HYDRODYNAMIC LIMIT FOR THE SIMPLE EXCLUSION PROCESS ON NON-HOMOGENEOUS GRAPHS

M.D. JARA

Abstract. We prove hydrodynamic limit for the symmetric simple exclusion process evolving in non-homogeneous systems, like finitely ramified fractals, triangulations of a bounded domain and random environment.

1. Introduction

The simple exclusion process is a system of random walks on the lattice, or more generally on a connected graph, that interact between them only by the exclusion rule: no more than a particle is allowed at each site of the graph. This process has been widely studied, and their scaling properties are well understood. In particular, the hydrodynamic limit is known to be the heat equation when the underlying random walk is symmetric, and the Burguers equation when the underlying random walk is asymmetric. In the other hand, the evolution of random walks in non-homogeneous structures, like random walks in random environment or random walks in prefractals have attracted some attention in the past years. By this reason, we have decided to introduce inhomogeneities also in the simple exclusion process by taking the underlying random walk evolving in a non-homogeneous structure. We consider in this article three examples of non-homogeneous structures: a triangulated domain, the Sierpinski pregasket and a random environment on the lattice. In all these cases we prove that the simple exclusion process when the underlying random walk is symmetric, has a hydrodynamic limit given by the heat equation. The main tool in order to obtain this hydrodynamic limit, is the introduction of the corrected empirical measure, that allow us to employ the by now standard techniques in this non-homogeneous setting. The hydrodynamic limit of interacting particle systems in random environment has been considered before in [Q2], [FM], [Q3] in a more general setting, where stronger interaction is allowed between particles. In these articles, the main tool is the generalization of the so called non-gradient method of Varadhan [V] to the situation in which bond disorder appears, and therefore they cover some of the results presented here. The method exposed in this article can be generalized to include models in which a gradient condition is satisfied [GJO], or where the environment is strongly degenerated [FJL]. As far as we know, there are no previous examples of hydrodynamic limit on triangulated domains or in fractals.

The article is organized as follows. In section 2 we introduce the corrected empirical measure and we prove the hydrodynamic limit for the exclusion process in an abstract framework (theorems 2.1, 2.4). In section 3 we check the hypothesis of theorems 2.1, 2.4 for the three examples mentioned before. In section 4 we discuss some further developments and extensions.

Note added in proof. At the time this article was being written, we learned of the work of Faggionato [Fag], where she have proved the same result considered here for a simple exclusion process in random environment.

2. Notations and results

2.1. The exclusion process. Let Γ_{n} = (V_{n}, E_{n}) be a graph, where the set V_{n} corresponds to the set of sites of Γ_{n} and E_{n} ⊆ V_{n} × V_{n} is the set of edges of the graph Γ_{n}. We say that the graph Γ_{n} is symmetric if E_{n} is a symmetric subset of V_{n} × V_{n}. We say that Γ_{n} has no loops if E_{n} does not intersect the diagonal of V_{n} × V_{n}. Two sites x, y ∈ V_{n} are neighbours if (x, y) ∈ E_{n}. We define the degree of Γ_{n} at site x as the number of neighbours of x. The degree of Γ_{n} is the supremum of the degree at its sites. We say that two points x, y ∈ Γ_{n} are connected if there exists a sequence x_{0} = x, x_{1}, ..., x_{l} = y of sites in Γ_{n} such that x_{i−1} and x_{i} are neighbours for any i. Γ_{n} is connected if any two sites x, y ∈ Γ_{n} are connected. Throughout this paper we assume that Γ_{n} is connected, symmetric, has no loops, has finite degree and it is numerable. Since the ordering is no longer important, we denote by ⟨xy⟩ the edge (x, y) ∈ E_{n}. Let c_{n} : E_{n} → ℝ_{+} be non-negative function. We define the simple exclusion process in Γ_{n} with jump rates c_{n} as follows. Let Ω_{n} = {0, 1}^{V_{n}} be the state space. We say that a function f : Ω_{n} → ℝ is local if it depends on the value of η ∈ Ω_{n} in x for a finite number of sites x. Consider the linear operator 𝕃_{n} whose action over local functions f is given by

𝕃_{n}f(η) = ∑_{⟨xy⟩∈E_{n}} c_{n}(⟨xy⟩)∇_{xy}f(η),

where ∇_{xy}f(η) = f(σ^{xy}η) − f(η) and

σ^{xy}η(z) = \begin{cases}η(y), & z = x \\ η(x), & z = y \\ η(z), & z ≠ x, y \end{cases}

Denote by η_{t}^{n} the Markov process generated by the operator 𝕃_{n}. If η(x) = 1 we say that the site x is occupied by a particle, and if η(x) = 0, we say that the site x is empty. With this convention, the process η_{t}^{n} corresponds to a dynamics on which particles jump only between sites connected by an edge. A particle at site x waits an exponential time of rate c(⟨xy⟩), at the end of which it tries to jump to the site y. If the site y is empty, the jump is accomplished, otherwise the jump is supressed, the particle stays at site x and a new exponential time begins. This is done independently for each particle and for each site of arrival. We further assume that the jump rates c_{n} are uniformly bounded in E_{n}. The process η_{t}^{n} is a well defined strong Markov process with generator 𝕃_{n}, and the set of local functions f : Ω_{n} → ℝ is a core for the generator 𝕃_{n}. For each n, fix a probability measure μ_{n} in Ω_{n}. We consider the process η_{t}^{n} starting from μ_{n}. To avoid uninteresting compactness issues, we take the evolution of the process η_{t}^{n} in a finite time interval [0, T]. We denote by ℙ_{n} the probability measure in the set D([0, T], Ω_{n}) of càdlàg paths corresponding to η_{t}^{n}, and by 𝔼_{n} the expectation with respect to this measure.

2.2. The empirical measure. Take a sequence {Γ_{n} = (V_{n}, E_{n})}_{n} of graphs and a sequence of rate functions {c_{n} : E_{n} → ℝ_{+}}_{n}. Denote by l_{0}(V_{n}) the set of functions f : V_{n} → ℝ such that f(x) ≠ 0 for a finite number of points in V_{n}. Let || · ||_{n} be a norm defined in l_{0}(V_{n}) and denote by H_{n} the closure of l_{0}(V_{n}) under this norm.

Assume that every element v in H_{n} can be represented as a function v : V_{n} → ℝ. Let (H, || · ||) be a separable, complete normed space, and let T_{n} : H → H_{n} be a family of linear bounded functions.

For each η ∈ Ω_{n} and each v ∈ l_{0}(V_{n}), define

(η, v)_{n} = \frac{1}{a_{n}} ∑_{x∈V_{n}} η(x)v(x).

Here a_{n} is a normalizing constant. We want to consider the process η_{t}^{n} as taking values in H_{n}^{∗}, the dual of H_{n}. For that, we make the following assumption:

(H1) There exists a constant K independent of n such that

|(η, v)_{n}| ≤ K||v||_{n}

for all v ∈ l_{0}(V_{n}).

In this case, by continuity, (η, v)_{n} can be extended to H_{n} and in fact η ∈ H_{n}^{∗}. Therefore, we can consider the process η_{t}^{n} as taking values in H_{n}^{∗}. Now we define the empirical process π_{t}^{n} as the image of η_{t}^{n} under T_{n}^{∗}: for f in H, π_{t}^{n} satisfies

π_{t}^{n}(f) = (η_{t}^{n}, T_{n}f)_{n}.

Since T_{n} is continuous, π_{t}^{n} is well defined as an element of H^{∗}. By construction, we are considering the weak-∗ topology in H^{∗}. In particular, it is not true that H^{∗} is a Polish space without further assumptions about H. However, what is true is that the weak-∗ topology restricted to a ball in H^{∗} is separable and metrizable. Moreover, the ball is complete under this metric. By this reason, we make the following assumption:

(H2) The operators T_{n} are uniformly bounded, that is, there exists constant K′ such that ||T_{n}f||_{n} ≤ K′||f|| for all f ∈ H and all n.

In this case, π_{t}^{n} is a process taking values in the ball of radius KK′ in H^{∗}. Take v ∈ l_{0}(V_{n}). By Itô’s formula, the process

(η_{t}^{n}, v)_{n} − (η_{0}^{n}, v)_{n} − ∫_{0}^{t}(η_{s}^{n}, L_{n}v)ds

is a martingale of quadatic variation

∫_{0}^{t} \frac{1}{a_{n}^{2}} ∑_{⟨xy⟩∈E_{n}} c_{n}(⟨xy⟩)[v(y) − v(x)]^{2}(η_{s}^{n}(y) − η_{s}^{n}(x))^{2}ds,

where L_{n} is the operator defined by

L_{n}v(x) = ∑_{y:⟨xy⟩∈E_{n}} c_{n}(⟨xy⟩)[v(y) − v(x)].

The following assumptions identify the hydrodynamic limit of η_{t}^{n} and ensure tightness for the sequence of empirical measures π_{t}^{n}.

(H3) There exist a closed operator L : D(L) ⊆ H → H, a kernel G ⊆ D(L) of L and a constant λ ≥ 0 such that for all f ∈ G we can find a sequence f_{n} ∈ H_{n} such that

i)||f_{n} − T_{n}f||_{n} ⟶^{n→∞} 0
ii)||(λ − L_{n})f_{n} − T_{n}(λ − L)f||_{n} ⟶^{n→∞} 0
iii)\frac{1}{a_{n}^{2}} ∑_{⟨xy⟩∈E_{n}} c_{n}(⟨xy⟩)(f_{n}(y) − f_{n}(x))^{2}.

(H4) There exists φ_{0} ∈ H^{∗} such that for every f ∈ G and every ϵ > 0,

lim_{n→∞} ℙ_{n}(|π_{0}^{n}(f) − (φ_{0}, f)| > ϵ) = 0,

and the Cauchy problem

\begin{cases}\frac{d}{dt}φ(t) = L^{∗}φ(t) \\ φ(0) = φ_{0} \end{cases}  (2.1)

admits a unique weak solution. Now we are ready to state the main result of this article:

Theorem 2.1. Assume that (H1)-(H4) hold. Then the empirical process π_{t}^{n} defined in D([0, T], H^{∗}) converges in probability to the deterministic path φ_{0}(t).

The proof of this theorem follows the usual lines of the proof of hydrodynamic limits for interacting particle systems. First we prove the tightness in D([0, T], H^{∗}) for the distributions of π_{t}^{n}. Then we prove that all their limit points are concentrated on weak solutions of the hydrodynamic equation 2.1. By the uniqueness of these solutions, we conclude that the limit point is unique and it is concentrated on the path φ_{t}. Since the topology of convergence in distribution is metrizable, we conclude that the whole sequence converges to this limit point. However, this plan can not be carried out for the empirical process π_{t}^{n}, but for another related process, the corrected empirical measure, that we introduce in the next section.

2.3. The corrected empirical measure. For each f ∈ G, take a sequence f_{n} ∈ H_{n} satisfying (H3). We define the corrected empirical measure π̂_{t}^{n} by π̂_{t}^{n}(f) = (η_{t}^{n}, f_{n})_{n}. Notice that a priori this relation does not define a continuous functional in H, so it is not clear if π̂_{t}^{n} actually belongs to H^{∗}. However, this point will not be relevant in our arguing. By Itô’s formula, the process

M_{t}^{n}(f) = π̂_{t}^{n}(f) − π̂_{0}^{n}(f) − ∫_{0}^{t}⟨η_{s}^{n}, L_{n}f_{n}⟩_{n}ds  (2.2)

is a martingale with quadratic variation given by

⟨M_{t}^{n}(f)⟩ = ∫_{0}^{t} \frac{1}{a_{n}^{2}} ∑_{⟨xy⟩∈E_{n}} c_{n}(⟨xy⟩)[f_{n}(y) − f_{n}(x)]^{2}(η_{s}^{n}(y) − η_{s}^{n}(x))^{2}ds.

Notice that the quadratic variation ⟨M_{t}^{n}(f)⟩ is uniformly bounded by

\frac{T}{a_{n}^{2}} ∑_{⟨xy⟩∈E_{n}} c_{n}(⟨xy⟩)[f_{n}(y) − f_{n}(x)]^{2}.

In particular, M_{t}^{n}(f) converges to 0 in L^{2}(ℙ_{n}) as n goes to ∞. Recall Prohorov’s theorem for tightness of a sequence of distributions in D([0, T], X), where X is a separable, complete metric space.

Proposition 2.2. A sequence {P_{n}} of probability measures in D([0, T], X) is tight if and only if:

i) For each t ∈ [0, T] and each ϵ > 0 there exists a compact K_{ϵ}(t) ⊆ X such

that

sup_{n} P_{n}[x(t) ∈ K_{ϵ}(t)] ≥ 1 − ϵ.

ii) For each ϵ > 0 there exists δ > 0 such that

sup_{n} P_{n}[w′_{δ}(x) > ϵ] ≤ ϵ.

Here w′_{δ}(x) corresponds to the modified modulus of continuity of x ∈ D([0, T], X) as defined in [Bil]. We can consider, instead of ii), the stronger condition

ii’) For each ϵ > 0 there exists δ > 0 such that

sup_{n} P_{n}[w_{δ}(x) > ϵ] ≤ ϵ,

where now w_{δ}(x) is the usual modulus of continuity. In that case, the limit points of P_{n} give full probability to the set C([0, T], X) of continuous paths. In our case X is the ball of radius KK′ in H^{∗} and P_{n} is the distribution of π_{t}^{n} in D([0, T], H^{∗}) (remember that P_{n} puts mass 1 in the ball of radius KK′). Condition i) is an immediate consequence of Banach-Alaoglu’s theorem. Let {g_{k}}_{k∈ℕ} be a numerable dense subset of H. A possible choice of a metric for the weak-∗ topology in X is the following:

d(π, ξ) = ∑_{k} \frac{1}{2^{k}} \frac{|⟨π, g_{k}⟩ − ⟨ξ, g_{k}⟩|}{1 + |⟨π, g_{k}⟩ − ⟨ξ, g_{k}⟩|}.

For each f ∈ H, let ℙ_{n}^{f} be the distribution in D([0, T], ℝ) of the process π_{t}^{n}(f). In order to verify the second condition, we use the following result (for a proof see [KL]):

Proposition 2.3. Condition ii) is verified by the sequence {ℙ_{n}}_{n} if and only if it is verified for {ℙ_{n}^{g}}_{n} for g on a dense subset of H.

Therefore, condition ii) is satisfied if we prove that the distributions of π_{t}^{n}(f) are tight in D([0, T], ℝ) for any f ∈ G. At this point the corrected empirical process comes into play. Instead of proving tightness for π_{t}^{n}(f), we will prove tightness for π̂_{t}^{n}(f). After this, we prove that π_{t}^{n}(f) and π̂_{t}^{n}(f) are close when n goes to ∞. By Doob’s inequality, convergence in L^{2}(ℙ_{n}) to 0 of ⟨M_{t}^{n}(f)⟩ implies tightness for M_{t}^{n}(f). Moreover, since the convergence to 0 of ⟨M_{t}^{n}(f) is uniform, the limit points are concentrated in continuous paths. By hypothesis (H3)−i and (H4), (η_{0}^{n}, f_{n})_{n} converges to (φ_{0}, f). The integral term in equation 2.2 is handled as follows. Notice that for all t ∈ [0, T],

|(η_{t}^{n}, L_{n}f_{n})_{n} − (η_{t}^{n}, λf_{n} − T_{n}(λ − L)f)_{n}| ≤ K||(λ − L_{n})f_{n} − T_{n}(λ − L)f||_{n}.

This last term goes to 0 as n goes to ∞. In the other hand, by hypothesis (H1)-(H2), |(η_{t}^{n}, λf_{n})| ≤ λK(||f_{n} − T_{n}f||_{n} + K′||f||) and |(η_{t}^{n}, T_{n}(λ − L)f)_{n}| ≤ KK′||(λ − L)f||. In other words, the integrand in 2.2 is uniformly bounded by a constant that depends only on f. In particular, the integral term is also tight, with limit points concentrated in continuous paths. The proof of tightness for the process π̂_{t}^{n}(f) is now complete. It is easy to see that

sup_{0≤t≤T} |π̂_{t}^{n}(f) − π_{t}^{n}(f)| ≤ K||f_{n} − T_{n}f||_{n}

and by (H3)-i this expression goes to 0 as n goes to ∞. A simple ϵ/3 argument shows that the sequence π_{t}^{n}(f) is also tight, and that π_{t}^{n} and π̂_{t}^{n} have the same limit points. This ends the proof of tightness for the whole process π_{t}^{n}, since G is dense in H.

Let π_{t} be a limit point of π_{t}^{n}. To avoid the introduction of more notation, we denote simply by n the subsequence for which π_{t}^{n} converges to π_{t}. Remember that π_{t} is a process defined in D([0, T], X) and for any f in G, π̂_{t}^{n}(f) also converges to π_{t}(f). By the previous estimates, the integral term in equation 2.2 conveges to ∫_{0}^{t} π_{s}(Lf)ds. Since M_{t}^{n}(f) vanishes in the limit, we conclude that

π_{t}(f) − π_{0}(f) − ∫_{0}^{t} π_{s}(Lf)ds = 0

for all f ∈ G. Since L is closed, this equation is still valid for any f in the domain of L. This is exactly the weak form of equation 2.1. By hypothesis (H4), this equation has a unique weak solution φ(t), so the distribution of π_{t} is a Dirac-δ measure concentrated in φ(t). Therefore, π_{t} is unique and the whole sequence π_{t}^{n} converges in distribution to φ(t). Since the limit is deterministic, we can improve the convergence to convergence in probability. The proof of theorem 2.1 is finished.

Remark. The crucial hypothesis in the proof of this theorem is (H3). However, a careful reading of the proof of theorem 2.1 shows that this hypothesis can be relaxed. For v : V_{n} → ℝ define

|v|_{n} = \frac{1}{a_{n}} ∑_{x∈V_{n}} |v(x)|.

Theorem 2.4. Theorem 2.1 remains true if we assume the following condition instead of (H3): (H3’) There exist a closed operator L : D(L) ⊆ H → H, a kernel G ⊆ D(L) of

L and a constant λ ≥ 0 such that for all f ∈ G we can find a sequence f_{n} ∈ H_{n} such that

i)|f_{n} − T_{n}f|_{n} ⟶^{n→∞} 0
ii)|(λ − L_{n})f_{n} − T_{n}(λ − L)f|_{n} ⟶^{n→∞} 0
iii)\frac{1}{a_{n}^{2}} ∑_{⟨xy⟩∈E_{n}} c_{n}(⟨xy⟩)(f_{n}(y) − f_{n}(x))^{2}.

Usually the norm | · |_{n} is weaker that the norm || · ||_{n}. We prefer to enunciate theorem 2.1 with hypothesis (H3) in detriment of (H3’) by two reasons. The first one is that the definition of the empirical measure π_{t}^{n} is made in terms of the norm || · ||_{n}. The second one is that convergence results for the norm || · ||_{n} are more naturally obtained, and therefore it is easier to obtain (H3’) as a consequence of convergence results for || · ||_{n}.

3. Applications

3.1. Exclusion process in a triangulated domain. Let Ω ⊆ ℝ^{d} a open and bounded domain. For simplicity, we restric ourselves to the case d = 2, and we assume that Ω is a convex poligon. A triangulation of Ω is a partition T = {△_{i}}_{i} of Ω in triangles such that: i) The union of the triangles in T is equal to \overline{Ω}.

i) If two triangles △_{i}, △_{j} ∈ T intersect at exactly one point, the intersection △_{i} ∩ △_{j} is a common vertex of △_{i}, △_{j}.

iii) If two different triangles △_{i}, △_{j} intersect at more than one point, the

intersection △_{i} ∩ △_{j} is a common edge of △_{i} and △_{j}. To a triangulation T of Ω we can associate a graph Γ = (V, E) by taking V as

the set of vertex of the triangles in T and E as the set of edges of the triangles in T . We define the size of T by

h(T ) = sup_{△∈T} diam(△).

For each △ ∈ T we define ρ(△) as the diameter of the inscribed circle in △, and define

γ(T ) = inf_{△∈T} \frac{ρ(△)}{h(T )}.

Take a sequence T_{n} of triangulations of Ω such that h_{n} = h(T_{n}) goes to 0 as n goes to ∞. We say that the sequence is uniform if there exists a constant γ_{0} > 0 such that γ(T_{n}) > γ_{0} for all n. Assume that T_{n} is uniform, and consider the graph Γ_{n} = (V_{n}, E_{n}) associated with T_{n}. Since T_{n} is uniform, Γ_{n} is a finite graph and the degree of Γ_{n} is uniformly bounded in n. Let G(Ω) be the set of infinitely differentiable functions f : Ω → ℝ such that ∫_{Ω} f(u)du = 0. In G(Ω) we define the norm

||g||^{2} = ∫_{Ω} (∇g(u))^{2}du

and we define H = H_{1}(Ω) as the closure of G(Ω) under this norm. Denote by a(·, ·) the inner product in H. Poincaré’s inequality states that there exists a constant C = C(Ω) such that for any g ∈ H,

∫_{Ω} g(u)^{2}du ≤ C ∫_{Ω} ||g||^{2}.

In particular, we have the inclusions H ⊆ L_{∗}^{2}(Ω) ⊆ H^{∗}, where L_{∗}^{2}(Ω) is the set of mean-zero, square integrable functions in Ω.

We define H_{n} as the set of functions f ∈ H such that the restriction of f to any △ ∈ T_{n} is linear. For each x ∈ V_{n}, define ψ_{x} : Ω → ℝ as the continuous function such that ψ_{x}(y) = δ_{xy} for all y ∈ V_{n} and such that the restriction of ψ_{x} to any △ ∈ T_{n} is linear. Notice that the set of functions {ψ_{x} − ∫ ψ_{x}; x ∈ V_{n}} form a basis of H_{n}. Therefore, we can represent v ∈ H_{n} as an element {v(x); x ∈ V_{n}} of l_{0}(V_{n}) via the identification

v = ∑_{x∈V_{n}} v(x)ψ_{x}.

For x ∈ V_{n} , denote by R_{x} the union of the triangles that have x as one of its vertex. We adopt the following convention: for any subset R of Ω, we also denote by R its area. For v, w ∈ H_{n} we simply define ||v||_{n} = ||v||, a(v, w)_{n} = a(v, w).

We choose a_{n} equal to the number of points in V_{n}. Define the jump rates c_{n} by c_{n}(⟨xy⟩) = −a_{n} ∫ ∇ψ_{x}∇ψ_{y}. If every triangle in T_{n} have angles less than π/2, c_{n}(⟨xy⟩) is non-negative. By the uniformity of T_{n}, the rates c_{n} are bounded above by Ca_{n}, with C independent of n. Now we consider the exclusion process η_{t}^{n} associated to the jump rates c_{n} and the graph Γ_{n}. We have the following result.

Theorem 3.1. Let Ω ⊆ ℝ^{2} be a polygonal domain. Let T_{n} be a family of uniform triangulations in Ω. Assume (H4) for some initial profile ρ_{0} : Ω → ℝ. Then the empirical process π_{t}^{n} associated to η_{t}^{n} converges in probability to the path ρ(t, u), where ρ(t, u) is the solution of the heat equation with Neumann boundary conditions

\begin{cases}∂_{t}ρ(t, x) & = Δρ(t, x) in Ω \\ ∂_{n}ρ(t, x) & = 0 in ∂Ω \\ ρ(0, x) & = ρ_{0}(x) in Ω. \end{cases}  (3.1)

Proof. By theorem 2.1, the proof of theorem 3.1 follows if we check hypothesis (H1)-(H3) for this case. We begin with (H1). For v ∈ H_{n}, a simple computation shows that

∫_{Ω} v(u)^{2}du = \frac{1}{3} ∑_{x∈V_{n}} R_{x}v(x)^{2} − \frac{1}{12} ∑_{⟨xy⟩∈E_{n}}(v(y) − v(x))^{2}R_{y} ∩ R_{x},

||v||_{n}^{2} = ∫_{Ω} (∇v(u))^{2}du = \frac{1}{a_{n}} ∑_{⟨xy⟩∈E_{n}} c_{n}(⟨xy⟩)(v(y) − v(x))^{2}.

We say that two sequences α_{n}, β_{n} are equivalent if the quotients α_{n}/β_{n} are bounded above and below in n. Two sequences of functions A_{n}, B_{n} : V_{n} (or E_{n}) → ℝ are equivalent if the quotients A_{n}(x)/B_{n}(y) are bounded above and below, uniformly in n and x, y. When A_{n} = B_{n} are equivalent, we say that A_{n} is uniform. Since we are assuming that the triangulations T_{n} are uniform, c_{n}/a_{n}^{2} : E_{n} → ℝ, R_{x} ∩ R_{y} : E_{n} → ℝ and R_{x} : V_{n} → ℝ are uniform and equivalent to a_{n}^{−1}. By Poincaré’s inequality applied to v ∈ H_{n}, we see that there exists a constant C = C(Ω) such that

∑_{x∈V_{n}} R_{x}v(x)^{2} ≤ C ∑_{⟨xy⟩∈E_{n}} c_{n}(⟨xy⟩)(v(y) − v(x))^{2}.

Therefore,

|(η, v)_{n}| ≤ \frac{1}{a_{n}} ∑_{x∈V_{n}} |v(x)| ≤ C ∑_{x∈V_{n}} R_{x}|v(x)| ≤ C(Ω, γ_{0})||v||_{n},

and hypothesis (H1) is verified. (H2) is automatically satisfied, since T_{n} is an orthogonal projection. Take now f ∈ L^{2}(Ω) and consider the solution u ∈ H of the Laplace problem

\begin{cases}Δu = f in Ω \\ ∂_{n}f = 0 in ∂Ω. \end{cases}

The variational formulation of this problem is

a(u, v) = ∫ fv for all v ∈ H.

In the same way, define u_{n} ∈ H_{n} as the solution of the variational problem

a(u, v)_{n} = ∫ fv for all v ∈ H_{n}.

It is easy to see that u_{n} = T_{n}u. Moreover, if f ∈ H_{n}, by construction of H_{n} we have the identity L_{n}u_{n} = f. For f ∈ H, we see that L_{n}u_{n} = T̂_{n}f, where T̂_{n} is the L^{2}-projector onto H_{n}, that is, T̂_{n} ∈ H_{n} satisfies

∫ T̂_{n}fv = ∫ fv for all v ∈ H_{n}.

We recall the following well known results from the theory of finite elements [Bra]:

Proposition 3.2. Let T_{n} a family of uniform triangulations of a polygonal domain Ω. There exist a constant c independent of n such that for all f ∈ H,

||f − T_{n}f|| ≤ ch_{n}||f||,
||T̂_{n}f||_{n} ≤ c||f||.

Notice that T̂_{n}T_{n}f = T_{n}f. Then, using the proposition,

||T̂_{n}f − T_{n}f||_{n} ≤ c||f − T_{n}f|| ≤ c^{2}h_{n}||f||.

Therefore, (H3)-i, (H3)-ii are satisfied for u. (H3)-iii is satisfied due to the fact that ||u_{n}||_{n}/a_{n} ≤ ||u||/a_{n} and a_{n} goes to 0 as n goes to ∞. In consequence, we have proved (H3) for the set G of functions u in H such that Δu also belongs to H. The proof of theorem 3.1 ends noticing that this set is a kernel for Δ. □

Remark. In theorem 3.1, we are taking (H4) as an hypothesis. But if we can not find some sequence of measures μ_{n} and some profile ρ_{0} satisfying (H4), theorem 3.1 is vacuous. Uniqueness for the equation 3.1 follows from standard arguments for any continuous initial condition ρ_{0}. For the initial configuration, take μ_{n} as the product measure in {0, 1}^{V_{n}} such that μ_{n}(η(x) = 1) = a_{n}ρ_{0}(x)R_{x}/3. Since R_{x} and a_{n} are equivalent, this number is always bounded by 1 provided that ρ_{0}(x) ≤ C for some constant C independent of n. By the law of large numbers, for any f smooth enough,

lim_{n→∞} μ_{n}\{|\frac{1}{a_{n}} ∑_{x∈V_{n}} η(x)T_{n}f(x) − \frac{1}{3} ∑_{x∈V_{n}} ρ_{0}(x)R_{x}T_{n}f(x)| ≥ ϵ\} = 0.

In the other hand,

| ∫ ρ_{0}(u)T_{n}f(u)du − \frac{1}{3} ∑_{x∈V_{n}} ρ_{0}(x)R_{x}T_{n}f(x)| ≤ Ω · w_{h_{n}}(ρ_{0}).

By the continuity of ρ_{0}, this difference goes to 0. Since T_{n}f converges to f in H, we conclude that μ_{n} and ρ_{0} satisfy hypothesis (H4).

3.2. Exclusion process on a pre-fractal. Now we will consider Γ_{n} as the nth step in the construction of the Sierpinski gasket. Let a_{0} = (0, 0), a_{1} = (1, 0), a_{2} = (1/2, \sqrt{3}/2) be the vertices of an equilateral triangle of side one. Define inductively

V_{n+1} = ⋃_{i=0,1,2} 1/2(V_{n} + a_{i}),

V^{∗} = ⋃_{n≥0} V_{n}

and denote by V = \overline{V}^{∗}, the closure of V^{∗}. We call V the Sierpinski gasket. We define the graph Γ_{n} = (V_{n}, E_{n}) in the canonical way: V_{n} is already defined, and E_{n} = {⟨xy⟩; x, y ∈ V_{n}, |y − x| = 1/2^{n}}. Denote by ν_{n} the measure that puts mass 3^{−n} on each point x of V_{n}. It is well known that ν_{n} converges in the vague topology to the Hausdorff measure ν in V of dimension d_{f} = log 3/ log 2. For this result and other results concerning the theory of harmonic analysis in V, see [Koz] and the reference therein. For each v : V_{n} → ℝ, define

E_{n}(v, v) = ∑_{⟨xy⟩∈E_{n}} (\frac{5}{3})^{n}(v(y) − v(x))^{2}.

For f : V^{∗} → ℝ, denote by f_{n} the restriction of f to V_{n}.

Proposition 3.3. For any f : V^{∗} → ℝ, E_{n}(f, f) is non-decreasing on n. Define E(f, f) = sup_{n} E_{n}(f_{n}, f_{n}). If E(f, f) is finite, then

sup_{x,y∈V^{∗}} \frac{|f(y) − f(x)|}{|y − x|^{α}} ≤ 6E(f, f)^{1/2},

where α = log(5/3)/2.

Notice that if E(f, f) < ∞, f is continuous in V^{∗}, so it can be extended to a function that we still denote by f defined in the whole set V. Denote by H the space obtained as the completion of ∪_{n}H_{n} under the norm ||f|| = E(f, f)^{1/2}. Clearly, H is a Hilbert space, and L^{2}(ν) ⊆ H. For f, g in H, we denote by E(f, g) the inner product in H. Denote by (f, g) the inner product in L^{2}(ν). We define the Laplacian Δ : H → H^{∗} via the variational relation

E(f, g) = (Δf, g)

for all g ∈ H.

Remark. The boundary of V is the set V_{0} = {a_{0}, a_{1}, a_{2}}. This is easy to see from the fact that any point in V_{n} \ V_{0} has exactly 4 neighbours. In order to simplify the choice of boundary conditions, we restrict the spaces H, H_{n} to functions f such that f(a_{0}) = f(a_{1}) = f(a_{2}). We also indentify the three points a_{0}, a_{1}, a_{2} as a unique point in V_{n}. We call this assumption periodic boundary conditions.

For each f in H, we define T_{n}f as the solution of the Laplace problem

\begin{cases}Δg = 0 in V \ V_{n} \\ g = f in V_{n}. \end{cases}

In other words, T_{n}f is the harmonic continuation of f|_{V_{n}}. we define H_{n} as the image of H under T_{n}. Notice that H_{n} is isometric to the space l_{0}(V_{n}) with the norm ||v||_{n} = E_{n}(v, v)^{1/2}. In fact, it is not hard to see that

ΔT_{n}f(x) = \begin{cases}L_{n}T_{n}f(x), & if x ∈ V_{n} \\ 0, & if x ∉ V_{n}, \end{cases}  (3.2)

where L_{n} is given by

L_{n}v(x) = ∑_{⟨xy⟩∈E_{n}} 5^{n}(v(y) − v(x)).

We define the simple exclusion process η_{t}^{n} in Γ_{n} as the process asssociated to the jump rates c_{n}(⟨xy⟩) = 5^{n} and the scaling constants a_{n} = 3^{n}. With these definitions, we have an explicit formula for the empirical measure:

π_{t}^{n}(f) = \frac{1}{3^{n}} ∑_{x∈V_{n}} η_{t}^{n}(x)f(x).

In particular, π_{t}^{n} can be understood as a measure in V:

π_{t}^{n}(du) = \frac{1}{3^{n}} ∑_{x∈V_{n}} η_{t}^{n}(x)δ_{x}(du),

where δ_{x} is the Dirac mass at x. In this case, hypothesis (H1)-(H3) are easy to verify. In fact, (H1) is an immediate consequence of Poincaré’s inequality, (H2) follows from the fact that E_{n}(f, f) is non-decreasing, and by 3.2, (H3) is satisfied trivially for f_{n} = T_{n}f. Uniqueness results for the hydrodynamic equation are well known. Take ρ_{0} : V → [0, 1] continuous. Since the elements in H are also continuous, by the law of large numbers hypothesis (H4) is satisfied for the product measures μ_{n} in {0, 1}^{V_{n}} with marginals μ_{n}{η(x) = 1} = ρ_{0}. In order to state the next theorem in full generality, we need a definition. We say that a sequence of measures μ_{n} in {0, 1}^{V_{n}} is associated to the profile ρ_{0} : V → [0, 1] if for every continuous function f : V → ℝ and every ϵ > 0 we have

lim_{n→∞} μ_{n}{η, |3^{−n} ∑_{x∈V_{n}} η(x)f(x) − ∫ ρ_{0}(u)f(u)ν(du)| > ϵ} = 0.

The next theorem follows from the previous observations.

Theorem 3.4. Let ρ_{0} : V → [0, 1] be a continuous profile, and let μ_{n} be associated to ρ_{0}. Consider the exclusion process η_{t}^{n} defined in Γ_{n} and starting from μ_{n}. Then the empirical measure π_{t}^{n} converges in probability to the deterministic path ρ_{0}(t, u), solution of the heat equation

\begin{cases}∂ρ(t, x) = Δρ(t, x) in V \ V_{0} \\ ρ(t, a_{0}) = ρ(t, a_{1}) = ρ(t, a_{2}) \\ ρ(0, x) = ρ_{0}(x). \end{cases}

This result is not surprising, since the motion of the particles in the lattice Γ_{n} is uniform. In order to generalize this theorem to the case on which the jump rates c_{n}(⟨xy⟩) are no longer uniform, we need to introduce the concept of Γ-convergence. A sequence of functionals \overline{E}_{n} : H → ℝ is Γ-convergent to a functional \overline{E} : H → ℝ if

i) for any sequence {f_{n} ∈ H}_{n} weakly convergent to f in H, we have

lim inf_{n→∞} \overline{E}_{n}(f_{n}) ≥ \overline{E}(f),

ii) there exists a sequence {f_{n}^{0} ∈ H}_{n} such that f_{n}^{0} converges weakly to f in

H and

lim_{n→∞} \overline{E}_{n}(f_{n}^{0}) = \overline{E}(f).

The sequence \{\overline{E}_{n}\}_{n} is said to be equicoercive if there exists a constant C > 0 independent of n such that \overline{E}_{n}(f) ≤ CE(f, f) for all f ∈ H.

The introduction of Γ-convergence is useful due to the following result [dM]:

Proposition 3.5. Let I_{n} be equicoercive and Γ-convergent to I. Assume that I has a unique minimizer f. Let f_{n} be a minimizer (not necessarily unique) of I_{n}. Then f_{n} converges weakly to f.

Take a function f ∈ H and take f_{n} ∈ H_{n} (strongly) converging to f in H. Define the functionals

I_{n}(g) = I_{n,f}(g) = \begin{cases}\overline{E}_{n}(g) − 2(f_{n}, g)_{n}, if g ∈ H_{n} \\ +∞, if g ∉ H_{n}, \end{cases}

I(g) = I_{f}(g) = \overline{E}(g) − 2(f, g),

where (v, w)_{n} denotes the inner product

(v, w)_{n} = \frac{1}{3^{n}} ∑_{x∈V_{n}} v(x)w(x).

If \overline{E}_{n} Γ-converges to \overline{E}, then I_{n} also Γ-converges to I. If \overline{E}_{n} is equicoercive, then I_{n} is also equicoercive. Assume that I_{n,f} (I_{f} resp.) has a unique minimizer u_{n} (u resp.). Define the operator L_{n} (L resp.) by L_{n}f = u_{n} (Lf = u resp.). By the previous proposition, L_{n}f_{n} converges weakly to Lf.

Now take a sequence c̄_{n}(⟨xy⟩) of jump rates and define the corresponding exclusion process ξ_{t}^{n} in {0, 1}^{V_{n}}. Define

\overline{E}_{n}(f) = \frac{1}{3^{n}} ∑_{⟨xy⟩∈E_{n}} c̄_{n}(⟨xy⟩)(f(y) − f(x))^{2}.

Notice that with this choice, the operator L_{n} is given by

L_{n}v(x) ∑_{y:⟨xy⟩∈E_{n}} c̄_{n}(⟨xy⟩)(v(y) − v(x)).

The sequence \overline{E}_{n} is equicoercive if c̄_{n}(⟨xy⟩) ≤ C · 5^{n} for all n, all ⟨xy⟩ ∈ E_{n}. And I_{n} has a unique minimizer if c̄_{n}(⟨xy⟩) > 0 for all ⟨xy⟩ ∈ E_{n}.

Theorem 3.6. Let {c̄_{n}}_{n} be a sequence of positive jump rates such that c̄_{n}(⟨xy⟩)/5^{n} is uniformly bounded in n. Take an initial profile ρ_{0} and μ_{n} associated to ρ_{0}. Assume that \overline{E}_{n} Γ-converges to a quadratic form \overline{E}. Assume further that the operator L corresponding to \overline{E} satisfies hypothesis (H4) for the initial condition ρ_{0}. Then, the empirical measure π_{t}^{n} associated to the exclusion process ξ_{t}^{n} converges in probability to ρ(t, u), solution of the Cauchy problem

\begin{cases}∂_{t}ρ = Lρ \\ ρ(t, a_{0}) = ρ(t, a_{1}) = ρ(t, a_{2}) \\ ρ(0, u) = ρ_{0}(u). \end{cases}

Proof. Hypothesis (H1) and (H2) were already verified for the previos theorem. This time we will verify hypothesis (H3’). In fact, take a sequence f_{n} ∈ H_{n} converging strongly to f in H. Of course, f_{n} = T_{n}f is a possible choice. Since \overline{E}_{n} is Γ-convergent, the sequence u_{n}, solution of the equation L_{n}u_{n} = f_{n} converges weakly to u, solution of the equation Lu = f. We affirm that u_{n} converges to u uniformly. In fact, {u_{n}}_{n} is bounded in H. Therefore, by proposition 3.3, the functions u_{n} are equicontinuous and bounded. Since V is compact, by Arzela-Ascoli theorem, u_{n} have a subsequence uniformly convergent. This subsequence has to converge to u, and by uniqueness of the limit, the whole sequence converges to u. Since V_{n} has a number of points bounded by C ·3^{n}, (H3’)-‘i, (H3’)-ii follows. Again by the boundedness of u_{n} in H, (H3’)-iii is true. The theorem then follows from theorem 2.4. □

An example of a sequence {c̄_{n}}_{n} of jump rates that have a Γ-limit is the following. Choose two positive constants a, b. Take c̄_{n}(⟨xy⟩) = 3^{n}a if the edge ⟨xy⟩ is parallel to the x-axis, and otherwise equal to 3^{n}b. Then, there exists a constant σ(a, b) such that \overline{E}_{n} Γ-converges to σ(a, b)E. For a proof of this result and a product formula for σ(a, b), see [Koz].

3.3. Simple exclusion process in a random environment. In this section we consider the simple exclusion process in a random environment. This is a exclusion process evolving in the lattice, but with jump rates given by a realization of some ergodic process. Let (Ω, F, P) some probability space of σ-algebra F and probability measure P. Let {τ_{x} : Ω → Ω; x ∈ ℤ^{d}} be a group of F-measurable transformations that preserve P:

i) τ_{x} : Ω → Ω is F-measurable for all x ∈ ℤ^{d},

ii) P (τ_{x}B) = P (B) for all B ∈ F and x ∈ ℤ^{d},
iii) τ_{0} = I, τ_{x}τ_{y} = τ_{x+y} for all x, y ∈ ℤ^{d}.

In what follows we assume that the group τ_{x} is ergodic, that is, any function f ∈ L^{1}(P) such that f(τ_{x}ω) = f(ω) P -a.s for all x ∈ ℤ^{d} is constant P -a.s. Take a function b = (b_{1}, ..., b_{d}) : Ω^{d} → ℝ and assume that there exists a constant C > 0 such that C^{−1} ≤ a_{i}(ω) ≤ C P -a.s. for i = 1, ...d. Define V_{n} = n^{−1}ℤ^{d} and E_{n} = (n^{−1}ℤ^{d})^{∗}, the set of nearest-neighbour edges in V_{n}. In l_{0}(V_{n}) we define the norms

||v||_{0,n}^{2} = \frac{1}{n^{d}} ∑_{x∈V_{n}} v(x)^{2},

||v||_{1,n}^{2} = ||v||_{0,n}^{2} + \frac{1}{n^{d}} ∑_{⟨xy⟩∈E_{n}} n^{2}(v(y) − v(x))^{2}.

Let H_{n}^{0} (H_{n}^{1} resp.) the Hilbert space obtained as the completion of l_{0}(V_{n}) under the norm || · ||_{0,n} (|| · ||_{1,n} resp.) and we denote by (·, ·)_{1,n} ((·, ·)_{0,n} resp.) the inner product in H_{n}^{1} (H_{n}^{0} resp.) Fix ω ∈ Ω and define the jump rates c_{n} by c_{n}(⟨xy⟩) = n^{2}a_{i}(τ_{nx}ω), where y = x+e_{i}/n. Notice that the operator L_{n} associated to these jump rates depends in ω. We denote by H^{1} the Hilbert space obtained as the completion of C_{c}^{∞}(ℝ^{d}) under the norm

||f||_{1}^{2} = ∫_{ℝ^{d}} dxf(x)^{2} + ∫_{ℝ^{d}} dx(∇f(x))^{2}

and by (·, ·)_{1} the corresponding inner product. We can take H_{n}^{1} as a closed subset of H^{1} using the following representation. First, ws define Γ_{n} as the define the set of edges E′_{n} adding the NE-diagonal edges to each square of Γ_{n} = (V_{n}, E_{n}). In this form we have defined a triangulation of ℝ^{d}. For each x in V_{n}, define ψ_{x}^{n} as the continuous function such that its restriction to each triangle of Γ_{n} is linear and ψ_{x}^{n}(y) = δ_{x}y for all y ∈ V_{n}. To each sequence {v(x); x ∈ V_{n}}_{n} ∈ H_{n}^{1} we associate the function v = ∑_{x∈V_{n}} v(x)ψ_{x}^{n}. A simple computation shows that

\begin{aligned}||v||_{1}^{2} = \frac{1}{n^{d}} & ∑_{x∈V_{n}} v(x)^{2} − \frac{1}{12n^{d}} ∑_{⟨xy⟩∈E′_{n}} (v(y) − v(x))^{2} \\ & + \frac{1}{n^{d}} ∑_{⟨xy⟩∈E_{n}} n^{2}(v(y) − v(x))^{2}. \end{aligned}

Therefore, 0 ≤ ||v||_{1,n}^{2} − ||v||_{1}^{2} ≤ ||v||_{1,n}^{2}/4n^{2}. In particular, for a sequence f_{n} ∈ H_{n}^{1}, the notions of weak and strong convergence are the same for the inner poducts (·, ·)_{1} and (·, ·)_{1,n}. We define T_{n} as the ortogonal projection from H^{1} to H_{n}^{1}.

We have the following result from the theory of homogenization. A proof in a finite-volume setting can be found in [PR]. The proof translates to this infinitevolume setting easily. We also refer to [PR] for precise definitions of weak and strong convergence in H_{n}^{1}.

Proposition 3.7. Fix λ > 0. Let f ∈ L^{2}(ℝ) a uniformly continuous function. There exists a matrix A, depending only on the distribution of {a_{i}}_{i}, such that for P -a.e. ω ∈ Ω, the weak solutions u_{n} of

λu_{n}(x) − L_{n}u_{n}(x) = f(x)

converges weakly in H_{n}^{1} to the weak solution u ∈ H^{1} of

λu(x) − ∇ · (A∇u(x)) = f(x).

Now we can state the corresponding result for the exclusion process η_{t}^{n} associated to the jump rates c_{n}(⟨xy⟩).

Theorem 3.8. Consider an uniformly continuos initial profile ρ_{0} : ℝ^{d} → [0, 1] in L^{1}(ℝ^{d}) and define μ_{n} as the product measure in {0, 1}^{V_{n}} with marginals μ_{n}{η(x) = 1} = ρ_{0}(x). Take the jump rates c_{n}(⟨xy⟩) as i the previous theorem. Then, the rmpirical measure π_{t}^{n} converges in probability to the deterministic path ρ_{0}(t, u), solution of the equation

\begin{cases}∂ρ = ∇ · (A∇ρ) \\ ρ(0, x) = ρ_{0}(x). \end{cases}

Proof. By theorem 3.1, we just need to verify hypothesis (H1), (H2), (H3’) and (H4). The principal difficuty here is that H^{1} is not contained in L^{1}(ℝ^{d}). Take for example f(x) = (1 + x^{2})^{−1/2}. The same is true for l^{1}(V_{n}). Therefore, an arbitrary configuration η does not belong to H_{n}^{−1} = (H_{n}^{1})^{∗}. But μ_{n}-a.s., η has a finite number of particles. In fact, define m = ∫_{ℝ^{d}} ρ_{0}(x)dx. By a large deviations argument, for any δ > 0 the μ_{n}-measure of the set of configurations η with more than n^{d}(m + δ) particles is exponentially small. Define b_{n} = μ_{n}{η; ∑_{x} η > n^{d}(m+δ)} and consider the measure μ_{n}^{δ} in Ω_{δ,n} = {η; ∑_{x} η(x) ≤ n^{d}(m + δ)} defined by μ_{n}^{δ} = μ_{n}|_{Ω_{δ,n}} + b_{n}δ(η ≡ 0). Since b_{n} goes to 0 as n → ∞, μ_{n}^{δ} is also associated to the profile ρ_{0}. Notice that

|(η, v)_{0,n}| =≤ \frac{1}{n^{d}} ∑_{x∈V_{n}} η(x)|v(x)| ≤ (\frac{1}{n^{d}} ∑_{xinV_{n}} v(x)^{2})^{1/2}(\frac{1}{n^{d}} ∑_{x∈V_{n}} η(x))^{1/2}.

Therefore, the empirical measure π_{t}^{n} is well defined for μ_{n}-a.e. η. However, it is not true that π_{t}^{n} belongs to a fixed ball in H^{−1}, the dual of H^{1}. But this is true for the process starting from the measures μ_{n}^{δ}, and the limit points of π_{t}^{n} starting from μ_{n} and starting from μ_{n}^{δ} coincide. By the conservation of the number of particles, the process π_{t}^{n} belongs to the ball of radius m + δ in H^{−1} for all t ∈ [0, T] with probability converging to 1. Since the balls are compact in the weak-∗ topology of H^{−1}, we conclude that the process π_{t}^{n} starting from μ_{n} is tight if the process startin from μ_{n}δ is tight. Resuming, it is enough to prove theorem 3.8 for μ_{n}^{δ}. Hypothesis (H1) holds for the process starting from μ_{n}^{δ}, and hypothesis (H2) holds trivially since T_{n} is an orthogonal projection. Hypothesis (H4) also holds, since the hydrodynamic equation has unique weak solutions. But in this case hypothesis (H3’) is not so easy to verify. We will repeat the strategy of the previous section, that is, take f_{n} ∈ H_{n}^{1} given by f_{n}(x) = f(x) for all x ∈ V_{n}. If f is smooth enough, f_{n} converges to f in H_{n}^{1}, therefore ||f_{n} − T_{n}f||_{1,n} goes to 0 as n → ∞. By proposition 3.7, u_{n} = (λ − L_{n})^{−1}f_{n} converges weakly in H_{n}^{1} to u = (λ − L_{n})^{−1}f. In the previous section we used a version of Rellich’s theorem, that guarantees that weak convergence in H_{n}^{1} implies strong convergence in L^{2}. But this is true only in finite volume, so we need another argument here. What is true is that u_{n} converges to u strongly in L_{loc}^{2}(ℝ^{d}), just restricting the convergence result to test functions with support in a bounded set of ℝ^{d} and using Rellich’s theorem. Now we use the fact that L_{n} is the generator of a simple random walk in V_{n}. Denote by P_{t}^{n} the semigroup associated to the random walk generated by L_{n}. Then, u_{n} is the Lapace transform of f_{n}:

(λ − L_{n})^{−1}f(x) = ∫_{0}^{∞} e^{−λt}P_{t}^{n}f(x)dt.  (3.3)

In particular, the l^{1} and l^{∞} norms of u_{n} can be estimated in terms of the ones for f:

\frac{1}{n^{d}} ∑_{x∈V_{n}} u_{n}(x) = \frac{1}{λn^{d}} ∑_{x∈V_{n}} f(x),

λ^{−1} inf_{x∈V_{n}} f(x) ≤ inf_{x∈V_{n}} u_{n}(x) ≤ sup_{x∈V_{n}} u_{n}(x) ≤ λ^{−1} sup_{x∈V_{n}} f(x).

Therefore, for u smooth enough, u_{n} converges to u in the sense of hypothesis (H3’)-ii. Taking inner product with respect to u_{n} in 3.3, we see that there exists a constant C depending only on a_{i} such that

\frac{1}{n^{d}} ∑_{⟨xy⟩∈E_{n}} n^{2}(u_{n}(y) − u_{n}(x))^{2} ≤ \frac{C(1 + λ)}{λ^{2}}||f||_{0,n}^{2}.

We can find a dense subset G ⊆ H^{1} such that for all f ∈ G, all these estimates can be taken uniformly in n. Therefore, theorem 3.8 is proved. □

4. Comments and remarks

As far as theorem 2.1 is concerned, the operator L can have a space dependence, provided that the corresponding hydrodynamic equation still has unique weak solutions. Therefore, in theorems 3.1, 3.8, the hydrodynamic equation can be of the form ∂_{t}ρ = ∇ · A(x)∇ρ, provided the corresponding convergence criteria for L_{n}. For a triangulated domain, when A(x) is smooth enough, we can find jump rates c̄_{n}(⟨xy⟩) such that L_{n} now converges to ∇ · A(x)∇ using the weak formulation of ∇ · A(x)∇u = f in terms of the basis {ψ_{x}; x ∈ V_{n}}.

The method exposed here can be used to obtain the hydrodynamic limit and the fluctuations from this limit for systems on which the interaction satisfies a gradient condition and a bond disorder is introduced. Notice that the non-gradient method of [Q3], [FM] does not work for systems on which the spectral gap estimate is not uniform in the density, like the zero-range process with bounded interaction rate or kinetically constrained interacting systems, but the method of the corrected empirical measure applies for these models when a bond disorder is introduced [GJO].

In the applications, we have assumed in one or another form, an ellipticity condition, that is, that the inhomogeneities are comparable in size to the homogeneous situation. This condition is not necessary, but it was assumed to simplify the proofs. In particular, we believe that a combination of theorem 2.1 with some results about homogenization on percolation clusters [MP] should permit to prove hydrodynamic limit for a simple exclusion process evolving in a percolation cluster.

Finally, we point out that the corrected empirical measure can be understood as a version of the compensated compactness lemma in the context of interacting particle systems.

Acknowledgements. The author is grateful to S.R.S. Varadhan for useful discussions, and for his hospitality at the Courant Institute of Mathematical Sciences, where the present work was initiated.

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