# Chaos propagation in mean field networks of FitzHugh-Nagumo neurons

Laetitia Colombani, Pierre Le Bris

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Laetitia Colombani, Pierre Le Bris. Chaos propagation in mean field networks of FitzHugh-Nagumo neurons. 2022. hal-03706842

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Preprint submitted on 28 Jun 2022

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# Propagation of chaos in mean field networks of FitzHugh-Nagumo neurons

Laetitia Colombani, Pierre Le Bris

► To cite this version:

Laetitia Colombani, Pierre Le Bris. Propagation of chaos in mean field networks of FitzHugh-Nagumo neurons. Mathematical Neuroscience and Applications, 2023, 3, 10.46298/mna.9748. hal-03706842v2

HAL Id: hal-03706842

https://hal.science/hal-03706842v2

Submitted on 7 Mar 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

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# Chaos propagation in mean field networks of FitzHugh-Nagumo neurons

Laetitia Colombani*, Pierre Le Bris†

## Abstract

In this article, we are interested in the behavior of a fully connected network of $N$ neurons, where $N$ tends to infinity. We assume that neurons follow the stochastic FitzHugh-Nagumo model, whose specificity is the non-linearity with a cubic term. We prove a result of uniform in time propagation of chaos of this model in a mean-field framework. We also exhibit explicit bounds. We use a coupling method initially suggested by A. Eberle [Ebe16], and recently extended by [DEGZ20], known as the *reflection coupling*. We simultaneously construct a solution of the $N$ particles system and $N$ independent copies of the non-linear McKean-Vlasov limit such that, by considering an appropriate semimetrics that takes into account the various possible behaviors of the processes, the two solutions tend to get closer together as $N$ increases, uniformly in time. The reflection coupling allows us to deal with the non-convexity of the underlying potential in the dynamics of the quantities defining our network, and show independence at the limit for the system in mean field interaction with sufficiently small Lipschitz continuous interactions.

# 1 Introduction

## 1.1 Understanding the model

Understanding the brain activity is both a complex and important challenge in current research. Of course, interests are plentiful: characterizing brain functions, understanding structures and links between them and figuring out some phenomena - such as cyclic heartbeat. A way of modeling this activity is by considering a very large number of individual neurons with interactions. Since the number of neurons in a human brain is around $10^{11}$, and even "small" parts of the brain are thus constituted of very large number of them, such a strategy can be considered coherent.

The main quantity we study is the membrane potential of the nerve cells: it can "easily" be observed and its modification characterizes a synapse (an interaction between neurons). Neurons regulate their electrical potential. In general, without interaction, the potential evolves with time but has quite small changes. Incoming potentials from other neurons are usually what make the neuron fire, i.e. send potential to other neurons. We will here focus on an homogeneous network of neurons and consider mean-field interactions. This way, each neuron will interact with every other one, as it can be the case in small regions of the brain. The parameters of the model will be considered the same for each neuron.

A classical model was introduced by Hodgkin and Huxley [HH52] using experimental data of the activity of the giant squid axon. It describes the ion exchanges $K^+$, $Na^+$ and $Cl^-$ through the membrane and their effects on the potential. A simplification of this model is the FitzHugh-Nagumo model, which reduces the dimension: from four-dimensional model (for one neuron) with Hodgkin-Huxley equations, we obtain a two-dimensional model. It's a compromise between the biological accuracy and the mathematical simplicity.

<footnote>*Institut Mathématiques de Toulouse - Université de Toulouse. Email: laetitia.colombani[AT]math.univ-toulouse.fr</footnote>

<footnote>†Laboratoire Jacques-Louis Lions - Sorbonne Université. Email: pierre.lebris[AT]sorbonne-universite.fr</footnote>
# Propagation of chaos in mean field networks of FitzHugh-Nagumo neurons.

Laetitia Colombani*

Pierre Le Bris†

## Abstract

In this article, we are interested in the behavior of a fully connected network of $N$ neurons, where $N$ tends to infinity. We assume that the neurons follow the stochastic FitzHugh-Nagumo model, whose specificity is the non-linearity with a cubic term. We prove a result of uniform in time propagation of chaos of this model in a mean-field framework. We also exhibit explicit bounds. We use a coupling method initially suggested by Eberle [9], and recently extended in [8], known as the *reflection coupling*. We simultaneously construct a solution of the $N$-particle system and $N$ independent copies of the non-linear McKean-Vlasov limit in such a way that, considering an appropriate semi-metric that takes into account the various possible behaviors of the processes, the two solutions tend to get closer together as $N$ increases, uniformly in time. The reflection coupling allows us to deal with the non-convexity of the underlying potential in the dynamics of the quantities defining our network, and show independence at the limit for the system in mean field interaction with sufficiently small Lipschitz continuous interactions.

**Keywords:** FitzHugh-Nagumo; Propagation of Chaos; Mean field; Reflection Coupling.

**MSC2020 subject classifications:** 92C20; 60H10; 60F99.

Submitted to MNA on June 28, 2022, final version accepted on May 3, 2023.

Supersedes arXiv:2206.13291.

Supersedes HAL:hal-03706842.

# 1 Introduction

## 1.1 Understanding the model

Understanding brain activity is both a complex and important challenge in current research. Of course, interests are plentiful: characterizing brain functions, unveiling structures and links between them, and understanding some phenomena such as cyclic heartbeat. A way of modeling this activity is by considering a very large number of individual neurons with interactions. Since the number of neurons in a human brain is around $10^{11}$, and even “small” parts of the brain are thus constituted of a very large number of them, such a strategy can be considered coherent.

The main quantity we study is the membrane potential of the nerve cells: it can “easily” be observed and its modification characterizes a synapse (an interaction between

<footnote>*Institut Mathématiques de Toulouse - Université de Toulouse 3, France.
E-mail: laetitia.colombani.math@gmail.com</footnote>

<footnote>†Laboratoire Jacques-Louis Lions, Sorbonne Université, France.
E-mail: pierre.lebris@sorbonne-universite.fr</footnote>
The deterministic FitzHugh-Nagumo model for one neuron (or one particle) is given by the following equations:

$$ \begin{cases} dX_t = (X_t - (X_t)^3 - C_t - \alpha)dt \\ dC_t = (\gamma X_t - C_t + \beta)dt, \end{cases} $$

where $X$ is the potential membrane and $C$ is a recovery variable, called the adaptation variable. The parameters $\gamma$ and $\beta$ are positive constants that determine the duration of an excitation and the position of the equilibrium point of this system. Finally $\alpha \in \mathbb{R}$ is the magnitude of a stimulus current (an entrance current in the system). Note that the variable $C$ isn't a physical quantity, and is used to allow $X$ to mimic the behavior of the potential. This variable $C$ has linear dynamics and provides a slower negative feedback.

This deterministic model has been largely studied. In Chapter 7 of [Thi13], Thieullen describes the behavior of the solution of one deterministic FitzHugh-Nagumo system. She also extends the result in the case of a stochastic FitzHugh-Nagumo system: she considers a noise on the dynamics of $X$.

In fact, noise can be introduced in both equations to model different types of randomness: when the noise is in the first equation (dynamics of $X$) with a standard deviation $\sigma_X > 0$, it models a noisy presynaptic current. When it is in the second equation (dynamics of $C$) with a standard deviation $\sigma_C > 0$, it describes a noisy conductance dynamic (a noise in the chemical behavior). In general, noise in this model is additive. Various mathematical questions can be studied. Some authors choose to focus on the properties of the natural macroscopic limit of the model as $N \to \infty$ when it is clearly defined (see system (1.2)), when others work on properties of the particles system for fixed $N$. These models can be quite complicated to study mathematically. The main objectives are to characterize the behavior of these models when the number of neurons $N$ tends to $+\infty$ in a mean-field limit, and to prove whether or not there exists an equilibrium, a stationary behavior, when $t$ tends to $+\infty$. The question of the synchronization of neurons can also be studied, since it is a phenomenon observed in different contexts, such as the generation of respiratory rhythm or complex neurological functionalities.

In [TRW03], the authors work on the determination of firing time. They consider a stochastic FitzHugh-Nagumo model for one neuron, with Brownian noise on $X$, obtain approximation of firing times and compare them with numerical simulations.

Even if the majority of authors consider a noise only on one equation of the model, some study stochastic models with two noises. Berglund and Landon describe the behavior of the deterministic FitzHugh-Nagumo model for one neuron in [BL12], and consider the stochastic model, with noise on both equations, to work on the behavior of the interspike interval and the distribution of oscillations of the solution.

In [TBSST13], Tatchim Bemmo, Siewe Siewe and Tchawoua focus on a quite different stochastic model by considering additive noise $\eta$ on the dynamics of $X$, and multiplicative noise $\xi$ on the dynamics of $C$, both defined as sinusoidal function of correlated Brownian motions. They choose to avoid Gaussian noises since it is an unbounded noise. They also consider a deterministic and periodic entrance signal in the first equation. They observe abrupt transitions of the potential membrane $X$ when the intensity of the noise is gradually changed.

In general, a lot of authors focus on a noise on only one variable. In [LS18], León and Samson consider a FitzHugh-Nagumo model with a noise on $C$ but not on $X$, i.e. $\sigma_X = 0$, and study the properties of the equations for one neuron. In particular, they focus on hypoellipticity of the model, the existence and uniqueness of an invariant probability and a mixing property by establishing a link between the model and the class of stochastic damping Hamiltonian systems. They also consider neuronal modeling questions and study the generation of spikes in function of parameters of the model. On the contrary, the article [Uda19] focuses on stochastic FitzHugh-Nagumo model with noise in the dynamics of $X$, and $\sigma_C = 0$. They study one neuron in a periodically forced regime. This study relies on the theory of Markovian Random Dynamical System. The model is driven by a cosinus signal, and Uda studies the spike rate and compares it with the probability of two-points motion of membrane potential.

As said above, we consider mean-field interactions. These interactions are described by two functions $K_X$ and $K_C$, applied on the difference between two states $((X_t^i, C_t^i) - (X_t^j, C_t^j))$. In particular, this type of interaction
neurons). Neurons regulate their electrical potential. In general, without interaction, the potential evolves with time but has quite small changes. Incoming potentials from other neurons are usually what make the neuron fire, i.e. send action potentials to other neurons. We will here focus on a homogeneous network of neurons and consider mean-field interactions. This way, each neuron will interact with every other one, as it can be the case in small regions of the brain. The parameters of the model will be considered the same for each neuron.

A classical model was introduced by Hodgkin and Huxley [11] using experimental data on the activity of the giant squid axon. It describes the ion exchanges K⁺, Na⁺, and Cl⁻ through the membrane and their effects on the potential. A simplification of this model is the FitzHugh-Nagumo model, which reduces the dimension: from a four-dimensional model (for one neuron) with the Hodgkin-Huxley equations, we obtain a two-dimensional model, thus yielding a compromise between biological accuracy and mathematical simplicity.

The deterministic FitzHugh-Nagumo model for one neuron (or one particle) is given by the following equations

$$ \begin{cases} dX_t = (X_t - (X_t)^3 - C_t - \alpha)dt \\ dC_t = (\gamma X_t - C_t + \beta)dt, \end{cases} $$

where X is the membrane potential and C is a recovery variable, called the adaptation variable. The parameters γ and β are positive constants that determine the duration of excitation and the position of the equilibrium point of this system. Finally, α ∈ ℝ is the magnitude of a stimulus current (an entrance current in the system). Note that the variable C isn't a physical quantity, and is used to allow X to mimic the behavior of the potential. This variable C has linear dynamics and provides slower negative feedback.

This deterministic model has been largely studied. In Chapter 7 of [30], Thieullen describes the behavior of the solution of one deterministic FitzHugh-Nagumo system. She also extends the result in the case of a stochastic FitzHugh-Nagumo system: she considers a noise on the dynamics of X.

In fact, noise can be introduced in both equations to model different types of randomness: when the noise is on the first equation (dynamics of X) with a standard deviation σ_X > 0, it models a noisy presynaptic current. When it is on the second equation (dynamics of C) with a standard deviation σ_C > 0, it describes a noisy conductance dynamic (a noise in the chemical behavior). In general, noise in this model is additive. Various mathematical questions can be studied. Some authors choose to focus on the properties of the natural macroscopic limit of the model as N → ∞ when it is clearly defined (see system (1.2)) while others work on properties of the particle system for fixed N. These models can be quite complicated to study mathematically. The main objectives are to characterize the behavior of these models when the number of neurons N tends to +∞ in a mean-field limit, and to prove whether or not there exists an equilibrium, a stationary behavior, when t tends to +∞. The question of the synchronization of neurons can also be studied, since it is a phenomenon observed in different contexts, such as the generation of respiratory rhythm or complex neurological functionalities. It can be characterized as the dissipation of the empirical variance of the system of neurons. We refer the reader to [6] for further discussion on the synchronization in neuron models, and especially in the Hodgkin-Huxley model.

In [31], the authors work on the determination of firing times. They consider a stochastic FitzHugh-Nagumo model for one neuron, with Brownian noise on X, obtain an approximation of firing times, and compare them with numerical simulations.

In [29], Tatchim Bemmo, Siewe Siewe, and Tchawoua focus on a quite different stochastic model by considering additive noise η on the dynamics of X, and multiplica-
models electrical synapses, which are a sort of synapses.

In their article [BFFT12], Baladron, Fasoli, Faugeras and Touboul study FitzHugh-Nagumo and Hodgkin-Huxley models with mean-field interaction, only on $X$. They consider more general interactions, not only applied on the difference between two states, modeling chemical synapses and electrical synapses. For the FitzHugh-Nagumo model, they consider a noise on $X$, and prove propagation of chaos, i.e. the convergence of the law of $k$ neurons towards the law of $k$ independent solutions of the mean-field equations. This article is completed and clarified by the work of Bossy, Faugeras and Talay in [BFO19].

Mischler, Quininao and Touboul consider a FitzHugh-Nagumo model in [MQT16], with a linear interaction on $X$, and a noise only on $X$, i.e. $\sigma_C = 0$ and $K_X(z) = \lambda x$. The drift on $X$ is not exactly the same as in the model above, but remains similar as it is a cubic function of $X$. They work on the properties of a solution of the McKean-Vlasov evolution PDE associated to this model and obtain the uniqueness of a global weak solution. Furthermore, they prove that there exists at least one stationary solution, and when the interaction is small, the stationary solution is unique and exponentially stable. They also exhibit numerical results with open problems, like attractive periodic solution in time. In a similar framework, Luçon and Poquet study the macroscopic limit of this mean-field model in [LP21], and in particular the periodicity of such a system. They analyze the influence of both noise and interaction on the emergence of periodic behavior, and prove the existence of periodic solution, exponentially attractive, when the parameters satisfy some assumptions and the drift is "small" enough with respect to interaction and noise. Their approach relies on a slow-fast analysis and Floquet theory.

This model can be complexified, by considering non mean-field interaction. In particular, Bayrak, Hövel and Vuksanović work on a stochastic FitzHugh-Nagumo model with a network interaction in [BHV21]. Their type of interaction take into account a connectivity coefficient between two neurons, and a propagation velocity.

Other authors choose to complexify the model by considering stochastic FitzHugh-Nagumo with a spatial model. A second spatial derivative of $X$ is added in the dynamics of $X$. Various authors study the behavior of such a model, and explore the notion of random attractors [LW10], [Li21], [LL19] and [LX21].

Various authors also study numerical schemes for the interacting particles system in the stochastic model. In [RS22], the authors adapt Euler-Maruyama scheme to approximate solution of the particles system.

## 1.2 Framework and results

Combining noise and interaction, we work specifically on the following equations, for $1 \le i \le N$, where $N$ is the number of neurons:

$$ \begin{cases} dX_t^{i,N} = (X_t^{i,N} - (X_t^{i,N})^3 - C_t^{i,N} - \alpha)dt + \frac{1}{N} \sum_{j=1}^N K_X(Z_t^{i,N} - Z_t^{j,N}) + \sigma_X dB_t^{i,X} \\ dC_t^{i,N} = (\gamma X_t^{i,N} - C_t^{i,N} + \beta)dt + \frac{1}{N} \sum_{j=1}^N K_C(Z_t^{i,N} - Z_t^{j,N}) + \sigma_C dB_t^{i,C}, \end{cases} \quad (1.1) $$

where we denote by $Z_t^i$ the couple $(X_t^i, C_t^i)$ to simplify the notations.

We assume $(B_t^{i,X})_i$ and $(B_t^{i,C})_i$ to be independent Brownian motions. Here, we consider two Brownian noises $B^X$ and $B^C$, one on each equation, and thus assume that each neuron has its own independent noises, and that there is no environmental (or shared) noise.

We also assume $K_X$ and $K_C$ to be Lipschitz continuous and respectively denote their Lipschitz constants by $L_X$ and $L_C$.

The goal of this article is to describe the behavior of this network as the number $N$ of neurons tends to infinity.

To describe its behavior, we consider the $\mathbb{R}^2$-valued process $(\bar{Z}_t)_{t \ge 0} = (\bar{X}_t, \bar{C}_t)_{t \ge 0}$ evolving according to the following non-linear stochastic differential equation of McKean-Vlasov type

$$ \begin{cases} d\bar{X}_t = (\bar{X}_t - (\bar{X}_t)^3 - \bar{C}_t - \alpha)dt + K_X * \bar{\mu}_t(\bar{Z}_t)dt + \sigma_x d\bar{B}_t^X \\ d\bar{C}_t = (\gamma \bar{X}_t - \bar{C}_t + \beta)dt + K_C * \bar{\mu}_t(\bar{Z}_t)dt + \sigma_c d\bar{B}_t^C, \end{cases} \quad (1.2) $$
tive noise $\xi$ on the dynamics of $C$, both defined as sinusoidal functions of correlated Brownian motions. They choose to avoid Gaussian noises since they are unbounded. They also consider a deterministic and periodic entrance signal in the first equation, and observe abrupt transitions of the membrane potential $X$ when the intensity of the noise is gradually changed.

In general, a lot of authors focus on noise on only one variable. In [15], León and Samson consider a FitzHugh-Nagumo model with noise on $C$ but not on $X$, i.e. $\sigma_X = 0$, and study the properties of the equations for one neuron. In particular, they focus on the hypoellipticity of the model, the existence and uniqueness of an invariant probability, and a mixing property by establishing a link between the model and the class of stochastic damping Hamiltonian systems. They also consider neuronal modeling questions and study the generation of spikes according to the parameters of the model. On the contrary, the article [32] focuses on the stochastic FitzHugh-Nagumo model with noise in the dynamics of $X$, and $\sigma_C = 0$. They study one neuron in a periodically forced regime. This study relies on the theory of Markovian Random Dynamical Systems. The model is driven by a cosine signal, and Uda studies the spike rate and compares it with the probability of a two-point motion of membrane potential.

However, some do study stochastic models with two noises. Berglund and Landon describe the behavior of the deterministic FitzHugh-Nagumo model for one neuron in [3], and consider the stochastic model, with noise on both equations, to work on the behavior of the interspike interval and the distribution of oscillations of the solution.

As said above, we consider mean-field interactions. These interactions are described by two functions $K_X$ and $K_C$, applied on the difference between two states ($(X_t^i, C_t^i) - (X_t^j, C_t^j)$). In particular, this type of interaction models electrical synapses.

In their article [1], Baladron, Fasoli, Faugeras, and Touboul study FitzHugh-Nagumo and Hodgkin-Huxley models with mean-field interaction, only on $X$. They consider more general interactions, not only applied to the difference between two states, modeling chemical synapses and electrical synapses. For the FitzHugh-Nagumo model, they consider a noise on $X$ and prove propagation of chaos, i.e. the convergence of the law of $k$ neurons towards the law of $k$ independent solutions of the mean-field equations. This article is completed and clarified by the work of Bossy, Faugeras, and Talay in [5]. Mishler, Quininao, and Touboul consider a FitzHugh-Nagumo model in [24], with a linear interaction on $X$, and a noise only on $X$, i.e. $\sigma_C = 0$ and $K_X(z) = \lambda x$. The drift on $X$ is not exactly the same as in the model above but remains similar as it is a cubic function of $X$. They work on the properties of a solution of the McKean-Vlasov PDE associated to this model and obtain the uniqueness of a global weak solution. Furthermore, they prove that there exists at least one stationary solution, and when the interaction is small, the stationary solution is unique and exponentially stable. They also exhibit numerical results with open problems, like attractive periodic solutions in time. In a similar framework, Luçon and Poquet study the macroscopic limit of this mean-field model in [20], and in particular the periodicity of such a system. They analyze the influence of both noise and interaction on the emergence of periodic behavior and prove the existence of a periodic solution, exponentially attractive, when the parameters satisfy some assumptions and the drift is “small” enough with respect to interaction and noise. Their approach relies on a slow-fast analysis and Floquet theory. Results of non-uniform propagation of chaos has also been obtained in [23] by Mehri et al., for stochastic spatially structured neuron networks, by applying the Euler approximation to the construction of a solution.

This model can be complexified, by considering non-mean-field interaction. In particular, Bayrak, Hövel, and Vuksanović work on a stochastic FitzHugh-Nagumo model with a network interaction in [2]. Their type of interaction takes into account a connectivity
where $\bar{\mu}_t = \text{Law}(\bar{Z}_t)$ is the law at time $t$ of the process $(\bar{X}_t, \bar{C}_t)$, and $*$ denotes the operation of convolution, i.e.

$$K_X * \bar{\mu}_t(u) = \int K_X(u - v)\mu_t(dv).$$

To some extent, (1.1) can be seen as an approximation of (1.2) in which the operation of convolution is applied to the empirical measure $\mu_{t,\text{emp}} = \frac{1}{N} \sum_{i=1}^N \delta_{Z_t^i}$, and what We wish to prove is that, indeed, the law $\mu_t^N$ of the network (1.1) converges in some sense to $\bar{\mu}_t^{\otimes N}$ (i.e the law of the solution of (1.2) tensorized $N$ times) as $N$ tends to infinity. This phenomenon has been stated under the name propagation of chaos -an idea motivated by M. Kac [Kac56]- as it amounts to saying that, as the number of particle increases in the system, two particles will become "more and more" independent, converging towards a tensorized law. The notion of "propagation" refers to the fact that proving such convergence at time 0 is sufficient to prove it at a later time $t$.

In order to prove the convergence of $\mu_t^N$ to $\bar{\mu}_t^{\otimes N}$, we follow the coupling method described in a recent work by one of the authors in [GLM21], the result of which cannot be applied directly here. This method has been put forward by Eberle, following earlier works by Lindvall and Rogers [LR86]. Before recalling the method, let us also mention the recent work [Sch22], which uses a coupling approach adapted to a well-chosen distance.

We consider $r_t^i = |\bar{X}_t^i - X_t^{i,N}| + \delta|\bar{C}_t^i - C_t^{i,N}|$ with $\delta > 0$, a constant not yet specified.

A natural distance between probability measures is the Wasserstein distance, linked to the theory of optimal transport. For $\mu$ and $\nu$ two probability measures on $\mathbb{R}^d$, we denote

$$W_p(\mu, \nu) = \inf_{X \sim \mu, Y \sim \nu} \mathbb{E} \left( \left\| X - Y \right\|_p^p \right)^{1/p}, \quad (1.3)$$

where $\| \cdot \|_p$ denotes the usual $L^p$ distance on $\mathbb{R}^d$. It is thus defined as the minimum over all possible choices of a pair $(X, Y)$, such that $X$ is distributed according to $\mu$ and $Y$ according to $\nu$, of the expectation of the distance between $X$ and $Y$. The basic idea behind a coupling method is then that an upper bound on the Wasserstein distance between $\mu$ and $\nu$ is given by the construction of any pair of random variables distributed according to these probability measures. Thus, instead of considering the minimum over all possible coupling, we construct simultaneously two solutions of (1.1) and (1.2) that will tend to get closer together as the number of neurons increases.

Have $(\bar{X}_t^i, \bar{C}_t^i)$, for $i$ between 1 and $N$, be $N$ independent copies of a solution of (1.2) driven by some independent Brownian motions $(\bar{B}_t^{i,X})_{t \ge 0}$ and $(\bar{B}_t^{i,C})_{t \ge 0}$. A coupling of $(\bar{X}_t^i, \bar{C}_t^i)$ and $(X_t^{i,N}, C_t^{i,N})$ then follows from a coupling of the Brownian motions $B$ and $\bar{B}$.

The first natural choice, popularized by Sznitman [Szn91], is the synchronous coupling and consists in choosing $B = \bar{B}$. By doing so, when considering the time evolution of $\bar{Z}_t^i - Z_t^{i,N} = (\bar{X}_t^i - X_t^{i,N}, \bar{C}_t^i - C_t^{i,N})$, the noise cancels out. The contraction of a distance between the processes can then only be induced by the deterministic drift, as in [BGM10], and this usually only holds under rather restrictive conditions (in particular the drift should be strongly convex). Nevertheless, in our case, the calculation of the evolution of $\bar{X}_t^i - X_t^{i,N}$ and $\bar{C}_t^i - C_t^{i,N}$ (see later) shows that there is still some deterministic contraction when $\bar{X}_t^i - X_t^{i,N} = 0$. We can therefore use a synchronous coupling in the vicinity of this subspace.

Outside of this subspace, we use the noise to get the processes closer together. In the direction orthogonal to the contracting space we consider $B = -\bar{B}$, as this maximizes the variance of the noise. This yields the reflection coupling. Notice however at this stage that, because of the symmetry of the noise, there is a priori no reason $r_t^i$ should decrease rather than increase. This invites us to consider $f(r_t^i)$, with $f$ a concave function, so that a random decrease has more effect than a random increase of the same value. We will define the function $f$ later.

Finally we construct a Lyapunov function $H$ to take into account the trend of each process to come back to some compact set of $\mathbb{R}^2$. We are then led to the study of a suitable distance between the two processes, which will be of the form $\rho_t := \frac{1}{N} \sum_{i=1}^N f(r_t^i)(1 + \epsilon H(\bar{Z}_t^i) + \epsilon H(Z_t^{i,N}))$, where $\epsilon > 0$. This quantity controls the usual $L^1$ and $L^2$ distances between the two systems, and is interesting as, when $r_t^i$ is small, $f(r_t^i)$ tends to decrease either because of the deterministic drift or the reflection coupling, and when $r_t^i$ is big, the Lyapunov functions $H$ will
coefficient between two neurons and a propagation velocity.

Other authors choose to complexify the model by considering stochastic FitzHugh-Nagumo with a spatial model. A second spatial derivative of X is added to the dynamics of X. Various authors study the behavior of such a model and explore the notion of random attractors [22, 16, 18, 17].

Various authors also study numerical schemes for the interacting particle system in the stochastic model. In [25], the authors adapt the Euler-Maruyama scheme to approximate the solution of the particle system.

## 1.2 Framework and results

Combining noise and interaction, we work specifically on the following equations, for $1 \le i \le N$, where N is the number of neurons

$$ \left\{ \begin{aligned} dX_t^{i,N} &= (X_t^{i,N} - (X_t^{i,N})^3 - C_t^{i,N} - \alpha)dt + \frac{1}{N} \sum_{j=1}^N K_X(Z_t^{i,N} - Z_t^{j,N})dt + \sigma_X dB_t^{i,N} \\ dC_t^{i,N} &= (\gamma X_t^{i,N} - C_t^{i,N} + \beta)dt + \frac{1}{N} \sum_{j=1}^N K_C(Z_t^{i,N} - Z_t^{j,N})dt + \sigma_C dB_t^{i,N}, \end{aligned} \right. \quad (1.1) $$

where we denote by $Z_t^{i,N}$ the couple $(X_t^{i,N}, C_t^{i,N})$ to simplify the notation.

We assume $(B_t^{i,N})_i$ and $(B_t^{i,C})_i$ to be independent Brownian motions. Here, we consider two Brownian noises $B^X$ and $B^C$, one on each equation, and thus assume that each neuron has its own independent noise and that there is no environmental (or shared) noise.

We also assume $K_X$ and $K_C$ to be Lipschitz continuous and respectively denote their Lipschitz constants by $L_X$ and $L_C$.

The goal of this article is to describe the behavior of this network as the number N of neurons tends to infinity.

To describe its behavior, we consider the $\mathbb{R}^2$-valued process $(\bar{Z}_t)_{t \ge 0} = (\bar{X}_t, \bar{C}_t)_{t \ge 0}$ evolving according to the following non-linear stochastic differential equation of McKean-Vlasov type

$$ \left\{ \begin{aligned} d\bar{X}_t &= (\bar{X}_t - (\bar{X}_t)^3 - \bar{C}_t - \alpha)dt + K_X * \bar{\mu}_t(\bar{Z}_t)dt + \sigma_X d\bar{B}_t^X \\ d\bar{C}_t &= (\gamma \bar{X}_t - \bar{C}_t + \beta)dt + K_C * \bar{\mu}_t(\bar{Z}_t)dt + \sigma_C d\bar{B}_t^C, \end{aligned} \right. \quad (1.2) $$

where $\bar{\mu}_t = \text{Law}(\bar{Z}_t)$ is the law at time t of the process $(\bar{X}_t, \bar{C}_t)$, and * denotes the operation of convolution, i.e.

$$ K_X * \bar{\mu}_t(u) = \int K_X(u - v) \bar{\mu}_t(dv). $$

To some extent, (1.1) can be seen as an approximation of (1.2) in which the operation of convolution is applied to the empirical measure $\mu_{t,\text{emp}} = \frac{1}{N} \sum_{i=1}^N \delta_{Z_t^{i,N}}$, and what we wish to prove is that, indeed, the law $\mu_t^N$ of the network (1.1) converges in some sense to $\bar{\mu}_t^N$ (i.e the law of the solution of (1.2) tensorized N times) as N tends to infinity. This phenomenon has been stated under the name propagation of chaos -an idea motivated by M. Kac [13]- as it amounts to saying that, as the number of particles increases in the system, two particles will become "more and more" independent, their joint law converging towards a tensorized law. The notion of "propagation" refers to the fact that proving such convergence at time 0 is sufficient to prove it at a later time t.

In order to prove the convergence of $\mu_t^N$ to $\bar{\mu}_t^{N}$, we follow the coupling method described in recent work by one of the authors in [10], the result of which cannot be applied directly here. This method has been put forward by Eberle, following earlier works by Lindvall and Rogers [19]. Before recalling the method, let us also mention the recent work [27], which uses a coupling approach adapted to a well-chosen distance.
tend to decrease. We thus show that $E\rho_t$ decays exponentially fast. This leads to several constraints on $\delta, \epsilon$ and on the parameters involved in the definition of $f$, and we have to prove that it is possible to meet all these conditions simultaneously. In reality, the quantity $\rho_t$ considered will be a slight twist of the one given above (see (2.23)) so as to take into account the non linearity of the process.

As explained, this method requires some noise in the direction orthogonal to the naturally contracting subspace. This means, in the description of the method above, that one should have $\sigma_X > 0$ (so that we can use a reflection coupling to bring $\bar{X}_t^i$ and $X_t^{i,N}$ closer together). In the case $\sigma_X = 0$ and $\sigma_C > 0$, a modification of the calculations is necessary. We describe this case and the resulting modifications of the computations in Appendix B.

**Assumption 1.** Let $L_{X,max}$ and $L_{C,max}$ be two universal constants

$$ \begin{aligned} & \exists L_X \in [0, L_{X,max}], \forall z, z' \in \mathbb{R}^2 \quad |K_X(z) - K_X(z')| \le L_X(\|z - z'\|_1) \\ & \exists L_C \in [0, L_{C,max}], \forall z, z' \in \mathbb{R}^2 \quad |K_C(z) - K_C(z')| \le L_C(\|z - z'\|_1) \\ & K_X(0, 0) = 0 \text{ and } K_C(0, 0) = 0 \end{aligned} $$

These constants $L_{X,max}$ and $L_{C,max}$ might seem at first glance off putting as they are not given. When we prove non uniform in time propagation of chaos, these constants can be chosen to be $+\infty$. When we prove uniform in time propagation of chaos however, $L_{X,max}$ and $L_{C,max}$ are a priori bounds : Theorem 2 below will be true for $L_X$ and $L_C$ sufficiently small. The condition $L_X \le L_{X,max}$ and $L_C \le L_{C,max}$ are therefore not restrictive conditions, and are useful in proving some parameters are independent of $L_X$ and $L_C$. Lemma 2.2 below shows that one can for instance consider $L_{X,max} = 4$ and $L_{C,max} = \frac{1}{5}$.

Before any result on propagation of chaos, we prove that both systems (1.1) and (1.2) have well-defined solutions:

**Proposition 1.1.** Let $K_X$ and $K_C$ satisfy Assumptions 1, with $L_{X,max} = \infty$ and $L_{C,max} = \infty$. There exists a unique strong solution for system (1.1) and a unique strong solution for system (1.2).

We denote $W_1$ and $W_2$ the usual $L^1$ and $L^2$ Wasserstein distances defined in (1.3).

**Theorem 1.** [Non uniform in time propagation of chaos] Let $K_X$ and $K_C$ satisfy Assumptions 1, with $L_{X,max} = \infty$ and $L_{C,max} = \infty$. There exist explicit $C_1, C_2 > 0$, such that for all probability measures $\mu_0$ on $\mathbb{R}^2$ with finite second moment,

$$ W_1(\mu_t^{k,N}, \bar{\mu}_t^{\otimes k}) \le C_1 e^{C_2 t} \frac{k}{\sqrt{N}}, $$

for all $k \in \mathbb{N}$, where $\mu_t^{k,N}$ is the marginal distribution at time $t$ of the first $k$ neurons $((X_t^1, C_t^1), \dots, (X_t^k, C_t^k))$ of an $N$ particles system (1.1) with initial distribution $(\mu_0)^{\otimes N}$, while $\bar{\mu}_t$ is a solution of (1.2) with initial distribution $\mu_0$.

This first theorem is in accordance with the theorem from [KNRS20], and explicits the dependency in $t$. Since its proof is rather quick, and provides a good entry point into coupling methods, we give it in Subsection 1.4.

Our main result consist in removing the time dependency in the previous upperbound. This uniform in time propagation of chaos however requires stronger assumptions on the interaction kernels.

**Theorem 2.** [Uniform in time propagation of chaos] Let $C_{init,exp} > 0$ and $\tilde{a} > 0$. There is an explicit $c^K > 0$ such that, for all $K_X$ and $K_C$ satisfying Assumptions 1 with $L_X, L_C < c^K$, there exist explicit $B_1, B_2 > 0$, such that for all probability measures $\mu_0$ on $\mathbb{R}^2$ satisfying $E_{\mu_0}(e^{\tilde{a}(|X|+|C|)}) \le C_{init,exp}$,

$$ W_1(\mu_t^{k,N}, \bar{\mu}_t^{\otimes k}) \le B_1 \frac{k}{\sqrt{N}}, \quad W_2(\mu_t^{k,N}, \bar{\mu}_t^{\otimes k}) \le B_2 \frac{k}{\sqrt{N}}, $$

for all $k \in \mathbb{N}$, where $\mu_t^{k,N}$ is the marginal distribution at time $t$ of the first $k$ neurons $((X_t^1, C_t^1), \dots, (X_t^k, C_t^k))$ of an $N$ particles system (1.1) with initial distribution $(\mu_0)^{\otimes N}$, while $\bar{\mu}_t$ is a solution of (1.2) with initial distribution $\mu_0$.
We consider $r_t^i = |\bar{X}_t^i - X_t^{i,N}| + \delta|\bar{C}_t^i - C_t^{i,N}|$ with $\delta > 0$, a constant not yet specified (to prove the first result Theorem 1.3, we will consider $\delta = 1$, but we will need a more specific one for Theorem 1.4).

A natural distance between probability measures is the Wasserstein distance, linked to the theory of optimal transport. For $\mu$ and $\nu$ two probability measures on $\mathbb{R}^d$, we denote

$$W_p(\mu, \nu) = \inf_{X \sim \mu, Y \sim \nu} \mathbb{E} (\|X - Y\|_p^p)^{1/p}, \quad (1.3)$$

where $\| \cdot \|_p$ denotes the usual $L^p$ distance on $\mathbb{R}^d$. It is thus defined as the minimum over all possible choices of a pair $(X, Y)$, such that $X$ is distributed according to $\mu$ and $Y$ according to $\nu$, of the expectation of the distance between $X$ and $Y$. The basic idea behind a coupling method is then that an upper bound on the Wasserstein distance between $\mu$ and $\nu$ is given by the construction of any pair of random variables distributed according to these probability measures. Thus, instead of considering the minimum over all possible couplings, we construct simultaneously two solutions of (1.1) and (1.2) that will tend to get closer together as the number of neurons increases.

Let $(\bar{X}_t^i, \bar{C}_t^i)$, for $i$ between 1 and $N$, be $N$ independent copies of a solution of (1.2) driven by some independent Brownian motions $(\bar{B}_t^{i,X})_{t \ge 0}$ and $(\bar{B}_t^{i,C})_{t \ge 0}$. A coupling of $(\bar{X}_t^i, \bar{C}_t^i)$ and $(X_t^{i,N}, C_t^{i,N})$ then follows from a coupling of the Brownian motions $B$ and $\bar{B}$.

The first natural choice, popularized by Sznitman [28], is the synchronous coupling and consists in choosing $B = \bar{B}$. By doing so, when considering the time evolution of $\bar{Z}_t^i - Z_t^{i,N} = (\bar{X}_t^i - X_t^{i,N}, \bar{C}_t^i - C_t^{i,N})$, the noise cancels out. The contraction of a distance between the processes can then only be induced by the deterministic drift, as in [4], and this usually only holds under rather restrictive conditions (in particular the drift should be strongly convex). Nevertheless, in our case, the calculation of the evolution of $\bar{X}_t^i - X_t^{i,N}$ and $\bar{C}_t^i - C_t^{i,N}$ (see later) shows that there is still some deterministic contraction when $\bar{X}_t^i - X_t^{i,N} = 0$. We can therefore use a synchronous coupling in the vicinity of this subspace.

Outside of this subspace, we use the noise to get the processes closer together. In the direction orthogonal to the contracting space we consider $B = -\bar{B}$, as this maximizes the variance of the noise. This yields the reflection coupling. Notice however at this stage that, because of the symmetry of the noise, there is a priori no reason why $r_t^i$ should decrease rather than increase. This invites us to consider $f(r_t^i)$, with $f$ a concave function, so that a random decrease has more effect than a random increase of the same value. We will define the function $f$ later.

Finally, we construct a Lyapunov function $H$ to take into account the trend of each process to come back to some compact set of $\mathbb{R}^2$. We are then led to the study of a suitable distance between the two processes, which will be of the form $\rho_t := \frac{1}{N} \sum_{i=1}^N f(r_t^i)(1+\epsilon H(\bar{Z}_t^i) + \epsilon H(Z_t^{i,N}))$, where $\epsilon > 0$. This quantity controls the usual $L^1$ and $L^2$ distances between the two systems and is interesting as, when $r_t^i$ is small, $f(r_t^i)$ tends to decrease either because of the deterministic drift or the reflection coupling, and when $r_t^i$ is big, the Lyapunov functions $H$ will tend to decrease. We thus show that $E\rho_t$ decays exponentially fast. This leads to several constraints on $\delta, \epsilon$ and on the parameters involved in the definition of $f$, and we have to prove that it is possible to meet all these conditions simultaneously. In reality, the quantity $\rho_t$ considered will be a slight twist of the one given above (see (2.25)) so as to take into account the non-linearity of the process.

As explained, this method requires some noise in the direction orthogonal to the naturally contracting subspace. This means, in the description of the method above, that one should have $\sigma_X > 0$ (so that we can use a reflection coupling to bring $\bar{X}_t^i$ and
As previously mentioned, this work follows the method described in [GLM21]. Beyond the result of uniform in time propagation of chaos for the FitzHugh-Nagumo model, which is in itself an interesting result, the present work is also a testament to the robustness of the coupling method.

The reader will find an index containing all notations, constants and parameters for reference at the end of the document.

## 1.3 Existence of solutions

First of all, we prove Proposition 1.1, i.e existence of strong solutions of systems (1.1) and (1.2), under Assumption 1, and with begin with (1.1).

Let's denote, for $K \in \mathbb{R}^+$,

$$g_K(x) = \begin{cases} -K^3 & \text{if } x < -K \\ x^3 & \text{if } x \in [-K, K] \\ K^3 & \text{if } x > K. \end{cases}$$

$g_K$ is locally Lipschitz and is bounded.

Thus, it's well known (see Chapter 3 [IW89]) that the following system (under Assumption 1)

$$\left\{ \begin{aligned} dX_t^{i,N} &= (X_t^{i,N} - g_K(X_t^{i,N}) - C_t^{i,N} - \alpha)dt + \frac{1}{N} \sum_{j=1}^N K_X(Z_t^{i,N} - Z_t^{j,N}) + \sigma_X dB_t^{i,N} \\ dC_t^{i,N} &= (\gamma X_t^{i,N} - C_t^{i,N} + \beta)dt + \frac{1}{N} \sum_{j=1}^N K_C(Z_t^{i,N} - Z_t^{j,N}) + \sigma_C dB_t^{i,C}, \end{aligned} \right. \quad (1.4)$$

for $1 \le i \le N$, have strong and unique solution.

In consequence, for a fixed $K \in \mathbb{R}^+$, there exists strong solution of system (1.1) until time $T_K = \sup\{t, \forall i, \forall s \le t, X_t^i \le K \text{ and } C_t^i \le K\}$, and the solution coincide with the solution of the system with $g_K$.

We have the following Proposition:

**Proposition 1.2.** If, for each $i \le N$, $\mathbb{E}(|X_0^{i,N}|^2) < +\infty$ and $\mathbb{E}(|C_0^{i,N}|^2) < +\infty$, then for all $t \ge 0$ there exists $C_t < \infty$ such that, for each $i \le N$:

$$\mathbb{E} \left( |X_t^{i,N}|^2 + |C_t^{i,N}|^2 \right) \le C_t. \quad (1.5)$$

The proof relies on the Lyapunov function defined in the next Section, and is given in Appendix A.2.

Then, by denoting $T_\infty$ the explosion time of a solution of system (1.1)

$$T_\infty = \inf\{t, \exists i, \forall A > 0, \exists \varepsilon > 0, \forall s \in (t - \varepsilon, t), X_s^{i,N} > A \text{ or } C_s^{i,N} > A\}$$

we deduce $\forall t \in \mathbb{R}^+, \mathbb{P}(T_\infty \le t) = 0$ and $\mathbb{P}(\bar{T}_\infty \le t) = 0$. Eventually, there exists unique and strong solution for system (1.1).

The existence and uniqueness of a solution of (1.2) is known from the Theorem 3.3 from [RST19]. We only have to prove that the Assumptions 3.2 [RST19] are verified. We define, for all $t \in \mathbb{R}^+$, $z = (x, c) \in \mathbb{R}^2$ and for all probability distribution $\nu$ with a finite variance:

$$b(t, z, \nu) = \begin{pmatrix} x - x^3 - c - \alpha + K_X * \nu(z) \\ \gamma x - c + \beta + K_C * \nu(z) \end{pmatrix} \quad \text{and} \quad \sigma(t, z, \nu) = \begin{pmatrix} \sigma_X \\ \sigma_C \end{pmatrix}.$$

$\sigma$ is a constant function, so it clearly satisfies the various conditions.

For $t \in \mathbb{R}^+$, $z, z'$ in $\mathbb{R}^2$, and $\nu$ a probability measure:

$$\langle z - z', b(t, z, \nu) - b(t, z', \nu) \rangle = (x - x') ((x - x') - (x^3 - x'^3) - (c - c') + K_X * \nu(z) - K_X * \nu(z'))$$
$X_t^{i,N}$ closer together). In the case $\sigma_X = 0$ and $\sigma_C > 0$, a modification of the calculations is necessary. We describe this case and the resulting modifications in the computations in Appendix B.

**Assumption 1.1.** $K_X$ and $K_C$ are Lipschitz continuous, i.e.

$$ \begin{aligned} \exists L_X \ge 0, \forall z, z' \in \mathbb{R}^2 & \quad |K_X(z) - K_X(z')| \le L_X(\|z - z'\|_1) \\ \exists L_C \ge 0, \forall z, z' \in \mathbb{R}^2 & \quad |K_C(z) - K_C(z')| \le L_C(\|z - z'\|_1) \\ K_X(0, 0) = 0 \quad \text{and} \quad K_C(0, 0) = 0. \end{aligned} $$

Before any result on the propagation of chaos, we prove that both systems (1.1) and (1.2) have well-defined solutions.

**Proposition 1.2 (Existence of solutions).** Let $K_X$ and $K_C$ satisfy Assumption 1.1. We assume the law of $((X_0^{1,N}, C_0^{1,N}), \dots, (X_0^{N,N}, C_0^{N,N}))$ and the law of $(\bar{X}_0, \bar{C}_0)$ have a moment of order 2. Then, there exists a unique strong solution for system (1.1) and a unique strong solution for system (1.2).

We denote $W_1$ and $W_2$ the usual $L^1$ and $L^2$ Wasserstein distances defined in (1.3).

**Theorem 1.3 (Non uniform in time propagation of chaos).** Let $K_X$ and $K_C$ satisfy Assumption 1.1. There exist explicit $C_1, C_2 > 0$, such that for all probability measures $\mu_0$ on $\mathbb{R}^2$ with finite second moment,

$$ W_1(\mu_t^{k,N}, \bar{\mu}_t^{k}) \le C_1 e^{C_2 t} \frac{k}{\sqrt{N}}, $$

for all $k \in \mathbb{N}$, where $\mu_t^{k,N}$ is the marginal distribution at time $t$ of the first $k$ neurons $((X_t^{1,N}, C_t^{1,N}), \dots, (X_t^{k,N}, C_t^{k,N}))$ of an $N$-particle system (1.1) with initial distribution $\mu_0^{\otimes N}$, while $\bar{\mu}_t$ is a solution of (1.2) with initial distribution $\mu_0$.

This first theorem is in accordance with the theorem from [14] and makes the dependence in $t$ explicit. Since its proof is rather quick and provides a good entry point into coupling methods, we give it in Subsection 1.4.

Our main result consists in removing the time dependency in the previous upper bound. This uniform in time propagation of chaos however requires stronger assumptions on the interaction kernels.

**Theorem 1.4 (Uniform in time propagation of chaos).** Let $L_{X,\max}$ and $L_{C,\max}$ be two (explicit) universal constants such that $L_X \le L_{X,\max}$ and $L_C \le L_{C,\max}$. Let $C_{init,exp} > 0$ and $\tilde{a} > 0$. There is an explicit $c^K > 0$ such that, for all $K_X$ and $K_C$ satisfying Assumptions 1.1 with $L_X, L_C < c^K$, there exist explicit $B_1, B_2 > 0$, such that for all probability measures $\mu_0$ on $\mathbb{R}^2$ satisfying $E_{\mu_0}(e^{\tilde{a}(|X|+|C|)}) \le C_{init,exp}$,

$$ W_1(\mu_t^{k,N}, \bar{\mu}_t^{k}) \le B_1 \frac{k}{\sqrt{N}}, \quad W_2(\mu_t^{k,N}, \bar{\mu}_t^{k}) \le B_2 \frac{k}{\sqrt{N}}, $$

for all $k \in \mathbb{N}$, where $\mu_t^{k,N}$ is the marginal distribution at time $t$ of the first $k$ neurons $((X_t^{1,N}, C_t^{1,N}), \dots, (X_t^{k,N}, C_t^{k,N}))$ of an $N$-particle system (1.1) with initial distribution $\mu_0^{\otimes N}$, while $\bar{\mu}_t$ is a solution of (1.2) with initial distribution $\mu_0$.

When we prove uniform in time propagation of chaos, $L_{X,\max}$ and $L_{C,\max}$ are a priori bounds. Theorem 1.4 above will be true for $L_X$ and $L_C$ sufficiently small: the condition $L_X \le L_{X,\max}$ and $L_C \le L_{C,\max}$ are therefore not restrictive conditions and are useful in proving some parameters are independent of $L_X$ and $L_C$. Lemma 2.2 below shows that one can for instance consider $L_{X,\max} = 4$ and $L_{C,\max} = \frac{1}{5}$. Furthermore $c^K$, that controls both interactions $K_X$ and $K_C$, is explained in Subsection 2.4.
$$ \begin{aligned} & + (c - c') (\gamma(x - x') - (c - c') + K_C * \nu(z) - K_C * \nu(z')) \\ &= (x - x')^2 - (x - x')^2 (x^2 + xx' + x'^2) + (\gamma - 1)(c - c')(x - x') - (c - c')^2 \\ & + (K_X * \nu(z) - K_X * \nu(z')) + (K_C * \nu(z) - K_C * \nu(z')). \end{aligned} $$

Since $x^2 + xx' + x'^2 \ge 0$, the second term is non-positive. $K_X$ and $K_C$ are Lipschitz function, so the last line is clearly bounded by $\|z - z'\|_2^2$ up to a multiplicative constant. Then, there exists a constant $L$ such that

$$ \langle z - z', b(t, z, \nu) - b(t, z', \nu) \rangle \le L \|z - z'\|_2^2. $$

Since $K_X$ and $K_C$ are Lipschitz function, we also have, for all probability distribution $\nu$ and $\nu'$ with a finite variance,

$$ \|b(t, z, \nu) - b(t, z, \nu')\|_2 \le L W_2(\nu, \nu'). $$

Eventually, since $b$ is Locally Lipschitz with polynomial growth, each Assumption is satisfied and Theorem 3.3 [RST19] can be applied. Note that we could also apply Proposition 2.19 from [LS14] : assumptions are the same, and it gives a result for interaction depending on a spatial position.

To complete the Proposition 1.2, we also give the following

**Proposition 1.3.** If $\mathbb{E}(|\bar{X}_0|^2) < +\infty$ and $\mathbb{E}(|\bar{C}_0|^2) < +\infty$, then there exists $C_{0,1}$ and $C_{0,2}$ such that:

$$ \mathbb{E} (|\bar{X}_t|^2 + |\bar{C}_t|^2) \le C_{0,1} e^{C_{0,2} t}. \quad (1.6) $$

The proof is very similar with Proposition 1.2 and is in Appendix A.2.

## 1.4 Quick result : non uniform in time propagation of chaos

We start by proving Theorem 1, a non uniform in time propagation of chaos, as it highlights the basic strategy behind a coupling argument. Some of the following expressions will be used in the proof of Theorem 2, in Section 3.

We consider a synchronous coupling between $(Z_t^{i,N})_i$ and $(\bar{Z}_t^i)_i$, i.e. $\tilde{B}_t^{i,X} = B_t^{i,X}$ and $\tilde{B}_t^{i,C} = B_t^{i,C}$. We have

$$ \left\{ \begin{aligned} dX_t^{i,N} &= (X_t^{i,N} - (X_t^{i,N})^3 - C_t^{i,N} - \alpha)dt + \frac{1}{N} \sum_{j=1}^N K_X(Z_t^i - Z_t^j)dt + \sigma_x dB_t^{i,X} \\ dC_t^{i,N} &= (\gamma X_t^{i,N} - C_t^{i,N} + \beta)dt + \frac{1}{N} \sum_{j=1}^N K_C(Z_t^i - Z_t^j)dt + \sigma_c dB_t^{i,C} \end{aligned} \right. $$

and

$$ \left\{ \begin{aligned} d\bar{X}_t^i &= (\bar{X}_t^i - (\bar{X}_t^i)^3 - \bar{C}_t^i - \alpha)dt + K_X * \bar{\mu}_t(\bar{Z}_t^i)dt + \sigma_x dB_t^{i,X} \\ d\bar{C}_t^i &= (\gamma \bar{X}_t^i - \bar{C}_t^i + \beta)dt + K_C * \bar{\mu}_t(\bar{Z}_t^i)dt + \sigma_c dB_t^{i,C}, \end{aligned} \right. $$

with $\bar{\mu}_t$ the law of $\bar{Z}_t^1$. The method is the following :

* we compute the time evolution of $\mathbb{E}r_t^i = \mathbb{E}(|X_t^{i,N} - \bar{X}_t^i| + |C_t^{i,N} - \bar{C}_t^i|)$ using Ito's formula,

* we control the difference between the drifts $\frac{1}{N} \sum_{j \ne i} K(\bar{Z}_t^i - \bar{Z}_t^j)$ and $K * \bar{\mu}_t(\bar{Z}_t^i)$ using some form of law of large number. This is where the convergence rate $\sqrt{N}$ appears,

* and we conclude using Gronwall's lemma.

**Time evolution :** We have,

$$ d(X_t^{i,N} - \bar{X}_t^i) = \left( (X_t^{i,N} - \bar{X}_t^i) - ((X_t^{i,N})^3 - (\bar{X}_t^i)^3) - (C_t^{i,N} - \bar{C}_t^i) + \frac{1}{N} \sum_{j=1}^N K_X(Z_t^i - Z_t^j) - K_X * \bar{\mu}_t(\bar{Z}_t^i) \right) dt. $$
The main interest of obtaining uniform in time estimates is that it allows the study and comparison of the long-time behavior of the particle system and its nonlinear limit. As previously mentioned, this work follows the method described in [10]. Beyond the result of uniform in time propagation of chaos for the FitzHugh-Nagumo model, which is in itself an interesting result, the present work is also a testimony to the robustness of the coupling method.

The reader will find an index containing the notation, constants, and parameters for reference at the end of the document.

## 1.3 Existence of solutions

First of all, we prove Proposition 1.2, i.e existence of strong solutions of systems (1.1) and (1.2), under Assumption 1.1. Let's denote, for $\kappa \in \mathbb{R}^+$,

$$g_{\kappa}(x) = \begin{cases} -\kappa^3 & \text{if } x < -\kappa \\ x^3 & \text{if } x \in [-\kappa, \kappa] \\ \kappa^3 & \text{if } x > \kappa. \end{cases}$$

$g_{\kappa}$ is Lipschitz and is bounded.

Thus, it's well known (see Chapter 4 [12]) that the following system (under Assumption 1.1 and the assumption that the initial condition has a moment of order 2)

$$\left\{ \begin{aligned} dX_t^{i,N,\kappa} &= (X_t^{i,N,\kappa} - g_{\kappa}(X_t^{i,N,\kappa}) - C_t^{i,N,\kappa} - \alpha)dt + \frac{1}{N} \sum_{j=1}^{N} K_X(Z_t^{i,N,\kappa} - Z_t^{j,N,\kappa}) \\ &\quad + \sigma_X dB_t^{i,X} \\ dC_t^{i,N,\kappa} &= (\gamma X_t^{i,N,\kappa} - C_t^{i,N,\kappa} + \beta)dt + \frac{1}{N} \sum_{j=1}^{N} K_C(Z_t^{i,N,\kappa} - Z_t^{j,N,\kappa}) + \sigma_C dB_t^{i,C}, \end{aligned} \right. \quad (1.4)$$

for $1 \le i \le N$, has a strong and unique solution that we denote $(X_t^{i,N,\kappa}, C_t^{i,N,\kappa})_{1 \le i \le N}$.

In consequence, for a fixed $\kappa \in \mathbb{R}^+$, there exists a strong solution of system (1.1) until time

$$T_{\kappa} = \sup \{t, \forall i, \forall s \le t, X_s^{i,N,\kappa} \le \kappa \text{ and } C_s^{i,N,\kappa} \le \kappa\},$$

and the solution coincides with the solution of the system with $g_{\kappa}$.

We have the following Lemma

**Lemma 1.5.** If, for each $i \le N$, $E(|X_0^{i,N,\kappa}|^2) < +\infty$ and $E(|C_0^{i,N,\kappa}|^2) < +\infty$, then for all $t \ge 0$ there exists $C_t < \infty$ such that, for each $i \le N$

$$E(|X_t^{i,N,\kappa}|^2 + |C_t^{i,N,\kappa}|^2) \le C_t. \quad (1.5)$$

The proof relies on the Lyapunov function defined in the next Section and is given in Appendix A.3.

Then, by denoting $T_{\infty}$ the explosion time of a solution of system (1.1)

$$T_{\infty} = \inf \{t, \exists i, \forall A > 0, X_t^{i,N,\kappa} > A \text{ or } C_t^{i,N,\kappa} > A\}$$

we obtain that $\forall t \in \mathbb{R}^+$, $P(T_{\infty} \le t) = 0$ and $P(\bar{T}_{\infty} \le t) = 0$. Eventually, there exists a unique and strong solution of system (1.1).

The existence and uniqueness of a solution of (1.2) is known from the Theorem 3.3 from [7], under the assumption that the law of the initial point $(\bar{X}_0, \bar{C}_0)$ has a moment of order 2. We only have to prove that Assumptions 3.2 [7] are satisfied. We define, for all $t \in \mathbb{R}^+$, $z = (x, c) \in \mathbb{R}^2$ and for all probability distribution $\nu$ with a finite variance

$$b(t, z, \nu) = \begin{pmatrix} x - x^3 - c - \alpha + K_X * \nu(z) \\ \gamma x - c + \beta + K_C * \nu(z) \end{pmatrix} \quad \text{and} \quad \sigma(t, z, \nu) = \begin{pmatrix} \sigma_X \\ \sigma_C \end{pmatrix}.$$
We denote

$$
\operatorname{sign}(x) = \begin{cases} \frac{x}{|x|} & \text{if } x \neq 0, \\ 0 & \text{otherwise,} \end{cases}
$$

and obtain, using Ito’s formula,

$$
\begin{align}
d|X_t^{i,N} - \bar{X}_t^i| &= \left( \operatorname{sign}(X_t^{i,N} - \bar{X}_t^i)(X_t^{i,N} - \bar{X}_t^i) - \operatorname{sign}(X_t^{i,N} - \bar{X}_t^i) \left( (X_t^{i,N})^3 - (\bar{X}_t^i)^3 \right) \right. \nonumber \\
&\qquad \left. - \operatorname{sign}(X_t^{i,N} - \bar{X}_t^i)(C_t^{i,N} - \bar{C}_t^i) + \operatorname{sign}(X_t^{i,N} - \bar{X}_t^i) \frac{1}{N} \sum_{j=1}^{N} K_X(Z_t^i - Z_t^j) - K_X * \bar{\mu}_t(\bar{Z}_t^i) \right) dt \nonumber \\
&\le \left( |X_t^{i,N} - \bar{X}_t^i| - \left| (X_t^{i,N})^3 - (\bar{X}_t^i)^3 \right| + |C_t^{i,N} - \bar{C}_t^i| + \left| \frac{1}{N} \sum_{j=1}^{N} K_X(Z_t^i - Z_t^j) - K_X * \bar{\mu}_t(\bar{Z}_t^i) \right| \right) dt. \tag{1.7}
\end{align}
$$

Similarly,

$$
d(C_t^{i,N} - \bar{C}_t^i) = \left( \gamma(X_t^{i,N} - \bar{X}_t^i) - (C_t^{i,N} - \bar{C}_t^i) + \frac{1}{N} \sum_{j=1}^{N} K_C(Z_t^i - Z_t^j) - K_C * \bar{\mu}_t(\bar{Z}_t^i) \right) dt,
$$

and we obtain

$$
d|C_t^{i,N} - \bar{C}_t^i| \le \left( \gamma |X_t^{i,N} - \bar{X}_t^i| - |C_t^{i,N} - \bar{C}_t^i| + \left| \frac{1}{N} \sum_{j=1}^{N} K_C(Z_t^i - Z_t^j) - K_C * \bar{\mu}_t(\bar{Z}_t^i) \right| \right) dt. \quad (1.8)
$$

Thus, denoting $r_t^i = |X_t^{i,N} - \bar{X}_t^i| + |C_t^{i,N} - \bar{C}_t^i|$

$$
\begin{align*}
dr_t^i \le & \left( (1+\gamma)|X_t^{i,N} - \bar{X}_t^i| - \left| (X_t^{i,N})^3 - (\bar{X}_t^i)^3 \right| \right. \\
& \left. + \left| \frac{1}{N} \sum_{j=1}^N K_X(Z_t^i - Z_t^j) - K_X * \bar{\mu}_t(\bar{Z}_t^i) \right| + \left| \frac{1}{N} \sum_{j=1}^N K_C(Z_t^i - Z_t^j) - K_C * \bar{\mu}_t(\bar{Z}_t^i) \right| \right) dt.
\end{align*}
$$

**Difference of the drifts :** Let us now consider these last two terms

$$
\left| \frac{1}{N} \sum_{j=1}^{N} K_X(Z_t^i - Z_t^j) - K_X * \bar{\mu}_t(\bar{Z}_t^i) \right| \le \left| \frac{1}{N} \sum_{j=1}^{N} K_X(Z_t^i - Z_t^j) - \frac{1}{N} \sum_{j=1}^{N} K_X(\bar{Z}_t^i - \bar{Z}_t^j) \right| + \left| \frac{1}{N} \sum_{j=1}^{N} K_X(\bar{Z}_t^i - \bar{Z}_t^j) - K_X * \bar{\mu}_t(\bar{Z}_t^i) \right|.
$$

The first sum can be decomposed, using Assumption 1,

$$
\frac{1}{N} \left| \sum_{j=1}^{N} K_X(Z_t^i - Z_t^j) - K_X(\bar{Z}_t^i - \bar{Z}_t^j) \right| \le \frac{L_X}{N} \sum_{j=1}^{N} \| Z_t^i - Z_t^j - (\bar{Z}_t^i - \bar{Z}_t^j) \|_1
$$
$\sigma$ is a constant function, so it clearly satisfies the various conditions.

For $t \in \mathbb{R}^+$, $z, z'$ in $\mathbb{R}^2$, and $\nu$ a probability measure

$$ \begin{aligned} \langle z - z', b(t, z, \nu) - b(t, z', \nu) \rangle &= (x - x') [(x - x') - (x^3 - x'^3) - (c - c') + K_X * \nu(z) - K_X * \nu(z')] \\ &\quad + (c - c') (\gamma(x - x') - (c - c') + K_C * \nu(z) - K_C * \nu(z')) \\ &= (x - x')^2 - (x - x')^2(x^2 + xx' + x'^2) + (\gamma - 1)(c - c')(x - x') - (c - c')^2 \\ &\quad + (x - x')(K_X * \nu(z) - K_X * \nu(z')) + (c - c')(K_C * \nu(z) - K_C * \nu(z')). \end{aligned} $$

Since $x^2 + xx' + x'^2 \ge 0$, the second term is non-positive. $K_X$ and $K_C$ are Lipschitz continuous functions, so the last line is clearly bounded by $\|z - z'\|_2^2$ up to a multiplicative constant. Then, there exists a constant $L$ such that

$$ \langle z - z', b(t, z, \nu) - b(t, z', \nu) \rangle \le L \|z - z'\|_2^2. $$

Since $K_X$ and $K_C$ are Lipschitz continuous functions, we also have, for all probability distributions $\nu$ and $\nu'$ with a finite variance,

$$ \|b(t, z, \nu) - b(t, z, \nu')\|_2 \le L W_2(\nu, \nu'). $$

Eventually, since $b$ is locally Lipschitz continuous with polynomial growth, each Assumption is satisfied and Theorem 3.3 [7] can be applied. Note that we could also apply Proposition 2.19 from [21]: assumptions are the same, and it gives a result for interaction depending on a spatial position.

To complete the Lemma 1.5, we also give the following

**Proposition 1.6.** If, for each $i \le N$, $E(|X_0^{i,N}|^2) < +\infty$ and $E(|C_0^{i,N}|^2) < +\infty$, then for all $t \ge 0$ there exists $C_t < \infty$ such that, for each $i \le N$

$$ E(|X_t^{i,N}|^2 + |C_t^{i,N}|^2) \le C_t. \quad (1.6) $$

and

**Proposition 1.7.** If $E(|\bar{X}_0|^2) < +\infty$ and $E(|\bar{C}_0|^2) < +\infty$, then there exist $C_{0,1}$ and $C_{0,2}$ such that

$$ E(|\bar{X}_t|^2 + |\bar{C}_t|^2) \le C_{0,1} e^{C_{0,2}t}. \quad (1.7) $$

The proof is very similar to the proof of Lemma 1.5 and can be found in Appendix A.3.

## 1.4 Quick result: non uniform in time propagation of chaos

We start by proving Theorem 1.3, a non uniform in time propagation of chaos, as it highlights the basic strategy behind a coupling argument. Some of the following expressions will be used in the proof of Theorem 1.4, in Section 3.

We consider a synchronous coupling between $(Z_t^{i,N})_i$ and $(\bar{Z}_t^i)_i$, i.e. for each $1 \le i \le N$, we choose $\bar{B}_t^{i,X} = B_t^{i,X}$ and $\bar{B}_t^{i,C} = B_t^{i,C}$. We have

$$ \left\{ \begin{aligned} dX_t^{i,N} &= (X_t^{i,N} - (X_t^{i,N})^3 - C_t^{i,N} - \alpha)dt + \frac{1}{N} \sum_{j=1}^N K_X(Z_t^{i,N} - Z_t^{j,N})dt + \sigma_X dB_t^{i,X} \\ dC_t^{i,N} &= (\gamma X_t^{i,N} - C_t^{i,N} + \beta)dt + \frac{1}{N} \sum_{j=1}^N K_C(Z_t^{i,N} - Z_t^{j,N})dt + \sigma_C dB_t^{i,C} \end{aligned} \right. $$

and

$$ \left\{ \begin{aligned} d\bar{X}_t^i &= (\bar{X}_t^i - (\bar{X}_t^i)^3 - \bar{C}_t^i - \alpha)dt + K_X * \bar{\mu}_t(\bar{Z}_t^i)dt + \sigma_X dB_t^{i,X} \\ d\bar{C}_t^i &= (\gamma \bar{X}_t^i - \bar{C}_t^i + \beta)dt + K_C * \bar{\mu}_t(\bar{Z}_t^i)dt + \sigma_C dB_t^{i,C}, \end{aligned} \right. $$

with $\bar{\mu}_t$ the law of $\bar{Z}_t^1$. The method is the following
$$ \begin{aligned} & \le \frac{L_X}{N} \sum_{j=1}^{N} \left( \|Z_t^i - \bar{Z}_t^i\|_1 + \|Z_t^j - \bar{Z}_t^j\|_1 \right) \\ & \le L_X r_t^i + \frac{L_X}{N} \sum_{j=1}^{N} r_t^j. \end{aligned} $$

Similarly, we obtain

$$ \left| \frac{1}{N} \sum_{j=1}^{N} K_C(Z_t^i - Z_t^j) - K_C * \bar{\mu}_t(\bar{Z}_t^i) \right| \le L_C r_t^i + \frac{L_C}{N} \sum_{j=1}^{N} r_t^j + \left| \frac{1}{N} \sum_{j=1}^{N} K_C(\bar{Z}_t^i - \bar{Z}_t^j) - K_C * \bar{\mu}_t(\bar{Z}_t^i) \right| $$

Hence, we get

$$ \begin{aligned} dr_t^i & \le \left( (1 + \gamma) |X_t^{i,N} - \bar{X}_t^i| - \left| (X_t^{i,N})^3 - (\bar{X}_t^i)^3 \right| + (L_X + L_C) \left( r_t^i + \frac{1}{N} \sum_{j=1}^N r_t^j \right) \right. \\ & \qquad \left. + \left| \frac{1}{N} \sum_{j=1}^N K_X(\bar{Z}_t^i - \bar{Z}_t^j) - K_X * \bar{\mu}_t(\bar{Z}_t^i) \right| + \left| \frac{1}{N} \sum_{j=1}^N K_C(\bar{Z}_t^i - \bar{Z}_t^j) - K_C * \bar{\mu}_t(\bar{Z}_t^i) \right| \right) dt \\ & \le \left( (1 + \gamma) r_t^i + (L_X + L_C) \left( r_t^i + \frac{1}{N} \sum_{j=1}^N r_t^j \right) \right. \\ & \qquad \left. + \left| \frac{1}{N} \sum_{j=1}^N K_X(\bar{Z}_t^i - \bar{Z}_t^j) - K_X * \bar{\mu}_t(\bar{Z}_t^i) \right| + \left| \frac{1}{N} \sum_{j=1}^N K_C(\bar{Z}_t^i - \bar{Z}_t^j) - K_C * \bar{\mu}_t(\bar{Z}_t^i) \right| \right) dt. \end{aligned} $$

By considering the expectation, since $\mathbb{E}(r_t^j) = \mathbb{E}(r_t^i)$ for each $j$, by exchangeability of the particles, we have

$$ d\mathbb{E}(r_t^i) \le \left( (1 + \gamma + 2L_X + 2L_C)\mathbb{E}(r_t^i) + \mathbb{E} \left[ \left| \frac{1}{N} \sum_{j=1}^N K_X(\bar{Z}_t^i - \bar{Z}_t^j) - K_X * \bar{\mu}_t(\bar{Z}_t^i) \right| \right] \right. \\ \left. + \mathbb{E} \left[ \left| \frac{1}{N} \sum_{j=1}^N K_C(\bar{Z}_t^i - \bar{Z}_t^j) - K_C * \bar{\mu}_t(\bar{Z}_t^i) \right| \right] \right) dt. $$

Now, we bound the interaction part. We begin with $K_X$. By Cauchy-Schwarz, we can write

$$ \mathbb{E} \left[ \left| \frac{1}{N} \sum_{j=1}^{N} K_X(\bar{Z}_t^i - \bar{Z}_t^j) - K_X * \bar{\mu}_t(\bar{Z}_t^i) \right| \right] \le \mathbb{E} \left( \left| \frac{1}{N} \sum_{j=1}^{N} K_X(\bar{Z}_t^i - \bar{Z}_t^j) - K_X * \bar{\mu}_t(\bar{Z}_t^i) \right|^2 \right)^{1/2} $$

We notice that $(\bar{Z}_t^j)_j$ are i.i.d with law $\bar{\mu}_t$. Let's denote $\bar{Z}_t$ a generic random variable of law $\bar{\mu}_t$ independent of $\bar{Z}_t^i$. What is more, $K_X * \bar{\mu}_t(\bar{Z}_t^i) = \int K_X(\bar{Z}_t^i - z)\bar{\mu}_t(dz) = \mathbb{E}[K_X(\bar{Z}_t^i - \bar{Z}_t)|\bar{Z}_t^i]$. Hence

$$ \begin{aligned} \mathbb{E} \left( \mathbb{E} \left( \left| \frac{1}{N-1} \sum_{j \neq i} K_X(\bar{Z}_t^i - \bar{Z}_t^j) - K_X * \bar{\mu}_t(\bar{Z}_t^i) \right|^2 \middle| \bar{Z}_t^i \right) \right) &= \mathbb{E} \left( \mathrm{Var} \left( \frac{1}{N-1} \sum_{j \neq i} K_X(\bar{Z}_t^i - \bar{Z}_t^j) \middle| \bar{Z}_t^i \right) \right) \\ &= \frac{1}{N-1} \mathbb{E} \left( \mathrm{Var} \left( K_X(\bar{Z}_t^i - \bar{Z}_t) \middle| \bar{Z}_t^i \right) \right) \end{aligned} $$
* we compute the time evolution of $E(|X_t^{i,N} - \bar{X}_t^i| + |C_t^{i,N} - \bar{C}_t^i|)$ using Ito's formula,

* we control the difference between the drifts $\frac{1}{N} \sum_{j \neq i} K(\bar{Z}_t^i - \bar{Z}_t^j)$ and $K * \bar{\mu}_t(\bar{Z}_t^i)$ using some form of the law of large numbers. This is where the convergence rate $\sqrt{N}$ appears,

* and we conclude using Gronwall's lemma.

**Time evolution:** We have,

$$d(X_t^{i,N} - \bar{X}_t^i) = \left( (X_t^{i,N} - \bar{X}_t^i) - ((X_t^{i,N})^3 - (\bar{X}_t^i)^3) - (C_t^{i,N} - \bar{C}_t^i) + \frac{1}{N} \sum_{j=1}^{N} K_X(Z_t^{i,N} - Z_t^{j,N}) - K_X * \bar{\mu}_t(\bar{Z}_t^i) \right) dt.$$

We denote

$$\mathrm{sign}(x) = \begin{cases} \frac{x}{|x|} & \text{if } x \neq 0, \\ 0 & \text{otherwise,} \end{cases}$$

and obtain, using Ito's formula for a twice continuously differentiable approximation of the absolute value and usual convergence lemmas (see Lemma A.1 below),

$$\begin{aligned} d|X_t^{i,N} - \bar{X}_t^i| \\ &= \left( \mathrm{sign}(X_t^{i,N} - \bar{X}_t^i)(X_t^{i,N} - \bar{X}_t^i) - \mathrm{sign}(X_t^{i,N} - \bar{X}_t^i) \left( (X_t^{i,N})^3 - (\bar{X}_t^i)^3 \right) \right. \\ &\qquad \left. - \mathrm{sign}(X_t^{i,N} - \bar{X}_t^i)(C_t^{i,N} - \bar{C}_t^i) \right. \\ &\qquad \left. + \mathrm{sign}(X_t^{i,N} - \bar{X}_t^i) \frac{1}{N} \sum_{j=1}^{N} K_X(Z_t^{i,N} - Z_t^{j,N}) - K_X * \bar{\mu}_t(\bar{Z}_t^i) \right) dt \\ &\le \left( |X_t^{i,N} - \bar{X}_t^i| - \left| (X_t^{i,N})^3 - (\bar{X}_t^i)^3 \right| + |C_t^{i,N} - \bar{C}_t^i| \right. \\ &\qquad \left. + \left| \frac{1}{N} \sum_{j=1}^{N} K_X(Z_t^{i,N} - Z_t^{j,N}) - K_X * \bar{\mu}_t(\bar{Z}_t^i) \right| \right) dt. \end{aligned} \quad (1.8)$$

Similarly,

$$\begin{aligned} d|C_t^{i,N} - \bar{C}_t^i| \\ \le \left( \gamma \left| X_t^{i,N} - \bar{X}_t^i \right| - |C_t^{i,N} - \bar{C}_t^i| + \left| \frac{1}{N} \sum_{j=1}^{N} K_C(Z_t^{i,N} - Z_t^{j,N}) - K_C * \bar{\mu}_t(\bar{Z}_t^i) \right| \right) dt. \end{aligned} \quad (1.9)$$

Thus, denoting $r_t^i = |X_t^{i,N} - \bar{X}_t^i| + |C_t^{i,N} - \bar{C}_t^i|$ (i.e considering $\delta = 1$) we obtain,

$$\begin{aligned} dr_t^i \le & \left( (1 + \gamma) |X_t^{i,N} - \bar{X}_t^i| - \left| (X_t^{i,N})^3 - (\bar{X}_t^i)^3 \right| \right. \\ & + \left| \frac{1}{N} \sum_{j=1}^{N} K_X(Z_t^{i,N} - Z_t^{j,N}) - K_X * \bar{\mu}_t(\bar{Z}_t^i) \right| \\ & + \left| \frac{1}{N} \sum_{j=1}^{N} K_C(Z_t^{i,N} - Z_t^{j,N}) - K_C * \bar{\mu}_t(\bar{Z}_t^i) \right| \end{aligned} dt.$$
$$ \leq \frac{L_X^2}{N-1} \mathbb{E} \left( \mathrm{Var} \left( \| \bar{Z}_t^i - \bar{Z}_t \|_1 | \bar{Z}_t^i \right) \right). $$

Since

$$ \mathbb{E} \left[ \mathrm{Var} \left( \| \bar{Z}_t^i - \bar{Z}_t \|_1 | \bar{Z}_t^i \right) \right] \leq \mathbb{E} \left[ \mathbb{E} \left( \| \bar{Z}_t^i - \bar{Z}_t \|_1^2 | \bar{Z}_t^i \right) \right] \leq \mathbb{E} \left[ \mathbb{E} \left( 2 \| \bar{Z}_t^i \|_1^2 + 2 \| \bar{Z}_t \|_1^2 | \bar{Z}_t^i \right) \right] \leq 4 \mathbb{E} (\| \bar{Z}_t \|_1^2), $$

we obtain

$$ \mathbb{E} \left( \mathbb{E} \left( \left| \frac{1}{N-1} \sum_{j \neq i} K_X(\bar{Z}_t^i - \bar{Z}_t^j) - K_X * \bar{\mu}_t(\bar{Z}_t^i) \right|^2 \middle| \bar{Z}_t^i \right) \right) \leq \frac{4L_X^2}{N-1} \mathbb{E}(\|\bar{Z}_t\|_1^2). $$

We now want to control $\mathbb{E} \left( \left| \frac{1}{N} \sum_{j=1}^N K_X(\bar{Z}_t^i - \bar{Z}_t^j) - K_X * \bar{\mu}_t(\bar{Z}_t^i) \right|^2 \right)$. We decompose it with

$$ \begin{aligned} & \mathbb{E} \left( \left| \frac{1}{N} \sum_{j=1}^{N} K_X(\bar{Z}_t^i - \bar{Z}_t^j) - K_X * \bar{\mu}_t(\bar{Z}_t^i) \right|^2 \right) \\ &= \mathbb{E} \left( \left| \frac{N-1}{N} \frac{1}{N-1} \sum_{j=1}^{N} K_X(\bar{Z}_t^i - \bar{Z}_t^j) - \left( \frac{N-1}{N} + \frac{1}{N} \right) K_X * \bar{\mu}_t(\bar{Z}_t^i) \right|^2 \right) \\ &\leq 2 \left( \frac{N-1}{N} \right)^2 \mathbb{E} \left( \left| \frac{1}{N-1} \sum_{j=1}^{N} K_X(\bar{Z}_t^i - \bar{Z}_t^j) - K_X * \bar{\mu}_t(\bar{Z}_t^i) \right|^2 \right) + \frac{2}{N^2} \mathbb{E} (|K_X * \bar{\mu}_t(\bar{Z}_t^i)|^2). \end{aligned} $$

Since

$$ \mathbb{E} (|K_X * \bar{\mu}_t(\bar{Z}_t^i)|^2) = \mathbb{E} \left( \left| \mathbb{E} \left( K_X(\bar{Z}_t^i - \bar{Z}_t) |\bar{Z}_t^i\right) \right|^2 \right) \leq L_X^2 \mathbb{E} \left( \mathbb{E} \left( \|\bar{Z}_t^i - \bar{Z}_t\|_1^2 |\bar{Z}_t^i\right) \right) \leq 4L_X^2 \mathbb{E} (\|\bar{Z}_t\|_1^2), $$

we obtain

$$ \mathbb{E} \left( \left| \frac{1}{N} \sum_{j=1}^{N} K_X(\bar{Z}_t^i - \bar{Z}_t^j) - K_X * \bar{\mu}_t(\bar{Z}_t^i) \right|^2 \right) \leq \left( \frac{N-1}{N} \right)^2 \frac{4L_X^2}{N-1} \mathbb{E}(\|\bar{Z}_t\|_1^2) + \frac{4L_X^2}{N^2} \mathbb{E}(\|\bar{Z}_t\|_1^2) \leq \frac{8L_X^2}{N} \mathbb{E}(\|\bar{Z}_t\|_1^2), \quad (1.9) $$

and finally

$$ \mathbb{E} \left[ \left| \frac{1}{N} \sum_{j=1}^{N} K_X(\bar{Z}_t^i - \bar{Z}_t^j) - K_X * \bar{\mu}_t(\bar{Z}_t^i) \right| \right] \leq \left( \frac{8L_X^2}{N} \mathbb{E}(\|\bar{Z}_t\|_1^2) \right)^{1/2}. $$

Similarly, we have

$$ \mathbb{E} \left[ \left| \frac{1}{N} \sum_{j=1}^{N} K_C(\bar{Z}_t^i - \bar{Z}_t^j) - K_C * \bar{\mu}_t(\bar{Z}_t^i) \right| \right] \leq \left( \frac{8L_C^2}{N} \mathbb{E}(\|\bar{Z}_t\|_1^2) \right)^{1/2}. $$

Finally,

$$ d\mathbb{E}(r_t^i) \leq \left( (1 + \gamma + 2L_X + 2L_C)\mathbb{E}(r_t^i) + \sqrt{8L_X^2 + 8L_C^2} \left( \frac{1}{N} \mathbb{E}(\|\bar{Z}_t\|_1^2) \right)^{1/2} \right) dt. $$
**Difference of the drifts:** Let us now consider these last two terms

$$ \left| \frac{1}{N} \sum_{j=1}^{N} K_X(Z_t^{i,N} - Z_t^{j,N}) - K_X * \bar{\mu}_t(\bar{Z}_t^i) \right| \le \left| \frac{1}{N} \sum_{j=1}^{N} K_X(\bar{Z}_t^i - \bar{Z}_t^j) - K_X * \bar{\mu}_t(\bar{Z}_t^i) \right| + \left| \frac{1}{N} \sum_{j=1}^{N} K_X(Z_t^{i,N} - Z_t^{j,N}) - \frac{1}{N} \sum_{j=1}^{N} K_X(\bar{Z}_t^i - \bar{Z}_t^j) \right| $$

The first sum can be decomposed, using Assumption 1.1, into

$$ \frac{1}{N} \left| \sum_{j=1}^{N} K_X(Z_t^{i,N} - Z_t^{j,N}) - K_X(\bar{Z}_t^i - \bar{Z}_t^j) \right| \le \frac{L_X}{N} \sum_{j=1}^{N} \| Z_t^{i,N} - Z_t^{j,N} - (\bar{Z}_t^i - \bar{Z}_t^j) \|_1 \\ \le L_X r_t^i + \frac{L_X}{N} \sum_{j=1}^{N} r_t^j. $$

Similarly, we obtain

$$ \left| \frac{1}{N} \sum_{j=1}^{N} K_C(Z_t^{i,N} - Z_t^{j,N}) - K_C * \bar{\mu}_t(\bar{Z}_t^i) \right| \le L_C r_t^i + \frac{L_C}{N} \sum_{j=1}^{N} r_t^j + \left| \frac{1}{N} \sum_{j=1}^{N} K_C(\bar{Z}_t^i - \bar{Z}_t^j) - K_C * \bar{\mu}_t(\bar{Z}_t^i) \right|. $$

Hence, we get

$$ dr_t^i \le \left( (1 + \gamma) r_t^i + (L_X + L_C) \left( r_t^i + \frac{1}{N} \sum_{j=1}^{N} r_t^j \right) + \left| \frac{1}{N} \sum_{j=1}^{N} K_X(\bar{Z}_t^i - \bar{Z}_t^j) - K_X * \bar{\mu}_t(\bar{Z}_t^i) \right| + \left| \frac{1}{N} \sum_{j=1}^{N} K_C(\bar{Z}_t^i - \bar{Z}_t^j) - K_C * \bar{\mu}_t(\bar{Z}_t^i) \right| \right) dt. $$

By considering the expectation, since $E(r_t^j) = E(r_t^i)$ for each $j$, by exchangeability of the particles, we have

$$ dE(r_t^i) \le \left( (1 + \gamma + 2L_X + 2L_C)E(r_t^i) + E \left[ \left| \frac{1}{N} \sum_{j=1}^{N} K_X(\bar{Z}_t^i - \bar{Z}_t^j) - K_X * \bar{\mu}_t(\bar{Z}_t^i) \right| \right] + E \left[ \left| \frac{1}{N} \sum_{j=1}^{N} K_C(\bar{Z}_t^i - \bar{Z}_t^j) - K_C * \bar{\mu}_t(\bar{Z}_t^i) \right| \right] \right) dt. $$

Now, we bound the interaction part. We begin with $K_X$. By Cauchy-Schwarz
Then using Proposition 1.3, we obtain

$$d\mathbb{E}(r_t^i) \le \left( (1 + \gamma + 2L_X + 2L_C)\mathbb{E}(r_t^i) + \frac{\sqrt{8L_X^2 + 8L_C^2\sqrt{2C_{0,1}}}}{\sqrt{N}} e^{\frac{1}{2}C_{0,2}t} \right) dt$$

**Conclusion**: We have thus obtained

$$d \left( \mathbb{E}(r_t^i) + \sqrt{\frac{16(L_X^2 + L_C^2)C_{0,1}}{N}} \frac{1}{1 + \gamma + 2L_X + 2L_C - \frac{C_{0,2}}{2}} e^{\frac{1}{2}C_{0,2}t} \right) \\ \le (1 + \gamma + 2L_X + 2L_C) \left( \mathbb{E}(r_t^i) \sqrt{\frac{16(L_X^2 + L_C^2)C_{0,1}}{N}} \frac{1}{1 + \gamma + 2L_X + 2L_C - \frac{C_{0,2}}{2}} e^{\frac{1}{2}C_{0,2}t} \right) dt$$

and Gronwall's lemma yields

$$\mathbb{E}(r_t^i) + \sqrt{\frac{16(L_X^2 + L_C^2)C_{0,1}}{N}} \frac{1}{1 + \gamma + 2L_X + 2L_C - \frac{C_{0,2}}{2}} e^{\frac{1}{2}C_{0,2}t} \\ \le e^{(1+\gamma+2L_X+2L_C)t} \left[ \mathbb{E}(r_0^i) + \sqrt{\frac{16(L_X^2 + L_C^2)C_{0,1}}{N}} \frac{1}{1 + \gamma + 2L_X + 2L_C - \frac{C_{0,2}}{2}} \right],$$

thus

$$\mathbb{E}(r_t^i) \le C_1 e^{C_2 t} \frac{1}{\sqrt{N}}.$$

Let $\mu_0$ a measure on $\mathbb{R}^2$, $\mu_t^{k,N}$ the marginal distribution at time $t$ of the first $k$ neurons $(Z_t^1, \dots, Z_t^k)$ of an $N$ particles system (1.1) with initial distribution $(\mu_0)^{\otimes N}$, and $\bar{\mu}_t$ is a solution of (1.2) with initial distribution $\mu_0$.

We obtain for Wasserstein 1 distance

$$\begin{align*}
W_1(\mu_t^{k,N}, \bar{\mu}_t^{\otimes k}) &= \inf \left\{ \mathbb{E}[\|Z^{(k)} - \bar{Z}^{(k)}\|_1], \mathbb{P}_{Z^{(k)}} = \mu_t^{k,N}, \mathbb{P}_{\bar{Z}^{(k)}} = \bar{\mu}_t^{\otimes k} \right\} \\
&\le \inf \left\{ \mathbb{E} \left[ \sum_{i=1}^k r_t^i \right], \mathbb{P}_{(Z_t^{i,N})_i} = \mu_t^{k,N}, \mathbb{P}_{(\bar{Z}_t^i)_i} = \bar{\mu}_t^{\otimes k} \right\} \\
&\le k \mathbb{E}(r_t^1) \\
&\le C_1 e^{C_2 t} \frac{k}{\sqrt{N}}.
\end{align*}$$

Hence Theorem 1.

# 2 Preliminaries

## 2.1 Notations

For $h: \mathbb{R}^{2N} \to \mathbb{R}$, for all $(z_i)_{1 \le i \le N} = (x_i, c_i)_{1 \le i \le N} \in \mathbb{R}^{2N}$, the generator of (1.1) is

$$\mathcal{L}^N h(z_1, \dots, z_N) = \sum_{i=1}^{N} \mathcal{L}^{i,N} h,$$
inequality, we can write

$$ E \left[ \left| \frac{1}{N} \sum_{j=1}^{N} K_X(\bar{Z}_t^i - \bar{Z}_t^j) - K_X * \bar{\mu}_t(\bar{Z}_t^i) \right| \right] \\ \le E \left( \left| \frac{1}{N} \sum_{j=1}^{N} K_X(\bar{Z}_t^i - \bar{Z}_t^j) - K_X * \bar{\mu}_t(\bar{Z}_t^i) \right|^2 \right)^{1/2} $$

We notice that $(\bar{Z}_t^j)_j$ are i.i.d with law $\bar{\mu}_t$. Let's denote $\bar{Z}_t$ a generic random variable of law $\bar{\mu}_t$ independent of $\bar{Z}_t^i$. What is more, $K_X * \bar{\mu}_t(\bar{Z}_t^i) = \int K_X(\bar{Z}_t^i - z) \bar{\mu}_t(dz) = E[K_X(\bar{Z}_t^i - \bar{Z}_t^i) | \bar{Z}_t^i]$. Hence

$$ E \left( E \left( \left| \frac{1}{N-1} \sum_{j \neq i} K_X(\bar{Z}_t^i - \bar{Z}_t^j) - K_X * \bar{\mu}_t(\bar{Z}_t^i) \right|^2 \middle| \bar{Z}_t^i \right) \right) \\ = E \left( \mathrm{Var} \left( \frac{1}{N-1} \sum_{j \neq i} K_X(\bar{Z}_t^i - \bar{Z}_t^j) \middle| \bar{Z}_t^i \right) \right) \\ \le \frac{L_X^2}{N-1} E \left( \mathrm{Var} \left( \|\bar{Z}_t^i - \bar{Z}_t^i\|_1 \middle| \bar{Z}_t^i \right) \right). $$

Since

$$ E \left[ \mathrm{Var} \left( \| \bar{Z}_t^i - \bar{Z}_t^i \|_1 \middle| \bar{Z}_t^i \right) \right] \le E \left[ E \left( \| \bar{Z}_t^i - \bar{Z}_t^i \|_1^2 \middle| \bar{Z}_t^i \right) \right] \\ \le E \left[ E \left( 2 \| \bar{Z}_t^i \|_1^2 + 2 \| \bar{Z}_t^i \|_1^2 \middle| \bar{Z}_t^i \right) \right] \le 4 E(\| \bar{Z}_t^i \|_1^2), $$

we obtain

$$ E \left( E \left( \left| \frac{1}{N-1} \sum_{j \neq i} K_X(\bar{Z}_t^i - \bar{Z}_t^j) - K_X * \bar{\mu}_t(\bar{Z}_t^i) \right|^2 \middle| \bar{Z}_t^i \right) \right) \le \frac{4L_X^2}{N-1} E(\|\bar{Z}_t^i\|_1^2). $$

We now want to control $E \left( \left| \frac{1}{N} \sum_{j=1}^{N} K_X(\bar{Z}_t^i - \bar{Z}_t^j) - K_X * \bar{\mu}_t(\bar{Z}_t^i) \right|^2 \right)$. We decompose it into

$$ E \left( \left| \frac{1}{N} \sum_{j=1}^{N} K_X(\bar{Z}_t^i - \bar{Z}_t^j) - K_X * \bar{\mu}_t(\bar{Z}_t^i) \right|^2 \right) \\ = E \left( \left| \frac{N-1}{N} \frac{1}{N-1} \sum_{j=1}^{N} K_X(\bar{Z}_t^i - \bar{Z}_t^j) - \left( \frac{N-1}{N} + \frac{1}{N} \right) K_X * \bar{\mu}_t(\bar{Z}_t^i) \right|^2 \right) \\ \le 2 \frac{(N-1)^2}{N^2} E \left( \left| \frac{1}{N-1} \sum_{j=1}^{N} K_X(\bar{Z}_t^i - \bar{Z}_t^j) - K_X * \bar{\mu}_t(\bar{Z}_t^i) \right|^2 \right) \\ + \frac{2}{N^2} E \left( |K_X * \bar{\mu}_t(\bar{Z}_t^i)|^2 \right). $$
where

$$ \mathcal{L}^{i,N}h(z_1, \dots, z_N) = \left( x_i - x_i^3 - c_i - \alpha + \frac{1}{N} \sum_{j=1}^{N} K_X(z_i - z_j) \right) \partial_{x_i} h \\ + \left( \gamma x_i - c_i + \beta + \frac{1}{N} \sum_{j=1}^{N} K_C(z_i - z_j) \right) \partial_{c_i} h \\ + \frac{\sigma_x^2}{2} \partial_{x_i, x_i}^2 h + \frac{\sigma_c^2}{2} \partial_{c_i, c_i}^2 h. $$

For $h : \mathbb{R}^2 \to \mathbb{R}$, for all $z = (x, y) \in \mathbb{R}^2$, the generator of (1.2) for a given distribution $\mu$ is

$$ \mathcal{L}_{\mu}h(x, c) = (x - x^3 - c - \alpha + K_X * \mu(z)) \partial_x h + (\gamma x - c + \beta + K_C * \mu(z)) \partial_c h \\ + \frac{\sigma_x^2}{2} \partial_{xx}^2 h + \frac{\sigma_c^2}{2} \partial_{cc}^2 h. $$

In particular, we notice that for fixed $(z_i)_{1 \le i \le N} \in (\mathbb{R}^2)^N$, if we consider the empirical measure $\{\mu_{\text{emp}} = \frac{1}{N} \sum_j \delta_{z_j}\}$, we have for all $h : \mathbb{R}^2 \to \mathbb{R}$ and $\bar{z} \in \mathbb{R}^2$,

$$ \mathcal{L}_{\mu_{\text{emp}}}h(\bar{z}) = (\bar{x} - \bar{x}^3 - \bar{c} - \alpha + K_X * \mu_{\text{emp}}(\bar{z})) \partial_x h + (\gamma\bar{x} - \bar{c} + \beta + K_C * \mu_{\text{emp}}(\bar{z})) \partial_c h + \frac{\sigma_x^2}{2} \partial_{xx}^2 h + \frac{\sigma_c^2}{2} \partial_{cc}^2 h \\ = \left( \bar{x} - \bar{x}^3 - \bar{c} - \alpha + \frac{1}{N} \sum_{j=1}^{N} K_X(\bar{z} - z_j) \right) \partial_x h + \left( \gamma\bar{x} - \bar{c} + \beta + \frac{1}{N} \sum_{j=1}^{N} K_C(\bar{z} - z_j) \right) \partial_c h \\ + \frac{\sigma_x^2}{2} \partial_{xx}^2 h + \frac{\sigma_c^2}{2} \partial_{cc}^2 h. $$

In this case, if we consider $\bar{z} = z_i$ for a specific $i$ and we denote $\bar{h}^i : (z_1, \dots, z_N) \to h(z_i)$, then

$$ \mathcal{L}_{\mu_{\text{emp}}}h(z_i) = \left( x_i - x_i^3 - c_i - \alpha + \frac{1}{N} \sum_{j=1}^{N} K_X(z_i - z_j) \right) \partial_x h \\ + \left( \gamma x_i - c_i + \beta + \frac{1}{N} \sum_{j=1}^{N} K_C(z_i - z_j) \right) \partial_c h + \frac{\sigma_x^2}{2} \partial_{xx}^2 h + \frac{\sigma_c^2}{2} \partial_{cc}^2 h \\ = \mathcal{L}^{i,N} \bar{h}^i(z_1, \dots, z_N). $$

## 2.2 First Lyapunov function

Let $H : \mathbb{R}^2 \to \mathbb{R}$ be defined by

$$ H(z) = H(x, c) = \frac{1}{2}\gamma x^2 + \beta x + \frac{1}{2}c^2 + \alpha c + H_0, \qquad (2.1) $$

with

$$ H_0 = \frac{\beta^2}{\gamma} + \alpha^2, $$

where $\gamma$, $\beta$ and $\alpha$ are the parameters of the system (1.1).

**Lemma 2.1.** We have
Since

$$
\begin{aligned}
\mathrm{E} (|K_X * \bar{\mu}_t(\bar{Z}_t^i)|^2) &= \mathrm{E} \left( \left| \mathrm{E} \left( K_X(\bar{Z}_t^i - \bar{Z}_t) |Z_t^i| \right) \right|^2 \right) \\
&\le L_X^2 \mathrm{E} \left( \mathrm{E} \left( \|\bar{Z}_t^i - \bar{Z}_t\|_1^2 |\bar{Z}_t^i| \right) \right) \le 4L_X^2 \mathrm{E} (\|\bar{Z}_t\|_1^2),
\end{aligned}
$$

we obtain

$$
\begin{aligned}
& \mathrm{E} \left( \left| \frac{1}{N} \sum_{j=1}^{N} K_X(\bar{Z}_t^i - \bar{Z}_t^j) - K_X * \bar{\mu}_t(\bar{Z}_t^i) \right|^2 \right) \\
& \le \left( \frac{N-1}{N} \right)^2 \frac{4L_X^2}{N-1} \mathrm{E}(\|\bar{Z}_t\|_1^2) + \frac{4L_X^2}{N^2} \mathrm{E}(\|\bar{Z}_t\|_1^2) \le \frac{8L_X^2}{N} \mathrm{E}(\|\bar{Z}_t\|_1^2),
\end{aligned}
\tag{1.10}
$$

and finally

$$
\mathrm{E} \left[ \left| \frac{1}{N} \sum_{j=1}^{N} K_X(\bar{Z}_t^i - \bar{Z}_t^j) - K_X * \bar{\mu}_t(\bar{Z}_t^i) \right| \right] \le \left( \frac{8L_X^2}{N} \mathrm{E}(\|\bar{Z}_t\|_1^2) \right)^{1/2} .
$$

Similarly, we have

$$
\mathrm{E} \left[ \left| \frac{1}{N} \sum_{j=1}^{N} K_C(\bar{Z}_t^i - \bar{Z}_t^j) - K_C * \bar{\mu}_t(\bar{Z}_t^i) \right| \right] \le \left( \frac{8L_C^2}{N} \mathrm{E}(\|\bar{Z}_t\|_1^2) \right)^{1/2},
$$

which yields

$$
d\mathrm{E}(r_t^i) \le \left( (1 + \gamma + 2L_X + 2L_C)\mathrm{E}(r_t^i) + \sqrt{8L_X^2 + 8L_C^2} \left( \frac{1}{N}\mathrm{E}(\|\bar{Z}_t\|_1^2) \right)^{1/2} \right) dt.
$$

Then using Proposition 1.7, we obtain

$$
d\mathrm{E}(r_t^i) \le \left( (1 + \gamma + 2L_X + 2L_C)\mathrm{E}(r_t^i) + \frac{\sqrt{8L_X^2 + 8L_C^2}\sqrt{2C_{0,1}}e^{\frac{1}{2}C_{0,2}t}}{\sqrt{N}} \right) dt
$$

**Conclusion:** We have thus obtained

$$
\begin{aligned}
d \left( \mathrm{E}(r_t^i) + \sqrt{\frac{16(L_X^2 + L_C^2)C_{0,1}}{N}} \frac{1}{1 + \gamma + 2L_X + 2L_C - \frac{C_{0,2}}{2}} e^{\frac{1}{2}C_{0,2}t} \right) & \\
\le (1 + \gamma + 2L_X + 2L_C) & \\
\times \left( \mathrm{E}(r_t^i) + \sqrt{\frac{16(L_X^2 + L_C^2)C_{0,1}}{N}} \frac{1}{1 + \gamma + 2L_X + 2L_C - \frac{C_{0,2}}{2}} e^{\frac{1}{2}C_{0,2}t} \right) dt,
\end{aligned}
$$

and Gronwall’s lemma yields

$$
\begin{aligned}
& \mathrm{E}(r_t^i) + \sqrt{\frac{16(L_X^2 + L_C^2)C_{0,1}}{N}} \frac{1}{1 + \gamma + 2L_X + 2L_C - \frac{C_{0,2}}{2}} e^{\frac{1}{2}C_{0,2}t} \\
\le & e^{(1+\gamma+2L_X+2L_C)t} \\
& \times \left[ \mathrm{E}(r_0^i) + \sqrt{\frac{16(L_X^2 + L_C^2)C_{0,1}}{N}} \frac{1}{1 + \gamma + 2L_X + 2L_C - \frac{C_{0,2}}{2}} \right],
\end{aligned}
$$
(i) For all $x, c \in \mathbb{R}$, we have $H(x, c) \ge \frac{\gamma}{4}x^2 + \frac{c^2}{4} \ge 0$,

(ii) For all $x, c \in \mathbb{R}$, we have $H(x, c) \ge \frac{1}{2\max(\gamma, 1)}((\gamma x + \beta)^2 + (c + \alpha)^2)$,

(iii) For all $\delta > 0$ there is $C_{r,H} > 0$ such that for all $x, x', c, c' \in \mathbb{R}$, we have

$$ (|x - x'| + \delta|c - c'|)^2 \le C_{r,H}(H(x, c) + H(x', c')), $$

(iv) A direct consequence of the previous point is that for all $B \in \mathbb{R}$, $\lambda > 0$ and $\delta > 0$, there is $R \ge 0$ such that, for $x, x', c, c' \in \mathbb{R}$ satisfying $|x - x'| + \delta|c - c'| \ge R$, we have $H(x, c) + H(x', c') \ge \frac{80B}{\lambda}$. An explicit value of $R$ is given by $R = \sqrt{\frac{1280(1+\delta^2)B}{\lambda \min(\gamma, 1)}}$.

The first two points are consequences of direct calculations. The last two points are proved in Appendix A.1.

**Lemma 2.2** (Lyapunov's property of $H$). $H$, defined in (2.1), is a Lyapunov function. In fact, let $\lambda \in \mathbb{R}$ such that

$$ \frac{L_X}{8} + L_C \left(2 + \frac{1}{8}\right) < 1 - \frac{\lambda}{2}, \qquad (2.2) $$

then there exists $B > 0$ such that for all $(\bar{x}, \bar{c}) \in \mathbb{R}^2$, for all probability distribution $\mu$ on $\mathbb{R}^2$,

$$ \mathcal{L}_{\mu}H(\bar{z}) \le B + (\alpha_X L_X + \beta_X L_C) (\mathbb{E}_{\mu}(|X|)^2 - \bar{x}^2) + (\alpha_C L_X + \beta_C L_C) (\mathbb{E}_{\mu}(|C|)^2 - \bar{c}^2) - \lambda H(\bar{z}). \quad (2.3) $$

Moreover, for all $(z_i)_{1 \le i \le N} \in \mathbb{R}^{2N}$, by denoting $H : (z_1, \dots, z_N) \mapsto H(z_i)$,

$$ \begin{aligned} \mathcal{L}^{i,N} H(z_1, \dots, z_N) \le B &+ (\alpha_X L_X + \beta_X L_C) \left( \left( \frac{1}{N} \sum_{j=1}^N |x_j| \right)^2 - x_i^2 \right) \\ &+ (\alpha_C L_X + \beta_C L_C) \left( \left( \frac{1}{N} \sum_{j=1}^N |c_j| \right)^2 - c_i^2 \right) - \lambda H(z_i), \end{aligned} \quad (2.4) $$

with

$$ \alpha_X = \frac{\gamma}{2} + \frac{1}{2}, \quad \beta_X = \frac{17}{2}, \quad \alpha_C = \frac{1}{16}, \quad \beta_C = \frac{1}{2} + \frac{1}{32} $$

We refer to Appendix A.2 for the proof of this lemma and of the following Proposition.

**Proposition 2.1.** We have

$$ \mathcal{L}^N \left( \frac{1}{N} \sum_{i=1}^N H(Z_t^{i,N}) \right) \le B - \lambda \left( \frac{1}{N} \sum_{i=1}^N H(Z_t^{i,N}) \right), \quad (2.5) $$

A direct consequence of (2.3) is

$$ \mathbb{E}H(\bar{Z}_t^i) \le \mathbb{E}H(\bar{Z}_0^i) + \int_0^t (B - \lambda \mathbb{E}H(\bar{Z}_s^i)) ds, \quad (2.6) $$

and a consequence of (2.5) is

$$ \left( \frac{1}{N} \sum_{i=1}^{N} \mathbb{E}H(Z_t^{i,N}) \right) \le \left( \frac{1}{N} \sum_{i=1}^{N} \mathbb{E}H(Z_0^{i,N}) \right) + \int_0^t \left( B - \lambda \frac{1}{N} \sum_{i=1}^{N} \mathbb{E}H(Z_s^{i,N}) \right) ds. \quad (2.7) $$

From (2.7) we obtain bounds on the moments of $|X_t^{i,N}|^2$ and $|C_t^{i,N}|^2$, and from (2.6) Proposition 1.3 on the second moments of $\bar{X}_t^i$ and $\bar{C}_t^i$. The proof is given in Appendix A.2. It also yields the following result
thus

$$E(r_t^i) \le C_1 e^{C_2 t} \frac{1}{\sqrt{N}}.$$

Let $\mu_0$ a measure on $\mathbb{R}^2$, $\mu_t^{k,N}$ the marginal distribution at time $t$ of the first $k$ neurons $(Z_t^{1,N}, \dots, Z_t^{k,N})$ of an $N$-particle system (1.1) with initial distribution $\mu_0^{\otimes N}$, and $\bar{\mu}_t$ is a solution of (1.2) with initial distribution $\mu_0$. We obtain for the $L^1$ Wasserstein distance

$$ \begin{aligned} W_1(\mu_t^{k,N}, \bar{\mu}_t^{\otimes k}) &= \inf \left\{ \mathbb{E}[\|Z^{(k)} - \bar{Z}^{(k)}\|_1], \mathbb{P}_{Z^{(k)}} = \mu_t^{k,N}, \mathbb{P}_{\bar{Z}^{(k)}} = \bar{\mu}_t^{\otimes k} \right\} \\ &\le \inf \left\{ \mathbb{E} \left[ \sum_{i=1}^k r_t^i \right], \mathbb{P}_{(Z_t^{i,N})_i} = \mu_t^{k,N}, \mathbb{P}_{(\bar{Z}_t^i)_i} = \bar{\mu}_t^{\otimes k} \right\} \\ &\le k \mathbb{E}(r_t^1) \\ &\le C_1 e^{C_2 t} \frac{k}{\sqrt{N}}. \end{aligned} $$

We hence obtain Theorem 1.3.

## 2 Preliminaries

In this section, before tackling the proof by the coupling method of the uniform in time propagation of chaos, we gather the various technical lemmas and construct the necessary objects.

### 2.1 Notation

To construct the Lyapunov functions (which allow us to bound the moments of the processes and show that they tend to come back to some compact set), we begin by introducing the generators of the processes.

For $h: \mathbb{R}^{2N} \to \mathbb{R}$, for all $(z_i)_{1 \le i \le N} = (x_i, c_i)_{1 \le i \le N} \in \mathbb{R}^{2N}$, the generator of (1.1) is

$$ \mathcal{L}^N h(z_1, \dots, z_N) = \sum_{i=1}^N \mathcal{L}^{i,N} h, $$

where

$$ \begin{aligned} \mathcal{L}^{i,N} h(z_1, \dots, z_N) = & \left( x_i - x_i^3 - c_i - \alpha + \frac{1}{N} \sum_{j=1}^N K_X(z_i - z_j) \right) \partial_{x_i} h \\ & + \left( \gamma x_i - c_i + \beta + \frac{1}{N} \sum_{j=1}^N K_C(z_i - z_j) \right) \partial_{c_i} h \\ & + \frac{\sigma_X^2}{2} \partial_{x_i x_i}^2 h + \frac{\sigma_C^2}{2} \partial_{c_i c_i}^2 h. \end{aligned} $$

For $h: \mathbb{R}^2 \to \mathbb{R}$, for all $z = (x, y) \in \mathbb{R}^2$, the generator of (1.2) for a given distribution $\mu$ is

$$ \begin{aligned} \mathcal{L}_{\mu} h(x, c) = & (x - x^3 - c - \alpha + K_X * \mu(z)) \partial_x h + (\gamma x - c + \beta + K_C * \mu(z)) \partial_c h \\ & + \frac{\sigma_X^2}{2} \partial_{xx}^2 h + \frac{\sigma_C^2}{2} \partial_{cc}^2 h. \end{aligned} $$
**Lemma 2.3.** Provided the interaction kernels satisfy

$$ \frac{L_X}{8} + L_C \left( 2 + \frac{1}{8} \right) < 1, $$

and that $\mathbb{E}(|\bar{X}_0|^2) < +\infty$ and $\mathbb{E}(|\bar{C}_0|^2) < +\infty$, then there exists $C_{init,2}$ such that for all $t \ge 0$:

$$ \mathbb{E} (|\bar{X}_t|^2 + |\bar{C}_t|^2) \le C_{init,2}. $$

From now on, we consider $\lambda > 0$ satisfying (2.2) (and use the a priori bounds $L_{X,max}$ and $L_{C,max}$ to ensure the existence of such a $\lambda$).

## 2.3 Modification of the function

Let $C_{init,exp} > 0$, $\tilde{a} > 0$ and consider an initial measure $\mu_0$ on $\mathbb{R}^2$ which satisfies $\mathbb{E}_{\mu_0}(e^{\tilde{a}(|X|+|C|)}) \le C_{init,exp}$.

For technical reasons, we need a greater restoring force by the Lyapunov function than the one given in Lemma 2.2. We thus modify it in order to obtain estimates such as (2.13) and (2.19) below.

Let $a > 0$, such that $a \le \tilde{a}/(4\sqrt{2}\max(\sqrt{\gamma}, 1))$. This choice of $a$ is only necessary for further Propositions and Lemmas, in Section 3.

Let us consider for all $z \in \mathbb{R}^2$,

$$ \tilde{H}(z) = \int_{0}^{H(z)} \exp(a\sqrt{u}) du = \frac{2}{a^2} \exp(a\sqrt{H(z)}) (a\sqrt{H(z)} - 1) + \frac{2}{a^2}. \quad (2.8) $$

Direct calculations yield the following technical lemma.

**Lemma 2.4.** We have, for all $z \in \mathbb{R}^2$

$$ H(z) \exp(a\sqrt{H(z)}) \ge \tilde{H}(z) \ge \exp(a\sqrt{H(z)}) - \frac{2}{a^2} \left(\exp\left(\frac{a^2}{2}\right) - 1\right), \quad (2.9) $$

$$ \frac{2}{a}\sqrt{H(z)} \exp(a\sqrt{H(z)}) \ge \tilde{H}(z) \ge \frac{1}{a}\sqrt{H(z)} \exp(a\sqrt{H(z)}) - \frac{1}{a^2}(e-2), \quad (2.10) $$

$$ \tilde{H}(z) \ge H(z) \quad (2.11) $$

We may calculate, using Lemma 2.1 and Equation (2.3)

$$
\begin{align*}
\mathcal{L}_{\mu}(\tilde{H}) &= \exp(a\sqrt{H}) \mathcal{L}_{\mu}H + \frac{1}{2} \frac{a}{2\sqrt{H}} \exp(a\sqrt{H}) (|\sigma_x \partial_x H|^2 + |\sigma_c \partial_c H|^2) \\
&= \exp(a\sqrt{H}) \mathcal{L}_{\mu}H + \frac{a}{4\sqrt{H}} \exp(a\sqrt{H}) (\sigma_x^2 (\gamma x + \beta)^2 + \sigma_c^2 (c + \alpha)^2) \\
&\le \exp(a\sqrt{H}) (B + (\alpha_X L_X + \beta_X L_C) \mathbb{E}_{\mu}(|X|)^2 + (\alpha_C L_X + \beta_C L_C) \mathbb{E}_{\mu}(|C|)^2 - \lambda H) \\
&\quad + \frac{1}{2} \max(\sigma_x^2, \sigma_c^2) \max(\gamma, 1) a\sqrt{H} \exp(a\sqrt{H}) \\
&\le \exp(a\sqrt{H}) \left( B + \frac{(\frac{1}{2} \max(\sigma_x^2, \sigma_c^2) \max(\gamma, 1))^2 a^2}{2\lambda} + (\alpha_X L_X + \beta_X L_C) \mathbb{E}_{\mu}(|X|)^2 \right. \\
&\qquad \left. + (\alpha_C L_X + \beta_C L_C) \mathbb{E}_{\mu}(|C|)^2 - \frac{\lambda}{2} H \right), \quad (2.12)
\end{align*}
$$

where for this last inequality we used Young's inequality

$$ \frac{1}{2} \max(\sigma_x^2, \sigma_c^2) \max(\gamma, 1) a\sqrt{H} \le \frac{\lambda}{2} H + \frac{(\frac{1}{2} \max(\sigma_x^2, \sigma_c^2) \max(\gamma, 1))^2 a^2}{2\lambda}. $$
In particular, we notice that for fixed $(z_i)_{1\le i\le N} \in (\mathbb{R}^2)^N$, if we consider the empirical measure $\{\mu_{\text{emp}} = \frac{1}{N}\sum_j \delta_{z_j}\}$, we have for all $h: \mathbb{R}^2 \to \mathbb{R}$ and $\bar{z} \in \mathbb{R}^2$,

$$ 
\begin{aligned}
\mathcal{L}_{\mu_{\text{emp}}} h(\bar{z}) &= (\bar{x} - \bar{x}^3 - \bar{c} - \alpha + K_X * \mu_{\text{emp}}(\bar{z})) \partial_x h + (\gamma \bar{x} - \bar{c} + \beta + K_C * \mu_{\text{emp}}(\bar{z})) \partial_c h \\
&\quad + \frac{\sigma_X^2}{2} \partial_{xx}^2 h + \frac{\sigma_C^2}{2} \partial_{cc}^2 h \\
&= \left( \bar{x} - \bar{x}^3 - \bar{c} - \alpha + \frac{1}{N} \sum_{j=1}^N K_X(\bar{z} - z_j) \right) \partial_x h \\
&\quad + \left( \gamma \bar{x} - \bar{c} + \beta + \frac{1}{N} \sum_{j=1}^N K_C(\bar{z} - z_j) \right) \partial_c h + \frac{\sigma_X^2}{2} \partial_{xx}^2 h + \frac{\sigma_C^2}{2} \partial_{cc}^2 h.
\end{aligned}
$$

In this case, if we consider $\bar{z} = z_i$ for a specific $i$ and we denote $\bar{h}^i: (z_1, \dots, z_N) \to h(z_i)$, then

$$ 
\begin{aligned}
\mathcal{L}_{\mu_{\text{emp}}} h(z_i) &= \left( x_i - x_i^3 - c_i - \alpha + \frac{1}{N} \sum_{j=1}^N K_X(z_i - z_j) \right) \partial_x h \\
&\quad + \left( \gamma x_i - c_i + \beta + \frac{1}{N} \sum_{j=1}^N K_C(z_i - z_j) \right) \partial_c h + \frac{\sigma_X^2}{2} \partial_{xx}^2 h + \frac{\sigma_C^2}{2} \partial_{cc}^2 h \\
&= \mathcal{L}^{i,N} \bar{h}^i(z_1, \dots, z_N).
\end{aligned}
$$

## 2.2 First Lyapunov function

Let $H: \mathbb{R}^2 \to \mathbb{R}$ be defined by

$$ H(z) = H(x, c) = \frac{1}{2}\gamma x^2 + \beta x + \frac{1}{2}c^2 + \alpha c + H_0, \quad (2.1) $$

with

$$ H_0 = \frac{\beta^2}{\gamma} + \alpha^2, $$

where $\gamma$, $\beta$ and $\alpha$ are the parameters of the system (1.1).

**Lemma 2.1. (i)** For all $x, c \in \mathbb{R}$, we have $H(x, c) \ge \frac{\gamma}{4}x^2 + \frac{c^2}{4} \ge 0$,

(ii) For all $x, c \in \mathbb{R}$, we have $H(x, c) \ge \frac{1}{2 \max(\gamma, 1)} ((\gamma x + \beta)^2 + (c + \alpha)^2)$,

(iii) For all $\delta > 0$ there is $C_{r,H} > 0$ such that for all $x, x', c, c' \in \mathbb{R}$, we have

$$ (|x - x'| + \delta|c - c'|)^2 \le C_{r,H} (H(x, c) + H(x', c')), $$

(iv) A direct consequence of the previous point is that for all $B \in \mathbb{R}$, $\lambda > 0$ and $\delta > 0$, there is $R \ge 0$ such that, for $x, x', c, c' \in \mathbb{R}$ satisfying $|x - x'| + \delta|c - c'| \ge R$, we have $H(x, c) + H(x', c') \ge \frac{80B}{\lambda}$. An explicit value of $R$ is given by $R = \sqrt{\frac{1280(1+\delta^2)B}{\lambda \min(\gamma, 1)}}$.

The first two points are consequences of direct calculations. The last two points are proved in Appendix A.2. The constant $C_{r,H}$ has been thus named because it ensures the control of the modified Euclidean distance $r$, precisely defined in (2.24), by the function $H$.
Notice that (2.12) ensures that this new Lyapunov function also tends to bring back particle which ventured at infinity, and at an even greater rate. This new rate $H \exp(\sqrt{H})$ however comes at a cost : the initial condition must have a finite exponential moment, and no longer just have a finite second moment. First, by Lemma 2.3, $\mathbb{E}(\bar{X}_t)^2 + \mathbb{E}(\bar{C}_t)^2 \le C_{init,2}$. Furthermore, the function $h \mapsto \exp(a\sqrt{h}) (B - \frac{\lambda}{4}h)$ is bounded from above for $h \ge 0$. We therefore obtain from (2.12) the existence of $\tilde{B}$ such that

$$ \mathcal{L}_{\bar{\mu}_t} (\tilde{H}(\bar{Z}_t^i)) \le \tilde{B} - \frac{\lambda}{4} \left( H(\bar{Z}_t^i) \exp \left( a\sqrt{H(\bar{Z}_t^i)} \right) \right) \quad (2.13) $$

$$ \frac{d}{dt} \mathbb{E} \tilde{H} (\bar{Z}_t^i) \le \tilde{B} - \frac{\lambda}{4} \mathbb{E} \left( H (\bar{Z}_t^i) \exp \left( a \sqrt{H (\bar{Z}_t^i)} \right) \right) \quad (2.14) $$

$$ \text{and} \quad \frac{d}{dt} \mathbb{E} \tilde{H} (\bar{Z}_t^i) \le \tilde{B} - \frac{\lambda}{4} \mathbb{E} \tilde{H} (\bar{Z}_t^i), \quad (2.15) $$

where for this last inequality, we used (2.9). While (2.13) and (2.14) will be useful in ensuring a sufficient restoring force, Equation (2.15) give us a uniform in time bound on $\mathbb{E}\tilde{H}(\bar{Z}_t^i)$, provided we have an initial bound. These inequalities are to be understood in the sense of SDEs, where (2.15) should for instance be rigorously written

$$ \mathbb{E} \tilde{H} (\bar{Z}_t^i) \le \mathbb{E} \tilde{H} (\bar{Z}_0^i) + \int_0^t \left( \tilde{B} - \frac{\lambda}{4} \mathbb{E} \tilde{H} (\bar{Z}_s^i) \right) ds. $$

Now, for the system of particle, we have, using (2.12), $\forall i, \forall x_i, v_i \in \mathbb{R}^d$,

$$ \mathcal{L}^N \tilde{H} (z_i) \le \exp \left( a \sqrt{H(z_i)} \right) \left( \tilde{B} + (\alpha_X L_X + \beta_X L_C) \left( \frac{1}{N} \sum_{j=1}^N |x_j| \right)^2 \right. \\ \left. + (\alpha_C L_X + \beta_C L_C) \left( \frac{1}{N} \sum_{j=1}^N |c_j| \right)^2 - \frac{\lambda}{2} H(z_i) \right). $$

Summing over $i \in \{1, \dots, N\}$, we may calculate

$$ \begin{aligned} & (\alpha_X L_X + \beta_X L_C) \sum_{j=1}^{N} \left( \frac{\sum_{j=1}^{N} |x_j|}{N} \right)^2 \sum_{i=1}^{N} \frac{\exp\left(\frac{a\sqrt{H(z_i)}}{N}\right)}{N} - \frac{\lambda}{16} \sum_{i=1}^{N} \frac{H(z_i) \exp\left(\frac{a\sqrt{H(z_i)}}{N}\right)}{N} \\ & \le \frac{\lambda}{16} \left( \sum_{i,j=1}^{N} \frac{H(z_i) \exp\left(\frac{a\sqrt{H(z_j)}}{N}\right)}{N} - \sum_{i=1}^{N} \frac{H(z_i) \exp\left(\frac{a\sqrt{H(z_i)}}{N}\right)}{N} \right) \\ & \le 0. \end{aligned} \quad (2.16) $$

Here, we used Lemma 2.1, the fact that $\forall x, y \ge 0 \ x e^{\sqrt{y}} + y e^{\sqrt{x}} - x e^{\sqrt{x}} - y e^{\sqrt{y}} = (e^{\sqrt{x}} - e^{\sqrt{y}})(y - x) \le 0$ and assumed

$$ (\alpha_X L_X + \beta_X L_C) \le \frac{\gamma \lambda}{64}. $$

Likewise,

$$ (\alpha_C L_X + \beta_C L_C) \sum_{j=1}^{N} \left( \frac{\sum_{j=1}^{N} |c_j|}{N} \right)^2 \sum_{i=1}^{N} \frac{\exp\left(\frac{a\sqrt{H(z_i)}}{N}\right)}{N} - \frac{\lambda}{16} \sum_{i=1}^{N} \frac{H(z_i) \exp\left(\frac{a\sqrt{H(z_i)}}{N}\right)}{N} \le 0, \quad (2.17) $$

provided

$$ (\alpha_C L_X + \beta_C L_C) \le \frac{\lambda}{64}. $$
**Lemma 2.2** (Lyapunov's property of H). Let $\lambda \in \mathbb{R}$ such that

$$ \frac{L_X}{8} + L_C \left(2 + \frac{1}{8}\right) < 1 - \frac{\lambda}{2}, \qquad (2.2) $$

then, for H defined in (2.1), there exists $B > 0$ such that for all $(\bar{x}, \bar{c}) \in \mathbb{R}^2$, for all probability distribution $\mu$ on $\mathbb{R}^2$,

$$ \mathcal{L}_{\mu} H(\bar{z}) \leq B + (\alpha_X L_X + \beta_X L_C) \left( \mathbb{E}_{\mu}(|X|^2) - \bar{x}^2 \right) \\ + (\alpha_C L_X + \beta_C L_C) \left( \mathbb{E}_{\mu}(|C|^2) - \bar{c}^2 \right) - \lambda H(\bar{z}). \qquad (2.3) $$

Moreover, for all $(z_i)_{1 \le i \le N} \in \mathbb{R}^{2N}$, by denoting $H : (z_1, \dots, z_N) \mapsto H(z_i)$,

$$ \mathcal{L}^{i,N} H(z_1, \dots, z_N) \leq B + (\alpha_X L_X + \beta_X L_C) \left( \left( \frac{1}{N} \sum_{j=1}^{N} |x_j| \right)^2 - x_i^2 \right) \\ + (\alpha_C L_X + \beta_C L_C) \left( \left( \frac{1}{N} \sum_{j=1}^{N} |c_j| \right)^2 - c_i^2 \right) - \lambda H(z_i), \qquad (2.4) $$

with

$$ \alpha_X = \frac{\gamma}{2} + \frac{1}{2}, \quad \beta_X = \frac{17}{2}, \quad \alpha_C = \frac{1}{16}, \quad \beta_C = \frac{1}{2} + \frac{1}{32}. $$

We refer to H as a Lyapunov function, as it ensures that the processes tend to come back to a compact set.

We refer to Appendix A.3 for the proof of this lemma and of the following Proposition.

**Proposition 2.3.** We have

$$ \mathcal{L}^N \left( \frac{1}{N} \sum_{i=1}^{N} H(Z_t^{i,N}) \right) \leq B - \lambda \left( \frac{1}{N} \sum_{i=1}^{N} H(Z_t^{i,N}) \right), \qquad (2.5) $$

A direct consequence of (2.3) is

$$ \mathbb{E}H(\bar{Z}_t^i) \leq \mathbb{E}H(\bar{Z}_0^i) + \int_0^t (B - \lambda \mathbb{E}H(\bar{Z}_s^i)) ds, \qquad (2.6) $$

and a consequence of (2.5) is

$$ \left( \frac{1}{N} \sum_{i=1}^{N} \mathbb{E}H(Z_t^{i,N}) \right) \leq \left( \frac{1}{N} \sum_{i=1}^{N} \mathbb{E}H(Z_0^{i,N}) \right) + \int_0^t \left( B - \lambda \frac{1}{N} \sum_{i=1}^{N} \mathbb{E}H(Z_s^{i,N}) \right) ds. \quad (2.7) $$

From (2.7) we obtain bounds on the moments of $|X_t^{i,N}|^2$ and $|C_t^{i,N}|^2$, and from (2.6) Proposition 1.7 on the second moments of $\bar{X}_t^i$ and $\bar{C}_t^i$. The proof is given in Appendix A.3. It also yields the following result

**Lemma 2.4.** Provided the interaction kernels satisfy (2.2), and that $\mathbb{E}(|\bar{X}_0|^2) < +\infty$ and $\mathbb{E}(|\bar{C}_0|^2) < +\infty$, then there exists $C_{init,2}$ such that for all $t \ge 0$

$$ \mathbb{E}(|\bar{X}_t|^2 + |\bar{C}_t|^2) \leq C_{init,2}. $$

From now on, we consider $\lambda > 0$ satisfying (2.2) (and use the a priori bounds $L_{X,max}$ and $L_{C,max}$ to ensure the existence of such a $\lambda$).
There is therefore a constant, which for the sake of clarity we will also denote $\tilde{B}$ (as we may take the maximum of the previous constants), such that we get

$$ \mathcal{L}^{i,N} \tilde{H}(Z_t^{i,N}) \leq \tilde{B} + (\alpha_X L_X + \beta_X L_C) \left( \frac{\sum_{j=1}^{N} |X_t^{j,N}|}{N} \right)^2 \exp \left( a \sqrt{H(Z_t^{i,N})} \right) \\ + (\alpha_C L_X + \beta_C L_C) \left( \frac{\sum_{j=1}^{N} |C_t^{j,N}|}{N} \right)^2 \exp \left( a \sqrt{H(Z_t^{i,N})} \right) \\ - \frac{\lambda}{4} H(Z_t^{i,N}) \exp \left( a \sqrt{H(Z_t^{i,N})} \right) \qquad (2.18) $$

$$ \mathcal{L}^N \left( \frac{1}{N} \sum_{i=1}^N \tilde{H}(Z_t^{i,N}) \right) \leq \tilde{B} - \frac{\lambda}{4} \left( \frac{1}{N} \sum_{i=1}^N H(Z_t^{i,N}) \exp \left( a \sqrt{H(Z_t^{i,N})} \right) \right) \qquad (2.19) $$

and

$$ \mathcal{L}^N \left( \frac{1}{N} \sum_{i=1}^N \tilde{H}(Z_t^{i,N}) \right) \leq \tilde{B} - \frac{\lambda}{4} \left( \frac{1}{N} \sum_{i=1}^N \tilde{H}(Z_t^{i,N}) \right) \qquad (2.20) $$

Once again, (2.18) and (2.19) will be useful in ensuring a sufficient restoring force, and (2.20) yields a uniform in time bound on the expectation of $\tilde{H}(Z_t^{i,N})$, since $\mathbb{E}(\frac{1}{N} \sum_{j=1}^N \tilde{H}(Z_t^{j,N})) = \mathbb{E}(\tilde{H}(Z_t^{i,N}))$ by exchangeability of the particles.

## 2.4 Parameters

We start by fixing the values of some parameters. The somewhat intricate expressions in this section are dictated by the computations arising in the proofs later on. They are somewhat roughly chosen and far from optimal as we only wish to convey the fact that every constant is explicit.

Recall $\alpha_X, \beta_X, \alpha_C$ and $\beta_C$ given in Lemma 2.2. $a > 0$ is fixed from the last Subsection and the definition of $\tilde{H}$, and $\lambda$ and $\tilde{B}$ are obtained from the same Subsection.

We consider the following parameters

**Assumption 2.** Given any $\eta > 4$ and $\tilde{\delta} > 0$, consider the following set of parameters

$$ \begin{align*} \delta &= (1 + \tilde{\delta}) \frac{1 + L_{X,max}}{1 - L_{C,max}}, \quad R_0 = \sqrt{\frac{1280 \tilde{B}}{\lambda \min(\gamma, 1)}}, \quad R = \sqrt{1 + \tilde{\delta}^2} R_0, \\ C_{f,1} &= 16 \left( \frac{1}{a^2} \left( \gamma + a \left( \beta + \frac{\alpha}{\tilde{\delta}} \right) \sqrt{2 \max(\gamma, 1)} \right) \left( \exp\left(\frac{a^2}{2}\right) - 1 \right) + \sqrt{2 \max(\gamma, 1)} \left( \sqrt{\gamma} + \frac{1}{\tilde{\delta}} \right) (e-2) \right) \\ C_{f,2} &= 4 \left( \gamma + \left( a \left( \beta + \frac{\alpha}{\tilde{\delta}} \right) + 2a^2 \left( \sqrt{\gamma} + \frac{1}{\tilde{\delta}} \right) \right) \sqrt{2 \max(\gamma, 1)} \right) \\ c &= \min \left\{ \begin{aligned} & \frac{2\tilde{B}}{\eta}, \frac{\lambda}{160} \frac{\eta-4}{\eta}, \\ & \frac{1}{2(1+\eta)} \min \left( \frac{\sigma_x}{\sqrt{\pi}R}, 1 - L_C - \frac{1+L_X}{\tilde{\delta}} \right) \exp \left( -\frac{1}{4\sigma_x^2} \left( 1 + \delta\gamma + L_{X,max} + \delta L_{C,max} + (C_{f,1} + C_{f,2})\sigma_x^2 \right) R^2 \right) \end{aligned} \right\}, \\ \epsilon &= \frac{\eta c}{2\tilde{B}}, \quad \phi_{min} = \exp \left( -\frac{1}{4\sigma_x^2} \left( 1 + \delta\gamma + L_{X,max} + \delta L_{C,max} + (\epsilon C_{f,1} + C_{f,2})\sigma_x^2 \right) R^2 \right) \\ C_1 &= \frac{1}{\min(\tilde{\delta}, 1)} \frac{2}{\phi_{min}} \max \left( \frac{16(1+\tilde{\delta}^2)}{\epsilon \min(\gamma, 1)}, 1 \right), \quad C_2 = \frac{1}{\min(\tilde{\delta}^2, 1)} \frac{2}{\phi_{min}} \max \left( \frac{16(1+\tilde{\delta}^2)}{\epsilon \min(\gamma, 1)}, 1 \right) \end{align*} $$
## 2.3 Modification of the function

Let $C_{init,exp} > 0$, $\tilde{a} > 0$ and consider an initial measure $\mu_0$ on $\mathbb{R}^2$ which satisfies $E_{\mu_0}(e^{\tilde{a}(|X|+|C|)}) \le C_{init,exp}$, where $(X, C)$ is distributed according to $\mu_0$.

For technical reasons, we need a greater restoring force by the Lyapunov function than the one given in Lemma 2.2. We thus modify it to obtain estimates such as (2.14) and (2.18) below.

Let $a > 0$, such that $a \le \tilde{a}/(4\sqrt{2}\max(\sqrt{\gamma}, 1))$. This choice of $a$ is only necessary for further Propositions and Lemmas, in Section 3.

Let us consider for all $z \in \mathbb{R}^2$,

$$ \tilde{H}(z) = \int_{0}^{H(z)} \exp(a\sqrt{u}) du = \frac{2}{a^2} \exp(a\sqrt{H(z)}) (a\sqrt{H(z)} - 1) + \frac{2}{a^2}. \quad (2.8) $$

Direct calculations yield the following technical lemma.

**Lemma 2.5.** We have, for all $z \in \mathbb{R}^2$

$$ H(z) \exp(a\sqrt{H(z)}) \ge \tilde{H}(z) \ge \exp(a\sqrt{H(z)}) - \frac{2}{a^2}\left(\exp\left(\frac{a^2}{2}\right) - 1\right), \quad (2.9) $$

$$ \frac{2}{a}\sqrt{H(z)} \exp(a\sqrt{H(z)}) \ge \tilde{H}(z) \ge \frac{1}{a}\sqrt{H(z)} \exp(a\sqrt{H(z)}) - \frac{1}{a^2}(e-2), \quad (2.10) $$

$$ \tilde{H}(z) \ge H(z). \quad (2.11) $$

We may calculate, using Lemma 2.1 and Equation (2.3)

$$ \mathcal{L}_{\mu}(\tilde{H}) = \exp(a\sqrt{H}) \mathcal{L}_{\mu}H + \frac{1}{2} \frac{a}{2\sqrt{H}} \exp(a\sqrt{H}) (|\sigma_X \partial_X H|^2 + |\sigma_C \partial_C H|^2) \quad (2.12) $$

$$ \le \exp(a\sqrt{H}) \left( B + (\alpha_X L_X + \beta_X L_C) \mathbb{E}_{\mu}(|X|)^2 \right. \\ \left. + (\alpha_C L_X + \beta_C L_C) \mathbb{E}_{\mu}(|C|^2 - \lambda H) \right) \\ + \frac{1}{2} \max(\sigma_X^2, \sigma_C^2) \max(\gamma, 1) a\sqrt{H} \exp(a\sqrt{H}) $$

$$ \le \exp(a\sqrt{H}) \left( B + \frac{\left(\frac{1}{2}\max(\sigma_X^2, \sigma_C^2)\max(\gamma, 1)\right)^2 a^2}{2\lambda} + (\alpha_X L_X + \beta_X L_C) \mathbb{E}_{\mu}(|X|)^2 \right. \\ \left. + (\alpha_C L_X + \beta_C L_C) \mathbb{E}_{\mu}(|C|^2 - \frac{\lambda}{2}H) \right), \quad (2.13) $$

where for this last inequality we used Young's inequality

$$ \frac{1}{2} \max(\sigma_X^2, \sigma_C^2) \max(\gamma, 1) a\sqrt{H} \le \frac{\lambda}{2}H + \frac{\left(\frac{1}{2} \max(\sigma_X^2, \sigma_C^2) \max(\gamma, 1)\right)^2 a^2}{2\lambda}. $$

Notice that (2.13) ensures that this new Lyapunov function also tends to bring back particles into a compact set, and at an even greater rate. This new rate $H \exp(\sqrt{H})$ however comes at a cost: the initial condition must have a finite exponential moment, and no longer just have a finite second moment. First, by Lemma 2.4, $E(\bar{X}_t)^2 + E(\bar{C}_t)^2 \le C_{init,2}$. Furthermore, the function $h \mapsto \exp(a\sqrt{h}) (B - \frac{\lambda}{4}h)$ is bounded from above for $h \ge 0$. We therefore obtain from (2.13) the existence of $\tilde{B}$ such that

$$ \mathcal{L}_{\bar{\mu}_t}(\tilde{H}(\bar{Z}_t^i)) \le \tilde{B} - \frac{\lambda}{4} \left( H(\bar{Z}_t^i) \exp\left(a\sqrt{H(\bar{Z}_t^i)}\right) \right) \quad (2.14) $$

$$ \frac{d}{dt} \mathbb{E} \tilde{H}(\bar{Z}_t^i) \le \tilde{B} - \frac{\lambda}{4} \mathbb{E} \left( H(\bar{Z}_t^i) \exp\left(a\sqrt{H(\bar{Z}_t^i)}\right) \right) \quad (2.15) $$

$$ \text{and} \quad \frac{d}{dt} \mathbb{E} \tilde{H}(\bar{Z}_t^i) \le \tilde{B} - \frac{\lambda}{4} \mathbb{E} \tilde{H}(\bar{Z}_t^i), \quad (2.16) $$
$$C_z = \frac{2}{\phi_{min}} \max \left( 1, \frac{4}{\epsilon} \max \left( \sqrt{\frac{1}{\gamma}}, 1 \right) \right).$$

We define $f$ as follows

$$f(r) = \int_{0}^{r \wedge R} \phi(s)g(s)ds, \qquad (2.21)$$

$$\phi(r) = \exp \left( -\frac{1}{4\sigma_x^2} \left( 1 + \delta\gamma + L_X + \delta L_C + (\epsilon C_{f,1} + C_{f,2}) \sigma_x^2 \right) r^2 \right),$$

$$\Phi(s) = \int_{0}^{s} \phi(u)du,$$

$$g(r) = 1 - \frac{c + 2\epsilon \tilde{B}}{\sigma_x^2} \int_{0}^{r} \Phi(s)\phi(s)^{-1}ds.$$

Assume furthermore that $L_X$ and $L_C$, the Lipschitz constants, satisfy

$$L_X \le \min \left( \frac{\lambda}{128C_z}, \frac{\lambda a}{512\epsilon C_z}, \frac{c}{2C_1} \right) \quad \text{and} \quad L_C \le \min \left( \frac{\lambda}{128\delta C_z}, \frac{\lambda a}{512\epsilon\delta C_z}, \frac{c}{2\delta C_1} \right)$$

$$\alpha_X L_X + \beta_X L_C \le \frac{\gamma\lambda}{128} \quad \text{and} \quad \alpha_C L_X + \beta_C L_C \le \frac{\lambda}{128},$$

$$\frac{L_X}{8} + L_C \left(2 + \frac{1}{8}\right) < 1 - \frac{\lambda}{2}.$$

Notice how the bounds on $L_X$ and $L_C$ depend on $c$. This is one of the reasons why we use the *a priori* bounds $L_X \in [0, L_{X,max}]$ and $L_C \in [0, L_{C,max}]$ given in Assumption 1: they allow us to bound $c$ and $\delta$ independently of $L_C$ and $L_X$. We are thus able to begin by choosing an acceptable values for those parameters, before then giving upper bounds on $L_X$ and $L_C$.

We quickly mention that the constants $C_1$, $C_2$ and $C_z$ above come from Lemma 2.6 later. We gather some properties required in the calculations of the proof of Theorem 2 in the following lemma. Again, these properties are the ones motivating the choice of parameters

**Lemma 2.5.** The set of parameters given in Assumption 2 satisfy

* $f$ is $C^2$ on $(0, R)$ such that $f'_+(0) = 1$ and $f'_-(R) > 0$, and constant on $[R, \infty)$. Moreover, $f$ is non-negative, non-decreasing and concave, and for all $s \ge 0$,

$$\min(s, R) f'_-(R) \le f(s) \le \min(s, f(R)) \le \min(s, R).$$

* For all $r \in [0, R]$, $\phi(r) \ge \phi_{min}$ and $g(r) \ge \frac{1}{2}$.

* We have the conditions

$$2f'(R) \ge \exp \left( -\frac{1}{4\sigma_x^2} \left( 1 + \delta\gamma + L_X + \delta L_C + (\epsilon C_{f,1} + C_{f,2}) \sigma_x^2 \right) R^2 \right),$$

$$2c + 4\epsilon \tilde{B} \le \left(1 - L_C - \frac{1 + L_X}{\delta}\right) \min_{r \in [0, R]} \frac{f'(r)r}{f(r)},$$

$$c \le \frac{\lambda}{160} \frac{80\epsilon\tilde{B}}{\frac{\lambda}{\lambda + \frac{80\epsilon\tilde{B}}{\lambda}}}, \quad \frac{1 + L_X}{1 - L_C} < \delta \quad \text{and} \quad \epsilon \le 1.$$

The proof of this lemma is done in Appendix A.3.
where for this last inequality, we used (2.9). While (2.14) and (2.15) will be useful in ensuring a sufficient restoring force, Equation (2.16) gives us a uniform in time bound on $E\tilde{H}(\bar{Z}_t^i)$, provided we have an initial bound. These inequalities are to be understood in the sense of SDEs, where (2.16) should for instance be rigorously written

$$E\tilde{H}(\bar{Z}_t^i) \le E\tilde{H}(\bar{Z}_0^i) + \int_0^t \left(\tilde{B} - \frac{\lambda}{4} E\tilde{H}(\bar{Z}_s^i)\right) ds.$$

Now, for the system of particles, we have, using (2.13), $\forall i, \forall x_i, v_i \in \mathbb{R}^d$,

$$\mathcal{L}^N \tilde{H}(z_i) \le \exp \left( a\sqrt{H(z_i)} \right) \left( \tilde{B} + (\alpha_X L_X + \beta_X L_C) \left( \frac{1}{N} \sum_{j=1}^N |x_j| \right)^2 \right. \\ \left. + (\alpha_C L_X + \beta_C L_C) \left( \frac{1}{N} \sum_{j=1}^N |c_j| \right)^2 - \frac{\lambda}{2} H(z_i) \right).$$

Summing over $i \in \{1, \dots, N\}$, we may calculate

$$\begin{aligned} & (\alpha_X L_X + \beta_X L_C) \sum_{j=1}^{N} \left( \frac{\sum_{j=1}^{N} |x_j|}{N} \right)^2 \sum_{i=1}^{N} \frac{\exp\left(a\sqrt{H(z_i)}\right)}{N} - \frac{\lambda}{16} \sum_{i=1}^{N} \frac{H(z_i) \exp\left(a\sqrt{H(z_i)}\right)}{N} \\ & \le \frac{\lambda}{16} \left( \sum_{i,j=1}^{N} \frac{H(z_i)}{N} \frac{\exp\left(a\sqrt{H(z_j)}\right)}{N} - \sum_{i=1}^{N} \frac{H(z_i) \exp\left(a\sqrt{H(z_i)}\right)}{N} \right) \\ & \le 0. \end{aligned}$$

Here, we used Lemma 2.1, the fact that $\forall x, y \ge 0, xe^{\sqrt{y}} + ye^{\sqrt{x}} - xe^{\sqrt{x}} - ye^{\sqrt{y}} = (e^{\sqrt{x}} - e^{\sqrt{y}})(y - x) \le 0$ and assumed

$$(\alpha_X L_X + \beta_X L_C) \le \frac{\gamma\lambda}{64}.$$

Likewise,

$$\begin{aligned} & (\alpha_C L_X + \beta_C L_C) \sum_{j=1}^{N} \left( \frac{\sum_{j=1}^{N} |c_j|}{N} \right)^2 \sum_{i=1}^{N} \frac{\exp\left(a\sqrt{H(z_i)}\right)}{N} \\ & \quad - \frac{\lambda}{16} \sum_{i=1}^{N} \frac{H(z_i) \exp\left(a\sqrt{H(z_i)}\right)}{N} \le 0, \end{aligned}$$

provided

$$(\alpha_C L_X + \beta_C L_C) \le \frac{\lambda}{64}.$$

There is therefore a constant, which for the sake of clarity we will also denote $\tilde{B}$ (as we may take the maximum of the previous constants), such that we get

$$\begin{aligned} \mathcal{L}^{i,N} \tilde{H}(Z_t^{i,N}) \le & \tilde{B} + (\alpha_X L_X + \beta_X L_C) \left( \frac{\sum_{j=1}^N |X_t^{j,N}|}{N} \right)^2 \exp \left( a \sqrt{H(Z_t^{i,N})} \right) \\ & + (\alpha_C L_X + \beta_C L_C) \left( \frac{\sum_{j=1}^N |C_t^{j,N}|}{N} \right)^2 \exp \left( a \sqrt{H(Z_t^{i,N})} \right) \\ & - \frac{\lambda}{4} H(Z_t^{i,N}) \exp \left( a \sqrt{H(Z_t^{i,N})} \right), \end{aligned} \quad (2.17)$$
## 2.5 Control of the usual distances

As explained previously, we consider a modified semimetrics. For $z = (x, c) \in \mathbb{R}^2$ and $z' = (x', c') \in \mathbb{R}^2$, define

$$r(z, z') = r(x, c, x', c') = |x - x'| + \delta|c - c'|, \qquad (2.22)$$

and let $\rho((z_j, z'_j)_{1 \le j \le N})$ be defined as follows:

$$\rho((z_j, z'_j)_{1 \le j \le N}) = \frac{1}{N} \sum_{i=1}^{N} f(r(z_i, z'_i)) G^i((z_j, z'_j)_j), \qquad (2.23)$$

where for each $i \in \{1, ..., N\}$,

$$G^i((z_j, z'_j)_j) = 1 + \epsilon \tilde{H}(z_i) + \epsilon \tilde{H}(z'_i) + \frac{\epsilon}{N} \sum_{j=1}^{N} \tilde{H}(z_j) + \frac{\epsilon}{N} \sum_{j=1}^{N} \tilde{H}(z'_j). \qquad (2.24)$$

An immediate corollary of the definition and properties of $H$ is that $\rho$ is a quantity on $\mathbb{R}^{4N}$ which controls the usual $L^1$ and $L^2$ distances.

**Lemma 2.6.** The constants $C_1, C_2, C_z > 0$, given in Assumption 2, are such that for all $z = (x, c) \in \mathbb{R}^2$ and $z' = (x', c') \in \mathbb{R}^2$

$$(i) \quad \|z - z'\|_1 \le C_1 f(r(z, z')) (1 + \epsilon \tilde{H}(z) + \epsilon \tilde{H}(z'))$$

$$(ii) \quad \|z - z'\|_2^2 \le C_2 f(r(z, z')) (1 + \epsilon \tilde{H}(z) + \epsilon \tilde{H}(z'))$$

$$(iii) \quad \|z - z'\|_1 \le C_z f(r(z, z')) (1 + \epsilon \sqrt{H(z)} + \epsilon \sqrt{H(z')})$$

The proof of this lemma is postponed to Appendix A.4.

# 3 Proof of Theorem 2 in the case $\sigma_X > 0$

Let $\xi > 0$ be a parameter destined to vanish, and let $\varphi_{sc} : \mathbb{R}^+ \mapsto \mathbb{R}^+$ and $\varphi_{rc} : \mathbb{R}^+ \mapsto \mathbb{R}^+$ be two Lipschitz continuous functions such that

$$\begin{aligned} & \forall x, \quad \varphi_{sc}^2(x) + \varphi_{rc}^2(x) = 1 \\ & \qquad \varphi_{rc}(x) = 1 \quad \text{if } \xi \le x \le R \\ & \qquad \varphi_{rc}(x) = 0 \quad \text{if } x \le \frac{\xi}{2} \quad \text{or } x \ge R + \xi. \end{aligned}$$

Intuitively, $\varphi_{rc}$ represents the region of space in which we consider a reflection coupling, and $\varphi_{sc}$ the one in which we consider a synchronous coupling. We thus simultaneously construct the following solutions

$$\left\{ \begin{aligned} dX_t^{i,N} &= (X_t^{i,N} - (X_t^{i,N})^3 - C_t^{i,N} - \alpha)dt + \frac{1}{N} \sum_{j=1}^N K_X(Z_t^i - Z_t^j)dt \\ & \qquad + \sigma_x \varphi_{sc} (|X_t^{i,N} - \bar{X}_t^i|) dB_t^{i,sc,X} + \sigma_x \varphi_{rc} (|X_t^{i,N} - \bar{X}_t^i|) dB_t^{i,rc,X} \\ dC_t^{i,N} &= (\gamma X_t^{i,N} - C_t^{i,N} + \beta)dt + \frac{1}{N} \sum_{j=1}^N K_C(Z_t^i - Z_t^j)dt + \sigma_c dB_t^{i,C}, \end{aligned} \right.$$

and

$$\left\{ \begin{aligned} d\bar{X}_t^i &= (\bar{X}_t^i - (\bar{X}_t^i)^3 - \bar{C}_t^i - \alpha)dt + K_X * \bar{\mu}_t(\bar{Z}_t^i)dt \\ & \qquad + \sigma_x \varphi_{sc} (|X_t^{i,N} - \bar{X}_t^i|) dB_t^{i,sc,X} - \sigma_x \varphi_{rc} (|X_t^{i,N} - \bar{X}_t^i|) dB_t^{i,rc,X} \\ d\bar{C}_t^i &= (\gamma \bar{X}_t^i - \bar{C}_t^i + \beta)dt + K_C * \bar{\mu}_t(\bar{Z}_t^i)dt + \sigma_c dB_t^{i,C}. \end{aligned} \right.$$

Notice that we consider a symmetric coupling on the dynamics of $C$.
$$ \mathcal{L}^N \left( \frac{1}{N} \sum_{i=1}^{N} \tilde{H}(Z_t^{i,N}) \right) \leq \tilde{B} - \frac{\lambda}{4} \left( \frac{1}{N} \sum_{i=1}^{N} H(Z_t^{i,N}) \exp \left( a \sqrt{H(Z_t^{i,N})} \right) \right), \quad (2.18) $$

and

$$ \mathcal{L}^N \left( \frac{1}{N} \sum_{i=1}^{N} \tilde{H}(Z_t^{i,N}) \right) \leq \tilde{B} - \frac{\lambda}{4} \left( \frac{1}{N} \sum_{i=1}^{N} \tilde{H}(Z_t^{i,N}) \right). \quad (2.19) $$

Once again, (2.17) and (2.18) will be useful in ensuring a sufficient restoring force, and (2.19) yields a uniform in time bound on the expectation of $\tilde{H}(Z_t^{i,N})$, since by ex-changeability of the particles, $\mathbb{E}(\frac{1}{N} \sum_{j=1}^{N} \tilde{H}(Z_t^{j,N})) = \mathbb{E}(\tilde{H}(Z_t^{i,N}))$.

## 2.4 Parameters

We start by fixing the values of some parameters. The somewhat intricate expressions in this section are dictated by the computations arising in the proofs later on. They are somewhat roughly chosen and far from optimal as we only wish to convey the fact that every constant is explicit. On first reading, the exact choice of parameters can and should be skipped, as they are only meant to satisfy Lemma 2.6, which is the crucial Lemma of this subsection.

Recall $\alpha_X, \beta_X, \alpha_C$ and $\beta_C$ given in Lemma 2.2. $a > 0$ is fixed from the last Subsection and the definition of $\tilde{H}$, and $\lambda$ and $\tilde{B}$ are obtained from the same Subsection.

Given any $\eta > 4$ and $\tilde{\delta} > 0$, consider the following set of parameters

$$ \begin{aligned} \delta &= (1 + \tilde{\delta}) \frac{1 + L_{X,\max}}{1 - L_{C,\max}}, \quad R_0 = \sqrt{\frac{1280 \tilde{B}}{\lambda \min(\gamma, 1)}}, \quad R = \sqrt{1 + \tilde{\delta}^2} R_0, \\ C_{f,1} &= 16 \left( \left( \gamma + a \left( \beta + \frac{\alpha}{\delta} \right) \sqrt{2 \max(\gamma, 1)} \right) \frac{e^{a^2/2} - 1}{a^2} + \sqrt{2 \max(\gamma, 1)} \left( \sqrt{\gamma} + \frac{1}{\delta} \right) (e-2) \right), \\ C_{f,2} &= 4 \left( \gamma + \left( a \left( \beta + \frac{\alpha}{\delta} \right) + 2a^2 \left( \sqrt{\gamma} + \frac{1}{\delta} \right) \right) \sqrt{2 \max(\gamma, 1)} \right), \\ c &= \min \left\{ \frac{2\tilde{B}}{\eta}, \frac{\lambda}{160} \frac{\eta - 4}{\eta}, \frac{\min\left(\frac{\sigma_X}{\sqrt{\pi R}}, 1 - L_{C,\max} - \frac{1+L_{X,\max}}{\delta}\right)}{2(1+\eta)} \times \exp\left(-\frac{1}{4\sigma_X^2}\left(1+\delta\gamma + L_{X,\max} + \delta L_{C,\max} + (C_{f,1} + C_{f,2})\sigma_X^2\right)R^2\right) \right\}, \\ \epsilon &= \frac{\eta c}{2\tilde{B}}, \phi_{\min} = \exp\left(-\frac{1}{4\sigma_X^2}(1+\delta\gamma + L_{X,\max} + \delta L_{C,\max} + (\epsilon C_{f,1} + C_{f,2})\sigma_X^2)R^2\right), \\ C_1 &= \frac{1}{\min(\delta, 1)} \frac{2}{\phi_{\min}} \max\left(\frac{16(1+\delta^2)}{\epsilon \min(\gamma, 1)}, 1\right), C_2 = \frac{1}{\min(\delta^2, 1)} \frac{2}{\phi_{\min}} \max\left(\frac{16(1+\delta^2)}{\epsilon \min(\gamma, 1)}, 1\right), \\ C_z &= \frac{2}{\phi_{\min}} \max\left(1, \frac{4}{\epsilon} \max\left(\sqrt{\frac{1}{\gamma}}, 1\right)\right). \end{aligned} $$

We define $f$ as follows

$$ \begin{aligned} f(r) &= \int_{0}^{r \wedge R} \phi(s) g(s) ds, \quad (2.20) \\ \phi(r) &= \exp\left(-\frac{1}{4\sigma_X^2}\left(1+\delta\gamma + L_X + \delta L_C + (\epsilon C_{f,1} + C_{f,2})\sigma_X^2\right) r^2\right), \end{aligned} $$
## 3.1 Main proof and results

**Proposition 3.1.** We denote $r_t^i = r(Z_t^{i,N}, \bar{Z}_t^i)$ and $G_t^i = G^i((Z_t^{j,N})_j, (\bar{Z}_t^j)_j)$. For all $c \in \mathbb{R}$, for each $i \in \{1, \dots, N\}$, we have

$$d(e^{ct}f(r_t^i)G_t^i) \le e^{ct}K_t^i dt + dM_t^i, \quad (3.1)$$

where $M_t^i$ is a continuous local martingale and $K_t^i$ can be written as

$$K_t^i = \tilde{K}_t^i + I_t^{1,i} + I_t^{2,i} + I_t^{3,i}. \quad (3.2)$$

We define $\tilde{K}_t^i, I_t^{1,i}, I_t^{2,i}, I_t^{3,i}$ and $I_t^{4,i}$ as followed:

$$\begin{aligned} \tilde{K}_t^i = & G_t^i \left[ 2cf(r_t^i) + \frac{1}{2} f''(r_t^i) \left( 2\sigma_x^2 \varphi_{rc} \left( |X_t^{i,N} - \bar{X}_t^i| \right)^2 \right. \right. \\ & \left. \left. + f'(r_t^i) \left( (1 + \gamma\delta + L_X + \delta L_C) |X_t^{i,N} - \bar{X}_t^i| - |(X_t^{i,N})^3 - (\bar{X}_t^i)^3| + (1 + L_X + \delta L_C - \delta) |C_t^{i,N} - \bar{C}_t^i| \right. \right. \\ & \left. \left. + (\epsilon C_{f,1} + C_{f,2}) \sigma_x^2 \varphi_{rc} \left( |X_t^{i,N} - \bar{X}_t^i| \right)^2 r_t^i \right) \right] \\ & + \epsilon f(r_t^i) \left( 4\tilde{B} - \frac{\lambda}{16} \tilde{H}(\bar{Z}_t^i) - \frac{\lambda}{16} \tilde{H}(Z_t^{i,N}) - \frac{\lambda}{16N} \sum_{j=1}^N \tilde{H}(\bar{Z}_t^j) - \frac{\lambda}{16N} \sum_{j=1}^N \tilde{H}(Z_t^{j,N}) \right), \end{aligned} \quad (3.3)$$

$$I_t^{1,i} = G_t^i f'(r_t^i) \left( \left| \frac{1}{N} \sum_{j=1}^N K_X(\bar{Z}_t^i - \bar{Z}_t^j) - K_X * \bar{\mu}_t(\bar{Z}_t^i) \right| \right) + \delta G_t^i f'(r_t^i) \left( \left| \frac{1}{N} \sum_{j=1}^N K_C(\bar{Z}_t^i - \bar{Z}_t^j) - K_C * \bar{\mu}_t(\bar{Z}_t^i) \right| \right), \quad (3.4)$$

$$\begin{aligned} I_t^{2,i} = & G_t^i f'(r_t^i) \left( \frac{L_X}{N} \left( \sum_{j=1}^N \|Z_t^{j,N} - \bar{Z}_t^j\|_1 \right) \right) + \delta G_t^i f'(r_t^i) \left( \frac{L_C}{N} \left( \sum_{j=1}^N \|Z_t^{j,N} - \bar{Z}_t^j\|_1 \right) \right) \\ & - cf(r_t^i)G_t^i - \epsilon f(r_t^i) \left[ \frac{\lambda}{16} H(\bar{Z}_t^i) \exp \left( a\sqrt{H(\bar{Z}_t^i)} \right) + \frac{\lambda}{16} H(Z_t^{i,N}) \exp \left( a\sqrt{H(Z_t^{i,N})} \right) \right] \\ & - \epsilon f(r_t^i) \left[ \frac{\lambda}{16N} \sum_{j=1}^N H(\bar{Z}_t^j) \exp \left( a\sqrt{H(\bar{Z}_t^j)} \right) + \frac{\lambda}{16N} \sum_{j=1}^N H(Z_t^{j,N}) \exp \left( a\sqrt{H(Z_t^{j,N})} \right) \right], \end{aligned} \quad (3.5)$$

$$\begin{aligned} \text{and } I_t^{3,i} = & \epsilon f(r_t^i) \left( (\alpha_X L_X + \beta_X L_C) \left( \frac{\sum_{j=1}^N |X_t^{j,N}|}{N} \right)^2 \exp \left( a\sqrt{H(Z_t^{i,N})} \right) \right. \\ & \left. + (\alpha_C L_X + \beta_C L_C) \left( \frac{\sum_{j=1}^N |C_t^{j,N}|}{N} \right)^2 \exp \left( a\sqrt{H(Z_t^{i,N})} \right) \right. \\ & \left. - \frac{\lambda}{16} H(Z_t^{i,N}) \exp \left( a\sqrt{H(Z_t^{i,N})} \right) - \frac{\lambda}{16N} \sum_{j=1}^N H(Z_t^{j,N}) \exp \left( a\sqrt{H(Z_t^{j,N})} \right) \right). \end{aligned} \quad (3.6)$$

We need a control on $\mathbb{E}(G_t^i)$, which is a consequence of Lyapunov's properties on $\tilde{H}$ and the initial assumption of the Theorem 2. A proof is given in Appendix A.5.

**Lemma 3.1.** There exists $C_{G,1}$ and $C_{G,2}$, such that for each $i \le N$, for all $t > 0$, we have

$$\mathbb{E}(G_t^i) \le C_{G,1} \quad \text{and} \quad \mathbb{E}[(G_t^i)^2] \le C_{G,2}.$$
$$
\begin{align*}
\Phi(s) &= \int_{0}^{s} \phi(u) du, \\
g(r) &= 1 - \frac{c + 2\epsilon \tilde{B}}{\sigma_{X}^{2}} \int_{0}^{r} \Phi(s) \phi(s)^{-1} ds.
\end{align*}
$$

Assume furthermore that $L_X$ and $L_C$, the Lipschitz constants, satisfy

$$
L_X \le \min \left( \frac{\lambda}{128C_z}, \frac{\lambda a}{512\epsilon C_z}, \frac{c}{2C_1} \right) \quad \text{and} \quad L_C \le \min \left( \frac{\lambda}{128\delta C_z}, \frac{\lambda a}{512\epsilon\delta C_z}, \frac{c}{2\delta C_1} \right), \quad (2.21)
$$

$$
\alpha_X L_X + \beta_X L_C \le \frac{\gamma \lambda}{128} \quad \text{and} \quad \alpha_C L_X + \beta_C L_C \le \frac{\lambda}{128}, \qquad (2.22)
$$

$$
\frac{L_X}{8} + L_C \left(2 + \frac{1}{8}\right) < 1 - \frac{\lambda}{2}. \qquad (2.23)
$$

Notice how the bounds on $L_X$ and $L_C$ depend on $c$. This is one of the reasons why we use the a priori bounds $L_X \in [0, L_{X,max}]$ and $L_C \in [0, L_{C,max}]$ given in the assumptions of Theorem 1.4: they allow us to bound $c$ and $\delta$ independently of $L_C$ and $L_X$. We are thus able to begin by choosing acceptable values for those parameters, before then giving upper bounds on $L_X$ and $L_C$. The condition of taking $L_X$ and $L_C$ small enough (the condition $L_X < c^K$ and $L_C < c^K$ for a well chosen $c^K$, given in Theorem 1.4) is necessary to satisfy the conditions of (2.21), (2.22) and (2.23).

We quickly mention that the constants $C_1$, $C_2$ and $C_z$ above come from Lemma 2.7 later. We gather some properties required in the calculations of the proof of Theo-
rem 1.4 in the following lemma. Again, these properties are the ones motivating the
choice of parameters

**Lemma 2.6.** The set of parameters given in Subsection 2.4 satisfy

* $f$ is $C^2$ on $(0, R)$ such that $f'_+(0) = 1$ and $f'_-(R) > 0$, and constant on $[R, \infty)$.
  Moreover, $f$ is non-negative, non-decreasing, and concave, and for all $s \ge 0$,

$$
\min(s, R) f'_{-}(R) \leq f(s) \leq \min(s, f(R)) \leq \min(s, R).
$$

* For all $r \in [0, R]$, $\phi(r) \ge \phi_{\min}$ and $g(r) \ge \frac{1}{2}$.

• We have the conditions

$$
\frac{2'}{f}(R) \ge \exp \left( -\frac{1}{4\sigma_X^2} \left( 1 + \delta\gamma + L_X + \delta L_C + (\epsilon C_{f,1} + C_{f,2}) \sigma_X^2 \right) R^2 \right),
$$

$$
2c + 4\epsilon\tilde{B} \le \left(1 - L_C - \frac{1+L_X}{\delta}\right) \min_{r \in (0, R]} \frac{f'(r)r}{f(r)},
$$

$$
c \le \frac{\lambda}{160} \frac{\frac{80\epsilon\tilde{B}}{\lambda}}{1 + \frac{80\epsilon\tilde{B}}{\lambda}}, \quad \frac{1+L_X}{1-L_C} < \delta \quad \text{and} \quad \epsilon \le 1.
$$

The proof of this lemma is done in Appendix A.4.

**2.5 Control of the usual distances**

As explained previously, we consider a modified semi-metric. For $z = (x, c) \in \mathbb{R}^2$ and $z' = (x', c') \in \mathbb{R}^2$, define

$$
r(z, z') = r(x, c, x', c') = |x - x'| + \delta|c - c'|, \quad (2.24)
$$

where $\delta$ is given in Subsection 2.4, and let $\rho((z_j, z'_j)_{1\le j \le N})$ be defined as follows

$$
\rho((z_j, z'_j)_{1 \le j \le N}) = \frac{1}{N} \sum_{i=1}^{N} f(r(z_i, z'_i)) G^i((z_j, z'_j)_j), \quad (2.25)
$$
Each of the terms given in Proposition 3.1 will be controlled in a different way. The following lemmas summarize it. The first term, $\tilde{K}_t^i$, contains the various behaviors we have previously identified: we deal with it either through a synchronous coupling (when the deterministic drift is contracting), or through a reflection coupling (notice the second derivative $f$" which will provide contraction provided $f$ is sufficiently concave). Finally, notice the effect of Lyapunov function $\tilde{H}$ which yields a restoring force.

**Lemma 3.2.** For each $i \le N$, for all $t > 0$,

$$ \mathbb{E}\tilde{K}_t^i \le \xi \left( 2 + \delta\gamma + L_X + \delta L_C - L_C - \frac{1+L_X}{\delta} \right) \mathbb{E}G_t^i. \quad (3.7) $$

The interaction term $\frac{1}{N}K_X(Z_t^{j,N} - Z_t^{i,N}) - K_X * \bar{\mu}_t(\bar{Z}_t^i)$ can be decomposed into $\frac{1}{N}K_X(\bar{Z}_t^j - \bar{Z}_t^i) - K_X * \bar{\mu}_t(\bar{Z}_t^i)$ and $\frac{1}{N}\sum [K_X(Z_t^{j,N} - Z_t^{i,N}) - K_X(\bar{Z}_t^j - \bar{Z}_t^i)]$. The first part, which is in $I_t^{1,i}$, is dealt with using some form of law of large number in a similar way as what has been done in the proof of Theorem 1.

**Lemma 3.3.** For each $i \le N$, for all $t > 0$,

$$ \mathbb{E}(I_t^{1,i}) \le 4\sqrt{\frac{C_{init,2}C_{G,2}}{N}}(L_X + L_C), \quad (3.8) $$

where $C_{G,2}$ is defined in Lemma 3.1 and $C_{init,2}$ is defined in Lemma 2.3.

$I_t^{2,i}$ contains the leftovers of this decomposition and some of the additional terms of the Lyapunov function.

**Lemma 3.4.** For all $t > 0$,

$$ \frac{1}{N} \sum_{i=1}^{N} I_t^{2,i} \le 0. \quad (3.9) $$

Finally, $I_t^{3,i}$ deals with the non linearity appearing in the dynamics of the Lyapunov function, and will be non positive for values of $L_X$ and $L_C$ sufficiently small. It is also here we justify adding the last two terms in (2.24).

**Lemma 3.5.** For each $i \le N$, for all $t > 0$,

$$ I_t^{3,i} \le 0. \quad (3.10) $$

*Proof of Theorem 2.* With these four Lemmas, we can calculate

$$ \begin{aligned} \frac{1}{N} \sum_{i=1}^{N} \mathbb{E}K_t^i &= \frac{1}{N} \sum_{i=1}^{N} \mathbb{E}\tilde{K}_t^i + \frac{1}{N} \sum_{i=1}^{N} \mathbb{E}I_t^{1,i} + \frac{1}{N} \sum_{i=1}^{N} \mathbb{E}I_t^{2,i} + \frac{1}{N} \sum_{i=1}^{N} \mathbb{E}I_t^{3,i} \\ &\le \frac{1}{N} \sum_{i=1}^{N} \xi \left( 2 + \delta\gamma + L_X + \delta L_C - L_C - \frac{1+L_X}{\delta} \right) \mathbb{E}G_t^i + \frac{1}{N} \sum_{i=1}^{N} 4\sqrt{\frac{C_{init,2}C_{G,2}}{N}}(L_X + L_C) \\ &\le \xi \left( 2 + \delta\gamma + L_X + \delta L_C - L_C - \frac{1+L_X}{\delta} \right) \frac{1}{N} \sum_{i=1}^{N} \mathbb{E}G_t^i + 4\sqrt{\frac{C_{init,2}C_{G,2}}{N}}(L_X + L_C) \end{aligned} $$

Since by Lemma 3.1, we have $\frac{1}{N} \sum_{i=1}^{N} \mathbb{E}G_t^i \le C_{G,1}$, we obtain

$$ \frac{1}{N} \sum_{i=1}^{N} \mathbb{E}K_t^i \le \xi A + (L_X + L_C) \frac{B}{\sqrt{N}} $$
where for each $i \in \{1, \dots, N\}$,

$$G^i((z_j, z'_j)_j) = 1 + \epsilon \tilde{H}(z_i) + \epsilon \tilde{H}(z'_i) + \frac{\epsilon}{N} \sum_{j=1}^{N} \tilde{H}(z_j) + \frac{\epsilon}{N} \sum_{j=1}^{N} \tilde{H}(z'_j). \quad (2.26)$$

An immediate corollary of the definition and properties of $H$ is that $\rho$ is a quantity on $\mathbb{R}^{4N}$ which controls the usual $L^1$ and $L^2$ distances.

**Lemma 2.7.** The constants $C_1, C_2, C_z > 0$, given in Subsection 2.4, are such that for all $z = (x, c) \in \mathbb{R}^2$ and $z' = (x', c') \in \mathbb{R}^2$

$$(i) \quad \|z - z'\|_1 \le C_1 f(r(z, z')) (1 + \epsilon \tilde{H}(z) + \epsilon \tilde{H}(z')),$$

$$(ii) \quad \|z - z'\|_2^2 \le C_2 f(r(z, z')) (1 + \epsilon \tilde{H}(z) + \epsilon \tilde{H}(z')),$$

$$(iii) \quad \|z - z'\|_1 \le C_z f(r(z, z')) (1 + \epsilon \sqrt{H(z)} + \epsilon \sqrt{H(z')}).$$

The proof of this lemma is postponed to Appendix A.5.

### 3 Proof of Theorem 1.4 in the case $\sigma_X > 0$

Let $\xi > 0$ be a parameter destined for vanishing, and let $\varphi_{sc} : \mathbb{R}^+ \mapsto \mathbb{R}^+$ and $\varphi_{rc} : \mathbb{R}^+ \mapsto \mathbb{R}^+$ be two Lipschitz continuous functions such that

$$\begin{align}
\forall x, \quad & \varphi_{sc}^2(x) + \varphi_{rc}^2(x) = 1, \tag{3.1} \\
& \varphi_{rc}(x) = 1 \text{ if } \xi \le x \le R, \nonumber \\
& \varphi_{rc}(x) = 0 \text{ if } x \le \frac{\xi}{2} \text{ or } x \ge R + \xi. \nonumber
\end{align}$$

Intuitively, $\varphi_{rc}$ represents the region of space in which we consider a reflection coupling, and $\varphi_{sc}$ the one in which we consider a synchronous coupling. In reality, we would like to consider $\varphi_{sc}$ and $\varphi_{rc}$ indicator functions of the regions of space. However, we need to consider a Lipschitz approximation of indicator functions to ensure continuity (to apply Itô's calculus) and the strong existence and uniqueness of the stochastic processes. We thus simultaneously construct the following solutions, for $1 \le i \le N$

$$\left\{
\begin{aligned}
dX_t^{i,N} &= (X_t^{i,N} - (X_t^{i,N})^3 - C_t^{i,N} - \alpha)dt + \frac{1}{N} \sum_{j=1}^{N} K_X(Z_t^{i,N} - Z_t^{j,N})dt \\
&\quad + \sigma_X \varphi_{sc} (|X_t^{i,N} - \bar{X}_t^i|) dB_t^{i,sc,X} + \sigma_X \varphi_{rc} (|X_t^{i,N} - \bar{X}_t^i|) dB_t^{i,rc,X} \\
dC_t^{i,N} &= (\gamma X_t^{i,N} - C_t^{i,N} + \beta)dt + \frac{1}{N} \sum_{j=1}^{N} K_C(Z_t^{i,N} - Z_t^{j,N})dt + \sigma_C dB_t^{i,C},
\end{aligned}
\right.$$

and

$$\left\{
\begin{aligned}
d\bar{X}_t^i &= (\bar{X}_t^i - (\bar{X}_t^i)^3 - \bar{C}_t^i - \alpha)dt + K_X * \bar{\mu}_t(\bar{Z}_t^i)dt \\
&\quad + \sigma_X \varphi_{sc} (|X_t^{i,N} - \bar{X}_t^i|) dB_t^{i,sc,X} - \sigma_X \varphi_{rc} (|X_t^{i,N} - \bar{X}_t^i|) dB_t^{i,rc,X} \\
d\bar{C}_t^i &= (\gamma \bar{X}_t^i - \bar{C}_t^i + \beta)dt + K_C * \bar{\mu}_t(\bar{Z}_t^i)dt + \sigma_C dB_t^{i,C},
\end{aligned}
\right.$$

where $(B^{i,sc,X})_i$ and $(B^{i,rc,X})_i$ are independent Brownian motions (also independent of $(B^{i,C})_i$). Notice that we consider a symmetric coupling on the dynamics of $C$. By Levy's characterization of Brownian motion, using (3.1), we thus construct a solution of (1.1) and $N$ independent copies of a solution of (1.2).
where A and B are constants.

For all initial couplings such that $\mathbb{E}\rho((Z_0^j, \bar{Z}_0^j)_{1\le j\le N}) < \infty$, by taking the expectation of (3.1) along a sequence of increasing localizing stopping times, we have thanks to Fatou's lemma

$$ 
\begin{aligned}
e^{ct}\mathbb{E}\left(\rho\left((Z_t^j, \bar{Z}_t^j)_{1\le j\le N}\right)\right) &\le \mathbb{E}\left(\rho\left((Z_0^j, \bar{Z}_0^j)_{1\le j\le N}\right)\right) + \xi A \int_0^t e^{cs} ds + (L_X + L_C) \frac{B}{\sqrt{N}} \int_0^t e^{cs} ds \\
&\le \mathbb{E}\left(\rho\left((Z_0^j, \bar{Z}_0^j)_{1\le j\le N}\right)\right) + \xi A \frac{e^{ct} - 1}{c} + (L_X + L_C) \frac{B}{\sqrt{N}} \frac{e^{ct} - 1}{c}.
\end{aligned}
$$

We obtain

$$ \mathbb{E}\left(\rho\left((Z_t^j, \bar{Z}_t^j)_{1\le j\le N}\right)\right) \le \mathbb{E}\left(\rho\left((Z_0^j, \bar{Z}_0^j)_{1\le j\le N}\right)\right) e^{-ct} + \frac{\xi A}{c} (1 - e^{-ct}) + \frac{(L_X + L_C)B}{c} \frac{1}{\sqrt{N}} (1 - e^{-ct}). $$

By using the exchangeability of the particles, we have $\mathbb{E}(\rho((Z_t^j, \bar{Z}_t^j)_{1\le j\le N})) = \mathbb{E}(\frac{1}{N}\sum_{i=1}^N f(r_t^i)G_t^i) = \mathbb{E}(\frac{1}{k}\sum_{i=1}^k f(r_t^i)G_t^i)$ for all $k \in \mathbb{N}$. Then

$$ \mathbb{E}\left(\sum_{i=1}^{k} f(r_t^i) G_t^i\right) = k \mathbb{E}\left(\rho\left((Z_t^j, \bar{Z}_t^j)_{1\le j\le N}\right)\right). $$

Let $\mu_0$ a measure on $\mathbb{R}^2$, $\mu_t^{k,N}$ the marginal distribution at time $t$ of the first $k$ neurons $((X_t^1, C_t^1), \dots, (X_t^k, C_t^k))$ of an $N$ particles system (1.1) with initial distribution $(\mu_0)^{\otimes N}$, and $\bar{\mu}_t$ is a solution of (1.2) with initial distribution $\mu_0$. This implies $\mathbb{E}(\rho((Z_0^j, \bar{Z}_0^j)_{1\le j\le N})) = 0$. By Lemma 2.6, we obtain for Wasserstein 1 distance

$$ 
\begin{aligned}
W_1(\mu_t^{k,N}, \bar{\mu}_t^{\otimes k}) &= \inf \left\{ \mathbb{E}[\|Z^{(k)} - \bar{Z}^{(k)}\|_1], \mathbb{P}_{Z^{(k)}} = \mu_t^{k,N}, \mathbb{P}_{\bar{Z}^{(k)}} = \bar{\mu}_t^{\otimes k} \right\} \\
&= \inf \left\{ \mathbb{E}\left[\sum_{i=1}^k \|Z_t^{i,N} - \bar{Z}_t^i\|_1\right], \mathbb{P}_{(Z_t^{i,N})_i} = \mu_t^{k,N}, \mathbb{P}_{(\bar{Z}_t^i)_i} = \bar{\mu}_t^{\otimes k} \right\} \\
&\le \inf \left\{ C_1 \mathbb{E}\left[\sum_{i=1}^k f(r_t^i) G_t^i\right], \mathbb{P}_{(Z_t^{i,N})_i} = \mu_t^{k,N}, \mathbb{P}_{(\bar{Z}_t^i)_i} = \bar{\mu}_t^{\otimes k} \right\} \\
&\le \inf \left\{ k C_1 \mathbb{E}\left(\rho\left((Z_t^j, \bar{Z}_t^j)_{1\le j\le N}\right)\right), \mathbb{P}_{(Z_t^{i,N})_i} = \mu_t^{k,N}, \mathbb{P}_{(\bar{Z}_t^i)_i} = \bar{\mu}_t^{\otimes k} \right\} \\
&\le \frac{\xi A k C_1}{c} (1 - e^{-ct}) + \frac{(L_X + L_C) B C_1}{c} \frac{k}{\sqrt{N}} (1 - e^{-ct})
\end{aligned}
$$

By taking the limit as $\xi \to 0$ uniformly in time, we obtain the desired result. The same lemma and the same type of calculations yield the result for Wasserstein 2

$$ W_2(\mu_t^{k,N}, \bar{\mu}_t^{\otimes k})^2 \le \frac{k}{\sqrt{N}} C_2 \frac{(L_X + L_C)B}{c}. \quad \square $$

## 3.2 Proof of the decomposition

*Proof of Proposition 3.1.* First, we need to calculate $d(e^{ct}f(r_t^i)G_t^i)$, where we recall

$$ r_t^i = |X_t^{i,N} - \bar{X}_t^i| + \delta |C_t^{i,N} - \bar{C}_t^i| $$
## 3.1 Main proof and results

**Proposition 3.1.** We denote $r_t^i = r(Z_t^{i,N}, \bar{Z}_t^i)$ and $G_t^i = G^i((Z_t^{j,N})_j, (\bar{Z}_t^j)_j)$. For all $c \in \mathbb{R}$, for each $i \in \{1, \dots, N\}$, we have

$$d(e^{ct}f(r_t^i)G_t^i) \le e^{ct}K_t^idt + dM_t^i, \quad (3.2)$$

where $M_t^i$ is a continuous local martingale and $K_t^i$ can be written as

$$K_t^i = \tilde{K}_t^i + I_t^{1,i} + I_t^{2,i} + I_t^{3,i}. \quad (3.3)$$

We define $\tilde{K}_t^i, I_t^{1,i}, I_t^{2,i}$ and $I_t^{3,i}$ as follows

$$\tilde{K}_t^i = G_t^i \left[ 2cf(r_t^i) + \frac{1}{2}f''(r_t^i) \left( 2\sigma_X^2 \varphi_{rc} (|X_t^{i,N} - \bar{X}_t^i|^2) \right. \right. \\ 
+ f'(r_t^i) \left( (1 + \gamma\delta + L_X + \delta L_C) |X_t^{i,N} - \bar{X}_t^i| - |(X_t^{i,N})^3 - (\bar{X}_t^i)^3| \right. \\ 
+ (1 + L_X + \delta L_C - \delta) |C_t^{i,N} - \bar{C}_t^i| + (\epsilon C_{f,1} + C_{f,2}) \sigma_X^2 \varphi_{rc} (|X_t^{i,N} - \bar{X}_t^i|^2 r_t^i) \left. \right] \\ 
+ \epsilon f(r_t^i) \left( 4\tilde{B} - \frac{\lambda}{16}\tilde{H}(\bar{Z}_t^i) - \frac{\lambda}{16}\tilde{H}(Z_t^{i,N}) - \frac{\lambda}{16N} \sum_{j=1}^N \tilde{H}(\bar{Z}_t^j) - \frac{\lambda}{16N} \sum_{j=1}^N \tilde{H}(Z_t^{j,N}) \right), \quad (3.4)$$

$$I_t^{1,i} = G_t^i f'(r_t^i) \left( \left| \frac{1}{N} \sum_{j=1}^N K_X(\bar{Z}_t^i - \bar{Z}_t^j) - K_X * \bar{\mu}_t(\bar{Z}_t^i) \right| \right) \\ 
+ \delta G_t^i f'(r_t^i) \left( \left| \frac{1}{N} \sum_{j=1}^N K_C(\bar{Z}_t^i - \bar{Z}_t^j) - K_C * \bar{\mu}_t(\bar{Z}_t^i) \right| \right), \quad (3.5)$$

$$I_t^{2,i} = G_t^i f'(r_t^i) \left( \frac{L_X}{N} \left( \sum_{j=1}^N \|Z_t^{j,N} - \bar{Z}_t^j\|_1 \right) \right) + \delta G_t^i f'(r_t^i) \left( \frac{L_C}{N} \left( \sum_{j=1}^N \|Z_t^{j,N} - \bar{Z}_t^j\|_1 \right) \right) \\ 
- c f(r_t^i) G_t^i - \epsilon f(r_t^i) \left[ \frac{\lambda}{16} H(\bar{Z}_t^i) \exp \left( a \sqrt{H(\bar{Z}_t^i)} \right) + \frac{\lambda}{16} H(Z_t^{i,N}) \exp \left( a \sqrt{H(Z_t^{i,N})} \right) \right] \\ 
- \epsilon f(r_t^i) \left[ \frac{\lambda}{16N} \sum_{j=1}^N H(\bar{Z}_t^j) \exp \left( a \sqrt{H(\bar{Z}_t^j)} \right) \right. \\ 
\left. + \frac{\lambda}{16N} \sum_{j=1}^N H(Z_t^{j,N}) \exp \left( a \sqrt{H(Z_t^{j,N})} \right) \right], \quad (3.6)$$

$$I_t^{3,i} = \epsilon f(r_t^i) \left( (\alpha_X L_X + \beta_X L_C) \left( \frac{\sum_{j=1}^N |X_t^{j,N}|}{N} \right)^2 \exp \left( a \sqrt{H(Z_t^{i,N})} \right) \right. \\ 
+ (\alpha_C L_X + \beta_C L_C) \left( \frac{\sum_{j=1}^N |C_t^{j,N}|}{N} \right)^2 \exp \left( a \sqrt{H(Z_t^{i,N})} \right) \\ 
\left. - \frac{\lambda}{16} H(Z_t^{i,N}) \exp \left( a \sqrt{H(Z_t^{i,N})} \right) - \frac{\lambda}{16N} \sum_{j=1}^N H(Z_t^{j,N}) \exp \left( a \sqrt{H(Z_t^{j,N})} \right) \right). \quad (3.7)$$

We need to control $E(G_t^i)$. This control is a consequence of Lyapunov's properties on $\tilde{H}$ and the initial assumption of the Theorem 1.4. A proof is given in Appendix A.6.
and

$$G_t^i = 1 + \epsilon \tilde{H}(\bar{Z}_t^i) + \epsilon \tilde{H}(Z_t^{i,N}) + \frac{\epsilon}{N} \sum_{j=1}^{N} \tilde{H}(Z_t^{j,N}) + \frac{\epsilon}{N} \sum_{j=1}^{N} \tilde{H}(\bar{Z}_t^j).$$

We have already calculated $d(X_t^{i,N} - \bar{X}_t^i)$ and $d|X_t^{i,N} - \bar{X}_t^i|$ in the case of symmetric coupling in Subsection 1.4 in (1.7). Here, we need to use Ito's formula and usual convergence lemmas (see Lemma 7 of [DEGZ20]) to take care of the Brownian term. We obtain

$$d|X_t^{i,N} - \bar{X}_t^i| = A_t^X dt + 2\operatorname{sign}(X_t^{i,N} - \bar{X}_t^i) \sigma_x \varphi_{rc} (|X_t^{i,N} - \bar{X}_t^i|) dB_t^{i,rc,X},$$

with

$$A_t^X \le |X_t^{i,N} - \bar{X}_t^i| - |(X_t^{i,N})^3 - (\bar{X}_t^i)^3| + |C_t^{i,N} - \bar{C}_t^i| + \left| \frac{1}{N} \sum_{j=1}^N K_X(Z_t^j - Z_t^i) - K_X * \bar{\mu}_t(\bar{Z}_t^i) \right|.$$

Likewise, as it has already been calculated in (1.8) in Subsection 1.4,

$$d|C_t^{i,N} - \bar{C}_t^i| = A_t^C dt, \quad (3.11)$$

with

$$A_t^C \le \gamma |X_t^{i,N} - \bar{X}_t^i| - |C_t^{i,N} - \bar{C}_t^i| + \left| \frac{1}{N} \sum_{j=1}^N K_C(Z_t^j - Z_t^i) - K_C * \bar{\mu}_t(\bar{Z}_t^i) \right|.$$

Now we have

$$dr_t^i = (A_t^X + \delta A_t^C) dt + 2\operatorname{sign}(X_t^{i,N} - \bar{X}_t^i) \sigma_x \varphi_{rc} (|X_t^{i,N} - \bar{X}_t^i|) dB_t^{i,rc,X}$$

and we deduce with the Ito's formula

$$df(r_t^i) = f'(r_t^i)dr_t^i + \frac{1}{2}f''(r_t^i) \left(2\sigma_x\varphi_{rc}\left(|X_t^{i,N} - \bar{X}_t^i|\right)\right)^2 dt.$$

Finally, for $c > 0$,

$$d(e^{ct} f(r_t^i)) = c e^{ct} f(r_t^i) dt + e^{ct} df(r_t^i).$$

Then, by Ito's formula,

$$\begin{aligned} \frac{1}{\epsilon} dG_t^i = & \left( \mathcal{L}_{\bar{\mu}_t} \tilde{H}(\bar{Z}_t^i) + \mathcal{L}^N \tilde{H}(Z_t^{i,N}) \right) dt \\ & + \sigma_x \varphi_{rc} \left( |X_t^{i,N} - \bar{X}_t^i| \right) \left( \partial_x \tilde{H}(Z_t^{i,N}) - \partial_x \tilde{H}(\bar{Z}_t^i) \right) dB_t^{i,rc,X} \\ & + \sigma_x \varphi_{sc} \left( |X_t^{i,N} - \bar{X}_t^i| \right) \left( \partial_x \tilde{H}(Z_t^{i,N}) + \partial_x \tilde{H}(\bar{Z}_t^i) \right) dB_t^{i,sc,X} \\ & + \sigma_c \left( \partial_c \tilde{H}(Z_t^{i,N}) + \partial_c \tilde{H}(\bar{Z}_t^i) \right) dB_t^{i,C} \\ & + \frac{1}{N} \sum_{j=1}^N \left( \mathcal{L}_{\bar{\mu}_t} \tilde{H}(\bar{Z}_t^j) + \mathcal{L}^N \tilde{H}(Z_t^{j,N}) \right) dt \\ & + \frac{\sigma_x}{N} \sum_{j=1}^N \varphi_{rc} \left( |X_t^{j,N} - \bar{X}_t^j| \right) \left( \partial_x \tilde{H}(Z_t^{j,N}) - \partial_x \tilde{H}(\bar{Z}_t^j) \right) dB_t^{j,rc,X} \end{aligned}$$
**Lemma 3.2.** There exist $C_{G,1}$ and $C_{G,2}$, such that for each $i \le N$, for all $t > 0$, we have

$$E(G_t^i) \le C_{G,1} \quad \text{and} \quad E[(G_t^i)^2] \le C_{G,2}.$$

The decomposition given in the Proposition 3.1 is true for all $c \in \mathbb{R}$. To control exactly the behavior of each term, we will now consider $c$, defined in Subsection 2.4.

Each term given in Proposition 3.1 will be controlled differently. The following lemmas summarize it. The first term, $\tilde{K}_t^i$, contains the various behaviors we have previously identified: we deal with it either through a synchronous coupling (when the deterministic drift is contracting) or through a reflection coupling (notice the second derivative $f''$ which will provide contraction provided $f$ is sufficiently concave). Finally, notice the effect of the Lyapunov function $\tilde{H}$ which yields a restoring force.

**Lemma 3.3.** With the parameters and functions given in Subsection 2.4, for each $i \le N$, for all $t > 0$,

$$E\tilde{K}_t^i \le \xi \left( 2 + \delta\gamma + L_X + \delta L_C - L_C - \frac{1 + L_X}{\delta} \right) EG_t^i. \quad (3.8)$$

The interaction term $\frac{1}{N}\sum_j K_X(Z_t^{j,N} - Z_t^{i,N}) - K_X * \bar{\mu}_t(\bar{Z}_t^i)$ can be decomposed into the following two parts: $\frac{1}{N}\sum_j K_X(\bar{Z}_t^j - \bar{Z}_t^i) - K_X * \bar{\mu}_t(\bar{Z}_t^i)$ and $\frac{1}{N}\sum_j[K_X(Z_t^{j,N} - Z_t^{i,N}) - K_X(\bar{Z}_t^j - \bar{Z}_t^i)]$. The first part, which is in $I_t^{1,i}$, is dealt with using some form of the law of large numbers in a similar way to what has been done in the proof of Theorem 1.3.

**Lemma 3.4.** With the parameters and functions given in Subsection 2.4, for each $i \le N$, for all $t > 0$,

$$E(I_t^{1,i}) \le 4\sqrt{\frac{C_{init,2}C_{G,2}}{N}}(L_X + L_C), \quad (3.9)$$

where $C_{G,2}$ is defined in Lemma 3.2 and $C_{init,2}$ is defined in Lemma 2.4.

$I_t^{2,i}$ contains the leftovers of this decomposition and some of the additional terms of the Lyapunov function.

**Lemma 3.5.** With the parameters and functions given in Subsection 2.4, for all $t > 0$,

$$\frac{1}{N} \sum_{i=1}^{N} I_t^{2,i} \le 0. \quad (3.10)$$

Finally, $I_t^{3,i}$ deals with the non-linearity appearing in the dynamics of the Lyapunov function, and will be non-positive for values of $L_X$ and $L_C$ sufficiently small. It is also here we justify adding the last two terms in (2.26).

**Lemma 3.6.** With the parameters and functions given in Subsection 2.4, for each $i \le N$, for all $t > 0$,

$$I_t^{3,i} \le 0. \quad (3.11)$$

**Proof of Theorem 1.4.** With these four Lemmas, we can calculate

$$\frac{1}{N} \sum_{i=1}^{N} E K_t^i = \frac{1}{N} \sum_{i=1}^{N} E \tilde{K}_t^i + \frac{1}{N} \sum_{i=1}^{N} E I_t^{1,i} + \frac{1}{N} \sum_{i=1}^{N} E I_t^{2,i} + \frac{1}{N} \sum_{i=1}^{N} E I_t^{3,i} \\ \le \xi \left( 2 + \delta\gamma + L_X + \delta L_C - L_C - \frac{1 + L_X}{\delta} \right) \frac{1}{N} \sum_{i=1}^{N} EG_t^i + 4\sqrt{\frac{C_{init,2}C_{G,2}}{N}}(L_X + L_C)$$
$$
\begin{aligned}
&+ \frac{\sigma_x}{N} \sum_{j=1}^{N} \varphi_{sc} (|X_t^{j,N} - \bar{X}_t^j|) (\partial_x \tilde{H}(Z_t^{j,N}) + \partial_x \tilde{H}(\bar{Z}_t^j)) dB_t^{j,sc,X} \\
&+ \frac{\sigma_c}{N} \sum_{j=1}^{N} (\partial_c \tilde{H}(Z_t^{j,N}) + \partial_c \tilde{H}(\bar{Z}_t^j)) dB_t^{j,C}.
\end{aligned}
$$

We finally get

$$
\begin{aligned}
d(e^{ct} f(r_t^i) G_t^i) = & G_t^i d(e^{ct} f(r_t^i)) + e^{ct} f(r_t^i) dG_t^i \\
& + 2\epsilon \left(1 + \frac{1}{N}\right) \sigma_x^2 \varphi_{rc} \left(|X_t^{i,N} - \bar{X}_t^i|\right)^2 \mathrm{sign}(X_t^{i,N} - \bar{X}_t^i) \left(\partial_x \tilde{H}(Z_t^{i,N}) - \partial_x \tilde{H}(\bar{Z}_t^i)\right) e^{ct} f'(r_t^i) dt.
\end{aligned}
$$

Now, we need to use the following Lemma, proven in Appendix A.5, to have a more tractable expression:

**Lemma 3.6. We have the upper bound**

$$
\begin{aligned}
& 2\epsilon \left(1 + \frac{1}{N}\right) \sigma_x^2 \varphi_{rc} \left(|X_t^{i,N} - \bar{X}_t^i|\right)^2 \mathrm{sign}(X_t^{i,N} - \bar{X}_t^i) \left(\partial_x \tilde{H}(Z_t^{i,N}) - \partial_x \tilde{H}(\bar{Z}_t^i)\right) \\
& \leq (\epsilon C_{f,1} + C_{f,2}) \sigma_x^2 \varphi_{rc} \left(|X_t^{i,N} - \bar{X}_t^i|\right)^2 r_t^i G_t^i.
\end{aligned}
$$

Eventually, by denoting the terms in $dB_t^{i,rc,X}$, $dB_t^{i,sc,X}$, $dB_t^{i,C}$, ... as the local martingale $dM_t^i$, we obtain

$$
\begin{aligned}
d(e^{ct} f(r_t^i) G_t^i) \le {}& G_t^i c e^{ct} f(r_t^i) dt + e^{ct} G_t^i f'(r_t^i) (A_t^X + \delta A_t^C) dt \\
& + e^{ct} G_t^i \frac{1}{2} f''(r_t^i) (2 \sigma_x \varphi_{rc} (|X_t^{i,N} - \bar{X}_t^i|))^2 dt \\
& + e^{ct} f(r_t^i) \left( (\mathcal{L}_{\bar{\mu}_t} \tilde{H}(\bar{Z}_t^i) + \mathcal{L}^N \tilde{H}(Z_t^{i,N})) dt + \frac{1}{N} \sum_{j=1}^N (\mathcal{L}_{\bar{\mu}_t} \tilde{H}(\bar{Z}_t^j) + \mathcal{L}^N \tilde{H}(Z_t^{j,N})) dt \right) \\
& + (\epsilon C_{f,1} + C_{f,2}) \sigma_x^2 \varphi_{rc} (|X_t^{i,N} - \bar{X}_t^i|)^2 r_t^i G_t^i e^{ct} f'(r_t^i) dt \\
& + dM_t^i.
\end{aligned}
$$

We use (2.13) to bound $\mathcal{L}_{\bar{\mu}_t} \tilde{H}(\bar{Z}_t^i)$ and (2.18) to bound $\mathcal{L}^N \tilde{H}(Z_t^{i,N})$. The interaction terms in $A_t^X$ and $A_t^C$ are decomposed and we define $I_t^{1,i}$ as follows

$$
I_t^{1,i} = G_t^i f'(r_t^i) \left( \left| \frac{1}{N} \sum_{j=1}^{N} K_X(\bar{Z}_t^i - \bar{Z}_t^j) - K_X * \bar{\mu}_t(\bar{Z}_t^i) \right| \right) + \delta G_t^i f'(r_t^i) \left( \left| \frac{1}{N} \sum_{j=1}^{N} K_C(\bar{Z}_t^i - \bar{Z}_t^j) - K_C * \bar{\mu}_t(\bar{Z}_t^i) \right| \right).
$$

The second part of the decomposition is grouped in $I_t^{2,i}$ with compensating terms, and in particular Lyapunov functions, which appears with the use of (2.13) and (2.18), to control the sum:

$$
\begin{aligned}
I_t^{2,i} = {}& G_t^i f'(r_t^i) \left( \frac{L_X}{N} \left( \sum_{j=1}^N |X_t^{j,N} - \bar{X}_t^j| + |C_t^{j,N} - \bar{C}_t^j| \right) \right) + \delta G_t^i f'(r_t^i) \left( \frac{L_C}{N} \left( \sum_{j=1}^N |X_t^{j,N} - \bar{X}_t^j| + |C_t^{j,N} - \bar{C}_t^j| \right) \right) \\
& - c f(r_t^i) G_t^i - \epsilon f(r_t^i) \left[ \frac{\lambda}{16} H(\bar{Z}_t^i) \exp \left( a \sqrt{H(\bar{Z}_t^i)} \right) + \frac{\lambda}{16} H(Z_t^{i,N}) \exp \left( a \sqrt{H(Z_t^{i,N})} \right) \right] \\
& - \epsilon f(r_t^i) \left[ \frac{\lambda}{16N} \sum_{j=1}^N H(\bar{Z}_t^j) \exp \left( a \sqrt{H(\bar{Z}_t^j)} \right) + \frac{\lambda}{16N} \sum_{j=1}^N H(Z_t^{j,N}) \exp \left( a \sqrt{H(Z_t^{j,N})} \right) \right].
\end{aligned}
$$
Since by Lemma 3.2, we have $\frac{1}{N} \sum_{i=1}^{N} E G_t^i \le C_{G,1}$, we obtain

$$ \frac{1}{N} \sum_{i=1}^{N} E K_t^i \le \xi A + (L_X + L_C) \frac{B}{\sqrt{N}} $$

where A and B are constants.

For all initial couplings such that $E \rho((Z_0^{j,N}, \bar{Z}_0^j)_{1\le j \le N}) < \infty$, by taking the expectation of (3.2) along a sequence of increasing localizing stopping times, we have thanks to Fatou's lemma

$$ e^{ct} E \left( \rho((Z_t^{j,N}, \bar{Z}_t^j)_{1\le j \le N}) \right) \le E \left( \rho((Z_0^{j,N}, \bar{Z}_0^j)_{1\le j \le N}) \right) + \xi A \int_0^t e^{cs} ds \\ + (L_X + L_C) \frac{B}{\sqrt{N}} \int_0^t e^{cs} ds. $$

We obtain

$$ E \left( \rho((Z_t^{j,N}, \bar{Z}_t^j)_{1\le j \le N}) \right) \le E \left( \rho((Z_0^{j,N}, \bar{Z}_0^j)_{1\le j \le N}) \right) e^{-ct} + \frac{\xi A}{c} (1 - e^{-ct}) \\ + \frac{(L_X + L_C)B}{c \sqrt{N}} (1 - e^{-ct}). $$

By using the exchangeability of the particles, we have $E \left( \rho((Z_t^{j,N}, \bar{Z}_t^j)_{1\le j \le N}) \right) = E \left( \frac{1}{N} \sum_{i=1}^{N} f(r_t^i) G_t^i \right) = E \left( \frac{1}{k} \sum_{i=1}^{k} f(r_t^i) G_t^i \right)$ for all $k \in \mathbb{N}$. Then

$$ E \left( \sum_{i=1}^{k} f(r_t^i) G_t^i \right) = k E \left( \rho((Z_t^{j,N}, \bar{Z}_t^j)_{1\le j \le N}) \right). $$

Let $\mu_0$ be a measure on $\mathbb{R}^2$, $\mu_t^{k,N}$ the marginal distribution at time $t$ of the first $k$ neurons $((X_t^{1,N}, C_t^{1,N}), \dots, (X_t^{k,N}, C_t^{k,N}))$ of an $N$-particle system (1.1) with initial distribution $\mu_0^{\otimes N}$, and $\bar{\mu}_t$ a solution of (1.2) with initial distribution $\mu_0$. This implies $E \left( \rho((Z_0^{j,N}, \bar{Z}_0^j)_{1\le j \le N}) \right) = 0$. By Lemma 2.7, we obtain for the $L^1$ Wasserstein distance

$$ \begin{aligned} W_1(\mu_t^{k,N}, \bar{\mu}_t^{\otimes k}) &= \inf \left\{ E[\|Z^{(k)} - \bar{Z}^{(k)}\|_1], P_{Z^{(k)}} = \mu_t^{k,N}, P_{\bar{Z}^{(k)}} = \bar{\mu}_t^{\otimes k} \right\} \\ &\le \inf \left\{ C_1 E \left[ \sum_{i=1}^k f(r_t^i) G_t^i \right], P_{(Z_t^{i,N})_i} = \mu_t^{k,N}, P_{(\bar{Z}_t^i)_i} = \bar{\mu}_t^{\otimes k} \right\} \\ &\le \inf \left\{ k C_1 E \left( \rho((Z_t^{j,N}, \bar{Z}_t^j)_{1\le j \le N}) \right), P_{(Z_t^{i,N})_i} = \mu_t^{k,N}, P_{(\bar{Z}_t^i)_i} = \bar{\mu}_t^{\otimes k} \right\} \\ &\le \frac{\xi A k C_1}{c} (1 - e^{-ct}) + \frac{(L_X + L_C) B C_1}{c \sqrt{N}} \frac{k}{\sqrt{N}} (1 - e^{-ct}) \end{aligned} $$

By taking the limit as $\xi \to 0$ uniformly in time, we obtain the desired result. The same lemma and the same type of calculations yield the result for the $L^2$ Wasserstein distance

$$ W_2(\mu_t^{k,N}, \bar{\mu}_t^{\otimes k})^2 \le \frac{k}{\sqrt{N}} C_2 \frac{(L_X + L_C)B}{c}. $$

<figure><img src="image_2.png" /></figure>
We gather the expectations terms, obtained with (2.18), in $I_t^{3,i}$, and we keep a fraction of Lyapunov function to control it:

$$I_t^{3,i} = \epsilon f(r_t^i) \left( (\alpha_X L_X + \beta_X L_C) \left( \frac{\sum_{j=1}^N |X_t^{j,N}|}{N} \right)^2 \exp \left( a \sqrt{H(Z_t^{i,N})} \right) \right. \\ \left. + (\alpha_C L_X + \beta_C L_C) \left( \frac{\sum_{j=1}^N |C_t^{j,N}|}{N} \right)^2 \exp \left( a \sqrt{H(Z_t^{i,N})} \right) \right. \\ \left. - \frac{\lambda}{16} H(Z_t^{i,N}) \exp \left( a \sqrt{H(Z_t^{i,N})} \right) - \frac{\lambda}{16N} \sum_{j=1}^N H(Z_t^{j,N}) \exp \left( a \sqrt{H(Z_t^{j,N})} \right) \right).$$

Finally, we define $\tilde{K}_t^i$ with the leftovers. It will, in particular, give the constraints on $f$ which explain its choice.

$$\tilde{K}_t^i = G_t^i \left[ 2cf(r_t^i) + \frac{1}{2} f''(r_t^i) \left( 2\sigma_x^2 \varphi_{rc} \left( |X_t^{i,N} - \bar{X}_t^i| \right)^2 \right) \right. \\ + f'(r_t^i) \left( (1 + \gamma\delta + L_X + \delta L_C) |X_t^{i,N} - \bar{X}_t^i| - |(X_t^{i,N})^3 - (\bar{X}_t^i)^3| + (1 + L_X + \delta L_C - \delta) |C_t^{i,N} - \bar{C}_t^i| \right. \\ + (\epsilon C_{f,1} + C_{f,2}) \sigma_x^2 \varphi_{rc} \left( |X_t^{i,N} - \bar{X}_t^i| \right)^2 r_t^i \left. \right] \\ + \epsilon f(r_t^i) \left( 4\tilde{B} - \frac{\lambda}{16}\tilde{H}(\bar{Z}_t^i) - \frac{\lambda}{16}\tilde{H}(Z_t^{i,N}) - \frac{\lambda}{16N} \sum_{j=1}^N \tilde{H}(\bar{Z}_t^j) - \frac{\lambda}{16N} \sum_{j=1}^N \tilde{H}(Z_t^{j,N}) \right).$$

### 3.3 Controls of $I_t^{1,i}$, $I_t^{2,i}$ and $I_t^{3,i}$

*Proof of Lemma 3.5.* Since we assume

$$\frac{4}{\gamma} (\alpha_X L_X + \beta_X L_C) \le \frac{\lambda}{32} \quad \text{and} \quad 4 (\alpha_C L_X + \beta_C L_C) \le \frac{\lambda}{32},$$

and since

$$H(Z_t^{j,N}) \exp \left( a \sqrt{H(Z_t^{i,N})} \right) \le H(Z_t^{i,N}) \exp \left( a \sqrt{H(Z_t^{i,N})} \right) + H(Z_t^{j,N}) \exp \left( a \sqrt{H(Z_t^{j,N})} \right)$$

we obtain

$$(\alpha_X L_X + \beta_X L_C) \left( \frac{\sum_{j=1}^N |X_t^{j,N}|}{N} \right)^2 \exp \left( a \sqrt{H(Z_t^{i,N})} \right) + (\alpha_C L_X + \beta_C L_C) \left( \frac{\sum_{j=1}^N |C_t^{j,N}|}{N} \right)^2 \exp \left( a \sqrt{H(Z_t^{i,N})} \right) \\ - \frac{\lambda}{16N} \left( N H(Z_t^{i,N}) \exp \left( a \sqrt{H(Z_t^{i,N})} \right) + \sum_{j=1}^N H(Z_t^{j,N}) \exp \left( a \sqrt{H(Z_t^{j,N})} \right) \right) \le 0.$$

Then, for each $i \le N$, and for all $t > 0$, $I_t^{3,i} \le 0$.

*Proof of Lemma 3.4.* We prove the non-positivity of $\frac{1}{N} \sum_{i=1}^N I_t^{2,i}$. First, since $f'(r_t^i) \le 1$, we have

$$\frac{1}{N} \sum_{i=1}^{N} \left( \frac{1}{N} f'(r_t^i) G_t^i \sum_{j=1}^{N} \|Z_t^{j,N} - \bar{Z}_t^j\|_1 \right) \le \frac{1}{N^2} \sum_{i,j=1}^{N} \|Z_t^{j,N} - \bar{Z}_t^j\|_1 G_t^i$$
## 3.2 Proof of the decomposition

*Proof of Proposition 3.1.* First, we need to calculate $d(e^{ct}f(r_t^i)G_t^i)$, where we recall

$$r_t^i = |X_t^{i,N} - \bar{X}_t^i| + \delta|C_t^{i,N} - \bar{C}_t^i|$$

and

$$G_t^i = 1 + \epsilon \tilde{H}(\bar{Z}_t^i) + \epsilon \tilde{H}(Z_t^{i,N}) + \frac{\epsilon}{N} \sum_{j=1}^{N} \tilde{H}(Z_t^{j,N}) + \frac{\epsilon}{N} \sum_{j=1}^{N} \tilde{H}(\bar{Z}_t^j).$$

We have already calculated $d(X_t^{i,N} - \bar{X}_t^i)$ and $d|X_t^{i,N} - \bar{X}_t^i|$ in the case of symmetric coupling in Subsection 1.4 in (1.8). Here, we need to use Ito's formula and usual convergence lemmas, see Lemma A.1 below, to take care of the Brownian term (recall that the coefficient in front of the Brownian vanishes in the vicinity of the singularity of the absolute value). We obtain

$$d|X_t^{i,N} - \bar{X}_t^i| = A_t^X dt + 2\text{sign}(X_t^{i,N} - \bar{X}_t^i)\sigma_X \varphi_{\text{rc}}(|X_t^{i,N} - \bar{X}_t^i|) dB_t^{i,\text{rc},X},$$

with

$$A_t^X \le |X_t^{i,N} - \bar{X}_t^i| - |(X_t^{i,N})^3 - (\bar{X}_t^i)^3| + |C_t^{i,N} - \bar{C}_t^i| + \left| \frac{1}{N} \sum_{j=1}^N K_X(Z_t^{i,N} - Z_t^{j,N}) - K_X * \bar{\mu}_t(\bar{Z}_t^i) \right|.$$

Likewise, as it has already been calculated in (1.9) in Subsection 1.4,

$$d|C_t^{i,N} - \bar{C}_t^i| = A_t^C dt,$$

with

$$A_t^C \le \gamma |X_t^{i,N} - \bar{X}_t^i| - |C_t^{i,N} - \bar{C}_t^i| + \left| \frac{1}{N} \sum_{j=1}^N K_C(Z_t^{i,N} - Z_t^{j,N}) - K_C * \bar{\mu}_t(\bar{Z}_t^i) \right|.$$

Now we have

$$dr_t^i = (A_t^X + \delta A_t^C) dt + 2\text{sign}(X_t^{i,N} - \bar{X}_t^i)\sigma_X \varphi_{\text{rc}}(|X_t^{i,N} - \bar{X}_t^i|) dB_t^{i,\text{rc},X}$$

and we deduce with Ito's formula

$$df(r_t^i) = f'(r_t^i)dr_t^i + \frac{1}{2}f''(r_t^i)(2\sigma_X \varphi_{\text{rc}}(|X_t^{i,N} - \bar{X}_t^i|))^2 dt.$$

Finally, for $c > 0$,

$$d(e^{ct}f(r_t^i)) = ce^{ct}f(r_t^i)dt + e^{ct}df(r_t^i).$$
$$ \leq \frac{1}{N} \sum_{i=1}^{N} \|Z_{t}^{i,N} - \bar{Z}_{t}^{i}\|_{1} + \frac{2\epsilon}{N^2} \sum_{i,j=1}^{N} \|Z_{t}^{i,N} - \bar{Z}_{t}^{i}\|_{1} (\tilde{H}(\bar{Z}_{t}^{j}) + \tilde{H}(Z_{t}^{j,N})) $$

and, using Lemma 2.6 (i)

$$ \frac{1}{N} \sum_{i=1}^{N} \|Z_{t}^{i,N} - \bar{Z}_{t}^{i}\|_{1} \leq \frac{C_{1}}{N} \sum_{i=1}^{N} f(r_{t}^{i}) G_{t}^{i} $$

and with Lemma 2.6 (iii)

$$ \begin{aligned} & \sum_{i,j=1}^{N} \|Z_{t}^{i,N} - \bar{Z}_{t}^{i}\|_{1} (\tilde{H}(\bar{Z}_{t}^{j}) + \tilde{H}(Z_{t}^{j,N})) \\ & \leq C_{z} \sum_{i,j=1}^{N} f(r_{t}^{i}) (1 + \epsilon\sqrt{H(Z_{t}^{i,N})} + \epsilon\sqrt{H(\bar{Z}_{t}^{i})}) (\tilde{H}(\bar{Z}_{t}^{j}) + \tilde{H}(Z_{t}^{j,N})) \\ & \leq C_{z} \sum_{i,j=1}^{N} f(r_{t}^{i}) (\tilde{H}(\bar{Z}_{t}^{j}) + \tilde{H}(Z_{t}^{j,N})) + \epsilon C_{z} \sum_{i,j=1}^{N} f(r_{t}^{i}) (\sqrt{H(Z_{t}^{i,N})} + \sqrt{H(\bar{Z}_{t}^{i})}) (\tilde{H}(\bar{Z}_{t}^{j}) + \tilde{H}(Z_{t}^{j,N})). \end{aligned} $$

Using (2.9) from Lemma 2.4, we obtain for the first sum:

$$ C_z \sum_{i,j=1}^{N} f(r_t^i) (\tilde{H}(\bar{Z}_t^j) + \tilde{H}(Z_t^{j,N})) \le C_z \sum_{i,j=1}^{N} f(r_t^i) \left( H(\bar{Z}_t^j) \exp \left( a\sqrt{H(\bar{Z}_t^j)} \right) + H(Z_t^{j,N}) \exp \left( a\sqrt{H(Z_t^{j,N})} \right) \right). $$

With (2.10) from the same Lemma, we obtain for the second sum:

$$ \begin{aligned} & \epsilon C_z \sum_{i,j=1}^{N} f(r_t^i) \left( \sqrt{H(Z_t^{i,N})} + \sqrt{H(\bar{Z}_t^i)} \right) \left( \tilde{H}(\bar{Z}_t^j) + \tilde{H}(Z_t^{j,N}) \right) \\ & \le \epsilon C_z \frac{2}{a} \sum_{i,j=1}^{N} f(r_t^i) \left( \sqrt{H(Z_t^{i,N})} + \sqrt{H(\bar{Z}_t^i)} \right) \left( \sqrt{H(\bar{Z}_t^j)} \exp \left( a\sqrt{H(\bar{Z}_t^j)} \right) + \sqrt{H(Z_t^{j,N})} \exp \left( a\sqrt{H(Z_t^{j,N})} \right) \right). \end{aligned} $$

Since for all $(y_1, y_2, y_3, y_4) \in (\mathbb{R}^+)^4$, we have

$$ (y_1 + y_2) (y_3 e^{ay_3} + y_4 e^{ay_4}) \le 2 (y_1^2 e^{ay_1} + y_2^2 e^{ay_2} + y_3^2 e^{ay_3} + y_4^2 e^{ay_4}), $$

we obtain for this last sum

$$ \begin{aligned} & \frac{2\epsilon C_z}{a} \sum_{i,j=1}^{N} f(r_t^i) \left( \sqrt{H(Z_t^{i,N})} + \sqrt{H(\bar{Z}_t^i)} \right) \left( \sqrt{H(\bar{Z}_t^j)} \exp \left( a\sqrt{H(\bar{Z}_t^j)} \right) + \sqrt{H(Z_t^{j,N})} \exp \left( a\sqrt{H(Z_t^{j,N})} \right) \right) \\ & \le \frac{4\epsilon C_z}{a} \sum_{i,j=1}^{N} f(r_t^i) \left( H(\bar{Z}_t^i) \exp \left( a\sqrt{H(\bar{Z}_t^i)} \right) + H(Z_t^{i,N}) \exp \left( a\sqrt{H(Z_t^{i,N})} \right) \right) \\ & \qquad + \frac{4\epsilon C_z}{a} \sum_{i,j=1}^{N} f(r_t^i) \left( H(\bar{Z}_t^j) \exp \left( a\sqrt{H(\bar{Z}_t^j)} \right) + H(Z_t^{j,N}) \exp \left( a\sqrt{H(Z_t^{j,N})} \right) \right) \\ & \le \frac{4\epsilon C_z}{a} N \sum_{i=1}^{N} f(r_t^i) \left( H(\bar{Z}_t^i) \exp \left( a\sqrt{H(\bar{Z}_t^i)} \right) + H(Z_t^{i,N}) \exp \left( a\sqrt{H(Z_t^{i,N})} \right) \right) \\ & \qquad + \frac{4\epsilon C_z}{a} \sum_{i,j=1}^{N} f(r_t^i) \left( H(\bar{Z}_t^j) \exp \left( a\sqrt{H(\bar{Z}_t^j)} \right) + H(Z_t^{j,N}) \exp \left( a\sqrt{H(Z_t^{j,N})} \right) \right). \end{aligned} $$
Then, by Ito's formula,

$$
\begin{aligned}
\frac{1}{\epsilon} dG_t^i = & \left( \mathcal{L}_{\bar{\mu}_t} \tilde{H}(\bar{Z}_t^i) + \mathcal{L}^N \tilde{H}(Z_t^{i,N}) \right) dt \\
& + \sigma_X \varphi_{rc} \left( |X_t^{i,N} - \bar{X}_t^i| \right) \left( \partial_X \tilde{H}(Z_t^{i,N}) - \partial_X \tilde{H}(\bar{Z}_t^i) \right) dB_t^{i,rc,X} \\
& + \sigma_X \varphi_{sc} \left( |X_t^{i,N} - \bar{X}_t^i| \right) \left( \partial_X \tilde{H}(Z_t^{i,N}) + \partial_X \tilde{H}(\bar{Z}_t^i) \right) dB_t^{i,sc,X} \\
& + \sigma_C \left( \partial_C \tilde{H}(Z_t^{i,N}) + \partial_C \tilde{H}(\bar{Z}_t^i) \right) dB_t^{i,C} \\
& + \frac{1}{N} \sum_{j=1}^N \left( \mathcal{L}_{\bar{\mu}_t} \tilde{H}(\bar{Z}_t^j) + \mathcal{L}^N \tilde{H}(Z_t^{j,N}) \right) dt \\
& + \frac{\sigma_X}{N} \sum_{j=1}^N \varphi_{rc} \left( |X_t^{j,N} - \bar{X}_t^j| \right) \left( \partial_X \tilde{H}(Z_t^{j,N}) - \partial_X \tilde{H}(\bar{Z}_t^j) \right) dB_t^{j,rc,X} \\
& + \frac{\sigma_X}{N} \sum_{j=1}^N \varphi_{sc} \left( |X_t^{j,N} - \bar{X}_t^j| \right) \left( \partial_X \tilde{H}(Z_t^{j,N}) + \partial_X \tilde{H}(\bar{Z}_t^j) \right) dB_t^{j,sc,X} \\
& + \frac{\sigma_C}{N} \sum_{j=1}^N \left( \partial_C \tilde{H}(Z_t^{j,N}) + \partial_C \tilde{H}(\bar{Z}_t^j) \right) dB_t^{j,C}.
\end{aligned}
$$

We finally get

$$
\begin{aligned}
d(e^{ct} f(r_t^i) G_t^i) ={}& G_t^i d(e^{ct} f(r_t^i)) + e^{ct} f(r_t^i) dG_t^i \\
& + 2\epsilon \left(1 + \frac{1}{N}\right) \sigma_X^2 \varphi_{rc} \left(|X_t^{i,N} - \bar{X}_t^i|\right)^2 \text{sign}(X_t^{i,N} - \bar{X}_t^i) \\
& \times \left(\partial_X \tilde{H}(Z_t^{i,N}) - \partial_X \tilde{H}(\bar{Z}_t^i)\right) e^{ct} f'(r_t^i) dt.
\end{aligned}
$$

Now, we need to use the following Lemma, proven in Appendix A.6, to have a more tractable expression

**Lemma 3.7.** We have the upper bound

$$
\begin{aligned}
& 2\epsilon \left(1 + \frac{1}{N}\right) \sigma_X^2 \varphi_{rc} \left(|X_t^{i,N} - \bar{X}_t^i|\right)^2 \text{sign}(X_t^{i,N} - \bar{X}_t^i) \left(\partial_X \tilde{H}(Z_t^{i,N}) - \partial_X \tilde{H}(\bar{Z}_t^i)\right) \\
& \leq (\epsilon C_{f,1} + C_{f,2}) \sigma_X^2 \varphi_{rc} \left(|X_t^{i,N} - \bar{X}_t^i|\right)^2 r_t^i G_t^i.
\end{aligned}
$$

Eventually, by denoting the terms in $dB_t^{i,rc,X}$, $dB_t^{i,sc,X}$, $dB_t^{i,C}$, ..., as the local martin-
gale $dM_t^i$, we obtain

$$
\begin{aligned}
d(e^{ct} f(r_t^i) G_t^i) \le{}& G_t^i c e^{ct} f(r_t^i) dt + e^{ct} G_t^i f'(r_t^i) (A_t^X + \delta A_t^C) dt \\
& + e^{ct} G_t^i \frac{1}{2} f''(r_t^i) \left(2 \sigma_X \varphi_{rc} \left(|X_t^{i,N} - \bar{X}_t^i|\right)\right)^2 dt \\
& + e^{ct} f(r_t^i) \left( \left(\mathcal{L}_{\bar{\mu}_t} \tilde{H}(\bar{Z}_t^i) + \mathcal{L}^N \tilde{H}(Z_t^{i,N})\right) dt + \frac{1}{N} \sum_{j=1}^N \left(\mathcal{L}_{\bar{\mu}_t} \tilde{H}(\bar{Z}_t^j) + \mathcal{L}^N \tilde{H}(Z_t^{j,N})\right) dt \right) \\
& + (\epsilon C_{f,1} + C_{f,2}) \sigma_X^2 \varphi_{rc} \left(|X_t^{i,N} - \bar{X}_t^i|\right)^2 r_t^i G_t^i e^{ct} f'(r_t^i) dt + dM_t^i.
\end{aligned}
$$

We use (2.14) to bound $\mathcal{L}_{\bar{\mu}_t} \tilde{H}(\bar{Z}_t^i)$ and (2.17) to bound $\mathcal{L}^N \tilde{H}(Z_t^{i,N})$. The interaction terms
Then, by reconsidering the first expression:

$$
\begin{aligned}
& \frac{1}{N} \sum_{i=1}^{N} \left( \frac{1}{N} f'(r_t^i) G_t^i \sum_{j=1}^{N} \|Z_t^{j,N} - \bar{Z}_t^j\|_1 \right) \\
& \le \frac{C_1}{N} \sum_{i=1}^{N} f(r_t^i) G_t^i + \frac{2\epsilon}{N^2} C_z \sum_{i,j=1}^{N} f(r_t^i) \left( H(\bar{Z}_t^j) \exp \left( a\sqrt{H(\bar{Z}_t^j)} \right) + H(Z_t^{j,N}) \exp \left( a\sqrt{H(Z_t^{j,N})} \right) \right) \\
& \quad + \frac{2\epsilon}{N^2} \frac{4\epsilon C_z}{a} N \sum_{i=1}^{N} f(r_t^i) \left( H(\bar{Z}_t^i) \exp \left( a\sqrt{H(\bar{Z}_t^i)} \right) + H(Z_t^{i,N}) \exp \left( a\sqrt{H(Z_t^{i,N})} \right) \right) \\
& \quad + \frac{2\epsilon}{N^2} \frac{4\epsilon C_z}{a} \sum_{i,j=1}^{N} f(r_t^i) \left( H(\bar{Z}_t^j) \exp \left( a\sqrt{H(\bar{Z}_t^j)} \right) + H(Z_t^{j,N}) \exp \left( a\sqrt{H(Z_t^{j,N})} \right) \right)
\end{aligned}
$$

This way, by Assumption 2 since

$$
L_X C_1 \le \frac{c}{2}, \quad 2C_z L_X \le \frac{\lambda}{64} \quad \text{and} \quad L_X \epsilon \frac{8C_z}{a} \le \frac{\lambda}{64},
$$

we get

$$
\begin{aligned}
& \frac{1}{N} \sum_{i=1}^{N} \left( \frac{1}{N} f'(r_t^i) G_t^i \sum_{j=1}^{N} \|Z_t^{j,N} - \bar{Z}_t^j\|_1 \right) \\
\le & \frac{1}{N} \frac{c}{2L_X} \sum_{i=1}^{N} f(r_t^i) G_t^i + \frac{\epsilon}{N^2} \frac{\lambda}{64L_X} \sum_{i,j=1}^{N} f(r_t^i) \left( H(\bar{Z}_t^j) \exp \left( a\sqrt{H(\bar{Z}_t^j)} \right) + H(Z_t^{j,N}) \exp \left( a\sqrt{H(Z_t^{j,N})} \right) \right) \\
& + \frac{\epsilon}{N} \frac{\lambda}{64L_X} \sum_{i=1}^{N} f(r_t^i) \left( H(\bar{Z}_t^i) \exp \left( a\sqrt{H(\bar{Z}_t^i)} \right) + H(Z_t^{i,N}) \exp \left( a\sqrt{H(Z_t^{i,N})} \right) \right) \\
& + \frac{\epsilon}{N^2} \frac{\lambda}{64L_X} \sum_{i,j=1}^{N} f(r_t^i) \left( H(\bar{Z}_t^j) \exp \left( a\sqrt{H(\bar{Z}_t^j)} \right) + H(Z_t^{j,N}) \exp \left( a\sqrt{H(Z_t^{j,N})} \right) \right)
\end{aligned}
$$

and we finally obtain "half" the result

$$
\begin{aligned}
& \frac{1}{N} \sum_{i=1}^{N} G_t^i f'(r_t^i) \left( \frac{L_X}{N} \left( \sum_{j=1}^{N} \|Z_t^{j,N} - \bar{Z}_t^j\|_1 \right) \right) \\
& - \frac{c}{2} \frac{1}{N} \sum_{i=1}^{N} f(r_t^i) G_t^i - \frac{\epsilon}{2} \frac{1}{N} \sum_{i=1}^{N} f(r_t^i) \left[ \frac{\lambda}{16} H(\bar{Z}_t^i) \exp \left( a\sqrt{H(\bar{Z}_t^i)} \right) + \frac{\lambda}{16} H(Z_t^{i,N}) \exp \left( a\sqrt{H(Z_t^{i,N})} \right) \right] \\
& - \frac{\epsilon}{2} \frac{1}{N} \sum_{i=1}^{N} f(r_t^i) \left[ \frac{\lambda}{16N} \sum_{j=1}^{N} H(\bar{Z}_t^j) \exp \left( a\sqrt{H(\bar{Z}_t^j)} \right) + \frac{\lambda}{16N} \sum_{j=1}^{N} H(Z_t^{j,N}) \exp \left( a\sqrt{H(Z_t^{j,N})} \right) \right] \le 0.
\end{aligned}
$$

Likewise, by Assumption 2, since

$$
\delta L_C C_1 \le \frac{c}{2}, \quad 2C_z \delta L_C \le \frac{\lambda}{64} \quad \text{and} \quad \delta L_C \epsilon \frac{8C_z}{a} \le \frac{\lambda}{64},
$$
in $A_t^X$ and $A_t^C$ are decomposed and we define $I_t^{1,i}$ as follows

$$I_t^{1,i} = G_t^i f'(r_t^i) \left( \left| \frac{1}{N} \sum_{j=1}^{N} K_X(\bar{Z}_t^i - \bar{Z}_t^j) - K_X * \bar{\mu}_t(\bar{Z}_t^i) \right| \right) + \delta G_t^i f'(r_t^i) \left( \left| \frac{1}{N} \sum_{j=1}^{N} K_C(\bar{Z}_t^i - \bar{Z}_t^j) - K_C * \bar{\mu}_t(\bar{Z}_t^i) \right| \right).$$

The second part of the decomposition is grouped in $I_t^{2,i}$, with compensating terms that appear with the use of (2.14) and (2.17), to control the sum

$$I_t^{2,i} = G_t^i f'(r_t^i) \left( \frac{L_X}{N} \left( \sum_{j=1}^{N} |X_t^{j,N} - \bar{X}_t^j| + |C_t^{j,N} - \bar{C}_t^j| \right) \right) \\ + \delta G_t^i f'(r_t^i) \left( \frac{L_C}{N} \left( \sum_{j=1}^{N} |X_t^{j,N} - \bar{X}_t^j| + |C_t^{j,N} - \bar{C}_t^j| \right) \right) \\ - c f(r_t^i) G_t^i - \epsilon f(r_t^i) \left[ \frac{\lambda}{16} H(\bar{Z}_t^i) \exp \left( a \sqrt{H(\bar{Z}_t^i)} \right) + \frac{\lambda}{16} H(Z_t^{i,N}) \exp \left( a \sqrt{H(Z_t^{i,N})} \right) \right] \\ - \epsilon f(r_t^i) \frac{\lambda}{16N} \left[ \sum_{j=1}^{N} H(\bar{Z}_t^j) \exp \left( a \sqrt{H(\bar{Z}_t^j)} \right) + \sum_{j=1}^{N} H(Z_t^{j,N}) \exp \left( a \sqrt{H(Z_t^{j,N})} \right) \right].$$

We gather the expectations terms, obtained with (2.17), in $I_t^{3,i}$, and we keep a fraction of the Lyapunov function to control it

$$I_t^{3,i} = \epsilon f(r_t^i) \left( (\alpha_X L_X + \beta_X L_C) \left( \frac{\sum_{j=1}^{N} |X_t^{j,N}|}{N} \right)^2 \exp \left( a \sqrt{H(Z_t^{i,N})} \right) \right. \\ + (\alpha_C L_X + \beta_C L_C) \left( \frac{\sum_{j=1}^{N} |C_t^{j,N}|}{N} \right)^2 \exp \left( a \sqrt{H(Z_t^{i,N})} \right) \\ \left. - \frac{\lambda}{16} H(Z_t^{i,N}) \exp \left( a \sqrt{H(Z_t^{i,N})} \right) - \frac{\lambda}{16N} \sum_{j=1}^{N} H(Z_t^{j,N}) \exp \left( a \sqrt{H(Z_t^{j,N})} \right) \right).$$

Finally, we define $\tilde{K}_t^i$ with the leftovers. It will, in particular, give the constraints on $f$ which explain its choice.

$$\tilde{K}_t^i = G_t^i \left[ 2c f(r_t^i) + \frac{1}{2} f''(r_t^i) \left( 2\sigma_X^2 \varphi_{rc} (|X_t^{i,N} - \bar{X}_t^i|)^2 \right) \right. \\ + f'(r_t^i) \left( (1 + \gamma\delta + L_X + \delta L_C) |X_t^{i,N} - \bar{X}_t^i| - |(X_t^{i,N})^3 - (\bar{X}_t^i)^3| \right. \\ + (1 + L_X + \delta L_C - \delta) |C_t^{i,N} - \bar{C}_t^i| + (\epsilon C_{f,1} + C_{f,2}) \sigma_X^2 \varphi_{rc} (|X_t^{i,N} - \bar{X}_t^i|^2 r_t^i) \left. \right] \\ + \epsilon f(r_t^i) \left( 4\tilde{B} - \frac{\lambda}{16} \tilde{H}(\bar{Z}_t^i) - \frac{\lambda}{16} \tilde{H}(Z_t^{i,N}) - \frac{\lambda}{16N} \sum_{j=1}^{N} \tilde{H}(\bar{Z}_t^j) - \frac{\lambda}{16N} \sum_{j=1}^{N} \tilde{H}(Z_t^{j,N}) \right).$$

<figure><img src="image_3.png" /></figure>
we obtain the second "half"

$$
\begin{align*}
& \frac{1}{N} \sum_{i=1}^{N} \delta G_t^i f'(r_t^i) \left( \frac{L_C}{N} \left( \sum_{j=1}^{N} \|Z_t^{j,N} - \bar{Z}_t^j\|_1 \right) \right) \\
& - \frac{c}{2} \frac{1}{N} \sum_{i=1}^{N} f(r_t^i) G_t^i - \frac{\epsilon}{2} \frac{1}{N} \sum_{i=1}^{N} f(r_t^i) \left[ \frac{\lambda}{16} H(\bar{Z}_t^i) \exp \left( a \sqrt{H(\bar{Z}_t^i)} \right) + \frac{\lambda}{16} H(Z_t^{i,N}) \exp \left( a \sqrt{H(Z_t^{i,N})} \right) \right] \\
& - \frac{\epsilon}{2} \frac{1}{N} \sum_{i=1}^{N} f(r_t^i) \left[ \frac{\lambda}{16N} \sum_{j=1}^{N} H(\bar{Z}_t^j) \exp \left( a \sqrt{H(\bar{Z}_t^j)} \right) + \frac{\lambda}{16N} \sum_{j=1}^{N} H(Z_t^{j,N}) \exp \left( a \sqrt{H(Z_t^{j,N})} \right) \right] \le 0.
\end{align*}
$$

Eventually, we have proved $\sum_{i=1}^{N} I_t^{2,i} \le 0$.

<figure><img src="image_1.png" /></figure>

Proof of Lemma 3.3. Since $f'(r) \le 1$, we have by Cauchy-Schwarz

$$
\begin{align*}
& \mathbb{E} \left( G_t^i f'(r_t^i) \left( \left| \frac{1}{N} \sum_{j=1}^N K_X(\bar{Z}_t^i - \bar{Z}_t^j) - K_X * \bar{\mu}_t(\bar{Z}_t^i) \right| \right) \right) \\
& \le \mathbb{E} \left( |G_t^i|^2 \right)^{1/2} \mathbb{E} \left( \left| \frac{1}{N} \sum_{j=1}^N K_X(\bar{Z}_t^i - \bar{Z}_t^j) - K_X * \bar{\mu}_t(\bar{Z}_t^i) \right|^2 \right)^{1/2} .
\end{align*}
$$

By Lemma 3.1, we have for each $i \le N$, for all $t \ge 0$, $\mathbb{E}[(G_t^i)^2] \le C_{G,2}$.

Moreover, we notice that $(\bar{Z}_t^j)_j$ are i.i.d with law $\bar{\mu}_t$. Let's denote $\bar{Z}_t$ a generic random variable of law $\bar{\mu}_t$
independent of $\bar{Z}_t^i$. The calculus of the right term of the product has already be done in Subsection 1.4, and we
have (1.9):

$$
\mathbb{E} \left( \left| \frac{1}{N} \sum_{j=1}^{N} K_X(\bar{Z}_t^i - \bar{Z}_t^j) - K_X * \bar{\mu}_t(\bar{Z}_t^i) \right|^2 \right) \le \frac{8L_X^2}{N} \mathbb{E}(\|\bar{Z}_t\|_1^2) \mathbb{E}(\|\bar{Z}_t\|_1^2).
$$

A similar calculation yields

$$
\mathbb{E} \left( \left| \frac{1}{N} \sum_{j=1}^{N} K_C(\bar{Z}_t^i - \bar{Z}_t^j) - K_C * \bar{\mu}_t(\bar{Z}_t^i) \right|^2 \right) \le \frac{8L_C^2}{N} \mathbb{E}(\|\bar{Z}_t\|_1^2).
$$

By Lemma 2.3, $\mathbb{E} (|\bar{X}_t|^2 + |\bar{C}_t|^2) \le C_{init,2}$. In particular,

$$
\mathbb{E} (\|\bar{Z}_t\|_1^2) = \mathbb{E} (\left[ |\bar{X}_t| + |\bar{C}_t| \right]^2) \le 2\mathbb{E} (|\bar{X}_t|^2 + |\bar{C}_t|^2) \le 2C_{\text{init},2}.
$$

Thus

$$
\mathbb{E} \left( G_t^i f'(r_t^i) \left( \left| \frac{1}{N} \sum_{j=1}^N K_X(\bar{Z}_t^i - \bar{Z}_t^j) - K_X * \bar{\mu}_t(\bar{Z}_t^i) \right| \right) \right) \le L_X C_{G,2}^{1/2} \sqrt{2C_{\text{init},2}} \sqrt{\frac{8}{N}}
$$

and likewise

$$
\mathbb{E} \left( G_t^i f'(r_t^i) \left( \left| \frac{1}{N} \sum_{j=1}^{N} K_C(\bar{Z}_t^i - \bar{Z}_t^j) - K_C * \bar{\mu}_t(\bar{Z}_t^i) \right| \right) \right) \le L_C C_{G,2}^{1/2} \sqrt{2C_{\text{init},2}} \sqrt{\frac{8}{N}}.
$$

<figure><img src="image_2.png" /></figure>
### 3.3 Controls of $I_t^{1,i}$, $I_t^{2,i}$ and $I_t^{3,i}$

*Proof of Lemma 3.6.* Since we assume

$$ \frac{4}{\gamma} (\alpha_X L_X + \beta_X L_C) \le \frac{\lambda}{32} \quad \text{and} \quad 4 (\alpha_C L_X + \beta_C L_C) \le \frac{\lambda}{32}, $$

and since

$$ H(Z_t^{j,N}) \exp \left( a \sqrt{H(Z_t^{i,N})} \right) \le H(Z_t^{i,N}) \exp \left( a \sqrt{H(Z_t^{i,N})} \right) + H(Z_t^{j,N}) \exp \left( a \sqrt{H(Z_t^{j,N})} \right) $$

we obtain

$$ (\alpha_X L_X + \beta_X L_C) \left( \frac{\sum_{j=1}^N |X_t^{j,N}|}{N} \right)^2 \exp \left( a \sqrt{H(Z_t^{i,N})} \right) \\ + (\alpha_C L_X + \beta_C L_C) \left( \frac{\sum_{j=1}^N |C_t^{j,N}|}{N} \right)^2 \exp \left( a \sqrt{H(Z_t^{i,N})} \right) \\ - \frac{\lambda}{16N} \left( N H(Z_t^{i,N}) \exp \left( a \sqrt{H(Z_t^{i,N})} \right) + \sum_{j=1}^N H(Z_t^{j,N}) \exp \left( a \sqrt{H(Z_t^{j,N})} \right) \right) \le 0. $$

Then, for each $i \le N$, and for all $t > 0$, $I_t^{3,i} \le 0$. □

*Proof of Lemma 3.5.* We prove the non-positivity of $\frac{1}{N} \sum_{i=1}^N I_t^{2,i}$. First, since $f'(r_t^i) \le 1$, we have

$$ \frac{1}{N} \sum_{i=1}^{N} \left( \frac{1}{N} f'(r_t^i) G_t^i \sum_{j=1}^{N} \|Z_t^{j,N} - \bar{Z}_t^j\|_1 \right) \\ \le \frac{1}{N} \sum_{i=1}^{N} \|Z_t^{i,N} - \bar{Z}_t^i\|_1 + \frac{2\epsilon}{N^2} \sum_{i,j=1}^{N} \|Z_t^{i,N} - \bar{Z}_t^i\|_1 (\tilde{H}(\bar{Z}_t^j) + \tilde{H}(Z_t^{j,N})), $$

and, using Lemma 2.7 (i)

$$ \frac{1}{N} \sum_{i=1}^{N} \|Z_t^{i,N} - \bar{Z}_t^i\|_1 \le \frac{C_1}{N} \sum_{i=1}^{N} f(r_t^i) G_t^i, $$

and with Lemma 2.7 (iii)

$$ \sum_{i,j=1}^{N} \|Z_t^{i,N} - \bar{Z}_t^i\|_1 (\tilde{H}(\bar{Z}_t^j) + \tilde{H}(Z_t^{j,N})) \le C_z \sum_{i,j=1}^{N} f(r_t^i) (\tilde{H}(\bar{Z}_t^j) + \tilde{H}(Z_t^{j,N})) \\ + \epsilon C_z \sum_{i,j=1}^{N} f(r_t^i) (\sqrt{H(Z_t^{i,N})} + \sqrt{H(\bar{Z}_t^j)}) (\tilde{H}(\bar{Z}_t^j) + \tilde{H}(Z_t^{j,N})). $$

Using (2.9) from Lemma 2.5, we obtain for the first sum

$$ C_z \sum_{i,j=1}^{N} f(r_t^i) (\tilde{H}(\bar{Z}_t^j) + \tilde{H}(Z_t^{j,N})) \\ \le C_z \sum_{i,j=1}^{N} f(r_t^i) (H(\bar{Z}_t^j) \exp \left( a \sqrt{H(\bar{Z}_t^j)} \right) + H(Z_t^{j,N}) \exp \left( a \sqrt{H(Z_t^{j,N})} \right)). $$
## 3.4 Contraction in various regions of space

The goal of this section is to prove the Lemma 3.2 i.e show that for each $i \le N$, for all $t > 0$, we have the following control

$$ \mathbb{E}\tilde{K}_t^i \le \xi \left( 2 + \delta\gamma + L_X + \delta L_C - L_C - \frac{1+L_X}{\delta} \right) \mathbb{E}G_t^i. $$

Recall

$$ \tilde{K}_t^i = G_t^i \left[ 2cf(r_t^i) + \frac{1}{2}f''(r_t^i) \left( 2\sigma_x^2 \varphi_{\text{rc}} \left( |X_t^{i,N} - \bar{X}_t^i| \right)^2 \right) \right. \\ \left. + f'(r_t^i) \left( (1+\gamma\delta + L_X + \delta L_C) |X_t^{i,N} - \bar{X}_t^i| - |(X_t^{i,N})^3 - (\bar{X}_t^i)^3| + (1+L_X + \delta L_C - \delta) |C_t^{i,N} - \bar{C}_t^i| \right. \right. \\ \left. + (\epsilon C_{f,1} + C_{f,2}) \sigma_x^2 \varphi_{\text{rc}} \left( |X_t^{i,N} - \bar{X}_t^i| \right)^2 r_t^i \right] \\ + \epsilon f(r_t^i) \left( 4\tilde{B} - \frac{\lambda}{16}\tilde{H}(\bar{Z}_t^i) - \frac{\lambda}{16}\tilde{H}(Z_t^{i,N}) - \frac{\lambda}{16N} \sum_{j=1}^N \tilde{H}(\bar{Z}_t^j) - \frac{\lambda}{16N} \sum_{j=1}^N \tilde{H}(Z_t^{j,N}) \right), $$

which is a quantity that contains every term we have not yet dealt with. To prove Lemma 3.2, we divide for each $i \in \{1, ..., N\}$ the space into three regions

$$ \text{Reg}_1^i = \{ (\bar{Z}_t^i, Z_t^{i,N}) \text{ s.t. } |\bar{X}_t^i - X_t^{i,N}| \ge \xi \text{ and } r_t^i \le R \}, $$

$$ \text{Reg}_2^i = \{ (\bar{Z}_t^i, Z_t^{i,N}) \text{ s.t. } |\bar{X}_t^i - X_t^{i,N}| < \xi \text{ and } r_t^i \le R_1 \}, $$

$$ \text{Reg}_3^i = \{ (\bar{Z}_t^i, Z_t^{i,N}) \text{ s.t. } r_t^i > R \}, $$

where $R$ was given in Lemma 2.1, and consider

$$ \frac{1}{N} \sum_{i=1}^{N} \mathbb{E} \tilde{K}_t^i = \frac{1}{N} \sum_{i=1}^{N} \left( \mathbb{E} \left( \tilde{K}_t^i \mathbf{1}_{\text{Reg}_1^i} \right) + \mathbb{E} \left( \tilde{K}_t^i \mathbf{1}_{\text{Reg}_2^i} \right) + \mathbb{E} \left( \tilde{K}_t^i \mathbf{1}_{\text{Reg}_3^i} \right) \right). $$

### 3.4.1 Region 1: $\xi \le |X_t^{i,N} - \bar{X}_t^i|$ and $r_t^i \le R$.

In this region of space, since $\varphi_{\text{rc}}(|X_t^{i,N} - \bar{X}_t^i|) = 1$, we have

$$ \tilde{K}_t^i \mathbf{1}_{\text{Reg}_1^i} = \mathbf{1}_{\text{Reg}_1^i} \left( G_t^i \left[ 2cf(r_t^i) + 2\sigma_x^2 f''(r_t^i) + f'(r_t^i) (\epsilon C_{f,1} + C_{f,2}) \sigma_x^2 r_t^i \right. \right. \\ \left. + f'(r_t^i)(1+\gamma\delta + L_X + \delta L_C) |X_t^{i,N} - \bar{X}_t^i| \right] \\ - G_t^i f'(r_t^i)(\delta - 1 - L_X - \delta L_C) |C_t^{i,N} - \bar{C}_t^i| - G_t^i f'(r_t^i) |(X_t^{i,N})^3 - (\bar{X}_t^i)^3| \\ \left. + \epsilon f(r_t^i) 4\tilde{B} - \epsilon f(r_t^i) \left( \frac{\lambda}{16}\tilde{H}(\bar{Z}_t^i) + \frac{\lambda}{16}\tilde{H}(Z_t^{i,N}) + \frac{\lambda}{16N} \sum_{j=1}^N \tilde{H}(\bar{Z}_t^j) + \frac{\lambda}{16N} \sum_{j=1}^N \tilde{H}(Z_t^{j,N}) \right) \right), $$

and since $\tilde{H}(z) \ge 0$, $|X_t^{i,N} - \bar{X}_t^i| \le r_t^i$, $\delta > \frac{1+L_X}{1-L_C}$ (by Assumption 2(iii)) and $1 \le G_t^i$ we have

$$ \tilde{K}_t^i \mathbf{1}_{\text{Reg}_1^i} \le \mathbf{1}_{\text{Reg}_1^i} G_t^i \left[ (2c + 4\epsilon \tilde{B}) f(r_t^i) + 2\sigma_x^2 f''(r_t^i) + f'(r_t^i) (1 + \delta\gamma + L_X + \delta L_C + (\epsilon C_{f,1} + C_{f,2}) \sigma_x^2) r_t^i \right]. $$

Using the definition $f$ given in (2.21) we get

$$ 2\sigma_x^2 f''(r_t^i) + f'(r_t^i) (1 + \delta\gamma + L_X + L_C + (\epsilon C_{f,1} + C_{f,2}) \sigma_x^2) r_t^i $$
With (2.10) from the same Lemma, we obtain for the second sum

$$
\epsilon C_z \sum_{i,j=1}^{N} f(r_t^i) \left( \sqrt{H(Z_t^{i,N})} + \sqrt{H(\bar{Z}_t^i)} \right) \left( \tilde{H}(\bar{Z}_t^j) + \tilde{H}(Z_t^{j,N}) \right) \\
\le \epsilon C_z \frac{2}{a} \sum_{i,j=1}^{N} f(r_t^i) \left( \sqrt{H(Z_t^{i,N})} + \sqrt{H(\bar{Z}_t^i)} \right) \\
\times \left( \sqrt{H(\bar{Z}_t^j)} \exp \left( a \sqrt{H(\bar{Z}_t^j)} \right) + \sqrt{H(Z_t^{j,N})} \exp \left( a \sqrt{H(Z_t^{j,N})} \right) \right).
$$

Since for all $(y_1, y_2, y_3, y_4) \in (\mathbb{R}^+)^4$, we have

$$
(y_1 + y_2) (y_3 e^{ay_3} + y_4 e^{ay_4}) \le 2 (y_1^2 e^{ay_1} + y_2^2 e^{ay_2} + y_3^2 e^{ay_3} + y_4^2 e^{ay_4}),
$$

we obtain for this last sum

$$
\frac{2\epsilon C_z}{a} \sum_{i,j=1}^{N} f(r_t^i) \left( \sqrt{H(Z_t^{i,N})} + \sqrt{H(\bar{Z}_t^i)} \right) \\
\times \left( \sqrt{H(\bar{Z}_t^j)} \exp \left( a \sqrt{H(\bar{Z}_t^j)} \right) + \sqrt{H(Z_t^{j,N})} \exp \left( a \sqrt{H(Z_t^{j,N})} \right) \right) \\
\le \frac{4\epsilon C_z}{a} N \sum_{i=1}^{N} f(r_t^i) \left( H(\bar{Z}_t^i) \exp \left( a \sqrt{H(\bar{Z}_t^i)} \right) + H(Z_t^{i,N}) \exp \left( a \sqrt{H(Z_t^{i,N})} \right) \right) \\
+ \frac{4\epsilon C_z}{a} \sum_{i,j=1}^{N} f(r_t^i) \left( H(\bar{Z}_t^j) \exp \left( a \sqrt{H(\bar{Z}_t^j)} \right) + H(Z_t^{j,N}) \exp \left( a \sqrt{H(Z_t^{j,N})} \right) \right).
$$

Then, by reconsidering the first expression

$$
\frac{1}{N} \sum_{i=1}^{N} \left( \frac{1}{N} f'(r_t^i) G_t^i \sum_{j=1}^{N} \| Z_t^{j,N} - \bar{Z}_t^j \|_1 \right) \\
\le \frac{C_1}{N} \sum_{i=1}^{N} f(r_t^i) G_t^i \\
+ \frac{2\epsilon}{N^2} C_z \sum_{i,j=1}^{N} f(r_t^i) \left( H(\bar{Z}_t^j) \exp \left( a \sqrt{H(\bar{Z}_t^j)} \right) + H(Z_t^{j,N}) \exp \left( a \sqrt{H(Z_t^{j,N})} \right) \right) \\
+ \frac{2\epsilon}{N^2} \frac{4\epsilon C_z}{a} N \sum_{i=1}^{N} f(r_t^i) \left( H(\bar{Z}_t^i) \exp \left( a \sqrt{H(\bar{Z}_t^i)} \right) + H(Z_t^{i,N}) \exp \left( a \sqrt{H(Z_t^{i,N})} \right) \right) \\
+ \frac{2\epsilon}{N^2} \frac{4\epsilon C_z}{a} \sum_{i,j=1}^{N} f(r_t^i) \left( H(\bar{Z}_t^j) \exp \left( a \sqrt{H(\bar{Z}_t^j)} \right) + H(Z_t^{j,N}) \exp \left( a \sqrt{H(Z_t^{j,N})} \right) \right)
$$

This way, by (2.21) since

$$
L_X C_1 \le \frac{c}{2}, \quad 2C_z L_X \le \frac{\lambda}{64} \quad \text{and} \quad L_X \epsilon \frac{8C_z}{a} \le \frac{\lambda}{64},
$$
$$ 
\begin{aligned}
&= 2\sigma_x^2 \phi'(r_t^i) g(r_t^i) + 2\sigma_x^2 \phi(r_t^i) g'(r_t^i) + \phi(r_t^i) g(r_t^i) (1 + \delta\gamma + L_X + \delta L_C + (\epsilon C_{f,1} + C_{f,2}) \sigma_x^2) r_t^i \\
&= 2\sigma_x^2 \phi(r_t^i) g'(r_t^i) = -(2c + 4\epsilon \tilde{B}) \Phi(r_t^i).
\end{aligned}
$$

Thus

$$ 
\begin{aligned}
& (2c + 4\epsilon \tilde{B}) f(r_t^i) + 2\sigma_x^2 f''(r_t^i) + f'(r_t^i) (1 + \delta\gamma + L_X + \delta L_C + (\epsilon C_{f,1} + C_{f,2}) \sigma_x^2) r_t^i \\
&= (2c + 4\epsilon \tilde{B}) f(r_t^i) - (2c + 4\epsilon \tilde{B}) \Phi(r_t^i) \quad (3.12) \\
&\le 0.
\end{aligned}
$$

Eventually, in this region of space

$$ \tilde{K}_t^i \mathbf{1}_{\text{Reg}_1^i} \le 0. $$

### 3.4.2 Region 2: $|X_t^{i,N} - \bar{X}_t^i| < \xi$ and $r_t^i \le R$.

In this region, we can write $\tilde{K}_t^i$ as

$$ 
\begin{aligned}
\tilde{K}_t^i \mathbf{1}_{\text{Reg}_2^i} ={}& \mathbf{1}_{\text{Reg}_2^i} G_t^i [2cf(r_t^i) + \varphi_{\text{rc}} (|X_t^{i,N} - \bar{X}_t^i|^2) [2\sigma_x^2 f''(r_t^i) + (\epsilon C_{f,1} + C_{f,2}) \sigma_x^2 r_t^i f'(r_t^i)] \\
& + f'(r_t^i) ((1 + \gamma\delta + L_X + \delta L_C)|X_t^{i,N} - \bar{X}_t^i| - (\delta - 1 - L_X - \delta L_C)|C_t^{i,N} - \bar{C}_t^i|)] \\
& - \mathbf{1}_{\text{Reg}_2^i} G_t^i f'(r_t^i) |(X_t^{i,N})^3 - (\bar{X}_t^i)^3| \\
& + \mathbf{1}_{\text{Reg}_2^i} \epsilon f(r_t^i) 4\tilde{B} - \epsilon f(r_t^i) \mathbf{1}_{\text{Reg}_2^i} \left( \frac{\lambda}{16} \tilde{H}(\bar{Z}_t^i) + \frac{\lambda}{16} \tilde{H}(Z_t^{i,N}) + \frac{\lambda}{16N} \sum_{j=1}^N \tilde{H}(\bar{Z}_t^j) + \frac{\lambda}{16N} \sum_{j=1}^N \tilde{H}(Z_t^{j,N}) \right).
\end{aligned}
$$

Since $r_t^i = |X_t^{i,N} - \bar{X}_t^i| + \delta|C_t^{i,N} - \bar{C}_t^i|$ and $|X_t^{i,N} - \bar{X}_t^i| < \xi$, we have $|C_t^{i,N} - \bar{C}_t^i| \ge (r_t^i - \xi)/\delta$. Since $\delta > \frac{1+L_X}{1-L_C}$, we obtain

$$ 
\begin{aligned}
\tilde{K}_t^i \mathbf{1}_{\text{Reg}_2^i} &\le \mathbf{1}_{\text{Reg}_2^i} G_t^i [2cf(r_t^i) + \varphi_{\text{rc}} (|X_t^{i,N} - \bar{X}_t^i|^2) [2\sigma_x^2 f''(r_t^i) + (\epsilon C_{f,1} + C_{f,2}) \sigma_x^2 r_t^i f'(r_t^i)] \\
&\quad + f'(r_t^i) \left( (1 + \gamma\delta + L_X + \delta L_C)\xi - (\delta - \delta L_C - 1 - L_X) \frac{r_t^i - \xi}{\delta} \right)] \\
&\quad + \epsilon f(r_t^i) \mathbf{1}_{\text{Reg}_2^i} 4\tilde{B} \\
&\le \varphi_{\text{rc}} (|X_t^{i,N} - \bar{X}_t^i|^2) G_t^i \mathbf{1}_{\text{Reg}_2^i} [2\sigma_x^2 f''(r_t^i) + (\epsilon C_{f,1} + C_{f,2}) \sigma_x^2 r_t^i f'(r_t^i)] \\
&\quad + \mathbf{1}_{\text{Reg}_2^i} G_t^i f'(r_t^i) \xi \left[ 1 + \gamma\delta + L_X + \delta L_C + 1 - L_C - \frac{1+L_X}{\delta} \right] \\
&\quad + \mathbf{1}_{\text{Reg}_2^i} G_t^i \left( (2c + 4\epsilon \tilde{B}) f(r_t^i) - r_t^i f'(r_t^i) \left( 1 - L_C - \frac{1+L_X}{\delta} \right) \right).
\end{aligned}
$$

By (3.12),

$$ 2\sigma_x^2 f''(r_t^i) + (\epsilon C_{f,1} + C_{f,2}) \sigma_x^2 r_t^i f'(r_t^i) = -(2c + 4\epsilon \tilde{B}) \Phi(r_t^i) - f'(r_t^i) r_t^i (1 + \delta\gamma + L_X + L_C) \le 0, $$

and by Lemma 2.5

$$ 2c + 4\epsilon \tilde{B} \le \left(1 - L_C - \frac{1+L_X}{\delta}\right) \min_{r \in [0, R]} \frac{f'(r)r}{f(r)}, $$
we get

$$
\begin{aligned}
& \frac{1}{N} \sum_{i=1}^{N} \left( \frac{1}{N} f'(r_t^i) G_t^i \sum_{j=1}^{N} \|Z_t^{j,N} - \bar{Z}_t^j\|_1 \right) \\
& \le \frac{1}{N} \frac{c}{2L_X} \sum_{i=1}^{N} f(r_t^i) G_t^i \\
& \quad + \frac{\epsilon}{N^2} \frac{\lambda}{64L_X} \sum_{i,j=1}^{N} f(r_t^i) \left( H(\bar{Z}_t^j) \exp \left( a\sqrt{H(\bar{Z}_t^j)} \right) + H(Z_t^{j,N}) \exp \left( a\sqrt{H(Z_t^{j,N})} \right) \right) \\
& \quad + \frac{\epsilon}{N} \frac{\lambda}{64L_X} \sum_{i=1}^{N} f(r_t^i) \left( H(\bar{Z}_t^i) \exp \left( a\sqrt{H(\bar{Z}_t^i)} \right) + H(Z_t^{i,N}) \exp \left( a\sqrt{H(Z_t^{i,N})} \right) \right) \\
& \quad + \frac{\epsilon}{N^2} \frac{\lambda}{64L_X} \sum_{i,j=1}^{N} f(r_t^i) \left( H(\bar{Z}_t^j) \exp \left( a\sqrt{H(\bar{Z}_t^j)} \right) + H(Z_t^{j,N}) \exp \left( a\sqrt{H(Z_t^{j,N})} \right) \right),
\end{aligned}
$$

and we finally obtain “half” the result

$$
\begin{aligned}
& \frac{1}{N} \sum_{i=1}^{N} G_t^i f'(r_t^i) \left( \frac{L_X}{N} \left( \sum_{j=1}^{N} \|Z_t^{j,N} - \bar{Z}_t^j\|_1 \right) \right) - \frac{c}{2N} \sum_{i=1}^{N} f(r_t^i) G_t^i \\
& - \frac{\epsilon}{2} \frac{\lambda}{16N} \sum_{i=1}^{N} f(r_t^i) \left[ H(\bar{Z}_t^i) \exp \left( a\sqrt{H(\bar{Z}_t^i)} \right) + H(Z_t^{i,N}) \exp \left( a\sqrt{H(Z_t^{i,N})} \right) \right] \\
& - \frac{\epsilon}{2} \frac{\lambda}{16N^2} \sum_{i=1}^{N} f(r_t^i) \left[ \sum_{j=1}^{N} H(\bar{Z}_t^j) \exp \left( a\sqrt{H(\bar{Z}_t^j)} \right) + \sum_{j=1}^{N} H(Z_t^{j,N}) \exp \left( a\sqrt{H(Z_t^{j,N})} \right) \right] \le 0.
\end{aligned}
$$

Likewise, by (2.21), since

$$
\delta L_C C_1 \le \frac{c}{2}, \quad 2C_z \delta L_C \le \frac{\lambda}{64} \quad \text{and} \quad \delta L_C \epsilon \frac{8C_z}{a} \le \frac{\lambda}{64},
$$

we obtain the second “half”

$$
\begin{aligned}
& \frac{1}{N} \sum_{i=1}^{N} \delta G_t^i f'(r_t^i) \left( \frac{L_C}{N} \left( \sum_{j=1}^{N} \|Z_t^{j,N} - \bar{Z}_t^j\|_1 \right) \right) - \frac{c}{2N} \sum_{i=1}^{N} f(r_t^i) G_t^i \\
& - \frac{\epsilon}{2} \frac{\lambda}{16N} \sum_{i=1}^{N} f(r_t^i) \left[ H(\bar{Z}_t^i) \exp \left( a\sqrt{H(\bar{Z}_t^i)} \right) + H(Z_t^{i,N}) \exp \left( a\sqrt{H(Z_t^{i,N})} \right) \right] \\
& - \frac{\epsilon}{2} \frac{\lambda}{16N^2} \sum_{i=1}^{N} f(r_t^i) \left[ \sum_{j=1}^{N} H(\bar{Z}_t^j) \exp \left( a\sqrt{H(\bar{Z}_t^j)} \right) + \sum_{j=1}^{N} H(Z_t^{j,N}) \exp \left( a\sqrt{H(Z_t^{j,N})} \right) \right] \le 0.
\end{aligned}
$$

Eventually, we have proved $\sum_{i=1}^{N} I_t^{2,i} \le 0$. $\square$

*Proof of Lemma* 3.4. Since $f'(r) \le 1$, we have by Cauchy-Schwarz inequality

$$
\begin{aligned}
& \mathrm{E} \left( G_t^i f'(r_t^i) \left( \left| \frac{1}{N} \sum_{j=1}^N K_X (\bar{Z}_t^i - \bar{Z}_t^j) - K_X * \bar{\mu}_t (\bar{Z}_t^i) \right| \right) \right) \\
\le & \mathrm{E} (|G_t^i|^2)^{1/2} \mathrm{E} \left( \left| \frac{1}{N} \sum_{j=1}^N K_X (\bar{Z}_t^i - \bar{Z}_t^j) - K_X * \bar{\mu}_t (\bar{Z}_t^i) \right|^2 \right)^{1/2} .
\end{aligned}
$$
we obtain

$$ \tilde{K}_t^i \mathbf{1}_{\text{Reg}_2^i} \le \mathbf{1}_{\text{Reg}_2^i} G_t^i f'(r_t^i) \xi \left[ 1 + \gamma \delta + L_X + \delta L_C + 1 - L_C - \frac{1 + L_X}{\delta} \right]. $$

Finally, since $f'(r) \le 1$,

$$ \mathbb{E} \tilde{K}_t^i \mathbf{1}_{\text{Reg}_2^i} \le \xi \left( 2 + \delta \gamma + L_X + \delta L_C - L_C - \frac{1 + L_X}{\delta} \right) \mathbb{E} G_t^i. $$

### 3.4.3 Region 3: $r_t^i \ge R$.

In this region of space $f' = f'' = 0$ and $f$ is constant, and we therefore have

$$ \tilde{K}_t^i \mathbf{1}_{\text{Reg}_3^i} = f(r_t^i) \mathbf{1}_{\text{Reg}_3^i} \left[ 2cG_t^i + 4\epsilon \tilde{B} - \frac{\lambda \epsilon}{16} \left( \tilde{H}(\bar{Z}_t^i) + \tilde{H}(Z_t^{i,N}) + \frac{1}{N} \sum_{j=1}^N \tilde{H}(\bar{Z}_t^j) + \frac{1}{N} \sum_{j=1}^N \tilde{H}(Z_t^{j,N}) \right) \right]. $$

Since $G_t^i = 1 + \epsilon \tilde{H}(\bar{Z}_t^i) + \epsilon \tilde{H}(Z_t^{i,N}) + \frac{\epsilon}{N} \sum_{j=1}^N \tilde{H}(Z_t^{j,N}) + \frac{\epsilon}{N} \sum_{j=1}^N \tilde{H}(\bar{Z}_t^j)$ by definition (2.24), we can write

$$ \tilde{K}_t^i \mathbf{1}_{\text{Reg}_3^i} = f(r_t^i) \mathbf{1}_{\text{Reg}_3^i} \left[ 2c + 4\epsilon \tilde{B} + \epsilon \left( 2c - \frac{\lambda}{16} \right) \left( \tilde{H}(\bar{Z}_t^i) + \tilde{H}(Z_t^{i,N}) + \frac{1}{N} \sum_{j=1}^N \tilde{H}(\bar{Z}_t^j) + \frac{1}{N} \sum_{j=1}^N \tilde{H}(Z_t^{j,N}) \right) \right] $$

Since $c \le \lambda/32$ from Assumption 2, we obtain

$$ \tilde{K}_t^i \mathbf{1}_{\text{Reg}_3^i} \le f(r_t^i) \mathbf{1}_{\text{Reg}_3^i} \left[ 2c + 4\epsilon \tilde{B} - \epsilon \left( \frac{\lambda}{16} - 2c \right) \left( H(\bar{Z}_t^i) + H(Z_t^{i,N}) \right) \right]. $$

We have chosen $R$ such that, for $z, z'$ satisfying $r \ge R$, we have $H(z) + H(z') \ge 80 \frac{\tilde{B}}{\lambda}$ by Lemma 2.1 (iv). Therefore

$$ \begin{aligned} \tilde{K}_t^i \mathbf{1}_{\text{Reg}_3^i} &\le f(r_t^i) \mathbf{1}_{\text{Reg}_3^i} \left( 2c + 4\epsilon \tilde{B} - \epsilon \left( \frac{\lambda}{16} - 2c \right) 80 \frac{\tilde{B}}{\lambda} \right) \\ &= f(r_t^i) \mathbf{1}_{\text{Reg}_3^i} \left( 2c \left( 1 + 80 \frac{\epsilon \tilde{B}}{\lambda} \right) - \epsilon \tilde{B} \right) \end{aligned} $$

Lemma 2.5 and more specifically the inequality

$$ c \le \frac{1}{2} \frac{\epsilon \tilde{B}}{1 + 80 \frac{\epsilon \tilde{B}}{\lambda}} = \frac{\lambda}{160} \frac{\frac{80 \epsilon \tilde{B}}{\lambda}}{1 + \frac{80 \epsilon \tilde{B}}{\lambda}} $$

yields the desired result: $\tilde{K}_t^i \mathbf{1}_{\text{Reg}_3^i} \le 0$.

# A Various technical lemmas

## A.1 On Lemma 2.1

**Lemma A.1.** For all $z = (x, c), z' = (x', c') \in \mathbb{R}^d$, denoting $r(z, z') = |x - x'| + \delta|c - c|$

$$ r(z, z')^2 \le \frac{16(1 + \delta^2)}{\min(\gamma, 1)} (H(z) + H(z')) , \quad (A.1) $$
By Lemma 3.2, we have for each $i \le N$, for all $t \ge 0$, $E[(G_t^i)^2] \le C_{G,2}$.

Moreover, we notice that the $(\bar{Z}_t^j)_j$ are i.i.d with law $\bar{\mu}_t$. Let's denote $\bar{Z}_t$ a generic random variable of law $\bar{\mu}_t$ independent of $\bar{Z}_t^i$. The calculus of the right term of the product has already been done in Subsection 1.4, and we have (1.10)

$$ E \left( \left| \frac{1}{N} \sum_{j=1}^{N} K_X(\bar{Z}_t^i - \bar{Z}_t^j) - K_X * \bar{\mu}_t(\bar{Z}_t^i) \right|^2 \right) \le \frac{8L_X^2}{N} E(\|\bar{Z}_t\|_1^2). $$

A similar calculation yields

$$ E \left( \left| \frac{1}{N} \sum_{j=1}^{N} K_C(\bar{Z}_t^i - \bar{Z}_t^j) - K_C * \bar{\mu}_t(\bar{Z}_t^i) \right|^2 \right) \le \frac{8L_C^2}{N} E(\|\bar{Z}_t\|_1^2). $$

By Lemma 2.4, $E(|\bar{X}_t|^2 + |\bar{C}_t|^2) \le C_{init,2}$. In particular,

$$ E (\|\bar{Z}_t\|_1^2) = E (\left[ |\bar{X}_t| + |\bar{C}_t| \right]^2) \le 2E (|\bar{X}_t|^2 + |\bar{C}_t|^2) \le 2C_{init,2}. $$

Thus

$$ E \left( G_t^i f'(r_t^i) \left( \left| \frac{1}{N} \sum_{j=1}^{N} K_X(\bar{Z}_t^i - \bar{Z}_t^j) - K_X * \bar{\mu}_t(\bar{Z}_t^i) \right| \right) \right) \le L_X C_{G,2}^{1/2} \sqrt{2C_{init,2}} \sqrt{\frac{8}{N}}, $$

and likewise

$$ E \left( G_t^i f'(r_t^i) \left( \left| \frac{1}{N} \sum_{j=1}^{N} K_C(\bar{Z}_t^i - \bar{Z}_t^j) - K_C * \bar{\mu}_t(\bar{Z}_t^i) \right| \right) \right) \le L_C C_{G,2}^{1/2} \sqrt{2C_{init,2}} \sqrt{\frac{8}{N}}. $$

□

## 3.4 Contraction in various regions of space

The goal of this section is to prove Lemma 3.3, i.e show that for each $i \le N$, for all $t > 0$, we have the following control

$$ E \tilde{K}_t^i \le \xi \left( 2 + \delta\gamma + L_X + \delta L_C - L_C - \frac{1 + L_X}{\delta} \right) E G_t^i. $$

Recall

$$ \tilde{K}_t^i = G_t^i \left[ 2cf(r_t^i) + \frac{1}{2}f''(r_t^i) \left( 2\sigma_X^2 \varphi_{rc} (|X_t^{i,N} - \bar{X}_t^i|^2) \right. \right. \\ + f'(r_t^i) \left( (1 + \gamma\delta + L_X + \delta L_C) |X_t^{i,N} - \bar{X}_t^i| - |(X_t^{i,N})^3 - (\bar{X}_t^i)^3| \right. \\ + (1 + L_X + \delta L_C - \delta) |C_t^{i,N} - \bar{C}_t^i| + (\epsilon C_{f,1} + C_{f,2}) \sigma_X^2 \varphi_{rc} (|X_t^{i,N} - \bar{X}_t^i|^2 r_t^i) \left. \right] \\ + \epsilon f(r_t^i) \left( 4\tilde{B} - \frac{\lambda}{16}\tilde{H}(\bar{Z}_t^i) - \frac{\lambda}{16}\tilde{H}(Z_t^{i,N}) - \frac{\lambda}{16N} \sum_{j=1}^{N} \tilde{H}(\bar{Z}_t^j) - \frac{\lambda}{16N} \sum_{j=1}^{N} \tilde{H}(Z_t^{j,N}) \right), $$

which is a quantity that contains every term we have not yet dealt with. To prove Lemma 3.3, we divide for each $i \in \{1, \dots, N\}$ the space into three regions

$$ \text{Reg}_1^i = \{ (\bar{Z}_t^i, Z_t^{i,N}) \text{ s.t. } |\bar{X}_t^i - X_t^{i,N}| \ge \xi \text{ and } r_t^i \le R \}, $$

$$ \text{Reg}_2^i = \{ (\bar{Z}_t^i, Z_t^{i,N}) \text{ s.t. } |\bar{X}_t^i - X_t^{i,N}| < \xi \text{ and } r_t^i \le R_1 \}, $$

$$ \text{Reg}_3^i = \{ (\bar{Z}_t^i, Z_t^{i,N}) \text{ s.t. } r_t^i > R \}, $$
so that, in particular, for any constant $B > 0$, if $r(z, z') \ge R = \sqrt{\frac{1280(1 + \delta^2)B}{\lambda \min(\gamma, 1)}}$, then
$$ \lambda H(z) + \lambda H(z') \ge 80B. $$

*Proof.* We have $H(z) \ge \frac{\gamma}{4}x^2 + \frac{c^2}{4} \ge \frac{1}{4}\min(\gamma, 1)(x^2 + c^2)$. Thus

$$ \begin{aligned} r(z, z')^2 &= (|x - x'| + \delta|c - c'|)^2 \\ &\le 2|x - x'|^2 + 2\delta^2|c - c'|^2 \\ &\le 4x^2 + 4x'^2 + 4\delta^2c^2 + 4\delta^2c'^2 \\ &\le 4(1 + \delta^2)(x^2 + c^2) + 4(1 + \delta^2)(x'^2 + c'^2) \\ &\le 16 \frac{(1 + \delta^2)}{\min(\gamma, 1)} (H(z) + H(z')) \end{aligned} $$

<figure><img src="image_3.png" /></figure>

## A.2 Proof of Lyapunov's property of H and its consequences

### Lyapunov's property

*Proof of Lemma 2.2.* We write the proof for (2.3), as it also yields (2.4) by considering $\mu$ to be the empirical measure. We notice

$$ \partial_c H = c + \alpha \quad \text{and} \quad \partial_x H = \gamma x + \beta, $$

so

$$ \begin{aligned} \mathcal{L}_{\mu} H(z) &= \partial_x H(z)(x - x^3) + \partial_x H(z) K_X * \mu(z) - c \partial_c H(z) + \partial_c H(z) K_C * \mu(z) + \frac{\sigma_x^2 \gamma}{2} + \frac{\sigma_c^2}{2} \\ &= (\gamma x + \beta)(x - x^3) - c(c + \alpha) + (\gamma x + \beta) K_X * \mu(z) + (c + \alpha) K_C * \mu(z) + \frac{\sigma_x^2 \gamma}{2} + \frac{\sigma_c^2}{2}. \end{aligned} $$

First, we focus on interaction terms. We have

$$ \begin{aligned} |K_X * \mu(z)| &\le \int_{\mathbb{R}^2} |K_X(z - z')| \mu(dz') \\ &\le \int_{\mathbb{R}^2} L_X (\|z\|_1 + \|z'\|_1) \mu(dz'). \end{aligned} $$

Hence,

$$ \begin{aligned} (\gamma x + \beta) K_X * \mu(z) &\le L_X (\gamma|x| + \beta)(|x| + |c| + \mathbb{E}_\mu(|X|) + \mathbb{E}_\mu(|C|)) \\ &\le L_X (\gamma|x|^2 + \gamma|x||c| + \gamma|x|\mathbb{E}_\mu(|X|) + \gamma|x|\mathbb{E}_\mu(|C|) + \beta|x| + \beta|c| + \beta\mathbb{E}_\mu(|X|) + \beta\mathbb{E}_\mu(|C|)) \end{aligned} $$

and using Young's inequality $ab \le \frac{\alpha}{2}a^2 + \frac{1}{2\alpha}b^2$ ($\alpha = 16$ when we separate $x$ term and $c$ term, and $\alpha = 1$ otherwise on the various terms we get

$$ (\gamma x + \beta) K_X * \mu(z) \le L_X \left( \gamma|x|^2 + 8\gamma^2|x|^2 + \frac{|c|^2}{32} + \frac{\gamma}{2}|x|^2 + \frac{\gamma}{2}\mathbb{E}_\mu(|X|)^2 + 8\gamma^2|x|^2 + \frac{\mathbb{E}_\mu(|C|)^2}{32} + \frac{\beta^2}{2} \right. \\ \left. + \frac{|x|^2}{2} + 8\beta^2 + \frac{|c|^2}{32} + \frac{\beta^2}{2} + \frac{1}{2}\mathbb{E}_\mu(|X|)^2 + 8\beta^2 + \frac{\mathbb{E}_\mu(|C|)^2}{32} \right) $$
where R was given in Lemma 2.1, and consider

$$ \frac{1}{N} \sum_{i=1}^{N} \mathbb{E} \tilde{K}_{t}^{i} = \frac{1}{N} \sum_{i=1}^{N} \left( \mathbb{E} \left( \tilde{K}_{t}^{i} \mathbb{1}_{\text{Reg}_{1}^{i}} \right) + \mathbb{E} \left( \tilde{K}_{t}^{i} \mathbb{1}_{\text{Reg}_{2}^{i}} \right) + \mathbb{E} \left( \tilde{K}_{t}^{i} \mathbb{1}_{\text{Reg}_{3}^{i}} \right) \right). $$

**3.4.1 Region 1: $\xi \le |X_t^{i,N} - \bar{X}_t^i|$ and $r_t^i \le R$**

In this region of space, since $\varphi_{\text{rc}}(|X_t^{i,N} - \bar{X}_t^i|) = 1$, we have

$$ \tilde{K}_t^i \mathbb{1}_{\text{Reg}_1^i} = \mathbb{1}_{\text{Reg}_1^i} \left( G_t^i \left[ 2cf(r_t^i) + 2\sigma_X^2 f''(r_t^i) + f'(r_t^i) (\epsilon C_{f,1} + C_{f,2}) \sigma_X^2 r_t^i + f'(r_t^i)(1 + \gamma\delta + L_X + \delta L_C) |X_t^{i,N} - \bar{X}_t^i| \right] \right. \\ \left. - G_t^i f'(r_t^i)(\delta - 1 - L_X - \delta L_C) |C_t^{i,N} - \bar{C}_t^i| - G_t^i f'(r_t^i) |(X_t^{i,N})^3 - (\bar{X}_t^i)^3| \right. \\ \left. + \epsilon f(r_t^i) 4\tilde{B} - \epsilon f(r_t^i) \left( \frac{\lambda}{16} \tilde{H}(\bar{Z}_t^i) + \frac{\lambda}{16} \tilde{H}(Z_t^{i,N}) + \frac{\lambda}{16N} \sum_{j=1}^N \tilde{H}(\bar{Z}_t^j) + \frac{\lambda}{16N} \sum_{j=1}^N \tilde{H}(Z_t^{j,N}) \right) \right), $$

and since $\tilde{H}(z) \ge 0$, $|X_t^{i,N} - \bar{X}_t^i| \le r_t^i$, $\delta > \frac{1+L_X}{1-L_C}$ (by the choice given in Subsection 2.4) and $1 \le G_t^i$ we have

$$ \tilde{K}_t^i \mathbb{1}_{\text{Reg}_1^i} \le \mathbb{1}_{\text{Reg}_1^i} G_t^i \left[ (2c + 4\epsilon \tilde{B}) f(r_t^i) + 2\sigma_X^2 f''(r_t^i) + f'(r_t^i) (1 + \delta\gamma + L_X + \delta L_C + (\epsilon C_{f,1} + C_{f,2}) \sigma_X^2) r_t^i \right]. $$

Using the definition $f$ given in (2.20) we get

$$ \begin{aligned} & 2\sigma_X^2 f''(r_t^i) + f'(r_t^i) (1 + \delta\gamma + L_X + L_C + (\epsilon C_{f,1} + C_{f,2}) \sigma_X^2) r_t^i \\ &= 2\sigma_X^2 \phi'(r_t^i) g(r_t^i) + 2\sigma_X^2 \phi(r_t^i) g'(r_t^i) \\ & \quad + \phi(r_t^i) g(r_t^i) (1 + \delta\gamma + L_X + \delta L_C + (\epsilon C_{f,1} + C_{f,2}) \sigma_X^2) r_t^i \\ &= 2\sigma_X^2 \phi(r_t^i) g'(r_t^i) = -(2c + 4\epsilon \tilde{B}) \Phi(r_t^i). \end{aligned} $$

Thus

$$ \begin{aligned} & (2c + 4\epsilon \tilde{B}) f(r_t^i) + 2\sigma_X^2 f''(r_t^i) + f'(r_t^i) (1 + \delta\gamma + L_X + \delta L_C + (\epsilon C_{f,1} + C_{f,2}) \sigma_X^2) r_t^i \\ &= (2c + 4\epsilon \tilde{B}) f(r_t^i) - (2c + 4\epsilon \tilde{B}) \Phi(r_t^i) \quad (3.12) \\ &\le 0. \end{aligned} $$

Eventually, in this region of space

$$ \tilde{K}_t^i \mathbb{1}_{\text{Reg}_1^i} \le 0. $$
$$ =L_X \left( 17\beta^2 + |x|^2 \left( \frac{1}{2} + \frac{3}{2}\gamma + 16\gamma^2 \right) + \frac{|c|^2}{16} + \mathbb{E}_{\mu}(|X|)^2 \left( \frac{\gamma}{2} + \frac{1}{2} \right) + \frac{\mathbb{E}_{\mu}(|C|)^2}{16} \right). $$

Likewise

$$ (c + \alpha) K_C * \mu(z) \le L_C \left( 17\alpha^2 + \frac{17}{2}|x|^2 + |c|^2 \left( \frac{3}{2} + \frac{3}{32} \right) + \mathbb{E}_{\mu}(|X|)^2 \frac{17}{2} + \mathbb{E}_{\mu}(|C|)^2 \left( \frac{1}{2} + \frac{1}{32} \right) \right). $$

The idea is to bound $\lambda H(z) + \mathcal{L}_{\mu}H(z)$, by distinguishing 3 types of terms: we isolate terms in $\mathbb{E}_{\mu}(|C|)^2 - c^2$, $\mathbb{E}_{\mu}(|X|)^2 - x^2$, and we group polynomials terms. Then, we notice the polynomial is upper bounded by a constant A. Thus

$$ \begin{aligned} & \lambda H(z) + \mathcal{L}_{\mu}H(z) - \frac{\sigma_x^2\gamma}{2} - \frac{\sigma_c^2}{2} \\ &= \lambda \left( \frac{1}{2}\gamma x^2 + \beta x + \frac{1}{2}c^2 + \alpha c + H_0 \right) + (\gamma x + \beta)(x - x^3) - c(c + \alpha) \\ & \quad + (\gamma x + \beta)K_X * \mu(z) + (c + \alpha)K_C * \mu(z) \\ & \le (\lambda H_0 + 17\beta^2 L_X + 17\alpha^2 L_C) - \gamma x^4 - \beta x^3 + (1 + \lambda)\beta x \\ & \quad + \left( (1 + \frac{\lambda}{2})\gamma + L_X (1 + 2\gamma + 16\gamma^2) + 17L_C \right) x^2 \\ & \quad + \left( \frac{L_C}{8} + L_C \left( 2 + \frac{1}{8} \right) - \left( 1 - \frac{\lambda}{2} \right) \right) c^2 - (1 - \lambda)\alpha c \\ & \quad + \left( \frac{L_X}{16} + \frac{L_C}{2} + \frac{L_C}{32} \right) (\mathbb{E}_{\mu}(|C|)^2 - c^2) \\ & \quad + \left( \frac{\gamma}{2}L_X + \frac{1}{2}L_X + \frac{17}{2}L_C \right) (\mathbb{E}_{\mu}(|X|)^2 - x^2) \end{aligned} $$

Provided

$$ \frac{L_X}{8} + L_C \left( 2 + \frac{1}{8} \right) < 1 - \frac{\lambda}{2}, $$

there is $A \ge 0$ such that

$$ -\gamma x^4 - \beta x^3 + (1 + \lambda)\beta x + \left( (1 + \frac{\lambda}{2})\gamma + L_X (1 + 2\gamma + 16\gamma^2) + 17L_C \right) x^2 \\ + \left( \frac{L_C}{8} + L_C \left( 2 + \frac{1}{8} \right) - \left( 1 - \frac{\lambda}{2} \right) \right) c^2 - (1 - \lambda)\alpha c \le A. $$

Hence the result:

$$ \mathcal{L}_{\mu}H(\bar{z}) \le B + (\alpha_X L_X + \beta_X L_C) (\mathbb{E}_{\mu}(|X|)^2 - \bar{x}^2) + (\alpha_C L_X + \beta_C L_C) (\mathbb{E}_{\mu}(|C|)^2 - \bar{c}^2) - \lambda H(\bar{z}). $$

□

## First consequences

*Proof of Proposition 2.1.* Inequality (2.5) simply relies on the sum of (2.4) for each $i$ and the fact that $\mathcal{L}^{j,N}(H(Z_t^{i,N})) = 0$ for $i \ne j$:

$$ \frac{1}{N} \sum_{i=1}^{N} \mathcal{L}^{N} \left( H \left( Z_{t}^{i,N} \right) \right) = \frac{1}{N} \sum_{i=1}^{N} \mathcal{L}^{i,N} \left( H \left( Z_{t}^{i,N} \right) \right) $$
**3.4.2 Region 2:** $|X_t^{i,N} - \bar{X}_t^i| < \xi$ and $r_t^i \le R$.

In this region, we can write $\tilde{K}_t^i$ as

$$ \tilde{K}_t^i \mathbf{1}_{\text{Reg}_2^i} = \mathbf{1}_{\text{Reg}_2^i} G_t^i \left[ \begin{aligned} & 2cf(r_t^i) + \varphi_{\text{rc}} (|X_t^{i,N} - \bar{X}_t^i|)^2 [2\sigma_X^2 f''(r_t^i) + (\epsilon C_{f,1} + C_{f,2}) \sigma_X^2 r_t^i f'(r_t^i)] \\ & + f'(r_t^i) \left( (1 + \gamma\delta + L_X + \delta L_C) |X_t^{i,N} - \bar{X}_t^i| - (\delta - 1 - L_X - \delta L_C) |C_t^{i,N} - \bar{C}_t^i| \right) \\ & - \mathbf{1}_{\text{Reg}_2^i} G_t^i f'(r_t^i) |(X_t^{i,N})^3 - (\bar{X}_t^i)^3| \\ & + \mathbf{1}_{\text{Reg}_2^i} \epsilon f(r_t^i) 4\tilde{B} \\ & - \epsilon f(r_t^i) \mathbf{1}_{\text{Reg}_2^i} \left( \frac{\lambda}{16} \tilde{H}(\bar{Z}_t^i) + \frac{\lambda}{16} \tilde{H}(Z_t^{j,N}) + \frac{\lambda}{16N} \sum_{j=1}^N \tilde{H}(\bar{Z}_t^j) + \frac{\lambda}{16N} \sum_{j=1}^N \tilde{H}(Z_t^{j,N}) \right) \end{aligned} \right]. $$

Since $r_t^i = |X_t^{i,N} - \bar{X}_t^i| + \delta|C_t^{i,N} - \bar{C}_t^i|$ and $|X_t^{i,N} - \bar{X}_t^i| < \xi$, we have $|C_t^{i,N} - \bar{C}_t^i| \ge (r_t^i - \xi)/\delta$. Since $\delta > \frac{1+L_X}{1-L_C}$, we obtain

$$ \begin{aligned} \tilde{K}_t^i \mathbf{1}_{\text{Reg}_2^i} &\le \mathbf{1}_{\text{Reg}_2^i} G_t^i \left[ \begin{aligned} & 2cf(r_t^i) + \varphi_{\text{rc}} (|X_t^{i,N} - \bar{X}_t^i|)^2 [2\sigma_X^2 f''(r_t^i) + (\epsilon C_{f,1} + C_{f,2}) \sigma_X^2 r_t^i f'(r_t^i)] \\ & + f'(r_t^i) \left( (1 + \gamma\delta + L_X + \delta L_C)\xi - (\delta - \delta L_C - 1 - L_X) \frac{r_t^i - \xi}{\delta} \right) \\ & + \epsilon f(r_t^i) \mathbf{1}_{\text{Reg}_2^i} 4\tilde{B} \\ & \le \varphi_{\text{rc}} (|X_t^{i,N} - \bar{X}_t^i|)^2 G_t^i \mathbf{1}_{\text{Reg}_2^i} [2\sigma_X^2 f''(r_t^i) + (\epsilon C_{f,1} + C_{f,2}) \sigma_X^2 r_t^i f'(r_t^i)] \\ & + \mathbf{1}_{\text{Reg}_2^i} G_t^i f'(r_t^i) \xi \left[ 1 + \gamma\delta + L_X + \delta L_C + 1 - L_C - \frac{1+L_X}{\delta} \right] \\ & + \mathbf{1}_{\text{Reg}_2^i} G_t^i \left( (2c + 4\epsilon\tilde{B})f(r_t^i) - r_t^i f'(r_t^i) \left( 1 - L_C - \frac{1+L_X}{\delta} \right) \right). \end{aligned} \right. $$

By (3.12),

$$ \begin{aligned} 2\sigma_X^2 f''(r_t^i) + (\epsilon C_{f,1} + C_{f,2}) \sigma_X^2 r_t^i f'(r_t^i) &= -(2c + 4\epsilon\tilde{B})\Phi(r_t^i) - f'(r_t^i)r_t^i(1 + \delta\gamma + L_X + L_C) \\ &\le 0, \end{aligned} $$

and by Lemma 2.6

$$ 2c + 4\epsilon\tilde{B} \le \left(1 - L_C - \frac{1+L_X}{\delta}\right) \min_{r \in (0, R]} \frac{f'(r)r}{f(r)}, $$

we obtain

$$ \tilde{K}_t^i \mathbf{1}_{\text{Reg}_2^i} \le \mathbf{1}_{\text{Reg}_2^i} G_t^i f'(r_t^i) \xi \left[ 1 + \gamma\delta + L_X + \delta L_C + 1 - L_C - \frac{1+L_X}{\delta} \right]. $$

Finally, since $f'(r) \le 1$,

$$ \mathbb{E} \tilde{K}_t^i \mathbf{1}_{\text{Reg}_2^i} \le \xi \left( 2 + \delta\gamma + L_X + \delta L_C - L_C - \frac{1+L_X}{\delta} \right) \mathbb{E} G_t^i. $$