Principal Floquet subspaces and exponential separations of type II with applications to random delay differential equations

This paper deals with the study of principal Lyapunov exponents, principal Floquet subspaces, and exponential separation for positive random linear dynamical systems in ordered Banach spaces. The main contribution lies in the introduction of a new type of exponential separation, called of type II, important for its application to nonautonomous random differential equations with delay. Under weakened assumptions, the existence of an exponential separation of type II in an abstract general setting is shown, and an illustration of its application to dynamical systems generated by scalar linear random delay differential equations with finite delay is given.


Introduction
This paper continues the study of the existence of principal Lyapunov exponents, principal Floquet subspaces and generalized exponential separations for positive random linear skew-product semiflows in ordered Banach spaces. In particular, the concept of generalized exponential separation of type II is introduced as a natural modification of the classical concept, to later show the applicability of this new theory in the context of nonautonomous functional differential equations with finite delay.
The largest finite Lyapunov exponent (or top Lyapunov exponent) and its associated invariant subspace play a relevant role in this theory. Classically, the top finite Lyapunov exponent of a positive deterministic or random dynamical system in an ordered Banach space is called the principal Lyapunov exponent if the associated invariant family of subspaces, where this Lyapunov exponent is reached, is one-dimensional and spanned by a positive vector. In this case the invariant subspace is called the principal Floquet subspace.
The exponential separation theory was initiated for positive discrete-time deterministic dynamical systems by Ruelle [31], and developed later by Poláčik and Tereščák [28,29]. Given a strongly ordered Banach space X and θ : Ω → Ω a homeomorphism of a compact set Ω, if Ω × X → Ω × X, (ω, u) → (θ(ω), T ω u) defines a vector bundle map with T ω compact and strongly positive for each ω ∈ Ω, then it admits a continuous decomposition X = E(ω) ⊕ F (ω) where E(ω) is the principal Floquet subspace, F (ω) does not contain any strictly positive vector and the bundle map exhibits an exponential separation on the sum. This statement was generalized by Shen and Yi [35] to continuous-time deterministic (topological) skew-product semiflows R + ×Ω×X → Ω×X, (t, ω, u) → (θ t ω, U ω (t) u), where U ω (t) is strongly positive for each ω ∈ Ω and t > 0. In a typical instance of application, Ω is the translation hull of the coefficients of some linear differential equation, that is, the closure, in an appropriate topology, of the set of all time-translates of the coefficients, and U ω (t) is the solution operator of the equation. Applications and extensions of this theory in the context of nonautonomous ordinary and parabolic partial differential equations can be found in Húska and Poláčik [11], Húska [10], Húska et al. [12], and Novo et al. [23,24], among other references. The theory of principal Floquet spaces and exponential separations was developed further by Mierczyński and Shen [17,18]. Among others, they consider random families of parabolic linear partial differential equations whose coefficients are evaluated along the trajectories of a measurable dynamical system on a probability space: a random family is of such a type that it is embedded into a continuous deterministic (nonautonomous) family of linear equations which in its turn generates a (topological) skew-product flow exhibiting an exponential separation.
Novo et al. [23] introduced a modification of the notion of exponential separation. A linear skew-product semiflow R + × Ω × X → Ω × X, (t, ω, u) → (θ t ω, U ω (t) u) is considered and the strong monotonicity condition is substituted by the following dichotomy behaviour: there exist times 0 < t 1 < T such that U ω (t) is a compact operator for t ≥ T − t 1 and ω ∈ Ω, and for each vector u ≥ 0 either U ω (t 1 ) u = 0 or U ω (t 1 ) u ≫ 0. Under these assumptions, the existence of a continuous decomposition X = E(ω) ⊕ F (ω) is proved, where E(ω) is the principal Floquet subspace and the semiflow exhibits an exponential separation on the sum, but now F (ω) ∩ X + is not void and contains those positive vectors u satisfying U ω (t 1 ) u = 0. This dynamical behaviour is called exponential separation (or continuous separation) of type II, implicity referring to the classical concept as exponential separation of type I. Novo et al. [24], Calzada et al. [5] and Obaya and Sanz [25,26] show the importance of the exponential separation of type II in the study of linear and nonlinear nonautonomous functional differential equations with finite delay.
In the case where the coefficients of the linear differential equation are driven by trajectories of a measurable flow θ on a probability space (Ω, F, P), the natural setting is that of positive measurable linear skew-product semiflows: U ω (t) is a positive linear operator depending measurably on ω ∈ Ω. The definition of the generalized exponential separation (of type I) is almost the same as the definition of the exponential separation for topological semiflows, the only difference being that E(ω) and F (ω) now depend measurably on ω. Also, the generalized principal Lyapunov exponent is the largest Lyapunov exponent for P-a.e. ω ∈ Ω.
Arnold et al. in [4] were the first to prove the existence of generalized exponential separation for discrete-time positive random dynamical systems generated by random families of positive matrices. Later, Mierczyński and Shen [19] provided the assumptions required for general random positive linear skew-product semiflows (with both discrete and continuous time) in order to admit generalized principal Floquet subspaces and generalized exponential separation of type I (in contrast to [17,18], no embedding into topological semiflows was used in the proofs). The application of this theory to a variety of random dynamical systems arising from Leslie matrix models, cooperative linear ordinary differential equations and linear parabolic partial differential equations can be found in Mierczyński and Shen [20,21]. In particular, the existence of generalized principal Floquet subspaces for random skew-product semiflows generated by cooperative families of delay differential equations is also obtained in [21].
In this paper, the positivity conditions satisfied by the linear random dynamical systems are weakened, in order to assure the existence of generalized principal Floquet subspaces and the existence of generalized exponential separations of type II.
The structure and main results of the paper is as follows. In Section 2 the notions and assumptions used throughout the paper are introduced. In particular, X will be an ordered separable Banach space with dual X * separable and positive cone X + normal and reproducing. Conditions of integrability and positivity for the random linear skew-product semiflow are imposed, and a new focusing assumption is considered. More precisely, there is a positive time T > 0 such that if u ∈ X + and ω ∈ Ω then U ω (T ) u = 0 or U ω (T ) u is strictly positive and satisfies a classical focusing inequality in the terms stated by Mierczyński and Shen [19]. For simplicity we fix the scale T = 1 throughout the paper. From these assumptions, the integrability, positivity and an alternative focusing property for the measurable dual skew-product semiflow are obtained.
Under the new focusing condition, in Section 3 the existence of a family of generalized principal Floquet subspaces is shown. Section 4 is devoted to the introduction of the new concept of generalized exponential separation of type II and the proof of the existence under the previously considered assumptions. Finally, Section 5 illustrates the application of the theory to random dynamical systems generated by scalar linear random delay differential equations with finite delay.

2.1.
Measurable linear skew-product semidynamical systems. We consider a separable Banach space X such that its dual X * is separable.
The family of projections associated with the decomposition E(ω)⊕F (ω) = X is called strongly measurable if for each u ∈ X the mapping [ Ω 0 ∋ ω → P (ω)u ∈ X ] is (F, B(X))-measurable.
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 .
(2.5) 2.2. Ordered Banach spaces. Let X be a Banach space with norm · . We say that X is an ordered Banach space if there is a closed convex cone, that is, a nonempty closed subset X + ⊂ X satisfying . Then a partial ordering in X is defined by The cone X + is said to be reproducing if X + − X + = X. The cone X + is said to be normal if the norm of the Banach space X is semimonotone, i.e., there is a positive constant k > 0 such that 0 ≤ u ≤ v implies u ≤ k v . In such a case, the Banach space can be renormed so that for any . Such a norm is called monotone.
For an ordered Banach space X denote by (X * ) + the set of all u * ∈ X * such that u, u * ≥ 0 for all u ∈ X + . The set (X * ) + has the properties of a cone, except that (X * ) + ∩ (−(X * ) + ) = {0} need not be satisfied (such sets are called wedges).
If (X * ) + is a cone we call it the dual cone. This happens, for instance, when X + is total (that is, X + − X + is dense in X, which in particular holds when X + is reproducing and this will be one of our hypothesis).
Nonzero elements of X + (resp. of (X * ) + ) are called positive. We say that two positive vectors u, v ∈ X + \ {0} are comparable, written u ∼ v, if there are positive numbers, α, α, such that α v ≤ u ≤ α v. For a nonzero vector u ∈ X + we call the component of u i.e., the equivalence class of u. We now recall the concept of the Hilbert projective metric. For the next result, see [19,Lemma 4.6].
Lemma 2.2. Assume that X + is normal. Then for any u, v ∈ X + , u ∼ v, with
A simple consequence is the following. As usual, we denote by ⌊t⌋ the integer part of the real number t. Lemma 2.5. Assume (A3). Then for any ω ∈ Ω, any t ≥ 0 and any nonzero u * ∈ (X * ) + there holds U * ω (t) u * = 0.

Generalized principal Floquet subspaces
The main result of the paper is the proof of the existence, under the new focusing assumption (A3), of a new version of the concept of generalized exponential separation for a measurable linear skew-product semidynamical system. We will preserve the structure and follow the arguments in [19], just introducing the modifications which are required in this new situation. This section will be devoted to the proof of the existence of a family of generalized principal Floquet subspaces.
As stated before, X is an ordered separable Banach space such that X * is separable, with positive cone X + normal and reproducing, and recall that, from Lemma 2.4, once (A3) is assumed, assumption (A3)* holds.
, for any ω ∈ Ω and any t > 0, λ is called the generalized principal Lyapunov exponent of Φ associated to the gen- We recall the definitions of oscillation ratio, Birkhoff contraction ratio and projective diameter, needed in the proofs of our main theorems. See Definition 2.1 and Subsection 2.2 for previous definitions and notations. (1) The oscillation ratio of U ω (1) is defined as (2) The Birkhoff contraction ratio of U ω (1) is defined as The functions p * , q * and τ * for the dual Φ * are defined in an analogous way.
As in Lemma 4.13 of [19], taking into account (3.1) and changing the dense countable subsets of the proof by (v j + e/n) j,n , the following result is proved.
Next we consider the set of positive vectors where C e denotes the component of e defined in (2.6).
Lemma 3.6. Assume (A3). Then d is a metric on Σ, and (Σ, d) is a complete metric space.
Proof. As stated in Eveson [7], since X + is normal (hence almost Archimedean) and the norm on X is monotone, any two vectors u, v of Σ ⊂ C e are regularly comparable, i.e., they are comparable and m(u/v) ≤ 1 ≤ M (u/v) (see Definition 2.1 for notation). Consequently, from [7, Theorem 1.2.1] the projective distance d defined in (2.7) is a metric on Σ and the metric space (Σ, d) is complete.
Proof. First we check that U ω (t) e ∈ C e for each t ≥ 0. From (A3) we know that U ω (1) e = 0, and then (2 In a recursive way we obtain that U ω (n) e ∈ C e for each ω ∈ Ω and n ∈ N .
and consequently, from U θ−1ω (t + 1) e = U ω (t) U θ−1ω (1) e and the monotonicity property (A2) we deduce that α e ≤ β U ω (t) e and β e ≥ α U ω (t) e i.e., U ω (t) e ∈ C e , as claimed. Finally, if u ∈ C e , it is immediate to check from U ω (t) e ∈ C e and (A2) that U ω (t) u ∈ C e , which finishes the proof.
As a consequence of this lemma, under assumption (A3), the map is well defined for each t ≥ 0 and ω ∈ Ω. Moreover, it is continuous for the , as can be easily deduced from Lemmas 4.5 and 4.9 of [19]. Furthermore, as a consequence of (2.2) We omit the proof of the next result which is completely analogous to the first part of the proof of Proposition 5.3 of [19]. It follows from (3.4), the properties of the distance, the definition of q (see Definition 3.2) and Lemma 3.3. As before, we denote by ⌊t⌋ the integer part of the real number t.
As a consequence, the following contraction property follows, whose proof is also omitted because it follows the arguments of Proposition 5.4 of [19]. Proposition 3.9. Under assumption (A3), let I := Ω ln q dP < 0. Then, there is Proposition 3.9(1) ensures that for any ω ∈Ω 1 the following exists where the limit is taken in d. Since, by Lemma 3.6, (Σ, d) is a complete metric space, w(ω) belongs to Σ. Further, it follows from Lemma 2.2 that the above limit can be taken in the X-norm. Moreover, since the functions [ω → U θ−nω (n) e] are (F, B(X))-measurable, the function w :Ω 1 → X is measurable. The next theorem shows the existence of generalized Floquet subspaces and principal Lyapunov exponent, and the uniqueness of entire positive orbits, which is the equivalent result to Theorem 3.6 of [19] for the new focusing.
is a positive entire orbit of U ω , unique up to multiplication by a positive scalar; for each ω ∈ Ω 1 ; is a family of generalized principal Floquet subspaces, andλ 1 is the generalized principal Lyapunov exponent. Proof.
(2) First notice that if t ≤ 0 and ω ∈Ω 1 Now we check that w ω is an entire orbit, i.e., w ω (s + t) = U θsω (t) w ω (s) for each s ∈ R and t ≥ 0. If t ≥ 0 and s ≥ 0 it is immediate. If t ≥ 0, s ≤ 0 and t + s ≥ 0, from (2.2) and (3.10) Next, since θ t+s ω = θ t (θ s ω), if t ≥ 0, s ≤ 0 and t + s ≤ 0, from and w ω is an entire positive orbit, as claimed. Finally we check the uniqueness. Let v ω be another entire positive orbit of U ω .
, and since v ω (s) ∈ X + \{0} for each s ∈ R, the focusing condition (A3) yields v ω (s) ∈ C e for each s ∈ R. Therefore, and from Proposition 3.9(1) and the definition of w(ω) it follows that for each t ∈ R and ω ∈Ω 1 . Since, by Lemma 3.6, d is a metric on Σ, v i.e., they coincide up to multiplication by a positive scalar, as stated.
Remark 3.12. Note that as in Proposition 2.2 of Mierczyński and Shen [21], from the existence of the family of generalized Floquet subspaces and the generalized principal Lyapunov exponentλ 1 obtained in the previous Theorem 3. 10, it follows that for each ω ∈ Ω 1 .
(2) The fact that w * ω is a positive entire orbit follows along the lines of the proof of Theorem 3.10(2).
We check the uniqueness. Let v * ω be another entire positive orbit of U * ω . It follows from Proposition 3.15(1) and the definition of w * that for each t ∈ R and ω ∈Ω * 1 . By Lemma 3.13, for s ≥ 1, and by the counterpart of Lemma 3.6, (C * i.e., they coincide up to multiplication by a positive scalar, as stated. (3) The proof goes along the lines of the proof of Theorem 3.10(3).

Generalized exponential separation
As stated before, X is an ordered separable Banach space such that X * is separable, with positive cone X + normal and reproducing, and recall that, once (A3) is assumed, assumptions (A2), (A2)* and (A3)* hold.
In this section we will prove the existence of a generalized exponential separation of type II, now introduced and important for cases in which the previous concept of generalized exponential separation does not apply, as measurable linear skew-product semidynamical systems induced by delay differential equations.
Note that the only difference with the definition of generalized exponential separation given in [19] is that in this case F (ω) contains those positive vectors u > 0 for which U ω (1) u = 0 because U ω (1) is not assumed to be injective.
We omit the proof of the next result, based in Lemma 5.10 of [19] with the corresponding modifications due to the different definition of the maps (3.3) and (3.5). See Definition 3.2 for the oscillation of two vectors. subset Ω 2 ⊂ Ω 1 ∩ Ω * 1 of full measure P( Ω 2 ) = 1, with the property that for each 0 > J > Ω ln p dP and each ω ∈ Ω 2 , there is a C 5 (ω, J) > 0 such that whenever u ∈ Σ = C e ∩ S 1 (X + ) and t ≥ 1. for each ω ∈ Ω 1 ∩ Ω * 1 . Proof. We prove the result for t → ∞, the other limit is completely analogous. Sinceλ 1 =λ * 1 , from inequality (3.25) we deduce that lim sup Assume now on the contrary to the assertion of the lemma that there is a sequence t n ↑ ∞ such that lim n→∞ 1 t n ln w(θ tn ω), w * (θ tn ω) = a < 0 .
We include part of the proof of the next result, although similar to the proof of Proposition 5.11 of [19], to remark the differences derived from the new assumption (A3) and the fact that we do not have a Banach lattice but an ordered separable Banach space with positive cone X + normal and reproducing.
The next theorem shows the existence of a generalized exponential separation of type II. We maintain the notation of the previous results.

Scalar linear random delay differential equations
This section is devoted to show the applications of the previous theory to random dynamical systems generated by scalar linear random delay differential equations of the form Ω ∋ ω → a(ω) ∈ R ∈ L 1 (Ω, F, P), and Ω ∋ ω → ln + 1 0 |b(θ r ω)| q dr ∈ R ∈ L 1 (Ω, F, P).
In order to define the measurable linear skew-product semidynamical system we are going to deal with, for each u = (u 1 , u 2 ) ∈ X and ω ∈ Ω we consider the initial value problem Its solution will be denoted by z(t, ω, u).
Since a ∈ L 1 (Ω, F, P) and the measure P is invariant, and an application of Fubini's theorem gives that the map for ω ∈ Ω 0 ⊂ Ω, invariant set of full measure. Then we can put the value of a(ω) for ω ∈ Ω \ Ω 0 to be equal to zero to obtain (5.3) for all ω ∈ Ω. Analogously, by changing the value of b to zero in a set of null measure, the map for all ω ∈ Ω. Therefore, for a fixed ω ∈ Ω and 0 ≤ t ≤ 1 the system (5. In a recursive way we obtain the formula for z(t, ω, u) for any t ∈ [−1, ∞).
Proof. Relation (2.1) is immediate and (2.2) follows from (5.8). Once that this cocycle property is shown, to prove that U ω (t) ∈ L(X) for t ≥ 0, it is enough to check that U ω (t) is a bounded operator for t ∈ [0, 1] and ω ∈ Ω, which is a consequence of Lemma 5.3 because ) for t ∈ [0, 1], which finishes the proof.
In order to show that Φ = ((U ω (t)) ω∈Ω,t∈R + , (θ t ) t∈R ) is a measurable linear skew-product semidynamical system we start with the following auxiliary lemma.
We finish this section showing that the skew-product semidynamical system, generated by the family of scalar linear random delay differential equations of the form (5.1), satisfies all the requirements for the existence of a generalized exponential separation. Notice that according to Remark 2.6 we can take time T = 2 instead of 1 to check conditions (A1) and (A3).
Before proceeding we formulate and prove the following auxiliary Lemma 5.6. Let x 1 , . . . , x n > 0. Then ln + x i + ln n.
Proof. Applying the Jensen inequality to the convex function f (x) = x ln x we obtain For i such that ln x i ≤ 0 we have whereas for i such that ln x i > 0 we have ln + x i + ln n.
As the right-hand side of the above inequality is nonnegative, we obtain the desired result.
Proof. Since for ω ∈ Ω and u ∈ X fixed the mapping R + ∋ t → U ω (t) u ∈ X is easily seen to be continuous, the fact that the mapping R + × Ω × X ∋ (t, ω, u) → U ω (t) u ∈ X is (B(R + ) ⊗ F ⊗ B(X), B(X))-measurable follows from Proposition 5.4, Lemma 5.5 and Aliprantis and Border [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.
To sum up, we have proved the following.