Exponentially Stable Stationary Solutions for Stochastic Evolution Equations and Their Perturbation

EXPONENTIALLY STABLE STATIONARY SOLUTIONS FOR STOCHASTIC EVOLUTION EQUATIONS AND THEIR PERTURBATION TOM´ AS CARABALLO, PETER E. KLOEDEN, AND BJ¨ ORN SCHMALFUSS Abstract. We consider the exponential stability of stochastic evolution equations with Lipschitz continuous non-linearities when zero is not a solution for these equations. We prove the existence of a non-trivial stationary solution which is exponentially stable, where the stationary solution is generated by the composition of a random variable and the Wiener shift. We also construct stationary solutions with the stronger property of attracting bounded sets uniformly. The existence of these stationary solutions follows from the theory of random dynamical systems and their attractors. In addition, we prove some perturbation results and formulate conditions for the existence of stationary solutions for semi-linear stochastic partial differential equations with Lipschitz continuous non-linearities. 1. Introduction The exponential stability of stochastic partial differential equations is an important problem, and it has received considerable attention during the recent decades as the vast literature on this topic shows. Our aim here is to study the exponential stability of non-trivial stationary solutions of these equations. The investigation of stability for constant stationary solutions for finite dimensional stochastic differential equations goes back to Has ´minski˘ı [14] using Lyapunov functions for the generator of the Markov semi-group. These ideas have been extended by Mao [20] in the finite dimensional context, while non-constant stationary solutions have been treated in Schmalfuß [24]. Here we will generalize some techniques from these last two publications. For infinite dimensional (parabolic) stochastic differential equations the problem of exponentially stable constant stationary solutions has been considered by Caraballo and Real [5] (see also [3], [4]), Liu and Mao [19], Chow [7], Haussmann [15] and Ichikawa [16] among others. In contrast to these constant stationary solutions we will investigate the asymptotic exponential stability of non-trivial stationary solutions. In this respect, we will consider semilinear stochastic evolution equations with Lipschitz continuous non-linearities. Under suitable assumptions we prove the existence of a unique stationary solution by using a fixed point argument based on the pullback technique. This stationary solution turns to be exponentially stable in mean square, and also Date: June, 2004. Key words and phrases. random dynamical systems, stationary solutions, exponential stability, stabilization. 1 2 TOM´ AS CARABALLO, PETER E. KLOEDEN, AND BJ ¨ ORN SCHMALFUSS in the almost sure sense. Although we prove almost sure convergence to the stationary solution, the exceptional sets depend on the initial condition, so it is not possible to consider uniform convergence with respect to a bounded set of initial conditions. However, this problem will be overcome by using a technique from random dynamical systems which allows us to study the problem for exceptional sets independent of the set of initial conditions. It is unknown in general if stochastic partial differential equations with general diffusion coefficients generate random dynamical systems. However, if we suppose some kind of commutativity on these coefficients, then we are able to prove the existence of such a random dynamical system. We prove the existence of a random fixed point, which is in fact a random variable. This random variable generates the exponentially stable stationary solution of the stochastic partial differential equation. Moreover, this stationary solution attracts bounded sets of initial conditions. Stationary solutions in this interpretation correspond with single point random attractors, an important object in the theory of random dynamical systems, see Crauel, Debussche and Flandoli [8], Flandoli and Schmalfuß [12] or [23]. Another aim of this paper is to analyze perturbations of stochastic partial differential equations and the relation between their stationary solutions. These perturbations will be given by modifying the non-linear part of the equation. Under the assumption that the perturbations approach the non-linear part of the original equation and that the Lipschitz constants of the perturbed operators are uniformly bounded and not too large, we obtain the existence of stationary solutions which converge to the stationary solution of our original problem in the mean square sense. For omega-wise convergence we formulate a theorem on the continuous dependence of random fixed points on a parameter. The paper is organized as follows. In Section 2 we introduce basic concepts of stochastic evolution equations and random dynamical systems. The third section deals with the exponential stability of stationary solutions of stochastic evolution equations in the mean square sense and almost surely. In the next section, this exponential stability is analyzed from the point of view of random dynamical systems. Then in Section 5, we consider the perturbation problems, and illustrate the results with an example in the final section. 2. Random dynamical systems and stochastic evolution equations In this section we will describe the concept of exponentially stable stationary solutions for stochastic non-linear evolution equations generated by random fixed points. To do this we start by describing 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 will consider a two-sided Wiener process Wwith values in some separable Hilbert space Uwhere the covariance Qis a symmetric operator on Uof trace class. For instance, for the above probability space we will choose for Ω the 3 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 defined to be the Wiener measure with respect to the covariance Q. For Ftwe set σ{ω(u)−ω(v) : v, u ≤t}. What we have introduced is the Brownian motion metric dynamical system which is the standard noise for random dynamical systems generated by stochastic differential equations. Note that the above probability space is not completed. The completion of this probability space is denoted by (Ω,¯ F,{¯ Ft}t∈R,P) where{¯ Ft}t∈Rhas to be a normal filtration, see Da Prato and Zabczyk [10] Page 75. We now introduce on the above non-completed probability space a measurable flow θ={θt}t∈Ron Ω: (1) θ: (R×Ω,F ⊗ B(R)) →(Ω,F), θt+τ=θt◦θτ, t, τ ∈R, θ0= idΩ. 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 θ. In addition, with respect to the filtration we have that (2) θ−1 uFt=Ft+u for any t, u ∈R, see Arnold [1] Page 72. Since the above probability space is canonical we have for a Wiener process and its shift operator W(t, ω) = ω(t), W (t, θsω) = ω(t+s)−ω(s) = W(t+s, ω)−W(s, ω). It is important to note that the measurability in (1) is not true if we replace F by its completion, see Arnold [1] Appendix A3. However, for fixed twe have the measurability of θt: (Ω,¯ F)→(Ω,¯ F). Moreover, the mapping R3t→θtω∈C(R, U) is continuous for a fixed ω∈Ω. We will study the qualitative behaviour of stochastic evolution equations on some separable Hilbert space Hwhich have the form (3) dX =AXdt +f(X)dt +B(X)dW, X(0) = u0, where Wis the Wiener process on the probability space (Ω,¯ F,{¯ Ft}t∈R,P) introduced above. Assume that there exists a Gelfand triplet V⊂H⊂V0of separable Hilbert spaces, where V0denotes the dual of V(see Temam [25] Page 55 for more details). We denote by k · k,k · kVthe norms in Hand Vrespectively. The inner product in Hwill be denoted by (·,·), and the duality mapping between V0and V by h·,·i. The random variable u0is supposed to be ( ¯ F0,B(H)) measurable. Let us denote by a1>0 the constant of the injection V⊂H, i.e. a1kuk2≤ kuk2 V,for v∈V, 4 TOM´ AS CARABALLO, PETER E. KLOEDEN, AND BJ ¨ ORN SCHMALFUSS and let −A:V→V0be a positive, linear and continuous operator for which there exists a a2<0 such that h−Au, ui ≥ −a2kuk2 V,for all u∈V. Then, it is well known (see, for instance, Dautray and Lions [9]) that Ais the generator of a strongly continuous semigroup {S(t)}t≥0in Hsatisfying that (4) kS(t)kL(H)≤eat, where a=a1a2<0. The operator fis supposed to be Lipschitz continuous from Hto H: kf(u1)−f(u2)k ≤ Lfku1−u2k, u1, u2∈H, and Bis supposed to be Lipschitz continuous with respect to the Hilbert-Schmidt norm LQ 2(U, H) of linear operators from Uto H: trH((B(u1)−B(u2))Q(B(u1)−B(u2))∗=: kB(u1)−B(u2)k2 LQ 2 ≤LBku1−u2k2 for u1, u2∈H. We now need the spaces L2,s := L2(Ω,¯ Fs,P;H), s ∈R. We have the following theorem about the existence, uniqueness and regularity of (3). Theorem 2.1. Suppose that u0∈L2,0. Then (3) has a unique (up to equivalence) mild solution X(·)on [0,∞)which has a continuous version. In addition, EZT 0 kX(t)k2 Vdt < ∞, and Esup t∈[0,T ] kX(t)k2<∞, for any T≥0. For the existence of a mild solution see Da Prato and Zabczyk [10] Theorem 7.4. The regularity assertion can be found in Krylov and Rozovskii [17] Chapter 2. We are interested in stationary solutions that are exponentially attracting in the L2sense or almost surely. Stationary solution means that the finite dimensional distributions of the solution Xare independent of shifts with respect to t. These exponentially stable stationary solutions will be generated by exponentially attracting fixed points given by an ( ¯ F0,B(H))-measurable random variable X∗with values in Hsuch that if we choose the initial condition u0(ω) = X∗(ω) we have X(t, ω) = X∗(θtω) almost surely for all t≥0,where the exceptional set may depend on t. This fixed point is said to be exponentially attracting if the process (t, ω)→X∗(θtω) (or a version of this process) attracts the solution of (3) for any (appropriate) initial condition exponentially fast in the L2-sense or almost surely. By the θtinvariance of Pwe have that P(X∗(θt1ω)∈B1,· · · , X∗(θtnω)∈Bn) =P(X∗(θt1+tω)∈B1,· · · , X∗(θtn+tω)∈Bn) for t≥0,0≤t1< t2<· · · , tnand B1,· · · , Bn∈ B(H). 5 Another tool that can be used to describe the stability behaviour of a stochastic evolution equation are random dynamical systems. A comprehensive presentation can be found in Arnold [1]. A random dynamical system is given by a measurable mapping φ: (R+×Ω×H, B(R+)⊗ F ⊗ B(H)) →(H, B(H)), satisfying the cocycle property: φ(t+τ, ω, x) = φ(t, θτω, φ(τ, ω, x)), t, τ ∈R+, ω ∈Ω, x ∈H, φ(0, ω, x) = x, (5) where θis the flow of shift operators (Wiener shift) introduced above. We emphasize that (5) has to be satisfied for any ω∈Ω. However it is sufficient to replace Ω by a {θt}t∈R-invariant set of full measure. Outside of this invariant set we can redefine φby the identity mapping on H. Later on we will replace Ω by a smaller {θt}t∈R-invariant set Ω0∈ F. The measurability of (1) remains true if we replace Fby its trace σ-algebra. The mapping φis related to the solution of a stochastic or random differential equation. For the following we will always suppose that the mapping H3x7→ φ(t, ω, x)∈H is continuous for any t, ω. Although it is known that finite dimensional stochastic differential equations generate random dynamical systems (see Arnold [1] Chapter 1), this is not true in general for infinite dimensional equations. However, for particular kinds of noise we can apply the following simple lemma to obtain a random dynamical system. Lemma 2.2. Let φbe a random dynamical system. Suppose that the mapping T: Ω ×H→Hhas the following properties: For fixed ω∈Ωthe mapping T(ω, ·) is a homeomorphism on H. For fixed x∈Hthe mappings T(·, x), T−1(·, x)are measurable. Then the mapping (6) (t, ω, x)→T−1(θtω, φ(t, ω, T(ω, x))) =: ψ(t, ω, x) satisfies (5). Hence ψis a random dynamical system. The measurability of ψfollows because of the properties of T. Later on we will transform a stochastic evolution equation containing a noise term into an evolution equation without noise but with random coefficients. A random variable Yon (Ω,F,P) with values in His called tempered if (7) lim t→±∞ log+kY(θtω)k |t|= 0, or equivalently if t→ kY(θtω)khas a sub-exponential growth for t→ ±∞, in other words, for ε > 0 and ω∈Ω there exists a t0(ε, ω)≥0 such that for |t| ≥ t0(ε, ω) it holds (8) kY(θtω)k ≤ eε|t|, which means that the exponential growth rate of t→ kY(θtω)kis zero. 6 TOM´ AS CARABALLO, PETER E. KLOEDEN, AND BJ ¨ ORN SCHMALFUSS Let ω→G(ω) be a set valued mapping from Ω into the space of non-empty closed subsets from H. Such a mapping is called a random set if for any y∈Hthe mapping ω→inf x∈G(ω)kx−yk is a random variable. If Gis a random set then there exists a random variable g with g(ω)∈G(ω) (see Castaing and Valadier [6] Chapter 3). A random set Gis called tempered if the random variable ω→sup x∈G(ω) kxk is tempered. It is easily seen that the set of ωfor which (7), (8) are satisfied is {θt}t∈T-invariant. An (F,B(H))-measurable random variable X∗is called a random fixed point in the sense of random dynamical systems if φ(t, ω, X∗(ω)) = X∗(θtω) for t > 0, where ωis contained in a {θt}t∈T-invariant set of full measure. X∗is an exponentially stable random fixed point with respect to a closed random set G containing X∗if for any random variable g∈Gwe have lim t→∞ kφ(t, ω, g(ω)) −X∗(θtω)k= 0 for all ωin the above {θt}t∈T-invariant set of full measure with exponential speed, such that the exceptional set is independent of t. If φis defined by the solution mapping of a stochastic/random differential equation then (t, ω)7→ X∗(θtω) is a stationary solution of a stochastic/random differential equation. Our strategy will be to prove that, under certain assumptions, a random dynamical system possesses a random attractor which is a single (random) point. The following definition can be found in Flandoli and Schmalfuß [12]. Definition 2.3. Let Dbe the set of all closed tempered random sets in H. A compact random set A ∈ D is called a random attractor if the invariance property (9) φ(t, ω, A(ω)) = A(θtω) is satisfied for ω∈Ω, t ≥0and if, in addition, the pullback convergence (10) lim t→∞ distH(φ(t, θ−tω, D(θ−tω),A(ω))) = 0 holds for D∈ D and ω∈Ω. We note that from this convergence it follows (l.i.p.) lim t→∞ distH(φ(t, ω, D(ω),A(θtω)) = 0, where by (l.i.p.) we denote limit in probability. However, in general it does not imply ωwise almost sure convergence. Sufficient conditions for the existence of a random attractor can be found in [12]. Theorem 2.4. Suppose that the mapping x→φ(t, ω, x)is continuous for t≥0, and completely continuous for t > 0(which means that the image of every bounded 7 set by the mapping x→φ(t, ω, x)is relatively compact). In addition, suppose there is a G∈ D such that for any D∈ D and ω∈Ωthere exists T(D, ω)>0such that (11) φ(t, θ−tω, D(θ−tω)) ⊂G(ω),for all t≥T(D, ω). Then there exists a unique random attractor A(in D). If the random attractor A(ω), ω ∈Ω,consists of a single point then Adefines a random fixed point which attracts tempered random sets. 3. Exponential stability stochastic evolution equations In this section we will prove the existence of exponentially stable (both in the mean square sense and almost surely) non-trivial stationary solutions to our stochastic semi-linear partial differential equation (3). The exponential stability of trivial stationary solutions (in particular, the null solution) of stochastic PDEs has been extensively analyzed (see, for instance, [4], [19], [16],... and the literature cited therein). However, when zero is not a solution to the equation, it is interesting to find out whether or not there exist other stationary solutions, which are generated by random variables chosen as initial values in our problem, and to analyze their stability properties. This fact, can also be considered as a connection between the classical method for the local analysis of the long-time behaviour of stochastic partial differential equations and the global one provided by the theory of random dynamical systems. In the sequel we consider the process θsW(·, ω) = W(·, θsω) = W(·+s, ω)−W(s, ω), for s∈R, which is also a Wiener process with covariance Q. For t≥0 this process is adapted to the filtration {¯ Fs+t}t≥0which follows from (2). We will denote by Φ(·, s, u0) the solution of (3), corresponding to the initial value u0∈L2,s, which is driven by θsW, and satisfying the assertions of Theorem 2.1. The following equality holds for u0∈L2,0: (12) Φ(·,0, u0)(θs·) = Φ(·, s, us)(·),almost surely, where us(·) := u0(θs·). Both sides of (12) are driven by the same Wiener process and the same initial condition and the fact that solutions of (3) are unique. Indeed, by (2) u0(θs·) is ¯ Fs-measurable. Lemma 3.1. i) For s∈R, τ ≥0, u0∈L2,s Φ(·, s +τ, Φ(τ, s, u0)) = Φ(·+τ, s, u0),almost surely. ii) Set (13) µ:= 2a+ 2Lf+LB. Then EkΦ(t, s, u1 0)−Φ(t, s, u2 0)k2≤eµtEku1 0−u2 0k2 for t≥0, s ∈R, u1 0, u2 0∈L2,s. Proof. i) Consider the process Y(r, ω) = ½Φ(r, s, u0)(ω) : 0 ≤r≤τ Φ(r−τ, s +τ, Φ(τ, s, u0))(ω) : r > τ , 8 TOM´ AS CARABALLO, PETER E. KLOEDEN, AND BJ ¨ ORN SCHMALFUSS which is continuous (almost surely) by the continuity of Φ. For r > τ we have Y(r) =S(r−τ)Φ(τ, s, u0) +Zr−τ 0 S(r−τ−q)f(Φ(q, s +τ, Φ(τ, s, u0)))dq +Zr−τ 0 S(r−τ−q)B(Φ(q, s +τ, Φ(τ, s, u0))d(W(q+s+τ)−W(s+τ)) =S(r)u0 +Zr τ S(r−q)f(Y(q))dq +S(r−τ)Zτ 0 S(τ−q)f(Y(q))dq +Zr τ S(r−q)B(Y(q))d(W(q+s)−W(s)) +S(r−τ)Zτ 0 S(τ−q)B(Y(q))d(W(q+s)−W(s)). The first conclusion follows if we concatenate the integrals and use the fact that for the increments of the Wiener process we have W(q1+s+τ)−W(s+τ)−(W(q2+s+τ)−W(s+τ)) =W(q1+s+τ)−W(s)−(W(q2+s+τ)−W(s)). ii) It is not hard to see by the properties of the coefficients that (3) has the trajectories in L2(0, T;V) (see Krylov and Rozovskii [17], or Grecksch and Tudor [13]) which allows us to apply the Ito formula for the process e−µtkX1(t)−X2(t)k2where we have denoted Xi(t) := Φ(t, s, ui 0), i = 1,2.Thus e−µtkX1(t)−X2(t)k2 =ku1 0−u2 0k2−µZt 0 e−µqkX1(q)−X2(q)k2dq + 2 Zt 0 e−µqhA(X1(q)−X2(q), X1(q)−X2(q)idq + 2 Zt 0 e−µq(f(X1(q)) −f(X2(q)), X1(q)−X2(q))dq +Zt 0 e−µqkB(X1(q)) −B(X2(q))k2 LQ 2 dq + 2 Zt 0 e−µq(X1(q)−X2(q),(B(X1(q)) −B(X2(q)))dW (q, θsω)) ≤ku1 0−u2 0k2+Zt 0 e−µq(−µ+ 2a+ 2Lf+LB)kX1(q)−X2(q)k2dq + 2 Zt 0 e−µq(X1(q)−X2(q),(B(X1(q)) −B(X2(q)))dW (q, θsω)). ≤ku1 0−u2 0k2+ 2 Zt 0 e−µq(X1(q)−X2(q),(B(X1(q)) −B(X2(q)))dW (q, θsω)). (14) 9 The result follows easily by calculating the expectation if we replace tby t∧TN, where TNis a family of stopping times TN(ω) = inf{t≥0 : kX1(t, ω)k2+kX2(t, ω)k2≥N} such that lim N→∞(TN∧t) = t, almost surely, since X1, X2have continuous paths. ¤ Corollary 3.2. Φ(t, s, ·)maps L2,s into L2,s+tcontinuously. We shall now show the existence of an exponentially stable solution for (3). Theorem 3.3. Suppose that the constant µappearing in (13) is negative. Then there exists an exponentially attracting fixed point X∗∈L2,0generating an exponentially stable stationary solution for (3). In particular, the process (t, ω)→X∗(θtω) has a continuous version given by Φ(·,0, X∗). Proof. We show that (Φ(k, −k, u0(θ−k·)))k∈Nis a Cauchy sequence in L2,0for u0∈ L2,0. Notice that for this u0we have that u0(θ−k·)∈L2,−k. Indeed, EkΦ(k, −k, u0(θ−k·)) −Φ(k−1,1−k, u0(θ1−k·))k2 =EkΦ(k−1,1−k, Φ(1,−k, u0(θ−k·))) −Φ(k−1,1−k, u0(θ1−k·))k2 ≤eµ(k−1)EkΦ(1,−k, u0(θ−k·)) −u0(θ1−k·)k2 =eµ(k−1)EkΦ(1,0, u0(·)) −u0(θ1·)k2 Here we have applied Lemma 3.1 i), the {θt}t∈R-invariance of Pand (12). The Cauchy sequence property follows since µ < 0. Let the limit of this sequence be denoted by X∗∈L2,0. X∗as an element in L2,0is independent of the choice of u0∈L2,0. Indeed, for v0∈L2,0we have that EkΦ(k, −k, u0(θ−k·)) −Φ(k, −k, v0(θ−k·))k2 =EkΦ(k, 0, u0(·)) −Φ(k, 0, v0(·))k2≤eµkEku0−v0k2, which goes to zero for k→ ∞ such that the limit of the above Cauchy sequence in L2,0is independent of u0. Since X∗∈L2,0the process Φ(·,0, X∗) satisfies all of the conclusions of Theorem 2.1. We now show that Φ(t, 0, X∗)(·) = X∗(θt·) almost surely for any t∈R+. By the definition of X∗(ω) the random variable X∗(θt·) is given by (L2) lim k→∞ Φ(k, −k, u0(θ−k·))(θtω) which is equal to (L2) lim k→∞ Φ(k, t −k, u0(θt−k·))(ω) 16 TOM´ AS CARABALLO, PETER E. KLOEDEN, AND BJ ¨ ORN SCHMALFUSS for a sufficiently small ε > 0. Comparing the coefficients of (29) we have kψ(t)k2≤ R2(t) if kψ(0)k2≤R2(0) where R2(t) is the solution of dR2 dt =2(a+ N X j=1 λjbj|z∗ j(θtω)|+kT(θtω)kkT−1(θtω)kLf+ε 2)R2 +1 εkT−1(θtω)k2kf(0)k2. (30) Under the assumptions of the theorem, this equation has the unique exponentially stable stationary solution t→R2(θtω) defined by the random variable R2(ω) : = Z0 −∞ 1 εkT−1(θtω)k2kf(0)k2× ×exp  Z0 t 2(a+ N X j=1 λjbj|z∗ j(θτω)|+kT(θτω)kkT−1(θτω)kLf+ε 2)dτ dt (31) for ω∈Ω, which in turn follows easily by the variation of constants formula. This random variable is tempered (and finite), see Lemma 4.6 below. In particular the ball in Hgiven by G(ω) := B(0,2R(ω)) is mapped into itself: ψ(t, ω, G(ω)) ⊂G(θtω) for ω∈Ω. Indeed this ball is tempered. In addition, this ball has the property (11). To see this we consider the differential equation (30) with some random tempered initial condition r2(ω). If we replace ωby θ−tωfor the solution of (30) at time twe have by the variation of constants formula r2(θ−tω) exp  Z0 t 2(a+ N X j=1 λjbj|z∗ j(θτω)|+kT(θτω)kkT−1(θτω)kLf+ε 2)dτ  +Z0 t 1 εkT−1(θsω)k2kf(0)k2× ×exp  Z0 s 2(a+ N X j=1 λjbj|z∗ j(θτω)|+kT(θτω)kkT−1(θτω)kLf+ε 2)dτ ds. This term tends to R2(ω) as t→ −∞ which follows by Birkhoff’s ergodic theorem, (23), (28), (25). In particular the first term tends to zero. From Theorem 2.4 we just obtain the existence of a random attractor A={A(ω)}ω∈Ω⊂G. Set ∆ψ(t, ω, x1, x2) = ψ(t, ω, x1)−ψ(t, ω, x2). Then we have dk∆ψ(t)k2 dt ≤2(a+ N X j=1 λjbj|z∗ j(θtω)|+kT(θtω)kkT−1(θtω)kLf)k∆ψ(t)k2. 17 We can conclude by the invariance property (9) sup y1,y2∈A(ω) ky1−y2k2≤sup x1,x2∈A(θ−tω) kx1−x2k2× ×exp  Z0 t 2(a+ N X j=1 λjbj|z∗ j(θτω)|+kT(θτω)kkT−1(θτω)kLf)dτ  (32) Since A ⊂ G, it is tempered, and we obtain from the properties of z∗ jand Tthat the right hand side tends to zero for ω∈Ω, hence A(ω) is a random fixed point denoted by Y∗. To see that Y∗is exponentially attracting we note that sup x∈D(ω) kψ(t, ω, x)−Y∗(θtω)k2= sup x∈D(ω) kψ(t, ω, x)−ψ(t, ω, Y ∗(ω))k2 ≤sup x∈D(ω) kx−Y∗(ω)k2× ×exp  Zt 0 2(a+ N X j=1 λjbj|z∗ j(θτω)|+kT(θτω)kkT−1(θτω)kLf)dτ , and the right hand side tends to zero exponentially fast for ω∈Ω. ¤ Since we now know that A(ω) is a single point we have similarly to (32) and (10) that Corollary 4.5. The random fixed point Y∗(hence X∗) attracts tempered random sets in the pullback sense. We now prove the temperedness of R2. Lemma 4.6. For ω∈Ωthe mapping t→R2(θtω)has subexponential growth, hence R2is tempered. Proof. We abbreviate α(ω) =a+ N X j=1 λjbj|z∗ j(ω)|+kT(ω)kkT−1(ω)kLf+ε 2,Eα=: ¯α < 0, β(ω) =1 εkT−1(ω)k2kf(0)k2. By the definition of Ω we have Z0 t α(θτω)dτ ∼¯α|t|for ω∈Ω, t → −∞. In addition t→β(θtω) has sub-exponential growth for t→ ±∞ such that R2(ω) = Z0 −∞ exp µZ0 t α(θτω)¶β(θtω)dt < ∞. 18 TOM´ AS CARABALLO, PETER E. KLOEDEN, AND BJ ¨ ORN SCHMALFUSS For an arbitrary c > 0, 0 < ε < min(−¯α, c 2)/4 and negative s < s0(ω, ε) we then have ecs Z0 −∞ eR0 tα(θτ+sω)dτ β(θs+tω)dt ≤ec 2sZ0 −∞ eR0 t+s(α(θτω)−¯α)dτ−R0 s(α(θτω)−¯α)dτ−¯αt+sc 2+log+β(θt+sω)dt ≤ec 2sZ0 −∞ e−3ε(t+s)−¯αt+c 2sdt ≤ec 2sZ0 −∞ eεtdt. But the right hand side tends to zero for s→ −∞. Similarly, we obtain the convergence for t→+∞, see also Arnold [1] Proposition 4.1.3. ¤ Corollary 4.7. Suppose that fand Bjcommute, i.e. it holds T(ω)−1f(T(ω)x) = f(x),for ω∈Ω, x ∈H. Then the conclusion of Theorem 4.4 also holds if instead of (28) we assume a+Lf<0. Proof. Since the expectation of z∗ jdepends on λjsuch that E|z∗ j| ≤ 1 pλj , we can choose λjsufficiently small so that PN j=1 bjλjE|z∗ j|is also arbitrarily small. ¤ Remark 4.8. Usually Bjand fcommute if they are diagonal in some orthogonal basis of H(see Kwiecinska [18] and our example in Section 6) . The commutativity assumptions for Biensure that (20) can be transformed into a random differential equation. However, there exist other transformations for very special classes of stochastic evolution equations to be transformed into random differential equations, see e.g. Flandoli and Lisei [11] and Mohammed et al. [21]. For the resulting random evolution equation our theory can be applied. 5. Non-linear perturbations We now want to show that if we change the nonlinear part of (3) continuously, then the fixed points also change continuously. For this we study the family of problems indexed by n∈Z+given by (33) dX =AXdt +fn(X)dt +B(X)dW, X(0) = u0. We suppose that the constants µngiven by (13) corresponding to the functions fn, satisfy µn<0 uniformly for n∈Z+. We will also denote by Φn(·,0, u0), n = 0,1,2, ..., the solution to (33), and by X∗ ntheir associated random fixed points. Then, our objective is to prove that the X∗ nare close to X∗ 0if the fnis close to f0 in some sense. 19 Theorem 5.1. Consider the family of stochastic evolution equations (33). Suppose that µ:= sup n∈Z+ (2a+ 2Lfn+LB)<0, and, in addition, that lim n→∞ fn(u) = f0(u),for u∈H. Then, for the corresponding fixed points we have (L2) lim n→∞ X∗ n=X∗ 0. Proof. One can find the idea of the proof in Zeidler [26] Proposition 1.2 . Denoting k · k2 L2=Ek·k2and taking into account (12), we have kX∗ n−X∗ 0kL2=kΦn(1,−1, X∗ n(θ−1·)) −Φ0(1,−1, X∗ 0(θ−1·))kL2 =kΦn(1,0, X∗ n)−Φ0(1,0, X∗ 0)kL2 ≤kΦn(1,0, X∗ n)−Φn(1,0, X∗ 0)kL2+kΦn(1,0, X∗ 0)−Φ0(1,0, X∗ 0)kL2 ≤eµkX∗ n−X∗ 0kL2+kΦn(1,0, X∗ 0)−Φ0(1,0, X∗ 0)kL2, so (34) kX∗ n−X∗ 0kL2≤1 1−eµkΦn(1,0, X∗ 0)−Φ0(1,0, X∗ 0)kL2. Now it is not hard to prove that the right hand side tends to zero. Indeed, we set Xn(t) = Φn(t, 0, X∗ 0), n ∈Z+, corresponding to the solution of (3) with f=fn. Then by the Ito formula we obtain d dtEkXn(t)−X0(t)k2≤µEkXn(t)−X0(t)k2+Ekfn(X0(t)−f0(X0(t))k2 EkXn(0) −X0(0)k2= 0. The inequality kfn(X0(t))k2≤2 sup n∈Z+ kfn(0)k2+ 2LkX0(t)k2, L := sup n∈Z+ Lfn<∞ allows us to find an integrable majorant for kfn(X0(t)) −f0(X0(t))k2such that pointwise convergence of fn(u) to f0(u) and the variation of constants formula yield the convergence for the right hand side of (34). ¤ We now consider a family of evolution equations (20) with f=fn. To obtain a family of equations of the form (26) we can apply the transformation Twhich is independent of n. Theorem 5.2. Suppose that the Lipschitz constants of fnare uniformly bounded by L, that a+ N X j=1 bjλjE|z∗ j|+LΠN j=1E(kSBj(−z∗ j)kkSBj(z∗ j)k)<0, and that lim n→∞ fn(x) = f0(x)for x∈H. Let X∗ n, n ∈Z+,be the random fixed points of (20) with fninstead of f. Then lim n→∞ X∗ n(ω) = X∗ 0(ω),for ω∈Ω. 20 TOM´ AS CARABALLO, PETER E. KLOEDEN, AND BJ ¨ ORN SCHMALFUSS Proof. As in the proof of Theorem 4.4 we investigate the random dynamical systems ψngenerated by (26) with f=fnand fixed points Y∗ n(ω) attracting tempered sets, see Corollary 4.5. We have supn∈Nkfn(0)k<∞, so, as in the proof of Theorem 4.4, there exists a tempered set G(ω) containing all fixed points Y∗ n(ω). This set is given by a ball B(0,2R(ω)) where R2is a stationary solution of the one-dimensional affine differential equation (30) where Lfhas to be replace by L= supn∈Z+Lfn and kf(0)kby supn∈Z+kfn(0)k. By the assumptions R2(ω) exists. We have kY∗ n(ω)−Y∗ 0(ω)k=kψn(t, θ−tω, Y ∗ n(θ−tω)) −ψ0(t, θ−tω, Y ∗ 0(θ−tω))k ≤ kψn(t, θ−tω, Y ∗ n(θ−tω)) −ψn(t, θ−tω, Y ∗ 0(θ−tω))k +kψn(t, θ−tω, Y ∗ 0(θ−tω)) −ψ0(t, θ−tω, Y ∗ 0(θ−tω))k. Since the second factor on the right hand side of (32) is independent of nif we replace Lfby L, and Y∗ 0, Y ∗ nare contained in the tempered ball G, we have for any ε > 0, ω ∈Ω and tthat the first term on the right hand side is less than ε/2. In addition, for this tsimilar to the proof of Theorem 5.1 the second term on the right hand side is also less than ε/2 if nis large. We obtain kY∗ n(ω)−Y∗ 0(ω)k< ε for large n. Since the transformation Tis a homeomorphism we have the conclusion. ¤ 6. An example Let us take H=L2(O) and V=H1 0(O) where Ois a bounded domain in Rd with a smooth boundary. Consider the Laplace operator A= ∆ with the Dirichlet boundary condition. Theorem IX.31 in Brezis [2] ensures the existence of a sequence of real numbers {νn}n≥1such that 0 < ν1< ν2<··· < νn<· · · , and νn→+∞ (namely, the eigenvalues of the Laplacian), and a sequence {en}n≥1⊂V∩C∞(O) of associated eigenvectors (i.e. −∆en=νnenon O) which is a complete orthonormal basis in H. On the other hand, let us consider operators Bi,i= 1,· · · , N such that they are diagonalizable in the same basis and are bounded from above and below (see Kwiecinska [18]), i.e., there exist constants dk i, k = 1, . . . , N such that (Bkei, ej) = dk iδij, k = 1, . . . , N;i, j = 1,2, . . . The assumption that the operators Bkare bounded from above and below is equivalent to the condition that there exist positive constants mk, Mksuch that 0< mk≤ |dk i| ≤ Mk, k = 1, . . . , N;i= 1,2, . . . Under these conditions the operators Bigenerate C0-groups SBi(t) = eBit. Finally, consider a Lipschitz continuous function ffrom Hinto Hdefined as f(u)(x) = F(u(x)), x∈ O with F:R→RLipschitz continuous with constant Lf. Recall that hAu, ui≤−ν1kuk2for all u∈V, and therefore the semigroup generated by Asatisfies kS(t)k ≤ e−ν1tfor all t≥0. We now study the stochastic evolution equation (35) dX = (AX +f(X))dt + N X i=1 BiXdwi. 21 It follows that the constant µin Theorem 3.3 satisfies µ=−2ν1+ 2Lf+LB≤ −2ν1+ 2Lf+ N X i=1 M2 i. If µ < 0, Theorem 3.3 implies the existence of a unique stationary solution to our problem which is exponentially stable in mean square. Also, thanks to Lemma 3.1 ii) and Theorem 3.5 the almost sure exponential stability of this stationary solution holds. Observe that µis negative only if the Lipschitz constants Lfand LBare sufficiently small. However, by using the technique of random dynamical systems we can prove stability behaviour even for larger values of the constant LB. Indeed, we will be able to apply Theorem 4.4. To this end, we consider our equation (35) in its Stratonovich equivalent form dX ="(A−1 2 N X i=1 B2 i)X+f(X)#dt + N X i=1 BiX◦dwi. Denote C=A−1 2PN i=1 B2 iwhich also generates a C0-semigroup SC(t) . If the operators A, B1, ..., BNcommute mutually, then this semigroup SC(t) is given as SC(t) = S(t)e−t 2PN i=1 B2 i. Now, by easy computations (see [18]) we can deduce e−t 2PN i=1 M2 i≤ ke−t 2PN i=1 B2 ik ≤ e−t 2PN i=1 m2 i and assumption (28) in Theorem 4.4 becomes eµ=−ν1−1 2 N X i=1 m2 i+ N X i=1 MiλiE|z∗ i|+Lf N Y i=1 E(kSBi(−z∗ i)kkSBi(z∗ i)k)<0. Consequently, if the operators Biare such that eµ < 0, we can apply Theorem 4.4 and ensure that there exists a unique exponentially stable stationary solution given by a random fixed point. Notice that this can provide better stability results than the ones obtained in Section 3. Indeed, assume for instance that Biu=miuwhere mi∈R+for i, ··· , N. This means that mi=Mi, and it is easy to check that eµ=−ν1−1 2 N X i=1 m2 i+ N X i=1 miλiE|z∗ i|+Lf, since now kSBi(−z∗ i)kkSBi(z∗ i)k= 1.Then, if −ν1−1 2 N X i=1 m2 i+Lf<0 (what happens if, for example, the noise intensities miare large enough), we can choose stationary process z∗ icorresponding to λisuch that eµ < 0. Thus, some kind of stabilization has been obtained for the non-trivial stationary solution. Acknowledgements. This work was partially supported by the DAAD (Germany), and the Ministerio de Ciencia y Tecnolog´ıa (Spain) under the projects HA2001-0075 and BFM2002-03068. 22 TOM´ AS CARABALLO, PETER E. KLOEDEN, AND BJ ¨ ORN SCHMALFUSS T. Caraballo would like to thank B. Schmalfuss and his family for the kind hospitality they all offered him during his visit in November 2002. References [1] L. Arnold. Random Dynamical Systems. Springer, New York, 1998. [2] H. Brezis. Analyse Fonctionnelle. Masson, Paris, 1983. [3] T. Caraballo and J. Langa. 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E-mail address, Tom´as Caraballo: [email protected] E-mail address, Peter E. Kloeden: [email protected] E-mail address, Bj¨orn Schmalfuß: [email protected] merseburg.de (Tom´as Caraballo) Dpto. Ecuaciones Diferenciales y An´ alisis Num´ erico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla (Spain) (Peter E. Kloeden) Fachbereich Mathematik, Johann Wolfgang Goethe-Universit¨ at, D60054 Frankfurt am Main, Germany (Bj¨orn Schmalfuß) Department of Applied Sciences, University of Technology and Applied Sciences, Geusaer Strasse, D–06217 Merseburg, Germany,