Tracking Properties of Trajectories On Random Attracting Sets

tracking properties of trajectories on random attracting sets Tom´as Caraballo and Jos´e A. Langa Dpto. Ecuaciones Diferenciales y An´alisis Num´erico. Universidad de Sevilla. Apartado de Correos 1160. 41080–SEVILLA (Spain) e-mails: [email protected], [email protected] Abstract The theory of random attracting sets highlights interesting properties of the asymptotic behaviour of some stochastic differential equations. In this paper some results on the relation between the dynamics on random attractors and stochastic inertial manifolds, and the dynamics in the associated random dynamical system are studied. In particular, some tracking properties of trajectories on random attractors and a general result on the asymptotic completeness of stochastic inertial manifolds are shown. 1 INTRODUCTION One of the main concepts for the study of the asymptotic behaviour of dissipative dynamical systems is the global attractor (see Constantin et al. [4], Hale [18], Temam [26] and the references therein). This is a compact, invariant set attracting uniformly every trajectory starting in a bounded set of the phase space. The understanding of the dynamics on the global attractor gives us relevant information about the asymptotic behaviour of the dynamical system. One of the most important results in the theory of global attractors claims that the fractal, and so the Hausdorff, dimension of this set is finite, even if the dynamical system is posed in an infinite-dimensional phase space. That is, although the trajectories depend on an infinite number of degrees of freedom, the finite dimensionality of the attractors seems to be describing the asymptotic behaviour of the dynamical system with a finite number of timedependent coordinates. This makes even more interesting the study of the dynamics on the global attractor, as we expect that this dynamics can be described by a system of ordinary differential equations (see Eden et al. [12], chapter 10, and Robinson [23],[24]). In fact, this is what we obtain in the theory of inertial manifolds (smooth invariant manifolds which attracts every trajectory exponentially fast, see Foias et al. [16]), that is, the dynamics on the inertial manifold, governed by a system of ordinary differential equations, is determining the asymptotic behaviour of the dynamical system. From a geometrical point of view, this fact can be seen when it is proved that every trajectory of the system can be followed arbitrarily close, as time grows to infinite, by another trajectory moving on the inertial manifold. We say in this case that the inertial manifold satisfies a tracking property (Foias et al. [17]) or that it is asymptotically complete (Robinson [22]). In recent years, Crauel and Flandoli [7] (see also Schmalfuss [25] and Crauel et al. [6]) have introduced a concept for the attractor of some stochastic differential equations. The generalization to this situation is not trivial, as in this case the system is non autonomous and the new forcing term may have very large fluctuations which make that solutions are pushed out from any bounded ball in the phase space. However, it is possible to define in these cases a generalized concept of global attractor as a moving (in the parameters time and omega of the random term) compact set, invariant with respect to the shift associated to the random dynamical system (see section 2) and attracting, backwards in time, all the trajectories starting in any bounded set. This set is called random attractor and, although the convergence property is proved, roughly speaking, from −∞, it is easy to show that we also have convergence in probability, forward in time, to the random attractor. On the other hand, there are some results on the finite dimensionality (with probability one) of the random attractor (see Debussche [10], [11]), which motivates again the study of the dynamics on these random attracting sets to know in what sense it is determining the asymptotic behaviour of these stochastic differential equations. In this direction, Flandoli and Langa [14] have proved a result on determining modes for random dynamical systems which generalizes the results of Foias and Prodi [15] to the stochastic case. In section 3, we present a generalization of a result already known in the deterministic case (see Langa and Robinson [21]). There are also some papers extending the concept of inertial manifold to the stochastic case (see Bensoussan and Flandoli [3], Chueshov and Girya [8] or Da Prato and Debussche [9]). A stochastic inertial manifold is a random invariant Lipschitz manifold which attracts every trajectory exponentially fast. As in the deterministic case, the inertial manifold is given as the graph of some (random) Lipschitz function. In section 4, a result that shows the asymptotic completeness property of these stochastic inertial manifolds is proved. In fact, it is even more general than classical results on asymptotic completeness on inertial manifolds in the deterministic case, since it is still true for general invariant exponentially attracting sets and not necessarily smooth manifolds given as graphs of some functions. The result can be applied to the problems studied in [3] and [8]. Finally, some conclusions and possible generalizations are presented in last section. 2 RANDOM DYNAMICAL SYSTEMS AND ATTRACTORS Let (Ω,F, P) be a probability space and {θt: Ω →Ω, t ∈} a family of measure preserving transformations such that (t, ω)7→ θtωis measurable, θ0= id, θt+s=θtθs, for all s, t ∈. The flow θttogether with the probability space (Ω,F, P, (θt)t∈) is called a (measurable) dynamical system. Furthermore, we suppose that the shift θtis ergodic. Arandom dynamical system (RDS) on a Polish space (X, d) with Borel σ-algebra Bover θton (Ω,F, P) is a measurable map ϕ:+×Ω×X→X (t, ω, x)7→ ϕ(t, ω)x such that P−a.s. i) ϕ(0, ω) = id (on X) ii) ϕ(t+s, ω) = ϕ(t, θsω)◦ϕ(s, ω),∀t, s ∈+(cocycle property). A RDS is continuous or differentiable if ϕ(t, ω) : X→Xis continuous or differentiable. The theory of random dynamical systems covers all systems with randomness, in particular random and stochastic difference and differential equations (see Arnold and Crauel [2]). A random set K(ω) is said to absorb the set B⊂Xif P−a.s. there exists tB(ω) such that for all t≥tB(ω) ϕ(t, θ−tω)B⊂K(ω). Finally, a random set A(ω) is a random attractor associated to the RDS ϕif P−a.s. i) A(ω) is a random compact set, ii) ϕ(t, ω)A(ω) = A(θtω),∀t≥0 (invariance) and iii) for all B⊂Xbounded (and nonrandom) lim t→+∞dist(ϕ(t, θ−tω)B, A(ω)) = 0, where dist( . , . ) denotes the Hausdorff semidistance dist(A, B) = sup a∈A inf b∈Bd(a, b), A, B ⊂X. In this situation, we have the following theorem about existence of random attractors due to Crauel and Flandoli ([7], theorem 3.11). Theorem 1 Suppose there exists a compact set D(ω)absorbing every bounded nonrandom set B⊂X. Then, the set A(ω) = [ B⊂X ΛB(ω) is a random attractor for ϕ, where the union is taken over all B⊂Xbounded, and ΛB(ω)denotes the omega-limit set of Bwhich is given by ΛB(ω) = \ n≥0[ t≥n ϕ(t, θ−tω)B. 2 Moreover, in Crauel [5] it is shown that random attractors are unique and, under the ergodicity assumption on θt, there exists a compact set K⊂X such that P−a.s. the random attractor is the omega limit set of K, that is, A(ω) = \ n≥0[ t≥n ϕ(t, θ−tω)K. 3 A TRACKING PROPERTY ON RANDOM ATTRACTORS In this section we shall prove a property on random attractors already known for (deterministic) global attractors (see Langa and Robinson [21]). It says, roughly speaking, that given a random dynamical system for which there exists a random attractor, every random trajectory can be followed arbitrarily closely by skipping from one solution to another on the random attractor. Let us consider the following partial differential equation on the Hilbert space H(norm |.|) perturbed by an additive H−valued white noise process on the probability space (Ω,F, P) with shift θton Ω du =−Audt +f(u)dt +dW(t),(1) where Ais a linear operator, selfadjoint, positive and with compact inverse, whose domain is D(A), and fis the nonlinear term. In these conditions, there exist an increasing sequence of eigenvalues of A,λn%+∞, and the corresponding sequence of eigenfunctions {en}∞ n=1, which forms an orthogonal basis in H. Let ϕ:+×Ω×H→Hbe a random dynamical system associated to problem (1) and assume the following property of continuity with respect to initial conditions: there exists L > 0 such that, given s≥0, it follows that, for all u, v ∈H, for all t≥0, and P−a.s. |ϕ(t, θsω)u−ϕ(t, θsω)v| ≤ eLt|u−v|.(2) In addition to the precedent hypotheses, suppose the ones for the existence of a random attractor are satisfied, so that there exists a random compact set A(ω), invariant, and such that, for all B⊂Hbounded and P−a.s. lim t→+∞dist(ϕ(t, θ−tω)B, A(ω)) = 0. Since the shift θtis measure preserving, it is known (see Crauel and Flandoli [7]) that this last convergence implies convergence in probability, that is, for all ² > 0, lim t→+∞P(dist(ϕ(t, ω)B, A(θtω)) < ²) = 1.(3) ¿From (2) and (3) the following result holds: Proposition 1 Writing a trajectory of problem (1) as ϕ(t, ω)u0,u0∈H, and given 0< ² < 1,0< δ < 1, and T > 0, there exists a time τ(², δ, T)>0 such that, for all τ≥τ(², δ, T), there exists e Ωwith P(e Ω) >1−δsuch that, if ω∈e Ω, there exists a point vτ∈ A(θτω)satisfying |ϕ(t+τ, ω)u0−ϕ(t, θτω)vτ|< ² , for all 0≤t≤T. Proof. Given 0 < ², δ < 1, T > 0, from (3) we deduce that there exists τ=τ(², δ, T)>0 such that, ∀t≥τ(²), P(dist(ϕ(t, ω)u0,A(θtω)) < ²e−LT )≥1−δ. In particular, for t=τ, there exists Ωτ,P(Ωτ)≥1−δsuch that, for all ω∈Ωτ, dist(ϕ(τ, ω)u0,A(θτω)) < ²e−LT . As A(θτω) is compact, there exists vτ∈ A(θτω) such that |ϕ(τ, ω)u0−vτ| ≤ ²e−LT , and, by (2), |ϕ(t, θτω)ϕ(τ, ω)u0−ϕ(t, θτω)vτ| ≤ |ϕ(τ, ω)u0−vτ|eLt ≤², for all t∈[0, T], and this gives the proposition by the cocycle property. 2 As a consequence of this result we obtain the following tracking property for trajectories of problem (1). Corollary 1 Given 0< δ < 1,T > 0, and {²n}∞ n=1,²n>0,²n&0, there exist a sequence of times {τn}∞ n=1,τn%+∞, τn+1 > τn∀n∈N, τn+1 −τn→ ∞ as n→ ∞, and a subset e Ω⊂Ωwith P(e Ω) >1−δ, such that, for each ω∈e Ω, there exists vτn∈ A(θτnω)satisfying |ϕ(t+τn, ω)u0−ϕ(t, θτnω)vτn|< ²n,0≤t≤nT, ∀n∈ Proof. Given 0 < δ < 1, we choose a decreasing sequence of {δn}∞ n=1 such that ∞ X n=1 δn< δ (for instance, δn=δ 2n+1 ).(4) Then, applying Proposition 1 for ²1,δ1and T, there exists τ1=τ(δ1, ²1, T) such that, for all τ≥τ(δ1, ²1, T),there exists Ωτ⊂Ω with P(Ωτ)>1−δ1 satisfying that, for each ω∈Ωτ, there exists vτ∈ A(θτω) with |ϕ(t+τ, ω)u0−ϕ(t, θτω)vτ|< ²1,for all 0 ≤t≤T. Now, again by Proposition 1 for ²2,δ2and 2T, we get that there exists τ2=τ(δ2, ²2, T)≥τ1such that, for all τ≥τ2,there exists Ωτ⊂Ω with P(Ωτ)>1−δ2such that, for each ω∈Ωτ, there exists vτ∈ A(θτω) with |ϕ(t+τ, ω)u0−ϕ(t, θτω)vτ|< ²2,for all 0 ≤t≤2T. In general, given ²n,δnand nT, there exists τn=τ(δn, ²n, T)≥τn−1such that, for all τ≥τn, there exists Ωτ⊂Ω with P(Ωτ)>1−δnsuch that, for each ω∈Ωτ, there exists vτ∈ A(θτω) with |ϕ(t+τ, ω)u0−ϕ(t, θτω)vτ|< ²n,for all 0 ≤t≤nT. Let us call Ωn⊂Ω the set with the property that, for all ω∈Ωn, there exists vτn∈ A(θτnω) satisfying |ϕ(t+τn, ω)u0−ϕ(t, θτnω)vτn|< ²n,0≤t≤nT. We have that P(Ωn)≥1−δn.Denoting e Ω = ∩∞ n=1Ωn, it is clear from (4) that P(e Ω) >1−δ. Now, taking ω∈e Ω we have that there exists vτn∈ A(θτnω) with |ϕ(t+τn, ω)u0−ϕ(t, θτnω)vτn|< ²n,0≤t≤nT, for all n∈,and thus P(|ϕ(t+τn, ω)u0−ϕ(t, θτnω)vτn|< ²n,0≤t≤nT, ∀n∈)>1−δ. 2 3.1 Application. A reaction-diffusion equation with additive noise Let D⊂nbe an open bounded set with regular boundary and f(u) = 2p−1 X k=0 akuk, a2p−1<0. We consider the following partial differential equation of reaction-diffusion type in Dwith an additive white noise process:          du = ∆udt +f(u)dt +Pd i=1 φidWi tin D u= 0 on ∂D u(0) = u0 (5) where Wi t: Ω →,t∈, are independent one dimensional two-sided Wiener processes on a probability space (Ω,F, P). As it is well known, (5) can be expressed as a differential equation in H= L2(D),    du =Audt +F(u)dt +Pd i=1 φidWi tin H u(0) = u0 (6) where A:D(A)⊂H→H,Au = ∆u,F:Z→Z0,Z=L2p(D) and Z0=L(2p)0(D),with (2p)0= (2p−1)/2p, and is defined as F(u) = f(u). We take φi∈D(A). We can define a random dynamical system ϕ(t, ω) : H→Hon (Ω,F, P, (θt) (t∈), with the shift θtergodic, for which the existence of a random attractor A(ω) has already been proved (for the study of the random atractor of this problem, see Crauel and Flandoli [7], Crauel et al. [6], and Debussche [10]). Due to the condition on the nonlinear term f, and under standard computations (see, for instance, Debussche [10]), it can be shown that P−a.s. |u(t, ω;u0)−u(t, ω;v0)| ≤ ekt|u0−v0|, so that the continuity property with respect to initial conditions (2) is satisfied and, consequently, the results in this section are true for this problem. Thus, returning to (11) |ϕ(t−(s+h), θs+hω)vs+h−ϕ(t−s, θsω)vs| ≤D(θsω)D(θs+hω)e−γ(t−s)eγhC(ω)e−νs(e−νh +eLh), and thus, for all h≤h0,(previously chosen), ≤D(θsω)D(θs+hω)e−γte−(ν−γ)sC(ω)K. (12) ¿From (12) we can conclude that (10) converges uniformly on bounded intervals of [0,+∞) since, for any τ > T, |ϕ(t−τ, θτω)vτ−ϕ(t−T, θTω)vT| ≤Ke−γtC(ω)e−(ν−γ)T ∞ X n=0 D(θT+nhω)D(θT+(n+1)hω)e−(ν−γ)nh and, by the condition on D(ω), the series above is convergent, so that the last expression is =K0e−γte−(ν−γ)T which tends to zero uniformly for t∈[0, t0], for all t0>0,as T→+∞. Therefore, the limit in (10) exists and satisfies the equation of the differential equation, since it is the uniform limit of solutions of the problem. Moreover, it is now clear that v∞(t, ω) satisfies the tracking property for ϕ(t, ω)u0,since |v∞(t, ω)−ϕ(t, ω)u0| ≤ |v∞(t, ω)−vt(ω)|+|vt(ω)−ϕ(t, ω)u0| ≤ | lim T→∞ ϕ(t−T, θTω)vT−ϕ(t−t, θtω)vt|+|vt(ω)−ϕ(t, ω)u0| ≤KC(ω)e−γt ∞ X n=0 D(θt+nhω)D(θt+(n+1)hω)e−(ν−γ)(t+nh)+C(ω)e−νt ≤K0C(ω)e−γte−(ν−γ)t+C(ω)e−νt ≤C(ω)˜ K0e−νt. 2 Remark. Note that in the proof we have not made any reference to the inertial manifold given as a graph of some Lipschitz function, so that the theorem is true for a general invariant exponentially attracting closed random set which satisfies the flow normally hyperbolic property. 4.1 Application. A Semilinear Stochastic Differential Equation with Additive Noise In Bensoussan and Flandoli [3] (see also Chueshov and Girya [8]) it is proved the existence of a stochastic inertial manifold for the following differential equation with additive noise du(t) + Au(t)dt =R(u(t))dt +dW(t) u(0) = u0, where Ais a selfadjoint positive linear operator with a discrete spectrum and compact inverse, so that there exists a sequence of eigenvalues 0< λ1≤λ2≤ · · · ≤ λn≤ · · · whose corresponding eigenvectors form an orthogonal basis of H. R is the nonlinear term which is Lipschitz continuous with constant LR. Under the hypotheses on R(u), it is clear that condition (2) is satisfied, and that the random variable D(ω) in (8) is independent of ω. Furthermore, in [3] it is proved that under the spectral gap condition λn+1 −λn>4LR(13) there exists a stochastic inertial manifold M(ω) given as the graph of some random function. In this case γ= (1 + M)LR+λn. Indeed, a bound on the separation of trajectories on M(ω) is given by a bound on the separation of trajectories of the ODE dp +Apdt =PmR(p+φt(ω)p)dt +dPmWt, and the Lipschitz constant of Ap +PmR(p+φt(ω)p) is (1 + M)LR+λn. On the other part, we have that the rate of attraction ν=λn+1 −LR(1 + M),where M=Lφ, and it can be chosen to be less or equal to one (see [3]). Thus, from (13) we obtain γ= (1 + M)LR+λn<2LR+λn< λn+1 −2LR< λn+1 −LR(1 + M) = ν, so that, as γ < ν,M(ω) is flow normally hyperbolic and, by theorem 2, it is asymptotically complete. CONCLUSIONS Some results on the relation between the dynamics on random attracting sets and dissipative random dynamical systems have been studied. On the other hand, it has been proved a general result for the asymptotic completeness of invariant exponentially attracting random sets, which can be successfully applied to some interesting problems in the literature for which the existence of stochastic inertial manifolds has been proved. 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