Nonequilibrium fluctuations for a tagged particle in one-dimensional sublinear zero-range processes

www.imstat.org/aihp 2013, Vol. 49, No. 3, 611–637

DOI:10.1214/12-AIHP478

© Association des Publications de l’Institut Henri Poincaré, 2013

Nonequilibrium fluctuations for a tagged particle in one-dimensional sublinear zero-range processes

Milton Jara

, Claudio Landim

and Sunder Sethuraman

aIMPA, Estrada Dona Castorina 110, CEP 22460 Rio de Janeiro, Brasil. E-mail:mjara@impa.br

bCNRS UMR 6085, Avenue de l’Université, BP.12, Technopôle du Madrillet, F76801 Saint-Étienne-du-Rouvray, France. E-mail:landim@impa.br cDepartment of Mathematics, University of Arizona, Tucson, AZ 85721, USA. E-mail:sethuram@iastate.edu

Received 7 July 2011; revised 11 January 2012; accepted 11 January 2012

Abstract. Nonequilibrium fluctuations of a tagged, or distinguished particle in a class of one dimensional mean-zero zero-range systems with sublinear, increasing rates are derived. In Jara–Landim–Sethuraman (Probab. Theory Related Fields 145(2009) 565–590), processes with at least linear rates are considered.

A different approach to establish a main “local replacement” limit is required for sublinear rate systems, given that their mixing properties are much different. The method discussed also allows to capture the fluctuations of a “second-class” particle in unit rate, symmetric zero-range models.

Résumé. Nous démontrons les fluctuations hors d’équilibre d’une particule marquée pour une classe de systèmes de particules à portée nulle uni-dimensionels de moyenne nulle dont le taux de sauts croit de manière sous-linéaire. Dans Jara–Landim–

Sethuraman (Probab. Theory Related Fields 145(2009) 565–590), ce résutat a été démontré pour des processus dont le taux croit au moins linéairement.

La démonstration du lemme de remplacement dans le cas sous-linéaire exige une nouvelle approche en conséquence des diffé- rences entre les propriétés de mélanges des deux processus. La méthode présentée permet également de démontrer les fluctuations d’une particule de deuxième classe dans le modèle à portée nulle symmétrique dont le taux de sauts est égal à 1.

MSC:60K35

Keywords:Interacting; Particle system; Zero-range; Tagged; Nonequilibrium; Diffusion

1. Introduction

Characterizing the motion of a distinguished, or tagged particle interacting with others is a long standing concern in statistical physics, and connects with the problem of establishing a rigorous physical basis of Brownian motion (cf.

[18], Chapters 8.I, 6.II). Because of the particle interaction, the tagged particle is not usually Markovian with respect to its own history, which complicates analysis. However, despite this difficulty, one expects its position to homogenize to a diffusion with parameters given in terms of the “bulk” hydrodynamic density.

Although fluctuations of Markov processes are much examined in the literature (cf. [9]), and there are many central limit theorems for types of tagged particles when the system is in “equilibrium” (cf. [8,16,17]), much less is understood when particlesbothinteract nontrivially, and begin in “nonequilibrium.” Virtually, the only fluctuations work in this case takes advantage of special features in types of exclusion and interacting Brownian motion models, namely [4,5],

1Supported in part by NSF 0906713 and NSA Q H982301010180.

and [3]. Note also “propagation of chaos” results yield homogenization limits for the average tagged particle position in simple exclusion [15].

The main goal of this paper is to give a general method, which takes into account the evolution of the hydro- dynamic density, to capture the “nonequilibrium” fluctuations of a tagged particle in zero-range interacting particle systems. Such systems are well-established and have long served as models for types of queuing, traffic, fluid, gran- ular flow etc. [2]. Informally, they follow a collection of random walks on a lattice which interact in the following way: A particle at a location withkparticles displaces byj with infinitesimal rate(g(k)/k)p(j )where the process rateg:N0→R₊, whereN0= {0,1, . . .}, is a function on the nonnegative integers, andp(·)is a translation-invariant single particle transition probability. The name ‘zero-range’ derives from the fact that the infinitesimal interaction is only with the particle number at a vertex.

As might be suspected, different behaviors may be found by varying the choice of the rateg. For instance, whenp is nearest-neighbor and symmetric, the spectral gap or mixing properties of the zero-range system defined on a cube of widthnwithkparticles depend strongly on the form ofg. For a class of models, whenggrows linearly, the gap is of ordern⁻²and does not depend onk[10]. However, whenggrows at most sublinearly, the gap depends on the number of particlesk. In particular, whengis of formg(x)=x^γ for 0< γ ≤1, the gap is of the ordern⁻²(1+ρ)^γ−1 whereρ=k/n[13]. Also, whengis the unit rate,g(x)=1{x≥1}, the gap is of ordern⁻²(1+ρ)⁻²[12].

In this context, we prove a “nonequilibrium” scaling limit for a tagged particle in a large class of ‘bounded’ or

‘sublinear’ rate one dimensional zero-range interacting particle systems. This article can be thought of as a companion to our previous work [6] which considered the problem in ‘linear growth’ rate zero-range systems. The proof in [6]

relies on an important estimate, a “local” hydrodynamic limit, which however makes strong use of the linear growth assumption on the process rate, in particular, as mentioned above, that spectral gap bounds on a localized cube do not depend on the number of particles in the cube. Unfortunately, this proof does not carry over to the bounded or sublinear rate situation, where the mixing behavior must be more carefully understood.

Our main contribution then is to give a different approach for the “local hydrodynamic replacement” (Theorem2.6) with respect to a class of sublinear rate zero-range models so that the nonequilibrium limit for the tagged particle can be established (Theorem2.2). As in [6], a consequence of the argument is that the limit of the empirical density in the reference frame of the tagged particle can be identified as the hydrodynamic density in the frame of the limit tagged particle diffusion (Theorem2.3).

We remark that the approach taken here with respect to the “local hydrodynamic replacement” is robust enough so that it can apply to determine the nonequilibrium fluctuations of a “second-class” particle, and associated reference frame empirical density, in the symmetric unit rate case, that is wheng(k)=1{k≥1}(Theorems2.4,2.5). This is the first work to address a nonequilibrium central limit theorem for a second-class particle.

We now give a sketch of the results, and discuss afterwards the differences in the argument with [6]. Define configurationsξ ∈N^Z₀ of the zero-range process, so thatξ(x), forx∈Z, denotes the number of particles at sitex in stateξ. With respect to a scale parameter,N ≥1, we diffusively rescale the process, that is space is scaled by N⁻¹and time is speeded up byN². Suppose this system{ξ_t^N: t≥0}begins from a local equilibrium measure with density profileρ₀:R→R₊ (cf. before Theorem2.1). The bulk density evolution is captured by the well-known

“hydrodynamic limit” (Theorem2.1), where the empirical densityπ_t^N,0, the measure found by assigning massN⁻¹ to each particle at timest≥0, converges in probability to an absolutely continuous measureρ(t, u)du, whereρ(t, u) is the solution of a non-linear parabolic equation with initial conditionρ₀.

The main point is to relate the “hydrodynamics” to the tagged particle behavior. LetX_t^Nbe the position of a tagged particle, initially at the origin, at timet. It isn’t difficult to see that the rescaled trajectory{X_t^N/N: 0≤t≤T}is tight in the uniform topology. We now identify the limit points.

It turns out that in mean-zero zero-range processes, as opposed to other models with different interactions,X_t^Nis a square integrable martingale with bounded quadratic variation given by

X^N

t=σ²N² t

0

g(η_s^N(0)) η^N_s (0) ds.

Here,σ²is the variance of the transition probabilityp(·),g(·)is the process rate already mentioned, andη^N_s =τ_XN s ξ_s^N is the state of the process as seen in the reference frame of the tagged particle, where{τx: x∈Z}are translations.

We now observe, if the rescaled position of the tagged particlex^N_t =X^N_t /N converges to some trajectoryx_t, this processx_t inherits the martingale property fromX^N_t . If in additionx_t is continuous, by Levy’s characterization, one needs only to examine the asymptotics of its quadratic variation to identify it.

Denote by{νρ: ρ≥0}the family of invariant measures, indexed by the density, for the process as seen from the tagged particle. Let π_t^N be the associated empirical densityπ_t^N=τ_XN

t π_t^N,0 and suppose that one can replace the integrandg(η^N_s (0))/η_s^N(0)by a function of the empirical density. If we assume “conservation of local equilibrium”

for the reference process, that isπ_t^N converges to a certain density (cf. [7], Chapters I, III, VIII), this function should be in the formh(λ(s,0)), whereh(ρ)is the expected value ofg(η(0))/η(0)under the invariant measureν_ρandλ(s,0) is the density of particles around the tagged particle.

Since we assume X^N_t /N converges to x_t,π_t^N=τ_XN

t π_t^N,0, andπ_t^N,0 converges to ρ(t, u)du, we conclude that λ(s,0)=ρ(s, x_s). Therefore, the quadratic variation of X^N_t /N converges to the quadratic variation of x_t, xt = σ²t

0h(ρ(s, x_s))ds. In particular, by the characterization of continuous martingales,x_tsatisfies dxt=σ

h

ρ(s, x_s) dBs,

whereρis the solution of the hydrodynamic equation,his defined above andBis a Brownian motion.

The main difficulty in this outline is to prove the “conservation of local equilibrium” around the tagged particle, or what we have called “local hydrodynamic replacement.” A major complication is that since the local function is not a space average, one cannot avoid pathologies, such as large densities near the tagged particle, by ‘averaging’ them away as is the case with the usual hydrodynamics. It would seem then that the only robust tools available are local central limit theorems, and spectral gap and localized Dirichlet form estimates. Our argument is in three steps.

Step 1, “local 1-block,” of our method is to replace the quadratic variation integral _t

0g(η_s^N(0))/η^N_s (0)ds by _t

0H (Avη_s^N). Here,H (ρ) is the mean-value ofg(η(0))/η(0)with respect to the reference frame invariant mea- sureν_ρwith densityρ, and Avη^N_s is the local density around the tagged particle in a window of size, an introduced intermediate scale. In step 2, “local 2-block,” we replace the integrand H (Avη^N_s )by(N ε)^{−1 N ε}_x=1τ_xH (Avη^N_s ), the average of its translates in a block of sizeN ε whereεis small. Finally, in step 3, having now brought in anN ε- block average into the integrand, which is a spatial average over O(N )translates of a local function onsites, more usual hydrodynamic techniques can be used to replace it by(N ε)^{−1 N ε}_x=1τ_xH¯_l(AvN κη^N_s ). Here,H¯_l(ρ)is the mean- value ofH_l(η)=H (Avη)with respect to the invariant measureμ_ρin the usual undistinguished particles frame, and AvN κη_s^N, with another introduced small parameterκε1, is an O(N )average which can be written in terms of the reference frame empirical density.

In [6], as mentioned, the main assumption is that the process rategis of “linear order,” a condition which ensures a sharp lower bound on the spectral gap in a box of widthn, independent of the number of particleskin the box, and also which allows for uniform local central limit theorems. Because of such spectral gap bounds,uniformapproximations, with respect to local densities, in the scheme above may be performed. Intuitively, there is a lot of ‘local’ mixing which can be exploited in this model.

However, in the current work, whengis assumed to be ‘sublinear,’ since the spectral gap depends onk, large local densities slow down the mixing behavior, and need to be estimated. This observation would in fact likely prevent the

‘local replacement’ if the integrand of the quadratic variation integral were different. However, under the ‘sublinear’

assumption ong, the functionH (ρ)vanishes as the densityρ diverges, which is useful to temper some of the large density effects. [This is not the case in [6] whereH (ρ)is bounded above and below by constants.] Even so, more con- trol on large densities is needed in all steps 1, 2, and 3. In particular, in step 3, which performs a “global” replacement, ifgis not linearly growing, truncations, which are essential, are untractable in general. For this reason, we assumeg is also increasing, or “attractive,” so that certain couplings can be used (cf. [1]).

Step 2, in which H (Avη^N_s )andτ_xH (Avη^N_s )for 1≤x ≤N ε, functions of averages over distant blocks, are compared, is the most difficult. With the idea that the process mixes faster within each block than between the blocks, each term can be replaced by its conditional expectation given the number of particles in its block. This reduces the between block dynamics to a birth–death process whose mixing properties, with some analysis and the assumption limk↑∞g(k)/k=0, can be estimated and found suitable to complete the step. We remark that this last point rules out the case in [6].

2. Notation and results

Letξ_t= {ξ_t(x): x∈TN}be the zero-range process on the discrete one dimensional torusTN=Z/NZwith single particle transition probabilityp(·)and process rateg:N0→R+. We will assume thatg(0)=0,g(1) >0 and, as is usual when dealing with sublinear rates, thatgis increasing (or “attractive”),g(k+1)≥g(k)fork≥1. In addition, throughout the paper, and in all results, we impose one of the following set of conditions (B) or (SL):

(B) gis bounded: Fork≥1, there are constants 0< a0≤a1such thata0≤g(k)≤a1.

To give the class of sublinear rates considered, letW (l, k)be the inverse of the spectral gap of the process, where pis nearest-neighbor and symmetric, defined on the cubeΛ_l= {−l, . . . , l}withk particles (cf. Section3for more definitions).

(SL1) g is sublinear: limk→∞g(k)= ∞,g(k)/k:N→R₊ is decreasing, and limk→∞g(k)/k=0. In particular, sincegis increasing, there exists constantsa₀, a₁>0 such thata₀≤g(k)andg(k)/k≤a₁,k≥1.

(SL2) gisLipschitz: There is a constanta₂such that|g(k+1)−g(k)| ≤a₂fork≥0.

(SL3) The spectral gap satisfies, for all constantsCandl≥1, that

Nlim↑∞N⁻¹ max

1≤k≤ClogNk²W (l, k)=0. (2.1)

It is proved in Lemma3.2that all processes with bounded rates gsatisfy (2.1). In addition, by the spectral gap estimate [13], processes with ratesg(k)=k^γ for 0< γ ≤1 satisfy (2.1). In addition, we will assume thatpis finite- range, irreducible, and mean-zero, that is:

(MZ) There existsR >0 such thatp(z)=0 for|z|> R, and zp(z)=0.

We also will take the scaling parameterN larger than the support ofp(·).

Denote by Ω_N=N^T₀^N the state space and by ξ the configurations ofΩ_N so thatξ(x),x ∈TN, stands for the number of particles in the sitex for the configurationξ. The zero-range process is a continuous-time Markov chain generated by

(LNf )(ξ )=

x∈TN

z

p(z)g ξ(x)

f ξ^x,x+z

−f (ξ )

, (2.2)

whereξ^x,yrepresents the configuration obtained fromξ by displacing a particle fromxtoy:

ξ^x,y(z)=

ξ(x)−1 forz=x, ξ(y)+1 forz=y, ξ(z) forz=x, y.

The zero-range processξ(t )has a well known explicit family product invariant measuresμ¯ϕ, 0≤ϕ <limg(k)=:

g(∞), onΩ_N defined on the nonnegative integers,

¯ μϕ

ξ(x)=k

= 1 Z_ϕ

ϕ^k

g(k)! fork≥1 and μ¯ϕ

ξ(x)=0

= 1 Z_ϕ,

whereg(k)! =g(1)· · ·g(k) and Z_ϕ is the normalization. Denote byρ(ϕ) the mean of the marginal μ¯_ϕ, ρ(ϕ)=

kkμ_ϕ(ξ(x)=k). Sinceg is increasing, the radius of convergence ofZ_ϕ isg(∞), and limϕ↑g(∞)ρ(ϕ)= ∞. As ρ(0)=0 andρ(ϕ)is strictly increasing, for a given 0≤ρ <∞, there is a unique inverseϕ=ϕ(ρ). Define then the family in terms of the densityρasμ_ρ= ¯μ_ϕ(ρ).

Now consider an initial configurationξ such thatξ(0)≥1, and letΩ_N^∗ ⊂ΩN be the set of such configurations.

Distinguish, or tag one of the particles initially at the origin, and follow its trajectoryXt, jointly with the evolution of the processξ_t. It will be convenient for our purposes to consider the process as seen by the tagged particle. This reference processηt(x)=ξt(x+Xt)is also Markovian and has generator in formLN=L^env_N +L^tp_N, whereL^env_N ,L^tp_N

are defined by L^env_N f

(η)=

x∈TN\{0}

z

p(z)g η(x)

f η^x,x+z

−f (η)

+

y

p(y)g

η(0)η(0)−1 η(0)

f η^0,y

−f (η)

, (2.3)

L^tp_Nf

(η)=

z

p(z)g(η(0)) η(0)

f (θ_zη)−f (η) .

In this formula, the translationθ_zis defined by

(θzη)(x)=

η(x+z) forx=0,−z, η(z)+1 forx=0, η(0)−1 forx= −z.

The operatorL^tp_N corresponds to jumps of the tagged particle, while the operatorL^env_N corresponds to jumps of the other particles, called environment.

A key feature of the tagged motion is that it can be written as a martingale in terms of the reference process:

Xt=

j

j Nt(j )=

j

j Mt(j )+m _t

0

g(η_s(0))

η_s(0) ds=

j

j Mt(j ), (2.4)

where m = _jjp(j )=0 is the mean drift, N_t(j ) is the counting process of translations of size j up to time t, and Mt(j )=Nt(j )−p(j )t

0g(ηs(0))/ηs(0)ds is its corresponding martingale. In addition, M_t²(j )− p(j )_t

0g(η_s(0))/ηs(0)ds are martingales which are orthogonal as jumps are not simultaneous a.s. Hence, the quadratic variation ofX_t isXt=σ²_t

0g(η_s(0))/ηs(0)dswhereσ²= j²p(j ).

For the reference processηt, the “Palm” or origin size biased measures given by dνρ=(η(0)/ρ)dμρ are invariant (cf. [14,16]). Note that ν_ρ is also a product measure whose marginal at the origin differs from that at other points x=0. Here, we takeν₀=δ_d0, the Dirac measure concentrated on the configurationd₀with exactly one particle at the origin, and note thatν_ρconverges toδd₀ asρ↓0.

The families{μ_ρ: ρ≥0}and{ν_ρ: ρ≥0}are stochastically ordered. Indeed, this follows as the marginals ofμ_ρ andνρ are stochastically ordered. Also, since we assume thatgis increasing, the system is “attractive,” that is by the

“basic coupling” (cf. [11]) if dRand dRare initial measures of two processesξ_t andξ_t, and dRdRin stochastic order, then the distributions ofξ_tandξ_tare similarly stochastically ordered [11]. We also note, whenpis symmetric, thatμ_ρandν_ρare reversible with respect toLN, andL_N andL^env_N respectively.

From this point, to avoid uninteresting compactness issues, we define every process in a finite time interval[0, T], whereT <∞is fixed. LetTbe the unit torus and letM₊(T)be the set of positive Radon measures inT.

For a continuous, positive function ρ₀:T→R₊, define μ^N=μ^N_ρ

0(·) as the product measure in Ω_N given by μ^N_ρ

0(·)(η(x)=k)=μρ₀(x/N )(η(x)=k).

Consider the processξ_t^N =:ξ_{t N}2, generated byN²LN starting from the initial measureμ^N. Define the process π_t^N,0inD([0, T],M₊(T)), the space ofM₊(T)valued right-continuous paths with left limits for times 0≤t≤T endowed with the Skorohod topology, as

π_t^N,0(du)= 1 N

x∈TN

ξ_t^N(x)δ_x/N(du),

whereδ_uis the Dirac distribution at the pointu.

The next result, “hydrodynamics,” under the assumptionp(·)is mean-zero, is well known (cf. [1,7]).

Theorem 2.1. For each 0≤t ≤T, π_t^N,0 converges in probability to the deterministic measure ρ(t, u)du,where ρ(t, u)is the solution of the hydrodynamic equation

∂_tρ=σ²∂_x²ϕ(ρ),

ρ(0, u)=ρ₀(u), (2.5)

andϕ(ρ)=

g(ξ(0))dμρ.

We now state results for the tagged particle motion. Define the product measureν^N =ν_ρ^N

0(·) inΩ_N^∗ given by ν_ρ^N

0(·)(η(x)=k)=νρ₀(x/N )(η(x)=k), and letη^N_t =:η_{t N}2 be the process generated byN²LN and starting from the initial measureν^N. Define the empirical measureπ_t^NinD([0, T],M+(T))by

π_t^N(du)= 1 N

x∈TN

η^N_t (x)δ_x/N(du).

Let alsoX^N_t =X_N2t be the position of the tagged particle at timeN²t.

Define also the continuous functionψ:R₊→R₊by ψ (ρ)=

g(η(0)) η(0) dνρ.

Noteψ (ρ)=ϕ(ρ)/ρforρ >0, andψ (0)=g(1). The first main result of the article is to identify the scaling limit of the tagged particle as a diffusion process:

Theorem 2.2. Letx^N_t =X_t^N/N be the rescaled position of the tagged particle for the processξ_t^N.Then,{x_t^N: t∈ [0, T]} converges in distribution in the uniform topology to the diffusion{x_t: t ∈ [0, T]}defined by the stochastic differential equation

dxt=σ

ψ

ρ(t, xt)

dBt, (2.6)

whereB_t is a standard Brownian motion onT,andρ(t, u)is the solution of the hydrodynamic Eq. (2.5)as in Theo- rem2.1.

In terms of this characterization, we can describe the evolution of the empirical measure as seen from the tagged particle:

Theorem 2.3. We have {π_t^N: t ∈ [0, T]} converges in distribution with respect to the Skorohod topology on D([0, T],M₊(T))to the measure-valued process{ρ(t, u+x_t)du: t ∈ [0, T]},whereρ(t, u) is the solution of the hydrodynamic Eq. (2.5)andx_tis given by(2.6).

When the rateg(k)=1{k≥1}, scaling limits of a “second-class” particleXt can also be captured. Informally, such a particle must wait until all the other particles, say “first-class” particles, have left its position before it can displace byj with rate p(j ). More precisely, its dynamics can be described in terms of its reference frame motion.

For an initial configurationξ such thatξ(0)≥1, letζ_t(x)=ξ_t(x+Xt)−δ0,x, whereδ_a,bis Kronecker’s delta, be the system of first-class particles in the reference frame of the second-class particle. The generatorL_N takes form L_N=L^env_N +L^tp_N, where

L^env_N f

(ζ )=

x∈TN

z

p(z)1

ζ (x)≥1 f

ζ^x,x+z

−f (ζ ) , L^tp_Nf

(ζ )=

z

p(z)1

ζ (0)=0

f (τ_zζ )−f (ζ ) ,

whereτ_zis the pure spatial translation byz,(τ_zζ )(y)=ζ (y+z)fory∈TN. Then, as for the regular tagged particle, we have

Xt=

j

jNt(j )=

j

jMt(j )+m _t

0

1

ζ_s(0)=0

ds=

j

jMt(j ),

wherem= jjp(j )=0 is the mean drift,Nt(j )is the counting process of translations of sizej up to timet, and Mt(j )=Nt(j )−p(j )t

01{ζs(0)=0}dsare the associated martingales. As before, sinceM²t(j )−p(j )t

01{ζs(0)= 0}dsare orthogonal martingales, the quadratic variation ofXt isXt=σ²_t

01{ζ_s(0)=0}ds.

For the second-class reference processζ_t, under the assumptionp(·)is symmetric, the family dχρ=(1+ρ)⁻¹× (ζ (0)+1)dμρ for ρ >0 are invariant. We remark that symmetry ofp(·)is needed to showχρ are invariant with respect to the second-class tagged process.

Let χ^N =χ_ρ(^N_·) be the product measure with χ_ρ^N

0(·)(ζ (x)=k) =χ_{ρ0(x/N )}(ζ (x)=k). Let also ζ_t^N =ζ_N2t

be the process generated by N²LN starting from χ^N. Correspondingly, define empirical measure π_t^N,1(du)= (1/N ) _x∈T

Nζ_t^N(x)δ_x/N(du). In addition, let υ(ρ)=

1

ζ (0)=0

dχρ= 1 1+ρ

1

ζ (0)=0

dμρ= 1 (1+ρ)².

In the caseg(k)=1{k≥1},ϕ(ρ)=ρ/[1+ρ]. Denote byρ¹the solution of (2.5) with such functionϕ. We may now state results for the second-class tagged motion.

Theorem 2.4. Suppose p(·) is symmetric. Let y^N_t =Xt^N/N be the rescaled position of the second-class tagged particle for the processζ_t^N.Then,{y_t^N: t∈ [0, T]}converges in distribution in the uniform topology to the diffusion {y_t: t∈ [0, T]}defined by the stochastic differential equation

dyt=σ

υ

ρ¹(t, y_t)

dBt, (2.7)

whereB_t is a standard Brownian motion onT.

Theorem 2.5. Suppose p(·) is symmetric. Then, {π_t^N,1: t ∈ [0, T]} converges in distribution with respect to the Skorohod topology onD([0, T],M+(T)) to the measure-valued process{ρ¹(t, u+y_t)du: t∈ [0, T]} wherey_t is given by(2.7).

The outline of the proofs of Theorems2.2,2.3,2.4and2.5are given at the end of this section. We now state the main replacement estimate with respect to the processη^N_t for a (regular) tagged particle. A similar estimate holds with respect to the processζ_t^Nand a “second-class” particle, stated in the proof of Theorem2.4. As remarked earlier, this replacement estimate is the main ingredient to show Theorems2.2and2.3.

Denote byPνthe probability measure inD([0, T], Ω_N)induced by the processη^N_t , starting from the initial measure ν, and by Eν the corresponding expectation. Whenν=ν^N, we abbreviatePν^N =P^N andEν^N=E^N. With respect to the processξ_t^N, denote byPμthe probability measure inD([0, T], Ω_N)starting from measureμ, and byEμthe associated expectation. Denote also byEμ[h]andhμthe expectation of a functionh:ΩN→Rwith respect to the measureμ; whenμ=ν_ρ, letE_ρ[h],hρ stand forE_νρ[h],hνρ. Define also the inner productf, gμ=E_μ[f g], and covariancef;gμ=E_μ[f g] −E_μ[f]E_μ[g]with the same convention whenμ=ν_ρ. To simplify notation, we will drop the superscriptN in the speeded-up processη^N_t .

Forl≥0, let η^l(x)= 1

2l+1

|y|≤l

η(x+y).

Theorem 2.6. Leth:N→R₊be a nonnegative,bounded,Lipschitz function such that there exists a constantCsuch thath(k)≤C[g(k)/k]fork≥1.Then,

lim sup

l→∞ lim sup

ε→0

lim sup

κ→0

lim sup

N→∞ E^N t

0

h ηs(0)

− 1 εN

εN

x=1

H¯l

η^κN_s (x) ds

=0,

whereH (ρ)=E_νρ[h(η(0))],H_l(η)=H (η^l(0)),andH¯_l(ρ)=E_μρ[H_l].

We now give the outlines of the proof of the main theorems.

Proofs of Theorems2.2and2.3. First, the replacement estimate, Theorem2.6, applies whenh(k)=g(k)/k: Under the assumptions ong, clearlyhis positive, bounded, and Lipschitz. Given Theorem2.6, the proof of the main theorems straightforwardly follow the same steps as in [6]. Namely, (1) tightness is proved for (x_t^N, A^N_t , π_t^N,0, π_t^N) where A^N_t = x_t^N is the quadratic variation of the martingale x_t^N. (2) Using the hydrodynamic limit, Theorem 2.1, one determines the limit points ofπ_t^N,0, andπ_t^N=τ_xN

t π_t^N,0. Limit points ofA^N_t are obtained through the replacement estimate, Theorem2.6. Finally, one obtains that all limits ofx_t^N are characterized as continuous martingales with certain quadratic variations. Theorems2.2and2.3follow now by Levy’s theorem. More details on these last points

can be found in [6].

Proofs of Theorems2.4and2.5. The proofs follow the same scheme as for Theorems2.2and2.3, given a replace- ment estimate. One can rewrite Theorem2.6in terms ofζ_s^N:

lim sup

l→∞ lim sup

ε→0

lim sup

κ→0

lim sup

N→∞ E^Nsec

t 0

h ζ_s(0)

− 1 εN

εN

x=1

H¯_l ζ_s^κN(x)

ds

=0.

Here,h(k)=1{k=0},H (ρ)=E_χρ[h(ζ (0))],H_l(ζ )=H (ζ^l(0)), and H¯_l(ρ)=E_μρ[H_l]. Also,E^N_secis the process expectation with respect toζ_s^N.

Given dχρ=(1+ρ)⁻¹dμρ+ρ(1+ρ)⁻¹dνρ, the proof of this replacement follows quite closely the proof of

Theorem2.6with straightforward modifications.

The plan of the paper now is to give some spectral gap estimates, “global,” “local 1-block” and “local 2-blocks”

estimates in Sections3,4,5, and6, which are used to give the proof of Theorem2.6in Section6.2.

For simplicity in the proofs, we will suppose thatp(·)is symmetric, and nearest-neighbor, but our results hold, with straightforward modifications, whenp(·)is finite-range, irreducible, and mean-zero, because mean-zero zero-range processes are gradient processes.

3. Spectral gap estimates

We discuss some spectral gap bounds which will be useful in the sequel. Forl≥0, letΛl= {x: |x| ≤l}be a cube of length 2l+1 around the origin, and letν_ρ^Λl andμ^Λ_ρ^l be the measuresν_ρandμ_ρrestricted toΛ_l.

Forj≥0, define the sets of configurationsΣ_Λl,j= {η∈N^Λ₀^l: _x∈Λlη(x)=j}, andΣ_Λ^∗

l,j= {η∈N^Λ₀^l: η(0)≥ 1, _x∈Λlη(x)=j}. Define also the canonical measuresν_Λl,j(·)=ν_ρ^Λl(·|Σ_Λ^∗

l,j), andμ_Λl,j(·)=μ^Λ_ρ^l(·|Σ_Λl,j). Note that bothν_Λl,j andμ_Λl,j do not depend onρ.

Denote byLΛ_l,L^env_Λ

l the restrictions of the generatorsLN,L^env_N onΣΛ_l,j,Σ_Λ^∗

l,j, respectively. These generators are obtained by restricting the sums overx, y, z in (2.2) and (2.3) to x,x +z, y∈Λ_l. Clearly, ν_Λl,j, μ_Λl,j are invariant with respect toL^env_Λ

l ,LΛl, respectively. Denote the Dirichlet formsD(μ_Λl,j, f )= f, (−LΛlf )μ_{Λl ,j} and D(ν_Λl,j, f )= f, (−L^env_Λ

lf )ν_{Λl ,j}. One can compute D(μΛ_l,j, f )=1

2

x,y∈Λl

p(y−x)Eμ_{Λl ,j}

g η(x)

f η^x,y

−f (η)2 ,

D(νΛ_l,j, f )=1 2

x∈Λl\{0}

y∈Λl

p(y−x)Eν_{Λl ,j}

g η(x)

f η^x,y

−f (η)2

+1 2

z∈Λ_l

p(z)E_ν

Λl ,j

g

η(0)η(0)−1 η(0)

f η^0,z

−f (η)2 .

LetW (l, j )andW^env(l, j )be the inverse of the spectral gaps of LΛ_l andL^env_Λ

l with respect toΣΛ_l,j andΣ_Λ^∗

l,j

respectively. In particular, the following Poincaré inequalities are satisfied: For allL²functions, f;fμ_{Λl ,j}≤W (l, j )D(μΛ_l,j, f ),

f;fν_{Λl ,j}≤W^env(l, j )D(ν_Λl,j, f ).

In the next two lemmas, we do not assume thatgis increasing. We first relate the environment spectral gap to the untagged process spectral gap.

Lemma 3.1. Suppose on Σ_Λ^∗

l,j that a₁⁻¹ ≤η(0)/g(η(0))≤j a₀⁻¹. Then, for j ≥1, we have that W^env(l, j )≤ (a₁a₀⁻¹j )²W (l, j−1).

Proof. Note E_μρ[g(η(0))f (η)] =ϕ(ρ)E_μρ[f(η)] withf(η)=f (η+d₀), where we recalld₀ is the configura- tion with exactly one particle at the origin. By a suitable change of variables one can show thatD(μ_Λl,j−1, f)≤ a₁a⁻₀¹j D(ν_Λl,j, f ).

By the assumption ong, for everyc∈R, Eν_{Λl ,j}

(f−Eν_{Λl ,j}f )²

≤ E_μρ[η(0)(f−c)²1{Σ_Λ^∗

l,j}]

E_μρ[η(0)1{Σ_Λ^∗

l,j}]

≤a₁a₀⁻¹jE_μρ[g(η(0))(f−c)²1{Σ_Λ^∗

l,j}]

E_μρ[g(η(0))1{Σ_Λ^∗

l,j}] .

The change of variablesη=η−d₀and an appropriate choice of the constantcpermits to rewrite last expression as a₁a⁻₀¹j E_{μΛl ,j−1}

f−E_{μΛl ,j−1}f2

≤a₁a₀⁻¹j W (l, j−1)D

μ_Λl,j−1, f ,

where the last inequality follows from the spectral gap for the zero range process. By the observation made at the beginning of the proof, this expression is bounded by

a₁a⁻₀¹j2

W (l, j−1)D(νΛl,j, f ),

which concludes the proof of the lemma.

Lemma 3.2. Supposegsatisfiesa₀≤g(k)≤a₁fork≥1,andlimk↑∞g(k)=L.For everyα >0,there is a constant B=B_αsuch thatW (l, j )≤B^l(1+α)^j(l+j )².

Proof. We need only to establish, for allL²functionsf, that f;fμ_{Λl ,j}≤B^l(1+α)^j(l+j )²D(μ_Λl,j, f ).

To argue the bound, we make a comparison with the measureμ_Λl,j wheng(k)=1{k≥1}. Denote this measure byμ¹_Λ

l,j, and recall, by conversion to the simple exclusion process (cf. [10], Example 1.1 and [12]), that E_μ1

Λl ,j

(f−E_μ1 Λl ,jf )²

≤b0(l+j )²D_μ1

Λl ,j(f ) (3.1)

for some finite constantb₀. Write E_μ

Λl ,j

(f−E_μ

Λl ,jf )²

=inf

c E_μ

Λl ,j

(f−c)²

=inf

c η

_l

x=−l(ϕ^η(x))/(g(η(x))!)(f (η)−c)²1{ _xη(x)=j}

η

l

x=−l(ϕ^η(x))/(g(η(x))!)1{ _xη(x)=j} .

Without loss of generality, we may now assume thatL=1 since we can replacegby its scaled version,g=g/L, in the above expression.

Forβ >0, letr₀be so large that 1−β≤g(z)≤1+βforz≥r₀. Then, a₁^−r0(1+β)^−η(x)≤ 1

g(η(x))!≤a₀^−r0(1−β)^−η(x).

This bound is achieved by overestimating the firstr₀factors by the bounda₀1{z≥1} ≤g(z)≤a₁, and the remaining factors by

(1+β)^−η(x)≤ 1 η(x)

z=r0+1g(z) ≤(1−β)^−η(x), where by convention an empty product is defined as 1.

As there are 2l+1 sites, we bound the right hand side of the displayed expression appearing just below (3.1) by a₀⁻¹a1

(2l+1)r0

(1+β)(1−β)⁻¹j

E_μ1 Λl ,j

(f −E_μ1 Λl ,jf )²

.

By the spectral gap estimate (3.1) and the same bounds on the Radon–Nikodym derivative of dμ¹_Λ

l,j/dμΛ_l,j, the previous expression is less than or equal to

a₀⁻¹a1

2(2l+1)r0

(1+β)(1−β)⁻¹2j

b0(l+j )²Dμ_{Λl ,j}(f ).

We may now chooseβ=β(α)appropriately to finish the proof.

We claim that for any constantC >0,

Nlim→∞

1

N max

1≤j≤CllogNW^env(l, j )=0. (3.2)

Indeed, under the conditions (SL) this follows by Lemma3.1and by assumption (SL3). On the other hand, under the condition (B), by Lemma3.2we may chooseαappropriately to have max1≤j≤CllogNW (l, j )≤C2()N^1/2. This proves (3.2) in view of Lemma3.1.

4. “Global” replacement

In this section, we replace the full, or “global” empirical average of a local, bounded and Lipschitz function, with respect to the process ηs, in terms of the density field π_s^N. By a local function r:Ω_N^∗ →R, we mean a function r supported on a finite number of occupation variables. In addition, we say that a local function r, supported on coordinatesA⊂Z, is Lipschitz if there exists a finite constantC₀such that

r(η)−r

η≤C₀

x∈A

η(x)−η(x)

for all configurationsη,ηofΩ_N^∗, andNlarger than the support size|A|.

The proof involves only a few changes to the hydrodynamics proof of [1], Theorem 3.2.1, and is similar to that in [6]. However, since the rategis bounded, some details with respect to the “2-blocks” lemma below are different.