Random Attractors for Stochastic Retarded Reaction-Diffusion Equations on Unbounded Domains

and Applied Analysis 3 where d is the Hausdorff semimetric given by d(E, F) = sup x∈E infy∈F‖x − y‖X for any E ⊆ X and F ⊆ X. We remark that if A ∈ D, then this attractor is unique [33]. Definition 9. φ is said to be D-pullback asymptotically compact in X if, for all B ∈ D and P a.e. ω ∈ Ω, {φ(tn, θ−t n ω, xn)} ∞ n=1 has a convergent subsequence in X whenever tn → ∞, and xn ∈ B(θ−t n ω). The following existence result on a random attractor for a continuous random dynamical system can be found in [2, 34]. First, recall that a collectionD of random subsets ofX is called inclusion closed if whenever E is an arbitrary random set and F is inD with E(ω) ⊂ F(ω) for P a.e. ω ∈ Ω, then E must belong toD. Proposition 10. Let D be an inclusion-closed collection of random subsets of X and φ a continuous random dynamical system on X over (Ω,F,P, (θt)t∈R). Suppose that K ∈ D is a closed random absorbing set for φ in D and φ is D-pullback asymptotically compact inX. Then φ has a uniqueD-random attractorA which is given by A (ω) = ⋂ τ≥0 ⋃ t≥τ φ (t, θ−tω,K (θ−t)). (9) In this paper, we will take D as the collection of all tempered random subsets of C and prove the stochastic retarded reaction-diffusion equation on R has aD-random attractor. 3. Stochastic Retarded Reaction-Diffusion Equations on R with Additive Noise In this section, we show that there is a continuous random dynamical system generated by the stochastic retarded reaction-diffusion equation on R with additive white noise: du + (λu − Δu) dt = (f (u t ) (x) + g (x)) dt + dW, x ∈ R d , t > 0, (10) with the initial condition u (t, x) = u 0 (t, x) , x ∈ R d , t ∈ [−], 0] . (11) Here λ is a positive constant, g is a given function in L(R), W is an L(R)-valued two-sided Wiener process with a symmetric nonnegative finite trace covariance operator Q defined on a probability space which will be specified below, and f : C → L(R) is a continuous mapping satisfying the following conditions: (A1) f(0) = 0; (A2) there exists a positive continuous function lf(r) with lim r→∞ lf (r) r0 = 0 (12) for some positive integer k0 such that, for all ξ, η ∈ C with ‖ξ‖ ≤ r and ‖η‖ ≤ r, 󵄩󵄩󵄩󵄩f (ξ) − f (η) 󵄩󵄩󵄩󵄩 ≤ lf (r) 󵄩󵄩󵄩󵄩ξ − η 󵄩󵄩󵄩󵄩C; (13) (A3) there exist positive constants α0 and cf such that, for all α ∈ (0, α0), t > 0, u ∈ C([−], t]; L 2 (R)), and x ∈ R,


Introduction
The study of stochastic functional differential equations is motivated by the fact that, when one wants to model some evolution phenomena arising in physics, chemistry, biology, and other sciences, some hereditary characteristics such as aftereffect, time-lag, and time delay can appear in the variables.On the other hand, one of the most interesting problems concerning stochastic functional differential equations is to understood the asymptotic behavior of the solutions when time grows to infinite, since it can provide useful information about the future of the phenomenon described in the model.
In this paper, we investigate the asymptotic behavior of solutions to the following stochastic retarded reaction-diffusion equation with additive noise defined in the entire space R  :  + ( − Δ)  = ( (  ) () +  ())  + , (1) where  is a positive constant,  is a given function defined on R  ,  is a nonlinear functional satisfying certain conditions, and  is a two-sided infinite dimensional Wiener process on a probability space which will be specified later.
We note that the asymptotic behavior of several deterministic retarded PDEs on bounded domains was studied in [21][22][23][24][25], and the case of retarded Navier-Stokes equations on some unbounded domains was treated in [26].The random attractor for retarded stochastic differential equations was considered in [27] by monotone methods.Recently, in the case of stochastic retarded lattice dynamical systems defined on the entire integer set, the existence of a random attractor was proved in [28,29].Here we prove the existence of a random attractor for the stochastic retarded reactiondiffusion equation defined in R  .It is worth mentioning that the asymptotic behavior of the nonretarded version of (1) was investigated recently in [9].
Notice that Sobolev embeddings are not compact when domains are unbounded.This introduces a major obstacle for proving existence of attractors for PDEs on unbounded domains.Under certain circumstances, the tail-estimates method can be used to deal with the problem caused by the unboundedness of domains.This approach was developed in [30,31] for deterministic nonretarded PDEs and used in [9-11, 13, 17, 18] for stochastic systems.At the same time, the present of delays is another obstacle, which makes phase spaces not reflexive and increases the difficulty of uniform estimates.In this paper, we will develop a tail-estimates approach for stochastic retarded PDEs on unbounded domains and prove the existence of a compact random attractor for the stochastic retarded reaction-diffusion equation (1), in particular, defined on the unbounded domain R  .The idea is based on the observation that the solutions of the equation are uniformly small when space and time variables are sufficiently large.It is clear that our method can be used for a variety of other equations, as it was for the nonretarded case.
The rest of the paper is organized as follows.In the next section, we introduce basic concepts concerning random dynamical systems and random attractors.In Section 3, we define a continuous random dynamical system for the stochastic retarded reaction-diffusion equation on R  .The existence of the random attractor is given in Section 4.

Preliminaries
In this section, we introduce some basic concepts related to random attractors for random dynamical systems.The reader is referred to [1,2,6,[32][33][34] for more details.
A random set  is called tempered if () is contained in a ball with center zero and tempered radius () for all  ∈ Ω.
Hereafter, we always assume that  is a continuous random dynamical system over (Ω, F, P, (  ) ∈R ), and D is a collection of random subsets of .Definition 7. A random set  is called a random absorbing set in D if, for every  ∈ D and P a.e. ∈ Ω, there exists Definition 8.A random set A is called a D-random attractor (D-pullback attractor) for  if the following hold: (i) A is a random compact set; (ii) A is strictly invariant; that is, for P a.e. ∈ Ω and all  ≥ 0,  (, , A ()) = A (  ) ; (iii) A attracts all sets in D; that is, for all  ∈ D and P a.e. ∈ Ω, where  is the Hausdorff semimetric given by (, ) = sup ∈ inf ∈ ‖ − ‖  for any  ⊆  and  ⊆ .
The following existence result on a random attractor for a continuous random dynamical system can be found in [2,34].First, recall that a collection D of random subsets of  is called inclusion closed if whenever  is an arbitrary random set and  is in D with () ⊂ () for P a.e. ∈ Ω, then  must belong to D. Proposition 10.Let D be an inclusion-closed collection of random subsets of  and  a continuous random dynamical system on  over (Ω, F, P, (  ) ∈R ).Suppose that  ∈ D is a closed random absorbing set for  in D and  is D-pullback asymptotically compact in .Then  has a unique D-random attractor A which is given by In this paper, we will take D as the collection of all tempered random subsets of C and prove the stochastic retarded reaction-diffusion equation on R  has a D-random attractor.

Stochastic Retarded Reaction-Diffusion
Equations on R  with Additive Noise In this section, we show that there is a continuous random dynamical system generated by the stochastic retarded reaction-diffusion equation on R  with additive white noise: with the initial condition Here  is a positive constant,  is a given function in  2 (R  ),  is an  2 (R  )-valued two-sided Wiener process with a symmetric nonnegative finite trace covariance operator  defined on a probability space which will be specified below, and  : C →  2 (R  ) is a continuous mapping satisfying the following conditions: In the sequel, we consider the probability space (Ω, F, P) where F is the Borel -algebra induced by the compact-open topology of Ω, and P the corresponding Wiener measure on (Ω, F) with respect to the covariance operator .Let Then (Ω, F, P, (  ) ∈R ) is an ergodic metric dynamical system.Since the above probability space is canonical, we have Similar to Proposition A.1 in [34], we can find that there exists a full P-measure {  } ∈R -invariant set Ω ∈ F such that for each  ∈ Ω Let F be the P-completion of F, and let with where {( 2 ) − ( 1 ) :  ≤  1 ≤  2 ≤ } is the smallest -algebra generated by the random variable ( 2 ) − ( 1 ) for all  1 ,  2 such that  ≤  1 ≤  2 ≤  and N is the collection of P-null sets of F. Note that so (Ω, F, P, (  ) ∈R , (F   ) ≤ ) is a filtered metric dynamical system (see [32, pages 72 and 91] for more details).In addition, it is important to note that the measurability of  is not true if we replace F by its completion; see [32, page 547] for details.
In this paper, the solution of problem ( 10)-( 11) is interpreted in a mild sense: P a.s.for any  0 ∈ C, where () is the analytic semigroup on  2 (R  ) generated by Δ − .By the theory in [35], we deal with (22) on the complete probability space (Ω, F, P).
We now associate a continuous random dynamical system with the stochastic retarded reaction-diffusion equation ( 10)-( 11) over (Ω, F, P, (  ) ∈R ).To this end, we introduce an auxiliary Ornstein-Uhlenbeck process on (Ω, F, P, (  ) ∈R ) and transform the stochastic retarded reaction-diffusion equation into a random one.Let Then by ( 18), ( 23) is well defined.The process (),  ∈ R, is a stationary, Gaussian process.By Lemma 5.13 in [35], we can see that it is a mild solution of the linear equation That is, for all  ∈ R and P a.s.
(1) By (A 1 )-(A 2 ), following the same lines of Theorem 6.1.4 in [36], one can show that, for each V 0 ∈ C, there exists a  max ≤ ∞ such that (26) has a unique solution V on [0,  max ).Moreover, if  max < ∞, then lim sup We prove now that this local solution is a global one.For fixed  ∈ (0,  max ), by regularity of mild solutions for an analytic semigroup [37, page 145], we inform that for any  ∈ (0, ), and ( 27) holds for a.e. ∈ [0, ].Then, taking the inner product of ( 27) with V in  2 (R  ), we get that By (A 4 ), we can choose  > 0 small enough such that 2 > 2  + .Using the Young inequality, we find that Then it follows from ( 34) and ( 35) that Choose  ∈ (0,  0 ) small enough such that 2 > 2  +  + .
Proof.By a classical successive approximation argument, one can easily show that, for fixed V 0 ∈ C, V(, , V 0 ) is an F  -adapted continuous process.Hence, for fixed V 0 ∈ C, (, , V 0 ) is also an F  -adapted continuous process.On the other hand, from property (2) of Theorem 11, it follows that, for fixed  ≥ 0 and  ∈ Ω, (, , ⋅ By (26) we have that, for ,  ≥ 0 and  ∈ Then again by (26) we get For each  ∈ Ω consider Then for  =  +  we have It follows from (56) that for all  ∈ [−], 0].By the uniqueness of the solution of ( 26) we find that while (58) implies Therefore,  is a continuous random dynamical system.

Existence of Random Attractors
In this section, we prove the existence of a D-random attractor for the random dynamical system  associated with the stochastic retarded reaction-diffusion equation ( 10)-( 11) on R  .We first establish the existence of a D-random attractor for its conjugated random dynamical system , then the existence of a D-random attractor for  follows from the conjugation relation between  and .To this end, we will derive uniform estimates on the mild solutions of problem ( 27)-( 28) when  → ∞ with the purpose of proving the existence of a bounded random absorbing set and the asymptotic compactness for .In particular, we will show that the tails of the solutions, that is, solutions evaluated at large value of ||, are uniformly small when time is sufficiently large.
From now on, we always assume that D is the collection of all tempered subsets of C with respect to (Ω, F, P, (  ) ∈R ).The next lemma shows that  has a random absorbing set in D.
Lemma 16.Let  ∈ D and V 0 () ∈ ().Then for every  > 0 and P a.e. ∈ Ω, there exist  * =  * (, , ) > 0 and  * =  * (, ) > 0 such that the solution V(, , V 0 ()) of problem ( 27)-( 28) satisfies, for all  ≥  * , sup Proof.Let  be a smooth function defined on R + such that 0 ≤ () ≤ 1 for all  ≥ 0, and Then there exists a positive deterministic constant  2 such that |  ()| ≤  2 for all  ≥ 0. Taking the inner product of (27) with Abstract and Applied Analysis 11 We now estimate the terms in (100).First, we have that Note that the second term on the right-hand side of ( 101) is bounded by By ( 101) and (102), we find that For the right-hand side of (100), applying the Young inequality, we obtain that Then it follows from (100), (103), and (104) that Consequently, Take over the interval [ 1 , ] leads to Using the Young inequality and (A 3 ), we obtain that By ( 107) and (108), we find that Then we have, for all  >  1 , If we take  ≥  1 + ], then by (110) we find that, for all  ∈ Then we have, for all  ≥  1 + ], sup Replacing  with  − , we find that sup Since () and ‖ 0 ()‖
We are now in a position to present our main result about the existence of a D-random attractor for  in C. Since  and  are conjugated by the random homeomorphism (, ) = + 0 () and  0 () ∈ C is tempered, then, by Proposition 1.8.3 in [33],  has a unique D-random attractor {A 2 ()} ∈Ω in C which is given by A 2 () = { () +  0 () :  () ∈ A 1 ()} . ( The proof is complete.

Theorem 19 .
The random dynamical system  has a unique D-random attractor in C. Proof.Notice that  has a closed absorbing set  in D by Lemma 13 and is D-pullback asymptotically compact in C by Lemma 18.Hence, the existence of a unique D-random attractor {A 1 ()} ∈Ω for  follows from Proposition 10 immediately.
It is said that the random set is bounded (resp., closed or compact) if () is bounded (resp., closed or compact) for P a.e. ∈ Ω.