Nonautonomous Young Differential Equations Revisited

In this paper we prove that under mild conditions a nonautonomous Young differential equation possesses a unique solution which depends continuously on initial conditions. The proofs use estimates in p-variation norms, the construction of greedy sequence of times, and Gronwall-type lemma with the help of Shauder theorem of fixed points.


Introduction
This paper deals with the Young differential equation of the form where f : R × R d → R d and g : R × R d → R d×m are continuous functions, ω is a R m -valued function of finite p-variation norm for some 1 < p < 2. Such type of system is generated from stochastic differential equations driven by fractional Brownian noises, as seen e.g. in [20]. Equation (1.1) is understood in the integral form where the first integral is of Riemannian type, meanwhile the second integral can be defined in the Young sense [23]. The existence and uniqueness of the solution of (1.2) are studied by several authors. When f, g are time-independent, system (1.2) is proved in [23] and [22] and [18] to have a unique solution in a certain space of continuous functions with bounded p-variation. The result is then generalized for the case 2 ≤ p < 3 in Lyons' seminal paper [19] in which rough path theory is introduced to define the second integral in (1.2) and also the integration equation (see [8], [16] and [17]). An alternative theory of controlled paths in Gubinelli's work [10] simplifies and generalizes the concept of integration and differential equations, leading to the concept of rough differential equations (see recent works in [3] and [2] for 2 ≤ p < 3, or [14] for controlled differential equations as Young integrals). According to their settings, f, g are time independent and g is often assumed to be differentiable upto a certain order and bounded in its derivatives.
A different approach following Zähle [24] by using fractional derivatives can be seen in [21] which derives very weak conditions for time varying f and g in (1.1), in particular g need to be only C 1 with bounded and Hölder continuous first derivative, to ensure the existence and uniqueness of the solution in the space of Hölder continuous functions. Later, one finds that there is a connection between the rough path approach and the techniques in fractional calculus, see e.g. [11] and [12].
Our aim in this paper is to close the gap between the two methods for nonautonomous Young equations by proving that, under similar assumptions to those of Nualart and Rascanu [21], the existence and uniqueness theorem for system (1.1) still holds in the space of continuous functions with bounded p-variation norm. For that to work, we construct the socalled greedy sequence of times (see [4,Definition 4.7]) such that the solution can be proved to exists uniquely in each interval of the consecutive times of the greedy sequence, and is then concatenated to form a global solution. It is important to note that since we are using estimates for p-variation norms, we do not apply the classical arguments of contraction mappings, but use Shauder-Tychonoff fixed point theorem as seen in [18] and a Gronwall-type lemma.
Another issue is the generation of flow which was asserted in [17] for the autonomous systems. The idea is to construct the shift dynamical system in the extended space of finite p-variation norm for the whole real line time, and to prove that the system generates a nonautonomous dynamical system satisfying the cocycle property (see [3]). When applying to stochastic differential equations driven by fractional Brownian motions, by considering an appropriate probability space, one can prove that the system generates a random dynamical system (see [3,5,9]). However in the nonautonomous situation, one only expects a generation of a two-parameter flow on the phase space.
The paper is organized as follows. In Sect. 2, the Young integral is introduced and a version of greedy sequence of times is presented. In Sect. 3, we prove the existence and uniqueness of the global solution of system (1.2) in Theorem 3.6, for this we need to formulate a Gronwalltype lemma. Proposition 3.7 gives an estimate of q-var norm of solution via p-var norm of the driver ω. We also prove the existence and uniqueness of the solution of the backward equation (3.23) in Theorem 3.8. In Sect. 4, the fact in Theorem 4.1 that two trajectories do not intersect helps to conclude that the Cauchy operator or the Ito map of (1.2) generates a continuous two parameter flow. In the autonomous case it generates a continuous nonautonomous dynamical system which helps to form a topological skew product flow.

Young Integral
In this section we recall some facts about Young integral, more details can be seen in [8].
where the supremum is taken over the whole class of finite partition of [a, b]. The subspace It is easy to prove (see [8,Corollary 5.33, p. 98]) that for 1 ≤ p < p we have Also, for 0 < α ≤ 1 denote by C α-Hol ([a, b], R d ) the Banach space of all Hölder continuous paths x : [a, b] → R d with exponential α, equipped with the norm x α-Hol, [a,b] := |x a | + |||x||| α-Hol, [a,b] = |x a | + sup Hence, for all p such that pα ≥ 1 we have Proof The proof is similar to the one in [8, p. 84], by using triangle inequality and power means inequality [s,t] , where x is of bounded p-variation norm on [a, b] and q ≥ p are some examples of control function. The following lemma gives a useful property of controls in relation with variations of a path (see [8] for more properties of control functions).

Lemma 2.3 Let ω j be a finite sequence of control functions on
This implies the conclusion of the lemma.
converges as the mesh | | := max 0≤i≤n−1 |t i+1 − t i | tends to zero, we call the limit is the It is well known that if p, q ≥ 1 and 1 p + 1 q > 1, the Young integral b a x t dω t exists (see [23, pp. 264-265]). Moreover, if x n and ω n are of bounded variation, uniformly bounded in C q ([a, b], R d×m ), C p ([a, b], R m ) and converges uniformly to x, ω respectively, then the sequence of the Riemann-Stieljes integral b a x n t dω n t approach b a x t dω t as n → ∞ (see [8]). This integral satisfies additive property by the construction, and the so-called Young-Loeve estimate [8,Theorem 6.8 [s,t] |||ω||| p-var, [s,t] , (2.6) where Proof The conclusion is a direct sequence of (2.6) and [8, Proposition 5.10(i), p. 83].
Due to Lemma 2.4, the integral t → t a x s dω s is a continuous bounded p-variation path. Note that the definition of Young integral does depend on the direction of integration in a simple way like the Riemann-Stieltjes integral. Namely, it is easy to see that

The Greedy Sequence of Times
The original idea of a greedy sequence was introduced in [4, Definition 4.7]. Given α > 0, a compact interval I ∈ R and a control ω : (I ) → R + , the construction of such a sequence aims to have a "greedy" approximation to the supremum in the definition of the so-called accummulated α-local ω-variation (see [4,Definition 4.1]) In particular, ω s,t is chosen to be |||·||| p p-var, [s,t] in [4]. A similar version for stopping times was developed before in [9] and then has been studied further recently by [7] for stability of the system. Here we propose another version of greedy sequence of times which matches with the nonautonomous setting.
Let n ∈ N, observe that metric d satisfies where the second inequality holds for any fixed n and ω 1 , ω 2 close enough such that 2 n d(ω 1 , ω 2 ) < 1. Hence every Cauchy sequence (ω k ) k w.r.t. metric d is also a Cauchy sequence when restricted to C p ([−n, n], R m ), thus converges to a limit ω * ∈ . For any given λ, μ > 0 we construct a strict increasing sequence of times {τ n }, To do so, first define τ : Observe that the function κ(t) := t λ + |||ω||| p-var,[0,t] is continuous and stricly increasing w.r.t. t with κ(0) = 0 and κ(∞) = ∞, therefore due to the continuity there exists a unique τ = τ (ω) > 0 such that Thus τ is well defined. Next, we construct the time sequence inductively as follows. Set τ 0 := 0, τ 1 (ω) := τ (ω). Suppose that we have defined τ n (ω) for n ≥ 1, looking at the following equality as an equation of δ n (ω) ∈ R + , like above we find an unique δ n (ω) such that where δ n (ω) is determined above. Thus we have defined a time sequence {τ n } for all n = 0, 1, 2, . . .. Such a sequence then satisfies (2.11). Now, we fix ω ∈ C p (R, R m ) and consider the number of times of the greedy sequence inside an arbitrary finite interval of R + . We write τ n for τ n (ω) to simplify the notation. For given T > 0, we introduce the notation (2.14) or more generally, for any 0 ≤ a < b < ∞, More generally, Consequently, we obtain Similarly, (2.17) holds.
Remark 2.7 (i) Since the left-hand side of (2.18) tends to infinity as n goes to ∞ its right hand side can not be bounded. This implies that τ n → ∞ as n → ∞. (ii) We can construct a time sequence starts at τ 0 = t 0 , an arbitrary point in R, and on (−∞, t 0 ] in a similar manner.

Existence and Uniqueness Theorem
In this section, we are working with the restriction of any trajectory ω in a given time interval [0, T ] by considering it as an element in C p ([0, T ], R m ), for a certain p ∈ (1, 2) (see Remark 2.5 for the relation between ω ∈ C p (R, R m ) and its restrictions). Consider the Young differential equation in the integral form as: We recall here a result in [21] on existence and uniqueness of solution of (3.1), which was proved using contraction mapping arguments with ω in a Besov-type space. In this paper we however would like to derive a proof in C p applying Shauder fixed point theorem and greedy sequence of times tool. First we need to formulate some assumptions on the coefficient functions f and g of (3.1).
x) is differentiable in x and there exist some constants 0 < β, δ ≤ 1, a control function h(s, t) defined on [0, T ] and for every N ≥ 0 there exists M N > 0 such that the following properties hold: (H 2 ) There exists a > 0 and b ∈ L where 1 2 ≤ α < 1, and for every N ≥ 0 there exists L N > 0 such that the following properties hold:

(H 3 ) The parameters in H 1 and H 2 statisfy the inequalities
We would like to study the existence and uniqueness of the solution of (3.1) under the given conditions that By the assumption p ∈ (1, 2) and the condition H 3 can choose consecutively constants q 0 , q such that Then, we have Define the mapping given by It can be seen from the above assumptions that |g(t, For the next proposition we need the following auxiliary lemma. Take the superemum over the set of all finite partition we get g(·, x . ) ∈ C q 0 ([t 0 , t 1 ], R d×m ) and (iii) Note that q 0 β > 1 and q 0 δ ≥ q hence The lemma is proved.
For a proof of our main theorem on existence and uniqueness of solutions of an Young differential equation, we need the following proposition.

Proposition 3.2
Assume that H 1 − H 3 are satisfied. Let 0 ≤ t 0 < t 1 ≤ T be arbitrary, q be chosen as above satisfying (3.3) and F be defined by (3.5). Then for any Moreover, the following statements hold
(ii) By virtue of (2.8), (3.10) and the condition x t 0 = y t 0 , we have Similarly,

Inequality (3.12) is a direct consequence of these estimates for I (x) and J (x).
Before proving the existence and uniqueness theorem, we need the following lemma of Gronwall type. where [s,t] ). [u,v]
Hence, where N (s, t, ω) is defined in (2.15). We have s ≤ t N < t N +1 < · · · < t N ≤ t and where C = 4 p c p ln 2. The proof is complete.

Remark 3.4 (i) Gronwall
Lemma is an important tool in the theory of ordinary differential equations, and the theory of Young differential equations as well. Some versions of Gronwall-type lemma can be seen in [21] and [6].  . (3.18) We are now at the position to state and prove the main theorem of this section.

Corollary 3.5 If in Lemma
with T being an arbitrary fixed positive number and x 0 ∈ R d being an arbitrary initial condition. Assume that the conditions H 1 −H 3 hold. Then, this equation has a unique solution where q is chosen as above satisfying (3.3). Moreover, the Proof The proof proceeds in several steps.
Step 1: In this step we will show the local existence and uniqueness of solution. Set , (3.19) where M is defined in (3.6) and K is defined in (2.7). Let s 0 ∈ [t 0 , T ) be arbitrary but fixed. We recall here the time sequence τ n with the parameters α, μ, i.e Put r 0 = min{n : τ n > s 0 } and define s 1 = min{τ r 0 , T }. Then, We will show that the Eq. (3.1) restricted to [s 0 , s 1 ] has a unique solution.

Existence of local solutions.
Recall the mapping F defined by the formula (3.5) with t 0 , t 1 replaced by s 0 , s 1 , respectively. By Proposition 3. We show furthermore that if [s,t] Taking into account (3.12), F : B 1 → B 1 is continuous. We show that B 1 is a closed convex set in the Banach space C q ([s 0 , s 1 ], R d ), and F is a compact operator on B 1 . Indeed, for the former observation, note that if z = λx + (1 − λ)y for some x, y ∈ B 1 , λ ∈ [0, 1] then Now, we prove that for any sequence y n ∈ F(B 1 ), there exists an subsequence converges in p-var norm to an element y ∈ B 1 , i.e. F(B 1 ) is relatively compact in B 1 . To do that, we will show that (y n ) are equicontinuous, bounded in (q − )-var norm. Namely, take the sequence y n = F(x n ) ∈ F(S), x n ∈ B 1 . Then, by virtue of Lemma 2.3 we have It means that y n are bounded in C([s 0 , s 1 ], R d ) with sup norm, as well as bounded in Moreover, for all n, s 0 ≤ s ≤ t ≤ s 1 , which implies that (y n ) is equicontinuous. Applying Proposition 5.28 of [8], we conclude that y n converges to some y along a subsequence in C q ([s 0 , s 1 ], R d ). This proves the compactness Step 2: Next, by virtue of the additivity of the Riemann and Young integrals, the solution can be concatenated. Namely, let 0 < t 1 < t 2 < t 3 ≤ T . Let x t be a solution of the Eq. Furthermore, for all such that q − ≥ p the solution x belongs to   [s,t] ) (t − s) α + |||ω||| p-var, [s,t] .
Use the arguments similar to that of the proof of Lemma 3.3 we conclude that there exist C 1 = C 1 (T ) and C 2 = C 2 (T ) such that (3.22) is satisfied.
In order to study the flow generated by the solution of system (3.1) in the next section, we need also to consider the backward version of (3.1) in the following form where x T ∈ R d is the initial value of the backward equation (3.23), the coefficient functions Furthermore, by virtue of the property (2.9) of the Young integral we have Therefore, the backward equation (3.23) is equivalent to the forward equation where y 0 = x T ∈ R d . Now, we verify the conditions of Theorem 3.6 for the forward equation (3.24).

Topological Flow Generated by Young Differential Equations
In this section we show that the solution of a nonautonomous Young differential equation generates a two-parameter flow on the phase space R d , thus we can study the long term behavior of the solution flow using the tools of the theory of dynamical systems. We also discuss the autonomous situation, in which we show that the solution then satisfies the cocycle property, thus generates a topological skew product flow. The reader is referred to the work [15] and [18], [17] for the smoothness and diffeomorphism property of the flow.
For simplicity of the presentation, we will assume from now on that for any given T > 0 all hypotheses H 1 − H 3 hold on [0, T ]. Following [13, p. 114], below we introduce the concept of two parameter flows.

Definition 4.5 (Two-parameter flow)
A family of mappings X s,t : R d → R d depending on two real variables s, t ∈ [a, b] ⊂ R is call a two-parameter flow of homeomorphisms of R d on [a, b] if it satisfies the following conditions: (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]; (iii) X −1 s,t = X t,s for any s, t ∈ [a, b]; (iv) X s,t = X u,t • X s,u for any s, t, u ∈ [a, b]. Theorem 4.6 (Two-parameter flow generated by Young differential equations) Assume that the conditions H 1 − H 3 hold on any compact interval of R. The family of Cauchy operators of (1.1) generates a two parameter flow of homeomorphisms of R d . Namely, for −∞ < The problem of generation of the random dynamical systems [1] from stochastic differential equations driven by fractional Brownian noise has been discussed in [3,5,9], to name a few, where they solve the stochastic equation in the path-wise sense as in (4.4) for each realization ω of the fractional Brownian motion. Here in our deterministic setting, due to the fact that the shift dynamical system θ and the solution Cauchy operator are continuous, it follows that the skew product flow defined by is a continuous mapping which satisfies the group property, i.e. t+s = t • s , for all t, s ∈ R. Therefore it is a topological skew-product dynamical system.