Random invariant manifolds and foliations for slow-fast PDEs with strong multiplicative noise

. This article is devoted to the dynamical behaviors of a class of slow-fast PDEs perturbed by strong multiplicative noise. We will accomplish the existence of random invariant manifolds and foliations, and show exponential tracking property of them. Moreover, the asymptotic approximation for both objects will be presented


Introduction
Various kinds of mathematical models arising from physics, engineering and biology not only involve random effects such as uncertain parameters, stochastic perturbation, but also relate to multiple disparate time or spatial scales [2,16,21].Many important physical models, such as Burger's equation, Ginzburg-Laudau equation, Swift-Hohenberg equation are highly referred in this field.In order to investigate a variety of equations in the context of random influences, by combining probability theory, functional analysis and the theory of partial differential equations, mathematicians gradually developed and perfected a systematic framework of stochastic partial differential equations (SPDEs) in recent decades [13,30].In terms of SPDEs evolving on multi-scales, there are many methods used to analyze the dynamical behaviours of SPDEs, such as averaging method [9,10], amplitude equations [4,5] and the theory of invariant manifolds [17,29].
Among these methods, the theory of invariant manifolds is considered as a practicable tool, which can provide a geometric structure of complex systems [1,35,37].For deterministic systems, the pioneering results were obtained by Hadamard [22], Lyapunov [23] and Perron [25].Duan et al. [14,15] extended this theory to random dynamic systems and show the existence of random invariant manifolds for SPDEs with simple multiplicative noise.Equations with more general multiplicative noise were studied by Caraballo et al. [7] and Mohammed et al. [24].Also other dynamical properties of SPDEs have been already addressed in the literature, just to list a few but far from being complete: random invariant foliations [26,32,34], asymptotic dynamical behaviors [19,33,34,36], geometric shape [6,11,18,20], etc.
Applying the property that the random invariant manifolds contribute to the reduction of SPDEs, mathematicians can eliminate the fast variable of slow-fast systems to reduce the original system to a lower dimensional system.At earlier stage of the research, Schmalfuß and Schneider [31] studied a class of slow-fast systems with noise in the finite dimensional case by Hadamard method, and obtained that inertial manifolds tend to slow manifolds if the scaling parameter ε tends to 0. Fu et al. [17] applied Lyapunov-Perron method to a class of stochastic evolution equations with slow and fast components, and proved that slow manifolds asymptotically approximate to critical manifolds.Qiao et al. [28,29] obtained a reduced system of a class of SDEs under slow-fast Gaussian noisy fluctuations on the random invariant manifolds, and showed the delicate error between the filter of the original system and that of the reduced system.The slow invariant foliation, another interesting object in this field, was originally studied by Chen et al. [12].They constructed random invariant foliations for a class of slow-fast stochastic evolutionary systems, and presented the approximation of slow foliations.Recently, slow-fast systems with non-Gaussian noise have gained substantial attention from researchers.For details, please see [27,38,39], etc.
In this paper, we investigate a class of slow-fast PDEs driven by strong multiplicative noise: where H 1 and H 2 are separable Hilbert spaces, ε is small parameter (0 < ε ≪ 1), W(t) is a two-sided Wiener process taking value in R, • means Stratonovich stochastic differential, and A, B, f , g will be introduced later.Briefly, the main goal of this paper is to construct the random invariant manifolds and foliations for (1.1)-(1.2) and to derive corresponding approximations for both.Compared with [17,28,29,31], the system we study is forced by multiplicative noise rather than additive noise.To the best of our knowledge, this is the first research to consider the slow manifolds and slow foliations for slow-fast SPDEs with multiplicative noise.

Notations and assumptions
Let H 1 and H 2 be separable Hilbert spaces in (1.1) and (1.2).Denote their norms by ∥ • ∥ 1 and ∥ • ∥ 2 , respectively.Set satisfy f (0, 0) = 0 and g(0, 0) = 0, and there exists a constant K > 0 such that We remark that our main theorems hold when H 1 and H 2 are real or complex separable Hilbert spaces.For simplicity, we ignore it.
(2) Assumption 3 and Assumption 4 will be imposed in Section 3 and Section 4, respectively.We would like to point out that condition (2.2) is sufficient for condition (2.1), which implies that the condition used for the study of the random invariant manifolds is weaker than that used for the study of the random invariant foliations.
(3) Moreover, we remark that there are other conditions, which can also play the same role as condition (2.2).For the details, please see Remark 4.3 and Remark 4.4 in [12].

Random dynamical systems
Referring to the literature [1,12,14,15,26], we introduce some concepts of random dynamical systems.Definition 2.2.Let (Ω, F , P) be a probability space, and a flow θ of mappings {θ t } t∈R be defined by θ : R × Ω → Ω such that Then (Ω, F , P, {θ t } t∈R ) is called a metric dynamical system.Definition 2.3.A random dynamical system on the topological space X over a metric dynamical system (Ω, F , P, {θ t } t∈R ) is a mapping In what follows, we consider φ(t, ω, •) as a random dynamical system on a complete separable metric space (H, d H ) over a metric dynamical system (Ω, F , P, {θ t } t∈R ).

Definition 2.4. A family of nonempty closed sets
Furthermore, if, for every ω ∈ Ω, we can represent M by a graph of a Lipschitz mapping (v) A foliation W βu (ω) is invariant with respect to random dynamical system φ if each fiber of it satisfies that φ(t, ω, W βu (x, ω)) ⊂ W βu (φ(t, ω, x), θ t ω).

Transformation from SPDEs to RPDEs
The motivation of this subsection is to transform SPDEs (1.1)-(1.2) into random partial differential equations (RPDEs), and show the relationship between them.For our applications, we introduce the metric dynamical system induced by Wiener process.Let W(t) be a two-sided Wiener process with trajectories in the space C 0 (R, R) which is the collection of continuous functions ω : R → R with ω(0) = 0. Set Ω := C 0 (R, R).This set is equipped with a compactopen topology (please see the Appendix in [1]).Let F be its Borel σ-field and P be the Wiener measure.Set Note that P is ergodic with respect to θ t .Then ( Ω, F , P, {θ t } t∈R ) is a metric dynamical system.In order ot obtain RPDEs, we need the following preparation.Consider the linear stochastic differential equation: 3) The solution of (2.3) is called an Ornstein-Uhlenbeck process.Following Lemma 2.1 in [14], we present the properties of z ε (t) as follows.
(1) There exists a {θ t } t∈R -invariant set Ω ∈ B(C 0 (R, R)) of full measure with sublinear growth: (2) For ω ∈ Ω the random variable exists and generates a unique stationary solution of (2.3) given by The mapping t → z ε (θ t ω) is continuous.
Since F and G are also Lipschitz functions with the same Lipschitz constant K for ω ∈ Ω, there exists a unique solution H))-measurable and generates a random dynamical system.We introduce the transform and its inverse transform for x ∈ H and ω ∈ Ω.

Random invariant manifolds and slow manifolds
In this section, we use Lyapunov-Perron's method to prove the existence of random invariant manifolds for (2.4)-(2.5),and state that any orbit can be exponentially attracted by random invariant manifolds.Moreover, we show slow manifolds can approach to random invariant manifolds as the parameter ε tends to 0.
with Zε (0) = Z0 and satisfies Proof.The proof can be completed by that of Theorem 3.1 in [15], so it is omitted here.
Step1.Claim that for ε > 0 sufficiently small, (3.1)-(3.2) will have a unique solution Zε ( . We will use Banach's Fixed Point Theorem to achieve the claim.
Define two operators J ε 1 : Consequently, we use Banach's Fixed Point Theorem to obtain the existence of the unique solution Zε (•) , and the standard a-priori estimate: Step 2. Construct the random invariant manifold M ε (ω).Define Then, the Lipschitz constant of H ε (ω, Y 0 ) is given by Step 3. We need to prove M ε (ω) is a random set.To this end, we show that is measurable, where P is the projection from H to H 2 .Let H c be a countable dense set of the separable space H.The continuity of H ε (ω, •) yields that the right-hand side of (3.8) is equivalent to Since ω → H ε (ω, P z ′ ) is measurable, we obtain that measurability of any expression under the infimum of (3.9).Then the fact that M ε is a random set follows from Theorem III.9 in [8].

Slow manifolds
In this subsection, we are going to present the approximation of where W(T) and ε − 1 2 W(εT) are identical in distribution.Replacing z ε (θ εT ω) by η(θ T ω) in (3.17)-(3.18),we have where Since z ε (θ εT ω) is the same as η(θ T ω) in distribution (please see Lemma 3.2 in [31]), the distribution of the solution (3.17 where Then, for fixed where d = denotes the equivalence in distribution.We proceed to want to explore the approximation form of the invariant manifold M ε (ω) as ε → 0. To achieve it, we observe (3.20)-(3.21)when ε = 0. Consider where Furthermore, using the idea coming from Theorem 5.1 in [17], we state that Theorem 3.4.Under Assumptions 1, 2, 4, for sufficiently small ε > 0, we have Proof.For T ≤ 0 and Y 0 ∈ D(B), * D(B) means the domain of operator B Then, where we use the estimation (3.7) in the third inequality.By (3.27), we obtain where Note that there exists T sup < 0 such that dΣ (T)  dT | T=T sup = 0, which implies that R = Σ(T sup ) = O(ε).Then we have We now show the better approximation of slow manifolds.According to Assumption 3, we know F(x, y) has the partial derivatives.Let where F x and F y are the partial derivatives of F(x, y) with respect to x and y, respectively.Equating the same degree of ε, we have Let us consider where with H 0 (Y 0 , ω) given in (3.26).
We state there exists a unique solution for (3.33)- (3.34) easy to obtain that (3.31)-(3.32)has the random invariant manifold represented as Then, we can formally show the first order approximation of H ε ( Y 0 , ω) as follow: We have, in fact, proved the following theorem.
Theorem 3.5 (First order approximation of slow manifold).Under Assumptions 1-4, for sufficiently small ε > 0, we obtain the approximation of the random invariant manifold for (2.4)-(2.5)as where the second equality holds in distribution, that means for fixed Y 0 ∈ D(B), H ε (ω, Y 0 ) and H ε (ω, Y 0 ) are identical in distribution, while the third equality holds for all ω ∈ Ω, H 0 (ω, Y 0 ) is the critical manifold as (3.26), and H 1 (ω, Y 0 ) is the first order manifold as (3.35).

Random invariant foliations and slow foliations
In the section, we are going to show there also exist random invariant foliations for RPDEs (2.4)-(2.5),and any two orbits start from the same fiber can approach to each other as exponential rate in backward time.Then, we prove that random invariant foliations converge to slow foliations as the parameter ε tends to 0.

Random invariant foliations
In the followings, we will prove ), ω is a fiber of the random invariant foliations for (2.4)-(2.5).
(2) There exists a unique solution ( where Proof.The proof of (i) follows from the variation of constants formula.With the help of Banach's Fixed Point Theorem, we can prove (2).Using the same techniques as in the proof of Theorem 3.3, we can obtain (3).
We proceed to verify that each fiber is invariant.
Then, the cocycle property leads to ( where the second inequality is from direct calculation and the last one is from Lemma 4.1. Theorem 4.3 (Exponential tracking property in backward time).Under Assumptions 1, 2, 5, for sufficiently small ε > 0, any two points ( X1 0 , Y1 0 ) and with t ≤ 0. Proof.Let )).
Applying Lemma 4.1, we know that For sufficiently small ε > 0, we have which implies (4.7).

Slow foliations
The motivation of this subsection is to investigate the approximation of the random invariant foliations for RPDEs (2.4)-(2.5) in slow time-scale T = t ε .As the arguments in Subsection 3.2, we will study the approximation of the random invariant foliations for RPDEs (2.4)-(2.5)via (3.20)- (3.21).
Let Z ε 1 (T, ω, ( X0 , Y0 )) and Z ε 2 (T, ω, ( X 0 , Y 0 )) be the solutions of (3.20)-(3.21)with initial data ( X0 , Y0 ) and ( X 0 , Y 0 ), respectively.Set According to the variation of constants formula, we state where By Banach's Fixed Point Theorem, we can prove there exists a unique solution 2 ,η of (4.8)-(4.9).Then, by the similar arguments as in Theorem 4.2, we can prove that there exists a random invariant foliation for (3.20)-(3.21), each fiber of which is represented as where Using the similar discussion as (3.22), we derive Based on (4.11), we turn to the study of the approximation of l ε (ζ, ( X 0 , Y 0 ), ω).Taking into account the critical system (3.23)-(3.24).Let Z 0 1 (T, ω, ( X0 , Y0 )) and Z 0 2 (T, ω, ( X 0 , Y 0 )) be the solutions of (3.23)- (3.24) with initial data ( X0 , Y0 ) and ( X 0 , Y 0 ), respectively.Set where Furthermore, we claim that (3.23)-(3.24)has a random invariant foliation, whose each fiber is represented as where Proof.Due to the representations of l ε and l 0 , we obtain Hence, if we can estimate the error between X ε (T) and X 0 (T), the proof will be done.For T ≤ 0, we have In order to show the bounds of ∥ and where Analogous argument as (3.28) yields Furthermore, combining the above estimates, we obtain that Our proof is completed.
Assumption 2. Suppose that nonlinear terms