ODD RANDOM ATTRACTORS FOR STOCHASTIC NON-AUTONOMOUS KURAMOTO-SIVASHINSKY EQUATIONS WITHOUT DISSIPATION

. We study the random dynamics for the stochastic non-autonomous Kuramoto-Sivashinsky equation in the possibly non-dissipative case. We ﬁrst prove the existence of a pullback attractor in the Lebesgue space of odd functions, then show that the ﬁber of the odd pullback attractor semi-converges to a nonempty compact set as the time-parameter goes to minus inﬁnity and ﬁ-nally prove the measurability of the attractor. In a word, we obtain a longtime stable random attractor formed from odd functions. A key tool is the existence of a bridge function between Lebesgue and Sobolev spaces of odd functions.

The deterministic form (f = g = 0) is deduced from the the original KS equation (Temam [22]) with an unknown variableũ and u = Dũ. Then the periodicity ofũ and the integration by parts yield However, it is not easy to obtain an attractor since the equation is possibly nondissipative. Indeed, the first eigenvalue of the differential operator νD 4 + D 2 is given by with two eigenvectors given by cos 2πx l and sin 2πx l . If the viscosity is small, e.g. ν < l 2 /(4π 2 ), then λ 1 < 0 and thus the equation is not dissipative (which is different from dissipative equations in [11]).
In this article, we will show that the cocycle has a pullback random attractor A(t, ω) even in the non-dissipative case. Such a bi-parametric attractor (depending on time and sample) seems to have first been introduced by Wang [23] with developments in [9,14,26,28] and the references therein.
In order to overcome the difficulty of non-dissipation, we fix attention on the Lebesgue space H o of odd functions, where By proving the existence of a bridge function between H o and V o (the Sobolev space of odd functions), see Lemma 2.2, we can establish the pullback absorption and the pullback asymptotic compactness of the cocycle in H o if the force is tempered, and thus obtain a pullback attractor. Of course, this attractor consists of odd functions and thus we call it an odd attractor.
A further topic is to study the longtime stability of the pullback attractor. More precisely, A is called backward stable if there is a nonempty compact set E(ω) in Such a backward stability indicates that the attractor is not explosive and the system has more strong attraction ability in the past. The criteria for the longtime stability are given in terms of backward uniform asymptotic compactness of the cocycle, see [25] in the stochastic case and see [12,13] in the deterministic case.
If we further assume that the force f is backward tempered, then we can prove the backward uniform asymptotic compactness, which leads to the longtime stability as in (3). We conveniently obtain the backward compactness of the pullback attractor.
Since the backward absorbing set is an uncountable union of random sets, the measurability of the absorbing set (and thus the attractor) seems to be unknown. In order to overcome this difficulty, we consider two universes, one is the usual tempered universe and another is backward tempered. We prove an important result that the pullback attractors are the same set with respect to different universes and thus the measurability of the tempered attractor implies the measurability of the backward tempered attractor.
In a word, the stochastic equation (1) has a longtime stable random attractor formed from odd functions.
2. Random dynamical systems in the space of odd functions. In this section, we prove the existence of a bridge function and define a cocycle in the Lebesgue or Sobolev spaces of odd functions.
2.1. Sobolev spaces of odd functions. Let H =L 2 (I) with the L 2 -norm · and equip V =Ḣ 2 per (I) = H 2 per (I) H with the following scalar product and norm: Let H o be the subset of H formed from all odd functions and Proof. (i) The linearity follows from the fact that the linear combination of two odd functions is still odd. We then prove the closedness. Let u n ∈ H o and u ∈ H such that u n − u → 0. Then there are an index subsequence n k and a set I 1 ⊂ I with Lebesgue measure zero such that u n k (x) → u(x), for all x ∈ I \ I 1 .
Let I 2 = {x ∈ I : −x ∈ I 1 }. Then, for all x ∈ I\(I 1 ∪I 2 ), we know −x ∈ I\(I 1 ∪I 2 ) and which means u is odd on I \ (I 1 ∪ I 2 ). Since the Lebesgue measure of I 1 ∪ I 2 is zero, u almost everywhere equals to an odd function and thus u ∈ H o .
(ii) Let {u n } be a bounded sequence in V o . By the compactness of the Sobolev embedding V → H, there is a subsequence such that u n k → u in H. By the closedness of H o in H as proved above, we know u ∈ H o . Hence {u n } is precompact in H o as desired.
The following result means the existence of a bridge function between H o and V o , which improves [22, lemma III 4.1] (see also [21]) and will be very useful.
Lemma 2.2. For any a, b > 0, there is an odd function ξ ∈Ċ ∞ [−l/2, l/2], given by Proof. From the definition of ξ it is easy to show that where f k is the kth Fourier coefficient of the function u 2 (·). Since u ∈ V , the Sobolev embedding H 2 (I) → C 1,1/2 (I) yields the continuity of u(·) and so is u 2 (·). Hence the Fourier series converges: Since u is odd, we have u(0) = 0 and thus k∈Z f k = 0, which further implies Since H 2 (I) is a Banach algebra in dimension one, it follows that All above inequalities further imply We can choose M ≥ (c 6 /b) 2/3 to obatin (4) as desired.
By the change (7) of variables, we can rewrite the equation (1) as a random equation (without the stochastic derivative): By the similar method as in [7,8,10,27], the equation (1) is well-posed in H and so is the equation (8).
We prove the assertion about odd functions. Let u be the solution of the equation (1) and defineû(x) = −u(−x) for all x ∈ I. Since f (s) and g are odd, it follows that u fulfills (1) too. Since v τ , ξ, g are odd, it follows that , which means the initial conditions are the same. By the uniqueness of solutions, we haveû(x) = u(x), i.e. −u(−x) = u(x). Hence, the solution u of the equation (1) is odd and so is v (by (7)).

By Lemma 2.3, under the assumptions of
where the F-measurability of Φ can be proved by the same method as given in [6] and the uniqueness of solutions implies the cocycle property: We then take the universe B by the collection of all backward tempered bi- In order to prove the existence of a B-pullback attractor, we further assume By [25], the assumption (12) implies that it holds true for all positive rates: 3. Backward-uniform estimates. We will frequently use the trilinear form By the Sobolev embedding H 4 (I) → C 2 (I), the assumption g ∈Ḣ 4 per (I) implies g ∈ C 2 (I) ⊂ W 2,∞ (I) and thus where C(ω) is an intrinsic positive random variable and Proof. Multiplying (8) by v(r, s − t, θ −s ω; u s−t ) and integrating over I yield By the integration by parts, we have Dv 2 ≤ v D 2 v and thus By (14), (vDv, v) = 0 and 2(D(ξv), v) = 2(vDξ, v) + 2(ξDv, v) = (vDξ, v).
We need the non-autonomous version of the uniform Gronwall lemma (see [19]).
Finally, by the change (26) of variables and (34), The proof is complete.

4.
Existence and backward stability of odd random attractors. We need the concept of a pullback random attractor as introduced by Wang [23].
Definition 4.1. Let Φ be a cocycle (fulfilling (10)) on a Polish space X over the quadruple (Ω, F, P, θ) and D a universe of some bi-parametric sets on X. Then A ∈ D is called a pullback attractor for Φ if it is compact, invariant under Φ, and D-pullback attracting. The pullback attractor is called a pullback random attractor if it is further random, i.e. each A(τ, ·) is a random set.
For a pullback attractor, we can consider its longtime stability, i.e. the limiting behavior of its fiber when the time-parameter goes to infinity, see [5,25] in the stochastic case and see [12,13] in the deterministic case.

Definition 4.2. A pullback attractor A for a cocycle is called backward stable
if, for each ω ∈ Ω, there is a nonempty compact set K(ω) such that While, the minimal compact set fulfilling (35) (if exists) is called a backward controller.
The backward controller means the minimal compact set controlled the attractor from the past. (ii) A is backward stable with a backward controller, given by Proof.
Step 1. Prove the B-pullback absorption. By Lemma 3.1, Φ has a Bpullback absorbing set given by where ρ and R f are defined by (19) and (20) respectively. In fact, by (16), K is backward absorbing in the following sense To prove K ∈ B, we first claim that is tempered with any rate α > 0. Indeed, by (5), |z(ω)| is tempered, i.e. e −αt |z(θ −t ω)| → 0 as t → +∞. If Dg ∞ = 0, then ρ(ω) is obviously tempered. If Dg ∞ > 0, then we assume without loss of the generality that α < Dg ∞ min (1, E|z|). Then, by the ergodic limit (6), there is T > 0 such that for all t ≥ T and r ≤ −t, Since β = Dg ∞ (E|z| + 1) + 1 > Dg ∞ (E|z| + which means the second term of ρ(ω) is tempered. By (38) and (5), which means that the third term of ρ(ω) is tempered. Hence ρ(ω) is tempered and thus backward tempered (since it is independent of τ ). We then claim is backward tempered. Let α be the rate mentioned above and τ ∈ R. Since R f (·, ω) is increasing, it follows from (38) and (13) that which means R f (·, ·) is backward tempered and thus K ∈ B. Note that R f (τ, ω) is the supremum of uncountable random variables, its measurability is unknown.
Step 2. Prove backward B-pullback asymptotic compactness. We need to prove that, for any s n ≤ τ , t n → +∞, u 0,n ∈ B(s n − t n , θ −tn ω), where τ ∈ R, B ∈ B and ω ∈ Ω are fixed, the solution sequence {u(s n , s n − t n , θ −sn ω, u 0,n )} has a convergent subsequence in H o . Indeed, by (32) in Lemma 3.3, we have D 2 u(s n , s n − t n , θ −sn ω, u 0,n ) 2 ≤ R V (τ, ω) < +∞ provided n is large enough. Then the sequence {u(s n , s n − t n , θ −sn ω, u 0,n )} is bounded in V o . By the compactness of the Sobolev embedding V → H, the sequence is pre-compact in H. By Lemma 2.1, {u(s n , s n − t n , θ −sn ω, u 0,n )} is pre-compact in H o .
Step 3. Prove the existence of a B-pullback attractor A ∈ B. By the abstract result as in [23], the existence of a B-pullback attractor follows from the B-pullback absorption (by taking s = τ in Step 1) and the B-pullback asymptotic compactness (by taking s n ≡ τ in Step 2). But the measurability of A is temporarily unknown since we cannot prove the measurability of the absorbing set K (it is an uncountable union of random sets).
Step 4. Prove the backward stability of A and the existence of a backward controller. By the backward asymptotic compactness in Step 2, we can prove that A is backward compact, i.e. ∪ s≤τ A(s, τ ) is pre-compact (cf. [18,24]). Then the theorem of nested compact sets implies that A(−∞, ω) (as defined by (36)) is nonempty compact. By the same method as in [25], we can prove For any w ∈ A(−∞, ω) , we can take a sequence w n ∈ A(τ n , ω), where τ n → −∞, such that w n → w. While, dist Ho (w n , E(ω)) ≤ dist Ho (A(τ n , ω), E(ω)) → 0 and thus w ∈ E(ω), which proves the minimality. Therefore, A(−∞, ω) is the backward controller.
Step 5. Prove the measurability of A. Let D be the usual universe forming from all tempered set in H o , i.e. D = {D(τ, ω)} ∈ D if and only if lim t→+∞ e −εt D(τ − t, θ −t ω) 2 = 0, ∀ε > 0, τ ∈ R, ω ∈ Ω, where we have omitted the supremum in the definition (11) of B. Then, by the same method as in Lemma 3.1, one can prove that Φ has a D-pullback absorbing set given by where R D (τ, ω) = 0 −∞ e βr+ Dg ∞ 0 r |z(θrω)|dr f (r + τ ) 2 dr such that sup s≤τ R D (s, ω) = R f (τ, ω) in view of the definition of R f in (20). As an integral of random variables, R D (τ, ·) is measurable (although we do not know the measurability of R f ). By the same method as in Step 1, we know K D ∈ D (it may not belong to B). By the same method as in Lemma 3.3 and Step 2, one can prove that Φ is D-pullback asymptotically compact in H o . Then the abstract result in [23] can be applied to obtain a D-pullback random attractor A D such that A D is just constructed by the omega-limit set of K D .
Since R D (τ, ω) ≤ R f (τ, ω), we have K D ⊂ K and thus their omega-limit sets fulfill A D ⊂ A. On the other hand, since A ∈ B ⊂ D, it follows that A can be attracted by A D , which, together with the invariance of A, implies A ⊂ A D . So, A = A D and thus A is random too. Remark 1. The method for proving K ∈ B (where the absolute value |z(θ s ω)| is involved) differs from those in the literature. The method for proving the measurability of A also differs from the usual.
On the other hand, every function of the (nonempty) attractor A(τ, ω) is odd and smooth, where the smoothness follows from the Sobolev embedding V → C 1 (I) and the invariance of the attractor.