∈V, ω ∈Ω. We emphasize that tor u0dependent exceptional sets of P-measure zero, what are typical for the classical theory of stochastic differential equations, are not allowed in the definition of a random dynamical system. The first problem that we will face below is to show that the retarded evolution system (1.1) forms a random dynamical system. Next we introduce with details the retarded stochastic system we are interested in. Let Hbe a separable Hilbert space with norm |·| and, for some fixed µ > 0, let Cµ=C([−µ, 0]; H) be the usual space of continuous functions. We consider the delayed stochastic partial differential equation (1.1) interpreted in a mild sense: we look for a mild solution to (1.1), which means that we aim at solving in C([−µ, T ]; H) the following operator equation (2.2) u(t) =    S(t)ξ(0) + Zt 0 S(t−r)F(ur)dr +Zt 0 S(t−r)G(ur)dW(r), t ∈[0, T ], ξ(t), t ∈[−µ, 0], for the initial data ξ∈Cµand for the U-valued Brownian motion W, defined over (Ω,¯ F,(¯ Ft)t≥0,¯ P) and with covariance given by a trace-class operator Q. Now we describe the assumptions on the coefficients of this equation. −Ais a strictly positive and symmetric operator with a compact inverse generating a C0analytic–semigroup S= (S(t))t∈R+on H. For γ≥0 we consider the spaces D((−A)γ), defined in the usual way, see Sell and You [35], Chapter II, which are compactly 4 HAKIMA BESSAIH, MAR´ IA J. GARRIDO-ATIENZA, AND BJ ¨ ORN SCHMALFUSS embedded in H. Note that under these conditions the following estimates are satisfied: there exists a λ > 0 such that |S(t)| ≤ Me−λt, M ≥1, |(−A)γS(t)|L(H)≤cγ 1 tγe−λt, γ ∈[0,1), |S(t)−id|L(D((−A)σ),D((−A)θ)) ≤cσ,θtσ−θ,for θ≥0, σ ∈[θ, 1 + θ] (2.3) for t > 0. For simplicity, the previous constants M,cγand cσ,θ will be assumed to be equal to 1. From these inequalities, for 0 ≤q≤r≤s≤t, we can derive that |S(t−r)−S(t−q)|L(D((−A)δ),D((−A)γ)) ≤cδ,γ(r−q)α(t−r)−α−γ+δ,(2.4) for γ≥0, α∈[0,1] and δ≥α. The constant cδ,γ is also assumed to be equal to 1. Now we describe the assumptions regarding the nonlinear terms Fand G. For F:Cµ7→ Hwe assume global Lipschitz continuity, that is, there exists LF>0 such that (2.5) |F(x)−F(y)| ≤ LFkx−ykµ,for x, y ∈Cµ. For G:Cµ7→ L2,Q(U;H) we also assume Lipschitz continuity in the corresponding spaces, i.e., (2.6) kG(x)−G(y)kL2,Q(U;H)≤LGkx−ykµ,for x, y ∈Cµ, where L2,Q(U;H) denotes the Hilbert-Schmidt space of linear operators from Uto Hrelated to the trace-class operator Q, see Da Prato and Zabczyk [15] Chapter 4. For any T > 0 let us consider the space C([−µ, T ]; H), and denote by |||u||| the norm on this space, avoiding the typing of Tin the previous norm for the sake of exposition. As usual, for u∈C([−µ, T ]; H) and t∈[0, T ], the term ut∈Cµis given by ut(s) = u(t+s) with sbelonging to the interval [−µ, 0]. This means that ut tracks the history of the process over the delay period. These assumptions allow us to conclude the existence and uniqueness of a mild solution of (2.2) in the sense of Da Prato and Zabczyk [15] Chapter 7. We refer to the paper of Taniguchi et al. [36], where the equation contains a delay similarly to ours, and [25]. Theorem 2.2. Assume that the operators Fand Gsatisfy the assumptions (2.5) and (2.6). Then, there exists a unique global stochastic process (u(t))t∈[−µ,T ]with paths in C([−µ, T ]; H)and u(s) = ξ(s)for s∈[−µ, 0] such that the stochastic process (ut)t∈[0,T ]∈Cµis (¯ Ft)t∈[0,T ]-predictable and (2.2) holds for any t∈[0, T ]and for any ¯ F0-measurable random variable ξin Cµalmost surely. Our purpose goes beyond this existence result. Precisely speaking, we want to show the existence of a cocycle version of the above solution. To do this we have to impose stronger conditions on G, in order to obtain a mild solution where exceptional sets do not appear, since these sets contradict the definition of a cocycle. We also assume that Gis smoothing in the following sense: for any u∈C([−µ, T ]; H) the mapping [0, T ]∋t7→ G(ut)∈C1([0, T ]; L(U;H)) such that the derivative of this mapping is given by another operator Kwith a special structure, that is (2.7) d dtG(ut) = K(ut) with K:Cµ7→ L(U;H) being Lipschitz continuous with Lipschitz constant denoted by LK. In addition to the above conditions, let us also impose the stronger condition that Gis Lipschitz continuous with values in L(U;D((−A)ν)), the space of bounded linear operators from Uto D((−A)ν), where ν∈(0,1): (2.8) kG(x)−G(y)kL(U;D((−A)ν)) ≤LG,νkx−ykµ,for x, y ∈Cµ. Note that from the above conditions we can just conclude that Gis a Lipschitz mapping from Cµinto L(U;H). For an example of non-linear terms Fand Gsatisfying the previous assumptions see the Example 2.7 below. In the following we shall be able to get rid of the stochastic integral by applying an integration by parts formula, which allows us to handle our equation in a pathwise way, and which will turn out to be essential when proving the cocycle property for the solution operator to the stochastic delayed system (2.2). PATHWISE SOLUTIONS AND ATTRACTORS FOR RETARDED SPDES 5 In what follows ω(t), t∈R, represents the canonical version of the Brownian motion W. Avoiding the stochastic integral is possible thanks to the existence of the above operator K, since then we can give the following interpretation to the stochastic integral Zt 0 S(t−r)G(ur)dω(r) = G(ut)ω(t) + Zt 0 S(t−r)AG(ur)ω(r)dr −Zt 0 S(t−r)K(ur)ω(r)dr, (2.9) which follows by an application of the integration by parts formula (note that ω(0) = 0 and that Sand the operator Acommute). For more motivations to the integration by parts formula we refer to Bensoussan and Frehse [3]. Now we are going to establish the existence and uniqueness of a mild solution to our delayed equation (1.1) by taking into account the expression of the stochastic integral given by (2.9). As we shall prove, in a first step we obtain a local mild solution, in the sense that there exists a random variable in R+, denoted by T(ω), such that there exists exactly one u∈C([−µ, T (ω)]; H) such that (2.2) is satisfied. Theorem 2.3. Assume that F, G satisfy the assumptions (2.5) and (2.8) such that Ghas the special structure (2.7) and Kis Lipschitz continuous. Then, for any ξ∈Cµand any ω∈Ωthere exists a mild local solution to (2.2), that is, there exists T(ω)>0such that this equation has a unique mild solution u∈C([−µ, T (ω)]; H), i.e., usatisfies (2.10) u(t) =            S(t)ξ(0) + Zt 0 S(t−τ)F(uτ)dτ +Zt 0 S(t−r)AG(ur)ω(r)dr +G(ut)ω(t)−Zt 0 S(t−r)K(ur)ω(r)dr, for t∈[0, T (ω)], ξ(t),for t∈[−µ, 0], and depends continuously on ξ. Proof. We consider the complete metric subspace Cξ([−µ, T ]; H) of functions u∈C([−µ, T ]; H) with u(s) = ξ(s) for s∈[−µ, 0]. Let us write for a while Tinstead of T(ω). For such a T > 0 to be determined later, consider the operator T:Cξ([−µ, T ]; H)→Cξ([−µ, T ]; H) defined, for t≥0, by T(u)(t) = S(t)ξ(0) + Zt 0 S(t−τ)F(uτ)dτ +G(ut)ω(t) + Zt 0 S(t−r)AG(ur)ω(r)dr −Zt 0 S(t−r)K(ur)ω(r)dr. We want to check that this operator is a self-mapping and in addition a contraction, and thus it has a unique fixed point in Cξ([−µ, T ]; H), where Twill be determined according to the Banach fixed point theorem. First, taking into account that ω∈Ω, the Lipschitz continuity of F,Gand K, and the property (2.3), we obtain the above integrals define continuous mappings from [0, T ] into Hwhich are zero for the time parameter zero, see [3]. In particular, for the second integral we note that [0, t]∋r7→ |AS(t−r)G(ur)ω(r)|is integrable by the regularity of G. Therefore, it follows easily that t7→ T (u)(t) is continuous such that Tmaps Cξ([−µ, T ]; H) into itself. 6 HAKIMA BESSAIH, MAR´ IA J. GARRIDO-ATIENZA, AND BJ ¨ ORN SCHMALFUSS Now we take u1, u2∈Cξ([−µ, T ]; H). Due to the Lipschitz regularity of F,Gand K, for t∈[0, T ] we get |(T(u1)−T (u2))(t)| ≤ Zt 0 S(t−τ)(F(u1 τ)−F(u2 τ))dτ +|(G(u1 t)−G(u2 t))ω(t)| +Zt 0 (−A)1−νS(t−r)(−A)ν(G(u1 r)−G(u2 r))ω(r)dr +Zt 0 S(t−r)(K(u1 r)−K(u2 r))ω(r)dr ≤LFsup r∈[0,t] ku1 r−u2 rkµt+LGku1 t−u2 tkµkωkβtβ +kωkβZt 0 LG,νku1 r−u2 rkµ (t−r)1−νrβdr +kωkβZt 0 LKku1 r−u2 rkµrβdr ≤LFsup r∈[0,t] ku1 r−u2 rkµt+LGku1 t−u2 tkµkωkβtβ + sup r∈[0,t] ku1 r−u2 rkµ(LG,ν tβ+ν+LKtβ+1)kωkβ, and therefore |||T (u1)− T (u2)||| ≤ C|||u1−u2|||T+C|||u1−u2|||(Tβ+T1+β+Tβ+ν)kωkβ, with Ca positive constant. Taking T:= T(ω) small enough such that CT +C(Tβ+T1+β+Tβ+ν)kωkβ≤1/2,(2.11) we have |||T (u1)− T (u2)||| ≤ 1/2|||u1−u2|||. Then the Banach fixed point theorem gives a solution to (2.2). Moreover, since the contraction constant is independent of ξwe have that the solution depends continuously on ξ. Let us show this statement with some details. In order to prove the continuous dependence of the solution in the delay input ξ, consider also ˜ ξ∈Cµand let ˜u∈C([−µ, T ]; H) be the unique solution to (2.2) with initial condition ˜ ξ. Then, in a similar manner as we have proved the contraction property, for t≥0 we obtain |u(t)−˜u(t)| ≤ kξ−˜ ξkµ+C|||u−˜u|||t+C|||u−˜u|||(tβ+tβ+ν+t1+β)kωkβ, therefore, taking Tsmall enough such that (2.11) holds, we obtain |||u−˜u||| ≤ 2kξ−˜ ξkµ+1 2|||u−˜u||| and thus the continuous dependence on the initial condition ξfollows.  Up to now we have been able to prove the existence of a unique local mild solution u∈C([−µ, T (ω)]; H) to (2.10), since, as we have seen, for instance the term G(ut)ω(t) produces a sort of Lipschitz constant depending on kωkβ. In what follows we will derive the existence of a global mild solution. Denote T1(ω) = T(ω) and let us build the solution in the next time interval, say [T1(ω), T2(ω)], i.e., we need to find T2(ω) such that we also have a local mild solution in the last interval. For t≥T1(ω) (which in the following computations will be denoted by T1for short), similarly to (2.9) we get PATHWISE SOLUTIONS AND ATTRACTORS FOR RETARDED SPDES 7 Zt T1 S(t−r)G(ur)dω(r) = G(ut)ω(t)−S(t−T1)G(uT1)ω(T1) +Zt T1 S(t−r)AG(ur)ω(r)dr −Zt T1 S(t−r)K(ur)ω(r)dr =G(ut)ω(t)−S(t−T1)G(uT1)ω(T1) + Zt−T1 0 S(t−T1−r)AG(uT1+r)ω(r+T1)dr −Zt−T1 0 S(t−T1−r)K(uT1+r)ω(r+T1)dr =G(ut)ω(t)−S(t−T1)G(uT1)ω(T1) + Zt−T1 0 AS(t−T1−r)G(uT1+r)θT1ω(r)dr +Zt−T1 0 AS(t−T1−r)G(uT1+r)ω(T1)dr −Zt−T1 0 S(t−T1−r)K(uT1+r)θT1ω(r)dr −Zt−T1 0 S(t−T1−r)K(uT1+r)ω(T1)dr =G(uT1+(t−T1))θT1ω(t−T1)−Zt−T1 0 S(t−T1−r)K(uT1+r)θT1ω(r)dr +Zt−T1 0 AS(t−T1−r)G(uT1+r)θT1ω(r)dr, where the last equality follows from the fact that −Zt−T1 0 S(t−T1−r)K(uT1+r)ω(T1)dr +Zt−T1 0 AS(t−T1−r)G(uT1+r)ω(T1)dr =−Zt−T1 0 d dr (S(t−T1−r)G(uT1+r))ω(T1)dr =−G(ut)ω(T1) + S(t−T1)G(uT1)ω(T1). (2.12) Therefore, we are interested in solving u(t) =                      S(t−T1(ω))u(T1(ω)) + Zt−T1 0 S(t−T1−r)F(ur+T1)dr +G(uT1+(t−T1))θT1ω(t−T1)−Zt−T1 0 S(t−T1−r)K(uT1+r)θT1ω(r)dr +Zt−T1 0 AS(t−T1−r)G(uT1+r)θT1ω(r)dr, t −T1(ω)≥0, u1(t), t −T1(ω)∈[−µ, 0], where u1denotes the solution obtained on [−µ, T1(ω)]. But solving the above system is equivalent to solve the problem for y(s) = u(s+T1(ω)) y(s) =            S(s)ˆ ξ(0) + Zs 0 S(s−r)F(yr)dr +G(ys)θT1ω(s) −Zs 0 S(s−r)K(yr)θT1ω(r)dr +Zs 0 AS(s−r)G(yr)θT1ω(r)dr, s ≥0, ˆ ξ(s) = u1(s+T1(ω)), s ∈[−µ, 0], where u1is the solution to (2.10) and sabove is given by s:= t−T1.This means that we have to solve the same problem than in the previous step, but with initial condition ˆ ξand noise θT1(ω)ω. Therefore, following 8 HAKIMA BESSAIH, MAR´ IA J. GARRIDO-ATIENZA, AND BJ ¨ ORN SCHMALFUSS the same steps than before, we obtain a new piece given by a local solution defined now in the interval [T1(ω)−µ, T1(ω) + T1(θT1(ω)ω)], and thus we define T2(ω) as T2(ω) = T1(ω) + T1(θT1(ω)ω). Finally, to get a global mild solution to (2.10) it suffices to define appropriate stopping times, in the following way: for i∈N, considering that T0(ω) = 0 and (2.13) Ti(ω) = Ti−1(ω) + T1(θTi−1(ω)ω), it can be proven that then limi→∞ Ti(ω) = ∞, see Lemma 2.6 below, which then concludes the proof of the existence of a global solution. Therefore we have proven the following result: Theorem 2.4. Under the assumptions of Theorem 2.3, for any ξ∈Cµand any ω∈Ωthere exists a unique mild global solution to (2.2). We now prove that under our particular assumptions the cocycle property holds. Although this result is expected we demonstrate the existence of an RDS when the equation is given in the sense of mild solutions including terms stemming from the integration by parts formula. Theorem 2.5. The global mild solution uof (2.9) generates a random dynamical system ϕ:R+×Ω×Cµ→Cµ given by ϕ(t, ω, ξ)(·) = ut(·), i.e., ϕ(t, ω, ξ)(·) =                  S(t+·)ξ(0) + Zt+· 0 S(t+· − r)F(ur)dr +G(ut+·)ω(t+·) +Zt+· 0 AS(t+· − r)G(ur)ω(r)dτ −Zt+· 0 S(t+· − r)K(ur)ω(r)dτ, for t+· ≥ 0, ξ(t+·),for t+· ≤ 0. Moreover, ξ7→ ϕ(t, ω, ξ)is continuous on Cµfor t≥0and ω∈Ω. Proof. In order to prove that ϕis a cocycle it is of great importance to have used (2.9), since when we try to use directly the Itˆo stochastic integral we know that exceptional sets depending on the initial condition may appear, which is in contradiction with the cocycle property. We should have to distinguish several cases, but we present here two cases. The first one is when we consider t, τ ≥µso that t+s, τ +s≥0, for all s∈[−µ, 0]. In that situation ϕ(t+τ, ω, ξ)(s) = S(t+s)S(τ)ξ(0) + Zτ 0 S(τ−r)F(ur)dr +Zτ 0 AS(τ−r)G(ur)ω(r)dr −Zτ 0 S(τ−r)K(ur)ω(r)dr +Zt+s 0 S(t+s−r)F(uτ+r)dr +G(ut+τ+s)ω(t+τ+s) +Zt+s 0 AS(t+s−r)G(uτ+r)ω(τ+r)dr −Zt+s 0 S(t+s−r)K(uτ+r)ω(τ+r)dr =S(t+s)ϕ(τ, ω, ξ)(0) −S(t+s)G(uτ)ω(τ) + G(ut+τ+s)ω(t+τ+s) +Zt+s 0 S(t+s−r)F(uτ+r)dr +Zt+s 0 AS(t+s−r)G(uτ+r)ω(τ+r)dr −Zt+s 0 S(t+s−r)K(uτ+r)ω(r)dr. (2.14) PATHWISE SOLUTIONS AND ATTRACTORS FOR RETARDED SPDES 9 Notice that Zt+s 0 AS(t+s−r)G(uτ+r)ω(τ+r)dr =Zt+s 0 AS(t+s−r)G(uτ+r)θτω(r)dr +Zt+s 0 AS(t+s−r)G(uτ+r)ω(τ)dr, Zt+s 0 S(t+s−r)K(uτ+r)ω(τ+r)dr =Zt+s 0 S(t+s−r)K(uτ+r)θτω(r)dr +Zt+s 0 S(t+s−r)K(uτ+r)ω(τ)dr. In addition, similar to (2.12), Zt+s 0 AS(t+s−r)G(uτ+r)ω(τ)dr −Zt+s 0 S(t+s−r)K(uτ+r)ω(τ)dr =−G(ut+s+τ)ω(τ) + S(t+s)G(uτ)ω(τ). Hence we can rewrite (2.14) as ϕ(t+τ, ω, ξ)(s) = S(t+s)ϕ(τ, ω, ξ)(0) −S(t+s)G(uτ)ω(τ) +Zt+s 0 S(t+s−r)F(uτ+r)dr +Zt+s 0 AS(t+s−r)G(uτ+r)θτω(r)dr −Zt+s 0 S(t+s−r)K(uτ+r)θτω(r)dr +G(ut+τ+s)ω(t+τ+s) +S(t+s)G(uτ)ω(τ)−G(ut+s+τ)ω(τ) =S(t+s)ϕ(τ, ω, ξ)(0) + Zt+s 0 S(t+s−r)F(uτ+r)dr +Zt+s 0 AS(t+s−r)G(uτ+r)θτω(r)dr −Zt+s 0 S(t+s−r)K(uτ+r)θτω(r)dr +G(ut+τ+s)θτω(t+s). Defining the auxiliary function yp=uτ+pthe previous expression can be rewritten as ϕ(t+τ, ω, ξ)(s) = S(t+s)y(0) + Zt+s 0 S(t+s−r)F(yr)dr +Zt+s 0 AS(t+s−r)G(yr)θτω(r)dr −Zt+s 0 S(t+s−r)K(yr)θτω(r)dr +G(yt+s)θτω(t+s) =ϕ(t, θτω, ϕ(τ, ω, ξ))(s) which proves the cocycle property in that situation by the uniqueness conclusion. Let us consider now the case in which t+s+τ≤0, for s∈[−µ, 0]. Then it is straightforward to see that ϕ(t+τ, ω, ξ)(s) = ξ(t+τ+s) = ϕ(τ, ω, ξ)(t+s) = ϕ(t, θτω, ϕ(τ, ω, ξ))(s). The rest of cases are left to the reader.  As we have seen, for the proof of existence of a global solution we need to define a sequence of stopping times having a particular limit behavior, which is analyzed in the following result. Lemma 2.6. Consider an ω∈Ωwhere Ωis defined at the beginning of this section. Suppose that the sequence (Ti(ω))i∈Nis defined by (2.13) such that similar to (2.11) T(ω) = inf{T > 0 : CT +C(Tβ+T1+β+Tβ+ν)kωkβ,0,T ≥1/2} 16 HAKIMA BESSAIH, MAR´ IA J. GARRIDO-ATIENZA, AND BJ ¨ ORN SCHMALFUSS Choose a 0 < κ < λ −CFeλµ, then for t≥0 e−2κ|t|Z0 −∞ e(λ−CFeλµ)τk(θ−tω)τkµdτ ≤e−2κt Z0 −∞ e(λ−CFeλµ)τkω−t+τkµdτ +e−2κt λ−CFeλµ kω−tkµ ≤e−κt Z0 −∞ eκ(−t+τ)kω−t+τkµdτ +e−2κt λ−CFeλµ kω−tkµ ≤e−κt Z−t −∞ eκτ kωτkµdτ +e−2κt λ−CFeλµ kω−tkµ tends to zero for t→ ∞, which easily follows from the above growth properties. Of course, we also have this convergence for κ≥λ−CFeλµ. We can also prove this convergence for t→ −∞ taking into account that the conditions on ωensure that limt→−∞ e−2κ|t|supτ∈[0,−t]kωτkµ= 0 such that for any κ > 0 lim t→±∞ e−2κ|t|R1(θtω) = 0. We obtain the same convergence for t7→ Z0 −∞ eλr (−r)1−νk(θ−tω)τkµdτ =: R2(θ−tω),for ω∈Ω by the fact that R0 −∞ eλr(−r)ν−1dτ < ∞. Hence the set Bintroduced in Lemma 3.6 is in D. Remark 3.8. Note that in the previous proof we have obtained that lim t→−∞ e2κtkωtkµ= 0 for κ > 0, which is deduced from the polynomial growth of the Brownian motion. Moreover, from the proof of the last lemma it follows that the integrals R1(ω), R2(ω)are finite, for any ω∈Ω. Lemma 3.9. Let ube a solution to (2.2) and suppose that t > µ, 0< ǫ < ν, and α≤min{ν, β}. Then sup s∈[−µ,0] |(−A)ǫut(s)| ≤ c(t, ω sup s∈[−µ,t] |u(s)|),kutkCα([−µ,0];H)≤c(t, ω, sup s∈[−µ,t] |u(s)|) where c(t, ω, x)is bounded if the nonnegative numbers xare bounded for any fixed t > µ, ω ∈Ω. Proof. We start with the first estimate. We have (−A)ǫut(s) = (−A)ǫS(t+s)ξ(0) + Zt+s 0 (−A)ǫS(t+s−r)F(ur)dr + (−A)ǫG(ut+s)ω(t+s) + Zt+s 0 (−A)ǫAS(t+s−r)G(ur)ω(r)dr −Zt+s 0 (−A)ǫS(t+s−r)K(ur)ω(r)dr =I1+···+I5. We here only consider I1, I2, I3, I4because I5can be handled in a similar manner to I4. We have sup −µ≤s≤0 |I1|= sup −µ≤s≤0 |(−A)ǫS(t+s)ξ(0)| ≤cǫ|ξ(0)|sup −µ≤s≤0 e−λ(t+s) (t+s)ǫ≤cǫe−λtkξkµsup −µ≤s≤0 e−λs (t+s)ǫ. PATHWISE SOLUTIONS AND ATTRACTORS FOR RETARDED SPDES 17 We have for I2: sup −µ≤s≤0 |I2|= sup −µ≤s≤0 |Zt+s 0 (−A)ǫS(t+s−r)F(ur)dr| ≤sup −µ≤s≤0Zt+s 0 e−λ(t+s−r) (t+s−r)ǫ(¯ CF+CFkurkµ)dr ≤sup −µ≤s≤0Zt −s e−λ(t−r) (t−r)ǫ(¯ CF+CFkur+skµ)dr ≤¯ CF+CFsup 0≤τ≤t kuτkµZt 0 e−λ(t−r) (t−r)ǫdr. In addition, by the continuous embedding D((−A)ν)⊂D((−A)ǫ), sup −µ≤s≤0 |I3|= sup −µ≤s≤0 |(−A)ǫG(ut+s)ω(t+s)| ≤sup −µ≤s≤0 |(−A)νG(ut+s)ω(t+s)| ≤ CG,νkωtkµ. Finally, by (2.3) we obtain sup −µ≤s≤0 |I4|= sup −µ≤s≤0 |Zt+s 0 (−A)ǫAS(t+s−r)G(ur)ω(r)dr| ≤CG,ν sup −µ≤s≤0Zt+s 0 e−λ(t+s−r) (t+s−r)1+ǫ−ν|ω(r)|Udr ≤CG,ν sup −µ≤s≤0Zt −s e−λ(t−r) (t−r)1+ǫ−ν|ω(r+s)|Udr ≤CG,ν Zt 0 e−λ(t−r) (t−r)1+ǫ−νkωrkµdr. Therefore, we have already proven the first statement of this result. In order to prove the second one, consider s1≤s2∈[−µ, 0]. We have that ut(s1)−ut(s2) = (S(t+s1)−S(t+s2))ξ(0) +G(ut+s1)ω(t+s1)−G(ut+s2)ω(t+s2) +Zt+s1 0 S(t+s1−r)F(ur)dr −Zt+s2 0 S(t+s2−r)F(ur)dr +Zt+s1 0 AS(t+s1−r)G(ur)ω(r)dr −Zt+s2 0 AS(t+s2−r)G(ur)ω(r)dr −Zt+s1 0 S(t+s1−r)K(ur)ω(r)dr +Zt+s2 0 S(t+s2−r)K(ur)ω(r)dr. (3.17) On account of (2.4), |(S(t+s1)−S(t+s2))ξ(0)| ≤ |s1−s2|α|t+s1|−α|ξ(0)| ≤ |s1−s2|α|t−µ|−αkξkµ. 18 HAKIMA BESSAIH, MAR´ IA J. GARRIDO-ATIENZA, AND BJ ¨ ORN SCHMALFUSS In addition Zt+s1 0 AS(t+s1−r)G(ur)ω(r)dr −Zt+s2 0 AS(t+s2−r)G(ur)ω(r)dr ≤Zt −s1 A(S(t−r)−S(t+s2−s1−r))G(ur+s1)ω(r+s1)dr +Zt+s2 t+s1 AS(t+s2−r)G(ur)ω(r)dr ≤CG,ν |s1−s2|αZt+s1 0 1 |t+s1−r|1+α−ν|ω(r)|Udr +CG,ν Zt+s2 t+s1 1 |t+s2−r|1−ν|ω(r)|Udr ≤CG,ν |s1−s2|αZt 0 dr |t−r|1+α−ν+|s1−s2|ν−αkωtkµ≤C(G, ν, µ)|s1−s2|α. The terms stemming from Fand Kin (3.17) can be estimated in a similar manner, taking the linear grow condition of Fand the boundedness of Kinto account. Now, let us focus on the second term in (3.17), which can be expressed in the following way |G(ut+s1)ω(t+s1)−G(ut+s2)ω(t+s2)| ≤ |(G(ut+s1)−G(ut+s2))ω(t+s1)|+|G(ut+s2)(ω(t+s1)−ω(t+s2))|=: J1+J2. It is easy to see that J2≤CGkωkCβ([−µ,t];U)|s1−s2|β. Furthermore, since the mapping t7→ G(ut)∈C1([0, T ]; L(U;H)), then, we can deduce that there exists τ∈[t+s1, t +s2] such that J1≤ |K(uτ)ω(t+s1)||s1−s2| ≤ CK|s1−s2|kωtkµ. Hence we can estimate kutkCα([−µ,0];H). Lemma 3.10. Let B∈ D be the absorbing set from Lemma 3.6. Then there exists a compact absorbing set C∈ D. Proof. Since B∈ D, from Lemma 3.6 we know that Babsorbs itself. Let tB(ω) be the absorbing time for B, that is, ϕ(t, θ−tω, B(θ−tω)) ⊂B(ω) for t≥tB(ω). Indeed we choose a t > 0 such that t≥tB(ω) + 2µ, t ≥tB(θ−µω) + µ, t ≥tB(θ−2µω) + 2µ. Then, due to the absorbing property, ϕ(t, θ−tω, B(θ−tω)) ⊂B(ω) ϕ(t−µ, θ−tω, B(θ−tω)) ⊂B(θ−µω) ϕ(t−2µ, θ−tω, B(θ−tω)) ⊂B(θ−2µω). Note that by the last inclusion we also get ϕ(t, θ−tω, B(θ−tω)) ⊂ϕ(2µ, θ−2µω, B(θ−2µω)). Let ube a solution to (2.2) with initial function ξ∈B(θ−2µω) and with noise path θ−2µω. We can also rewrite the previous expressions as sup r∈[µ,2µ] |u(r)| ≤ ρ(ω),sup r∈[0,µ] |u(r)| ≤ ρ(θ−µω),sup r∈[−µ,0] |u(r)| ≤ ρ(θ−2µω). PATHWISE SOLUTIONS AND ATTRACTORS FOR RETARDED SPDES 19 As a consequence, on the one hand, according to Lemma 3.9, if now we consider c(2µ, θ−2µω, maxi∈{0,1,2}ρ(θ−iµω)), we can apply Arzela-Ascoli’s theorem to obtain the compactness of ϕ(2µ, θ−2µω, B(θ−2µω))Cµ. Therefore we can consider C(ω) = ϕ(t, θ−tω, B(θ−tω))Cµ which is also compact. On the other hand, C(ω)⊂B(ω) such that C∈ D. Note that we can apply Arzela-Ascoli’s theorem, because the H¨older continuity estimate obtained in Lemma 3.9 implies the equicontinuity, and the boundeness of ut(s) in the norm of D((−A)ǫ) implies the boundedness of the norm of ut(s) in H, since D((−A)ǫ) is compactly embedded into H, see [14], Chapter 7. Furthermore Cis pullback absorbing because of ϕ(s, θ−s−tω, D(θ−s−tω)) ⊂B(θ−tω) for s≥tD(θ−tω) and D∈ D. Finally, as a direct application of Theorem 3.4, we have Theorem 3.11. The random dynamical system generated by (2.2) has a random attractor. References [1] L. Arnold, Random Dynamical Systems, Springer Monographs in Mathematics, Springer–Verlag, Berlin, 1998. [2] P. R. Beesack, Gronwall Inequalities, Carleton Mathematical Lecture Notes, 1975. [3] A. Bensoussan and J. Frehse,Local solutions for stochastic Navier Stokes equations, Mathematical Modelling and Numerical Analysis (Mod´elisation Math´ematique et Analyse Num´erique.) 34(2) 2000, 241–273. [4] T. Caraballo, M. J. Garrido-Atienza and J. Real,Existence and uniqueness of solutions for delay stochastic evolution equations, Stochastic Anal. Appl. 20(6) (2002),1225–1256. [5] T. Caraballo, M. J. Garrido-Atienza and B. Schmalfuss,Existence of exponentially attracting stationary solutions for delay evolution equations, Disc. Contin. Dyn. Syst. 18 (2007) 271–293. [6] T. Caraballo, M. J. Garrido-Atienza, B. Schmalfuß and J. Valero,Non-autonomous and random attractors for delay random semilinear equations without uniqueness, Disc. Contin. Dyn. Syst. 21 (2008) 415–443. [7] T. Caraballo, M. J. Garrido-Atienza, B. Schmalfuß and J. Valero,Asymptotic behavior of a stochastic semilinear dissipative functional equation without uniqueness of solutions, Discrete and continuous dynamical systems, series B, 14(2) (2010), 439–455. [8] T. Caraballo, P. Kloeden and B. Schmalfuß,Exponentially stable stationary solutions for stochastic evolution equations and their perturbation, Appl. Math. Optimization, 50 (2004), 183207. [9] T. Caraballo, K. Liu and A. Truman,Stochastic functional partial differential equations: existence, uniqueness and asymptotic decay property, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 456(1999) (2000), 1775–1802. [10] T. Caraballo, J. Real and I.D. Chueshov, Pullback attractors for stochastic heat equations in materials with memory, Discrete Contin. Dyn. Syst. Ser. B 9(3-4) (2008) 525–539. [11] D. Cheban and B. Schmalfuß,Invariant manifolds, global attractors, almost automorphic and almost periodic solutions of non-autonomous differential equations, J. Math. Anal. Appl. 340 (2008) 374–393. [12] Y. Chen, H. Gao, M. J. Garrido-Atienza and B. Schmalfuß,Pathwise solutions of SPDEs and random dynamical systems, submitted. [13] Y. Chen, H. Gao, M. J. Garrido-Atienza and B. Schmalfuß,Random attractors for SPDEs driven by a fBm, in preparation. [14] J.A. Dieudonn´ e, Foundations of modern analysis, New York,Academic Press, 1964. [15] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, University Press, Cambridge, 1992. [16] J. Duan, K. Lu and B. Schmalfuß,Invariant manifolds for stochastic partial differential equations, Ann. Prob., 31 (2003), 2109–2135. [17] J. Duan, K. Lu and B. Schmalfuß,Smooth stable and unstable manifolds for stochastic evolutionary equations, J. Dynam. Differential Equations, 16 (2004), 949–972. [18] F. Flandoli and H. Lisei,Stationary conjugation of flows for parabolic SPDEs with multiplicative noise and some applications, Stochastic Anal. Appl., 22 (2004), 13851420. [19] F. Flandoli and B. Schmalfuß,Random attractors for the 3D stochastic NavierStokes equation with multiplicative white noise, Stochast. Rep. 59 (1996), 21–45. [20] M.J. Garrido-Atienza and J. Real,Existence and uniqueness of solutions for delay stochastic evolution equations of second order in time, Stoch. Dyn. 3 (2) (2003) 141–167. [21] J.K. Hale and S.M.Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, 1995. [22] P. Imkeller and B. Schmalfuß,The conjugacy of stochastic and random differential equations and the existence of global attractors, J. Dynam. Differential Equations 13 (2001) 215–249. [23] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics , Academic Press, 1993. [24] H. Kunita, Stochastic Flows and Stochastic Differential Equations, University Press, Cambridge, 1990. [25] Kai Liu, Personal Communication, 2013. 20 HAKIMA BESSAIH, MAR´ IA J. GARRIDO-ATIENZA, AND BJ ¨ ORN SCHMALFUSS [26] X. Mao, Stability of stochastic differential equations with respect to semimartingales, Pitman Research Notes in Mathematics Series, 251. Longman Scientific & Technical, Harlow, 1991. [27] X. Mao, Stochastic differential equations and applications, Second edition. Horwood Publishing Limited, Chichester, 2008. [28] W. Mather, M. R. Bennett, J. Hasty and L. S. Tsimring,Delay-induced degradeand-fire oscillations in small genetics circuits, Phys. Rev. Lett. 102 (2009) 1–4. [29] S.-E.A. Mohammed, Stochastic Functional Differential Equations, Research Notes in Mathematics, Vol. 99, Pitman Advanced Publishing Program, BostonLondonMelbourne, 1984. [30] S.-E. A. Mohammed and M. K.R. Scheutzow,The stable manifold theorem for non-linear stochastic systems with memory. I. Existence of the semiflow, Journal of Functional Analysis 205 (2003) 271–305. [31] J. D. Murray, Mathematical Biology , Springer, 1993. [32] J. Real,Stochastic partial differential equations with delays,. Stochastics 8(2) (1982-83), 81–102. [33] B. Schmalfuß, Backward cocycles and attractors of stochastic differential equations, in Int. Seminar on Applied MathematicsNonlinear Dynamics: Attractor Approximation and Global Behaviour, eds. Reitmann, V., Riedrich, T. & Koksch, N., pp. 185192, 1992. [34] B. Schmalfuß,Attractors for the nonautonomous dynamical systems, Int. Conf. Differential Equations, Vols. 1, 2 (Berlin, 1999), 684–689. [35] G.R. Sell and Y. You,Dynamics of evolutionary equations. Applied Mathematical Sciences, 143, Springer-Verlag, New York, 2002. [36] T. Taniguchi, K. Liu and A. Truman,Existence, uniqueness and asymptotic behavior of mild solutions to stochastic functional differential equations in Hilbert spaces, Journal of Differential Equations 181 (2002) 72–91. [37] K. Yoshida,Functional Analysis. Grundlehren der mathematischen Wissenschaften, 123, Springer-Verlag, 6. Edition. New York, 1980. (Hakima Bessaih) University of Wyoming Laramie, WY 82071-3036, US E-mail address, Hakima Bessaih: bessai[email protected]u (Mar´ıa J. Garrido-Atienza) Dpto. Ecuaciones Diferenciales y An´ alisis Num´ erico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla, Spain E-mail address, Mar´ıa J. Garrido-Atienza: [email protected] (Bj¨orn Schmalfuß) Institut f¨ ur Mathematik, Institut f¨ ur Stochastik, Ernst Abbe Platz 2, 07737, Jena,Germany, E-mail address, Bj¨orn Schmalfuß: [email protected]