Hydrodynamic limit of a <span class="mathjax-formula">$d$</span>-dimensional exclusion process with conductances

www.imstat.org/aihp 2012, Vol. 48, No. 1, 188–211

DOI:10.1214/10-AIHP397

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

Hydrodynamic limit of a d -dimensional exclusion process with conductances ¹

Fábio Júlio Valentim

Departamento de Matemática, Universidade Federal do Espírito Santo, Av. Fernando Ferrari, 514, Goiabeiras, 29075-910, Vitória – ES, Brasil.

E-mail:[email protected]

Received 10 February 2010; revised 21 October 2010; accepted 23 October 2010

Abstract. Fix a polynomialΦ of the formΦ(α)=α+

2≤j≤ma_jα^j withΦ(1) >0. We prove that the evolution, on the diffusive scale, of the empirical density of exclusion processes onT^d, with conductances given by special class of functionsW, is described by the unique weak solution of the non-linear parabolic partial differential equation∂_tρ=_d

k=1∂_xk∂_WkΦ(ρ). We also derive some properties of the operator_d

k=1∂x_k∂_Wk.

Résumé. Étant donné un polynômeΦ de la formeΦ(α)=α+

2≤j≤ma_jα^j respectantΦ(1) >0, nous démontrons que l’évolution, sur une échelle diffusive, de la densité empirique des processus d’exclusion surT^d, dont les conductances sont données par une classe spéciale de fonctionsW, est décrite par l’unique solution faible de l’équation aux dérivées partielles parabolique :

∂tρ=_d

k=1∂x_k∂_WkΦ(ρ). Nous dérivons également certaines propriétés de l’opérateur_d

k=1∂x_k∂_Wk. MSC:60K35; 26A24; 35K55

Keywords:Exclusion processes; Random conductances; Hydrodynamic limit

1. Introduction

The evolution of one-dimensional exclusion processes with random conductances has attracted some attention recently [2,3,6,7]. The purpose of this paper is to extend this analysis to higher dimension.

LetW:R^d→Rbe a function such thatW (x₁, . . . , x_d)=_d

k=1W_k(x_k), whered≥1 and each functionW_k:R→ Ris strictly increasing, right continuous with left limits (càdlàg), and periodic in the sense thatW_k(u+1)−W_k(u)= W_k(1)−W_k(0)for allu∈R. Informally, the exclusion process with conductances associated toW is an interacting particle systems on thed-dimensional discrete torusN⁻¹T^d_N, in which at most one particle per site is allowed, and only nearest-neighbor jumps are permitted. Moreover, the jump rate in the directionej is given by the reciprocal of the increments ofW with respect to thejth coordinate.

We show that, on the diffusive scale, the macroscopic evolution of the empirical density of exclusion processes with conductancesWis described by the nonlinear differential equation

∂_tρ= d k=1

∂_xk∂_WkΦ(ρ), (1.1)

1Research supported by CNPq.

whereΦis a polynomial of the formΦ(α)=α+

2≤j≤ma_jα^j withΦ(1) >0. Furthermore, we denote by∂_Wkthe generalized derivative with respect toWk, see [1,6] and a revision in Section3. The partial differential equation (1.1) appears naturally as, for instance, scaling limits of interacting particle systems in inhomogeneous media. It may model diffusions in which permeable membranes, at the points of discontinuities ofW, tend to reflect particles, creating space discontinuities in the density profiles.

The proof of hydrodynamic limit relies strongly on some properties of the differential operator_d

k=1∂_xk∂_Wk pre- sented in Theorem2.1. We prove, among other properties: that the operator_d

k=1∂_xk∂_Wk, defined on an appropriate domain, is non-positive, self-adjoint and dissipative; that its eigenvalues are countable and have finite multiplicity;

and that the associated eigenvectors form a complete orthonormal system.

There is a wide literature on the so-called Feller’s generalized diffusion operator(d/du)(d/dv). Where, typically, uandvare strictly increasing functions withv(but not necessarilyu) being continuous. It provides general diffusions operators and an appreciable simplification of the theory of second-order differential operators (see, for instance, [4,5,9]). The operator(d/dx)(d/du), considered in [6], is the formal adjoint of(d/du)(d/dv) in the particular case v(x)=x (as in [5]). The goal of this work is to extend this adjoint operator to higher dimensions and provide some results regarding this extension. The assumption that the functionW is a direct sum of one-dimensional functions is technically essential, so that the limit equation (1.1) has a special form. More details, see Sections4and5.

The article is organized as follows: in Section2we state the main results of the article; in Section3we prove the main properties of the operatorLW =_d

k=1∂_xk∂_Wk; in Section4we prove the convergence of random walks with random conductances to Markov processes with generator given byLW; in Section5we prove the scaling limit of the exclusion process with conductances given byW; and, finally, in Section6we show that the unique solution of (1.1) has finite energy.

2. Notation and results

We examine the hydrodynamic behavior of ad-dimensional exclusion process, withd≥1, with conductances given by a special class of functionsW:R^d→Rsuch that:

W (x₁, . . . , x_d)= d k=1

W_k(x_k), (2.1)

whereW_k:R→Rarestrictly increasingright continuous functions with left limits (càdlàg), and periodic in the sense that

W_k(u+1)−W_k(u)=W_k(1)−W_k(0)

for allu∈Randk=1, . . . , d. To keep notation simple, we assume thatWk vanishes at the origin, that is,Wk(0)=0.

Denote byT^d = [0,1)^d thed-dimensional torus and bye₁, . . . , e_d the canonical basis of R^d. For this class of functions we have:

• W (0)=0;

• W is strictly increasing on each coordinate:

W (x+ae_j) > W (x) for all 1≤j≤d,a >0, x∈R^d;

• W is continuous from above:

W (x)= lim

y→x,y≥xW (y),

where we say thaty≥xifyj≥xj for all 1≤j≤d;

• W is defined on the torusT^d:

W (x₁, . . . , x_j−1,0, xj+1, . . . , x_d)=W (x₁, . . . , x_j−1,1, xj+1, . . . , x_d)−W (e_j) for all 1≤j≤d,(x1, . . . , x_j−1, x_j+1, . . . , x_d)∈T^d−1.

Unless explicitly stated W belongs to this class. LetT^d_N be the d-dimensional discrete torus with N^d points.

Distribute particles throughoutT^d_Nin such a way that each site ofT^d_Nis occupied at most by one particle. Denote byη the configurations of the state space{0,1}^TdN, so thatη(x)=0 if sitexis vacant andη(x)=1 if sitexis occupied.

Fixa >−1/2 andW. Forx=(x₁, . . . , x_d)∈T^d_Nlet c_x,x+ej(η)=1+a

η(x−e_j)+η(x+2ej) ,

where all sums are moduloN, and let

ξ_x,x+ej = 1

N[W ((x+e_j)/N )−W (x/N )]= 1

N[W_j((x_j+1)/N )−W_j(x_j/N )].

We now describe the stochastic evolution of the process. Letx=(x₁, . . . , x_d)∈T^d_N. At rate ξ_x,x+ejc_x,x+ej(η) the occupation variablesη(x),η(x+e_j)are exchanged. IfW is differentiable atx/N∈ [0,1)^d, the rate at which particles are exchanged is of order 1 for each direction, but if someW_j is discontinuous atx_j/N, it no longer holds.

In fact, assume, to fix ideas, thatW_j is discontinuous atx_j/N, and smooth on the segments(x_j/N, x_j/N+εe_j)and (xj/N−εej, xj/N ). Assume, also, thatWk is differentiable in a neighborhood ofxk/N fork=j. In this case, the rate at which particles jump over the bonds{y−e_j, y}, withy_j=x_j, is of order 1/N, whereas in a neighborhood of sizeN of these bonds, particles jump at rate 1. Thus, note that a particle at sitey−e_j jumps toy at rate 1/N and jumps at rate 1 to each one of the 2d−1 other options. Particles, therefore, tend to avoid the bonds{y−e_j, y}.

However, since time will be scaled diffusively, and since on a time interval of lengthN²a particle spends a time of orderN at each sitey, particles will be able to cross the slower bond{y−e_j, y}.

Then, this process models membranes that obstruct passages of particles. Note that these membranes are(d−1)- dimensional hyperplanes embedded in ad-dimensional environment. Moreover, if we considerW_j having more than one discontinuity point for more than onej, these membranes will be more sophisticated manifolds, for instance, unions of(d−1)-dimensional boxes.

The effect of the factorc_x,x+ej(η)is analogous to the one-dimensional case. If the parameter a is positive, the presence of particles in the neighboring sites of the bond{x, x+ej}speeds up the exchange rate by a factor of order one.

The dynamics informally presented describes a Markov evolution. The generatorL_N of this Markov process acts on functionsf:{0,1}^TdN →Ras

L_Nf (η)= d j=1

x∈T^d_N

ξ_x,x+ejc_x,x+ej(η) f

σ^x,x+ejη

−f (η)

, (2.2)

whereσ^x,x+ejηis the configuration obtained fromηby exchanging the variablesη(x)andη(x+e_j):

σ^x,x+ejη (y)=

η(x+e_j) ify=x, η(x) ify=x+e_j, η(y) otherwise.

(2.3) A straightforward computation shows that the Bernoulli product measures{ν_α^N: 0≤α≤1}are invariant, and in fact reversible, for the dynamics. The measureν^N_α is obtained by placing a particle at each site, independently from the other sites, with probabilityα. Thus,ν_α^N is a product measure over{0,1}^TdN with marginals given by

ν_α^N

η: η(x)=1

=α

forxinT^d_N. For more details see [8], Chapter 2. We will often omit the indexNonν_α^N.

Denote by {η_t: t≥0}the Markov process on {0,1}^TdN associated to the generatorL_N speeded up byN². Let D(R₊,{0,1}^TdN)be the path space of càdlàg trajectories with values in {0,1}^TdN. For a measureμ_N on{0,1}^TdN, denote byPμ_N the probability measure onD(R+,{0,1}^TdN)induced by the initial stateμN, and the Markov process {ηt: t≥0}. Expectation with respect toPμ_N is denoted byEμ_N.

2.1. The operatorLW

FixW=d

k=1Wkas in (2.1). In [6] it was shown that there exist self-adjoint operatorsLW_k:DW_k⊂L²(T)→L²(T).

Further, the setAW_k of the eigenvectors ofLW_k forms a complete orthonormal system inL²(T). Let AW=

f:T^d→R;f (x₁, . . . , x_d)=

d

k=1

f_k(x_k), f_k∈AWk, k=1, . . . , d

, (2.4)

and denote by span(A)the space of finite linear combinations of the set A, and letDW :=span(AW). Define the operatorLW:DW→L²(T^d)as follows: forf =_d

k=1f_k∈AW, we have LW(f )(x1, . . . , xd)=

d k=1

d

j=1,j=k

fj(xj)LW_kfk(xk), (2.5)

and then extend toDW by linearity.

Lemma3.2, in Section3, shows that:LW is symmetric and non-positive;DW is dense inL²(T^d); and the setAW

forms a complete, orthonormal, countable system of eigenvectors for the operatorLW. LetAW = {hk}k≥0,{αk}k≥0

be the corresponding eigenvalues of−LW, and consider DW=

v=^∞

k=1

v_kh_k∈L² T^d

;^∞

k=1

v_k²α²_k<+∞

. (2.6)

Define the operatorLW:DW→L²(T^d)by

−LWv=

+∞

k=1

α_kv_kh_k. (2.7)

The operatorLW is clearly an extension of the operatorLW, and we present in Theorem2.1some properties of this operator.

Theorem 2.1. The operatorLW:DW→L²(T^d)enjoys the following properties:

(a) the domain DW is dense inL²(T^d).In particular,the set of eigenvectorsAW= {h_k}k≥0 forms a complete orthonormal system;

(b) the eigenvalues of the operator−LW form a countable set{α_k}k≥0.All eigenvalues have finite multiplicity,and it is possible to obtain a re-enumeration{α_k}k≥0such that

0=α₀≤α₁≤ · · · and lim

n→∞α_n= ∞;

(c) the operatorI−LW:DW→L²(T^d)is bijective;

(d) LW:DW →L²(T^d)is self-adjoint and non-positive:

−LWf, f ≥0;

(e) LWis dissipative.

In view of (a), (b) and (d), we may use Hille–Yosida theorem to conclude thatLW is the generator of a strongly continuous contraction semigroup{P_t:L²(T^d)→L²(T^d)}t≥0.

Denote by{G_λ:L²(T^d)→L²(T^d)}λ>0the semigroup of resolvents associated to the operatorLW: G_λ=(λ− LW)⁻¹.G_λcan also be written in terms of the semigroup{P_t;t≥0}:

G_λ= _∞

0

e^−λtP_tdt.

In Section4we derive some properties and obtain some results for these operators.

2.2. The hydrodynamic equation

A sequence of probability measures{μ_N: N≥1}on{0,1}^TdN is said to be associated to a profileρ₀:T^d→ [0,1]if

Nlim→∞μ_N 1

N^d

x∈T^dN

H (x/N )η(x)−

H (u)ρ₀(u)du > δ

=0 (2.8)

for everyδ >0 and every continuous functionH:T^d→R. For details, see [8], Chapter 3.

Fix a polynomialΦof the form Φ(α)=α+

m j=2

ajα^j

withΦ(1) >0 andma positive integer. Letγ:T^d→ [0,1]be a bounded density profile, and consider the parabolic differential equation

∂_tρ=LWΦ(ρ),

ρ(0,·)=γ (·). (2.9)

A bounded functionρ:R₊×T^d→ [0,1]is said to be a weak solution of the parabolic differential equation (2.9) if ρ_t, G_λH − γ , G_λH =

t 0

Φ(ρ_s),LWG_λH ds

for every continuous functionH:T^d→R, allt >0 and allλ >0.

Existence of these weak solutions follows from tightness of the sequence of probability measuresQ^W,NμN introduced in Section5. The proof of uniquenesses of weak solutions is analogous to [6].

Theorem 2.2. Fix a continuous initial profileρ0:T^d→ [0,1],and consider a sequence of probability measuresμ_N on{0,1}^TdN associated toρ₀,in the sense of(2.8).Then,for anyt≥0,

Nlim→∞PμN

1 N^d

x∈T^dN

H (x/N )η_t(x)−

H (u)ρ(t, u)du > δ

=0

for everyδ >0and every continuous functionH.Here,ρis the unique weak solution of the non-linear equation(2.9) withγ=ρ₀,andΦ(α)=α+aα².

Remark 2.3. As noted in[6],Remark2.3,the specific form of the ratescx,x+e_i is not important,but two conditions must be fulfilled:the rates must be strictly positive,although they may not depend on the occupation variablesη(x), η(x+ei);but they have to be chosen in such a way that the resulting process isgradient (cf.Chapter7in[8]for the definition of gradient processes).

We may define rates c_x,x+ei to obtain any polynomialΦ of the form Φ(α)=α+

2≤j≤ma_jα^j,m≥1, with 1+

2≤j≤mj a_j>0.Let,for instance,m=3.Then the rates ˆ

c_x,x+ei(η)=c_x,x+ei(η)+b

η(x−2ei)η(x−e_i)+η(x−e_i)η(x+2ei)+η(x+2ei)η(x+3ei) ,

satisfy the above three conditions,wherec_x,x+ei is the rate defined at the beginning of Section2anda,b are such that1+2a+3b >0.An elementary computation shows thatΦ(α)=α+aα²+bα³.

In Section6we prove that any limit pointQ^∗_W of the sequenceQ^W,N_μN is concentrated on trajectoriesρ(t, u)du, with finite energy in the following sense: for each 1≤j≤d, there is a Hilbert spaceL²_x

j⊗W_j, associated toW_j, such that

_t

0

ds∂_WjΦ

ρ(s,·)²

xj⊗Wj<∞, where · xj⊗Wj is the norm inL²_x

j⊗W_j, and∂_Wj is the derivative, which must be understood in the generalized sense.

3. The operatorLW

The operator LW:DW ⊂L²(T^d)→L²(T^d)is a natural extension, for thed-dimensional case, of the self-adjoint operator obtained for the one-dimensional case in [6]. We begin by presenting one of the main results obtained in [6], and we then present the necessary modifications to conclude similar results for thed-dimensional case.

3.1. Some remarks on the one-dimensional case

LetT⊂Rbe the one-dimensional torus. Denote by·,·the inner product ofL²(T):

f, g =

Tf (u)g(u)du.

LetW1:R→Rbe astrictly increasingright continuous function with left limits (càdlàg), and periodic in the sense thatW1(u+1)−W1(u)=W1(1)−W1(0)for alluinR.

LetDW₁ be the set of functionsf inL²(T)such that f (x)=a+bW1(x)+

(0,x]W1(dy) _y

0

f(z)dz for some functionfinL²(T)that satisfies:

₁

0

f(z)dz=0,

(0,1]W₁(dy)

b+ _y

0

f(z)dz

=0.

Define the operatorLW₁:DW₁→L²(T)byLW₁f=f. Formally LW1f = d

dx d dW1

f, (3.1)

where the generalized derivative d/dW1is defined as df

dW1

(x)=lim

→0

f (x+)−f (x)

W₁(x+)−W₁(x) (3.2)

if the above limit exists and is finite.

Theorem 3.1. Denote by Ithe identity operator inL²(T).The operatorLW1:DW1 →L²(T)enjoys the following properties:

(a) DW1 is dense inL²(T);

(b) the operatorI−LW₁:DW₁→L²(T)is bijective;

(c) LW₁:DW₁ →L²(T)is self-adjoint and non-positive:

−LW₁f, f ≥0;

(d) LW1 is dissipative i.e.,for allg∈DW andλ >0,we have λg ≤(λI−LW₁)g;

(e) the eigenvalues of the operator−LW form a countable set{λ_n: n≥0}.All eigenvalues have finite multiplicity, 0=λ₀≤λ₁≤ · · ·,andlimn→∞λ_n= ∞;

(f) the eigenvectors{fn}n≥0of the operatorLWform a complete orthonormal system.

The proof can be found in [6].

3.2. Thed-dimensional case

Consider W as in (2.1). Let AW_k be the countable complete orthonormal system of eigenvectors of the operator LWk:DWk⊂L²(T)→Rgiven in Theorem3.1.

LetAW be as in (2.4), and let the operatorLW:DW :=span(AW)→L²(T^d)be as in (2.5). By Fubini’s theorem, the setAW is orthonormal inL²(T^d), and the constant functions are eigenvectors of the operatorLW_k. Moreover, AW_k⊂AW, in the sense thatf_k(x₁, . . . , x_d)=f_k(x_k),f_k∈AW_k.

By (3.1), the operators LWk can be formally extended to functions defined on T^d as follows: given a function f:T^d→R, we defineLWkf as

LW_kf=∂_xk∂_Wkf, (3.3)

where the generalized derivative∂_Wkis defined by

∂_Wkf (x₁, . . . , x_k, . . . , x_d)=lim

→0

f (x1, . . . , xk+, . . . , xd)−f (x1, . . . , xk, . . . , xd)

Wk(xk+)−Wk(xk) , (3.4)

if the above limit exists and is finite. Hence, by (2.5), iff∈DW

LWf= d k=1

LWkf. (3.5)

Note that iff =_d

k=1f_k, wheref_k∈AW_k is an eigenvector ofLW_k associated to the eigenvalueλ_k, thenf is an eigenvector ofLW, with eigenvalued

k=1λk. Lemma 3.2. The following statements hold:

(a) the setDW is dense inL²(T^d);

(b) the operatorLW:DW→L²(T^d)is symmetric and non-positive:

−LWf, f ≥0.

Proof. The strategy to prove the above lemma is the following. We begin by showing that the set S=span

f∈L²

T^d

;f (x1, . . . , xd)=

d

k=1

fk(xk), fk∈DW_k

is dense in S=span

f ∈L²

T^d

;f (x₁, . . . , x_d)=

d

k=1

f_k(x_k), f_k∈L²(T)

.

We then show thatDW is dense inS. SinceSis dense inL²(T^d), item (a) follows.

We now prove item (a) rigorously. SinceSis a vector space, we only have to show that we can approximate the functions_d

k=1f_k∈L²(T^d), wheref_k∈DW_k, by functions ofDW. By Theorem3.1, the setDW_k is dense inL²(T), thus, there exists a sequence(f_n^k)_n∈Nconverging tof_kinL²(T). Thus, let

f_n(x₁, . . . , x_d)=

d

k=1

f_n^k(x_k).

By the triangle inequality and Fubini’s theorem, the sequence(f_n)converges to_d

k=1f_k. Fix >0, and let h(x₁, . . . , x_d)=

d

k=1

h_k(x_k), h_k∈DW_k.

Since, for eachk=1, . . . , d, AW_k ⊂DW_k is a complete orthonormal set, there exist sequencesg^k_j ∈AW_k, and α^k_j∈R, such that

h_k−

n(k) j=1

α_j^kg_j^k L²(T)

< δ,

whereδ=/dM^d−1andM:=1+sup_k=1:nhk. Let g(x₁, . . . , x_d)=

d

k=1

n(k) j=1

α_j^kg^k_j(x_k)∈DW.

An application of the triangle inequality, and Fubini’s theorem, yieldsh−g< . This proves (a).

To prove (b), let f (x₁, . . . , x_d)=

d

k=1

f_k(x_k) and g(x₁, . . . , x_d)=

d

k=1

g_k(x_k)

be functions belonging toAW. We have that f,LWg =

_d

k=1

f_k, d k=1

d

j=1,j=k

g_jLWkg_k

= d k=1

_d

j=1,j=k

f_jg_j, f_kLWkg_k

,

where·,·denotes the inner product inL²(T^d). Since, by Theorem3.1,LW_k is self-adjoint, we have d

k=1

_d

j=1,j=k

f_jg_j, g_kLWkf_k

= LWf, g.

In particular, the operatorLW_k is non-positive, and, therefore, f,LWf =

d k=1

_d

j=1,j=k

f_j², f_kLW_kf_k

≤0.

Item (b) follows by linearity.

Lemma3.2implies that the setAW forms a complete, orthonormal, countable, system of eigenvectors for the operatorLW.

LetLW:DW→L²(T^d)be the operator defined in (2.7). The operatorLW is clearly an extension of the opera- torLW. Formally, by (3.5),

LWf = d k=1

LWkf, (3.6)

where

LWkf= d dxk

d dWk

f.

We are now in conditions to prove Theorem2.1.

Proof of Theorem2.1. By Lemma3.2,DW is dense inL²(T^d). SinceDW⊂DW, we conclude thatDW is dense in L²(T^d).

Ifα_kare eigenvalues of−LW, we may find eigenvaluesλ_j, associated to somef_j∈AWj, such thatα_k=_d

j=1λ_j. By item (e) of Theorem3.1, (b) follows.

Let{αk}k≥0be the set of eigenvalues of−LW. Then, the set of eigenvalues ofI−LWis{γk}k≥0, whereγk=αk+1, and the eigenvectors are the same as the ones ofLW. By item (b), we have

1=γ0≤γ1≤ · · · and lim

n→∞γn= ∞. Thus,I−LW is injective. For

v= +∞

k=1

vkhk∈L² T^d

, such that ∞ k=1

v²_k<+∞,

let

u=^+∞

k=1

vk

γk

h_k.

Thenu∈DW and(I−LW)u=v. Hence, item (c) follows.

LetL^∗_W:DW^∗⊂L²(T^d)→L²(T^d)be the adjoint ofLW. SinceLW is symmetric, we haveDW⊂DW^∗. So, to show the equality of the operators it suffices to show thatDW^∗⊂DW. Given

ϕ=

+∞

k=1

ϕ_kh_k∈DW^∗,

letLW∗ϕ=ψ∈L²(T^d). Therefore, for allv=_+∞

k=1v_kh_k∈DW, v, ψ = v,LW∗ϕ = LWv, ϕ =

+∞

k=1

−αkvkϕk.

Hence ψ=

+∞

k=1

−αkϕkhk.

In particular, +∞

k=1

α_k²ϕ_k²<+∞ and ϕ∈DW.

Thus,LW is self-adjoint. Letv=_+∞

k=1v_kh_k∈DW. From item (b),α_k≥0, and

−LWv, v =

+∞

k=1

α_kv_k²≥0.

ThereforeLW is non-positive, and item (d) follows.

Fix a functionginDW,λ >0, and letf =(λI−LW)g. Taking inner product, with respect tog, on both sides of this equation, we obtain

λg, g + −LWg, g = g, f ≤ g, g^1/2f, f^1/2.

Sincegbelongs toDW, by (d), the second term on the left-hand side is non-negative. Thus,λg ≤ f = (λI−

LW)g.

4. Random walk with conductances

Recall the decomposition obtained in (3.6) for the operatorLW. In next subsection, we present the discrete versionLN

ofLW and we describe, informally, the Markovian dynamics generated byLN. 4.1. Discrete approximation of the operatorLW

Consider the random walk{X^N_t }t≥0in _N¹T^d_N, which jumps fromx/N(resp.(x+e_j)/N) to(x+e_j)/N(resp.x/N) with rate

N²ξ_x,x+ej =N/

W_j

(x_j+1)/N

−W_j(x_j/N ) .

The generatorLNof this Markov process acts on local functionsf:_N¹T^d_N→Ras LNf (x/N )=

d j=1

L^j_Nf (x/N ), (4.1)

where

L^j_Nf (x/N )=N² ξ_x,x+ej

f

(x+e_j)/N

−f (x/N )

+ξ_x−ej,x f

(x−e_j)/N

−f (x/N ) .

Note that L^j_Nf (x/N )is, in fact, a discrete version of the operator LWj. The counting measure m_N on T^d_N is reversible for this process. The following estimate is a key ingredient for proving the results in Section5:

Lemma 4.1. Letf be a function on _N¹T^d_N.Then,for eachj=1, . . . , d:

1 N^d

x∈T^d_N

L^j_Nf (x/N )2

≤ 1 N^d

x∈T^d_N

LNf (x/N )2

.

Proof. LetX_Nd be the linear space of functionsf on _N¹dT^d_Nover the fieldR. Note that the dimension ofX_Nd isN^d. Denote by·,·N^d the following inner product inX_Nd:

f, gN^d = 1 N^d

x∈T^dN

f (x/N )g(x/N ).

For eachj=1, . . . , d, consider the linear operatorsL^j_N onX_N(i.e.,d=1) given by L^j_Nf =∂_x^N∂_W^N

jf, where∂_x^N and∂_W^N

j are the difference operators:

∂_x^Nf (x/N )=N f

(x+1)/N

−f (x/N ) and

∂_W^N

jf (x/N )= f ((x+1)/N )−f (x/N ) W_j((x+1)/N )−W_j(x/N ).

The operatorsL^j_Nare symmetric and non-positive. In fact, a simple computation shows that L^j_Nf, g

N= −

x∈TN

Wj

(x+1)/N

−Wj(x/N )

∂_W^N

jf (x/N ) ∂_W^N

jg(x/N ).

Using the spectral theorem, we obtain an orthonormal basisA^j_N= {h^j₁, . . . , h^j_N}ofX_Nformed by the eigenvectors ofL^j_N, i.e.,

L^j_Nh^j_i =α_i^jh^j_i and h^j_i, h^j_k

N=δ_i,k,

whereδi,kis the Kronecker’s delta, which equals 0 ifi=k, and equals 1 ifi=k. SinceL^j_Nis non-positive, we have that the eigenvaluesα_i^jare non-positive:α_i^j≤0,j=1, . . . , dandi=1, . . . , N.

LetAN= {φ₁, . . . , φ_Nd} ⊂X_Nd be set of functions of the formφ_i(x₁, . . . , x_d)=_d

j=1h^j(x_j), withh^j∈A^j_N. Letα^j be the eigenvalue of h^j, i.e., L^j_Nh^j =α^jh^j. The linear operator LN on X_Nd, defined in (4.1), is such that L^j_Nφ_i =α^jφ_i and LNφ_i =_d

j=1α^jφ_i. Furthermore, if φ_i(x₁, . . . , x_d)=_d

j=1h^j(x_j)and φ_k(x₁, . . . , x_d)= _d

j=1g^j(xj), φi, φk∈AN, we have that φ_i, φ_kN^d =

d

j=1

h^j, g^j

N=δ_i,k

fori, k=1, . . . , N^d. So, the setAN is an orthonormal basis ofX_Nd formed by the eigenvectors ofLN andL^j_N. In particular, for eachf ∈X_Nd, there existβ_i∈Rsuch thatf =_N^d

i=1β_iφ_i. Thus, 1

N^d

x∈T^d_N

L^j_Nf (x/N )2

=L^j_Nf²

N^d = L^j_N

N^d

i=1

β_iφ_i

2

N^d

=

N^d

i=1

α^j_iβ_i2

≤

N^d

i=1

_d

j=1

α^j_i 2

(β_i)²= LNf²_Nd = 1 N^d

x∈T^d_N

LNf (x/N )2

,

whereα_i^j≤0 is the eigenvalue of the operatorL^j_Nassociated to the eigenvectorφi. This concludes the proof of the

lemma.