Long Time Behavior of Stochastic Nonlocal Partial Differential Equations and Wong--Zakai Approximations

LONG TIME BEHAVIOR OF STOCHASTIC NONLOCAL PARTIAL DIFFERENTIAL1 EQUATIONS AND WONG-ZAKAI APPROXIMATIONS∗ 2 JIAOHUI XU†AND TOM´ AS CARABALLO‡ 3 Abstract. This paper is devoted to investigating the well-posedness and asymptotic behavior of a class of stochastic4 nonlocal partial differential equations driven by nonlinear noise. First, the existence of a weak martingale solution is estab-5 lished by using the Faedo-Galerkin approximation and an idea analogous to Da Prato and Zabczyk [12]. Second, we show6 the uniqueness and continuous dependence on initial values of solutions to the above stochastic nonlocal problem when there7 exist some variational solutions. Third, the asymptotic local stability of steady-state solutions is analyzed either when the8 steady-state solutions of the deterministic problem is also solution of the stochastic one, or when this does not happen. Next,9 to study the global asymptotic behavior, namely, the existence of attracting sets of solutions, we consider an approximation10 of the noise given by Wong-Zakai’s technique using the so called colored noise. For this model, we can use the power of11 the theory of random dynamical systems and prove the existence of random attractors. Eventually, particularizing in the12 cases of additive and multiplicative noise, it is proved that the Wong-Zakai approximation models possess random attractors13 which converge upper-semicontinuously to the respective random attractors of the stochastic equations driven by standard14 Brownian motions. This fact justifies the use of this colored noise technique to approximate the asymptotic behavior of the15 models with general nonlinear noises, although the convergence of attractors and solutions is still an open problem.16 Key words. Nonlinear stochastic term, colored noise, variational solutions, steady-state solution, attractors, upper17 semi-continuity.18 AMS subject classifications. 60H15, 35B40.19 1. Introduction. Nowadays, a big amount of researchers develop stochastic systems to model phe-20 nomena from real world in a more realistic way, as can be seen in the published literature (for instance,21 [6,8,17,19,21,25,31] and references therein). In this paper, we are concerned with a stochastic version22 of a nonlocal partial differential equation, which has been well studied by M. Chipot and his collaborators23 (see [9,10,11]), to model the behavior of a migrating population in a bounded habitat or problems with24 magneto-elastic interactions. Precisely, we are interested in performing a rigorous study of well-posedness25 and dynamics of the following stochastic nonlocal reaction-diffusion equation,26      ∂u ∂t −a(l(u))∆u=f(u) + h(t) + g(t, u)dW dt , u= 0, u(x, τ) = u0(x), in O × (τ, ∞), on ∂O × (τ, ∞), in O, (1.1)27 28 where τ∈R, function a∈C(R;R+) and there exist two positive constants mand em, such that29 (1.2) m≤a(s)≤em, ∀s∈R.30 Moreover, let l∈ L(L2(O); R), f∈C(R) and there exist positive constants α1,α2,η,κand p > 2, such31 that32 (1.3) (f(s)−f(r))(s−r)≤η(s−r)2,∀s, r ∈R,33 34 (1.4) −κ−α1|s|p≤f(s)s≤κ−α2|s|p,∀s∈R.35 From (1.4), we can deduce that there exists β > 0, such that36 (1.5) |f(s)| ≤ β(|s|p−1+ 1),∀s∈R.37 ∗Submitted to the editors DATE. Funding: The research 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-I00, and by Junta de Andaluc´ıa and FEDER under the projects US1254251 and P18-FR-4509. †Dpto. Ecuaciones Diferenciales y An´alisis Num´erico, Universidad de Sevilla, Facultad de Matem´aticas, c/ Tarfia s/n, 41012-Sevilla, Spain ([email protected]). ‡Dpto. Ecuaciones Diferenciales y An´alisis Num´erico, Universidad de Sevilla, Facultad de Matem´aticas, c/ Tarfia s/n, 41012-Sevilla, Spain ([email protected]). 1 This manuscript is for review purposes only. 2J.H. XU, AND T. CARABALLO In addition, let (Ω,F,{Ft}t≥0,P) be a stochastic basis with expectation E,Kand Ube two separable38 Hilbert spaces. Let W(t) be a cylindrical Wiener process with values in Kdefined on the stochastic39 basis. Denote by L2(K, U) the set of Hilbert-Schmidt operators from Kto U. Eventually, let the initial40 value u0∈L2(O) and non-autonomous term h∈L2 loc(R;H−1(O)). The operator lacting on umust be41 understood as (l, u), but for short we keep the notation l(u).42 Now, we impose smoothness condition on the domain, namely, we require O ⊂ RNto be a bounded43 open set of class Ck, with k≥2 such that k≥N(p−2)/(2p).44 Initially, our intention was to prove the well-posedness of problem (1.1) in the sense of Definition 2.645 by following the variational technique which was originally introduced by Pardoux in his thesis [23], and46 subsequently in many other papers dealing with stochastic partial differential equations in the variational47 framework (see, e.g. [5,7,8,24]). However, on the one hand, the appearance of the nonlocal term a(·) in48 our problem makes the analysis more involved, since the main operator, a(l(u))∆u, does not satisfy the49 standard assumptions of monotonicity which are required in the aforementioned variational set-up. On the50 other hand, In the deterministic case (cf. [32]), the compactness method for nonlinear partial differential51 equations is somehow easier: when Lpbounds on the approximating solutions have been proved, the52 approximating equations readily give us estimates on the derivatives, and this implies strong convergence53 of some subsequence, while this strategy does not extend to the stochastic case since the solutions are54 not differentiable (cf. [14]). Therefore, by carrying out a careful analysis in a satisfactory way, some55 conclusions are obtained as follows:56 •When l∈ L(L2(O); R), we are able to prove the existence of a solution (see Theorem 2.8) in a57 weaker sense, the so called martingale solution (see Definition 2.7).58 •One should expect some positive answers, in some particular cases, about existence of variational59 solution to problem (1.1). In fact, when lis not a bounded linear operator as in our current case,60 for instance, when the functional lis given by l(u) = kuk2 H1 0, the existence and uniqueness of61 solution of the following problem62      ut−a(kuk2 H1 0)∆u=f(u) + h(t),(t, x)∈(0,∞)× O, u= 0,(t, x)∈(0,∞)×∂O, u(0, x) = u0(x), x ∈ O, 63 were shown in [3]. Moreover, recently, the authors studied in [4] the existence and uniqueness of64 variational solution to the stochastic version of the above problem,65      ut=a(kuk2 H1 0)∆u+f(u) + h(t, x) + σ(u)dw(t),(t, x)∈(τ, ∞)× O, u= 0,(t, x)∈(τ, ∞)×∂O, u(τ, x) = uτ(x), x ∈ O, 66 by using a monotone iterative approach. Let us point out the key point in the proof is to show that the nonlocal term −a(kuk2 H1 0)∆uis monotone. This holds true because in [4] it is imposed that s→a(s2)sis non-decreasing. However, in our case, it is not possible to prove the monotonicity of the operator −a(l(u))∆usince67 l∈ L(L2(O); R).68 •If we adopted a Picard scheme as in [18, Chapter 3], defining operator A(v) := −a(l(un−1))∆v, we could construct a sequence {un}∞ n=1, whose limit could be the solution of our problem. In this way, we would overcome the difficulty of proving monotonicity. However, in the last step to prove {un}∞ n=1 is a Cauchy sequence, we would not have enough regularity to ensure the stopping time tn N:= {τ≤t≤T:kun(t)k ≥ N}, is well defined, since un∈L2(Ω; L∞(τ, T;L2(O)))∩L2(Ω; L2(τ, T;H1 0(O)))∩Lp(Ω; Lp(τ, T;Lp(O)))69 for p > 2 by the Itˆo formula. As a result, we are not able to use a monotone iterative approach70 This manuscript is for review purposes only. STOCHASTIC NONLOCAL PDES AND WONG-ZAKAI APPROXIMATIONS 3 method when l∈ L(L2(O); R). As an alternative, we will show the existence of martingale solu-71 tions to problem (1.1).72 Next, we study the asymptotic local stability when there exist variational solutions to (1.1). Our73 analysis is intended in two directions: (i) We study the behavior of the solutions to the stochastic problem74 around steady-state solutions (equilibria) of the deterministic one (i.e. g≡0), when the latter are not75 necessarily equilibria of the stochastic problem. In this case, we prove exponential convergence (in mean76 square and almost surely) of solutions to (1.1) towards some steady-state solution to the deterministic77 problem; (ii) When the deterministic and stochastic problems have a common steady-state solution, we78 prove a sufficient condition ensuring its asymptotic exponential stability in mean square. However, the79 global asymptotic dynamics cannot be carried out by applying the well-established theory of random80 dynamical systems in the case of nonlinear noisy terms. This leads us to proceed in a different way as we81 will describe below.82 Notice that, for the particular case in which the noise term is linear (additive or multiplicative), the83 existence of random attractors of (1.1) has been analyzed in [33] by exploiting the tools of the theory84 of random dynamical systems. However, when the noise is nonlinear, this theory cannot be applied in a85 suitable way because it is not proved yet that the stochastic problem (1.1) generates a random dynamical86 system. Recently, B. X. Wang and his collaborators (see, e.g., [15,17,22,30]) have initiated a new87 approach to tackle the problem with nonlinear noise. The idea is to replace the noise in (1.1) by a Wong-88 Zakai approximation, denoted by ζδ(θtω), δ ∈(0,1] (see details in Section 4), whose integral Rt 0ζδ(θsω)ds89 converges to the Brownian motion Wt(ω), uniformly for tin bounded intervals of time, as δgoes to zero.90 Therefore, we will analyze the following random non-autonomous problem driven by colored noise,91      ∂u ∂t −a(l(u))∆u=f(u) + h(t) + g(t, u)ζδ(θtω), u= 0, u(x, τ) = u0(x), in O × (τ, ∞), on ∂O × (τ, ∞), in O. (1.6)92 93 Observe that the above random problem can be analyzed for each fixed ω, therefore it generates a random94 dynamical system. Hence, the deterministic techniques can be adopted here to state the well-posedness95 and the existence of a random attractor.96 Naturally, one should expect, at least formally, that the random attractor of (1.6) converges in some97 sense to a random attractor of the limit problem when δgoes to zero. This is a hard problem, there are98 answers only in some special cases. Motived by the previous work, for instance [30], we will particularize99 our study in the cases of additive and multiplicative noise. Indeed, we first study the dynamics of100      ∂u ∂t −a(l(u))∆u=f(u) + φdW dt , u= 0, u(x, τ) = u0, in O × (τ, ∞), on ∂O × (τ, ∞), in O, (1.7)101 102 where, for simplicity, we consider an autonomous version, i.e., h= 0 and g(t, u) = φ∈H1 0(O)∩H2(O).103 The corresponding approximate problem is104      ∂uδ ∂t −a(l(uδ))∆uδ=f(uδ) + φζδ(θtω), uδ= 0, uδ(x, τ) = u0,δ, in O × (τ, ∞), on ∂O × (τ, ∞), in O, (1.8)105 106 where functions aand fsatisfy conditions (1.2)-(1.4) with p= 2 and β=Cf. Then, by using appropriate changes of variable given by Ornstein-Uhlenbeck processes, we prove that both problems generate random dynamical systems which possess random attractors, denoted by Aand Aδ, respectively. Furthermore, it is shown that Aδconverges upper-semicontinuously to Aas δgoes to zero, and the solutions of problem (1.8) converge to solutions of (1.7). More precisely, if {δn}∞ n=1 is a sequence satisfying δn→0 as n→+∞, uδnand uare the solutions of (1.8) and (1.7) with initial values u0,δnand u0, respectively, and if u0,δn→u0 strongly in L2(O) as n→+∞, then for almost all ω∈Ω and t≥τ, uδn(t;τ, ω, u0,δn)→u(t;τ, ω, u0) strongly in L2(O) as n→+∞. This manuscript is for review purposes only. 4J.H. XU, AND T. CARABALLO Finally, we carry out a similar analysis in the case of multiplicative noise, i.e.,107      ∂u ∂t −a(l(u))∆u=f(u) + σu ◦dW dt , u= 0, u(x, τ) = u0, in O × (τ, ∞), on ∂O × (τ, ∞), in O, (1.9)108 109 and the corresponding approximate problem is110      ∂uδ ∂t −a(l(uδ))∆uδ=f(uδ) + σu ◦ζδ(θtω), uδ= 0, uδ(x, τ) = u0,δ, in O × (τ, ∞), on ∂O × (τ, ∞), in O, (1.10)111 112 where ◦denotes the Stratonovich sense in stochastic term.113 The analysis described above is developed in the following sections. Section 2is devoted to proving the114 main theorem about existence and construction of a martingale solution. In Section 3, the local asymptotic115 behavior of solutions is considered, proving some exponential decay of solutions of the stochastic problem116 to the steady-state solutions of the deterministic one (i.e., g≡0). The global asymptotic behavior of117 solutions is studied in Section 4by considering the Wong-Zakai approximate problem of our original one118 (cf. (1.1)). The theory of random non-autonomous dynamical systems is carried out to prove the existence119 of a random non-autonomous attractor for the approximate system (cf. (1.6)), which can be considered120 as a reasonable approximation of the dynamics for our original problem. This claim is justified with121 the analysis developed in sections 5and 6, where one can check that the attractors and solutions of the122 approximate problems converge, in appropriate sense.123 2. Existence of martingale solutions to problem (1.1).In this section, we use the Faedo-124 Galerkin approximation and an idea analogous to Da Prato and Zabczyk [12] showing the existence of125 a martingale solution to stochastic nonlocal problem (1.1). This theory has received increasing attention126 over the last years (see, e.g. [12,13,14,26]). In what follows, we introduce some necessary notations and127 most of the hypotheses relevant to our analysis.128 2.1. Stochastic setting. Let {Ω,F,P}be a complete probability space and F={Ft}t∈[0,T ]an129 increasing and right continuous family of sub σ-algebras of F, such that F0contains all of P-null sets of F.130 In this manuscript, all stochastic integrals are defined in the sense of Itˆo and EXdenotes the mathematical131 expectation of the stochastic process X=X(t, ω) with respect to P. Given Kand Utwo separable Hilbert132 spaces, W(t) a cylindrical Wiener process with values in K, we denote by L(K, U) the space of continuous133 linear mapping from Kto U. By L2(K, U), which is a subspace of L(K, U) consisting of Hilbert-Schmidt134 operators from Kto U. It is known that L2(K, U) is a Hilbert space and its norm is denoted by k·kL2(K,U).135 Given p > 1, α∈(0,1), let Wα,p(0, T;U) be the Sobolev space of all functions u∈Lp(0, T;U) such that ZT 0ZT 0 |u(t)−u(s)|p |t−s|1+αp dtds < ∞, endowed with the norm kukp Wα,p(0,T ;U)=ZT 0 |u(t)|pdt +ZT 0ZT 0 |u(t)−u(s)|p |t−s|1+αp dtds. For any progressively measurable process f∈L2(Ω ×[0, T ]; L2(K, U)), we denote by I(f) the Itˆo integral defined as I(f)(t) = Zt 0 f(s)dW(s), t ∈[0, T]. Clearly, I(f) is a progressively measurable process in L2(Ω ×[0, T ]; U).136 This manuscript is for review purposes only. STOCHASTIC NONLOCAL PDES AND WONG-ZAKAI APPROXIMATIONS 5 Lemma 2.1. ([14, Lemma 2.1])Let p≥2,0< α < 1 2. Then, for any progressively measurable process f∈Lp(Ω ×[0, T]; L2(K, U)), we have I(f)∈Lp(Ω; Wα,p(0, T;U)), and there exists a constant C(p, α)>0, independent of f, such that EkI(f)kp Wα,p(0,T ;U)≤C(p, α)EZT 0 kf(t)kp L2(K,U)dt. 2.2. Notations. We also introduce additional notations frequently used throughout the work, for137 simplicity, denote by H=L2(O), V=H1 0(O) and V∗=H−1(O). Identifying Hwith its dual, we have138 the usual chain of dense and compact embeddings V⊂H⊂V∗. We denote by | · |pthe norm in Lp(O),139 |·|and k·k∗the norms in Hand V∗, by (·,·) and ((·,·)) the scalar products in Hand V, respectively,140 and by <·,·>the duality product between Vand V∗. At last, let C∞ c(O) be the space of all functions141 of class C∞with compact supports contained in O.142 Given real numbers a < b and p > 1, we will denote by Ip(a, b;H) the space of all processes X∈143 Lp(Ω ×(a, b),F ⊗ B((a, b)), dP⊗dt;H), where B((a, b)) denotes the Borel σ-algebra on (a, b), such that144 X(t) is Ft-measurable for a.e. t∈(a, b). Moreover, the space Ip(a, b;H) is a closed subspace of Lp(Ω ×145 (a, b),F ⊗ B((a, b)), dP⊗dt;H).146 Denote by A=−∆ with Dirichlet boundary condition in our problem, and let D(A) be the domain of A. In this way, the linear operator A:D(A) := V∩H2(O)⊂V→His positive, self-adjoint with compact resolvent. We denote by 0 < λ1≤λ2≤ · · · the eigenvalues of A, and by e1,e2,· · · , a corresponding complete orthonormal system in L2(O) of eigenvectors of A. Recall that for every v∈V, the Poincar´e inequality λ1(O)|v|2≤ kvk2 holds. In the sequel, unless otherwise specified, we write λ1instead of λ1(O).147 2.3. Assumptions on g.Let g: (τ, T)×H→L2(H, H) satisfy:148 g1)g(t, 0) = 0 and kg(t, u)−g(t, v)kL2(H,H)≤Lg|u−v|,∀u, v ∈H, a.e. t∈(τ, T);149 g2) For every ρ∈C∞ c(O), the mapping H3u→< g(t, u), ρ >:= g(t, u)(ρ)∈His continuous for a.e.150 t∈(τ, T).151 Remark 2.2. We will show detailedly the proof of existence of martingale solutions to problem (1.1)152 in the next theorem. To present ideas clearly, we simply do estimations on g(u) instead of g(t, u). Indeed,153 the idea and procedures to obtain existence of martingale solutions to (1.1) with g(t, u) are similar, we154 only need to consider for every t∈(τ, T], ˜u(t) is ˜ Ft-measurable, for more details, see [13].155 2.4. Preliminaries. We now recall the following results which will be needed to prove the existence156 of martingale solutions.157 Lemma 2.3. ([14, Theorem 2.1])Let B0⊂B⊂B1be Banach spaces, B0and B1be reflexive, with compact embedding of B0in B. Let p∈(1,∞)and α∈(0,1), let Xbe the space X=Lp(0, T;B0)∩Wα,p(0, T;B1) endowed with the natural norm. Then the embedding of Xin Lp(0, T;B)is compact.158 Lemma 2.4. ([12, Skorohod theorem])Let Xbe a complete, separable metric space. For an arbitrary159 sequence {µn}, which is tight on (X, B(X)), there exists a subsequence {µnk}which converges weakly to a160 probability measure µ, and a probability space (Ω,F,P)with X-valued Borel measurable random variables161 xnand x, such that µnis the distribution of xn,µis the distribution of xand xn→x,P-a.s.162 Lemma 2.5. ([26, Vitali’s convergence theorem])Let p∈[1,∞),xn∈Lp(Ω), and xnconverge to xin163 probability. Then the following statements are equivalent:164 1. limn→∞ xn=xin Lp(Ω);165 2. |xn|pis uniformly integrable;166 This manuscript is for review purposes only. 6J.H. XU, AND T. CARABALLO 3. limn→∞ E[|xn|p] = E[|x|p].167 Particularly, if supnE[|xn|q]<∞for some p < q < ∞, or if there exists a y∈Lp(Ω) such that |xn|< y168 for all n, then the above properties hold true.169 2.5. Definitions of solutions. We introduce the concepts of solution of problem (1.1).170 Definition 2.6. (Variational solution) A solution of (1.1)is a stochastic process u∈I2(τ, T;V)∩171 L2(Ω; C(τ, T;H)) ∩Ip(τ, T;Lp(O)) for all T≥τ, with the initial value u(τ) = u0∈L2(Ω; H), such that172 u(t) = u0+Zt τ a(l(u(s)))∆u(s)ds +Zt τ f(u(s))ds +Zt τ h(s)ds +Zt τ g(u(s))dW(s),P-a.s. ∀t∈(τ, T], 173 where the above integro-equality should be understood in V∗+Lq(O), and qis the conjugate exponent of p.174 Definition 2.7. (Martingale solution) We say there exists a martingale solution of equation (1.1)if175 there exist176 •a stochastic basis (˜ Ω,˜ F,˜ {Ft}t≥0,˜ P);177 •a cylindrical Wiener process ˜ Won the space H;178 •a progressively measurable process ˜u: [τ, T]ט Ω→Hwith ˜ P-a.e. paths ˜u(·, ω)∈L2(τ, T;V)∩L∞(τ, T;H)∩Lp(τ, T;Lp(O)), such that for all t∈[τ, T]and v∈V∩Lp(O),179 (˜u(t), v) + Zt τ a(l(˜u(s))) < A˜u(s), v > ds = (˜u0, v) + Zt τ (f(˜u(s)), v)ds +Zt τ < h(s), v > ds +Zt τ g(˜u(s))d˜ W(s), v, (2.1) 180 where the identity holds ˜ P-a.s.181 2.6. Main results. We now prove the existence of martingale solutions to problem (1.1) after pre-182 senting all the required conditions, lemmas and techniques.183 Theorem 2.8. Assume that a∈C(R;R+)satisfies (1.2), f∈C(R)fulfills (1.3)-(1.4), g:H→184 L2(H, H)satisfies g1)-g2). Moreover, h∈L2 loc(R;V∗)and l∈ L(L2(O); R). Then, for every initial datum185 u0∈H, there exists at least one martingale solution to problem (1.1).186 Proof. We split the proof into several steps.187 Step 1. Faedo-Galerkin approximation. Making use of spectral theory, we recall that {ei}∞ i=1 188 is the orthonormal basis of Hconsisting of the eigenfunctions of Ain V. Observe that, thanks to the189 regularity imposed on the domain O, each eigenfunction ei∈Lp(O).190 Before going further, we first define two projection operators related to191 Pn:H−→ Vn:= span[e1,· · · , en], φ−→ n X i=1 (φ, ei)ei. 192 The first one is given by193 P1 n:V∗−→ V∗, v−→ [φ∈V→< P1 nv, φ >:=< v, Pnφ >]. 194 This manuscript is for review purposes only. STOCHASTIC NONLOCAL PDES AND WONG-ZAKAI APPROXIMATIONS 7 To define the second one, we recall that A=−∆ with homogeneous Dirichlet boundary condition, i.e. the isomorphism from Vinto V∗, which can be also seen as an unbounded operator in H. Let us consider the domains of fractional powers of A, D(Ak/2) = {u∈H:X i≥1 λk i(u, ei)2<∞}. Now we are ready to define the second projection operator, which is given by195 P2 n:Lq(O)−→ D(A−k/2), v−→ [φ∈D(Ak/2)→< P2 nv, φ >D(A−k/2),D(Ak/2):= (v, Pnφ)]. 196 Observe that P1 nand P2 nare the continuous extensions in V∗and Lq(O) of Pn, respectively. Therefore,197 from now on we will denote both projections by Pnmaking an abuse of notation.198 Let us consider the classical Faedo-Galerkin approximation in the space Vn,199 (dun(t)=[−a(l(un(t)))Aun(t) + Pnf(un(t)) + Pnh(t)] dt +Png(un(t))dW(t), t ∈(τ, T], un(τ) = Pnu0. (2.2)200 201 In what follows, we will show for all n∈N, there exist three positive constants C1,C2and C3, such that202 (2.3) Esup τ≤t≤T |un(t)|2≤C1,203 204 (2.4) EZT τ kun(t)k2dt ≤C2,205 and206 (2.5) EZT τ |un(t)|p pdt ≤C3.207 Applying the Itˆo formula to |un|2(n≥1) and integrating from τto T, we have208 |un(t)|2=|Pnu0|2+ 2 Zt τ a(l(un(s))) <−Aun(s), un(s)> ds + 2 Zt τ (Pnf(un(s)), un(s))ds + 2 Zt τ < Pnh(s), un(s)> ds + 2 Zt τ (un(s), Png(un(s))dW(s)) +Zt τ kPng(un(s))k2 L2(H,H)ds, a.e. t∈(τ, T ]. 209 Making use of (1.2) and (1.4), we obtain210 |un(t)|2+ 2mZt τ kun(s)k2ds + 2α2Zt τ |un(s)|p pds ≤ |u0|2+ 2κ|O|(T−τ) + 2 Zt τ kh(s)k∗kun(s)kds + 2 Zt τ (un(s), Png(un(s))dW(s)) +Zt τ kPng(un(s))k2 L2(H,H)ds, a.e. t∈(τ, T ]. 211 Applying the Young inequality and taking into account of g1) to the above inequality, we arrive at212 |un(t)|2+mZt τ kun(s)k2ds + 2α2Zt τ |un(s)|p pds ≤ |u0|2+ 2κ|O|(T−τ) + 1 mZt τ kh(s)k2 ∗ds +LgZt τ |un(s)|2ds + 2 Zt τ (un(s), Png(un(s))dW(s)) . (2.6) 213 This manuscript is for review purposes only. 8J.H. XU, AND T. CARABALLO Taking supremum and expectation on both sides of (2.6), by means of the Burkholder-Davis-Gundy in-214 equality, we derive215 Esup τ≤s≤t |un(s)|2≤2E|u0|2+ 4κ|O|(T−τ) + 2 mEZt τ kh(s)k2 ∗ds + 2 1+2C2 bLgZt τ Esup τ≤r≤s |un(r)|2ds, 216 where Cbis the constant derived from Burkholder-Davis-Gundy estimate. By iterating the preceding217 inequality, we obtain218 Esup τ≤s≤t |un(s)|2≤2E|u0|2+ 4κ|O|(T−τ) + 2 mEZt τ kh(s)k2 ∗ds × n−1 X i=0 (2(1 + 2C2 b)Lg)i(t−τ)i i!≤e(2+4C2 b)Lg(T−τ)≤const. 219 Moreover, it follows from (2.6) that220 mEZt τ kun(s)k2ds ≤E|u0|2+ 2κ|O|(T−τ) + 1 mEZt τ kh(s)k2 ∗ds +LgZt τ Esup τ≤r≤s |un(r)|2ds, 221 and222 2α2EZt τ |un(s)|p pds ≤E|u0|2+ 2κ|O|(T−τ) + 1 mEZt τ kh(s)k2 ∗ds +LgZt τ Esup τ≤r≤s |un(r)|2ds. 223 Thus, the desired results (2.3)-(2.5) are proved.224 Step 2. Tightness. For each n∈N, the solution unof the Galerkin equation defines a measure225 L(un) on L2(τ, T;V)∩L∞(τ, T;H)∩Lp(τ, T;Lp(O)). Using lemmas 2.1 and 2.3, together with estimates226 (2.3)-(2.5), we will prove the tightness of this set of measures.227 Decompose now unas228 un(t) = Pnu0−Zt τ a(l(un(s)))Aun(s)ds +Zt τ Pnf(un(s))ds +Zt τ Pnh(s)ds +Zt τ Png(un(s))dW(s) = I1 n+I2 n+I3 n+I4 n+I5 n. (2.7) 229 We will estimate each term of (2.7). Since u0∈H, it is easy to check there exists a constant C4, such that E|I1 n|2≤C4. For I2 n, by (1.2), (2.4), the H¨older inequality and Fubini Theorem, there exists a constant C5, such that230 EkI2 nk2 W1,2(τ,T ;V∗)=EkI2 nk2 L2(τ,T ;V∗)+EkdI2 n dt k2 L2(τ,T ;V∗) =EZT τ   Zt τ −a(l(un(s)))Aun(s)ds    2 ∗ dt +EZT τ k − a(l(un(s)))Aun(s)k2 ∗ds ≤˜m2(T−τ)EZT τZt τ k − Aun(s)k2 ∗dsdt + ˜m2EZT τ k − Aun(t)k2 ∗dt ≤C˜m2(T−τ)2+ ˜m2EZT τ kun(t)k2dt ≤C5. 231 This manuscript is for review purposes only. STOCHASTIC NONLOCAL PDES AND WONG-ZAKAI APPROXIMATIONS 9 For I3 n, let q=p p−1∈(1,2) be the conjugate of p, we first derive the following estimate by (1.5),232 |f(un)|q q=ZO |f(un)|qdx ≤βqZO (|un|p−1+ 1)qdx ≤2q−1βqZO |un|q(p−1)dx + 2q−1βq|O| := 2q−1βq|un|p p+ 2q−1βq|O|. 233 Observe that Pnf(un)∈Lq(τ, T;H−k(O)) since f(un)∈Lq(τ, T;Lq(O)). By the above inequality, (2.5),234 the H¨older inequality and Fubini Theorem, there exists a constant C6, such that235 EkI3 nkq W1,q(τ,T ;H−k(O)) =EkI3 nkq Lq(τ,T ;H−k(O)) +EkdI3 n dt kq Lq(τ,T ;H−k(O)) =EZT τZt τ Pnf(un(s))ds q H−k(O) dt +EZT τ |Pnf(un(t))|q H−k(O)dt ≤EZT τZt τ |Pnf(un(s))|H−k(O)dsq dt +EZT τ |f(un(t))|q qdt ≤(T−τ)1 p−1+1 + 1EZT τ |f(un(t))|q qdt ≤C6. 236 For I4 n, by the H¨older inequality and Fubini Theorem, there exists a constant C7, such that237 EkI4 nk2 W1,2(τ,T ;V∗)=EkI4 nk2 L2(τ,T ;V∗)+EkdI4 n dt k2 L2(τ,T ;V∗) =EZT τ   Zt τ Pnh(s)ds    2 ∗ dt +EZT τ kPnh(t)k2 ∗dt ≤((T−τ)2+ 1)Ekhk2 L2(τ,T ;V∗)≤C7. 238 As for the last term I5 n, by Lemma 2.1, assumption g1) and (2.3), we know there exists a constant C8(α), such that for every α∈(0,1 2), we have EkI5 nk2 Wα,2(τ,T ;H)≤C8(α). Obviously, for α∈(0,1 2), the natural continuous embedding D(Ak/2),→Hk(O),→Lp(O) implies W1,2(τ, T;V∗)⊂W1,q(τ, T;V∗)⊂Wα,q(τ, T;V∗)⊂Wα,q(τ, T;D(A−k/2)), Wα,2(τ, T;H)⊂Wα,q(τ, T;H)⊂Wα,q(τ, T;V∗)⊂Wα,q(τ, T;D(A−k/2)), and W1,q(τ, T;H−k(O)) ⊂Wα,q(τ, T;H−k(O)) ⊂Wα,q(τ, T;D(A−k/2)). Collecting all the previous estimates for I1 n-I5 n, together with the above natural embedding results, we obtain EkunkWα,q(τ,T ;D(A−k/2)) ≤C(α), for all α∈(0,1 2) and C(α)>0. Actually, thanks to (2.4), we deduce that the laws L(un) are bounded in probability in L2(τ, T;V)∩Wα,q(τ, T;D(A−k/2)). Additionally, L2(τ, T ;V)⊂Lq(τ, T;V), hence, it follows from Lemma 2.3 that L(un) is tight in Lq(τ, T;H).239 Step 3. Pass to limit. By Step 2, we obtain the set of measures L(un) is tight on the space240 Lq(τ, T;H). Moreover, Lemma 2.4 implies there exists a stochastic basis (˜ Ω,˜ F,{˜ Ft}t≥0,˜ P), and on this241 basis, there exist Lq(τ, T;H)-valued random variables {˜unk}(k≥1) and ˜u, such that242 (2.8) ˜unkhas the same law as unkon Lq(τ, T;H) and ˜unk→˜uin Lq(τ, T ;H),˜ P-a.s.243 This manuscript is for review purposes only. 16 J.H. XU, AND T. CARABALLO and358 ˜ Eh˜ M(t), ψih ˜ M(t), ζi−h ˜ M(s), ψih ˜ M(s), ζi −Zt s (g(˜u(σ))∗Pnψ, g(˜u(σ))∗Pnζ)ϕ(˜u|[τ,s])= 0, (2.31) 359 where ˜ Mis a D(A−k/2)-valued process defined by (2.19).360 Continuation of the proof of Theorem 2.8.Eventually, we apply an idea analogous to the361 reasoning used by Da Prato and Zabczyk, see [12, Section 8.3]. Consider the operator A:D(A)⊂V→H,362 the inverse operator A−1:H→D(A)⊂V, which is everywhere well-defined, bounded and compact,363 and the dual operator (A−1)∗:V∗→H. Since V∗is a dense subspace of D(A−k/2), we can extend the364 continuous operator (A−1)∗:D(A−k/2)→H. By (2.30) and (2.31) with ψ:= A−1αand ζ:= A−1β,365 where α,β∈H, we infer that (A−1)∗˜ M(t), t∈(τ, T] is a continuous square integrable martingale in H,366 whose dual is itself, with respect to the filtration ˜ Ft:= σ{˜u(s) : τ≤s≤t}, having the quadratic variation367 hh(A−1)∗˜ Miit=Zt τ (A−1)∗g(˜u(s))(g(˜u(s))A−1)∗ds, t ∈(τ, T].368 In particular, the continuity of the process (A−1)∗˜ Mfollows from the fact that ˜u∈C(τ, T;H). By the369 representation theorem [12, Theorem 8.2], there exist370 •a stochastic basis (˜ ˜ Ω,˜ ˜ F,˜ ˜ {Ft}t≥0,˜ ˜ P);371 •a cylindrical Wiener process ˜ ˜ Wdefined on this basis;372 •a progressively measurable process ˜ ˜usuch that373 (A−1)∗˜ ˜u(t)−(A−1)∗˜ ˜u0+ (A−1)∗Zt τ a(l(˜ ˜u(s)))A˜ ˜u(s)ds −(A−1)∗Zt τ f(˜ ˜u(s))ds −(A−1)∗Zt τ h(s)ds =Zt 0 (A−1)∗g(˜ ˜u(s))d˜ ˜ W(s). 374 However, Zt τ (A−1)∗g(˜ ˜u(s))d˜ ˜ W(s)=(A−1)∗Zt τ g(˜ ˜u(s))d˜ ˜ W(s). Hence, it follows from (2.12) that ˜ ˜u: [τ, T ]ט ˜ Ω→Hwith ˜ ˜ P-a.s. paths, ˜ ˜u(·, ω)∈L2(τ, T;V)∩L∞(τ, T;H)∩Lp(τ, T;Lp(O)), satisfies for all t∈[τ, T] and for all v∈V∩Lp(O),375 (˜ ˜u(t), v) + Zt τ a(l(˜ ˜u(s))) < A˜ ˜u(s), v > ds = (˜ ˜u0, v) + Zt τ (f(˜ ˜u(s)), v)ds +Zt τ < h(s), v > ds +Zt τ g(˜ ˜u(s))d˜ ˜ W(s), v, 376 where the identity holds ˜ ˜ P-a.s.377 The proof of this theorem is finished.378 Although we are not able to prove the existence of variational solutions to problem (1.1), we can show379 that there exists at most one solution when the coefficient a(·) is locally Lipschitz.380 Theorem 2.13. Assume a∈C(R;R+)is locally Lipschitz and satisfies (1.2),f∈C(R;R+)fulfills381 (1.3)-(1.4),g:H→L2(H, H)satisfies g1)and l∈L2(O). In addition, let h∈L2(Ω; L2 loc(R+;V∗)) and382 u0∈L2(Ω; H). Then, there exists at most one solution to problem (1.1)in the sense of Definition 2.6.383 This manuscript is for review purposes only. STOCHASTIC NONLOCAL PDES AND WONG-ZAKAI APPROXIMATIONS 17 Proof. Suppose there are two solutions uand vof problem (1.1) in the sense of Definition 2.6. Let384 σ(t) = exp(−µRt τ||u(s)||2ds) for all τ≤t≤T, which is positive and well-defined (cf. Step 1 of Theorem385 2.8), where µis a proper constant to be chosen later. Applying the Itˆo formula to σ(t)|u(t)−v(t)|2, by386 (1.2) and (1.3), we have387 σ(t)|u(t)−v(t)|2+ 2mZt τ σ(s)ku(s)−v(s)k2ds ≤2Zt τ σ(s)|a(l(u(s))) −a(l(v(s)))|ku(s)kku(s)−v(s)kds + 2ηZt τ σ(s)|u(s)−v(s)|2ds + 2 Zt τ σ(s) (u(s)−v(s), g(u(s))dW(s)−g(v(s))dW(s)) + Zt τ σ(s)kg(u(s)) −g(v(s))k2 L2(H,H)ds −µZt τ σ(s)ku(s)k2|u(s)−v(s)|2ds (2.32) 388 Since ais Locally Lipschitz, denote this Lipschitz constant by La, by the Young inequality, we have389 2σ(s)|a(l(u(s))) −a(l(v(s)))|ku(s)kku(s)−v(s)k ≤2La|l|σ(s)|u(s)−v(s)|ku(s)kku(s)−v(s)k ≤µσ(s)ku(s)k2|u(s)−v(s)|2+L2 a|l|2σ(s) µku(s)−v(s)k2. 390 Thus, by g1) and the above inequality, (2.32) becomes391 σ(t)|u(t)−v(t)|2+ 2mZt τ σ(s)ku(s)−v(s)k2ds ≤L2 a|l|2 µZt τ σ(s)ku(s)−v(s)k2ds + (2η+Lg)Zt τ σ(s)|u(s)−v(s)|2ds + 2 Zt τ σ(s) (u(s)−v(s), g(u(s))dW(s)−g(v(s))dW(s)) . 392 Taking the supremum (w.r.t. t) and expectation on both sides of the above inequality, by (1.2), we obtain393 Esup τ≤s≤t σ(s)|u(s)−v(s)|2≤L2 a|l|2 µEsup τ≤s≤tZs τ σ(r)ku(r)−v(r)k2dr + (2η+Lg)Esup τ≤s≤tZs τ σ(r)|u(r)−v(r)|2dr + 2Esup τ≤s≤tZs τ σ(r) (u(r)−v(r), g(u(r))dW(r)−g(v(r))dW(r)), (2.33) 394 and395 2mEZt τ σ(s)ku(s)−v(s)k2ds ≤L2 a|l|2 µEsup τ≤s≤tZs τ σ(r)ku(r)−v(r)k2dr + (2η+Lg)Esup τ≤s≤tZt τ σ(r)|u(r)−v(r)|2dr + 2Esup τ≤s≤tZs τ σ(r) (u(r)−v(r), g(u(r))dW(r)−g(v(r))dW(r)). (2.34) 396 This manuscript is for review purposes only. 18 J.H. XU, AND T. CARABALLO For the first term of the right hand side of (2.33), since µis positive, we have397 (2.35) L2 a|l|2 µEsup τ≤s≤tZs τ σ(r)ku(r)−v(r)k2dr=L2 a|l|2 µEZt τ σ(s)ku(s)−v(s)k2ds.398 For the second term of the right hand side of (2.33), by the same arguments as above, we obtain399 (2.36) (2η+Lg)Esup τ≤s≤tZs τ σ(r)|u(r)−v(r)|2dr≤(2η+Lg)EZt τ sup τ≤r≤s σ(r)|u(r)−v(r)|2ds.400 Next, assumption g1), the Burkholder-Davis-Gundy and Young inequalities imply401 2Esup τ≤s≤tZs τ σ(r) (u(r)−v(r), g(u(r))dW(r)−g(v(r))dW(r)) ≤2cEsup τ≤s≤t σ(s)|u(s)−v(s)|2Zt τ σ(s)kg(u(s)) −g(v(s))k2 L2(H,H)ds1 2 ≤1 4Esup τ≤s≤t σ(s)|u(s)−v(s)|2+ 4c2LgEZt τ sup τ≤r≤s σ(r)|u(r)−v(r)|2ds. (2.37) 402 Consequently, substituting (2.35)-(2.37) into (2.33)-(2.34), letting mµ =L2 a|l|2, we deduce403 Esup τ≤s≤t σ(s)|u(s)−v(s)|2≤42η+Lg+ 4c2LgZt τ Esup τ≤r≤s σ(r)|u(r)−v(r)|2ds. 404 It follows from the Gronwall lemma that Esup τ≤s≤t σ(s)|u(s)−v(s)|2= 0,∀t∈(τ, T]. Thus, we have u(t) = v(t) for a.a. ω∈Ω and a.e. t∈(τ, T] since σ(t) is positive. The proof of this405 theorem is complete.406 For the rest of this manuscript, to carry out the analysis of asymptotic behavior of solutions to (1.1)407 in the sense of Definition 2.6 and their Wong-Zakai approximation, we will assume, for simplicity, W(t) is408 a standard 1D Brownian motion. Moreover, let g: (τ, T)×H→Hbe a nonlinear operator, satisfying:409 g1) The mapping t∈(τ, T )→g(t, u)∈His Lebesgue measurable, for all u∈H;410 g2) g(t, 0) = 0,a.e. t∈(τ, T);411 g3) There exists a positive constant Lg(we use the same constant when no confusion is possible), such that |g(t, u)−g(t, v)|2≤Lg|u−v|2,∀u, v ∈H, a.e. t∈(τ, T). 3. Asymptotic behavior of solutions to problem (1.1)around steady-state solutions of412 the deterministic problem. In this section, we are interested in analyzing the long time behavior of413 solutions to problem (1.1) with respect to equilibria of the deterministic elliptic problem,414 (−a(l(u))∆u=f(u) + h u= 0, in O, on ∂O. (3.1)415 416 Since we are dealing with stationary solutions, the assumption imposed on function hdoes not depend417 on time, i.e., h∈V∗. The solutions to (3.1) are the so called steady-state solutions or equilibria and the418 formal definition is the following.419 Definition 3.1. A stationary or steady-state solution to problem (3.1)(also called equilibrium) is a function u∗∈V∩Lp(O)which fulfills a(l(u∗))((u∗, v)) = (f(u∗), v)+ < h, v >, ∀v∈V∩Lp(O), This manuscript is for review purposes only. STOCHASTIC NONLOCAL PDES AND WONG-ZAKAI APPROXIMATIONS 19 or, in other words, is a solution of the elliptic equation,420 (3.2) a(l(u∗))∆u∗=f(u∗) + h, in V∗+Lq(O).421 Observe that a steady-state solution u∗to problem (3.1) can only be solution to the stochastic problem422 (1.1) (with h(t) = h∈V∗) if g(t, u∗) = 0 for all t∈[τ, +∞), which is a very particular situation. Thus, our423 main interest is to study how the solutions to stochastic problem (1.1) behave around the equilibria of the424 deterministic problem (3.1). In this way, to establish some sufficient conditions ensuring the exponential425 decay of solutions to (1.1) towards some solutions of (3.1), we assume the existence of stationary solutions426 to (3.1) (see, for instance, [18, Theorem 3.8] ). Notice that, when function fis more general, namely, which427 satisfies the conditions (1.3)-(1.4), it is not easy to argue. Therefore, in order to prove the existence of at428 least one nontrivial stationary solution to problem (3.1), the authors in [18] studied one particular, but429 very interesting case when f: [0,1] →Ris given by f(s) = s−s3, for s∈[0,1], the arguments were based430 on a fixed point theorem. Whereas, considering again the general form function fand under new suitable431 assumptions, the authors in [18] showed that any stationary solution is positive provided its existence is432 guaranteed [18, Chapter 3.2].433 In the sequel, our goal is to establish sufficient conditions to prove exponential decay of variational434 solutions in mean square.435 Definition 3.2. A solution uto (1.1)is said to converge to (or to decay to) u∗∈V∩Lp(O)exponentially in mean square, if there exist α > 0and M=M(u0)>0such that E|u(t)−u∗|2≤Me−α(t−τ),∀t≥τ. Definition 3.3. A solution uto equation (1.1)is said to converge exponentially to u∗∈V∩Lp(O) almost surely, if there exists γ > 0such that lim sup t→+∞ 1 tlog |u(t)−u∗|≤−γ, almost surely. In order to prove the exponential stability results, the following condition as in [6] is considered.436 Assume there exists a steady-state solution u∗of (3.1) such that gsatisfies437 g4) |g(t, u)|2≤β(t)+(ξ+δ(t))|u−u∗|2, for all u∈H, where ξis a positive constant, β(t), δ(t) are nonnegative integrable functions, such that there exist real numbers θ > α,Mβ≥1 and Mδ≥1 with β(t)≤Mβe−θt and δ(t)≤Mδe−θt,∀t≥0. We will present in the next theorem that, any variational solution to (1.1) converges exponentially to438 u∗in mean square, showing that u∗is the only relevant stationary solution for the stochastic system. No439 matter how many steady-state solutions (3.1) may have, this u∗is attracting in mean square any other440 solution of the stochastic problem.441 Theorem 3.4. Assume (1.2)-(1.4)and g4) hold with442 (3.3) (2η+ξ)λ−1 1+ 2La|l|ku∗kλ−1/2 1< m,443 where a(·)is supposed to be globally Lipschitz, the Lipschitz constant is still denoted the same by La. Then:444 (i) Any variational solution u(·)of problem (1.1)converges to the stationary solution u∗of (3.1) exponentially in the mean square. That is, there exist α > 0and M=M(u0)such that, E|u(t)−u∗|2≤Me−α(t−τ), t ≥τ; (ii) Any variational solution u(t)of problem (1.1)converges to the stationary solution u∗of (3.1)445 almost surely exponentially.446 This manuscript is for review purposes only. 20 J.H. XU, AND T. CARABALLO Proof. (i) Since (2η+ξ)λ−1 1+ 2La|l|ku∗kλ−1/2 1< m, we can choose 0 < α < θ such that, (α+ 2η+ξ)λ−1 1+ 2La|l|ku∗kλ−1/2 1−2m < 0. By applying the Itˆo formula to eαt|u(t)−u∗|2and taking expectation, we obtain447 eαtE|u(t)−u∗|2=eατ E|u0−u∗|2+αEZt τ eαs|u(s)−u∗|2ds + 2EZt τ eαs < a(l(u))∆u(s), u(s)−u∗> ds + 2EZt τ eαs(f(u(s)), u(s)−u∗)ds + 2EZt τ eαs < h, u(s)−u∗> ds +EZt τ eαs|g(s, u(s))|2ds. 448 As u∗is the stationary solution to problem (3.1), we have449 −EZt τ eαs < a(l(u∗))∆u∗, u(s)−u∗> ds =EZt τ eαs(f(u∗), u(s)−u∗)ds+EZt τ eαs < h, u(s)−u∗> ds.450 It follows from the two above equalities that,451 eαtE|u(t)−u∗|2=eατ E|u0−u∗|2+αEZt τ eαs|u(s)−u∗|2ds + 2EZt τ eαs < a(l(u(s)))∆u(s)−a(l(u∗))∆u∗, u(s)−u∗> ds + 2EZt τ eαs(f(u(s)) −f(u∗), u(s)−u∗)ds +EZt τ eαs|g(s, u(s))|2ds. 452 By means of assumptions (1.2), (1.4) and g4), together with the fact that ais Lipschitz and the Poincar´e453 inequality, we derive454 eαtE|u(t)−u∗|2≤eατ E|u0−u∗|2+EZt τ eαs β(s) + δ(s)|u(s)−u∗|2ds +(α+ 2η+ξ)λ−1 1+ 2La|l|ku∗kλ−1/2 1−2mEZt τ eαsku(s)−u∗k2ds. (3.4) 455 Thanks to the fact that (α+ 2η+ξ)λ−1 1+ 2La|l|ku∗kλ−1/2 1−2m<0, the last term of (3.4) is negative,456 we obtain457 eαtE|u(t)−u∗|2≤eατ E|u0−u∗|2+Zt τ eαsβ(s)ds +Zt τ δ(s)eαsE|u(s)−u∗|2ds.458 Since θ > α, applying the Gronwall lemma to the above inequality, the result (i) is proved.459 (ii) We now move to the second assertion, let Nbe a natural number, by applying the Itˆo formula to460 |u(t)−u∗|2and using fact that u∗is a steady-state solution, it follows that461 |u(t)−u∗|2=|u(N)−u∗|2+ 2 Zt N < a(l(u(s)))∆u(s)−a(l(u∗))∆u∗, u(s)−u∗> ds + 2 Zt N (f(u(s)) −f(u∗), u(s)−u∗)ds + 2 Zt N (g(s, u(s)), u(s)−u∗)dW(s) + Zt N |g(s, u(s))|2ds. 462 This manuscript is for review purposes only. STOCHASTIC NONLOCAL PDES AND WONG-ZAKAI APPROXIMATIONS 21 Therefore, by (1.2)-(1.3), we have463 |u(t)−u∗|2+ 2mZt N ku(s)−u∗k2ds ≤2Zt N |<(a(l(u(s))) −a(l(u∗)))∆u∗, u(s)−u∗>|ds +|u(N)−u∗|2+ 2ηZt N |u(s)−u∗|2ds + 2 Zt N (g(s, u(s)), u(s)−u∗)dW(s)+Zt N |g(s, u(s))|2ds. 464 Consequently,465 Esup N≤t≤N+1 |u(t)−u∗|2+ 2mEZN+1 N ku(s)−u∗k2ds ≤4E"ZN+1 N |<(a(l(u(s))) −a(l(u∗)))∆u∗, u(s)−u∗>|ds# + 2E|u(N)−u∗|2+ 4ηEZN+1 N |u(s)−u∗|2ds + 4Esup N≤t≤N+1 Zt N (g(s, u(s)), u(s)−u∗)dW(s)+ 2E"ZN+1 N |g(s, u(s))|2ds#. (3.5) 466 With the help of the Burkholder-Davis-Gundy and Young inequalities, we have467 4Esup N≤t≤N+1 Zt N (g(s, u(s)), u(s)−u∗)dW(s) ≤4C2E"ZN+1 N |g(s, u(s))|2|u(s)−u∗|2ds#1 2 ≤4C2E"sup N≤t≤N+1 |u(t)−u∗|2ZN+1 N |g(s, u(s))|2ds#1 2 ≤1 2Esup N≤t≤N+1 |u(s)−u∗|2+8C2 2E"ZN+1 N |g(s, u(s))|2ds#. (3.6) 468 Proceeding now as in the proof of the previous theorem and substituting (3.6) into (3.5), it yields469 1 2Esup N≤t≤N+1 |u(t)−u∗|2 ≤2E|u(N)−u∗|2+−2m+ 4La|l|ku∗kλ−1/2 1+ 4ηλ−1 1EZN+1 N ku(s)−u∗k2ds + (8C2 2+ 2)EZN+1 Nβ(s)+(ξ+δ(s))|u(s)−u∗|2ds ≤2E|u(N)−u∗|2+ (8C2 2+ 2) ZN+1 Nβ(s)+(ξ+δ(s))E|u(s)−u∗|2ds 470 The last step of above inequality is true thanks to assumption (2η+ξ)λ−1 1+ 2La|l|ku∗kλ−1/2 1< m. Moreover, it follows from condition g4) that β(t)≤Mβe−θt and δ(t)≤Mδe−θt, 0 < α < θ,Mβ≥1 and This manuscript is for review purposes only. 22 J.H. XU, AND T. CARABALLO Mδ≥1. Thus, taking into account the exponential decay in mean square stated in Theorem 3.4, there exists M:= M(τ, u0)>0, such that Esup N≤t≤N+1 |u(t)−u∗|2≤Me−αN . The proof is completed by using the Borel-Cantelli lemma (see [8] for a detailed explanation).471 Remark 3.5. Notice that it is enough to assume that (2η+ξ)λ−1 1+2La|l|ku∗kλ−1/2 1<2min Theorem472 3.4 instead of (2η+ξ)λ−1 1+ 2La|l|ku∗kλ−1/2 1< m. However, in the next theorem it will be necessary to473 impose the latter, so we prefer to impose this one in both theorems.474 We conclude this section with a result on the exponential stability of the steady-state solution in mean475 square, when this becomes also a solution of the stochastic equation.476 Theorem 3.6. Assume (1.2)-(1.4)hold with477 (3.7) 2La|l|ku∗kλ−1/2 1+ 2ηλ−1 1+Lgλ−1 1<2m.478 where a(·)is supposed to be globally Lipschitz, the Lipschitz constant is still denoted the same by La. Additionally, assume the nonlinear stochastic term gfulfills g3), and g(t, u∗) = 0 for all t≥τ. Then the solution to problem (1.1)converges to the stationary solution of (3.1)u∗exponentially in the mean square. Namely, there exists a real number γ > 0, such that E|u(t)−u∗|2≤E|u0−u∗|2e−γ(t−τ),∀t≥τ. Proof. Since u∗is the stationary solution of (3.1), combined with (1.1), we derive479 u(t)−u∗=u0−u∗+Zt τ (a(l(u(s)))∆u(s)−a(l(u∗)) ∆u∗)ds +Zt τ (f(u(s)) −f(u∗))ds +Zt τ (g(s, u(s)) −g(s, u∗))dW(s). 480 Thanks to (3.7), we can choose a sufficiently small γ > 0, such that γλ−1 1+ 2La|l|ku∗kλ−1/2 1+ 2ηλ−1 1+Lgλ−1 1−2m < 0. Applying now the Itˆo formula to eγt|u(t)−u∗|2, taking expectation and using the same arguments as in481 Theorem 3.4, we obtain482 eγtE|u(t)−u∗|2=eγτ E|u0−u∗|2+γEZt τ |u(s)−u∗|2ds + 2EZt τ eγs < a(l(u(s)))∆u(s)−a(l(u∗))∆u∗, u(s)−u∗> ds + 2EZt τ eγs(f(u(s)) −f(u∗), u(s)−u∗)ds +EZt τ eγs|g(s, u(s)) −g(s, u∗)|2ds ≤eγτ E|u0−u∗|2+γλ−1 1EZt τ eγsku(s)−u∗k2ds +−2m+ 2La|l|ku∗kλ−1/2 1+ 2ηλ−1 1+Lgλ−1 1EZt τ eγsku(s)−u∗k2ds. 483 Due to the choice of γ, the result follows immediately. 484 This manuscript is for review purposes only. STOCHASTIC NONLOCAL PDES AND WONG-ZAKAI APPROXIMATIONS 23 4. Attractors of nonlocal stochastic PDEs driven by colored noise. Our aim now is to study485 the existence of attractors for the solution of problem (1.1). However, as it is well known, the theory of486 random dynamical systems has only been applied successfully to problems modeled by partial differential487 equations when the noise possesses a particular form: additive or multiplicative noise. These two cases488 have already been analyzed in [33]. Recently, B. X. Wang and his collaborators (see [17,15,22]) have489 been using an idea to approximate the nonlinear noise by a stochastic process (called colored noise), which490 basically is a Wong-Zakai approximation of the derivative of the Wiener process, providing a rigorous491 approximation of the cases with additive and multiplicative noise (as we explained in the Introduction).492 This is why, in this section, we study the long time behavior of the following non-autonomous nonlocal493 partial differential equations driven by colored noise,494      ∂u ∂t −a(l(u))∆u=f(u) + h(t) + g(t, u)ζδ(θtω), u= 0, u(x, τ) = uτ(x), in O × (τ, ∞), on ∂O × (τ, ∞), in O, (4.1)495 496 where ζδ(θtω) is the colored noise with correlation time δ > 0, functions a,f,hand gfulfill the same497 assumptions as in Section 2.498 4.1. Cocycles for nonlocal PDEs. To describe the global long time behavior of problem (4.1),499 it is necessary to establish the existence of a continuous non-autonomous cocycle for (4.1). Let us first500 recall some notions, definitions and lemmas which furnish the essential tools used throughout this section501 ([15,17,29,31]).502 Let (Ω,F,P) be a standard probability space, where Ω = C0(R,R) := {ω∈C(R,R) : ω(0) = 0}with the open compact topology, Fis its Borel σ-algebra, and Pis the Wiener measure on (Ω,F). In what follows, we will consider the Wiener shift {θt}t∈Rdefined on the probability space (Ω,F,P) by θtω(·) = ω(t+·)−ω(t),for all ω∈Ω, t ∈R. It is known that Pis an ergodic invariant measure for {θt}t∈R, and the quadruple (Ω,F,P,{θt}t∈R) forms503 a metric dynamical system (see [1]).504 In the sequel, we use (X, d) to denote a complete separable metric space. If Aand Bare two nonempty505 subsets of X, then we use distX(A, B) := supa∈Ainfb∈Bd(a, b) to denote their Hausdorff semidistance.506 Definition 4.1. ([28, Definition 2.6])Let D:R×Ω→2Xbe a set-valued mapping with closed507 nonempty images. We say Dis measurable with respect to Fin Ω, if the mapping ω∈Ω→d(x, D(τ, ω))508 is (F,B(R))-measurable for every fixed x∈Xand τ∈R.509 Definition 4.2. ([28, Definition 2.7])Let Dbe a collection of some families of nonempty subsets of Xand B={B(τ, ω) : τ∈R, ω ∈Ω}∈D. Then Bis called a D-pullback absorbing set for Φ, if for all τ∈R,ω∈Ωand for every B∈ D, there exists T=T(B, τ, ω)>0such that Φ(t, τ −t, θ−tω, B(τ−t, θ−tω)) ⊂B(τ, ω)for all t≥T. Definition 4.3. ([28, Definition 2.8])Let Dbe a collection of some families of nonempty subsets of X. Then Φis said to be D-pullback asymptotically compact in Xif for all τ∈Rand ω∈Ω, the sequence {Φ(tn, τ −tn, θ−tnω, xn)}∞ n=1 has a convergent subsequence in X, whenever tn→ ∞ and xn∈D(τ−tn, θ−tnω)with {D(τ, ω) : τ∈R, ω ∈Ω}∈D.510 Definition 4.4. ([28, Definition 2.9])Let Dbe a collection of some families of nonempty subsets of511 Xand A={A(τ, ω) : τ∈R, ω ∈Ω} ∈ D. Then Ais called a D-pullback attractor for Φif the following512 conditions (i)-(iii)are fulfilled:513 (i)Ais measurable in the sense of Definition 4.1, and A(τ, ω)is compact for all τ∈Rand ω∈Ω.514 (ii)Ais invariant, that is, for every τ∈Rand ω∈Ω, Φ(t, τ, ω, A(τ, ω)) = A(τ+t, θtω),∀t≥0. This manuscript is for review purposes only. 24 J.H. XU, AND T. CARABALLO (iii)Aattracts every member of D, that is, for every D={D(τ, ω) : τ∈R, ω ∈Ω} ∈ D and for every τ∈R,ω∈Ω, lim t→∞ d(Φ(t, τ −t, θ−tω, D(τ−t, θ−tω)),A(τ, ω)) = 0. We have introduced all required definitions of stochastic dynamical systems, which later on will allow515 us to define a cocycle Φ : R+×R×Ω×H→Hfor equation (4.1), such that for all t∈R+,τ∈R,ω∈Ω516 and uτ∈H,517 (4.2) Φ(t, τ, ω, uτ) = u(t+τ;τ, θ−τω, uτ),518 where u(·;τ, ω, uτ) denotes the solution to (4.1) which will be proved to exist in Section 4.3. Thus, Φ will519 be a continuous cocycle on Hover (Ω,F,P,{θt}t∈R). Moreover, let D={D(τ, ω) : τ∈R, ω ∈Ω}be a520 tempered family of bounded nonempty subsets of H, that is, for every γ > 0, τ∈Rand ω∈Ω,521 (4.3) lim t→−∞ eγt|D(τ+t, θtω)|= 0,522 where |D|= supu∈D|u|. Throughout this section, we will use Dto denote the collection of all tempered523 families of bounded nonempty subsets of H, i.e.,524 (4.4) D={D={D(τ, ω) : τ∈R, ω ∈Ω}:Dsatifies (4.3)}.525 Remark 4.5. Although the cocycle generated by (4.1) depends on the parameter δ, we will omit this526 dependence in this section since it will be fixed from the beginning. Hence, we will use Φ instead of using527 the notation Φδ.528 4.2. Properties of white and colored noises. We recall some known results for the Wiener process529 W(t, ω) = ω(t) in [1] and the colored noise ζδ(θtω) in [17,15], since they play important roles in the proof530 of the main theorems.531 Lemma 4.6. Let the correlation time δ∈(0,1]. There exists a {θt}t∈R-invariant subset (still denoted532 by) Ωof full measure, such that for all ω∈Ω,533 (i) (4.5) lim t→±∞ ω(t) t= 0;534 (ii)The mapping535 (4.6) (t, ω)→ζδ(θtω) = −1 δ2Z0 −∞ es δθtω(s)ds536 is a stationary solution (also called an Ornstein-Uhlenbeck process or a colored noise) of the one-537 dimensional stochastic differential equation dζδ+1 δζδdt =1 δdW with continuous trajectories, sat-538 isfying539 (4.7) lim t→±∞ ζδ(θtω) t= 0 for all 0< δ ≤1,540 541 (4.8) lim t→±∞ 1 tZt 0 ζδ(θsω)ds =Eζδ= 0,uniformly for 0< δ ≤1;542 (iii)For arbitrary T > 0,ε > 0, there exists δ0=δ0(τ, ω, T, ε)>0, such that for all 0< δ < δ0and543 t∈[τ, τ +T],544 (4.9) Zt 0 ζδ(θsω)ds −ω(t)< ε.545 Remark 4.7. Notice that, from (4.9), we can derive that there exist δ0=δ0(τ, ω, T) and ˜c= ˜c(τ, ω, T)>546 0 such that, for all 0 < δ < δ0and t∈[τ, τ +T],547 (4.10) Zt 0 ζδ(θsω)ds≤˜c.548 This manuscript is for review purposes only. STOCHASTIC NONLOCAL PDES AND WONG-ZAKAI APPROXIMATIONS 25 4.3. Well-posedness of problem (4.1).We are now in a position to show the existence and unique-549 ness of solution to equation (4.1) in the following sense.550 Definition 4.8. A weak solution to problem (4.1)is a mapping u(·;τ, ω, uτ):[τ, T)→H, for all T > τ with u(τ) = uτ, satisfying for any τ∈R,ω∈Ω, u(·;τ, ω, uτ)∈C(τ, T;H)∩L2(τ, T;V)∩Lp(τ, T;Lp(O)). Moreover, for every t>τ and v∈V+Lp(O),551 (u, v)=(uτ, v) + Zt τ a(l(u))((u, v))ds +Zt τ (f(u), v)ds +Zt τ < h, v > ds +Zt τ (g(s, u(s))ζδ(θsω), v)ds. 552 Note that, if we denote by Athe operator −∆with homogeneous boundary condition, then the above equality can be written as du dt +a(l(u))Au =f(u) + h(t) + g(t, u)ζδ(θtω),in V∗+Lq(O). Theorem 4.9. Assume that function ais locally Lipschitz and satisfies (1.2),f∈C(R)fulfills (1.3)-553 (1.4),h∈L2 loc(R+;V∗)and l∈L2(O). Additionally, function gsatisfies g1)-g3). Then, for each initial554 datum u0∈H, there exists a unique weak solution to problem (4.1)in the sense of Definition 4.8. Moreover,555 this solution behaves continuously in Hwith respect to the initial values.556 Proof. Since equation (4.1) can be viewed as a deterministic problem parametrized by ω(cf. [22]), for every T > τ and ω∈Ω, we can prove (4.1) has a unique solution, u(·;τ, ω, uτ)∈C(τ, T;H)∩L2(τ, T;V)∩Lp(τ, T;Lp(O)), by applying the Galerkin method and energy estimations [18, Chapter 3, Theorem 3.3].557 558 In this subsection, we first derive uniform estimations on the solution of (4.1) and then prove D-559 pullback asymptotic compactness by using the idea introduced by Ball in [2]. To this end, we need the560 following assumptions:561 h1) Suppose that Zτ −∞ emλ1skh(s)k2 ∗ds < ∞,∀τ∈R. For the existence of tempered random attractors, we need the assumption below:562 h2) For every γ > 0, it holds lim t→−∞ eγt Z0 −∞ emλ1skh(s+t)k2 ∗ds = 0. It is worth stressing that h1) and h2) do not require h(t) is bounded in V∗as t→ ±∞.563 Lemma 4.10. Assume conditions of Theorem 4.9 and h1) hold. Then, for every δ∈(0,1],τ∈R,564 ω∈Ωand D={D(τ, ω) : τ∈R, ω ∈Ω} ∈ D, there exists T=T(τ, ω, δ, D)>0such that for all t≥T565 and σ≥τ−t, the solution of problem (4.1)satisfies,566 |u(σ;τ−t, θ−τω, uτ−t)|2≤e−mλ1(σ−τ) +Zσ−τ −∞ emλ1(s−σ+t)2 mkh(s+τ)k2 ∗+2κ+c|ζδ(θsω)|p/(p−2)|O|ds, 567 This manuscript is for review purposes only. 32 J.H. XU, AND T. CARABALLO Proof. The idea to prove the existence of random DF-attractor to (5.3) is the same as [33, Theorem 9].701 Namely, looking for a random compact absorbing set Bδ(ω) (which will be given by the ball of center 0702 and radius Rδ(ω) in V) absorbing every bounded deterministic set D⊂H, together with the compact703 embedding V ,→H, we achieve the goal. Firstly, multiplying (5.11) by vδ(t) := vδ(t;τ, ω, v0,δ) in H, by704 (1.2), we obtain705 d dt|vδ(t)|2+ 2mkvδ(t)k2≤2(f(vδ(t) + φyδ(θtω)), vδ(t)) + 2yδ(θtω)(φ, vδ(t)) + 2 emkφkkvδ(t)k,706 with the help of (1.5), the Young and Poincar´e inequalities, we have707 d dt|vδ(t)|2+mkvδ(t)k2≤(−mλ1+ 2Cf(µ1+ 1) + µ2)|vδ(t)|2+Cf|O| µ1 +Cf µ1λ1 +1 µ2λ1|yδ(θtω)|2kφk2+em2 µ3 kφk2+µ3kvδ(t)k2. (5.14) 708 Letting µ1=1 2,µ2=Cfand µ3=m 2in (5.14), we derive709 d dt|vδ(t)|2≤ −(mλ1−4Cf)|vδ(t)|2+ 2Cf|O| +|yδ(θtω)|2 λ1Cf +2Cf|yδ(θtω)|2 λ1 +2em2 mkφk2−m 2kvδ(t)k2. (5.15) 710 Neglecting the last term of (5.15) and integrating in [t0,−1] with t0≤ −1, we have711 |vδ(−1)|2≤e−(mλ1−4Cf)(−1−t0)Z−1 t02Cf|O| +|yδ(θtω)|2 λ1Cf +2Cf|yδ(θtω)|2 λ1 +2em2 mkφk2 ×e(mλ1−4Cf)(t−t0)dt +|vδ(t0)|2 ≤e−(mλ1−4Cf)(−1−t0)|vδ(t0)|2 +Z−1 t0 e−(mλ1−4Cf)(−t−1)2Cf|O| +|yδ(θtω)|2 λ1Cf +2Cf|yδ(θtω)|2 λ1 +2em2 mkφk2dt ≤e(mλ1−4Cf)e(mλ1−4Cf)t0|vδ(t0)|2 +Z−1 t0 e(mλ1−4Cf)t2Cf|O| +|yδ(θtω)|2 λ1Cf +2Cf|yδ(θtω)|2 λ1 +2em2 mkφk2dt. 712 Consequently, for a given B(0, ρδ)⊂H, there exists T(ω, ρδ)≤ −1, such that for all t0≤T(ω, ρδ) and for all u0∈B(0, ρδ), |vδ(−1; t0, ω, uδ(t0)−φyδ(θt0ω))|2≤r2 3,δ(ω), with713 r2 3,δ(ω) = 1 + 2Cf|O| mλ1−4Cf +Z−1 −∞ e(mλ1−4Cf)(t+1) |yδ(θtω)|2 λ1Cf +2Cf|yδ(θtω)|2 λ1 +2em2 mkφk2dt,714 which is well defined. Indeed, it is enough to choose T(ω, ρδ) such that, for any t0≤T(ω, ρδ), we have715 e(mλ1−4Cf)(t0+1)|vδ(t0)|2=e(mλ1−4Cf)(t0+1)|uδ(t0)−φyδ(θt0ω)|2 ≤2e(mλ1−4Cf)(t0+1) ρ2 δ+|φ|2|yδ(θt0ω)|2 ≤1. 716 This manuscript is for review purposes only. STOCHASTIC NONLOCAL PDES AND WONG-ZAKAI APPROXIMATIONS 33 From (5.15), for t∈[−1,0], we have717 |vδ(t)|2≤e−(mλ1−4Cf)(t+1)Zt −12Cf|O| +|yδ(θsω)|2 λ1Cf +2Cf|yδ(θsω)|2 λ1 +2em2 mkφk2 −m 2kvδ(s)k2e(mλ1−4Cf)(s+1)ds +|vδ(−1)|2. 718 Therefore,719 |vδ(t)|2≤e−(mλ1−4Cf)(t+1)|vδ(−1)|2+2Cf|O| mλ1−4Cf +Zt −1 e−(mλ1−4Cf)(t−s)|yδ(θsω)|2 λ1Cf +2Cf|yδ(θsω)|2 λ1 +2em2 mkφk2ds, 720 and721 Z0 −1 e(mλ1−4Cf)skvδ(s)k2ds ≤2 me−(mλ1−4Cf)|vδ(−1)|2+4Cf|O| m(mλ1−4Cf) +2 mZ0 −1 e(mλ1−4Cf)s|yδ(θsω)|2 λ1Cf +2Cf|yδ(θsω)|2 λ1 +2em2 mkφk2ds. 722 Thus, we conclude for a given B(0, ρδ)⊂H, there exists T(ω, ρδ)≤ −1, such that for all t0≤T(ω, ρδ)723 and for all u0∈B(0, ρδ),724 |vδ(t)|2≤e−(mλ1−4Cf)(t+1)r2 3,δ(ω) + 2Cf|O| mλ1−4Cf +Zt −1 e−(mλ1−4Cf)(t−s)|yδ(θsω)|2 λ1Cf +2Cf|yδ(θsω)|2 λ1 +2em2 mkφk2ds, 725 and726 Z0 −1 e(mλ1−4Cf)skvδ(s)k2ds ≤2 me−(mλ1−4Cf)r2 3,δ(ω) + 4Cf|O| m(mλ1−4Cf) +2 mZ0 −1 e(mλ1−4Cf)s|yδ(θsω)|2 λ1Cf +2Cf|yδ(θsω)|2 λ1 +2em2 mkφk2ds. (5.16) 727 To obtain a bounded absorbing set in V, multiplying (5.11) by −∆vδ(t), making use of (1.2), (1.5), the728 Poincar´e and Young inequalities, we have729 d dtkvδ(t)k2≤ −(mλ1−4Cf)kvδ(t)k2+λ1Cf|O| +λ1Cf|vδ(t)|2 +Cfλ1+λ1 Cf|yδ(θtω)|2|φ|2+em2 m|∆φ|2. 730 Integrating the above inequality between sand 0, where s∈[−1,0], we have731 kvδ(0)k2≤e(mλ1−4Cf)skvδ(s)k2+Z0 s e(mλ1−4Cf)tλ1Cf|O| +λ1Cf|vδ(t)|2 + (Cfλ1+λ1C−1 f)|yδ(θtω)|2|φ|2+em2 m|∆φ|2dt. 732 This manuscript is for review purposes only. 34 J.H. XU, AND T. CARABALLO Integrating again the above inequality in [−1,0], together with the above inequality, it follows733 kvδ(0)k2≤2 me−(mλ1−4Cf)r2 3,δ(ω) + 4Cf|O| m(mλ1−4Cf)+2 mZ0 −1 e(mλ1−4Cf)s ×|yδ(θsω)|2 λ1Cf +2Cf|yδ(θsω)|2 λ1 +2em2 mkφk2ds +Z0 −1 e(mλ1−4Cf)t ×λ1Cf|O| +λ1Cf|v(t)|2+ (Cfλ1+λ1C−1 f)|yδ(θtω)|2|φ|2+em2 m|∆φ|2dt. 734 Consequently, there exists r4,δ(ω) satisfying, for a given ρδ>0, there exists T(ω, ρδ)≤ −1, such that for all t0≤T(ω, ρδ) and |u0,δ| ≤ ρδ, kuδ(0; t0, ω, u0,δ)k2=kvδ(0; t0, ω, u0,δ −φyδ(θt0ω)) + φyδ(ω)k2≤r2 4,δ(ω), where735 r2 4,δ(ω)=2kφk2|yδ(ω)|2+4m−1+ 2λ1Cfr2 3,δ(ω) + 8Cf|O| m(mλ1−4Cf)+4λ1C2 f|O| (mλ1−4Cf)2 +4m−1+ 2λ1CfZ0 −∞ e(mλ1−4Cf)s|yδ(θsω)|2 λ1Cf +2Cf|yδ(θsω)|2 λ1 +2em2 mkφk2ds + 2 Z0 −1 e(mλ1−4Cf)sλ1Cf|O| + (Cfλ1+λ1C−1 f)|yδ(θsω)|2|φ|2+em2 m|∆φ|2ds. 736 Thus, we conclude from [33, Theorem 1] that, there exists a unique random attractor Aδ(ω) to equation737 (5.3) with respect to deterministic bounded sets.738 Theorem 5.5. Let conditions of Theorem 5.3 hold. Then, for almost all ω∈Ω, we have lim δ→0Rδ(ω) = R0(ω), where R0(ω)and Rδ(ω)are given in theorems 5.3 and 5.4, respectively.739 Proof. From (5.6), we obtain740 (5.17) lim δ→0yδ(ω) = z∗(ω).741 On the one hand, (5.7) implies that there exist r < 0 and δ0>0, such that for all 0 < δ < δ0,742 (5.18) |yδ(θtω)|≤|t|,∀t≤r.743 Notice that,744 Z0 −∞ e(mλ1−4Cf)t|yδ(θtω)|2 λ1Cf +2Cf|yδ(θtω)|2 λ1||φk2dt =Zr −∞ e(mλ1−4Cf)t|yδ(θtω)|2 λ1Cf +2Cf|yδ(θtω)|2 λ1||φk2dt +Z0 r e(mλ1−4Cf)t|yδ(θtω)|2 λ1Cf +2Cf|yδ(θtω)|2 λ1||φk2dt. 745 Therefore, for all 0 < δ < δ0, it follows from (5.18) that746 Zr −∞ e(mλ1−4Cf)t|yδ(θtω)|2 λ1Cf +2Cf|yδ(θtω)|2 λ1||φk2dt ≤Zr −∞ e(mλ1−4Cf)t|t|2 λ1Cf +2Cf|t|2 λ1kφk2dt < ∞. 747 This manuscript is for review purposes only. STOCHASTIC NONLOCAL PDES AND WONG-ZAKAI APPROXIMATIONS 35 By means of the above inequality, (5.6) and dominated convergence theorem, we have748 lim δ→0Zr −∞ e(mλ1−4Cf)t|yδ(θtω)|2 λ1Cf +2Cf|yδ(θtω)|2 λ1||φk2dt =Zr −∞ e(mλ1−4Cf)t|z∗(θtω)|2 λ1Cf +2Cf|z∗(θtω)|2 λ1||φk2dt. (5.19) 749 On the other hand, by (5.6), the continuity of yδ(θtω) and the dominated convergence theorem, it follows750 lim δ→0Z0 r e(mλ1−4Cf)t|yδ(θtω)|2 λ1Cf +2Cf|yδ(θtω)|2 λ1||φk2dt =Z0 r e(mλ1−4Cf)t|z∗(θtω)|2 λ1Cf +2Cf|z∗(θtω)|2 λ1||φk2dt. (5.20) 751 By similar arguments to (5.20), it is easy to check752 lim δ→0Z0 −1 e(mλ1−4Cf)tCfλ1+λ1C−1 f|yδ(θtω)|2||φk2dt =Z0 −1 e(mλ1−4Cf)tCfλ1+λ1C−1 f|z∗(θtω)|kφk2dt. (5.21) 753 The conclusion of this theorem follows from (5.19)-(5.21). The proof is complete.754 Lemma 5.6. Under assumptions of Theorem 5.3, let {δn}∞ n=1 be a sequence satisfying δn→0as n→+∞. Let uδnand ube the solutions of (5.3)and (5.1)with initial values u0,δnand u0, respectively. If u0,δn→u0strongly in Has n→+∞, then for almost all ω∈Ωand t≥τ, uδn(t;τ, ω, u0,δn)→u(t;τ, ω, u0)strongly in Has n→+∞. Proof. The proof is similar to [16, Lemma 4.4] and we omit the details here.755 Lemma 5.7. Assume conditions of Theorem 5.3 hold, let {δn}∞ n=1 be a sequence so that δn→0as756 n→+∞. Let vδnand vbe the solutions of problems (5.11)and (5.2)with initial data v0,δnand v0,757 respectively. If v0,δn→v0weakly in Has n→+∞, then for almost all ω∈Ω,758 (5.22) vδn(r;τ, ω, v0,δn)→v(r;τ, ω, v0)weakly in H, ∀r≥τ,759 and760 (5.23) vδn(·;τ, ω, v0,δn)→v(·;τ, ω, v0)weakly in L2(τ, τ +T;V),∀T > 0.761 Proof. The results follow analogously to the proof of existence of solutions to problem (5.11) [15,762 Lemma 3.5]. We therefore omit the details.763 Lemma 5.8. Suppose conditions of Theorem 5.3 hold, let ω∈Ωbe fixed. If δn→0as n→+∞and764 uδn∈ Aδn(ω), then the sequence {uδn}∞ n=1 has a convergent subsequence in H.765 Proof. Since δn→0 as n→+∞, by Theorem 5.5, we obtain for almost all ω∈Ω, there exists766 N=N(ω), such that for all n≥N767 (5.24) Rδn(ω)≤2R0(ω).768 Thanks to un:= uδn(t;τ, ω, u0,δn)∈ Aδn(ω) and Aδn(ω)⊂Rδn(ω), hence for all n≥N, we have769 (5.25) |un|2≤2λ−1 1R0(ω).770 This manuscript is for review purposes only. 36 J.H. XU, AND T. CARABALLO In fact, (5.25) implies unis bounded in H, thus, up to a subsequence (relabeled the same), we have771 (5.26) un→˜uweakly in H.772 In what follows, we prove that the weak convergence in (5.26) is actually a strong one. On the one hand,773 since un∈ Aδn(ω), making use of the invariance of Aδn, for every k≥1, there exists un,k(ω)∈ Aδn(θ−kω)774 such that775 (5.27) un= Ψδn(k, θ−kω, un,k) = uδn(0; −k, ω, un,k).776 Since un,k ∈ Aδn(θ−kω) and Aδn(θ−kω)⊂Bδn(θ−kω), by (5.24), we infer that for each k≥1 and777 n≥N:= N(θ−kω),778 (5.28) |un,k|2≤2λ−1 1R0(θ−kω).779 On the other hand, by (5.10), we have780 (5.29) vδn(0; −k, ω, vn,k) = uδn(0; −k, ω, un,k)−φyδn(ω),781 where vn,k =un,k −φyδn(θ−kω). Therefore, (5.27) and (5.29) imply782 (5.30) un=vδn(0; −k, ω, vn,k) + φyδn(ω).783 By (5.28), we have784 (5.31) |vn,k|2≤2|un,k|2+ 2|φ|2|yδn(ω)|2≤4λ−1 1R0(θ−kω)+2|φ|2|yδn(ω)|2.785 It follows from (5.6) and (5.31) that there exists N1:= N1(ω, k) such that for every k≥1 and n≥N1,786 (5.32) |vn,k|2≤4λ−1 1R0(θ−kω)+4|φ|2(1 + |z∗(ω)|2).787 Notice that (5.6), (5.28) and (5.30) imply, as n→+∞,788 (5.33) vδn(0; −k, ω, vn,k)→˜vweakly in Hwith ˜v= ˜u−φz∗(ω).789 Next, using energy estimations, we evaluate the limit of norm |vδn(0; −k, ω, vn,k)|for each kas n→790 +∞. By (5.32) we know that for each k≥1, the sequence {vn,k}∞ n=1 is bounded in H, hence by a diagonal791 process, we can find a subsequence (relabeled the same) such that for each k≥1, there exists ¯vk∈Hsuch792 that793 (5.34) vn,k →¯vkweakly in Has n→+∞.794 Lemma 5.7 and (5.34) conclude, as n→+∞,795 (5.35) vδn(0; −k, ω, vn,k)→v(0; −k, ω, ¯vk) weakly in H,796 and797 (5.36) vδn(·;−k, ω, vn,k)→v(·;−k, ω, ¯vk) weakly in L2(τ, τ +T;V).798 By the uniqueness of limit, from (5.33) and (5.36), we obtain799 (5.37) v(0; −k, ω, ¯vk) = ˜v.800 By energy equality and (5.11), we have801 d dt|vδn(t)|2+ 2a(l(vδn) + yδn(θtω)l(φ))kvδn(t)k2= 2(f(vδn+φyδn(θtω)), vδn(t)) + 2yδn(θtω)(φ, vδn(t)) −2a(l(vδn) + yδn(θtω)l(φ))((φ, vδn)), 802 This manuscript is for review purposes only. STOCHASTIC NONLOCAL PDES AND WONG-ZAKAI APPROXIMATIONS 37 i.e.,803 d dt|vδn(t)|2+mλ1|vδn(t)|2+ Θ(vδn(t)) = 2(f(vδn+φyδn(θtω)), vδn(t)) + 2yδn(θtω)(φ, vδn(t)) −2a(l(vδn) + yδn(θtω)l(φ))((φ, vδn)), (5.38) 804 where Θ(vδn(t)) = 2a(l(vδn)+yδn(θtω)l(φ))kvδn(t)k2−mλ1|vδn(t)|2,which is a functional in V. Multiplying805 (5.38) by emλ1tand integrating it from −kto 0, we obtain806 |vδn(0; −k, ω, vn,k)|2=e−mλ1k|vn,k|2−Z0 −k emλ1tΘ(vδn(t;−k, ω, vn,k))dt + 2 Z0 −k emλ1t(f(vδn(t;−k, ω, vn,k) + φyδn(θtω)), vδn(t;−k, ω, vn,k))dt + 2 Z0 −k emλ1tyδn(θtω)(φ, vδn(t;−k, ω, vn,k))dt −2Z0 −k emλ1ta(l(vδn(t;−k, ω, vn,k)) + yδn(θtω)l(φ))((φ, vδn(t;−k, ω, vn,k)))dt. 807 Similarly, by (5.2),(5.33) and (5.37), we have808 |˜v|2:= |˜v(0; −k, ω, ¯vk)|2=e−mλ1k|¯vk|2−Z0 −k emλ1tΘ(v(t;−k, ω, ¯vk))dt + 2 Z0 −k emλ1t(f(v(t;−k, ω, ¯vk) + φz∗(θtω)), v(t;−k, ω, ¯vk))dt + 2 Z0 −k emλ1tz∗(θtω)(φ, v(t;−k, ω, ¯vk))dt −2Z0 −k emλ1ta(l(v(t;−k, ω, ¯vk)) + z∗(θtω)l(φ))((φ, v(t;−k, ω, ¯vk)))dt. (5.39) 809 It is obvious that810 lim sup n→∞ |vδn(0; −k, ω, vn,k)|2 ≤e−mλ1k4λ−1 1R0(θ−kω)+4|φ|21 + |z∗(ω)|2+|˜v|2−e−mλ1k|¯vk|2 ≤e−mλ1k4λ−1 1R0(θ−kω)+4|φ|21 + |z∗(ω)|2+|v(0; −k, ω, ¯vk)|2. (5.40) 811 Notice that, from (5.37) we know for n→+∞,812 (5.41) v(0; −k, ω, ¯vk) = ˜v=u(0; −k, ω, ¯uk)−φz∗(ω) := ˜u−φz∗(ω).813 By (5.30), we find814 (5.42) vδn(0; −k, ω, vn,k) = un−φyδn(ω).815 It follows from (5.40)-(5.42) that816 (5.43) lim sup n→∞ |un−φyn(ω)| ≤ e−mλ1k4λ−1 1R0(θ−kω)+4|φ|21 + |z∗(ω)|2+|˜u−φz∗(ω)|2.817 Since R0and z∗are tempered, we have818 lim sup k→∞ e−mλ1k4λ−1 1R0(θ−kω)+4|φ|21 + |z∗(ω)|2= 0.819 This manuscript is for review purposes only. 38 J.H. XU, AND T. CARABALLO Let k→+∞in (5.43), we obtain820 (5.44) lim sup n→∞ |un−φyn(ω)| ≤ |˜u−φz∗(ω)|.821 (5.26) and (5.6) lead us to822 un−φyn(ω)→˜u−φz∗(ω) weakly in H,823 together with (5.44), we have824 (5.45) un−φyn(ω)→˜u−φz∗(ω) strongly in H.825 Therefore, by (5.6), we conclude that un→˜ustrongly in H, as desired. This completes the proof.826 We are now ready to establish the upper semicontinuity of random attractors as δ→0.827 Theorem 5.9. Suppose that ais locally Lipschitz and fulfills (1.2),f∈C(R)satisfies (1.3)and (1.5) with p= 2 and β=Cf, respectively, φ∈V∩H2(O)and l∈L2(O). Also, let mλ1>4Cf. Then for almost all ω∈Ω, lim δ→0distH(Aδ(ω),A(ω)) = 0. Proof. For every fixed ω∈Ω, define828 ¯ B(ω) = u∈H:|u|2≤λ−1 12kφk2|z∗(ω)|2+8Cf|O| m(mλ1−4Cf)+4λ1C2 f|O| (mλ1−4Cf)2 +4+2λ1Cfm+mλ1−4Cf+ 2Cf|O| m(mλ1−4Cf) +4m−1+ 2λ1CfZ0 −∞ e(mλ1−4Cf)t|z∗(θtω)|2 λ1Cf +2Cf|z∗(θtω)|2 λ1 +2em2 mkφk2dt + 2 Z0 −1 e(mλ1−4Cf)tλ1Cf|O| +Cfλ1+λ1C−1 f|z∗(θtω)|2|φ|2+em2 m|∆φ|2dt. 829 By Theorem 5.3, we know ¯ B:= {¯ B(ω) : ω∈Ω}is also a DF-(pullback) random absorbing set for Ψ. Let Bδbe the DF-(pullback) random absorbing set of Ψδgiven by Theorem 5.4, it follows from Theorem 5.5 that lim δ→0|Bδ(ω)|=|¯ B(ω)|for almost all ω∈Ω. Which, together with Lemmas 5.6 and 5.8, completes the proof by applying [27, Theorem 3.1].830 Remark 5.10. Notice that, if for every ω∈Ω, the set Sδ∈(0,1] Aδ(ω) is precompact in H, the results831 of Lemma 5.8 hold true automatically [27]. Indeed, in our case, we define the absorbing set Bδ(ω) =832 {u∈H:|u| ≤ λ−1 1Rδ(ω)}(Theorem 5.4) for every δ∈(0,1], it is clear that the upper bound of Bδ(ω)833 is uniform with respect to δ. In fact, using the similar arguments as Theorem 5.5, with the help of the834 properties of yδ(θtω) (cf. (5.6)-(5.8)), it is enough to show that |Bδ(ω)| ≤ C(ω), where C(ω) is a positive835 constant which does not depend on δ. Therefore, we can replace the complicated proof of Lemma 5.6 by836 this conclusion to prove the upper semicontinuity of random attractors (cf. Theorem 5.9).837 This manuscript is for review purposes only. STOCHASTIC NONLOCAL PDES AND WONG-ZAKAI APPROXIMATIONS 39 6. Convergence of random attractors for stochastic nonlocal PDEs with multiplicative838 noise. We conclude our paper with studying the following stochastic nonlocal partial differential equations839 driven by colored noise,840      ∂uδ ∂t −a(l(uδ))∆uδ=f(uδ) + σuζδ(θtω), uδ= 0, uδ(x, τ) = u0,δ, in O × (τ, ∞), on ∂O × (τ, ∞), in O, (6.1)841 842 which is an approximation of the following one studied in [33],843      ∂u ∂t −a(l(u))∆u=f(u) + σu ◦dW dt , u= 0, u(x, τ) = u0, in O × (τ, ∞), on ∂O × (τ, ∞), in O, (6.2)844 845 where ◦denotes the Stratonovich sense in stochastic term. On account of the change of variable v(t) =846 e−σz∗(θtω)u(t), (6.2) can be written as,847 (6.3) dv dt −al(v)eσz∗(θtω)∆v=e−σz∗(θtω)f(veσz∗(θtω)) + vσz∗(θtω).848 Analogously, to study the pathwise dynamics of problem (6.1), we need to transform the stochastic equa-849 tions into random ones parameterized by ω∈Ω. Let850 (6.4) vδ(t) = uδ(t)e−σyδ(θtω).851 Then, (6.1) and (6.4) imply that852 (6.5) dvδ dt −al(vδ)eσyδ(θtω)∆vδ=e−σyδ(θtω)f(vδeσyδ(θtω)) + vδ(t)σyδ(θtω),853 with initial value v0,δ := vδ(τ) = u0e−σyδ(θτω).854 Proposition 6.1. Suppose assumptions (1.2)-(1.5)are true with p= 2 and β=Cf, respectively.855 Then, for almost all ω∈Ω, function a(ω, ·) = al(·)eσyδ(θtω)∈C(R;R+)is locally Lipschitz and satisfies856 (1.2). Furthermore, there exists a constant CF,δ depending on ω,σ,Cfand η, such that,857 |F(ω, s)| ≤ CF,δ(1 + |s|)and (F(ω, s)−F(ω, r))(s−r)≤η|s−r|2,∀s, r ∈R,858 where F(ω, s) = e−σyδ(ω)f(eσyδ(ω)s) + σyδ(ω)s.859 In what follows, we will use vδ(·;τ, ω, v0,δ) to denote the solution of equation (6.5). In a similar way as860 [33, Theorem 3], we deduce (6.5) has a unique weak solution in the sense of [33, Definition 7] which belongs861 to L2(τ, T;V)∩L∞(τ, T;H) for every T≥τ. At this point, thanks to the transformation (6.4), there862 exists a unique weak solution uδ(·;τ, ω, u0,δ)∈L2(τ, T ;V)∩L∞(τ, T;H) for every T≥τ. In addition,863 this solution behaves continuously in Hwith respect to the initial value.864 Define a mapping Σδ:R+×Ω×H→H, such that for every t∈R+, Σδ(t, ω, v0,δ) = vδ(t; 0, ω, v0,δ),∀v0,δ ∈H, ∀ω∈Ω. Thanks to the conjugation [33, Lemma 1], there is a mapping Φδ:R+×Ω×H→Hsuch that for all t∈R+, Φδ(t, ω, u0,δ) = uδ(t; 0, ω, u0,δ) := vδ(t; 0, ω, e−σyδ(ω)v0,δ)eσyδ(θtω),∀u0,δ ∈H, ∀ω∈Ω. Theorem 6.2. ([33, Theorem 5])Assume that function a∈C(R;R+)fulfills (1.2), function fsatisfies (1.3)and (1.5)with p= 2 and β=Cf, respectively, l∈L2(O). Also, let mλ1>3Cf. Then there exists a This manuscript is for review purposes only. 40 J.H. XU, AND T. CARABALLO unique random attractor A(ω)for the dynamical system Φ(t, ω, u)associated to problem (6.2). Additionally, this DF-pullback absorbing set B0:= {B0(ω) : ω∈Ω}in His given by B0(ω) = {u∈H:|u|2≤λ−1 1R0(ω)}, with865 R0(ω) = 1 meR0 −12σz∗(θsω)ds+2σz∗(ω) ×1 + Cf|O| Z−1 −∞ e−2σz∗(θsω)+(mλ1−3Cf)s+R−1 s2σz∗(θτω)dτ ds +1 mCf|O| +2 mC2 f|O|Z0 −1 e−2σz∗(θsω)+(mλ1−3Cf)s+2σz∗(ω)+R0 s2σz∗(θrω)drds. 866 Theorem 6.3. Under assumptions of Theorem 6.2, there exists δ0>0such that for all 0< δ < δ0, equation (6.1)generates a random dynamical system Φδ(t, ω, u0,δ), which possesses a unique random attractor Aδ(ω). Additionally, the DF-pullback absorbing set Bδ:= {Bδ(ω) : ω∈Ω}in His given by Bδ(ω) = {u∈H:|u|2≤λ−1 1Rδ(ω)}, with867 Rδ(ω) = 1 meR0 −12σyδ(θsω)ds+2σyδ(ω) ×1 + Cf|O| Z−1 −∞ e−2σyδ(θsω)+(mλ1−3Cf)s+R−1 s2σyδ(θτω)dτ ds +1 mCf|O| +2 mC2 f|O|Z0 −1 e−2σyδ(θsω)+(mλ1−3Cf)s+2σyδ(ω)+R0 s2σyδ(θrω)drds. 868 Proof. The same method as [33, Theorem 5] will be used to prove this result. We first derive the869 boundedness of vδ(·) := vδ(·;t0, ω, v0,δ) in Hfor all t∈[t0,−1] with t0≤ −1, where v0,δ =e−σyδ(θt0ω)u0 870 and u0∈D(a deterministic bounded set). Firstly, multiplying (6.5) by vδin H, thanks to (1.5) and the871 Young inequality, we have872 1 2 d dt|vδ(t)|2+a(eσyδ(θtω)l(vδ))kvδ(t)k2 ≤1 2e−2σyδ(θtω)Cf|O| +3Cf 2+σyδ(θtω)|vδ(t)|2, 873 thanks to the Poincar´e inequality and (1.2), we have874 (6.6) d dt|vδ(t)|2+mkvδ(t)k2≤(−mλ1+ 3Cf+ 2σyδ(θtω))|vδ(t)|2+e−2σyδ(θtω)Cf|O|.875 Integrating (6.6) between t0and −1, it follows876 |vδ(−1)|2≤e(mλ1−3Cf)e(mλ1−3Cf)t0+R−1 t02σyδ(θsω)ds|vδ(t0)|2 +Cf|O| Z−1 t0 e−2σyδ(θsω)e(mλ1−3Cf)s+R−1 s2σyδ(θτω)dτ ds. 877 Consequently, for a given deterministic bounded set D⊂H, there exist a constant ρδ>0 and T(ω, ρδ)≤ −1, P-a.e., such that, for any u0,δ ∈D⊂B(0, ρδ), for all t0≤T(ω, ρδ), we have vδ−1; t0, ω, e−σyδ(θt0ω)u0,δ 2≤r2 1,δ(ω), This manuscript is for review purposes only. STOCHASTIC NONLOCAL PDES AND WONG-ZAKAI APPROXIMATIONS 41 with878 r2 1,δ(ω) = e(mλ1−3Cf)1 + Cf|O| Z−1 −∞ e−2σyδ(θsω)+(mλ1−3Cf)s+R−1 s2σyδ(θτω)dτ ds.879 Secondly, we show v∈L∞(−1, t;H)∩L2(−1, t;V) with t∈[−1,0] by energy estimations. Integrating880 (6.6) from −1 to twith t∈[−1,0], we obtain881 |vδ(t)|2≤e−(mλ1−3Cf)(t+1)+Rt −12σyδ(θsω)ds|vδ(−1)|2 +Cf|O| Zt −1 e−2σyδ(θsω)+(3Cf−mλ1)(t−s)+Rt s2σyδ(θτω)dτ ds −mZt −1 e(3Cf−mλ1)(t−s)+Rt s2σyδ(θτω)dτ kvδ(s)k2ds. (6.7) 882 Therefore, by similar arguments, we conclude that for a given deterministic subset D⊂B(0, ρδ)⊂H,883 there exists T(ω, ρδ)≤ −1, P-a.e., such that for all t0≤T(ω, ρδ), for all u0,δ ∈D, we have884 |vδ(t)|2≤e−(mλ1−3Cf)(t+1)+Rt −12σyδ(θsω)dsr2 1,δ(ω) +Cf|O| Zt −1 e−2σyδ(θsω)+(3Cf−mλ1)(t−s)+Rt s2σyδ(θτω)dτ ds, 885 and886 Z0 −1 e(mλ1−3Cf)s+R0 s2σyδ(θτω)dτ kvδ(s)k2ds ≤1 me−(mλ1−3Cf)+R0 −12σyδ(θsω)dsr2 1,δ(ω) +Cf|O| mZ0 −1 e−2σyδ(θsω)+(mλ1−3Cf)s+R0 s2σyδ(θτω)dτ ds. (6.8) 887 Thirdly, the boundedness of vδ(·) in Vfor all t∈[−1,0] and the compact embedding V ,→Hensure the888 existence of a compact absorbing ball in H.To obtain a bound in V, we first need to ensure the existence889 of strong solutions, by slightly improving the regularity of initial value, namely, u0,δ ∈V, but assumptions890 imposed on functions aand fare the same, this result holds, for more details, see [32, Theorem 2.9].891 Multiplying (6.5) by −∆vδ(t), with the help of (1.3) and the Young inequality, we derive892 1 2 d dtkvδ(t)k2+a(eσyδ(θtω)l(vδ))| − ∆vδ(t)|2 ≤1 me−2σyδ(θtω)C2 f|O| +C2 f m|vδ(t)|2+m 2|∆v(t)|2+σyδ(θtω)kv(t)k2. (6.9) 893 Using the Poincar´e inequality, (6.9) can be bounded by894 d dtkvδ(t)k2≤ −m|∆vδ(t)|2+2 mC2 f|O|e−2σyδ(θtω)+2C2 f m|v(t)|2+ 2σyδ(θtω)kvδ(t)k2 ≤ −mλ1+2C2 f mλ1 + 2σyδ(θtω)!kvδ(t)k2+2 mC2 f|O|e−2σyδ(θtω). (6.10) 895 Integrating (6.10) between sand 0 with s∈[−1,0], we obtain896 kvδ(0)k2≤e(mλ1−2C2 f/mλ1)s+R0 s2σyδ(θτω)dτ kvδ(s)k2 +2 mC2 f|O| Z0 s e−2σyδ(θτω)+(mλ1−2C2 f/mλ1)τ+R0 τ2σyδ(θtω)dtdτ. 897 This manuscript is for review purposes only.