# Fluctuations for Spatially Extended Hawkes Processes

Julien Chevallier<EMAIL_ADDRESS>Univ. Grenoble
Alpes, CNRS, Grenoble INP111Institute of Engineering Univ. Grenoble Alpes,
LJK, 38000 Grenoble, France  Guilherme Ost<EMAIL_ADDRESS>Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brazil

###### Abstract

In a previous paper [10], it has been shown that the mean-field limit of
spatially extended Hawkes processes is characterized as the unique solution
$u(t,x)$ of a neural field equation (NFE). The value $u(t,x)$ represents the
membrane potential at time $t$ of a typical neuron located in position $x$,
embedded in an infinite network of neurons. In the present paper, we
complement this result by studying the fluctuations of such a stochastic
system around its mean field limit $u(t,x)$. Our first main result is a
central limit theorem stating that the spatial distribution associated to
these fluctuations converges to the unique solution of some stochastic
differential equation driven by a Gaussian noise. In our second main result we
show that the solutions of this stochastic differential equation can be well
approximated by a stochastic version of the neural field equation satisfied by
$u(t,x)$. To the best of our knowledge, this result appears to be new in the
literature.

Keywords: Hawkes Processes, Central Limit Theorem, Neural Field Equations,
network of neurons

##  1 Introduction

We consider multivariate point processes $(N^{1},\ldots,N^{n})$ on
$[0,\infty)$ representing the time occurrences of action potentials (often
called _spikes_) of a network of $n$ neurons. We assume that the intensity
process of $N^{i}$ is of the form

$\begin{cases}\lambda^{i}_{t}=f\left(U^{i}_{t-}\right),\\\
U^{i}_{t}=e^{-\alpha
t}u_{0}(x_{i})+\frac{1}{n}\sum_{j=1}^{n}w(x_{j},x_{i})\int_{0}^{t}e^{-\alpha(t-s)}dN^{j}_{s}.\end{cases}$
(1)

In the above formula, $U^{i}_{t}$ describes the membrane potential of neuron
$i$ at time $t\geq 0$ and $x_{i}=i/n$ represents the position of neuron $i$ in
the network. The function $f:{\mathbb{R}}\to{\mathbb{R}}_{+}$ is the _firing
rate_ of each neuron. The function $w:[0,1]\times[0,1]\to{\mathbb{R}}$ is the
matrix of _synaptic strengths_. It introduces a spatial structure in the
model; the value $w(x_{j},x_{i})$ models the influence of a spike of neuron
$j$ on neuron $i$, as a function of their positions. On the one hand, when the
sign of $w(x_{j},x_{i})$ is positive, neuron $j$ excites neuron $i$. On the
other hand, if the sign of $w(x_{j},x_{i})$ is negative, neuron $j$ inhibits
neuron $i$. The _leakage rate_ is modeled by the parameter $\alpha\geq 0.$ The
function $u_{0}:[0,1]\to{\mathbb{R}}$ describes the membrane potential of all
neurons in the network at time $t=0$. We refer to $f,w,u_{0}$ and $\alpha$ as
parameters of the multivariate point process $(N^{1},\ldots,N^{n}).$

Such point processes are known as nonlinear Hawkes Processes, named after the
pioneer work of A. G. Hawkes [20] where the model has been introduced in the
linear case (i.e., for $f$ linear). Their defining characteristic is that past
events (spikes in our framework) can affect the probability of future events
to occur. The literature of neuronal modeling via Hawkes processes is vast. To
cite just a few articles, see for instance [7, 9, 11, 15, 19, 22, 25, 31, 34]
and the references therein.

Recently, in [10], the authors have established a connection between solutions
of (scalar) neural field equations(NFE) and mean field limits of nonlinear
Hawkes processes. Specifically, it has been proved that the multivariate
process $(U^{1}_{t},\ldots,U^{n}_{t})_{t}$ defined in (1) converges as
$n\to\infty$, under some assumptions on the parameters of the model, to a
deterministic function $u(t,x)$ which solves the neural field equation:

$\begin{cases}\dfrac{\partial}{\partial t}u(t,x)=-\alpha
u(t,x)+\int_{0}^{1}w(y,x)f(u(t,y))dy,\ t>0\ \mbox{and}\ x\in[0,1],\\\
u(0,x)=u_{0}(x).\end{cases}$ (2)

Here, $u(t,x)$ represents the membrane potential at time $t$ of a typical
neuron located in position $x$, embedded in an infinite network of neurons.
Neural field equations have been widely studied in the literature since the
pioneer works of Wilson, Conwan [38, 39] and Amari [1] in the 1970s. Such
models have attracted a great interest from the scientific community, due to
its wide range of applications and mathematical tractability; see [6] for a
recent and comprehensive review.

The goal of the present paper is to complement the results in [10] by
describing the fluctuations of the process $(U^{1}_{t},\ldots,U^{n}_{t})_{t}$
around its mean field limit $u(t,x).$ More precisely, by writing
$\eta^{i}_{t}=n^{1/2}(U^{i}_{t}-u(t,x_{i}))$ to denote the _individual
fluctuations_ , the purpose of this paper is to study the convergence of the
sequence of stochastic processes $(\Gamma^{n}_{t})_{t}$ as $n\to\infty$, where
$\Gamma^{n}_{t}$ is the random signed measure on $\mathcal{S}^{\prime}$
(representing the _spatial fluctuations_) defined as

$\Gamma^{n}_{t}(dx)=\frac{1}{n}\sum_{i=1}^{n}\eta^{i}_{t}\delta_{x_{i}}(dx).$
(3)

Here, the set $\mathcal{S}^{\prime}$ denotes the dual space of the Fréchet
space $S=\mathcal{C}^{\infty}([0,1])$, the space of all real-valued functions
on $[0,1]$ with continuous derivatives of all orders. Fix $T\geq 0$, denote
$\Gamma^{n}=(\Gamma^{n}_{t})_{0\leq t\leq T}$ and observe that
$\Gamma^{n}\in\mathcal{D}([0,T],\mathcal{S}^{\prime}),$ the space of càdlàg
functions from $[0,T]$ to $\mathcal{S}^{\prime}$. Our first main result,
namely Theorem 1, is a Central Limit Theorem saying that under some
assumptions on the parameters of the model, the sequence of processes
$(\Gamma^{n})_{n\geq 1}$ converges in law to a limit process
$\Gamma=(\Gamma_{t})_{0\leq t\leq T}$ as $n\to\infty$. Moreover, the limit
process $\Gamma$ belongs to $\mathcal{C}([0,T],\mathcal{S}^{\prime})$, the set
of continuous functions from $[0,T]$ to $\mathcal{S}^{\prime}$, and for each
$t\geq 0$, the measure $\Gamma_{t}\in\mathcal{S}^{\prime}$ is characterized by
the following identity: for all $\varphi\in\mathcal{S}$,

$\Gamma_{t}(\varphi)=e^{-\alpha
t}M_{t}(\varphi)+\int_{0}^{t}e^{-\alpha(t-s)}\Gamma_{s}\left(\int_{0}^{1}\varphi(x)w(\cdot,x)f^{\prime}(u(s,\cdot))dx\right)ds,$
(4)

where $M=(M_{t})_{t\geq 0}$ is a continuous centered Gaussian process taking
values in $\mathcal{S}^{\prime}$ with covariance function given, for all
$t_{1},t_{2}\geq 0$ and $\varphi_{1},\varphi_{2}\in\mathcal{S}$, by

$\begin{cases}{\mathbb{E}}(M_{t_{1}}(\varphi_{1})M_{t_{2}}(\varphi_{2}))=\int_{0}^{t_{1}\wedge
t_{2}}\int_{0}^{1}e^{2\alpha
s}I[\varphi_{1}](y)I[\varphi_{2}](y)f(u(s,y))dyds,\\\
I[\varphi](y)=\int_{0}^{1}\varphi(x)w(y,x)dx,\quad y\in[0,1],\end{cases}$ (5)

and $u(t,x)$ is the solution of (2). The interested reader is referred to [26,
$\Phi^{\prime}$-Wiener processes] for details on such Gaussian processes.

Let us give some intuition about Equation (4). The first term in the RHS of
(4), namely $e^{-\alpha t}M_{t}(\varphi)$, comes from the error one makes when
replacing the point measure $dN^{i}_{t}$ by the intensity measure
$f(U^{i}_{t})dt$. It is the diffusion approximation for point processes:
formally taking $\varphi_{1}=\varphi_{2}=\delta_{x}$ the Dirac mass at
position $x$, one obtains in Equation (5) the product $w(y,x)^{2}f(u(s,y))$
which is the limit variance of the jumps induced by spiking neurons in
position $y$ onto neurons in position $x$, at time $s$. The second term in the
RHS of (4) comes from the error one makes when replacing the intensity
$f(U^{i}_{t})$ by the limit one $f(u(t,x_{i}))$: the linearization of $f$
gives the product of the derivative $f^{\prime}$ times the difference between
$U^{i}_{t}$ and $u(t,x_{i})$ (which is encapsulated in $\eta^{i}_{t}$ and so
in the spatial fluctuation $\Gamma^{n}_{t}$).

The study of the fluctuations is a natural follow-up to the study of the mean-
field limits for interacting particle systems (see for instance [5, 8, 13, 14,
21, 27, 28, 29, 36]). These results are not only interesting per se, they are
also relevant from an applied point of view. Indeed, in the mean-field limit,
typically one can show that the so-called propagation of chaos property holds,
meaning that evolution of any finite number of particles (the neurons in our
framework) become independent (see for instance [2, 4]). In other terms, mean
field limits neglect the correlations between particles which are present in
finite (but large) systems. In contrast, the correlations do appear in the
fluctuations, in particular in the covariance kernel (5).

With slight abuse of terminology, the mean field limit $u_{t}=u(t,\cdot)$,
which can be seen as an element of $\mathcal{S}^{\prime}$ given by
$u_{t}(\varphi)=\int_{0}^{1}\varphi(x)u(t,x)dx$, can be thought of as a
_zeroth-order approximation_ of the finite size system
$(U^{1}_{t},\ldots,U^{n}_{t})_{t}$. In that respect, we say that the following
process with values in $\mathcal{S}^{\prime}$,

$(u_{t}+n^{-1/2}\Gamma_{t})_{t},$ (6)

is a _first-order approximation_ of the finite size system, this last
definition being justified by our Central Limit Theorem. In addition to the
Central Limit Theorem, we also investigate here the link between the first-
order approximation and the solution of the following stochastic neural field
equation

$\begin{cases}dV^{n}_{t}(x)=\left(-\alpha
V^{n}_{t}(x)+\int_{0}^{1}w(y,x)f(V^{n}_{t}(y))dy\right)dt+\int_{0}^{1}w(y,x)\frac{\sqrt{f(V^{n}_{t}(y))}}{\sqrt{n}}W(dt,dy),\\\
V^{n}_{0}(x)=u_{0}(x),\end{cases}$ (7)

where $W$ is a Gaussian white noise on ${\mathbb{R}}_{+}\times[0,1]$. Loosely
speaking, in our second main result, namely Theorem 6, we show that the
process $(u_{t}+n^{-1/2}\Gamma_{t})_{t}$ is an “almost” solution of (7). To
the best of our knowledge, this result appears to be new in the literature and
is of independent interest. To some extent, the solutions of (7) can be
interpreted as an intermediate modeling scale, sometimes called _mesoscopic
scale_ , between the microscopic scale given by Hawkes process (1) and the
macroscopic scale one given by neural field equation (2). In order to give
sense to solutions of (7) we follow the approach developed by Walsh (see for
instance [17], [12] and the seminal lecture notes [37]). Some heuristics
arguments leading to the stochastic neural field equation (7) are provided in
Section 8.1. Let us mention the article [8] which discusses similar results in
a non rigorous way in the context of non linear stochastic partial
differential equations.

The literature devoted to mean-field limits is usually concerned with the
convergence of an empirical measure towards a probability measure which is
characterized as the solution of some partial differential equation. It is
worth mentioning that it is not the case here: the mean-field equation (2) is
not satisfied by a probability density of the potential but by the value of
the potential itself. This difference makes the study of (7) simpler: the
square root term, namely $\sqrt{f(V_{t}(y))}$, is trivially well-defined which
is not the case when the mean field limit concerns an empirical measure (see
[8] for instance).

The results of the present paper are stated in the distribution space
$\mathcal{S}^{\prime}$ so the parameters of the model ($f$, $w$ and $u_{0}$)
are assumed to be smooth. Concerning the rate function $f$, we also assume
that its first and second derivatives are bounded (in particular, $f$ is
Lipschitz) and that it is lower-bounded by a positive constant (only in the
last section). No additional assumptions on the model are needed and, in
particular, the function $f$ could be unbounded.

The present paper is organized as follows. In Section 2, the notation used
throughout the paper is introduced, the model is described and our first main
result, Theorem 1, is stated. In Section 3, some regularity properties of
solutions of the neural field equation are derived. Uniform estimates on the
second moment of the individual fluctuations (used all along the paper) are
provided in Section 4. Section 5 is devoted to the proof of the tightness of
the sequence $(\Gamma^{n})_{n}$ defined in (3). In Section 6, we show that the
limit of any converging sub-sequence of $(\Gamma^{n})_{n}$ solves the limit
equation (4). In Section 7, the uniqueness of solutions of the limit equation
(4) is proved which concludes the proof of the Central Limit Theorem (Theorem
1). In Section 8, we first develop the mathematical framework required to
study the stochastic neural field equation (7) and then we prove our second
main result, Theorem 6, which makes the link between the first-order
approximation (6) and the stochastic neural field equation (7). Some technical
results used in the previous sections are collected in the A. We include in B
some basic definitions about Fréchet spaces.

##  2 General notation, model definition and the central limit theorem

###  2.1 General notation

Let $E$ and $F$ be some metric spaces. The space of continuous (respectively
càdlàg) functions from $E$ to $F$ is denoted by $\mathcal{C}(E,F)$ (resp.
$\mathcal{D}(E,F)$). When $F={\mathbb{R}}$, we write $\mathcal{C}(E)$ (resp.
$\mathcal{D}(E)$) instead of $\mathcal{C}(E,{\mathbb{R}})$ (resp.
$\mathcal{D}(E,{\mathbb{R}})$). For each integer $n\geq 1,$ let
$[n]=\\{1,\ldots,n\\}$. We write $\mathcal{C}^{\infty}([0,1])$ (resp.
$\mathcal{C}^{\infty}({\mathbb{R}})$) to denote the set of all functions
$\varphi:[0,1]\to{\mathbb{R}}$ (resp. $\varphi:{\mathbb{R}}\to{\mathbb{R}}$ )
with continuous derivatives of all orders. Similarly, we write
$\mathcal{C}^{\infty}([0,1]\times[0,1])$ to denote the set of all functions
$\psi:[0,1]\times[0,1]\to{\mathbb{R}}$ with continuous partial derivatives of
all orders. To ease the notation, the partial derivatives with respect to the
first and second variable of a differentiable function $\psi:[0,1]\times[0,1]$
are respectively denoted by $\partial_{1}\psi$ and $\partial_{2}\psi$.
Throughout the paper, the letter $C$ denotes a positive constant. Most of the
time, the dependence of $C$ with respect to the parameters of the model is
specified.

We equip the space $\mathcal{C}^{0}([0,1])=\mathcal{C}([0,1])$ with the sup
norm

$||f||_{0}=\sup_{x\in[0,1]}|f(x)|.$

The space $\mathcal{C}^{k}([0,1])$ of functions with continuous derivatives up
to order $k$ is equipped with the norm

$||f||_{k}=\sum_{i=0}^{k}||f^{(i)}||_{0},$ (8)

where $f^{(0)}=f$ and $f^{(i)}$ denotes the $i$-th derivative of $f$ for
$i\in[k]$. The space $\mathcal{S}=\mathcal{C}^{\infty}([0,1])$ is a Fréchet
space [35] with the filtering family of semi-norms $(||f||_{k})_{k\geq 0}$.
Hence it is equipped with the metric $d_{\mathcal{S}}$ defined for all $f,g$
in $\mathcal{S}$ by,

$d_{\mathcal{S}}(f,g):=\sum_{k\geq 0}2^{-k}\frac{||f-g||_{k}}{1+||f-g||_{k}}.$
(9)

For a reader not familiar with these notions, some details about Fréchet
spaces are gathered in B.

Let $N$ be a point process in $[0,\infty)$, defined on a filtered probability
space $(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\geq 0},{\mathbb{P}})$. We say
that $N$ is locally finite if for all $t\geq 0$, the random variable
$N_{t}=N((0,t])$ counting the number of points of $N$ in the interval $(0,t]$
is finite almost surely. We say that the $(\mathcal{F}_{t})_{t\geq
0}$-predictable process $(\lambda_{t})_{t\geq 0}$ is the intensity process of
$N$ if the process $(N_{t}-\int_{0}^{t}\lambda_{s}ds)_{t\geq 0}$ is a
$(\mathcal{F}_{t})_{t\geq 0}$-local martingale. For bounded measurable
functions $g:[0,\infty)\to{\mathbb{R}}$ and a locally finite point process
$N$, we define $\int_{0}^{t}g(s)dN_{s}=\sum_{s\in N\cap(0,t]}g(s)$ for any
$t>0.$

For any locally square integrable martingale $(M_{t})_{t\geq 0}$, the Doob-
Meyer decomposition gives rise to the angle bracket, usually denoted by
$(\langle M\rangle_{t})_{t\geq 0}$, which is the unique non-decreasing
predictable process such that $\langle M\rangle_{0}=0$ and $(M_{t}^{2}-\langle
M\rangle_{t})_{t\geq 0}$ is local martingale.

###  2.2 Model definition and the central limit theorem

Throughout the paper we work on a filtered probability space
$(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\geq 0},{\mathbb{P}})$. We assume
that this filtered probability space is rich enough so that all the processes
we shall consider may be defined on it. We consider a nonlinear Hawkes process
$(N^{1},\ldots N^{n})$ in $[0,\infty)$ representing the spiking activity of
$n$ interacting neurons. We assume that neuron $i\in[n]$ is located at
position $x_{i}=i/n.$ The dynamics of the Hawkes process
$(N^{1},\ldots,N^{n})$ is described as follows.

######  Definition 1.

Let $f:{\mathbb{R}}\to{\mathbb{R}}_{+}$, $w:[0,1]\times[0,1]\to{\mathbb{R}}$
and $u_{0}:[0,1]\to{\mathbb{R}}$ be measurable functions and $\alpha\geq 0$ be
a fixed parameter. We say $(N^{1},\ldots,N^{n})$ is a Hawkes process with
parameters $(f,w,u_{0},\alpha)$ if

  1. 1.

${\mathbb{P}}-$ almost surely, for all pairs $i,j\in[n]$ with $i\neq j$, the
point processes $N^{i}$ and $N^{j}$ never jump simultaneously.

  2. 2.

For each $i\in[n]$, the intensity process $(\lambda^{i}_{t})_{t\geq 0}$ of
$N^{i}$ is given by $\lambda^{i}_{t}=f(U^{i}_{t-})$, where $U^{i}_{t}$ is
defined by

$U^{i}_{t}=e^{-\alpha
t}u_{0}(x_{i})+\frac{1}{n}\sum_{j=1}^{n}w(x_{j},x_{i})\int_{0}^{t}e^{-\alpha(t-s)}dN^{j}_{s}.$
(10)

We shall work under the following assumption on the parameters
$(f,w,u_{0},\alpha)$ of the model.

######  Assumption 1.

The function $f:{\mathbb{R}}\to{\mathbb{R}}_{+}$ belongs to
$\mathcal{C}^{\infty}({\mathbb{R}})$. Moreover, the first and second
derivatives of $f$ are both bounded, that is $\|f^{\prime}\|_{0}<\infty$ and
$\|f^{(2)}\|_{0}<\infty$. Furthermore, the functions
$u_{0}:[0,1]\to{\mathbb{R}}$ and $w:[0,1]\times[0,1]\to{\mathbb{R}}$ are both
smooth, that is, $u_{0}\in\mathcal{C}^{\infty}([0,1])$ and
$w\in\mathcal{C}^{\infty}([0,1]\times[0,1])$.

Note that under the assumption $\|f^{\prime}\|_{0}<\infty$, the function $f$
is Lipschitz continuous.

######  Remark 1.

Here we briefly discuss some examples of functions $f$, $w$ and $u_{0}$
satisfying Assumption 1. They are widely used in the literature (see the
reviews [6, 16] for instance).

  * •

firing rate $f$: the sigmoid rate $f(u)=f_{0}/(1+e^{-(u-\kappa)})$ and the
gaussian rate $f(u)=f_{0}(1+\operatorname{erf}(u-\kappa))/2$, with
$\operatorname{erf}(x)=(2/\sqrt{\pi})\int_{0}^{x}e^{-t^{2}/2}dt$, where
$\kappa\in\mathbb{R}$ can be thought as a threshold and $f_{0}>0$ a maximal
firing capacity of the neurons;

  * •

synaptic strength $w$: the standard form is $w(x,y)=w(|x-y|)$. In that
framework, the gaussian $w(x)=e^{-x^{2}}$, the exponential $w(x)=e^{-x}$ and
the mexican hat function are widely used. The latter writes as the difference
of two gaussians or two exponentials: for instance
$w(x)=e^{-x^{2}}-Ae^{-x^{2}/\sigma}$ with $A<1$ and $\sigma>1$ describes short
range excitation and long range inhibition;

  * •

initial condition $u_{0}$: a constant function or a smooth interpolation
between $u_{0}=0$ and $u_{0}=a>0$.

######  Remark 2.

Throughout the paper we work with smooth functions. We do this partly in order
to avoid some technicalities which make our proofs less transparent. Following
the approach adopted in [8, 29, 28] it is possible to state the central limit
theorem in some Hilbert space and weaken the assumptions to consider functions
that are only twice differentiable with bounded derivatives.

######  Remark 3.

Note that the positions $x_{i}$’s are regularly spaced in the compact set
$[0,1].$ We stress that our results do not rely on this specific choice. They
can be easily extended to the case in which the positions $x_{i}$ belong to a
regular grid of some compact set $K\subset{\mathbb{R}}^{d}$ for some integer
$d\geq 1$, at the cost of more complicated notation.

For each $t\geq 0$, let $U_{t}=(U^{1}_{t},\ldots,U^{n}_{t})$. Under some
assumptions on the functions $f,w$ and $u_{0}$ (much weaker than those of
Assumption 1), it has been proved [10, Corollary 2] that the process
$(U_{t})_{t\geq 0}$ converges (in some sense) to the unique solution $u(t,x)$
of the scalar neural field equation (2).

Recall that we write $\mathcal{S}$ to denote $\mathcal{C}^{\infty}([0,1])$ and
$\mathcal{S}^{\prime}$ to denote its dual space. The main goal of this paper
is to describe the fluctuations of $(U_{t})_{t\geq 0}$ around its limit, the
continuous deterministic solution $u(t,x)$ of the neural field equation (2).
For this reason, for each $i\in[n]$ and $t\geq 0$, we define the individual
fluctuations $\eta^{i}_{t}=n^{1/2}(U^{i}_{t}-u(t,x_{i}))$ and consider the
random signed measures $\Gamma^{n}_{t}$ on $\mathcal{S}^{\prime}$
(representing the spatial fluctuations) defined as

$\Gamma^{n}_{t}(dx)=\frac{1}{n}\sum_{i=1}^{n}\eta^{i}_{t}\delta_{x_{i}}(dx).$

For some fixed $T>0$, denote $\Gamma^{n}=(\Gamma^{n}_{t})_{0\leq t\leq T}$ and
observe that $\Gamma^{n}\in\mathcal{D}([0,T],\mathcal{S}^{\prime}).$ Our first
main result is the following.

######  Theorem 1.

Under Assumption 1, the sequence $(\Gamma^{n})_{n\geq 1}$ converges in law in
$\mathcal{D}([0,T],\mathcal{S}^{\prime})$ to the unique solution
$\Gamma=(\Gamma_{t})_{0\leq t\leq
T}\in\mathcal{C}([0,T],\mathcal{S}^{\prime})$ of equation (4).

The proof of Theorem 1 is divided in several steps. We first derive some
regularity properties of solutions of the NFE (2) - see Proposition 1 (its
proof is based mainly on results provided in [10]). Next we prove tightness of
the sequence $(\Gamma^{n})_{n\geq 1}$ in
$\mathcal{D}([0,T],\mathcal{S}^{\prime})$. To that end, we rely on [30,
Theorem 4.1], according to which the tightness of the sequence
$(\Gamma^{n})_{n\geq 1}$ in $\mathcal{D}([0,T],\mathcal{S}^{\prime})$ follows
from the tightness of the sequence $(\Gamma^{n}(\varphi))_{n\geq 1}$ in
$\mathcal{D}([0,T],{\mathbb{R}})$ for each $\varphi\in\mathcal{S}$, where
$\Gamma^{n}(\varphi)=(\Gamma^{n}_{t}(\varphi))_{0\leq t\leq T}$ and for each
$0\leq t\leq T$,

$\Gamma^{n}_{t}(\varphi)=\frac{1}{n}\sum_{i=1}^{n}\eta^{i}_{t}\varphi(x_{i}).$

To show the tightness of $(\Gamma^{n}(\varphi))_{n\geq 1}$ in
$\mathcal{D}([0,T],{\mathbb{R}})$, we first decompose
$\Gamma^{n}_{t}(\varphi)$ as

$\Gamma^{n}_{t}(\varphi)=e^{-\alpha
t}M^{n}_{t}(\varphi)+B^{n}_{t}(\varphi)+C^{n}_{t}(\varphi),$ (11)

where $M^{n}(\varphi)=(M^{n}_{t}(\varphi))_{0\leq t\leq T}$ is a local
martingale, $B^{n}(\varphi)=(B^{n}_{t}(\varphi))_{0\leq t\leq T}$ is a
continuous stochastic process and $C^{n}(\varphi)=(C^{n}_{t}(\varphi))_{0\leq
t\leq T}$ is a continuous function: all these quantities are carefully defined
in Equation (25). We then show (see Proposition 3) that the sequence of
functions $(C^{n}(\varphi))_{n\geq 1}$ goes to $0$ and use Aldous criterion to
show that both sequences $(M^{n}(\varphi))_{n\geq 1}$ and
$(B^{n}(\varphi))_{n\geq 1}$ are tight. From that it is easy to conclude the
tightness of $(\Gamma^{n}(\varphi))_{n\geq 1}$ in
$\mathcal{D}([0,T],{\mathbb{R}})$ \- see Corollary 3.

Once established the tightness of the sequence $(\Gamma^{n})_{n\geq 1}$, we
show that its limit points belong to $\mathcal{C}([0,T],{\mathbb{R}})$ and
satisfy equation (4) - see Proposition 4 and Theorem 2 respectively. To
conclude the proof of Theorem 1, we then prove that solutions of equation (4)
are unique - see Theorem 3.

##  3 Solutions of the Neural Field Equation

The purpose of this section is to show regularity properties for the solution
$u(t,x)$ of the NFE involved in the definition of the individual fluctuations
$(\eta^{i}_{t})_{0\leq t\leq T}$. In the preliminary study made in [10], some
regularity properties of $u(t,x)$ are shown. Then, using this a priori
regularity we are able to show that $u(t,x)$ is in fact smooth.

In [10], the function of interest is not the limit potential $u(t,x)$ but the
limit intensity $\lambda(t,x)$ which is proven to be continuous and uniquely
characterized as the unique _physical solution_ 222By physical solution, we
mean a solution which satisfies some a priori property inherited from the
microscopic model (see [10, equation above Proposition 5]) of some fixed point
equation. Nevertheless these two functions are closely linked by [10, Equation
(3.20)]:

$\begin{cases}u(t,x)=e^{-\alpha
t}u_{0}(x)+\int_{0}^{t}e^{-\alpha(t-s)}\int_{0}^{1}w(y,x)\lambda(s,y)dyds,\
t>0\ \mbox{and}\ x\in[0,1],\\\ \lambda(t,x)=f(u(t,x)).\end{cases}$ (12)

Since $\lambda(t,x)$ belongs to
$\mathcal{C}([0,T]\times[0,1],{\mathbb{R}}_{+})$ [10, Proposition 5], it then
follows that $u(t,x)$ belongs to $\mathcal{C}([0,T]\times[0,1],{\mathbb{R}})$
(which can be identified to $\mathcal{C}([0,T],\mathcal{C}([0,1])))$. In the
following, the evaluation of a function
$u(t,x)\in\mathcal{C}([0,T],\mathcal{C}([0,1]))$ is rather denoted by
$u_{t}(x)$.

In particular, Equation (12) means that $u_{t}(x)$ is a fixed point of the map
$F$ defined by : for all $v\in\mathcal{C}([0,T],\mathcal{C}([0,1]))$, for all
$t\geq 0$ and $x\in[0,1]$,

$F(v)_{t}(x):=e^{-\alpha
t}u_{0}(x)+\int_{0}^{t}e^{-\alpha(t-s)}\int_{0}^{1}w(y,x)f(v_{s}(y))dyds,$
(13)

where $u_{0}$ is the initial condition.

######  Proposition 1.

Under Assumption 1, for all $v\in\mathcal{C}([0,T],\mathcal{C}([0,1]))$,
$F(v)$ belongs to the smaller space $\mathcal{C}([0,T],\mathcal{S})$. In
particular, there is a unique physical solution of the NFE and existence of a
smooth solution.

###### Proof.

Let $v$ be in $\mathcal{C}([0,T],\mathcal{C}([0,1]))$. In particular, $v$ is
locally bounded in time ($\sup_{t\leq T,x\in[0,1]}v_{t}(x)<+\infty$) so, using
the Lipschitz continuity of $f$ and the smoothness of $w$ and $u_{0}$, it is
clear that for all $t\leq T$, $F(v)_{t}\in\mathcal{S}$ and that

$F(v)_{t}^{(k)}(x)=e^{-\alpha
t}u_{0}^{(k)}(x)+\int_{0}^{t}e^{-\alpha(t-s)}\int_{0}^{1}\partial_{2}^{k}w(y,x)f(v_{s}(y))dyds.$

Let $s\leq t\leq T$, using the Lipschitz continuity of $f$ and the fact that
$||\partial_{2}^{k}w(y,x)||_{0}<\infty$, we have for all $k\geq 1$,

$\displaystyle||(F(v)_{t})^{(k)}-(F(v)_{s})^{(k)}||_{0}$ $\displaystyle\leq$
$\displaystyle e^{-\alpha s}|e^{\alpha(t-s)}-1|\,||u_{0}^{(k)}||_{0}$
$\displaystyle+\int_{s}^{t}\left|\int_{0}^{1}\partial_{2}^{k}w(y,x)f(v_{h}(y))dy\right|dh$
$\displaystyle+\int_{0}^{s}e^{\alpha h}\left|e^{-\alpha t}-e^{-\alpha
s}\right|\left|\int_{0}^{1}\partial_{2}^{k}w(y,x)f(v_{h}(y))dy\right|dh$
$\displaystyle\leq$ $\displaystyle
C(t-s)\,||u_{0}^{(k)}||_{0}+CT(t-s)e^{\alpha T}(1+\sup_{t\leq
T}||v_{t}||_{0}).$

Summing up, we get $||F(v)_{t}-F(v)_{s}||_{k}\leq
kC_{T}(t-s)(1+||u_{0}||_{k}+\sup_{t\leq T}||v_{t}||_{0})$.

Let $\varepsilon>0$ and $k_{0}$ be such that
$\sum_{k=k_{0}+1}^{+\infty}2^{-k}<\varepsilon$. It suffices then to take $s$
and $t$ close enough such that

$k_{0}C_{T}(t-s)(1+||u_{0}||_{k_{0}}+\sup_{t\leq
T}||v_{t}||_{0})<\varepsilon,$

to get $d_{\infty}(F(v)_{t},F(v)_{s})\leq 2\varepsilon$ and so
$F(v)\in\mathcal{C}([0,T],\mathcal{S})$.

Assume that $u$ and $\tilde{u}$ are two physical solutions of the NFE. Then,
$\lambda(t,x)=f(u(t,x))$ and $\tilde{\lambda}(t,x)=f(\tilde{u}(t,x))$ define
two physical solutions of the fixed point equation [10, Equation (3.10)].
Hence, uniqueness for $\lambda$ proved in [10] implies uniqueness for $u$.
Existence is already proven in [10]. ∎

##  4 First estimates

In the sequel, for each $t\geq 0$ and $i\in[n]$, we write

$\begin{cases}M^{i}_{t}=N^{i}_{t}-\int_{0}^{t}f(U^{i}_{s})ds,\\\
g(s,x_{i})=\frac{1}{n}\sum_{j=1}^{n}w(x_{j},x_{i})f(u(s,x_{j})).\end{cases}$

Recall (see Section 2.1) that $(M^{i}_{t})_{t\geq 0}$ is the local martingale
associated with neuron $i$. With this notation, by using (10) and (2), we can
rewrite $\eta^{i}_{t}=n^{1/2}(U^{i}_{t}-u(t,x_{i}))$ as follows:

$\eta^{i}_{t}=A^{i}_{t}+B^{i}_{t}+C^{i}_{t},$

where $A^{i}_{t}$, $B^{i}_{t}$ and $C^{i}_{t}$ are given respectively by

$\begin{cases}A^{i}_{t}=e^{-\alpha
t}n^{-1/2}\sum_{j=1}^{n}\int_{0}^{t}e^{\alpha s}w(x_{j},x_{i})dM^{j}_{s},\\\
B^{i}_{t}=n^{-1/2}\sum_{j=1}^{n}\int_{0}^{t}e^{-\alpha(t-s)}w(x_{j},x_{i})\left(f(U^{j}_{s}))-f(u(s,x_{j}))\right)ds,\\\
C^{i}_{t}=n^{1/2}\int_{0}^{t}e^{-\alpha(t-s)}\left(g(s,x_{i})-\int_{0}^{1}w(y,x_{i})f(u(s,y))dy\right)ds.\end{cases}$
(14)

Note that $(C^{i}_{t})_{t\geq 0}$ is deterministic, while both
$(A^{i}_{t})_{t\geq 0}$ and $(B^{i}_{t})_{t\geq 0}$ are stochastic.
Furthermore, $(A^{i}_{t})_{t\geq 0}$ belongs to
$\mathcal{D}({\mathbb{R}}_{+})$ (but is not a local martingale even if
$(M^{i}_{t})_{t\geq 0}$ is) and $(B^{i}_{t})_{t\geq 0}$ belongs to
$\mathcal{C}({\mathbb{R}}_{+})$. Although every object defined above depends
on $n$, we omit this dependence to ease the notation.

We start this section with the following result.

######  Proposition 2.

Assume that $f\in\mathcal{C}^{1}({\mathbb{R}})$ is Lipschitz continuous,
$u_{0}$ is Lipschitz continuous,
$u(t,x)\in\mathcal{C}([0,T],\mathcal{C}^{1}([0,1]))$ and $w$ is bounded such
that $y\to\partial_{1}w(y,x)$ exists for all $x\in[0,1]$ and
$\sup_{x\in[0,1]}\|\partial_{1}w(\cdot,x)\|_{0}<\infty$. Then, for each $T>0,$

$\sup_{n\geq 1}\sup_{0\leq t\leq
T}\max_{i\in[n]}{\mathbb{E}}\left[(\eta^{i}_{t})^{2}\right]<\infty.$ (15)

###### Proof.

By Jensen inequality, we have that

${\mathbb{E}}\left[(\eta^{i}_{t})^{2}\right]\leq
3\left({\mathbb{E}}\left[(A^{i}_{t})^{2}\right]+{\mathbb{E}}\left[(B^{i}_{t})^{2}\right]+(C^{i}_{t})^{2}\right).$

Now, we will bound from above each term on the RHS of the inequality above. We
will start with ${\mathbb{E}}\left[(A^{i}_{t})^{2}\right].$ To that end, we
use [18, Proposition II.4.1.] and the fact that $w$ is bounded to obtain that
for all $0\leq t\leq T$,

$\displaystyle{\mathbb{E}}\left[(A^{i}_{t})^{2}\right]$
$\displaystyle=e^{-2\alpha
t}\frac{1}{n}\sum_{j=1}^{n}{\mathbb{E}}\left[\int_{0}^{t}e^{2\alpha
s}w^{2}(x_{j},x_{i})f(U^{j}_{s})ds\right]$
$\displaystyle\leq\|w\|_{0}^{2}\frac{1}{n}\sum_{j=1}^{n}{\mathbb{E}}\left[\int_{0}^{t}f(U^{j}_{s})ds\right]$
$\displaystyle=\|w\|_{0}^{2}\frac{1}{n}\sum_{j=1}^{n}{\mathbb{E}}\left[N^{j}_{t}\right]\leq\|w\|_{0}^{2}\frac{1}{n}\sum_{j=1}^{n}{\mathbb{E}}\left[N^{j}_{T}\right].$
(16)

Since $w$ is bounded and $u_{0}$ is Lipschitz continuous on $[0,1]$ (hence
bounded as well), [10, Proposition 3] implies that not only the RHS of (4) is
finite, but also that

$\sup_{n\geq 1}\sup_{0\leq t\leq
T}\max_{i\in[n]}{\mathbb{E}}[(A^{i}_{t})^{2}]<\infty.$ (17)

Next, we will deal with the term $(C^{i}_{t})^{2}.$ From the classical Riemann
approximation, we have for each $i\in[n]$ and $s\geq 0$,

$\left|g(s,x_{i})-\int_{0}^{1}w(y,x_{i})f(u(s,y))dy\right|\leq\frac{1}{2n}\sup_{y\in[0,1]}\left|\partial_{1}w(y,x_{i})f(u(s,y)\right.\\\
\left.+w(y,x)f^{\prime}(u(s,y))\partial_{2}u(s,y)\right|,$ (18)

and thus we obtain for $t\geq 0$,

$|C^{i}_{t}|\leq\frac{1}{2n^{1/2}}\max_{j\in[n]}\sup_{0\leq s\leq
t,y\in[0,1]}\left|\partial_{1}w(y,x_{j})f(u(s,y))+w(y,x)f^{\prime}(u(s,y))\partial_{2}u(s,y)\right|.$

As a consequence of the inequality above, we have for all $0\leq t\leq T$,

$\displaystyle|C^{i}_{t}|^{2}$ $\displaystyle\leq\frac{1}{4n}\left(\sup_{0\leq
s\leq
T,x,y\in[0,1]}\left|\partial_{1}w(y,x)f(u(s,y)+w(y,x)f^{\prime}(u(s,y))\partial_{2}u(s,y)\right|\right)^{2}.$

Since $M_{T}=\sup_{0\leq s\leq T}\|u(s,\cdot)\|_{0}<\infty$ and $f$ is
Lipschitz continuous, we have that $f$ is locally bounded, implying that
$\sup_{s\leq T,y,\in[0,1]}|f(u(s,y))|<\infty$. The assumptions on the
functions $u$ and $w$ ensure that both
$\sup_{x,y\in[0,1]}\|\partial_{1}w(y,x)\|_{0}$ and $\sup_{s\leq
T,y\in[0,1]}\|\partial_{2}w(s,y)\|_{0}$ are finite, so that

$\sup_{n\geq 1}\sup_{t\leq T}\max_{i\in[n]}|C^{i}_{t}|^{2}<\infty.$ (19)

It remains to deal with the term ${\mathbb{E}}\left[(B^{i}_{t})^{2}\right].$
In what follows, fix an integer $k\geq 1$, and consider $\tau_{k}=\inf\\{0\leq
t\leq T:\max_{i\in[n]}|\eta^{i}_{t}|\geq k\\}$. By applying Jensen inequality
twice and using the fact that $f$ is Lipschitz continuous, we deduce that

$\displaystyle{\mathbb{E}}\left[(B^{i}_{t\wedge\tau_{k}})^{2}\right]$
$\displaystyle\leq\sum_{j=1}^{n}{\mathbb{E}}\left[\left(\int_{0}^{t\wedge\tau_{k}}e^{-\alpha((t\wedge\tau_{k})-s)}w(x_{j},x_{i})(f(U^{j}_{s})-f(u(s,x_{j})))ds\right)^{2}\right]$
$\displaystyle\leq
t\|f^{\prime}\|^{2}_{0}\frac{1}{n}\sum_{j=1}^{n}\int_{0}^{t}w^{2}(x_{j},x_{i}){\mathbb{E}}\left[(\eta^{j}_{s\wedge\tau_{k}})^{2}\right]ds.$

Now, since $w$ is a bounded function, it follows then that for any $0\leq
t\leq T$,

$\displaystyle{\mathbb{E}}\left[(B^{i}_{t\wedge\tau_{k}})^{2}\right]$
$\displaystyle\leq
T\|f^{\prime}\|^{2}_{0}\|w\|^{2}_{0}\int_{0}^{t}\max_{j\in[n]}{\mathbb{E}}\left[(\eta^{j}_{s\wedge\tau_{k}})^{2}\right]ds.$
(20)

Combining (17), (20) and (19), we have that there exists a finite positive
constant $C=C(T,w,f,u)$ such that for all $0\leq t\leq T$ and any integer
$n\geq 1$,

$\max_{i\in[n]}{\mathbb{E}}\left[(\eta^{i}_{t\wedge\tau_{k}})^{2}\right]\leq
C\left(1+\int_{(0,t]}\max_{i\in[n]}{\mathbb{E}}\left[(\eta^{i}_{s\wedge\tau_{k}})^{2}\right]ds\right).$

Since $t\to{\mathbb{E}}[(\eta^{i}_{t\wedge\tau_{k}})^{2}]$ is locally bounded,
we may apply Gronwall inequality to conclude that for all $0\leq t\leq T$ and
any integer $n\geq 1$

$\max_{i\in[n]}{\mathbb{E}}\left[(\eta^{i}_{t\wedge\tau_{k}})^{2}\right]<C,$

for some finite positive constant $C=C(T,w,f,u).$ By Lemma 2, which is in A,
we know that $\tau_{k}\to T$ a.s. as $k\to\infty$, and hence, by Fatou’s
lemma, for all $t\leq T$ and integer $n\geq 1$,

$\max_{i\in[n]}{\mathbb{E}}\left[(\eta^{i}_{t})^{2}\right]<C,$

implying the result. ∎

######  Corollary 1.

Under the assumptions of Proposition 2, for all bounded functions
$\varphi:[0,1]\to{\mathbb{R}}$ and $T>0,$

$\sup_{n\geq 1}\sup_{0\leq t\leq
T}{\mathbb{E}}[(\Gamma^{n}_{t}(\varphi))^{2}]<\infty.$ (21)

###### Proof.

Apply Jensen inequality to deduce that

$\displaystyle{\mathbb{E}}((\Gamma^{n}_{t}(\varphi))^{2})$
$\displaystyle={\mathbb{E}}\left(\left(\frac{1}{n}\sum_{i=1}^{n}\eta^{i}_{t}\varphi(x_{i})\right)^{2}\right)$
$\displaystyle\leq\frac{1}{n}\sum_{i=1}^{n}\varphi^{2}(x_{i}){\mathbb{E}}((\eta^{i}_{t})^{2}).$

Since $\varphi$ is bounded, the result then follows from Proposition 2. ∎

##  5 Tightness

The goal of this section is to prove that the sequence of
$\mathcal{S}^{\prime}$-valued stochastic processes $(\Gamma^{n})_{n\geq 1}$ is
tight in $\mathcal{D}([0,T],\mathcal{S}^{\prime}).$ According to Mitoma [30,
Theorem 4.1], it suffices to show that the sequence of stochastic processes
$(\Gamma^{n}(\varphi))_{n\geq 1}$ is tight in
$\mathcal{D}([0,T],{\mathbb{R}})$, for each fixed $\varphi\in\mathcal{S}.$

In what follows, we fix $\varphi\in\mathcal{S}$ and consider the sequence of
stochastic processes $(\Gamma^{n}(\varphi))_{n\geq 1}.$ Our goal is to show
that this sequence is tight in $\mathcal{D}([0,T],{\mathbb{R}})$. To show
this, we use Aldous’ tightness criterion. According to Aldous [3, Theorem
16.10], a sequence of stochastic processes $(X^{n})_{n\geq 1}$ in
$\mathcal{D}([0,T],{\mathbb{R}})$ is tight if both conditions below are
satisfied:

  1. 1.

for any $0\leq t\leq T$ and $\epsilon>0$, there exist an integer $n_{0}\geq 1$
and $K>0$ such that

$\sup_{n\geq n_{0}}{\mathbb{P}}(|X^{n}_{t}|>K)\leq\epsilon.$ (22)

  2. 2.

for any $\epsilon_{1},\epsilon_{2}>0$, there exist $\delta>0$ and an integer
$n_{0}\geq 1$ such that

$\sup_{n\geq n_{0}}\sup_{(\tau^{\prime},\tau)\in
St_{\delta}}{\mathbb{P}}(|X^{n}_{\tau^{\prime}}-X^{n}_{\tau}|>\epsilon_{1})\leq\epsilon_{2},$
(23)

where $St_{\delta}$ is the set of all pairs $(\tau^{\prime},\tau)$ of
$(\mathcal{F}_{t})_{t\geq 0}$-stopping times such that ${\mathbb{P}}$-a.s we
have $\tau\leq\tau^{\prime}\leq\tau+\delta\leq T.$

To verify that $(\Gamma^{n}(\varphi))_{n\geq 1}$ satisfies Aldous’s criterion,
it will be convenient to introduce some new notation. Note that for each
$t\geq 0$ and all $n\geq 1$, the spatial fluctuation $\Gamma_{t}^{n}(\varphi)$
can be rewritten as follows:

$\Gamma^{n}_{t}(\varphi)=A^{n}_{t}(\varphi)+B^{n}_{t}(\varphi)+C^{n}_{t}(\varphi),$
(24)

with $A^{n}_{t}(\varphi)$, $B^{n}_{t}(\varphi)$ and $C^{n}_{t}(\varphi)$ given
respectively by

$\begin{cases}A^{n}_{t}(\varphi)=\frac{1}{n}\sum_{i=1}^{n}\varphi(x_{i})A^{i}_{t},\\\
B^{n}_{t}(\varphi)=\frac{1}{n}\sum_{i=1}^{n}\varphi(x_{i})B^{i}_{t},\\\
C^{n}_{t}(\varphi)=\frac{1}{n}\sum_{i=1}^{n}\varphi(x_{i})C^{i}_{t}.\end{cases}$
(25)

where $A^{i}_{t}$, $B^{i}_{t}$ and $C^{i}_{t}$ for $1\leq i\leq n$ are defined
in (14).

For later use, it will be useful to write $A^{n}_{t}(\varphi)=e^{-\alpha
t}M^{n}_{t}(\varphi)$, where $M^{n}(\varphi)$ is a local martingale given for
all $t\geq 0$ by

$M^{n}_{t}(\varphi)=\frac{1}{n^{1/2}}\sum_{j=1}^{n}\int_{0}^{t}e^{\alpha
s}\frac{1}{n}\sum_{i=1}^{n}w(x_{j},x_{i})\varphi(x_{i})dM^{j}_{s}.$ (26)

By writing $I^{n}[\varphi](y):=n^{-1}\sum_{i=1}^{n}w(y,x_{i})\varphi(x_{i})$,
one can check (see for instance [18]) that its angle bracket is given by

$\langle M^{n}(\varphi)\rangle_{t}=n^{-1}\sum_{j=1}^{n}\int_{0}^{t}e^{2\alpha
s}I^{n}[\varphi](x_{j})^{2}f(U^{j}_{s})ds,$ (27)

and by the polarization identity,

$\langle
M^{n}(\varphi_{1}),M^{n}(\varphi_{2})\rangle_{t}=n^{-1}\sum_{j=1}^{n}\int_{0}^{t}e^{2\alpha
s}I^{n}[\varphi_{1}](x_{j})I^{n}[\varphi_{2}](x_{j})f(U^{j}_{s})ds.$ (28)

With this notation, we can prove the following result.

######  Proposition 3.

Let us make the same assumptions as in Proposition 2. Then, for each fixed
$\varphi$ in $\mathcal{S}$, the sequences of the stochastic processes
$(M^{n}(\varphi))_{n\geq 1}$ and $(A^{n}(\varphi))_{n\geq 1}$ are tight in the
space $\mathcal{D}([0,T])$ and $(B^{n}(\varphi))_{n\geq 1}$ is tight in
$\mathcal{C}([0,T])$. Moreover, the sequence of functions
$(C^{n}(\varphi))_{n\geq 1}$ satisfies

$\sup_{t\in[0,T]}|C^{n}_{t}(\varphi)|\leq Cn^{-1/2}||\varphi||_{0},$ (29)

where $C=C(T,w,f,u)$ is some finite positive constant.

###### Proof.

We need to show that the sequences $(M^{n}(\varphi))_{n\geq 1}$,
$(A^{n}(\varphi))_{n\geq 1}$ and $(B^{n}(\varphi))_{n\geq 1}$ satisfy Aldous’
criterion. We start with $(M^{n}(\varphi))_{n\geq 1}$. All along the proof, we
write $L$ to denote a constant which might change from line to line and
depends only on $T,\|w\|_{0},\|\varphi\|_{0},\alpha$ and $\|f\|_{0}.$

Tightness of $(M^{n}(\varphi))_{n\geq 1}$. Fix $0\leq t\leq T$ and
$\epsilon>0$. By Markov’s inequality, for all $n\geq 1$ and $K>0$, we have

${\mathbb{P}}(|M^{n}_{t}(\varphi)|>K)\leq\frac{1}{K^{2}}{\mathbb{E}}(|M^{n}_{t}(\varphi)|^{2}),$

so that by (27), we can deduce from the above inequality that

${\mathbb{P}}(|M^{n}_{t}(\varphi)|>K)\leq\frac{1}{K^{2}}\frac{1}{n}\sum_{j=1}^{n}{\mathbb{E}}\left(\int_{0}^{t}\left(n^{-1}\sum_{i=1}^{n}\varphi(x_{i})w(x_{j},x_{i})\right)^{2}f(U^{j}_{s})e^{2\alpha
s}ds\right).$

Since $w$ and $\varphi$ are bounded, it follows from this last inequality that

${\mathbb{P}}(|M^{n}_{t}(\varphi)|>K)\leq\frac{L}{K^{2}}\frac{1}{n}\sum_{j=1}^{n}{\mathbb{E}}\left(\int_{0}^{t}f(U^{j}_{s})ds\right)\leq\frac{L}{K^{2}}\sup_{n\geq
1}\frac{1}{n}\left\\{\sum_{j=1}^{n}{\mathbb{E}}\left(N^{j}_{T}\right)\right\\}.$

Then, [10, Proposition 3] together with the inequality above implies that for
all $n\geq 1$,

${\mathbb{P}}(|M^{n}_{t}(\varphi)|>K)\leq\frac{L}{K^{2}},$

and the condition (22) holds whenever $K\geq\sqrt{L/\epsilon}.$

It remains to show that $(M^{n}(\varphi))_{n\geq 1}$ satisfies the condition
(23). By [32, Theorem 2.3.2], it suffices to show that $(\langle
M^{n}(\varphi)\rangle)_{n\geq 1}$ satisfies the condition (23). Thus, take
stopping times $(\tau,\tau^{\prime})\in St_{\delta}$ and observe that by (27),

$\displaystyle{\mathbb{E}}\left(\left|\langle
M^{n}(\varphi)\rangle_{\tau^{\prime}}-\langle
M^{n}(\varphi)\rangle_{\tau}\right|\right)$
$\displaystyle=n^{-1}\sum_{j=1}^{n}{\mathbb{E}}\left(\left|\int_{\tau}^{\tau^{\prime}}\left(n^{-1}\sum_{i=1}^{n}\varphi(x_{i})w(x_{j},x_{i})\right)^{2}f(U^{j}_{s})e^{2\alpha
s}ds.\right|\right)$ $\displaystyle\leq
L\frac{1}{n}\sum_{j=1}^{n}{\mathbb{E}}\left(\int_{\tau}^{\tau^{\prime}}f(U^{j}_{s})e^{2\alpha
s}ds\right).$

Now, note that for all $j\in[n]$ and $t\geq 0$,

$f(U^{j}_{s})\leq
f(u(t,x_{j}))+\frac{\|f^{\prime}\|_{0}}{\sqrt{n}}|\eta^{j}_{t}|\leq\sup_{t\leq
T}\|f(u(t,\cdot))\|+\frac{\|f^{\prime}\|_{0}}{\sqrt{n}}|\eta^{i}_{t}|.$

The local boundedness of both $f$ and $u$, and the fact that
$0\leq\tau\leq\tau^{\prime}\leq T$, imply then that

${\mathbb{E}}\left(\left|\langle M^{n}(\varphi)\rangle_{\tau^{\prime}}-\langle
M^{n}(\varphi)\rangle_{\tau}\right|\right)\leq
L\left({\mathbb{E}}(\tau^{\prime}-\tau)+\frac{1}{n^{3/2}}\sum_{j=1}^{n}{\mathbb{E}}\left(\int_{\tau}^{\tau^{\prime}}e^{2\alpha
s}|\eta^{j}_{s}|ds\right)\right)$

By applying Young inequality, we have for all $s\geq 0$, $j\in[n]$ and
$\xi>0$,

$e^{2\alpha s}|\eta^{j}_{s}|\leq\frac{1}{2\xi}e^{4\alpha
s}+\frac{\xi}{2}|\eta^{j}_{s}|^{2},$

so that

${\mathbb{E}}\left(\int_{\tau}^{\tau^{\prime}}e^{2\alpha
s}|\eta^{j}_{s}|ds\right)\leq\frac{1}{8\xi\alpha}{\mathbb{E}}(e^{4\alpha\tau^{\prime}}-e^{4\alpha\tau})+\frac{\xi}{2}\int_{0}^{T}{\mathbb{E}}|\eta^{j}_{s}|^{2}ds,$

where in the last inequality we have also used the fact that
$0\leq\tau\leq\tau^{\prime}\leq T.$ By using that
$|e^{x}-e^{y}|\leq|x-y|\frac{1}{2}(e^{x}+e^{y})$ in the previous inequality
and the fact that $\tau^{\prime}-\tau\leq\delta$, it follows that

${\mathbb{E}}(\tau^{\prime}-\tau)+\frac{1}{n^{3/2}}\sum_{j=1}{\mathbb{E}}\left(\int_{\tau}^{\tau^{\prime}}e^{2\alpha
s}|\eta^{j}_{s}|ds\right)\leq\delta+\frac{\delta}{2\xi}e^{4\alpha
T}+\frac{T\xi}{2n^{1/2}}\sup_{t\leq
T}\max_{j\in[n]}{\mathbb{E}}(|\eta^{j}_{s}|^{2}).$

By using Proposition 2 and taking $\xi=\sqrt{\delta}$ in the inequality above,
we deduce that

${\mathbb{E}}\left(\left|\langle M^{n}(\varphi)\rangle_{\tau^{\prime}}-\langle
M^{n}(\varphi)\rangle_{\tau}\right|\right)\leq L(\delta+\sqrt{\delta}),$

showing that $(\langle M^{n}(\varphi)\rangle)_{n\geq 1}$ satisfies the
condition (23) and thus the tightness of $(M^{n}(\varphi))_{n\geq 1}$.

Tightness of $(A^{n}(\varphi))_{n\geq 1}$. Since
$|A^{n}_{t}(\varphi)|\leq|M^{n}_{t}(\varphi)|$ for all $n\geq 1$ and $0\leq
t\leq T$, and $(M^{n}(\varphi))_{n\geq 1}$ satisfies (22), we have that
$(A^{n}(\varphi))_{n\geq 1}$ satisfies (22) as well.

To show that tightness of $(A^{n}(\varphi))_{n\geq 1}$, it remains to show
that $(A^{n}(\varphi))_{n\geq 1}$ satisfies the condition (23). To that end,
take stopping times $(\tau,\tau^{\prime})\in St_{\delta}$, and note that

$\displaystyle{\mathbb{P}}(|A^{n}_{\tau^{\prime}}(\varphi)-A^{n}_{\tau}(\varphi)|>\epsilon_{1})$
$\displaystyle\leq$
$\displaystyle{\mathbb{P}}(|M^{n}_{\tau^{\prime}}(\varphi)(e^{-\alpha(\tau^{\prime}-\tau)}-1)|>\epsilon_{1}/2)$
$\displaystyle+{\mathbb{P}}(|M^{n}_{\tau^{\prime}}(\varphi)-M^{n}_{\tau}(\varphi)|>\epsilon_{1}/2)$
$\displaystyle\leq$
$\displaystyle\frac{4}{\epsilon^{2}_{1}}{\mathbb{E}}\left((M^{n}_{\tau^{\prime}}(\varphi))^{2}(e^{-\alpha(\tau^{\prime}-\tau)}-1)^{2}\right)$
$\displaystyle+{\mathbb{P}}(|M^{n}_{\tau^{\prime}}(\varphi)-M^{n}_{\tau}(\varphi)|>\epsilon_{1}/2)$
$\displaystyle\leq$
$\displaystyle\frac{4(\alpha\delta)^{2}}{\epsilon^{2}_{1}}{\mathbb{E}}\left((M^{n}_{\tau^{\prime}}(\varphi))^{2}\right)+{\mathbb{P}}(|M^{n}_{\tau^{\prime}}(\varphi)-M^{n}_{\tau}(\varphi)|>\epsilon_{1}/2)$
$\displaystyle\leq$
$\displaystyle\frac{4(\alpha\delta)^{2}}{\epsilon^{2}_{1}}{\mathbb{E}}\left((M^{n}_{T}(\varphi))^{2}\right)+{\mathbb{P}}(|M^{n}_{\tau^{\prime}}(\varphi)-M^{n}_{\tau}(\varphi)|>\epsilon_{1}/2).$

Now, we can proceed as in the proof of the tightness of
$(M^{n}(\varphi))_{n\geq 1}$ to conclude that $(A^{n}(\varphi))_{n\geq 1}$
satisfies the condition (23), thus establishing the tightness of
$(A^{n}(\varphi))_{n\geq 1}.$

Tightness of $(B^{n}(\varphi))_{n\geq 1}$. Fix $0\leq t\leq T$ and
$\epsilon>0$. By using Markov inequality and then Jensen inequality, we have
that for all $n\geq 1$ and $K>0$, (recall the definition of
$B^{n}_{t}(\varphi)$),

${\mathbb{P}}(|B^{n}_{t}(\varphi)|>K)\leq\frac{1}{K^{2}}\frac{1}{n}\sum_{i=1}^{n}\varphi^{2}(x_{i}){\mathbb{E}}(\left(B^{i}_{t})^{2}\right).$

The inequality above and (20) yield

${\mathbb{P}}(|B^{n}_{t}(\varphi)|>K)\leq\frac{L}{K^{2}}\sup_{n\geq
1}\sup_{0\leq s\leq
T}\max_{j\in[n]}{\mathbb{E}}\left((\eta^{j}_{s})^{2}\right).$

Since $\sup_{n\geq 1}\sup_{0\leq s\leq
T}\max_{j\in[n]}{\mathbb{E}}\left((\eta^{j}_{s})^{2}\right)$ is finite by
Proposition 2, we deduce from inequality above that condition (22) holds.

We will now check that condition (23) holds as well. In the sequel, let
$\Delta^{j}_{s}(f)=f(U^{j}_{s})-f(u(s,x_{j}))$ for $1\leq j\leq n$, $0\leq
s\leq T$ and $x_{j}\in[0,1].$ Take stopping times $(\tau,\tau^{\prime})\in
St_{\delta}$, and note that

${\mathbb{E}}|B^{n}_{\tau^{\prime}}(\varphi)-B^{n}_{\tau}(\varphi)|\leq{\mathbb{E}}\left|\frac{1}{n}\sum_{i=1}^{n}\varphi(x_{i})\frac{1}{n^{1/2}}\sum_{j=1}^{n}\int_{\tau}^{\tau^{\prime}}e^{-\alpha(\tau^{\prime}-s)}w(x_{j},x_{i})\Delta^{j}_{s}(f)ds\right|\\\
+{\mathbb{E}}\left|\frac{1}{n}\sum_{i=1}^{n}\varphi(x_{i})\frac{1}{n^{1/2}}\sum_{j=1}^{n}\int_{0}^{\tau}(e^{-\alpha(\tau-s)}-e^{-\alpha(\tau^{\prime}-s)})w(x_{j},x_{i})\Delta^{j}_{s}(f)ds\right|.$
(30)

Since $w$ and $\varphi$ are bounded functions, $f$ is Lipschitz continuous and
$|e^{-x}-e^{-y}|\leq|x-y|$ for all $x,y\geq 0$, we have that

${\mathbb{E}}\left|\frac{1}{n}\sum_{i=1}^{n}\varphi(x_{i})\frac{1}{n^{1/2}}\sum_{j=1}^{n}\int_{0}^{\tau}(e^{-\alpha(\tau-s)}-e^{-\alpha(\tau^{\prime}-s)})w(x_{j},x_{i})\Delta^{j}_{s}(f)ds\right|\\\
\leq\alpha\|w\|_{0}\|\varphi\|_{0}L_{f}\frac{1}{n}\sum_{j=1}^{n}{\mathbb{E}}\int_{0}^{\tau}|\tau^{\prime}-\tau||\eta^{j}_{s}|ds\leq\delta
L\int_{0}^{T}\frac{1}{n}\sum_{j=1}^{n}{\mathbb{E}}|\eta^{j}_{s}|ds$

so that Proposition 2 implies that

$\sup_{n\geq 1}\sup_{(\tau^{\prime},\tau)\in
St_{\delta}}{\mathbb{E}}\left|\frac{1}{n}\sum_{i=1}^{n}\varphi(x_{i})\frac{1}{n^{1/2}}\sum_{j=1}^{n}\int_{0}^{\tau}(e^{-\alpha(\tau-s)}-e^{-\alpha(\tau^{\prime}-s)})w(x_{j},x_{i})\Delta^{j}_{s}(f)ds\right|\leq
L\delta.$ (31)

Similarly, one can check that

${\mathbb{E}}\left|\frac{1}{n}\sum_{i=1}^{n}\varphi(x_{i})\frac{1}{n^{1/2}}\sum_{j=1}^{n}\int_{\tau}^{\tau^{\prime}}e^{-\alpha(\tau^{\prime}-s)}w(x_{j},x_{i})\Delta^{j}_{s}(f)ds\right|\\\
\leq\|f^{\prime}\|_{0}\|w\|_{0}\|\varphi\|_{0}\frac{1}{n}\sum_{j=1}^{n}{\mathbb{E}}\int_{\tau}^{\tau^{\prime}}e^{-\alpha(\tau^{\prime}-s)}|\eta^{j}_{s}|ds.$
(32)

By applying Young’s inequality we have that for all $\xi>0$,

$e^{-\alpha(\tau^{\prime}-s)}|\eta^{j}_{s}|\leq\frac{1}{2\xi}e^{-2\alpha(\tau^{\prime}-s)}+\frac{\xi}{2}|\eta^{j}_{s}|^{2},$

so that

${\mathbb{E}}\int_{\tau}^{\tau^{\prime}}e^{-\alpha(\tau^{\prime}-s)}|\eta^{j}_{s}|ds\leq\frac{1}{2\xi}{\mathbb{E}}\int_{\tau}^{\tau+\delta}e^{-2\alpha(\tau^{\prime}-s)}ds+\frac{\xi}{2}\int_{0}^{T}{\mathbb{E}}|\eta^{j}_{s}|^{2}ds\leq\frac{\delta}{2\xi}+\frac{\xi}{2}\int_{0}^{T}{\mathbb{E}}|\eta^{j}_{s}|^{2}ds.$

As a consequence, by taking $\xi=\sqrt{\delta}$, it follows from (32) and
inequality above that

${\mathbb{E}}\left|\frac{1}{n}\sum_{i=1}^{n}\varphi(x_{i})\frac{1}{n^{1/2}}\sum_{j=1}^{n}\int_{\tau}^{\tau^{\prime}}e^{-\alpha(\tau^{\prime}-s)}w(x_{j},x_{i})\Delta^{j}_{s}(f)ds\right|\leq\sqrt{\delta}L(1+\int_{0}^{T}\sup_{n\geq
1}\frac{1}{n}\sum_{j=1}^{n}{\mathbb{E}}|\eta^{j}_{s}|^{2}ds).$

By Proposition (2), inequality above and (31), it follows then that

$\sup_{n\geq 1}\sup_{(\tau^{\prime},\tau)\in
St_{\delta}}{\mathbb{E}}|B^{n}_{\tau^{\prime}}(\varphi)-B^{n}_{\tau}(\varphi)|\leq
L(\delta+\sqrt{\delta}),$

proving that (23) holds.

Proof of (29). In the proof of Proposition (2), it has been proved that for
any $0\leq t\leq T$ and $1\leq i\leq n$,

$|C^{i}_{t}|\leq\frac{1}{2n^{1/2}}\max_{j\in[n]}\sup_{0\leq s\leq
t,y\in[0,1]}\left|\partial_{1}w(y,x_{j})f(u(s,y))+w(y,x)f^{\prime}(u(s,y))\partial_{2}u(s,y)\right|,$

so that

$\sup_{0\leq t\leq
T}|C^{n}_{t}(\varphi)|\leq\frac{\|\varphi\|_{0}}{2n^{1/2}}\max_{j\in[n]}\sup_{0\leq
s\leq
T,y\in[0,1]}\left|\partial_{1}w(y,x_{j})f(u(s,y))+w(y,x)f^{\prime}(u(s,y))\partial_{2}u(s,y)\right|,$

and the result follows. ∎

Since $(A^{n}(\varphi))_{n\geq 1}$ is càdlàg tight, $(B^{n}(\varphi))_{n\geq
1}$ is continuous tight and $(C^{n}(\varphi))$ goes to $0$, the following
result holds (see [23, Corollary VI.3.33.]). Furthermore, Corollary 3 is
granted by [30, Theorem 4.1].

######  Corollary 2.

Let us make the same assumptions as in Proposition 3. Then, for each fixed
$\varphi$ in $\mathcal{S}$, the sequence of stochastic processes
$(\Gamma^{n}(\varphi))_{n\geq 1}$ is tight in
$\mathcal{D}([0,T],{\mathbb{R}})$.

######  Corollary 3.

Let us make the same assumptions as in Proposition 3. Then, the sequences of
the laws of $(\Gamma^{n})_{n\geq 1}$ and $(M^{n})_{n\geq 1}$ are tight in
$\mathcal{D}([0,T],\mathcal{S}^{\prime})$.

Furthermore, the limit trajectories of $(\Gamma^{n})_{n\geq 1}$ and
$(M^{n})_{n\geq 1}$ are continuous as stated below.

######  Proposition 4.

Suppose that $w$ is bounded. Then the limit points of $(\Gamma^{n})_{n}$ and
$(M^{n})_{n\geq 1}$ are supported by
$\mathcal{C}([0,T],\mathcal{S}^{\prime})$.

###### Proof.

We only give here the proof for the sequence $(\Gamma^{n})_{n}$ but the same
argument can be applied to $(M^{n})_{n}$. For completeness, let us mention
that the result for $(M^{n})_{n}$ is also a byproduct of Proposition 5.

By [3, Theorem 13.4.] it suffices to show that the maximum jump size of
$\Gamma^{n}$ goes to $0$ in probability. This means that we need to prove
that, for any bounded set $B\subset\mathcal{S}$, the random variable
$J_{B}(\Gamma^{n})=\sup_{0\leq t\leq T}\sup_{\varphi\in
B}|\Gamma^{n}_{t}(\varphi)-\Gamma^{n}_{t-}(\varphi)|$ converges in probability
to $0$ as $n\to\infty.$ Recall that $B\subset\mathcal{S}$ is bounded if
$\sup_{\varphi\in B}\|\varphi\|_{k}<\infty$ for any $k\geq 0$ (which is
different from boundedness with respect to $d_{\mathcal{S}}$, see [35,
Definition 2.9]). In particular, we have $\sup_{\varphi\in
B}\|\varphi\|_{0}<\infty.$ Now, observe that (see also proof of condition 2 in
the proof of Proposition 5)

$\sup_{0\leq t\leq T}\sup_{\varphi\in
B}|M^{n}_{t}(\varphi)-M^{n}_{t-}(\varphi)|\leq e^{\alpha
T}\frac{\|w\|_{0}}{\sqrt{n}}\sup_{\varphi\in B}\|\varphi\|_{0}\sup_{t\leq
T}\sum_{j=1}^{n}\Delta N^{j}_{s},$

where $\Delta N^{j}_{s}=N^{j}_{s}-N^{j}_{s-}$ for each $j\in[n]$ and $s\geq
0$. Almost surely for all $1\leq i,j\leq n$ with $i\neq j$, the counting
processes $N^{j}$ and $N^{i}$ never jump simultaneously, so that

$\sup_{0\leq t\leq T}\sum_{j=1}^{n}\Delta N^{j}_{s}\leq 1\ \mbox{almost
surely,}$

and therefore almost surely

$\sup_{0\leq t\leq T}\sup_{\varphi\in
B}|M^{n}_{t}(\varphi)-M^{n}_{t-}(\varphi)|\leq e^{\alpha
T}\frac{\|w\|_{0}}{\sqrt{n}}\sup_{\varphi\in B}\|\varphi\|_{0}.$

Finally, since
$\Gamma^{n}_{t}(\varphi)=e^{-t\alpha}M^{n}_{t}(\varphi)+B^{n}_{t}(\varphi)+C^{n}_{t}(\varphi)$
and both $B^{n}_{t}(\varphi)$ and $C^{n}_{t}(\varphi)$ are continuous
functions of time, we deduce that
$|\Gamma^{n}_{t}(\varphi)-\Gamma^{n}_{t-}(\varphi)|\leq|M^{n}_{t}(\varphi)-M^{n}_{t-}(\varphi)|$.
Thus, it follows that almost surely,

$J_{B}(\Gamma^{n})\leq e^{\alpha T}\frac{\|w\|_{0}}{\sqrt{n}}\sup_{\varphi\in
B}\|\varphi\|_{0},$

implying the result. ∎

Since the limit trajectories of $(M^{n})_{n\geq 1}$ and $(\Gamma^{n})_{n\geq
1}$ are continuous, we have the joint tightness [24, Corollary VI.3.33].

######  Corollary 4.

Under assumptions of Proposition 3, the sequence of the laws of
$(\Gamma^{n},M^{n})_{n\geq 1}$ is tight in the space
$\mathcal{D}([0,T],\mathcal{S}^{\prime}\times\mathcal{S}^{\prime})$ with limit
points in $\mathcal{C}([0,T],\mathcal{S}^{\prime}\times\mathcal{S}^{\prime})$.

##  6 Limit equation

In this section we first show the convergence of the local martingale
$(M^{n})_{n\geq 1}$ in order to state the limit equation (38) satisfied by the
limit points of $(\Gamma^{n})_{n\geq 1}$.

######  Definition 2.

Let $M$ be a continuous centered Gaussian process with values in
$\mathcal{S}^{\prime}$ with covariance given, for all $\varphi_{1}$ and
$\varphi_{2}$ in $\mathcal{S}$, for all $t_{1}$ and $t_{2}\geq 0$, by

${\mathbb{E}}\left[M_{t_{1}}(\varphi_{1})M_{t_{2}}(\varphi_{2})\right]=\int_{0}^{t_{1}\wedge
t_{2}}\int_{0}^{1}e^{2\alpha
s}I[\varphi_{1}](y)I[\varphi_{2}](y)f(u(s,y))dyds,$ (33)

where for each $y\in[0,1]$,

$I[\varphi](y)=\int_{0}^{1}w(y,x)\varphi(x)dx.$

######  Proposition 5.

Under assumptions of Proposition 3, the sequence $(M^{n})_{n\geq 1}$ of
processes in $\mathcal{D}({\mathbb{R}}_{+},\mathcal{S}^{\prime})$ converges in
law to $M$ defined above.

###### Proof.

By Corollary 3 and [30, Theorem 5.3], it suffices to show that for $0\leq
t_{1}\leq t_{2}\leq\ldots\leq t_{m}\leq T$ and
$\varphi_{1},\ldots,\varphi_{m}\in\mathcal{S}$, the sequence of random vectors
$(M^{n}_{t_{1}}(\varphi_{1}),\ldots,M^{n}_{t_{m}}(\varphi_{m}))_{n\geq 1}$
converges in law to a Gaussian random vector $\mathcal{N}(0,\Sigma)$ with
covariance matrix $\Sigma=(\Sigma_{ij})_{1\leq i,j\leq m}$ given by

$(\Sigma)_{ij}=\int_{0}^{t_{i}\wedge t_{j}}\int_{0}^{1}e^{2\alpha
s}I[\varphi_{i}](y)I[\varphi_{j}](y)f(u(s,y))dyds.$

To show this convergence holds, by Cramér–Wold Theorem, it is enough to show
that for all $\beta=(\beta_{1},\ldots,\beta_{m})$, the sequence of random
variables $(\sum_{p=1}^{m}\beta_{p}M^{n}_{t_{p}}(\varphi_{p}))_{n\geq 1}$
converges in law to a Gaussian random variable $\mathcal{N}(0,\sigma^{2})$
with variance $\sigma^{2}=\beta^{T}\Sigma\beta.$ To that end, we will resort
to the central limit theorem for local martingales from [32].

In what follows, for each $n\geq 1$, let $(\tilde{M}^{n}_{t})_{t\geq 0}$ be
the local martingales defined by

$\tilde{M}^{n}_{t}=\sum_{p=1}^{m}\beta_{p}M^{n}_{t_{p}\wedge t}(\varphi_{p}).$

Its angle bracket can be written in the following form:

$\langle\tilde{M}^{n}\rangle_{t}=\sum_{1\leq p,q\leq
m}\beta_{p}\beta_{q}\langle
M^{n}(\varphi_{p}),M^{n}(\varphi_{q})\rangle_{(t_{p}\wedge t_{q})\wedge t}.$

According to [32], the sequence $(\tilde{M}^{n})_{n\geq 1}$ converges in law
to a centered Gaussian process $\tilde{M}$ with covariance function
$\mbox{Cov}(\tilde{M}_{t},\tilde{M}_{s})=C(t\wedge s)$, if the following
conditions are verified:

  1. 1.

$\langle\tilde{M}^{n}\rangle_{t}$ converges to $C(t)$ in probability as
$n\to\infty$, for each $t>0.$

  2. 2.

For each $\epsilon>0$ and $t>0$, the sequence of random variables

$\sum_{s\leq
t}|\Delta\tilde{M}^{n}_{s}|1_{\\{|\Delta\tilde{M}^{n}_{s}|>\epsilon\\}},$

converges to $0$ in probability as $n\to\infty.$

Let us assume that these two conditions have been checked with $(C(t))_{t\geq
0}$ such that $C(T)=\sigma^{2}$. In that case, we would have that
$\sum_{p=1}^{m}\beta_{p}M^{n}_{t_{p}}(\varphi_{p})=\tilde{M}^{n}_{T}$
converges to $\mathcal{N}(0,\sigma^{2})$ in law as $n\to\infty$, concluding
the proof of the proposition. In the remaining part of the proof, we will
check conditions 1 and 2 above are satisfied with $(C(t))_{t\geq 0}$ such that
$C(T)=\sigma^{2}.$

Proof of condition 1. For each $t\geq 0$, write

$C(t)=\sum_{1\leq p,q\leq m}\beta_{p}\beta_{q}\int_{0}^{(t_{q}\wedge
t_{p})\wedge t}I[\varphi_{p}](y)I[\varphi_{q}](y)e^{2\alpha s}f(u(s,y))dyds,$

and for each $n\geq 1$, $y\in[0,1]$ and $\varphi\in\mathcal{S}$, let

$I^{n}[\varphi](y)=\frac{1}{n}\sum_{i=1}^{n}\varphi(x_{i})w(y,x_{i}).$

Observe that $C(T)=\sigma^{2}$ and
$\|I^{n}[\varphi]\|_{0}\leq\|\varphi\|_{0}\|w\|_{0}$. Now, from the Riemann
sum approximation, we also have that

$\|I^{n}[\varphi]-I[\varphi]\|_{0}\leq\frac{1}{2n}\sup_{x,y\in[0,1]}|\varphi^{\prime}(x)w(y,x)+\varphi(x)\partial_{2}w(y,x)|.$

Moreover, with this notation, it follows from (28) that

$\langle\tilde{M}^{n}\rangle_{t}=\sum_{1\leq p,q\leq
m}\beta_{q}\beta_{p}\frac{1}{n}\sum_{j=1}^{n}\int_{0}^{t\wedge(t_{p}\wedge
t_{q})}I^{n}[\varphi_{p}](x_{j})I^{n}[\varphi_{q}](x_{j})e^{2\alpha
s}f(U^{j}_{s})ds,$

so that

${\mathbb{E}}|\langle\tilde{M}^{n}\rangle_{t}-C(t)|\leq\sum_{1\leq p,q\leq
m}\beta_{q}\beta_{p}{\mathbb{E}}\left|\frac{1}{n}\sum_{j=1}^{n}\int_{0}^{t\wedge(t_{p}\wedge
t_{q})}I^{n}[\varphi_{p}](x_{j})I^{n}[\varphi_{q}](x_{j})e^{2\alpha
s}f(U^{j}_{s})ds\right.\\\ \left.-\int_{0}^{t\wedge(t_{p}\wedge
t_{q})}\int_{0}^{1}I[\varphi_{p}](y)I[\varphi_{q}](y)e^{2\alpha
s}f(u(s,y))dyds\right|.$ (34)

Now, for any $1\leq p,q\leq m$, one can check that

$\left|\frac{1}{n}\sum_{j=1}^{n}(I^{n}[\varphi_{p}](x_{j})I^{n}[\varphi_{q}](x_{j})-I[\varphi_{p}](x_{j})I[\varphi_{q}](x_{j}))\right|\\\
\leq\frac{\|w\|_{0}}{n}\max_{u,v\in\\{p,q\\}}\left\\{\|\varphi_{v}\|_{0}\sup_{x,y\in[0,1]}|\frac{\partial}{\partial
x}(\varphi_{u}(x)w(y,x))|\right\\}.$ (35)

Similarly, from Riemann sum approximation, we have for any $s\leq t$ fixed,

$\left|\frac{1}{n}\sum_{j=1}^{n}I[\varphi_{p}](x_{j})I[\varphi_{q}](x_{j})f(u(s,x_{j}))-\int_{0}^{1}I[\varphi_{p}](y)I[\varphi_{q}](y)f(u(s,y))dy\right|\\\
\leq\frac{1}{2n}\sup_{y\in[0,1],h\leq
t}\left|\frac{d}{dy}(I[\varphi_{p}](y)I[\varphi_{q}](y)f(u(h,y)))\right|.$
(36)

The local boundedness of both $f$ and $u$ implies $\sup_{t\leq
T}\|f(u(h,\cdot)\|_{0}<\infty$. Combining this fact with the boundedness of
$f^{(}1)$, and the smoothness of both $\varphi_{p}$ and $\varphi_{q}$, one can
show that

$\sup_{y\in[0,1],h\leq
T}\left|\frac{d}{dy}(I[\varphi_{p}](y)I[\varphi_{q}](y)f(u(h,y)))\right|<\infty$

Furthermore, we have that

${\mathbb{E}}\left|\frac{1}{n}\sum_{j=1}^{n}\int_{0}^{t\wedge(t_{p}\wedge
t_{q})}e^{2\alpha
2}I[\varphi_{p}](x_{j})I[\varphi_{q}](x_{j})(f(U^{j}_{s})-f(u(s,x_{j})))ds\right|\\\
\leq\|f^{\prime}\|_{0}\frac{\|I[\varphi_{p}]\|_{0}\|I[\varphi_{q}]\|_{0}}{n^{1/2}}\int_{0}^{t}e^{2\alpha
s}\sup_{n\geq 1}\frac{1}{n}\sum_{j=1}^{n}{\mathbb{E}}|\eta^{j}_{s}|ds.$ (37)

Combining the inequalities (35), (36) and (37) with (34), and using that the
function $s\mapsto\sup_{n\geq
1}\frac{1}{n}\sum_{j=1}^{n}{\mathbb{E}}|\eta^{j}_{s}|$ is locally bounded, we
then have that there exists a constant $C$ not depending on $n$ such that for
all $n$ sufficiently large

${\mathbb{E}}|\langle\tilde{M}^{n}\rangle_{t}-C(t)|\leq C\frac{1}{n^{1/2}}.$

The proof of condition 1 follows now from Markov inequality.

Proof of condition 2. It is enough to prove that there exists a positive
constant $C$ not depending on $n\geq 1$ such that for each $t>0$ and $n\geq
1$,

$\sup_{0\leq s\leq t}|\Delta\tilde{M}^{n}_{s}|\leq\frac{C}{n^{1/2}}\
\mbox{almost surely.}$

To prove that, observe that for each $s\geq 0$,

$|\Delta\tilde{M}^{n}_{s}|\leq m\max_{1\leq p\leq
m}\\{\|\varphi_{p}\|_{0}\\}\|w\|_{0}\frac{1}{n^{1/2}}\sum_{j=1}^{n}\Delta
N^{j}_{s},$

implying that (with $C=m\max_{1\leq p\leq
m}\\{\|\varphi_{p}\|_{0}\\}\|w\|_{0}$)

$\sup_{0\leq s\leq t}|\Delta\tilde{M}^{n}_{s}|\leq\frac{C}{n^{1/2}}\sup_{0\leq
s\leq t}\sum_{j=1}^{n}\Delta N^{j}_{s}.$

Since almost surely for all $1\leq i,j\leq n$ with $i\neq j$, the counting
processes $N^{j}$ and $N^{i}$ never jump simultaneously, it follows that

$\sup_{0\leq s\leq t}\sum_{j=1}^{n}\Delta N^{j}_{s}\leq 1\ \mbox{almost
surely,}$

and the result follows. ∎

We are now in position to state the limit equation satisfied by the limit
points, generically denoted by $\Gamma$, of the sequence $(\Gamma^{n})_{n}$.
The limit equation is:

$\forall\varphi\in\mathcal{S},\quad\Gamma_{t}(\varphi)=e^{-\alpha
t}M_{t}(\varphi)+\int_{0}^{t}e^{-\alpha(t-s)}\Gamma_{s}\left(\int_{0}^{1}\varphi(x)w(\cdot,x)f^{\prime}(u(s,\cdot))dx\right)ds,$
(38)

where $M$ is the Gaussian process of Definition 2.

For notational simplicity, let us introduce the following maps
$F_{\varphi}:\mathcal{D}([0,T],\mathcal{S}^{\prime}\times\mathcal{S}^{\prime})\to\mathcal{D}([0,T],{\mathbb{R}})$
defined, for all $(g_{t},m_{t})_{t\in[0,T]}$ in
$\mathcal{D}([0,T],\mathcal{S}^{\prime}\times\mathcal{S}^{\prime})$, by

$F_{\varphi}(g,m)_{t}=g_{t}(\varphi)-e^{-\alpha
t}m_{t}(\varphi)-\int_{0}^{t}e^{-\alpha(t-s)}g_{s}\left(\int_{0}^{1}\varphi(x)w(\cdot,x)f^{\prime}(u(s,\cdot))dx\right)ds,$

so that (38) is equivalent to : for all $\varphi\in\mathcal{S}$,
$F_{\varphi}(\Gamma,M)=0$.

######  Remark 4.

Assumption 1 ensures that the function
$y\mapsto\int_{0}^{1}\varphi(x)w(y,x)f^{\prime}(u(s,y))dx$ is in $S$ so that
the RHS of (38) is well defined.

About the space
$\mathcal{D}([0,T],\mathcal{S}^{\prime}\times\mathcal{S}^{\prime})$, the
product $\mathcal{S}^{\prime}\times\mathcal{S}^{\prime}$ is endowed with the
product topology of the strong topology on $\mathcal{S}^{\prime}$. Then,
$\mathcal{D}([0,T],\mathcal{S}^{\prime}\times\mathcal{S}^{\prime})$ is endowed
with some projective topology inspired by Skorokhod topology (see B for more
insight). Finally, the maps $F_{\varphi}$ are continuous with respect to that
topology.

######  Proposition 6.

Under Assumption 1, for all $\varphi\in\mathcal{S}$,
$F_{\varphi}(\Gamma^{n},M^{n})\to 0$ in probability.

###### Proof.

Observe that for each $t\geq 0,$

$F_{\varphi}(\Gamma^{n},M^{n})_{t}=B^{n}_{t}(\varphi)+C^{n}_{t}(\varphi)-\frac{1}{n}\sum_{j=1}^{n}\int_{0}^{1}\varphi(x)w(x_{j},x)dx\int_{0}^{t}e^{-\alpha(t-s)}\eta^{j}_{s}f^{\prime}(u(s,x_{j}))ds.$

Recall that $\sup_{t\leq T}|C^{n}_{t}(\varphi)|\leq
C\|\varphi\|_{\infty}n^{-1/2}$ by Proposition 3. Thus given $\epsilon>0$, we
have that $\sup_{t\leq T}|C^{n}_{t}(\varphi)|\leq\epsilon/2$ for all
$n\geq(2C\|\varphi\|_{0}\epsilon^{-1})^{2},$ so that the event
$\left\\{\sup_{t\leq T}|F_{\varphi}(\Gamma^{n},M^{n})_{t}|>\epsilon\right\\}$
is contained in event

$\left\\{\sup_{t\leq
T}\left|B^{n}_{t}(\varphi)-\frac{1}{n}\sum_{j=1}^{n}\int_{0}^{1}\varphi(x)w(x_{j},x)dx\int_{0}^{t}e^{-\alpha(t-s)}\eta^{j}_{s}f^{\prime}(u(s,x_{j}))ds\right|>\epsilon/2\right\\}.$

We will show in the remaining part of the proof that

$\sup_{0\leq t\leq
T}\left|B^{n}_{t}(\varphi)-\frac{1}{n}\sum_{j=1}^{n}\int_{0}^{1}\varphi(x)w(x_{j},x)dx\int_{0}^{t}e^{-\alpha(t-s)}\eta^{j}_{s}f^{\prime}(u(s,x_{j}))ds\right|\to
0$

in $L^{1}$ as $n\to\infty$, implying the result.

To that end, note that for all $t\leq T$ and $n\geq 1,$

$\left|B^{n}_{t}(\varphi)-\frac{1}{n}\sum_{j=1}^{n}\int_{0}^{1}\varphi(x)w(x_{j},x)dx\int_{0}^{t}e^{-\alpha(t-s)}\eta^{j}_{s}f^{\prime}(u(s,x_{j}))ds\right|\leq
I^{n}+II^{n},$

where (remember that $\Delta^{j}_{s}(f)=f(U^{j}_{s})-f(u(s,x_{j}))$)

$I^{n}=\left|\frac{1}{\sqrt{n}}\sum_{j=1}^{n}\int_{0}^{t}e^{-\alpha(t-s)}\left[\frac{1}{n}\sum_{i=1}^{n}\varphi(x_{i})w(x_{j},x_{i})-\int_{0}^{1}\varphi(x)w(x_{j},x)dx\right]\Delta_{s}^{j}(f)ds\right|,$

and

$II^{n}=\left|\frac{1}{\sqrt{n}}\sum_{j=1}^{n}\int_{0}^{1}\varphi(x)w(x_{j},x)dx\int_{0}^{t}e^{-\alpha(t-s)}\left[\Delta_{s}^{j}(f)-f^{\prime}(u(s,x_{j}))\frac{\eta^{j}_{s}}{\sqrt{n}}\right]\right|.$
(39)

Now, using the fact that $f$ is Lipschitz and classical Riemann estimates, it
follows that

$I^{n}\leq\frac{\|f^{\prime}\|_{0}}{2n}\sup_{0\leq x,y\leq
1}|\frac{\partial}{\partial
x}(\varphi(x)w(y,x))|\int_{0}^{t}e^{-\alpha(t-s)}\frac{1}{n}\sum_{j=1}^{n}|\eta^{j}_{s}|ds.$

Since the function $s\to\sup_{n\geq
1}\frac{1}{n}\sum_{j=1}^{n}{\mathbb{E}}|\eta^{j}_{s}|$ is locally bounded by
Proposition 2), we have that $I^{n}\to 0$ in $L^{1}$ as $n\to\infty$.

To deal with $II^{n}$, we recall that the Taylor approximation of order 2
yields

$|f(x)-f(y)-f^{\prime}(y)(x-y)|\leq\frac{|x-y|^{2}}{2}\|f^{\prime\prime}\|_{0},\
\mbox{for all}\ x,y\in{\mathbb{R}}.$

Hence, we have for all $s\geq 0$ and $j\in[n],$

$|\Delta^{j}_{s}(f)-f^{\prime}(u(s,x_{j}))\frac{\eta^{j}_{s}}{\sqrt{n}}|\leq\frac{(U^{j}_{s}-u(s,x_{j}))^{2}}{2}\|f^{\prime\prime}\|_{0}=\frac{\|f^{\prime\prime}\|_{0}}{2n}(\eta^{j}_{s})^{2},$

so that

${\mathbb{E}}(II^{n})\leq\sup_{0\leq y\leq
1}\left|\int_{0}^{1}\varphi(x)w(y,x)dx\right|\frac{\|f^{\prime\prime}\|_{0}}{2\sqrt{n}}\int_{0}^{t}e^{-\alpha(t-s)}\sup_{m\geq
1}\frac{1}{m}\sum_{j=1}^{m}{\mathbb{E}}((\eta_{s}^{j})^{2})ds.$

The local boundedness of $s\to\sup_{n\geq
1}\frac{1}{n}\sum_{j=1}^{n}{\mathbb{E}}((\eta^{j}_{s})^{2})$ implies that
$II^{n}\to$ in $L^{1}$ as $n\to\infty$ as well, concluding the proof of the
proposition. ∎

We are now in position to state the main result of this section.

######  Theorem 2.

Under Assumption 1, any limit point $\Gamma$ of the sequence
$(\Gamma^{n})_{n\geq 1}$ is a solution of (38) in
$\mathcal{C}({\mathbb{R}}_{+},\mathcal{S}^{\prime})$.

###### Proof.

Let $\Gamma$ be a limit point of $(\Gamma^{n})_{n}$ and $(n_{k})_{k}$ be such
that $\Gamma^{n_{k}}\to\Gamma$ in distribution. Like in Corollary 4, we
obviously have joint tightness of $(\Gamma^{n_{k}},M^{n_{k}})$. Hence let $M$
be such that $(\Gamma,M)$ is a limit point of $(\Gamma^{n_{k}},M^{n_{k}})$.
The convergence result of Proposition 6 and continuous mapping theorem imply
that for all $\varphi$ in $\mathcal{S}$, $F_{\varphi}(\Gamma,M)=0$. Hence
$\Gamma$ satisfies (38).

Finally, the continuity of $\Gamma$ follows from Proposition 4. ∎

##  7 Convergence

######  Proposition 7.

Under Assumption 1, there is path-wise uniqueness of the solutions of limit
equation (38): if $\Gamma$ and $\tilde{\Gamma}$ are two solutions in
$\mathcal{C}({\mathbb{R}}_{+},\mathcal{S}^{\prime})$ constructed on the same
probability space as $M$, then $\Gamma$ and $\tilde{\Gamma}$ are
indistinguishable.

###### Proof.

Let $\Gamma$ and $\tilde{\Gamma}$ be two solutions and take $T>0$. In the
following, consider the restrictions of $\Gamma$ and $\tilde{\Gamma}$ to
$[0,T]$. For almost every $\omega\in\Omega$, $\Gamma(\omega)$ and
$\tilde{\Gamma}(\omega)$ are continuous and
$F_{\varphi}(\Gamma(\omega)-\tilde{\Gamma}(\omega),M)=0$ for all
$\varphi\in\mathcal{S}$, i.e.

$(\Gamma(\omega)-\tilde{\Gamma}(\omega))_{t}(\varphi)=\int_{0}^{t}e^{-\alpha(t-s)}(\Gamma(\omega)-\tilde{\Gamma}(\omega))_{s}\left(\int_{0}^{1}\varphi(x)w(\cdot,x)f^{\prime}(u(s,\cdot))dx\right)ds.$
(40)

In the following, we assume that such a generic $\omega$ is fixed and omit to
write the dependence.

Let $\varphi$ be in $\mathcal{S}$. Since $\Gamma$ and $\tilde{\Gamma}$ are
continuous, $\Gamma(\varphi)$ and $\tilde{\Gamma}(\varphi)$ belong to
$\mathcal{C}([0,T],{\mathbb{R}})$ and in particular,
$\sup_{t\in[0,T]}|(\Gamma_{t}-\tilde{\Gamma}_{t})(\varphi)|<+\infty$. The
uniform boundedness principle [35, Theorem 10.11.] implies that there exists
$k$ and $c>0$ such that

$\forall\varphi\in\mathcal{S},\,\sup_{s\in[0,T]}|(\Gamma_{s}-\tilde{\Gamma}_{s})(\varphi)|\leq
c||\varphi||_{k}.$

Therefore, let us define for all $t$ in $[0,T]$,

$L_{t}=\sup_{\varphi\in\mathcal{S}}\sup_{s\in[0,t]}\frac{|(\Gamma_{s}-\tilde{\Gamma}_{s})(\varphi)|}{||\varphi||_{k}}\leq
c<+\infty.$

In order to use boundedness for $f$ and its derivatives, let us remark that
$K_{T}=\sup_{t\in[0,T]}||u(t,\cdot)||_{0}<+\infty$ so that without loss of
generality, $f$ could be restricted to the compact interval $[-K_{T},K_{T}]$
and so considered to be $\mathcal{C}^{\infty}$ with bounded derivatives of any
order.

Then, using Lemma 1 and the fact that $w$, $u$ and $f$ are smooth,

$\left\|\int_{0}^{1}\varphi(x)w(\cdot,x)f^{\prime}(u(s,\cdot))dx\right\|_{k}\leq||\varphi||_{0}\sup_{s\in[0,T]}\sup_{x\in[0,1]}||w(\cdot,x)f^{\prime}(u(s,\cdot))||_{k}\\\
\leq
C||\varphi||_{k}\left(\sup_{x\in[0,1]}||w(\cdot,x)||_{k}\right)||f_{|[-K_{T},K_{T}]}||_{k+1}\left(\sup_{s\in[0,T]}||u(s,\cdot)||_{k}^{k}\right)\leq
C_{T}||\varphi||_{k}.$

Going back to (40), we have

$|(\Gamma-\tilde{\Gamma})_{t}(\varphi)|\leq\int_{0}^{t}L_{s}\left\|\int_{0}^{1}\varphi(x)w(\cdot,x)f^{\prime}(u(s,\cdot))dx\right\|_{k}ds.$

and so $L_{t}\leq C_{T}\int_{0}^{t}L_{s}ds$. Since $L_{t}$ is a priori bounded
by $c$, Gronwall’s lemma implies that for all $t\in[0,T]$, $L_{t}=0$ which
means that $\Gamma=\tilde{\Gamma}$.

The argument above holds for almost every $\omega$ so path-wise uniqueness is
proven. ∎

######  Theorem 3.

Under Assumption 1, the sequence $(\Gamma_{n})_{n}$ converges in law in
$\mathcal{D}({\mathbb{R}}_{+},\mathcal{S}^{\prime})$ to the unique solution of
(38) in $\mathcal{C}({\mathbb{R}}_{+},\mathcal{S}^{\prime})$.

###### Proof.

Let $\Gamma$ be a limit point of $(\Gamma_{n})_{n}$. According to Theorem 2,
$\Gamma$ is a solution of the limit system (38) in
$\mathcal{C}({\mathbb{R}}_{+},\mathcal{S}^{\prime})$. Yet, path-wise
uniqueness given by Proposition 7 and Yamada–Watanabe theorem gives weak
uniqueness by the same argument as in [33, Theorem IX.1.7(i)]. Finally, weak
uniqueness gives uniqueness of the limit point $\Gamma$ and so convergence. ∎

##  8 Connection with a stochastic NFE

Let us begin this section with some discussion about the standard central
limit theorem. Let $\bar{X}_{n}$ be the empirical mean of some i.i.d. square
integrable centered and normalized random variables $X_{1},\dots,X_{n}$. The
law of large numbers and central limit theorem respectively tells that
$\bar{X}_{n}=0+o(1)$ and $\bar{X}_{n}=0+n^{-1/2}Z+o(n^{-1/2})$ where $Z$ is a
standard Gaussian random variable. Of course, the second statement is purely
informal but gives the flavor of the result.

With this description in mind, we provide here an informal overview of the
mean-field limit and central limit theorem stated in this paper. Assume for
ease of the presentation that the limit $\Gamma_{t}$ is in fact a function,
namely there exists $(t,x)\mapsto\Gamma_{t}(x)$ such that for all $\varphi$,
$\Gamma_{t}(\varphi)=\int_{0}^{1}\varphi(x)\Gamma_{t}(x)dx$. Then, the take-
away message until this point of the paper is: in order to approximate the
microscopic system $(U^{i}_{t})_{t\geq 0,i\in[n]}$, there are two steps,

  1. 1.

the mean-field limit $u(t,x)$ makes an error of order $n^{-1/2}$ since the
renormalized error $\Gamma^{n}$ goes to something non trivial;

  2. 2.

the mean-field combined with the fluctuations, namely
$Y^{n}(t,x):=u(t,x)+n^{-1/2}\Gamma_{t}(x)$ makes an error of order
$o(n^{-1/2})$.

With this in mind, the goal of this section is to find an approximation
$V^{n}$ of $Y^{n}$ with an error of order $o(n^{-1/2})$ justifying that
$V^{n}$ is a better approximation than the standard mean-field limit. This
approximation $V^{n}$ will be characterized as the unique solution of a
particular stochastic version of the neural field equation and the discussion
above justifies that it is the _Stochastic Neural Field Equation_ (SNFE)
naturally associated with the Hawkes processes given by (1).

In the following we are interested in the following SNFE

$\begin{cases}dV^{n}_{t}(x)=\left(-\alpha
V^{n}_{t}(x)+\int_{0}^{1}w(y,x)f(V^{n}_{t}(y))dy\right)dt+\int_{0}^{1}w(y,x)\frac{\sqrt{f(V^{n}_{t}(y))}}{\sqrt{n}}W(dt,dy),\\\
V^{n}_{0}(x)=u_{0}(x).\end{cases}$ (41)

where $W$ is a Gaussian white noise. The mathematical arguments used below are
highly inspired from [17] where other kinds of stochastic neural field
equations can be found.

First we need to specify what we mean by Gaussian white noise. Here, we use
the Gaussian random field

$W=(W(A))_{A\in\mathcal{B}({\mathbb{R}}_{+}\times[0,1])},$ (42)

with covariance function

${\mathbb{E}}(W(A)W(B))=|A\cap B|,$

where $|A\cap B|$ denotes the Lebesgue measure of $A\cap B$. Then, the SNFE
(41) has to be understood in the weak sense.

######  Definition 3.

By a solution to (41) we mean a real-valued random field
$(V^{n}_{t}(x))_{t\geq 0,x\in[0,1]}$ such that

$V^{n}_{t}(x)=e^{-t}V^{n}_{0}(x)+\int_{0}^{t}e^{-(t-s)}\int_{0}^{1}w(y,x)f(V^{n}_{s}(y))dyds\\\
+\int_{0}^{t}e^{-(t-s)}\int_{0}^{1}w(y,x)\frac{\sqrt{f(V^{n}_{s}(y))}}{\sqrt{n}}W(ds,dy),$
(43)

almost surely for all $t\geq 0$ and $x\in[0,1].$

Now it suffices to give sense to the stochastic term,

$\int_{0}^{t}e^{-(t-s)}\int_{0}^{1}w(y,x)\frac{\sqrt{f(V^{n}_{s}(y))}}{\sqrt{n}}W(ds,dy).$

Walsh’s theory of stochastic integration provides a nice framework to give it
a sense, see for instance [17, Theorem 3.1] or [12] for details. We use this
theory in the rest of the paper.

Before stating the well posedness of the SFNE and the approximation result, we
first provide some heuristics leading to (41).

###  8.1 Heuristic motivation for (41)

Assume for now that $\Gamma_{t}$ and $M_{t}$ defined as distributions in the
previous sections are in fact functions (this will be precised later on).
Then, let us make the following abuse of notation:
$\Gamma_{t}(x)=\Gamma_{t}(\delta_{x})$ and $M_{t}(x)=M_{t}(\delta_{x})$ in
such a way that, for any $\varphi\in\mathcal{S}$,

$\Gamma_{t}(\varphi)=\int_{0}^{1}\varphi(x)\Gamma_{t}(x)dx\ \mbox{and}\
M_{t}(\varphi)=\int_{0}^{1}\varphi(x)M_{t}(x)dx.$

To guess what equation $\Gamma_{t}(x)$ solves, we take (informally)
$\varphi=\delta_{x}$ in (4) to get

$\Gamma_{t}(x)=e^{-\alpha
t}M_{t}(x)+\int_{0}^{t}e^{-\alpha(t-s)}\int_{0}^{1}w(y,x)f^{\prime}(u(s,y))\Gamma_{s}(y)dyds,\
t\geq 0\ \mbox{and}\ x\in[0,1].$ (44)

As briefly discussed in the Introduction, the first term on the RHS of the
equation above, namely $M_{t}(x)$, is the limit in distribution as
$n\to\infty$ of

$\frac{1}{\sqrt{n}}\sum_{j=1}^{n}w(x_{j},x)\int_{0}^{t}e^{\alpha
s}(dN^{j}_{s}-f(U^{j}_{s}))ds,$

which has mean zero and limit variance

$\int_{0}^{t}e^{2\alpha s}\int_{0}^{1}w^{2}(y,x)f(u(s,y))dyds.$

Besides, the Martingale Central Limit theorem ensures that $M_{t}(x)$ is
Gaussian so a suitable description for $M_{t}(x)$ should be:

$M_{t}(x)=\int_{0}^{t}\int_{0}^{1}e^{\alpha s}w(y,x)\sqrt{f(u(s,y))}W(ds,dy),\
t\geq 0\ \mbox{and}\ x\in[0,1],$ (45)

where $W$ is the white noise process defined in (42).

Now, we are interested in $Y^{n}_{t}(x)=u(t,x)+n^{-1/2}\Gamma_{t}(x)$ which,
by summing (12) and (44) (where $M$ is replaced according to (45)), is given
by

$Y^{n}_{t}(x)=e^{-\alpha
t}u_{0}(x_{i})+\int_{0}^{t}e^{-\alpha(t-s)}\int_{0}^{1}w(y,x)\frac{\sqrt{f(u(s,y))}}{\sqrt{n}}W(ds,dy)\\\
+\int_{0}^{t}e^{-\alpha(t-s)}\int_{0}^{1}w(y,x)\left(f(u(s,y))+n^{-1/2}f^{\prime}(u(s,y))\Gamma_{s}(y)\right)dyds.$
(46)

According to Taylor approximation, the error made when replacing
$f(Y^{n}_{s}(y))$ by $f(u(s,y))+n^{-1/2}f^{\prime}(u(s,y))\Gamma_{s}(y)$ is of
order $o(n^{-1/2})$ and replacing $f(Y^{n}_{s}(y))$ by $f(u(s,y))$ is of order
$o(1)$. When making these replacements, Equation (46) is exactly the equation
satisfied by the solution of our SNFE. For this reason, we expect that the
difference between $Y^{n}$ and the solution of the SFNE is of order
$o(n^{-1/2})$. This result is confirmed below in Theorem 6.

###  8.2 Results on the SNFE

######  Theorem 4.

Under Assumption 1, assume that $f$ is lower bounded by $m>0$. Then, for all
$n\geq 1$ there exists a unique (up to modification) solution $V$ of (41) (in
the sense of Definition 3) such that for all $T>0$,

$\sup_{t\leq T,x\in[0,1]}\mathbb{E}\left[|V_{t}(x)|^{2}\right]<+\infty.$ (47)

###### Proof.

Without loss of generality we assume in this proof that $n=1$.

We begin with uniqueness. Suppose that $V$ and $\tilde{V}$ are two solutions
and define $\Delta=V-\tilde{V}$. We have

$\Delta(t,x)=\int_{0}^{t}e^{-(t-s)}\int_{0}^{1}w(y,x)[f(V_{s}(y))-f(\tilde{V}_{s}(y))]dyds\\\
+\int_{0}^{t}e^{-(t-s)}\int_{0}^{1}w(y,x)\left[\sqrt{f(V_{s}(y))}-\sqrt{f(\tilde{V}_{s}(y))}\right]W(ds,dy).$

By Jensen and Burkhölder inequalities it follows that

$\mathbb{E}[|\Delta(t,x)|^{2}]\leq
2t\|w\|^{2}_{0}\int_{0}^{t}e^{-2(t-s)}\int_{0}^{1}\mathbb{E}\left[\left(f(V_{s}(y))-f(\tilde{V}_{s}(y))\right)^{2}\right]dyds\\\
+2\|w\|^{2}_{0}\int_{0}^{t}e^{-2(t-s)}\int_{0}^{1}{\mathbb{E}}\left[\left(\sqrt{f(V_{s}(y))}-\sqrt{f(\tilde{V}_{s}(y))}\right)^{2}\right]dyds.$

Then, we use the fact that $f$ and $\sqrt{f}$ are Lipschitz (since $f$ is
lower bounded) and it follows that

$\mathbb{E}[|\Delta(t,x)|^{2}]\leq
2t\|w\|^{2}_{0}\|f^{\prime}\|^{2}_{0}\int_{0}^{t}e^{-2(t-s)}\int_{0}^{1}\mathbb{E}\left[|\Delta(s,y)|^{2}\right]dyds\\\
+2\|w\|^{2}_{0}C\|f^{\prime}\|^{2}_{0}\int_{0}^{t}e^{-2(t-s)}\int_{0}^{1}{\mathbb{E}}\left[|\Delta(s,y)|^{2}\right]dyds.$

Writing $G(t):=\sup_{x\in[0,1]}\mathbb{E}[|\Delta(t,x)|^{2}]$, we get

$G(t)\leq C(t+1)\int_{0}^{t}G(s)ds,$ (48)

and Gronwall’s lemma implies $G(t)=0$ for all $t\geq 0$ which grants the
uniqueness property.

For the existence of a solution, we can proceed with Picard iteration. Let
$V^{(0)}_{t}(x)=u_{0}(x)$ for all $t\geq 0$, and define iteratively on $k$,

$V^{(k+1)}_{t}(x):=e^{-t}u_{0}(x)+\int_{0}^{t}e^{-(t-s)}\int_{0}^{1}w(y,x)f(V^{(k)}_{s}(y))dyds\\\
+\int_{0}^{t}e^{-(t-s)}\int_{0}^{1}w(y,x)\sqrt{f(V^{(k)}_{s}(y))}W(ds,dy).$
(49)

The proof of convergence of the Picard iteration is pretty classic and follows
computations which are similar to the ones given above to show the uniqueness
property. We give below a sketch of proof with the main steps whereas the
interested reader is referred to [17, Theorem 3.7.] for the missing
computations.

Using the fact that $\sup_{x\in[0,1]}u_{0}(x)^{2}<+\infty$ (which is granted
by Assumption 1), one can show by induction that $\sup_{t\leq
T,x\in[0,1]}{\mathbb{E}}\left[|V^{(k)}_{t}(x)|^{2}\right]<+\infty$ so that the
stochastic integral in (49) is well defined in Walsh’s sense. Then, defining
$\Delta^{(k)}:=V^{(k+1)}-V^{(k)}$ and
$G_{k}(t):=\sup_{x\in[0,1]}\mathbb{E}[|\Delta^{(k)}_{t}(x)|^{2}]$ and applying
induction on the same computations as the one leading to Equation (48) give

$G_{k}(t)\leq
C^{k}(t+1)^{k}\int_{0}^{t}\dots\int_{0}^{t_{k-1}}G_{0}(t_{k})dt_{k}k\dots
dt_{1},$

for $k\geq 1$ and $G_{0}(t)\leq C_{t}(1+\sup_{x\in[0,1]}u_{0}(x)^{2})$ for
some time dependent constant $C_{t}$. Using both inequalities above, one can
show that

$\sup_{k}\sup_{t\leq
T,x\in[0,1]}{\mathbb{E}}\left[|V^{(k)}_{t}(x)|^{2}\right]<+\infty.$

Finally, this implies the existence of the limit $V_{t}(x)$ in $L^{2}$ and
that the convergence is uniform, i.e.

$\sup_{t\leq
T,x\in[0,1]}{\mathbb{E}}\left[|V^{(k)}_{t}(x)-V_{t}(x)|^{2}\right]\to 0.$

Hence, Equation (47) is satisfied and the uniform convergence justifies taking
the limit as $k$ goes to infinity in (49) in order to prove that the limit $V$
is indeed a solution in the sense of Definition 3. ∎

Furthermore, the mean-field limit is an approximation of the SFNE in the
following sense.

######  Proposition 8.

Under the assumptions of Theorem 4, there exists a constant $C=C(T,w,f,u_{0})$
such that the unique solution $V^{n}$ of (41) satisfies

$\sup_{t\leq
T,x\in[0,1]}\mathbb{E}\left[|V^{n}_{t}(x)-u(t,x)|^{2}\right]\leq\frac{C}{n}.$
(50)

###### Proof.

The proof is pretty similar to the proof of Theorem 4 but here we must keep
track of the index $n$. Let us denote $\Delta^{n}:=V^{n}-u$ and
$G^{n}(t):=\sup_{x\in[0,1]}\mathbb{E}[|\Delta^{n}_{t}(x)|^{2}]$. We have

$\Delta^{n}(t,x)=\int_{0}^{t}e^{-(t-s)}\int_{0}^{1}w(y,x)[f(V^{n}_{s}(y))-f(u(s,y))]dyds\\\
+n^{-1/2}\int_{0}^{t}e^{-(t-s)}\int_{0}^{1}w(y,x)\sqrt{f(V^{n}_{s}(y))}W(ds,dy).$

The same arguments (Jensen and Burkhölder inequalities) imply the existence of
a constant $C$ such that

$G^{n}(t)\leq C\left(n^{-1}+t\int_{0}^{t}G^{n}(s)ds\right),$

which grants the result according to Gronwall’s lemma. ∎

The following result gives the space-time regularity of the solution of the
SFNE (41).

######  Theorem 5.

Under the assumptions of Theorem 4, the unique solution $V$ of (41) is $L^{p}$
bounded,

$\sup_{t\leq T,x\in[0,1]}\mathbb{E}\left[|V_{t}(x)|^{p}\right]<+\infty,$ (51)

for all $p\geq 2$, and there is a modification of $V$ such that $(t,x)\mapsto
V_{t}(x)$ is $(\eta_{1},\eta_{2})$-Hölder continuous for any $\eta_{1}<1/2$
and $\eta_{2}<1$.

###### Proof.

Using Jensen and Burkhölder inequalities in a similar way as in the proof of
Theorem 4, one can get, from (43), the following inequality

${\mathbb{E}}\left[|V_{t}(x)|^{p}\right]\leq
3^{p-1}\left[|u_{0}(x)|^{p}+||w||_{0}^{p}||f^{\prime}||_{0}^{p}\left(t^{p-1}+c_{p}t^{p/2-1}\right)\int_{0}^{t}{\mathbb{E}}\left[\left(\sup_{x}|V_{s}(x)|+f(0)\right)^{p}\right]ds\right],$

and so there is a constant $C_{t}$ such that
$H_{p}(t):=\sup_{x\in[0,1]}\mathbb{E}\left[|V_{t}(x)|^{p}\right]$ satisfies

$H_{p}(t)\leq
C_{t}\left(1+\sup_{x\in[0,1]}u_{0}(x)^{p}+\int_{0}^{t}H_{p}(s)ds\right).$

Since $u_{0}$ is bounded, Gronwall’s lemma applied to the last inequality
gives (51).

For the time regularity, the same kind of computations gives for any $p$ and
times $0\leq s\leq t$, the existence of a constant $C_{t}$ such that

${\mathbb{E}}\left[|V_{t}(x)-V_{t^{\prime}}(x)|^{p}\right]\leq
C_{t}\left(1+H_{p}(t)\right)(t-t^{\prime})^{p/2}.$

Finally, Kolmogorov’s continuity theorem gives the stated regularity (remember
that $p$ can be as large as one needs).

For the spatial regularity, let us write
$V_{t}(x):=I_{1}(t,x)+I_{2}(t,x)+I_{3}(t,x)$ as given in Definition 3 by

$\begin{cases}I_{1}(t,x):=e^{-t}V_{0}(x),\\\
I_{2}(t,x):=\int_{0}^{t}e^{-(t-s)}\int_{0}^{1}w(y,x)f(V_{s}(y))dyds,\\\
I_{3}(t,x):=\int_{0}^{t}e^{-(t-s)}\int_{0}^{1}w(y,x)\frac{\sqrt{f(V_{s}(y))}}{\sqrt{n}}W(ds,dy).\end{cases}$

For all $x$ and $x^{\prime}$ in $[0,1]$, one can show that for any $p\geq 2$,
and $j=1,2,3$,

${\mathbb{E}}\left[|I_{j}(t,x)-I_{j}(t,x^{\prime})|^{p}\right]\leq
C_{t}|x-x^{\prime}|^{p}.$ (52)

Instead of giving the full computations, we only give the arguments here (once
again, similar computations can be found in [17, Theorem 3.10]. For $j=1$, it
is a direct consequence of the Lipschitz continuity of $u_{0}$. For $j=2$ and
$3$, it comes from the Lipschitz continuity of $w$.

Once again, the stated regularity is obtained thanks to Kolmogorov’s
continuity theorem. ∎

###  8.3 Approximation result

We start this section discussing the well-posedness of both processes
$(\Gamma_{t}(x))_{t,x}$ and $(M_{t}(x))_{t,x}.$

######  Proposition 9.

Fix $T>0$ and assume assumptions of Theorem 4.

  1. 1.

The process $(M_{t}(x))_{t\leq T,x\in[0,1]}$ given by (45) is well-defined in
$L^{2}$. Moreover, there is a modification of $(M_{t}(x))_{t,x}$ such that
$(t,x)\mapsto M_{t}(x)$ is $(\eta_{1},\eta_{2})$-Hölder continuous for any
$\eta_{1}<1/2$ and $\eta_{2}<1$. If we define for such a modification

$M_{t}(\varphi)=\int_{0}^{1}\varphi(x)M_{t}(x)dx,\ \mbox{for}\
\varphi\in\mathcal{S}\ \mbox{and}\ t\leq T,$

then $M=(M_{t})_{t\leq T}$ is a $\mathcal{S}^{\prime}$-valued centered
Gaussian process with covariance function defined in (5).

  2. 2.

There exists a unique solution $(\Gamma_{t}(x))_{t\leq T,x\in[0,1]}$ of (44)
such that for all $T>0$,

$\sup_{t\leq T,x\in[0,1]}\mathbb{E}\left[|\Gamma_{t}(x)|^{2}\right]<+\infty.$

Moreover, the solution is bounded in $L^{p}$ for any $p\geq 2$,

$\sup_{t\leq T}\sup_{x\in[0,1]}{\mathbb{E}}(|\Gamma_{t}(x)|^{p})<\infty,$

and there exists a modification such that $(t,x)\mapsto\Gamma_{t}(x)$ is
$(\eta_{1},\eta_{2})$-Hölder continuous for any $\eta_{1}<1/2$ and
$\eta_{2}<1$. Furthermore, if we define for such a modification

$\Gamma_{t}(\varphi)=\int_{0}^{1}\varphi(x)\Gamma_{t}(x)dx,\ \mbox{for}\
\varphi\in\mathcal{S}\ \mbox{and}\ t\leq T,$

then the resulting $\mathcal{S}^{\prime}$-valued process $(\Gamma_{t})_{t\leq
T}$ is the unique solution of (4).

###### Proof.

By Proposition 1, we have that $\sup_{t\leq T}\|u(t,\cdot)\|_{0}<\infty$.
Using this fact and recalling that $f$ is Lipschitz continuous and $w$ is
bounded, one can easily deduce that for any $t\leq T$ and $x\in[0,1],$

$\int_{0}^{t}\int_{0}^{1}e^{2\alpha s}w^{2}(y,x)f(u(s,y))dsdy<\infty.$

Then, by [17, Theorem 3.1] the process $M_{t}(x)$ given by (45) is well-
defined for any $t\leq T$ and $x\in[0,1].$ The proof of the regularity
properties of $(M_{t}(x))_{t,x}$ is omitted here since it follows along the
same lines as in the proof of Theorem 5.

Now, fix $t\leq T$ and take $\varphi\in\mathcal{S}$. Under assumptions of
Theorem 4 we can apply Fubini–Tonelli property for martingale integrals (see
for instance [12, Theorem 5.30]) to deduce that

$M_{t}(\varphi)=\int_{0}^{1}\varphi(x)M_{t}(x)dx=\int_{0}^{t}e^{\alpha
s}\int_{0}^{1}I(\varphi)(y)\sqrt{f(u(s,y))}W(dsdy),$

where $y\mapsto I(\varphi)(y)$ is defined in (5). By using the properties of
the white noise $W$ and [17, Theorem 3.1], one can easily check that $M$ is
indeed a $\mathcal{S}^{\prime}$-valued centered Gaussian process with
covariance function defined in (5).

It remains to prove Item 2. The proof of existence and uniqueness of solutions
of (44) is similar to that of Theorem 4, whereas the boundedness in $L^{p}$
and the regularity properties of $(\Gamma_{t}(x))_{t,x}$ follow along the same
lines as in the proof of Theorem 5. For this reason the details are not given
here. To conclude the proof of the theorem, fix $t\leq T$, take $\varphi\in S$
and observe that by the definition of $\Gamma_{t}(x)$ we have

$\Gamma_{t}(\varphi)=e^{-\alpha
t}M_{t}(\varphi)+\int_{0}^{1}\varphi(x)\int_{0}^{t}e^{-\alpha(t-s)}\int_{0}^{1}w(y,x)f^{\prime}(u(s,y))\Gamma_{s}(y)dydsdx.$

By Fubini Theorem, we can interchange the order of integration of the the term
in the RHS of the equation above to deduce that

$\Gamma_{t}(\varphi)=e^{-\alpha
t}M_{t}(\varphi)+\int_{0}^{t}e^{-\alpha(t-s)}\int_{0}^{1}\left(\int_{0}^{1}\varphi(x)w(y,x)f^{\prime}(u(s,y))dx\right)\Gamma_{s}(y)dyds.$

Since for any $s\geq 0$,

$\Gamma_{s}\left(\int_{0}^{1}\varphi(x)w(\cdot,x)f^{\prime}(u(s,\cdot))dx\right)=\int_{0}^{1}\left(\int_{0}^{1}\varphi(x)w(y,x)f^{\prime}(u(s,y))dx\right)\Gamma_{s}(y)dy,$

we have that $\Gamma_{t}(\varphi)$ solves (4) and the result follows from
uniqueness of Proposition 7. ∎

We are now ready to state and prove the approximation result: that is the
control of $D^{n}$ defined by, for all $t\in[0,T]$ and $x\in[0,1]$,

$D^{n}(t,x):=Y^{n}_{t}(x)-V^{n}_{t}(x)$ (53)

where $Y^{n}_{t}(x)$ and $V^{n}_{t}(x)$ are respectively given by (46) and
(43).

######  Theorem 6.

Under assumptions of Theorem 4, for any $T>0$ there exists a constant
$C=C(T,w,f^{\prime},f^{(2)},\alpha)$ such that for all $n\geq 1$,

$\sup_{0\leq t\leq
T}\sup_{x\in[0,1]}{\mathbb{E}}((D^{n}(t,x))^{2})\leq\frac{C}{n^{2}}.$

###### Proof.

By Jensen and Burkhölder inequalities it follows that

${\mathbb{E}}(D^{n}(t,x)^{2})\leq\frac{2\|w\|^{2}_{0}}{n}\int_{0}^{t}e^{-2\alpha(t-s)}\int_{0}^{1}{\mathbb{E}}\left(\left(\sqrt{f(u(s,y))}-\sqrt{f(V^{n}_{s}(y))}\right)^{2}\right)dyds\\\
+t\int_{0}^{t}e^{-2\alpha(t-s)}\int_{0}^{1}w^{2}(y,x){\mathbb{E}}\left(\left|f(u(s,y))+f^{\prime}(u(s,y))n^{-1/2}\Gamma_{s}(y)-f(V^{n}_{s}(y))\right|^{2}\right)dyds.$
(54)

By applying Taylor approximation of order 2 and then the inequality
$(a+b)^{2}\leq 2(a^{2}+b^{2})$, we obtain that

${\mathbb{E}}\left(\left|f(u(s,y))+f^{\prime}(u(s,y))n^{-1/2}\Gamma_{s}(y)-f(V^{n}_{s}(y))\right|^{2}\right)\leq
2\|f^{\prime}\|^{2}_{0}{\mathbb{E}}((D^{n}(s,y))^{2})\\\
+\frac{\|f^{(2)}\|_{0}}{2n^{2}}{\mathbb{E}}(\Gamma^{4}_{s}(y)).$

Now, by using first that $f$ is lower bounded by $m$ and then the fact that
$f$ is Lipschitz, it follows that

${\mathbb{E}}\left(\left(\sqrt{f(u(s,y))}-\sqrt{f(V^{n}_{s}(y))}\right)^{2}\right)\leq\frac{\|f^{\prime}\|^{2}_{0}}{4m}{\mathbb{E}}\left(\left|u(s,y)-V^{n}_{s}(y)\right|^{2}\right).$

Combining these last two inequalities with inequality (54) yields

${\mathbb{E}}(D^{n}(t,x))^{2}\leq\frac{\|w\|^{2}_{0}\|f^{\prime}\|_{0}}{2nm}\int_{0}^{t}e^{-2\alpha(t-s)}\int_{0}^{1}{\mathbb{E}}\left(\left|u(s,y)-V^{n}_{s}(y)\right|\right)dyds\\\
+2t\|w\|^{2}_{0}\|f^{\prime}\|_{0}^{2}\int_{0}^{t}e^{-2\alpha(t-s)}\int_{0}^{1}{\mathbb{E}}((D^{n}(s,y))^{2})dyds\\\
+\frac{t\|w\|^{2}_{0}\|f^{(2)}\|_{0}^{2}}{2n^{2}}\int_{0}^{t}e^{-2\alpha(t-s)}\int_{0}^{1}{\mathbb{E}}(\Gamma^{4}_{s}(y))dyds.$

Let $H(t)=\sup_{x\in[0,1]}{\mathbb{E}}((D^{n}(t,x))^{2})$. By Proposition 8
and Theorem 9, it follows that for all $t\leq T$,

$H(t)\leq\frac{C_{1}}{n^{2}}+C_{2}\int_{0}^{t}H(s)ds,$

for positive constants $C_{1}$ and $C_{2}$ depending only on
$T,w,f^{\prime},f^{(2)}$ and $\alpha$. Proposition 1 together with Theorems 4
and 9 imply that the function $t\mapsto H(t)$ is locally bounded, so that the
result follows from Gronwall inequality. ∎

##  9 Acknowledgments

This research has been conducted as part of FAPESP project Research,
Innovation and Dissemination Center for Neuromathematics (grant 2013/07699-0).
We also acknowledge the support of CNRS under the grant PEPS JCJC _MaNHawkes_.

##  Appendix A Lemmas

######  Lemma 1.

Let $f$ and $g$ be in $\mathcal{C}^{k}_{b}({\mathbb{R}},{\mathbb{R}})$. Then
$fg$ and $f\circ g$ are in $\mathcal{C}^{k}_{b}({\mathbb{R}},{\mathbb{R}})$
and there exists $C>0$ such that

$||fg||_{k}\leq C||f||_{k}||g||_{k}\text{ and }||f\circ g||_{k}\leq
C||f||_{k}(1+||g||_{k}^{k}).$

###### Proof.

From Leibniz rule, it is clear that $||fg||_{k}\leq C||f||_{k}||g||_{k}$. For
the second statement, one can proceed by induction on $k$ using $||f\circ
g||_{k}\leq||f||_{0}+||g^{\prime}\,f^{\prime}\circ g||_{k-1}$ and the first
statement. ∎

######  Lemma 2.

For $T>0$ and integers $n,k\geq 1$, let $\tau_{k}=\inf\\{0\leq t\leq
T:\max_{i\in[n]}|\eta^{i}_{t}|\geq k\\}$. If $f$ is Lipschitz continuous, both
$u_{0}$ and $w_{0}$ are bounded, and
$u(t,x)\in\mathcal{C}([0,T],\mathcal{C}[0,1])$, then $\tau_{k}\to T$ almost
surely as $k\to\infty$.

###### Proof.

By Markov inequality, we have

$\displaystyle{\mathbb{P}}(\tau_{k}<t)$
$\displaystyle={\mathbb{P}}(\sup_{s\leq t}\max_{i\in[n]}|\eta^{i}_{s}|\geq k)$
$\displaystyle\leq\frac{1}{k}{\mathbb{E}}(\sup_{s\leq
t}\max_{i\in[n]}|\eta^{i}_{s}|)$
$\displaystyle\leq\frac{1}{k}{\mathbb{E}}(\sup_{s\leq
T}\max_{i\in[n]}|\eta^{i}_{s}|)$

For each $s\geq 0$ and $i\in[n]$, since
$\eta^{i}_{s}=\sqrt{n}(U^{i}_{s}-u(s,x_{i}))$ and by assumption
$M_{T}=\sup_{s\leq T}\|u(s,\cdot)\|_{0}<\infty$, it follows that

$\displaystyle|\eta^{i}_{s}|\leq\sqrt{n}(|U^{i}_{s}|+M_{T}).$

Now, one can check that for all $s\leq T$ and $i\in[n]$,

$|U^{i}_{s}|\leq\|u_{0}\|_{0}+\|w\|_{0}\frac{1}{n}\sum_{j=1}^{n}N^{j}_{T},$

so that [10, Proposition 3] ensures that

${\mathbb{E}}\left(\sup_{s\leq
T}\max_{i\in[n]}|U^{i}_{s}|\right)\leq\|u_{0}\|_{0}+\|w\|_{0}\frac{1}{n}\sum_{j=1}^{n}{\mathbb{E}}(N^{j}_{T})<\infty.$

Collecting the estimates above, we deduce that for some positive constant
$C=C(T,w,u,n)$, it holds ${\mathbb{E}}\left(\sup_{t\leq
T}\max_{i\in[n]}|\eta^{i}_{t}|\right)<C$, implying if $\tau=\lim_{k}\tau_{k}$,
then for all $t\leq T,$

${\mathbb{P}}(\tau<t)=\lim_{k\to\infty}{\mathbb{P}}(\tau_{k}<t)=0,$

and the result follows. ∎

##  Appendix B Fréchet spaces

Here are gathered some technical definitions and results about semi-normed
spaces and Fréchet spaces in particular. Most of what appears here is taken
from [35].

In the following, let $E$ denote a separated semi-normed space equipped with
the family of semi-norms $\\{N_{\nu},\nu\in\mathcal{N}_{E}\\}$ and $e$ denote
a generic element of $E$. The family is said to be _filtering_ if for all
finite subset $N\subset\mathcal{N}_{E}$, there exists $\mu$ in
$\mathcal{N}_{E}$ such that, for all $e\in E$,

$\sup_{\nu\in N}||u||_{\nu}\leq||u||_{\mu}.$

The dual space $E^{\prime}$ is the space of continuous linear forms $\xi$ on
$E$. If the family of norms is filtering then there is a simple
characterization of $E^{\prime}$:

$\xi\in
E^{\prime}\quad\Leftrightarrow\quad\exists\nu\in\mathcal{N}_{E},\,c>0,\
\sup_{e\in E}\frac{|\xi(e)|}{||e||_{\nu}}\leq c.$

######  Definition 4.

Any $B\subset E$ is called bounded if for every $\nu\in\mathcal{N}_{E}$,
$\sup_{e\in B}||e||_{\nu}<+\infty$.

In this paper, we endow $E^{\prime}$ with the _strong topology_ defined by the
family of semi-norms indexed by the bounded sets $B$ of $E$,

$||\xi||_{B}:=\sup_{e\in B}|\xi(e)|.$

Hence, $\xi_{n}\to\xi$ in $E^{\prime}$ is equivalent to
$||\xi_{n}-\xi||_{B}\to 0$ for every bounded set $B$. In particular,
$\xi_{n}\to\xi$ implies $\xi_{n}(e)-\xi(e)$ for all $e$ in $E$.

######  Definition 5.

A Fréchet space is any sequentially complete metrizable semi-normed space.

All through the paper, the Fréchet space of interest is $\mathcal{S}$ equipped
with its natural filtering family of semi-norms. Hence, for instance, the
space $\mathcal{C}([0,T],\mathcal{S}^{\prime})$ is understood as the space of
continuous functions $\gamma$ from $[0,T]$ to $\mathcal{S}^{\prime}$ equipped
with the strong topology. Its topology is given by the projective limit
topology of $\\{\xi\mapsto\sup_{0\leq t\leq T}||\xi_{t}||_{B},\,B\text{
bounded set of }E\\}$. In particular,
$\gamma\in\mathcal{C}([0,T],\mathcal{S}^{\prime})$ implies
$\gamma(\varphi)\in\mathcal{C}([0,T],{\mathbb{R}})$ for all $\varphi$ in
$\mathcal{S}$. The construction of $\mathcal{D}([0,T],\mathcal{S}^{\prime})$
follows the same idea where the sup norm over $t\in[0,T]$ is replaced by
Skorokhod metric.

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