Lyapunov spectrum of nonautonomous linear Young differential equations

We show that a linear Young differential equation generates a topological two-parameter flow, thus the notions of Lyapunov exponents and Lyapunov spectrum are well-defined. The spectrum can be computed using the discretized flow and is independent of the driving path for triangular systems which are regular in the sense of Lyapunov. In the stochastic setting, the system generates a stochastic two-parameter flow which satisfies the integrability condition, hence the Lyapunov exponents are random variables of finite moments. Finally, we prove a Millionshchikov theorem stating that almost all, in a sense of an invariant measure, linear nonautonomous Young differential equations are Lyapunov regular.


Introduction
In this article we study the Lyapunov spectrum of the nonautonomous linear Young differential equation (abbreviated by YDE) where A, C ∈ C([0, ∞), R d×d ), the space of all continuous matrix valued functions on [0, ∞), and ω is a continuous path in C p−var for some p ∈ (1, 2). Such system (1.1) appears, for instance, when considering the linearization of the autonomous Young differential equation dy(t) = f (y(t))dt + g(y(t))dω(t) (1.2) along any reference solution y(t, y 0 , ω). An example is when we would like to solve in the pathwise sense stochastic differential equations driven by fractional Brownian motions with Hurst index H ∈ ( 1 2 , 1) defined on a complete probability space (Ω, F, P) [24]. In fact it follows from [4] that (1.2) under the stochastic setting also satisfies the integrability condition.
The equation (1.1) can be rewritten in the integral form where the second integral is understood in the Young sense [28], which can also be presented in terms of fractional derivatives [29]. Under some mild conditions, the unique solution of (1.1) generates a two-parameter flow Φ ω (t 0 , t), as seen in [8]. Under a certain stochastic setting, (1.1) actually generates a stochastic two-parameter flow in the sense of Kunita [15]. Our aim is to study the Lyapunov exponents and Lyapunov spectrum of the linear two-parameter flow generated from Young equation (1.1). Notice that Lyapunov spectrums and its splitting are the main content of the celebrated multiplicative ergodic theorem (MET) by Oseledets [25]. It was also investigated by Millionshchikov in [18,19,20,21] for linear nonautonomous differential equations. In the stochastic setting, the MET is also formulated for random dynamical systems in [1,Chapter 3]. Further investigations can be found in [5,6,7,9] for stochastic flows generated by nonautonomous linear stochastic differential equations driven by standard Brownian motion. For Young equations, we show that Lyapunov exponents can be computed based on the discretization scheme. Moreover, if the driving path ω satisfies certain conditions, the Lyapunov spectrum of triangular systems which are Lyapunov regular can be computed independently of ω.
One important issue is the non-randomness of Lyapunov exponents when the system is considered under a certain stochastic setting, namely if the driving path ω is a realization of a certain stochastic noise. In case the system is driven by standard Brownian noises, a filtration of independent σ− algebras can be constructed and the argument of Kolmogorov's zero-one law can be applied to prove the non-randomness of Lyapunov exponents, which are measurable to tail events, see [6]. In general, the stochastic noise might be a fractional Brownian motion which is not Markov, hence it is difficult to construct such a filtration and to apply the Kolmogorov's zero-one law. The question of non-randomness of Lyapunov spectrum is therefore still open. However, the answer is affirmative for some special cases. For example, autonomous and periodic systems can generate random dynamical systems satisfying the integrability condition, thus the Lyapunov spectrum is non-random by the multiplicative ergodic theorem [1]. Our investigation shows that the Lyapunov spectrum of triangular systems that are Lyapunov regular are also non-random. In general, we expect that the statement of non-randomness of Lyapunov spectrum is still true for any Lyapunov regular system, although finding a counter-example of a nonautonomous system with random Lyapunov spectrum also attracts our interest.
The paper is organized as follow. In Section 2, we briefly prove in Theorem 2.1 the existence and uniqueness of solution of (1.1), and in Theorem 2.7 the generation of a topological two-parameter flow. The concepts of Lyapunov exponents and Lyapunov spectrum of system (1.1) are then defined in Section 3. Under the assumptions on the driving path ω and on the coefficient functions, we prove in Theorem 3.6 that Lyapunov spectrum can be computed using the discretized flow and give an explicit formula of the spectrum in Theorem 3.10 in case of triangular systems which are regular in the sense of Lyapunov. Theorem 3.15 provides a criterion for a triangular system of YDE to be Lyapunov regular. In Section 4, we consider the system under random perspectives in which the driving path acts as a realization of a stochastic stationary process in a function space equipped with a probabilistic framework. The system is then proved to generate a stochastic two-parameter flow which satisfies the integrability condition, hence the Lyapunov exponents are proved in Theorem 4.6 to be random variables of finite moments. Subsection 4.2 is devoted to study the regularity of the system, where we prove a Millionshchikov Theorem 4.9 stating that almost all, in a sense of an invariant measure, nonautonomous linear Young differential equations are Lyapunov regular. We end up with a discussion on the non-randomness of Lyapunov spectrum in some special cases, and raise this interesting question in general.
We finish the introduction part by presenting some well-known fact of Young integral. Let 0 ≤ T 1 < T 2 < ∞, C r−var ([T 1 , T 2 ], R d ) be the Banach space of bounded r-variation continuous functions on [T 1 , T 2 ] having values in R d with finite norm where the supremum is taken over the whole class of finite partition Π(T 1 , with α = 1/r. Moreover, we have the following estimate, whose proof follows directly from the definitions of p−var norm and sup norm and will be omitted here. Now, consider x ∈ C q−var ([T 1 , T 2 ], R d×m ) and ω ∈ C p−var ([T 1 , T 2 ], R m ), p, q ≥ 1 and 1 p + 1 q > 1, the Young integral b a x(t)dω(t) can be defined as the limit b a x(t)dω(t) := lim where the limit is taken on all the finite partition Π = {T 1 = t 0 < t 1 < . . . < t n = T 2 } with |Π| := max [28, p. 264-265]). This integral satisfies additive property by the construction, and the so-called Young-Loeve estimate [11,Theorem 6.8 where 2 Topological two-parameter flows generated by YDE In this section, we fix ω ∈ C p−var ([t 0 , t 0 + T ], R) with some 1 < p < 2 and consider the deterministic Young equation.
We first show that under mild conditions on coefficient functions A, C, (2.1) has a unique solution in Proof: The results for general system were stated for example in [24] and also in [8], but the conditions are stronger and can not be applied to the linear case here. For the benefit of the readers, we would like to give a detailed proof in the appendix, which use the same approach and technique as in [8].
Remark 2.2 Fix [t 0 , t 0 +T ], by consider the backward equation similar to that of [8], the conclusion is still true if the initial value is set at an arbitrary point a ∈ [t 0 , t 0 + T ].
Next, we consider ω varying as an element of C p−var ([t 0 , t 0 + T ]) with p−var norm and define the solution mapping
Proof: See the appendix.
then it is easy to see that the solution is 1/p−Hölder continuous and is continuous. Definition 2.5 (Cauchy operator) Suppose that the assumptions of Theorem 2.1 are satisfied. For any t 0 ≤ t 1 ≤ t 2 ≤ t 0 + T the Cauchy operator Φ ω (t 1 , t 2 ) of the YDE (1.1) is defined as the mapping along trajectories of (1.1) from time moment t 1 to time moment t 2 , i.e., for any vector x t 1 ∈ R d we define Φ ω (t 1 , t 2 )x t 1 to be the vector x t 2 ∈ R d which is the value of the solution x of (1.1) evaluated at time t 2 .
Following [1, p. 551], below we introduce the concept of two-parameter flows. (i) For any s, t ∈ [a, b] the mapping X s,t is a homeomorphism of R d ; (ii) X s,s = id for any s ∈ [a, b]; Proof: We may use arguments similar to those of the proof of Theorem [8,Theorem 4.4] for our theorem. However, we give here a direct proof using linearity of the system. First note that the same method as in Theorem 2.1 can be applied to prove the existence and uniqueness of solution Φ ω (t 0 , t) of the matrix-valued differential equation (2.5) It is easy to show that the solution Φ ω (·, ·) : ∆ 2 → R d×d , with ∆ 2 := {(s, t) ∈ [t 0 , t 0 +T ]×[t 0 , t 0 +T ] : s ≤ t}, has properties that Φ ω (s, s) = I d×d for all s ≥ 0 and The solution Φ ω (·, ·) is the mapping along trajectories of (2.1) in forward time since YDE is directed. Like the ODE case, in our setting, the solution of the matrix equation (2.5) is the Cauchy operator of the vector equation (2.1). Next, consider the adjoint matrix-valued pathwise differential equation with initial value Ψ(t 0 , t 0 ) = I, and A T (·), C T (·) are the transpose matrices of A(·) and C(·), respectively. By similar arguments we can prove that there exists a unique solution Ψ ω (t 0 , t) of (2.7). Introduce the transformation u(t) = Ψ ω (t 0 , t) T x(t). By the formula of integration by parts (see [11,Proposition 6.12 and Exercise 6.13] or a fractional version in Zähle [29]), we conclude that = 0.
In other words, As a direct consequence, for any x 0 = 0 we have Φ ω (t 0 , t)x 0 = 0 for all t ≥ t 0 . Thus we showed that the linear operator Φ ω (t 0 , t), t ≥ t 0 , is nondegenerate. It is easily seen we may change t 0 to s, i.e. for all , and it is clearly a continuous two-parameter flow generated by (2.1).

Lyapunov spectrum for nonautonomous linear system of YDEs
The classical Lyapunov spectrum of linear system of differential equation is a powerful tool in investigation of qualitative behavior of the system, see e.g. [3] or [23]. Since (1.3) generate a twoparameter flow of homeomorphisms, we can instead study Lyapunov spectrum of the flow generated by the equation.

Exponents and spectrum
We aim to follow the technique in Cong [6] and Millionshchikov [18,19]. From now on, let us consider the following assumptions on the coefficients of (1.3).
In (H 2 ) we can assume, without loss of generality that δ = 1. Put where K given by (1.5). It is obvious from (2.4) that, for any t 0 ∈ R + , Note that condition (H 1 ), (H 2 ) and Theorem 2.1 assure the existence and uniqueness of solution of (1.3) on the whole R + . Moreover, Theorem 2.7 asserts that (1.3) generates a two-parameter flow on R d by means of its Cauchy operators Φ ω (·, ·), and Φ ω (s, t)x 0 represents the value at time t ∈ R + of the solution of (1.3) started from x 0 ∈ R d at time s ∈ R + . We follow Cong [6] and introduce the notion of Lyapunov exponents of two-parameter flow of linear operators first, and then use it to define the Lyapunov exponents. We shall denote by G k the Grassmannian manifold of all linear k-dimensional subspaces of R d .
Recall that for a real function h : R + → R d the Lyapunov exponent of h is the number (which could be ∞ or −∞) (We make the convention that log is the logarithm of natural base and log 0 := −∞.)

2)
are called Lyapunov exponents of the flow Φ ω (s, t). The collection are called Lyapunov subspaces at time u of the flow Φ ω (s, t). The flag of nonincreasing linear It is easily seen that the Lyapunov exponents in Definition 3.1 are ordered: Moreover, follow Cong [6, Theorems 2.5, 2.7, 2.8], for any u ∈ [t 0 , ∞), the Lyapunov subspaces The classical definition of Lyapunov spectrum of a linear system of ODE is based on the normal basis of the solution of the system (see [10]). Millionshchikov [18] pointed out that these definitions are equivalent. In the following remark we restate some facts in [10]).

Note that if Lyapunov exponents of the columns of a fundamental solution matrix
(but the inverse is not true). Now we turn to the YDE (1.3). By Theorem 2.7, (1.3) generates a two-parameter flow by means of its Cauchy operators, hence the following definition is natural.  Proof: Use the two-parameter flow property of Φ ω (s, t) and the fact that the norm of product of matrices is less than or equal to the product of the norms of the multipliers. Now we are able to formulate one of the main results of this paper on the Lyapunov spectrum of the equation (1.3). Let us consider the following assumptions on the driving path ω.
is an increasing sequence of positive real numbers on which the upper limit lim sup Let n m denotes the largest natural number which is smaller than or equal to t m . Using the flow property of Φ ω (s, t) and the assumption (H 3 ) we have On the other hand, clearly Consequently, we have for all k ∈ {1, . . . , d} and y ∈ R d the equality Hence, either which then follows that where the last inequality can be proved similarly to the one in (3.7). Hence (3.5) holds.
Remark 3.7 The discretization scheme in Theorem 3.6 can be formulated for any step size h > 0.

Lyapunov spectrum of triangular systems
It is well known in the theory of ODE that a linear triangular system can be solved successively and its Lyapunov spectrum is easily computed via its coefficients. A similar situation holds also for triangular linear systems of YDE, however some additional conditions are to be assumed. In this subsection we present our result for linear triangular systems of YDE. Let us consider the system ( As a motivation of our ideas, (H 4 ) is satisfied for almost all realization ω of a fractional Brownian motion (see Lemma 5.3 in Section 5 for the proof and see Mishura [22] for details on fractional Brownian motions); another situation, in which (H 4 ) holds is the case ω(t) = t α with 0 < α < 1 and C is continuous and bounded. To see how the assumption (H 4 ) can be applied, we first consider the equation (3.8) in the one dimensional case Using integration by part formula (see Zähle [29, Theorem 3.1]), (3.9) can be solved explicitly as Moreover, we have the following lemma.
Lemma 3.8 The following estimates hold for any nontrivial solution z ≡ 0 of (3.9) (ii) Due to linearity it suffices to prove for the case z 0 = 1 what we will assume here. Introduce and χ(e f (t) ) = a, χ(e g(t) ) = 0.
Let ε be arbitrary, there exists D 1 such that On the other hand, by mean value theorem and the continuity of f , for any which implies and then by using Minkowski inequality we get

Note that condition (H 3 ) implies the boundedness of
, t 0 ∈ R + . Therefore, there exists a constant D 3 such that This proves the second claim.
Next we will show by induction that the Lyapunov spectrum of system (3.8) is {a kk , 1 ≤ k ≤ d} with a kk := lim t→∞ t 0 a kk (s)ds t provided that the limit is well-defined and exact.
Lemma 3.9 Assume that a finite sequence of functions g i : R + → R, g i are continuous and have finite q-var norm on any compact interval of R + , i = 1, . . . , n, satisfies Proof: (i) See [10, Theorem 1, p. 127], note that (ii) The first statement is known due to [10,Theorem 2,p 19]. For the second one, we just need to prove for k = 2, the general case is obtained by induction. From Lemma 1.1, it can be seen that Therefore the the second statement is a consequence of the first one and (i).
By similar arguments using integration by part formula, the non-homogeneous one dimensional linear equation can be solved explicitly as provided that h 1 , h 2 are in C q−var ([0, t], R) for all t > 0. This allow us to solve triangular systems by substitution as seen in the following theorem.
We construct a fundamental solution matrix X(t) = (x ij (t)) d×d of (3.8) as follows.
in which, t ik = 0, if a kk − a ii ≥ 0 +∞, if a kk − a ii < 0. Now we consider the d th collumn of X and prove by inductive that Namely, by Lemma 3.8 the statement is true for m = d. Assume that χ(x jd (t)), χ(|||x jd ||| q−var,[t,t+1] ) ≤ a dd for all i + 1 ≤ j ≤ d, we will point out that Firstly, since A is bounded, due to [10, Corollary of Theorem 2, p. 129] we have In addition, we can prove that χ(x jd (t)), χ(|||x jd ||| q−var,[t,t+1] ) ≤ a dd , ∀i + 1 ≤ j ≤ d and applying Lemma 3.9 we obtain Due to Lemma 5.1 and 5.2 we have Now, apply Lemma 3.9 again for Y i , I, and J we arrive at Hence, the Lyapunov exponent of the column d th , X d , of matrix X does not exceed a dd , combining this with χ(x dd (t)) = a dd we conclude that χ(X d (t)) = a dd . Similarly, χ(X i (t)) = a ii for i = 1, 2, . . . , d, in which X i is the column i th of X. Finally, since is a normal matrix solution to (3.8) and the Lyapunov spectrum of (3.8) is {a 11 , a 22 , . . . , a dd }.

Remark 3.11
In the theory of ODEs, Theorem Perron states that a linear equation can be reduced to a linear triangular system (see [10, p. 180]). However, we do not know if it is true for linear Young differential equations. That is because for a linear YDE, besides the drift term A corresponding to dt we do have also the diffusion term C corresponding to dω, hence it is difficult to tranform the original system to a triangular form whose coefficient matrices only depend on t.

Lyapunov regularity
The concept regularity has been introduced by Lyapunov for linear ODEs, and since then has attracted lots of interests (see e.g. [1], [7], [20]). For a linear YDE, we define the concept of Lyapunov regularity via the generated two-parameter flow of linear operators in R d .
The coefficient of nonregularity of the linear YDE (1.3) is, by definition, the coefficient of nonregularity of the two-parameter flow generated by (1.3). A two-parameter flow is called Lyapunov regular if its coefficient of nonregularity equals 0 identically. A linear YDE is called Lyapunov regular if its coefficient of nonregularity equals 0.
The following conclusion can be derived with similar proof to the one in [6]. We define the adjoint equation of (1.1) (and also of the equivalent integral equation (1.3)) by

dy(t) = −A T (t)y(t)dt − C T (t)y(t)dω(t).
(3.14) The following lemma is a version of Perron Theorem from the classical ODE case.
We have

Lyapunov spectrum for linear stochastic differential equations
In this section, we would like to investigate the same question in the random perspective, i.e. the driving path ω is a realization of a stochastic process Z with stationary increments. System (1.1) can then be embedded into a stochastic differential equation, or precisely a random differential equation which can be solved pathwise. Such a system can then be proved to generate a stochastic two-parameter flow, hence it makes sense to study its Lyapunov spectrum and also to raise the question on the non-randomness of the spectrum.

Generation of stochastic two-parameter flows
More precisely, recall that C 0,p−var ([a, b], R d ) is the closure of smooth path from [a, b] to R d in pvariation norm and C 0,p−var (R, R d ) is the space of all x : R → R d such that x| I ∈ C 0,p−var (I, R d ) for each compact interval I ⊂ R, equipped with the compact open topology given by the p−var norm, i.e topo generated by the metric: Note that for x ∈ C 0,p−var It is easy to check that θ forms a metric dynamical system (θ t ) t∈R on (C 0,p−var 0 (R, R), B). Moreover, the Young integral satisfies the shift property with respect to θ, i.e.   R), B) that is invariant under θ, and the so-called diagonal process Z : R × Ω → R, Z(t,ω) =ω(t) for all t ∈ R,ω ∈ Ω, such that Z has the same law withZ and satisfies the helix property: Z t+s (ω) = Z s (ω) + Z t (θ s ω), ∀ω ∈ Ω, t, s ∈ R.
Such stochastic process Z has also stationary increments and its every realization in C 0,p−var 0 (R, R). It is important to note that the existence ofZ is necessary to construct the diagonal process Z. For example ifZ is a fractional Brownian motion then the corresponding probability space (Ω,F ,P) can be constructed explicitly as in [12].
Next, we consider the stochastic differential equation where the second differential is understood in the path-wise sense as Young differential.
Proof: These statements are direct consequences of Theorem 2.1 and Proposition 2.3.
Now, we give a definition of the (random) Cauchy operator of the SDE (4.2). We recall the definition by Kunita (see [15, p. 114].

Definition 4.2 (Stochastic two-parameter flow)
A family of continuous R d −value random mappings, define on the probability space (Ω, F, P), Φ ω (s, t) : R d → R d depending on two real variables s, t ∈ [a, b] ⊂ R is call a stochastic two-parameter flow of homeomorphisms of R d on [a, b] if it satisfies following properties for any ω from a subset Ω ′ ⊂ Ω of full P-measure: u) holds for all s, t, u ∈ R + , where • denotes the composition of maps; (ii) Φ ω (s, s) is the identity map for all s ∈ R + ; (iii) the map Φ ω (s, t) : R d → R d is an onto homeomorphism for all s, t ∈ R + .  (ii) For any u ∈ [t 0 , ∞), the Lyapunov subspaces E u k (ω), k = 1, . . . , d, of Φ ω (s, t) are measurable with respect to ω ∈ Ω, and invariant with respect to the flow in the following sense  Then under assumption (H 1 ) and (H 2 ), Φ ω satisfies the following integrability condition for any where M 0 is determined by (3.1), 0 < µ < min{1, M 0 } and η = − log(1−µ), and we use the notation Proof: The proof follows directly from (3.6) for Φ and Ψ, and from (4.3), with note that for the inverse flow Ψ ω (s, t) and that (4.3) is still satisfied for all s, t ∈ [t 0 , t 0 + 1] due to the increment stationary property of Z.

Theorem 4.6
Under assumption (H 1 ) and (H 2 ) and condition (4.3), for each k = 1, . . . , d the Lyapunov exponent λ k (ω) is of finite moments in any order r > 0. More precisely, In particular, if (Ω, F, P, (θ t ) t∈R ) is ergodic, the Lyapunov spectrum is bounded a.s. by non-random constants as follow, Moreover, it is known in [12] that Z can be defined on a metric dynamical system (Ω, F, P, (θ t ) t∈R ) which is ergodic.

Almost sure Lyapunov regularity
In this subsection, for simplicity of presentation we consider all the equations in the whole time line R. The half-line case R + can be easily treated in a similar manner. We start the subsection with a very special situation in which the coefficient functions are autonomous, i.e. A(·) ≡ A, C(·) ≡ C. In this case, the stochastic two-parameter flow Φ ω (s, t) of (4.2) generates a linear random dynamical system Φ ′ (see e.g. Arnold [1,Chapter 1] for the definition of random dynamical systems). Indeed, from (4.1) and the fact that it follows due to the autonomy that Φ ω (s, t) = Φ(t − s, θ s ω). Hence Φ ′ (t, ω) := Φ ω (0, t) satisfies the cocycle property .
By applying the multiplicative ergodic theorem (see Oseledets [25] and Arnold [1, Chapter 3]) for Φ ′ generated from (4.2), there exists a Lyapunov spectrum consisting of exact Lyapunov exponents provided by the multiplicative ergodic theorem and it coincides with the Lyapunov spectrum defined in Definition 3.3. In addition, the flag of Oseledets' spaces coincides with the flag of Lyapunov spaces defined in Definition 3.3.
In general, it might not be true that system (4.2) is regular for almost sure ω. However, under the further assumptions of A, C, we can construct a linear random dynamical system such that almost sure the pathwise system is Lyapunov regular. The construction uses the so-called Bebutov flow, as investigated by Millionshchikov [20], [21] (see also [13], [26], [27]). Specifically, assume A satisfies a stronger condition that Consider the shift dynamical system S A t (A)(·) := A(· + t) in the space C b = C b (R, R d×d ) of bounded and uniformly continuous matrix-valued continuous function on R with the supremum norm. The closed hull H A := ∪ t S t (A) in C b is then compact, hence we can construct on H A a probability structure such that (H A , F A , µ A , S A ) is a probability space where µ A is a S-invariant probability measure, see e.g. [14,Theorem 4.9,p. 63].
When applying Millionshchikov's approach of using Bebutov flows to our system (4.2), we need to construct not only (H A , F A , µ A , S A ), but also (H C , F C , µ C , S C ), with a little more regularity condition for C. Recall that C 0,α−Hol ([a, b], R d×d ) is the closure of smooth paths from [a, b] to R d×d in α-Hölder norm and C 0,α−Hol (R, R d×d ) is the space of all x : R → R d×d such that x| I ∈ C 0,α−Hol (I, R d×d ) for each compact interval I ⊂ R, equipped with the compact open topology given by the Hölder norm, i.e the topology generated by metric  To construct a Bebutov flow for C, assume there exists α > 1 q such that C ∈ C 0,α−Hol (R, R d×d ) satisfies a condition stronger than (H 2 ): Consider the set of translations C r (·) := C(r + ·) ∈ C 0,α−Hol (R, R d×d ). Under conditions (4.7), Lemma 4.8 concludes that the closure set H C := {C r : r ∈ R} is compact on the separable complete metric space C 0,α−Hol (R, R d×d ), in fact θ t also preserves the norm on C 0,α−Hol (R, R d×d ). The shift dynamical system S C t c(·) = c(t + ·) maps H C into itself, hence by Krylov-Bogoliubov theorem [23,Chapter VI,§9], there exists at least one probability measure µ C on H C that is invariant under S C , i.e. µ C (S C t ·) = µ C (·), for all t ∈ R. It makes sense then to construct the product probability space B = H A × H C × Ω with the product sigma field F A × F C × F, the product measure µ B := µ A × µ C × P and the product dynamical system Θ = S A × S C × θ given by being the value at t of the (vector) solution x(·) which starts at x(0) = x 0 , satisfies the cocycle property due to the existence and uniqueness theorem and the fact that Therefore the nonautonomous linear YDE (4.8) generates a cocycle (random dynamical system) Φ * : R × B × R d → R d over the metric dynamical system (B, µ B ). Thus, starting from investigation of one linear stochastic nonautonomous YDE (4.2) we consider it ω-wise and embed to a Bebutov flow using Millionshchikov's approach [21], henceforth construct a random dynamical system over the product probability space for which we can apply the multiplicative ergodic theorem to get the following main result of this section. Proof: The integrability condition for the product probability measure µ B is a direct consequence of (4.4). Hence all the conclusions of the multiplicative ergodic theorem hold for almost all b ∈ B, which implies the Lyapunov regularity of (4.8) for almost all b ∈ B in the sense of the probability measure µ B . Remark 4.10 (i) In [20] and [21], Millionshchikov proved the Lyapunov regularity (almost surely with respect to an arbitrary invariant measure of the Bebutov flow on H A generated by the ordinary differential equationẋ = A(t)x), using the triangularization scheme provided by the Perron theorem for ordinary differential equations. In other words, Millionshchikov obtained an alternative proof of the multiplicative ergodic theorem (see also Arnold [1,p. 112], Johnson, Palmer and Sell [13]). In fact, Millionshchikov proved a bit stronger property than Lyapunov regularity that, almost all such systems are statistically regular.
(ii) Theorem 4.9 can be viewed as a version of multiplicative ergodic theorem for a nonautonomous linear stochastic Young differential equation which uses combination of Millionshchikov [21] approach (topological setting using Bebutov flow for differential equation) and Oseledets [25] approach (measurable setting with probability space (Ω, F, P)).
(iii) It is important to note that, although for almost all b ∈ B the nonautonomous linear stochastic (ω-wise) Young equation (4.8) is Lyapunov regular, it does not follow that the original system (4.2) is Lyapunov regular.

Discussions on the non-randomness of Lyapunov exponents
Since we are dealing with stochastic equation YDE (4.2) it is important and interesting to know whether its Lyapunov spectrum is nonrandom. We give here a brief discussion on this problem.
We remind the readers of the non-randomness of Lyapunov exponents λ 1 (ω), . . . , λ d (ω) for systems driven by standard Brownian noises (see e.g. [6,9]). Since (3.4) and Proposition 3.4 still hold in that situation, it follows that λ k (ω) is measurable with respect to the sigma algebra generate by {W (n + 1) − W (n) : n ≥ m} for any m ≥ 0, thus measurable w.r.t. the tail sigma field ∩ m σ({W (n + 1) − W (n) : n ≥ m}). Due to pairwise independence of all variables of the form W (n + 1) − W (n), one can apply Kolmogorov's zero-one law [14] to conclude that Lyapunov exponents are in fact non-random constants. Thus the first case we have nonrandomness of the Lyapunov spectrum is the case of nonautonomous linear stochastic differential equations driven by standard Brownian motions. Note that here the Lyapunov exponents of the systems can be nonexact.
In general, a stochastic process Z does not have independent increments, thus it is difficult to construct such a filtration and to apply the Kolmogorov's zero-one law. However, the Lyapunov spectrum is non-random for autonomous or periodic systems, provided that θ is ergodic. The second case of nonrandom Lyapunov spectrum is the case of autonomous or periodic linear stochastic Young equations discussed at the beginning of this subsection where we may apply the classical Oseledets MET by exploiting autonomy or periodicity of the system. Note that in this case the probability measure is the probability measure of the process Z and the Lyapunov exponents of the systems are exact.
The third case we have in hand is triangular nonautonomous linear stochastic Young differential equation treated in Section 3. In this case, due to the triangular form of the system we may solve it successively and use explicit formula of the solution to derive Theorem 3.10 describing the Lyapunov spectrum and to show that the Lyapunov spectrum consists of exact Lyapunov exponents and nonrandom. Thus, Lyapunov spectrum of a regular nonautonomous linear triangular system is nonrandom. Note that in this case the system in nonautonomous, the measure is the probability measure of the process Z and the Lyapunov exponents of the systems are exact. For a general system (4.2) which satisfies assumptions (H ′ 1 ), (H ′ 2 ) and (4.3), the statement on the non-randomness of Lyapunov spectrum depends on whether the product dynamical system Θ is ergodic on the product probability measure µ B , as a consequence of the Birkhorff ergodic theorem. The answer is then affirmative in case S A and S C are weakly mixing and θ is ergodic, i.e. S A (respectively S C ) satisfies the condition (respectively for S C ). It is well known (see e.g. Mañé [17, p. 147]) that the weak mixing of S A and S C implies the weak mixing of the product dynamical system S A × S C which, together with the ergodicity of θ, implies the ergodicity of the product flow Θ. The problem on non-randomness of Lyapunov spectrum can therefore be translated into the question on the weak-mixing of dynamical systems S A and S C .

Acknowledgments
This research is partly funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number FWO.101.2017.01.
For the "only if" part, assume (4.5) and (4.6) and prove the compactness ofH. SinceC is a complete metric space, it suffices to prove that every sequence {c n } ∞ n=1 ⊂ H has a convergent subsequence. Now following the arguments of [14, Theorem 4.9, p. 63] line by line, we can construct a convergent subsequence {c n } ∞ n=1 by the "diagonal sequence" such thatc n (r) → c(r) as n → ∞ for any rational number r ∈ Q. With  ≤ 4ε This implies |||c n − c||| α−Hol, [a,b] converge to 0 as n → ∞. This complete the proof.