Time averaging for nonautonomous/random linear parabolic equations

Linear nonautonomous/random parabolic partial differential equations are considered under the Dirichlet, Neumann or Robin boundary conditions, where both the zero order coefficients in the equation and the coefficients in the boundary conditions are allowed to depend on time. The theory of the principal spectrum/principal Lyapunov exponents is shown to apply to those equations. In the nonautonomous case, the main result states that the principal eigenvalue of any time-averaged equation is not larger than the supremum of the principal spectrum and that there is a time-averaged equation whose principal eigenvalue is not larger than the infimum of the principal spectrum. In the random case, the main result states that the principal eigenvalue of the time-averaged equation is not larger than the principal Lyapunov exponent.


Introduction
It is well known that parabolic equations can be used to model many evolution processes in science and engineering. Parabolic equations with general time dependence are gaining more and more attention since they can take various time variations of the underlying processes into account in modeling the processes. A great amount of research work has been carried out toward the existence, uniqueness, and regularity of solutions of general linear, semilinear, quasilinear parabolic equations (see [2], [3], [11], [12], [14], [26], [27], [39], etc.). As a basic tool for nonlinear problems, it is of great significance to study the spectral theory for linear parabolic equations.
Spectral theory, in particular, principal spectrum theory (i.e., principal eigenvalues and principal eigenfunctions theory) for time independent and time periodic parabolic equations is well understood (see, for example, [16]). For such an equation, its principal eigenvalue provides the growth rate of the evolution operator and hence a least upper bound of the growth rates of all the solutions. Recently much effort has been devoted to the extension of principal eigenvalue and principal eigenfunction theory of time independent and periodic parabolic equations to general time dependent and random parabolic equations. See, for example, [17], [18], [19], [20], [21], [22], [28], [29], [31], [35], [36], [37], etc.
In the current paper, we focus on time dependent parabolic equations of the form and random parabolic equations of the form and ((Ω, F , P), {θ t } t∈R ) is an ergodic metric dynamical system (see Section 2 for definition).
Our objective is to study the influence of time variations of the zeroth order terms on the so-called principal spectrum and principal Lyapunov exponent of (1.1) and (1.2) (which are analogs of principal eigenvalue of time independent and periodic parabolic equations), respectively. To do so, we first study the existence and uniqueness of globally positive solutions via the skew-product semiflows on (a subspace of) C 1 (D) generated by (1.1) and (1.2). Next we define the principal spectrum and principal Lyapunov exponent of (1.1) and (1.2) in terms of the globally positive solutions. We then compare the principal spectrum and principal Lyapunov exponent of (1.1) and (1.2) with those of their averaged equations.
To be more precise, we first introduce some notations and state some basic assumptions.
For m 1 , m 2 ∈ N ∪ {0} and β ∈ [0, 1) the symbol C m1+β,m2+β (R ×D) denotes the Banach space consisting of functions h : R×D → R whose mixed derivatives of order up to m 1 in t and up to m 2 in x are bounded, and whose mixed derivatives of order m 1 in t and m 2 in x are globally Hölder continuous with exponent β, uniformly in (t, x) ∈ R ×D (provided that the boundary ∂D of D is of class C m2+β , at least).
Similarly, for m 1 , m 2 ∈ N ∪ {0} and β ∈ [0, 1) the symbol C m1+β,m2+β (R × ∂D) denotes the Banach space consisting of functions h : R × ∂D → R whose mixed derivatives of order up to m 1 in t and up to m 2 in x are bounded, and whose mixed derivatives of order m 1 in t and m 2 in x are globally Hölder continuous with exponent β, uniformly in (t, x) ∈ R × ∂D (provided that ∂D is of class C m2+β , at least).
Throughout the paper, we assume the following smoothness conditions on the domain and the coefficients in (1.1) and (1.2) (the nonsmooth case will be considered in the monograph [32]).
(A1) D ⊂ R N is a bounded domain, with boundary ∂D of class C 3+α , for some α > 0.
(A2) The functions a ij , a i belong to C 2 (D) and the functions b i belong to C 2 (∂D).
In the case of (1.1) let be equipped with the open-compact topology, where the closure is also taken in the open-compact topology. We will write Y instead of Y (c, d) (for the case of (1.1)) or instead of Y (Ω) (for the case of (1.2)). For given (c,d) ∈ Y and u 0 ∈ L p (D), consider B(t)u = 0, t > 0, x ∈ ∂D, (1.5) whereB with the initial condition (1. 6) Applying the theory presented by H. Amann in [2], we have that (1.5)+(1.6) has a unique L p (D)-solution U (c,d),p (·, 0)u 0 : [0, ∞) → L p (D) (p > 1) (see Proposition 3.2). Note that U (c,d),p (·, 0)u 0 is also a classical solution of (1.5)+(1.6) (see Section 3 for more detail). We may therefore write U (c,d),p (t, 0)u 0 as U (c,d) (t, 0)u 0 for u 0 ∈ L p (D). In the present paper we further assume the following continuous dependence.
(A7) For any T > 0 the mapping is continuous, where L(L 2 (D), L 2 (D)) represents the space of all bounded linear operators from L 2 (D) into itself, endowed with the norm topology, and B(·, ·) stands for the Banach space of bounded functions, endowed with the supremum norm.
It should be pointed out that in [4] and [34] conditions, for some special cases (for example, the Dirichlet boundary condition case and the case with infinitely differentiable coefficients), are given that guarantee the continuous dependence of , L 2 (D))) on the coefficients is not covered in [4] and [34]. We will not investigate the conditions under which (A7) is satisfied in this paper.
Then (1.1) ((1.2)) generates the following skew-product semiflow (see Section 3 for detail) Throughout the paper, we denote · as the norm in L 2 (D) (see Section 2 for other notations).
3) Consider (1.1). Then the set Σ(c, d) consisting of all limits where T n − S n → ∞ as n → ∞, is a compact interval (see Theorem 5.1).
Observe that if c(t, x) and d(t, x) in (1.1) are independent of t or are periodic in t, then λ inf (c, d)(= λ sup (c, d)) is the principal eigenvalue of (1.1) and v(t, ·; c, d) is an eigenfunction associated with λ inf (c, d) (called a principal eigenfunction). As in the time independent and periodic cases, the principal spectrum of (1.1) and principal Lyapunov exponent of (1.2) provide upper bounds of growth rates of the solutions of (1.1) and (1.2), respectively. This can indeed be easily seen from the fact that for any nontrivial u 0 ∈ X with u 0 (x) ≥ 0 for x ∈ D as long as the limits exist (the existence of one of the limits implies the existence of the others), and for any nontrivial u 0 ∈ X with u 0 (x) ≥ 0 for x ∈ D as long as the limits exist (again the existence of one of the limits implies the existence of the others) (this fact follows from Theorem 4.2). We remark that the existence and uniqueness of globally positive solutions to nonautonomous parabolic equations with time independent boundary conditions were studied in [28], [29], [35]. In [17] the author studied the uniqueness of globally positive solutions to nonautonomous parabolic equations with time dependent boundary conditions. When the boundary conditions are time independent, the results 3) and 4) are proved in [31]. The results 3), 4), and the existence part of 2) for time dependent boundary conditions are new. The strong monotonicity result 1) basically follows from [5,Theorem 11.6] and strongly maximum principal and the Hopf boundary point principle for classical solutions of parabolic equations.
We now consider the averaged equations of (1.1) and (1.2) in the following sense: In the case of (1.1) we call (ĉ(·),d(·)) an averaged function of (c, d) if for some T n − S n → ∞, where the limit is uniform in x ∈D (resp. in x ∈ ∂D).
In the case of (1.2) we call (ĉ(·),d(·)) the averaged function of (c, d) if The equation is called an averaged equation of (1.1) (the averaged equation of (1.2)) if (ĉ,d) is an averaged function of (c, d) (the averaged function of (c, d)).
Denote λ(ĉ,d) to be the principal eigenvalue of (1.10). We then have the following main results of the paper.
Hence time variations cannot reduce the principal spectrum and principal Lyapunov exponent (or the principal eigenvalues of the time averaged equations give lower bounds of principal spectrum and principal Lyapunov exponent of non-averaged equations). Indeed, the time variations increase the principal spectrum and principal Lyapunov exponents except in the degenerate cases. In the biological context these results mean that invasion by a new species (see [10], p. 220) is always easier in the time-dependent case or that time variations favor persistence (viewing both (1.1) and (1.10) as linear population growth models, then by 5), positive solutions of all averaged equations (1.10) of (1.1) bounded away from zero implies positive solutions of (1.1) also bounded away from zero, but not vice versa in general).
It should be pointed out that the results 5), 6) have been proved in [22] and [31] when the boundary conditions are time independent. They are new when the boundary conditions are time dependent and the proof presented in this paper is not the same as those in [22] and [31].
It should be also pointed out that the results 1)-4) apply to fully time dependent/random parabolic equations (i.e., equations in which all the coefficients can depend on t/θ t ω). But 5) and 6) are mainly for equations of form (1.1) and (1.2), respectively.
The rest of the paper is organized as follows. In Section 2 we collect several elementary lemmas and introduce some standing notations for future reference. We review some existence and regularity theorems and construct the skewproduct semiflow generated by (1.1) and (1.2) in Section 3. Section 4 is devoted to the study of the monotonicity of the skew-product semiflow constructed in Section 3 and the existence of global positive solutions of (1.5). Definition and basic properties of principal spectrum and principal Lyapunov exponents are discussed in Section 5. We prove the time averaging results in Section 6.
The authors are grateful to the referees for their remarks.

Elementary Lemmas and notations
We collect first, for further reference, some elementary results.
First of all, let Z be a compact metric space and B(Z) be the Borel σ-algebra of Z. (Z, R) := (Z, {σ t } t∈R ) is called a compact flow if σ t : Z → Z (t ∈ R) satisfies: [ (t, z) → σ t z ] is jointly continuous in (t, z) ∈ R × Z, σ 0 = id, and σ s • σ t = σ s+t for any s, t ∈ R. We may write z · t or (z, t) for σ t z. A probability measure µ on (Z, B(Z)) is called an invariant measure for (Z, {σ t } t∈R ) if for any E ∈ B(Z) and any t ∈ R, µ(σ t (E)) = µ(E). An invariant measure µ for (Z, {σ t } t∈R ) is said to be ergodic if for any E ∈ B(Z) satisfying µ(σ −1 t (E)△E) = 0 for all t ∈ R, µ(E) = 1 or µ(E) = 0. The compact flow (Z, {σ t } t∈R ) is said to be uniquely ergodic if it has a unique invariant measure (in such case, the unique invariant measure is necessarily ergodic). We say that (Z, {σ t } t∈R ) is minimal or recurrent if for any z ∈ Z, the orbit { σ t z : t ∈ R } is dense in Z.
Proof. Part (a) follows easily by the fact that the continuity is uniform in ω ∈ Ω.
Denote, for each x ∈ D (resp. x ∈D), Then (a) for each x ∈ D (resp. each x ∈D) the derivative (∂ĥ/∂x i )(x) exists, and the equality ∂ĥ for any y ∈ D (resp. any y ∈D) with |x − y| < δ 0 , for all ω ∈ Ω ′ and all x ∈ D (resp. x ∈D). Moreover, the convergence is uniform in x ∈ D 0 , for any compact D 0 ⋐ D (resp. uniform in x ∈D).
Proof. Parts (a) and (b) follow in a standard way. Part (c) follows by an application of Lemma 2.3(b) to the function (∂h/∂x i )(ω, x).
From now on we assume that (A1)-(A6) are satisfied. Consider the space H consisting of (c(·, ·),d(·, ·)), wherec : R ×D → R and d : R × ∂D → R are bounded continuous. The set H endowed with the topology of uniform convergence on compact sets (the open-compact topology) becomes a Fréchet space.
The following result is a consequence of the Ascoli-Arzelà theorem.
Moreover, the C 2+α,1+α (R×D)norms are bounded uniformly in Y by the same bound as in (A3).
(v) For a sequence (c (n) ,d (n) ) → (c,d) in Y , the mixed derivatives ofc (n) of order up to 2 in t and up to 1 in x converge to the respective derivatives ofc, uniformly on compact subsets of R ×D.
(vi) For a sequence (c (n) ,d (n) ) → (c,d) in Y , the mixed derivatives ofd (n) of order up to 2 in t and up to 3 in x converge to the respective derivatives ofd, uniformly on compact subsets of R × ∂D.
Similarly, for x ∈ ∂D and S < T we denotē The following result is a consequence of the Ascoli-Arzelà theorem (compare Lemma 2.5).
(2) Let a be as in x) and d(t, x) are almost periodic in t uniformly with respect to x ∈D and x ∈ ∂D, respectively, then (c, d) is both uniquely ergodic and minimal.
(2) If c(t, x) and d(t, x) are almost automorphic in t uniformly with respect to x ∈D and x ∈ ∂D, respectively, then (c, d) is minimal, but it may not be uniquely ergodic (see [23] for examples).
uniformly for x ∈ D, and uniformly for x ∈ ∂D.
Proof. We prove only (2.2), the other proof being similar. It follows via the Ascoli-Arzelà theorem that the set { (1/T ) T 0 c(t, ·) dt : T > 0 } = {c(·; 0, T ) : T > 0 } has compact closure in C(D), consequently from any sequence (T n ) with lim n→∞ T n = ∞ one can extract a subsequence (T n k ) such thatc(·; 0, T n k ) converges uniformly in x ∈D to someč (depending perhaps on the subsequence).
On the other hand, as (Y (c, d), R) is uniquely ergodic, for each continuous (compare, e.g., Oxtoby [33]). Fix x ∈D and take g((c,d)) :=c 0 (x). We have thus obtained that ifc(x; 0, T n ) converges, for some T n → ∞, uniformly in x ∈D, then the limit is always equal We introduce the following standing notations (X 1 , X 2 are Banach spaces): L(X 1 , X 2 ) represents the space of all bounded linear operators from X 1 to X 2 , endowed with the norm topology; · X1 denotes the norm in X 1 ; X * 1 denotes the Banach space dual to X 1 ; (·, ·) X1,X * 1 stands for the duality pairing between X 1 and X * 1 ; · denotes the norm in L 2 (D) or the norm in L(L 2 (D), L 2 (D)); ·, · stands for the standard inner product in L 2 (D); · X1,X2 indicates the norm in L(X 1 , X 2 ); [·, ·] θ is a complex interpolation functor; (·, ·) θ,p is a real interpolation functor (see [9], [38] for more detail); Z denotes the set of integers; N denotes the set of nonnegative integers.

Skew-product semiflows
We construct in this section a linear skew-product semiflow on X generated by (1.1) or by (1.2), where X is as in (1.7).
To do so, we first use the theory presented by H. Amann in [2] to consider the existence of solution of (1.5)+(1.6) for any (c,d) ∈ Y and any u 0 ∈ L p (D). Recall that we assume (A1)-(A6) throughout.
Let A(c) denote the operator given by and let B(d) denote the boundary operator given by Proof.
Recall the following compact embedding: . Then (1.5)+(1.6) can be written as and all x ∈ D, and it satisfies the boundary conditions for all t ∈ (t 0 , t 1 ) and all x ∈ ∂D.
The following existence result follows from [2, Theorem 15.1]. It follows from the uniqueness of L p -solutions that the following cocycle property for the solution operator holds: for any (c,d) ∈ Y, s, t ≥ 0.
(3.5) We collect now the regularity properties of the L p (D)-solutions which will be useful in the sequel.

Proof. It follows from Proposition 3.3 and [2, Corollary 15.3]).
Proposition 3.5 allows us to write for any t ∈ R and any s ≥ 0.
From now on, we assume (A7). For any sequence (c (here the convergence is uniform in the space variable and uniform on compact sets in the time variable). We then present various continuous dependence propositions.
where t > 0, and lim n→∞ u n = u 0 in L 2 (D), then the following holds.
(2) The mapping is continuous. Moreover, for any t > 0 and any (c,d) ∈ Y the linear operator U (c,d) (t, 0) is compact (completely continuous). Proof.
(1) Assume that (c (n) ,d (n) ) converges to (c,d) in Y and that t n converges to t > 0. Suppose to the contrary that Then there are ǫ 0 > 0 and a sequence (u n ) ∞ n=1 ⊂ L 2 (D) with u n = 1 such that for all n. By Proposition 3.3, there are u * , u * * ∈ V θ p such that (after possibly extracting a subsequence) in V θ p , as n → ∞. Without loss of generality, we may assume that there is u * ∈ V θ p such that U (c,d) (t/2, 0)u n →ũ * in V θ p as n → ∞. Then by Proposition 3.6, we have as n → ∞. By the property (A7) we have as n → ∞. Then we must have u * = u * * , hence (2) It follows by (1) and Eq. (3.1).
We are now ready to construct the skew-product semiflow on X (X is as in (1.7)) generated by (1.1) (3.10) Π = { Π t } t≥0 satisfies the usual algebraic properties of a semiflow on X : Π 0 equals the identity on X, and Π t • Π s = Π s+t for any s, t ≥ 0. Moreover, the continuity of Π restricted to (0, ∞) × X × Y follows by Proposition 3.6 and the embedding X ֒→ L 2 (D). (However, we need not have continuity at t = 0.) The semigroup property Π t • Π s = Π s+t takes in that notation the following form (see the cocycle property (3.5)): Proposition 3.8 (Continuity in C(D) at t = 0). Let θ ∈ (1/2, 1) and p > 1 be such that 2θ − p ∈ N and V θ p ֒→ C(D). Then for any (c,d) ∈ Y and u 0 ∈ V θ p (d), Proof. It follows from [2, Theorem 15.1] and Eq. (3.1).

Strong monotonicity and globally positive solutions
In this section, we first show that the skew-product semiflow Π t constructed in the previous section is strongly monotone and then show that (1.5) has a unique globally positive solution, which will be used in next section to define the principal spectrum and principal Lyapunov exponent of (1.1) and (1.2).
Let X be as in (1.7). The Banach space X is ordered by the standard cone The interior X ++ of X + is nonempty, where for the Dirichlet boundary conditions, and for the Neumann or Robin boundary conditions. For u 1 , u 2 ∈ X, we write u 1 ≤ u 2 if u 2 − u 1 ∈ X + , u 1 < u 2 if u 1 ≤ u 2 and u 1 = u 2 , and u 1 ≪ u 2 if u 2 − u 1 ∈ X ++ . The symbols ≥, > and ≫ are used in the standard way.
We proceed now to investigate the strong monotonicity property of the solution operator U (c,d) (t, 0). When the equations (1.1) and (1.2) are in divergence form, the monotonicity of U (c,d) (t, 0) follows from [5,Theorem 11.6]. But the strong monotonicity is not included in [5,Theorem 11.6]. Though the monotonicity for equations in non-divergence form can also be proved by [5,Theorem 11.6] after verifying certain conditions, however for convenience we will give a proof for the monotonicity directly. We will prove the strong monotonicity by using the strong maximum principle and the Hopf boundary point principle for classical solutions. But before we do that we have to analyze whether the existing theory (as presented, e.g., in [14]) can be applied: notice that in the Robin cased may change sign. We show that coefficient can be made nonnegative by an appropriate change of variables.

First note that there is a sequence (v
and v (n) and v (n) 0 → v 0 uniformly on any set on which v 0 is bounded. It is therefore sufficient to prove the claim for the case that v 0 = χ E , where E ⊂ D is a Lebesgue measurable set. Now assume v 0 = χ E , where E ⊂ D is a Lebesgue measurable set. For ǫ n := 1 4n 2 , choose a compact set K ⊂ E and an open set U ⊃ K such that U ⋐ D, |E \ K| < ǫ n and |U \ K| < ǫ n , where here |·| denotes the Lebesgue measure of a set. Then, by the C ∞ Urysohn Lemma (see [13,Lemma 8.18 . The claim is thus proved. Denote by v(t, ·; v 0 ) and v(t, ·; v (n) ) the solutions of (4.2) with v(0, ·; v 0 ) = v 0 (·) and v(0, ·; v (n) ) = v (n) (·) (n = 1, 2, . . . ), respectively.
By Proposition 3.5, for any n = 2, 3, . . . the function v(·, ·; v 0 ) is continuous on [T /n, T ], satisfies the equation in (4.2) pointwise on (T /n, T ]×D and satisfies the boundary condition in (4.2) pointwise on (T /n, T ] × ∂D. Further, from Proposition 3.2 and the nonnegativity of v it follows that for n sufficiently large there is x n ∈ D such that v(T /n, x n ; v 0 ) > 0. An application of the strong maximum principle for parabolic equations gives v(t, x; v 0 ) > 0 for each t ∈ (0, T ] and each x ∈ D.
Suppose to the contrary that, in the Neumann or Robin boundary condition case, there are t * ∈ (0, T ] and x * ∈ ∂D such that v(t * , x * ; v 0 ) = 0. It follows from the Hopf boundary point principle (applied to v restricted to which is incompatible with the boundary condition. Hence v(t, x; v 0 ) > 0 for all t ∈ (0, T ] and all x ∈D.
Since v(t, x; v 0 ) = e Mu * (t,x) ((Uã(t, 0)u 2 )(x) − (Uã(t, 0)u 1 )(x)) for any t ∈ (0, T ] and x ∈D, this completes the proof. (It is to be remarked that in Eqs. (4.1) and (4.2) their coefficients may not belong to Y , so formally we cannot apply propositions from Section 3 in those cases. This should not cause any misunderstanding.) By Theorem 4.1 we have the following strong monotonicity: The theory of existence and uniqueness of globally positive solutions can then be extended to our case. Below, we collect its basic concepts and facts.
is a globally positive solution of (1.5).
(ii) Let, for some (c,d) ∈ Y , v = v(t, x) be a globally positive solution of (1.5).
Then there exists a constant β > 0 such that v(t, x) = βv(t, x;c,d) for each t ∈ R and each x ∈ D.
We prove now that w * ((c,d))(x) > 0 for a.e. x ∈ D, or, which is equivalent, thatw * ((c,d))(x) > 0 for a.e. x ∈ D. Suppose first that for some (c,d) ∈ X there are D + , D − ⊂ D of positive Lebesgue measure such that w * ((c,d))(x) > 0 for x ∈ D + ,w * ((c,d))(x) < 0 for x ∈ D − , andw * ((c,d))(x) = 0 for x ∈ D \ (D + ∪ D − ). Define v ∈ L 2 (D) to be the simple function equal to 1/ D+w * ((c,d))(x) dx on D + , equal to −1/ D−w * ((c,d))(x) dx on D − , and equal to zero elsewhere. We have This contradicts (b). Suppose now that for some (c,d) ∈ X there are D + , D 0 ⊂ D of positive Lebesgue measure such thatw * ((c,d))(x) > 0 for x ∈ D + andw * ((c,d))(x) = 0 for x ∈ D 0 , and the complement of the union D + ∪ D 0 in D has Lebesgue measure zero. We repeat the above construction, this time with v equal to zero on D + and equal to one The fact that if there exists a globally positive solution of (1.5) then it is unique up to multiplication by a positive constant is proved for the Dirichlet case in [21], and for the Neumann and Robin case in [17]. We proceed now to the construction of a globally positive solution.
We define first the trace of a positive solution v(t, x;c,d) on Z: It follows from (a) and (c) that U (c,d) (l + k, k)v(k, ·;c,d) = v(k + l, ·;c,d) The fact that v(t, ·;c,d) ∈ X ++ for each t ∈ R is a consequence of the construction of v and of Theorem 4.1.

Formula (4.3) for t ≥ 0 is straightforward. It follows from the uniqueness of globally positive solutions that
From (4.10) we obtain, for any t < 0, that which concludes the proof of formula (4.3).
It is a consequence of the continuity of w, the compactness of Y and Proposition 3.6 that the denominators on the right-hand side are positive and bounded away from zero, uniformly in Y . Now we apply the parabolic regularity estimates [14, Theorem 5, Chapter 3].

Principal spectrum and principal Lyapunov exponent
In this section, we collect the basic concepts and facts about the principal spectrum and principal Lyapunov exponent of (1.1) and (1.2).
Definition 5.1. In case of (1.1) we define its principal spectrum to be the set of all limits where T n − S n → ∞ as n → ∞.
Definition 5.2. The λ(c, d) as in Theorem 5.2 is called the principal Lyapunov exponent of (1.2).
Remark 5.1. In the existing literature, the principal spectrum is either defined precisely as in Definition 5.1 (see [30] ) or with the L 2 (D)-norm replaced by the norm in some fractional power space that embeds continuously into C 1 (D) (see, e.g., [31] ). In our setting, as X 1 is a one-dimensional invariant subbundle spanned by a continuous function from Y into X, we can replace the L 2 (D)norm in Definition 5.1 with the X-norm.
Remark 5.2. Similarly, in the Definition 5.2 the L 2 (D)-norm can be replaced with the X-norm. Further, in [31] the principal Lyapunov exponent was introduced as the (a.e. constant ) limit where X is some fractional power space that embeds continuously into C 1 (D).
With the help of (4.4) one can prove that for those ω ∈ Ω for which λ(c, d) = lim T →∞  We introduce now a useful concept. d))(x) dx. so κ((c,d)) is well defined. The function κ : Y → R is continuous. Indeed, notice that applying integration by parts we can write As w : Y → X is continuous, the above expression depends continuously on (c,d), too. We point out that the function κ((c,d)) introduced in (5.1) is a very useful quantity in the investigation of various properties of principal spectrum and principal Lyapunov exponents. This quantity will be heavily used in next section. In the rest of this section, we discuss how to use the function κ to characterize the principal spectrum and principal Lyapunov exponents. Let Taking the inner product of the above equation with w((c,d) · t) and observing that w((c,d) · t), w((c,d) · t) ≡ 1 and ∂ ∂t w((c,d) · t), w((c,d) · t) ≡ 0 we get for any (c,d) ∈ Y and any t ≥ 0.
for any (c,d) ∈ Y and S < T . Then following from Definition 5.1 we have be the principal spectrum interval of (1.1). Then In the case of (1.2) we write κ(ω) instead of κ((c ω , d ω )). We have for a.e. ω ∈ Ω. We remark that if c(t, x) and d(t, x) are independent of t, then (c, d) = (ĉ,d) and λ inf (c, d) = λ sup (c, d) = λ(c, d). Moreover, we have the following easy theorem about the continuous dependence of λ(c, d) on (c, d).
Theorem 5.5. If c (n) converges in C(D) to c and d (n) converges in C(∂D) to d then λ(c (n) , d (n) ) → λ(c, d).

Time averaging
In this section we state and prove our results on the influence of time variations on principal spectrum and principal Lyapunov exponent of (1.1) and (1.  ( for any ω ∈ Ω * , t ∈ R and x ∈D, and d(θ t ω, x) = d(x) for any ω ∈ Ω * , t ∈ R and x ∈ ∂D.
In the case that the boundary condition is of the Dirichlet or Neumann type or of the Robin type with d independent of t, the above theorems have been proved in [31]. For completeness, we will provide proofs of the theorems including the case that the boundary condition is of the Robin type with d depending on t. We note that the proof in the following for Theorem 6.1 is not the same as that in [31] even in the case d is independent of t.
Note that for given S, T > 0, By Theorem 5.3 there are (S n ), (T n ) with T n → ∞ such that 1 T n Tn+Sn Sn κ((c, d) · t) dt = ln η(T n ; c, d, S n ) T n → λ inf (c, d).
Without loss of generality we may assume that the limits lim n→∞ We may thus assume that there is w * = w * (x) such that lim n→∞ŵ (x; c, d, S n , T n ) = w * (x) (6.9) for i, j = 1, 2, . . . , N and x ∈ D. In the Dirichlet boundary conditions case, it follows from Theorems 4.3 and 4.4 that w * can be extended to a function continuous onD by putting w * (x) = 0 for x ∈ ∂D. Moreover, by Theorem 4.3, Regarding the uniformity of convergence, in the Dirichlet case, the limit in (6.9) is uniform for x inD and the limits in (6.10) and (6.11) are uniform for x in any compact subset D 0 ⋐ D, and in the Neumann and Robin cases, the limits in (6.9) and (6.10) are uniform for x ∈D and the limit (6.11) is uniform for x in any compact subset D 0 ⋐ D.
By arguments as in the proof of Part (1), we must have λ ≥λ.
Therefore we must have for all x ∈ D.