Invariant measures for stochastic 3D Lagrangian-averaged Navier–Stokes equations with infinite delay

Invariant measures for stochastic 3D Lagrangian-averaged Navier-Stokes equations with infinite delay Shuang Yanga,b, Tom´as Caraballob,∗, Yangrong Lia aSchool of Mathematics and Statistics, Southwest University, Chongqing 400715, China bDpto. Ecuaciones Diferenciales y An´alisis Num´erico, Facultad de Matem´aticas, Universidad de Sevilla, C/Tarfia s/n, 41012-Sevilla, Spain Abstract In this paper we investigate stochastic dynamics and invariant measures for stochastic 3D Lagrangian-averaged Navier- Stokes (LANS) equations driven by infinite delay and additive noise. We first use the Galerkin approximations, a priori estimates and standard Gronwall lemma to show the well-posedness for the corresponding random equation, whose solution operators lead to the existence of a random dynamical system. Next, the asymptotic compactness for the random dynamical system is established via the Ascoli-Arzel`a theorem. Besides, we derive the existence of a global random attractor for the random dynamical system. Moreover, we prove that the random dynamical system is bounded and continuous with respect to the initial time. Eventually, we construct a family of invariant Borel probability measures, which is supported by the global random attractor. Keywords: Stochastic 3D Lagrangian-averaged Navier-Stokes equations; Infinite delay; Random attractors; Invariant measures; Generalized Banach limit 2020 MSC: 35B41, 35Q30, 76F20 1. Introduction As we know, the LANS model arises from a one-dimensional model of nonlinear shallow-water wave dynamics. It is the first to use the Lagrangian averaging technique to deal with the turbulence closure problem. The main reason is that such a model requires lower computational cost than a usual Navier-Stokes equation (see Holm [27] for more details). It is also worth pointing out that delay effects play a significant role in the modeling of physical, biological, engineering phenomena and in other real world applications. To describe better a realistic model, we should consider some hereditary characteristics such as aftereffect, time lag, memory and time delay. It seems natural to impose an external force which may take into account not only the current state of the system, but also some part of its history (bounded delay), sometimes even the whole history (unbounded or infinite delay). Inspired by this fact, Caraballo and Real initiated Navier-Stokes models with some hereditary features in [12], in which the existence of solutions was established in both two and three-dimensional spaces. Besides, the uniqueness of solutions was proved in the two-dimensional case. Later on, the asymptotic behavior of those solutions and the existence of a pullback attractor were carried out in [13, 14]. In this article, we investigate the following stochastic 3D LANS equations driven by infinite delay and additive noise:                ∂t(u−α∆u) + ν(Au −α∆(Au)) + (u·∇)(u−α∆u) −α∇u∗·∆u+∇p=f(x) + g(ut) + κ(x)˙ W, in (s, +∞)×O, div u= 0,in (s, +∞)×O, u= 0, Au = 0 on (s, +∞)×∂O, u(s+r, x) = φ(r, x), r ∈(−∞,0], x ∈ O, (1.1) where O ⊆ R3is a bounded open set with sufficiently regular boundary ∂O, s ∈R,Ais the Stokes operator, the positive constants νand αdenote the kinematic viscosity of the fluid and the square of the spatial scale at ∗Corresponding author. Email addresses: [email protected] (Shuang Yang), [email protected] (Tom´as Caraballo), [email protected] (Yangrong Li) Preprint submitted to October 23, 2022 which fluid motion is filtered respectively, the symbol ∗denotes the transpose of a matrix, u= (u1, u2, u3) and pare the averaged (or large-scale) velocity and pressure of the fluid respectively, fand κare given functions defined on O, and the term gis the external force containing some hereditary terms, such as memory, unbounded variable or infinite distributed delay, etc, utdenotes the segment of solutions up to time t, i.e. ut(s) = u(t+s) for all s≤0, φis an initial velocity field defined in (−∞,0], and Wis a two-sided real-valued Wiener process on a complete probability space which will be specified later. Notice that when αconverges to zero, system (1.1) goes to the classical 3D Navier-Stokes equation whose dynamical behavior has been widely investigated in [7, 12, 17, 18, 26, 28, 37]. Problem (1.1) in the deterministic case without delay (i.e., κ= 0 and gis independent of u) has been studied in [19, 35], where several issues have been investigated: the global existence, uniqueness, regularity and asymptotic stability of solutions as well as the existence and finite dimensionality of global attractors. For the stochastic case but without delay, Caraballo et al. in [15] have investigated stochastic dynamics of the 3D LANS equations for the first time. Concerning this model with finite delays or memory, the analysis was discussed by Caraballo et al. in [10, 11]. In the first paper, the authors proved the existence of a unique solution to the stochastic 3D LANS equations. In the second one, the existence and exponential stability of stationary solutions were established. To our knowledge, unbounded or infinite delay effects have been considered in other equations, such as reaction-diffusion equations, globally modified Navier-Stokes equations and usual Navier-Stokes models (see [30, 31, 33, 34, 43]). However, they have not yet been thoroughly investigated for LANS models. Motivated by the above references, we may choose several state spaces to deal with the infinite delay case in the stochastic 3D LANS system. The first one is the Banach space Cγ(H) = {ϕ∈C((−∞,0]; H) : lim r→−∞ eγrϕ(r) exists in H},where γ > 0,(1.2) where His the 3D Lebesgue-type Hilbert space. The second one is C−∞(H) = {ϕ∈C((−∞,0]; H) : lim r→−∞ ϕ(r) exists in H}.(1.3) Moreover, we also use Cγ(V) and C−∞(V), where Vis another Sobolev-type subspace instead of Hin (1.2) and (1.3). In the present article, our main goal is to prove the existence of invariant Borel probability measures for the stochastic 3D LANS system driven by infinite delay and additive noise. For the research of infinite dimensional evolution equations, especially for invariant probability measures, there are a number of different mathematical methods applied to ergodicity. One approach is to consider some kinds of simple linear or semi-linear first order PDEs, see [22, 29]. Although the authors discussed linear systems where solutions are able to depict chaotic or ‘turbulent’ behavior, the method is limited to a very specific class of equations. Another different method, i.e. the classical Krylov-Bogolyubov method, focuses on the study of stochastic PDEs, one can define ergodicity via the Markov semigroup induced by the stochastic semiflow. Thanks to Krylov-Bogolyubov’s method, the existence of invariant measures for stochastic PDEs has been extensively investigated, we refer the reader to [2–4, 21, 41] and [5, 6, 23, 36, 39] for bounded and unbounded domains, respectively. However, by using this approach, we find that invariant measures of the Markov semigroup are deterministic probability measures and thus they are not supported by global random attractors. Based on the previous discussion, in this paper we introduce a different technique to construct invariant measures, that is, the so called ‘generalized Banach limit’, denoted by LIMt→+∞, pioneered in [25] and developed in [8, 16, 24, 32, 40, 42]. The use of such limit allows us to relate time averages with ensemble averages in the state space. To our knowledge, this work seems to be the first one to discuss invariant Borel probability measures for the stochastic 3D LANS system with infinite delay and additive noise, even there is not any published work on studying this issue for stochastic delay equations. We mainly apply the abstract theory for autonomous random dynamical systems in [42, Theorem 2.1] to our model (1.1). To this end, we shall transform (1.1) into a deterministic equation with a random parameter (see Eq. (3.8)) and prove that the solution operators associated to Eq. (1.1) generate a random dynamical system {ϕ(t, s, ω)}t≥s,ω∈Ωin the state space Cγ(V). In addition, (1) the random dynamical system {ϕ(t, s, ω)}t≥s,ω∈Ωpossesses a global random attractor A(ω) in Cγ(V); (2) for each given t∈R, almost all ω∈Ω and φ∈Cγ(V), the Cγ(V)-valued function s7→ ϕ(t, s, θ−tω)φis continuous and bounded on (−∞, t]. Using the classical Galerkin method, a priori estimates and standard Gronwall lemma, we first derive the existence of a unique weak solution to system (3.8). We then obtain some uniform estimates of the solutions 2 to system (3.8) ensuring the existence of a global random absorbing set in Cγ(V) (see Lemma 4.2). Next, we establish the asymptotic compactness for ϕin Cγ(V) via the Ascoli-Arzel`a theorem (see Lemma 4.4). Moreover, we show the existence of global random attractors for ϕ(see Theorem 4.6). Finally, for each given t∈R, almost all ω∈Ω and φ∈Cγ(V), the continuity and boundedness of Cγ(V)-valued function ϕ(t, s, ω)φwith respect to son (−∞, t] are proved in Lemmas 5.1 and 5.4, respectively. This paper is organized as follows. In Section 2, we describe some preliminaries, including some definitions relative to the random dynamical system, some notations and linear operators, some suitable assumptions about the non-delayed external force f, delay term gand additive noise κ. In Section 3, we prove the well-posedness of the stochastic 3D LANS equation with infinite delay and additive noise (1.1). Section 4 is devoted to the existence of a global random attractor in Cγ(V) for the stochastic equation (1.1). In the last section, we construct a family of invariant Borel probability measures of Eq. (1.1) by using the method of generalized Banach limit. 2. Preliminaries 2.1. Random dynamical system Let (X, k·kX) be a separable Banach space equipped with a Borel σ-algebra B(X). Let (Ω,F,P) be a complete probability space with a group {θt}t∈Rsuch that Pis the Wiener distribution, Ω is a subset of {ω∈C(R,R) : ω(0) = 0}with P(Ω) = 1, Fis a σ-algebra, and each θt: Ω →Ω is measure-preserving. If {θt}t∈R fulfills the group property and the mapping (t, ω)7→ θtωis B(R×F,F) measurable, then (Ω,F,P;{θt}t∈R) is called a measurable dynamical system and {θt}t∈Ris said to be the metric dynamical system over the complete probability space (Ω,F,P). For the reader convenience, we need to introduce the following definitions relative to the random dynamical system (see [20, 42]). Definition 2.1. A family of mappings ϕ(t, s, ω) : X7→ X, −∞ <s<t<+∞, parameterized by ω∈Ω, is said to be a random dynamical system over the measurable dynamical system (Ω,F,P;{θt}t∈R), if it holds for almost all ω∈Ω, (i) ϕ(t, r, ω)ϕ(r, s, ω)x=ϕ(t, s, ω)xfor all s≤r≤tand x∈X; (ii) ϕ(t, s, ω)·is continuous on X; (iii) for all t∈R, x ∈X, the mapping (s, ω)7→ ϕ(t, s, ω)xis measurable from ((−∞, t]×Ω,B((−∞, t]) ⊗F) to (X, B(X)); (iv) for all s < t, x ∈X, the mapping ω7→ ϕ(t, s, ω)xis measurable from (Ω,F) to (X, B(X)). Set Ψ(t−s, θsω) = ϕ(t, s, ω) and Φ(t):(ω, φ)7→ (θtω, Ψ(t, ω)φ). If Ψ(t, ω) satisfies the cocycle property, that is, Ψ(t+s, ω) = Ψ(t, θsω)Ψ(s, ω), then {Φ(t)}t∈Rfulfills the semigroup property Φ(t+s) = Φ(t)Φ(s). The mapping {Φ(t)}t∈Ris a skew product flow on Ω ×X. A family D={D(ω) : ω∈Ω}is said to be random (or measurable) if the mapping ω→distX(x, D(ω)) is (F,B(R+)) measurable for each x∈X. Denote by P(X) the family of all nonempty subsets of X. Let Dbe given a family of nonempty random sets D={D(ω) : ω∈Ω}⊆P(X). The class Dis said to be a universe in P(X). Definition 2.2. The random dynamical system ϕis said to be D-asymptotically compact if, for any (t, ω, D)∈ R×Ω×Dand any sequences {sn},{xn} ⊂ Xwith sn≤t, limn→+∞sn=−∞ and xn∈ D(θsn−tω), the sequence {ϕ(t, sn, θ−tω)xn}is relatively compact in X. Definition 2.3. Let {ϕ(t, s, ω)}t≥s,ω∈Ωbe a random dynamical system with a universe Dover the measurable dynamical system (Ω,F,P;{θt}t∈R). A random subset {A(ω)}ω∈Ωof Xis called a global D-random attractor for {ϕ(t, s, ω)}t≥s,ω∈Ω, if (i) Ais compact, that is, each A(ω) is compact in X; (ii) Ais invariant, that is, for all (t, ω)∈R×Ω, ϕ(t, s, ω)A(θsω) = A(θtω),∀s≤t; (iii) Ais D-attracting, that is, for each (t, ω, D)∈R×Ω×D, lim s→−∞ distX(ϕ(t, s, θ−tω)D(θs−tω),A(ω)) = 0.(2.1) 3 2.2. Notations and hypotheses Denote by L2(O) := (L2(O))3,H1 0(O) := (H1 0(O))3,C∞ 0(O) := (C∞ 0(O))3and V={u∈C∞ 0(O) : ∇·u= 0 in O}. Let Hand Vbe the closure of Vin L2(O) and H1 0(O), respectively. Denote by |·|,k·kand k·k∗the norms in H, V and V∗, respectively, where V∗is the dual space of V. Let (·,·) and ((·,·)) be the scalar products in H and V, respectively. For all u, v ∈H1 0(O), we set ((u, v)) = (u, v) + α(∇u, ∇v) = 3 X j=1 ZO uj(x)vj(x)dx +α 3 X i,j=1 ZO ∂uj ∂xi ∂vj ∂xi dx. (2.2) We now consider the Stokes operator A, defined by Aw =−P(∆w),∀w∈D(A) = H2(O)∩V, where H2(O)=(H2(O))3,Pis the Leray operator from L2(O) onto H. We deduce (Au, v) = ((u, v)),kukH2(O)≤C1|Au|,∀u∈D(A), v ∈V, (2.3) where C1is a positive constant. In particular, D(A) is a Hilbert space. Denote by k·k(D(A))∗the norm in (D(A))∗, where (D(A))∗is the dual space of D(A). Identifying Vwith its dual space V∗, in view of D(A)⊂V∗, we identify v∈D(A) with the element hv∈V∗, defined by hv(u) = ((v, u)), u ∈V. Let h·,·i be the duality product between (D(A))∗and D(A). We then define a continuous linear operator e A∈ L(D(A),(D(A))∗) defined dy he Au, vi=ν(Au, v) + να(Au, Av),∀u, v ∈D(A).(2.4) Let v=u, we infer from (2.4) that 2he Au, ui= 2ν(Au, u)+2να(Au, Au)≥2ναkuk2 D(A)=: eαkuk2 D(A).(2.5) For all k≥1,let ξk, λkbe the eigenvectors and eigenvalues of the Stokes operator A, respectively, by the definition of operator e Aand (2.2), we obtain he Aξk, vi=νλk((ξk, v)),(2.6) which implies that, the eigenvalues of the operator e Aare given by e λk:= νλk. Due to the properties of e A, we define ((u, v)) e A=he Au, vi, for all u, v ∈D(A), it is clear that ((·,·)) e Ais the inner product in D(A), and its norm is equivalent to k·kD(A). Without loss of generality, we can assume that for all u, v ∈D(A), (u, v)D(A)=he Au, vi,and so e λ1kuk2≤ kuk2 D(A),∀u, v ∈D(A).(2.7) Denote by D(e A) = {u∈D(A) : e Au ∈V}the domain of the operator e A, it is a subspace of D(A) with the inner product (u, v)D(e A)= (( e Au, e Av)), for all u, v ∈D(e A) , and norm |u|D(e A)=ke Auk. Note that D(e A) is a Hilbert space, and the injection D(e A)⊂D(A) is continuous and e λ1kuk2 D(A)≤ |u|2 D(e A),∀u∈D(e A).(2.8) We then consider the following trilinear operator: b#(u, v, w) = h(u·∇)v, wi−1+h∇u∗·v, wi−1,∀(u, v, w)∈D(A)×L2(O)×H1 0(O), where h·,·i−1denotes the duality product between H−1(O) and H1 0(O) or between H−1(O) and H1 0(O). Thanks to [15, Proposition 2.2], we find b#(u, v, w) = −b#(w, v, u),∀(u, v, w)∈D(A)×L2(O)×D(A),(2.9) 4 which implies that b#(u, v, u)=0,∀(u, v)∈D(A)×L2(O). Define a bilinear mapping e B:D(A)×D(A)→(D(A))∗, denoted by he B(u, v), wi=b#(u, v −α∆v, w),∀(u, v, w)∈D(A)×D(A)×D(A), e B(u) will be used to denote e B(u, u) for all u∈D(A). By the definition and properties of b#, it follows that there exists a positive constant ec:= ec(O) such that for all (u, v, w)∈D(A)×D(A)×D(A), he B(u, v), ui= 0,he B(u), vi=−he B(v, u), ui; (2.10) ke B(u, v)k(D(A))∗≤eckukkvkD(A); (2.11) |he B(u, v), wi| ≤ eckukD(A)kvkD(A)kwk.(2.12) Taking the state space X=Cγ(V), which is a Banach space with sup norm, that is, kukCγ(V)= sup r∈(−∞,0] eγrku(r)k. In order to analyze our problem, we need to establish some suitable assumptions. We first suppose that there exists a constant µsuch that 0< µ < 1 and (1 −µ)e λ1< γ. (2.13) Let a:= 2(1 −µ)e λ1, then 0 <a<2γ. Recall that P(Cγ(V)) is the family of all subsets of Cγ(V). Denote by Dathe tempered universe of nonempty random subsets D={D(ω) : ω∈Ω}⊆P(Cγ(V)), that is, D ∈ Daif and only if, lim s→−∞ ea(s−t)kD(θs−tω)k2 Cγ(V)= 0,∀(t, ω)∈R×Ω.(2.14) We then assume that the non-delayed external force fand the additive noise κsatisfy: fand κ∈H−1(O).(2.15) Then, we require some assumptions on the delay term g. Let g:Cγ(V)→H−1(O) satisfy the following conditions: (G1) For any η∈Cγ(V), g(η) is measurable; (G2) g(0) = 0; (G3) There exists a constant Lg>0 such that for all η, ζ ∈Cγ(V), kg(η)−g(ζ)kH−1(O)≤Lgkη−ζkCγ(V); (G4) There exists a constant Cg>0 such that, for all s∈R, t ≥sand u, v ∈C0((−∞, t); V), Zt skg(ur)−g(vr)k2 H−1(O)dr ≤C2 gZt −∞ ku(r)−v(r)k2dr; (G5) There exists a constant e Cg>0 such that, for all s∈R, t ≥s, all decreasing function $∈C0([s, t]), Zt s $(r)kg(ur)−g(vr)k2 H−1(O)dr ≤e C2 gZt s $(r)ku(r)−v(r)k2dr; (G6) If the sequence {vm}converges weakly to vin L2(−∞, T;D(A)), weakly star in L∞(s, T ;V) and strongly in L2(−∞, T;V), then g(vm ·) converges weakly to g(v·) in L2(s, T ;H−1(O)),∀T > s. Next, let us define e f, eκ∈Vand eg:Cγ(V)→Vas (( e f, w)) = hf, wi−1,∀w∈V, (2.16) ((eκ, w)) = hκ, wi−1,∀w∈V, (2.17) ((eg(η), w)) = hg(η), wi−1,∀(η, w)∈Cγ(V)×V. (2.18) 5 Besides, eg:Cγ(V)→Valso satisfies the following conditions: (H1) For any η∈Cγ(V), eg(η) is measurable; (H2) eg(0) = 0; (H3) Setting Leg=Lg, we obtain, for all η, ζ ∈Cγ(V), keg(η)−eg(ζ)k ≤ Legkη−ζkCγ(V); It follows from (H2) and (H3) that, for all η∈Cγ(V), keg(η)k ≤ LegkηkCγ(V).(2.19) (H4) Letting Ceg=Cg, for all s∈R, t ≥sand u, v ∈C0((−∞, t); V), Zt skeg(ur)−eg(vr)k2dr ≤C2 egZt −∞ ku(r)−v(r)k2dr; (H5) Taking e Ceg=e Cg, for all s∈R, t ≥sand all decreasing function $∈C0([s, t]), Zt s $(r)keg(ur)−eg(vr)k2dr ≤e CegZt s $(r)ku(r)−v(r)k2dr; (H6) If the sequence {vm}converges weakly to vin L2(−∞, T;D(A)), weakly star in L∞(s, T ;V) and strongly in L2(−∞, T;V), then eg(vm ·) converges weakly to eg(v·) in L2(s, T;V),∀T > s. 3. Well-posedness of stochastic 3D LANS equations with infinite delay and additive noise Based on the previous operators and assumptions, we focus on the stochastic dynamics and invariant measures of the following stochastic 3D LANS equations with infinite delay and additive noise:    du dt +e Au(t) + e B(u(t)) = e f+eg(ut) + eκdW dt ,∀t > s, us=φ, (3.1) which is satisfied in (D(A))∗, a.s. for all t>s. In order to define a random dynamical system for Eq. (3.1), we need to transform the stochastic equation into a random system. As usual, let z(θtω) = −R0 −∞ es(θtω)(s)ds with t∈R, which is the solution of the stochastic Ornstein-Uhlenbeck equation dz +zdt =dW (t). Thanks to [1], we obtain that there exists a θt-invariant set e Ω of full measure such that z(θtω) is continuous with respect to t, and the following results hold: lim t→±∞ z(θtw) t= lim t→±∞ 1 tZ0 −t z(θsw)ds = 0,(3.2) lim t→±∞ 1 tZ0 −t|z(θsw)|mds =Γ(1+m 2) √π,∀m > 0,(3.3) for all ω∈e Ω, where Γ denotes the Gamma function. Note that t→z(θtω) is continuous and tempered for all ω∈e Ω, where e Ω is a θ-invariant full-measure subspace of Ω but we do not distinguish them below. Therefore, it follows from [1, Proposition 4.3.3] that there exists a tempered function r(ω)>0 such that for P-a.e. ω∈Ω, |z(ω)|2≤r(ω),(3.4) where r(ω) satisfies, for P-a.e. ω∈Ω, r(θtω)≤ea 2|t|r(ω), t ∈R.(3.5) Combining (3.4) and (3.5), for P-a.e. ω∈Ω, |z(θtω)|2≤ea 2|t|r(ω), t ∈R.(3.6) 6 Let v(t, s, ω, ψ) = u(t, s, ω, φ)−eκz(θtω), t ≥s, (3.7) then by (3.1) and (3.7), we deduce          dv dt +e Av(t) + e B(v(t)) = −z(θtω)e Aeκ−z(θtω)e B(eκ) +e f+eg(ut) + z(θtω)eκ, ∀t > s, vs=ψ, (3.8) where ψ(ι) = φ(ι)−eκz(θι+sω) with ι≤0. Definition 3.1. Suppose that ψ∈Cγ(V). A stochastic process vdefined on Ris called a solution to system (3.8) if v∈L2(s, T;D(A)) ∩L∞(s, T;V),∀T > s, vs=ψand the system (3.8) is satisfied in (D(A))∗, that is, for almost all ω∈Ω, ((v(t), w)) + Zt she Av(r) + e B(v(r)), widr +Zt s z(θrω)he Aeκ+e B(eκ), widr = ((ψ(0), w)) + Zt se f+eg(ur), wdr +Zt s z(θrω)((eκ, w))dr, (3.9) for all t≥sand w∈D(A). Theorem 3.2. Suppose (H1)-(H6) and (2.16)-(2.18) hold. For each (ω, s, ψ)∈Ω×R×Cγ(V), the system (3.8) has a unique weak solution v(·, s, ω, ψ)in the sense of Definition 3.1 defined on [s, +∞). Proof. Using a Galerkin method and a priori estimates given in [12], one can similarly prove the existence of weak solutions to Eq. (3.8), while, the uniqueness follows from a standard Gronwall lemma. The following result shows that the solution to system (3.8) is continuous with respect to initial data. Theorem 3.3. Suppose that (H1)-(H6) and (2.16)-(2.18) are satisfied. Let ψ, e ψ∈Cγ(V)be two initial values to problem (3.8). Let v(·) = v(·, s, ω, ψ)and ev(·) = ev(·, s, ω, e ψ)be two solutions to system (3.8) at the initial time s, respectively. Then, we obtain, for all ι≥s, max r∈[s,ι]kv(r, s, ω, ψ)−ev(r, s, ω, e ψ)k2 ≤1 + L2 eg 2γkψ−e ψk2 Cγ(V)exp Zι sec2 2kv(σ)k2 D(A)+1+L2 egdσ.(3.10) Proof. Setting u(ι) = u(ι, s, ω, φ) = v(ι) + eκz(θιω), eu(ι) = eu(ι, s, ω, e φ) = ev(ι) + eκz(θιω), we have v(ι)−ev(ι) = u(ι)−eu(ι) and kvι−evιk2 Cγ(V)=kuι−euιk2 Cγ(V). Then it follows from (3.8) that d dιkv(ι)−ev(ι)k2=−2he A(v(ι)−ev(ι)) + e B(v)−e B(ev), v(ι)−ev(ι)i+ 2((eg(uι)−eg(euι), v(ι)−ev(ι))),(3.11) where ι≥s. By (2.12), (H3) and the fact that he B(u)−e B(v), u −vi=he B(v, u −v), u −vifor all u, v ∈D(A), we easily obtain d dιkv(ι)−ev(ι)k2=ec2 2kvk2 D(A)+ 1kv(ι)−ev(ι)k2+L2 egkvι−evιk2 Cγ(V).(3.12) Note that for σ∈[s, ι], we find kvσ−evσk2 Cγ(V)≤max nsup ϑ≤s−σ e2γϑkv(σ+ϑ)−ev(σ+ϑ)k2,sup s−σ≤ϑ≤0 e2γϑkv(σ+ϑ)−ev(σ+ϑ)k2o 7 ≤max nsup ϑ≤s−σ e2γϑkψ(σ+ϑ−s)−e ψ(σ+ϑ−s)k2,sup s−σ≤ϑ≤0 e2γϑkv(σ+ϑ)−ev(σ+ϑ)k2o ≤max (sup ϑ≤0 e2γ(ϑ−σ+s)kψ(ϑ)−e ψ(ϑ)k2,sup s≤ϑ≤σkv(ϑ)−ev(ϑ)k2) ≤max ne2γ(s−σ)kψ−e ψk2 Cγ(V),max s≤ϑ≤σkv(ϑ)−ev(ϑ)k2o.(3.13) Combining (3.12) and (3.13), we have kv(ι)−ev(ι)k2≤ kψ(0) −e ψ(0)k2+Zι sec2 2kv(σ)k2 D(A)+ 1kv(σ)−ev(σ)k2dσ +L2 egkψ−e ψk2 Cγ(V)Zι s e2γ(s−σ)dσ +L2 egZι s max s≤ϑ≤σkv(ϑ)−ev(ϑ)k2dσ. (3.14) Taking supremum of (3.14), max r∈[s,ι]kv(r, s, ω, ψ)−ev(r, s, ω, e ψ)k2 ≤ kψ(0) −e ψ(0)k2+Zι sec2 2kv(σ)k2 D(A)+ 1kv(σ)−ev(σ)k2dσ +L2 egkψ−e ψk2 Cγ(V)Zι s e2γ(s−σ)dσ +L2 egZι s max s≤ϑ≤σkv(ϑ)−ev(ϑ)k2dσ (3.15) ≤1 + L2 eg 2γkψ−e ψk2 Cγ(V)+Zι sec2 2kv(σ)k2 D(A)+1+L2 egmax s≤ϑ≤σkv(ϑ)−ev(ϑ)k2dσ, which, together with the Gronwall inequality, yields (3.10) as desired. Let v(t, s, ω, ψ) be the solution to system (3.8) with ψ(ι) = φ(ι)−eκz(θι+sω), where ι≤0, and sis the initial time. Then u(t, s, ω, φ) = v(t, s, ω, ψ) + eκz(θtω) is the solution to Eq. (3.1) corresponding to the initial value φ. Next, we can define the family of operators {ϕ(t, s, ω)}t≥s,ω∈Ωby ϕ(t, s, ω)φ=ut(·, s, ω, φ) =vt(·, s, ω, ψ) + eκz(θt+·ω). By Theorems 3.2 and 3.3, one can prove the above mapping is a continuous random dynamical system over the measurable dynamical system (Ω,F,P;{θt}t∈R) with state space Cγ(V) by using the same method in [20]. Moreover, for P-a.s. ω∈Ω, ϕ(t, s, ω)φ=ϕ(t−s, 0, θsω)φ, ∀s < t, φ ∈Cγ(V). 4. Existence of global random attractors in Cγ(V) In this section, we first obtain uniform estimates on the solutions of problem (3.8), we then prove that the random dynamical system ϕassociated with problem (1.1) has a global Da-random absorbing set in Cγ(V), and further prove it is Da-asymptotically compact in Cγ(V) via the Ascoli-Arzel`a theorem. Finally, the existence of global Da-random attractors for ϕis proved. 4.1. Uniform estimates of solutions In this section, we first obtain uniform estimates on the solutions of problem (3.8), we then prove that the random dynamical system ϕassociated with problem (1.1) has a global Da-random absorbing set in Cγ(V), and further prove it is Da-asymptotically compact in Cγ(V) via the Ascoli-Arzel`a theorem. Finally, the existence of global Da-random attractors for ϕis proved. Lemma 4.1. Let (H1)-(H6),(2.13) and (2.16)-(2.18),eκ∈D(e A)and ψ∈Cγ(V)hold. Then, for each (t, ω, D)∈ R×Ω×Daand ψ∈ D(θs−tω), there exists an s0:= s0(t, ω, D)<0such that for all s≤s0+t, kvt(·, s, θ−tω, ψ)k2 Cγ(V)≤R(ω),(4.1) where R(ω) = M1 + r(ω)with the positive constant Mbeing independent of t, s and ω. 8 Proof. Let r∈Rbe fixed. Taking the inner product of the first equation in (3.8) with v(r) := v(r, s, ω, ψ), s ≤r, we obtain d dr kvk2+ 2kvk2 D(A)=−2z(θrω)he Aeκ+e B(eκ), vi+ 2(( e f+eg(ur), v)) + 2z(θrω)((eκ, v)).(4.2) Consider µ, which is given by (2.13). Since eκ∈D(e A),→D(A), (2.7) and (2.12), we deduce that, there exists a positive constant c1such that 2|z(θrω)he Aeκ+e B(eκ), vi| ≤ 2|z(θrω)||he Aeκ+e B(eκ), vi| ≤2|z(θrω)|(keκkD(A)kvkD(A)+e λ−1 2 1eckeκk2 D(A)kvkD(A)) ≤c1|z(θrω)|2+1 4µkvk2 D(A).(4.3) Thanks to (2.7) and (2.19), we find 2(( e f+eg(ur), v)) ≤2e λ−1 2 1(ke fk+keg(ur)k)kvkD(A) ≤c2ke fk2+c2keg(ur)k2+1 2µkvk2 D(A) ≤c2ke fk2+c3kurk2 Cγ(V)+1 2µkvk2 D(A),(4.4) where c2= 4e λ−1 1µ−1, c3=c2L2 eg. Since v(r, s, ω, ψ) = u(r, s, ω, φ)−eκz(θrω) with s≤r, we deduce kurk2 Cγ(V)= sup ι≤0 e2γιku(r+ι)k2 = sup ι≤0 e2γιkv(r+ι) + eκz(θr+ιω)k2 ≤2kvrk2 Cγ(V)+ 2keκk2sup ι≤0 e2γι|z(θr+ιω)|2.(4.5) By (2.7), we obtain 2|z(θrω)||((eκ, v))| ≤ 2e λ−1 2 1|z(θrω)|keκkkvkD(A) ≤c4|z(θrω)|2+1 4µkvk2 D(A),(4.6) where c4=c2keκk2<∞on account of eκ∈D(e A),→V. Substituting (4.3)-(4.6) into (4.2), we have d dr kvk2+ (2 −µ)kvk2 D(A)≤c2ke fk2+ 2c3kvrk2 Cγ(V)+c5sup ι≤0 e2γι|z(θr+ιω)|2, where c5=c1+ 2c3keκk2+c4. By a= 2(1 −µ)e λ1∈(0,2γ) and (2.7), we can rewrite the above inequality as d dr kvk2+akvk2+µkvk2 D(A)≤c2ke fk2+ 2c3kvrk2 Cγ(V)+c5sup ι≤0 e2γι|z(θr+ιω)|2.(4.7) Multiplying (4.7) by ear and integrating the inequality on [s, r], we obtain kv(r)k2+µZr s ea(σ−r)kv(σ, s, ω, ψ)k2 D(A)dσ ≤ea(s−r)kψk2 Cγ(V)+c2a−1ke fk2+ 2c3Zr s ea(σ−r)kvσk2 Cγ(V)dσ +c5Zr s ea(σ−r)sup ι≤0 e2γι|z(θσ+ιω)|2dσ. (4.8) For all t∈Rwith t≥s, we replace ωby θ−tωin (4.8), then by (3.6), kv(r, s, θ−tω, ψ)k2+µZr s ea(σ−r)kv(σ, s, θ−tω, ψ)k2 D(A)dσ 9 ≤c3r(ω) + 1 6ky(n)(r)k2 D(A).(4.49) Substituting (4.44)-(4.49) into (4.43), d dr ky(n)(r)k2+ky(n)(r)k2 D(A)≤6kv(n)(r∗)k2 D(A)+eR(ω)+R(ω).(4.50) Integrating (4.50) from r∗to r(i) n, we have ky(n)(r(i) n)k2≤eρ0+eR(ω)+R(ω)|r(i) n−r∗|,(4.51) where eρ0= 6 supr∈[s0−T ,s0]{kv(n)(r)k2 D(A):n≥n0, ω ∈Ω}is bounded by eR(ω)due to (4.27). Letting n→+∞ in (4.51), we derive kz(i)−z∗k2= lim n→+∞kv(n)(r(i) n)−v(n)(r∗)k2= 0, i = 1,2, which contradicts (4.42). Step 3:We establish the asymptotic compactness of solutions to problem (3.8) in Cγ(V). Recall that v(n)(·) = v(·, sn, θ−tω, ψn). By steps 1-2, it follows from the Ascoli-Arzel`a theorem that {v(n)(·)}n∈N+,ω∈Ωis pre-compact in C([s0−T , s0]; V) with each T > 0, and thus there exists a function ξ(·)∈C([−T, 0]; V) and a subsequence of v(n) s0(·) such that v(n) s0(·)[−T ,0] →ξ(·) in C([−T, 0]; V). Repeating the procedure for nT with n= 2,3,··· ,and using the diagonal procedure (relabeled the same), we can obtain a function ξ(·)∈C((−∞,0]; V) satisfying v(n) s0(·)[−T ,0] →ξ(·) in C([−T, 0]; V). Moreover, by the estimate (4.26), we obtain kξ(r)k2≤R(ω),∀r∈[−T, 0],for any T > 0.(4.52) In the following, we prove that in fact v(n) s0(·)→ξ(·) in Cγ(V). It suffices to prove that, for every  > 0, there exists some integer n>0 such that sup r∈(−∞,0] e2γrkv(n) s0(r)−ξ(r)k2< , ∀t∈[s0−T, s0], n ≥n.(4.53) Let T>0 be fixed such that max{Me−2γT, Me−2γTr(ω), Me−(2γ−a 2)Tea 2Tr(ω)}< 8. Taking n≥n0such that e2γrkv(n) s0(r)−ξ(r)k2<  for all r∈[−T,0], and sn≤s0−Tfor all n≥n. Therefore, to prove (4.53), we only need to check the following conclusion holds: sup r∈(−∞,−T] e2γrkv(n) s0(r)−ξ(r)k2< , ∀t∈[s0−T, s0], n ≥n. By (4.52) and the fact that 0 <a<2γ, we find, for all m≥0, and r∈[−(T+m+ 1),−(T+m)], e2γrkξ(r)k2≤Me−2γ(T+m)1 + r(ω) ≤Me−2γT+Me−2γTr(ω) < 4. Note that v(n) s0(r) = (ψn(r+s0−sn),if r∈(−∞, sn−s0), v(n)(r+s0),if r∈[sn−s0,0]. Therefore, this proof is finished if we prove that max nsup r∈(−∞,sn−s0) e2γrkψn(r+s0−sn)k2,sup r∈[sn−s0,−T] e2γrkv(n)(r+s0)k2o< 4. On the one hand, by ψn∈ K(θsn−tω), (3.5) and the fact that 0 < a < 2γ, we deduce, for all t∈[s0−T, s0], sup r≤sn−s0 e2γrkψn(r+s0−sn)k2= sup r≤sn−s0 e2γ(r−sn+s0)e2γ(sn−s0)kψn(r+s0−sn)k2 16 ≤e2γ(sn−s0)kψnk2 Cγ(V) ≤Me2γ(sn−s0)(1 + r(θsn−tω)) ≤Me2γ(sn−s0)(1 + ea 2|sn−s0|ea 2|s0−t|r(ω)) ≤Me−2γT+Me−(2γ−a 2)Tea 2Tr(ω) ≤ 8+ 8 = 4. On the other hand, by (4.26) with T=T, we deduce sup r∈[sn−s0,−T] e2γrkv(n)(r+s0)k2= sup r∈[sn−s0+T,0] e2γ(r−T)kv(n)(r−T+s0)k2 ≤e−2γTkv(n) −T+s0k2 Cγ(V) ≤e−2γTR(ω) =Me−2γT(1 + r(ω)) ≤ 4. Therefore, the proof of Lemma 4.4 is complete. Let us now prove the Da-asymptotic compactness of solutions to problem (1.1) in Cγ(V). Lemma 4.5. Assume that (H1)-(H6),(2.13),(2.16)-(2.18) and eκ∈D(e A)hold. Let ϕbe the random dynamical system generated by problem (1.1). For each (t, s0, ω)∈R×R×Ω, if sn→ −∞ as n→+∞and sn≤s0and φn∈ D(θsn−tω), then ϕis Da-asymptotically compact in Cγ(V). Proof. Notice that ϕ(s0, sn, θ−tω)φn=us0(·, sn, θ−tω, φn) =vs0(·, sn, θ−tω, ψn) + eκz(θs0−t+·ω),∀sn≤s0, t ≤s0. By φn∈ K(θsn−tω), we easily imply ψn∈ K(θsn−tω). Thanks to Lemma 4.4, we find that the sequence vs0(·, sn, θ−tω, ψn) of solutions to problem (3.8) has a convergent subsequence in Cγ(V). Thus, ϕ(s0, sn, θ−tω)φn has also a convergent subsequence in Cγ(V). 4.4. Existence of global random attractors Based on the previous results, one can derive the existence of a global Da-random attractor for ϕin Cγ(V) which is stated below. Theorem 4.6. Suppose that the same hypotheses and notations in Lemmas 4.2 and 4.5 hold. Then, the random dynamical system ϕpossesses a global Da-random attractor A={A(ω) : ω∈Ω}in Cγ(V). Proof. It follows from Lemma 4.2 that the random dynamical system ϕhas a global Da-random absorbing set in Cγ(V). By Lemma 4.5, ϕis Da-asymptotically compact in Cγ(V). Thanks to [9, Theorem 7], we finally obtain the existence of a global Da-random attractor A={A(ω) : ω∈Ω}. 5. Existence of invariant measures in Cγ(V) In the rest of this paper, we prove the existence of invariant measures in Cγ(V). To this end, we need to show that, for each given t∈R,ω∈Ω and φ∈Cγ(V), the Cγ(V)-valued function s7→ ϕ(t, s, θ−tω)φis bounded and continuous on (−∞, t]. Lemma 5.1. Let (H1)-(H6),(2.13),(2.16)-(2.18) and eκ∈D(e A)hold. Then, for each (t, ω, φ)∈R×Ω×Cγ(V), the Cγ(V)-valued function s7→ ϕ(t, s, θ−tω)φis bounded on (−∞, t]. 17 Proof. Let t∈R,ω∈Ω and φ∈Cγ(V) be given. By (4.15) in Lemma 4.2, we deduce that, for all s∈(−∞, t], kϕ(t, s, θ−tω)φk2 Cγ(V)=kut(·, s, θ−tω, φ)k2 Cγ(V)≤R(ω),(5.1) where we recall that R(ω) = M1 + r(ω)with the positive constant Mbeing independent of t, s and ω. This shows R(ω) is bounded independently of s∈(−∞, t]. Lemma 5.2. If ψ∈Cγ(V)is given, then for each  > 0, there exists δ=δ(ψ, )>0such that, for all s1, s2∈(−∞,0] with |s1−s2|< δ, eγs2kψ(s1)−ψ(s2)k< . (5.2) Proof. Let ψ∞:= lim s→−∞ eγsψ(s)∈V. By the definition of Cγ(V), we find that, for any  > 0 there exists s0<0 such that keγsψ(s)−ψ∞k< 4,∀s≤s0,(5.3) which implies keγs1ψ(s1)−eγs2ψ(s2)k ≤ keγs1ψ(s1)−ψ∞k+keγs2ψ(s2)−ψ∞k< 2,∀s1, s2≤s0.(5.4) Due to the uniform continuity of the V-valued function s7→ eγsψ(s) on the interval [s0,0], there exists δ0∈(0,1) such that, for all s1, s2∈[s0,0] with |s1−s2|< δ0, keγs1ψ(s1)−eγs2ψ(s2)k< 2.(5.5) Let δ= min nδ0,1 γln(1+  2kψkCγ(V))o, we infer from (5.4) and (5.5) that, for all s1, s2∈(−∞,0] with |s1−s2|< δ, eγs2kψ(s1)−ψ(s2)k≤keγs1ψ(s1)−eγs2ψ(s2)k+|eγs1−eγs2|kψ(s1)k ≤ 2+eγ(s2−s1)−1eγs1kψ(s1)k ≤ 2+eγ(s2−s1)−1kψkCγ(V)< . (5.6) This yields (5.2) as desired. Lemma 5.3. Let (H1)-(H6),(2.13),(2.16)-(2.18) and eκ∈D(e A)hold. For given s∗∈R,ψ∈Cγ(V), almost all ω∈Ω, and for any  > 0, there exists δ0=δ0(s∗, ψ, ω, )>0such that for all s∈(s∗−δ0, s∗),r, t ∈[s, s∗], kv(r, s, θ−tω, ψ)−ψ(0)k< . (5.7) Proof. Firstly, we prove that there exists a constant χ > 0 such that Zs∗ s  d dr v(r, s, θ−tω, ψ)   2 (D(A))∗ dr ≤χ, ∀s∈[s∗−1, s∗], t ∈[s, s∗].(5.8) Indeed, we infer from (3.8), V ,→(D(A))∗, (2.11), (2.7) and eκ∈D(e A) that   d dr v(r, s, θ−tω, ψ)   2 (D(A))∗≤cke Avk2+cke B(v)k2 (D(A))∗+c|z(θr−tω)|2(ke Aeκk2+ke B(eκ)k2 (D(A))∗) +cke fk2+ckeg(ur)k2+c|z(θr−tω)|2keκk2 ≤cke Avk2+ckvk2 D(A)+c|z(θr−tω)|2+cke fk2+ckurk2 Cγ(V).(5.9) Integrating (5.9) over [s, s∗], we deduce, for all s∈[s∗−1, s∗], t ∈[s, s∗], Zs∗ s  d dr v(r, s, θ−tω, ψ)   2 (D(A))∗ dr ≤cZs∗ s∗−1ke Av(r)k2dr +cZs∗ s∗−1kv(r)k2 D(A)dr +cke fk2+Zs∗ s∗−1kurk2 Cγ(V)dr +cZs∗ s∗−1|z(θr−tω)|2dr. (5.10) 18 By similar arguments to those in (4.28) and (4.32), we deduce the first two terms on the right-hand side of (5.10) are bounded, that is, cZs∗ s∗−1ke Av(r)k2dr +cZs∗ s∗−1kv(r)k2 D(A)dr < ∞. Thanks to (4.5) and (4.12) in Lemma 4.1, for all s∈[s∗−1, s∗], t ∈[s, s∗], we have Zs∗ s∗−1kurk2 Cγ(V)dr ≤MZs∗ s∗−1 (1 + ea 2|r−t|r(ω))dr ≤M(1 + ea 2r(ω)). By (3.6), the last term in (5.10) is bounded by Zs∗ s∗−1|z(θr−tω)|2dr ≤r(ω)Zs∗ s∗−1 ea 2|r−t|dr ≤ea 2r(ω),∀s∈[s∗−1, s∗], t ∈[s, s∗],(5.11) which, together with e f∈V, shows (5.10) is finite. Therefore, (5.8) holds. Note that, for all s∗−1≤s≤r≤s∗and t∈[s, s∗], it follows, kv(r, s, θ−tω, ψ)−ψ(0)k2 =kv(r, s, θ−tω, ψ)k2−kψ(0)k2−2v(r, s, θ−tω, ψ)−ψ(0), ψ(0) =Zr s d dσ   v(σ, s, θ−tω, ψ)   2 dσ −2v(r, s, θ−tω, ψ)−ψ(0), ψ(0).(5.12) We now estimate the last two terms of (5.12). On the one hand, by (3.6), (3.7), (4.7) and (4.12), we have Zr s d dσ   v(σ, s, θ−tω, ψ)   2 dσ≤c(s∗−s)ke fk2+cZs∗ skvσk2 Cγ(V)dσ +cZs∗ s sup ι≤0 e2γι|z(θσ+ι−tω)|2dσ ≤c(s∗−s)ke fk2+cZs∗ skvσk2 Cγ(V)dσ +cZs∗ s ea 2|σ−t|r(ω)dr ≤c(s∗−s)ke fk2+c(s∗−s)1 + sup σ∈[s∗−1,s∗] ea 2|σ−t|r(ω) ≤c(s∗−s)ke fk2+c(s∗−s)1 + ea 2r(ω), t ∈[s, s∗].(5.13) As e f∈V, there exists some δ0 0=δ0 0(, s∗, ψ)∈(0,1) such that Zr s d dσ   v(σ, s, θ−tω, ψ)   2 dσ<2 2, s∗−δ0 0< s < t ≤s∗.(5.14) On the other hand, since ψ(0) ∈Vand D(A) is dense in V, there exists some ψ∗∈D(A) such that kψ∗−ψ(0)k<2 8R(ω).(5.15) Thanks to (4.12) in Lemma 4.1, we find, for all s∈[s∗−1, s∗], t ∈[s, s∗], r ∈[s∗−1, s∗], kv(r, s, θ−tω, ψ)k2≤M1 + ea 2|r−t|r(ω) ≤M(1 + ea 2r(ω)) = R(ω).(5.16) By (5.8), (5.15) and (5.16), we deduce 2v(r, s, θ−tω, ψ)−ψ(0), ψ(0) ≤2hv(r, s, θ−tω, ψ)−ψ(0), ψ∗i+ 2v(r, s, θ−tω, ψ)−ψ(0), ψ(0) −ψ∗ 19 ≤2DZr s d dσ v(σ, s, θ−tω, ψ)dσ, ψ∗E+ 2R(ω)kψ(0) −ψ∗k ≤2√s∗−skψ∗kD(A)Zs∗ s  d dσ v(σ, s, θ−tω, ψ)   2 (D(A))∗ dσ1 2 +2 4 ≤2pχ(s∗−s)kψ∗kD(A)+2 4,∀r∈[s, s∗], t ∈[s, s∗].(5.17) This implies that there exists some δ00 0=δ00 0(, s∗, ψ)∈(0,1) such that 2v(r, s, θ−tω, ψ)−ψ(0), ψ(0)≤2 2, s∗−δ00 0< s ≤r≤s∗, s ≤t≤s∗.(5.18) Letting δ0= min{δ0 0, δ00 0}, we infer from (5.12), (5.14) and (5.18) that (5.7) holds. Lemma 5.4. Let (H1)-(H6),(2.13),(2.16)-(2.18) and eκ∈D(e A)hold. Then, for given t∈R, almost all ω∈Ω and φ∈Cγ(V), the Cγ(V)-valued function s7→ ϕ(t, s, θ−tω)φis continuous on (−∞, t]. Proof. Note that (ϕ(t, s, θ−tω)φ)(r) = ut(r, s, θ−tω, φ) =vt(r, s, θ−tω, ψ) + eκz(θrω), r ≤0,(5.19) where ψ∈Cγ(V) due to φ∈Cγ(V). The above equality, together with the continuity of |z(θtω)|with respect to t∈Rfor P-a.s. ω∈Ω, shows that we only need to prove that vtr, ·, θ−tω, ψis both left and right continuous on (−∞, t]. We start with the left continuity. Without loss of generality, for given s∗∈R,ω∈Ω and ψ∈Cγ(V), we just need to prove that for any  > 0, there exists a positive constant δ=δ(, s∗, ω, ψ) such that for all s∈(s∗−δ, s∗), t∈[s, s∗] and almost all ω∈Ω, kvt(·, s, θ−tω, ψ)−vt(·, s∗, θ−tω, ψ)kCγ(V)< . By Lemma 5.2, we find that, for every  > 0, there exists some δ0=δ0(, ψ)>0 such that for all s1, s2∈(−∞,0] with |s1−s2|< δ0, eγs2kψ(s1)−ψ(s2)k< 2.(5.20) Thanks to Lemma 5.3, there exists δ00 =δ00(, s∗, ω, ψ)>0 such that, for all s∈(s∗−δ00, s∗), r∈[s, s∗], t∈[s, s∗] and almost all ω∈Ω, it follows kv(r, s, θ−tω, ψ)−ψ(0)k< 2.(5.21) By (5.20) and (5.21), for the above , there exists some δ=δ(, s∗, ω, ψ) = min{δ0, δ00}>0 such that, for all s∈(s∗−δ, s∗), r∈[s, s∗], t∈[s, s∗] and almost all ω∈Ω, kv∗(r, s∗, θ−tω, ψ)−v(r, s, θ−tω, ψ)k=kψ(r−s∗)−v(r, s, θ−tω, ψ)k ≤ kψ(r−s∗)−ψ(0)k+kψ(0) −v(r, s, θ−tω, ψ)k < , (5.22) where v∗(r, s∗, θ−tω, ψ) is the solution with the initial datum ψat the initial time s∗. The above inequality implies max r∈[s,s∗]kv∗(r, s∗, θ−tω, ψ)−v(r, s, θ−tω, ψ)k ≤ , s ∈(s∗−δ, s∗), t ∈[s, s∗],(5.23) which implies, for all t∈[s, s∗], max r∈[s,t]kv∗(r, s∗, θ−tω, ψ)−v(r, s, θ−tω, ψ)k ≤ , s ∈(s∗−δ, s∗).(5.24) By (5.20) and (5.24), we find that, for the above , for all s∈(s∗−δ, s∗), t∈[s, s∗] and almost all ω∈Ω, kvt(r, s, θ−tω, ψ)−vt(r, s∗, θ−tω, ψ)kCγ(V)= sup r≤0 eγrkv(r+t, s, θ−tω, ψ)−v∗(r+t, s∗, θ−tω, ψ)k 20 ≤max nsup r≤s−t eγrkψ(r+t−s)−ψ(r+t−s∗)k, sup s−t≤r≤0 eγrkv(r+t, s, θ−tω, ψ)−v∗(r+t, s∗, θ−tω, ψ)ko ≤max nsup r≤0 eγ(r+s−t)kψ(r)−ψ(r+s−s∗)k, sup s≤r≤tkv(r, s, θ−tω, ψ)−v∗(r, s∗, θ−tω, ψ)ko≤. (5.25) This implies that the Cγ(V)-valued function s7→ ϕ(t, s, θ−tω)φis left continuous at s=s∗. It is similar to prove the right continuity of ϕ(t, s, θ−tω)φat s=s∗, and we omit the details. Recall the definition of generalized Banach limit, which plays an important role in constructing the invariant measures for ϕ(see [24, 25, 32, 42] for more details). Definition 5.5. A generalized Banach limit is any linear functional, denoted by LIMt→+∞, defined on the space of all bounded real-valued functions on [0,+∞) and satisfying (1) LIMt→+∞ξ(t)≥0 for nonnegative functions ξ(·) on [0,+∞); (2) LIMt→+∞ξ(t) = limt→+∞ξ(t) if the usual limit limt→+∞ξ(t) exists. Remark 5.6. Note that we will discuss the asymptotic behavior s→ −∞ of ϕ(t, s, ω)•, and thus, we require generalized limits as s→ −∞. For a given real-valued function ξdefined on (−∞,0] and a given Banach limit LIMt→+∞, we define LIMt→−∞ξ(t)=LIMt→+∞ξ(−t). Theorem 5.7. Let (H1)-(H6),(2.13),(2.16)-(2.18) and eκ∈D(e A)hold. Let {ϕ(t, s, ω)}t≥s,ω∈Ωbe the random dynamical system associated with problem (1.1) over the measurable dynamical system (Ω,F,P;{θt}t∈R)with the state space Cγ(V). Let A(ω)be the global Da-random attractor obtained in Theorem 4.6. Then for a given continuous mapping ζs:R7→ Cγ(V)with ζs(·)∈Daand a generalized Banach limit LIMt→+∞, there exists for almost all ω∈Ω, a family of Borel probability measures {µθtω}t∈Ron Cγ(V)such that the support of µθtωis contained in A(θtω)and LIMs→−∞ 1 t−sZt s Υ(ϕ(t, r, ω)ζr)dr =ZA(θtω) Υ(u)dµθtω(u) =ZCγ(V) Υ(u)dµθtω(u) = LIMs→−∞ 1 t−sZt sZCγ(V) Υ(ϕ(t, r, ω)u)dµθrω(u)dr, for every real-valued continuous functional Υon Cγ(V). Moreover, µθtωis invariant in the sense that ZA(θtω) Υ(u)dµθtω(u) = ZA(θsω) Υ(ϕ(t, s, ω)u)dµθsω(u),∀t≥s. Proof. For the random dynamical system {ϕ(t, s, ω)}t≥s,ω∈Ωon the space Cγ(V), we need to verify the conditions (i) and (ii) in [42, Theorem 2.1]. By Theorem 4.6, we obtain that {ϕ(t, s, θ−tω)}t≥s,ω∈Ωpossesses a global Da-random attractor A(ω) in Cγ(V). Thus, (i) has been proved. Moreover, by Lemmas 5.1 and 5.4, we deduce that for each t∈R, almost all ω∈Ω and φ∈Cγ(V), the Cγ(V)-valued function s7→ ϕ(t, s, θ−tω)φis continuous and bounded on (−∞, t], and thus (ii) holds true. Therefore, we obtain the results of Theorem 5.7. 6. Acknowledgments Shuang Yang was supported by the China Scholarship Council (CSC No. 202006990054). This work was done when Shuang Yang visited the Department of Differential Equations and Numerical Analysis at the University of Sevilla. She would like to express her thanks to all people there for their kind hospitality. The research of the first two authors has been partially supported by the Spanish Ministerio de Ciencia, Innovaci´on y Universidades (MCIU), Agencia Estatal de Investigaci´on (AEI) and Fondo Europeo de Desarrollo Regional (FEDER) under the project PGC2018-096540-B-I00, and by Junta de Andaluc´ıa (Consejer´ıa de Econom´ıa y Conocimiento) under projects US-1254251 and P18-FR-4509. 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