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Vol. 15 (2010), Paper no. 1, pages 1–43. Journal URL

http://www.math.washington.edu/~ejpecp/

Strong hydrodynamic limit for attractive particle systems

on

Z

C. Bahadoran

∗ †

, H. Guiol

‡ §

, K. Ravishankar

¶ k

, E. Saada

∗∗†

Abstract

We prove almost sure Euler hydrodynamics for a large class of attractive particle systems onZ

starting from an arbitrary initial profile. We generalize earlier works by Seppäläinen (1999) and Andjel et al. (2004). Our constructive approach requires new ideas since the subadditive ergodic theorem (central to previous works) is no longer effective in our setting.

Key words: Strong (a.s.) hydrodynamics, attractive particle system, graphical construc-tion, non-explicit invariant measures, non-convex or non-concave flux, entropy soluconstruc-tion, Glimm scheme.

AMS 2000 Subject Classification:Primary 60K35; Secondary: 82C22.

Submitted to EJP on June 16, 2009, final version accepted December 17, 2009.

∗_{Laboratoire de Mathématiques, Université Clermont 2, 63177 Aubière, France, bahadora@math.univ-bpclermont.fr} †_{Supported by ANR grant BLAN07-2184264}

‡_{TIMC - TIMB, INP Grenoble, Faculté de Médecine, 38706 La Tronche Cedex, France} §_{Supported by ANR grant BLAN07-2184264 and IXXI Rhône-Alpes}

¶_{Dep. of Mathematics, SUNY, College at New Paltz, NY, 12561, USA}

k_{Supported by NSF grant DMS 0104278, NFS grant DMS-06-06696 and by ANR grant BLAN07-2184264} ∗∗_{CNRS, UMR 6085, Univ. Rouen, 76801 Saint Etienne du Rouvray Cedex, France}

1

Introduction

Hydrodynamic limit ([14; 40; 25; 42]) is a law of large numbers for the time evolution (usually described by a limiting PDE, called the hydrodynamic equation) of empirical density fields in interacting particle systems (IPS). In most known results, only a weak law of large numbers is established. In this description one need not have an explicit construction of the dynamics: the limit is shown in probability with respect to the law of the process, which is characterized in an abstract way by its Markov generator and Hille-Yosida’s theorem ([30]). Nevertheless, when simulating particle systems, one naturally uses a pathwise construction of the process on a Poisson space-time random graph (the so-called “graphical construction”). In this description the dynamics is deterministically driven by a random space-time measure which tells when and where the configuration has a chance of being modified. It is of special interest to show that the hydrodynamic limit holds for almost every realization of the space-time measure, as this means a single simulation is enough to approximate solutions of the limiting PDE. In a sense this is comparable to the interest of proving strong (rather than weak) consistency for statistical estimators, by which one knows that a single path observation is enough to estimate parameters.

Most usual IPS can be divided into two groups, diffusive and hyperbolic. In the first group, which contains for instance the symmetric or mean-zero asymmetric exclusion process, the macroscopic→microscopic space-time scaling is(x,t)7→(N x,N2t)with N → ∞, and the limiting PDE is a diffusive equation. In the second group, which contains for instance the nonzero mean asymmetric exclusion process, the scaling is (x,t) 7→ (N x,N t), and the limiting PDE is of Euler type. In both groups this PDE often exhibits nonlinearity, either via the diffusion coefficient in the first group, or via the flux function in the second one. This raises special difficulties in the hyperbolic case, due to shocks and non-uniqueness for the solution of the PDE, in which case the natural problem is to establish convergence to the so-called “entropy solution" ([39]). In the diffusive class it is not so necessary to specifically establish strong laws of large numbers, because one has a fairly robust method ([26]) to establish large deviations from the hydrodynamic limit. Large deviation upper bound and Borel Cantelli’s lemma imply a strong law of large numbers as long as the large deviation functional is lower-semicontinuous and has a single zero. The situation is quite different in the hyperbolic class, where large deviation principles are much more difficult to obtain. So far only the remarkable result of Jensen and Varadhan ([22]and[43]) is available, and it applies only to the one-dimensional totally asymmetric simple exclusion process. Besides, the fact that the resulting large deviation functional has a single zero is not at all obvious: it follows only from recent and refined work on conservation laws ([13]). An even more difficult situation occurs for particle systems with nonconvex and nonconcave flux function, for which the Jensen-Varadhan large deviation functional does not have a unique zero, due to the existence of non-entropic solutions satisfying a single entropy inequality ([21]). In this case a more complicated rate functional can actually be conjectured from[5; 32].

The derivation of hyperbolic equations as hydrodynamic limits began with the seminal paper

[36], which established a strong law of large numbers for the totally asymmetric simple exclusion process onZ, starting with 1’s to the left of the origin and 0’s to the right. This result was extended by [6] and [1] to nonzero mean exclusion process starting from product Bernoulli distributions with arbitrary densitiesλto the left andρ to the right (the so-called “Riemann” initial condition). The Bernoulli distribution at time 0 is related to the fact that uniform Bernoulli measures are

invariant for the process. For the one-dimensional totally asymmetric K-exclusion process, which does not have explicit invariant measures, a strong hydrodynamic limit was established in [37], starting from arbitrary initial profiles, by means of the so-called “variational coupling”. These are the only strong laws available so far. A common feature of these works is the use of sub-additive ergodic theorem to exhibit some a.s. limit, which is then identified by additional arguments. On the other hand, many weak laws of large numbers were established for attractive particle systems. A first series of results treated systems with product invariant measures and product initial distributions. In [2], for a particular zero-range model, a weak law was deduced from conservation of local equilibrium under Riemann initial condition. It was then extended in

[3] to the misanthrope’s process of [12] under an additional convexity assumption on the flux function. These were substantially generalized (using Kružkov’s entropy inequalities, see [28]) in [34] to multidimensional attractive systems with arbitrary Cauchy data, without any convexity requirement on the flux. For systems with unknown invariant measures, the result of[37]was later extended to other models, though only through weak laws. In[35], using semigroup point of view, hydrodynamic limit was established for the one-dimensional nearest-neighborK-exclusion process. In [8] we studied fairly general attractive systems, employing a constructive approach we had initiated in[7]. This method is based on an approximation scheme (to go from Riemann to Cauchy initial data) and control of some distance between the particle system and the entropy solution to the hydrodynamic equation.

In this paper we prove a strong law of large numbers, starting from arbitrary initial profiles, for finite-range attractive particle systems on Z _{with bounded occupation number. We need no}

assumption on the flux, invariant measures or microscopic structure of initial distributions. This includes finite-range exclusion and K-exclusion processes, more general misanthrope-type models and k-step exclusion processes. We proceed in the constructive spirit of [7; 8], which is adapted to a.s. convergence, though this requires a novel approach for the Riemann part, and new error analysis for the Cauchy part. The implementation of an approximation scheme involves Riemann hydrodynamics from any space-time shifted initial position, which cannot be obtained through subadditive ergodic theorem, and thus prevents us from using the a.s. result of [1]. This can be explained as follows (see Appendix B for more details). Assume that on a probability space we have a shift operator θ that preserves the probability measure, and a sequence (X_n)_n∈N of real-valued

random variables converging a.s. to a constant x. Then we can conclude that the sequence

(Y_n :=X_n◦θn)_n convergesin probabilityto x, but not necessarily almost surely. Indeed, for fixed

n, Y_n has the same distribution asX_n, but the sequence(Y_n)_n need not have the same distribution as(Xn)n. So the derivation of a.s. convergence forYn is a case by case problem depending on how

it was obtained for X_n. In particular, if convergence ofX_n was established from the subadditivity property

Xn+m≤Xn+Xm◦θn

this property is no longer satisfied byYn. In contrast, if convergence ofXn was established by large

deviation estimates forX_n, these estimates carry over toY_n, thus also implying a.s. convergence of

(Y_n)_n. In our context the random variableX_nis a current in the Riemann setting. In order to handle the Cauchy problem we have to establish a.s. convergence for a shifted version Yn of this current.

The paper is organized as follows. In Section 2, we define the model, give its graphical rep-resentation, its monotonicity properties, and state strong hydrodynamics. In Section 3, we derive almost sure Riemann hydrodynamics. In Section 4 we prove the result starting from any initial profile.

2

Notation and results

Throughout this paperN={1, 2, ...}will denote the set of natural numbers,Z+={0, 1, 2, ...}the set of non-negative integers, andR+∗=R+− {0}the set of positive real numbers. The integer part of

x ∈R_{, denoted by[x]∈}Z_{, is uniquely defined by[x]≤ x<[x]+1. We consider particle systems on} Zwith at mostK particles per site,K∈N. Thus the state space of the process isX={0, 1,· · ·,K}Z

, which we endow with the product topology, that makesXa compact metrisable space. A function defined onXis calledlocalif it depends on the variableη∈Xonly through(η(x),x ∈Λ)for some finite subsetΛ ofZ. We denote by τx, either the spatial translation operator on the real line for

x ∈R_{, defined byτxy}= x+y, or its restriction toZ_forx ∈Z_{. By extension, we setτxf} = f ◦τx

if f is a function defined onR_{;τxη=η◦τx, for x ∈}Z_{, ifη∈Xis a particle configuration (that is a}

particular function onZ);τxµ=µ◦τx−1ifµis a measure onRor onX. We letM

+_{(R)denote the set}

of positive measures onR_{equipped with the metrizable topology of vague convergence, defined by}

convergence on continuous test functions with compact support. The set of probability measures on Xis denoted byP(X). Ifηis anX-valued random variable andν∈ P(X), we writeη∼ν to specify thatηhas distributionν. The notationν(f), where f is a real-valued function andν ∈ P(X), will be an alternative forR_Xf dν. We say a sequence(νn,n∈N)of probability measures onXconverges

weakly to someν ∈ P(X), if and only ifνn(f)→ν(f)as n→ ∞ for every continuous function f

onX. The topology of weak convergence is metrizable and makesP(X)compact.

2.1

The system and its graphical construction

We consider Feller processes onXwhose transitions consist of particle jumps encoded by the Markov generator

L f(η) = X

x,y∈Z

p(y−x)b(η(x),η(y))f ηx,y−f(η) (1) for any local function f, whereηx,y denotes the new state after a particle has jumped from x to y

(that isηx,y_{(x) =η(x)−1,η}x,y_{(y) =η(y) +1,η}x,y_{(z) =η(z)otherwise),pis the particles’ jump}

kernel, that is a probability distribution onZ_{, andb :} Z+_×Z+_→R+_{is the jump rate. We assume}

thatpand bsatisfy :

(A1)The semigroup ofZ_{generated by the support ofpis}Z_{itself (irreducibility);} (A2) phas a finite first moment, that isP_z∈Z|z|p(z)<+∞;

(A3) b(0, .) =0, b(.,K) =0 (no more thanKparticles per site), andb(1,K−1)>0;

(A4) bis nondecreasing (nonincreasing) in its first (second) argument.

We denote by(S(t),t∈R+_{)the semigroup generated by L. Without additional algebraic relations}

satisfied by b(see[12]), the system in general has no explicit invariant measures. This is the case even forK-exclusion processwithK≥2 (see[37]), whereb(n,m) =1{n>0,m<K}.

Remark 2.1. For the sake of simplicity, we restrict our attention to processes of “misanthrope type”, for which particles leave more likely crowded sites for less occupied ones (cf. [12]). But our method can be applied to a much larger class of attractive models that can be constructed through a graphical representation, such as k-step exclusion (see[18]and[7]).

We now describe the graphical construction of the system given by (1), which uses the Harris rep-resentation ([19; 20],[30, p. 172], [9, p. 119],[31, p. 215]); see for instance[1]for details and justifications. This enables us to define the evolution from different initial configurations simultane-ously on the same probability space. We consider the probability space(Ω,F, IP)of measuresωon

R+_×Z2_{×[0, 1]of the form}

ω(d t,d x,dz,du) = X

m∈N

δ(tm,xm,zm,um)

whereδ(·)denotes Dirac measure, and(tm,xm,zm,um)m≥0 are pairwise distinct and form a locally finite set. Theσ-fieldF is generated by the mappingsω7→ω(S)for Borel setsS. The probability measure IP onΩis the one that makesωa Poisson process with intensity

m(d t,d x,dz,du) =||b||∞λR+(d t)×λZ(d x)×p(dz)×λ[0,1](du)

whereλdenotes either the Lebesgue or the counting measure. We denote by IE the corresponding expectation. Thanks to assumption(A2), for IP-a.e.ω, there exists a unique mapping

(η0,t)∈X×R+7→ηt=ηt(η0,ω)∈X (2) satisfying: (a) t 7→ηt(η0,ω)is right-continuous; (b)η0(η0,ω) = η0; (c) for t ∈R+, (x,z) ∈Z2,

η_t=ηx_t−,x+z if

∃u∈[0, 1]: ω{(t,x,z,u)}=1 andu≤ b(ηt−(x),ηt−(x+z))

||b||_∞ (3)

and (d) for alls,t∈R+∗_andx ∈Z_,

ω{[s,t]×Zx×(0, 1)}=0⇒ ∀v∈[s,t],ηv(x) =ηs(x) (4)

where

Zx :=

¦

(y,z)∈Z2:y =x or y+z= x©

In short, (3) tells how the state of the system can be modified by an “ω-event”, and (4) says that the system cannot be modified outsideω-events. This defines a Feller process with generator (1): that is for any t ∈R+ _{and f ∈C(X) (the set of continuous functions onX), S(t)f ∈C(X) where} S(t)f(η0) =IE[f(ηt(η0,ω))].

An equivalent formulation is the following. For each (x,z) ∈ Z2_{, let {T}x,z

n ,n ≥ 1} be the

ar-rival times of mutually independent rate||b||_∞p(z)Poisson processes, let{U_nx,z,n≥1}be mutually independent (and independent of the Poisson processes) random variables, uniform on [0, 1]. At time t = T_nx,z, the configuration ηt− becomes ηx,x+z

t− if U x,z

n ≤

b(ηt−(x),η_t−(x+z))

||b||∞

, and stays unchanged otherwise.

Remark 2.2. For the K-exclusion process, b takes its values in{0, 1}, the probability space can be reduced to measuresω(d t,d x,dz)onR+×Z2_{, and}(3)toηt=ηx,x

+z

t− ifηt−(x)>0andη_t−(x+z)<

K. One recovers exactly the graphical construction presented in e.g. [38]or[1].

One may further introduce an “initial” probability space (Ω₀,F₀, IP0), large enough to construct random initial configurationsη0 = η0(ω0)forω0 ∈Ω0. The general process with random initial configurations is constructed on the enlarged space(Ω = Ωe ₀×Ω,Fe=σ(F₀× F),IPe=IP₀⊗IP)by setting

ηt(ωe) =ηt(η0(ω0),ω)

for ωe = (ω0,ω) ∈ Ωe. If η0 has distribution µ0, then the process thus constructed is Feller with generator (1) and initial distribution µ0. By a coupling of two systems, we mean a process

(ηt,ζt)t≥0 defined on Ωe, where each component evolves according to (2)–(4), and the random variablesη0andζ0 are defined simultaneously onΩ0.

We define onΩthespace-time shiftθx0,t0: for anyω∈Ω, for any(t,x,z,u)

(t,x,z,u)∈θx0,t0ωif and only if(t0+t,x0+x,z,u)∈ω

where (t,x,z,u) ∈ω meansω{(t,x,z,u)}=1. By its very definition, the mapping introduced in (2) enjoys the following properties, for alls,t≥0,x ∈Z_{and(η,ω)∈X×Ω:}

ηs(ηt(η,ω),θ0,tω) =ηt+s(η,ω) (5)

which implies Markov property, and

τxηt(η,ω) =ηt(τxη,θx,0ω) (6)

which implies thatS(t)andτx commute.

2.2

Main result

We give here a precise statement of strong hydrodynamics. LetN∈N_{be the scaling parameter for}

the hydrodynamic limit, that is the inverse of the macroscopic distance between two consecutive sites. The empirical measure of a configurationηviewed on scaleN is given by

αN(η)(d x) =N−1X

y∈Z

η(y)δy/N(d x)∈ M+(R) (7)

We now state our main result.

Theorem 2.1. Assume p(.)is finite range, that is there exists M >0such that p(x) =0for all|x|>M . Let(ηN₀,N ∈N_{)be a sequence ofX-valued random variables onΩ0. Assume there exists a measurable}

[0,K]-valued profile u₀(.)onR_{such that}

lim

N→∞α

N_(ηN

0)(d x) =u0(.)d x, IP0-a.s. (8)

that is,

lim

N→∞

Z

R

ψ(x)αN(η₀N)(d x) =

Z

for every continuous functionψ onR _{with compact support. Let(x,t)7→ u(x,t) denote the unique} entropy solution to the scalar conservation law

∂tu+∂x[G(u)] =0 (10)

with initial condition u₀, where G is a Lipschitz-continuous flux function (defined in(13)below) deter-mined by p(.)and b(., .). Then, withIPe-probability one, the convergence

lim

N→∞α

N_(η

N t(ηN0(ω0),ω))(d x) =u(.,t)d x (11)

holds uniformly on all bounded time intervals. That is, for every continuous function ψ on R _with compact support, the convergence

lim

N→∞

Z

R

ψ(x)αN(ηN_{N t})(d x) =

Z

ψ(x)u(x,t)d x (12)

holds uniformly on all bounded time intervals.

Remark 2.3. This result has been recently extended to kernels with infinite range and finite third moment (see[33]).

As a byproduct we derive in Appendix A the main result of[8]:

Corollary 2.1. The strong law of large numbers(11)implies the weak law of large numbers established in[8].

We recall from[8, pp.1346–1347 and Lemma 4.1]the definition of the Lipschitz-continuous macro-scopic flux functionG. In Proposition 2.1 below, we define a closed subsetR of[0,K]depending onp(.)and b(., .)(the set of possible equilibrium densities for the system). We let

G(ρ) =νρ

 X

z∈Z

z p(z)b(η(0),η(z))



 (13)

This represents the expectation, under the shift invariant equilibrium measure with densityρ(see Proposition 2.1), of the microscopic current through site 0. On the complement ofR, which is a finite or countably infinite union of disjoint open intervals,G is interpolated linearly.

Remark 2.4. Very little is known about the setR. That{0,K} ⊂ R is a trivial consequence of Assump-tion (A3). When p(.)and b(., .)satisfy the algebraic relations of[12], the system has a whole contin-uum of product equilibrium measures, andR= [0,K]. WhetherR= [0,K]outside this case is a diffi-cult open problem. We do not know of any definite conjecture regarding this question. The only known result is for the totally asymmetric K-exclusion process, that is p(1) =1and b(n,m) =1n>0,m<K. It

is established in[8, Corollary 2.1]thatR contains at least one point in[1/3,K−1/3], and0and K are limit points ofR. In particular,Ris infinite. This result is derived from the (weak) hydrodynamic limit.

A Lipschitz constantV ofG is determined by the ratesb,pin (1):

V = 2BX

z∈Z

|z|p(z), with

B = sup

0≤a≤K,0≤k<K

This constant will be a key tool for the finite propagation property of the particle system (see Propo-sition 4.1 below).

Remark 2.5. When the system has product invariant measures (see [12]), (13)can be decomposed into a mean drift and a scalar flux:

G(ρ) =

 X

z∈Z

z p(z)



νρ[b(η(0),η(1))] (14)

In Assumption (A2) we only assumed integrability of the jump kernel p(.), but no non-zero mean or even asymmetry condition. Thus our result applies also whenP_zz p(z) =0, but then the flux is identically zero and the hydrodynamic limit is trivial, u(x,t) =u0(x)for all t >0and x ∈R. In this case the

relevant time scale to observe nontrivial behavior is the diffusive scale. When the system has no explicit invariant measures, there is a priori no reason for(14)to hold, so the driftP_zz p(z)does not play a special role. In particular, a mean-zero or even symmetry property for p(.)need not imply a trivial flux.

In[8, Section 2.2]we discussed the different definitions ofentropy solutions(loosely speaking : the physically relevant solutions) to an equation such as (10), and the ways to prove their existence and uniqueness. Therefore we just briefly recall the definition of entropy solutions based on shock admissibility conditions (Ole˘ınik’s entropy condition), valid only for solutions with locally bounded space-time variation ([44]). A weak solution to (10) is an entropy solution if and only if, for a.e. t > 0, all discontinuities of u(.,t) are entropy shocks. A discontinuity (u−,u+), with

u±:=u(x±0,t), is called an entropy shock, if and only if:

The chord of the graph ofG betweenu−andu+lies below the graph ifu−<u+, above the graph ifu+<u−.

In the above condition, “below" or “above" is meant in wide sense, that is the graph and chord may coincide at some points betweenu−andu+.

If the initial datumu0(.)has locally bounded space-variation, there exists a unique entropy solution to (10), within the class of functions of locally bounded space-time variation ([44]).

2.3

Monotonicity and invariant measures

Assumption (A4) implies the following monotonicity property, crucial in our approach. For the coordinatewise partial order onX, defined byη≤ζif and only if η(x)≤ζ(x)for every x ∈Z, we have

(η0,t)7→ηt(η0,ω)is nondecreasing w.r.t.η0 (15) for every ω such that this mapping is well defined. The partial order on X induces a partial stochastic order onP(X); namely, forµ1,µ2∈ P(X), we writeµ1≤µ2 if the following equivalent conditions hold (seee.g.[30],[41]):

i) For every non-decreasing nonnegative function f onX,µ1(f)≤µ2(f).

ii) There exists a coupling measure µe on eX = X× X with marginals µ1 and µ2, such that e

It follows from this and (15) that

µ1≤µ2⇒ ∀t∈R+,µ1S(t)≤µ2S(t) (16)

Either property (15) or (16) is usually calledattractiveness.

Let I and S denote respectively the set of invariant probability measures for L, and the set of shift-invariant probability measures onX. We derived in[8, Proposition 3.1], that

Proposition 2.1.

(I ∩ S)_e=νρ,ρ∈ R (17)

with R a closed subset of [0,K] containing 0 and K, and νρ a shift-invariant measure such that

νρ[η(0)] =ρ. (The index e denotes extremal elements.) The measuresνρare stochastically ordered:

ρ≤ρ′⇒νρ≤νρ′ (18)

Moreover we have, quoting[35, Lemma 4.5]:

Proposition 2.2. The measureνρ hasa.s.densityρ, that is

lim

l→∞ 1 2l+1

l

X

x=−l

η(x) =ρ, νρ−a.s.

By[23, Theorem 6], (18) implies existence of a probability space on which one can define random variables satisfying

ηρ∼νρ (19)

and, with probability one,

ηρ≤ηρ′,∀ρ,ρ′∈ Rsuch thatρ≤ρ′ (20)

Proceeding as in the proof of [24, Theorem 7], one can also require (but we shall not use this property) the joint distribution of(ηρ:ρ∈ R)to be invariant by the spatial shiftτ_x. In the special case whereνρ are product measures, that is when the function b(., .)satisfies assumptions of[12], such a family of random variables can be constructed explicitely: if U_xx∈Z is a family of i.i.d.

random variables uniformly distributed on(0, 1), one defines

ηρ(x) =F_ρ−1(Ux) (21)

where F_ρ is the cumulative distribution function of the single site marginal distribution of νρ. We will assume without loss of generality (by proper enlargement) that the “initial” probability space

Ω₀ is large enough to define a family of random variables satisfying (19)–(20).

An important result for our approach is a space-time ergodic theorem for particle systems mentioned in[35], which we state here in a general form, and prove in Appendix C.

Proposition 2.3. Let(ηt)t≥0be a Feller process onXwith a translation invariant generator L, that is

τ1Lτ−1= L (22)

Assume further that

µ∈(I_L∩ S)_e

where I_L denotes the set of invariant measures for L. Then, for any local function f on X, and any a>0

lim

ℓ→∞ 1

aℓ2 Z aℓ

0

ℓ

X

i=0

τif(ηt)d t=

Z

f dµ= lim

ℓ→∞ 1

aℓ2 Z aℓ

0

−1 X

i=−ℓ

τif(ηt)d t (23)

a.s. with respect to the law of the process with initial distributionµ.

Remark 2.6. It follows from(23)that, more generally,

lim

ℓ→∞

1

(b−a)(d−c)ℓ2 Z bℓ

aℓ

X

i∈Z∩[cl,d l]

τif(ηt)d t=

Z

f dµ (24)

for every0≤a<b and c<d inR_{, as can be seen by decomposing the space-time rectangle}[al,bl]×

[cl,d l]into rectangles containing the origin.

3

Almost sure Riemann hydrodynamics

By definition, the Riemann problem with valuesλ,ρ∈[0,K]for (10) is the Cauchy problem for the particular initial condition

R0_λ,ρ(x) =λ1{x<0}+ρ1{x≥0} (25) The entropy solution for this Cauchy datum will be denoted in the sequel by Rλ,ρ(x,t). In this

section, we derive the corresponding almost sure hydrodynamic limit whenλ,ρ∈ R. We will use the following variational representation of the Riemann problem. We henceforth assumeλ < ρ(for the caseλ > ρ, replace minimum with maximum below and make subsequent changes everywhere in the section).

Proposition 3.1. ([8, Proposition 4.1]). Letλ,ρ∈[0,K],λ < ρ.

i) There is a countable setΣ_{l ow}⊂[λ,ρ](depending only on the differentiability properties of the convex envelope of G on[λ,ρ]) such that, for every v∈R\Σ_{l ow}, G(·)−v·achieves its minimum over[λ,ρ]

at a unique point h_c(v). Then Rλ,ρ(x,t) =hc(x/t)whenever x/t6∈Σl ow.

ii) Supposeλ,ρ∈ R. Then the previous minimum is unchanged if restricted to[λ,ρ]∩ R. As a result, the Riemann entropy solution isa.e. R-valued.

Remark 3.1. Property ii) holds if we replace R by any closed subset of [0,K]on the complement of which G is affine. We stated it forR because it is the set of densities relevant to the particle system.

To state Riemann hydrodynamics, we define a particular initial distribution for the particle system. We introduce a transformationT:X2→Xby

We define νλ,ρ as the distribution of T(ηλ,ηρ) =: ηλ,ρ, and ¯νλ,ρ as the coupling distribution of

(ηλ,ηρ). Note that, by (20),

¯

νλ,ρ¦(η,ξ)∈X2: η≤ξ©=1 (26)

The measureνλ,ρis non-explicit unless we are in the special case of[12]where theνρare product, one can use (21), andνλ,ρ_{is itself a product measure. In all cases,ν}λ,ρ _{enjoys the properties:}

P1) Negative (nonnegative) sites are distributed as underνλ (νρ); P2)τ1νλ,ρ≥νλ,ρ(τ1νλ,ρ≤νλ,ρ) ifλ≤ρ(λ≥ρ);

P3)νλ,ρis stochastically increasing with respect toλandρ.

Let us also define an extended shift θ′ on the compound probability space Ω′ = X2 × Ω. This is a particular case ofΩewhen the “initial” probability spaceΩ₀is the setX2 of coupled particle configurations(η,ξ). Let

ω′= (η,ξ,ω) (27)

denote a generic element of this space. We set

θ_x′_,tω′= (τxηt(η,ω),τxηt(ξ,ω),θx,tω) (28)

We can now state and prove the main results of this section. Proposition 3.2. Set

N_tv,w(ω′):= X

[v t]<x≤[w t]

ηt(T(η,ξ),ω)(x) (29)

Then, for every t>0,α∈R+_,β∈R_{and v,w∈}R_{\Σl ow,}

lim

t→∞ 1

tN

v,w

t ◦θ[′βt],αt(ω

′₎

= [G(Rλ,ρ(v, 1))−v Rλ,ρ(v, 1)]−[G(Rλ,ρ(w, 1))−w Rλ,ρ(w, 1)] (30)

¯

νλ,ρ⊗IP-a.s.

Remark 3.2. This result is an almost sure version of[3, Lemma 3.2], where the limit of the corre-sponding expectation was derived.

Corollary 3.1. Set

β_tN(ω′)(d x):=αN(ηt(T(η,ξ),ω))(d x) (31)

(i) For every t>0, s0≥0and x0∈R, we have theν¯λ,ρ⊗IP-a.s. convergence lim

N→∞β

N N t(θ

′

[N x0],N s0ω

′_{)(d x) =R}

λ,ρ(.,t)d x

(ii) In particular, for an initial sequence(ηN

0)N such thatηN0 =η

λ,ρ_{for every N ∈N, the conclusion of}

For the asymmetric exclusion process, [1] proved a statement equivalent to the particular case

α=β=0 of Proposition 3.2. Their argument, which is a correction of[6], is based on subadditivity. As will appear in Section 4 (in the proof of Lemma 4.4 for example, that is the main step of that section), we do need to consider nonzeroαandβ in order to prove a.s. hydrodynamics for general (non-Riemann) Cauchy datum. Using arguments similar to those in [1], it would be possible to prove Proposition 3.2 with α = β = 0 for our model (1). However this no longer works for nonzero α and β (see Appendix B). Therefore we construct a new approach for a.s. Riemann hydrodynamics, that does not use subadditivity.

To prove Proposition 3.2, we first rewrite (in Subsection 3.1) the quantity N_tv,w in terms of particle currents for which we then state (in Subsection 3.2) a series of lemmas (proven in Subsection 3.4), and finally obtain the desired limits for the currents, by deriving upper and lower bounds. For one bound (the upper bound if λ < ρ or lower bound if λ > ρ), we derive an “almost-sure proof” inspired by the ideas of[3]and their extension in[8]. For the other bound we use new ideas based on large deviations of the empirical measure.

3.1

Currents

Let us define particle currents in a system (ηt)t≥0 governed by (3)–(4). Let x. = (xt,t ≥ 0) be

a Z_{-valued cadlagpath with xt}−x_t−

≤ 1. In the sequel this path will be either deterministic, or a random path independent of the Poisson measure ω. We define the particle current as seen by an observer travelling along this path. We first consider a semi-infinite system, that is with P

x>0η0(x)<+∞: in this case, we set

ϕx.

t (η0,ω):= X

y>xt

ηt(η0,ω)(y)− X

y>x0

η0(y) (32)

where ηt(η0,ω)is the mapping introduced in (2). In the sequel we shall most often omit ωand writeηt forηt(η0,ω)when this creates no ambiguity. We have

ϕx.

t (η0,ω) =ϕ

x.,+

t (η0,ω)−ϕ

x.,−

t (η0,ω) +ϕ˜

x.

t (η0,ω) (33)

where

ϕx.,+

t (η0,ω) = ω

(s,y,z,u): 0≤s≤t, y≤ xs< y+z,

u≤ b(ηs−(y),ηs−(y+z)) ||b||_∞

ϕx.,−

t (η0,ω) = ω

(s,y,z,u): 0≤s≤t, y+z≤xs< y,

u≤ b(ηs−(y),ηs−(y+z)) ||b||∞

(34)

count the number of rightward/leftward crossings due to particle jumps, and ˜

ϕx.

t (η0,ω) =− Z

[0,t]

ηs−(x_s∨x_s−)d x_s (35)

is the current due to the self-motion of the observer. For an infinite system, we may still define

ϕx.

The following identities are immediate from (32) in the semi-infinite case, and extend to the infinite case:

ϕx.

t (η0,ω)−ϕ

y.

t (η0,ω)

≤ K€xt−yt

+x0−y0

Š (36)

[_Xw t]

x=1+[v t]

ηt(η0,ω)(x) = ϕvt(η0,ω)−ϕwt(η0,ω) (37)

Following (37), the quantityN_tv,w(ω′)defined in (29) can be written as

N_tv,w(ω′) =φv_t(ω′)−φw_t (ω′) (38)

where we define the current

φ_tv(ω′):=ϕ_tv(T(η,ξ),ω)

Notice that

φ_tv(η,η,ω) =ϕv_t(η,ω)

The proof of the existence of the limit in Proposition 3.2 is thus reduced to lim

t→∞ t −1_φv

t

θ_[′_βt],αtω′ exists ¯νλ,ρ⊗IP-a.s. (39)

3.2

Lemmas

We fixα∈R+_andβ∈R_{. The first lemma deals with equilibrium processes.}

Lemma 3.1. For all r∈[λ,ρ]∩ R,ς∈X, v∈R_{\Σl ow,}

lim

t→∞ t −1

φv_t ◦θ_[′_βt],αt(ς,ς,ω) =G(r)−v r, ν¯r,r⊗IP-a.s.

The second lemma relates the current of the process under study with equilibrium currents; here it plays the role of Lemma 3.3 in [3]. It is connected to the “finite propagation property” of the particle model (see Lemma 4.3):

Lemma 3.2. There exist¯v and v (depending on b and p) such that we have,ν¯λ,ρ_⊗IP-a.s.,

lim

t→∞

h

t−1φ_tv◦θ_[′_βt],αt(η,ξ,ω)−t−1φv_t ◦θ_[′_βt],αt(ξ,ξ,ω)i=0, (40)

for all v>¯v, and

lim

t→∞

h

t−1φv_t ◦θ_[′_βt],αt(η,ξ,ω)−t−1φ_tv◦θ_[′_βt],αt(η,η,ω)i=0, (41)

for all v<v.

For the next lemmas we need some more notation and definitions. Let v ∈ R. We consider a probability space Ω+, whose generic element is denoted by ω+, on which is defined a Poisson processN_t=N_t(ω+)with intensity|v|. We denote by IP+ the associated probability. We set

x_st(ω+) := (sgn(v))”N_αt+s(ω+)−N_αt(ω+)— (42) e

ηt_s(η0,ω,ω+) := τxt s(ω

Thus(η_est)_s≥0 is a Feller process with generator

L_v=L+S_v, S_vf(ζ) =|v|[f(τsgn(v)ζ)−f(ζ)] (44)

(for f local andζ∈X) for which the set of local functions is a core, as it is known to be ([30]) for

L. We denote byI_v the set of invariant measures for L_v. Since any translation invariant measure onXis stationary for the pure shift generatorSv, we have

I ∩ S =I_v∩ S (45)

It can be shown (see [16, Theorem 3.1] and[17, Corollary 9.6]) that I_v ⊂ S for v 6= 0, which implies

I ∩ S =I_v∩ S =I_v

but we shall not use this fact here. Define the time empirical measure

mt(ω′,ω+):=t−1

Z t

0

δη_et

s(T(η,ξ),ω,ω+)ds (46)

and space-time empirical measure (whereǫ >0) by

m_t,ǫ(ω′,ω+):=|Z∩[−ǫt,ǫt]|−1

X

x∈Z:|x|≤ǫt

τxmt(ω′,ω+) (47)

We introduce the set

M_λ,ρ:=¦µ∈ P(X): νλ≤µ≤νρ©

Notice that this is a closed (thus compact) subset of the compact spaceP(X).

Lemma 3.3. (i) Withν¯λ,ρ⊗IP⊗IP+-probability one, every subsequential limit of m_t,ǫ(θ_[′_βt],αtω′,ω+)

as t→ ∞lies inI_v∩ S ∩ M_λ,ρ=I ∩ S ∩ M_λ,ρ.

(ii) I ∩ S ∩ M_λ,ρ is the set of probability measures ν of the form ν = Rνr_{γ(d r), where γ is a}

probability measure supported onR ∩[λ,ρ].

The proof of Lemma 3.3 will be based on the following large deviation result in the spirit of[15]. Lemma 3.4. (i) The functionalD_v defined onP(X)by

D_v(µ):= sup

flocal

−

Z

L_vef

ef (ηe)dµ(ηe) (48)

is nonnegative, lower-semicontinuous, andD−1

v (0) =Iv.

(ii) Let ξe. be a Markov process with generator Lv and distribution denoted by P. Define the

empirical measures

πt(ξe.):=t−1 Z t

0

δ_ξe

sds, πt,ǫ:=|

Z_{∩[−ǫt,ǫt]|}−1 X

x∈Z∩[−ǫt,ǫt]

τxπt (49)

whereǫ >0. Then, for every closed subset F ofP(X)and every t ≥0,

lim sup

t→∞

t−1logP€πt,ǫ(ξe.)∈F Š

≤ −inf

3.3

Proofs of Lemma 3.1, Proposition 3.2 and Corollary 3.1

Proof of Lemma 3.1. The result is a particular case, forλ=ρ, of (51) proven below. ƒ Proof of Proposition 3.2. We denote byω′= (η,ξ,ω)a generic element ofΩ′. We will establish the following limits. First,

inf

r∈[λ,ρ]∩R[G(r)−v r] ≤ lim inft→∞ t −1_φv

t ◦θ

′

[βt],αt(ω

′₎

≤ lim sup

t→∞ t −1_φv

t ◦θ

′

[βt],αt(ω

′₎

≤ sup

r∈[λ,ρ]∩R

[G(r)−v r], ν¯λ,ρ⊗IP-a.s. (51)

and then

lim sup

t→∞

t−1φv_t ◦θ_[′_βt],αt(ω′)≤ inf

r∈[λ,ρ]∩R[G(r)−v r], ν¯

λ,ρ_{⊗IP-a.s. (52)}

which will imply the result, when combined with Proposition 3.1 and the expression (38) of

N_tv,w(ω′). Though only the first inequality in (51) (together with (52)) seems relevant to Proposi-tion 3.2, we will need the whole set of inequalities: Indeed, writing (51) forλ=ρ=r∈ R proves the equilibrium result of Lemma 3.1.

To obtain the bounds in (51) we proceed as follows. First we replace the deterministic path

v tin the currentφ_tv byx_tt(ω+). Then we consider a spatial average ofϕx

t

.(ω +_)+x

t forx∈[−ǫt,ǫt],

and we introduce, for ζ ∈ X, the martingale M_tx,v(ζ,ω,ω+) associated to ϕx

t

.(ω +_)+x

t (ζ,ω) (see

below (57)). An exponential bound on the martingale reduces the derivation of (51) to bounds (deduced thanks to Lemma 3.3) on

Z

[f(η)−vη(1)]m_t,ǫ(θ_[′_βt],αtω′,ω+)(dη)

(see below (62)), where [f(η)−vη(1)] corresponds to the compensator of ϕx

t

.(ω +_)+x

t (ζ,ω) in

M_tx,v(ζ,ω,ω+). The bound (52) relies on Lemmas 3.1 and 3.2 combined with the monotonicity of the process.

Step one: proof of (51).We have

t−1φ_tv◦θ_[′_βt],αt(ω′) =t−1ϕ_tv(̟αt,θ[βt],αtω) (53)

where the configuration

̟αt=̟αt(η,ξ,ω):=T

€

τ[βt]ηαt(η,ω),τ[βt]ηαt(ξ,ω)

Š

(54) depends only on the restriction ofω to[0,αt]×Z_{. Thus, sinceω is a Poisson measure under IP,}

θ[βt],αtωis independent of̟αt(η,ξ,ω)under ¯νλ,ρ⊗IP, and

Define

ψv_t,ǫ(ζ,ω,ω+):=|Z∩[−ǫt,ǫt]|−1 X y∈Z:|y|≤ǫt

ϕx

t

.(ω +_)+y

t (ζ,ω)

for every(ζ,ω,ω+)∈X×Ω×Ω+, with x_.t(ω+)given by (42). By (36),

_ϕv

t(ζ,ω)−ψ v,ǫ

t (ζ,ω,ω+)

≤K€2ǫt+xt_t(ω+)−v tŠ

Sincet−1xt_t(ω+)→vwith IP+-probability one, the proof of (51) can be reduced to that of the same inequalities with the l.h.s. of (53) replaced by

t−1ψvt,ǫ(̟αt,θ[βt],αtω,ω+) (56)

and ¯νλ,ρ⊗IP replaced by ¯νλ,ρ⊗IP⊗IP+. Let f(η):= f+(η)− f−(η), with

f+(η) = X

y,z∈Z:y≤0<y+z

p(z)b(η(y),η(y+z))

f−(η) = X

y,z∈Z_:y+z≤0<y

p(z)b(η(y),η(y+z))

andn(η):=η(δ), whereδis defined to be 1 ifv >0, 0 if v<0, and any integer if v=0. By the definition of particle current (33)–(35), we have that, for anyζ∈X,

M_tx,v(ζ,ω,ω+) := ϕx

t

.(ω +_)+x

t (ζ,ω) −

Z t

0

τx[f −vn](ηe_st−(ζ,ω,ω+)) (57)

E_tx,v,θ(ζ,ω,ω+) := exp §

θ ϕx

t

.(ω+)+x

t (ζ,ω)

−€eθ −1Š

Z t

0

τxf+(ηet_s−(ζ,ω,ω

+_))ds

−€e−θ −1Š

Z t

0

τxf−(ηet_s−(ζ,ω,ω+))ds

−

Z t

0

|v|

e−sgn(v)θ τxn

e

ηt s−(ζ,ω,ω

+₎ −1 ds « (58)

= exp¦θM_tx,v(ζ,ω,ω+)

−€eθ −1−θŠ

Z t

0

τxf+(ηet_s−(ζ,ω,ω

+_))ds

−€e−θ −1+θŠ

Z t

0

τxf−(ηet_s−(ζ,ω,ω+))ds

−

Z t

0

|v|

e−sgn(v)θ τxn

e

ηt

s−(ζ,ω,ω

+₎

−1

+sgn(v)θ τxn(ηe_st−(ζ,ω,ω

+₎₎Š_ds©

are martingales under IP⊗IP+, with respective means 0 and 1. Notice that ηs− and η_s can be replaced with each other in the above martingales, because, by the graphical construction of Section

2.1,s7→ηs(x)is IP⊗IP+-a.s. locally piecewise constant for everyζ∈Xandx ∈Z. It follows from

(58) that

IEeθMtx,v

≤eC t(eK|θ|−1−K|θ|) (59)

for anyζ∈X, where expectation is w.r.t. IP⊗IP+, and the constantC depends only onp(.), b(.)and

vbut not onζ. Cramer’s inequality and (59) imply the large deviation bound

IP⊗IP+({M_tx,v≥ y})≤2e−tIC(y) ₍₆₀₎

for anyζ∈Xand y ≥0, with the rate function

I_C(y) = y

K log

 1+ y

C K

‹

−C

• _y

C K −log

 1+ y

C K

‹˜

Since(60) holds for everyζ∈X, the independence property (55) also implies ¯

νλ,ρ⊗IP⊗IP+€{M_tx,v(̟αt,θ[βt],αtω,ω+))

≥ y}Š≤2e−tIC(y)

This and Borel Cantelli’s lemma imply that lim

t→∞t

−1_|Z∩[

−ǫt,ǫt]|−1 X

x∈Z:|x|≤ǫt

M_tx,v(̟αt,θ[βt],αtω,ω+) =0, (61)

¯

νλ,ρ_⊗IP⊗IP+_{-a.s. In view of (57) and (61), the proof of (51) is now reduced to that of the same set}

of inequalities with (56) replaced by

t−1|Z_{∩[−ǫt,ǫt]|}−1 Z t

0 X

x∈Z:|x|≤ǫt

τx[f −vn](ηest(̟αt,θ[βt],αtω,ω+))ds

withδas in (57), which is exactly, because of (54), Z

[f(η)−vη(δ)]mt,ǫ(θ[′βt],αtω

′_,ω+_{)(dη) (62)}

with the empirical measure m_t,ǫ defined in (47). By Lemma 3.3, every subsequential limit ν as

t→ ∞ofmt,ǫ(θ_[′_βt],αtω′,ω+)is of the form

ν=

Z

νrγ(d r)

for some measureγsupported onR ∩[λ,ρ]. Then the corresponding subsequential limit ast→ ∞

of (62) is of the form _Z

[G(r)−v r]γ(d r)

as one verifies, using shift invariance ofνr, that Z

This concludes the proof.

Step two: proof of (52). Let u1 ≤ v ≤ v ≤ ¯v ≤ v1, λ < ρ ∈ R, and r ∈ [λ,ρ]∩ R. We set

ς=ηr, and we define ¯νλ,r,ρas the coupling distribution of(ηλ,ηr,ηρ). Note that, by the stochastic ordering property (20),

¯

νλ,r,ρ¦(η,ς,ξ)∈X3:η≤ς≤ξ©=1 (63)

The following limits are true for ¯νλ,r,ρ-a.e. (η,ς,ξ). By the expression (38) ofN_t.,.and the equilib-rium limit Lemma 3.1,

lim

t→∞ 1

tN

u1,v

t ◦θ[′βt],αt(ς,ς,ω) =r(v−u1) (64) By attractiveness,

lim inf

t→∞ 1

tN

u1,v

t ◦θ[′βt],αt(ς,ξ,ω)≥_tlim_→∞

1

tN

u1,v

t ◦θ[′βt],αt(ς,ς,ω) (65)

Putting together (64) and (65), lim inf

t→∞ 1

tN

u1,v

t ◦θ[′βt],αt(ς,ξ,ω)≥r(v−u1) (66) Now, by (38), Lemma 3.2 and Lemma 3.1 respectively forr andρ,

lim

t→∞ 1

tN

u1,v1

t ◦θ[′βt],αt(ς,ξ,ω) = (G(r)−u1r)−(G(ρ)−v1ρ) (67)

Subtracting (66) from (67), we get lim sup

t→∞ 1

tN

v,v1

t ◦θ[′βt],αt(ς,ξ,ω)≤(G(r)−v r)−(G(ρ)−v1ρ) (68) By attractiveness, (63) and (68), we have

lim sup

t→∞ 1

tN

v,v1

t ◦θ[′βt],αt(η,ξ,ω) ≤ lim sup t→∞

1

tN

v,v1

t ◦θ[′βt],αt(ς,ξ,ω)

≤ (G(r)−v r)−(G(ρ)−v₁ρ) (69)

Using (38), (69), Lemma 3.2, and Lemma 3.1 forρ, we obtain lim sup

t→∞ 1

tφ

v t ◦θ

′

[βt],αt(ω

′₎

= lim sup

t→∞

₁

tφ

v t ◦θ

′

[βt],αt(ω

′₎₋1

tφ

v1

t ◦θ[′βt],αt(ω

′_{) +}1

tφ

v1

t ◦θ[′βt],αt(ω

′₎

≤ lim sup

t→∞ 1

tN

v,v1

t ◦θ[′βt],αt(ω

′_{) +lim sup}

t→∞ 1

tφ

v1

t ◦θ[′βt],αt(ω

′₎

≤ ((G(r)−v r)−(G(ρ)−v₁ρ)) + (G(ρ)−v₁ρ) = G(r)−v r

for every r ∈ [λ,ρ]∩ R. Since ς is no longer involved in the above inequalities, we obtain a ¯

a common exceptional set of ¯νλ,ρ-probability 0 for allr ∈[λ,ρ]∩ R. This proves (52). ƒ Proof of Corollary 3.1.

(i).By Proposition 3.1 (cf.[8, (28)]),

d

d v[G(hc(v))−vhc(v)] =−hc(v)

weakly with respect tov. Thus, settingu=Rλ,ρ, we have

[G(u(v, 1))−vu(v, 1)]−[G(u(w, 1))−wu(w, 1)] =

Z w v

u(x, 1)d x (70) for allv,w∈R_{. Leta<bin}R_{. Setting}

̟=T€τ[N x0]ηN s0(η,ω),τ[N x0]ηN s0(ξ,ω)

Š we have

β_{N t}N (θ_[′_{N x}

0],N s0(η,ξ,ω))((a,b]) = N

−1 X

[N a]<x≤[N b]

ηN t(̟,θ[N x0],N s0ω)(x)

= t(N t)−1N_{N t}a/t,b/t(θ_[′_{N x}

0],N s0ω ′₎ Thus, by Proposition 3.2 and (70),

lim

N→∞β

N N t(θ

′

[N x0],N s0ω

′_{)((a,b]) =t}

Z b/t a/t

u(x, 1)d x=

Z b a

u(x,t)d x (71)

¯

νλ,ρ⊗IP-a.s., where the last equality follows from Proposition 3.1. Now (70) implies that the r.h.s. of (30) is a continuous function of(v,w), while the l.h.s. is a uniformly Lipschitz function of(v,w), since the number of particles per site is bounded. It follows that one can find a single exceptional set of ¯νλ,ρ⊗IP-probability 0 outside which (71) holds simultaneously for alla,b, which proves the claim.

(ii).Sinceηλ,ρhas distribution ¯νλ,ρ under IP0, the statement follows from(i)withx0=s0=0. ƒ

3.4

Proofs of remaining lemmas

Proof of Lemma 3.2. Letǫ > 0. We consider the probability spaceΩ′×(Z+₎Z _{equipped with the}

product measure

IP′_ǫ:=ν¯λ,ρ⊗IP⊗Pǫ

where Pǫis the product measure onZwhose marginal at each site is Poisson with meanK(1+ǫ).

A generic element of this space is denoted by (ω′,χ), with ω′ = (η,ξ,ω) and χ ∈ (Z+₎Z_{. We}

first prove (40). Because of (26) (that is, coupled configurations are ordered under ¯νλ,ρ), by the attractiveness property (15), we may define

fors≥0. Thereforeγ-particles represent the discrepancies between the system starting fromξand the system starting fromT(η,ξ). We set

p(z) = (p(z) +p(−z))||b||_∞ (73) and fix ¯vsuch that

¯

v>X z∈Z

z p(z) (74)

Letv>¯v. Becauseγ0(x) =0 for allx >0, by the definition of current (32),

φ_tv(ξ,ξ,ω) =φ_tv(ω′) + X

y>[v t]

γt(y)

Therefore, to prove (40), we want to obtain lim

t→∞t

−1 X

y>¯v t

γt◦θ[′βt],αt(ω

′_{)(y) =0, ¯}

νλ,ρ⊗IP-a.s. (75)

To this end, we follow the proof of [1, Proposition 5], with minor modifications. We emphasize that even if the latter proof corresponds toα = β = 0, we will see that the arguments extend to

(α,β) 6= (0, 0). Eachγ0-particle andχ-particle is given an integer label j ≤0, and we denote by

R₀j (resp. Z₀j) the position of theγ0-particle (resp. χ-particle) with label j. That is,R.0 andZ . 0must

satisfy _X

j≤0

δ_Rj

0

=X

x∈Z

γ0(x)δx,

X

j≤0

δ_Zj

0

=X

x∈Z

χ(x)δx (76)

There is a unique way to choose suchR.₀(resp.Z₀.) if we impose moreover thatR₀j ≤R₀j+1(resp.Z₀j≤

Z₀j+1) for all j<0, which we will assume. By the definition ofχ, the number ofχ-particles between

−nand 0 will be eventually larger thannK. Let

W(χ):=inf n

n∈Z+_{: Z}j 0≥ −[

j/K]for every j≤ −n

o

By Poisson large deviation bounds, the random variableW is IP′_ǫ-a.s. finite with exponentially de-caying distribution. Sinceγ0(x)≤K for every x ∈Z, we haveR

j

0≤ −[

j/K]for all j ≤0, hence

Z₀j≥R₀j for every j≤ −W(χ). The dynamics of(R._t)_t≥0is defined as follows. By (72) and the graph-ical construction, at most one discrepancy can jump at each event time ofω. If aγ-particle jumps at timet from x to y, we setRj_t= y andRk_t =Rk_t− for allk6= j, where j is the largest label l such thatRl_t−=x. The choice of the largest label is arbitrary and could be replaced by any deterministic or random choice procedure, provided the corresponding source of randomness is independent of

(ω′,χ). Note that we do not in general haveR_tj ≤Rj_t+1 fort >0, but this does not matter for our purpose. The dynamics ofZ_.j is defined as follows: ifRj_t

−=x and, for somez>0 andu∈[0, 1],

{(t,x,z,u),(t,x+z,−z,u)} ∩ω6=; (77)

then Z_tj = Z_tj

−+z. In other words, χ-particles evolve as mutually independent (given their initial positions) random walks, that jump from y to y+z at ratep(z)given by (73) for all y ∈Z_,z≥0.

Since a jump forR_.jfromRj_t

−= x toR

j

t =x+zis possible only under (77),

In view of (75), (78), we estimate X

x>¯v t

γt(x) =

X

j≤0

1_{Rj

t(ω′)>¯v t} =

X

j≤−W(χ)

1_{Rj

t(ω′)>¯v t}+ X

−W(χ)<j≤0

1_{Rj t(ω′)>¯v t}

≤ X

j≤−W(χ)

1_{Zj

t(ω′)>¯v t}+W(χ)

≤ Z¯t(ω,χ) +W(χ) (79)

where ¯Z_t:=P_i≤01_{Zi

t>¯v t} is a Poisson random variable with mean

IE′_ǫZ¯_t=K(1+ǫ)X

j≥¯v t

IP′_ǫ(Y_t> j) (80)

forY_t a random walk starting at 0 that jumps from y to y+zwith ratep(z). Since ¯v satisfies (74), repeating[1, (43)–(45)]gives that

lim

t→∞IE ′

ǫZ¯t/t=0 (81)

Letδ >0. Since ¯Zt is a Poisson variable,

IP′_ǫ(|Z¯_t−IE′_ǫZ¯_t|> δt) ≤ IE

′

ǫ(Z¯t−IE′ǫZ¯t)4

(δt)4

≤ [IE

′

ǫ(Z¯t−IE′ǫZ¯t)2]2

(δt)4

≤ (IE

′

ǫZ¯t/t)2t2

(δt)4 (82)

Therefore, by Borel Cantelli lemma lim

t→∞( ¯

Zt−IE′ǫZ¯t)/t =0, IP′ǫ-a.s. (83)

Since IP is invariant by θ[βt],αt, ¯Zt(θ[βt],αtω,χ) has the same distribution as ¯Zt(ω,χ) under IP′ǫ.

Thus (80)–(83) still hold with ¯Zt(θ[βt],αtω,χ)instead of ¯Zt(ω,χ), and

lim

t→∞t −1_Z¯

t(θ[βt],αtω,χ) =0, IP′ǫ-a.s. (84)

Because the random variable W in (79) does not depend onω′, (84) and (79) imply (75). This concludes the proof of (40).

If we now defineγs as

γs(ω′):=ηs(T(η,ξ),ω)−ηs(η,ω)

and replaceφv_t(ω′) by−φ_tv(ω′) (so that the current, which was rightwards, becomes leftwards), then the proof of (41) can be obtained by repeating the same steps as in the previous argument.ƒ

Then,

F_∗∗n(x)−f_∗∗(x) ≤ 1

n n X j=1 ”

f_∗n(Tjx)−f_∗(Tjx)—1_{X \A} k(T

j_x)

+ 1 n n X j=1 ”

f_∗n(Tjx)−f∗(Tjx) —

1Ak(T

j_x)

+ 1 n n X j=1

f_∗(Tjx)−f_∗∗(x) ≤ sup X \Ak

f_∗n− f_∗+2M g_An k(x) +

f_∗∗n(x)− f_∗∗(x)=:B_n,k(x)

whereM :=sup_Xfand, by ergodic theorem,

g_An k(x):=

1 n

n

X

j=1

1_Ak(Tjx)n→∞→ g_Ak(x), P-a.s. (143)

for some bounded, nonnegative,F-measurable g_A

k such that Z

gAkdν=ν(Ak)

k→∞

→ 0 (144)

By uniform convergence of f_∗n onX \A_k, a.s. convergence (142), and (143), lim sup_n→∞B_n,k(x)≤

2M gAk(x) holds P-a.s. By (144), gAk goes to 0 in L

1_{(P) as k → ∞. Thus it has a subsequence} converging to 0P-a.s. Lettingk→ ∞along this subsequence concludes the proof. ƒ

Proof of proposition 2.3.

Existence of the limit. Define the random variables X_i,j := R₍ja

j−1)af(τ i−1_η

s)ds, where i,j ∈ N

and (ηs)s≥0 is the stationary Markov process with generator L and initial distribution µ. Take

X = RN×N_{, F the product Borel σ-field, P the distribution of the X-valued random variable}

(X_i,j)_i,j∈N, T”(x_i,j)_i,j∈N —

= (x_i,j+1)_i,j∈N, S”(x_i,j)_i,j∈N —

= (x_i+1,j)_i,j∈N. We have P◦ T−1 = P because (ηs)s≥0 is stationary, and P◦S−1 = S because µ and L are invariant by τ. Then the existence of the limit follows from Proposition C.1.

Identification of the limit. Let now X be the Skorokhod space of X-valued paths, and P = Pµ the law of the Markov process with generator L and initial distribution µ. We consider on X

the space shifts (τx)x∈Z and time shifts (Tt)t≥0 defined as follows: if η. = (ηs)s≥0 ∈ X, then

τxη. := (τxηs)s≥0, where τx on the r.h.s. is the spatial shift on particle configurations defined

in Section 2, and T_tη. := (ηt+s)s≥0. What follows is a generalization of a standard result for one-parameter Markov processes (see e.g. [11, Chapter 7]). By the above existence step, we can define

P_µ-a.s., where

F_∗∗n(η.) = 1 an

Z an

0 1 n

n

X

i=1

τif(ηt)d t

As a limit of measurable functions, f_∗∗ is measurable. For every t >0, T_tF_∗∗n −F_∗∗n andτF_∗∗n −F_∗∗n consist of space-time sums over boundary domains of order O(n) = o(n2_{), hence in the limit} n→ ∞, f∗∗ is invariant by(Tt)t≥0 andτ:=τ1. To show that this implies f∗∗is a P-a.s. constant function, we will prove that any measurable subsetF ofX which is invariant by(T_t)_t≥0 andτhas Pµ-probability 0 or 1. Taking expectations in (145), the constant value of f∗∗must be

R

f dµ, and Proposition 2.3 is thus established.

LetF⊂ X be measurable, and invariant by(T_t)_t≥0 andτ. Set

g(η) =Pµ(F|η0=η) =:Pη(F) (146) which is defined forµ-a.e. η. Here,P_ηdenotes the law of the Markov process starting from deter-ministic stateη∈X. We are going to prove that

g≡0 org≡1 (147)

µ-a.s., which will implyP(F) =R_Xg(η)µ(dη)∈ {0, 1}. We have

g(τη) =Pτη(F) =Pη(τ−1F) =Pη(F) =g(η)

where the second equality follows from translation invariance (22) ofL(which impliesPτη=τPη), and the third from τ-invariance of F. Therefore g is µ-a.s. invariant by the spatial shift τ. We claim that g=1G µ-a.s. for someG ⊂X. Indeed, letFt denote theσ-field ofX generated by the

mappingsη.7→ηs fors≤t. WithPµ-probability one, g(ηt) =Pηt(F) =Pµ(T

−1

t F|Ft) =Pµ(F|Ft)

where the second equality follows from Markov property, and the third from theT_t invariance ofF. By the martingale convergence theorem, we have thePµ-a.s. limit

lim

t→∞g(ηt) =1F(η.) (148)

Sinceηt∼µfor allt≥0, for everyǫ >0,

Pµ(η.: ǫ≤ g(ηt)≤1−ǫ) =µ(η:ǫ≤ g(η)≤1−ǫ) (149)

we conclude from (148)–(149) that the law of g(η_t) is Bernoulli, hence g = 1_G µ-a.s. for some G⊂X. The desired conclusion (147) is thus equivalent toµ(G)∈ {0, 1}, which we now establish. First we claim that, with Pµ-probability one, we have g(ηt) = g(η0) for all t > 0. Indeed,

byT_t invariance ofF, Markov property and definition ofG, P_µ {ηt6∈G} ∩F

= P€{ηt 6∈G} ∩Tt−1F

Š

= Z

{ηt6∈G}⊂X

Pηt(F)Pµ(dη.)

= Z

{ηt6∈G}⊂X

g(ηt)Pµ(dη.)

= 0 (150)

Similarly we have

Pµ {ηt ∈G} ∩(X \F)

=0 (151)

It follows from (150)–(151) that{ηt ∈G}=F up to a set of Pµ-probability 0. In other words, for everyt>0, we have g(ηt) =g(η0) =1F(η.)withPµ-probability one.

Now, assume µ(G) 6∈ {0, 1}, and define the conditioned measures µG(dη) := µ(dη|η ∈ G)

andµX \G(dη):=µ(dη|η∈ X \G), so thatµ=µ(G)µG+µ(X \G)µX \G. The invariance ofLandµ

with respect to space shift imply the same forµGandµX \G. A simple computation, using invariance

ofµforL, and the fact that g(ηt)isPµa.s. constant, shows thatµGandµX \Gare also invariant for

L. This contradicts the fact thatµ∈(I ∩ S)_e. ƒ

Acknowledgments: We thank Tom Mountford for helpful discussions. We would like to

thank an anonymous referee for carefully reading the manuscript and offering useful suggestions which improved the clarity of the paper. K.R. and E.S. thank TIMC - TIMB, INP Grenoble. K.R. thanks Université de Rouen for hospitality. Part of this work was done during the semester “Interacting Particle Systems, Statistical Mechanics and Probability Theory” at IHP, where K.R. benefited from a CNRS “Poste Rouge”.

References

[1] Andjel, E., Ferrari, P.A., Siqueira, A. Law of large numbers for the asymmetric exclusion pro-cess.Stoch. Process. Appl.132(2004) no. 2, 217–233.

[2] Andjel, E.D., Kipnis, C. Derivation of the hydrodynamical equation for the zero range interac-tion process.Ann. Probab.12(1984), 325–334. MR0735841

[3] Andjel, E.D., Vares, M.E. Hydrodynamic equations for attractive particle systems onZ_{.J. Stat.}

Phys.47(1987) no. 1/2, 265–288. MR0892931

[4] Andjel, E.D., Vares, M.E. Correction to : “Hydrodynamic equations for attractive particle sys-tems onZ_{”.J. Stat. Phys.113(2003) 379-380.}

[5] Belletini, G., Bertini, L., Mariani, M., Novaga, M.Γ-entropy cost for scalar conservation laws. Archive Rational Mechanics Anal.195(2010), 261–309.

[6] Benassi, A., Fouque, J.P. Hydrodynamical limit for the asymmetric exclusion process. Ann. Probab.15(1987), 546–560. MR0885130

[7] Bahadoran, C., Guiol, H., Ravishankhar, K., Saada, E. A constructive approach to Euler hydro-dynamics for attractive particle systems. Application to k-step exclusion. Stoch. Process. Appl.

99(2002) no. 1, 1–30. MR1894249

[8] Bahadoran, C., Guiol, H., Ravishankhar, K., Saada, E. Euler hydrodynamics of one-dimensional attractive particle systems.Ann. Probab.34(2006), 1339–1369. MR2257649

[9] Biane, P., Durrett, R. Lectures on Probability Theory, Ecole d’Eté de Probabilités de Saint-Flour XXIII-1993.LN in Math. 1608, Springer, 1995. MR1383120

[10] Bramson, M., Mountford, T. Existence of blocking measures for one-dimensional nonzero mean exclusion processes.Ann. Probab.30(2002), 1082–1130. MR1920102

[11] Breiman, L.Probability.Addison-Wesley, 1968. MR0229267

[12] Cocozza-Thivent, C. Processus des misanthropes.Z. Wahrsch. Verw. Gebiete 70 (1985) no. 4, 509–523. MR0807334

[13] De Lellis, C., Otto, F., Westdickenberg, M. Minimal entropy conditions for Burgers Equa-tion.Quarterly of Applied Math.62(2004), 687–700. MR2104269

[14] De Masi A., Presutti E. Mathematical Methods for Hydrodynamic Limits. LN in Math. 1501, Springer, 1991. MR1175626

[15] Donsker, M.D., Varadhan, S.R.S. Asymptotic evaluation of certain Markov process expectations for large time I.Commun. Pure Appl. Math.28(1975), 1–47. MR0386024

[16] Ferrari, P.A. Shock fluctuations in asymmetric simple exclusion. Probab. Theory Rel. Fields91

(1992), 81–101. MR1142763

[17] Ferrari, P.A. Shocks in the Burgers equation and the asymmetric simple exclusion process; in Automata Networks, Dynamical Systems and Statistical Physics, edited by E. Goles, S. Martinez, Kluwer Math. and its Appl. (1992), 25–64. MR1263704

[18] Guiol, H. Some properties of k-step exclusion processes. J. Stat. Phys.94 (1999) no. (3/4), 495–511. MR1675362

[19] Harris, T.E. Nearest neighbor Markov interaction processes on multidimensional lattices.Adv. in Math.9(1972), 66–89. MR0307392

[20] Harris, T.E. Additive set-valued Markov processes and graphical methods. Ann. Probab. 6

(1978), 355–378. MR0488377

[21] Hayes, B.T., le Floch, P.G. Non-Classical Shocks and Kinetic Relations: Scalar Conservation Laws.Archive for Rational Mech. Anal.139(1997), 1–56. MR1475777

[22] Jensen, L. Large deviations of the asymmetric simple exclusion process in one dimension.PhD dissertation, New York University(2000).

[23] Kamae, T., Krengel, U. Stochastic partial ordering. Ann. Probab. 6 (1978), 1044–1049. MR0512419

[24] Kamae, T., Krengel, U., O’Brien, L.G. Stochastic inequalities on partially ordered spaces.Ann. Probab.5(1977), 899–912. MR0494447

[25] Kipnis, C., Landim, C.Scaling limits of interacting particle systems. Grundlehren der Mathema-tischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 320. Springer-Verlag, Berlin, 1999. MR1707314

[26] Kipnis, C., Olla, S., Varadhan, S.R.S. Hydrodynamics and large deviations for simple exclusion process.Comm. Pure. Appl. Math.XLII(1989), 115–137. MR0978701

[27] Krengel, U.Ergodic theory.De Gruyter Studies in Mathematics6, 1985. MR0797411

[28] Kružkov, N. First order quasilinear equations in several independent variables.Math. USSR Sb.

10(1970), 217–243.

[29] Lax, P.D. Hyperbolic systems of conservation laws II. Comm. Pure. Appl. Math. 10 (1957), 537–566. MR0093653

[30] Liggett, T.M. Interacting particle systems. Classics in Mathematics (Reprint of first edition), Springer-Verlag, New York, 2005. MR2108619

[31] Liggett, T.M.Stochastic Interacting Systems: Contact, Voter and Exclusion Processes.Grundlehren der Mathematischen Wissenschaften[Fundamental Principles of Mathematical Sciences],324, Springer-Verlag, New York, 1999. MR1717346

[32] Mariani, M. Large deviations principle for stochastic conservation laws. M. Large deviations principle for stochastic scalar conservation laws. Probab. Theory Related Fields (2009) (Pub-lished online)

[33] Mountford, T.S., Ravishankar, K., Saada, E. Macroscopic stability for nonfinite range kernels. http://fr.arxiv.org/abs/0903.0498,Braz. J. Probab. Stat.(2009).

[34] Rezakhanlou, F. Hydrodynamic limit for attractive particle systems onZd_{.Comm. Math. Phys.}

140(1991), no. 3, 417–448. MR1130693

[35] Rezakhanlou, F. Continuum limit for some growth models. II.Ann. Probab.29 (2001) no. 3, 1329–1372. MR1872745

[36] Rost, H. Nonequilibrium behaviour of a many particle process: density profile and local equi-libria,Z. Wahrsch. Verw. Gebiete58(1981) no. 1, 41–43.

[37] Seppäläinen, T. Existence of hydrodynamics for the totally asymmetric simple K-exclusion process.Ann. Probab.27(1999) no. 1, 361–415. MR1681094

[38] Seppäläinen, T. Translation Invariant Exclusion Processes. In preparation.http://www.math. wisc.edu/~seppalai/excl-book/etusivu.html.

[39] Serre, D.Systèmes de lois de conservation. Diderot Ed., 1996.

[41] Strassen, V. The existence of probability measures with given marginals.Ann. Math. Statist.36

(1965), 423–439. MR0177430

[42] Varadhan, S.R.S. Lectures on Hydrodynamic Scaling; inHydrodynamic Limits and Related Top-ics;edited by S. Feng, A. Lawniczak, S.R.S. Varadhan, Fields Institute Communications, AMS, 2000. MR1798641

[43] Varadhan, S.R.S. Large deviations for the asymmetric simple exclusion process.Stochastic anal-ysis on large scale interacting systems, 1–27,Adv. Stud. Pure Math.39, Math. Soc. Japan, Tokyo, 2004. MR2073328

[44] Vol’pert, A.I. The spaces BV and quasilinear equations. Math. USSR Sbornik2 (1967) no. 2, 225–267.Transl. from(Russian)Mat. Sb. (N.S.)73(1967) no. 115, 255–302. MR0216338