The main results of this dissertation include:
1) The existence and uniqueness of the solution of the nonautonomous stochastic differential equations driven by fractional Brownian motions and the properties 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 discretization scheme to compute the Lyapunov spectrum, the formula for the
spectrum for regular triangular equation, the regularity almost sure of the nonautonomous 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.

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− 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
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