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PETER W. BATES, HANNELORE LISEI, AND KENING LU. The study of global random attractors was initiated by Ruelle [35]. The fundamental theory of global ...
ATTRACTORS FOR STOCHASTIC LATTICE DYNAMICAL SYSTEMS PETER W. BATES, HANNELORE LISEI, AND KENING LU Abstract. We consider a one-dimensional lattice with diffusive nearest neighbor interaction, a dissipative nonlinear reaction term and additive independent white noise at each node. We prove the existence of a compact global random attractor within the set of tempered random bounded sets. An interesting feature of this is that, even though the spatial domain is unbounded and the solution operator is not smoothing or compact, pulled back bounded sets of initial data converge under the forward flow to a random compact invariant set.

1. Introduction In this paper, we study the long term behavior for the following stochastic lattice differential equation dui (t) dwi (t) = ν(ui−1 − 2ui + ui+1 ) − λui − f (ui ) + gi + ai , i ∈ Z, dt dt where u = (ui )i∈Z ∈ ℓ2 , Z denotes the integer set, ν and λ are positive constants, f is a smooth function satisfying a dissipative condition, g = (gi )i∈Z ∈ ℓ2 , a = (ai )i∈Z ∈ ℓ2 , and {wi : i ∈ Z} are independent Brownian motions. (1)

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 random effects are not only introduced to compensate for the defects in some deterministic models, but also are often rather intrinsic phenomena. Equation (1) is a one-dimensional lattice system with diffusive nearest neighbor interaction, a dissipative nonlinear reaction term and additive independent white noise at each node. Higher dimensional lattices could also be considered Without considering noise in systems, deterministic lattice differential equations are used to model such systems as cellular neural networks with applications to image processing, pattern recognition, and brain science [12, 13, 14, 15]. They are also used to model the propagation of pulses in myclinated axons where the membrane is excitable only at spatially discrete sites. In this case, ui represents the potential at the i-th active site; see for example, [6], [7], [37], [34], [27, 28]. Lattice differential equations can also be found in chemical reaction theory [21, 25, 30]. Recently, there are many works on deterministic lattice dynamical systems. For traveling waves, we refer the readers to [8, 31, 32, 9, 40, 1, 4] and the references therein. The chaotic properties of solutions for such systems have been investigated by [8, 31] and [11, 38, 10, 19]. In the absence of the white noise, the existence of a global attractor for lattice differential equation (1) was established in [5]. 1991 Mathematics Subject Classification. Primary: 60H15; Secondary: 34C35, 58F11, 58F15, 58F36. Key words and phrases. Stochastic Lattice Differential Equations, Random Attractors. This work was partially supported by NSF0200961. 1

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PETER W. BATES, HANNELORE LISEI, AND KENING LU

The study of global random attractors was initiated by Ruelle [35]. The fundamental theory of global random attractors for stochastic partial differential equations was developed by Crauel, Debussche, and Flandoli [17], Crauel and Flandoli [18], Flandoli and Schmalfuss [23], Imkeller and Schmalfuss [24], 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). 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, pulled back bounded sets of initial data converge under the forward flow to a random compact invariant set. The noise involved here is an infinite dimensional white noise. The domain of attraction is the set of all tempered sets as used in [23] instead of all bounded deterministic sets. 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) generates a infinite dimensional random dynamical system. The existence of the global random attractor is given in Section 4. Finally, a basic property of white noise is given in Section 5. 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 [23]. 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 → Ω is (B(R) × F, F)-measurable, θ0 is the identity on Ω, θs+t = θt ◦ θs for all s, t ∈ R and θt P = P for all t ∈ R.  Definition 2.2. A stochastic process ϕ(t) t≥0 is a continuous random dynamical system  over Ω, F, P, (θt )t∈R if ϕ is (B[0, ∞) × F × B(H), B(H))-measurable, and for all ω ∈ Ω (D1) the mapping ϕ(·, ω, ·) : [0, ∞] × H → H is continuous, (D2) ϕ(0, ω, ·) is the identity on H, (D3) ϕ(s + t, ω, ·) = ϕ(t, θs ω, ·) ◦ ϕ(s, ω, ·) f or all s, t ≥ 0 (cocycle property). Definition 2.3. A random bounded set B(ω) ⊂ H is called tempered with respect to (θt )t≥0 if for a.e. ω ∈ Ω lim e−βt d(B(θ−t ω)) = 0 f or all β > 0, t→∞

where d(B) = sup kxkH . x∈B

  We consider a continuous random dynamical system ϕ(t) t≥0 over Ω, F, P, (θt )t∈R and D a collection of random subsets of H. Definition 2.4. A random set K is called an absorbing set in D if for all B ∈ D and a.e. ω ∈ Ω there exists tB (ω) > 0 such that ϕ(t, θ−t ω, B(θ−t ω)) ⊂ K(ω) f or all t ≥ tB (ω).

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Definition 2.5. A random set A is called a global random D attractor (pullback D attractor ) for ϕ if the following hold: (A1) A is a random compact set, i.e. ω 7→ d(x, A(ω)) is measurable for every x ∈ H and A(ω) is compact for a.e. ω ∈ Ω; (A2) A is strictly invariant, i.e. for a.e. ω ∈ Ω and all t ≥ 0 one has ϕ(t, ω, A(ω)) = A(θt ω); (A3) A attracts all sets in D, i.e., for all B ∈ D and a.e. ω ∈ Ω we have lim d(ϕ(t, θ−t ω, B(θ−t ω)), A(ω)) = 0,

t→∞

where d(X, Y ) = sup inf kx − ykH is the Hausdorff semi-metric (here X ⊆ H, Y ⊆ H). x∈X y∈Y

The collection D is called domain of attraction of A. Domains D of attraction used in the investigation of random attractors are for example: 1) the collection of all finite deterministic subsets of H (in this case the random D attractor is called a point attractor ); 2) the collection of all compact deterministic subsets of H (in this case the random D attractor is called a set attractor ); 3) the collection of all tempered random subsets of H. Examples of point attractors and set attractors can be found in [36], [16] and also in [2] (Theorem 9.3.3 pp. 484, Lemma 9.3.5 pp. 485). The results on random D attractors, where D is the collection of tempered random sets, can be found in [22], [23], [24]. 3. Stochastic Lattice Differential Equations We consider a stochastic lattice differential equation (2)

dwi (t) dui (t) = ν(ui−1 − 2ui + ui+1 ) − λui − f (ui ) + gi + ai , dt dt

i ∈ Z,

where u = (ui )i∈Z ∈ ℓ2 , Z denotes the integer set, ν and λ are positive constants, f is a smooth function satisfying a dissipative condition, g = (gi )i∈Z ∈ ℓ2 , a = (ai )i∈Z ∈ ℓ2 , {wi : i ∈ Z} are independent two-side Brownian motions. We note that equation (2) is interpreted as an system of integral equations Z t  (3) ui (t) = ui (0) + ν(ui−1 (s) − 2ui (s) + ui+1 (s)) − λui (s) − f (ui (s)) + gi ds + ai wi (t),

i ∈ Z,

0

Let ei ∈ ℓ2 denote the element having 1 at position i and all the other components 0. Let X (4) W (t) ≡ W (t, ω) ≡ ai wi (t)ei with (ai )i∈Z ∈ ℓ2 , i∈Z

be white noise with values in

ℓ2

defined on the probability space (Ω, F, P), where Ω = {ω ∈ C(R, ℓ2 ) : ω(0) = 0}

is endowed with the compact open topology (see [2], Appendix A.2), P is the corresponding Wiener measure and F is the P-completion of the Borel σ-algebra on Ω. Let θt ω(·) = ω (· + t) − ω (t) ,

t ∈ R,

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PETER W. BATES, HANNELORE LISEI, AND KENING LU

then (Ω, F, P, (θt )t∈R )) is a metric dynamical system with the filtration _ Ft := Fst , t ∈ R s≤t

where

Fst = 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 , τ2 such that s ≤ τ1 ≤ τ2 ≤ t. t+τ , so (θt )t∈R is filtered with respect to Fst (see [2], pp. 91 and 546 for Note that θτ−1 Fst = Fs+τ more details). X For any T > 0 the series ai wi from (4) converges to W in the space C([0, T ], ℓ2 ) for a.e. ω ∈ Ω i∈Z

(see [20] Theorem 4.3 pp. 88). We can and will assume without loss of generality that W (·, ω) is ˜ 1 with P(Ω ˜ 1 ) = 1. continuous for all Ω Assumptions on the nonlinearity f : Let f ∈ C 1 (R) satisfy (5)

f (0) = 0 and (f (x) − f (y)) (x − y) ≥ 0

for all x, y ∈ R

and the polynomial growth condition |f (x)| ≤ cf |x|(1 + x2p ) for all x ∈ R,

(6)

where p is a positive integer. p X aj s2j+1 with aj ≥ 0 for each j = 0, . . . , p, then conditions (5) and (6) are satisfied. If f (s) = j=0

This kind of nonlinearity was considered in [8] and [19]. For convenience, we now formulate system (2) as an abstract ordinary differential equation in ℓ2 . Denote by B, B ∗ and A the linear operators from ℓ2 to ℓ2 defined as follows. For u = (ui )i∈Z ∈ ℓ2 , (B ∗ u)i = ui−1 − ui ,

(Bu)i = ui+1 − ui , 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 ∈ ℓ2 . Therefore (Au, u) ≥ 0 for all u ∈ ℓ2 . Let f˜ be the Nemytski operator associated with f , that is, for u = (ui )i∈Z ∈ ℓ2 , let f˜(u) = (f (ui ))i∈Z . Then we have X X X (7) kf˜(u)k2 = |f (ui )|2 = |f (ui ) − f (0)|2 = |f ′ (ξi )|2 |ui |2 , i∈Z

i∈Z

i∈Z

with ξi = τi ui for some τi ∈ (0, 1). Since

|ξi | ≤ |ui | ≤ kuk and f is a smooth function, it follows that there exists a constant µ (depending on kuk) such that X kf˜(u)k2 ≤ µ |ui |2 = µkuk2 , i∈Z

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which means f˜(u) ∈ ℓ2 . Similar to (7), one can see that f˜ is locally Lipschitz from ℓ2 to ℓ2 , more precisely, for every bounded set Y in ℓ2 , there exists a constant CY (depending only on Y ) such that kf˜(x) − f˜(y)k ≤ CY kx − yk for all x, y ∈ Y. In the sequel, when no confusion arises we identify f˜ with f . The system (2) with initial values u0 ≡ (u0,i )i∈Z ∈ ℓ2 may be rewritten as an equation in ℓ2 (8)

u(t) = u0 +

Zt  0

 − νAu(s) − λu(s) − f (u(s)) + g ds + W (t),

t ≥ 0,

a.e. ω ∈ Ω.

˜ 1, By excluding a set measure zero, we can take W (t, ω) to be continuous on [0, ∞) for all ω ∈ Ω ˜ 1 has full measure. We will show that equation (8) has a solution for all ω ∈ Ω ˜ 1 and u0 ∈ ℓ2 . where Ω Theorem 3.1. Let T > 0. Then the following properties hold: (1) Equation (8) admits a solution u ∈ L2 (Ω, C([0, T ], ℓ2 )) which is almost surely unique. (2) For a.e. ω ∈ Ω we have the following estimate  ZT    2 2 kW (s)k2 + kW (s)k4p+2 + kgk2 ds , sup ku(t)k ≤ c ku0 k + sup kW (t)k + 2

t∈[0,T ]

t∈[0,T ]

0

where c > 0 is a constant. (3) The solution u of (8) depends continuously on the initial data u0 , i.e., for each ω ∈ Ω such that (8) has a solution, the mapping u0 ∈ ℓ2 7→ u(·, ω, u0 ) ∈ C([0, T ], ℓ2 ) is continuous. ˜1 Proof. Denote z(t) = u(t)− W (t). Equation (8) has a solution u ∈ L2 (Ω, C([0, T ], ℓ2 )) for all ω ∈ Ω if and only if the following equation (9)

z(t) = u0 +

Zt  0

 − νAz(s) − λz(s) − f (z(s) + W (s)) + g − νAW (s) − λW (s) ds

˜ 1 has a solution z ∈ L2 (Ω, C([0, T ], ℓ2 )). For each fixed ω ∈ Ω ˜ 1, for all t ∈ [0, T ] and all ω ∈ Ω equation (9) is a deterministic equation. By the standard argument, equation (9) has a local solution z ∈ C([0, Tmax ], ℓ2 ), where [0, Tmax ] is the maximal interval of existence of the solution of ˜ 1 . From (9) it follows (9). We prove now that this local solution is a global solution. Let ω ∈ Ω that Zt h 2 2 kz(t)k = ku0 k + 2 − ν(Az(s), z(s)) − λkz(s)k2 − (f (z(s) + W (s)) − f (W (s)), z(s)) 0

(10)

i + (g − νAW (s) − λW (s) − f (W (s)), z(s)) ds

Zt   kW (s)k2 + kW (s)k4p+2 + kgk2 ds, ≤ ku0 k + c0 2

0

where c0 is a positive constant depending on ν, λ, cf , and A. Then we obtain that kz(t)k is bounded by a continuous function, so there exists a global solution on any interval [0, T ].

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PETER W. BATES, HANNELORE LISEI, AND KENING LU

˜ 1 we have Using (10) it follows that for all ω ∈ Ω ZT    2 4p+2 2 sup kz(t)k ≤ ku0 k + c0 kW (s)k + kW (s)k + kgk ds 2

(11)

2

t∈[0,T ]

0

and by taking expectation on both sides of the above inequality it follows by the properties of the white noise that z ∈ L2 (Ω, C([0, T ], ℓ2 )). Therefore equation (8) has a global solution u ∈ L2 (Ω, C([0, T ], ℓ2 )) and by (11) it follows that for a.e. ω ∈ Ω the inequality  ZT    2 2 2 4p+2 2 sup ku(t)k ≤ c ku0 k + sup kW (t)k + kW (s)k + kW (s)k + kgk ds 2

t∈[0,T ]

t∈[0,T ]

0

holds, where we can take c = 2 + c0 . Let u0 , v0 ∈ ℓ2 and X(t) := u(t, u0 ), Y (t) := u(t, v0 ) the corresponding solutions of (8). Then 2

kX(t) − Y (t)k

Zt  = ku0 − v0 k + 2 − ν(A(X(s) − Y (s)), X(s) − Y (s)) − λkX(s) − Y (s)k2 2

0

 −(f (X(s)) − f (Y (s)), X(s) − Y (s)) ds.

By using the properties of A and f we obtain

kX(t) − Y (t)k2 ≤ ku0 − v0 k2 for t ∈ [0, T ]. Therefore sup kX(t) − Y (t)k2 ≤ ku0 − v0 k2 . t∈[0,T ]

If u0 = v0 , then the above inequality shows the almost surely uniqueness and continuous dependence on the initial data of the solution of (8). So, the properties (1), (2) and (3) of this theorem hold.    Theorem 3.2. Equation (8) generates a continuous random dynamical system ϕ(t) over t≥0   Ω, F, P, (θt )t∈R , where ϕ(t, ω, u0 ) = u(t, ω, u0 )

f or all t ≥ 0 and f or a.e. ω ∈ Ω.

˜ 1 (see Theorem 3.1), Proof. Since u(·, u0 ) depends continuously on the initial data u0 for all ω ∈ Ω there exists a continuous modification of this process denoted by ϕ(t) such that ϕ(·, ω, ·) : t≥0

[0, ∞] × ℓ2 → ℓ2 is continuous for all ω and ϕ satisfies (8) for a.e. ω (see also [3] concerning the perfection of cocycles). ¿From the definition of (θt )t∈R we have the property (12)

W (t + s, ω) = W (t, θs ω) + W (s, ω)

for all s, t ∈ R.

For simplicity we denote G(x) = −νAx − λx − f (x) + g for each x ∈ ℓ2 . By (8) we have for s, t ≥ 0 ϕ(t, θs ω, ϕ(s, ω, u0 )) = ϕ(s, ω, u0 ) +

Zt 0

G(ϕ(r, θs ω, ϕ(s, ω, u0 )))dr + W (t, θs ω).

ATTRACTORS FOR STOCHASTIC LATTICE DYNAMICAL SYSTEMS

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Then again by (8) we get (13)

ϕ(t, θs ω, ϕ(s, ω, u0 )) = u0 +

Zs 0

Zt+s G(ϕ(r, ω, u0 ))dr + G(ϕ(r − s, θs ω, ϕ(s, ω, u0 )))dr s

+W (t, θs ω) + W (s, ω). For each ω ∈ Ω consider Φ(r, ω, u0 ) =



ϕ(r, ω, u0 ), ϕ(r − s, θs ω, ϕ(s, ω, u0 )),

if 0 ≤ r ≤ s if s < r ≤ t + s.

Then for r = t + s we have (14)

Φ(t + s, ω, u0 ) = ϕ(t, θs ω, ϕ(s, ω, u0 )) for s, t ≥ 0.

By (12) and (13) it follows that Zt+s Φ(t + s, ω, u0 ) = u0 + G(Φ(r, ω, u0 ))dr + W (t + s, ω). 0

But ϕ is a solution to (8), then by the uniqueness of the solution of (8) it follows that Φ(t+s, ω, u0 ) = ϕ(t + s, ω, u0 ), while (14) implies ϕ(t + s, ω, u0 ) = ϕ(t, θs ω, ϕ(s, ω, u0 )) for s, t ≥ 0. 

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 (2). Our main result is Theorem 4.1. The random lattice dynamical system ϕ has a unique global random attractor. The proof of this theorem is based on an abstract result on the global attractor and a priori estimates. In what follows we consider D to be the universe of all tempered sets from ℓ2 . 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 [23]. The proof is based on the ideas given in [23]. Proposition 4.2. Let K ∈ D be an absorbing set for the continuous random dynamical system  ϕ(t) t≥0 which is closed and which satisfies for a.e. ω ∈ Ω the following asymptotic compactness condition: each sequence xn ∈ ϕ(tn , θtn , K(θtn ω)) with tn → ∞ has a convergent subsequence in H. Then the cocycle ϕ has a unique global random attractor \ [ ϕ(t, θ−t ω, K(θ−t ω)). A(ω) = τ ≥tK (ω) t≥τ

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PETER W. BATES, HANNELORE LISEI, AND KENING LU

Proof. We denote for all τ ≥ tK (ω) the set [ ϕ(t, θ−t ω, K(θ−t ω)). Fτ (ω) := t≥τ

First we prove that for a.e. ω ∈ Ω we have (15)

lim d(ϕ(t, θ−t ω, K(θ−t ω)), A(ω)) = 0.

t→∞

If this were not so there would be an increasing sequence (tn )n∈N with lim tn = ∞ and sequence n→∞

(xn )n∈N with xn ∈ ϕ(tn , θ−tn ω, K(θ−tn ω)) and an ε > 0 such that for any n ∈ N (16)

d(xn , A(ω)) > ε.

¿From the asymptotic compactness of K it follows that there exist x0 ∈ H and a subsequence (xnj )j∈N such that lim xnj = x0 . Since for all τ > tK (ω) we have j→∞

xnj ∈ ϕ(tnj , θ−tnj ω, K(θ−tnj ω)) ⊆

[

ϕ(t, θ−t ω, K(θ−t ω)) = Fτ

t≥τ

with sufficiently large j. Then by the definition of A it follows that x0 ∈ A(ω) which contradicts (16). Next, we show that (A3) in Definition 2.5 holds. By (15) for any ε > 0 we have t2 = t2 (ε, ω) such that d(ϕ(t2 , θ−t2 ω, K(θ−t2 ω)), A(ω)) < ε. Let B ∈ D. Then by property (D3) from Definition 2.2 and the absorption property for K we can write for any t1 > tB (θ−t2 ω) ϕ(t1 + t2 , θ−t1 −t2 ω, B(θ−t1 −t2 ω)) = ϕ(t2 , θ−t2 ω, ϕ(t1 , θ−t1 −t2 ω, B(θ−t1 −t2 ω))) ⊆ ϕ(t2 , θ−t2 ω, K(θ−t2 ω)). Then by using (15) we obtain for a.e. ω ∈ Ω (17)

lim d(ϕ(t, θ−t ω, B(θ−t ω)), A(ω)) = 0.

t→∞

Now, we prove the invariance property (A2) in Definition 2.5. From the definition of A we have \ A(ω) = Fτ (ω). τ ≥tK (ω)

We see that (18)

 ϕ(s, ω, A(ω)) = ϕ s, ω,

\

τ ≥tK (ω)

 Fτ (ω) ⊆

\

ϕ(s, ω, Fτ (ω)) for s ≥ 0.

τ ≥tK (ω)

We now show that \

(19)

ϕ(s, ω, Fτ (ω)) ⊆ ϕ(s, ω, A(ω))

τ ≥tK (ω)

Let y ∈

\

τ ≥tK (ω)

ϕ(s, ω, xτ ).

ϕ(s, ω, Fτ (ω)). Then for any τ ≥ tK (ω) there exists xτ ∈ Fτ (ω) such that y =

ATTRACTORS FOR STOCHASTIC LATTICE DYNAMICAL SYSTEMS

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For a sequence (tn )n∈N with lim tn = ∞ we then have xtn ∈ Ftn (ω) ⊆ FtK (ω) (ω) for all n such n→∞

that tn ≥ τK (ω). Using the asymptotic compactness of K, therefore there exist x ∈ FtK (ω) (ω) and a convergent subsequence (xtnj )j∈N such that lim xtnj = x. Because xtnj ∈ Ftnj (ω) and j→∞

{Fτ (ω) : τ ≥ tK (ω)} is a decreasing family of closed sets, we have x ∈ Ftnj (ω) for all tnj ≥ τK (ω). Hence \ \ x∈ Fτ (ω) = A(ω). Ftnj (ω) = τ ≥tK (ω)

j∈N

tnj ≥tK (ω)

The continuity of ϕ then implies y = ϕ(s, ω, x) ∈ ϕ(s, ω, A(ω)). So (19) holds and by (18) we then have \ (20) ϕ(s, ω, A(ω)) = ϕ(s, ω, Fτ (ω)). τ ≥tK (ω)

We write ϕ(s, ω, A(ω)) =

\

ϕ(s, ω, Fτ (ω)) =

\

ϕ(s, ω,

\

[

τ ≥tK (ω)



ϕ(s, ω, Fτ (ω))

τ ≥tK (ω)

[

ϕ(t, θ−t ω, K(θ−t ω))) =

t≥τ

τ ≥tK (ω)



\

\

[

ϕ(s, ω, ϕ(t, θ−t ω, K(θ−t ω)))

τ ≥tK (ω) t≥τ

ϕ(t + s, θ−t−s θs ω, K(θ−t−s θs ω))) = A(θs ω),

τ ≥tK (ω) t≥τ

here we again used the asymptotic compactness of K and the continuity of ϕ in x and the cocycle property. Thus, we obtained (21)

A(θs ω) ⊆ ϕ(s, ω, A(ω))

for s ≥ 0.

This property implies A(ω) ⊆ ϕ(r, θ−r ω, A(θ−r ω))

for r ≥ 0,

and so for r ≥ 0 we have (22)

ϕ(s, ω, A(ω)) ⊆ ϕ(s, ω, ϕ(r, θ−r ω, A(θ−r ω))) = ϕ(s + r, θ−s−r θs ω, A(θ−s−r θs ω)).

¿From the definition of A and the absorption property and closeness of K it follows that \ A(ω) = Fτ (ω) ⊆ K(ω) = K(ω) for a.e. ω ∈ Ω. τ ≥tK (ω)

The asymptotic compactness of K yields that A(ω) is a compact set for a.e. ω ∈ Ω. Then by (17) it follows that lim d(ϕ(T, θ−T ω, A(θ−T ω)), A(ω)) = 0. T →∞

Taking s + r = T in (22) and letting r → ∞ we have d(ϕ(s, ω, A(ω)), A(θs ω)) = 0. This implies (23)

ϕ(s, ω, A(ω)) ⊆ A(θs ω).

Then together with (21) we get the invariance property (A2) in Definition 2.5.

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PETER W. BATES, HANNELORE LISEI, AND KENING LU

We now prove the uniqueness. Assume that A1 and A2 are two attractors from D. Then by the invariance property (A2) and attraction property (A3) in Definition 2.5 we get t→∞

d(A1 (ω), A2 (ω)) = d(ϕ(t, θ−t ω, A1 (θ−t ω)), A2 (ω)) −→ 0, which implies A1 (ω) ⊆ A2 (ω). The symmetry of the argument implies A1 (ω) = A2 (ω).



For the remainder of this section, we will establish the existence of a absorbing set K for the random dynamical system ϕ generated by the stochastic lattice differential equation (2) and the asymptotic compactness of it. To establish the existence of an absorbing random set, we need the following lemma. Lemma 4.3. There exists a random ball B(0, R(ω)) = {x ∈ ℓ2 : kxk ≤ R(ω)} ∈ D such that for all B ∈ D and a.e. ω ∈ Ω there exists tB (ω) > 1 such that ϕ(t − 1, θ−t ω, B(θ−t ω)) ⊂ B(0, R(ω))

f or all t ≥ tB (ω).

Proof. For u0 ∈ ℓ2 consider the Ornstein-Uhlenbeck process (with values in ℓ2 ) Zt y(t) = u0 − α y(s)ds + W (t),

(24)

t ≥ 0,

0

where the value of the parameter α > 0 will be fixed later. Then we have −αt

yi (t) = u0i e

+ ai

Zt

eα(τ −t) dwi (τ ),

i ∈ Z,

0

and −α(s+t)

yi (s + t, θ−t ω) = u0i e

+ ai

Zs

eατ dwi (τ ),

s ≥ −t,

i ∈ Z.

−t

By Ito’s formula we have Zs Zs ατ αs αr α e wi (τ )dτ = e wi (s) − e wi (r) − eατ dwi (τ ),

s ≥ r.

r

r

Then, 

yi (s + t, θ−t ω) = u0i e−α(s+t) + ai eαs wi (s) − e−αt wi (−t) − α

Zs

−t



eατ wi (τ )dτ  ,

By using some Cauchy-Schwartz’ and Young’s inequalities we have the estimates (25)

ky(s + t, θ−t ω)k4p+2 ≤ c1 (p) e−(4p+2)α(s+t) ku0 k4p+2 + e(4p+2)αs kW (s)k4p+2 + e−(4p+2)αt kW (−t)k4p+2  Zs 2p+1 ! e(2p+1)αs ατ 2 , e kW (τ )k dτ + α2p+1 −t

i ∈ Z.

ATTRACTORS FOR STOCHASTIC LATTICE DYNAMICAL SYSTEMS

11

where c1 (p) is a positive constant depending on p. For p = 0 we obtain  2 (26) ky(s + t, θ−t ω)k ≤ c1 (0) e−2α(s+t) ku0 k2 + e2αs kW (s)k2 −2αt

+ e

eαs kW (−t)k + α 2

Zs

2

ατ



e kW (τ )k dτ ,

−t

and for s = −1 and t ≥ 1 2

(27)

ky(t − 1, θ−t ω)k

 ≤ c1 (0) e−2α(t−1) ku0 k2 + e−2α kW (−1)k2 −2αt

+ e

e−α kW (−t)k + α 2

 Z−1 eατ kW (τ )k2 dτ .

−t

By choosing α :=

λ , adding (25) and (26), multiplying by eλs and integrating, we get the estimates 2

Z−1 Z−1  h λs 2 λs 4p+2 ds ≤ c2 e ky(s + t, θ−t ω)k + e ky(s + t, θ−t ω)k e−λt ku0 k2

−t

−t

i

h

i

h

+ ku0 k4p+2 + e2λs kW (s)k2 + kW (s)k4p+2 + eλs−λt kW (−t)k2 + kW (−t)k4p+2  Zs  !  Zs 2p+1

λ

λ

e 2 τ kW (τ )k2 dτ +

+ eλs

e 2 τ kW (τ )k2 dτ

ds,

−t

−t

where c2 is a positive constant depending on λ and p. Then, using H¨older’s inequality, Z−1  eλs ky(s + t, θ−t ω)k2 + eλs ky(s + t, θ−t ω)k4p+2 ds

(28)

−t

i h ≤ c3 e−λt (t − 1) ku0 k2 + ku0 k4p+2 + kW (−t)k2 + kW (−t)k4p+2 +

Z−1

e

−∞

λ τ 4

h

i

!

kW (τ )k2 + kW (τ )k4p+2 dτ ,

where c3 is a positive constant depending on λ and p. ¿From Property 5.1 (from Appendix) we have kW (r)k2 = 0 for a.e. ω ∈ Ω, r→±∞ r2

(29)

lim

which implies λ

(30)

e4t lim 3 t→∞ t

−t−1 Z

−∞

  λ e 4 τ kW (τ )k2 + kW (τ )k4p+2 dτ = 0 for a.e. ω ∈ Ω.

i

12

PETER W. BATES, HANNELORE LISEI, AND KENING LU

˜ 2 ⊆ Ω with P(Ω ˜ 2 ) = 1 be such that (29) and (30) hold for all ω ∈ Ω ˜ 2 . It is easy to verify that Let Ω ˜ Ω2 is a (θt )t∈R invariant set. ˜B ⊆ Ω ˜ 2 with Now suppose B ∈ D and u0 ∈ B(θ−t ω). Then ku0 k ≤ d(B(θ−t ω)). There exists Ω ˜ ˜ P(ΩB ) = 1 and ΩB is a (θt )t∈R invariant set such that we have h i lim e−λt (t − 1) d2 (B(θ−t ω)) + d4p+2 (B(θ−t ω)) + kW (−t)k2 + kW (−t)k4p+2 = 0, t→∞

˜ B , since B is a tempered set. for all ω ∈ Ω We obtain from (27) and (28) (31)

Z−1   ˜ 2 (ω), ky(t − 1, θ−t ω)k + eλs ky(s + t, θ−t ω)k2 + ky(s + t, θ−t ω)k4p+2 ds ≤ R 2

−t

˜ B , where (for example) for all t ≥ tB (ω), ω ∈ Ω λ

e− 2 c3 + 2c1 (0) λ

˜ 2 (ω) := 1 + c1 (0)e−λ kW (−1)k2 + R

! Z−1 −∞

i h λ e 4 τ kW (τ )k2 + kW (τ )k4p+2 dτ < ∞

˜ 2. for all ω ∈ Ω Using (8) and (24) we find λ d kϕ(t) − y(t)k2 = −2ν(Aϕ(s), ϕ(s) − y(s)) − 2(λϕ(s) − y(s), ϕ(s) − y(s)) dt 2 − 2(f (ϕ(s)), ϕ(s) − y(s)) + 2(g, ϕ(s) − y(s)). It follows from the properties of A and from Cauchy-Schwartz’ and Young’s inequalities that there exist positive constants c4 , c5 , c6 , c7 depending on ν, λ, cf such that the following estimates hold −2ν(Aϕ(s), ϕ(s) − y(s)) ≤ −2ν(Ay(s), ϕ(s) − y(s)) ≤ c4 ky(s)k2 +

λ kϕ(s) − y(s)k2 , 4

λ kϕ(s) − y(s)k2 , 4 λ 7λ −2(λϕ(s) − y(s), ϕ(s) − y(s)) ≤ c7 ky(s)k2 − kϕ(s) − y(s)k2 , 2 4 λ −2(f (ϕ(s)), ϕ(s) − y(s)) ≤ −2(f (y(s)), ϕ(s) − y(s)) ≤ c5 (1 + ky(s)k4p+2 ) + kϕ(s) − y(s)k2 , 4 for the last inequality we used the polynomial growth property for f and the inequality X |yi (s)|4p+2 ≤ ky(s)k4p+2 . 2(g, ϕ(s) − y(s)) ≤ c6 kgk2 +

i∈Z

Therefore, d kϕ(t) − y(t)k2 ≤ −λkϕ(t) − y(t)k2 + (c4 + c7 )ky(t)k2 + c5 ky(t)k4p+2 + c6 kgk2 + c5 , dt which implies 2

−λ(t−1)

kϕ(t − 1) − y(t − 1)k ≤ e

Zt−1   eλs (c4 + c7 )ky(s)k2 + c5 ky(s)k4p+2 + c6 kgk2 + c5 ds. 0

In the case ω 7→ θ−t ω we obtain

ATTRACTORS FOR STOCHASTIC LATTICE DYNAMICAL SYSTEMS

13

kϕ(t − 1, θ−t ω)k2 ≤ 2ky(t − 1, θ−t ω)k2 Zt−1   + 2e eλ(s−t) (c4 + c7 )ky(s, θ−t ω)k2 + c5 ky(s, θ−t ω)k4p+2 + c6 kgk2 + c5 ds λ

0

 2 c6 kgk2 + c5 + c8 ≤ 2ky(t − 1, θ−t ω)k + λ 2

Z−1  eλs ky(s + t, θ−t ω)k2 + ky(s + t, θ−t ω)k4p+2 ds,

−t

˜ B and all t ≥ tB (ω) where (for example) c8 = 2(c4 + c5 + c7 )eλ . Then by (31) we have for all ω ∈ Ω that ϕ(t − 1, θ−t ω, B(θ−t ω)) ⊂ B(0, R(ω)),   ˜ 2 (ω) for all ω ∈ Ω ˜ 2 . Finally we where for example we take R2 (ω) = λ2 c6 kgk2 + c5 + (2 + c8 )R ˜ 2 we have prove that B(0, R(ω)) is a tempered set, i.e. for all ω ∈ Ω lim e−βt R(θ−t ω) = 0 for all β > 0.

t→∞

˜ that is Obviously this is true, if the same property holds for R, ˜ −t ω) = 0 for all β > 0 and ω ∈ Ω ˜ 2. lim e−βt R(θ

(32)

t→∞

But there exists a positive constant c9 depending on λ and p such that  2 ˜ R (θ−t ω) ≤ 1 + c9 kW (−t − 1)k2 + kW (−t)k2 + kW (−t)k4p+2 + e

λ t 4

−t−1 Z

e

λ τ 4

−∞

   2 4p+2 ˜ 2. dτ < ∞ for all ω ∈ Ω kW (τ )k + kW (τ )k

Property (32) follows by using (30).



The next result is about the existence of a tempered absorbing set. Theorem 4.4. The set K(ω) = ϕ(1, θ−1 ω, B(0, R(ω))) is an absorbing set, i.e., for all B ∈ D and a.e. ω ∈ Ω there exists tB (ω) > 0 such that ϕ(t, θ−t ω, B(θ−t ω)) ⊂ K(ω) f or all t ≥ tB (ω). Moreover K ∈ D. Proof. Let B ∈ D. We use the cocycle property for the continuous random dynamical system ϕ (see Theorem 3.2) and use Lemma 4.3, to write for a.e. ω ∈ Ω ϕ(t, θ−t ω, B(θ−t ω)) = ϕ(1, θ−1 ω, ϕ(t − 1, θ−t ω, B(θ−t ω))) ⊂ ϕ(1, θ−1 ω, B(0, R(ω))), for all t ≥ tB (ω). As in the proof of Lemma 4.3 we have the estimate 2

2

kϕ(1)k ≤ 2ky(1)k + 2

Z1 0

  eλ(s−1) (c4 + c7 )ky(s)k2 + c5 ky(s)k4p+2 + c6 kgk2 + c5 ds.

14

PETER W. BATES, HANNELORE LISEI, AND KENING LU

Now we take ω 7→ θ−1 ω and obtain  2 c6 kgk2 + c5 + 2ky(1, θ−1 ω)k2 kϕ(1, θ−1 ω)k2 ≤ λ Z0   + 2(c4 + c7 + c5 ) eλs ky(s + 1, θ−1 ω)k2 + ky(s + 1, θ−1 ω)k4p+2 ds. −1

Then ˆ ϕ(1, θ−1 ω, B(0, R(ω))) ⊂ B(0, R(ω)), where Z0     λ 2 2 2 ˆ R (ω) = c10 R (ω) + kW (−1)k + 1 + e 4 τ kW (τ )k2 + kW (τ )k4p+2 dτ < ∞

a.e. ω ∈ Ω

−1

with c10 a positive constant depending on ν, λ, cf , and g. It follows that K is a tempered set, ˆ because as in the proof of Lemma 4.3 (see (32)) we can verify that B(0, R(ω)) is also tempered set.  Since K is a tempered absorbing set it follows that for a.e. ω ∈ Ω there exists tK (ω) > 0 such that (33)

ϕ(t, θ−t ω, K(θ−t ω)) ⊂ K(ω) for all t ≥ tK (ω).

˜ 3 the set of measure 1 which is (θt )t∈R invariant such that ϕ satisfies the equation We denote by Ω ˜ (8) for all ω ∈ Ω3 and the inequality holds (34)

ZT     2 ku0 k + sup kϕ(t, u0 )k ≤ c sup kW (t)k + kW (s)k2 + kW (s)k4p+2 ds < ∞ 2

2

t∈[0,T ]

t∈[0,T ]

0

˜ 3 (the set Ω ˜ 3 does not depend on the initial condition). Here we used the results from for all ω ∈ Ω Theorem 3.1. In order to establish the asymptotic compactness, we need the following lemma on the weak continuities of the random dynamical system ϕ. ˜ 3 we have ϕ(T, un ) ⇀ ϕ(T, u0 ) in ℓ2 and Lemma 4.5. If un ⇀ u0 in ℓ2 , then for each fixed ω ∈ Ω ϕ(·, un ) ⇀ ϕ(·, u0 ) in L2 ([0, T ], ℓ2 ), where T > 0 is arbitrary. ˜ 3 and all t ∈ [0, T ] we have Proof. For all ω ∈ Ω (35)

ϕ(t, un ) = un +

Zt  0

 − νAϕ(s, un ) − λϕ(s, un ) − f (ϕ(s, un )) + g ds + W (t).

By Theorem 3.1 and (34) we have (36)

 sup kϕ(t, un )k2 ≤ c kun k2 + sup kW (t)k2

t∈[0,T ]

t∈[0,T ]

+

ZT 0

   2 4p+2 2 kW (s)k + kW (s)k + kgk + 1 ds < ∞.

ATTRACTORS FOR STOCHASTIC LATTICE DYNAMICAL SYSTEMS

15

This implies that (ϕ(·, un ))n∈N is a bounded sequence in L2 ([0, T ], ℓ2 ), since (un )n∈N is also bounded (by the weak convergence from the hypothesis). ˜ 3 be fixed. It follows by (36) that (ϕ(·, un ))n∈N is a bounded sequence in L2 ([0, T ], ℓ2 ), Let ω ∈ Ω since (un )n∈N is also bounded (by the weak convergence from the hypothesis). Then there exist ϕ∗ , f ∗ ∈ L2 ([0, T ], ℓ2 ) and a subsequence of indices (nk )k∈N (which depends on the fixed ω) such that ϕ(·, unk ) ⇀ ϕ∗

(37)

in L2 ([0, T ], ℓ2 )

and f (ϕ(·, unk )) ⇀ f ∗

(38)

in L2 ([0, T ], ℓ2 ).

We let k → ∞ in equation (35) and by the weak convergence in L2 ([0, T ], ℓ2 ) we get (39)



ϕ (t) = u0 +

Zt 

 − νAϕ∗ (s) − λϕ∗ (s) − f ∗ (s) + g ds + W (t),

0

for a.e. t ∈ [0, T ] (we also used the linearity and continuity of A). We define ϕˆ : [0, T ] → ℓ2 by (40)

ϕ(t) ˆ = u0 +

Zt  0

Then by (39) we get (41)

ϕ∗ (t)

 − νAϕ∗ (s) − λϕ∗ (s) − f ∗ (s) + g ds + W (t).

= ϕ(t) ˆ for a.e. t ∈ [0, T ] and (40) implies

ϕ(t) ˆ = u0 +

Zt  0

 − νAϕ(s) ˆ − λϕ(s) ˆ − f ∗ (s) + g ds + W (t),

for all t ∈ [0, T ]. Let t ∈ [0, T ] be fixed in (35) and take the inner product with an arbitrary element of ℓ2 . Using the weak convergence in L2 ([0, t], ℓ2 ) on the right side of (35) and noting (41) we deduce that (42)

ϕ(t, unk ) ⇀ ϕ(t) ˆ

in ℓ2 .

This implies also the convergence (in R) of each coordinate i ∈ Z ϕi (t, unk ) → ϕˆi (t) . By the assumption f ∈

C 1 (R)

we then have, for each fixed t and i,

f (ϕi (t, unk )) → f (ϕˆi (t)) . The definition of f (more exactly f˜) as an operator from ℓ2 to ℓ2 and a dominated convergence argument for sums implies (43)

f (ϕ(t, unk )) ⇀ f (ϕ(t)) ˆ

in ℓ2 .

But t ∈ [0, T ] was fixed arbitrary, so the convergence in (42) and (43) hold for all t ∈ [0, T ]. Using Lebesgue’s Dominated Convergence Theorem it also follows that ZT

(f (ϕ(t, unk )), ξ(t))dt →

0

ZT

(f (ϕ(t)), ˆ ξ(t))dt

0

for any ξ ∈ L2 ([0, t], ℓ2 ). Therefore, (44)

f (ϕ(·, unk )) ⇀ f (ϕ) ˆ in L2 ([0, T ], ℓ2 ).

16

PETER W. BATES, HANNELORE LISEI, AND KENING LU

Then by (38) we get f (ϕ(t)) ˆ = f ∗ (t) for a.e. t ∈ [0, T ]. Using (41) we obtain (45)

ϕ(t) ˆ = u0 +

Zt  0

 − νAϕ(s) ˆ − λϕ(s) ˆ − f (ϕ(s)) ˆ + g ds + W (t),

for all t ∈ [0, T ]. So, ϕ(t) ˆ = ϕ(t, u0 ) is the (unique) solution of equation (8) with initial condition u0 . Then (37) and (42) imply ϕ(·, unk ) ⇀ ϕ(·, u0 ) in L2 ([0, T ], ℓ2 ). But, in addition, every weakly convergent subsequence of (ϕ(·, un ))n∈N has the same limit ϕ(·, u0 ) in L2 ([0, T ], ℓ2 ) (which is the unique solution of (8)), then the whole sequence (ϕ(·, un ))n∈N converges weakly to ϕ(·, u0 ). So, the weak convergence holds for the whole sequence ϕ(·, un ) ⇀ ϕ(·, u0 ) in L2 ([0, T ], ℓ2 ).

(46)

We take the weak limit (n → ∞) in L2 ([0, t], ℓ2 ) in the right side of (35) (for t = T ) and use (46) to get ϕ(T, un ) ⇀ ϕ(T, u0 ) in ℓ2 . Here we used again that ϕ(·, u0 ) is the unique solution of equation (8) with initial condition u0 .  We are now ready to show the asymptotic compactness of K. Theorem 4.6. For a.e. ω ∈ Ω the set K(ω) is asymptotically compact: each sequence pn ∈ ϕ(tn , θ−tn ω, K(θ−tn ω)) with tn → ∞ has a convergent subsequence in H. ˜ 3 . Consider (tn )n∈N with lim tn = ∞ and pn ∈ ϕ(tn , θ−tn ω, K(θ−tn ω)). We Proof. Let ω ∈ Ω n→∞

consider pn = ϕ(tn , θ−tn ω, xn ), where xn ∈ K(θ−tn ω). We want to show that (ϕ(tn , θ−tn ω, xn ))n∈N has a convergent subsequence. Let T > 0 arbitrary. Since pn ∈ ϕ(tn , θ−tn ω, K(θ−tn ω)) and the inclusion (33) holds, it follows that ϕ(tn , θ−tn ω, xn ) ∈ K(ω) for all n with tn ≥ tK (ω) and ϕ(tn − T, θ−(tn −T ) θ−T ω, xn ) ∈ K(θ−T ω) for all n with tn − T ≥ tK (θ−T ω). For simplicity we denote zn = ϕ(tn − T, θ−tn ω, xn ). Using the boundedness of K(ω) it follows that there exist v, vT ∈ ℓ2 and subsequences (the indices we denote also by n) such that ϕ(tn , θ−tn ω, xn ) ⇀ v in ℓ2

(47) and

zn = ϕ(tn − T, θ−tn ω, xn ) ⇀ vT in ℓ2 .

(48)

To show that ϕ(tn , θ−tn ω, xn ) converges strongly to v, we show that kϕ(tn , θ−tn ω, xn )k converges to kvk. First, from the weak lower semi-continuity of the norm it follows that kvk2 ≤ lim inf kϕ(tn , θ−tn ω, xn )k2 .

(49)

n→∞

Next, we prove that kvk2 ≥ lim sup kϕ(tn , θ−tn ω, xn )k2 . n→∞

Since ϕ is a continuous random dynamical system (see Theorem 3.2), we have (50)

ϕ(tn , θ−tn ω, xn ) = ϕ(T, θ−T ω, ·) ◦ ϕ(tn − T, θ−tn ω, xn ),

ATTRACTORS FOR STOCHASTIC LATTICE DYNAMICAL SYSTEMS

then (47), (48), and Lemma 4.5 imply (51)

v = ϕ(T, θ−T ω, vT ).

Using (8) and Ito’s formula we have 2λT

e

ZT kϕ(T ) − W (T )k = kϕ(0)k + 2 e2λs (−νAϕ(s) − λW (s) − f (ϕ(s)) + g, ϕ(s) − W (s))ds, 2

2

0

which implies e2λT kϕ(T )k2 = kϕ(0)k2 + 2e2λT (ϕ(T ), W (T )) − e2λT kW (T )k2 ZT h + 2 e2λs − ν(Aϕ(s), ϕ(s)) + ν(Aϕ(s), W (s)) 0

i − (f (ϕ(s)), ϕ(s) − W (s)) + (g − λW (s), ϕ(s) − W (s)) ds.

We take ω 7→ θ−T ω and the initial condition ϕ(0) = zn and use (50) ϕ(T, θ−T ω, zn ) = ϕ(tn , θ−tn ω, xn ) to get e2λT kϕ(tn , θ−tn ω, xn )k2

= kzn k2 + 2e2λT (ϕ(tn , θ−tn ω, xn ), W (T, θ−T ω)) − e2λT kW (T, θ−T ω)k2 ZT h + 2 e2λs − ν(Aϕ(s, θ−T ω, zn ), ϕ(s, θ−T ω, zn )) + ν(Aϕ(s, θ−T ω, zn ), W (s, θ−T ω)) 0

i − (f (ϕ(s, θ−T ω, zn )), ϕ(s, θ−T ω, zn ) − W (s)) + (g − λW (s), ϕ(s, θ−T ω, zn ) − W (s, θ−T ω)) ds.

Using the properties of A we can write

−(Ayn , yn ) = −(A(yn − y), yn − y) − (Ay, yn − y) − (A(yn − y), y) − (Ay, y) ≤ −(Ay, yn − y) − (A(yn − y), y) − (Ay, y) and by the properties of f we have − (f (yn ), yn − W ) = −(f (yn ) − f (y), y − W ) − (f (yn ) − f (y), yn − y) − (f (y), yn − y) − (f (y), y − W ) ≤ −(f (yn ) − f (y), y − W ) − (f (y), yn − y) − (f (y), y − W ) By the weak convergence (see (48) and weak continuity from Lemma 4.5) for yn (·) = ϕ(·, θ−T ω, zn ) ⇀ y(·) = ϕ(·, θ−T ω, vT ) in

L2 ([0, T ], ℓ2 )

and yn (s) = ϕ(s, θ−T ω, zn ) ⇀ y(s) = ϕ(s, θ−T ω, vT ) in

ℓ2

for all s ∈ [0, T ].

17

18

PETER W. BATES, HANNELORE LISEI, AND KENING LU

Taking W = W (s), using the above inequalities and (44) we get

n→∞

n

≤ −2ν

ZT

lim sup

− 2ν

ZT

e2λs (Aϕ(s, θ−T ω, zn ), ϕ(s, θ−T ω, zn ))ds

0

o

e2λs (Aϕ(s, θ−T ω, vT ), ϕ(s, θ−T ω, vT ))ds

0

and lim sup n→∞

n

ZT o − 2 e2λs (f (ϕ(s, θ−T ω, zn )), ϕ(s, θ−T ω, zn ) − W (s, θ−T ω))ds 0

ZT

≤ −2 e2λs (f (ϕ(s, θ−T ω, vT )), ϕ(s, θ−T ω, vT ) − W (s, θ−T ω))ds. 0

Because for sufficiently large n we have ˆ −T ω)) zn = ϕ(tn − T, θ−tn ω, xn ) ∈ K(θ−T ω) ⊆ B(0, R(θ (see the proof of Theorem 4.4) we then have ˆ 2 (θ−T ω). lim sup kϕ(tn − T, θ−tn ω, xn )k2 ≤ R n→∞

Hence, e2λT lim sup kϕ(tn , θ−tn ω, xn )k2 n→∞

ˆ 2 (θ−T ω) + 2e2λT (v, W (T, θ−T ω)) − e2λT kW (T, θ−T ω)k2 ≤R ZT h + 2 e2λs − ν(Aϕ(s, θ−T ω, vT ), ϕ(s, θ−T ω, vT )) + ν(Aϕ(s, θ−T ω, vT ), W (s, θ−T ω)) 0

− (f (ϕ(s, θ−T ω, vT )), ϕ(s, θ−T ω, vT ) − W (s, θ−T ω)) i + (g − λW (s, θ−T ω), ϕ(s, θ−T ω, vT ) − W (s, θ−T ω)) ds.

Therefore

ˆ 2 (θ−T ω) − e2λT kv − W (T, θ−T ω)k2 + e2λT kvk2 e2λT lim sup kϕ(tn , θ−tn ω, xn )k2 ≤ R n→∞

+ e2λT kϕ(T, θ−T ω, vT ) − W (T, θ−T ω)k2 . By (51) we have v = ϕ(T, θ−T ω, vT ), so ˆ 2 (θ−T ω) + kvk2 . lim sup kϕ(tn , θ−tn ω, xn )k2 ≤ e−2λT R n→∞

ˆ −T ω)) is a tempered set R(θ ˆ −T ω) behaves subexponentially, then by taking T → ∞ Since B(0, R(θ we have lim sup kϕ(tn , θ−tn ω, xn )k2 ≤ kvk2 . n→∞

ATTRACTORS FOR STOCHASTIC LATTICE DYNAMICAL SYSTEMS

19

Hence, (47) and (49) give strong convergence ϕ(tn , θ−tn ω, xn ) −→ v

in ℓ2 .

Therefore, K is asymptotically compact.



Putting Proposition 4.2, Theorem 4.4 and Theorem 4.6 together gives Theorem 4.1. 5. Appendix Proposition 5.1. The following strong law of large numbers holds: lim

t→∞

kW (t)k =0 t

a.e. ω ∈ Ω.

Proof. Since {W (t) : t ≥ 0} is an ℓ2 -valued martingale, it follows that kW (t)k is a submartingale (see [20], Proposition 3.7, pg. 78) and by Doob’s maximal inequality (see [26] Theorem 3.8, pg. 13ff) we get E sup kW (t)k2 ≤ 4EkW (τ )k2 . η≤t≤τ

We write E

kW (t)k2 sup t2 η≤t≤τ

!



4 4kak2 τ 1 2 2 E sup kW (t)k ≤ EkW (τ )k = . η 2 η≤t≤τ η2 η2

Then by Markov’s inequality we have ! 1 4kak2 τ kW (t)k2 · > ε ≤ . P sup t2 ε η2 η≤t≤τ We choose τ = 2n+1 , η = 2n , with n ∈ N, to get P

sup 2n ≤t≤2n+1

! kak2 kW (t)k2 > ε ≤ . t2 ε2n−3

Then, by the Borel–Cantelli Lemma it follows that for a.e. ω ∈ Ω kW (t)k2 = 0. t→∞ t2 lim



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ATTRACTORS FOR STOCHASTIC LATTICE DYNAMICAL SYSTEMS

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(Peter W. Bates) Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA E-mail address, P. Bates: [email protected] (Hannelore Lisei) Faculty of Mathematics and Computer Science, Babes¸-Bolyai University, Str. Kogalniceanu Nr. 1, RO - 400084 Cluj-Napoca, Romania E-mail address, H. Lisei: [email protected] (K. Lu) Department of Mathematics, Brigham Young University, Provo, Utah 84602, USA, and, Department of Mathematics, Michigan State University, East Lansing, Michigan 48824, USA E-mail address, K. Lu: [email protected] or [email protected]