getdoc9c3b. 2138KB Jun 04 2011 12:04:45 AM

El e c t ro n ic

Jo ur

n a l o

f P

r o

b a b i_{l i t y} Vol. 16 (2011), Paper no. 26, pages 792–829.

Journal URL

http://www.math.washington.edu/~ejpecp/

Quenched limits and fluctuations of the empirical measure

for plane rotators in random media

Eric Luçon

∗

eric.lucon@etu.upmc.fr

Abstract

The Kuramoto model has been introduced to describe synchronization phenomena observed in groups of cells, individuals, circuits, etc. The model consists ofN interacting oscillators on the one dimensional sphereS1, driven by independent Brownian Motions with constant drift chosen at random. This quenched disorder is chosen independently for each oscillator according to the same lawµ. The behaviour of the system for largeN can be understood via its empirical measure: we prove here the convergence asN _{→ ∞}of the quenched empirical measure to the unique solution of coupled McKean-Vlasov equations, under weak assumptions on the disorder µand general hypotheses on the interaction. The main purpose of this work is to address the issue of quenched fluctuations around this limit, motivated by the dynamical properties of the disordered system for large but fixed N: hence, the main result of this paper is a quenched Central Limit Theorem for the empirical measure. Whereas we observe a self-averaging for the law of large numbers, this no longer holds for the corresponding central limit theorem: the trajectories of the fluctuations process are sample-dependent.

Key words:Synchronization - quenched fluctuations - central limit theorem - disordered systems - Kuramoto model.

AMS 2000 Subject Classification:Primary 60F05, 60K37, 82C44, 92D25. Submitted to EJP on July 23, 2010, final version accepted March 17, 2011.

∗_{Laboratoire de Probabilités et Modèles Aléatoires (CNRS U.M.R. 7599) and Université Paris 6 – Pierre et Marie Curie,} U.F.R. Mathematiques, Case 188, 4 place Jussieu, 75252 Paris cedex 05, France

1

Introduction

In this work, we study the fluctuations in the Kuramoto model, which is a particular case of interact-ing diffusions with a mean field Hamiltonian that depends on a random disorder. The Kuramoto model was first introduced in the 70’s by Yoshiki Kuramoto ([17], see also [1] and references therein) to describe the phenomenon of synchronization in biological or physical systems. More precisely, the Kuramoto model is a particular case of a system of N oscillators (considered as ele-ments of the one-dimensional sphereS1:=R/2πZ) solutions to the following SDE:

dxi,N_t = 1

N

N

X

j=1

b(xi,N,xj,N,ωj)dt+c(xi,N,ωi)dt+dBti, t∈[0,T], i=1 . . .N, (1)

where T >0 is a fixed (but arbitrary) time, bandc are smooth periodic functions. The Kuramoto case corresponds to a sine interaction (b(x,y,ω) =Ksin(y₋x)andc(x,ω) =ω). This case which has the particularity of being rotationally invariant (namely, if(xi,N)_i is a solution of the Kuramoto model,(xi,N+c)_i,ca constant, is also a solution), will be referred to in this work as thesine-model. The parameter K > 0 is the coupling strength and (ωj) is a sequence of randomly chosen reals

(being i.i.d. realizations of a lawµ). The sequence(ω_j)_j is calleddisorderand represents the fact that the behaviour of each rotator xj,N depends on its own local frequencyωj.

Due to the mean field character of (1), the behaviour of the system can be understood via its empirical measureνN, process with values in_M1(S1_×R), that is the set of probability measures on oscillators and disorder:

∀(ω)_∈RN, _∀t_∈[0,T], νtN,(ω):=

1

N

N

X

j=1

δ_(xj,N t ,ωj),

where(ω) = (ωj)j≥1is a fixed sequence of disorder inRN.

The purpose of this paper is to address the issue of both convergence and fluctuations of the empir-ical measure, asN _{→ ∞}; thus the main theorem of this paper (Theorem 2.10) concerns a Central Limit Theorem in a quenched set-up (namely the quenched fluctuations ofνN around its limit). Some heuristic results have been obtained in the physical literature ([1] and references therein) concerning the convergence of the empirical measure, as N → ∞, to a time-dependent measure

(P_t(dx, dω))_t∈[0,T], whose density w.r.t. Lebesgue measure at time t, q_t(x,ω) is the solution of a deterministic non-linear McKean-Vlasov equation (see Eq.(5)). It is well understood ([1], [11]) that crucial features of this equation are captured in the sine-model by order parameters rt andψt

defined by:

r_teiψt₌

Z

S1_×R

ei xq_t(x,ω)dxµ(dω).

The quantity r_t captures in fact the degree of synchronization of a solution (the profile q_{t ≡ 2}1_π

for example corresponds to r =0 and represents a total lack of synchronization) andψt identifies

the center of synchronization: this is true and rather intuitive for unimodal profiles. Moreover ([22], see also [11], p.118) it turns out that if µis symmetric, all the stationary solutions can be parameterized (up to rotation) by the stationary version r of r_t which must satisfy a fixed point

relation r= Ψ_K,µ(r), withΨ_K,µ(_·)an explicit function such thatΨ_K,µ(0) =0. ForK small, r=0 is the only solution of such an equation and the system is not synchronized, but forKlarge, non-trivial solutions appear (synchronization). In the easiest instances, such a non-trivial solution is unique (in the sense thatr = Ψ_K,µ(r)admits a unique non-zero solution but of course one obtains an infinite number of solutions by rotation invariance that isψcan be chosen arbitrarily).

In[9], Dai Pra and den Hollander have rigorously shown the convergence of the averaged empirical measure LN ∈ M1(_C([0,T],S1)_× R) (probability measure on the whole trajectories and the disorder):

LN = 1

N

N

X

j=1

δ(xj,N_,ω j).

This convergence of the law of LN under the joint law of the oscillators and the disorder is shown via an averaged large deviations principle in the case where b(x,y,ω) = K · f(y − x) and

c(x,ω) = g(x,ω) for f and g smooth and bounded functions. As a corollary, it is deduced in

[9]the convergence of L_N and of νN, via a contraction principle, in the averaged set-up. In the case of unbounded disorder, the same proof can be generalized (thesis in preparation) under the following assumption:

∀t>0,

Z

R

et|ω|µ(dω)<∞. (H_µA)

One aim of this paper is to obtain the limit of νN,(ω) in the quenched model, namely for a fixed

realization of the disorder(ω). This result can be deduced from the large deviations estimates in

[9], via a Borel-Cantelli argument, but our result is more direct and works under the much weaker assumption onµ,R

R|ω|µ(dω)<∞.

A crucial aspect of the quenched convergence result, which is a law of large numbers, is that it shows theself-averagingcharacter of this limit: every typical disorder configuration leads asN → ∞to the same evolution equation.

However, it seems quite clear even at a superficial level that if we consider the central limit theorem associated to this convergence, self-averaging does not hold since the fluctuations of the disorder compete with the dynamical fluctuations. This leads for example to a remarkable phenomenon (pointed out e.g. in[2]on the basis of numerical simulations): even if the distributionµ is sym-metric, the fluctuations of a fixed chosen sample of the disorder makes it not symmetric and thus the center of the synchronization of the systemslowly(i.e. with a speed of order 1/pN) rotates in one direction and with a speed that depends on the sample of the disorder (Fig. 1 and 2). This non-self averaging phenomenon can be tackled in the sine-model by computing the finite-size order parameters (Fig. 2):

r_tN,(ω)eiψNt,(ω) ₌ 1

N

N

X

j=1

ei xtj,N ₌

D

νtN,(ω),ei x

E

. (2)

As a step toward understanding this non self-averaging phenomenon, the second and main goal of this paper is to establish a fluctuations result around the McKean-Vlasov limit in the quenched set-up (see Theorem 2.10). A central limit theorem for the averaged model is addressed in[9], applying techniques introduced by Bolthausen [6]. A fluctuations theorem may also be found in [8] for a

Figure 1: We plot here the evolution of the marginal on S1 of νN,(ω) for N = 600 oscillators in the sine-model (µ= 1

2(δ−1+δ1), K = 6). The oscillators are initially chosen independently and

uniformly on [0, 2π] independently of the disorder. First the dynamics leads to synchronization of the oscillators (t =6) to a profile which is close to a non-trivial stationary solution of McKean-Vlasov equation. We then observe that the center ψN,_t (ω) of this density moves to the right with an approximately constant speed; what is more, this speed of fluctuation turns out to be sample-dependent (see Fig. 2).

Figure 2: Trajectories of the center of synchronizationψN,(ω)in the sine-model for different realiza-tions of the disorder (µ= 1₂(δ₋0.5+δ0.5),K=4,N=400). We observe here the non self-averaging

phenomenon: direction and speed of the center depend on the choice of the initialN-sample of the disorder. Moreover, these simulations are compatible with speeds of order 1/pN.

model of social interaction in an averaged set-up. Here we prove convergence in law of thequenched

fluctuations process

ηN,(ω):=pN€νN,(ω)−PŠ,

seen as a continuous process in the Schwartz spaceS′ of tempered distributions onS1×R to the solution of an Ornstein-Uhlenbeck process. The quenched convergence is here understood as a weak convergencein law w.r.t. the disorderand is more technically involved than the convergence in the averaged system. The main techniques we exploit have been introduced by Fernandez and Méléard

[12]and Hitsuda and Mitoma [15], who studied similar fluctuations in the case without disorder. In[7], A Large Deviation Principle is also proved. We refer to Section 4 for detailed definitions. While numerical computations of the trajectories of the limit process of fluctuations clearly show a non self-averaging phenomenon, the dynamical properties of the fluctuations process that we find are not completely understood so far. Progress in this direction requires a good understanding of the spectral properties of the linearized operator of McKean-Vlasov equation around its non-trivial stationary solution; the stability of the non-synchronized solution q ≡ ₂1_π has been treated by Strogatz and Mirollo in[24]. In the particular case without disorder, spectral properties of the evolution operator linearized around the non-trivial stationary solution are obtained in[3], but the case with a general distributionµneeds further investigations.

This work is organized as follows: Section 2 introduces the model and the main results. Section 3 focuses on the quenched convergence ofνN. In Section 4, the quenched Central Limit Theorem is

proved. The last section 5 applies the fluctuations result to the behaviour of the order parameters in the sine-model.

2

Notations and main results

2.1

Notations

• ifX is a metric space,BX will be its Borelσ-field,

• Cb(X)(resp. C_bp(X), p=1, . . . ,∞), the set of bounded continuous functions (resp. bounded continuous with bounded continuous derivatives up to orderp) onX, (Xwill be oftenS1_×R),

• Cc(X) (resp. Ccp(X), p = 1, . . . ,∞), the set of continuous functions with compact support

(resp. continuous with compact support with continuous derivatives up to orderp) onX,

• D([0,T],X), the set of right-continuous with left limits functions with values onX, endowed with the Skorokhod topology,

• M1(Y), the set of probability measures onY (Y topological space, with a regularσ-field_B),

• MF(Y), the set of finite measures onY,

• (_M1(Y),w): _M1(Y) endowed with the topology of weak convergence, namely the coars-est topology on M1(Y) such that the evaluations ν 7→R f dν are continuous, where f are bounded continuous,

• (_M1(Y),v): _M1(Y) endowed with the topology of vague convergence, namely the coarsest topology on_M1(Y)such that the evaluationsν7→R fdν are continuous, where f are contin-uous with compact support.

We will useC as a constant which may change from a line to another.

2.2

The model

We consider the solutions of the following system of SDEs: fori=1, . . . ,N, forT>0, for all t≤T,

xi,N_t =ξi+ 1

N

N

X

j=1

Z t

0

b(x_si,N,x_sj,N,ωj)ds+

Z t

0

c(x_si,N,ωi)ds+Bit, (3)

where the initial conditionsξiare independent and identically distributed with lawλ, and indepen-dent of the Brownian motion(B) = (Bi)_i≥1, and whereb(resp.c) is a smooth function, 2π-periodic w.r.t. the two first (resp. first) variables. The disorder(ω) = (ωi)i≥1is a realization of i.i.d. random

variables with lawµ.

Remark 2.1.

The assumption that the random variables(ωi)are independent will not always be necessary and

will be weakened when possible.

Instead of consideringxi,N as elements ofR, we will consider their projection onS1. For simplicity, we will keep the same notationxi,N for this projection1.

We introduce the empirical measureνN (on the trajectories and disorder):

Definition 2.2.

For all t _≤ T , for a fixed trajectory(x1, . . . ,xN)_{∈ C}([0,T],(S1)N)and a fixed sequence of disorder

(ω), we define an element ofM1(S1_×R)by:

ν_tN,(ω)= 1

N

N

X

i=1

δ_(xi,N t ,ωi).

Finally, we introduce the fluctuations processηN,(ω)ofνN,(ω)around its limitP(see Th. 2.5):

Definition 2.3.

For all t_≤T , for fixed(ω)_∈RN, we define:

ηN,_t (ω)=pNν_tN,(ω)−P_t.

Throughout this article, we will denote asPthe law of the sequence of Brownian Motions and as_P the law of the sequence of the disorder. The corresponding expectations will be denoted asEand_E respectively.

2.3

Main results

2.3.1 Quenched convergence of the empirical measure

In[9], Dai Pra and den Hollander are interested in theaveragedmodel, i.e. in the convergence in law of the empirical measureunder the joint law of both oscillators and disorder. The model studied here, which is more interesting as far as the biological applications are concerned isquenched: for a fixed realization of the disorder (ω), do we have the convergence of the empirical measure? Moreover the convergence is shown under weaker assumptions on the moments of the disorder. We consider here the general case where b(x,y,ω) is bounded, Lipschitz-continuous, and 2π -periodic w.r.t. the two first variables. cis assumed to be Lipschitz-continuous w.r.t. its first variable, but not necessarily bounded (see the sine-model, where c(x,ω) =ω). We also suppose that the function ω 7→ S(ω) := sup_x∈S1|c(x,ω)| is continuous (this is in particular true if c is uniformly

continuous w.r.t. to both variables (x,ω), and obvious in the sine-model whereS(ω) = _|ω|). The Lipschitz bounds forbandc are supposed to be uniform inω.

The disorder(ω), is assumed to be a sequence of identically distributed random variables (but not necessarily independent), such that the law of each ωi is µ. We suppose that the sequence (ω)

satisfies the following property: for_P-almost every sequence(ω), 1

N

N

X

i=1

sup

x∈S1|

c(x,ωi)| →N→∞ Z

sup

x∈S1|

c(x,ω)_|µ(dω)<∞. (HQ_µ)

We make the following hypothesis on the initial empirical measure:

ν₀N,(ω)N−→→∞ν0, in law, in(M1(S1×R),w). (H0)

Remark 2.4. 1. The required hypotheses about the disorder and the initial conditions are weaker than for the large deviation principle:

• the (identically distributed) variables(ωi)need not be independent: we simply need a

convergence (similar to a law of large numbers) only concerning the functionS,

• Condition (H_µQ) is weaker than (H_µA) on page 796; for the sine-model, (H_µQ) reduces to

R

|ω|µ(dω)<∞,

• the initial values need not be independent, we only assume a convergence of the empir-ical measure.

2. The hypothesis (HQ_µ) is verified, for example, in the case of i.i.d. random variables, or in the case of an ergodic stationary Markov process.

3. Under (H₀), the second marginal ofν0 isµ.

In Section 3, we show the following:

Theorem 2.5.

Under the hypothesis(H₀)and (HQ_µ), for _P-almost every sequence(ωi), the random variableνN,(ω)

following weak equation (for every f continuous bounded onS1_×R, twice differentiable, with bounded derivatives):

Pt, f

=ν0, f

+1

2

Z t

0

dsPs, f′′

+

Z t

0

dsPs, f′(b[·,Ps] +c)

, (4)

where

b[x,m] =

Z

b(x,y,π)m(dy, dπ).

Moreover, with the same hypotheses, the law ofνN _{under the joint law of the oscillators and disorder}

(averaged model) converges weakly to P as well. Remark 2.6.

An easy calculation shows that P can be considered as a weak solution to the family of coupled McKean-Vlasov equations (see[9]):

1. Pcan be written asP(dx, dω) =µ(ω)Pω(dx),

2. if we defineqω_t through P_t(dx, dω) = µ(ω)qω_t (dx), qω_t is the unique weak solution of the McKean-Vlasov equation:

d dtq

ω

t =L

ω_qω

t , q

ω

0 =λ. (5)

where,_Lωis the following differential operator:

Lωqω_t =₋ ∂

∂x

–‚Z

R

b(x,y,π)qπ_t(dy)µ(dπ) +c(x,ω)

Œ

qω_t

™

+1

2 ∂2 ∂x2q

ω

t . (6)

We insist on the fact that Eq. (5) is indeed a (possibly) infinite system of coupled non-linear PDEs. To fix ideas, one may consider the simple case where µ= 1

2(δ−1+δ1). Then (5) reduces to two

equations (one for+1, the other for₋1) which are coupled via the averaged measure 1

2(q

+1

t +q−

1 t ).

But for more general situations (µ=_N(0, 1)say) this would consist of an infinite number of coupled equations.

Remark 2.7 (Generalization to the non compact case).

The assumption that the state variables are inS1, although motivated by the Kuramoto model, is not absolutely essential: Theorem 2.5 still holds in the non-compact case (e.g. whenS1 is replaced by

Rd), under the additional assumptions of boundedness of x _{7→ |}c(x,ω)_|and the first finite moment of the initial condition:R_|x|λ(dx)<∞.

2.3.2 Quenched fluctuations of the empirical measure

Theorem 2.5 says that for_P-almost every realization(ω)of the disorder, we have the convergence ofνN,(ω)_{towards P, which is a law of large numbers. We are now interested in the corresponding} Central Limit Theorem associated to this convergence, namely, for afixedrealization of the disorder

(ω), in the asymptotic behaviour, asN → ∞of the fluctuations fieldηN,(ω)taking values in the set of signed measures:

∀t∈[0,T], η_tN,(ω):=pN

ν_tN,(ω)−Pt

.

In the case with no disorder, such fluctuations have already been studied by numerous authors (eg. Sznitman[25], Fernandez-Méléard[12], Hitsuda-Mitoma[15]). More particularly, Fernandez and Méléard show the convergence of the fluctuations field in an appropriate Sobolev space to an Ornstein-Uhlenbeck process.

Here, we are interested in thequenchedfluctuations, in the sense that the fluctuations are studied for fixed realizations of the disorder. We will prove a weak convergence of the law of the process ηN,(ω)_{,in law w.r.t. the disorder.}

In addition to the hypothesis made in §2.3.1, we make the following assumptions about b and c

(where_Dpis the set of all differential operators of the form∂uk∂_πl withk+l≤p):







b∈ C_b∞(S1×R), c∈ C∞(S1×R),

∃α >0, sup

D∈D6 Z

R

sup_u∈S1|Dc(u,π)|2

1+_|π|2α dπ <∞,

(H_b,c)

Furthermore, we make the following assumption about the law of the disorder (α is defined in (H_b,c)):

the(ω_j)are i.i.d. and

Z

R

|ω|4αµ(dω)<∞. (H_µF)

Remark 2.8.

The regularity hypothesis aboutbandccan be weakened (namelyb∈ Cbn(S 1

×R)andc∈ Cm(S1_× R)for sufficiently largenandm) but we have keptm=n=_∞for the sake of clarity.

Remark 2.9.

In the case of the sine-model, Hypothesis (H_b,c) is satisfied withα=2 for example.

In order to state the fluctuations theorem, we need some further notations: for alls_≤T, let_Ls be the second order differential operator defined by

Ls(ϕ)(y,π):= 1

2ϕ

′′_{(y,π) +ϕ}′_(y,π)(b[y,P

s] +c(y,π)) +

P_s,ϕ′(_·,_·)b(_·,y,π). LetW the Gaussian process with covariance:

E(Wt(ϕ1)Ws(ϕ2)) =

Z s∧t 0

¬

Pu,ϕ′1ϕ2′

¶

For allϕ1,ϕ2bounded and continuous onS1×R, let

Γ₁(ϕ1,ϕ2) =

Z

R

Cov_λ ϕ1(·,ω),ϕ2(·,ω)

µ(dω), (8)

=

Z

S1_×R ‚

ϕ1−

Z

S1

ϕ1(·,ω)dλ

Œ ‚

ϕ2−

Z

S1

ϕ2(·,ω)dλ

Œ

λ(dx)µ(dω),

and

Γ₂(ϕ1,ϕ2) =Covµ ‚Z

S1

ϕ1dλ,

Z

S1

ϕ2dλ

Œ

, (9)

=

Z

R ‚Z

S1

ϕ1dλ−

Z

S1_×R

ϕ1dλdµ

Œ ‚Z S1

ϕ2dλ−

Z

S1_×R

ϕ2dλdµ

Œ

dµ.

For fixed (ω), we may consider _HN(ω), the law of the process ηN,(ω); _HN(ω) belongs to

M1(_C([0,T],_S′)), where _S′ is the usual Schwartz space of tempered distributions on S1_×R. We are here interested in the law of the random variable(ω)_{7→ HN}(ω)which is hence an element of_M1(_M1(_C([0,T],_S′))).

The main theorem (which is proved in Section 4) is the following:

Theorem 2.10 (Fluctuations in the quenched model).

Under(HF_µ),(H_b,c), the sequence(ω)_{7→ HN}(ω)converges in law to the random variableω7→ H(ω), where H(ω) is the law of the solution to the Ornstein-Uhlenbeck process ηω solution in S′ of the following equation:

ηω_t =X(ω) +

Z t

0 Ls∗η

ω

s ds+Wt, (10)

where,Ls∗ is the formal adjoint operator ofLs and for all fixedω, X(ω) is a non-centered Gaussian

process with covarianceΓ₁ and with mean value C(ω). As a random variable inω, ω7→C(ω)is a Gaussian process with covarianceΓ₂. Moreover, W is independent on the initial value X .

Remark 2.11.

In the evolution (10), the linear operatorLs∗ is deterministic ; the only dependence in ω lies in

the initial conditionX(ω), through its non trivial meansC(ω). However, numerical simulations of trajectories ofηω(see Fig. 3) clearly show a non self-averaging phenomenon, analogous to the one observed in Fig 2: ηω_t not only depends on ω through its initial condition X(ω), but also for all positive timet>0.

Understanding how the deterministic operatorLs∗ propagates the initial dependence in ωon the

whole trajectory is an intriguing question which requires further investigations (work in progress). In that sense, one would like to have a precise understanding of the spectral properties of_Ls∗, which appears to be deeply linked to the differential operator in McKean-Vlasov equation (6) linearized around its non-trivial stationary solution.

Remark 2.12 (Generalization to the non-compact case).

whereS1is replaced byRd. To this purpose, one has to introduce an additional weight(1+_|x_|α)−1in the definition of the Sobolev norms in Section 4 and to suppose appropriate hypothesis concerning the first moments of the initial conditionλ(R|x|βλ(dx)<∞for a sufficiently largeβ).

2.3.3 Fluctuations of the order parameters in the Kuramoto model

For givenN_≥1, t_∈[0,T]and disorder(ω)_∈RN, let rN_t ,(ω)>0 andζ_tN,(ω)∈S1 such that

r_tN,(ω)ζN,t (ω)=

1

N

N

X

j=1

ei xtj,N ₌

D

νtN,(ω),ei x

E

.

Proposition 2.13 (Convergence and fluctuations forr_tN,(ω)).

We have the following:

1. Convergence of r_tN,(ω): ForP-almost every realization of the disorder, rN,(ω)converges in law in

C([0,T],R), to r defined by

t∈[0,T]_7→rt:=

€

Pt, cos(·)

2

+Pt, sin(·)

2Š12_.

2. If r₀>0then

∀t∈[0,T], rt>0. (Hr)

3. Fluctuations of r_tN,(ω)around its limit: Let

t_{7→ Rt}N,(ω):=pNr_tN,(ω)₋r_t

be the fluctuations process. For fixed disorder (ω), let RN,(ω) _{∈ M1}(_C([0,T],R))

be the law of _RN,(ω). Then, under (H_r), the random variable (ω) _7→ RN,(ω) con-verges in law to the random variable ω 7→ Rω, where Rω is the law of Rω :=

1

r 〈P, cos(·)〉 ·

ηω, cos(_·)+_〈P, sin(_·)_{〉 ·}ηω, sin(_·). Remark 2.14.

In simpler terms, this double convergence in law corresponds for example to the convergence in law of the corresponding characteristic functions (since the tightness is a direct consequence of the tightness of the process η); i.e. for t1, . . . ,tp ∈ [0,T] (p ≥ 1) the characteristic function of

(_RtN,(ω)

1 , . . . ,R

N,(ω)

tp ) for fixed (ω) converges in law, as a random variable in (ω), to the random

characteristic function of(_Rtω

1, . . . ,R ω

tp).

Proposition 2.15 (Convergence and fluctuations forζN,(ω)_).

We have the following:

1. Convergence of ζN,(ω): Under(Hr), for P-almost every realization of the disorder (ω), ζN,(ω)

converges in law toζ:t∈[0,T]_7→ζt:= 〈 Pt,ei x〉

Figure 3: We plot here the evolution of the imaginary part of the processZω, for different realiza-tions ofω; the trajectories are sample-dependent, as in Fig 2.

2. Fluctuations ofζN,(ω)around its limit: Let

t_{7→ Zt}N,(ω):=pNζN,_t (ω)−ζt

be the fluctuations process. For fixed disorder(ω), letZN,(ω)_{∈ M1}(_C([0,T],R))be the law of

ZN,(ω). Then, under(Hr), the random variable (ω)7→ZN,(ω)converges in law to the random

variableω7→Zω, whereZωis the law of_Zω:= _r12 €

rηω, cos(_·)+¬P,ei x¶_RωŠ.

In the sine-model, we haveζN,(ω)=eiψN,(ω) whereψN,(ω)is defined in (2) and is plotted in Fig. 2. Some trajectories of the process_Zωare plotted in Fig. 3.

This fluctuations result is proved in Section 5.

3

Proof of the quenched convergence result

In this section we prove Theorem 2.5. Reformulating (3) in terms ofνN,(ω), we have:

∀i=1 . . .N, _∀t_∈[0,T], xi,N_t =ξi+

Z t

0

b[x_si,N,ν_sN]ds+

Z t

0

c(x_si,N,ωi)ds+Bti, (11)

where we recall that b[x,m]:=R b(x,y,π)m(dy, dπ).

The idea of the proof of Theorem 2.5 is the following: we show the tightness of the sequence(νN,(ω))

separable and by an argument of boundedness of the second marginal of any accumulation point, thanks to (H_µQ), we show the tightness inD([0,T],(_MF,w)). The proof is complete when we prove the uniqueness of any accumulation point.

3.1

Proof of the tightness result

We want to show successively:

1. Tightness of_L(νN,(ω))inD([0,T],(_MF,v)), 2. Equation verified by any accumulation point, 3. Characterization of the marginals of any limit, 4. Convergence inD([0,T],(_MF,w)).

3.1.1 Equation verified byνN,(ω)

For f _{∈ Cb}2(S1_×R), we denote by f′, f′′the first and second derivative off with respect to the first variable. Moreover, ifm∈ M1(S1_×R), thenm, fstands forR

S1_×Rf(x,π)m(dx, dπ).

Applying Ito’s formula to (11), we get, for all f _{∈ Cb}2(S1_×R),

D

νtN,(ω), f

E

=

D

ν₀N,(ω), f

E

+1

2

Z t

0

ds¬ν_sN,(ω), f′′¶

+

Z t

0

ds¬ν_sN,(ω), f′·(b[_·,ν_sN,(ω)] +c)¶+MN,f(t),

whereMN,f(t):= _N1

PN j=1

Rt 0 f

′_(xN,(ω)

j ,ωj)dBj(s)is a martingale (f′bounded).

3.1.2 Tightness of_L(νN,(ω))in D([0,T],(_MF,v))

Cc(S1×R)is separable: let(fk)k≥1(elements ofC∞(S1×R)) a dense sequence inCc(S1×R), and

let f_0≡1. We defineΩ:=D([0,T],(_M1,v))and the applicationsΠ_f, f _{∈ Cc}(S1_×R)by:

Π_f : Ω _→ D([0,T],R)

m 7→ m, f.

Let (P_n)_n a sequence of probabilities on Ω and (Π_fP_n) = P_n◦Π−_f1 _∈D([0,T],R). We recall the following result:

Lemma 3.1.

If for all k_≥0, the sequence(Π_f

kPn)n is tight inM1(D([0,T],R)), then the sequence(Pn)n is tight in M1(D([0,T],(_M1,v))).

Hence, it suffices to have a criterion for tightness in D([0,T],R). Let Xn_t be a sequence of processes in D([0,T],R) and _Ftn a sequence of filtrations such that Xn is _Fn-adapted. Let φn_{={stopping times forF}n_{}. We have (cf. Billingsley[4]):}

Lemma 3.2 (Aldous’ criterion).

If the following holds, 1. L€sup_t≤TXn

t

Š

n is tight,

2. For allǫ >0andη >0, there existsδ >0such that

lim sup

n

sup

S,S′_∈φn_;S≤S′_{≤(S+δ)∧T}

P€X_Sn₋X_Sn′

_{> η}Š_≤ǫ,

then_L(Xn)is tight.

Proposition 3.3.

The sequence_L(νN,(ω))is tight inD([0,T],(_MF,v)).

Proof. For allǫ >0, for allk_≥1 (the casek=0 is straightforward),

P ‚

sup

t≤T

D

ν_tN,(ω), f_k

E > 1 ǫ Œ

≤ǫf_k ∞E



  sup_t≤T

D

ν_tN,(ω), 1

E

| {z } =1   

, (Markov Inequality).

The tightness follows.

For allk≥1, we have the following decomposition:

D

ν_tN,(ω), fk

E

=Dν₀N,(ω), fk

E

+AN,_t (ω)(fk) +M N,(ω)

t (fk),

whereAN_t,(ω)(f_k)is a process of bounded variations, and M_tN,(ω)(f_k)is a square-integrable martin-gale. Then it suffices to verify Lemma 3.2, (2) forAandM separately. For allǫ >0 andη >0, for all stopping timesS,S′∈φN_;S≤S′_{≤(S+δ)∧T, we have:}

a_N:=P 

A

N,(ω)

S′ (fk)−A

N,(ω)

S (fk)

> η ‹ , ≤ 1 ηE   Z S′

S ds ¬

ν_sN,(ω), f_k′_·(b[_·,ν_sN,(ω)] +c)¶

 + 1 ηE   1 2

Z S′

S ds ¬

ν_sN,(ω), f_k′′¶



,

≤ C

ηE

S′−S≤ǫ, forδsufficiently small.

(we use here that f_k are of compact support for k _≥ 1; in particular the function (x,π) _7→

f_k′(x,π)c(x,π)is bounded). Furthermore,

P 

M

N,(ω)

S′ (fk)−M

N,(ω)

S (fk)

> η ‹ =P M

N,(ω)

S′ (fk)−M

N,(ω)

S (fk)

2

> η2

,

≤ 1

η2E

M

N,(ω)

S′ (fk)−M

N,(ω)

S (fk)

2 , ≤ 1

(Nη)2E





N

X

i=1

Z S′

S

f_k′2(x_si,ωi)ds



≤ C

At this point,_L(νN,(ω))is tight inD([0,T],(_MF,v)).

3.1.3 Equation satisfied by any accumulation point in D([0,T],(_MF,v))

Using hypothesis (H0), it is easy to show that the following equation is satisfied for every

accumu-lation pointν, for everyf _{∈ Cc}2(S1_×R)(we use here thatS1is compact):

ν_t, f=ν0, f

+

Z t

0

dsν_s, f′·(b[_·,ν_s] +c)+1

2

Z t

0

dsν_s, f′′. (12)

For any accumulation pointν, the following lemma gives a uniform bound for the second marginal ofν:

Lemma 3.4.

Let Q be an accumulation point of L(νN,(ω))_N in D([0,T],(_M1,v)) and let beν ∼Q. For all t ∈

[0,T], we define by(νt,2)the second marginal ofνt:

∀A∈ B(R), (νt,2)(A) =

Z

S1_×A

ν_t(dx, dπ).

Then, for all t_∈[0,T],

Z

R sup

x∈S1|

c(x,π)_|(νt,2)(dπ)≤

Z

R sup

x∈S1|

c(x,π)_|µ(dπ).

Proof. Letφbe a_C2positive function such thatφ≡1 on[₋1, 1],φ≡0 on[₋2, 2]and_φ ∞≤1. Let,

∀k≥1, φk:=π7→φ

_π

k

‹

.

Thenφk ∈ Cc2(R) and φk(π) →k→∞ 1, for allπ. We have also for all π ∈R, |φk(π)| ≤

_φ

∞,

|φ_k′(π)_{| ≤φ}′

∞,|φ′′k(π)| ≤

_φ′′

∞.

We have successively, denotingS(π):=sup_x∈S1|c(x,π)|, Z

S1_×R

S(π)νt(dx, dπ) =

Z

S1_×R

lim inf

k→∞ φk(π)S(π)νt(dx, dπ),

≤lim inf

k→∞

Z

S1_×R

φk(π)S(π)νt(dx, dπ), (Fatou’s lemma),

=lim inf

k→∞ Nlim→∞

Z

S1_×R

φk(π)S(π)ν N,(ω)

t (dx, dπ), (13)

≤ lim

N→∞

Z

S1_×R

S(π)νtN,(ω)(dx, dπ), (since

_φ

∞≤1).

The equality (13) is true since(x,π)_7→φk(π)S(π)is of compact support inS1×R(recall thatS is

But, by definition ofν_tN,(ω), and using the hypothesis (HQ_µ) concerningµ, we have,

lim

N→∞

Z

S1_×R

S(π)ν_tN,(ω)(dx, dπ) =

Z

R

S(π)µ(dπ). (14)

The result follows.

Remark 3.5.

(14) is only true for_P-almost every sequence(ω). We assume that the sequence(ω)given at the beginning satisfies this property.

3.1.4 Tightness in D([0,T],(_MF,w))

We have the following lemma (cf. [21]):

Lemma 3.6.

Let (Xn) be a sequence of processes in D([0,T],(MF,w)) and X a process belonging to C([0,T],(_MF,w)). Then,

Xn

L

→X⇔

(

X_n→L X inD([0,T],(_MF,v)),

X_n, 1_{→ 〈}L X, 1_〉 inD([0,T],R). So, it suffices to show, for any accumulation pointν:

1. 〈ν, 1〉 = 1: Eq. (12) is true for all f ∈ C2

c(S1×R), so in particular for fk(x,π):=φk(π).

Using the boundedness shown in lemma 3.4, we can apply dominated convergence theorem in Eq. (12). We then haveνt, 1

=1, for all t∈[0,T]. The fact that Eq. (12) is verified for all f ∈ C2

b(S

1_{×R)can be shown in the same way.}

2. Continuity of the limit: For all 0_≤s≤t≤T, for all f ∈ C_b2(S1_×R),

_νt, f₋νs, f

_≤K

Z t

s

_νu, f′_·b[_·,νu]

du+1

2

Z t

s

_νu, f′′du

+

Z t

s

_{νu, f}′_{·cdu≤C× |t−s|, for some constantC.}

Noticing that we used again Lemma 3.4 to bound the last term, we have the result.

Remark 3.7.

It is then easy to see that the second marginal (on the disorder) of any accumulation point isµ. At this point _L(νN,(ω)) is tight in D([0,T],(_MF,w)). It remains to show the uniqueness of any accumulation point: it shows firstly that the sequence effectively converges and that the limit does not depend on the given sequence(ω).

3.2

Uniqueness of the limit

Proposition 3.8.

There exists a unique element P ofD([0,T],MF(S1×R))which satisfies equation(12), P0∈ M1 and

P_0,2=µ.

Remark 3.9.

Since this proof is an adaptation of Oelschläger[20]Lemma 10, p.474, we only sketch the proof: We can rewrite Eq. (12) in a more compact way:

P_t, f=P₀, f+

Z t

0

P_s, L(P_s)(f)ds, _∀f _{∈ Cb}2(S1_×R), 0_≤t_≤T, (15)

where,

L(P)(f)(x,ω):= 1

2f

′′_{(x,ω) +h(x,ω,P)f}′_{(x,ω), ∀x ∈S}1_,

∀ω∈R, and,

h(x,π,P) =b[x,P] +c(x,π).

Lett7→Ptbe any solution of Eq. (15). One can then introduce the following SDE (whereξ∈Rand

˜

ξ∈S1 is its projection onS1):

¨

dξt = h(ξ˜t,ωt,Pt)dt+dWt, ξ0=ξ˜0, (ξ˜0,ω0)∼P0,

dωt = 0.

(16) Eq. (16) has a unique (strong) solution(ξt,ωt)t∈[0,T]= (ξt,ω0)t∈[0,T]. The proof of uniqueness in Eq. (12) consists in two steps:

1. P_t=_L(ξ˜_t,ω_t), for all t∈[0,T], 2. Uniqueness of ˜ξ.

We refer to Oelschläger[20]for the details. Another proof of uniqueness can be found in[9]or in

[13]via a martingale argument.

4

Proof of the fluctuations result

Now we turn to the proof of Theorem 2.10. To that purpose, we need to introduce some distribution spaces:

4.1

Distribution spaces

Let _S := _S(S1_×R) be the usual Schwartz space of rapidly decreasing infinitely differentiable functions. Let_Dp be the set of all differential operators of the form∂uk∂_πl withk+l ≤p. We know

from Gelfand and Vilenkin[14]p. 82-84, that we can introduce on_S a nuclear Fréchet topology by the system of seminorms_{k · kp},p=1, 2, . . . , defined by

φ2

p=

p

X

k=0

Z

S1_×R

(1+_|π|2)2p X

D∈Dk

|Dφ(y,π)_|2dydπ.

Let_S′ be the corresponding dual space of tempered distributions. Although, for the sake of sim-plicity, we will mainly consider ηN,(ω) as a process in _C([0,T],_S′), we need some more precise estimations to prove tightness and convergence. We need here the following norms:

For every integer j, α∈R+, we consider the space of all real functions ϕ defined on S1×R with derivative up to order jsuch that

_ϕ

j,α:= 

 

X

k1+k2≤j Z

S1_×R

|∂_xk₁∂_πk₂ϕ(x,π)|2

1+_|π|2α dxdπ 

 

1/2

<∞.

LetW₀j,α be the completion of_Cc∞(S1_×R)for this norm;(W₀j,α,_{k · kj,}α)is a Hilbert space. LetW−

j,α

0

be its dual space.

LetCj,α be the space of functionsϕwith continuous partial derivatives up to order j such that lim

|π|→∞xsup∈S1

|∂_xk₁∂_πk₂ϕ(x,π)|

1+_|π|α =0, for allk1+k2≤ j,

with norm

_ϕ

Cj,α= X

k1+k2≤j

sup

x∈S1

sup

π∈R

|∂xk1∂πk2ϕ(x,π)|

1+_|π|α .

We have the following embeddings:

W₀m+j,α,→Cj,α,m>1,j≥0,α≥0, i.e. there exists some constantC such that

_ϕ

Cj,α≤C _ϕ

m+j,α. (17)

Moreover,

W₀m+j,α,→W₀j,α+β,m>1,j_≥0,α≥0,β >1. Thus there exists some constantC such that

_ϕ

j,α+β≤C _ϕ

m+j,α.

We then have the following dual continuous embedding:

W₀−j,α+β ,→W₀−(m+j),α, m>1,α≥0,β >1. (18) It is quite clear thatS ,→W₀j,α for any jandα, with a continuous injection.

Lemma 4.1.

For every x,y∈S1,ω∈R, for allα, the linear mappings W₀3,α→Rdefined by Dx,y,ω(ϕ):=ϕ(x,ω)−ϕ(y,ω);Dx,ω:=ϕ(x,ω);Hx,ω=ϕ′(x,ω),

are continuous and

Dx,y,ω

−3,α≤C|x−y|(1+|ω|

α_{), (19)}

D_x,ω

−3,α≤C(1+|ω|

α_{), (20)}

H_x,ω

−3,α≤C(1+|ω|

α_{). (21)}

Proof. Letϕbe a function of class_C∞with compact support onS1_×R, then,

|ϕ(x,ω)₋ϕ(y,ω)_{| ≤ |}x− y|sup

u

_ϕ′(u,ω)_,

≤ |x− y|(1+_|ω|α)sup

u,ω

_ϕ′(u,ω)

1+_|ω|α !

,

≤ |x₋ y_|(1+_|ω|α)_ϕ

1,α,

≤C_|x₋y_|(1+_|ω|α)_ϕ

3,α,

following (17) with j =1 andm=2>1. Then, (19) follows from a density argument. (20) and (21) are proved in the same way.

4.2

The non-linear process

The proof of convergence is based on the existence of the non-linear process associated to McKean-Vlasov equation. Such existence has been studied by numerous authors (eg. Dawson[10], Jourdain-Méléard[16], Malrieu[18], Shiga-Tanaka[23], Sznitman[25],[26]) mostly in order to prove some propagation of chaos properties in systems without disorder. We consider the present similar case where disorder is present. Let us give some intuition of this process. One can replace the non-linearity in Eq. (5) by an arbitrary measurem(dx, dω):

∂_tqω_t = 1

2∂x xq

ω

t −∂x

”

(b[x,m] +c(x,ω))qω_t —.

In this particular case, it is usual to interpretqω_t as the time marginals of the following diffusion:

d xω_t =d Bt+b[xωt ,m]d t+c(x

ω

t ,ω)d t, ω∼µ.

It is then natural to consider the following problem, wheremis replaced by the proper measureP: on a filtered probability space(Ω,_F,_Ft,B,x₀,Q), endowed with a Brownian motionB and with a

F0 measurable random variablex₀, we introduce the following system:







xω_t = x0+

Rt 0 b[x

ω

s ,Ps]ds+

Rt 0c(x

ω

s ,ω)ds+Bt,

ω ∼ µ,

Pt = L(xt,ω),∀t∈[0,T].

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 onC([0,T],S1×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 asN_{→ ∞}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. oscillatorsand notw.r.t. the disorder, one has to deal with remaining terms, seeZ_N

in Proposition 4.3, to compare with[12], Lemma 3.2, that would have disappeared in theaveraged 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 processηN,(ω), see Propositions 4.4 and 4.8, based on the generalized stochastic differential equation verified byηN(ω), see (30).

4.3.1 Preliminary results

We consider here a fixed realization of the disorder (ω) = (ω1,ω2, . . .). On a common filtered

probability space (Ω,_F,_Ft,(Bi)_i≥1,Q), endowed with a sequence of i.i.d. _Ft-adapted Brownian motions (B_i) and with a sequence of i.i.d. _F0 measurable random variables(ξi) with law λ, we define asxi,N the solution of (11), and as xωi _{the solution of (22), with the same Brownian motion}

Bi and with the same initial valueξi.

The main technical proposition, from which every norm estimation ofηN,(ω)follows is the following:

Proposition 4.3.

E –

sup

t≤T

x

i,N

t −x

ωi t

2™

≤C/N+Z_N(ω1, . . . ,ωN), (23)

where the random variable(ω)_7→Z_N(ω)is such that:

lim

A→∞lim supN→∞ 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 termZ_N would have disappeared in the averaged model. The first norm estimation of the process ηN,(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_µF)onµ, the processηN,(ω)satisfies the following property: for all T >0,

sup

t≤T

E

ηN

,(ω)

t

2

−3,2α

where

lim

A→∞lim supN→∞ P

AN>A

=0.

Proof. For allϕ∈W₀3,2α, writing

D

ηN,_t (ω),ϕE= _p1

N

N

X

i=1

¦

ϕ(xi,N_t ,ωi)−ϕ(x

ω_i

t ,ωi)

©

+_p1

N

N

X

i=1

¦

ϕ(xωi

t ,ωi)−

P_s,ϕ©,

=:S_tN,(ω)(ϕ) +T_tN,(ω)(ϕ), we have:

D

ηN,_t (ω),ϕ

E2 ≤2

SN,_t (ω)(ϕ)2+T_tN,(ω)(ϕ)2

. (26)

But, by convexity,

S_tN,(ω)(ϕ)2_≤

N

X

i=1

D2

xit,N,x

ωi t ,ωi

(ϕ).

Then, applying the latter equation to an orthonormal system (ϕp)p≥1 in the Hilbert spaceW3,2

α

0 ,

summing onp, we have by Parseval’s identity on the continuous functionalD_xi,N

t ,x ωi t , E S

N,(ω)

t

2

−3,2α ≤E   N X

i=1

Dxi_t,N,xωi t ,ωi

2

−3,2α 

,

≤C

N

X

i=1

(1+_|ωi|4α)E

x

i,N

t −x

ωi t 2 , (27) ≤C N X

i=1

€

1+_|ω_i|4አC/N+Z_N(ω1, . . . ,ωN)

, (28)

where we used (19) in (27), and (23) in (28). On the other hand,

E h

T_tN,(ω)(ϕ)2

i = 1 NE    ( _N X

i=1

(ϕ(xωi

t ,ωi)−

P_t,ϕ)

)2

 , = 1 NE   N X

i=1

(ϕ(xωi

t ,ωi)−

P_t,ϕ)2

  + 1 NE    X

i6=j

(ϕ(xωi

t ,ωi)−

Pt,ϕ

)(ϕ(xω_t j,ωj)−

Pt,ϕ

)   , ≤ 2 NE   N X

i=1

(ϕ(xωi

t ,ωi)2+

Pt,ϕ

2

)



+ 1

N

X

i6=j

G(ϕ)(ωi)G(ϕ)(ωj),

≤ 2 NE   N X

i=1

ϕ(xωi t ,ωi)2



+2P_t,ϕ2+ _p1

N

N

X

i=1

G(ϕ)(ωi)

!2

whereG(ϕ)(ω):=Rϕ(y,ωi)P

ωi

t (dy)−

P_t,ϕ. If we apply the same Hilbertian argument as for

SN,(ω), we see

E

T

N,(ω)

t

2

−3,2α

≤ 2C

N E





N

X

i=1

(1+_|ωi|4α)



+C+

φ7→ p1

N

N

X

i=1

G(φ)(ωi)

!

2

−3,2α

, (29)

It is easy to see that the last term in (29) can be reformulated asBN(ω1, . . . ,ωN), with the property

that lim_A→∞lim sup_N→∞P(B_N >A) =0. Combining (24), (26), (28) and (29), Proposition 4.4 is proved.

4.3.2 Tightness of the fluctuations process

Applying Ito’s formula to (11), we obtain, for allϕbounded function onS1×R, with two bounded derivatives w.r.t. x, for every sequence(ω), for all t_≤T:

D

ηN,_t (ω),ϕE=Dη₀N,(ω),ϕE+

Z t

0

D

ηN,_s (ω),_LsνN(ϕ)Eds+M_tN,(ω)(ϕ), (30)

where, for all y ∈S1,π∈R,

LsνN(ϕ)(y,π) =

1 2ϕ

′′_{(y,π) +ϕ}′_(y,π)€_b[y,νN

s ] +c(y,π)

Š

+P_s,ϕ′(_·,_·)b(_·,y,π),

andM_tN,(ω)(ϕ)is a real continuous martingale with quadratic variation process

¬

MN,(ω)(ϕ)¶

t=

Z t

0

¬

ν_sN,(ω),ϕ′(y,π)2¶ds.

Lemma 4.5.

For every N , the operator_LsνN defines a linear mapping from_S into_S and for allϕ∈ S,

Lν

N

s (ϕ)

2 3,2α≤C

_ϕ2

6,α.

Proof. The terms 1₂ϕ′′(y,π) and ϕ′(y,π)b[y,ν_sN] clearly satisfy the lemma. We study the two remaining terms:

P_s,ϕ′b(_·,y,π) 2

3,2α= X

k1+k2≤3 Z

S1_×R ¬

Ps,ϕ′∂yk₁∂_πk₂b(·,y,π)

¶2

1+_|π|4α dydπ,

≤C

Z

R 1 1+_|π|4αdπ

Z

S1_×R

ϕ′(y,π)2Ps(dy, dπ),

≤C_ϕ2

C3,α Z

R 1 1+_|π|4α dπ

Z

S1_×R

(1+_|π|α)2P_s(dy, dπ),

≤C_ϕ2

6,α Z

R 1 1+_|π|4αdπ

Z

R

2. The following sequences are well defined: _∀k_≥0,

uk(t):= Z

S1_×R

e−|ω|ωkcos(θ)Pt(dθ, dω),

v_k(t):= Z

S1_×R

e−|ω|ωksin(θ)P_t(dθ, dω).

LetE= (ℓ_∞(N),_{k · k∞})be the Banach space of real bounded sequences endowed with its usual k · k∞ norm, (kuk∞ =supk≥0|uk|). For all t > 0, let At : E×E → E×E, be the following linear operator (where(u,v)is a typical element ofE×E): For allk≥0

¨

At(u, 0)_k = ₋1

2uk−αk(t)v0+βk(t)u0−K vk+1, At(0,v)_k = ₋1

2vk+γk(t)v0−αk(t)u0+Kuk+1, where,

αk(t) = ¬

Pt,e−|ω|ωkcos(·)sin(·) ¶

,

βk(t) = ¬

P_t,e−|ω|ωksin2(_·)¶,

γk(t) = ¬

P_t,e−|ω|ωkcos2(_·)¶.

(t,u,v) _{7→ At·}(u,v) is globally Lipschitz-continuous map from [0,T]_×E_×E into E_×E

and one easily verifies considering (4) (in the case of the sine-model) and developing the sine interaction thatt 7→(u(t),v(t))satisfies in E×Ethe following linear non-homogeneous Cauchy Problem:

  

d

dt(u(t),v(t)) = At·(u(t),v(t)),

u_k(0) = ¬P₀,e−|ω|ωkcos(_·)¶, _∀k_≥0

v_k(0) = ¬P₀,e−|ω|ωksin(_·)¶, _∀k_≥0.

Let us suppose that there exists somet0∈[0,T]such thatrt0=0, namelyu0(t0) =v0(t0) =0.

Then, if(˜u, ˜v)is the constant function on[0,T]such that for allk_≥0, ˜u_k≡u_k(t₀), ˜v_k≡v_k(t₀), then (˜u, ˜v) satisfy the same Cauchy Problem as (u,v) with initial condition at time t₀. By Cauchy-Lipschitz theorem, both functions coincide on [0,T]. In particular, u0 and v0 are always zero and thusr_≡0.

3. We suppose (Hr). A simple calculation shows that the fluctuations processRN verifies for all

t_∈[0,T],

RtN,(ω)= D

η_tN,(ω), cos(_·)E Dν_tN,(ω)+Pt, cos(·) E

+DηN,_t (ω), sin(_·)E Dν_tN,(ω)+Pt, sin(·) E

rN_t ,(ω)+r_t

,

= ℜ

D

ηN,t (ω),ei x E D

νtN,(ω)+Pt,ei x E

D

νtN,(ω),ei x E

+rt

.

Let uN,(ω) := ¬νN,(ω), ei x¶, vN,(ω) := ¬ηN,(ω), ei x¶ and u := ¬P,ei x¶, vω := ¬ηω,ei x¶

the random variables (ω) _{7→ L}€uN,(ω),vN,(ω)Š converges in law to the random variable

ω 7→ L(u,vω). The tightness of this random variable follows from the convergence of both empirical measure and fluctuations process. As already said in Remark 2.14, it suffices to prove the convergence of the finite-dimensional marginals (uN,_t (ω),vN,_t (ω)) =

(uN_t,(ω) 1 , . . . ,u

N,(ω)

tp ),(v

N,(ω)

t1 , . . . ,v

N,(ω)

tp )

, for all element of[0,T], t₁, . . . ,t_p, p_≥1.

Since the limit of(uN,(ω))is a constant, this is mainly a consequence of Slutsky’s theorem. But since this is a convergencein law with respect to the disorder, one has to adapt the proof. We prove the following:_∀G∈ C_b1(R),_∀r=€r1, . . . ,rp

Š

∈Rp,_∀s=€s1, . . . ,sp Š

∈Rp, E

•

G



ϕ_(uN,(ω)

t ,v

N,(ω)

t )(r,s)

‹˜ → N→∞E h G

ϕ(u_t,vω

t)(r,s)

i , whereϕ(X,Y)(r,s) =E

”

ei r·X+is·Y—is the characteristic function of the couple(X,Y). Indeed, we have successively:

aN:= E • G 

ϕ_(uN,(ω)

t ,v

N,(ω)

t )

(r,s) ‹˜

−E h

G

ϕ(u_t,vω

t)(r,s)

i , ≤ E • G 

ϕ_(uN,(ω)

t ,v

N,(ω)

t )

(r,s) ‹˜

−E

G

ϕ_(u

t,v N,(ω)

t )

(r,s) + E G

ϕ_(u

t,v N,(ω)

t )

(r,s)

−E h

G

ϕ(u_t,vω

t)(r,s)

i , ≤CE

ϕ(uNt,(ω),v N,(ω)

t )

(r,s)₋ϕ_(u

t,v N,(ω)

t )

(r,s) + E G

ϕ_(u

t,v N,(ω)

t )

(r,s)

−E h

G

ϕ(u_t,vω

t)(r,s)

i , ≤CEE

e

i r·uN_t,(ω)

−ei r·ut

+ E G

ϕ_(u

t,v N,(ω)

t )

(r,s)

−E h

G

ϕ(u_t,vω

t)(r,s)

i .

But, we haveE

e

i r_t·uNt,(ω)

−ei r·ut

≤min

2,_|r_||uN,_t (ω)₋u_t|

. So, for allǫ >0,

E

e

i r·uN_t,(ω)

−ei r·ut

≤ǫ|r|+2P 

u

N,(ω)

t −ut > ǫ

‹ . Taking lim sup_N→∞, and lettingǫ→0, we get lima_N=0. The result follows.

5.2

Proof of Proposition 2.15

The proof is similar to the previous one and relies on the two following equalities :

ζN,(ω)= ¬

P,ei x¶ rN,(ω) ,

p

N€ζN,(ω)−ζŠ= 1 · N,(ω)

€

A

Proof of Proposition 4.3

Thanks to the Lipschitz continuity ofbandc, introducingνas the empirical measure corresponding to(xωi_,ω

i), we have, (inserting b[x

ωi

s ,νsN]−b[x

ωi

s ,νs]in thebterm),

E

– sup

s≤t

x_si,N₋xωi

s 2 ™ ≤C ‚Z t 0

Eh€b[x_si,N,ν_sN]₋b[xωi

s ,Ps] Š2i

ds

+ Z t

0

Eh€c(x_si,N,ωi)−c(xsωi,ωi) Š2i ds Œ , ≤C ‚ 2 Z t 0 E – sup

u≤s _xi,N

u −x

ωi

u 2

™ ds+

Z t 0 sup 1≤j≤N E – sup

u≤s x

ωj

u −xuj,N 2™ ds + Z t 0

Eh€b[xωi

s ,νs]−b[xsωi,Ps] Š2i

ds

Π.

Applying Gronwall’s Lemma to sup_1≤j≤NE

sup_u≤t

x

ωj

u −xuj,N

2

, it suffices to prove that for some

Z_N:

Z t

0

Eh€b[xωi

s ,νs]−b[xsωi,Ps] Š2i

ds≤C/N+Z_N(ω1, . . . ,ωN). Indeed, for all 1≤i≤N, (we writexi instead of xωi _{to simplify notations):}

u_i,N:=€b[x_si,νs]−b[xis,Ps] Š2 = 1 N2    N X

j=1

T(xi,xj)2+X k6=l

T(xi,xk)T(xi,xl)   ,

where T(xi,xj):=b(x_si,xsj,ωj)− R

b(x_si,y,π)P_s(dy, dπ). Sincebis bounded, we see that the first term is of order(1/N). We only have to study the remaining term:

E

 

X k6=l

T(xi,xk)T(xi,xl) 

≤C N+E    

X k6=i,l6=i

k6=l

T(xi,xk)T(xi,xl)    .

Since the(xi)are independent, if we take conditional expectation w.r.t.(xr,r₆=l)in the last term, we get: E     X k6=i,l6=i

k6=l

T(xi,xk)T(xi,xl)    =E

•

E• XT(xi,xk)T(xi,xl)

xr,r6=l ˜˜ , =E     X k6=i,l6=i

k6=l

T(xi,xk)Gl(xi)    =E

   

X k6=i,l6=i

k6=l

Gk(xi)Gl(xi)    ,

whereG_l(x) =G(x,ωl) = R

b(x,y,ωl)P

ωl

s (dy)− R

b(x,y,π)P_s(dy, dπ). Defining

Z_N(ω1, . . . ,ωN):=

C N

Z T

0

E

  

1 p

N

N X l=1

G(x_si,ωl) !2

 ds, in order to prove (24) it suffices to show that for some constantC,

E   

Z T

0

E

  

1 p

N

N X

l=1

G(x_si,ωl) !2

 ds

  ≤C.

The rest of the proof is devoted to prove this last assertion: we have successively (setting

UN(x

ωi

s ,ω):= p1_N PN

l=1G(x

ωi

s ,ωl)) E

 

Z T

0

E”UN(xsωi,ω) 2—_ds

 ≤

Z T

0

E”_E”UN(xsωi,ω) 2——_ds,

≤ 1

N

Z T

0

E

 _E

 

N X k=1

N X l=1

G(xωi

s ,ωk)G(xsωi,ωl)  

 ds,

≤ _N1 Z T

0

E

 E

 

N X l=1

G(xωi

s ,ωl)2    ds+C

+ 1

N

Z T

0

E

 _E

 

X l6=k,l6=i,k6=i

G(xωi

s ,ωk)G(xsωi,ωl)  

 ds. The first term of the RHS of the last inequality is bounded, since bis bounded. But, if we condition w.r.t.ωrforr6=i,r6=k, we see that the second term is zero. The result follows.

Acknowledgements

This is a part of my PhD thesis. I would like to thank my PhD supervisors Giambattista Giacomin and Lorenzo Zambotti for introducing this subject, for their useful advice, and for their encouragement. I would also like to thank the referee for useful suggestions and comments.

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