Asymptotic Dynamics of Young Differential Equations

We provide a unified analytic approach to study the asymptotic dynamics of Young differential equations, using the framework of random dynamical systems and random attractors. Our method helps to generalize recent results (Duc et al. in J Differ Equ 264:1119–1145, 2018, SIAM J Control Optim 57(4):3046–3071, 2019; Garrido-Atienza et al. in Int J Bifurc Chaos 20(9):2761–2782, 2010) on the existence of the global pullback attractors for the generated random dynamical systems. We also prove sufficient conditions for the attractor to be a singleton, thus the pathwise convergence is in both pullback and forward senses.


Introduction
This paper studies the asymptotic behavior of the stochastic differential equation dy t = [Ay t + f (y t )]dt + g(y t )d Z t (1.1) where A ∈ R d×d , f : R d → R d , g : R d → R d×m are globally Lipschitz continuous functions, and Z is a two-sided stochastic process with stationary increments such that almost sure all realizations of Z are in the space C p−var (R, R m ) of continuous paths with finite p -variation norm, for some 1 ≤ p < 2. An example for such a process Z is a fractional Brownian motion B H [20] with Hurst index H > 1 2 . It is well known that Eq. (1.1) can be solved in the path-wise approach by taking a realization x ∈ C p−var (R, R m ) (which is also B Luu Hoang Duc duc.luu@mis.mpg.de ; lhduc@math.ac.vn Phan Thanh Hong hongpt@thanglong.edu.vn 1 Max-Planck-Institute for Mathematics in the Sciences, Leipzig, Germany 2 Institute of Mathematics, Viet Nam Academy of Science and Technology, Hanoi, Vietnam 3 Thang Long University, Hanoi, Vietnam called a driving path) and considering the Young differential equation dy t = [Ay t + f (y t )]dt + g(y t )dx t , t ∈ R + , y 0 ∈ R d .
(1.2) This way, system (1.2) is understood in the integral form where the second integral is understood in the Young sense [24]. The existence and uniqueness theorem for Young differential equations is proved in many versions, e.g. [6,[17][18][19]21,25].
Our aim is to investigate the role of the driving noise in the longterm behavior of system (1.1). Namely we impose assumptions for the drift coefficient so that there exists a unique equilibrium for the deterministic systemμ = Aμ+ f (μ) which is asymptotically stable; and then raise the questions on the asymptotic dynamics of the perturbed system, in particular the existence of stationary states and their asymptotic (stochastic) stability [14] with respect to almost sure convergence.
These questions could be studied in the framework of random dynamical systems [3]. Specifically, results in [11] and recently in [4,9] reveal that the stochastic Young system (1.1) generates a random dynamical system, hence asymptotic structures like random attractors are well-understood. In this scenarios, system (1.1) has no deterministic equilibrium but is expected to possess a random attractor, although little is known on the inside structure of the attractor and much less on whether or not the attractor is a (random) singleton.
We remind the reader of a well-known technique in [15,16,23] to generate RDS and to study random attractors of system (1.2) by a conjugacy transformation y t = ψ g (η t , z t ), where the semigroup ψ g generated by the equationu = g(u) and η is the unique stationary solution of the Langevin equation dη = −ηdt + d Z t . The transformed systeṁ (η t , z t ) Aψ g (η t , z t ) + f (ψ g (η t , z t )) + ηCψ g (η t , z t ) (1.4) can then be solved in the pathwise sense and the existence of random attractor for (1.4) is equivalent to the existence of random attractor for the original system. This conjugacy method works in some special cases, particularly if g(·) is the identity matrix, or more general if g(y) = Cy for some matrix C that commutes with A (see further details in Remark 3.10). For more general cases, the reader is refered to [8,9] and the references therein for recent methods in studying the asymptotic behavior of Young differential equations. Another approach in [8,11] uses the semigroup technique to estimate the solution norms, which then proves the existence of a random absorbing set that contains a random attractor for the generated random dynamical system. Specifically, thanks to the rule of integration by parts for Young integrals, the "variation of constants" formula for Young differential equations holds (see e.g. [25] or [8]), so that y t in Eq. (1.3) satisfies y t = (t)y 0 + t 0 (t − s) f (y s )ds + t 0 (t − s)g(y s )dx s , ∀t ≥ 0, (1.5) where (t) is the semigroup generated by A. By constructing a suitable stopping times {τ k } k∈N , one can estimate the Hölder norm of y on interval [τ n , τ n+1 ] in the left hand side of (1.5) by the same norm on previous intervals [τ k , τ k+1 ] for k < n following a recurrent relation, thereby can apply the discrete Gronwall Lemma (see Lemma 3.12 in the Appendix). However this construction of stopping times only works under the assumption that the noise is small in the sense that its Hölder seminorm is integrable and can be controlled to be sufficiently small.
In this paper, we propose a different approach in Lemma 3.3, which first estimates the Euclidean norm y n of y at time n in (1.5) by applying the continuous Gronwall Lemma, and then estimate Young integrals in the right hand side by the p-variation norms y p−var, [k,k+1] using Proposition 3.2. Thanks to Theorem 2.4 and its corollaries, these norms y p−var, [k,k+1] are estimated by y k , which leads to a recurrent relation between y n and previous terms y k . As a consequence, one can apply the discrete Gronwall Lemma 3.12 and yield a stability criterion in Theorem 3.4. Therefore, the method works for a general source of noises, and the stability criterion matches the classical one for ordinary differential equations when the effect of driving noise is cleared. Moreover, the same arguments can be applied for stochastic process Z with lower regularity (for instance Z is a fractional Brownian motion B H with 1 3 < H < 1 2 ), in that case equation (1.3) is no longer a Young equation but should be understood as a rough differential equation and can be solved by Lyon's rough path theory [18] (see also [10,22]).
The paper is organized as follows. Section 2 is devoted to present preliminaries and main results of the paper, where the norm estimates of the solution of (1.2) is then presented in Sect. 2.1. In Sect. 3.1, we introduce the generation of random dynamical system from the equation (1.1). Using Lemma 3.3, we prove the existence of a global random pullback attractor in Theorem 3.4. Finally in Sect. 3.3, we prove that the attractor is both a pullback and forward singleton attractor if g is a linear map in Theorem 3.9, or if g ∈ C 2 b for small enough Lipschitz constant C g in Theorem 3.11.

Preliminaries and Main Results
Let us first briefly make a survey on Young integrals. Denote by C([a, b], R r ), for r ≥ 1, the space of all continuous paths x : [a, b] → R r equipped with supremum norm x ∞,[a,b] = sup t∈ [a,b] x t , where · is the Euclidean norm of a vector in R r . For p ≥ 1 and [a, b] ⊂ R, denote by C p−var ([a, b], R r ) the space of all continuous paths x ∈ C([a, b], R r ) which is of finite p−variation, i.e. |||x||| p−var, [a,b] where the supremum is taken over the whole class (a, b) of finite partitions = {a = t 0 < t 1 < · · · < t n = b} of [a, b] (see e.g. [10]). Then C p−var ([a, b], R r ), equipped with the p−var norm x p−var, [a,b] := x a + |||x||| p−var, [a,b] , is a nonseparable Banach space [10,Theorem 5.25,p. 92]. Also for each 0 < α < 1, denote by C α−Hol ([a, b], R r ) the space of Hölder continuous paths with exponent α on [a, b], and equip it with the norm x α−Hol, [a,b] We recall here a result from [6, Lemma 2.1].
From now on, we only consider q = p for convenience. We impose the following assumptions on the coefficients A, f and g and the driving path x. Assumptions (H 1 ) A ∈ R d×d is a matrix which has all eigenvalues of negative real parts; (H 2 ) f : R d → R d and g : R d → R d×m , are globally Lipschitz continuous functions. In addition, g ∈ C 1 such that D g is also globally Lipschitz continuous. Denote by C f , C g the Lipschitz constants of f and g respectively; (H 3 ) for a given p ∈ (1, 2), Z t is a two-sided stochastic process with stationary increments such that almost sure all realizations belong to the space C p−var (R, R m ) and that For instance, Z could be an m−dimensional fractional Brownian motion B H [20] with Hurst exponent H ∈ ( 1 2 , 1), i.e. a family of centered Gaussian processes B H = {B H t }, t ∈ R or R + with continuous sample paths and the covariance function Assumption (H 1 ) ensures that the semigroup (t) = e At , t ∈ R generated by A satisfies the following properties.

Proposition 2.2
Assume that A has all eigenvalues of negative real parts. Then there exist constants C A ≥ 1, λ A > 0 such that the generated semigroup (t) = e At satisfies Proof The first inequality is due to [1, Chapter 1, §3]. The second one is followed from the mean value theorem where e −λ A · is a decreasing function.

5)
where ( p) is defined in (2.2) and K in (2.7), then the generated random dynamical system ϕ of (1.1) possesses a pullback attractor A(x). Moreover, in case g(y) = Cy + g(0) is a linear map satisfying (2.5) or in case g ∈ C 2 b with the Lipschitz constant C g small enough, this attractor is a singleton, i.e. A(x) = {a(x)} a.s., thus the pathwise convergence is in both the pullback and forward directions.
For the convenience of the readers, we introduce some notations and constants which are used throughout the paper. (2.8) (2.10) (2.11)

Solution Estimates
In this preparatory subsection we are going to estimate several norms of the solution. To do that, the idea is to evaluate the norms of the solution on a number of consecutive small intervals. Here we would like to construct, for any γ > 0 and any given interval [a, b], a sequence of greedy times {τ k (γ )} k∈N as follows (see e.g. [5,6,8])  [a,b] . (2.14) From now on, we fix p ∈ (1, 2) and γ := We assume throughout this section that the assumption (H 2 ), (H 3 ) are satisfied. The following theorem presents a standard method to estimate the p−variation and the supremum norms of the solution of (1.2), by using the continuous Gronwall lemma and a discretization scheme with the greedy times (2.12).

Theorem 2.4
There exists a unique solution to (1.2) for any initial value, whose supremum and p−variation norms are estimated as follows As a result,  (2.12), it follows from induction that On the other hand, it follows from inequality of p-variation seminorm in Lemmas 2.1 and (2.18) that for all which proves (2.16).
By the same arguments, we can prove the following results.

Corollary 2.5 If in addition g is bounded by g
Corollary 2. 6 The following estimate holds The lemma below is useful in evaluating the difference of two solutions of Eq. (1.2). The proof is similar to [6, Lemma 3.1] and will be omitted here.

i) If in addition, D g is of Lipschitz continuity with Lipschitz constant C g , then
Thanks to Lemma 2.7, the difference of two solutions of (1.2) can be estimated in p-var norm as follows.
Corollary 2.8 Let y 1 , y 2 be two solutions of (1.2) and assign z t = y 2 t − y 1 t for all t ≥ 0. (i) If D g is of Lipschitz continuity with Lipschitz constant C g then (

2.24)
Proof The proof use similar arguments to the proof of Theorem 2.4, thus it will be omitted here. The the readers are referred to [19,Proposition 1], [6, Theorem 3.9] for similar versions.
In the following, let y be a solution of (1.2) on [a, b] ⊂ R + and μ be the solution of the corresponding deterministic system, i.ė with the same initial condition μ a = y a . Assign h t := y t − μ t . The following result, which is used in studying singleton attractors in Theorem 3.11, estimates the norms of h with the initial condition y a , up to a fractional order.

Corollary 2.10 Assume that g is bounded. Then for a fixed constant
The proof follows similar steps to [13,Proposition 4.6] with only small modifications in estimates for p -variation norms and in usage of the continuous Gronwall lemma. To sketch out the proof, we first observe from (2.26) with (2.29) Next, it follows from H 2 and the boundedness of g by g ∞ that (2.30) Observe that due to the boundedness of g, where the last estimate is due to (2.29) and D is a generic constant depending on b − a. This leads to  [s,u] [s,t] +K C g |||x||| p−var, [s,t] |||h||| p−var, [s,t] which is similar to (2.17). Using similar arguments to the proof of Theorem 2.4 and taking into account (2.19), we conclude that for a generic constant D. Finally, (2.27) is derived since h a = 0. The estimate (2.28) is obtained similarly.

Generation of Random Dynamical Systems
In this subsection we would like to present the generation of a random dynamical system from Young equation (1.1). Let ( , F , P) be a probability space equipped with a so-called metric dynamical system θ , which is a measurable mapping θ : ω)y 0 is then defined as a measurable mapping which is also continuous in t and y 0 such that the cocycle property is satisfied [3]. In our scenario, denote by with the compact open topology given by the p−variation norm, i.e. the topology generated by the metric: Assign and equip with the Borel σ − algebra F . Note that for x ∈ C 0, p−var 0 (R, R m ), |||x||| p−var,I and x p−var,I are equivalent norms for every compact interval I containing 0. To equip this measurable space ( , F ) with a metric dynamical system, consider a stochastic processZ defined on a probability space (¯ ,F,P) with realizations in (C Assume further thatZ has stationary increments. Denote by θ the Wiener shift It is easy to check that θ forms a continuous (and thus measurable) dynamical system (θ t ) t∈R on (C 0, p−var 0 (R, R), F ). Moreover, the Young integral satisfies the shift property with respect (see details in [6, p. 1941]). It follows from [4,Theorem 5] that, there exists a probability P on ( , F ) = (C 0, p−var 0 (R, R m ), F ) that is invariant under θ , and the so-called diagonal process Z : R × → R m , Z (t, x) = x t for all t ∈ R, x ∈ , such that Z has the same law withZ and satisfies the helix property: Such stochastic process Z has also stationary increments and almost all of its realizations belongs to C 0, p−var 0 (R, R m ). It is important to note that the existence ofZ is necessary to construct the diagonal process Z . When dealing with fractional Brownian motion [20], we can start with the space C 0 (R, R m ) of continuous functions on R vanishing at zero, with the Borel σ −algebra F , and the Wiener shift and the Wiener probability P, and then follow [12, Theorem 1] to construct an invariant probability measure P H = B H P on the subspace C ν such that B H • θ = θ • B H . It can be proved that θ is ergodic (see [12]). Under this circumstance, if we assume further that (2.2) is satisfied, then it follows from Birkhorff ergodic theorem that (see details in [6,Theorem 3.8]). Define the mapping ϕ(t, x)y 0 := y t (x, y 0 ), for t ∈ R, x ∈ , y 0 ∈ R d , which is the pathwise solution of (1.1), then it follows from the existence and uniqueness theorem and the Wiener shift property (3.2) that ϕ satisfies the cocycle property (3.1) (see [6,Subsection 4.2] and [11] for more details). Also, it is proved in [6,Theorem 3.9] that the solution y t (x, y 0 ) is continuous w.r. t. (t, x, y 0 ), hence given a probability structure on , ϕ is a continuous random dynamical system.

Existence of Pullback Attractors
Given a random dynamical system ϕ on R d , we follow [7], [3,Chapter 9] to present the notion of random pullback attractor. Recall that a setM := {M(x)} x∈ a random set, if y → d(y|M(x)) is F -measurable for each y ∈ R d , where d(E|F) = sup{inf{d(y, z)|z ∈ F}|y ∈ E} for E, F are nonempty subset of R d and d(y|E) = d({y}|E). An universe D is a family of random sets which is closed w.r.t. inclusions (i.e. ifD 1 ∈ D andD 2 ⊂D 1 then D 2 ∈ D).
In our setting, we define the universe D to be a family of tempered random sets D(x), which means the following: A random variable ρ(x) > 0 is called tempered if it satisfies An invariant random compact set A ∈ D is called a pullback random attractor in D, if A attracts any closed random setD ∈ D in the pullback sense, i.e.

5)
A is called a forward random attractor in D, if A is invariant and attracts any closed random setD ∈ D in the forward sense, i.e.
The existence of a random pullback attractor follows from the existence of a random pullback absorbing set (see [7,Theorem 3]). A random set B ∈ D is called pullback absorbing in a universe D if B absorbs all sets in D, i.e. for anyD ∈ D, there exists a time t 0 = t 0 (x,D) such that Given a universe D and a random compact pullback absorbing set B ∈ D, there exists a unique random pullback attractor in D, given by Since the rule of integration by parts for Young integral is proved in [25], the "variation of constants" formula for Young differential equations holds (see e.g. [8]), i.e. y t satisfies We need the following auxiliary results.
The following lemma is the crucial technique of this paper. (3.9). Then for any n ≥ 0,
Proof First, for any t ∈ [n, n + 1), it follows from (2.3) and the global Lipschitz continuity of f that where β t := t 0 (t − s)g(y s )dx s . Multiplying both sides of the above inequality with e λ A t yields By applying the continuous Gronwall Lemma [2, Lemma 6.1, p 89], we obtain Once again, multiplying both sides of the above inequality with e −L f t yields where we use the fact that (3.14) The continuity of y at t = n + 1 then proves (3.11).
We now formulate the first main result of the paper.
holds, where ( p) is defined in (2.2), then the generated random dynamical system ϕ of system (1.1) possesses a pullback attractor A.
Step 3. Notice that ( , we obtain the temperedness ofb(x) in the sense of (3.4). We conclude that there exists a compact absorbing set B(x) =B(0,b(x)) and thus a pullback attractor A(x) for system (1.2) in the form of (3.8).
To complete the proof of Theorem 3.4, we now formulate and prove Proposition 3.5.
Hence the supremum in the definition of b(x) in (3.24) can be taken for rational , which proves b(x) to be a random variable on .
To prove the temperdness of b, observe that for each t ∈ [n, n for t ∈ n . It follows that all other solutions converge exponentially in the pullback sense to the trivial solution, which plays the role of the global pullback attractor.

Remark 3.7
In [8,11] the authors consider a Hilbert space V together with a covariance operator Q on V such that Q is of a trace-class, i.e. for a complete orthonormal basis (e i ) i∈N of V , there exists a sequence of nonnegative numbers (q i ) i∈N such that tr(Q) : i ) i∈N are stochastically independent scalar fractional Brownian motions of the same Hurst exponent H . The authors then develop the semigroup method to estimate the Hölder norm of y on intervals τ k , τ k+1 where τ k is a sequence of stopping times for some μ ∈ (0, 1) and β > 1 p , which leads to the estimate of the exponent as where C(C A , μ) is a constant depending on C A , μ. It is then proved that there exists lim inf n→∞ τ n n = 1 d , where d = d(μ, tr(Q)) is a constant depending on the moment of the stochastic noise. As such the exponent is estimated as − λ A − C(C A , μ)e λ A μ max{C f , C g }d(μ, tr(Q)) .
(3.31) However, it is technically required from the stopping time analysis (see [8,Section 4]) that the stochastic noise has to be small in the sense that the trace tr(Q) = ∞ i=1 q i must be controlled as small as possible. In addition, in case the noise is diminished, i.e. g ≡ 0, (3.31) reduces to a very rough criterion for exponential stability of the ordinary differential equation In contrast, our method uses the greedy time sequence in Theorem 2.4, so that later we can work with the simpler (regular) discretization scheme without constructing additional stopping time sequence. Also in Lemma 3.3 we apply first the continuous Gronwall lemma in (3.12) in order to clear the role of the drift coefficient f . Then by using (2.15) to give a direct estimate of y k , we are able apply the discrete Gronwall Lemma directly and obtain a very explicit criterion. The left and the right hand sides of criterion (2.5) can be interpreted as, respectively, the decay rate of the drift term and the intensity of the diffusion term, where the term e λ A +4(|A|+C f ) is the unavoidable effect of the discretization scheme. Criterion (2.5) is therefore a better generalization of the classical criterion for stability of ordinary differential equations, and is satisfied if either C g or ( p) is sufficiently small. In particular, when g ≡ 0, (2.5) reduces to λ A > C A C f , which matches to the classical result.

Singleton Attractors
In the rest of the paper, we would like to study sufficient conditions for the global attractor to consist of only one point, as seen, for instance, in Corrollary 3.6. First, the answer is affirmative for g of linear form, as proved in [9] for dissipative systems. Here we also present a similar version using the semigroup method.
To begin, let y 1 , y 2 be two solutions of (1.2) and assign z t = y 2 t − y 1 t for all t ≥ 0. Similar to (3.9), z satisfies Observe that by similar computations to (3.10), it is (3.33) We need the following auxiliary result.
where β t = t 0 (t −s)Q(s, z s )dx s is estimated using (3.33). The rest will be omitted.

Theorem 3.9
Assume that g(y) = Cy + g(0) is a linear map so that C g = |C|. Then under the condition (2.5), the pullback attractor is a singleton, i.e. A(x) = {a(x)} almost surely. Moreover, it is also a forward singleton attractor.
Proof The existence of the pullback attractor A is followed by Theorem 3.4. Take any two points a 1 , a 2 ∈ A(x). For a given n ∈ N, assign x * := θ −n x and consider the equation where G is defined in (3.18). Now applying the discrete Gronwall Lemma 3.12, we conclude that e λn z n ≤ C A z 0 (3.38) Similar to (3.25) under the condition (2.5). This follows that lim n→∞ a 1 − a 2 = 0 and A is a one point set almost surely. Similar arguments in the forward direction (x * is replaced by x) also prove that A is a forward singleton attractor almost surely.

Remark 3.10
As pointed out in the Introduction section, if we use the conjugacy transformation (developed in [15,16,23]) of the form y t = e Cη t z t , where the semigroup e Ct is generated by the equationu = Cu and η is the unique stationary solution of the Langevin equation dη = −ηdt + d Z t (with Z is a scalar stochastic process), then the transformed system has the formż t = e −Cη t Ae Cη t z t + f (e Cη t z t ) + η t Ce Cη t z t .
However, even in the simplest case f ≡ 0, there is no effective method to study the asymptotic stability of the non-autonomous linear stochastic systeṁ An exception is when A and C are commute, since we could reduce system (3.39) in the formż thereby solve it explicitly as In this case, the exponential stability is proved using the fact that exp{−C(η t −η 0 − Z t + Z 0 )} is tempered. However, since A and C are in general not commute, we can not apply the conjugacy transformation but should instead use our method described in Theorems 3.4 and 3.9.
Next, motivated by [13], we consider the case in which g ∈ C 2 b and C g is also the Lipschitz constant of Dg. Notice that our conditions for A and f can be compared similarly to the dissipativity condition in [13]. However, unlike the probabilistic conclusion of existence and uniqueness of a stationary measure in [13], we go a further step by proving that for C g small enough, the random attractor is indeed a singleton, thus the convergence to the attractor is in the pathwise sense and of exponential rate. and (3.40). By choosing r > 0 large enough and ∈ (0, 1) small enough such that C A e −λr < 1 and η := (1 + ) 2(2 p−1) (C A e −λr ) 2 p + β < 1 we obtain (3.41).
Step 2. Next, for simplicity we only estimate y at the discrete times nr for n ∈ N, the estimate for t ∈ [nr, (n +1)r ] is similar to (3.21). From (3.41), it is easy to prove by induction that y nr (x, y 0 ) 2 p ≤ η n y 0 2 p + n−1 i=0 η i ξ r (θ (n−i)r x), ∀n ≥ 1; thus for n large enough y nr (θ −nr x, y 0 ) 2 p ≤ η n y 0 In this case we could chooseb(x) in (3.27) to beb(x) = R r (x) 1 2 p so that there exists a pullback absorbing set B(x) = B(0,b(x)) containing our random attractor A(x). Moreover, due to the integrability of ξ r (x), R r (x) is also integrable with ER r = 1 + Eξ r 1−η .
Step 3. Now back to the arguments in the proof of Theorem 3.9 and note that Dg is also globally Lipschitz with the same constant C g . Using Lemma 2.7 (i) and rewriting (3.34  All together, I k is bounded from above by where the right hand side of (3.48) is a function of θ (k−n) x. (1 + α j ).