# The voter model with a slow membrane

Xiaofeng Xue  School of Mathematics and Statistics, Beijing Jiaotong
University, Beijing 100044, China<EMAIL_ADDRESS>and  Linjie Zhao  School
of Mathematics and Statistics, Huazhong University of Science & Technology,
Wuhan 430074, China<EMAIL_ADDRESS>
###### Abstract.

We introduce the voter model on the infinite lattice with a slow membrane and
investigate its hydrodynamic behavior and nonequilibrium fluctuations. The
model is defined as follows: a voter adopts one of its neighbors’ opinion at
rate one except for neighbors crossing the hyperplane $\\{x:x_{1}=1/2\\}$,
where the rate is $\alpha N^{-\beta}$. Above, $\alpha>0,\,\beta\geq 0$ are two
parameters and $N$ is the scaling parameter. The hydrodynamic equation turns
out to be heat equation with various boundary conditions depending on the
value of $\beta$. For the nonequilibrium fluctuations, the limit is described
by generalized Ornstein-Uhlenbeck process with certain boundary condition
corresponding to the hydrodynamic equation.

###### Key words and phrases:

voter model; slow membrane; hydrodynamic limits; nonequilibrium fluctuations

##  1\. Introduction

One of the main issues in statistical physics is to derive partial
differential equations from microscopic systems. Roughly speaking, for
symmetric systems the macroscopic behaviors are usually governed by diffusive
equations, and for asymmetric systems usually by hyperbolic equations. We
refer the readers to [10, 15] for a comprehensive understanding of the above
subject. Recently, it has been a popular topic to establish PDEs with various
boundary conditions from interacting particle systems. For example, Gonçalves,
Franco and their collaborators have obtained the heat equation with
Dirichlet/Robin/Neumann boundary conditions from the symmetric simple
exclusion processes in one dimension [1, 6, 7]. The microscopic models they
considered are either defined on a ring with a slow site/slow bond, or defined
on a segment with slow boundaries. The results have also been extended to
higher dimensions in [8, 14, 16].

The voter model also plays an important role in the theory of interacting
particle systems, and its hydrodynamic behavior has been considered by
Presutti and Spohn in [13]. For each $x\in\mathbb{Z}^{d}$, imagine there is a
voter at site $x$. Each voter has one of two possible opinions on an issue,
and the possible opinions are denoted by $0$ or $1$. The dynamics is quite
simple: a voter adopts one of its neighbors’ opinion at rate one. Although the
voter model has the same macroscopic behavior as the symmetric exclusion
process, it differs from the later in essential points: the magnetization of
the voter model is not locally conserved and the static correlations decay
slowly. We refer the readers to [13, 12] for details of the above properties.
A natural question is to consider the impact of the slow dynamics introduced
in [1, 6, 7] on the voter model, which is the main aim of this article.

In order to obtain boundary conditions, we let the voters at sites $x$ and
$x+e_{1}$ adopt the other’s opinion at rate $\alpha N^{-\beta}$ if $x_{1}=0$.
Above, $x_{1}$ is the first coordinate of site $x$, and $\\{e_{i}\\}_{1\leq
i\leq d}$ is the canonical basis of $\mathbb{Z}^{d}$. The two parameters
$\alpha>0,\beta\geq 0$ denote the strength of the boundary interaction, and
$N\in\mathbb{N}$ is a scaling parameter. We consider the voter model in
dimensions $d\geq 4$, and derive Robin boundary conditions if $\beta=1$ and
Neumann boundary conditions if $\beta>1$. There are no boundary conditions in
the case $0\leq\beta<1$. Moreover, for nonequilibrium fluctuations, we prove
that if $\beta=1$, the limit is given by generalized Ornstein–Uhlenbeck
process with Robin boundary condition, and if $\beta>3/2$, the limit has
Neumann boundary condition. We remark that the other cases remain open.

Although the results are similar to the exclusion process, the proof differs a
lot from the latter because of the two properties we addressed above. The
standard approach to prove a hydrodynamic limit result is as follows: we first
write down a martingale formula for the empirical measure of the process by
using Dynkin’s formula, and then prove the so called replacement lemmas to
close the equation. To ensure uniqueness of solution to hydrodynamic
equations, an energy estimate is usually also needed. For exclusion process
with boundary dynamics, the replacement lemmas and energy estimates are proved
by investigating the entropy production of the process. This is not an easy
task for the voter model with boundary terms since the invariant measures of
the voter model in this case are not explicitly known. Instead, we adopt the
duality method and investigate the correlation functions of the model. It is
well known that the duality of the voter model is coalescing random walks,
_cf._[12] for example. For the voter model with boundary dynamics we
introduced above, the duality turns out to be coalescing random walks with
slow bonds. Along the proof, we exploit an invariance principle for the random
walk with slow bonds, proved very recently by Erhard et al. in [3]. We also
remark that no replacement lemmas are needed in Presutti and Spohn’s paper
[13] since the process is linear.

We believe the results should also hold for dimension $d=3$. The main issue is
to investigate the meeting probabilities of two independent random walks with
slow bonds in dimension $d=3$, which has long been a hard problem and has its
own interest. This case is presently out of our reach and we leave it open.
For dimensions $d\leq 2$, locally the voter model is in a superposition of the
configuration with all sites occupied and the one with all sites empty, see
[13, Section 2] for details.

The rest of the paper is organized as follows. In Section 2, we define the
model rigorously and state our main results. We state duality of the voter
model and invariance principle for the random walk with slow bounds in Section
3. Hydrodynamic limits and fluctuations for the voter model are proved
respectively in Sections 4 and 5.

##  2\. Notation and Results

###  2.1. The voter model.

The state space of the voter model is
$\Omega_{d}:=\\{0,1\\}^{\mathbb{Z}^{d}}$, $d\geq 1$. In this article, we focus
on the case $d\geq 4$. We use $x,y$ (resp. $u,v$) to denote points in
$\mathbb{Z}^{d}$ (resp. $\mathbb{R}^{d}$), and $x_{i}$ (resp. $u_{i}$), $1\leq
i\leq d$, to denote the $i$-th component of the point $x$ (resp. $u$). Let
$\\{e_{i}\\}_{1\leq i\leq d}$ be the canonical basis of $\mathbb{Z}^{d}$,
_i.e._ , $e_{i}$ is the vector in $\mathbb{Z}^{d}$ with the $i$–th component
equal to one and the others equal to zero. For a point $x\in\mathbb{Z}^{d}$,
denote $|x|=\sum_{i=1}^{d}|x_{i}|$. Fix parameters $\alpha>0,\,\beta\geq 0$.
We use $N\in\mathbb{N}$ to denote the scaling parameter. The generator of the
process $\mathcal{L}_{N}$, which acts on local functions
$f:\Omega_{d}\rightarrow\mathbb{R}$, is given by

$\mathcal{L}_{N}f(\eta)=\sum_{x,y\in\mathbb{Z}^{d},\atop|x-y|=1}\xi^{N}_{x,y}\big{(}f(\eta^{x,y})-f(\eta)\big{)},$
(2.1)

where $\eta^{x,y}$ is the configuration obtained from $\eta$ by flipping the
value of $\eta(x)$ to $\eta(y)$, _i.e._ , $\eta^{x,y}(x)=\eta(y)$ and
$\eta^{x,y}(z)=\eta(z)$ for $z\neq x$. The flipping rates $\xi^{N}_{x,y}$ are
symmetric, $\xi^{N}_{x,y}=\xi^{N}_{y,x}$ for $|x-y|=1$, and are given by

$\xi^{N}_{x,y}=\begin{cases}\alpha N^{-\beta}&\text{if\;}x_{1}=0,y_{1}=1\\\
1\quad&\text{otherwise}\end{cases}$ (2.2)

We refer the readers to [12] for a construction of the process.

###  2.2. Hydrodynamic limit.

In this subsection, we state the first main result of the article. To this
aim, we first introduce weak solutions of heat equations with/without boundary
conditions. Hereafter, we fix a time horizon $T>0$.

######  Definition 2.1 (Hydrodynamic equation for $0\leq\beta<1$).

Let $\rho_{0}:{\mathbb{R}}^{d}\rightarrow[0,1]$ be continuous. A bounded
function $\rho:[0,T]\times\mathbb{R}^{d}\rightarrow\mathbb{R}$ is said to be a
weak solution of the heat equation

$\begin{cases}\partial_{t}\rho(t,u)=\Delta\rho(t,u),\quad&t>0,\;u\in\mathbb{R}^{d}\\\
\rho(0,u)=\rho_{0}(u),\quad&u\in\mathbb{R}^{d}\end{cases}$ (2.3)

if for any $H\in C_{c}^{2}(\mathbb{R}^{d})$ and any $t\in[0,T]$,

$\int_{\mathbb{R}^{d}}\rho(t,u)H(u)du=\int_{\mathbb{R}^{d}}\rho_{0}(u)H(u)du+\int_{0}^{t}\int_{\mathbb{R}^{d}}\rho(s,u)\Delta
H(u)\,du\,ds.$

For the case $\beta\geq 1$, we first introduce the notion of Sobolev spaces.
For an open subset $U\subset\mathbb{R}^{d}$, let ${\mathcal{H}^{1}}(U)$ be the
Sobolev space which consists of locally integrable functions with _weak
derivatives_ in $L^{2}(U)$, _i.e._ , for any $\varrho\in{\mathcal{H}^{1}}(U)$,
there exist $d$ elements in $L^{2}(U)$, denoted by
$\partial_{u_{1}}\varrho,\ldots,\partial_{u_{d}}\varrho$, such that

$\int_{U}\varrho(u)\partial_{u_{i}}H(u)du=-\int_{U}\partial_{u_{i}}\varrho(u)H(u)du$

for any $H\in C_{c}^{\infty}(U)$ and for any $1\leq i\leq d$. Denote by
$L^{2}([0,T],{\mathcal{H}}^{1}(U))$ the space of measurable functions
$\rho:[0,T]\rightarrow{\mathcal{H}}^{1}(U)$ such that the norm defined by

$\|\rho\|^{2}_{L^{2}([0,T],{\mathcal{H}}^{1}(U))}:=\int_{0}^{T}\int_{U}\sum_{i=1}^{d}\big{(}\partial_{u_{i}}\rho(t,u)\big{)}^{2}\,du\,dt$

is finite. We refer the readers to [5, Chapter 5] for properties of Sobolev
spaces.

In this article, we are concerned with the case where
$U=\mathbb{R}^{d}\backslash\\{u\in\mathbb{R}^{d}:u_{1}=0\\}$. For simplicity,
from now on we write $\\{u\in\mathbb{R}^{d}:~{}u_{1}\in A\\}$ as $\\{u_{1}\in
A\\}$ for any subset $A\subset\mathbb{R}$. For any
$f\in{\mathcal{H}}^{1}\left(\mathbb{R}^{d}\backslash\\{u_{1}=0\\}\right)$, we
have that $f\big{|}_{u_{1}>0}\in{\mathcal{H}}^{1}\left(\\{u_{1}>0\\}\right)$
and $f\big{|}_{u_{1}<0}\in{\mathcal{H}}^{1}\left(\\{u_{1}<0\\}\right)$. Then,
we use $f(\cdot^{+})$ to denote the trace of $f\big{|}_{u_{1}>0}$ on
$\\{u_{1}=0\\}$ and use $f(\cdot^{-})$ to denote the trace of
$f\big{|}_{u_{1}<0}$ on $\\{u_{1}=0\\}$. The following property of
$f(\cdot^{+})$ and $f(\cdot^{-})$ is crucial. For any bounded
$D\subset\\{u_{1}=0\\}$, according to the trace theorem (_cf._ [5, Theorem 1,
Section 5.5] for example), it is easy to check that

${\rm Aver}_{\varepsilon,+}f\big{|}_{D}\rightarrow
f(\cdot^{+})\big{|}_{D}\text{~{}and~{}}{\rm
Aver}_{\varepsilon,-}f\big{|}_{D}\rightarrow f(\cdot^{-})\big{|}_{D}$ (2.4)

in $L^{2}(D)$ as $\varepsilon\rightarrow 0$, where

${\rm
Aver}_{\varepsilon,+}f(u)=\frac{1}{(2\varepsilon)^{d-1}\varepsilon}\int\limits_{-\varepsilon<v_{i}<\varepsilon\text{~{}for~{}}2\leq
i\leq d,\atop 0<v_{1}<\varepsilon}f(u+v)dv$

and

${\rm
Aver}_{\varepsilon,-}f(u)=\frac{1}{(2\varepsilon)^{d-1}\varepsilon}\int\limits_{-\varepsilon<v_{i}<\varepsilon\text{~{}for~{}}2\leq
i\leq d,\atop-\varepsilon<v_{1}<0}f(u+v)dv$

for any $u\in\\{u_{1}=0\\}$.

We use $\mathcal{C}$ to denote the set of functions $H$ such that

$H(u)=H^{+}(u)1_{\\{u_{1}>0\\}}+H^{-}(u)1_{\\{u_{1}\leq 0\\}}$

for some $H^{+},H^{-}\in C^{2}_{c}(\mathbb{R}^{d})$. Then, for any
$H\in\mathcal{C}$ and $u\in\\{u_{1}=0\\}$, define

$H(u^{+})=H^{+}(u),\;H(u^{-})=H^{-}(u),\;\partial_{u_{1}}^{+}H(u)=\partial_{u_{1}}H^{+}(u),\;\partial_{u_{1}}^{-}H(u)=\partial_{u_{1}}H^{-}(u).$

Note that functions in $\mathcal{C}$ may be discontinuous at the boundary
$\\{u_{1}=0\\}$.

######  Definition 2.2 (Hydrodynamic equation for $\beta=1$).

Let $\rho_{0}:{\mathbb{R}}^{d}\rightarrow[0,1]$ be continuous. A bounded
function $\rho:[0,T]\times\mathbb{R}^{d}\rightarrow\mathbb{R}$ is said to be a
weak solution of the heat equation with Robin boundary condition

$\begin{cases}\partial_{t}\rho(t,u)=\Delta\rho(t,u),\quad&t>0,\;u\in\mathbb{R}^{d}\backslash\\{u_{1}=0\\}\\\
\partial_{u_{1}}^{+}\rho(t,u)=\partial_{u_{1}}^{-}\rho(t,u)=\alpha(\rho(t,u^{+})-\rho(t,u^{-})),\quad&u_{1}=0,\\\
\rho(0,u)=\rho_{0}(u),\quad&u\in\mathbb{R}^{d}\end{cases}$ (2.5)

if $\rho\in
L^{2}([0,T],{\mathcal{H}}^{1}(\mathbb{R}^{d}\backslash\\{u_{1}=0\\}))$, and
for any function $H\in\mathcal{C}$, for any $t\in[0,T]$,

$\int_{\mathbb{R}^{d}}\rho(t,u)H(u)du=\int_{\mathbb{R}^{d}}\rho_{0}(u)H(u)du\\\
+\int_{0}^{t}\int_{\mathbb{R}^{d}}\rho(s,u)\sum_{i=2}^{d}\partial_{u_{i}}^{2}H(u)du\,ds+\int_{0}^{t}\int_{\\{u_{1}\neq
0\\}}\rho(s,u)\partial_{u_{1}}^{2}H(u)du\,ds\\\
+\int_{0}^{t}\int_{\\{u_{1}=0\\}}\big{\\{}\rho_{s}(u^{+})\partial_{u_{1}}^{+}H(u)-\rho_{s}(u^{-})\partial_{u_{1}}^{-}H(u)+\alpha[\rho_{s}(u^{-})-\rho_{s}(u^{+})][H(u^{+})-H(u^{-})]\big{\\}}\,dS\,ds.$

######  Definition 2.3 (Hydrodynamic equation for $\beta>1$).

Let $\rho_{0}:{\mathbb{R}}^{d}\rightarrow[0,1]$ be continuous. A bounded
function $\rho:[0,T]\times\mathbb{R}^{d}\rightarrow\mathbb{R}$ is said to be a
weak solution of the heat equation with Neumann boundary condition

$\begin{cases}\partial_{t}\rho(t,u)=\Delta\rho(t,u),\quad&t>0,\;u\in\mathbb{R}^{d}\backslash\\{u_{1}=0\\}\\\
\partial_{u_{1}}^{+}\rho(t,u)=\partial_{u_{1}}^{-}\rho(t,u)=0,\quad&u_{1}=0,\\\
\rho(0,u)=\rho_{0}(u),\quad&u\in\mathbb{R}^{d}\end{cases}$ (2.6)

if $\rho\in
L^{2}([0,T],{\mathcal{H}}^{1}(\mathbb{R}^{d}\backslash\\{u_{1}=0\\}))$, and
for any function $H\in\mathcal{C}$, for any $t\in[0,T]$,

$\int_{\mathbb{R}^{d}}\rho(t,u)H(u)du=\int_{\mathbb{R}^{d}}\rho_{0}(u)H(u)du\\\
+\int_{0}^{t}\int_{\mathbb{R}^{d}}\rho(s,u)\sum_{i=2}^{d}\partial_{u_{i}}^{2}H(u)du\,ds+\int_{0}^{t}\int_{\\{u_{1}\neq
0\\}}\rho(s,u)\partial_{u_{1}}^{2}H(u)du\,ds\\\
+\int_{0}^{t}\int_{\\{u_{1}=0\\}}\big{\\{}\rho_{s}(u^{+})\partial_{u_{1}}^{+}H(u)-\rho_{s}(u^{-})\partial_{u_{1}}^{-}H(u)\big{\\}}\,dS\,ds.$

######  Remark 2.4 (Uniqueness of weak solutions).

It is well known that the weak solution to the heat equation (2.3) is unique.
Following the ideas in [6, Section 7.2], it is easy to prove the weak
solutions to the above PDEs (2.5) and (2.6) are unique. We also refer the
readers to [8, Section 7] for the uniqueness of weak solutions to the above
PDEs if the underlying space is the torus ${\mathbb{T}}^{d}$ instead of
$\mathbb{R}^{d}$.

Throughout the article, we assume the initial distribution $\mu_{N}$ of the
process satisfies the following condition.

######  Assumption (A).

Fix an initial density profile $\rho_{0}:\mathbb{R}^{d}\rightarrow[0,1]$ such
that $\rho_{0}$ has continuous and bounded partial derivative with respect to
the first coordinate $u_{1}$, and that $\rho_{0}$ is a global Lipschitz
function on $\mathbb{R}^{d}$. Assume that under $\mu_{N}$,
$\\{\eta(x)\\}_{x\in\mathbb{Z}^{d}}$ are independent, and that

$\mu_{N}(\eta(x)=1)=\rho_{0}\left(\tfrac{x}{N}\right)$

for any $x\in\mathbb{Z}^{d}$.

Let $\\{\eta_{t}\\}_{t\geq 0}$ be the accelerated process with generator
$N^{2}\mathcal{L}_{N}$ and with initial condition $\mu_{N}$. To make notations
short, we omit the dependence of the process $\\{\eta_{t}\\}_{t\geq 0}$ on
$N$. Denote by $\mathbb{P}$ the probability measure on the path space
$D([0,T],\Omega_{d})$ associated with the process $\\{\eta_{t}\\}_{t\geq 0}$
and the initial distribution $\mu_{N}$, and by $\mathbb{E}$ the corresponding
expectation. Note that we also omit the dependence of $\mathbb{P}$ (resp.
$\mathbb{E}$) on $N$ and $\mu_{N}$ for short.

Now we are ready to state the hydrodynamic limits for the voter model with a
slow membrane.

######  Theorem 2.5.

Assume $d\geq 4$. For any $t\in[0,T]$, any $\varepsilon>0$ and any $H\in
C_{c}(\mathbb{R}^{d})$,

$\lim_{N\rightarrow\infty}\mathbb{P}\Big{(}\Big{|}\frac{1}{N^{d}}\sum_{x\in\mathbb{Z}^{d}}\eta_{t}(x)H(\tfrac{x}{N})-\int_{\mathbb{R}^{d}}\rho(t,u)H(u)du\Big{|}>\varepsilon\Big{)}=0,$

where $\rho(t,u)$ is the unique weak solution

  1. (i)

to the heat equation (2.3) if $0\leq\beta<1$;

  2. (ii)

to the heat equation with Robin boundary condition (2.5) if $\beta=1$;

  3. (iii)

to the heat equation with Neumann boundary condition (2.6) if $\beta>1$.

###  2.3. Fluctuations.

In this subsection, we state nonequilibrium fluctuations for the voter model
with a slow membrane. We first introduce the space of test functions for the
density fluctuation field.

######  Definition 2.6 (The space of test functions).

(1) For $0<\beta<1$, let $\mathcal{S}_{\beta}(\mathbb{R}^{d})$ be the usual
Schwartz space $\mathcal{S}(\mathbb{R}^{d})$.

(2) For $\beta=1$, let $\mathcal{S}_{\beta}(\mathbb{R}^{d})$ be the family of
functions $H\in{\mathcal{S}}(\mathbb{R}^{d}\backslash\\{u_{1}=0\\})$ such that

$\partial_{u_{1}}^{+,2k+1}H(t,u)=\partial_{u_{1}}^{-,2k+1}H(t,u)=\alpha\big{[}\partial_{u_{1}}^{+,2k}H(t,u)-\partial_{u_{1}}^{-,2k}H(t,u)\big{]},\;\forall
u_{1}=0,\;\forall k\geq 0.$

(3) For $\beta>1$, let $\mathcal{S}_{\beta}(\mathbb{R}^{d})$ be the family of
functions $H\in{\mathcal{S}}(\mathbb{R}^{d}\backslash\\{u_{1}=0\\})$ such that

$\partial_{u_{1}}^{+,2k+1}H(t,u)=\partial_{u_{1}}^{-,2k+1}H(t,u)=0,\;\forall
u_{1}=0,\;\forall k\geq 0.$

Note that, according to above definitions, the semigroup $T_{\beta,t}$
associated to the corresponding hydrodynamic equations can be considered as an
operator from $\mathcal{S}_{\beta}(\mathbb{R}^{d})$ to
$\mathcal{S}_{\beta}(\mathbb{R}^{d})$ for any $\beta>0$ (_cf._ [4, Proposition
2.3]), which is crucial for us later to define a generalized Ornstein-
Uhlenbeck process as the fluctuation limit of our process.

Define the fluctuation field ${\mathcal{Y}}^{N}_{t}$, which acts on functions
$H\in{\mathcal{S}}_{\beta}(\mathbb{R}^{d})$, as

${\mathcal{Y}}^{N}_{t}(H)=\frac{1}{N^{1+d/2}}\sum_{x\in\mathbb{Z}^{d}}\bar{\eta}_{t}(x)H(\tfrac{x}{N})=N^{d/2-1}\Big{(}\pi^{N}_{t}(H)-\mathbb{E}\big{[}\pi^{N}_{t}(H)\big{]}\Big{)},$

where $\bar{\eta}_{t}(x)=\eta_{t}(x)-\mathbb{E}[\eta_{t}(x)]$. Next, we
introduce the limit of the fluctuation field, which is given by a martingale
problem.

######  Definition 2.7 (The limiting fluctuation field).

We say $\mathcal{Y}_{\cdot}\in C([0,T],\mathcal{S}^{\prime}(\mathbb{R}^{d}))$
is a solution to the following generalized Ornstein-Uhlenbeck process

$\begin{cases}\partial_{t}{\mathcal{Y}}_{t}=\Delta{\mathcal{Y}}_{t}+\dot{{\mathcal{W}}}_{t},\\\
{\mathcal{Y}}_{0}=0,\end{cases}$ (2.7)

if for any $H\in\mathcal{S}_{\beta}(\mathbb{R}^{d})$,

$\displaystyle{\mathcal{M}}_{t}(H)$
$\displaystyle={\mathcal{Y}}_{t}(H)-\int_{0}^{t}{\mathcal{Y}}_{s}(\Delta
H)ds,$ $\displaystyle{\mathcal{N}}_{t}(H)$
$\displaystyle=\big{[}{\mathcal{M}}_{t}(H)\big{]}^{2}-4d(1-\gamma_{d})\int_{0}^{t}\int_{\mathbb{R}^{d}}\rho(s,u)(1-\rho(s,u))H^{2}(u)duds$

are both martingales. Above, $\rho(t,u)$ is the unique solution to the
corresponding hydrodynamic equation defined in the last subsection, and
$\dot{{\mathcal{W}}}_{t}$ is inhomogeneous space-time white noise with mean
zero and covariance given by

$E[{\mathcal{W}_{t}}(H){\mathcal{W}}_{s}(G)]=4d(1-\gamma_{d})\int_{0}^{t\wedge
s}\int_{\mathbb{R}^{d}}\rho(\tau,u)(1-\rho(\tau,u)H(u)G(u)dud\tau$

for $H,G\in L^{2}(\mathbb{R}^{d})$, where $\gamma_{d}$ is the probability that
the simple random walk on $\mathbb{Z}^{d}$ starting at $0$ returns to $0$
again.

######  Remark 2.8.

We refer the readers to [4, Proposition 2.7] for a detailed proof of the
uniqueness of the solution to the generalized Ornstein-Uhlenbeck process
(2.7). Here we give an outline. Roughly speaking, since
$\\{{\mathcal{M}}_{t}(H)\\}_{t\geq 0},\\{{\mathcal{N}}_{t}(H)\\}_{t\geq 0}$
are martingales, via Itô’s formula we can deduce that

$\displaystyle
E\left(e^{\sqrt{-1}{\mathcal{Y}}_{t}(H)}\Big{|}{\mathcal{Y}}_{r},r\leq
s\right)$
$\displaystyle=\exp\left(\sqrt{-1}{\mathcal{Y}}_{s}\left(T_{\beta,t-s}H\right)-\frac{4d(1-\gamma_{d})}{2}\int_{s}^{t}\int_{\mathbb{R}^{d}}\rho(\tau,u)(1-\rho(\tau,u))\left(T_{\beta,t-\tau}H(u)\right)^{2}dud\tau\right)$

for any $s<t$, where $\\{T_{\beta,t}H\\}_{t\geq 0}$ is the solution to the
heat equation with corresponding boundary condition and initial condition $H$.
Hence $\\{{\mathcal{Y}}_{t}\\}_{t\geq 0}$ is uniquely distributed.

The following is the second main result of the article.

######  Theorem 2.9.

Assume $\beta=1$ or $\beta>3/2$. The sequence of the processes
$\\{{\mathcal{Y}}^{N}_{t},0\leq t\leq T\\}_{N\geq 1}$, converges in
distribution, as $N\rightarrow\infty$, with respect to the Skorohod topology
of $D([0,T],\mathcal{S}_{\beta}^{\prime}(\mathbb{R}^{d}))$ to
$\\{{\mathcal{Y}}_{t},0\leq t\leq T\\}$, which is the unique solution to the
generalized Ornstein-Uhlenbeck process (2.7).

######  Remark 2.10.

The above results should also hold for $0<\beta<1$ and $1<\beta\leq 3/2$.
However, this is beyond the techniques of the present article. See Remark 5.1
and 5.3 for details.

According to Remark 2.8, we have the following corollary of Theorem 2.9.

######  Corollary 2.11.

Assume $\beta=1$ or $\beta>3/2$. For any
$H\in\mathcal{S}_{\beta}(\mathbb{R}^{d})$ and $t>0$,
${\mathcal{Y}}^{N}_{t}(H)$ converges in distribution to

${\mathcal{N}}\left(0,4d(1-\gamma_{d})\int_{0}^{t}\int_{\mathbb{R}^{d}}\rho(\tau,u)(1-\rho(\tau,u))\left(T_{\beta,t-\tau}H(u)\right)^{2}dud\tau\right)$

as $N\rightarrow+\infty$, where ${\mathcal{N}}(\mu,\sigma^{2})$ is the normal
distribution with mean $\mu$ and variance $\sigma^{2}$.

##  3\. Preliminaries

###  3.1. Duality

In this subsection we recall the duality relationship introduced in [9]. Let
$\\{Y_{t,\beta}^{N}\\}_{t\geq 0}$ be the random walk on the one dimensional
integer lattice $\mathbb{Z}$ with generator $\Omega_{N,\beta}$ given by

$\Omega_{N,\beta}f(i)=\begin{cases}f(i+1)-f(i)+f(i-1)-f(i)&\text{~{}if~{}}i\neq
0,1,\\\ \alpha
N^{-\beta}\left[f(1)-f(0)\right]+f(-1)-f(0)&\text{~{}if~{}}i=0,\\\ \alpha
N^{-\beta}\left[f(0)-f(1)\right]+f(2)-f(1)&\text{~{}if~{}}i=1\end{cases}$

for every $f:{\mathbb{Z}}\rightarrow{\mathbb{R}}$. Roughly speaking, the above
random walk jumps to its neighbor at rate one everywhere except across the
bond $(0,1)$, where the rate is $\alpha N^{-\beta}$. We further denote by
$\\{U_{t}\\}_{t\geq 0}$ the usual simple symmetric random walk on
$\mathbb{Z}$, i.e., the generator of $U_{t}$ is given by $\Omega_{N,\beta}$
with $(\alpha,\beta)$ replaced by $(1,0)$. Let $\\{X_{t,\beta}^{N}\\}_{t\geq
0}$ be the random walk on $\mathbb{Z}^{d}$ that
$\\{\\{X_{t,\beta}^{N}(i)\\}_{t\geq 0}\\}_{1\leq i\leq d}$ are independent,
$\\{X_{t,\beta}^{N}(1)\\}_{t\geq 0}$ is a copy of
$\\{Y_{t,\beta}^{N}\\}_{t\geq 0}$ and $\\{X_{t,\beta}^{N}(i)\\}_{t\geq 0}$ is
a copy of $\\{U_{t}\\}_{t\geq 0}$ for $i=2,3,\ldots,d$, where
$X_{t,\beta}^{N}(i)$ is the $i$-th coordinate of $X_{t,\beta}^{N}$. We write
$X_{t,\beta}^{N}$ as $X_{t,\beta}^{N,x}$ when $X_{0,\beta}^{N}=x$.

For given $s>0$ and $x\in\mathbb{Z}^{d}$, we denote by
$\\{\hat{X}_{t,\beta}^{N,x,s}\\}_{t\geq 0}$ the _frozen_ random walk on
$\mathbb{Z}^{d}$ that $\hat{X}_{u,\beta}^{N,x,s}=x$ for $u\leq s$ and
$\\{\hat{X}_{s+t,\beta}^{N,x,s}\\}_{t\geq 0}$ is an independent copy of
$\\{{X}_{t,\beta}^{N,x}\\}_{t\geq 0}$. Intuitively, the random walk
$\\{\hat{X}_{t,\beta}^{N,x,s}\\}_{t\geq 0}$ stays frozen at site $x$ until
time $s$, and then performs the random walk as
$\\{{X}_{t,\beta}^{N,x}\\}_{t\geq 0}$ after time $s$. For given
$x,y\in\mathbb{Z}^{d},s>0$ and $\\{X_{t,\beta}^{N,x}\\}_{t\geq
0},\\{\hat{X}_{t,\beta}^{N,y,s}\\}_{t\geq 0}$ which are independent, we define

$\tau_{x,y}^{N,\beta,s}=\inf\\{u\geq
s:~{}X_{u,\beta}^{N,x}=\hat{X}^{N,y,s}_{u,\beta}\\}-s,$

i.e., $\tau_{x,y}^{N,\beta,s}$ is the time it takes for
$X^{N,x}_{\cdot,\beta}$ and $\hat{X}^{N,y,s}_{\cdot,\beta}$ to meet after
$\hat{X}^{N,y,s}_{\cdot,\beta}$ is unfrozen from $y$.

For given $x,y\in\mathbb{Z}^{d}$ and $s>0$, we denote by
$\\{\tilde{X}_{t,\beta,x}^{N,y,s}\\}_{t\geq 0}$ the random walk on
$\mathbb{Z}^{d}$ that

$\tilde{X}_{t,\beta,x}^{N,y,s}=\begin{cases}\hat{X}_{t,\beta}^{N,y,s}&\text{~{}if~{}}t\leq
s+\tau^{N,\beta,s}_{x,y},\\\
X_{t,\beta}^{N,x}&\text{~{}if~{}}t>s+\tau^{N,\beta,s}_{x,y}.\end{cases}$

Note that $\\{\tilde{X}_{t,\beta,x}^{N,y,s}\\}_{t\geq 0}$ and
$\\{\hat{X}_{t,\beta}^{N,y,s}\\}_{t\geq 0}$ have the same distribution but
$\\{\tilde{X}_{t,\beta,x}^{N,y,s}\\}_{t\geq 0}$ is not independent of
$\\{X_{t,\beta}^{N,x}\\}_{t\geq 0}$. The process
$\left\\{\left(X_{t,\beta}^{N,x},\tilde{X}_{t,\beta,x}^{N,y,s}\right)\right\\}_{t\geq
0}$ is the so-called _coalescing_ random walk.

To distinguish from the accelerated voter model, let
$\\{\tilde{\eta}_{t}\\}_{t\geq 0}$ be the process with generator given by
$\mathcal{L}_{N}$. According to the duality-relationship between the voter
model and the coalescing random walk given in [9], for given
$x,y\in\mathbb{Z}^{d}$ and $t>s$,

${\mathbb{P}}\left(\tilde{\eta}_{t}(x)=1\right)=\sum_{u\in\mathbb{Z}^{d}}{\mathbb{P}}\left(X_{t,\beta}^{N,x}=u\right){\mathbb{P}}(\tilde{\eta}_{0}(u)=1),$
(3.1)

and

${\mathbb{P}}\left(\tilde{\eta}_{t}(x)=\tilde{\eta}_{s}(y)=1\right)=\sum_{u\in\mathbb{Z}^{d}}\sum_{v\in\mathbb{Z}^{d}}{\mathbb{P}}\left(X_{t,\beta}^{N,x}=u,\tilde{X}_{t,\beta,x}^{N,y,t-s}=v\right){\mathbb{P}}(\tilde{\eta}_{0}(u)=\tilde{\eta}_{0}(v)=1)\\\
=\sum_{u\in\mathbb{Z}^{d}}{\mathbb{P}}\left(X_{t,\beta}^{N,x}=u,\tau_{x,y}^{N,\beta,t-s}\leq
s\right)P(\tilde{\eta}_{0}(u)=1)\\\ +\sum_{u\in\mathbb{Z}^{d}}\sum_{v\neq
u}{\mathbb{P}}\left(X_{t,\beta}^{N,x}=u,\hat{X}_{t,\beta}^{N,y,t-s}=v,\tau_{x,y}^{N,\beta,t-s}>s\right){\mathbb{P}}(\tilde{\eta}_{0}(u)=\tilde{\eta}_{0}(v)=1).$
(3.2)

Above, we also use ${\mathbb{P}}$ to denote the law of the process
$\\{\tilde{\eta}_{t}\\}_{t\geq 0}$ and the random walks since this will not
cause confusion. We refer the readers to [9] or [12, Section 3.4] for proofs
of the above duality relations.

###  3.2. Invariance principle

In this subsection we recall the invariance principle given in [3] of the
random walk $\\{Y_{t,\beta}^{N}\\}_{t\geq 0}$ on $\mathbb{Z}$ with slow bond
$(0,1)$. We denote by $\\{B_{t}\\}_{t\geq 0}$ the $1$-dimensional standard
Brownian motion and write $B_{t}$ as $B_{t}^{u}$ when $B_{0}=u$. For any
$\beta\geq 0$, we define $\\{B_{t,\beta}\\}_{t\geq 0}$ as follows. If
$0\leq\beta<1$, then $B_{t,\beta}=B_{t}$, i.e., the standard Brownian motion.
When $\beta>1$, then $B_{t,\beta}=|B_{t}^{u}|$ when
$B_{0,\beta}=u\in[0^{+},\infty)$ and $B_{t,\beta}=-|B_{t}^{u}|$ when
$B_{0,\beta}=u\in(-\infty,0^{-}]$, i.e., $B_{t,\beta}$ is the reflected
Brownian motion. When $\beta=1$, then $B_{t,\beta}$ is the _snapping out_
Brownian motion with parameter $2\alpha$ introduced in [11]. More precisely,
for every $u\in(-\infty,0^{-}]\cup[0^{+},+\infty)$ and $f\in
C_{b}\left((-\infty,0^{-}]\cup[0^{+},+\infty)\right)$,

$\mathbb{E}_{u}[f(B_{t,\beta})]=\mathbb{E}\left[\left(\frac{1+e^{-2\alpha
L_{t}}}{2}\right)f\left({\rm
sgn}(u)|B_{t}^{u}|\right)+\left(\frac{1-e^{-2\alpha
L_{t}}}{2}\right)f\left(-{\rm sgn}(u)|B_{t}^{u}|\right)\right],$

where $L_{t}$ is the local time of $\\{B_{t}\\}_{t\geq 0}$ at point $0$. We
write $B_{t,\beta}$ (resp. $Y_{t,\beta}^{N}$) as $B_{t,\beta}^{u}$ (resp.
$Y_{t,\beta}^{N,u}$) when $B_{0,\beta}=u$ (resp. $Y_{0,\beta}^{N}=u$). The
following invariance principle is proved by Erhard et al. in [3].

######  Theorem 3.1 ([3, Theorem 2.2]).

The following invariance principle holds for the one dimensional symmetric
random walk $\\{Y_{t,\beta}^{N}\\}_{t\geq 0}$ with a slow bond $(0,1)$,

  1. (i)

for every $u\neq 0$ and $t\geq 0$, $\frac{Y_{tN^{2},\beta}^{N,\lfloor
uN\rfloor}}{N}$ converges weakly to $B_{2t,\beta}^{u}$ as
$N\rightarrow+\infty$;

  2. (ii)

for every $x\in\\{1,2,\ldots\\}$ and $t\geq 0$,
$\frac{Y_{tN^{2},\beta}^{N,x}}{N}$ converges weakly to $B_{2t,\beta}^{0^{+}}$
as $N\rightarrow+\infty$;

  3. (iii)

for every $x\in\\{-1,-2,\ldots\\}$ and $t\geq 0$,
$\frac{Y_{tN^{2},\beta}^{N,x}}{N}$ converges weakly to $B_{2t,\beta}^{0^{-}}$
as $N\rightarrow+\infty$.

Note that although statements $(ii)$ and $(iii)$ in the above theorem are not
listed in the main theorem of [3], they are direct corollaries of Lemma 5.2
and Eq. (5.11) of [3].

##  4\. Hydrodynamic limits

###  4.1. Proof Outline

In this subsection, we outline the proof of Theorem 2.5. The procedures are
quite standard (_cf._ [10, Chapter 4] for example), and the main ingredients
are the replacement lemmas proved in Subsection 4.2.

Denote by ${\mathcal{M}}_{+}(\mathbb{R}^{d})$ the space of Radon measures on
$\mathbb{R}^{d}$ endowed with the vague topology, that is, for a sequence
$\\{\nu^{N}\\}_{N\geq 1},\,\nu\in{\mathcal{M}}_{+}(\mathbb{R}^{d})$,
$\nu^{N}\rightarrow\nu$ as $N\rightarrow\infty$ if for every $f\in
C_{c}(\mathbb{R}^{d})$,

$\lim_{N\rightarrow\infty}\big{\langle}\nu^{N},f\big{\rangle}=\big{\langle}\nu,f\big{\rangle}.$

For a configuration $\eta\in\Omega_{d}$, define the empirical measure
$\pi^{N}(\eta)\in{\mathcal{M}}_{+}(\mathbb{R}^{d})$ as

$\pi^{N}(\eta;du)=\frac{1}{N^{d}}\sum_{x\in\mathbb{Z}^{d}}\eta(x)\delta_{x/N}(du).$

Put $\pi^{N}_{t}(du)=\pi^{N}(\eta_{t};du)$. Let $Q^{N}$ be the distribution on
the path space $D([0,T],{\mathcal{M}}_{+}(\mathbb{R}^{d}))$ associated to the
process $\pi^{N}_{t}$ and the initial measure $\mu_{N}$.

By Lemma 4.2, the sequence $\\{Q^{N}\\}_{N\geq 1}$ is tight. Whence, any
subsequence of $Q^{N}$ further has a subsequence that converges as
$N\rightarrow\infty$, whose limit is denoted by $Q^{*}$. Moreover, $Q^{*}$ is
concentrated on trajectories which are absolutely continuous with respect to
the Lebesgue measure. Denote by $\rho(t,u)$ the corresponding density.

Now we characterize the limit. Fix a test function
$H:\mathbb{R}^{d}\rightarrow\mathbb{R}$ which will be specified later
depending on whether $\beta<1$ or $\beta\geq 1$. By Dynkin’s martingale
formula,

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

where $M^{N}_{t}(H)$ is a martingale, whose quadratic variation is given by

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

A simple computation shows that the last line is bounded by $C_{H}N^{2-d}$ for
some finite constant $C_{H}$, hence, by Doob’s inequality,

$\lim_{N\rightarrow\infty}\,\mathbb{E}\Big{[}\sup_{0\leq t\leq
T}\big{(}M^{N}_{t}(H)\big{)}^{2}\Big{]}=0.$ (4.2)

A tedious but elementary computation shows that the integrand in Eq. (4.1) is
equal to

$\displaystyle
N^{-d}\sum_{x\in\mathbb{Z}^{d}}\sum_{i=2}^{d}\eta_{s}(x)\partial_{u_{i}}^{2}H\big{(}\tfrac{x}{N}\big{)}+N^{-d}\sum_{x_{1}\neq
0,1}\eta_{s}(x)\partial_{u_{1}}^{2}H\big{(}\tfrac{x}{N}\big{)}+o_{N}(1)$ (4.3)
$\displaystyle+N^{1-d}\sum_{x_{1}=1}\eta_{s}(x)\partial_{u_{1}}^{+}H\big{(}\tfrac{x}{N}\big{)}-N^{1-d}\sum_{x_{1}=0}\eta_{s}(x)\partial_{u_{1}}^{-}H\big{(}\tfrac{x}{N}\big{)}$
(4.4) $\displaystyle+\alpha
N^{2-d-\beta}\sum_{x_{1}=0}\eta_{s}(x)\Big{(}H\big{(}\tfrac{x+e_{1}}{N}\big{)}-H\big{(}\tfrac{x}{N}\big{)}\Big{)}+\alpha
N^{2-d-\beta}\sum_{x_{1}=1}\eta_{s}(x)\Big{(}H\big{(}\tfrac{x-e_{1}}{N}\big{)}-H\big{(}\tfrac{x}{N}\big{)}\Big{)}.$
(4.5)

For a configuration $\eta$ and a positive integer $k\in\mathbb{N}$, define the
space average of $\eta$ over a box of size $k$ centered at $x$ as

$\bar{\eta}^{k}(x)=\frac{1}{(2k+1)^{d}}\sum_{|y-x|\leq k}\eta(y).$

Similarly, the space averages over the right/left boxes are defined as

$\bar{\eta}^{k,+}(x)=\frac{1}{|\Lambda^{+}_{x,k}|}\sum_{y\in\Lambda^{+}_{x,k}}\eta(y),\quad\bar{\eta}^{k,-}(x)=\frac{1}{|\Lambda^{-}_{x,k}|}\sum_{y\in\Lambda^{-}_{x,k}}\eta(y),$

where

$\Lambda^{+}_{x,k}=\\{y:|y-x|\leq k,y_{1}\geq
x_{1}\\},\quad\Lambda^{-}_{x,k}=\\{y:|y-x|\leq k,y_{1}\leq x_{1}\\}.$

Next we discuss the three cases respectively.

####  4.1.1. The case $0\leq\beta<1$

Take $H\in C_{c}^{2}(\mathbb{R}^{d})$. Whence

$\partial_{u_{1}}^{+}H\big{(}\tfrac{x}{N}\big{)}=\partial_{u_{1}}^{-}H\big{(}\tfrac{x}{N}\big{)}.$

By Lemma 4.1 $(i)$, we could replace $\eta_{s}(x)$ by
$\bar{\eta}_{s}^{\varepsilon N}(x)$ in (4.4). Note also that if $|x-y|=1$,
then

$|\bar{\eta}^{\varepsilon N}(x)-\bar{\eta}^{\varepsilon
N}(y)|\leq\frac{C}{\varepsilon N}$

for some finite constant $C$. Whence, the time integral of the term (4.4)
converges in probability to zero as $N\rightarrow\infty$. The term (4.5)
vanishes in the limit in the same way. Therefore, the limit $Q^{*}$
concentrates on trajectories whose densities $\rho(t,u)$ satisfy

$\int_{\mathbb{R}^{d}}\rho(t,u)H(u)du=\int_{\mathbb{R}^{d}}\rho_{0}(u)H(u)du+\int_{0}^{t}\int_{\mathbb{R}^{d}}\rho(s,u)\Delta
H(u)du\,ds$

####  4.1.2. The case $\beta=1$

Take $H\in\mathcal{C}$. By Lemma 4.1 $(ii)$, we could replace the time
integral of the term (4.4) with the time integral of

$N^{1-d}\sum_{x_{1}=1}\bar{\eta}_{s}^{\varepsilon
N,+}(x)\partial_{u_{1}}^{+}H\big{(}\tfrac{x}{N}\big{)}-N^{1-d}\sum_{x_{1}=0}\bar{\eta}_{s}^{\varepsilon
N,-}(x)\partial_{u_{1}}^{-}H\big{(}\tfrac{x}{N}\big{)}.$

The term (4.5) is handled in the same way. Observe that

$\bar{\eta}_{s}^{\varepsilon
N,+}(x)=\big{\langle}\pi^{N}_{s},\iota_{\varepsilon,x/N}^{+}\big{\rangle}+o_{N}(1),\quad\bar{\eta}_{s}^{\varepsilon
N,-}(x)=\big{\langle}\pi^{N}_{s},\iota_{\varepsilon,x/N}^{-}\big{\rangle}+o_{N}(1),$

where

$\displaystyle\iota_{\varepsilon,u}^{+}(v)=2^{-d+1}\varepsilon^{-d}\mathbbm{1}\\{0\leq
v_{1}-u_{1}\leq\varepsilon,|v_{i}-u_{i}|\leq\varepsilon,2\leq i\leq d\\},$
$\displaystyle\iota_{\varepsilon,u}^{-}(v)=2^{-d+1}\varepsilon^{-d}\mathbbm{1}\\{-\varepsilon\leq
v_{1}-u_{1}\leq 0,|v_{i}-u_{i}|\leq\varepsilon,2\leq i\leq d\\}.$

Therefore, by Eq. (2.4), as $N\rightarrow\infty$ and $\varepsilon\rightarrow
0$, the limit density $\rho(t,u)$ satisfies

$\int_{\mathbb{R}^{d}}\rho(t,u)H(u)du=\int_{\mathbb{R}^{d}}\rho_{0}(u)H(u)du\\\
+\int_{0}^{t}\int_{\mathbb{R}^{d}}\rho(s,u)\sum_{i=2}^{d}\partial_{u_{i}}^{2}H(u)du\,ds+\int_{0}^{t}\int_{u_{1}\neq
0}\rho(s,u)\partial_{u_{1}}^{2}H(u)du\,ds\\\
+\int_{0}^{t}\int_{u_{1}=0}\rho_{s}(u^{+})\partial_{u_{1}}^{+}H(u)-\rho_{s}(u^{-})\partial_{u_{1}}^{-}H(u)+\alpha(\rho_{s}(u^{-})-\rho_{s}(u^{+}))(H(u^{+})-H(u^{-}))\,dS\,ds.$

####  4.1.3. The case $\beta>1$

Take $H\in\mathcal{C}$. This case is similar to the case $\beta=1$. The
difference is that the term (4.5) converges in probability to zero as
$N\rightarrow\infty$ since $\beta>1$. Therefore, the limit density $\rho(t,u)$
satisfies

$\int_{\mathbb{R}^{d}}\rho(t,u)H(u)du=\int_{\mathbb{R}^{d}}\rho_{0}(u)H(u)du\\\
+\int_{0}^{t}\int_{\mathbb{R}^{d}}\rho(s,u)\sum_{i=2}^{d}\partial_{u_{i}}^{2}H(u)du\,ds+\int_{0}^{t}\int_{u_{1}\neq
0}\rho(s,u)\partial_{u_{1}}^{2}H(u)du\,ds\\\
+\int_{0}^{t}\int_{u_{1}=0}\rho_{s}(u^{+})\partial_{u_{1}}^{+}H(u)-\rho_{s}(u^{-})\partial_{u_{1}}^{-}H(u)\,dS\,ds.$

In Lemma 4.3, we show that any limit $Q^{*}$ of the sequence
$\\{Q^{N}\\}_{N\geq 1}$ is concentrated on trajectories whose densities belong
to the space
$L^{2}([0,T],{\mathcal{H}}^{1}(\mathbb{R}^{d}\backslash\\{u_{1}=0\\}))$ if
$\beta\geq 1$. Together with the above observations, $Q^{*}$ is concentrated
on trajectories whose densities are weak solutions to the corresponding
hydrodynamic equations. Since the weak solution is unique, the limit $Q^{*}$
is uniquely determined. This concludes the proof.

###  4.2. Replacement Lemma

The aim of this subsection is to prove the following replacement lemma.

######  Lemma 4.1 (Replacement Lemma).

For every $\delta>0$,

  1. $(i)$

if $0\leq\beta<1$, then for every $H\in C_{c}^{2}(\mathbb{R}^{d})$,

$\lim_{\varepsilon\rightarrow
0}\,\limsup_{N\rightarrow\infty}\,\mathbb{P}\Big{(}\Big{|}\int_{0}^{t}N^{1-d}\sum_{x_{1}=1}\big{(}\eta_{s}(x)-\bar{\eta}_{s}^{\varepsilon
N}(x)\big{)}H\left(\tfrac{x}{N}\right)\,ds\Big{|}>\delta\Big{)}=0.$

The same result holds with the summation over $\\{x_{1}=1\\}$ replaced by over
$\\{x_{1}=0\\}$.

  2. $(ii)$

if $\beta\geq 1$, then for every $H\in\mathcal{C}$

$\lim_{\varepsilon\rightarrow
0}\,\limsup_{N\rightarrow\infty}\,\mathbb{P}\Big{(}\Big{|}\int_{0}^{t}N^{1-d}\sum_{x_{1}=1}\big{(}\eta_{s}(x)-\bar{\eta}_{s}^{\varepsilon
N,+}(x)\big{)}H\left(\tfrac{x}{N}\right)\,ds\Big{|}>\delta\Big{)}=0.$

The same result holds with the summation over $\\{x_{1}=1\\}$ replaced by over
$\\{x_{1}=0\\}$, and with $\bar{\eta}_{s}^{\varepsilon N,+}(x)$ replaced by
$\bar{\eta}_{s}^{\varepsilon N,-}(x)$.

We only prove statement $(ii)$ since $(i)$ could be handled with in the same
way.

###### Proof of Lemma 4.1, $(ii)$.

By Chebyshev’s inequality, we only need to show that

$\lim_{\varepsilon\rightarrow
0}\limsup_{N\rightarrow+\infty}\mathbb{E}\left[\left(N^{1-d}\int_{0}^{t}\sum_{x_{1}=1}\left(\eta_{s}(x)-\bar{\eta}_{s}^{\varepsilon
N,+}(x)\right)H\left(\tfrac{x}{N}\right)ds\right)^{2}\right]=0.$ (4.6)

We start with writing the expectation above as ${\rm I}^{2}+{\rm II}$, where

${\rm
I}=\mathbb{E}\left[N^{1-d}\int_{0}^{t}\sum_{x_{1}=1}\left(\eta_{s}(x)-\bar{\eta}_{s}^{\varepsilon
N,+}(x)\right)H\left(\tfrac{x}{N}\right)ds\right]$

and

${\rm II}={\rm
Var}\left(N^{1-d}\int_{0}^{t}\sum_{x_{1}=1}\left(\eta_{s}(x)-\bar{\eta}_{s}^{\varepsilon
N,+}(x)\right)H\left(\tfrac{x}{N}\right)ds\right).$

Then, Eq. (4.6) holds if we can check that

$\lim_{\varepsilon\rightarrow 0}\limsup_{N\rightarrow+\infty}{\rm I}=0$ (4.7)

and

$\lim_{N\rightarrow+\infty}{\rm II}=0$ (4.8)

for every $\varepsilon>0$.

To check Eq. (4.7), we define

$W_{t,\beta}=\left(W_{t,\beta}(1),W_{t,\beta}(2),\ldots,W_{t,\beta}(d)\right),$

as the
$\left((-\infty,0^{-}]\cup[0^{+},+\infty)\right)\times\mathbb{R}^{d-1}$-valued
stochastic process, where $\\{\\{W_{t,\beta}(i)\\}_{t\geq 0}\\}_{1\leq i\leq
d}$ are independent, $\\{W_{t,\beta}(1)\\}_{t\geq 0}$ is an independent copy
of $B_{2t,\beta}$ introduced in the last subsection, and $W_{t,\beta}(i)$ is
an independent copy of $\\{B_{2t}\\}_{t\geq 0}$ for $2\leq i\leq d$. We write
$W_{t,\beta}$ as $W_{t,\beta}^{u}$ when $W_{0,\beta}=u$. By Eq. (3.1) and
Assumption (A),

$\mathbb{E}[\eta_{s}(x)]=\mathbb{E}\left[\rho_{0}\left(\tfrac{X_{sN^{2},\beta}^{N,x}}{N}\right)\right]\quad\text{and}\quad\mathbb{E}[\bar{\eta}_{s}^{\varepsilon
N,+}(x)]=\frac{1}{|\Lambda^{+}_{x,\varepsilon
N}|}\sum_{y\in\Lambda^{+}_{x,\varepsilon
N}}\mathbb{E}\left[\rho_{0}\left(\tfrac{X_{sN^{2},\beta}^{N,y}}{N}\right)\right]$

for every $x\in\mathbb{Z}^{d}$. Hence, by Theorem 3.1,

$\limsup_{N\rightarrow+\infty}{\rm
I}=\int_{0}^{t}\int_{u\in\mathbb{R}^{d-1}}H(0,u)\mathbb{E}\left[\rho_{0}\left(W_{s,\beta}^{(0^{+},u)}\right)\right]duds\\\
-\int_{0}^{t}\int_{u\in\mathbb{R}^{d-1}}H(0,u)\left\\{\frac{1}{\varepsilon(2\varepsilon)^{d-1}}\int_{v\in[0,\varepsilon]\times[-\varepsilon,\varepsilon]^{d-1}}\mathbb{E}\left[\rho_{0}\left(W_{s,\beta}^{(0^{+},u)+v}\right)\right]dv\right\\}\,du\,ds,$

According to the definition of $W_{t,\beta}$,

$\lim_{v_{1}\downarrow 0,\atop v_{i}\rightarrow 0\text{~{}for~{}}2\leq i\leq
d}\mathbb{E}\left[\rho_{0}\left(W_{s,\beta}^{(0^{+},u)+v}\right)\right]=\mathbb{E}\left[\rho_{0}\left(W_{s,\beta}^{(0^{+},u)}\right)\right]$

for every $u\in\mathbb{R}^{d-1}$ and hence Eq. (4.7) holds.

Now we check Eq. (4.8). According to the fact that the covariance operator is
bilinear and $\eta_{t}=\tilde{\eta}_{tN^{2}}$, it is easy to check that Eq.
(4.8) is a direct corollary of the following claim.

Claim. For every $s>0$,

$\lim_{t\rightarrow+\infty}\sup_{N\geq 1,x,y\in Z^{d}}{\rm
Cov}\left(\tilde{\eta}_{t}(x),\tilde{\eta}_{s}(y)\right)=0.$ (4.9)

Now we prove the claim. By Eq. (3.1) and Assumption (A), for $s<t$,

${\mathbb{P}}\left(\tilde{\eta}_{t}(x)=1\right){\mathbb{P}}\left(\tilde{\eta}_{s}(y)=1\right)=\sum_{u\in\mathbb{Z}^{d}}\sum_{v\in\mathbb{Z}^{d}}{\mathbb{P}}\left(X_{t,\beta}^{N,x}=u\right){\mathbb{P}}\left(X_{s,\beta}^{N,y}=v\right)\rho_{0}\left(\tfrac{u}{N}\right)\rho_{0}\left(\tfrac{v}{N}\right)\\\
=\sum_{u\in\mathbb{Z}^{d}}\sum_{v\in\mathbb{Z}^{d}}{\mathbb{P}}\left(X_{t,\beta}^{N,x}=u\right){\mathbb{P}}\left(\hat{X}_{t,\beta}^{N,y,t-s}=v\right)\rho_{0}\left(\tfrac{u}{N}\right)\rho_{0}\left(\tfrac{v}{N}\right)\\\
=\sum_{u\in\mathbb{Z}^{d}}\sum_{v\in\mathbb{Z}^{d}}{\mathbb{P}}\left(X_{t,\beta}^{N,x}=u,\hat{X}_{t,\beta}^{N,y,t-s}=v\right)\rho_{0}\left(\tfrac{u}{N}\right)\rho_{0}\left(\tfrac{v}{N}\right).$

Then, by Eq. (3.2),

${\rm Cov}\left(\tilde{\eta}_{t}(x),\tilde{\eta}_{s}(y)\right)={\rm III}+{\rm
IV}+{\rm V},$

where

${\rm
III}=\sum_{u\in\mathbb{Z}^{d}}{\mathbb{P}}\left(X_{t,\beta}^{N,x}=u,\tau_{x,y}^{N,\beta,t-s}\leq
s\right)\rho_{0}\left(\tfrac{u}{N}\right),$ ${\rm
IV}=-\sum_{u\in\mathbb{Z}^{d}}\sum_{v\neq
u}{\mathbb{P}}\left(X_{t,\beta}^{N,x}=u,\hat{X}_{t,\beta}^{N,y,t-s}=v,\tau_{x,y}^{N,\beta,t-s}\leq
s\right)\rho_{0}\left(\tfrac{u}{N}\right)\rho_{0}\left(\tfrac{v}{N}\right),$

and

${\rm
V}=-\sum_{u\in\mathbb{Z}^{d}}{\mathbb{P}}\left(X_{t,\beta}^{N,x}=\hat{X}_{t,\beta}^{N,y,t-s}=u\right)\rho^{2}_{0}\left(\tfrac{u}{N}\right).$

Since $\tilde{\eta}_{t}(x),\tilde{\eta}_{s}(y)$ are positive correlated under
Assumption (A) and ${\rm IV},{\rm V}\leq 0$,

$0\leq{\rm Cov}\left(\tilde{\eta}_{t}(x),\tilde{\eta}_{s}(y)\right)\leq{\rm
III}\leq{\mathbb{P}}\left(\tau_{x,y}^{N,\beta,t-s}<+\infty\right).$

According to Markov property,

${\mathbb{P}}\left(\tau_{x,y}^{N,\beta,t-s}<+\infty\right)=\sum_{u\in\mathbb{Z}^{d}}{\mathbb{P}}\left(X_{t-s,\beta}^{N,x}=u\right){\mathbb{P}}\left(\tau_{u,y}^{N,\beta,0}<+\infty\right).$

For any $x=(x_{1},x_{2},\ldots,x_{d})\in\mathbb{Z}^{d}$, we define
$x^{\bot}=(x_{2},\ldots,x_{d})\in\mathbb{Z}^{d-1}$. Denote by
$\\{V_{t}\\}_{t\geq 0}$ the symmetric simple random walk on
$\mathbb{Z}^{d-1}$. For $\omega,\varpi\in\mathbb{Z}^{d-1}$, let

$p_{t}(\omega,\varpi)={\mathbb{P}}\left(V_{t}=\varpi\big{|}V_{0}=\omega\right),\quad\Gamma(\omega)={\mathbb{P}}\left(V_{t}=0\text{~{}for
some~{}}t\geq 0\big{|}V_{0}=\omega\right).$

According to the definition of $\\{X_{t,\beta}^{N,x}\\}_{t\geq 0}$,
$\left\\{\left(X_{t,\beta}^{N,x}-\hat{X}_{t,\beta}^{N,y,0}\right)^{\bot}\right\\}_{t\geq
0}$ is a copy of $\\{V_{2t}\\}_{t\geq 0}$ with $V_{0}=(x-y)^{\bot}$. As a
result,
${\mathbb{P}}\left(\tau_{u,y}^{N,\beta,0}<+\infty\right)\leq\Gamma\left((u-y)^{\bot}\right)$
and hence

${\mathbb{P}}\left(\tau_{x,y}^{N,\beta,t-s}<+\infty\right)\leq\sum_{u\in\mathbb{Z}^{d}}{\mathbb{P}}\left(X_{t-s,\beta}^{N,x}=u\right)\Gamma\left((u-y)^{\bot}\right).$

Without confusion, we also use $|\cdot|$ to denote the $l^{1}$-norm on
$\mathbb{R}^{d-1}$. Then $\lim_{|\omega|\rightarrow+\infty}\Gamma(\omega)=0$
since $d-1\geq 3$. Therefore, for every $\delta>0$, there exists $M=M(\delta)$
such that $\Gamma(\omega)\leq\delta$ if $|\omega|>M$. As a result,

${\mathbb{P}}\left(\tau_{x,y}^{N,\beta,t-s}<+\infty\right)\leq\delta+\sum_{u:|(u-y)^{\bot}|\leq
M}{\mathbb{P}}\left(X_{t-s,\beta}^{N,x}=u\right).$

Since $\left\\{\left(X_{t,\beta}^{N,x}\right)^{\bot}\right\\}_{t\geq 0}$ is a
copy of $\\{V_{t}\\}_{t\geq 0}$ with $V_{0}=x^{\bot}$,

${\mathbb{P}}\left(\tau_{x,y}^{N,\beta,t-s}<+\infty\right)\leq\delta+{\mathbb{P}}\left(|V_{t-s}-y^{\bot}|\leq
M\big{|}V_{0}=x^{\bot}\right)\leq\delta+(2M+1)^{d-1}p_{t-s}(0,0).$

As a result,

$\limsup_{t\rightarrow+\infty}\sup_{N\geq 1,x,y\in\mathbb{Z}^{d}}{\rm
Cov}\left(\tilde{\eta}_{t}(x),\tilde{\eta}_{s}(y)\right)\leq\delta+(2M+1)^{d-1}\lim_{t\rightarrow+\infty}p_{t-s}(0,0)=\delta.$

Since $\delta$ is arbitrary, Eq. (4.9) holds. As we have pointed out above,
Eq. (4.9) implies Eq. (4.8) and the proof is completed. ∎

###  4.3. Tightness.

The aim of this subsection is to prove the following tightness result.

######  Lemma 4.2 (Tightness).

The sequence $\\{Q^{N}\\}_{N\geq 1}$ is tight. Moreover, any limit point of
$Q^{N}$ is concentrated on trajectories which are absolutely continuous with
respect to the Lebesgue measure.

###### Proof.

The proof of tightness is standard, and so we only sketch the proof. The
second statement in the lemma follows directly from the fact that
$|\eta(x)|\leq 1$ for all $x$. To prove tightness, by [10, Chapter 4], we only
need to prove for any $H\in C^{2}_{c}(\mathbb{R}^{d})$,

$\lim_{M\rightarrow\infty}\limsup_{N\rightarrow\infty}\mathbb{P}\Big{(}\sup_{0\leq
t\leq T}|\big{\langle}\pi^{N}_{t},H\big{\rangle}|\geq M\Big{)}=0$ (4.10)

and for every $\varepsilon>0$,

$\lim_{\delta\rightarrow
0}\limsup_{N\rightarrow\infty}\mathbb{P}\Big{(}\sup_{0\leq
t-s\leq\delta}\Big{|}\int_{s}^{t}\big{\langle}\pi^{N}_{\tau},H\big{\rangle}d\tau\Big{|}\geq\varepsilon\Big{)}=0.$
(4.11)

Eq. (4.10) follows from Chebyshev’s inequality and the fact that
$|\big{\langle}\pi^{N}_{t},H\big{\rangle}|\leq C_{H}$ for some finite constant
$C_{H}$. To prove Eq. (4.11), we only need to prove it separately for the
martingale $M^{N}_{t}(H)$ defined in Eq. (4.1) and the terms (4.3)-(4.5). The
martingale term satisfies (4.11) by Cauchy-Schwarz inequality and Eq. (4.2).
Observe that all the terms (4.3)-(4.5) are bounded by $C_{H}$ for some finite
constant $C_{H}$, whence satisfy Eq. (4.11). This completes the proof. ∎

###  4.4. Energy estimates.

In this subsection, we shall prove the following result.

######  Lemma 4.3 (Energy estimate).

Fix $\beta\geq 1$. Any limit $Q^{*}$ of the sequence $\\{Q^{N}\\}_{N\geq 1}$
is concentrated on paths $\pi_{t}(du)=\rho(t,u)du$ such that $\rho(t,u)\in
L^{2}([0,T],{\mathcal{H}}^{1}(\mathbb{R}^{d}\backslash\\{u_{1}=0\\}))$.

By Lemma 4.2,

$Q^{*}(\pi_{t}(du)={\rho(t,u)}\,du\;\text{for all $0\leq t\leq T$})=1.$

By [8, Section 5.3], to prove Lemma 4.3, we only need to prove the following
result.

######  Lemma 4.4.

For any $M>1$, there exists a constant $K=K(M)<+\infty$ such that

$E_{Q^{*}}\left[\sup_{H}\left(\int_{0}^{T}\int_{\mathbb{R}^{d}}(\partial_{u_{1}}H)(s,u)\rho(s,u)duds-\frac{1}{2}\int_{0}^{T}\int_{\mathbb{R}^{d}}H^{2}(s,u)duds\right)\right]\leq
K,$

where the supremum is carried over all functions $H\in
C^{0,1}([0,T]\times\mathbb{R}^{d})$ with compact support contained in
$[0,T]\times([-M,M]^{d}\backslash\\{|u_{1}|\leq\frac{1}{M}\\})$.

Based on the above lemma, the proof of Lemma 4.3 follows from the same
analysis as that given in the proof of [8, Lemma 5.7], where a crucial step is
the utilization of Riesz’s representation theorem, the details of which we
omit here.

To prove Lemma 4.4, we need the following lemma.

######  Lemma 4.5.

$Q^{*}\left(\rho(t,u)=E_{Q^{*}}[\rho(t,u)]\text{~{}for all~{}}0\leq t\leq
T\right)=1.$

###### Proof of Lemma 4.5.

Since $Q^{*}$ is concentrated on càdlàg paths, we only need to show that

$Q^{*}\left(\pi_{t}(H)=E_{Q^{*}}[\pi_{t}(H)]\right)=1$ (4.12)

for every $0<t\leq T$ and $H\in C_{c}(\mathbb{R}^{d})$. To prove Eq. (4.12),
we claim that

$\lim_{N\rightarrow+\infty}{\rm Var}\left(\pi_{t}^{N}(H)\right)=0$ (4.13)

for every $0<t\leq T$ and $H\in C_{c}(\mathbb{R}^{d})$.

We first use Eq. (4.13) to prove Eq. (4.12) and then check Eq. (4.13). Note
that $\pi_{\cdot}$ is the weak limit of a subsequence of
$\\{\pi^{N}_{\cdot}\\}_{N\geq 1}$ but we still write this subsequence as
$\\{\pi^{N}\\}_{N\geq 1}$ for simplicity. Since $\eta_{t}(x)\leq 1$,

$|\pi_{t}^{N}(H)|\leq C\|H\|_{\infty}$

for every $N\geq 1$, where $\|H\|_{\infty}=\sup_{u\in\mathbb{Z}^{d}}|H(u)|$
and $C<+\infty$ is a constant only depending on the compact support of $H$. As
a result, by dominated convergence theorem,

$\lim_{N\rightarrow+\infty}\mathbb{E}[\pi_{t}^{N}(H)]=E_{Q^{*}}[\pi_{t}(H)].$
(4.14)

Then, Fatou’s lemma and Eq. (4.13) imply that

${\rm
Var}_{Q^{*}}\left(\pi_{t}(H)\right)\leq\liminf_{N\rightarrow+\infty}\mathbb{E}\left[\left(\pi_{t}^{N}(H)\right)^{2}\right]-\left(E_{Q^{*}}[\pi_{t}(H)]\right)^{2}\\\
=\lim_{N\rightarrow+\infty}\left(\mathbb{E}\left[\left(\pi_{t}^{N}(H)\right)^{2}\right]-\left(\mathbb{E}[\pi_{t}^{N}(H)]\right)^{2}\right)=\lim_{N\rightarrow+\infty}{\rm
Var}(\pi_{t}^{N}(H))=0$

and hence Eq. (4.12) holds.

Now we check Eq. (4.13). According to an analysis similar with that leading to
Eq. (4.9),

${\rm
Var}\left(\pi_{t}^{N}(H)\right)\leq\frac{1}{N^{2d}}\sum_{x\in\mathbb{Z}^{d}}\sum_{y\in\mathbb{Z}^{d}}\left|H\left(\frac{x}{N}\right)\right|\left|H\left(\frac{y}{N}\right)\right|{\rm
Cov}(\eta_{t}(x),\eta_{t}(y))\\\
\leq\frac{1}{N^{2d}}\sum_{x\in\mathbb{Z}^{d}}\sum_{y\in\mathbb{Z}^{d}}\left|H\left(\frac{x}{N}\right)\right|\left|H\left(\frac{y}{N}\right)\right|P\left(\tau_{x,y}^{N,\beta,0}<+\infty\right)\\\
\leq\frac{1}{N^{2d}}\sum_{x\in\mathbb{Z}^{d}}\sum_{y\in\mathbb{Z}^{d}}\left|H\left(\frac{x}{N}\right)\right|\left|H\left(\frac{y}{N}\right)\right|\Gamma\left((y-x)^{\bot}\right).$

Since $d-1\geq 3$, for any $\varepsilon>0$, there exists
$M=M(\varepsilon)<+\infty$ that $\Gamma(u)<\varepsilon$ when $|u|>M$. Since
$H$ is with compact support, there exists $C=C(H)\in[1,+\infty)$ that $H(u)=0$
when $|u_{1}|>C$. As a result, for sufficiently large $N$, the last formula is
bounded by

$\frac{\varepsilon}{N^{2d}}\sum_{x\in\mathbb{Z}^{d}}\sum_{y:|(y-x)^{\bot}|>M}\left|H\left(\frac{x}{N}\right)\right|\left|H\left(\frac{y}{N}\right)\right|+\frac{1}{N^{2d}}\sum_{x\in\mathbb{Z}^{d}}\sum_{y:|(y-x)^{\bot}|\leq
M,\|y_{1}\|\leq
CN}\left|H\left(\frac{x}{N}\right)\right|\left|H\left(\frac{y}{N}\right)\right|\\\
\leq C_{H}\varepsilon+C_{H}(2M+1)^{d-1}N^{1-d}.$

This is enough to prove Eq. (4.13). ∎

At last, we prove Lemma 4.4.

###### Proof of Lemma 4.4.

By Lemma 4.5, we only need to show that

$\sup_{H}\left(\int_{0}^{T}\int_{\mathbb{R}^{d}}(\partial_{u_{1}}H)(s,u)E_{Q^{*}}[\rho(s,u)]duds-\frac{1}{2}\int_{0}^{T}\int_{\mathbb{R}^{d}}H^{2}(s,u)duds\right)<+\infty$
(4.15)

for each $M>0$, where the supremum is carried over all functions $H\in
C^{0,1}([0,T]\times\mathbb{R}^{d})$ with compact support contained in
$[0,T]\times([-M,M]^{d}\backslash\\{|u_{1}|\leq\frac{1}{M}\\})$.

Since $Q^{*}$ is any weak limit of the sequence $\\{Q^{N}\\}_{N\geq 1}$ along
some subsequence, which we still denote by $\\{Q^{N}\\}_{N\geq 1}$ for
simplicity,

$\int_{0}^{T}\int_{\mathbb{R}^{d}}(\partial_{u_{1}}H)(s,u)E_{Q^{*}}[\rho(s,u)]duds=\lim_{\delta\rightarrow
0}\lim_{N\rightarrow+\infty}\int_{0}^{T}\frac{1}{N^{d}}\sum_{x\in\mathbb{Z}^{d}}\frac{H\left(s,\frac{x}{N}+\delta
e_{1}\right)-H\left(s,\tfrac{x}{N}\right)}{\delta}\mathbb{E}[\eta_{s}(x)]ds\\\
=\lim_{\delta\rightarrow
0}\lim_{N\rightarrow+\infty}\int_{0}^{T}\frac{1}{N^{d}}\sum_{x\in\mathbb{Z}^{d}}H\left(s,\tfrac{x}{N}\right)\frac{\mathbb{E}[\eta_{s}(x-N\delta
e_{1})]-\mathbb{E}[\eta_{s}(x)]}{\delta}ds,$

where $e_{1}=(1,0,\ldots,0)$. Above, the first identify follows from Eq.
(4.14) and the second from summation by parts. By Eq. (3.1),

$\mathbb{E}[\eta_{s}(x)]=\mathbb{E}\left[\rho_{0}\left(\tfrac{X_{sN^{2},\beta}^{N,x}}{N}\right)\right].$

By Theorem 3.1,

$\int_{0}^{T}\int_{\mathbb{R}^{d}}(\partial_{u_{1}}H)(s,u)E_{Q^{*}}[\rho(s,u)]duds=\lim_{\delta\rightarrow
0}\int_{0}^{T}\int_{\mathbb{R}^{d}}H(s,u)\frac{\mathbb{E}[\rho_{0}(W^{u-\delta
e_{1}}_{s,\beta})]-\mathbb{E}[\rho_{0}(W^{u}_{s,\beta})]}{\delta}duds,$ (4.16)

where $W_{s,\beta}$ is defined as in the proof of Lemma 4.1.

For all $\beta\geq 1$, we claim that there exists
$C=C(\beta,\rho_{0},M)<+\infty$ that

$\left|\frac{\mathbb{E}[\rho_{0}(W^{u-\delta
e_{1}}_{s,\beta})]-\mathbb{E}[\rho_{0}(W^{u}_{s,\beta})]}{\delta}\right|\leq
C$ (4.17)

for every $(u,s,\delta)$ satisfying
$|u_{1}|\in\left[\frac{1}{M},M\right],0\leq s\leq T$ and
$\delta<\frac{1}{2M}$. This is enough to prove Eq. (4.15) since by Cauchy-
Schwarz inequality, we may bound the right-hand side of (4.16) by

$\frac{1}{2}\int_{0}^{T}\int_{\mathbb{R}^{d}}H^{2}(s,u)duds+\frac{C^{2}}{2}T(2M)^{d}.$

It remains to prove Eq. (4.17). Without loss of generality, we assume that
$u_{1}\in\left[\frac{1}{M},M\right]$ since the other part can be checked in
the same way. If $\beta>1$, since $u_{1},u_{1}-\delta>0$ when
$\delta<\frac{1}{2M}$,

$W_{s,\beta}^{u}=\left(|B_{2s}^{u_{1}}(1)|,B_{2s}^{u_{2}}(2),\ldots,B_{2s}^{u_{d}}(d)\right),\quad
W_{s,\beta}^{u-\delta
e_{1}}=\left(|B_{2s}^{u_{1}-\delta}(1)|,B_{2s}^{u_{2}}(2),\ldots,B_{2s}^{u_{d}}(d)\right)$

according to the definition of $W_{s,\beta}$ given in Section 4.2, where
$\\{B_{t}(i):t\geq 0\\}_{1\leq i\leq d}$ are independent copies of standard
Brownian motions and $B_{t}^{a}(i)$ means $B_{0}(i)=a$ for $1\leq i\leq d$. By
translation invariance,

$B_{2s}^{u_{1}}(1)=_{d}B_{2s}^{0}(1)+u_{1},\quad
B_{2s}^{u_{1}-\delta}(1)=_{d}B_{2s}^{0}(1)+u_{1}-\delta.$

By Lagrange’s mean value theorem,

$\left|\frac{\mathbb{E}[\rho_{0}(W^{u-\delta
e_{1}}_{s,\beta})]-\mathbb{E}[\rho_{0}(W^{u}_{s,\beta})]}{\delta}\right|\leq\|\partial_{u_{1}}\rho_{0}\|_{\infty}$

and hence Eq. (4.17) holds for $\beta>1$.

When $\beta=1$,

$W_{s,\beta}^{u}=\left(B_{2s,1}^{u_{1}},B_{2s}^{u_{2}}(2),\ldots,B_{2s}^{u_{d}}(d)\right),\quad
W_{s,\beta}^{u-\delta
e_{1}}=\left(B_{2s,1}^{u_{1}-\delta},B_{2s}^{u_{2}}(2),\ldots,B_{2s}^{u_{d}}(d)\right),$

where $\\{B_{t,1}\\}_{t\geq 0}$ is the snapping out Browning motion with
parameter $2\alpha$. For any $u\in\mathbb{R}^{d},v\in\mathbb{R}^{d-1}$, let
$f_{u^{\bot}}(t,v)$ be the density function of
$\left(B_{t}^{u_{2}}(2),B_{t}^{u_{3}}(3),\ldots,B_{t}^{u_{d}}(d)\right)$ at
$v$, then

$\frac{\mathbb{E}[\rho_{0}(W^{u-\delta
e_{1}}_{s,\beta})]-\mathbb{E}[\rho_{0}(W^{u}_{s,\beta})]}{\delta}=\int_{\mathbb{R}^{d-1}}\frac{\mathbb{E}\left[\rho_{0}\left(B_{2s,1}^{u_{1}-\delta},v\right)\right]-\mathbb{E}\left[\rho_{0}\left(B_{2s,1}^{u_{1}},v\right)\right]}{\delta}f_{u^{\bot}}(2s,v)dv.$
(4.18)

As we have recalled in Section 4.2,

$\mathbb{E}[\rho_{0}(B_{2s,1}^{u_{1}},v)]=\mathbb{E}\left[\left(\frac{1+e^{-2\alpha
L_{2s}}}{2}\right)\rho_{0}\left(|B_{2s}^{u_{1}}|,v\right)+\left(\frac{1-e^{-2\alpha
L_{2s}}}{2}\right)\rho_{0}\left(-|B_{2s}^{u_{1}}|,v\right)\right]\\\
=\mathbb{E}\left[\left(\frac{1+e^{-2\alpha
L_{2s}(-u_{1})}}{2}\right)\rho_{0}\left(|B_{2s}^{0}+u_{1}|,v\right)+\left(\frac{1-e^{-2\alpha
L_{2s}(-u_{1})}}{2}\right)\rho_{0}\left(-|B_{2s}^{0}+u_{1}|,v\right)\right]$

where $L_{t}$ is the local time of $\\{B_{t}\\}_{t\geq 0}$ at point $0$ and
$L_{t}(a)$ is the local time of $\\{B_{t}\\}_{t\geq 0}$ at point $a$. Note
that in the above equations we utilize the fact that $B_{t}^{a}$ hits $0$ if
and only if $B_{t}^{0}$ hits $-a$ when we write $B_{t}^{a}$ as $B_{t}^{0}+a$
in the sense of coupling. Similarly, we may write
$\mathbb{E}[\rho_{0}(B_{2s,1}^{u_{1}-\delta},v)]$ as

$\mathbb{E}\Bigg{[}\left(\frac{1+e^{-2\alpha
L_{2s}(-u_{1}+\delta)}}{2}\right)\rho_{0}\left(|B_{2s}^{0}+u_{1}-\delta|,v\right)+\left(\frac{1-e^{-2\alpha
L_{2s}(-u_{1}+\delta)}}{2}\right)\rho_{0}\left(-|B_{2s}^{0}+u_{1}-\delta|,v\right)\Bigg{]}.$

Hence,

$\mathbb{E}\left[\rho_{0}\left(B_{2s,1}^{u_{1}-\delta},v\right)\right]-\mathbb{E}\left[\rho_{0}\left(B_{2s,1}^{u_{1}},v\right)\right]={\rm
I}+{\rm II},$

where

${\rm I}=-\mathbb{E}\left[\left(\frac{1+e^{-2\alpha
L_{2s}(-u_{1})}}{2}\right)\rho_{0}\left(|B_{2s}^{0}+u_{1}|,v\right)\right]+\mathbb{E}\left[\left(\frac{1+e^{-2\alpha
L_{2s}(-u_{1}+\delta)}}{2}\right)\rho_{0}\left(|B_{2s}^{0}+u_{1}-\delta|,v\right)\right]$

and

${\rm II}=-\mathbb{E}\left[\left(\frac{1-e^{-2\alpha
L_{2s}(-u_{1})}}{2}\right)\rho_{0}\left(-|B_{2s}^{0}+u_{1}|,v\right)\right]+\mathbb{E}\left[\left(\frac{1-e^{-2\alpha
L_{2s}(-u_{1}+\delta)}}{2}\right)\rho_{0}\left(-|B_{2s}^{0}+u_{1}-\delta|,v\right)\right].$

We only calculate ${\rm I}$ since ${\rm II}$ can be calculated in the same
way. For any $a\neq 0$, we define $\tau_{a}=\inf\\{t:~{}B_{t}^{0}=a\\}.$ Let
$p(t,a)$ be the density function of $\tau_{a}$, then by strong Markov
property,

$\mathbb{E}\left[\left(\frac{1+e^{-2\alpha
L_{2s}(-u_{1})}}{2}\right)\rho_{0}\left(|B_{2s}^{0}+u_{1}|,v\right)\right]=\mathbb{E}\left[\left(\frac{1+e^{-2\alpha
L_{2s}(-u_{1})}}{2}\right)\rho_{0}\left(|B_{2s}^{0}+u_{1}|,v\right)\mathbf{1}_{\\{\tau_{-u_{1}}\leq
2s\\}}\right]\\\ +\mathbb{E}\left[\left(\frac{1+e^{-2\alpha
L_{2s}(-u_{1})}}{2}\right)\rho_{0}\left(|B_{2s}^{0}+u_{1}|,v\right)\mathbf{1}_{\\{\tau_{-u_{1}}>2s\\}}\right]\\\
=\int_{0}^{2s}p(\theta,-u_{1})\mathbb{E}\left[\left(\frac{1+e^{-2\alpha
L_{2s-\theta}}}{2}\right)\rho_{0}\left(|B_{2s-\theta}^{0}|,v\right)\right]d\theta+\mathbb{E}\left[\rho_{0}\left(|B_{2s}^{0}+u_{1}|,v\right)\mathbf{1}_{\\{\tau_{-u_{1}}>2s\\}}\right],$

where we utilize the facts that $L_{t}(a)=0$ when $t<\tau_{a}$ and that
$(L_{t}(-a),B_{t}^{-a}+a)$ and $(L_{t},B_{t}^{0})$ have the same distribution.
As a result,

${\rm I}={\rm III}+{\rm IV},$

where

${\rm
III}=\int_{0}^{2s}\left(p(\theta,-u_{1}+\delta)-p(\theta,-u_{1})\right)\mathbb{E}\left[\left(\frac{1+e^{-2\alpha
L_{2s-\theta}}}{2}\right)\rho_{0}\left(|B_{2s-\theta}^{0}|,v\right)\right]d\theta$

and

${\rm
IV}=\mathbb{E}\left[\rho_{0}\left(|B_{2s}^{0}+u_{1}-\delta|,v\right)\mathbf{1}_{\\{\tau_{-u_{1}+\delta}>2s\\}}\right]-\mathbb{E}\left[\rho_{0}\left(|B_{2s}^{0}+u_{1}|,v\right)\mathbf{1}_{\\{\tau_{-u_{1}}>2s\\}}\right].$

It is shown in [2, Chapter 8] that

$p(\theta,a)=\frac{a}{\sqrt{2\pi}}e^{-\frac{a^{2}}{2\theta}}\theta^{-\frac{3}{2}}.$

As a result, there exists $C_{1}=C_{1}(M)<+\infty$ independent of $0\leq s\leq
T$ that

$\int_{0}^{r}\left|\frac{d}{da}p(\theta,a)\right|d\theta<C_{1}$

for any $0\leq r\leq 2T$ and $\frac{1}{2M}\leq a\leq M$. Then, by Lagrange’s
mean value theorem, for $\delta<\frac{1}{2M}$,

$\frac{\left|{\rm III}\right|}{\delta}\leq C_{1}\|\rho_{0}\|_{\infty}$

Now we deal with ${\rm IV}$. Since $\tau_{-u_{1}+\delta}<\tau_{-u_{1}}$ for
$\\{B_{t}^{0}\\}_{t\geq 0}$,

${\rm IV}={\rm V}+{\rm VI},$

where

${\rm
V}=-\mathbb{E}\left[\rho_{0}\left(|B_{2s}^{0}+u_{1}-\delta|,v\right)\mathbf{1}_{\\{\tau_{-u_{1}}>2s>\tau_{-u_{1}+\delta}\\}}\right]$

and

${\rm
VI}=\mathbb{E}\left[\left(\rho_{0}\left(|B_{2s}^{0}+u_{1}-\delta|,v\right)-\rho_{0}\left(|B_{2s}^{0}+u_{1}|,v\right)\right)\mathbf{1}_{\\{\tau_{-u_{1}+\delta}>2s\\}}\right].$

According to the expression of ${\rm V}$ and the definition of $p(t,a)$,

$|{\rm
V}|\leq\|\rho_{0}\|_{\infty}\int_{0}^{2s}\left|p(\theta,-u_{1}+\delta)-p(\theta,-u_{1})\right|d\theta$

and hence $\frac{|{\rm V}|}{\delta}\leq C_{1}\|\rho_{0}\|_{\infty}$ when
$\delta<\frac{1}{2M}$ by Lagrange’s mean value theorem. Since
$\left||x|-|y|\right|\leq|x-y|$, by Lagrange’s mean value theorem,

$\frac{|{\rm VI}|}{\delta}\leq\|\partial_{u_{1}}\rho_{0}\|_{\infty}.$

As a result,

$\frac{|{\rm I}|}{\delta}\leq\frac{|{\rm III}|+|{\rm IV}|}{\delta}\leq
2C_{1}\|\rho_{0}\|_{\infty}+\|\partial_{u_{1}}\rho_{0}\|_{\infty}$ (4.19)

when $\delta<\frac{1}{2M}$. Note that Eq. (4.19) still holds when $|{\rm I}|$
is replaced by $|{\rm II}|$ according to the same analysis. Therefore,

$\left|\frac{\mathbb{E}\Big{[}\rho_{0}\left(B_{2s,1}^{u_{1}-\delta},v\right)\Big{]}-\mathbb{E}\Big{[}\rho_{0}\left(B_{2s,1}^{u_{1}},v\right)\Big{]}}{\delta}\right|\leq
2\big{(}2C_{1}\|\rho_{0}\|_{\infty}+\|\partial_{u_{1}}\rho_{0}\|_{\infty}\big{)}$

when $\delta<\frac{1}{2M}$ and hence Eq. (4.17) follows from Eq. (4.18) when
$\beta=1$. This completes the proof. ∎

##  5\. Nonequilibrium fluctuations

###  5.1. Proof outline.

By Dynkin’s martingale formula,

$\displaystyle{\mathcal{M}}^{N}_{t}(H)$
$\displaystyle={\mathcal{Y}}^{N}_{t}(H)-{\mathcal{Y}}^{N}_{0}(H)-\int_{0}^{t}(N^{2}\mathcal{L}_{N}+\partial_{s}){\mathcal{Y}}^{N}_{s}(H)ds,$
(5.1) $\displaystyle{\mathcal{N}}^{N}_{t}(H)$
$\displaystyle=\big{[}{\mathcal{M}}^{N}_{t}(H)\big{]}^{2}-\int_{0}^{t}\Big{\\{}N^{2}\mathcal{L}_{N}\big{[}{\mathcal{Y}}^{N}_{s}(H)\big{]}^{2}-2{\mathcal{Y}}^{N}_{s}(H)N^{2}\mathcal{L}_{N}{\mathcal{Y}}^{N}_{s}(H)\Big{\\}}ds$
(5.2)

are both martingales. Similar to (4.3), (4.4) and (4.5), the integrand in the
martingale ${\mathcal{M}}^{N}_{t}(H)$ equals

$\displaystyle(N^{2}\mathcal{L}_{N}+\partial_{s}){\mathcal{Y}}^{N}_{s}(H)$
$\displaystyle=\frac{1}{N^{1+d/2}}\sum_{x\in\mathbb{Z}^{d}}\sum_{i=2}^{d}\bar{\eta}_{s}(x)\partial_{u_{i}}^{2}H(\tfrac{x}{N})+\frac{1}{N^{1+d/2}}\sum_{x_{1}\neq
0,1}\bar{\eta}_{s}(x)\partial_{u_{1}}^{2}H(\tfrac{x}{N})+o_{N,p}(1)$ (5.3)
$\displaystyle+\frac{N}{N^{d/2}}\sum_{x_{1}=1}\bar{\eta}_{s}(x)\big{[}H(\tfrac{x+e_{1}}{N})-H(\tfrac{x}{N})\big{]}+\frac{N}{N^{d/2}}\sum_{x_{1}=0}\bar{\eta}_{s}(x)\big{[}H(\tfrac{x-e_{1}}{N})-H(\tfrac{x}{N})\big{]}$
(5.4) $\displaystyle+\frac{\alpha
N^{1-\beta}}{N^{d/2}}\sum_{x_{1}=0}\big{[}\bar{\eta}_{s}(x)-\bar{\eta}_{s}(x+e_{1})\big{]}\big{[}H(\tfrac{x+e_{1}}{N})-H(\tfrac{x}{N})\big{]}.$
(5.5)

Direct calculations show that the integrand in the expression of the
martingale ${\mathcal{N}}^{N}_{t}(H)$ equals

$N^{2}\mathcal{L}_{N}\big{[}{\mathcal{Y}}^{N}_{s}(H)\big{]}^{2}-2{\mathcal{Y}}^{N}_{s}(H)N^{2}\mathcal{L}_{N}{\mathcal{Y}}^{N}_{s}(H)=\frac{1}{N^{d}}\sum_{x,y\in\mathbb{Z}^{d},\atop|x-y|=1}\xi^{N}_{x,y}(\eta_{s}(x)-\eta_{s}(y))^{2}H^{2}\big{(}\tfrac{x}{N}\big{)}.$

In Subsection 5.3, we prove the tightness of the sequences
$\\{{\mathcal{M}}^{N}_{t},0\leq t\leq T\\}_{N\geq 1}$ and
$\\{{\mathcal{Y}}^{N}_{t},0\leq t\leq T\\}_{N\geq 1}$. Denote by
$\mathcal{M}_{\cdot}$ and $\mathcal{Y}_{\cdot}$ any limit points of
${\mathcal{M}}^{N}_{\cdot}$ and ${\mathcal{Y}}^{N}_{\cdot}$ along some
subsequence respectively. For convenience, we still denote this subsequence by
$\\{N\\}$. Next, we characterize the limit point $\mathcal{M}_{\cdot}$ in
different regimes of $\beta$.

####  5.1.1. The case $\beta=1$

Recall in this case, the function $H$ satisfies

$\partial_{u_{1}}^{+,2k+1}H(u)=\partial_{u_{1}}^{-,2k+1}H(u)=\alpha[\partial_{u_{1}}^{+,2k}H(u)-\partial_{u_{1}}^{-,2k}H(u)],\quad\forall
u_{1}=0,\;\forall k\geq 0.$

We claim that the time integral of the sum of (5.4) and (5.5) converges to
zero in $L^{2}$. Indeed, by Taylor’s expansion,

$\frac{N}{N^{d/2}}\sum_{x_{1}=1}\bar{\eta}_{s}(x)\big{[}H(\tfrac{x+e_{1}}{N})-H(\tfrac{x}{N})\big{]}-\frac{\alpha}{N^{d/2}}\sum_{x_{1}=0}\bar{\eta}_{s}(x+e_{1})\big{[}H(\tfrac{x+e_{1}}{N})-H(\tfrac{x}{N})\big{]}\\\
=\frac{N}{N^{d/2}}\sum_{x_{1}=1}\bar{\eta}_{s}(x)\big{[}H(\tfrac{x+e_{1}}{N})-H(\tfrac{x}{N})\big{]}-\frac{\alpha}{N^{d/2}}\sum_{x_{1}=1}\bar{\eta}_{s}(x)\big{[}H(\tfrac{x}{N})-H(\tfrac{x-e_{1}}{N})\big{]}\\\
=\frac{1}{N^{d/2}}\sum_{x_{1}=1}\bar{\eta}_{s}(x)\Big{[}\partial^{+}_{u_{1}}H(\tfrac{x-e_{1}}{N})-\alpha\big{(}H^{+}(\tfrac{x-e_{1}}{N})-H^{-}(\tfrac{x-e_{1}}{N})\big{)}\Big{]}+\frac{1}{2N^{d/2+1}}\sum_{x_{1}=1}\bar{\eta}_{s}(x)R_{1,H}(x),$

where
$R_{1,H}(x)=[4\partial_{u_{1}}^{2}H(\tfrac{\theta(x-e_{1},x+e_{1})}{N})-\partial_{u_{1}}^{2}H(\tfrac{\theta(x-e_{1},x)}{N})]-\alpha
N[H(\tfrac{x}{N})-H^{+}(\tfrac{x-e_{1}}{N})]$ for $x_{1}=0$. In the expression
of $R_{1,H}$, we use $\theta(x,y)$ to denote some point between $x$ and $y$.
Note that $R_{1,H}$ is of order one. With the above choices of $H$, the first
term in the last equation is zero, and by (5.8), the last term converges to
zero in $L^{2}$. This proves the claim. Moreover, since $\eta(x)$ is bounded,
by (5.2),

$\sup_{N\geq 1}\mathbb{E}\Big{[}\sup_{0\leq t\leq
T}\big{(}{\mathcal{M}}^{N}_{t}\big{)}^{2}\Big{]}\leq C$

for some constant $C$. Thus, the martingale ${\mathcal{M}}^{N}_{t}$ is
uniformly integrable. This implies the limit ${\mathcal{M}}_{t}$ is a
martingale. Moreover, passing to the limit, we have

${\mathcal{M}}_{t}(H)={\mathcal{Y}}_{t}(H)-{\mathcal{Y}}_{0}(H)-\int_{0}^{t}{\mathcal{Y}}_{s}(\Delta
H)ds.$ (5.6)

For the martingale ${\mathcal{N}}^{N}_{t}(H)$, in Theorem 5.4, we prove that

$\lim_{N\rightarrow\infty}\int_{0}^{t}\frac{1}{N^{d}}\sum_{x,y\in\mathbb{Z}^{d},\atop|x-y|=1}\xi^{N}_{x,y}(\eta_{s}(x)-\eta_{s}(y))^{2}H^{2}\big{(}\tfrac{x}{N}\big{)}ds\\\
=4d(1-\gamma_{d})\int_{0}^{t}\int_{\mathbb{R}^{d}}\rho(s,u)(1-\rho(s,u))H^{2}(u)duds$

in $L^{2}$. Since ${\mathcal{M}}^{N}_{t}(H)$ converges to $\mathcal{M}_{t}(H)$
in distribution, ${\mathcal{N}}_{t}^{N}(H)$ also converges in distribution,
and denote the limit by ${\mathcal{N}}_{t}$. Passing to the limit,

${\mathcal{N}}_{t}(H)=\big{[}{\mathcal{M}}_{t}(H)\big{]}^{2}-4d(1-\gamma_{d})\int_{0}^{t}\int_{\mathbb{R}^{d}}\rho(s,u)(1-\rho(s,u))H^{2}(u)duds.$
(5.7)

Similarly, one could also prove that ${\mathcal{N}}_{t}^{N}(H)$ is uniformly
integrable, implying ${\mathcal{N}}_{t}(H)$ is a martingale. This concludes
the proof for the case $\beta=1$.

####  5.1.2. The case $\beta>3/2$

Recall in this case we take $H$ to be such that

$\partial_{u_{1}}^{+,2k+1}H(t,u)=\partial_{u_{1}}^{-,2k+1}H(t,u)=0,\quad\forall
u_{1}=0,\;\forall k\geq 0.$

Then, by Taylor’s expansion, we could write (5.4) as

$\frac{{1}}{2N^{1+d/2}}\sum_{x_{1}=1}\bar{\eta}_{s}(x)R_{2,H}(x)+\frac{1}{2N^{1+d/2}}\sum_{x_{1}=0}\bar{\eta}_{s}(x)R_{3,H}(x),$

where
$R_{2,H}(x)=[4\partial_{u_{1}}^{2}H(\tfrac{\theta(x-e_{1},x+e_{1})}{N})-\partial_{u_{1}}^{2}H(\tfrac{\theta(x-e_{1},x)}{N})]$
for $x_{1}=1$, and
$R_{3,H}(x)=\partial_{u_{1}}^{2}H(\tfrac{\theta(x-e_{1},x)}{N})$ for
$x_{1}=0$. By (5.8), the term (5.4) vanishes in the limit. Using (5.8) again,
the variance of the time integral of the term (5.5) is bounded by
$CN^{3-2\beta}$. Thus, the term (5.5) also vanishes in the limit since
$\beta>3/2$. Thus, passing to the limit, ${\mathcal{M}}_{t}(H)$ and
${\mathcal{N}}_{t}(H)$ are still given by (5.6) and (5.7) respectively.
Similar to the case $\beta=1$, one could also check directly that
${\mathcal{M}}_{t}^{N}(H)$ and ${\mathcal{N}}_{t}^{N}(H)$ are uniformly
integrable, implying ${\mathcal{M}}_{t}(H)$ and ${\mathcal{N}}_{t}(H)$ are
martingales. This concludes the proof for the case $\beta>3/2$.

######  Remark 5.1.

For the case $0<\beta<1$, one needs to prove that for any
$H\in{\mathcal{S}}(\mathbb{R}^{d})$,

$\lim_{N\rightarrow\infty}\frac{1}{N^{d/2}}\sum_{x_{1}=0}[\bar{\eta}_{s}(x+e_{1})-\bar{\eta}_{s}(x)]H\big{(}\tfrac{x}{N}\big{)}=0$

in probability, which is beyond the technique in this article. We leave it as
a further investigation.

###  5.2. Limit theorems.

The main aim of this subsection is to prove Theorems 5.2 and 5.4.

######  Theorem 5.2.

Under Assumption (A),

${\rm
Var}\Big{(}\int_{0}^{t}\sum_{x_{1}=0}\bar{\eta}_{s}(x)H\big{(}\tfrac{x}{N}\big{)}ds\Big{)}\leq
CN^{d+1}.$ (5.8)

######  Remark 5.3.

We believe the above bound is not sharp, and the correct bound should be
$CN^{d}$. If this is the case, Theorem 2.9 also holds for $1<\beta\leq 3/2$.

###### Proof.

We first develop the left side in (5.8) as

$2\int_{0}^{t}\int_{0}^{s}\sum_{x,y:x_{1}=0,y_{1}=0}H(\tfrac{x}{N})H(\tfrac{y}{N})\mathbb{E}\big{[}\bar{\eta}_{s}(x)\bar{\eta}_{s^{\prime}}(y)\big{]}\,ds^{\prime}\,ds.$

For $s^{\prime}<s$, following the proof of (4.9),

$\mathbb{E}\big{[}\bar{\eta}_{s}(x)\bar{\eta}_{s^{\prime}}(y)\big{]}\leq\mathbb{P}\big{(}\tau_{x,y}^{N,\beta,(s-s^{\prime})N^{2}}<+\infty\big{)}\leq\Gamma\big{(}(x-y)^{\bot}\big{)}.$

Recall for $z\in\mathbb{Z}^{d-1}$, $\Gamma(z)$ is the probability of the
simple random walk on $\mathbb{Z}^{d-1}$ ever hitting the origin when starting
from $z$. It is well known that $\Gamma(z)\leq C|z|^{2-(d-1)}$ for some
constant $C$. Thus, we could bound the left side in (5.8) by

$Ct^{2}\sum_{x,y:x_{1}=0,y_{1}=0}H(\tfrac{x}{N})H(\tfrac{y}{N})|(x-y)^{\bot}|^{3-d}\leq
Ct^{2}N^{d+1}$

for some constant $C=C(H)$. This concludes the proof. ∎

######  Theorem 5.4.

Under Assumption (A), for any $\beta\geq 0$,

$\lim_{N\rightarrow+\infty}\frac{1}{N^{d}}\sum_{x,y\in\mathbb{Z}^{d},|x-y|=1}\xi_{x,y}^{N}\left(\eta_{s}(x)-\eta_{s}(y)\right)^{2}H^{2}\left(\tfrac{x}{N}\right)=4d(1-\gamma_{d})\int_{\mathbb{R}^{d}}\rho(s,u)(1-\rho(s,u))H^{2}(u)du$

in $L^{2}$ for any $s>0$, where $\gamma_{d}$ is the probability that the
simple random walk on $\mathbb{Z}^{d}$ starting at $0$ returns to $0$ again,
and $\rho(s,u)$ is the solution to the corresponding hydrodynamic equation
introduced in Subsection 2.2.

It is easy to check that Theorem 5.4 is a direct corollary of the following
two Lemmas.

######  Lemma 5.5.

Under Assumption (A), for any $\beta\geq 0$,

$\lim_{N\rightarrow+\infty}\frac{1}{N^{d}}\sum_{x,y\in\mathbb{Z}^{d},|x-y|=1}\mathbb{E}\left(\eta_{s}(x)\eta_{s}(y)\right)H^{2}\left(\tfrac{x}{N}\right)=2d\int_{\mathbb{R}^{d}}\left(\gamma_{d}\rho(s,u)+(1-\gamma_{d})\rho(s,u)^{2}\right)H^{2}(u)du.$

######  Lemma 5.6.

Under Assumption (A), for any $\beta\geq 0$,

$\lim_{N\rightarrow+\infty}{\rm
Var}\left(\frac{1}{N^{d}}\sum_{x,y\in\mathbb{Z}^{d},|x-y|=1}\eta_{s}(x)\eta_{s}(y)H^{2}\left(\tfrac{x}{N}\right)\right)=0.$

###### Proof of Lemma 5.5.

According to Assumption (A) and Equation (3.2),

$\mathbb{E}\left[\eta_{s}(x)\eta_{s}(y)\right]=\mathbb{E}\left[\rho_{0}\Big{(}\tfrac{X^{N,x}_{sN^{2},\beta}}{N}\Big{)}1_{\\{\tau_{x,y}^{N,\beta,0}\leq
sN^{2}\\}}\right]+\mathbb{E}\left[\rho_{0}\left(\tfrac{X^{N,x}_{sN^{2},\beta}}{N}\right)\rho_{0}\left(\tfrac{\hat{X}^{N,y,0}_{sN^{2},\beta}}{N}\right)1_{\\{\tau_{x,y}^{N,\beta,0}>sN^{2}\\}}\right].$

Thus,

$\frac{1}{N^{d}}\sum_{x,y\in\mathbb{Z}^{d},|x-y|=1}\mathbb{E}[\eta_{s}(x)\eta_{s}(y)]H^{2}\left(\tfrac{x}{N}\right)={\rm
I}+{\rm II},$ (5.9)

where

${\rm
I}=\frac{1}{N^{d}}\sum_{x,y\in\mathbb{Z}^{d},|x-y|=1}\mathbb{E}\left[\rho_{0}\Big{(}\tfrac{X^{N,x}_{sN^{2},\beta}}{N}\Big{)}1_{\\{\tau_{x,y}^{N,\beta,0}\leq
sN^{2}\\}}\right]H^{2}\left(\tfrac{x}{N}\right)$

and

${\rm
II}=\frac{1}{N^{d}}\sum_{x,y\in\mathbb{Z}^{d},|x-y|=1}\mathbb{E}\left[\rho_{0}\left(\tfrac{X^{N,x}_{sN^{2},\beta}}{N}\right)\rho_{0}\left(\tfrac{\hat{X}^{N,y,0}_{sN^{2},\beta}}{N}\right)1_{\\{\tau_{x,y}^{N,\beta,0}>sN^{2}\\}}\right]H^{2}\left(\tfrac{x}{N}\right).$

As a result, to complete this proof, we only need to show that

$\lim_{N\rightarrow+\infty}{\rm
I}=2d\int_{\mathbb{R}^{d}}\gamma_{d}\rho_{s}(u)H^{2}(u)du$ (5.10)

and

$\lim_{N\rightarrow+\infty}{\rm
II}=2d\int_{\mathbb{R}^{d}}(1-\gamma_{d})\rho_{s}^{2}(u)H^{2}(u)du.$ (5.11)

We first check Equation (5.10). As we have explained in Section 4.2,
$\left(X_{t,\beta}^{N,x}-\hat{X}_{t,\beta}^{N,y,0}\right)^{\perp}=V_{2t}^{(x-y)^{\perp}}$,
where $\\{V_{t}^{v}\\}_{t\geq 0}$ is the simple random walk on
$\mathbb{Z}^{d-1}$ with $V_{0}=v$ for each $v\in\mathbb{Z}^{d-1}$. As a
result, for any $K>0$,

$\sup_{x\in\mathbb{Z}^{d},|y-x|=1}{\mathbb{P}}\left(K\leq\tau_{x,y}^{N,\beta,0}<+\infty\right)\leq\sup_{|v|\leq
1}{\mathbb{P}}\left(V_{2t}^{v}=0\text{~{}for some~{}}t\geq K\right).$

Since $d-1\geq 3$, $\\{V_{2t}\\}_{t\geq 0}$ visits $0$ for finite times with
probability $1$ and hence

$\lim_{K\rightarrow+\infty}\sup_{|v|\leq
1}{\mathbb{P}}\left(V_{2t}^{v}=0\text{~{}for some~{}}t\geq K\right)=0,$

which implies

$\lim_{K\rightarrow+\infty}\sup_{x\in\mathbb{Z}^{d},|y-x|=1}{\mathbb{P}}\left(K\leq\tau_{x,y}^{N,\beta,0}<+\infty\right)=0.$

Therefore, for any given $\varepsilon>0$, there exists
$K_{1}=K_{1}(\varepsilon)>0$ such that

$\sup_{x\in\mathbb{Z}^{d},|y-x|=1}\mathbb{P}\left(K_{1}\leq\tau_{x,y}^{N,\beta,0}<+\infty\right)\leq\varepsilon.$

Then, for sufficiently large $N$,

${\rm
I}=\frac{1}{N^{d}}\sum_{x,y\in\mathbb{Z}^{d},|x-y|=1}\mathbb{E}\left[\rho_{0}\Big{(}\tfrac{X^{N,x}_{sN^{2},\beta}}{N}\Big{)}1_{\\{\tau_{x,y}^{N,\beta,0}\leq
K_{1}\\}}\right]H^{2}\left(\tfrac{x}{N}\right)+\varepsilon_{1},$

where

$|\varepsilon_{1}|\leq 4d\varepsilon\int_{\mathbb{R}^{d}}H^{2}(u)du.$

Since $X^{N,x}_{t,\beta}$ jumps at rate at most $2d+\alpha$, for any integer
$K>0$,

${\mathbb{P}}\left(\sup_{0\leq t\leq
K_{1}}\left|X^{N,x}_{t,\beta}-x\right|\geq
K\right)\leq{\mathbb{P}}\left(\sum_{i=1}^{K}T_{i}\leq K_{1}\right),$

where $\\{T_{i}\\}_{i=1}^{K}$ are independent exponential times with rate
$2d+\alpha$. Therefore,

$\lim_{K\rightarrow+\infty}\sup_{x\in\mathbb{Z}^{d}}{\mathbb{P}}\left(\sup_{0\leq
t\leq K_{1}}\left|X^{N,x}_{t,\beta}-x\right|\geq K\right)=0.$

Equivalently, there exists $K_{2}=K_{2}(\varepsilon)>0$ such that

$\sup_{x\in\mathbb{Z}^{d}}{\mathbb{P}}\left(\sup_{0\leq t\leq
K_{1}}\left|X^{N,x}_{t,\beta}-x\right|\geq K_{2}\right)\leq\varepsilon.$

On the event $\\{\tau_{x,y}^{N,\beta,0}\leq K_{1}\\}$,
$\left|X^{N,x}_{\tau_{x,y}^{N,\beta,0},\beta}-x\right|\geq K_{2}$ implies that
$\sup_{0\leq t\leq K_{1}}\left|X^{N,x}_{t,\beta}-x\right|\geq K_{2}$. Thus,

${\rm
I}=\frac{1}{N^{d}}\sum_{x,y\in\mathbb{Z}^{d},|x-y|=1}\mathbb{E}\left[\rho_{0}\left(\tfrac{X^{N,x}_{sN^{2},\beta}}{N}\right)1_{\left\\{\tau_{x,y}^{N,\beta,0}\leq
K_{1}\right\\}}1_{\left\\{\left|X^{N,x}_{\tau_{x,y}^{N,\beta,0},\beta}-x\right|\leq
K_{2}\right\\}}\right]H^{2}\left(\tfrac{x}{N}\right)+\varepsilon_{1}+\varepsilon_{2}$
(5.12)

when $N$ is sufficiently large, where

$|\varepsilon_{2}|\leq 4d\varepsilon\int_{\mathbb{R}^{d}}H^{2}(u)du.$

Since $H^{2}$ is integrable, we can choose $K_{3}>0$ such that $\int_{|u|\geq
K_{3}}H^{2}(u)du<\varepsilon$. Since $\rho_{0}$ is global Lipschitz, according
to Theorem 3.1,

$\lim_{N\rightarrow+\infty}\sup_{x,w\in\mathbb{Z}^{d},\theta>0:\atop\varepsilon
N\leq|x|\leq K_{3}N,|w-x|\leq
K_{2},\theta<K_{1}}\left|\mathbb{E}\rho_{0}\left(\frac{X_{sN^{2},\beta}^{N,x}}{N}\right)-\mathbb{E}\rho_{0}\left(\frac{X_{sN^{2}-\theta,\beta}^{N,w}}{N}\right)\right|=0.$
(5.13)

According to the strong Markov property,

$\displaystyle\mathbb{E}\left[\rho_{0}\left(\tfrac{X^{N,x}_{sN^{2},\beta}}{N}\right)1_{\left\\{\tau_{x,y}^{N,\beta,0}\leq
K_{1}\right\\}}1_{\left\\{\left|X^{N,x}_{\tau_{x,y}^{N,\beta,0},\beta}-x\right|\leq
K_{2}\right\\}}\right]$
$\displaystyle=\mathbb{E}\left[1_{\left\\{\tau_{x,y}^{N,\beta,0}\leq
K_{1}\right\\}}1_{\left\\{\left|X^{N,x}_{\tau_{x,y}^{N,\beta,0},\beta}-x\right|\leq
K_{2}\right\\}}\mathbb{E}\rho_{0}\bigg{(}\frac{X_{sN^{2}-\tau_{x,y}^{N,\beta,0},\beta}^{N,X^{N,x}_{\tau_{x,y}^{N,\beta,0},\beta}}}{N}\bigg{)}\right].$

By Equation (5.13), for sufficiently large $N$ and $x\in\mathbb{Z}^{d}$ such
that $\varepsilon N\leq|x|\leq NK_{3}$,

$\left|\mathbb{E}\rho_{0}\left(\frac{X_{sN^{2}-\tau_{x,y}^{N,\beta,0},\beta}^{N,X^{N,x}_{\tau_{x,y}^{N,\beta,0},\beta}}}{N}\right)-\mathbb{E}\rho_{0}\left(\frac{X_{sN^{2},\beta}^{N,x}}{N}\right)\right|\leq\varepsilon$

conditioned on $\left|X^{N,x}_{\tau_{x,y}^{N,\beta,0},\beta}-x\right|\leq
K_{2}$ and $\tau_{x,y}^{N,\beta,0}\leq K_{1}$. Then by Equation (5.12),

${\rm I}=\frac{1}{N^{d}}\sum_{x,y\in\mathbb{Z}^{d}:\atop\varepsilon
N\leq|x|\leq
K_{3}N,|x-y|=1}{\mathbb{P}}\left(\left|X^{N,x}_{\tau_{x,y}^{N,\beta,0},\beta}-x\right|\leq
K_{2},\tau_{x,y}^{N,\beta,0}\leq
K_{1}\right)\mathbb{E}\left[\rho_{0}\left(\tfrac{X^{N,x}_{sN^{2},\beta}}{N}\right)\right]H^{2}\left(\tfrac{x}{N}\right)+\sum_{j=1}^{3}\varepsilon_{j}$

for sufficiently large $N$, where

$|\varepsilon_{3}|\leq
4d\varepsilon+4d\int_{|u|\leq\varepsilon}H^{2}(u)du+4d\varepsilon\int_{\mathbb{R}^{d}}H^{2}(u)du.$

Then, according to the definition of $K_{3}$,

${\rm
I}=\frac{1}{N^{d}}\sum_{x,y\in\mathbb{Z}^{d},|x-y|=1}{\mathbb{P}}\left(\left|X^{N,x}_{\tau_{x,y}^{N,\beta,0},\beta}-x\right|\leq
K_{2},\tau_{x,y}^{N,\beta,0}\leq
K_{1}\right)\mathbb{E}\left[\rho_{0}\left(\tfrac{X^{N,x}_{sN^{2},\beta}}{N}\right)\right]H^{2}\left(\tfrac{x}{N}\right)+\sum_{j=1}^{4}\varepsilon_{j}$

for sufficiently large $N$, where

$|\varepsilon_{4}|\leq 4d\varepsilon+4d\int_{|u|\leq\varepsilon}H^{2}(u)du.$

Then, according to definitions of $K_{1}$ and $K_{2}$,

${\rm
I}=\frac{1}{N^{d}}\sum_{x,y\in\mathbb{Z}^{d},|x-y|=1}{\mathbb{P}}\left(\tau_{x,y}^{N,\beta,0}<+\infty\right)\mathbb{E}\left[\rho_{0}\left(\tfrac{X^{N,x}_{sN^{2},\beta}}{N}\right)\right]H^{2}\left(\tfrac{x}{N}\right)+\sum_{j=1}^{5}\varepsilon_{j}$
(5.14)

for sufficiently large $N$, where

$|\varepsilon_{5}|\leq 8d\varepsilon\int_{\mathbb{R}^{d}}H^{2}(u)du.$

We claim that

$\limsup_{|x_{1}|\rightarrow+\infty}\sup_{|y-x|=1,N\geq
1}\left|{\mathbb{P}}\left(\tau_{x,y}^{N,\beta,0}<+\infty\right)-\gamma_{d}\right|=0,$
(5.15)

where $x_{1}$ is the first coordinate of $x$. We prove Equation (5.15) later.
By Equation (5.15), there exists $K_{4}=K_{4}(\varepsilon)>0$ such that

$\left|{\mathbb{P}}\left(\tau_{x,y}^{N,\beta,0}<+\infty\right)-\gamma_{d}\right|\leq\varepsilon$

for any $N\geq 1$ and $x,y\in\mathbb{Z}^{d}$ such that $|x_{1}|\geq K_{4}$ and
$|x-y|=1$. Since $\varepsilon N\geq K_{4}$ when $N$ is sufficiently large, by
Equation (5.14),

${\rm I}=\frac{1}{N^{d}}\sum_{|x_{1}|\geq\varepsilon
N,|x-y|=1}\gamma_{d}\mathbb{E}\left[\rho_{0}\left(\tfrac{X^{N,x}_{sN^{2},\beta}}{N}\right)\right]H^{2}\left(\tfrac{x}{N}\right)+\sum_{j=1}^{6}\varepsilon_{j}$

for sufficiently large $N$, where

$|\varepsilon_{6}|\leq
4d\varepsilon\int_{\mathbb{R}^{d}}H^{2}(u)du+4d\int_{|u_{1}|\leq\varepsilon}H^{2}(u)du.$

Then,

${\rm
I}=\frac{1}{N^{d}}\sum_{x\in\mathbb{Z}^{d},|x-y|=1}\gamma_{d}\mathbb{E}\left[\rho_{0}\left(\tfrac{X^{N,x}_{sN^{2},\beta}}{N}\right)\right]H^{2}\left(\tfrac{x}{N}\right)+\sum_{j=1}^{7}\varepsilon_{j}$

for sufficiently large $N$, where

$|\varepsilon_{7}|\leq 4d\int_{|u_{1}|\leq\varepsilon}H^{2}(u)du.$

Then, by Theorem 3.1,

$\limsup_{N\rightarrow+\infty}\left|{\rm
I}-2d\gamma_{d}\int_{\mathbb{R}^{d}}\mathbb{E}[\rho_{0}\left(B_{2s,\beta}^{u}\right)]H^{2}(u)du\right|\leq\sum_{j=1}^{7}|\varepsilon_{j}|.$

Since $\varepsilon$ is arbitrary, let $\varepsilon\rightarrow 0$, then
$\sum_{j=1}^{7}|\varepsilon_{j}|\rightarrow 0$ and hence

$\lim_{N\rightarrow+\infty}{\rm
I}=2d\gamma_{d}\int_{\mathbb{R}^{d}}\mathbb{E}\big{[}\rho_{0}(B_{2s,\beta}^{u})\big{]}H^{2}(u)du.$

According to [3, Proposition 2.3],
$\mathbb{E}\big{[}\rho_{0}(B_{2s,\beta}^{u})\big{]}=\rho(s,u)$ and hence
Equation (5.10) holds.

Now we check Equation (5.15) to complete the proof of Equation (5.10). For
$x,y\in\mathbb{Z}^{d}$ such that $|x-y|=1$, let
$X_{t,0}^{N,x},\hat{X}_{t,0}^{N,y,0}$ be defined as in Section 4.2 with
$\alpha=1$, i.e., $\\{X_{t,0}^{N,x}\\}_{t\geq 0}$ and
$\\{\hat{X}_{t,0}^{N,y,0}\\}_{t\geq 0}$ are independent simple random walks on
$\mathbb{Z}^{d}$ starting at $x$ and $y$ respectively. Since
$\\{X_{t,0}^{N,x}-\hat{X}_{t,0}^{N,y,0}\\}_{t\geq 0}$ is an accelerated
version of the simple random walk on $\mathbb{Z}^{d}$ starting at a neighbor
of $0$, we have

${\mathbb{P}}\left(\tau_{x,y}^{N,0,0}<+\infty\right)=\gamma_{d},$

where $\tau_{x,y}^{N,0,0}$ is the collision time of $X_{t,0}^{N,x}$ and
$\hat{X}_{t,0}^{N,y,0}$ as defined in Section 4.2. Hence,

$\left|\mathbb{P}\left(\tau_{x,y}^{N,\beta,0}<+\infty\right)-\gamma_{d}\right|=\left|\mathbb{P}\left(\tau_{x,y}^{N,\beta,0}<+\infty\right)-\mathbb{P}\left(\tau_{x,y}^{N,0,0}<+\infty\right)\right|.$
(5.16)

Let

$\psi_{x,y}^{N}=\inf\left\\{t:~{}\left(X_{t,0}^{N,x}\right)_{1}\in\\{0,1\\}\text{~{}or~{}}\left(\tilde{X}^{N,y,0}_{t,0,x}\right)_{1}\in\\{0,1\\}\right\\},$

then
$\left\\{\left(X_{t,\beta}^{N,x},\tilde{X}_{t,\beta,x}^{N,y,0}\right)\right\\}_{t\geq
0}$ and
$\left\\{\left(X_{t,0}^{N,x},\tilde{X}_{t,0,x}^{N,y,0}\right)\right\\}_{t\geq
0}$ can be coupled in the same probability space such that

$\left(X_{t,\beta}^{N,x},\tilde{X}_{t,\beta,x}^{N,y,0}\right)=\left(X_{t,0}^{N,x},\tilde{X}_{t,0,x}^{N,y,0}\right)$

for $0\leq t\leq\psi_{x,y}^{N}$. As a result, for any $K>0$,

$\mathbb{P}\left(\tau_{x,y}^{N,\beta,0}<K,\psi_{x,y}^{N}\geq
K\right)=\mathbb{P}\left(\tau_{x,y}^{N,0,0}<K,\psi_{x,y}^{N}\geq K\right).$

Therefore,

$\left|\mathbb{P}\left(\tau_{x,y}^{N,\beta,0}<+\infty\right)-\mathbb{P}\left(\tau_{x,y}^{N,0,0}<+\infty\right)\right|\leq{\rm
III}+{\rm IV},$ (5.17)

where

${\rm
III}=\mathbb{P}\left(K\leq\tau_{x,y}^{N,\beta,0}<+\infty\right)+\mathbb{P}\left(K\leq\tau_{x,y}^{N,0,0}<+\infty\right)$

and

${\rm IV}=2\mathbb{P}\left(\psi_{x,y}^{N}<K\right).$

Let $\\{U_{t}\\}_{t\geq 0}$ be the simple random walk on $\mathbb{Z}^{1}$
defined as in Section 3 with $U_{0}=0$, then for $x,y$ such that $|x-y|=1$,

$\mathbb{P}\left(\psi_{x,y}^{N}<K\right)\leq 2\mathbb{P}\left(\max_{0\leq
t\leq K}|U_{t}|\geq|x_{1}|-2\right),$

which converges to $0$ as $|x_{1}|\rightarrow+\infty$. Hence,

$\lim_{|x_{1}|\rightarrow+\infty}\sup_{N\geq
1,x,y\in\mathbb{Z}^{d},|x-y|=1}{\rm IV}=0.$ (5.18)

Since
$\left(X_{t,\beta}^{N,x}-\hat{X}_{t,\beta}^{N,y,0}\right)^{\perp}=V_{2t}^{(x-y)^{\perp}}$
for any $\beta\geq 0$ as we have explained,

${\rm III}\leq 2\sup_{|v|\leq 1}\mathbb{P}\left(V_{2t}^{v}=0\text{~{}for
some~{}}t\geq K\right).$

As a result, by Equations (5.16)-(5.18),

$\limsup_{|x_{1}|\rightarrow+\infty}\sup_{|y-x|=1,N\geq
1}\left|\mathbb{P}\left(\tau_{x,y}^{N,\beta,0}<+\infty\right)-\gamma_{d}\right|\leq
2\sup_{|v|\leq 1}\mathbb{P}\left(V_{2t}=0\text{~{}for some~{}}t\geq K\right)$
(5.19)

for any $K>0$. As we have explained, $\\{V_{2t}\\}_{t\geq 0}$ visits $0$ for
finite times with probability $1$ and hence
$\lim_{K\rightarrow+\infty}\sup_{|v|\leq
1}\mathbb{P}\left(V_{2t}=0\text{~{}for some~{}}t\geq K\right)=0$. Therefore,
let $K\rightarrow+\infty$ in Equation (5.19) and then Equation (5.15) holds.

Now we check Equation (5.11). The proof of Equation (5.11) is similar with
that of Equation (5.10) and hence we only give an outline. According to the
definition of ${\rm II}$,

${\rm II}={\rm V}-{\rm VI},$

where

${\rm
V}=\frac{1}{N^{d}}\sum_{x,y\in\mathbb{Z}^{d},|x-y|=1}\mathbb{E}\left[\rho_{0}\left(\frac{X^{N,x}_{sN^{2},\beta}}{N}\right)\rho_{0}\left(\frac{\hat{X}^{N,y,0}_{sN^{2},\beta}}{N}\right)\right]H^{2}\left(\tfrac{x}{N}\right)$

and

${\rm
VI}=\frac{1}{N^{d}}\sum_{x,y\in\mathbb{Z}^{d},|x-y|=1}\mathbb{E}\left[\rho_{0}\left(\frac{X^{N,x}_{sN^{2},\beta}}{N}\right)\rho_{0}\left(\frac{\hat{X}^{N,y,0}_{sN^{2},\beta}}{N}\right)1_{\\{\tau_{x,y}^{N,\beta,0}\leq
sN^{2}\\}}\right]H^{2}\left(\tfrac{x}{N}\right).$

Since $X^{N,x}_{sN^{2},\beta}$ and $\hat{X}^{N,y,0}_{sN^{2},\beta}$ are
independent, it is easy to show that

$\lim_{N\rightarrow+\infty}{\rm
V}=2d\int_{\mathbb{R}^{d}}\rho^{2}(s,u)H^{2}(u)du$

according to Theorem 3.1 and [3, Proposition 2.3 ]. Hence, to prove Equation
(5.11) we only need to show that

$\lim_{N\rightarrow+\infty}{\rm
VI}=2d\gamma_{d}\int_{\mathbb{R}^{d}}\rho^{2}(s,u)H^{2}(u)du.$ (5.20)

Let $K_{1},K_{2},K_{3}$ be defined as in the proof of Equation (5.10), then
according to the strong Markov property and an analysis similar with that
given in the proof of Equation (5.10),

$\displaystyle{\rm
VI}=\frac{1}{N^{d}}\sum_{x,y\in\mathbb{Z}^{d}:\atop\varepsilon N\leq|x|\leq
K_{3}N,|x-y|=1}\mathbb{P}\left(\left|X^{N,x}_{\tau_{x,y}^{N,\beta,0},\beta}-x\right|\leq
K_{2},\tau_{x,y}^{N,\beta,0}\leq K_{1}\right)$
$\displaystyle\text{\quad\quad\quad\quad}\times\mathbb{E}\left[\rho_{0}\left(\frac{X^{N,x}_{sN^{2},\beta}}{N}\right)\right]\mathbb{E}\left[\rho_{0}\left(\frac{X^{N,x}_{sN^{2},\beta}}{N}\right)\right]H^{2}\left(\tfrac{x}{N}\right)+\varepsilon_{8},$

where $\varepsilon_{8}=\varepsilon_{8}(\varepsilon)$ such that
$\lim_{\varepsilon\rightarrow 0}\varepsilon_{8}=0$. Then, according to
definitions of $K_{1},K_{2},K_{3}$, Equation (5.15) and an analysis similar
with that given in the proof of Equation (5.10),

$\displaystyle{\rm
VI}=\frac{1}{N^{d}}\sum_{x,y\in\mathbb{Z}^{d},|x-y|=1}\gamma_{d}\mathbb{E}\left[\rho_{0}\left(\frac{X^{N,x}_{sN^{2},\beta}}{N}\right)\right]^{2}H^{2}\left(\tfrac{x}{N}\right)+\varepsilon_{9},$

where $\lim_{\varepsilon\rightarrow+\infty}\varepsilon_{9}=0$. Equation (5.20)
follows from Theorem 3.1 and [3, Proposition 2.3 ] and then the proof of
Equation (5.11) is complete. ∎

###### Proof of Lemma 5.6.

We claim that

$\lim_{K\rightarrow+\infty}\sup_{N\geq
1,x,y,z,w\in\mathbb{Z}^{d}:\atop|x-y|=1,|z-w|=1,|(x-z)^{\perp}|>K}\left|{\rm
Cov}\left(\eta_{s}(x)\eta_{s}(y),\eta_{s}(z)\eta_{s}(w)\right)\right|=0,$
(5.21)

where $x^{\perp}=(x_{2},\ldots,x_{d})$ as we have defined. We prove Equation
(5.21) at the end of this proof. By Equation (5.21), for any $\varepsilon>0$,
there exists $K_{5}=K_{5}(\varepsilon)>0$ such that

$\left|{\rm
Cov}\left(\eta_{s}(x)\eta_{s}(y),\eta_{s}(z)\eta_{s}(w)\right)\right|\leq\varepsilon$

for any $N\geq 1$ and $x,y,z,w\in\mathbb{Z}^{d}$ such that
$|x-y|=|z-w|=1,|(x-z)^{\perp}|>K_{5}$. Then, since $\varepsilon N>K_{5}$ when
$N$ is sufficiently large,

$\displaystyle{\rm
Var}\left(\frac{1}{N^{d}}\sum_{x,y\in\mathbb{Z}^{d},|x-y|=1}\eta_{s}(x)\eta_{s}(y)H^{2}\left(\tfrac{x}{N}\right)\right)$
$\displaystyle\leq\varepsilon\frac{1}{N^{2d}}\sum_{x,y\in\mathbb{Z}^{d},|x-y|=1}\sum_{z,w\in\mathbb{Z}^{d},|z-w|=1,|(z-x)^{\perp}|>\varepsilon
N}H^{2}\left(\tfrac{x}{N}\right)H^{2}\left(\tfrac{z}{N}\right)$
$\displaystyle\text{\quad}+\frac{1}{N^{2d}}\sum_{x,y\in\mathbb{Z}^{d},|x-y|=1}\sum_{z,w\in\mathbb{Z}^{d},|z-w|=1,|(z-x)^{\perp}|\leq\varepsilon
N}H^{2}\left(\tfrac{x}{N}\right)H^{2}\left(\tfrac{z}{N}\right)$
$\displaystyle\leq\varepsilon_{10}$

for sufficiently large $N$, where

$\varepsilon_{10}=\varepsilon_{10}(\varepsilon)=8\varepsilon
d^{2}\left(\int_{u\in\mathbb{R}^{d}}H^{2}(u)du\right)^{2}+8d^{2}\int_{u\in\mathbb{R}^{d}}\int_{v\in\mathbb{R}^{d},|(v-u)^{\perp}|\leq\varepsilon}H^{2}(u)H^{2}(v)dudv.$

As a result,

$\limsup_{N\rightarrow+\infty}{\rm
Var}\left(\frac{1}{N^{d}}\sum_{x,y\in\mathbb{Z}^{d},|x-y|=1}\eta_{s}(x)\eta_{s}(y)H^{2}\left(\tfrac{x}{N}\right)\right)\leq\varepsilon_{10}(\varepsilon).$

Since $\varepsilon$ is arbitrary and $\lim_{\varepsilon\rightarrow
0}\varepsilon_{10}(\varepsilon)=0$, let $\varepsilon\rightarrow 0$ and then
Lemma 5.6 holds.

At last we only need to check Equation (5.21). For any $N\geq 1$ and
$x,y,z,w\in\mathbb{Z}^{d}$, let

$\left\\{\left\\{\pi_{t,\beta}^{N,x},\pi_{t,\beta}^{N,y},\pi_{t,\beta}^{N,z},\pi_{t,\beta}^{N,w}\right\\}\right\\}_{t\geq
0}$

be the coalescing random walk on $\mathbb{Z}^{d}$ such that

$\left\\{\pi_{0,\beta}^{N,x},\pi_{0,\beta}^{N,y},\pi_{0,\beta}^{N,z},\pi_{0,\beta}^{N,w}\right\\}=\\{x,y,z,w\\}$

and
$\left\\{\pi_{t,\beta}^{N,x},\pi_{t,\beta}^{N,y},\pi_{t,\beta}^{N,z},\pi_{t,\beta}^{N,w}\right\\}$
jumps to
$\left\\{\pi_{t,\beta}^{N,x},\pi_{t,\beta}^{N,y},\pi_{t,\beta}^{N,z},\pi_{t,\beta}^{N,w}\right\\}\cup\\{b\\}\setminus\\{a\\}$
at rate $\xi^{N}_{a,b}$ given in (2.2) for any $t\geq 0$ and
$a\in\\{\pi_{t,\beta}^{N,x},\pi_{t,\beta}^{N,y},\pi_{t,\beta}^{N,z},\pi_{t,\beta}^{N,w}\\}$,
$|b-a|=1$. Then, according to Assumption (A) and the duality relationship
between the voter model and the coalescing random walk,

$\displaystyle{\rm
Cov}\left(\eta_{s}(x)\eta_{s}(y),\eta_{s}(z)\eta_{s}(w)\right)$
$\displaystyle=\mathbb{E}\left(\prod_{a\in\\{\pi_{sN^{2},\beta}^{N,x},\pi_{sN^{2},\beta}^{N,y},\pi_{sN^{2},\beta}^{N,z},\pi_{sN^{2},\beta}^{N,w}\\}}\rho_{0}(a)\right)$
$\displaystyle\text{\quad}-\mathbb{E}\left(\prod_{a\in\\{X^{N,x}_{sN^{2},\beta},\tilde{X}^{N,y,0}_{sN^{2},0,x}\\}}\rho_{0}(a)\right)\mathbb{E}\left(\prod_{a\in\\{X^{N,z}_{sN^{2},\beta},\tilde{X}^{N,w,0}_{sN^{2},0,z}\\}}\rho_{0}(a)\right).$
(5.22)

We couple
$\left\\{\\{X^{N,x}_{t,\beta},\tilde{X}^{N,y,0}_{t,0,x}\\}\right\\}_{t\geq 0}$
and $\left\\{\\{X^{N,z}_{t,\beta},\tilde{X}^{N,w,0}_{t,0,z}\\}\right\\}_{t\geq
0}$ in the same probability space such that
$\left\\{\\{X^{N,x}_{t,\beta},\tilde{X}^{N,y,0}_{t,0,x}\\}\right\\}_{t\geq 0}$
and $\left\\{\\{X^{N,z}_{t,\beta},\tilde{X}^{N,w,0}_{t,0,z}\\}\right\\}_{t\geq
0}$ are independent. We define

$\displaystyle\mathcal{T}^{x,y,z,w}_{N,\beta}=\inf\left\\{t:~{}\left\\{X^{N,x}_{t,\beta},\tilde{X}^{N,y,0}_{t,0,x}\right\\}\cap\left\\{X^{N,z}_{t,\beta},\tilde{X}^{N,w,0}_{t,0,z}\right\\}\neq\emptyset\right\\},$

then we can further couple
$\left\\{\\{\pi_{t,\beta}^{N,x},\pi_{t,\beta}^{N,y},\pi_{t,\beta}^{N,z},\pi_{t,\beta}^{N,w}\\}\right\\}_{t\geq
0}$ and
$\left\\{\\{X^{N,x}_{t,\beta},\tilde{X}^{N,y,0}_{t,0,x},X^{N,z}_{t,\beta},\tilde{X}^{N,w,0}_{t,0,z}\\}\right\\}_{t\geq
0}$ in the same probability space such that

$\\{\pi_{t,\beta}^{N,x},\pi_{t,\beta}^{N,y},\pi_{t,\beta}^{N,z},\pi_{t,\beta}^{N,w}\\}=\\{X^{N,x}_{t,\beta},\tilde{X}^{N,y,0}_{t,0,x},X^{N,z}_{t,\beta},\tilde{X}^{N,w,0}_{t,0,z}\\}$

for $0\leq t\leq\mathcal{T}^{x,y,z,w}_{N,\beta}$. Therefore, by Equation
(5.2),

$\left|{\rm
Cov}\left(\eta_{s}(x)\eta_{s}(y),\eta_{s}(z)\eta_{s}(w)\right)\right|\leq
2\mathbb{P}\left(\mathcal{T}^{x,y,z,w}_{N,\beta}\leq sN^{2}\right)\leq
2\mathbb{P}\left(\mathcal{T}^{x,y,z,w}_{N,\beta}<+\infty\right).$ (5.23)

Since $\left\\{\left(X^{N,a}_{t,\beta}\right)^{\perp}\right\\}_{t\geq 0}$ is a
simple random walk on $\mathbb{Z}^{d-1}$ starting at $a^{\perp}$ for any
$a\in\mathbb{Z}^{d}$,

$\mathbb{P}\left(\mathcal{T}^{x,y,z,w}_{N,\beta}<+\infty\right)\leq
4\sup_{\omega\in\mathbb{Z}^{d-1},|\omega|\geq K-2}\Gamma(\omega)$ (5.24)

for any $x,y,z,w\in\mathbb{Z}^{d}$ such that
$|x-y|=1,|z-w|=1,|(x-z)^{\perp}|\geq K$, where $\Gamma(\omega)$ is defined as
in Subsection 4.2. As we have shown in Subsection 4.2,

$\lim_{|\omega|\rightarrow+\infty}\Gamma(\omega)=0$

and hence Equation (5.21) follows from Equation (5.24). This completes the
proof. ∎

###  5.3. Tightness.

In this subsection, we prove the tightness of the sequence of processes
$\\{{\mathcal{Y}}^{N}_{t},0\leq t\leq T\\}_{N\geq 1}$. To this end, we only
need to prove the tightness of the sequence of processes
$\\{{\mathcal{Y}}^{N}_{t}(H),0\leq t\leq T\\}_{N\geq 1}$ for any
$H\in{\mathcal{S}}_{\beta}(\mathbb{R}^{d})$.

As in the proof of Lemma 4.2, we only need to check (4.10) and (4.11) also
hold for $\\{{\mathcal{Y}}^{N}_{t}(H),0\leq t\leq T\\}_{N\geq 1}$. Since

$\sup_{N\geq
1}\mathbb{E}\Big{[}\big{(}{\mathcal{Y}}^{N}_{t}(H)\big{)}^{2}\Big{]}<\infty,$
(5.25)

by Chebyshev’s inequality, (4.10) holds for $\\{{\mathcal{Y}}^{N}_{t}(H),0\leq
t\leq T\\}_{N\geq 1}$. By (5.1), it suffices to prove (4.11) holds for
$\\{{\mathcal{M}}^{N}_{t}(H),0\leq t\leq T\\}_{N\geq 1}$ and
$\\{\int_{0}^{t}N^{2}\mathcal{L}_{N}{\mathcal{Y}}_{s}^{N}(H)ds,0\leq t\leq
T\\}_{N\geq 1}$. By (5.2) and Theorem 5.4,

$\sup_{N\geq 1}\mathbb{E}\Big{[}\sup_{0\leq t\leq
T}\big{(}{\mathcal{M}}^{N}_{t}(H)\big{)}^{2}\Big{]}<\infty.$

From the above inequality, as in the proof of Lemma 4.2, one could easily
prove that $\\{{\mathcal{M}}^{N}_{t}(H),0\leq t\leq T\\}_{N\geq 1}$ satisfies
(4.11). To conclude the proof of the tightness, we only need to prove that for
any $\varepsilon>0$,

$\lim_{\delta\rightarrow
0}\limsup_{N\rightarrow\infty}\mathbb{P}\Big{(}\sup_{0\leq
t-s\leq\delta}\Big{|}\int_{s}^{t}N^{2}\mathcal{L}_{N}{\mathcal{Y}}_{\tau}^{N}(H)d\tau\Big{|}>\varepsilon\Big{)}=0$
(5.26)

From the computations in Subsection 5.1, we only need to prove the above
equation for the three terms (5.3), (5.4) and (5.5). By (5.25), it is true for
(5.3). In Subsection 5.1, we have shown that the sum of the two terms (5.4)
and (5.5) converges to zero in $L^{2}$, thus (5.26) is also true for (5.4) and
(5.5). This concludes the proof of the tightness.

Acknowledgments. Xue thanks the financial support from the Fundamental
Research Funds for the Central Universities with grant number 2022JBMC039.
Zhao thanks the financial support from the Fundamental Research Funds for the
Central Universities in China, and from the ANR grant MICMOV
(ANR-19-CE40-0012) of the French National Research Agency (ANR).

## References

  * [1] R. Baldasso, O. Menezes, A. Neumann, and R. Souza.  Exclusion process with slow boundary.  Journal of Statistical Physics, 167(5):1112–1142, 2017.
  * [2] R. Durrett.  Probability: theory and examples, volume 49.  Cambridge university press, 2019.
  * [3] D. Erhard, T. Franco, and D. Da Silva.  The slow bond random walk and the snapping out Brownian motion.  The Annals of Applied Probability, 31:99–127, 2021.
  * [4] D. Erhard, T. Franco, P. Gonçalves, A. Neumann, and M. Tavares.  Non-equilibrium fluctuations for the SSEP with a slow bond.  In Annales de l’Institut Henri Poincaré, Probabilités et Statistiques, volume 56, pages 1099–1128. Institut Henri Poincaré, 2020\.
  * [5] L. C. Evans.  Partial differential equations.  American Mathematical Society, 2010.
  * [6] T. Franco, P. Gonçalves, and A. Neumann.  Hydrodynamical behavior of symmetric exclusion with slow bonds.  In Annales de l’IHP Probabilités et statistiques, volume 49, pages 402–427, 2013.
  * [7] T. Franco, P. Gonçalves, and G. M. Schütz.  Scaling limits for the exclusion process with a slow site.  Stochastic Processes and their applications, 126(3):800–831, 2016\.
  * [8] T. Franco and M. Tavares.  Hydrodynamic limit for the SSEP with a slow membrane.  Journal of Statistical Physics, 175(2):233–268, 2019.
  * [9] R. Holley and T. M. Liggett.  Ergodic theorems for weakly interacting systems and the voter model.  The Annals of Probability, 3:643–663, 1975.
  * [10] C. Kipnis and C. Landim.  Scaling limits of interacting particle systems, volume 320.  Springer Science & Business Media, 2013.
  * [11] A. Lejay.  The snapping out Brownian motion.  The Annals of Applied Probability, 26:1727–1742, 2016.
  * [12] T. M. Liggett.  Interacting particle systems, volume 276.  Springer Science & Business Media, 2012.
  * [13] E. Presutti and H. Spohn.  Hydrodynamics of the voter model.  The Annals of Probability, 11(4):867–875, 1983.
  * [14] L. D. Schiavo, L. Portinale, and F. Sau.  Scaling limits of random walks, harmonic profiles, and stationary non-equilibrium states in lipschitz domains.  arXiv preprint arXiv:2112.14196, 2021.
  * [15] H. Spohn.  Large scale dynamics of interacting particles.  Springer Science & Business Media, 2012.
  * [16] T. Xu.  Hydrodynamic limit of exclusion processes with slow boundaries on hypercubes.  arXiv preprint arXiv:2109.14491, 2021.