ASYMPTOTIC AUTONOMY OF RANDOM ATTRACTORS FOR BBM EQUATIONS WITH LAPLACE-MULTIPLIER NOISE∗

We study asymptotic autonomy of random attractors for possibly non-autonomous Benjamin-Bona-Mahony equations perturbed by Laplacemultiplier noise. We assume that the time-indexed force converges to the time-independent force as the time-parameter tends to negative infinity, and then show that the time-indexed force is backward tempered and backward tail-small. These properties allow us to show that the asymptotic compactness of the non-autonomous system is uniform in the past, and then obtain a backward compact random attractor when the attracted universe consists of all backward tempered sets. More importantly, we prove backward convergence from time-fibers of the non-autonomous attractor to the autonomous attractor. Measurability of solution mapping, absorbing set and attractor is rigorously proved by using Egoroff, Lusin and Riesz theorems.


Introduction
We develop a new subject on asymptotic autonomy of random attractors for the following non-autonomous stochastic Benjamin-Bona-Mahony (BBM) equation: where ν > 0, S = I − ∆ and Q is an unbounded 3D-channel: When the equation is deterministic (S = 0) and autonomous (g(t) ≡ g ∞ ∈ L 2 (Q)), it was first proposed in [3] as a nonlinear dispersive model to describe the physical phenomenon of long waves in shallow water. Both well-posedness and global attractor had been extensively investigated (cf. [1,10,12,14,29,33]). Wang [31] obtained a random attractor for the BBM equation with additive noise (Su = h).
We take the Laplace-multiplier noise (S = I − ∆) instead of the usual multiplicative noise (S = I, see [5,9,13,18,21,22,44]). From the viewpoint of physics, this operator-type noise vibrates in resonance with the dispersive wave (d(∆u)). From the viewpoint of mathematics, it is possible to translate the stochastic equation with Laplace-multiplier noise such that the differential of the Wiener process W disappears. Therefore, we can obtain a non-autonomous random dynamical system (NRDS) Φ in the sense of Wang [32], where, measurability of the system is rigorously proved by showing Lusin continuity in the sample, see Proposition 2.1.
The main purpose of this paper is to consider not only existence of a nonautonomous random attractor A = {A(τ, ω)}, but also upper semi-continuity from A(τ, ω) to A ∞ (ω) as τ → −∞, that is, where, Ω is a probability space, and A ∞ = {A ∞ (ω)} is the random attractor (obtained by [25]) for the RDS Φ ∞ generated from the autonomous BBM equation with the time-independent force g ∞ (x) instead of g(t, x) in (1.1).
Such an asymptotically autonomous problem in the non-random case (omitting the sample in (1.2)) had been investigated by Kloeden et al. [15][16][17] or [6,11]. They established some abstract results by using the uniform convergence of the system and the uniform compactness of the pullback attractor. Two uniformness conditions had been reduced by Li et al. [23], in which, it was shown that the asymptotic autonomy only relates to backward or forward compactness of a pullback attractor.
The above abstract results can be partly generalized to the random case, where we must consider variety of the sample. In fact, in order to establish the asymptotic autonomy as given in (1.2), on one hand, we need to show the convergence from the NRDS Φ to the RDS Φ ∞ , on the other hand, we need to show that the NRDS Φ is backward asymptotically compact, which means that the asymptotic compactness is uniform in the past, see Theorem 5.1.
Interestingly, the above two properties can be available by using only one assumption on two forces. Hypothesis G. (Convergence condition). g ∈ L 2 loc (R, L 2 (Q)) and g ∞ ∈ L 2 (Q) such that In fact, under the hypothesis G, we can prove that Φ backward converges to Φ ∞ , see Lemma 2.2. Moreover, we can show that the hypothesis G can imply that the time-dependent force g is backward tempered and backward tail-small (see Lemma 2.1). These properties are enough to ensure that the random attractor A(τ, ω) is backward compact, which means that ∪ s≤τ A(s, ω) is pre-compact.
A difficulty arises from proving measurability of the absorbing set, which is a union of some random sets over an uncountable index set. Fortunately, both Egoroff and Lusin theorems can solve the problem, see Proposition 3.1.
The tail-estimates can be realized by using square of the usual cut-off function and by treating carefully the biquadrate of solutions. Those tail-estimates are further proved to be uniform in the past.
Final application results are summarized in Theorem 5.2, where we show backward compactness and asymptotic autonomy of the random attractor for Eq.(1.1). Furthermore, Riesz theorem and measure-preserving property imply the convergence of A(s n , θ sn ω) as s n → −∞, where the sample is varying.

Two backward properties of the time-indexed force
We show that the hypothesis G can imply the following conditions.
(I) g is tempered: 0 −∞ e ar g(r) 2 dr < +∞ for all a > 0. This is a common condition to ensure the existence of a pullback attractor, see [20,28] and the references therein.
(II) g is backward tempered: for all a > 0 and τ ∈ R, This is a basic condition to guarantee existence of a backward compact attractor, see [7,38] This is a condition to ensure the existence of a backward compact attractor when a PDE is defined on an unbounded domain, see [24,26,34,40] for some deterministic PDEs.
Lemma 2.1. Let the time-indexed force g satisfy the hypothesis G. Then, (i) g is backward tempered, which obviously implies that g is tempered.
(ii) g is backward tail-small.
Therefore, by e a(r−s) ≤ 1 for all r ≤ s, we have (ii) It is similar to the above proof that for each k ∈ N, By the absolute continuity of the integral, we have Q(|x3|≥k) |g ∞ (x)| 2 dx → 0 as k → ∞. Thereby, (2.2) implies that g is backward tail-small as required.
We give the nonlinearity assumption as follows.

Lusin continuity and measurability of systems
We identify the Wiener process W (·, ω) with ω(·) on the metric dynamical system (Ω, F, P, θ t ), where, equipped with the Frechét metric: given ω 1 , ω 2 ∈ Ω, F is the Borel sigma-algebra on (Ω, ρ), P is the two-sided Wiener measure on (Ω, F) and θ t is a group defined by In order to deal with Laplace-multiplier noise, we make an exponential change of variables: In this case, we have We substitute the above equality into Eq.(1.1) to find that with the initial conditions: v(τ, τ, ω, v τ ) = v τ = e −z(θτ ω) u τ . We establish some energy inequalities, which will be useful frequently.
Proof. Taking the inner product of Eq.
By the boundary condition in (1.1) and f i (0) = 0 for i = 1, 2, 3, where − → n is the outer unit normal vector. Then, by the Poincaré inequality ∇v 2 ≥ λ 0 v 2 , the energy inequality (2.9) follows immediately. Based on the above estimate, the similar argument as given in [29] shows the well-posedness.
Now, we can show the Lusin continuity of the solution mapping in samples.
where the nonlinear term I k are defined and split into three parts: By (2.4), H 1 (Q) → L 4 (Q) and Lemmas 2.5, 2.4, (2.5) in Lemma 2.2 and the Young inequality, Then, the rest term on the right-hand side of (2.13) is bounded by . We substitute all estimates into (2.13) to see that Noting that V k (τ ) = 0, then applying the Gronwall inequality to (2.14) over (τ, t), By the energy inequality (2.9) with v k instead of v, it follows from Lemma 2.5 that Then, the Gronwall lemma gives By Lemma 2.4, we can define a mapping Φ: where X = H 1 0 (Q). The Lusin continuity in Proposition 2.1 gives the F-measurability of Φ. Therefore, we obtain Theorem 2.1. The mapping Φ as given by (2.16) is a non-autonomous random dynamical system (NRDS) on X in the following sense.

Backward convergence of NRDS
We consider the autonomous BBM equation with Laplace-multiplier noise: Taking the inner product of (2.21) with V τ in L 2 (Q), we have, The Young inequality implies that (2.25) Applying the Gronwall inequality to (2.25) over (0, T ), we have By the hypothesis G, we have where C 1 , C 2 are independent of τ . The Gronwall inequality implies that for all

Increasing random absorbing sets
In this section, we show existence of a D-random absorbing set, where D is the backward tempered universe as given in (1.4). The main difficult is to verify measurability of the absorbing set because the absorbing radius is a supremum of some random functions over an uncountable index set.

1)
where R(τ, ω) is given by Proof. We rewritten the energy inequality (2. Applying the Gronwall inequality to (3.4) with respect to r ∈ (s − t,ŝ), we obtain and D is backward tempered, it follows from (2.6) and (1.4) that there exists a T = T (τ, ω, D) such that for all t ≥ T , Therefore, by taking the maximum on s ∈ (−∞, τ ] in (3.5), we show (3.1) as required.
Recall that a bi-parametric set K is said to be a D-pullback absorbing set (briefly, an absorbing set) if for each (D, τ, ω) ∈ D × R × Ω there is a T := T (D, τ, ω) such that Proposition 3.1. There is an increasing random absorbing set K given by where R(τ, ω) is defined by (3.2). Moreover, K is backward tempered, i.e. K ∈ D.
Proof. By Lemma 2.1, g is backward tempered. So, it follows from the convergence (2.6) that It is easy to show that K is tempered. Since τ → R(τ, ω) is obviously an increasing function, K(τ, ω) is increasing. Then, K is an increasing tempered set and thus backward tempered, that is, K ∈ D. The absorption follows from Lemma 3.1 immediately. Next, we prove the measurability of the absorbing set K. It suffices to prove that ω → R(τ, ω) is a measurable function for each τ ∈ R, where we need to carefully treat the supremum when s ∈ (−∞, τ ], this interval is an uncountable set. For this end, we actually prove that ω → R(τ, ω) is Lusin continuous.
By the Egoroff theorem, the convergence given in (2.6) is basically uniform on Ω, that is, for each N ∈ N, there is a measurable setΩ N ⊂ Ω such that P (Ω \Ω N ) < 1/N and Given ε > 0, we take r 1 ≤ r 0 < 0 such that e δ 4 r1 < ε. Then, the above inequality implies that By the same method as given in [8,Corollary 22] (see (2.11) in Lemma 2.5), we have Hence, the above two estimates yield which tends to zero as k → ∞, in view of (3.8) and that g is backward tempered. Therefore, ω → R(τ, ω) is continuous in E N and thus Lusin continuous in Ω, which further implies the measurablity. For the later purpose, we need an auxiliary estimate, which is similar to the autonomous case given by [25,Lemma 5.1], and so we omit the proof.

Backward tail-estimates and backward flattening
Now, we intend to give the backward tail-estimate when the third component of space-variable is large enough. We will use the square of the usual cut-off function: where I 1 , I 2 , I 3 are defined later. By (2.4), ∇ρ 2 k ∞ ≤ c k and H 1 (Q) → L 3 (Q), we have Similarly, by ∇ρ 2 k ∞ ≤ c k and Lemma 3.2, we have .
Applying the Poincaré inequality on ρ k v, we have Substituting all above estimates into (4.2) and recalling δ : Applying the Gronwall inequality to (4.3) over (s − t, s) and replacing ω by θ −s ω, we have, where J 1 , J 2 , J 3 , J 4 are given and estimated as follows. Since v 0 ∈ D(s − t, θ −t ω) for all s ≤ τ , by (2.6) and (1.4), we have as t → +∞. By Lemma 2.1, the hypothesis G implies that g is backward tail-small. So, by (2.6), we have as k → +∞. Since g is backward tempered as given in Lemma 2.1, it follows that the following term is finite, and so cJ 3 /k → 0 as k → ∞. It suffices to prove finiteness of the following term: where we need to deal with the biquadrate. By using (3.3) in Lemma 3.1, we can split J 4 ≤Ĵ 4 +J 4 witĥ The first integral in the last line is obviously finite. Also, by v 0 ∈ D(s − t, θ −t ω), as t → +∞. So,Ĵ 4 < +∞. Another termJ 4 is given bỹ s z(θσω)dσ dŝ < +∞. Then, since g is backward tempered, it easily follows that . The proof is completed. Next, we give backward flattening estimates in the bounded domain. For each k ≥ 1, we let where, P i : It is easy to calculate that ξ k ∆v = ∆v − v∆ξ k − 2∇ξ k · ∇v and ξ k ∆v s = ∆v s − v s ∆ξ k − 2∇ξ k · ∇v s . Hence, we multiply (2.8) by ξ k , the equation can be rewritten as

Proof.
Applying I − P i to Eq.(4.8) and taking the inner product of the resulting equation withv i,2 in L 2 (Q 2k ), it yields from the orthogonal decomposition (4.7) that where I 1 , I 2 , I 3 are defined and estimated as follows. By v 2 3 ≤ c ∇v v and ∇v i, 2 2 ≥ λ i+1 v i,2 2 , we have Similarly, the Young inequality implies that By Lemma 3.2 and ∆ξ k ∞ ≤ c, We assume without loss of generality that λ i ≥ 1, then λ −1 . Substituting all above estimates into (4.9) yields where δ := min( ν 2 , νλ0 4 ). Hence, the Gronwall lemma over (4.10) implies where J 3 and J 4 are given by (4.5) and (4.6) respectively. By the same method as given in the proof of Lemma 4.1, both J 3 and J 4 are finite.
On the other hand, it is obvious that Hence, by (2.6) and (1.4), Therefore, (4.11) implies the needed convergence.

Abstract results
Let Φ be a general NRDS on a Banach space X over (Ω, F, P, θ), as defined in Theorem 2.1. Let D = {D(τ, ω)} be a universe of some bi-parametric sets. We assume that D is backward-closed, which means D ∈ D provided D ∈ D and D(τ, ω) = ∪ s≤τ D(s, ω). Also, D is inclusion-closed (see [32]).

Definition 5.1. A bi-parametric set
The backward compact random attractor has been studied in [27,35,36]. In this article, we use it as one of the criteria for asymptotic autonomy of pullback random attractors.

Definition 5.2.
A non-autonomous cocycle Φ on X is said to be D-backward asymptotically compact if for each (τ, ω, D) ∈ R × Ω × D, the sequence {Φ(t n , s n − t n , θ −tn ω)x n } ∞ n=1 has a convergent subsequence in X, whenever s n ≤ τ , t n → +∞ and x n ∈ D(s n − t n , θ −tn ω).
We then introduce a concept of a backward limiting set : given a bi-parametric set D, which generalizes the usual omega-limit set W(τ, ω, D) (see [32]).
The following results are crucial for finding a backward compact attractor.
Proposition 5.1. Let D be a bi-parametric set, τ ∈ R and ω ∈ Ω. Then, The backward limit-set contains the backward union of the usual limit-set, that is, (iv) If Φ is D-backward asymptotically compact in X, then, both limit-sets W(τ, ω, D) and W b (τ, ω, D) are backward compact for each D ∈ D.

Proof.
The assertion (i) is similar to the deterministic case (see e.g. [26]), while the assertion (ii) follows from the definition (5.2) immediately.
The assertion (iii) follows from the following inclusion: We prove (iv). Assume that Φ is backward asymptotically compact, then it is asymptotically compact. It is well-known that W(τ, ω, D) is nonempty. Hence, by (iii), W b (τ, ω, D) is nonempty.
By the backward asymptotical compactness of Φ, passing to a subsequence, we have By (i), y 0 ∈ W b (τ, ω, D). Also, we have y n k → y 0 in X, and so W b (τ, ω, D) is compact. Hence, it follows from (5.4) that both limit-sets W and W b are backward compact. Now, we give a unified result for asymptotic autonomy and backward compactness of a non-autonomous random attractor. Let D ∞ = {D(ω)} be an inclusionclosed universe of some single-parametric sets.
Theorem 5.1. Suppose that a NRDS Φ satisfies the following two conditions: (a) Φ has a closed random absorbing set K ∈ D; (b) Φ is D-backward asymptotically compact. Then, Φ has a unique backward compact random attractor A ∈ D.
Let Φ ∞ be an RDS with a D ∞ -random attractor A ∞ , and further assume that (c) Φ backward converges to Φ ∞ in the following sense: Moreover, for any sequence τ n → −∞, there is a subsequence {τ n k } such that P -a.s.
Proof. Existence. Note that backward asymptotic compactness obviously implies asymptotic compactness. By [32,Theorem 2.23], both conditions (a) and (b) imply that Φ has a unique D-random attractor A ∈ D given by the omega-limit set of K, that is, A(τ, ω) = W(τ, ω, K). Since Φ is D-backward asymptotically compact, by (iv) of Proposition 5.1, we know A is backward compact. Asymptotic autonomy. In order to show the asymptotic autonomy as given in (5.6), it suffices to prove P (Ω 1 ) = 1, where where Ω 0 is the θ-invariant full-measure set given in (5.5). Then, P (Ω 2 ) > 0, and θ s Ω 2 ⊂ Ω 0 for all s ∈ R.
By the compactness of A(τ n , ω), for each n ∈ N, we can take a x n ∈ A(τ n , ω) such that A(τ, ω). Indeed, by invariance of A ∈ D and absorption of K, we know, for large t > 0, By (d) and by the inclusion-closedness of D ∞ , we have A τ0 ∈ D ∞ .
Since A τ0 ∈ D ∞ can be attracted by the attractor A ∞ , there is an n 0 ∈ N such that τ n0 ≤ τ 0 ≤ 0 and Furthermore, by the continuity of Φ ∞ : X → X, we have On the other hand, by the invariance of A, we know Hence, we can rewrite x n ∈ A(τ n , ω) as If n ≥ n 0 , then τ n − |τ n0 | ≤ τ n ≤ τ n0 ≤ τ 0 , and thus Because we have proved that A is backward compact, we know A τ0 (θ τn 0 ω) is a pre-compact set, which further implies that {y n } has a convergent subsequence: y n k → y 0 as k → ∞ for some y 0 ∈ A τ0 (θ τn 0 ω).
By the Lebesgue theorem (because P (Ω) is finite), any almost everywhere convergent sequence of measurable functions must be convergent in probability. Hence, Note that each θ τn is measure preserving, it follows the following convergence in probability: Therefore, by the Riesz theorem, the above convergence in probability implies that there is a subsequence satisfying (5.7) as required.
Proof. By Propostion 3.1, Φ has an increasing random absorbing set K such that K is backward tempered, i.e. K ∈ D. We then show that Φ is backward asymptotically compact in H 1 0 (Q). For this end, we let (τ, ω, D) ∈ R × Ω × D be fixed, and take arbitrary sequences s n ≤ τ , t n → +∞ and v 0,n ∈ D(s n − t n , θ −tn ω). We need to show the precompactness of the following sequence: v n = Φ(t n , s n − t n , θ −tn ω)v 0,n = v(s n , s n − t n , θ −sn ω, v 0,n ).
For each η > 0, by Lemma 4.1, there are N 1 ∈ N and K ≥ 1 such that v n H 1 (Q c K ) ≤ η, for all n ≥ N 1 , (5.14) where we recall Q c K = Q \ Q K and Q K = {x ∈ Q : |x 3 | ≤ K}. By Lemma 4.2, there are i ∈ N and N 2 ≥ N 1 such that (5.15) By Lemma 3.1, the set B N2 is bounded in H 1 0 (Q). Then, {ξ K v n : n ≥ N 2 } is bounded in H 1 0 (Q 2K ), hence, P i {ξ K v n : n ≥ N 2 } is pre-compact in H 1 0 (Q 2k ) due to the finitely dimensional range of P i . In a conclusion, which along with (5.15) implies that Since B N2 ⊂ B N1 , we deduce from (5.14) and (5.16) that . So far, we have verified both conditions (a) and (b) in Theorem 5.1. Therefore, Φ possesses a backward compact random attractor A ∈ D, where the measurability of A follows from the measurability of the NRDS and the absorbing set.
On the other hand, by Proposition 2.2, the NRDS Φ backward converges to the RDS Φ ∞ , that is, the condition (c) in Theorem 5.1 holds true.