Lyapunov exponents and Oseledets decomposition in random dynamical systems generated by systems of delay differential equations

Linear skew-product semidynamical systems generated by random systems of delay differential equations are considered, both on a space of continuous functions as~well as on a space of $p$-summable functions. The main result states that in both cases, the Lyapunov exponents are identical, and that the Oseledets decompositions are related by natural embeddings.


Introduction
The theory of linear random skew-product semidynamical systems has become a powerful tool in the investigation of random linear parabolic PDEs of second order driven by a measurable dynamical system on a probability space. In particular, when the solution operator is compact (and that holds if the domain is bounded) then, assuming the summability of the coefficients of the PDE, we have an Oseledets decomposition: the separable Banach space decomposes into a countable direct sum of invariant measurable families of finite-dimensional vector subspaces which can be characterized as corresponding to solutions defined on the whole real line having given logarithmic growth rates (Lyapunov exponents) both in the future and in the past (plus, possibly, an invariant measurable family corresponding to solutions having logarithmic growth rates −∞). See Lian and Lu's monograph [12].
When we consider systems of linear random delay differential equations z ′ (t) = A(θ t ω) z(t) + B(θ t ω) z(t − 1), any "natural" space on which we define a linear skew-product semidynamical system must contain (or, at least, be equal to) a space consisting of functions defined on [−1, 0] and taking values on R N . In general, there is no hope that the solution operator is compact. But it is compact after some time, so this is not a big obstacle. A more important thing is that, generally, the solution operator is not injective. This makes it impossible to directly apply the results contained in [12]. It is natural to work in the framework of semi-invertible Oseledets theorems: the metric dynamical system on the base space is invertible, but the operators between fibers are not necessarily injective.
For such systems, Doan proved in his dissertation [7] the existence of an Oseledets filtration: an invariant measurable filtration by finite-codimensional vector subspaces such that the solutions corresponding to the set difference of two subsequent subspaces have logarithmic rates of growth equal to a given Lyapunov exponent. Starting from Doan's results, González-Tokman and Quas [9] proved that there exists an Oseledets splitting, provided only that the fibers are separable Banach spaces and that the base space is a Lebesgue space. Indeed, in an earlier paper by Froyland et al. [8] an Oseledets splitting was obtained, but under an additional assumption that the base space is a Borel subset of a separable complete metric space with the σ-algebra of Borel sets and with a Borel probability measure.
The above result should be considered sufficient for our purposes: C([−1, 0], R N ) appears to be the natural, at first sight, Banach space for which the solution operator satisfies all the axioms of a skew-product random semidynamical system. Such a Banach space is separable, and there are no difficulties.
However, one should remember that we sometimes need to calculate (at least, to estimate) the Lyapunov exponents. As shown in Calzada et al. [3], one needs a Hilbert space, more precisely, the space L 2 ([−1, 0], R N , µ 0 ), with µ 0 = δ 0 + l, where l is the Lebesgue measure on [−1, 0], is a natural choice here. In such a case, one can use results from González-Tokman and Quas [10]: an Oseledets decomposition is proved there for reflexive separable Banach spaces. In general, good geometric properties for the Banach spaces provide a more constructive version of the theory. Mierczyński and Shen [13] and Mierczyński et al. [15] prove, under adequate dynamical assumptions, the existence of a principal Floquet subspace and a generalized exponential separation decomposition when the fiber is a separable Banach space with separable dual.
This has to do with the dual skew-product semidynamical systems. In the case of ordinary differential equations, or parabolic partial differential equations of second order, such dual skew-product systems are generated by adjoint equations. Then, the adjoint equation has the same properties as the original equation, and in many cases one needs only to prove "one half" of a theorem (for example, the existence of an Oseledets filtration, whereas the other half can be given by applying the theorem to the skew-product system generated by the adjoint equation; for a similar approach see Section 3 in Mierczyński and Shen [14]).
However, this is not the case for delay differential equations. To be sure, there exists a well-defined "abstract" dual skew-product semidynamical system, but, at least in the case of C([−1, 0], R N ), it is not generated by anything resembling an adjoint equation. For generation of the dual system by an adjoint equation (sort of), see Delfour and Mitter [5].
The paper is organized as follows. Section 2 contains preliminaries and explains notions used throughout the rest of the sections. In Section 3 a definition of an Oseledets decomposition for a measurable linear skew-product semidynamical system is given, and, under appropriate assumptions for our purposes, some theorems of existence are explained Section 4 is devoted to showing that linear systems of delay differential equations generate measurable linear skew-product semidynamical systems when we take as our fiber both C([−1, 0], R n ) and R N × L p ([−1, 0], R N ), and some measurability and summability assumptions on the coefficients are considered.
The main results of the paper are contained in Section 5. It is shown that for both spaces, the Lyapunov exponents are the same, and that the Oseledets decomposition are related by natural embeddings. The importance of these results is that the geometrical methods of construction for the Oseledets subspaces, obtained in [10] for reflexive separable Banach spaces, as well as the estimates of Lyapunov exponents, can be applied on R N × L p ([−1, 0], R N ) and then translated to C([−1, 0], R N ) by embedding.

Preliminaries
Let (Y, d) be a metric space, B(y; ǫ) denotes the closed ball in Y centered at y ∈ Y with radius ǫ > 0, and B(Y ) stands for the σ-algebra of all Borel subsets of Y . For a compact metric space Z and a Banach space X, C(Z, X) denotes the Banach space of continuous functions from Z into X, with the supremum norm. For Banach spaces X 1 , X 2 , L(X 1 , X 2 ) stands for the Banach space of bounded linear mappings from X 1 into X 2 , endowed with the standard norm. Instead of L(X, X) we write L(X).
A probability space is a triple (Ω, F, P), where Ω is a nonempty set, F is a σ-algebra of subsets of Ω, and P is a probability measure defined for all F ∈ F. We always assume that the measure P is complete.
A metric dynamical system is a measurable dynamical system ((Ω, F, P), (θ t ) t∈R ) such that for each t ∈ R the mapping θ t : Ω → Ω is P-preserving (i.e., P(θ −1 t (F )) = P(F ) for any F ∈ F and t ∈ R). It is said to be ergodic if for any invariant F ∈ F, either P(F ) = 1 or P(F ) = 0.
We say that the decomposition A strongly measurable family of projections associated with the invariant decomposition E(ω) ⊕ F (ω) = X is referred to as tempered if lim t→±∞ ln P (θ t ω) t = 0 P-a.e. on Ω 0 .

Oseledets decomposition
Let Φ = ((U ω (t)), (θ t )) be a measurable linear skew-product semidynamical system covering an ergodic metric dynamical system ((Ω, F, P), (θ t ) t∈R ) with P complete. We assume throughout the present section that (O1) the functions Then it follows from the Kingman subadditive ergodic theorem that there exists for P-a.e. ω ∈ Ω, which is referred to as the top Lyapunov exponent of Φ.
We will also assume that Definition 3.1. Φ admits an Oseledets decomposition if there exists an invariant subset Ω 0 ⊂ Ω, P(Ω 0 ) = 1, with the property that one of the following mutually exclusive cases, (I) or (II), holds: of finite dimensional vector subspaces, and a family {F ∞ (ω)} ω∈Ω0 of closed vector subspaces of finite codimension such that (a) for i = 1, . . . , k, any ω ∈ Ω 0 and t ≥ 0 In particular, are strongly measurable and tempered; (d) for i = 1, . . . , k , any ω ∈ Ω 0 and any nonzero u ∈ E i (ω) (e) for i = 1, . . . , k , any ω ∈ Ω 0 and any u ∈ if and only if there exists a negative semiorbitũ : In this case, (II) There are a decreasing sequence of real numbers λ 1 = λ top > · · · > λ i > λ i+1 > · · · with limit −∞, called the Lyapunov exponents for Φ, countably many measurable families {E i (ω)} ω∈Ω0 , i ∈ N, of finite dimensional vector subspaces, and countably many families {F i (ω)} ω∈Ω0 , i ∈ N, of closed vector subspaces of finite codimensions, called the Oseledets filtration for Φ, such that (a) for i ∈ N, any ω ∈ Ω 0 and t ≥ 0 (c) for i ∈ N, the families of projections associated with the decompositions only if there exists a negative semiorbitũ : As a consequence of the existence of an Oseledets decomposition, we easily deduce the following properties.
Proposition 3.2. Assume that Φ admits an Oseledets decomposition. Then for each ω ∈ Ω 0 and each nonzero u ∈ X the limit exists and equals some Lyapunov exponent λ i or −∞.
Proof. Assume case (I) and fix ω ∈ Ω 0 and a nonzero u ∈ X. If u belongs to F ∞ (ω) then (3.1) equals −∞. If u does not belong to F ∞ (ω) then there is a j ∈ {1, . . . , k} such that in the decomposition Assume case (II) and fix ω ∈ Ω 0 and a nonzero u ∈ X. If in each decomposition Proof. The necessity is a consequence of (e) or (f), and (g) for the case (I), and (e) and (f) for the case (II). Assume now that for some nonzero u ∈ X there exists a full orbit with the above properties. It follows from (e) or (f) in the first case, or (e) in the second one, that u belongs to F i−1 (ω) for some i = 1, . . . , k, or some i ∈ N respectively. Now we need to apply (I)(g) or (II)(f) to finish the proof.
It follows from the results in [12] that if X is separable and U ω (1) is an injective and compact operator for all ω ∈ Ω then there exists an Oseledets decomposition.
The papers [8] and [9] give a proof of the Oseledets decomposition for semiinvertible discrete ergodic transformations. We want to state a version of these results valid for continuous linear skew-product semidynamical systems. We omit the details of the proof that follow standard arguments from the theory mentioned above. The conclusions of this theorem will be relevant in the applications to the theory of delay differential equations.
Theorem 3.4. Assume Φ is a measurable linear skew-product semidynamical system satisfying the following: Then Φ admits an Oseledets decomposition.
Indication of proof. The existence of a discrete-time Oseledets decomposition for systems satisfying (a), (b) and (c) was proved in [9, Theorem. A]. To pass to the continuous-time decomposition we proceed along the lines of the proof of [12,Theorem. 3.3].
Remark 3.5. Analogously, from a discrete-time result in [10, Corollary 17], the existence of an Oseledets decomposition for Φ is obtained when X is a separable and reflexive Banach space and (a) is not assumed. The importance of this approach is its constructive nature. More precisely, a way of approximating the Oseledets splitting is provided, which is important in applications.

Semiflows generated by linear random delay differential equations
This section is devoted to show the applications of the previous theory to random dynamical systems generated by systems of linear random delay differential equations of the form where z(t) ∈ R N , N ≥ 2, A(ω), B(ω) are N × N real matrices: and ((Ω, F, P), (θ t ) t∈R ) is an ergodic metric dynamical system, with P complete.
From now on, the Euclidean norm on R N will be denote by · , R N ×N will stand for the algebra of N × N real matrices with the operator or matricial norm induced by the Euclidean norm, i.e., A := sup{ A u | u ≤ 1}, for any A ∈ R N ×N .
We denote by J the linear mapping from C to L which belongs to L(C, L) and J = 2. In the following, p will be fixed and q ∈ (0, ∞) is such that 1/p + 1/q = 1. and Remark 4.1. The following is sufficient for the fulfillment of the second condition in (S2): Indeed, since |b| q ∈ L 1 (Ω, F, P) and the measure P is invariant, for any t ∈ R Ω |b(θ t ω ′ )| q dP(ω ′ ) = Ω |b(ω ′ )| q dP(ω ′ ) and an application of Fubini's theorem gives that the map belongs to L 1 (Ω, F, P), from which the required statement follows immediately.

4.1.
Linear skew-product semiflows on C and L. Before proceeding to the existence of solutions, notice that the coefficients A and B are defined only P-a.e.
on Ω, whereas the theory of Lyapunov exponents requires us to have the solution operator defined on the whole of Ω. In particular, we need to have for each ω ∈ Ω. Since a ∈ L 1 (Ω, F, P) and the measure P is invariant, and an application of Fubini's theorem gives (4.4) for ω ∈ Ω 1 ⊂ Ω, invariant set of full measure. Analogously, from the second condition in (S2), there is an invariant set Ω 2 ⊂ Ω of full measure such that Then we can put the value of A(ω) and B(ω) for ω ∈ Ω \ (Ω 1 ∩ Ω 2 ) to be equal to zero to obtain (4.4) and (4.5) for all ω ∈ Ω, as needed.
where the initial datum u is assumed to belong to C = C([−1, 0], R N ) and assumptions (S1) and (S2) hold.

Connections between semidynamical systems on C and on L.
First of all, observe that for any t ≥ 0 and any ω ∈ Ω one has where J is defined in (4.3). We can also define for t ≥ 1 the linear operator (1) ∈ L(L, C), as claimed. For the compactness, fix ω ∈ Ω, 0 ≤ s 1 ≤ s 2 ≤ 0, u = (u 1 , u 2 ) ∈ L and note that Then denoting t 1 = 1 + s 1 and t 2 = 1 + s 2 , we deduce that Moreover, again Lemma 4.3, (4.5) and Hölder inequality provide which together with (4.4), (4.5) and u 2 p ≤ u L imply the equicontinuity of the set U (L,C) ω (1) u | u L ≤ k , and by Ascoli-Arzelà theorem, the precompactness, as needed.
Finally, for t ≥ 1, from z t (ω, u) = z t−1 (θ 1 ω, z 1 (ω, u)) we deduce that which shows, from Proposition 4.4, that for t ≥ 1, U (L,C) ω (t) is the composition of a compact operator and a linear operator. Hence, U (L,C) ω (t) ∈ L(L, C) and it is a compact operator for t ≥ 1 and each ω ∈ Ω, as stated.  Proof. Since J is a bounded operator from C to L, the result is a consequence of the previous proposition and the relations for any t ≥ 1 and any ω ∈ Ω.
Proof. It is well known (see, e.g., [2, Example 2.2.8]) that the mapping As a consequence, for each t ∈ (0, 1] the mapping B(R N ×N ))-measurable, and hence, from (S1) and taking a Borel representation of the function u 2 ∈ L p ([−1, 0], R N ), we deduce that , B(R N ))-measurable. Therefore, an application of Fubini's theorem show that for each t ≥ 1 which together with (4.22) and formula (4.13) finishes the proof.
Proof. Since for ω ∈ Ω and u ∈ C fixed the mapping R + ∋ t → U (C) ω (t) u ∈ C is easily seen to be continuous, the fact that the mapping is (B(R + )⊗F⊗B(C), B(C))-measurable follows from Proposition 4.4, Lemma 4.11 and [1, Lemma 4.51 on pp. 153]. The rest of the properties have been already checked, so that Φ is a measurable linear skew-product semidynamical system, as claimed.
Lemma 4.13. Assume (S1) and (S2). For u ∈ L and t > 0 fixed, the mapping Proof. It follows from (2.2) that it suffices to prove the result for t ∈ (0, 1] only. Since L is separable, from Pettis' Theorem (see Hille and Phillips [11,Theorem 3.5.3 and Corollary 2 on pp. 72-73]) the weak and strong measurability notions are equivalent and therefore, it is enough to check that for each u * ∈ L * the mapping Hence, Lemma 4.10 and similar arguments prove that (4.23) holds, as stated.
Proof. Since for ω ∈ Ω and u ∈ L the mapping R + ∋ t → U Remark 4.15. It can also be proved that for u ∈ L and t ≥ 1 fixed, the mapping

Lyapunov exponents
In this section it is shown that the Lyapunov exponents for the two cases considered in the previous section are the same, and that the Oseledets decomposition are related by natural embeddings.
Throughout the whole section we assume that (S1) and (S2) holds.
In addition, for t > 0 it can be checked from the definition of c and the cocycle property (2.2) satisfied by U 0 ω that c(θ t ω) ≤ c(θ ⌊t⌋+1 ω) c(θ ⌊t⌋ ω), where ⌊t⌋ denotes the integer part of the real number t. From this which we conclude that lim sup t→∞ ln c(θ t ω) t ≤ 0 for any ω ∈Ω. In order to proof the second inequality it is enough to notice that, again from (S2), we deduce that ln + d ∈ L 1 (Ω, F, P) and in this case d(θ t ω) ≤ d(θ ⌊t⌋ ω)+ d(θ ⌊t⌋+1 ω), which is an easy consequence of the definition of d. The other two inequalities follow from the application of the previous ones to the reversed flow The next result shows that we have the same Lyapunov exponents, independent of the choice of the Banach space, C or L. (1) Assume that for some ω ∈ Ω 1 , u ∈ C and λ ∈ [−∞, ∞) there holds (2) Assume that for some ω ∈ Ω 1 , u ∈ L and λ ∈ [−∞, ∞) there holds On the other hand, from (4.21), (4.20), (4.19) and J = 2 we obtain θ1ω (t) , which together with the invariance of the set satisfying (5.3) implies that λ top , and finishes the proof.
Remark 5.4. In view of the above we will denote the common value of λ From now on, we assume that both Φ (C) and Φ (L) admit an Oseledets decomposition. From Proposition 3.2 we can find a common invariant set Ω 0 ⊂ Ω 1 with P(Ω 0 ) = 1, such that for any ω ∈ Ω 0 and each nonzero u ∈ C the limit (ω)), P-a.e. on Ω.