Principal Lyapunov exponents and principal Floquet spaces of positive random dynamical systems. III. Parabolic equations and delay systems

This is the third part in a series of papers concerned with principal Lyapunov exponents and principal Floquet subspaces of positive random dynamical systems in ordered Banach spaces. The current part focuses on applications of general theory, developed in the authors' paper"Principal Lyapunov exponents and principal Floquet spaces of positive random dynamical systems. I. General theory,"Trans. Amer. Math. Soc. 365 (2013), pp. 5329-5365, to positive continuous-time random dynamical systems on infinite dimensional ordered Banach spaces arising from random parabolic equations and random delay systems. It is shown under some quite general assumptions that measurable linear skew-product semidynamical systems generated by random parabolic equations and by cooperative systems of linear delay differential equations admit measurable families of generalized principal Floquet subspaces, and generalized principal Lyapunov exponents.


Introduction
This is the third part of a series of several 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.
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 the associated invariant family of subspaces corresponding to it consists of one-dimensional subspaces spanned by a positive vector (in such case, invariant subspaces are called the principal Floquet subspaces). For more on those subjects see [23].
In the first part of the series, [23], we introduced the notions of generalized principal Floquet subspaces, generalized principal Lyapunov exponents, and generalized exponential separations, which extend the corresponding classical notions. The classical theory of principal Lyapunov exponents, principal Floquet subspaces, and exponential separations for strongly positive and compact deterministic systems is extended to quite general positive random dynamical systems in ordered Banach spaces.
In the present, third part of the series, we consider applications of the general theory developed in [23] to positive random dynamical systems arising from random parabolic equations and systems of delay differential equations. To be more specific, let ((Ω, F, P), θ t ) be an ergodic metric dynamical system. We consider a family, indexed by ω ∈ Ω, of second order partial differential equations where s ∈ R is an initial time and D ⊂ R N is a bounded domain with boundary ∂D, complemented with boundary condition B θtω u = 0, t > s, x ∈ ∂D, (1.2) where a ij (ω, x) ∂u ∂x j + a i (ω, x)u ν i + d 0 (ω, x)u (Robin).
Among others, we obtain the following results. We remark that (1)-(3) are analogs of principal eigenvalue and principal eigenfunction theory for elliptic and periodic parabolic equations. Our main assumptions on (1.1)+(1.2) are the boundedness of a ij , a i , b i and d 0 . No boundedness of a 0 is assumed. The results of the current paper hence extend those corresponding ones in [22] (it is assumed in [22] that c 0 is also bounded). In addition to the cooperative assumption, our main assumptions on (1.3) are the irreducibility of B(ω) or the positivity of B(ω). Such assumptions are also used in [26]. No boundedness of A(ω) and B(ω) is assumed in the current paper and the results of the current paper extend those in [26] and [32] for cooperative systems of delay differential equations.
It should be pointed out that the generalized principal Lyapunov exponents in (2) may be −∞. In such a case, when generalized exponential separation holds, the (nontrivial) invariant measurable decomposition associated with the generalized exponential separation is essentially finer than the (trivial) decomposition in the Oseledets multiplicative ergodic theorem.
The results obtained in this paper would have important applications to the study of asymptotic dynamics of nonlinear random parabolic equations and systems of random delay differential equations.
The rest of this paper is organized as follows. First, for the reader's convenience, in Section 2 we recall some notions, assumptions, definitions, and main results established in Part I ( [23]). We then consider random systems arising from parabolic equations and cooperative systems of delay differential equations in Sections 3 and 4, respectively.

General Theory
In this section, we recall some general theory established in part I to be applied in this paper. To do so, we first introduce some notions, assumptions, and definitions introduced in part I. Then we recall some of the main results in part I.

Notions, assumptions, and definitions
In this subsection, we introduce some notions, assumptions, and definitions introduced in part I. The reader is referred to part I [23] for detail.
If f is a real function defined on a set Y , we define its nonnegative (resp. nonpositive) part f + (f − ) as For a metric space (Y, d), B(y; ǫ) denotes the closed ball in Y centered at y ∈ Y and with radius ǫ > 0. Further, B(Y ) stands for the σ-algebra of all Borel subsets of Y .
A probability space is a triple (Ω, F, P), where Ω is a 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.
For a Banach space X, with norm · , 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 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).
(ii) For any ω ∈ Ω the mapping is continuous. To prove (ii), fix ω ∈ Ω and T > 0 and observe that for any u ∈ X the set { U ω (t)u : t ∈ [0, T ] } is bounded. Hence, by the Uniform Boundedness Theorem, the set { U ω (t) : t ∈ [0, T ] } is bounded (by M > 0, say). Take a sequence (t n ) ∞ n=1 ⊂ [0, T ] convergent to t and a sequence (u n ) ∞ n=1 ⊂ X convergent to u. We estimate which goes to 0 as n → ∞.
By a cone in a Banach space X we understand a closed convex set X + such that • α ≥ 0 and u ∈ X + imply αu ∈ X + , and 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 an analogous way.
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 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).
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.
For application purposes, we give some examples of Banach lattices.
is a Banach lattice, and the norm · p is a lattice norm. The dual cone in L p (D) * = L q (D), where 1 p + 1 q = 1, equals L q (D) + .
We introduce now our assumptions.
Definition 2.1 (Entire positive orbit). For ω ∈ Ω, by an entire positive orbit of U ω we understand a mapping v ω : R → X + such that v ω (s + t) = U θsω (t)v ω (s) for any s ∈ R and t ∈ R + . The function constantly equal to zero is referred to as the trivial entire orbit.
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.
Observe that if {Ẽ(ω)}ω ∈Ω is a family of generalized principal Floquet subspaces of ((U ω (t)) ω∈Ω,t∈R + , (θ t ) t∈R ), then for any ω ∈Ω, v ω (·) is an entire positive orbit, where In the literature on random linear skew-product dynamical systems the concept of the top (or the largest ) Lyapunov exponent is introduced. It can be defined either as the largest exponential growth rate of the norms of the individual vectors (in such a case, when Φ has a family of generalized principal Floquet subspaces then the generalized principal Lyapunov exponent is, by definition, the top Lyapunov exponent), or as the exponential growth rate of the norms of the operators. These definitions are equivalent, however we have been unable to locate a concise proof in the existing literature. This is the reason why we decided to formulate and prove the result below (the proof is patterned after the proof of [13, Theorem 2.2], in the light of the first and second remarks on p. 528 of [13]. Proposition 2.2. Assume that Φ = ((U ω (t)), (θ t )) has a family of generalized principal Floquet subspaces, with the principal Lyapunov exponentλ. Assume moreover (C1)(i). Then for any ω ∈Ω, whereΩ is as in the Definition 2.2.
Proof. We start by proving that for any ω ∈Ω. Fix some ω ∈Ω and λ >λ, and define functions p n : X → [0, ∞), n = 1, 2, . . . , and p : For m = 1, 2, . . . put The sets W m are closed and their union equals the whole of X (by Definition 2.2(iv)). By the Baire theorem, there is m 0 ∈ N such that W m0 has nonempty interior. In other words, there exist v ∈ X and ǫ > 0 such that B(v; ǫ) ⊂ W m0 . From this it follows that for all w ∈ X with w ≤ ǫ and all n = 1, 2, . . . . In particular, by taking w = 0 we have that U ω (n)v ≤ m 0 e λn for all n. By the triangle inequality, for all w ∈ X with w ≤ ǫ and all n = 1, 2, . . . . As λ >λ is arbitrary, we have that Definition 2.3 (Generalized exponential separation). Φ = ((U ω (t)), (θ t )) admits a generalized exponential separation if it has 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 for any ω ∈Ω, where the decomposition is invariant, and the family of projections associated with this decomposition is strongly measurable and tempered, We say that {Ẽ(·),F (·),σ} generates a generalized exponential separation.
We end this subsection with the following proposition which follows from the Oseledets-type theorems proved in [12] (we do not assume (C0) or (C2) now).
be a measurable linear skew-product semidynamical system satisfying (C1)(i)-(iii). Then there exist: Moreover, if λ 1 > −∞ then there are a measurable family {E 1 (ω)} ω∈Ω0 of vector subspaces of finite dimension, and a family {F 1 (ω)} ω∈Ω0 of closed vector subspaces of finite codimension such that where the decomposition is invariant, and the family of projections associated with this decomposition is strongly measurable and tempered, Proof. See [23, Theorem 3.4, and (3.1) on p. 5342].

General theorems
In this subsection, we state some general theorems, most of which are established in part I. The first theorem is on the existence of entire positive orbits.
Next theorem shows the existence of generalized Floquet subspaces and principal Lyapunov exponent.

Linear Random Parabolic Equations
In this section, we consider applications of the general results stated in Section 2 to linear random parabolic equations. Let ((Ω, F, P), (θ t ) t∈R ) be an ergodic metric dynamical system, with P complete. Consider (1.1)+(1.2), that is, a family, indexed by ω ∈ Ω, of second order partial differential equations, where s ∈ R is an initial time and D ⊂ R N is a bounded domain with boundary ∂D, complemented with boundary condition Above, ν = (ν 1 , . . . , ν N ) denotes the unit normal vector pointing out of ∂D. When d 0 ≡ 0 in the Robin case, B ω u = 0 is also referred to as the Neumann boundary condition.
In addition, we consider also the adjoint problem to (3.1)+(3.2), that is, where t ∈ R is a final time, complemented with boundary condition where B * ω = B ω in the Dirichlet boundary conditions case, or in the Robin case.
When we want to emphasize that (3.1) Throughout the present section, · stands for the norm in L 2 (D) or for the norm in L(L 2 (D)), depending on the context. Sometimes we use summation convention. For example, we can write (3.1) as When speaking of properties satisfied by points in D, we use the expression "for a.e. x ∈ D" to indicate that the N -dimensional Lebesgue measure of the set of points not satisfying the property is zero. Similarly, when speaking of properties satisfied by points in ∂D, we use the expression "for a.e. x ∈ ∂D" to indicate that the (N − 1)-dimensional Lebesgue measure of the set of points not satisfying the property is zero. The expressions "for a.e. (t, x) ∈ R × D," "for a.e. (t, x) ∈ R × ∂D" are used in an analogous way.

Measurable linear skew-product semiflows
In this subsection, we give a sketch of the existence theory for (weak) L 2 (D)-solutions of (3.1)+(3.2) (or of (3.3)+(3.4)). It is an appropriate modification of the proof presented in the authors' monograph (PA0) (Boundary regularity) D ⊂ R N is a bounded domain with Lipschitz boundary ∂D.
In the case of Robin boundary conditions the function d 0 : (i) (Boundedness of second and first order terms) For each ω ∈ Ω the functions In the Robin case, for each ω ∈ Ω the functions (ii) (Local boundedness of zero order terms) For each ω ∈ Ω, c 0 (ω, ·) ∈ L ∞ (D). Moreover, there are (F, B(R))-measurable functions c For ω ∈ Ω define functions a ω ij : From now on we assume that (PA0) through (PA3) are satisfied. For any s ∈ R, T > 0 and ω ∈ Ω, the restriction For any s ∈ R, M > 0 and T > 0 we write Y s,M,T for the closure of { a ω s,T : ω ∈ Ω s,M,T } in the weak-* topology of L ∞ ((s, complemented with boundary conditions where Bãu = u in the Dirichlet case, or in the Robin case. Recall that, ifd 0 ≡ 0 in the Robin case, Bãu = 0 is also referred to as the Neumann boundary condition. To emphasize the dependence of the equation on the parameterã we write (3.7)ã+(3.8)ã. Let V be defined as follows wherev := dv/dt is the time derivative in the sense of distributions taking vales in V * (see [5, Chapter XVIII] for definitions). Forã ∈ Y s,M,T denote by Bã = Bã(t, ·, ·) the bilinear form on V associated withã, (3.12) in the Dirichlet and Neumann boundary condition cases, and for all v ∈ V and φ ∈ D([s, s + T )), where D([s, s + T )) is the space of all smooth real functions having compact support in [s, s + T ).
Our next assumptions will guarantee continuous dependence of solutions on parameters.

(3.15)
Indication of proof. The proof goes by appropriately rewriting the proofs of Propositions 2.1.6 through 2.1.8 in [22].

Proof. See [22, Proposition 2.3.2].
We define From now on, we assume additionally that (PA4) is satisfied.
is (B(R + ) ⊗ F ⊗ B(L 2 (D)), B(L 2 (D)))-measurable. In order to check that Φ * is indeed the dual of Φ, observe that We will call Φ as above the measurable linear skew-product semiflow on L 2 (D) generated by (3.1)+(3.2). The above construction of the measurable linear skew-product semiflow on L 2 (D), as well as its dual, can be repeated for the case when the zero-order term c 0 (·, ·) is put to be constantly equal to zero, that is, for the problem complemented with boundary condition where B ω is the same as in (3.2), and its adjoint complemented with boundary condition where B * ω is the same as in (3.4). For ω ∈ Ω, s ∈ R and u 0 ∈ L 2 (D) let u 0 (·; s, ω, u 0 ) stand for the global weak solution of (3.18) ω +(3.19) ω satisfying the initial condition u(s) = u 0 . Similarly, for ω ∈ Ω, t ∈ R and u 0 ∈ L 2 (D) let u 0 * (·; t, ω, u 0 ) stand for the global weak solution of (3.20) ω +(3.21) ω satisfying the final condition u(t) = u 0 . We write for all ω ∈ Ω and t > 0.

Generalized Floquet subspaces, Lyapunov exponent, and exponential separation
In this subsection, we investigate the existence of generalized Floquet subspaces, Lyapunov exponent, and exponential separation. Throughout this subsection, we assume (PA0)-(PA3). We first introduce some further assumptions on D and the coefficients of the problem (3.1)+(3.2).
At the end of this section, we give two sets of sufficient assumptions on the first and second-order coefficients, (R)(i), and (R)(ii), for the satisfaction of (PA4) and (PA5).
Before we prove Theorems 3.1-3.3, we first prove some propositions.
In the Robin boundary condition case: • D is a bounded domain, where its boundary ∂D is an (N − 1)-dimensional manifold of class C 2 .
• The function d 0 depends on x only, and belongs to C 1 (∂D).
• In the Robin boundary condition case, for each ω ∈ Ω the function d ω 0 belongs to C 2+α,2+α (R× ∂D). Moreover, their C 2+α,2+α (R × ∂D)-norms are bounded uniformly in ω ∈ Ω. Proof. First, assume (R)(i). Regarding (PA4) there is nothing to prove. As the coefficients are independent of ω, we write the global solution of (3.1)+(3.2) (in the case c 0 ≡ 0) with initial value u(0) = u 0 ∈ L 2 (D) as u 0 (·; u 0 ). By Proposition 3.1(iii), (3.29) We claim that there are e ∈ L 2 (D) + , C > 0, andβ : for any nonzero u 0 ∈ L 2 (D) + . In fact, it follows from [8] and [9] that there is a simple eigenvalue λ princ (the principal eigenvalue) of the problem which is real and larger than and bounded away from the real parts of the remaining eigenvalues, and that an eigenfunction corresponding to it (a principal eigenfunction) can be chosen so to take positive values on D (note that B ω in (3.31) is independent of ω).
As e we take the principal eigenfunction, normalized so that e = 1. In the Dirichlet boundary condition case, by [9, Theorem 2.1], there is a constant C > 1 such that is positive, since otherwise, by (3.32), u 0 (1; u 0 ) would be constantly equal to zero, which contradicts (3.29). Therefore, (3.30) holds, which implies that (3.25) is satisfied. (3.26) is proved in an analogous way. Regarding (3.27) and (3.28), there is nothing to prove.
Second, we assume (R)(ii). We embed problem into a family of problems as in [22].
For ω ∈ Ω put a 0,ω : (in the Dirichlet case we put d ω 0 constantly equal to zero). Let Y 0 be the closure of the set { a 0,ω : with the corresponding norms bounded uniformly inã 0 ∈ Y 0 . Also, on Y 0 the weak-* and open-compact topologies coincide. Consequently, (PA4) holds.
This is a contradiction. Hence C − exists. Similarly, we can prove that C + exists. We also claim that there is C > 0 such that  This is a contradiction. Hence (3.27) is satisfied. The fulfillment of (3.26) and (3.28) follows by applying analogous reasoning to the adjoint problem.

Systems of Linear Random Delay Differential Equations
In this section, we consider applications of the general results stated in Section 2 to (1.3), that is, the following systems of linear random delay differential equation, where u ∈ R N , N ≥ 2, and A(ω), B(ω) are N by N real matrices (we write A(ω), B(ω) ∈ R N ×N ): Again, let ((Ω, F, P), (θ t ) t∈R ) be an ergodic metric dynamical system, with P complete. In this section, as a Banach space X we will consider the space C Similarly, · 1 stands either for the ℓ 1 -norm on R N or for the corresponding maximum norm on C([−1, 0], R N ): We will use the notation ≤ (and ≥) to denote the order relations generated by the standard cone (R N ) + in R N as well as the standard cone C([−1, 0], R N ) + in C([−1, 0], R N ).
Throughout this section, we make the following standing assumption.
We give now a useful representation of the solution of (4.1)+(4.2). Namely, for ω ∈ Ω and u 0 0 ∈ R N denote by U 0 ω (t)u 0 0 the value at time t ≥ 0 of the solution of u ′ = A(θ t ω)u satisfying the initial condition u(0) = u 0 0 . For each ω ∈ Ω and each u 0 ∈ C([−1, 0], R N ) the function u(·; ω, u 0 ) is the solution of the integral equation  1,0]. Consequently, the problem reduces to proving the (F, B(R N ))-measurability of the mapping Ω ∋ ω → u(t; ω, u 0 ) ∈ R N for each t ∈ (0, 1]. Observe that for such t (4.3) takes the form By repeated application of the variant of Pettis' Measurability Theorem mentioned above, together with (OA0) and the fact that From now until the end of the present section we assume that (OA0) holds. We will call (U ω (t)) ω∈Ω,t∈[0,∞) , (θ t ) t∈R the measurable linear skew-product semiflow on C([−1, 0], R N ) generated by (4.1).
In systems of delay differential equations the choice of C([−1, 0], R N ) as the "state space" is not the only possible: observe that, since the dual space is not separable, we are unable to apply the theory of generalized exponential separation as presented in [23]. It appears that applying the (separable and reflexive) space L 2 ((−1, 0), R N ) ⊕ R N (as in [3]) could be useful in proving such properties.
For some linear time-periodic (systems of) delay differential equations with an additional structure invariant decompositions into countably many finite-dimensional subbundles (labelled by a lap number) were proved in [15] and [14].