Luận án Some qualitative problems of nonautonomous stochastic differential equations driven by fractional brownian motions

− Cg|zu|. We choose t0 = n0r, n0 ∈ Z−. Repeat the arguments in Lemma 4.1 |zt|eλ(t−t0) ≤ CA|zn0|+ eλAKCA(1+ CA‖A‖r) n ∑ k=n0 |||ω|||p−var,∆rk e λ(kr−n0r) × ×Cg‖z‖p−var,∆rk [ 1+ r + ∣∣∣∣∣∣∣∣∣x1∣∣∣∣∣∣∣∣∣ p−var,∆rk + ∣∣∣∣∣∣∣∣∣x2∣∣∣∣∣∣∣∣∣ p−var,∆rk ] , ∀t ∈ ∆rn. (4.43) Now we follow Proposition 2.1 to estimate ‖z‖p−var,∆rk . Namely, for s, t ∈ ∆rk |zt − zs| ≤ L ∫ t s |zu|du +KCg |||ω|||p-var,[s,t] [ 1+ r + ∣∣∣∣∣∣∣∣∣x1∣∣∣∣∣∣∣∣∣ p−var,∆rk + ∣∣∣∣∣∣∣∣∣x2∣∣∣∣∣∣∣∣∣ p−var,∆rk ] ‖z‖p−var,[s,t], which yields ‖z‖p−var,[s,t] ≤ |zs|+ ∫ t s L‖z‖p−var,[s,u]du +KCg |||ω|||p−var,[s,t] [ 1+ r + ∣∣∣∣∣∣∣∣∣x1∣∣∣∣∣∣∣∣∣ p−var,∆k + ∣∣∣∣∣∣∣∣∣x2∣∣∣∣∣∣∣∣∣ p−var,∆k ] ‖z‖p−var,[s,t]. 91 Hence ‖z‖p−var,[s,t] ≤ ∫ t s 2L‖z‖p−var,[s,u]du + 2|zs|, and by using Gronwall lemma A1, it follows that ‖z‖p−var,[s,t] ≤ 2|zs|+ ∫ t s 2Le2L(t−u)2|zs|du ≤ 2e2L(t−s)|zs| whenever Λ′(ω, [s, t]) := 2KCg |||ω|||p−var,[s,t] [ 1+ r + ∣∣∣∣∣∣∣∣∣x1∣∣∣∣∣∣∣∣∣ p−var,[s,t] + ∣∣∣∣∣∣∣∣∣x2∣∣∣∣∣∣∣∣∣ p−var,[s,t] ] ≤ 1. Again, similar arguments to the proof of Proposition 2.1 show that ‖z‖p−var,∆rk ≤ (N ′ ∆rk (ω)) p−1 p 2 N′∆rk (ω) e2Lr|zkr|, (4.44) where N′[a,b](ω) is the maximal index of the greedy sequence of times τ0 = a, τi+1 := inf{t > τi : Λ′(ω, [τi, t]) = 1} ∧ b that lies in interval [a, b]. Observe that N′[a,b](ω) can be estimated as N′[a,b](ω) ≤ 1+Λ′p(ω, [a, b]). (4.45) Combining (4.44) with (4.43) we obtain eλ(n−n0)r|zt| ≤ CA|zn0|+ D n−1 ∑ k=n0 Λ′(ω,∆rk)(N ′ k(ω)) p−1 p 2N ′ k(ω)eλ(k−n0)r|zkr| ≤ CA|zn0|+ n−1 ∑ k=n0 Ikeλ(k−n0)r|zkr| in which N′k(ω) := N ′ ∆rk (ω) and Ik := DΛ′(ω,∆rk)[1+Λ ′p−1(ω,∆rk)]2 Λ′p(ω,∆rk). (4.46) Applying Lemma A2, we obtain |znr| ≤ CA|zn0r|e−λ(n−n0)r n−1 ∏ k=n0 (1 + Ik) and then log |znr| (n− n0)r ≤ log ( 2CAR 1 m (θ(n0−1)rω) ) (n− n0)r − λ+ 1 (n− n0)r n−1 ∑ k=n0 log(1+ Ik). (4.47) Step 2: Now we are going to estimate Ik. Note that due to Corollary 2.1, i = 1, 2 ‖xi‖p−var,∆rk ≤ D ( 1+ |||ω|||2p−1p−var,∆rk )( R 1 m (θ(k−1)rω) + 1+ |||ω|||p−var,∆rk ) , 92 we then have Λ′(ω,∆rk) ≤ DCg [ |||ω|||p−var,∆rk + |||ω||| 2 p−var,∆rk + |||ω||| 2p p−var,∆rk + |||ω||| 2p+1 p−var,∆rk ] ×[1+ R 1m (θ(k−1)rω)] =: Cg Fˆ(θkrω), with a generic constant D. Therefore Ik is bounded by Ik ≤ D [ Cg Fˆ(θkrω) + C p g Fˆp(θkrω) ] eC p g Fˆp(θkrω) log 2. Hence using the ergodic Birkhorff theorem, we obtain from (4.47) that almost sure lim n0→−∞ 1 (n− n0)r log |znr| ≤ −λ+ 1 r E log { 1+ D [ Cg Fˆ(ω) + C p g Fˆp(ω) ] eC p g Fˆp(ω) log 2 } . (4.48) Using the inequalities log(1 + x + y) ≤ log(1 + x) + log(1 + y), log(1 + xy) ≤ log(1+ x) + log(1+ y), log(1+ xey) ≤ x + y, for x, y ≥ 0 we obtain log { 1+ D [ Cg Fˆ(ω) + C p g Fˆp(ω) ] eC p g Fˆp(ω) log 2 } ≤ D + log [ 1+ Cg Fˆ(ω) ] + log [ 1+ Cpg Fˆp(ω) ] + Cpg Fˆp(ω) log 2. (4.49) Using Cauchy and Young inequality we obtain, up to a generic constant D > 0 Fˆp(ω) ≤ D [ 1+ |||ω|||2p−var,[0,r] + |||ω|||4p−var,[0,r] + + |||ω|||4pp−var,[0,r] + |||ω||| 4p+2 p−var,[0,r] + R 1 m (θ(k−1)rω) ]p ≤ D [ 1+ |||ω|||2pp−var,[0,r] + |||ω||| 4p p−var,[0,r] + + |||ω|||4p2p−var,[0,r] + |||ω||| 4p2+2p p−var,[0,r] + R(ω) ] ≤ D [ 1+ |||ω|||4p2+2pp−var,[0,r] + R(ω) ] . Hence the right hand side in the last line of (4.49) is integrable due to the in- tegrability of |||ω|||2p(2p+1)p−var,[0,r] and of R(ω). On the other hand, the expression under the expectation of (4.48) tends to zero a.s. as Cg tends to zero. Due to the Lebesgue’s dominated convergence theorem, the expectation converges to zero as Cg tends to zero. As a result, there exists δ small enough such that for Cg < δ we have |znr| = |a1 − a2| → 0 as n0 tends to −∞ exponentially with the uniform convergence rate in (4.48), this proves a1 ≡ a2 a.s. Hence At is a singleton {at(ω)}. The fact of exponential convergence rate is then obvious. 93 Finally, let x1t = x(t, t0, at0(ω),ω), x 2 t = x(t, t0, xt0,ω) be the solutions starting from at0(x), xt0 respectively at t0. Since At(ω) is invariant, x1t = at(ω). By repeating the arguments in Step 1 we conclude that At(ω) is also a forward attractor. 2 Example 4.3. Following [26], [90], we consider the forced damped general pendulum equation. mϑ¨+ 2bϑ˙− m L g sin ϑ+ kϑ = Q0 cosωt, where m is the tip mass, b is the viscous damping coefficient, L is the bar’s length, g is the gravitational constant and k is the excitation intensity and Q0 cosωt, is the periodic driving force. The equation can be rewrite in the form dϑt = ϑ¯tdt dϑ¯t = [ − k m ϑt − 2bm ϑ¯t + g L sin ϑt + Q0 cosωt ] dt. (4.50) Following [79] we suppose that system (4.50) is under influence of stochastic perturba- tions. Particularly, we consider the following version of (4.50) dϑt = ϑ¯tdt dϑ¯t = [ − k m ϑt − 2bm ϑ¯t + g L sin ϑt + Q0 cosωt ] dt + e sin ϑtdBHt (4.51) Then (4.51) has the form of (4.1) with x = (x1, x2)T := (ϑ, ϑ¯)T A = ( 0 1 − km −2bm ) , f (t, x) = ( 0 g L sin x 1 + Q0 cosωt ) , g(t, x) = ( 0 e sin x1 ) . It is easy to check that matrix A has two eigenvalues of negative real parts, f is globally Lipschitz continuous with C f = g L , and g is smooth and bounded together with its derivatives, with Cg = e. If 1/L, e are small enough, (4.51) possesses a singleton attractor which is both pullback and forward one. 4.5 Bebutov flow and its generation Motivated by [81], in this section we show that (4.1) generates a random dy- namical system in the extended space by using Bebutov flow. Moreover, the RDS possesses a random pullback attractor and in case small Cg the attractor is one point and also is the forward one. Recall from [81] that for each h defined on R×Rd denote by ht-the translate of h given by ht(s, x) = h(t + s, x), (s, x) ∈ R×Rd for each t ∈ R. We also 94 denote by S the shift mapping on (C(R×Rd,Rd), d1)-the space of continuous functions on R×Rd equipped with the compact open topology (see details in Appendix). St f = S(t, f ) = ft, ∀ f ∈ C(R×Rd,Rd). Then define the hull of f , which denoted byH f - the closure of the sets {Sτ f |τ ∈ R} in (C(R×Rd,Rd), d1). Similarly, as introduced in Chapter 3, for A ∈ (Cb(R,Rd×d), ‖ · ‖∞,R) -the space of continuous bounded functions from R to Rd×d with supermum norm we recall the notation SA-shift mapping and consider HA- the hull of A, is the the closure of the sets {SAτ A|τ ∈ R}. The following theorems are due to [81, Theorem 1, 12, 14, p. 258]. Theorem 4.6. ( [81, Theorem 1]) (i) SA defines a dynamical system on C(R,Rd×d). (ii)S defines a dynamical system on C(R×Rd,Rd). Theorem 4.7. ( [81, Theorem 14]) (i) The hull HA is compact in C(R,Rd) if (and only if) A is bounded and uniformly continuous on R. (ii) The hull H f is compact in C(R×Rd,Rd) if (and only if) f is bounded and uni- formly continuous on every set R× K, where K is a compact set in Rd. These results and Theorems 4.4, 4.5 promote the following additional condi- tion to (H1)− (H5). (H6) A is uniformly continuous, f is uniformly continuous onR×K for each K compact in Rd; f , g are bounded by ‖ f ‖∞, ‖g‖∞ respectively. Moreover, ∂xg is of Lipchitz continuity w.r.t. x with Lipchitz constant Cg. In the similar manner, we consider the hull of g. Firstly, we fix 1 − 1p < β0 < β. Denote by Hg the closure of {Sτg|τ ∈ R} in the space (Cβ0;1,0(R× Rd,Rd×m), d2) (see detail in the Appendix). The similar results hold for hull of g in the two following Theorems. Proposition 4.4. If g satisfies (H3) and (H6), then so does each g∗ ∈ Hg. Moreover, Hg is a compact set in (Cβ0;1,0(R×Rd,Rd×m), d2). Proof. It can be seen from the assumption of g that g together with ∂xg sat- isfies the condition boundedness and equicontinuous on R× K for each K ⊂ B¯(0, N) compact in Rd. Due to [81, Theorem 16] Hg is compact in C1,0(R × Rd,Rd×m) with metric ρ. Hence, for g∗ ∈ Hg, ∂xg∗ exists and is continuous and 95 there exists tn such that limn→∞ d2(g ∗, gtn) = 0. It is evident that g∗ is bounded by ‖g‖∞. Moreover |g∗(t, x)− g∗(t, y)| = lim n→∞ |gtn(t, x)− gtn(t, y)| = lim n→∞ |g(tn + t, x)− g(tn + t, y)| ≤ Cg|x− y|, ‖∂xg∗(t, x)− ∂xg∗(t, y)‖ = limn→∞ ‖∂xgtn(t, x)− ∂xgtn(t, y)‖ = lim n→∞ ‖∂xg(tn + t, x)− ∂xg(tn + t, y)‖ ≤ Cg|x− y|, |g∗(t, x)− g∗(s, x)|+ ‖∂xg∗(t, x)− ∂xg∗(s, x)‖ = lim n→∞ |gtn(t, x)− gtn(s, x)|+ ‖∂xgtn(t, x)− ∂xgtn(s, x)‖ ≤ C ′ g|t− s|β. The first statement is proved. For the second one, due to the compactness ofHg in the sense of ρ, from a sequence f n ∈ Hg there exists a subsequence f nk that converges (in ρ) to f ∈ C1,0(R×Rd,Rd×m). One may choose the subsequence in the form gtn . Applying the above arguments for g ∗ = f and the sequence gtn we have f ∈ Cβ;1,0(R×Rd,Rd×m). Moreover, ||| f nk |||β,K1×K2, ||| f |||β,K1×K2 are less than C′g for K1, K2 are compact sets in R,Rd respectively. Finally, put hk = f nk − f . Since β0 < β, for s, t ∈ K1, x ∈ K2 |hk(t, x)− hk(s, x)| |t− s|β0 = ( |hk(t, x)− hk(s, x)| |t− s|β )β0/β .|hk(t, x)− hk(s, x)|1−β0/β ≤ |||hk|||β0/ββ,K1×K2 (|hk(t, x)|+ |hk(s, x)|) 1−β0/β , hence |||hk|||β0,K1×K2 ≤ 4C ′β0/β g ‖hk‖1−β0/β∞,K1×K2 → 0 as k→ ∞. To sum up, f nk converges to f in d2. The proof is completed. 2 Since the space Cβ0;1,0(R×Rd,Rd×m) is not separable we directly prove that under the assumption of Theorem 4.4, S defines a dynamical system onHg. Theorem 4.8. S defines a dynamical system on hull of g. Proof. Due to [81, Theorem 12], S defined a dynamical system on C1,0(R× Rd,Rd×m). We just need to check that for fixed (t0, f 0) ∈ R×Hg, if t ∈ R, f ∈ Hg such that |t− t0|, d2( f , f 0) → 0 then ∣∣∣∣∣∣∣∣∣ ft(·, x)− f 0t0(·, x)∣∣∣∣∣∣∣∣∣β0−Hol,[a,b]×K → 0 for each a, b, each compact interval K in Rd. Namely, by choosing appropriate [a′, b′] we have∣∣∣∣∣∣∣∣∣ ft(·, x)− f 0t0(·, x)∣∣∣∣∣∣∣∣∣β0−Hol,[a,b]×K ≤ ∣∣∣∣∣∣∣∣∣ ft(·, x)− f 0t (·, x)∣∣∣∣∣∣∣∣∣ β0−Hol,[a,b]×K + ∣∣∣∣∣∣∣∣∣ f 0t (·, x)− f 0t0(·, x)∣∣∣∣∣∣∣∣∣β0−Hol,[a,b]×K 96 ≤ ∣∣∣∣∣∣∣∣∣ f (·, x)− f 0(·, x)∣∣∣∣∣∣∣∣∣ β0−Hol,[a′,b′]×K +2 ∣∣∣∣∣∣∣∣∣ f 0(·, x)∣∣∣∣∣∣∣∣∣β0/β β−Hol,[a′,b′]×K .‖ f 0t − f 0t0‖ 1−β0/β ∞,[a,b]×K → 0, as |t− t0| → 0, d2( f , f 0)→ 0. This shows the continuity of S on Hg. Since Hg is compact, S is measurable w.r.t. the σ−algebra generated by d2. The proof is completed. 2 Since the hull of A, f , g are compact sets with appropriate metrics, we de- duce from Krylov-Bogoliubov theorem [72, Chapter VI, §9] that there are prob- ability measures PA,P f ,Pg on measurable space (HA,BA), (H f ,B f ), (Hg,Bg) with Borel σ−algebras BA,B f ,Bg, that are invariant under the shifts mapping introduced above respectively. Denote by Ω¯ the Catersian product HA ×H f × Hg × Ω with the product Borel σ−field denoted by B¯ and the product mea- sure P¯ = PA × P f × Pg × PH and consider the product dynamical system θ¯ : R× Ω¯→ Ω¯ given by θ¯(t, A˜, f˜ , g˜,ω) = (SAt A˜, St f˜ , St g˜, θtω), (A˜, f˜ , g˜,ω) ∈ Ω¯. The following Lemma is evident. Lemma 4.3. (Ω¯, B¯, P¯, θ¯) is a metric dynamical system. Remark 4.3. (i) In (H6) we assume f , g to be bounded to simplify in presentation. One may follow Theorem 4.7 to work with more general setting. (ii) One may reorganize the equation to dxt = A(t)xtdt + G(t, xt)dZt where G(t, x) := ( f (t, x) g(t, x) ) and Zt := ( t BHt ) and consider the hull for A and G. The product space is then constructed similarly. Proposition 4.5. Each A˜, f˜ , g˜ inHA,H f ,Hg satisfy (H1)− (H6). Proof. We have proved for g˜ in Proposition 4.4. The statement for f˜ is evident with the note that f is bounded hence one may replace b(t) in (H2) by ‖ f ‖∞. We now check the properties of A˜. Since A is bounded, SAτ (A) and then A˜ are bounded by ‖A‖. Denote by Φ˜(t, s) the Cauchy matrix of the system dxt = A˜(t)xtdt. We just need to prove that Φ˜(t, s) satisfies (4.2), i.e. ‖Φ˜(t, s)‖ ≤ CAe−λA(t−s)∀s ≤ t. 97 Firstly, due to [31, p. 75] Φ˜(t, s) = E + ∫ t s A˜(t1)dt1 + ∫ t s A˜(t1)dt1 ∫ t1 s A˜(t2)dt2 + · · · , where E is the unit matrix of order d. Hence if A˜ = Aτ for some τ ∈ R Φτ(t, s) := Φ˜(t, s) = E + ∫ t s A(τ + t1)dt1 + ∫ t s A(τ + t1)dt1 ∫ t1 s A(τ + t2)dt2 + · · · , = E + ∫ t+τ s+τ A(t1)dt1 + ∫ t+τ s+τ A(t1)dt1 ∫ t1 s+τ A(t2)dt2 + · · · , = Φ(s + τ, t + τ) ≤ CAe−λA(t−s). Now for A˜ ∈ HA, for each ε > 0 there exists Aτ for some τ ∈ R such that ‖A˜− Aτ‖∞,R ≤ ε. Hence with B = A˜− Aτ Φ˜(t, s) = E + ∫ t s [Aτ + B](t1)dt1 + ∫ t s [Aτ + B](t1)dt1 ∫ t1 s [Aτ + B](t2)dt2 + · · · , and then ‖Φ˜(t, s)−Φτ(t, s)‖ ≤ ε(t− s) + (2ε‖A‖+ ε2)(t− s) 2 2! + (3ε2‖A‖+ 3ε‖A‖2 + ε3)(t− s) 3 3! + · · · ≤ ∑ k≥0 (ε+ ‖A‖)k(t− s)k k! − ∑ k≥0 ‖A‖k(t− s)k k! ≤ e(ε+‖A‖)(t−s) − e‖A‖(t−s) ≤ e‖A‖(t−s) [ eε(t−s) − 1 ] . This implies that ‖Φ˜(t, s)‖ ≤ CAe−λA(t−s) + e‖A‖(t−s) [ eε(t−s) − 1 ] , ∀s ≤ t for arbitrary ε > 0. We deduce that ‖Φ˜(t, s)‖ ≤ CAe−λA(t−s) ∀s ≤ t. 2 Theorem 4.9. For each ω¯ = (A˜, f˜ , g˜,ω) ∈ Ω¯, the equation dxt = [A˜(t)xt + f˜ (t, xt)]dt + g˜(t, xt)dBHt (ω), t ∈ R, x(0) = x0 ∈ Rd, (4.52) possesses a unique solution x(t, x0, ω¯) := X(t, 0, x0, ω¯) that inherits all the properties as introduced in Chapter 2. Moreover, x(t, x0, ·) is measurable. 98 Proof. The first statement is evident due to Proposition 4.5. For the second one, we prove that the solution is continuous w.r.t. ω¯ as an element in the product of separable metric spaces HA,H f ,Hg,Ω. We fix t, x0 and [−T, T] contains t and consider ω¯1 = (A1, f 1, g1,ω1), ω¯2 = (A2, f 2, g2,ω2) in Ω¯. Put y1t := x(t, x0, ω¯ 1), y2t := x(t, x0, ω¯ 2) then we have y1t = x0 + ∫ t 0 [A1(s)y1s + f 1(s, y1s )]ds + ∫ t 0 g1(s, y1s )dω 1 s , y2t = x0 + ∫ t 0 [A2(s)y2s + f 2(s, y2s )]ds + ∫ t 0 g2(s, y2s )dω 2 s . Therefore, zt := y1t − y2t satisfies the equation zt = y1t − y2t = ∫ t 0 [A1(s)y1s + f 1(s, y1s )− A2(s)y2s − f 2(s, y2s )]ds + ∫ t 0 g1(s, y1s )dω 1 s − ∫ t 0 g2(s, y2s )dω 2 s = ∫ t 0 [A1(s)− A2(s)]y1s ds + ∫ t 0 A2(s)(y1s − y2s )ds + ∫ t 0 [ f 2(s, y1s )− f 2(s, y2s )]ds + ∫ t 0 [ f 1(s, y1s )− f 2(s, y1s )]ds + ∫ t 0 g1(s, y1s )d(ω 1 s −ω2s ) + ∫ t 0 [g1(s, y1s )− g2(s, y1s )]dω2s + ∫ t 0 [g2(s, y1s )− g2(s, y2s )]dω2s . Fixing ω¯1, due to Proposition 2.1 one can find R depends on ω¯1 such that ‖x(·, x0, ω¯)‖p−var,[−T,T] ≤ R for all ω¯ lies in the neibourgh of ω¯1 of radius 1. We choose a upper bound for the norms of Ai, f i, gi,ωi on K¯ := [−T, T] × B¯(0, R) and reuse the notation R for convenient. We will show that z is near 0 when ‖A1 − A2‖∞,[−T,T], ‖ f 1 − f 2‖∞,K¯, ‖g1 − g2‖∞,K¯, ‖∂xg1 − ∂xg2‖∞,K¯, and∣∣∣∣∣∣g1 − g2∣∣∣∣∣∣ β0,K¯ less than ε small enough. For 0 ≤ u < v ∈ [−T, T] |zu − zv| = ∣∣∣∣∫ vu A2(s)(y1s − y2s )ds ∣∣∣∣+ ∣∣∣∣∫ vu [A1(s)− A2(s)]y1s ds ∣∣∣∣ + ∣∣∣∣∫ vu [ f 2(s, y1s )− f 2(s, y2s )]ds ∣∣∣∣+ ∣∣∣∣∫ vu [ f 1(s, y1s )− f 2(s, y1s )]ds ∣∣∣∣ + ∣∣∣∣∫ vu [g2(s, y1s )− g2(s, y2s )]dω2s ∣∣∣∣ + ∣∣∣∣∫ vu g1(s, y1s )d(ω1s −ω2s ) ∣∣∣∣+ ∣∣∣∣∫ vu [g1(s, y1s )− g2(s, y1s )]dω2s ∣∣∣∣ 99 in which ∣∣∣∣∫ vu A2(s)(y1s − y2s )ds ∣∣∣∣ ≤ R ∫ vu |zs|ds∣∣∣∣∫ vu [ f 2(s, y1s )− f 2(s, y2s )]ds ∣∣∣∣ ≤ C f ∫ vu |zs|ds∣∣∣∣∫ vu [g2(s, y1s )− g2(s, y2s )]dω2s ∣∣∣∣ ≤ 2(K + 1)M′R(1+ R) × ∣∣∣∣∣∣∣∣∣ω2∣∣∣∣∣∣∣∣∣ p−var,[u,v] ‖z‖q−var,[u,v] where the final estimate due to Lemma 2.2 (ii) (the estimate on J). And∣∣∣∣∫ vu [A1(s)− A2(s)]y1s ds ∣∣∣∣ ≤ ‖A1 − A2‖∞,[−T,T]‖y1‖∞,[−T,T](v− u)∣∣∣∣∫ vu [ f 1(s, y1s )− f 2(s, y1s )]ds ∣∣∣∣ ≤ ‖ f 1 − f 2‖∞,K¯(v− u)∣∣∣∣∫ vu g1(s, y1s )d(ω1s −ω2s ) ∣∣∣∣ ≤ ∣∣∣∣∣∣∣∣∣ω1 −ω2∣∣∣∣∣∣∣∣∣p−var,[u,v] × [ KCg‖y1‖q−var,[u,v] + KCg(v− u)β + ‖g‖∞ ] ∣∣∣∣∫ vu [g1(s, y1s )− g2(s, y1s )]dω2s ∣∣∣∣ ≤ ∣∣∣∣∣∣∣∣∣ω2∣∣∣∣∣∣∣∣∣ p−var,[u,v] [ ‖g1 − g2‖∞,K¯ + K ∣∣∣∣∣∣∣∣∣g1 − g2∣∣∣∣∣∣∣∣∣ q−var,[u,v] ] ≤ ∣∣∣∣∣∣∣∣∣ω2∣∣∣∣∣∣∣∣∣ p−var,[u,v] [ ‖g1 − g2‖∞,K¯ +K ∣∣∣∣∣∣∣∣∣g1 − g2∣∣∣∣∣∣∣∣∣ β−Hol,[u,v] (v− u)β + K‖∂xg1 − ∂xg2‖∞,K¯ ∣∣∣∣∣∣∣∣∣y1∣∣∣∣∣∣∣∣∣ q−var,[u,v] . In the final estimate we use the mean value theorem (see [74, Lemma 7.1]), namely for s, t ∈ [u, v] |g1(t, y1t )− g2(t, y1t )− g1(s, y1s ) + g2(s, y1s )| ≤ |(g1 − g2)(t, y1t )− |(g1 − g2)(s, y1t )|+ |(g1 − g2)(s, y1t )− |(g1 − g2)(s, y1s )| ≤ ∣∣∣∣∣∣∣∣∣g1 − g2∣∣∣∣∣∣∣∣∣ β0,K¯ (v− u)β0 + ‖∂xg1 − ∂xg2‖∞,K¯|y1t − y1s |. Therefore |||z|||q−var,[u,v] ≤ D (∫ v u |zs|ds + ‖z‖q−var,[u,v] + A1/qu,v ) where D is a constant depending on R and A is a control function defined by A1/qu,v := e(v− u) + ∣∣∣∣∣∣∣∣∣ω1 −ω2∣∣∣∣∣∣∣∣∣ p−var,[u,v] + e ∣∣∣∣∣∣∣∣∣ω2∣∣∣∣∣∣∣∣∣ p−var,[u,v] . 100 Apply Lemma 2.3, we obtain ‖z‖q−var,[0,T] ≤ D(‖z0‖+ ε) = Dε→ 0 as ε→ 0. This completes the proof. 2 Now for each ω¯ = (A˜, f˜ , g˜,ω) ∈ Ω¯ denote by Φ∗(t, ω¯)x0 the value of the of the solution of (4.52) at the time t ∈ R with the initial time s = 0, i.e. X(t, 0, x0,ω). We have xt+s = ∫ t+s 0 [A˜(u)xu + f˜ (u, xu)]du + ∫ t+s 0 g˜(u, xu)dωu = ∫ s 0 [A˜(u)xu + f˜ (u, xu)]du + ∫ s 0 g˜(u, xu)dωu + ∫ t+s s [A˜(u)xu + f˜ (u, xu)]du + ∫ t+s s g˜(u, xu)dωu = xs + ∫ t 0 [A˜(s + u)xs+u + f˜ (s + u, xs+u)]du + ∫ t 0 g˜(s + u, xs+u)dθsωu = xs + ∫ t 0 [SAs A˜(u)xs+u + Ss f˜ (u, xs+u)]du + ∫ t 0 Ss g˜(u, xs+u)dθsωu. It means that Φ∗ satisfies the cocycle property Φ∗(t + s, ω¯)x0 = Φ∗(t, θ¯sω¯) ◦Φ∗(s, ω¯)x0. Therefore, we have proved the following theorem. Theorem 4.10. The system (4.52) generates a random dynamical system over the met- ric dynamical system (Ω¯, B¯, P¯, θ¯). Theorem 4.11. The system (4.52) possesses a random pullback attractor. Moreover, if Cg, C′g is small enough the attractor is a singleton almost surely, thus the path wise convergence is in both the pullback and forward directions. Proof. Come back to dxt = [A˜(t)xt + f˜ (t, xt)]dt + g˜(t, xt)dωt, t ∈ R, x(0) = x0 ∈ Rd. Due to the cocycle property of Φ∗, for each t > 0 IdRd(·) = Φ∗(t, θ−tω¯) ◦Φ∗(−t, ω¯)(·) = Φ∗(t, θ−tω¯) ◦ X(−t, 0, ·,ω), where X is the flow generate by (4.52). On the other hand, base on the flow property of X(t, s, ·ω) IdRd(·) = X(0,−t, X(−t, 0, ·,ω),ω). 101 Since X is convertable,Φ∗(t, θ−tω¯)x0 = X(0,−t, x0,ω). Moreover, due to Lemma A6, R(ω) = 1+∑∞i=0 η iξr(θ−irω) is a tempered random variable. Therefore the conclusion is directed from Theorem 4.4 and Theorem 4.5. 4.6 Conclusions and discussions In this chapter we obtain a criterion for the existence of the non autonomous pullback attractor of the system (4.1) in Theorem 4.1. In the case g is linear, the attractor is proved to be one point and is also forward attractor. The case g is bounded is treated in Theorem 4.4, in which the attractor exists for any Cg not neccesarily small. Moreover, when Cg, C′g are small enough the attractor is one point and is the forward one as in the linear case. As a consequence of these results, in Theorem 4.11 under some additional conditions on regularity of coefficient functions A, f , g we show that the system generates a random dynamical system in the sense of Bebutov flow. As such the generated RDS possesses a one point random attractor in both pullback and forward direction. As presented in this chapter, the existence of one point attractor is established for the system where the diffusion is linear or bounded. In the general case, the need of evaluating the solution in term of its p−variation may causes the over estimate. We have developed the problems for the delay equation (see [22]) and here we meet the similar difficulty. We expect to be able to deal with the prob- lems in the future works. We also raise the question for the dissipative system and the problem on the construction the Bebutov flow for the nonautonomous delay equation driven by fBm. 102 Conclusions The main results of this dissertation include: 1) The existence and uniqueness of the solution of the nonautonomous stochas- tic differential equations driven by fractional Brownian motions and the prop- erties of the solution. 2) The generation of the stochastic two-parameter flow by the equation, and particularly the random dynamical system in the case of autonomy. 3) Three theorems on the Lyapunov spectrum of the linear systems: the dis- cretization scheme to compute the Lyapunov spectrum, the formula for the spectrum for regular triangular equation, the regularity almost sure of the nonau- tonomous equation in the sense of an probability measure. 4) The criterion for the existence of global pullback attractor for the generated flow. If the diffusion part is linear or bounded then system possesses a singleton attractor provided that the noise intensity is small. 5) The construction of the Bebutov flow for nonautonomous fSDE which is a random dynamical system with an appropriate metric dynamical system. 103 List of Author’s Related Papers 1. N. D. Cong, L. H. Duc, P. T. Hong, Nonautonomous Young differential equations revisited, Journal of Dynamics and Differential Equations 30, (2018), 1921-1943. 2. L. H. Duc, P. T. Hong, Young differential delay equations driven by Ho¨lder continuous paths, book chapter. Modern Mathematics and Mechanics, Springer International Publishing AG., (2019), 313-333. 3. N. D. Cong, L. H. Duc, P. T. Hong, Lyapunov spectrum of nonautonomous linear Young differential equations. Journal of Dynamics and Differential Equations 32, (2020), 1749–1777. 4. L. H. Duc, P. T. Hong, N. D. Cong, Asymptotic stability for stochastic dis- sipative systems with a Holder noise. SIAM Journal on Control and Opti- mization 57 (4) (2019), 3046–3071. 5. L. H. Duc, P. T. Hong, Asymptotic dynamics of Young differential equa- tions. Accepted by Journal of Dynamics and Differential Equations. https://link.springer.com/article/10.1007%2Fs10884-021-10095-1. (2021). 6. N. D. Cong, L. H. Duc, P. T. Hong, Pullback attractors for stochastic Young differential delay equations, Journal of Dynamics and Differential Equa- tions 34, (2022), 605–636. 104 Appendix Lemma A1 (Continuous Gronwall Lemma). Assume that ut, αt are nonnegative continuous function on [a,∞) and β is a nonengative real number such that ut ≤ αt + ∫ t a βusds, ∀t ≥ a. Then ut ≤ αt + ∫ t a βeβ(t−s)αsds, ∀t ≥ a. Proof. See [2, Lemma 6.1, p. 89]. Lemma A2 (Discrete Gronwall Lemma). Let a be a non negative constant and un, αn, βn be nonnegative sequences satisfying un ≤ a + n−1 ∑ k=0 αkuk + n−1 ∑ k=0 βk, ∀n ≥ 1 then un ≤ max{a, u0} n−1 ∏ k=0 (1+ αk) + n−1 ∑ k=0 βk n−1 ∏ j=k+1 (1+ αj) for all n ≥ 1. Proof. Put Sn := a + n−1 ∑ k=0 αkuk + n−1 ∑ k=0 βk Tn := max{a, u0} n−1 ∏ k=0 (1+ αk) + n−1 ∑ k=0 βk n−1 ∏ j=k+1 (1+ αj). We will prove by induction that Sn ≤ Tn for all n ≥ 1. Namely, the statement holds for n = 1 since S1 = a + α0u0 + β0 ≤ max{a, u0}(1+ α0) + β0 = T1. We assume that Sn ≤ Tn for n ≥ 1, then due to the fact that un ≤ Sn we obtain Sn+1 = a + n−1 ∑ k=0 αkuk + n−1 ∑ k=0 βk + αnun + βn 105 = Sn + αnun + βn ≤ Sn + αnSn + βn ≤ Tn(1+ αn) + βn ≤ [ max{a, u0} n−1 ∏ k=0 (1+ αk) + n−1 ∑ k=0 βk n−1 ∏ j=k+1 (1+ αj) ] (1+ αn) + βn ≤ max{a, u0} n ∏ k=0 (1+ αk) + n ∑ k=0 βk n−1 ∏ j=k+1 (1+ αj) = Tn+1. Since un ≤ Sn, (A2) holds. Spaces of functions Variation and Ho¨lder spaces The content in this part is from the book [42]. Let C([a, b],Rd) denote the space of all continuous paths x : [a, b]→ Rd, t 7→ xt equipped with the supremum norm ‖ · ‖∞,[a,b] given by ‖x‖∞,[a,b] = supt∈[a,b] |xt|, where | · | is the Euclidean norm in Rd. For p ≥ 1 and [a, b] ⊂ R, a continuous path x : [a, b]→ Rd is called of finite p-variation if |||x|||p-var,[a,b] := ( sup Π[a,b] n ∑ i=1 |xti+1 − xti |p )1/p < ∞, where the supremum is taken over the whole class of finite partitions Π[a, b] = {a = t0 < t1 < · · · < tn = b} of [a, b]. The subspace C p−var([a, b],Rd) ⊂ C([a, b],Rd) consists of all paths x with finite p-variation and equipped with the p-var norm ‖x‖p-var,[a,b] := |xa|+ |||x|||p-var,[a,b] , is a nonseparable Banach space [42, Theorem 5.25, p. 92]. In the following defi- nition, the notion of control or control function is defined on the simplex ∆[a, b] := {(s, t) : a ≤ s ≤ t ≤ b}. Definition A.4. ( [42, Definition 1.7]) A continuous map ω : ∆[a, b] −→ R+ is called a control (on [a, b]) if it is zero on the diagonal and superadditive, i.e. (i) For all t ∈ [a, b], ω(t, t) = 0, (ii) For all s ≤ t ≤ u in [a, b], ω(s, t) +ω(t, u) ≤ ω(s, u). Example A.4. The following functions are controls on [a, b]. 106 1. ω(s, t) = (t− s)θ, θ ≥ 1, 2. ω(s, t) = |||x|||qp−var,[s,t] where x ∈ C p−var([a, b],Rd), p ≥ 1 is given and q ≥ p, 3. ω(s, t) = ∫ t s budu where b is a nonnegative integrable function on [a, b], 4. (φ ◦ ω)(s, t) where ω(s, t) is a control and φ : [0,∞) → [0,∞) is increasing, convex and vanish at 0. The following lemmas, which are more general results of Propostion 5.10 and Exercise 5.11 in [42], give us useful properties of controls in relation with the variations of a path. Lemma A3. Let ω j be a finite sequence of control functions on [a, b], Cj > 0, j = 1, k, p ≥ 1 and x : [a, b]→ Rd be a continuous path satisfying |xt − xs| ≤ k ∑ i=j Cjω 1/p j (s, t), ∀s < t ∈ [a, b], then |||x|||p−var,[a,b] ≤ k ∑ j=1 Cjω1/p(a, b). Proof. Consider an arbitrary finite partitionΠ = (si), i = 0, . . . , n+ 1 of [a, b]. By assumption and Minskowski inequality we have( n ∑ i=0 |xsi+1 − xsi |p )1/p ≤ [ n ∑ i=0 ( k ∑ j=1 Cjω 1/p j (si, si+1) )p]1/p ≤ k ∑ j=1 ( n ∑ i=0 Cpj ω j(si, si+1) )1/p ≤ k ∑ j=1 Cjω 1/p j (a, b). This implies the conclusion of the lemma. 2 Lemma A4. Let x ∈ C p−var([a, b],Rd), p ≥ 1. If a = a1 < a2 < · · · < ak = b, then k−1 ∑ i=1 |||x|||pp-var,[ai,ai+1] ≤ |||x||| p p-var,[a,b] ≤ (k− 1)p−1 k−1 ∑ i=1 |||x|||pp-var,[ai,ai+1] . Proof. The proof is similar to that in [42, p. 84], by using triangle inequality and power means inequality 1 n n ∑ i=1 zi ≤ ( 1 n n ∑ i=1 zri )1/r , ∀zi ≥ 0, r ≥ 1. 2 107 For 0 < α ≤ 1 denote by Cα-Hol([a, b],Rd) the space of all Ho¨lder continuous paths x : [a, b]→ Rd with exponential α, equipped with the norm ‖x‖α-Hol,[a,b] := |xa|+ |||x|||α-Hol,[a,b] = |xa|+ sup a≤s<t≤b |xt − xs| (t− s)α < ∞. Then Cα-Hol([a, b],Rd) is a nonseparable Banach space (see [42]). Clearly, if x ∈ Cα-Hol([a, b],Rd) then for all s, t ∈ [a, b] we have |xt − xs| ≤ |||x|||α-Hol,[a,b] |t− s|α. Hence, for all p such that pα ≥ 1 we have |||x|||p-var,[a,b] ≤ |||x|||α-Hol,[a,b] (b− a)α < ∞. Therefore, C1/p-Hol([a, b],Rd) ⊂ C p−var([a, b],Rd). 2 Compactness In the following we recall the facts on the compactness of a subset in C p−var([a, b],Rd) and Cα−Hol([a, b],Rd). Proposition A.6. ( [42, Proposition 5.28]) Assume that (xn) is a sequence in C([a, b],Rd). (i) If (xn) is equicontinuous, bounded and supn |||x|||p−var,[a,b] < ∞ then for each p′ > p, xn converges along a subsequence to some x ∈ C p−var([a, b],Rd). (ii) If (xn) is bounded and supn |||x|||α−Hol,[a,b] < ∞ then for each α′ < α, xn converges along a subsequence to some x ∈ Cα−Hol([a, b],Rd). Closure of smooth paths in variation norm, Ho¨lder norm Define C0,p−var([a, b],Rd), C0,α−Hol([a, b],Rd) as the closure of C∞([a, b],Rd)-the space of smooth functions on [a, b], in C p−var([a, b],Rd) and Cα−Hol([a, b],Rd) respectively. Thus they are Banach space and so are the spaces C0,p−var0 ([a, b],Rd) := {x ∈ C0,p−var([a, b],Rd)| x0 = 0}, and C0,α−Hol0 ([a, b],Rd) := {x ∈ C0,α−Hol([a, b],Rd)| x0 = 0}. Moreover, these are separable spaces which can be defined as C0,p−var([a, b],Rd) = { x ∈ C p−var([a, b],Rd) | lim δ→0 sup Π(a,b), |Π|≤δ ∑ ti∈Π |xti+1 − xti |p = 0 } , and C0,α−Hol([a, b],Rd) = { x ∈ Cα−Hol([a, b],Rd) | lim δ→0 sup a≤s<t≤b, |t−s|<δ |xt − xs| |t− s|α = 0 } . 108 Due to [42, Corollary 5.33, p. 98], for 1 ≤ p < p′ we have C p−var([a, b],Rd) ⊂ C0,p′−var([a, b],Rd). Similarly, for all β > α Cβ−Hol([a, b],Rd) ⊂ C0,α−Hol([a, b],Rd). Other spaces of functions Define C(R×Rd,Rn) is the space of continuous functions on R×Rd, valued in Rn. Equip this space with compact open topol- ogy, i.e. topo generated by metric d1 d1( f , g) := ∞ ∑ n=1 1 2n ‖ f − g‖∞,Kn 1+ ‖ f − g‖∞,Kn , f , g ∈ C(R×Rd,Rn) where Kn = [−n, n]× B¯(0, n) ⊂ R×Rd. Denote by C1,0(R×Rd,Rd×m) the subspace of C(R×Rd,Rd×m) contains all functions h which is continuously differential w.r.t. x and of which ∂xh contin- uous w.r.t. (t, x) with seminorms ‖h‖1,0;K := ‖h‖∞,K + ‖∂xh‖∞,K, where K is a compact subset in R × Rd. Then a complete metric is given by (see [4, Appendix B.2, p. 552-553]) ρ( f , g) := ∞ ∑ n=1 1 2n ‖ f − g‖1,0;Kn 1+ ‖ f − g‖1,0;Kn , f , g ∈ C1,0(R×Rd,Rd×m). For 0 < α < 1, consider the subspace Cα;1,0(R × Rd,Rd×m) ⊂ C1,0(R × Rd,Rd×m) containing functions h which is of local α−Holder w.r.t. t for each x ∈ Rd and moreover for each compact set K in Rd sup x∈K |||h(·, x)|||α−Hol,[a,b] < ∞, ∀[a, b] ⊂ Rd. We consider the following metric on Cα;1,0(R×Rd,Rd×m) which is denoted by d2 d2( f , g) := ∞ ∑ n=1 1 2n ‖ f − g‖α,1,0;Kn 1+ ‖ f − g‖α,1,0;Kn , where ‖ f − g‖α,1,0;K1×K2 := ‖ f − g‖1,0;K1×K2 + ||| f − g|||α,K1×K2 ||| f − g|||α,K1×K2 := sup x∈K2 ||| f (·, x)− g(·, x)|||α−Hol,K1 with K1, K2 are compact sets in R, Rd respectively. 109 Proposition A.7. (Cα;1,0(R×Rd,Rd ×m), d2) is a complete metric space. Proof. That d2 is a metric on Cα;1,0(R × Rd,Rd × m) is evident due to the seminorm properties of the Ho¨lder norm. We only need to prove the complete- ness. Let f n be a Cauchy sequence in Cα;1,0(R × Rd,Rd×m). Since (C1,0(R × Rd,Rd × m), ρ) is complete, there exists a subsequence, which we still use the notation f n, converges to f in C1,0(R×Rd,Rd×m), i.e. lim n→∞ ρ( f n, f ) = 0. We will prove that for each K1, K2 compact sets in R,Rd, ||| f n − f |||α,K1×K2 → 0 as n → ∞. Fix K ⊂ Rd compact, we have for each [a, b] ⊂ R there exist a constant M such that supn supx∈K ||| f n(·, x)|||α−Hol,[a,b] ≤ M. For each x ∈ K | f (t, x)− f (s, x)| = lim n→∞ | f n(t, x)− f n(s, x)| ≤ M|t− s|α, this implies that supx∈K ||| f (·, x)|||α−Hol,[a,b] < ∞ or f ∈ Cα;1,0(R×Rd,Rd×m). Now to complete the proof we show that f n converges to f , in α−Ho¨lder norm on each K compact in Rd. For each s < t ∈ [a, b], x ∈ K |( f n − f )(t, x)− ( f n − f )(s, x)| |t− s|α = lim m→∞ |( f n − f m)(t, x)− ( f n − f m)(s, x)| |t− s|α ≤ lim m→∞ supx∈K sup a≤v<u≤b |( f n − f m)(u, x)− ( f n − f m)(v, x)| |u− v|α ≤ lim m→∞ ||| f n − f m|||α,[a,b]×K , which implies ||| f n − f |||α,[a,b]×K ≤ limm→∞ ||| f n − f m|||α,[a,b]×K → 0, as n→ ∞ The proof is completed. Proof.[Lemma 3.6] The if part is obvious since it can be proved that lim δ→0 m[a,b](c, δ) = 0, |m[a,b](c, δ)−m[a,b](c′, δ)| ≤ ∣∣∣∣∣∣c− c′∣∣∣∣∣∣ α,[a,b] , which shows the continuity of m on C0,α−Hol(R,Rk). Hence m is uniformly continuous on a compact set, which shows (3.32) and (3.33). To be more precise, denote by C˜ the space C0,α−Hol(R,Rk), assume that H is compact in C˜, we prove (3.32) and (3.33) are fulfilled. For each n ∈N∗, put Gn = {c ∈ C˜ | |c(0)| < n}. 110 Then Gn is open in C˜. Since H ⊂ ⋃∞n=1 Gn and Gn is an increasing sequence of open sets, there exists n0 such thatH ⊂ Gn0, which proves (3.32). To prove (3.33), first note that for each c ∈ C˜ and [a, b] ⊂ R, lim δ→0 m[a,b](c, δ) = 0 (see [42, Theorem 5.31,p. 96]). Secondly |m[a,b](c, δ)−m[a,b](c′, δ)| ≤ ∣∣∣∣∣∣c− c′∣∣∣∣∣∣ α−Hol,[a,b] . Indeed, fix [a, b], δ due to the definition of m[a,b](c, δ) for each ε > 0 there exists s0, t0 ∈ [a, b], 0 < |s0 − t0| ≤ δ such that m[a,b](c, δ) ≤ |c(t0)− c(s0)||t0 − s0|α + ε. On the other hand, m[a,b](c′, δ) ≥ |c′(t0)−c′(s0)||t0−s0|α , therefore m[a,b](c, δ)−m[a,b](c′, δ) ≤ |c(t0)− c(s0)| − |c ′(t0)− c′(s0)| |t0 − s0|α + ε ≤ |c(t0)− c(s0)− c ′(t0) + c′(s0)| |t0 − s0|α + ε ≤ ∣∣∣∣∣∣c− c′∣∣∣∣∣∣ α−Hol,[a,b] + ε. Exchange the role of c and c′ we obtain |m[a,b](c, δ)−m[a,b](c′, δ)| ≤ ∣∣∣∣∣∣c− c′∣∣∣∣∣∣ α−Hol,[a,b] since ε is arbitrary. This implies the continuity of the map m[a,b](· , δ) : (C˜, d)→ R. In fact, fix [−n, n] contains [a, b]. For each c0 ∈ C˜ and ε ∈ (0, 1) choose η = ε/2n. If d(c, c0) < η we have ‖c− c0‖α,[−n,n] ∧ 1 ≤ 2nd(c, c0) < ε. Therefore |m[a,b](c, δ)−m[a,b](c′, δ)| ≤ ∣∣∣∣∣∣c− c′∣∣∣∣∣∣ α,[−n,n] ≤ ε. Now, fix ε > 0 and put Kδ := {c ∈ A¯ | m[a,b](c, δ) ≥ ε}. The Kδ are closed for all δ. Due to the fact that lim δ→0 m[a,b](c, δ) = 0 for all c ∈ C˜ we have ⋂ δ>0 Kδ = ∅. Then there exists δ = δ(ε) > 0 such that Kδ = ∅, which proves (3.33). 111 For the ”only if” part, assume (3.32) and (3.33) and prove the compactness of H¯. Since C˜ is a complete metric space, it suffices to prove that every sequence {cn}∞n=1 ⊂ H has a convergent subsequence. Now following the arguments of [55, Theorem 4.9, p. 63] line by line, we can construct a convergent subse- quence {c˜n}∞n=1 by the ”diagonal sequence” such that c˜n(r) → c(r) as n → ∞ for any rational number r ∈ Q. With (3.32) and (3.33), H satisfies the condition in [55, Theorem 4.9, p. 63], hence c˜n converge uniformly to a continuous func- tion c in every [a, b] ⊂ R. Fix [a, b], by (3.33) for each ε > 0 there exist δ0 > 0 such that if δ ≤ δ0, sup s,t∈[a,b] |s−t|≤δ |c˜n(t)−c˜n(s)| |t−s|α ≤ ε for all n. Hence sup s,t∈[a,b] |s−t|≤δ |c(t)− c(s)| |t− s|α ≤ ε and then c ∈ C˜. Finally, we prove that c˜n converge to c in the Ho¨lder seminorm on every compact interval [a, b]. Namely, with ε, δ0 given, there exist n0 such that for all n ≥ n0, ‖c˜n − c‖∞,[a,b] ≤ δα0 ε. We then have for n ≥ n0 sup s,t∈[a,b] |(c˜n − c)(t)− (c˜n − c)(s)| |t− s|α ≤ sups,t∈[a,b] |t−s|≤δ0 |(c˜n − c)(t)− (c˜n − c)(s)| |t− s|α + sup s,t∈[a,b] |t−s|≥δ0 |(c˜n − c)(t)− (c˜n − c)(s)| |t− s|α ≤ m[a,b](c˜n, δ0) + m[a,b](c, δ0) + 2‖c˜n − c‖∞,[a,b] δα0 ≤ 4ε. This implies |||c˜n − c|||α−Hol,[a,b] converge to 0 as n→ ∞. This complete the proof. 2 Tempered variables Let (Ω,F ,P) be a probability space equipped with an ergodic metric dynam- ical system θ, which is aPmeasurable mapping θ : T×Ω→ Ω,T is eitherR or Z, and θt+s = θt ◦ θs for all t, s ∈ T. Recall that a random variable ρ : Ω→ [0,∞) is called tempered if lim t→±∞ 1 t log+ ρ(θtω) = 0, a.s. 112 which, as shown in [52, p. 220], [44], is equivalent to the sub-exponential growth lim t→±∞ e −c|t|ρ(θtω) = 0 a.s. ∀c > 0. Note that our definition of temperedness corresponds to the notion of tempered- ness from above given in [4, Definition 4.1.1(ii)]. Lemma A5. (i) If h1, h2 ≥ 0 are tempered random variables then h1 + h2 and h1h2 are tempered random variables. (ii) If h1 ≥ 0 is a tempered random variable, h2 ≥ 0 is a measurable random variable and h2 ≤ h1 almost surely, then h2 is a tempered random variable. (iii) Let h1 be a nonnegative measurable function. If log+ h1 ∈ L1 then h1 is tempered. Proof. (i) See [4, Lemma 4.1.2, p. 164]. (ii) Immediate from the definition of tempered random variable. (iii) See [4, Proposition 4.1.3, p. 165]. 2 Lemma A6. Let c : Ω → [0,∞) be a tempered random variable, and δ > 0 be an arbitrary fixed positive number. Put d(ω) := ∞ ∑ k=1 e−δkc(θ−kω). Then d(·) is a nonnegative almost everywhere finite and tempered random variable. Proof. Put dn(ω) := ∑nk=1 e −δkc(θ−kω). Then dn(·), n ∈N, is an increasing sequence of nonnegative random variable, hence converges to the nonnegative random variable d(·). By temperedness of c(·) we can find a measurable set Ω˜ ⊂ Ω of full measure such that for all ω ∈ Ω˜ there exists n0(ω) > 0 such that for all n ≥ n0(ω) we have c(θ−nω) ≤ enδ/2. Hence dn(ω), n ∈ N, is an increasing sequence of positive numbers tending to finite value d(ω). Thus d(·) is finite almost everywhere. Furthermore, for m ∈N and x ∈ Ω˜ we have d(θ−mω) = ∞ ∑ k=1 e−δkc(θ−kθ−mω) ≤ eδm ∞ ∑ l=1 e−δlc(θ−lω) = eδmd(ω). This implies that lim supm→∞ 1 m log + d(θ−mω) ≤ δ. Following [4, Proposition 4.1.3], d(·) is tempered. 2 113 References [1] E. Alos, O. Mazet, D. Nualart. Stochastic calculus with respect to Gaussian processes. The Annals of Probability, 29, No. 2, (2001), 766–801. [2] H. Amann Ordinary Differential Equations: An Introduction to Nonlinear Anal- ysis. Walter de Gruyter, Berlin New York, (1990). [3] V. I. Arnold. Ordinary Differential Equations. Springer-Verlag Berlin Heidel- berg, (1992). [4] L. Arnold. Random Dynamical Systems. Springer, Berlin Heidelberg New York, (1998). [5] I. Bailleul, S. Riedel, M. Scheutzow. Random dynamical systems, rough paths and rough flows. J. Diff. Equat., 262, Iss. 12, (2017), 5792–5823. [6] F. Baudoin. Rough Paths Theory. Lecture note. [7] B. Boufoussi, S. Hajji Functional differential equations driven by a frac- tional Brownian motion. Computers & Mathematics with Applications, 62, Iss. 2, (2011), 746–754. [8] L. Barreira. Lyapunov exponents. Birkha¨user, (2017). [9] L. Barreira, C. Valls Stability of Nonautonomous Differential Equations. Springer-Verlag Berlin Heidelberg , (2008). [10] B. F. Bylov, R. E. Vinograd, D. M. Grobman and V. V. Nemytskii. Theory of Lyapunov Exponents, Nauka, Moscow, (1966), in Russian. [11] T. Cass, C. Litterer, T. Lyon. Integrability and tail estimates for Gaussian rough differential equations. Annal of Probability, 41(4), (2013), 3026–3050. [12] T. Caraballo, M. J. Garrido-Atienza, B. Schmalfuß, and J. Valero, Asymp- totic behaviour of a stochastic semilinear dissipative functional equation without uniqueness of solutions. Dis. and Cont. Dyn. Sys..Series B, 14, (2010), 439–455. 114 [13] T. Caraballo, S. Keraani Analysis of a stochastic SIR model with fractional Brownian motion. Stochastic Analysis and Applications, 36(5), (2018), 1–14, [14] C. Castaing, M. Valadier. Convex Analysis and Measurable Multifunctions. Lecture Notes in Math. No 580, Springer-Verlag, Berlin, Heidelberg, New York, (1977). [15] D. N. Cheban, Uniform exponential stability of linear almost periodic systems in Banach spaces. Electronic Journal of Differential Equations, 2000, No. 29, (2000), 1–18. [16] Y. Chen, H. Gao, M. J. Garrido-Atienza, B. Schmalfuß. Pathwise solutions of SPDEs driven by Ho¨lder-continuous integrators with exponent larger than 1/2 and random dynamical systems. Discrete Contin. Dyn. Syst., 34(1), (2014), 79–98. [17] L.Coutin. An introduction to (stochastic) calculus with respect to fractional Brownian motion. Se´minaire de Probabilite´s XL. Lecture Notes in Mathematics, vol 1899. Springer, Berlin, Heidelberg, (2007). [18] F. Comte and E. Renault. Long memory continuous time models. J. Econo- metrics, 73(1), (1996), 101–149. [19] N. D. Cong. Lyapunov spectrum of nonautonomous linear stochastic dif- ferential equations. Stoch. Dyn., 1, No. 1, (2001), 1–31. [20] N. D. Cong. Almost all nonautonomous linear stochastic differential equa- tions are regular. Stoch. Dyn., 4, No. 3, (2004), 351–371. [21] N. D. Cong, L. H. Duc, P. T. Hong. Young differential equations revisited. J. Dyn. Diff. Equat., Vol. 30, Iss. 4, (2018), 1921–1943. [22] N. D. Cong, L. H. Duc, P. T. Hong. Pullback attractors for stochastic Young differential delay equations. J. Dyn. Diff. Equat., to appear. [23] N. D. Cong, L. H. Duc, P. T. Hong. Lyapunov spectrum of nonautonomous linear Young differential equations. J. Dyn. Diff. Equat, (2020), 1749–1777. [24] H. Crauel, A. Debussche, F. Flandoli. Random attractors J. Dyn. Diff. Equat. 9, (1997), 307–341. [25] H. Crauel, F. Flandoli Attractors for random dynamical systems. Probab. Theory Relat. Fields 100, 365–393, (1994). 115 [26] C. Floris. Stochastic stability of the inverted pendulum subjected to delta- correlated base excitation. Advances in Engineering Software, 120, (2018), 4– 13. [27] H. Crauel, P. Kloeden, Nonautonomous and random attractors. Jahresber Dtsch. Math-Ver. 117 (2015), 173–206. [28] H. Cui, P. E.Kloeden Invariant forward attractors of non-autonomous ran- dom dynamical systems. J. Diff. Equat., 265, (2018), 6166–6186. [29] Deya, A., Gubinelli, M., Hofmanova´, M.Tindel, S. A priori estimates for rough PDEs with application to rough conservation laws. Journal of Func- tional Analysis, 276, Iss. 12, (2019), 3577–3645. [30] A. M. Davie Differential equations driven by rough paths: An approach via discrete approximation. Applied Mathematics Research Express, (2008). [31] B. P. Demidovich. Lectures on Mathematical Theory of Stability. Nauka, (1967). In Russian. [32] A. Deya, M. Gubinelli, M. Hofmanova, S. Tindel. A priori estimates for rough PDEs with application to rough conservation laws. Journal of Func- tional Analysis, 276, (2019), 3577–3645. [33] L. H. Duc, M. J. Garrido-Atienza, A. Neuenkirch, B. Schmalfuß. Exponen- tial stability of stochastic evolution equations driven by small fractional Brownian motion with Hurst parameter in (12 , 1). J. Diff. Equat., 264, Iss. 2, (2018), 1119–1145. [34] L. H. Duc, P. T. Hong. Young differential delay equations driven by Ho¨lder continuous paths, book chapter. Modern Mathematics and Mechanics, Springer International Publishing AG., (2019), 313–333. [35] L. H. Duc, P. T. Hong, N. D. Cong. Asymptotic stability for stochastic dissipative systems with a Ho¨lder noise. SIAM J. Control Optim., 57, No. 4, (2019), 3046 - 3071. [36] L. H. Duc, P. T. Hong. Asymptotic stability of controlled differential equations. Part I: Young integrals. Preprint. ArXiv: https://arxiv.org/abs/1905.04945, (2020). [37] L. H. Duc, J. Jost. How signatures affect expected return and volatility: a rough model under transaction cost. Preprint. https://www.mis.mpg.de/preprints/2019/preprint2019 101.pdf, (2019). 116 [38] L. H. Duc. Stability theory for Gaussian rough differential equations. Part I. Preprint. ArXiv: https://arxiv.org/abs/1901.00315, (2019). [39] L. H. Duc. Stability theory for Gaussian rough differential equations. Part II. Preprint. ArXiv: https://arxiv.org/abs/1901.01586, (2019). [40] L. H. Duc. Controlled differential equations as rough integrals. Preprint. https://arxiv.org/abs/2007.06295v1, (2020) [41] L. H. Duc. Random attractors for dissipative systems with rough noises. Preprint. https://www.mis.mpg.de/publications/preprints/2020/prepr2020- 97.html, (2020). [42] P. Friz, N. Victoir. Multidimensional Stochastic Processes as Rough Paths: The- ory and Applications. Cambridge Studies in Advanced Mathematics, 120. Cambridge Unversity Press, Cambridge, (2010). [43] L. Galeati Nonlinear Young differential equations: a review J. Dyn. Diff. Equat., (2021). [44] M. Garrido-Atienza, B. Maslowski, B. Schmalfuß. Random attractors for stochastic equations driven by a fractional Brownian motion. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 20, (2010), 2761–2782. [45] M. J. Garrido-Atienza, B. Schmalfuß. Ergodicity of the infinite dimensional fractional Brownian motion. J. Dyn. Dif. Equat., 23, (2011), 671-681. [46] M. Hairer, A. Ohashi. Ergodic theory for sdes with extrinsic memory. The Annals of Probability, 35, (2007), 1950–1977. [47] Y. Hu. Analysis on Gaussian Spaces. World scientic Publishing, (2016). [48] H.E. Hurst. Long term storage capacity of reservoir (with discussion). Trans. Am. Soc. Civ. Eng., (1951), 770–808. [49] P. E. Kloedena, C. Po¨tzscheb Nonautonomous bifurcation scenarios in SIR models. Mathematical methods in the Applied Sciences, 38, Iss. 16, (2015), 3495–3518. [50] L.C.G. Rogers. Arbitrage with fractional Brownian motion. Math Finance, 7(1), (1997), 95–105. [51] M. Garrido-Atienza, B. Maslowski, B. Schmalfuß. Random attractors for stochastic equations driven by a fractional Brownian motion. International Journal of Bifurcation and Chaos, 20, No. 9, (2010), 2761–2782. 117 [52] P. Imkeller, B. Schmalfuss. The conjugacy of stochastic and random differ- ential equations and the existence of global attractors. J. Dyn. Diff. Equat. 13, No. 2, (2001), 215–249. [53] R. A. Johnson, K. J. Palmer, G.R. Sell. Ergodic properties of linear dynami- cal systems. SIAM J. Math. Anal., 18, No.1, (1987), 1–33. [54] A. L. Jose , J. C. Robinson, A. Suarez. Stability, instability, and bifurca- tion phenomena in non-autonomous differential equations Nonlinearity, 15, (2002), 887–903. [55] I. Karatzas, S. Shreve. Brownian motion and Stochastics Caculus. Springer- Verlag, Second Edition (1991). [56] Peter E. Kloeden, Martin Rasmussen Nonautonomous Dynamical Systems. American Mathematical Society, (2011). [57] A. N. Kolmogorov Wienersche Spiralen und einige andere interessante Kur- ven im Hilbertschen Raum. Comptes Rendus (Doklady) de l’Acade´mie des Sciences de l’URSS (N.S.), 26, (1940), 115–118. [58] H. Kunita. Stochastic Flows and Stochastic Differential Equations. Cambridge University Press, (1990). [59] A. Lejay. Controlled differential equations as Young integrals: A simple approach. J. Diff. Equat., 249, (2010), 1777–1798. [60] X. Li, T. Lyons. Smoothness of Ito maps and diffusion processes on path spaces (I). Ann. Scient. E´c. Norm. Sup., 39, Iss. 4, (2006), 649–677. [61] T. Lyons. Differential equations driven by rough signals, I, An extension of an inequality of L.C. Young. Math. Res. Lett. 1, (1994), 451–464. [62] T. Lyons. Differential equations driven by rough signals. Revista Matema´tica Iberamaricana 14(2), (1998), 215–310. [63] T. Lyons, Zh. Qian. System Control and Rough Paths. Oxford Mathematical Monographs, (2002). [64] T. Lyons, M. Caruana, T. Le´vy. Differential equations driven by rough paths. Lecture Notes in Math. 1908. Springer-Verlag, (2007). [65] B. Mandelbrot, J. van Ness. Fractional Brownian motion, fractional noises and applications. SIAM Review, 4, No. 10, (1968), 422–437. 118 [66] B. Maslowskia, D. Nualart Evolution equations driven by a fractional Brownian motion. Journal of Functional Analysis, 202, (2003), 277–305. [67] V. M. Millionshchikov. Formulae for Lyapunov exponents of a family of en- domorphisms of a metrized vector bundle. Mat. Zametki, 39, 29–51. English translation in Math. Notes, 39 (1986), 17–30. [68] V. M. Millionshchikov. Formulae for Lyapunov exponents of linear systems of differential equations. Trans. I. N. Vekya Institute of Applied Mathematics, 22, (1987), 150–179, in Russian. [69] V. M. Millionshchikov. Statistically regular systems. Math. USSR-Sbornik, 4, No. 1 (1968), 125–135. [70] V. M. Millionshchikov. Metric theory of linear systems of differential equa- tions. Math. USSR-Sbornik, Vol. 4, No. 2 (1968), 149–158. [71] Y. Mishura. Stochastic Calculus for Fractional Brownian Motion and Related Processes. Lecture notes in Mathematics, Springer, (2008). [72] V. V. Nemytskii, V. V. Stepanov. Qualitative Theory of Differential Equations. GITTL, Moscow–Leningrad. (1949). English translation, Princeton Univer- sity Press, (1960). [73] I. Nourdin. Selected Aspects of Fractional Brownian Motion. Bocconi Univer- sity Press, Springer, (2012). [74] D. Nualart, A. Ra˘s¸canu. Differential equations driven by fractional Brow- nian motion. Collect. Math. 53, No. 1, (2002), 55–81. [75] V. I. Oseledets, A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems. Trans. Moscow Math. Soc., 19, (1968), 197- 231. [76] D. Revuz, M. Yor. Continuous Martingales and Brownian Motion, volume 293 of Grundlehren der Mathematischen Wissenschaften. Fundamental Principles of Mathematical Sciences. Springer Berlin Heidelberg New York, third ed., (1999). [77] S. Riedel, M. Scheutzow. Rough differential equations with unbounded drift terms. J. Diff. Equat., 262, 283–312, (2017). [78] A. Ruzmaikina. Stieltjes integrals of Ho¨lder continuous functions with ap- plications to fractional Brownian motion. J. Statist. Phys. 100, (2000), 1049– 1069. 119 [79] L. Shaikhet. Improved condition for stabilization of controlled inverted pendulum under stochastic perturbations. Discret. Contin. Dyn. Syst., 24, No. 4, (2009), 1335–1343. [80] R. J. Sacker, G. R. Sell. Lifting properties in skew-product flows with ap- plications to differential equations. Memoirs of the American Mathematical Society, 11, No. 190 (1977). [81] G. R. Sell. Nonautonomous differential equations and topological dynam- ics. I. The Basic theory Transactions of the American Mathematical Society, 127, No. 2 (1967), 241-262. [82] B. Schmalfuß. Attractors for the nonautonomous dynamical systems. In K. Groger, B. Fiedler and J. Sprekels, editors, Proceedings EQUADIFF99, World Scientific, (2000), 684–690. [83] B. Schmalfuß. Attractors for the nonautonomous and random dynamical systems perturbed by impulses. Discret. Contin. Dyn. Syst., 9, No. 3, (2003), 727–744. [84] W. Willinger, W. Leland, M. Taqqu, D. Wilson. On self-similar nature of ethernet traffic. IEEE/ACM Trans. Networking, (1994), 1–15. [85] L. C. Young. An inequality of the Ho¨lder type, connected with Stieltjes integration. Acta Math., 67, (1936), 251–282. [86] M. Za¨hle. Integration with respect to fractal functions and stochastic cal- culus. I. Probab. Theory Related Fields, 111 (1998), 333–374. [87] E. Zeidler. Nonlinear Functional Analysis and its Applications I. Springer - Verlag, (1986). [88] S.G. Samko, A.A. Kilbas, and O.I. Marichev. Fractional Integrals and Deriva- tives, Theory and Applications. Gordon and Breach Science Publishers, Yven- don, (1993). [89] Y. G. Sinai, Dynamical System II. Ergodic Theory with Applications to Dy- namical Systems and Statistical Mechanics. Springer-Verlag Berlin New York, (1989). [90] A. E. Zuniga, O. M. Romero, Equivalent Mathematical Representation of Second-Order Damped, Driven Nonlinear Oscillators Mathematical Prob- lems in Engineering, (2013), 11 pages. 120