Olivier Benois
email: olivier.benois@univ-rouen.fr

Mustapha Mourragui
email: mustapha.mourragui@univ-rouen.fr

Enza Orlandi
email: orlandi@mat.uniroma3.it

Ellen Saada
email: ellen.saada@mi.parisdescartes.fr

Livio Triolo
email: triolo@mat.uniroma2.it

Quenched large deviations for Glauber evolution with Kac interaction and random field. *

Keywords: Interacting particle systems, random environment, Kac Interaction, large deviations. 2000 Mathematics Subject Classification : 60K35, 82C22, 60F10

We study a spin-flip model with Kac type interaction, in the presence of a random field given by i.i.d. bounded random variables. The system, spatially inhomogeneous, evolves according to a non conservative (Glauber) dynamics. We show an almost sure (with respect to the random field) large deviation principle for the empirical magnetizations of this process. The rate functional associated with the large deviation principle depends on the statistical properties of the external random field, it is lower semicontinuous with compact level sets.

Introduction

We consider interacting spin-flip systems, in dimension d, with Kac type interaction in the presence of a random field given by i.i.d. bounded random variables. Kac potentials J γ are two-body interactions with range γ -1 and strength γ d , where γ is a dimensionless scaling parameter. When γ → 0, i.e. very long range compared with the inter particle distance, the strength of the interaction becomes very weak, but in such a way that the total interaction between one particle and all the others is finite. Kac potentials were introduced in [KUH], and then generalized in [LP], to present a rigorous derivation of the van der Waals theory of a gas-liquid phase transition. There has been in the last decades an increasing interest in them. Indeed they induce the intermediate scale of interaction γ -1 (called mesoscopic) between the microscopic (lattice) one and a macroscopic one much bigger than the latter. They are suitable to interpolate not only between short and long range interactions, but, scaling space and time as functions of γ, one can hope to obtain more insights into the physics of the model. Recently they have been considered as models to describe social interactions and more general complex social systems, see for example [CDS] and references therein.

There has been several results on Kac Ising spin systems (without random field) in equilibrium and in non equilibrium statistical mechanics. We refer for a survey to the book [P]. The papers [C], [CE] were among the first dealing with dynamics issues. They considered spin systems in a torus evolving according to 1 a reversible and non conservative (Glauber) dynamics, with Kac interactions. In [C] the long time analysis of the spin system is studied, using large deviations techniques. In [CE] the main results are the infinite particle limits of the non-critical and critical fluctuation processes. In [DOPT] and [KS] (see also references therein), a complete description of the development and motion of interfaces (long time behaviour) has been derived: it is governed by the law of motion by mean curvature.

A natural extension of this analysis is its application to disordered systems. One of the simplest prototype models is obtained by adding a random magnetic field to an interacting spin system. Equilibrium statistical properties of these systems have been widely studied in the last decades, see [Bo] for a survey of results in this direction. The case of Kac type interaction has been investigated in d = 1 by [COP], [COPV] and [OP].

In this paper we study a reversible, nonconservative (Glauber) dynamics of ±1 valued spins, interacting via a Kac potential and under the influence of an external random field. We assume the latter given by i.i.d. random variables taking values a i ∈ IR with probability p i , for i = 1, . . . , N , with N a fixed integer. We do not require the Kac potential to be positive (that is we do not restrict the model to the ferromagnetic case).

Our main result is a quenched large deviation principle, almost sure with respect to the random field, for the empirical random magnetizations of this spin-flip process. The rate functional associated with the large deviation principle, which depends on the distribution of the random field, is lower semicontinuous, positive, with compact level sets. In contrast with the non random case studied in [C], the magnetization m of our spin model is not of mean field type. Nevertheless, this difficulty is overcome by coloring the sites according to the random external force, so that the colored magnetizations become a mean field system. The large deviation rate functional is then obtained via a contraction principle from the rate functional associated with the large deviation principle of the empirical colored magnetizations m i (i.e. the magnetization over the sites where the random field takes value a i ), i = 1, . . . , N ; we have m = N i=1 m i . As usual, the rate functional is determined by two distinct types of large deviations of the same order. The first one corresponds to large deviations from the initial state, the second one to the stochasticity of the evolution. Suppose A = {π γ (•, dr) ≃ v(•, r)dr, t ∈ [0, T ]} where π γ (•, dr) is the local magnetization density, ≃ denotes closeness in some norm and v is a profile different from the solution of the nonlinear macroscopic equation giving the law of large numbers. We need to modify the measure of the process over the magnetization profiles so that event A becomes typical. One possible choice is to drive the spin system by weak, slowly varying, space-time dependent external forces. This is the standard choice for spin systems evolving according to non conservative (Glauber) evolutions without randomness involved, or to conserved (Kawasaki) evolutions with gradient type interactions. For conserved non gradient systems, the force must be configuration dependent (see [Sp], p. 248), to take into account that for these systems the response in the current to an external force field is partially delayed. Namely, when an external random field is added to the Hamiltonian, in the conservative, non gradient case (see [MO]), one needs to take the external force weakly dependent on the field randomness. In the non conservative case, it turns out that the external force strongly depends on the field randomness. In other words, in dynamics with a conserved quantity, there is less freedom in choosing the class of perturbations than in non conservative dynamics.

We distinguish between sites where the random field takes different values; on each of them we take a deterministic space-time dependent external force. This allows to write the rate functional associated to the large deviation principle in a closed form with respect to the local colored magnetization. We carried out explicitly the computations for a couple (N = 2); the general case follows. The simplest case to have in mind is a 1 = 1, a 2 = -1, p 1 = p 2 = 1/2 and J ≥ 0; then, when β (which is proportional to the inverse temperature) is such that β ≥ β c = ( J(r) dr) -1 , interesting phenomena appear when studying the long time behaviour of the spin system. This is related to the fact that the underlying spin systems at equilibrium undergo to phase transition, even in one dimension in the limit γ ↓ 0. In this paper we will study the dynamics of the spin system for finite time: in this regime, the evolution does not depend crucially on the value of β. We will then set β = 1.

The random Curie-Weiss model (RCW), which describes a mean field interaction, has given rise to many results on short and long time dynamics. In [DD], short time dynamics has been studied. More precisely the large deviations for the empirical measures in the product space of magnetization trajectories and realizations of the random field are given. From this result one could derive annealed large deviations for the RCW but not quenched ones. In [MP] and [FMP], long time dynamics, convergence to equilibrium when the random field takes only the two values ±ǫ are considered. In [BEGK], the RCW model is analyzed when the random field takes finitely many values, as an example of the use of the potential theoretical approach to metastability. Furthermore, in [BBI], the previous results are extended to continuous distributions of the field, and precise asymptotics of metastable characteristics are derived.

There are no available results for short and long time dynamics of the random field Kac model. We make here a first step in addressing this problem.

In Section 2 we present the model, main definitions and results. In Section 3 we define the rate functional associated to the large deviation principle, we exhibit different representations for it, and we give its main properties (lower semicontinuity, compactness). There, we follow the scheme of [C], Section III, but working with the couple (m 1 , m 2 ) induces intricate computations. Since the spins have value ±1, the local and colored magnetizations are always between +1 and -1. A consequence of the randomness is that the functional becomes infinite for the colored particle system at the boundary p i ≤ |m i | ≤ 1 (i = 1, 2) of the coupled magnetization. Thus these boundaries are not rare enough in the large deviations regime, and we have to deal with this lack of regularity. This is different from the non random case [C], where the boundary is reduced to the two values ±1 of the magnetization. A preliminary step to derive the large deviation principle (LDP) is the hydrodynamic behavior for the colored particle process, sketched in Section 4. The class of time dependent, random perturbations needed to derive the LDP lower bound is introduced in Section 5, where the perturbed process is studied. In Section 6 we derive the upper bound and in Section 7 the lower bound of the LDP. The lower bound is obtained first for trajectories that are smooth is space and time, and outside the boundaries p i ≤ |m i | ≤ 1 (i = 1, 2). Then it is extended to a larger class P of paths, by smoothing by successive steps trajectories with a finite rate functional, using techniques introduced in [QRV]. In this context, P consists in trajectories absolutely continuous with respect to the Lebesgue measure and absolutely continuous in time. Then, in order for the usual martingale technique to be effective to obtain the upper bound, we need to show that the process concentrates on P. To this aim, we introduce an energy functional via an exponential martingale which excludes the trajectories not in P (in the spirit of [QRV], [MO], [FLM]). The appendices (Sections 8 and 9) gather the most technical proofs.

The model and the main results

The space of configurations: Let Λ be the d-dimensional torus of diameter 1. For 0

< γ < 1 such that γ -1 ∈ IN , Λ γ = Z Z d /γ -1 Z Z d is the d-dimensional discrete torus of diameter γ -1
. We denote by S γ ≡ {-1, +1} Λγ the configuration space and by σ = (σ(x)) x∈Λγ a spin configuration, where for each

x ∈ Λ γ , σ(x) ∈ {-1, 1}.
The disorder: It is described by a collection of i.i.d. random variables α = {α(x), x ∈ Z Z d } taking two values, i.e. α(x) ∈ {a 1 , a 2 }. The corresponding product measure on Ω = {a 1 , a 2 } Z Z d is denoted by IP (and IE is the expectation with respect to IP ),

IP {α(x) = a i } = p i , i = 1, 2.
(2.2)

For γ -1 an odd integer, α induces in a natural way a random field on Λ γ , also denoted by α.

The Kac potential: We consider a pair interaction among particles given by a Kac potential of the form

J γ (x, y) ≡ γ d J(γ(x -y)), (x, y) ∈ Λ γ × Λ γ , (2.3)
where J : Λ → IR is a symmetric function, that is J(r) = J(-r), such that J(r)dr = 1 (normalization). The interaction J might have any sign. Denote by C(Λ) (resp. C 1 (Λ), C 2 (Λ)) the space of continuous (resp. continuously differentiable, twice continuously differentiable) real functions on Λ. We assume J ∈ C 1 (Λ).

The Energy: Given a realization α of the magnetic field, define for all γ, θ > 0, σ ∈ S γ , the Hamiltonian

H γ,α (σ) = - 1 2 (x,y)∈Λγ ×Λγ J γ (x, y)σ(x)σ(y) -θ x∈Λγ α(x)σ(x), (2.4)
and the Gibbs measure µ γ,α,β associated to H γ,α at inverse temperature β, with normalization constant Z γ,α,β :

µ γ,α,β (σ) = 1 Z γ,α,β exp -βH γ,α (σ) .
The Glauber dynamics: Denote by σ x the configuration obtained from σ by flipping the spin at site x:

σ x (z) = -σ(x) if z = x, σ(z) otherwise,
so that the energy difference resulting from a spin flip at x is

H γ,α (σ x ) -H γ,α (σ) = 2σ(x) [(J γ ⋆ σ)(x) + θα(x)] , (2.5)
where without loss of generality we have assumed J(0) = 0, and we define the discrete convolution ⋆ between function J γ and a configuration σ by

(J γ ⋆ σ)(x) = γ d y∈Λγ J(γ(x -y))σ(y).
(2.6)

We consider a Markovian evolution on S γ , whose generator L γ,α acts on cylinder functions f as

L γ,α f (σ) = x∈Λγ c γ,α x (σ)[f (σ x ) -f (σ)], (2.7)
where, for x ∈ Λ γ ,

c γ,α x (σ) = exp[-(β/2)(H γ,α (σ x ) -H γ,α (σ))] 2 cosh[(β/2)(H γ,α (σ x ) -H γ,α (σ))] . (2.8)
Then L γ,α viewed as an operator on L 2 (µ γ,α,β ) is self-adjoint. Since temperature is kept fixed in all the paper and does not play any role we set for simplicity β = 1. We fix a time T > 0, and we will study the process (σ t ) t∈[0,T ] with infinitesimal generator given in (2.7).

The measure spaces: Let M 1 be the set of signed Borel measures µ on the Borel σ-field of Λ with total variation norm bounded by 1. We equip M 1 with the weak τ * topology induced by C(Λ) via < µ, G >= Gdµ (for G ∈ C(Λ)). We denote by ρ(•, •) the distance which makes (M 1 , τ * ) a metrizable compact space, see [Bill]: that is, given (H k ) k∈IN a dense subset in the unit ball of C(Λ)

for µ i ∈ M 1 , i = 1, 2, ρ(µ 1 , µ 2 ) = k≥0 2 -k | < µ 1 -µ 2 , H k > |.
(2.9) Let 0 < q ≤ 1, and

M ac q = µ ∈ M 1 : µ << λ and dµ dλ ≤ q λ -a.s. , (2.10)
where λ is the Lebesgue measure on Λ. We identify µ ∈ M ac q with its Radon-Nikodym derivative dµ dλ , and, by an abuse of notation, we write

< µ, G >=< dµ dλ , G >. Since M ac q is a closed ball of M 1 , it is τ * compact. If σ ∈ S γ we define the empirical measure π γ (σ) ∈ M 1 by π γ (σ)(dr) = γ d x∈Λγ σ(x)δ γx (dr), (2.11)
where δ γx is the Dirac measure concentrated on point γx. Remark that if we denote by µ * G the convolution of a measure µ and a function G over Λ, namely (µ * G)(r ′ ) = Λ G(r ′ -r) µ(dr), then we can rewrite

(J γ ⋆ σ)(x) = (π γ (σ) * J)(γx).
(2.12)

We denote by D([0, T ], M 1 ) (resp. D([0, T ], S γ )) the space of functions from [0, T ] to M 1 (resp. to S γ ) that are right continuous with left limits, endowed with the Skorohod topology, see [Bill].

The initial condition: Let (σ γ ) γ be a sequence of configurations such that π γ (σ γ ) converges when γ → 0 in the weak topology to the measure m 0 λ, for a continuous function

m 0 : Λ → [-1, 1]. This means that lim γ→0 ρ(π γ (σ γ ), m 0 λ) = 0. (2.13)
We denote by P γ,α σ γ the law (and by E γ,α σ γ the expectation) of the process (σ t ) t∈[0,T ] on D([0, T ], S γ ) starting at time t = 0 from the deterministic initial configuration σ γ , and by Q γ,α σ γ the law on D([0, T ], M 1 ) of the corresponding empirical measure process (π γ t ) t∈[0,T ] , where π γ t stands for π γ (σ t ).

We first obtain the "law of large numbers".

Theorem 2.2 Assume (σ γ ) γ , m 0 satisfy (2.13). For all t ≥ 0,

lim γ→0 ρ(π γ t , m(t, •)λ) = 0, IP -a.s., (2.14)
where m(•, •) is the unique weak solution of

     ∂ t m(t, r) = -m(t, r) + i=1,2 p i tanh [(J * m(t, •))(r) + a i θ] m(0, •) = m 0 (•).
(2.15)

Furthermore, for all G ∈ C 0,1 ([0, T ] × Λ) (that is, continuous in its first variable, and continuously differentiable in its second variable), δ > 0,

lim γ→0 P γ,α σ γ sup t∈[0,T ] |< π γ t , G(t, •) > -< m(t, •), G(t, •) >| ≥ δ = 0. (2.16)
By an abuse of notation we write from now on (J * m)(t, r) instead of (J * m(t, •))(r).

Remark 2.3 . The Cauchy problem (2.15) in this setup is well posed with a unique global solution, because the right hand side of (2.15) is uniformly Lipschitz, and because the set {m ∈ L ∞ (Λ) : m ∞ ≤ 1} is left invariant, since | tanh z| ≤ 1 for all z. Furthermore the solution is differentiable in time.

Next we state the quenched large deviation principle for Q γ,α σ γ . Different choices of initial conditions could be treated as well. The only difference would be an extra term to add to the rate functional associated with the large deviation principle I m0 (•), taking into account the deviation from the initial profile at time t = 0. The functional I m0 (•) depends on the distribution of the random field but not on its realization; it is obtained through a contraction principle, as explained in the introduction. Its explicit formulation relies on several intermediate steps. Let D( I m0 ) = {π ∈ D([0, T ], M 1 ) : I m0 (π) < ∞}.

(2.17)

Theorem 2.4 Assume (σ γ ) γ , m 0 satisfy (2.13). For all closed subsets F ⊂ D([0, T ], M 1 ) and open subsets A ⊂ D([0, T ], M 1 ), we have

lim sup γ→0 γ d log Q γ,α σ γ (F ) ≤ -inf π∈F I m0 (π), IP -a.s., (2.18) lim inf γ→0 γ d log Q γ,α σ γ (A) ≥ -inf π∈A I m0 (π), IP -a.s. (2.19)
The functional I m0 (•), defined in (2.28) below, is non-negative for π ∈ D([0, T ], M 1 ), lower semicontinuous with compact level sets and, see Definition 3.1 later on,

D( I m0 ) ⊂ {m ∈ C([0, T ], M ac 1 ) : m(t, .
) absolutely continuous for t ∈ [0, T ]}.

The colored particle system: To derive the rate functional associated with the large deviation principle we introduce random empirical measures

π γ = π γ 1 , π γ 2 ). For α ∈ Ω, x ∈ Λ γ , i = 1, 2, set α i (x) = 1I {α(x)=ai} ,
(2.20)

π γ i (σ)(dr) = γ d x∈Λγ α i (x)σ(x)δ γx (dr).
(2.21)

Though we do not write it explicitly, π γ i (σ) ∈ M 1 depends on the randomness. Moreover the knowledge of

π γ i (σ) for i = 1, 2 determines π γ (σ) = π γ 1 (σ)+π γ 2 (σ). We denote by Q γ,α σ γ the law on D([0, T ], M 1 ×M 1 ) of the empirical measure process (π γ t ) t∈[0,T ] = (π γ 1,t , π γ 2,t ) t∈[0,T ] under P γ,α σ γ . We denote, for G = (G 1 , G 2 ) ∈ (C(Λ)) 2 , < π γ t , G >= i=1,2 γ d x∈Λγ G i (γx)α i (x)σ t (x) (2.22) and, for m = (m 1 , m 2 ) ∈ (L ∞ (Λ)) 2 , by an abuse of notation, < m, G >=< (m 1 λ, m 2 λ), G >= 2 i=1 Λ G i (r)m i (r) dr.
(2.23)

Theorem 2.5 Assume (σ γ ) γ , m 0 satisfy (2.13). For all t ∈ [0, T ], δ > 0 and G ∈ C 1 (Λ) 2 , lim γ→0 P γ,α σ γ < π γ t , G > -< m(t, •), G > ≥ δ = 0 IP -a.s. ,
where m = (m 1 , m 2 ) is the unique weak solution of

∂ t m i (t, r) = -m i (t, r) + p i tanh [β((J * m)(t, r) + a i θ)] , m = m 1 + m 2 ; m i (0, •) = p i m 0 (•), i = 1, 2.
(2.24)

Remark 2.6 . Similarly to Remark 2.3, the Cauchy problem (2.24) in this setup is well posed with a unique global solution; here, the set {m ∈ (L ∞ (Λ)

) 2 : m i ∞ ≤ p i , i = 1, 2} is left invariant. The solution is differentiable in time. The case J ≥ 0, a 1 = 1, a 2 = -1, p 1 = p 2 = 1/2 is analyzed in [COP4].
To derive Theorem 2.9 below, we need a stronger type of convergence:

Corollary 2.7 For all G ∈ C 0,1 ([0, T ] × Λ) 2 , δ > 0, lim γ→0 P γ,α σ γ sup t∈[0,T ] < π γ t , G(t, •) > -< m(t, •), G(t, •) > ≥ δ = 0.
Remark 2.8 . Theorem 2.5 and Corollary 2.7 imply Theorem 2.

2 since if G = (G, G), < π γ t , G >=< π γ 1,t , G > + < π γ 2,t , G >=< π γ t , G > .
Next theorem states the large deviation principle for the colored particle system. Theorem 2.4 is based on this important intermediate result, interesting for itself.

Theorem 2.9 Assume (σ γ ) γ , m 0 satisfy (2.13). We have, for all open subset A and closed subset

F in D([0, T ], M 1 × M 1 ), lim inf γ→0 γ d log Q γ,α σ γ (A) ≥ -inf π∈A I m0 (π), IP -a.s. (2.25) lim sup γ→0 γ d log Q γ,α σ γ (F ) ≤ -inf π∈F I m0 (π), IP -a.s.
(2.26)

where

I m0 (π) = I 0 (π) if π i (0, •) = p i m 0 (•)λ, i = 1, 2, + ∞ otherwise, (2.27)
and I 0 (•), defined in (3.6) below, is lower semicontinuous with compact level sets.

Define, for a path π ∈ D([0, T ], M 1 ),

I m0 (π) = inf I m0 (π), π = (π 1 , π 2 ), π i ∈ D([0, T ], M 1 ), i = 1, 2, π 1 + π 2 = π . (2.28) Since the map (π 1 , π 2 ) → π 1 + π 2 is continuous in D([0, T ], M 1 × M 1 )
, by the contraction principle, see [V], [DZ], Theorem 2.9 proves Theorem 2.4. Therefore in the following sections we will focus on the colored particle system.

Rate functional

In this section we define the rate functional I 0 (•) of the colored particle system and state its main properties. Proofs of the latter, quite technical, are carried out in Section 8. Heuristics to define I 0 (•) consists in finding, for any path φ on [0, T ] smooth enough, an exponential change of probability under which the process (σ t ) t∈[0,T ] is uniformly close to φ on [0, T ]. When there exists some potential V (t, •), t ∈ [0, T ] smooth enough for φ to be the solution of a perturbed equation (obtained by the law of large numbers from the process (σ t ) t∈[0,T ] , see (5.5) later on), then I 0 (•) is related to the Radon-Nikodym derivative of the distribution of (σ t ) t∈[0,T ] with respect to the distribution of the original process, see Theorem 5.4. In the general case, i.e. when there is no such V (t, •), we are still able to provide an explicit representation of I 0 (•) (this is similar to the results of [C]). We will then show that this representation of I 0 (•) is equivalent to the usual definition of the rate functional, given through the macroscopic functional associated to the Radon-Nikodym derivative, see (3.8). We start by specifying the functional spaces on which we will define I 0 (•). For (p 1 , p 2 ) ∈ [0, 1] 2 , we identify the set

B p1,p2 = {u = (u 1 , u 2 ) : u i ∈ L ∞ (Λ), u i ∞ ≤ p i , i = 1, 2} (3.1)
with M ac p1 × M ac p2 , see (2.10), and extend the distance ρ (see (2.9)) to elements of M ac p1 × M ac p2 by

ρ(µ, ν) = i=1,2 ρ(µ i , ν i ). (3.2) Definition 3.1 Let AC([0, T ], B 1,1 ) ⊂ C([0, T ], B 1,1
) be the subset of absolutely continuous functions φ = (φ 1 , φ 2 ), that is, for j = 1, 2:

for all t ′ ∈ [0, T ], t ∈ [t ′ , T ], there exists φj ∈ L 1 ([0, T ] × Λ) such that φ j (t)(r) -φ j (t ′ )(r) = t t ′ φj (s, r) ds , λ -a.s.
By an abuse of notation, from now on we write φ j (t, r) instead of φ j (t)(r).

To write I 0 (•), we start by defining, for each t ∈ [0, T ], the following functionals, in which time is kept fixed, therefore we omit to write it. For π

= (π 1 , π 2 ) ∈ M 1 × M 1 (we write π = π 1 + π 2 ), µ = (µ 1 , µ 2 ) ∈ M 1 × M 1 and V = (V 1 , V 2 ) ∈ (L ∞ (Λ)) 2 denote F V (µ, π) = i=1,2 < µ i , tanh(π * J + a i θ) sinh(2V i ) + cosh(2V i ) -1 > - i=1,2 < π i , tanh(π * J + a i θ)[cosh(2V i ) -1] + sinh(2V i ) >, (3.3) and for u = (u 1 , u 2 ) ∈ B 1,1 , g ∈ L 1 (Λ) 2 , Γ V (u) = F V (p 1 λ, p 2 λ), (u 1 λ, u 2 λ) , (3.4) H * (u, g) = sup [< V , g > - 1 2 Γ V (u)]. (3.5) The function g → H * (u, g) is convex. Next lemma ensures that H * (u, •) is the Fenchel-Legendre transform of Γ (•) (u) when u ∈ B p1,p2
, and we will derive in that case an explicit formula for H * (u, g).

Lemma 3.2 As a function of V ∈ (L ∞ (Λ)) 2 , Γ V (u) is convex differentiable for u ∈ B p1,p2 .
Definition 3.3The dynamical rate functional

I 0 : D([0, T ], M 1 × M 1 ) → IR ∪ {∞} is given by I 0 (π) =      I 0 (φ) = T 0 H * (φ(s, •), φ(s, •))ds, for π = (φ 1 λ, φ 2 λ), φ = (φ 1 , φ 2 ) ∈ AC([0, T ], B 1,1 ), ∞ otherwise. (3.6)
To derive properties of the rate functional associated with the large deviation principle it is convenient to have different representations of I 0 . To this aim let

V = (V 1 , V 2 ) ∈ (L ∞ ([0, T ] × Λ)) 2 . We define, for π ∈ D([0, T ], M 1 × M 1 ) (cf. (3.4)), K V (π) = K V (φ), for π = (φ 1 λ, φ 2 λ), φ = (φ 1 , φ 2 ) ∈ AC([0, T ], B 1,1 ), ∞ otherwise, (3.7)
where

K V (φ) = T 0 < V (s, •), φ(s, •) > ds - 1 2 T 0 Γ V (s,•) (φ(s, •))ds, J 0 (π) = sup V ∈(L ∞ ([0,T ]×Λ)) 2 K V (π). (3.8) J 1 (π) =      J 1 (φ) = T 0 Λ H(φ(t, r), φ(t, r)) dr dt, for π = (φ 1 λ, φ 2 λ), φ = (φ 1 , φ 2 ) ∈ AC([0, T ], B 1,1 ), ∞ otherwise, (3.9) where for u = (u 1 , u 2 ) ∈ B 1,1 , g = (g 1 , g 2 ) ∈ (L 1 ([0, T ] × Λ)) 2 , (t, r) ∈ [0, T ] × Λ, H(u, g)(t, r) = H(u(t, r), g(t, r)) = 2 i=1 H i (u, g i )(t, r), (3.10)
H i (u, g i )(t, r) = sup vi∈IR g i (t, r)v i - 1 2 B i (u(t, r), v i ) , i = 1, 2, (3.11) B i (u(t, r), v i ) =(p i -u i (t, r)) e (J * u)(t,r)+aiθ 2 cosh[(J * u)(t, r) + a i θ] e 2vi -1 + (p i + u i (t, r)) e -[(J * u)(t,r)+aiθ] 2 cosh[(J * u)(t, r) + a i θ] e -2vi -1 . (3.12) When u(t, r) ∈ [-p 1 , p 1 ] × [-p 2 , p 2 ], 2 i=1 B i (u(t, r), •) is convex so that H(u(t, r),
•) is its Fenchel-Legendre transform. We now give an explicit representation of H(•, •). To simplify notations denote

A i = A i (u, θ)(t, r) = (J * u)(t, r) + a i θ, R i = R i (u, g i , θ)(t, r) = (g i (t, r) cosh[A i (u, θ)(t, r)]) 2 + p 2 i -u 2 i (t, r), D i = D i (u, g i , θ)(t, r) = g i (t, r) cosh[A i (u, θ)(t, r)] + R i (u, g i , θ)(t, r). (3.13) Note that D i (u, g i , θ)(t, r) ≥ 0 regardless of the sign of g i (t, r). When (t, r) is kept fixed we omit to write it. The function sgn : IR → IR is given by sgn(x) =    x |x| if x = 0 , 0 if x = 0 . (3.14) Proposition 3.4 (a) If |u 1 | > p 1 or |u 2 | > p 2 , then H(u, g) = +∞. (b) For i = 1, 2, when |u i | < p i , then H i (u, g i ) = g i 2 log D i p i -u i -A i + p i 2 - u i 2 tanh A i - R i 2 cosh A i . (3.15)
(c) For i = 1, 2, when either (u i = p i and g i ≤ 0) or (u i = -p i and g i ≥ 0), then

H i (u, g i ) = 1I {gi =0} |g i | 2 log |g i | cosh A i p i e -sgn(ui)Ai -1 + p i e -sgn(ui)Ai 2 cosh A i . (3.16) (d) For i = 1, 2, when either (u i = p i and g i > 0) or (u i = -p i and g i < 0), then H i (u, g i ) = +∞.
The following proposition shows that the order of supremum and the integrals can be reversed. In particular we can compute the supremum for each point (t, r) ∈ [0, T ] × Λ.

Proposition 3.5 For π = (φ 1 λ, φ 2 λ), φ = (φ 1 , φ 2 ) ∈ AC([0, T ], B 1,1 ), we have I 0 (π) = J 0 (π) = J 1 (π). Furthermore if φ ∈ AC([0, T ], B 1,1 ) \ AC([0, T ], B p1,p2 ), then I 0 (φ) = +∞.
Proof. This follows and extends the proof in [START_REF] Comets | Nucleation for a long range magnetic model[END_REF]p. 171,Properties III(a). By their respective Definitions (3.6), (3.8), (3.9) (see also (3.5), (3.7), (3.10)), we have J 0 (π) ≤ I 0 (π) ≤ J 1 (π). We now prove that we have equalities. In all cases, for i = 1, 2, we denote by ϑ i the value of v i that realizes the extremum of

H i (u, g i ). From Proposition 3.4, ϑ i belongs to IR ∪ {+∞, -∞}. Let ϑ m i = sgn(ϑ i ) × [|ϑ i | ∧ m] and b m i be the corresponding (finite) value of H i (u, g i ). Then as m → ∞, ϑ m i → ϑ i and b m i → H i (u, g i ) ∈ IR + ∪ {∞}.
According to the case we consider, either a i and/or b i are finite, and there is no problem, or b i = +∞ thus b m i > 0 for m large enough, or, when u ∈ B p1,p2 , b m i is non-negative because ϑ m i is between 0 and ϑ i , and

v i → B i (u, v i ) is a convex function.
Therefore in all cases we apply Fatou's Lemma to get

J 1 (π) ≤ lim inf m→∞ T 0 Λ i=1,2 b m i (t, r) drdt
which is smaller than J 0 (π), whence the result. Notice that this implies that I 0 (π) is infinite when u / ∈ B p1,p2 .

Next we characterize the finite energy trajectories.

Proposition 3.6 Take (t, r) ∈ [0, T ] × Λ. (a) Let u be such that for i = 1, 2, |u i | < p i . There exist positive constants K 1 ,K 2 and C such that H(u, g)(t, r) ≤ i=1,2 |g i | 2 (log |g i |) + + 1I {gi>0} log 1 p i -u i + + 1I {gi<0} log 1 p i + u i + + K i (t, r) + C (3.17) H(u, g)(t, r) ≥ i=1,2 |g i | 2 [log |g i | -K i ] (t, r) -C. (3.18) (b) I 0 (φ) < ∞ if and only if for i = 1, 2, φi log | φi |, φi log 1 p i -φ i 1I { φi>0} , φi log 1 p i + φ i 1I { φi<0} belong to L 1 ([0, T ] × Λ). Proposition 3.7 (1) The functional I 0 (•) is lower semicontinuous on D([0, T ], B 1,1 ). (2) The set D L0 = {π; I 0 (π) ≤ L 0 } is compact in D([0, T ], B 1,1
) for all L 0 > 0.

(3) I 0 (φ) ≥ 0, and I 0 (φ) = 0 if and only if φ is the solution of equation (2.24).

Hydrodynamic behavior for the colored system

In this section, we prove Theorem 2.5, through a by now standard scheme. Nevertheless, we detail it since many of its parts will also appear in the following sections.

We first highlight that throughout the paper, one of the key ingredients to deal with the randomness of the interaction will be the following applications of the ergodic theorem and strong law of large numbers. For all function h on Λ γ , integer l, we denote by h (l) the averaged function

h (l) (x) = 1 (2l + 1) d y∈Λγ ,|y-x|≤l h(y) , x ∈ Λ γ .
(4.1)

Lemma 4.1 (ergodic theorem for local functions) Let Θ(α) be a bounded measurable cylinder function on Ω and G ∈ C(Λ). Then, for almost any disorder configuration α,

lim γ→0 γ d x∈Λγ G(γx)τ x Θ(α) = IE Θ Λ G(r)dr . Proof. Write γ d x∈Λγ G(γx)τ x Θ(α) = γ d x∈Λγ G(γx) τ x Θ(α) -IE[Θ] + IE[Θ]γ d x∈Λγ G(γx).
For any l ∈ IN , by the regularity of G,

γ d x∈Λγ G(γx) τ x Θ(α) -IE[Θ ] ≤ G ∞ γ d x∈Λγ (τ • Θ(α)) (l) (x) -IE Θ] + ǫ(γl),
where lim s→0 ǫ(s) = 0. Keeping l fixed, by the ergodic theorem,

lim γ→0 γ d x∈Λγ (τ • Θ(α)) (l) (x) -IE[Θ] = IE (τ • Θ) (l) (x) -IE[Θ] .
The law of large numbers (letting l → ∞) gives the result.

We introduce (cf. [K]), for i = 1, 2 and δ > 0,

A l,δ (x, i) = α ∈ Ω : α (l) i (x) -IE(α i (x)) ≤ δ , x ∈ Λ γ , (4.2) E i (δ, l, γ, α) = γ d x∈Λγ 1I A c l,δ (x,i) (α). (4.3) Lemma 4.2 For any δ > 0, for i = 1, 2, lim l→∞ lim γ→0 E i (δ, l, γ, α) = 0, IP -a.s. Proof. Applying Lemma 4.1 to the function Θ = 1I A c l,δ (0,i) gives lim γ→0 E i (δ, l, γ, α) = IP (A c l,δ (0, i))
, IPa.s. Then by the strong law of large numbers lim l→∞ IP (A c l,δ (0, i)) = 0.

In the following it is convenient to define the random discrete measures λ γ (α) = (λ γ 1 (α), λ γ 2 (α)), where

λ γ i (α) = γ d x∈Λγ α i (x)δ γx , i = 1, 2, λ γ = λ γ 1 (α) + λ γ 2 (α) = γ d x∈Λγ δ γx . (4.4)
Proof of Theorem 2.5 We follow the general scheme introduced in [KL] chap. 4. We have to show: (i) For any α, the sequence (Q

γ,α σ γ ) γ is tight. (ii) Any limit point Q α of (Q γ,α σ γ ) γ is IP -a.s. concentrated on measures (π t ) t∈[0,T ] ∈ C([0, T ], M ac p1 × M ac p2 ). (iii) For IP -a.s. α, any limit point Q α of (Q γ,α σ γ ) γ is concentrated on trajectories (π t ) t∈[0,T ] such that π t (dr) = m(t, r)dr,
where the density m is a weak solution of (2.24). (iv) Equation ( 2.24) has a unique weak solution.

For (ii) we use that the spins are finite-valued (cf. [KL]). Namely, fix G ∈ C(Λ),

sup 0≤t≤T < π γ i,t , G > ≤ γ d x∈Λγ |G(γx)|α i (x), i = 1, 2,
because there is at the most one spin per site and α i (x) ≥ 0. As in the case without random field, the application (π i,t ) t∈[0,T ] → sup t∈[0,T ] < π i,t , G > is continuous in the weak topology. Thus by weak convergence and Lemma 4.1 (by the independence of the r.v. α's) all limits points are concentrated on trajectories

(π i,t ) t∈[0,T ] such that |< π i,t , G >| ≤ Λ |G(r)|p i dr, IP -a.s.
Point (iv) is derived similarly to the proof of the Cauchy-Lipschitz theorem. For Points (i) and (iii), let

G = (G 1 , G 2 ) ∈ C 1,0 ([0, T ] × Λ) 2 . For π ∈ D([0, T ], M 1 × M 1 ), let ℓ t (π, G) =< π t , G(t, •) > -< π 0 , G(0, •) > - t 0 < π s , ∂ s G(s, •) > ds. (4.5) We have, for x ∈ Λ γ , L γ,α (σ(x)) = -σ(x) + σ(x)(1 -2c γ,α x (σ)) = -σ(x) + tanh[(J γ ⋆ σ)(x) + θα(x)].
(4.6)

The P γ,α σ γ -martingale N G γ ≡ (N G γ (t)) t∈[0,T ] with respect to the natural filtration associated to (σ t ) t∈[0,T ] (cf. (2.20)) given by N G γ (t) = ℓ t (π γ , G) -γ d t 0 i=1,2 x∈Λγ G i (s, γx)α i (x)L γ,α (σ s (x)) ds = ℓ t (π γ , G) + t 0 < π γ s , G(s, •) > ds - i=1,2 t 0 < λ γ i (α), G i (s, .) tanh[π γ s * J + a i θ] > ds , (4.7) has quadratic variation < N G γ , N G γ > (t) = -2γ 2d i=1,2 x∈Λγ α i (x)G 2 i (s, γx) t 0 -1 + σ s (x) tanh[(J γ ⋆ σ s )(x) + a i θ] ds. (4.8)
Hence, for any α ∈ Ω, since tanh is a smooth function and (J γ ⋆ σ)(x) + a i θ is uniformly bounded in x, σ, by Doob's inequality,

lim γ→0 P γ,α σ γ sup t∈[0,T ] N G γ (t) > δ ≤ lim γ→0 1 δ 2 E γ,α σ γ N G γ (T ) 2 ≤ lim γ→0 1 δ 2 C(G, T )γ d = 0. (4.9)
Bound (4.9) yields Point (i), by Prohorov's criterion. Point (iii) will consist in identification of the limit. To obtain a closed form for the limiting equation, we only need to average over the disorder, that is to replace in the limit γ → 0 the random discrete measures λ γ 1 (α) and λ γ 2 (α) by their expectations p 1 λ and p 2 λ with respect to the environment. Denote

ℓ t (π γ , G) = t 0 π γ s , G(s, •) - 2 i=1 p i λ γ , G i (s, •) tanh(π γ s * J + a i θ ds. (4.10)
Putting together (4.7), (4.9), and applying Lemma 4.3 below, we get that for all subsequences

lim inf k→+∞ Q γ k ,α σ γ k sup t≤T ℓ t (π γ k , G) + ℓ t (π γ k , G) > δ 2 = 0. (4.11) Denoting m = m 1 + m 2 , for m = (m 1 , m 2 ), see (2.24
), this gives, for almost any α,

Q α sup t≤T 2 i=1 Λ {G i (t, r)m i (t, r) -G i (0, r)m i (0, r)} - t 0 ∂ s G i (s, r)m i (s, r) ds dr + i=1,2 t 0 Λ [G i (s, r)m i (s, r) -IE(α i (0)) tanh[(J * m)(s, r) + a i θ]] drds > δ 2 = Q α sup t≤T ℓ t (m, G) + ℓ t (m, G) > δ 2 = 0, (4.12)
where we set by an abuse of notation

ℓ t (m, G) =< m(t, •), G(t, •) > -< m(0, •), G(0, •) > - t 0 < m(s, •), ∂ s G(s, •) > ds, (4.13) ℓ t (m, G) = t 0 m(s, •), G(s, •) - 2 i=1 p i λ, G i (s, •) tanh(J * m(s, •) + a i θ ds. (4.14)
This leads to identification of the limit (iii), that is to equation (2.24).

Lemma 4.3 For i = 1, 2, G i ∈ C 1,0 ([0, T ]×Λ)
, there exists a positive function ǫ on IR + with lim s→0 ǫ(s) = 0 such that for all l ∈ IN \ {0}, δ > 0, and E i (δ, l, γ, α) defined in (4.3), the quantity

∆ γ i (α, σ, T ) = T 0 < λ γ i (α) -p i λ, G i (s, .) tanh[π γ s * J + a i θ] > ds (4.15) satisfies ∆ γ i (α, σ, T ) ≤ δ 2 T G i (s, •) 1 + 2T G i (s, •) ∞ E i (δ, l, γ, α) + ǫ(γl)T + ǫ(γ)T, (4.16) lim γ→0 ∆ γ i (α, σ, T ) = 0, IP -a.s. (4.17)
Proof. We introduce averages over large microscopic boxes of size l but small w.r.t. the range γ -1 of the interaction (l will go to infinity but after the limit γ → 0). To keep notation readable, the function ǫ may vary from one line to another but keeping the same property that lim s→0 ǫ(s) = 0. Since J and tanh are uniformly Lipschitz, and G i is uniformly continuous (in space), there are constants

c 1 > 0, c 2 > 0 such that, see (4.1), for x ∈ Λ γ , sup σ∈Sγ (π γ (σ) * J)(γx) -(π γ (σ) * J)(γ.) (l) (x) ≤ c 1 γl, sup σ∈Sγ tanh[β((π γ (σ) * J)(γx) + a i θ)] -tanh[(π γ (σ) * J)(γ.) + a i θ] (l) (x) ≤ c 2 γl. sup 0≤s≤T G i (s, γx) -G i (s, γ.) (l) (x) ≤ ǫ(γl).
Recalling notation (4.4), by summation by parts we get

T 0 < λ γ i (α) -λ γ i (α (l) ), G i (s, .) tanh[π γ s * J + a i θ] > ds ≤ ǫ(γl)T
and by uniform continuity or Lipschitz condition

T 0 |< p i λ γ -p i λ, G i (s, .) tanh[π γ s * J + a i θ] >| ds ≤ ǫ(γ)T.
Therefore, we have

∆ γ i (α, σ, T ) ≤ ǫ(γl)T + ǫ(γ)T + T 0 < λ γ i (α (l) ) -p i λ γ , G i (s, •) tanh[π γ s * J + a i θ] > ds.
To derive (4.16), we take into account definitions (4.2), (4.3), and that |α

(l) i (x) -p i | ≤ 2, to write | < λ γ i (α (l) ) -p i λ γ , G i (s, •) tanh[π γ s * J + a i θ] > | = γ d x∈Λγ G i (s, γx) tanh[(J γ ⋆ σ s )(x) + a i θ] 1I A l,δ (x,i) (α) + 1I A c l,δ (x,i) (α) α (l) i (x) -IE(α i (x)) ≤ γ d x∈Λγ |G i (s, γx) tanh[(J γ ⋆ σ s )(x) + a i θ]| δ + 21I A c l,δ (x,i) (α) ≤ δ G i (s, •) 1 + 2 G i (s, •) ∞ E i (δ, l, γ, α).
Applying Lemma 4.2 to (4.16) we get (4.17).

Proof of Corollary 2.7: First notice that applying Lebesgue dominated convergence Theorem in the time integral, Theorem 2.5 implies that for any

G i ∈ C 0,1 ([0, T ] × Λ) we have lim γ→0 P γ,α σ γ T 0 < π γ i,s , G i (s, •) > -< m i (s, •), G i (s, •) > ds ≥ δ = 0. (4.18)
Now remark that integrating in time (2.24),

< m i (t, •), G i (t, •) >=< m i (0, •), G i (0, •) > + t 0 < m i (s, •), ∂ s G i (s, •) -G i (s, •) > ds + p i t 0 < tanh[(J * m)(s, •) + a i θ], G i (s, •) > ds.
Introducing the martingale N G γ , see (4.7), and using (4.9) we get

P γ,α σ γ sup t∈[0,T ] < π γ i,t , G i (t, •) >)-< m i (t, •), G i (t, •) > ≥ δ ≤ A γ + B γ + C γ + ǫ(γ),
with lim γ→0 ǫ(γ) = 0 and

A γ = P γ,α σ γ T 0 < π γ i,s -m i (s, •), ∂ s G i (s, •) -G i (s, •) > ds ≥ δ 4 , C γ = P γ,α σ γ < π γ i,0 -m i (0, •), G i (0, •) > ≥ δ 4 , B γ = P γ,α σ γ T 0 < λ γ i (α), G i (s, •) tanh[π γ s * J + a i θ] > -< p i λ, G i (s, •) tanh[(J * m)(s, •) + a i θ] > ds ≥ δ 4 ≤ P γ,α σ γ T 0 |< λ γ i (α) -p i λ, G i (s, •) tanh[π γ s * J + a i θ] >| ds ≥ δ 8 + P γ,α σ γ T 0 < p i λ, G i (s, •) tanh[π γ s * J + a i θ] -tanh[(J * m i )(s, •) + a i θ] > ds ≥ δ 8 .
(4.19) From (2.13) and (4.18), lim γ→0 A γ = lim γ→0 C γ = 0. For B γ , from Lemma 4.3, the limit when γ → 0 of the first term in the right hand side of (4.19) is equal to zero; the second term vanishes from (4.18) since the function tanh is Lipschitz continuous.

The perturbed dynamics and Radon-Nikodym derivative

The general strategy to derive the large deviation principle prescribes to find a family of mean one positive martingales that can be expressed as functions of the empirical measures. Following [DV], the relevant martingales are obtained as Markovian perturbations of the original process. In this section we define a class of time dependent, random external potentials, the perturbations, to which we can associate a trajectory (m(t, •)) t∈[0,T ] smooth in time. We show the law of large numbers for the empirical measures of the dynamics associated to these perturbations and derive the Radon-Nikodym derivative of the perturbed process with respect to the unperturbed one.

Given a realization α of the magnetic field,

V = (V 1 , V 2 ) ∈ C 1,0 ([0, T ] × Λ) 2 , let V (t, γx, α(x)) = i=1,2 α i (x)V i (t, γx) (5.1)
be the full external random perturbation for the magnetization trajectories π γ (σ) (not colored). As pointed out in the introduction this perturbation strongly depends on the randomness. It is therefore convenient to consider a Glauber evolution for the colored particle system, associated to the Hamiltonian obtained by summing up (2.4) and

H V ,γ,α (σ) = - x∈Λγ i=1,2 α i (x)V i (t, γx)σ(x).
(5.2)

To this aim we define time dependent rates, for all x ∈ Λ γ , σ ∈ S γ ,

c V ,γ,α x (σ, t) = e -σ(x)2V (t,γx,α(x)) c γ,α x (σ) = e -σ(x)[(Jγ ⋆σ)(x)+θα(x)+2V (t,γx,α(x))] 2 cosh[(J γ ⋆ σ)(x) + θα(x)] .
(5.3) Assume (σ γ ) γ , m 0 satisfy (2.13). We denote by P V ,γ,α σ γ the law (and by E V ,γ,α σ γ the expectation) of the corresponding inhomogeneous Glauber process (σ t ) t∈[0,T ] on S γ , that is the unique probability measure on

D([0, T ], S γ ) with initial condition σ γ under which f (σ t ) -f (σ 0 ) - t 0 L V ,γ,α s (f )(σ s )ds is a martingale w.r.t.
the canonical filtration, for all cylinder function f , where

L V ,γ,α s (f )(σ s ) = x∈Λγ c V ,γ,α x (σ s , s)[f (σ x s ) -f (σ s )]. (5.4) Let Q V ,γ,α σ γ
be the law of the corresponding empirical measures.

Theorem 5.1 Assume (σ γ ) γ , m 0 satisfy (2.13). For all t ∈ [0, T ], G = (G 1 , G 2 ) ∈ C 1 (Λ) 2 , and δ > 0, lim γ→0 Q V ,γ,α σ γ < π γ t , G > -< m V (t, •), G > ≥ δ = 0, IP -a.s.,
where

m V = (m V 1 , m V 2 ) is the solution of, for i = 1, 2,      ∂ t m i (t, r) = {-m i (t, r) + p i tanh [(J * m)(t, r) + a i θ + 2V i (t, r)]} cosh [(J * m)(t, r) + a i θ + 2V i (t, r)] cosh [(J * m)(t, r) + a i θ] , m i (0, •) = p i m 0 (•), m = m 1 + m 2 .
(5.5)

Remark 5.2 . For existence and uniqueness of the solution m

V ∈ (C([0, T ], L ∞ (Λ))) 2 , we refer to Remark 2.6. Notice that the set {m ∈ (L ∞ (Λ)) 2 : m i ∞ ≤ p i , i = 1, 2} is still left invariant.
Proof. We proceed as for Theorem 2.5. We use

1 = (1 -σ s (x)) 2 + (1 + σ s (x)) 2 = 1I {σs(x)=-1} + 1I {σs(x)=1} .
(5.6)

For i ∈ {1, 2} we have

L V ,γ,α s (α i (x)σ(x)) = -2α i (x)σ(x)c V ,γ,α x (σ, s) = -α i (x)e -σ(x)2Vi (s,γx) 2σ(x) e -σ(x)[(Jγ ⋆σ)(x)+aiθ] 2 cosh[J γ ⋆ σ)(x) + a i θ] = -α i (x) (σ(x) + 1) 2 e -[2Vi(s,γx)+(Jγ⋆σ)(x)+aiθ] cosh[(J γ ⋆ σ)(x) + a i θ] + (σ(x) -1) 2 e [2Vi(s,γx)+(Jγ⋆σ)(x)+aiθ] cosh[(J γ ⋆ σ)(x) + a i θ] = -α i (x)σ(x) cosh[2V i (s, γx) + (J γ ⋆ σ)(x) + a i θ] cosh[(J γ ⋆ σ)(x) + a i θ] -α i (x) sinh[2V i (s, γx) + (J γ ⋆ σ)(x) + a i θ] cosh[(J γ ⋆ σ)(x) + a i θ] .
(5.7)

We have the analogous result to Corollary 2.7:

Corollary 5.3 For all G = (G 1 , G 2 ) ∈ C 1 (Λ) 2 , and δ > 0, lim γ→0 Q V ,γ,α σ γ sup t∈[0,T ] < π γ t , G > -< m V (t, •), G > ≥ δ = 0. Theorem 5.4 Let V = (V 1 , V 2 ) ∈ C 1,0 ([0, T ] × Λ) 2 .
The Radon-Nikodym derivative is given by

dP V ,γ,α σ γ dP γ,α σ γ (σ [0,T ] ) = exp γ -d ℓ T (π γ (σ), V ) - 1 2 T 0 F V (s) (λ γ (α), π γ s ) ds , (5.8)
where ℓ T was defined in (4.5),

λ γ (α) in (4.4), F V (s) (•, •) in (3.
3), and we have abbreviated

σ [0,T ] = (σ t ) t∈[0,T ] .
Proof. The Radon-Nikodym derivative associated with rates (5.3) is given by (see [HS] or [KL], Appendix 1, Proposition 7.3)

dP V ,γ,α σ γ dP γ,α σ γ (σ [0,t] ) = exp -H V ,γ,α (σ t ) + H V ,γ,α (σ 0 ) - t 0 exp H V ,γ,α (σ s ) ∂ s + L γ exp -H V ,γ,α (σ s ) ds = exp ℓ t (π γ (σ), V ) - t 0 x∈Λγ c γ,α x (σ s ) e -2σs(x)V (s,γx,α(x)) -1 ds ,
because of (4.5), (5.2). To get (5.8), we use trigonometric formulas to write (remember (2.8), (2.20), (5.7))

2γ d x∈Λγ c γ,α x (σ s ) e -2σs(x)V (s,γx,α(x)) -1 =2γ d x∈Λγ (1 -σ s (x)) 2 + (1 + σ s (x)) 2    i=1,2 α i (x)    c γ,α x (σ s ) e -2σs(x)V (s,γx,α(x)) -1 =γ d i=1,2 x∈Λγ (1 -σ s (x)) 2 α i (x) exp[(J γ ⋆ σ s )(x) + a i θ] cosh[(J γ ⋆ σ s )(x) + a i θ] e 2Vi(s,γx) -1 + γ d i=1,2 x∈Λγ (1 + σ s (x)) 2 α i (x) exp[-(J γ ⋆ σ s )(x) -a i θ] cosh[(J γ ⋆ σ s )(x) + a i θ] e -2Vi(s,γx) -1 =γ d i=1,2 x∈Λγ α i (x) {cosh[2V i (s, γx)] -1 + tanh[(J γ ⋆ σ s )(x) + a i θ] sinh[2V i (s, γx)]} -γ d i=1,2 x∈Λγ α i (x)σ s (x) {tanh[(J γ ⋆ σ s )(x) + a i θ](cosh[2V i (s, γx)] -1) + sinh[2V i (s, γx)]} =F V (s) (λ γ (α), π γ s ).
Note that the Radon-Nikodym derivative depends on the randomness through π γ and λ γ (α). By next proposition, which is proved in Appendix B, we can replace λ γ (α) in F V (s) (•, π γ s ) with (p 1 λ γ , p 2 λ γ ), making an error which goes uniformly (for all σ ∈ S γ and IP -a.s.) to zero as γ → 0.

Proposition 5.5 Let V = (V 1 , V 2 ) ∈ C 1,0 ([0, T ] × Λ)
2 . There exists a positive function ǫ on IR + with lim s→0 ǫ(s) = 0 such that for any δ > 0, l ∈ IN \ {0}, we have

T 0 F V (s) (λ γ (α), π γ s ) -F V (s) ((p 1 λ γ , p 2 λ γ ), π γ s ) ds ≤ ǫ(γl)T + T C(V 1 , V 2 )[δ + i=1,2 E i (δ, l, γ, α)]
where the positive constant

C(V 1 , V 2 ) depends on the L ∞ norm of (V 1 , V 2 ).

Upper Bound

In this section we investigate the upper bound of the large deviation principle for compact sets and then closed sets of the topological space D([0, T ), M 1 ×M 1 ). Notice that in [C] the result was stated for closed sets in C([0, T ), M 1 × M 1 ). We first prove exponential tightness, so that it is enough to derive the upper bound of the large deviation principle for compact subsets. The strategy then follows the martingale approach introduced by [DV]: we need to show that trajectories which are not absolutely continuous with respect to the Lebesgue measure and not absolutely continuous in time can be neglected in the large deviations regime. To exclude these "bad" paths, as in [FLM], we introduce an energy functional via an exponential martingale. With this we prove an upper bound with an auxiliary rate functional which is infinite on the set of bad trajectories. Proposition 6.1 For any ℓ ≥ 1, there exists a compact subset

K ℓ ⊂ D([0, T ], M 1 × M 1 ) such that for any σ γ ∈ S γ , lim sup γ→0 γ d log Q γ,α σ γ (K c ℓ ) ≤ -ℓ.
The proof is standard, however the main lines are recalled in Appendix B.

For π ∈ D [0, T ], M 1 × M 1 , G = (G 1 , G 2 ) ∈ (C 1,0 ([0, T ] × Λ)) 2 denote I J G (π) = ℓ T (π, G) + ℓ T (π, G) -2 2 i=1 T 0 p i λ, G 2 i (s, •) -π i,s , G 2 i (s, •) tanh(π s * J + a i θ ds , (6.1)
for ℓ T , lT given in (4.13), (4.14). We define the auxiliary rate functional J :

D [0, T ], M 1 × M 1 → IR as J (π) =      sup G∈(C 1,0 ([0,T ]×Λ)) 2 I J G (π) if π ∈ D([0, T ], M ac 1 × M ac 1 ) , +∞ otherwise . (6.2) Lemma 6.2 For all π ∈ D [0, T ], M 1 × M 1 , if J (π) < ∞, then π ∈ C [0, T ], M ac 1 × M ac 1 . Proof. Fix π ∈ D [0, T ], M 1 × M 1 such that J (π) < ∞. By definition of J (•), π ∈ D [0, T ], M ac 1 × M ac 1 . Let g = (g 1 , g 2 ) ∈ (C(Λ)
) 2 and 0 ≤ s < t < T . For each δ > 0, let ψ δ s,t : [0, T ] → IR be the function given by

ψ δ s,t (τ ) =                    0 if 0 ≤ τ ≤ s or t ≤ τ ≤ T , τ -s δ if s ≤ τ ≤ s + δ , 1 if s + δ ≤ τ ≤ t -δ , t -τ δ if t -δ ≤ τ ≤ t . (6.3) Denote G δ (τ, r) = ψ δ s,t (τ )g(r). Since G δ can be approximated by functions in (C 1,0 ([0, T ] × Λ)) 2 , considering G δ √ t -s
as a test function and performing the limit δ → 0, we obtain

√ t -s lim δ→0 I J G δ √ t-s (π) = < π t , g > -< π s , g > + t s < π τ , g > - 2 i=1 < p i λ, g i tanh(π τ * J + a i θ > dτ -2 1 √ t -s 2 i=1 t s < p i λ, g 2 i > -< π i,τ , g 2 i tanh(π τ * J + a i θ > dτ . (6.4) Since √ t -s lim δ→0 I J G δ √ t-s (π) ≤ √ t -s J (π), we get < π t , g > -< π s , g > ≤ C 0 (t -s) 2 i=1 g i 1 + 1 √ t -s g i 2 2 + √ t -sJ (π) = C 0 (t -s) 2 i=1 g i 1 + √ t -s C 0 2 i=1 g i 2 2 + J (π) ,
for some positive constant C 0 . This implies that π ∈ C [0, T ], M ac 1 × M ac 1 .

To prove next Lemma, we will use the following characterization of absolutely continuous functions, see [DS].

Proposition 6.3 A function φ belongs to AC([0, T ], B 1,1 ) if and only if: for all ǫ > 0, there exists ∆ > 0 such that for all integer k > 0, rectangles A 1 , . . . , A k of Λ and

{(s i , t i ), 1 ≤ i ≤ k} nonempty disjoint intervals of [0, T ], k i=1 |t i -s i |λ(A i ) < ∆ ⇒ k i=1 Ai (φ j (t i , r) -φ j (s i , r)) dr < ǫ , j = 1, 2 . Lemma 6.4 Let π = (φ 1 (s, r)dr, φ 2 (s, r)dr) ∈ D [0, T ], M ac 1 × M ac 1 such that J (π) < ∞, then (a) for i = 1, 2, F i (s, r) := p i -φ i (s, r) tanh(π s (r) * J + a i θ ≥ 0 for almost all (s, r) ∈ [0, T ] × Λ, (b) (φ 1 , φ 2 ) ∈ AC [0, T ], B 1,1 .
Proof. (a) Taking as a test function AG(•, •), for all G = (G 1 , G 2 ) ∈ (C 1,0 ([0, T ] × Λ)) 2 and A > 0, we obtain from Definition (6.2) of the rate function J ,

- 2 i=1 T 0 Λ G 2 i (s, r) F i (s, r) dr ds ≤ - 1 2A ℓ T (π, G) + ℓ T (π, G) + 1 2A 2 J (π) . (6.5) Letting A ↑ ∞, we get 2 i=1 T 0 Λ G 2 i (s, r) F i (s, r)drds ≥ 0 .
Since G is arbitrary, we conclude that F i (s, r) ≥ 0 for 1 ≤ i ≤ 2, almost everywhere. (b) We show the absolute continuity in time for φ 1 . The proof for φ 2 is similar. We apply the characterization of AC [0, T ], B 1,1 given in Proposition 6.3. For all positive integer k let {A i , 1 ≤ i ≤ k} be rectangles of Λ and {(s i , t i ) , 1 ≤ i ≤ k} be nonempty disjoint intervals of [0, T ]. For i = 1, . . . , k, denote

η i := sgn Ai [φ 1 (t i , r) -φ 1 (s i , r)] dr , For each 1 ≤ i ≤ k, 0 < δ < 1 4 min 1≤i≤k t i -s i , we set, see (6.3), V 1 (t, r) = k i=1 η i × ψ δ si,ti (t) × 1I Ai (r), V 2 (t, r) = 0, V = (V 1 , V 2 ). (6.6)
Since V 1 can be approximated by functions in C 1,0 ([0, T ] × Λ), proceeding as in (6.4), we obtain for any b > 0, see (6.5),

k i=1 η i Ai [φ 1 (t i , r) -φ 1 (s i , r)] dr ≤ - k i=1 ti si < η i 1I Ai , φ 1 (s, •) -p 1 tanh(π s * J + a 1 θ > ds + 2b k i=1 ti si Λ |η i | 1I Ai (r) F 1 (s, r) drds + J (π) b .
Minimizing over b yields

k i=1 Ai [φ 1 (t i , r) -φ 1 (s i , r)] dr ≤ 2 k i=1 (t i -s i )λ A i + 2 2J (π) k i=1 ti si Λ |η i | 1I Ai (r) F 1 (s, r) drds 1/2 ≤ 2 k i=1 (t i -s i )λ A i + 4 J (π) k i=1 (t i -s i )λ A i . (6.7) For all ε > 0 denote ∆ = min ε/4 , ε 2 /(64J (π)) . It follows from (6.7) that k i=1 (t i -s i )λ A i ≤ ∆ implies k i=1
Ai [φ 1 (t i , r) -φ 1 (s i , r)] dr ≤ ε. This concludes the proof.

For π ∈ D [0, T ], M 1 × M 1 , ℓ T as in (4.13) and F V (s,.) defined in (3.3) let

J V (π) = ℓ T (π, V ) - 1 2 T 0 F V (s,.
) (p 1 λ, p 2 λ), π s ds, (6.8)

J(π) = sup V ∈(C 1,0 ([0,T ]×Λ)) 2 J V (π) . (6.9)
Remark that when π = (φ 1 λ, φ 2 λ), with φ = (φ 1 , φ 2 ) ∈ AC([0, T ], B 1,1 ), J coincides with the functional I 0 = J 0 = J 1 (cf. Proposition 3.5). The proof of the upper bound of the large deviation principle relies on the following proposition.

Proposition 6.5 Let K be a compact set of D [0, T ], M 1 × M 1 . For any 0 < b < 1, lim sup γ→0 γ d log Q γ,α σ γ (K) ≤ - 1 1 + b inf π∈K J(π) + bJ (π) .
Proof. For ε > 0, µ ∈ M 1 , g ∈ C(Λ) denote by ι ε the approximation of the identity

ι ε (r) = 1 (2ε) d 1I -ε,ε d (r), r ∈ Λ
and by µ * ι ε the measure defined by µ * ι ε , g = µ, g * ι ε . It is absolutely continuous with respect to the Lebesgue measure with density

d(µ * ι ε ) dλ (r) = µ, ι ε (r -•) , r ∈ Λ .
In general, we can only bound this density by ι ε ∞ which is of order ε -d . Nevertheless, in the case of the empirical measure, we have

| π γ s , ι ε (r -•) | = γ d (2ε) d x : γx∈[r-ε,r+ε] σ s (x) ≤ 1 ,
for almost all 0 ≤ s ≤ T ,

which means that π γ * ι ε ∈ M ac 1 , when 0 < γ < ε. Furthermore for any π ∈ D [0, T ], M 1 × M 1 , denote by π s * ι ε := π 1,s * ι ε , π 2,s * ι ε , 0 ≤ s ≤ T the trajectory in D [0, T ], M ac 1 × M ac 1 . Fix a function G ∈ (C 1,0 ([0, T ] × Λ)) 2 . Consider the mean one exponential martingale Z G,γ t t≥0 Z G,γ t = exp γ -d N G γ (t) - γ -2d 2 N G γ , N G γ (t) ,
where the martingale N

γ d log Q γ,α σ γ (K) = γ d log E Q γ,α σ γ 1I K (π γ ) Z G,γ t b 1+b × Z G,γ t -b 1+b ≤ b 1 + b γ d log E Q γ,α σ γ 1I K (π γ )Z G,γ t + 1 1 + b γ d log E Q γ,α σ γ 1I K (π γ ) Z G,γ t -b ≤ 1 1 + b γ d log E Q γ,α σ γ 1I K (π γ ) Z G,γ t -b .
(6.12)

We now exclude paths whose densities are not absolutely continuous with respect to the Lebesgue measure. Fix a sequence {F k : k ≥ 1} of smooth nonnegative functions dense in C(Λ) for the uniform topology. For k ≥ 1, ̺ > 0 and δ > 0, let

D k,̺ = π ∈ D([0, T ], M 1 × M 1 ) : 0 ≤< |π i,t |, F k >≤ Λ F k (x) dx + C k ̺ , 0 ≤ t ≤ T, i = 1, 2 , where C k = C( ∇F k ∞ ) is a constant depending on the gradient ∇F k of F k . The sets D k,̺ , k ≥ 1 are closed subsets of D([0, T ], M 1 × M 1 ), as well as E m,̺ = m k=1 D k,̺ , m ≥ 1 .
Note that the empirical measure π γ belongs to E m,̺ for γ sufficiently small. We have that

D([0, T ], M ac 1 × M ac 1 ) = ∩ n≥1 ∩ m≥1 E m,1/n . (6.13) Fix 0 < b < 1. For G, V ∈ (C 1,0 ([0, T ] × Λ)) 2 , ε > 0 and m, n ∈ Z Z + , let I J b,m,n V ,G,ε : D([0, T ], M 1 × M 1 ) → IR ∪ {∞} be the functional given by I J b,m,n V ,G,ε (π) = J V (π * ι ε ) + bI J G (π * ι ε ) if π ∈ E m, 1 n , +∞ otherwise . (6.14) It is lower semicontinuous because so is π → J V (π * ι ǫ ) + bI J G (π * ι ǫ ),
and because E m,1/n is closed.

We now return to inequality (6.12). By Proposition 5.5, the exponential martingale M V ,γ t defined by the Girsanov formula (5.8) satisfies , γ, ε, l, δ, α) . (6.15)

M V ,γ T := dP V ,γ,α σ γ dP γ,α σ γ (σ [0,T ] ) = exp γ -d J V (π γ * ι ε ) + γ -d r(V
We rewrite (6.12) as

γ d log Q γ,α σ γ (K) ≤ 1 1 + b γ d log E Q γ,α σ γ 1I K (π γ )M V ,γ T × M V ,γ T -1 × Z G,γ t -b . Since M V ,γ
T is a mean one positive martingale, taking into account (6.10) and (6.15) and optimizing over π in K, we obtain, for all positive integers m, n, , G, γ, ε, l, δ, α) .

lim sup γ→0 γ d log Q γ,α σ γ (K) ≤ 1 1 + b sup π∈K∩E m, 1 n -J V (π * ι ǫ ) -bI J G (π * ι ǫ ) + lim sup γ→0 r(V , G, γ, ε, l, δ, α) = 1 1 + b sup π∈K -I J b,m,n V ,G,ε (π) + lim sup γ→0 r(V
Optimizing the previous expression with respect to V , G, ε, l, δ, m, n, taking into account (6.11), we get lim sup

γ→0 γ d log Q γ,α σ γ (K) ≤ inf V ,G,ε,l,δ,m,n 1 1 + b sup π∈K -I J b,m,n V ,G,ε (π) . (6.16) Since K is compact and π → 1 1 + b sup π∈K -I J b,m,n V ,G,ε (π) is lower semi-continuous for all V , G and ε, l, δ, m, n,
we may apply the arguments presented in [V], Lemma 11.3 to exchange the supremum with the infimum. In this way we obtain that the right hand side of (6.16) is bounded above by

sup π∈K inf V ,G,ε,l,δ,m,n - 1 1 + b I J b,m,n V ,G,ε (π) .
By (6.13) we have lim sup

ε→0 lim sup m→∞ lim sup n→∞ I J b,m,n V ,G,ε (π) := I J b V ,G (π) = J V (π) + bI J G (π) if π ∈ D([0, T ], M ac 1 × M ac 1 ), +∞ otherwise .
By (6.2) and (6.9) we have that sup

V ,G I J b V ,G (π) = J(π) + bJ (π). Therefore, lim sup γ→0 γ d log Q γ,α σ γ (K) ≤ sup π∈K inf V ,G - 1 1 + b I J b V ,G (π) = - 1 1 + b inf π∈K sup V ,G I J b V ,G (π) = - 1 1 + b inf π∈K J(π) + bJ (π) .
(6.17)

Proof of the upper bound. Let K be a compact set of D [0, T ], M 1 × M 1 . By Proposition 6.5, if J ≡ +∞ on K, then the upper bound of the large deviation principle is satisfied. Otherwise, there exists π ∈ K such that J (π) < ∞. By semicontinuity of the functional π → J (π), we obtain from (6.17) for any 0

< b < 1, lim sup γ→0 γ d log Q γ,α σ γ (K) ≤ - 1 1 + b inf π∈K ,J (π)<∞ J(π) - b 1 + b inf π∈K J (π) .
Letting b → 0, we get lim sup

γ→0 γ d log Q γ,α σ γ (K) ≤ - inf π∈K ,J (π)<∞ J (π) ≤ -inf π∈K I 0 (π) .
For the last inequality we used Lemma 6.4. By Proposition 6.1 the proof of the upper bound of the large deviation principle is completed.

Lower Bound

We first get in Lemma 7.1 a lower estimate for the probability of a neighborhood of suitable trajectories. We perform the computation with the uniform metric on the time interval [0, T ] defined as following: for µ and ν in D([0, T ], M 1 ) and ρ(•, •) defined in (2.9),

ρ [0,T ] (µ, ν) = sup t∈[0,T ] ρ(µ t , ν t ) and ρ [0,T ] (µ, ν) = i=1,2 ρ [0,T ] (µ i , ν i ). (7.1)
Taking into account that if d S [0,T ] (µ, ν) denotes the Skorohod distance, then

d S [0,T ] (µ, ν) ≤ ρ [0,T ] (µ, ν) (7.2),
the result holds for the Skorohod topology as well, see [Bill].

To conclude the proof of the lower bound of the large deviation principle in Theorem 2.9, it will remain to show that all π's such that I 0 (π) < ∞ can be approximated by a sequence (π n ) n of smooth trajectories, for which Lemma 7.1 holds with lim n→∞ I 0 (π n ) = I 0 (π). For this, in Lemma 7.2 we prove that any trajectory m smooth enough and far away from the boundaries (±p 1 , ±p 2 ) is associated to a function V (•, •).

Then, given m 0 ∈ B p1,p2 , denote by R(t, •), t ∈ [0, T ] the solution of (2.24) with R(0, •) = m 0 (•): for i = 1, 2, R i (t, •) = e -t m i (0, •) + p i t 0 e -(t-s) tanh[(J * R)(s, •) + a i θ] ds, (7.3) where R = R 1 + R 2 . It is continuously differentiable in time, actually it is C ∞ in time for t ≥ t 0 > 0,
|R i (t, •)| ≤ e -t m i (0, •) + p i (1 -d) t 0 e -(t-s) ds ≤ p i [1 -d(1 -e -t )]. (7.4)
Recall that I 0 (R) = 0, see (3) of Proposition 3.7. Define the sets:

C 0 = C 0 (m 0 ) = {φ ∈ AC([0, T ], B p1,p2 ) : φ(0) = m 0 , I 0 (φ) < ∞}, (7.5) C 1 = {φ ∈ C 0 : ∃ 0 < η < T, φ(t) = R(t), t ∈ [0, η]}, (7.6) C 2 = {φ ∈ C 1 : ∀η ∈ (0, T ], ∃δ i = δ i (φ) > 0, i = 1, 2 : φ i (t) ∞ ≤ p i -δ i , t ∈ [η, T ]}, (7.7) C 3 = {φ ∈ C 2 : φ i ∈ C 2 ((0, T ], B p1,p2 ), i = 1, 2, φ i (t) ∈ C(Λ), ∀t ∈ (0, T ]}. (7.8) By construction C 3 ⊂ C 2 ⊂ C 1 ⊂ C 0
. By Lemma 7.2 below we can associate a function V to φ ∈ C 3 . To extend the lower bound, we show that for i ∈ {1, 2, 3},

C i is (ρ [0,T ] , I 0 )-dense in C i-1 , that is, for all φ ∈ C i-1 there exists a sequence (φ n ) n ⊂ C i such that lim n→∞ ρ [0,T ] (φ n , φ) = 0, lim n→∞ I 0 (φ n ) = I 0 (φ). (7.9)
This method has been inspired by a similar strategy in [QRV].

Lemma 7.1 Assume (σ γ ) γ , m 0 satisfy (2.13). Let δ > 0 and µ = m V λ, where m V is the solution of (5.5) for

V = (V 1 , V 2 ) ∈ C 1,0 ([0, T ] × Λ) 2 and m V i (0, •) = p i m 0 (•) for i = 1, 2. Then we have, for V δ (µ) = {µ ′ ∈ D([0, T ], M 1 × M 1 ) : ρ [0,T ] (µ, µ ′ ) < δ}, and I m0 given in (2.27), lim inf γ→0 γ d log Q γ,α σ γ (V δ (µ)) ≥ -I m0 (µ), IP -a.s.
Proof. We introduce the perturbed process. By Jensen inequality we get

log Q γ,α σ γ (V δ (µ)) ≥ E V ,γ,α σ γ 1I V δ (µ) π γ [0,T ] log dP γ,α σ γ dP V ,γ,α σ γ (σ [0,T ] ) Q V ,γ,α σ γ (V δ (µ)) -1 + log Q V ,γ,α σ γ (V δ (µ)). By Corollary 5.3, lim γ→0 Q V ,γ,α σ γ (V δ (µ)) = 1. By Lebesgue dominated convergence Theorem, lim inf γ→0 γ d log Q γ,α σ γ (V δ (µ)) ≥ lim inf γ→0 E V ,γ,α σ γ γ d log dP γ,α σ γ dP V ,γ,α σ γ (σ [0,T ] ) .
By Radon-Nikodym formula, see Theorem 5.4, and Proposition 5.5 we have

γ d log dP γ,α σ γ dP V ,γ,α σ γ (σ [0,T ] ) ≥ -ℓ T π γ (σ), V + 1 2 T 0 Γ V (s,•) (π γ s ) ds -ǫ(γl)T -T C(V 1 , V 2 )[δ + i=1,2 E i (δ, l, γ, α)].
From Theorem 5.1, recalling the definition of K V (•) given in (3.8), we get that for any l,

lim inf γ→0 γ d log Q γ,α σ γ (V δ (µ)) ≥ -K V (µ) -T C(V 1 , V 2 )[δ + lim γ→0 i=1,2 E i (δ, l, γ, α)],
which yields the result letting l → ∞ by Lemma 4.2 and Proposition 3.5.

Lemma 7.2 Given m = (m 1 , m 2 ) ∈ C 2,0 ([0, T ] × Λ) 2 , with, for i = 1, 2, |m i (t, r)| < p i , for all t ∈ [0, T ], r ∈ Λ, there exists V = (V 1 , V 2 ) ∈ C 1,0 ([0, T ] × Λ)
2 such that m = m V is the solution of (5.5). For

(t, r) ∈ (0, T ] × Λ, 2V i (t, r) = log ∂ t m i (t, r) cosh [(J * m)(t, r) + a i θ] + (∂ t m i (t, r) cosh [(J * m)(t, r) + a i θ]) 2 + p 2 i -m 2 i (t, r) -[(J * m)(t, r) + a i θ] -log {p i -m i (t, r)}, (7.10)
and for t = 0 we set lim t→0 V i (t, r) = V i (0, r).

Proof. By (5.5), for t ∈ (0, T ], we determine

V (t, •) = (V 1 (t, •), V 2 (t, •)) with V i ∈ C 1,0 ((0, T ] × Λ) for i = 1, 2, such that m = m V . Namely, for (t, r) ∈ (0, T ]× Λ, denoting A i = (J * m)(t, r)+ a i θ, Y i = (cosh A i )∂ t m i (t, r), Z i = -m i (t, r
), (5.5) is written as

Y i = Z i cosh [A i + 2V i (t, r)] + p i sinh [A i + 2V i (t, r)] .
We multiply both sides by X i = e 2Vi(t,r) and obtain

e Ai (Z i + p i )X 2 i -2Y i X i + e -Ai (Z i -p i ) = 0.
Its positive solution is

X i = Y i + Y 2 i -(Z 2 i -p 2 i ) e Ai (Z i + p i ) ,
which gives (7.10). Note that V = (V 1 , V 2 ) has the same spatial regularity as m, namely the argument of the square root is always strictly positive.

Corollary 7.3 If m is solution of (2.24) then V = 0 in (7.10).

Remark 7.4 . Lemma 7.2 could have been stated requiring m ∈ AC([0, T ], B 1,1 ). In this case one would get

V ∈ L 1 ([0, T ], C(Λ))
2 . We prefer to obtain more regularity in time for V , so that uniformity and other technical needs become straightforward.

Lemma 7.5 C 1 is (ρ [0,T ] , I 0 )-dense in C 0 . Proof. Fix m ∈ C 0 . Let R(t, •), t ∈ [0, T ],
be the solution of (2.24) with initial datum R(0, •) = m 0 (•). For any η ∈ (0, T ), define

m η (t, •) =      R(t, •) for t ∈ [0, η], R(2η -t, •) for t ∈ (η, 2η], m(t -2η, •) for t ∈ (2η, T ].
We have m η ∈ C 1 for any 0 < η < T and lim η→0 ρ [0,T ] (m η , m) = 0. Since I 0 is lower semicontinuous it remains to show lim

η→0 I 0 (m η ) ≤ I 0 (m). (7.11)
We split [0, T ] into [0, 2η] and [2η, T ] in the integration. We have that

T 2η Λ H(m η , ṁη )(t, r) drdt = T -2η 0 Λ H(m, ṁ)(t, r) drdt ≤ I 0 (m).
Next we show that lim

η→0 2η 0 Λ H(m η , ṁη )(t, r) drdt = 0. Since m η = R for t ∈ [0, η] solves (2.24), by (3) of Proposition 3.7, η 0 Λ H(R, Ṙ)(t, r) drdt = 0. (7.12)
Since the profile m η in (η, 2η] is the profile in (0, η] backwards in time, we have

2η η Λ H(m η , ṁη )(t, r) drdt = η 0 Λ H(R, -Ṙ)(t, r) drdt.
Because R solves (2.24) and for t > 0,

|R i (t, •)| ≤ p i -δ i , for i = 1, 2, H(R, Ṙ) belongs to L 1 ([0, T ] × Λ), as well as H(R, -Ṙ), see explicit formula (3.15). By dominated convergence, lim η→0 2η η Λ H(m η , ṁη )(t, r) drdt = 0.
In this way we prove (7.11).

Lemma 7.6 C 2 is (ρ [0,T ] , I 0 )-dense in C 1 . Proof. Let m ∈ C 1 and η ∈ (0, T ) so that m(t, •) = R(t, •) for t ∈ [0, η]. By (7.4), m i (η, •) ∞ ≤ p i -δ i for some δ i > 0 and i = 1, 2. Define m n i (t, r) =      m i (t, r) for t ∈ [0, η], m i (η, r) + 1 - 1 n (m i (t, r) -m i (η, r)) for t ∈ (η, T ]. (7.13)
By construction and from (7.12), To apply the Lebesgue dominated convergence Theorem we give an upper bound, uniformly with respect to n, of |H(m n , ∂m n ∂t )(t, r)| (see also [C] p. 174). For that we combine (3.17) with the facts that,

I 0 (m n ) = T η Λ H(m n (t, r), ∂m n ∂t (t, r)) drdt. Moreover, since I 0 (m) < ∞, by Proposition 3.4 we have m i (t) ∞ ≤ p i for t ∈ [η, T ], then m n i (t) ∞ ≤ p i - δ i n , ∀t ∈ [η, T ]. (7.14) Hence m n ∈ C 2 for all n. Furthermore lim n→∞ m n i (t, r) = m i (t, r) and ∂m n i ∂t (t, r) = (1 - 1 n ) ∂m i ∂t → ∂m i ∂t (t, r) for almost all (t, r) ∈ [η, T ] × Λ.
{(t, r) : ∂m n i ∂t > 0} = {(t, r) : ∂m i ∂t > 0},
and on the set {(t, r) :

m n i (t, r) ≥ p i -δ i } we have m i (t, r) -m i (0, r) ≥ 0 and p i -m n i (t, r) ≥ p i -m i (t, r). To get shorter notation, we denote for φ, ψ ∈ AC([0, T ], B p1,p2 ) Υ(φ i , ψ i ) = 1I { φi>0; pi-δi≤ψi} log 1 p i -ψ i + + 1I { φi<0; -pi+δi≥ψi} log 1 p i + ψ i + . We have 2H(m n , ∂m n ∂t )(t, r) ≤ i=1,2 | ṁi | (log | ṁi |) + + Υ( ṁi , m n i ) + K i (t, r) + i=1,2 | ṁi | 1I { ṁi>0; m n i <pi-δi} log 1 p i -m n i + + 1I { ṁi<0; m n i >-pi+δi} log 1 p i + m n i + (t, r) + C ≤ i=1,2 | ṁi | (log | ṁi |) + + Υ( ṁi , m i ) + K i (t, r) + i=1,2 | ṁi | 1I { ṁi>0; m n i <pi-δi} log 1 δ i + 1I { ṁi<0; m n i >-pi+δi} log 1 δ i (t, r) + C ≤ i=1,2 | ṁi | (log | ṁi |) + + 1I { ṁi>0} log 1 p i -m i + + 1I { ṁi<0} log 1 p i + m i + + K i + log 1 δ i (t, r) + C.
Since by assumption I 0 (m) < ∞, by Proposition 3.6, part (b), the above upper bound is integrable. By Lebesgue dominated convergence Theorem we then have

lim n→∞ I 0 (m n ) = I 0 (m). (7.15) Obviously m n → m in the metric (3.2). Lemma 7.7 C 3 is (ρ [0,T ] , I 0 )-dense in C 2 .
Proof. Take ψ ∈ C 2 . To get more regularity we convolve with a smooth kernel the function both in time and space. To perform the convolution in time we extend the definition of ψ to [T, T + 1] by setting, for each s ∈ [0, 1], if u = (u 1 , u 2 ) is the solution of equation (2.24) with initial condition ψ(T, •),

ψ(T + s, r) = u(s, r). (7.16)
Since ψ ∈ C 2 there exist δ i , i = 1, 2, such that |ψ i (T, r)| ≤ p i -δ i . It follows from (7.3) that ψ i (T +s, r) ≤ p i -δi for all s ∈ [0, 1], for some δi smaller than δ i . In the following we will denote it always by δ i . Denote by θ s ψ the time translation of ψ, (θ s ψ)(t, r) = ψ(t + s, r) for (t, r) ∈ [0, T ] × Λ. Let Φ ǫ1 be a smooth non-negative kernel, Φ ǫ1 ∈ C ∞ (Λ) with support in a ball of radius ǫ 1 and integral one which we use as spatial mollifier.

For ǫ 0 > 0, let Ψ ǫ0 be the C ∞ (IR) non-negative temporal mollifier with support [0, ǫ 0 ] and integral one. Set ǫ ≡ (ǫ 0 , ǫ 1 ), ǫ ↓ 0 stands for ǫ 0 ↓ 0 and ǫ 1 ↓ 0. Let η > 0 be such that ψ(t,

•) = R(t, •) for t ∈ [0, 3η]. Let χ 1 (t)
, χ 2 (t) be a C 2 partition of the unity enjoying the properties:

     χ 1 (t) = 1 for t ∈ [0, η], χ 1 (t) = 0 for t ∈ [2η, T ], χ 2 (t) = 0 for t ∈ [0, η], χ 2 (t) = 1 for t ∈ [2η, T ], χ 1 (t) + χ 2 (t) = 1, ∀t ∈ [0, T ]. Let ψ ǫ i (t, •) = χ 1 (t)ψ i (t, •) + χ 2 (t) IR Ψ ǫ0 (s)(Φ ǫ1 * θ s ψ i )(t, •) ds. (7.17)
By construction ψ i (•, •), i = 1, 2, satisfies all the regularity requirements to be in C 3 . Furthermore, since |ψ i (t, r)| ≤ p i -δ i , δ i > 0, for all ǫ > 0 and t ∈ [0, T ], we still have that By using the expression (3.9) of I 0 , see Proposition 3.5, we have (7.22) where the first equality comes from a change of variables, the second one from the definition of ψ in [T, T + 1] (see (7.16)), the third one from Λ dyΦ ǫ1 (y) = 1 and Λ dxH(ψ, ∂ψ ∂t )(t, x -y) = Λ dxH(ψ, ∂ψ ∂t )(t, x), and the last one from IR dsΨ ǫ0 (s) = 1. Therefore

|ψ ǫ i (t, r)| ≤ p i -δ i , i = 1, 2, ( 7
I 0 (ψ ǫ ) -I 0 (ψ) = T 0 Λ H(ψ ǫ , ∂ψ ǫ ∂t )(t,
T 2η Λ H(ψ ǫ , ∂ψ ǫ ∂t )(t, x) -H(ψ, ∂ψ ∂t )(t, x) dxdt = T 2η Λ H(ψ ǫ , ∂ψ ǫ ∂t )(t, x) dxdt - 2η+s 2η Λ H(ψ, ∂ψ ∂t )(t, x) dxdt - T 2η+s Λ H(ψ, ∂ψ ∂t )(t, x) dxdt ≤ IR Ψ ǫ0 (s) T 2η Λ Φ ǫ1 (y) Λ H(ψ ǫ (t,

Appendix A

In this Appendix we give the proofs of the properties of the rate functional stated in Section 3. Proof of Lemma 3.2 The differentiability of Γ V (u) in (L ∞ (Λ)) 2 is easily verified. For the convexity we compute first the Hessian of Γ V (u) with respect to V 1 and V 2 . Since the Hessian is a diagonal matrix, it is enough to study separately the convexity with respect to V 1 and V 2 , we do it for V 1 . For r ∈ Λ, we set V 1 (r) = x, u 1 (r) = m and denote by

f 1 (x) = (p 1 tanh ϑ -m) sinh(2x) + (p 1 -m tanh ϑ)(cosh(2x) -1)
the integrand term in Γ V (u) which depends only on V with ϑ varying in some bounded interval of IR, x ∈ IR, |m| ≤ p 1 . We then study the sign of the second derivative of f 1 .

1 4 f ′′ 1 (x) = cosh(2x)[p 1 -m tanh ϑ] + sinh(2x)[p 1 tanh ϑ -m] .
Notice that p 1 -m tanh ϑ ≥ 0, and p 1 -m tanh ϑ ≥ p 1 tanh ϑ -m ≥ -(p 1 -m tanh ϑ). Since cosh(2x) > | sinh(2x)| and | tanh ϑ| < 1 when ϑ varies in a bounded interval we obtain that f 1 is convex.

Proof of Proposition 3.4 Recalling (3.12), for i = 1, 2, denote

F i (v i ) = g i v i - 1 2 B i (u, v i ) = g i v i -(p i -u i ) e Ai 4 cosh A i [e 2vi -1] -(p i + u i ) e -Ai 4 cosh A i [e -2vi -1].
(8.8)

Hence ∂F i ∂v i = g i -(p i -u i ) e Ai 2 cosh A i e 2vi + (p i + u i ) e -Ai 2 cosh A i e -2vi .
(8.9)

First assume that u i > p i . By (8.8), since -(p i -u i ) > 0, we have lim vi→+∞ F i (v i ) = +∞. In the same way, we get lim vi→-

∞ F i (v i ) = +∞ if u i < -p i because then -(p i + u i ) > 0.
Therefore, (a) holds.

For the remaining cases, we exploit that for u ∈ B p1,p2 , the function

v i → B i (u, v i ) is convex differentiable on IR. (b)
To compute the Legendre transform of B i (u, v i ), when |u i | < p i , by (8.9), the maximum in (3.11) is obtained for (remember (3.13))

e 2vi = e -Ai D i p i -u i , hence v i = 1 2 log D i p i -u i -A i . (8.10)
Inserting (8.10) in (8.8) we have

H i (u, g i ) = g i 2 log D i p i -u i -A i + p i e Ai + e -Ai 4 cosh A i -u i e Ai -e -Ai 4 cosh A i - 1 4 cosh A i D i + p 2 i -u 2 i D i ,
which yields (3.15) since, using (3.13), we write

D i + p 2 i -u 2 i D i = g i cosh A i + R i + (p 2 i -u 2 i ) (g i cosh A i -R i ) (g i cosh A i ) 2 -(g i cosh A i ) 2 -p 2 i + u 2 i = 2R i . (c) When u i = p i (resp. u i = -p i ), ∂F i ∂v i = g i + p i e -sgn(ui)Ai cosh A i e -2sgn(ui)vi
and to solve ∂F i ∂v i = 0 (that is to find a finite extremum) we need g i < 0 (resp. g i > 0), namely

g i = -p i e -sgn(ui)Ai cosh A i e -2sgn(ui)vi .
Inserting this value in (8.8) we get (3.16) when g i = 0. When u i = p i (resp. u i = -p i ) and g i = 0, (8.8) becomes

F i (v i ) = p i e -sgn(ui)Ai 2 cosh A i [1 -e -2sgn(ui)vi ].
It is an increasing (resp. decreasing) function with a finite maximal limit:

lim vi→+∞ F i (v i ) = p i e -sgn(ui)Ai 2 cosh A i = H i (u, g i ), resp. lim vi→-∞ F i (v i ) = p i e -sgn(ui)Ai 2 cosh A i = H i (u, g i ).
(d) When u i = p i and g i > 0 (resp. u i = -p i and g i < 0), (8.8) becomes

F i (v i ) = g i v i + p i e -sgn(ui)Ai 2 cosh A i [1 -e -2sgn(ui)vi ].
Hence lim

vi→+∞ F i (v i ) = +∞ = H i (u, g i ), resp. lim vi→-∞ F i (v i ) = +∞ = H i (u, g i ).
Proof of Proposition 3.6 We use the explicit representation of H(•, •) given in Proposition 3.4.

(a) We give an upper bound of expression (3.15). The difficulty comes from the term (8.11) where D i (u, g i , θ) is defined in (3.13). Let -u = (-u 1 , -u 2 ). We have

F (u, g i , θ) = g i log D i (u, g i , θ) p i -u i ,
F (-u, -g i , θ) = g i log (p i + u i ) g i cosh[(J * u) -a i θ] + (g i cosh[(J * u) -a i θ]) 2 + p 2 i -u 2 i -(g i cosh[(J * u) -a i θ]) 2 + (g i cosh[(J * u) -a i θ]) 2 + p 2 i -u 2 i = g i log g i cosh[(J * u) -a i θ] + (g i cosh[(J * u) -a i θ]) 2 + p 2 i -u 2 i p i -u i = F (u, g i , -θ).
We write F (u, g i , θ) = F (u, g i , θ)1I {gi≥0} + F (-u, -g i , -θ)1I {gi<0} . (8.12)

Hence it suffices to estimate F (u, g i , θ) for g i > 0 and θ ∈ IR. We get

F (u, g i , θ) ≤ |g i | log D i (u, g i , θ) + log 1 p i -u i + 1I {gi>0} + log 1 p i + u i + 1I {gi<0} .
We obtain (3.17) by the upper bound D i (u, g i , θ) ≤ 2|g i | + 1. The lower bound (3.18) is obtained as in [C], p. 171. We rely on formulas (3.11), (3.12). Since e βa ≤ e β|a| , there exists a constant C such that

B i (u, v i ) ≤ 2C[e 2|vi| -1] 2p i ≥ 2C ≥ max (p i -u i ) e Ai 2 cosh A i , (p i + u i ) e -Ai 2 cosh A i .
Then

H i (u, g i ) ≥ sup vi∈IR g i v i -C e 2|vi| -1 = max |g i | 2 log |g i | 2C -1 + C, 0 .
(b) If (3.17) holds then I 0 (φ) < ∞. For the converse, by (3.18), it is necessary to have φi log

| φi | ∈ L 1 ([0, T ] × Λ). To conclude, notice that when g i > 0, uniformly in θ ∈ IR, F (u, g i , θ) -g i log{g i cosh A i } ≥ 2g i 1I {gi>0} log 1 p i -u i .
Proof of Proposition 3.7 For (1), (2) we refer to the similar proof of [C], Theorem III.4, p. 148 (indeed, the rate functional in infinite outside C([0, T ], B 1,1 )). To show the first part of (3), notice that for V = 0 the r.h.s.

of the argument of the sup in (3.5) is equal to zero. This implies that for s ∈ [0, T ], H * (φ(s, •), φ(s, •)) ≥ 0 in (3.6), therefore I 0 (π) ≥ 0. For the second half of (3), we start by proving that if I 0 (π) = 0, then π = (φ 1 λ, φ 2 λ) with φ = (φ 1 , φ 2 ) ∈ AC([0, T ], B 1,1 ) is the solution of equation (2.24). From Proposition 3.5, we know that J 0 (π) = 0 (see (3.7), (3.8)), that is, for any

V = (V 1 , V 2 ) ∈ (L ∞ ([0, T ] × Λ)) 2 , we have T 0 < V (s, •), φ(s, •) > ds ≤ 1 2 T 0 Γ V (s,•) (φ(s, •)) ds.
Now take V 2 = 0 and ηV 1 instead of V 1 , where η > 0. Denote φ = φ 1 + φ 2 , then recalling definitions (3.3) and (3.4), we get 2η

T 0 < V 1 (s, •), φ1 (s, •) > ds ≤ p 1 T 0 < tanh(φ(s, •) * J + a 1 θ) sinh(2ηV 1 (s, •)) + cosh(2ηV 1 (s, •)) -1 > ds - T 0 < φ 1 (s, •) tanh(φ(s, •) * J + a 1 θ)[cosh(2ηV 1 (s, •)) -1] + sinh(2ηV 1 (s, •)) > ds.
Using Taylor expansion in η when η → 0, dividing by η and letting η → 0, we obtain

T 0 < V 1 (s, •), φ1 (s, •) > ds ≤ p 1 T 0 < tanh(φ(s, •) * J + a 1 θ)V 1 (s, •) > ds - T 0 < φ 1 (s, •)V 1 (s, •) > ds.
Since all terms in the previous expression are linear in V 1 , we may change V 1 into -V 1 to obtain the converse inequality. Then, exchanging the roles of indices 1 and 2, we have, for i = 1, 2,

T 0 < V i (s, •), φi (s, •) > ds = p i T 0 < tanh(φ(s, •) * J + a i θ)V i (s, •) > ds - T 0 < φ i (s, •)V i (s, •) > ds.
This means that φ is the (unique) weak solution of (2.24), since by definition (2.27) of the rate functional the initial condition is fulfilled.

For the reverse, we prove that if φ ∈ AC([0, T ], B 1,1 ) is the solution of equation (2.24), then π = (φ 1 λ, φ 2 λ) is such that J 1 (π) = 0; hence, by Proposition 3.5, I 0 (π) = 0. We insert equation (2.24) into the explicit representation (3.15). Namely if π solves (2.24) then, by Corollary 7.3,

log D i p i -φ i -A i = 0, (8.13) R i = D i -(∂ t φ i ) cosh A i = e Ai (p i -φ i ) + (φ i -p i tanh A i ) cosh A i . Hence p i -φ i tanh A i - R i cosh A i = (p i -φ i )(1 + tanh A i - e Ai cosh A i ) = 0. (8.14)
By (8.13), (8.14), the right hand side of (3.15) is equal to zero, which completes the proof of (3).

Appendix B

This appendix is devoted to proofs postponed from Sections 5, 6 and 7.

Proof of Proposition 5.5 Let s ∈ [0, T ],

C(V 1 , V 2 ) = i=1,2 C(V i ) = i=1,2 sup s∈[0,T ] sup r∈Λ sinh 2 [V i (s, r)] + | sinh[2V i (s, r)]| . Then F V (s) (λ γ (α), π γ s ) -F V (s) ((p 1 λ γ , p 2 λ γ ), π γ s ) ≤ γ d 2 x∈Λγ [α i (x) -p i ] B i (x, σ, s) , with B i (x, σ, s) = cosh[2V i (s, γx)] -1 + tanh[(J γ ⋆ σ s )(x) + a i θ] sinh[2V i (s, γx)] |B i (x, σ, s)| ≤ sinh 2 [V i (s, γx)] + | sinh[2V i (s, γx)]| ≤ C(V i ). Take l ∈ Z Z, l = 0. Since IE[α i (x)] = p i for all x ∈ Λ γ , γ d 2 x∈Λγ [α i (x) -p i ] B i (x, σ, s) = γ d 2 x∈Λγ 1 (2l + 1) d |y|≤l [B i (x + y, σ, s) -B i (x, σ, s)] α i (x + y) - γ d 2 x∈Λγ B i (x, σ, s) α (l) i (x) -IE[α i (x)] . (9.1)
Using uniform continuity as in the proof of Lemma 4.3, there exists a positive function ǫ on IR + with lim s→0 ǫ(s) = 0 (depending only on T , J and V ) such that the first term on the r.h.s. of (9.1) is bounded uniformly in α and σ; for the second term, let δ > 0 and E i (δ, l, γ, α) defined in (4.3). We conclude by

γ d 2 x∈Λγ [α i (x) -p i ] B i (x, σ, s) ≤ ǫ(γl) + C(V i ) [δ + E i (δ, l, γ, α)] .
Proof of Proposition 6.1 Consider a sequence of functions {H k } k≥1 in C 2 (Λ) dense in C(Λ) for the uniform topology with H k ∞ ≤ 1. Denote for all integers m ≥ 1, ℓ ≥ 1, and δ > 0,

A m,δ,ℓ = {π ∈ D([0, T ], M 1 × M 1 ) : inf {t ′ i } max i sup t ′ i ≤s<t ′ i+1 m j=1 | < π t , H j > -< π s , H j > | 4 j ≤ ℓ + 1 m },
where the infimum extends over all positive integers K and all finite sets of points {t

′ i , 0 ≤ i ≤ K} satisfying 0 = t ′ 0 < t ′ 1 < . . . < t ′ K = T , t ′ i+1 -t ′ i > δ.
We first show that for m ≥ 1 and for ℓ ≥ 1 there exists δ(m, ℓ) and γ 0 (m, ℓ) so that for all γ ≤ γ 0 ,

Q γ,α σ γ [π / ∈ A m,δ(m,ℓ),ℓ ] ≤ e -ℓ+1 γ d .
This is done taking into account that

{ inf {t ′ i } max i sup t ′ i ≤s<t ′ i+1 | < π t , H > -< π s , H > | ≥ ℓ + 1 m } ⊂ ∪ T δ k=0 { sup kδ≤t<(k+1)δ | < π t , H > -< π kδ , H > | ≥ ℓ + 1 4m },
and estimating the right hand side as in [KL] (p. 271, after formula (4.6)). Then the construction of the compact K ℓ is obtained by a general procedure as explained in [Bill] and Section 8 of [QRV].

Next lemma states some technical results needed in the extension of the lower bound of the large deviation principle.

Lemma 9.1 For (t, t ′ ) ∈ [0, T ] 2 , (x, y) ∈ Λ 2 , δ 0 > 0, let τ ∈ {t, t ′ }, ζ ∈ {x, y}. We assume (u(τ, •), v(τ, •)) ∈ (B p1-δ0,p2-δ0 ) 2 , g i (τ, ζ) ∈ IR and h i (τ, ζ) ∈ IR, for i = 1, 2; we have |H i (u(t, x), g i (t ′ , y)) -H i (v(t, x), g i (t ′ , y))| ≤ |g i (t ′ , y)| 1 + K δ 0 ( v(t, •) -u(t, •) 1 + |u i (t, x) -v i (t, x)|) , (9.2) |H i (v(t, x), g i (t ′ , y)) -H i (v(t ′ , y), g i (t ′ , y))| ≤ |g i (t ′ , y)| 1 + K δ 0 (|x -y| + |v i (t, x) -v i (t ′ , y)|) , (9.3) |H i (u(t, x), g i (t, x)) -H i (u(t, x), h i (t, x))| ≤ K + log 1 δ 0 + K|g i (t, x)| |g i (t, x) -h i (t, x)|, (9.4)
where the constant K = K(J, θ) may change from one occurrence to the other.

Proof. The assumptions enable to prove (9.2)-(9.4) by writing formula (3.15) for H i , using (3.13) for A i , D i , R i . The latter depend on u or v, g i or h i , (t, x) or (t ′ , y). In each computation, we stress the dependence on the involved quantities, writing e.g. A i (u) for (9.2), R i (g i ) for (9.4). Notice that, unlike in (3.15), those functions depend not only on (t, x), but on (t, x) and (t ′ , y); this does not change the expression of H i , since (3.15) was established pointwise in the proof of Proposition 3.4. In the intermediate computations, we omit to write (t, x), (t ′ , y).

We begin with auxiliary estimates. For (9.4), notice that |(J * u)(t, x)| ≤ 1 (since J(r) dr = 1), and (9.9) (9.11) as well as its consequence analogous to (9.9). The proofs of (9.2), (9.3) go along the same scheme. Namely all the estimates are done pointwise, they rely respectively on (9.7) to (9.11), the other changes being straightforward starting from expressions analogous to (9.12) below. Hence we detail only the proof of (9.2). We have 2 [H i (u(t, x), g i (t ′ , y)) -H i (v(t, x), g i (t ′ , y))]

|A i (u)| ≤ K(J, θ). (9.5) When |v i | ≤ p i -δ 0 and g i > 0, we have p 2 i -v 2 i = (p i -|v i |)(p i + |v i |) ≥ p i δ 0 , hence g i K(J, θ) + 1 ≥ R i (u, g i ) ≥ g 2 i + p i δ 0 ≥ max(g i , p i δ 0 ), (9.6) g i K(J, θ) + 1 ≥ D i (v, g i ) ≥ g i +
|cosh[A i (u)] -cosh[A i (v)]| ≤ K(J, θ) u(t, •) -v(t, •) 1 . ( 9
= g i (t (9.13) To this aim, see (8.11)-(8.12), it is enough to estimate, when g i (t ′ , y) > 0 and uniformly for θ ∈ IR, |F (u(t, x), g i (t ′ , y), θ) -F (v(t, x), g i (t ′ , y), θ)| = g i (t ′ , y) log D i (u)(t, x) p i -u i (t, x) -log D i (v)(t, x) p i -v i (t, x)

≤ g i log p i -v i p i -u i + log D i (u) D i (v) ≤ g i u i -v i p i -u i + g i D i (u) -D i (v) D i (v)
, (9.14) because | log(1 + a)| ≤ log(1 + |a|) ≤ |a|. By (9.7), g i ≤ D i (v). Using also (9.10) we get

g i D i (u) -D i (v) D i (v) ≤ g 2 i (cosh[A i (u)] -cosh[A i (v)]) D i (v) + g i R i (u) -R i (v) D i (v)
≤ g i K(J, θ) u(t, .) -v(t, .) 1 + |R i (u) -R i (v)| .

(9.15)

To estimate the second term on the right hand side of (9.15), we apply (9.5), (9.6), (9.10) and obtain

|R i (u) -R i (v)| = [R i (u)] 2 -[R i (v)] 2 R i (u) + R i (v) ≤ g 2 i cosh 2 [A i (u)] -cosh 2 [A i (v)] R i (u) + R i (v) + v 2 i -u 2 i R i (u) + R i (v) ≤ g 2 i K(J, θ) u(t, •) -v(t, •) 1 R i (u) + R i (v) + v 2 i -u 2 i R i (u) + R i (v) ≤ |g i |K(J, θ) u(t, •) -v(t, •) 1 + p i √ p i δ 0 |u i -v i |.
(9.16)

Combining (9.14), (9.16) we obtain (9.13). Next we estimate the last term of (9.12). Taking into account (9.10) and (9.16) we have

R i (v) cosh[A i (v)] - R i (u) cosh[A i (u)] = R i (v) -R i (u) cosh[A i (v)] + R i (u) cosh[A i (u)] -cosh[A i (v)] cosh[A i (v)] cosh[A i (u)] ≤ |R i (v) -R i (u)| + R i (u) |cosh[A i (u)] -cosh[A i (v)]| ≤ p i δ 0 |u i -v i | + (K(J, θ)|g i | + K(J, θ)[|g i | + 1]) v(t, •) -u(t, •) 1 .
(9.17)

Finally, combining (9.8), (9.9), (9.13), (9.17) yields (9.2). We now derive (9.4) in a similar way:

2 [H i (u(t, x), g i (t, x)) -H i (u(t, x), h i (t, x))]

= [h i (t, x) -g i (t, x)] A i (u)(t, x) + log 1 p i -u i (t, x) + g i (t, x) log D i (u, g i )(t, x) -h i (t, x) log D i (u, h i )(t, x)

+ R i (u, h i )(t, x) -R i (u, g i )(t, x) cosh[A i (u)(t, x)] .
We have, restricting ourselves to g i > 0, h i > 0, see (8.11)-(8.12), and using first | log(1 + a)| ≤ |a| as in (9.14), then (9.7),

|g i log D i (g i ) -h i log D i (h i )| ≤ |g i -h i | |log D i (g i )| + h i |log D i (g i ) -log D i (h i )| ≤ |g i -h i | |log D i (g i )| + h i D i (g i ) -D i (h i ) D i (h i ) ≤ |g i -h i | |log D i (g i )| + h i cosh[A i (u)] D i (h i ) + h i R i (g i ) -R i (h i ) D i (h i ) ≤ |g i -h i |(g i + 1)K(J, θ) + |R i (g i ) -R i (h i )|.
Then, as in (9.16),

R i (g i ) -R i (h i ) = R 2 i (g i ) -R 2 i (h i ) R i (g i ) + R i (h i ) = g 2 i -h 2 i (cosh[A i (u)]) 2 R i (g i ) + R i (h i ) .
Therefore, using that |u i | ≤ p i -δ 0 , and (9.6), Therefore for all η letting ǫ → 0 the term in the right hand side of (9.19) goes to zero. For the second term in the r.h.s. of (9.18), applying (9.4) in Lemma 9.1 and taking into account (9.20), we get the result.

2 |H i (u, g i ) -H i (u, h i )| ≤ |h i -g i | K(J, θ) + log 1 δ 0 + (g i + 1)K(J, θ) + 2[K(J, θ)]

=

  t≥0 were given in (4.7) and (4.8). Using the same arguments as in Proposition 5.5, by smoothness of G and π γ * J, a spatial summation by parts and Taylor expansion permit to rewrite the martingale Z exp γ -d I J G (π γ * ι ε ) + γ -d r(G, γ, ε, l, δ, α) , (6.10) where 0 < γ, ε, δ < 1, l is a positive integer. Here and in the sequel, r(G, γ, ε, l, δ, α) (resp. r(G, V , γ, ε, l, δ, α) later on) stands for some random variable γ, ε, l, δ, α) = 0, IP -a-e . (6.11)Let K be a compact set of D [0, T ], M 1 × M 1 . By Hölder inequality,

  and there exists δ i which depends on T such that |R i (t, •)| ≤ p i -δ i for t ∈ [t 0 , T ]. Namely, since | tanh z| ≤ 1 -d, for |z| ≤ K(β, θ) with 1 > d = d(β, θ) > 0, we have, for t ∈ [0, T ],

  .10) Respectively for (9.3), we need, since |v(•, •)| ≤ 1,|(J * v)(t, x) -(J * v)(t ′ , y)| ≤ |(J * v)(t, x) -(J * v)(t, y)| + |(J * v)(t, y) -(J * v)(t ′ , y)| ≤ J ′ ∞ |x -y| + | Λ J(y -z)[v(t ′ , z) -v(t, z)] dz| ≤ J ′ ∞ (|x -y| + v(t ′ , •) -v(t, •) 1 ) .

2 .

 2 Proof of Lemma 7.8 We exploit that ψ ∈ AC([0, T ], B p1,p2 ) and ψ is differentiable in time in (T, T + 1] (see (7.16)), hence ∂ψ i ∂t ∈ L 1 ([0, T + 1] × Λ). Therefore for A > 0 andD A = {x ∈ Λ : sup s, x)|1I DA (x) dxdt = 0 .By (9.2) of Lemma 9.1, we obtain, for 2ν 0 = min{δ 1 , δ 2 }, splitting Λ = D A ∪ D c A ,

  The inequality holds by (7.21), and because H(ψ, ∂ψ ∂t )(t, x) ≥ 0. Finally we use (7.22). To estimate the last difference in (7.23) we add and subtract to it the term The proofs of the next two Lemmas are postponed to Appendix B.

					IR	dsΨ ǫ0 (s)	T 2η	dt	Λ	Φ ǫ1 (y)dy	Λ	dxH(ψ(t, x),	∂ψ ∂t	(t + s, x -y)),
	which gives
						T 2η Λ		H(ψ	ǫ ,	∂ψ ∂t ǫ	)(t, x) -H(ψ,	∂ψ ∂t	)(t, x) dxdt ≤ W 1 + W 2 ,
	where
	W 1 =	IR	Ψ ǫ0 (s)	T 2η Λ	Φ ǫ1 (y)	Λ		H(ψ	ǫ (t, x),	∂ψ ∂t	(t + s, x -y)) -H(ψ(t, x),	∂ψ ∂t	(t + s, x -y)) dxdydtds,
	W 2 =	IR	Ψ ǫ0 (s)	T 2η Λ	Φ ǫ1 (y)	Λ		H(ψ(t, x),	∂ψ ∂t	(t + s, x -y)) -H(ψ,	∂ψ ∂t	)(t + s, x -y) dxdydtds.
													(7.24)
	Taking into account Lemma 7.8 below we get the result.
	Lemma 7.8
													lim ǫ→0	|W i | = 0,	i = 1, 2.
	Lemma 7.9
							lim ǫ→0	η	2η		Λ	H(ψ	ǫ ,	∂ψ ∂t ǫ	)(t, x) -H(ψ,	∂ψ ∂t	)(t, x) dxdt = 0.
													(7.23)
													x),	∂ψ ∂t	(t + s, x -y)) dxdydtds
	-	IR	Ψ ǫ0 (s)	T 2η Λ	Φ ǫ1 (y)	Λ	H(ψ,	∂ψ ∂t	)(t + s, x -y) dxdydtds.

  max(g i , p i δ 0 ).≤ |v i (t, x) -u i (t, x)|| tanh[A i (v)(t, x)]| + |u i (t, x)|| tanh[A i (v)(t, x)] -tanh[A i (u)(t, x)]| ≤ |v i (t, x) -u i (t, x)| + J ∞ u(t, •) -v(t, •) 1 .

	(9.7)
	For (9.2), we need

|(J * u)(t, x) -(J * v)(t, x)| ≤ J ∞ u(t, •) -v(t, •) 1 (9.8)

and its consequences

|v i (t, x) tanh[A i (v)(t, x)] -u i (t, x) tanh[A i (u)(t, x)]|

  ′ , y)[(J * v)(t, x) -(J * u)(t, x)] + g i (t ′ , y) log D i (u)(t, x) p i -u i (t, x) -log D i (v)(t, x) p i -v i (t, x) + v i (t, x) tanh[A i (v)(t, x)] -u i (t, x) tanh[A i (u)(t, x)] + R i (v)(t, x) cosh[A i (v)(t, x)] -R i (u)(t, x) cosh[A i (u)(t, x)] . (t, x)-v i (t, x)|(1+|g i (t ′ , y)|)+|g i (t ′ , y)| u(t, .)-v(t, .) 1 .

							(9.12)
	Next we show
	g i (t ′ , y) log	D i (u)(t, x) p i -u i (t, x)	-log	D i (v)(t, x) p i -v i (t, x)	≤	K δ 0	|u i

  -ψ i (t, x)| + ψ ǫ i (t, •) -ψ i (t, •) 1 } dxdtdyds where we noticed that {|ψ ǫ i (t, x) -ψ i (t, x)| + ψ ǫ i (t, •) -ψ i (t, •) 1 } ≤ 4. Letting first ǫ → 0 then A → ∞, we get lim ǫ→0 |W 1 | = 0.Next we estimate W 2 . We apply (9.3) of Lemma 9.1. More preciselyH(ψ(t + s, x -y), ∂ψ ∂t (t + s, x -y)) -H(ψ(t, x), ∂ψ ∂t (t + s, x -y)) {(|ψ i (t + s, x -y) -ψ i (t, x)|) + |y|} . i (t + s, x -y) -ψ i (t, x)|) + |y|} dxdydtds -ψ i (t, x)| dxds = 0, letting ǫ → 0 and then A → ∞ we obtain lim ǫ→0 |W 2 | = 0. (t, •) -ψ(t, •) 1 + |ψ i (t, x) -ψ ǫ i (t, x)| dxdt,(9.19)where ν 0 = min{δ 1 , δ 2 }. Note that

					≤	2 i=1	∂ψ i ∂t	(t + s, x -y) 1 +	K ν 0
	As before take A > 0 large enough, split Λ = D A ∪ D c A , to get
	|W 2 | ≤A 1 +	K ν 0	2 i=1 IR	Ψ ǫ0 (s)	T 2η Λ	Φ ǫ1 (y)
			+ C	2 i=1	T 2η Λ IR	Ψ ǫ0 (s)1I DA (x)	∂ψ i ∂t	(t + s, x) dsdxdt.
	Since
			lim ǫ1→0 Λ	Φ ǫ1 (y)|y|dy = 0 , lim ǫ0→0 IR	xΨ ǫ0 (s)
	Proof of Lemma 7.9 We have
									η	2η	Λ	H(ψ	ǫ ,	∂ψ ∂t ǫ	)(t, x) -H(ψ,	∂ψ ∂t	)(t, x) dxdt
									=	η	2η	Λ	H(ψ	ǫ ,	∂ψ ∂t ǫ	)(t, x) -H(ψ,	ǫ ∂t ∂ψ	)(t, x) dxdt	(9.18)
									+	η	2η	Λ	H(ψ,	∂ψ ∂t ǫ	)(t, x) -H(ψ,	∂ψ ∂t	)(t, x) dxdt.
	The first term is estimated by applying (9.2) of Lemma 9.1. We have
	η	2η	Λ		H(ψ	ǫ ,	∂ψ ∂t ǫ	)(t, x) -H(ψ,	ǫ ∂t ∂ψ	)(t, x) dxdt
	≤	i=1,2	η	2η	Λ	1 +	K ν 0	|	∂ψ ǫ i (t, x) ∂t	| ψ
												T 2η Λ	2 i=1	|	∂ψ i ∂t	(t + s, x -y)|(1I DA (x -y) + 1I D c A (x -y))(1 +	K ν 0	)
	{|ψ ǫ i (t, x) ≤ A(1 + (t, •) = χ 1 (t) ∂ψ i ∂t (t, •) + χ 2 (t) 2 K ν 0 ) sup t∈[0,T ] i=1 + 4 where we denote χ ′ ∂ψ ǫ i ∂t i (t) = d dt χ i (t), for i = 1, 2, and we use that χ ′ IR Ψ ǫ0 (s)(Φ ǫ1 * (θ s ∂ψ i ∂t )(t, •) + χ ′ 1 (t)[ψ i (t, •) -ψ ǫ i (t, •)] ds, t ∈ [0, T ], Λ |ψ ǫ i (t, x) -ψ i (t, x)| dx + ψ ǫ i (t, •) -ψ i (t, •) 1 2 (t) = -χ ′ 1 (t). Since for t ∈ (0, 3η), ψ(t) = R(t) solves (2.24) we have 2 i=1 T 2η Λ | ∂ψ i (t + s, x)|1I DA (x) dxdt, sup ∂t t∈[η,2η]

Λ

{(|ψ

Λ |ψ i (t + s, x -y) ǫ

Acknowledgements. We thank Errico Presutti for helpful discussions. E. O. acknowledges the warm hospitality of Université de Rouen and Université Paris Descartes. Part of this work was done during the authors' stay at Institut Henri Poincaré, Centre Émile Borel (whose hospitality is acknowledged), for the semester "Interacting Particle Systems, Statistical Mechanics and Probability Theory".

* work supported by ANR-07-BLAN-0230, ANR-2010-BLAN-0108, GDRE GREFI-MEFI, INDAM-CNRS, Prin07: 20078XY-