Existence of exponentially attracting stationary solutions for delay evolution equations

Manuscript submitted to Website: http://AIMsciences.org AIMS’ Journals Volume X, Number 0X, XX 200X pp. X–XX EXISTENCE OF EXPONENTIALLY ATTRACTING STATIONARY SOLUTIONS FOR DELAY EVOLUTION EQUATIONS T. Caraballo1, M.J. Garrido-Atienza1, & B. Schmalfuß2 1Departamento de Ecuaciones Diferenciales y An´alisis Num´erico, Universidad de Sevilla, Apdo. de Correos 1160, 41080–Sevilla, Spain 2Institut f¨ur Mathematik Fakult¨at EIM, Universit¨at Paderborn, Warburger Strasse 100, 33098 Paderborn, Germany (Communicated by Aim Sciences) Abstract. We consider the exponential stability of semilinear stochastic evolution equations with delays when zero is not a solution for these equations. We prove the existence of a non-trivial stationary solution exponentially stable, for which we use a general random fixed point theorem for general cocycles. We also construct stationary solutions with the stronger property of attracting bounded sets uniformly, by means of the theory of random dynamical systems and their conjugation properties. 1. Introduction. The asymptotic behaviour of stochastic partial differential equations is an important task which has been receiving much attention during the last decades. In particular, stochastic evolution equations containing some sort of delay or retarded argument have also been extensively studied due to their importance in applications (see, for example, [2], [4], [5], [11], [15], [24] and [25]). However, even in the non-delay framework, most results in the literature are concerned with the exponential stability of constant stationary solutions, mainly the trivial one (see [12] and [18] in the finite dimensional context, and [7], [16], [13], [14] among others in the infinite dimensional framework). In the recent work [6], the asymptotic behaviour of semilinear stochastic partial differential equations has been analysed, focusing on the exponential stability of their non-constant stationary solutions in mean square and pathwise. Our aim in this work is to prove analogous results in the case in which the non-linear term can eventually contain some hereditary features. Although it may be possible to develop a parallel analysis to that one in [6], but with necessary modifications due to the different nature of the problem, we will use a different technique in this paper. In fact, our results will be deduced as consequences of a general fixed point 2000 Mathematics Subject Classification. Primary: 60H15, 35K40; Key words and phrases. cocycles, random dynamical systems, stationary solutions, delay equations, exponential stability. Partially supported by Junta de Andaluc´ıa Project FQM314, and by Ministerio de Educaci´on y Ciencia (Spain) and FEDER (European Community) under the projects MTM2005-01412, HA2005-0082, and by Deutscher Akademischer Austauschdienst DAAD D/05/25674. 1 2 T. CARABALLO, M.J. GARRIDO-ATIENZA & B. SCHMALFUSS theorem for general cocycles. In this way, by simply choosing appropriate phase spaces we can deduce both types of stability (in mean square and pathwise) under a unified treatment. These generalized fixed points will provide stationary solutions of our problem having different properties, depending on the chosen setup. In the first case, the stationary solution will be exponentially attracting in mean square, while in the second it will be pathwise exponentially attracting. The content of the paper is as follows. In Section 2 we state a general theorem ensuring the existence of generalized fixed points for cocycles. In Section 3, a semilinear stochastic evolution equation with finite delay is considered. First, we construct a cocycle associated to our model and prove the existence of a mean square exponentially attracting solution. In addition, this stationary solution is proved to attract with probability one. However, the exceptional set does depend on the initial value. To overcome this disadvantage we exploit the tools from the theory of random dynamical systems and construct another cocycle which possesses a random fixed point exponentially attracting for every path. To do this, we consider a more specific noisy term in our equation since it is not always possible to generate a random dynamical system from general stochastic PDEs. 2. A generalized fixed point theorem for cocycles. In this section we will establish a theorem which ensures the existence and uniqueness of generalized fixed points for nonautonomous and random dynamical systems. The main property describing the dynamics of such a system is termed cocycle property, which is a generalization of the semigroup property for autonomous systems. Let X= (X(r),k·kr) for r∈Rbe a family of Banach spaces. We say that ψ=ψ(·,·,·) is a cocycle on Xif, for any t∈R+and r∈R,the mapping ψ(t, r, ·) : X(r)→X(r+t) satisfies ψ(0, r, ·) = idX(r),(1) ψ(t+s, r, ·) = ψ(t, r +s, ψ(s, r, ·)). Indeed, deleting all the r’s in the above formula one gets the well known semigroup property. A family ψ∗= (ψ∗(r))r∈R,where ψ∗(r)∈X(r) for all r∈R, is called a generalized fixed point for the cocycle ψif ψ(t, r, ψ∗(r)) = ψ∗(t+r),for all t≥0 and r∈R. Let κbe a positive constant. A family ξ= (ξ(r))r∈R, ξ(r)∈X(r) is called κ-growing if lim r→−∞ kξ(r)kre(κ−ε)r= 0 for every ε > 0. The following theorem formulates sufficient conditions for the existence of a pullback attracting generalized fixed point for a cocycle. Theorem 1. We consider the family of Banach spaces X= (X(r),k · kr)r∈R. Let ψbe a cocycle defined on Xand Pbe some non-empty subset of κ-growing families on Xfor a κ > 0. In addition, (A1) Suppose that there exists a mapping K:R+×R→R+ such that kψ(t, r, x)−ψ(t, r, y)k2 r+t≤K(t, r)kx−yk2 rfor all x, y ∈X(r), STATIONARY SOLUTIONS FOR DELAY SPDE 3 and for every ε > 0 lim t→∞ K(t, r)e(κ−ε)t= 0,lim t→∞ K(t, r −t)e(κ−ε)t= 0. (A2) Suppose that (ψ(N, r −N, ξ(r−N)))N∈Nis a Cauchy sequence in X(r)for every r∈Rand ξin P. Then the limit family is contained in P. (A3) We have that for x∈ P, r ∈Rand for every ε > 0 sup q∈[0,1] kψ(q, r −q−t, x(r−q−t)) −x(r−t)kr−t=o(e(κ−ε)t) for t→ ∞. Then there exists a generalized fixed point ψ∗∈ P for ψwhich is the unique one (in P). Moreover, for all families ξ∈ P the following convergence lim t→∞ kψ(t, r −t, ξ(r−t)) −ψ∗(r)kr= 0 (pullback convergence) holds exponentially fast for all r∈R. In addition, we have lim t→∞ kψ(t, r, ξ(r)) −ψ∗(r+t)kr+t= 0 for all families ξ∈ P exponentially fast. Proof. This theorem is a version of similar results in Duan et al. [10], Schmalfuß [21], [23]. Although we do not prefer to include the complete proof, we would like to mention that the idea is to show that, for every ξ∈ P, the Cauchy sequence {ψ(N, r −N, ξ(r−N))}N∈Nhas a limit in X(r) where its limit does not depend on the family ξ. This follows from (A1) and (A2). According to (A3) (ψ(t, r−t, ξ(r−t))) has the same limit for t→ ∞ as the above sequence which causes the existence of a generalized fixed point. 3. Mean square exponentially attracting stationary solutions for a delay model. In this section we deal with the concept of exponentially attracting stationary solutions for a kind of delay stochastic non-linear evolution equation generated by random fixed points. 3.1. Preliminaries on a semilinear stochastic evolution equation with delays. We start to describe the noise driving the differential equation. Let (Ω,F,{Ft}t∈R,P) be a filtered probability space such that Fs⊂ Ft⊂ F for s≤t. In what follows, we consider a two-sided Wiener process Wtaking values in some separable Hilbert space Uwith covariance Qbeing a trace class symmetric operator on U. For instance, as it is usual in the literature, the previous probability space (Ω,F,P) is to be chosen as the set of continuous paths C0(R, U) which are zero at zero equipped with the compact open topology. Fis supposed to be the associated Borel σ-algebra and Pis the Wiener measure with respect to the covariance Q. We also set Ft=σ{ω(u)−ω(v) : v, u ≤t}. Note that the above probability space is not complete. Its completion is denoted by (Ω,¯ F,{¯ Ft}t∈R,P) where {¯ Ft}t∈Ris a normal filtration, see Da Prato and Zabczyk [8], Page 75. For the extension of Pto ¯ Fwe choose the same notation P. We now introduce a measurable flow θ={θt}t∈Ron the above non-completed probability space (Ω,F,P): θ: (R×Ω,F ⊗ B(R)) →(Ω,F), θt+τ=θt◦θτ, θ0= idΩ,(2) 4 T. CARABALLO, M.J. GARRIDO-ATIENZA & B. SCHMALFUSS where by means of B(S) we denote the Borel σ-algebra of open subsets in the topological space S. The Wiener shift operators which form the flow θ θtω(·) = ω(·+t)−ω(t), t ∈R, ω ∈Ω leave the Wiener measure Pinvariant. More precisely, Pis ergodic with respect to θand, in addition, we have that θ−1 uFt=Ft+u(3) for any t, u ∈R, see Arnold [1], Page 72. In particular, we mention that the quadruple (Ω,F,P, θ) is called a metric dynamical system. As this probability space is canonical, we have for a Wiener process Wand its shift operators W(t, ω) = ω(t), W (t, θsω) = ω(t+s)−ω(s) = W(t+s, ω)−W(s, ω). Given two real numbers a < b and a separable Banach space H, we denote by I2(a, b;H) the closed subspace of L2(Ω ×[a, b],¯ F ⊗ B ([a, b]) ,dP⊗dt;H) of all stochastic processes (t, ω)→X(t, ω)∈ H such that X(t) is ¯ Ft-adapted. We denote by L2 s(Ω; C(a, b;H)) the space of processes u∈L2(Ω,¯ F,dP;C(a, b;H)) such that u(t) is ¯ Ft+s-measurable for each tin [a, b], where s∈Rand C(a, b;H) denotes the space of all continuous functions from [a, b] into Hequipped with supremum norm, which will be denoted by || · ||C(a,b;H).|| · ||L2is defined to be the norm in the spaces L2 s(Ω; C(a, b;H)) for all s∈R. We also write L2(Ω; C(a, b;H)) instead of L2 0(Ω; C(a, b;H)). Let us fix a h > 0 and consider T > 0.For brevity we denote CH=C(−h, 0; H).If we have a function u∈C(−h, T;H),for each t∈[0, T ] we denote by ut∈CHthe function defined by ut(s) = u(t+s),−h≤s≤0.Moreover, if y∈L2(−h, T;H) we also denote by yt∈L2(−h, 0; H),for almost every (in the sequel, a.e.) t∈(0, T), the function defined by yt(s) = y(t+s),a.e. s∈(−h, 0). For the following let Wbe the Wiener process on the probability space (Ω,¯ F,{¯ Ft}t∈R,P) introduced above. In addition, assume that there exists a Gelfand triplet V⊂H⊂ V0of separable Hilbert spaces, where V0denotes the dual of V. We denote by |·| ,k · kVthe norms in Hand Vrespectively. The inner product in Hwill be denoted by (·,·), and the duality mapping between V0and Vby h·,·i. We will study the qualitative behaviour of the following delay stochastic evolution equations      du =Audt +f(ut)dt +B(u)dW, t ≥0, u(t) = ξ(t), t ∈[−h, 0], u∈I2(0, T;V)∩L2(Ω; C(−h, T;H)),for all T > 0. (P) We mention that the equation in (P) has to be satisfied in V0so that we consider our solutions from a variational point of view. The random variable ξis supposed to be ( ¯ F0,B(CH))–measurable. Let us denote by a1>0 the constant of the injection V⊂H, i.e. a1|u|2≤ kuk2 V,for v∈V, and let −A:V→V0be a positive, linear and continuous operator for which there exists a constant a2<0 such that h−Au, ui ≥ −a2kuk2 V,for all u∈V. STATIONARY SOLUTIONS FOR DELAY SPDE 5 Therefore, −Agenerates a norm in Vwhich is equivalent to the previous one. In the sequel, we suppose that k·kVdenotes this norm in V, which is then defined by kuk2 V=h−Au, ui,for all u∈V. Moreover, it is also well-known (see, for instance, Dautray and Lions [9]) that Ais the generator of a strongly continuous semigroup {S(t)}t≥0in Hsatisfying kS(t)kL(H)≤eat, where a=a1a2<0. Also, suppose that f:CH→His a mapping satisfying the following properties: (f.1) there exists a constant Cf>0 such that for all ξ, e ξ∈CH |f(ξ)−f(e ξ)| ≤ Cf||ξ−e ξ||CH, (f.2) for any positive continuous function ρover (−h, +∞) there exists a constant Kf=Kf(ρ, h)≥0 such that for all u, eu∈C(−h, +∞;H),and all t≥0 Zt 0 ρ(s)|f(us)−f(eus)|2ds≤K2 fZt −h ρ(s)|u(s)−eu(s)|2ds. Assumption (f.2) is often assumed in the context of delay partial differential equations. See [6], [3], [4], [17] for some particular examples. Finally, B:H→ LQ 2(U, H) is supposed to be Lipschitz continuous with respect to the Hilbert-Schmidt norm LQ 2(U, H) of linear operators from Uto H(see Da Prato and Zabczyk [8] Chapter 4): trH((B(u)−B(v))Q(B(u)−B(v))∗) := kB(u)−B(v)k2 LQ 2 ≤L2 B|u−v|2 for u, v ∈H. We now recall the following theorem on the existence, uniqueness and regularity of solutions to (P) (cf. M´arquez-Dur´an [17] and Caraballo et al. [6]). Theorem 2. For any s∈Rand ξ∈L2 s:= L2 s(Ω; CH)there exists a unique (variational and mild) solution to problem (P) with continuous trajectories in CH, denoted by u(t, s, ξ),defined for t≥ −h, and such that u(t, s, ξ) = ξ(t)for t∈ [−h, 0],where the Wiener process is θsWinstead of W. Moreover, for any τ≥ 0, s ∈Rit holds that u(t, s +τ, u(τ, s, ξ)) = u(t+τ, s, ξ),P−a.s. for all t≥0.(4) Notice that the process θsW(·, ω) = W(·, θsω) = W(·+s, ω)−W(s, ω), for s∈R, is also a Wiener process with covariance Q. For t≥0,this process is adapted to the filtration {¯ Fs+t}t≥0which follows from (3). Moreover, observe that the following equality holds for ξ∈L2 0: u(·,0, ξ)(θs·) = u(·, s, ξs),P−a.s., (5) where ξs(·,·) := ξ(·, θs·). Indeed, by (3) the random variable ξ(·, θsω) is ¯ Fsmeasurable, and both sides of (5) are driven by the same Wiener process (θsW) with the same initial condition. Then, the uniqueness of solutions of (P) proves (5). As we have already mentioned, the intention of this article is to find mean square and/or omega-wise exponentially attracting stationary solutions to problem (P), which means to find a solution process for which the finite dimensional distributions do not depend on time shifts. Suppose, we can find an ( ¯ F0,B(CH))-measurable random variable ξ∗with values in CH(i.e. a measurable mapping ξ∗: Ω →CH) 6 T. CARABALLO, M.J. GARRIDO-ATIENZA & B. SCHMALFUSS such that, if we choose the initial condition ξ∗(·, ω),then the mapping ut(·, ω) := ξ∗(·, θtω) solves problem (P). More precisely, the process given by v(t, ω) = ξ∗(t, ω), if t∈[−h, 0] ξ∗(0, θtω),if t > 0 is a solution of (P). Then we have for t1< t2<· · · < tn P(ξ∗(·, θt1ω)∈B1,· · · , ξ∗(·, θtnω)∈Bn) =P(ξ∗(·, θt1+tω)∈B1,· · · , ξ∗(·, θtn+tω)∈Bn) for any t∈Rand Borel sets B1,· · · , Bnfrom B(CH),which follows directly from the θt-invariance of P. Hence ξ∗generates a stationary solution. 3.2. Mean square exponentially attracting stationary solutions. In this paragraph we establish the mean square exponential attractivity for the stationary solutions to problem (P). Now, we define an appropriate cocycle and will prove that it possesses a unique generalized fixed point which generates a mean square exponentially attracting stationary solution to our problem (P). Lemma 1. Consider the Banach space X(r) = L2(Ω,¯ Fr,P;CH), for r∈R, with its usual norm || · ||L2=qEk·k2 CHwhich is independent of r. For t≥0, r ∈R, define ψ(t, r, ·) : X(r)→X(t+r)by ψ(t, r, ξ) = ut(·, r, ξ),for ξ∈X(r).(6) Then, ψsatisfies the cocycle property (1). Proof. The proof follows from Theorem 2, in particular from the equality (4). Our next aim is to prove the existence of a generalized fixed point for this cocycle ψ. Let Pintroduced in Section 2 be given by the families of CH-valued random variables {ξ(r)}r∈Rsuch that ξ(r) is ¯ Fr−measurable and kξ(r)kL2is uniformly bounded in r. To check that assumptions in Theorem 1 are fulfilled we need some preliminary results concerning the solutions of problem (P). From now on, we denote µ0= 2a+L2 B+2Kfand suppose it is a negative constant. We first establish a result concerning the mean square attractivity for the solution to problem (P). Theorem 3. Suppose all the assumptions on A, B and fhold. Then, for each µ∈[µ0,0] there exists a K1=K1(µ, h)>0such that E|u(t, s, ξ)−u(t, s, η)|2≤K1||ξ−η||2 L2eµt,(7) for all t≥0, s ∈R,and ξ, η ∈L2 s. Moreover, for each µ∈(µ0,0] there exists a constant K2=K2(µ, h)>0such that for any solution uof problem (P)we have that EZt 0 e−µr |u(r, s, ξ)−u(r, s, η)|2dr≤K2||ξ−η||2 L2,(8) for all t≥0, s ∈R,and ξ, η ∈L2 s. STATIONARY SOLUTIONS FOR DELAY SPDE 7 Proof. Let µ∈[µ0,0], s ∈Rbe fixed and denote u(t) := u(t, s, ξ), v(t) := u(t, s, η). Then, for any t≥0 it follows from (f.2) that 2Zt 0 e−µr(f(ur)−f(vr), u(r)−v(r))dr ≤2Zt 0 e−µr |f(ur)−f(vr)|2dr1/2Zt 0 e−µr |u(r)−v(r)|2dr1/2 ≤2KfZt −h e−µr |u(r)−v(r)|2dr. Applying now Itˆo’s formula to the process e−µt |u(t)−v(t)|2, and taking into account the assumptions on Aand B, we obtain for every t≥0 e−µt |u(t)−v(t)|2=|ξ(0) −η(0)|2−µZt 0 e−µr |u(r)−v(r)|2dr + 2 Zt 0 e−µr hA(u(r)−v(r)), u(r)−v(r)idr + 2 Zt 0 e−µr(f(ur)−f(vr), u(r)−v(r))dr(9) +Zt 0 e−µr||B(u(r)) −B(v(r))||2 LQ 2 dr + 2 Zt 0 e−µr(u(r)−v(r),(B(u(r)) −B(v(r)))dW(r, θsω)) ≤ |ξ(0) −η(0)|2+ 2KfZ0 −h e−µr |u(r)−v(r)|2dr + (−µ+ 2a+L2 B+ 2Kf)Zt 0 e−µr |u(r)−v(r)|2dr + 2 Zt 0 e−µr(u(r)−v(r),(B(u(r)) −B(v(r)))dW(r, θsω)) ≤ |ξ(0) −η(0)|2+ 2Kf||ξ−η||2 L2Z0 −h e−µrdr + 2 Zt 0 e−µr(u(r)−v(r),(B(u(r)) −B(v(r)))dW(r, θsω)) P−a.s. Then (7) follows easily by taking the expectation after having replaced tby t∧TN, where TNis the family of stopping times TN(ω) = inf{t≥0 : |u(t)|2+|v(t)|2≥N} such that lim N→∞(t∧TN) = t, P−a.s., since u, v have continuous paths. 8 T. CARABALLO, M.J. GARRIDO-ATIENZA & B. SCHMALFUSS Next, we prove (8). To this end, we consider µ∈(µ0,0]. Thanks to (9) we know that e−µtE|u(t, s, ξ)−u(t, s, η)|2 ≤E|ξ(0,·)−η(0,·)|2+ 2Kfh||ξ−η||2 L2 + (−µ+ 2a+L2 B+ 2Kf)EZt 0 e−µr |u(r, s, ξ)−u(r, s, η)|2dr. Taking into account that −µ+ 2a+L2 B+ 2Kf<0,we obtain EZt 0 e−µr |u(r, s, ξ)−u(r, s, η)|2dr ≤ −(1 + 2Kfh)(−µ+ 2a+L2 B+ 2Kf)−1||ξ−η||2 L2, which proves (8). Now, using this lemma, we can prove the following result. Lemma 2. For each µ∈(µ0,0) there exists a K3=K3(µ, h)>0such that ||ψ(t, s, ξ)−ψ(t, s, η)||2 L2=||ut(·, s, ξ)−ut(·, s, η)||2 L2≤K3||ξ−η||2 L2eµt,(10) for all t≥0, s ∈R,and ξ, η ∈L2 s. Proof. We use the same notation as in the previous proof. Let us assume that t≥h. Applying Itˆo’s formula on the intervals [0, t +σ],for σ∈[−h, 0],and [0, t −h],and then subtracting the obtained equalities, we can deduce that e−µ(t+σ)|ut(σ)−vt(σ)|2 = e−µ(t−h)|u(t−h)−v(t−h)|2−µZt+σ t−h e−µr |u(r)−v(r)|2dr + 2 Zt+σ t−h e−µr hA(u(r)−v(r)), u(r)−v(r)idr + 2 Zt+σ t−h e−µr(f(ur)−f(vr), u(r)−v(r))dr +Zt+σ t−h e−µr||B(u(r)) −B(v(r))||2 LQ 2 dr + 2 Zt+σ t−h e−µr(u(r)−v(r),(B(u(r)) −B(v(r)))dW(r, θsω)). Thanks to hypothesis (f.2) and using the assumptions on A, B and the fact that −µ+µ0<0,we obtain e−µ(t+σ)|ut(σ)−vt(σ)|2 ≤e−µ(t−h)|u(t−h)−v(t−h)|2 + 2KfZ0 −h e−µr |u(r)−v(r)|2dr+ 2KfZt−h 0 e−µr |u(r)−v(r)|2dr + 2 Zt+σ t−h e−µr(u(r)−v(r),(B(u(r)) −B(v(r)))dW(r, θsω)). STATIONARY SOLUTIONS FOR DELAY SPDE 9 Taking into account (8), it follows e−µtEsup −h≤σ≤0 |ut(σ)−vt(σ)|2 ≤e−µtE|u(t−h)−v(t−h)|2+ 2e−µhKf(h+K2)||ξ−η||2 L2 + 2e−µhEsup −h≤σ≤0Zt+σ t−h e−µr(u(r)−v(r),(B(u(r)) −B(v(r)))dW(r, θsω)). On the other hand, Burkholder-Davis-Gundy’s inequality yields that 2e−µhEsup −h≤σ≤0Zt+σ t−h e−µr(u(r)−v(r),(B(u(r)) −B(v(r)))dW(r, θsω)) ≤6e−µhEsup −h≤σ≤0e −µ(t+σ) 2|u(t+σ)−v(t+σ)| Zt t−h e−µr||B(u(r)) −B(v(r))||2 LQ 2 dr1/2# ≤1 2e−µtEsup −h≤σ≤0 |ut(σ)−vt(σ)|2 + 18e−2µhL2 BEZt t−h e−µr |u(r)−v(r)|2dr. Therefore, applying Theorem 3 we obtain e−µt 2Esup −h≤σ≤0 |ut(σ)−vt(σ)|2 ≤e−µtE|u(t−h)−v(t−h)|2+ 2e−µhKf(h+K2)||ξ−η||2 L2 + 18e−2µhL2 BEZt+σ t−h e−µr |u(r)−v(r)|2dr ≤K1e−µh||ξ−η||2 L2+ 2Kfe−µh(h+K2)||ξ−η||2 L2 + 18K2e−2µhL2 B||ξ−η||2 L2. The case 0 ≤t < h can be easily deduced from the previous analysis by noticing that ||ut(·, s, ξ)−ut(·, s, η)||2 CH≤sup σ∈[−h,−t] |u(t+σ, s, ξ)−u(t+σ, s, η)|2 + sup σ∈[−t,0] |u(t+σ, s, ξ)−u(t+σ, s, η)|2. Recall that the family Pis formed by the families of CH−valued random variables {ξ(r)}r∈Rsuch that ξ(r) is ¯ Fr−measurable and that ||ξ(r)||L2is uniformly bounded in r∈R. Then a similar analysis to the one carried out in the previous lemma allows to prove the following consequence. Corollary 1. For any given {ξ(r)}r∈R∈ P it holds that ||ψ(t, r, ξ(r))||2 L2is uniformly bounded for t≥0, r ∈R, and thus {ψ(t, r, ξ(r))}r∈R∈ P, for t≥0and {ξ(r)}r∈R∈ P. 16 T. CARABALLO, M.J. GARRIDO-ATIENZA & B. SCHMALFUSS contained in a {θt}t∈R–invariant set of full measure. However, by our definition of the metric dynamical system we can state that these properties will hold for every ω∈Ω. We now formulate an evolution equation with random coefficients but without white noise    dv dt =A+ N P i=1 λiz∗ i(θtω)Biv+T−1(θtω)f(e T(θtω)vt), v(t) = ξ(t), t ∈[−h, 0], (22) where ξbelongs to CH,which will be our phase space. Lemma 5. Suppose that A, B1,· · · , BN, λ1, . . . , λNsatisfy the preceding assumptions. Then i) the random evolution equation (22) possesses a unique solution, and this solution generates a random dynamical system χ:R+×Ω×CH→CHdefined for t≥0, ω∈Ωand ξ∈CHby χ(t, ω, ξ) = vt(·, ω, ξ). ii) the process φ:R+×Ω×CH→CHdefined for t≥0, ω ∈Ωand ξ∈CHby φ(t, ω, ξ) = e T(θtω)χ(t, ω, e T−1(ω)ξ) is another random dynamical system such that φ(t, ω, ξ) = ut(·, ω, ξ), being ua solution version to problem (17), unique modulo P. Proof. i) We mention only that by the continuity and linearity of Tand e Tthe mapping ξ∈CH→T−1(θtω)f(e T(θtω)ξ)∈His Lipschitz continuous, where the Lipschitz constant is uniformly bounded on any interval [0, T]. Hence we can prove the existence of a mild solution of (22) for every ω. The proof of measurability is straightforward. ii) This part follows in a standard way by applying the chain rule to the function ytdefined as yt(s) := e T(θtω)vt(·, ω, e T−1(ω)ξ)(s) = T(θt+sω)v(t+s, ω, e T−1(ω)ξ), s ∈[−h, 0], and taking into account the commutativity of the operators involved (see [6] for a similar situation in the non-delay case). On account of Lemma 3, φand χare conjugated random dynamical systems. 4.3. Existence of exponentially stationary solutions. In the following let us apply the general method from Section 2 to find an exponentially attracting random fixed point ξ∗. Theorem 6. Suppose A, B1,· · · , BNmutually commute and W= (w1,· · · , wN) satisfies the assumptions at the beginning of this section. In addition, there are positive constants λ1,· · · , λNsuch that a+ N X i=1 biλiE|z∗ i|+Kf 2 N Y i=1 E||SBi(z∗ i)||2+ N Y i=1 E||SBi(−z∗ i)||2!<0 (23) holds. Then the random dynamical systems χand φpossess, respectively, a tempered random fixed point χ∗and φ∗,which are unique under all tempered random variables in CHand which attract exponentially fast every random variable in CH. STATIONARY SOLUTIONS FOR DELAY SPDE 17 Proof. First of all, we point out that to prove the theorem it is sufficient to show that (22) has a unique exponentially attracting generalized fixed point. The conjugation technique then gives the existence of a fixed point for (17). First step: Let us consider, for δ > 0 small enough, the mapping γ(ω) = −2a−2 N X i=1 biλi|z∗ i(ω)| − (1 + δ)1 2Kf||T(ω)||2+||T−1(ω)||2, which satisfies, by (23) and (21), Eγ=: ¯γ > 0. According to Remark 1 where we introduced the metric dynamical system (Ω,F,P, θ) we have that lim t→±∞ 1 tZt 0 γ(θτω)dτ= ¯γ, for all ω∈Ω. We are interested in calculating a priori estimate for the solution of (22). Denote v(t) = vt(0, ω, ξ), t ∈R+. On account of the properties of the coefficients of (22) and the chain rule the following estimate holds: eRt 0γ(θlω)dl|v(t)|2≤ |v(0)|2 −Zt 0 (1 + δ)1 2Kf||T(θsω)||2+||T−1(θsω)||2eRs 0γ(θlω)dl|v(s)|2ds (24) + 2 Zt 0 eRs 0γ(θlω)dl(T−1(θsω)f(e T(θsω)vs), v(s))ds. The terms in the above integrals are locally integrable in s, l such that these integrals exist. Now we evaluate the last term in (24) by means of condition (f.2). 2Zt 0 eRs 0γ(θlω)dl(T−1(θsω)f(e T(θsω)vs), v(s))ds(25) ≤εZt 0 ||T−1(θsω)||2eRs 0γ(θlω)dl|v(s)|2ds +ε−1(1 + δ)K2 fZt −h ||T(θsω)||2eRs 0γ(θlω)dl|v(s)|2ds +ε−1(1 + δ−1)|f(0)|2Zt 0 eRs 0γ(θlω)dlds, where ε > 0.Choosing ε= (1 + δ)1 2Kfin the last inequality,if we suppose t≥σ we obtain from (24), (25) eRt+σ 0γ(θlω)dl|v(t+σ)|2≤ |v(0)|2 + (1 + δ)1 2Kf||ξ||2 CHZ0 −h ||T(θsω)||2eRs 0γ(θlω)dlds + (1 + δ−1)(1 + δ)−1 2K−1 f|f(0)|2Zt+σ 0 eRs 0γ(θlω)dlds. 18 T. CARABALLO, M.J. GARRIDO-ATIENZA & B. SCHMALFUSS Now, if we take the supremum in the last inequality, we have sup σ∈[−h,0] |v(t+σ)|2(26) ≤e−Rt 0γ(θlω)dlkξk2 CHe−2ah +cδZ0 −h ||T(θsω)||2eRs 0γ(θlω)dlds + ˜cδZt 0 eRs 0γ(θlω)dlds, for all t≥h, where we have denoted cδ=e−2ah(1 + δ)1 2Kfand ˜cδ=e−2ah(1 + δ−1)(1 + δ)−1 2K−1 f|f(0)|2.A similar estimate is true for t∈[0, h). Second step: We check the contraction condition which we need to get (A1) from Theorem 1. Consider ξ1, ξ2∈CH.Let ω∈Ω and denote vi(t) = vt(0, ω, ξi),for i= 1,2,and v(t) = v1(t)−v2(t), t ∈R+. Taking into account that vi(t) is a solution to the equation dvi(t) dt = A+ N X i=1 λiz∗ i(θtω)Bi!vi(t) + T−1(θtω)f(e T(θtω)vi t), we have that d|v(t)|2 dt =2 * A+ N X i=1 λiz∗ i(θtω)Bi!v(t), v(t)+ + 2(T−1(θtω)(f(e T(θtω)v1 t)−f(e T(θtω)v2 t)), v(t)) ≤ 2a+ 2 N X i=1 λibi|z∗ i(θtω)|+Kf||T−1(θtω)||2!|v(t)|2 +1 Kf |f(e T(θtω)v1 t)−f(e T(θtω)v2 t)|2. Therefore, using condition (f.2), e−2at|v(t)|2 ≤|v(0)|2+KfZ0 −h ||T(θsω)||2e−2as|v(s)|2ds +Zt 0 2 N X i=1 λibi|z∗ i(θsω)|+Kf(||T−1(θsω)||2+||T(θsω)||2)!e−2as|v(s)|2ds, and by Gronwall’s lemma, |v(t)|2≤1 + KfZ0 −h ||T(θsω)||2ds||ξ1−ξ2||2 CH ×e2at exp Zt 0 2 N X i=1 λibi|z∗ i(θsω)|+Kf||T−1(θsω)||2+||T(θsω)||2ds.! STATIONARY SOLUTIONS FOR DELAY SPDE 19 Now, setting t≥hand σ∈[−h, 0], taking supremum, sup σ∈[−h,0] |v(t+σ)|2 ≤1 + KfZ0 −h ||T(θsω)||2ds||ξ1−ξ2||2 CH ×e−2ah exp Zt 0 2a+ 2 N X i=1 λibi|z∗ i(θsω)|+Kf||T−1(θsω)||2+||T(θsω)||2ds!, since sup σ∈[−h,0] e2a(t+σ)=e−2ahe2at.Therefore, if we set K(t, ω) := e−2ah 1 + KfZ0 −h ||T(θsω)||2ds ×exp Zt 0 2a+ 2 N X i=1 λibi|z∗ i(θsω)|+Kf||T−1(θsω)||2+||T(θsω)||2ds!, (27) then, for t≥hit holds ||vt(·, ω, ξ1)−vt(·, ω, ξ2)||2 CH≤K(t, ω)||ξ1−ξ2||2 CH. Third step: Now we apply Theorem 1. For each fixed ω∈Ω,we consider the family of Banach spaces given by X(r) = CHfor r∈R. In particular, we fix some ¯ω∈Ω and define ψ(t, r, ξ(r)) := χ(t, θr¯ω, ξ(θr¯ω)). Then (A1) from Theorem 1 follows from the second step, since by the definition of Ω (see (19) and (21)) we can set κ=−a− N X i=1 biλiE|z∗ i| − Kf 2 N Y i=1 E||SBi(z∗ i)||2+ N Y i=1 E||SBi(−z∗ i)||2! and K(t, r) := K(t, θr¯ω) from (27). To see that K(t, r −t) := K(t, θr−t¯ω) has the growth condition from (A1) we need that R0 −h||T(θsω)||2dsis tempered what follows from the methods in the proof of Lemma 6. For the fixed ¯ωwe choose Pto be the families ξ= (ξ(r))r∈Rwith values in CH such that kξ(r)kCHhas a subexponential growth for r→ ±∞. These families are κ-growing. In particular, let Ξ be the set of random variables with values in CH such that kξkCHis tempered for ξ∈Ξ.Then the orbits ξ= (ξ(θr¯ω))r∈Rare elements from P. Define the random variable R(ω) := 2˜cδZ0 −∞ e−R0 sγ(θlω)dlds. (28) 20 T. CARABALLO, M.J. GARRIDO-ATIENZA & B. SCHMALFUSS From (26) we obtain kψ(t, r −t, ξ(r−t))k2 CH≤e−R0 −tγ(θr+l¯ω)dlkξ(θr−t¯ω)k2 CH× ×e−2ah +cδZ0 −∞ kT(θs+r−t¯ω)k2eRs 0γ(θl+r−t¯ω)dlds +1 2R(θr¯ω)≤R(θr¯ω) for |t|sufficiently large and r≥0 and ξ∈Ξ because t→e−R0 −tγ(θl+r¯ω)dl tends to zero exponentially fast for t→ ∞.Rand the infinite integral are tempered what follows from Lemma 6 below. The Cauchy sequence has a limit by the completeness of the phase space. Its norm is bounded by pR(θr¯ω). Hence the limit family is in Pwhich gives (A2). To see (A3) we note that from (26) by some straightforward integral transforms sup q∈[0,1] kψ(q, r −q, ξ(r−q))k2 CH ≤e−2ah sup q∈[0,1] e−Rq 0γ(θr+l−q¯ω)dlsup q∈[0,1] kξ(θr−q¯ω)k2 CH(29) +cδsup q∈[0,1] e−Rq 0γ(θr+l−q¯ω)dlZ0 −∞ kT(θr+s−q¯ω)k2eRs 0γ(θr+l−q¯ω)dlds+R(θr¯ω) ≤e−2aheR0 −12PN i=1 biλi|z∗ i(θl+r¯ω)|+(1+δ)1 2Kf(||T(θr+l¯ω)||2+||T−1(θr+l¯ω)||2)dl ×sup q∈[0,1] kξ(θr−q¯ω)k2 CH +cδZ0 −∞ kT(θr+s¯ω)k2eRs 0γ(θr+l¯ω)dlds+R(θr¯ω). The sum/product of tempered random variables is tempered. In addition, it is easy to see that supp∈[−1,0] kξ(θpω)kCHis tempered if kξ(ω)kCHis. Moreover, the first term on the right hand side of (29) is tempered by the Birkhoff ergodic theorem: lim t→∞ 1 tZ0 −2 2 N X i=1 biλi|z∗ i(θl−[t]ω)|+(1+δ)1 2Kf||T(θl−[t]ω)||2+||T−1(θl−[t]ω)||2dl= 0 for ω∈Ω,being [t] the integer part of t. The integral and the last term on the right hand side of (29) is defined by a tempered random variable what follows by Lemma 6 below. Hence we can apply Theorem 1 which gives us a generalized fixed point ψ∗. Fourth step: We set χ∗(¯ω) := ψ∗(0) = ψ∗(0,¯ω)∈BCH(0, R(¯ω)) for every ¯ω∈Ω. Note that by the cocycle property, this definition is correct. Hence χ∗satisfies the fixed point relation χ(t, ω, χ∗(ω)) = χ∗(θtω) for t≥0 and ω∈Ω. STATIONARY SOLUTIONS FOR DELAY SPDE 21 According to the contraction condition given in the Second step we obtain (16) for every random variable ξ∈CH. In particular, χ∗can be generated as a pointwise limit of random variables such that χ∗is a random variable. If we define φ∗(ω) := e T(ω)χ∗(ω), as φand χare conjugated random dynamical systems, it follows φ(t, ω, φ∗(ω)) = e T(θtω)χ(t, ω, e T−1(ω)φ∗(ω)) =e T(θtω)χ(t, ω, χ∗(ω)) =e T(θtω)χ∗(θtω) = φ∗(θtω). Since ke TkL(CH)and ke T−1kL(CH)are tempered we have for φ∗the same uniqueness and convergence conclusion as for χ∗. Now we prove the temperedness statements formulated in the proof of the last theorem. Lemma 6. Under the assumptions of Theorem 6 the random variable Rdefined in (28) and Z0 −∞ kT(θsω)k2eRs 0γ(θlω)dlds are tempered. Proof. We note that by the definition of Ω in Remark 1 we have Zt 0 (γ(θτ+rω)−¯γ)dτ≤ε|t|,log+(kT(θtω)k2+ 1) ≤ε|t| for every ω∈Ω, r ∈R, ε > 0 if |t| ≥ t0(ω, r, ε). It is clear that it is sufficient to prove that Z0 −∞ (kT(θτω)k2+ 1)e−R0 −τγ(θlω)dlds is tempered. For every ω∈Ω and c < 0,consider ε∈(0,min(−c 6,¯γ 4)).Then, lim s→−∞ e−cs Z0 −∞ (kT(θτ+sω)k2+ 1)e−R0 τγ(θl+sω)dldτ = lim s→−∞ e−cs Z0 −∞ (kT(θτ+sω)k2+ 1)e−Rs τ+sγ(θlω)dldτ = lim s→−∞ e−cs Z0 −∞ eR0 τ+s(−γ(θlω)+¯γ)dl+¯γ(τ+s)−R0 s(−γ(θlω)+¯γ)dl−¯γs+log+(kT(θτ+sω)k2+1)dτ ≤lim s→−∞ e−c 2sZ0 −∞ e−c 2s−(τ+s)−s+¯γτ−ε(τ+s)dτ ≤lim s→−∞ e−c 2sZ0 −∞ e¯γ 2τdτ= 0. We consider now the case s→+∞. First of all, note that ec|s|Z0 −∞ (kT(θτ+sω)k2+ 1)e−R0 τγ(θl+sω)dldτ 22 T. CARABALLO, M.J. GARRIDO-ATIENZA & B. SCHMALFUSS can be written as ecse−Rs 0γ(θlω)dlZs 0 (kT(θτω)k2+1)eRτ 0γ(θlω)dldτ+Z0 −∞ (kT(θτω)k2+1)eRτ 0γ(θlω)dldτ. We obtain that lim s→+∞ecse−Rs 0γ(θlω)dlZ0 −∞ (kT(θτω)k2+ 1)eRτ 0γ(θlω)dldτ= 0. 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