The boundary driven zero-range process
aa r X i v : . [ m a t h . P R ] J un THE BOUNDARY DRIVEN ZERO-RANGE PROCESS
SUSANA FR ´OMETA, RICARDO MISTURINI, AND ADRIANA NEUMANN
Abstract.
We study the asymptotic behaviour of the symmetric zero-rangeprocess in the finite lattice { , . . . , N − } with slow boundary, in which parti-cles are created at site 1 or annihilated at site N − N − θ , for θ ≥
1. We present the invariant measure for this model and obtainthe hydrostatic limit. In order to understand the asymptotic behaviour of thespatial-temporal evolution of this model under the diffusive scaling, we startto analyze the hydrodynamic limit, exploiting attractiveness as an essential in-gredient. We obtain that the hydrodynamic equation has boundary conditionsthat depend on the value of θ . Introduction
The zero-range process, originally introduced in 1970 by Spitzer [27], is a modelthat describes the behaviour of interacting particles moving on a lattice withoutrestriction on the total number of particles per site. In this model, a particle leavesa site according to a jump rate g ( k ) that only depends on the number of particles, k , in that site. The zero-range process has been mostly studied in infinite lattices(see [2, 3, 18, 19, 26]) and in discrete torus (see [20, 21, 4, 9, 22] and the referencestherein). In the present work we consider the process defined in the finite lattice I N = { , . . . , N − } with creation and annihilation of particles at the boundary.One of the main interest in the study of interacting particle systems is thederivation of partial differential equations (PDE) to describe the time evolutionof the macroscopic density of particles as the lattice is rescaled to the continuum.Such classical scaling limit is called hydrodynamic limit and the associated PDE iscalled hydrodynamic equation . In recent years there has been an increasing inter-est in models that leads to hydrodynamic equations with boundary conditions (see[8, 12, 13, 17]). This has been done, for example, for the exclusion process in [5]and for the porous medium model in [10]. In both cases, the lattice I N is connectedto reservoirs so that particles can be inserted into or removed from the system withrate proportional to N − θ , and the obtained hydrodynamic equations have bound-ary conditions that depend on the value of θ . One common characteristic of themodels in [5, 10] is that the exclusion rule only allows one particle per site, whichprovides a natural control for the number of particles in the system.For the classical zero-range process in the discrete torus, see [22, Chapter 5],conservation of particles is an extensively used property in the proof of hydrody-namic limit, together with a hypothesis that controls the relative entropy of theinitial distribution with respect to some invariant measure. In the open zero-rangeprocess, considered in the present work, the number of particles in the system isnot conserved as it was in the process in the discrete torus and neither bounded Key words and phrases.
Zero-range process, Slow boundary, Invariant measure, Hydrostaticlimit, Hydrodynamic limit, Boundary conditions. as it was in the exclusion process and porous medium model. To overcome thisdifficulty, instead of assuming a relative entropy hypothesis, we exploit the attrac-tiveness present in our model under the assumption that the jump rate function g is non decreasing an that the initial distribution is bounded above by the in-variant measure. Attractiveness was also an essential ingredient in [3], where theauthors obtained the hydrodynamic limit through preservation of local equilibriumfor the asymmetric zero-range process on Z under Euler scaling. The same wasdone, for example, for the symmetric zero-range process in the discrete torus underthe diffusive scaling, see [22, Chapter 9].In this work we consider a symmetric nearest-neighbour zero-range process in I N with the following dynamics at the boundary: a particle is inserted into the systemat site 1 with rate αN θ and removed from the system through site N − g ( k ) N θ ,if there are k particles at site N − , where α ≥ θ ≥ g is same jump ratefunction used in I N . Computing analytically the stationary distribution of a non-equilibrium stochastic model is usually a very challenging task, see [14, 15, 16, 7].However, an important general aspect of the zero-range process, that is not presentin the models considered in [5, 10], is that its invariant distribution is a productmeasure that can be explicitly computed, see [27, 1]. This is also true in our case,despite of the boundary conditions, as already considered in [11, 23, 6], and theresulting steady-state, when it exists, is a product measure imitating the periodiccase, but now it is characterized by a non homogeneous space-dependent fugacitywhich is a function of the boundary rates. In Section 3, we present the invariantmeasure for our model obtained through elementary computations involving thejump rates. Having the explicit form of the invariant measure, we obtain thestationary density profile, the so called hydrostatic limit .Our main goal is to describe the asymptotic behaviour for the time evolution ofthe spacial density of particles for zero-range process with slow boundary introducedabove. More precisely, we want to prove that, if we start our evolution with aninitial configuration of particles that converges to a macroscopic density profile γ : [0 , → R + , as N → ∞ , then, under the diffusive scaling, and in a fixed timeinterval [0 , T ], the time trajectory of the spatial density of particles, { π Nt : t ∈ [0 , T ] } , converges to a deterministic limit, { π t : t ∈ [0 , T ] } . In the present work weprove relative compactness for the sequence { π Nt : t ∈ [0 , T ] } and that the limitpoints, { π t : t ∈ [0 , T ] } , are trajectories of absolutely continuous measures on [0 , π t ( du ) = ρ ( t, u ) du , for t ∈ [0 , T ] and u ∈ [0 , ρ ( t, · ), we get, through some heuristicarguments, that ρ is the weak solution of the following non-linear diffusion equationwith boundary conditions: ∂ t ρ ( t, u ) = ∆Φ( ρ ( t, u )) , for u ∈ (0 ,
1) and t ∈ (0 , T ] ,∂ u Φ( ρ ( t, − κ α , for t ∈ (0 , T ] ,∂ u Φ( ρ ( t, − κ Φ( ρ ( t, , for t ∈ (0 , T ] ,ρ (0 , u ) = γ ( u ) , for u ∈ [0 , , where κ = 1, if θ = 1, and κ = 0, if θ >
1. The function Φ will be defined in (4.3),in terms of the jumps rate g . See Remark 3.4 for a more general dynamics allowing creation and annihilation of particlesat both sides of the boundary.
HE BOUNDARY DRIVEN ZERO-RANGE PROCESS 31 2 3 ... x − x x +1 ... N − N − g ( η (1)) γN θ g ( η (1)) αN θ g ( η ( N − N θ g ( η ( N − βN θ g ( η ( x )) g ( η ( x )) Figure 1.
The boundary driven zero-range process.
The paper is organized as follows. In Section 2, we introduce some notationsand define precisely the zero-range process with the boundary dynamics that weare considering. In Section 3, we present the invariant measure and observe thedifferent asymptotic behaviour of the fugacity profile, depending on the value of θ .We also provide the invariant measure for a more general dynamics that allowscreation and annihilation of particles in both sides of I N . In Section 4, we definethe notion of measures associated to a density profile and present the hydrostaticlimit for our model. The small Section 5 is devoted to recall the essential propertyof attractiveness for the zero-range process. In Section 6, we prove tightness for thesequence of probabilities of interest. For that, we introduce the related martingalesthat will be very useful also in the derivation of the hydrodynamic equation. InSection 7, we start the characterization of the limit points by showing concentra-tion on absolutely continuous measures. In Section 8, we present the hydrodynamicequation that we conjecture for this model, together with the necessary steps fora complete proof the of hydrodynamic limit. In Appendix A, we show how to ob-tain the integral form of the hydrodynamic equation from the Dynkin martingalespresented in Section 6. We use some heuristic arguments that can be formalizedthrough some fundamental replacement lemmas, whose proof is postponed to a fu-ture work. Finally, in Appendix B, we present the hydrodynamic equation obtainedif we consider the general model presented in Remark 3.4, in which particles arecreated and annihilated in both sides of I N .2. Definition of the model
Let I N = { , . . . , N − } be the finite lattice where the distinguishable particleswill be moving around, we called it by bulk. For x ∈ I N , the occupation variable η ( x ) stands for the number of particles at site x . The zero-range process is aevolution without restriction on the total number of particles per site, and thereforethe state space for the configurations η is the set Ω N = N I N .The process is defined through a function g : N → R + , with g (0) = 0. Weassume, throughout this work, that g has bounded variation in the following sense: g ∗ = sup k | g ( k + 1) − g ( k ) | < ∞ . (2.1) SUSANA FR ´OMETA, RICARDO MISTURINI, AND ADRIANA NEUMANN
The bulk dynamics can be described as: a particle leaves a site x ∈ { , . . . , N − } with rate 2 g ( η ( x )), and jumps to one of the neighbouring sites ( x − x +1) chosenuniformly. A particle jumps from the sites x = 1 and x = N − I N with rate g ( η ( x )). The boundary dynamics is given by the following birthand death processes at the sites x = 1 and x = N − α and θ , a particle is inserted into the system with rate αN θ at site 1 and removed with rate g ( η ( N − N θ through the site N − We can entirely characterize the continuous time Markov process { η t : t ≥ } by its infinitesimal generator L N given by L N = L N, + L N,b , (2.2)where L N, and L N,b represent the infinitesimal generators of the bulk dynamics andthe boundary dynamics, respectively. The generators act on functions f : Ω N → R as ( L N, f )( η ) = N − X x =1 X y ∈{ x − ,x +1 }∩ I N g ( η ( x )) [ f ( η x,y ) − f ( η )] , (2.3)( L N,b f )( η ) = αN θ [ f ( η ) − f ( η )] + g ( η ( N − N θ [ f ( η ( N − − ) − f ( η )] , (2.4)where η x,y represents the configuration obtained when, in the configuration η , aparticle jumps from site x to y , i.e, η x,y ( z ) = η ( z ) , if z = x, y,η ( z ) − , if z = x,η ( z ) + 1 , if z = y ; (2.5)and η ω ± represents a configuration obtained from η adding or subtracting oneparticle at site ω , that is, η ω ± ( z ) = ( η ( z ) , if z = ω,η ( z ) ± , if z = ω. (2.6) Remark 2.1.
Contrary to the classical zero-range process on the torus, see forexample [22] , the process with these boundary conditions is not reversible, and donot conserve the number of particles. Invariant measure
Since we do not have conservation of particles, the Markov process with gener-ator L N is irreducible in Ω N . If the process is non-explosive and has an invariantdistribution, then the invariant measure is unique and the process is positive recur-rent (see [25, Proposition 3.5.3]). Coupling with a birth and death processes, we cansee that if g is such that P ∞ k =1 1max ≤ i ≤ k g ( i ) = ∞ , then the process is non-explosive.This condition is satisfied, since we are assuming that g has bounded variation, asstated in (2.1).A particular aspect of the zero-range process is that its invariant measure canbe explicitly computed (see [27, 1]). This can also be done in our case, despite ofthe boundary conditions, as already considered in [11, 23]. For the convenience ofthe reader, we will present the calculations in the following. See Remark 3.4 for a more general boundary dynamics.
HE BOUNDARY DRIVEN ZERO-RANGE PROCESS 5
Inspired by the periodic case, we look for an invariant probability ¯ ν N which is aproduct measure on Ω N with marginals given by¯ ν N { η : η ( x ) = k } = 1 Z ( ϕ ( x )) ( ϕ ( x )) k g ( k )! , (3.1)for x ∈ I N . Here g ( k )! stands for Π ≤ j ≤ k g ( j ), and g (0)! = 1, ϕ : I N → R + is afunction to be determined, and Z is the normalizing partition function Z ( ϕ ) = X k ≥ ϕ k g ( k )! . (3.2)Denote by ϕ ∗ the radius o of convergence of the partition function (3.2). Lemma 3.1.
For α , θ and N satisfying α ( N θ − − N θ + 1) < ϕ ∗ , the measure ¯ ν N defined in (3.1) with fugacity profile ϕ ( x ) = ϕ N ( x ) = − αN θ ( x + 1) + αN θ − + α , x ∈ I N , (3.3) is the unique invariant distribution for the Markov process on Ω N with infinitesimalgenerator L N , defined in (2.2) .Proof. Let η ∈ Ω N be an arbitrary configuration. We have to prove that X ˜ η = η ¯ ν N (˜ η )¯ ν N ( η ) R (˜ η, η ) = λ ( η ) , (3.4)where R (˜ η, η ) is the rate at which the process jumps from ˜ η to η and λ ( η ) = g ( η (1)) + 2 N − X x =2 g ( η ( x )) + g ( η ( N − αN θ + g ( η ( N − N θ (3.5)is the rate at which the process jumps from the configuration η . In the left-hand sideof the equation (3.4), there are four types of configurations ˜ η for which R (˜ η, η ) = 0:˜ η = η x,x +1 and ˜ η = η x +1 ,x , for x ∈ { , . . . , N − } , ˜ η = η − and ˜ η = η ( N − .Decomposing the summation in these types of configurations, using the definitionof ¯ ν N in (3.1) and the jump rates in (2.3) and (2.4), we can rewrite the left-handside of (3.4) as N − X x =1 ϕ ( x + 1) ϕ ( x ) g ( η ( x )) + N − X x =1 ϕ ( x ) ϕ ( x + 1) g ( η ( x + 1)) + αg ( η (1)) N θ ϕ (1) + ϕ ( N − N θ . Thus, changing the index in the second sum above, the last expression becomes N − X x =2 ϕ ( x +1)+ ϕ ( x − ϕ ( x ) g ( η ( x ))+ ϕ (2)+ αN θ ϕ (1) g ( η (1))+ ϕ ( N − ϕ ( N − g ( η ( N − ϕ ( N − N θ . (3.6)In order to (3.6) be equal to (3.5) we must require ϕ ( x +1)+ ϕ ( x − ϕ ( x ) = 2, for all x ∈ { , . . . , N − } . To get that, choose ϕ a linear function, let us say ϕ ( x ) = ax + b .The other required conditions: ϕ (2)+ αNθ ϕ (1) = 1, ϕ ( N − ϕ ( N − = 1 + N θ and ϕ ( N −
1) = α ,are satisfied with the choice a = − αN θ and b = αN θ ( N −
1) + α , which leads to (3.3). (cid:3) SUSANA FR ´OMETA, RICARDO MISTURINI, AND ADRIANA NEUMANN α α α θ = 1 θ > Figure 2.
The asymptotic fugacity profile ¯ ϕ : [0 , → R + Remark 3.2.
The condition α ( N θ − − N θ +1) < ϕ ∗ imposed in Lemma 3.1 ensuresthat the fugacity function satisfies ϕ ( x ) < ϕ ∗ for all x ∈ I N . Note that if ϕ ∗ isfinite (which occurs, for instance, when g is bounded), then the probability measure ¯ ν N is not well defined if α is too big. This is quite intuitive, since large α (manyparticles entering the system) and small g (few particles leaving the system) wouldimply transience of the process. A simple computation shows that E ¯ ν N [ g ( η ( x ))] = ϕ N ( x ), for x ∈ I N , where E ¯ ν N denotes expectation with respect to the measure ¯ ν N . That is why ϕ N ( x ) iscalled the fugacity at the site x . Remark 3.3.
We observe that, depending on the value of θ ∈ [0 , ∞ ) , we havedifferent asymptotic behaviours of the fugacity, see Figure 2: • For θ = 1 , for x ∈ I N , ϕ N ( x ) = ¯ ϕ ( x +1 N ) , where the asymptotic fugacityprofile ¯ ϕ : [0 , → R is given by ¯ ϕ ( u ) = α (2 − u ) . • For θ > , ϕ N ( x ) = α + r N ( x ) , where lim N →∞ sup x ∈ I N | r N ( x ) | = 0 . Inthis case, the asymptotic fugacity profile ¯ ϕ is equal to the constant α . • For θ < , we must look at the two different situations: ϕ ∗ < ∞ and ϕ ∗ = ∞ . If ϕ ∗ < ∞ , the partition function will not be defined for largevalues of N . If ϕ ∗ = ∞ , it would make sense to consider N → ∞ , however,we will have ϕ N (1) → ∞ . Thus, as I N is rescaled to the continuum, ϕ N can not be rescaled to a macroscopic profile ¯ ϕ : [0 , → R , as in the previouscases. Remark 3.4.
It is possible also to consider a more general model allowing creationand annihilation of particles at both sides of the boundary (see Figure 3), let us say:at site , particles are inserted into the system with rate αN θ and removed from thesystem with rate γN θ g ( η (1)) ; at site N − , particles are inserted into the system withrate βN θ and removed from the system with rate δN θ g ( η ( N − . Following the linesof Lemma 3.1 we found that the invariant probability is also a product measure withmarginals given by (3.1) for a linear fugacity profile ϕ ( x ) = ϕ N ( x ) = − ( αδ − βγ )( x −
1) + αδ ( N −
2) + ( α + β ) N θ γδ ( N −
2) + ( γ + δ ) N θ , HE BOUNDARY DRIVEN ZERO-RANGE PROCESS 71 2 3 ... x − x x +1 ... N − N − g ( η (1)) γN θ g ( η (1) ) αN θ g ( η ( N − δN θ g ( η ( N − βN θ g ( η ( x )) g ( η ( x )) Figure 3.
The general slow boundary driven zero-range process. for x ∈ I N and α, β, δ, γ, θ ≥ . In the case θ = 0 , this formula coincides with theone presented in [23] . All results obtained in the present work, including the deduc-tion of the hydrodynamic limit (see Appendix B), can be straightforward adaptedto this general case. Nevertheless, in order to avoid too much notation, we choose δ = 1 , γ = β = 0 . Also, since the creation of particles at one side has an analogouseffect to the creation of particles at the other side, and the same holds for annihila-tion, the case studied in the present paper captures the essence of the macroscopiceffect of the boundary dynamics. Moreover, the dynamic presented in this work hasa natural interpretation as a flux of particles from a reservoir at the left-hand sideof the bulk toward the one at the right-hand side. Hydrostatic limit
Definition 4.1.
A sequence { µ N } N ∈ N of probabilities on Ω N is said to be associatedto the profile ρ : [0 , → R + if, for any δ > and any continuous function H : [0 , → R the following limit holds: lim N →∞ µ N h η ∈ Ω N : (cid:12)(cid:12)(cid:12) N N − X x =1 H ( xN ) η ( x ) − Z H ( u ) ρ ( u ) du (cid:12)(cid:12)(cid:12) > δ i = 0 . (4.1)Recall that ϕ ∗ denotes the radius of convergence of the partition function Z ( ϕ ) = P k ≥ ϕ k g ( k )! . The average particle density corresponding to the fugacity ϕ is a func-tion R : [0 , ϕ ∗ ) → R + , given by R ( ϕ ) = 1 Z ( ϕ ) X k ≥ k ϕ k g ( k )! . (4.2)As shown in [22, Section 2.3], R is strictly increasing, and, if we assume thatlim ϕ ↑ ϕ ∗ Z ( ϕ ) = ∞ , then the range of R is all R + , i.e, lim ϕ ↑ ϕ ∗ R ( ϕ ) = ∞ . Therefore, the inverse of R is well defined. Let Φ : R + → [0 , ϕ ∗ ) be the inverse function of R. (4.3) SUSANA FR ´OMETA, RICARDO MISTURINI, AND ADRIANA NEUMANN
Definition 4.2.
For a continuous function ρ : [0 , → R + , denote by ν Nρ ( · ) theproduct measure with slowly varying parameter associated to ρ , this is the productmeasure on Ω N with marginals given by ν Nρ ( · ) { η : η ( x ) = k } = 1 Z (Φ( ρ ( xN ))) Φ( ρ ( xN )) k g ( k )! , for k ≥ . (4.4)From (4.2), we have E ν Nρ · ) [ η ( x )] = ρ ( xN ) , for all x ∈ I N . (4.5)The sequence { ν Nρ ( · ) } N ∈ N is a particular case of a sequence of probabilities asso-ciated to the profile ρ in the sense of Definition 4.1, as stated in Proposition 4.4.To prove this, we begin with the following lemma. Lemma 4.3. If ρ : [0 , → R + is a continuous profile, then for each positiveinteger ℓ , sup N sup x ∈ I N E ν Nρ · ) [( η ( x )) ℓ ] < ∞ . Proof.
First of all, note that E ν Nρ · ) [( η ( x )) ℓ ] = R ℓ (Φ( ρ ( xN ))), where R ℓ is definedfor ϕ ∈ [0 , ϕ ∗ ) as R ℓ ( ϕ ) = Z ( ϕ ) P k ≥ k ℓ ϕ k g ( k )! . The function Φ = R − is strictlyincreasing and lim v →∞ Φ( v ) = ϕ ∗ . Denote by ϕ ∗∗ = sup u ∈ [0 , Φ( ρ ( u )). Since ρ is bounded, we have ϕ ∗∗ < ϕ ∗ . Therefore,sup N sup x ∈ I N E ν Nρ · ) [( η ( x )) ℓ ] = sup N sup x ∈ I N R ℓ (Φ( ρ ( x/N ))) ≤ sup ≤ ϕ ≤ ϕ ∗∗ R ℓ ( ϕ ) . In order to conclude that the last expression above is finite we observe that thefunction R ℓ is analytic on [0 , ϕ ∗ ). To see this, we write R ℓ ( ϕ ) = A ℓ ( ϕ ) Z ( ϕ ) , where A ℓ isdefined inductively by A ( ϕ ) = Z ( ϕ ) and A n ( ϕ ) = ϕA ′ n − ( ϕ ). (cid:3) Proposition 4.4. If ρ : [0 , → R + is continuous, then the product measure ν Nρ ( · ) defined in (4.4) is associated to the profile ρ in the sense of Definition 4.1.Proof. Fix a continuous test function H . Observing that1 N N − X x =1 H ( xN ) ρ ( xN ) → Z H ( u ) ρ ( u ) du, it is enough to show that, for each δ > ν Nρ ( · ) " η : (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N N − X x =1 H ( xN )[ η ( x ) − ρ ( xN )] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) > δ (4.6)goes to zero as N → ∞ . By Chebyshev’s inequality, (4.5) and independence, theexpression in (4.6) is bounded above by1 δ N N − X x =1 H ( xN ) E ν Nρ · ) (cid:2) ( η ( x ) − ρ ( xN )) (cid:3) ≤ δ N N − X x =1 H ( xN ) E ν Nρ · ) (cid:2) η ( x ) (cid:3) . By Lemma 4.3 and since H is bounded, there exists some constant C such that H ( xN ) E ν Nρ · ) (cid:2) η ( x ) (cid:3) < C for every N and x ∈ I N . Therefore, the right-hand sideof the last displayed inequality goes to 0 when N → ∞ . (cid:3) HE BOUNDARY DRIVEN ZERO-RANGE PROCESS 9
Since we have the explicit formula for the fugacity profile of the invariant measure¯ ν N , it is straightforward to obtain, in terms of the function R , an expression for thestationary density profile ¯ ρ : [0 , → R + . Such result is usually called hydrostaticlimit . Recalling Remark 3.3, note that, when θ = 1, the invariant measure ¯ ν N satisfies ¯ ν N { η : η ( x ) = k } = ν N ¯ ρ ( · ) { η : η ( x + 1) = k } , (4.7)where ¯ ρ ( u ) = R ( α (2 − u )). Also, when θ > ϕ N ( x ) − α goes to zero uniformly in x ∈ I N , as N → ∞ . Therefore, the next result is derived following the lines of theproof of Proposition 4.4. Proposition 4.5 (Hydrostatic Limit) . Let ¯ ν N be the invariant measure in Ω N for the Markov process with infinitesimal generator L N . Then the sequence ¯ ν N isassociated to the profile ¯ ρ : [0 , → R + given by ¯ ρ ( u ) = ( R ( α (2 − u )) , if θ = 1 ,R ( α ) , if θ > , (4.8) for all u ∈ [0 , . Notice that the linear fugacity profile does not imply a linear density profile,except in the special case of non-interacting particles where g ( k ) = k .5. Attractiveness
This small section is devoted to recall the essential property of attractiveness forthe zero-range process.Consider in Ω N the partial order: η ≤ ξ if and only if η ( x ) ≤ ξ ( x ) for every x ∈ I N . A function f : Ω N → R is called monotone if f ( η ) ≤ f ( ξ ) for all η ≤ ξ .This partial order extends to measures on Ω N . We say that µ ≤ µ , if Z f dµ ≤ Z f dµ , (5.1)for all monotone functions f : Ω N → R .An interacting particle system { η t } t ≥ is said to be attractive if its semigroup S ( t ), defined by S ( t ) f ( η ) = E η [ f ( η t )], preserves the partial order: µ ≤ µ ⇒ µ S ( t ) ≤ µ S ( t ) , for all t ≥
0. Here E η [ f ( η t )] stands for the expectation of f ( η t ) when the processstarts at η (0) = η .It is well known, see [22, Theorem 2.5.2], that the zero-range process is attractiveif g is non decreasing. 6. Tightness
Let us denote by { η t = η Nt : t ≥ } the continuous-time Markov process onΩ N with generator N L N . Let M + be the space of positive measures on [0 , π N : Ω N → M + the function thatassociates to each configuration η the measure obtained by assigning mass 1 /N toeach particle: π N ( η, du ) = 1 N N − X x =1 η ( x ) δ xN ( du ) , where δ u denotes the Dirac mass at u . Thus, the empirical process π N ( η t ) is aMarkov process in the space M + . By abuse of notation, in this section we willsimply write π Nt instead of π N ( η t ). For a function G : [0 , → R , we denote by h π Nt , G i the integral of G with respect to the measure π Nt : h π Nt , G i = 1 N X x ∈ I N G ( xN ) η t ( x ) . For a measure µ N on Ω N we denote by P µ N the probability on D ([0 , T ] , Ω N ), theSkorohod space of c`adl`ag trajectories, corresponding to the jump process { η t : t ≥ } with generator N L N and initial distribution µ N . Expectations with respect to P µ N will be denoted by E µ N . We denote by Q N the probability on D ([0 , T ] , M + )defined by Q N = P µ N ( π N ) − .In the next proposition we state the tightness of the sequence { Q N } N ≥ underthe hypothesis g ( · ) is non decreasing , (6.1)which implies attractiveness of the process.The conservation of particles is an extensively used property in the proof oftightness for the classical zero-range process in the torus, together with a hypothesisthat controls the relative entropy of the initial distribution µ N with respect to theinvariant measure; see [22, Lemma 5.1.5]. Since we do not have conservation in ourcase, a different approach is necessary. Instead of a relative entropy hypothesis, weassume µ N ≤ ¯ ν N , (6.2)in the sense of (5.1), where ¯ ν N is the invariant measure. Hypothesis (6.2), alongwith attractiveness, provide us a way to control the number of particles in thesystem, as time evolves.As a consequence of [22, Lemma 2.3.5] the limitation (6.2) holds if, for instance, µ N is a product measure of the form (3.1) associated to a fugacity function boundedabove by the fugacity of the stationary measure obtained in (3.3).Tightness of the sequence { Q N } N ≥ is also true if we require that the function g is bounded, instead of the hypothesis (6.1) and (6.2). See Remark 6.4 for moredetails. Proposition 6.1.
Let us consider θ ≥ . Suppose that the rate function g satisfies (6.1) . Assume that the sequence { µ N } N ∈ N is associated to an integrable initialprofile ρ : [0 , → R + , in the sense of (4.1) and satisfies (6.2) . Then the sequenceof measures { Q N } N ≥ is tight. Remark 6.2.
Because of assumption (6.2) , the profile ρ in the above proposi-tion needs to be bounded above by the profile ¯ ρ given in (4.8) . A natural se-quence { µ N } N ∈ N satisfying the hypothesis is the sequence ν Nρ ( · ) of product mea-sures with slowly varying parameter associated to a profile ρ : [0 , → R + , suchthat ρ ( u ) + ε ≤ ¯ ρ ( u ) for all u ∈ [0 , , for some ε > . Proof of Proposition 6.1 will be postponed to Subsection 6.2. We will introducenow the related martingales of the process studied in this work, which will be veryimportant not only in tightness as in the whole proof of hydrodynamic limit as well.
HE BOUNDARY DRIVEN ZERO-RANGE PROCESS 11
Related Martingales.
For G ∈ C [0 , , M Gt , defined as M Gt = (cid:10) π Nt , G (cid:11) − (cid:10) π N , G (cid:11) − Z t N L N (cid:10) π Ns , G (cid:11) ds, (6.3)is a martingale. Recalling the definition of the generator (2.2), we write N L N (cid:10) π Ns , G (cid:11) = 1 N N − X x =2 g ( η s ( x ))∆ N G (cid:0) xN (cid:1) (6.4)+ g ( η s (1)) ∇ + N G (cid:0) N (cid:1) − g ( η s ( N − ∇ − N G (cid:0) N − N (cid:1) + αN θ − G (cid:0) N (cid:1) − g ( η s ( N − N θ − G (cid:0) N − N (cid:1) , where ∆ N G (cid:0) xN (cid:1) = N (cid:2) G (cid:0) x +1 N (cid:1) + G (cid:0) x − N (cid:1) − G (cid:0) xN (cid:1)(cid:3) , ∇ + N G ( xN ) = N (cid:2) G ( x +1 N ) − G ( xN ) (cid:3) , (6.5) ∇ − N G ( xN ) = N (cid:2) G ( xN ) − G ( x − N ) (cid:3) . The quadratic variation of the martingale M Gt is (cid:10) M G (cid:11) t = Z t h N L N (cid:10) π Ns , G (cid:11) − N (cid:10) π Ns , G (cid:11) L N (cid:10) π Ns , G (cid:11)i ds. (6.6)After standard calculations we can see that (cid:10) M G (cid:11) t = R t B N ( s ) ds , where B N ( s ) = N − X x =1 X y ∈{ x − ,x +1 }∩ I N g ( η s ( x ))[ G ( yN ) − G ( xN )] (6.7)+ αN θ G ( N ) + g ( η s ( N − N θ G ( N − N ) . Proof of Tightness.
By [22, Proposition 4.1.7], to prove Proposition 6.1 it issufficient to show the tightness of the measures corresponding to the real processes h π Nt , G i for every G in C ([0 , Condition 1:
For every t ∈ [0 , T ],lim A →∞ lim sup N →∞ P µ N " N N − X x =1 η t ( x ) ≥ A = 0 . Condition 2:
For every δ > γ → lim sup N →∞ sup τ ∈ T Tω ≤ γ P µ N "(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N N − X x =1 G ( xN ) η τ + ω ( x ) − N N − X x =1 G ( xN ) η τ ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) > δ = 0 , where T T is the family of all stopping times bounded by T . Proof of Condition 1.
For η ∈ D ([0 , T ] , Ω N ) define Y t ( η ) = number of particles created up to time t. (6.8)We have the following natural bound N − X x =1 η t ( x ) ≤ N − X x =1 η ( x ) + Y t , (6.9)and then P µ N " N N − X x =1 η t ( x ) ≥ A ≤ P µ N " N N − X x =1 η ( x ) ≥ A + P µ N (cid:20) N Y t ≥ A (cid:21) =: A N + B N . Not that lim A →∞ lim sup N →∞ A N = 0, since µ N is associated to and integrableprofile ρ . On the other hand, since the process is accelerated by N , under P µ N , Y t is a Poisson process with intensity N − θ α , and then B N ≤ AN E µ N [ Y t ] = 2 AN · N − θ αt ≤ αtA , which goes to zero when A → ∞ . Proof of Condition 2.
By (6.3), it is enough to show that
Condition 2.1:
For every δ > γ → lim sup N →∞ sup τ ∈ T Tω ≤ γ P µ N (cid:20)(cid:12)(cid:12)(cid:12)(cid:12)Z τ + ωτ N L N h π Ns , G i ds (cid:12)(cid:12)(cid:12)(cid:12) > δ (cid:21) = 0 . Condition 2.2:
For every δ > γ → lim sup N →∞ sup τ ∈ T Tω ≤ γ P µ N (cid:2)(cid:12)(cid:12) M Gτ + ω − M Gτ (cid:12)(cid:12) > δ (cid:3) = 0 . By (6.4), to show Condition 2.1 it is sufficient to show that, for all δ > γ → lim sup N →∞ sup τ ∈ T Tω ≤ γ P µ N "(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z τ + ωτ N N − X x =2 g ( η s ( x ))∆ N G ( xN ) ds (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) > δ = 0 (6.10)lim γ → lim sup N →∞ sup τ ∈ T Tω ≤ γ P µ N (cid:20)(cid:12)(cid:12)(cid:12)(cid:12)Z τ + ωτ g ( η s (1)) ∇ + N G ( N ) ds (cid:12)(cid:12)(cid:12)(cid:12) > δ (cid:21) = 0 (6.11)lim γ → lim sup N →∞ sup τ ∈ T Tω ≤ γ P µ N (cid:20)(cid:12)(cid:12)(cid:12)(cid:12)Z τ + ωτ g ( η s ( N − ∇ − N G ( N − N ) ds (cid:12)(cid:12)(cid:12)(cid:12) > δ (cid:21) = 0 (6.12)lim γ → lim sup N →∞ sup τ ∈ T Tω ≤ γ P µ N (cid:20)(cid:12)(cid:12)(cid:12)(cid:12)Z τ + ωτ αN θ − G ( N ) ds (cid:12)(cid:12)(cid:12)(cid:12) > δ (cid:21) = 0 (6.13)lim γ → lim sup N →∞ sup τ ∈ T Tω ≤ γ P µ N (cid:20)(cid:12)(cid:12)(cid:12)(cid:12)Z τ + ωτ g ( η s ( N − N θ − G ( N − N ) ds (cid:12)(cid:12)(cid:12)(cid:12) > δ (cid:21) = 0 (6.14)Condition (6.13) is immediate, since G ∈ C ([0 , HE BOUNDARY DRIVEN ZERO-RANGE PROCESS 13
Proof of (6.10) . Since G is of class C and g increases at most linearly (recallhypothesis (2.1)), the integral in (6.10) is bounded by C ( g ∗ , G ) Z τ + ωτ N N − X x =2 η s ( x ) ds. By (6.9), this is bounded above by C ( g ∗ , G ) " ωN N − X x =1 η ( x ) + Z τ + ωτ N Y s ds . Then, observing that Y s is non decreasing, it is enough to show that, for any δ > ω → lim sup N →∞ P µ N " ωN N − X x =1 η ( x ) > δ = 0 (6.15)and lim ω → lim sup N →∞ P µ N h ωN Y T + ω > δ i = 0 . (6.16)As in the proof of Condition 1, (6.15) holds because µ N is associated to an integrableprofile ρ , and (6.16) follows from P µ N h ωN Y T + ω > δ i ≤ ωδN E µ N [ Y T + ω ] = ωα ( T + ω ) δN θ − ≤ ωα ( T + ω ) δ , which goes to zero as ω → (cid:3) For the proof of (6.11), (6.12) and (6.14) we will use the following lemma.
Lemma 6.3.
Under the conditions (6.1) and (6.2) , for every s ≥ and x ∈ I N , itholds E µ N [ g ( η s ( x ))] ≤ ϕ N ( x ) , (6.17) E µ N (cid:2) g ( η s ( x )) (cid:3) ≤ g ∗ ϕ N ( x ) + ( ϕ N ( x )) . (6.18) And consequently, for ℓ = 1 , , E µ N (cid:2) g ( η s ( x )) ℓ (cid:3) ≤ C ( α ) , (6.19) where C ( α ) is a positive constant that only depends on α . Remark 6.4.
In the proof of Proposition 6.1, the hypotheses (6.1) and (6.2) areonly used in Lemma 6.3 above. Since this result is trivial when g is bounded, in thiscase such hypotheses are not needed to prove tightness.Proof. For every x ∈ I N , by (6.1) the function h x : Ω N → R , given by h x ( η ) =[ g ( η ( x ))] ℓ , is monotone. So, by attractiveness and hypothesis (6.2), we have E µ N (cid:2) g ( η s ( x )) ℓ (cid:3) ≤ E ¯ ν N (cid:2) g ( η s ( x )) ℓ (cid:3) = E ¯ ν N (cid:2) g ( η ( x )) ℓ (cid:3) = E ¯ ν N (cid:2) g ( η ( x )) ℓ (cid:3) . To conclude the proof of (6.17), we recall that E ¯ ν N [ g ( η ( x ))] = ϕ N ( x ).For the proof of (6.18), we write E ¯ ν N (cid:2) g ( η ( x )) (cid:3) = 1 Z ( ϕ N ( x )) ∞ X k =0 g ( k ) ϕ N ( x ) k g ( k )!= ϕ N ( x ) Z ( ϕ N ( x )) ∞ X k =1 g ( k ) ϕ N ( x ) k − g ( k − . By (2.1), we have that g ( k ) ≤ g ∗ + g ( k − . Then E ¯ ν N (cid:2) g ( η ( x )) (cid:3) ≤ g ∗ ϕ N ( x ) + ϕ N ( x ) Z ( ϕ N ( x )) ∞ X k =1 g ( k − ϕ N ( x ) k − g ( k − g ∗ ϕ N ( x ) + ϕ N ( x ) Z ( ϕ N ( x )) ∞ X k =2 ϕ N ( x ) k − g ( k − g ∗ ϕ N ( x ) + ϕ N ( x ) . Since ϕ N is a linear function satisfying ϕ N ( N −
1) = α and, for every θ ≥ ϕ N (1) ≤ α , the proof Lemma 6.3 is concluded. (cid:3) Proof of (6.11) , (6.12) and (6.14) . Since G is of class C , the integrals in (6.11),(6.12) and (6.14) are bounded above by C ( g ∗ , G ) Z τ + ωτ g ( η s ( x )) ds, for x = 1 or x = N − x ∈ I N , we have P µ N (cid:20)Z τ + ωτ g ( η s ( x )) ds > δ (cid:21) ≤ δ E µ N (cid:20)Z τ + ωτ g ( η s ( x )) ds (cid:21) . By Cauchy-Schwarz’s inequality E µ N (cid:20)Z τ + ωτ g ( η s ( x )) ds (cid:21) = E µ N "Z T [ τ,τ + ω ] ( s ) g ( η s ( x )) ds ≤ √ ω " E µ N "Z T g ( η s ( x )) ds / = √ ω "Z T E µ N (cid:2) g ( η s ( x )) (cid:3) ds / . Then, using Lemma 6.3, we obtain E µ N (cid:20)Z τ + ωτ g ( η s ( x )) ds (cid:21) ≤ ( ωT C ( α )) / . (6.20)Sending ω →
0, we conclude the proof. (cid:3)
Proof of Condition 2.2.
Using Chebychev’s inequality and the explicit formula forthe quadratic variation given in (6.6), we have P µ N (cid:2)(cid:12)(cid:12) M Gτ + ω − M Gτ (cid:12)(cid:12) > δ (cid:3) ≤ δ E µ N (cid:2) ( M Gτ + ω − M Gτ ) (cid:3) = 1 δ E µ N (cid:20)Z τ + ωτ B N ( s ) ds (cid:21) , (6.21)where B N ( s ) was defined in (6.7).Using that G and its derivative are bounded functions, and then, using (6.20),we can see that (6.21) is bounded above by CωN , where C is a constant that doesnot depends on N and ω . Thus the proof is concluded. (cid:3) HE BOUNDARY DRIVEN ZERO-RANGE PROCESS 15
Remark 6.5.
Considering a model in which a particle is removed from the systemthroughout site N − with rate g ( η ( N − instead of the slow boundary assumption g ( η ( N − N θ made in this work, our proof can be adapted and tightness will also holdif we assume that particles are inserted into the system at site with rate αN θ with θ > . In this case the fugacity profile ϕ N goes to zero uniformly as N → ∞ . Limit points are concentrated on absolutely continuous measures
The next step to characterize the limit points of { Q N } is to show that theyare concentrated on trajectories of measures that are absolutely continuous withrespect to the Lebesgue measure.Next lemma states that for any sequence µ N of probabilities on Ω N bounded bythe invariant measure ¯ ν N , the corresponding sequence of empirical measures, ob-tained via π N : Ω N → M + , if converges, must converge to an absolutely continuousmeasures with respect to Lesbegue. Lemma 7.1.
Let µ N be a sequence of probabilities on Ω N bounded by the invariantmeasure ¯ ν N , i.e., µ N ≤ ¯ ν N . Let R µ N be the probability measure µ N ( π N ) − on M + ,defined by R µ N ( A ) = µ N { η : π N ( η ) ∈ A} for every Borel subset A ∈ M + . Then,all limit points R ∗ of the sequence R µ N are concentrated on absolutely continuousmeasures with respect to the Lebesgue measure: R ∗ [ π : π ( du ) = ρ ( u ) du ] = 1 . Proof.
Let R ∗ be a limit point of the sequence R µ N . Recall from (4.8) that wedenoted by ¯ ρ : [0 , → R + the density profile associated to the sequence of invariantmeasures ¯ ν N . Fix some ε >
0, it is enough to prove that, for every non negativecontinuous function G : [0 , → R , R ∗ (cid:20) π : h π, G i ≤ Z G ( u )(¯ ρ ( u ) + ε ) du (cid:21) = 1 . Let R µ Nk be a subsequence converging to R ∗ , then R ∗ (cid:20) π : h π, G i ≤ Z G ( u )(¯ ρ ( u ) + ε ) du (cid:21) ≥ lim sup k →∞ R µ Nk (cid:20) π : h π, G i ≤ Z G ( u )(¯ ρ ( u ) + ε ) du (cid:21) (7.1)= lim sup k →∞ µ N k (cid:20) η : h π N ( η ) , G i ≤ Z G ( u )(¯ ρ ( u ) + ε ) du (cid:21) . Since µ N ≤ ¯ ν N , by [24, Theorem 2.2.4] there exist a coupling ¯ µ N , i.e, a probabilitymeasure on Ω N × Ω N , with marginals µ N and ¯ ν N respectively, such that¯ µ N [( η, ξ ) : η ≤ ξ ] = 1 , and consequently ¯ µ N (cid:2) ( η, ξ ) : h π N ( η ) , G i ≤ h π N ( ξ ) , G i (cid:3) = 1 . (7.2) By (7.1) and (7.2), R ∗ (cid:20) π : h π, G i ≤ Z G ( u )(¯ ρ ( u ) + ε ) du (cid:21) ≥ lim sup k →∞ ¯ ν N k (cid:20) η : h π N ( η ) , G i ≤ Z G ( u )(¯ ρ ( u ) + ε ) du (cid:21) = 1 , (7.3)by Proposition 4.5. (cid:3) Assuming that the rate function g is non decreasing, by attractiveness, the semi-group S N ( t ) associated to the generator N L N preserves the partial order µ N ≤ ¯ ν N ,that is, µ N S N ( t ) ≤ ¯ ν N S N ( t ) = ¯ ν N for each 0 ≤ t ≤ T . Therefore, Lemma 7.1,when applied to the marginal at time t of the measure Q N = P µ N ( π N ) − , which is µ N S N ( t ), says that for every limit point Q ∗ , and every t ∈ [0 , T ], Q ∗ [ π : π t ( du ) = ρ t ( u ) du ] = 1 . To short notations, we write ρ t ( u ) instead of ρ ( t, u ). Now consider the functional J : M + → R + ∪ {∞} defined by J ( π ) = ( , if π ( du ) = ρ ( u ) du, ∞ , otherwise.By Fubini’s lemma, E Q ∗ "Z T J ( π t ) dt = Z T E Q ∗ [ J ( π t )] dt = T. In particular, changing, if necessary, π t ( du ) in a time set of measure zero, all limitpoints Q ∗ are concentrated on absolutely continuous trajectories: Q ∗ [ π · : π t ( du ) = ρ t ( u ) du, ≤ t ≤ T ] = 1 . Hydrodynamic limit
In this section we will present the hydrodynamic limit that we expect in thismodel, together with the structure of the proof. Since some elements of the proofare not yet completed, we present it as a conjecture.Let us recall the hypotheses assumed in Sections 6 and 7, that is, θ ≥ g is anon decreasing function with bounded variation, as stated in (2.1). Also recall that,for T > P µ N denotes the probability on the space D ([0 , T ] , Ω N ) corresponding tothe process { η t : t ∈ [0 , T ] } on Ω N with infinitesimal generator N L N , where L N is defined in (2.2). Conjecture 8.1 (Hydrodynamic limit) . Let { µ N } N ∈ N be a sequence of probabilitymeasures on Ω N , bounded by the invariant measure, i.e., µ N ≤ ¯ ν N . Assume thatthe sequence { µ N } N ∈ N is associated to a continuous profile γ : [0 , → R + in thesense of the Definition 4.1. Then, for all t ∈ [0 , T ] , for all continuous function G : [0 , → R and δ > , lim N → + ∞ P µ N " η · : (cid:12)(cid:12)(cid:12) N X x ∈ I N G (cid:0) xN (cid:1) η t ( x ) − Z G ( u ) ρ t ( u ) du (cid:12)(cid:12)(cid:12) > δ = 0 , As discussed in Remark 6.2. the assumption (6.2) naturally imposes the initial profile γ tobe bounded above by the profile ¯ ρ of the hydrostatic limit, given in (4.8). HE BOUNDARY DRIVEN ZERO-RANGE PROCESS 17 where • for θ = 1 , ρ t ( u ) is a weak solution of (8.2) with Robin boundary condition( κ = 1 ); • for θ > , ρ t ( u ) is a weak solution of (8.2) with Neumann boundary condi-tion ( κ = 0 ). Before introducing the hydrodynamic equation (8.2), we need to define somefunction spaces. The bracket h · , · i means the inner product in L [0 ,
1] and k F k = h F, F i , for all F ∈ L [0 , ρ t ( u ) and G s ( u ) instead of ρ ( t, u ) and G ( s, u ), respectively. The reader must not misunderstand this notationwith the time derivative, denoted by ∂ s . Definition 8.2.
Let H (0 , be the set of all locally summable functions ξ : [0 , → R such that there exists a function ∂ u ξ ∈ L [0 , satisfying h ∂ u G, ξ i = −h G, ∂ u ξ i , for all C ∞ function G : (0 , → R with compact support.Let L (0 , T ; H (0 , be the set of all measurable functions ¯ ξ : [0 , T ] → L [0 , such that ¯ ξ t ∈ H (0 , , for almost t ∈ [0 , T ] , and k ¯ ξ k L (0 ,T ; H (0 , := Z T {k ¯ ξ t k + k ∂ u ¯ ξ t k } dt < ∞ . (8.1)Denote by C , ([0 , T ] × [0 , , T ] × [0 ,
1] that are differentiable on the first variable and twice differentiable on thesecond variable.Recall that the function Φ : R + → [0 , ϕ ∗ ) is the inverse function of R , defined in(4.2). Definition 8.3 (Hydrodynamic equation) . Let γ : [0 , → R + be a continuousfunction. Consider the parameter κ equal to or . We say that a function ρ :[0 , T ] × [0 , → R + is a weak solution of the equation ∂ t ρ t ( u ) = ∆Φ( ρ t ( u )) , for u ∈ (0 , and t ∈ (0 , T ] ,∂ u Φ( ρ t (0)) = − κ α, for t ∈ (0 , T ] ,∂ u Φ( ρ t (1)) = − κ Φ( ρ t (1)) , for t ∈ (0 , T ] ,ρ ( u ) = γ ( u ) , for u ∈ [0 , , (8.2) if Φ( ρ ) ∈ L (0 , T ; H (0 , and h ρ t , G i − h γ, G i − Z t (cid:8) h ρ s , ∂ s G s i + h Φ( ρ s ) , ∆ G s i (cid:9) ds − Z t (cid:8) Φ( ρ s (0)) ∂ u G s (0) − Φ( ρ s (1)) ∂ u G s (1) (cid:9) ds − κ Z t (cid:8) αG s (0) − Φ( ρ s (1)) G s (1) (cid:9) ds = 0 , (8.3) for all t ∈ [0 , T ] and G ∈ C , ([0 , T ] × [0 , . • When κ = 0 , we say that the PDE (8.2) has Neumann boundary condition; • When κ = 1 , we say that the PDE (8.2) has Robin boundary condition. We consider the PDE (8.2) with more general boundary conditions in the Ap-pendix B, see equation (B.1).Conjecture 8.1 is a consequence of the conjecture we will state below. Letus recall, from the beginning of Section 6, that Q N denotes the probability on D ([0 , T ] , M + ), corresponding to the empirical process { π Nt : t ∈ [0 , T ] } . Conjecture 8.4. As N → ∞ , the sequence of probabilities { Q N } N ∈ N convergesweakly to Q , the probability measure on D ([0 , T ] , M + ) that gives mass one to thetrajectory π t ( du ) = ρ t ( u ) du , where ρ : [0 , T ] × [0 , → R is the weak solution of thehydrodynamic equation (8.2) , with κ = 1 if θ = 1 , and κ = 0 if θ > . We call ρ t ( u ) the hydrodynamic profile. The proof of Conjecture 8.4 may be divided into three steps.The first step is to show tightness, which is done in Section 6. This implies thatthe sequence { Q N } N ∈ N has limit points.The second step is the characterization of these limit points, which we split intwo parts: The first part is the subject of Section 7, where we proved that the limitpoints of the sequence { Q N } are concentrated on trajectories of measures that areabsolutely continuous with respect to the Lebesgue measure, so that for each t , π t ( du ) = ρ t ( u ) du for some function ρ : [0 , T ] × [0 , → R + . The second part isto show that ρ is a solution of the corresponding hydrodynamic equation. Thispart we postpone to a future work, however, in the Appendix A, we present someheuristics of this proof.The third step, which will be also postponed to a future work, is to show theuniqueness of solution for the hydrodynamic equations. This uniqueness wouldguarantee that the sequence { Q N } has a unique limit point, and thus the proof ofConjecture 8.4 would be concluded. Appendix A. Heuristics of the hydrodynamic equation
Let { µ N } N ∈ N be a sequence of probability measures on Ω N , bounded by theinvariant measure, as stated in (6.2), and associated with to a continuous profile γ : [0 , → R + in the sense of Definition 4.1. Recall that { Q N } is a sequence ofprobabilities on D ([0 , T ] , M + ) defined by Q N = P µ N ( π N ) − .Let Q ∗ be a limit point of { Q N } . In Section 7, we proved that Q ∗ is a probabilitymeasure on D ([0 , T ] , M + ) which gives mass one to paths of absolutely continuousmeasures: π t ( du ) = ρ t ( u ) du . In this appendix we present some heuristics to obtainthat ρ t ( u ) is a weak solution of the corresponding hydrodynamic equation. For thispurpose, we will assume, heuristically , that (cid:10) π Ns , H (cid:11) → Z H ( u ) ρ s ( u ) du , (A.1)when N → ∞ , for all s ∈ [0 , T ] and H ∈ C [0 , ρ t ( u ) satisfies the hydrodynamic equation, we evoke theDynkin martingale, introduced in Subsection 6.1, M Gt for G ∈ C ([0 , HE BOUNDARY DRIVEN ZERO-RANGE PROCESS 19 (6.3) and (6.4), we rewrite this martingale as M Gt = (cid:10) π Nt , G (cid:11) − (cid:10) π N , G (cid:11) − Z t N N − X x =2 g ( η s ( x ))∆ N G (cid:0) xN (cid:1) ds − Z t (cid:16) g ( η s (1)) ∇ + N G (cid:0) N (cid:1) − g ( η s ( N − ∇ − N G (cid:0) N − N (cid:1) (cid:17) ds (A.2) − Z t (cid:16) αN θ − G (cid:0) N (cid:1) − g ( η s ( N − N θ − G (cid:0) N − N (cid:1) (cid:17) ds. By the definition of ∆ N G , ∇ + N G and ∇ − N G in (6.5) and the fact that G ∈ C ([0 , M Gt as (cid:10) π Nt , G (cid:11) − (cid:10) π N , G (cid:11) − Z t N N − X x =2 g ( η s ( x ))∆ G (cid:0) xN (cid:1) ds − Z t (cid:16) g ( η s (1)) ∂ u G (0) − g ( η s ( N − ∂ u G (1) (cid:17) ds (A.3) − Z t (cid:16) αN θ − G (0) − g ( η s ( N − N θ − G (1) (cid:17) ds + R ,θN ( G, t ) , where E µ N [ R ,θN ( G, t )] goes to zero, as N → ∞ , for all θ ≥
1, and uniformly on t ∈ [0 , T ], because of Lemma 6.3 and Taylor’s expansion. By (A.1), as N → ∞ , (cid:10) π Nt , G (cid:11) − (cid:10) π N , G (cid:11) → h ρ t , G i − h ρ , G i . Then we need to study the bulk and boundary terms of the expression A.3. Westart by the bulk term: R t N P N − x =2 g ( η s ( x ))∆ G (cid:0) xN (cid:1) ds . In order to do this, wewill introduce some notation.For ε >
0, consider the set I εN := { εN, . . . , N − − εN } . (A.4)Above and in all text εN must be understood as ⌊ εN ⌋ .Then the bulk term becomes Z t N X x ∈ I εN g ( η s ( x ))∆ G (cid:0) xN (cid:1) ds + R N,ε ( G, t ) , (A.5)where R N,ε ( G, t ) = Z t N εN X x =2 g ( η s ( x ))∆ G (cid:0) xN (cid:1) + N − X x = N − εN g ( η s ( x ))∆ G (cid:0) xN (cid:1)! ds . Using (6.19) from Lemma 6.3, we have E µ N [ R N,ε ( G, t )] ≤ εCT k ∆ G k ∞ . Thus, E µ N [ R N,ε ( G, t )] goes to zero, when N → ∞ and ε →
0. Adding and subtractingsuitable terms, we can see that the integral in (A.5) is equal to Z t N X x ∈ I εN ∆ G (cid:0) xN (cid:1) εN x + εN X y = x +1 g ( η s ( y )) ds + R N,ε ( G, t ) , (A.6)where R N,ε ( G, t ) = Z t N X x ∈ I εN n g ( η s ( x )) − εN x + εN X y = x +1 g ( η s ( y )) o ∆ G (cid:0) xN (cid:1) ds . Changing variables R N,ε ( G, t ) can be rewritten as Z t N ( X x ∈ I εN g ( η s ( x ))∆ G (cid:0) xN (cid:1) − X y ∈ I N g ( η s ( y )) 1 εN εN X z =1 ∆ G (cid:0) y − zN (cid:1) ) ds = Z t N X x ∈ I εN g ( η s ( x )) n ∆ G (cid:0) xN (cid:1) − εN εN X z =1 ∆ G (cid:0) x − zN (cid:1) o ds + Z t N X x ∈ I N \ I εN g ( η s ( x )) 1 εN εN X z =1 ∆ G (cid:0) x − zN (cid:1) ds . Then E µ N [ R N,ε ( G, t )] →
0, as N → ∞ and ε →
0. To handle the integral termin (A.6), we use the function Φ : R + → [0 , ϕ ∗ ), which is the inverse function of R ,defined in (4.2). We will need to introduce some more notation. Let −→ η εNs ( x ) bethe empirical density in the box of size εN , which is given on, x ∈ I εN , by −→ η εNs ( x ) = 1 εN x + εN X y = x +1 η s ( y ) . (A.7)Then the integral term in (A.6) is equal to Z t N X x ∈ I εN ∆ G (cid:0) xN (cid:1) Φ( −→ η εNs ( x )) ds + R N,ε ( G, t ) . (A.8)The last term above is the important expression: R N,ε ( G, t ) = Z t N X x ∈ I εN ∆ G (cid:0) xN (cid:1) n εN x + εN X y = x +1 g ( η s ( y )) − Φ( −→ η εNs ( x )) o ds, (A.9)which to prove that it is negligible we need to evoke the following very importantresult: Lemma A.1 (Replacement lemma for the bulk) . For every δ > , lim ε → lim N →∞ P µ N h η · : (cid:12)(cid:12)(cid:12) R N,ε ( G, T ) (cid:12)(cid:12)(cid:12) > δ i = 0 , where R N,ε ( G, t ) was defined in (A.9) . Note that, −→ η εNs ( x ) = h π Ns , ι x/Nε i , where ι uε ( v ) = ε ( u,u + ε ) ( v ) , for u, v ∈ [0 , Z t N X x ∈ I εN ∆ G (cid:0) xN (cid:1) Φ (cid:0) h π Ns , ι x/Nε i (cid:1) ds . Since the function inside the summation above is integrable, it is possible to provethat the last integral is asymptotically (when N → ∞ and ε →
0) equal to Z t Z ∆ G ( u ) Φ (cid:0) h π Ns , ι uε i (cid:1) du ds . The proof of this lemma is a future work.
HE BOUNDARY DRIVEN ZERO-RANGE PROCESS 21
By (A.1), we have that, for u ∈ [0 , h π Ns , ι uε i → Z ρ s ( v ) ι uε ( v ) dv, as N → ∞ . Finally, taking ε →
0, the last integral converges to ρ s ( u ). Then thebulk term of the expression A.3 converges to Z t Z ∆ G ( u )Φ( ρ s ( u )) du ds = Z t h Φ( ρ s ) , ∆ G i ds, when N → ∞ and ε → P µ N of Z t (cid:16) αN θ − G (0) − g ( η s ( N − N θ − G (1) (cid:17) ds, goes to zero, as N → ∞ , in the case θ >
1, because of Lemma 6.3. Then, when θ >
1, we only need to analyze the term − Z t (cid:16) g ( η s (1)) ∂ u G (0) − g ( η s ( N − ∂ u G (1) (cid:17) ds. (A.10)In the case θ = 1, rewriting the boundary terms of (A.3), we have − Z t (cid:16) g ( η s (1)) ∂ u G (0) − αG (0) − g ( η s ( N − ∂ u G (1) + G (1)) (cid:17) ds. (A.11)In both cases we need to replace g ( η s (1)) and g ( η s ( N − g in a box of size εN in a neighborhood of x = 1 or x = N − I N , thatis εN P εNy =2 g ( η s ( y )) and εN P N − y = N − − εN g ( η s ( y )), respectively. Note that thisis similar to what we did above in (A.6). The next step is to use the followingreplacement lemma with a suitable choice of f and f . Lemma A.2 (Replacement lemma for the boundary) . For θ ≥ and for allcontinuous functions f i : [0 , T ] → R , with i = 1 , , and every δ > , we have lim ε → lim N →∞ P µ N h η · : (cid:12)(cid:12)(cid:12) R bN,ε ( f , f , T ) (cid:12)(cid:12)(cid:12) > δ i = 0 , where R bN,ε ( f , f , T ) = Z T f ( s ) n εN εN X y =2 g ( η s ( y )) − Φ( −→ η εNs (1)) o ds + Z T f ( s ) n εN N − X y = N − − εN g ( η s ( y )) − Φ( ←− η εNs ( N − o ds where −→ η εNs (1) was defined in (A.7) and ←− η εNs ( N −
1) = 1 εN N − X y = N − − εN η s ( y ) . The proof of this lemma is a future work.
As we said above −→ η εNs (1) ∼ ρ s (0) and ←− η εNs ( N − ∼ ρ s (1), then the expressionsin (A.10) and in (A.11) converge, as N → ∞ and ε →
0, to − Z t (cid:16) Φ( ρ s (0)) ∂ u G (0) − Φ( ρ s (1)) ∂ u G (1) (cid:17) ds, in case θ >
1, and − Z t (cid:16) Φ( ρ s (0)) ∂ u G (0) − Φ( ρ s (1)) ( ∂ u G (1) + G (1)) − αG (0) (cid:17) ds, in case θ = 1. These expressions are the boundary terms in the integral equations(8.3), with κ = 0 and κ = 1, respectively.Summarizing, the expression of the Dynkin martingale, in (A.2), converges tothe left-hand side of the integral equation (8.3) with κ = 0 and κ = 1, for θ > θ = 1, respectively. Since we are only providing an idea of the proof, to makeclear the notation, up to this point we have been assuming that the test G doesnot depends on the time, that is G ∈ C [0 , P µ N [ | M Gt | > δ ] vanishes as N → ∞ . Recall that Q ∗ is a limit point of the sequence { Q N } , which is defined by Q N = P µ N ( π N ) − .Then, using Portmanteau Theorem, we can conclude that, in the case θ = 1, Q ∗ satisfies Q ∗ " π · : h ρ t , G i − h γ, G i − Z t (cid:8) h ρ s , ∂ s G s i + h Φ( ρ s ) , ∆ G s i (cid:9) ds − Z t (cid:8) Φ( ρ s (0)) ∂ u G s (0) − Φ( ρ s (1)) ∂ u G s (1) (cid:9) ds − Z t (cid:8) αG s (0) − Φ( ρ s (1)) G s (1) (cid:9) ds = 0 , ∀ t ∈ [0 , T ] , ∀ G ∈ C , ([0 , T ] × [0 , = 1 . Note that the expression inside the probability above is the integral equation (8.3)with κ = 1. In the case θ >
1, using the same argument, we obtain a similarexpression as the one above with the integral equation (8.3) with κ = 0 instead of κ = 1. Remark A.3.
In order to show that the boundary terms Φ( ρ s (0)) and Φ( ρ s (1)) inthe integral equation (8.3) are well defined we need to assure that Φ( ρ ) belongs to L (0 , T ; H (0 , . To obtain it, using Riesz representation theorem, it is enough toprove the Energy Estimate, that is: E Q ∗ " sup H n Z T h Φ( ρ s ) , ∂ u H s i ds − c Z T h H s , H s i ds o ≤ M < ∞ , for some constants M and c . The notation E Q ∗ means the expectation with respectto the measure Q ∗ , which is the limit point of Q N . HE BOUNDARY DRIVEN ZERO-RANGE PROCESS 23
Appendix B. Heuristics for hydrodynamics of the general model
If we had considered the more general slow boundary introduced in Remark 3.4,see Figure 3, the Dynkin martingale (A.2) would be M Gt = (cid:10) π Nt , G (cid:11) − (cid:10) π N , G (cid:11) − Z t N N − X x =2 g ( η s ( x ))∆ N G (cid:0) xN (cid:1) ds − Z t (cid:16) g ( η s (1)) ∇ + N G (cid:0) N (cid:1) − g ( η s ( N − ∇ − N G (cid:0) N − N (cid:1) (cid:17) ds − Z t ((cid:16) α − γg ( η s (1)) N θ − (cid:17) G (cid:0) N (cid:1) + (cid:16) β − δg ( η s ( N − N θ − (cid:17) G (cid:0) N − N (cid:1) ) ds, for G ∈ C [0 , Q ∗ satisfies Q ∗ " π · : h ρ t , G i − h γ, G i − Z t (cid:8) h ρ s , ∂ s G s i + h Φ( ρ s ) , ∆ G s i (cid:9) ds − Z t (cid:8) Φ( ρ s (0)) ∂ u G s (0) − Φ( ρ s (1)) ∂ u G s (1) (cid:9) ds − κ Z t n(cid:0) α − γ Φ( ρ s (0)) (cid:1) G s (0) + (cid:0) β − δ Φ( ρ s (1)) (cid:1) G s (1) o ds = 0 , ∀ t ∈ [0 , T ] , ∀ G ∈ C , ([0 , T ] × [0 , = 1 . Above, κ = 1 in the case θ = 1 and κ = 0 in the case θ >
1. Therefore, thehydrodynamic equation is ∂ t ρ t ( u ) = ∆Φ( ρ t ( u )) , for u ∈ (0 ,
1) and t ∈ (0 , T ] ,∂ u Φ( ρ t (0)) = − κ (cid:0) α − γ Φ( ρ s (0)) (cid:1) , for t ∈ (0 , T ] ,∂ u Φ( ρ t (1)) = κ (cid:0) β − δ Φ( ρ s (1)) (cid:1) , for t ∈ (0 , T ] ,ρ ( u ) = γ ( u ) , for u ∈ [0 , . (B.1) Acknowledgements
A.N. was supported through a grant “L’OR´EAL - ABC - UNESCO Para Mul-heres na Ciˆencia”.
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