Propagation of Gibbsianness for infinite-dimensional gradient Brownian diffusions

HAL Id: hal-00003585

https://hal.archives-ouvertes.fr/hal-00003585

Preprint submitted on 15 Dec 2004

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires

Propagation of Gibbsianness for infinite-dimensional gradient Brownian diffusions

David Dereudre, Sylvie Roelly

To cite this version:

David Dereudre, Sylvie Roelly. Propagation of Gibbsianness for infinite-dimensional gradient Brown- ian diffusions. 2004. �hal-00003585�

Propagation of Gibbsianness for infinite-dimensional gradient Brownian diffusions.

DavidDereudre* Sylvie Rœlly**

Abstract

We study the (strong-)Gibbsian character on R^Zd of the law at timet of an infinite- dimensional gradient Brownian diffusion , when the initial distribution is Gibbsian.

AMS Classifications: 60G15 - 60G60 - 60H10 - 60J60.

KEY-WORDS: infinite-dimensional Brownian diffusion, Gibbs measure, cluster expansion.

∗Universit´e de Valenciennes et du Hainaut-Cambr´esis, Le Mont Houy, 59313 Valenciennes Cedex 09, France

∗∗ Institut f¨ur Mathematik der Universit¨at Potsdam, Postfach 601553, 14415 Potsdam, Germany;

e-mail : [email protected] ; phone number : (49) 331 977 1478 ; fax number : (49) 331 977 1001 On leave of absence Centre de Math´ematiques Appliqu´ees, UMR C.N.R.S. 7641, ´Ecole Polytechnique, 91128 Palaiseau Cedex, France.

1 Introduction

Let I be a finite index set and X = (X_i(t), i ∈ I, t ∈ [0, T]) be an R^I-valued diffusion which is the solution of the following finite-dimensional stochastic differential equation (s.d.e.)

dX_i(t) =dB_i(t)−1

2∇ih(X(t))dt , i∈I, t∈[0, T] (1) where h is a smooth function from R^I into R and (B_i)_i∈I is a sequence of independent, real-valued Brownian motions. This stochastic dynamics corresponds to a perturbation by gradient interactions in the form of drift terms of a sequence of finitely many Brownian free dynamics.

The stationary measures for (1) are proportional to the measure µ(dx) = exp(−h(x))⊗i∈Idxi.

It is well known that, for h sufficiently regular, if the initial distribution ν of the system (1) is not the stationary one but absolutely continuous with respect to Lebesgue measure, this property propagates, that is : at each time t > 0 the law ν^t of X(t) ∈ R^I remains absolutely continuous (with density given by exp−h^tfor some functionh^t).

When I is replaced byZ^d and the dynamics (1) is generalised in a natural way (see (4)), the question whether the global absolute continuity of the initial distribution propagates is irrelevant since stationary measures like Gibbs measures are no more globally absolutely continuous with respect to the infinite product of Lebesgue measure, but only locally absolutely continuous. Now, the question which is relevant is the following :

does Gibbsianness of the initial measure propagate?

In fact, at each fixed time t the law ofX(t) can behave badly in the sense that the sum of the interactions between the (infinitely many) components can explode. So, to obtain a positive answer to the above question, we restrict our study to two particular regimes which can be better controlled. In Section 3, we present the propagation of Gibbsianness for small times t, and in Section 4, we analyse the case of small interactions between the coordinates - but arbitrary times.

To our knowledge, these results are new ( but see also [18]) which are related to the propagation of Gibbsianness under a stochastic evolution like a diffusion with values in a continuous space (here the state space is the infinite-dimensional vector space R^Zd) . We were inspired by the nice work of van Enter, Fernandez, den Hollander and Redig, who consider in [8] the question of possible loss and recovery of Gibbsianness in the context of Interacting Particle Systems with values in {−1,+1}^Zd which follow a high-temperature Glauber dynamics. They treat several cases and can exhibit situations where the pro- cess at time t is strong Gibbsian (in a sense to be defined below), and other situations where it is not. See also [19] for related results for Kawasaki dynamics. Unfortunately, since our state space R^Zd is unbounded, we cannot use all the criteria they have at their disposition (in particular, the criterion of non-Gibbsianness contained in [11]) to test the Gibbsianness/non-Gibbsianness of ν^t. So our present results only concern situations for which the Gibbsianness is conserved. We hope to extend them soon to some non-Gibbsian example.

2 Gibbs measures and infinite-dimensional gradient diffu- sions : the framework of our study

Let us first introduce some definitions and notations.

An interaction potential - or interaction - φ on R^Zd is a collection of functions φ_Λ from R^Zd into R∪ {+∞} where Λ varies in the set of finite subsets of Z^d. Each φ_Λ is supposed to be measurable with respect to FΛ, the σ-algebra generated by the canonical projection on R^Λ; that is for any x∈R^Zd,

φ_Λ(x) =φ_Λ(x_Λ) where x_Λ is the projection onR^Λ of x.

The interaction φis said to be of finite rangeif it satisfies : (FR) ∃r >0, diameter Λ> r=⇒φ_Λ≡0

The interaction φis said to beregular boundedif it satisfies : (RB) ∀Λ, φ_Λ is C³, bounded with bounded derivatives.

The interaction φis said to beabsolutely summable if it satisfies : (AS) ∀i∈Z^d,P

Λ∋ikφ_Λk∞=P

Λ∋isup_x∈R^Zd|φ_Λ(x)|<+∞

When an interactionφis (AS) one can define the collectionh^φ= (h^φ_Λ)_Λ⊂Zdof associated Hamiltonian functions on R^Zd by

h^φ_Λ= X

Λ^′:Λ^′∩Λ6=∅

φ_Λ^′. (2)

More generally, we note for x, y∈R^Zd and Λ,∆⊂Z^d h^φ_Λ,∆(x, y) = X

Λ^′:Λ^′∩Λ6=∅ Λ^′⊂Λ∪∆

φ_Λ^′(x_Λy_∆\Λ),

where x_Λy_∆\Λ is the element inR^Zd equal tox on Λ,y on ∆\Λ and 0 outside of Λ∪∆.

For example, h^φ_Λ,Zd(x, x) coincides withh^φ_Λ(x). Furthermore,h^φ_Λ,∅(x) =P

Λ^′⊂Λφ_Λ^′(x_Λ^′) is a function of x_Λ. To recall this property, we will sometimes denote it byh^φ_Λ,∅(x_Λ).

In fact, as soon as the series on the right-hand side of (2) converges pointwise, one can define a Hamiltonian function associated to a (possibly non absolutely) summable interaction. To simplify we will always denote by h^φ_i the function h^φ_{i} (i ∈ Z^d), by h^φ_Λ,∆(x) the functionh^φ_Λ,∆(x, x) (Λ,∆∈Z^d, x∈R^Zd).

We call ρ a Gibbsian measure on R^Zd associated to the reference measure m and to an interaction φfor which the series (2) converges if it satisfies the system of Dobrushin- Lanford-Ruelle (DLR) equations :

ρ(dx_i/x_j, j6=i) = 1 zi

exp− h^φ_i(x)

m(dx_i), i∈Z^d.

The set of such measures will be denoted by G(φ, m). (For general references on Gibbs measures, see [13] and [24].)

The measureρ will be called strong Gibbsianif the associated interaction is absolutely summable, i.e. satisfies (AS).

Let ϕ be a so-called dynamical interaction on R^Zd, having a C²-regularity and satisfying (FR). The associated hamilton function h^ϕ_i, denoted by hi to simplify, is also C². We can now consider the following infinite-dimensional system given by :

dX_i(t) =dB_i(t)− ¹₂∇ih_i(X(t))dt , i∈Z^d, t∈[0, T],

X(0)≃ν (3)

whereν is a probability measure onR^Zd. We will precise in Section 3 (resp. Section 4 ) the exact assumptions on h and ν we take to assure that the infinite-dimensional stochastic system (3) has a unique strong Markovian solution X with values in the infinite product of continuous trajectories ΩT =C([0, T],R)^Zd. Deuschel described in ([5],[6]) the Gibbsian structure on the path space Ω_T of the law Q^ν, when the initial distribution ν itself is Gibbsian (associated to an initial interaction ϕ). Later, this result was completed and˜ generalised in [2], by showing a bijection between the set of Gibbs measures associated to the initial interaction ˜ϕ on R^Zd and a set of Gibbs measures on the path space Ω_T describing the full dynamics. Having a Gibbs representation ofQ^ν on the path level (even a strong Gibbsian one, see [4] Corollary 2) , we would like to know if at each time t, the law ν^t ofX(t), a probability measure on R^Zd, remains strong Gibbsian. Clearly, ν^t is the projection at time t of Q^ν, but projections are maps which do not conserve a priori the Gibbsianness (see the famous examples of [9], and also [10], [11] amoung others). In [2], we remarked that, projecting at time 0 a general strong Gibbs measure on the path space, the image measure which is obtained on the state space preserves a Gibbsian form in the following weak sense : it is associated to a modification (cf. [13] Section 1.3, for the exact definition), roughly speaking to a family of compatible local densities with respect to a reference measure. But the regularity of the density and the existence of an underlying nice interaction potential is completely unclear. In the Remarks after Proposition 2.5 in [2], we referred the reader to the work of Kozlov to clarify this question. This is the object of this paper, not only for the projection at time 0 but also at time t >0.

The challenge is to control the evolution of an initial absolutely summable interaction

˜

ϕunder the dynamics (3). It is clear that, even if ˜ϕis of finite range this property imme- diately disappears for any time t > 0 since the Brownian motions carry instantaneously information between all the coordinates. So, to assure that at time t, the process is still Gibbsian and associated to a ”good” interaction, i.e. an absolutely summable one, we are obliged to restrict our study to two cases; first for small times t, which implies that the process stays close to the initial Gibbsian condition. Secondly, for small dynamical interactionϕbetween the coordinates, which assures that the sum of the initial interaction and the interaction induced by the dynamics does not explode.

3 Propagation of Gibbsianness during a short stochastic diffusive evolution

Let us consider the infinite-dimensional gradient system (3) introduced in Section 2 where the initial distribution is Gibbsian. We have the following result.

Theorem 1 Let Q^ν be the law on Ω = C(R⁺,R)^Zd of the infinite-dimensional diffusion solution of

dX_i(t) =dB_i(t)−¹₂∇ih_i(X(t))dt , i∈Z^d, t >0,

X(0)≃ν (4)

where ν ∈ G( ˜ϕ, dx) with support included in l¹(γ), with γ = (e^−α|i|)_i∈Z^d, α > 0. Let us moreover suppose that

• the initial interaction ϕ˜is of finite range (FR) and each ϕ˜Λ is Lipschitz continuous (uniformly in Λ)

• the dynamical interaction ϕis of finite range (FR), and each ϕ_Λ isC² with bounded derivatives of order 1 and 2 (uniformly in Λ).

Then, there exists a time t₀ >0 depending only on ϕ˜ andϕ such that, for any t≤t₀, {ν^t=Q^ν◦X(t)⁻¹ :ν ∈ G( ˜ϕ, dx)} ⊂ G(ϕ^t, dx)

where ϕ^t is an absolutely summable (AS) interaction depending only on the initial and dynamical interactions ϕ˜ and ϕ.

Remark 1 One can make explicit some additional assumptions on ϕ˜ in order to assure that G( ˜ϕ, dx) contains at least one measure with support included in l¹(γ). For example suppose there exists a >0, b≥0 such that for eachi∈Z^d

(i) ∀x∈R, x∇iϕ˜_i(x)≥a|x| −b (ii) a > X

#Λ>1Λ∋i

k∇iϕ˜Λk∞.

Then there existsν∈ G( ˜ϕ, dx)satisfyingR

kxkγν(dx)<+∞wherekxkγ=:P

i∈Z^d|x_i|e^−α|i|. This obviously implies that ν{x:kxkγ<+∞}= 1.

The Proof of Remark 1 is postponed to the end of the section.

Proof of Theorem 1 :

The proof is based on an approximation of ν^t by a sequence of probability measuresν_Λ^t, which are the laws at time t of finite-dimensional systems. It will be relatively easy to obtain a Gibbs representation for each ν_Λ^t. But the delicate point will be the convergence of their associated Hamiltonian functions towards a limiting function, which will be a good candidate as Hamiltonian function associated to ν^t.

Let us first recall a representation theorem, which we will use for the initial Gibbs measure ν (Theorems 7.12 and 7.26 in [13]).

Lemma 1 The probability measure ν, like every element of G( ˜ϕ, dx), is a mixture of elements of exG( ˜ϕ, dx), where exG( ˜ϕ, dx) is the set of extremal Gibbs measures µwhich are characterized by the following property : there exists y∈R^Zd such that

µ= lim

ΛրZ^d

µ_Λ,y⊗δ_yΛc where µ_Λ,y(dx_Λ) = 1 Z˜Λ,y

e^{−˜hΛ,Λc(x,y)}dx_Λ. (5) The family ofµ_Λ,y is in fact the family of finite volume specifications with fixed boundary condition y.

The limit in the above Lemma is taken in the following sense: for any increasing sequence Λ_nof finite subsets inZ^dconverging toZ^dwhenngoes to infinity, µ_Λn,y⊗δ_yΛc

n converges in the local convergence topology towards µ.

We first prove the theorem in the case whereν ∈exG( ˜ϕ, dx).

Let (ν_Λ,y)_Λ⊂Z^d be the approximating sequence of ν defined by (5). For Λ⊂Z^d fixed and any i∈Λ, we introduce thei-decoupled infinite measureν_Λ,yⁱ as follows :

ν_Λ,yⁱ (dx_Λ) = 1 Z˜Λ,y

e^{−˜hΛ\i,Λc(x,y)}dx_Λ\idx_i.

Since ˜h_Λ,Λ^c(x, y) = ˜h_Λ\i,Λ^c(x, y) + ˜h_i(x_Λy_Λ^c), we obtain

ν_Λ,yⁱ (dx_Λ) =e^{˜hi(xΛyΛc)}ν_Λ,y(dx_Λ). (6) Let us remark that ν_Λ,yⁱ ⊗δy_Λc converges in Λ towards a measure νⁱ on R^Zd which is absolutely continuous with respect toν and satisfies :

νⁱ(dx) =e^˜hi(x)ν(dx).

In the same way, we denote by µ_Λ and µⁱ_Λthe following measures (not necessary finite) : µ_Λ(dx_Λ) =e^{−hΛ,∅(xΛ)}dx_Λ, µⁱ_Λ(dx_Λ) =e^{−hΛ\i,∅(xΛ\i)}dx_Λ\idx_i. (7) Then

µⁱ_Λ(dx_Λ) =e^hi,Λ(xΛ)µ_Λ(dx_Λ). (8) Let us now introduce the following finite-dimensional approximation of the dynamics (4) :

dXi(t) =dBi(t)−1

2∇ihi,Λ(XΛ(t))dt, ∀i∈Λ, t >0. (9) Remark that we also could write the drift as −¹₂∇ih_Λ,∅(XΛ(t)). Under this form, µΛ is clearly a reversible measure associated to this dynamics.

We denote by Q^x_Λ^Λ the law on C(R⁺,R)^Λ of the solution of (9) when the initial condition is xΛ∈R^Λ.

We now introduce, in the same way as above, some decoupled (infinite- and finite-dimensional) dynamics at the site i:

dX_j(t) =dB_j(t)−¹₂∇jh_j,Z^d\i(X(t))dt, ∀j∈Z^d\{i}, t >0

dX_i(t) =dB_i(t), t >0. (10)

Q^x,i denotes the law of the solution of (10) with deterministic initial condition x ∈R^Zd. These dynamics are useful since they are simpler than the undecoupled ones, and we will prove that the lawν^tof the gradient system at timetis absolutely continuous with respect to the law at time t of the above decoupled system.

We also consider the finite-dimensional approximation of (10) :

dX_j(t) =dB_j(t)−¹₂∇jh_j,Λ\i(X_Λ\i(t))dt, ∀j∈Λ\i, t >0

dX_i(t) =dB_i(t). (11)

We denote byQ^x_Λ^Λ,ithe law onC(R⁺,R)^Λof the solution of (11) when the initial condition is x_Λ∈R^Λ. µⁱ_Λ is a reversible measure associated to this dynamics.

Since the solution of (4) (when it exists) is Markovian, one has : Q^ν = R

Q^xν(dx).

More generally, for any measure µ, we denote byQ^µ (resp. Q^µ,i, Q^µ_Λ orQ^µ,i_Λ ) the mixture of Q^x under µ: Q^µ=R

Q^xµ(dx) (resp. R

Q^x,iµ(dx),R

Q^x_Λµ(dx) or R

Q^x,i_Λ µ(dx)) . We also define the projections at time tof these measures :

ν^t=Q^ν◦X(t)⁻¹, ν^t,i=Q^νi,i◦X(t)⁻¹, ν_Λ,y^t =Q^ν_Λ^Λ,y◦X(t)⁻¹, ν_Λ,y^t,i =Q^ν

i Λ,y,i

Λ ◦X(t)⁻¹. Lemma 2 For each t∈[0, T]and i∈Z^d, the following weak convergences hold :

ΛրlimZdν_Λ,y^t =ν^t and lim

ΛրZdν_Λ,y^t,i =ν^t,i. Proof :

We only prove the first convergence. The proof of the second one is analogous.

Under the assumptions satisfied by ϕ in Theorem 1, it is simple to verify that for any initial deterministic condition x ∈ l¹(γ) = {y = (y_i)_i∈Z^d ∈ R^Zd : kykγ <+∞}, a strong solution of (4) exists inC(R⁺, l¹(γ)). It is obtained as limit of finite dimensional diffusions solution of (9). More precisely, let Λn be an increasing sequence of finite subsets in Z^d converging to Z^d when n goes to infinity. To clarify the notations, instead of using the canonical processes, for x∈R^Zd we denote by X^x the solution of (4) with ν =δ_x and by X^(n),x the (infinite-dimensional) process with initial condition x whose restriction on Λ_n solves (9) with Λ = Λ_n and whose coordinates outside Λ_n are frozen inx_Λ^c_n. So the law of X^x is equal to Q^x and the law of X^(n),x is equal toQ^x_Λ^Λn_n ⊗δ_xΛc

n. Following analogous techniques as the one used in [27] Theorem 4.1 (or [7], [26] if the interaction is reduced to a pair interaction), we now prove that, for any T >0,X^(n),x is a Cauchy sequence in L¹(C([0, T], l¹(γ))).

Let r the range ofϕand k >0 the supremum of a Lipschitz constant for ∇jh_j (uniform in j) and a bound for sup_x|∇jh_j(x)|. Let m < nand let Λ^◦_m denote ther-interior of Λ_m defined by Λ^◦_m={j∈Λm:∀kwith|k−j| ≤r, k∈Λm}. So Λ^◦_m ⊂Λm ⊂Λn.

For i∈Λ^◦_m,

|X_i^(n),x(t)−X_i^(m),x(t)| = 1 2|

Z _t

0 ∇ih_i,Λn(X^(n),x(s))− ∇ih_i,Λm(X^(m),x(s))ds|

≤ k 2

X

{j:|j−i|≤r}

Z t

0 |X_j^(n),x(s)−X_j^(m),x(s)|ds

Thus, E(sup

s≤t|X_i^(n),x(s)−X_i^(m),x(s)|)≤ k 2

X

{j:|j−i|≤r}

Z _t

0

E(sup

u≤s|X_j^(n),x(u)−X_j^(m),x(u)|)ds.

For i∈Λ_m\Λ^◦_m,

|X_i^(n),x(t)−X_i^(m),x(t)|

= 1

2| Z _t

0 ∇ih_i,Λn(X^(n),x(s))− ∇ih_i,Λm(X^(m),x(s))ds|

≤ k 2

X

{j:|j−i|≤r}∩Λm

Z t

0 |X_j^(n),x(s)−X_j^(m),x(s)|ds+1 2|

Z t 0

X

Λ^′:Λ^′6⊂Λm

Λ^′⊂Λn

∇iϕ_Λ^′(X^(m),x(s))ds|

Thus, E(sup

s≤t|X_i^(n),x(s)−X_i^(m),x(s)|)≤ k 2

X

{j:|j−i|≤r}

Z _t

0

E(sup

r≤s|X_j^(n),x(r)−X_j^(m),x(r)|)ds+k 2t.

For i∈Λ_n\Λ_m, we have

|X_i^(n),x(t)−X_i^(m),x(t)|=|B_i(t)−1 2

Z t

0 ∇ih_i,Λn(X^(n),x(s))ds|; thus, using Doob inequality for B_i,

E(sup

s≤t|X_i^(n),x(s)−X_i^(m),x(s)|)≤2√ t+k

2t.

For i∈Λ^c_n,|X_i^(n),x(t)−X_i^(m),x(t)| ≡0.

To obtain the desired Cauchy property, we apply an infinite-dimensional version of Gron- wall’s Lemma. It is simple to verify (see for example [25] Lemme1, page 197) that, if C is a positive constant which satisfies : for any j∈Z^d,^k₂P

{i:|j−i|≤r}γi ≤Cγj, ( one can take C =e^αr#{k∈Z^d,|k| ≤r}), then,

E(sup

s≤t kX^(n),x(s)−X^(m),x(s)kγ) = E(sup

s≤t

X

i∈Zd

γ_i|X_i^(n),x(s)−X_i^(m),x(s)|)

≤ (2√ t+ k

2t)( X

i∈Λn\Λ^◦m

γ_i) expkC

2 t. (12) Since γ ∈l¹(Z^d) the right hand side of this inequality goes to 0 whenn and m are large enough - uniformly in x - we conclude that X^(n),x converges in L¹(C([0, T], l¹(γ))) uni- formly in x towards X^x.

Moreover, with similar computations as above, one obtains the following Lipschitz regu- larity of X^x as a function ofx :

∃C^′>0, E(sup

s≤t kX^x(s)−X^y(s)kγ)≤kx−ykγe^C′t.

Now, the law of the solution of (4) is just a mixture under ν of the laws Q^x. We may claim this since the support ofν is included inl¹(γ).

Let us now prove the local convergence ofν_Λ^t_n,ytowardsν^t. Letgbe a ∆-local Lipschitz function onR^Zdwith Lipschitz constantK_gand fixm < nlarge enough to have ∆⊂Λ_m(⊂ Λn). We have

| Z

g(x_∆)ν^t(dx)− Z

g(x_∆)ν_Λ^t_n,y(dx_Λn)|

=| Z

g(X_∆(t))Q^ν(dX)− Z

g(X_∆(t))Q^ν_Λ^{Λn ,y}

n (dX_Λn)|

=| Z

R^Zd

Z

g(X_∆(t))Q^x(dX)ν(dx)− Z

RΛn

g(X_∆(t))Q^x_Λ^Λn_n (dX_Λn)ν_Λn,y(dx_Λn)|

≤C₁+C₂+C₃ where

C₁ = Z

R^Zd| Z

g(X_∆(t))Q^x(dX)− Z

g(X_∆(t))Q^x_Λ^Λm

m (dX)|ν(dx) C₂ = |

Z Z

g(X_∆(t))Q^x_Λ^Λm

m (dX)ν(dx)− Z Z

g(X_∆(t))Q^x_Λ^Λm

m (dX)ν_Λn,y(dx_Λn)| C₃ =

Z

RΛn| Z

g(X_∆(t))Q^x_Λ^Λm

m (dX)− Z

g(X_∆(t))Q^x_Λ^Λn

n (dX)|ν_Λn,y(dx_Λn).

With the above notations, C₁ =

Z

R^Zd

E|g(X_∆^x(t))−g(X_∆^(m),x(t))|ν(dx)

≤ K_g Z

R^Zd

E|X_∆^x(t)−X_∆^(m),x(t)|ν(dx)

≤ Kg

Z

R^Zd

EX

i∈∆

|X_i^x(t)−X_i^(m),x(t)|ν(dx)

≤ K_g( inf

i∈∆γ_i)⁻¹ Z

R^Zd

EX

i∈∆

γ_i|X_i^x(t)−X_i^(m),x(t)|ν(dx)

≤ K_g( inf

i∈∆γ_i)⁻¹ Z

R^Zd

E(kX^x(t)−X^(m),x(t)kγ)ν(dx)

≤ c₁( X

i∈(Λ^◦_m)^c

γ_i),

using (12) for the last inequality, wherec₁ >0 is a constant depending only ont, ϕ, g, r,∆, γ.

In the same way, one has C₃ =

Z

RΛn

E|g(X_∆^(m),x(t))−g(X_∆^(n),x(t))|ν_Λn,y(dx_Λn)

≤ K_g Z

RΛn

E|X_∆^(m),x(t)−X_∆^(n),x(t)|ν_Λn,y(dx_Λn)

≤ K_g( inf

i∈∆γ_i)⁻¹ Z

E(kX^(m),x(t)−X^(n),x(t)kγ)ν_Λn,y(dx_Λn)

≤ c₁( X

i∈(Λ^◦_m)^c

γ_i).

The second term is controlled in the following way : C₂=|

Z

E(g(X_∆^(m),x(t)))ν(dx)− Z

E(g(X_∆^(m),x(t)))ν_Λn,y(dx_Λn)|.

But, formfixed,X_∆^(m),x(t) and thenE(g(X_∆^(m),x(t)) are Λm-local functions inxdepending continuously on x_Λm. So, thanks the local convergence of the finite-volume specifications ν_Λn,y towards ν when n goes to infinity, this term vanishes for m fixed and n going to infinity.

To complete the proof of the convergence of ν_Λ^t_n,y towards ν^t it remains now to take m large enough in such a way that (P

i∈(Λ^◦_m)^cγ_i) (and thus C₁+C₃) stays as small as nec-

essary.

To prove thatν^tis Gibbsian, we will use the fact thatν^tis absolutely continuous with respect toν^t,i, which itself is a consequence of the absolute continuity ofν_Λ,y^t with respect to ν_Λ,y^t,i . Let us start with a nice representation of this density ^dν

t Λ,y

dν_Λ,y^t,i . Lemma 3 For each t >0, Λ⊂Z^d and any i∈Λ,

dν_Λ,y^t

dν_Λ,y^t,i (x_Λ) =e^{−h˜i(xΛyΛc)} E_Q^xΛ

Λ

e^{fΛ,y(XΛ(t))−fΛ,y(xΛ)}

E_Q^xΛ Λ,i

e^{fΛ\i,y(XΛ(t))−fΛ\i,y(xΛ)}, (13) where

f_Λ,y(x) =h_Λ,∅(x_Λ)−h˜_Λ,Λ^c(x, y).

Proof : We have :

dν_Λ,y^t

dν_Λ,y^t,i (x_Λ) = dν_Λ,y^t dµΛ

(x_Λ)dµ_Λ

dµⁱ_Λ(x_Λ) dµⁱ_Λ dν_Λ,y^t,i (x_Λ).

Using the reversibility ofQ^µ_Λ^Λ (resp. Q^µ_Λ^iΛ,i) the first term of the right hand side is obtained as follows : for any regular bounded local function g,

Z

g(x_Λ)ν_Λ,y^t (dx_Λ) = Z

g((X_Λ(t))Q^ν_Λ^Λ,y(dX)

= Z Z

g((X_Λ(t))dν_Λ,y

dµ_Λ (X_Λ(0))Q^µ_Λ^Λ(dX)

= Z Z

g((X_Λ(0))dν_Λ,y

dµ_Λ (X_Λ(t))Q^µ_Λ^Λ(dX)

= Z

g(x_Λ)

Z dνΛ,y

dµ_Λ (X_Λ(t))Q^x_Λ^Λ(dX)µ_Λ(dx_Λ).

Then dν_Λ,y^t

dµ_Λ (x_Λ) =E_Q^xΛ Λ

dν_Λ,y

dµ_Λ (X_Λ(t)) .

Doing a similar computation for the decoupled dynamics one obtains dν_Λ,y^t

dν_Λ,y^t,i (x_Λ) = E_Q^xΛ Λ

e^{fΛ,y(XΛ(t))}

e^{−hi,Λ(xΛ)}E_Q^xΛ Λ,i

e^{fΛ\i,y(XΛ(t))}₋1

= e^{−hi,Λ(xΛ)+ϕi(xi)} e^fΛy(xΛ) e^fΛ\i,y(xΛ)

E_Q^xΛ Λ

e^{fΛ,y(XΛ(t))−fΛ,y(xΛ)}

E_Q^xΛ Λ,i

e^{fΛ\i,y(XΛ(t))−fΛ\i,y(xΛ)}, which is the same as the expression (13).

Let us first remark that, since ˜ϕ is of finite range, the expression e^{−˜hi(xΛyΛΛc)} does not depend on y and on Λ for Λ large enough. We will now prove, using cluster expansions, that the last ratio in (13) is a function ofx indexed by Λ which converges uniformly in y when Λ tends to Z^d.

Thanks to Girsanov theorem, the probability measures Q^x_Λ^Λ and Q^x_Λ,i^Λ have a Gibbs representation on the path space C(R⁺,R)^Λ, that is, if one restricts them to the the canonical filtration at any time t they have both an explicit density with respect to the Wiener measure with deterministic initial conditionx_Λ, denoted by ⊗i∈Λρ^xi. The density of Q^x_Λ^Λ is the following :

FΛ(XΛ) = expX

i∈Λ

Z t 0 −1

2∇ihi,Λ(XΛ(s))dXi(s)−1 2

Z t 0

1

4(∇ihi,Λ)²(XΛ(s))ds ,

which becomes, using Ito formula, F_Λ(X_Λ)

= exp

− 1

2h_Λ,∅(X_Λ(t)) + 1

2h_Λ,∅(X_Λ(0)) +X

i∈Λ

Z _t

0

1

4∆_ih_Λ,∅−1

8(∇ih_Λ,∅)²

(X_Λ(s))ds

= exp

− X

A⊂Λ

Φ_A(X_Λ)

, (14)

with

Φ_A(X_Λ) = 1

2ϕ_A(X_Λ(t))− 1

2ϕ_A(X_Λ(0)) (15)

− Z t

0

1 4

X

j∈A

∆_jϕ_A(X_Λ(s))−1 8

X

B,C⊂A B∪C=A B∩C6=∅

X

j∈B∩C

∇jϕ_B(X_Λ(s))∇jϕ_C(X_Λ(s)) ds.

The family Φ = (Φ_A)_A⊂Z^d is an interaction potential on Ω; sinceϕis of finite range (FR), Φ is of finite range too. Denoting byH the hamiltonian function associated to Φ, we then obtained that, for any Λ⊂Z^dand i∈Λ, on the events depending only on times between 0 and t,

Q^x_Λ^Λ(dX_Λ) =e^{−HΛ,∅(XΛ)}⊗j∈Λρ^xj(dX_j). (16) In the same way, one proves that

Q^x_Λ,i^Λ(dX_Λ) =e^{−HΛ\i,∅(XΛ\i)}⊗j∈Λ\iρ^xj(dX_j) ⊗ρ^xi(dX_i). (17) Let us describe some properties of the interaction potential Φ.

Lemma 4 There exists a constant C > 0 independent of the time t such that for any X ∈Ω and any A⊂Z^d

|Φ_A(X)| ≤ C t+ sup

j∈A|X_j(t)−X_j(0)| .

Proof :

It is clear, due to the equality (15) and the assumptions given on ϕ.

We can now expend the termsE_Q^xΛ Λ

e^{fΛ,y(XΛ(t))−fΛ,y(xΛ)}

andE_Q^xΛ Λ,i

e^{fΛ\i,y(XΛ(t))−fΛ\i,y(xΛ)} . We give the detailed computations only for the first expansion, since the second one is obtained in a similar way.

E_Q^xΛ Λ

e^{fΛ,y(XΛ(t))−fΛ,y(xΛ)}

=E_⊗j∈Λρxj exp

− X

A⊂Λ

Ψ^y,Λ_A (X_Λ)

!

(18) where Ψ^y,Λ is the following interaction potential on C([0, T],R)^Λ :

Ψ^y,Λ_A (X) = Φ_A(X)−ϕ_A(X(t)) +ϕ_A(X(0)) + X

B⊂Z^d B∩Λ=A

ϕ˜_B(X_Λ(t)y_Λ^c)−ϕ˜_B(X_Λ(0)y_Λ^c) .(19)

We also denote by Ψ the interaction potential onC([0, T],R)^Zd : Ψ_A(X) = Φ_A(X)−ϕ_A(X(t)) +ϕ_A(X(0)) +

˜

ϕ_A(X(t))−ϕ˜_A(X(0))

(20) and immediately remark that, as soon as Λ is large enough with respect to the index set A, Ψ^y,Λ_A ≡Ψ_A.

As in the Lemma 4, we obtain the following estimates for the interactions Ψ^y,Λ and Ψ.

Lemma 5 There exists a constant C > 0 independent of the time t such that for any y ∈R^Zd,Λ⊂Z^d, X∈Ωand any A⊂Λ,

|Ψ^y,Λ_A (X)| ≤ C t+ sup

j∈A|X_j(t)−X_j(0)| and

|Ψ_A(X)| ≤ C t+ sup

j∈A|X_j(t)−X_j(0)| .

Proof :

It is a direct consequence of Lemma 4 and the assumptions given on ϕ and ˜ϕ. The uni- formity of the first upperbound with respect to y and Λ is then clear.

Let us now introduce the main notations and tools we need for the cluster representa- tion. Let N ∈Nlarge enough is such a way that for #A > N,Ψ^y,Λ_A ≡0. Such a number N exists since Ψ^y,Λ_A is of finite range uniformly in y and Λ. Let V a finite subset of Z^d such that for any A⊂Z^d with Ψ^y,Λ_A 6= 0 thenA⊂ ∩j∈A(V+j).

Let us define a subclass of finite volumes inZ^d : D=

A⊂Z^d: 1≤#A≤N and A⊂ ∩j∈A(V+j)

.

A finite setγ ={A₁, A₂, . . . , A_n}, n≥1,of elements ofDis acluster. It is called connected if for anyA_i, A_j ∈γ, there exists a sequencei=i₁, i₂, . . . , i_m =j such thatA_i1∩A_i2 6=∅, A_i2 ∩A_i3 6= ∅, . . ., A_im−1 ∩A_im 6= ∅. We call support of the cluster γ the subset of Z^d equal to Sn

l=1A_l and denote it by supp(γ). The entire number|γ|is the cardinality of the support of γ.

We denote by A the set of connected clusters and AΛ the subset of A which contains the clusters whose supports are included in Λ. A collection of clusters γ1, γ2, . . . , γn is called compatible if their associated supports are disjoint. We denote by LΛ the set of all compatible collections of non empty clusters belonging to AΛ.

We can now start the expansion of the expression (18).

E_Q^xΛ Λ

e^{fΛ,y(XΛ(t))−fΛ,y(xΛ)}

= E_⊗j∈Λρxj

Y

A⊂Λ

e^{−Ψy,ΛA (XΛ)}−1 + 1

!

= E_⊗j∈Λρxj 1 + X∞ n=1

X

{γ1,...γn}∈LΛ

K^y,Λ(γ₁)(X)K^y,Λ(γ₂)(X). . .K^y,Λ(γ_n)(X)

! ,

where

K^y,Λ(γ)(X) = Y

A∈γ

e^{−Ψy,ΛA (XΛ)}−1 We then obtain the below cluster decomposition :

E_Q^xΛ Λ

e^{fΛ,y(XΛ(t))−fΛ,y(xΛ)}

= 1 +

∞

X

n=1

X

{γ1,...γn}∈LΛ

K_x^y,Λ(γ₁)K_x^y,Λ(γ₂). . . K_x^y,Λ(γ_n) (21) where

K_x^y,Λ(γ) = E_⊗

j∈Zdρ^xj

K^y,Λ(γ)(X) .

In a similar way, we obtain for any i∈Λ : E_Q^xΛ

Λ,i

e^{fΛ\i,y(XΛ(t))−fΛ\i,y(xΛ)}

= 1 + X∞ n=1

X

{γ1,...γn}∈LΛ\i

K_x^y,Λ(γ₁)K_x^y,Λ(γ₂). . . K_x^y,Λ(γ_n).

Let also define the coefficients related to the interaction Ψ (instead of Ψ^y,Λ) by : K(γ)(X) = Y

A∈γ

e^−ΨA(X)−1

, K_x(γ) =E_⊗

j∈Zdρ^xj

K(γ)

. (22)

In the next lemma, we provide estimates for the weight of the clusters (uniformly in x,y and Λ).

Lemma 6 There exists some strictly positive constant λ(t) which tends to 0 as t goes to 0 such that, for any x∈R^Zd, any y∈R^Zd,Λ⊂Z^d and any γ ∈ A

|K_x^y,Λ(γ)| ≤λ(t)^|γ| and |K_x(γ)| ≤λ(t)^|γ|.

Proof :

We need to commute several times integration and products. To this aim, the following abstract integration lemma, which generalizes H¨older inequalities, is very useful. It is proved in [23] Lemma 5.2 :

Lemma 7 Let (µ_x)_x∈X be a family of probability measures, each one defined on a space E_x, where the elements x belong to some finite set X. Let us also define a finite family (g_k)_k of functions on E_X =×x∈XE_x such that eachg_k isXk-local for a certain Xk⊂ X , in the sense that

g_k(e) =g_k(e_Xk), for e= (e_x)_x∈X ∈E_X. Let ρ_k>1 be numbers satisfying the following conditions :

∀x∈ X, X

Xk∋x

1

ρ_k ≤1. (23)

Then

Z

EX

Y

k

g_k⊗x∈X dµx

≤Y

k

Z

EXk

|g_k|^ρk⊗x∈Xk dµx

1/ρk

. (24)

Let γ = {A1, A2, . . . , An} ∈ A; we apply Lemma 7 with X =supp(γ), Xk = A_k, g_k = e^−Ψy,ΛAk −1 and ρ_k= _#¹_V for all k. We get

|K_x^y,Λ(γ)| ≤

n

Y

k=1

E_⊗j∈

Akρ^xj

|e^−Ψy,ΛAk −1|^ρk#V

.

Using the bound of Lemma 5, we obtain

|K_x^y,Λ(γ)| ≤

n

Y

k=1

E_⊗j∈

Akρ^xj

e^{C(t+supj∈Ak|Xj(t)−xj|)}−1ρk_#V .

Now, due to the existence of exponential moment for the N-dimensional Brownian motion, E_⊗j∈

Akρ^xj

(e^{C(t+supj∈Ak|Xj(t)−xj|)}−1)^ρk

≤λ(t),

where the constant λ(t) tends to 0 as t goes to 0, uniformly in x and A_k ∈ D. So, since n/ρ_k=n#V ≥ |γ|, one obtains the first desired upperbound :

|K_x^y,Λ(γ)| ≤λ(t)^|γ|. The second upperbound is then obvious.

One can then deduce from Lemma 6 the following upper bound : for any cluster γ ∈ A and for t small enough,

sup

x,y∈R^Zd

sup

Λ⊂Z^d

X

γ^′∈A: supp(γ)∩supp(γ^′)6=∅

|K_x^y,Λ(γ^′)|e^|γ′|≤ |γ|. (25)