A dynamical theory for singular stochastic delay differential equations I: Linear equations and a Multiplicative Ergodic Theorem on fields of Banach spaces
aa r X i v : . [ m a t h . P R ] D ec A DYNAMICAL THEORY FOR SINGULAR STOCHASTIC DELAYDIFFERENTIAL EQUATIONS I: LINEAR EQUATIONS AND AMULTIPLICATIVE ERGODIC THEOREM ON FIELDS OF BANACH SPACES
M. GHANI VARZANEH, S. RIEDEL, AND M. SCHEUTZOW
Abstract.
We show that singular stochastic delay differential equations (SDDEs) induce cocyclemaps on a field of Banach spaces. A general Multiplicative Ergodic Theorem on fields of Banachspaces is proved and applied to linear SDDEs. In Part II of this article, we use our results toprove a stable manifold theorem for non-linear singular SDDEs.
Introduction
Stochastic delay differential equations (SDDEs) describe stochastic processes for which the dy-namics do not only depend on the present state, but may depend on the whole past of the process.In its simplest formulation, an SDDE takes the form dy t = b ( y t , y t − r ) dt + σ ( y t , y t − r ) dB t ( ω )(0.1)for some delay r > B is a Brownian motion, b is the drift and σ the diffusion coefficient,both depending on the present and a delayed state of the system. In this case, we speak of a(single) discrete time delay . SDDE appear frequently in practice. For instance, they can be usedto model cell population growth and neural control mechanisms, cf. [Buc00] and the referencestherein, they are applied in financial modeling [Sto05] and for climate models [BTR07]. To be ableto solve (0.1) uniquely, an initial condition has to be given which is a path or, more generally, astochastic process. This means that we are led to solve an equation on an infinite dimensional(path) space. Popular choices for spaces of initial conditions are continuous paths or L paths.However, standard It¯o theory can be applied without too much effort to solve (0.1) for such initialconditions, cf. [Mao08, Moh84].To analyse the qualitative behaviour of solutions to (0.1), in particular its long-time behaviour,it is natural to use a dynamical systems approach. Maybe the most popular concept, which wassuccessfully applied to stochastic differential equations (SDEs) in both finite and infinite dimensions,was developed by L. Arnold and is called the theory of Random Dynamical Systems (RDS), cf.[Arn98] for an exposition. Examples for which the language and theory of RDS are used includerandom attractors [Sch92, CF94, CDF97], random stable and unstable manifolds [MS99, MS04]and different concepts of stochastic bifurcation [Arn98, Chapter 9]. The crucial result on whichArnold’s theory is built is a
Multiplicative Ergodic Theorem (MET), originally proved by Oseledec[Ose68]. This theorem has attracted much attention by different researchers and has been provenwith different techniques and increasing generality, cf. [Rag79, Rue79, Rue82, Mn83, Thi87, LL10,GTQ15, Blu16]. Under certain conditions, the MET shows that linear and linearised Random
Mathematics Subject Classification.
Key words and phrases. multiplicative ergodic theorem, random dynamical systems, rough paths, stochastic delaydifferential equation.
Dynamical Systems possess a
Lyapunov spectrum which can be seen as an analogue to the spectrumof eigenvalues of a matrix. Studying the behaviour of the possibly complex RDS can often be reducedto study its Lyapunov spectrum, which is a huge simplification. For a long time, it was believedthat the RDS approach can not be used to study SDDE of the form (0.1). This article claims, thatindeed, it is possible.Let us explain why it was believed that RDS are not applicable for general SDDEs. The ideais to show that certain equations do not generate a continuous stochastic semi-flow which is anecessary condition for generating an RDS and to apply the MET. Recall that given a probabilityspace (Ω , F , P ), a continuous stochastic semi-flow on a topological space E is a measurable map φ : { ( s, t ) ∈ [0 , ∞ ) | s ≤ t } × Ω × E → E such that on a set of full measure ˜Ω, we have φ ( t, t, ω, x ) = x and φ ( s, u, ω, x ) = φ ( t, u, ω, φ ( s, t, ω, x ))for every s, t, u ∈ [0 , ∞ ), s ≤ t ≤ u , every x ∈ E and every ω ∈ ˜Ω and x φ ( s, t, ω, x ) is assumed tobe continuous for every choice of s, t ∈ [0 , ∞ ), s ≤ t , and every ω ∈ ˜Ω. Consider the linear equation dy t = y t − dB t ( ω ); t ≥ y t = ξ t ; t ∈ [ − , ,
1] shouldbe given by y t = ξ + Z t ξ s − dB s ( ω )whenever the stochastic integral makes sense. However, Mohammed proved in [Moh86] that thereis no modification of the process y which depends continuously on ξ in the supremum norm. Thisrules out the choice of E = C ([ − , , R ) on which a possible semi-flow φ induced by (0.2) couldbe defined. At this stage, one might still hope that another choice of E could be a possible statespace for our semi-flow. However, we will prove now that there is in fact no such choice. Inspiredby [LCL07, Section 1.5.1], we make the following definition: Definition 0.1.
Let E be a Banach space of functions mapping from [ − ,
0] to R . We say that E carries the Wiener measure if the functions t sin[( n − / πt ] are contained in E for every n ≥ ∞ X n =1 Z n ( ω ) sin[( n − / πt ]( n − / π , t ∈ [ − , E almost surely for every sequence ( Z n ) of independent, N (0 , carrying the Wiener measure is indeed a minimum requirement for the state space E ofa possible semi-flow induced by (0.2), otherwise we would not even be able to choose constant pathsas initial conditions. However, this assumption already rules out the possibility of the existence ofa continuous semi-flow, as the following theorem shows. Theorem 0.2.
There is no space E carrying the Wiener measure for which the equation (0.2) induces a continuous mapping I : E → R , I ( ξ ) = y , on a set of full measure, which extends thepathwise defined mapping for smooth initial conditions. YNAMICAL THEORY FOR SDDE 3
Proof.
Let ( Z n ) be a sequence of independent standard normal random variables. Set B Nt ( ω ) = N X n =1 Z n ( ω ) sin[( n − / πt ]( n − / π . Then B N → B as N → ∞ in α -H¨older norm, α < /
2, on a set of full measure Ω , cf. (0.4) wherewe recall the definition of the H¨older norm and [Bog98, 3.5.1. Theorem] for a general result aboutGaussian sequences from which the convergence above follows. Assume that E carries the Wienermeasure. Then there is a set of full measure Ω such that the limit ∞ X n =1 ˜ Z n ( ω ) sin[( n − / πt ]( n − / π =: lim N →∞ ˜ B Nt ( ω ) =: ˜ B t ( ω )exists in E for every ω ∈ Ω where ˜ Z n := ( − n Z n . The theory of Young integration [You36]implies that Z ˜ B Nt ( ω ) dB Mt ( ω ) → Z ˜ B Nt ( ω ) dB t ( ω )as M → ∞ for every ω ∈ Ω ∩ Ω . Noting that ˜ Z n sin[( n − / πt ] = Z n cos[( n − / π (1 + t )], weobtain that Z ˜ B Nt ( ω ) dB Mt ( ω ) = N X n =1 Z n ( ω )( n − / π for all M ≥ N . Therefore, Z ˜ B Nt ( ω ) dB t ( ω ) = N X n =1 Z n ( ω )( n − / π → ∞ as N → ∞ on a set of full measure Ω ⊂ Ω ∩ Ω . Now we can argue by contradiction. Assumethat there is a set of full measure Ω such that for every ω ∈ Ω , the map E ∋ ξ ξ + Z ξ t − dB t ( ω )is continuous. Since Ω ∩ Ω has full measure, the set is nonempty and we can choose ω ∈ Ω ∩ Ω .Set ξ n := ˜ B n ( ω ) and ξ := ˜ B ( ω ). Then we have ξ n → ξ in E as n → ∞ , but R ξ nt − dB t ( ω ) divergesas n → ∞ which leads to a contradiction. (cid:3) This theorem shows that there is no reasonable space of functions on which the SDDE (0.2) induces a continuous semi-flow , and using RDS to study such equations seems indeed hopeless. Letus however mention here that only a delay in the diffusion part causes the trouble, a delay in apossible drift part would be harmless. For this reason, we will discard the drift in our article andstudy equations of the form (0.1) with b = 0 only. We also remark that studying delay equationswhere the diffusion coefficient may depend on a whole path segment of the solution, so-called continuous delay , can lead to easier equations since in that case, the diffusion coefficient might havea smoothing effect. Such equations are called regular stochastic delay differential equations, andthey can indeed be studied using RDS, cf. [MS96] and [MS97]. The equation (0.2) is an exampleof singular stochastic delay differential equation. M. GHANI VARZANEH, S. RIEDEL, AND M. SCHEUTZOW
Let us now explain the idea of the present article. We have seen that there is no space of paths E on which E ∋ ξ R ξ s dB s ( ω ) is a continuous map on a set of full measure. However, in roughpath theory, one knows that there is a family of Banach spaces { E ω } ω ∈ Ω and a set of full measure˜Ω such that the maps E ω ∋ ξ Z ξ s d B s ( ω )are continuous for every ω ∈ ˜Ω where the integral has to be interpreted as a rough paths integral .Indeed, the spaces E ω are nothing but the usual spaces of controlled paths introduced by Gubinelliin [Gub04] for which we will recall the definition below. Therefore, we can hope to establish a semi-flow property for solutions to (0.2) (and even more general equations) if we allow the state spacesto be random and by interpreting the equation as a delay differential equation driven by a randomrough path. Fortunately, Neuenkirch, Nourdin and Tindel already studied delay equations drivenby rough paths in [NNT08], and we can build on their results. Having established such a semi-flow property, the corresponding RDS will involve random spaces as well. This seems hopelesslycomplicated and maybe unnatural at first sight, but we argue that it is not. It turns out that thestructure of such RDS is similar to that which appears when studying the linearisation of an RDSwhich is induced by an SDE defined on a Riemannian manifold, cf. [Arn98, Chapter 4]. TheseRDS act on measurable bundles and are therefore called bundle RDS , cf. [Arn98, Section 1.9]. Ina sense, we will see that SDDE induce bundle RDS with the fibres being (infinite dimensional!)spaces of controlled paths. However, it turns out that defining a bundle structure is not necessarysince we are only interested in the fibres. Therefore, instead of studying RDS defined on an infinitedimensional bundle, we will study RDS which are defined on measurable fields of Banach spaces .After having defined such a structure, the crucial point to ask is whether an MET holds on it.Fortunately, this is indeed the case, and we provide a full proof of such a theorem in the presentwork. With the MET at hand, we can indeed deduce the existence of a Lyapunov spectrum forlinear SDDE. Our main result, which is a combination of Theorem 5.1 and Corollary 5.2 to befound in Section 5, can loosely be formulated as follows: Theorem 0.3.
Linear stochastic delay differential equations of the form dy t = σ ( y t , y t − r ) dB t ( ω )(0.3) induce linear RDS on measurable fields of Banach spaces given by the spaces of controlled pathsdefined by B ( ω ) . Furthermore, an MET applies and provides the existence of a Lyapunov spectrumfor the linear RDS. In Part II of our paper, we will show that also non-linear equations linearized around equilibriumpoints induce linear RDS, and prove a stable manifold theorem for such equations.Let us finally remark that stochastic differential equations on infinite dimensional spaces fre-quently lack the semi-flow property. For instance, this is often the case for stochastic partialdifferential equations (SPDEs), too, cf. e.g. [Fla95] and the references therein. We believe thatthe approach we present here can be applied also in the context of SPDEs to provide a dynamicalsystems approach to equations for which the semi-flow property is known not to hold.The article is structured as follows. In Section 1, we introduce the techniques to study delayequations driven by rough paths and prove some basic properties. The content of Section 2 isto show that the Brownian motion can drive rough delay equations and to prove a Wong-Zakaitheorem, also in the non-linear case, which might be of independent interest. In Section 3, we
YNAMICAL THEORY FOR SDDE 5 establish the connection to Arnold’s theory and define RDS on measurable fields of Banach spaces.Section 4 provides the formulation and the proof of an MET on a field of Banach spaces. The mainresults of the present paper and a discussion of them are contained in Section 5. Finally, we comeback to the example (0.2) and discuss it in more detail in Section 6.
Preliminaries and notation.
In this section we collect some notations which will be used through-out the paper. • If not stated differently, U , V , W and ¯ W will always denote finite-dimensional, normedvector spaces over the real numbers, with norm denoted by | · | . By L ( U, W ) we mean theset of linear and continuous functions from U to W equipped with usual operator norm. • Let I be an interval in R . A map m : I → U will also be called a path . For a path m , wedenote its increment by m s,t = m t − m s where by m t we mean m ( t ). We set k m k ∞ ; I := sup s ∈ I | m s | and define the γ -H¨older seminorm, γ ∈ (0 , k m k γ ; I := sup s,t ∈ I ; s = t | m s,t || t − s | γ . For a general 2-parameter function m : I × I → U , the same notation is used. We willsometimes omit I as subindex if the domain is clear from the context. The space C ( I, U )consists of all continuous paths m : I → U equipped with the uniform norm, C γ ( I, U )denotes the space of all γ -H¨older continuous functions equipped with the norm k · k C γ ; I := k · k ∞ ; I + k · k γ ; I . (0.4) C ∞ ( I, U ) is the space of all arbitrarily often differentiable functions. If 0 ∈ I , using 0 assubindex such as for C γ ( I, U ) denotes the subspace of functions for which x = 0. An upperindex such as C ,γ ( I, U ) means taking the closure of smooth functions in the correspondingnorms.Next, we introduce some basic objects from rough paths theory needed in this article. We referthe reader to [FH14] for a general overview. • Let X : R → U be a locally γ -H¨older path, γ ∈ (0 , L´evy area for X is a continuousfunction X : R × R → U ⊗ U for which the algebraic identity X s,t = X s,u + X u,t + X s,u ⊗ X u,t is true for every s, u, t ∈ R and for which k X k γ ; I < ∞ holds on every compact interval I ⊂ R . If γ ∈ (1 / , /
2] and X admits L´evy area X , we call X = (cid:0) X, X (cid:1) a γ -rough path . If X and Y are γ -rough paths, one defines ̺ γ ; I ( X , Y ) := sup s,t ∈ I ; s = t | X s,t − Y s,t || t − s | γ + sup s,t ∈ I ; s = t | X s,t − Y s,t || t − s | γ . M. GHANI VARZANEH, S. RIEDEL, AND M. SCHEUTZOW • Let I = [ a, b ] be a compact interval. A path m : I → ¯ W is a controlled path based on X onthe interval I if there exists a γ -H¨older path m ′ : I → L ( U, ¯ W ) such that m s,t = m ′ s X s,t + m s,t for all s, t ∈ I where m : I × I → ¯ W satisfies k m k γ ; I < ∞ . The path m ′ is called a Gubinelli derivative of m . We use D γX ( I, ¯ W ) to denote the space of controlled paths basedon X on the interval I . It can be shown that this space is a Banach space with norm k m k D γX := k ( m, m ′ ) k D γX := | m a | + | m ′ a | + k m ′ k γ ; I + k m k γ ; I . If X and ˜ X are γ -H¨older paths, ( m, m ′ ) ∈ D γX ( I, ¯ W ) and ( ˜ m, ˜ m ′ ) ∈ D γ ˜ X ( I, ¯ W ), we set d γ ; I (( m, m ′ ) , ( ˜ m, ˜ m ′ )) := k m ′ − ˜ m ′ k γ ; I + k m − ˜ m k γ ; I . If ¯ W = R , we will also use the noation D γX ( I ) instead of D γX ( I, R ).We finally recall the definition of a random dynamical system introduced by L. Arnold [Arn98]. • Let (Ω , F ) and ( X, B ) be measurable spaces. Let T be either R or Z , equipped with a σ -algebra I given by the Borel σ -algebra B ( R ) in the case of T = R and by P ( Z ) in the caseof T = Z . A family θ = ( θ t ) t ∈ T of maps from Ω to itself is called a measurable dynamicalsystem if(i) ( ω, t ) θ t ω is F ⊗ I / F -measurable,(ii) θ = Id,(iii) θ s + t = θ s ◦ θ t , for all s, t ∈ T .If T = Z , we will also use the notation θ := θ , θ n := θ n and θ − n := θ − n for n ≥
1. If P isfurthermore a probability on (Ω , F ) that is invariant under any of the elements of θ , P ◦ θ − t = P for every t ∈ T , we call the tuple (cid:0) Ω , F , P , θ (cid:1) a measurable metric dynamical system . Thesystem is called ergodic if every θ -invariant set has probability 0 or 1. • Let T + := { t ∈ T : t ≥ } , equipped with the trace σ -algebra. An (ergodic) measurablerandom dynamical system on ( X, B ) is an (ergodic) measurable metric dynamical system (cid:0) Ω , F , P , θ (cid:1) with a measurable map ϕ : T + × Ω × X → X that enjoys the cocycle property , i.e. ϕ (0 , ω, · ) = Id X , for all ω ∈ Ω, and ϕ ( t + s, ω, · ) = ϕ ( t, θ s ω, · ) ◦ ϕ ( s, ω, · )for all s, t ∈ T + and ω ∈ Ω. The map ϕ is called cocycle . If X is a topological space with B being the Borel σ -algebra and the map ϕ · ( ω, · ) : T + × X → X is continuous for every ω ∈ Ω, it is called a continuous (ergodic) random dynamical system . In general, we saythat ϕ has property P if and only if ϕ ( t, ω, · ) : X → X has property P for every t ∈ T + and ω ∈ Ω whenever the latter statement makes sense.1.
Basic properties of rough delay equations
In this section, we show how to solve rough delay differential equations and present some basicproperties of the solution.
YNAMICAL THEORY FOR SDDE 7
Basic objects, existence, uniqueness and stability.
This section basically summarizesthe concepts and results from [NNT08]. We start by introducing “delayed” versions of rough pathsand controlled paths. Note that, as already mentioned in the introduction, we restrict ourselvesto the case of one time delay only. We refer to [NNT08] for corresponding definitions for a finitenumber of delays.
Definition 1.1.
Let X : R → U be a locally γ -H¨older path and r >
0. A delayed L´evy area for X is a continuous function X ( − r ) : R × R → U ⊗ U for which the algebraic identity X s,t ( − r ) = X s,u ( − r ) + X u,t ( − r ) + X s − r,u − r ⊗ X u,t is true for every s, u, t ∈ R and for which k X ( − r ) k γ ; I < ∞ holds on every compact interval I ⊂ R . If γ ∈ (1 / , /
2] and X admits L´evy- and delayed L´evy area X and X ( − r ), we call X = (cid:0) X, X , X ( − r ) (cid:1) a delayed γ -rough path with delay r >
0. If X and Y are delayed γ -rough paths, we set ̺ γ ; I ( X , Y ) := sup s,t ∈ I ; s = t | X s,t − Y s,t || t − s | γ + sup s,t ∈ I ; s = t | X s,t − Y s,t || t − s | γ + sup s,t ∈ I ; s = t | X ( − r ) s,t − Y ( − r ) s,t || t − s | γ . Remark . For X as in the former definition, set Z := ( X, X ·− r ) ∈ U ⊕ U. If X admits a L´evy- and delayed L´evy area, also Z admits a L´evy area Z given by Z = (cid:18) X ¯ X ( − r ) X ( − r ) X ·− r, ·− r (cid:19) where ¯ X ij ( − r ) := X is,t X js − r,t − r − X jis,t ( − r ). Conversely, if Z admits a L´evy area, the path X admitsboth L´evy- and delayed L´evy area. The delayed L´evy area can therefore be understood as the usualL´evy area of a path enriched with its delayed path.Next, we recall what is a delayed controlled path. Definition 1.3.
Let I = [ a, b ] be a compact interval. A path m : I → ¯ W is a delayed controlledpath based on X on the interval I if there exist γ -H¨older paths ζ , ζ : I → L ( U, ¯ W ) such that m s,t = ζ s X s,t + ζ s X s − r,t − r + m s,t (1.1)for all s, t ∈ I where m : I × I → ¯ W satisfies k m k γ ; I < ∞ . The path ( ζ , ζ ) will again be called Gubinelli derivative of m . We use D γX ( I, ¯ W ) to denote the space of delayed controlled paths basedon X on the interval I . A norm on this space can be defined by k m k D γX := k ( m, ζ , ζ ) k D γX := | m a | + | ζ a | + | ζ a | + k ζ k γ ; I + k ζ k γ ; I + k m k γ ; I . (1.2) Remark . Note that any controlled path is also a delayed controlled path (by the choice ζ = 0),but the converse might not be true. However, considering again the enhanced path Z = ( X, X ·− r ) ∈ U ⊕ U, the identity (1.1) shows that m is a usual ¯ W -valued controlled path based on Z with Gubinelliderivative ¯ ζ : I → L ( U ⊕ U, ¯ W ) given by ¯ ζ t ( v, w ) := ζ t v + ζ t w .With these objects, we can define an integral as follows. M. GHANI VARZANEH, S. RIEDEL, AND M. SCHEUTZOW
Theorem 1.5.
Let X = (cid:0) X, X , X ( − r ) (cid:1) be a delayed γ -rough path and m an L ( U, W ) -valued delayedcontrolled path based on X with decomposition as in (1.1) on the interval [ a, b ] . Then the limit Z ba m s d X s := lim | Π |→ X t j ∈ Π m t j X t j,tj +1 + ζ t j X t j ,t j +1 + ζ t j X t j ,t j +1 ( − r )(1.3) exists where Π denotes a partition of [ a, b ] . Moreover, there is a constant C depending on γ and ( b − a ) only such that for all s < t ∈ [ a, b ] , the estimate (cid:12)(cid:12)(cid:12)(cid:12)Z ts m u d X u − m s X s,t − ζ s X s,t − ζ s X s,t ( − r ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:0) k m k γ k X k γ + k ζ k γ k X k γ + k ζ k γ k X ( − r ) k γ (cid:1) | t − s | γ holds. In particular, t Z ts m u d X u is controlled by X with Gubinelli derivative m .Proof. This is just an application of the Sewing lemma, cf. e.g. [FH14, Lemma 4.2], applied toΞ s,t = m s X s,t + ζ s X s,t + ζ s X s,t ( − r ) . (cid:3) Example 1.6.
Let U = W = R and X = (cid:0) X, X , X ( − (cid:1) be a delayed γ -rough path. We aim tosolve the equation dy t = y t − d X t ; t ≥ y t = ξ t ; t ∈ [ − , . (1.4)If ξ ∈ D γX ([ − , , ∋ t ξ t − is a delayed controlled path, thus the integral[0 , ∋ t Z t ξ s − d X s exists. Therefore, the path y t := ( ξ t if t ∈ [ − , R t ξ s − d X s + ξ if t ∈ [0 , − , D γX ([0 , Definition 1.7. By C b ( W , L ( U, W )), we denote the space of bounded functions σ : W ⊕ W → L ( U, W ) possessing 3 bounded derivatives.We can now state the first existence and uniqueness result for rough delay equations.
YNAMICAL THEORY FOR SDDE 9
Theorem 1.8 (Neuenkirch, Nourdin, Tindel) . For r > , let X be a delayed γ -rough path for γ ∈ (1 / , / , σ ∈ C b ( W , L ( U, W )) and ( ξ, ξ ′ ) ∈ D βX ([ − r, , W ) for some β ∈ (1 / , γ ) . Then theequation y t = ξ + Z t σ ( y s , y s − r ) d X s ; t ∈ [0 , r ] y t = ξ t ; t ∈ [ − r, has a unique solution ( y, y ′ ) ∈ D βX ([0 , T ] , W ) with Gubinelli derivative given by y ′ t = σ ( y t , y t − r ) .Proof. The theorem was proved in [NNT08, Theorem 4.2], we quickly sketch the idea here: First,it can be shown that for an element ζ ∈ D βX ([0 , r ] , W ), the path σ ( ζ · , ξ ·− r ) is a delayed controlledpath. Therefore, one can consider the map ζ ξ + Z · σ ( ζ u , ξ u − r ) d X u and prove that it has a fixed point in the space D βX ([0 , r ] , W ) to obtain a solution on [0 , r ]. Theclaimed Gubinelli derivative can be deduced using the estimate provided in Theorem 1.5. (cid:3) We proceed with a theorem which shows that the solution map induced by (1.5) is continuous.Unfortunately, the corresponding result stated in [NNT08, Theorem 4.2] is not correct, therefore wecan not cite it directly. We will first formulate the correct statement and then discuss the differencecompared to [NNT08, Theorem 4.2].
Theorem 1.9.
Let X and ˜ X be a delayed γ -rough paths with γ ∈ (1 / , / , σ ∈ C b ( W , L ( U, W )) and choose ( ξ, ξ ′ ) ∈ D βX ([ − r, , W ) and ( ˜ ξ, ˜ ξ ′ ) ∈ D β ˜ X ([ − r, , W ) for some β ∈ (1 / , γ ) . Considerthe solutions ( y, y ′ ) and (˜ y, ˜ y ′ ) to dy t = σ ( y t , y t − r ) d X ; t ∈ [0 , r ] y t = ξ t ; t ∈ [ − r, resp. d ˜ y t = σ (˜ y t , ˜ y t − r ) d ˜ X ; t ∈ [0 , r ]˜ y t = ˜ ξ t ; t ∈ [ − r, . Then d β ;[0 ,r ] (( y, y ′ ) , (˜ y, ˜ y ′ )) ≤ C (cid:16) | ξ − r − ˜ ξ − r | + | ξ ′− r − ˜ ξ ′− r | + d β ;[ − r, (( ξ, ξ ′ ) , ( ˜ ξ, ˜ ξ ′ )) + ̺ γ ;[0 ,r ] ( X , ˜ X ) (cid:17) (1.6) holds for some constant C > depending on r , γ , β and M , where M is chosen such that M ≥k ξ k D βX + k ˜ ξ k D β ˜ X + k X k γ + k X k γ + k X ( − r ) k γ + k ˜ X k γ + k ˜ X k γ + k ˜ X ( − r ) k γ . Remark . In [NNT08, Theorem 4.2], the authors state that the estimate k y − ˜ y k β ;[0 ,r ] ≤ C ( | ξ − r − ˜ ξ − r | + k ξ − ˜ ξ k β ;[ − r, + ρ γ ( X , ˜ X ))(1.7) holds for the usual H¨older norm. However, this estimate can not be true in general. To see this,assume X = X =: X and consider the equation in Example 1.6. If (1.7) was true, the map ξ Z ξ d X would be continuous in the β -H¨older norm, which is clearly not the case for a genuine rough path X .The proof of Theorem 1.9 is a bit lengthy, but mostly straightforward. We sketch it in theappendix, cf. page 43.1.2. Linear equations.
In this section, we consider the case where σ is linear, i.e. σ ∈ L (cid:0) W , L ( U, W ) (cid:1) .Note that in this case, there are σ , σ ∈ L (cid:0) W, L ( U, W ) (cid:1) such that σ ( y , y ) = σ ( y ) + σ ( y ) forall y , y ∈ W . Since linear vector fields are unbounded, we cannot directly apply Theorem 1.8.However, we can prove an a priori bound for any solution of the equation and then deduce existence,uniqueness and stability for linear equations from Theorem 1.8 and 1.9 by truncating the vectorfield σ . Theorem 1.11.
Let X be a delayed γ -rough path over X with γ ∈ (1 / , / and σ ∈ L ( W , L ( U, W )) .Then any solution y : [0 , r ] → W of dy t = σ ( y t , y t − r ) d X ; t ∈ [0 , r ] y t = ξ t ; t ∈ [ − r, satisfies, for ( y, y ′ ) = ( y, σ ( y, ξ ·− r )) , the bound k y k D βX ([0 ,r ] ,W ) C (cid:0) r γ − β k X k γ ;[0 ,r ] (cid:1) k ξ k D βX ([ − r, ,W ) exp n C ( k X k γ ;[0 ,r ] + k X k γ ;[0 ,r ] + k X ( − r ) k γ ;[0 ,r ] ) γ − β o (1.9) where C depends on r , k σ k , γ and β .Proof. For s, t ∈ [0 , r ], we have y s,t = y ′ s X s,t + y s,t where y ′ s = σ ( y s , ξ s − r )(1.10)and y s,t = Z ts σ ( y u , y u − r ) d X u − σ ( y s , ξ s − r ) X s,t = ˜ ρ s,t + σ y ′ s X s,t + σ ξ ′ s − r X s,t ( − r )with ˜ ρ given by˜ ρ s,t = Z ts σ ( y u , y u − r ) d X u − σ ( y s , ξ s − r ) X s,t − σ y ′ s X s,t − σ ξ ′ s − r X s,t ( − r ) . Note that u σ ( y u , ξ u − r ) is a delayed controlled path with Gubinelli derivative u ( σ y ′ u , σ ξ ′ u − r ).Therefore, we can use the estimate provided in Theorem 1.5 to see that for a constant M = M ( β, r ) YNAMICAL THEORY FOR SDDE 11 and I = [ a, b ] ⊂ [0 , r ] : k y k β ; I k σ k (cid:0) k y ′ k ∞ ; I k X k γ ; I + k ξ ′ k ∞ ;[ − r, k X ( − r ) k γ ; I (cid:1) ( b − a ) γ − β + M k σ k (cid:0) k y k β ; I k X k γ ; I + k ξ k β ;[ − r, k X k γ ; I (cid:1) ( b − a ) γ + M k σ k (cid:0) k y ′ k β ; I k X k γ ; I + k ξ ′ k β ;[ − r, k X ( − r ) k γ ; I (cid:1) ( b − a ) γ − β (1.11)and by relation (1.10) : k y k β ; I k σ k (cid:0) k y k ∞ ,I + k ξ k ∞ , [ − r, (cid:1) k X k γ ; I ( b − a ) γ − β + k y k β ; I ( b − a ) β and k y ′ k β ; I k σ k (cid:0) k y k β ; I + k ξ k β ;[ − r, (cid:1) . Now assume that b − a = θ < ∧ r for a given θ and set A := 1 + k X k γ ;[0 ,r ] + k X k γ ;[0 ,r ] + k X ( − r ) k γ ;[0 ,r ] . Our former estimates imply that there are constants ˜
M , ˜ N depending on k σ k such that k y k β ; I + k y k ∞ ; I + k y ′ k β ; I + k y k β ; I + k y ′ k ∞ ; I ˜ M Aθ γ − β (cid:0) k y k β ; I + k y k ∞ ; I + k y ′ k β ; I + k y k β ; I + k y ′ k ∞ ; I (cid:1) +(1.12) ˜ N A (cid:0) k ξ k β ;[ − r, + k ξ k ∞ ;[ − r, + k ξ ′ k ∞ ;[ − r, + k ξ k β ;[ − r, (cid:1) + (cid:0) k σ k (cid:1) k y k ∞ ; I . Choose θ small enough such that˜ M Aθ γ − β
14 and θ β (1 + k σ k ) . (1.13)For n > nθ r , set I n := [( n − θ, nθ ] and B n = k y k β ; I n + k y k ∞ ; I n + k y ′ k β ; I n + k y k β ; I n + k y ′ k ∞ ; I n B = k ξ k β ;[ − r, + k ξ k ∞ ;[ − r, + k ξ ′ k ∞ ;[ − r, + k ξ k β ;[ − r, . Note that k y k ∞ ; I n B n − + θ β B n . By (1.12) and (1.13), B n NAB + 2 (cid:0) k σ k (cid:1) B n − . Set C = 2 ˜ N A and ˜ C = 2 (cid:0) k σ k (cid:1) . By a simple induction argument, it is not hard to verify thatfor k ≤ n , B n C (1 + ˜ C + ˜ C + ... + ˜ C k − ) B + ˜ C k B n − k which implies B n ˜ C n (1 + C ) B for k = n . Note that since y s,t = y s,u + y u,t + y ′ s,u X u,t , k y k β ;[0 ,r ] X n m k y k β ; I n + r γ − β k X k γ ;[0 ,r ] X n m k y ′ k β ; I n (1.14)Now set m = [ rθ ] + 1. By (1.14) and subadditivity of the H¨older norm, k y k β ;[0 ,r ] + k y k ∞ ;[0 ,r ] + k y ′ k β ;[0 ,r ] + k y k β ;[0 ,r ] + k y ′ k ∞ ;[0 ,r ] (cid:0) r γ − β k X k γ ;[0 ,r ] (cid:1) X n m B n (cid:0) r γ − β k X k γ ;[0 ,r ] (cid:1) ˜ C m +1 (1 + C ) B . Note that an appropriate choice for θ is θ = 1 (cid:0) M A (cid:1) γ − β + (cid:0) k σ k ) (cid:1) β + 1 + r (1.15)which implies the claimed bound. (cid:3) From Theorem 1.11, it follows that in the case of linear vector fields σ , the solution map inducedby (1.8) is a bounded linear map. We now prove that it is even compact. Proposition 1.12.
Under the same assumptions as in Theorem 1.11, the solution map induced by (1.8) is a compact linear map for every / < β < γ .Proof. Fix β < γ . Let { ξ ( n ) } n > be a bounded sequence in D βX ([ − r, , W ), i.e. ξ ( n ) u,v = ( ξ ( n ) ) ′ u X u,v + ( ξ ( n ) ) u,v with uniformly bounded β -H¨older norm of ξ ( n ) and ( ξ ( n ) ) ′ and uniformly bounded 2 β -H¨older normof ( ξ ( n ) ) . From the Arzel`a-Ascoli theorem, there are continuous functions ξ and ξ ′ such that( ξ ( n ) , ( ξ ( n ) ) ′ ) → ( ξ, ξ ′ )uniformly along a subsequence, which we will henceforth denote by ( ξ ( n ) , ( ξ ( n ) ) ′ ) n itself. It followsthat ( ξ ( n ) , ( ξ ( n ) ) ′ ) → ( ξ, ξ ′ ) in δ -H¨older norm for every δ < β . Define ξ u,v := ξ u,v − ξ ′ u X u,v . Clearly,( ξ ( n ) ) → ξ uniformly, and since | ξ u,v | ≤ sup n k ( ξ ( n ) ) k β ;[ − r, | v − u | β for every − r ≤ u ≤ v ≤
0, it follows that ( ξ ( n ) ) → ξ in 2 δ -H¨older norm for every δ < β . Thisimplies that ( ξ ( n ) , ( ξ ( n ) ) ′ ) → ( ξ, ξ ′ ) in the space D δX ([ − r, , W ) for every δ < β . Let ( y n , ( y n ) ′ )denotes the solutions to (1.8) for the initial conditions ( ξ n , ( ξ n ) ′ ). Fix some 1 / < δ < β . Fromcontinuity, the solutions ( y n , ( y n ) ′ ) converge in the space D δX ([0 , r ] , W ), too. Choose β < β ′ < γ .Using a similar estimate as (1.11) in Theorem 1.11 where we apply the estimate in Theorem 1.5 for δ shows that we can bound k ( y n , ( y n ) ′ ) k D β ′ X ([0 ,r ] ,W ) uniformly over n where the bound depends, inparticular, on sup n k ( y n , ( y n ) ′ ) k D δX ([0 ,r ] ,W ) . This implies convergence also in the space D βX ([0 , r ] , W )and therefore proves compactness. (cid:3) A semi-flow property.
In this section, we discuss the flow property induced by a roughdelay equation. Recall that a flow on some set M is a mapping φ : [0 , ∞ ) × [0 , ∞ ) × M → M such that φ ( t, t, ξ ) = ξ and φ ( s, t, ξ ) = φ ( u, t, φ ( s, u, ξ ))(1.16)hold for every ξ ∈ M and s, t, u ∈ [0 , ∞ ). Our prime example of a flow is a differential equationin which case ξ ∈ M denotes an initial condition at time point s and φ ( s, t, ξ ) denotes the solutionat time t . In the setting of a delay equation, we can only expect to solve the equation forward intime, i.e. φ ( s, t, ξ ) will only be defined for s ≤ t . If (1.16) is assumed to hold only for s ≤ u ≤ t , wewill speak of a semi-flow . In case of a rough delay equation, we will give up the idea of choosing a YNAMICAL THEORY FOR SDDE 13 common set of admissible initial conditions M which will work for all time instances. Instead, oursemi-flow will actually consist of a family of maps φ ( s, t, · ) : M s → M t where ( M t ) t ≥ are sets (later: spaces) indexed by time. Note that the semi-flow property (1.16)still makes perfect sense in this setting, and this is what we are going to prove. Note also that thephenomenon of time-varying spaces is already visible in Example 1.6: admissible initial conditionsare controlled paths defined on intervals depending on the time when we start to solve the equation. Theorem 1.13.
Let X be a delayed γ -rough path over X with γ ∈ (1 / , / and σ ∈ C b ( W , L ( U, W )) .Consider the equation dy t = σ ( y t , y t − r ) d X ; t ∈ [ s, ∞ ) y t = ξ t ; t ∈ [ s − r, s ](1.17) for s ∈ R . Let β ∈ (1 / , γ ) . If ξ ∈ D βX ([ s − r, s ] , W ) , the equation (1.17) has a unique solution y : [ s, ∞ ) → W and for t ≥ s , we denote by φ ( s, t, ξ ) the solution path segment φ ( s, t, ξ ) = ( y u ) t − r ≤ u ≤ t . If r t − s , we have that φ ( s, t, ξ ) ∈ D βX ([ t − r, t ] , W ) with Gubinelli derivative φ ′ ( s, t, ξ ) =( σ ( y u , y u − r )) t − r ≤ u ≤ t and φ ( s, t, · ) : D βX ([ s − r, s ] , W ) → D βX ([ t − r, t ] , W )(1.18) ξ φ ( s, t, ξ ) is a continuous map. In case that ξ ′ s = σ (cid:0) ξ s , ξ s − r (cid:1) , we have φ ( s, t, ξ ) ∈ D βX ([ − r + t, t ] , W ) for all s ≤ t with Gubinelli derivative given by φ ( s, t, ξ )( u ) = ( ξ ′ u for t − r ≤ u ≤ sσ ( y u , y u − r ) for s ≤ u ≤ t for r > t − s . For s ≤ u ≤ t and r u − s , we have the semi-flow property φ ( s, s, · ) = Id D pX ([ − r + s,s ] ,W ) and φ ( u, t, · ) ◦ φ ( s, u, ξ ) = φ ( s, t, ξ ) . (1.19) Again, if ξ ′ s = σ (cid:0) ξ s , ξ − r + s (cid:1) , (1.19) is true for all s ≤ u ≤ t .Proof. As in Theorem 1.8, we can first solve (1.17) on the time interval [ s, s + r ]. This can nowbe iterated to obtain a solution on [ s, ∞ ). The claimed Gubinelli derivatives on every interval[ s + kr, s + ( k + 1) r ], k ∈ N , are a consequence of Theorem 1.5. Since the derivatives agree onthe boundary points of the intervals, we can “glue them together” to obtain a controlled path onarbitrary intervals [ u, v ] ⊂ [ s, ∞ ). If the assumption ξ ′ s = σ (cid:0) ξ s , ξ s − r (cid:1) holds, this can even be donefor every interval [ u, v ] ⊂ [ s − r, ∞ ). Continuity of the map (1.18) is a consequence of Theorem 1.9.The semi-flow property (1.19) is a consequence of existence and uniqueness of solutions: Let y s,ξτ be the solution of (1.17) for τ ≥ s − r where y s,ξτ = ξ τ for s − r τ s . Let s ≤ u ≤ t and assumeeither r u − s or ξ ′ s = σ (cid:0) ξ s , ξ s − r (cid:1) . For τ < u , it is not hard to verify that y s,ξτ = y u,φ ( s,u,ξ ) τ . If u τ by definition: y s,ξτ = ξ s + Z τs σ ( y s,ξz , y s,ξz − r ) d X z = y s,ξu + Z τu σ ( y s,ξz , y s,ξz − r ) d X z and y u,φ ( s,u,ξ ) τ = y s,ξu + Z τu σ ( y u,φ ( s,u,ξ ) z , y u,φ ( s,u,ξ ) z − r ) d X z . Given the uniqueness of the solution, y s,ξτ = y u,φ ( s,u,ξ ) τ which indeed implies (1.19). (cid:3) Existence of delayed L´evy areas for the Brownian motion and a Wong-Zakaitheorem
In order to apply the results from Section 1 to stochastic delay differential equations, we needto make sure that the Brownian motion can be ”lifted” to a process taking values in the space ofdelayed rough paths. In this section, B = ( B , . . . , B d ) : R → R d will always denote an R d -valuedtwo-sided Brownian motion defined on a probability space (Ω , F , P ) adapted to some two-parameterfiltration ( F ts ) s ≤ t , i.e. ( B t + s − B s ) t ≥ is a usual ( F t + ss ) t ≥ -Brownian motion for every s ∈ R and B = 0 almost surely (cf. [Arn98, Section 2.3.2] for a more detailed discussion about two-sidedstochastic processes). Definition 2.1.
For r >
0, set B It¯o s,t := (cid:0) B s,t , B It¯o s,t , B It¯o s,t ( − r ) (cid:1) := (cid:18) B t − B s , Z ts ( B u − B s ) ⊗ dB u , Z ts ( B u − r − B s − r ) ⊗ dB u (cid:19) for s ≤ t ∈ R where the stochastic integrals are understood in It¯o-sense. We furthermore define B Strat s,t := (cid:18) B s,t , B It¯o s,t + 12 ( t − s ) I d , B It¯o s,t ( − r ) (cid:19) where I d denotes the identity matrix in R d . Proposition 2.2.
Both processes B It¯o and B Strat have modifications, henceforth denoted with thesame symbols, with sample paths being delayed γ -rough paths for every γ < / almost surely.Moreover, the γ -H¨older norms of both processes have finite p -th moment for every p > on anycompact interval.Proof. The assertion follows by considering the usual It¯o- and Stratonovich lifts of the enhancedprocess (
B, B ·− r ) as in [FH14, Section 3.2 and 3.], using the Kolmogorov criterion for rough pathsstated in [FH14, Theorem 3.1] (cf. also Remark 1.2). (cid:3) The next proposition justifies the names of the processes defined above.
Proposition 2.3.
Let ( m ( ω ) , ζ ( ω ) , ζ ( ω )) ∈ D γB ( ω ) almost surely. Furthermore, assume that theprocess ( m t , ζ t , ζ t ) t ≥ is ( F t ) t ≥ -adapted. Then Z T m s dB s = Z T m s d B It¯o s and Z T m s ◦ dB s = Z T m s d B Strat almost surely for every
T > . YNAMICAL THEORY FOR SDDE 15
Proof.
We will first consider the It¯o-case which is similar to [FH14, Proposition 5.1]. Set F t := F t .To simplify notation, assume W = R . Let ( τ j ) be a partition of [0 , T ]. We first prove that E (cid:20)(cid:0) ζ τ j B τ j ,τ j +1 + ζ τ j B τ j ,τ j +1 ( − r ) (cid:1)(cid:0) ζ τ k B τ k ,τ k +1 + ζ τ k B τ k ,τ k +1 ( − r ) (cid:1)(cid:21) = 0(2.1)for j < k . To see this, note that E (cid:2)(cid:0) ζ τ j B τ j ,τ j +1 ( − r ) (cid:1)(cid:0) ζ τ k B τ k ,τ k +1 ( − r ) (cid:1)(cid:3) = E (cid:20) E (cid:2)(cid:0) ζ τ j B τ j ,τ j +1 ( − r ) (cid:1)(cid:0) ζ τ k B τ k ,τ k +1 ( − r ) (cid:1)(cid:12)(cid:12) F τ k (cid:3)(cid:21) = E (cid:20) ζ τ j B τ j ,τ j +1 ( − r ) ζ τ k E (cid:2) B τ k ,τ k +1 ( − r ) (cid:12)(cid:12) F τ k (cid:3)(cid:21) . We show that E (cid:2) B s,u ( − r ) (cid:12)(cid:12) F s (cid:3) = 0 for s ≤ u . By definition, B s,u ( − r ) = lim | Π |→ X t k ∈ Π B s − r,t k − r ⊗ B t k ,t k +1 where Π is a partition for [ s, u ] and the limit is understood in L (Ω)-sense. Consequently, E (cid:2) B s,u ( − r ) (cid:12)(cid:12) F s (cid:3) = lim | Π |→ X t k ∈ Π E (cid:2) B s − r,t k − r ⊗ B t k ,t k +1 (cid:12)(cid:12) F s (cid:3) again in L . Note that E (cid:2) B s − r,t k − r ⊗ B t k ,t k +1 (cid:12)(cid:12) F s (cid:3) = ( B s − r,t k − r ⊗ E (cid:2) B t k ,t k +1 (cid:12)(cid:12) F s (cid:3) = 0 , if t k − r sB s − r,s ⊗ E (cid:2) B t k ,t k +1 (cid:12)(cid:12) F s (cid:3) + E (cid:2) B s,t k − r ⊗ B t k ,t k +1 (cid:12)(cid:12) F s (cid:3) = 0 , if s < t k − r. Other cases are similar and (2.1) can be deduced. Using a stopping argument, we may assume thatthere is a deterministic
M > t ∈ [0 ,T ] k ζ t ( ω ) k ∨ k ζ t ( ω ) k ≤ M almost surely. Then, E (cid:20)(cid:0) X j ζ t j B τ j ,τ j +1 + ζ τ j B τ j ,τ j +1 ( − r ) (cid:1) (cid:21) M X j ( τ j +1 − τ j ) ≤ M T max j | τ j +1 − τ j | which converges to 0 when the mesh size of the partition gets small. The claim now follows usingthe definition of the It¯o integral as a limit of Riemann sums. The proof for the Stratonovich integralis similar to [FH14, Corollary 5.2]. (cid:3) The following corollary is immediate.
Corollary 2.4.
The solution to the It¯o equation dY t = σ ( Y t , Y t − r ) dB t is almost surely equal to the solution to the random rough delay equation dY t = σ ( Y t , Y t − r ) d B It¯o t if the initial condition is F − -measurable and almost surely controlled by B . The same statementholds in the Stratonovich case. Next, we prove an approximation result.
Definition 2.5.
Let ρ : R → [0 ,
2] be a smooth function such that supp( ρ ) ⊂ [0 ,
1] and whichintegrates to 1. We set B εt := Z R B − εz,t − εz ρ ( z ) dz, ε ∈ (0 , . It is not hard to see that E | B εs,t | M ( t − s ) and lim ε → E | B εs,t − B s,t | = 0(2.2)where M is independent of ε . Lemma 2.6.
We have the following pathwise identity: Z ts B εs − r,u − r ⊗ dB εu = Z R ρ ( z ) Z t − εzs − εz B εs − r,u + εz − r ⊗ dB u dz. (2.3) Proof.
Note that both integrals in (2.3) are indeed pathwise defined since B ε is smooth and B isH¨older continuous. Using integration by parts, for i, j ∈ { , . . . , d } , Z t − εzs − εz ( B ε ) is − r,u + εz − r dB ju = ( B ε ) is − r,t − r B jt − εz − Z t − rs − r B ju + r − εz d ( B ε ) iu . Consequently, Z R ρ ( z ) Z t − εzs − εz ( B ε ) is − r,u + εz − r dB ju dz = Z R ( B ε ) is − r,t − r ρ ( z ) B jt − εz dz − Z R Z t − rs − r ρ ( z ) B ju + r − εz d ( B ε ) iu dz = ( B ε ) is − r,t − r ( B ε ) jt − Z t − rs − r ( B ε ) ju + r d ( B ε ) iu . Using integration by parts again, we have( B ε ) is − r,t − r ( B ε ) jt − Z t − rs − r ( B ε ) ju + r d ( B ε ) iu = Z ts ( B ε ) is − r,u − r d ( B ε ) ju which implies the claim. (cid:3) Lemma 2.7.
For B s,t ( − r ) = R ts B s − r,u − r ⊗ dB u and B εs,t ( − r ) = R ts B εs − r,u − r ⊗ dB εu , E (cid:12)(cid:12) B s,t ( − r ) (cid:12)(cid:12) ≤ M ( t − s ) and E (cid:12)(cid:12) B εs,t ( − r ) (cid:12)(cid:12) ≤ M ( t − s ) (2.4) for a constant M > independent of s, t and ε .Proof. An easy consequence of the Cauchy-Schwarz inequality and Lemma 2.6. (cid:3)
Lemma 2.8.
We have lim ε → E (cid:12)(cid:12) B s,t ( − r ) − B εs,t ( − r ) (cid:12)(cid:12) = 0 . (2.5) Proof.
A direct consequence of Lemma 2.6 and (2.2). (cid:3)
Theorem 2.9.
Setting B εs,t := (cid:0) B εs,t , B εs,t , B εs,t ( − r ) (cid:1) := (cid:18) B εs,t , Z ts B εs,u ⊗ dB εu , Z ts B εs − r,u − r ⊗ dB εu (cid:19) , YNAMICAL THEORY FOR SDDE 17 we have lim ε →∞ sup q > (cid:13)(cid:13) d γ ; I (cid:0) B ε , B Strat (cid:1)(cid:13)(cid:13) L q √ q = 0 for every γ < / and every compact interval I ⊂ R where d γ ; I denotes the homogeneous metric d γ ; I ( X , Y ) = sup s,t ∈ I ; s = t | X s,t − Y s,t || t − s | γ + s sup s,t ∈ I ; s = t | X s,t − Y s,t || t − s | γ + s sup s,t ∈ I ; s = t | X s,t ( − r ) − Y s,t ( − r ) || t − s | γ . Proof.
The strategy of the proof is standard, cf. [FV10, Chapter 15], we only sketch the mainarguments. First, the uniform bounds (2.4) and the convergence (2.5) hold for B ε and B Strat , too,cf. [FV10, Theorem 15.33 and Theorem 15.37]. Since all objects are elements in the second Wienerchaos, the results even hold in the L q -norm for any q ≥
1. We can now argue as in the proof of[FV10, Proposition 15.24] to conclude. (cid:3)
As an application, we can prove a Wong-Zakai theorem for stochastic delay equations.
Theorem 2.10.
Let σ ∈ C b ( W , L ( U, W )) and B ε be defined as above. Assume that there is a setof full measure ˜Ω ⊂ Ω such that ( ξ ( ω ) , ξ ′ ( ω )) ∈ D B ε ( ω ) ([ − r, , W ) ∩ D B ( ω ) ([ − r, , W )(2.6) holds for every ε ∈ (0 , and every ω ∈ ˜Ω . Then the solutions to random delay ordinary differentialequations dY εt = σ ( Y εt , Y εt − r ) dB εt ; t ≥ Y εt = ξ t ; t ∈ [ − r, converge in probability as ε → in γ -H¨older norm on compact sets for every γ < / to the solution Y of dY t = σ ( Y t , Y t − r ) d B Strat t ; t ≥ Y t = ξ t ; t ∈ [ − r, . Moreover, if ( ξ t , ξ ′ t ) is F − -measurable for every t ∈ [ − r, , the solution Y coincides almost surelywith the solution of the Stratonovich delay equation dY t = σ ( Y t , Y t − r ) ◦ dB t ; t ≥ Y t = ξ t ; t ∈ [ − r, . Proof.
A combination of the stability result in Theorem 1.9, Theorem 2.9 and Corollary 2.4. (cid:3)
Remark . Note that (2.6) is satisfied, for instance, if ξ has almost surely differentiable samplepaths, in which case we can choose ξ ′ ≡ Random Dynamical Systems induced by stochastic delay equations
This section establishes the connection between stochastic delay equations and Arnold’s conceptof a random dynamical system.
Delayed rough path cocycles.
We start by describing the object which will drive ourequation. The following definition is an analogue of a rough paths cocycle defined in [BRS17] fordelay equations.
Definition 3.1.
Let (Ω , F , P , ( θ t ) t ∈ R ) be a measurable metric dynamical system and r >
0. A delayed γ -rough path cocycle X (with delay r > ) is a delayed γ -rough path valued stochasticprocess X ( ω ) = ( X ( ω ) , X ( ω ) , X ( − r )( ω )) such that X s,s + t ( ω ) = X ,t ( θ s ω )(3.1)holds for every ω ∈ Ω and every s, t ∈ R .Our goal is to prove that Brownian motion together with L´evy- and delayed L´evy area can beunderstood as delayed rough path cocycles. Definition 3.2.
For a finite-dimensional vector space U , set˜ T ( U ) := (cid:8)(cid:0) ⊕ ( α, β ) ⊕ ( γ, θ ) (cid:1) | α, β ∈ U and γ, θ ∈ U ⊗ U (cid:9) . We define projections Π ji byΠ ji (cid:0) ⊕ ( α, β ) ⊕ ( γ, θ ) (cid:1) := α if i = 1 , j = 1 β if i = 1 , j = 2 γ if i = 2 , j = 1 θ if i = 2 , j = 2 . Furthermore, we set (cid:0) ⊕ ( α , β ) ⊕ ( γ , θ ) (cid:1) ⊛ (cid:0) ⊕ ( α , β ) ⊕ ( γ , θ ) (cid:1) := (cid:0) ⊕ ( α + α , β + β ) ⊕ ( γ + γ + α ⊗ α , θ + θ + β ⊗ α ) (cid:1) and := (1 , (0 , , (0 , T ( U ) , ⊛ ) is a topological group with identity . For a continuous U -valued path of bounded variation x , we can define the following natural lifting map˜ S ( x ) u,v := (cid:18) ⊕ (cid:0) x u,v , x u − r,v − r (cid:1) ⊕ (cid:0) Z vu x u,τ ⊗ dx τ , Z vu x u − r,τ − r ⊗ dx τ (cid:1)(cid:19) ∈ ˜ T ( U ) . Definition 3.3.
Assume I ⊂ R and 0 ∈ I . We define C , − var0 ( I, U ) as the closure of the set ofarbitrarily often differentiable paths x from I to U with x = 0 with respect to the 1-variationnorm. Furthermore, C ,p − var ( I, ˜ T ( U )) is defined as the set of continuous maps x : I → ˜ T ( U )such that x = and for which there exists a sequence x n ∈ C , − var0 ( I, U ) with d p − var (cid:0) x , ˜ S ( x n ) (cid:1) := sup i,j ∈{ , } (cid:18) sup P⊂ I X t k ∈P (cid:12)(cid:12) Π ji (cid:0) x t k ,t k +1 − ˜ S ( x n ) t k ,t k +1 (cid:1)(cid:12)(cid:12) pi (cid:19) p −→ n → ∞ . We use the notation x s,t := x − s ⊛ x t here. The space C ,p − var ( R , ˜ T ( U )) consists ofall continuous paths x : R → ˜ T ( U ) for which x | I ∈ C ,p − var ( I, ˜ T ( U )) for every I as above.We can now state the following results: YNAMICAL THEORY FOR SDDE 19
Theorem 3.4.
Let p > and let X be an C ,p − var ( R , ˜ T ( U )) -valued random variable on a prob-ability space (Ω , F , P ) . Assume that X has stationary increments, i.e. the law of the process ( X t ,t + h ) h ∈ R does not depend on t ∈ R . Then we can define a metric dynamical system (Ω , F , P , θ ) and a C ,p − var ( R , ˜ T ( U )) -valued random variable X on Ω with the same law as X which satisfiesthe cocycle property (3.1) .Proof. The proof in all lines is similar to Theorem 5 in [BRS17] by setting Ω = C ,p − var ( R , ˜ T ( U )), F being the Borel σ -algebra, P the law of ¯ X and for ω ∈ Ω, we define( θ s ω )( t ) := ω ( s ) − ⊛ ω ( t + s ) , X t ( ω ) = ω ( t ) . (cid:3) Remark . Note that the cocycle property (3.1) is equivalent to X t ( θ s ( ω )) = X − s ( ω ) ⊛ X t + s ( ω )for every s, t ∈ R and every ω ∈ Ω.We will also ask for ergodicity of rough cocycles. The following lemma will be useful.
Lemma 3.6.
Let (Ω , F , P , ( θ t ) t ∈ R ) and ( ˜Ω , ˜ F , ˜ P , (˜ θ t ) t ∈ R ) be two measurable metric dynamical sys-tems and let Φ : Ω → ˜Ω be a measurable map such that ˜ P = P ◦ Φ − . Assume that for every t ∈ R ,there is a set of full P -measure Ω t ⊂ Ω on which Φ ◦ θ t = ˜ θ t ◦ Φ holds. Then, if P is ergodic, ˜ P isergodic, too.Proof. The reader will have no difficulties to check that the assertion is just a slight generalizationof [GAS11, Lemma 3]. (cid:3)
Theorem 3.7.
Consider the processes B It¯o and B Strat defined in Section 2. Then for each process,we can find an ergodic metric dynamical system (Ω , F , P , θ ) on which we can define a new processwith the same law, satisfying the cocycle property (3.1) , i.e. both processes are delayed γ -rough pathcocycles for every γ ∈ (1 / , / .Proof. We will first consider B Strat . From the approximation result in Theorem 2.9, we see that B Strat takes values in C ,p − var ( R , ˜ T ( R d )) for every p ∈ (2 , C ,p − var ( R , ˜ T ( R d )), F is the Borel σ -algebra and P = ˆ P ◦ S − where ( ˆΩ , ˆ F , ˆ P , ˆ θ )is the measurable metric dynamical system given by ˆΩ = C ( R , R d ), ˆ F the corresponding Borel σ -algebra, ˆ P the Wiener measure and ˆ θ = (ˆ θ t ) t ∈ R the Wiener shift. The map S : ˆΩ → Ω is definedas follows: For x ∈ ˆΩ, set S ( x ) = (cid:18) ⊕ (cid:0) x s,t , x s − r,t − r (cid:1) ⊕ (cid:0) Z ts x s,τ ⊗ dx τ , Z ts x s − r,τ − r ⊗ dx τ (cid:1)(cid:19) s ≤ t if the integrals exist as limits of Riemann sums, in Statonovich sense, on compact sets for thesequence of partitions given by Π n = { k/ n : k ∈ Z } as n → ∞ , and S ( x ) = (1 , ,
0) otherwise. Itis not hard to see that there is a set of full ˆ P -measure on which the limits do exist. It follows thatfor every t ∈ R , there is a set of full measure ˆΩ t such that for every x ∈ ˆΩ t , S (ˆ θ t x ) = θ ( S ( x )) . Since ˆ P is ergodic, ergodicity of P follows by Lemma 3.6 which completes the proof for theStratonovich case. For the It¯o-case, we can argue analogously: First, we define a mapˆ S ( x ) s,t := (cid:18) ⊕ (cid:0) x s,t , x s − r,t − r (cid:1) ⊕ (cid:0) Z ts x s,τ ⊗ dx τ −
12 ( t − s ) I d , Z ts x s − r,τ − r ⊗ dx τ (cid:1)(cid:19) ∈ ˜ T ( U )for smooth paths and a corresponding (separable!) space ˆ C ,p − var ( R , ˜ T ( R d )) in which, using againthe approximation result for the Stratonovich lift, the random variable B It¯o takes its values. Thena version of [BRS17, Theorem 5] applies and shows the claim. Ergodicity is proven analogously tothe Stratonvich case. (cid:3)
Cocycle property of the solution map.
Let I ⊂ R be a compact interval and X : I → U a γ -H¨older continuous path. It is easy to see that for α ≤ β ≤ γ , i α,β : D βX ( I, W ) → D αX ( I, W ) , ( ξ, ξ ′ ) ( ξ, ξ ′ )is a continuous embedding. We make the following definition: Definition 3.8.
We define D α,βX ( I, W ) as the closure of D βX ( I, W ) in the space D αX ( I, W ).The reason why we introduce these spaces is their separability, which we will prove in the nextlemma.
Lemma 3.9.
For all α < β , the spaces D α,βX ( I, W ) are separable.Proof. The space D α,βX ( I, W )) can be viewed as a subset of C α,β ( I, W ) × C α,β ( I, L ( U, W )) × C α, β ( I, W )where C α,β again means taking the closure of β -H¨older functions in the α -H¨older norm. Since allspaces above are separable, the result follows. (cid:3) If the parameters α < β < γ satisfy a certain condition, we can find a very explicit dense subset.This is the content of the next theorem, which has far reaching consequences, as we will see.
Theorem 3.10.
Let α < β < γ ≤ / . Assume that there is a κ ∈ (0 , γ ) such that β − α > (1 − α )(1 − β − γ + κ )(1 − β )(1 − α + κ )(3.2) holds. Then the set (cid:26) ( ψ, ψ ′ ) | ψ s,t = Z ts f ( τ ) dX τ + R s,t , ψ ′ s = f ( s ) where f ∈ C ∞ (cid:0) I, L ( U, W ) (cid:1) and R ∈ C ∞ ( I, W ) (cid:27) is dense in D α,βX ( I, W ) , the integral being understood as a Young-integral here. In particular, D α,βX ( I, W ) does not depend on β in this case.Proof. Let I = [ a, b ] and r = b − a . Take ( ξ, ξ ′ ) ∈ D α,βX ( I, W )), i.e. ξ s,t = ξ ′ s X s,t + ξ s,t , and assumethat k ξ ′ k β ; I , k ξ k β ; I < ∞ . Let φ : R → R be a mollifier. Note that we can extend ξ ′ to a β -H¨older YNAMICAL THEORY FOR SDDE 21 function on R by setting ξ ′ t = ξ ′ b for t > b and ξ ′ t = ξ ′ a for t a . For given n ∈ N , we define asmooth function f : I → L ( U, W ) by setting f s := Z R ξ ′ s − n z φ ( z ) dz = n Z R ξ ′ z φ (cid:0) n ( s − z ) (cid:1) dz. Our goal is to find a smooth function R such that for ψ s,t := Z ts f ( τ ) dX τ + R s,t and ψ ′ s := f ( s ) , (3.3)we have k ( ξ, ξ ′ ) − ( ψ, ψ ′ ) k D αX ( I,W )) < ε for any given ε > n large enough. Notethat ψ s,t = Z ts f ( τ ) − f ( s ) dX τ + R s,t is finite 2 β -H¨older continuous by standard Young estimates, which implies that ( ψ, ψ ′ ) is indeed anelement in D α,βX ( I, W ).Note first that for η s,t = f s,t − ξ ′ s,t , (cid:12)(cid:12) η s,t (cid:12)(cid:12) = (cid:12)(cid:12) f s,t − ξ ′ s,t (cid:12)(cid:12) Z R φ ( z ) (cid:12)(cid:12) ξ ′ s − n z,t − n z − ξ ′ s,t (cid:12)(cid:12) dz = Z R φ ( z ) (cid:12)(cid:12) ξ ′ s − n z,t − n z − ξ ′ s,t (cid:12)(cid:12) αβ (cid:12)(cid:12) ξ ′ s − n z,t − n z − ξ ′ s,t (cid:12)(cid:12) − αβ dz . ( t − s ) α ( 1 n ) β − α (3.4)which implies that k ξ ′ − ψ ′ k α ; I + | ξ ′ a − ψ ′ a | → n → ∞ . Since dds f s = n Z R ξ ′ s − n z ddz φ ( z ) dz and from β -H¨older continuity of ξ ′ , (cid:12)(cid:12) f s,t (cid:12)(cid:12) . ( t − s ) β and (cid:12)(cid:12) f s,t (cid:12)(cid:12) . n ( t − s ) . By polarization, this implies that | f s,t | . n θ ( t − s ) θ + β (1 − θ ) holds for every 0 θ
1. Setting θ = − γ − β + κ − β , we obtain (cid:12)(cid:12) f s,t (cid:12)(cid:12) . n − γ − β + κ − β ( t − s ) − γ + κ . (3.5)Let ρ s,t := R ts f τ dX τ − f s X s,t . From the Young inequality and relation (3.5), (cid:12)(cid:12) ρ s,t (cid:12)(cid:12) . k f k − γ + κ k X k γ ( t − s ) κ . n − γ − β + κ − β ( t − s ) κ . (3.6)Let ǫ > ρ := ξ s,t − ρ s,t . Since k ˜ ρ k β < ∞ , we can choose δ > s,t ∈ I, | t − s | δ | ˜ ρ s,t | ( t − s ) α k ξ k β δ β − α ) + sup s,t ∈ I, | t − s | δ | ρ s,t | ( t − s ) α ǫ. (3.7)More precisely, we see from (3.6) that δ has to be chosen such that δ . ǫ β − α ) and δ . (cid:0) ǫn − β − γ + κ − β (cid:1) − α + κ . (3.8) We will determine n and consequently δ in the future. Now, it is easy to verify that˜ ρ s,t − ˜ ρ s,u − ˜ ρ u,t = ( ξ ′ s,u − f s,u ) X u,t = η s,u X u,t . (3.9)Let P = { a = t < t < ... < t m = b } be a partition of I such that t i − t i − = δ for 1 i m − t m − t m − δ . We can define a piecewise linear function ˜ R by˜ R τ,υ := ˜ ρ t k ,t k +1 υ − τt k +1 − t k , τ, υ ∈ [ t k , t k +1 ]and consequently for τ, υ ∈ I with t k τ t k +1 ... t j υ t j +1 ,˜ R τ,υ = t k +1 − τt k +1 − t k ˜ ρ t k ,t k +1 + ˜ ρ t k +1 ,t k +2 + .... + υ − t j t j +1 − t j ˜ ρ t j ,t j +1 . From (3.9),˜ R τ,υ − ˜ ρ τ,υ = (cid:0) t k +1 − τt k +1 − t k ˜ ρ t k ,t k +1 + .... + υ − t j t j +1 − t j ˜ ρ t j ,t j +1 (cid:1) − (cid:0) ˜ ρ τ,t k +1 + ... + ˜ ρ t j − ,t j + ˜ ρ t j ,υ (cid:1) − (cid:0) η τ,t k +1 X t k +1 ,υ + η t k +1 ,t k +2 X t k +2 ,υ + ... + η t j − ,t j X t j ,υ (cid:1) = (cid:0) t k +1 − τt k +1 − t k ˜ ρ t k ,t k +1 + υ − t j t j +1 − t j ˜ ρ t j ,t j +1 − ˜ ρ τ,t k +1 − ˜ ρ t j ,υ (cid:1) − (cid:0) η τ,t k +1 X t k +1 ,υ + ... + η t j − ,t j X t j ,υ (cid:1) =: A − B. From (3.7), it is not hard to verify that k A k ǫ ( υ − τ ) α . (3.10)By (3.4) and our assumptions on X , k B k . ( 1 n ) β − α (cid:2) ( t k +1 − τ ) α ( υ − t k +1 ) γ + ... + ( t j − t j − ) α ( υ − t j ) γ (cid:3) δ α + γ n β − α (cid:2) ( j − k ) γ + ... + 1 γ (cid:3) . Since ( j − k − δ υ − τ and mδ r , k B k ( υ − τ ) α . δ γ − α n β − α m γ +1 − α . n β − α δ − α . (3.11)Now from (3.8), if β − α > (1 − α )(1 − β − γ + κ )(1 − β )(1 − α + κ ) , we can find n and δ such that k B k ( υ − τ ) α ǫ and therefore k ˜ R − ˜ ρ k α ; I ≤ ǫ. Since ˜ R is a piecewise linear function, we can find an R ∈ C ∞ ( I, W ) such thatsup s,t ∈ I | R s,t − ˜ R s,t | ( t − s ) α ǫ. Using this R in (3.3), we obtain k ξ − ψ k α ; I ≤ k ˜ ρ − ˜ R k α ; I + k ˜ R − R k α ; I ≤ ǫ, thus the stated set is indeed dense. (cid:3) YNAMICAL THEORY FOR SDDE 23
Remark . In the applications we have in mind, γ < / / α < / α < β < γ < / < κ < γ such that2(1 − β − γ + κ ) < (1 − α + κ )( β − α ) . Note that this can always be achieved by choosing β and γ close to 1 / κ close to 0. Since1 − α < − β > /
2, this implies that(1 − α )(1 − β − γ + κ )(1 − β )(1 − α + κ ) < β − α, i.e. in this case, we can always find parameters such that the condition in Theorem 3.10 is satisfied. Theorem 3.12.
Let X be a delayed γ -rough path cocycle for some γ ∈ (1 / , / . Under theassumptions of Theorem 1.13, the map ϕ ( n, ω, · ) := φ (0 , nr, ω, · )(3.12) is a continuous map ϕ ( n, ω, · ) : D βX ( ω ) ([ − r, , W ) → D βX ( θ nr ω ) ([ − r, , W ) and the cocycle property ϕ ( n + m, ω, · ) = ϕ ( n, θ mr ω, · ) ◦ ϕ ( m, ω, · )(3.13) holds for every s, t ∈ [0 , ∞ ) . If σ is linear, the cocycle is compact linear. Furthermore, all assertionsremain true if we replace the spaces D β by D α,β for / < α < β < γ .Proof. Note that D βX ( ω ) ([ − r + nr, nr ] , W ) ∼ = D βX ( θ nr ω ) ([ − r, , W ) by the natural linear mapΨ : D βX ( ω ) ([ − r + nr, nr ] , V ) −→ D βX ( θ nr ω ) ([ − r, , V )( ξ τ ) − r + nr τ nr ( ˜ ξ τ = ξ τ + nr ) − r τ . Continuity of ϕ is a consequence of Theorem 1.13. Regarding the cocycle property, by the semi-flowproperty (1.19) it is enough to show that φ (0 , nr, θ mr ω, · ) = φ (cid:0) mr, ( m + n ) r, ω, · (cid:1) . Using again the semi-flow property (1.19), it is enough to show the equality for n = 1 only. Finally,by the definition of the integral in (1.3) and the cocycle property of a rough cocycle, this caneasily be verified. The statements about linearity and compactness are a consequence of 1.11 andProposition 1.12. The claim that all spaces D β can be replaced by D α,β follows from the invariance ϕ (cid:0) n, ω, D α,βX ( ω ) ([ − r, , W ) (cid:1) ⊂ D α,βX ( θ nr ω ) ([ − r, , W )(3.14)which is a consequence of the continuity of ϕ . (cid:3) Note that so far, we worked with delayed rough path cocycles X which are defined on acontinuous-time metric dynamical system (Ω , F , P , ( θ t ) t ∈ R ). In Theorem 3.12, we saw that stochas-tic delay equations a priori induce discrete-time RDS only. The reason is that we cannot expectthat the semi-flow property (1.16) holds in full generality for all times, cf. Theorem 1.13. There-fore, in what follows, we will continue working with discrete time only. From now on, whenever weconsider cocycles induced by delay equations with delay r >
0, our underlying discrete-time metricdynamical system is given by (Ω , F , P , θ ) with θ := θ r . We also use the notation ϕ ( ω, · ) := ϕ (1 , ω, · )for the cocycle ϕ defined in (3.12).Next, we describe a structure which will be useful for us. Definition 3.13.
Let (Ω , F ) be a measurable space. A family of Banach spaces { E ω } ω ∈ Ω is calleda measurable field of Banach spaces if there is a set of sections∆ ⊂ Y ω ∈ Ω E ω with the following properties:(i) ∆ is a linear subspace of Q ω ∈ Ω E ω .(ii) There is a countable subset ∆ ⊂ ∆ such that for every ω ∈ Ω, the set { g ( ω ) : g ∈ ∆ } isdense in E ω .(iii) For every g ∈ ∆, the map ω
7→ k g ( ω ) k E ω is measurable. Remark . Let us remark here that the former definition originates from us, we did not encountera description of a measurable field of Banach spaces elsewhere in the literature. In fact, it is a mixof a measurable field of Hilbert spaces to be found e.g. in [Fol95, page 220] and a continuous fieldof Banach spaces , cf. e.g. [Dix77, page 211]. Since the stated properties in Definition 3.13 areexactly what we need for proving the Multiplicative Ergodic Theorem in the next section, it is alsoa pragmatic definition.
Proposition 3.15.
Let X : Ω → C γ ( I, U ) be a stochastic process. Assume that there are α < β < γ and some κ ∈ (0 , γ ) such that (3.2) is satisfied. Then { D α,βX ( ω ) ( I, W ) } ω ∈ Ω is a measurable field ofBanach spaces.Proof. For s = ( v, f, R ) ∈ R × C ∞ ( I, L ( U, W )) × C ∞ ( I, W ), define g s ( ω ) := (cid:18) v + Z ·− r f ( τ ) dX τ ( ω ) + R, f (cid:19) ∈ D α,βX ( ω ) ( I, W )and set ∆ := { g s : s ∈ R × C ∞ ( I, L ( U, W )) × C ∞ ( I, W ) } . (3.15)It is clear that (i) holds for ∆. Let S be a countable and dense subset of R × C ∞ ( I, L ( U, W )) × C ∞ ( I, W ) and define ∆ := { g s : s ∈ S } . By definition, ∆ is countable, and { g s ( ω ) : s ∈ S } isdense in D α,βX ( ω ) ( I, W ) for fixed ω ∈ Ω by Theorem 3.10. It remains to prove (iii). Let I = [ a, b ] andchoose s = ( v, f, R ). Then k g s ( ω ) k = | v | + | f ( a ) | + sup s,t ∈ I ∩ Q ,s Let (Ω , F , P , θ ) be a measurable metric dynamical system and ( { E ω } ω ∈ Ω , ∆)a measurable field of Banach spaces. A continuous cocycle on { E ω } ω ∈ Ω consists of a family ofcontinuous maps ϕ ( ω, · ) : E ω → E θω . (3.16) YNAMICAL THEORY FOR SDDE 25 If ϕ is a continuous cocycle, we define ϕ ( n, ω, · ) : E ω → E θ n ω as ϕ ( n, ω, · ) := ϕ ( θ n − ω, · ) ◦ · · · ◦ ϕ ( ω, · ) . We say that ϕ acts on { E ω } ω ∈ Ω if the maps ω 7→ k ϕ ( n, ω, g ( ω )) k E θnω , n ∈ N are measurable for every g ∈ ∆. In this case, we will speak of a continuous random dynamicalsystem on a field of Banach spaces . If the map (3.16) is bounded linear/compact, we call ϕ abounded linear/compact cocycle. Theorem 3.17. The continuous cocycle ϕ ( ω, · ) : D α,βX ( ω ) ([ − r, , W ) → D α,βX ( θ r ω ) ([ − r, , W ) defined in Theorem 3.12 induces a random dynamical system on the field of Banach spaces { D α,βX ( ω ) ([ − r, , W ) } ω ∈ Ω .Proof. Let ∆ be defined as (3.15) and take g ∈ ∆. Consider the solution y to y t ( ω ) = g ( ω ) + Z t σ ( y τ ( ω ) , y τ − r ( ω )) d X τ ( ω ) , t ≥ y t ( ω ) = g t ( ω ) , t ∈ [ − r, . To simplify notation, set k·k D X ( ω ) ([0 ,r ]) := k·k D α,βX ( ω ) ([0 ,r ] ,W ) . We will prove that ω 7→ k y ( ω ) k D X ( ω ) ([0 ,r ]) is measurable. Define y t ( ω ) := g ( ω ) + Z t σ (cid:0) g ( ω ) , g τ − r ( ω ) (cid:1) d X τ ( ω )and recursively for n > y n +1 t ( ω ) := g ( ω ) + Z t σ (cid:0) y nτ ( ω ) , g τ − r ( ω ) (cid:1) d X τ ( ω ) . By induction, one can show that ω y nt ( ω ) is measurable for every t ∈ [0 , r ] and n ≥ 1. By asimilar strategy for proving continuity of the It¯o-Lyons map, one can show that y n ( ω ) → y ( ω ) inthe space D α,βX ( ω ) ([0 , T ( A ( ω ))] , W ) as n → ∞ where A ( ω ) = k X ( ω ) k γ ;[0 ,r ] + k X ( ω ) k γ ;[0 ,r ] + k X ( ω )( − r ) k γ ;[0 ,r ] and T : [0 , ∞ ) → (0 , r ] is a decreasing function. DefineΩ m := n ω ∈ Ω : T ( A ( ω )) ≤ rm o . Then Ω m is a measurable subset and Ω = S m > Ω m . Fix m ∈ N and choose ω ∈ Ω m . Then( y n ( ω )) n is a Cauchy sequence in the space D α,βX ( ω ) ([0 , r/m ] , W ) and, consequently, converges tosome element ˜ y ( ω ) for which we can conclude that ω ˜ y t ( ω ) is measurable for every t ∈ [0 , r/m ].Now we can repeat this argument in [ jrm , ( j +1) rm ] for j = 0 , . . . , m − y j ( ω ) ∈ D α,βX ( ω ) ([ jr/m, ( j + 1) r/m ] , W ) with the properties that ω ˜ y jt ( ω ) is measurablefor every t ∈ [ jr/m, ( j + 1) r/m ] and y t ( ω ) = m − X j =0 ˜ y jt ( ω ) χ [ jrm , ( j +1) rm ) ( t ) . This implies that ω y t ( ω ) is measurable for every t ∈ [0 , r ] on the subspace Ω m . Since m wasarbitrary, measurability follows also on the space Ω. Note that y ′ t ( ω ) = σ ( y t ( ω ) , g t − r ( ω )), thus k y ( ω ) k D X ( ω ) ([0 ,r ]) = | y ( ω ) | + | y ′ ( ω ) | + sup s 7→ k y ( ω ) k D X ( ω ) ([0 ,r ]) follows. We can now repeat this argument to see that ω 7→ k y ( ω ) k D X ( ω ) ([ nr, ( n +1) r ]) is measurable for every n ≥ (cid:3) A Multiplicative Ergodic Theorem on a measurable field of Banach spaces In this section, (Ω , F , P , θ ) will denote a measurable metric dynamical system, ( { E ω } ω ∈ Ω , ∆ , ∆ )will be a measurable field of Banach spaces as in Definition 3.13 and ϕ a bounded linear cocycleacting on it, cf. Definition 3.16. Our goal is to prove a Multiplicative Ergodic Theorem (MET)in this abstract setting. The strategy we use is close to the one introduced in two recent works,both proving an MET on a Banach space. The first one is due to Blumenthal [Blu16], the secondwas written by Gonz´alez-Tokman and Quas [GTQ15]. Note, however, that none of them gives aproof of the MET for cocycles acting on fields of Banach spaces. For that reason, the measurabilityassumption in these works is very different from ours, and we have to prove measurability for ourobjects in a completely different way. Furthermore, we do not assume reflexivity of the Banachspaces as in [GTQ15].We start with an easy observation. Lemma 4.1. For every n ∈ N , the map ω 7→ k ϕ ( n, ω, · ) k L ( E ω ,E θnω ) is measurable.Proof. Using properties of ∆ and continuity of ϕ , k ϕ ( n, ω, · ) k L ( E ω ,E θnω ) = sup ξ ∈ E ω \{ } k ϕ ( n, ω, ξ ) kk ξ k = sup g ∈ ∆ k ϕ ( n, ω, g ( ω )) kk g ( ω ) k χ {k g k > } ( ω )with the convention ∞ · (cid:3) Definition 4.2. Let V be a vector space. If we can write V as a direct sum V = F ⊕ H of vectorspaces, we call it an algebraic splitting . We also say that F is a complement of H and vice versa.The projection operator π F k H ( v ) = f with v = f + h , f ∈ F , h ∈ H , is called the projectionoperator onto F parallel to H . If V is a normed space and π F k H is bounded linear, i.e. k π F k H k = sup f ∈ F,e ∈ H,f + h =0 k f kk f + h k < ∞ , we call V = F ⊕ H a topological splitting .The next lemma proves a further measurability result. The assumptions will be justified in thesequel. YNAMICAL THEORY FOR SDDE 27 Lemma 4.3. For ω ∈ Ω and µ ∈ R , define the subspace F µ ( ω ) := (cid:26) ξ ∈ E ω : lim sup n →∞ n log k ϕ ( n, ω, ξ ) k ≤ µ (cid:27) . Assume that there is a strictly decreasing sequence ( µ j ) j N , N ∞ , and a θ -invariant, measur-able set Ω ⊂ Ω of full measure with the following properties: (i) F µ ( ω ) = E ω for every ω ∈ Ω . (ii) For every j < N , there is a number m j ∈ N such that F µ j +1 ( ω ) is closed and m j -codimensional in F µ j ( ω ) for every ω ∈ Ω . (iii) For every j < N , lim n →∞ n log k ϕ ( n, ω, · ) | F µj ( ω ) k = µ j (4.1) for every ω ∈ Ω . (iv) For every j < N , if H jω is any complement of F µ j +1 ( ω ) in F µ j ( ω ) , lim n →∞ n log inf h ∈ H jω \{ } k ϕ ( n, ω, h ) kk h k = µ j (4.2) for every ω ∈ Ω . (v) lim sup n →∞ n log k ϕ ( n, ω, · ) | F µN ( ω ) k ≤ µ N (4.3) for every ω ∈ Ω .Then for every n ∈ N and j N , the map ω 7→ k ϕ ( n, ω, · ) | F µj ( ω ) k χ Ω ( ω )(4.4) is measurable.Proof. First we claim that for every g ∈ ∆ and j N the map ω d (cid:0) g ( ω ) , F µ j ( ω ) (cid:1) (4.5)is measurable. To see this, it suffices to show measurability of the function d (cid:0) g ( ω ) , S F µj ( ω ) (cid:1) := inf ξ ∈ F µj ( ω ) k ξ k =1 k g ( ω ) − ξ k where S F µj ( ω ) is the unit sphere in F µ j ( ω ). We use induction to prove the claim. The statementis clear for j = 1, so let j > 2. For every 1 i < j , since dim (cid:2) F µi ( ω ) F µi +1 ( ω ) (cid:3) < ∞ , we can find afinite-dimensional subspace H i ( ω ) such that for a constant M , F µ i ( ω ) = H i ( ω ) ⊕ F µ i +1 ( ω ) and k π H i ( ω ) || F µi +1 ( ω ) k < M. (4.6) The existence of this complement with the given bound for the projection is a classical result and follows e.g.from [Woj91, III.B.11], cf. also [Blu16, Lemma 2.3]. For µ := µ and l, k ≥ B l,kω ( µ j ) = (cid:26) ξ ∈ E ω : k ξ k = 1 , k ϕ ( k, ω, ξ ) k < exp (cid:0) k ( µ j + 1 l ) (cid:1) and d (cid:0) ξ, F µ i ( ω ) (cid:1) < exp (cid:0) k ( µ j − µ i − ) (cid:1) , i < j (cid:27) . We claim that d (cid:0) g ( ω ) , S F µj ( ω ) (cid:1) = lim k →∞ lim inf l →∞ d (cid:0) g ( ω ) , B l,kω ( µ j ) (cid:1) . (4.7)Set the right side equal to A . By definition, it is straightforward to show that d (cid:0) g ( ω ) , S F µj ( ω ) (cid:1) > A . For the opposite direction, let ǫ > 0. For large k, l we can find ξ l,k ∈ B l,kω ( µ j ) such that k g ( ω ) − ξ l,k k A + ǫ . By our assumptions on B k,lω ( µ j ), we have a decomposition of the form ξ l,k = X i From (4.2), choosing k larger if necessary, we obtain that for a given δ > (cid:0) k ( µ j − − δ ) (cid:1) k h l,kj − k k ϕ ( k, ω, h l,kj − ) k k ϕ ( k, ω, ξ l,k ) k + X i 1. Note that d (cid:0) g ( ω ) , S F µj ( ω ) (cid:1) = inf ˜ g ∈ ∆ J ˜ g ( ω )where(4.8) J ˜ g ( ω ) = ( ∞ if ω / ∈ C l,k,j (˜ g ) k g ( ω ) − ˜ g ( ω ) k ˜ g ( ω ) k k otherwise. YNAMICAL THEORY FOR SDDE 29 Since J ˜ g ( ω ) is measurable, this proves the claim. Therefore, we have also shown measurability of C l,k,j ( g ) for every j, k, l ≥ g ∈ ∆. Next, with the same argument as above, we can show that k ( ϕ ( n, ω, · ) | F µj ( ω ) k χ Ω ( ω ) = lim l →∞ lim inf k →∞ (cid:20) sup ξ ∈ B l,kω ( µ j ) k ϕ ( n, ω, ξ ) k (cid:21) χ Ω ( ω )for every j ≥ 2. Since sup ξ ∈ B l,kω ( µ j ) k ϕ ( n, ω, ξ ) k = sup g ∈ ∆ k ϕ ( n, ω, g ( ω )) kk g ( ω ) k χ C l,k,j ( g ) ( ω ) , measurability of (4.4) follows. (cid:3) The next lemma is a version of [Blu16, Lemma 3.7]. Unfortunately, there was a gap in proofwhich, however, was corrected in a subsequent erratum . We present a full proof here, using thestrategy of the above mentioned erratum. Lemma 4.4. Let the same assumptions as in Lemma 4.3 be satisfied. Then there exists a θ -invariant, measurable set Ω ⊂ Ω of full measure such that for every ω ∈ Ω , if H ω is a complementof F µ ( ω ) in E ω , we have lim n →∞ n log k π ϕ ( n,ω,H ω ) k F µ ( θ n ω ) k = 0 . (4.9) Proof. It is enough to show thatlim sup n →∞ log k π ϕ ( n,ω,H ω ) k F µ ( θ n ω ) k . (4.10)Define φ ( ω ) = sup p > exp (cid:0) − p ( µ + δ ) (cid:1) k ϕ ( p, ω, · ) k φ ( ω ) = sup p > exp (cid:0) − p ( µ + δ ) (cid:1) k ϕ ( p, ω, · ) | F µ ( ω ) k From Lemma 4.3, φ and φ are measurable functions and bounded on a set of full measure Ω . Sofrom [Mn83, Lemma III.8], there exists a measurable subset Ω of full measure such that for any ω ∈ Ω , lim n →∞ n log + φ ( θ n ω ) = 0(4.11)where φ ( ω ) = max { φ ( ω ) , φ ( ω ) } . Note that we can assume that Ω is also θ -invariant, otherwisewe can replace it by T j ∈ Z ( θ j ) − (Ω ). Fix ω ∈ Ω and assume that H ω ⊕ F µ ( ω ) = E ω . Let ǫ > N ∈ N such that for n > N , k ϕ ( n, ω, · ) k exp (cid:0) n ( µ + δ ) (cid:1) , inf h ∈ H ω \{ } k ϕ ( n, ω, h ) kk h k > exp (cid:0) n ( µ − δ ) (cid:1) φ ( θ n ω ) exp( nǫ ) . (4.12)We prove (4.10) by contradiction. Assume there is a γ > (cid:0) n k , h k , f k (cid:1) ∈ (cid:0) N , H ω , F µ ( θ n k ω ) (cid:1) such that n k → ∞ , k h k k = 1 and k ϕ ( n k , ω, h k ) kk ϕ ( n k , ω, h k ) − f k k > 12 exp( n k γ ) for all k ≥ . (4.13) Private communication with A. Blumenthal. For p ≥ k ϕ ( n k + p, ω, h k ) k = k ϕ ( p, θ n k ω, ϕ ( n k , ω, h k )) k k ϕ ( p, θ n k ω, · ) kk ϕ ( n k , ω, h k ) − f k k + k ϕ ( p, θ n k ω, · ) | F θnk ω kk f k k (4.14)From (4.13), it follows that k f k k k ϕ ( n k , ω, h k ) k . Now for large n k , from (4.12) and (4.14),exp (cid:0) ( n k + p )( µ − δ ) (cid:1) (cid:18) n k ǫ + p ( µ + δ ) + n k ( µ + δ ) − n k γ (cid:19) + 3 exp (cid:18) p ( µ + δ ) + n k ǫ + n k ( µ + δ ) (cid:19) . Choosing p = n k and δ, ǫ small, we will have a contradiction. (cid:3) The following definition is taken from [GTQ15]. Definition 4.5. Let X, Y be Banach spaces. For x , ..., x k ∈ X , we defineVol( x , x , ..., x k ) := k x k k Y i =2 d ( x i , h x j i j
1, set D k ( T ) := sup k x i k =1; i =1 ,...,k Vol (cid:0) T ( x ) , T ( x ) , ..., T ( x k ) (cid:1) We summarize some basic properties of D k in the next lemma. Lemma 4.6. Let X, Y, Z be Banach spaces and T : X → Y , S : Y → Z bounded linear maps. (i) D ( T ) = k T k and D k ( T ) k T k k for k ≥ . (ii) D k ( S ◦ T ) D k ( S ) D k ( T ) for k ≥ .Proof. The proof of (i) is straightforward, (ii) is proven in [GTQ15, Lemma 1]. (cid:3) Lemma 4.7. Let T : X → Y be a bounded linear map between two Banach spaces, x ∈ h x i i i k and k x i k = 1 . Then there exists a constant α k which only depends on k such that Vol (cid:0) T ( x ) , T ( x ) , ..., T ( x k ) (cid:1) α k k T k k − k T x kk x k Proof. Assume x k x k = P j k β j x j . Consequently, there exists 1 t k such that β t > k . Define y = ( y , . . . , y k ) as y i = x i for i = t, n,x n for i = t,x t for i = n. By definition, Vol (cid:0) T ( y ) , T ( y ) , ..., T ( y n ) (cid:1) k T k k − d (cid:0) T ( y n ) , h T ( y i ) i i n − (cid:1) k k T k k − k T x kk x k . (4.16) YNAMICAL THEORY FOR SDDE 31 From [Blu16, Proposition 2.14], there is an inner product ( · , · ) V on V = h T ( x i ) i i k such that1 √ k k T ( x ) k V k T ( x ) k √ k ∀ x ∈ h x i i i k . It is not hard to see that this implies that √ k d V (cid:0) T ( x j ) , h T ( x i ) i i Assume that X, Y are Banach spaces and that T : X → Y is a linear map. Let V ⊂ X be a closed subspace of codimension m . Then for k > m , there exists a constant C whichonly depends on k and m such that D k ( T ) CD m ( T ) D k − m ( T | V )(4.18) Proof. [GTQ15, Lemma 8]. (cid:3) Proposition 4.9. Let ϕ be a bounded linear cocycle acting on a measurable field of Banach spaces ( { E ω } ω ∈ Ω , ∆ , ∆ ) . Then for every n, k > , the map Ψ kn : Ω → R ω D k ( ϕ ( n, ω, · )) is measurable.Proof. For k = 1, the claim follows from Lemma 4.6 and Lemma 4.1. Note that for ω ∈ Ω,Ψ kn ( ω ) = sup g ,...,g k ∈ ∆ Vol( ϕ ( n, ω, ˜ g ( ω )) , . . . , ϕ ( n, ω, ˜ g k ( ω ))) χ {k g k > ,..., k g k k > } ( ω )where we used the notation ˜ g i ( ω ) = g i ( ω ) / k g i ( ω ) k , i = 1 , . . . , k . It is therefore sufficient to provethat for fixed g , . . . , g k ∈ ∆, ω Vol( ϕ ( n, ω, ˜ g ( ω )) , . . . , ϕ ( n, ω, ˜ g k ( ω ))) χ {k g k > ,..., k g k k > } ( ω )is measurable. For i > 2, we have d (cid:18) ϕ (cid:0) n, ω, ˜ g i ( ω ) (cid:1) , h ϕ (cid:0) n, ω, ˜ g t ( ω ) (cid:1) i t
Lemma 4.10. Under the same setting as in Proposition 4.9, let χ kn ( ω ) = log(Ψ kn ( ω )) . Assume that log + k ϕ (1 , ω, · ) k ∈ L (Ω) . Then there exists a measurable forward invariant set Ω ⊂ Ω of full measure such that the limit Λ k ( ω ) := lim n →∞ χ kn ( ω ) n ∈ [ −∞ , ∞ )(4.19) exists for every ω ∈ Ω and k ≥ . Furthermore, Λ k ( θω ) = Λ k ( ω ) for every k ≥ , ω ∈ Ω and Λ k ( ω ) is constant on Ω in case the underlying metric dynamical system is ergodic.Proof. From Lemma 4.6 and the cocycle property, χ kn + m ( ω ) χ kn ( θ m ω ) + χ km ( ω ) . (4.20)By assumption and Lemma 4.6, it follows that χ k ;+1 ∈ L (Ω). Therefore, we can directly applyKingman’s Subadditive Ergodic Theorem [Arn98, 3.3.2 Theorem] to conclude. (cid:3) Remark . (i) From Birkhoff’s Ergodic Theorem, we can furthermore assume thatlim n →∞ log + k ϕ (1 , θ n ω, · ) k n = 0(4.21) for all ω ∈ Ω .(ii) From Lemma 4.8, it follows thatΛ k ≤ Λ m + Λ k − m for every k > m . In particular, if Λ m = −∞ , it follows that Λ k = −∞ for every k > m . Definition 4.12. If the assumptions of Lemma 4.10 are satisfied, we define λ k ( ω ) := ( Λ k ( ω ) − Λ k − ( ω ) if Λ k ( ω ) , Λ k ( ω ) ∈ R −∞ if Λ k ( ω ) = −∞ for k ≥ 1, where we set Λ ( ω ) := 0. We call λ k the k -th Lyapunov exponent of ϕ . Note that theyare deterministic almost surely in case the underlying system is ergodic. Remark . Following the same strategy as in [GTQ15, Theorem 13], one can show that ( λ k ) k ≥ is a decreasing sequence.The next lemma shows that the sequence ( λ k ) does not have real cluster points in case thecocycle is compact. Lemma 4.14. Let ϕ be as in Lemma 4.10. Furthermore, assume that it is compact. Then there isa measurable forward invariant subset ˜Ω ⊂ Ω with full measure such that for any ω ∈ ˜Ω and ρ ∈ R ,there are only finitely many exponents λ k ( ω ) that exceed ρ .Proof. Let Ω be the set provided in Lemma 4.10. For ω ∈ Ω, let B ω be the unit ball in E ω . Set G ( ϑ, ν ) := (cid:26) ω ∈ Ω : ϕ (1 , ω, B ω ) can be covered by e ϑ balls with sizes less than e ν (cid:27) . (4.22)We claim that G ( ϑ, ν ) is a measurable subset. To see this, define S ( ω ) := (cid:26) s ∈ B ω : s = r g ( ω ) k g ( ω ) k χ {k g k > } ( ω ) , g ∈ ∆ , r ∈ Q ∩ [0 , (cid:27) . YNAMICAL THEORY FOR SDDE 33 One can easily check that S ( ω ) is dense in B ω . Let p = e ϑ and define H ( ω ) = inf s ,...,s p ∈ S ( ω ) (cid:18) sup s ∈ S ( ω ) min i p (cid:0) k ϕ (1 , ω, s ) − ϕ (1 , ω, s i ) k (cid:1)(cid:19) . It is not hard to see that G ( ϑ, ν ) = (cid:8) ω ∈ Ω : H ( ω ) < e ν (cid:9) and consequently G ( ϑ, ν ) is indeed measurable. Since ϕ is compact, for any ν ∈ R ,lim ϑ →∞ P (cid:0) G ( ϑ, ν ) (cid:1) = 1 . Let ω ∈ Ω . With the same argument as on [GTQ15, page 247], we can say that ϕ ( m, ω, B ω ) canbe covered by N m = e mϑ balls of size R ϑ,νm = e mγ ϑ,νm where γ ϑ,νm ( ω ) = 1 m (cid:20) ν X j m χ G ( ϑ,ν ) ( θ j ω ) + X j m χ G ( ϑ,ν ) c log + k ϕ (1 , θ j ω, · ) k (cid:21) =: νA ϑ,νm ( ω ) + B ϑ,νm ( ω ) . Let λ k ( ω ) > ρ . For large m , we must have k ( ρ − γ ϑ,νm ) ϑ . If we can show that ρ − γ ϑ,νm > m, ϑ, µ , the proof is finished since in that case, k < ϑρ − γ ϑ,νm .Let ǫ > ν < ν < ρ − ǫǫ . From integrability of log + k ϕ (1 , ω, · ) k , there existsa δ > P ( E ) < δ , Z E log + k ( ϕ (1 , ω, · ) k d P ǫ . (4.23)Now we choose ϑ > P (cid:0) G ( ϑ, ν ) c (cid:1) ǫ ∧ δ. (4.24)Since 0 A ϑ,νm ( ω ) Z Ω A ϑ,νm d P P ( A ν,rm > ǫ ) + ǫ and P ( B ϑ,νm > ǫ ) ǫ Z Ω B ϑ,νm d P . Now from (4.23), (4.24) and Birkhoff’s Ergodic theorem, for large m , P ( A ϑ,νm > ǫ ) > − ǫ and P ( B ϑ,νm > ǫ ) ǫ. Set A := { A ϑ,νm > ǫ } and B := { B ϑ,νm ǫ } and note that P ( A ∩ B ) > − ǫ . For ω ∈ A ∩ B , γ ϑ,νm < ρ. Since ǫ is arbitrary, we can find a set Ω ⊂ Ω of full measure with the desired property. Finallywe put Ω := T ∞ j =0 ( θ j ) − Ω . (cid:3) The following proposition, a trajectory-wise version of the Multiplicative Ergodic Theorem, willplay a central role in the proof of our main result. It is a slight reformulation of [Blu16, Proposition3.4]. The proof is very similar to Blumenthal’s original proof, but because of its importance, wedecided to sketch it in the appendix, cf. page 44. Proposition 4.15. Let { V j } j > be a sequence of Banach spaces and T i : V i → V i +1 a sequence ofbounded linear operators. Set T n = T n − ◦ ... ◦ T . Assume that: (i) lim sup n →∞ n log + k T n k = 0 . (ii) For any k ≥ , the following limits exists: L k = lim n →∞ n log D k ( T n ) . (iii) Setting L := 0 and l k := L k − L k − for k ≥ , assume that there is a number m < ∞ forwhich l := l = . . . = l m > l m +1 =: l .Then the subspace F := (cid:8) v ∈ V : lim sup n →∞ n log k T n v k l (cid:9) is closed and m -codimensional. Also, for v ∈ V \ F , lim n →∞ n log k T n v k = l. (4.25) Furthermore, for any complement H of F , lim n →∞ n log inf v ∈ H \{ } k T n v kk v k = l. (4.26) Finally, if h , . . . , h m ∈ V are linearly independent and H = h h , ..., h m i , lim n →∞ n log Vol ( T n h , T n h , ..., T n h m ) = ml. (4.27) Remark . In the proof of the proposition above, we will also see thatlim sup n →∞ n log k T n | F k l (4.28)holds.We finally state the main result of this section, a Multiplicative Ergodic Theorem for cocyclesacting on measurable fields of Banach spaces. Theorem 4.17. Let (Ω , F , P , θ ) be an ergodic measurable metric dynamical system and ϕ be acompact linear cocycle acting on a measurable field of Banach spaces { E ω } ω ∈ Ω in the sense ofDefinition 3.16. For λ ∈ R ∪ {−∞} and ω ∈ Ω , define F λ ( ω ) := (cid:8) x ∈ E ω : lim sup n →∞ n log k ϕ ( n, ω, x ) k λ (cid:9) . Assume that log + k ϕ (1 , ω, · ) k ∈ L (Ω) . Then there is a measurable forward invariant set ˜Ω ⊂ Ω of full measure such that: (i) For any ω ∈ ˜Ω and k > , the limit Λ k := lim n →∞ n log D k ( ϕ ( n, ω, · )) ∈ [ −∞ , ∞ )(4.29) exists and is independent of ω . (ii) Setting Λ := 0 and λ k := Λ k − Λ k − with λ k = −∞ if Λ k = −∞ , the sequence ( λ k ) is de-creasing. If the number of distinct values of this sequence is infinite, then lim k →∞ λ k = −∞ .We denote the decreasing subsequence of distinct values by ( µ j ) j > , which can be a finiteor an infinite sequence, and m j will denote the multiplicity of µ j in the sequence ( λ j ) . If µ j ∈ R , m j is finite. YNAMICAL THEORY FOR SDDE 35 (iii) For λ i > λ i +1 and ω ∈ ˜Ω , x ∈ F λ i ( ω ) \ F λ i +1 ( ω ) if and only if lim n →∞ n log k ϕ ( n, ω, x ) k = λ i . (4.30)(iv) For any µ j , codim F µ j ( ω ) = m + . . . + m j − for every ω ∈ ˜Ω . (v) For ω ∈ ˜Ω , if h , . . . , h k ∈ E ω are linearly independent and H ω = h h , ..., h k i is a comple-ment subspace for F µ j ( ω ) in E ω , then lim n →∞ n log Vol (cid:0) ϕ ( n, ω, h ) , ..., ϕ ( n, ω, h k ) (cid:1) = X i j m i µ i . (4.31) Remark . The sequence ( µ j ) is called the Lyapunov spectrum , the filtration of spaces F µ ( ω ) ⊃ F µ ( ω ) ⊃ · · · is called Oseledets filtration . Proof. Note that (i) and (ii) are direct consequences of Lemma 4.10 and Lemma 4.14, hence we onlyhave to prove (iii), (iv) and (v). The idea is to prove the consecutive statements for each Lyapunovexponent by induction, where Proposition 4.15 will play a central role. We will only give the proofin case that the Lyapunov spectrum is infinite, the case of a finite Lyapunov spectrum is similar.Let us start to formulate a result for the first Lyapunov exponent µ . Consider Ω ⊂ Ω as inLemma 4.10. We may assume that (4.21) is also satisfied for every ω ∈ Ω . Fix some ω ∈ Ω and define V j := E θ j ω and T j := ϕ (1 , θ j ω, · ). Note that, by definition, µ = λ = ... = λ m >λ m +1 = µ and µ = Λ , therefore F µ ( ω ) = E ω = V . Proposition 4.15 now implies that for x ∈ F µ ( ω ) \ F µ ( ω ), we have lim n →∞ n log k ϕ ( n, ω, x ) k = µ and that F µ ( ω ) is m -codimensional.Furthermore, if H ω = h h , . . . , h m i is a complement for F µ ( ω ),lim n →∞ n log Vol (cid:0) ϕ ( n, ω, h ) , ..., ϕ ( n, ω, h k ) (cid:1) = m µ . (4.32)(4.31)For the next step, we set V j := F µ ( θ j ω ) and T j := ϕ (1 , θ j ω, · ) | F µ ( θ j ω ) . Note that from thecocycle property, T j : V j → V j +1 . We claim that there is a measurable and θ -invariant subsetΩ ⊂ Ω with full measure such that for any ω ∈ Ω and k > n →∞ n log D k (cid:2) ϕ ( n, ω, · ) | F µ ( ω ) (cid:3) = Λ k + m − Λ m (4.33)where we set m := m for simplicity. Let Ω ⊂ Ω be a measurable subset with the propertiesstated in Lemma 4.4. Fix some ω ∈ Ω . As a consequence of Lemma 4.8,Λ k + m Λ m + lim inf n →∞ n log D k (cid:2) ϕ ( n, ω, · ) | F µ ( ω ) (cid:3) . (4.34)For n ∈ N to be specified later, let { f i } i k ⊂ F µ ( ω ) be chosen such that k f i k = 1 for every i and Vol (cid:18) ϕ ( n, ω, f ) , ..., ϕ ( n, ω, f k ) (cid:19) > D k (cid:2) ϕ ( n, ω, · ) | F µ ( ω ) (cid:3) . (4.35) Let H ω = h h , h , ..., h m i be a complement subspace for F µ ( ω ). We can assume that k h i k = 1 forall i . To ease notation, set ϕ nω ( · ) := ϕ ( n, ω, · ). By definition, D k + m ( ϕ nω ( · )) > Vol (cid:0) ϕ nω ( h ) , ..., ϕ nω ( h m ) , ϕ nω ( f ) , ..., ϕ nω ( f k ) (cid:1) = Vol (cid:0) ϕ nω ( h ) , ..., ϕ nω ( h m ) (cid:1) m Y j =1 d (cid:18) ϕ nω ( f jω ) , (cid:10) ϕ nω ( h ) , ..., ϕ nω ( h m ) , ϕ nω ( f ) , ..., ϕ nω ( f j − ) (cid:11)(cid:19) . (4.36)It is not hard to see that d (cid:18) ϕ nω ( f j ) , (cid:10) ϕ nω ( f ) , ..., ϕ nω ( f j − ) (cid:11)(cid:19) d (cid:18) ϕ nω ( f j ) , (cid:10) ϕ nω ( h ) , ..., ϕ nω ( h m ) , ϕ nω ( f ) , ..., ϕ nω ( f j − ) (cid:11)(cid:19) k Π F µ ( θ n ω ) || ϕ ( n,ω,H ω ) k . Consequently, by (4.35) and (4.36), D k + m ( ϕ nω ( · )) > k Π F µ ( θ n ω ) || ϕ ( n,ω,H ω ) k − m Vol (cid:0) ϕ nω ( h ) , ..., ϕ nω ( h m ) (cid:1) Vol( ϕ nω ( f ) , ..., ϕ nω ( f n )) > k Π F µ ( θ n ω ) || ϕ ( n,ω,H ω ) k − m Vol (cid:0) ϕ nω ( h ) , ..., ϕ nω ( h m ) (cid:1) D k (cid:2) ϕ ( n, ω, · ) | F µ ( ω ) (cid:3) . Note that, by definition of the projection operator,1 k Π F µ ( θ n ω ) || ϕ ( n,ω,H ω ) k k Π ϕ ( n,ω,H ω ) || F µ ( θ n ω ) k + 1 . Choosing n large, using (4.32) and Lemma 4.4, we see thatlim sup n →∞ n log D k (cid:2) ϕ ( n, ω, · ) | F µ ( ω ) (cid:3) + Λ m Λ k + m (4.37)and (4.33) is shown. We can now use Proposition 4.15 again with l = µ , l = µ and m = m whichproves that for ω ∈ Ω and x ∈ F µ ( ω ) \ F µ ( ω ),lim n →∞ n log k ϕ ( n, ω, x ) k = µ . Moreover, F µ ( ω ) is m -codimensional in F µ ( ω ). Using that F µ ( ω ) is m -codimensional in E ω implies that F µ ( ω ) has codimension m + m in E ω .It remains to prove (v). Let h h , ..., h m + m i be a complement subspace for F µ ( ω ). Notethat Vol (cid:0) ϕ nω ( h ) , ..., ϕ nω ( h m + m ) (cid:1) is not invariant under permutation, but all permutations areequivalent up to a constant which only depends on m + m , cf. the proof of Lemma 4.7. We mayassume that H ω = h h , ..., h m i is a complement subspace for F µ ( ω ) and that for m + 1 j m + m , we have h j = g j − m + f j − m where g j − m ∈ F µ ( ω ) and f j − m ∈ H ω . It is not hard tosee that G ω := h g , ..., g m i is a complement subspace for F µ ( ω ) in F µ ( ω ). By definition,Vol (cid:0) ϕ nω ( g ) , ..., ϕ nω ( g m ) , ϕ nω ( h ) , ..., ϕ nω ( h m ) (cid:1) = Vol (cid:0) ϕ nω ( g ) , ..., ϕ nω ( g m ) (cid:1) m Y j =1 d (cid:0) ϕ nω ( h j ) , h ϕ nω ( g ) , ..., ϕ nω ( g m ) , ϕ nω ( h ) , ..., ϕ nω ( h j − ) i (cid:1) . Note that1 d (cid:0) ϕ nω ( h j ) , h ϕ nω ( h ) , ..., ϕ nω ( h j − ) i (cid:1) d (cid:0) ϕ nω ( h j ) , h ϕ nω ( g ) , ..., ϕ nω ( g m ) , ϕ nω ( h ) , .., ϕ nω ( h j − ) i (cid:1) k Π ϕ ( n,ω,H ω ) || F µ ( θ n ω ) k . YNAMICAL THEORY FOR SDDE 37 Together with Lemma 4.4 and (4.27) in Proposition 4.15, this implies thatlim n →∞ n log Vol (cid:0) ϕ nω ( h ) , ..., ϕ nω ( h m ) , ϕ nω ( g ) , ..., ϕ nω ( g m ) (cid:1) = lim n →∞ n log Vol (cid:0) ϕ nω ( g ) , ..., ϕ nω ( g m ) , ϕ nω ( h ) , ..., ϕ nω ( h m ) (cid:1) = m µ + m µ . (4.38)Since f k ∈ H ω for 1 j m , d (cid:0) ϕ nω ( g j ) , h ϕ nω ( h ) , ..., ϕ nω ( h m ) , ϕ nω ( g ) , ..., ϕ nω ( g j − ) i (cid:1) = d (cid:0) ϕ nω ( h m + j ) , h ϕ nω ( h ) , ..., ϕ nω ( h m ) , ϕ nω ( h m +1 ) , .., ϕ nω ( h m + j − ) i (cid:1) . Consequently, by (4.38),lim n →∞ n log Vol (cid:0) ϕ nω ( h ) , ..., ϕ nω ( h m ) , ϕ nω ( h m +1 ) , ..., ϕ nω ( h m + m ) (cid:1) = lim n →∞ n log Vol (cid:0) ϕ nω ( h m +1 ) , ..., ϕ nω ( h m + m ) , ϕ nω ( h ) , ..., ϕ nω ( h m ) (cid:1)(cid:3) = m µ + m µ . This finishes step 2. We can now iterate the procedure and the general result follows by induction. (cid:3) The Lyapunov spectrum for linear equations In this section, we formulate the main results of the article. Theorem 5.1. Let (Ω , F , P , ( θ ) t ∈ R ) be an ergodic measurable metric dynamical system and X adelayed γ -rough path cocycle for some γ ∈ (1 / , / and some delay r > . Assume that there are α < β < γ such that (3.2) holds for some κ ∈ (0 , γ ) . In addition, we assume that k X k γ ;[0 ,r ] + k X k γ ;[0 ,r ] + k X ( − r ) k γ ;[0 ,r ] ∈ L γ − β (Ω) . (5.1) Let σ ∈ L ( W , L ( U, W )) . Then we have the following: (i) The equation dy t = σ ( y t , y t − r ) d X t ( ω ); t ≥ y t = ξ t ; t ∈ [ − r, has a unique solution y : [0 , ∞ ) → W for every initial condition ( ξ, ξ ′ ) ∈ D α,βX ( ω ) ([ − r, , W ) with ( y t + n ( ω ) , y ′ t + n ( ω )) t ∈ [ − r, ∈ D α,βX ( θ nr ω ) ([ − r, , W ) for every n ≥ where y ′ t ( ω ) = ( σ ( y t ( ω ) , y t − r ( ω )) for t ≥ ξ ′ t for t ∈ [ − r, . (ii) Set ϕ ( n, ω, ξ ) := ( y t + n ( ω ) , y ′ t + n ( ω )) t ∈ [ − r, and E ω := D α,βX ( ω ) ([ − r, , W ) . Then ϕ is acompact linear cocycle defined on the discrete ergodic measurable metric dynamical system (Ω , F , P , θ r ) acting on the measurable field of Banach spaces { E } ω ∈ Ω and all statementsof the Multiplicative Ergodic Theorem 4.17 hold. In particular, a deterministic Lyapunovspectrum ( µ j ) j ≥ exists and induces an Oseledets filtration of the space of admissible initialconditions D α,βX ( ω ) ([ − r, , W ) on a set of full measure. Proof. Theorem 3.12 together with Theorem 3.17 show that (5.2) induces a cocycle acting ona measurable field of Banach spaces given by the spaces of controlled paths. The estimate inTheorem 1.11 together with our assumption (5.1) show that the moment condition of the MET4.17 is satisfied and the theorem follows. (cid:3) Finally, we apply our results for the Brownian motion. Corollary 5.2. Theorem 5.1 can be applied for X being a two-sided Brownian motion B adaptedto a two-paramter filtration ( F ts ) and X being either B It¯o or B Strat . In the former case, the solutionto (5.2) coincides with the usual It¯o-solution and in the later case it coincides with the Stratonovichsolution of a stochastic differential equation almost surely in case the initial condition is F − -measurable.Proof. The fact that B It¯o and B Strat are delayed γ -rough path cocycles on an ergodic measurablemetric dynamical system for every γ ∈ (1 / , / 2) was shown in Theorem 3.7. Choosing γ closeenough to 1 / 2, we can find α and β such that (3.2) holds. In Proposition 2.2, we saw that theintegrability condition (5.1) is satisfied in the Brownian case, and we can indeed apply Theorem5.1. The fact that the solution to (5.2) coincides with the usual It¯o resp. Stratonovich solution wasshown in Corollary 2.4. (cid:3) We close this section with a few remarks. Remark . (1) As already mentioned, it is not hard to prove Theorem 5.1 for a vector ofdelays 0 < r < . . . < r m in which case the equation reads dy t = σ ( y t , y t − r , . . . , y t − r m ) d X t . In that case, the largest delay r m will play the role of r . It is also straightforward to includea smooth and bounded drift term in the equation by adding the function t t as a smoothcomponent to the process X . Including unbounded drifts is more challenging, cf. [RS17]for a discussion regarding equations without delay.(2) Theorem 5.1 is formulated in a generality which opens the possibility to apply the results fora much larger class of driving processes X . For instance, [NNT08] prove that the fractionalBrownian motion possess a “canonical” delayed L´evy area using the Russo-Vallois integral[RV93]. However, this approach does not directly show that the fractional Brownian motionhas a canonical lift to a delayed rough path cocycle since we used that such lifts are limitsof smooth convolutions, cf. the proof of Theorem 3.7 where we used Theorem 2.9. However,it is possible to show that the delayed L´evy area for the fractional Brownian motion definedthrough the Russo-Vallois integral is also a limit of smooth convolutions. This fact evenholds for a significantly larger class of Gaussian processes and will be discussed in anotherfuture work. Other possible drivers in Theorem 5.1 are semimartingales with stationaryincrements and good integrability properties.(3) It is possible to use the language of Hairer’s Regularity Structures [Hai14] to reformulateour results. In that case, the space of controlled paths has to be replaced by the space of modelled distributions . We decided to use the language of rough paths here because lesstheory is needed and we can directly rely on prior work such as [NNT08]. However, it mightbe useful to use regularity structures in the future. YNAMICAL THEORY FOR SDDE 39 An example In view of our main results obtained in the former section, we now come back to the previousexample already discussed in the introduction: we consider the stochastic delay equation dy t = y t − d B It¯o t ; t ≥ y t = ξ t ; t ∈ [ − , . (6.1)This equation can be considered as the prototype of a singular stochastic delay equation. In itsclassical It¯o formulation, it was studied by one of us in [Sch13]. In that work, it was shown thatthere exists a deterministic real number Λ such thatΛ = lim t →∞ t log k ϕ ( t, ω, ξ ) k (6.2)almost surely for any initial condition ξ ∈ C ([ − , , R ) \ { } . In (6.2), the norm k ·k may denote theuniform norm or the M -norm which we will define below. It is a natural question to ask whetherΛ coincides with the top Lyapunov exponent provided by the Multiplicative Ergodic Theorem 4.17.We will give an affirmative answer in this section.Set E ω = D α,βB ( ω ) ([ − , α , β chosen such that (3.2) holds. Take ( ξ, ξ ′ ) ∈ E ω . On the timeinterval [ − , y t , y ′ t ) = ( ( ξ t , ξ ′ t ) if t ∈ [ − , (cid:16)R t ξ s − d B It¯o + ξ , ξ t − (cid:17) if t ∈ [0 , . (6.3)Note that C ([ − , , R ) ⊂ E ω for every ω ∈ Ω by the embedding η ( η, M := R × L ([ − , , R ) furnished with the norm k ( ν, η ) k M := (cid:0) | ν | + k η k L (cid:1) for ( ν, η ) ∈ M . Note that C ([ − , , R ) ⊂ M using the embedding η ( η , η ). Recall thedefinition of Vol given in Definition 4.5. Our main result in this section is the following. Theorem 6.1. For every η , ..., η k ∈ C ([ − , , R ) \ { } , the limit lim n →∞ n log Vol (cid:0) ϕ ( n, ω, η ) , ..., ϕ ( n, ω, η k ) (cid:1) (6.4) exists almost surely in [ −∞ , ∞ ) . Moreover, the limit is independent of the choice of the norm whenwe take k · k E θnω , k · k C α , k · k ∞ or k · k M in the definition of Vol . For k = 1 , if k · k denotes anyof the norms above, the limit lim n →∞ n log k ϕ ( n, ω, η ) k is independent of the choice of η ∈ C ([ − , , R ) \ { } and coincides with the largest Lyapunovexponent provided by the Multiplicative Ergodic Theorem 4.17. Before proving Theorem 6.1, we need two classical inequalities: Lemma 6.2. Let α < , p > and let ξ : [ − , → R be an α -H¨older path. Then there is aconstant A p such that k ξ k α = sup − s Cf. [Gub04, Corollary 4]. (cid:3) Proof of Theorem 6.1. First, we claim that the limit (6.4) exists for any choice of η , . . . , η k for thenorm k · k E θnω . Indeed, if η , . . . , η k are linearly dependent, the limit (6.4) clearly exists and equals −∞ . Also if for every j > h η , ..., η k i ∩ F µ j ( ω ) = { } , since µ j → −∞ , Lemma 4.7 impliesthat (6.4) exists and equals −∞ . So we can assume that for some j > h η , ..., η k i∩ F µ j +1 ( ω ) = { } .For i j we can find a finite-dimensional subspace H i ( ω ) such that H i ( ω ) L F µ i +1 ( ω ) = F µ i ( ω ).Furthermore, for each i j , there is a subspace ˜ H i ( ω ) ⊂ H i ( ω ) with dim (cid:2) ˜ H i ( ω ) (cid:3) = n i such that h η , ..., η k i F µ j +1 ( ω ) = L i j ˜ H i ( ω ) F µ j +1 ( ω ) . Now as a consequence of item (v) in the Multiplicative Ergodic Theorem 4.17,lim n →∞ n log Vol (cid:0) ϕ ( n, ω, η ) , ..., ϕ ( n, ω, η k ) (cid:1) = X i j n i µ i which shows the claim.The strategy of the proof now is to compare all norms against one another. For − t ξ, η ∈ C ([ − , , R ) \ { } and n ∈ N set ξ nt = y ξn + t and η nt = y ηn + t where y ξ and y η are solutionsto (6.1) starting from ξ , η respectively. By definition,( ξ n ) ′ t = ξ n − t , ( ξ n ) s,t = Z ts ξ n − s,u dB n + u (6.7)for − ≤ s ≤ t ≤ n ≥ ξ − ≡ 0. Set F t := F t . From Lemma 6.2, for any C > P (cid:0) k ξ n k α > C | F n − (cid:1) P Z Z [ − , | ξ nv,u | p | u − v | pα du dv > C p ( A p ) p (cid:12)(cid:12) F n − ! = P Z Z [ − , | R [ u,v ] ξ n − τ dB n + τ | p | u − v | pα +2 du dv > C p ( A p ) p |F n − ! almost surely. Similarly, P (cid:18) inf β ∈ Q k η n − βξ n k α > C (cid:12)(cid:12) F n − (cid:19) inf β ∈ Q P Z Z [ − , | R [ u,v ] η n − τ − βξ n − τ dB n + τ | p | u − v | pα +2 du dv > C p ( A p ) p (cid:12)(cid:12) F n − ! almost surely. Set p = 2 m for m chosen such that m (1 − α ) > 1. From the Burkholder-Davis-Gundy inequality, it follows that E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z [ u,v ] ξ n − τ dB n + τ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m (cid:12)(cid:12) F n − B m | u − v | m k ξ n − k m ∞ YNAMICAL THEORY FOR SDDE 41 almost surely for some constant B m > 0. Consequently, P ( k ξ n k α > C |F n − ) ˜ A m k ξ n − k m ∞ C m and(6.8) P (cid:18) inf β ∈ Q k η n − βξ n k α > C |F n − (cid:19) ˜ A m inf β ∈ Q k η n − − βξ n − k m ∞ C m (6.9)for a general constant ˜ A m . Now for any ε > 0, (6.8) implies that P (cid:18) n log k ξ n k α > ε + 1 n − k ξ n − k ∞ ] (cid:19) P (cid:0) k ξ n − k α > k ξ n − k ∞ exp[ ε ( n − (cid:1) ˜ A m exp [2 mε ( n − −→ n → ∞ . Similarly, P (cid:18) n log inf β ∈ Q k η n − βξ n k α > ε + 1 n − β ∈ Q k η n − − βξ n − k ∞ (cid:19) −→ n → ∞ . Now from (6.6) and (6.7), P sup − s 1. Then E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z [ u,v ] ξ n − u,τ dB n + τ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m (cid:12)(cid:12) F n − B m ( v − u ) m (2 α +1) k ξ n − k mα almost surely. Consequently, for general constants M and ˜ M , P sup − s 1. Let ξ , η ∈ C ∞ ([ − , , R ) \ { } , a ∈ R and set ˜ ξ t := R t − ξ τ dB τ . Using (6.3), we have γ t := ˜ ξ t + η t + a = ϕ (1 , θ − ω, ξ )[ t ] + η t + a − ξ (6.15)and (6.14) implies that lim n →∞ n log k ϕ ( n, ω, γ ) k µ . From Theorem 3.10, we know that elementsof the form γ are dense in E ω . Choose ξ ω ∈ F µ ( ω ) \ F µ ( ω ). Since F µ ( ω ) is a closed subspace, YNAMICAL THEORY FOR SDDE 43 we can find a neighborhood B ( ξ ω , δ ) ⊂ F µ ( ω ) \ F µ ( ω ) and an element γ ∈ B ( ξ ω , δ ) of the form(6.15). Therefore, µ ≤ µ , thus µ = µ . (cid:3) Remark . Taking the Hilbert space norm k · k M in the definition of Vol, we actually haveVol( X , ..., X k ) = k X ∧ X ∧ ... ∧ X k k M . We conjecture that the limitlim n →∞ n log k ϕ ( n, ω, η ) ∧ · · · ∧ ϕ ( n, ω, η k ) k M is independent of the choice of η , . . . , η k whenever these vectors are linearly independent, and thatthe limit coincides with Λ k almost surely. This would be in good accordance with the classicaldefinition of Lyapunov exponents in the finite dimensional case, cf. [Arn98, Chapter 3]. Appendix A. Stability for rough delay equations In the following, we sketch the proof of Theorem 1.9. The strategy is the same as in [NNT08,Theorem 4.2]. Proof of Theorem 1.9 (sketch). For simplicity, we assume that U = W = R . By definition, y s,t = Z ts σ ( y τ , ξ τ − r ) d X τ = Λ s,t + ρ s,t = σ ( y s , ξ s − r ) X s,t + ρ s,t + ρ s,t (A.1)where Λ s,t = σ ( y s , ξ s − r ) X s,t + σ ( y s , ξ s − r ) y ′ s X s,t + σ ( y s , ξ s − r ) ξ ′ s − r X s,t ( − r ) ,ρ s,t = σ ( y s , ξ s − r ) y ′ s X s,t + σ ( y s , ξ s − r ) ξ ′ s − r X s,t ( − r ) and ρ s,t = Z ts σ ( y τ , ξ τ − r ) d X τ − Λ s,t , using the notation σ ( x, y ) = ∂ x σ ( x, y ), σ ( x, y ) = ∂ y σ ( x, y ). Analogously, one defines ˜Λ, ˜ ρ and˜ ρ such that ˜ y s,t = ˜Λ s,t + ˜ ρ s,t = σ (˜ y s , ˜ ξ s − r ) ˜ X s,t + ˜ ρ s,t + ˜ ρ s,t . Note that y ′ s = σ ( y s , ξ s − r ) and y s,t = ρ s,t + ρ s,t . It is not hard to see thatΛ s,t − Λ s,u − Λ u,t = [ σ ( y s , ξ s − r ) y s,u + σ ( y s , ξ s − r ) ξ s − r,u − r ] X u,t + [ σ ( y u , ξ u − r ) y ′ u − σ ( y s , ξ s − r ) y ′ s ] X u,t + [ σ ( y u , ξ u − r ) ξ ′ u − r − σ ( y s , ξ s − r ) ξ ′ s − r ] X u,t ( − r )+ Z (1 − τ ) (cid:2) σ , ( z τs,u , ¯ z τs,u )( y s,u ) + 2 σ , ( z τs,u , ¯ z τs,u ) y s,y ξ s − r,u − r + σ , ( z τs,u , ¯ z τs,u )( ξ s − r,u − r ) (cid:3) dτ X u,t (A.2) where z τs,u = τ y u + (1 − τ ) y s , ¯ z τs,u = τ ξ u − r + (1 − τ ) ξ s − r and σ , ( x, y ) = ∂ x σ ( x, y ), σ , ( x, y ) = ∂ x ∂ y σ ( x, y ) and σ , ( x, y ) = ∂ y σ ( x, y ). Set R := k X − ˜ X k γ, [0 ,r ] + k X − ˜ X k γ, [0 ,r ] + k X ( − r ) − ˜ X ( − r ) k γ, [0 ,r ] + k ξ ′ − ˜ ξ ′ k β, [0 ,r ] + k ξ − ˜ ξ k β, [0 ,r ] + k ξ − ˜ ξ k β, [0 ,r ] ,C ( y ) := k X k γ + k X k γ, [0 ,r ] + k X ( − r ) k γ, [0 ,r ] + k y k D βX ([0 ,r ] ,W ) + k ξ k D βX ([ − r, ,W ) and D ( X ) := k X k γ + k X k γ + k X ( − r ) k γ + k ξ k D βX ([0 ,r ] ,W ) with an analogous definition of C (˜ y ) and D ( ˜ X ). It is not hard to see that there is a continuousfunction g : (0 , ∞ ) → [0 , ∞ ), increasing in every of its arguments, such that k ρ − ˜ ρ k β ;[ a,b ] ( b − a ) γ − β g (cid:2) D ( X ) , D ( ˜ X ) , C ( y ) , C (˜ y ) (cid:3)(cid:2) R + k y − ˜ y k β ;[ a,b ] + k y ′ − ˜ y ′ k β ;[ a,b ] + k y − ˜ y k β ;[ a,b ] (cid:3) for every [ a, b ] ⊆ [0 , r ]. From the Sewing lemma [FH14, Lemma 4.2], k ρ − ˜ ρ k β ;[ a,b ] M sup s,u,t ∈ [ a,b ] (cid:12)(cid:12) (Λ s,t − ˜Λ s,t ) − (Λ s,u − ˜Λ s,u ) − (Λ u,t − ˜Λ u,t ) (cid:12)(cid:12) ( t − s ) β for some constant M > 0. Using (A.2), one can deduce thatsup s,u,t ∈ [ a,b ] (cid:12)(cid:12) (Λ s,t − ˜Λ s,t ) − (Λ s,u − ˜Λ s,u ) − (Λ u,t − ˜Λ u,t ) (cid:12)(cid:12) ( t − s ) β ( b − a ) γ − β g (cid:2) D ( X ) , D ( ˜ X ) , C ( y ) , C (˜ y ) (cid:3)(cid:2) R + k y − ˜ y k β ;[ a,b ] + k y ′ − ˜ y ′ k β ;[ a,b ] + k y − ˜ y k β ;[ a,b ] (cid:3) . Now, along with ( A. k y − ˜ y k β ;[ a,b ] + k y ′ − ˜ y ′ k β ;[ a,b ] + k y − ˜ y k β ;[ a,b ] ( b − a ) γ − β ˜ g (cid:2) D ( X ) , D ( ˜ X ) , C ( y ) , C (˜ y ) (cid:3)(cid:2) R + k y − ˜ y k β ;[ a,b ] + k y ′ − ˜ y ′ k β ;[ a,b ] + k y − ˜ y k β ;[ a,b ] (cid:3) with ˜ g being a continuous increasing function. Using the bounds for the norm of y and ˜ y providedin [NNT08, Equation (62)], we can find an increasing continuous function H : (0 , ∞ ) → [0 , ∞ )such that k y − ˜ y k β ;[ a,b ] + k y ′ − ˜ y ′ k β ;[ a,b ] + k y − ˜ y k β ;[ a,b ] ( b − a ) γ − β H [ D ( X ) , D ( ˜ X )][ R + k y − ˜ y k β ;[ a,b ] + k y ′ − ˜ y ′ k β ;[ a,b ] + k y − ˜ y k β ;[ a,b ] ] . Now by the same argument as for the linear case, cf. the proof of Theorem 1.11, one sees that k y − ˜ y k β ;[0 ,r ] + k y ′ − ˜ y ′ k β ;[0 ,r ] + k y − ˜ y k β ;[0 ,r ] F [ D ( X ) + D ( ˜ X )] R (A.3)holds for an increasing continous function F . The claim follows from (A.3) . (cid:3) Appendix B. A pathwise MET Proof of Proposition 4.15. For given n ∈ N , let E n := h e n , . . . , e mn i be an m -dimensional subspaceof V with k e in k = 1 and Vol( T n e n , ..., T n e mn ) > D m ( T n ) . (B.1) YNAMICAL THEORY FOR SDDE 45 By [Blu16, Lemma 2.3], we can find a closed complement subspace F n to E n := T n E n in V n suchthat for P n := Π E n || F n , k P n k √ m. Let F n := { v ∈ V : T n v ∈ F n } . One can check that F n is a closed complement subspace to E n .Set P n := Π E n || F n . From Lemma 4.7 and (B.1), it follows that there is a constant α m such that forany v ∈ E n , k T n v kk v k > D m ( T n )2 α m k T n k m − . (B.2)From P n = ( T n | E n ) − P n T n , (B.2) implies that k P n k ( m + 1) k T n kk ( T n | E n ) − k α m k T n k m D m ( T n ) . (B.3)Let v ∈ F n with k v k = 1. ThenVol( T n e n , ..., T n e mn , T n v ) = Vol( T n e n , ..., T n e mn ) d ( T n v, h T n e n , ..., T n e mn i ) . (B.4)Since d ( T n v, h T n e n , ..., T n e mn i ) = inf β j ∈ R k T n v − P j m β j T n e jn k , we see that k T n v k d ( T n v, h T n e n , ..., T n e mn i ) k P n k + 1 √ m + 1 . Consequently, from (B.1) and (B.4), k T n v k √ m + 1) D m +1 ( T n ) D m ( T n ) . (B.5)The rest of the proof is almost identical to the original proof of [Blu16, Proposition 3.4]. First,one can show that the sequence of subspaces ( F n ) converge to F in the Hausdorff distance at asufficiently fast exponential rate, cf. [Blu16, Claim 3 on page 2396]. Together with (B.5), thisimplies the bound lim sup n →∞ n log k T n | F k l which was announced in Remark (4.16). From the convergence, we can also deduce that F is closedand m -codimensional. The identities (4.25) and (4.26) can be proved exactly as in [Blu16]. To see(4.27), let H = h h , ..., h m i be a complement subspace to F . Note that, from (4.26) and assumption(ii), for any δ > 0, we can choose n large enough such thatexp (cid:0) n ( l − δ ) (cid:1) k T n v kk v k exp (cid:0) n ( l + δ ) (cid:1) holds for all v ∈ H . Consequently,exp (cid:0) n ( l − δ ) (cid:1) d (cid:0) T n h j , h T n h i i i MGV acknowledges a scholarship from the Berlin Mathematical School(BMS). SR and MS acknowledge financial support by the DFG via Research Unit FOR 2402. Allauthors would like to thank A. Blumenthal for sending us a corrected version of the proof of [Blu16,Lemma 3.7] and A. Schmeding for valuable discussions and comments during the preparation ofthe manuscript. References [Arn98] Ludwig Arnold. Random dynamical systems . Springer Monographs in Mathematics. Springer-Verlag,Berlin, 1998.[Blu16] Alex Blumenthal. A volume-based approach to the multiplicative ergodic theorem on Banach spaces. Dis-crete Contin. Dyn. Syst. , 36(5):2377–2403, 2016.[Bog98] Vladimir I. Bogachev. Gaussian measures , volume 62 of Mathematical Surveys and Monographs . AmericanMathematical Society, Providence, RI, 1998.[BRS17] Isma¨el Bailleul, Sebastian Riedel, and Michael Scheutzow. Random dynamical systems, rough paths andrough flows. J. Differential Equations , 262(12):5792–5823, 2017.[BTR07] Ian Boutle, Richard HS Taylor, and Rudolf A R¨omer. El ni˜no and the delayed action oscillator. AmericanJournal of Physics , 75(1):15–24, 2007.[Buc00] Evelyn Buckwar. Introduction to the numerical analysis of stochastic delay differential equations. J. Com-put. Appl. Math. , 125(1-2):297–307, 2000. Numerical analysis 2000, Vol. VI, Ordinary differential equationsand integral equations.[CDF97] Hans Crauel, Arnaud Debussche, and Franco Flandoli. Random attractors. J. Dynam. Differential Equa-tions , 9(2):307–341, 1997.[CF94] Hans Crauel and Franco Flandoli. Attractors for random dynamical systems. Probab. Theory Related Fields ,100(3):365–393, 1994.[Dix77] Jacques Dixmier. C ∗ -algebras . North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. Trans-lated from the French by Francis Jellett, North-Holland Mathematical Library, Vol. 15.[FH14] Peter K. Friz and Martin Hairer. A Course on Rough Paths with an introduction to regularity structures ,volume XIV of Universitext . Springer, Berlin, 2014.[Fla95] Franco Flandoli. Regularity theory and stochastic flows for parabolic SPDEs , volume 9 of StochasticsMonographs . Gordon and Breach Science Publishers, Yverdon, 1995.[Fol95] Gerald B. Folland. A course in abstract harmonic analysis . Studies in Advanced Mathematics. CRC Press,Boca Raton, FL, 1995.[FV10] Peter K. Friz and Nicolas B. Victoir. Multidimensional stochastic processes as rough paths , volume 120 of Cambridge Studies in Advanced Mathematics . Cambridge University Press, Cambridge, 2010. Theory andapplications.[GAS11] Mar´ıa J. Garrido-Atienza and Bj¨orn Schmalfuß. Ergodicity of the infinite dimensional fractional Brownianmotion. J. Dynam. Differential Equations , 23(3):671–681, 2011.[GTQ15] Cecilia Gonz´alez-Tokman and Anthony Quas. A concise proof of the multiplicative ergodic theorem onBanach spaces. J. Mod. Dyn. , 9:237–255, 2015.[Gub04] Massimiliano Gubinelli. Controlling rough paths. J. Funct. Anal. , 216(1):86–140, 2004.[Hai14] Martin Hairer. A theory of regularity structures. Invent. Math. , 198(2):269–504, 2014.[LCL07] Terry J. Lyons, Michael Caruana, and Thierry L´evy. Differential equations driven by rough paths , volume1908 of Lecture Notes in Mathematics . Springer, Berlin, 2007. Lectures from the 34th Summer School onProbability Theory held in Saint-Flour, July 6–24, 2004, With an introduction concerning the SummerSchool by Jean Picard.[LL10] Zeng Lian and Kening Lu. Lyapunov exponents and invariant manifolds for random dynamical systems ina Banach space. Mem. Amer. Math. Soc. , 206(967):vi+106, 2010.[Mao08] Xuerong Mao. Stochastic differential equations and applications . Horwood Publishing Limited, Chichester,second edition, 2008.[Mn83] Ricardo Ma˜n´e. Lyapounov exponents and stable manifolds for compact transformations. In Geometricdynamics (Rio de Janeiro, 1981) , volume 1007 of Lecture Notes in Math. , pages 522–577. Springer, Berlin,1983.[Moh84] Salah-Eldin A. Mohammed. Stochastic functional differential equations , volume 99 of Research Notes inMathematics . Pitman (Advanced Publishing Program), Boston, MA, 1984. YNAMICAL THEORY FOR SDDE 47 [Moh86] Salah-Eldin A. Mohammed. Nonlinear flows of stochastic linear delay equations. Stochastics , 17(3):207–213,1986.[MS96] Salah-Eldin A. Mohammed and Michael Scheutzow. Lyapunov exponents of linear stochastic functional dif-ferential equations driven by semimartingales. I. The multiplicative ergodic theory. Ann. Inst. H. Poincar´eProbab. Statist. , 32(1):69–105, 1996.[MS97] Salah-Eldin A. Mohammed and Michael Scheutzow. Lyapunov exponents of linear stochastic functional-differential equations. II. Examples and case studies. Ann. Probab. , 25(3):1210–1240, 1997.[MS99] Salah-Eldin A. Mohammed and Michael Scheutzow. The stable manifold theorem for stochastic differentialequations. Ann. Probab. , 27(2):615–652, 1999.[MS04] Salah-Eldin A. Mohammed and Michael Scheutzow. The stable manifold theorem for non-linear stochasticsystems with memory. II. The local stable manifold theorem. J. Funct. Anal. , 206(2):253–306, 2004.[NNT08] Andreas Neuenkirch, Ivan Nourdin, and Samy Tindel. Delay equations driven by rough paths. Electron. J.Probab. , 13:no. 67, 2031–2068, 2008.[Ose68] Valery I. Oseledec. A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamicalsystems. Trudy Moskov. Mat. Obˇsˇc. , 19:179–210, 1968.[Rag79] Madabusi S. Raghunathan. A proof of Oseledec’s multiplicative ergodic theorem. Israel J. Math. , 32(4):356–362, 1979.[RS17] Sebastian Riedel and Michael Scheutzow. Rough differential equations with unbounded drift term. J.Differential Equations , 262(1):283–312, 2017.[Rue79] David Ruelle. Ergodic theory of differentiable dynamical systems. Inst. Hautes ´Etudes Sci. Publ. Math. ,(50):27–58, 1979.[Rue82] David Ruelle. Characteristic exponents and invariant manifolds in Hilbert space. Ann. of Math. (2) ,115(2):243–290, 1982.[RV93] Francesco Russo and Pierre Vallois. Forward, backward and symmetric stochastic integration. Probab.Theory Related Fields , 97(3):403–421, 1993.[Sch92] Bj¨orn Schmalfuss. Backward cocycle and attractors of stochastic differential equations. In V. Reitmann,T. Riedrich, and N. Koksch, editors, International Seminar on Applied Mathematics - Nonlinear Dynamics:Attractor Approximation and Global Behavior , pages 185–192. Technische Universit¨at Dresden, 1992.[Sch13] Michael Scheutzow. Exponential growth rate for a singular linear stochastic delay differential equation. Discrete Contin. Dyn. Syst. Ser. B , 18(6):1683–1696, 2013.[Sto05] George Stoica. A stochastic delay financial model. Proc. Amer. Math. Soc. , 133(6):1837–1841, 2005.[Thi87] Philippe Thieullen. Fibr´es dynamiques asymptotiquement compacts. Exposants de Lyapounov. Entropie.Dimension. Ann. Inst. H. Poincar´e Anal. Non Lin´eaire , 4(1):49–97, 1987.[Woj91] Przemys law Wojtaszczyk. Banach spaces for analysts , volume 25 of Cambridge Studies in Advanced Math-ematics . Cambridge University Press, Cambridge, 1991.[You36] Laurence C. Young. An inequality of the H¨older type, connected with Stieltjes integration. Acta Math. ,67(1):251–282, 1936. Mazyar Ghani Varzaneh, Institut f¨ur Mathematik, Technische Universit¨at Berlin, Germany and De-partment of Mathematical Sciences, Sharif University of Technology, Tehran, Iran E-mail address : [email protected] Sebastian Riedel, Institut f¨ur Mathematik, Technische Universit¨at Berlin, Germany E-mail address : [email protected] Michael Scheutzow, Institut f¨ur Mathematik, Technische Universit¨at Berlin, Germany E-mail address ::