Hydrodynamic limit in a particle system with topological interactions

We study a system of particles in the interval $[0,\epsilon^{-1}] \cap \mathbb Z$, $\epsilon^{-1}$ a positive integer. The particles move as symmetric independent random walks (with reflections at the endpoints); simultaneously new particles are injected at site 0 at rate $j\epsilon$ ($j>0$) and removed at same rate from the rightmost occupied site. The removal mechanism is therefore of topological rather than metric nature. The determination of the rightmost occupied site requires a knowledge of the entire configuration and prevents from using correlation functions techniques. We prove using stochastic inequalities that the system has a hydrodynamic limit, namely that under suitable assumptions on the initial configurations, the law of the density fields $\epsilon \sum \phi(\epsilon x) \xi_{\epsilon^{-2}t}(x)$ ($\phi$ a test function, $\xi_t(x)$ the number of particles at site $x$ at time $t$) concentrates in the limit $\epsilon\to 0$ on the deterministic value $\int \phi \rho_t$, $\rho_t$ interpreted as the limit density at time $t$. We characterize the limit $\rho_t$ as a weak solution in terms of barriers of a limit free boundary problem.


Introduction and model definition
This paper is inspired by the analysis in [12] and we are indebted to Pablo Ferrari for discussions and in particular for suggesting the inequalities in Section 6. This is a first in a series of three papers where we study a particle system whose hydrodynamic limit is described by a free boundary problem.
Our system is made of particles confined to the lattice [0, −1 ] ∩ Z, for brevity in the sequel we shall just write [0, −1 ]. In this notation −1 is a positive integer denoting the system size and we will be eventually interested in the asymptotics as → 0. The evolution is a Markov process {ξ t , t ≥ 0} on the space of particles configurations ξ = (ξ(x)) x∈[0, −1 ] , the component ξ(x) ∈ N is interpreted as the number of particles at site x. The generator is denoted by (the dependence on is not made explicit). L 0 is the generator of the independent random walks process, it is defined on functions f by where ξ x,y denotes the configuration obtained from ξ by removing one particle from site x and putting it at site y, i.e.
Namely L 0 describes independent symmetric random walks which jump with equal probability after an exponential time of mean 1 to the nearest neighbor sites, the jumps leading outside [0, −1 ] being suppressed (reflecting boundary conditions).
The term L b in (1.1) is It describes the action of throwing into the system new particles at rate j, j > 0, which then land at site 0; instead L a removes particles: namely a particle is taken out from the edge R ξ of the configuration ξ defined as R ξ is such that: ξ(y) > 0 for y = R ξ ξ(y) = 0 for y > R ξ (1.6) L a f (ξ) = 0 if R ξ does not exist, i.e. if ξ ≡ 0. We interpret L as the generator of a system of independent walkers with "current reservoirs" which impose a positive current j at site 0 and at the edge of the configuration. See [9,10] for a comparison with the density reservoirs used in the analysis of the Fourier law.
Here is a list of the main issues which are studied in this and in the other papers in this series.
• The interaction described by L a is highly non local as R ξ depends on the positions of all the particles. This spoils any attempt to use the BBGKY hierarchy of equations for the correlation functions, as customary in perturbations of the independent system, see for instance [8].
• The L a interaction is "topological rather than metric", as the influence on a particle i of a particle j only depends on whether j is to the right or left of i and not on their distance. Topological interactions appear often in natural sciences as in population dynamics, in particular the motion of crowds of people [6], or of animals [1]. Our result shows that there are natural examples in physical systems as well. The relative simplicity of our model allows a rigorous analysis of such an interaction.
• To the left of R ξ the particles do not feel the L a interaction and move freely, but R ξ depends on the configuration of particles and hence on time as well. Ours therefore is a microscopic model for a free boundary problem and one may thus guess that the hydrodynamic limit is also ruled by a free boundary problem. In such a case the hydrodynamic equations would be the linear heat equation in an open, time dependent space interval with suitable boundary conditions complemented by a law for the speed of the right boundary.
• The action of L b and L a is to add from the left and respectively remove from the right particles at rate j. They act therefore as "current reservoirs" [11,9,10] because they are imposing a current j (recall that for density reservoirs [7,4] the particles current scales by ). Supposing the validity of Fick's law the stationary macroscopic profiles are then linear functions with slope −2j: there are therefore infinitely many such profiles (as here the boundary densities are not fixed). Two scenarios are then possible: either there is a preferential profile or there is a second time scale beyond the hydrodynamical one, where we see that such profiles are not stationary.
We shall give answers to most of the above issues, our main results being stated in the next section. We then say that u ∈ L ∞ ([0, 1], R + ) has "an edge" R(u) if R(u) = inf{r : F (r; u) = 0} < 1 (2.2)

Main results
The definition extends naturally to Borel positive measures µ on [0, 1].
Remark. For some results we will need extra assumptions, namely that ρ init ∈ C([0, 1], R + ) and/or that it has an "edge".
We shall next discuss in which way particle systems and evolution of macroscopic profiles are related.

Hydrodynamic limit.
Particle configurations ξ are elements of N [0, −1 ] which may be regarded as positive measures µ ξ on the real interval [0, −1 ] by setting where D x , the Dirac delta at x, is the probability measure supported by the point x. Analogously to (2.1) we set and, as for the macroscopic profiles, we say that ξ has an edge R ξ if R ξ = inf{x : F (x; ξ) = 0} < −1 (2.4) which means that R ξ < −1 is the largest integer x such that ξ(x) > 0, in agreement with (1.6). To compare macroscopic profiles and particles configurations we shall use the functions F (x; ξ) and F (r; u). We define in particular the local averages: A (x, ξ) := 1 F (x; ξ) − F (x + − 1; ξ) = 1 with a positive integer and x ∈ [0, −1 − + 1]. The corresponding quantity for macroscopic profiles u ∈ L ∞ ([0, 1], R + ) is Definition 2.2 (Assumptions on the initial particle configuration). We fix b < 1 suitably close to 1 and a > 0 suitably small, for the sake of definiteness we set b = 9/10 and a = 1/20. We then denote by the integer part of −b and suppose that for any the initial configuration ξ verifies max and moreover that if ρ init has an edge R(ρ init ), see (2.2), then with R ξ as in (2.4). We shall denote by P ( ) ξ the law of the process {ξ t , t ≥ 0} in the interval [0, −1 ] with generator L given in (1.1) and started at time 0 from a configuration ξ as above.
Thus the initial configuration ξ converges to ρ init as → 0 in the sense of (2.7). Our first result proves that the convergence extends to all positive times (but in a weaker sense).
Theorem 2.1 states the existence and some regularity properties of the hydrodynamic limit, but does not say about its qualitative features: in particular which equation is satisfied by the limit and which equation rules the motion of the edge, if it exists. The continuum analogue of our particle evolution is where the first term (on the right hand side) corresponds to the random walk evolution, jD 0 to the addition of particles at the origin and jD Rt to the removal of the rightmost particles.
In [2] a suitable notion of quasi-solutions for (2.11) in R + is given and it is proved that the limit of such quasi-solutions coincide with the hydrodynamic limits found in Theorem 2.1. The main ingredient in the proof is established here and it is based on the notion of upper and lower barriers. These are "approximate solutions" of (2.11) which bound from below and from above the hydrodynamic limit ρ(r, t), the inequalities being in the sense of mass transport. This is defined as follows: two positive Borel measures µ and ν on [0, 1] are ordered with We shall apply the notion to measures in U defined as follows: The set U and the partial order). U is the set of all positive Borel measures u on [0, 1] which have the form u = c u D 0 + ρ u (r)dr, c u ≥ 0, ρ u ∈ L ∞ ([0, 1], R + ). By an abuse of notation we shall also write the elements of U as u = c u D 0 + ρ u . For any u, v ∈ U we then set is the total variation of the measure u − v.
Definition 2.4. (The cut and paste operator). We define for any δ > 0 the subset U δ ⊂ U as (2.14) and the cut-and-paste operator In the following definition of barriers we use the Green function G neum δ (r, r ) (for the heat equation in [0, 1] with Neumann boundary conditions): r k being the images of r under repeated reflections of the interval [0, 1] to its right and left (see for instance [14] pag. 97 for details).
We denote by Definition 2.5 (Barriers). Let u ∈ L ∞ ([0, 1], R + ) be such that F (0; u) > 0. Then for all δ small enough u ∈ U δ and for such δ we define the "barriers" S (δ,±) nδ (u) ∈ U δ , n ∈ N, as follows: we set S (δ,±) 0 (u) = u, and, for n ≥ 1, The families {S The functions S (δ,±) nδ are obtained by alternating the map G neum δ (i.e. a diffusion) and the cut and paste map K (δ) (which takes out a mass jδ from the right and put it back at the origin, the macroscopic counterpart of L b and L a ). It can be easily seen that unlike the original process ξ t the evolutions S (δ,±) nδ conserve the total mass, that S (δ,+) nδ has a singular component (jδD 0 ) plus a L ∞ component (which is C ∞ inside its support).
The name "upper and lower barriers" is justified by the following theorem: (u) for all δ, δ , t such that u ∈ U δ ∩ U δ and t = kδ = k δ , with k, k ∈ N (2.18) where the inequality is in the sense of Definition 2.3.
It thus looks natural to look for elements which separate the barriers: Definition 2.6 (Separating elements). For a given non negative u ∈ L ∞ , the function u = u(r, t), r ∈ [0, 1], t ≥ 0, is below the upper barriers {S It is above the lower barriers {S

Super-hydrodynamic limit and further results.
In [3] we shall study the stationary solutions of (2.11), they are linear functions with slope −2j. We shall prove that any weak solution (in the sense of barriers) converges as t → ∞ to a linear profile, the one with the same total mass as the initial state. We shall also prove that at super-hydrodynamic times, i.e. times of order −3 the particle processes is "close" to the manifold of linear profiles performing a brownian motion on such a set.
We conclude the list of results in this paper by a last theorem where we identify the limit equation for ρ(·, t) when ρ init (·) has no edge: Theorem 2.5 (Hydrodynamic limit in the absence of an edge). Let ρ init such that F (r; ρ init ) ≥ α(1 − r), α > 0, then there exists T > 0 such that ρ(1, t) > 0 for t ∈ [0, T ] and ρ(r, t) is given by

Strategy of proof.
The key observation is that if we anticipate/posticipate the addition and removal of the particles which occur in the true process in a given time interval then we stochastically increase/decrease the final configuration (in the sense of mass transport to the right, i.e. the microscopic version of (2.12)).
To implement this we introduce the processes ξ (δ,±) k −2 δ , k ∈ N. If for the true process the number of added and removed particles in the time interval [k −2 δ, (k + 1) −2 δ] is equal to N k;± then ξ (δ,−) (k+1) −2 δ is obtained from ξ (δ,−) k −2 δ by letting it evolve with generator L 0 and at the end adding N k;+ particles at 0 and then removing the rightmost N k;− particles. In a similar fashion ξ (δ,+) (k+1) −2 δ is obtained by reversing the order of the operations: first the addition/removal and then after the free evolution. We then have for all δ > 0 and all k ∈ N ξ (δ,−) (see Section 6 for details, in particular the definition of microscopic notion of partial order). The probabilistic part of the paper is essentially concentrated in the analysis of the hydrodynamic limit of the process ξ (δ,±) k −2 δ : in Section 4 we prove that it converges to S (δ,±) kδ (u) (if the initial ξ "approximates" u) where convergence is in the sense of (2.10). This is important because it implies that the inequalities are preserved in the limit.
The hydrodynamic limit for the independent random walks process is easy and well known in the literature, but in our case there is an extra difficulty related to a macroscopic occupation at the origin, ξ(0) ≈ −1 , due to the cut and paste operations. This severely limits the choice of the parameters (b close to 1, a close to 0 which in normal situations have a much larger range of values) but luckily some room is left. Instead the convergence of the microscopic cut and paste to its macroscopic counterpart is easy, as the variables N k;± are modulo negligible deviations independent Poisson variables with mean j −1 δ.
Once we have convergence to S (δ,±) kδ (u) we are left with the analytic problem of studying the limits of the latter as δ → 0. We first prove some regularity properties uniform in δ, see Section 7, and then complete the proof of all theorems.

Sections content.
In Section 3 we introduce the δ-approximate processes {ξ (δ,±) t } and prove that the law of the total particles number process |ξ t | is a symmetric random walk on N with reflection at the origin (a result which follows directly from the definition of the process ξ t ). We then state some consequences of such a result which will be used in the sequel.
In Section 4 we prove that if the initial configuration ξ approximates a profile u ∈ U then ξ (δ,±) −2 kδ converges in law to S (δ,±) kδ (u) as → 0. The proof exploits duality for the independent process but is not a consequence of well known results on the hydrodynamic limit for independent particles because we need to take into account the case when there is a macroscopic occupation number at the origin. As a consequence the bounds are not as strong as those which appear in the literature.
In Section 5 we introduce a probability space (Ω, P ) where we can realize simultaneously all the processes ξ t and ξ (δ,±) −2 kδ for all . In Section 6 we relate the true process ξ −2 kδ and the auxiliary ones ξ (δ,±) −2 kδ by stochastic inequalities, in the sense of mass transport theory, exploiting the realization of the process of Section 5. By using the convergence proved in Section 4 the inequalities extend to flows S (δ,±) kδ , thus proving Theorem 2.2.
In Section 7 we prove regularity properties of the flows S (δ,±) kδ which are uniform in δ. In Section 8 we prove we first prove existence and uniqueness of the separating element of barriers (Theorem 2.3) and then deduce our main results (Theorems 2.1 and 2.4). We conclude by giving the proof of Theorem 2.5.
3 The δ-approximate particle processes In this Section we define the stochastic processes ξ (δ,±) k −2 δ , k ∈ N which are analogous to the barriers S (δ,±) kδ of Definition 2.5. As we shall explain below, these processes are defined in such a way that the number of added and removed particles in the time interval [k −2 δ, (k+1) −2 δ], denoted by N k;± , are the same as those in the true process {ξ t }.
The variables N k;± , k ∈ N are determined by the increments of process |ξ t | yielding the particles' number at time t. This last process, despite the complexity of the full process ξ t , is very simple: Theorem 3.1 (Distribution of the particles' number). |ξ t | has the law of a random walk (n t ) t≥0 on N which jumps with equal probability by ±1 after an exponential time of parameter 2j , the jumps leading to −1 being suppressed.
Proof. For any bounded function f on N we have which coincides with the action of the generator of the random walk (n t ) t≥0 on the function f (n). This proves that the law of |ξ t | is the same as that of the random walk.
Thus in the ξ (δ,±) t processes births and deaths are concentrated at the times k −2 δ, in between such times the particles are independent random walks. While the analysis of the true process (ξ t ) t≥0 is rather complex due to the non local nature of L a , the study of the hydrodynamical limit for ξ (δ,±) t is much simpler because the number of rightmost particles to delete is macroscopic and becomes deterministic, the analysis will be carried out in the next section.
We shall often use in the sequel the following explicit realization of the random walk process (n t ) t≥0 .
are infinite sequences of increasing positive "times" t h;0 and of symmetric "jumps", σ h;0 = ±1. (Ω 0 , P 0 ) is the product of a Poisson process of intensity 2j for the increments of the time sequence t 0 and of a Bernoulli process with parameter 1/2 for the jump sequence σ 0 .
Given n 0 ∈ N and ω 0 ∈ Ω 0 we define (n t ) t≥0 , iteratively: we set n t = n t h;0 in the time It is readily seen that the law of (n t ) t≥0 as a process on (Ω 0 , P 0 ) (for a given initial value n 0 ) is the same as the Markov process of Theorem 3.1 and hence of the particles' number |ξ t | in our original process once n 0 = |ξ 0 |.
Under the assumptions on the initial datum ξ, see Definition 2.2, the process of adding and removing particles becomes quite simple. For any integer k > 0 define on Ω 0 B 0 k and A 0 k are independent Poisson distributed variables with average −1 jδ.

Definition 3.3 (Good sets)
. Given T > 0 and γ > 0 we define for any δ and positive Theorem 3.2 (Reduction to Poisson variables). Given ξ as in Definition 2.2, T > 0 and γ > 0 there is δ * > 0 so that for any δ < δ * and any > 0 small enough the following holds. For any ω 0 ∈ G (see (3.6)) and any k such that kδ ≤ T , where N k,± (ω 0 , |ξ|)) denote the variables N k,± when realized on Ω 0 . Finally, for any n there is c n so that Proof. By Definition 2.2 the initial number of particles |ξ| is bounded from below by We choose δ * := C/(2j) and shall prove by induction that for any δ < δ * and all small enough we have in G Suppose that the inequality holds for k and let us prove it for k + 1.
which is strictly positive for any k ≤ T /δ if is small enough. Thus (3.7) holds and This proves the induction hypothesis and for what seen in the proof, (3.7) holds as well.
Once restricted to G the processes ξ (δ,±) t , 0 ≤ t ≤ −2 T , become quite simple. The particles move as independent random walks in the finitely many intervals [k −2 δ, (k + 1) −2 δ], while births and deaths at the times k −2 δ are "essentially deterministic" like in the δ-approximated evolutions S (δ,±) t of Definition 2.5. Such considerations are made precise in Section 4 where we prove convergence of ξ (δ,±) t to S (δ,±) t (ρ init ) in the hydrodynamic limit.

Hydrodynamic limit for the approximating processes
The main result in this section is in Theorem 4.1 below. It states that the δ-approximate processes ξ (δ,±) t of Definition 3.1 converge in the hydrodynamic limit to the evolutions S (δ,±) t (·) of Definition 2.5.
Here we exploit duality to prove convergence in a very strong form of the independent system to the heat equation.
For any fixed δ and T > 0, the processes ξ (δ,±) t , t ≤ −2 T are obtained by alternating independent random walk evolutions to cut and paste operations. The latter involve macroscopic quantities and can be controlled by means of Theorem 3.2 once we have the hydrodynamic limit for the independent process. This is well studied and very detailed estimates are available but in the present case we have the extra difficulty that the initial configurations may have a macroscopic occupation number at the origin ξ(0) ≈ −1 . This is because in the the cut and paste we actually paste ≈ jδ −1 particles at the origin. This is not a case studied in the literature (as far as we know) and indeed it affects greatly the decay of correlations in the hydrodynamic limit.
As in our iterative procedure we have initial data with macroscopic occupation at the origin, we may as well take more general initial conditions (than those in Definition 2.2) with macroscopic occupation at the origin, this will be actually useful in the sequel. Thus the "macroscopic initial profile v 0 " is here taken in U, namely it is the sum of a non negative L ∞ function plus cD 0 , with c either equal to 0 or to jδ, we suppose that v 0 = F (0; v 0 ) > 0. Analogously to (2.7) for any > 0 we choose the initial configuration ξ 0 so that Theorem 4.1. Given any T > 0 for any δ > 0 small enough, any k : kδ ≤ T and any ζ > 0 where v 0 and ξ 0 are as above; P ( ) ξ 0 as in Definition 2.2; F and F as in (2.1).
The theorem is proved at the end of the section, as we shall see stronger results actually hold but what stated is what needed for Theorem 2.1. In the course of the proof we shall introduce several positive parameters: b, a, a * , γ: b should be close to 1 and the others close to 0, for the sake of definiteness we take: We prove the theorem only for the process ξ is similar and omitted. The first step is a spatial discretization of the flow S (δ,−) kδ : , the transition probability of a continuous time, simple symmetric random walk with reflections at 0 and −1 (i.e. the random walker jumps by ±1 with equal probability after an exponential time of mean 1, the jumps which would lead outside [0, −1 ] are suppressed). For δ small enough we define functions u k (x), x ∈ [0, −1 ] ∩ Z, with the property that mass is conserved: The definition is iterative, we set u 0 (x) := v 0 ( x); then supposing that u k−1 has been already defined and that F (0; u k−1 ) = F (0; u 0 ) we define u k as follows. We first call u k is then obtained from u 0 k by adding particles at 0 and removing particles on the right. To make this precise let R k be an integer such that where 1 0 is the Krönecker delta at 0. To complete the induction we observe that In the next proposition we show that in (4.2) we can replace S (δ,−) kδ (v 0 ) by the sequence u k with a negligible error: In the same context as in Theorem 4.1, Proof. In this proof we shorthand by g(r, r ) the Green function G neum δ (r, r ), r, r ∈ [0, 1], defined in (2.16) and also write for brevity p(x, y) := p 0 −2 δ (x, y), as in Definition 4.1. Let u k , u 0 k and R k be as in Definition 4.1. We define for any real r between 0 and −1 , Analogously to (1.6) we denote by R k the real number in [0, −1 ] such that ψ k (r) > 0 for r < R k and ψ k (r) = 0 for r > R k . We also call Claim. There are strictly positive constants C ± which depend on δ so that for all k, The proof of the claim follows from classical estimates on random walks and Green functions: The crucial step in the proof of the proposition is the following statement: We prove (4.9) by induction. We thus suppose that it holds for k − 1. Calling R * k the largest integer smaller or equal than R k and R k We use the local central limit theorem to bound: and get using the induction assumption As a consequence: , thus completing the proof of (4.9). Using (4.11) we then conclude the proof of the proposition, details are omitted.
The proof of Theorem 4.1 is thus reduced to showing that: for all n so that nδ ≤ T , which will be done in the sequel. Both sequences {ξ (δ,−) n −2 δ } and {u n } are determined by alternating free evolution and a cut and paste procedure. We first study the free evolution part proving that the independent random walk configuration ξ 0 −2 δ is well approximated by its average. Call P ξ and E ξ law and expectation of the independent process starting from ξ,  ξ(x) ≤ −a * (4.14) Then for any ξ ∈ X c * ,a * max (c 2 as in (4.8)). Moreover let c * , a * and b be strictly positive and such that (a condition which is satisfied by the choice (4.3)). Let be the integer part of −b and A be as in (2.5), then for any integer n there is c n so that Finally there is a constant c so that Proof. For brevity in this proof we shall write w(x) instead of w(x|ξ). Recalling that p(x, y) is defined in (4.4) and bounded in (4.8), we have for any ξ ∈ X c * ,a * hence (4.15). The proof of (4.17) and (4.18) uses in a crucial way duality: Duality. Given ξ ∈ N [0,N ] and a labeled configuration are called Poisson polynomials. We then have: where x 0 t is the independent random walks evolution. • Proof of (4.17). Call x = (x 1 , .., x 2k ) with x i = x for all i = 1, .., 2k. Then by (4.21) and (4.19) Moreover by the Chebishev inequality and (4.22) which proves (4.17) because |ξ 0 −2 δ | = |ξ| ≤ −1 c * . To prove (4.18) we shall use again duality but also several maybe non totally straightforward algebraic manipulations. We start by expanding the product in the expectation: Call B (i) , i = 1, 2, 3, 4, the set of x ∈ B such that there are i mutually distinct sites. We then have for i ≤ 2: as the expectation of products of ξ 0 −2 δ (·) is bounded, which is proved using (4.23). We are thus left with the sum over x ∈ B and using duality: Suppose there is a singleton h, namely such that y h = y j for all j = h, then Indeed let σ a sequence with σ h = 1 and σ the one obtained from σ by changing only σ h , then We have thus proved that calling X n.s. the set of all y with no singletons then A similar property holds also when x ∈ B (3, * ) which is the set of all x such that x 1 = x 2 , x 3 = x 4 , x 1 and x 4 = x 1 (modulo permutation of labels all x ∈ B (3) are in B (3, * ) ). We write Then analogously to (4.27) but with x ∈ B (3, * ) , with Φ 4 (x) as in (4.30). Going back to (4.25), using (4.26) and (4.15) Let us bound one by one the functions Φ i starting from Φ 4 . Recalling (4.30) the condition y ∈ X n.s. is realized (modulo label permutations) in only two cases: (i) y 1 = y 2 = y 3 = y 4 ; so that from (4.8) and since ξ ∈ X c * ,a * The condition (y 2 , y 3 , y 4 ) ∈ X n.s. in Φ 3 implies y 2 = y 3 = y 4 and for such a y: so that from (4.8) and since ξ ∈ X c * ,a * Finally if (y 3 , y 4 ) ∈ X n.s. then y 3 = y 4 and for such a y, Thus (4.18) follows from (4.33) together with the above inequalities.
The cut and paste sequence of operations which appear in the definition of {ξ (δ,−) t k , k ≤ k * }, k * the largest integer such that δk * ≤ T , t k = k −2 δ, is independent of the motion of the particles so that we have a rather explicit expression for the law of the variables {ξ (δ,−) t k , k ≤ k * }, see (4.42) below. We first write (with ξ 0 below the initial condition in Theorem 4.1) where N k,± are defined in (3.3) and (3.2), their law depends only on |ξ 0 |. We also write π(ξ |ξ) = P ξ ξ 0 −2 δ = ξ , |ξ| = |ξ |, a.s. (4.40) (ξ 0 t the independent random walk process). We finally denote by K (n − ,n + ) ξ the configuration obtained from ξ by adding n + particles at 0 and then removing the n − rightmost particles (the definition requires that |ξ| + n + − n − ≥ 0, condition automatically satisfied below as the variables n ± are the increments of the particles' number n t ). Then, writing with n ± 0 := 0, we have By (3.8) for any n there is c n so that The strategy now is to fix {n ± k , k = 1, .., k * } ∈ G and prove estimates uniform in the choice of {n ± k , k = 1, .., k * }, as the contribution to (4.2) of the complement of G has negligible probability. We have Recalling (4.41) for notation and that w is defined in (4.13), having fixed {n ± k , k = 1, .., k * } ∈ G, see (4.43), with n ± 0 ≡ 0, we call Then by Proposition 4.3 and (4.43) after using Chebishev with the fourth power, The proof of (4.12) continues by showing that in the set G ∩ C, ξ k (as defined in (4.41)) is "close" to u k (as in Definition 4.1). More precisely call X k and R k the integers such that k ) (see again Definition 4.1 for notation). Then the analogue of (4.9) holds: Proof. By (4.45) Supposing for instance that R k−1 ≤ X k−1 we get We decompose the interval [1, R k−1 − 1] into consecutive intervals [z i , z i ] of length with the last interval which may have length < and get using (4.8) We also have By (4.8) and (4.15) and decomposing as before the interval [R k−1 + 1, X k−1 ] into consecutive intervals of length , By collecting the above bounds and using the induction hypothesis: For small enough C a ≤ 1, hence, recalling (4.7), which proves (4.47) with β = C −1 − (5 + C + ).
Proof of Theorem 4.1. We need to prove (4.12). By (4.46) we can reduce to configurations in G ∩ C and want to prove that in such a set |F (x; ξ k ) − F (x; u k )| = 0 (4.52) Let us suppose for the sake of definiteness that R k ≤ X k . Then for CallingR k ≤ R k the largest integer so thatR k − x is a multiple integer of , we get from (4.47): CallX k the smallest integer ≥ X k such thatX k − R k is a multiple integer of , then Analogous bounds hold for x > R k and (4.52) then follows.  N (x), so that x i = −1 for i > N (x) and x i ≥ 0 for i ≤ N (x). We also define M (x) as the largest integer n such that

Realization of the process
Viceversa, given any ξ ∈ X we define x ξ by labeling the particles of ξ consecutively starting from the right. Finally, given a sequence y with finitely many entries in [0, −1 + 1], say y i 1 ...y i k , its re-ordering is the sequence x where x 1 is the largest element in y i 1 ...y i k , x 2 the second largest and so on; x n = −1 for n ≥ k + 1.
We shall be exploiting the fact that the physically relevant quantities are the unlabeled configurations and we are therefore free to label the particles as we like.
Definition 5.2 (The probability space (Ω, P )). We set ..) are infinite sequences of increasing positive "times" t k;i and σ i = (σ 1;i , σ 2;i , ...) infinite sequences of symmetric "jumps", σ k;i = ±1. For i ≥ 1 P i is the product probability law of a Poisson process of intensity 1 for the time sequences t i and of a Bernoulli process with parameter 1/2 for the jump sequences σ i . (Ω 0 , P 0 ) is the probability space introduced in Definition 3.2.
Graphical representation. For each label i ≥ 0 we draw a vertical time axis R + (called the i-th time axis) and on each of them we put "marks" (with values ±) as described below. For any element ω i ∈ Ω i , i ≥ 1, we draw on the i-th time axis a sequence of arrows, at heights t k;i pointing to right or left if σ k;i = ±1 respectively (the σ k;i are called marks). The marks on the 0-time axis are specified by ω 0 : they are + or − crosses which are put at the times t k;0 with ± being the value of σ k;0 . To each arrow we associate a displacement operator and to each cross a creation or annihilation operator. Roughly speaking an arrow on the i-th axis indicates the displacement at that time of the i-th particle, provided it is in [0, −1 ] before and after the displacement (otherwise the displacement is canceled). The creation operator moves a particle from −1 to 0, while the annihilation operator takes to −1 + 1 the rightmost particle in [0, −1 ] (if such a particle exists, otherwise the operation aborts). The precise definitions are given below: Definition 5.3. Creation, annihilation and displacement operators on X ord , denoted respectively by a ± 0 and a ± i , i ≥ 1.
x is the re-ordering (see Definition 5.1) of y where y j = x j for j = i and • a + 0 x =: y + is defined as follows: y + j = x j for j = k ≡ N (x) + 1 and y k = 0, (see Definition 5.1). Thus N (a + 0 x) = N (x) + 1.
• a − 0 x =: y − is defined as follows: The enlarged space has been introduced to make simpler the proof of the inequalities of the next section, but in the end what is relevant is the restriction x ∩ [0, −1 ] of the configuration to the physical space. To this end we shall use the following lemma:  Definition 5.4. Fix t > 0. Then with P probability 1 t 0 ∩ [0, t] has finitely many elements which are all mutually distinct. We define C t (ω 0 ) = card t k,0 ∈ t 0 : t k,0 ≤ t, σ k;0 = + (5.5) and given x ∈ X ord let n ≥ C t (ω 0 ) + N (x). Thus it is well defined (with P probability 1) the sequence t = (t 1 , .., t k ), 0 ≤ t j < t j+1 ≤ t of all times t k;i ∈ [0, t], k ≥ 1, i = 0, .., n. We call i j , j = 1, .., k, the label of the time axis to which t j belongs and σ j the corresponding ± mark.
Definition 5.5 (The time flows). T 0 t (x, ω) and T t (x, ω), t > 0, x ∈ X ord and ω ∈ Ω, are defined (P almost surely) as follows. Let t be as in the previous definition, then using the same notation, To define T (δ,±) N δ −2 (x, ω), N a positive integer, we split t (defined as in Definition 5.4 with t → N δ −2 ) in N groups: t (1) , .., t (N ) where t (h) = t ∩ [(h − 1) −2 δ, h −2 δ] (with P probability 1 we may suppose that all such times are mutually distinct). We then set We finally define T In other words in T (δ,+) N δ −2 the creation-annihilation operators of the N -th group occur all at time (N − 1)δ −2 , while in T (δ,−) N δ −2 they occur at time N δ −2 , thus the above rule for defining T (δ,±) t (x, ω) means that we drop all the operators which appear at times larger than t.
It is easy to see that the marginal over unlabeled configurations of each one of the processes ω)} has the law respectively of the free process ξ 0 t , the interacting process ξ t and the auxiliary processes ξ (δ,±) t . It also follows from (5.4) that

Mass transport inequalities
In this section we introduce a partial order among measures based on moving mass to the right, we are evidently in the context of mass transport theory from where we are borrowing the notions used in this section. We work first in the space of particle configurations ξ regarding ξ as a distribution of masses and then in the space U, considering u ∈ U as a mass density (which may have a Dirac delta at 0), the notions are the same except for a change of language.
The main goal is to prove inequalities between ξ t and the auxiliary processes ξ (δ,±) t (recall that the hydrodynamic limit of the latter is known since Section 4) and then derive analogous inequalities for S (δ,±) t (u) and their limit as δ → 0. We tacitly suppose in the sequel that the configurations ξ are in X as specified in the beginning of Section 5. Definition 6.1 (Partial order). For any ξ, ξ ∈ X , we say that ξ ≤ ξ iff Observe that ξ ≤ ξ has not the usual meaning, i.e. ξ(x) ≤ ξ (x) for all x ! The notion of order has rather to be interpreted in the sense of "the interfaces" F (x; ξ) = y≥x ξ(y), see Definition 2.3 and Figure 2 for a visual illustration. One can easily check that the above "≤" relation has indeed all the properties of a partial order. Same considerations apply to the case of continuous mass distributions as in (2.12) where the notion is well known and much used in mass transport theory.
The equivalence with the previous statement about moving mass to the right is established next. We first introduce a partial order in X ord by saying that x ≤ x iff x i ≤ x i for all i. Since there is a one-to-one correspondence between X (see Definition 6.1) and X ord this defines a priori a new order in X , but the two orders are the same as proved in the following Proposition.
Suppose (2) holds, then hence (2) ⇒ (1). Suppose (1) holds and let x = (x 1 , .., x m ) and x = (x 1 , .., x n ). Then n ≥ m because otherwise F (0; ξ) > F (0; ξ ). We also have that x i ≤ x i for i ≤ m: suppose by contradiction that x k > x k then F (x k ; ξ) ≥ k while F (x k ; ξ ) < k, hence the contradiction. Thus (1) ⇒ (2). Let x = (x 1 , .., x m ) and x = (x 1 , .., x n ) be sequences with values in [0, −1 + 1] such that n ≥ m and with a one to one map i j as in the text of the proposition. Then   hence ξ x ≤ ξ x . To prove the converse statement we suppose that x = (x 1 , .., x m ) and x = (x 1 , .., x n ) are such that ξ := ξ x ≤ ξ := ξ x . Then y := x ξ ≤ y = x ξ , and there are one to one maps j : {1, .., m} onto itself and j : {1, .., n} onto itself so that y j = x j and x h = y h . Then x j ≤ x i j with i j = j .
As a corollary we have Proof. The inequality x ≤ a ± 0 x holds trivially because a ± 0 x does not decrease the entries of Let next y = a ± i x and y = a ± i x with i ≥ 1 and for the sake of definiteness let us just consider the + case. y = x if i ≤ M (x) and i > N (x). In the former case x i = −1 + 1 is also unchanged, in the latter x i = −1 and again the inequality holds trivially. Let us then suppose that M (x) < i ≤ N (x) and suppose that this holds as well for x (otherwise x i = −1 + 1). Then min{x i + 1, −1 } ≤ min{x i + 1, −1 } hence the desired inequality applying the last statement in Proposition 6.1.
As already mentioned we ultimately need inequalities for the restrictions x ∩ [0, −1 ] of the configurations to the physical space. We shall use the following simple observation: Definition 6.2 (Stochastic order). A process (ξ t ) t≥0 is stochastically smaller than a process (ξ t ) t≥0 , writing in short ξ t ≤ ξ t (stochastically), if they can be both realized on a same space where the inequality holds pointwise almost surely.
We shall prove stochastic order by realizing the processes on the same space (Ω, P ) of Definition 5.2.
The first inequality in (6.4) proves that all the maps a ± i preserve order and since all the flows have been defined in terms of products of such maps: Theorem 6.4 (Stochastic inequalities). All the maps T (δ,±) m −2 δ (·, ω), T 0 t (·, ω) and T t (·, ω), preserve order.
To compare the flows T (δ,±) t and T t we shall use the following lemma: Proof. Let σ 0 = +. Call y = a + 0 x, then by the second inequality in (6.4), x ≤ y. Since a σ i i preserves order: a σ i i x ≤ a σ i i y and since N (y) = N (x) + 1 we have (6.6) (having used the third inequality in (6.4)). Let σ 0 = −. Call y = a − 0 x, then by the second inequality in (6.4), x ≤ y. Since a σ i i preserves order: a σ i i x ≤ a σ i i y and since M (y) = M (x) + 1 we have again (6.6) (having used the fourth inequality in (6.4)).
We then say that an exchange at (h, h + 1) is "allowed" if i h = 0 and i h+1 > 0.
Then if π is a permutation obtained by applying repeatedly allowed exchanges starting from Call {(a j , σ j )} the sequence associated to T m −2 δ (x, ω) Also the sequence {(a j , σ j )} associated to T m −2 δ (x, ω) is obtained by repeated allowed exchanges from {(a j , σ j )}, hence The sequence {(a j , σ j )} associated to T (δ ,+) m −2 δ (x, ω) is obtained by repeated allowed exchanges from {(a j , σ j )}, hence We have thus proved: Proof. We have already proved the inequality for the configurations on [−1, −1 + 1], thus the proof of (6.8) follows from (6.5) and (5.9).
The theorem has its continuum analogue which can be proved directly, see Section 4 of [3], but it can also be deduced from Theorem 6.7, as we shall see. Proof. (6.9) follows from (6.8) and (4.2). Proof that K (δ) u ≤ K (δ) v, u, v ∈ U δ . We have The property that G neum t * preserves the order is inherited from the same property for the independent flow T 0 t . As a consequence of the two previous statements we have that also S (δ,±) t preserves the order (see the definition in (2.14)).

Regularity properties of the barriers
In this section we shall prove some regularity properties of the barriers S By the smoothness of G neum t (r, r ), t > 0, it is easy to prove that for any n > 0, S nδ (u) is equal to jδD 0 plus a function which is C ∞ in the interior of its support. Such a smoothness however, being inherited from G neum δ , depends on δ, while we want properties which hold uniformly as δ → 0.
The properties of the Green functions that we use in this section are: and then bounding |r −r|>X Such bounds are verified also by the Green function for the Neumann problem in [0, ] for any > 0 and = ∞ as well, so that the analysis in this section extends to all such cases. Observe that if is finite and positive the bound on the derivative is much better: • There is a constant c so that for any δ > 0: F (0; u) > jδ S (δ,+) t (u) ∞ ≤ c j + u ∞ for all t ∈ δN, t ≤ 1 j + F (0; u) for all t ∈ δN, t > 1 (7.4)
• Proof of (7.4). Let t = nδ, n a positive integer, then The inequality is because we are not taking into account the "loss part" in the action of K (δ) . Iterating we get for s = mδ, m < n a non negative integer, Let n δ be the smallest integer such that δn δ ≥ 1 and suppose that in (7.7) t < δn δ and s = 0. By (7.2) the integral in (7.7) is bounded by u ∞ whereas by (7.1) the sum is bounded by c j √ nδ ≤ c j. Thus (7.4) is proved for t ≤ 1. Let us next take t = δn δ and s = 0 in (7.7). Then using (7.1) we bound the integral in (7.7) by c F (0; u)(δn δ ) −1/2 ≤ c F (0; u). As before the last term in (7.7) is bounded by c j √ δn δ ≤ 2c j so that (we may suppose c < 2c ) n δ δ (u) ∞ ≤ 2c (F (0; u) + j) By the same argument for any integer k ≥ 1 the last equality because we have already proved that mass is conserved. Thus (7.4) is proved for t ∈ (δn δ )N. Let now m = kn δ and kn δ < n ≤ (k + 1)n δ k a positive integer. The last term in (7.7) is bounded again by 2c j, whereas the integral is smaller than S (δ,+) kn δ δ (u) ∞ . Thus (7.4) follows from (7.8) when t ≥ 1.
We next prove the analogue of (7. which has the same structure as (7.7). The analysis after (7.7) extends to the present case and yields the proof of (7.4) for ρ The proof of (7.5) and (7.6) will be given after the following lemma.
so that v (δ,+) s,t ∞ ≤ cj √ t − s and the second inequality in (7.14) is proved. To prove the first one we use (7.17), (7.12) and (7.2) to write where R is such that Since d ζ = cζ 3 (see the proof of space continuity) for a small enough the above integral is < ζ as well.
By (6.9) we then have for all r ∈ [0, 1] and all δ and t in {k2 −n τ , k ∈ N + , n ∈ N}, F (r; ψ(·, t)) ≥ F (r; S (δ,−) t (u)), F (r; ψ(·, t)) ≤ F (r; S (δ,+) t (u)) (8.5) (8.5) does not yet prove that ψ separates the barriers because we have to consider all t and δ and not only those above. To this end we observe that the function ψ(r, t) that we have defined so far actually depends on the initial choice of τ , to make this explicit we write ψ τ (r, t).