# Hydrodynamic limit of the Schelling model with spontaneous Glauber and
Kawasaki dynamics

Florent BARRET MODAL’X, UMR 9023, UPL, Univ. Paris Nanterre, F92000 Nanterre
France. Email<EMAIL_ADDRESS>and
<EMAIL_ADDRESS>Niccolò TORRI 11footnotemark: 1

###### Abstract

In the present article we consider the Schelling model, an agent-based model
describing a segregation dynamics when we have a cohabitation of two social
groups. As for several social models, the behavior of the Schelling model was
analyzed by exploiting theoretical physics tools and computer simulations.
This approach led to conjecture a phase diagram in which either different
social groups were segregated in two large clusters or they were mixed. As far
as the authors know, this model has never been consider by the mathematical
community, a rigorous mathematical analysis of the model is one of the main
novelties of the paper. We describe the Schelling model as a particle system
model and, adding a Glauber and Kawasaki dynamics to the original Schelling
dynamics, we prove the existence of an hydrodynamic limit described by a
reaction-diffusion equation with a discontinuous non-linear reaction term. The
existence and uniqueness of the solution is non trivial and the analysis of
the limit PDE is interesting in its own. Based on our results, we conjecture
the existence of a phase diagram in which we have a mixed, a segregated and a
metastable segregation phase.
2010 Mathematics Subject Classification: 60K35, 82C22, 82D99.
Keywords: Schelling model, particle systems, hydrodynamics limit, reaction-
diffusion equation, Ising model.

##  1 Introduction

Schelling’s model of segregation was introduced by Thomas Schelling in 1971
[21, 22]. The original model is defined on a square grid of $N$ sites (or,
more generally, on a regular graph with $N$ sites) where agents (individuals)
belonging to two groups are randomly disposed. A local dynamic is defined by
each agent, the agent located at a site of the grid compares its group with
the group of its neighbors. More precisely, we fix a tolerance threshold
$T\in[0,1]$. We call $r_{x}$ the fraction of neighbors belonging to the
agent’s group at site $x$ and we say that the agent is satisfied if $r_{x}\geq
T$. If the agent is unsatisfied, then he _moves_ on a site that makes him
satisfied. The dynamics is not unique, we briefly discuss this point here
below. The process ends when all the agents are either satisfied or blocked
(no move makes the agent satisfied). This model is thus characterized by a
local dynamics which generates a global dynamics of the system. The study of
this model enters naturally in the field of statistical physics which provides
a large number of tools to study the properties of the model. The parameters
which characterize the model are

  1. (1)

The size of the neighborhood,

  2. (2)

The initial distribution of the agents,

  3. (3)

The choice of the satisfaction condition,

  4. (4)

The local dynamics between agents.

Several variants of Schelling’s model have been considered in the recent
literature in order to study the behavior of the model when the fundamental
parameters are modified. We refer to [19] for a complete overview on the
subject. The common result of the considered variants is the existence of
three stationary states separated by two critical thresholds $T_{f}$ and
$T_{c}$ (a frozen state $T<T_{f}$, a segregated state $T_{f}<T<T_{c}$ and a
mixed state $T>T_{c}$) towards which the system evolves, suggesting an
universal behavior of the model. When the system is open (agents can change
the group) the behaviour of the system is expected to be symmetric with
respect to $T=1/2$, see [13].

In the present paper, we approach the model from a physical point of view, by
interpreting the agent dynamics as a particle systems in interaction. This
approach was adopted by the physical community to study this model, see for
instance [3, 9, 12, 13]. In particular we consider the setting where the
points of the grid (a discrete torus
${\mathbb{T}}^{d}_{N}:=\left(\mathbb{Z}/N\mathbb{Z}\right)^{d}$, $d\geq 1$)
are fully occupied and unsatisfied agents flip their state if it makes them
satisfied (Glauber dynamics). The size of the neighborhood taken into account
to compute the fraction $r_{x}$, grows at most logarithmically with $N$.
Moreover we introduce random perturbations, either by flipping a state of an
agent at rate $\beta$ (spontaneous Glauber dynamics) or exchanging the
position of two agents at rate $\alpha N^{2}$ (accelerated Kawasaki dynamics).

Our main result (Theorem 3.1) is to prove an hydrodynamic limit described by a
reaction-diffusion equation for the system and to give a complete description
of the limit PDE that we get. In the case where the size of the neighborhood
stays finite in the limit, we obtain a classical reaction-diffusion equation,
this is the case where the size of the interaction term stays finite and thus
microscopic. However, when the size of the neighborhood goes to infinity we
get a non-linearity (the reaction term) which is discontinuous at two points.
In this case, the interaction of the Schelling dynamics takes into account
more and more agents but in the limit, the reaction term is still purely
local. In this “mesoscopic” limit, the existence and uniqueness of the
solution of the reaction-diffusion equation with discontinuities is one of the
major technical points of the paper. More precisely, the system does not have
a unique solution for some class of initial condition, and some values of
$\beta$, the parameter tuning the spontaneous Glauber dynamics. Finally, we
conjecture the existence of a rich phase diagram in which beyond a mixed and a
segregated phase there is a transition in between. In this phase, the system
has the potential to show a metastable segregation: the mixing phase should be
the most stable one but symmetrical stable segregation phases also exist. The
critical points depend on the parameters $\beta$ and $T$ but not on $\alpha$,
see Figure 1. A rigorous proof of the phase diagram requires a delicate
analysis of the local dynamics that goes beyond the techniques used in the
present paper. We reserve this for future work.

Our method of proof is based on the technique of the relative entropy method
in the framework developed by Jara and Mezenes in [14], [15], and also used by
Funaki and Tsunoda in [8] for a finite number of particle in the interaction.
However, in our setting, we need to improve their bounds to cover the case
where the number of particles in the interaction is going to infinity. More
precisely, in order to use the relative entropy method, a central step is the
control of $\partial_{t}{\mathcal{H}}_{N}(t)$, the derivative in time of the
relative entropy between the law of the process $\mu_{N}(t)$ and a discrete
measure which approximates the density solution of the reaction-diffusion
equation, see Proposition 4.3 and (4.13). Since the number of the particles in
the interaction grows with $N$, the size of the system, we need to retrace the
bounds obtained in [14], [15] and [8] by taking into account the size of the
interactions. This is done in Theorem 4.5. With our bound (4.13), we get that
as soon as the diameter of the interaction grows at most as
$\delta(\log(N))^{1/d}$ (see Assumption 1), the relative entropy is
$O(N^{d-\varepsilon})$ for some $\varepsilon>0$ (see Equation (4.14)) which
entails that the empirical measure is close in probability to a deterministic
discrete process defined by Equation (4.10).

To complete the proof of the main result (Theorem 3.1), we also prove that
this deterministic discrete process, defined by Equation (4.10), converges to
a solution of a limiting reaction-diffusion equation (Equation (3.3)). This is
done in two steps : we first prove that the limiting PDE has a solution (in
Proposition 6.3), and for some class of initial conditions, this solution is
locally unique (in Proposition 6.4). The existence result is done via an
approximating sequence of smooth non-linearities which are natural in our
framework (defined by Equation (6.13)). Note that we do not use the
deterministic process defined by Equation (4.10) which is discrete in space.
The second step is therefore to prove that the deterministic process, defined
by Equation (4.10), has accumulation points in a uniform norm on compact set
which are all solutions of Equation (3.3) (this is Theorem 7.1). If we have
uniqueness for solutions of Equation (3.3), we have the main result.

We establish local uniqueness for (3.3) only for a class of initial conditions
(in Proposition 6.4), we use and adapt arguments of Gianni [11] and Deguchi
[4] (which proves existence and uniqueness with only one discontinuity). Note
that for some initial conditions, (3.3) does not have a unique solution, see
Remark 6.2 for a concrete example. It would be therefore quite interesting to
understand if for such initial conditions, the empirical measure process
converges in some sense. We also reserve this for future work.

A detailed plan of the method of proof and the article is given at the end of
Section 3 containing the main result and assumptions. In the following
Section, we define the model.

### Acknowledgments

The authors thank Antoine Lucquiaud and Alessandra Faggionato for the fruitful
discussion on the model and on the article. This research has been conducted
within the FP2M federation (CNRS FR 2036) and as part of the project Labex
MME-DII (ANR11-LBX-0023-01).

##  2 The model

###  2.1 Configurations

For $N\in\mathbb{N}=\\{1,2,3,\ldots\\}$ we let
${\mathbb{T}}^{d}_{N}=(\mathbb{Z}/N\mathbb{Z})^{d}$ be the discrete torus and
let $\Omega_{N}=\\{0,1\\}^{{\mathbb{T}}^{d}_{N}}$ be the space of all possible
configurations. We call $\eta\in\Omega_{N}$ a configuration and
$i\in{\mathbb{T}}^{d}_{N}$ a site. Let
$\mathcal{V}_{N}\subset{\mathbb{T}}^{d}_{N}\setminus\\{0\\}$ be a bounded set.
We say that two sites $i,j\in{\mathbb{T}}^{d}_{N}$ are neighbors if
$i-j\in\mathcal{V}_{N}$. We denote $K_{N}=|\mathcal{V}_{N}|$ its cardinality.
For a configuration $\eta\in\Omega_{N}$ and a site $i\in{\mathbb{T}}^{d}_{N}$
we let

$r_{i}(\eta)=\frac{1}{K_{N}}\sum_{j\in\mathcal{V}_{N}}\mathbbm{1}_{\\{\eta_{i}=\eta_{i+j}\\}}\qquad\text{and}\qquad\rho_{i}(\eta)=\frac{1}{K_{N}}\sum_{j\in\mathcal{V}_{N}}\eta_{i+j}.$
(2.1)

The quantity $\rho_{i}(\eta)$ is the mean field of $\eta$ on the neighborhood
$\mathcal{V}_{N}+i$. Let us observe that $\rho_{i}(\eta)$ is independent of
$\eta_{i}$. We note that

$r_{i}(\eta)=\rho_{i}(\eta)\mathbbm{1}_{\\{\eta_{i}=1\\}}+(1-\rho_{i}(\eta))\mathbbm{1}_{\\{\eta_{i}=0\\}}.$
(2.2)

For a given configuration, we now introduce the definition of stable, unstable
and potentially stable site.

######  Definition 2.1.

For a given site $i$, let us denote $\eta^{i}$ the configuration where we
change $\eta_{i}$ to $1-\eta_{i}$.

Let $T\in[0,1]$. If $r_{i}(\eta)<T$, the site $i$ is said unstable for $\eta$,
otherwise if $r_{i}(\eta)\geqslant T$ the site is said stable for $\eta$. An
unstable site $i$ for $\eta$ which is stable for $\eta^{i}$ is said
potentially stable.

Note that $r_{i}(\eta^{i})=1-r_{i}(\eta)$. Thus

  1. 1.

a site $i$ is potentially stable if and only if $r_{i}(\eta)<T$ and
$r_{i}(\eta)\leq 1-T$. In particular if $T\leqslant\frac{1}{2}$ an unstable
site for $\eta$ is automatically potentially stable.

  2. 2.

if $T>\frac{1}{2}$ and $1-T<r_{i}(\eta)<T$, we have $r_{i}(\eta^{i})<T$ and
the site $i$ is unstable for $\eta$ and $\eta^{i}$.

###  2.2 Infinitesimal generator, construction of the process

Fix $\alpha>0$ and $\beta>0$. Let us consider the following dynamics :
starting from a configuration $\eta$

  1. 1.

if a site $i$ is potentially stable, we flip the value at $i$ with rate $1$,

  2. 2.

two nearest-neighbors $i$ and $j$ exchange their values with rate $\alpha
N^{2}$ (accelerated Kawasaki dynamics),

  3. 3.

a site $i$ can change its value at rate $\beta>0$ (spontaneous Glauber
dynamics).

This dynamics defines an infinitesimal generator ${\mathcal{L}}_{N}$ defined
for $F$ cylinder function on $\Omega_{N}$ by

$\displaystyle{\mathcal{L}}_{N}F(\eta)=$
$\displaystyle\sum_{i\in{\mathbb{T}}_{N}^{d}}(F(\eta^{i})-F(\eta))(\mathbbm{1}_{\\{r_{i}(\eta)<T,r_{i}(\eta)\leqslant
1-T\\}}+\beta)+\alpha
N^{2}\sum_{\begin{subarray}{c}i,j\in{\mathbb{T}}_{N}^{d}\\\
|i-j|=1,\end{subarray}}(F(\eta^{ij})-F(\eta)),$ (2.3)

where $\eta^{ij}$ is the configuration where the values at site $i$ and $j$
have been exchanged.

The following proposition states that the process is well defined, since the
state space is finite.

######  Proposition 2.1.

Given an initial configuration $\eta_{0}$, ${\mathcal{L}}_{N}$ is the
infinitesimal generator of a Feller process, denoted $(\eta^{N}(t))_{t\geq
0}$.

We let $\mu^{N}_{t}$ be the distribution of $\eta^{N}(t)$.

######  Remark 2.1.

In this article we focus on the compact setting (torus) because a non compact
framework, as ${\mathbb{R}}^{d}$, presents technical problems for the
convergence of the process, nevertheless the discrete model can be well
defined on ${\mathbb{Z}}^{d}$ (cf. (2.3) and Proposition 2.1).

##  3 Main results

We let $u^{N}_{0}(i):={\mathbb{E}}_{\mu_{N}}\big{[}\eta_{i}^{N}(0)\big{]}$,
$i\in{\mathbb{T}}^{d}_{N}$ be the initial distribution of our process, that
is, $\eta_{i}^{N}(0)$ is distributed as a Bernoulli of parameter
$u_{0}^{N}(i)$.

For a vector $v$, $|v|$ denotes its euclidean norm and $|v|_{\infty}$ its
uniform norm.

######  Assumption 1.

Assumptions on ${\mathcal{V}}_{N}$.

  1. 1.

Let $\ell_{\mathcal{V}}$ be the diameter of ${\mathcal{V}}$. Then,
$\ell_{\mathcal{V}}\leq\delta\big{(}\log N\big{)}^{\frac{1}{d}}$ for some
$\delta>0$.

  2. 2.

If $d=1$, suppose that
${\mathcal{V}}_{N}\subset\mathbb{Z}\setminus\mathbb{N}$.

######  Assumption 2.

Assumptions on $u_{0}^{N}$.

  1. 1.

There exists $\varepsilon>0$ such that $\varepsilon\leq u_{0}^{N}(i)\leq
1-\varepsilon$ uniformly on $i\in{\mathbb{T}}^{d}_{N}$ and $N\in{\mathbb{N}}$.

  2. 2.

There exists $C_{0}>0$ independent of $N$ such that $|\nabla
u_{0}^{N}(i)|_{\infty}\leq\frac{C_{0}}{N}$, where $\nabla
u_{0}^{N}(i)=(u_{0}^{N}(i+e_{k})-u_{0}^{N}(i))_{k=1}^{d},$ with
$e_{k}\in{\mathbb{Z}}^{d}$ the unit vector of direction $k$.

  3. 3.

Let
$\upsilon_{u_{0}^{N}}^{N}=\bigotimes_{i\in{\mathbb{T}}^{d}_{N}}\mathrm{B}\big{(}u_{0}^{N}(i)\big{)}$
be the law of a sequence of independent Bernoulli of parameter $u_{0}^{N}(i)$.
Suppose that
${\mathcal{H}}(\mu_{0}^{N}\,|\,\upsilon_{u^{N}(0)}^{N})=O(N^{d-\varepsilon_{0}})$
for some $\varepsilon_{0}>0$ small, where ${\mathcal{H}}(\mu\,|\,\upsilon)$ is
the relative entropy of $\mu\ll\upsilon$,

${\mathcal{H}}(\mu\,|\,\upsilon):=\int\frac{{\mathrm{d}}\mu}{{\mathrm{d}}\upsilon}\log\bigg{(}\frac{{\mathrm{d}}\mu}{{\mathrm{d}}\upsilon}\bigg{)}{\mathrm{d}}\upsilon$
(3.1)

  4. 4.

Let $\widetilde{u}_{0}^{N}(x)$ be the linear interpolation on
${\mathbb{T}}^{d}=({\mathbb{R}}/{\mathbb{Z}})^{d}$, the $d$ dimensional torus,
of $u_{0}^{N}(i)$ such that $\widetilde{u}_{0}^{N}(i/N)=u_{0}^{N}(i)$. Then,
there exists $u_{0}\in\mathcal{C}({\mathbb{T}}^{d})$ such that
$\widetilde{u}_{0}^{N}$ converges to $u_{0}$ in
$\mathcal{C}({\mathbb{T}}^{d})$.

######  Remark 3.1.

The assumption 1(2.) is only technical and it could be removed by considering
the dimension $d=1$ separately from the rest of the dimensions, cf. Remark
4.3.

Define

$\pi^{N}_{t}=\pi^{N}_{t}(\eta,{\mathrm{d}}v)=\frac{1}{N^{d}}\sum_{i\in{\mathbb{T}}^{d}_{N}}\,\eta_{i}(t)\delta_{\frac{i}{N}}({\mathrm{d}}v)$
(3.2)

the empirical measure associated to the Markov process $\eta$ where the space
is rescaled by $\frac{1}{N}$, which is a positive measure on
${\mathbb{T}}^{d}$.

We now state our main result, which concerns the convergence in probability of
the empirical measure.

######  Theorem 3.1.

Under Assumptions 1 and 2, if $u_{0}$ (the limit of the initial condition,
according to Assumption 2-4) is such that

  1. 1.

$u_{0}\in\mathcal{C}^{1}({\mathbb{T}}^{d})$, with $\nabla u_{0}$ Lipschitz,

  2. 2.

$\nabla u_{0}(p)\neq 0$ for $p=\min(T,1-T)$ and $p=1-\min(T,1-T)$,

then the reaction-diffusion equation

$\begin{cases}\partial_{t}u(t,x)=2\alpha\Delta
u(t,x)+\beta(1-2u(t,x))+g_{\infty}(u(t,x)),\\\ u(0,x)=u_{0}(x),\end{cases}$
(3.3)

has an unique density solution $u=u(t,x)$ with
$(t,x)\in[0,\tau]\times{\mathbb{T}}^{d}$ for some $\tau>0$, where
${\mathbb{T}}^{d}$ is the $d$-dimension torus and
$g_{\infty}(p):=(1-p)\mathbbm{1}_{\\{1-p<\min(T,1-T)\\}}-p\mathbbm{1}_{\\{p\leqslant\min(T,1-T)\\}}$.
Moreover, for every test function $\varphi:{\mathbb{T}}^{d}\to{\mathbb{R}}$
and for every $\varepsilon>0$,

$\lim_{N\to+\infty}\mu^{N}\bigg{(}\,\Big{|}\langle\pi^{N},\varphi\rangle-\langle
u,\varphi\rangle\Big{|}>\varepsilon\bigg{)}=0\,,\qquad\forall\,t\in[0,\tau]\,,$
(3.4)

where $\langle\pi^{N},\varphi\rangle$ and $\langle u,\varphi\rangle$ denote
the integral of $\varphi$ with respect to the measure $\pi^{N}$ or
$u(x){\mathrm{d}}x$ respectively.

######  Remark 3.2.

If the solution of (3.3) is not unique, which is not technical difficulty but
a real possibility for some initial conditions (see Remark 6.2 for a concrete
example), then any accumulation point of the sequence of empirical measure is
a solution of (3.3).

###  3.1 Organisation of the paper

To prove Theorem 3.1 we first prove that the empirical measure is close to a
discrete measure $u^{N}$ which is a solution of a discrete analogous of (3.3),
this is Theorem 4.1. Its proof is based on a entropy method approach in which
the relative entropy between $\mu^{N}$ and $\upsilon_{u^{N}}^{N}$, see Theorem
4.2. Even if this technique is quite standard in the particle system theory,
some new technical estimations arising from the geometry of the system are
needed, this is Theorem 4.5. In Section 5 we discuss some central technical
estimations about $u^{N}$, in order to describe the behaviour of the discrete
model. Then, in Section 6 we discuss the existence and uniqueness of (3.3) and
in Section 7 we show the convergence of the $u^{N}$ toward the density $u$ by
completing the proof of Theorem 3.1. We stress that the proof of the existence
and uniqueness is not standard and the analysis of this PDE is interesting in
its own.

In Section 3.2 we state our conjecture on the phase diagram of the model.

###  3.2 Conjecture on the phase diagram

In this Section, we discuss the phase diagram that describes the mixed and
segregated phases. We start by setting the Equation (3.3) in a more convenient
form. Set $p_{0}(T):=\min(T,1-T)\in[0,\frac{1}{2}]$. For $p\in[0,1]$ we define

$\displaystyle\gamma_{\infty,\beta}(p)$ $\displaystyle=\left\\{\begin{aligned}
&\beta\left(p-\frac{1}{2}\right)^{2}+\frac{1}{2}\left(p^{2}-p_{0}(T)^{2}\right)&\text{
for $0\leqslant p<p_{0}(T)$},\\\ &\beta\left(p-\frac{1}{2}\right)^{2}&\text{
for $p_{0}(T)\leqslant p<1-p_{0}(T)$},\\\
&\beta\left(p-\frac{1}{2}\right)^{2}+\frac{1}{2}\left((1-p)^{2}-p_{0}(T)^{2}\right)&\text{
for $1-p_{0}(T)\leqslant p\leq 1$}.\end{aligned}\right.$ (3.5)

Our conjecture is based on the analysis of $\gamma_{\infty,\beta}$ and it is
represented in Figure 1. We observe that $\gamma_{\infty,\beta}$ is continuous
and satisfies $\gamma_{\infty,\beta}(p)=\gamma_{\infty,\beta}(1-p)$. For
$p\neq p_{0}(T),1-p_{0}(T)$, we have that
$\gamma^{\prime}_{\infty,\beta}(p)=-\beta(1-2p)-g_{\infty}(p)$ and (3.3) can
be written as

$\partial_{t}u(t,x)=2\alpha\Delta
u(t,x)-\gamma^{\prime}_{\infty,\beta}(u(t,x)).$ (3.6)

To discuss the phase transition we can look at the structure of
$\gamma_{\infty,\beta}(p)$. The function
$p\mapsto\beta\left(p-\frac{1}{2}\right)^{2}+\frac{1}{2}\left(p^{2}-p_{0}(T)^{2}\right)$
has a unique minimum at $p=p^{\ell}:=\frac{\beta}{1+2\beta}$. Therefore, if
$0\leq p^{\ell}<p_{0}(T)<\frac{1}{2}$, the function $\gamma_{\infty,\beta}$
has three regular minima : $p^{c}:=\frac{1}{2}$, $p^{\ell}$ and
$p^{r}:=1-p^{\ell}$. Note that

$\gamma_{\infty,\beta}(p^{c})=0\quad\text{ and
}\quad\gamma_{\infty,\beta}(p^{r})=\gamma_{\infty,\beta}(p^{\ell})=\frac{\beta}{4(1+2\beta)}-\frac{p_{0}(T)^{2}}{2}\,.$

Then we get that if $p_{0}(T)<p^{m}:=\sqrt{\frac{\beta}{2(1+2\beta)}}$, we
have $\gamma_{\infty,\beta}(p^{c})<\gamma_{\infty,\beta}(p^{\ell})$ and if
$p^{m}<p_{0}(T)$, we have that
$\gamma_{\infty,\beta}(p^{c})>\gamma_{\infty}(p^{\ell})$. If
$p^{\ell}>p_{0}(T)$, $p^{c}$ is the only minimum.

The two thresholds for $p_{0}(T)$ are then $p^{\ell}$ and $p^{m}$, see Figure
1. Since $p^{\ell}<p^{m}$, we have the following picture: as $T$ is close to
$0$ and below $p^{\ell}$, we have a unique minimum of $\gamma_{\infty,\beta}$,
so that typical configurations are close to $p=1/2$ which is of lowest energy
$\gamma_{\infty,\beta}$. It means that, at equilibrium, we expect a
configuration balanced between $0$ and $1$ and we do not have segregation.
Then, as $T$ goes above the threshold $p^{\ell}$ but stays below $p^{m}$,
other minima at $p=p^{\ell}$ and $p=p^{r}$ appear, and these two
configurations are metastable since their energy is higher, so we can have
segregation for a small proportion of the time. The next threshold is $p^{m}$,
at which the two metastable configurations become stable and $p=1/2$ is the
metastable one so that we expect stable segregation. For $T$ above
$\frac{1}{2}$ the picture is symmetric.

Figure 1: Representation of the different phases of the system as function of
the parameter $p_{0}(T)\in[0,\frac{1}{2}]$. When $T$ is close to $0$ or $1$
($p_{0}(T)\in(0,p^{\ell})$) we do not have segregation (red parts) and typical
configurations are provided by a mixing of $0$ and $1$. If $T$ is close to
$1/2$ ($p_{0}(T)\in(p^{m},\frac{1}{2})$) we have segregation (green parts): a
very large parts of the configuration are composed of $0$ or $1$. We have also
intermediate values of $T$ ($p_{0}(T)\in(p^{\ell},p^{m})$) for which the
segregation is metastable (yellow parts).

######  Remark 3.3.

The hydrodynamic limit (3.3) has, at least, two different formulations as a
gradient flow :

  1. 1.

in the classical $L^{2}({\mathbb{T}}^{d})$ setting, with the potential
$\mathcal{F}$ defined for $u:{\mathbb{T}}^{d}\to[0,1]$

$\displaystyle\mathcal{F}(u)=\int\alpha\|\nabla
u\|^{2}+\gamma_{\infty,\beta}(u),$ (3.7)

Equation (6.2) can be written as $\partial_{t}u=-\delta\mathcal{F}(u)$ where
$\delta\mathcal{F}$ denotes the Fréchet derivative of $\mathcal{F}$.

  2. 2.

in a Wasserstein-like setting defined in [18] with the entropy potential
$\mathcal{H}$, for $u:{\mathbb{T}}^{d}\to]0,1]$, and
$\xi:{\mathbb{T}}^{d}\to{\mathbb{R}}$ (seen as an element in the tangent
bundle)

$\displaystyle\mathcal{H}(u)$
$\displaystyle=\frac{1}{2}\int(2u\log(2u)-2u+1)\geqslant 0,$
$\displaystyle\mathcal{K}(u)\xi$
$\displaystyle=-2\alpha\nabla\cdot(u\nabla\xi)+\frac{\gamma^{\prime}_{\infty,\beta}(u)}{\log(2u)}\xi,$
(3.8)

Equation (3.3) can be written as

$\partial_{t}u=-\mathcal{K}(u)(\delta\mathcal{H}).$

Since $\delta\mathcal{H}=\log(2u)$ and
$\mathcal{K}(u)(\delta\mathcal{H})=-2\alpha\Delta
u+\gamma^{\prime}_{\beta,\infty}(u).$

Both formulations could be useful to establish a rigorous proof of the phase
diagram given in Figure 1. In particular, along a gradient flow the potential
is non-increasing, thus for all $t>0$, along a solution $u$ we have
$\mathcal{F}(u(t))\leq\mathcal{F}(u(t=0))$ and if $u$ converges to a
stationary solution $v$, it must be a stationary point of ${\mathcal{F}}$
(i.e. $\delta{\mathcal{F}}(v)=0$).

For the second formulation, note that

$\frac{\gamma^{\prime}_{\infty,\beta}(u)}{\log(2u)}>0,u\in]0,1]\Longleftrightarrow
p_{0}(T)<p^{\ell}=\frac{\beta}{1+2\beta}\Longleftrightarrow\beta>\frac{p_{0}(T)}{1-2p_{0}(T)}.$
(3.9)

We are thus in the mixing phase of the diagram and $\mathcal{K}(u)$ is
positive definite in the sense that

$\displaystyle\int\xi\mathcal{K}(u)\xi=\int 2\alpha
u|\nabla\xi|^{2}+\frac{\gamma^{\prime}_{\infty,\beta}(u)}{\log(2u)}\xi^{2}\geqslant
0.$ (3.10)

Then, one can prove that along a solution $u$, we get that :

$\displaystyle\frac{{\mathrm{d}}}{{\mathrm{d}}t}\mathcal{H}(u)=\int\partial_{t}u\delta\mathcal{H}(u)\leqslant-\int\gamma^{\prime}_{\infty,\beta}(u)\log(2u)\leqslant-c\mathcal{H}(u)$
(3.11)

where $c=\beta-\frac{p_{0}(T)}{1-2p_{0}(T)}>0$, and we get that
$\mathcal{H}(u(t))\leqslant\mathcal{H}(u(0))e^{-ct}$. Thus $\mathcal{H}(u(t))$
goes to $0$ as $t\to\infty$, this entails that $u$ converges to the only
stationary point of $\mathcal{H}$ which is the constant $\frac{1}{2}$.
Heuristically, it suggests that an exponential relaxation is taking place in
the mixing phase of Figure 1.

##  4 Relative entropy method

Using the relative entropy method, in this section we prove that the empirical
measure $\pi^{N}(t)$ is close to
$u^{N}(t)=(u^{N}(t,i))_{i\in{\mathbb{T}}^{d}_{N}}$ the solution of a suitable
discrete PDE.

In order to state the main result of this section, we observe that the
generator ${\mathcal{L}}_{N}$ in (2.3) can be written as
${\mathcal{L}}_{N}={\mathcal{G}}_{N}+2\alpha N^{2}{\mathcal{K}}_{N}$ where
${\mathcal{G}}_{N}$ is the generator which describes the Glauber-Schelling
dynamics and ${\mathcal{K}}_{N}$ the generator which describes the Kawasaki
dynamics, that is,

$\displaystyle{\mathcal{G}}_{N}F(\eta)=$
$\displaystyle\sum_{i\in{\mathbb{T}}_{N}^{d}}\Big{(}\mathbbm{1}_{\\{r_{i}(\eta)<T,r_{i}(\eta)\leqslant
1-T\\}}+\beta\Big{)}(F(\eta^{i})-F(\eta)),$ (4.1)
$\displaystyle{\mathcal{K}}_{N}F(\eta)=$
$\displaystyle\frac{1}{2}\sum_{\begin{subarray}{c}i,j\in{\mathbb{T}}_{N}^{d}\\\
|i-j|=1\end{subarray}}(F(\eta^{ij})-F(\eta)).$ (4.2)

Let us stress that ${\mathcal{G}}_{N}$ can be written as follows

${\mathcal{G}}_{N}F(\eta)=\sum_{i\in{\mathbb{T}}^{d}_{N}}(c_{i}(\eta)+\beta)\big{(}F(\eta^{i})-F(\eta)\big{)},$
(4.3)

where $c_{i}(\eta)$ is a local function which describes the dynamics (Glauber-
Schelling dynamics). To be more precise, we write
$c_{i}(\eta)=c_{0}(\tau_{i}\eta)$ where $c_{0}$ is the flipping rate of a
particle at the origin, that is,

$c_{0}(\eta):=\mathbbm{1}_{\\{r_{0}(\eta)<T,r_{0}(\eta)\leqslant 1-T\\}}$
(4.4)

and $(\tau_{i}\eta)_{j}=\eta_{i+j}$, likewise $\tau_{i}$ acts on $u$. Note
that $c_{0}$ is a random variable and take the value $0$ or $1$.

We let

$\kappa_{N}=\kappa_{N}(T):=\min\Big{\\{}\lfloor K_{N}T\rfloor-1;\lfloor
K_{N}(1-T)\rfloor\Big{\\}}.$ (4.5)

We observe that
$\lim\limits_{n\to+\infty}\frac{\kappa_{N}}{K_{N}}=\min(T,1-T)$. Since
$c_{0}(\eta)=\mathbbm{1}_{\\{r_{0}(\eta)\leqslant\frac{\kappa_{N}}{K_{N}}\\}}=\mathbbm{1}_{\\{r_{0}(\eta)\leqslant\frac{\kappa_{N}}{K_{N}},\eta_{0}=0\\}}+\mathbbm{1}_{\\{r_{0}(\eta)\leqslant\frac{\kappa_{N}}{K_{N}},\eta_{0}=1\\}}$,
we define

$c_{0}^{+}(\eta):=\mathbbm{1}_{\\{\rho_{0}(\eta)\geqslant
1-\frac{\kappa_{N}}{K_{N}}\\}}\qquad\text{and}\qquad
c_{0}^{-}(\eta):=\mathbbm{1}_{\\{\rho_{0}(\eta)\leqslant\frac{\kappa_{N}}{K_{N}}\\}},$
(4.6)

so that $c_{0}(\eta)=c_{0}^{+}(\eta)(1-\eta_{0})+c_{0}^{-}(\eta)\eta_{0},$ by
(2.2). The functions $c^{+}$ and $c^{-}$ can be viewed as the rate of creation
and annihilation of a particle at $i=0$ respectively.

For any function $u=(u_{i})_{i\in{\mathbb{T}}^{d}_{N}}$ we define

$\upsilon_{u}({\mathrm{d}}\eta)=\upsilon_{u}^{N}({\mathrm{d}}\eta):=\bigotimes_{i\in{\mathbb{T}}^{d}_{N}}\mathrm{B}\big{(}u_{i})$
(4.7)

and we let $c_{0}^{+}(u)$ and $c_{0}^{-}(u)$ be the expectation of
$c_{0}^{+}(\eta)$ and $c_{0}^{-}(\eta)$ under $\upsilon_{u}$, that is,

$\displaystyle
c_{0}^{+}(u):=\mathbb{P}_{\upsilon_{u}}\Bigg{(}\rho_{0}(\eta)\geqslant
1-\frac{\kappa_{N}}{K_{N}}\Bigg{)}\qquad\text{and}\qquad
c_{0}^{-}(u):=\mathbb{P}_{\upsilon_{u}}\Bigg{(}\rho_{0}(\eta)\leqslant\frac{\kappa_{N}}{K_{N}}\Bigg{)}.$
(4.8)

We finally define

$G(u):=c_{0}^{+}(u)(1-u_{0})-c_{0}^{-}(u)u_{0}\quad\text{and}\quad
G(i,u):=G(\tau_{i}u).$ (4.9)

Let $u^{N}(t)=(u^{N}(t,i))_{i\in{\mathbb{T}}^{d}_{N}}$ be the solution of

$\begin{cases}\partial_{t}u^{N}(t,i)=2\alpha N^{2}\Delta
u^{N}(t,i)+\beta(1-2u^{N}(t,i))+G(i,u^{N})\\\
u^{N}(t,i)=u^{N}_{0}(i),\end{cases}$ (4.10)

where $\Delta u^{N}(t,i)$ is the discrete Laplacian on the torus. Note that
(4.10) can be interpreted as a discretized version of (3.3).

To prove Theorem 3.1 we first define a discrete approximation of $u$ and we
show an equivalent of Theorem 3.1 for $u^{N}$ defined in (4.10). More
precisely, we consider $u^{N}$ as a measure on ${\mathbb{T}}^{d}$, that is,

$u^{N}(t,{\mathrm{d}}v):=\frac{1}{N^{d}}\sum_{i\in{\mathbb{T}}^{d}_{N}}u^{N}(t,i)\delta_{\frac{i}{N}}({\mathrm{d}}v)\,,$
(4.11)

The main result of this section is the following theorem.

######  Theorem 4.1.

Under Assumptions 1 and 2, for every test function
$\varphi:{\mathbb{T}}^{d}\to{\mathbb{R}}$ and for every $\delta>0$ there
exists $\tau>0$ such that

$\lim_{N\to+\infty}\mu^{N}\bigg{(}\,\Big{|}\langle\pi^{N},\varphi\rangle-\langle
u^{N},\varphi\rangle\Big{|}>\delta\bigg{)}=0\,,\qquad\forall\,t\in[0,\tau]\,.$

To prove Theorem 4.1, the main ingredient is that the relative entropy of
$\upsilon^{N}_{u^{N}(t)}$ (cf. (4.7)) with respect to $\mu_{t}^{N}$ stays
small in time, if it is small at $t=0$.

######  Theorem 4.2.

Under Assumptions 1 and 2, with $\delta>0$ sufficiently small, we have that
for some $\varepsilon>0$ small

${\mathcal{H}}_{N}(t):={\mathcal{H}}(\mu_{t}^{N}\,|\,\upsilon^{N}_{u^{N}(t)})=O(N^{d-\varepsilon}),\quad\forall\,t\in[0,\tau]$
(4.12)

with $\tau=\tau(\delta)>0$.

To prove Theorem 4.2 it is enough to show that

$\partial_{t}{\mathcal{H}}_{N}(t)\leq
C\ell_{\mathcal{V}}^{d}\Big{(}{\mathcal{H}}_{N}(t)+O(N^{d-a})\Big{)}$ (4.13)

with $a\in(0,1)$ if $d\geq 2$ and $a\in(0,\frac{1}{2})$ if $d=1$, indeed in
this case Gronwall’s inequality gives

$\displaystyle{\mathcal{H}}_{N}(t)\leq\Big{(}{\mathcal{H}}_{N}(0)+tO(N^{d-a})\Big{)}e^{C\ell_{\mathcal{V}}^{d}t}.$
(4.14)

Since $e^{C\ell_{\mathcal{V}}^{d}t}\leq N^{C\delta t}$ by Assumption 1, the
proof of Theorem 4.2 is complete. We prove (4.13) in Section 4.1

Let us observe that Theorem 4.2 implies Theorem 4.1. Indeed, we recall the
entropy inequality stated for a set ${\mathcal{A}}$ and two measures
$\upsilon\ll\mu$, cf. A1.8.2 of [16] or Section 2.2 of [8],

$\mu({\mathcal{A}})\leq\frac{\log
2+{\mathcal{H}}(\mu\,|\,\upsilon)}{\log\big{(}1+\frac{1}{\upsilon({\mathcal{A}})}\big{)}}.$
(4.15)

For a given test function $\varphi$ and $\delta>0$, we let

${\mathcal{A}}_{N,t,\varphi}^{\delta}:=\big{\\{}\eta\in\Omega_{N}\colon\big{|}\langle\pi^{N},\varphi\rangle-\langle
u^{N},\varphi\rangle\big{|}>\delta\big{\\}},$ (4.16)

so that the proof follows by Theorem 4.2 and (4.15) if

$\upsilon_{u^{N}(t)}^{N}({\mathcal{A}}_{N,t,\varphi}^{\delta})\leq
e^{-C_{\delta}N^{d}}.$ (4.17)

Since $u^{N}\in(0,1)$ (cf. Proposition 5.2), the proof of (4.17) is model
independent and follows line by line the proof of Proposition 2.2 in [8], we
omit the details.

######  Remark 4.1.

Let us note that $c_{0}$ can be expressed as a polynomial on the variables
$\eta_{j}$’s, as in relation (1.5) of [8]. This remark will be useful in the
sequel of the paper. For this purpose, let $A\subset{\mathcal{V}}_{N}$, we
denote :

$\displaystyle c_{A}^{+}(\eta)=\prod_{j\in
A}(1-\eta_{j})\prod_{j\in\bar{A}\cap{\mathcal{V}}_{N}}\eta_{j}\quad\text{and}\quad
c_{A}^{-}(\eta)=\prod_{j\in
A}\eta_{j}\prod_{j\in\bar{A}\cap{\mathcal{V}}_{N}}(1-\eta_{j})=c_{A}^{+}(1-\eta)$
(4.18)

where $1-\eta$ is the configuration with $(1-\eta)_{i}=1-\eta_{i}$ since
$r_{0}(\eta)=r_{0}(1-\eta)$. Note that

$c_{A}^{+}(\eta)=\left\\{\begin{aligned} 1&\text{ if $\eta_{j}=0$ for $j\in A$
and $\eta_{j}=1$ for $j\in\bar{A}\cap{\mathcal{V}}_{N}$}\\\ 0&\text{
otherwise}.\end{aligned}\right.$ (4.19)

By an abuse of notation, for a function $u=(u_{j})_{j\in{\mathbb{T}}^{d}_{N}}$
we let let also $c_{A}^{+}(u)=\prod_{j\in
A}(1-u_{j})\prod_{j\in\bar{A}\cap{\mathcal{V}}_{N}}u_{j}$ and accordingly for
$c_{A}^{-}(u)$. By (4.6) we have that

$c_{0}^{+}(\eta)=\sum_{k=0}^{\kappa_{N}}\mathbbm{1}_{\\{r_{0}(\eta)=\frac{k}{K_{N}},\eta_{0}=0\\}}=(1-\eta_{0})\sum_{\begin{subarray}{c}A\subset{\mathcal{V}}_{N}\\\
|A|\leqslant\kappa_{N}\end{subarray}}c_{A}^{+}(\eta).$ (4.20)

Accordingly, we have

$c_{0}^{-}(\eta)=\sum_{k=0}^{\kappa_{N}}\mathbbm{1}_{\\{r_{0}(\eta)=\frac{k}{K_{N}},\eta_{0}=1\\}}=\eta_{0}\sum_{\begin{subarray}{c}A\subset{\mathcal{V}}_{N}\\\
|A|\leqslant\kappa_{N}\end{subarray}}c_{A}^{-}(\eta).$ (4.21)

In the rest of this section we prove Theorem 4.2. The strategy that we use
follows the one used to prove the analogous result in [8] and [15], but some
extra-technicality is required due to the geometry of our problem.

###  4.1 Proof of Theorem 4.2

To make the notation lighter, for $t\geq 0$, $j\in{\mathbb{T}}^{d}_{N}$,
$p\in(0,1)$ and $u^{N}$ the solution of (4.10), we let

$\displaystyle
u_{j}(t):=u^{N}(t,j),\qquad\chi(p):=p(1-p),\qquad\omega_{j}:=\frac{\eta_{j}-u_{j}}{\chi(u_{j})}.$
(4.22)

and, more generally, whenever the context is clear we omit the superscript
$N$, so that $\upsilon_{u^{N}(t)}^{N}$ and $\mu^{N}_{t}$ will be denoted
simply by $\upsilon_{u(t)}$ and $\mu_{t}$ respectively.

To compare $\mu$ and $\upsilon_{u}$ we introduce

$\vartheta_{\alpha}({\mathrm{d}}\eta):=\bigotimes_{i\in{\mathbb{T}}^{d}_{N}}\mathrm{B}\big{(}\alpha\big{)}$
(4.23)

be a sequence of independent Bernoulli of parameter $\alpha\in(0,1)$ defined
on the space of configurations. We define

$f_{t}:=\frac{{\mathrm{d}}\mu_{t}}{{\mathrm{d}}\upsilon_{u(t)}}\qquad\text{and}\qquad\psi_{t}:=\frac{{\mathrm{d}}\upsilon_{u(t)}}{{\mathrm{d}}\vartheta_{\alpha}}.$
(4.24)

We have all the ingredients to state Yau’s inequality in our context. The
proof is quite standard (cf. proof of Lemma A.1 in [15]), so that it is
omitted.

######  Proposition 4.3.

For any $t\geq 0$ we have that

$\partial_{t}{\mathcal{H}}_{N}(t)\leq-\int\,\Gamma_{N}\Big{(}\sqrt{f_{t}(\eta)}\Big{)}\,\upsilon_{u(t)}({\mathrm{d}}\eta)+\int\,f_{t}(\eta)\,\Big{[}{\mathcal{L}}_{N}^{*,\upsilon_{u(t)}}\mathbf{1}-\partial_{t}\log\psi_{t}\Big{]}\,\upsilon_{u(t)}({\mathrm{d}}\eta),$
(4.25)

where ${\mathcal{L}}_{N}^{*,\upsilon_{u(t)}}$ is the adjoint of
${\mathcal{L}}_{N}$ with respect to the measure $\upsilon_{u(t)}$ and
$\Gamma_{N}(h)(\eta)={\mathcal{L}}_{N}h^{2}(\eta)-2h(\eta){\mathcal{L}}_{N}h(\eta)$
is the carré du champ operator.

We define the current ${\mathcal{J}}_{t}={\mathcal{J}}_{t}^{N}(\eta)$ as

${\mathcal{J}}_{t}:={\mathcal{L}}_{N}^{*,\upsilon_{u(t)}}\mathbf{1}-\partial_{t}\log\psi_{t}.$
(4.26)

Our main goal is to estimate the current ${\mathcal{J}}_{t}$ to control the
right hand side of (4.25) and get (4.13).

####  4.1.1 The current ${\mathcal{J}}_{t}^{N}$

To control ${\mathcal{L}}_{N}^{*,\upsilon_{u(t)}}\mathbf{1}$ we have to
compute the adjoint of ${\mathcal{K}}_{N}$ and of ${\mathcal{G}}_{N}$, cf.
(4.1) ans (4.2). We follow the computations done in [8]. By Lemma 2.4 of [8],
we get

${\mathcal{K}}_{N}^{*,\upsilon_{u(t)}}\mathbf{1}=-\frac{1}{2}\sum_{\begin{subarray}{c}i,j\in{\mathbb{T}}^{d}_{N},\\\
|i-j|=1\end{subarray}}(u_{i}-u_{j})^{2}\omega_{j}\omega_{i}+\sum_{i\in{\mathbb{T}}^{d}_{N}}(\Delta
u)_{i}\omega_{i}$ (4.27)

where $(\Delta u)_{i}=\sum_{j\in{\mathbb{T}}^{d}_{N},|i-j|=1}(u_{j}-u_{i})$ is
the discrete Laplacian. Since $c_{0}$ satisfies the condition (1.5) of [8]
(see Remark 4.1), by Lemma 2.5 of [8] we get

$\begin{split}{\mathcal{G}}_{N}^{*,\upsilon_{u(t)}}\mathbf{1}&=\sum_{i\in{\mathbb{T}}^{d}_{N}}\left(c_{i}^{+}(\eta)(1-u_{i})-c_{i}^{-}(\eta)u_{i}+\beta(1-2u_{i})\right)\omega_{i}\\\
&=\sum_{i\in{\mathbb{T}}^{d}_{N}}\left((c_{i}^{+}(\eta)-c_{i}^{+}(u))(1-u_{i})-(c_{i}^{-}(\eta)-c_{i}^{-}(u))u_{i}\right)\omega_{i}\\\
&\hskip
71.13188pt+\sum_{i\in{\mathbb{T}}^{d}_{N}}\left(c_{i}^{+}(u)(1-u_{i})-c_{i}^{-}(u)u_{i}+\beta(1-2u_{i})\right)\omega_{i}.\end{split}$
(4.28)

In the second equality we centered the variables $c_{i}^{+}(\eta)$
$c_{i}^{-}(\eta)$ since under $\upsilon_{u}$, $c_{i}^{+}(\eta)$ and
$c_{i}^{-}(\eta)$ are Bernouilli random variables with expectation
$c_{i}^{+}(u)$ and $c_{i}^{-}(u)$ respectively. Finally, by Lemma 2.6 of [8]
we get

$\partial_{t}\log\psi_{t}(\eta)=\sum_{i\in{\mathbb{T}}^{d}_{N}}(\partial_{t}u_{i})\omega_{i}.$
(4.29)

Summarizing, we obtain the following result.

######  Proposition 4.4.

The current ${\mathcal{J}}_{t}(\eta)$ satisfies

$\begin{split}{\mathcal{J}}_{t}(\eta)=\sum_{i\in{\mathbb{T}}^{d}_{N}}(-\partial_{t}u_{i}+2\alpha
N^{2}(\Delta u)_{i}+\beta(1-2u_{i})+&G(i,u))\omega_{i}\\\
&-V(u,\eta)+V^{+}(u,\eta)-V^{-}(u,\eta),\end{split}$ (4.30)

where $G$ was defined in (4.9) and

$\displaystyle V^{+}(u,\eta)$
$\displaystyle=\sum_{i\in{\mathbb{T}}^{d}_{N}}\left(c_{i}^{+}(\eta)-c_{i}^{+}(u)\right)(1-u_{i})\omega_{i},$
(4.31) $\displaystyle V^{-}(u,\eta)$
$\displaystyle=\sum_{i\in{\mathbb{T}}^{d}_{N}}\left(c_{i}^{-}(\eta)-c_{i}^{-}(u)\right)u_{i}\omega_{i},$
(4.32) $\displaystyle V(u,\eta)$ $\displaystyle=-\alpha
N^{2}\sum_{\begin{subarray}{c}i,j\in{\mathbb{T}}^{d}_{N},\\\
|i-j|=1\end{subarray}}(u_{i}-u_{j})^{2}\omega_{j}\omega_{i}.$ (4.33)

In particular, if $u$ satisfies (4.10) the current reduces to the second line.

In the rest of the section we provide estimates of $V^{+}$, $V^{-}$ and $V$.

####  4.1.2 Estimates of $V^{+}$ and $V^{-}$

Let us denote $\\{e_{1},e_{2},\dots,e_{d}\\}$ the canonical basis of
$\mathbb{Z}^{d}$. For $\varphi:{\mathbb{T}}^{d}_{N}\to{\mathbb{R}}$ and
$i\in{\mathbb{T}}^{d}_{N}$ and $k\in\\{1,\dots,d\\}$, let
$\nabla_{k}\varphi(i)=\varphi(i+e_{k})-\varphi(i)$. We denote
$\|\nabla\varphi\|_{\infty}=\max_{i,k}|\nabla_{k}\varphi(i)|$.

We note that

$V^{-}(u,\eta)=-V^{+}(1-u,1-\eta),$ (4.34)

so that the bound for $V^{+}$ can be transferred to $V^{-}$, see Remark 4.2.
In the following we get an upper-bound for $V^{+}$. Denote
$\vartheta_{i}^{+}=c_{i}^{+}(\eta)-c_{i}^{+}(u)$ and
$\omega^{+}_{i}=(1-u_{i})\omega_{i}=\frac{\eta_{i}-u_{i}}{u_{i}}$.

Then

$\displaystyle V^{+}(u,\eta)$
$\displaystyle=\sum_{i\in{\mathbb{T}}^{d}_{N}}\vartheta^{+}_{i}\omega^{+}_{i}$
(4.35)

To bound $V^{+}(u,\eta)$ we follow the method used by Jara and Menezes [15]
and by Funaki and Tsuneda [8]. For this purpose let us observe that the carré
du champ referred to the generator ${\mathcal{L}}_{N}$, namely,
$\Gamma_{N}(h)={\mathcal{L}}_{N}h^{2}-2h{\mathcal{L}}_{N}h$ can be decomposed
as

$\Gamma_{N}(h)=\Gamma_{N}^{\mathcal{G}}(h)+2\alpha
N^{2}\Gamma_{N}^{\mathcal{K}}(h),$ (4.36)

where $\Gamma_{N}^{\mathcal{G}}(h)$ and $\Gamma_{N}^{\mathcal{K}}(h)$ are the
carré du champ related to the generator ${\mathcal{G}}_{N}$ and
${\mathcal{K}}_{N}$ respectively (cf. (4.1) and (4.2)). In particular,

$\Gamma_{N}^{\mathcal{K}}(h)=\frac{1}{2}\sum_{\begin{subarray}{c}i,j\in{\mathbb{T}}^{d}_{N}\colon\\\
|i-j|=1\end{subarray}}(h(\eta^{i,j})-h(\eta))^{2}\,.$ (4.37)

In the next result we provide the control that we need for $V^{+}$.

######  Theorem 4.5.

Under Assumption 1, we have that for any $u:{\mathbb{T}}^{d}_{N}\to[0,1]$ such
that

  1. 1.

$\varepsilon:=1-|||u|||_{\infty}>0$ with
$|||u|||_{\infty}:=\min\Big{\\{}\|u\|_{\infty},\|1-u\|_{\infty}\Big{\\}}$,

  2. 2.

$\|\nabla u(i)\|_{\infty}\leq\frac{C_{0}}{N}$ with $C_{0}$ independent of $N$,

and for any density $f$ with respect to $\upsilon_{u}$ we have that

$\int V^{+}(u,\eta)f(\eta)\upsilon_{u}({\mathrm{d}}\eta)\leqslant\delta
N^{2}\int\Gamma^{\mathcal{K}}_{N}\big{(}\sqrt{f}\big{)}(\eta)\upsilon_{u}({\mathrm{d}}\eta)+C_{1}\ell_{\mathcal{V}}^{d}{\mathcal{H}}(f\upsilon_{u}\,|\,\upsilon_{u})+C_{2}N^{d-a},$
(4.38)

for any $\delta>0$ and
$C_{1}=3^{d}d(d+1)\frac{6}{\varepsilon^{2}}\left(\frac{1}{\delta}+\frac{C_{0}}{\varepsilon^{2}}\right)$,
$C_{2}=\frac{6C_{0}}{\varepsilon^{4}}$ for $N>\frac{2C_{0}}{\varepsilon^{2}}$.

######  Remark 4.2.

Note that if the estimate (4.38) holds, then it holds for $V^{-}$. Indeed,
using (4.34), the fact that
$|||u|||_{\infty}=|||1-u|||_{\infty}=1-\varepsilon$ and that, under
$\upsilon_{1-u}({\mathrm{d}}\eta)$, $1-\eta$ has for law $\upsilon_{u}$ we get

$\displaystyle\int V^{-}(u,\eta)f(\eta)\upsilon_{u}({\mathrm{d}}\eta)$
$\displaystyle=-\int
V^{+}(1-u,1-\eta)f(\eta)\upsilon_{u}({\mathrm{d}}\eta)=\int
V^{+}(1-u,\eta)f(1-\eta)\upsilon_{1-u}({\mathrm{d}}\eta)$
$\displaystyle\leqslant\delta
N^{2}\int\Gamma^{\mathcal{K}}_{N}\big{(}\sqrt{f}\big{)}(1-\eta)\upsilon_{1-u}({\mathrm{d}}\eta)+C_{1}\ell_{\mathcal{V}}^{d}{\mathcal{H}}(f(1-\cdot)\upsilon_{1-u}\,|\,\upsilon_{1-u})+C_{2}N^{d-a}$
$\displaystyle=\delta
N^{2}\int\Gamma^{\mathcal{K}}_{N}\big{(}\sqrt{f}\big{)}(\eta)\upsilon_{u}({\mathrm{d}}\eta)+C_{1}\ell_{\mathcal{V}}^{d}{\mathcal{H}}(f\upsilon_{u}\,|\,\upsilon_{u})+C_{2}N^{d-a}.$

###### Proof of Theorem 4.5.

The proof will proceed in several steps.

Note that, according to Hoeffeding Inequality (Lemma A.4), under the
probability measure $\upsilon_{u}$, $\vartheta^{+}_{i}$ is sub-Gaussian with
variance parameter $\frac{1}{4}$ and $\omega^{+}_{i}$ is sub-Gaussian with
variance parameter $\frac{1}{4u_{i}^{2}}$.

As in Jara and Menezes [15], we proceed to use an averaged version of $V^{+}$.
Let $\ell>0$ and consider $\Lambda_{\ell}=\\{0\dots,\ell-1\\}^{d}$ the cube of
size $\ell$ starting at $0$. Let
$p_{\ell}(i)=\ell^{-d}\mathbbm{1}_{\\{i\in\Lambda_{\ell}\\}}$ and
$\hat{p}_{\ell}(i)=\ell^{-d}\mathbbm{1}_{\\{i\in-\Lambda_{\ell}\\}}=p_{\ell}(-i)$.
Then for $\varphi$ defined on ${\mathbb{T}}^{d}_{N}$, we set

$\displaystyle\overset{\xleftarrow{\phantom{\varphi}}}{\varphi}(i)$
$\displaystyle:=p_{\ell}*\varphi(i)=\sum_{j+k=i}p_{\ell}(j)\varphi(k)=\sum_{j}p_{\ell}(j)\varphi(i-j)=\ell^{-d}\sum_{j\in\Lambda_{\ell}}\varphi(i-j)$
(4.39) $\displaystyle\overset{\xrightarrow{\phantom{\varphi}}}{\varphi}(i)$
$\displaystyle:=\hat{p}_{\ell}*\varphi(i)=\sum_{j+k=i}p_{\ell}(-j)\varphi(k)=\sum_{j}p_{\ell}(-j)\varphi(i-j)=\ell^{-d}\sum_{j\in\Lambda_{\ell}}\varphi(i+j).$
(4.40)

Let us denote $q_{\ell}=p_{\ell}*p_{\ell}$. We have
$q_{\ell}(i)=\ell^{-2d}|\Lambda_{\ell}\cap(i-\Lambda_{\ell})|.$ Thus,
$0\leqslant q_{\ell}(i)\leqslant\ell^{-d}$ and $q_{\ell}(i)=0$ if and only if
$i\notin\Lambda_{2\ell-2}$. All sum being finite, we get for $\varphi$ and
$\psi$ defined on ${\mathbb{T}}^{d}_{N}$ :

$\displaystyle\sum_{i,j}\varphi(i)\psi(i+j)q_{\ell}(j)$
$\displaystyle=\sum_{i,j,k}\varphi(i)\psi(i+j)p_{\ell}(k)p_{\ell}(j-k)=\sum_{i,k}\varphi(i)p_{\ell}(k)\sum_{j}\psi(i+j)p_{\ell}(j-k)$
$\displaystyle=\sum_{i,k}\varphi(i)p_{\ell}(k)(\hat{p}_{\ell}*\psi)(k+i)=\sum_{j}\sum_{i}\varphi(i)p_{\ell}(j-i)(\hat{p}_{\ell}*\psi)(j)$
$\displaystyle=\sum_{j}(p_{\ell}*\varphi)(j)(\hat{p}_{\ell}*\psi)(j)=\sum_{j}\overset{\xleftarrow{\phantom{\varphi}}}{\varphi}(j)\overset{\xrightarrow{\phantom{\psi}}}{\psi}(j).$

Thus let
$V^{+,\ell}:=\sum_{i\in{\mathbb{T}}^{d}_{N}}\vartheta^{+}_{i}\omega^{+,\ell}_{i}$
with $\omega^{+,\ell}_{i}=\sum_{j}\omega^{+}_{i+j}q_{\ell}(j)$. We have that

$V^{+,\ell}=\sum_{i,j\in{\mathbb{T}}^{d}_{N}}\vartheta^{+}_{i}\omega^{+}_{i+j}q_{\ell}(j)=\sum_{i\in{\mathbb{T}}^{d}_{N}}\overset{\xleftarrow{\phantom{\vartheta^{+}}}}{\vartheta^{+}}_{i}\overset{\xrightarrow{\phantom{\omega^{+}}}}{\omega^{+}}_{i}.$
(4.41)

Then Theorem 4.5 is proved with the following two estimates.

######  Lemma 4.6.

Suppose that $|||u|||_{\infty}<1$ and let $\varepsilon=1-|||u|||_{\infty}$.
Then for any $\ell>\ell_{\mathcal{V}}$

$\int V^{+,\ell}f\upsilon_{u}({\mathrm{d}}\eta)\leqslant
C\varepsilon^{-1}\ell_{\mathcal{V}}^{d/2}\left({\mathcal{H}}(f\upsilon_{u}\,|\,\upsilon_{u})+\frac{N^{d}}{\ell^{d}}\right).$
(4.42)

The constant $C$ depends only on the dimension, namely it can be taken as
$C=3^{d}2^{d/2+2}$.

######  Lemma 4.7.

Suppose $\varepsilon=1-|||u|||_{\infty}>0$ and $\|\nabla
u(i)\|_{\infty}\leq\frac{C_{0}}{N}$ for some $C_{0}>0$ independent of $N$,
where $\nabla u(i)=(\nabla_{k}u(i))_{k=1}^{d}$. Then, for any
$\ell>\ell_{\mathcal{V}}$, with $\ell=N^{\kappa}$ for some $\kappa>0$, and
$\delta>0$

$\int(V^{+}-V^{+,\ell})f\upsilon_{u}({\mathrm{d}}\eta)\leqslant\delta
N^{2}\int\Gamma_{N}^{\mathcal{K}}(\sqrt{f})\,\upsilon_{u}({\mathrm{d}}\eta)\\\
+C_{1}\ell_{\mathcal{V}}^{d}\frac{g_{d}(\ell)\ell^{d}}{N}\Bigg{(}\Big{(}2+\frac{\ell_{\mathcal{V}}}{\ell}\Big{)}^{d}{\mathcal{H}}\big{(}f\upsilon_{u}\,\big{|}\,\upsilon_{u}\big{)}+\frac{N^{d}}{\ell^{d}}\Bigg{)}+C_{2}N^{d-1}$
(4.43)

where
$C_{2}=\frac{3C_{0}}{\varepsilon^{4}}\left(1+\frac{2C_{0}}{N\varepsilon^{2}}\right)$
and
$C_{1}=3^{d}d(d+1)\left(\frac{2}{\delta\varepsilon^{2}}\left(1+\frac{C_{0}}{N\varepsilon^{2}}\right)+C_{2}\right)$
depends only on the dimension and $g_{d}(\ell)$ is defined in (4.47).

Indeed, we choose $\ell$ such that $\frac{\ell^{d}g_{d}(\ell)}{N}\leq C_{0}$:
more precisely, for any $\delta_{0}\in(0,1)$ we take

$\displaystyle\ell=N^{\frac{1}{2}(1-\delta_{0})}\quad\text{for}\quad
d=1,\qquad\text{and}\qquad\ell=N^{\frac{1}{d}(1-\delta_{0})}\quad\text{for}\quad
d\geq 2.$ (4.44)

Since $\ell_{\mathcal{V}}^{d}$ has a $\log$-growth (cf. Assumption 1), this
choice of $\ell$ together with Lemmas 4.6 and 4.7 concludes the proof of
Theorem 4.5. ∎

We now prove Lemma 4.6 and Lemma 4.7.

###### Proof of Lemma 4.6.

We start by recalling that
$V^{+,\ell}=\sum_{i\in{\mathbb{T}}^{d}_{N}}\overset{\xleftarrow{\phantom{\vartheta^{+}}}}{\vartheta^{+}}_{i}\overset{\xrightarrow{\phantom{\omega^{+}}}}{\omega^{+}}_{i}$.
Note that, under $\upsilon_{u}$, using Lemma A.3,
$\overset{\xrightarrow{\phantom{\omega^{+}}}}{\omega^{+}}_{i}$ is a sub-
Gaussian variable with variance parameter
$\sum_{i\in\Lambda_{\ell}}\frac{1}{4u_{i}^{2}\ell^{2d}}\leqslant\frac{1}{4\varepsilon^{2}\ell^{d}}.$

Since $c_{i}^{+}(\eta)$ is a function of
$(\eta_{i+j})_{j\in{\mathcal{V}}_{N}}$, then for $i$ and $j$ such that
$|i-j|_{\infty}>\ell_{\mathcal{V}}$, $\vartheta^{+}_{i}$ and
$\vartheta^{+}_{j}$ are independent, and sub-Gaussian with variance parameter
$\frac{1}{4}$. Using Lemma A.3,
$\overset{\xleftarrow{\phantom{\vartheta^{+}}}}{\vartheta^{+}}_{i}$ is a sub-
Gaussian variable with variance parameter
$\frac{\ell_{\mathcal{V}}^{d}}{4\ell^{d}}\left(1+\frac{\ell_{\mathcal{V}}}{\ell}\right)^{d}\leqslant
2^{d-2}\frac{\ell_{\mathcal{V}}^{d}}{\ell^{d}}.$

We note that all the sites involved in the averages
$\overset{\xleftarrow{\phantom{\vartheta^{+}}}}{\vartheta^{+}}_{i}\overset{\xrightarrow{\phantom{\omega^{+}}}}{\omega^{+}}_{i}$
are in $i+Q_{\ell_{\mathcal{V}}/2+\ell}$, where $Q_{m}=\\{-m,\ldots,m\\}^{d}$
is the $d$-dimensional cube centered at $0$. In particular, for $i$ and $j$
such that $|i-j|_{\infty}>\ell_{\mathcal{V}}+2\ell$, the corresponding
averages are independent (under $\upsilon_{u}$). Then, we can take a partition
of ${\mathbb{T}}^{d}_{N}$ into independent sites by letting
$i=j+(\ell_{\mathcal{V}}+2\ell)k$ where
$j\in\Lambda_{\ell_{\mathcal{V}}+2\ell}$ and $k\in\Lambda_{\lceil
N/(\ell_{\mathcal{V}}+2\ell)\rceil}$. The entropy inequality (cf. (B.3) in
[15]) gives

$\displaystyle\int V^{+,\ell}f\upsilon_{u}({\mathrm{d}}\eta)$
$\displaystyle=\sum_{j\in\Lambda_{\ell_{\mathcal{V}}+2\ell}}\int\sum_{k\in\Lambda_{\lceil
N/(\ell_{\mathcal{V}}+2\ell)\rceil}}\overset{\xleftarrow{\phantom{\vartheta^{+}}}}{\vartheta^{+}}_{j+(\ell_{\mathcal{V}}+2\ell)k}\overset{\xrightarrow{\phantom{\omega^{+}}}}{\omega^{+}}_{j+(\ell_{\mathcal{V}}+2\ell)k}f\upsilon_{u}({\mathrm{d}}\eta)$
$\displaystyle\leqslant\sum_{j\in\Lambda_{\ell_{\mathcal{V}}+2\ell}}\frac{1}{\gamma}\left({\mathcal{H}}(f\upsilon_{u}\,|\,\upsilon_{u})+\sum_{k\in\Lambda_{\lceil
N/(\ell_{\mathcal{V}}+2\ell)\rceil}}\log\int\exp\Bigg{\\{}\gamma\overset{\xleftarrow{\phantom{\vartheta^{+}}}}{\vartheta^{+}}_{j+(\ell_{\mathcal{V}}+2\ell)k}\overset{\xrightarrow{\phantom{\omega^{+}}}}{\omega^{+}}_{j+(\ell_{\mathcal{V}}+2\ell)k}\Bigg{\\}}\upsilon_{u}({\mathrm{d}}\eta)\right)$
$\displaystyle\leqslant\frac{(\ell_{\mathcal{V}}+2\ell)^{d}}{\gamma}{\mathcal{H}}(f\upsilon_{u}\,|\,\upsilon_{u})+\frac{1}{\gamma}\sum_{i\in{\mathbb{T}}^{d}_{N}}\log\int\exp\Big{\\{}\gamma\overset{\xleftarrow{\phantom{\vartheta^{+}}}}{\vartheta^{+}}_{i}\overset{\xrightarrow{\phantom{\omega^{+}}}}{\omega^{+}}_{i}\Big{\\}}\upsilon_{u}({\mathrm{d}}\eta)$

According to Lemma A.2, for
$\gamma^{-1}=2^{d/2+2}\ell^{-d}\varepsilon^{-1}\ell_{\mathcal{V}}^{d/2}$, we
have
$\log\displaystyle\int\exp\Big{\\{}\gamma\overset{\xleftarrow{\phantom{\vartheta^{+}}}}{\vartheta^{+}}_{i}\overset{\xrightarrow{\phantom{\omega^{+}}}}{\omega^{+}}_{i}\Big{\\}}\upsilon_{u}({\mathrm{d}}\eta)\leqslant\log
3.$ Thus we obtain

$\int V^{+,\ell}f\upsilon_{u}({\mathrm{d}}\eta)\leqslant
2^{d/2+2}\varepsilon^{-1}\ell_{\mathcal{V}}^{d/2}\left(\left(2+\frac{\ell_{\mathcal{V}}}{\ell}\right)^{d}{\mathcal{H}}(f\upsilon_{u}\,|\,\upsilon_{u})+\frac{N^{d}}{\ell^{d}}\log
3\right).$

∎

###### Proof of Lemma 4.7.

We first prove the Lemma by also assuming that
${\mathcal{V}}_{N}\subset\mathbb{Z}^{d}\setminus\mathbb{N}^{d}$. In Remark 4.3
at the end of the proof we show how to remove this assumption in dimension
$d>1$.

We then proceed as in Jara and Menezes [15]. We use the fact that

$\displaystyle V^{+}-V^{+,\ell}$
$\displaystyle=\sum_{i\in{\mathbb{T}}^{d}_{N}}\vartheta^{+}_{i}(\omega^{+}_{i}-\omega^{+,\ell}_{i})$
$\displaystyle=\sum_{i,j\in{\mathbb{T}}^{d}_{N}}\vartheta^{+}_{i}\omega^{+}_{i+j}(\mathbbm{1}_{\\{0\\}}(j)-q_{\ell}(j)).$
(4.45)

We now use Lemma 3.2 in [15] stating that there exists a function
$\Phi_{\ell}:\mathbb{Z}^{d}\times\mathbb{Z}^{d}\to{\mathbb{R}}$ which is a
flow connecting the distribution $\mathbbm{1}_{\\{0\\}}$ to $q_{\ell}$, i.e :

  * •

$\Phi_{\ell}(i,j)=-\Phi_{\ell}(j,i)$

  * •

$\sum_{j\colon|i-j|=1}\Phi_{\ell}(i,j)=\mathbbm{1}_{\\{0\\}}(i)-q_{\ell}(i)$

  * •

$\Phi_{\ell}(i,j)=0$ for $i,j\notin\Lambda_{2\ell-1}$

  * •

there is a constant $C=C(d)$ independent of $\ell$ such that

$\displaystyle\sum_{|i-j|=1}|\Phi_{\ell}(i,j)|^{2}<Cg_{d}(\ell)\qquad\text{and}\qquad\sum_{|i-j|=1}|\Phi_{\ell}(i,j)|<C\ell$
(4.46)

where

$g_{d}(\ell)=\left\\{\begin{aligned} \ell&\text{ for $d=1$}\\\
\log(\ell)&\text{ for $d=2$}\\\ 1&\text{ for $d\geqslant
3$}\end{aligned}\right.$ (4.47)

Using the flow $\Phi_{\ell}$ we therefore have

$\displaystyle V^{+}-V^{+,\ell}$
$\displaystyle=\sum_{i,j\in{\mathbb{T}}^{d}_{N}}\vartheta^{+}_{i}\omega^{+}_{i+j}\sum_{k\colon|j-k|=1}\Phi_{\ell}(j,k)$
$\displaystyle=\sum_{i,j\in{\mathbb{T}}^{d}_{N}}\vartheta^{+}_{i}\omega^{+}_{i+j}\sum_{k=1}^{d}\Big{\\{}\Phi_{\ell}(j,j+e_{k})-\Phi_{\ell}(j-e_{k},j)\Big{\\}}$
$\displaystyle=\sum_{k=1}^{d}\sum_{i,j\in{\mathbb{T}}^{d}_{N}}\vartheta^{+}_{i}\Phi_{\ell}(j,j+e_{k})(\omega^{+}_{i+j}-\omega^{+}_{i+j+e_{k}})$
$\displaystyle=\sum_{k=1}^{d}\sum_{i,j\in{\mathbb{T}}^{d}_{N}}\vartheta^{+}_{i-j}\Phi_{\ell}(j,j+e_{k})(\omega^{+}_{i}-\omega^{+}_{i+e_{k}})$
$\displaystyle=\sum_{k=1}^{d}\sum_{i\in{\mathbb{T}}^{d}_{N}}h^{k}_{i}(\omega^{+}_{i}-\omega^{+}_{i+e_{k}})$
(4.48)

where

$h_{i}^{k}=h_{i}^{\ell,k}:=\sum_{j\in{\mathbb{T}}^{d}_{N}}\vartheta^{+}_{i-j}\Phi_{\ell}(j,j+e_{k})=\sum_{j\in\Lambda_{2\ell-1}}\vartheta^{+}_{i-j}\Phi_{\ell}(j,j+e_{k}).$
(4.49)

To complete the result we need to estimate (4.48). We apply Lemma 3.5 of [8].
Note that the hypothesis of Lemma 3.5 are satisfied in our case:
$u_{-}=\varepsilon$ and $u_{+}=1-\varepsilon$ and
$h_{i}^{k}(\eta^{i,i+e_{k}})=h_{i}^{k}(\eta)$ for any configuration $\eta$
inasmuch $h_{i}^{k}$ is only a function of the sites $\eta_{i-j+s}$, for
$s\in{\mathcal{V}}_{N}$, $j\in\Lambda_{2\ell-1}$, so that it does not depend
on $\eta_{i}$ and $\eta_{i+e_{k}}$ since
${\mathcal{V}}_{N}\subset{\mathbb{Z}}^{d}\setminus{\mathbb{N}}^{d}$. Moreover,
we observe that Lemmas 3.4 and 3.5 apply to our case replacing $\chi(u_{i})$
by $u_{i}$. Therefore, for any $\alpha^{\prime}>0$ we have that

$\displaystyle\int
h^{k}_{i}(\omega^{+}_{i}-\omega^{+}_{i+e_{k}})f\upsilon_{u}({\mathrm{d}}\eta)\leq\frac{\alpha^{\prime}}{2}\int\big{(}\sqrt{f(\eta^{i,i+e_{k}})}-\sqrt{f(\eta})\big{)}^{2}\upsilon_{u}({\mathrm{d}}\eta)+\frac{C_{1,\varepsilon}}{\alpha^{\prime}}\int(h_{i}^{k})^{2}f\upsilon_{u}({\mathrm{d}}\eta)+R_{i}^{k}.$
(4.50)

with
$C_{1,\varepsilon}=\frac{2}{\varepsilon^{2}}\left(1+\frac{C_{0}}{N\varepsilon^{2}}\right)$
and where the rest term $R_{i}^{k}$ is controlled as

$\displaystyle R_{i}^{k}\leq
C_{2,\varepsilon}\big{|}\,\nabla_{k}u_{i}\,\big{|}\int\big{|}h_{i}^{k}(\eta)\big{|}f\upsilon_{u}({\mathrm{d}}\eta)\,,$
(4.51)

with, for $\varepsilon<\frac{1}{2}$,
$C_{2,\varepsilon}=\frac{3}{\varepsilon^{4}}\left(1+\frac{2C_{0}}{N\varepsilon^{2}}\right)$.
We now take $\alpha^{\prime}=\delta N^{2}$, with $\delta>0$. We get (cf.
(4.37))

$\displaystyle\int(V^{+}-V^{+,\ell})f\upsilon_{u}({\mathrm{d}}\eta)$
$\displaystyle=\sum_{k=1}^{d}\sum_{i\in{\mathbb{T}}^{d}_{N}}\int
h^{k}_{i}(\omega^{+}_{i}-\omega^{+}_{i+e_{k}})f\upsilon_{u}({\mathrm{d}}\eta)$
$\displaystyle\leq\delta
N^{2}\int\Gamma_{N}^{\mathcal{K}}(\sqrt{f})\,\upsilon_{u}({\mathrm{d}}\eta)+\frac{C_{1,\varepsilon}}{\delta
N^{2}}\sum_{k=1}^{d}\sum_{i\in{\mathbb{T}}^{d}_{N}}\int(h_{i}^{k})^{2}f\upsilon_{u}({\mathrm{d}}\eta)+\sum_{k=1}^{d}\sum_{i\in{\mathbb{T}}^{d}_{N}}R_{i}^{k}.$
(4.52)

The first term is the same of (4.43), then to conclude the proof we have to
upper bound the second and the third term of (4.52).

Let us start from the third one. Using that
$\big{|}\,\nabla_{k}u_{i}\,\big{|}\leq\frac{C_{0}}{N}$ and $|h_{i}^{k}|\leq
1+(h_{i}^{k})^{2}$ in (4.51) we get

$\displaystyle
R_{i}^{k}\leq\frac{C_{0}C_{2,\varepsilon}}{N}\int\big{(}1+(h_{i}^{k})^{2}\big{)}f\upsilon_{u}({\mathrm{d}}\eta)\,.$

So that, the last term of (4.52) is bounded by

$C_{0}C_{2,\varepsilon}\left(N^{d-1}+\frac{1}{N}\sum_{k=1}^{d}\sum_{i\in{\mathbb{T}}^{d}_{N}}\int(h_{i}^{k})^{2}f\upsilon_{u}({\mathrm{d}}\eta)\right).$

Let us note that this last term dominates the second term of (4.52). To
conclude the proof we need to upper-bound

$\sum_{k=1}^{d}\sum_{i\in{\mathbb{T}}^{d}_{N}}\int(h_{i}^{k})^{2}f\upsilon_{u}({\mathrm{d}}\eta).$

As we noted above, $h_{i}^{k}$ depends only on the sites $\eta_{i-j+s}$, for
$s\in{\mathcal{V}}_{N}$ and $j\in\Lambda_{2\ell-1}$, so that the random
variables $h_{i}^{k}$ and $h_{i^{\prime}}^{k}$ are independent if
$|i-i^{\prime}|_{\infty}>(2\ell+\ell_{\mathcal{V}})$. We then decompose
${\mathbb{T}}^{d}_{N}$ as a disjoint union of cubes of size
$2\ell+\ell_{\mathcal{V}}$, that is, we write
$i=j+(\ell_{\mathcal{V}}+2\ell)z$ where
$j\in\Lambda_{\ell_{\mathcal{V}}+2\ell}$ and $z\in\Lambda_{\lceil
N/(\ell_{\mathcal{V}}+2\ell)\rceil}$,

$\displaystyle\sum_{i\in{\mathbb{T}}^{d}_{N}}\int(h_{i}^{k})^{2}f\upsilon_{u}({\mathrm{d}}\eta)=\sum_{j\in\Lambda_{2\ell+\ell_{\mathcal{V}}}}\sum_{z\in\Lambda_{\lceil
N/(\ell_{\mathcal{V}}+2\ell)\rceil}}\int(h_{j+z}^{k})^{2}f\upsilon_{u}({\mathrm{d}}\eta).$

We then apply the entropy inequality and we get

$\begin{split}\sum_{i\in{\mathbb{T}}^{d}_{N}}\int(h_{i}^{k})^{2}f\upsilon_{u}({\mathrm{d}}\eta)&\leq\frac{1}{\gamma}\sum_{j\in\Lambda_{2\ell+\ell_{\mathcal{V}}}}\left({\mathcal{H}}\big{(}f\upsilon_{u}\,\big{|}\,\upsilon_{u}\big{)}+\log\int\exp\Bigg{\\{}\gamma\sum_{z\in\Lambda_{\lceil
N/(\ell_{\mathcal{V}}+2\ell)\rceil}}(h_{j+z}^{k})^{2}\Bigg{\\}}\,\upsilon_{u}({\mathrm{d}}\eta)\right)\\\
&=\frac{(2\ell+\ell_{\mathcal{V}})^{d}}{\gamma}{\mathcal{H}}\big{(}f\upsilon_{u}\,\big{|}\,\upsilon_{u}\big{)}+\frac{1}{\gamma}\sum_{i\in{\mathbb{T}}^{d}_{N}}\log\int\exp\Big{\\{}\gamma(h_{i}^{k})^{2}\Big{\\}}\,\upsilon_{u}({\mathrm{d}}\eta)\end{split}$
(4.53)

To conclude we use a concentration inequality. We have that $h_{i}^{k}$ is
sub-Gaussian random variable, let $\sigma^{2}$ be its variance parameter. By
Proposition A.1 we have that for any $\gamma\leq\frac{1}{4\sigma^{2}}$,

$\int\exp\Big{\\{}\gamma(h_{i}^{k})^{2}\Big{\\}}\,\upsilon_{u}({\mathrm{d}}\eta)\leq\log
3\,.$

Moreover, to get an upper bound on the variance parameter, we use the same
decomposition of the sum (4.49) into subsets which are independent (since
$\vartheta_{j}^{+}$ is a function of $\eta_{i+j}$, for $i\in{\mathcal{V}}_{N}$
and is sub-Gaussian with variance parameter $\frac{1}{4}$). This is done in
Lemma F.12 in [15]. We then have that $\sigma^{2}\leq
C_{d}\ell_{\mathcal{V}}^{d}g_{d}(\ell)$, where $C_{d}$ is a constant which
depends only on the dimension. By getting $\gamma$ as large as possible,
namely $\gamma^{-1}=(d+1)\ell_{\mathcal{V}}^{d}\,g_{d}(\ell)$ we obtain that

$\displaystyle\sum_{i\in{\mathbb{T}}^{d}_{N}}\int(h_{i}^{k})^{2}f\upsilon_{u}({\mathrm{d}}\eta)\leq(d+1)\ell_{\mathcal{V}}^{d}\,g_{d}(\ell)\ell^{d}\Bigg{(}\Big{(}2+\frac{\ell_{\mathcal{V}}}{\ell}\Big{)}^{d}{\mathcal{H}}\big{(}f\upsilon_{u}\,\big{|}\,\upsilon_{u}\big{)}+\frac{N^{d}}{\ell^{d}}\log
3\Bigg{)}.$ (4.54)

To conclude the proof we have to remove the assumption
${\mathcal{V}}_{N}\subset{\mathbb{Z}}^{d}\setminus{\mathbb{N}}^{d}$. We show
that in Remark 4.3. ∎

######  Remark 4.3.

By (4.48) we recall that
$V^{+}-V^{+,\ell}=\sum_{k=1}^{d}\sum_{i\in{\mathbb{T}}^{d}_{N}}h^{k}_{i}(\omega^{+}_{i}-\omega^{+}_{i+e_{k}})$
where $h_{i}^{k}$ is defined in (4.49). We also observe that, by assumption,
$\ell>\ell_{\mathcal{V}}$. Since $h_{i}^{k}$ is a function of the sites
$\eta_{i-j+s}$, for $s\in{\mathcal{V}}_{N}$, $j\in\Lambda_{2\ell-1}$, whenever
$|j|>\ell_{\mathcal{V}}$, then $\vartheta^{+}_{i-j}$ does not depend on
$\eta_{i}$ and $\eta_{i+e_{k}}$. Therefore we split $h_{i}^{k}$ into the sum
of two functions $h_{i}^{\prime}$ and $h_{i}^{\second}$. The first one is
independent of $\eta_{i}$ and $\eta_{i+e_{k}}$, while the second one depends
on these sites,

$\displaystyle
h_{i}^{\prime}:=\sum_{\begin{subarray}{c}j\in\Lambda_{2\ell-1}\\\
|j|>2\ell_{\mathcal{V}}\end{subarray}}\vartheta^{+}_{i-j}\Phi_{\ell}(j,j+e_{k}),\qquad\text{and}\qquad
h_{i}^{\second}:=\sum_{\begin{subarray}{c}j\in\Lambda_{2\ell-1}\\\ |j|\leq
2\ell_{\mathcal{V}}\end{subarray}}\vartheta^{+}_{i-j}\Phi_{\ell}(j,j+e_{k}).$
(4.55)

In such a way

$\displaystyle V^{+}-V^{+,\ell}$
$\displaystyle=(V^{+}-V^{+,\ell})^{\prime}+(V^{+}-V^{+,\ell})^{\second},$
(4.56)

where

$\displaystyle(V^{+}-V^{+,\ell})^{\prime}:=\sum_{k=1}^{d}\sum_{i\in{\mathbb{T}}^{d}_{N}}h^{\prime}_{i}(\omega^{+}_{i}-\omega^{+}_{i+e_{k}})\qquad\text{and}\qquad(V^{+}-V^{+,\ell})^{\second}:=\sum_{k=1}^{d}\sum_{i\in{\mathbb{T}}^{d}_{N}}h^{\second}_{i}(\omega^{+}_{i}-\omega^{+}_{i+e_{k}})$
(4.57)

We can apply the method used above to $(V^{+}-V^{+,\ell})^{\prime}$ obtaining
that (4.54) holds. We control $(V^{+}-V^{+,\ell})^{\second}$ by showing that
$\int(V^{+}-V^{+,\ell})^{\second}f\upsilon_{u}({\mathrm{d}}\eta)=O(N^{d-\varepsilon})$,
for some $\varepsilon>0$. For this purpose we apply Cauchy-Swartz inequality
to the measure $f\upsilon_{u}({\mathrm{d}}\eta)$, which gives

$\displaystyle\int
h^{\second}_{i}(\omega^{+}_{i}-\omega^{+}_{i+e_{k}})f\upsilon_{u}({\mathrm{d}}\eta)$
$\displaystyle\leq\Big{(}\int(h^{\second}_{i})^{2}f\upsilon_{u}({\mathrm{d}}\eta)\Big{)}^{\frac{1}{2}}\Big{(}\int(\omega^{+}_{i}-\omega^{+}_{i+e_{k}})^{2}f\upsilon_{u}({\mathrm{d}}\eta)\Big{)}^{\frac{1}{2}}$
(4.58)

Therefore

$\displaystyle\int(V^{+}-V^{+,\ell})^{\second}f\upsilon_{u}({\mathrm{d}}\eta)$
$\displaystyle=\sum_{k=1}^{d}\sum_{i\in{\mathbb{T}}^{d}_{N}}\int
h^{\second}_{i}(\omega^{+}_{i}-\omega^{+}_{i+e_{k}})f\upsilon_{u}({\mathrm{d}}\eta)$
$\displaystyle\leq\sum_{k=1}^{d}\sum_{i\in{\mathbb{T}}^{d}_{N}}\Big{(}\int(h^{\second}_{i})^{2}f\upsilon_{u}({\mathrm{d}}\eta)\Big{)}^{\frac{1}{2}}\Big{(}\int(\omega^{+}_{i}-\omega^{+}_{i+e_{k}})^{2}f\upsilon_{u}({\mathrm{d}}\eta)\Big{)}^{\frac{1}{2}}$
$\displaystyle\leq\left(\sum_{k=1}^{d}\sum_{i\in{\mathbb{T}}^{d}_{N}}\Big{(}\int(h^{\second}_{i})^{2}f\upsilon_{u}({\mathrm{d}}\eta)\Big{)}\sum_{k=1}^{d}\sum_{i\in{\mathbb{T}}^{d}_{N}}\Big{(}\int(\omega^{+}_{i}-\omega^{+}_{i+e_{k}})^{2}f\upsilon_{u}({\mathrm{d}}\eta)\Big{)}\right)^{\frac{1}{2}},$
(4.59)

where in the last inequality we used again Cauchy-Swartz inequality.

We observe that
$(\omega^{+}_{i}-\omega^{+}_{i+e_{k}})=(\frac{\eta_{i}}{u_{i}}-\frac{\eta_{i+e_{k}}}{u_{i+e_{k}}})=\eta_{i}(\frac{1}{u_{i}}-\frac{1}{u_{i+e_{k}}})+\frac{1}{u_{i+e_{k}}}(\eta_{i}-\eta_{i+e_{k}}),$
so that

$\displaystyle\int(\omega^{+}_{i}-\omega^{+}_{i+e_{k}})^{2}f\upsilon_{u}({\mathrm{d}}\eta)$
$\displaystyle\leq
2\int\eta_{i}\Big{(}\frac{1}{u_{i}}-\frac{1}{u_{i+e_{k}}}\Big{)}^{2}f\upsilon_{u}({\mathrm{d}}\eta)+2\int\frac{1}{u_{i+e_{k}}^{2}}(\eta_{i}-\eta_{i+e_{k}})^{2}f\upsilon_{u}({\mathrm{d}}\eta)$
$\displaystyle\leq C_{\varepsilon}|\nabla
u_{k}|_{\infty}^{2}+C_{\varepsilon}\int(\eta_{i}-\eta_{i+e_{k}})^{2}f\upsilon_{u}({\mathrm{d}}\eta)\leq
C^{\prime}_{\varepsilon},$

uniformly on $i$ and $N$. So that,

$\sum_{k=1}^{d}\sum_{i\in{\mathbb{T}}^{d}_{N}}\Big{(}\int(\omega^{+}_{i}-\omega^{+}_{i+e_{k}})^{2}f\upsilon_{u}({\mathrm{d}}\eta)\Big{)}\leq
C^{\prime}_{\varepsilon}N^{d}.$ (4.60)

Therefore, by (4.59), to conclude the proof it is enough to show that

$\sum_{k=1}^{d}\sum_{i\in{\mathbb{T}}^{d}_{N}}\Big{(}\int(h^{\second}_{i})^{2}f\upsilon_{u}({\mathrm{d}}\eta)\Big{)}=O(N^{d-\varepsilon}),$
(4.61)

for some $\varepsilon>0$ small. For this purpose we have to look more
carefully at the function $\Phi_{\ell}(j,j+e_{k})$ which defines the flow. We
recall (cf. Appendix G of [15]) that in the construction of the flow
connecting $\mathbbm{1}_{\\{0\\}}$ and $p_{\ell}*p_{\ell}$, we first define a
flow $\Psi_{\ell}$ connecting $\mathbbm{1}_{\\{0\\}}$ and $p_{\ell}$ supported
in $\Lambda_{\ell}$ which satisfies (4.46) and then we define

$\displaystyle\Phi_{\ell}(j,j+e_{k}):=\sum_{i\in{\mathbb{T}}^{d}_{N}}\Psi_{\ell}(i,i+e_{k})p_{\ell}(j-i),$

therefore

$\displaystyle\sum_{\begin{subarray}{c}j\in\Lambda_{2\ell-1}\\\ |j|\leq
2\ell_{\mathcal{V}}\end{subarray}}\big{|}\Phi_{\ell}(j,j+e_{k})\big{|}$
$\displaystyle\leq\sum_{\begin{subarray}{c}j\in\Lambda_{2\ell-1}\\\ |j|\leq
2\ell_{\mathcal{V}}\end{subarray}}\sum_{i\in{\mathbb{T}}^{d}_{N}}\big{|}\Psi_{\ell}(i,i+e_{k})\big{|}p_{\ell}(j-i)$
$\displaystyle\leq\sum_{i\in{\mathbb{T}}^{d}_{N}}\big{|}\Psi_{\ell}(i,i+e_{k})\big{|}\sum_{\begin{subarray}{c}j\in\Lambda_{2\ell-1}\\\
|j|\leq 2\ell_{\mathcal{V}}\end{subarray}}p_{\ell}(j-i)\leq
C\ell\frac{\ell_{\mathcal{V}}^{d}}{\ell^{d}},$

where we used that $\sum_{\begin{subarray}{c}j\in\Lambda_{2\ell-1}\\\ |j|\leq
2\ell_{\mathcal{V}}\end{subarray}}p_{\ell}(j-i)\leq\frac{\ell_{\mathcal{V}}^{d}}{\ell^{d}}$
uniformly on $i$ and (4.46) applied to $\Psi_{\ell}$. We deduce that, since
$|\vartheta^{+}_{i-j}|\leq 2$,

$\displaystyle\sum_{k=1}^{d}\sum_{i\in{\mathbb{T}}^{d}_{N}}\Big{(}\int(h^{\second}_{i})^{2}f\upsilon_{u}({\mathrm{d}}\eta)\Big{)}\leq
CN^{d}\frac{\ell_{\mathcal{V}}^{2d}}{\ell^{2(d-1)}}.$

Since $\ell=N^{\kappa}$ for some $\kappa>0$ and $\ell_{\mathcal{V}}^{d}$ grows
logarithmically, the result follows (in dimension $d\geq 2$).

####  4.1.3 Estimate of $V$

We show that under the hypothesis of Theorem 4.5, $V(u,\eta)$ satisfies
(4.38). For $i\in{\mathbb{T}}^{d}_{N}$ and $k\in\\{1,\ldots,d\\}$ we let

$\displaystyle\widetilde{\omega}_{i}^{k}:=-\alpha
N^{2}(u_{i}-u_{i+e_{k}})^{2}\omega_{i}.$

In such a way we get

$\displaystyle V(u,\eta)$ $\displaystyle=-\alpha
N^{2}\sum_{\begin{subarray}{c}i,j\in{\mathbb{T}}^{d}_{N}\\\
|i-j|=1\end{subarray}}(u_{i}-u_{j})^{2}\omega_{i}\omega_{j}$
$\displaystyle=-\alpha
N^{2}\sum_{k=1}^{d}\sum_{i\in{\mathbb{T}}^{d}_{N}}\Bigg{\\{}(u_{i}-u_{i+e_{k}})^{2}\omega_{i}\omega_{i+e_{k}}+(u_{i}-u_{i-e_{k}})^{2}\omega_{i}\omega_{i-e_{k}}\Bigg{\\}}$
$\displaystyle=\sum_{k=1}^{d}\sum_{i\in{\mathbb{T}}^{d}_{N}}\Bigg{\\{}\widetilde{\omega}_{i}^{k}\omega_{i+e_{k}}+\widetilde{\omega}_{i-e_{k}}^{k}\omega_{i}\Bigg{\\}}=2\sum_{k=1}^{d}\sum_{i\in{\mathbb{T}}^{d}_{N}}\widetilde{\omega}_{i-e_{k}}^{k}\omega_{i}.$

For $k\in\\{1,\ldots,d\\}$ we let
$\widetilde{V}^{k}:=\sum_{i\in{\mathbb{T}}^{d}_{N}}\widetilde{\omega}_{i-e_{k}}^{k}\omega_{i}$
and
$\widetilde{V}^{k,\ell}:=\sum_{i\in{\mathbb{T}}^{d}_{N}}\overset{\xleftarrow{\phantom{\widetilde{\omega}_{i-e_{k}}^{k}}}}{\widetilde{\omega}_{i-e_{k}}^{k}}\overset{\xrightarrow{\phantom{\omega_{i}}}}{\omega_{i}}$.
We have that $\widetilde{V}^{k,\ell}$ and
$\widetilde{V}^{k}-\widetilde{V}^{k,\ell}$ satisfy (4.42) and (4.43)
respectively, which implies that $V(u,\eta)$ satisfies (4.38). The proof
follows the same ideas of $V^{+}$ and $V^{-}$ and it is actually simpler since
we do not have to deal with the diameter of ${\mathcal{V}}_{N}$. We omit the
details.

####  4.1.4 Conclusion: Gronwall’s inequality (4.13)

We observe that since
$\Gamma_{N}^{\mathcal{K}}(\sqrt{f})\leq\Gamma_{N}(\sqrt{f})$, cf. (4.36) and
that the carré du champ operator is non-negative, (4.13) is a consequence of
Proposition 4.3 and Theorem 4.5 with $u(t)=u^{N}(t)$, the solutions of (4.10),
$f=\frac{{\mathrm{d}}\mu_{t}}{{\mathrm{d}}\upsilon_{u(t)}}$ if we show that
$u^{N}$ satisfies Assumption 2 for any $t\in[0,\tau]$. This is one of the goal
of the Section 5, see Propositions 5.2 and 5.3.

##  5 Estimates on the solutions $(u^{N})$ of (4.10)

Note that (4.10) is a first order ordinary differential equation in
${\mathbb{R}}^{{\mathbb{T}}^{d}_{N}}$. By standard theory, we have a solution,
locally in time, starting from every initial condition.

We say that $u$ is supersolution of (4.10) if $\partial_{t}u_{i}\geqslant
2\alpha N^{2}(\Delta u)_{i}+\beta(1-2u_{i})+G(i,u)$ and that it is a
subsolution if $\partial_{t}u_{i}\leqslant 2\alpha N^{2}(\Delta
u)_{i}+\beta(1-2u_{i})+G(i,u)$.

Note that any solution is both a super- and a subsolution. We have a
comparaison lemma between super and subsolutions.

######  Proposition 5.1.

Let $u$ be a supersolution, and $v$ be a subsolution such that
$u(0,i)\geqslant v(0,i)$ for all $i\in{\mathbb{T}}^{d}_{N}$. Then, for all
$t\geqslant 0$ and all $i\in{\mathbb{T}}^{d}_{N}$, $u(t,i)\geqslant v(t,i)$.

###### Proof.

Since $u$ and $v$ have derivative in time they are continuous, consider $i$
and $t$ such that $u(t,i)=v(t,i)$ and $u(t,j)\geqslant v(t,j)$. Then, we have
$G(i,u(t))\geqslant G(i,v(t))$ by Proposition B.1, and thus

$\displaystyle\partial_{t}(u(i,t)-v(i,t))$ $\displaystyle=2\alpha
N^{2}\sum_{j,|i-j|=1}(u(j,t)-v(j,t))+G(i,u(t))-G(i,v(t))\geqslant 0.$ (5.1)

This proves that $u$ stays above $v$ at all time. ∎

From the two previous propositions we conclude :

######  Proposition 5.2.

Let $\delta=\frac{\beta}{1+4\beta}$ and suppose $0<\delta<\frac{1}{4e}$. Let
also $4\delta\leqslant T\leqslant 1-4\delta$ and
$0\leqslant\varepsilon<2\delta$. For all
$K_{N}>\frac{|\log(\varepsilon/2)|}{\varepsilon^{2}}$, and all $N$, consider
the solution $(u^{N}(t,i))_{i\in{\mathbb{T}}^{d}_{N}}$ of Equation (4.10)
starting from $u_{0}\in[\varepsilon,1-\varepsilon]$, then we have that
$u^{N}(t,i)\in[\varepsilon,1-\varepsilon]$ for all $i$ and $t\geq 0$.

Note that the proposition holds for $\varepsilon=0$.

###### Proof.

Let $p\in[0,1]$, and set $u(i)=p$, for all $i$. Using that
$g_{K_{N}}(p)=G(i,p)$, we have that

  * •

if $g_{K_{N}}(p)+\beta(1-2p)\geqslant 0$, then $u$ is a subsolution;

  * •

if $g_{K_{N}}(p)+\beta(1-2p)\leqslant 0$, then $u$ is a supersolution.

Then, by using the result of Proposition B.2 on the analysis of $g_{K}(p)$
close to $p=0$ and $p=1$, it is easy to check that $u(i)=\varepsilon$ is a
subsolution and $u(i)=1-\varepsilon$ is supersolution, for $\varepsilon$
satisfying the hypothesis of the proposition.

In particular, for $p=1$, $g_{K_{N}}(1)=0$ , we have a supersolution. For
$p=0$, $g_{K_{N}}(0)=0$, we have a subsolution. ∎

In the next result, we show that if $u$ solves (4.10) and $\|\nabla
u(0,i)\|_{\infty}\leq\frac{C}{N}$, then $\|\nabla
u(t,i)\|_{\infty}\leq\frac{C\sqrt{t}}{N}$ for any $t>0$.

######  Proposition 5.3.

Let $u$ be a solution of (4.10) with $u(0)=u_{0}$ such that, there exists
$C_{0}>0$ for which $|\nabla u_{0}|_{\infty}\leqslant\frac{C_{0}}{N}$. Then,
there exists $C>0$ such that $\|\nabla
u(t,i)\|_{\infty}\leq\frac{C_{0}+C\sqrt{t}}{N}$ for all $t\geqslant 0$.

To prove Proposition 5.3 we follow [8] and the reference therein, in
particular [7]. Using the notation of [7] we let $p(t,x,z)$ be the heat kernel
of discrete Laplacian

$\Delta
u(t,i)=\sum\limits_{j\in\mathbb{Z}^{d}}a(t,i,i+j)\big{[}u(t,i+j)-u(t,i)\big{]},$
(5.2)

with $a(t,i,j)=\mathbbm{1}_{\\{j-i\in\Gamma\\}}$ and $\Gamma=\\{\pm
e_{i},\,i=1,\dots,d\\}$, that is,

$\begin{cases}p(\cdot,i_{0},i)=\delta_{i_{0}}(\cdot),\\\
\partial_{t}p(t,i_{0},i)=\Delta p(t,i_{0},i).\end{cases}$

In this case (comments below (1.2) of [7]) we have that
$a^{*}(t,i,j)=a(t,i,j)$ so that $p^{*}(t,i,j)=p(t,i,j)$. Let
$\nabla_{k}u(t,i)=u(t,i+e_{k})-u(t,i)$, then since $p$ is a uniform
transitions function, there exist $c,C>0$ independent of $t,k$ such that (cf.
(1.3) of [7])

$\big{|}\nabla_{k}p(t,0,i)\big{|}\leq C\frac{p(ct,0,i)}{\sqrt{1\vee t}}.$
(5.3)

We refer to [7], Section 4 and the reference therein for a proof. We stress
that the delicate point is to extend the classical theory of E. De Giorgi, J.
Nash and J. Moser to discrete operators. The authors follow mainly [6], but
similar results can be also found in [10] (Appendix B) and [23].

###### Proof of Proposition 5.3.

Using Duhamel’s formula (i.e., variation of constant) we get that

$u(t,i)=\sum_{j\in{\mathbb{T}}^{d}_{N}}u(0,j)p_{N}(t,i,j)+\int_{0}^{t}{\mathrm{d}}s\sum_{j\in{\mathbb{T}}^{d}_{N}}(\beta(1-2u_{j})+G_{j}(u))p_{N}(t-s,i,j),$
(5.4)

where $p_{N}(t,i,j)=\sum_{z\in N\mathbb{Z}^{d}}p(2\alpha N^{2}t,i,j+z)$ is the
heat kernel of discrete Laplacian on the torus speeds up by a factor $2\alpha
N^{2}$ and $p(t,i,j)$ is the heat kernel introduced above. We observe that
(5.3) gives

$\big{|}\nabla_{k}p_{N}(t,0,i)\big{|}\leq\frac{C}{N}\frac{p_{N}(ct,0,i)}{\sqrt{t}}.$
(5.5)

For the first term, using that $p_{N}(t,i,j)=p_{N}(t,i+z,j+z)$ for any
$i,j,z\in{\mathbb{T}}^{d}_{N}$ and the assumption on $\nabla u_{0}$ (cf.
Assumptions 2), an integration by parts gives

$\displaystyle\bigg{|}\nabla_{k}\bigg{\\{}\sum_{j\in{\mathbb{T}}^{d}_{N}}u(0,j)p_{N}(t,i,j)\bigg{\\}}\bigg{|}$
$\displaystyle=\bigg{|}\sum_{j\in{\mathbb{T}}^{d}_{N}}u(0,j)\big{(}p_{N}(t,i+e_{k},j)-p_{N}(t,i,j)\big{)}\bigg{|}$
$\displaystyle=\bigg{|}\sum_{j\in{\mathbb{T}}^{d}_{N}}u(0,j)p_{N}(t,i,j-e_{k})-\sum_{j\in{\mathbb{T}}^{d}_{N}}u(0,j)p_{N}(t,i,j)\bigg{|}$
$\displaystyle=\bigg{|}\sum_{j\in{\mathbb{T}}^{d}_{N}}\Big{(}u(0,j+e_{k})-u(0,j)\Big{)}p_{N}(t,i,j)\bigg{|}\leq\frac{C}{N}.$

By using that $(\beta(1-2u_{j})+G_{j}(u))$ is bounded and (5.5) we get that
for any $k=1,\dots,d$,

$\displaystyle\bigg{|}\nabla_{k}\bigg{\\{}\int_{0}^{t}{\mathrm{d}}s\sum_{j\in{\mathbb{T}}^{d}_{N}}(\beta(1-2u_{j})+G_{j}(u))p_{N}(t-s,i,j)\bigg{\\}}\bigg{|}\leq\frac{C}{N}\int_{0}^{t}\frac{1}{{\sqrt{t-s}}}{\mathrm{d}}s=\frac{C}{N}\sqrt{t}\,.$

∎

##  6 Existence and uniqueness of reaction-diffusion PDE

In Section 7, we prove that in the limit $N\to\infty$, the solution of the
discretized Equation 4.10, with
$u^{N}(t=0)=u^{N}_{0}\in[0,1]^{{\mathbb{T}}^{d}_{N}}$ satisfying Assumption 2,
converges to a solution of the scalar nonlinear reaction diffusion equation

  * •

when $K_{N}\to K$ as $N\to\infty$ :

$\partial_{t}u(t,x)=2\alpha\Delta u(t,x)+\beta(1-2u(t,x))+g_{K}(u(t,x)),$
(6.1)

  * •

when $K_{N}\to\infty$ as $N\to\infty$ :

$\partial_{t}u(t,x)=2\alpha\Delta u(t,x)+\beta(1-2u(t,x))+g_{\infty}(u(t,x)).$
(6.2)

The main difference between the two equations is that in the first case
($K<+\infty$), $g_{K}$ is a $C^{1}$ function on $[0,1]$ (thus Lipschitz) so
the reaction diffusion equation (6.1) is very classical, whereas for the
second case ($K=+\infty$), since $g_{\infty}$ is not even continuous we need
to consider (6.2) as a subdifferential inclusion.

The main results of this section are

  1. (i)

Proposition 6.3 which proves existence of a solution, in a suitable sense, for
the equation (6.2),

  2. (ii)

Proposition 6.4 which proves local uniqueness of the solution, in a suitable
sense, for the equation (6.2), starting from a suitable class of initial
conditions,

  3. (iii)

and Theorem 7.1 which proves that all accumulation points of $(u^{N})_{N}$ is
a solution, in a suitable sense, for the equation (6.2).

In the rest of this section we change our notations and define $v=2u-1$. We
center the solution around the constant steady state $u=\frac{1}{2}$. It
simplifies the presentation and proofs of our results. The original form of
our equations can be retrieved by letting $u=\frac{1}{2}(v+1)$. In such a way
(6.1) takes the form

$\partial_{t}v(t,x)-2\alpha\Delta v(t,x)+2\beta
v-2g_{K}\left(\frac{1}{2}(v+1)\right)=0.$ (6.3)

###  6.1 Solution of (6.1)

Let us denote $s(t,x,y)$ the semigroup of the operator $\frac{1}{2}\Delta$ on
${\mathbb{T}}^{d}$, that is,

$s(t,x,y)=\frac{1}{(2\pi
t)^{d/2}}\sum_{k\in\mathbb{Z}^{d}}\exp\left(-\frac{\|x-y-k\|^{2}}{2t}\right).$
(6.4)

Denote also $s_{0}(t,x,y)$ the semigroup of the operator $\frac{1}{2}\Delta$
on $\mathbb{R}^{d}$,

$s_{0}(t,x,y)=\frac{1}{(2\pi
t)^{d/2}}\exp\left(-\frac{\|x-y\|^{2}}{2t}\right).$ (6.5)

Note that $\xi\mapsto s_{0}(t,0,\xi)$ is the density of $d$ independent normal
random variables with variance $t$.

Let us consider $(S^{\lambda,\gamma}_{t})$ the semigroup on
$L^{1}({\mathbb{T}}^{d})$ defined by, for $f\in L^{1}({\mathbb{T}}^{d})$,
$\lambda\geqslant 0$ and $\gamma>0$

$S^{\lambda,\gamma}_{t}f(x)=\int_{{\mathbb{T}}^{d}}e^{-\lambda t}s(\gamma
t,x,y)f(y){\mathrm{d}}y=\int_{{\mathbb{R}}^{d}}e^{-\lambda t}s_{0}(\gamma
t,x,y)\widetilde{f}(y){\mathrm{d}}y,$ (6.6)

where for a measurable function $f$ on $\mathbb{T}^{d}$, we denoted
$\widetilde{f}$ its extension on $\mathbb{R}^{d}$ defined by
$\widetilde{f}(x)=f(x-\lfloor x\rfloor)$.

Another way to define $S^{\lambda,\gamma}_{t}$ is to use the Brownian motion:
denote by $X$ a Brownian motion on $\mathbb{R}^{d}$ starting from $x$ on some
probability space $(\Omega,{\mathcal{F}},\mathbb{P}_{x})$, indeed we have
$S^{\lambda,\gamma}_{t}f(x)=e^{-\lambda
t}\mathbb{E}_{x}(\widetilde{f}(X_{\gamma t}))$, and for all $\lambda\geqslant
0$, and $\gamma>0$, $S^{\lambda,\gamma}$ is a $C_{0}$-contraction semigroup on
$L^{p}({\mathbb{T}}^{d})$ for $p\in[1,+\infty]$.

As we will look at (6.1) in its mild form, the following result is crucial to
study the regularity of the solution.

######  Proposition 6.1.

For $v_{0}\in L^{\infty}({\mathbb{T}}^{d})$ and $g\in
L^{\infty}([0,\tau]\times\mathbb{T}^{d})$, define

$v(t,x):=S^{\lambda,\gamma}_{t}v_{0}(x)+\int_{0}^{t}S^{\lambda,\gamma}_{t-s}(g(s,\cdot))(x){\mathrm{d}}s.$
(6.7)

then $v\in C([0,\tau],{\mathbb{T}}^{d})$.We have the following estimates, for
all $(t,x)\in\mathbb{R}^{+}\times{\mathbb{T}}^{d}$,

$\displaystyle|v(t,x)|$ $\displaystyle\leqslant e^{-\lambda
t}\|v_{0}\|_{\infty}+\frac{1}{\lambda}(1-e^{-\lambda
t})\|g\|_{\infty}\leqslant\|v_{0}\|_{\infty}+\frac{1}{\lambda}\|g\|_{\infty}$
(6.8)

and for all $\tau>0$, there exists a constant $C$ depending only on
$\tau,\gamma,\lambda$ and $d$, such that for all
$(t,x),(s,y)\in[1/\tau,\tau]\times\mathbb{T}^{d}$ with $s<t$

$\displaystyle|v(t,x)-v(s,y)|$ $\displaystyle\leqslant
C((t-s)|\log(t-s)|+\|x-y\|)(\|g\|_{\infty}+\|v_{0}\|_{\infty}).$ (6.9)

######  Remark 6.1.

$v$ is called a mild solution of the equation
$\partial_{t}v-\frac{\gamma}{2}\Delta v+\lambda v=g$ with initial value
$v_{0}$. The fact that a mild solution is a classical solution if $g$ is
sufficiently regular is a result from Pazy ([20], Corollary 4.2.5).

Estimates (6.8) and (6.9) are quite standard but we include the proof for the
sake of completeness.

###### Proof of Proposition 6.1.

The fact that $v\in C([0,\tau],L^{\infty}({\mathbb{T}}^{d}))$ is a consequence
of the fact that $S^{\lambda,\gamma}$ is a $C_{0}$ contraction semigroups on
$L^{\infty}$. For the first estimate (6.8), we have that :

$\displaystyle|v(t,x)|$
$\displaystyle\leqslant|S^{\lambda,\gamma}_{t}v_{0}(x)|+\left|\int_{0}^{t}S^{\lambda,\gamma}_{t-u}(g(u,\cdot))(x){\mathrm{d}}u\right|$
$\displaystyle\leqslant e^{-\lambda
t}\|v_{0}\|_{\infty}\int_{{\mathbb{R}}^{d}}s_{0}(\gamma
t,x,z){\mathrm{d}}z+\|g\|_{\infty}\int_{0}^{s}e^{-\lambda(t-u)}\int_{{\mathbb{R}}^{d}}s_{0}(\gamma(t-u),x,z){\mathrm{d}}z{\mathrm{d}}u$
$\displaystyle=e^{-\lambda
t}\|v_{0}\|_{\infty}+\frac{1}{\lambda}(1-e^{-\lambda t})\|g\|_{\infty}.$

For the estimate (6.9), we start by letting $s<t$, we have

$\displaystyle|v(t,x)-v(s,y)|$
$\displaystyle\leqslant|S^{\lambda,\gamma}_{t}v_{0}(x)-S^{\lambda,\gamma}_{s}v_{0}(y)|+\left|\int_{0}^{t}S^{\lambda,\gamma}_{t-u}(g(u,\cdot))(x){\mathrm{d}}u-\int_{0}^{s}S^{\lambda,\gamma}_{s-u}(g(u,\cdot))(y){\mathrm{d}}u\right|$
$\displaystyle\leqslant I_{1}\|v_{0}\|_{\infty}+(I_{2}+I_{3})\|g\|_{\infty}$

where

$\displaystyle I_{1}$ $\displaystyle=\int_{{\mathbb{R}}^{d}}\left|e^{-\lambda
t}s_{0}(\gamma t,x,z)-e^{-\lambda s}s_{0}(\gamma s,y,z)\right|{\mathrm{d}}z,$
$\displaystyle I_{2}$
$\displaystyle=\int_{0}^{s}\int_{{\mathbb{R}}^{d}}\left|e^{-\lambda(t-u)}s_{0}(\gamma(t-u),x,z)-e^{-\lambda(s-u)}s_{0}(\gamma(s-u),y,z)\right|{\mathrm{d}}z{\mathrm{d}}u,$
$\displaystyle I_{3}$
$\displaystyle=\int_{s}^{t}\int_{{\mathbb{R}}^{d}}e^{-\lambda(t-u)}s_{0}(\gamma(t-u),x,z){\mathrm{d}}z{\mathrm{d}}u.$

We have that $I_{3}\leqslant t-s$. For $I_{1}$ and $I_{2}$, we use the fact
that, for $i=1\dots d$

$\displaystyle\partial_{t}s_{0}(t,0,\xi)$
$\displaystyle=\frac{1}{2}\Delta_{\xi}s_{0}(t,0,\xi)=\frac{1}{2}\left(\frac{\|\xi\|^{2}}{t^{2}}-\frac{d}{t}\right)s_{0}(t,0,\xi)$
$\displaystyle\partial_{\xi_{i}}s_{0}(t,0,\xi)$
$\displaystyle=-\frac{\xi_{i}}{t}s_{0}(t,0,\xi).$

First for $I_{1}$, denote $c_{1}(r)=rt+(1-r)s$ and
$c_{2}(r)=r(z-x)+(1-r)(z-y))=z-(rx+(1-r)y)$ for $r\in[0,1]$, We have that

$\displaystyle|(\partial_{t}s_{0})(\gamma c_{1}(r),0,c_{2}(r))|$
$\displaystyle\leqslant\frac{1}{2}\left(\frac{\|c_{2}(r)\|^{2}}{\gamma^{2}c_{1}(r)^{2}}+\frac{d}{\gamma
c_{1}(r)}\right)s_{0}(\gamma c_{1}(r),0,c_{2}(r))$
$\displaystyle|(\partial_{\xi_{i}}s_{0})(\gamma c_{1}(r),0,c_{2}(r))|$
$\displaystyle\leqslant\frac{|c_{2}(r)_{i}|}{\gamma c_{1}(r)}s_{0}(\gamma
c_{1}(r),0,c_{2}(r))$

Therefore

$\displaystyle I_{1}$ $\displaystyle=\int_{{\mathbb{R}}^{d}}\left|e^{-\lambda
t}s_{0}(\gamma t,0,z-x)-e^{-\lambda s}s_{0}(\gamma
s,0,z-y)\right|{\mathrm{d}}z$
$\displaystyle\leqslant\int_{{\mathbb{R}}^{d}}\int_{0}^{1}|c^{\prime}_{1}(r)\partial_{t}(e^{-\lambda
t}s_{0}(\gamma
t,0,\xi))_{|t=c_{1}(r),\xi=c_{2}(r)}|{\mathrm{d}}r{\mathrm{d}}z$
$\displaystyle\quad+\int_{{\mathbb{R}}^{d}}\int_{0}^{1}\sum_{i=1}^{d}|c^{\prime}_{2}(r)_{i}\partial_{\xi_{i}}(e^{-\lambda
t}s_{0}(\gamma
t,0,\xi))_{|t=c_{1}(r),\xi=c_{2}(r)}|{\mathrm{d}}r{\mathrm{d}}z$
$\displaystyle\leqslant I_{1,1}+I_{1,2},$

where we have, by applying Fubini and a change of variable

$\displaystyle I_{1,1}$
$\displaystyle=(t-s)\int_{{\mathbb{R}}^{d}}\int_{0}^{1}\left[\lambda
e^{-\lambda
c_{1}(r)}+\frac{\gamma}{2}\left(\frac{\|c_{2}(r)\|^{2}}{c_{1}(r)^{2}}+\frac{d}{c_{1}(r)}\right)\right]s_{0}(\gamma
c_{1}(r),0,c_{2}(r)){\mathrm{d}}r{\mathrm{d}}z$
$\displaystyle\leqslant(t-s)\int_{0}^{1}\left[\lambda e^{-\lambda
c_{1}(r)}+\frac{\gamma}{2}\left(\frac{d\gamma
c_{1}(r)}{\gamma^{2}c_{1}(r)^{2}}+\frac{d}{\gamma
c_{1}(r)}\right)\right]{\mathrm{d}}r$
$\displaystyle\leqslant(t-s)\int_{0}^{1}\left[\lambda e^{-\lambda
c_{1}(r)}+\frac{d}{c_{1}(r)}\right]{\mathrm{d}}r=\int_{s}^{t}\left(\lambda
e^{-\lambda u}+\frac{d}{u}\right){\mathrm{d}}u=e^{-\lambda
s}(1-e^{-\lambda(t-s)})+d\log\left(\frac{t}{s}\right)$
$\displaystyle\leqslant\left(\lambda+\frac{d}{s}\right)(t-s).$

For the term $I_{1,2}$, we get, using Cauchy-Schwarz, Fubini and a change of
variable

$\displaystyle I_{1,2}$
$\displaystyle=\int_{{\mathbb{R}}^{d}}\int_{0}^{1}\sum_{i=1}^{d}|y_{i}-x_{i}|e^{-\lambda
c_{1}(r)}\frac{|c_{2}(r)_{i}|}{\gamma c_{1}(r)}s_{0}(\gamma
c_{1}(r),0,c_{2}(r)){\mathrm{d}}r{\mathrm{d}}z$
$\displaystyle\leqslant\|x-y\|\int_{0}^{1}\frac{e^{-\lambda c_{1}(r)}}{\gamma
c_{1}(r)}\int_{{\mathbb{R}}^{d}}\|c_{2}(r)\|s_{0}(\gamma
c_{1}(r),0,c_{2}(r)){\mathrm{d}}z{\mathrm{d}}r=\|x-y\|\int_{0}^{1}\frac{e^{-\lambda
c_{1}(r)}}{\sqrt{\gamma c_{1}(r)}}{\mathrm{d}}rC_{1}$ $\displaystyle\leqslant
C_{1}\|x-y\|\int_{s}^{t}\frac{e^{-\lambda u}}{\sqrt{\gamma
u}}\frac{{\mathrm{d}}u}{t-s}\leqslant\frac{C_{1}}{\sqrt{\gamma s}}\|x-y\|$

where $C_{1}$ is the expectation of the quadratic norm of
$X=(X_{1},X_{2},\dots X_{d})$ of $d$ independent standard normal variables :
$C_{1}=\mathbb{E}(\|X\|)\leqslant\mathbb{E}(\|X\|^{2})^{1/2}=\sqrt{d}$ (we
also have $C_{1}=\sqrt{2}\frac{\Gamma((d+1)/2)}{\Gamma(d/2)}\sim\sqrt{d}$). We
get that, for $s\geqslant\frac{1}{T}$,

$\displaystyle I_{1}$
$\displaystyle\leqslant\left(\lambda+\frac{d}{s}\right)(t-s)+\sqrt{\frac{d}{\gamma
s}}\|x-y\|\leqslant\left(\lambda+d\tau\right)(t-s)+\sqrt{\frac{d\tau}{\gamma}}\|x-y\|.$

For $I_{2}$, we make the same computations with
$c_{1}(r)=r(t-u)+(1-r)(s-u)=rt+(1-r)s-u$ where $u\in[0,s]$, we have, since
$c_{1}^{\prime}(r)=t-s$,

$\displaystyle I_{2}$
$\displaystyle\leqslant\int_{0}^{s}\int_{s-u}^{t-u}\left(\lambda e^{-\lambda
v}+\frac{d}{v}\right){\mathrm{d}}v{\mathrm{d}}u+C_{1}\|x-y\|\int_{0}^{s}\int_{s-u}^{t-u}\frac{e^{-\lambda
v}}{\sqrt{\gamma v}}\frac{{\mathrm{d}}v}{t-s}{\mathrm{d}}u$

For the first integral, we have

$\displaystyle\int_{0}^{s}\int_{s-u}^{t-u}\left(\lambda e^{-\lambda
v}+\frac{d}{v}\right){\mathrm{d}}v{\mathrm{d}}u$
$\displaystyle=(1-e^{-\lambda(t-s)})\int_{0}^{s}e^{-\lambda(s-u)}{\mathrm{d}}u+d\int_{0}^{s}\log\left(\frac{t-u}{s-u}\right){\mathrm{d}}u$
$\displaystyle=\frac{1}{\lambda}(1-e^{-\lambda(t-s)})(1-e^{-\lambda
s})+d\left[t\log(t)-s\log(s)-(t-s)\log(t-s)\right]$
$\displaystyle\leqslant\lambda
s(t-s)+d\left[s(\log(t)-\log(s))+(t-s)(\log(t)-\log(t-s))\right]$
$\displaystyle\leqslant\lambda s(t-s)+d(t-s)+d(t-s)(|\log(t)|+|\log(t-s))|)$
$\displaystyle\leqslant(\lambda s+d+|\log(t)|)(t-s)+d(t-s)|\log(t-s)|.$

For the second integral, we get

$\displaystyle\int_{0}^{s}\int_{s-u}^{t-u}\frac{e^{-\lambda v}}{\sqrt{\gamma
v}}{\mathrm{d}}v{\mathrm{d}}u$
$\displaystyle\leqslant(t-s)\int_{0}^{s}\frac{1}{\sqrt{\gamma(s-u)}}{\mathrm{d}}u=(t-s)\sqrt{\frac{2s}{\gamma}}.$

Then we obtain, for $1/\tau\leqslant s<t\leqslant\tau$

$\displaystyle I_{2}$ $\displaystyle\leqslant(\lambda
s+d+|\log(t)|)(t-s)+d(t-s)|\log(t-s)|+C_{1}\|x-y\|\sqrt{\frac{2s}{\gamma}}$
$\displaystyle\leqslant(\lambda\tau+d+|\log(\tau)|)(t-s)+d(t-s)|\log(t-s)|+\sqrt{\frac{2d\tau}{\gamma}}\|x-y\|.$

At last, we get the following estimate

$\displaystyle|v(t,x)-v(s,y)|$
$\displaystyle\leqslant\left[\left(\lambda+d\tau\right)(t-s)+\sqrt{\frac{d\tau}{\gamma}}\|x-y\|\right]\|v_{0}\|_{\infty}$
$\displaystyle+\left[(\lambda\tau+|\log(\tau)|+d+1)(t-s)+d(t-s)|\log(t-s)|+\sqrt{\frac{2d\tau}{\gamma}}\|x-y\|\right]\|g\|_{\infty}.$

∎

We modify a little our equation (6.1), both in order to obtain a sharper
estimate on the uniform norm of the solution and to get a coherent notation
with the solution of the limit equation when $K\to+\infty$.

We define for $q\in[-1,1]$,

$\displaystyle r_{K}(q)$
$\displaystyle:=-\int_{\frac{1}{2}}^{\frac{1}{2}(q+1)}4g_{K}(s)ds.$ (6.10)

We have that

$\displaystyle
r^{\prime}_{K}(q)=-2g_{K}\left(\frac{1}{2}(q+1)\right)=-(1-q)\mathbb{P}_{\frac{1-q}{2}}\big{[}X<\kappa(K,T)\big{]}+(1+q)\mathbb{P}_{\frac{1+q}{2}}\big{[}X\leqslant\kappa(K,T)\big{]}.$
(6.11)

So we let $h_{K}$ be

$h_{K}(q)=\left\\{\begin{aligned} -1&\text{ for $q<-1$}\\\
-r^{\prime}_{K}(q)+q&\text{ for $q\in[-1,1]$}\\\ 1&\text{ for
$q>1$}\end{aligned}\right.$ (6.12)

Since $r^{\prime}_{K}(1)=0$ and $r^{\prime}_{K}(-1)=0$, $h_{K}$ is continuous
on $\mathbb{R}$. We have that, for $q\in[-1,1]$,

$\displaystyle
h_{K}(q)=-r^{\prime}_{K}(q)+q=(1-q)\mathbb{P}_{\frac{1-q}{2}}\big{[}X<\kappa(K,T)\big{]}-(1+q)\mathbb{P}_{\frac{1+q}{2}}\big{[}X\leqslant\kappa(K,T)\big{]}+q.$

We now solve the following equation

$\partial_{t}v(t,x)-2\alpha\Delta v(t,x)+(2\beta+1)v=h_{K}(v(t,x)).$ (6.13)

Note that, this equation and (6.3) are exactly the same with the term $v$
added on both sides if $\|v\|_{\infty}\leqslant 1$, since for $q\notin[-1,1]$,
$h_{K}(q)\neq q$. Thus, a solution $v$ of (6.3) with $\|v\|_{\infty}\leqslant
1$ will also be a solution of (6.13) and reciprocally.

######  Proposition 6.2.

For $v_{0}\in L^{\infty}({\mathbb{T}}^{d})$ with $\|v_{0}\|_{\infty}\leqslant
1$, there exists a unique solution $(v(t,x),t\geq 0,x\in{\mathbb{T}}^{d})$ to
the problem

  * •

$v$ is continuous from $\mathbb{R}_{+}^{*}$ to $L^{\infty}({\mathbb{T}}^{d})$

  * •

$v$ satisfies, for all $t>0$ and $x\in{\mathbb{T}}^{d}$

$v(t,x)=S^{2\beta+1,4\alpha}_{t}v_{0}(x)+\int_{0}^{t}S^{2\beta+1,4\alpha}_{t-s}[h_{K}(v(s,\cdot))](x){\mathrm{d}}s.$
(6.14)

We say that $v$ is a mild solution to (6.13). We have also that
$\|v\|_{\infty}\leqslant 1$ and $v$ satisfies

$v(t,x)=S^{2\beta,4\alpha}_{t}v_{0}(x)-\int_{0}^{t}S^{2\beta,4\alpha}_{t-s}[r^{\prime}_{K}(v(s,\cdot))](x){\mathrm{d}}s$
(6.15)

and thus is a mild solution of (6.3).

###### Proof.

The first part of the proposition comes from a fixed point argument (see also
Pazy [20], Theorem 6.1.2) applied to the following functional let $\tau>0$ and
define the functional $F:C(]0,\tau],L^{\infty}({\mathbb{T}}^{d}))\to
C(]0,\tau],L^{\infty}({\mathbb{T}}^{d}))$ defined by

$F(v)(t,x):=S^{2\beta+1,4\alpha}_{t}v_{0}(x)+\int_{0}^{t}S^{2\beta+1,4\alpha}_{t-s}[h_{K}(v(s,\cdot))](x){\mathrm{d}}s.$
(6.16)

We equip $C(]0,\tau],L^{\infty}({\mathbb{T}}^{d}))$ with the uniform topology
on all compact subset. We can apply the Banach fixed point Theorem to $F$ (see
the proof of Pazy [20], Theorem 6.1.2. Moreover, the mapping $v_{0}\mapsto v$
is Lipschitz continuous from $L^{\infty}$ to
$C(]0,\tau],L^{\infty}({\mathbb{T}}^{d}))$.

An application of Proposition 6.1 proves that $\|v\|_{\infty}\leqslant 1$.

Since $h_{K}$ is differentiable, then, if $v_{0}\in C^{2}({\mathbb{T}}^{d})$,
$v_{0}$ is in the domain of $\Delta$ and thus $v$ is classical solution of
(6.13) (Theorem 6.1.5 [20]). Thus $v$ is a classical solution of (6.3) since
$\|v\|_{\infty}\leqslant 1$ and a mild solution of (6.3). Now consider an
approximating sequence $(v_{0,n})$ in $C^{2}({\mathbb{T}}^{d})$ of $v_{0}\in
L^{\infty}$, and $(v_{n})$ the sequence of mild solutions with initial value
$v_{0,n}$ and $v$ the mild solution of (6.13) with initial value $v$. Then,
since $v_{0}\mapsto v$ is Lipschitz continuous, by the dominated convergence
theorem, we get that, uniformly on $[t_{0},\tau]\times{\mathbb{T}}^{d}$ for
all $t_{0}>0$, the right hand side of

$v_{n}(t,x)=S^{2\beta,4\alpha}_{t}v_{0,n}(x)-\int_{0}^{t}S^{2\beta,4\alpha}_{t-s}[r^{\prime}_{K}(v_{n}(s,\cdot))](x){\mathrm{d}}s$
(6.17)

converges to
$S^{2\beta,4\alpha}_{t}v_{0}(x)-\int_{0}^{t}S^{2\beta,4\alpha}_{t-s}[r^{\prime}_{K}(v(s,\cdot))](x){\mathrm{d}}s$,
whereas the left hand side converges to $v$. So we obtain that $v$ is a mild
solution of (6.3). ∎

###  6.2 Solution of (6.2)

####  6.2.1 Existence of a solution

We use the same transform as before, and let $h_{\infty}$ be the pointwise
limit of $h_{K}$. The equation (6.2) is now formally

$\partial_{t}v(t,x)-2\alpha\Delta v(t,x)+(2\beta+1)v=h_{\infty}(v(t,x)).$
(6.18)

For the limiting equation, we prove first that the family $(v_{K})_{K}$ of
solutions associated to $h_{K}$ with common initial value $v_{0}\in
L^{\infty}({\mathbb{T}}^{d})$ in
$C(\mathbb{R}^{+}_{*}\times\mathbb{{\mathbb{T}}^{d}})$ is compact (with
uniform norm on all compact subset). Then, by taking the limit, any
accumulation point $v_{\infty}$ of the sequence satisfy the mild formulation
of the limiting equation relaxed as a subdifferential inclusion.

In order to prove this, we set some notations :

$\displaystyle r_{\infty}(q)$
$\displaystyle:=-\int_{\frac{1}{2}}^{\frac{1}{2}(q+1)}4g_{\infty}(s){\mathrm{d}}s$
(6.19)

$h_{\infty}$, the pointwise limit of $h_{K}$, is the function on $\mathbb{R}$

$\displaystyle h_{\infty}(q)$
$\displaystyle=2g_{\infty}\left(\frac{1+q}{2}\right)+q=-\mathbbm{1}_{q\leqslant-2\rho}+q\mathbbm{1}_{-2\rho<q\leqslant
2\rho}+\mathbbm{1}_{q>2\rho}.$ (6.20)

Then $h_{\infty}$ is non-decreasing and is the left-derivative of the convex
function $H_{\infty}$

$\displaystyle H_{\infty}(q)$ $\displaystyle:=-r_{\infty}(q)+\frac{q^{2}}{2}$
(6.21)
$\displaystyle=[-q+2\rho+2\rho^{2}]\mathbbm{1}_{\\{q\leqslant-2\rho\\}}+\frac{1}{2}q^{2}\mathbbm{1}_{\\{-2\rho<q\leqslant
2\rho\\}}+[q-2\rho+2\rho^{2}]\mathbbm{1}_{\\{2\rho<q\\}}$

The subdifferential of $H_{\infty}$ at $q$ is defined as

$\partial
H_{\infty}(q)=\\{p\in\mathbb{R},H_{\infty}(q^{\prime})-H_{\infty}(q)\geqslant
p(q^{\prime}-q),\text{ for all $q^{\prime}\in[-1,1]$}\\}.$

In particular we have,

$\displaystyle\partial H_{\infty}(q)$ $\displaystyle=\left\\{\begin{aligned}
&\\{-1\\}&&\text{ for $q<-2\rho$},\\\ &[-1,-2\rho]&&\text{ for $q=-2\rho$},\\\
&\\{q\\}&&\text{ for $-2\rho<q<2\rho$},\\\ &[2\rho,1]&&\text{ for
$q=2\rho$},\\\ &\\{1\\}&&\text{ for $q>2\rho$}.\\\ \end{aligned}\right.$
(6.22)

We adopt the following definition for a solution of the equation :

$\partial_{t}v(t,x)-2\alpha\Delta v(t,x)+(2\beta+1)v\in\partial
H_{\infty}(v(t,x)).$ (6.23)

######  Definition 6.1.

We say that $v$ is a mild solution of Equation (6.23) if it satisfies, for
some $\tau>0$ and all $t\leqslant\tau$

$v(t,x)=S^{2\beta+1,4\alpha}_{t}v_{0}(x)+\int_{0}^{t}S^{2\beta+1,4\alpha}_{t-s}[w(s,\cdot)](x){\mathrm{d}}s.$
(6.24)

where $w\in L^{2}([0,\tau]\times{\mathbb{T}}^{d})$, with $w(t,x)\in\partial
H_{\infty}(v(t,x))$ almost everywhere.

######  Proposition 6.3.

For $v_{0}\in L^{\infty}({\mathbb{T}}^{d})$, any accumulation point (in
$C(\mathbb{R}_{+}^{*}\times{\mathbb{T}}^{d})$ equipped with uniform norm on
each compact set), of the sequence $(v_{K})$ of solutions given by Proposition
6.2 is a mild solution of (6.23).

As a consequence of the proposition, there exists $(v(t,x),t\geq
0,x\in{\mathbb{T}}^{d})$ a mild solution of (6.23) such that $v$ is continuous
from $\mathbb{R}_{+}^{*}$ to $L^{\infty}({\mathbb{T}}^{d})$, and
$\|v\|_{\infty}\leqslant 1$.

The existence of a solution for a given initial condition $v_{0}$ is not
difficult and can be prove in different ways. Here we adopt some kind of
regularization procedure, since we have a natural family of differentiable
functions (namely the $(h_{K})$) approximating $h_{\infty}$ and we use the
convergence of the sequence $(v_{K})$ in the next section to prove the
convergence of the stochastic process. We also present the proof because we
need its arguments in order to prove the Theorem 7.1.

In the Remark 6.3, we present another construction of solution(s) using the
monotonicity of $h_{\infty}$ which is interesting since it also gives an
insight on the problem of non-uniqueness.

###### Proof.

For each $K$, we have a mild solution $v_{K}$ from Proposition 6.2. From
Proposition 6.1, we have that each solution is uniformly bounded, uniformly
continuous on $[1/\tau,\tau]\times{\mathbb{T}}^{d}$, and the modulus of
continuity only depends on $\tau>1$ (since the others parameters are fixed).

Therefore, by the Arzela-Ascoli Theorem, the sequence $(v_{K})_{K}$ is compact
on $C([1/\tau,\tau]\times{\mathbb{R}}^{d})$ and we can extract a subsequence
converging uniformly in $C([1/\tau,\tau]\times{\mathbb{R}}^{d})$, and then by
a diagonal argument, a sequence converging to a limit $v_{\infty}$ in
$C(]0,\infty[\times{\mathbb{R}}^{d})$, uniformly on each compact. Note that
since $\|v_{K}\|\leqslant 1$ for all $K$, we also have
$\|v_{\infty}\|\leqslant 1$. We show that $v_{\infty}$ satisfies (6.18).

Let us assume that $\rho>0$. Denote for any $s>0$,
${\mathcal{A}}_{\rho}(s):=\\{y\in{\mathbb{R}}^{d}\colon|v_{\infty}(s,y)\pm
2\rho|=0\\}$. Note that since $h_{\infty}$ is uniformly continuous on
$[-1,-2\rho[$, $]-2\rho,2\rho[$ and $]2\rho,1]$, we have that for any
$y\notin{\mathcal{A}}_{\rho}(s)$,
$\lim_{K\to\infty}h_{K}(v_{K}(s,y))=h_{\infty}(v_{\infty}(s,y))$. Let
$y\in{\mathcal{A}}_{\rho}(s)$ and assume $v_{\infty}(s,y)=2\rho$ without loss
of generality, since $v_{K}(s,y)$ converges to $v_{\infty}(s,y)$, for all
$\varepsilon>0$ such that $2\rho-\varepsilon>0$, there exists $K_{0}$ such
that for all $K\geqslant K_{0}$,
$2\rho-\varepsilon<v_{K}(s,y)<2\rho+\varepsilon$. Thus, using Lemma B.5, we
have that, for all $K\geqslant K_{0}$, $2\rho-2\varepsilon\leqslant
h_{K}(v_{K}(s,y))\leqslant 1$. Then taking the limits in $K$ and
$\varepsilon\to 0$, we get

$\displaystyle
h_{\infty}(2\rho)=2\rho\leqslant\liminf_{K\to\infty}h_{K}(v_{K}(s,y))\leqslant\limsup_{K\to\infty}h_{K}(v_{K}(s,y))\leqslant
1=h_{\infty}(2\rho^{+}).$ (6.25)

For $\rho=0$, we have the same inequality since then $h_{\infty}(0)=-1$ and
$h_{\infty}(0+)=1$.

Therefore, let $w_{+}(s,y)=\limsup_{K\to\infty}h_{K}(v_{K}(s,y))$ and
$w_{-}(s,y)=\liminf_{K\to\infty}h_{K}(v_{K}(s,y))$, we have that for all
$(s,y)\in]0,+\infty[\times{\mathbb{T}}^{d}$:

$\displaystyle h_{\infty}(v_{\infty}(s,y))\leqslant w_{-}(s,y)\leqslant
w_{+}(s,y)\leqslant h_{\infty}(v_{\infty}(s,y)^{+}).$ (6.26)

Since $h_{K}(v_{K})$ is bounded, by the Banach Alaoglu Theorem, the sequence
is weakly compact in $L^{2}(]0,\tau[\times{\mathbb{T}}^{d})$, and we have a
subsequence of $(h_{K}(v_{K}))_{K}$ converging weakly to $w\in
L_{loc}^{2}(]0,+\infty[\times{\mathbb{T}}^{d})$. Since the density of the
semigroup $S^{2\beta+1,4\alpha}$ is in $L^{2}(]0,\tau]\times{\mathbb{R}}^{d})$
for all $T>0$, we have as $K\to+\infty$,

$\displaystyle v_{\infty}(t,x)$
$\displaystyle=S^{2\beta+1,4\alpha}_{t}v_{0}(x)+\int_{0}^{t}S^{2\beta+1,4\alpha}_{t-s}w(s,\cdot)(x){\mathrm{d}}s.$
(6.27)

Moreover, $w_{-}$ and $w_{+}$ are bounded and therefore in
$L^{2}(]0,\tau]\times{\mathbb{T}}^{d})$. Let $\varphi\in
L^{2}(]0,\tau]\times{\mathbb{T}}^{d})$ and $\varphi\geqslant 0$, by the Fatou
Lemma we get

$\displaystyle
0=\int_{]0,\tau]\times{\mathbb{T}}^{d}}\liminf_{K\to+\infty}(h_{K}(v_{K})-w_{-})\varphi\leqslant\liminf_{K\to+\infty}\int_{]0,\tau]\times{\mathbb{T}}^{d}}(h_{K}(v_{K})-w_{-})\varphi=\int_{]0,\tau]\times{\mathbb{T}}^{d}}(w-w_{-})\varphi.$
(6.28)

We also have

$\displaystyle
0=\int_{]0,\tau]\times{\mathbb{T}}^{d}}\liminf_{K\to+\infty}(w_{+}-h_{K}(v_{K}))\varphi\leqslant\liminf_{K\to+\infty}\int_{]0,\tau]\times{\mathbb{T}}^{d}}(w_{+}-h_{K}(v_{K}))\varphi=\int_{]0,\tau]\times{\mathbb{T}}^{d}}(w_{+}-w)\varphi.$
(6.29)

Thus, almost everywhere on $]0,+\infty[\times{\mathbb{T}}^{d}$, we have that

$\displaystyle h_{\infty}(v_{\infty})\leqslant w_{-}\leqslant w\leqslant
w_{+}\leqslant h_{\infty}(v_{\infty}^{+}).$ (6.30)

Therefore, $w\in\partial H_{\infty}(v_{\infty})$ a.e. ∎

####  6.2.2 Uniqueness of solution of (6.23)

The main problem concerns the uniqueness of a solution. We prove first that,
we do not have uniqueness for a constant initial condition $v_{0}(x)=2\rho$
when $2\rho<\frac{1}{1+2\beta}$, so we are in the case of segregation or
metastable segragation described by Figure 1.

######  Remark 6.2.

We describe three possible solutions starting from the initial condition
$v_{0}(x)=2\rho$ when $2\rho<\frac{1}{1+2\beta}$.

Note that

$\displaystyle
S^{2\beta+1,4\alpha}_{t}v_{0}(x)=\int_{\mathbb{R}^{d}}e^{-(2\beta+1)t}s_{0}(t,x,y)2\rho{\mathrm{d}}y=2\rho
e^{-(2\beta+1)t}.$ (6.31)

Suppose that $v$ does not depend on $x$, $v(t,x)=c(t)$ for all
$x\in{\mathbb{T}}^{d}$, we have

$\displaystyle\int_{0}^{t}S^{2\beta+1,4\alpha}_{t-s}[h_{\infty}(v(s,\cdot))](x){\mathrm{d}}s$
$\displaystyle=\int_{0}^{t}h_{\infty}(c(s))e^{-(2\beta+1)(t-s)}{\mathrm{d}}s.$
(6.32)

Let us consider the functions

$\displaystyle v^{1}(t,x)=c^{1}(t)$ $\displaystyle=2\rho e^{-2\beta t}$ (6.33)
$\displaystyle v^{2}(t,x)=c^{2}(t)$ $\displaystyle=2\rho
e^{-(2\beta+1)t}+\frac{1}{1+2\beta}\left(1-e^{-(2\beta+1)t}\right)=2\rho+\left(\frac{1}{1+2\beta}-2\rho\right)\left(1-e^{-(2\beta+1)t}\right).$
(6.34)

Since, $c^{1}(t)\in[0,2\rho[$, for $t>0$, we have
$h_{\infty}(c^{1}(t))=c^{1}(t)$ and then

$\displaystyle\int_{0}^{t}h_{\infty}(c^{1}(s))e^{-(2\beta+1)(t-s)}{\mathrm{d}}s$
$\displaystyle=\int_{0}^{t}2\rho e^{-2\beta
s}e^{-(2\beta+1)(t-s)}{\mathrm{d}}s=2\rho e^{-(2\beta+1)t}(e^{t}-1).$ (6.35)

Therefore

$\displaystyle
S^{2\beta+1,4\alpha}_{t}v_{0}(x)+\int_{0}^{t}S^{2\beta+1,4\alpha}_{t-s}[h_{\infty}(v^{1}(s,\cdot))](x){\mathrm{d}}s$
$\displaystyle=2\rho e^{-(2\beta+1)t}+2\rho e^{-(2\beta+1)t}(e^{t}-1)=2\rho
e^{-2\beta t}=v^{1}(t,x).$ (6.36)

Since $2\rho<\frac{1}{1+2\beta}$, $c^{2}(t)\in]2\rho,1]$, for $t>0$, we have
$h_{\infty}(c^{2}(t))=1$ and then by the same computation,

$\displaystyle\int_{0}^{t}h_{\infty}(c^{2}(s))e^{-(2\beta+1)(t-s)}{\mathrm{d}}s$
$\displaystyle=\frac{1}{1+2\beta}(1-e^{-(2\beta+1)t}).$ (6.37)

Therefore, we also have

$\displaystyle
S^{2\beta+1,4\alpha}_{t}v_{0}(x)+\int_{0}^{t}S^{2\beta+1,4\alpha}_{t-s}[h_{\infty}(v^{2}(s,\cdot))](x){\mathrm{d}}s$
$\displaystyle=v^{2}(t,x).$ (6.38)

Thus, both $v^{1}$ and $v^{2}$ are mild solutions to (6.18) and thus to (6.23)
with the same initial conditions. Note that at $t=0$, we have
$v^{1}(0,x)=v^{2}(0,x)=2\rho$ and

$\displaystyle\partial_{t}v^{1}(0,x)$
$\displaystyle=-4\beta\rho=-(2\beta+1)2\rho+2\rho=-(2\beta+1)2\rho+h_{\infty}(2\rho^{-})$
(6.39) $\displaystyle\partial_{t}v^{2}(0,x)$
$\displaystyle=-2\rho(2\beta+1)+1=-(2\beta+1)2\rho+h_{\infty}(2\rho^{+}).$
(6.40)

We see that non uniqueness comes from the fact that at $t=0$, where
$v(t,x)=2\rho$, we have at least two choices for the derivative due to the
fact that $h_{\infty}$ is not continuous.

Note that if we consider the mild solution to the subdiffrential inclusion
(6.23), then we have at least a third solution : $v^{3}(t,x)=2\rho$. We
consider $w(t,x)=2\rho(2\beta+1)$, we have

$\displaystyle
S^{2\beta+1,4\alpha}_{t}v_{0}(x)+\int_{0}^{t}S^{2\beta+1,4\alpha}_{t-s}[w(s,\cdot)](x){\mathrm{d}}s$
$\displaystyle=2\rho
e^{-(2\beta+1)t}+2\rho(2\beta+1)\int_{0}^{t}e^{-(2\beta+1)(t-s)}{\mathrm{d}}s$
(6.41) $\displaystyle=2\rho.$

Since, $2\rho<\frac{1}{1+2\beta}$, we have that $2\rho<2\rho(2\beta+1)<1$ so
$w(t,x)\in\partial H_{\infty}(2\rho)$.

Therefore, we cannot expect uniqueness for all initial condition, we have to
impose some condition on the initial condition if we want a unique solution.

In the literature, we can find different conditions ensuring that the solution
of Equation (6.23) is unique. Adapting [11] and [4], we prove that the
regularity of the initial condition at the levels where the non-linearity
$h_{\infty}$ is not continuous is sufficient.

######  Definition 6.2.

A function $v_{0}:{\mathbb{T}}^{d}\to[-1,1]$ in $C^{1}({\mathbb{T}}^{d})$ is
_regular at level $q\in]-1,1[$_ if for all $x\in{\mathbb{T}}^{d}$, such that
$v_{0}(x)=p$, we have $\nabla v_{0}(x)\neq 0$.

######  Proposition 6.4.

For $v_{0}\in C^{1}({\mathbb{T}}^{d})$, such that $\nabla v_{0}$ is Lipschitz
on ${\mathbb{T}}^{d}$ and regular at levels $2\rho$ and $-2\rho$, the solution
$v$ to Equation (6.23) is locally unique. Moreover, the Lebesgue measure of
the set ${\mathcal{A}}_{\rho}(s):=\\{y\in{\mathbb{R}}^{d}\colon|v(s,y)\pm
2\rho|=0\\}$ is zero.

We adapt two arguments by [11] and [4].

######  Lemma 6.5.

If $v$ is a mild solution of (6.23) with $v_{0}\in C^{1}({\mathbb{T}}^{d})$,
and such that $\nabla v_{0}$ is Lipschitz on ${\mathbb{T}}^{d}$, then, for all
$\tau>0$, there exists a constant $C>0$ such that, for all $t\in[0,\tau]$

$\displaystyle\|v(t)-v_{0}\|_{\infty}$ $\displaystyle\leqslant Ct^{1/2},$
$\displaystyle\|\nabla v(t)-\nabla v_{0}\|_{\infty}$ $\displaystyle\leqslant
Ct^{1/2}.$ (6.42)

###### Proof.

Since $v$ is a mild solution of (6.23) there exists $w\in
L^{\infty}_{loc}([0+\infty[\times{\mathbb{T}}^{d})$ with
$\|w\|_{\infty}\leqslant 1$ since $w\in\partial H_{\infty}(v)$. Thus we get

$\displaystyle
v(t)-v_{0}=S_{t}^{2\beta+1,4\alpha}v_{0}-v_{0}+\int_{0}^{t}S_{t-s}^{2\beta+1,4\alpha}w(s){\mathrm{d}}s.$
(6.43)

Then, we have for the last integral

$\displaystyle\left\|\int_{0}^{t}S_{t-s}^{2\beta+1,4\alpha}w(s){\mathrm{d}}s\right\|_{\infty}\leqslant\int_{0}^{t}e^{-(2\beta+1)(t-s)}{\mathrm{d}}s\leqslant
t.$

We have also

$\displaystyle|S_{t}^{2\beta+1,4\alpha}v_{0}(x)-v_{0}(x)|$
$\displaystyle\leqslant e^{-(2\beta+1)t}\int_{{\mathbb{T}}^{d}}s(4\alpha
t,x,y)|v_{0}(y)-v_{0}(x)|{\mathrm{d}}y$ $\displaystyle\leqslant
Le^{-(2\beta+1)t}\int_{{\mathbb{R}}^{d}}s_{0}(4\alpha
t,x,y)\|y-x\|_{2}{\mathrm{d}}y$ $\displaystyle\leqslant
Le^{-(2\beta+1)t}\sqrt{4\alpha td}\leqslant L\sqrt{4\alpha d}t^{1/2}$

where $L$ is the Lipschitz constant of $v_{0}$ and the third inequality comes
from the computation of the upper bound of the quadratic norm of $d$
independent random variables with common variance $4\alpha t$. Therefore we
obtain

$\displaystyle\|v(t)-v_{0}\|_{\infty}$ $\displaystyle\leqslant(L\sqrt{4\alpha
d}+\sqrt{\tau})t^{1/2}$ (6.44)

For the second bound, we use the fact that
$\frac{{\mathrm{d}}}{{\mathrm{d}}x_{i}}s_{0}(t,x,y)=-\frac{{\mathrm{d}}}{{\mathrm{d}}y_{i}}s_{0}(t,x,y)$,
thus we have, using an integration by parts

$\displaystyle\partial_{x_{i}}S_{t}^{2\beta+1,4\alpha}(v_{0})=S_{t}^{2\beta+1,4\alpha}(\partial_{x_{i}}v_{0}).$
(6.45)

Then, we have

$\displaystyle\partial_{x_{i}}v(t)-\partial_{x_{i}}v_{0}=S_{t}^{2\beta+1,4\alpha}(\partial_{x_{i}}v_{0})-(\partial_{x_{i}}v_{0})+\int_{0}^{t}\int_{\mathbb{R}^{d}}e^{-(2\beta+1)(t-s)}\partial_{x_{i}}s_{0}(4\alpha(t-s),x,y)w(s,y){\mathrm{d}}y{\mathrm{d}}s.$
(6.46)

We can treat both integrals as before, for the last integral :

$\displaystyle\left|\int_{0}^{t}\int_{\mathbb{R}^{d}}e^{-(2\beta+1)(t-s)}\partial_{x_{i}}s_{0}(4\alpha(t-s),x,y)w(s,y){\mathrm{d}}y{\mathrm{d}}s\right|$
$\displaystyle\leqslant\int_{0}^{t}\frac{e^{-(2\beta+1)(t-s)}}{4\alpha(t-s)}\int_{\mathbb{R}^{d}}|x_{i}-y_{i}|s_{0}(4\alpha(t-s),x,y){\mathrm{d}}y{\mathrm{d}}s$
$\displaystyle\leqslant\int_{0}^{t}\frac{e^{-(2\beta+1)(t-s)}}{4\alpha(t-s)}\sqrt{2\alpha(t-s)}{\mathrm{d}}s$
$\displaystyle\leqslant\frac{1}{\sqrt{2\alpha}}\int_{0}^{\sqrt{t}}e^{-(2\beta+1)u^{2}}{\mathrm{d}}u\leqslant\frac{\sqrt{t}}{\sqrt{2\alpha}}.$

For the first integral, we have the same estimates as before

$\displaystyle|S_{t}^{2\beta+1,4\alpha}(\partial_{x_{i}}v_{0})(x)-\partial_{x_{i}}v_{0}(x)|\leqslant
L^{\prime}\sqrt{4\alpha d}t^{1/2}$ (6.47)

where $L^{\prime}$ is the maximum of the Lipschitz constants of
$(\partial_{x_{i}}v_{0})_{i}$. Thus, we get

$\displaystyle\|\nabla v(t)-\nabla v_{0}\|_{\infty}$
$\displaystyle\leqslant\left(L^{\prime}\sqrt{4\alpha
d}+\frac{1}{\sqrt{2\alpha}}\right)t^{1/2}$ (6.48)

∎

We now prove Proposition 6.4.

###### Proof.

Let us assume that we have two solutions, $v_{1}$ and $v_{2}$ and let
$e(t)=\|v_{1}-v_{2}\|_{L^{\infty}([0,t]\times\mathbb{R}^{d})}$. Note that the
previous Lemma entails that, for all $\tau>0$, there exists $C$, such that for
$t<\tau$, we have $e(t)\leqslant C\sqrt{t}$.

We define $I^{+}_{s,t}=\\{(s,y),s\leqslant t,|v_{1}(s,y)-2\rho|\leqslant
e(t)\\}$ and $I^{-}_{s,t}=\\{(s,y),s\leqslant t,|v_{1}(s,y)+2\rho|\leqslant
e(t)\\}$.

Since $v_{1}$ and $v_{2}$ are solutions of (6.23), there exists $w_{1}$ and
$w_{2}$ such that $w_{1}\in\partial H_{\infty}(v_{1})$ a.e. and
$w_{2}\in\partial H_{\infty}(v_{2})$. We can decompose each $w_{i}$ as
$w_{i}(t,x)=f_{\infty}(v_{i}(t,x))+g_{i}(t,x)$ where $f_{\infty}$ is the
continuous part of $\partial H_{\infty}$ :

$\displaystyle f_{\infty}(q)=\left\\{\begin{aligned} -2\rho&\text{ for
$q\in[-1,-2\rho]$}\\\ q&\text{ for $q\in[-2\rho,2\rho]$}\\\ 2\rho&\text{ for
$q\in[2\rho,1]$}\end{aligned}\right.$ (6.49)

and $g_{i}(t,x)=w_{i}(t,x)-f_{\infty}(v_{i}(t,x))$. Note that

$\displaystyle g_{i}(t,x)$ $\displaystyle=-1+2\rho\text{ a.e. on
$\\{(t,x),v_{i}(t,x)<-2\rho\\}$},$ $\displaystyle g_{i}(t,x)$
$\displaystyle=0\text{ a.e. on $\\{(t,x),-2\rho<v_{i}(t,x)<2\rho\\}$},$ (6.50)
$\displaystyle g_{i}(t,x)$ $\displaystyle=1-2\rho\text{ a.e. on
$\\{(t,x),v_{i}(t,x)>2\rho\\}$},$

since $w_{i}=h_{\infty}(v_{i})$ a.e. on $\\{(t,x),|v_{i}(t,x)|=2\rho\\}$.

As a consequence we have that, up to a negligible set, $\\{(s,y),s\leqslant
t,g_{1}(s,y)\neq g_{2}(s,y)\\}\subset I^{+}_{s,t}\cup I^{-}_{s,t}$ since,
$g_{1}(s,y)\neq g_{2}(s,y)$ entails that one of the following inequalities is
true $v_{2}(s,y)<2\rho<v_{1}(s,y)$ or $v_{2}(s,y)<-2\rho<v_{1}(s,y)$ or
$v_{1}(s,y)<2\rho<v_{2}(s,y)$ or $v_{1}(s,y)<-2\rho<v_{2}(s,y)$. For each
case, the inclusion is true : for the first one for example, if $s\leqslant t$
and $v_{2}(s,y)<2\rho<v_{1}(s,y)$

$\displaystyle e(t)\geqslant v_{1}(s,y)-v_{2}(s,y)=v_{1}(s,y)-2\rho+2\rho-
v_{2}(s,y)\geqslant v_{1}(s,y)-2\rho\geqslant 0$ (6.51)

thus $(s,y)\in I^{+}_{s,t}$. The same is true for the other cases.

Therefore we obtain the following expression for the difference $v_{1}-v_{2}$:

$\begin{split}v_{1}(t,x)-v_{2}(t,x)=&\int_{0}^{t}\int_{\mathbb{T}^{d}}e^{-(2\beta+1)(t-s)}s(4\alpha(t-s),x,y)(f_{\infty}(v_{1}(s,y))-f_{\infty}(v_{2}(s,y))){\mathrm{d}}y{\mathrm{d}}s\\\
&+\int_{0}^{t}\int_{I^{+}_{s,t}\cup
I^{-}_{s,t}}e^{-(2\beta+1)(t-s)}s(4\alpha(t-s),x,y)(g_{1}(s,y)-g_{2}(s,y)){\mathrm{d}}y{\mathrm{d}}s.\end{split}$
(6.52)

For the first integral in (6.52) we note that $f_{\infty}$ is $1$-Lipschitz,
thus

$\displaystyle\left|\int_{0}^{t}\int_{\mathbb{T}^{d}}e^{-(2\beta+1)(t-s)}s(4\alpha(t-s),x,y)(f_{\infty}(v_{1}(s,y))-f_{\infty}(v_{2}(s,y))){\mathrm{d}}y{\mathrm{d}}s\right|$
$\displaystyle\leqslant\int_{0}^{t}\int_{\mathbb{T}^{d}}e^{-(2\beta+1)(t-s)}s(4\alpha(t-s),x,y)e(t){\mathrm{d}}y{\mathrm{d}}s\leqslant
te(t).$

For the second integral in (6.52) we note that $|g_{i}|\leqslant 1-2\rho$,
then we first have

$\displaystyle\left|\int_{0}^{t}\int_{I^{+}_{s,t}\cup
I^{-}_{s,t}}e^{-(2\beta+1)(t-s)}s(4\alpha(t-s),x,y)(g_{1}(s,y)-g_{2}(s,y)){\mathrm{d}}y{\mathrm{d}}s\right|$
$\displaystyle\leqslant 2(1-2\rho)\int_{0}^{t}\int_{I^{+}_{s,t}\cup
I^{-}_{s,t}}e^{-(2\beta+1)(t-s)}s(4\alpha(t-s),x,y){\mathrm{d}}y{\mathrm{d}}s.$

Let $s\leqslant t$, since $v_{0}$ is regular on the level set
$\\{v_{0}=2\rho\\}$ which is compact (since ${\mathbb{T}}^{d}$ is) and $\nabla
v_{0}$ is a Lipschitz function, we can find $\delta,\eta>0$ such that on
$\\{v_{0}=2\rho\\}+B_{\delta}(0)$, $|\nabla v_{0}(x)|>\eta$. Using the second
part of Lemma 6.5, and since $e(t)\leqslant C\sqrt{t}$, there exists $T>0$
such that for $s\leqslant t\leqslant T$,
$I^{+}_{s,t}\subset\\{v_{0}=2\rho\\}+B_{\delta}(0)$ and on $I^{+}_{s,t}$,
$|\nabla v_{1}(s)|>\eta/2$.

Since $I^{+}_{s,t}$ is compact and $\nabla v_{1}(s)\neq 0$, by the implicit
function theorem, we can find a finite cover by open balls
$(B_{i})_{1\leqslant i\leqslant N}$ centered on points on $I^{+}_{s,t}$ such
that locally on each ball $B_{i}$, the level set $\\{v_{1}(s,y)=2\rho\\}$ is
the graph of a function, e.g $y_{1}=\varphi(y_{2},\dots y_{d})$. Note that
since $\\{v_{0}=2\rho\\}$ is compact, $N$ is uniform in $s\leqslant T$, since
by the lemma, we can make the cover of open balls on $\\{v_{0}=2\rho\\}$ and
take their traces on $\\{v_{1}(s,y)=2\rho\\}$.

By the mean value theorem on the first coordinate $y_{1}$ of $v_{1}(s)$, we
have $I^{+}_{s,t}\cap B_{i}\subset[-2e(t)/\nu,2e(t)/\nu]\times\Pi_{1}(B_{i})$,
where $\Pi_{1}$ is the projection along the first coordinate. Thus,

$\displaystyle\int_{0}^{t}$ $\displaystyle\int_{I^{+}_{s,t}\cap
B_{i}}e^{-(2\beta+1)(t-s)}s(4\alpha(t-s),x,y){\mathrm{d}}y{\mathrm{d}}s$
$\displaystyle\leqslant\int_{0}^{t}e^{-(2\beta+1)(t-s)}\frac{4e(t)}{\nu\sqrt{4\alpha(t-s)}}\int_{\Pi_{1}(B_{i})}s(4\alpha(t-s),0,(0,y_{2},\dots
y_{d})){\mathrm{d}}y_{2}\cdots{\mathrm{d}}y_{d}{\mathrm{d}}s$
$\displaystyle\leqslant\int_{0}^{t}e^{-(2\beta+1)(t-s)}\frac{2e(t)}{\nu\sqrt{\alpha(t-s)}}{\mathrm{d}}s\leqslant\frac{2e(t)t^{1/2}}{\nu\sqrt{\alpha}}.$

Thus

$\displaystyle\int_{0}^{t}\int_{I^{+}_{s,t}}e^{-(2\beta+1)(t-s)}s(4\alpha(t-s),x,y){\mathrm{d}}y{\mathrm{d}}s$
$\displaystyle\leqslant\frac{2Ne(t)t^{1/2}}{\nu\sqrt{\alpha}}.$ (6.53)

Since the same holds for $I^{-}_{s,t}$, we obtain, that for some constant
$C>0$, and all $t<\tau$

$\displaystyle|v_{1}(t,x)-v_{2}(t,x)|$
$\displaystyle\leqslant(t+Ct^{1/2})e(t)\leqslant(\tau+C\tau^{1/2})e(\tau).$
(6.54)

Then $e(\tau)\leqslant(\tau+C\tau^{1/2})e(\tau)$, and taking $\tau$ small
enough, we obtain $e(\tau)=0$ thus $v_{1}=v_{2}$ on
$[0,\tau]\times{\mathbb{T}}^{d}$. ∎

######  Remark 6.3.

Maximal and minimal solutions. Another approach to (6.2) is to use a monotone
construction of solutions, which arises from a comparison principle close to
the one developed in Proposition 5.1. This was done initially in [2] and also
in [5] Define $\overline{h_{\infty}}$ the right continuous version of
$h_{\infty}$ (Equation 6.20) by

$\displaystyle\overline{h_{\infty}}(q)$
$\displaystyle:=-\mathbbm{1}_{q<-2\rho}+q\mathbbm{1}_{-2\rho\leqslant
q<2\rho}+\mathbbm{1}_{q\geqslant 2\rho}.$ (6.55)

Note that $h_{\infty}$ and $\overline{h_{\infty}}$ are non decreasing (recall
that $2\rho\in[0,1]$). Recall that $(S_{t})$ is the semigroup on
$L^{1}(\mathbb{T}^{d})$ associated to $-2\alpha\Delta$, we denote

$\displaystyle\underline{F}(v)(t,x):=e^{-(2\beta+1)t}S_{t}v_{0}(x)+\int_{0}^{t}e^{-(2\beta+1)(t-s)}S_{t-s}[h_{\infty}(v(s,\cdot))](x){\mathrm{d}}s$
(6.56)
$\displaystyle\overline{F}(v)(t,x):=e^{-(2\beta+1)t}S_{t}v_{0}(x)+\int_{0}^{t}e^{-(2\beta+1)(t-s)}S_{t-s}[\overline{h_{\infty}}(v(s,\cdot))](x){\mathrm{d}}s.$
(6.57)

Then, fixed points of the maps above are mild solutions of these two
formulations of our subdifferential inclusion:

$\displaystyle\partial_{t}v(t,x)-2\alpha\Delta v(t,x)+(2\beta+1)v(t,x)$
$\displaystyle=h_{\infty}(v(t,x))$ (6.58)
$\displaystyle\partial_{t}v(t,x)-2\alpha\Delta v(t,x)+(2\beta+1)v(t,x)$
$\displaystyle=\overline{h_{\infty}}(v(t,x))$ (6.59)

Since $h_{\infty}$ (resp. $\overline{h_{\infty}}$) is non decreasing and that
$h_{\infty}(p)\leqslant\overline{h_{\infty}}(p)$ for all $p\in[-1,1]$, we have
that, for $u,v$ two functions such that $-1\leqslant v\leqslant u\leqslant 1$,

  1. 1.

$\underline{F}(v)(t,x)\leqslant\underline{F}(u)(t,x)$

  2. 2.

$\overline{F}(v)(t,x)\leqslant\overline{F}(u)(t,x)$

  3. 3.

$\underline{F}(u)(t,x)\leqslant\overline{F}(u)(t,x)$

We define the sequences $(V^{n})_{n}$ and $(W^{n})_{n}$ of functions on
$\mathbb{R}^{+}\times\mathbb{T}^{d}$ : $V^{0}(t,x)=1$, $W^{0}(t,x)=-1$ and for
all $n\geqslant 1$

$\displaystyle V^{n}(t,x)$
$\displaystyle=e^{-(2\beta+1)t}S_{t}v_{0}(x)+\int_{0}^{t}e^{-(2\beta+1)(t-s)}S_{t-s}[\overline{f}(V^{n-1}(s,\cdot))](x){\mathrm{d}}s$
(6.60) $\displaystyle W^{n}(t,x)$
$\displaystyle=e^{-(2\beta+1)t}S_{t}v_{0}(x)+\int_{0}^{t}e^{-(2\beta+1)(t-s)}S_{t-s}[\underline{f}(W^{n-1}(s,\cdot))](x){\mathrm{d}}s.$
(6.61)

Thus, for $-1\leqslant v_{0}(x)\leqslant 1$, we can prove by induction that
the sequences $(V^{n})$ and $(W^{n})$ satisfy, for all $n$

$-1\leqslant W^{1}\leqslant W^{2}\cdots\leqslant W^{n}\leqslant
V^{n}\leqslant\cdots\leqslant V^{2}\leqslant V^{1}\leqslant 1.$ (6.62)

By a compactness and monotony argument, one can prove that $(W^{n})$ and
$(V^{n})$ converge to functions $w$ and $w$ which are mild solutions of the
subdifferential inclusion. These are the minimal and maximal solutions of the
subdifferential inclusion, in the sense that any other solution (Definition
6.1) must be bounded below by $w$ and above by $v$. Uniqueness follows if one
can prove that $w=v$.

##  7 Convergence of the discrete PDE

In analogy with the continuous setting, we define $v^{N}=2u^{N}-1$ where
$u^{N}$ is the solution of the discretized Equation (4.10) and
$H(i,v^{N})=2G(i,\frac{v^{N}+1}{2})+v^{N}(i)$.

In such a way (4.10) becomes

$\begin{cases}\partial_{t}v^{N}(t,i)=2\alpha N^{2}\Delta
u^{N}(t,i)-(2\beta+1)v^{N}(t,i)+H(i,v^{N})\\\
v^{N}(0,i)=2u^{N}_{0}(i)-1,\end{cases}$ (7.1)

We main goal of this section is to prove the following result which states the
convergence of $v^{N}$.

######  Theorem 7.1.

Let $v^{N}$ be the solution of (7.1). Then $(v^{N})$ is pre-compact for the
uniform convergence on each compact sets of
${\mathbb{T}}^{d}\times]0,+\infty[$ and any accumulation points $v_{\infty}$
is a solution of (6.23). In particular, whenever the solution of (7.1) is a.e.
unique, $v_{\infty}$ is also (the) mild solution of (6.2) and the whole
sequence $v^{N}$ converges to $v_{\infty}$, uniformly on all compact sets of
${\mathbb{T}}^{d}\times]0,+\infty[$.

To prove Theorem 7.1 we need some technical results. Let consider the
semigroup of the discrete Laplacian $\frac{1}{2}N^{2}\Delta^{N}$ on
${\mathbb{T}}^{d}_{N}$ and $\mathbb{Z}^{d}$, denoted by $s^{N}(t,i,j)$ and
$s_{0}^{N}(\gamma t,i,j)$ respectively. In particular we have that
$s^{N}(t,i,j)=p_{N}(t,i,j)$, where $p_{N}(t,i,j)$ is heat kernel of discrete
Laplacian on the discrete torus, cf. (5.2).

For any $\lambda,\gamma\geq 0$ and
$f:\frac{1}{N}{\mathbb{T}}^{d}_{N}\to\mathbb{R}$, we let
$(S^{N,\lambda,\gamma}_{t})$ be the semigroup defined by

$S^{N,\lambda,\gamma}_{t}f(x)=\sum_{y\in\frac{1}{N}{\mathbb{T}}^{d}_{N}}e^{-\lambda
t}s^{N}(\gamma t,Nx,Ny)f(y)=\sum_{y\in\frac{1}{N}\mathbb{Z}^{d}}e^{-\lambda
t}s_{0}^{N}(\gamma t,Nx,Ny)\widetilde{f}(y)$ (7.2)

where, as in (6.6), $\widetilde{f}$ is the periodic extension of $f$ on
$\frac{1}{N}\mathbb{Z}^{d}$.

In the remaining part of this article we will consider
$S^{N,\lambda,\gamma}_{t}f(x)$ with $f\in\mathcal{C}({\mathbb{T}}^{d})$. In
that case we mean that the function $f$ is restricted on
$\frac{1}{N}{\mathbb{T}}^{d}_{N}\subset{\mathbb{T}}^{d}$, which is equivalent
to consider $f^{N}(x):=f(\lfloor Nx\rfloor/N)$. We observe that if $f$ is also
Lipschitz, then $\|f-f^{N}\|_{{\mathbb{T}}^{d}}\leq\frac{c}{N^{d}}$ for some
$c>0$ and $\|f^{N}\|_{{\mathbb{T}}^{d}}\leq\|f\|_{{\mathbb{T}}^{d}}$. Then,
with the same extension to ${\mathbb{T}}^{d}$ for $s^{N}$ we can write, for
any $x\in{\mathbb{T}}^{d}$

$S^{N,\lambda,\gamma}_{t}f(x)=\int_{{\mathbb{T}}^{d}}e^{-\lambda
t}N^{d}s^{N}(\gamma t,Nx,Ny)f^{N}(y){\mathrm{d}}y.$ (7.3)

By a slight abuse of notation, we still denote by $v^{N}$ the linear
interpolation on ${\mathbb{T}}^{d}$ such that
$v^{N}(t,\frac{i}{N})=v^{N}_{i}(t)$. We also redefine the function $H$ on the
torus ${\mathbb{T}}^{d}$ by the linear interpolation such that
$H(\frac{i}{N},v^{N})=H(i,v^{N})$ and we define $H^{N}$ as $f^{N}$ in (7.3).

######  Definition 7.1.

Let $N\in\mathbb{N}$ and $v_{0}^{N}\in L^{\infty}({\mathbb{T}}^{d})$. We say
that $(v^{N}(t,x),t\geq 0,x\in{\mathbb{T}}^{d})$ is a mild solution of (7.1)
if

  * •

for any $N$, $v^{N}$ is continuous from $\mathbb{R}_{+}^{*}$ to
$L^{\infty}({\mathbb{T}}^{d})$,

  * •

for all $t>0$ and $x\in{\mathbb{T}}^{d}$

$v^{N}(t,x)=S^{N,2\beta+1,4N^{2}\alpha}_{t}(v_{0}^{N})(x)+\int_{0}^{t}S^{N,2\beta+1,4N^{2}\alpha}_{t-s}\Big{[}H^{N}(\cdot,v^{N}(s,\cdot))\Big{]}(x){\mathrm{d}}s\,.$
(7.4)

Let $u^{N}$ be the unique solution of (4.10), so that $v^{N}=2u^{N}-1$
satisfies (7.1). Of course, for any $N$ the solution $v^{N}$ of (7.4) exists
and it is unique.

The proof of Theorem 7.1 is based on the representation of $v^{N}$ as in
(7.4). We define $\widetilde{v}^{N}$ as a slight modification of (7.4), that
is,

$\widetilde{v}^{N}(t,x):=S^{2\beta+1,4\alpha}_{t}(v_{0})(x)+\int_{0}^{t}S^{2\beta+1,4\alpha}_{t-s}\Big{[}H(\cdot,v^{N}(s,\cdot))\Big{]}(x){\mathrm{d}}s\,.$
(7.5)

######  Lemma 7.2.

For any $\tau>0$

$\lim_{N\to+\infty}\|\widetilde{v}^{N}-v^{N}\|_{[1/\tau,\tau]\times{\mathbb{T}}^{d}}=0.$
(7.6)

###### Proof.

Let $\tau>0$, then

$\sup_{t\in[\frac{1}{\tau},\tau],\,x\in{\mathbb{T}}^{d}}\Big{|}\widetilde{v}^{N}(t,x)-v^{N}(t,x)\Big{|}\leq\sup_{t\in[\frac{1}{\tau},\tau],\,x\in{\mathbb{T}}^{d}}\Big{|}S^{N,\lambda,N^{2}\gamma}_{t}v_{0}^{N}(x)-S^{\lambda,\gamma}_{t}v_{0}(x)\Big{|}\\\
+\sup_{t\in[\frac{1}{\tau},\tau],\,x\in{\mathbb{T}}^{d}}\Big{|}\int_{0}^{t}\Big{\\{}S^{N,2\beta+1,4N^{2}\alpha}_{t-s}\Big{[}H^{N}(\cdot,v^{N}(s,\cdot))\Big{]}(x)-S^{2\beta+1,4\alpha}_{t-s}\Big{[}H(\cdot,v^{N}(s,\cdot))\Big{]}(x)\Big{\\}}{\mathrm{d}}s\Big{|}.$
(7.7)

We show that the right hand side of (7.7) converges to $0$. We detail the
convergence of the second term, which is more delicate. The argument can be
adapted to the first term by using Assumption 2 which ensures that $v_{0}^{N}$
converges to $v_{0}$ in $C({\mathbb{T}}^{d})$.

We fix $\varepsilon\in(0,\frac{1}{\tau})$ and we get that

$\Big{|}\int_{0}^{t}S^{N,2\beta+1,4N^{2}\alpha}_{t-s}\Big{[}H^{N}(\cdot,v^{N}(s,\cdot))\Big{]}(x)-S^{2\beta+1,4\alpha}_{t-s}\Big{[}H(\cdot,v^{N}(s,\cdot))\Big{]}(x){\mathrm{d}}s\Big{|}\\\
\leq\Big{|}\int_{0}^{t-\varepsilon}S^{N,2\beta+1,4N^{2}\alpha}_{t-s}\Big{[}H^{N}(\cdot,v^{N}(s,\cdot))\Big{]}(x)-S^{2\beta+1,4\alpha}_{t-s}\Big{[}H(\cdot,v^{N}(s,\cdot))\Big{]}(x){\mathrm{d}}s\Big{|}+C\varepsilon\,,$
(7.8)

where we used that Lemma B.6 which implies that $H(\cdot,v^{N})$ is bounded by
$1$ uniformly on $N$. The integral on the right hand side of (7.8) is bounded
from above by

$\int_{0}^{t-\varepsilon}e^{-(2\beta+1)(t-s)}\Big{|}\int_{{\mathbb{T}}^{d}}\Big{(}N^{d}s^{N}(4N^{2}\alpha(t-s),Nx,Ny)-s(4\alpha(t-s),x,y)\Big{)}H^{N}(y,v^{N}(s,y))\,{\mathrm{d}}y\,\Big{|}\,{\mathrm{d}}s\,\\\
+\int_{0}^{t-\varepsilon}e^{-(2\beta+1)(t-s)}\int_{{\mathbb{T}}^{d}}s(4\alpha(t-s),x,y)\Big{|}H^{N}(y,v^{N}(s,y))-H(y,v^{N}(s,y))\Big{|}\,{\mathrm{d}}y{\mathrm{d}}s\,.$
(7.9)

Since $\sup_{s\geq
0,\,y\in{\mathbb{T}}^{d}}\Big{|}H^{N}(y,v^{N}(s,y))-H(y,v^{N}(s,y))\Big{|}\leq\frac{1}{N}$,
the second integral is smaller than $\frac{c_{\alpha,\beta}}{N}$. For the
first integral, we first use that $H(\cdot,v^{N})$ is bounded by $1$ uniformly
on $N$ and then we operate the change of variable $u=t-s$ which gives that it
is bounded from above by

$\displaystyle\int_{\varepsilon}^{t}e^{-(2\beta+1)u}\int_{{\mathbb{T}}^{d}}\Big{|}\,N^{d}s^{N}(4N^{2}\alpha
u,Nx,Ny)-s(4\alpha u,x,y)\Big{|}\,{\mathrm{d}}y\,{\mathrm{d}}u\,$ (7.10)

We now use the local central limit theorem (cf. Theorem 2.1.1 and (2.5) in
[17]): let $\rho$ be the Gaussian Kernel and
$\rho(u,x,y)=\frac{1}{u^{d/2}}\rho\Big{(}\frac{x-y}{u^{1/2}}\Big{)}$, then

$\psi_{\varepsilon,\tau,N}:=\sup_{u\in[\varepsilon,\tau],\,y\in\frac{1}{N}\mathbb{Z}^{d}}\Big{|}N^{d}p(uN^{2},Nx,Ny)-\rho(u,x,y)\Big{|}\xrightarrow[]{N\to+\infty}0.$
(7.11)

We also observe that by symmetry the supremium in (7.11) is independent of
$x$. Moreover, by Proposition 2.4.6 in [17] we have that there exist
$c_{1},c_{2}>0$ independent of $x,y,u$ such that

$\Big{|}N^{d}p(uN^{2},Nx,Ny)-\rho(u,x,y)\Big{|}\leq\frac{c_{1}}{u^{\frac{d}{2}}}e^{-c_{2}\frac{\|x-y\|^{2}}{u}}.$
(7.12)

In such a way, for any $M>0$ fixed we write ${\mathbb{T}}^{d}=B_{x}(M)\cup
B_{x}(M)^{c}$ and we get that (7.10) is smaller than

$c_{d}\int_{\varepsilon}^{t}e^{-(2\beta+1)u}\Big{(}M^{d}\psi_{N,\varepsilon,\tau}+\frac{c_{1}}{M^{d}u^{\frac{d}{2}}}e^{-c_{2}\frac{M^{2d}}{u}}\Big{)}\,{\mathrm{d}}s\leq
C_{d}\Big{(}M^{d}\psi_{N,\varepsilon,\tau}+\frac{1}{M^{d}}\Big{)},$ (7.13)

where $c_{d},C_{d}>0$ are two positive constants that depend only on the
dimension $d$.

We conclude that the right hand side of (7.8) is bounded by
$\frac{c_{\alpha,\beta}}{N}+cM^{d}\psi_{N,\varepsilon,\tau}+\frac{c}{M^{d}}+C\varepsilon$,
uniformly on $x\in{\mathbb{T}}^{d}$ and $t\in[\frac{1}{\tau},\tau]$.
Therefore, by taking the limit on $N\to+\infty$ and then on $M\to+\infty$ and
$\varepsilon\to 0$ we conclude the proof. ∎

###### Proof of Theorem 7.1.

We control $\widetilde{v}^{N}$ to get the convergence of $v^{N}$. We observe
that since $H(\cdot,v^{N}(s,\cdot))$ is uniformly bounded so that by
Proposition 6.1 $\widetilde{v}^{N}$ is uniformly bounded in $N$, uniformly
continuous on $[1/\tau,\tau]\times{\mathbb{T}}^{d}$, and the modulus of
continuity only depends on $\tau>1$. By the Ascoli-Arzela Theorem, the
sequence $(\widetilde{v}^{N})_{N}$ is pre-compact on
$C([1/\tau,\tau]\times{\mathbb{R}}^{d})$ and therefore, by Lemma (7.2),
$(v^{N})$ also. By a diagonal argument, we can extract from $(v^{N})$ a
subsequence converging uniformly to a limit $v_{\infty}$ in
$C(]0,\infty[\times{\mathbb{R}}^{d})$, uniformly on each compact.

Using Corollary B.4, we can adapt the argument used in the proof of
Proposition 6.3 (6.25–6.30) to get that each accumulation point $v_{\infty}$
is a mild solution of (6.23), we omit the details. ∎

###  7.1 Proof of Theorem 3.1

Theorem 3.1 is now a consequence of Theorems 4.1 and 7.1.

##  Appendix A Concentration inequalities

We follow the definitions in Jara and Menezes [15] and [14] and Boucheron,
Lugosi, Massart [1], Section 2.3. We omit the proofs since there are present
in the references.

######  Definition A.1 ([15] and [1], Section 2.3).

Let $X$ be a real random variable. $X$ is said to be sub-Gaussian with
variance parameter $\sigma^{2}$ if, for all $t\in{\mathbb{R}}$

$\psi_{X}(t):=\log\mathbb{E}(\exp(tX))\leqslant\sigma^{2}\frac{t^{2}}{2}.$
(A.1)

We denote ${\mathcal{G}}(\sigma^{2})$ the set of real sub-Gaussian random
variables with variance parameter $\sigma^{2}$.

######  Proposition A.1.

[[1] and [14], Proposition F.7] The following statements are equivalent :

  1. 1.

$X\in{\mathcal{G}}(\sigma^{2})$

  2. 2.

$\mathbb{P}(|X|>t)\leqslant 2\exp(-\frac{t^{2}}{2\sigma^{2}})$

  3. 3.

$\mathbb{E}(\exp(\gamma X^{2}))\leqslant 3$ for all
$0\leqslant\gamma<\frac{1}{4\sigma^{2}}$.

Let us complete our family of inequality :

######  Lemma A.2.

[[14], Proposition F.8] Let $X\in{\mathcal{G}}(\sigma_{1}^{2})$ and
$Y\in{\mathcal{G}}(\sigma_{2}^{2})$, then for all
$0\leqslant\gamma<\frac{1}{4\sigma_{1}\sigma_{2}}$,

$\mathbb{E}(\exp(\gamma XY))\leqslant 3$

######  Lemma A.3.

[[14], Proposition F.12] Let $X_{1},\dots,X_{n}$ be random variables with
$X_{i}\in{\mathcal{G}}(\sigma_{i}^{2})$, such that there is a partition of $K$
subsets $P_{1},\dots,P_{K}$ each containing $L$ variables that
$\Sigma_{k}=\sigma(X_{i},i\in P_{k})$ are independent $\sigma$-algebra then,
for all real $\alpha_{1},\dots,\alpha_{n}$, the random variable
$Y=\sum_{i}\alpha_{i}X_{i}$ is sub-Gaussian with variance parameter
$L\sum_{i}\alpha_{i}^{2}\sigma_{i}^{2}$.

Note that if $L=1$, the variables are independent.

######  Lemma A.4 (Hoeffeding Inequality, [1], Section 2.3).

Let $X$ be a bounded random variable with $X\in[a,b]$, then
$X-\mathbb{E}X\in{\mathcal{G}}\left(\frac{(b-a)^{2}}{4}\right)$.

##  Appendix B Controls for the non-linearities

In this section, we collect some results about the specifics of our model.
Recall the notations: for $\eta\in\\{0,1\\}^{{\mathbb{T}}^{d}_{N}}$

$\displaystyle c_{0}^{+}(\eta)$
$\displaystyle:=\mathbbm{1}_{\\{\rho_{0}(\eta)\geqslant
1-\frac{\kappa_{N}}{K_{N}}\\}},$ $\displaystyle c_{0}^{-}(\eta)$
$\displaystyle:=\mathbbm{1}_{\\{\rho_{0}(\eta)\leqslant\frac{\kappa_{N}}{K_{N}}\\}},$

and $\kappa_{N}=\kappa_{N}(T):=\min\Big{\\{}\lfloor K_{N}T\rfloor-1;\lfloor
K_{N}(1-T)\rfloor\Big{\\}}$. For any function
$u=(u_{i})_{i\in{\mathbb{T}}^{d}_{N}}$, we define
$\upsilon_{u}({\mathrm{d}}\eta)=\upsilon_{u}^{N}({\mathrm{d}}\eta):=\bigotimes_{i\in{\mathbb{T}}^{d}_{N}}\mathrm{B}\big{(}u_{i})$
where $B(u_{i})$ denote a Bernoulli distribution with parameter $u_{i}$. and
we let $c_{0}^{+}(u)$ and $c_{0}^{-}(u)$ be the expectations of
$c_{0}^{+}(\eta)$ and $c_{0}^{-}(\eta)$ under $\upsilon_{u}$. We set
$(\tau_{i}\eta)_{j}=\eta_{i+j}$, and likewise $\tau_{i}$ acts on $u$. Then,

$\displaystyle G(u):=(1-u_{0})c_{0}^{+}(u)-u_{0}c_{0}^{-}(u)$
$\displaystyle=(1-u_{0})\mathbb{P}_{\upsilon_{u}}\left[\rho_{0}(\eta)\geqslant
1-\frac{\kappa_{N}}{K_{N}}\right]-u_{0}\mathbb{P}_{\upsilon_{u}}\left[\rho_{0}(\eta)\leqslant\frac{\kappa_{N}}{K_{N}}\right]$
(B.1)

and $G(i,u):=G(\tau_{i}u)$. We start with some results on the non linearity
$G$ :

######  Proposition B.1.

Let $u,v\in[0,1]^{{\mathbb{T}}^{d}_{N}}$ such that $u_{i}\geqslant v_{i}$ for
all $i\in{\mathcal{V}}_{N}$ and $u_{0}=v_{0}$. Then $G(0,u)\geqslant G(0,v)$.

###### Proof.

We construct a coupling between $\upsilon_{u}$ and $\upsilon_{v}$: let
$(U_{i})$ be independent and identically distributed random variables uniform
on $[0,1]$. Define $\eta_{i}=\mathbbm{1}_{\\{U_{i}\leqslant u_{i}\\}}$ and
$\eta^{\prime}_{i}=\mathbbm{1}_{\\{U_{i}\leqslant v_{i}\\}}$. We have
$\eta_{i}\geqslant\eta^{\prime}_{i}$ for all $i\in{\mathcal{V}}_{N}$,
therefore $\rho_{0}(\eta)\geqslant\rho_{0}(\eta^{\prime})$. This proves that
$c^{+}_{0}(u)\geqslant c^{+}_{0}(v)$ and $c^{-}_{0}(u)\leqslant c^{-}_{0}(v)$.
The results follows since $u_{0}=v_{0}$. ∎

For $p\in[0,1]$, $T\in[0,1]$, $K\in\mathbb{N}^{*}$, we let
$\kappa(K,T)=\min\Big{\\{}\lceil KT\rceil-1;\lfloor
K(1-T)\rfloor\Big{\\}}\leqslant K/2$, and we define $g_{K}(p)$ as

$g_{K}(p):=(1-p)\mathbb{P}_{p}\big{[}X>K-\kappa(K,T)\big{]}-p\mathbb{P}_{p}\big{[}X\leqslant\kappa(K,T)\big{]}$

where $X$ is a random variable with binomial distribution with parameter
$(K,p)$ under $\mathbb{P}_{p}$. In particular, we have that
$g_{K_{N}}(p)=G(i,u)$ for $u(t,i)=p$ for all $i\in{\mathbb{T}}^{d}_{N}$. Note
that $g_{K}$ is $C^{\infty}([0,1])$. We also have

$\begin{split}g_{K}(p)&=(1-p)\mathbb{P}_{1-p}\big{[}X<\kappa(K,T)\big{]}-p\mathbb{P}_{p}\big{[}X\leqslant\kappa(K,T)\big{]}\\\
&=\sum_{k=0}^{\kappa(K,T)-1}\binom{K}{k}\left[(1-p)^{k+1}p^{K-k}-p^{k+1}(1-p)^{K-k}\right]-\binom{K}{\kappa(K,T)}p^{\kappa(K,T)+1}(1-p)^{K-\kappa(K,T)}.\end{split}$
(B.2)

We recall
$g_{\infty}(p):=(1-p)\mathbbm{1}_{\\{1-p<p_{0}(T)\\}}-p\mathbbm{1}_{\\{p\leqslant
p_{0}(T)\\}}$, where $p_{0}(T)=\min(T,1-T)$.

The following proposition estimates the convergence of $g_{K}$ to
$g_{\infty}$, in particular we prove that close to $p=0$ (resp. $p=1$),
$g_{K}$ is negative (resp. positive).

######  Proposition B.2.

For all $K\in\mathbb{N}^{*}\cup\\{\infty\\}$, we have $g_{K}(0)=g_{K}(1)=0$.
For all $K\in\mathbb{N}^{*}$ and $p\in[0,1]$, we have, for
$|p-p_{0}(T)|>\frac{1}{K}$ and $|(1-p)-p_{0}(T)|>\frac{1}{K}$

$|g_{K}(p)-g_{\infty}(p)|\leqslant
2e\exp(-2K(p_{0}(T)-p)^{2})+2e\exp(-2K(p_{0}(T)-(1-p))^{2})$ (B.3)

In particular, for any $0<\delta<\frac{1}{4e}$, $T\in[4\delta,1-4\delta]$, and
$K>\frac{|\log(\delta)|}{4\delta^{2}}$, we have that $g_{K}(2\delta)<-\delta$
and $g_{K}(1-2\delta)>\delta$.

###### Proof.

We consider $g_{K}$ written as in (B.2). Then, the values at $p=0$ and $p=1$
are obvious.

Note that, under $\mathbb{P}_{p}$, $\frac{X}{K}-p$ converges to $0$ (in
$L^{2}(\Omega_{N})$) and is sub-Gaussian with variance parameter
$\frac{1}{4K}$, thus, for any $t>0$,

$\displaystyle\mathbb{P}_{p}\left(\left|\frac{X}{K}-p\right|>t\right)\leqslant
2\exp(-2Kt^{2}).$

We also have that

$\displaystyle 0\leqslant p_{0}(T)-\frac{\kappa(K,T)}{K}\leqslant\frac{1}{K}.$

Then, for $p>p_{0}(T)$,

$\begin{split}\mathbb{P}_{p}\big{[}X\leqslant\kappa(K,T)\big{]}&=\mathbb{P}_{p}\left[p-\frac{X}{K}\geqslant
p-p_{0}(T)+p_{0}(T)-\frac{\kappa(K,T)}{K}\right]\leqslant\mathbb{P}_{p}\left[p-\frac{X}{K}\geqslant
p-p_{0}(T)\right]\\\ &\leqslant 2\exp(-2K(p-p_{0}(T))^{2}),\end{split}$

and for $p<p_{0}(T)-\frac{1}{K}$,

$\begin{split}\mathbb{P}_{p}[X>\kappa(K,T)]&=\mathbb{P}_{p}\left[\frac{X}{K}-p>\frac{\kappa(K,T)}{K}-p_{0}(T)+p_{0}(T)-p\right]\leqslant\mathbb{P}_{p}\left[\frac{X}{K}-p\geqslant
p_{0}(T)-p-\frac{1}{K}\right]\\\ &\leqslant
2\exp(-2K(p_{0}(T)-p-1/K)^{2})\leqslant
2e^{2(p_{0}(T)-p)}\exp(-2K(p_{0}(T)-p)^{2})\\\ &\leqslant
2e\exp(-2K(p_{0}(T)-p)^{2}).\end{split}$

Thus, we have the following, for $|p-p_{0}(T)|>\frac{1}{K}$,

$\displaystyle\left|\mathbb{P}_{p}[X\leqslant\kappa(K,T)]-\mathbbm{1}_{\\{p\leqslant
p_{0}(T)\\}}\right|<2e\exp(-2K(p_{0}(T)-p)^{2}).$

The results follows with the same estimates with $(1-p)$ instead of $p$. This
prove (B.3).

Let $\delta>0$ and set $p=2\delta$, and $T\in[4\delta,1-4\delta]$, then
$|2\delta-p_{0}(T)|\leqslant 2\delta$ for $K>\frac{1}{2\delta}$ and we have
the same for $1-p=1-2\delta$. Applying the result, we have

$\displaystyle|g_{K}(2\delta)-g_{\infty}(2\delta)|$
$\displaystyle=|g_{K}(2\delta)+2\delta|\leqslant 4e\exp(-8K\delta^{2})$

Then, if $4e\exp(-8K\delta^{2})<\delta$, we get the result. This happens if
$\exp(-8K\delta^{2})<\delta^{2}$ and $\delta<\frac{1}{4e}$, which gives the
condition on $K$. ∎

######  Proposition B.3.

Let $v:{\mathbb{T}}^{d}\to[0,1]$ be a continuous fixed density on the torus.
For any $i\in{\mathbb{T}}^{d}_{N}$ we let $u_{i}:=v_{i/N}$. Then as $i/N\to
x$, we have that $u_{i}$ converges to $v_{x}$. Let
$\upsilon_{u}=\bigotimes_{i\in{\mathbb{T}}^{d}_{N}}\mathrm{B}\big{(}u_{i})$.
Then, $\rho_{0}(\eta)$ converges to $v_{0}$ in probability.

###### Proof.

We use the coupling introduced in Lemma B.1: we let $(U_{i})$ be i.i.d uniform