Probabilistic Continuity of a Pullback Random Attractor in Time-Sample

DISCRETE AND CONTINUOUS doi:10.3934/dcdsb.2020028 DYNAMICAL SYSTEMS SERIES B Volume 25, Number 7, July 2020 pp. 2699–2722

PROBABILISTIC CONTINUITY OF A PULLBACK RANDOM ATTRACTOR IN TIME-SAMPLE

Shulin Wang Faculty of Eduction Southwest University Chongqing 400715, China Yangrong Li∗ School of Mathematics and Statistics Southwest University Chongqing 400715, China

(Communicated by Bj¨ornSchmalfuß)

Abstract. Given a time-sample dependent attractor of a random dynamical system, we study its lower semi-continuity in probability along the time axis, and the criteria are established by using the local-sample asymptotically compactness for a triple-continuous system. The abstract results are applied to the non-autonomous stochastic p-Laplace equation on an unbounded domain with weakly dissipative nonlinearity. Without any additional hypotheses, we prove that the pullback random attractor is probabilistically continuous in both time and sample parameters.

1. Introduction. A core concept in the theory of non-autonomous random dynamical systems is the so-called pullback random attractor (PRA), which seems to be first involved in Crauel et al. [8] and systemically studied by Wang [39]. A PRA is formulated by a bi-parametric set A = {A(τ, ω)}, where τ ∈ R is the current time and ω is the sample in a measurable dynamical system (Ω, F, P, θt). Its properties for a fixed time-fiber had been researched, see [10, 11, 12, 29,42, 45]. Recently, Cui, Kloeden and Wu[14] have studied the upper semi-continuity of a PRA in time-parameter:

lim dist(A(τ, ω), A(τ0, ω)) = 0, ∀τ0 ∈ R, ω ∈ Ω. (1) τ→τ0 In this paper, we consider a more challenging problem on the lower semi-continuity:

lim P {ω ∈ Ω : dist(A(τ0, ω), A(τ, ω)) ≥ δ} = 0, ∀δ > 0, τ0 ∈ R. (2) τ→τ0 In general, it is impossible to obtain the lower semi-continuity even if the upper semi-continuity holds true, many examples can be found in [19, 32, 33]. Only in the special case, such as, the number of equilibria is ﬁnite, the lower semi-continuity was

2010 Mathematics Subject Classiﬁcation. Primary: 37L55; Secondary: 35B41, 60H15. Key words and phrases. Random attractor, probabilistic lower semicontinuity, local-sample compactness, stochastic p-Laplace equation, weak dissipation, pullback attractor. Li was supported by Natural Science Foundation of China grant 11571283, Wang was supported by National Social Science Foundation of China, 13th Five-year plan of Education grant BJA160059. ∗ Corresponding author: Yangrong Li.

2699 2700 SHULIN WANG AND YANGRONG LI established in [2,5, 15, 16, 21, 27], in these references, the target of investigation is a family of attractors {Aα} rather than a single attractor A as in (2). It is hard to directly deal with the lower semi-continuity (2). We will convert the lower semi-continuity into an upper semi-continuity. In fact, we will prove in Theorem 2.4 that the lower semi-continuity (2) can be deduced from the upper semi-continuity in sample:

lim dist(A(τ, θsω), A(τ, θs0 ω)) = 0, ∀s0, τ ∈ R, ω ∈ Ω. (3) s→s0 For an (autonomous) random attractor A = {A(ω)} ([4,6,7,9, 36, 40]), both

(1) and (2) are trivially true, however, (3) is given by dist(A(θsω),A(θs0 ω)) → 0, which is nontrivial. Maybe (3) is more meaningful than (2). Next, our main task is to establish criteria for (3) (and thus for (2)). An essential criterion for (3) is local-sample compactness of the attractor (Theorem 2.11), which means that ∪|s|≤1A(τ, θsω) is pre-compact for each τ ∈ R and ω ∈ Ω. In this case, we need that the system (cocycle) is triple-continuous (Deﬁnition 2.10) in three variables: time, sample and space, where the continuity in sample seems to be a new idea in the literature. We also require that the attracted universe is local-sample closed (Deﬁnition 2.6). Furthermore, we establish the criteria in terms of the cocycle. A main criterion is that the cocycle is local-sample limit-set compact, which means the usual limit-set compactness is uniform in local samples. Also, we show that it is equivalent to the local-sample asymptotic compactness, see Theorem 2.9. All abstract results are summarized in Theorem 2.12. In the application part, we consider the p-Laplace equation perturbed by additive noise and generally dissipative nonlinearity on Rm: du + (λu − div(|∇u|p−2∇u) + f(x, u) − g(t, x))dt = h(x)dW, (4) where p > 2, λ > 0. The equation is said to be strongly dissipative if p ≤ q and weakly dissipative if p > q, where q > 2 such that q − 1 is the growth order of f. It seems to be open even for the existence of a random attractor in the weakly dissipative case, although the strongly dissipative case had been considered in [23, 24, 25, 26, 30, 37, 43, 44]. In a bounded domain, it is not necessary to distinguish the strong or weak dissipation, see [17, 18]. In this paper, the nonlinearity can be both strongly and weakly dissipative. Now, let us describe technique and novelty how to prove both existence and probabilistic lower semi-continuity of a random D-pullback attractor A in L2(Rm), where D is the usual universe forming from all tempered sets. In section3, we mainly prove the solution operator is triple-continuous, especially in sample. So, the equation (4) generates a triple-continuous cocycle Φ. In section4, we take a local-sample closed sub-universe D0 of D. By an inductive method, we prove the existence of a D0-pullback (local-sample) absorbing set in 2 q Xk = L ∩ L k , where qk := k(q − 2) + 2. Let N = {(p − 2)/(q − 2)}, then qN ≥ p and so the absorption can reach the Lp-level. By the same inductive method, we can prove a triple-compactness result of the cocycle on the localization of space, time and sample, where the usual localization method [25, 26] can be generalized from one space-variable to three variables. In section5, we take a sequence of local-sample closed universes Dk in Xk. We then present the local-uniform tail-estimate when the initial data belong to DN−1. PROBABILISTIC CONTINUITY OF A RANDOM ATTRACTOR 2701

In section6, we show our main application results. On one hand, we can prove that the cocycle is local-sample (and local-time) asymptotically compact on DN−1 and then on D0. So, we obtain a D0-pullback attractor A0, which is local-sample compact and local-time compact. On the other hand, the usual method can show the existence of a D-pullback random attractor A. We then prove A0 = A and so A0 is still measurable. There- fore, we obtain a D-pullback random attractor A ∈ D0 such that A is local-sample (and local-time) compact, and thus it satisﬁes (2), (3) and conveniently (1).

2. Abstract results on probabilistic lower semi-continuity.

2.1. A bi-parametric attractor for a cocycle. Let X be a separable Banach space with the Borel algebra B(X). Let (Ω, F,P ) be a probability space with a group {θs}s∈R such that each θs :Ω → Ω is measure-preserving. A measurable mapping Φ : R+ × R × Ω × X 7→ X is said to be a cocycle if

Φ(0, τ, ω) = IX , Φ(t + s, τ, ω) = Φ(t, τ + s, θsω)Φ(s, τ, ω) (5) for all t, s ≥ 0, τ ∈ R, ω ∈ Ω. We always assume that Φ(t, τ, ω)x is continuous in t ≥ 0, τ ∈ R and x ∈ X respectively. We use the terminology of a bi-parametric set B = {B(τ, ω)} to denote a set- valued map B : R × Ω 7→ 2X \{∅}. A bi-parametric set B is called compact if each component B(τ, ω) is compact. Let D be a universe of some bi-parametric sets in X. Deﬁnition 2.1. A ∈ D is called a D-pullback bi-parametric attractor for a cocycle Φ if it is compact, invariant, that is,

Φ(t, τ, ω)A(τ, ω) = A(t + τ, θtω), ∀t ≥ 0, and D-pullback attracting, that is, for each D ∈ D,

lim dist(Φ(t, τ − t, θ−tω)D(τ − t, θ−tω), A(τ, ω)) = 0. t→+∞ A bi-parametric attractor A is called a D-pullback random attractor (PRA) if it is measurable, which means that the mapping ω → d(x, A(τ, ω)) is (F, B(R+)) measurable for each x ∈ X and τ ∈ R. For the details of a PRA, one can refer to Wang [39].

2.2. Probabilistic lower semi-continuity of a PRA. We denote the Hausdorﬀ distance by

disth(A, B) = max{dist(A, B), dist(B,A)}, ∀A, B ⊂ X. Lemma 2.2. Suppose a random attractor A is upper semi-continuous in sample:

lim dist(A(τ, θsω), A(τ, ω)) = 0, ∀τ ∈ , ω ∈ Ω, (6) s→0 R which is equivalent to (3). Then, we have the probabilistic continuity in sample, that is, for each δ > 0,

lim P {ω ∈ Ω: disth(A(τ, θs0 ω), A(τ, θsn ω)) ≥ δ} = 0, ∀τ, s0 ∈ R. (7) sn→s0 2702 SHULIN WANG AND YANGRONG LI

Proof. Since an almost everywhere convergent sequence is convergent in probability, (6) implies the probabilistic upper semi-continuity in sample:

lim P {ω ∈ Ω : dist(A(τ, θsn ω), A(τ, θs0 ω)) ≥ δ} = 0, ∀δ > 0, τ ∈ R. (8) sn→s0

Notice that each θsn :Ω → Ω is measure-preserving, we have

P {ω ∈ Ω : dist(A(τ, θs0 ω), A(τ, θsn ω)) ≥ δ}

=P {θsn ω ∈ Ω : dist(A(τ, θs0 ω), A(τ, θsn ω)) ≥ δ}

=P {ω ∈ Ω : dist(A(τ, θs0−sn ω), A(τ, ω)) ≥ δ} (9)

By (8), the last term in (9) tends to zero as s0 − sn → 0. Therefore, both (8) and (9) imply the probabilistic continuity (7).

We also need a type of (automatic) continuity for a PRA. Lemma 2.3. [41, Theorem 2.3]. Suppose a cocycle has a PRA A, then

lim disth(A(τ0, θτ0 ω), A(τ, θτ ω)) = 0, ∀ω ∈ Ω. (10) τ→τ0 Now, we convert the lower semi-continuity (2) into the upper semi-continuity (6). Theorem 2.4. Let a cocycle Φ have a pullback random attractor A. (i) If A is upper semi-continuous in sample (i.e.(6) holds true), then A is probabilistic lower semi-continuous in time:

lim P {ω ∈ Ω: dist(A(τ0, ω), A(τ, ω)) ≥ δ} = 0, ∀δ > 0, τ0 ∈ R. (11) τ→τ0 (ii) Conversely, the lower semi-continuity (11) implies the probabilistic continuity in sample (i.e.(7) holds true).

Proof. (i) It suﬃces to prove that (11) holds true for any sequence {τn} such that τn → τ0. By the triangle inequality of Hausdorﬀ semi-distance, for P -a.s. ω ∈ Ω,

dist(A(τ0, ω), A(τn, ω))

≤dist(A(τ0, ω), A(τ0, θτ0−τn ω)) + dist(A(τ0, θτ0−τn ω), A(τn, ω)). By the sub-additivity of the probability, we have, for each δ > 0,

P {ω ∈ Ω : dist(A(τ0, ω), A(τn, ω)) ≥ δ} δ ≤P {ω ∈ Ω : dist(A(τ , ω), A(τ , θ ω)) ≥ } 0 0 τ0−τn 2 δ + P {ω ∈ Ω : dist(A(τ , θ ω), A(τ , ω)) ≥ }. (12) 0 τ0−τn n 2

Since A is upper semi-continuous in sample and τ0 − τn → 0, it follows from Lemma 2.2 that δ lim P {ω ∈ Ω : dist(A(τ0, ω), A(τ0, θτ −τ ω)) ≥ } = 0. (13) n→∞ 0 n 2 On the other hand, by (10) in Lemma 2.3, we have

dist(A(τ0, θτ0 ω), A(τn, θτn ω)) → 0, for a.e. ω ∈ Ω. PROBABILISTIC CONTINUITY OF A RANDOM ATTRACTOR 2703

Hence, the sequence {dist(A(τ0, θτ0 ·), A(τn, θτn ·))}n of random variables is convergent in probability. Furthermore, by using the measure-preserving property of each

θ−τn :Ω → Ω, we have δ P {ω ∈ Ω : dist(A(τ , θ ω), A(τ , ω)) ≥ } 0 τ0−τn n 2 δ =P {θ ω ∈ Ω : dist(A(τ , θ ω), A(τ , ω)) ≥ } −τn 0 τ0−τn n 2 δ =P {ω ∈ Ω : dist(A(τ , θ ω), A(τ , θ ω)) ≥ } → 0 (14) 0 τ0 n τn 2 as n → ∞. Now, we substitute (13) and (14) into (12) to obtain

P {ω ∈ Ω : dist(A(τ0, ω), A(τn, ω)) ≥ δ} → 0 as n → ∞, which proves (11) for any sequence {τn} with τn → τ0. (ii) Suppose (11) is true. Then, for sn → 0 and τ ∈ R, δ P {ω ∈ Ω : dist(A(τ, ω), A(τ − s , ω)) ≥ } → 0. n 2 Since τ − sn → τ, it follows from (10) in Lemma 2.3 that

dist(A(τ − sn, θτ−sn ω), A(τ, θτ ω)) → 0, which holds true in probability. Noting that θsn−τ is measure-preserving, we have δ P {ω ∈ Ω : dist(A(τ − s , ω), A(τ, θ ω)) ≥ } n sn 2 δ =P {ω ∈ Ω : dist(A(τ − s , θ ω), A(τ, θ ω)) ≥ } → 0. n τ−sn τ 2 By the same method as given in (12), we have

P {ω ∈ Ω : dist(A(τ, ω), A(τ, θsn ω)) ≥ δ} δ ≤P {ω ∈ Ω : dist(A(τ, ω), A(τ − s , ω)) ≥ } n 2 δ + P {ω ∈ Ω : dist(A(τ − s , ω), A(τ, θ ω)) ≥ } → 0 n sn 2 as n → ∞. Since θsn is measure-preserving, it further implies that

P {ω ∈ Ω : dist(A(τ, θsn ω), A(τ, ω)) ≥ δ} → 0 as n → ∞. We obtain the probabilistic continuity in sample, i.e.(7) is true. 2.3. Local-sample compactness of a bi-parametric attractor. In this sub- section, we do not require the measurability of the attractor. Deﬁnition 2.5. A bi-parametric attractor A is called local-sample compact if the local-sample union ALS is compact in X, where

ALS(τ, ω) := ∪|s|≤1A(τ, θsω), ∀τ ∈ R, ω ∈ Ω. (15)

A universe D is inclusion-closed [34] if De ∈ D whenever D ∈ D and De ⊂ D. This condition ensures A ∈ D. We also need a special property of D.

Deﬁnition 2.6. A universe D is called local-sample closed if DLS ∈ D whenever D ∈ D, where DLS is the local-sample union of D, as deﬁned by (15). We then generalize the concepts of limit-set compactness [22] and asymptotic compactness [3] to the local-sample version. 2704 SHULIN WANG AND YANGRONG LI

Definition 2.7. A cocycle Φ is called local-sample limit-set compact on D if [ [ lim κX Φ(t, τ − t, θs−tω)D(τ − t, θ−tω) = 0 T →+∞ t≥T |s|≤1 for each D ∈ D, τ ∈ R and ω ∈ Ω, where, the Kuratowski measure κX (·) is defined by the minimal one among diameters of sets forming a finite cover. Definition 2.8. A cocycle Φ is called local-sample asymptotically compact on D if, for each (τ, ω, D) ∈ R × Ω × D, the sequence

{Φ(tn, τ − tn, θsn−tn ω)xn}n is pre-compact whenever |sn| ≤ 1, tn → +∞ and xn ∈ D(τ − tn, θ−tn ω). We recall that a bi-parametric set B is called a D-pullback absorbing set for a cocycle Φ if, for each D ∈ D, τ ∈ R and ω ∈ Ω, there is a T := T (D, τ, ω) > 0 such that

Φ(t, τ − t, θ−tω)D(τ − t, θ−tω) ⊂ B(τ, ω), ∀t ≥ T. Theorem 2.9. Let D be a local-sample closed and inclusion-closed universe. Then, the cocycle Φ possesses a unique D-pullback bi-parametric attractor A with local- sample compactness if (i) Φ has a closed D-pullback absorbing set B ∈ D, and (ii) Φ is local-sample limit-set compact on D, or equivalently, (ii*) Φ is local-sample asymptotically compact on D. Proof. Suppose (i) and (ii) hold true. By taking s = 0 in Deﬁnition 2.7, we know Φ is (usual) limit-set compact. By [35, theorem 2.2], there is a unique D-pullback attractor given by \ [ A(τ, ω) = αB(τ, ω) := Φ(t, τ − t, θ−tω)B(τ − t, θ−tω). T >0 t≥T We then introduce the notation of local-sample limit-set by \ [ [ ΓD(τ, ω) := Φ(t, τ − t, θs−tω)D(τ − t, θ−tω). (16) T >0 t≥T |s|≤1

We claim that ΓD is nonempty compact for D ∈ D. Indeed, let [ [ KT (τ, ω) := Φ(t, τ − t, θs−tω)D(τ − t, θ−tω). (17) t≥T |s|≤1

By the condition (ii), {KT (τ, ω)}T >0 is a decreasing family of closed sets in X such that

κX (KT (τ, ω)) = κX (KT (τ, ω)) → 0 as T → +∞.

By [31, lemma 2.5], the intersection ∩T >0KT (τ, ω) (which is just ΓD(τ, ω)) is nonempty compact.

Since D is local-sample closed and B ∈ D, we know BLS ∈ D and thus ΓBLS is compact as proved above. On the other hand, [ [ [ \ [ A(τ, θsω) = αB(τ, θsω) = Φ(t, τ − t, θs−tω)B(τ − t, θs−tω) |s|≤1 |s|≤1 |s|≤1 T >0 t≥T \ [ [ ⊂ Φ(t, τ − t, θs−tω)BLS(τ − t, θ−tω). T >0 t≥T |s|≤1 PROBABILISTIC CONTINUITY OF A RANDOM ATTRACTOR 2705

The last set is just the compact set ΓBLS (τ, ω) and so A is local-sample compact. Next, we show (ii) ⇔ (ii*). It is relatively easy to show (ii) ⇒ (ii*). Indeed, let

|sn| ≤ 1, tn ↑ +∞ and xn ∈ D(τ − tn, θ−tn ω) with D ∈ D. By (ii), we have ∞ κX ({Φ(tn, τ − tn, θsn−tn ω)xn}n=k) [ [ ≤κX Φ(t, τ − t, θs−tω)D(τ − t, θ−tω) → 0

t≥tk |s|≤1

as k → ∞. By [30, lemma 2.3], {Φ(tn, τ − tn, θsn−tn ω)xn} is pre-compact. In order to show (ii*) ⇒ (ii), we ﬁrst notice that y ∈ ΓD(τ, ω) if and only if there are |sn| ≤ 1, tn ↑ +∞ and zn ∈ D(τ − tn, θ−tn ω) such that

Φ(tn, τ − tn, θsn−tn ω)zn → y in X.

We then show that ΓD(τ, ω) is still compact under the condition (ii*). Indeed, let ∞ {yn}n=1 be a sequence taken from ΓD(τ, ω). We can sequentially choose |sn| ≤ 1, tn ≥ max{n, tn−1} (ensuring tn ↑ +∞) and zn ∈ D(τ − tn, θ−tn ω) such that 1 ky − Φ(t , τ − t , θ ω)z k ≤ , ∀n ∈ . n n n sn−tn n n N

∗ ∗ ∗ By the condition (ii*), passing to a subsequence, Φ(tn , τ − tn , θsn∗ −tn∗ ω)zn → y0 in X and y0 ∈ ΓD(τ, ω). Moreover,

∗ ∗ ∗ ∗ ∗ kyn − y0k ≤ kyn − Φ(tn , τ − tn , θsn∗ −tn∗ ω)zn k

∗ ∗ ∗ + kΦ(tn , τ − tn , θsn∗ −tn∗ ω)zn − y0k → 0

∗ as n → ∞. So, ΓD(τ, ω) is compact as desired. We prove that ΓD local-sample uniformly attracts D ∈ D in the following sense:

lim sup distX (Φ(t, τ − t, θs−tω)D(τ − t, θ−tω), ΓD(τ, ω)) = 0. (18) t→+∞ |s|≤1

Indeed, if (18) is not true, then there are η > 0, |sn| ≤ 1, tn ↑ +∞ and zn ∈

D(τ − tn, θ−tn ω) such that

distX (Φ(tn, τ − tn, θsn−tn ω)zn, ΓD(τ, ω)) ≥ η, ∀n ∈ N.

By the condition (ii*), passing to a subsequence, Φ(tn, τ − tn, θsn−tn ω)zn → y ∈ ΓD(τ, ω), which gives a contradiction. So (18) holds true. Given now ε > 0, by (18), we can choose a T1 > 0 such that for all T ≥ T1,

distX (KT (τ, ω), ΓD(τ, ω)) < ε, where KT is deﬁned by (17). Hence, KT (τ, ω) ⊂ Nε(ΓD(τ, ω)) for all T ≥ T1, where Nε(·) denotes the ε-neighborhood. On the other hand, by the compactness of ΓD as proved above, we can ﬁnd m y1, y2, ··· , ym ∈ ΓD(τ, ω) such that ΓD(τ, ω) ⊂ ∪k=1Nε(yk). Hence, m [ KT (τ, ω) ⊂ Nε(ΓD(τ, ω)) ⊂ N2ε(yk), k=1 which implies that κX (KT (τ, ω)) ≤ 4ε for all T ≥ T1. Therefore, κX (KT (τ, ω)) → 0 as T → +∞, which means the condition (ii). 2706 SHULIN WANG AND YANGRONG LI

2.4. Upper semi-continuity in sample. Now, we establish the criterion for the upper semi-continuity (3), which leads to the lower semi-continuity (2) in view of Theorem 2.4. In this case, we need the triple-continuity of the cocycle. Deﬁnition 2.10. A cocycle Φ is called triple-continuous if Φ is continuous in time, sample and space, more precisely,

kΦ(tn, τ, θsn ω)xn − Φ(t0, τ, θs0 ω)x0k → 0, ∀τ ∈ R, ω ∈ Ω (19) whenever tn → t0, sn → s0 and kxn − x0k → 0.

Since {θs}s∈R is a group, it is easy to show that each θs :Ω → Ω is bijective. Thereby, we only require that (19) holds true at s0 = 0. Theorem 2.11. Let a triple-continuous cocycle Φ have a D-pullback attractor A, where D is local-sample closed. Then, A is upper semi-continuous in sample if and only if A is local-sample compact.

Proof. Since each θs :Ω → Ω is bijective, it is easy to show that (3) is equivalent to

lim dist(A(τ, θsω), A(τ, ω)) = 0, ∀τ ∈ , ω ∈ Ω. (20) s→0 R Now, suppose A is local-sample compact, we argue (20) by contradiction. If (20) is not true, then, there are η > 0, sn → 0, yn ∈ A(τ, θsn ω) with τ ∈ R and ω ∈ Ω such that

dist(yn, A(τ, ω)) ≥ η, ∀n ∈ N. (21)

Since D is local-sample closed, the local-sample union ALS ∈ D. So, the attractor A can attract ALS, and thus there is a T > 0 such that

dist(Φ(T, τ − T, θ−T ω)ALS(τ − T, θ−T ω), A(τ, ω)) < η.

By the invariance of A (at θsn ω), there is zn ∈ A(τ − T, θsn−T ω) such that

yn = Φ(T, τ − T, θsn−T ω)zn, ∀n ∈ N.

We assume without loss of generality that |sn| ≤ 1, then,

zn ∈ A(τ − T, θsn−T ω) ⊂ ALS(τ − T, θ−T ω), ∀n ∈ N.

By the assumption, ALS(τ −T, θ−T ω) is compact, and so, passing to a subsequence,

zn → z ∈ ALS(τ − T, θ−T ω). By the triple-continuity of Φ, we have

Φ(T, τ − T, θsn−T ω)zn → Φ(T, τ − T, θ−T ω)z in X. Therefore, if n is large enough,

dist(yn, A(τ, ω)) ≤ kΦ(T, τ − T, θsn−T ω)zn − Φ(T, τ − T, θ−T ω)zk

+ dist(Φ(T, τ − T, θ−T ω)ALS(τ − T, θ−T ω), A(τ, ω)) < η, which contradicts with (21). Therefore, A is upper semi-continuous in sample. On the contrary, suppose A is upper semi-continuous in sample. We take a

∗ sequence yn ∈ A(τ, θsn ω) with |sn| ≤ 1. Passing to a subsequence, sn → s0 and so

∗ dist(yn , A(τ, θs0 ω)) ≤ dist(A(τ, θsn∗ ω), A(τ, θs0 ω)) → 0 ∗ ∗ ∗ as n → ∞. On the other hand, for each n , there is a zn ∈ A(τ, θs0 ω) such that 1 d(y ∗ , z ∗ ) ≤ dist(y ∗ , A(τ, θ ω)) + → 0. n n n s0 n∗ PROBABILISTIC CONTINUITY OF A RANDOM ATTRACTOR 2707

∗ ∗ Since A(τ, θs0 ω) is compact, the sequence {zn } is pre-compact and so is {yn }. Therefore, A is local-sample compact. By combining Theorems 2.4, 2.9 and 2.11, we summarize our abstract results. Theorem 2.12. Let D be a local-sample closed and inclusion-closed universe. As- sume a cocycle Φ satisﬁes the following three conditions: (i) it is triple-continuous; (ii) it has a closed D-pullback absorbing set B ∈ D; (iii) it is local-sample limit-set (or asymptotically) compact on D. Then, there is a unique D-pullback bi-parametric attractor A with the following properties: (a) A is local-sample compact. (b) A is upper semi-continuous in sample (i.e. (3) holds true). Moreover, A is random if B is random. In this case, we have (c) A is probabilistically continuous in sample (i.e. (7) holds true). (d) A is probabilistically lower semi-continuous in time (i.e. (2) holds true).

3. Stochastic non-autonomous p-Laplace equations. The equation can be read as ( du + (λu − div(|∇u|p−2∇u) + f(x, u) − g(t, x))dt = h(x)dW, t ≥ τ, m (22) u(τ, x) = uτ (x), x ∈ R , where p > 2 and λ > 0.

3.1. Hypotheses and transformation for the equation. We use k·kr to denote the norm in Lr(Rm) and delete the subscript when r = 2. Hypothesis F. f : Rm × R → R is continuously diﬀerential, and for all x ∈ Rm, s ∈ R, q q−1 f(x, s)s ≥ α1|s| + ψ1(x), |f(x, s)| ≤ α2|s| + ψ2(x), (23) ∂f ∂f (x, s) ≥ −α , (x, s) ≤ α |s|q−2 + ψ (x), (24) ∂s 3 ∂s 4 3 1 N 2 N 2 where q > 2, αi > 0, ψ1 ∈ L ∩ L , ψ2 ∈ L ∩ L , ψ3 ∈ L and np − 2o n p − 2o N = := min n ∈ : n ≥ . q − 2 Z q − 2 Hypothesis H. The density function h ∈ W 1,p ∩ W 1,Np ∩ L1 ∩ L∞. 2 2 m N N Hypothesis G. The force g ∈ Lloc(R; L (R )) ∩ Lloc(R; L ) satisﬁes: Z 0 λs 2 N G(τ) := e (kg(s + τ)k + kg(s + τ)kN )ds < +∞, ∀τ ∈ R. (25) −∞ We will frequently use a local estimate of G(·): 2λ sup G(τ + s) ≤ e G(τ + 1) < +∞, ∀τ ∈ R. (26) |s|≤1

We consider a measurable dynamical system (Ω, F, P, θs), where Ω = {ω ∈ C(R, R): ω(0) = 0}, F = B(Ω) is the Borel algebra on Ω equipping with the Frech´etmetric, P is the two-sided Wiener measure and θsω(·) = ω(s + ·) − ω(s). Note that {θs : s ∈ R} is a group of measure-preserving transformations on Ω. As usual, we consider a special Wiener process W (s, ω) = ω(s). The pathwise continuous solution of dz + λz = dW is denoted by z(θsω). It is easy to prove that 2708 SHULIN WANG AND YANGRONG LI there is a θ-invariant full-measure set (still denoted by Ω) such that ω(s)/s → 0 and z(θsω)/s → 0 as s → ±∞ for all ω ∈ Ω. Moreover, by [3, (3.8)] (where [1, Proposition 4.3.3] was imposed), we have M λ |s| Z(θsω) : = |z(θsω)| + |z(θsω)| ≤ e 2 ρ(ω) for all s ∈ R, ω ∈ Ω, (27) where M = max{Nq, Np} and ρ(·) is a tempered random variable. By setting

v(s, τ, ω) = u(s, τ, ω) − hz(θsω), we can translate the equation (22) into a random equation: dv + λv + A(v + hz(θ ω)) + f(x, v + hz(θ ω)) = g(s), s ≥ τ, (28) ds s s where the Laplace operator A : W 1,p → W −1,pˆ is deﬁned by Z p−2 1,p hAu1, u2i = |∇u1| ∇u1 · ∇u2dx, u1, u2 ∈ W . m R 3.2. Basic estimate of the solution. We will frequently use a priori estimate.

2 Lemma 3.1. For each s ≥ τ, ω ∈ Ω and vτ ∈ L , we have 2 kv(s, τ, ω, vτ )k ≤ cΥ(s, τ, ω, vτ ), (29) Z s λ(ς−s) p q e (k∇u(ς, τ, ω, vτ )kp + kv(ς)kq)dς ≤ cΥ(s, τ, ω, vτ ), (30) τ where Z s λ(τ−s) 2 λ(ς−s) 2 Υ(s, τ, ω, vτ ) := e kvτ k + e (1 + kg(ς)k + Z(θς ω))dς. τ

Proof. By the inner product of (28) with v = v(s, τ, ω, vτ ), we obtain 1 d kvk2 + λkvk2 + hAu, vi + (f(x, u), v) = (g(s), v). (31) 2 ds By (23) and the inequality |a|q ≥ 2−q|b|q − |a − b|q, we have

f(x, u)v = f(x, u)u − f(x, u)hz(θsω) q q−1 ≥ α1|u| − |ψ1| − c|hz(θsω)||u| − |ψ2hz(θsω)| α ≥ 1 |v|q − |ψ | − c|hz(θ ω)|q − |ψ hz(θ ω)|. (32) 2q 1 s 2 s We integrate (32) over Rm to obtain α (f(x, u), v) ≥ 1 kvkq − c(1 + Z(θ ω)), (33) 2q q s where Z is given in (27). By the Young inequality, α |(g(s), v)| ≤ 1 kvkq + ckg(s)k2. 2q+1 q

Since ∇u = ∇v + z(θsω)∇h, we have Z hAu, vi = |∇u|p−2∇u · ∇vdx m ZR Z p p−2 = |∇u| dx − |∇u| ∇u · ∇hz(θrω)dx m m RZ R 1 p ≥ |∇u| dx − cZ(θsω). (34) 2 m R PROBABILISTIC CONTINUITY OF A RANDOM ATTRACTOR 2709

q We substitute (33)-(34) into (31) to obtain (for c1 = α1/2 ) d kvk2 + λkvk2 + k∇ukp + c kvkq ≤ c(1 + kg(s)k2 + Z(θ ω)). ds p 1 q s Therefore, both (29) and (30) follow from the Gronwall inequality over [τ, s].

By a standard method, Lemma 3.1 implies the well-posedness (cf. [25, 26]).

2 m Lemma 3.2. For each τ ∈ R, ω ∈ Ω and vτ ∈ X0 := L (R ), Eq.(28) has a unique weak solution v(t, τ, ω; vτ ) such that v(τ, τ, ω; vτ ) = vτ and

p 1,p q q v ∈ C([τ, +∞); X0) ∩ Lloc(τ, +∞; W ) ∩ Lloc(τ, +∞; L ).

Furthermore, for each ﬁxed ω ∈ Ω, the solution v(s, τ, ω; vτ ) is continuous in s, τ, vτ ∈ X0, where s ≥ τ.

+ By Lemma 3.2, we obtain a cocycle Φ : R × R × Ω × X0 → X0 deﬁned by

Φ(t, τ, ω)vτ = v(τ + t, τ, θ−τ ω; vτ ), ∀t ≥ 0, τ ∈ R, (35) where the F-measurability of Φ can be proved by the same method as given in [13].

3.3. Triple-continuity of the cocycle. We need to show Φ is triple-continuous.

Proposition 1. Let sn → s0, τn → 0 and vσ,n → vσ as n → ∞. Then, for any σ < s0 and ω ∈ Ω, we have

2 v(sn, σ, θτn ω, vσ,n) → v(s0, σ, ω, vσ) in L . Proof. By the continuity as given in Lemma 3.2, it suﬃces to prove that

kv(sn, σ, θτn ω, vσ,n) − v(sn, σ, ω, vσ,n)k → 0. (36)

1 0 We assume that σ < sn ≤ s0 +1, and deﬁne Vn(s) := vn(s)−vn(s) for s ∈ [σ, s0 +1], 1 0 where vn(s) := v(s, σ, θτn ω, vσ,n) and vn(s) := v(s, σ, ω, vσ,n). By the diﬀerence of Eq.(28), we have dV n + λV + A(v1 + hz(θ ω)) − A(v0 + hz(θ ω)) ds n n s+τn n s 1 0 + f(x, vn + hz(θs+τn ω)) − f(x, vn + hz(θsω)) = 0. (37)

By multiplying (37) with Vn, we obtain 1 d kV k2 + λkV k2 + I + I = 0, where (38) 2 ds n n A f 1 0 IA := hA(vn + hz(θs+τn ω)) − A(vn + hz(θsω)),Vni, 1 0 If := hf(x, vn + hz(θs+τn ω)) − f(x, vn + hz(θsω)),Vni.

Since s → z(θsω) is continuous, we have the uniform continuity:

Zn := sup |z(θs+τn ω) − z(θsω)| → 0. s∈[σ,s0+1] 2710 SHULIN WANG AND YANGRONG LI

By (24) and the boundedness of Zn, it follows from the mean value theorem that ∂f I =h (x, r)(V + (z(θ ω) − z(θ ω))h),V i f ∂r n s+τn s n ∂f ≥ − α kV k2 − Z |hh (x, r),V i| 3 n n ∂r n Z 2 1 q−2 0 q−2 q−2 ≥ − α3kVnk − cZn |h|(|vn| + |vn| + |h| + |ψ3|)|Vn|dx m ZR 2 ≥ − α3kVnk − cZn |ψ3||Vn|dx m Z R 1 q−2 0 q−2 q−2 1 0 − cZn |h|(|vn| + |vn| + |h| )(|vn| + |vn|)dx m R 2 2 2 q 1 q 0 q ≥ − 2α3kVnk − cZnkψ3k − cZn(khkq + kvnkq + kvnkq) 2 1 q 0 q ≥ − 2α3kVnk − cZn(kvnkq + kvnkq + 1), (39) where we have used h ∈ L∞ ∩ L1 (so h ∈ Lq) and the triple Young inequality with q exponents (q, q−2 , q). We split IA as IA = IA1 − IA2, then the monotonicity of A implies 1 0 p IA1 := hA(vn + hz(θs+τn ω)) − A(vn + hz(θs+τn ω)),Vni ≥ γk∇Vnkp. By the mean value theorem again, 0 0 IA2 :=hA(vn + hz(θsω)) − A(vn + hz(θs+τn ω)),Vni Z 0 p−2 0 = |∇(vn + hz(θsω))| ∇(vn + hz(θsω)) m R 0 p−2 0 − |∇(vn + hz(θs+τn ω))| ∇(vn + hz(θs+τn ω)) · ∇Vndx Z 0 p−2 p−1 ≤cZn (|∇vn| |∇h||∇Vn| + |∇h| |∇Vn|).

p−2 1 1 Since p + p + p = 1, the triple Young inequality gives Z Z p−2 2 0 p−2 0 p−2 p p cZn (|∇vn| |∇h||∇Vn| = c |∇Vn|(|∇vn| Zn )(|∇h|Zn ) γ γ ≤ k∇V kp + cZ k∇v0 kp + cZ2k∇hkp ≤ k∇V kp + cZ (k∇v0 kp + 1), 2 n p n n p n p 2 n p n n p where we use h ∈ W 1,p in Hypothesis H. By the Young inequality and the boundedness of Zn, Z γ p γ cZ |∇h|p−1|∇V | ≤ k∇V kp + cZ p−1 ≤ k∇V kp + cZ . n n 2 n p n 2 n p n By all estimates mentioned above, we have 0 p IA = IA1 − IA2 ≥ −cZn(k∇vnkp + 1). (40) We substitute (39)-(40) into (38) to obtain d kV k2 ≤ ckV k2 + cZ (kv1 kq + kv0 kq + k∇v0 kp + 1). ds n n n n q n q n p

Note that Vn(σ) = 0. The Gronwall inequality over [σ, sn] gives

Z s0+1 2 1 q 0 q 0 p kVn(sn)k ≤ cZn (kvn(s)kq + kvn(s)kq + k∇vn(s)kp + 1)ds. σ PROBABILISTIC CONTINUITY OF A RANDOM ATTRACTOR 2711

By the basic estimate (30) at the sample θτn ω, we have

Z s0+1 Z s0+1 1 q 2 kvn(s, σ, θτn ω, vσ,n)kqds ≤ ckvσ,nk + c Z(θs+τn ω)ds + c, σ σ which is bounded, since kvσ,nk and Z(θs+τn ω) are bounded. By (30) again,

Z s0+1 Z s0+1 0 q 0 p 2 (kvn(s, σ, ω, vσ,n)kq + k∇vn(s)kp)ds ≤ ckvσ,nk + c Z(θsω)ds + c, σ σ which is still bounded in n. Therefore, kVn(sn)k → 0 and thus (36) holds true.

3.4. Attracted universes. We use D to denote the usual universe of all tempered 2 m sets in X0 = L (R ), where D = {D(τ, ω): τ ∈ R, ω ∈ Ω} is called tempered if −λt 2 lim e kD(τ − t, θ−tω)k = 0, ∀τ ∈ , ω ∈ Ω. (41) t→+∞ R Unfortunately, the universe D is not local-sample closed (Def.2.6). So, we introduce a sub-universe D0 ⊂ D such that D0 ∈ D0 if and only if

−λt 2 lim e sup kD0(τ + s − t, θ−tω)k = 0, ∀τ ∈ R, ω ∈ Ω. (42) t→+∞ |s|≤1

0 It is easy to show that D0 is local-time closed ([41]), which means DLT ∈ D0 as 0 long as D0 ∈ D0, where DLT is the local-time union given by

0 DLT (τ, ω) := ∪|s|≤1D0(τ + s, ω).

Moreover, D0 is local-sample closed as proved below.

Lemma 3.3. The universe D0 is local-sample closed.

0 0 Proof. Let D0 ∈ D0. Then its local-time union DLT ∈ D0. Let DLS be the local- sample union as given in (15). Let tˆ= t−σ with |σ| ≤ 1. If t → +∞, then tˆ→ +∞ uniformly in |σ| ≤ 1. Hence, as t → +∞,

−λt 0 2 −λt 2 e sup kDLS(τ + s − t, θ−tω)k = e sup sup kD0(τ + s − t, θσ−tω)k |s|≤1 |s|≤1 |σ|≤1 λ −λtˆ ˆ 2 ≤e e sup sup kD0(τ + s − σ − t, θ−tˆω)k |s|≤1 |−σ|≤1 λ −λtˆ 0 ˆ 2 =e e sup kDLT (τ + s − t, θ−tˆω)k → 0, |s|≤1

0 in view of DLT ∈ D0. Therefore, D0 is local-sample closed.

In order to discuss the weakly dissipative case (q < p), we also introduce a 2 q sequence of universes Dk in Xk = L ∩ L k such that Dk ∈ Dk if and only if

lim e−λt sup (kD (τ + s − t, θ ω)k2 + kD (τ + s − t, θ ω)kqk ) = 0 (43) k −t k −t qk t→+∞ |s|≤1 for all τ ∈ R and ω ∈ Ω, where qk = k(q − 2) + 2. We have DN−1 ⊂ · · · ⊂ D0 ⊂ D, which follows from the following interpolation:

r2 r1 r3 a ≤ a + c()a , for a > 0, 0 < r1 < r2 ≤ r3. (44) 2712 SHULIN WANG AND YANGRONG LI

4. Inductive estimates and triple-compactness for localization.

Proposition 2. For D0 ∈ D0, τ ∈ R and ω ∈ Ω, there is T = T (D0, τ, ω) ≥ 1 such that for all t ≥ T and vτ−t ∈ D0(τ − t, θ−tω),

2 sup sup kv(s, τ − t, θσ−τ ω; vτ−t)k ≤ c0R(τ, ω), (45) s∈[τ−1,τ] |σ|≤1 Z τ p q sup (k∇u(ς, τ − t, θσ−τ ω; vτ−t)kp + kv(ς)kq)dς ≤ c0R(τ, ω), (46) |σ|≤1 τ−t where R(τ, ω) = 1+G(τ)+ρ(ω) with G(τ), ρ(ω) as given by (25), (27) respectively.

Proof. We take the sample θσ−τ ω in (29) of Lemma 3.1 to obtain Z s 2 λ(ς−s) 2 sup sup kv(s, τ − t, θσ−τ ω; vτ−t)k ≤ c sup e (1 + kg(ς)k )dς s∈[τ−1,τ] |σ|≤1 s∈[τ−1,τ] τ−t Z s λ(ς−s) λ(τ−t−s) 2 + c sup sup e Z(θς+σ−τ ω)dς + sup e kvτ−tk s∈[τ−1,τ] |σ|≤1 τ−t s∈[τ−1,τ]

=: I1 + I2 + I3.

By Hypothesis G, I1 ≤ c(1 + G(τ)) < ∞. By (27), Z 0 Z 0 λ λς λς λ (|ς|+|σ|) I2 ≤ c sup e e Z(θς+σω)dς ≤ cρ(ω) sup e e 2 dς ≤ cρ(ω). |σ|≤1 −t |σ|≤1 −∞

λ −λt 2 While, I3 ≤ e e kD0(τ − t, θ−tω)k → 0 as t → +∞. Hence, (45) holds true. Similarly, (46) follows from (30) at the sample θσ−τ ω.

Corollary 1. The cocycle Φ has a D0-pullback, local-sample absorbing set K0 ∈ D0, given by

2 K0(τ, ω) := {u ∈ X0 : kuk ≤ c0R(τ, ω)}. (47) Proof. By taking s = τ in (45), we obtain the local-sample absorption:

2 sup kv(τ, τ − t, θσ−τ ω; vτ−t)k ≤ c0R(τ, ω), ∀t ≥ T. (48) |σ|≤1

Next, we prove K0 ∈ D0. Indeed, by (26),

−λt −λt −λt −λt e sup R(τ + s − t, θ−tω) = e + e sup G(τ + s − t) + e ρ(θ−tω) |s|≤1 |s|≤1 −λt −λt 2λ −λt ≤ e + e e G(τ + 1 − t) + e ρ(θ−tω) → 0 (49) as t → +∞, in view of the fact that G(τ + 1) and ρ(ω) are tempered.

q We need to show the local-sample absorption in L k , where qk = k(q − 2) + 2, k = 0, ··· ,N − 1. This is unnecessary in the strongly dissipative case (N = 1).

Proposition 3. For each k ∈ {0, 1, ··· ,N − 1}, D0 ∈ D0, τ ∈ R, ω ∈ Ω, there exists Tk = Tk(D0, τ, ω) such that for all t ≥ Tk and vτ−t ∈ D0(τ − t, θ−tω),

sup kv(τ, τ − t, θ ω; v )kqk ≤ c R(τ, ω). (50) σ−τ τ−t qk k |σ|≤1 PROBABILISTIC CONTINUITY OF A RANDOM ATTRACTOR 2713

Proof. By using an induction method, we prove the following stronger conclusions: sup sup kv(s, τ − t, θ ω; v )kqk ≤ c R(τ, ω), (51) σ−τ τ−t qk k 1 |σ|≤1 s∈[τ− k+1 ,τ] Z τ sup kv(ς, τ − t, θ ω; v )kqk+1 dς ≤ c R(τ, ω) (52) σ−τ τ−t qk+1 k 1 |σ|≤1 τ− k+1 for all t ≥ Tk and vτ−t ∈ D0(τ − t, θ−tω). By Proposition2, the case of k = 0 is true. We assume that (51) and (52) hold true for k − 1 with 1 ≤ k ≤ N − 1. By taking the inner product of Eq.(28) with |v|qk−2v, where v(s) = v(s, τ − t, θσ−τ ω; vτ−t), we have 1 d kvkqk + λkvkqk + hAu, |v|qk−2vi + (f(x, u), |v|qk−2v) = (g(s), |v|qk−2v). (53) qk qk qk ds q −2 By the production of (32) with |v| k and by q + qk − 2 = qk+1, we have α1 f(x, u)v|v|qk−2 ≥ |v|qk+1 − c(|ψ | + |hz(θ ω)|q + |ψ hz(θ ω)|)|v|qk−2. 2q 1 s+σ−τ 2 s+σ−τ

qk−2 2k Let µk = 1/(1 − ) = k + 1 − , then, the Young inequality gives qk+1 q α f(x, u)v|v|qk−2 ≥ |v|qk+1 2q+1 µk qµk µk − c(|ψ1| + |hz(θs+σ−τ ω)| + |ψ2hz(θs+σ−τ ω)| ). µ N Obviously, 1 < µk ≤ N for 1 ≤ k ≤ N − 1. By (44), |ψ1| k ≤ c(|ψ1| + |ψ1| ). Since 2 < qµk ≤ qN, the interpolation (44) implies

µkN µk N N−µ |ψ2hz(θs+σ−τ ω)| ≤ |ψ2| + c|hz(θs+σ−τ ω)| k N qN ≤ |ψ2| + c(|hz(θs+σ−τ ω)| + |hz(θs+σ−τ ω)| ). By integrating over Rm and using (27), we have α qk−2 1 qk+1 (f(x, u), v|v| ) ≥ kvk − c(1 + Z(θs+σ−τ ω)). (54) 2q+1 qk+1 q −2 q −2 Since ∇(|v| k v) = (qk − 1)|v| k ∇v, we can estimate the Laplace term: Z qk−2 p−2 qk−2 hAu, |v| vi = (qk − 1) |∇u| ∇u · |v| ∇vdx m R Z qk−2 p p−2 =(qk − 1) |v| |∇u| − |∇u| ∇u · ∇hz(θs+σ−τ ω) dx m Z R qk−2 p ≥ − c |v| |∇hz(θs+σ−τ ω)| dx. m R qk+1 Since and µk are conjugate exponents, and p < pµk ≤ pN, we have qk−2 Z α1 hAu, |v|qk−2vi ≥ − kvkqk+1 − c |∇hz(θ ω)|pµk dx q+3 qk+1 s+σ−τ 2 m R α 1 qk+1 p pN ≥ − kvk − c(k∇hk + k∇hk )Z(θs+σ−τ ω) 2q+3 qk+1 p pN α 1 qk+1 ≥ − kvk − cZ(θs+σ−τ ω). (55) 2q+3 qk+1 2714 SHULIN WANG AND YANGRONG LI

qk−1 k−1 Let νk := 1/(1 − ) = k + 1 − and so 2 ≤ νk ≤ N. The interpolation (44) qk+1 q−1 gives α1 |(g(s), |v|qk−2v)| ≤ kvkqk+1 + ckg(s)kνk 2q+3 qk+1 νk α1 ≤ kvkqk+1 + c(kg(s)k2 + kg(s)kN ). (56) 2q+3 qk+1 N α1qk Substituting (54)-(56) into (53), we obtain, forc ˆ = 2q+2 , d qk qk qk+1 2 N kvk + λkvk +c ˆkvk ≤ c(1 + kg(s)k + kg(s)k + Z(θs+σ−τ ω)). (57) ds qk qk qk+1 N 1 1 1 Integrating (57) froms ˆ ∈ [τ − k , τ − k+1 ] to s ∈ [τ − k+1 , τ], we obtain that for |σ| ≤ 1, Z τ kv(s)kqk ≤ kv(ˆs)kqk + c (1 + kg(ς)k2 + kg(ς)kN + Z(θ ω))dς. qk qk N ς+σ−τ τ−1 1 1 Integrating it ins ˆ over [τ − k , τ − k+1 ], we obtain that for all t ≥ 1, sup sup kv(s, τ − t, θ ω; v )kqk σ−τ τ−t qk 1 |σ|≤1 s∈[τ− k+1 ,τ] Z τ ≤ sup kv(ς, τ − t, θ ω; v )kqk dς σ−τ τ−t qk 1 |σ|≤1 τ− k Z τ + c(1 + G(τ)) + c sup Z(θς+σ−τ ω)dς. |σ|≤1 τ−1 By (27), the last term is bounded by Z τ λ (ς−τ) λ |σ| cρ(ω) e 2 dς sup e 2 ≤ cρ(ω). τ−1 |σ|≤1 Therefore, by the inductive hypothesis (52) for k − 1, we have proved (51) for k. 1 On the other hand, we integrate (57) over [τ − k+1 , τ] to obtain, for all t ≥ Tk and vτ−t ∈ D0(τ − t, θ−tω), Z τ sup kv(s, τ − t, θ ω; v )kqk+1 ds σ−τ τ−t qk+1 1 |σ|≤1 τ− k+1 1 ≤c sup kv(τ − , τ − t, θ ω)kqk + c(1 + G(τ)) σ−τ qk |σ|≤1 k + 1 Z τ + c sup Z(θς+σ−τ ω)dς ≤ cR(τ, ω), |σ|≤1 τ−1 which completes the inductive proof.

We restrict the initial data in L2 ∩ LqN−1 to obtain a useful result for the tail- estimate later.

Corollary 2. Given DN−1 ∈ DN−1, for each τ ∈ R and ω ∈ Ω, there exists T := T (DN−1, τ, ω) such that Z τ λ(ς−τ) p sup e kv(ς, τ − t, θσ−τ ω, vτ−t)kpdς ≤ cpR(τ, ω) (58) |σ|≤1 τ−t for all t ≥ T and vτ−t ∈ DN−1(τ − t, θ−tω). PROBABILISTIC CONTINUITY OF A RANDOM ATTRACTOR 2715

Proof. We rewrite (57) for k = N − 1 as follows. d qN−1 qN−1 qN 2 N kvk + λkvk +c ˆkvk ≤ c(1 + kg(s)k + kg(s)k + Z(θs+σ−τ ω)). ds qN−1 qN−1 qN N

By the Gronwall lemma and by DN−1 ∈ DN−1, we have, as t → +∞, Z τ sup eλ(ς−τ)kv(ς, τ − t, θ ω, v )kqN dς σ−τ τ−t qN |σ|≤1 τ−t Z τ ≤ ce−λtkv kqN−1 + c(1 + G(τ)) + c sup eλ(ς−τ)Z(θ ω)dς τ−t qN−1 ς+σ−τ |σ|≤1 τ−t ≤ ce−λtkD (τ − t, θ ω)kqN−1 + c(1 + G(τ) + ρ(ω)) ≤ cR(τ, ω). (59) N−1 −t qN−1 p Note that qN = N(q − 2) + 2 ≥ p > 2. The interpolation (44) gives kvkp ≤ ckvk2 + ckvkqN . Therefore, we obtain (58) from (59) and (45) immediately. qN Next, we prove a triple-compactness result on the time-space-sample localization. m p−2 We denote by Or = {x ∈ R ; |x| < r}, r ∈ N and recall that N = { q−2 } ≥ 1.

Proposition 4. (Triple-compactness) Let {v0,n}n be a bounded sequence in X0 and 1 ∗ let σn → σ0, τ ∈ R, ω ∈ Ω. Then, there are s0 ∈ (τ − N , τ), a subsequence {n } of ∗ {n} and v0 ∈ X0 such that, as n → ∞, 2 ∗ v(s0, τ − 1, θσn∗ ω; v0,n ) → v0 strongly in L (Or) for each r ∈ N.

Proof. We assume without lose of generality that |σn − σ0| < 1 and let

vn := vn(s) = v(s, τ − 1, θσn ω; v0,n), ∀s ∈ [τ − 1, τ].

By Lemma 3.1 and using boundedness of {v0,n}, we have Z τ 2 p sup kvn(s)k + k∇vn(ς)kpdς s∈[τ−1,τ] τ−1 Z τ λ 2 ≤ce kv0,nk + c(1 + G(τ)) + c Z(θς+σn ω)dς ≤ c. (60) τ−1 By the same induction method as given in (52) for k = N − 1, we have Z τ Z τ kv (ς)kqN dς ≤ c + c Z(θ ω)dς ≤ c. (61) n qN ς+σn 1 τ− N τ−1

Since qN ≥ p > 2, by the interpolation (44), it follows from (60) and (61) that Z τ Z τ kv (ς)kpdς ≤ (kv (ς)k2 + kv (ς)kqN ))dς ≤ c. (62) n p n n qN 1 1 τ− N τ− N p p p Noting that kvkW 1,p = kvkp + k∇vkp, we see from (60) and (62) that 1 1 {v } bounded in L∞(τ − , τ; X ) ∩ Lp(τ − , τ; W 1,p( m)). (63) n N 0 N R On the other hand, it is easy from (63) to prove that for each r ∈ N, 1 {A(v + hz(θ ω)} bounded in Lpˆ(τ − , τ; W −1,pˆ(O )) n ·+σn N r 1 {f(x, v + hz(θ ω)} bounded in Lqˆ(τ − , τ; Lqˆ(O )), n ·+σn N r 2716 SHULIN WANG AND YANGRONG LI wherep ˆ,q ˆ are conjugate exponents of p, q. Given µ = min{p,ˆ qˆ}, by Eq.(28), we obtain that for each r ∈ N, dv 1 { n } bounded in Lµ(τ − , τ; W −1,pˆ(O ) + Lqˆ(O )). (64) ds N r r 2 1 By (63), there isv ˜ ∈ L (τ − N , τ; X0) such that, passing to a subsequence, 1 v * v˜ weakly in L2(τ − , τ; L2( m)). n N R Let r ∈ N be ﬁxed. By applying the Aubin’s compactness theorem [38] on three 1,p 2 −1,pˆ qˆ spaces Y0 = W (Or), Y = L (Or) and Y1 = W (Or) + L (Or), we see from r r−1 (63) and (64) that there is a subsequence {vn} of {vn } such that 1 vr → v˜ strongly in L2(τ − , τ; L2(O )). n N r n We still use {vn} to denote the diagonal subsequence {vn}, then 1 v → v˜ strongly in L2(τ − , τ; L2(O )), ∀r ∈ . (65) n N r N 1 For each r ∈ N, by (65), we can subsequently choose a set Ir ⊂ [τ − N , τ] of measure 1 r r−1 N and a subsequence {vn} of {vn } such that r r 2 vn(s) = v(s, τ − 1, θσn ω; v0,n) → v˜(s) strongly in L (Or) for all s ∈ Ir.

n∗ ∗ Therefore, the diagonal subsequence {vn∗ } = {vn∗ } satisﬁes: as n → ∞, 2 ∗ ∗ vn (s) = v(s, τ − 1, θσn∗ ω; v0,n ) → v˜(s) strongly in L (Or) ∞ ∞ for each r ∈ N and s ∈ ∩r=1Ir. Since ∩r=1Ir has a positive measure, there is at ∞ least one s0 ∈ ∩r=1Ir, which ﬁnishes the proof.

5. Local-sample uniform estimate of the tail. We show that the tail of the solution is uniformly small in local samples when the initial data belong to XN−1.

Proposition 5. If DN−1 ∈ DN−1, τ ∈ R and ω ∈ Ω are fixed, then, for each ε > 0, there exist T = T (ε) and r0 = r0(ε) such that Z 2 sup sup |v(s, τ − t, θσ−τ ω; vτ−t)| dx ≤ ε, (66) s∈[τ−1,τ] |σ|≤1 |x|≥r0 whenever t ≥ T and vτ−t ∈ DN−1(τ − t, θ−tω). Proof. We consider a cut-off function defined by |x|2 ξ (x) = ξ( ), x ∈ m, r > 0, r r2 R where ξ : R+ → [0, 1] is a smooth function such that ξ ≡ 0 on [0, 1] and ξ ≡ 1 on [4, +∞). It is easy to verify that kξrk∞ ≤ 1 and k∇ξrk∞ ≤ C/r for all r > 0. We then multiply Eq.(28) with ξrv to find Z Z 1 d 2 2 ξr|v| dx + λ ξr|v| dx + hAu, ξrvi + (f(x, u), ξrv) = (g(s), ξrv). (67) 2 ds m m R R PROBABILISTIC CONTINUITY OF A RANDOM ATTRACTOR 2717

By (27), we have Z Z p−2 p−2 hAu, ξrvi = v|∇u| ∇u · ∇ξrdx + ξr|∇u| ∇u · ∇vdx m m R Z ZR c p p p p−2 ≥ − (kvkp + k∇ukp) + ξr|∇u| dx − ξr|∇u| ∇u · ∇hzdx r m m R Z R c p p p ≥ − (kvkp + k∇ukp) − cZ(θs+σ−τ ω) |∇h| dx. (68) r |x|≥r

For the nonlinearity, since ξr ≥ 0, by (32), we have Z q (f(x, u), ξrv) ≥ −c ξr(|ψ1| + |hz| + |ψ2hz|)dx m ZR 2 2 q ≥ −c (|ψ1| + ψ3 + Z(θs+σ−τ ω)(h + |h| ))dx. |x|≥r The Young inequality implies Z Z λ 2 2 |(g(s), ξrv)| ≤ ξr|v| dx + c g (s, x)dx. (69) 4 m R |x|≥r We substitute (68)-(69) into (67) to obtain Z Z d 2 3λ 2 ξr|v| dx + ξr|v| dx ds m 2 m R Z R Z c p p 2 2 ≤ (kvkp + k∇ukp) + (|ψ1| + ψ3) + c g (s) r |x|≥r |x|≥r Z 2 q p + cZ(θs+σ−τ ω) (h + |h| + |∇h| ). |x|≥r

Given ε > 0, there is a r1 > 0 such that for all r ≥ r1, Z Z Z d 2 2 λ 2 ξr|v| dx + λ ξr|v| dx + ξr|v| dx ds m m 2 m R R R Z p p 2 ≤εkvkp + εk∇ukp + ε(1 + Z(θs+σ−τ ω)) + c g (s, x)dx. (70) |x|≥r

Applying the Gronwall lemma on (70) over [τ − t, τ], we have, for all r ≥ r1, Z τ Z λ(s−τ) 2 sup e ξr|v(s, τ − t, θσ−τ ω; vτ−t)| dxds m |σ|≤1 τ−t R Z −λt 2 ≤ce ξr|vτ−t| dx + ε(J1 + J2 + J3) + J4(r), (71) m R where J1,J2,J3,J4(r) are deﬁned and estimated as follows. Since DN−1 ∈ DN−1, by Corollary2, there is a T > 0 such that for all t ≥ T and vτ−t ∈ DN−1(τ − t, θ−tω), Z τ λ(s−τ) p J1 := c sup e kv(s, τ − t, θσ−τ ω; vτ−t)kpds ≤ cR(τ, ω). (72) |σ|≤1 τ−t By (46) in Proposition2, Z τ λ(s−τ) p J2 := c sup e k∇u(s, τ − t, θσ−τ ω; vτ−t)kpds ≤ cR(τ, ω). (73) |σ|≤1 τ−t 2718 SHULIN WANG AND YANGRONG LI

By (27), we have, for all t > 0, Z τ λ(s−τ) J3 := c sup e (1 + Z(θs+σ−τ ω))ds ≤ c(1 + ρ(ω)). (74) |σ|≤1 τ−t The Lebesgue controlled convergence theorem implies that as r → +∞, Z τ Z λ(s−τ) 2 J4(r) := c e g (s, x)dxds → 0. (75) τ−t |x|≥r

Theorem 6.1. Let Φ be the cocycle generated from the p-Laplace equation, D0 is the local-uniform tempered universe and D is the tempered universe. (i) Φ has a unique D0-pullback attractor A0 ∈ D0 such that A0 is local-sample 2 m compact in X0 = L (R ). (ii) Φ has a unique D-pullback random attractor A ∈ D such that A = A0. (iii) A is upper semi-continuous in sample:

lim dist(A(τ, θsω), A(τ, θs0 ω)) = 0, ∀τ ∈ R, ω ∈ Ω. (79) s→s0 (iv) A is probabilistically continuous in sample, i.e. (7) holds true. PROBABILISTIC CONTINUITY OF A RANDOM ATTRACTOR 2719

(v) A is probabilistically lower semi-continuous in time:

lim P {ω ∈ Ω: dist(A(τ0, ω), A(τ, ω)) ≥ δ} = 0, ∀δ > 0, τ0 ∈ R. (80) τ→τ0 Proof. We mainly prove (i) in three steps.

Step 1. The cocycle Φ is local-sample asymptotically compact on DN−1. More precisely, the sequence

{Φ(tn, τ − tn, θσn−tn ω)vτ−tn }n = {v(τ, τ − tn, θσn−τ ω; vτ−tn )}n (81) is pre-compact in X0, whenever

|σn| ≤ 1, tn → +∞, vτ−tn ∈ DN−1(τ − tn, θ−tn ω) with DN−1 ∈ DN−1.

Indeed, passing to a subsequence, we can assume that σn → σ0. Since DN−1 ∈ DN−1 ⊂ D0, it follows from Proposition2 that

{v0,n}n := {v(τ − 1, τ − tn, θσn−τ ω; vτ−tn )}n bounded in X0. 1 By the triple-compactness in Proposition4, there are s0 ∈ (τ − N , τ) and v0 ∈ X0 such that, passing to a subsequence,

v(s0, τ − tn, θσn−τ ω; vτ−tn ) = v(s0, τ − 1, θσn−τ ω; v0,n) 2 → v0 strongly in L (Or) for each r ∈ N. (82)

Given ε > 0, by Proposition5, there are r1 ∈ N and n1 ∈ N such that Z 2 |v(s0, τ − tn, θσn−τ ω; vτ−tn )| dx < ε, ∀n ≥ n1. (83) |x|≥r1

Since v0 ∈ X0, one can choose r2 ∈ N with r2 ≥ r1 such that Z 2 |v0(x)| dx < ε. (84) |x|≥r2 2 By (82), v(s0, τ −tn, θσn−τ ω; vτ−tn ) → v0 in L (Or2 ). We choose n2 ≥ n1 such that Z 2 |v(s0, τ − tn, θσn−τ ω; vτ−tn ) − v0| dx < ε, ∀n ≥ n2. (85) |x|

Hence, by (83)-(85), for all n ≥ n2, 2 kv(s , τ − t , θ ω; v ) − v k 2 m 0 n σn−τ τ−tn 0 L (R ) Z 2 ≤ |v(s0, τ − tn, θσn−τ ω; vτ−tn ) − v0| dx |x|

In a conclusion, v(s0, τ − tn, θσn−τ ω; vτ−tn ) → v0 strongly in X0. By the triple-continuity as proved in Proposition1, we obtain

v(τ, τ − tn, θσn−τ ω; vτ−tn )

= v(τ, s0, θσn−τ ω; v(s0, τ − tn, θσn−τ ω; vτ−tn ))

→ v(τ, s0, θσ0−τ ω; v0), strongly in X0.

Step 2. The cocycle Φ is local-sample asymptotically compact on D0. It suﬃces to prove {v(τ, τ − tn, θσn−τn ω; vτ−tn )}n is pre-compact, whenever |σn| ≤ 1, tn → +∞ and vτ−tn ∈ D0(τ − tn, θ−tn ω) with D0 ∈ D0. 2720 SHULIN WANG AND YANGRONG LI

qN−1 For this end, we deﬁne a bi-parametric set KN−1 in XN−1 = X0 ∩ L by \ K (τ, ω) = K (τ, ω) {u ∈ LqN−1 : kukqN−1 ≤ c R(τ, ω)}, N−1 0 qN−1 N−1 where K0 is the absorbing set given in (47). By (49), we know KN−1 ∈ DN−1. By Proposition3, KN−1 locally-uniformly absorbs D0 ∈ D0 in the sense of (50). In particular, for each ﬁxed i ∈ N, there exists a large time Ti = Ti(D0, τ, ω) such that

v(τ − i, τ − i − t, θσ−(τ−i)θ−iω; vτ−i−t) ∈ KN−1(τ − i, θ−iω) (86) whenever |σ| ≤ 1, t ≥ Ti and vτ−i−t ∈ D0(τ − i − t, θ−tθ−iω).

Since tn → +∞, we can choose a subsequence {ni} of {n} such that tni − i ≥ Ti for each i ∈ . Since v ∈ D (τ − t , θ ω), we can rewrite N τ−tni 0 ni −tni

vτ−t ∈ D0((τ − i) − (tn − i), θ θ−iω). ni i −(tni −i)

Note that |σni | ≤ 1, it follows from (86) that

yi : = v(τ − i, τ − i − (tn − i), θ θ−iω; vτ−t ) i σni −(τ−i) ni ∈ KN−1(τ − i, θ−iω). (87)

By Step 1, Φ is local-sample asymptotically compact on KN−1 ∈ DN−1. Hence, {v(τ, τ − t , θ ω; v )}∞ = {v(τ, τ − i, θ ω; y )}∞ ni σni −τ τ−tni i=1 σni −τ i i=1 has a convergent subsequence.

Step 3. Φ has a unique D0-pullback attractor A0 ∈ D0 such that A0 is local-sample compact. Indeed, by Corollary1, Φ has a D0-pullback absorbing set K0 ∈ D0. By Step 2, Φ is D0-pullback asymptotically compact. By [39, Theorem 2.23], Φ has a unique D0-pullback attractor A0 = αK0 ∈ D0. On the other hand, by Lemma 3.3, the universe D0 is local-sample closed. By Step 2, Φ is local-sample asymptotically compact on D0. So, it follows from Theo- rem 2.9 that A0 is local-sample compact. (ii) By the usual method, one can show that K0 is a D-pullback random absorbing set and Φ is D-pullback asymptotically compact. So, Φ has a unique random D- pullback attractor A ∈ D. In addition, both A and A0 are the α-limit set of the absorbing set K0, that is,

A(τ, ω) = αK0 (τ, ω) = A0(τ, ω). (iii)-(v) By Proposition1, the cocycle is triple-continuous. By Lemma 3.3, the universe D0 is local-sample closed. By Step 2, Φ is D0-pullback local-sample asymptotically compact. By (ii), A = A0 ∈ D0. Therefore, the assertions (iii)-(v) follow from the abstract Theorem 2.12 immediately.

Notice that D0 is also local-time closed. Hence, by using the same method as given in [14, 41], one can show that A0 is local-time compact, that is, ∪|s|≤1A0(τ + s, ω) is pre-compact. By combining Theorem 6.1 and [41, theorem 2.2], we obtain the probabilistic continuity of the attractor A.

Corollary 3. The attractor A = A0 ∈ D0 in Theorem 6.1 has the following properties. (I) A is both local-time compact and local-sample compact. (II) A is upper semi-continuous in time-sample:

lim dist(A(τn, θs ω), A(τ0, θs ω)) = 0, whenever τn → τ0, sn → s0. n→∞ n 0 PROBABILISTIC CONTINUITY OF A RANDOM ATTRACTOR 2721

(III) A is probabilistically continuous in time-sample: for δ > 0, τn → τ0 and sn → s0,

lim P {ω ∈ Ω: disth(A(τ0, θs ω), A(τn, θs ω)) ≥ δ} = 0. (88) n→∞ 0 n Remark 1. It remains open even for the existence of an attractor in the most weakly dissipative case that q = 2 < p. In this case, qk ≡ 2, the induction method in Proposition3 loses eﬀectiveness and thus the tail-estimate in Proposition 5 cannot be established.

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