In combination with 7.4 the convexity of κ 7→ λ
κ follows. To show that
λ κ is nonincreasing in κ ∈ 0, ∞ assume to the contrary that there is 0 κ
1
κ
2
such that λ
κ
1
λ κ
2
. The convexity of λ would then imply lim
κ→∞
λ κ = ∞ which is
impossible since we clearly have λ
κ ≤ 0 for all κ ∈ 0, ∞. b From Theorem 2.1 we deduce
λ κ ∈ [−β
c r
, 0], 7.5
and using 1.8 the claim follows. c 2.7 follows from 7.4 and the fact that lim
x ↓0
Ψx = 0, which is due to the boundedness of ξ from below. 2.8 follows using Ψ1 = λ
1 0, the fact that Ψx ∈ [−1, 0] due to Theorem 2.1 and 1.8, as well as 7.4 and the monotonicity of Ψ.
7.2 Annealed regime
In this subsection we primarily deal with proofs concerning the annealed Lyapunov exponents, i.e. in particular we assume 2.9.
Proof of Proposition 2.11. a For p
0 we directly obtain λ ≤ λ
p
from the corresponding for- mulae given in Theorems 2.1 and 2.7. If 0
p q, then Jensen’s inequality supplies us with 〈ut, 0
p
〉
1 p
≤ 〈ut, 0
q
〉
1 q
and the statement follows from the definition of λ
p
. b For β ∈ 0, 1 and 0 p q we get
〈ut, 0
β p+1−βq
〉 ≤ 〈ut, 0
p
〉
β
〈ut, 0
q
〉
1 −β
by Hölder’s inequality, which implies the desired convexity on 0, ∞.
c For p = 0 this follows from Proposition 2.6 a; for p ∈ 0, ∞ the proof proceeds in complete analogy to the corresponding part of the proof of Proposition 2.6 a.
d Assume to the contrary that u is p-intermittent but not q-intermittent for some q p. Then, by the definition of p-intermittency and part a of this same proposition we have
λ
p
λ
p+ ǫ
for all ǫ 0 and there exists ǫ
∗
0 such that λ
q
= λ
q+ ǫ
∗
. Fixing ǫ := q − p2 ∧ ǫ
∗
, we get using the convexity statement of part b and
λ
p
λ
q
: q
λ
q
≤ ǫ
q + ǫ − p
p λ
p
+ q
− p q +
ǫ − p q + ǫλ
q+ ǫ
ǫ q +
ǫ − p p
λ
q
+ q
− p q +
ǫ − p q + ǫλ
q+ ǫ
= ǫpq
q + ǫ − p
q λ
q
+ q − pq + ǫq
q + ǫ − p
q λ
q
= qλ
q
, a contradiction. Hence, u must be q-intermittent as well.
Proof of Proposition 2.9. We first show that L
sup p
has a zero in 0, β
c r
for p 0 large enough and then invoke Lemma 5.10 to conclude the proof.
To show the existence of such a zero, let µ ∈ M
1
[b, 0] such that Hµ|η ∞ and µ[−β
c r
3, 0] = 1. Then due to 5.1 and Proposition 5.2 we have I µ
N
= lim
n →∞
H µ
n
|µ
n −1
⊗ η = Hµ|η ∞ 2325
as well as L
β
c r
2, µ
N
= Z
Σ
+ b
log E
1
exp n
Z
T
ζY
s
+ β
c r
2 ds o
µ
N
dζ ≥ log E
1
exp {−β
c r
3 + β
c r
2T } 0.
We deduce L
sup p
β
c r
2 ≥ Lβ
c r
2, µ
N
− I µ
N
p 0 for p
0 large enough, in which case L
sup p
has zero −λ
p
∈ 0, β
c r
, cf. Theorem 2.7. Lemma 5.10 now tells us that we find
ν
p
∈ M
s 1
Σ
+ b
with L
sup p
−λ
p
= L−λ
p
, ν
p
− I ν
p
p. Since Prob can be assumed to be non-degenerate, one can show that for p large enough we have
ν
p
6= Prob . We then have I ν
p
∈ 0, ∞ and for ǫ 0 we obtain L
sup p+
ǫ
−λ
p
≥ L−λ
p
, ν
p
− I ν
p
p + ǫ L−λ
p
, ν
p
− I ν
p
p = L
sup p
−λ
p
= 0. Therefore, L
sup p+
ǫ
has a zero in 0, −λ
p
, whence due to Theorem 2.7 we have λ
p+ ǫ
λ
p
and u is p-intermittent.
The following claim is employed in the proof of Theorem 2.10.
Claim. For each neighbourhood U of Prob = η
N
in M
1
Σ
+ b
, there exists ǫ 0 such that {I ≤ ǫ} ⊆ U. Proof. Indeed, if this was not the case, we would find an open neighbourhood U of Prob such that
{I ≤ ǫ} 6⊆ U for all ǫ 0. Now since I is a good rate function cf. Corollary 6.5.15 in [DZ98] {I ≤ ǫ}∩ U
c
is compact and non-empty whence there exists ν ∈ M
1
Σ
+ b
with I ν = 0 and ν 6∈ U. But due to Corollary 5.3,
η
N
is the only zero of I , contradicting ν
∈ U.
Proof of Theorem 2.10. The continuity on 0,
∞ follows from Proposition 2.11 b. It therefore remains to show the continuity in 0.
For this purpose, we first show that L
sup p
↓ L pointwise as p ↓ 0 on 0, β
c r
. Fix
β ∈ 0, β
c r
. Then M := sup
ν∈M
s 1
Σ
+ b
L β, ν ∞ due to Corollary 5.7 and for ǫ 0 we may
therefore find a neighbourhood UProb of Prob such that |Lβ, ν − Lβ| ǫ for all ν ∈ UProb.
Choosing δ 0 small enough such that {I ≤ δ} ⊂ UProb which is possible due to the above
claim, we set p
ǫ
:= δM − Lβ. Then for p ∈ 0, p
ǫ
, we have |L
sup p
β − Lβ| ≤ ǫ. This proves the above convergence.
The continuity of p 7→ λ
p
in zero now follows from Theorems 2.1 and 2.7 where we may distinguish the cases that L does or does not have a zero in 0,
β
c r
.
8 The case of maximal drift
In subsection 8.1 we will give the modifications necessary to adapt the proofs leading to the results of section 2 to the case h = 1.
Subsequently, in subsection 8.2 we will provide an alternative approach to establish the existence of the first annealed Lyapunov exponent using a modified subadditivity argument. By means of the
2326
Laplace transform we will then retrieve an easy formula for the p-th annealed Lyapunov exponent for p
∈ N. Note that there have been some initial investigations of the first annealed Lyapunov exponent in the
case h = 1 using a large deviations approach to establish its existence cf. [Sch05].
8.1 Modifications in proofs for maximal drift
