# Long-time behavior of SSEP with slow boundary

Linjie Zhao  Inria, Univ. Lille, CNRS, UMR 8524 - Laboratoire Paul Painlevé,
F-59000 Lille<EMAIL_ADDRESS>
###### Abstract.

We consider the symmetric simple exclusion process with slow boundary first
introduced in [Baldasso et al., Journal of Statistical Physics, 167(5), 2017].
We prove a law of large number for the empirical measure of the process under
a longer time scaling instead of the usual diffusive time scaling.

###### Key words and phrases:

Exclusion process; slow boundary; empirical measure, law of large numbers

##  1\. Introduction

Interacting particle systems in contact with reservoirs have been investigated
in various literature [5, 7, 8]. We study in this article the symmetric simple
exclusion process (SSEP) on a line segment $\\{1,\ldots,N-1\\}$ with slow
boundary, which is first introduced by Baldasso et al. [1]. Here, $N$ is the
scaling parameter. There is at most one particle per site. In the bulk, a
particle jumps to one of its neighbors at rate one provided the target site is
empty. Fix parameters $c>0,\theta\geq 0$ and $\alpha,\,\beta\in(0,1)$. At the
boundary site $1$ (resp. $N-1$), a particle is created at rate $c\alpha
N^{-\theta}$ (resp. $c\beta N^{-\theta}$) if site $1$ (resp. $N-1$) is empty,
and a particle is destroyed at rate $c(1-\alpha)N^{-\theta}$ (resp.
$c(1-\beta)N^{-\theta}$) if site $1$ (resp. $N-1$) is occupied. Therefore, the
particle density of the left (resp. right) reservoir is $\alpha$ (resp.
$\beta$), and the interaction strength between the bulk and the reservoirs is
$cN^{-\theta}$. The hydrodynamic equation of the model turns out to be the
heat equation with Dirichlet boundary conditions if $\theta<1$, with Robin
boundary conditions if $\theta=1$ and with Neumann boundary conditions if
$\theta>1$. We refer the readers to [1] for more background of the model.

The hydrodynamic limit of the model is considered under the diffusive time
scaling, i.e., with time speeded up by $N^{2}$ and space divided by $N$. The
aim of this article is to consider the behavior of the process under a longer
time scaling $N^{2+\gamma},\,\gamma>0$. Since the process is irreducible, it
has a unique invariant measure. Under the invariant measure, the empirical
measure converges in probability to the stationary solution of the
corresponding hydrodynamic equations as $N\rightarrow\infty$. This is called
hydrostatic limit [1, 13]. The hydrostatic limit could be formally interpreted
as taking $\gamma=\infty$. For $0<\gamma<\infty$, it is natural to expect that
the limit of the empirical measure should coincide with the hydrostatic limit.
This is indeed true for $\theta\leq 1$, since the stationary solution of the
corresponding hydrodynamic equation is unique in this case. For $\theta>1$,
since the stationary solution is not unique, three regimes appear depending on
whether $\gamma<\theta-1$, $\gamma=\theta-1$ or $\gamma>\theta-1$. See Theorem
2.3 for details.

Despite the simple structure of the model, it has attracted a lot of attention
since then. The equilibrium/non-equilibrium fluctuations from the hydrodynamic
limit are considered in [9, 11]. The large deviation of the SSEP with slow
boundary are investigated in [2, 6, 10].

The paper is organized as follows. In Section 2 we define the model rigorously
via its infinitesimal generator, review the hydrodynamic/hydrostatic limit
already proven in [1, 13], and state the main result of the article. In
Section 3 we introduce the notation of Dirichlet forms and prove the so-called
replacement lemmas under a longer time scaling. The estimates involving the
Dirichlet forms are mostly borrowed from [1]. The proof of Theorem 2.3 is
presented in Section 4.

##  2\. Notation and Results

###  2.1. The model.

The state space of the process $(\eta_{t})_{t\geq 0}$ is
$\Omega_{N}:=\\{0,1\\}^{I_{N}}$, where $I_{N}:=\\{1,\ldots,N-1\\}$ . Here, $N$
is the scaling parameter. For a configuration $\eta\in\Omega_{N}$, $\eta(x)=1$
if and only if there is a particle at site $x$. Fix parameters $c>0$,
$\theta\geq 0$ and $\alpha,\beta\in(0,1)$. The parameter $\theta$ denotes the
strength of interaction with reservoirs and $\alpha,\,\beta$ are the particle
densities of reservoirs. The generator $L_{N}$ of the process
$(\eta_{t})_{t\geq 0}$ is given by

$L_{N}=L_{N,0}+L_{N,b}^{\alpha}+L_{N,b}^{\beta}.$

Above, the generator $L_{N,0}$ of the bulk dynamics acting on functions
$f:\Omega_{N}\rightarrow\mathbb{R}$ is

$\left(L_{N,0}f\right)(\eta)=\sum_{x=1}^{N-2}\left[f\left(\eta^{x,x+1}\right)-f(\eta)\right],$

where $\eta^{x,y}$ is the configuration obtained from $\eta$ by exchanging the
values of $\eta(x)$ and $\eta(y)$, i.e., $\eta^{x,y}(x)=\eta(y)$,
$\eta^{x,y}(y)=\eta(x)$ and $\eta^{x,y}(z)=\eta(z)$ for $z\neq x,\,y$. The
generators $L_{N,b}^{\alpha}$ and $L_{N,b}^{\beta}$ correspond to the boundary
effects, and are given by

$\displaystyle(L_{N,b}^{\alpha}f)(\eta):=cN^{-\theta}r_{\alpha}(\eta)\left[f\left(\eta^{1}\right)-f(\eta)\right],\quad(L_{N,b}^{\beta}f)(\eta):=cN^{-\theta}r_{\beta}(\eta)\left[f\left(\eta^{N-1}\right)-f(\eta)\right],$

where $\eta^{x}$ is the configuration obtained from $\eta$ by flipping the
value of $\eta(x)$, i.e., $\eta^{x}(x)=1-\eta(x)$ and $\eta^{x}(z)=\eta(z)$
for $z\neq x$, and

$r_{\alpha}(\eta)=\alpha(1-\eta(1))+(1-\alpha)\eta(1),\quad
r_{\beta}(\eta)=\beta(1-\eta(N-1))+(1-\beta)\eta(N-1).$

Denote by $\mu_{N}$ the initial measure of the process. For any positive
integer $k$, let $C^{k}[0,1]$ be the family of functions on $[0,1]$ such that
the $m$-th derivative is uniformly continuous in $(0,1)$ for any $m\leq k$.

###  2.2. Hydrodynamic limit.

It has been proven in [1] that phase transitions occur for the SSEP with slow
boundary, depending on whether $\theta<1$, $\theta=1$ or $\theta>1$. To state
the hydrodynamic limit, we impose the following assumptions on the initial
measure $\mu_{N}$: there exists a measurable initial density profile
$\rho_{0}:[0,1]\rightarrow[0,1]$ such that for any $G\in C[0,1]$,

$\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{x=1}^{N-1}\eta(x)G\big{(}\tfrac{x}{N}\big{)}=\int_{0}^{1}\rho_{0}(u)G(u)\,du$

in probability with respect to $\mu_{N}$. The following result characterize
the macroscopic density profile under the diffusive time scaling.

######  Theorem 2.1 (Cf. [1, Theorem 2.8]).

For any $t\geq 0$ and for any $G\in C[0,1]$,

$\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{x=1}^{N-1}\eta_{tN^{2}}(x)G\big{(}\tfrac{x}{N}\big{)}=\int_{0}^{1}\rho(t,u)G(u)\,du$

in probability, where

$(i)$ if $0\leq\theta<1$, then $\rho(t,u)$ is the unique weak solution to the
heat equation with Dirichlet boundary

$\left\\{\begin{array}[]{ll}\partial_{t}\rho(t,u)=\Delta\rho(t,u),&u\in(0,1),\quad
t\geq 0\\\ \rho(t,0)=\alpha,\rho(t,1)=\beta,&t\geq 0,\\\
\rho(0,u)=\rho_{0}(u),&u\in[0,1]\end{array}\right.$ (2.1)

$(ii)$ if $\theta=1$, then $\rho(t,u)$ is the unique weak solution to the heat
equation with Robin boundary

$\left\\{\begin{array}[]{ll}\partial\rho(t,u)=\Delta\rho(t,u),&u\in(0,1),\quad
t\geq 0\\\ \partial_{u}\rho(t,0)=c\left(\rho(t,0)-\alpha\right),&t\geq 0,\\\
\partial_{u}\rho(t,1)=c\left(\beta-\rho(t,1)\right),&t\geq 0\\\
\rho(0,u)=\rho_{0}(u),&u\in[0,1]\end{array}\right.$ (2.2)

$(iii)$ if $\theta>1$, then $\rho(t,u)$ is the unique weak solution to the
heat equation with Neumann boundary

$\left\\{\begin{array}[]{ll}\partial_{t}\rho(t,u)=\Delta\rho(t,u),&u\in(0,1),\quad
t\geq 0\\\ \partial_{u}\rho(t,0)=0,\partial_{u}\rho(t,1)=0,&t\geq 0\\\
\rho(0,u)=\rho_{0}(u),&u\in[0,1]\end{array}\right.$ (2.3)

We refer the readers to [1] for rigorous definitions of weak solutions the
above PDEs.

###  2.3. Hydrostatic limit.

Since the process $(\eta_{t})_{t\geq 0}$ is irreducible, it has a unique
invariant measure denoted by $\mu_{N}^{ss}$. The following result characterize
the macroscopic density profile under the invariant measure $\mu_{N}^{ss}$.

######  Theorem 2.2 (Cf. [1, Theorem 2.2] and [13]).

For any $G\in C[0,1]$,

$\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{x=1}^{N-1}\eta(x)G\big{(}\tfrac{x}{N}\big{)}=\int_{0}^{1}\bar{\rho}(u)G(u)\,du$

in probability with respect to $\mu_{N}^{ss}$, where

$\bar{\rho}(u)=\left\\{\begin{array}[]{ll}(\beta-\alpha)u+\alpha,&\text{ if
}\theta\in[0,1)\\\
\frac{c(\beta-\alpha)}{2+c}u+\alpha+\frac{\beta-\alpha}{2+c},&\text{ if
}\theta=1\\\ \frac{\beta+\alpha}{2},&\text{ if
}\theta\in(1,\infty)\end{array}\right.$

We say $\rho:[0,1]\rightarrow[0,1]$ is a stationary solution to (2.1) if

$\Delta\rho(u)=0,\;u\in(0,1),\quad\rho(0)=\alpha,\quad\rho(1)=\beta.$

Stationary solutions to (2.2) and (2.3) could be defined in the same way. Note
that the hydrostatic limits $\bar{\rho}$ are stationary solutions to the
corresponding hydrodynamic equations as stated in Theorem 2.1. We underline
that the stationary solution to (2.1) and (2.2) is unique, while the
stationary solution to (2.3) is not unique. Indeed, any constant function is a
stationary solution to (2.3). The above theorem tells us that the correct
choice is $(\alpha+\beta)/2$.

###  2.4. Long-time limit.

In this subsection, we state the main result of the article. Fix $\gamma>0$.
We shall consider the process speeded up by $N^{2+\gamma}$. Denote by
$(\eta^{N}_{t})_{t\geq 0}$ the process with generator $N^{2+\gamma}L_{N}$.
Then $(\eta^{N}_{t})_{t\geq 0}$ and $(\eta_{tN^{2+\gamma}})_{t\geq 0}$ have
the same distribution. We are interested in the long time behavior of the
empirical measure $\pi^{N}_{t}$ of the process defined as

$\pi^{N}_{t}(du)=\frac{1}{N}\sum_{x=1}^{N-1}\eta^{N}_{t}(x)\delta_{x/N}(du),$

where $\delta_{x/N}(du)$ is the Dirac measure on the point $x/N$. Whence,
$\pi^{N}_{t}$ is a random measure on $[0,1]$ with total mass bounded by one.
With this notation, for any $G\in C[0,1]$,

$\big{\langle}\pi^{N}_{t},G\big{\rangle}=\frac{1}{N}\sum_{x=1}^{N-1}\eta^{N}_{t}(x)G\big{(}\tfrac{x}{N}\big{)}.$

For $\theta>1$ and $0<\gamma\leq\theta-1$, we assume that the average number
of particles converges in the following sense: there exists $m_{0}\in[0,1]$
such that

$\lim_{N\rightarrow\infty}E_{\mu_{N}}\Big{[}\Big{|}\frac{1}{N-1}\sum_{x=1}^{N-1}\eta(x)-m_{0}\Big{|}\Big{]}=0.$
(2.4)

We underline that we impose no restrictions on the initial measure $\mu_{N}$
in the rest of the cases.

Denote by $\mathbb{P}^{N}_{\mu_{N}}$ the probability measure on
$D([0,\infty),\Omega_{N})$ associated to the process $(\eta^{N}_{t})_{t\geq
0}$ and the initial measure $\mu_{N}$, and by $\mathbb{E}^{N}_{\mu_{N}}$ the
corresponding expectation.

We now state the law of large numbers for the empirical measure $\pi^{N}_{t}$.

######  Theorem 2.3.

For any $t>0$ and for any $G\in C[0,1]$,

$\lim_{N\rightarrow\infty}{\mathbb{E}}^{N}_{\mu_{N}}\Big{[}\big{|}\int_{0}^{t}\Big{\\{}\big{\langle}\pi^{N}_{s},G\big{\rangle}-\int_{0}^{1}\rho_{\theta,\gamma}(s,u)G(u)du\Big{\\}}ds\big{|}\Big{]}=0,$

where

$\rho_{\theta,\gamma}(t,u)=\left\\{\begin{array}[]{ll}(\beta-\alpha)u+\alpha,&\text{
if }0\leq\theta<1\\\
\frac{c(\beta-\alpha)}{2+c}u+\alpha+\frac{\beta-\alpha}{2+c},&\text{ if
}\theta=1,\\\ m_{0},&\text{ if }\theta>1,\,0<\gamma<\theta-1,\\\
\frac{\beta+\alpha}{2},&\text{ if }\theta>1,\,\gamma>\theta-1,\\\
\frac{\beta+\alpha}{2}+\Big{(}m_{0}-\frac{\alpha+\beta}{2}\Big{)}e^{-2ct},&\text{
if }\theta>1,\,\gamma=\theta-1.\end{array}\right.$ (2.5)

######  Remark 2.4.

Note that $\rho_{\theta,\infty}=\bar{\rho}$. This is not surprising since the
hydrostatic limit stated in Theorem 2.2 could be formally interpreted as
taking $\gamma=\infty$. Compared with Theorem 2.2, the above theorem states
that phase transition occurs even in the supercritical case $\theta>1$.

######  Remark 2.5.

The result should also hold if the density reservoirs vary slowly with time,
i.e., if replacing $\alpha$ (resp. $\beta$) with some smooth function
$\rho_{-}(t):\mathbb{R}_{+}\rightarrow(0,1)$ (resp.
$\rho_{+}(t):\mathbb{R}_{+}\rightarrow(0,1)$). This is called quasi-static
hydrodynamic limit [3, 4].

##  3\. Preliminary results

In this section, we introduce the notion of Dirichlet forms and prove several
replacement lemmas in different regimes, which are crucial in the proof of
Theorem 2.3.

###  3.1. Dirichlet form.

For a probability measure $\mu$ on $\Omega_{N}$ and a function
$g:\Omega_{N}\rightarrow\mathbb{R}$, the Dirichlet forms corresponding to the
bulk/boundary dynamics are defined as

$\displaystyle D_{N,0}(g,\mu)$
$\displaystyle:=\frac{1}{2}\sum_{x=1}^{N-2}\sum_{\eta\in\Omega_{N}}\left(g\left(\eta^{x,x+1}\right)-g(\eta)\right)^{2}\mu(\eta),$
$\displaystyle D_{N,b}^{\alpha}(g,\mu)$
$\displaystyle:=\frac{1}{2}\sum_{\eta\in\Omega_{N}}cN^{-\theta}r_{\alpha}(\eta)\left(g\left(\eta^{1}\right)-g(\eta)\right)^{2}\mu(\eta),$
$\displaystyle D_{N,b}^{\beta}(g,\mu)$
$\displaystyle:=\frac{1}{2}\sum_{\eta\in\Omega_{N}}cN^{-\theta}r_{\beta}(\eta)\left(g\left(\eta^{N-1}\right)-g(\eta)\right)^{2}\mu(\eta).$

For any two functions $f,g:\Omega_{N}\rightarrow\mathbb{R}$, denote

$\big{\langle}f,g\big{\rangle}_{\mu}=\sum_{\eta\in\Omega_{N}}f(\eta)g(\eta)\mu(\eta).$

For any density profile $\lambda:[0,1]\rightarrow[0,1]$, let
$\nu^{N}_{\lambda(\cdot)}$ be the product measure on $\Omega_{N}$ with
marginals given by

$\nu_{\lambda(\cdot)}^{N}\\{\eta:\eta(x)=1\\}=\lambda\left(\tfrac{x}{N}\right),\quad
x\in I_{N}.$

In particular, if $\lambda(\cdot)\equiv\rho$ for some $\rho\in[0,1]$, we
simply write $\nu^{N}_{\rho}$.

The following lemma compares $\big{\langle}L_{N}g,g\big{\rangle}_{\mu}$ with
$D_{N}(g,\mu)$.

######  Lemma 3.1.

$(i)$ Let $\lambda:[0,1]\rightarrow(0,1)$ be a smooth density profile such
that there exists a neighborhood of $0$ where $\gamma(\cdot)=\alpha$, and a
neighborhood of $1$ where $\gamma(\cdot)=\beta$. Let $f$ be a
$\nu^{N}_{\lambda(\cdot)}$-density,

$f\geq
0,\quad\sum_{\eta\in\Omega_{N}}f(\eta)\nu^{N}_{\lambda(\cdot)}(\eta)=1.$

Then

$\big{\langle}L_{N}\sqrt{f},\sqrt{f}\big{\rangle}_{\nu^{N}_{\lambda(\cdot)}}=-(1/2)D_{N,0}(\sqrt{f},\nu^{N}_{\lambda(\cdot)})-D_{N,b}^{\alpha}(\sqrt{f},\nu^{N}_{\lambda(\cdot)})-D_{N,b}^{\beta}(\sqrt{f},\nu^{N}_{\lambda(\cdot)})+\mathcal{O}(N^{-1}),$

where $|\mathcal{O}(N^{-1})|\leq CN^{-1}$ for some finite constant $C$.

$(ii)$ Let $\rho\in(0,1)$ be a constant. Let $f$ be a
$\nu^{N}_{\rho}$-density. Then

$\big{\langle}L_{N}\sqrt{f},\sqrt{f}\big{\rangle}_{\nu^{N}_{\rho}}=-D_{N,0}(\sqrt{f},\nu^{N}_{\rho})-D_{N,b}^{\alpha}(\sqrt{f},\nu^{N}_{\rho})-D_{N,b}^{\beta}(\sqrt{f},\nu^{N}_{\rho})+\mathcal{O}(N^{-\theta}).$

The first statement $(i)$ is a direct consequence of [1, Lemma 5.1 $(ii)$ and
Lemma 5.2]. The second statement $(ii)$ follows directly from [1, Lemma 5.1
$(i)$ and Corrollary 5.3]. For this reason, we omit the proof here.

The following lemma bound the occupation variables at the boundary sites by
the corresponding Dirichlet forms.

######  Lemma 3.2.

Let $\lambda:[0,1]\rightarrow(0,1)$ be a smooth density profile such that
there exists a neighborhood of $0$ where $\gamma(\cdot)=\alpha$, and a
neighborhood of $1$ where $\gamma(\cdot)=\beta$. Let $f$ be a
$\nu^{N}_{\lambda(\cdot)}$-density. Then there exists a finite constant
$C_{\alpha}$ such that for any $B>0$,

$\big{|}\sum_{\eta\in\Omega_{N}}(\eta(1)-\alpha)f(\eta)\nu^{N}_{\lambda(\cdot)}(\eta)\big{|}\leq
C_{\alpha}B+C_{\alpha}N^{\theta}B^{-1}D^{\alpha}_{N,b}(\sqrt{f},\nu^{N}_{\lambda(\cdot)}).$

The same result holds at the right boundary: there exists a finite constant
$C_{\beta}$ such that for any $B>0$,

$\big{|}\sum_{\eta\in\Omega_{N}}(\eta(N-1)-\beta)f(\eta)\nu^{N}_{\lambda(\cdot)}(\eta)\big{|}\leq
C_{\beta}B+C_{\beta}N^{\theta}B^{-1}D^{\beta}_{N,b}(\sqrt{f},\nu^{N}_{\lambda(\cdot)}).$

The above lemma is a direct consequence of [1, Lemmas 5.6 and 5.7]. For that
reason, we omit the proof here.

###  3.2. Replacement lemmas.

In this subsection, we prove several replacement lemmas under the longer time
scaling $N^{2+\gamma}$, $\gamma>0$.

The following lemma states that in the subcritical regime $0\leq\theta<1$, we
could replace the occupation variable at the boundary sites with the
corresponding particle density of reservoirs.

######  Lemma 3.3 (Replacement lemma for the case $0\leq\theta<1$.).

Suppose $0\leq\theta<1$. Then for any $t>0$,

$\lim_{N\rightarrow\infty}{\mathbb{E}}^{N}_{\mu_{N}}\Big{[}\big{|}\int_{0}^{t}(\eta_{s}^{N}(1)-\alpha)\,ds\big{|}\Big{]}=0.$

The same result holds with $\eta_{s}^{N}(1)$ replaced with $\eta_{s}^{N}(N-1)$
and $\alpha$ with $\beta$.

###### Proof.

Let $\lambda:[0,1]\rightarrow(0,1)$ be a smooth density profile such that
there exists a neighborhood of $0$ where $\gamma(\cdot)=\alpha$, and a
neighborhood of $1$ where $\gamma(\cdot)=\beta$. By the relative entropy
inequality (cf. [12, A.1.8]), for any $A>0$, the expectation in the lemma
could be bounded from above by

$\frac{H(\mu_{N}|\nu^{N}_{\lambda(\cdot)})}{AN}+\frac{1}{AN}\log{\mathbb{E}}^{N}_{\nu^{N}_{\lambda(\cdot)}}\Big{[}\exp\Big{\\{}AN\big{|}\int_{0}^{t}(\eta_{s}^{N}(1)-\alpha)\,ds\big{|}\Big{\\}}\Big{]},$
(3.1)

where for any probability measures $\mu,\,\nu$ on $\Omega_{N}$ such that $\mu$
is absolutely continuous with respect to $\nu$, $H(\mu|\nu)$ is the relative
entropy of $\mu$ with respect to $\nu$ defined as

$H(\mu|\nu)=\sum_{\eta\in\Omega_{N}}\mu(\eta)\log\frac{\mu(\eta)}{\nu(\eta)}.$

It is not hard to prove that $H(\mu_{N}|\nu^{N}_{\lambda(\cdot)})\leq
C_{\lambda}N$ for some finite constant $C_{\lambda}$. Therefore, the first
term in (3.1) is bounded by $C_{\lambda}/A$. In the sequel, we shall take
$A=A(N)\rightarrow\infty$ as $N\rightarrow\infty$. Whence the first term in
(3.1) vanishes in the limit. Since

$\lim_{N\rightarrow\infty}r_{N}^{-1}\log(a_{N}+b_{N})=\max\\{\lim_{N\rightarrow\infty}r_{N}^{-1}\log
a_{N},\lim_{N\rightarrow\infty}r_{N}^{-1}\log b_{N}\\}$

for any positive sequences $\\{a_{N}\\}_{N\geq 1},\,\\{b_{N}\\}_{N\geq 1}$ and
$\\{r_{N}\\}_{N\geq 1}$ such that $\lim_{N\rightarrow\infty}r_{N}=\infty$, we
could remove the modulus inside the exponential for the second term in (3.1).
By the Feynman-Kac formula (cf. [12, Lemma A.1.7.2]), the second term in (3.1)
is bounded by

$t\sup_{f\,\text{density}}\Big{\\{}\sum_{\eta\in\Omega_{N}}(\eta(1)-\alpha)f(\eta)\nu^{N}_{\lambda(\cdot)}(\eta)+\frac{N^{1+\gamma}}{A}\big{\langle}L_{N}\sqrt{f},\sqrt{f}\big{\rangle}_{\nu^{N}_{\lambda(\cdot)}}\Big{\\}}.$
(3.2)

By Lemma 3.1 $(i)$,

$\big{\langle}L_{N}\sqrt{f},\sqrt{f}\big{\rangle}_{\nu^{N}_{\lambda(\cdot)}}\leq-
D_{N,b}^{\alpha}(\sqrt{f},\nu^{N}_{\lambda(\cdot)})+\mathcal{O}(N^{-1}).$

Together with Lemma 3.2, for any $B>0$, we may bound (3.2) by

$t\sup_{f:\,\text{$\nu^{N}_{\lambda(\cdot)}$-density}}\Big{\\{}C_{\alpha}B+C_{\alpha}N^{\theta}B^{-1}D^{\alpha}_{N,b}(\sqrt{f},\nu^{N}_{\lambda(\cdot)})-\frac{N^{1+\gamma}}{A}D_{N,b}^{\alpha}(\sqrt{f},\nu^{N}_{\lambda(\cdot)})+\mathcal{O}(N^{\gamma}/A)\Big{\\}}$

for some finite constant $C_{\alpha}$. Taking
$B=C_{\alpha}AN^{\theta-1-\gamma}$ and $A=N^{\gamma}\log N$, the above term is
bounded by $t\big{(}C_{\alpha}^{2}N^{\theta-1}\log N+\mathcal{O}(1/\log
N)\big{)}$, which converges to zero as $N\rightarrow\infty$ since $\theta<1$.
This concludes the proof. ∎

Let $m^{N}(\eta)$ be the average number of particles in the system

$m^{N}(\eta)=\frac{1}{N-1}\sum_{x=1}^{N-1}\eta(x).$

Denote $m^{N}_{t}=m^{N}(\eta^{N}_{t})$. The next result states that in the
supercritical regime $\theta>1$, we could replace the occupation variables at
the boundary sites with the average number of particles in the system.

######  Lemma 3.4 (Replacement lemma for the case $\theta>1$.).

Suppose $\theta>1$. Then for any $t>0$,

$\displaystyle\lim_{N\rightarrow\infty}\,{\mathbb{E}}^{N}_{\mu_{N}}\Big{[}\big{|}\int_{0}^{t}(\eta_{s}^{N}(1)-m^{N}_{s})\,ds\big{|}\Big{]}=0.$

The same result holds with $\eta_{s}^{N}(1)$ replaced with $\eta_{s}^{N}(N-1)$

###### Proof.

The proof is similar to that of Lemma 3.3, and we only sketch the proof here.
Fix a constant $\rho\in(0,1)$. By the relative entropy inequality (cf. [12,
A.1.8]), for any $A>0$, the expectation in the lemma could be bounded from
above by

$\frac{H(\mu_{N}|\nu^{N}_{\rho})}{AN}+\frac{1}{AN}\log{\mathbb{E}}^{N}_{\nu^{N}_{\rho}}\Big{[}\exp\Big{\\{}AN\big{|}\int_{0}^{t}(\eta_{s}^{N}(1)-m^{N}_{s})\,ds\big{|}\Big{\\}}\Big{]}.$
(3.3)

As in Lemma 3.3, the first term is bounded by $C/A$ for some finite constant
$C$. By the Feynman-Kac formula (cf. [12, Lemma A.1.7.2]), the second term in
(3.3) is bounded by

$t\sup_{f:\,\nu^{N}_{\rho}\text{-density}}\Big{\\{}\sum_{\eta\in\Omega_{N}}(\eta(1)-m^{N}(\eta))f(\eta)\nu^{N}_{\rho}(\eta)+\frac{N^{1+\gamma}}{A}\big{\langle}L_{N}\sqrt{f},\sqrt{f}\big{\rangle}_{\nu^{N}_{\rho}}\Big{\\}}.$
(3.4)

We may rewrite $\eta(1)-m^{N}(\eta)$ as a telescope sum

$\frac{1}{N-1}\sum_{x=1}^{N-1}\sum_{y=1}^{x-1}(\eta(y)-\eta(y+1)).$

Making the change of variables $\eta\mapsto\eta^{y,y+1}$,

$\sum_{\eta\in\Omega_{N}}(\eta(1)-m^{N}(\eta))f(\eta)\nu^{N}_{\rho}(\eta)=\frac{1}{2(N-1)}\sum_{x=1}^{N-1}\sum_{y=1}^{x-1}\sum_{\eta\in\Omega_{N}}(\eta(y)-\eta(y+1))\big{(}f(\eta)-f(\eta^{y,y+1})\big{)}\nu^{N}_{\rho}(\eta).$

By Cauchy-Schwarz inequality, for any $B>0$, we may bound the last term by

$\displaystyle\frac{B}{4(N-1)}\sum_{x=1}^{N-1}\sum_{y=1}^{x-1}\sum_{\eta\in\Omega_{N}}\big{(}\sqrt{f}(\eta)-\sqrt{f}(\eta^{y,y+1})\big{)}^{2}\nu^{N}_{\rho}(\eta)$
$\displaystyle\qquad\qquad+\frac{1}{4B(N-1)}\sum_{x=1}^{N-1}\sum_{y=1}^{x-1}\sum_{\eta\in\Omega_{N}}\big{(}\sqrt{f}(\eta)+\sqrt{f}(\eta^{y,y+1})\big{)}^{2}\nu^{N}_{\rho}(\eta)$
$\displaystyle\leq\frac{B}{2}D_{N,0}(\sqrt{f},\nu^{N}_{\rho})+\frac{N}{B}.$

The last inequality follows from the basic inequality $(a+b)^{2}\leq
2(a^{2}+b^{2})$ and the fact that $f$ is a density with respect to
$\nu^{N}_{\rho}$. By Lemma 3.1 $(ii)$,

$\big{\langle}L_{N}\sqrt{f},\sqrt{f}\big{\rangle}_{\nu^{N}_{\rho}}\leq-
D_{N,0}(\sqrt{f},\nu^{N}_{\rho})+\mathcal{O}(N^{-\theta}).$

Whence, (3.4) is bounded by

$t\sup_{f:\,\nu^{N}_{\rho}\text{-density}}\Big{\\{}\frac{B}{2}D_{N,0}(\sqrt{f},\nu^{N}_{\rho})+\frac{N}{B}-\frac{N^{1+\gamma}}{A}D_{N,0}(\sqrt{f},\nu^{N}_{\rho})+\mathcal{O}(N^{1+\gamma-\theta}/A).\Big{\\}}$

Taking $B=2N^{1+\gamma}/A$ and $A=N^{\gamma}/(\log N)$, the above term is
bounded by $1/(2\log N)+\mathcal{O}(N^{1-\theta}\log N)$, which converges to
zero as $N\rightarrow\infty$ since $\theta>1$. This concludes the proof. ∎

The next lemma concerns about the long time behavior of the average number of
particles in the supercritical case.

######  Lemma 3.5 (Replacement lemma for the average particle number.).

Suppose $\theta>1$. Recall $m_{0}$ defined in (2.4) is the average number of
particles at the initial time. For any $t>0$,

  1. (i)

if $0\leq\gamma<\theta-1$, then

$\lim_{N\rightarrow\infty}{\mathbb{E}}^{N}_{\mu_{N}}\Big{[}\big{|}\int_{0}^{t}\big{(}m^{N}_{s}-m_{0}\big{)}\,ds\big{|}\Big{]}=0,$

  2. (ii)

if $\gamma=\theta-1$, then

$\lim_{N\rightarrow\infty}{\mathbb{E}}^{N}_{\mu_{N}}\Big{[}\big{|}\int_{0}^{t}\big{(}m^{N}_{s}-m_{s}\big{)}\,ds\big{|}\Big{]}=0,$

where

$m_{s}=\frac{\alpha+\beta}{2}+\Big{(}m_{0}-\frac{\alpha+\beta}{2}\Big{)}e^{-2cs}.$

  3. (iii)

if $\gamma>\theta-1$, then

$\lim_{N\rightarrow\infty}{\mathbb{E}}^{N}_{\mu_{N}}\Big{[}\big{|}\int_{0}^{t}\big{(}m^{N}_{s}-(\alpha+\beta)/2\big{)}\,ds\big{|}\Big{]}=0.$

###### Proof.

The statement $(ii)$ is a direct consequence of [13, Proposition 4.5]. For the
rest of the statements, consider the martingale $\mathfrak{m}^{N}_{t}$ defined
as

$\mathfrak{m}^{N}_{t}:=m^{N}_{t}-m^{N}_{0}-\int_{0}^{t}N^{2+\gamma}L_{N}m^{N}_{s}\,ds,$
(3.5)

whose quadratic variation at time $t$ is given by

$\int_{0}^{t}\big{\\{}N^{2+\gamma}L_{N}(m^{N}_{s})^{2}-2m^{N}_{s}N^{2+\gamma}L_{N}m^{N}_{s}\big{\\}}ds.$

A simple calculation shows that the quadratic variation of
$\mathfrak{m}^{N}_{t}$ is bounded by $CN^{\gamma-\theta}$ for some finite
constant $C$, and that the integral term in (3.5) equals

$\frac{cN^{2+\gamma-\theta}}{N-1}\int_{0}^{t}\big{(}\alpha-\eta_{s}^{N}(1)+\beta-\eta_{s}^{N}(N-1)\big{)}\,ds.$

If $0\leq\gamma<\theta-1$, by Doob’s inequality, for any $T>0$,

$\lim_{N\rightarrow\infty}\mathbb{E}^{N}_{\mu_{N}}\Big{[}\sup_{0\leq t\leq
T}\big{(}\mathfrak{m}^{N}_{t}\big{)}^{2}\Big{]}=0.$

The integral term in (3.5) is of order $N^{1+\gamma-\theta}$, which converges
to zero as $N\rightarrow\infty$ uniformly in a bounded time interval.
Therefore,

$\lim_{N\rightarrow\infty}{\mathbb{E}}^{N}_{\mu_{N}}\big{[}\sup_{0\leq t\leq
T}|m^{N}_{t}-m_{0}|\big{]}=0$

This proves the first statement $(i)$.

If $\gamma>\theta-1$, divided by $N^{1+\gamma-\theta}$ in (3.5), we have

$\lim_{N\rightarrow\infty}\mathbb{E}^{N}_{\mu_{N}}\Big{[}\sup_{0\leq t\leq
T}\big{(}N^{\theta-\gamma-1}\mathfrak{m}^{N}_{t}\big{)}^{2}\Big{]}=0.$

Since $m^{N}_{t}\leq 1$, by (3.5),

$\lim_{N\rightarrow\infty}{\mathbb{E}}^{N}_{\mu_{N}}\Big{[}\big{|}\int_{0}^{t}\big{(}\alpha-\eta_{s}^{N}(1)+\beta-\eta_{s}^{N}(N-1)\big{)}\,ds\big{|}\Big{]}=0.$

By Lemma 3.4, we could replace $\eta_{s}^{N}(1)$ and $\eta_{s}^{N}(N-1)$ in
the time integral with $m^{N}_{s}$. This concludes the proof. ∎

##  4\. Proof of Theorem 2.3

In this section, we prove Theorem 2.3 depending on whether $0\leq\theta<1$,
$\theta=1$ or $\theta>1$. For $H\in C^{2}[0,1]$, consider the martingale
defined as

$\displaystyle
M^{N}_{t}(H)=\big{\langle}\pi^{N}_{t},H\big{\rangle}-\big{\langle}\pi^{N}_{0},H\big{\rangle}-\int_{0}^{t}N^{2+\gamma}L_{N}\big{\langle}\pi^{N}_{s},H\big{\rangle}ds,$
(4.1)

whose quadratic variation at time $t$ is given by

$\int_{0}^{t}\big{\\{}N^{2+\gamma}L_{N}\big{\langle}\pi^{N}_{s},H\big{\rangle}^{2}-2\big{\langle}\pi^{N}_{s},H\big{\rangle}N^{2+\gamma}L_{N}\big{\langle}\pi^{N}_{s},H\big{\rangle}\big{\\}}ds.$

Direct calculations show that the quadratic variation of $M^{N}_{t}(H)$ is
bounded by $C_{H}(N^{\gamma-1}+N^{\gamma-\theta})$ for some finite constant
$C_{H}$. Therefore, for any $T>0$,

$\lim_{N\rightarrow\infty}\mathbb{E}^{N}_{\mu_{N}}\Big{[}\sup_{0\leq t\leq
T}\big{(}N^{-\gamma}M^{N}_{t}(H)\big{)}^{2}\Big{]}=0.$

Since there is at most one particle per site,
$|\big{\langle}\pi^{N}_{t},H\big{\rangle}|\leq||H||_{\infty}$ uniformly in
$t$, where $||H||_{\infty}:=\max_{u\in[0,1]}|H(u)|$ is the uniform norm.
Divided by $N^{\gamma}$ in (4.1),

$\lim_{N\rightarrow\infty}{\mathbb{E}}^{N}_{\mu_{N}}\Big{[}\big{|}\int_{0}^{t}N^{2}L_{N}\big{\langle}\pi^{N}_{s},H\big{\rangle}ds\big{|}\Big{]}=0.$

Direct calculations yield that

$\displaystyle
N^{2}L_{N}\big{\langle}\pi^{N}_{s},H\big{\rangle}=\big{\langle}\pi^{N}_{s},H^{\prime\prime}\big{\rangle}-\eta_{s}^{N}(N-1)H^{\prime}(1)+\eta_{s}^{N}(1)H^{\prime}(0)$
$\displaystyle+cN^{1-\theta}\left(\alpha-\eta_{s}^{N}(1)\right)H(0)+cN^{1-\theta}\left(\beta-\eta_{s}^{N}(N-1)\right)H(1)+\mathcal{O}(N^{-\theta}+N^{-1}).$

Whence,

$\displaystyle\lim_{N\rightarrow\infty}{\mathbb{E}}^{N}_{\mu_{N}}\Big{[}$
$\displaystyle\big{|}\int_{0}^{t}\Big{\\{}\big{\langle}\pi^{N}_{s},H^{\prime\prime}\big{\rangle}-\eta_{s}^{N}(N-1)H^{\prime}(1)+\eta_{s}^{N}(1)H^{\prime}(0)$
(4.2)
$\displaystyle+cN^{1-\theta}\left(\alpha-\eta_{s}^{N}(1)\right)H(0)+cN^{1-\theta}\left(\beta-\eta_{s}^{N}(N-1)\right)H(1)\Big{\\}}ds\big{|}\Big{]}=0.$

###  4.1. The case $0\leq\theta<1$.

In this subsection, we prove Theorem 2.3 for the case $0\leq\theta<1$. Fix
$G\in C[0,1]$. The main technique here is to find an appropriate function $H$
such that $H^{\prime\prime}=G$ on (0,1) and that $H(0)=H(1)=0$. With such a
function $H$, the second line in (4.2) vanishes and the result follows by the
corresponding replacement lemmas.

###### Proof of Theorem 2.3 in the case $0\leq\theta<1$..

For $G\in C[0,1]$, let

$H(u)=H_{G}(u)=\int_{0}^{u}\int_{0}^{v}G(w)\,dw\,dv+u\int_{0}^{1}(v-1)G(v)\,dv.$
(4.3)

It is easy to check that $H\in C^{2}[0,1]$, $H^{\prime\prime}=G$ on $(0,1)$,
and that

$\displaystyle H(0)=H(1)=0,\quad H^{\prime}(0)=\int_{0}^{1}(u-1)G(u)\,du,\quad
H^{\prime}(1)=\int_{0}^{1}uG(u)\,du.$

Substituting the function $H$ into (4.2),

$\lim_{N\rightarrow\infty}{\mathbb{E}}^{N}_{\mu_{N}}\Big{[}\big{|}\int_{0}^{t}\Big{\\{}\big{\langle}\pi^{N}_{s},G\big{\rangle}-\int_{0}^{1}\big{[}(\eta^{N}_{s}(N-1)-\eta^{N}_{s}(1))u+\eta^{N}_{s}(1)\big{]}G(u)\,du\Big{\\}}ds\big{|}\Big{]}=0.$

By Lemma 3.3, we could replace $\eta^{N}_{s}(1)$ (resp. $\eta^{N}_{s}(N-1)$)
with $\alpha$ (resp. $\beta$). This concludes the proof for the case
$0\leq\theta<1$. ∎

###  4.2. The case $\theta=1$.

In this subsection, we prove Theorem 2.3 for the case $\theta=1$. Fix $G\in
C[0,1]$. We need to find an appropriate function $H$ such that
$H^{\prime\prime}=G$ on (0,1) and that the coefficients of $\eta(1)$ and
$\eta(N-1)$ vanish in (4.2). Note that in this case we do not need any
replacement lemma.

###### Proof of Theorem 2.3 in the case $\theta=1$..

For $G\in C[0,1]$, let

$\displaystyle
H(u)=\int_{0}^{1}\Big{(}\frac{v}{2+c}-\frac{1+c}{c(2+c)}\Big{)}G(v)dv+u\int_{0}^{1}\Big{(}\frac{cv}{2+c}-\frac{1+c}{2+c}\Big{)}G(v)dv+\int_{0}^{u}\int_{0}^{v}G(w)\,dw\,dv.$

It is easy to check that $H\in C^{2}[0,1]$, $H^{\prime\prime}=G$ on $(0,1)$,
and that

$\displaystyle
H(0)=\int_{0}^{1}\Big{(}\frac{v}{2+c}-\frac{1+c}{c(2+c)}\Big{)}G(v)dv,$
$\displaystyle\quad
H(1)=\int_{0}^{1}\Big{(}-\frac{v}{2+c}-\frac{1}{c(2+c)}\Big{)}G(v)dv,$
$\displaystyle
H^{\prime}(0)=\int_{0}^{1}\Big{(}\frac{cv}{2+c}-\frac{1+c}{2+c}\Big{)}G(v)dv,$
$\displaystyle\quad
H^{\prime}(1)=\int_{0}^{1}\Big{(}\frac{cv}{2+c}+\frac{1}{2+c}\Big{)}G(v)dv.$

Taking the function $H$ into (4.2) and calculating the coefficients of
$\eta(1)$ and $\eta(N-1)$, we have

$\displaystyle-H^{\prime}(1)-cH(1)=0,\quad H^{\prime}(0)-cH(0)=0.$

The constant term in (4.2) is given by

$c\alpha H(0)+c\beta
H(1)=\int_{0}^{1}\Big{(}\frac{c(\alpha-\beta)}{2+c}v-\alpha-\frac{\beta-\alpha}{2+c}\Big{)}G(v)dv.$

Whence, the integrand in (4.2) is equal to

$\big{\langle}\pi^{N}_{s},G\big{\rangle}-\int_{0}^{1}\Big{(}\frac{c(\beta-\alpha)}{2+c}v+\alpha+\frac{\beta-\alpha}{2+c}\Big{)}G(v)dv.$

By (4.2), the time integral of the above term converges in
$L^{1}(\mathbb{P}^{N}_{\mu_{N}})$ to zero as $N\rightarrow\infty$. This
concludes the proof for the case $\theta=1$. ∎

###  4.3. The case $\theta>1$.

In this subsection, we prove Theorem 2.3 for the case $\theta>1$. In this
case, the last line in (4.2) converges to zero as $N\rightarrow\infty$.
Whence, we do not need special properties of the function $H$.

###### Proof of Theorem 2.3 in the case $\theta>1$..

For $G\in C[0,1]$, let

$\displaystyle H(u)=\int_{0}^{u}\int_{0}^{v}G(w)\,dw\,dv.$

Taking the function $H$ into (4.2) and by Lemma 3.4,

$\lim_{N\rightarrow\infty}{\mathbb{E}}^{N}_{\mu_{N}}\Big{[}\big{|}\int_{0}^{t}\Big{\\{}\big{\langle}\pi^{N}_{s},G\big{\rangle}-m^{N}_{s}(H^{\prime}(1)-H^{\prime}(0))\Big{\\}}ds\big{|}\Big{]}=0.$

It is easy to check

$H^{\prime}(1)-H^{\prime}(0)=\int_{0}^{1}G(u)du.$

Therefore,

$\lim_{N\rightarrow\infty}{\mathbb{E}}^{N}_{\mu_{N}}\Big{[}\big{|}\int_{0}^{t}\Big{\\{}\big{\langle}\pi^{N}_{s},G\big{\rangle}-m^{N}_{s}\int_{0}^{1}G(u)du\Big{\\}}ds\big{|}\Big{]}=0.$

By Lemma 3.5, we conclude the proof for the case $\theta>1$. ∎

Acknowledgments. Zhao thanks the financial support from the ANR grant MICMOV
(ANR-19-CE40-0012) of the French National Research Agency (ANR).

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