Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 21, 1996, 309–352

STOCHASTIC DYNAMICS MACROSCOPICALLY GOVERNED BY THE POROUS MEDIUM EQUATION FOR ISOTHERMAL FLOW

Michael Ekhaus and Timo Seppäläinen

Secure Computing Corporation, 2675 Long Lake Road Roseville, MN 55113, U.S.A.; ekhaus@sctc.com Iowa State University, Department of Mathematics Ames, Iowa 50011-2066, U.S.A.; seppalai@pollux.math.iastate.edu

Abstract. We describe interacting lattice models on the torus whose special feature is that the macroscopic equation of the empirical density is a degenerate parabolic equation, namely the equation of an ideal gas flowing isothermally through a porous medium. The models come in two versions: one with continuous variables and one with particles on the sites. In the particle model a degenerate equation is obtained only if the size of the particle vanishes in the limit, otherwise the limiting equation is a nondegenerate equation that also governs the densities of certain exclusion processes with speed change. We establish basic properties of these models such as attractiveness and reversibility, and prove the hydrodynamic scaling limits for the empirical densities.

1. Introduction and results

The porous medium equation ∂_{t}u = Δ(u^{m}), m > 1, has for some time been among the most intensely studied partial differential equations. A rich theory has developed since the fundamental solutions were found in the early 1950’s in Russia, but results connecting this equation with interesting stochastic dynamics are few. This equation is a degenerate parabolic equation in the sense that when written in the form ∂_{t}u = ∇ · (D(u) ∇u), the diffusion matrix D(u) vanishes for u = 0. In this paper we describe some simple interacting lattice models whose empirical densities obey the porous medium equation ∂_{t}u = Δ(u^{2}) in a hydrodynamical scaling limit.

The stochastic model we study comes in two versions, one with continuous variables (the stick model) and one with discrete variables (the particle model). The stick model is a relative of the linear models discussed in Chapter IX of Liggett’s monograph [L], in the sense that when an event takes place at some time t, the new configuration η_{t} is a linear function of the old one η_{t−}. But the rates are not uniform as in the linear models of [L], for an event takes place at a site x at a rate proportional to the size of the variable η(x). This model is not new. It has been studied earlier in [SU], where H. Tanaka is credited for suggesting the model. The particle model resembles the zero-range process introduced by Spitzer [S], in that particles jump from a site with an intensity determined by the number of particles at that site. The difference is that now more than one particle may jump simultaneously. The size of an individual particle will be an additional parameter of the model, and only if this size vanishes in the limit do we get a degenerate macroscopic equation.

1991 Mathematics Subject Classification: Primary 60K35, 82C22.

Physically the porous medium equation with m = 2 represents the density of an ideal gas flowing isothermally through a porous medium. Whether these stochastic models can be given natural physical interpretations has yet to be determined.

For background on hydrodynamical scaling limits and their context in statistical physics we suggest the monograph of Spohn [Sp], and the lectures of De Masi and Presutti [DP] for a survey of part of the mathematical theory. The papers of Aronson [A] and Vasquez [V] present overviews of the theory of the porous medium equation.

Our paper is intended as the first of a series of studies of interacting particle systems that have zero range of interaction and lead to a degenerate macroscopic equation. These stochastic models will be generalized in a future paper so that porous medium equations with m ≠ 2 appear. We present one proof here in fairly complete technical detail so that similar arguments in later work can do with sketchier proofs, but also with the hope of making our exposition an accessible entry point into the mathematics of hydrodynamic limits. With a wider audience in mind we have provided some explanations that a probabilist may find tedious.

The paper is organized as follows: In this section we describe the models and state the scaling limits in Theorems 1, 2 and 3. Theorem 1 is about the stick model with short-range interactions and the limit comes by diffusion scaling. In Theorem 2 the stick model interacts over an intermediate range and a different scaling is needed for the limit. It is possible to adjust the range of interaction so that the scaling limit comes by hyperbolic scaling. Theorem 3 is about the particle model. After presenting the theorems we describe earlier work relating the porous medium equation to stochastic models. Section 1 concludes with a sketch of the proofs. In Section 2 we study the stochastic models more closely. We show that they are attractive in the interacting particle systems sense and ergodic on certain hyperplanes of the state space. We describe the invariant measures, which are also reversible for the process. In Section 3 we prove the scaling limits for the stick model and in Section 4 for the particle model.

1.1. The short-range stick process. We begin with an informal description of the stick process. Fix a dimension d for the remainder of the paper. Let the scaling parameter N be a large natural number. Write x = (x_{1}, …, x_{d}) for the sites of the lattice Z^{d}. Our process lives on the sites of the cube

Z_{N}^{d} = {x ∈ Z^{d} : 0 ≤ x_{i} < N for i = 1, …, N}

with periodic boundary conditions, that is, coordinatewise addition in Z_{N}^{d} is performed modulo N. A state of the process is an assignment η = (η(x) : x ∈ Z_{N}^{d}) of nonnegative real numbers 0 ≤ η(x) < ∞ to each site x, which could be thought of as the lengths of vertical sticks sitting on the sites. Thus the state space of the process is Ω_{N} = [0, ∞)^{Z_{N}^{d}} . Fix a probability distribution p(z) on Z^{d} such that p(z) = p(−z), thinking of p(y − x) as the step distribution of a random walk on Z^{d}. Let p_{N}(x, y) be the step distribution induced on Z_{N}^{d} given by

p_{N}(x, y) = ∑_{z:z≡ymodN} p(z − x)

where z ≡ y mod N is understood coordinatewise. The symmetry is preserved: p_{N}(x, y) = p_{N}(y, x).

The state evolves in time through events of the following type: When the state of the process is η, each site x ∈ Z_{N}^{d} has an independent exponential clock with rate η(x). When the clock at site x rings, pick a site y with probability p_{N}(x, y), break off a uniformly distributed random piece from the stick at x, and add this piece to the stick at y.

Here is a more precise description for the benefit of the reader not familiar with the jargon of interacting particle systems: Suppose the state of the process is η at some time t. To determine when the process moves from η to another state, imagine given a collection {T_{z} : z ∈ Z_{N}^{d}} of independent random variables, where T_{z} has exponential distribution with expectation 1/η(z). These are the random clocks. Let x be the site whose clock rings first: T_{x} = min_{z∈Z_{N}^{d}} T_{z}. (Since Z_{N}^{d} is finite and the T_{z} ’s have continuous distributions, this description is not problematic: The infimum of the T_{z} ’s is positive and it is realized at exactly one site, almost surely.) It is conventional to construct processes so that their paths are right-continuous in time. Thus we declare that the process remains at η for the time interval [t, t + T_{x}), and at time t′ = t + T_{x} it resides at a new state η′ determined as follows: Pick y as indicated above and pick a random quantity U uniformly distributed on [0, η(x)], both independently of everything else. Then set η′(x) = η(x) − U, η′(y) = η(y) + U, and η′(w) = η(w) for other sites w. Now repeat this cycle, starting with the state η′ at time t′ and with new random clocks independent of the past.

To achieve the correct parabolic or diffusion scaling, we speed up the dynamics by a factor N^{2} . All this is codified in the generator ℒ_{N} that acts on bounded continuous functions f on Ω_{N}:

(1)  ℒ_{N}f(η) = N^{2} ∑_{x∈Z_{N}^{d}} ∫_{0}^{η(x)} ∑_{y∈Z_{N}^{d}} p_{N}(x, y) [f(η^{u,x,y}) − f(η)] du,

where the configuration η^{u,x,y} is defined, for x, y, w ∈ Z_{N}^{d} and 0 ≤ u ≤ η(x), by

(2)  η^{u,x,y}(w) = \{ \begin{matrix}η(x) − u, & w = x, \\ η(y) + u, & w = y, \\ η(w), & w ≠ x, y. \end{matrix}

For a comprehensive treatment of the theory and machinery of interacting particle systems we refer the reader to [L].

Let η_{t} = (η_{t}(x) : x ∈ Z_{N}^{d}), with 0 ≤ t < ∞ the time variable, denote the Markov process we have described, for a fixed N. Instead of running the process on larger and larger cubes Z_{N}^{d} as N increases, we shrink space by a factor of N and imagine that the stick configuration approximates a density on the d-dimensional torus T^{d} = R^{d}/Z^{d}. This notion is captured by the empirical measure α_{t}^{N} determined by the heights of the sticks. Let ℳ denote the space of finite nonnegative Borel measures on T^{d}, topologized weakly by C(T^{d}). Then α_{t}^{N} is the ℳ -valued random variable defined by

α_{t}^{N} = N^{−d} ∑_{x∈Z_{N}^{d}} η_{t}(x) δ_{x/N},

in other words, the integral of a bounded Borel function ϕ on T^{d} against α_{t}^{N} is given by

α_{t}^{N}(ϕ) = N^{−d} ∑_{x∈Z_{N}^{d}} η_{t}(x) ϕ(\frac{x}{N} ).

In general we will replace ‘t’ with ‘ •’ to denote the whole path as a function of t as opposed to a value at a particular time.

As initial data we assume given a nonnegative, bounded Borel function u_{0} on T^{d} that serves as the initial macroscopic density of the sticks. This means that, for large N, the empirical density of the sticks at time 0 approximates the measure u_{0}(ξ) dξ with high probability, where dξ denotes Lebesgue measure on T^{d}, for integration purposes identifiable with [0, 1)^{d}. For this and certain technical reasons we make the following precise assumption on the initial distributions μ_{0}^{N} of the processes:

Assumption 1. The probability distributions μ_{0}^{N} on Ω_{N} satisfy these conditions: The variables η(x), x ∈ Z_{N}^{d}, are independent exponential random variables under μ_{0}^{N}. There is a constant K_{0} bounding the expectations:

μ_{0}^{N}{η(x)} ≤ K_{0} for all N and for all x ∈ Z_{N}^{d}.

The expectations are chosen so that, as N → ∞, α_{0}^{N} → u_{0}(ξ) dξ in μ_{0}^{N} - probability, in the topology of ℳ .

The precise choice of the expectations is immaterial. One can take

μ_{0}^{N}{η(x)} = N^{d} ∫_{(x/N)+[0,1/N)^{d}} u_{0}(ξ) dξ,

or simply μ_{0}^{N}{η(x)} = u_{0}(x/N) if u_{0} is continuous. We also need to make further assumptions about p. Let p^{∗n} denote the nth convolution power of p. In terms of the random walk specified by p, p^{∗n}(x−y) is the probability that, after starting at x and taking n steps, the walker is at y.

Assumption 2. The probability vector p satisfies ∑_{z} p(z)∥z∥^{4} < ∞ and is irreducible in the sense that for each z ∈ Z^{d} there is an n such that p^{∗n}(z) > 0.

Fix a final time T < ∞ and set Q_{T} = [0, T] × T^{d}. Let P^{N} denote the distribution of the process on the path space 𝒟_{Ω_{N}} = 𝒟([0, T], Ω_{N}), with the initial distribution μ_{0}^{N}. 𝒟_{Ω_{N}} is the space of right-continuous functions from [0, T] into Ω_{N} that have a left limit at each point. It is a Polish space under the so-called Skorokhod topology. This is a standard setting for stochastic processes, developed for example in [B] and [EK]. Similarly, 𝒟_{ℳ} = 𝒟([0, T], ℳ) is the space of ℳ - valued paths where αN_{•} takes its values. Set

a_{i,j} = ∑_{z∈Z^{d}} p(z)z_{i}z_{j} for 1 ≤ i, j ≤ d,

where z = (z_{1}, …, z_{d}) denotes a vector in Z^{d}. Write ξ = (ξ_{1}, …, ξ_{d}) for an element of T^{d}. Here is the first scaling limit:

Theorem 1. Assume Assumptions 1 and 2. Then there is a jointly measurable function u(t, ξ) on Q_{T} such that, as N → ∞, αN_{•} → u( • , ξ)dξ in P^{N} - probability, in the topology of 𝒟_{ℳ} .

Furthermore, 0 ≤ u(t, ξ) ≤ ∥u_{0}∥_{∞}, the measure u(t, ξ)dξ is continuous in t, and u(t, ξ) is the unique weak solution of

\frac{∂u}{∂t} = \frac{1}{2} ∑_{i,j} a_{i,j} \frac{∂^{2}}{∂ξ_{i}∂ξ_{j}} (u^{2})

on T^{d} with initial condition u(0, ξ) = u_{0}(ξ).

The porous medium equation as a special case. Let p be the step probability of symmetric nearest-neighbor random walk: p(±e_{i}) = 1/2d for i = 1, …, d, where e_{1} = (1, 0, …, 0), e_{2} = (0, 1, 0, …, 0), etc. Then the limiting equation is the porous medium equation ∂_{t}u = (2d)^{−1}Δ(u^{2}). The constant (2d)^{−1} can be scaled away by multiplying the generator by 2d, in other words, by letting the clocks ring at rate 2d η(x) instead of η(x).

1.2. The long-range stick process. To specify the long range of interaction fix a parameter 0 < α < 1 and let

V_{N} = {x ∈ Z^{d} : |x_{i}| ≤ N^{α} for all i}.

To avoid unnecessary technicalities we consider only the simplest step distribution: The receiving site y is chosen from x + V_{N} uniformly at random, again observing periodic boundary conditions. Otherwise the dynamics of the N th process follow the description given above for the short-range model. To get a meaningful scaling limit we have to compensate for the increased range by decreasing the time speedup: N^{β} with β = 2(1 − α) turns out to be the correct time scaling. Thus the generator of the N th process of the long-range model is

(3)  ℒ_{N}f(η) = N^{β} ∑_{x∈Z_{N}^{d}} ∫_{0}^{η(x)} \frac{1}{|V_{N}|} ∑_{y∈x+V_{N}} [f(η^{u,x,y}) − f(η)] du,

with the same conventions as in (1) above. Define the empirical measure α_{t}^{N} and the distributions μ_{0}^{N} and P^{N} of the process as before. For this model we have the following scaling limit:

Theorem 2. Assume Assumption 1. Then the conclusion of Theorem 1 holds for the sequence of processes with generators (3) and initial distributions μ_{0}^{N}, and the limiting equation is

\frac{∂u}{∂t} = \frac{1}{6} Δ(u^{2}).

Remark on hyperbolic scaling. Setting α = \frac{1}{2} gives β = 1, so in this case the hydrodynamic limit comes by hyperbolic scaling: both space and time are scaled by the factor N. This is potentially of interest for the following reason: Suppose the step distribution p(z) has nonzero expectation. We expect such an asymmetric version of the stick model to obey a nonlinear conservation law under hyperbolic scaling, as is the case for the simple exclusion and zero-range processes (see [R]). Assuming this happens, we can then superimpose on the asymmetric dynamics the long-range dynamics described by the generator (3) with α = \frac{1}{2} and β = 1, and obtain a system that obeys a viscous conservation law. Though not with the usual diffusive part of the linear heat equation, but instead with a nonlinear diffusive part from the porous medium equation. This superposition of short-range asymmetric and long-range symmetric stochastic dynamics would furnish a method for manufacturing stochastic models for viscous conservation laws. An alternative way is to combine the symmetric dynamics with a weakly asymmetric part, as has been done in [DPS], [G], and [KOV].

1.3. The particle process. For the particle models we shall consider only the short-range case. The setting is identical to that of the stick model except for the changes brought about by the discretization of the state space. Fix again N for the moment and a parameter κ_{N} > 0, the particle size. Let η(x) denote the height of the stack of particles at site x, hence an element of the set {kκ_{N} : k = 0, 1, 2, …}. The state space is

Ω_{N}^{(κ_{N} )} = {kκ_{N} : k = 0, 1, 2, …}^{Z_{N}^{d}} .

The clock at site x rings at rate η(x), and then the number k of particles to move is picked uniformly at random from {1, 2, …, η(x)/κ_{N} }. The generator becomes

(4)  ℒ_{N}f(η) = N^{2} ∑_{x,y∈Z_{N}^{d}} p_{N}(x, y) κ_{N} ∑_{k=1}^{η(x)/κ_{N}} [f(η^{kκ_{N} ,x,y}) − f(η)].

Concerning the particle sizes and initial distributions we make the following assumption:

Assumption 1′. There is a number κ ≥ 0 such that κ_{N} → κ as N → ∞.

The probability distributions μ_{0}^{N} on Ω_{N}^{(κ_{N} )} satisfy these conditions: The variables η(x), x ∈ Z_{N}^{d}, are independent geometrically distributed κ_{N} Z_{+} -valued random variables under μ_{0}^{N}, there is a constant K_{0} such that μ_{0}^{N}{η(x)} ≤ K_{0} for all N and for all x ∈ Z_{N}^{d}, and the expectations are chosen so that, as N → ∞, α_{0}^{N} → u_{0}(ξ) dξ in μ_{0}^{N} -probability, in the topology of ℳ .

More precisely, the initial probabilities of individual stack heights are given by

μ_{0}^{N}\{η : η(x) = kκ_{N} \} = \frac{κ_{N} (μ_{0}^{N}\{η(x)\})^{k}}{(κ_{N} + μ_{0}^{N}\{η(x)\})^{k+1}} for x ∈ Z_{N}^{d} and k ≥ 0.

Though the state space of the particle process changes with N, the empirical measure α_{t}^{N} is ℳ -valued for each N, so it makes again sense to ask about the convergence of the random ℳ -valued path αN_{•}.

Theorem 3. Assume Assumptions 1′ and 2. Then the statement of Theorem 1 holds for the particle processes with generators (4) and initial distributions μ_{0}^{N}, with the limiting equation

(5)  \frac{∂u}{∂t} = \frac{1}{2} ∑_{i,j} a_{i,j} \frac{∂^{2}}{∂ξ_{i}∂ξ_{j}} (κu + u^{2}).

In particular, in the case κ = 0 we get the same equation as in Theorem 1.

1.4. Earlier related work. Let us first point out that the hydrodynamic limit of the zero-range process does not yield a degenerate equation, at least under the assumptions employed in [DP]. The diffusion constant of the macroscopic equation is given by D(u) = z′(u) where z denotes fugacity (see Theorem 3.2.1 in [DP]). A computation shows that z′(u) > 0 for all u ≥ 0, and in particular z′(0) = c(1) which is positive by assumption.

As far as we know, the earliest constructions of stochastic models for the porous medium equation are by M. Inoue with an approach completely different from ours. He applied difference schemes to construct diffusion processes whose probability densities are solutions to the porous medium equation [I1] and particle systems converging to solutions of the equation [I2].

The hydrodynamics of the basic stick model with symmetric nearest-neighbor exchanges was studied by Y. Suzuki and K. Uchiyama [SU]. In their approach this model is embedded in a family of not necessarily attractive processes. Hence they do not make use of attractiveness and are led to arguments different from ours. Both approaches have their advantages: The result in [SU] allows for more general initial distributions than our Theorem 1, subject to the restriction that this initial distribution be absolutely continuous with respect to an i.i.d. exponential distribution on the sticks. In other words, the initial stick heights must all be strictly positive almost surely. Our Theorem 1 does not have this restriction. The equality μ_{0}^{N}{η(x)} = 0 is permitted, so the stochastic model may start with some sites initially empty. This follows naturally from attractiveness, as the reader will see in Section 3.6.

It is also believed that the porous medium equation is connected with lattice gas dynamics at critical temperature, see Section II.3.3 in [Sp]. Rigorous results in this direction have been obtained in [LOP], where the actual model studied is a symmetric exclusion process with a weak asymmetry that involves interactions over a microscopically long range. In the critical case the density profile f obeys the macroscopic equation ∂_{t}f = ∂_{ξ}[(\frac{1}{2} − 2f(1 − f))∂_{ξ}f] provided that the initial profile f_{0} satisfies either 0 ≤ f_{0} < \frac{1}{2} or \frac{1}{2} < f_{0} ≤ 1. If we now set f = \frac{1}{2} − u or f = \frac{1}{2} + u, depending on the case, then u satisfies the porous medium equation.

For results about deterministic particles whose empirical density obeys the porous medium equation in an infinite particle limit, see [O], [U], and their references.

The equation (5) with κ > 0 also governs the densities of certain exclusion processes with speed change. This example is one of a class treated by T. Funaki, K. Handa, and K. Uchiyama [FHU], also presented in Section II.3.2 of [Sp]. Consider the configuration space {0, 1}^{N} and the generator

L_{N}f(η) = N^{2} ∑_{x=0}^{N−1} c(x, x + 1, η) [f(η^{x,x+1}) − f(η)]

with exchange rates

c(x, x + 1, η) = (η(x) − η(x + 1))^{2}[1 + α(η(x − 1) + η(x + 2))],

where α is a constant satisfying 1 + 2α > 0, η^{x,x+1} is the configuration got from η by interchanging η(x) and η(x + 1), and x + 1 is taken modulo N. The macroscopic equation for the empirical density of this process is ∂_{t}u = ∂_{ξ}^{2}(u+αu^{2}) on the unit circle. Taking α > 0 and multiplying the generator by α^{−1} then gives (5) with κ = α^{−1} .

1.5. The proofs at a glance. Here is a sketch of the proof of Theorem 1. Writing

Aϕ = ∑_{1≤i,j≤d} a_{i,j}∂_{ξ_{i}}∂_{ξ_{j}}ϕ

for a smooth test function ϕ, the usual martingale arguments show that the equality

(6)  α_{t}^{N}(ϕ) − α_{0}^{N}(ϕ) = ∫_{0}^{t} \frac{1}{4}N^{−d} ∑_{x∈Z_{N}^{d}} Aϕ(\frac{x}{N} ) η_{s}^{2}(x) ds

holds approximately, with high probability. In a cube x + Λ_{Nε} of intermediate scale ε, N^{−1} ≪ ε ≪ 1, the sticks are almost in equilibrium. They behave approximately as i.i.d. exponential random variables with expectation given by the local empirical mean

\frac{1}{|Λ_{Nε}|} ∑_{y∈x+Λ_{Nε}} η_{t}(y).

Since Aϕ is nearly constant across Λ_{Nε}, this turns (6) into

α_{t}^{N}(ϕ) − α_{0}^{N}(ϕ) = ∫_{0}^{t} \frac{1}{2}N^{−d} ∑_{x∈Z_{N}^{d}} Aϕ(\frac{x}{N} ) (\frac{1}{|Λ_{Nε}|} ∑_{y∈x+Λ_{Nε}} η_{s}(y))^{2} ds,

and upon introducing χ_{ε,ξ}, the characteristic function of the cube ξ + [0, ε)^{d} in T^{d} normalized by the volume ε^{d}, we have that

α_{t}^{N}(ϕ) − α_{0}^{N}(ϕ) = \frac{1}{2} ∫_{0}^{t} ∫_{T^{d}} Aϕ(ξ) [α_{s}^{N}(χ_{ε,ξ})]^{2} dξ ds,

again approximately and with high probability. In the limit N → ∞, α_{t}^{N} has a density u(t, ξ), and after also letting ε ↘ 0 we recover the weak form of the differential equation:

∫_{T^{d}} u(t, ξ) ϕ(ξ) dξ − ∫_{T^{d}} u_{0}(ξ) ϕ(ξ) dξ = \frac{1}{2} ∫_{0}^{t} ∫_{T^{d}} Aϕ(ξ) u^{2}(s, ξ) dξ ds.

To prove the local equilibrium we use the method of entropy estimates developed by Guo, Papanicolau and Varadhan [GPV]. However, since the equilibrium distribution is exponential, an entropy estimate alone cannot control higher moments. That is, if ν is an exponential distribution and μ some other probability measure on [0, ∞), an entropy bound H(μ | ν) ≤ C does not guarantee that ∫ s^{1+ε} μ(ds) < ∞. To control moments we use a priori bounds that come from the attractiveness of the stochastic process. The degeneracy presents another source of trouble for the entropy estimate: Suppose the initial density u_{0} equals zero on a set with nonempty interior. Then there will be sites at which the initial distribution is a unit mass at zero. But δ_{0} is not absolutely continuous with respect to a nondegenerate exponential distribution, and we would have infinite entropy. Thus we prove Theorem 1 in two parts: We first assume that u_{0} is bounded away from zero, and remove this assumption only at the very end.

The proof of Theorem 2 for the long-range model does not introduce anything conceptually new, but some estimates need different proofs for the two cases. The proof of Theorem 3 proceeds along similar lines, and in Section 4 we briefly touch on those aspects of its proof that differ from the earlier arguments.

Acknowledgements. We gratefully acknowledge valuable advice from D. Aronson and from S.R.S. Varadhan, especially for suggesting the argument utilized in Section 3.6. The second author also thanks the IMA at the University of Minnesota and the Institut Mittag-Leffler for their hospitality.

2. The stochastic processes

2.1. The stick process. We now take a closer look at the properties of the Markov processes and begin with a concrete construction of the stick process simultaneously for all initial states η on a single probability space. Such a construction is called a coupling in the probability literature. Fix N and the distribution p. The time scale factor plays no role here, so we leave out N^{2} and N^{β} from (1) and (3), respectively, and then (3) becomes a special case of (1).

The probability space is constructed by first giving each site x ∈ Z_{N}^{d} an independent copy 𝒜^{x} of a Poisson point process on the positive quadrant {(t, b) : t ≥ 0, b ≥ 0} of the plane, with Lebesgue measure as the intensity measure. (Think of the coordinate t as time and b as the height of the stick at x.) Next, for each point (t, b) ∈ 𝒜^{x}, pick a site y = y_{(t,b)}^{x} with probability p_{N}(x, y) independently of everything else. These are the random choices needed.

Given an initial state η, the process η_{t} is defined as a 𝒟([0, ∞), Ω_{N})-valued function of η and the random variables {𝒜^{x}}, {y_{(t,b)}^{x}}. The distribution P^{η} of the process starting at η is then the probability measure on 𝒟([0, ∞), Ω_{N}) induced by this function. Let

T_{x} = inf{t : (t, b) ∈ 𝒜^{x}, b ≤ η(x)}

be the first time t that a point of 𝒜^{x} is contained in the rectangle [0, t]×[0, η(x)]. Take this to mean that the clock rang at x. T_{x} is exponentially distributed with rate η(x) by the definition of a Poisson point process. Let x be the site whose clock rang first and (t, b) ∈ 𝒜^{x} the point that triggered the event. Let y = y_{(t,b)}^{x} be the receiving site chosen for this point. Declare η_{s} = η for 0 ≤ s < t, and the new state η_{t} at time t is obtained by setting η_{t}(x) = b, η_{t}(y) = η(y) + η(x) − b, and by leaving the other sticks intact. Now start over again with η_{t} the current state and consider the point processes on [t, ∞)×[0, ∞). Note that a.s. there is no other point with the same t-coordinate. Note also that, conditioned on b ≤ η(x), b is uniformly distributed on [0, η(x)], and hence the piece we broke off η(x) was uniformly distributed. Each point process has only finitely many points in each rectangle [0, t] × [0, ∑_{x} η(x)], so to determine the state at time t a new state needs to be computed only finitely many times. Thus this construction defines the process for all times 0 ≤ t < ∞. It is also not hard to derive the generator from this description and arrive at (1) without the N^{2} time scale factor.

With this coupling, proof of attractiveness reduces to a mere observation, and the reader who already made this observation is invited to skip the next proof. Given η and ζ in Ω_{N}, let η_{t} and ζ_{t} denote the processes with initial conditions η and ζ, respectively, constructed as above as functions on the probability space of the point processes {𝒜^{x}} and the receiving sites {y_{(t,b)}^{x}}. Let P^{η,ζ} be the probability measure on 𝒟([0, ∞), Ω_{N} × Ω_{N}) induced by the function (η_{t}, ζ_{t}). P^{η,ζ} is the distribution of the joint process (η_{t}, ζ_{t}) starting at (η, ζ).

Lemma 2.1. Assume η ≤ ζ (pointwise for all x). Then the joint process constructed above satisfies P^{η,ζ}{η_{t} ≤ ζ_{t}} = 1 for all t ≥ 0.

Proof. Pick and fix a realization of {𝒜^{x}} and {y_{(t,b)}^{x}}. We shall argue that η_{t} ≤ ζ_{t}, while true for t = 0 by assumption, continues to hold for all t > 0. It is true for 0 ≤ t < ε for some ε > 0 because there is a strictly positive time before the process jumps. Assume it is true for all times 0 ≤ t < s, and a clock rings at x at time s. By assumption, η_{s−} ≤ ζ_{s−}, so if (s, b) is the point that triggered the event, either b ≤ η_{s−}(x) ≤ ζ_{s−}(x) or η_{s−}(x) < b ≤ ζ_{s−}(x). In the first case η_{s}(x) = ζ_{s}(x) = b and η_{s}(y) = η_{s−}(y) + η_{s−}(x) − b ≤ ζ_{s−}(y) + ζ_{s−}(x) − b = ζ_{s}(y), where y = y_{(t,b)}^{x} is the receiving site; and similarly the inequality is preserved in the second case. Thus η_{t} ≤ ζ_{t} holds for 0 ≤ t < s + ε where ε > 0 is the time to the next jump after s. Again because only finitely many jumps take place in a finite time interval, η_{t} ≤ ζ_{t} holds for all t and for each realization of {𝒜^{x}} and {y_{(t,b)}^{x}}, hence in particular with probability 1.

Next we characterize the invariant measures of the stick process and prove that these measures are reversible for the dynamics. The total length ∑_{x} η(x) is obviously conserved under the dynamics, hence it is natural to study the process on the hyperplanes

Ω_{N,λ} = {η ∈ Ω_{N} : ∑_{x} η(x) = λ}

for λ ≥ 0. Each Ω_{N,λ} supports a probability measure m_{λ} that could be heuristically described as ‘the conditional distribution of Lebesgue measure, given that ∑_{x} η(x) = λ’. Precisely speaking, we define the integral of a bounded Borel function g on Ω_{N} against m_{λ}, for λ > 0, by

\begin{aligned}m_{λ}(g) = & \frac{(N^{d} − 1)!}{λ^{N^{d}−1}} ∫_{0}^{λ} dη(x_{1}) ∫_{0}^{λ−η(x_{1})} dη(x_{2}) ⋯ ∫_{0}^{λ−∑_{i=1}^{N^{d}−2} η(x_{i})} dη(x_{N^{d}−1}) \\ & × g(η(x_{1}), η(x_{2}), …, η(x_{N^{d}−1}), λ − ∑_{i=1}^{N^{d}−1} η(x_{i})). \end{aligned}

Here {x_{1}, x_{2}, …, x_{N^{d}}} is an arbitrary ordering of the sites in Z_{N}^{d}. In the course of the proof of the next lemma we show that the process is ergodic on each hyperplane Ω_{N,λ} and m_{λ} is the unique invariant measure on Ω_{N,λ}.

Lemma 2.2. A probability measure μ on Ω_{N} is invariant for the process if and only if it is a mixture of the m_{λ} ’s. Furthermore, each such mixture is reversible for the process.

Before proving Lemma 2.2, we wish to point out that i.i.d. exponential distributions are mixtures of m_{λ} ’s. Let

(7)  γ_{r}(dw) = I_{\{w>0\}} \frac{1}{r} e^{−w/r}dw

denote the exponential distribution with expectation r. Then

γ_{r}^{⊗Z_{N}^{d}} (dη) = \frac{r^{−N^{d}}}{(N^{d} − 1)!} ∫_{0}^{∞} m_{λ}(dη) λ^{N^{d}−1}e^{−λ/r} dλ.

The mixing measure is the law of ∑_{x} η(x) under γ_{r}^{⊗Z_{N}^{d}} , namely a gamma distribution with parameters 1/r and N^{d}. Furthermore, if λ/N^{d} → r as N → ∞, then m_{λ} converges weakly to γ_{r}^{⊗Z^{d}}. This suggests that as we proceed towards the scaling limit, the only relevant equilibria are the i.i.d. exponentials and their mixtures. This becomes clear in the course of the proof: see Proposition 3.7 where the local equilibrium is established.

Now for the proof of Lemma 2.2. First we show that on Ω_{N,λ} any two initial states eventually couple.

Lemma 2.3. Suppose η, ζ ∈ Ω_{N,λ}. Then the joint process (η_{t}, ζ_{t}) can be defined so that P^{η,ζ}{η_{t} = ζ_{t} for all large enough t} = 1.

Proof. We ask the reader to consider another equivalent way of constructing the process: Instead of giving each site an individual clock, we take only one clock whose rate is the sum of the stick lengths. When it rings, we pick the site x that gives off a piece with probability proportional to its stick length, and then proceed as before. Except that to couple two processes, the clock needs to ring at a rate \overline{λ}_{t} = ∑_{x} \overline{η}_{t}(x), where \overline{η}_{t}(x) = η_{t}(x) ∨ ζ_{t}(x). If the clock rings at time t, pick a site x with probability \overline{η}_{t−}(x)/\overline{λ}_{t−}, pick a random quantity B uniformly distributed on [0, \overline{η}_{t−}(x)], and pick a site y with probability p_{N}(x, y). The new state is defined by letting η_{t}(x) = η_{t−}(x)∧B and η_{t}(y) = η_{t−}(y)+[η_{t−}(x)−η_{t}(x)], and similarly for ζ_{t}. Then \overline{η}_{t} and \overline{λ}_{t} are updated appropriately. In other words, the taller one of η_{t−}(x) and ζ_{t−}(x) gives off a piece to the site y, and the shorter may or may not, depending on whether it is above or below the cutoff height B. We leave it to the reader to verify that the marginals of this joint process on the η and ζ coordinates are indeed again processes with generator (1), without the N^{2} . It is obvious that, if η_{t} = ζ_{t} for some time t, then η_{t} = ζ_{t} for all later times t too. Note that earlier we denoted by U the piece of stick that was moved and now by B the piece that remains.

The argument that η_{t} and ζ_{t} eventually couple proceeds as follows. Fix a sequence x_{0}, x_{1}, x_{2}, …, x_{s} = x_{0} of sites such that every site appears at least once and p_{N}(x_{i}, x_{i+1}) > 0 for each i. Let γ = λ/4s^{2} . Rotate the labels of the x_{0}, …, x_{s} so that η(x_{0}) ≥ 4sγ. Such a site x_{0} must exist since ∑_{x} η(x) = λ. Let E_{i} be the following event: When the ith clock rings, x = x_{i−1} , y = x_{i} and B ∈ [γ, 2γ]. (We continue to use x to denote the site that gives off a piece and y to denote the receiving site.) Consider what happens to η on the intersection E_{1} ∩ ⋯ ∩ E_{s}: For i = 1, …, s, as the ith clock rings, the stick at x_{i−1} is cut and the piece is passed on to x_{i}. At the ith ring site x_{i} receives a piece of length at least (4s − 2i)γ, hence the stick left at x_{i} after the (i + 1)st ring has length at least γ. Finally, after s rings, all sticks for η have length at least γ.

While we went through this cycle, somewhere a ζ -stick of length at least 4sγ was cut. The piece was passed along, and each time it lost at most 2γ of its length, hence after the s first rings, the ζ -stick at x_{0} has length at least 2sγ. Now repeat the cycle, replacing γ with \frac{1}{2}γ. In other words, let E_{s+j} be the event: When the (s + j)th clock rings, x = x_{j−1} , y = x_{j} and B ∈ [\frac{1}{2}γ, γ]. Let E = E_{1} ∩ ⋯ ∩ E_{s} ∩ E_{s+1} ∩ ⋯ ∩ E_{2s}. The point to notice is that after 2s rings η = ζ on the event E. This is because the second round equalizes the η - and ζ -sticks at each site in turn. Before the (s + j + 1)st ring, both sticks at x_{j} have length at least γ, and after the (s + j + 1)st ring they have a common length B ∈ [\frac{1}{2}γ, γ], for j = 0, …, s − 1; the (2s)th ring puts the remaining total length to the site x_{0} .

Let the superscript (i) denote a value after the (i − 1)st but before the ith ring. At each ring 1 ≤ i ≤ s, assuming that E_{1} ∩ ⋯ ∩ E_{i−1} occurred, the probability of choosing x = x_{i−1} , y = x_{i}, and B ∈ [γ, 2γ] equals

(8)  \frac{\overline{η}^{(i)}(x_{i−1})}{\overline{λ}^{(i)}} · p_{N}(x_{i−1}, x_{i}) · \frac{γ}{\overline{η}^{(i)}(x_{i−1})} ≥ \frac{γ}{2λ} · p_{N}(x_{i−1}, x_{i}).

Note that B ∈ [γ, 2γ] has probability γ/\overline{η}^{(i)}(x_{i−1}) at the ith ring due to the conditioning on E_{1} ∩ ⋯ ∩ E_{i−1} because then η^{(i)}(x_{i−1}) ≥ (4s − 2(i − 1))γ ≥ 2γ. Bound (8) works for s + 1 ≤ i ≤ 2s upon replacing γ by \frac{1}{2}γ. Thus there is a number ε_{0} > 0 such that, conditioned on E_{1} ∩ ⋯ ∩ E_{i−1} and the ith ring, E_{i} occurs with at least probability ε_{0} .

Now we include the clocks. Fix R > 0, and let G_{i} be the event: The clock rings exactly once in the time interval [(i−1)R, iR) and when it rings E_{i} happens.

Let T_{i} denote an exponential clock with rate \overline{λ}^{(i)} . Conditioned on G_{1} ∩⋯∩G_{i−1} , the probability of G_{i} equals

\begin{aligned}∫_{0}^{R} P^{η^{(i)},ζ^{(i)}} & (E_{i} ∩ \{T_{i+1} ≥ R − t\})P^{η^{(i)},ζ^{(i)}}(T_{i} ∈ dt) \\ & ≥ ∫_{0}^{R} ε_{0} exp[−\overline{λ}^{(i+1)}(R − t)]\overline{λ}^{(i)} exp[−\overline{λ}^{(i)}t] dt ≥ ε_{1} \end{aligned}

for a constant ε_{1} > 0, uniformly over 1 ≤ i ≤ s, η^{(i)} , and ζ^{(i)} , because λ ≤ \overline{λ}^{(i)} ≤ 2λ holds for all i. G_{i} is measurable with respect to the process on the time interval [(i − 1)R, iR), so the Markov property gives, with T = 2sR,

P^{η,ζ}{η_{t} = ζ_{t} for t ≥ T } ≥ P^{η,ζ}(G_{1} ∩ ⋯ ∩ G_{2s}) ≥ ε_{1}^{2s} ≡ ε_{2}.

This bound holds uniformly over all starting states (η, ζ) ∈ Ω_{N,λ} × Ω_{N,λ}, hence again by the Markov property

P^{η,ζ}{η_{kT} ≠ ζ_{kT} } ≤ (1 − ε_{2})P^{η,ζ}{η_{(k−1)T} ≠ ζ_{(k−1)T} } ≤ ⋯ ≤ (1 − ε_{2})^{k}.

An application of the Borel–Cantelli lemma completes the proof.

Proof of Lemma 2.2. We start by verifying the formula

(9)  m_{λ}{∫_{0}^{η(x)} h(u, η^{u,x,y}) du} = m_{λ}{∫_{0}^{η(y)} h(u, η) du},

valid for bounded Borel functions h on R × Ω_{N} and all sites x and y, with η^{u,x,y} defined by (2). The left-hand side equals

∫_{0}^{λ} dη(x) ∫_{0}^{λ−η(x)} dη(y) ⋯ ∫_{0}^{η(x)} du h(u, η^{u,x,y})

= ∫_{0}^{λ} du ∫_{0}^{λ−u} dη(y) ∫_{u}^{λ−η(y)} dη(x) ⋯ h(u, η^{u,x,y})

where we changed the integration order and ‘⋯’ denotes the integrals over all the other sticks except η(x) and η(y). Now do the change of variable

\{ \begin{matrix}ω(x) = η(x) − u, \\ ω(y) = η(y) + u \end{matrix}

and again change integration order. This yields the right-hand side of (9). To prove that m_{λ} is reversible, we need to show that

(10)  m_{λ}(f ℒ_{N}g) = m_{λ}(g ℒ_{N}f).

The left-hand side equals

∑_{x,y} p_{N}(x, y)m_{λ}{∫_{0}^{η(x)} f(η)g(η^{u,x,y}) du} − ∑_{x,y} p_{N}(x, y)m_{λ}{∫_{0}^{η(x)} f(η)g(η) du}.

Consider a term in the first sum for fixed x and y. Define

h(u, η) = f(η^{u,y,x})g(η) I_{{η(y)≥u}}.

Then the integrand equals h(u, η^{u,x,y}), and applying (9) shows that the first sum equals

∑_{x,y} p_{N}(x, y)m_{λ}{∫_{0}^{η(y)} f(η^{u,y,x})g(η) du}

which in turn equals

∑_{x,y} p_{N}(x, y)m_{λ}{∫_{0}^{η(x)} f(η^{u,x,y})g(η) du}

by the symmetry of p_{N}. The equation (10) follows. We have proved that the m_{λ} ’s, and consequently their mixtures too, are reversible invariant measures for the dynamics.

Let now μ be an arbitrary invariant probability measure for the process on Ω_{N,λ} and f a bounded Borel function on Ω_{N,λ}. By the invariance,

\begin{aligned}|m_{λ}(f) − μ(f)| & = |∫ E^{η,ζ}\{f(η_{t}) − f(ζ_{t})\} m_{λ}(dη) μ(dζ)| \\ & ≤ 2∥f∥ ∫ P^{η,ζ}\{η_{t} ≠ ζ_{t}\} m_{λ}(dη) μ(dζ) ⟶ 0 \end{aligned}

as t → ∞, by Lemma 2.3. Thus μ = m_{λ}.

Finally, let μ be an invariant probability measure on Ω_{N}. Let ν be the distribution of ∑_{x} η(x) under μ, and μ_{λ} its conditional distribution on Ω_{N,λ}. To show that μ = ∫ m_{λ} ν(dλ), it suffices to show, by the previous paragraph, that ν -almost every μ_{λ} is invariant. Let f be a function on Ω_{N} and g a function on [0, ∞). In the next calculation use the definitions of ν and μ_{λ}, the invariance of μ and the fact that ∑_{x} η(x) is conserved:

\begin{aligned}ν\{μ_{λ}(f) g(λ)\} & = μ\{f(η) g(∑_{x} η(x))\} = μ\{E^{η}[f(η_{t})g(∑_{x} η_{t}(x))]\} \\ & = μ\{E^{η}f(η_{t}) g(∑_{x} η(x))\} = ν\{μ_{λ}(E^{η}f(η_{t}))g(λ)\}. \end{aligned}

Thus μ_{λ}(f) = μ_{λ}(E^{η}f(η_{t})) for ν -a.e. λ, and letting f vary over a countable collection of functions that separates measures shows that almost every μ_{λ} is invariant. This shows that μ is a mixture of m_{λ} ’s and completes the proof of Lemma 2.2.

2.2. The particle process. This analysis is considerably easier for the particle models whose state spaces are countable, and we simply record the facts, leaving the details to the reader. The particle processes are attractive. The invariant hyperplanes

Ω_{N,ℓ} = {η ∈ Ω_{N}^{(κ_{N} )} : ∑_{x} η(x) = ℓκ_{N} }

for integral ℓ are finite sets, and the unique invariant measure on Ω_{N,ℓ} is the uniform distribution m_{ℓ} that gives equal probability to each configuration. The formula corresponding to (10) now reads

(11)  m_{ℓ}{ ∑_{k=1}^{η(x)/κ_{N}} h(kκ_{N} , η^{kκ_{N} ,x,y})} = m_{ℓ}{ ∑_{k=1}^{η(y)/κ_{N}} h(kκ_{N} , η)}.

The asymptotic analysis as N → ∞ will be slightly different for the cases κ = 0 and κ > 0. The former case returns to the stick model with the relevant equilibria given by i.i.d. exponential distributions. For the latter case the equilibria will be i.i.d. geometric distributions.

3. Proofs of Theorems 1 and 2

Throughout this section, for each N at a time, the process η_{t} denotes either the short-range or the long-range stick process with generator (1) or (3), respectively, initial distribution μ_{0}^{N} on Ω_{N} and distribution P^{N} on the path space 𝒟_{Ω_{N}} . The distribution of the process on Ω_{N} at time t is denoted by μ_{t}^{N}. The proofs of Theorems 1 and 2 are carried out simultaneously with unified notation, and with separate arguments furnished only when necessary. We first treat the case where u_{0} is bounded away from zero:

Assumption 3. For some constant ε_{0} > 0 and all ξ ∈ T^{d}, N ∈ N, and x ∈ Z_{N}^{d}, u_{0}(ξ) ≥ ε_{0} and μ_{0}^{N}{η(x)} ≥ ε_{0} .

Here is an outline of the proof:

3.1. Preliminaries. The topology of ℳ . The a priori estimate. The martingales governing the evolution of α_{t}^{N}.

3.2. The distributions of αN_{•}. The distributions of αN_{•} are tight in 𝒟_{ℳ} , and every limit point 𝒫 is supported by elements ω_{•} ∈ C([0, T], ℳ) such that 𝒫(dω) ⊗ dt-almost every ω_{t} has a density with respect to Lebesgue measure. These are consequences of the a priori estimate alone.

3.3. The local equilibrium. We introduce an intermediate scale ε so that N^{−1} ≪ ε ≪ 1 and prove that in a cube of size Nε the sticks behave almost like i.i.d. exponential random variables. The entropy bound needed for this step utilizes Assumption 3.

3.4. Further technicalities. The final steps needed for proving that weak limits of the distributions of αN_{•} are supported by weak solutions of the differential equation.

3.5. Uniqueness. The weak convergence is upgraded to convergence in probability by showing that a solution to the differential equation is unique. This step is independent of Assumption 3.

3.6. Removing Assumption 3. We let ε_{0} ↘ 0 in Assumption 3 and argue that in the limit we recover Theorems 1 and 2 as stated.

3.1. Preliminaries. Let {ϕ_{k}} be a countable set of smooth functions on T^{d} such that ϕ_{1} ≡ 1, ∥ϕ_{k}∥_{∞} ≤ 1 for all k and the span of {ϕ_{k}} is dense in the space C(T^{d}). Then the C(T^{d})-topology of ℳ can be metrized by

(12)  r(μ, ν) = ∑_{k=1}^{∞} 2^{−k−1}|μ(ϕ_{k}) − ν(ϕ_{k})|.

Lemma 3.1. (ℳ , r) is a complete separable metric space.

Proof. Let {μ_{n}} be a Cauchy sequence in the metric r. The sequence {μ_{n}(ϕ_{1})} is Cauchy, hence converges. If lim_{n→∞} μ_{n}(ϕ_{1}) = 0, then μ_{n} → 0 in the metric r. So suppose lim_{n→∞} μ_{n}(ϕ_{1}) = a > 0. An easy computation shows that the sequence {μ̃_{n}}, μ̃_{n} = μ_{n}/μ_{n}(ϕ_{1}), is also Cauchy in the metric r. The measures μ̃_{n} are elements of the space ℳ_{1} of probability measures, which is known to be compact since T^{d} is compact (see [B] or [EK]). Thus r is complete as a metric on ℳ_{1} , and we have a measure μ̃ ∈ ℳ_{1} such that μ̃_{n} → μ̃ in the r-metric. It follows that μ_{n} → aμ̃ in the r-metric.

Next we turn to the a priori estimates that give bounds over the moments of the sticks, uniformly over t and N.

Lemma 3.2. Under Assumption 1, there are constants C_{k} < ∞ such that E^{N}{η_{t}^{k}(x)} ≤ C_{k} for all t, N, and k.

First we show that products of exponential distributions dominate each other stochastically if the sitewise expectations dominate each other (see Section II.2 in [L] for the definitions). We continue the convention of letting (η, ζ) denote an element of Ω_{N} × Ω_{N}. Recall (7) for the definition of γ_{r}.

Lemma 3.3. Suppose μ = ⨂_{x∈Z_{N}^{d}} γ_{r(x)} and ν = ⨂_{x∈Z_{N}^{d}} γ_{s(x)} are two products of exponential distributions on Ω_{N}. If r(x) ≤ s(x) for all x, then there exists a probability measure Q on Ω_{N} × Ω_{N} such that Q{(η, ζ) : η ≤ ζ} = 1, the η -marginal of Q is μ, and the ζ -marginal of Q is ν.

Proof. For each x, define the distribution Q_{x} of the pair (η(x), ζ(x)) so that ζ(x) is distributed according to γ_{s(x)} and η(x) = (r(x)/s(x))ζ(x) almost surely. Take Q = ⊗_{x∈Z_{N}^{d}} Q_{x}.

Proof of Lemma 3.2. Set ν = ⨂_{x∈Z_{N}^{d}} γ_{K_{0}} where K_{0} is the constant appearing in Assumption 1. Let Q be the measure given by Lemma 3.3 with marginals μ_{0}^{N} and ν and supported by the set {(η, ζ) : η ≤ ζ}. For each such pair (η, ζ), construct the process P^{η,ζ} as in Lemma 2.1, and define a joint process P^{Q} with initial distribution Q by

E^{Q}{f(η_{•}, ζ_{•})} = ∫∫ E^{η,ζ}{f(η_{•}, ζ_{•})} Q(dη, dζ).

Let Q_{t} be the distribution of the joint process at time t, similarly μ_{t}^{N} and ν_{t} for the processes started with μ_{0}^{N} and ν, respectively. Then

Q_{t}{η ≤ ζ} = ∫∫ P^{η,ζ}{η_{t} ≤ ζ_{t}} Q(dη, dζ) = 1

by Lemma 2.1. The marginals of Q_{t} are μ_{t}^{N} and ν_{t} by construction, and ν_{t} = ν because ν is invariant by Lemma 2.2 and the remark following it. Again by Theorem 2.4 on p. 74 in [L],

μ_{t}^{N}{η^{k}(x)} ≤ ν{η^{k}(x)} = K_{0}^{k}k! ≡ C_{k}.

For the remainder of this subsection fix a smooth test function ϕ on T^{d}. We look for two processes z_{1}(t) = z_{1}^{N,ϕ}(t) and z_{2}(t) = z_{2}^{N,ϕ}(t) such that the processes

M_{t} = M_{t}^{N,ϕ} = α_{t}^{N}(ϕ) − α_{0}^{N}(ϕ) − ∫_{0}^{t} z_{1}(s) ds

and

V_{t} = V_{t}^{N,ϕ} = M_{t}^{2} − ∫_{0}^{t} z_{2}(s) ds

are martingales. It is well-known, and not hard to verify, that these are given by z_{1}(t) = ℒ_{N}f(η_{t}) and z_{2}(t) = ℒ_{N}(f^{2})(η_{t}) −2f(η_{t})ℒ_{N}f(η_{t}), where f(η) = N^{−d} ∑_{x∈Z_{N}^{d}} ϕ(x/N)η(x). We start with the short-range model, so let ℒ_{N} be given by (1). Thinking of ϕ as a periodic function on R^{d}, using the symmetry of p and by Taylor expanding, we may write (13)

z_{1}(t) = N^{2−d} ∑_{x∈Z_{N}^{d}} ∑_{z∈Z^{d}} p(z)\frac{1}{2}η_{t}^{2}(x)\{ϕ(\frac{x + z}{N} ) − ϕ(\frac{x}{N} )\}

= \frac{1}{4}N^{−d} ∑_{x∈Z_{N}^{d}} η_{t}^{2}(x)∑_{z∈Z^{d}} p(z)N^{2}\{ϕ(\frac{x + z}{N} ) + ϕ(\frac{x − z}{N} ) − 2ϕ(\frac{x}{N} )\}

= \frac{1}{4}N^{−d} ∑_{x∈Z_{N}^{d}} η_{t}^{2}(x)∑_{z∈Z^{d}} p(z)\{ ∑_{1≤i,j≤d}∂_{ξ_{i}}∂_{ξ_{j}}ϕ(\frac{x}{N} )z_{i}z_{j} + O(N^{−1}∥z∥^{3})\}

= \frac{1}{4}N^{−d} ∑_{x∈Z_{N}^{d}} η_{t}^{2}(x)Aϕ(\frac{x}{N} ) + O(N^{−1}) · σ_{3} \frac{1}{4}N^{−d} ∑_{x∈Z_{N}^{d}} η_{t}^{2}(x),

where

Aϕ = ∑_{1≤i,j≤d} a_{i,j}∂_{ξ_{i}}∂_{ξ_{j}}ϕ

with a_{i,j} = ∑_{z∈Z^{d}} p(z)z_{i}z_{j}, and σ_{3} = ∑_{z∈Z^{d}} p(z)∥z∥^{3} , a finite constant by Assumption 2. The constant hidden in the error term O(N^{−1}) depends on the size of the third derivatives of ϕ. In particular, this estimate holds uniformly in t. Unraveling the definition of z_{2}(t) gives, also uniformly in t,

(14) \begin{aligned}z_{2}(t) & = N^{−2d} ∑_{x∈Z_{N}^{d}} \frac{1}{3}η_{t}^{3}(x) ∑_{z∈Z^{d}} p(z)N^{2}\{ϕ(\frac{x + z}{N} ) − ϕ(\frac{x}{N} )\}^{2} \\ & = N^{−2d} ∑_{x∈Z_{N}^{d}} \frac{1}{3}η_{t}^{3}(x) ∑_{z∈Z^{d}} p(z)\{⟨∇ϕ(\frac{x}{N} ), z⟩ + O(N^{−1}∥z∥^{2})\}^{2} \\ & = O(N^{−d}) · σ_{4}N^{−d} ∑_{x∈Z_{N}^{d}} η_{t}^{3}(x), \end{aligned}

where σ_{4} = ∑_{z∈Z^{d}} p(z)∥z∥^{4} , another finite constant by Assumption 2. These estimates and the a priori bound are basic for everything that follows.

For the long-range model, the same computations give

(15) z_{1}(t) = \frac{1}{12}N^{−d} ∑_{x∈Z_{N}^{d}} η_{t}^{2}(x)Δϕ(\frac{x}{N} ) + O(N^{−α}) · N^{−d} ∑_{x∈Z_{N}^{d}} η_{t}^{2}(x)

and

(16)  z_{2}(t) = O(N^{−d}) · N^{−d} ∑_{x∈Z_{N}^{d}} η_{t}^{3}(x).

Let us declare c = \frac{1}{4} for the short-range model, c = \frac{1}{12} for the long-range model, and A = Δ for the long-range model. Then the leading part of z_{1}(t) is given by

c N^{−d} ∑_{x∈Z_{N}^{d}} η_{t}^{2}(x) Aϕ(\frac{x}{N} )

for both models.

3.2. The distributions of αN_{•}. The distributions of αN_{•} are probability measures 𝒫^{N} on 𝒟_{ℳ} , defined by 𝒫^{N}(B) = P^{N}{αN_{•} ∈ B} for Borel subsets B of 𝒟_{ℳ} . We write ω_{•} for a generic element of 𝒟_{ℳ} .

Lemma 3.4. The sequence {𝒫^{N}} is tight.

Proof. By Theorem 15.3 in [B] it is enough to show (i) compact containment, that for any ε > 0 there exists a compact K ⊂ ℳ such that

(17)  P^{N}{α_{t}^{N} ∈ K for all 0 ≤ t ≤ T } ≥ 1 − ε,

and (ii) that, given positive ε_{1} and ε_{2} , there exists a 0 < δ < 1 and N_{0} such that

(18)  P^{N}{ sup_{|s−t|≤δ} r(α_{t}^{N}, α_{s}^{N}) ≥ ε_{1}} ≤ ε_{2}

for all N ≥ N_{0} . (Theorem 15.3 in [B] is stated only for real valued processes, but it is not hard to see that its proof and the proof of Theorem 14.4 on which it is based both apply to complete metrics on arbitrary Polish spaces.)

Compact containment is immediate from conservation of total stick length: α_{t}^{N}(T^{d}) = N^{−d} ∑_{x} η_{t}(x) = N^{−d} ∑_{x} η_{0}(x) P^{N} -a.s., and so

P^{N}{ sup_{0≤t≤T} α_{t}^{N}(T^{d}) ≥ B} ≤ B^{−1}N^{−d} ∑_{x} μ_{0}^{N}{η(x)} ≤ C/B,

where the last inequality comes from Assumption 1. The sets ℳ^{B} = {ν ∈ ℳ : ν(T^{d}) ≤ B} are compact for B < ∞ and (17) follows by taking K = ℳ^{B} for B > C/ε.

Next we show that

(19)  lim sup_{δ↘0} lim sup_{N→∞} E^{N}{ sup_{|s−t|≤δ} |α_{t}^{N}(ϕ) − α_{s}^{N}(ϕ)|^{2}} = 0

for an arbitrary smooth function ϕ on T^{d}. Then (18) follows by Markov’s inequality and the definition (12) of the metric r. From

α_{t}^{N}(ϕ) − α_{s}^{N}(ϕ) = M_{t} − M_{s} + ∫_{s}^{t} z_{1}(u) du

we see that the integrand in (19) is bounded above by

(20)  sup_{|s−t|≤δ} 2(M_{t} − M_{s})^{2} + sup_{|s−t|≤δ} 2(∫_{s}^{t} z_{1}(u) du)^{2}.

Bound the first term by sup_{0≤t≤T} 8M_{t}^{2} , then apply Doob’s inequality, the definition of the martingale V_{t}, the bound (14) on z_{2}(u), and finally the a priori estimate Lemma 3.2:

(21)  E^{N}{ sup_{0≤t≤T} M_{t}^{2}} ≤ CE^{N}{M_{T}^{2}} = C E^{N}{∫_{0}^{T} z_{2}(u) du} ≤ CT N^{−d}.

(C denotes a constant whose value changes from line to line.) For the second term in (20) apply Schwarz’s inequality and (13) to get

\begin{aligned}sup_{|s−t|≤δ}(∫_{s}^{t} z_{1}(u) du)^{2} & ≤ sup_{0≤s≤t≤(s+δ)∧T}(t − s) ∫_{s}^{t} z_{1}^{2}(u) du ≤ δ ∫_{0}^{T} z_{1}^{2}(u) du \\ & ≤ CδN^{−d} ∑_{x} ∫_{0}^{T} η_{u}^{4}(x) du. \end{aligned}

Now integrate and use the a priori bound to bound the expectation by CT δ. This together with (21) gives

E^{N}{ sup_{|s−t|≤δ} |α_{t}^{N}(ϕ) − α_{s}^{N}(ϕ)|^{2}} ≤ CT (N^{−d} + δ),

which implies (19). For the long-range model, just use (15)–(16) instead of (13)– (14).

Secondly we establish some properties of the limit points of {𝒫^{N}}.

Lemma 3.5. Let 𝒫 be a limit point of {𝒫^{N}}. Then 𝒫 is supported by continuous paths, ω_{t} ≪ dξ for 𝒫(dω_{•}) ⊗ dt-a.e. measure ω_{t}, and for all 1 ≤ p < ∞,

(22)  ℰ {∫_{0}^{T} ∫_{T^{d}} u^{p}(t, ξ) dξ dt} < ∞,

where the 𝒫(dω_{•}) ⊗ dt ⊗ dξ -a.e. defined derivative u(ω_{•}, t, ξ) = (dω_{t}/dξ)(ξ) is jointly measurable in the variable (ω_{•}, t, ξ), and ℰ denotes expectation under 𝒫.

Proof. Let Δ(ω_{•}) = sup_{t} r(ω_{t}, ω_{t−}) be the maximal jump of a path ω_{•} ∈ 𝒟_{ℳ} . It is a continuous function in the Skorokhod topology, hence

ℰ {Δ(ω_{•})} ≤ lim sup_{N→∞} E^{N}{Δ(αN_{•})}.

It is obvious that Δ(ω_{•}) ≤ sup_{|s−t|≤δ} r(ω_{s}, ω_{t}) for all δ > 0, hence (19) implies that ℰ {Δ(ω_{•})} = 0. In other words, 𝒫 -a.e. ω_{•} is continuous.

For the second part, let ϕ ≥ 0 be a bounded continuous function on T^{d}, and p > 0 an integer.

\begin{aligned}∫_{0}^{T}E^{N}\{α_{t}^{N}(ϕ)^{p}\} dt & = N^{−pd} ∑_{x_{1},...,x_{p}}ϕ(\frac{x_{1}}{N} ) ⋯ ϕ(\frac{x_{p}}{N} )∫_{0}^{T}E^{N}\{η_{t}(x_{1}) ⋯ η_{t}(x_{p})\} dt \\ & ≤ CT N^{−pd} ∑_{x_{1},...,x_{p}}ϕ(\frac{x_{1}}{N} ) ⋯ ϕ(\frac{x_{p}}{N} ) = CT (N^{−d} ∑_{x} ϕ(\frac{x}{N} ))^{p}, \end{aligned}

where the inequality comes from the a priori bound. Letting N → ∞ along a suitable subsequence gives

\frac{1}{T} ∫_{0}^{T}∫ ω_{t}(ϕ)^{p} 𝒫(dω) dt ≤ C(∫_{T^{d}} ϕ(ξ) dξ)^{p}.

The proof is then completed by an application of the following lemma.

Lemma 3.6. Let X and Y be two Polish spaces, μ and κ probability measures on X and Y , respectively, and let y ↦ ν^{y} be a measurable map from Y into the space of finite real-valued Borel measures on X, topologized weakly by bounded continuous functions on X. Suppose that for some constants 1 < p < ∞ and C < ∞ and all bounded continuous f ≥ 0 on X,

(23)  ∫_{Y} ν^{y}(f)^{p} κ(dy) ≤ Cμ(f)^{p}.

Then ν^{y} ≪ μ for κ-a.e. y, the derivative φ^{y}(x) = (dν^{y}/dμ)(x) is jointly measurable in (x, y) and satisfies

(24)  ∫_{Y} μ{(φ^{y})^{p}} κ(dy) ≤ C.

Remark. To complete the proof of Lemma 3.5, take X = T^{d}, Y = 𝒟_{ℳ} × [0, T], μ = dξ, κ = 𝒫 ⊗ T^{−1} dt, and ν^{y} = ω_{t} for y = (ω_{•}, t).

Proof. We give a probabilistic proof, based on constructing a Radon–Nikodym derivative from the martingale convergence theorem. Since (23) is preserved under bounded pointwise convergence, it holds for all bounded Borel functions f ≥ 0 on X. Let {ℬ_{k}} be an increasing sequence of finite partitions that generate ℬ_{X}, the Borel field of X. By taking f = I_{A} for A ∈ ⋃_{k} ℬ_{k} in (23) we see that ν^{y} ≪ μ on each ℬ_{k}, outside a single κ-null set of y ’s. Set

φ_{k}(x, y) = ∑_{{A:A∈ℬ_{k} \atop μ(A)>0}} I_{A}(x)\frac{ν^{y}(A)}{μ(A)}.

Then φ_{k}^{y} = (dν^{y}/dμ)|_{ℬ_{k}} and also φ_{k} = (dν/dμ ⊗ κ)|_{ℬ_{k}⊗ℬ_{Y}} , where the measure ν on X × Y is defined by ν(dx, dy) = ν^{y}(dx) κ(dy) and ℬ_{Y} is the Borel field of Y . Under the measure μ ⊗ κ, φ_{k} is a nonnegative martingale with respect to the filtration {ℬ_{k} ⊗ℬ_{Y} }, hence there is a μ⊗κ-a.s. limit φ(x, y) = lim_{k→∞} φ_{k}(x, y).

If {φ_{k}} were uniformly μ ⊗ κ-integrable, then φ_{k} → φ would also hold in L^{1} (μ ⊗ κ), and consequently, for any m and A ∈ ℬ_{m} ⊗ ℬ_{Y} ,

ν(A) = lim_{k→∞} ∫_{A} φ_{k} dμ ⊗ κ = ∫_{A} φ dμ ⊗ κ.

The filtration {ℬ_{k}⊗ℬ_{Y} } generates ℬ_{X} ⊗ℬ_{Y} , which in turn equals ℬ_{X×Y} by the second countability of the topologies, hence the above implies that φ = dν/dμ ⊗ κ. In particular, since ν and μ ⊗ κ have a common Y -marginal, φ^{y} = dν^{y}/dμ and ν^{y} ≪ μ for κ-a.e. y.

It remains to prove the uniform integrability of {φ_{k}} and (24). For both it suffices to show that μ ⊗ κ(φ_{k}^{p}) ≤ C for all k.

\begin{aligned}μ ⊗ κ(φ_{k}^{p}) & = ν(φ_{k}^{p−1}) = ∑_{{A:A∈ℬ_{k} \atop μ(A)>0}} ∫∫ I_{A}(x) \frac{ν^{y}(A)^{p−1}}{μ(A)^{p−1}} ν(dx, dy) \\ & = ∑_{{A:A∈ℬ_{k} \atop μ(A)>0}} \frac{1}{μ(A)^{p−1}} ∫ ν^{y}(A)^{p} κ(dy) ≤ C ∑_{A∈ℬ_{k}} μ(A) = C. \end{aligned}

3.3. The local equilibrium. Now fix a smooth function ϕ on T^{d}. We start by showing that, for any δ > 0, (25)

lim_{N→∞} P^{N}\{ sup_{0≤t≤T} | α_{t}^{N}(ϕ) − α_{0}^{N}(ϕ) −∫_{0}^{t}cN^{−d} ∑_{x∈Z_{N}^{d}} η_{s}^{2}(x) Aϕ(\frac{x}{N} ) ds |≥ δ\} = 0.

Using estimates (13) and (15) (for the short-range and long-range models, respectively) the expression in | |’s can be bounded by

sup_{0≤t≤T}(|M_{t}|+CN^{−1}∫_{0}^{t}N^{−d} ∑_{x} η_{s}^{2}(x) ds) ≤ sup_{0≤t≤T} |M_{t}|+CN^{−1}∫_{0}^{T}N^{−d} ∑_{x} η_{s}^{2}(x) ds

for a constant C that depends on ϕ alone. The expectation of this quantity is bounded by

E^{N}{ sup_{0≤t≤T} M_{t}^{2}}^{1/2} + O(N^{−1}),

which vanishes as N → ∞ as was shown in (21) above. This establishes (25).

Let Λ_{Nε} = {z ∈ Z^{d} : 0 ≤ z_{i} < Nε for i = 1, …, d} for ε > 0. Next we turn (25) into

(26)

lim_{ε→0} lim sup_{N→∞} P^{N}\{ sup_{0≤t≤T} | α_{t}^{N}(ϕ) − α_{0}^{N}(ϕ) − ∫_{0}^{t} cN^{−d} ∑_{x∈Z_{N}^{d}} Aϕ(\frac{x}{N} )

× (\frac{1}{|Λ_{Nε}|} ∑_{y∈x+Λ_{Nε}} η_{s}^{2}(y)) ds |≥ δ\} = 0.

Of course, y ∈ x + Λ_{Nε} is again interpreted with periodic boundary conditions of Z_{N}^{d} in mind. To prove (26), notice first that by changing the order of summation

\begin{aligned}N^{−d} & ∑_{x} Aϕ(\frac{x}{N} )(\frac{1}{|Λ_{Nε}|} ∑_{y∈x+Λ_{Nε}} η_{s}^{2}(y)) = N^{−d} ∑_{x} η_{s}^{2}(x)Aϕ(\frac{x}{N} ) \\ & + O( sup_{∥ξ−ξ′∥≤\sqrt{d}ε} |Aϕ(ξ) − Aϕ(ξ′)|) · N^{−d} ∑_{x} η_{s}^{2}(x). \end{aligned}

By the a priori bound the expectation of the error term vanishes as ε → 0, uniformly in N and t, so (26) follows from (25).

The next proposition establishes a weak form of local equilibrium, sufficient for our needs: The empirical second moment of the sticks in the cube Λ_{Nε} is asymptotically the same as for i.i.d. exponential random variables with expectation given by the empirical mean:

Proposition 3.7.

(27)

lim_{ε→0} lim sup_{N→∞} E^{N}\{∫_{0}^{T} N^{−d} ∑_{x∈Z_{N}^{d}} | \frac{1}{|Λ_{Nε}|} ∑_{y∈x+Λ_{Nε}} η_{t}^{2}(y)

− 2(\frac{1}{|Λ_{Nε}|} ∑_{y∈x+Λ_{Nε}} η_{t}(y))^{2} | dt \} = 0.

Proposition 3.7 will be proved via a number of intermediate results. Write S_{Λ}(η) = |Λ|^{−1} ∑_{x∈Λ} η(x) for any set Λ ⊂ Z_{N}^{d}, and similarly S_{Λ}(η^{2}) for the average of squares. Define the probability measure \overline{μ}^{N} on Ω_{N} by

(28)  \overline{μ}^{N} = \frac{1}{T} ∫_{0}^{T} N^{−d} ∑_{x∈Z_{N}^{d}} μ_{t}^{N} ∘ θ_{x} dt,

where μ_{t}^{N} is the distribution of the stick process at time t and θ_{x} are the translations defined on Ω_{N} by (θ_{x}η)_{y} = η_{x+y}, again modulo the cube Z_{N}^{d}. Then \overline{μ}^{N} is a translation invariant measure that continues, by Lemma 3.2, to satisfy the a priori bounds:

(29)  \overline{μ}^{N}\{η^{k}(x)\} ≤ T C_{k} for all k, x, and N.

The claim (27) of the proposition now reads

(30)  lim_{ε→0} lim sup_{N→∞} \overline{μ}^{N}\{|S_{Λ_{Nε}}(η^{2}) − 2 S_{Λ_{Nε}}^{2}(η)|\} = 0.

As a standing notational convention, ν always denotes a product probability measure under which the sticks are i.i.d. exponential random variables with expectation K_{0} . For example, on each Ω_{N}, ν = γ_{K_{0}}^{⊗Z_{N}^{d}} , but we shall have occasion to consider other sets of sites too besides the Z_{N}^{d} ’s. The choice of K_{0} for the expectation is technically convenient for then we have derivatives f_{t}^{N}(η) = (dμ_{t}^{N}/dν)(η) on Ω_{N} that are bounded uniformly over both η and t. This holds for t = 0 by Assumption 1, and for t > 0 by general principles: Suppose ν is invariant for a Markov process and f_{0} = dμ/dν. If 1 ≤ p ≤ ∞ and q is the dual exponent, then for 0 ≤ g ∈ L^{q} (ν)

μ_{t}(g) = ν{f_{0}E^{•}g(η_{t})} ≤ ∥f_{0}∥_{L^{p}(ν)}∥E^{•}g(η_{t})∥_{L^{q}(ν)}.

First with p = 1 this shows that μ_{t} ≪ ν. Letting f_{t} = dμ_{t}/dν we see for all p that ∥f_{t}∥_{L^{p}(ν)} is nonincreasing in t.

The relative entropy or Kullback–Leibler information H(Q | P) of two probability measures Q and P is defined by

H(Q | P) = \{ \begin{matrix}P \{\frac{dQ}{dP} log \frac{dQ}{dP}\} & if Q ≪ P, \\ ∞ & otherwise. \end{matrix}

H(Q | P) measures a certain statistical distance between Q and P. H(Q | P) ≥ 0 holds always and H(Q | P) = 0 if and only if Q = P. A straightforward computation shows that Assumptions 1 and 3 imply that the entropy bound

(31)  H(μ_{0}^{N} | ν) ≤ CN^{d}

holds for all N with a constant C = C(K_{0}, ε_{0}) that depends only on the constants of Assumptions 1 and 3. It is for the sake of (31) that we need to make Assumption 3.

To handle simultaneously both the short-range and the long-range model, let p_{N}(x, y) be the uniform distribution on x + V_{N} for the long-range model, and let β = 2 for the short-range model. For Borel functions f ≥ 0 on Ω_{N}, set (32)

σ_{N}(f) = ∑_{x,y} p_{N}(x, y) ν{∫_{0}^{η(x)}[f(η^{u,x,y}) − f(η)] [log f(η^{u,x,y}) − log f(η)] du}.

The functional σ_{N} is nonnegative, convex, and translation invariant: σ_{N}(f) = σ_{N}(f ∘ θ_{x}). Set H_{t}^{N} = H(μ_{t}^{N} | ν).

Lemma 3.8. (d/dt)H_{t}^{N} = −\frac{1}{2}N^{β}σ_{N}(f_{t}^{N}).

Proof. First let ε > 0 and deduce the conclusion for g_{t}^{ε} = (1 − ε)f_{t}^{N} + ε by direct calculation:

\frac{d}{dt}ν\{g_{t}^{ε} log g_{t}^{ε}\} = ν\{(1 + log g_{t}^{ε}) ℒ_{N}g_{t}^{ε}\}

= ν\{log g_{t}^{ε} ℒ_{N}g_{t}^{ε} + g_{t}^{ε} ℒ_{N}(log g_{t}^{ε})\} = −\frac{1}{2}N^{β}σ_{N}(g_{t}^{ε})

by using reversibility, by substituting in (1) or (3), and by applying (9) to one of the resulting terms. Nothing is problematic here since everything in sight is bounded. Then write, for s < t,

ν\{g_{t}^{ε} log g_{t}^{ε}\} − ν\{g_{s}^{ε} log g_{s}^{ε}\} = −\frac{1}{2}N^{β} ∫_{s}^{t} σ_{N}(g_{u}^{ε}) du

and let ε ↘ 0. Calculus shows that the nonnegative integrand of σ_{N}(g_{t}^{ε}) increases as ε decreases, hence σ_{N}(g_{t}^{ε}) ↗ σ_{N}(f_{t}^{N}) by monotone convergence as ε ↘ 0. On the left-hand side we may apply dominated convergence as g_{t}^{ε} is bounded above uniformly in ε and ξ log ξ ≥ −1/e holds for all ξ ≥ 0.

Let \overline{f}^{N} = d\overline{μ}^{N}/dν. By the convexity of F (a, b) = (a − b)(log a − log b) and the translation invariance of σ_{N},

σ_{N}(\overline{f}^{N}) ≤ \frac{1}{T} ∫_{0}^{T} σ_{N}(f_{t}^{N}) dt.

Hence by Lemma 3.8

(33)  σ_{N}(\overline{f}^{N}) ≤ \frac{2N^{−β}}{T} (H_{0}^{N} − H_{T}^{N}) ≤ CN^{d−β},

where we used (31) and H_{T}^{N} ≥ 0. For x ∈ Z_{N}^{d} and functions g ≥ 0 on Ω_{N} define

(34)  D_{x}^{N}(g) = ∑_{y∈Z_{N}^{d}} p_{N}(x, y) ν{∫_{0}^{η(x)}[g(η^{u,x,y}) − g(η)]^{2} du}.

Let g^{N} = \sqrt{\overline{f}^{N}}. By the translation invariance of ν and \overline{f}^{N}, D_{x}^{N}(g^{N}) does not depend on the site x. Hence, utilizing the inequality

(\sqrt{u} − \sqrt{v})^{2} ≤ (u − v)(log u − log v),

we get

(35)  D_{x}^{N}(g^{N}) = N^{−d} ∑_{z} D_{z}^{N}(g^{N}) ≤ N^{−d}σ_{N}(\overline{f}^{N}),

and then from (33) our next fundamental bound

(36)  D_{x}^{N}(g^{N}) ≤ CN^{−β}

that controls the exchange of stick pieces between sites.

Now fix a positive integer L, large but much smaller than Nε, and choose K = K(N, L, ε) so that Nε−L < KL ≤ Nε. Then our goal (30) is not affected by replacing Λ_{Nε} by a union Λ_{KL} = Δ_{1}∪⋯∪Δ_{K^{d}} of disjoint translates Δ_{i} = x_{i}+Λ_{L} of Λ_{L} = {z ∈ Z^{d} : 0 ≤ z_{i} < L for i = 1, …, d}.

Lemma 3.9. For a constant C = C(K_{0}) coming from the a priori bound and for all N, K, and L,

(37)  \begin{matrix}\overline{μ}^{N}\{ & |S_{Λ_{KL}}(η^{2}) − 2S_{Λ_{KL}}^{2}(η)|\} ≤ \overline{μ}^{N}\{|S_{Λ_{L}}(η^{2}) − 2 S_{Λ_{L}}^{2}(η)|\} \\ + C( K^{−2d} ∑_{1≤i,j≤K^{d}} \overline{μ}^{N}\{|S_{Δ_{i}}(η) − S_{Δ_{j}}(η)|^{2}\})^{1/2}. \end{matrix}

Proof. Write

\begin{aligned}|S_{Λ_{KL}}(η^{2}) − 2S_{Λ_{KL}}^{2}(η)| & ≤ |K^{−d} ∑_{i=1}^{K^{d}} S_{Δ_{i}}(η^{2}) − 2K^{−d} ∑_{i=1}^{K^{d}} S_{Δ_{i}}^{2}(η)| \\ & + 2|K^{−d} ∑_{i=1}^{K^{d}} S_{Δ_{i}}^{2}(η) − (K^{−d} ∑_{i=1}^{K^{d}} S_{Δ_{i}}(η))^{2}|. \end{aligned}

By translation invariance, the \overline{μ}^{N} -expectation of the first right-hand-side term above is bounded by the first term on the right-hand side of (37). For the second term, apply the inequality

(38) |\frac{1}{n} ∑_{i} ξ_{i}^{2} − (\frac{1}{n} ∑_{i} ξ_{i})^{2}| ≤ (\frac{1}{n} ∑_{i} ξ_{i}^{2})^{1/2}(\frac{1}{n^{2}} ∑_{i,j}(ξ_{i} − ξ_{j})^{2})^{1/2},

take expectations and apply Schwarz’s inequality. (38) is true because

0 ≤ \frac{1}{n} ∑_{i} ξ_{i}^{2} − (\frac{1}{n} ∑_{i} ξ_{i})^{2} = \frac{1}{n^{2}} ∑_{i,j} ξ_{i}(ξ_{i} − ξ_{j})

and Schwarz’s inequality again.

Showing that the two terms on the right-hand side of (37) vanish as first N → ∞, then ε → 0, and then L → ∞ are called the one-block and the twoblock estimate, respectively. These are our next tasks.

3.3.1. One-block estimate. Here the goal is to show that

(39)  lim_{L→∞} lim sup_{N→∞} \overline{μ}^{N}\{|S_{Λ_{L}}(η^{2}) − 2S_{Λ_{L}}^{2}(η)|\} = 0.

The first observation is that it suffices to show that

(40)  lim_{L→∞} μ_{L}{|S_{Λ_{L}}(η^{2}) − 2S_{Λ_{L}}^{2}(η)|} = 0

for an arbitrary collection \{μ_{L}\} of limit points of \{\overline{μ}^{N}\} on Ω = [0, ∞)^{Z_{+}^{d}}. The justification is twofold: Firstly, the a priori bound (29) implies that \{\overline{μ}^{N}\} is tight, so any subsequence \overline{μ}^{N_{j}} has a further convergent subsequence. Secondly, since the a priori bound is uniform in N for each fixed L, the following general fact applies: If ν_{n} → ν weakly, f ≥ 0 is a continuous function and sup_{n} ν_{n}(f^{2}) < ∞, then ν_{n}(f) → ν(f) (proof elementary). This implies also that a limit point μ_{L} continues to satisfy the a priori bound (29).

Fix a limit point μ of \{\overline{μ}^{N}\}. For η ∈ Ω, let η_{Γ} denote its restriction to a subset Γ ⊂ Zd_{+} . Recall also (2).

Lemma 3.10. Suppose Γ is a finite subset of Zd_{+} and q = (q(x) : x ∈ Γ) is a fixed stick configuration on Γ. Then whenever x, y ∈ Γ and 0 ≤ u ≤ q(x),

(41)  μ{η : η_{Γ} ≥ q} = μ{η : η_{Γ} ≥ q^{u,x,y}}.

Proof. We start by proving that, for 0 ≤ a < b ≤ q(x),

(42)  μ\{η : η_{Γ} ≥ q\} = \frac{1}{b − a} ∫_{a}^{b} μ\{η : η_{Γ} ≥ q^{u,x,y}\} du.

Pick N large enough so that Γ ⊂ Z_{N}^{d}. As before g^{N} = \sqrt{\overline{f}^{N}}. By a change of variable,

(43) \begin{aligned}|∫_{a}^{b} \overline{μ}^{N}\{η : η_{Γ} ≥ q\} & − \overline{μ}^{N}\{η : η_{Γ} ≥ q^{u,x,y}\} du| \\ & = |∫_{a}^{b} ∫_{Ω_{N}} I_{\{η_{Γ}≥q\}}\{\overline{f}^{N}(η) − \overline{f}^{N}(η^{u,x,y})\} ν(dη) du| \\ & ≤ ∫_{Ω_{N}} I_{\{η_{Γ}≥q\}}\{∫_{a}^{b}|\overline{f}^{N}(η) − \overline{f}^{N}(η^{u,x,y})| du\} ν(dη) \\ & ≤ ν\{∫_{0}^{η(x)}|\overline{f}^{N}(η^{u,x,y}) − \overline{f}^{N}(η)| du\} \\ & ≤ (ν\{∫_{0}^{η(x)}[g^{N}(η^{u,x,y}) − g^{N}(η)]^{2} du\})^{1/2} \\ & × (ν\{∫_{0}^{η(x)}[g^{N}(η^{u,x,y}) + g^{N}(η)]^{2} du\})^{1/2}. \end{aligned}

The last factor on the last line contributes a constant which we ignore in the sequel, as can be seen by bringing the square inside the [ ]’s and using the a priori bound. The first factor is controlled by the single-site Dirichlet form D_{x}^{N}. For the short-range model, pick a path x = x_{0} , x_{1} , …, x_{R} = y in Z^{d} such that

(44)  A = inf_{1≤i≤R} p(x_{i} − x_{i−1}) > 0.

This induces a path on Z_{N}^{d}, again denoted by x = x_{0}, x_{1}, …, x_{R} = y, with p_{N}(x_{i−1}, x_{i}) ≥ A for each i. Take

h(u, η) = [g^{N}(η^{u,x_{1},x_{R}}) − g^{N}(η^{u,x_{1},x_{0}})]^{2}I_{{η(x_{1})≥u}}

in (9). Then

ν{∫_{0}^{η(x_{0})}[g^{N}(η^{u,x_{0},x_{R}}) − g^{N}(η)]^{2} du}
= ν{∫_{0}^{η(x_{1})}[g^{N}(η^{u,x_{1},x_{R}}) − g^{N}(η^{u,x_{1},x_{0}})]^{2} du}
≤ 2ν{∫_{0}^{η(x_{1})}[g^{N}(η^{u,x_{1},x_{R}}) − g^{N}(η)]^{2} du}
+ 2ν{∫_{0}^{η(x_{1})}[g^{N}(η^{u,x_{1},x_{0}}) − g^{N}(η)]^{2} du}.

Iterating this R times gives the bound

ν{∫_{0}^{η(x)}[g^{N}(η^{u,x,y}) − g^{N}(η)]^{2} du}

≤ C ∑_{i=1}^{R} ν{∫_{0}^{η(x_{i})}[g^{N}(η^{u,x_{i},x_{i−1}}) − g^{N}(η)]^{2} du}

≤ CA^{−1} ∑_{i=1}^{R} ∑_{w} p_{N}(x_{i}, w)ν{∫_{0}^{η(x_{i})}[g^{N}(η^{u,x_{i},w}) − g^{N}(η)]^{2} du}

≤ CD_{x}^{N}(g^{N}) ≤ CN^{−β}.

Substituting this back into (43) proves for the short-range model that

(45)  lim_{N→∞}|\overline{μ}^{N}\{η : η_{Γ} ≥ q\} − \frac{1}{b − a} ∫_{a}^{b} \overline{μ}^{N}\{η : η_{Γ} ≥ q^{u,x,y}\} du| = 0.

For the long-range model we have to proceed differently because the constant A defined in (44) vanishes as N → ∞. Let W_{N} = (x + V_{N}) ∩ (y + V_{N}). Then |W_{N}|/|V_{N}| → 1 as N → ∞.

ν{∫_{0}^{η(x)}[g^{N}(η^{u,x,y}) − g^{N}(η)]^{2} du}

= ν{∫_{0}^{η(x)}(|W_{N}|^{−1} ∑_{w∈W_{N}}[g^{N}(η^{u,x,y}) − g^{N}(η^{u,x,w})

+ g^{N}(η^{u,x,w}) − g^{N}(η)] )^{2} du}

≤ \frac{2}{|W_{N}|} ∑_{w∈W_{N}} ν\{∫_{0}^{η(x)}[g^{N}(η^{u,x,y}) − g^{N}(η^{u,x,w})]^{2} du\}

+ ν{∫_{0}^{η(x)}[g^{N}(η^{u,x,w}) − g^{N}(η)]^{2} du}

≤ C \frac{1}{|V_{N}|} ∑_{z∈V_{N}} ν\{∫_{0}^{η(y)}[g^{N}(η^{u,y,y+z}) − g^{N}(η)]^{2} du\}

+ ν{∫_{0}^{η(x)}[g^{N}(η^{u,x,x+z}) − g^{N}(η)]^{2} du}

≤ CD_{x}^{N}(g^{N}) ≤ CN^{−β},

where the passage to the second last line involved two applications of (9). Thus (45) holds also for the long-range model.

From (45) we argue to (42) as follows: For all but countably many q ’s, μ{η : η_{Γ} = q} = 0. For such q ’s (45) implies (42). Since (42) is preserved by increasing limits q^{(n)} ↗ q, it holds for all q.

Lastly we go from (42) to (41). If 0 ≤ u < q(x), then let 0 ≤ a < b < q(x)−u. By (42),

\begin{aligned}μ\{η ≥ q^{u,x,y}\} & = \frac{1}{b − a} ∫_{a}^{b} μ\{η ≥ q^{u+w,x,y}\} dw \\ & = \frac{1}{b − a} ∫_{a+u}^{b+u} μ\{η ≥ q^{w,x,y}\} dw = μ\{η ≥ q\}. \end{aligned}

If u = q(x) > 0, pick 0 ≤ a < b ≤ q(x) so that

\begin{aligned}μ\{η ≥ q^{q(x),x,y}\} & = \frac{1}{b − a} ∫_{a}^{b} μ\{η ≥ (q^{q(x),x,y})^{w,y,x}\} dw \\ & = \frac{1}{b − a} ∫_{q(x)−b}^{q(x)−a} μ\{η ≥ q^{w,x,y}\} dw = μ\{η ≥ q\}. \end{aligned}

Corollary 3.11. μ is a mixture of i.i.d. exponential distributions.

Proof. Since interchanging the sticks q(x) and q(y) in Γ turns q into q^{q(x)−q(y),x,y} (if q(x) ≥ q(y)) it is clear from Lemma 3.10 that μ is exchangeable. Thus there is a probability measure Q on the probability measures on [0, ∞) such that μ = ∫ ρ^{⊗Z_{+}^{d}} Q(dρ). By Lemma 3.10,

(46)  \begin{aligned}∫ ρ[a, ∞)^{k} Q(dρ) & = μ\{η_{\{x_{1},...,x_{k}\}} ≥ (a, …, a)\} \\ & = μ\{η_{x_{1}} ≥ ka\} = ∫ ρ[ka, ∞) Q(dρ). \end{aligned}

It follows that ρ[1/n, ∞) < 1 for all n Q-a.s., for if Q{ρ : ρ[a, ∞) = 1} = δ > 0 for some a > 0, then

δ ≤ ∫ ρ[a, ∞)^{k} Q(dρ) = μ{η_{x_{1}} ≥ ka} ↘ 0

as k ↗ ∞, a contradiction. Hence there is a finite function r_{n}(ρ) such that for Q-a.e. ρ,

ρ[\frac{1}{n}, ∞) = exp\{−\frac{1}{n r_{n}(ρ)}\}.

Then by (46)

(47)  \begin{aligned}∫ ρ[\frac{k}{n}, \frac{k + 1}{n}) Q(dρ) & = ∫ ρ[\frac{1}{n}, ∞)^{k} − ρ[\frac{1}{n}, ∞)^{k+1} Q(dρ) \\ & = ∫ e^{−k(n r_{n}(ρ))^{−1}} − e^{−(k+1)(n r_{n}(ρ))^{−1}} Q(dρ) \\ & = ∫ \{∫_{k/n}^{(k+1)/n} \frac{e^{−w/r_{n}(ρ)}}{r_{n}(ρ)} dw \} Q(dρ). \end{aligned}

Setting Q_{n}(B) = Q{ρ : γ_{r_{n}(ρ)} ∈ B} for Borel sets B of probability measures defines a sequence of measures Q_{n} supported by exponential distributions. (Recall that γ_{r} was defined as the exponential distribution with expectation r, see (7).) Let f be a bounded uniformly continuous function on [0, ∞) and

δ_{n}(f) = sup{|f(w) − f(w′)| : |w − w′| ≤ 1/n}.

Utilizing (47) we get

(48) \begin{aligned}∫ ρ(f) Q(dρ) & = ∑_{k=0}^{∞} f(\frac{k}{n}) ∫ ρ[\frac{k}{n}, \frac{k + 1}{n}) Q(dρ) + O(δ_{n}(f)) \\ & = ∫ \{ ∑_{k=0}^{∞} f(\frac{k}{n}) ∫_{k/n}^{(k+1)/n} \frac{e^{−w/r_{n}(ρ)}}{r_{n}(ρ)} dw\} Q(dρ) + O(δ_{n}(f)) \\ & = ∫ γ_{r_{n}(ρ)}(f) Q(dρ) + O(δ_{n}(f)) \\ & = ∫ γ(f) Q_{n}(dγ) + O(δ_{n}(f)). \end{aligned}

Taking f(w) = w (truncate and pass to a limit in (48)) gives

∫ γ(f) Q_{n}(dγ) = μ\{η(x)\} + O(\frac{1}{n}) ≤ C

by the a priori bound, consequently

Q_{n}{γ : γ(f) ≥ A} ≤ C/A

for all A > 0 and so {Q_{n}} is tight, because γ ↦ γ(f) is a homeomorphism from the set of exponential distributions onto [0, ∞). Let Q̃ be a limit point of {Q_{n}}, still a measure supported by exponential distributions. Letting n → ∞ in (48) along a suitable subsequence shows that, at least on a single site, μ behaves as a Q̃-mixture of exponentials. But then for any finite set Γ ⊂ Zd_{+} and x ∈ Γ, Lemma 3.10 implies that

\begin{aligned}μ\{η : η_{Γ} ≥ q\} & = μ\{η : η_{x} ≥ ∑_{y∈Γ} q(y)\} = ∫ γ[∑_{y∈Γ} q(y), ∞) Q̃(dγ) \\ & = ∫ γ^{⊗Z_{+}^{d}}\{η : η_{Γ} ≥ q\} Q̃(dγ) \end{aligned}

where the last equality used an elementary property of exponential distributions. Since the class of sets {η : η_{Γ} ≥ q} is rich enough to determine a probability measure, μ = ∫ γ^{⊗Z_{+}^{d}} Q̃(dγ) and the corollary is proved.

We are in a position to finish off the proof of the one-block estimate. Let

ν_{r} = γ_{r}^{⊗Z_{+}^{d}} be the i.i.d. exponential distribution on Ω with expectation r. Since

E\{|\frac{1}{n} ∑_{i=1}^{n} X_{i} − EX_{1}|^{2}\} = \frac{1}{n}E\{|X_{1} − EX_{1}|^{2}\}

holds for any square-integrable i.i.d. random variables, and exponential variables satisfy the formulas E{(X−EX)^{2}} = (EX)^{2} and E{(X^{2}−E(X^{2}))^{2}} = 20(EX)^{4} , we can estimate as follows:

ν_{r}{|S_{Λ_{L}}(η^{2}) − 2 S_{Λ_{L}}^{2}(η)|} ≤ ν_{r}{|S_{Λ_{L}}(η^{2}) − 2r^{2}|}
+ 2ν_{r}{|r + S_{Λ_{L}}(η)| · |r − S_{Λ_{L}}(η)|}

≤ ∥S_{Λ_{L}}(η^{2}) − 2r^{2}∥_{L^{2}(ν_{r})} + Cr∥S_{Λ_{L}}(η) − r∥_{L^{2}(ν_{r})}

≤ CL^{−d/2} r^{2} ≤ CL^{−d/2}ν_{r}{η^{2}(x)}.

Let μ = ∫ ν_{r} Q(dr) be the decomposition of the limit point μ we have been considering. Then the above gives, together with the a priori bound,

μ{|S_{Λ_{L}}(η^{2}) − 2 S_{Λ_{L}}^{2}(η)|} ≤ CL^{−d/2}μ{η^{2}(x)} ≤ CL^{−d/2}.

This estimate holds for all limit points μ with the same constant C; hence it holds uniformly over L in (40). We have established (39) and completed the one-block estimate.

3.3.2. Two-block estimate. In this subsection we prove

(49)  lim_{L→∞} lim sup_{ε→0} lim sup_{N→∞} K^{−2d} ∑_{1≤i,j≤d} \overline{μ}^{N}\{|S_{Δ_{i}}(η) − S_{Δ_{j}}(η)|^{2}\} = 0.

We remind the reader that K^{d} is the maximal number of disjoint translates of Λ_{L} that fit inside Λ_{Nε}, and Δ_{1}, …, Δ_{K^{d}} is a tiling of Λ_{KL} with a maximal collection of such translates. The first task is to get rid of the i, j -dependence in the integral appearing in (49). Define a two-site Dirichlet form by

D_{x,y}(g) = ν{∫_{0}^{η(x)}[g(η^{u,x,y}) − g(η)]^{2} du}.

Set D_{x,y} = D_{x,y}(g^{N}) where as before g^{N} = \sqrt{\overline{f}^{N}}.

Lemma 3.12. There is a constant C such that D_{x,y} ≤ Cε^{2} for all N, ε, and x, y ∈ Λ_{Nε}.

Proof. We begin with the short-range model. Fix R > 0 and for all pairs z, w ∈ Λ_{R} pick and fix a path z = z_{0}, z_{1}, …, z_{b(z,w)} = w such that p(z_{i} − z_{i−1}) > 0 for i = 1, …, b(z, w). Set

A = inf_{z,w∈Λ_{R}} inf_{1≤i≤b(z,w)} p(z_{i} − z_{i−1})  and  B = max_{z,w∈Λ_{R}} b(z, w).

Then A > 0 and B < ∞. For each pair z, w ∈ Λ_{R} and for each N > R there is a path z = \overline{z}_{0}, \overline{z}_{1}, …, \overline{z}_{b(z,w)} = w inside Z_{N}^{d} such that p_{N}(\overline{z}_{i−1}, \overline{z}_{i}) ≥ p(z_{i} −z_{i−1}) ≥ A.

Given x, y ∈ Λ_{Nε}, construct first a path x = w_{0}, w_{1}, …, w_{ℓ} = y so that each consecutive pair w_{i−1} , w_{i} is contained in a translate of Λ_{R}. By proceeding along each coordinate axis in turn, this can be achieved with

(50)  ℓ ≤ 3dNε/R.

Now fill in between each pair w_{i−1}, w_{i} with translates of the paths constructed earlier. This results in the path x = x_{0}, x_{1}, …, x_{m} = y with m ≤ Bℓ and p(x_{i} − x_{i−1}) ≥ A > 0 for each i.

\begin{aligned}[g^{N}(η^{u,x,y}) − g^{N}(η)]^{2} & = \{∑_{i=1}^{m}[g^{N}(η^{u,x_{0},x_{i}}) − g^{N}(η^{u,x_{0},x_{i−1}})]\}^{2} \\ & ≤ m ∑_{i=1}^{m}[g^{N}(η^{u,x_{0},x_{i}}) − g^{N}(η^{u,x_{0},x_{i−1}})]^{2}, \end{aligned}

hence an application of (9) and the above development gives

ν{∫_{0}^{η(x)}[g^{N}(η^{u,x,y}) − g^{N}(η)]^{2} du}

≤ m ∑_{i=1}^{m} ν{∫_{0}^{η(x_{i−1})}[g^{N}(η^{u,x_{i−1},x_{i}}) − g^{N}(η)]^{2} du}

≤ m ∑_{i=1}^{m} \frac{1}{p(x_{i} − x_{i−1})} ∑_{z} p_{N}(x_{i−1}, z)

× ν{∫_{0}^{η(x_{i−1})}[g^{N}(η^{u,x_{i−1},z}) − g^{N}(η)]^{2} du}

≤ A^{−1}m^{2} D_{x}^{N}(g^{N}) ≤ Cε^{2}

by (36) and (50), remembering that β = 2 for this model. This proves the lemma for the short-range model.

For the long-range model, arrange a sequence E_{0}, E_{1}, …, E_{m} of cubes with side length \frac{1}{4}N^{α} so that x ∈ E_{0} , y ∈ E_{m}, and E_{i} ⊂ z + V_{N} for each z ∈ E_{i−1} for all i. This can be achieved with m ≤ CN^{1−α}ε. Consider a path x = x_{0}, x_{1}, …, x_{m} = y with x_{i} ∈ E_{i} for all i. Reasoning as above, we get

D_{x,y} ≤ m ∑_{i=1}^{m} D_{x_{i−1},x_{i}}.

Sum over x_{m} ∈ E_{m} to get

\begin{aligned}D_{x,y} & ≤ m ∑_{i=1}^{m−1} D_{x_{i−1},x_{i}} + 4^{d}N^{−αd}m ∑_{z∈E_{m}} D_{x_{m−1},z} \\ & ≤ m ∑_{i=1}^{m−1} D_{x_{i−1},x_{i}} + CmD_{x_{m−1}}^{N}(g^{N}). \end{aligned}

Now iterate: sum over x_{m−1} ∈ E_{m−1}, …, x_{1} ∈ E_{1} in turn, and use (36) together with 2α + β = 2.

As far as this estimate and the a priori bound goes, the relative positions of x and y, and consequenly of Δ_{i} and Δ_{j}, are immaterial, so we can simply think of two disjoint cubes Λ_{L} and Λ′_{L} and a probability measure on the sticks in the union Λ_{L} ∪ Λ′_{L}. N disappears, and instead of (49) we prove

(51)  lim_{L→∞} lim_{ε→0} sup_{μ∈𝒩_{L,ε}} μ{|S_{Λ_{L}}(η) − S_{Λ′_{L}}(η)|^{2}} = 0,

where 𝒩_{L,ε} is the class of probability measures μ on Ω_{Λ_{L}∪Λ′_{L}} that satisfy μ ≪ ν, D_{x,y}(\sqrt{f} ) ≤ Cε^{2} for f = (dμ/dν), and the a priori bound μ\{η^{k}(x)\} ≤ C_{k}, for all x, y ∈ Λ_{L} ∪ Λ′_{L}.

For each L, let μ_{L} be a measure that satisfies

μ_{L}{|S_{Λ_{L}}(η) − S_{Λ′_{L}}(η)|^{2}} = lim_{ε→0} sup_{μ∈𝒩_{L,ε}} μ{|S_{Λ_{L}}(η) − S_{Λ′_{L}}(η)|^{2}}.

Existence of the μ_{L} ’s is justified as in the discussion following (40). Take Γ = Λ_{L} ∪ Λ′_{L} in Lemma 3.10, and observe that the last bound in (43) is precisely what we have control over with Lemma 3.12. Thus the proof of Lemma 3.10 goes through again, and (41) holds for μ_{L}. In particular, the variables (η(x) : x ∈ Λ_{L} ∪ Λ′_{L}) are exchangeable under μ. (But finite exchangeability does not imply a decomposition into product measures, so Corollary 3.11 does not follow yet.)

Fix R, and for L ≫ R replace Λ_{L} with a disjoint union Γ_{1} ∪ ⋯ ∪ Γ_{K^{d}} of translates of Λ_{R}, maximal with respect to the property of fitting inside Λ_{L}. Let Γ′_{i} be the translate inside Λ′_{L} that sits relative to Λ′_{L} as Γ_{i} sits relative to Λ_{L}. Then (51) follows from

(52)  lim_{R→∞} lim sup_{L→∞} K^{−d} ∑_{i} μ_{L}{|S_{Γ_{i}}(η) − S_{Γ′_{i}}(η)|^{2}} = 0,

which by exchangeability is equivalent to

(53)  lim_{R→∞} lim sup_{L→∞} μ_{L}{|S_{Λ_{R}}(η) − S_{Λ′_{R}}(η)|^{2}} = 0.

But any limit point of the μ_{L} ’s is infinitely exchangeable, hence we may prove

(54)  lim_{R→∞} sup_{μ} μ{|S_{Λ_{R}}(η) − S_{Λ′_{R}}(η)|^{2}} = 0,

where the supremum is over infinitely exchangeable μ that satisfy the moment conditions μ{η^{k}(x)} ≤ C_{k} for all k. The proof can be completed as was done for the one-block estimate.

We are ready to prove the local equilibrium:

Proof of Proposition 3.7. Combine (30), (37), (39), and (49).

Before utilizing (27) we wish to change it slightly. For ξ, θ ∈ T^{d}, let

(55)  χ_{ε,ξ}(θ) = ε^{−d}I_{ξ+[0,ε)^{d}}(θ),

i.e. the indicator function of the ε-cube on T^{d} with lower left corner at ξ, normalized by the volume. Since y ∈ x + Λ_{Nε} if and only if y/N ∈ x/N + [0, ε)^{d},

(27) is equivalent to (56)

lim_{ε→0} lim sup_{N→∞} E^{N}\{∫_{0}^{T}N^{−d} ∑_{x∈Z_{N}^{d}} |\frac{1}{|Λ_{Nε}|} ∑_{y∈x+Λ_{Nε}}η_{t}^{2}(y) − 2[α_{t}^{N}(χ_{ε,x/N})]^{2}| dt\} = 0.

By inserting (56) into (26) we get as conclusion of this subsection:

(57)

lim_{ε→0} lim sup_{N→∞} P^{N}{ sup_{0≤t≤T} |α_{t}^{N}(ϕ) − α_{0}^{N}(ϕ)

− 2c ∫_{0}^{t} N^{−d} ∑_{x∈Z_{N}^{d}} Aϕ(\frac{x}{N} )[α_{s}^{N}(χ_{ε,x/N})]^{2} ds| ≥ δ\} = 0.

3.4. Further technicalities. First we turn (57) into

(58)

lim_{ε→0} lim sup_{N→∞} P^{N}{ sup_{0≤t≤T} |α_{t}^{N}(ϕ) − α_{0}^{N}(ϕ)

− 2c ∫_{0}^{t} ∫_{T^{d}} Aϕ(ξ) [α_{s}^{N}(χ_{ε,ξ})]^{2} dξ ds| ≥ δ} = 0.

This step requires the a priori estimate and the continuity of Aϕ: For ξ ∈ x/N + [0, N^{−1})^{d},

\begin{matrix}|Aϕ(ξ) & [α_{s}^{N}(χ_{ε,ξ})]^{2} − Aϕ(\frac{x}{N} )[α_{s}^{N}(χ_{ε,x/N})]^{2}| ≤ 2∥Aϕ∥_{∞}(N^{−d} ∑_{y} η(y)) \\ × (N^{−d} ∑_{y∈Λ(x,Nξ,Nε)} η(y)) + (N^{−d} ∑_{y} η(y))|Aϕ(ξ) − Aϕ(\frac{x}{N} )|, \end{matrix}

where Λ(x, y, K) = (x + Λ_{K})Δ(y + Λ_{K}). Integrate this bound over T^{d} and note that |Λ(x, Nξ, Nε)| = O((Nε)^{d−1}). Then (58) follows from (57).

Lemma 3.13. For ψ ∈ C(T^{d}) and fixed ε > 0, ν ↦ ∫_{T^{d}} ψ(ξ)ν(χ_{ε,ξ})^{2} dξ is a continuous function of ν ∈ ℳ .

Proof. Suppose ν_{n} → ν in the topology of ℳ . Let 0 ≤ f_{k} ≤ g_{k} ≤ ε^{−d} be bounded continuous functions such that f_{k} ↗ ε^{−d}I_{(0,ε)^{d}} and g_{k} ↘ ε^{−d}I_{[0,ε]^{d}} on T^{d}, and let f_{k}^{ξ}(θ) = f_{k}(θ − ξ), similarly for g_{k}^{ξ}. Then for all k,

∫ ψ(ξ)ν(f_{k}^{ξ})^{2} dξ ≤ lim inf_{n→∞} ∫ ψ(ξ)ν_{n}(χ_{ε,ξ})^{2} dξ

≤ lim sup_{n→∞} ∫ ψ(ξ)ν_{n}(χ_{ε,ξ})^{2} dξ ≤ ∫ ψ(ξ)ν(g_{k}^{ξ})^{2} dξ.

Thus it suffices to show that lim_{k→∞}[ν(g_{k}^{ξ})−ν(f_{k}^{ξ})] = 0 for Lebesgue a.e. ξ. This comes by a simple Fubini argument:

\begin{aligned}∫ lim_{k→∞}[ν(g_{k}^{ξ}) − ν(f_{k}^{ξ})] dξ & = ∫ \{∫ I_{[0,ε]^{d}}(θ − ξ) − I_{(0,ε)^{d}}(θ − ξ) ν(dθ) \} dξ \\ & = ∫ \{∫ I_{[0,ε]^{d}}(θ − ξ) − I_{(0,ε)^{d}}(θ − ξ) dξ \}ν(dθ) = 0. \end{aligned}

Let 𝒫 be any limit point of {𝒫^{N}}. The probability in (58) equals

𝒫^{N}{ω_{•} ∈ 𝒟_{ℳ} : Φ_{ε}(ω_{•}) ≥ δ}

for a certain function Φ_{ε} that is continuous on the support of 𝒫.

Since (58) holds for all δ > 0, it implies that

(59)  lim_{ε→0} 𝒫{ω_{•} : Φ_{ε}(ω_{•}) ≥ δ} = 0,

again for all δ > 0. Now recall that according to Lemma 3.5 there exists a 𝒫(dω_{•}) ⊗ dt ⊗ dξ -a.e. defined jointly measurable function u(ω_{•}, t, ξ) such that ω_{t}(dξ) = u(ω_{•}, t, ξ) dξ. We suppress the ω_{•} -argument from u(ω_{•}, t, ξ), and then (59) may be written as

(60) \begin{aligned}lim_{ε→0} 𝒫\{ & sup_{0≤t≤T} |ω_{t}(ϕ) − ω_{0}(ϕ) \\ & − 2c ∫_{0}^{t} ∫_{T^{d}} Aϕ(ξ)(ε^{−d} ∫_{ξ+[0,ε)^{d}} u(s, θ) dθ)^{2} dξ ds| ≥ δ\} = 0. \end{aligned}

For a.e. ω_{•} and s,

lim_{ε→0}(ε^{−d} ∫_{ξ+[0,ε)^{d}} u(s, θ) dθ)^{2} = u^{2}(s, ξ)

for a.e. ξ by Lebesgue’s differentiation theorem and (22). To extend this convergence to the integral inside (60), introduce the maximal function

M_{p}(s, ξ) = sup_{ε>0} ε^{−d} ∫_{ξ+[0,ε)^{d}} u^{p}(s, θ) dθ.

Write ℰ for expectation under the measure 𝒫.

Lemma 3.14.

ℰ {∫_{0}^{T} ∫_{T^{d}} M_{2}(s, ξ) dξ ds} < ∞.

Proof. Since M_{2}^{2}(s, ξ) ≤ M_{4}(s, ξ), the maximal theorem gives (see p. 91 in [Fo]):

|{ξ : M_{2}(s, ξ) > r}| ≤ |{ξ : M_{4}(s, ξ) > r^{2}}| ≤ Cr^{−2} ∫_{T^{d}} u^{4}(s, ξ) dξ,

where | · | denotes Lebesgue measure on T^{d}. Hence by (22)

\begin{aligned}ℰ \{∫_{0}^{T} ∫_{T^{d}} & M_{2}(s, ξ) dξ ds\} ≤ ℰ \{∫_{0}^{T} ∫_{1}^{∞}|\{ξ : M_{2}(s, ξ) > r\}| dr ds\} + T \\ & ≤ C ∫_{1}^{∞} r^{−2} dr · ℰ \{∫_{0}^{T} ∫_{T^{d}} u^{4}(s, ξ) dξ ds\} + T < ∞. \end{aligned}

This lemma and the dominated convergence theorem imply that

lim_{ε→0} ℰ { sup_{0≤t≤T} |∫_{0}^{t} ∫_{T^{d}} Aϕ(ξ)(ε^{−d} ∫_{ξ+[0,ε)^{d}} u(s, θ) dθ)^{2} dξ ds

− ∫_{0}^{t} ∫_{T^{d}} Aϕ(ξ)u^{2}(s, ξ) dξ ds| } = 0,

and consequently (60) turns into the statement

(61)  𝒫{ sup_{0≤t≤T} |ω_{t}(ϕ) − ω_{0}(ϕ) − 2c ∫_{0}^{t} ∫_{T^{d}} Aϕ(ξ)u^{2}(s, ξ) dξ ds| ≥ δ} = 0.

Letting δ ↘ 0 shows that 𝒫 -a.e. ω_{•} is a continuous ℳ -valued path with a derivative for a.e. t, and a weak solution of the equation ∂_{t}u = 2cA(u^{2}). By the uniqueness lemma of the next subsection 𝒫 is supported by a single path. This has two consequences: (i) It implies that ω_{t} has a density u(t, ξ) for all t. For we can reprove Lemma 3.5 for a fixed time t and conclude that ω_{t} ≪ dξ for 𝒫 -a.e. ω_{•}, in particular, for the unique ω_{•} supporting 𝒫. (ii) It promotes the weak convergence 𝒫^{N} → 𝒫 to convergence in probability of the 𝒟_{ℳ} -valued random variables αN_{•} to the path u(•, ξ) dξ as stated in Theorem 1. This comes from a general fact: Weak convergence to a degenerate distribution implies convergence in probability.

For Theorems 1 and 2, it remains to prove that u(t, ξ) ≤ ∥u_{0}∥_{∞} for all (t, ξ) ∈ Q_{T} . For any 0 ≤ ϕ ∈ C(T^{d}),

∫_{T^{d}} ϕ u(t) = lim_{N→∞} E^{N}\{α_{t}^{N}(ϕ)\} = lim_{N→∞} N^{−d} ∑_{x} ϕ(\frac{x}{N} )E^{N}\{η_{t}(x)\} ≤ K_{0} ∫_{T^{d}} ϕ,

where the precise constant K_{0} of Assumption 1 comes from the last line of the proof of the a priori estimate Lemma 3.2. But now note that it is perfectly possible to choose the initial distribution so that K_{0} ≤ ∥u_{0}∥_{∞}.

Thus with the uniqueness lemma we have proved Theorems 1 and 2 under Assumption 3, that the initial density is bounded away from 0. This assumption will be lifted after the uniqueness proof.

3.5. Uniqueness lemma.

Lemma 3.15. Let (a_{i,j}) be a symmetric positive semidefinite matrix and set

Aϕ = ∑_{i,j} a_{i,j}∂_{ξ_{i}}∂_{ξ_{j}}ϕ.

Suppose t ↦ ω(t, dξ) is a continuous ℳ -valued path on 0 ≤ t ≤ T such that (i) there exists a jointly measurable function u(t, ξ) on Q_{T} such that ω(t, dξ) =

u(t, ξ) dξ for a.e. t, (ii) ∫_{0}^{T} ∫_{T^{d}} u^{3}(t, ξ) dξ dt < ∞ , and (iii) ∫_{T^{d}} ϕ(ξ) ω(t, dξ) − ∫_{T^{d}} ϕ(ξ) ω(0, dξ) = ∫_{0}^{t} ∫_{T^{d}} Aϕ(ξ) u^{2}(s, ξ) dξ ds for all smooth

ϕ on T^{d} and for all 0 ≤ t ≤ T. Then if σ(t, dξ) is another continuous ℳ -valued path that satisfies the analogues of (i)–(iii) and σ(0) = ω(0), then σ(t) = ω(t) for all 0 ≤ t ≤ T.

Proof. Let σ(t, dξ) = v(t, ξ) dξ when the density exists. Let f_{ε}(ξ) = ε^{−d}f(ε^{−1}ξ) be a compactly supported, symmetric, smooth approximation to the identity. Set

u_{ε}(t, ξ) = [ω(t) ∗ f_{ε}](ξ) = ∫_{T^{d}} f_{ε}(ξ − θ) ω(t, dθ)

and similarly define v_{ε}(t, ξ). Then u_{ε} (and also v_{ε}) satisfies

∫_{T^{d}} ψ(ξ)u_{ε}(t, ξ) dξ − ∫_{T^{d}} ψ(ξ)u_{ε}(0, ξ) dξ = ∫_{0}^{t} ∫_{T^{d}} Aψ(ξ)[u^{2}(s) ∗ f_{ε}](ξ) dξ ds

for all smooth ψ and all t. We wrote u^{2}(s) for the function u^{2}(s)(θ) = u^{2}(s, θ). Subtracting the equation for v_{ε} from the equation for u_{ε} and writing ϕ_{ε}^{s} = u^{2}(s)∗ f_{ε} − v^{2}(s) ∗ f_{ε} (a well-defined smooth function on T^{d} for a.e. s) gives

(62)  ∫_{T^{d}} ψ[u_{ε}(t) − v_{ε}(t)] = ∫_{T^{d}} ∑_{i,j} a_{i,j}(∂_{ξ_{i}}∂_{ξ_{j}}ψ)(∫_{0}^{t} ϕ_{ε}^{s} ds).

Now take ψ = ϕ_{ε}^{t} for those a.e. t for which this makes sense and integrate over 0 ≤ t ≤ T to render the exceptional set of t’s harmless. After an integration by parts on the right-hand side we have

\begin{aligned}∫_{0}^{T} dt ∫_{T^{d}} ϕ_{ε}^{t}[u_{ε}(t) − v_{ε}(t)] & = − ∫_{T^{d}} ∑_{i,j} a_{i,j} ∫_{0}^{T} (∂_{ξ_{i}}ϕ_{ε}^{t})(∫_{0}^{t} ∂_{ξ_{j}}ϕ_{ε}^{s} ds) dt \\ & = − ∫_{T^{d}} ∑_{i,j} a_{i,j} ∫_{0}^{T} ∫_{0}^{T} (∂_{ξ_{i}}ϕ_{ε}^{t})(∂_{ξ_{j}}ϕ_{ε}^{s})I_{\{t≥s\}} dt ds. \end{aligned}

By the symmetry of (a_{i,j})

2 ∫_{0}^{T} dt ∫_{T^{d}} ϕ_{ε}^{t}[u(t) ∗ f_{ε} − v(t) ∗ f_{ε}] = − ∫_{T^{d}} ∑_{i,j} a_{i,j}(∫_{0}^{T} ∂_{ξ_{i}}ϕ_{ε}^{t} dt)(∫_{0}^{T} ∂_{ξ_{j}}ϕ_{ε}^{t} dt).

The conclusion we derive from this, by (a_{i,j})’s positive semidefiniteness, is that for all ε > 0,

∫_{0}^{T} dt ∫_{T^{d}}[u^{2}(t) ∗ f_{ε} − v^{2}(t) ∗ f_{ε}] [u(t) ∗ f_{ε} − v(t) ∗ f_{ε}] ≤ 0.

Hypothesis (ii) gives sufficient integrability to let ε ↘ 0 and recover

∫_{Q_{T}} [u^{2} − v^{2}] [u − v] ≤ 0

which implies that u(t, ξ) = v(t, ξ) almost everywhere. Hence ω(t) = σ(t) for a.e. t, and by continuity for all t.

3.6. Removing Assumption 3. Assume given an initial density u_{0} , not necessarily bounded away fom zero, and initial distributions μ_{0}^{N} satisfying Assumption 1. For ε > 0, the versions of the theorems thus far proved apply to the initial densities u_{0}^{ε} = u_{0} + ε with initial distributions μ_{0}^{ε,N} arranged to satisfy μ_{0}^{ε,N}{η(x)} = μ_{0}^{N}{η(x)} + ε. Fix ε > 0 for the moment. Let Q^{N} be the coupling of μ_{0}^{N} and μ_{0}^{ε,N} given in Lemma 3.3. Let P^{N} be the distribution of the joint process with initial distribution Q^{N}, constructed as in the proof of Lemma 3.2 so that P^{N}{(η_{•}, ζ_{•}) : η_{•} ≤ ζ_{•}} = 1, and the η_{•} and ζ_{•} marginals of P^{N} are the processes P^{N} and P^{ε,N} with initial distributions μ_{0}^{N} and μ_{0}^{ε,N}, respectively. Let 𝔓^{N} be the distribution of

(αN_{•}, αε,N_{•}) = (N^{−d} ∑_{x} η_{•}(x)δ_{x/N}, N^{−d} ∑_{x} ζ_{•}(x)δ_{x/N})

on the space 𝒟_{ℳ} × 𝒟_{ℳ} . The tightness proof did not depend on Assumption 3, hence {𝔓^{N}} has tight marginals and consequently is itself tight. Let 𝔓 be a limit point with marginals 𝒫 and 𝒫^{ε}. We know that 𝒫^{ε} is supported by the unique path u^{ε}(t, ξ) dξ described in Theorem 1 or 2, whichever model we are talking about. Nor did Lemma 3.5 depend on Assumption 3, and so 𝒫(dω_{•}) ⊗ dt-a.e. ω_{t} has a density v(t, ξ). For all 0 ≤ a < b ≤ T and 0 ≤ ϕ ∈ C(T^{d}), the coupling implies

P^{N}{∫_{a}^{b} α_{t}^{N}(ϕ) dt ≤ ∫_{a}^{b} α_{t}^{ε,N}(ϕ) dt} = 1,

hence in the limit

(63)  𝒫{∫_{a}^{b} ∫_{T^{d}} v(t, ξ)ϕ(ξ) dξ dt ≤ ∫_{a}^{b} ∫_{T^{d}} u^{ε}(t, ξ)ϕ(ξ) dξ dt} = 1.

(The random variable inside the probability is v(t, ξ).) By considering all rational a, b and a suitable countable set of functions ϕ we see that, for 𝒫 -almost every ω_{•}, v(t, ξ) ≤ u^{ε}(t, ξ) almost everywhere on Q_{T} .

Coupling processes for different values of ε shows that u^{ε} is increasing in ε, hence the limit u^{ε}(t, ξ) ↘ u(t, ξ) exists as ε ↘ 0, and by applying dominated convergence to the weak form of the differential equation we see that u(t, ξ) satisfies the equation with initial data u_{0}(ξ). It remains to show that

𝒫{ω_{t} = u(t, ξ) dξ} = 1

for all t. Since ω_{t} and u(t, ξ) dξ are continuous in t (by Lemma 3.5 which is valid for 𝒫), it suffices to get this for a.e. t. Letting ε ↘ 0 through a countable set of values gives a shrinking sequence of events in (63), hence 𝒫 -a.s.,

(64)  v(t, ξ) ≤ u(t, ξ)  a.e. on Q_{T} .

Conversely, since the process preserves total stick length, 𝒫 -a.s.

∫_{T^{d}} v(t) = ω_{t}(T^{d}) = ω_{0}(T^{d}) = ∫_{T^{d}} u_{0} = ∫_{T^{d}} u(t)

for those t for which the density v(t) exists. This and (64) force

v(t, ξ) = u(t, ξ) a.e. on Q_{T} ,

and we are done. In conclusion, 𝒫 is supported by the unique path u( • , ξ) dξ, and Theorems 1 and 2 follow for the density u_{0} and initial distributions μ_{0}^{N}.

4. Proof of Theorem 3

The proof for the particle models is basically the same as for the stick models. The a priori estimate is again a consequence of attractiveness. Proceeding as in Section 3.1 gives

\begin{aligned}z_{1}(t) & = \frac{1}{4}N^{−d} ∑_{x∈Z_{N}^{d}} η_{t}(x)(κ_{N} + η_{t}(x))Aϕ(\frac{x}{N} ) \\ & + O(N^{−1}) · N^{−d} ∑_{x∈Z_{N}^{d}} η_{t}(x)(κ_{N} + η_{t}(x)) \end{aligned}

and E^{N}{z_{2}(t)} = O(N^{−d}). The results of Section 3.2 on tightness and the properties of the limit points of the distributions of αN_{•} follow as before.

For each N, ν^{N} denotes a product measure under which the particle stacks at different sites are {kκ_{N} : k = 0, 1, 2, …}-valued, i.i.d. geometric random variables with expectation K_{0} . Again we have derivatives f_{t}^{N}(η) = dμ_{t}^{N}/dν^{N}(η), bounded by some constant K_{1} uniformly over N, η and t.

The entropy bound (31) is valid as long as κ_{N} + μ_{0}^{N}{η(x)} ≥ ε_{0} > 0 holds for some constant ε_{0} , uniformly over N and x ∈ Z_{N}^{d}. Thus for the case κ = 0 we need to proceed as for the stick model, by first assuming u_{0} bounded away from zero and then removing this assumption in the end, but for the case κ > 0 no such assumption is needed. The quantities σ_{N}(g) and D_{x}^{N}(g) are defined as in (32) and (34) except that the integral ∫_{0}^{η(x)} du and u are replaced by the sum

κ_{N} ∑_{k=1}^{η(x)/κ_{N}} and kκ_{N} , respectively. Then (36) follows from the entropy bound as before, with β = 2.

For the local equilibrium we need to prove

(65)

lim_{ε→0} lim sup_{N→∞} E^{N}\{∫_{0}^{T} N^{−d} ∑_{x∈Z_{N}^{d}} |\frac{1}{|Λ_{Nε}|} ∑_{y∈x+Λ_{Nε}} η_{t}(y)(κ_{N} + η_{t}(y))

− 2(κS_{x+Λ_{Nε}}(η_{t}) + S_{x+Λ_{Nε}}^{2}(η_{t}))| dt} = 0.

As before S_{Γ}(η) = |Γ|^{−1} ∑_{y∈Γ} η(y). Equation (65) follows from showing

(66)

lim_{ε→0} lim sup_{N→∞} E^{N}{∫_{0}^{T} N^{−d} ∑_{x∈Z_{N}^{d}} |S_{x+Λ_{Nε}}(η_{t}^{2})

− (κS_{x+Λ_{Nε}}(η_{t}) + 2S_{x+Λ_{Nε}}^{2}(η_{t}))| dt} = 0.

Repeating Lemma 3.9, our tasks are again to prove the one-block estimate

(67)  lim_{L→∞} lim sup_{N→∞} \overline{μ}^{N}\{|S_{Λ_{L}}(η^{2}) − κS_{Λ_{L}}(η) − 2S_{Λ_{L}}^{2}(η)|\} = 0

and the two-block estimate

(68)  lim_{L→∞} lim sup_{ε→0} lim sup_{N→∞} K^{−2d} ∑_{1≤i,j≤d} \overline{μ}^{N}\{|S_{Δ_{i}}(η) − S_{Δ_{j}}(η)|^{2}\} = 0.

The measure \overline{μ}^{N} was defined by (28).

4.1. One-block estimate, case κ = 0. Letting μ_{L} denote a limit point of a subsequence of \{\overline{μ}^{N}\} that realizes the lim sup_{N→∞} in (67), we need to prove

(69)  lim_{L→∞} μ_{L}{|S_{Λ_{L}}(η^{2}) − 2S_{Λ_{L}}^{2}(η)|} = 0

exactly as in Section 3.3.1. Thus it suffices to show that Lemma 3.10 holds for an arbitrary limit point μ of \{\overline{μ}^{N}\}. Given q = (q(x) : x ∈ Γ) ∈ [0, ∞)^{Γ} , define the configuration q_{N} by q_{N}(x) = [q(x)/κ_{N}]κ_{N} , where [ ·] denotes integer part. Then we get

∫_{a}^{b} \overline{μ}^{N}\{η_{Γ} ≥ q^{u,x,y}\} du = κ_{N} ∑_{k=[a/κ_{N} ]}^{[b/κ_{N} ]−1} \overline{μ}^{N}\{η_{Γ} ≥ q_{N}^{kκ_{N} ,x,y}\} + O(κ_{N}).

The error is O(κ_{N}), because for any x and any fixed constants a_{0} and a_{1} ,

\overline{μ}^{N}\{a_{0} ≤ η(x) < a_{0} + a_{1}κ_{N} \} ≤ K_{1}a_{1} sup_{k} ν^{N}\{η(x) = kκ_{N} \} = O(κ_{N}).

Reasoning as in (43) yields

(70)  \begin{aligned}|∫_{a}^{b} & \overline{μ}^{N}\{η : η_{Γ} ≥ q\} − \overline{μ}^{N}\{η : η_{Γ} ≥ q^{u,x,y}\} du| \\ & ≤ C(ν^{N}\{κ_{N} ∑_{k=1}^{η(x)/κ_{N}}[g^{N}(η^{kκ_{N} ,x,y}) − g^{N}(η)]^{2}\})^{1/2} + O(κ_{N}). \end{aligned}

Now proceed as in the proof of Lemma 3.10.

4.2. One-block estimate, case κ > 0. Let μ be any limit point of \{\overline{μ}^{N}\}. It is a probability measure on {kκ : k = 0, 1, 2, …}^{Z_{+}^{d}}. Let Γ be a finite subset of Zd_{+} and q = (k(x)κ : x ∈ Γ) a particle configuration on Γ, where k(x) are nonnegative integers.

Lemma 4.1. For any x, y ∈ Γ and k ≤ k(x),

(71)  μ{η : η_{Γ} = q} = μ{η : η_{Γ} = q^{kκ,x,y}}.

Proof. Let q_{N} = (k(x)κ_{N} : x ∈ Γ) be the corresponding configuration for the N th process. Proceeding as in (43), then constructing a suitable path from x to y and using the bound (36) give, as in the proof of Lemma 3.10,

\begin{aligned}|\overline{μ}^{N} & \{η_{Γ} = q_{N}\} − \overline{μ}^{N}\{η_{Γ} = q_{N}^{kκ_{N} ,x,y}\}| \\ & ≤ Cκ_{N}^{−1}(ν^{N}\{κ_{N} ∑_{k=1}^{η(x)/κ_{N}}[g^{N}(η^{kκ_{N} ,x,y}) − g^{N}(η)]^{2}\})^{1/2} ≤ CN^{−1}. \end{aligned}

This suffices for (71).

Corresponding to Corollary 3.11 we now have:

Corollary 4.2. μ is a mixture of i.i.d. geometric distributions.

Proof. Exchangeability is immediate from (71), so there is a representation

μ = ∫ ρ^{⊗Z_{+}^{d}} Q(dρ)

of μ in terms of i.i.d. distributions. The integration variable ρ is a probability measure on {kκ : k = 0, 1, 2, …}. Consider first a single site x, and a set {x_{1}, …, x_{m}} of mutually distinct sites distinct from x. By (71),

μ{η(x) = mκ} = ∑_{k_{1},...,k_{m}≥0} μ{η(x) = mκ, η(x_{1}) = k_{1}κ, …, η(x_{m}) = k_{m}κ}

= ∑_{k_{1},...,k_{m}≥0}μ{η(x) = 0, η(x_{1}) = (k_{1} + 1)κ, …, η(x_{m}) = (k_{m} + 1)κ}

= μ{η(x) = 0, η(x_{1}) ≥ κ, …, η(x_{m}) ≥ κ}

= ∫ ρ(0)(1 − ρ(0))^{m} Q(dρ),

so the distribution μ{η(x) ∈ · } is a mixture of geometric distributions. But then

\begin{aligned}μ\{η_{Γ} ≥ q\} & = μ\{η(x) ≥ κ ∑_{y∈Γ} k(y)\} = ∫ (1 − ρ(0))^{∑_{y∈Γ} k(y)} Q(dρ) \\ & = ∫ ∏_{y∈Γ}(1 − ρ(0))^{k(y)} Q(dρ), \end{aligned}

and we see that μ is indeed a mixture of i.i.d. geometrics.

The one-block estimate is now completed as was done in Section 3.1.1, utilizing the fact that if X is a κZ_{+} -valued geometric random variable, then E(X^{2}) = κEX + 2(EX)^{2} .

4.3. Two-block estimates. This time we have to define the two-site Dirichlet form separately for each N:

D_{x,y}^{N} = ν^{N}{κ_{N} ∑_{k=1}^{η(x)/κ_{N}} [g^{N}(η^{kκ_{N} ,x,y}) − g^{N}(η)]^{2}}.

The argument of Lemma 3.12 works again to give

D_{x,y}^{N} ≤ Cε^{2}

for all N, ε, and x, y ∈ Λ_{Nε}. As in Section 3.3.2, this bound allows us to deduce (68) by proving

(72)  lim_{L→∞} lim sup_{ε→0} lim sup_{N→∞} sup_{μ} μ{|S_{Λ_{L}}(η) − S_{Λ′_{L}}(η)|^{2}} = 0,

where the supremum is over the class of probability measures μ on Ω_{Λ_{L}∪Λ′_{L}}^{(κ_{N} )} that satisfy μ ≪ ν^{N}, D_{x,y}^{N} ≤ Cε^{2} , and μ{η^{k}(x)} ≤ C_{k} for all x, y ∈ Λ_{L} ∪ Λ′_{L}.

For each L, let μ_{L} be a measure that satisfies

μ_{L}{|S_{Λ_{L}}(η) − S_{Λ′_{L}}(η)|^{2}} = lim sup_{ε→0} lim sup_{N→∞} sup_{μ} μ{|S_{Λ_{L}}(η) − S_{Λ′_{L}}(η)|^{2}}.

For the case κ = 0 the computation done in (70) shows that

|∫_{a}^{b} \overline{μ}^{N}\{η : η_{Γ} ≥ q\} − \overline{μ}^{N}\{η : η_{Γ} ≥ q^{u,x,y}\} du| ≤ Cε + O(κ_{N}).

Thus, as we let first N → ∞ and then ε → 0, Lemma 3.10 holds for μ_{L} and we can complete the proof of the two-block estimate for the case κ = 0 as was done in Section 3.3.2.

The pattern is clear by now so we leave the details of the two-block estimate for the case κ > 0 to the reader.

After establishing the local equilibrium we have the analogue of (57) for the particle model:

lim_{ε→0} lim sup_{N→∞} P^{N}\{ sup_{0≤t≤T} |α_{t}^{N}(ϕ) − α_{0}^{N}(ϕ) − 2c ∫_{0}^{t} N^{−d} ∑_{x∈Z_{N}^{d}} Aϕ(\frac{x}{N} )(κα_{s}^{N}(χ_{ε,x/N})

+ [α_{s}^{N}(χ_{ε,x/N})]^{2}) ds| ≥ δ} = 0.

The remaining technical steps follow as before, and with this we consider Theorem 3 proved.

352 M. Ekhaus and T. Seppäläinen

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Received 29 March 1995