Attractors for Stochastic Lattice Dynamical Systems with a Multiplicative Noise

ATTRACTORS FOR STOCHASTIC LATTICE DYNAMICAL SYSTEMS WITH A MULTIPLICATIVE NOISE TOM´ AS CARABALLO AND KENING LU Abstract. In this paper, we consider a stochastic lattice differential equation with diffusive nearest neighbor interaction, a dissipative nonlinear reaction term, and a multiplicative white noise at each node. We prove the existence of a compact global random attractor which pulled back attracts tempered random bounded sets. 1. Introduction This paper is devoted to the long term behavior of the following stochastic lattice differential equation (1.1) dui(t) dt =ν(ui−1−2ui+ui+1)−λui−fi(ui) + gi+ N X j=1 cjui◦dwj(t) dt , i ∈Z, where u= (ui)i∈Z∈`2,Zdenotes the integer set, νand λare positive constants, fiis a smooth function satisfying a dissipative condition, g= (gi)i∈Z∈`2, cj∈R, for j= 1, ..., N, and wjare mutually independent Brownian motions, where ◦denotes the Stratonovich sense in the stochastic term. Stochastic lattice differential equations arise naturally in a wide variety of applications where the spatial structure has a discrete character and uncertainties or random influences, called noises, are taken into account. These systems are used to model such systems as cellular neural networks with applications to image processing, pattern recognition, and brain science [15, 16, 17, 18]. They are also used to model the propagation of pulses in myelinated axons where the membrane is excitable only at spatially discrete sites. In this case, uirepresents the potential at the i-th active site; see for example, [7], [8], [39], [36], [30, 31]. Lattice differential equations can also be found in chemical reaction theory [24, 28, 33]. Equation (1.1) is a one-dimensional lattice system with diffusive nearest neighbor interaction, a dissipative nonlinear reaction term and a multiplicative white noise at each node. This may be the result of an environmental effect on the whole domain of the system. The system with an additive noise was studied in Bates et al. [5]. Recently, there are many works on deterministic lattice dynamical systems. For traveling waves, we refer the readers to [11, 34, 12, 42, 1, 4] and the references therein. The chaotic properties of solutions for such systems have been investigated by [11] and [14, 40, 13, 22]. In the absence of the white noise, the existence of a global attractor for lattice differential equation (1.1) was established in [6]. 1991 Mathematics Subject Classification. Primary: 60H15; Secondary: 34C35, 58F11, 58F15, 58F36. Key words and phrases. stochastic lattice differential equations, random attractors, multiplicative noise. This work was partially supported by Ministerio de Educaci´on y Ciencia (Spain) MTM2005-01412 (T. Caraballo) and NSF0200961, NSF0401708, and NSFC10371083 (K. Lu). 1 2 TOM´ AS CARABALLO AND KENING LU The study of global random attractors was initiated by Ruelle [37]. The fundamental theory of global random attractors for stochastic partial differential equations was developed by Crauel, Debussche, and Flandoli [20], Crauel and Flandoli [21], Flandoli and Schmalfuss [26], Imkeller and Schmalfuss [27], and others. Due to the unbounded fluctuations in the systems caused by the white noise, the concept of pull-back global random attractor was introduced to capture the essential dynamics with possibly extremely wide fluctuations. This is significantly different from the deterministic case. In the present paper, we prove the existence of a global random attractor for the infinite dimensional random dynamical system generated by the stochastic lattice differential equation (1.1). An interesting feature of this is that, even though the spatial domain is unbounded and the solution operator is not smoothing or compact, unlike parabolic type of partial differential equations on bounded domains, bounded sets of initial data converge, in the pullback sense, under the forward flow to a random compact invariant set. The noise involved here is multiplicative. The domain of attraction is the set of all tempered sets as used in [26] instead of all bounded deterministic sets. It is worth mentioning that, although it is known that finite dimensional stochastic differential equations generate random dynamical systems (see Arnold [2] Chapter 1), this is not true in general for infinite dimensional equations. However, for particular kinds of noise it is possible to transform the stochastic differential equation into a random one so that it generates a random dynamical system. For general types of noise it is unknown whether this is possible. In Section 2, we introduce basic concepts concerning random dynamical systems and global random attractors. In Section 3, we show that the stochastic lattice differential equation (1.1) generates a infinite dimensional random dynamical system. The existence of the global random attractor is given in Section 4. 2. Random Dynamical Systems In this section, we introduce basic concepts related to random dynamical systems and the concept of attractor, which are taken from [2] and [26]. Let (H, k·kH) be a Hilbert space and (Ω,F,P) be a probability space. Definition 2.1. ¡Ω,F,P,(θt)t∈R¢is called a metric dynamical system, if θ:R×Ω→Ωis (B(R)× F,F)-measurable, θ0is the identity on Ω,θs+t=θt◦θsfor all s, t ∈Rand θtP=Pfor all t∈R. Definition 2.2. A stochastic process ¡ϕ(t)¢t≥0is a continuous random dynamical system over ¡Ω,F,P,(θt)t∈R¢if ϕis (B[0,∞)× F × B(H),B(H))-measurable, and for all ω∈Ω (D1) the mapping ϕ(·, ω, ·) : [0,∞)×H→His continuous, (D2) ϕ(0, ω, ·)is the identity on H, (D3) ϕ(s+t, ω, ·) = ϕ(t, θsω, ·)◦ϕ(s, ω, ·)for all s, t ≥0(cocycle property). As we mentioned in the introduction, it is known that finite dimensional stochastic differential equations generate random dynamical systems (see Arnold [2] Chapter 1). This is not true in general for infinite dimensional equations. However, for particular kinds of noise, as will be our case, we can apply the following simple lemma to obtain a random dynamical system (see Caraballo et al. [9]). ATTRACTORS FOR STOCHASTIC LATTICE DYNAMICAL SYSTEMS 3 Lemma 2.3. Let ψbe a random dynamical system. Suppose that the mapping T: Ω ×H→H possesses the following properties: For fixed ω∈Ω, the mapping T(ω, ·)is a homeomorphism on H, and for fixed x∈H, the mappings T(·, x), T−1(·, x)are measurable. Then the mapping (2.1) (t, ω, x)→ϕ(t, ω, x) := T−1(θtω, ψ(t, ω, T (ω, x))) is a (conjugated) random dynamical system. The measurability of ϕfollows because of the properties of T. Later on we will transform our stochastic evolution equation containing a noisy term into an evolution equation without noise but with random coefficients. Definition 2.4. A set A⊂Ωis called invariant with respect to (θt)t∈Rif for all t∈Rit holds (2.2) θ−1 tA=A. Definition 2.5. Arandom set A(ω)is a multi-valued map A: Ω →2H\ ∅ such that, for every x∈H, the map ω7→ d(x, A(ω)) is measurable. It is said that the random set is bounded (resp. closed or compact) if A(ω)is bounded (resp. closed or compact) for almost all ω∈Ω. Definition 2.6. A random bounded set B(ω)⊂His called tempered with respect to (θt)t∈Rif for a.e. ω∈Ω (2.3) lim t→∞ e−βtd(B(θ−tω)) = 0 for all β > 0, where d(B) = sup x∈B kxkH. We consider a continuous random dynamical system (ϕ(t))t≥0over (Ω,F,P,(θt)t∈R) and Da collection of random subsets of H. Definition 2.7. A random set Kis called an absorbing set in Dif for all B∈ D and a.e. ω∈Ω there exists tB(ω)>0such that (2.4) ϕ(t, θ−tω, B(θ−tω)) ⊂K(ω)for all t≥tB(ω). Definition 2.8. A random set Ais called a global random Dattractor (pullback Dattractor) for ϕif the following hold: (A1) Ais a random compact set; (A2) Ais strictly invariant, i.e. for a.e. ω∈Ωand all t≥0one has ϕ(t, ω, A(ω)) = A(θtω); (A3) Aattracts all sets in D, i.e., for all B∈ D and a.e. ω∈Ωwe have (2.5) lim t→∞ d(ϕ(t, θ−tω, B(θ−tω)),A(ω)) = 0, where d(X, Y ) = sup x∈X inf y∈Ykx−ykHis the Hausdorff semi-metric (here X⊆H, Y ⊆H). The collection Dis called domain of attraction of A. Some possibilities for domains Dof attraction used very often in the investigation of random attractors are the following: 1) the collection of all finite deterministic subsets of H(in this case the random Dattractor is called a point attractor); 2) the collection of all compact deterministic subsets of H(in this case the random Dattractor is called a set attractor); 4 TOM´ AS CARABALLO AND KENING LU 3) the collection of all tempered random subsets of H. Examples of point attractors and set attractors can be found in [38], [19] and also in [2] (Theorem 9.3.3 pp. 484, Lemma 9.3.5 pp. 485). The results on random Dattractors, where Dis the collection of tempered random sets, can be found in [25], [26], [27]. The next proposition is an abstract result on the existence of global random attractor, which is a slight generalization of Theorem 3.5 on pg. 27 in [26] (see [5] for the proof). Although it may be possible to apply a more general result from Caraballo et al. [10], the following one will fit our purposes. Proposition 2.9. Let K∈ D be an absorbing set for the continuous random dynamical system (ϕ(t))t≥0which is closed, and which satisfies for a.e. ω∈Ωthe following asymptotic compactness condition: each sequence xn∈ϕ(tn, θ−tn, K(θ−tnω)) with tn→ ∞ has a convergent subsequence in H. Then the cocycle ϕhas a unique global random attractor A(ω) = \ τ≥tK(ω)[ t≥τ ϕ(t, θ−tω, K(θ−tω)). 3. Stochastic Lattice Differential Equations We consider a stochastic lattice differential equation (3.1) dui(t) dt =ν(ui−1−2ui+ui+1)−λui−fi(ui) + gi+ N X j=1 cjui◦dwj(t) dt , i ∈Z, where u= (ui)i∈Z∈`2,Zdenotes the integer set, νand λare positive constants, fiis a smooth function satisfying a dissipative condition, g= (gi)i∈Z∈`2, cj∈R, for j= 1, ..., N, and wjare mutually independent two-sided Brownian motions on the same probability space (Ω,F,P). We note that equation (3.1) is interpreted as a system of integral equations ui(t) = ui(0) + Zt 0 (ν(ui−1(s)−2ui(s) + ui+1(s)) −λui(s)−fi(ui(s)) + gi)ds(3.2) +Zt 0 N X j=1 cjui(s)◦dwj(t), i ∈Z, where the stochastic integral is understood in the sense of Stratonovich. Assumptions on the nonlinearity fi: Let fi∈ C1(R) satisfy that supi∈Z|fi0(u)|is bounded for uin bounded sets and (3.3) fi(x)x≥0 for all x∈R. We note that condition (3.3) implies fi(0) = 0.If fi(s) = p X j=0 ajs2j+1 with aj≥0 for each j= 0, . . . , p, then conditions (3.3) is satisfied. This kind of nonlinearity was considered in [11] and [22]. ATTRACTORS FOR STOCHASTIC LATTICE DYNAMICAL SYSTEMS 5 For convenience, we now formulate system (3.1) as a stochastic differential equation in `2. Denote by || · || the norm in the space `2,and by B, B∗, Cj, j = 1, ..., N, and Athe linear operators from `2 to `2defined as follows. For u= (ui)i∈Z∈`2, (Bu)i=ui+1 −ui,(B∗u)i=ui−1−ui,(Cju)i=cjui and (Au)i=−ui−1+ 2ui−ui+1 for each i∈Z. Then we find that A=BB∗=B∗B, and (B∗u, v) = (u, Bv) for all u, v ∈`2. Therefore (Au, u)≥0 for all u∈`2. Let ˜ fbe the Nemytski operator associated with fi, that is, for u= (ui)i∈Z∈`2, let ˜ f(u) = (fi(ui))i∈Z. Then we have (3.4) k˜ f(u)k2=X i∈Z |fi(ui)|2=X i∈Z |fi(ui)−fi(0)|2=X i∈Z |f0 i(ξi)|2|ui|2, with ξi=τiuifor some τi∈(0,1). Since |ξi| ≤ |ui| ≤ kuk and supi∈Z|f0 i|is bounded on a bounded set, it follows that there exists a constant µ(depending on kuk) such that k˜ f(u)k2≤µX i∈Z |ui|2=µkuk2, which means ˜ f(u)∈`2. Similar to (3.4), one can see that ˜ fis locally Lipschitz from `2to `2, more precisely, for every bounded set Yin `2, there exists a constant CY(depending only on Y) such that k˜ f(x)−˜ f(y)k ≤ CYkx−ykfor all x, y ∈Y. In the sequel, when no confusion arises we identify ˜ fwith f. The system (3.1) with initial values u0≡(u0,i)i∈Z∈`2may be rewritten as an equation in `2 for t≥0 and ω∈Ω, (3.5) u(t) = u0+Zt 0 (−νAu(s)−λu(s)−˜ f(u(s)) + g)ds + N X j=1 Zt 0 Cju(s)◦dwj(t). To prove that this stochastic equation (3.5) generates a random dynamical system, we will transform it into a random differential equation in `2. This can be done thanks to the special form of the stochastic term. Before performing this transformation, we need to recall some properties of the Ornstein-Uhlenbeck processes. Let us start by describing a probability space (Ω,F,P) which will be useful for our analysis. Consider Ω = {ω∈ C(R,R) : ω(0) = 0}=C0(R,R) 6 TOM´ AS CARABALLO AND KENING LU endowed with the compact open topology (see [2], Appendix A.2 and A.3), Pis the corresponding Wiener measure and F0is the Borel σ-algebra on Ω. Let (3.6) θtω(·) = ω(·+t)−ω(t), t ∈R, then (Ω,F0,P,(θt)t∈R) is a metric dynamical system. Let Fbe the P-completion of F0and let Ft:= _ s≤t Ft s, t ∈R with Ft s=σ{w(τ2)−w(τ1) : s≤τ1≤τ2≤t}_N, where σ{w(τ2)−w(τ1) : s≤τ1≤τ2≤t}is the smallest σ-algebra generated by the random variable w(τ2)−w(τ1) for all τ1, τ2such that s≤τ1≤τ2≤tand Nare the null sets of F. Note that θ−1 τFt s=Ft+τ s+τ, so (Ω,F0,P,(θt)t∈R,(Ft s)s≤t) is a filtered dynamical system (see [2], pp. 72, 91 and 546 for more details). Let us consider the one-dimensional stochastic differential equation (3.7) dz =−α z dt +dw(t), for α > 0.This equation has a random fixed point in the sense of random dynamical systems generating a stationary solution known as the stationary Ornstein-Uhlenbeck process (see Caraballo et al. [9] for more details). In fact, we have Lemma 3.1. (Caraballo et al. [9]) There exists a {θt}t∈R-invariant subset ¯ Ω∈ F of Ω = C0(R,R) of full measure such that lim t→±∞ |ω(t)| t= 0 for ω∈¯ Ω, and, for such ω, the random variable given by z∗(ω) := −αZ0 −∞ eατ ω(τ)dτ is well defined. Moreover, for ω∈¯ Ω, the mapping (t, ω)→z∗(θtω) = −αZ0 −∞ eατ θtω(τ)dτ =−αZ0 −∞ eατ ω(t+τ)dτ +ω(t) is a stationary solution of (3.7) with continuous trajectories. In addition, for ω∈¯ Ω lim t→±∞ |z∗(θtω)| |t|= 0,lim t→±∞ 1 tZt 0 z∗(θτω)dτ = 0, lim t→±∞ 1 tZt 0 |z∗(θτω)|dτ =E|z∗|<∞. (3.8) Remark 3.2. We now consider θdefined in (3.6) on ¯ Ωinstead of Ω. This mapping possesses the same properties as the original one if we choose for Fthe trace σ-algebra with respect to ¯ Ωdenoted also by F. ATTRACTORS FOR STOCHASTIC LATTICE DYNAMICAL SYSTEMS 7 Let us consider α= 1 and denote by z∗ jits associated Ornstein-Uhlenbeck process corresponding to (3.7) with wjinstead of w. Then for any j= 1, ..., N we have a stationary Ornstein-Uhlenbeck process generated by a random variable z∗ j(ω) on ¯ Ωjwith properties formulated in Lemma 3.1 defined on the metric dynamical system (¯ Ωj,Fj,Pj, θ). We set (3.9) (Ω,F,P, θ), where Ω = ¯ Ω1× · · · × ¯ ΩN,F= N O i=1 Fi,P=P1×P2× · · · × PN, and θis the flow of Wiener shifts. Now, let us note that operator Cjgenerates a strongly continuous semigroup (in fact, a uniformly continuous group) of operators SCj (t).More precisely, SCj(t) is given by SCj(t)u=ecjtu, for u∈`2. Then we denote T(ω) := SC1(z∗ 1(ω)) ◦... ◦SCN(z∗ N(ω)) = e N P j=1 cjz∗ j(ω) Id`2 which is clearly a homeomorphism in H=`2. The inverse operator is well defined by T−1(ω) := SCN(−z∗ N(ω)) ◦... ◦SC1(−z∗ 1(ω)) = e − N P j=1 cjz∗ j(ω) Id`2. From simplicity, let us denote δ(ω) = N P j=1 cjz∗ j(ω). It easily follows that kT−1(θtω)khas subexponential growth as t→ ±∞ for any ω∈Ω. Hence kT−1kis tempered. According to Remark 3.2 we can change our metric dynamical system with respect to ¯ Ω. However the new metric dynamical system will be denoted by the same symbols (Ω,F,P, θ). We now argue in a heuristic informal way. Let us consider the change of variable v(t) = T−1(θtω)u(t) = e−δ(θtω)u(t), where uis a solution to (3.5). Then, dv(t) = e−δ(θtω)du(t)− N X j=1 cje−δ(θtω)u(t)◦dz∗ j(θtω) =e−δ(θtω)(−νAu(t)−λu(t)−˜ f(u(t)) + g+δ(θtω)u(t))dt =³−νAv(t)−λv(t)−e−δ(θtω)˜ f(eδ(θtω)v(t)) + e−δ(θtω)g+δ(θtω)v(t)´dt. So we can consider the following evolution equation with random coefficients but without white noise dv dt =−νAv −(λ−δ(θtω)) v−e−δ(θtω)˜ f³eδ(θtω)v´+e−δ(θtω)g (3.10) and initial condition v(0) = v0∈H. Now we establish the following result. 8 TOM´ AS CARABALLO AND KENING LU Theorem 3.3. Let T > 0and v0∈Hbe fixed. Then the following properties hold: (1) For every ω∈Ω,equation (3.10) admits a unique solution v(·, ω, v0)∈ C([0, T], `2), (2) The solution vof (3.10) depends continuously on the initial data v0, i.e., for each ω∈Ω the mapping v0∈`27→ v(·, ω, v0)∈ C([0, T], `2)is continuous. Proof. (1) For any fixed T > 0 and ω∈Ω,and thanks to standard arguments (notice that the mapping F(t, v) = e−δ(θtω)f¡eδ(θtω)v¢is locally Lipschitz with respect to its second variable and the Lipschitz constant is uniformly bounded in [0, T]) (3.10) possesses a local solution v(·;ω, v0)∈ C([0, Tmax), `2),where [0, Tmax) is the maximal interval of existence of the solution of (3.10). We prove now that this local solution is a global one. From (3.10) it follows that d dt||v(t)||2= 2 ³−ν(Av, v)−λ||v||2−³e−δ(θtω)˜ f³eδ(θtω)v´, v´+δ(θtω)||v||2´ (3.11) + 2 ³e−δ(θtω)g, v´ ≤ −2λ||v||2+ 2δ(θtω)||v||2+λ||v||2+1 λe−2δ(θtω)||g||2 ≤(−λ+ 2δ(θtω)) ||v||2+1 λe−2δ(θtω)||g||2, whence ||v(t)||2≤e−λt+2 R t 0δ(θsω)ds||v0||2 +||g||2 λe−λt+2 R t 0δ(θsω)ds Zt 0 e−2δ(θsω)+λs−2 R s 0δ(θrω)drds. If we denote β(ω) = maxt∈[0,T ]³||g||2 λe−λt+2 R t 0δ(θsω)ds Rt 0e−2δ(θsω)+λs−2 R s 0δ(θrω)drds´and α(ω) = 2RT 0|δ(θsω)|ds, then we have that (3.12) ||v(t)||2≤ ||v0||2eα(ω)+β(ω) which implies that the solution vis defined in any interval [0, T ]. (2) Let u0, v0∈`2and X(t) := v(t, ω, u0), Y(t) := v(t, ω, v0) the corresponding solutions of (3.10). Then, if we denote Z(t) = X(t)−Y(t) we have dZ(t) dt =−νAZ(t)−(λ−δ(θtω)) Z(t)−e−δ(θtω)³˜ f³eδ(θtω)X(t)´−˜ f³eδ(θtω)Y(t)´´, and, d dt||Z(t)||2= 2 ³−ν(AZ, Z)−λ||Z||2−e−δ(θtω)³˜ f³eδ(θtω)X(t)´−˜ f³eδ(θtω)Y(t)´, Z´´ + 2δ(θtω)||Z||2 ≤2³e−δ(θtω)° ° °˜ f³eδ(θtω)X(t)´−˜ f³eδ(θtω)Y(t)´° ° °||Z|| +δ(θtω)||Z||2´ ≤2¡L||Z||2+δ(θtω)||Z||2¢ ≤γ||Z||2, where γ= 2 ¡L+ maxt∈[0,T ]|δ(θtω)|¢,and Ldenotes the Lipschitz constant of fcorresponding to a bounded set where Xand Ybelong to (see (3.12)). ATTRACTORS FOR STOCHASTIC LATTICE DYNAMICAL SYSTEMS 9 Now, by standard computations, we obtain that ||Z(t)||2≤eγt||Z(0)||2, and, consequently, sup t∈[0,T ] ||X(t)−Y(t)||2≤eγT ||u0−v0||2. If u0=v0, then the above inequality shows the uniqueness and continuous dependence on the initial data of the solution of (3.10). So, the properties (1) and (2) of this theorem hold. ¤ Theorem 3.4. Equation (3.10) generates a continuous random dynamical system (ψ(t))t≥0over (Ω,F0, P, (θt)t∈R), where ψ(t, ω, u0) = v(t, ω, u0)for u0∈`2, t ≥0and for all ω∈Ω. Moreover, if we define ϕby (3.13) ϕ(t, ω, u0) = T(θtω)ψ(t, ω, T−1(ω)u0)for u0∈`2, t ≥0and for all ω∈Ω, then ϕis another random dynamical system for which the process (ω, t)→ϕ(t, ω, u0) solves (3.5) for any initial condition u0∈`2. Proof. The fact that ψis a continuous random dynamical system follows from Theorem 3.3. The measurability of ϕfollows by the properties of T. To prove that ϕand ψare conjugated random dynamical systems, we will use the chain rule. Then, omitting the arguments for simplicity, we have that d(Tψ) = Tdψ + N X j=1 cjTψ ◦dz∗ j =T(−νAψ −(λ−δ)ψ−T−1˜ f(Tψ) + T−1g)dt −δTψdt + N X j=1 cjTψ ◦dwj = (−νA (Tψ)−λT ψ −˜ f(Tψ) + g)dt + N X j=1 cjTψ ◦dwj, and the proof is complete. ¤ 4. Existence of Global Random Attractors In this section, we prove the existence of a global random attractor for the random lattice dynamical system generated by equation (3.1). Our main result is Theorem 4.1. The random lattice dynamical system ϕgenerated by Eq. (3.5) has a unique global random attractor. To prove this theorem we will use Proposition 2.9. In order to prove that our lattice random dynamical system ϕsatisfies the assumptions in the proposition above, we will proceed in the following way. We will prove first that there exists an absorbing set for ψin D. Next we will construct a corresponding closed absorbing one in Dfor its conjugated lattice random dynamical system ϕ. Finally, we will prove the asymptotic compactness. 16 TOM´ AS CARABALLO AND KENING LU [12] S.-N. Chow, J. Mallet-Paret, and W. Shen, Traveling waves in lattice dynamical systems. J. Diff. Eq. 149 (1998), 248–291. [13] S.-N. Chow, J. Mallet-Paret, and E. S. 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Caraballo: [email protected] E-mail address, K. Lu: [email protected] or [email protected] (T. Caraballo) Depto. Ecuaciones Diferenciales y An´ alisis Num´ erico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla (SPAIN) (K. Lu) Department of Mathematics, Brigham Young University, Provo, Utah 84602, USA