Homogenization results for a linear dynamics in random Glauber type environment

www.imstat.org/aihp 2012, Vol. 48, No. 3, 792–818

DOI:10.1214/11-AIHP424

© Association des Publications de l’Institut Henri Poincaré, 2012

Homogenization results for a linear dynamics in random Glauber type environment

Cédric Bernardin

Université de Lyon and CNRS, UMPA, UMR-CNRS 5669, ENS-Lyon, 46 allée d’Italie, 69364 Lyon Cedex 07, France.

E-mail:cbernard@umpa.ens-lyon.fr

Received 28 May 2010; revised 24 January 2011; accepted 11 March 2011

Abstract. We consider an energy conserving linear dynamics that we perturb by a Glauber dynamics with random site dependent intensity. We prove hydrodynamic limits for this non-reversible system in random media. The diffusion coefficient turns out to depend on the random field only by its statistics. The diffusion coefficient defined through the Green–Kubo formula is also studied and its convergence to some homogenized diffusion coefficient is proved.

Résumé. On considère un système d’équations differentielles linéaires couplées conservant une certaine énergie et l’on perturbe ce système par une dynamique de type Glauber dont l’intensité varie aléatoirement site par site. Nous prouvons les limites hydro- dyanmiques pour ce système non réversible en milieu aléatoire. Le coefficient de diffusion dépend de l’aléa uniquement par sa loi.

Nous étudions aussi le coefficient de diffusion défini par la formule de Green–Kubo et montrons la convergence de celle-ci vers un coefficient de diffusion homogénéisé.

MSC:60K35; 82C22; 82C44

Keywords:Hydrodynamic limits; Random media; Green–Kubo formula; Homogenization

1. Introduction

The derivation of hydrodynamic limits for interacting particle diffusive systems in random environment has attracted a lot of interest in the last decade. One of the first paper to consider such question is probably [10] where hydrody- namic behavior of a one-dimensional Ginzburg–Landau model in the presence of random conductivities is studied.

In [19], a lattice gas with random rates is considered and a complete proof of hydrodynamic limits has been given in [8,20]. Other systems have been investigated such as exclusion processes and zero-range processes [6,7,9,12,14,16].

Interacting particle systems evolving in random media are in general of non-gradient. Roughly speaking the gradi- ent condition means that the microscopic current associated to the conserved quantity is already of gradient form.

Otherwise the general non-gradient techniques [15,22] consists in establishing a microscopic fluctuation-dissipation equation which permits to replace the current by a gradient plus a fluctuation term. But, if the system evolves in a random medium, such a decomposition does not hold microscopically because the fluctuations induced by the ran- dom medium are too large, and it is only in a mesoscopic scale that this fluctuation-dissipation equation makes sense [8,20].

In [12,14], by extending some ideas of [16], a simpler approach is proposed. The idea is to introduce a functional transformation of the empirical measure, which turns the system into a gradient-model, in such a way that the trans- formed empirical measure is very close to the original empirical measure. The advantage of the method is that it avoids the heavy machinery of the non-gradient tools but is unfortunately restricted to specific models. Even if the techniques

developed in [8,20] seem to be more robust than the precedent approach, it is not clear that in some situations, as in the situation considered here, they can be applied without a substantial modification.

The interacting particle system we consider is the following. To a simple energy conserving linear dynamics, flips with site dependent rates are superposed. Fix a sequence (γ_x)_x of positive numbers and denote by(η(t))_t≥0 the Markov process with state spaceR^Zand generator given by

(Lf )(η)=(Af )(η)+(Sf )(η), f:R^Z→R, (1)

where

(Af )(η)=

x∈Z

(η_x+1−η_x−1) ∂_ηxf and

(Sf )(η)=

x∈Z

γx

f η^x

−f (η)

withη^x the configuration obtained fromηby flippingηx:(η^x)z=ηzifz=x,(η^x)x= −ηx. This system conserves the energy

xe_x,e_x=η_x²/2, and the product of centered Gaussian probability measures with variance T >0 are invariant for the dynamics.

Let(γ_x)_x be a sequence satisfying (3) and (11). For example, the sequence(γ_x)_x is a realization of i.i.d. positive bounded below and above random variables with positive finite mean. We show (cf. Theorem1) that, starting from a local equilibrium state with temperature profileT₀=1/β0, the system evolves in a diffusive time scale following a temperature profileT, which is a solution of the heat equation

∂_tT = ¯γ⁻¹T ,

T (0,·)=β₀⁻¹(·), (2)

whereγ¯ is the average of the flip ratesγ_xdefined by (11).

One of the main interest of the model is its non-reversibility. To the best of our knowledge, it is the first time that hydrodynamic limits are established for a non-reversible interacting particle system evolving in a random medium.

In fact, our first motivation was to work with a simplified version of the energy conserving model of heat conduction with random masses [2] and we think that some of the methods developed in this paper could be useful to study this model.

The derivation of the hydrodynamic limits presents three difficulties: the first is that the system is non-gradient.

The second one is that it is non-reversible and that the symmetric partSof the generator is very degenerate and gives only few pieces of information on the ergodic properties of the system. The third difficulty is more technical. The state space is non-compact and the control of high energies is non-trivial. The first problem is solved by using the

“corrected empirical measure” method introduced in [12,14] and some special features of the model. For the second one, we apply in this context some deep ideas introduced in [11] (see also [18]). The third problem is solved by observing that the set of convex combinations of Gaussian measures is preserved by the dynamics. The control of large energies is then reduced to the control of large covariances.

In the perspective to study heat conduction models with random masses our main interest lies in the properties of the diffusion coefficient (given here by 1/γ¯).

The diffusion coefficient is also often expressed by the Green–Kubo formula, which is nothing but the space–time variance of the current at equilibrium. The Green–Kubo expression is only formal in the sense that a double limit (in space and time) has to be taken. For reversible systems, the existence is not difficult to establish. But for non-reversible systems even the convergence of the formula is challenging [17]. Let us remark that a priori the Green–Kubo formula depends on the particular realization of the disorder.

If we let aside the existence problem, widely accepted heuristic arguments predict the equality between the diffu- sion coefficient defined through hydrodynamics and the diffusion coefficient defined by the Green–Kubo formula.

The second main theorem of our paper shows that the homogenization effect also occurs for the Green–Kubo for- mula (see Theorem2): for almost every realization of the disorder, the Green–Kubo formula exists and is independent of the disorder. Unfortunately we did not succeed to prove that the value of the Green–Kubo formula is 1/γ¯.

The paper is organized as follows. In Section2we define the system. The proof of hydrodynamic limits is given in Section3. The two main technical steps which are the derivation of a one block lemma and the control of high energies are postponed to Sections4and5. The study of the Green–Kubo formula is the content of the last section.

2. The model

For anyα >0, letΩ_αbe the set composed of configurationsη=(η_x)_x∈Zsuch thatηα<+∞where η²α=

x∈Z

e^−α|x|η_x².

LetΩ= α>0Ωα be equipped with its natural product topology and its Borelσ-field. The set of Borel probability measures on Ω will be denoted byP(Ω). We also introduce the set C₀^k(Ω), k≥1, composed of bounded local functions onΩwhich are differentiable up to orderkwith bounded partial derivatives.

The time evolution of the process(η(t))_t≥0can be defined as follows. Let{Nx;x∈Z}be a sequence of independent Poisson processes. We shall denote byγx>0 the intensity ofNx. We assume there exist positive constantsγ₋ and γ₊such that

∀x∈Z, γ₋≤γ_x≤γ₊. (3)

For every realization of the random elementN =(Nx)_x∈Z, consider the set of integral equations:

η_x(t)=(−1)^Nx(t)

η_x(0)− _t

0

(−1)^Nx(s)

η_x+1(s)−η_x−1(s) ds

. (4)

For each initial conditionσ∈Ω the Eq. (4) can be solved by a classical iterative scheme. The solutionη(·):=

η(·, σ )defines a strong Markov process with càdlàg trajectories. Moreover each path η(·, σ )is a continuous and differentiable function of the initial dataσ[5,10,11]. We define the corresponding semigroup(P_t)_t≥0by(P_tf )(σ )= E_N(f (η(t, σ )))whereE_N denotes the expectation with respect to the Poisson clocks andf is a bounded measurable function onΩ.

Since the state space is not compact Hille–Yosida theory cannot be applied directly. Nevertheless, the differentia- bility with respect to initial conditions and stochastic calculus show that the Chapman–Kolmogorov equations

(P_tf )(σ )=f (σ )+ t

0

(LP_sf )(σ )ds, f∈C₀¹(Ω), and

(P_tf )(σ )=f (σ )+ _t

0

(P_sLf )(σ )ds, f∈C₀¹(Ω), are valid withLthe formal generator defined by (1).

The two Chapman–Kolmogorov equations permit to deduce that the probability measuresν∈P(Ω), which are invariant for(η(t))t≥0, are characterized by the stationary Kolmogorov equation

(Lf )(η)dν(η)=0 for allf ∈C₀¹(Ω).

In particular, every Gibbs measureμ_βwith inverse temperatureβ >0 is a stationary probability measure. Observe that μ_β is nothing but the product of centered Gaussian probability measures onR with variance β⁻¹. It is easy to show that (P_t)_t≥0 defines a strongly continuous contraction semigroup inL²(μ_β) whose generator is a closed extension ofL.

In fact, the infinite volume dynamics is well approximated by the finite dimensional dynamicsηⁿ(t)= {ηⁿ_x(t);x∈ Z},n≥2. It is defined by the generatorLn=An+Snwhere, for any functionf ∈C₀¹(Ω),

Anf =

n−1

x=−n+1

(η_x+1−η_x−1) ∂_ηxf −η_n−1∂_ηnf+η_−(n−1)∂_η−nf

and

(Snf )(η)= n x=−n

γ_x f

η^x

−f (η) .

Observe that η_xⁿ(t),|x|> n, do not change in time. Moreover, the total energy

x∈Zex is conserved by the finite dimensional dynamics. We denote by(P_tⁿ)_t≥0the corresponding semigroup. Let us fix a positive timeT >0, a pa- rameterα >0 and a functionφ∈C₀¹(Ω). One can prove there exist constantsC_n:=C(n, α, T , φ),n≥2, such that

sup

t∈[0,T]

P_tⁿφ

(η)−(P_tφ)(η)≤C_nη²α (5)

and

nlim→∞Cn=0.

This approximation is only used in the proof of Lemma9. The proof of (5) in a similar context can be found in [3], Chapter 2 (see also [11]).

3. Hydrodynamic limits

For any functionu:Z→R, the discrete gradient∇uofuis the function defined onZby

∀x∈Z, (∇u)(x)=u(x+1)−u(x).

The hydrodynamic limits are established in a diffusive scale. This means that we perform the time acceleration t→N²tand the space dilatationx→x/N. In the rest of the paper, apart from Section6, the process(η(t))_t≥0is the Markov process defined above with this time change. The corresponding generator isN²L.

The local conservation of energyex=η²_x/2 is expressed by the following microscopic continuity equation e_x(t)−e_x(0)= −N²

_t

0

(∇j_x−1,x) η(s)

ds, where the currentj_x,x+1:=j_x,x+1(η)is defined by

j_x,x+1(η)= −η_xη_x+1.

We denote byC₀(R)the space of continuous functions onRwith compact support and byC₀^k(R),k≥1, the space of compactly supported functions which are differentiable up to orderk. LetM(resp.M⁺) be the space of Radon measures (resp. positive Radon measures) on Rendowed with the weak topology. If G∈C₀²(R)andm∈Mthen m, Gdenotes the integral ofGwith respect tom.

The empirical positive Radon measureπ_t^N∈M⁺, associated to the processe(t):= {ex(t);x∈Z}, is defined by π_t^N(du)= 1

N

x∈Z

e_x(t)δ_x/N(du).

Fix a strictly positive inverse temperature profileβ₀:R→(0,+∞)and a positive constantβ¯such that

Nlim→∞

1 N²

x∈Z

1

β0(x/N )− 1 β¯

2

=0. (6)

Denote byμ^N=μ^N_β

0(·)∈P(Ω)the product probability measure defined by μ^N_β

0(·)(dη)=

x∈Z

g_{β0(x/N )}(η_x)dηx,

whereg_β(u)duis the centered Gaussian probability measure onRwith varianceβ⁻¹. We assume that the initial state satisfies

H μ^N|μ_β¯

≤C0N (7)

for a positive constantC0independent ofN. HereH (·|·)is the relative entropy, which is defined, for two probability measuresP , Q∈P(Ω), by

H (P|Q)=sup

φ

φdP−log

e^φdQ

(8) with the supremum carried over all bounded measurable functionsφonΩ. Let us recall the entropy inequality, which states that for every positive constanta >0 and every bounded measurable functionφ,

φdP ≤a⁻¹

log

e^aφdQ

+H (P|Q)

. (9)

Fix a positive timeT >0. The law of the process on the path spaceD([0, T], Ω), induced by the Markov process (η(t))t≥0starting fromμ^N, is denoted byPμ^N. For any times≥0, the probability measure onΩgiven by the law of η(s)is denoted byμ^N_s .

Since entropy is decreasing in time, (7) implies that

∀s≥0, H μ^N_s |μ_β¯

≤C₀N. (10)

The conditions (6) and (7) are introduced to get some moment bounds (see Section5). They are satisfied by any continuous functionβ₀⁻¹going toβ¯⁻¹at infinity sufficiently fast.

Theorem 1. Let(γx)_x∈Zbe a sequence of positive numbers satisfying(3)and such that

Klim→∞

1 K

K x=1

γ_x= ¯γ , lim

K→∞

1 K

0 x=−K

γ_x= ¯γ (11)

for someγ¯∈(0,∞).Assume that the initial stateμ^N=μ^N_β

0(·)satisfies(7)andβ₀satisfies(6).

Then,underPμ^N, π_t^N converges in probability to T_t/2 where T_t is the unique weak solution of (2): For every G∈C₀(R),everyt >0,and everyδ >0,

Nlim→∞Pμ^N

π_t^N, G

−1 2Tt, G

≥δ

=0.

We follow the method of the “corrected empirical measure” introduced in [12,14]. Since the state space is not compact, technical adaptations are necessary. In particular, it is not given for free that the corrected empirical measure and the empirical measure have the same limit points for the weak convergence. It would be trivial if the state space was compact. Moreover, a replacement lemma, reduced to a one-block estimate, has to be established (see Section4).

For anyG∈C₀(R), we defineT_γG:Z→Rby (T_γG)(x)=

j <x

(γ_j+γ_j+1)

G j+1

N

−G j

N

. Observe that

N 1

γ_x+γ_x+1

(T_γG)(x+1)−(T_γG)(x)

=(∇NG)(x/N ),

where∇N stands for the discrete derivative:(∇NG)(x/N )=N{G((x+1)/N )−G(x/N )}.

SinceT_γGmay not belong to₁(Z), we modifyT_γGin order to integrate it with respect to the empirical measure.

Fix 0< θ <1/2 and consider aC²increasing non-negative functiong˜defined onRsuch thatg(q)˜ =0 forq≤0,

˜

g(q)=1 forq≥1 andg(q)˜ =qforq∈ [θ,1−θ].

Fix an arbitrary integer >0 and letg=g_θ,:R→Rbe given by g(q)= ˜g(q/).

We define

(T_{γ ,}G)(x)=(T_γG)(x)−T_{γ ,G} Tγ ,g

(T_γg)(x), where

T_{γ ,h}=

x∈Z

(γ_x+γ_x+1) h

(x+1)/N

−h(x/N ) . In the rest of the paper we make the choice:=(N )=N^1/4.

Lemma 1. For each functionG∈C₀²(R),and each environmentγ satisfying(3)and(11),

Nlim→∞N^1/4sup

x∈Z

T_{γ ,}G(x)− ¯γ G(x/N )=0

and

Nlim→∞N^1/4Tγ ,G=0.

Proof. This is a slight modification of Lemma 4.1 in [14].

We shall denote byX_t^N∈Mthe corrected empirical measure defined by X^N_t (G)=X^N,γ_t (G)= 1

N

x∈Z

Tγ ,G(x)et(x).

The system is non-gradient but we have j_x,x+1= − 1

γx+γx+1∇

e_x+1

2η_x−1η_x+1

+L

1

2(γx+γx+1)η_xη_x+1

. This implies that

N²L

X^N(G)

= 1 N

x∈Z

(NG)(x/N )−T_{γ ,G}

T_{γ ,g}(Ng)(x/N )

ex+1

2ηx−1ηx+1

+1

2L

x∈Z

(∇NG)(x/N )−Tγ ,G

T_{γ ,g}(∇Ng)(x/N )

η_xη_x+1

,

where_N stands for the discrete Laplacian:

(_NG)(x/N )=N² G

(x+1)/N +G

(x−1)/N

−2G(x/N ) . Therefore, we have

X^N_t (G)−X^N₀(G)=U_t^N(G)+V_t^N(G)+M_t^N(G) (12) withM^N(G)a martingale andU^N(G),V^N(G), which are given by

U_t^N(G)= _t

0

ds1 N

x∈Z

B_N^G(x/N )

e_x(s)+1

2η_x−1(s)η_x+1(s)

,

where

B_N^G(x/N )=

(NG)(x/N )−T_{γ ,G}

T_{γ ,g}(Ng)(x/N )

and

V_t^N(G)= 1 N²

x

(∇NG)(x/N )−T_{γ ,G} Tγ ,g

(∇Ng)(x/N )

η_x(t )η_x+1(t )−η_x(0)ηx+1(0) .

Lemma 2. The sequence{(X_·^N,_·

0π_s^Nds)∈D([0, T],M)×D([0, T],M⁺);N≥1}is tight.

Proof. It is well known that the sequence

X_·^N, _·

0

π_s^Nds

∈D

[0, T],M

×D

[0, T],M⁺

;N≥1

is tight if and only if the sequence

X_·^N(G), _·

0

π_s^N(H )ds

∈D

[0, T],R

×D

[0, T],R

;N≥1

is tight for everyG, H ∈C²₀(R).

By Aldous criterion for tightness inD([0, T],R)², it is sufficient to show that (1) for everyt∈ [0, T]and everyε >0, there exists a finite constantA >0 such that

sup

N

P_μ^NY_t^N(G)≥A

≤ε, (2) for everyδ >0,

εlim→0lim sup

N→∞ sup

τ∈Θ,θ≤ε

Pμ^NY_τ^N_+θ(G)−Y_τ^N(G)≥δ

=0, whereΘis the set of all stopping times bounded byT forY_·^N(G)=X_·^N(G)andY_·^N(G)=_·

0π_s^N(G)ds.

SinceGhas compact support, there exists a constantK >0 (independent oftandN) such that Eμ^Nγ¯

π_t^N, G

−X^N_t (G)

≤

N^1/4sup

x∈Z

γ G(x./N )¯ −(T_{γ ,}G)(x)Eμ^N

1 N^5/4

|x|≤KN^5/4

e_x(t )

(13)

and consequently Eμ^N

T 0

dtγ¯ π_t^N, G

−X_t^N(G)

≤

N^1/4sup

x∈Z

γ G(x./N )¯ −(Tγ ,G)(x) _T

0

dtEμ^N

1 N^5/4

|x|≤KN^5/4

ex(t)

. (14)

By Lemmas1and10, the right-hand side of (13) (resp. of (14)) vanishes asN→ ∞. Hence, it is sufficient to show Aldous criterion forY_·^N(G)=X_·^N(G)and forY_·^N(G)=_·

0X^N_s (G)ds.

From the definition of the Skorohod topology, it is easy to show that the application Φ fromD([0, T],R)onto itself defined by

Φ:x:=

x(t);0≤t≤T

→Φ(x):=

t 0

x(s)ds;0≤t≤T

is continuous. Thus, if(X_·^N(G))_Nis tight, then(_·

0X_s^N(G))_Nis tight.

Therefore it just remains to show Aldous criterion forY_·^N(G)=X_·^N(G).

Proof of (1)forX^N_· (G):

Pμ^NX_t^N(G)≥A

≤ 1 AEμ^N

1 N

x∈Z

(Tγ ,G)(x)ex(t)

.

We write(T_{γ ,}G)(x)=((T_{γ ,}G)(x)− ¯γ G(x/N ))+ ¯γ G(x/N )and we get that Pμ^NX_t^N(G)≥A

≤ 1 AEμ^N

1 N

|x|≤KN^5/4

(T_{γ ,}G)(x)− ¯γ G(x/N )e_x(t)

+γ¯ AEμ^N

1 N

|x|≤KN

G(x/N )e_x(t)

.

The first term on the right-hand side of the previous inequality can be bounded above by the right-hand side of (13), which vanishes. By Lemma10, the second term is bounded above by C/Awith a constantC independent ofN. Therefore, the first condition is satisfied.

Proof of (2)forX^N_· (G): Recall the decomposition (12). In order to estimate the term E_μ^NU_τ^N_+ε(G)−U_τ(G)

we observe that|B_N^G(x/N )|is bounded above by C

1_|x|≤KN +|Tγ ,G|

|T_{γ ,g}|

⁻²+ N ⁻³

1_{x/(N )∈[1−2θ,1+2θ]∪[−2θ,2θ]}

,

whereC, Kare constants depending onθandGbut not onN. By Schwarz inequality, we are reduced to estimate Eμ^N

T+ε 0

ds1 N

|x|≤2KN

e_x(s)

and

|T_{γ ,G}|

|T_{γ ,g}| 1

²+ 1 N ³

Eμ^N

T+ε 0

ds1 N

|x|≤(1+3θ )N

ex(s)

.

By Lemma10, the first term is of order one. It is not difficult to show that lim infN→∞T_{γ ,g}>0, and Lemma1gives T_{γ ,G}→0. Thus, by Lemma10, the second one vanishes asNgoes to infinity.

The two last terms of (12) are given by t

0

dsL

x

(∇NG)(x/N )−T_{γ ,G}

T_{γ ,g}(∇Ng)(x/N )

η_xη_x+1

(s)

=V_t^N(G)+M_t^N(G).

By using Lemmas1and10, similar estimates as before show that

Nlim→∞ sup

t∈[0,T+ε]Eμ^NV_t^N(G)=0.

By computing the quadratic variation of the martingaleM^N(G), one obtains that (we recall that sup_xγ_x≤γ₊) Eμ^N

M_τ^N_+ε(G)−M_τ^N(G)2

≤2γ₊ N²Eμ^N

τ+ε τ

ds

x

(∇NG)(x/N )−T_{γ ,G}

T_{γ ,g}(∇Ng)(x/N ) 2

e_s(x)e_s(x+1)

. Observe that

(∇NG)(x/N )−T_{γ ,G} Tγ ,g

(∇Ng)(x/N ) 2

≤C1_|x|≤KN+⁻²(T_{γ ,G}/T_{γ ,g})²1_|x|≤KN5/4. Since lim infN→∞T_{γ ,g}>0 andT_{γ ,G}→0, from Lemma10, we get

sup

s≥0

Eμ^N

1 N²

|x|≤KN

e²_x(s)

and sup

s≥0

Eμ^N

1 N³

|x|≤KN^5/4

e²_x(s)

go to 0 withN(and are in particular bounded above by a constant independent ofN).

Lemma 3. Let(α, β)∈M×M⁺be a limit point of the sequence

X_·^N, _·

0

π_s^Nds

∈D

[0, T],M

×D

[0, T],M⁺

;N≥1

. For everyG∈C₀²(R)and everyt∈ [0, T],we have

α_t(G)−α₀(G)= ¯γ⁻¹ _t

0

α_s(G)ds, β_t= ¯γ⁻¹ _t

0

α_sds.

Proof. In the proof of the tightness ofX_·^Nwe have seen that the term Eμ^N

_t

0

dsL

x

(∇NG)(x/N )−T_{γ ,G} Tγ ,g

(∇Ng)(x/N )

η_xη_x+1

(s)

and the term Eμ^N

_t

0

ds 1 N

x∈Z

T_{γ ,G}

T_{γ ,g}(_Ng)(x/N )

e_x(s)+1

2η_x−1(s)η_x+1(s)

vanish asN→ ∞. By using Lemma5, it implies that αt(G)−α0(G)=βt(G).

Moreover, by (14), we have β_t= ¯γ⁻¹

t 0

α_sds.

Lemma 4. Any limit pointβ of the sequence{_·

0π_s^Nds∈D([0, T],M⁺);N≥1}is such that,for anyt∈ [0, T],βt

is absolutely continuous with respect to the Lebesgue measure onR.

Proof. Fix a positive timetand letR_μN be the probability measure onM⁺given by R_μN(A)=Pμ^N

1 t

t 0

π_s^Nds∈A

for every Borel subset Aof M⁺. LetJ:M⁺→ [0,+∞)be a continuous and bounded function. By the entropy inequality (9) and by using (10) we have

J (π )dR_μN(π )≤C₀+ 1 Nlog

e^{N J (π )}dRμ_β¯(π )

. (15)

By the Laplace–Varadhan theorem, the second term on the right-hand side converges asN goes to infinity to sup

π∈M⁺

J (π )−I₀(π ) ,

whereI₀is the large deviations rate function for the random measureπunderR_μ¯

β. It is a simple exercise to compute the rate functionI₀. We have

I₀(π )= sup

f∈C₀(R)

f (u)π(du)−

logM_β¯ f (u)

du

,

whereM_β¯(α)is the Laplace transform ofη²₀/2 underμ_β¯: M_β¯(α)=μ_β¯

e^αη20/2

=

β/(¯ β¯−α) ifα <β¯, and+∞otherwise.

The functionI0also takes the simple form I₀(π )= ^Rh

π(u)

du ifπ(du)=π(u)du,

+∞ otherwise,

where the Legendre transformhofM_β¯is given byh(α)= ¯βα−1/2−1/2 log(2αβ)¯ ≥0 ifα >0, and+∞otherwise.

Let(f_k)_k≥1be a dense sequence inC₀(R)withf₁being the function identically equal to 0. ThenI₀is the increasing limit ofJ_k≥0 defined by

J_k(π )= sup

1≤j≤k

f_j(u)π(du)−

logM_β¯ f_j(u)

du

∧k.

By using (15) we have lim sup

N→∞

J_k(π )dR_μ^N(π )≤C0

for eachk. SinceJ_k is a lower semi-continuous function, any limit pointR^∗ofR_μN is such that

Jk(π )dR^∗(π )≤C0.

By the monotone convergence theorem, we have

I0(π )dR^∗(π )≤C0<+∞. SinceI0(π )is equal to+∞ifπ is not absolutely continuous with respect to the Lebesgue measure, it implies that

R^∗

π;π(du)=π(u)du

=1

and the lemma is proved.

We conclude as follows. Let(α, β)be a limit point of(X^N_· (G),_·

0π_s^N(G))N≥1. From the equation α_·(G)−α₀(G)= ¯γ⁻¹

_·

0

α_s(G)ds

we see thatαis time continuous. Moreover, ifAis a subset ofRwith zero Lebesgue measure, thenβ_t(A)=0 for anyt∈ [0, T]. This implies thatα_t(A)=0 for anyt∈ [0, T], i.e. thatα_t is absolutely continuous with respect to the Lebesgue measure onR.

By uniqueness of weak solution to the heat equation, we have that 2α is the Dirac mass concentrated on the (smooth) solution of the heat equation(t, u)∈ [0, T] ×R→ ˜T_t(u)starting fromγ β¯ ₀⁻¹:∂_tT˜_t= ¯γ⁻¹T˜_t,T˜₀= ¯γ β₀⁻¹.

Hence we conclude that{X_·^N∈D([0, T],M);N≥1}converges in distribution to(T˜_·(u)/2)du. Since the limit is continuous in time we have that{X_t^N;N≥1}converges in distribution to the deterministic limit(T˜_t(u)/2)du. Since convergence in distribution to a deterministic variable implies convergence in probability, this implies that

Nlim→∞Pμ^N

X^N_t (G)−1 2

T˜_t(u)G(u)du ≥ε

=0.

We use again (13) and the fact thatγ¯⁻¹T˜t=Tt to get

Nlim→∞Pμ^N

π_t^N(G)−1 2

Tt(u)G(u)du ≥ε

=0 and the theorem is proved.

4. One-block estimate

The aim of this section is to prove the following so-called one block estimate [15].

Lemma 5 (One block estimate). For anyG∈C₀²(R),anyt≥0,and anyδ >0,

Nlim→∞Pμ^N

1 N

x∈Z

(_NG)(x/N ) _t

0

dsηx−1(s)η_x+1(s) ≥δ

=0.

SinceG∈C₀²(R), we can replace(NG)(x/N )by 1

2k+1

|y−x|≤k

(NG)(y/N )

as soon askN and we are left to prove that

klim→∞ lim

N→∞Eμ^N

1 2N+1

|x|≤N

t 0

ds 1

2k+1

|x−y|≤k

ηy(s)ηy+1(s)

=0.

Given two probability measuresP , QonΩ andΛa finite subset ofZ,HΛ(P|Q)denotes the relative entropy of the projection ofP onR^Λwith respect to the projection ofQonR^Λ. We shall denote the projection ofP onR^Λby P|Λ. IfΛ=Λ_k= {−k, . . . , k}, we use the short notationP_k.

We define the space–time average of(μ^N_s )_0≤s≤t by ν^N= 1

(2N+1)t

|x|≤N

_t

0

τxμ^N_s ds.

Hereτ_xdenotes the shift byx: for anyη∈Ω, the configurationτ_xηis defined by(τ_xη)_z=η_x+z; for any function gonΩ,τ_xgis the function onΩ given by(τ_xg)(η)=g(τ_xη); for anyp∈P(Ω),τ_xpis the push-forward ofpby τ_x. The probability measurepis said to be translation invariant ifτ_xp=pfor anyx∈Z.

We have to show

klim→∞ lim

N→∞

R^Λkdν_k^N 1

2k+1

|y|≤k

ηyηy+1

=0. (16)

Lemma 6. For each fixedk,the sequence of probability measure(ν_k^N)_N≥k onR^Λk is tight.

Proof. It is enough to prove that there exists a constantC_k<∞independent ofN such that

i∈Λk

e_idν_k^N≤C_k. (17)

We begin to prove that H_Λk

ν^N|μ_β¯

=H

ν_k^Nμ_β¯|Λk

≤C₀|Λ_k|. (18) Fix a bounded measurable functionφdepending only on the sites inΛ:=Λ_k= {−k, . . . , k}. Assume for simplicity that 2N+1=(2k+1)(2p+1)for somep≥1. Then we can index the elements of the set {−N, . . . , N}in the following way

{−N, . . . , N} = {x_j+y;j= −p, . . . , p;y∈Λ_k},

wherexj =2kj +1. Sinceφdepends only on the sites inΛ, it is clear that underμ_β¯, for eachy∈Λ, the random variables(τ_xj+yφ)_j=−p,...,pare i.i.d.

Letμ¯^N_t =t⁻¹_t

0μ^N_s ds. By convexity of the entropy and (10), we haveH (μ¯^N_t |μ_β¯)≤C0N.

We write

φdν_k^N= 1 2N+1

|x|≤N

τxφ

dμ¯^N_t

≤ |Λ| 2N+1H

¯ μ^N_t |μ_β¯

+ |Λ| 2N+1log

dμ_β¯e^|Λ|−1

|x|≤Nτxφ

≤C₀|Λ| + |Λ| 2N+1log

dμ_β¯e^|Λ|−1

y∈Λ(

|j|≤pτy+xjφ)

≤C0|Λ| + |Λ|

2N+1|Λ|⁻¹

y∈Λ

log

dμ_β¯e

|j|≤pτ_y+xjφ ,

where we used the entropy inequality (9) and the convexity of the applicationf→log(

dμ_β¯e^f). By independence, for eachy, of(τ_xj+yφ)_j=−p,...,pand the translation invariance ofμ_β¯, we get

φdν_k^N≤C0|Λ| +log

e^φdμ_β¯

. (19)

This implies (18) and, by the entropy inequality (9), the inequality (17).

For anyk, letν_k^∗be a limit point of the sequence(ν_k^N)_N≥k. The sequence of probability measures(ν_k^∗)_k≥0forms a consistent family and, by Kolmogorov theorem, there exists a unique probability measureνonΩ such thatν_k=ν_k^∗. By construction, the probability measureνis invariant by translations.

Lemma 7. There existsC0such that for any boxΛk= {−k, . . . , k},k≥0,

HΛ_k(ν|μ_β¯)≤C0|Λk|. (20)

Proof. We have seen in the proof of the previous lemma that H_Λk

ν^N|μ_β¯

=H

ν_k^Nμ_β¯|Λ_k

≤C₀|Λ_k|. (21)

Since the entropy is lower semicontinuous, it follows that

H_Λk(ν|μ_β¯)≤C₀|Λ_k|.

A translation invariant probability measureνonΩsuch that (20) is satisfied is said to have a finite entropy density.

By a super-additivity argument (see [3,11]), the following limit H (ν¯ |μ_β¯)= lim

k→∞

H_Λk(ν|μ_β¯)

|Λk| (22)

exists and is finite. For any bounded local measurable functionφonΩ, we define the limit F (φ)¯ = lim

k→∞

1

2k+1F¯_k(φ), F¯_k(φ)=log

e^ki=−kτiφdμ_β¯.

The entropy densityH (ν¯ |μ_β¯)can be expressed by the variational formula H (ν¯ |μ_β¯)=sup

φ

φdν− ¯F (φ)

, (23)

where the supremum is taken over all bounded local measurable functionsφonΩ. We now show the following lemma

Lemma 8. For any functionF∈C₀¹(Ω),we have

LFdν=0.

Proof. Assume thatF ∈C₀¹(Ω)has a support included inR^Λk−1. We have

LFdν=

LFdνk= lim

N→∞

LFdν_k^N.