1 Introduction

A free boundary problem in its simplest version is given by the linear heat equation in a domain \(\varOmega \) which itself changes in time with a law which depends on the same solution. In the Stefan problem for instance the heat equation is complemented by Dirichlet boundary conditions while the local velocity of the points of the boundary are specified in terms of the local gradient of the solution.

The purpose of this paper is to study a particle version of such free boundary problems. The linear heat equation is in our case replaced by the one dimensional symmetric simple exclusion process, SSEP, in the set of configurations having a rightmost particle and a leftmost hole. Furthermore, at a given rate the rightmost particle dies and at the same rate, independently, a birth of a particle occurs at the position of the leftmost hole. Since the leftmost hole and the rightmost particle move, we call them free boundaries.

More precisely, Call \(\eta \in \{0,1\}^{\mathbb {Z}}\) a particle configuration, think of \(\eta \) as the subset of \({\mathbb {Z}}\) occupied by particles and consider those \(\eta \) having a rightmost particle located at \(\textsc {r}(\eta ):=\max (\eta )\) and a leftmost hole located at \(\textsc {l}(\eta ):=\min ({\mathbb {Z}}{\setminus }\eta )\). Let \((\eta _t)\) be the Markov process performing symmetric simple exclusion process at rate \(\frac{1}{2}\) and such that the rightmost particle and the leftmost hole are killed at rate \(\textsc {j}\):

$$\begin{aligned} \eta \rightarrow \eta {\setminus }\{\textsc {r}(\eta )\} \text { and } \eta \rightarrow \eta \cup \{\textsc {l}(\eta )\} \text { at rate }\textsc {j}\text { each}. \end{aligned}$$

Since particles are injected to the left and extracted from the right, \(\textsc {j}\) can be seen as the average current of particles through the system. It is well known that under a diffusive space and time scaling the collective behavior of the SSEP is ruled by the linear heat equation [4]. We perform the same scaling with a parameter \(\varepsilon \) such that time is \(\varepsilon ^{-2}t\), space \(\varepsilon ^{-1}r\) and the killing is \(\textsc {j}=\textsc {j}(\varepsilon ) = \varepsilon j\), where \(j\) is the macroscopic current.

Consider a function \(\rho :{\mathbb {R}}\rightarrow [0,1]\) identically zero to the right of \(\textsc {r}(\rho ):=\sup \{r:\rho (r)>0\}<\infty \) and identically one to the left of \(\textsc {l}(\rho ):=\inf \{r:\rho (r)<1\}>-\infty \) and continuous in \((\textsc {l}(\rho ),\textsc {r}(\rho ))\). Call \({\mathcal {R}}\) the set of functions with those properties. We consider a macroscopic density \(\rho \in {\mathcal {R}}\) and ask the initial configuration \(\eta ^{(\varepsilon )}\) indexed by \(\varepsilon \) to approach the density \(\rho \) as follows:

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0} \sup _{a\le b}\left| \varepsilon \sum _{\varepsilon x\in [a,b]}\eta ^{(\varepsilon )}(x) - \int _{a}^{b} \rho (r)\,dr\right| = 0 \end{aligned}$$
(1.1)

Our main result is

Theorem 1

Let \((\eta ^{(\varepsilon )}_t)\) be the process with killing at rate \(j\varepsilon \) and with initial configuration \(\eta ^{(\varepsilon )}\) satisfying (1.1). Then for \(t\ge 0\) there exists a function \(\rho _t\in {\mathcal {R}}\) such that \(\rho _0=\rho \) and

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0} P\left( \sup _{a\le b}\left| \varepsilon \sum _{\varepsilon x\in [a,b]}\eta ^{(\varepsilon )}_{t\varepsilon ^{-2}}(x) - \int _{a}^{b} \rho _t(r)\,dr\right| >\gamma \right) =0\qquad \text {for all }\gamma >0.\qquad \end{aligned}$$
(1.2)

We characterize the limit \(\rho _t\) in terms of “lower and upper barriers”, the inequalities being in the sense of mass transport. The notion is defined by introducing first a map from density functions \(\rho \in {\mathcal {R}}\) to functions \(\phi (r|\rho ),\,r\in {\mathbb {R}}\), that we call “interfaces” and then by saying that \(\rho \le \rho '\) if \(\phi (\cdot |\rho )\le \phi (\cdot |\rho ')\). See Sect. 4 where the notion is first defined at the particles level and then for densities in \({\mathcal {R}}\).

The barriers are defined by functions \(\rho ^{\delta ,\pm }_t,\,\delta >0\), which satisfy “discretized free boundary problems”. More specifically \(\rho ^{\delta ,-}_t\) evolves according to the heat equation in the intervals \([n\delta ,(n+1)\delta )\) and at times \(n\delta \) takes the rightmost portion \(j\delta \) of mass from the right and puts it on the left. Namely, define the \(\delta \)-quantiles \(\textsc {r}^\delta (\rho )\) and \(\textsc {l}^\delta (\rho )\) by

$$\begin{aligned} \int \limits _{\textsc {r}^\delta (\rho )}^{\infty }\rho (r)\,dr =\delta , \qquad \int \limits ^{\textsc {l}^\delta (\rho )}_{-\infty }(1-\rho (r))\,dr =\delta , \end{aligned}$$
(1.3)

Define also

$$\begin{aligned} (\Gamma ^{\delta }\rho )(r):= {\left\{ \begin{array}{ll} 1&{}\text {if } r\le \textsc {l}^{\delta }(\rho )\\ \rho (r) &{}\text {if } \textsc {l}^{\delta }(\rho )< r< \textsc {r}^{\delta }(\rho )\\ 0&{}\text {if } r\ge \textsc {r}^{\delta }(\rho ), \end{array}\right. } \end{aligned}$$
(1.4)

and let \(G_t(r,r')\) be the Gaussian kernel (see (5.11)). Set \(\rho ^{\delta ,-}_0=\rho \) and iteratively

$$\begin{aligned} \rho ^{\delta ,-}_t:= {\left\{ \begin{array}{ll} G_{t-n\delta }\rho ^{\delta ,-}_{n\delta },&{}\text {if } t\in [n\delta ,(n+1)\delta ),\quad n=0,1,\dots \\ \Gamma ^{j\delta } \rho ^{\delta ,-}_{n\delta -},&{}\text {if } t= n\delta ,\quad n=1,2,\dots . \end{array}\right. } \end{aligned}$$
(1.5)

which is well defined for \(\delta \) small enough. Define \(\rho ^{\delta ,+}_t\) with the same evolution but with initial profile \(\rho ^{\delta ,+}_0=\Gamma ^{j\delta }\rho \). We prove that

$$\begin{aligned} \rho ^{\delta ,-}_{t} \le \rho _{t} \le \rho ^{\delta ,+}_t \end{aligned}$$

for any \(\delta \) and any \(t\in \delta {\mathbb { N}}\). We also prove that any function \(\tilde{\rho }_t\) which satisfies the above inequality (for all \(\delta \) and \(t\) as above) must necessarily be equal to \(\rho _t\) (uniqueness of separating elements). The precise statement is in Theorem 5. In particular this allows to show that:

Theorem 2

Let \(\rho \in {\mathcal {R}}\) and \(\rho _t\) be the evolution of Theorem 1 with initial datum \(\rho \). Let \(\rho ^{\delta ,-}_t\) be the evolution (1.5) with the same initial datum. Then for any \(a<b\) real numbers and for any \(\delta >0\),

$$\begin{aligned} \left| \int _a^b\rho ^{\delta ,-}_t(r)\,dr -\int _a^b\rho _t(r)\,dr\right| \le 2 j\delta ,\quad \forall t\ge 0 \end{aligned}$$
(1.6)

Theorems 1 and 2 are proved at the end of Sect. 6.1.

A formal limit of our particle system leads to conjecture that \(\rho _t\) solves

$$\begin{aligned}&\frac{\partial \rho }{\partial t}= \frac{1}{2} \frac{\partial ^2 \rho }{\partial r^{2}}, \quad r\in (\textsc {l}_t,\textsc {r}_t),\\&\textsc {r}_0, \textsc {l}_0, \rho (r,0) \;\text {given} \nonumber \\&\rho (\textsc {l}_t ,t)=1,\;\;\rho (\textsc {r}_t ,t)=0;\quad \frac{\partial \rho }{\partial r}(\textsc {l}_t ,t)= \frac{\partial \rho }{\partial r} (\textsc {r}_t ,t)=-2j \nonumber \end{aligned}$$
(1.7)

where \(\textsc {r}_t:=\textsc {r}(\rho (\cdot ,t))\) and \(\textsc {l}_t:=\textsc {l}(\rho (\cdot ,t))\). Indeed the density flux \(J(r)\) associated to the equation \(\frac{\partial \rho }{\partial t}= \frac{1}{2} \frac{\partial ^{2} \rho }{\partial r^{2}}\) is equal to \(J(r):=-\frac{1}{2} \frac{\partial \rho }{\partial r}\) so that the last two conditions in (1.7) just state that the outgoing flux at \(\textsc {r}_t\) is equal to the killing rate \(j\) and that the incoming flux at \(\textsc {l}_t\) is equal to the birth rate \(j\).

The free boundary problem (1.7) is not of Stefan type. In fact in (1.7) we impose both Dirichlet and Neumann conditions as we prescribe the values of the function and of its derivative at the boundaries, while in the classical Stefan problem the Dirichlet boundary conditions are complemented by assigning the speed of the boundary (in terms of the derivative of the solution at the boundaries). We can obviously recover the velocity of the boundaries from a smooth solution \(\rho (r,t)\) of (1.7) by differentiating the identities \(\rho (\textsc {l}_t ,t)=1,\,\rho (\textsc {r}_t ,t)=0\), thus obtaining:

$$\begin{aligned} \frac{d \textsc {l}_t }{dt} = \frac{1}{2j} \frac{\partial ^2 \rho }{\partial r^2}(\textsc {l}_t ,t), \;\; \frac{d \textsc {r}_t }{dt} = \frac{1}{2j} \frac{\partial ^2 \rho }{\partial r^2}(\textsc {r}_t ,t) \end{aligned}$$
(1.8)

The traditional way to study the hydrodynamic limit of particle systems is to prove that the limit law of the system is supported by weak solutions of a PDE for which existence and uniqueness of weak solutions holds true. In our case this approach is problematic. While we know closeness to the heat equation away from the boundaries, we do not control the motion of the boundaries: we only know that they do not escape to infinity. We are not aware of existence and uniqueness theorems for (1.7), however in general in free boundary problems some assumptions of regularity on the motion of the boundaries is required, for instance Lipschitz continuity.

Our proof of hydrodynamic limit avoids this pattern as we prove directly existence of the limit by squeezing the particles density between lower and upper barriers which have a unique separating element. This suggests a variational approach to the analysis of (1.7) based on a proof that its classical solutions are also squeezed by the barriers, or more generally that any limit of “approximate solutions” of (1.7) lies in between the barriers. Such an approach has been carried in [1] for a simpler version of (1.7) where particles are in the semi-infinite line with reflections at the origin, so that there is a single free boundary \(\textsc {r}_t\) where particles are killed at rate \(j\), while births occur at the origin, at same rate \(j\). An extension to our case is in preparation.

A similar equation has been derived in [9] for a different interface process. In that case an existence and uniqueness theorem for the limit equation is proved to hold (see also reference therein). Regularity of the free boundary motions in [9] follows from some monotonicity properties intrinsic to the model and absent in our case.

Free boundary problems have also been derived in [7] for particles evolutions via branching and in [10] for a variant of the simple exclusion process.

We have a proof that the limit evolution \(\rho _t\) in Theorem 1 satisfies (1.7) in the very special case when \(\rho _t\) is a time-independent linear profile with slope \(-2j\). Stationary solutions for free boundary problems have also been studied in [5] for non local evolutions in systems which undergo a phase transition.

The model we study in this paper is inspired by previous works on Fourier law with current reservoirs [36]; its actual formulation came out from discussions with Stefano Olla to whom we are indebted. We are also indebted to F. Comets and H. Lacoin for helpful comments and discussions and to a referee of PTRF for useful comments.

In Sect. 2 we define the particle process and in Sect. 3 we prove the existence of a unique invariant measure for the process as seen from the median.

Due to the non local nature of the birth-death process the usual techniques for hydrodynamic limit fail. To overcome this problem we use inequalities based on imbedding the particle process in an interface process. The relationship between the one dimensional nearest neighbors simple exclusion process and the interface process is known since the seminal paper by Rost [12], where he established the hydrodynamics of the asymmetric simple exclusion process. In Sect. 4 we define the interface dynamics and show that our particle process can be realized in terms of the interface dynamics. We also introduce the delta interface processes which correspond to the delta particle processes where the killing of particles and holes are grouped together and occur only a finite number of times (uniformly in the hydrodynamic limit) and give as limit the delta macroscopic evolution defined in (1.5).

In Sect. 4 we also establish basic inequalities between the true and the delta interface dynamics by giving a simultaneous explicit graphical construction of all of them; that is, a coupling. In Sect. 5 we prove convergence in the hydrodynamic limit. The proof uses that any limit point of the true interface dynamics is squeezed in between two approximate evolutions which depend on an approximating parameter \(\delta >0\). In Sect. 6 we establish basic properties of the macroscopic evolution. In particular, we prove the existence of a stationary solution for the limit evolution and use this result to prove that at any positive time the particle density (in the limit evolution) is identically 0 and 1 outside of a compact. In the last section we summarize the results.

2 The free boundary ssep

The space of particle configurations is

$$\begin{aligned} {\mathcal {X}}:=\left\{ \eta \in \{0,1\}^{\mathbb {Z}}:\sum _{x\ge 0} \eta (x) <\infty ,\; \sum _{x\le 0}(1-\eta (x))<\infty \right\} \end{aligned}$$

that is, for any \(\eta \in {\mathcal {X}}\) the number of particles to the right of the origin and the number of holes to its left are both finite. A configuration \(\eta \in {\mathcal {X}}\) has a rightmost particle located at \(\textsc {r}(\eta )\) and a leftmost hole located at \(\textsc {l}(\eta )\), where

$$\begin{aligned} \textsc {r}(\eta ):= \max \{x\in {\mathbb {Z}}:\eta (x)=1\};\qquad \textsc {l}(\eta ):= \min \{x\in {\mathbb {Z}}:\eta (x)=0\}. \end{aligned}$$
(2.1)

We define the median \( \textsc {m}(\eta )\) of a configuration \(\eta \in {\mathcal {X}}\) as the unique \(m\in {\mathbb {Z}}+\frac{1}{2}\) such that

$$\begin{aligned} \sum _{x>m} \eta (x) - \sum _{x<m}(1-\eta (x))=0, \end{aligned}$$
(2.2)

that is, the number of particles to the right of \(\textsc {m}(\eta )\) is the same as the number of holes to its left (Fig. 1). The definition is well-posed because for \(\eta \in {\mathcal {X}}\), as \(m\) increases by one (2.2) increases by one and goes to \(\pm \infty \) as \(m\) goes to \(\mp \infty \).

Fig. 1
figure 1

A typical configuration in \({\mathcal {X}}\), black and white circles represent respectively particles and holes. \(\textsc {r}\) is the position of the rightmost particle, \(\textsc {l}\) the position of the leftmost hole and \(M\) is the median

We next define a family of dynamics indexed by a current \(\textsc {j}>0\). The particle dynamics is a (countable state) Markov process on \({\mathcal {X}}\) whose generator is

$$\begin{aligned} {L^{{j}}_{\mathrm{part}}}=L_0+{L^{{j}}_{{r}}}+{L^{{j}}_{{l}}}, \end{aligned}$$
(2.3)

where

$$\begin{aligned} L_{0}f(\eta ) := \sum _{x\in {\mathbb {Z}}} \frac{1}{2}\left[ f(\eta ^{x,x+1})-f(\eta )\right] \end{aligned}$$
(2.4)

with \(\eta ^{x,x+1}(y)=\eta (y)\) if \(y\ne x,x+1,\,\eta ^{x,x+1}(x)=\eta (x+1),\,\eta ^{x,x+1}(x+1)=\eta (x)\); and

$$\begin{aligned} {L^{{j}}_{{l}}}f(\eta ):= \textsc {j}(f(\eta \cup \{\textsc {l}(\eta )\})-f(\eta ));\quad {L^{{j}}_{{r}}}f(\eta ):= \textsc {j}\left( f(\eta {\setminus }\{\textsc {r}(\eta )\})-f(\eta )\right) ,\quad \qquad \end{aligned}$$
(2.5)

where \(\eta \) is identified with the set of occupied sites \(\{x\in {\mathbb {Z}}:\eta (x)=1\}\).

In other words, particles perform symmetric simple exclusion with generator \(L_{0}\) and, at rate \(\textsc {j}\), the rightmost particle is “killed” and replaced by a hole and at the same rate independently the leftmost hole is killed and a particle is born at its place (Fig. 2). We omit the proof that the process is well defined at all times. Denote by \((\eta _t)\) the process with generator \({L^{{j}}_{\mathrm{part}}}\).

Fig. 2
figure 2

The effect of three SSEP jumps: the upper line is before and the bottom line is after the jumps. The jumps are between the positions \(\textsc {l}\) and \(\textsc {l}+1,\,\textsc {m}-\frac{1}{2}\) and \(\textsc {m}+\frac{1}{2}\) and \(x\) and \(x+1\). None of these jumps change the position of \(\textsc {m}\)

Denote by \(B_t\) and \(A_t\) the number of particles killed, respectively born, in the time interval \([0,t]\). By definition, \(A_t\) and \(B_t\) are independent Poisson processes with rate \(\textsc {j}\).

Lemma 1

For any \(\eta _0\in {\mathcal {X}}\),

$$\begin{aligned} \textsc {m}(\eta _t)=\textsc {m}(\eta _0)+A_{t}-B_{t} \end{aligned}$$
(2.6)

That is, the marginal distribution of the median of \(\eta _t\) is a continuous time symmetric nearest neighbor random walk on \({\mathbb {Z}}+\frac{1}{2}\) with rate \(\textsc {j}\) to jump right and \(\textsc {j}\) to jump left.

Proof

Note that (a) jumps due to the exclusion dynamics do not change \(\textsc {m}(\cdot )\), (b) when the rightmost particle dies, \(\textsc {m}(\cdot )\) decreases by 1 and (c) when the leftmost hole dies (and a particle appears in its place), \(\textsc {m}(\cdot )\) increases by 1 (Fig. 3). This shows (2.6). \(\qquad \square \)

Fig. 3
figure 3

The effect of killing the rightmost particle: the upperline is before the killing and the bottom line is after the killing. The position \(\textsc {r}\) of the rightmost particle moves to the left by 3 (for this configuration) and \(\textsc {m}\) moves by 1 to the left (for any configuration). Analogously, the killing of the leftmost hole moves \(\textsc {m}\) one unit to the right (not in the picture)

3 The process as seen from the median

Since \(\textsc {m}(\eta _t)\) is a symmetric simple random walk the law of \(\eta _t\) cannot be tight but we show below that the process as seen from the median has a unique invariant measure. Let

$$\begin{aligned} \tilde{\eta }_{t}:= \theta _{\textsc {m}(\eta _t)-1/2}\eta _{t}, \end{aligned}$$

where for \(y\in {\mathbb {Z}}\), the translation \(\theta _y:{\mathcal {X}}\rightarrow {\mathcal {X}}\) is the map \((\theta _y\eta )(x) = \eta (x-y)\). Clearly \(\tilde{\eta }_t\in {\mathcal {X}}^0:=\{\eta \in {\mathcal {X}}: \textsc {m}(\eta )=1/2\}\) and we have that \((\tilde{\eta }_{t})\) is a Markov process on \({\mathcal {X}}^0\) with generator

$$\begin{aligned} {\tilde{ L}^{{j}}_{}\mathrm{part}}= L_0+{\tilde{L}^{{j}}_{{l}}}+{\tilde{L}^{{j}}_{{r}}}\end{aligned}$$
(3.1)

where for any \(\eta \in {\mathcal {X}}^0\),

$$\begin{aligned} {\tilde{L}^{{j}}_{{r}}}f(\eta ) :=\textsc {j}\left( f(\theta _{-1}(\eta {\setminus }\{\textsc {r}(\eta )\}))\!-\!f(\eta )\right) ;\!\quad {\tilde{L}^{{j}}_{{l}}}f(\eta ):= \textsc {j}\left( f(\theta _1(\eta \cup \{\textsc {l}(\eta )\}))\!-\!f(\eta )]\right) \end{aligned}$$

Before stating the result we introduce some notation. For \(\eta \in {\mathcal {X}}\), define

$$\begin{aligned} N^{0}(\eta ):=\sum _{x<\textsc {r}(\eta )}(1-\eta (x)),\qquad N^{1}(\eta ):=\sum _{x>\textsc {l}(\eta )}\eta (x) \end{aligned}$$
(3.2)

the number of holes in \(\eta \) to the left of the rightmost particle and the number of particles in \(\eta \) to the right of leftmost hole, respectively. Clearly

$$\begin{aligned} N^{0}+N^{1}=\textsc {r}-\textsc {l}+1. \end{aligned}$$
(3.3)

Let \(\eta ^0\in {\mathcal {X}}^0\) be the Heaviside configuration given by

$$\begin{aligned} \eta ^0(x) = {\mathbf { 1}}\{x\le 0\}. \end{aligned}$$
(3.4)

Define \(\psi \) on \({\mathcal {X}}\) by

$$\begin{aligned} \psi (\eta ):= \sum _{x<y}\; (1-\eta (x))\eta (y) \end{aligned}$$
(3.5)

Observe that \(\psi (\eta )<\infty \) for all \(\eta \in {\mathcal {X}}\). If \(\eta \in {\mathcal {X}}^0,\,\psi (\eta )\) is the number of jumps \(10\rightarrow 01\) needed to get from \(\eta ^0\) to \(\eta \). In particular, \(\psi (\eta ^0)=0\).

Theorem 3

For any \(\textsc {j}>0\) the process \((\tilde{\eta }_t)\) has a unique invariant measure \(\mu _{\textsc {j}}\) on \({\mathcal {X}}^0\) and

$$\begin{aligned} {\mu _{\textsc {j}}}[ \textsc {r}-\textsc {l}+1] = \frac{1}{2\textsc {j}} \end{aligned}$$
(3.6)

Proof

We show that \(\psi \) is a Lyapunov function for \((\tilde{\eta }_t)\) by computing \({L^{{j}}_{\mathrm{part}}}\psi \). Since \(\psi (\theta _x\eta )=\psi (\eta )\) for all \(x\) and \(\eta \),

$$\begin{aligned} ({\tilde{L}^{{j}}_{{l}}}\!+\!{\tilde{L}^{{j}}_{{r}}})\psi (\eta ) \!=\!({L^{{j}}_{{l}}}+{L^{{j}}_{{r}}})\psi (\eta ) \!=\! -\! {\textsc {j}}(N^{0}(\eta ) \!-\! N^{1}(\eta )) \!=\! -\! {\textsc {j}} \left( \textsc {r}(\eta )\!-\!\textsc {l}(\eta )\!+\!1\right) \qquad \end{aligned}$$
(3.7)

the second identity holds because when the rightmost particle disappears, \(\psi \) decreases by \(N^{0}\), the number of holes to the left of \(\textsc {r}\). Analogously, when the leftmost hole disappears, \(\psi \) decreases by \(N^{1}\), the number of particles to the right of \(\textsc {l}\). The third identity follows from (3.3).

Since \(\psi \) increases by one when there is a transition \(10\rightarrow 01\) while it decreases by one due to the opposite transition:

$$\begin{aligned} L_0\psi (\eta )= \frac{1}{2} \sum _{x} \{\eta (x)(1-\eta (x+1))-(1-\eta (x))\eta (x+1)\} = \frac{1}{2}, \end{aligned}$$
(3.8)

because for any configuration \(\eta \in {\mathcal {X}}\), the number of pairs \(10\) exceeds by one the number of pairs 01. The sums in (3.8) are finite for \(\eta \in {\mathcal {X}}\).

Call \(\nu _t\) the law of \(\tilde{\eta }_t\) starting from a configuration \(\eta \in {\mathcal {X}}^0\). For any \(t\ge 0\) we have \({\nu _t}\psi <\infty \) and

$$\begin{aligned} \frac{1}{t}({\nu _t}\psi -\nu _0\psi ) = \frac{1}{t}\int _0^t \nu _s[{L^{{j}}_{\mathrm{part}}}\psi ]\, ds= \frac{1}{2} - \frac{\textsc {j}}{t} \int _0^t {\nu _s}[\textsc {r}-\textsc {l}+1]\,ds. \end{aligned}$$
(3.9)

Hence, calling \(\mu _t:= \frac{1}{t} \int _0^t \nu _s\,ds\),

$$\begin{aligned} {\mu _t} [\textsc {r}-\textsc {l}+1] \le \frac{1}{2\textsc {j}}+ \frac{1}{\textsc {j}t}{\nu _0}\psi . \end{aligned}$$
(3.10)

Since for any \(c\) we have \(\# \{\eta \in {\mathcal {X}}^0:\textsc {r}-\textsc {l}+1\le c\} <\infty \), then the family \(\{\mu _t,t\ge 0\}\) of probabilities on \({\mathcal {X}}^0\) is tight on \({\mathcal {X}}^0\) and it has therefore a limit point \(\mu \). \(\mu \) is stationary by construction and unique because the process is irreducible.

Identity (3.6) follows from (3.9) by replacing \(\nu _t\) by \(\mu \), but we need first to show that \(\mu \psi <\infty \). Introduce a new Lyapunov function \(\psi _2\) on \({\mathcal {X}}\) by setting

$$\begin{aligned} \psi _2(\eta ):= \sum _{x<y<z}\; [1-\eta (x)][1-\eta (y)]\eta (z) \end{aligned}$$

Then,

$$\begin{aligned} L_0\psi _2(\eta )&= \frac{1}{2} \left( \sum _{x<y}\; [1\!-\!\eta (x)]\eta (y)[1\!-\!\eta (y\!+\!1)] \!-\! \sum _{x<y}\; [1\!-\!\eta (x)][1\!-\!\eta (y)]\eta (y\!+\!1)\right) \\&= \frac{1}{2} \sum _{x<y}\; [1\!-\!\eta (x)][\eta (y)\!-\!\eta (y\!+\!1)] \!=\!\frac{1}{2} \sum _{x}\; [1\!-\!\eta (x)]\eta (x\!+\!1) \end{aligned}$$

and

$$\begin{aligned} {L^{{j}}_{{r}}}\psi _2(\eta )&= -\textsc {j}\sum _{x<y<\textsc {r}}\; [1-\eta (x)] [1-\eta (y)] = -\frac{\textsc {j}}{2} N^{0}(\eta )[N^{0}(\eta )-1] \\ {L^{{j}}_{{l}}}\psi _2(\eta )&= -\textsc {j}\sum _{\textsc {l}<y< z} [1-\eta (y)] \eta (z) =-\textsc {j}\sum _{y<z}\; [1-\eta (y)] \eta (z) + \textsc {j}\sum _{\textsc {l}<z}\; \eta (z) \\&= -\textsc {j}\psi (\eta )+ \textsc {j}N^{1}(\eta ) \end{aligned}$$

Let \(\nu _t\) as defined before (3.9) and \(\mu _t\) after it. Then \({\nu _t}\psi _2<\infty \) and since \(\sum _{x}\; [1-\eta (x)]\eta (x+1)\le N^{0}(\eta )\),

$$\begin{aligned} \frac{1}{t} \left( {\nu _t}\psi _2-{\nu _0}\psi _2\right) \le - \frac{\textsc {j}}{t} \int _0^t {\nu _s}\left[ -\frac{N^{0}}{2\textsc {j}} +\psi -N^{1}+ \frac{N^{0}}{2}(N^{0}-1)\right] \,ds \end{aligned}$$

so that, since \(N^{0} (N^{0}-1) \ge 0\),

$$\begin{aligned} \mu _t\psi \le \frac{1}{\textsc {j}t} {\nu _0}\psi _2 + {\mu _t}\left[ \frac{N^{0}}{2\textsc {j}} +N^{1}\right] \le \text {Constant}, \end{aligned}$$

by (3.3) and (3.10). Then \(\mu \psi <\infty \) and setting \(\nu _0=\mu \) in (3.9) we get (3.6). \(\square \)

The theorem says that under the invariant measure the average distance between the rightmost particle and the leftmost hole is \((2\textsc {j})^{-1}\). This is in agreement with the Fick’s law (the analogue of the Fourier law for mass densities). In fact Fick’s law states that the stationary current \(J\) flowing in a system of length \(\ell \) when at the endpoints the densities are \(\rho _{\pm }\) is:

$$\begin{aligned} J=- \frac{1}{2}\; \frac{\rho _+-\rho _-}{\ell }\end{aligned}$$

\(1/2\) being the particle mobility. In our case \(J=\textsc {j},\,\rho _+=1\) and \(\rho _-=0\) hence \(\ell =(2\textsc {j})^{-1}\). The validity of Fick’s law in our case is however not completely obvious as the endpoints \(\textsc {r}(\eta _t)\) and \(\textsc {l}(\eta _t)\) depend on time.

4 The interface process

The interfaces Let the set of vertices be

$$\begin{aligned} {\mathcal {V}}:=\{v= (v_1,v_2)\in {\mathbb {Z}}\times {\mathbb {Z}}^+ \text { with }v_1+v_2\text { even}\}. \end{aligned}$$
(4.1)

and for a vertex \(v=(v_1,v_2)\in {\mathcal {V}}\) let \(V_v:{\mathbb {Z}}\rightarrow {\mathbb {Z}}\) be the cone with vertex \(v\) defined by

$$\begin{aligned} V_v(x):= |x-v_1| + v_{2}. \end{aligned}$$
(4.2)

Define the space of interfaces as

$$\begin{aligned} {\mathcal {Y}}_v&:= \left\{ \xi \in {\mathbb {Z}}^{\mathbb {Z}}\,:\,|\xi (x)-\xi (x+1)|=1,\;x\in {\mathbb {Z}};\;\#\{x:\xi (x) \ne V_v(x)\}<\infty \right\} \nonumber \\ {\mathcal {Y}}&:= \cup _{v\in {\mathcal {V}}} {\mathcal {Y}}_v \end{aligned}$$
(4.3)

That is, an interface in \({\mathcal {Y}}_v\) coincides with the cone \(V_v\) for all but a finite number of sites. Interfaces share the property “\(x+\xi (x)\) even” which is conserved by the dynamics defined later. For an interface \(\xi \in {\mathcal {Y}}\) define

$$\begin{aligned} \textsc {l}(\xi )&:= \sup \{x\in {\mathbb {Z}}\,:\,\xi (x-y)=\xi (x) +y\;\text { for all} y \ge 0 \} \end{aligned}$$
(4.4)
$$\begin{aligned} \textsc {r}(\xi )&:= \inf \{x\in {\mathbb {Z}}\,:\,\xi (x+y)=\xi (x) +y\;\text { for all} y\ge 0 \}. \end{aligned}$$
(4.5)

Any interface \(\xi \) coincides with a cone \(V_v\) outside the finite interval \((\textsc {l}(\xi ),\textsc {r}(\xi ))\). The vertex \(v=v(\xi )\) is the following function of \(\textsc {r}=\textsc {r}(\xi )),\,\textsc {l}=\textsc {l}(\xi ),\,\xi (\textsc {l})\) and \(\xi (\textsc {r})\):

$$\begin{aligned} v_1= v_1(\xi ) :=\frac{\xi (\textsc {l})-\xi (\textsc {r})}{2}+ \frac{\textsc {l}+\textsc {r}}{2},\;\; v_2= v_2(\xi ):= \frac{\xi (\textsc {l})+\xi (\textsc {r})}{2}+ \frac{\textsc {l}-\textsc {r}}{2}.\quad \qquad \end{aligned}$$
(4.6)

Correspondence between interfaces and particle configurations Given an interface \(\xi \) we say that there is a particle at \(x\) if \(\xi (x+1)<\xi (x)\) and that there is a hole at \(x\) if \(\xi (x+1)>\xi (x)\). This defines the map \(D:{\mathcal {Y}}\rightarrow {\mathcal {X}}\) given by

$$\begin{aligned} \eta (x)\equiv D(\xi )(x) = \frac{1}{2} - \frac{\xi (x+1)-\xi (x)}{2} \end{aligned}$$
(4.7)

The map is clearly surjective but it is not injective as \(D\) is invariant under uniform vertical shifts: \(D(\xi +n) = D(\xi )\) for all \(n\in {\mathbb { Z}}\). However the extremes of the non conic part of \(\xi \) correspond to the leftmost hole and the rightmost particle of \(D(\xi )\) and the absise of the vertex of the cone containing \(\xi \) corresponds to the median of \(D(\xi )\). More precisely:

Lemma 2

For any \(\xi \in {\mathcal {Y}}\),

$$\begin{aligned} \textsc {l}(D(\xi ))=\textsc {l}(\xi ), \quad \textsc {r}(D(\xi ))=\textsc {r}(\xi )-1,\quad \textsc {m}(D(\xi ))=v_1(\xi )-1/2. \end{aligned}$$

Proof

The first and second identities follows directly from the definitions. The third identity is trivially satisfied by any cone and its mapped particle configuration: \(v_1(V_{(a,b)})=a\) and \(\textsc {m}(D(V_{(a,b)}))=a-\frac{1}{2}\). Any interface \(\xi \) in the cone \(V_{(a,b)}\) can be attained from \(V_{(a,b)}\) by a finite number of moves of the type:

$$\begin{aligned} \xi =( \dots ,z,z-1,z,\dots ) \rightarrow (\dots ,z,z+1,z,\dots )=\xi ' \end{aligned}$$
(4.8)

The corresponding particle moves are

$$\begin{aligned} \eta =( \dots ,1,0,\dots ) \rightarrow (\dots ,0,1,\dots )=\eta ' \end{aligned}$$
(4.9)

and

$$\begin{aligned} D(\xi )=\eta \quad \text { if and only if } \quad D(\xi ')=\eta '. \end{aligned}$$
(4.10)

Observing that \(v(\xi )=v(\xi ')\) and that \(\textsc {m}(D(\xi ))=\textsc {m}(D(\xi '))\), we conclude that \(D(\xi )\) has median \(a-\frac{1}{2}=v_1(\xi )\) for all \(\xi \in {\mathcal {Y}}_{(a,b)}\). \(\square \)

Interface dynamics Let \((\xi _t)\) be the Markov process on \({\mathcal {Y}}\) with generator

$$\begin{aligned} {{\mathcal {L}}^{{j}}_{\mathrm{inter}}}&:= {{\mathcal {L}}_{\Delta }}+ {{\mathcal {L}}^{{j}}_{{r}}}+ {{\mathcal {L}}^{{j}}_{{l}}},\quad \text { with} \nonumber \\ {{\mathcal {L}}_{\Delta }}f(\xi )&:= \frac{1}{2}\sum _{x\in {\mathbb {Z}}}\{f(\xi +\Delta _x\xi ) -f(\xi )\}, \\ {{\mathcal {L}}^{{j}}_{{r}}}f(\xi )&:= \textsc {j}\left[ f(\max \{ \xi ,V_{v(\xi )+(-1,1)}\} )-f(\xi )\right] ,\nonumber \\ {{\mathcal {L}}^{{j}}_{{l}}}f(\xi )&:= \textsc {j}\left[ f(\max \{\xi , V_{v(\xi )+(1,1)}\} )-f(\xi ) \right] ,\nonumber \end{aligned}$$
(4.11)

where \(\Delta _x\xi (y):= (\xi (x+1)+\xi (x-1)-2\xi (x))\,{\mathbf { 1}}\{y=x\}\). The jumps of \(\xi (x)\) due to \({{\mathcal {L}}_{\Delta }}\) occur only when \(\xi (x)\) has the two neighbors at equal height. The jump is up by 2 if the neighboring heights are both above \(\xi (x)\) and down by 2 if they are both below \(\xi (x)\). \({{\mathcal {L}}^{{j}}_{{r}}}\) acts by changing the rightmost downward variation of \(\xi \) into an upward one with the interface to its right being a straight line with slope 1. The cone containing the updated interface is obtained from the previous cone by a translation up by 1 and left by 1; see Fig. 4. A symmetrical picture describes the action of \( {{\mathcal {L}}^{{j}}_{{l}}}\). We denote by

$$\begin{aligned} A_t=\#(\text {jumps due to} {{\mathcal {L}}^{{j}}_{{l}}}\hbox {in} [0,t]);\;\;\;\;B_t=\#(\text {jumps due to} {{\mathcal {L}}^{{j}}_{{r}}}\hbox {in} [0,t]) \end{aligned}$$
(4.12)

\(A=(A_t)\) and \(B=(B_t)\) are independent Poisson processes of intensity \(\textsc {j}\).

The interface evolution induces via the map \(D\) the particle evolution described in Sect. 2.

Let \((\xi _t)\) be the interface process with generator \({{\mathcal {L}}^{{j}}_{\mathrm{inter}}}\) and starting interface \(\xi \) and define

$$\begin{aligned} \eta _t:=D(\xi _t) \end{aligned}$$
(4.13)

Lemma 3

The particle process \((\eta _t)\) defined by (4.13) is Markov with generator \({L^{{j}}_{\mathrm{part}}}\), defined in (2.3). Moreover, if \(\xi _0\in {\mathcal {Y}}_{(0,0)}\), then

$$\begin{aligned} \xi _t(0)= 2 B_t + 2 \sum _{x\ge 0} \eta _t(x) \end{aligned}$$
(4.14)
Fig. 4
figure 4

The thick red line represents the interface \(\xi _t\); its corresponding particle configuration is \(\eta _t=D(\xi _t)\). The narrow blue line represents \(\xi _0\), the interface at time \(0; \eta _0=D(\xi _0)\). Two particles and one hole have been killed and several particles have moved due to the exclusion dynamics; in particular the third particle moved to the place originally occupied by the second one and the second hole has moved three units to the left. Due to the killings the vertex of the cone containing \(\xi _t\) has moved by \(2(-1,1)+1(1,1)\) which amounts to three units up and one unit left so that \(\xi _0\in {\mathcal {Y}}_{(0,0)}\) while \(\xi _t\in {\mathcal {Y}}_{(-1,3)}\) (color figure online)

Proof

It just follows from the definitions of \((\xi _t)\) and \(D\) that \((D(\xi _t))\) is Markov with generator \({L^{{j}}_{\mathrm{part}}}\). Calling \(\textsc {r}\equiv \textsc {r}(\xi _t)=\textsc {r}(\eta _t)+1\) we get from (4.7)

$$\begin{aligned} \xi _t(0)=\xi _t(\textsc {r})-\textsc {r}+2\sum _{x\ge 0}\eta _t(x) \end{aligned}$$

From (4.6), \(\xi _t(\textsc {r})=\textsc {r}-(v_1-v_2)\) and since \(v_1=A_t-B_t\) and \(v_2=A_t+B_t\), we get (4.14). \(\square \)

Harris graphical construction We construct explicitly the interface process \((\xi _t)\) as a function of the initial interface and of the Poisson processes governing the different jumps.

We construct first the process with generator \({{\mathcal {L}}_{\Delta }}\) and later use it to define the process with the moving boundary conditions. The probability space \(( \varOmega ,P)\) is the product of independent rate-\(\frac{1}{2}\) Poisson processes on \({\mathbb { R}}_+\) indexed by \({\mathbb {Z}}\times \{\uparrow ,\downarrow \}\). A typical element of \(\Omega \) is \(\omega = (\omega ^\uparrow _x,\, \omega ^\downarrow _x,\,x\in {\mathbb {Z}}\)). The Poisson points in \(\omega ^\uparrow _x\) and respectively \(\omega ^\downarrow _x\) are called up-arrows and down-arrows, respectively.

We define operators \(T_t:\Omega \times {\mathcal {Y}}\rightarrow {\mathcal {Y}}\) with \(t\ge 0\) where \((T_t(\omega ,\xi ))\) is the process with initial interface \(\xi \) using the arrows of \(\omega \), as follows. We drop the dependence on \(\omega \) and write just \(T_t\xi \), instead. We can take \(\omega \) such that at most one arrow occurs at any given time.

Set \(T_0\xi =\xi \). Assume that \(\xi _{t'}:=T_{t'}\xi \) is defined for all \(t'\in [0,s]\).

Let \(t\) be the first arrow after \(s\) belonging to \(\omega ^\uparrow _x\cup \omega ^\downarrow _x\) for some \(x\) such that \(\xi _s(x+1)=\xi _s(x-1)\). Since \(\xi \in {\mathcal {Y}}\), there are a finite number of such \(x\) and \(t-s>0\) a.s. These are the arrows involved in the evolution at time \(s\).

Set

$$\begin{aligned} T_{t'}\xi = T_s\xi \text { for }t'\in [s,t) \end{aligned}$$

and (1) If \(t\) is an up-arrow, then the interface at \(x\) is set to \(\xi _t(x-1)+1\) no matter its value at \(t-\) and does not change at the other sites:

$$\begin{aligned} T_t\xi (x):=T_{t-}\xi (x-1)+1; \quad T_t\xi (y) :=T_{t-}\xi (y),\;y\ne x. \end{aligned}$$

(2) Analogously, if \(t\) is a down-arrow,

$$\begin{aligned} T_t\xi (x):=T_{t-}\xi (x-1)-1; \quad T_t\xi (y) :=T_{t-}\xi (y),\;y\ne x. \end{aligned}$$

The reader can show that the process \(T_t\xi \) so defined is Markov and evolves with the generator \({{\mathcal {L}}_{\Delta }}\) with initial interface \(\xi \) at time \(t=0\).

In the next definition we need to use the operator \(T_t\) in different time intervals. With this in mind we define

$$\begin{aligned} T_{[s,t]}(\omega ,\xi ):= T_{t-s}(\theta _{-s}\omega ,\xi ), \end{aligned}$$
(4.15)

where \(\theta _s\omega \) is the translation by \(s\) of the arrows in \(\omega \). That is, \(T_{[s,t]}\) has the same distribution as \(T_{t-s}\) but uses the arrows in \(\omega \) belonging to the interval \([s,t]\). We drop the dependence on \(\omega \) in the notation and write simply \(T_{[s,t]}\xi \).

Generalizing the boundary conditions Consider the partial order in the vertex space \({\mathcal {V}}\) given by

$$\begin{aligned} v\le v'\quad \text { if }V_{v}(x)\le V_{v'}(x)\text { for all }x\in {\mathbb {Z}}, \end{aligned}$$
(4.16)

so that vertex order corresponds to cone order. Let \(z=(z_t)\) with \(z_t\in {\mathcal {V}}\) be a non decreasing path of vertices with finite number of finite jumps in finite time intervals: \(\Vert z_{t}-z_{t'}\Vert <\infty \) for all \(t<t'\). Let \(T_{[s,t]}\) be the family of random operators governing the \({{\mathcal {L}}_{\Delta }}\) motion, defined in (4.15). Typically \(z\) will be a function of the Poisson processes \(A\) and \(B\) which are independent of the arrows \(\omega \) used to define the operators \(T_{[s,t]}\). We abuse notation and call \(P\) the probability associated to \(\omega ,\,A\) and \(B\).

Let \(0= s_{0}< s_{1}<\cdots \) be the times of jumps of \(z\). Define iteratively \(T^z_t\xi :=\xi \) for \(t<0\) and

$$\begin{aligned} T^z_t\xi := {\left\{ \begin{array}{ll} \max \{T^z_{t-}\xi ,V_{z_t}\}&{}\text {if } t=s_n,\\ T_{[s_n,t]}T^z_{s_n}\xi &{}\text {if } t\in (s_n,s_{n+1}). \end{array}\right. } \end{aligned}$$
(4.17)

So \((\xi ^z_t)\) evolves with \({{\mathcal {L}}_{\Delta }}\) in the intervals \((s_n,s_{n+1})\) and at times \(s_n\) is updated to the maximum between the interface at time \(s_n-\) and the cone with vertex \(z_{s_n}\).

Given an initial interface \(\xi \in {\mathcal {Y}}\) and a number \(\delta >0\), we will consider the following choices for \(z_t\), denoted by \(O_t,\,R_t\) and \(z^{\delta ,-}_t\) and \(z^{\delta ,+}_t\):

$$\begin{aligned} O_t&:= v(\xi ),\quad \text { for all }t; \end{aligned}$$
(4.18)
$$\begin{aligned} R_t&:= v(\xi ) +(A_t-B_t, A_t+B_t) \end{aligned}$$
(4.19)
$$\begin{aligned} z^{\delta ,-}_t&:= R_{n\delta }, \qquad z^{\delta ,+}_t:= R_{(n+1)\delta }, \qquad t\in [n\delta ,(n+1)\delta ),\;\; n\ge 0. \end{aligned}$$
(4.20)

When the path is \((O_t)\), the cone does not move and the resulting process has generator \({{\mathcal {L}}_{\Delta }}\). When the path is \((R_t)\) the process has generator \({{\mathcal {L}}^{{j}}_{\mathrm{inter}}}\). The process with path \((z^{\delta ,-}_t)\) records the increments of \((R_t)\) in the intervals \([n\delta ,(n+1)\delta )\) and takes the maximum of the interface at the end of this interval and the cone with vertex \(R_{n\delta }\). The process with path \((z^{\delta ,+}_t)\) records the increments of \((R_t)\) in the intervals \([n\delta ,(n+1)\delta )\) but takes the maximum between the interface at the beginning of this interval and the cone with center \(R_{(n+1)\delta }\). The processes with paths (4.20) will be used later.

Monotonicity and attractivity Consider the natural partial order in \({\mathcal {Y}}\) given by: \(\xi \le \xi '\) if and only if \(\xi (x)\le \xi '(x)\) for all \(x\in {\mathbb {Z}}\). Use this order to define stochastic order for random interfaces in \({\mathcal {Y}}\). If \(\xi ,\xi '\) are random, \(\xi \) is stochastically dominated by \(\xi '\) if \(Ef(\xi )\le Ef(\xi ')\) for non decreasing \(f:{\mathcal {Y}}\rightarrow {\mathbb {R}}\). This is equivalent to the existence of a coupling \((\hat{\xi },\hat{\xi }')\) whose marginals have the same distribution as \(\xi \) and \(\xi '\), respectively, such that \(P(\hat{\xi }\le \hat{\xi }') =1\).

Let \((\xi ^1_t)\) and \((\xi ^2_t)\) be two realizations of a stochastic process \((\xi _t)\) on \({\mathcal {Y}}\) with initial random interfaces \(\xi ^1\) and \(\xi ^2\), respectively. We say that the process \((\xi _t)\) is attractive if the following holds:

$$\begin{aligned} \text {If } \xi ^1\le \xi ^2 \text { stochastically, then } \xi ^1_t\le \xi ^2_t \text { stochastically, for all }t\ge 0. \end{aligned}$$
(4.21)

Lemma 4

If \(\xi \le \xi '\) then \(T_t\xi \le T_t\xi '\) almost surely. As a consequence, the process with generator \({{\mathcal {L}}_{\Delta }}\) is attractive.

Proof

Consider \(\xi \le \xi '\) and call \(\xi _t=T_t\xi ,\,\xi '_t=T_t\xi '\), assume that \(\xi _{t-}\le \xi '_{t-},\,\xi _{t-}(x\pm 1) = \xi '_{t-}(x\pm 1)\) and that \(t\in \omega ^\uparrow _x\). Then \(\xi _{t}(x)= \xi '_{t}(x)=\xi _{t-}(x-1)+1\) no matter the values \(\xi _{t-}(x)\) and \( \xi '_{t-}(x)\). Analogous argument applies when \(t\in \omega ^\downarrow _x\). Hence an arrow does not change the order if the two interfaces coincide at \(x\pm 1\). If at least one of the neighbors of \(x\) in \(\xi _{t-}\) is different of the corresponding neighbor in \(\xi '_{t-}\), a similar argument shows that no jump can break the domination. We have proven that if the \(\xi \) process is dominated by the \(\xi '\) process just before an arrow, then the domination persist after the jump(s) produced by the arrow. Since the set of involved arrows is finite in any time interval, an iterative argument concludes the proof. \(\square \)

Remark

It is usual to realize the process with generator \({{\mathcal {L}}_{\Delta }}\) by introducing only one rate-\(\frac{1}{2}\) Poisson process of marks \(\omega _x, x\in {\mathbb {Z}}\) associated to each \(x\) and updating \(\xi (x)\rightarrow \xi (x-1)\pm 1\ne \xi (x)\) whenever a \(\omega _x\)-mark appears provided \(\xi (x-1)=\xi (x+1)\). This is indeed a realization of the process, but order is not preserved: the jumps cross if two interfaces coincide at \(x\pm 1\) and differ by 2 at \(x\) at the updating time of \(x\). Using the up-arrows and down-arrows, only one of the interfaces jumps, and order is preserved.

Lemma 5

If \(\xi \in {\mathcal {Y}}\), then \(v(T_t\xi )=v(\xi )\) a.s. As a consequence, \(V_{v(\xi )}\le T_t\xi \). Furthermore, if \(z\) is a non decreasing path with \(v(\xi )=z_0\), then \(v(T^z_t\xi )=z_t\), which implies \(V_{z_t}\le T^z_t\xi \).

Proof

By the definition of \((T_t\xi )\), any time \(\textsc {l}\) jumps due to an arrow, the opposite jump is performed by \(\xi (\textsc {l})\) and analogously, any jump of \(\textsc {r}\) is replicated by \(\xi (\textsc {r})\). This implies

$$\begin{aligned} \textsc {l}(T_t\xi )+\xi (\textsc {l}(T_t\xi ))&= \textsc {l}(\xi )+\xi (\textsc {l}(\xi )),\nonumber \\ \textsc {r}(T_t\xi )-\xi (\textsc {r}(T_t\xi ))&= \textsc {r}(\xi )-\xi (\textsc {r}(\xi )) \end{aligned}$$
(4.22)

Putting these identities in the definition (4.6) of \(v(\xi )\) and noting that the total number of jumps of \(\textsc {l}\) and \(\textsc {r}\) is a.s. finite (it is dominated by a Poisson process of rate \(2\)), we get \(v(T_t\xi )=v(\xi )\).

The fact that \(v(T^z_t\xi )=z_t\) is true by definition at the \(z\)-events and by the first part of the lemma, it is true for \(t\in [s,s')\) where \(s\) and \(s'\) are successive \(z\)-events. \(\square \)

Proposition 1

Let \(z\) and \(z'\) be non decreasing paths in \({\mathcal {V}}\).

$$\begin{aligned} \text {If }{z_t}\le {z'_t} \text { for all }t\ge 0 \text { and }\xi \le \xi ',\text { then }T^{z}_{t}\xi \le T^{z'}_{t}\xi ' \text { for all }t\ge 0\quad \text {a.s.}.\qquad \end{aligned}$$
(4.23)

In particular, by taking \(z=z'\), Proposition 1 says that for any non decreasing path \(z\), the process \((T^z_t\xi )\) is attractive.

Proof

Since the domination is preserved in intervals with no \(z\) or \(z'\) events by Lemma 4, we only need to check that the inequality \(T_t^{z}\xi \le T_t^{z'}\xi \) is preserved when \(z\) and \(z'\) events occur. We thus suppose the inequality is satisfied for all \(s<t\) and this is evidently still true if \(t\) is a \(z'\)-event. If instead \(t\) is a \(z\)-event and not a \(z'\)-event, \(T^{z}_{t-}\xi \le T^{z'}_{t-}\xi '\) implies

$$\begin{aligned} T^{z}_t\xi =\max \{ T^{z}_{t-}\xi , V_{z_t}\}\le \max \{ T^{z'}_{t-}\xi ', V_{z_t}\} \le \max \{ T^{z'}_{t-}\xi ', V_{z'_t}\} = T^{z'}_t\xi , \end{aligned}$$

where the last inequality follows from \(V_{z_t}\le V_{z'_t}\) and the last identity from the last inequality in Lemma 5. \(\square \)

Let \(\xi \in {\mathcal {Y}}\) and \(v=(v_1,v_2)\in {\mathcal {V}}\). Define \(\theta _v\xi \), the translation by \(v\) of \(\xi \), by

$$\begin{aligned} \theta _v\xi (x):= \xi (x-v_1)-v_2. \end{aligned}$$

Recall the order (4.16) in \({\mathcal {V}}\). Taking \(v,v'\in {\mathcal {V}}\) and \(\xi ,\xi '\in {\mathcal {Y}}\), the following statement is immediate.

$$\begin{aligned} \hbox { If }v\ge v'\quad \hbox { and }\quad \xi \le \xi ',\hbox { then } \theta _v\xi \le \theta _{v'}\xi '. \end{aligned}$$
(4.24)

Call \(o:=(0,0)\) and take \(v\in {\mathcal {V}}\) satisfying \(v\ge o\) and \(\xi \in {\mathcal {Y}}_o\), then

$$\begin{aligned} \theta _v\xi \le \xi \quad \text { and } \quad \max \{\theta _v\xi ,V_o\}\le \xi . \end{aligned}$$
(4.25)

The interface process as seen from the vertex Take \(\xi \in {\mathcal {Y}}_o\) and let \(z\) be a non decreasing path of vertices. Define \((\tilde{T}^z_t\xi )\), the interface process as seen from the vertex, by

$$\begin{aligned} \tilde{T}^z_t\xi (x) := \theta _{z_t}T^z_t\xi (x), \end{aligned}$$
(4.26)

Of course \(\tilde{T}^z_t\xi \in {\mathcal {Y}}_o\).

Monotonicity We show that if the initial interface \(\xi '\) dominates \(\xi \) and any jump of \(z'\) is dominated by a jump of \(z\), then the interface process as seen from \(z'\) dominates the one as seen from \(z\). More precisely,

Proposition 2

Let \(z\) and \(z'\) be non decreasing paths on \({\mathcal {V}}\) and \(\xi \le \xi '\) be interfaces in \({\mathcal {Y}}_o\).

$$\begin{aligned} \text {If }z_t-z_{t-}\ge z'_t-z'_{t-} \text { for all }t\ge 0, \text { then }\tilde{T}^{z}_{t}\xi \le \tilde{T}^{z'}_{t}\xi ' \text { for all }t\ge 0\quad \text {a.s.}.\qquad \end{aligned}$$
(4.27)

Proof

Since by Lemma 4, the domination is preserved in intervals with no \(z\) or \(z'\) events, it suffices to take care of those events. Assume that \(\tilde{T}^z_{t-}\xi \le \tilde{T}^{z'}_{t-}\xi '\) and that \(t\) is a \(z\)-event, then

$$\begin{aligned} \tilde{T}^z_{t}\xi = \theta _{z_t-z_{t-}}\tilde{T}^z_{t-}\xi \le \theta _{z'_t-z'_{t-}}\tilde{T}^{z'}_{t-}\xi '=\tilde{T}^{z'}_{t}\xi ', \end{aligned}$$
(4.28)

where the inequality holds by (4.24). \(\square \)

Let

$$\begin{aligned} z_t(\textsc {j}) := R_t= (A_t-B_t, A_t+B_t), \end{aligned}$$
(4.29)

recalling that \(A\) and \(B\) are independent Poisson processes of rate \(\textsc {j}\). For each \(\textsc {j}>0\) define the interface process \((\tilde{\xi }_t)\) by

$$\begin{aligned} \tilde{\xi }_t:=\tilde{T}^{z(\textsc {j})}_{t}\xi \end{aligned}$$
(4.30)

Then, \((\tilde{\xi }_t)\) has generator is \({\tilde{{\mathcal {L}}}^{{j}}_{\mathrm{inter}}}\) given by

$$\begin{aligned} {\tilde{{\mathcal {L}}}^{{j}}_{\mathrm{inter}}}= {{\mathcal {L}}_{\Delta }}+ {\tilde{\mathcal {L}}^{{j}}_{{r}}}+ {\tilde{{\mathcal {L}}}^{{j}}_{{l}}}\end{aligned}$$
(4.31)

where \({{\mathcal {L}}_{\Delta }}\) was defined in (4.11) and the other generators govern the updating with the maximum of the cone and the corresponding translation of the origin:

$$\begin{aligned} \begin{aligned} {\tilde{\mathcal {L}}^{{j}}_{{r}}}f(\xi )&:= \textsc {j}\left[ f(\max \{\theta _{(-1,1)}\xi , V_{o}\} )-f(\xi )\right] ,\\ {\tilde{{\mathcal {L}}}^{{j}}_{{l}}}f(\xi )&:= \textsc {j}\left[ f(\max \{\theta _{(1,1)}\xi , V_{o}\} )-f(\xi )\right] . \end{aligned} \end{aligned}$$
(4.32)

For \(\xi \in {\mathcal {Y}}_o\), the process \((D(\tilde{\xi }_t))\) has the same law as the particle process \((\tilde{\eta }_t)\) defined in Sect. 3 with initial particle configuration \(D(\xi )\in {\mathcal {X}}_0\). The map \(D:{\mathcal {Y}}_0\rightarrow {\mathcal {X}}_0\) (interfaces with vertex in the origin to particle configurations with median \(-\frac{1}{2}\)) is bijective. Since the process \((\tilde{\eta }_t)\) has a unique invariant measure on \({\mathcal {X}}_0,\,(\tilde{\xi }^z_t)\) has a unique invariant measure \(\tilde{\mu }^\textsc {j}\) on \({\mathcal {Y}}_0\).

Corollary 1

The process \((\tilde{\xi }_t)\) is attractive. Furthermore, if \(\tilde{\xi }_0=V_o\) the law of \(\tilde{\xi }_t\) is non decreasing in \(t\), is stochastically dominated by the invariant measure \(\tilde{\mu }^{\textsc {j}}\) and converges to \(\tilde{\mu }^{\textsc {j}}\) as \(t\rightarrow \infty \).

Proof

Attractivity follows by taking \(z=z'\) in (4.27). As before use the notation \( \tilde{T}_{[s,t]}^z\xi \) to indicate that the evolution uses the Poisson processes of the Harris construction in the time interval \([s,t]\) for both the interface evolution and the vertex evolution \(z\). So that, for \(s,t\ge 0\), \(\tilde{T}_{t+s}^z\xi \) has the same law as \(\tilde{T}_{[-s,t]}^z\xi \). Since \(V_o\) is minimal in \({\mathcal {Y}}_o\), using (4.27) we get almost surely

$$\begin{aligned} \tilde{T}_{[0,t]}^zV_o\le \tilde{T}_{[0,t]}^z\tilde{T}_{[-s,0]}^zV_o = \tilde{T}_{[-s,t]}^zV_o . \end{aligned}$$

This shows that the law of \(\tilde{\xi }_t\) is stochastically non decreasing. Take a random \(\xi \) with law \(\tilde{\mu }^{\textsc {j}}\). Then,

$$\begin{aligned} \tilde{T}_{[0,t]}^zV_o\le \tilde{T}_{[0,t]}^z \xi \sim \tilde{\mu }^{\textsc {j}}. \end{aligned}$$

by invariance of \(\tilde{\mu }^{\textsc {j}}\). The convergence of the law of \(\tilde{\xi }_t\) to the unique invariant measure \(\tilde{\mu }^{\textsc {j}}\) is routine for countable state irreducible Markov processes. \(\square \)

Corollary 2

The invariant measures \(\tilde{\mu }^\textsc {j}\) for the interface processes \((\tilde{\xi }_{t})\) are stochastically ordered:

$$\begin{aligned} \text {If }\textsc {j}\ge \textsc {j}', \text { then } \tilde{\mu }^\textsc {j}\le \tilde{\mu }^{\textsc {j}'} \end{aligned}$$

Proof

Take Poisson processes \((A,A',B,B')\) such that \((A,A')\) and \((B,B')\) are independent. \(A\) and \(B\) have rate \(\textsc {j}\) while \(A'\) and \(B'\) have rate \(\textsc {j}'\) and \(A\supset A',\,B\supset B'\). In this way the vertex paths \(z\) and \(z'\) defined by (4.29) with \((A,B)\) and \((A',B')\), respectively, satisfy the conditions of Proposition 2. This implies that \(\tilde{T}_t^zV_o\le \tilde{T}_t^{z'}V_o\) almost surely for all \(t\). Like in Corollary 1, the coupled process \((\tilde{T}_t^zV_o, \tilde{T}_t^{z'}V_o)\) is stochastically non decreasing for the (partial) coordinatewise order and each coordinate is dominated by the respective invariant measure. This implies the existence of \(\lim _{t\rightarrow \infty }(\tilde{T}_t^zV_o, \tilde{T}_t^{z'}V_o)\) in distribution. The coordinates of the limit are ordered and its marginals have distributions \(\tilde{\mu }^\textsc {j}, \tilde{\mu }^{\textsc {j}'}\), respectively. \(\square \)

5 Hydrodynamic limit

It is well known that in the diffusive scaling limit (space scaled as \(\varepsilon ^{-1}\) and time as \(\varepsilon ^{-2}\)) the hydrodynamic limit of the SSEP process alone converges to the linear heat equation:

$$\begin{aligned} \frac{\partial \rho _t}{\partial t} = \frac{1}{2} \frac{\partial ^2 \rho _t}{\partial r^2},\quad \rho _0=\rho \end{aligned}$$
(5.1)

Since the distance between the first hole and last particle is random, it is not clear a-priori that our model should be scaled diffusively as well. From Theorem 3 we know that at equilibrium the mean distance between the rightmost particle and the leftmost hole is of order \(\textsc {j}^{-1}\) and this suggest that together with the above diffusive scaling we should scale \(\textsc {j}\) proportionally to \(\varepsilon \). Indeed we will prove that under such scaling limit the density of the process converges as \(\varepsilon \rightarrow 0\) to a deterministic evolution.

5.1 Results

Hydrodynamics of interfaces Initial configurations For each \(\varepsilon >0\) the interface evolution starts from an interface \(\xi ^{(\varepsilon )}\) such that:

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0} \sup _{x\in {\mathbb {Z}}} |\varepsilon \xi ^{(\varepsilon )}(x)- \phi _0(\varepsilon x)| = 0 \end{aligned}$$
(5.2)

where \(\phi _0(r)=r\) for all \(r\ge \textsc {r}(\phi ) >0; \, \phi _0(r)=-r\) for all \(r\le \textsc {l}(\phi )<0\) and \(\phi _0\) is differentiable in \((\textsc {l}(\phi ),\textsc {r}(\phi ))\) with derivative \(\phi '_0\) such that \({\sup \nolimits _{r\in (\textsc {l}(\phi ),\textsc {r}(\phi )) }|\phi '_0(r)| <1}\).

Fix the macroscopic current \(j>0\), define \(\textsc {j}(\varepsilon ):=j\varepsilon \) and call

$$\begin{aligned} (\xi ^{(\varepsilon )}_t)&:= \text { interface process with generator } {{\mathcal {L}}_{\Delta }}+{{\mathcal {L}}^{j\varepsilon }_{\text {\textsc {r}}}}+{{\mathcal {L}}^{j\varepsilon }_{\text {\textsc {l}}}}, \text { starting from }\xi ^{(\varepsilon )}.\qquad \end{aligned}$$
(5.3)

The following is the hydrodynamic limit for the interface process.

Theorem 4

There is a function \(\phi _t(r),\,t\ge 0,\,r\in {\mathbb { R}}\), so that for any \(\gamma >0\) and \(t>0\):

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0} P\left[ \sup _{x\in {\mathbb {Z}}} |\varepsilon \xi ^{(\varepsilon )}_{\varepsilon ^{-2}t}(x)- \phi _t(\varepsilon x)| \ge \gamma \right] = 0 \end{aligned}$$
(5.4)

The proof will be given in Sect. 5.5 and properties of \(\phi _t(r)\) will be discussed in Sect. 6.

Hydrodynamics of particles As we shall prove at the end of Sect. 6.1, the hydrodynamic limit for the particle process Theorem 1 is a corollary of Theorem 4.

The proof that (5.1) is the hydrodynamic limit for the SSEP is quite simple because the correlation functions obey closed equations. The addition of birth-death processes spoils such a property but if the rates are cylinder functions and are “small” (births and deaths happen at rate \(\varepsilon ^{2}\)) the proofs carry over and the limit is a reaction diffusion equation as in [2]. In our case the birth-death rates are not as small (because the killing rates are of order \(\varepsilon \)) but the main difficulty is that the killings are highly non local functions of the configuration, since births and deaths occur at the position of the leftmost hole and the rightmost particle, respectively. This spoils completely an analysis based on the study of the hierarchy of the correlation functions.

The way out is to use inequalities namely to sandwich the interface process between two delta processes, using Proposition 2. The corresponding delta particle process behave as the exclusion dynamics in macroscopic time intervals of length \(\delta \) and the (accumulated) killings occur at the extremes of those intervals.

5.2 The delta processes

The delta interface process \((\xi ^{\delta ,\pm }_t)\) is defined via (4.17) and (4.20) by

$$\begin{aligned} \xi _t^{\delta ,\pm } := T^{z\pm }_t\xi ,\text { with } z\pm :=z^{\pm \delta }_{t} \end{aligned}$$
(5.5)

This process is obtained by patching together finitely many pieces of the evolution with generator \({{\mathcal {L}}_{\Delta }}\) as explained after (4.20). By Proposition 1, as

$$\begin{aligned} z_t^{\delta ,-}&\le R_t\le z_t^{\delta ,+},\quad \hbox {for all }t\ge 0, \nonumber \\ \text {then, }\xi _t^{\delta ,-}&\le \xi _t\le \xi _t^{\delta ,+},\quad \hbox {for all }t\ge 0. \end{aligned}$$
(5.6)

The delta particle processes are defined by using the map (4.7), setting \(\eta _t^{\delta ,\pm }= D(\xi _t^{\delta ,\pm })\). In other words, \((\eta _t^{\delta ,-})\) evolves with the generator \(L_0\) in the intervals \([n\delta ,(n+1)\delta )\), and \(\eta _{n\delta }^{\delta ,-}\) is obtained from \(\eta _{n\delta -}\) by removing its \(B_{n\delta }-B_{(n-1)\delta }\) rightmost particles and its \(A_{n\delta }-A_{ (n-1)\delta }\) leftmost holes, where \(A\) and \(B\) are the independent Poisson processes with intensity \(\textsc {j}\). The interpretation of the process \((\eta _t^{\delta ,+})\) is analogous but the removal of particles and holes is done at the beginning rather than at the end of each time interval. For a particle configuration \(\eta \in {\mathcal {X}}\) and positive integers \(a\) and \(b\) define the quantiles \(\textsc {l}^a(\eta )\) and \( \textsc {r}^b(\eta )\) as the lattice points satisfying

$$\begin{aligned} \sum _{x\ge \textsc {r}^b(\eta )}\eta (x)=b, \qquad \sum _{x\le \textsc {l}^a(\eta )}(1-\eta (x))=a. \end{aligned}$$
(5.7)

and define

$$\begin{aligned} \Gamma ^{a,b}(\eta )(x) = \eta (x)(1-{\mathbf { 1}}_{x\ge \textsc {r}^b})+ (1-\eta (x)){\mathbf { 1}}_{x\le \textsc {l}^a}; \end{aligned}$$
(5.8)

this is the configuration obtained from \(\eta \) by erasing particles and holes as explained above.

We abuse notation and write \(T_{[s,t]}\eta \) the evolution \(D(T_{[s,t]}\xi )\), with \(\eta =D\xi \). Then the delta particle processes satisfy the following: for \(n=0,1,\dots \),

$$\begin{aligned} \eta ^{\delta ,\pm }_t&= T_{[n\delta ,t]}\eta ^{\delta ,\pm }_{n\delta },\qquad \hbox {for }t\in [n\delta ,(n+1)\delta ) \nonumber \\ \eta ^{\delta ,-}_t&= \Gamma ^{A_{n\delta },B_{n\delta }}\eta ^{\delta ,-}_{t-},\qquad \hbox {for } t=n\delta \\ \eta ^{\delta ,+}_t&= \Gamma ^{A_{(n+1)\delta },B_{(n+1)\delta }}\eta ^{\delta ,+}_{t-},\qquad \hbox {for } t=n\delta \nonumber \end{aligned}$$
(5.9)

where recall \(A\) and \(B\) are independent Poisson processes of parameter \(\textsc {j}\).

The rescaled delta processes We consider a family of processes indexed by \(\varepsilon \) by considering \(\varepsilon ^{-2}\delta \) instead of \(\delta \) and \(\varepsilon j\) instead of \(\textsc {j}\). We call \((\xi ^{\varepsilon ,\delta ,\pm }_t)\) and \((\eta ^{\varepsilon ,\delta ,\pm }_t)\) the interface and particle processes so obtained. Later we consider those processes with time rescaled by a factor \(\varepsilon ^{-2}\) and space by a factor \(\varepsilon ^{-1}\).

For any fixed (macroscopic) \(\delta >0\), we will prove the existence of \(\phi ^{\delta ,\pm }_t\), the limit as \(\varepsilon \rightarrow 0\) of \(\varepsilon \xi ^{\varepsilon ,\delta ,\pm }_{t\varepsilon ^{-2}}\). We also prove that \(\phi ^{\delta ,\pm }_t\) are close to each other and that their difference vanishes as \(\delta \rightarrow 0\). Taking \(\delta \) to zero, their common limit \(\phi _t\) is then the hydrodynamic limit of the rescaled evolution \(\xi ^{(\varepsilon )}_t\) —as this is squeezed between \(\xi ^{\varepsilon ,\delta ,-}_t\) and \(\xi ^{\varepsilon ,\delta ,+}_t\).

While the above outline involves the interface process alone (which then implies convergence for the particle process as well), yet the analysis of the limit as \(\varepsilon \rightarrow 0\) of \((\xi ^{\varepsilon ,\delta ,\pm }_t)\) is more conveniently studied by looking at the delta particle process \((\eta ^{\varepsilon ,\delta ,\pm }_t)\) and then translating the results to the delta interface process. We start from the particle model defining first the corresponding approximate macroscopic density delta evolutions and then prove existence of the hydrodynamic limit for the delta particle process.

5.3 The macroscopic delta evolutions

In this subsection we fix \(\delta >0\) and define the macroscopic delta evolutions of densities \(\rho _t^{\delta ,\pm }\) and interfaces \(\phi ^{\delta ,\pm }_t\).

Preliminary results For any density \(\rho :{\mathbb {R}}\rightarrow [0,1]\) define the \({\mathbb { R}}_+ \cup \{+\infty \}\) valued functions

$$\begin{aligned} {F}(r;\rho ):=\int _r^\infty \rho (r')\,dr',\quad {\hat{F}}(r;\rho ):=\int _{-\infty }^r (1-\rho (r'))\,dr' \end{aligned}$$

representing the mass of \(\rho \) to the right of \(r\) and the antimass to its left. These are the macroscopic analogues of the number of particles, respectively holes, to the right, respectively left, of \(r\). We introduce two disjoint subsets of densities called \({\mathcal { R}}\) and \({\mathcal { U}}\).

Let \({\mathcal { R}}\) be the set of densities \(\rho \in L^\infty (\mathbb R, [0,1])\) satisfying the following conditions:

  1. (i)

    \({F}(0,\rho )= {\hat{F}}(0,\rho )<\infty \), that is mass to the right and antimass to the left are finite and the origin is the median of \(\rho \).

  2. (ii)

    \(\rho \) has finite boundaries. That is, there exist \(-\infty <\textsc {l}(\rho )\le \textsc {r}(\rho )<\infty \) such that \(\rho \) has finite support \([\textsc {l}(\rho ),\textsc {r}(\rho )]\): \(\rho \) put no mass to the right of \(\textsc {r}(\rho )\) and no antimass to the left of \(\textsc {l}(\rho )\).

  3. (iii)

    \(\rho \) is continuous in the interior of the support. That is, if \(\textsc {l}(\rho )<\textsc {r}(\rho )\) then \(\rho (r)\) is continuous in \((\textsc {l}(\rho ),\textsc {r}(\rho ))\) with values in \( (0,1)\).

We call \({\mathcal { R}}^*\) the set of \(\rho \in {\mathcal { R}}\) such that \(\textsc {r}(\rho )>0\).

Let \({\mathcal { U}}=\{u\in C({\mathbb { R}}, (0,1)):{F}(0,u)= {\hat{F}}(0,u)<\infty \}\). This is the set of continuous densities with median 0.

Let \(h\in {\mathcal {R}}\) be the Heaviside density defined by \(h(r) = {\mathbf { 1}}_{\{r\le 0\}}\). Clearly \(\textsc {l}(h)=\textsc {r}(h)=0\).

Lemma 6

\({F}(r;u)<\infty \) and \({\hat{F}}(r;u)<\infty \) for all \(r\in {\mathbb { R}}\) and \(u\in {\mathcal { U}}\cup {\mathcal { R}}\).

If \(\rho \in {\mathcal { R}}\) then \(\textsc {r}(\rho )\ge 0\) and \(\textsc {l}(\rho )\le 0\); the two inequalities are strict unless \(\rho =h\).

If \(\textsc {r}(\rho )>0\) then \(\textsc {l}(\rho )<0\) and the derivatives \({F}'(r;\rho )\) and \({\hat{F}}'(r;\rho )\) exist in \((\textsc {l}(\rho ),\textsc {r}(\rho ))\) where they are respectively strictly negative and positive.

If \(u\in {\mathcal { U}}\) then \({F},\,{\hat{F}}\in C^{1}({\mathbb { R}})\) and \({F}'(r;u)<0\) and \({\hat{F}}'(r;u)\ge 0\) for all \(r\in {\mathbb { R}}\).

For any \(u\in {\mathcal U}\cup {\mathcal R}\) and for any \(\delta >0\) there are unique points \(\textsc {r}^\delta (u),\textsc {l}^\delta (u)\) such that

$$\begin{aligned} {\hat{F}}(\textsc {l}^\delta ;u) = \delta ;\qquad {F}(\textsc {r}^\delta ;u) = \delta . \end{aligned}$$
(5.10)

If \({F}(0;u) \gtreqless \delta \) then \(\textsc {r}^{\delta }(u) \gtreqless 0\) and \(\textsc {l}^{\delta }(u) \lesseqgtr 0\).

Proof

Let \(u\in {\mathcal U}\cup {\mathcal R}\) then for any \(r\in {\mathbb R}\)

$$\begin{aligned} {F}(r;u) =\int _0^\infty u(r') \,dr' - \int _0^r u(r')\,dr'\le {F}(0;u) +|r| <\infty , \end{aligned}$$

with an analogous argument showing that also \({\hat{F}}(r;u)<\infty \). If \(\rho \in {\mathcal R}\) then \(\textsc {r}(\rho )\ge 0\) because if \(\textsc {r}(\rho )< 0\) then \({F}(0;\rho )=0\). Since \({F}(0;\rho )={\hat{F}}(0;\rho )\), then \({\hat{F}}(0;\rho )=0\) and this gives a contradiction since \(\textsc {r}(\rho )< 0\) and \(\rho \equiv 1\) is not allowed. Moreover \(\textsc {r}(\rho )=0\) if and only if \(\rho \) is the Heaviside density because \({F}(0;\rho )={\hat{F}}(0;\rho )\).

If \(u\in {\mathcal U}\) then \({F}'(r;u)=-u(r)<0\) and \({\hat{F}}'(r;u)=1-u(r)>0\), by the definition of \({\mathcal U}\). If \(\rho \in {\mathcal R}\) and \(\textsc {r}(\rho )>0\) then \(\textsc {l}(\rho )<0\) and by the definition of \({\mathcal R},\,\rho \) is continuous in \((\textsc {l}(\rho ),\textsc {r}(\rho ))\) and away from \(0\) and \(1\). Hence \({F}'(r;\rho )=-\rho (r)<0\) and \({\hat{F}}'(r;\rho )=1-\rho (r)>0\) for all \(r\in (\textsc {l}(\rho ),\textsc {r}(\rho ))\).

Let \(u\in {\mathcal U}\cup {\mathcal R}\). By the monotonicity of \({F}(r;u)\) if \({F}(0;u)>\delta \) then \(\textsc {r}^\delta (u)>0\), while if \({F}(0;u)<\delta \) then \(\textsc {r}^\delta (u)<0\) with the analogous property for \(\textsc {l}^\delta (u)\). \(\square \)

Definition of \(\Gamma ^\delta \) and \(G_t\). We call \(\Gamma ^\delta :{\mathcal U} \cup {\mathcal R}\rightarrow {\mathcal R}\) the following map. If \( \textsc {r}^{\delta }(u)\le 0\) and therefore \(\textsc {l}^{\delta }(u)\ge 0\) we set \(\Gamma ^\delta (u)=h\), the Heaviside density. If instead \( \textsc {r}^{\delta }(u)> 0\) and hence \( \textsc {l}^{\delta }(u)< 0\) we set \(\rho =\Gamma ^\delta (u)\) equal to \(0\) for \(r> \textsc {r}^{\delta }(u)\), equal to 1 for \(r< \textsc {l}^{\delta }(u)\) and equal to \(u\) elsewhere. In this latter case \(\rho =\Gamma ^\delta (u)\in {\mathcal R}^*\) (that is, \(\textsc {r}(\rho )=\textsc {r}^{\delta }(u)>0\)). Thus \(\Gamma ^\delta \) acts by removing a portion \(\delta \) of mass from the right of \(\textsc {r}^{\delta }(u)\) and put it back to the left of \(\textsc {l}^{\delta }(u)\).

Denote by \(G_t\) the Gaussian kernel:

$$\begin{aligned} G_t(r,r'):= \frac{1}{\sqrt{2\pi t}}{e^{-(r-r')^2/2t}} \end{aligned}$$
(5.11)

And write \(G_t\rho (r)= \int dr' G_t(r,r')\rho (r')\). Recall that \(G_t\rho \) is the solution of the heat equation (5.1) with initial data \(\rho \).

Lemma 7

Let \(\rho \in {\mathcal R}\cup {\mathcal U}\). Then \(G_t\rho \in {\mathcal U}\) for any \(t>0\). Moreover, calling

$$\begin{aligned} {\mathcal U}_{\delta }=\left\{ u\in {\mathcal U}: \textsc {l}^{\delta }(u)<0< \textsc {r}^{\delta }(u)\right\} \end{aligned}$$
(5.12)

for any \(j>0\) there is \(\delta (j)>0\) so that for any \(u \in {\mathcal R}\cup {\mathcal U}\) and \(\delta <\delta (j),\,G_\delta u\in {\mathcal U}_{j\delta }\) and therefore \(\Gamma ^{j\delta }(G_\delta u)\in {\mathcal R}^{*}\).

Proof

Since \(\rho \in {\mathcal R}\cup {\mathcal U}\), we have \({F}(0;\rho ) <\infty \). Then for any \(t>0\):

$$\begin{aligned} {F}(0;G_t\rho )&= \int _0^{\infty } dr \int _{-\infty }^{+\infty } dr' G_t(r,r') \rho (r') \\&\le \int _0^{\infty } dr \int _{0}^{+\infty } dr' G_t(r,r') \rho (r') + \int _0^{\infty } dr \int _{-\infty }^{0} dr' G_t(r,r') \le {F}(0;\rho ) + c, \end{aligned}$$

where we used Fubini and \(\int G_t(r,r') \,dr' =1\) to bound the first term. Since \(G_t\rho \in [0,1],\,{F}(r;G_t\rho ) <\infty \) for all \(r\) (see the beginning of the proof of Lemma 6). An analogous argument shows that \({\hat{F}}(r;G_t\rho ) <\infty \) for all \(r\) and \(t>0\) and, being the solution of the heat equation, \(G_t\rho \in C^{\infty }\) for all \(t>0\). To prove that \(G_t\rho \in {\mathcal U}\) it remains to show that \({F}(0;G_t\rho )={\hat{F}}(0;G_t\rho )\) for all \(t>0\). Using the symmetry properties of \(G_t\) and Fubini, for any \(t\ge 0\) we have

$$\begin{aligned} {F}(0;\rho _t)-{\hat{F}}(0;\rho _t)=\int _0^{\infty }\,dr\,G_t\rho (r) - \int _{-\infty }^0\,dr\,[1-G_t\rho (r)]= 0. \end{aligned}$$

The last statement in Lemma 7 follows from the following inequalities

$$\begin{aligned} \int _{\sqrt{\delta }}^\infty \,dr \,G_\delta u(r) \ge \int _{\sqrt{\delta }}^\infty \,dr \,G_\delta h(r) \ge C\, \sqrt{\delta },\quad C>0 \end{aligned}$$

which is larger than \(j\delta \) for \(\delta \) small enough. The first inequality follows from the fact that \(u\) is stochastically larger than the Heaviside density \(h\), in the sense that \(u\) is obtained from \(h\) by moving mass to the right. The last inequality follows from direct computations. \(\square \)

Delta density evolutions We are finally ready to define the delta density evolutions \(\rho ^{\delta ,\pm }_t\). Restrict to \(\delta \le \delta (j)\) as defined in Lemma 7, take the initial density \(\rho \in {\mathcal R}\cup {\mathcal U}\) and define iteratively \(\rho ^{\delta ,-}_0=\rho \) and

$$\begin{aligned} \rho ^{\delta ,-}_t:= {\left\{ \begin{array}{ll} G_{t-n\delta }\rho ^{\delta ,-}_{n\delta }, &{}t\in [n\delta ,(n+1)\delta ) \\ \Gamma ^{j\delta }\rho ^{\delta ,-}_{t-},&{} t=n\delta . \end{array}\right. } \end{aligned}$$
(5.13)

The evolution \(\rho _t^{\delta ,+}\) is defined as \(\rho ^{\delta ,-}_t\) but with initial datum \(\rho _0^{\delta ,+}:=\Gamma ^{j\delta }(\rho )\).

Delta interface evolutions The delta interface evolutions are defined as follows. Fix an initial interface \(\phi \) belonging to the cone with vertex at the origin and for \(t\ge 0\) define iteratively \(\phi ^{\delta ,-}_0=\phi ,\,\phi ^{\delta ,+}_0=\max \{\phi ,V_{(0,\delta j)}\}\) and for \(n\ge 0\),

$$\begin{aligned} \phi ^{\delta ,\pm }_t&:= G_{t-n\delta }\phi ^{\delta ,-}_{n\delta },\qquad \text {if}\ t\in [n\delta ,(n+1)\delta ) \nonumber \\ \phi ^{\delta ,-}_t&:= \max \{\phi ^{\delta ,-}_{t-},V_{(0,n \delta j)}\},\qquad \text {if}\ t=n\delta .\\ \phi ^{\delta ,+}_t&:= \max \{\phi ^{\delta ,+}_{t-},V_{(0,(n+1) \delta j)}\},\qquad \text {if}\ t=n\delta . \nonumber \end{aligned}$$
(5.14)

We leave the proof of the following Lemma to the reader. It relates both definitions.

Lemma 8

The delta density evolutions \(\rho ^{\delta ,\pm }_t\) defined in (5.13) and the delta interface evolutions \(\phi ^{\delta ,\pm }_t\) defined in (5.14) are related by

$$\begin{aligned} \phi ^{\delta ,\pm }_{t}(r)- \phi ^{\delta ,\pm }_{t}(r') = 2 (r-r') - 2 \int _{r'}^r \rho ^{\delta ,\pm }_{t}(r'')\,dr''. \end{aligned}$$
(5.15)

The initial data are related by \(\phi (0) = \int _{-\infty }^0 (1-\rho (r))\,dr\) (this is the same as \(\int ^{\infty }_0 \rho (r) dr\) as \(\rho \in {\mathcal {R}})\); this fixes the vertex of the cone of \(\phi \) at the origin.

5.4 Hydrodynamic limit of the delta particle process

We now study the hydrodynamic limit for the delta particle process defined in Sect. 5.2.

We introduce partitions \({\mathcal D}^{(\ell )}\) of \({\mathbb {Z}}\) into intervals \(I^{(\ell )}\) of length \(\ell \) where, denoting by \(I^{(\ell )}_x\) the interval which contains \(x,\,I^{(\ell )}_0=[0,\ell -1]\) (\({\mathcal D}^{(\ell )}\) is now completely specified). We take \(\ell \) equal to the integer part of \(\varepsilon ^{-\beta }\) with \(\beta \in (0,1),\,\rho \in L^\infty ( {\mathbb R},[0,1])\) and (by an abuse of notation) we write

$$\begin{aligned} {\mathcal A}^{(\ell )}_x(\eta )= \frac{1}{\ell } \sum _{y\in I^{(\ell )}_x} \eta (y),\quad {\mathcal A}^{(\ell )}_x(\rho )= \frac{1}{\varepsilon \ell } \int _{\varepsilon I^{(\ell )}_x} \rho (r)\,dr \end{aligned}$$
(5.16)

not making explicit the dependence on \(\varepsilon \).

Introduce an accuracy parameter of the form \(\varepsilon ^{\alpha },\,0<\alpha <\beta \); the parameter \(\beta \in (0,1)\) is fixed while \(\alpha \) will change at each step of the iteration scheme used in the sequel. Let \({\mathcal G}_{\varepsilon ,\alpha ,\beta } (\rho )\)be the set of particle configurations which \((\varepsilon ,\alpha ,\beta )\)-recognize the macroscopic density \(\rho \in {\mathcal R}\) defined by:

$$\begin{aligned} {\mathcal G}_{\varepsilon ,\alpha ,\beta } (\rho )&= \left\{ \eta \in {\mathcal {X}}: |\varepsilon \textsc {l}(\eta )-\textsc {l}(\rho )| + |\varepsilon \textsc {r}(\eta )-\textsc {r}(\rho )|\right. \nonumber \\&\quad \left. \le \varepsilon ^{\alpha }, \sup _{x\in {\mathbb {Z}}{\setminus }\{I_\textsc {r}\cup I_\textsc {l}\}}| {\mathcal A}^{(\ell )}_x(\eta )- {\mathcal A}^{(\ell )}_x(\rho )| \le \varepsilon ^{\alpha } \right\} ,\nonumber \\ \qquad \qquad \qquad \ell&= \text { integer part of }\varepsilon ^{-\beta },\;\;0<\alpha <\beta <1, \end{aligned}$$
(5.17)

where \( \textsc {l}(\eta )\) and \(\textsc {r}(\eta )\) are defined in (2.1), and \(I_\textsc {l}\) is the smallest \({\mathcal D}^{(\ell )}\) measurable interval which contains both \(\textsc {l}(\eta )\) and \(\varepsilon ^{-1} \textsc {l}(\rho ); \, I_\textsc {r}\) is defined analogously with reference to \(\textsc {r}(\eta )\) and \(\textsc {r}(\rho )\).

Proposition 3

Let \(\rho \in {\mathcal R}\) and \(\eta ^{\varepsilon ,\delta ,\pm }_0\in {\mathcal G}_{\varepsilon ,\alpha ,\beta }(\rho )\), then for any \(\alpha '\in (0,\alpha )\) such that \(\alpha '< \min \{\frac{\beta }{2},1-\beta ,\frac{1}{4}\}\) the following holds: for any \(k\ge 1\) there are coefficients \(c_k\) so that

$$\begin{aligned} P\left[ \eta ^{\varepsilon ,\delta ,\pm }_{\varepsilon ^{-2}\delta } \in {\mathcal G}_{\varepsilon ,\alpha ',\beta }(\rho ^{\delta ,\pm }_\delta )\right] \ge 1 -c_k\varepsilon ^k \end{aligned}$$
(5.18)

Remark

By iteration the result extends to any finite macroscopic time interval and we also have:

Corollary 3

Under the same assumptions of Proposition 3, for any integer \(m\ge 1\) and for any \(k\ge 1\) there are coefficients \(c_k\) so that

$$\begin{aligned} P\left[ \bigcap _{n=1}^m\{\eta ^{\varepsilon ,\delta ,\pm }_{\varepsilon ^{-2}n\delta } \in {\mathcal G}_{\varepsilon ,\alpha ',\beta }(\rho ^{\delta ,\pm }_{n\delta })\} \right] \ge 1 -c_k\varepsilon ^k \end{aligned}$$
(5.19)

To show Proposition 3 we need to control the position of the quantiles \(\textsc {r}^a\) and \(\textsc {l}^b\) of the process evolving with the exclusion by the macroscopic time \(\delta \varepsilon ^{-2}\). Here \(a= A_{\varepsilon ^{-2}t},\,b= B_{\varepsilon ^{-2}t}\) which are Poisson processes of parameter \(\varepsilon j\). These bonds only depend on the exclusion dynamics governed by \(L_0\).

Sharp convergence of the exclusion process to the solution of the heat equation Abusing notation denote \((T_t\eta )\) the process in \({\mathcal {X}}\) with initial configuration \(\eta \) evolving only with the exclusion generator \(L_0\). The SSEP evolution is close to a linear diffusion in the following sense: for any \(n\ge 2\) there is \(c_n\) so that for any \(t>0\)

$$\begin{aligned} \sup _{(x_1,\dots ,x_n)\in {\mathbb {Z}}_{\ne }^n} | v(x_1,\dots ,x_n;t)| \le c_n t^{-n/8} \end{aligned}$$
(5.20)

where \({\mathbb {Z}}_{\ne }^n\) is the set of all \(n\)-tuple of mutually distinct elements of \({\mathbb {Z}}\),

$$\begin{aligned} v(x_1,\dots ,x_n;t) = E \left[ \prod _{i=1}^n \{T_t\eta (x_i) - u_t(x_i)\}\right] \end{aligned}$$
(5.21)

hereafter called \(v\)-functions, and \(u_t(x)\) solves the discretized heat equation

$$\begin{aligned} \frac{du_t(x)}{dt}= \frac{1}{2}\, \Delta u_t(x)= \frac{1}{2}\left( u_t(x+1)+u_t(x-1)-2u_t(x)\right) ,\quad u_0=\eta \qquad \end{aligned}$$
(5.22)

(5.20) is proved in [4]. The solution of (5.22) is

$$\begin{aligned} u_{t}(x) = \sum _{y\in {\mathbb {Z}}} p_{t}(x,y) \eta (y) \end{aligned}$$

\(p_t(x,y)\) the transition probability kernel of the symmetric nearest-neighbors random walk. The solution of the heat equation starting from \(\rho \) is \(G_t\rho \) (recall (5.11)). Thus, since \(\eta \in {\mathcal G}_{\varepsilon ,\alpha ,\beta }(\rho )\)

$$\begin{aligned} |u_{\varepsilon ^{-2}t}(x)-G_t\rho (\varepsilon x)|&\le c' \left( \varepsilon t^{-1/2} +\varepsilon ^{1-\beta }t^{-1/2}+\varepsilon ^\alpha + \varepsilon t^{-1/2}\varepsilon ^{-(1-\alpha )} \right) \nonumber \\&\le c \left( \varepsilon ^{1-\beta }t^{-1/2}+\varepsilon ^\alpha \right) \end{aligned}$$
(5.23)

The proof of the first inequality is done by changing \(u_{\varepsilon ^{-2}t}(x)\) into \(G_t\rho (\varepsilon x)\) in successive steps:

  • Replace \(p_{\varepsilon ^{-2}t}(x,y)\) by \(G_t(\varepsilon x, \varepsilon y)\). By the local central limit theorem the error is bounded by the first term on the right hand side of (5.23).

  • Replace \(G_t(\varepsilon x, \varepsilon y)\) by its average in the intervals \(\varepsilon I_z^{(\ell )}\) of length \(\varepsilon ^{1-\beta }\), hence the second term on the right hand side of (5.23).

  • The contribution of the difference between averages of \(\eta \) and \(\rho \) in good intervals (i.e those not in \(I_\textsc {l}\cup I_\textsc {r}\)) is bounded by \(\varepsilon ^\alpha \), the contribution of the intervals in \(I_\textsc {l}\cup I_\textsc {r}\) by \(\varepsilon t^{-1/2} \varepsilon ^{-(1-\alpha )}\).

  • We finally reconstruct in each interval \(\varepsilon I_z^{(\ell )}\) the correct term from \(G_t\rho (\varepsilon x)\) with an error given again by the second term on the right hand side of (5.23).

Bounds on \(|{\mathcal A}^{(\ell )}_x(T_{\varepsilon ^{-2}t}\eta )- {\mathcal A}^{(\ell )}_x(\rho _t)|\). For any \(x\in {\mathbb Z}\)

$$\begin{aligned} |{\mathcal A}^{(\ell )}_x(T_{\varepsilon ^{-2}t}\eta )\!-\! {\mathcal A}^{(\ell )}_x(\rho _t)|&\!\le \!&|{\mathcal A}^{(\ell )}_x(T_{\varepsilon ^{-2}t}\eta )\!-\! {\mathcal A}^{(\ell )}_x(u_{\varepsilon ^{-2}t})|\!+\!|{\mathcal A}^{(\ell )}_x(u_{\varepsilon ^{-2}t})- {\mathcal A}^{(\ell )}_x(\rho _t)| \nonumber \\&\!\le \!&|{\mathcal A}^{(\ell )}_x(T_{\varepsilon ^{-2}t}\eta )\!-\! {\mathcal A}^{(\ell )}_x(u_{\varepsilon ^{-2}t})|\!+\! c \left( \varepsilon ^{1-\beta }t^{-1/2}\!+\!\varepsilon ^\alpha \right) ,\nonumber \\ \end{aligned}$$
(5.24)

by (5.23). We are going to show that for any integer \(n\),

$$\begin{aligned} E\left[ |{\mathcal A}^{(\ell )}_x(T_{\varepsilon ^{-2}t}\eta )- {\mathcal A}^{(\ell )}_x(u_{\varepsilon ^{-2}t})|^{2n}\right] \le c \left( \varepsilon ^{\beta n} + [t\varepsilon ^{-2}]^{-n/4}\right) \end{aligned}$$
(5.25)

Proof of (5.25):

  • We expand \(|{\mathcal A}^{(\ell )}_x(T_{\varepsilon ^{-2}t}\eta )- {\mathcal A}^{(\ell )}_x(u_{\varepsilon ^{-2}t})|^{2n}\) getting a sum of products of factors \(\eta _{\varepsilon ^{-2} t}(z) - u_{\varepsilon ^{-2}t}(z)\).

  • Each term of the form \((\eta _{\varepsilon ^{-2} t}(x) - u_{\varepsilon ^{-2}t}(x))^k\) with \(k>1\) can be rewritten as \(c + c'(\eta _{\varepsilon ^{-2} t}(x) - u_{\varepsilon ^{-2}t}(x))\) with constants \(c\) and \(c'\) not depending on \(\eta \). \(c\) and \(c'\) depend on the value of \(u_{\varepsilon ^{-2}t}(x)\) but that each of them is always smaller (in absolute value) than one.

  • Thus \(E[|{\mathcal A}^{(\ell )}_x(T_{\varepsilon ^{-2}t}\eta )- {\mathcal A}^{(\ell )}_x(u_{\varepsilon ^{-2}t})|^{2n}]\) is a sum of product of constants times \(v\)-functions. We then use (5.20) to get (5.25).

Let \(\gamma >0\), then since \(\eta \in {\mathcal G}_{\varepsilon ,\alpha ,\beta }(\rho )\),

$$\begin{aligned} \sum _{x\ge \varepsilon ^{-1-\gamma }}P[ T_{\varepsilon ^{-2}t}\eta (x)=1] \le c'_k\varepsilon ^k,\quad \sum _{x\le -\varepsilon ^{-1-\gamma }}P [ T_{\varepsilon ^{-2}t}\eta (x)=0] \le c'_k\varepsilon ^k \end{aligned}$$

As a consequence

$$\begin{aligned} P\left[ \textsc {r}(T_{\varepsilon ^{-2}t}\eta )\le \varepsilon ^{-1-\gamma }; \;\;\textsc {l}(T_{\varepsilon ^{-2}t}\eta )\ge -\varepsilon ^{-1-\gamma }\right] \ge 1- c''_k\varepsilon ^k \end{aligned}$$
(5.26)

By the hypotheses on \(\rho \),

$$\begin{aligned} \int _{r\ge \varepsilon ^{-\gamma }}G_t\rho (r)\,dr \le c'_k\varepsilon ^k,\quad \int _{r\le -\varepsilon ^{-\gamma }}\,dr [1- G_t\rho (r)] \le c'_k\varepsilon ^k. \end{aligned}$$
(5.27)

which proves that

$$\begin{aligned} P\left[ \sup _{|x|\ge \varepsilon ^{-1-\gamma }} | {\mathcal A}^{(\ell )}_x(T_{\varepsilon ^{-2}t}\eta )- {\mathcal A}^{(\ell )}_x(G_t\rho )| \le c'_k\varepsilon ^k\right] \ge 1 -c'''_k\varepsilon ^k. \end{aligned}$$
(5.28)

We shall use (5.28) to prove that for any \(\alpha '\) as in Proposition 3

$$\begin{aligned} P\left[ \sup _{x\in {\mathbb {Z}}}| {\mathcal A}^{(\ell )}_x(T_{\varepsilon ^{-2}t}\eta )- {\mathcal A}^{(\ell )}_x(G_t\rho )| \le \varepsilon ^{\alpha '}\right] \ge 1 -c_k\varepsilon ^k. \end{aligned}$$
(5.29)

By (5.28) and (5.24) it suffices to prove that for any \(\alpha '\) as above

$$\begin{aligned} P\left[ \sup _{|x|\le \varepsilon ^{-1-\gamma }}| {\mathcal A}^{(\ell )}_x(T_{\varepsilon ^{-2}t}\eta )- {\mathcal A}^{(\ell )}_x(u_{\varepsilon ^{-2}t})| \le \varepsilon ^{\alpha '}\right] \ge 1 -c_k\varepsilon ^k. \end{aligned}$$
(5.30)

which follows using the Chebishev’s inequality with power \(2n\) for \(n\) sufficiently large and (5.25), because \(\alpha '< \min \{\frac{\beta }{2},\frac{1}{4}\}\).

Quantile bounds To complete the proof of (5.18) we fix \(\alpha '\) as in Proposition 3 and take \(\alpha '' < \min \{\frac{\beta }{2},1-\beta ,\alpha ,\frac{1}{4}\}\) such that \(\alpha '' >\alpha '\). Then there is a positive \(\gamma \) such that \(\alpha ''-2\gamma >\alpha '\).

We now fix \(t=\delta \) and use (5.29) (with \(\alpha ''\)) to obtain bounds for the quantiles defined in (5.10) and (5.7). Recalling (5.10) let

$$\begin{aligned} \textsc {r}':= \varepsilon ^{-1}\textsc {r}^{j\delta }(G_\delta \rho ) + \varepsilon ^{-1+\alpha ''-2\gamma }. \end{aligned}$$

Then

$$\begin{aligned} P\left( \left| \sum _{\textsc {r}'\le x \le \varepsilon ^{-1-\gamma }} T_{\varepsilon ^{-2}\delta }\eta (x) - \varepsilon ^{-1}\int _{\varepsilon \textsc {r}'}^ {\varepsilon ^{-\gamma }}G_\delta \rho ( r)\,dr \right| \le c \varepsilon ^{\alpha ''} \varepsilon ^{-1-\gamma }\right) \ge 1 - c_k\varepsilon ^k, \end{aligned}$$

On the other hand by the definition of the quantile \(\textsc {r}^{j\delta }(G_\delta \rho )\) and by (5.27)

$$\begin{aligned} \int _{\varepsilon \textsc {r}'}^ {\varepsilon ^{-\gamma }}G_\delta \rho ( r)\,dr&= \int _{\textsc {r}^{j\delta }(G_\delta \rho )}^ {\infty }G_\delta \rho ( r)\,dr- \int _{\textsc {r}^{j\delta }(G_\delta \rho )}^ {\varepsilon \textsc {r}'}G_\delta \rho ( r)\,dr-\int _{\varepsilon ^{-\gamma }}^ {\infty }G_\delta \rho ( r)\,dr \\&\le j\delta -c' \varepsilon ^{\alpha ''-2\gamma } \end{aligned}$$

with \(c'=\min \{G_\delta \rho (r):|r-\textsc {r}^{j\delta }(G_\delta \rho )|\le 1 \}>0\). Hence with probability \(\ge 1 - c_k\varepsilon ^k\),

$$\begin{aligned} \varepsilon \sum _{\textsc {r}'\le x \le \varepsilon ^{-1-\gamma }} T_{\varepsilon ^{-2}\delta }\eta (x) \le j\delta -c' \varepsilon ^{\alpha ''-2\gamma } + c \varepsilon ^{\alpha ''} \varepsilon ^{-\gamma } < j\delta -\frac{c'}{2} \varepsilon ^{\alpha ''-2\gamma } \end{aligned}$$

Let \(\textsc {r}^b(\eta )\) be the quantile defined in (5.7) with \(b=B_{\varepsilon ^{-2}\delta },\,B\) a Poisson process of rate \(j\varepsilon \). Observing that for any \(\kappa >0\)

$$\begin{aligned} P\left( \left| \varepsilon B_{\varepsilon ^{-2}\delta }-j\delta \right| \le \varepsilon ^{\frac{1}{2} -\kappa }\right) \ge 1-c_k\varepsilon ^k \end{aligned}$$

we get that for \(\kappa \) small enough and with probability \(\ge 1 - c_k\varepsilon ^k\),

$$\begin{aligned} \varepsilon \sum _{ x \ge \textsc {r}^b(T_{\varepsilon ^{-2}\delta }\eta )} T_{\varepsilon ^{-2}\delta }\eta (x)&= \varepsilon B_{\varepsilon ^{-2}\delta } \ge j\delta - \varepsilon ^{\frac{1}{2}-\kappa } \ge j\delta -\frac{c'}{2} \varepsilon ^{\alpha ''-2\gamma }\\&\ge \varepsilon \sum _{\textsc {r}'\le x \le \varepsilon ^{-1-\gamma }} T_{\varepsilon ^{-2}\delta }\eta (x) \end{aligned}$$

that implies \(\textsc {r}^b(T_{\varepsilon ^{-2}\delta }\eta )\ge R'= \varepsilon ^{-1}\textsc {r}^{j\delta }(G_\delta \rho ) + \varepsilon ^{-1+\alpha ''-2\gamma }\). Using an analogous argument for the lower bound we get

$$\begin{aligned} \begin{aligned}&P(|\varepsilon \textsc {r}^b(T_{\varepsilon ^{-2}\delta }\eta ) - \textsc {r}^{j\delta }(G_\delta \rho )|\le \varepsilon ^{\alpha ''-2\gamma }) > 1-c_k\varepsilon ^k, \\&P(|\varepsilon \textsc {l}^a(T_{\varepsilon ^{-2}\delta }\eta ) - \textsc {l}^{j\delta }(G_\delta \rho )|\le \varepsilon ^{\alpha ''-2\gamma }) > 1-c_k\varepsilon ^k; \end{aligned} \end{aligned}$$
(5.31)

the second inequality is proved by using the same arguments for \(a=A_{\varepsilon ^{-2}\delta },\,A\) being a Poisson process of rate \(\varepsilon j\).

Proof of Proposition 3

By the definitions (5.9) and (5.13),

$$\begin{aligned} \eta ^{\varepsilon ,\delta ,-}_{\varepsilon ^{-2}\delta } = \Gamma ^{A_{\varepsilon ^{-2}\delta },B_{\varepsilon ^{-2}\delta }}T_{\varepsilon ^{-2}\delta }\eta ,\qquad \rho ^{\delta ,-}_\delta = \Gamma ^{j\delta }G_\delta \rho . \end{aligned}$$

Since the left and right boundaries after applying \(\Gamma \) are the quantiles before applying it, inequality (5.18) for the delta\(-\) processes follows from (5.29) and (5.31). The same argument applies for the delta\(+\) processes. \(\square \)

5.5 Hydrodynamic limit of interfaces

Proof of Theorem 4

We call \(\tau >0\) the time \(t\) fixed in Theorem 4. For each \(n\in {\mathbb N}\) we let \(\delta \in \{ \tau 2^{-n} \}\) and consider the evolutions \((\xi _t^{\varepsilon ,\delta ,\pm })\) in a bounded time interval, \(t\le T= 2^{N+n}\delta =2^N\tau ,\,N\) an arbitrary, fixed non negative integer. We have by (4.14),

$$\begin{aligned}&\varepsilon \xi ^{\varepsilon ,\delta ,\pm }_{\varepsilon ^{-2}(k+1)\delta }(0)-\varepsilon \xi ^{\varepsilon ,\delta ,\pm }_{\varepsilon ^{-2}k\delta }(0) \nonumber \\&\qquad = 2 \varepsilon B_{\varepsilon ^{-2}(k+1)\delta } -2\varepsilon B_ {\varepsilon ^{-2}k\delta } + 2 \varepsilon \sum _{x\ge 0} \eta ^{\varepsilon ,\delta ,\pm }_{\varepsilon ^{-2}(k+1)\delta }(x) - 2 \varepsilon \sum _{x\ge 0} \eta ^{\varepsilon ,\delta ,\pm }_{\varepsilon ^{-2}k\delta }(x)\qquad \qquad \end{aligned}$$
(5.32)

By (4.7) for \(k=1,\dots ,2^{n+N}\) and \(x>y\),

$$\begin{aligned} \varepsilon \xi ^{\varepsilon ,\delta ,\pm }_{\varepsilon ^{-2}k\delta }(x)-\varepsilon \xi ^{\varepsilon ,\delta ,\pm }_{\varepsilon ^{-2}k\delta }(y) = 2 {\varepsilon (x-y)} - 2\varepsilon \sum _{z=y}^{x-1}\eta ^{\varepsilon ,\delta ,\pm }_{\varepsilon ^{-2}k\delta }(z) \end{aligned}$$
(5.33)

By (5.19) and (5.32)–(5.33)–(5.15) we then get that for any \(\gamma >0\) and any \(t\in \{k\delta : k\le 2^{N+n}\}\)

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0}P\left[ \sup _{x\in {\mathbb Z}} |\varepsilon \xi ^{\varepsilon ,\delta ,\pm }_{\varepsilon ^{-2}t}(x)- \phi ^{\delta ,\pm }_{t}(\varepsilon x)| \ge \gamma \right] =0 \end{aligned}$$
(5.34)

In the next section we shall prove that for any \(t\):

$$\begin{aligned} \lim _{n\rightarrow \infty } \sup _{r\in {\mathbb R}} \left| \phi ^{\tau 2^{-n},+}_{t}(r)-\phi ^{\tau 2^{-n},-}_{t}(r)\right| =0 \end{aligned}$$
(5.35)

and that there is a function \(\phi ^{(\tau )}_t(r),\,r\in {\mathbb R}, t\ge 0\), so that

$$\begin{aligned} \lim _{n\rightarrow \infty } \sup _{r\in {\mathbb R}} \left| \phi ^{(\tau )}_t(r)-\phi ^{\tau 2^{-n},-}_{t}(r)\right| =0 \end{aligned}$$
(5.36)

Then by (5.34), (5.35), (5.36) and (5.6), for all \(\gamma >0\),

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0} P\left[ \sup _{x\in {\mathbb {Z}}} |\varepsilon \xi ^{(\varepsilon )}_{\varepsilon ^{-2}t}(x)- \phi ^{(\tau )}_t(\varepsilon x)| \ge \gamma \right] = 0, \quad t\in {\mathcal T}(\tau ) \end{aligned}$$
(5.37)

where

$$\begin{aligned} {\mathcal T}(\tau )=\{k2^{-n}\tau , k\in {\mathbb N}, n\in {\mathbb N}\} \end{aligned}$$
(5.38)

Since \(\tau \in {\mathcal T}(\tau )\) we have proved (5.4) for \(t=\tau \) and since \(\tau \) was arbitrary, Theorem 4 is proved. \(\square \)

6 The macroscopic evolution

In Sect. 6.1 we prove that as \(\delta \rightarrow 0\) the macroscopic delta processes converge—that is, we prove (5.35) and (5.36)—and that \(\phi _t\) is well defined by

$$\begin{aligned} \phi _{t} =\lim _{\delta \rightarrow 0} \phi ^{\delta ,\pm }_{t},\qquad t\ge 0 \end{aligned}$$

We also collect some properties of the macroscopic evolutions \(\phi _t\) and \(\rho _t\), in particular monotonicity properties of \(\phi _t\) and existence of boundary points for both motions. In Sect. 6.2 we construct macroscopic stationary profiles.

6.1 Existence and regularity of the macroscopic profiles

Proof of (5.35) and (5.36) Let \(\tau >0\) and \(\delta \in \{\tau 2^{-n}, n\in {\mathbb N}\}\). We shall first prove by induction on \(k\) that for any such \(\delta \),

$$\begin{aligned} \sup _{r\in {\mathbb R}} \left| \phi ^{\delta ,+}_{k\delta }(r)-\phi ^{\delta ,-}_{k\delta }(r)\right| \le j\delta . \end{aligned}$$
(6.1)

(6.1) holds for \(k=0\) because

$$\begin{aligned} \phi ^{\delta ,+}_{0}(r) = \max \{ \phi _0(r),j\delta +|r|\},\quad \phi ^{\delta ,-}_{0}(r) = \phi _0(r). \end{aligned}$$

Suppose next that (6.1) holds for \(k-1\), then by the maximum principle (for the linear heat equation),

$$\begin{aligned} \left| \phi ^{\delta ,+}_{(k\delta )-}(r)-\phi ^{\delta ,+}_{(k\delta )-}(r)\right| \le j\delta \end{aligned}$$

hence (6.1) holds for \(k\) because

$$\begin{aligned} \phi ^{\delta ,+}_{k \delta }(r) = \max \left\{ \phi ^{\delta ,+}_{(k \delta ){-}}(r),j(k+1)\delta +|r|\right\} ,\quad \phi ^{\delta ,-}_{k \delta }(r) = \max \left\{ \phi ^{\delta ,-}_{(k \delta ){-}}(r),jk\delta + |r|\right\} \end{aligned}$$

(6.1) and (5.35) are thus proved. It is not difficult to see that

$$\begin{aligned} \phi ^{\delta ,-}_{t}(r)\le \phi ^{\delta ',-}_{t}(r),\quad \phi ^{\delta ,+}_{t}(r)\ge \phi ^{\delta ',-}_{t}(r),\qquad \delta =k\delta ' { \text{ for } \text{ some } \text{ integer } }k>0 \end{aligned}$$

Thus for any \(n\in {\mathbb N}\) and \(t\ge 0\)

$$\begin{aligned} \phi ^{\tau 2^{-n},-}_{t}(r)\le \phi ^{\tau 2^{-(n+1)},-}_{t}(r) \le \phi ^{\tau 2^{-n} ,+}_{t}(r) \end{aligned}$$
(6.2)

Hence for any fixed \(t,\,\phi ^{\tau 2^{-n},-}_{t}(r)\) converges pointwise to a function that we call \(\phi _t^{(\tau )}(r)\) (it may depend on \(\tau \)). Since by definition \(|\phi ^{\delta ,\pm }_t(r)-\phi ^{\delta ,\pm }_t(r')|\le |r-r'|\), the convergence is in sup norm and (5.36) is then proved with \(\phi _t^{(\tau )}(r)\) a Lipschitz function with Lipschitz constant 1. \(\square \)

In the next Theorem we prove that \(\phi _t^{(\tau )}\) is independent of \(\tau \) and also regularity properties of this function.

Theorem 5

The function \(\phi _t^{(\tau )}\) is independent of \(\tau \) and will be denoted by \(\phi _t\) (the same as in Theorem 4). \(\phi _t\) is continuous in \(r\) and \(t\), more precisely there is \(c>0\) so that for all \(t,\,t'\) such that \(|t-t'|\le 1\) and all \(r\) and \(r'\),

$$\begin{aligned} |\phi _{t}(r) -\phi _{t'}(r) |\le c \sqrt{|t-t'|},\;\;\; |\phi _{t}(r) -\phi _{t}(r ') |\le |r-r'| \end{aligned}$$
(6.3)

Denoting by \(\delta _n\) any sequence of positive numbers such that \(\delta _{n+1}=\delta _n/2\) then

$$\begin{aligned} \phi _{t} =\lim _{n\rightarrow \infty } \phi ^{\delta _n,\pm }_{t},\qquad \forall t\ge 0, \end{aligned}$$
(6.4)

with \(\phi ^{\delta _n,-}_{t}\) monotonically increasing and \(\phi ^{\delta _n,+}_{t}\) monotonically decreasing.

Proof

Let \(t\ge 0\) and \(s> 0\), recalling (5.11) we have

$$\begin{aligned} G_s\phi ^{\delta ,-}_{t}(r) \le \phi ^{\delta ,-}_{t+s}(r)\le G_s\phi ^{\delta ,-}_{t}(r) + j (s+\delta ). \end{aligned}$$
(6.5)

The first inequality is obvious. We have

$$\begin{aligned} \phi _{(k+1)\delta }^{\delta ,-}&= \max \{ G_\delta \phi _{k\delta }^{\delta ,-}, j(k+1)\delta +|r|\} \le \max \{ G_\delta \phi _{k\delta }^{\delta ,-}+j\delta , j(k+1)\delta +|r|\} \\&= j\delta + G_\delta \phi _{k\delta }^{\delta ,-}, \end{aligned}$$

because \(\phi _{k\delta }^{\delta ,-} \ge jk\delta +|r|\). We then get the last inequality (without the term \(j\delta \)) for \(t=h\delta ,\,h\in {\mathbb N}\). If instead \(t\in (h\delta , (h+1)\delta )\), then

$$\begin{aligned} \phi ^{\delta ,-}_{(h+1)\delta } \le G_{(h+1)\delta -t} \phi ^{\delta ,-}_t + j\delta \end{aligned}$$

hence (6.5).

Since \( \phi ^{\delta ,-}_t(r)\) is Lipschitz, it follows from (6.5) that \(| \phi ^{\delta ,-}_{t+s}(r)- \phi ^{\delta ,-}_{t}(r)| \le c \sqrt{s} + j (s+j\delta )\) and, by taking \(\delta \rightarrow 0\),

$$\begin{aligned} |\phi ^{(\tau )}_{t}(r) -\phi ^{(\tau )}_{t'}(r) |\le c \sqrt{|t-t'|},\;\;\;\text {for all}\;r\;\text {and}\;|t-t'|\le 1 \end{aligned}$$
(6.6)

We shall next prove that \(\phi ^{(\tau )}_{t}(r)\) is independent of \(\tau \). Obviously \(\phi ^{(\tau )}_{t}(r)=\phi ^{(\tau ')}_{t}(r)\) if, recalling (5.38), \(\tau '\in {\mathcal T}(\tau )\) (or viceversa). We next suppose that \(\tau \) and \(\tau '\) are not related in such a way. We fix \(T>0\) and want to prove that \(\phi ^{(\tau )}_T(r)=\phi ^{(\tau ')}_T(r)\). Let \(\delta \le \delta '\) and \(k\) such that \(k\delta <\delta ' <(k+1)\delta \). Then

$$\begin{aligned} \phi _{\delta '}^{\delta ',-} (r)&= \max \{ G_{\delta '-k\delta } \phi _{k\delta }^{\delta ',-} (r), j\delta ' +|r|\}\\&\le \max \{ G_{\delta '-k\delta } \phi _{k\delta }^{\delta ,-} (r)+j(\delta '-k\delta ), j\delta ' +|r|\} \le \phi _{\delta '}^{\delta ,-} (r)+j(\delta '-k\delta ) \end{aligned}$$

because \(\phi _{k\delta }^{\delta ,-} (r) \ge jk\delta +|r|\).

By iteration \(\phi _{T}^{\delta ',-} \le \phi _{T}^{\delta ,-} + j N \delta \) if \(N\) is the cardinality of \(\{k: k\delta ' \le T\}\). Thus

$$\begin{aligned} \phi _{T}^{\delta ',-} \le \phi _{T}^{\delta ,-} + c T \frac{\delta }{\delta '} \end{aligned}$$

Take \(\delta '= \tau ' 2^{-n'}\) and \(\delta = \tau 2^{-n}\). Take first \(n\rightarrow \infty \) and then \(n'\rightarrow \infty \) to get \(\phi _T^{(\tau ')} \le \phi _T^{(\tau )}\). The opposite inequality holds as well by interchanging \(\delta \) and \(\delta '\) in the previous argument.\(\square \)

Proof of Theorem 1

From (5.4) we get for all \(\gamma >0\)

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0} P\left( \sup _{a< b}\left| \varepsilon \sum _{\varepsilon x\in [a,b]}\eta ^{(\varepsilon )}_{t\varepsilon ^{-2}}(x) -\frac{1}{2}\left\{ \varepsilon (b-a)- [\phi _t(\varepsilon b)-\phi _t(\varepsilon a)]\right\} \right| >\gamma \right) =0 \end{aligned}$$

Since \(\phi _t\) is Lipschitz there is \(\rho _t \in L^1\) such that, given any \(r_0\in {\mathbb {R}}\),

$$\begin{aligned} \phi _{t}(r) =\phi _{t}(r_0) + \int _{r_0}^r \left( 1-2 \rho _t(r') \right) \,dr' \end{aligned}$$
(6.7)

and since by (6.3) the Lipschitz constant is \(1,\,\rho _t\) has (almost surely) values in \([0,1]\) and this proves (1.2). We prove in Theorem 7 later that \(\rho _t\in {\mathcal {R}}\), i.e. \(-\infty <\textsc {l}(\rho _t)\le \textsc {r}(\rho _t)<\infty \). \(\square \)

Proof of Theorem 2

For any \(t\) and \(\delta \),

$$\begin{aligned} \int _{a}^b \rho ^{\delta ,-}_t(r)\,dr-\int _{a}^b \rho _t(r)\,dr= \frac{1}{2} \left( \phi _t(b)-\phi ^{\delta ,-}_t(b)+\phi ^{\delta ,-}_t(a)-\phi _t(a)\right) \end{aligned}$$
(6.8)

Let \(\delta _n:= 2^{-n}\delta \), then from (6.4), for all \(r\)

$$\begin{aligned} \phi ^{\delta _n,-}_t(r)\le \phi _t(r)\le \phi ^{\delta _n,+}_t(r) \end{aligned}$$
(6.9)

so that from (6.1) and (6.8)

$$\begin{aligned} \int _{a}^b \rho ^{\delta _n,-}_t(r)\,dr\!-\!\int _{a}^b \rho _t(r)\,dr\!\le \! \frac{1}{2} \left( \phi ^{\delta _n,+}_t(b)\!-\!\phi ^{\delta },\!-\!_t(b)\!+\!\phi ^{\delta ,-}_t(a)\!-\!\phi ^{\delta _n,-}_t(a)\right) \!\le \! j\delta \nonumber \\ \end{aligned}$$
(6.10)

\(\square \)

6.2 Stationary solutions

We say that a macroscopic interface \(\phi \in V_{0}\) is stationary if, \(\phi _0=\phi \) implies \(\phi _t = \phi + 2jt\). A macroscopic density \(\rho \in {\mathcal {R}}\) is stationary if \(\rho _0=\rho \) implies \(\rho _t=\rho \). Here \(\phi _t\) and \(\rho _t\) are the dynamics given by Theorems 4 and 1, respectively.

If \(\phi \) is stationary, then the density \(\rho \) associated to \(\phi \) via (6.7) is stationary because by (6.7)

$$\begin{aligned} \int _{r_0}^r \left( 1-2 \rho _t(r')\right) \,dr'= \int _{r_0}^r \left( 1-2 \rho _0(r') \right) \,dr',\quad \text { for all } r_0,\,r \hbox { and } t\ge 0 \end{aligned}$$

Let

$$\begin{aligned} \bar{\rho }(r) := {\left\{ \begin{array}{ll} 0, &{}\text {for } r\ge \frac{1}{4j}\\ \frac{1}{2} - 2 jr, &{}\text {for } |r|\le \frac{1}{4j}\\ 1, &{}\text {for } r\le -\frac{1}{4j} \end{array}\right. } \qquad \bar{\phi }(r) := {\left\{ \begin{array}{ll} 2j r^2 + \frac{1}{8j} ,&{} \text {for } |r|\le \frac{1}{4j} \\ |r|,&{}\text {for } r\ge \frac{1}{4j}. \end{array}\right. }\qquad \qquad \end{aligned}$$
(6.11)

Theorem 6

The macroscopic interface \(\bar{\phi }\) and the associated macroscopic density \(\bar{\rho }\) are stationary.

If \(\xi ^{(\varepsilon )}_0\) approximates \(\bar{\phi }\) in the sense of (5.2), then the theorem says that the rescaled interface process as seen from its vertex converges in the sense of Theorem 4 at any macroscopic time \(t\) to the initial value \(\bar{\phi }\) shifted by \(2jt\). An analogous statement holds for the particle process (but the stationary density profile does not move).

Theorem 6 is proven in the next section by introducing a deterministic harness process on \({\mathbb {R}}^{\mathbb {Z}}\), a discrete time process that approaches \(\phi _t\) and whose stationary solution is directly computable.

6.3 Monotonicity

We collect some monotonicity properties of the macroscopic interface inherited from the microscopic dynamics. We tacitly suppose hereafter that the initial data \(\phi \in V_{o}\), namely that \(\phi (r)=V_{o}(r)\equiv |r|\) for all \(|r|\) large enough.

The following lemma is a direct consequence of the definition of \(\phi _t\).

Lemma 9

For any \(t>0,\,\phi _t(r) \ge (|r|+jt),\,r\in {\mathbb R}\), and

$$\begin{aligned} \lim _{|r|\rightarrow \infty } |\phi _t(r) - (|r|+jt)|=0 \end{aligned}$$

Proof

Let \(t = \tau k2^{-n}\). It follows from the inequality \(\phi _t^{\tau 2^{-n},-} \le \phi _t \le \phi _t^{\tau 2^{-n},+}\) that there is \(R=R_{n,\tau }\) so that

$$\begin{aligned} |r|+ jt \le \phi _t(r)\le |r|+ j(t+ 2^{-n}),\;\; |r|\ge R. \end{aligned}$$

\(\square \)

We shall next establish inequalities relating evolutions with different values of \(j\), we thus add a superscript \(j\) writing \(\phi ^{(j)}_t\) and \(\phi ^{(j,\delta ,\pm )}_t\).

Lemma 10

Let \(j \le j'\) then for all \(t\ge 0\)

$$\begin{aligned} \phi ^{(j)}_t -jt \ge \phi ^{(j')}_t -j't. \end{aligned}$$
(6.12)

Proof

It is enough to prove that

$$\begin{aligned} \phi ^{(j,\delta ,-)}_\delta (r) - j\delta \ge \psi ^{(j',\delta ,-)}_\delta (r) -j'\delta ,\quad \text {if}\; \phi \ge \psi \end{aligned}$$

Let \(\textsc {l}',\textsc {r}'\) be such that

$$\begin{aligned} \psi ^{(j',\delta ,-)}_\delta (r) = G_\delta \psi (r), \;\;\; \textsc {l}'\le r \le \textsc {r}',\quad \text {and } \psi ^{(j',\delta ,-)}_\delta (r)= |r|+j'\delta \text {elsewhere} \end{aligned}$$

By the maximum principle \(G_\delta \phi \ge G_\delta \psi \), then

$$\begin{aligned} \phi ^{(j,\delta ,-)}_\delta (r) \ge G_\delta \phi (r)\ge \psi ^{(j',\delta ,-)}_\delta (r), \;\;\; \textsc {l}'\le r \le \textsc {r}' \end{aligned}$$

and a fortiori:

$$\begin{aligned} \phi ^{(j,\delta ,-)}_\delta (r) \ge \psi ^{(j',\delta ,-)}_\delta (r) -(j'-j)\delta , \;\;\; \textsc {l}'\le r \le \textsc {r}' \end{aligned}$$

By definition \(\phi ^{(j,\delta ,-)}_\delta (r) \ge |r| + j\delta , \;\;\; r \in {\mathbb R}\) so that

$$\begin{aligned} \phi ^{(j,\delta ,-)}_\delta (r) \ge |r| + j\delta = \psi ^{(j',\delta ,-)}_\delta (r) -(j'-j)\delta , \;\;\; r \notin (\textsc {l}', \textsc {r}') \end{aligned}$$

which concludes the proof. \(\square \)

6.4 Existence of boundaries

Recall the definition in Sect. 5.3 of the boundaries \(\textsc {l}(\rho ),\textsc {r}(\rho )\) of a density \(\rho \in {\mathcal {R}}\). They are also the boundaries of the interface \(\phi _t\) which corresponds to \(\rho _t\).

Theorem 7

The boundaries \(\textsc {l}(\rho _t),\;\textsc {r}(\rho _t)\) of a density \(\rho _t\) as defined in Theorem 1 starting from \(\rho \in {\mathcal R}\) are finite. In other words, \(\rho \in {\mathcal {R}}\) implies \(\rho _t\in {\mathcal {R}}\).

Proof

We shall prove the theorem in the framework of interfaces. We thus want to prove that the boundary points of the interface are finite, that is, \(-\infty <\textsc {l}(\phi ^{(j)}_t),\,\textsc {r}(\phi ^{(j)}_t)<\infty \) for initial \(\rho \in {\mathcal {R}}\) and all \(t\ge 0\). Let \({\bar{\phi }}^{(j')}\) be the stationary interface for the \(j'\)-evolution. If \(j'<j\) is small enough, \(\phi \le {\bar{\phi }}^{(j')}\) so that, by Lemma 10,

$$\begin{aligned} \phi ^{(j)}_t -jt \le {\bar{\phi }}^{(j')}_t - j't= {\bar{\phi }}^{(j')}. \end{aligned}$$

This implies that the boundary points of \(\phi ^{(j)}_t\) are bounded by those of \({\bar{\phi }}^{(j')}\) and the theorem is proved. \(\square \)

7 The harness process

We consider now the (deterministic) Harness Process proposed by Hammersley [8] with “moving cone” boundary conditions and prove that with the diffusive scaling this process also converges to the macroscopic evolution \(\phi \).

Let \(H:{\mathbb {Z}}\rightarrow {\mathbb {R}}\) and define the operator \(\varTheta \) by

$$\begin{aligned} (\varTheta H)(x) = \frac{H(x-1)+H(x+1)}{2} \end{aligned}$$
(7.1)

but to keep notation light we drop the parentheses and write \(\varTheta H(x)\). Let \((H_n(x),\,x\in {\mathbb {Z}}, \,n\in {\mathbb {Z}}^+),\,H_n(x)\in {\mathbb {R}}\) be the deterministic process satisfying the discrete heat equation:

$$\begin{aligned} H_{n+1}(x):= \varTheta H_n(x) =\varTheta ^{n+1}H_0(x) \end{aligned}$$
(7.2)

Here \(n\) is time and \(x\) is space.

Duality Let \(X^x_n\) be a symmetric nearest neighbors random walk on \({\mathbb {Z}}\) with \(X^x_0=x\) and \(p_n(x,y):= P(X^x_n=y)\) the probability that the walk goes from \(x\) to \(y\) in \(n\) steps. Then, a simple recurrence shows that

$$\begin{aligned} H_n(x) =\sum _{y\in {\mathbb {Z}}} p_n(x,y)H_0(y) = E(H_0(X^x_n)) \end{aligned}$$
(7.3)

Traveling wave solutions A family of traveling wave solutions of this process are

$$\begin{aligned} \bar{H}(x) := a x^2 + b, \end{aligned}$$

where \(a\) and \(b\) are arbitrary constants. Indeed,

$$\begin{aligned} \bar{H}_n:=\varTheta ^n\bar{H} = \bar{H}+2an. \end{aligned}$$
(7.4)

Moving boundaries For \(v=(v_1,v_2)\in {\mathbb {Z}}\times {\mathbb {R}}\), let \(V_v:{\mathbb {Z}}\rightarrow {\mathbb {R}}\) be the cone defined by \(V_{v}(x)=|x-v_1|+v_2\); call \(v\) the vertex of \(V_v\). Let

$$\begin{aligned} {\mathcal {H}}_v&:= \left\{ K:{\mathbb {Z}}\rightarrow {\mathbb {R}}\,:\,K(x)=V_v(x)\;\text {for all but a finite number of }x\in {\mathbb {Z}}\right\} \\ {\mathcal {H}}&:= \cup _{v\in {\mathbb {Z}}\times {\mathbb {R}}} {\mathcal {H}}_v \end{aligned}$$

For \(K\in {\mathcal {H}}_v\) define

$$\begin{aligned} \textsc {l}(K)&:= \min \{\ell \in {\mathbb {Z}}\,:\,K(\ell +1)\ne V_v(\ell +1)\}\\ \textsc {r}(K)&:= \max \{\ell \in {\mathbb {Z}}\,:\,K(\ell -1)\ne V_v(\ell -1)\} \end{aligned}$$

The harness process with moving cone boundary conditions and initial interface \(K^\textsc {j}_0\in {\mathcal {H}}_{(0,0)}\) is defined for \(n\ge 1\) by

$$\begin{aligned} K^\textsc {j}_{n}(x):=\max \{\varTheta K^\textsc {j}_{n-1}(x), V_{(0,2\textsc {j}n)}(x)\}. \end{aligned}$$
(7.5)

So that \(K^\textsc {j}_n\in {\mathcal {H}}_{(0,2\textsc {j}n)}\).

A travelling wave solution \(\bar{K}\) of this process is associated to \(\bar{H}\): if the initial interface is given by

$$\begin{aligned} \bar{K}^\textsc {j}_{0}(x) := \left\{ \begin{array}{ll} \frac{1}{8\textsc {j}}+2\textsc {j}x^2, &{}|x|\le \frac{1}{4\textsc {j}}\\ |x|, &{}|x|\ge \frac{1}{4\textsc {j}} \end{array} \right. \end{aligned}$$
(7.6)

then the evolution (7.5) at time \(n\) gives a translation of the initial interface:

$$\begin{aligned} \bar{K}^\textsc {j}_{n}(x) = \bar{K}^\textsc {j}_0(x) + 2\textsc {j}n . \end{aligned}$$
(7.7)

The sides of the cone \(y=|x|\) are tangent to the parabola \(y=\frac{1}{8\textsc {j}}+2\textsc {j}x^2\) at the points \((\frac{-1}{4\textsc {j}},\frac{1}{4 \textsc {j}})\) and \((\frac{1}{4\textsc {j}},\frac{1}{4 \textsc {j}})\).

Hydrodynamic limit Let \(K\) be a Lipschitz function on \({\mathcal {H}}\) and define

$$\begin{aligned} \Phi ^{(\varepsilon )}_t(r) := \varepsilon \left( K^{\varepsilon j}_{[\varepsilon ^{-2}t]}([\varepsilon ^{-1}r])\right) \end{aligned}$$
(7.8)

Proposition 4

(Hydrodynamic Limit) Let \(\phi _t\) be the evolution of Theorem 4 with initial condition \(\phi \) and let \(K^{\varepsilon j}_n\) the evolution (7.5) with initial interface \(K^{(\varepsilon )}_0(x)= \phi (\varepsilon x)\). Then,

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0} \sup _{r\in {\mathbb {R}}}|\Phi ^{(\varepsilon )}_t(r) - \phi _t(r)|=0. \end{aligned}$$
(7.9)

where the rescaled process \(\Phi ^{(\varepsilon )}_t(r)\) is defined in (7.8).

The proof of Theorem 6 follows from the above proposition:

Proof of Theorem 6

Taking \(\bar{K}^{\varepsilon j}\) as the explicit stationary solution in (7.6) with \(\textsc {j}=\varepsilon j\) and \(\bar{\Phi }^{(\varepsilon )}\) the corresponding renormalization as in (7.8),

$$\begin{aligned} \phi _t(r) = \lim _{\varepsilon \rightarrow 0} \bar{\Phi }^{(\varepsilon )}_t(r) = \Phi ^{(\varepsilon )}_0(r) + 2jt = \bar{\phi }(r)+2jt \end{aligned}$$
(7.10)

where the first identity is consequence of (7.9), the second one comes from the stationarity of \(\bar{K}\) given by (7.7) and the last one is a computation based on the explicit expressions of \(\bar{K}^{(\varepsilon )}\) and \(\bar{\phi }\). \(\square \)

To prove Proposition 4 we introduce the delta processes associated to \(K\).

The delta harness processes We define the delta harness processes \(K^{\textsc {j},\delta ,-}_\ell ,\,K^{\textsc {j},\delta ,+}_\ell \) as follows. Take \(\delta \ge 1\), fix an initial condition \(K^{\textsc {j},\delta ,-}_{0}\in {\mathcal {H}}_{(0,0)},\,K^{\textsc {j},\delta ,+}_{0}= \max \{K^{\textsc {j},\delta ,-}_{0},V_{(0,\delta \textsc {j})}\}\) and define iteratively

$$\begin{aligned} K^{\textsc {j},\delta ,\pm }_{\ell }&:= \varTheta ^{\ell -[n\delta ]}K^{\textsc {j},\delta ,\pm }_{[n\delta ]}, \;\text {if } \ell \in [[n\delta ],[(n+1)\delta ]-1],\;n\ge 0 \nonumber \\ K^{\textsc {j},\delta ,-}_{[n\delta ]}&:= \max \{ K^{\textsc {j},\delta ,-}_{[n\delta ]-1}, V_{(0,n\delta 2\textsc {j})}\}, \;\text {if } \;n\ge 1 \\ K^{\textsc {j},\delta ,+}_{[n\delta ]}&:= \max \{ K^{\textsc {j},\delta ,+}_{[n\delta ]-1}, V_{(0,(n+1)\delta 2\textsc {j})}\}, \;\text {if } \;n\ge 1.\nonumber \end{aligned}$$
(7.11)

Both processes evolve with (7.1) in the time intervals \([[n\delta ],[(n+1)\delta ]-1]\cap {\mathbb {Z}}\) and update the interface at times \([n\delta ]\): the process delta\(-\) takes the max with the cone with vertex \((0,n\delta 2\textsc {j})\) while the process delta\(+\) takes the max with the cone with vertex \((0,(n+1)\delta 2\textsc {j})\). The following dominating Lemma follows immediately.

Lemma 11

If \(K^{\textsc {j},\delta ,-}_0\le K^\textsc {j}_0\le K^{\textsc {j},\delta ,+}_0\), then \(K^{\textsc {j},\delta ,-}_\ell \le K^\textsc {j}_\ell \le K^{\textsc {j},\delta ,+}_\ell \), for all \(\ell \ge 1\); for all \(\delta \ge 1, \textsc {j}\ge 0\). Furthermore,

$$\begin{aligned} \sup _{x\in {\mathbb {Z}},\ell \ge 0}\left\{ K^{\textsc {j},\delta ,+}_\ell (x)- K^{\textsc {j},\delta ,-}_\ell (x)\right\} \le \delta 2\textsc {j}. \end{aligned}$$
(7.12)

Hydrodynamic limit of the delta processes Take a macroscopic initial condition \(\phi \) as in Theorem 4. Fix \(j>0\) and take \(\textsc {j}=\varepsilon j\), fix \(\delta >0\) and \(\varepsilon \) small such that \(\delta \varepsilon ^{-2}>1\), take \(\delta \varepsilon ^{-2}\) in the place of \(\delta \) and define

$$\begin{aligned} \Phi ^{\varepsilon ,\delta ,\pm }_t(r) := \varepsilon K^{\varepsilon j,\varepsilon ^{-2}\delta ,\pm }_{[\varepsilon ^{-2}t]}([\varepsilon ^{-1}r]) \end{aligned}$$

with initial \(K^{(\varepsilon )}_0 (x)=\phi (\varepsilon x)\), which implies

$$\begin{aligned} \Phi ^{\varepsilon ,\delta ,\pm }_0(r,0) = \Phi ^\varepsilon _0(r) = \phi (\varepsilon [r\varepsilon ^{-1}]). \end{aligned}$$

From Lemma 11 we have

$$\begin{aligned} \Phi ^{\varepsilon ,\delta ,-}_t(r)\le \Phi ^\varepsilon _t(r)\le \Phi ^{\varepsilon ,\delta ,+}_t(r) \end{aligned}$$
(7.13)
$$\begin{aligned} \sup _{r,t,\varepsilon }\left( \Phi ^{\varepsilon ,\delta ,+}_t(r)- \Phi ^{\varepsilon ,\delta ,-}_t(r)\right) \le 2\delta j.\qquad \quad \end{aligned}$$
(7.14)

Proposition 5

(hydrodynamics) Let \(\phi \) be Lipschitz. Then, there exists a constant \(C>0\) such that,

$$\begin{aligned} \sup _r\left| \Phi ^{\varepsilon ,\delta ,\pm }_t(r)- \phi _t^{\delta ,\pm }(r)\right| \le {Ct}{\delta ^{\beta -1}}\varepsilon ^{1-2\beta }, \end{aligned}$$
(7.15)

for any \(\beta >0\).

Proof

Take \(t<\delta \). In this case \(\phi _t\) obeys the heat equation. Let \(W^r_t\) be Brownian motion with starting point \(r\); then the solution \(\phi _t\) is given by \(\phi _t(r) = G_t\phi (r)= E\phi (W^r_t)\). By duality (7.3) and assuming \(X_n\) and \(W_t\) are defined in the same probability space,

$$\begin{aligned}&\left| \Phi ^{\varepsilon ,\delta ,\pm }_t(r)- \phi _t^{\delta ,\pm }(r)\right| = \left| {\mathbb {E}}\left( \phi (\varepsilon X_{[\varepsilon ^{-2}t]}^{[\varepsilon ^{-1}r]}) -\phi (W^r_t)\right) \right| \nonumber \\&\quad \le \;{\mathbb {E}}\left| \phi (\varepsilon X_{[\varepsilon ^{-2}t]}^{[\varepsilon ^{-1}r]}) -\phi (W^r_t)\right| \le {\mathbb {E}}\left| \varepsilon X_{[\varepsilon ^{-2}t]}^{[\varepsilon ^{-1}r]} -W^r_t \right| \end{aligned}$$
(7.16)
$$\begin{aligned}&\quad \le \; C \delta ^ {\beta }\varepsilon ^{1-2\beta },\qquad \text {for any }\beta >0,\;\;t<\delta , \end{aligned}$$
(7.17)

where in (7.16) we used that \(\phi \) is Lipschitz and in (7.17) the dyadic KMT coupling between the Brownian motion and the random walk [11], Theorem 7.1.

At \(t=\delta \) we have

$$\begin{aligned} \left| \max \left\{ \Phi ^{\varepsilon ,\delta ,\pm }_{\delta -},V_{(0,[\varepsilon ^{-2}\delta ]2j\varepsilon ^2)}\right\} - \max \left\{ \phi _{\delta -}^{\delta ,\pm },V_{(0,\delta 2j)}\right\} \right| \le C \delta ^{\beta }\varepsilon ^{1-2\beta } +\varepsilon ,\qquad \end{aligned}$$
(7.18)

because the two cones differ at most by \(\varepsilon \). Changing the constant \(C\), (7.18) is bounded by \(C \delta ^{\beta }\varepsilon ^{1-2\beta }\), so that iterating (7.18) \([(t+1)/\delta ]\) times we get (7.15). \(\square \)

Proof of Proposition 4

As a consequence of (7.13) and (7.14),

$$\begin{aligned} \left| \phi _t-\Phi ^{(\varepsilon )}_t(r)\right|&\le \left| \phi _t-\phi _t^{\delta ,\pm }\right| +\left| \phi _t^{\delta ,\pm }-\Phi ^{\varepsilon ,\delta ,\pm }_t\right| +\left| \Phi ^{\varepsilon ,\delta ,\pm }_t - \Phi _t^{(\varepsilon )}\right| \\&\le 2\delta + Ct\delta ^{1-\beta } \varepsilon ^{1-2\beta } + 2\delta . \end{aligned}$$

Taking first \(\varepsilon \rightarrow 0\) and then \(\delta \rightarrow 0\), we get (7.9). \(\square \)

8 Conclusions

In this section we summarize the results we have obtained so far. In Theorems 4 and 1 we have proved convergence in the hydrodynamic limit to a deterministic evolution for the interface and, respectively, the density. The limit interface \(\phi _t\) is Lipschitz continuous in space and continuous in time, see Theorem 5. At each time \(t>0\) it “belongs” to a cone, in the sense that there are real numbers \(\textsc {l}(\phi _t)\) and \(\textsc {r}(\phi _t)\) so that \(\phi _t(r)= |r| + jt\) for \(r\notin (\textsc {l}(\phi _t),\textsc {r}(\phi _t))\), Theorem 7. The limit particle densities inherit analogous properties from the interface.

The interface evolution \(\phi _t\) is characterized in terms of a sequence of upper and lower bounds \(\phi _t^{\delta ,\pm }\) which in the limit \(\delta \rightarrow 0\) become identical. \(\phi _t^{\delta ,\pm }\) are solutions of time discretized free boundary problems obtained by alternating linear heat diffusion and motion of the boundaries.

We miss however a proof that the limit evolution satisfies the free boundary problem described in the introduction for the particle density. We do know however that the stationary solution of (1.7)–(1.8) is indeed stationary for the limit evolution, Theorem 6. The formula for the velocity of the boundaries is quite natural once we observe that the levels of the solution of the heat equation have velocity \(-\rho ''_t/(2\rho '_t)\) (\(\rho '\) and \(\rho ''\) the space derivatives of \(\rho \)). To get (1.8) we need to add the information that at the endpoints \(\rho ' = -2j\) which is consistent with the analysis of the stationary solution.

We have proved in Theorem 3 that there is a stationary measure for the particle process at fixed \(\textsc {j}=j\varepsilon >0\); we miss however a proof that in the limit \(\varepsilon \rightarrow 0\) it becomes supported by the stationary solution of (1.7)–(1.8), even though this is stationary for the limit evolution.