THE STABLE MANIFOLD THEOREM FOR STOCHASTIC DIFFERENTIAL EQUATIONS

Transcription

1 The Annals of Probability 999, Vol. 27, No. 2, THE STABLE MANIFOLD THEOREM FOR STOCHASTIC DIFFERENTIAL EQUATIONS By Salah-Eldin A. Mohammed and Michael K. R. Scheutzow 2 Southern Illinois University at Carbondale and Mathematical Sciences Research Institute and Technical University of Berlin We formulate and prove a local stable manifold theorem for stochastic differential equations (SDEs) that are driven by spatial Kunita-type semimartingales with stationary ergodic increments. Both Stratonovich and Itôtype equations are treated. Starting with the existence of a stochastic flow for a SDE, we introduce the notion of a hyperbolic stationary trajectory. We prove the existence of invariant random stable and unstable manifolds in the neighborhood of the hyperbolic stationary solution. For Stratonovich SDEs, the stable and unstable manifolds are dynamically characterized using forward and backward solutions of the anticipating SDE. The proof of the stable manifold theorem is based on Ruelle Oseledec multiplicative ergodic theory.. Introduction. Consider the following Stratonovich and Itô stochastic differential equations (SDEs) on R d : S dφt = F dt φt φs =x t>s I dφt =Fdt φt φs =x t>s defined on a filtered probability space F Fs t s tp. Equation (S) is driven by a continuous forward backward spatial semimartingale F R R d R d, and equation (I) is driven by a continuous forward spatial semimartingale F R R d R d [0]. Both F and F have stationary ergodic increments. It is known that, under suitable regularity conditions on the driving spatial semimartingale F, the SDE (S) admits a continuous (forward) stochastic flow φ s t R d R d <s t< [0]. The inverse flow is denoted by φ t s = φ s t Rd R d <s t<. This flow is generated by Received October 997; revised November 998. Supported in part by NSF Grants DMS , DMS and by MSRI, Berkeley, California. 2 Supported in part by MSRI, Berkeley, California. AMS 99 subject classifications. Primary 60H0, 60H20; secondary 60H25, 60H05. Key words and phrases. Stochastic flow, spatial semimartingale, local characteristics, stochastic differential equation (SDE), ( perfect) cocycle, Lyapunov exponents, hyperbolic stationary trajectory, local stable/unstable manifolds, asymptotic invariance. 65

2 66 S.-E. A. MOHAMMED AND M. K. R. SCHEUTZOW Kunita s backward Stratonovich SDE, S dφs = F ds ˆ φs φt =x s<t Similarly, the inverse flow φ t s = φ s t Rd R d, <s t< of the Itô equation (I) solves a backward Itô SDE with a suitable correction term ([0], page 7). The main objective of the present article is to establish a local stablemanifold theorem for the SDEs (S) and (I) when the driving semimartingales F and F have stationary ergodic increments. Our main result is Theorem 3.. It gives a random flow-invariant local splitting of R d into stable and unstable differentiable submanifolds in the neighborhood of each hyperbolic ( possibly anticipating) stationary solution. The method we use to establish these results is based on a nonlinear discrete-time multiplicative ergodic theorem due to Ruelle [24] (cf. [25]). Although the article is largely self-contained, familiarity with the arguments in [24] will sometimes be needed. Key ingredients of this approach are Ruelle Oseledec integrability conditions which we prove in Lemma 3. under a very mild integrability hypothesis on the stationary solution. The proof of this lemma is in turn based on spatial estimates on the flow and its derivatives [0], [7]. These estimates are stated in Theorem 2. for easy reference. Several authors have contributed to the development of the stable-manifold theorem for nonlinear SDEs. The first successful attempt was carried out by Carverhill [6] for SDEs on compact manifolds. In [6], a stable manifold theorem is obtained in the globally asymptotically stable case where the Lyapunov exponents of the linearized flow are all negative. The general hyperbolic case with positive Lyapunov exponents is not treated in [6]. The work by Boxler [5] focuses on the existence of a (global) center manifold under small (white) noise. Wanner [27] deals with the existence of global and local invariant manifolds for continuous and discrete-time smooth cocycles. For an account of Wanner s results the reader may look at [27] and [], Chapter 7. The results in [27] and [] are obtained under stringent conditions on the spatial growth of the cocycle, namely, almost sure global boundedness of its spatial derivatives. These conditions clearly cover the case of compact state space (cf. [6]), the case of discrete-time cocycles and the case of random differential equations driven by real-noise ([], Theorems 7.3., 7.3.0, 7.3.4, 7.3.7, 7.5.5). However, they do not apply in our present context of smooth cocycles generated by SDEs in Euclidean space. Typically such cocycles may have almost surely globally unbounded spatial derivatives even if the driving vector fields are smooth with all derivatives globally bounded. See [7] and the examples therein and also [8]. In this paper, we prove the existence of local stable and unstable manifolds for smooth cocycles in Euclidean space that are generated by a large class of SDEs of the form (S) or (I). The regularity conditions imposed on the local characteristics of the driving noise in (S) or (I) are such that the SDE admits

3 STABLE MANIFOLD THEOREM FOR SDEs 67 a global flow for all time. The local stable and unstable manifolds are dynamically characterized in two ways: first, using the cocycle and then through anticipating versions of the underlying forward and backward SDEs. This is done using an approach based on classical work by Ruelle [24] and anticipating stochastic calculus [8 20]. In addition, using a standard imbedding argument, the method of construction of the stable and unstable manifolds also works if the state space R d is replaced by a (possibly non-compact) finitedimensional Riemannian manifold; compare [6]. The multiplicative ergodic theory of linear finite-dimensional systems was initiated by Oseledec in his fundamental work [2]. An infinite-dimensional stable-manifold theorem for linear stochastic delay equations was developed by Mohammed [5] in the white noise case, and by Mohammed and Scheutzow for general semimartingales with stationary ergodic increments [6]. 2. Basic setting and preliminary results. Let F P be a probability space. Let θ R be a P-preserving flow on, namely:. θ is jointly measurable; 2. θt + s = θt θs, s t R; 3. θ0 = I, the identity map on ; 4. P θt = P, t R. Denote by F the P-completion of F. Let Fs t <s t< be a family of sub-σ-algebras of F satisfying the following conditions:. θ r Fs t s+r for all r R, <s t<. 2. For each s R, both F Fs s+u u 0 P and F Fs u s u 0P are filtered probability spaces satisfying the usual conditions [23]. =F t+r A random field F R R d R d is called a (continuous forward) spatial semimartingale helix if it satisfies the following:. For every s R, there exists a sure event s F such that Ft + s x ω =Ft x θs ω + Fs x ω for all t R, all ω s and all x R d. 2. For almost all ω, the mapping R R d t x Ft x ω R d is continuous. 3. For any fixed s R and x R d, the process Fs + t x ω Fs x ω, t 0 is an Fs s+t t 0 -semimartingale. Similarly, a random field F R R d R d is called a continuous backward spatial semimartingale helix if it satisfies () and (2) and has the property that for fixed s R and x R d, the process Fs t x ω Fs x ω, t 0, is an Fs t s t 0 -semimartingale. In (3) above, it is enough to require that the semimartingale property holds for some fixed s (e.g., s = 0); then it will hold automatically for every s R ([3], Theorem 4).

4 68 S.-E. A. MOHAMMED AND M. K. R. SCHEUTZOW Note that a semimartingale helix F always satisfies F0xω=0 for a.a. ω and all x R d. It is also possible to select a suitable perfect version of F such that the helix property () holds for every ω. See [3] for further details, and [22] for other general properties of semimartingale helices. Suppose that the continuous forward semimartingale helix F R R d R d is decomposed as Ft x =Vt x+mt x t R + x R d where V x = V xv d x is a continuous bounded variation process, M x=m xm d x is a continuous local martingale with respect to F0 t t 0, and V0x = M0x = 0 for each x R d. Let M i xm j y be the joint quadratic variation of M i xm j y, for x y R d, i j d. Throughout this paper, assume that F0 has forward local characteristics at x ybt x that satisfy the relations M i xm j yt = t 0 a i j u x y du V i t x = t 0 b i u x du for all i, j d, 0 t T, and where at x y =a ij t x y i j=d, bt x =b t xb d t x. Further measurability properties of the local characteristics are given in [0], pages Note that the local characteristics are uniquely determined by F up to null sets. In what follows, let denote the diagonal =x x x R d in R d R d, and let c be its complement. The space R d carries the usual Euclidean norm. We shall use the notation α = α α 2 α d D α x = α x α xd α d d α = α i i= for α i nonnegative integers, i = d. Following [0], we shall say that the spatial forward semimartingale F has forward local characteristics of class B m δ ub Bk δ ub for nonnegative integers m k and δ 0, if for all T>0, its characteristics satisfy where at m+δ = ess ω x y R d 0 t T [ at m+δ +bt k+δ ] < at x y +x +y + + D α x Dα y at ˆ δ α=m α m D α x Dα yat x y x y R d

5 STABLE MANIFOLD THEOREM FOR SDEs 69 bt x bt k+δ = x R +x + d Dα α=k xy c + α k D α xbt x x R d x bt x Dα ybt y x y δ and { fx y fx y fx y +fx y } fˆ δ = x x y y c x x δ y y δ for any δ-hölder continuous function f R d R d R d. Similar definitions hold for the backward local characteristics of a backward spatial semimartingale. The local characteristics of F are said to be of class B mδ loc Bk δ loc if for any compact set K R d, and any finite positive T, one has [ ] ess at m+δ K +bt k+δ K < ω 0 t T where m+δ K, bt k+δ K are defined by similar expressions to the above with the rema taken over the compact set K. Now consider the Stratonovich and Itô stochastic differential equations S dφt = F dt φt φs =x t>s I dφt =Fdt φt φs =x t>s The SDE (S) is driven by a continuous forward( backward) spatial helix semimartingale Ft x=f t xf d t x, x R d. In the SDE (I), F denotes a spatial continuous forward helix semimartingale. It is known that, under suitable regularity hypotheses on the local characteristics of F (or F), the SDEs (S) and (I) generate the same stochastic flow. Throughout this article, these flows will be denoted by the same symbol φ s t s t R. More precisely, we will need the following hypotheses. Hypothesis [STk δ]. F is a continuous spatial helix forward semimartingale with forward local characteristics of class B k+δ ub B k δ ub. The function d 0 R d a j t x y t x x R d j y=x belongs to B kδ ub. j=

6 620 S.-E. A. MOHAMMED AND M. K. R. SCHEUTZOW Hypothesis [ST k δ]. F is a continuous helix backward semimartingale with backward local characteristics of class B k+δ ub B k δ ub Hypothesis [ITk δ]. F R R d R d is a continuous spatial helix forward semimartingale with forward local characteristics of class B k δ ub Bk δ ub. The following proposition establishes a relationship between the SDEs (S) and (I). Proposition 2.. Suppose the helix semimartingale F satisfies Hypothesis [ST(k δ)] for some positive integer k and δ 0. Let the following relation hold: Ft x ω = Ft x ω+ t d a j u x y 2 0 x du t R x R d j= j y=x Then F is a helix semimartingale which satisfies Hypothesis [IT(k δ)]. In this case, the SDEs (S) and (I) generate the same stochastic flow φ s t st R,onR d. Proof. The assertion of the proposition follows from Theorem in [0], except for the helix property. The helix property of F follows from that of F and the fact that the R d d -valued process F x F yt = F xf yt, t R is a helix for any x y R d [22]. Proposition 2. shows that for given k δ, Hypothesis [ST(k δ)] is stronger than [IT(k δ)]. Although our results will cover both the Stratonovich and Itô cases, the reader may note that the Stratonovich SDE (S) allows for a complete and more aesthetically pleasing dynamic characterization of the stochastic flow φ st and its inverse. Indeed, under [ST(k δ)] and [ST k δ)], φ s t solves the backward Stratonovich SDE based on F and hence provides a natural dynamical representation of the local unstable manifold in terms of trajectories of the backward Stratonovich SDE. Such a dynamical characterization is not available for the Itô SDE (I). See Section 3. From now on, we will implicitly assume that the spatial semimartingales F and F are related by the formula in Proposition 2.. In this context, all our results will be derived under both sets of hypotheses [ST(k δ)] and [IT(k δ)], although the conclusions pertain invariably to the generated flow φ s t. The following proposition is elementary. Its proof is an easy induction argument using the chain rule. Proposition 2.2. Let f = f f 2 f d R d R d be a C k diffeomorphism for some integer k. Then, for each α with α k, 2 D α x f i x = p αi f x detdff x n α i = d

7 STABLE MANIFOLD THEOREM FOR SDEs 62 for all x R d, and some integer n α. In the above identity, p α i y is a polynomial in the partial derivatives of f of order up to α evaluated at y R d. Proof. We use induction on α. Forα =, the chain rule gives Df x = Dff x. By Cramer s rule, this implies (2.) with n α =. Assume by induction that for some integer n<k, (2.) holds for all α such that α n and all i = d. Take α such that α =n and fix i j 2d. Taking partial derivatives with respect to x j in both sides of (2.) shows that the right-hand side of the resulting equation is again of the same form with α replaced by α = α α 2 α d, where α i = α i + δ i j. This completes the proof of the proposition. The next proposition allows the selection of sure θt -invariant events in F from corresponding ones in F. Proposition 2.3. Let F be a sure event such that θt for all t 0. Then there is a sure event 2 F such that 2 and θt 2 = 2 for all t R. Proof. Define ˆ = k=0 θk. Then ˆ is a sure event, ˆ and θt ˆ = ˆ for all t R. Since F is the completion of F,wemay pick a sure event 0 ˆ such that 0 F. Define 2 = { ω ω θt ω 0 for Lebesgue-a.e. t R } Using Fubini s theorem and the P-preserving property of θ, it is easy to check that 2 satisfies all the conclusions of the proposition. Theorem 2.. Let F satisfy Hypothesis [STk δ] (resp., F satisfies [ITk δ]) for some k and δ 0. Then there exists a jointly measurable modification of the trajectory random field of (S) [resp., (I)] also denoted by φ s t x <st<, x R d, with the following properties. If φ R R d R d is defined by φt x ω =φ 0t x ω x R d ω t R then the following is true for all ω : (i) For each x R d and s t R, φ s t x ω =φt s x θs ω. (ii) φ θ is a perfect cocycle: φt + s ω=φt θs ω φs ω for all s t R. (iii) For each t R, φt ω R d R d is a C k diffeomorphism. (iv) The mapping R 2 s t φ s t ω Diff k R d is continuous, where Diff k R d denotes the group of all C k diffeomorphisms of R d, given the C k - topology.

8 622 S.-E. A. MOHAMMED AND M. K. R. SCHEUTZOW (v) For every ε 0δγρT>0, and α k, the quantities 0 s t T x R d x R d φ s t x ω +xlog + x γ 0 s t T 0<x x ρ 0 s t T x R d D α x φ s tx ω +x γ D α x φ s tx ω D α x φ s tx ω x x ε +x γ are finite. Furthermore, the random variables defined by the above expressions have pth moments for all p. Proof. The cocycle property stated in (ii) is proved in [9] for the white noise case using an approximation argument (cf. [4], [5]). Assertions (iii) and (iv) are well known to hold for a.a. ω ([0], Theorem 4.6.5). A perfect version of φ s t satisfying (i) (iv) for all ω is established in [3]. The arguments in [3] use perfection techniques and Theorem of [0] (cf. also [6]). Assume that for every ε 0δ, γ, T, ρ>0 the random variables in (v) have finite moments of all orders. Let T γ ρ ε be the set of all ω for which all random variables in (v) are finite. Define the set 0 by 0 = θs T /n /l /k k l T n N s R Then θs 0 = 0 for all s R. Furthermore, it is not hard to see that θmt 2T /n /l /k 0 k l T n N m Z Therefore 0 is a sure event in F. By Proposition 2.3, 0 contains a sure invariant event 0 F. Hence we can redefine φ s t ω and φt ω to be the identity map R d R d for all ω \ 0. This can be done without violating properties (i) (iv). By Proposition 2.2, Theorem in [7] and the remark following its proof, it follows that the two random variables, X = X 2 = 0 s t T x R d 0 s t T x R d φ s t x +xlog + x γ x +φ s t x log + x γ have pth moments for all p. To complete the proof of the first assertion in (v), it is sufficient to show that the random variable ˆX = 0 s t T x R d φ t s x +xlog + x γ

9 STABLE MANIFOLD THEOREM FOR SDEs 623 has pth moments for all p. To do this, assume (without loss of generality) that γ 0. From the definition of X 2,wehave y X 2 +φ s t y log + y γ for all 0 s t T y R d. Use the substitution, y = φ t s x ω =φ s tx ω φ s t y ω =x 0 s t T ω x R d to rewrite the above inequality in the form y X 2 +xlog + y γ By an elementary computation, the above inequality may be solved for log + y. This gives a positive nonrandom constant K (possibly dependent on ε and T) such that y K X 2 +x +log + X 2 γ +log + x γ Since X 2 has moments of all orders, the above inequality implies that ˆX also has pth moments for all p. We now prove the second assertion in (v). First, note that the following two random variables, X 3 = 0 s t T x R d D α x φ s tx +x γ Dφ s t x α k X 4 = 0 s t T +x γ x R d have pth moments for all p ([0], Exercise 4.6.9, page 76; [7], Remark (i) following Theorem 2). We must show that the random variables, ˆX 3 = 0 s t T x R d D α x φ s t x +x γ α k have pth moments for all p. Note that there is a positive constant C such that for any nonsingular matrix A, one has det A =deta CA d Using this fact and applying Proposition 2.2 with f = φ s t, s t T, shows that for every δ > 0, any i 2d and any α k, there exists a random variable K δ p L p R such that D α x φ s t ix K δ +x δ m α i for all x R d and some positive integer m αi. Now for any given ε>0, choose δ = γ/m α i to obtain D α x φ s t ix K δ +x γ/m α i m α i 2 m α i K δ +x γ for all x R d. This shows that ˆX 3 has pth moments for all p.

10 624 S.-E. A. MOHAMMED AND M. K. R. SCHEUTZOW The last estimate in (v) follows from a somewhat lengthy argument. We will only sketch it. First note that for every p, there exists a constant c 0 such that ED α x φ s tx D α x φ s t x 2p cx x 2pδ +s s p +t t p uniformly for all x x R d,0 s t T ([0], Theorem 4.6.4, pages 72 and 73). Using the above estimate, we can employ the inequality of Garsia Rodemich Rumsey in its majorizing measure version in order to show that the expression x R d 0 s t T 0<x x ρ D α x φ s tx ω D α x φ s tx ω x x ε +x γ has moments of all orders. The argument used to show this is similar to the one used in [8]. The application of the Garsia Rodemich Rumsey inequality is effected using the following metric on the space 0T 0T R d : ds t x s t x = x x δ +s s /2 +t t /2 Finally, we extend the estimate to cover the over all s t 0T 0T by appealing to Proposition 2.2 and the argument used above to establish the existence of pth moments of ˆX 3. This completes the proof of the theorem. 3. The local stable manifold theorem. In this section, we shall maintain the general setting and hypotheses of Section 2. Furthermore, we shall assume from now on that the P-preserving flow θ R is ergodic. For any ρ>0 and x R d, denote by Bx ρ the open ball with center x and radius ρ in R d. Denote by Bx ρ the corresponding closed ball. Recall that φ θ is the perfect cocycle associated with the trajectories φ s t x of (S) or (I) (Theorem 2.). Definition 3.. Say that the cocycle φ has a stationary trajectory if there exists an F -measurable random variable Y R d such that 3 φt Yωω=Yθt ω for all t R and every ω. In the sequel, we will always refer to the stationary trajectory (3.) by φt Y. If (3.) is known to hold on a sure event t that may depend on t, then there are perfect versions of the stationary random variable Y and of the flow φ such that (3.) and the conclusions of Theorem 2. hold for all ω (under the hypotheses therein) [26]. We may replace ω in (3.) by θs ω, s R, toget 32 for all s t R and every ω. φt Yθs ωθs ω = Yθt + s ω

11 STABLE MANIFOLD THEOREM FOR SDEs 625 To illustrate the concept of a stationary trajectory, we give a few simple examples. Examples. (i) Consider the Itô SDE, m dφt =hφt dt + g i φt dw i t where W = W W m is an m-dimensional Brownian motion on Wiener space F P. Namely, is the space of all continuous paths ω R R m given the topology of uniform convergence on compacta, F = Borel, P is Wiener measure on and W is defined by evaluations Wt ω =ωt ω t R + The vector fields h g i R d R d i = m are in C k δ b for some k δ 0. Let θ R denote the canonical Brownian shift θt ωs =ωt + s ωt ts R ω Suppose hx 0 =g i x 0 =0, i m for some fixed x 0 R d. Take Yω =x 0 for all ω. Then Y is a stationary trajectory of the above SDE. (ii) Consider the affine linear one-dimensional SDE, dφt =λφt dt + dwt where λ > 0 is fixed and W is one-dimensional Brownian motion. Take Yω = 0 i= e λu dwu and let θ denote the canonical Brownian shift in Example (i) above. Using integration by parts and variation of parameters, the reader may check that there is a version of Y such that φt Yωω=Yθt ω for all t ω R. (iii) Consider the two-dimensional affine linear SDE, dφt =Aφt dt + GdWt where A is a fixed hyperbolic 2 2-diagonal matrix, ( ) λ 0 A = λ 2 < 0 <λ 0 λ 2 and G is a constant matrix, for example, ( ) g g 2 G = g 3 g 4 with g i R, i = Let W = ( ) W W be two-dimensional Brownian motion. Set Y = ( ) 2 Y Y where 2 [ ] [ ] Y = g exp λ dw u g 2 exp λ u dw 2 u 0 0

12 626 S.-E. A. MOHAMMED AND M. K. R. SCHEUTZOW and [ 0 ] [ 0 ] Y 2 = g 3 exp λ 2 u dw u + g 4 exp λ 2 u dw 2 u Using variation of parameters and integration by parts [as in (ii)], it is easy to see that Y has a measurable version Y R 2 which gives a stationary trajectory of the SDE in the sense of Definition 3.. In the general white noise case in Example (i) above, one can generate a large class of stationary trajectories as follows. Let ρ be an invariant probability measure on R d for the Markov process associated with the solution of the SDE in Example (i). Then ρ gives rise to a stationary trajectory by suitably enlarging the underlying probability space using the following procedure. If P t C b R d R C b R d R t 0, is the Markov semigroup associated with the SDE, then P t fx dρx = fx dρx t 0 R d R d where P t fx =Efφt x t 0 x R d for all f C b R d R. Define = R d F = F BR d P = P ρ ω = ω x θt ω =θt ωφt x ω t R + ω x R d φt x ω =φt x ω t R + x R d Ỹ ω =x ω =ω x ω The group θt t R +,is P-preserving (and ergodic) [6]. Furthermore, it is easy to check that φt ω θt ω is a perfect cocycle on R d and Ỹ R d satisfies φt Ỹ ω ω =Ỹ θt ω for all t R + ω. Hence Ỹ is a stationary trajectory for the cocycle φ θ and ρ = P Ỹ. Conversely, let Y R d be a stationary trajectory satisfying the identity (3.) and independent of the Brownian motion Wt, t 0. Then ρ = P Y is an invariant measure for the one-point motion. For related issues on statistical equilibrium and invariant measures for stochastic flows, the reader may consult [], [2], [7], [4] and [], Chapter. Lemma 3.. Let the conditions of Theorem 2. hold. Assume also that log + Y is integrable. Then the cocycle φ satisfies 33 log + φt 2 Yθt ω + θt ω k ε dpω < T t t 2 T

13 STABLE MANIFOLD THEOREM FOR SDEs 627 for any fixed 0 <Tρ< and any ε 0δ. The symbol kε denotes the C kε -norm on C kε mappings B0ρ R d. Furthermore, the linearized flow D 2 φt Yωωθt ω t 0 is an LR d -valued perfect cocycle and 34 log + T t t 2 T D 2 φt 2 Yθt ωθt ω LR d dpω < for any fixed 0 < T <. The forward cocycle D 2 φt Yωωθt ω t>0 has a nonrandom finite Lyapunov spectrum λ m < <λ i+ <λ i < <λ 2 <λ. Each Lyapunov exponent λ i has a nonrandom (finite) multiplicity q i, i m and m i= q i = d. The backward linearized cocycle D 2 φt Yωωθt ωt < 0, admits a backward nonrandom finite Lyapunov spectrum defined by lim t t log D 2φt Yωωv v R d and taking values in λ i m i= with nonrandom (finite) multiplicities q i, i m and m i= q i = d. Note that Lemma 3. stipulates regularity only on the forward characteristics of F and F. Proof. φ θ, 35 We first prove (3.4). Start with the perfect cocycle property for φt + t 2 ω=φt 2 θt ω φt ω for all t t 2 R and all ω. The perfect cocycle property for D 2 φt Yω ωθt ω follows directly by taking Fréchet derivatives at Yω on both sides of (3.5), namely, 36 D 2 φt + t 2 Yωω = D 2 φt 2 φt Yωωθt ω D 2 φt Yωω = D 2 φt 2 Yθt ωθt ω D 2 φt Yωω for all ω 0 t t 2 R. The existence of a fixed discrete spectrum for the linearized cocycle follows the analysis in [5] and [6]. This analysis uses the integrability property (3.4) and the ergodicity of θ. Although (3.4) is an easy consequence of (3.6) and Theorem 2.(v), it is clear that (3.3) implies (3.4). Therefore it is sufficient to establish (3.3). In view of (3.) and the identity φ t t +t 2 x ω =φt 2 xθt ω x R d t t 2 R

14 628 S.-E. A. MOHAMMED AND M. K. R. SCHEUTZOW [Theorem 2.(i)], (3.3) will follow if we show that the following integrals are finite for 0 α k: 37 log + D α x φ s tφ 0s Yωω+x ω dpω 0 s t T x ρ 37 log + 0 s t T xx B0ρx x [ D α x φ s t φ 0s Yωω+x ω D α x φ s tφ 0s Yωω+x ω ] x x ε dpω For simplicity of notation, we shall denote random constants by the letters K i, i = Each K i, i = 2 3 4, has pth moments for all p and may depend on ρ and T. The following string of inequalities follows easily from Theorem 2.(v). 38 log + s t 0T x ρ log + D α x φ s tφ 0s Yωω+x ω s 0T { K ω +ρ +φ 0s Yωω 2 } log + K 2 ω+log + + 2ρ 2 + K 3 ω +Yω 4 log + K 4 ω+log + 2ρ 2 +4 log + Yω for all ω. Now (3.8) and the integrability hypothesis on Y imply that the integral (3.7) is finite. The finiteness of (3.7 ) follows in a similar manner using Theorem 2.(v). This completes the proof of the lemma. Definition 3.2. A stationary trajectory φt Y of φ is said to be hyperbolic if E log + Y < and the linearized cocycle D 2 φt Yωω θt ω t 0 has a Lyapunov spectrum λ m < <λ i+ <λ i < <λ 2 <λ which does not contain 0. Let ω ω ω denote the unstable and stable subspaces for the linearized cocycle D 2 φt Y θt as given by Theorem 5.3 in [6]. See also [5]. This requires the integrability property (3.4). The following discussion is devoted to the Stratonovich SDE (S) and the linearization of the stochastic flow around a stationary trajectory. The Linearization. In (S), pose F is a forward backward semimartingale helix satisfying Hypotheses [ST(k δ)] and [ST (k δ)] for some k 2 and δ 0. Then it follows from Theorem 4.2(i) that the (possibly anticipating)

15 STABLE MANIFOLD THEOREM FOR SDEs 629 process φt Yωω is a trajectory of the anticipating Stratonovich SDE, SII dφt Y = F dt φt Y φ0y=y t > 0 In the above SDE, the Stratonovich differential F dt is defined as in Section 4, Definition 4. (cf. [0], page 86). The above SDE follows immediately by substituting x = Yω in SI dφt x = F dt φt x φ0 =x R d t > 0 [Theorem 4.2(i)]. This substitution works in spite of the anticipating nature of φt Yωω=Yθt ω, because the Stratonovich integral is stable under random anticipating substitutions (Theorem 4.). Furthermore, we can linearize the SDE (S) along the stationary trajectory and then match the solution of the linearized equation with the linearized cocycle D 2 φt Yωω. That is to say, the (possibly nonadapted) process yt =D 2 φt Yωω, t 0 satisfies the associated Stratonovich linearized SDE, SIII dyt =D 2 F dt Yθtyt y0 =I LR d t > 0 In (SIII), the symbol D 2 denotes the spatial (Fréchet) derivative of the driving semimartingale along the stationary trajectory φt Yωω=Yθt ω [Theorem 4.2(ii)]. In view of Hypothesis [ST (k δ)] for k 2δ 0, and Theorem 4.2(iii), (iv), it follows that the backward trajectories φt Y, ŷt = D 2 φt Y, t<0, satisfy the backward SDEs, SII dφt Y = F dt ˆ φt Y φ0y=y t < 0 SIII dŷt = D 2 F ˆ dt φt Yŷt ŷ0y=i LR d t < 0 Note however that the significance of (SIII) is to provide a direct link between the linearized flow D 2 φt Yωω and the linearized SDE. The Stratonovich equation (SII) does not play a direct role in the construction of the stable and unstable manifolds (cf. [27], Section 4.2). On the other hand, (SII) and (SII ) provide a dynamic characterization of the stable and unstable manifolds in Theorem 3.(a), (d).

16 630 S.-E. A. MOHAMMED AND M. K. R. SCHEUTZOW In order to apply Ruelle s discrete theorem, [24], Theorem 5., page 292, we will introduce the following auxiliary cocycle Z R R d R d, which is essentially a centering of the flow φ about the stationary solution, 39 for t R x R d ω. Zt x ω =φt x + Yωω Yθt ω Lemma 3.2. Assume the hypotheses of Theorem 2.. Then Z θ is a perfect cocycle on R d and Zt 0ω=0 for all t R and all ω. Proof. Let t t 2 R ω x R d. Then by the cocycle property for φ and Definition 3., we have Zt 2 Zt xωθt ω = φt 2 Zt xω+yθt ωθt ω Yθt 2 θt ω = φt 2 φt x+ Yωωθt ω Yθt 2 + t ω = Zt + t 2 xω The assertion Zt 0ω=0, t R, ω, follows directly from the definition of Z and Definition 3.. The next lemma will be needed in order to construct the shift-invariant sure events appearing in the statement of the local stable manifold theorem. The lemma essentially gives perfect versions of the ergodic theorem and Kingman s subadditive ergodic theorem. Lemma 3.3. (i) Let h R + be F -measurable and such that hθu ω dpω < 0 u Then there is a sure event F such that θt = for all t R, and lim t hθt ω = 0 t for all ω. (ii) Suppose f R + R is a measurable process on F P satisfying the following conditions: (a) f + u ω dpω < 0 u f + u θu ω dpω < 0 u (b) ft + t 2 ω ft ω+ft 2 θt ω for all t t 2 0 and all ω.

17 STABLE MANIFOLD THEOREM FOR SDEs 63 Then there is sure event 2 F such that θt 2 = 2 for all t R, and a fixed number f R such that for all ω 2. lim t ft ω =f t Proof. A proof of (i) is given in [5], Lemma 5(iii), with a sure event F such that θt for all t 0. Proposition 2.3 now gives a sure event such that F and satisfies assertion (i) of the lemma. Assertion (ii) follows from [5], Lemma 7 and Proposition 2.3. The proof of the local stable-manifold theorem (Theorem 3.) uses a discretization argument that requires the following lemma. Lemma 3.4. Assume the hypotheses of Lemma 3.2 and pose that log + Y is integrable. Then there is a sure event 3 F with the following properties: (i) θt 3 = 3 for all t R. (ii) For every ω 3 and any x R d, the statement 30 lim n log Zn x ω < 0 n implies 3 Proof. 32 lim t log Zt x ω = lim log Zn x ω t n n The integrability condition (3.3) of Lemma 3. implies that log + 0 t t 2 x B0 D 2 Zt x θt 2 ω LR d dpω < Therefore by (the perfect version of) the ergodic theorem [Lemma 3.3(i)], there is a sure event 3 F such that θt 3 = 3 for all t R, and 33 lim t t log+ D 2 Zu x θt ω LR d = 0 0 u x B0 for all ω 3. Let ω 3 and pose x R d satisfies (3.0). Then (3.0) implies that there exists a positive integer N 0 x ω such that Zn x ω B0 for all

18 632 S.-E. A. MOHAMMED AND M. K. R. SCHEUTZOW n N 0. Let n t<n+ where n N 0. Then by the cocycle property for Z θ and the mean value theorem, we have n t n+ t log Zt x ω n log+ + D 2 Zu x θn ω LR d 0 u x B0 n n + log Zn x ω n Take lim n in the above relation and use (3.3) to get The inequality lim t lim n log Zt x ω lim log Zn x ω t n n log Zn x ω lim log Zt x ω n t t is obvious. Hence () holds, and the proof of the lemma is complete. In order to formulate the measurability properties of the stable and unstable manifolds, we will consider the class R d of all nonempty compact subsets of R d. Give R d the Hausdorff metric d, d A A 2 =dx A x A 2 dy A 2 y A where A A 2 R d and dx A i =inf x y y A i x R d i= 2. Denote by B R d the Borel σ-algebra on R d with respect to the metric d. Then R d d is a complete separable metric space. Morevover, it is not hard to see that finite nonempty intersections are jointly measurable and translations are jointly continuous on R d. These facts are used in the proof of Theorem 3.(h). We now state the local stable manifold theorem for the SDEs (S) and (I) around a hyperbolic stationary solution. Theorem 3. (Local stable and unstable manifolds). Assume that F satisfies Hypothesis [ST(k δ)] (resp., F satisfies [IT(k δ)]) for some k and δ 0. Suppose φt Y is a hyperbolic stationary trajectory of (S) [resp., (I)] with E log + Y <. Suppose the linearized cocycle D 2 φt Yωωθt ω t 0 has a Lyapunov spectrum λ m < <λ i+ <λ i < <λ 2 <λ. Define λ i0 = maxλ i λ i < 0 if at least one λ i < 0. Ifallλ i > 0, set λ i0 =. (This implies that λ i0 is the smallest positive Lyapunov exponent of the linearized flow, if at least one λ i > 0; in case all λ i are negative, set λ i0 =.) Fix ε 0 λ i0 and ε 2 0λ i0. Then there exist: (i) A sure event F with θt = for all t R.

19 STABLE MANIFOLD THEOREM FOR SDEs 633 (ii) F -measurable random variables ρ i β i 0, β i >ρ i > 0, i = 2, such that for each ω, the following is true: there are C k ε (ε 0δ) submanifolds ω ω of BYωρ ω and BYωρ 2 ω (resp.) with the following properties: (a) ω is the set of all x BYωρ ω such that φn x ω Yθn ω β ω expλ i0 + ε n for all integers n 0. Furthermore, 34 lim t t log φt x ω Yθt ω λ i 0 for all x ω. Each stable subspace ω of the linearized flow D 2 φ is tangent at Yω to the submanifold ω, namely, T Yω ω =ω. In particular, dim ω =dim ω and is nonrandom. (b) lim t [ { φt t log x ω φt x 2 ω x x 2 x x 2 x x 2 }] ω λ i0 (c) (Cocycle-invariance of the stable manifolds). There exists τ ω 0 such that 35 φt ω ω θt ω t τ ω Also 36 D 2 φt Yωω ω = θt ω t 0 (d) ω is the set of all x BYωρ 2 ω with the property that 37 φ n x ω Yθ n ω β 2 ω exp λ i0 + ε 2 n for all integers n 0. Also 38 lim t t log φ t x ω Yθ t ω λ i 0 for all x ω. Furthermore, ω is the tangent space to ω at Yω. In particular, dim ω =dim ω and is nonrandom. (e) lim t [ { φ t t log x ω φ t x 2 ω x x 2 x x 2 x x 2 }] ω λ i0

20 634 S.-E. A. MOHAMMED AND M. K. R. SCHEUTZOW (f) (Cocycle-invariance of the unstable manifolds). There exists τ 2 ω 0 such that 39 Also φ t ωω θ t ω t τ 2 ω 320 D 2 φ t Yωωω = θ t ω t 0 (g) The submanifolds ω and ω are transversal, namely, 32 R d = T Yω ω T Yω ω (h) The mappings R d ω are F B R d -measurable. R d ω ω ω Assume, in addition, that F satisfies Hypothesis [ST(k δ)] (resp., F satisfies [IT(k δ)]) for every k and δ 0. Then the local stable and unstable manifolds ω ω are C. The following corollary follows from Theorem 3.. See [24], Section (5.3), page 49. Corollary 3.. (White noise, Itô case). Consider the Itô SDE m V dxt =hxt dt + g i xt dw i t Suppose that for some k, δ 0 hg i, i m, are C k δ b vector fields on R d, and W = W W m is an m-dimensional Brownian motion on Wiener space F P. Let θ R denote the canonical Brownian shift 322 θt ωs =ωt + s ωt ts R ω Suppose φt Y is a hyperbolic stationary trajectory of (V) with E log + Y <. Then the conclusions of Theorem 3. hold. Furthermore, if the vector fields h g i i m, are C b, then the conclusions of Theorem 3. hold, where ω, ω are C manifolds. Remarks. (i) A similar statement to that of Corollary 3.. holds for the corresponding Stratonovich SDE driven by finite-dimensional Brownian motion, namely m SIV dxt =hxt dt + g i xt dw i t i= i=

21 STABLE MANIFOLD THEOREM FOR SDEs 635 However, in this case one needs stronger conditions to ensure that Hypothesis [ST(k δ)] holds for (SIV). In fact, such hypotheses will hold if we assume that the functions R d x m l= g i l x g j l x R x j are in C k δ b for each i j d and some k, δ 0. Compare the conditions expressed in [2]. For example, this holds if for some k, δ 0, the vector field h is of class C k δ b and g i, i m, are globally bounded and of class C k+δ b. We conjecture that the the global boundedness condition is not needed. This conjecture is not hard to check if the vector fields g i, i m, are C b and generate a finite-dimensional solvable Lie algebra. See [0], Theorem 4.9.0, page 22. (ii) Recall that if F is a forward backward semimartingale helix satisfying Hypotheses [ST(k δ)] and [ST (k δ)] for some k 2 and δ 0, then the inverse φt θ t ω x =φ t x ω, t>0 corresponds to a solution of the SDE (S ). Furthermore, φ t Y and D 2 φ t Y, t>0, satisfy the anticipating SDEs (SII ) and (SIII ), respectively. See Theorem 4.2(iii), (iv), of Section 4. (iii) In Corollary 3.., let ρ be an invariant probability measure for the one-point motion in R d. Assume that log + x dρx < R d Recall the discussion and the notation preceding Lemma 3.. More specifically, we will work on the enlarged probability space = R d F = F BR d P = P ρ. Define the process W = W W m R R m by W i t ω =W i t ω, ω = ω x i m. Then W is a Brownian motion on F P and the perfect cocycle φ θ solves a SDE similar to (V), with the same coefficients but driven by the Brownian motion W. Assuming hyperbolicity of the linearized cocycle D 2 φt Ỹ ω ω θt ω, we may apply Corollary 3.. to obtain stable and unstable manifolds that are defined for all pairs ω x in a θt -invariant set of full P ρ-measure; compare [6] for the globally asymptotically stable case on a compact manifold. Note also that the local stable/unstable manifolds are asymptotically invariant with repect to φ θ and the corresponding backward flow. The reader may fill in the details. (iv) In Corollary 3.., one can allow for infinitely many Brownian motions (cf. [0], pages 06 and 07). Details are left to the reader. (v) Consider the SDE (S) and assume that the helix forward semimartingale F has local characteristics of class B k+δ loc B k δ loc for some k, δ 0. See Section 2 for the definition of B k δ loc. Suppose that Ft 0ω=0 for all t Rω. Then (S) admits a local (possibly explosive) forward flow φt x ω x R d τ + x ω ω, where τ + R d 0 denotes

22 636 S.-E. A. MOHAMMED AND M. K. R. SCHEUTZOW the forward explosion time random field. A similar statement holds for the local backward flow with backward explosion time random field τ R d 0 ([0], pages 76 85). Furthermore, τ + 0ω= τ 0ω= and φt 0ω=0for all t 0, ω. Suppose further that the Lyapunov spectrum of the linearized SDE (SIII) (with Y 0) does not vanish. We claim that for ε ε 2 as in Theorem 3., there exist ρ i β i i = 2 and C k ε 0 <ε<δ local stable and unstable manifolds ω B0ρ ω, ω B0ρ 2 ω satisfying assertions (3.4), (b), (c), (3.8), (e), (g) and (h) of Theorem 3. with Y 0. To see this, pick a smooth funtion ψ R d 0 with compact port and such that ψ B0. Define F 0 = F ψ. Then F 0 is a helix semimartingale satisfying Hypothesis [ST(k δ)], and F 0 t ω B0 = Ft ω B0 for all t R ω. Now in (S) replace F by F 0 and denote by φ 0 the C k ε cocycle of the resulting (truncated) SDE. Apply Theorem 3. to φ 0. This gives local stable and unstable manifolds 0 ω 0 ω for φ 0. These manifolds will also serve as local stable/unstable manifolds for φ and satisfy our claim above. Indeed, observe that we may define ω to be the set of all x B0ρ ω with τ + x ω = and for which the first assertion in (a) holds. Define ω in a similar fashion. Hence ω = 0 ω and ω = 0 ω. This follows directly from the fact that for all x 0 ω, one has τ + x ω = and φt x ω =φ 0 t x ω for every t R +. A similar observation holds for x 0 ω. The local stable/unstable manifolds for (S) depend on the choice of truncation, but for different truncations these manifolds agree within a sufficiently small neighborhood of 0; compare [], 7.5. The truncation argument may be adapted to cover the case of an essentially bounded stationary trajectory. Proof of Theorem 3.. Assume the hypotheses of the theorem. Consider the cocycle Z θ defined by (3.9). Define the family of maps F ω R d R d by F ω x =Zxω for all ω and x R d. Let τ = θ. Following [24], page 292, define F n ω = F τ n ω F τω F ω. Then by the cocycle property for Z, wegetf n ω = Zn ω for each n. Clearly, each F ω is C k ε ε 0δ and DF ω 0 =D 2 φyωω. By measurability of the flow φ, it follows that the map ω DF ω 0 is F -measurable. By (3.4) of Lemma 3., it is clear that the map ω log + D 2 φyωω LR d is integrable. Furthermore, the discrete cocycle DF n ω0θn ω n 0 has a nonrandom Lyapunov spectrum which coincides with that of the linearized continuous cocycle D 2 φt Yωωθt ω t 0, namely, λ m < <λ i+ <λ i < <λ 2 <λ, where each λ i has fixed multiplicity q i, i m (Lemma 3.). Note that λ i0 (and λ i0 ) are well defined by hyperbolicity of the stationary trajectory. If λ i > 0 for all i m, then take ω =Yω for all ω. The assertions of the theorem are trivial in this case. From now on pose that at least one λ i < 0. We use Theorem 5. of [24], page 292, and its proof to obtain a sure event F such that θt = for all t R, F -measurable positive random variables ρ β 0, ρ <β, and a random family of C kε ε 0δ

23 STABLE MANIFOLD THEOREM FOR SDEs 637 submanifolds of B0ρ ω denoted by d ω, ω following properties for each ω : 323 and satisfying the d ω = { x B0ρ ω Zn x ω β ω expλ i0 + ε n for all integers n 0 } Each d ω is tangent at 0 to the stable subspace ω of the linearized flow D 2 φ, namely, T 0 d ω = ω. In particular, dim d ω is nonrandom by the ergodicity of θ. Furthermore, [ ] 324 lim n n log Zn x ω Zn x 2 ω λ x x 2 i0 x x 2 x x 2 d ω Before we proceed with the proof, we will indicate how one may arrive at the above θt -invariant sure event F from Ruelle s proof. Consider the proof of Theorem 5. in [24], page 293. In the notation of [24], set T t ω = D 2Zt 0ω T n ω =D 2 Z 0θn ω τ t ω =θt ω, for t R + n = 2 3 By the integrability condition (3.4) (Lemma 3.) and Lemma 3.3(i), (ii), there is a sure event F such that θt = for all t R, with the property that continuous-time analogues of equations (5.2), (5.3), (5.4) in [24], page 45, hold. In particular, lim D 2Zt 0ω D 2 Zt 0ω /2t = ω t 325 lim t t log+ Z θt ω ε = 0 for all ω ε 0δ. See Theorem (B.3), [24], page 304. The rest of the proof of Theorem 5. works for a fixed choice of ω. In particular, (the proof of) the perturbation theorem, [24], Theorem 4., does not affect the choice of the sure event because it works pointwise in ω and hence does not involve the selection of a sure event ([24], pages ). For each ω, let ω be the set defined in part (a) of the theorem. Then it is easy to see from (3.23) and the definition of Z that 326 ω = d ω+yω Since d ω is a C k ε ε 0δ submanifold of B0ρ ω, it follows from (3.26) that ω is a C kε ε 0δ submanifold of BYωρ ω. Furthermore, T Yω ω =T 0 d ω =ω. In particular, dim ω =dim ω = mi=i0 q i and is nonrandom. Now (3.24) implies that 327 lim n n log Zn x ω λ i 0 for all ω in the shift-invariant sure event and all x d ω. Therefore, by Lemma 3.4, there is a sure event 2 such that θt 2 = 2 for all

24 638 S.-E. A. MOHAMMED AND M. K. R. SCHEUTZOW t R, and 328 lim t t log Zt x ω λ i 0 for all ω 2 and all x d ω. This immediately implies assertion (3.4) of the theorem. To prove assertion (b) of the theorem, let ω. By (3.24), there is a positive integer N 0 = N 0 ω [independent of x d ω] such that Zn x ω B0 for all n N 0. Let 4 = 2 3, where 3 is the shift-invariant sure event defined in the proof of Lemma 3.4. Then 4 is a sure event and θt 4 = 4 for all t R. Using an argument similar to the one used in the proof of Lemma 3.4, it follows that [ ] n t n+ t log φt x ω φt x 2 ω x x 2 x x 2 x x 2 ω = n t n+ t log n log+ + n n + [ 0 u x B0 x x 2 x x 2 d ω n log ] Zt x ω Zt x 2 ω x x 2 D 2 Zu x θn ω LR d [ x x 2 x x 2 d ω ] Zn x ω Zn x 2 ω x x 2 for all ω 4 all n N 0ω and sufficiently large. Taking lim n in the above inequality and using (3.24), immediately gives assertion (b) of the theorem. To prove the invariance property (3.6), we apply the Oseledec theorem to the linearized cocycle D 2 φt Yωωθt ω ([5], Theorem 4, Corollary 2). This gives a sure θt -invariant event, also denoted by, such that D 2 φt Yωω ω θt ω for all t 0 and all ω. Equality holds because D 2 φt Yωω is injective and dim ω =dim θt ω for all t 0 and all ω. To prove the asymptotic invariance property (3.5), we will need to take a closer look at the proofs of Theorems 5. and 4. in [24], pages We will first show that ρ β and a sure event (also denoted by) may be chosen such that θt = for all t R, and 329 ρ θt ω ρ ω expλ i0 + ε t β θt ω β ω expλ i0 + ε t for every ω and all t 0. The above inequalities hold in the discrete case (when t = n, a positive integer) from Ruelle s theorem ([24], Remark (c), page 297, following the proof of Theorem 5.). We claim that the relations

25 STABLE MANIFOLD THEOREM FOR SDEs 639 (3.29) hold also for continuous time. To see this, we will use the method of proof of Theorems 5. and 4. in [24]. In the notation of the proof of Theorem 5., [24], page 293, observe that the random variable G in (5.5) may be replaced by the larger one, 330 Gω = F τ ω t θ exp tη λθ < + θ 0 t 0 for 0 <η< λ i0 + ε/4, and Ruelle s λ corresponds to λ i0 + ε in our notation. Now β may be chosen using δ A from Theorem (4.) [24] and replacing G with G in (5.0), [24], page 293. Note that ρ = β /B ε, where B ε is given by (4.5) in Theorem (4.) of [24], page 285. Therefore, given any fixed ω,we need to determine how the choices of Ruelle s constants δ, A and B ε are affected if ω is replaced by θl ω =τ l ω, where l is any positive real number. Since T n τ l ω = D 2 Z 0θn θl ω = T n+l ω, for all positive integers n, it is sufficient to apply Theorem 4. [24] to the sequence T n+l ω n=. Hence we may follow the discussion in Section (4.7), [24], pages 29 and 292. We claim that the argument therein still works for positive real l. We will indicate the reasoning for δ and leave the rest of the details to the reader. Consider the definition of δ in (4.5) in the proof of Lemma (4.2), [24], page 288. Set δω =δ, Dω =D, Cω =C given by (4.5), (4.), (4.3), respectively. Redefine D and C by larger constants which we will denote by the symbols, (3.3) D η ω = exp tηξ t < t 0 C η ω = 0<s<t< h k m T t ξ 0 h T s ξ 0 h Ts ξ 0 k Tt ξ 0 (3.32) expλ rk λ rh t s+λ rk λ rµ 2ηt < where ξ t = ξ t ξt m, ξ t k = Tt ξ 0 k /Tt ξ 0 k k m. The λrk are the eigenvalues of log ω with multiplicities. Observe that D η ω is finite because the following continuous-time version of (4.9), [24], page 287, m 333 lim t t log Tt ξ 0 T t ξm 0 = λ rk holds everywhere on a θt -invariant sure event in F also denoted by. This is an immediate consequence of Lemma 3.3(ii). Compare [24], pages 287 and 303. The constant C η ω satisfies the inequality (4.3) of [24], page 288, because [by choice of t n k = t n k expλrµ λ rk ] one has N t n =T N ξ 0 expnλrµ λ rk n= k k for all positive integers N and k µ m. Indeed, C η ω is finite because lim t t log Tt ξ 0 k =λrk k k=

26 640 S.-E. A. MOHAMMED AND M. K. R. SCHEUTZOW on a θt -invariant sure event in F (also denoted by ). Now replace ω in (3.3) and (3.32) by θl ω. This changes ξ t to ξ t+l and T t ξ 0 h to T t+l ξ 0 h. Hence we get D ε θl ω e εl/2 D ε ω C ε θl ω e εl/2 C ε ω for sufficiently small ε 0ε and all l R +. From (4.5) in [24], page 288, we obtain δθl ω e εl δω for all l R + and all sufficiently small ε. The behavior of the constants A and B ε in Theorem 4., [24], page 285, can be analyzed in a similar fashion. See [24], Section 4.7. This yields the inequalities (3.29). We now prove (3.5). Use (b) to obtain a sure event 5 4 such that θt 5 = 5 for all t R, and for any 0 <ε<ε and ω 4, there exists βε ω > 0 (independent of x) with 334 φt x ω Yθt ω β ε ω expλ i0 + εt for all x ω, t 0. Fix any real t 0, ω 5 and x ω. Let n be a nonnegative integer. Then the cocycle property and (3.34) imply that 335 φn φt x ωθt ω Yθn θt ω =φn + t x ω Yθn + t ω β ε ωe λ i 0 +εn+t β ε ωe λ i +εt 0 e λ i 0 +ε n If ω 5, then it follows from (3.29), (3.34), (3.35) and the definition of θt ω that there exists τ ω > 0 such that φt x ω θt ω for all t τ ω. This proves (3.5) and completes the proof of assertion (c) of the theorem. Note that assertions (a), (b) and (c) still hold for all ω 5. We now prove assertion (d) of the theorem, regarding the existence of the local unstable manifolds ω. We do this by running both the flow φ and the shift θ backward in time. Define φt x ω =φ t x ω Zt x ω =Z t x ω θt ω =θ t ω for all t 0 and all ω. Clearly, Zt ω θt ω t 0 is a smooth cocycle, with Zt 0ω = 0 for all t 0. By the hypothesis on F and Y, it follows that the linearized flow D 2 φt Yωω θt ω t 0 is an LR d -valued perfect cocycle with a nonrandom finite Lyapunov spectrum λ < λ 2 < < λ i < λ i+ < < λ m where λ m < <λ i+ < λ i < <λ 2 <λ is the Lyapunov spectrum of the forward linearized flow D 2 φt Yωωθt ω t 0. Now apply the first part of the proof of this

27 STABLE MANIFOLD THEOREM FOR SDEs 64 theorem. This gives stable manifolds for the backward flow φ satisfying assertions (a), (b), (c). This immediately translates into the existence of unstable manifolds for the original flow φ, and assertions (d), (e), (f) automatically hold. In particular, we get a sure event 6 F such that θ t 6 = 6 for all t R, and with the property that assertions (d), (e) and (f) hold for all ω 6. Define the sure event = 6 5. Then θt = for all t R. Furthermore, assertions (a) (f) hold for all ω. Assertion (g) follows directly from the following facts: T Yω ω =ω T Yω ω =ω for all ω. We shall now prove assertion (h). Recall that by (3.26), 336 ω = Yω d ω R d = ω ω for all ω, where Rd R d R d denotes the translation map x A =x + A x R d A R d Hence, by joint continuity of and measurability of Y, the F -measurability of the mapping ω ω R d would follow from (3.36) if we can show that the map ω d ω R d is F -measurable. The rest of the argument will demonstrate this. Define the sequence of random diffeomorphisms, f n x ω =β ω exp λ i0 + ε n Zn x ω x R d ω for all integers n 0. Let HomR d be the topological group of all homeomorphisms of R d onto itself. HomR d carries the topology of uniform convergence of sequences of maps and their inverses on compacta. The joint measurability of f n implies that for each positive integer n, the map ω f n ω HomR d is measurable into the Borel field of HomR d. Using (3.23), d ω can be expressed in the form 337 d ω = lim m B0ρ ω m f i ω B0 for all ω. In (3.37), the limit is taken in the metric d on R d. The F -measurability of the map ω d ω follows directly from (3.37), the measurability of f i, ρ, that of finite intersections and the continuity of the maps i= R + r B0r R d HomR d f f B0 R d Hence the mapping ω ω R d is F B R d -measurable. A similar argument yields the measurability of ω ω R d. This completes the proof of assertion (h) of the theorem. If F (resp. F) satisfies Hypothesis [ST(k δ)] (resp., [IT(k δ)]) for every k and δ 0, then a simple adaptation of the argument in [24], Section 5.3,