Principal Lyapunov exponents and principal Floquet spaces of positive random dynamical systems. I. General theory

This is the first of a series of papers concerned with principal Lyapunov exponents and principal Floquet subspaces of positive random dynamical systems in ordered Banach spaces. It focuses on the development of general theory. First, the notions of generalized principal Floquet subspaces, generalized principal Lyapunov exponents, and generalized exponential separations for general positive random dynamical systems in ordered Banach spaces are introduced, which extend the classical notions of principal Floquet subspaces, principal Lyapunov exponents, and exponential separations for strongly positive deterministic systems in strongly ordered Banach to general positive random dynamical systems in ordered Banach spaces. Under some quite general assumptions, it is then shown that a positive random dynamical system in an ordered Banach space admits a family of generalized principal Floquet subspaces, a generalized principal Lyapunov exponent, and a generalized exponential separation. We will consider in the forthcoming parts applications of the general theory developed in this part to positive random dynamical systems arising from a variety of random mappings and differential equations, including random Leslie matrix models, random cooperative systems of ordinary differential equations, and random parabolic equations.


Introduction
This is the first part of a series of papers. The series is devoted to the study of principal Lyapunov exponents and principal Floquet subspaces of positive random dynamical systems in ordered Banach spaces. This first part focuses on the development of general theory of principal Lyapunov exponents and principal Floquet subspaces of general positive random dynamical systems in ordered Banach spaces. The forthcoming part(s) of the series will concern the applications of the general theory developed in Part I to positive random dynamical systems arising from a variety of random mappings and differential equations, including random Leslie matrix models, random cooperative systems of ordinary differential equations, and random parabolic equations.
Lyapunov exponents play an important role in the study of asymptotic dynamics of linear and nonlinear random evolution systems. The study of Lyapunov exponents traces back to Lyapunov [16]. Oseledets in [26] obtained some important results on Lyapunov exponents for finite dimensional systems, which is called now the Oseledets multiplicative ergodic theorem. Since then, a huge amount of research has been carried out toward alternative proofs of the Osedelets multiplicative ergodic theorem and extensions of the Osedelets multiplicative theorem for finite dimensional systems to certain infinite dimensional ones (see [1], [10], [13], [14], [17], [25], [30], [31], [33], [35], and references therein). In the recent work [14], Lian and Lu studied Lyapunov exponents of general infinite dimensional random dynamical systems in a Banach space and established a multiplicative ergodic theorem for such systems.
The largest finite Lyapunov exponents (or top Lyapunov exponents) and the associated invariant subspaces of both deterministic and random dynamical systems play special roles in the applications to nonlinear systems. 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 its associated invariant subspace is one dimensional and is spanned by a positive vector (in such case, the invariant subspace is called the principal Floquet subspace). Principal Lyapunov exponents and principal Floquet subspaces are the analog of principal eigenvalues and principal eigenfunctions of elliptic and time periodic parabolic operators. Numerous works have also been carried out toward principal Lyapunov exponents and principal Floquet subspaces for certain positive deterministic as well as random dynamical systems in ordered Banach spaces, in particular, for deterministic and random dynamical systems generated by nonautonomous and random parabolic equations with bounded coefficients (see [6], [7], [8], [9], [18], [19], [21], [22], [23], [27], [37], [38], and references therein).
Many strongly positive deterministic as well as random dynamical systems in strongly ordered Banach space are shown to have principal Lyapunov exponents (and hence principal Floquet subspaces and entire positive orbits). Moreover, the so called exponential separations are admitted in such systems. For example, let (Z, (σ t ) t∈R ) be a compact uniquely ergodic minimal flow and X be a strongly ordered Banach space with the positive cone X + (see 2.3 for detail). Let Π = (Π t ) t≥0 , Π t : X × Z → X × Z be a skew-product semiflow over (Z, (σ t ) t∈R ), where Φ(t, z) ∈ L(X, X). If Π is strongly positive (i.e. Φ(t, z)x ∈ Int (X + ) for any t > 0, z ∈ Z, and x ∈ X + \{0}) and completely continuous (i.e., { Φ(t, z)B : z ∈ Z } is a relatively compact subset of X for any t > 0 and any bounded subset B of X), then there are λ 1 ∈ R, M, γ > 0, a subspace E(z) ⊂ X with E(z) = span {v(z)} for some v(z) ∈ Int (X + ), v(z) = 1, and a subspace F (z) ⊂ X with F (z) ∩ X + = {0} such that X = E(z) ⊕ F (z) for any z ∈ Z, E(z) and F (z) are continuous in z ∈ Z, and (i) Φ(t, z)E(z) = E(σ t z) for any t > 0 and z ∈ Z; (ii) Φ(t, z)F (z) ⊂ F (σ t z) for any t > 0 and z ∈ Z; (iii) lim t→∞ ln Φ(t, z)v(z) t = λ 1 ; (iv) Φ(t, z)w Φ(t, z)v(z) ≤ M e −γt for any w ∈ F (z) with w = 1, t > 0, and z ∈ Z (see [21], [29]). Here λ 1 and {E(z)} z∈Z are the principal Lyapunov exponent and principal Floquet subspaces of Π, respectively, and the property (iv) is referred to the exponential separation of Π. Note that the above results extend the classical Kreȋn-Rutman theorem for strongly positive and compact operators in strongly order Banach spaces to strongly positive and compact deterministic skew-product semiflows in strongly ordered Banach spaces. For a general positive random dynamical system, there may be no finite Lyapunov exponents (and hence no principal Lyapunov exponent in classical sense); if the top Lyapunov exponent is finite, its associated invariant subspace may not be one dimensional (and hence there is no principal Lyapunov exponent in the classical sense either). It is not known whether a general positive random dynamical system admits positive entire orbits and/or invariant subspaces spanned by positive vectors.
The objective of the current part of the series is to investigate the extent to which the principal Lyapunov exponents and principal Floquet subspaces theory for strongly positive and compact deterministic dynamical systems may be generalized to general positive random dynamical systems. The classical Kreȋn-Rutman theorem for strongly positive and compact operators in strongly ordered Banach spaces is extended to quite general positive random dynamical systems in ordered Banach spaces. In particular, the existence of entire positive orbits is shown without the assumption of strong positivity (see Theorem 3.5); the existence of one dimensional invariant measurable subspaces which are spanned by positive vectors and whose associated Lyapunov exponent is the largest (such invariant subspaces and the associated Lyapunov exponent are called generalized principal Floquet subspaces and generalized principal Lyapunov exponents, respectively, see Definition 3.2) is proved without the assumption of the existence of finite Lyapunov exponents (see Theorem 3.6), and the existence of a generalized exponential separation (see Definition 3.3) is proved too without the assumption of the existence of finite Lyapunov exponents (see Theorem 3.8).
In the forthcoming part(s) of this series we will study the applications of the general results established in this part to random Leslie matrix models, random cooperative systems of ordinary differential equations, and random parabolic equations.
The rest of the current part is organized as follows. In Section 2, we introduce standing notions and assumptions. We introduce the concepts of generalized principal Floquet subspaces, generalized principal Lyapunov exponents, and generalized exponential separations and state the main results of this part in Section 3. In Section 4, we present some preliminary materials to be used in the proofs of the main results, including some classical ergodic theorems and fundamental properties of Hilbert projective metric. We prove the main results in the last section.

Standing Notions and Assumptions
In this section, we introduce standing notions and assumptions. .
For a metric space Y , B(Y ) stands for the σ-algebra of all Borel subsets of Y .
When T = R we call a (measurable, metric) dynamical system a (measurable, metric) flow . To emphasize the situation when T = Z, we speak of (measurable, metric) discrete-time dynamical system. When we use the symbol "lim n→∞ " it is implied that n is considered for (perhaps sufficiently large) n ∈ N. Similarly, when we use the symbol "lim n→−∞ " it is implied that n is considered for (perhaps sufficiently large) negative integers n.

2.2.
Measurable Linear Skew-Product Semidynamical Systems. Let X be a real Banach space, with norm · . Let L(X) stand for the Banach space of bounded linear mappings from X into X. The standard norm in L(X) will be also denoted by · .
For a Banach space X, we will denote by X * its dual and by ·, · the standard duality pairing (that is, for u ∈ X and u * ∈ X * the symbol u, u * denotes the value of the bounded linear functional u * at u). Without further mention, we understand that the norm in X * is given by Let ((Ω, F, P), (θ t ) t∈T ) be a measurable dynamical system. By a measurable linear skew-product semidynamical system Φ = ((U ω (t)) ω∈Ω,t∈T + , • for each ω ∈ Ω and t ∈ T + , [ X ∋ u → U ω (t)u ∈ X ] ∈ L(X).
If the Banach space X is separable, by Pettis' theorem (see, e.g., [36, Theorem 1.1.6]), the measurability of the mapping [ (t, ω, u) → U ω (t)u ] is equivalent to the fact that for each u * ∈ X * the mapping For ω ∈ Ω, t ∈ T + and u * ∈ X * we define U * ω (t)u * by for any ω ∈ Ω and any t, s ∈ T + .
In case where the mapping For instance, if we assume that X is separable and reflexive, then X * is separable, hence, by Pettis' theorem, the measurability of the mapping [ (t, ω, u * ) → U * ω (t)u * ] is equivalent to the fact that for each u ∈ X the mapping , B(R))-measurable, which in its turn follows from the facts that the composition [ (t, ω) → (t, θ −t ω) → U θ−tω (t)u ] is (B(T + ) ⊗ F, B(X))-measurable and that the ·, · operation is continuous.
Since now till the end of the subsection we assume that ((U ω (t)) ω∈Ω,t∈T + , (θ t ) t∈T + ) is a measurable linear skew-product semidynamical system.
Let l be a positive integer. By a family of l-dimensional vector subspaces of X we understand a mapping E, defined on some Ω 0 ⊂ Ω with P(Ω 0 ) = 1, assigning to each ω ∈ Ω 0 an l-dimensional vector subspace E(ω) of X. Similarly, by a family of l-codimensional closed vector subspaces of X we understand a mapping F , defined on some Ω 0 ⊂ Ω with P(Ω 0 ) = 1, assigning to each ω ∈ Ω 0 an l-codimensional closed vector subspace F (ω) of X.
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→±∞ t∈T ln P (θ t ω) t = 0 P-a.s. on Ω 0 .

2.3.
Ordered Banach Spaces. Let X be a real Banach space, with norm · . By a cone in X we understand a closed convex set X + such that (C1) α ≥ 0 and u ∈ X + imply αu ∈ X + , and (C2) A pair (X, X + ), where X is a Banach space and X + is a cone in X, is referred to as an ordered Banach space.
The symbols ≥ and > are used in analogous way.
We say that u, v ∈ X + \ {0} are comparable, written u ∼ v, if there are positive numbers α, α such that αv ≤ u ≤ αv. The ∼ relation is clearly an equivalence relation. For a nonzero u ∈ X + we call the component of u, denoted by C u , the equivalence class of u, An ordered Banach space (X, X + ) is called strongly ordered if X + is solid. A cone X + in a Banach space X is called normal if there exists K > 0 such that for any u, v ∈ X satisfying 0 ≤ u ≤ v there holds u ≤ K v .
If X + is a normal cone we say that (X, X + ) is a normally ordered Banach space. In such a case, the Banach space X can be renormed so that for any u, v ∈ X, . Such a norm is called monotonic.
From now on, when speaking of a normally ordered Banach space we assume that the norm on X is monotonic.
For an ordered Banach space (X, X + ) denote by (X * ) + the set of all u * ∈ X * such that u, u * ≥ 0 for all u ∈ X + . The closed subset (X * ) + of X * is convex and satisfies (C1), however it need not satisfy (C2). Nevertheless, if the cone X + is total then (X * ) + satisfies (C2) (therefore is a cone).
It is a classical result that X + is normal if and only if (X * ) + is reproducing, and that X + is reproducing if and only if (X * ) + is normal, see e.g. [34, V.3.5].
Sometimes an ordered Banach space (X, X + ) is a lattice: any two u, v ∈ X have a least upper bound u ∨ v and a greatest lower bound u ∧ v. In such a case we write u + := u ∧ 0, u − := (−u) ∨ 0, and |u| := u + + u − . We have u = u + − u − for any u ∈ X.
An ordered Banach space (X, X + ) being a lattice is a Banach lattice if there is a norm · on X (a lattice norm) such that for any u, v ∈ X, if |u| ≤ |v| then u ≤ v . From now on, when speaking of a Banach lattice we assume that the norm on X is a lattice norm.
It is straightforward that in a Banach lattice the cone is normal and reproducing. Moreover, if (X, X + ) is a Banach lattice then (X * , (X * ) + ) is a Banach lattice, too (see [34, V.7.4]).
The reader is referred to forthcoming papers in the current series for a variety of examples of ordered Banach spaces and ordered Banach lattices.

Assumptions.
We list now assumptions we will make at various points in the sequel.
is a measurable linear skew-product semidynamical system on X covering an ergodic metric dynamical system (θ t ) t∈T on (Ω, F, P), satisfying the following: Similarly, we write is a measurable linear skew-product semidynamical system on X * covering an ergodic metric dynamical system (θ −t ) t∈T on (Ω, F, P), satisfying the following: Observe that if (A0) * (i) is satisfied and the measurability in the definition of Φ * holds then (A2) implies (A2) * .

Definitions and Main Results
In this section, we state the definitions and main results of the paper. We first state the definitions in 3.1, then recall an Oseledets-type Theorem proved in [14] in 3.2, and finally state the main results in 3.3. Throughout this section, we assume that ((U ω (t)) ω∈Ω,t∈T + , (θ t ) t∈T + ) is a measurable linear skew-product semidynamical system on a Banach space X covering (θ t ) t∈T .

3.1.
Definitions. In this subsection, we introduce the concepts of entire solutions and extend the notions of principal eigenfunction and exponential separation of strongly positive and compact operators. Throughout this subsection, we assume (A0)(i) and (A2).
The function constantly equal to zero is referred to the trivial entire orbit .
Entire orbits of Φ * are defined in a similar way.
Note that the notions of generalized principal Floquet subspaces and principal Lyapunov exponent are the extensions of principal eigenspaces and principal eigenvalues of strongly positive and compact operators.
is said to admit a generalized exponential separation if there are a family of generalized principal Floquet subspaces {Ẽ(ω)} ω∈Ω and a family of one-codimensional subspaces {F (ω)} ω∈Ω of X satisfying the following where the decomposition is invariant, and the family of projections associated with this decomposition is strongly measurable and tempered, for each ω ∈Ω. We say that {Ẽ(·),F (·),σ} generates a generalized exponential separation.
We remark that in general the generalized principal Lyapunov exponentλ associated to the generalized principal Floquet subspaces {Ẽ(ω)} ω∈Ω may be −∞. The limit in Definition 3.3 may not be uniform in ω ∈Ω. The generalized exponential separation is the extension of the classical exponential separation.
In the above, for t = −s for some s ∈ T + and u ∈ E i (ω) the symbol U ω (t)u stands In literature, λ i 's in the cases (2) and (3) are called Lyapunov exponents and E i (ω)'s are called the Oseledets spaces associated to λ i 's.

3.3.
Main results. We state the main results of the paper in this subsection. The first theorem is on the existence of entire positive orbits.
Theorem 3.5 (Entire positive orbits). Assume Φ is a continuous measurable linear skew-product semidynamical system satisfying (A0)(i), (A1)(i)-(iii) and (A2). If Theorem 3.4(2) or (3) occurs and X + is total then the set Ω 1 of those ω ∈ Ω 0 such that E 1 (ω) ∩ X + {0} has P-measure one, and for each ω ∈ Ω 1 there exists an entire positive orbit v ω : The above theorem shows the existence of an entire positive orbit of U ω for a.e. ω ∈ Ω without the assumption that U ω is strongly positive, which extends the principal eigenfunction theory for strongly positive and compact operators. Note that in general E 1 (ω) = span{v ω (0)} in the case that Theorem 3.4 (2) or (3) occurs.
Next theorem shows the existence of generalized Floquet subspaces and principal Lyapunov exponent and the uniqueness of entire positive orbits.
Observe that U ω (t)Ẽ 1 (ω) =Ẽ 1 (θ t ω), for any ω ∈Ω 1 and any t ∈ T + . Since For ω ∈Ω 1 , the function w ω : T → X + is a nontrivial entire orbit of U ω . By Theorem 3.6(2), a nontrivial entire orbit of U ω is unique up to multiplication by positive scalar, which extends the fundamental property on the existence and uniqueness of positive eigenvectors of compact u 0 -positive linear operators (see [11] and [12]). Note that Theorem 3.4(1) may occur under the assumptions of Theorem 3.6.

Preliminaries
In this section, we present some preliminary materials for the use in the proofs of the main results, including Birkhoff Ergodic Theorem, Kingman Subadditive Ergodic Theorem, Hilbert projective metric in ordered Banach spaces and basic properties, and oscillation ratio, Birkhoff contraction ratio, and projective diameter of positive operators in ordered Banach spaces and basic properties.  If ((Ω, F, P), (θ n ) n∈Z ) is ergodic then f av is constantly equal to Ω f dP.

4.2.
Hilbert Projective Metric. Throughout this subsection, we assume that (X, X + ) is an ordered Banach space. We recall the concept of Hilbert projective metric and present some basic properties.  It should be noted that for comparable u, v ∈ X + \ {0} we have the following alternative: The following lemma follows easily. ( (4) For any two u, v ∈ X + \ {0}, d(u, v) = 0 implies the existence of α > 0 such that v = αu.
Lemma 4.6. Assume that X + is normal. Then for any u, v ∈ X + , u ∼ v, with If (X, X + ) is a Banach lattice then for any Proof. First, let M > 1 be such that By (C2), α k ≤ α k . Without loss of generality, we may assume that α k → α and α k → α as n → ∞. Then 0 < α ≤ α and αv ≤ u ≤ αv.
Assume that k l → ∞ is such that for any ǫ > 0, and hence
Proof. By (A3), for any ω ∈ Ω and u ∈ X + \ {0} we have It suffices now to apply the definition of d(·, ·) and Lemma 4.5(3).  Similarly, for any ω ∈ Ω, if u * , v * ∈ (X * ) + \{0} are such that u * ∼ v * but v * = αu * for any positive real α then Proof. For u, v as in the assumption we have which gives that The first inequality follows immediately. The proof of the second inequality is similar.
Proof. By Lemma 4.11, it suffices to prove that Then Ω k is the inverse image of Similarly, it can be proved that [ ω → m(U ω (1)u/U ω (1)v) ] is (F, B(R))-measurable, for fixed u and v in X + \ {0}.
In this subsection, we investigate the existence of generalized principal Floquet subspaces and principal Lyapunov exponents and prove Theorems 3.6 and 3.7. Throughout this subsection, we assume additionally (A0)(ii) and (A3).
Before proving Theorems 3.6 and 3.7, we first prove some propositions.
Proof. (1) Consider first the discrete-time case.
A proof of Part (2) is similar: we find an invariant setΩ 1 ) = 1 having the corresponding properties.
Proof of Theorem 3.6.
The next result gives the formula for the projection of X onF 1 (ω) alongẼ 1 (ω).
Proof of Theorem 3.8. (1) The strong measurability follows, through the formula (5.18), by the measurability of w and w * .
(2) It is a consequence of Lemma 5.8.

5.4.
Monotonicity. In this subsection, we prove Theorem 3.9, which shows that the monotonicity of two measurable skew-product semiflows at some time implies the monotonicity of the associated generalized principal Lyapunov exponents. We assume Proof of Theorem 3.9. LetΩ ω (nt 0 )u n =λ (2) 1 .