Regular measurable dynamics for reaction-diffusion equations on narrow domains with rough noise

This paper is concerned with the regular random dynamics for the reaction-diffusion equation defined on a thin domain and perturbed by rough noise, where the usual Winner process is replaced by a general stochastic process satisfied the basic convergence. A bi-spatial attractor is obtained when the non-initial space is $p$-times Lebesgue space or Sobolev space. The measurability of the solution operator is proved, which leads to the measurability of the attractor in both state spaces. Finally, the upper semi-continuity of attractors under the $p$-norm is established when the narrow domain degenerates onto a lower dimensional set. Both methods of symbolical truncation and spectral decomposition provide all required uniform estimates in both Lebesgue and Sobolev spaces.

In this paper, we assume that W is a general stochastic process defined on a general probability space (Ω, F, P ) (see Hypothesis W in section 2), which contains the special case of usual Wiener process used in [18,19].
The first purpose is to prove F-measurability of the solution map for Eq.(1) in three spaces: L 2 , L p and H 1 . This measurability (even for Winner process when the state space is L 2 ) seems not to be proved although claimed in the literatures, but it is a foundation to prove F-measurability of an attractor.
The second purpose is to discuss strong attraction of the (L 2 , L 2 )-attractors (obtained by [19]) for Eq.(1) when the terminate space becomes more regular such as L p , H 1 . More precisely, we will prove the existence of a bi-spatial (L 2 , L p ∩ H 1 ) random attractor A for Eq.(1) by using the theory of bi-spatial random attractors developed by Li et al [21]. Although the theory of bi-spatial attractors aimed at a non-thin domain (see [12,34,35]), it is possible to adapt the thin-domain problem if we make a transformation of the thin domain as done in section 2.
The third purpose is to prove the upper semi-continuity of A as → 0, not only in L 2 but also in L p . More precisely, as → 0, the n + 1-dimensional thin domain O degenerates onto the n-dimensional domain Q. We consider the limiting equation defined on Q as follows.
When providing some uniform estimates in L p and H 1 respectively, our techniques are to impose symbolic truncation of the solution and decomposition of the spectrum.
2. Random cocycle from the equation.
Hypothesis T. The above functions satisfy some tempered conditions: for any τ ∈ R and σ > 0, where we use · ∞ to denote the norm in L ∞ (Õ). Also, we assume that the restrictions G 0 , ψ 1,0 , ψ 4,0 defined on Q satisfy the same conditions as given in (8) and (9).
Remark 1. Comparing with [18,19], we have expanded the defining fields of f and . This is necessary to ensure that the restrictions f 0 , G 0 are well-defined, and satisfy some growth conditions on Q. In this case, the limiting equation (2) is well-defined and solvable. On the other hand, u ∈ L ∞ (Ô) if and only if u ∈ L ∞ ( O) with the same norm. However, we do not have u 0 ∈ L ∞ (Q) even if u ∈ L ∞ (Ô), where u 0 (y * ) = u(y * , 0).

Transformation of the thin domain. Let
It is easy to see that T is bijective with the Jacobian matrix: where I is the n-dimensional unit matrix, and the determinant |J| = 1 g(y * ) . Let ∇ x , ∇ y , ∆ x and div y be the gradient, Laplace and divergence operators in x ∈ O or y ∈ O. Then they are related by (see [16,19]): ∇ xũ (x) = J * ∇ y u(y) and where we denote by u(y) =ũ(x) (y = T x ∈ O), J * is the transport of J and Υ is the operator given by Now, we rewrite f (t, x, u), G(t, x) as some functions in y = (y * , y n+1 ) ∈ O: G (t, y * , y n+1 ) = G(t, y * , g(y * )y n+1 ), f (t, y * , y n+1 , u) = f (t, y * , g(y * )y n+1 , u).

FUZHI LI, YANGRONG LI AND RENHAI WANG
Then, problem (1) is equivalent to the following system defined on O: where ν is the unit outward normal vector on ∂O. Also, we take three state spaces defined by On X and Y , the original L 2 -norm (denoted by · ) and L p -norm are respectively equivalent to the new norms: On Z, we consider a family of new norms and bilinear forms defined by It is easy to prove the equivalence of norms with small (cf. [15,16]).
In the sequel, we always assume that ∈ (0, 0 ] instead of ∈ (0, 1]. Denote by A the unbounded operator on X with domain D(A ) = {u ∈ H 2 (O); Υ u · ν = 0 on ∂O}, given by Hence, we can rewritten Equ. (11) as an abstract evolution equation on X.
2.3. Continuity and measurability of the solution mapping in samples.
We construct the stochastic process W in the following way.
Remark 2. In the above construction, different probability measures determine different stochastic processes. By [3,7,9], if P is a Wiener measure, then, the corresponding process is the usual Wiener process, which is widely used in literatures (see [1,6,8,13,32]).
for every T > 0. Moreover, this solution continuously depends on v τ ∈ X and t ≥ τ .
Next, we prove F-measurability (even continuity) of the solution mapping from Ω to X.
where c and C are positive constants which are independent of k and t, but depend on τ, T, ω 0 .
Proof. Suppose [τ, τ + T ] ⊂ [−n 0 , n 0 ] with n 0 ∈ N. Then, by the continuity of t → ω 0 (t), uniformly in k ∈ N, where we have used the fact: (ω k , ω 0 ) → 0 if and only if n (ω k , ω 0 ) → 0 for all n ∈ N. Then (17) follows from the above estimate immediately. Note that (e −x ) = −e −x , by the mean valued theorem, we have, for all t ∈ [−n 0 , n 0 ], which tends to zero as k → ∞. We have proved the first part and similarly the second part in (16).
where v is the solution of Equ. (14) with the initial value v τ ∈ X.
Proof. We omit the superscript when there is no ambiguity. Let ω 0 ∈ Ω be fixed and suppose ω k ∈ Ω such that ρ(ω k , ω 0 ) → 0 as k → ∞. We denote by (14) and the linearity of A , we have dV k dt

FUZHI LI, YANGRONG LI AND RENHAI WANG
with the initial data V k (τ ) = v τ − v τ = 0. We multiply (18) with gV k and then integrate over O to obtain where I(f ) and I(G) are defined and estimated as follows.
where we recall that · ∞ denotes the norm in L ∞ (Õ). We then split the nonlinear term into three terms By the first condition in (6), it follows from the mean valued theorem that I 1 ≤ β V k 2 g . By the second condition in (6), we have, (17) in Lemma 2.3. By the condition (5) we have We substitute all above estimates into (19) to find, for all We then apply the Gronwall inequality on (20) over [τ, t] to find where we have used (15) in Lemma 2.2 and Hypotheses G, F in the last step. Therefore, in order to prove V k (t) 2 g → 0, it suffices to prove For this end, we use the following energy inequality (cf. [19, (3.6)]): Since t lies in a finite interval, it follows from (17) in Lemma 2.3 that (22) can be rewritten as Since e λ(s−τ −T ) ≥ e −λT for s ∈ [τ, τ + T ], (21) follows immediately and the whole proof is complete.
As a conclusion of Lemma 2.4, the solution mapping is (F, B(X)) measurable. We need to prove that it is (F, B(Y )) and (F, B(Z)) measurable. In fact, this measurability (in more regular spaces) can be deduced from some abstract results on the concept of quasi-continuity, which seems to be first introduced by Li and Guo [22]. While the relationship between measurability and quasi-continuity has been discussed recently by Cui, Langa and Li [14].
Proof. By Lemma 2.4, the mapping is continuous (thus quasi-continuous) from (Ω, ) to X. By Lemma 2.2, v(t, τ, ω, v τ ) ∈ Y for t > τ and v τ ∈ X. Since Y → X and X * → Y * densely, it follows from inheritability of the quasi-continuity (see [14,Prop.3]) that the mapping is quasi-continuous from (Ω, ) to Y . Then, by the measurability of a quasi-continuous mapping (see [14,Prop.5]), the solution mapping is (F, B(Y )) measurable as required. It is similar to show that the solution mapping is (F, B(Z)) measurable.
By Lemma 2.2, we can define a family of mappings φ : where v τ = z(τ, θ −τ ω)u τ . Recall that a concept of random cocycle is given by Wang [31].
Theorem 2.7. For each ∈ (0, 0 ], φ is a continuous random cocycle on X. Its restriction on Y is a quasi-continuous random cocycle on Y , and its restriction on Z is a quasi-continuous random cocycle on Z.
In order to study random attractors, we take some universes where D denote the supremum of norms for all elements, and 3. Random attractors in p-times Lebesgue space. We need the following basic estimates for the solution v (s, τ − t, θ −τ ω, v 0 ) in X. When P is a Winner measure, the following lemma was proved in [19] only by using the convergence: ω(t)/t → 0 as t → ±∞. However, this convergence is assumed in Hypothesis W, and so the following lemma holds true.
where ρ 0 is tempered and given by (23), we have the following estimates. The proof is standard and so omitted.
Now, we intend to provide further estimates in more regular spaces. The following Gronwall-type lemma will be used frequently, which can be founded in [23,24,37].
Lemma 3.2. Let z, z 1 and z 2 be nonnegative, locally integrable such that dz/ds is locally integrable and where a > 0. If τ ∈ R and µ > 0, then , where c > 0 and ρ 1 is a tempered variable given by Proof. We drop the superscript for convenience. Multiply (14) with g|v| p−2 v and integrating over O to obtain We first prove non-negativity of the Laplace term. Indeed, We then estimate the nonlinearity in (29). By the condition (4), it is shown that which implies that The final term of (29) is controlled by
Proof. Let D, τ, ω be fixed, and so the entry time T is fixed. We claim that where the Lebesgue measure |O K | decreases as K increases. Indeed, by Lemma 3.3, we know that where and below, we denote by C = C(τ, ω) and denote by c a constant. Note that C is independent of ∈ (0, 0 ] and t ≥ T . Letting K → +∞ in the above inequality yields (32).
As B (T 0 ) is pre-compact in L 2 (O), it has a finite (2 −p K 2−p η p ) 1/2 -net in L 2 such that the centers v i ∈ B (T ), i = 1, 2, · · · , m. Then, for any v ∈ B (T 0 ), we can It is easy to see that |v| ≥ K ≥ |v i | on O 1 , and |v| ≤ K ≤ |v i | on O 2 , then, by (41), On the other hand, Therefore, v − v i p p ≤ η p , which implies that, for any η > 0, B (T 0 ) has a finite η-net in L p (O), and thus it is pre-compact in L p (O). It is obvious that B (T ) decreases as T increase, then B (T ) is pre-compact in L p for all T ≥ T 0 . (1) For each τ ∈ R, ω → A (τ, ω) is F-measurable in X and in Y respectively; (2) A ∈ D 1 , and A (τ, ω) is compact in X ∩ Y ; (3) A is invariant, i.e. φ (s, τ, ω)A (τ, ω) = A (τ + s, θ s ω) for s ≥ 0; (4) A is pullback attracting in Y , i.e. for every D ∈ D 1 , Proof. By Lemma 3.1, φ has a random absorbing set E from D 1 , which is defined by By Proposition 1, φ is (X, Y )-eventually compact and thus (X, Y )-omega-limit compact. Then, by an abstract result on bi-spatial attractors given in [21,23], it is easy to prove the existence of a (X, Y )-attractor A (except for F-measurability). By Lemma 2.4, the cocycle φ is F-measurable in X, which, together with an abstract result given in [31], implies F-measurability of A in X. By Lemma 2.5, the cocycle φ is F-measurable in Y . By Lemma (3.3), φ has a random absorbing set E p in L p (O) given by , ω ∈ Ω. Therefore, it follows from [14,Theorem 19] that the attractor A is F-measurable in L p (O).

Remark 4.
In the definition of a bi-spatial random attractor given above, we require the F-measurability holds true in both state spaces, while the bi-spatial attractor given in [21] is F-measurable in the initial space only, and the bi-spatial attractor given in [14] is F-measurable in the terminate space only. The above definition may be more reasonable. 4. Random attractors in Sobolev space. We need to consider the eigenvalue problem: which is equivalent to Then, it is well known that each unbounded operator A on X has countable eigenvalues λ j , j = 1, 2, · · · such that 0 < λ 1 ≤ · · · ≤ λ j → ∞ as j → ∞ for each ∈ (0, 0 ).
By using the method of symbolical truncation given in Lemma 3.4, we can provide an auxiliary estimate.
is an entry time, independent of and j.
Proof. By applying the Gronwall-type inequality (27) to the energy inequality (38), we have By Lemma 3.3, we know that for all ∈ (0, 0 ], t ≥ T and v 0 ∈ D(τ − t, θ −t ω), By the assumptions G and F, Next, we prove the main convergence (42). Let δ j := λ j + 2λ → +∞. We make the decomposition: where I 1 , I 2 , I 3 are defined and estimated as follows.
Similarly, on O −K , we can rewrite By (28) in Lemma 3.3, we have sup Let η > 0. Then, by (43) and (44), we can choose a large K such that for all j ∈ N, Finally, I 3 is given by Since δ j → ∞, we can take j 0 large enough such that δ j ≥ η −1 (K p−2 + K 2p−2 ) for all j ≥ j 0 , which implies that for all j ≥ j 0 , uniformly in t ≥ T . The proof is complete.
It is known that the eigenvectors {e j } ∞ j=1 consist of a complete orthonormal basis of X and {e j } ∞ j=1 ⊂ H 1 (O) := Z. Let H m = span{e 1 , e 2 , · · · , e m } ⊂ Z and P m : Z → H m be the canonical projector and I be the identity. Then for every v ∈ Y there exists a unique decomposition where H ,⊥ m is the orthogonal complement of H m .
Proof. We drop the superscript for convenience. Taking the inner product of (14) with A v 2 in (L 2 (O), · g ), we find that, We first estimate the nonlinear term. By the condition (5), we have For the last term in (46), we have Since λ m+1 is the first eigenvalue of A restricted on the subspace H ,⊥ m , we have Therefore, (46)-(48) imply a differential inequality of v 2 (s, τ − t, θ −τ ω, v 2,0 ) as follows.
For each ∈ (0, 0 ], the cocycle φ generated by the problem (14) has a unique (X, Z) random attractor A , this attractor is the same set as given in Let T 0 be an entry time for absorption. We will prove κ Z B (T 0 ) = 0, where the Kuratowski measure κ(·) denotes the minimal diameter of all sets constituted a finite cover. Indeed, by Lemma 4.2, for each η > 0, there is an m = m(η) ∈ N such that By Lemma 3.1, we have which means that P m B (T 0 ) is bounded in the finitely dimensional subspace of Z, and thus it is pre-compact in Z with the Kuratowski measure zero. Hence, We have, κ Z B (T 0 ) = 0, which implies that B (T 0 ) is pre-compact in Z, and so φ is eventually compact. Therefore, by an abstract result on bi-spatial attractors given in [21], the random cocycle φ has a (X, Z) attractor A , which is the same set as an attractor in X or in Y given in Theorem 3.5. By Lemma 2.5, the cocycle φ is F-measurable in Z. By Lemma 3.1, φ has a random absorbing set E in H 1 (O) given by H 1 ≤ cρ 0 (τ, ω)}, ∀τ ∈ R, ω ∈ Ω. Therefore, it follows from [14,Theorem 19] that the attractor A is F-measurable in Z.
In order to consider the limiting equation (2) on Q, we define an operator A 0 by D(A 0 ) = {u ∈ H 2 (Q), ∂u ∂ν0 = 0 on ∂Q}, and for u ∈ D(A 0 )
The solution determines a continuous random cocycle φ 0 (t, τ, ω, u 0 τ ) on L 2 (Q). Analogous results on random attractors for this cocycle are easily obtained when the n-dimensional domain Q replaces the n + 1-dimensional domain O.

5.
Upper semicontinuity of bi-spatial random attractors. In order to consider convergence for random attractors, we assume some convergence for both source and force.
In the above hypothesis, we regard that a function u defined on Q is identical to the functionû(y * , y n+1 ) = u(y * ), (y * , y n+1 ) ∈ O. It is easy to see that u ∈ L 2 (Q) if and only ifû ∈ L 2 (O). Conversely, for a function defined on O, we consider its average function on Q by using the average operator M : Under the hypothesis C, the following convergence of the cocycle φ can be found in [19,Theorem 5.1].
The following convergence of the random attractor A in L 2 (O) can be found in [19,Theorem 5.2].

Lemma 5.3. [19]
The random attractor A is upper semicontinuous in L 2 (O) at = 0, that is, for every τ ∈ R, ω ∈ Ω, Next, we show that upper semicontinuity of the random attractor holds true under the more strong topology.
Step 1. We show that any sequence {z k } ∞ k=1 is pre-compact in L 2 (O), where z k ∈ A k (τ, ω), k → 0, and (τ, ω) ∈ R × Ω is fixed. Indeed, by the convergence of the attractors given in Lemma 5.3, we have As A 0 (τ, ω) is a compact set in L 2 (Q), passing to a subsequence, we have z k − z L 2 (Q) → 0 for somez ∈ L 2 (Q). Therefore, as k → ∞, Step 2. We prove that any sequence z k ∈ A k (τ, ω) is pre-compact in L p (O), where k → 0, and we assume without lose of generality that k ∈ (0, 0 ] for all k ∈ N. By Lemma 3.1, each cocycle φ k has a collective absorbing set E defined by Then, the invariance of A k and the absorption of E implies that k∈N A k (τ, ω) ⊂ E(τ, ω).
Let T be an entry time when E ∈ D 1 is absorbed by itself. By the invariance of A k again, we know that for each k ∈ N, Since E ∈ D 1 and { k } ⊂ (0, 0 ], it follows from (31) in Lemma 3.4 that for each η > 0 there is a K = K(η) such that By Step 1, {z k } ∞ k=1 is pre-compact in L 2 (O), and so it has a finite (K 2−p η p ) 1/2 -net in L 2 (O) such that the finite centers are taken from the sequence {z k }. Then, for each z k , there is a center z i such that It is similar to the proof of Proposition 2 to split the domain O =

FUZHI LI, YANGRONG LI AND RENHAI WANG
Hence, z k −z i p p ≤ 2 p+3 η p and so z k −z i p ≤ 8η. Therefore, the sequence {z k } ∞ k=1 has a finite η-net in L p (O) for any η > 0, which implies that the sequence {z k } ∞ k=1 is pre-compact in L p (O) as required.
This gives a contradiction with (57) and finishes the whole proof.
Remark 5. The upper continuity of random attractors in H 1 (O) remains open, although we have shown the existence of an attractor in H 1 . The main difficulty arises from that the equivalence between · H 1 and · H 1 is not uniform in (see Lemma 2.1), and that the eigenvalue λ m depends on . Maybe, the spectral continuity given by Arrieta and Carvalho [4] can provide some new insights for this question.