Normally Hyperbolic Invariant Manifolds for Random Dynamical Systems: Part I - Persistence

TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 365, Number 11, November 2013, Pages 5933–5966 S 0002-9947(2013)05825-4 Article electronically published on July 10, 2013

NORMALLY HYPERBOLIC INVARIANT MANIFOLDS FOR RANDOM DYNAMICAL SYSTEMS: PART I - PERSISTENCE

JI LI, KENING LU, AND PETER BATES

Abstract. In this paper, we prove the persistence of smooth normally hyperbolic invariant manifolds for dynamical systems under random perturbations.

1. Introduction Invariant manifolds provide large-scale structures that serve to organize the global phase space of a dynamical system. Persistence of invariant manifolds under small perturbation, when that occurs, gives a more global robustness or structural stability, allowing one to classify the behavior of families of systems, even those with the vector field known only approximately. Persistence results are particu- larly important in geometric singular perturbation theory, which provides a pow- erful technique for analyzing systems involving multiple time scales which arise in applications. The theory gives a rigorous and well-defined way to reduce systems with small parameters to lower-dimensional systems that are more easily analyzed. That theory is well developed for deterministic systems [F1, F2, F3, JK, B, S2] and has been enormously effective in a wide variety of applications (see for instance [J], [BFGJ], [S1], and [KSS]). This work is the first step in a program to build geometric singular perturbation theory under stochastic perturbations by providing the persistence theory for random normally hyperbolic invariant manifolds. We also give persistence results for random overflowing and random inflowing invariant manifolds, thus extending Fenichel’s fundamental results [F1] to randomly perturbed dynamical systems. t Let (Ω, F,P) be a probability space and (θ )t∈R be a measurable P -measure preserving dynamical system on Ω. A discrete time random dynamical system (or acocycle)onspaceRn over the dynamical system θt is a measurable map φ(·, ·, ·):R × Ω × Rn → Rn, (ω, x) → φ(t, ω, x), for t ∈ R such that the map φ(t, ω):=φ(t, ω, ·)formsacocycleoverθt: φ(0,ω)=Id, for all ω ∈ Ω, φ(t + s, ω)=φ(t, θsω)φ(s, ω), for all t, s ∈ R,ω∈ Ω.

Received by the editors August 30, 2011 and, in revised form, February 28, 2012. 2010 Mathematics Subject Classification. Primary 34C37, 34C45, 34F05, 37H10. Key words and phrases. Random dynamical systems, random normally hyperbolic invariant manifolds, overflowing manifolds, inflowing manifolds, and random stable and unstable foliations. The second author was partially supported by NSF0908348. The third author was partially supported by NSF0909400.

c 2013 American Mathematical Society Reverts to public domain 28 years from publication 5933

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 5934 JI LI, KENING LU, AND PETER BATES

When φ(·,ω,·):R × Rn → Rn is continuous for each ω ∈ Ω, φ(t, ω, x) is called a continuous random dynamical system. An RDS φ is called a smooth RDS of class Ck,k≥ 1, if for each (t, ω) ∈ R × Ω, φ(t, ω, ·):Rn → Rn is Ck–smooth. Namely, φ(t, ω, x)isk times differentiable in x and the derivatives are continuous in (t, x) for each ω ∈ Ω. Random dynamical systems arise in the modeling of many phenomena in physics, biology, climatology, economics, etc. which are often subject to uncertainty or random influences, such as stochastic forcing, uncertain parameters, random sources or inputs, and random boundary conditions. An example is the solution operator for a random differential equation driven by a real noise: dx = f(θ ω, x), dt t d d d where x ∈ R , f :Ω× R → R is a measurable function and fω(t, ·) ≡ f(θtω, ·) ∈ R 0,1 Lloc( ,Cb ), Here, (Ω, F, P) is the classic Wiener space, i.e., Ω = {ω : ω(·) ∈ C(R, Rd),ω(0) = 0} endowed with the open compact topology so that Ω is a Polish space and P is the Wiener measure. Define a measurable dynamical system θt on the probability space (Ω, F,P) by the Wiener shift (θtω)(·)=ω(t + ·) − ω(t)fort>0. It is well known that P is invariant and ergodic under θt. This measurable dynamical system θt is also called a metric dynamical system. This models the noise in the system; see [A], page 60, for details. We consider a deterministic flow ψ(t)(x) ≡ ψ(t, x)inRn and its randomly perturbed flow (cocycle) φ(t, ω)(x) ≡ φ(t, ω, x). Our main results can be summarized as Theorem. Assume that ψ(t) is a Cr flow, r ≥ 1, and has compact, connected Cr normally hyperbolic invariant manifold M⊂Rn. Then there exists ρ>0 such that for any C1 random flow φ(t, ω) in Rn,if

||φ(t, ω) − ψ(t)||C1 <ρ, for t ∈ [0, 1],ω ∈ Ω, then (i) Persistence: φ(t, ω) has a C1 normally hyperbolic random invariant manifold M˜ (ω). (ii) Smoothness: If φ(t, ω) is Cr and the normal hyperbolicity is sufficiently strong (see Theorem 2.1), then M˜ (ω) is a Cr manifold diffeomorphic to M for each ω ∈ Ω. (iii) Existence of stable manifolds: φ(t, ω) has a stable manifold W˜ s(ω) at M˜ (ω). (iv) Existence of unstable manifolds: φ(t, ω) has an unstable manifold W˜ u(ω) at M˜ (ω). (v) Persistence of overflowing and inflowing manifold: If ψ(t) has an overflowing/inflowing manifold M,thenφ(t, ω) has an overflowing/inflowing manifold M˜ (ω). As was stated earlier, when the perturbed flow φ(t, ω) is a deterministic flow, i.e., φ(t, ω) is independent of ω, this theorem was proved by Fenichel [F1], and we follow the approach given there. The basic idea is due to Hadamard [H] and involves a graph transform to first construct a center-unstable manifold. One takes a Lipschitz graph over the unstable bundle at the deterministic normally hyperbolic invariant manifold for ψ(t) and maps it forward under φ(t, ω) for large but fixed t for each

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use NH INVARIANT MANIFOLD FOR RDS 5935

ω ∈ Ω. Because each φ(t, ω)isC1 close to ψ(t) and because the latter stretches the graph in the unstable direction while compressing in the normal direction, the image (for each ω) is again a Lipschitz graph. Furthermore, this mapping is a contraction on the space of such graphs and the resulting fixed graph (for each ω) is the center- unstable manifold. The center-stable manifold is obtained in the same way, by reversing time. The desired manifold is then the intersection of these two (for each ω). There are important technical hurdles that must be overcome. The fact that the manifolds are global forces us to use local charts, which is a messy but harmless complication. The fact that we are dealing with a random dynamical system also leads to some complicated notation and bookkeeping. The main difficulty, however, is the need to show certain measurability properties, and this is the crux of the extension. Of course, a random dynamical system is also nonautonomous, and so naturally the concept of an invariant manifold must be extended. Here, one is faced with the need to construct random invariant manifolds that change with time according to a metric dynamical system on a probability space (see the next section for definitions). It should be noted that there are other approaches to proving perturbation results for normally hyperbolic invariant manifolds, most notably the Lyapunov-Perron method, but we believe that the obstacles are equally challenging and so take this more geometric route. We mentioned the fact that persistence of invariant manifolds under perturbation is the basis for geometric singular perturbation theory. This paper is to be followed with another which gives the next step, that of constructing random invariant foliations (the counterpart of the Fenichel fibration) of the center-stable and center- unstable manifolds for φ(t, ω). We also expect to develop an exchange lemma (see [JK]) for random dynamical systems. We hope that these will lay the groundwork for applications and provide tools for studying stochastic systems. At present we do not have results along the lines of this paper for infinite- dimensional random semiflows, as one encounters with stochastic parabolic PDEs. The deterministic case was treated in [BLZ1, BLZ2, BLZ3], but the extension to random systems is elusive. The results we present here are closely related to the theory of local random stable, unstable, and center manifolds of stationary solutions. For finite-dimensional randomdynamicalsystems,wereferto[C,BK,Bx,D,LQ,S,W,A,LiL].The results on local invariant manifolds for infinite-dimensional random dynamical sys- temscanbefoundin[R,CLR,DSL1,DSL2,MS,MZZ,LS,GLS,CDLS,WD]. Random inertial manifolds were obtained in [BF, GC, DD]. Included in the pre- viously cited work are discrete time dynamics with products of diffeomorphisms, in Euclidean space, on Riemannian manifolds, or in infinite-dimensional spaces, where applications to PDEs are considered. Various approaches have been used, the Lyapunov-Perron method in particular.

2. Main results In this section, we ﬁrst introduce some basic concepts on random dynamical systems. Then, we state our main results.

2.1. Random dynamical systems. In this subsection, we review some of the basic concepts related to random dynamical systems in a Banach space that are

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 5936 JI LI, KENING LU, AND PETER BATES

taken from Arnold [A]. Let (Ω, F,P) be a probability space and X be a Banach space. Let T = R or Z endowed with their Borel σ-algebra.

t Definition 2.1. A family (θ )t∈T of mappings from Ω into itself is called a metric dynamical system if (1) (ω, t) → θtω is F⊗B(T) measurable; 0 t+s t s (2) θ = idΩ,theidentityonΩ,θ = θ ◦ θ for all t, s ∈ T; (3) θt preserves the probability measure P . Definition 2.2. Amap φ : T × Ω × X → X, (t, ω, x) → φ(t, ω, x), is called a random dynamical system (or a cocycle) on the Banach space X over a t metric dynamical system (Ω, F,P,θ )t∈T if (1) φ is B(T) ⊗F⊗B(X)-measurable; (2) the mappings φ(t, ω):=φ(t, ω, ·):X → X form a cocycle over θt: φ(0,ω)=Id, for all ω ∈ Ω, φ(t + s, ω)=φ(t, θsω) ◦ φ(s, ω), for all t, s ∈ T,ω∈ Ω. An example is the solution operator for a random differential equation driven by arealnoise, dx = f(θ ω, x), dt t d d d where x ∈ R , f :Ω× R → R is a measurable function and fω(t, ·) ≡ f(θtω, ·) ∈ R 0,1 Lloc( ,Cb ). 2.2. Normally hyperbolic random invariant manifolds. We first recall that a multifunction M =(M(ω))ω∈Ω of nonempty closed sets M(ω),ω∈ Ω, contained in a separable Banach space X is called a random set if ω → inf ||x − y|| y∈M(ω) is a random variable for any x ∈ X. When each of the sets M(ω)isamanifold,we call M a random manifold. Definition 2.3. A random manifold M is called a random invariant manifold for a random dynamical system φ(t, ω)if φ(t, ω, M(ω)) = M(θtω) for all t ∈ R,ω ∈ Ω. Definition 2.4. A random invariant manifold M is said to be normally hyperbolic if for almost every ω ∈ Ωandx ∈M(ω), there exists a splitting which is C0 in x and measurable in ω: X = Eu(ω, x) ⊕ Ec(ω, x) ⊕ Es(ω, x) of closed subspaces with associated projections Πu(ω, x), Πc(ω, x), and Πs(ω, x) such that (i) The splitting is invariant: i i Dxφ(t, ω)(x)E (ω, x)=E (θtω, φ(t, ω)(x)), for i = u, c, and s s Dxφ(t, ω)(x)E (ω, x) ⊂ E (θtω, φ(t, ω)(x)).

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use NH INVARIANT MANIFOLD FOR RDS 5937 i → i (ii) Dxφ(t, ω)(x) Ei(ω,x) : E (ω, x) E (θtω, φ(t, ω)(x)) is an isomorphism for i = u, c. Ec(ω, x) is the tangent space of M(ω)atx. (iii) There are (θ, φ)-invariant random variablesα, ¯ β¯ : M→(0, ∞),α< ¯ β¯,and a tempered random variable K : M→[1, ∞) such that

−βω,x¯ )t (2.1) ||Dxφ(t, ω)(x)Πs(ω, x)|| ≤ K(ω, x)e for t ≥ 0, β¯(ω,x)t (2.2) ||Dxφ(t, ω)(x)Πu(ω, x)|| ≤ K(ω, x)e for t ≤ 0, α¯(ω,x)|t| (2.3) ||Dxφ(t, ω)(x)Πc(ω, x)|| ≤ K(ω, x)e for −∞0 such that for any random C1 ﬂow φ(t, ω) in Rn,if

||φ(t, ω) − ψ(t)||C1 <ρ, for t ∈ [0, 1],ω ∈ Ω, then (i) Persistence: φ(t, ω) has a C1 normally hyperbolic random invariant manifold M˜ (ω). (ii) Smoothness: If α

2.3. Normally hyperbolic random overflowing and inflowing invariant manifolds. Definition 2.5. A random manifold M¯ (ω)=M(ω) ∪ ∂M(ω)iscalledarandom overflowing invariant manifold for a random dynamical system φ(t, ω)if φ(t, ω, M(ω)) ⊃ M¯ (θtω) for all t>0,ω ∈ Ω. Definition 2.6. A random manifold M¯ (ω)=M(ω) ∪ ∂M(ω)iscalledarandom inflowing invariant manifold for a random dynamical system φ(t, ω)if φ(t, ω, M(ω)) ⊃ M¯ (θtω) for all t<0,ω ∈ Ω. Definition 2.7. A random overflowing invariant manifold M¯ (ω)=M(ω)∪∂M(ω) is said to be normally (stably) hyperbolic if for almost every ω ∈ Ωandx ∈M(ω), there exists a splitting which is C0 in x and measurable in ω: X = Ec(ω, x) ⊕ Es(ω, x)

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 5938 JI LI, KENING LU, AND PETER BATES

of closed subspaces with associated projections Πc(ω, x)andΠs(ω, x) such that (i) The splitting is invariant: Dφ(t, θ−tω)(φ(−t, ω)(x))Ec(θ−tω, φ(−t, ω)(x)) = Ec(ω, x) and Dφ(t, θ−tω)(φ(−t, ω)(x))Es(θ−tω, φ(−t, ω)(x)) ⊂ Es(ω, x). (ii) Dφ(t, θ−tω)(φ(−t, ω)(x))|Ec(θ−tω, φ(−t, ω)(x)) : Ec(θ−tω, φ(−t, ω)(x)) → Ec(ω, x) is an isomorphism. Ec(ω, x)isthetangentspaceofM(ω)atx. (iii) There are (θ, φ)-invariant random variables α, β : M→(0, ∞), α<β,and a tempered random variable K(ω, x):M→[1, ∞) such that −t −t −β(ω,x)t (2.4) ||Dφ(t, θ ω)(φ(−t, ω)(x))Πs|| ≤ K(θ ω, φ(−t, ω)(x))e for t ≥ 0, (2.5) α(ω,x)|t| t ||Dxφ(t, ω)(x)Πc(ω, x)|| ≤ K(ω, x)e for t such that φ(t, ω, x) ∈M(θ ω). Definition 2.8. A random inflowing invariant manifold M¯ (ω)=M(ω)∪∂M(ω)is said to be normally (unstably) hyperbolic if for almost every ω ∈ Ωandx ∈M(ω), there exists a splitting which is C0 in x and measurable in ω: X = Eu(ω, x) ⊕ Ec(ω, x) of closed subspaces with associated projections Πu(ω, x)andΠc(ω, x) such that (i) The splitting is invariant: Dφ(t, ω, x)Ei(ω, x)) = Ei(θtω, φ(t, ω)(x), for i = u, c. (ii) Dφ(t, θ−tω)(φ(−t, ω)(x))|Ei(θ−tω, φ(−t, ω)(x)) : Ei(θ−tω, φ(−t, ω)(x)) → Ei(ω, x) is an isomorphism for i = u, c. Ec(ω, x) is the tangent space of M(ω)atx. (iii) There are (θ, φ)-invariant random variablesα, ¯ β¯ : M→(0, ∞),α< ¯ β¯,and a tempered random variable K(ω, x):M→[1, ∞) such that −t −t β¯(ω,x)t (2.6) ||Dφ(t, θ ω)(φ(−t, ω, x))Πu|| ≤ K(θ ω, φ(−t, ω, x))e for t ≤ 0, (2.7) α¯(ω,x)|t| t ||Dxφ(t, ω)(x)Πc(ω, x)|| ≤ K(ω, x)e for t such that φ(t, ω, x) ∈M(θ ω). Theorem 2.2. Assume that ψ(t) is a Cr flow, r ≥ 1, and has a compact, connected Cr normally hyperbolic overflowing invariant manifold M¯ = M∪∂M⊂Rn with positive deterministic constant hyperbolicity exponents α<β. Then there exists ρ>0 such that for any random Cr flow φ(t, ω) in Rn,if

||φ(t, ω) − ψ(t)||C1 <ρ, for t ∈ [0, 1],ω ∈ Ω, and if α0 such that for any random Cr ﬂow φ(t, ω) in Rn,if

||φ(t, ω) − ψ(t)||C1 <ρ, for t ∈ [0, 1],ω ∈ Ω, and if α

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use NH INVARIANT MANIFOLD FOR RDS 5939

3. Basic lemmas In this section, we first recall some facts for deterministic system taken from [F1]. Then we deduce the corresponding properties for random systems. We consider a compact connected Cr normally hyperbolic invariant manifold M under the deterministic Cr flow ψ(t) and let T Rn|M = T M⊕Es ⊕ Eu be the associated splitting. We will sometimes use the notation Ec to denote the bundle M i i T , and we will sometimes use E to denote the subspace Em (i = c, s, u)at apointm ∈Mwhen the context makes the meaning obvious. We denote the associated positive constant hyperbolicity exponents by α<β. The first two propositions concern the hyperbolicity of the invariant manifold M. Proposition 3.1. Normal hyperbolicity is independent of the metric. Proposition 3.2. If M is normally hyperbolic under the flow ψ(t):

(1) There exist positive constants a<1 and c1 such that s t ||Dψ(t)(ψ(−t)(m))|E || 0 and r >rsuch that s c r ||Dψ(t)(ψ(−t)(m))|E || ||D(ψ(−t))(m)|E ||

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 5940 JI LI, KENING LU, AND PETER BATES

{ ˜ j } M { j j } M j and U still cover . Suppose (σj,U5 (= U )) is an atlas on and σ(U5 )= Dj ⊂ Rm Rm 5 is an open neighborhood of the origin in . We may take open subsets Dj Dj Dj Dj Dj Dj ⊂Dj ˜ j ⊂ −1 Dj 1, 2, 3, 4 of 5 such that i−1 i for i =2, 3, 4, 5andU σj ( 1). j −1 Dj j ˜ j ⊂ j Taking Ui = σj ( i ), Ui is an open neighborhood of pj and U Ui for all i =1, 2, 3, 4, 5. So {U j } cover M.Infact,wehave i j M j ⊂ j ⊂···⊂ j M = U1 U2 U5 = . j j j Deﬁne for small >0 ˜ ˜s ⊕ ˜u { s u ∈ ˜ || s|| || u|| } E = E E := (m, ν ,ν ) E : ν < , ν < .

It follows from Proposition 3.4 that there exists 0 > 0 such that for 0 < < 0, r n E˜ is C diffeomorphic to a neighborhood V of M in R .Inotherwords,E˜ is a tubular neighborhood of M. From now on, we view both ψ(t, x)andφ(t, ω, x)as flows on E˜ as well as on V . r ˜| j ∈ j Choose a C orthonormal basis on E U5 for each j.Thus,form U5 and s ∈ ˜s s s ··· s (m, ν ) E , ν has coordinates (x1, ,xk) under the chosen orthonormal basis, s ˜s| j → Rk and these depend smoothly on m. Define τj : E U5 by s s s s ··· s ∈ Rk τj (m, ν )=x =(x1, ,xk) u and define τj similarly by u u u u ··· u ∈ Rl τj (m, ν )=x =(x1 , ,xl ) . We have s s u u ||x ||Rk = ||ν ||, ||x ||Rl = ||ν ||. r ˜ | j Rn Rm ⊕ Rk ⊕ Rl ThesegiveusC diffeomorphisms γj from E U5 to = ,where s u c s u s s u u γj (m, ν ,ν )=(x ,x ,x )=(σj (m),τj (m, ν ),τj (m, ν )). { ˜ | j } ˜ So (E U5 ,γj) gives us an atlas on E. M −1 Since is compact, there exists L1 > 0 such that Dσj and Dσj are bounded by L1 for all m and j. Then for sufficiently large T>0, we represent the random diffeomorphism φ(T,ω)(·) in the local charts by c s u → c c s u s c s u u c s u (x ,x ,x ) (fij(ω, x ,x ,x ),fij(ω, x ,x ,x ),fij(ω, x ,x ,x )), where l c s u ◦ ◦ · ◦ −1 c s u fij (ω, x ,x ,x )=Pl γj φ(T,ω, ) γi (x ,x ,x ) c s u ∈ ◦ ˜ | i ∩ − j c s u l for (x ,x ,x ) γi E U4 ψ( T,U4 )andPl(x ,x ,x )=x for l = c, s, u.Note c s u c s u that given i and (x ,x ,x ), fij(ω, x ,x ,x ) is defined only for some j’s. Denoting the differential operators D1 = Dxc ,D2 = Dxs ,D3 = Dxu ,wehave s c s u D2fij(ω, x ,x ,x ) −1 c s u = DPlDγj Dφ(T,ω)D2γ (x ,x ,x ) ˜ s | ˜s −1 c s u = DPlDγj (Π Dφ(T,ω) E )D2γi (x ,x ,x ). −1 Since M is compact, DPl, Dγj ,andD2γ , along with their inverses, are uniformly bounded by a constant L2 > 0. Then || s c s u || ≤ 3||˜ s | ˜s|| D2fij(ω, x ,x ,x ) L2 Π Dφ(T,ω) E .

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use NH INVARIANT MANIFOLD FOR RDS 5941

Similarly, we get for k =1, 2, 3andl = c, s, u, || l c s u ±1|| ≤ ||˜ l | ˜l|| (Dkfij(ω, x ,x ,x )) L Π Dφ(T,ω) E , for some L>0. We have the following lemma: Lemma 3.1. There exists a T>0 such that for any η>0,if and ρ are small enough, then 1 (3.1) ||(D f u (ω, xc,xs,xu))−1|| < , 3 ij 4 1 (3.2) ||D f s (ω, xc,xs,xu)|| < , 2 ij 2

1 (3.3) ||(D f c (ω, xc,xs,xu))−1||k||D f s (ω, xc ,xs ,xu )|| < , 1 ij 2 ij 4 for 0 ≤ k ≤ r, xc close to xc,and || c c s u −1|| (3.4) (D3fij(ω, x ,x ,x )) <η, || s c s u || || u c s u || (3.5) fij (ω, x ,x ,x ) <η, fij(ω, x ,x ,x ) <η, || s c s u || || s c s u || (3.6) D1fij (ω, x ,x ,x ) <η, D3fij(ω, x ,x ,x ) <η, || u c s u || || u c s u || (3.7) D1fij (ω, x ,x ,x ) <η, D2fij(ω, x ,x ,x ) <η.

Moreover, the norm of all first partial derivatives of fij and their inverses are bounded by some Q>0. Proof. By Proposition 3.2, there exists a T>0 such that 1 Lr||Dψ(−T )(m)|Eu|| < , 16 1 Lr||Dψ(T )(ψ(−T )(m))|Es|| < , 8 1 Lr+1||Dψ(−T )(m)|T M||k||Dψ(T )(ψ(−T,m))|Es|| < , 0 ≤ k ≤ r. 16 ˜ Since E and E are arbitrarily close, and ||φ(t, ω) − ψ(t)||C1 <ρfor all t ∈ [0, 1] and ω ∈ Ω, if ρ is small enough, we have 1 (3.8) Lr||Π˜ uDφ(−T,ω)(m)|E˜u|| < , 8 1 (3.9) Lr||Π˜ sDφ(T,ω)(ψ(−T,m))|E˜s|| < , 4 (3.10) 1 Lr+1||Dφ(−T,ω)(m)|T M||k||Π˜ sDφ(T,ω)(ψ(−T )(m))|E˜s|| < , 0 ≤ k ≤ r. 8 Note that we identified V with E( ) and viewed ψ(t)andφ(t, ω)asflowsinthe l bundles. So in terms of fij, 1 (3.11) ||(D f u (ω, xc, 0, 0))−1|| < , 3 ij 8 1 (3.12) ||D f s (ω, xc, 0, 0)|| < , 2 ij 4

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 5942 JI LI, KENING LU, AND PETER BATES

1 (3.13) ||(D f c (ω, xc, 0, 0))−1||k||D f s (ω, xc, 0, 0)|| < , 0 ≤ k ≤ r. 1 ij 2 ij 8 We also have 1 (3.14) ||(D f c (ω, xc, 0, 0))−1|| < η. 3 ij 2 Choose and ρ small enough, we have 1 ||(D f u (ω, xc,xs,xu))−1|| < , 3 ij 4 1 ||D f s (ω, xc,xs,xu)|| < , 2 ij 2 1 ||(D f c (ω, xc,xs,xu))−1||k||D f s (ω, xc ,xs ,xu )|| < , 1 ij 2 ij 4 for 0 ≤ k ≤ r, xc close to xc,and || c c s u −1|| (D3fij(ω, x ,x ,x )) <η. By the invariance of M under ψ(T ), the invariance of Eu, Es, T M under Dψ(T ) and the closeness of E˜ and E,if and ρ are small enough, we have || s c s u || || u c s u || fij (ω, x ,x ,x ) <η, fij(ω, x ,x ,x ) <η, || s c s u || || s c s u || D1fij (ω, x ,x ,x ) <η, D3fij(ω, x ,x ,x ) <η, || u c s u || || u c s u || D1fij (ω, x ,x ,x ) <η, D2fij(ω, x ,x ,x ) <η. This completes the proof of the lemma.

4. Existence of the random unstable manifold In this section, we will prove the existence of a random unstable manifold using the graph transform. The random unstable manifold will be constructed as a section ˜ ˜u of E over E . We will define a transform on the space of all Lipschitz sections and prove that the transform has a fixed point which gives the random unstable manifold. ˜ ˜u ∈ Let X denote the space of sections of E over E .Foru(ω) X,itislocally represented by ui(ω): c u s ◦ s ◦ ◦ × u −1 c u ui(ω, x ,x )=τi P u(ω) (σi τi ) (x ,x ), s ¯ ⊂Di where P is the fiber projection. Let >0befixedsothatB(0) 5 for all j. Define Lip u(ω):=max sup i c c∈Di || u|| || u||≤ c u c u x ,x 3, x , x ,(x ,x )=(x ,x ) (4.1) || c u − c u || × ui(ω, x ,x ) ui(ω, x ,x ) max{||xc − xc||, ||xu − xu||}

if it exists. Denote Xδ = {u(ω) ∈ X|Lip u(ω) ≤ δ}. Denote a random section by u = {u(ω) ∈ X|ω ∈ Ω} and denote by S the set of all such u = {u(ω):ω ∈ Ω}. Deﬁne Lip u := sup Lip u(ω) ω and let Sδ = {u ∈ S|Lip u ≤ δ}.

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use NH INVARIANT MANIFOLD FOR RDS 5943

Deﬁne the norm on Sδ by c u ||u|| := sup max sup |ui(ω, x ,x )|. ω i xc,xu

Then the induced metric on Sδ makes it into a complete metric space. Let T be given by Lemma 3.1. Proposition 4.1. There is a unique u ∈ S such that φ(t, ω, graph u(ω)) ⊃ graph u(θtω)

for all t>T and ω ∈ Ω. Furthermore, u ∈ Sδ. We need several lemmas to prove this proposition. We occasionaly write u(ω)in place of graph u(ω). In the local coordinates, the mapping u(ω) → φ(T,ω)(u(ω)) has the form c c u u → c c c u u s c c u u (x ,ui(ω, x ,x ),x ) (fij(ω, x ,ui(ω, x ,x ),x ),fij(ω, x ,ui(ω, x ,x ),x ), u c c u u fij(ω, x ,ui(ω, x ,x ),x )).

Deﬁne an operator G on Sδ by c u s −T c −T c u u (Gu)j(ω, ξ ,ξ )=fij (θ ω, x ,ui(θ ω, x ,x ),x ), where c c −T c −T c u u ξ = fij(θ ω, x ,ui(θ ω, x ,x ),x ), u u −T c −T c u u ξ = fij(θ ω, x ,ui(θ ω, x ,x ),x ). In the following, we will show G is well deﬁned. For convenience, we denote xcu =(xc,xu),ξcu =(ξc,ξu), and write s cu s s c s u fij(ω, x ,x )=fij(ω, x ,x ,x ) and cu cu s c c s u u c s u T fij (ω, x ,x )=(fij(ω, x ,x ,x ),fij(ω, x ,x ,x )) . We also use the norms ||xcu|| =max(||xc||, ||xu||), ||ξcu|| =max(||ξc||, ||ξu||), || cu|| || c || || u || fij =max( fij , fij ). Thus, we may write the local representatives as cu c u ui(ω, x )=ui(ω, x ,x ), cu s cu cu (Gu)j(ξ )=fij (ω, x ,u(ω, x )) with cu cu cu cu ξ = fij (ω, x ,u(ω, x )). Denote ≡ s cu s ≡ s c s u s c s u A D1,3fij(ω, x ,x ) (D1fij(ω, x ,x ,x ),D3fij(ω, x ,x ,x )), B ≡ D f s (ω, xcu,xs)=D f s (ω, xc,xs,xu), 2 ij 2 ij c c s u c c s u ≡ cu cu s D1fij(ω, x ,x ,x ) D3fij(ω, x ,x ,x ) C D1,3fij (ω, x ,x )= u c s u u c s u , D1fij(ω, x ,x ,x ) D3fij(ω, x ,x ,x ) and ≡ cu cu s c c s u u c s u E D2fij (ω, x ,x )=(D2fij (ω, x ,x ,x ),D2fij (ω, x ,x ,x )).

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 5944 JI LI, KENING LU, AND PETER BATES

Lemma 4.1. For δ>0 small enough and any 0 < < 0,thereexistsρ( ) > 0 such that for 0 <ρ<ρ( ), G is well defined on Sδ. cu cu ∈Di × Proof. For fixed ω and i and any x ,x 3 B(0, ), we have ||ξcu − ξcu || || cu −T cu −T cu − cu −T cu −T cu || = fij (θ ω, x ,ui(θ ω, x )) fij (θ ω, x ,ui(θ ω, x )) 1 ≥ ||xcu − xcu || − Qδ||xcu − xcu || Q 1 > ||xcu − xcu ||, 2Q cu −T · −T · which implies that fij (θ ω, ,ui(θ ω, )) is one-to-one and thus is a homeomor- Di × phism from 3 B(0, ) to its range. Note that although the Mean Value Theorem does not hold in multi-dimensional space, we still have the first inequality above since the matrix of partial derivatives obtained by using the Mean Value Theorem componentwise is close to the Jacobian. Such estimates will be used repeatedly in our analysis. Letting xc = xc and using Lemma 3.1, we have (4.2) ||ξu − ξu || || u −T c −T c u u − u −T c −T c u u || = fij(θ ω, x ,ui(θ ω, x ,x ),x ) fij(θ ω, x ,ui(θ ω, x ,x ),x ) ≥|| u −1||−1|| u − u|| − || u |||| c − c|| (D3fij) x x D1fij x x −|| u |||| c u − c u || || cu − cu|| D2fij ui(x ,x ) ui(x ,x ) + o( x x ) ≥ 4||xu − xu || − 2Qδ||xcu − xcu || =(4− 2Qδ)||xu − xu || > 3||xu − xu ||, where by taking small enough, we have from ||xu − xu|| < 2 that o(||xcu − xcu ||) 0, there exists ρ( ) > 0 such that for 0 <ρ<ρ( ), we have 1 (4.3) ||f u (θ−T ω, xc,u (θ−T ω, xc, 0), 0)|| < , ij i 3 which together with (4.2) yield that || u −T c −T c u u || fij (θ ω, x ,ui(θ ω, x ,x ),x ) > 2 || u|| cu −T · −T · for 3 < x < . From this and the fact that fij (θ ω, ,ui(θ ω, )) is a Dk × homeomorphism, we conclude that k 3 B(0, ) is contained in the range of − cu T · T · fij (θ ω, ,ui(θ ω, )) for i, j running all possible choices. This shows G is well defined and completes the proof of the lemma.

Lemma 4.2. For all 0 ≤ k ≤ r, 1 (4.4) ||B|| ||C−1||k < . 2

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use NH INVARIANT MANIFOLD FOR RDS 5945

Proof. Using Lemma 3.1, we have 1 (4.5) ||B|| < . 2 Writing c c s u c c s u D1fij(ω, x ,x ,x ) D3fij (ω, x ,x ,x ) ab C = u c s u u c s u = , D1fij(ω, x ,x ,x ) D3fij (ω, x ,x ,x ) cd we have −1 −1 −1 −1 −1 − (a − bd c) −(a − bd c) bd C 1 = . (ca−1b − d)−1ca−1 −(ca−1b − d)−1 From Lemma 3.1, we know that ||a||

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 5946 JI LI, KENING LU, AND PETER BATES

Lemma 4.3. For ﬁxed δ small enough and any 0 < < 0,thereexistsaρ( ) > 0 such that for all 0 <ρ<ρ( ), G maps Sδ to Sδ.

Proof. It is sufficient to show that if u ∈ Sδ,then cu cu cu cu (4.7) ||(Gu)j(ω, ξ ) − (Gu)j(ω, ξ )|| ≤ δ||ξ − ξ || for ξcu and ξcu in any sufficiently small neighborhood for all ω ∈ Ωandj.This cu cu ∈Dj × is because any two points ξ ,ξ 3 B(0, ) are connected by a line segment which is covered by finitely many such neighborhoods and one can always choose a cu cu sequence of points ξ0 = ξ ,ξ1, ··· ,ξm = ξ along the line segment such that for any 1 ≤ k ≤ m, ξ and ξ are located in the same small neighborhood. Then k k+1 cu cu cu cu ||(Gu)j(ω, ξ ) − (Gu)j(ω, ξ )|| ≤ δ||ξk − ξk−1|| = δ||ξm − ξ0|| = δ||ξ − ξ ||. k cu cu cu ∈Di cu Using Lemma 4.1, we have for ξ near ξ that there exists x 3 near x such that cu cu −T cu −T cu ξ = fij (θ ω, x ,ui(θ ω, x )) and cu cu −T cu −T cu ξ = fij (θ ω, x ,ui(θ ω, x )). Thus, we have cu cu ||(Gu)j(ω, ξ ) − (Gu)j(ω, ξ )|| || s −T cu −T cu − s −T cu −T cu || = fij(θ ω, x ,ui(θ ω, x )) fij(θ ω, x ,ui(θ ω, x )) ≤||A||||xcu − xcu || + ||B||δ||xcu − xcu || + o(||xcu − xcu ||) ≤ (2η + δ||B||)||xcu − xcu ||. Here, we have taken ||xcu − xcu|| small enough so that |o(||xcu − xcu||)| <η||xcu − xcu||. Next we need to estimate ||ξcu − ξcu || in terms of ||xcu − xcu||: || cu − cu|| || cu −T cu −T cu − cu −T cu −T cu || ξ ξ = fij (θ ω, x ,ui(θ ω, x )) fij (θ ω, x ,ui(θ ω, x )) ≥|| cu −T cu −T cu − cu −T cu −T cu || fij (θ ω, x ,ui(θ ω, x )) fij (θ ω, x ,ui(θ ω, x )) −|| cu −T cu −T cu − cu −T cu −T cu || fij (θ ω, x ,ui(θ ω, x )) fij (θ ω, x ,ui(θ ω, x )) ≥||C−1||−1||xcu − xcu || + o(||xcu − xcu ||) −||E||δ||xcu − xcu || ≥||C−1||−1(1 − 2δQ2)||xcu − xcu ||. Here again we have taken ||xcu − xcu|| small enough so that |o(||xcu − xcu||)| < δQ||xcu − xcu||. Then, we have cu cu cu cu ||(Gu)j(ω, ξ ) − (Gu)j(ω, ξ )|| ≤ (2η + δ||B||)||x − x || −1 ≤ || || ||C || || cu − cu|| (2η + δ B ) (1−2δQ2) ξ ξ 2ηQ+ 1 δ ≤ 2 || cu − cu || ≤ 3 || cu − cu || 1−2δQ2 ξ ξ 4 δ ξ ξ , 1 δ provided δ<12Q2 and η<16Q .SoG maps Sδ to Sδ. This completes the proof of the lemma. Lemma 4.4. If δ, η > 0 are small enough, there exists (η) > 0 such that for any 0 < < (η) there exists ρ( ) > 0 such that for 0 <ρ<ρ( ), G is a contraction on Sδ.

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use NH INVARIANT MANIFOLD FOR RDS 5947

∈ ∈ cu ∈ × u ˜u| j Proof. Let u, u Sδ.Fixω Ω and let ξ (σj τj )(E U3 ) be given. By cu cu ∈Di × Lemma 4.1, there exist i and x ,x 3 B(0, ) such that cu c −T cu −T cu c −T cu −T cu ξ = fij(θ ω, x ,ui(θ ω, x )) = fij (θ ω, x ,ui(θ ω, x )). cu cu Di × Here, as long as is small, x ,x are close enough to be in the same 3 B(0, ). Then if is small enough, we have that o(||xcu −xcu||)ando(||u−u||) are so small that cu cu ||(Gu)j(ω, ξ ) − (Gu )j (ω, ξ )|| || s −T cu −T cu − s −T cu −T cu || = fij(θ ω, x ,ui(θ ω, x )) fij(θ ω, x ,ui(θ ω, x )) ≤|| |||| cu − cu|| || |||| −T cu − −T cu || A x x + B ui(θ ω, x ) ui(θ ω, x ) || −T cu − −T cu || || cu − cu|| || − || +Q ui(θ ω, x ) ui(θ ω, x ) + o( x x )+o( u u ) ≤ || cu − cu|| 2 || −T − −T || (η +2δQ) x x + 3 ui(θ ω) ui(θ ω) . || cu − cu|| || −T − −T || Next, we need to estimate x x in terms of ui(θ ω) ui(θ ω) .We ﬁrst have || cu −T cu −T cu − cu −T cu −T cu || fij (θ ω, x ,ui(θ ω, x )) fij (θ ω, x ,ui(θ ω, x )) ≥ Q−1||xcu − xcu || and || cu −T cu −T cu − cu −T cu −T cu || fij (θ ω, x ,ui(θ ω, x )) fij (θ ω, x ,ui(θ ω, x )) ≤ || cu − cu|| || −T − −T || Q(δ x x + ui(θ ω) ui(θ ω) ). Noting that cu cu −T cu −T cu cu −T cu −T cu ξ = fij (θ ω, x ,ui(θ ω, x )) = fij (θ ω, x ,ui(θ ω, x )), we combine the above estimates to obtain 2 ||xcu − xcu || ≤ (2Q2(η +2δQ)+ )||u (θ−T ω) − u (θ−T ω)||. 3 i i Taking δ, η small enough, then we get 3 ||(Gu) (ω, ξcu) − (Gu) (ω, ξcu)|| ≤ ||u (θ−T ω) − u (θ−T ω)||. j j 4 i i

Proof of Proposition 4.1. Using Lemma 4.4, by the uniform contraction principle, G has a unique ﬁxed point in Sδ,callitu.Ifu ∈ S is such that φ(T,ω)(graph u(ω)) ⊃ graph u(θT ω), cu then the proof of Lemma 4.4 can be applied: Fix ω ∈ Ω and let ξ ∈ (σj × u ˜u| j cu ∈Di × τj )(E U3 ) be given. Since u is invariant, there exist i and x 3 B(0, ) such that cu cu −T cu −T cu cu −T cu −T cu ξ = fij (θ ω, x ,ui(θ ω, x )) = fij (θ ω, x ,ui(θ ω, x )). cu cu Di × Here, as long as is small, x ,x are close enough to be in the same 3 B(0, ). Then || cu − cu || uj (ω, ξ ) uj (ω, ξ ) || s −T cu −T cu − s −T cu −T cu || = fij(θ ω, x ,ui(θ ω, x )) fij(θ ω, x ,ui(θ ω, x )) ≤ || cu − cu|| 2 || −T − −T || (η +2δQ) x x + 3 ui(θ ω) ui(θ ω) .

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 5948 JI LI, KENING LU, AND PETER BATES

Similarly, as in Lemma 4.4 we get 2 ||xcu − xcu || ≤ (2Q2(η +2δQ)+ )||u (θ−T ω) − u (θ−T ω)||. 3 i i So 3 ||u (ω, ξcu) − u (ω, ξcu)|| ≤ ||u (θ−T ω) − u (θ−T ω)||. j j 4 i i This shows that u = u, giving the uniqueness in S. To show φ(t, ω)(graph u(ω)) ⊃ graph u(θtω), we use the uniqueness of u. For any fixed t>T, we can define G on Sδ based on t just as G is based on T . G is well defined. Then we have that φ(t, ω)(graph u(ω)) ∩ E˜ is the graph of an t elementu ¯ ∈ Sδ at the θ ω fiber. Since for each ω (4.8) φ(T,θ−T ω)(graph u(θ−T ω)) ⊃ graph u(ω), we have φ(t, ω)(graph u(ω)) ⊂ φ(t, ω)φ(T,θ−T ω)(graph u(θ−T ω)) = φ(T,θt−T ω)φ(t, θ−T ω)(graph u(θ−T ω)), which gives graphu ¯(θtω) ⊂ φ(T,θ−T (θtω))(graphu ¯(θ−T (θtω))), or equivalently, (4.9) graphu ¯(˜ω) ⊂ φ(T,θ−T (˜ω))(graphu ¯(θ−T (˜ω))). Comparing (4.8) and (4.9) and using the uniqueness of the fixed point of G,we obtain u¯ = u. This completes the proof of the proposition.

5. Smoothness of the random unstable manifolds In this section, we prove the smoothness of the random unstable manifold. Let u ∈ Sδ denote the unique fixed point of G. So the graph of u(ω) is the unique random unstable manifold W˜ u(ω). We prove u is Cr. We construct linear functions in the local coordinates as the candidates for Dui(ω) and prove they are indeed the derivatives of ui(ω). Di × In the local coordinates, for any fixed ω, u(ω) is represented by ui(ω):( 3 k 1 B(0, )) → R ,fori =1, ··· ,s.Ifu(ω) ∈ C , Dui(ω) assigns to each point in Di × Rn−k Rk ( 3 B(0, )) a linear map from to .ThusDu(ω)isrepresentedby ∈ 0 Di × Rn−k Rk vi(ω) C (( 3 B(0, )),L( , )), for i =1, ··· ,s. The candidates for Du(ω) are of the form s ··· ∈ 0 Di × Rn−k Rk v(ω)=(v1(ω), ,vs(ω)) C (( 3 B(0, )),L( , )), i=1 and we denote the space of such mappings by TS¯ .Ifv(ω)issuchans-tuple, define cu (5.1) ||v(ω)|| =max sup ||vi(ω, x )||, 1≤i≤s cu∈ Di × x ( 3 B(0,))

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use NH INVARIANT MANIFOLD FOR RDS 5949

if this exists, where ||·|| is the operator norm. For v = {v(ω) ∈ TS¯ : ω ∈ Ω}, define (5.2) ||v|| := sup ||v(ω)||. ω Let TS be the space of all such random linear mappings v. To be more clear, any element v of TS canfirstbeviewedasafunctionof ω ∈ Ω, and for any fixed ··· ∈ s 0 Di × Rn−k Rk ω, v(ω)isans-tuple (v1(ω), ,vs(ω)) i=1 C (( 3 B(0, )),L( , )). Define the norm ||·||on TS by (5.1) and (5.2). Under this norm, TS is complete. In terms of the local coordinates, u(ω) satisfies T cu s cu cu uj (θ ω, ξ )=fij(ω, x ,ui(ω, x )), cu cu cu cu where ξ = fij (ω, x ,ui(ω, x )). Differentiating these formally, we have T cu cu cu vj (θ ω, ξ )[C + Evi(ω, x )] = A + Bvi(ω, x ), s s cu cu where A = D1,3fij,B= D2fij,C= D1,3fij and E = D2fij , the arguments of cu cu A, B, C, E,areall(ω, x ,ui(ω, x )). So T cu cu vj (θ ω, ξ )=Hij(ω)vi(ω, x ), where cu cu cu −1 (5.3) Hij(ω)vi(ω, x )=[A + Bvi(ω, x )][C + Evi(ω, x )] . Thus, (5.3) induces an operator H on TS.

cu Remark. In the definition of Hij, the argument of all A, B, C, E are all (ω, x , cu ui(ω, x )), while later in the computation of contraction, the argument of A, B, C, E may be different, and in fact the matrices themselves may be different, being small perturbations of these Jacobians. However, this does not affect the estimates. We want to prove that H has a fixed point v(ω). For technical reasons, we need r ˆ i → a smooth partition of unity. Choose C functions: hi : U3 [0, 1] with support of s s ⊂ ˆ i ˆ i ˆ i × u −1 Di × hi U2 and hi =1on U1. Here, Uk := (σi τi ) ( k B(0, )). Define i=1 i 0 cu ··· vi (ω, x )=0,i=1, 2, ,s, and s n+1 T cu n cu vj (θ ω, ξ )= hi(m−)Hij(ω)vi (ω, x ), i=1 where T −1 cu φ(T,ω,u(ω, m−)) = u(θ ω, σj (ξ )). s s ∈ ˆ i { n} Then m− U1 and hi(m−) = 1. We will show v converges to a fixed i i=1 point of (5.3). Proposition 5.1. The sequence {vn(ω)} converges to a solution of the equations s T (5.4) vj (θ ω)= hi · Hij(ω)vi(ω). i=1 First, we claim || n cu || Lemma 5.1. vi (ω, x ) <δfor all n and ω.

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 5950 JI LI, KENING LU, AND PETER BATES

Proof. It is suﬃcient to show || n || Hij(ω)vi (ω) <δ. 2 || n || Letting δ>0 be such that δQ 1, for vi (ω) <δwe have || n −1|| || −1 n −1|| [C + Evi (ω)] = [C (1 + C Evi )] ∞ || − − −1 n −1 −1|| || − −1 n k −1|| = [1 ( C Evi )] C = ( C Evi ) C k=0 ∞ ∞ ||C−1|| ≤ ||C−1Evn||k||C−1|| ≤ ||δQ2||k||C−1|| = . i 1 − δQ2 k=0 k=0 Thus, using (5.3), by induction we have ||C−1|| Qη + 1 δ ||H (ω)vn(ω)|| ≤ (η + ||B||δ) ≤ 2 ≤ δ, ij i 1 − δQ2 1 − δQ2 provided η is small enough. This completes the proof of the lemma.

Lemma 5.2. ||vn+1(θT ω) − vn(θT ω)|| ≤ λ||vn(ω) − vn−1(ω)|| for some 0 <λ<1. Proof. It is suﬃcient to show || n − n−1 || ≤ || n − n−1 || Hij(ω)vi (ω) Hij(ω)vi (ω) λ v (ω) v (ω) for all i, j. First, we note that n − n−1 Hij(ω)vi (ω) Hij(ω)vi (ω) n n −1 =[A + Bvi (ω)][C + Evi (ω)] − n−1 n−1 −1 [A + Bvi (ω)][C + Evi (ω)] n n −1 n−1 − n n−1 −1 =(A + Bvi )(C + Evi ) [(C + Evi ) (C + Evi )](C + Evi ) n − n−1 n−1 −1 +[(A + Bvi ) (A + Bvi )](C + Evi ) . Estimating the above and using Lemma 4.2, we have || n − n−1 || Hij(ω)vi (ω) Hij(ω)vi (ω) ||C−1|| ||C−1|| ≤ (η + ||B||δ) Q||vn − vn−1|| 1 − δQ2 i i 1 − δQ2 ||C−1|| +||B||||vn − vn−1|| i i 1 − δQ2

||C−1||2 1 = (η + ||B||δ) Q + 2 ||vn(ω) − vn−1(ω)|| (1 − δQ2)2 1 − δQ2 = λ||vn(ω) − vn−1(ω)||, for all i, j. By choosing η and δ small enough, we have λ<1. The proof is complete.

By the contraction principle, {vn} converges to v, which satisﬁes (5.4). We are ready to prove that v is the derivative of u. cu ∈ Dj × T cu Proposition 5.2. For each ξ ( 3 B(0, )), Duj (θ ω, ξ ) exists and equals T cu 1 T cu cu vj (θ ω, ξ ).Hence,u ∈ C and vj (θ ω, ξ )=Hij(ω)vi(ω, x ).

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use NH INVARIANT MANIFOLD FOR RDS 5951

Proof. For a ﬁxed ω ∈ Ω, we deﬁne an increasing function γω :(0, 1) → R:

γθT ω(a)=max sup i cu cu∈ Di × || cu− cu|| ξ ,ξ ( 3 B(0,)),0< ξ ξ

Note that γθT ω is bounded by 2δ. We want to show γω(a) → 0asa → 0. To prove this, we claim

Claim. γω(a) satisﬁes

γθT ω(a) ≤ sγω(za)+r(ω, a) for small a,forsome0≤ s<1, z>1, where r(ω, a) is a decreasing function which approaches zero as a → 0 uniformly in ω ∈ Ω.

cu ∈ Dj × cu Proof of the Claim. Let ξ ( 3 B(0, )). By Lemma 4.1, we have ξ = cu cu cu cu ∈ Di × ∈ fij (ω, x ,ui(ω, x )) for some x ( 2 B(0, )). We choose d (0,a)so cu ∈ Dj × || cu − cu|| cu ∈ small that if ξ ( 3 B(0, )) with ξ ξ

γθT ω(a) ≤ sγω(za)+r(ω, a), it is suﬃcient to show that

T cu T cu T cu cu cu ||uj (θ ω, ξ ) − uj (θ ω, ξ ) − Hij(ω)vi(θ ω, ξ ) · (ξ − ξ )|| cu cu ≤ [sγω(za)+r(a)]||ξ − ξ ||

for all ξcu,ξcu ,i,j as above and ||ξcu − ξcu|| ≤ a

cu − cu cu cu cu − cu cu cu ξ ξ = fij (ω, x ,ui(ω, x )) fij (ω, x ,ui(ω, x )) cu cu cu cu cu cu = C(x − x )+E(ui(ω, x ) − ui(ω, x )) + o(||x − x ||),

T cu − T cu s cu cu − s cu cu uj (θ ω, ξ ) uj (θ ω, ξ )=fij(ω, x ,ui(ω, x )) fij(ω, x ,ui(ω, x )) cu cu cu cu = A(x − x )+B(ui(ω, x ) − ui(ω, x )) +o(||xcu − xcu||).

Note that the quantity o(||xcu − xcu||) is uniformly in ω ∈ Ω. From the ﬁrst equation it follows that

cu cu ||ξ − ξ || ||xcu − xcu|| ≤ ||C−1|| 1 − 2δQ2 and that

cu cu cu cu cu ξ − ξ =[C + Evi(ω, x )](x − x ) cu cu cu cu cu cu cu + E[ui(ω, x )−ui(ω, x )−vi(ω, x )(x −x )] + o(||x −x ||).

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 5952 JI LI, KENING LU, AND PETER BATES

Hence, T cu T cu T cu cu cu ||uj (θ ω, ξ ) − uj (θ ω, ξ ) − Hij(ω)vi(θ ω, ξ )(ξ − ξ )|| cu cu cu cu cu cu = ||A(x − x )+B(ui(ω, x ) − ui(ω, x )) + o(||x − x ||) cu cu −1 cu cu − [A + Bvi(ω, x )][C + Evi(ω, x )] · (ξ − ξ )|| cu cu cu cu cu cu cu = ||A(x − x )+B[ui(ω, x ) − ui(ω, x )] − [A + Bvi(ω, x )](x − x ) cu cu −1 cu cu − [A + Bvi(ω, x )][C + Evi(ω, x )] E[ui(ω, x ) − ui(ω, x ) cu cu cu cu cu − vi(ω, x )(x − x )] + o(||x − x ||)|| cu cu −1 cu cu = ||{B − [A + Bvi(ω, x )][C + Evi(ω, x )] · E}[ui(ω, x ) − ui(ω, x ) cu cu cu cu cu − vi(ω, x )(x − x )] + o(||x − x ||)|| cu cu cu cu cu cu ≤ [||B|| + O(η + δ)]γω(||x − x ||)||x − x || + ||o(||x − x ||)|| −1 [||B|| + O(η + δ)]||C || ≤ γ (||xcu − xcu||)||ξcu − ξcu|| + ||o(||xcu − xcu||)||. 1 − 2δQ2 ω

||B||+O(η+δ)||C−1 || Taking s = 1−2δQ2 ,ifη and δ are small enough, then s<1. Let Q z = 1−2δQ2 .Then ||xcu − xcu|| ≤ z||ξcu − ξcu||

(5.5) γθT ω(a) ≤ sγω(za)+r(ω, a).

Note that z can be taken to be larger than 1 because γω(a) is increasing in a.This completes the proof of the Claim.

We are ready to show γω(a) → 0asa → 0. Replace a successively by az−1,az−2, ··· ,az−n, replace ω successively by ω, θT ω, ··· ,θ(n−1)T ω, weigh the terms with sn−1,sn−2, ··· , 1 and add them together to get

−n n (n−1)Tω −n (n−2)Tω −n+1 γθnT ω(az ) ≤ s γω(a)+r(θ ,az )+sr(θ ,az ) + ···+ sn−1r(ω, az−1) 1 ≤ sn · 2δ + sup max r(ω, t). 1 − s ω 0≤t≤a Since here ω is arbitrary, we get

−n n 1 γω(sz ) ≤ 2δs + sup max r(t). 1 − s ω 0≤t≤a

It follows that γω(a) → 0asa → 0. So u is diﬀerentiable. Then from the deﬁnition of vn we have vn ∈ C0.Since vn → v uniformly, v ∈ C0.SinceDu = v, u ∈ C1.

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use NH INVARIANT MANIFOLD FOR RDS 5953

Next, we show u ∈ Cr. Knowing u ∈ C1, by the construction of vn, and using the partition of unity {h },wehavevn ∈ C1 and i n+1 T cu n cu n cu −1 vj (θ ω, ξ )= hi[A + Bvi (ω, x )][C + Evi (ω, x )] . Hence, Dvn+1(θT ω, ξcu)[C + Ev(ω, xcu)] j i n cu n cu −1 = hi[BDvi (ω, x )(C + Evi (ω, x )) − n cu n cu −1 n cu n cu −1 (A + Bvi (ω, x ))(C + Evi (ω, x )) EDvi (ω, x )(C + Evi (ω, x )) ] + (terms not involving derivatives of vk or v), which yields Dvn+1(θT ω, ξcu) j n cu n cu −1 = hi BDvi (ω, x )(C + Evi (ω, x ))

− n cu n cu −1 n cu n cu −1 (A + Bvi (ω, x ))(C + Evi (ω, x )) EDvi (ω, x )(C + Evi (ω, x ))

−1 cu × C + Evi(ω, x ) + (terms not involving derivatives of vk or v). Then, using Lemma 4.2 and arguments similar to those in Lemma 5.1 and 5.2, one can show Dvn is a Cauchy sequence. By the definition of vn, Dvn ∈ C0. Hence Dv exists and equals the limit of Dvn.Sov ∈ C1 and u ∈ C2. By using induction and Lemma 4.2, one can prove that Dk−1vn is a Cauchy sequence in C0 and converges to Dk−1v uniformly. So v ∈ Ck−1, i.e., u ∈ Ck. Thus, we have Proposition 5.3. u ∈ Cr. This completes the proof of the smoothness of the random unstable manifold W˜ u. From the proof, we also obtain the following property, which will be used in the next section: cu Proposition 5.4. Each vi is measurable in (ω, x ) jointly. n+1 n Proof. From the definition of the sequence vn.Eachv is defined by H(ω, v )ina local chart. H(ω, v) has the following Carathéodory property: H(·,v)ismeasurable for any fixed v,andH(ω, ·)isCr−1 for almost every ω ∈ Ω. So as long as v0 is measurable in ω and Cr−1 in xcu,eachvn is measurable in ω and Cr−1 in xcu. Since vn is measurable in ω and Cr−1 in xcu,itismeasurablein(ω, xcu)jointly (see [CV]). Since v0 is chosen to be 0, all the above hold. Then the limit v of vn is jointly measurable in (ω, xcu).

6. Measurability of the unstable manifold and its tangent space In this section, we prove that the unique random unstable manifold is a random set and the tangent space of the random unstable manifold is measurable. Let u = {u(ω):ω ∈ Ω} be the unique ﬁxed point of G in Sδ. For simplicity, we use u(ω) to denote both the section and the graph of the section. Proposition 6.1. u(ω) is a random set.

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 5954 JI LI, KENING LU, AND PETER BATES

Proof. Recall ˆ i ≡ × u −1 Di × Uk (σi τi ) ( k B(0, ) and let s 0 ˆ i u (ω)= U3, i=1 u1(ω)=φ(T,θ−T ω, u0(θ−T ω)) ∩ V, un+1(ω)=φ(T,θ−T ω, un(θ−T ω)) ∩ V, ··· un = {un(ω):ω ∈ Ω}. Recall that V is a tubular neighborhood of M in Rn which is Cr diffeomorphic 0 to E˜. Since we identify V with E˜( ), u (ω) can be viewed as a set or the zero section. Here, we view u0(ω)asaset.WehaveV ∈B(Rn). Since G is a uniform contraction on Sδ,wehave un → u, i.e., un(ω) → u(ω) uniformly in ω ∈ Ω. n { } s ˆ i We will show that all u (ω) are random sets. Let mk be dense in i=1 U3. Define 0 yk(ω)=mk and let 1 −T 0 −T ··· yk(ω)=φ(T,θ ω, yk(θ ω)), , n+1 −T n −T yk (ω)=φ(T,θ ω, yk (θ ω)). B R ⊗F⊗BRn n Since φ(t, ω, m)is ( ) ( )-measurable, we know that each yk (ω)is B Rn { n } ∩ n n ( )-measurable. We also know that yk (ω) k u (ω)isdenseinu (ω), since φ(−T,θT ω, ·)iscontinuous. For any x = xcu ∈ Rn, define n | − | rx (ω) = inf x y ; y∈un(ω) n ∈ R then rx is a random variable. This is because for each fixed a , { n ≥ } ω : rx (ω) a = {ω :inf|x − y|≥a} ∈ n y u (ω) { | n ∈ | − n |≥ }∪{ | n ∈ c} = ( ω yk (ω) V, x yk (ω) a ω yk (ω) V ), k F n which is -measurable because of the measurability of yk (ω)andV .Sowehave n n proved rx (ω) is a random variable. Since n, x are arbitrary, u (ω) are random sets. To show that u(ω)isarandomset,letx ∈ Rn and define

rx(ω) = inf |x − y|. y∈u(ω)

We want to show rx(ω) is a random variable. Since un(ω) → u(ω) uniformly for ω ∈ ΩinC0 norm, we have n → rx (ω) rx(ω)

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use NH INVARIANT MANIFOLD FOR RDS 5955

pointwise, which implies that rx(ω) is a random variable, hence, uω is a random set. This completes the proof of the proposition.

To show the measurability of the tangent space T W˜ u, we need to show ˜ u ˜ u ˜ u T W :(ω, m) ∈ W → T(ω,m)W

˜ u n is measurable. Since each T(ω,m)W is an (n − k)-dimensional subspace of R ,the map from W˜ u to the tangent space can be represented by

u u u T W˜ : W˜ = {(ω, m):ω ∈ Ω,m∈ W˜ (ω)}→Kn−k,

˜ u (ω, m) → T(ω,m)W ∈ Kn−k, n where Kn−k is the set of all (n − k)-dimensional subspaces of R . n Consider the metric introduced by [K]. Let N1, N2 be a linear subspace of R { } and let SN1 be the unit sphere in N1. Deﬁne, for N1 = 0 = N2,

d1(N1,N2)= sup dist(x, SN2 ), ∈ x SN1

d2(N1,N2)=max(d1(N1,N2),d2(N1,N2)). Also deﬁne

d2(0,N)=d2(N,0) = 1. Then the set of all closed linear subspaces of Rn is a complete metric space with metric d2 and Kn−k is also a complete metric space with this metric. See also [LL]. Di × × Rk ∈ Corresponding to the local chart 3 B(0, ) ,foreachω Ω, there is an u open set Uˆi(ω)onW˜ (ω). We denote

Uî := {(ω, m):ω ∈ Ω,m∈ Uî(ω)}. Then u W˜ = Uˆ1 ∪ Uˆ2 ∪···∪Uˆs. u Each Uî is a measurable subset of W˜ . Di × × Rk ˆ In the local chart 3 B(0, ) , Ui is constructed as a collection of Lipschitz cu Di × cu functions ui(ω, x )over 3 B(0, ). vi(ω, x ) is the derivative of ui(ω, x)which n−k k cu has values in L(R , R ). We usev î(ω, x ) to denote the tangent space in the cu local chart. Sov î(ω, x ) has values in Kn−k. By an elementary computation, for ∈ cu ∈Di × cu Rn−k ×{ } any ω Ω, x 3 B(0, ), the d2 distance betweenv î(ω, x )and 0 is at most 2 − √ 2 , which, for convenience, will be denoted by δ∗. Then the 1+δ2 range ofv î can be denoted by

Rn−k ×{ } ∗ Bd2 ( 0 ,δ ), n the closed set containing all (n−k)-dimensional subspaces of R ,whosed2 distance Rn−k ×{ } ∗ Di × × to 0 are no more than δ . Remember this set is in the chart 3 B(0, ) Rk. u u In order for T W˜ to be a well-defined map, the range of all T W˜ |Uî should be in the same metric space. In other words, all values ofv î should be measured in a single chart.

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 5956 JI LI, KENING LU, AND PETER BATES

Di × × Rk D1 × × Rk Suppose the coordinate change from 3 B(0, ) to 3 B(0, ) is D1 × × Rk represented by the invertible matrix Ai1. Then in the chart 3 B(0, ) ,ˆv is represented by (A11vˆ1, ··· ,As1vˆs). It has range s Rn−k ×{ } ∗ Ai1Bd2 ( 0 ,δ ). i=1 To prove the measurability ofv ˆ = T W˜ u, it is enough to prove the measurability u of each T W˜ |Uî or, equivalently, the measurability of each Ai1vî as a function Di × × Rk defined in the chart 3 B(0, ) and taking value in Kn−k measured in the D1 × × Rk chart 3 B(0, ) : ×Di × → Rn−k ×{ } ∗ Ω 3 B(0, ) Ai1Bd2 ( 0 ,δ ).

Since Ai,1 is an invertible matrix, it is a diﬀeomorphism from Rn−k ×{ } ∗ Bd2 ( 0 ,δ ) to Rn−k ×{ } ∗ Ai1Bd2 ( 0 ,δ ).

So it suffices to prove the measurability ofv î as a function defined in the chart Di × ×Rk Di × ×Rk 3 B(0, ) and taking value in Kn−k measured in the chart 3 B(0, ) .

Proposition 6.2. vˆi is measurable.

Proof. From Section 5 we know vi is measurable. We use the measurability of vi to prove the measurability ofv ˆi. The operator norm of each vi(ω, x)isatmostδ.LetB(0,δ) be the closed ball n−k k centered at 0 with radius δ in the space L(R , R ). It induces a subset Kδ of Kn−k: n−k Kδ := {(a, La˜ ):a ∈ R , L˜ ∈ B(0,δ)}. We also have the representation Rn−k ×{ } ∗ Kδ = Bd2 ( 0 ,δ ). This is because for any L˜ ∈ B(0,δ), n−k n−k 2 d2(R ×{0}, (I × L˜)(R )) = 2 − , 1+||L˜||2

which is no more than δ∗ = 2 − √ 2 . On the other hand, for any 0 <δ ≤ δ, 1+δ2 1 n n−k and J an (n − k)-dimensional subspace of R whose d2 distance to R ×{0} is n−k n−k δ1,thereexistsanL˜ such that ||L˜|| = δ1 and R × L˜(R )=J.Sowehave Rn−k ×{ } ∗ Kδ = Bd2 ( 0 ,δ ).

Deﬁne a metric d3 on Kδ by n−k n−k n−k n−k d3(R × L1(R ), R × L2(R )) := ||L1 − L2||L(Rn−k,Rk).

This makes Kδ into a metric space, and obviously we have Rn−k ×{ } Kδ = Bd3 ( 0 ,δ).

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use NH INVARIANT MANIFOLD FOR RDS 5957

Lemma 6.1. d2 and d3 generate the same topology on Kδ.

Proof. Let Dn ⊂ Kδ.Itisobviousthatd2(Dn,D0) → 0 if and only if d3(Dn,D0) → 0.

We use the metric d3 on Kδ.ThismakesKδ isomorphic to B(0,δ). So the measurability ofv ˆi as a function to Kδ is equivalent to the measurability of vi as a function to B(0,δ). By Proposition 5.4, vi is measurable, sov ˆi is measurable. From the discussion before Proposition 6.2, we have: Proposition 6.3. The tangent space of the random unstable manifold W˜ u is measurable.

7. Existence of the random stable manifold Since the random flow φ(t, ω) is invertible, by reversing the time and applying the results on unstable manifold in Sections 4, 5, and 6, we have Proposition 7.1. There exists a unique Cr random stable manifold W˜ s(ω) in the neighborhood V of M. Moreover, it is a random set and its tangent space is measurable. Taking the intersection of W˜ u(ω)andW˜ s(ω), we get an invariant manifold M˜ (ω) of the random flow φ(t, ω). From the proof of the existence of W˜ s(ω)andW˜ u(ω)we can find this intersection by taking a measurable m-dimensional random manifold M0(ω) on the random unstable manifold as a graph over M and mapping it under the inverse flow. For example, we can take the random set which corresponds to cu c ui(ω, x )=ui(ω, x , 0) in a local chart. Iterates of this random set approach the random stable manifold uniformly under the inverse random flow. On the other hand, the random set stays in the random unstable manifold. So it converges to the intersection of the random stable and unstable manifolds. In other words, the invariant manifold of intersection is the limit of M0(ω) under the inverse random c flow. Since we have the measurability and smoothness of ui(ω, x , 0) in local charts, we know M0(ω) is measurable and smooth, and also since the derivatives (tangent spaces) converge uniformly, then the limit M˜ (ω)isalsomeasurableandsmooth. M˜ (ω) is obviously Cr diffeomorphic to M for each ω ∈ Ωbecauseitisthegraph of a Cr section over the tangent bundle of M. The measurability of the tangent space of M˜ (ω) is also proved from the measurability and smoothness of the tangent space of M0(ω). From the measurability and smoothness of the tangent space of M0(ω), we get the measurability of the derivative map of M0(ω, m) in local charts using the Carathéodory property as we did in Proposition 5.4. Then from uniform convergence of the tangent spaces, we get the measurability of the derivative map of M(ω) in local charts. Applying the method of proving the measurability of the tangent space of the unstable manifold, which shows the equivalence between the measurability of the tangent space and the measurability of the tangent map, we obtain the measurability of the tangent space of the random invariant manifold M(ω). Summarizing the above, we get Proposition 7.2. There exists a unique Cr random invariant manifold M˜ (ω) in a neighborhood of M. For each fixed ω ∈ Ω, M˜ (ω) is Cr diffeomorphic to M. Moreover, it is a random set and its tangent space is measurable.

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 5958 JI LI, KENING LU, AND PETER BATES

8. Persistence of normal hyperbolicity In this section we show that M˜ is normally hyperbolic. We prove the following proposition: Proposition 8.1. For each x ∈ M˜ (ω) there exists a splitting Rn = Eu(ω, x) ⊕ Ec(ω, x) ⊕ Es(ω, x) of closed subspaces with associated projections Πu(ω, x), Πc(ω, x),andΠs(ω, x) such that (i) The splitting is invariant:

i i Dxφ(t, ω)(x)E (ω, x)=E (θtω, φ(t, ω)(x)), for i = u, c, s. i → i (ii) Dxφ(t, ω)(x) Ei(ω,x) : E (ω, x) E (θtω, φ(t, ω)(x)) is an isomorphism for i = u, c, s. Ec(ω, x) is the tangent space of M(ω) at x. (iii) There are (θ, φ)-invariant random variables α, β : M→(0, ∞), 0 <α<β, and a tempered random variable K(ω, x):M→[1, ∞) such that

s −β(ω,x)t (8.1) ||Dxφ(t, ω)(x)Π (ω, x)|| ≤ K(ω, x)e for t ≥ 0, u β(ω,x)t (8.2) ||Dxφ(t, ω)(x)Π (ω, x)|| ≤ K(ω, x)e for t ≤ 0, c α(ω,x)|t| (8.3) ||Dxφ(t, ω)(x)Π (ω, x)|| ≤ K(ω, x)e for −∞

Since E˜i is arbitrarily close to Ei for i = s, u,wehavethatE˜i(ω) is uniformly close to Ei. Since there is no invariance condition on E˜i(ω), we can modify E˜i(ω, m) such that it is Cr in m for each ﬁxed ω ∈ Ω and still stays close to Ei.Thisisfrom [W]. In any case, E˜i(ω, m) is not necessarily measurable. Deﬁne E˜c(ω, m) ≡ T M˜ (ω), and for i = s, u, c, Π˜ i ≡ Π˜ i(ω, m) the projections onto E˜i(ω, m). Recall that 0 <α<βare constants associated with the normal hyperbolicity of M with respect to ψ(t).

Lemma 8.2. (1) There exist positive constants 0

s −t s −t t ||Π˜ Dφ(t, θ ω)(φ(−t, ω)(m))|E˜ (θ ω)||

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use NH INVARIANT MANIFOLD FOR RDS 5959

for all m ∈ M˜ (ω) and t ≥ 0,and u −t u −t t ||Π˜ Dφ(t, θ ω)(φ(−t, ω)(m))|E˜ (θ ω)|| 0 and r >rsuch that s −t s −t c r ||Π˜ Dφ(t, θ ω)(φ(−t, ω)(m))|E˜ (θ ω)|| ||Dφ(−t, ω)(m)|E˜ (ω)|| 0 and r >rsuch that ||Π˜ sDφ(t, θ−tω)φ(−t, ω)(m)|E˜s(θ−tω)|| ||Dφ(−t, ω)(m)|E˜c(ω)|| −t c −t r−1 ||Dφ(t, θ ω)φ(−t, ω)(m)|E˜ (θ ω)|| 0 such that ||Dψ(T (m))ψ(−T (m))(m)|Es|| < 1. Hence, there exists a neighborhood U(m)ofm in M such that for m ∈ U(m), ||Dψ(T (m))ψ(−T (m))(m)|Es|| < 1.

Since M is compact, we may choose ﬁnitely many points m1,m2, ··· ,mN such that

M⊂U(m1) ∪ U(m2) ∪···∪U(mN ). Choose 0

s T (mi) ||Dψ(T (mi))ψ(−T (mi))(m )|E ||

s −T (mi) s −T (mi) T (mi) ||Π˜ Dφ(T (mi),θ ω)φ(−T (mi),ω)(u(ω, m ))|E˜ (θ ω)||

M˜ (ω) ⊂ U(ω, m1) ∪ U(ω, m2) ∪···∪U(ω, mN ) and

s −T (mi) s −T (mi) T (mi) ||Π˜ Dφ(T (mi),θ ω)φ(−T (mi),ω)(m ˜ )|E˜ (θ ω)||

for all ω ∈ Ωand˜m ∈ U(ω, mi). Fix an arbitrary ω ∈ Ω and let m ∈ M˜ (ω) be given. Choose a sequence of integers i(1),i(2), ···, as follows. Choose i(1) such that m ∈ U(ω, mi(1)). If i(1),i(2), ··· ,i(j) have been chosen, let τ(j)=T (mi(1))+···+ T (mi(j)). Choose

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 5960 JI LI, KENING LU, AND PETER BATES

−τ(j) i(j+1) such that φ(−τ(j),ω)(m) ∈ U(θ ω, mi(j+1)). Letting t>0, it is possible to write t = τ(j)+r for some j and 0 ≤ r

and the supremum is taken over all ω ∈ Ω,m ∈ M˜ (ω)and0≤ r

Lemma 8.3. There exist unique subbundles Es(ω) and Eu(ω) of T Rn|M˜ (ω) such that Es(ω) is complementary to T M˜ (ω) in T W˜ s(ω)|M˜ (ω) and Eu(ω) is complementary to T M˜ (ω) in T W˜ u(ω)|M˜ (ω). Moreover, Ei(ω)(i = s, u) is Cr−1 and invariant under Dφ(t, ω) for any t ∈ R and ω ∈ Ω. Proof. We will show that for some K to be determined later, there exists a unique random subbundle Eu(ω)ofT W˜ s(ω)|M˜ (ω) that is invariant under Dφ(K, ω). This will be suﬃcient to conclude the invariance under Dφ(t, ω) for any t ∈ R.Thisis because once we proved the ﬁrst, we know that Dφ(t, ω)(Eu(ω)) is also invariant under Dφ(K, ω). Then by the uniqueness we are done. To start, we notice that any bundle complementary to T M˜ (ω)inT W˜ u(ω)|M˜ (ω) is the graph of a family of linear maps h(ω, m):E˜u(ω, m) → T M˜ (ω, m), and the bundle is invariant under Dφ(K, ω) if and only if Dφ(K, θ−K ω)h(θ−Kω, φ(−K, ω, m)) = h(ω, m) for all ω ∈ Ωandm ∈ M˜ (ω). Equivalently, h(ω, m)Π˜ uDφ(K, θ−Kω, φ(−K, ω, m)) · (ξcu + h(θ−K ω, φ(−K, ω, m)))ξcu = Π˜ cDφ(K, θ−Kω, φ(−K, ω, m)) · (ξ + h(θ−K ω, φ(−K, ω, m)))ξ

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use NH INVARIANT MANIFOLD FOR RDS 5961

||Dφ(K, θ−K ω, φ(−K, ω, m))|E˜c(θ−K ω)|| ||Dφ(−K, ω, m)|E˜c(ω)||k (8.5) 1 ||Π˜ uDφ(−K, ω, m)|E˜u|| ≤ 4 for 1 ≤ k ≤ r − 1. Deﬁne a space S whose element h ∈ S is of the form h = {h(ω, m)|ω ∈ Ω,m∈ M˜ (ω)}.

Deﬁne the norm ||·||S by || || || || h S := sup max h(ω, m) L(E˜u(ω,m),T M˜ (ω,m)). ω m∈M˜ (ω) Under this norm, S is a complete metric space. Let h0(ω, m) ≡ 0 ∈ L(E˜u(ω, m),TM˜ (ω, m)), and deﬁne hn+1(ω, m) = Π˜ cDφ(K, θ−Kω, φ(−K, ω, m)) ◦ Π˜ uDφ(−K, ω, m)|E˜u + Π˜ cDφ(K, θ−K ω, φ(−K, ω, m)) ◦ hn(θ−K ω, φ(−K, ω, m)) ◦ Π˜ uDφ(−K, ω, m)|E˜u.

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 5962 JI LI, KENING LU, AND PETER BATES

Then for all n ≥ 0 and for any fixed ω, hn(ω, m)isCr−1 in m . By (8.4), hn is a Cauchy sequence. Suppose h is the unique limit of hn.Thenh satisfies h(ω, m) = Π˜ cDφ(K, θ−K ω, φ(−K, ω, m)) ◦ Π˜ uDφ(−K, ω, m)|E˜u + Π˜ cDφ(K, θ−Kω, φ(−K, ω, m)) ◦ h(θ−K ω, φ(−K, ω, m)) ◦ Π˜ uDφ(−K, ω, m)|E˜u, so h(ω, m) represents the unique invariant bundle E(ω, m). Since hn(ω, m) → h(ω, m) uniformly, h(ω, m)isC0 in m. k n ≤ − By (8.5), Dmh (ω, m)fork r 1 is a Cauchy sequence in the corresponding k n ∞ k space. So Dmh (ω, m) converges uniformly as n goes to . Therefore, Dmh(ω, m) k n r−1 exists and equals the limit of Dmh (ω, m), which means h(ω, m)isC in m for any fixed ω. Hence, we obtain a unique Cr−1 invariant bundle Eu(ω). Similarly, we have a unique Cr−1 invariant bundle Es(ω). From now on, we use Es(ω)andEu(ω) to denote the unique invariant bundle of Lemma 8.3. Since ψ(t)andφ(t, ω) are uniformly close, and M and M˜ (ω) is uniformly close, we conclude that E(ω) is uniformly close to E.SinceM is compact, the angle between E(m)andTmM is uniformly bounded away from 0. So we also have that the angle between E(ω, m)andTmM˜ (ω) is uniformly bounded || s|| || u|| 1 1 away from 0. So Π , Π , ||Πs|| and ||Πu|| are bounded. Suppose they are all bounded by c4 > 0. Lemma 8.4. (8.1), (8.2) and (8.3) are satisfied for Es(ω), Eu(ω) and Ec(ω)= T M˜ (ω). Proof. For any ν ∈ T W˜ s(ω)|M˜ (ω), we have ν − Πsν ∈ T M˜ (ω). So Π˜ sΠs = Π˜ s. Similarly, ΠsΠ˜ s =Πs. For each fixed ω ∈ Ω, m ∈ M˜ (ω)andν ∈ Es(ω, m), letν ˜ = Π˜ sν.Then ν˜ − ν ∈ T M˜ (m)and −1|˜ s − | c4 Π Dφ( t, ω)(˜ν) −1|˜ s − ˜ s | −1|˜ s − |≤| s ˜ s − | = c4 Π Dφ( t, ω)(Π ν) = c4 Π Dφ( t, ω)(ν) Π Π Dφ( t, ω)(ν) s s s s = |Π Dφ(−t, ω)(ν)|≤c4|Π˜ Π Dφ(−t, ω)(ν)| = c4|Π˜ Dφ(−t, ω)(ν)| s s s = c4|Π˜ Dφ(−t, ω)(Π˜ ν)| = c4|Π˜ Dφ(−t, ω)(˜ν)|, which gives us −1|˜ s − |≤| s − |≤ |˜ s − | (8.6) c4 Π Dφ( t, ω)(˜ν) Π Dφ( t, ω)(ν) c4 Π Dφ( t, ω)(˜ν) . From (8.6), all properties listed in Lemma 8.2 persist if we change E˜s(ω)and E˜u(ω)toEs(ω)andEu(ω). From (1) of Lemma 8.2, if we take K(ω, x)=c1 and β(ω, x)=− log a,then β>0and(θ, φ)-invariant, K(ω, x) is tempered, and (8.1) and (8.2) hold. From the uniform closeness of φ(t, ω)andψ(t), the normal hyperbolicity property of M under ψ(t) and the uniform closeness of M˜ (ω)andM,wehaveforsome (θ, φ)-invariant random variable α : M˜ → (0, ∞) and tempered random variable K(ω, x): M→˜ [1, ∞) such that (8.3) holds.

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use NH INVARIANT MANIFOLD FOR RDS 5963

What we need to prove is that rα < β. From (2) of Lemma 8.2, α and β can be chosen such that rα < β. Lemma 8.5. Ei(ω, m) is measurable. Proof. In Sections 6 and 7, we proved the measurability of Ec.Hereweonlyprove the measurability of Eu.ForEs, just follow exactly the same argument for Eu. ˜ ˜ u Take the orthogonal complement of T(ω,m)M in T(ω,m)W to get a normal bundle Eû(ω, m). Since T M˜ and T W˜ u are both measurable, Eû(ω, m) is also measurable. û ˜ Moreover, the angle between E (ω, m)andT(ω,m)M is uniformly bounded away from 0. So by Lemma 8.4, all properties listed in Lemma 8.2 hold, even though Eˆ(ω, m)maynotclosetoEu. So we can replace the bundle E˜u(ω, m) in Lemma 8.3 by Eû(ω, m) and still get the unique invariant bundle Eu(ω, m) by the same argument. In this way, the contraction mapping in the functional equation of h(ω, x)is measurable in ω and Cr−1 in x. As we obtained the measurability of v in Propo- sition 5.4, we obtain the measurability of h(ω, x)–the representation of Eu(ω, m). This completes the proof of the lemma. By using Lemmas 8.3, 8.4 and 8.5, we have Proposition 8.1. Summarizing Propositions 4.1, 5.3, 6.1, 6.3, 7.1, 7.2 and 8.1, gives Theorem 2.1.

9. Persistence for overflowing and inflowing invariant manifolds In this section, we give the persistence of overflowing and inflowing manifolds. The proof of these results follows in the same fashion as the persistence of normally hyperbolic invariant manifolds with slight modifications. Theorem 9.1. Assume that ψ(t)(x) is a Cr flow, r ≥ 1, and has a compact, connected Cr normally hyperbolic overflowing invariant manifold M¯ = M∪∂M⊂ Rn. Then there exists ρ>0 such that for any Cr random flow φ(t, ω, x) in Rn,if

||φ(t, ω) − ψ(t)||C1 <ρ, for t ∈ [0, 1],ω ∈ Ω, then if α

M⊂M¯ 1 ⊂ M¯ 1 ⊂M2 ⊂ M¯ 2,

where Mi, i =1, 2, are the interiors of M¯ i. Following the same argument as in the proof of Theorem 2.1, one can construct M˜ (ω)nearM1 in a tubular neighborhood V of M¯ 2 as a section of a normal bundle. Since the normal direction contains only the stable direction, there is no need to combine the center and the unstable direction. So the proof for the persistence actually is shorter. We have the following theorem for the inflowing manifolds. Theorem 9.2. Assume that ψ(t)(x) is a Cr flow, r ≥ 1, and has a compact, connected Cr normally hyperbolic inflowing invariant manifold M¯ = M∪∂M⊂ Rn. Then there exists ρ>0 such that for any Cr random flow φ(t, ω, x) in Rn,if

||φ(t, ω) − ψ(t)||C1 <ρ, for t ∈ [0, 1],ω ∈ Ω,

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 5964 JI LI, KENING LU, AND PETER BATES

then if α0 such that for any random Cr ﬂow φ(t, ω) in Rn,if

||φ(t, ω) − ψ(t)||C1 <ρ, for t ∈ [0, 1],ω ∈ Ω, as long as φ(t, ω) has a compact, connected Cr random overﬂowing (inﬂowing) invariant manifold M˜¯ (ω) with stable and unstable manifold W˜ s(ω) and W˜ u(ω) such that M˜¯ (ω), W˜ s(ω) and W˜ u(ω) are C1 close to M¯ , Ws and Wu, respectively, then M˜¯ (ω) is normally hyperbolic with constant α

[A] L. Arnold, Random Dynamical Systems, Springer, New York (1998). MR1723992 (2000m:37087) [BFGJ] P. Bates, P. Fife, R. Gardner, and C.K.R.T. Jones, Phase field models for hypercooled solidification,PhysicaD104 (1997), 1-31. MR1447948 (97m:80011) [BLZ1] P. Bates, K. Lu, and C. Zeng, Existence and persistence of invariant manifolds for semiflows in Banach space, Memoirs of the AMS, 135 (1998). MR1445489 (99b:58210) [BLZ2] P. Bates, K. Lu, and C. Zeng, Persistence of overflowing manifold for semiflow, Comm. Pure Appl. Math. 52 (1999), 983–1046. MR1686965 (2000f:37116) [BLZ3] P. Bates, K. Lu, and C. Zeng, Invariant foliations for semiflows near a normally hyperbolic invariant manifold, Trans. Amer. Math. Soc., 352 (2000), 4641–4676. MR1675237 (2001b:37031) [BF] A. Bensoussan and F. Flandoli, Stochastic inertial manifold, Stochastics Stochastics Rep., 53 (1995), no. 1–2, 13–39. MR1380488 (97d:60102) [Bx] P. Boxler, A stochastic version of center manifold theory, Probab. Theory Related Fields 83 (1989), no. 4, 509–545. MR1022628 (91d:58115) [BK] M. Brin and Y. Kifer, Dynamics of Markov chains and stable manifolds for random diffeomorphisms, Ergodic Theory Dynam. Systems 7 (1987), no. 3, 351–374. MR912374 (88k:58162) [B] P. Brunovsky, Tracking invariant manifolds without differential forms,ActaMath.Univ. Comenianae Vol. 65, (1996), no. 1, 23-32. MR1422292 (97m:34100) [CDLS] T. Caraballo, J. Duan, K. Lu, and B. Schmalfuss, Invariant manifolds for random and stochastic partial differential equations, Adv. Nonlinear Stud. 10 (2010), no. 1, 23–52. MR2574373 (2011f:60118)

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use NH INVARIANT MANIFOLD FOR RDS 5965

[CLR] T. Caraballo, J. Langa and J. C. Robinson, A stochastic pitchfork bifurcation in a reaction-diffusion equation, Proc. R. Soc. Lond. A 457 (2001), 2441-2453 MR1857922 (2003a:60106) [C] A. Carverhill, A formula for the Lyapunov numbers of a stochastic flow. Application to a perturbation theorem, Stochastics 14 (1985), no. 3, 209–226. MR800244 (86k:93141) [CV] C. Castaing and M. Valadier. Convex Analysis and Measurable Multifunctions.LNM 580. Springer–Verlag, Berlin–Heidelberg–New York, 1977. MR0467310 (57:7169) [DD] G. Da Prato and A. Debussche. Construction of stochastic inertial manifolds using back- ward integration, Stochastics Stochastics Rep. 59 (1996): no. 3–4, 305–324. MR1427743 (98d:60121) [D] S. Dahlke, Invariant manifolds for products of random diffeomorphisms, J. Dynam. Differential Equations 9 (1997), no. 2, 157–210. MR1451289 (98m:58078) [PJ] Pavel Drabek and Jaroslav Milota, Methods of Nonlinear Analysis. Applications to Dif- ferential Equations ,Birkhäuser Verlag, Basel, 2007. MR2323436 (2008i:47134) [DSL1] J. Duan, K. Lu, and B. Schmalfuss. Invariant manifolds for stochastic partial differential equations, Annals of Probability 31(2003), 2109-2135. MR2016614 (2004m:60136) [DSL2] J. Duan, K. Lu and B. Schmalfuss, Smooth stable and unstable manifolds for stochastic evolutionary equations, J. Dynamics and Diff. Eqns, 16 (2004), 949-972. MR2110052 (2005j:60124) [F1] N. Fenichel, Persistence and smoothness of invariant manifolds for flows, Indiana Univ. Math. Journal 21 (1971), 193–226. MR0287106 (44:4313) [F2] N. Fenichel, Asymptotic stability with rate conditions, Indiana Univ. Math. Journal 23 (1974), 1109–1137. MR0339276 (49:4036) [F3] N. Fenichel, Asymptotic stability with rate conditions II, Indiana Univ. Math. Journal 26 (1977), 81–93. MR0426056 (54:14002) [GLS] M.Garrido-Atienza,K.Lu,andB.Schmalfuss,Unstable invariant manifolds for stochastic PDEs driven by a fractional Brownian motion, J. Differential Equations 248 (2010), no. 7, 1637–1667. MR2593602 (2011a:60226) [GC] T. V. Girya and I. D. Chueshov, Inertial manifolds and stationary measures for stochas- tically perturbed dissipative dynamical systems, Sb. Math. 186 (1995), no. 1, 29–45. MR1641664 (99k:34134) [H] J. Hadamard, Sur l’iteration et les solutions asymptotiques des equations differentielles, Bull. Soc. Math. France 29 (1901), 224-228. [HPS] M. W. Hirsch, C.C. Pugh, and M. Shub, Invariant manifolds, Lecture Notes in Mathe- matics, 583, Springer-Verlag, New York, 1977. MR0501173 (58:18595) [JK] C. K. R. T. Jones and N. Kopell, Tracking invariant manifolds with differential forms in singularly perturbed systems, J. Differential Equations 108 (1994), 64-88. MR1268351 (95c:34085) [J] C.K.R.T.Jones,Stability of the travelling wave solution of the FitzHugh-Nagumo system,Trans.Amer.Math.Soc.286 (1984), 431–469. MR760971 (86b:35011) [K] T. Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132 Springer-Verlag New York, Inc., New York, 1966. MR0203473 (34:3324) [KSS] M. Krupa, B. Sandstede, and P. Szmolyan, Fast and Slow Waves in the FitzHugh- Nagumo Equation, J. Differential Equations 133 (1997) 49-97. MR1426757 (98e:35165) [LiL] W. Li and K. Lu, Sternberg theorems for random dynamical systems, Comm. Pure Appl. Math. 58 (2005), no. 7, 941–988. MR2142880 (2006b:37099) [LL] Z. Lian and K. Lu, Lyapunov Exponents and Invariant Manifolds for Infinite Dimen- sional Random Dynamical Systems in a Banach Space, Memoirs of the AMS. 206 (2010), no. 967, 106 pp. MR2674952 (2011g:37145) [LQ] P-D. Liu and M. Qian, Smooth ergodic theory of random dynamical systems, Lecture Notes in Mathematics, 1606. Springer-Verlag, Berlin, 1995. MR1369243 (96m:58139) [LS] K. Lu and B. Schmalfuss, Invariant manifolds for stochastic wave equations,J.Differ- ential Equations 236 (2007), no. 2, 460–492. MR2322020 (2009c:37063) [MS] S.-E. A. Mohammed and M. K. R. Scheutzow, The stable manifold theorem for stochastic differential equations, The Annals of Probability 27 (1999), no. 2, 615–652. MR1698943 (2001e:60121)

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 5966 JI LI, KENING LU, AND PETER BATES

[MZZ] S.-E. A. Mohammed, T. Zhang, and H. Zhao, Thestablemanifoldtheoremforsemi- linear stochastic evolution equations and stochastic partial differential equations, Mem- oirs of the American Mathematical Society 196 (2008), no. 917, 105 pp. MR2459571 (2010b:60175) [R] D. Ruelle, Characteristic exponents and invariant manifolds in Hilbert spaces, Ann. of Math. 115 (1982), 243–290. MR647807 (83j:58097) [S1] S. Schecter, Existence of Dafermos profiles for singular shocks, J. Differential Eqs. 205 (2004), 185-210. MR2094383 (2005k:35269) [S2] S. Schecter, Exchange lemmas. II. General exchange lemma, J. Differential Equations 245 (2008), no. 2, 411–441. MR2428005 (2009h:37052) [S] B. Schmalfuss, A random fixed point theorem and the random graph transforma- tion, Journal of Mathematical Analysis and Applications, 225 (1998), no. 1, 91–113. MR1639297 (99i:47118) [WD] W. Wang and J. Duan, A dynamical approximation for stochastic partial differential equations, J. Math. Phys. 48 (2007), no. 10, 102701, 14pp. MR2362785 (2009d:60211) [W] T. Wanner, Linearization random dynamical systems. In C. Jones, U. Kirchgraber and H. O. Walther, editors, Dynamics Reported, Vol. 4, 203-269, Springer-Verlag, New York, 1995. MR1346499 (96m:34084) [W] H. Whitney, Differential manifolds, Ann. of Math. 37 (1936), 645-680. MR1503303

Department of Mathematics, Brigham Young University, Provo, Utah 84602 E-mail address: [email protected] Current address: Institute for Mathematics and its Application, University of Minnesota, Minneapolis, Minnesota 55455 E-mail address: [email protected] Department of Mathematics, Brigham Young University, Provo, Utah 84602 – and – School of Mathematics, Sichuan University, Chengdu, People’s Republic of China E-mail address: [email protected] Department of Mathematics, Michigan State University, East Lansing, Michigan 48824 E-mail address: [email protected]

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use