Stabilisation of differential inclusions and PDEs without uniqueness by noise

Stabilisation of differential inclusions and PDEs without uniqueness by noise∗ T. Caraballo1, J.A. Langa1, J. Valero2 1Dpto. Ecuaciones Diferenciales y An´alisis Num´erico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla, Spain. E-mail addresses: [email protected]; [email protected] 2Centro de Investigaci´on Operativa, Universidad Miguel Hern´andez, Avda Universidad s/n, 03202 Elche, Alicante, Spain. E-mail address: jv[email protected] Abstract We prove that the asymptotic behaviour of partial differential inclusions and partial differential equations without uniqueness of solutions can be stabilised by adding some suitable Itˆo noise as an external perturbation. We show how the theory previously developed for the single-valued cases can be successfully applied to handle these set-valued cases. The theory of random dynamical systems is used as an appropriate tool to solve the problem. 1 Introduction The stabilising and destabilising effects produced by noisy terms in the evolution of single valued deterministic systems is now very well known as the literature on this topic reveals (see [1], [4], [5], [13], [18], [20], [22], and the references therein). The importance of these effects in the understanding of the long time behaviour of real systems is now out of any doubt. Indeed, if we assume that the real world is non-deterministic (what seems to be a very sensible fact) and we approximate a real model by a deterministic one, we can find that on some occasions the appearance of noise in the deterministic models could produce dramatic changes in the behaviour (see, e.g. [1], [4], [20]). However, in many real situations, the real models are better described if we consider some multi-valued or set-valued features. For instance, many systems described by differential equations does not have uniqueness of solutions, or even the problem is better modelled by a differential inclusion. So far, we still have not seen in the literature any paper dealing with the ∗Partly supported by Ministerio de Educaci´on y Ciencia (Spain) under grants MTM2005-01412, MTM200503868, Consejer´ıa de Innovaci´on, Ciencia y Empresa (Junta de Andaluc´ıa, Spain) under the Proyecto de Excelencia FQM-02468, and Consejer´ıa de Cultura y Educaci´on (Comunidad Aut´onoma de Murcia) grant 00684/PI/04. 1 effects produced by noise in the asymptotic behaviour of multivalued dynamical systems (in the direction developed in the papers mentioned above). Our aim in this paper is to start an investigation on this topic. In fact, we aim to show how some of the techniques already developed and successfully applied in the single-valued case can also be adapted to handle with some multivalued situations. The objective of our investigation in this paper is twofold. On the one hand, we want to show how a very simple multiplicative noise (in the Itˆo sense) can produce a stabilising effect on the solutions of a deterministic partial differential inclusion. This is an important fact when we want to stabilise an unstable system by acting on it with an external forcing term. Although there exists a controversy on the use of different kinds of noise (Itˆo versus Stratonovich), we will not go deeper in this discussion and simply show, in Section 2, a first easy way to produce stabilisation of a deterministic partial differential inclusion. It may be possible to obtain the same result with a much more complicate noisy term, and even one may try to get stabilisation by using linear Stratonovich noise (as in [13]), or much general additive noise (as in [4]), but these will be the topics for some future papers. On the other hand, we will show in Section 3 how the theory of random dynamical systems can be a suitable and helpful tool in the analysis of the effects produced by noise in some deterministic multivalued dynamical systems. In fact, we will consider a partial differential equation (of reaction-diffusion type) without uniqueness of solutions, and which generates a multi-valued or set-valued dynamical system but without having a global attractor. Then, if we add a high-intensity multiplicative linear Itˆo noise, we will prove that the stochastic model generates a random dynamical system which possesses a random attractor. This reflects a regularising/stabilising effect of the noise. Even more, in some situations and for a higher intensity of the noise this random attractor becomes a single random point (random equilibria). 2 Stabilising evolution inclusions 2.1 Setting of the problem Let Vbe a separable and reflexive Banach space (with norm || · || and inner product ((·,·))), and consider a Hilbert space H(with norm | · | and inner product (·,·)). If we identify Hwith its dual space, then we can identify Hwith a subspace of V0, so that we have V ,→H ,→V0, where the previous inclusions are continuous and dense. We will denote by || · ||∗the norm in V0and by h·,·i the duality product between Vand V0. Now, let us consider the following stochastic evolution inclusion in the Ito sense    du (t) dt ∈Au (t) + F(u(t)) + Pd i=1 Biu(t)dwi(t) dt ,0≤t < +∞, u(0) = u0∈H, (1) where w1, w2, ..., wdare mutually independent standard Wiener processes over the same filtered probability space (Ω,F,{Ft}t≥0,P), Bi:H→His a linear operator for i= 1, ..., d, A :V→V0 2 is a linear Aoperator which is the infinitesimal generator of a strongly continuous semigroup (i.e. of class C0) denoted by S(t). As we are interested in analysing the behaviour of the variational solutions of (1) (see below for the definition), we need to assume some additional hypotheses ensuring their existence. To be more precise, we need the following assumptions: Coercivity: There exist α > 0, λ ∈Rsuch that −2hAu, ui+λ|u|2≥α||u||p,for all u∈V, (2) where p > 1 is fixed. Boundedness: There exists β > 0 such that ||Au||∗≤β||u||p−1,for all u∈V. (3) Notice that, in the case p= 2, condition (2) implies that operator Ais the generator of a strongly continuous semigroup (see Dautray and Lions [15, page 388]). On the other hand, we assume that F:H→2Hsatisfies: (F1) Fhas closed, bounded, convex, non-empty values. (F2) There exists C > 0 such that distH(F(u), F (v)) ≤C|u−v|,∀u, v ∈H, where distH(·,·) denotes the Hausdorff distance between bounded sets. (F3) F(0) = 0. Under the preceding assumptions (in fact without assuming (2), (3) and (F3)), Theorem 2.1 in Da Prato and Frankowska [16] ensures the existence of at least one solution u(·) of (1) for any random variable u0∈Lp(Ω,F0,P, H) with some p > 2. By such a solution we mean an adapted process u(·) taking values in Hand such that: 1. u(·, ω) is continuous for P-a.a. ω∈Ω. 2. For any T > 0, u(·) is a mild solution, on the interval [0, T],of the problem ½du (t) = Au(t)dt +f(t)dt +Pd i=1 Biu(t)dwi(t), u(0) = u0,(4) in other words, we have for all t∈[0, T], u(t) = S(t)u0+Zt 0 S(t−s)f(s)ds + d X i=1 Zt 0 S(t−s)Biu(s)dwi(s), being f(·) an adapted process such that f(s, ω)∈F(u(s, ω)) , for a.a. (s, ω)∈(0, T)×Ω, EµZT 0 |f(s)|2ds¶<∞. 3 Observe that, for the selection f(·),the unique mild solution to (4) is given by u(·). Remark 2.1 Although we have imposed that the initial value u0belongs to the space Lp(Ω,F0,P, H) for some p > 2, there are some interesting cases in which it is possible to take p= 2 (see Da Prato and Frankowska [16] for more details on this point). However, in order to apply Itˆo’s formula (or to make an appropriate change of variable) we need to handle a stronger concept of solution, say, either the so-called strong solution or the variational concept of solution. In this paper we consider the latter one. In addition to the previous assumptions we do assume now (2) and (3). Then (see Pardoux [21]), for any u0∈Lp(Ω,F0,P, H) (p > 2), there exists a unique variational solution of problem (4). In other words, there exists a stochastic process v(·) which belongs to Lp(Ω ×(0, T); V)∩ L2(Ω; C(0, T;H)) and that satisfies the equation in (4) in the sense of V0,i.e., it follows that, for all t∈[0, T], v(t) = u0+Zt 0 (Av(s) + f(s)) ds + d X i=1 Zt 0 Biv(s)dwi(s),for P−a.a. ω∈Ω,(5) where the equality is understood in the sense of V0. Now, taking into account that the variational solution (when it exists) is also a mild solution (see, e.g. Caraballo [3]) we have that, for an initial datum u0∈Lp(Ω,F0,P, H),and for the selection f(·) in (4), we can ensure the existence of a unique variational solution which is also a solution of (1) in the sense of Da Prato and Frankowska. So, from now on, when we deal with a solution of (1) we will be always referring to this one. 2.2 Stabilisation by a linear one-dimensional noise An important task in the asymptotic behaviour of dynamical systems is the analysis of the stability properties. As it has been mentioned in the Introduction, on some occasions it might be very important to act on a system so that its stability properties can be improved. For instance, the original (deterministic or stochastic) system may be unstable and after our action it becomes stable (or being already stable, we might be able of improving their stability, say, we increase the approaching speed of solutions, etc....). Although this problem has been extensively studied in the literature in the single-valued case, as far as we know, it still has not been considered in a set-valued framework. It is our aim here to establish some preliminary results which can serve as the basis for further investigations in this field. As it has been noted in the single-valued case, in order to produce a stabilization effect on deterministic (and even stochastic) systems one does not need to perturb the model with a very general noise (provided it is considered in the Itˆo sense). In fact, a very simple multiplicative one is enough. However, if we consider the noise in the sense of Stratonovich, the analysis requires of a much more complicate structure in the noise (see, e.g. [13] and [4] for a detailed discussion on this topic).Nevertheless, we do not aim to go deeper in this direction in the present paper, since this will be the topic of some future work. Now we can establish our main stabilisation result 4 Theorem 2.2 Assume that B2=· · · =Bd= 0 and B1is given by B1v=σv for v∈H, and σ∈R.Under the preceding hypotheses, for a large enough σ2such that γ=σ2−λ−2C > 0(λ and Care the constants appearing in (2) and (F2)), there exists Ω0⊂Ωwith P(Ω0)=0and a random variable T(ω)≥0, such that for any initial datum u0∈Lp(Ω,F0,P, H)(p > 2), any of its corresponding solutions u(·)of problem (1) satisfies |u(t, ω)|2≤e−γt/2|u0(ω)|2for all t≥T(ω), a.s. (6) Proof. Let us fix u0∈L2(Ω,F0,P, H).Then, there exists at least a variational solution u(·) of (1). This means that there exists a selection f(t)∈F(u(t)) such that u(·) is solution of (5), which can be rewritten as½du(t) = (Au(t) + f(t)) dt +σu(t)dw1(t) u(0) = u0.(7) Then, if we perform the change of variable v(t) =e−σw1(t)u(t), thanks to our assumptions, it follows that the process vsolves the random problem    dv(t) dt =Av(t) + e f(t)−σ2 2v(t) v(0) = u0, where e f(t) =e−σw1(t)f(eσw1(t)v(t)) ∈e−σw1(t)F(eσw1(t)v(t)).Then, taking into account that (F2)- (F3) imply |e f(t)| ≤ C|v(t)|, it follows from (2) that d|v(t)|2 dt = 2 ¿v(t), Av(t) + e f(t)−σ2 2v(t)À ≤(λ−σ2)|v(t)|2+ 2(v(t),e f(t)) ≤(λ−σ2+ 2C)|v(t)|2 =−γ|v(t)|2, and, consequently, |v(t, ω)|2≤ |u0(ω)|2e−γt,for all t≥0,and all ω∈Ω. Thus, |u(t, ω)|2≤ |u0(ω)|2e−γt+σw1(t),for all t≥0,and all ω∈Ω. Noticing now that limt→∞ w1(t) t= 0,P−a.s., there exists Ω0⊂Ω,P(Ω0) = 0,such that for ω /∈Ω0there exists T(ω) satisfying, for all t≥T(ω), σw1(t) t≤γ 2, and, therefore, |u(t, ω)|2≤ |u0(ω)|2e−γt/2,for all t≥T(ω),and all ω /∈Ω0. 5 Remark 2.3 As we have already mentioned, this result illustrates how we can stabilise a possible unstable system by adding a linear multiplicative noise. It is also possible to develop a similar theory to the existing one in the single-valued case in order to establish sufficient conditions ensuring the stability properties of the stochastic inclusion (1) with even nonlinear operators in the diffusion term (see, e.g. [5]). Moreover, we could also analyse the existence of random attractors for the set-valued dynamical system generated by the stochastic evolution inclusion (with very special kinds of noise: multiplicative or additive) and compare with the deterministic situation. However, instead of doing this with a differential inclusion we will carry out the problem in the next section but working with a partial differential equation without uniqueness of solutions, which also yields to a set-valued dynamical system. 3 Stabilising a PDE without uniqueness properties As far as we know, the analysis carried out in the literature concerning the effects of noise in the asymptotic behaviour of deterministic systems has been done in several directions: stabilization of constant solutions (equilibria) by different types of noise (see, e.g., [1], [7], [5], [13]), improvement of the stability of already stable solutions (see, e.g., [5]), existence of exponentially stable stationary solutions for stochastic perturbations of deterministic models ([6]), existence of random attractors for special stochastic perturbations (additive or multiplicative linear noise) of deterministic models possessing global (deterministic) attractors ([8], [4]), and also comparing the structure of the deterministic attractor with the corresponding random one, which can yield to prove that both attractors have “more or less” the same complexity or structure ([9]), or the random one can be somehow simpler ([8], [4]). But the common fact in all the cases we know is that the deterministic problem already possesses a global attractor. However, there is a research line which remains unexplored, and which is very important in our opinion. We are referring to the problem of analyzing if the appearance (or addition) of certain kind of noise in a deterministic model which does not have a global attractor, could ensure the existence of a non-trivial random attractor for the stochastically perturbed one. Later, it could be also interesting to investigate when this random attractor becomes a single (fixed) equilibrium. Although we could have developed our theory in a single-valued framework, we have preferred to proceed directly in a more general set-valued context in the case of a reaction-diffusion equation (eventually) without uniqueness of solutions. In what follows we will first recall some definitions and results from the theory of random attractors for set-valued random dynamical systems. Next, and before setting our reactiondiffusion model, we will exhibit a “simple” example of an ordinary differential equation which does not have a global attractor, and will show how the addition of a high intensity linear noise in the sense of Itˆo ensures the existence of a non-trivial random attractor. If the noise is considered in the sense of Stratonovich, the random perturbed model will not have a random attractor (same behaviour than the deterministic equation) no matter how large is the intensity of the noise. Finally, we will establish the general results for our reaction-diffusion equation without uniqueness. Of course, these are the first results on this direction and much more analysis has to be done in the future. We hope that this paper could be considered, at least, as a stimulating reason to 6 continue investigating on this topic. 3.1 Preliminaries on set-valued random dynamical systems and random attractors We summarize the main concepts and results from the theory of random attractors of set-valued (or multi-valued) random dynamical systems developed in the papers [10], [11], [12]. Let (X, dX) be a complete and separable metric space with the Borel σ-algebra B(X). Let (Ω,F,P) be a probability space and θt: Ω →Ω a measure preserving group of transformations in Ω such that the map (t, ω)7→ θtωis measurable and satisfying θ0=Id;θt+s=θt◦θs=θs◦θt,for t, s ∈R. The set Ris endowed with its Borel σ-algebra B(R). Definition 3.1 A set valued map G:R+×Ω×X→C(X)(C(X)denotes the set of non-empty closed subsets of X) is called a set-valued or multi-valued random dynamical system (MRDS) if is measurable (see Aubin and Frankowska [2], definition 8.1.1) and satisfies i) G(0, ω) = Id on X; ii) G(t+s, ω)x=G(t, θsω)G(s, ω)x(cocycle property) for all t, s ∈R+, x ∈X, ω ∈Ω Remark 3.2 Observe that we will use the notation G(t, ω)xinstead of G(t, ω, x). Remark 3.3 Throughout this paper all assertions about ωare assumed to hold on a θ-invariant set of full measure (unless otherwise stated). Using the notation and assumptions from [10] and [11], we have the following definition. Definition 3.4 A closed random set ω7→ A(ω)is said to be a global random attractor of the MRDS Gif: i) G(t, ω)A(ω)⊇ A(θtω),for all t≥0,P−a.s (that is, it is negatively invariant); ii) for all D⊂Xbounded, lim t→+∞dist(G(t, θ−tω)D, A(ω)) = 0; iii) A(ω)is compact P−a.s. Let us now establish two assumptions which will be crucial in the following theorem. (H1) There exists an absorbing random compact set B(ω), that is, for P−almost all ω∈Ω and every bounded set D⊂X, there exists t(ω, D) such that for all t≥t(ω, D) G(t, θ−tω)D⊂B(ω).(8) 7 (H2) G(t, ω) : X→C(X) is upper semicontinuous, for all t∈R+and ω∈Ω. Theorem 3.5 (see [10], [11]) Let assumptions (H1) −(H2) hold , the map (t, ω)7→ G(t, ω)D be measurable for all deterministic bounded sets D⊂X, and the map x∈X7→ G(t, ω)xhave compact values. Then, A(ω) := [ D⊂X bounded ΛD(ω) (9) is a global random attractor for G(measurable with respect to F). It is unique and the minimal closed attracting set. Moreover, if the map x7−→ G(t, ω)xis lower semicontinuous for each fixed (t, ω), then the global random attractor A(ω)is strictly invariant, i.e., G(t, ω)A(ω) = A(θtω),for all t≥0. Remark 3.6 Although it is possible to refer the attractor to attract a universe of random sets instead of attracting only deterministic bounded sets (see, e.g. [6]), this last universe is enough for our purposes. In what follows, we will consider as our base probability space (Ω,F,P) the canonical one generated by a standard two-sided real Wiener process Wt(t∈R).In other words, we consider the Wiener probability space (Ω,F,P) defined by Ω = {ω∈C(R,R)|ω(0) = 0}, equipped with the Borel σ−algebra F, the Wiener measure P,and the usual uniform convergence on bounded sets of R. Recall that Wt(ω) := ω(t) and that the flow θtis defined as (θtω)(s) = ω(t+s)−ω(t),for t, s ∈R. 3.2 A motivating single-valued example: Stabilisation to a random attractor To illustrate our later analysis, we would like to consider now an example given by an ordinary differential equation which does not possess a global attractor and show the different effects that a very simple noise can produce on the systems depending on the interpretation given to the noise (Itˆo or Stratonovich). Consider the following initial value (autonomous) problem ½˙x(t) = x(t)+1 x(0) = x0,(10) with x0∈R. As the solution of (10) is x(t; 0, x0) = −1+(x0+ 1)et,(11) it is clear that the dynamical system generated by the equation in (10) does not possess a global attractor. 8 First, let us assume that a linear Itˆo noise appears in the equation, i.e, let us consider the following problem (˙x(t) = x(t)+1+σx(t)dWt dt x(s) = x0, (12) where σ∈Rrepresents the intensity of the noise and s∈Ris the initial time. It can be easily seen (see, e.g., [14], [8]) that this equation generates a single-valued random dynamical system. More precisely, we can first transform (12) into an equivalent problem but for a stochastic equation in the sense of Stratonovich, namely (˙x(t) = (1 −σ2 2)x(t)+1+σx(t)◦dWt dt x(s) = x0, (13) where the ◦denotes the Stratonovich sense for the stochastic term. Now, in order to obtain the expression for the random dynamical system generated by this equation, we perform a suitable change of variables which transforms our stochastic equation into a random one. Indeed, for a fixed realisation of our Wiener process, i.e., for a fixed ω∈Ω,and setting y(t) = e−σWt(ω)x(t), we obtain ½˙y(t) = (1 −σ2 2)y(t) + e−σWt(ω) y(s) = ys=e−σWs(ω)x0,(14) whose solution is explicitly given by y(t;s, ω, ys) = e(1−σ2 2)(t−s)e−σWs(ω)x0+e(1−σ2 2)tZt s e−(1−σ2 2)re−σWr(ω)dr. Therefore, the random dynamical system G(t, ω) generated by our problem is defined as G(t, ω)x0=eσWt(ω)y(t; 0, ω, x0). If we now choose σwith absolute value large enough so that 1 −σ2 2<0,then it is possible to take limits when s→ −∞ (observe that the resulting improper integral below is well defined) yielding lim s→−∞ y(t;s, ω, ys) = e(1−σ2 2)tZt −∞ e−(1−σ2 2)re−σWr(ω)dr, and, consequently lim s→−∞ eσWt(ω)y(t;s, ω, ys) = e(1−σ2 2)teσWt(ω)Zt −∞ e−(1−σ2 2)re−σWr(ω)dr. (15) Denoting now A(ω) = Z0 −∞ e−(1−σ2 2)re−σWr(ω)dr, it is straightforward to check that x(t, ω) := A(θtω) is a stationary solution of the stochastic equation in (13), and the following equality holds A(θtω) = e(1−σ2 2)teσWt(ω)Zt −∞ e−(1−σ2 2)re−σWr(ω)dr. 9 Proposition 3.12 Given un→u0weakly in L2(O),tn→t0and ωn→ω0,with yn∈ G(tn, ωn)un,there exists a subsequence ynkconverging to y0∈G(t0, ω0)u0.As a consequence, the maps (t, ω, x)7→ G(t, ω)x, (t, ω)7→ G(t, ω)D(where Dis any bounded set of L2(O)) are measurable, Gpossesses an absorbing random compact set, has compact values, and G(t, ω)is upper semicontinuous for all (t, ω)∈R×Ω. Proof. Take T > 0 such that tn, t0∈[0, T].For yn∈G(tn, ωn)unthere exist solutions of (20) vn(·) with vn(0) = unsuch that yn=eσWtnvn(tn). If we multiply (20) by vn(see (26)) using (21) we get, for Wn t=Wt(ωn), 1 2 d dt|vn(t)|2+kvn(t)k2≤M1e−2σWn t(|h|2+ 1) + (−α−σ2 2+1 2)|vn|2. As ωn→ω0and t∈[0, T] we have a uniform bound |Wn t| ≤ K, ∀t∈[0, T], so that d dt|vn(t)|2+ 2kvn(t)k2≤M2+M3|vn|2 and so |vn(t)|2+ 2 Zt s kvn(s)k2≤M2(t−s) + M3Zt s |vn(s)|2ds. (36) Now, by Gronwall’s lemma, we obtain that vnare bounded in L∞(0, T;L2(O))∩L2(0, T;H1 0(O)). Now, taking subsequences if necessary, and using the compactness Lemma [19], we have that vn→vweakly in L2(0, T ;H1 0(O)),and strongly in L2(0, T;L2(O)), vn→vweakly star in L∞(0, T ;L2(O)), vn(t)→v(t) strongly in L2(O), for almost all t∈[0, T], dvn dt →dv dt weakly in L2(0, T ;H−1(O)), vn(t, x)→v(t, x) for almost all (t, x)∈[0, T]× O. Thus, gn(t, x) = e−σWn tf(eσWn tvn(t, x)) →g(t, x) = e−σWtf(eσWtv(t, x)) for a.a. (t, x)∈[0, T]× O. Moreover, we also have that |gn|L2(0,T ;H)≤C, so that arguning as in the proof of Theorem 3.10 we have gn→gweakly in L2(0, T;L2(O)).Thus, v(·) is a weak solution of (20). Let us see that v(0) = v0.Since L2(O),→H−1(O) compactly, from the Ascoli-Arzel`a theorem we get that vn→vin C([0, T]; H−1(O)).Then, for every tn→t0we have that v(tn)→v(t0) strongly in H−1(O).Thanks to the uniform bound |vn(tn)| ≤ C, a standard argument gives v(tn)→v(t0) weakly in L2(O). In particular, vn(0) →v(0) = v0=u0weakly in L2(O).(37) If we finally prove that v(tn)→v(t0) strongly in L2(O) we would finish the proof, since then yn→eσWt0v(t0)∈G(t0, ω0)u0. 16 Now, from (37) we get that |v(t0)| ≤ lim inf n→+∞|vn(tn)|. Let us finally prove that lim sup n→+∞ |vn(tn)| ≤ |v(t0)|.(38) Let us define the non-increasing (see (36)) and continuous functions Jn(t) = |vn(t)|2−M2t−M3Zt 0 |vn(s)|ds, J(t) = |v(t)|2−M2t−M3Zt 0 |v(s)|ds. As vn(t)→v(t) in L2(O), for almost all t∈[0, T], and vn→vstrongly in L2(0, T ;L2(O)), we have that Jn(t)→J(t) for almost all t∈[0, T]. Moreover, given 0 < tm< t0such that vn(tm)→v(t0) in L2(O),using that Jnare non-increasing and Jis continuous, we get that given ε > 0 there exist tmand N(ε, tm) such that Jn(tn)−J(t0)≤Jn(tn)−Jn(tm) + Jn(tm)−J(tm) + J(tm)−J(t0) ≤Jn(tn)−Jn(tm) + |Jn(tm)−J(tm)|+ε ≤0 + ε+ε, if n≥N(note that tn> tmfor nlarge). Thus, lim sup n→+∞ Jn(tn) = lim sup n→+∞ |vn(tn)|2−M2t0−M3Zt0 0 |v(s)|2ds ≤J(t0), from which we have (38), and so the result holds. As a consequence, G(t, ω) has compact values. A standard argument (see, for instance, Corollary 7 in [24]) implies that the map x→G(t, ω)xis upper semicontinuous. Finally, the measurability of the maps (t, ω, x)7→ G(t, ω)x, (t, ω)7→ G(t, ω)Dand the existence of an absorbing random compact set follow from Lemma 2 in Kapustyan [17] and Proposition 3.11. We finally have, as a consequence of Proposition 3.12 and Theorem 3.5, the following theorem: Theorem 3.13 There exists a compact random attractor associated to (17). As we remarked before, we have obtained that the random equation (23) possesses a random attractor, despite the fact that equation (17) in the deterministic case (i.e. with σ= 0) may not possess a global attractor, which can be interpreted as a regularizing effect produced by the noise on the deterministic problem. This result is also new in the single-valued setting (which appears assuming condition (31)), which is a particular case of our theorem. It is interesting to point out that in the single-valued framework the existence of the global random attractor could be established by using for example the arguments of [14]. However, the proof of the existence of a compact absorbing set is not suitable for the set-valued case, as we do not have enough regularity of solutions in order to justify the estimates in H1spaces. 17 3.5 Stabilisation to a single equilibrium We have just proved that the addition of noise can imply the existence of a random attractor for a stochastic perturbation of a deterministic model which could not have it previously. Although we can interpret this result as a kind of stabilisation or regularization for the deterministic model, it does not say anything about the stability or attractivity of the equilibria for the deterministic system (if they exist). It may happens, as we will show below, that a deterministic equation possesses an equilibrium (or more than one) which is not (or not known to be) stable, then the appearance of a high intensity noisy term can ensure the existence of a random attractor which is given by a single deterministic point (which is a steady state solution) and which attracts any other solution. In other words, some improvement has been produced in the behaviour of the system. Let us illustrate this with the following example. Let us assume that the constant c= 0 in assumption f.2) and that h= 0 (for simplicity). Then, it follows that f(0) = 0, α =−γand M= 0,and therefore u≡0 is an equilibrium of the deterministic and the stochastic problems, i.e. (17) with σ= 0 and σ6= 0. Then, arguing as in the proof of Proposition 3.11, we obtain from (32) d dt|v(t)|2+ 2 µλ1−γ+1 2σ2¶|v(t)|2≤0. Denoting β1= 2 ¡λ1−γ+1 2σ2¢,we obtain for any u0∈Hand any t0≤t |v(t)|2≤e−β1(t−t0)−2σWt0|u0|2 and |u(t)|2≤e−β1(t−t0)+2σ(Wt−Wt0)|u0|2. Then, observe the following facts: •If λ1−γ > 0,the null solution of the deterministic problem attracts any other solution starting in any initial point u0.Futhermore, there exists a global attractor which is given by A={0}.Then for any σ∈R, the stochastically perturbed system also possesses a random attractor which is given by A(ω) = {0}. •If λ1−γ < 0,then the deterministic problem may not have a global attractor, and even the steady state null solution can be unstable (in fact, it is known to be unstable when, for instance, we are in the linear case, i.e., f(u) = −γu). 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