The effect of a small bounded noise on the hyperbolicity for autonomous semilinear differential equations

THE EFFECT OF A SMALL BOUNDED NOISE ON THE HYPERBOLICITY FOR AUTONOMOUS SEMILINEAR DIFFERENTIAL EQUATIONS TOM´ AS CARABALLO1, ALEXANDRE N. CARVALHO2, JOS´ E A. LANGA1, AND ALEXANDRE N. OLIVEIRA-SOUSA2 Abstract. In this work, we study permanence of hyperbolicity for autonomous differential equations under nonautonomous random/stochastic perturbations. For the linear case, we study robustness and existence of exponential dichotomies for nonautonomous random dynamical systems. Next, we establish a result on the persistence of hyperbolic equilibria for nonlinear differential equations. We show that for each nonautonomous random perturbation of an autonomous semilinear problem with a hyperbolic equilibrium there exists a bounded random hyperbolic solution for the associated nonlinear nonautonomous random dynamical systems. Moreover, we show that these random hyperbolic solutions converge to the autonomous equilibrium. As an application, we consider a semilinear differential equation with a small nonautonomous multiplicative white noise, and as an example, we apply the abstract results to a strongly damped wave equation. 1. Introduction The study of permanence properties in dynamical systems has been widely developed in the past decades (see [2, 3, 4, 7, 12, 14, 15, 19, 20, 21, 26, 24, 28, 27, 29] and references therein). Some of these works dealt with exponential dichotomies, which corresponds to the notion of hyperbolicity in the non-autonomous framework and gives, for each time, a decomposition of the space into two parts, one along which solutions decay exponentially to zero forwards, and another along which solutions decay exponentially to zero backwards. In several of these papers they proved robustness of exponential dichotomies for deterministic nonautonomous dynamical systems [20, 21, 23], and random dynamical systems [6, 29]. Our purpose is to establish robustness results of exponential dichotomies for nonautonomous random dynamical systems. First, we extend the concept of exponential dichotomies to encompass random and nonautonomous dynamical systems and provide conditions to guarantee that it persists under perturbation. Then, 2010 Mathematics Subject Classification. Primary 37H99, 34D09, 34D30, 34D10. Key words and phrases. nonautonomous random dynamical systems, nonautonomous random differential equations, exponential dichotomies, persistence of hyperbolic equilibria, bounded noise, damped wave equation. 1Departamento de Ecuaciones Diferenciales y An´alisis Num´erico, Universidad de Sevilla, Campus Reina Mercedes, 41012, Sevilla, Spain. 2Instituto de Ciˆencias Matem´aticas e de Computa¸c˜ao, Universidade de S˜ao Paulo, Av. Trab. S˜ao Carlense, 400 - Centro, S˜ao Carlos - SP, 13566-590 Brazil. 1 2 T. CARABALLO, A. N. CARVALHO, J. A. LANGA, AND A. N. OLIVEIRA-SOUSA we apply these abstracts results to study permanence of hyperbolicity on nonautonomous random semilinear differential equations obtained by small perturbations of an autonomous problem. In this way, we consider a autonomous semilinear problem in a Banach space X ˙y=Ay+f0(y), t > 0, y(0) = y∈X, (1.1) and nonautonomous random perturbations of it ˙y=Ay+fη(t, θtω, y), t > τ, y(τ) = yτ∈X, η ∈(0,1],(1.2) where Agenerates a strongly continuous semigroup {eAt:t≥0}⊂L(X), θt: Ω → Ω is a random flow defined in a probability space (Ω,F,P). We assume that there exist a hyperbolic equilibrium for (1.1) y∗ 0, i.e., y∗ 0is such that f(y∗ 0) = −Ay, and the linearized problem ˙z=Az+f0(y∗ 0)zadmits an exponential dichotomy. Then, we provide conditions to prove existence and continuity of “hyperbolic equilibria” for (1.2). In fact, we show that for each small perturbation fηof (1.1), there exists a global solution of (1.2) ξ∗ ηthat presents an hyperbolic behavior (which means that the linear nonautonomous random dynamical system generated by ˙y=Ay+Dyfη(t, θtω, ξ∗ η(t, θtω))y, t ≥τ, admits an exponential dichotomy), and that these hyperbolic solutions ξ∗ ηconverges to y∗ 0, as η→0. To prove this result on the existence and continuity of “hyperbolic equilibria” for semilinear differential equations, we need first to guarantee permanence of hyperbolicity for linear nonautonomous random dynamical systems, which can be interpreted as follows: suppose that fη(t, θtω, ·) := Bη(t, θtω)∈ L(X) is a linear perturbation of ˙x=Ax, and that the autonomous evolution process generated by Aadmits an exponential dichotomy, then we prove the linear co-cycle generated by ˙x=Ax+Bη(t, θtω)x, (1.3) admits an exponential dichotomy, under the assumption that B(t, θtω) is uniformly small (on time). We prove the robustness result via “discretization”, i.e., we first prove the robustness for the discrete case, then we provide some “connecting” results between discrete and continuous dynamical systems, and we use these results to establish the robustness result for continuous nonautonomous random dynamical systems. As an application of these results, we consider a family of stochastic differential equations with a nonautonomous multiplicative white noise dy =Aydt +f0(y)dt +ηκty◦dWt, t ≥τ, y(τ) = yτ∈X, (1.4) where η∈[0,1], and the mapping R3t7→ κt∈Ris a real function. We use a formal change of variables to obtain a family of nonautonomous random differential equations like (1.2). As an example, we consider a damped wave equation with a nonautonomous multiplicative white noise    utt +βut−∆u=f(u) + ηκtu◦˙ W, x ∈D, u(0, x) = h1(x), ut(0, x) = h2(x), x ∈D, u(t, x) = 0, x ∈∂D, t > 0. (1.5) EFFECT OF A SMALL BOUNDED NOISE ON HYPERBOLICITY 3 This work was motivated by the following stochastic differential equation, with a small (autonomous) multiplicative white noise, dyt=Aydt +y ◦dWt, t ≥τ, y(τ) = yτ∈X. (1.6) where ˙y=Ay admits an exponential dichotomy, the family {Wt:t∈R}is the standard Wiener process, and  > 0. Consider a stochastic process z(t, ω) = z∗(θtω) known as the Orstein-Uhlenbeck process (which will be describe later), where ωis a parameter in the probability space (Ω,F,P), and {θt:t∈R}is a group on Ω. By a standard procedure, we define v(t, ω) = e−z∗(θtω)y(t, ω), that satisfies ˙v=Av +z∗(θtω)v, (1.7) which is a (autonomous) random differential equation as a perturbation of an autonomous problem. However, for each random parameter ω∈Ω, our perturbation B(θtω) := z∗(θtω) have some sub-linear growth on t, which brings some fundamental difficulties to prove robustness, because we cannot assume that operator B(θtω) is uniformly bounded in time. Therefore, when ˙v=Av is hyperbolic it is not possible to guarantee that (1.7) admits an exponential dichotomy, in the sense of [6, 29], for  > 0 small enough. To deal with this issue we propose “to bound” the noise with respect to time. Bounded noise is very sensible in real life applications, [12, 13, 16, 22, 28]. For instance, in [13] the authors consider an Ornstein-Uhlenbeck process depending on a control parameter that allows them to ensure that the noise is inside a bounded interval, which is fixed by practitioners based on experiments. They use such a bounded noise to model perturbations on the input flow of a chemostat. This idea is generalized in [12], where several examples in population dynamics with bounded random fluctuations are given. In this work, we “bound” the noise in the following way: Since the stochastic process z∗satisfies limt→±∞ |z∗(θtω)/t|= 0, for almost all ω, see [14], it is possible to choose any differentiable positive real function κsuch that there is a random variable m > 0 such that m(ω) := sup t∈R|κtz∗(θtω)|+|˙κtz∗(θtω)|<+∞. Hence, instead of (1.6), we consider dyt=Aydt +κty◦dWt,(1.8) where κtis a real function that allows us to control the white noise on time. By the change of variables v(t, ω) = e−κtz∗(θtω)y(t, ω), we obtain ˙v=Av −[˙κtz∗(θtω)−κtz∗(θtω)]v, (1.9) which is a linear nonautonomous random differential equation. Since the perturbation B(t, θtω) := [˙κt−κt]z∗(θtω) is uniformly bounded on time, we can develop a theory of exponential dichotomies to guarantee existence of hyperbolicity for (1.9). Historically, to study permanence of properties under perturbations it is sensible to assume that the perturbation is uniformly bounded on time, see [2, 3, 4, 6, 10, 19, 23]. For instance, if ˙y=Ay is hyperbolic, and B:R→ L(X) is uniformly bounded with respect of t, then the hyperbolicity persists on ˙y=Ay +B(t)y, see [23]. Also, for a differential equation on a Banach space, driven by a group θt: Σ →Σ on a compact Hausdorff set Σ, ˙y=A(θtσ)y, exhibiting an exponential dichotomy, [20] proved that the hyperbolicity persists for B: Σ → L(X), which is naturally 4 T. CARABALLO, A. N. CARVALHO, J. A. LANGA, AND A. N. OLIVEIRA-SOUSA uniformly bounded by the compactness of Σ, and in [21] they considered an the case that Bis a unbounded operator, but also with some uniformly boundedness condition on time. More recently, they consider the random case, in [6, 29] it was studied stability of tempered exponential dichotomies for random difference equations, yn+1 = A(θω)yn, where θ: Ω →Ω is a random flow defined on a probability space (Ω,F,P), but usually in the situation the assumption on the perturbation Bare more restricted because has to be exponentially small, typically kB(θnω)k ≤ δ(ω)e−ν|n|. Therefore, all these results on robustness assume that the perturbation are uniformly bounded on time. However, as seen before, we cannot assume that even a small noise is satisfy such property. For this reason, to prove permanence of hyperbolicity we have to consider a time dependent scalar function κtto compensate the the growth of the noise in time, (1). Accordingly, we consider a small nonautonomous random perturbation B:R× Ω→ L(X) of a hyperbolic problem ˙y=Ay. Our perturbation B(t, ω) depends on two parameters, the time tof deterministic nature, and another ωvarying in a probability set (Ω,F,P). This leads us to establish robustness results of exponential dichotomies for nonautonomous random dynamical systems, which is a co-cycle (ϕ, Θ) driven by Θ : R×Ω→R×Ω. Hence, using similar techniques to those in [29] (for the discrete case), and [20] (for the continuous case), we establish stability results, and applied these results to guarantee existence of exponential dichotomy for ˙y=Ay +B(Θt(τ, ω))y, where Bis uniformly small (with respect to t), and ˙x=Ax is hyperbolic. For the nonlinear case, inspired by [19], considering a small nonautonomous bounded noise as in (1.4) we provide conditions that allow us to prove existence and continuity of random hyperbolic solutions for (1.2). In [19] they study existence and continuity of hyperbolic solutions, invariant manifolds, and provide conditions to prove continuity of pullback attractors for evolution processes. This work was crucial for further development on continuity and stability of attractors in a deterministic scenario [10, 11, 17]. However, for random dynamical systems the theory is still under development. There are many works on the existence and upper semi continuity of random attractors, including nonautonomous random dynamical systems, see [9, 15, 25, 24, 27], and references therein. Nevertheless, as it is suggested in [19, 17], the lack of theorems on the persistence of random hyperbolic solutions makes very difficult to prove results on the lower semi-continuity of random attractors for problems under small random perturbations. The study of these special solutions are the core on the problem of continuity and stability of attractors, and we believe that this paper provides a direction to study continuity and stability of random attractors. For random dynamical systems, Arnold, Boxler, Crauel and Kloeden [2, 3, 4], studied the effect of the noise on hyperbolic dynamical systems. In [2] they proved existence of a stationary solution for a nonlinear random differential equation. More recently, in [26], they studied stability of tempered stationary solutions, and tempered attractors for small random perturbations of nonautonomous problems. However, differently of our work, in none of these works it is possible to conclude that their stationary solutions exhibit hyperbolicity, because the noise presents some growth in time that do not allow us to apply any robustness result. EFFECT OF A SMALL BOUNDED NOISE ON HYPERBOLICITY 5 We organize our paper as follows. In Section 2, we present a notion of exponential dichotomy, simultaneously, for discrete and continuous nonautonomous random dynamical systems. Then, for the discrete case, we prove a robustness result, uniqueness, and continuous dependence of projections associated with the exponential dichotomy for the discrete case in Subsection 2.1. In Subsection 2.2 we establish some theorems to compare exponential dichotomies of discrete and continuous nonautonomous random dynamical systems which allows to conclude the same stability results for the continuous case. In Section 3, we apply the abstract robustness results for nonautonomous random/stochastic differential equations. In Subsection 3.1, we show existence of exponential dichotomy for (1.3) for the linear case. Later, in Subsection 3.2, we consider a family of nonautonomous random semilinear differential equations, we establish the existence and continuity of random hyperbolic solutions for (1.2). Finally, in Subsection 3.3, we show how to apply our result on persistence of hyperbolic equilibria for (1.4), and as an example, we consider a damped wave equation with a nonautonomous multiplicative white noise (1.5). 2. Exponential dichotomies for Nonautonomous RDS In this section, we introduce the notion of nonautonomous random dynamical systems in a Banach space X. We consider a driving flow {Θt}t∈Tover T×Ω, where (Ω,F,P) is a probability space, and T=Zor T=R. Definition 2.1. Let (Ω,F,P)be a probability space. We say that a family of maps {θt: Ω →Ω : t∈T}is a random flow if it satisfies •θ0=IdΩ; •θt+s=θt◦θs, for all t, s ∈T; •θt: Ω →Ωis measurable for all t∈T. Definition 2.2. Let θ:= {θt: Ω →Ω : t∈T}be a random flow. Define Θt(τ, ω) := (t+τ, θtω)for each (τ, ω)∈T×Ω, and t∈T. We say that a family of maps {ϕ(t, τ, ω) : X→X; (t, τ, ω)∈T+×T×Ω}is a nonautonomous random dynamical system (co-cycle) driven by Θif (1) the mapping T+×Ω×X3(t, ω, x)7→ ϕ(t, τ, ω)x∈Xis measurable for each fixed τ∈T; (2) ϕ(0, τ, ω) = IdX, for each (τ, ω)∈T×Ω; (3) ϕ(t+s, τ, ω) = ϕ(t, Θs(τ, ω)) ◦ϕ(s, τ, ω), for every t, s ≥0in T, and (τ, ω)∈T×Ω; (4) ϕ(t, τ, ω) : X→Xis a continuous map for each (t, τ, ω)∈T+×T×Ω. We usually denote the pair (ϕ, Θ)(X,T×Ω), or (ϕ, Θ), to denote the co-cycle ϕdriven by Θ. Remark 2.3. To simplify the notation we will write ωτ:= (τ, ω)∈T×Ω, and Θt(ωτ) := (θtω)τ+t. Remark 2.4. Let (ϕ, Θ)(X,T×Ω) be a nonautonomous random dynamical system such that the mapping T+×X3(t, x)7→ ϕ(t, ωτ)x∈Xis continuous, for each ωτ∈T×Ω.(2.1) 6 T. CARABALLO, A. N. CARVALHO, J. A. LANGA, AND A. N. OLIVEIRA-SOUSA Then (ϕ, Θ) can be associate with a family of continuous evolutions processes. In fact, for each ωτ∈T×Ω, we define the following evolution process Φωτ:= {ϕt,s(ωτ) := ϕ(t−s, Θsωτ) ; t≥s}. This means that, for each ωp∈T×Ωthe family of maps Φωτsatisfies (1) φt,t(ωτ) = IdX, for each t∈T; (2) φt,s(ωτ)◦φs,r =ϕt,r(ωτ), for each t≥s≥r; (3) the mapping {(t, s)∈T2:t≥s} × X3(t, s, x)7→ ϕt,s(ωτ)x∈Xis continuous. Throughout this work we will assume that a nonautonomous random dynamical system (ϕ, Θ) satisfies condition (2.1). This assumption is sensible in the applications, for instance, when the co-cycle is induced by a well-posed stochastic/random differential equation. In this paper, we will use the concept of global solutions for the evolution process Φωp, for each ωp∈T×Ω, associated with a given co-cycle (ϕ, Θ). Definition 2.5. Let S={S(t, s) : t≥s, t, s ∈T}a evolution process on a Banach space X. We say that a map ξ:R→Xis a global solution for Sif S(t, s)ξ(s) = ξ(t)for every t, s ∈Twith t≥s. A global solution ξis backwards bounded if ξ(−∞,0]) = {ξ(t) : t≤0}is a bounded subset of X. We set L(X) as the space of all bounded linear maps between Xand X. Recall the definition of strongly measurable: Definition 2.6. Let Ωbe a measurable space, and Xa Banach space. A map P: Ω → L(X)is said to be strongly measurable if for every x∈Xthe map Ω3ω7→ P(ω)x∈Xis measurable. Definition 2.7. A map D:T×Ω→Ris said to be Θ-invariant if for each ωτ∈R×Ωwe have that D(Θtωτ) = D(ωτ), for every t∈T. Now, we define the notion of exponential dichotomy for linear nonautonomous random dynamical systems. Definition 2.8. A nonautonomous random dynamical system (ϕ, Θ) such that ϕ(t, τ, ω)∈ L(X), for all (t, τ, ω)∈T+×T×Ω, is said to admit an (uniform) exponential dichotomy if there exists a θ-invariant subset ˜ Ωof Ωwith full measure, P(˜ Ω) = 1, and a family of projections, Πs:= {Πs(ωτ) : ωτ∈Tט Ω}such that (1) for each τ∈Tthe map Πs τ(·) := Πs(τ, ·) : ˜ Ω→ L(X)is strongly measurable; (2) Πs(Θtωτ)ϕ(t, ωτ) = ϕ(t, ωτ)Πs(ωτ), for every t∈T+and ωτ∈Tט Ω; (3) ϕ(t, ωτ) : R(Πu(ωτ)) →R(Πs(Θtωτ)) is an isomorphism, where Πu τ:= IdX−Πs τfor all τ∈T; (4) there exist Θ-invariant maps α:T×Ω→(0,+∞)and K:T×Ω→[1,+∞) such that kϕ(t, ωτ)Πs(ωτ)kL(X)≤K(ωτ)e−α(ωτ)t,for every t≥0; kϕ(t, ωτ)Πu(ωτ)kL(X)≤K(ωτ)eα(ωτ)t,for every t≤0, for every ωτ∈Tט Ω EFFECT OF A SMALL BOUNDED NOISE ON HYPERBOLICITY 7 In this case, the function Kis called a bound and αan exponent for the exponential dichotomy. We refer to the exponential dichotomy as: continuous if T:= R, and discrete if T:= N. Remark 2.9. If for each ωpthe map R3t→K(Θtωp)is not constant we say that (ϕ, Θ) admits an nonuniform (with respect to t) exponential dichotomy. In the special case when the mapping R3t→K(Θtωp)is tempered we say that (ϕ, Θ) admits a tempered exponential dichotomy In this work, we do not deal with the case of nonuniform exponential dichotomies. For more information on this topic we recommend [6, 8, 29]. Remark 2.10. If a co-cycle (ϕ, Θ) admits an exponential dichotomy, then for each fixed ωp∈Tט Ωthe associated evolution process Φωpalso admits it in the sense of Henry [23, Section 7.6]. The interpretation of a nonautonomous random dynamical system as a family of evolution processes with a parameter provides also a special function usually called as Green functions: Definition 2.11. Let (ϕ, Θ) be a co-cycle which admits an exponential dichotomy with family of projections Πs. A Green function associated to (ϕ, Θ) and family of projection Πsis given by Gωp(t, s) = ϕt,s(ωp)Πs(Θsωp),if t≥s, −ϕt,s(ωp)Πu(Θsωp),if t < s, for each ωpfixed. 2.1. Exponential dichotomy for nonautonomous random co-cycles: discrete case. In this subsection, we study a discrete nonautonomous random dynamical system with exponential dichotomy. In this paper, the purpose of the discrete case is to works as a tool to obtain results for the continuous case of exponential dichotomies and hence, for differential equations. The goal is to presented a summary of results concerning exponential dichotomy for discrete co-cycles driving by flows over noncompact symbols spaces that we are going to need in order to establish robustness results of hyperbolicity for differential equations. We prove that the property of admitting an discrete exponential dichotomy is stable under perturbation (Theorem 2.14), a type of admissibility result (Theorem 2.12), and uniqueness and continuous dependence of projections (see Corollary 2.13 and Theorem 2.23, respectively). The techniques used in this subsection are the same introduced by Henry [23] (for deterministic dynamical systems) and by Zhou et al. in [29] (for random dynamical systems). A linear discrete nonautonomous random dynamical systems (ϕ, Θ) can be associated with nonautonomous random difference equations. In fact, for each ωp∈ Z×Ω we study xn+1 =A(Θnωp)xn, xn∈Xand n∈Z,(2.2) where A:T×Ω→ L(X) and ϕ(n, ωp) := A(Θn−1ωp)◦ · · · ◦ A(ωp) for n > 0 and ϕ(0, ωp) = IdX. We prove existence of solutions for the non-homogeneous problem xn+1 =A(Θnωp)xn+B(Θnωp)xn+fn,for every n∈Z.(2.3) 8 T. CARABALLO, A. N. CARVALHO, J. A. LANGA, AND A. N. OLIVEIRA-SOUSA As in the deterministic case, see Henry [23], this plays an important role in the proof of robustness of exponential dichotomy. Theorem 2.12. Let (ϕ, Θ) be a co-cycle generated by A:Z×Ω→ L(X)and assume that it admits an exponential dichotomy with bound Kand exponent α. Then there exists a Θ-invariant map δwith 0≤δ(ωp)<1−e−α(ωp) 1 + e−α(ωp),for each ωp∈Z×Ω, for which, if B:Z×Ω→ L(X)satisfies kB(Θkωp)kL(X)≤δ(ωp)K(ωp)−1,∀k∈Z, then, for each ωpfixed and {fn} ∈ l∞(Z)the difference equation xn+1 =A(Θnωp)xn+B(Θnωp)xn+fn,for every n∈Z,(2.4) possesses a unique bounded solution x(·, ωp). The proof of Theorem 2.12 follows the same line of arguments presented by Zhou et al. in [29, Lemma 2]. Some details of the proof are included for they will be needed in forthcoming results. Proof. Let ωp∈Zט Ω and f∈l∞(Z). As a standard procedure, see for instance [23, 29], we only need to prove that the operator (Γfx)(n, ωp) := +∞ X k=−∞ Gωp(n, k + 1)(B(Θkωp)xk+fk),∀n∈Z has a unique fixed point x(·, ωp) in l∞(Z). First, let us prove that Γfx(·, ωp)∈l∞(Z), for x∈l∞(Z). k(Γfx)(n, ωp)kX≤ +∞ X k=−∞ kGωp(n, k + 1)kL(X)(kB(Θkωp)kL(X)kxkkX+kfkkX) ≤ +∞ X k=−∞ K(Θk+1ωp)e−α(ωp)|n−1−k|(δ(ωp)K(ω)−1kxkkX+kfkkX) ≤ +∞ X k=−∞ e−α(ωp)|n−1−k|(δ(ωp)kxkl∞+kfkl∞K(ωp)) ≤1 + e−α(ωp) 1−e−α(ωp)(δ(ωp)kxkl∞+kfkl∞K(ωp)) <+∞. Then, Γf(·, ωp)(l∞(Z)) ⊂l∞(Z). Finally, if x, y ∈l∞(Z), we have that k(Γfx)(n, ωp)−(Γfy)(n, ωp)kX ≤ +∞ X k=−∞ kGωp(n, k + 1)kL(X)kB(ΘkωpkL(X)kxk−ykkX ≤ +∞ X k=−∞ K(Θk+1ωp)e−α(ωp)|n−1−k|δ(ωp)K(ωp)−1kxk−ykkX ≤1 + e−α(ωp) 1−e−α(ωp)δ(ωp)kx−ykl∞. EFFECT OF A SMALL BOUNDED NOISE ON HYPERBOLICITY 9 Therefore, kΓfx(·, ωp)−Γfy(·, ωp)kl∞≤1 + e−α(ωp) 1−e−α(ωp)δ(ωp)kx−ykl∞, thus, we choose δ(ωp)<1−e−α(ωp) 1+e−α(ωp), thus Γf(·, ωp) is a contraction in l∞(Z). In this way, we obtain x∈l∞(Z) such that xn(ωp) = (Γfx)(n, ωp) for each n∈Z, in other words, x(·, ωp) is the only solution for (2.4).  The following corollary establishes uniqueness for the family of projections. The proof follows the ideas of [18, Corollary 7.5] and it is included for the readers convenience. Corollary 2.13. If (ϕ, Θ) admits an exponential dichotomy, then the family of projections are uniquely determined. Proof. Let Πu,(i), for i= 1,2, be projections associated with an exponential dichotomy of (ϕ, Θ). Given ωp∈Zט Ω and z∈X, define fn= 0, for all n6=−1, and f−1=z. From Theorem 2.12 with B= 0, there exists {x(n, ωp) : n∈Z}the unique bounded solution of xn+1(ωp) = A(Θnωp)xn+fn, n ∈Z. From the proof of Theorem 2.12 (with B= 0), it is possible to see that this solution is given by xn(ωp) = +∞ X k=−∞ G(i) ωp(n, k + 1)fk,for i= 1,2,(2.5) where G(i)is the Green function associated with Π(i), for i= 1,2. By uniqueness of the solution, we must have that x0(ωp) = P+∞ −∞ G(i) ωp(0, k + 1)fk=G(i) ωp(0,0)f−1= Πu,(i)(ωp)z, for i= 1,2. Therefore, Πu,1(ωp)=Πu,2(ωp) for all ωp∈Tט Ω.  Now, we state a robustness result of exponential dichotomies for nonautonomous random dynamical systems. Theorem 2.14. Let (ψ, Θ) be a discrete nonautonomous random dynamical system with an exponential dichotomy with bound Kand exponent α. There exists a Θ- invariant map with 0≤δ(ωp)<1−e−α(ωp) 1 + e−α(ωp),for each ωp∈Z×Ω, for which, if (ϕ, Θ) is a discrete nonautonomous random dynamical system such that sup n∈N {K(ωp)kψ(1,Θnωp)−ϕ(1,Θnωp)kL(X)} ≤ δ(ωp),(2.6) then (ϕ, Θ) admits an exponential dichotomy with bound M(ωp) := K(ωp)1 + δ(ωp) (1 −ρ(ωp))(1 −e−α(ωp))max{D1(ωp), D2(ωp)}, and exponent ˜α(ωp) := −ln(cosh α(ωp)−[cosh2α(ωp)−1−2δ(ωp) sinh α(ωp)]1/2), 16 T. CARABALLO, A. N. CARVALHO, J. A. LANGA, AND A. N. OLIVEIRA-SOUSA Assume that for almost all ω∈Ω the solutions of (3.1) are associated with a nonlinear evolution process Sω:= {Sω(t, s) : t≥s}. More precisely, if for every τ∈Rand y0∈Xthere exists [τ, +∞)3t7→ y(t, τ, ω;y0) a solution for (3.1), then we define Sω(t, τ, ω)u0:= y(t, τ, ω;u0). Another equivalent way to generate a dynamical system from problem (3.1) is the following: define x(t) := y(t+τ, τ, ω;u0), for every t≥0 and some fixed τ∈R. Hence we obtain the initial value problem ˙x=f(t+τ, θt+τω, x), t > 0 and x(0) = x0. Now, the relation f(t+τ, θt+τω) = f(Θt(t, θτω)), where Θ is the flow {Θt:R×Ω→ R×Ω}t∈Rdefined as Θt(τ, ω) = (t+τ, θtω), leads us to consider a nonautonomous random differential with a nonlinearity driven by the flow Θ i.e. ˙z=f(Θt(τ, ω), z), t > 0 and z(0) = z0,for each (τ, ω)∈R×Ω.(3.2) Thus, the solutions of (3.2) defines a nonautonomous random dynamical system ϕ(t, τ, ω)z0:= z(t, (τ, ω); z0). Therefore, we rewrite problem (3.1) using formulation of (3.2) as follows ϕ(t, τ, ω)y0:= y(t+τ, τ, θ−τω;y0). Or equivalently, for almost all ω∈Ω we have that ϕ(t, τ, ω) = Sθ−τω(t+τ, τ), t ≥0, τ ∈R. Consequently, we study asymptotic behavior of both dynamical systems: the cocycle (ϕ, Θ)(X,R×Ω) generated by (3.2), and the family of evolution processes {Sω;ω∈ Ω}associated with (3.1). 3.1. Linear nonautonomous random differential equations. In this subsection we shall study linear nonautonomous random differential equations on a Banach space X. We provide conditions to guarantee the existence of an exponential dichotomy for a nonautonomous random perturbation of a hyperbolic autonomous problem. The results contained here were inspired by those of Chow and Leiva [20] in a deterministic context, where the base flow is a group over a Hausdorff compact set. Let Abe the generator of a strongly continuous semigroup {eAt :t≥0}and B:R×Ω→ L(X) be a bounded operator depending on parameters on R×Ω. We study the linear problem ˙x=Ax +B(Θtωτ)x, t > 0 and x(0) = x0.(3.3) where ωτ:= (τ, ω) and for every t∈Rthe map Θt:R×Ω→R×Ω is defined by Θtωτ:= (t+τ, θtω). To study equation (3.3), we consider the following family of integral equations x(t, τ, ω;x0) = eAtx0+Zt 0 eA(t−s)B(Θsωτ)x(s)ds, x0∈X, t ≥0, ωτ∈R×Ω. We have the following result on the robustness of exponential dichotomies for linear nonautonomous random differential equations. Theorem 3.1. Let (ϕ, Θ) be a linear nonautonomous random dynamical system with L(ωτ) := sup t∈R kϕ(t, ωτ)kL(X)<+∞,for each ωτ∈R×Ω,(3.4) EFFECT OF A SMALL BOUNDED NOISE ON HYPERBOLICITY 17 where supt∈RL(Θtωτ)≤L(ωτ). Suppose that (ϕ, Θ) admits an exponential dichotomy with exponent αand bound K. Then there exists a Θ-invariant map :R×Ω→(0,+∞)such that for every B:R×Ω→ L(X)with sup 0≤t≤1 kZt 0 B(Θsωτ)xdskX< (ωτ)kxkX a nonautonomous random dynamical system satisfying ψ(t, ωτ) = ϕ(t, ωτ) + Zt 0 ϕ(t−s, Θsωτ)B(Θsωτ)ψ(s, ωτ)ds (3.5) admits an exponential dichotomy with bound ˆ Mand exponent ˜αprovided in Theorem 2.21. Proof. Let (ψ, Θ) be a nonautonomous random dynamical system satisfying (3.5). Then kψ(t, ωτ)xkX≤L(ωτ)kxkX+Zt 0 L(Θsωτ)kB(Θsωτ)kL(X)kψ(s, ωτ)xkXds, 0≤t≤1. From Gr¨onwall’s inequality we obtain kψ(t, ωτ)kL(X)≤L1(ωτ) := L(ωτ)eL(ωτ)(ωτ), for every 0 ≤t≤1. Hence, for every 0 ≤t≤1 kϕ(t, ωτ)x−ψ(t, ωτ)xkX≤(ωτ)L(ωτ)L1(ωτ)kxkX. Finally, since (ϕ, Θ) admits an exponential dichotomy, there exists a Θ-invariant measurable map δ > 0 as in Theorem 2.21. Therefore, for each ωτchoose = (ωτ)>0 such that (ωτ)L(ωτ)L1(ωτ)< δ(ωτ). Note that, for every t∈R L2(Θtωτ) := L(Θtωτ)L1(Θtωτ)≤L2(ωτ), therefore we choose (Θtωτ) = (ωτ), and the proof is complete.  Remark 3.2. Theorem 3.1 is saying that if problem ˙x=A(Θtωτ)xgenerates a nonautonomous random dynamical system with an exponential dichotomy then for the class of bounded linear perturbation Bgiven in the Theorem the perturbed nonautonomous random dynamical system generated by problem ˙x=A(Θtωτ)x+B(Θtωτ)x, x(0) = x0∈X, t ≥0,(3.6) admits an exponential dichotomy. Now, as a corollary we have the following robustness result for an autonomous problem under nonautonomous random perturbation. Theorem 3.3. Assume that Agenerates a analytic semigroup {eAt:t≥0}, and that the spectrum of A,σ(A)does not intersect the imaginary axis and that the set σ+:= {λ∈σ(A) : Reλ > 0}is compact. Then there exists a Θ-invariant map :R×Ω→(0,+∞)such that, if B:R×Ω→ L(X)satisfies sup 0≤t≤1 kZt 0 B(Θsωτ)xdskX< (ωτ)kxkX, then any nonautonomous random dynamical system satisfying ϕ(t, ωτ) = eAt+Zt 0 eA(t−s)B(Θsωτ)ϕ(s, ωτ)ds (3.7) 18 T. CARABALLO, A. N. CARVALHO, J. A. LANGA, AND A. N. OLIVEIRA-SOUSA admits an exponential dichotomy with bound ˆ Mand exponent ˜αprovided in Theorem 2.21. Proof. These assumptions on Aimplies the existence of an exponential dichotomy for {eAt:t≥0}, see for instance [23]. In fact, if γis a smooth closed simple curve in ρ(A)∩ {λ∈C: Reλ > 0}oriented counterclockwise and enclosing σ+let Q=Q(σ+) = 1 2πi Zγ (λ− A)−1dλ, and define X+=QX,X−= (I−Q)X, and A±:= A±. At this scenario, A− generates a strongly continuous semigroup on X−,A+∈ L(X+), and there are M≥1, β > 0 such that keA+tkL(X+)≤Meβt, t ≤0; keA−tkL(X−)≤Me−βt, t ≥0. Now we are ready to apply Theorem 3.1 for {eAt:t≥0}. Remark 3.4. Similar as in Henry [23], Theorem 3.3 can be proved in the parabolic case, when −A is a sectorial operator with A ∈ L(Xα, X), for a unbounded perturbation B:R×Ω→ L(Xα, X), where Xαis a fractional power of X, with 0≤α < 1. In this situation there exists a Θ-invariant such that if Bsatisfies kB(Θtωτ)xkX≤b(ωτ)kxkXα, sup0≤t≤1kRt 0B(Θsωτ)x dskX< q(ωτ)kxkXα, and q(ωτ)δb(ωτ)1−δ≤(ωτ)with 0< δ < (1 −α)/2, then any nonautonomous random dynamical system satisfying (3.7) admits an exponential dichotomy in Xα. Remark 3.5. Note that, Theorem 3.3 can be proved independently of Theorem 3.1 and any nonautonomous random dynamical systems (ϕ,Θ) satisfying (3.7) will satisfy the hypotheses of Theorem 3.1. 3.2. Existence and continuity of random hyperbolic solutions. Now we study a semilinear problem under a nonautonomous random perturbation. We provide conditions to obtain existence of a bounded random hyperbolic solution for a co-cycle. We consider the semilinear problem on a Banach space X ˙y=By+f0(y), y(0) = y0(3.8) and a nonautonomous random perturbation of it ˙y=By+fη(Θtωτ, y), y(0) = y0,(3.9) where (θ, Ω) is a random flow, and {Θt:t∈R}is a driving flow given by Θt(ωτ) = (t+τ, θtω) for every ωτ∈R×Ω, and η∈(0,1] is a real parameter. We assume that problem (3.8) generates a (nonlinear) semigroup {T(t) : t≥0}, and that (3.9) generates a (nonlinear) nonautonomous random dynamical system (ψη,Θ), for each η∈[0,1]. Our goal is to provide conditions on the limit “fη→f0”, as η→0, such that, if {T(t) : t≥0}has a hyperbolic equilibrium y∗ 0, then there exists a (unique) random hyperbolic equilibrium for (ψη,Θ) near y∗ 0, for η > 0 “small enough”. We first prove existence and continuity of global solutions for (3.9) and then show that these solutions exhibit a hyperbolic behavior. EFFECT OF A SMALL BOUNDED NOISE ON HYPERBOLICITY 19 Definition 3.6. Let (ψ, Θ) be a nonautonomous random dynamical system. We say that a map ζ:R×Ω→Xis a global solution for (ψ, Θ) if ψ(t, ωτ)ζ(ωτ) = ζ(Θtωτ),for every t≥0. Remark 3.7. For random dynamical systems some authors called these global solutions of equilibria. In our work, we will only refer as equilibrium to these kind of global solutions when it exhibits a hyperbolic behavior. In this situation, we will call hyperbolic equilibria or hyperbolic solutions. Remark 3.8. Let (ψ, Θ) be a nonautonomous random dynamical system and a global solution ζ. Then, for each ωτfixed, the mapping R3t7→ ξ(t, ωτ) := ζ(Θtωτ) defines a global solution for the evolution process {ψ(t−s, Θsωτ) : t≥s}. Suppose that y∗ 0∈Xis a hyperbolic equilibrium for (3.8), i.e., the linear operator A:= B+f0 0(y∗ 0) generates a C0-semigroup {et:t≥0}that admits an exponential dichotomy, see for instance Henry [23]. Let Ube an open neighborhood of y∗ 0in Xsuch that, fη(ωτ,·)∈C1(U, X), for all η∈[0,1] and ωτ∈R×Ω. Define λ(η, ωτ) := sup (t,x)∈R×Unkfη(Θt(ωτ), x)−f0(x)kX+k(fη)x(Θt(ωτ), x)−f0 0(x)kL(X)o, and we assume that lim η→0sup t∈R λ(η, Θtωτ)=0.(3.10) Also suppose ρ() := sup x∈×U sup khk≤kf0(x+h)−f0(x)−f0 0(x)hkX khkX→0,as →0.(3.11) Theorem 3.9. Let y∗ 0be a hyperbolic equilibrium for (3.8) and assume that (3.10) and (3.11) hold. Given  > 0small enough, there exists a Θ-invariant map η: R×Ω→(0,1] such that there exists R3t7→ ξ∗ η(t, ωτ)∈X, for every η∈(0, η(ωτ)], a global solution of {ψη(t−s, Θsωτ) : t≥s}such that sup t∈R kξ∗ η(t, ωτ)−y∗ 0kX< , for every η∈(0, η(ωτ)]. Proof. Let ybe a global solution of (3.9). Then, if we define φ=y−y∗ 0, it satisfies ˙ φ=Aφ+gη(Θtωτ, φ), where gη(Θtωτ, φ) = fη(Θtωτ, y∗ 0+φ)−f0(y∗ 0)−f0 0(y∗ 0)φ, so that φ(t) = eA(t−τ)φ(τ) + Zt τ eA(t−s)gη(Θsωτ, φ(s))ds (3.12) Hence, if we project Qand I−Qand take limits we obtain φ(t) = Z+∞ −∞ GA(t, s)gη(Θsωτ, φ(s))ds, where Gis the Green function associated with the semigroup {eAt:t≥0}and projection Q. 20 T. CARABALLO, A. N. CARVALHO, J. A. LANGA, AND A. N. OLIVEIRA-SOUSA Consequently, a complete bounded solution to (3.12) exists in a small neighborhood of x= 0, if and only if, the operator Iωτ,η(φ)(t) = Z+∞ −∞ GA(t, s)gη(Θsωτ, φ(s))ds has a unique fixed point in the set X:= nφ:R→X: sup t∈R kφ(t)kX≤o for a given  > 0 small. This follows by a fixed point argument for Iωτ,η for each ω∈Ω fixed. Indeed, let 1>0 such that kf0 0(y∗ 0+h)−f0 0(y∗ 0)kL(X)<1 6Mβ−1,for every khkX< 1,(3.13) and 2∈(0,1/2) be such that ρ0()<1/6Mβ−1, for every 0 <  < 2, where M > 1 is the bound, and β > 0 is the exponent of the exponential dichotomy of {eAt:t≥0}. Define 0= min{1, 2/2}and for a given ωτ∈R×Ω fixed and ∈(0, 0), define η(ωτ)>0 such that λ(η, ωτ)< 6Mβ−1,for every η∈(0, η(ωτ)]. Then, it is possible to prove that Iωτ,η maps Xinto itself. In fact, for φ∈X kgη(Θtωτ, φ(t))kX≤ kfη(Θtωτ, y∗ 0+φ(t)) −f0(y∗ 0+φ(t))kX+ρ() ≤λ(η, ωτ) + ρ0(), hence kIη,ωφ(t)kX≤2β−1Mkgη(Θtωτ, φ(t))kX< , and that Iωτ,η is a contraction. Let φ1, φ2∈X kgη(Θtωτ, φ1(t)) −gη(Θtωτ, φ2(t))kX ≤ kfη(Θtωτ, y∗ 0+φ1(t)) −fη(Θtωτ, y∗ 0+φ2(t)) −f0 0(y∗ 0)(φ1(t)−φ2(t))kX ≤λ(η, ωτ) + ρ() + kf0 0(y∗ 0+φ1)−f0 0(y∗ 0)kL(X)k(φ1(t)−φ2(t))kX. Then kIη,ωφ1(t)− Iη,ωφ1(t)kX≤1 2kφ1(t)−φ2(t)kX. Therefore, there exists a fixed point φ∗ η(·, ωτ) of Iωτ,η in X, and the global solution of (3.9) is given by ξ∗ η(·, ωτ) = φ∗ η(·, ωτ) + y∗ 0. As in the deterministic case, see Carvalho and Langa [19], these solutions {ξ∗ η} play the role of an hyperbolic equilibrium for (3.9). Given  > 0 define, for each ωτ fixed and η∈(0, η(ωτ)], ζ∗ η(τ, ω) := ξ∗ η(0, ωτ). Note that, for each ωτfixed, there exists η(ωτ)>0 such that the mapping R3 t7→ ξ∗ η(t, ωτ) := ζ∗ η(Θtωτ), t∈Ris a complete solution for ˙x=Bx+fη(Θtωτ, x), η ∈(0, η(ωτ)].(3.14) Then, to ensure that ξ∗ ηexhibits a hyperbolic behavior, we linearized problem (3.14) over ζ∗ ηand guarantee that the associated linear nonautonomous random dynamical system admits an exponential dichotomy. EFFECT OF A SMALL BOUNDED NOISE ON HYPERBOLICITY 21 Remark 3.10. Let ωτ∈R×Ωbe fixed, xη(·, ωτ)a solution of (3.9) and define zη(t) = xη(t, ω)−ζ∗ η(Θtωτ), for each t≥0and η∈(0, η(ωτ)]. Then ˙z=Az+Bη(Θtωτ)z+hη(Θtωτ, z),(3.15) where Bη(Θtωτ)=(fη)z(Θtωτ, ζ∗ η(Θtωτ)) −f0 0(y∗ 0), and hη(Θtωτ, z) := fη(Θtωτ, ζ∗ η(Θtωτ)+z)−fη(Θtωτ, ζ∗ η(Θtωτ))−(fη)z(Θtωτ, ζ∗ η(Θtωτ))z. Thus, 0is a globally defined bounded solution for (3.15) and hη(Θtωτ,0) = 0, (hη)z(Θtωτ,0) = 0 ∈ L(X). We consider the linearization of problem (3.9) about ξ∗ η ˙z=Bz+ (fη)z(Θtωτ, ζ∗ η(Θtωτ))z(3.16) =Az+Bη(Θtωτ)z. (3.17) Note that for each ωτ∈R×Ωfixed lim η→0sup t∈R kBη(Θtωτ)kL(X)= 0. Then we choose ˜η≤ηΘ-invariant and consider a linear co-cycle ϕ(t, ωτ) = ϕ˜η(ωτ)(t, ωτ)satisfying ϕ(t, τ, ω)x0=eAtx0+Zt 0 eA(t−s)B˜η(ωτ)(Θsωτ)ϕ(s, τ, ω)x0ds. Since {eAt:t≥0}admits an exponential dichotomy, we apply Theorem 3.1 to guarantee that the linear co-cycle (ϕ,Θ) admits an exponential dichotomy for each suitable small  > 0. Thus ζ∗ := ζ∗ ˜ηis a global solution that exhibits a hyperbolic behavior. The discussion above suggests a proper definition of random hyperbolic solution for a nonautonomous random differential equation. Definition 3.11. Let Bbe a generator of a strongly continuous semigroup, and f:R×Ω×X→Xsuch that for each (t, ω)fixed X3x7→ f(t, ω, x)is differentiable, and ζ:R×Ω→Xbe a bounded global solution of ˙x=Bx +f(Θtωτ, x), t ≥0, x(0) = x0∈X. (3.18) We say that ζis a random hyperbolic solution of (3.18) if there exists a linear nonautonomous random dynamical system (ϕ, Θ) satisfying ϕ(t, ωτ) = eBt +Zt 0 eB(t−s)Dxf(Θsωτ, ζ(Θsωτ))ϕ(s, ωτ)ds, for all ωτ∈R×Ω, and (ϕ, Θ) admits an exponential dichotomy. Joining the results of Theorem 3.9 and Remark 3.10 we obtain: Theorem 3.12 (Existence and continuity of random hyperbolic solutions).Let y∗ 0 be a hyperbolic equilibrium for (3.8) and assume that (3.10) and (3.11) hold. Given  > 0small enough, there exists a Θ-invariant map η:R×Ω→(0,1] and a global solution ζ∗ :R×Ω→Xof (ψη,Θ) defined by ζ∗ (ωτ) := ζ∗ η(ωτ)(ωτ) such that ζ∗ is a random hyperbolic solution of (ψη,Θ), with sup t∈R kζ∗ (Θtωτ)−y∗ 0kX< . 22 T. CARABALLO, A. N. CARVALHO, J. A. LANGA, AND A. N. OLIVEIRA-SOUSA Remark 3.13. Theorem 3.12 provides existence and continuity of random hyperbolic solutions from nonautonomous random perturbations of a hyperbolic problem. However, this result of persistence can be proved in a general context. In other words, following similar steps, it is possible to prove that random hyperbolic solutions are stable under (random nonautonomous) perturbations. Remark 3.14. In the parabolic case, when −A is sectorial, with A ∈ L(Xδ, X), 0< δ < 1, where Xδis a fractional power of X, we cannot assume that the nonlinearity f0:U⊂X→Xis differentiable, see [10]. We have to assume that the hyperbolic equilibrium y∗ 0is in Xδand that Uis a open neighborhood of y∗ 0in Xδsuch that f0:U⊂Xδ→Xis differentiable with derivative f0(y∗ 0)∈ L(Xδ, X). Also, we have to use a slightly different estimative on the Green function of {eAt: t≥0} kGA(t, s)kL(X,Xδ)≤D(M, δ)(t−s)−δe−β|t−s|,0< t −s≤1 kGA(t, s)kL(X,Xδ)≤D(M, δ)e−β|t−s|,otherwise, where D=D(M, δ)>0is a constant, see [23]. Under these conditions the proof of existence and continuity of bounded random hyperbolic equilibrium on Xδis analogous to the argument used in Theorem 3.12. 3.3. Stochastic perturbations of autonomous problems. In this subsection we apply the results established in this work to study stability under a nonautonomous stochastic perturbation of a hyperbolic autonomous problem. We consider the following family of Stratonovich stochastic differential equations with a nonautonomous multiplicative white noise dy =Bydt +f(y)dt +ηκty◦dWt, t ≥τ, y(τ) = yτ,(3.19) where Bis a generator of a strongly continuous semigroup {eBt:t≥0}on X, the family {Wt:t∈R}is the standard Wiener process, see [1], and κ:R→R+is continuously differentiable, η > 0. We apply the results of Subsection 3.2, by a formal change of variable we show how to modify problem (3.19) to a nonautonomous random differential equations. We prove existence and continuity of random hyperbolic solutions for the associated nonautonomous differential equation. The canonical sample space of a Wiener process is Ω := C0(R) the set of continuous functions over Rwhich are 0 at 0 equipped with the compact open topology. We denote Fthe associated Borel σ-algebra. Let Pbe the Wiener probability measure on Fwhich is given by the distribution of a two-sided Wiener process with trajectories in C0(R). The flow θis given by the Wiener shifts θtω(·) = ω(t+·)−ω(t), t ∈R, ω ∈Ω. In order to obtain a nonautonomous random differential equation from (3.19) we consider an auxiliary scalar stochastic differential equation dzt+zdt =dWt.(3.20) This problem has a stationary solution known as the Ornstein-Uhlenbeck process. EFFECT OF A SMALL BOUNDED NOISE ON HYPERBOLICITY 23 Lemma 3.15. There exists a θ-invariant subset ˜ Ω∈ F of full measure such that limt→±∞ |ω(t)| t= 0, ω ∈˜ Ωand, for such ω, the random variable given by z∗(ω) = −Z0 −∞ esω(s)ds is well defined. Moreover, for ω∈˜ Ω, the mapping (t, ω)7→ z∗(θtω)is a stationary solution of (3.20) with continuous trajectories, and lim t→±∞ |z∗(θtω)| t= 0,∀ω∈˜ Ω. The proof of the following Lemma can be found on [14]. Let ybe a solution for (3.19) and consider v(t, ω) := e−ηκtz∗(θtω)y(t, ω). Hence, vhas to satisfy the following nonautonomous random differential equation ˙v=Bv+e−ηκtz∗(θtω)f(eηκtz∗(θtω)v) + η[κt−˙κt]z∗(θtω)v, Define fη(t, ω, v) := e−ηκtz∗(ω)f(eηκtz∗(ω)v), and Bη(t, ω)v:= η[κt−˙κt]z∗(ω)v. In order to obtain hyperbolic solutions for (3.19) we study existence of random hyperbolic solutions for ˙v=Bv+fη(Θtωτ, v) + Bη(Θtωτ)v, t ≥0, v(0) = v0∈X. (3.21) To apply our results, we have to check hypotheses of Theorem 3.12 to guarantee existence of random hyperbolic solutions (3.21). To that end, we choose any differentiable positive real function κfor which there are random variables m1, m2>0 such that m1(ω) := sup t∈R {|κtz∗(θtω)|} <∞,and m2(ω) := sup t∈R {|[κt−˙κt]z∗(θtω)|} <∞. Define, for each ωτ∈Rט Ω, c(τ, ω) := (κτ−˙κτ)z∗(ω), M(τ, ω) := sup t∈R {|c(Θt(ωτ))|} ≤ m2(θ−τω). Note that supt∈RM(Θtωτ) = M(ωτ).Then supt∈RkBη(Θtωτ)kL(X)=ηM(ωτ). Let y∗ 0be a hyperbolic equilibrium of ˙y=By+f(y), and Ube a neighborhood of y∗ 0such that fη(ωτ,·)∈C1(U, X), for every η∈[0,1] and ωτ∈Rט Ω. We claim that for each ωτ∈Rט Ω sup (t,x)∈R×Unkfη(Θtωτ, x)−f(x)kX+kDxfη(Θtωτ, x)−f0(x)+Bη(Θtωτ)kL(X)o→0, (3.22) as η→0. The derivative of fη(Θtωτ,·) on xis Dxfη(Θtωτ, x) = f0(eηκt+τz∗(θtω)x). Since, for every (τ, ω)∈Rט Ω the limit ηsupt∈R|κt+τz∗(θtω)|goes to 0, as η→0, by continuity supt∈R|eηκt+τz∗(θtω)−1| → 0,as η→0.Similarly, for f0we have that sup t∈R kf0(eηκt+τz∗(θtω)x)−f0(x)kL(X)→0,as η→0, kfη(Θtωτ, x)−f(x)kX≤e−ηκt+τz∗(θtω)kf(eηκt+τz∗(θtω)x)−f(x)kX +|e−ηκt+τz∗(θtω)−1|kf(x)kX, 24 T. CARABALLO, A. N. CARVALHO, J. A. LANGA, AND A. N. OLIVEIRA-SOUSA and both terms on the right side go to zero when η→0, uniformly on t. Therefore, for each ωτ∈Rט Ω we have that sups∈Rλ(η, Θsωτ) = λ(η, ωτ)→0,as η→0. Hence, we can apply Theorem 3.12 to (3.21) to prove existence and continuity of hyperbolic solutions. Theorem 3.16. Let y∗ 0be a hyperbolic equilibrium for ˙y=By+f(y). Given  > 0 small enough, there exists a Θ-invariant map η:R×Ω→(0,1] and a global solution ζ∗ :R×Ω→Xof (ψη,Θ) defined by ζ∗ (ωτ) := ζ∗ η(ωτ)(ωτ) such that ζ∗ is a random hyperbolic solution of (ψη,Θ), with sup t∈R kζ∗ (Θtωτ)−y∗ 0kX< . Moreover, ζ∗ defines a stochastic process, i.e., the mapping ˜ Ω3ω7→ ζ∗ (t, ω)∈X is (˜ F,BX)-measurable, for each t∈R. Proof. Given  > 0 small enough, thanks to Theorem 3.12, we know that for every ωτ∈Rט Ω there exists a Θ-invariant map η:Rט Ω→(0,1] such that ζ∗ :Rט Ω→ Xis a random hyperbolic solution for (3.21). Let us prove that ζ∗ (t, ·) is ( ˜ F,BX)-measurable. Consider φη(ωτ)∈Xthe fixed point of operator Iωτ,η(ωτ), see the proof of Theorem 3.9. We claim that the mapping ˜ Ω3ω7→ φη(ωτ)(0, ωτ) is ( ˜ F,BX)-measurable, where ˜ Fis the σ-algebra of ˜ Ω. Indeed, since φη(ωτ)(0) = Iωτ,η(ωτ)(φη(ωτ))(0),we know that φη(ωτ)(0) = lim n→+∞In ωτ,η(ωτ)(˜ 0)(0) where ˜ 0∈X. Thus, the claim follows from the fact that, for each τ∈R, the mapping ω7→ Iωτ,η(ωτ)(˜ 0)(0) is ( ˜ F,BX)-measurable. Therefore, for each τ∈Rfixed, the mapping ˜ Ω3ω7→ ζ∗ (ωτ) := φ∗ ηωτ(0, ωτ) + y∗ 0is ( ˜ F,BX)-measurable, and the proof is complete.  To end this section we present two special examples: the linear case, and an application in partial differential equations. Example 3.17. If we assume that Agenerates a C0-semigroup satisfying conditions of Theorem 3.3, then y= 0 is a random hyperbolic solution for dy =Aydt +ηκty◦dWt, t ≥τ, for suitable small η > 0. In fact, the associated nonautonomous random differential equation is ˙v=Av+η[κt−˙κt]z∗(θtω)v, t ≥τ. Thus, by Theorem 3.3, for every  > 0suitable small there exists a Θ-invariant η:R×Ω→(0,1] such that A(ωτ) = A+η(ωτ)[κτ−˙κτ]z∗(ω)vgenerates a linear nonautonomous random dynamical system that admits an exponential dichotomy. We now apply Theorem 3.16 to a strongly damped wave equation. Example 3.18. Let Dbe a bounded smooth domain in R3,A:D(A)⊂L2(D)→ L2(D)is −∆with Dirichlet boundary condition, and f:R→Ris twice differentiable. Consider the damped wave equation utt +βut−∆u=f(u),in D EFFECT OF A SMALL BOUNDED NOISE ON HYPERBOLICITY 25 with boundary condition u= 0, in ∂D. The initial data will be taken in the space X=H1 0(D)×L2(D). Hence we can consider an abstract evolutionary equation in X: ˙y=By+F(y),(3.23) where y=y1 y2∈X, B=0I −A−β, F(y) = 0 fe(y1), with fe:H1 0(D)→L2(D)is given by fe(y1)(x) = f(y1(x)) for x∈D. Moreover, feis continuously differentiable, see [5]. The hyperbolic equilibrium points of (3.23) are of the form y∗ 0= (u∗ 0,0) where u∗ 0is a solution of −∆u=g0(u), such that 0/∈σ(−∆ + Dxfe(u∗ 0)IdX). Now we consider a nonautonomous multiplicative white noise on (3.23), i.e., ˙y=By+F(y) + ˜ηκty◦dWt,(3.24) where ˜η=η0 0 0,0 0 0ηor η0 0η, and we may choose where to consider the noise, on the position u, or velocity utor both. Let Abe a 2×2- matrix with real coefficients, the exponential of Ais represented by eA. Let ybe a solution of (3.24), then apply the change of variables v(t, ω) = e−˜ηκtz∗(θtω)yto obtain ˙v=Bv+Fη(Θtωτ, v) + ˜η(κt−˙κt)z∗(θtω)v, (3.25) where Fη(ωτ, v) := e˜ηκτz∗(ω)F(e−˜ηκτz∗(ω)v). Note that, for each η∈[0,1] and ωτ∈ R×Ωfixed, problem (3.25) is a well-posed nonautonomous differential equation, see Arrieta et al. [5]. Hence induces a nonautonomous random dynamical system that satisfy condition (2.1). Now, we apply Theorem 3.16 for (3.25) to ensure that for each hyperbolic equilibrium y∗ 0and  > 0there exists a random hyperbolic solution that defines stochastic processes {ξ∗ (t, ·) : ˜ Ω→X:t∈R}for each  > 0such that supt∈Rkξ∗ (Θtωτ)− y∗ 0k → 0as →0. Remark 3.19. We note that it is not clear yet if the dynamical systems generated by (3.3) and (3.19) are conjugated. This is not a simple task because our change of variables is not stationary because of the presence of κt. However, for every solution of (3.21) it is possible to find a corresponding solution for the nonautonomous random differential equations (3.3) and we analyze properties of the latter. More precisely, we know now under which conditions it is possible to guarantee that global solutions of (3.21) exhibit a hyperbolic behavior. Remark 3.20. The study of this subsection could be applied to Itˆo stochastic differential equations with multiplicative white noise, by modifying to a Stratonovich stochastic differential equation, whenever is possible.