Proposition 4.2. There is pathwise existence and uniqueness for Equation
22. Proof.
The proof is the same as given in Sznitman [26], Th 1.1, p.172, up to minor modifications. The main idea consists in using a Picard iteration in the space of probabilities on
C [0, T ], S
1
× R
endowed with an appropriate Wasserstein metric. We refer to it for details.
4.3 Fluctuations in the quenched model
The key argument of the proof is to explicit the speed of convergence as N → ∞ for the rotators to
the non-linear process see Prop. 4.3. A major difference between this work and [12] is that, since in our quenched model, we only
integrate w.r.t. oscillators and not w.r.t. the disorder, one has to deal with remaining terms, see Z
N
in Proposition 4.3, to compare with [12], Lemma 3.2, that would have disappeared in the averaged model
. The main technical difficulty of Proposition 4.3 is to control the asymptotic behaviour of such terms, see 24. As in [12], having proved Prop. 4.3, the key argument of the proof is a uniform
estimation of the norm of the process η
N ,
ω
, see Propositions 4.4 and 4.8, based on the generalized stochastic differential equation verified by
η
N ω
, see 30.
4.3.1 Preliminary results
We consider here a fixed realization of the disorder ω = ω
1
, ω
2
, . . . . On a common filtered probability space Ω,
F , F
t
, B
i i
≥1
, Q, endowed with a sequence of i.i.d. F
t
-adapted Brownian motions B
i
and with a sequence of i.i.d. F measurable random variables
ξ
i
with law λ, we define as x
i ,N
the solution of 11, and as x
ω
i
the solution of 22, with the same Brownian motion B
i
and with the same initial value ξ
i
. The main technical proposition, from which every norm estimation of
η
N ,
ω
follows is the following:
Proposition 4.3. E
sup
t ≤T
x
i ,N
t
− x
ω
i
t 2
≤ CN + Z
N
ω
1
, . . . , ω
N
, 23
where the random variable ω 7→ Z
N
ω is such that: lim
A →∞
lim sup
N →∞
P N Z
N
ω A = 0.
24 The rather technical proof of Proposition 4.3 is postponed to the end of the document see §A.
Once again, we stress the fact that the term Z
N
would have disappeared in the averaged model. The first norm estimation of the process
η
N ,
ω
which will be used to prove tightness is a direct consequence of Proposition 4.3 and of a Hilbertian argument:
Proposition 4.4. Under the hypothesis
H
F µ
on µ, the process η
N ,
ω
satisfies the following property: for all T 0,
sup
t ≤T
E
η
N ,
ω t
2 −3,2α
≤ A
N
ω
1
, . . . , ω
N
, 25
812
where lim
A →∞
lim sup
N →∞
P A
N
A = 0.
Proof. For all
ϕ ∈ W
3,2 α
, writing D
η
N ,
ω t
, ϕ
E =
1 p
N
N
X
i= 1
¦ ϕx
i ,N
t
, ω
i
− ϕx
ω
i
t
, ω
i
© +
1 p
N
N
X
i= 1
¦ ϕx
ω
i
t
, ω
i
− P
s
, ϕ
© ,
=: S
N ,
ω t
ϕ + T
N ,
ω t
ϕ, we have:
D η
N ,
ω t
, ϕ
E
2
≤ 2 S
N ,
ω t
ϕ
2
+ T
N ,
ω t
ϕ
2
. 26
But, by convexity, S
N ,
ω t
ϕ
2
≤
N
X
i= 1
D
2 x
i ,N
t
,x
ωi t
, ω
i
ϕ. Then, applying the latter equation to an orthonormal system
ϕ
p p
≥1
in the Hilbert space W
3,2 α
, summing on p, we have by Parseval’s identity on the continuous functional D
x
i ,N
t
,x
ωi t
, E
S
N ,
ω t
2 −3,2α
≤ E
N
X
i= 1
D
x
i ,N
t
,x
ωi t
, ω
i
2 −3,2α
, ≤ C
N
X
i= 1
1 + |ω
i
|
4 α
E
x
i ,N
t
− x
ω
i
t 2
, 27
≤ C
N
X
i= 1
1 +
|ω
i
|
4 α
C
N + Z
N
ω
1
, . . . , ω
N
, 28
where we used 19 in 27, and 23 in 28. On the other hand,
E
h T
N ,
ω t
ϕ
2
i =
1 N
E
N
X
i= 1
ϕx
ω
i
t
, ω
i
− P
t
, ϕ
2
,
= 1
N E
N
X
i= 1
ϕx
ω
i
t
, ω
i
− P
t
, ϕ
2
+ 1
N E
X
i 6= j
ϕx
ω
i
t
, ω
i
− P
t
, ϕ
ϕx
ω
j
t
, ω
j
− P
t
, ϕ
,
≤ 2
N E
N
X
i= 1
ϕx
ω
i
t
, ω
i 2
+ P
t
, ϕ
2
+ 1
N X
i 6= j
G ϕω
i
Gϕω
j
,
≤ 2
N E
N
X
i= 1
ϕx
ω
i
t
, ω
i 2
+ 2 P
t
, ϕ
2
+ 1
p N
N
X
i= 1
G ϕω
i 2
,
813
where G ϕω :=
R ϕ y, ω
i
P
ω
i
t
d y − P
t
, ϕ
. If we apply the same Hilbertian argument as for S
N ,
ω
, we see
E T
N ,
ω t
2 −3,2α
≤ 2C
N E
N
X
i= 1
1 + |ω
i
|
4 α
+ C + φ 7→
1 p
N
N
X
i= 1
G φω
i 2
−3,2α
, 29
It is easy to see that the last term in 29 can be reformulated as B
N
ω
1
, . . . , ω
N
, with the property that lim
A →∞
lim sup
N →∞
PB
N
A = 0. Combining 24, 26, 28 and 29, Proposition 4.4 is proved.
4.3.2 Tightness of the fluctuations process
