$$
\le \frac{1}{N} \sum_{i=1}^{N} \left[ B + (\alpha_X L_X + \beta_X L_C) \left( \left( \frac{1}{N} \sum_{k=1}^{N} |X_t^{k,N}| \right)^2 - (X_t^{i,N})^2 \right) \right. \\
\left. \qquad + (\alpha_C L_X + \beta_C L_C) \left( \left( \frac{1}{N} \sum_{k=1}^{N} |C_t^{k,N}| \right)^2 - (C_t^{i,N})^2 \right) - \lambda H(Z_t^{i,N}) \right] \\
\le B - \lambda \frac{1}{N} \sum_{i=1}^{N} H(Z_t^{i,N}).
$$

The last inequality uses the fact that $\left(\frac{1}{N} \sum_{i=1}^{N} |y_i|\right)^2 - \frac{1}{N} \sum_{i=1}^{N} (y_i)^2 \le 0$ for all $(y_i)_{1\le i\le N} \in \mathbb{R}^N$. $\square$

**Bounds on the second moments of processes** We can now prove the uniform in bounds on the second moments of $X_t^{i,N}$, $C_t^{i,N}$, $\bar{X}_t^i$ and $\bar{C}_t^i$ from (2.3) and (2.4).

*Proof of Proposition 1.2.* $K_X$ and $K_C$ are Lipschitz with constants $L_X$ and $L_C$ respectively. We do not assume any bound on these constants. We assume for each $i \le N$, $\mathbb{E}(|X_0^{i,N}|^2) < +\infty$ and $\mathbb{E}(|C_0^{i,N}|^2) < +\infty$. We have

$$
d \left( \frac{e^{\lambda t}}{N} \sum_{i=1}^{N} H(Z_t^{i,N}) \right) = \lambda \frac{e^{\lambda t}}{N} \sum_{i=1}^{N} H(Z_t^{i,N}) dt + e^{\lambda t} \mathcal{L}^N \left( \frac{1}{N} \sum_{i=1}^{N} H(Z_t^{i,N}) \right) dt + dM_t,
$$

where $M_t$ is a local martingale. Using (2.4)

$$
d \left( \frac{e^{\lambda t}}{N} \sum_{i=1}^{N} H(Z_t^{i,N}) \right) = A_t dt + dM_t,
$$

where $A_t \le Be^{\lambda t}$. Let $\tau_n$ be an increasing sequence of localizing stopping times converging to $\infty$ for $M_t$

$$
\begin{align*}
\mathbb{E} \left( \frac{e^{\lambda t \wedge \tau_n}}{N} \sum_{i=1}^{N} H(Z_{t \wedge \tau_n}^{i,N}) \right) &\le \mathbb{E} \left( \frac{1}{N} \sum_{i=1}^{N} H(Z_0^{i,N}) \right) + \mathbb{E} \left( \int_0^{t \wedge \tau_n} B e^{\lambda s} ds \right) \\
&\le \mathbb{E} \left( \frac{1}{N} \sum_{i=1}^{N} H(Z_0^{i,N}) \right) + B \frac{\mathbb{E} (e^{\lambda t \wedge \tau_n}) - 1}{\lambda} \\
&\le \mathbb{E} \left( \frac{1}{N} \sum_{i=1}^{N} H(Z_0^{i,N}) \right) + B \max \left( \frac{e^{\lambda t} - 1}{\lambda}, \frac{1}{|\lambda|} \right),
\end{align*}
$$

where the maximum on this last inequality depends on the sign of $\lambda$. By Fatou's lemma, we obtain

$$
\begin{align*}
e^{\lambda t} \mathbb{E} \left( \frac{1}{N} \sum_{i=1}^{N} H(Z_t^{i,N}) \right) &= \mathbb{E} \left( \liminf_{n \to \infty} \frac{e^{\lambda t \wedge \tau_n}}{N} \sum_{i=1}^{N} H(Z_{t \wedge \tau_n}^{i,N}) \right) \\
&\le \liminf_{n \to \infty} \mathbb{E} \left( \frac{e^{\lambda t \wedge \tau_n}}{N} \sum_{i=1}^{N} H(Z_{t \wedge \tau_n}^{i,N}) \right) \\
&\le \mathbb{E} \left( \frac{1}{N} \sum_{i=1}^{N} H(Z_0^{i,N}) \right) + B \max \left( \frac{e^{\lambda t} - 1}{\lambda}, \frac{1}{|\lambda|} \right).
\end{align*}
$$

Hence the various bounds on $\mathbb{E}(|X_t^{i,N}|^2)$ and $\mathbb{E}(|C_t^{i,N}|^2)$, since by Lemma 2.1 (i) we have

$$
\mathbb{E}H(Z_t^{i,N}) \ge \frac{\gamma}{4}\mathbb{E}(|X_t^{i,N}|^2) + \frac{1}{4}\mathbb{E}(|C_t^{i,N}|^2) \quad \text{and} \quad \mathbb{E}H(Z_0^{i,N}) \le \gamma\mathbb{E}(|X_0^{i,N}|^2) + \mathbb{E}(|C_0^{i,N}|^2) + \frac{3}{2}H_0.
$$
### 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 (2.26), 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 $c \le \lambda/32$ by the choice given in Subsection 2.4, 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 2.6 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 Itô's formula for the $L^1$ norm

Let us here detail the calculations leading to the use of Itô's formula to derive the dynamics of the $L^1$ norm of the processes. At first glance it should not be possible, as the absolute value is not a twice continuously differentiable function. However, in our case, we consider a diffusion coefficient which is zero around the point of discontinuity of the function. The following lemma is based on the calculations done in Lemma 7 of [8], and relies on an approximation of the absolute value function and usual convergence lemmas. We here give a quite general result.

**Lemma A.1.** Let $(X_t, X'_t, C_t, C'_t)$ be continuous processes and $F, G: \mathbb{R}^+ \times \mathbb{R}^4 \to \mathbb{R}$ be two continuous functions. Assume furthermore that there is $R_G > 0$ such that $G(t, x, x', c, c') = 0$ if $|x - x'| < R_G$ and that $G$ is bounded. Consider the dynamics

$$ d(X_t - X'_t) = F(t, X_t, X'_t, C_t, C'_t)dt + G(t, X_t, X'_t, C_t, C'_t)dB_t, $$
These bounds are uniform in time provided $\lambda > 0$, i.e. $\frac{L_X}{8} + L_C (2 + \frac{3}{32}) < 1$.

*Proof of Proposition 1.3 and Lemma 2.3.* The proof are done in exactly the same fashion as the proof of Proposition 1.2 above using (2.3).

### A.3 Proof of Lemma 2.5

We now prove that there are constants $c, \epsilon$ and $\delta$ such that

$$c + 2\epsilon \tilde{B} \le \frac{\sigma_x^2}{2} \left( \int_0^R \Phi(s) \phi(s)^{-1} ds \right)^{-1} \qquad (A.2)$$

$$2c + 4\epsilon \tilde{B} \le \left(1 - L_C - \frac{1 + L_X}{\delta}\right) \min_{r \in [0, R]} \frac{f'(r)r}{f(r)} \qquad (A.3)$$

$$c \le \frac{\lambda \frac{80\epsilon \tilde{B}}{\lambda}}{160 \left(1 + \frac{80\epsilon \tilde{B}}{\lambda}\right)} \qquad (A.4)$$

$$\delta > \frac{1 + L_X}{1 - L_C} \qquad (A.5)$$

* Since for all $u \ge 0$, $0 < \phi(u) \le 1$, we have $0 < \Phi(s) = \int_0^s \phi(u) du \le s$, i.e $s/\Phi(s) \ge 1$. Therefore

$$\inf_{r \in [0, R]} \frac{r\phi(r)}{\Phi(r)} \ge \inf_{r \in [0, R]} \phi(r) = \phi(R).$$

It is thus sufficient for (A.3) to have

$$2c + 4\epsilon \tilde{B} \le \frac{1}{2} \left( 1 - L_C - \frac{1 + L_X}{\delta} \right) \phi(R).$$

* We have

$$\phi(r) \le \exp\left(-\frac{1}{4\sigma_x^2}r^2\right).$$

So

$$\Phi(r) \le \int_0^\infty \exp\left(-\frac{r^2}{4\sigma_x^2}\right) dr = \sigma_x \sqrt{\pi}.$$

Then

$$\int_0^R \frac{\Phi(r)}{\phi(r)} dr \le \sigma_x \sqrt{\pi} R \frac{1}{\phi(R)}.$$

It is thus sufficient for (A.2) that

$$c + 2\epsilon \tilde{B} \le \frac{\sigma_x \phi(R)}{2\sqrt{\pi} R}.$$

* The various conditions involving $c$ invite us to consider $2\epsilon \tilde{B} = \eta c$. Then

$$\begin{align*} c \le \frac{\lambda \frac{80\epsilon \tilde{B}}{\lambda}}{160 \left(1 + \frac{80\epsilon \tilde{B}}{\lambda}\right)} &\iff c \le \frac{40\eta c}{160 \left(\lambda + 40\eta c\right)} \\ &\iff 1 \le \lambda \frac{\eta}{4\lambda + 160\eta c} \quad (\text{since } c \ge 0) \\ &\iff c \le \frac{\lambda \eta - 4}{160 \eta}. \end{align*}$$
where B is a Brownian motion. Then almost surely for all $t \ge 0$

$$d|X_t - X_t'| = \text{sign}(X_t - X_t')F(t, X_t, X_t', C_t, C_t')dt \\ + \text{sign}(X_t - X_t')G(t, X_t, X_t', C_t, C_t')dB_t, \qquad (A.1)$$

where

$$\text{sign}(x) = \begin{cases} 1 & \text{if } x > 0 \\ 0 & \text{if } x = 0 \\ -1 & \text{if } x < 0. \end{cases}$$

*Proof.* By the standard Itô's formula for twice continuously differentiable functions, we have

$$d(X_t - X_t')^2 = 2(X_t - X_t')F(t, X_t, X_t', C_t, C_t')dt + 2(X_t - X_t')G(t, X_t, X_t', C_t, C_t')dB_t \\ + G^2(t, X_t, X_t', C_t, C_t')dt.$$

Consider, for $\eta > 0$ (which in the end will go to 0), the function $\psi_{\eta}(r) = (r + \eta)^{1/2}$ which is smooth on $[0, \infty[$ and satisfies

$$\forall r > 0, \lim_{\eta \to 0} \psi_{\eta}(r) = r^{1/2}, \lim_{\eta \to 0} 2\psi'_{\eta}(r) = r^{-1/2}, \lim_{\eta \to 0} 4\psi''_{\eta}(r) = -r^{-3/2}$$

and thus $\lim_{\eta \to 0} 2r\psi''_{\eta}(r) + \psi'_{\eta}(r) = 0$ and $\forall r \in \mathbb{R}$, $\lim_{\eta \to 0} 2r\psi'_{\eta}(r^2) = \text{sign}(r)$.

Then

$$d\psi_{\eta}((X_t - X_t')^2) = 2(X_t - X_t')\psi'_{\eta}((X_t - X_t')^2)F(t, X_t, X_t', C_t, C_t')dt \\ + 2(X_t - X_t')\psi'_{\eta}((X_t - X_t')^2)G(t, X_t, X_t', C_t, C_t')dB_t \\ + \psi'_{\eta}((X_t - X_t')^2)G^2(t, X_t, X_t', C_t, C_t')dt \\ + 2(X_t - X_t')^2\psi''_{\eta}((X_t - X_t')^2)G^2(t, X_t, X_t', C_t, C_t')dt,$$

which is just another way of writing that for all $t \ge 0$

$$\psi_{\eta}((X_t - X_t')^2) = \psi_{\eta}((X_0 - X_0')^2) \\ + \int_0^t 2(X_s - X_s')\psi'_{\eta}((X_s - X_s')^2)F(s, X_s, X_s', C_s, C_s')ds \\ + \int_0^t 2(X_s - X_s')\psi'_{\eta}((X_s - X_s')^2)G(s, X_s, X_s', C_s, C_s')dB_s \\ + \int_0^t (\psi'_{\eta}((X_s - X_s')^2) + 2(X_s - X_s')^2\psi''_{\eta}((X_s - X_s')^2))G^2(s, X_s, X_s', C_s, C_s')ds$$

We now compute the limit of each term. First

$$\psi_{\eta}((X_t - X_t')^2) \underset{\eta \to 0}{\longrightarrow} |X_t - X_t'| \quad \text{and} \quad \psi_{\eta}((X_0 - X_0')^2) \underset{\eta \to 0}{\longrightarrow} |X_0 - X_0'|.$$

Then

$$|2(X_s - X_s')\psi'_{\eta}((X_s - X_s')^2)| = \frac{|X_s - X_s'|}{\sqrt{(X_s - X_s')^2 + \eta}} \le 1.$$
• We choose to write

$$ \delta = (1 + \tilde{\delta}) \frac{1 + L_{X,max}}{1 - L_{C,max}} > \frac{1 + L_X}{1 - L_C} $$

• Let us assume, for simplicity, that $\epsilon \le 1$. It is sufficient for this later condition to have

$$ c \le \frac{2\tilde{B}}{\eta}. $$

• The appearance of $\phi(R)$ suggests we should try to minimize it. We recall

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

It is therefore sufficient for (A.2) to have

$$ c \le \frac{1}{1 + \eta} \frac{\sigma_x}{2\sqrt{\pi}} \frac{1}{R} \exp \left( -\frac{1}{4\sigma_x^2} \left( 1 + \delta\gamma + L_X + \delta L_C + (C_{f,1} + C_{f,2}) \sigma_x^2 \right) R^2 \right), $$

and for (A.3) to have

$$ c \le \frac{1}{2(1 + \eta)} \left( 1 - L_C - \frac{1 + L_X}{\delta} \right) \exp \left( -\frac{1}{4\sigma_x^2} \left( 1 + \delta\gamma + L_X + \delta L_C + (C_{f,1} + C_{f,2}) \sigma_x^2 \right) R^2 \right). $$

• Finally, we bound $L_X$ and $L_C$ by either 0 or $L_{X,max}$ and $L_{C,max}$, to obtain bounds on $c$ independent of $L_X$ and $L_C$.

## A.4 Proof of Lemma 2.6

Let $z, z' \in \mathbb{R}^2$.

**Proof of control of the $L^1$ distance :** We have

$$ \|z - z'\|_1 = |x - x'| + |c - c'| \le \frac{1}{\min(\delta, 1)} (|x - x'| + \delta|c - c'|) = \frac{1}{\min(\delta, 1)} r(z, z'). $$

If $r(z, z') \le 1 \le R$, we have, using Lemma 2.5

$$ r(z, z') \le \frac{f(r)}{f'_-(R)} \le \frac{f(r)}{\phi(R)g(R)} \left( 1 + \epsilon \tilde{H}(z) + \epsilon \tilde{H}(z') \right). $$

If $r(z, z') \ge 1$, we have, using (A.1)

$$ \begin{aligned} r(z, z') & \le r(z, z')^2 \\ & \le \frac{16(1 + \delta^2)}{\epsilon \min(\gamma, 1)} (\epsilon H(z) + \epsilon H(z')) \\ & \le \frac{16(1 + \delta^2)}{\epsilon \min(\gamma, 1)} \frac{f(r)}{f(1)} (1 + \epsilon H(z) + \epsilon H(z')) \\ & \le \frac{16(1 + \delta^2)}{\epsilon \min(\gamma, 1)} \frac{f(r)}{\phi(R)g(R)} (1 + \epsilon \tilde{H}(z) + \epsilon \tilde{H}(z')). \end{aligned} $$

Thus

$$ \|z - z'\|_1 \le \frac{1}{\min(\delta, 1)} \frac{1}{\phi(R)g(R)} \max \left( \frac{16(1 + \delta^2)}{\epsilon \min(\gamma, 1)}, 1 \right) f(r(z, z')) (1 + \epsilon \tilde{H}(z) + \epsilon \tilde{H}(z')). $$
Thus for all $t \ge 0$, by dominated convergence (recall $F$ is a continuous function, thus integrable on $[0, t]$), we obtain almost surely

$$ \int_0^t 2(X_s - X_s')\psi_\eta'((X_s - X_s')^2)F(s, X_s, X_s', C_s, C_s')ds \\ \xrightarrow{\eta \to 0} \int_0^t \text{sign}(X_s - X_s')F(s, X_s, X_s', C_s, C_s')ds, $$

and by Theorem 2.12 Chapter 4 of [26], almost surely we have

$$ \int_0^t 2(X_s - X_s')\psi_\eta'((X_s - X_s')^2)G(s, X_s, X_s', C_s, C_s')dB_s \\ \xrightarrow{\eta \to 0} \int_0^t \text{sign}(X_s - X_s')G(s, X_s, X_s', C_s, C_s')dB_s, $$

Finally, since $G(s, X_s, X_s', C_s, C_s') = 0$ if $|X_s - X_s'| < R_G$ and

$$ \begin{aligned} & \psi_\eta'((X_s - X_s')^2) + 2(X_s - X_s')^2 \psi_\eta''((X_s - X_s')^2) \\ &= \frac{1}{2} \left( \frac{1}{\sqrt{(X_s - X_s')^2 + \eta}} - \frac{(X_s - X_s')^2}{((X_s - X_s')^2 + \eta)^{3/2}} \right) \\ &= \frac{\eta}{2 \sqrt{(X_s - X_s')^2 + \eta}}^{3/2} \\ &\le \frac{\eta}{2 |X_s - X_s'|^3}, \end{aligned} $$

by dominated convergence we almost surely have

$$ \int_0^t \left( \psi_\eta'((X_s - X_s')^2) + 2(X_s - X_s')^2 \psi_\eta''((X_s - X_s')^2) \right) G^2(s, X_s, X_s', C_s, C_s')ds \xrightarrow{\eta \to 0} 0 $$

Thus for all $t \ge 0$ we almost surely have (A.1), and continuity allows us to conclude that we almost surely have for all $t \ge 0$ (A.1). □

## A.2 On Lemma 2.1

**Lemma A.2.** For all $z = (x, c), z' = (x', c') \in \mathbb{R}^d$, denoting $r(z, z') = |x - x'| + \delta|c - c'|$

$$ r(z, z')^2 \le \frac{16(1 + \delta^2)}{\min(\gamma, 1)} (H(z) + H(z')), \quad (A.2) $$

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

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

□
**Proof of control of the L2 distance :** We have

$$r(z, z')^2 = (|x - x'| + \delta|c - c'|)^2 \geq |x - x'|^2 + \delta^2|c - c'|^2 \geq \min(1, \delta^2) (|x - x'|^2 + |c - c'|^2).$$

If $r(z, z') \geq 1$, we have, using (A.1)

$$ \begin{aligned} r(z, z')^2 &\leq \frac{16(1 + \delta^2)}{\epsilon \min(\gamma, 1)} (\epsilon H(z) + \epsilon H(z')) \\ &\leq \frac{16(1 + \delta^2)}{\epsilon \min(\gamma, 1)} \frac{f(r)}{f(1)} (1 + \epsilon H(z) + \epsilon H(z')) \\ &\leq \frac{16(1 + \delta^2)}{\epsilon \min(\gamma, 1)} \frac{f(r)}{\phi(R)g(R)} (1 + \epsilon \tilde{H}(z) + \epsilon \tilde{H}(z')). \end{aligned} $$

If $r(z, z') \leq 1 \leq R$, we have, using Lemma 2.5

$$r(z, z')^2 \leq r(z, z') \leq \frac{f(r)}{f'_{-}(R)} \leq \frac{f(r)}{\phi(R)g(R)} (1 + \epsilon \tilde{H}(z) + \epsilon \tilde{H}(z')).$$

Thus

$$\|z - z'\|_2^2 \leq \frac{1}{\min(\delta^2, 1)} \frac{1}{\phi(R)g(R)} \max\left(\frac{16(1 + \delta^2)}{\epsilon \min(\gamma, 1)}, 1\right) f(r(z, z')) (1 + \epsilon \tilde{H}(z) + \epsilon \tilde{H}(z')).$$

**Proof of the second control of the L¹ distance :** We have, if $r(z, z') \leq 1 \leq R$

$$r(z, z') \leq \frac{f(r)}{f'_{-}(R)} \leq \frac{f(r)}{\phi(R)g(R)} \left(1 + \epsilon \sqrt{H(z)} + \epsilon \sqrt{H(z')}\right).$$

and, if $r(z, z') \geq 1$, recall Lemma 2.1.

$$ \begin{aligned} \|z - z'\|_1 &\leq \sqrt{\frac{4}{\gamma}H(z)} + \sqrt{\frac{4}{\gamma}H(z')} + \sqrt{4H(z)} + \sqrt{4H(z')} \\ &\leq 4 \max\left(\sqrt{\frac{1}{\gamma}}, 1\right) \left(\sqrt{H(z)} + \sqrt{H(z')}\right) \\ &\leq \frac{4}{\epsilon} \max\left(\sqrt{\frac{1}{\gamma}}, 1\right) \frac{f(r)}{\phi(R)g(R)} \left(1 + \epsilon\sqrt{H(z)} + \epsilon\sqrt{H(z')}\right), \end{aligned} $$

and thus

$$\|z - z'\|_1 \leq \frac{1}{\phi(R)g(R)} \max\left(1, \frac{4}{\epsilon}\max\left(\sqrt{\frac{1}{\gamma}}, 1\right)\right) f(r(z, z')) \left(1 + \epsilon\sqrt{H(z)} + \epsilon\sqrt{H(z')}\right).$$

**Independence with respect to $L_X$ and $L_C$** The a priori bounds $L_X \in [0, L_{X,max}]$ and $L_C \in [0, L_{C,max}]$ allow us to bound $\phi(R)$ independently of $L_C$ and $L_X$ by $\phi_{\min}$ (and we also use $g(R) \geq \frac{1}{2}$), thus giving us constant $C_1$, $C_2$ and $C_z$ independent of $L_C$ and $L_X$.

## A.5 Proof of Lemmas 3.1 and 3.6

*Proof of Lemma 3.1.* Let's prove there exists a uniform in time bound on $E(G_t^i)$ and $E[(G_t^i)^2]$. First, let's recall the definition of $G$ from(2.24):

$$G_t^i = 1 + \epsilon \tilde{H}(\bar{Z}_t^i) + \epsilon \tilde{H}(Z_t^{i,N}) + \frac{\epsilon}{N} \sum_{j=1}^{N} \tilde{H}(Z_t^{j,N}) + \frac{\epsilon}{N} \sum_{j=1}^{N} \tilde{H}(\bar{Z}_t^j).$$
### A.3 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

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

First, we focus on the 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,

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

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

$$ \begin{aligned} & (\gamma x + \beta) K_X * \mu(z) \\ &\le L_X \left( \gamma|x|^2 + 8\gamma^2|x|^2 + \frac{|c|^2}{32} + \frac{\gamma}{2}|x|^2 + \frac{\gamma}{2}\mathbb{E}_\mu(|X|)^2 + 8\gamma^2|x|^2 + \frac{\mathbb{E}_\mu(|C|)^2}{32} + \frac{\beta^2}{2} \right. \\ &\qquad \left. + \frac{|x|^2}{2} + 8\beta^2 + \frac{|c|^2}{32} + \frac{\beta^2}{2} + \frac{1}{2}\mathbb{E}_\mu(|X|)^2 + 8\beta^2 + \frac{\mathbb{E}_\mu(|C|)^2}{32} \right) \\ &\quad = L_X \left( 17\beta^2 + |x|^2 \left( \frac{1}{2} + \frac{3}{2}\gamma + 16\gamma^2 \right) + \frac{|c|^2}{16} + \mathbb{E}_\mu(|X|)^2 \left( \frac{\gamma}{2} + \frac{1}{2} \right) + \frac{\mathbb{E}_\mu(|C|)^2}{16} \right). \end{aligned} $$

Likewise

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

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 polynomial terms. Then, we notice
The idea is to bound the different expectation in terms of the expectation at time $t = 0$. Since $\mathbb{E}(e^{\tilde{a}(|X_0|+|C_0|)})$ is finite, we know that for each $k \in \mathbb{N}$, $\mathbb{E}(|X_0|^k)$ and $\mathbb{E}(|C_0|^k)$ are also finite. We deduce that for each $k \in \mathbb{N}$, for each $j \le N$, $\mathbb{E}[H(\bar{Z}_0^j)]$ and $\mathbb{E}[H(Z_0^{j,N})]$ are finite.

In fact, to bound uniformly in time the first moment, we only have to bound $\mathbb{E}(\tilde{H}(Z_t^{j,N}))$ and $\mathbb{E}(\tilde{H}(\bar{Z}_t^j))$ for each $j \le N$. Let's begin with $\bar{Z}^j$. By (2.15), we have

$$ \frac{d}{dt} \mathbb{E} [\tilde{H}(\bar{Z}_t^j)] \le \tilde{B} - \frac{\lambda}{4} \mathbb{E} [\tilde{H}(\bar{Z}_t^j)]. $$

By using Itô's formula on $e^{\lambda t/4} \tilde{H}(\bar{Z}_t^j)$ and the bound above, we obtain

$$ \begin{aligned} \mathbb{E}[\tilde{H}(\bar{Z}_t^j)] &\le \frac{4\tilde{B}}{\lambda} + e^{-\frac{\lambda}{4}t} \left( \mathbb{E}[\tilde{H}(\bar{Z}_0^j)] - \frac{4\tilde{B}}{\lambda} \right) \\ &\le \max \left( \mathbb{E}[\tilde{H}(\bar{Z}_0^j)], \frac{4\tilde{B}}{\lambda} \right). \end{aligned} $$

By (2.9), in Lemma 2.4, we deduce the following inequality and we apply Cauchy-Schwarz

$$ \begin{aligned} \mathbb{E}[\tilde{H}(\bar{Z}_0^j)] &\le \mathbb{E} \left[ H(\bar{Z}_0^j) \exp \left( a \sqrt{H(\bar{Z}_0^j)} \right) \right] \\ &\le \mathbb{E} \left[ H(\bar{Z}_0^j)^2 \right]^{1/2} \mathbb{E} \left[ \exp \left( 2a \sqrt{H(\bar{Z}_0^j)} \right) \right]^{1/2} \end{aligned} \quad (A.6) $$

We already know $\mathbb{E}[H(\bar{Z}_0^j)^2]$ is bounded. Now, it is enough to prove that there exist $C$ such that for all $z \in \mathbb{R}^2$

$$ \exp(2a\sqrt{H(z)}) \le C \times e^{\tilde{a}(|x|+|c|)}. $$

In fact, from the definition of $H$ in (2.1), we have

$$ \begin{aligned} 2\sqrt{H(z)} &= \sqrt{2}\sqrt{\gamma \left(x + \frac{\beta}{\gamma}\right)^2 + (c + \alpha)^2 + H_0} \\ &\le \sqrt{2\gamma} \left|x + \frac{\beta}{\gamma}\right| + \sqrt{2}|c + \alpha| + \sqrt{H_0} \\ &\le \sqrt{2\gamma}|x| + \sqrt{2}|c| + \frac{1}{a} \ln C, \end{aligned} $$

where $C$ is a constant independent of $z$. Finally, since $\max(a\sqrt{2\gamma}, a\sqrt{2}) \le \tilde{a}$, we have

$$ \exp(2a\sqrt{H(z)}) \le C \times e^{\tilde{a}(|x|+|c|)}. $$

Then, $\mathbb{E}[\exp(2a\sqrt{H(\bar{Z}_0^j)})]$ is bounded and we deduce $\mathbb{E}(\tilde{H}(\bar{Z}_t^j))$ is bounded for each $j \le N$ and all $t \ge 0$.

The same calculations can be done for $Z_t^{j,N}$. By (2.20), we have

$$ \mathcal{L}^N \left( \frac{1}{N} \sum_{i=1}^{N} \tilde{H}(Z_t^{i,N}) \right) \le \tilde{B} - \frac{\lambda}{4} \left( \frac{1}{N} \sum_{i=1}^{N} \tilde{H}(Z_t^{i,N}) \right). $$
the polynomial is upper bounded by a constant A. Thus

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

Provided that

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

there is $A \ge 0$ such that

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

Hence the result

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

**First consequences**

*Proof of Proposition 2.3.* 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 $i \neq j$

$$
\begin{align*}
\frac{1}{N} \sum_{i=1}^{N} \mathcal{L}^{N} (H(Z_t^{i,N})) &= \frac{1}{N} \sum_{i=1}^{N} \mathcal{L}^{i,N} (H(Z_t^{i,N})) \\
&\leq \frac{1}{N} \sum_{i=1}^{N} \left[ B + (\alpha_X L_X + \beta_X L_C) \left( \left( \frac{1}{N} \sum_{k=1}^{N} |X_t^{k,N}| \right)^2 - (X_t^{i,N})^2 \right) \right. \\
&\qquad \left. + (\alpha_C L_X + \beta_C L_C) \left( \left( \frac{1}{N} \sum_{k=1}^{N} |C_t^{k,N}| \right)^2 - (C_t^{i,N})^2 \right) - \lambda H(Z_t^{i,N}) \right] \\
&\leq B - \lambda \frac{1}{N} \sum_{i=1}^{N} H(Z_t^{i,N}).
\end{align*}
$$
In particular,

$$ \frac{d}{dt} \left[ \mathbb{E} \left( \frac{1}{N} \sum_{i=1}^{N} \tilde{H}(Z_t^{i,N}) \right) \right] \le \mathbb{E} \left[ \mathcal{L}^N \left( \frac{1}{N} \sum_{i=1}^{N} \tilde{H}(Z_t^{i,N}) \right) \right] \le \tilde{B} - \frac{\lambda}{4} \mathbb{E} \left( \frac{1}{N} \sum_{i=1}^{N} \tilde{H}(Z_t^{i,N}) \right), $$

and we can use the same method as above.

Finally, we have proved that for each $j \le N$, $\mathbb{E}(\tilde{H}(Z_t^{j,N}))$ and $\mathbb{E}(\tilde{H}(\bar{Z}_t^j))$ are bounded uniformly in time. Thus, $\mathbb{E}(G_t^i)$ is bounded uniformly in time (and in $N$).

To bound the second moment of $G_t^i$, we have to bound each type of the following expectations: $\mathbb{E}[\tilde{H}(Z_t^{j_1,N})\tilde{H}(Z_t^{j_2,N})]$, $\mathbb{E}[\tilde{H}(Z_t^{j_1,N})\tilde{H}(\bar{Z}_t^{j_2})]$, $\mathbb{E}[\tilde{H}(\bar{Z}_t^{j_1})\tilde{H}(\bar{Z}_t^{j_2})]$, $\mathbb{E}[\tilde{H}(Z_t^{j,N})^2]$ and $\mathbb{E}[\tilde{H}(\bar{Z}_t^j)^2]$. By Cauchy-Schwarz, it is in fact enough to bound $\mathbb{E}[\tilde{H}(Z_t^{j,N})^2]$ and $\mathbb{E}[\tilde{H}(\bar{Z}_t^j)^2]$.

First, by the definition of $\tilde{H}$ in (2.8),

$$ \begin{aligned} \tilde{H}(z)^2 &= \left( \frac{2}{a^2} \exp \left( a\sqrt{H(z)} \right) \left( a\sqrt{H(z)} - 1 \right) + \frac{2}{a^2} \right)^2 \\ &\le 2 \frac{2^2}{a^4} \exp \left( 2a\sqrt{H(z)} \right) \left( a\sqrt{H(z)} - 1 \right)^2 + 2 \frac{2^2}{a^4} \\ &\le \frac{8}{a^4} \exp \left( 2a\sqrt{H(z)} \right) \left( 2a^2 H(z) + 2 \right) + \frac{8}{a^4}. \end{aligned} $$

As for the first moment, the study of $Z_t^{j,N}$ is very similar to the one of $\bar{Z}_t^j$. Here, we only focus on the second one.

To bound $\mathbb{E}[\tilde{H}(\bar{Z}_t^j)^2]$, by Cauchy-Schwarz, it is sufficient to bound $\mathbb{E}[\exp(4a\sqrt{H(\bar{Z}_t^j)})]$ and $\mathbb{E}[H(\bar{Z}_t^j)^2]$. The latter has already been bounded uniformly in time, and the former can be obtained by the same calculations as previously, replacing $a$ by $4a$ (and thus assuming $\tilde{a} \ge 4\sqrt{2a} \max(\sqrt{\gamma}, 1)$, which we do). Finally, we deduce $\mathbb{E}((G_t^i)^2)$ is bounded uniformly in time. $\square$

**Proof of Lemma 3.6.** Using $\partial_x H(z) = \gamma x + \beta$, we have

$$ \begin{aligned} \left| \partial_x \tilde{H}(Z_t^{i,N}) - \partial_x \tilde{H}(\bar{Z}_t^i) \right| &= \left| (\gamma X_t^{i,N} + \beta) \exp \left( a\sqrt{H(Z_t^{i,N})} \right) - (\gamma \bar{X}_t^i + \beta) \exp \left( a\sqrt{H(\bar{Z}_t^i)} \right) \right| \\ &\le \left| \gamma X_t^{i,N} - \gamma \bar{X}_t^i \right| \left( \exp \left( a\sqrt{H(Z_t^{i,N})} \right) + \exp \left( a\sqrt{H(\bar{Z}_t^i)} \right) \right) \\ &\quad + \left| \gamma \bar{X}_t^i + \beta \right| \left| \exp \left( a\sqrt{H(Z_t^{i,N})} \right) - \exp \left( a\sqrt{H(\bar{Z}_t^i)} \right) \right|. \end{aligned} $$

$$ \text{Since } |X_t^{i,N} - \bar{X}_t^i| \le r_t^i, $$

$$ \left| \gamma X_t^{i,N} - \gamma \bar{X}_t^i \right| \left( \exp \left( a\sqrt{H(Z_t^{i,N})} \right) + \exp \left( a\sqrt{H(\bar{Z}_t^i)} \right) \right) \le \gamma r_t^i \left( \exp \left( a\sqrt{H(Z_t^{i,N})} \right) + \exp \left( a\sqrt{H(\bar{Z}_t^i)} \right) \right). $$

By Lemma 2.1 (ii), we have $H(z) \ge \frac{1}{2} \min(\frac{1}{\gamma}, 1)(\gamma x + \beta)^2$. By the mean value theorem, for all $y_1 \le y_2$ in $\mathbb{R}$, there exists $y_3 \in [y_1, y_2]$ such that: $e^{ay_1} - e^{ay_2} = a(y_1 - y_2)e^{ay_3}$. In particular, $|e^{ay_1} - e^{ay_2}| \le a|y_1 - y_2|(e^{ay_1} + e^{ay_2})$. Thus

$$ \left| \gamma \bar{X}_t^i + \beta \right| \left| \exp \left( a \sqrt{H(Z_t^{i,N})} \right) - \exp \left( a \sqrt{H(\bar{Z}_t^i)} \right) \right| $$
The last inequality uses the fact that $(\frac{1}{N} \sum_{i=1}^{N} |y_i|)^2 - \frac{1}{N} \sum_{i=1}^{N} (y_i)^2 \le 0$ for all $(y_i)_{1 \le i \le N} \in \mathbb{R}^N$. □

**Bounds on the second moments of processes** We can now prove the uniform in bounds on the second moments of $X_t^{i,N}$, $C_t^{i,N}$, $\bar{X}_t^i$ and $\bar{C}_t^i$ from (2.3) and (2.4). Let's notice that $(X_t^{i,N,\kappa}, C_t^{i,N,\kappa})_i$ coincides with $(X_t^{i,N}, C_t^{i,N})_i$ before the time $T_\kappa$ defined in Subsection 1.3. Since our interest is in $(X_t^{i,N}, C_t^{i,N})_i$, we chose to give the proof of the Proposition 1.6. The proof of the Lemma 1.5 is very similar.

**Proof of Proposition 1.6.** $K_X$ and $K_C$ are Lipschitz with constants $L_X$ and $L_C$ respectively. We do not assume any bounds on these constants. We assume for each $i \le N$, $E(|X_0^{i,N}|^2) < +\infty$ and $E(|C_0^{i,N}|^2) < +\infty$. We have

$$d\left(\frac{e^{\lambda t}}{N} \sum_{i=1}^{N} H(Z_t^{i,N})\right) = \lambda \frac{e^{\lambda t}}{N} \sum_{i=1}^{N} H(Z_t^{i,N}) dt + e^{\lambda t} \mathcal{L}^N \left(\frac{1}{N} \sum_{i=1}^{N} H(Z_t^{i,N})\right) dt + dM_t,$$

where $M_t$ is a local martingale. Using (2.4)

$$d\left(\frac{e^{\lambda t}}{N} \sum_{i=1}^{N} H(Z_t^{i,N})\right) = A_t dt + dM_t,$$

where $A_t \le B e^{\lambda t}$. Let $\tau_n$ be an increasing sequence of localizing stopping times converging to $\infty$ for $M_t$

$$\begin{aligned}
E\left(\frac{e^{\lambda t \wedge \tau_n}}{N} \sum_{i=1}^{N} H(Z_{t \wedge \tau_n}^{i,N})\right) &\le E\left(\frac{1}{N} \sum_{i=1}^{N} H(Z_0^{i,N})\right) + E\left(\int_0^{t \wedge \tau_n} B e^{\lambda s} ds\right) \\
&\le E\left(\frac{1}{N} \sum_{i=1}^{N} H(Z_0^{i,N})\right) + B \frac{E(e^{\lambda t \wedge \tau_n}) - 1}{\lambda} \\
&\le E\left(\frac{1}{N} \sum_{i=1}^{N} H(Z_0^{i,N})\right) + B \max\left(\frac{e^{\lambda t} - 1}{\lambda}, \frac{1}{|\lambda|}\right),
\end{aligned}$$

where the maximum on this last inequality depends on the sign of $\lambda$. By Fatou's lemma, we obtain

$$\begin{aligned}
e^{\lambda t} E\left(\frac{1}{N} \sum_{i=1}^{N} H(Z_t^{i,N})\right) &= E\left(\liminf_{n \to \infty} \frac{e^{\lambda t \wedge \tau_n}}{N} \sum_{i=1}^{N} H(Z_{t \wedge \tau_n}^{i,N})\right) \\
&\le \liminf_{n \to \infty} E\left(\frac{e^{\lambda t \wedge \tau_n}}{N} \sum_{i=1}^{N} H(Z_{t \wedge \tau_n}^{i,N})\right) \\
&\le E\left(\frac{1}{N} \sum_{i=1}^{N} H(Z_0^{i,N})\right) + B \max\left(\frac{e^{\lambda t} - 1}{\lambda}, \frac{1}{|\lambda|}\right).
\end{aligned}$$

Hence the various bounds on $E(|X_t^{i,N}|^2)$ and $E(|C_t^{i,N}|^2)$, since by Lemma 2.1 (i) we have

$$\begin{aligned}
\mathrm{EH}(Z_t^{i,N}) &\ge \frac{\gamma}{4} \mathrm{E}(|X_t^{i,N}|^2) + \frac{1}{4} \mathrm{E}(|C_t^{i,N}|^2) \\
\mathrm{and} \quad \mathrm{EH}(Z_0^{i,N}) &\le \gamma \mathrm{E}(|X_0^{i,N}|^2) + \mathrm{E}(|C_0^{i,N}|^2) + \frac{3}{2} H_0.
\end{aligned}$$

These bounds are uniform in time provided $\lambda > 0$, i.e $\frac{L_X}{8} + L_C(2 + \frac{3}{32}) < 1$. □
$$
\begin{aligned}
&\le a \sqrt{\frac{2H(\bar{Z}_t^i)}{\min\left(\frac{1}{\gamma}, 1\right)}} \left|\sqrt{H(Z_t^{i,N})} - \sqrt{H(\bar{Z}_t^i)}\right| \left(\exp\left(a\sqrt{H(Z_t^{i,N})}\right) + \exp\left(a\sqrt{H(\bar{Z}_t^i)}\right)\right) \\
&\le a\sqrt{2\max(\gamma, 1)} \left|H(Z_t^{i,N}) - H(\bar{Z}_t^i)\right| \left(\exp\left(a\sqrt{H(Z_t^{i,N})}\right) + \exp\left(a\sqrt{H(\bar{Z}_t^i)}\right)\right).
\end{aligned}
$$

Then by the definition of H we get

$$
\begin{aligned}
|H(Z_t^{i,N}) - H(\bar{Z}_t^i)| &= \left| \frac{1}{2}\gamma \left( (X_t^{i,N})^2 - (\bar{X}_t^i)^2 \right) + \beta(X_t^{i,N} - \bar{X}_t^i) + \frac{1}{2} \left( (C_t^{i,N})^2 - (\bar{C}_t^i)^2 \right) + \alpha(C_t^{i,N} - \bar{C}_t^i) \right| \\
&\le \frac{1}{2}\gamma |X_t^{i,N} - \bar{X}_t^i| |X_t^{i,N} + \bar{X}_t^i| + \beta |X_t^{i,N} - \bar{X}_t^i| + \frac{1}{2} |C_t^{i,N} - \bar{C}_t^i| |C_t^{i,N} + \bar{C}_t^i| + \alpha |C_t^{i,N} - \bar{C}_t^i|.
\end{aligned}
$$

Now, by Lemma 2.1 (i), we have $H(z) \ge \frac{\gamma}{4}x^2 + \frac{1}{4}c^2$ and since $|X_t^{i,N} - \bar{X}_t^i| \le r_t^i$ and $|C_t^{i,N} - \bar{C}_t^i| \le r_t^i/\delta$, we get

$$
|X_t^{i,N} - \bar{X}_t^i| \left( \frac{1}{2}\gamma |X_t^{i,N} + \bar{X}_t^i| + \beta \right) \le r_t^i \left( \sqrt{\gamma} \left( \sqrt{H(Z_t^{i,N})} + \sqrt{H(\bar{Z}_t^i)} \right) + \beta \right)
$$

and

$$
|C_t^{i,N} - \bar{C}_t^i| \left( \frac{1}{2} |C_t^{i,N} + \bar{C}_t^i| + \alpha \right) \le \frac{r_t^i}{\delta} \left( \sqrt{H(Z_t^{i,N})} + \sqrt{H(\bar{Z}_t^i)} + \alpha \right).
$$

Thus

$$
|H(Z_t^{i,N}) - H(\bar{Z}_t^i)| \le \left(\beta + \frac{\alpha}{\delta}\right) r_t^i + \left(\sqrt{\gamma} + \frac{1}{\delta}\right) r_t^i \left(\sqrt{H(Z_t^{i,N})} + \sqrt{H(\bar{Z}_t^i)}\right).
$$

Finally,

$$
\begin{aligned}
& |\partial_x \tilde{H}(Z_t^{i,N}) - \partial_x \tilde{H}(\bar{Z}_t^i)| \\
&\le \gamma r_t^i \left( \exp\left(a\sqrt{H(Z_t^{i,N})}\right) + \exp\left(a\sqrt{H(\bar{Z}_t^i)}\right) \right) \\
&\quad + a\sqrt{2\max(\gamma, 1)} \left(\beta + \frac{\alpha}{\delta}\right) r_t^i \left( \exp\left(a\sqrt{H(Z_t^{i,N})}\right) + \exp\left(a\sqrt{H(\bar{Z}_t^i)}\right) \right) \\
&\quad + a\sqrt{2\max(\gamma, 1)} \left(\sqrt{\gamma} + \frac{1}{\delta}\right) r_t^i \left( \sqrt{H(Z_t^{i,N})} + \sqrt{H(\bar{Z}_t^i)} \right) \left( \exp\left(a\sqrt{H(Z_t^{i,N})}\right) + \exp\left(a\sqrt{H(\bar{Z}_t^i)}\right) \right) \\
&\le r_t^i \left( \gamma + a\sqrt{2\max(\gamma, 1)} \left(\beta + \frac{\alpha}{\delta}\right) \right) \left( \exp\left(a\sqrt{H(Z_t^{i,N})}\right) + \exp\left(a\sqrt{H(\bar{Z}_t^i)}\right) \right) \\
&\quad + ar_t^i \sqrt{2\max(\gamma, 1)} \left(\sqrt{\gamma} + \frac{1}{\delta}\right) \left( 2\sqrt{H(Z_t^{i,N})} \exp\left(a\sqrt{H(Z_t^{i,N})}\right) + 2\sqrt{H(\bar{Z}_t^i)} \exp\left(a\sqrt{H(\bar{Z}_t^i)}\right) \right).
\end{aligned}
$$

Now, we can finally use Lemma 2.4, and more precisely (2.9) and (2.10), we obtain

$$
\begin{aligned}
& |\partial_x \tilde{H}(Z_t^{i,N}) - \partial_x \tilde{H}(\bar{Z}_t^i)| \\
&\le r_t^i \left( \gamma + a\sqrt{2\max(\gamma, 1)} \left(\beta + \frac{\alpha}{\delta}\right) \right) \left( \tilde{H}(Z_t^{i,N}) + \tilde{H}(\bar{Z}_t^i) + \frac{4}{a^2} \left( \exp\left(\frac{a^2}{2}\right) - 1 \right) \right) \\
&\quad + ar_t^i \sqrt{2\max(\gamma, 1)} \left( \sqrt{\gamma} + \frac{1}{\delta} \right) \left( 2a\tilde{H}(Z_t^{i,N}) + \frac{2}{a}(e-2) + 2a\tilde{H}(\bar{Z}_t^i) + \frac{2}{a}(e-2) \right) \\
&\le r_t^i \left( \tilde{H}(Z_t^{i,N}) + \tilde{H}(\bar{Z}_t^i) \right) \left[ \gamma + a\sqrt{2\max(\gamma, 1)} \left( \beta + \frac{\alpha}{\delta} \right) + 2a^2\sqrt{2\max(\gamma, 1)} \left( \sqrt{\gamma} + \frac{1}{\delta} \right) \right]
\end{aligned}
$$
Proof of Proposition 1.7 and Lemma 2.4. The proof is done in exactly the same way as the proof of Proposition 1.6 above using (2.3). □

## A.4 Proof of Lemma 2.6

We now prove that there are constants $c, \epsilon$ and $\delta$ such that

$$c + 2\epsilon \tilde{B} \le \frac{\sigma_X^2}{2} \left( \int_0^R \Phi(s) \phi(s)^{-1} ds \right)^{-1} \quad (A.3)$$

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

$$c \le \frac{\lambda \frac{80\epsilon \tilde{B}}{\lambda}}{160 \left(1 + \frac{80\epsilon \tilde{B}}{\lambda}\right)} \quad (A.5)$$

$$\delta > \frac{1 + L_X}{1 - L_C} \quad (A.6)$$

* Since for all $u \ge 0$, $0 < \phi(u) \le 1$, we have $0 < \Phi(s) = \int_0^s \phi(u) du \le s$, i.e $s/\Phi(s) \ge 1$. Therefore

$$\inf_{r \in (0, R]} \frac{r \phi(r)}{\Phi(r)} \ge \inf_{r \in (0, R]} \phi(r) = \phi(R).$$

It is thus sufficient for (A.4) to have

$$2c + 4\epsilon \tilde{B} \le \frac{1}{2} \left(1 - L_C - \frac{1 + L_X}{\delta}\right) \phi(R).$$

* We have

$$\phi(r) \le \exp\left(-\frac{1}{4\sigma_X^2}r^2\right).$$

So

$$\Phi(r) \le \int_0^\infty \exp\left(-\frac{r^2}{4\sigma_X^2}\right) dr = \sigma_X \sqrt{\pi}.$$

Then

$$\int_0^R \frac{\Phi(r)}{\phi(r)} dr \le \sigma_X \sqrt{\pi} R \frac{1}{\phi(R)}.$$

It is thus sufficient for (A.3) that

$$c + 2\epsilon \tilde{B} \le \frac{\sigma_X \phi(R)}{2\sqrt{\pi} R}.$$

* The various conditions involving $c$ invite us to consider $2\epsilon \tilde{B} = \eta c$. Then

$$\begin{align*} c \le \frac{\lambda \frac{80\epsilon \tilde{B}}{\lambda}}{160 \left(1 + \frac{80\epsilon \tilde{B}}{\lambda}\right)} &\iff c \le \frac{\lambda 40\eta c}{160 \lambda + 40\eta c} \\ &\iff 1 \le \lambda \frac{\eta}{4\lambda + 160\eta c} \quad (\text{since } c \ge 0) \\ &\iff c \le \frac{\lambda \eta - 4}{160 \eta}. \end{align*}$$
$$+ r_t^i \left[ \left( \gamma + a \sqrt{2 \max(\gamma, 1)} \left( \beta + \frac{\alpha}{\delta} \right) \right) \frac{4}{a^2} \left( \exp\left(\frac{a^2}{2}\right) - 1 \right) + 4 \sqrt{2 \max(\gamma, 1)} \left( \sqrt{\gamma} + \frac{1}{\delta} \right) (e-2) \right].$$

We denote by $C_{f,1}$ and $C_{f,2}$ (given in Lemma 2.5) the following constants

$$C_{f,1} = 4 \left[ \left( \gamma + a \sqrt{2 \max(\gamma, 1)} \left( \beta + \frac{\alpha}{\delta} \right) \right) \frac{4}{a^2} \left( \exp\left(\frac{a^2}{2}\right) - 1 \right) + 4 \sqrt{2 \max(\gamma, 1)} \left( \sqrt{\gamma} + \frac{1}{\delta} \right) (e-2) \right]$$

$$C_{f,2} = 4 \left[ \gamma + a \sqrt{2 \max(\gamma, 1)} \left( \beta + \frac{\alpha}{\delta} \right) + 2a^2 \sqrt{2 \max(\gamma, 1)} \left( \sqrt{\gamma} + \frac{1}{\delta} \right) \right]$$

By the definition of $G_t^i$ and since $G_t^i \ge 1$, we obtain

$$|\partial_x \tilde{H}(Z_t^{i,N}) - \partial_x \tilde{H}(\bar{Z}_t^i)| \le r_t^i \frac{G_t^i C_{f,2}}{\epsilon} + r_t^i G_t^i \frac{C_{f,1}}{4},$$

and eventually

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

□

## B Proof of Theorem 2 in the case $\sigma_X = 0$ and $\sigma_C > 0$

We quickly explain in this section how we may also deal with the case $\sigma_X = 0$ and $\sigma_C > 0$. Recall how the choice of the coupling method was motivated by the observation in (1.8) that the difference of potentials $|C_t^{i,N} - \bar{C}_t^i|$ was naturally contracting when $|X_t^{i,N} - \bar{X}_t^i|$ was close to 0. This lead us to using a reflection coupling on the Brownian motions acting on the potential $X$, to bring the difference close to 0, and it was thus necessary for $\sigma_X$ to be positive ($\sigma_C$ however did not hold any importance). In the case $\sigma_X = 0$, we then have to assume $\sigma_C > 0$, and we do a change of variable, motivated by the following observation. We have, when $\sigma_X = 0$

$$d(X_t^i - \bar{X}_t^i) = ((X_t^i - \bar{X}_t^i) - ((X_t^i)^3 - (\bar{X}_t^i)^3) - (C_t^i - \bar{C}_t^i)) dt + \left( \frac{1}{N} \sum_{j=1}^N K_X(Z_t^i - Z_t^j) - K_X * \bar{\rho}(\bar{Z}_t^i) \right) dt, \\ = (2(X_t^i - \bar{X}_t^i) - (C_t^i - \bar{C}_t^i) - (X_t^i - \bar{X}_t^i) - ((X_t^i)^3 - (\bar{X}_t^i)^3)) dt \\ + \left( \frac{1}{N} \sum_{j=1}^N K_X(Z_t^i - Z_t^j) - K_X * \bar{\rho}(\bar{Z}_t^i) \right) dt.$$

Thus

$$d|X_t^i - \bar{X}_t^i| = (\text{sign}(X_t^i - \bar{X}_t^i) (2(X_t^i - \bar{X}_t^i) - (C_t^i - \bar{C}_t^i)) - |(X_t^i)^3 - (\bar{X}_t^i)^3| - |X_t^i - \bar{X}_t^i|) dt \\ + \text{sign}(X_t^i - \bar{X}_t^i) \left( \frac{1}{N} \sum_{j=1}^N K_X(Z_t^i - Z_t^j) - K_X * \bar{\rho}(\bar{Z}_t^i) \right) dt.$$

The quantity $|X_t^i - \bar{X}_t^i|$ is therefore naturally contracting when $|2(X_t^i - \bar{X}_t^i) - (C_t^i - \bar{C}_t^i)|$ is close to 0. Thanks to the presence of a Brownian motion in the stochastic differential equations defining the potential $C$, we can now
* We choose to write

$$ \delta = (1 + \tilde{\delta}) \frac{1 + L_{X,\max}}{1 - L_{C,\max}} > \frac{1 + L_X}{1 - L_C} $$

* Let us assume, for simplicity, that $\epsilon \le 1$. It is sufficient for this later condition to have

$$ c \le \frac{2\tilde{B}}{\eta}. $$

* The appearance of $\phi(R)$ suggests we should try to minimize it. We recall

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

It is therefore sufficient for (A.3) to have

$$ c \le \frac{1}{1 + \eta} \frac{\sigma_X}{2\sqrt{\pi}} \frac{1}{R} \exp \left( -\frac{1}{4\sigma_X^2} \left( 1 + \delta\gamma + L_X + \delta L_C + (C_{f,1} + C_{f,2}) \sigma_X^2 \right) R^2 \right), $$

and for (A.4) to have

$$ c \le \frac{1}{2(1 + \eta)} \left( 1 - L_C - \frac{1 + L_X}{\delta} \right) \times \exp \left( -\frac{R^2}{4\sigma_X^2} \left( 1 + \delta\gamma + L_X + \delta L_C + (C_{f,1} + C_{f,2}) \sigma_X^2 \right) \right). $$

* Finally, we bound $L_X$ and $L_C$ by either 0 or $L_{X,\max}$ and $L_{C,\max}$, to obtain bounds on $c$ independent of $L_X$ and $L_C$.

## A.5 Proof of Lemma 2.7

Let $z, z' \in \mathbb{R}^2$.

**Proof of control of the $L^1$ distance:** We have

$$ \|z - z'\|_1 = |x - x'| + |c - c'| \le \frac{1}{\min(\delta, 1)} (|x - x'| + \delta|c - c'|) = \frac{1}{\min(\delta, 1)} r(z, z'). $$

If $r(z, z') \le 1 \le R$, we have, using Lemma 2.6

$$ r(z, z') \le \frac{f(r)}{f'_{-}(R)} \le \frac{f(r)}{\phi(R)g(R)} \left( 1 + \epsilon \tilde{H}(z) + \epsilon \tilde{H}(z') \right). $$

If $r(z, z') \ge 1$, we have, using (A.2)

$$ \begin{aligned} r(z, z') &\le r(z, z')^2 \\ &\le \frac{16(1 + \delta^2)}{\epsilon \min(\gamma, 1)} (\epsilon H(z) + \epsilon H(z')) \\ &\le \frac{16(1 + \delta^2)}{\epsilon \min(\gamma, 1)} \frac{f(r)}{f(1)} (1 + \epsilon H(z) + \epsilon H(z')) \\ &\le \frac{16(1 + \delta^2)}{\epsilon \min(\gamma, 1)} \frac{f(r)}{\phi(R)g(R)} (1 + \epsilon \tilde{H}(z) + \epsilon \tilde{H}(z')). \end{aligned} $$

Thus

$$ \|z - z'\|_1 \le \frac{1}{\min(\delta, 1)} \frac{1}{\phi(R)g(R)} \max \left( \frac{16(1 + \delta^2)}{\epsilon \min(\gamma, 1)}, 1 \right) f(r(z, z')) \left( 1 + \epsilon \tilde{H}(z) + \epsilon \tilde{H}(z') \right). $$
use a reflection coupling to have $|2(X_t^i - \bar{X}_t^i) - (C_t^i - \bar{C}_t^i)|$ go to 0. Consider the following coupling

$$ egin{cases} dX_t^i = (X_t^i - (X_t^i)^3 - C_t^i - \alpha)dt + \frac{1}{N} \sum_{j=1}^N K_X(Z_t^i - Z_t^j)dt \\ dC_t^i = (\gamma X_t^i - C_t^i + \beta)dt + \frac{1}{N} \sum_{j=1}^N K_C(Z_t^i - Z_t^j)dt + \sigma_c \varphi_{sc} (|2(X_t^i - \bar{X}_t^i) - (C_t^i - \bar{C}_t^i)|) dB_t^{i,sc,C} \\ + \sigma_c \varphi_{rc} (|2(X_t^i - \bar{X}_t^i) - (C_t^i - \bar{C}_t^i)|) dB_t^{i,rc,C} \end{cases} \quad (B.1) $$

and

$$ egin{cases} d\bar{X}_t^i = (\bar{X}_t^i - (\bar{X}_t^i)^3 - \bar{C}_t^i - \alpha)dt + K_X * \bar{\rho}(\bar{Z}_t^i)dt \\ d\bar{C}_t^i = (\gamma\bar{X}_t^i - \bar{C}_t^i + \beta)dt + K_C * \bar{\rho}(\bar{Z}_t^i)dt + \sigma_c\varphi_{sc}(|2(\bar{X}_t^i - \bar{X}_t^i) - (\bar{C}_t^i - \bar{C}_t^i)|) dB_t^{i,sc,C} \\ - \sigma_c\varphi_{rc}(|2(\bar{X}_t^i - \bar{X}_t^i) - (\bar{C}_t^i - \bar{C}_t^i)|) dB_t^{i,rc,C} \\ \bar{\rho} = \mathcal{L}((\bar{X}_t^1, \bar{C}_t^1)) \end{cases} \quad (B.2) $$

and for $\delta > 0$, the following modified distance $r_t^i = \delta|X_t^i - \bar{X}_t^i| + |2(X_t^i - \bar{X}_t^i) - (C_t^i - \bar{C}_t^i)|$. Like previously, we consider a modified semimetric of the form $\frac{1}{N} \sum f(r_t^i)G_t^i$, and similar calculations yield

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

where $M_t^i$ is a continuous local martingale and

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

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

$$ \tilde{K}_t^i = G_t^i \left[ 2cf(r_t^i) + 2f''(r_t^i)\sigma_c^2\varphi_{rc} (|2(X_t^i - \bar{X}_t^i) - (C_t^i - \bar{C}_t^i)|)^2 \right. \\ + f'(r_t^i) (|2(X_t^i - \bar{X}_t^i) - (C_t^i - \bar{C}_t^i)| (\delta + 1) - |(X_t^i)^3 - (\bar{X}_t^i)^3|(\delta - 2) \\ + |X_t^i - \bar{X}_t^i| (-\delta + \gamma + L_X(\delta + 2) + L_C) + |C_t^i - \bar{C}_t^i|(L_X(\delta + 2) + L_C) \\ + \sigma_c^2\varphi_{rc} (|2(X_t^i - \bar{X}_t^i) - (C_t^i - \bar{C}_t^i)|)^2 (\epsilon_{C,1} + \epsilon_{C,2}) r_t^i \right] \\ + \epsilon f(r_t^i) \left( 4\tilde{B} - \frac{\lambda}{8}\tilde{H}(\bar{Z}_t^i) - \frac{\lambda}{8}\tilde{H}(Z_t^{i,N}) - \frac{\lambda}{8N} \sum_{j=1}^N \tilde{H}(\bar{Z}_t^j) - \frac{\lambda}{32N} \sum_{j=1}^N \tilde{H}(Z_t^{j,N}) \right), $$

$$ I_t^{1,i} = G_t^i f'(r_t^i) \left[ (\delta + 2) \left( \left| \frac{1}{N} \sum_{j=1}^N K_X(\bar{Z}_t^i - \bar{Z}_t^j) - K_X * \bar{\mu}_t(\bar{Z}_t^i) \right| \right) + \left( \left| \frac{1}{N} \sum_{j=1}^N K_C(\bar{Z}_t^i - \bar{Z}_t^j) - K_C * \bar{\mu}_t(\bar{Z}_t^i) \right| \right) \right], $$

$$ I_t^{2,i} = G_t^i f'(r_t^i) \left[ (\delta + 2) \left( \frac{L_X}{N} \left( \sum_{j=1}^N |X_t^{j,N} - \bar{X}_t^j| + |C_t^{j,N} - \bar{C}_t^j| \right) \right) + \left( \frac{L_C}{N} \left( \sum_{j=1}^N |X_t^{j,N} - \bar{X}_t^j| + |C_t^{j,N} - \bar{C}_t^j| \right) \right) \right] \\ - cf(r_t^i)G_t^i - \epsilon f(r_t^i) \left[ \frac{\lambda}{32} H(\bar{Z}_t^i) \exp \left( a\sqrt{H(\bar{Z}_t^i)} \right) + \frac{\lambda}{32} H(Z_t^{i,N}) \exp \left( a\sqrt{H(Z_t^{i,N})} \right) \right] \\ - \epsilon f(r_t^i) \left[ \frac{\lambda}{32N} \sum_{j=1}^N H(\bar{Z}_t^j) \exp \left( a\sqrt{H(\bar{Z}_t^j)} \right) + \frac{\lambda}{32N} \sum_{j=1}^N H(Z_t^{j,N}) \exp \left( a\sqrt{H(Z_t^{j,N})} \right) \right], $$

$$ I_t^{3,i} = \epsilon f(r_t^i) \left( (\alpha_X L_X + \beta_X L_C) \left( \frac{\sum_{j=1}^N |X_t^{j,N}|}{N} \right)^2 \exp \left( a\sqrt{H(Z_t^{i,N})} \right) \right. \\ \left. + (\alpha_C L_X + \beta_C L_C) \left( \frac{\sum_{j=1}^N |C_t^{j,N}|}{N} \right)^2 \exp \left( a\sqrt{H(Z_t^{i,N})} \right) \right) $$
**Proof of control of the L² distance:** We have

$$r(z, z')^2 = (|x - x'| + \delta|c - c'|)^2 \geq |x - x'|^2 + \delta^2|c - c'|^2 \\ \geq \min(1, \delta^2) (|x - x'|^2 + |c - c'|^2).$$

If $r(z, z') \geq 1$, we have, using (A.2)

$$r(z, z')^2 \leq \frac{16(1 + \delta^2)}{\epsilon \min(\gamma, 1)} \frac{f(r)}{\phi(R)g(R)} (1 + \epsilon \tilde{H}(z) + \epsilon \tilde{H}(z')).$$

If $r(z, z') \leq 1 \leq R$, we have, using Lemma 2.6

$$r(z, z')^2 \leq r(z, z') \leq \frac{f(r)}{f'_{-}(R)} \leq \frac{f(r)}{\phi(R)g(R)} (1 + \epsilon \tilde{H}(z) + \epsilon \tilde{H}(z')).$$

Thus

$$\|z - z'\|_2^2 \leq \frac{1}{\min(\delta^2, 1)} \frac{1}{\phi(R)g(R)} \max \left( \frac{16(1 + \delta^2)}{\epsilon \min(\gamma, 1)}, 1 \right) f(r(z, z')) (1 + \epsilon \tilde{H}(z) + \epsilon \tilde{H}(z')).$$

**Proof of the second control of the L¹ distance:** We have, if $r(z, z') \leq 1 \leq R$

$$r(z, z') \leq \frac{f(r)}{f'_{-}(R)} \leq \frac{f(r)}{\phi(R)g(R)} (1 + \epsilon \sqrt{H(z)} + \epsilon \sqrt{H(z')}),$$

and, if $r(z, z') \geq 1$, recall Lemma 2.1.

$$\begin{aligned} \|z - z'\|_1 &\leq \sqrt{\frac{4}{\gamma}H(z)} + \sqrt{\frac{4}{\gamma}H(z')} + \sqrt{4H(z)} + \sqrt{4H(z')} \\ &\leq 4 \max \left( \sqrt{\frac{1}{\gamma}}, 1 \right) \left( \sqrt{H(z)} + \sqrt{H(z')} \right) \\ &\leq \frac{4}{\epsilon} \max \left( \sqrt{\frac{1}{\gamma}}, 1 \right) \frac{f(r)}{\phi(R)g(R)} \left( 1 + \epsilon \sqrt{H(z)} + \epsilon \sqrt{H(z')} \right), \end{aligned}$$

and thus

$$\|z - z'\|_1 \leq \frac{1}{\phi(R)g(R)} \max \left( 1, \frac{4}{\epsilon} \max \left( \sqrt{\frac{1}{\gamma}}, 1 \right) \right) f(r(z, z')) (1 + \epsilon \sqrt{H(z)} + \epsilon \sqrt{H(z')}),$$

**Independence with respect to $L_X$ and $L_C$** The a priori bounds $L_X \in [0, L_{X,\max}]$ and $L_C \in [0, L_{C,\max}]$ allow us to bound $\phi(R)$ independently of $L_C$ and $L_X$ by $\phi_{\min}$ (and we also use $g(R) \geq \frac{1}{2}$), thus giving us constant $C_1, C_2$ and $C_z$ independent of $L_C$ and $L_X$.

### A.6 Proof of Lemmas 3.2 and 3.7

**Proof of Lemma 3.2.** Let's prove there exists a uniform in time bound on $E(G_t^i)$ and $E[(G_t^i)^2]$. First, let's recall the definition of $G$ from(2.26)

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

The idea is to bound the different expectations in terms of the expectations at time $t = 0$. Since $E(e^{\bar{a}(|X_0|+|C_0|)})$ is finite, we know that for each $k \in \mathbb{N}$, $E(|X_0|^k)$ and $E(|C_0|^k)$ are
$$ -\frac{\lambda}{16} H(Z_t^{i,N}) \exp \left( a \sqrt{H(Z_t^{i,N})} \right) - \frac{\lambda}{16N} \sum_{j=1}^{N} H(Z_t^{j,N}) \exp \left( a \sqrt{H(Z_t^{j,N})} \right) . $$

We then have the additional constraint of $\delta > 2$ (so that the coefficient appearing in front of $|(X_t^i)^3 - (\bar{X}_t^i)^3|$ in the expression of $\tilde{K}_t^i$ is non positive). Otherwise, we deal with the various terms exactly as previously, through the choice of a sufficiently concave function $f$ and a law of large numbers, and by considering the regions of space

$$ \mathrm{Reg}_1^i = \{ (\bar{Z}_t^i, Z_t^{i,N}) \text{ s.t. } |2(X_t^i - \bar{X}_t^i) - (C_t^i - \bar{C}_t^i)| \ge \xi \text{ and } r_t^i \le R \}, $$

$$ \mathrm{Reg}_2^i = \{ (\bar{Z}_t^i, Z_t^{i,N}) \text{ s.t. } |2(X_t^i - \bar{X}_t^i) - (C_t^i - \bar{C}_t^i)| < \xi \text{ and } r_t^i \le R_1 \}, $$

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

## Acknowledgements

Laetitia Colombani is a PhD student under the supervision of Patrick Cattiaux and Manon Costa, and Pierre Le Bris is a PhD student under the supervision of Arnaud Guillin and Pierre Monmarché. The authors would like to thank them, as well as Samir Salem, for their help throughout the redaction of the present article.

This work has been (partially) supported by the Project EFI ANR-17-CE40-0030 of the French National Research Agency.

## Index

Throughout this article, we define many parameters and constants. For the sake of clarity, we list the main ones here so as to give the reader an index to refer to.

* $X, C, Z$: $X$ and $C$ are the processes we consider (see (1.1) and (1.2)) and we often refer to $Z = (X, C)$,

* $\bar{\mu}_t = \mathrm{Law}(\bar{Z}_t)$: the density of the non linear limit (see (1.2)),

* $\alpha, \beta, \gamma, \sigma_X, \sigma_C$: parameters of the problem (see (1.1)),

* $K_X, K_C, L_X, L_C, L_{X,max}, L_{C,max}$: $K_X$ (resp. $K_C$) is an Lipschitz continuous interaction kernel, with Lipschitz constant $L_X \in [0, L_{X,max}]$ (resp. $L_C \in [0, L_{C,max}]$), as given in Assumption 1. In the case of uniform in time propagation of chaos, the inequalities $L_X$ and $L_C$ must satisfy are listed in Assumption 2,

* $W_p$: the usual Wasserstein distance associated to the $L^p$ distance (see (1.3)),

* $a, \tilde{a}, C_{init,exp}$: constants used to give an exponential initial moment to the problem (see the assumptions of Theorem 2 and Section 2.3),

* $\lambda, B, \tilde{B}, H, \tilde{H}, \alpha_X, \alpha_C, \beta_X, \beta_C$: $H$ (resp. $\tilde{H}$) is a Lyapunov functions given in (2.1) (resp. (2.8)). Its main property involves parameters $\lambda$ and $B$ (resp. $\lambda$ and $\tilde{B}$), as can for instance be seen in (2.3) (resp. (2.13)). $\alpha_X, \alpha_C, \beta_X$ and $\beta_C$ are intermediate constants given in Lemma 2.2,

* $c$: a contraction rate (see Assumption 2),

* $r, f, g, \phi, \Phi, G, \rho, \delta, R, \epsilon, C_{f,1}, C_{f,2}$: $f$ (see (2.21)) is a concave function, the definition of which involves $g, \phi, \Phi$ (see Assumption 2). Function $G$ (see (2.24)) is then used to define $\rho$ (see (2.23)), the semimetrics we consider in the end. All those notations thus refer to the modified distance we consider. These functions will be applied to $r$ a modification of the usual $L^1$ distance (see equation (2.22)). Then, parameters $\delta, R, \epsilon, C_{f,1}$, and $C_{f,2}$ are used to define such functions (see Assumption 2 for some explicit values),
also finite. We deduce that for each $k \in \mathbb{N}$, for each $j \le N$, $E[H(\bar{Z}_0^j)^k]$ and $E[H(Z_0^{j,N})^k]$ are finite.

In fact, to bound uniformly in time the first moment, we only have to bound $E(\tilde{H}(Z_t^{j,N}))$ and $E(\tilde{H}(\bar{Z}_t^j))$ for each $j \le N$. Let's begin with $\bar{Z}^j$. By (2.16), we have

$$ \frac{d}{dt} \mathbb{E}[\tilde{H}(\bar{Z}_t^j)] \le \tilde{B} - \frac{\lambda}{4} \mathbb{E}[\tilde{H}(\bar{Z}_t^j)]. $$

By using Itô's formula on $e^{\lambda t/4} \tilde{H}(\bar{Z}_t^j)$ and the bound above, we obtain

$$ \begin{aligned} \mathbb{E}[\tilde{H}(\bar{Z}_t^j)] &\le \frac{4\tilde{B}}{\lambda} + e^{-\frac{4}{3}t} \left( \mathbb{E}[\tilde{H}(\bar{Z}_0^j)] - \frac{4\tilde{B}}{\lambda} \right) \\ &\le \max \left( \mathbb{E}[\tilde{H}(\bar{Z}_0^j)], \frac{4\tilde{B}}{\lambda} \right). \end{aligned} $$

By (2.9), in Lemma 2.5, we deduce the following inequality and we apply Cauchy-Schwarz inequality

$$ \begin{aligned} \mathbb{E}[\tilde{H}(\bar{Z}_0^j)] &\le \mathbb{E} \left[ H(\bar{Z}_0^j) \exp \left( a \sqrt{H(\bar{Z}_0^j)} \right) \right] \\ &\le \mathbb{E} \left[ H(\bar{Z}_0^j)^2 \right]^{1/2} \mathbb{E} \left[ \exp \left( 2a \sqrt{H(\bar{Z}_0^j)} \right) \right]^{1/2}. \end{aligned} \quad (A.7) $$

We already know $\mathbb{E}[H(\bar{Z}_0^j)^2]$ is bounded. Now, it is enough to prove that there exist $C$ such that for all $z \in \mathbb{R}^2$

$$ \exp(2a\sqrt{H(z)}) \le C \times e^{\tilde{a}(|x|+|c|)}. $$

In fact, from the definition of $H$ in (2.1), we have

$$ \begin{aligned} 2\sqrt{H(z)} &= \sqrt{2}\sqrt{\gamma\left(x+\frac{\beta}{\gamma}\right)^2 + (c+\alpha)^2 + H_0} \\ &\le \sqrt{2\gamma}\left|x+\frac{\beta}{\gamma}\right| + \sqrt{2}|c+\alpha| + \sqrt{H_0} \\ &\le \sqrt{2\gamma}|x| + \sqrt{2}|c| + \frac{1}{a}\ln C, \end{aligned} $$

where $C$ is a constant independent of $z$. Finally, since $\max(a\sqrt{2\gamma}, a\sqrt{2}) \le \tilde{a}$, we have

$$ \exp(2a\sqrt{H(z)}) \le C \times e^{\tilde{a}(|x|+|c|)}. $$

Then, $\mathbb{E}[\exp(2a\sqrt{H(\bar{Z}_0^j)})]$ is bounded and we deduce $\mathbb{E}(\tilde{H}(\bar{Z}_t^j))$ is bounded for each $j \le N$ and all $t \ge 0$.

The same calculations can be done for $Z_t^{j,N}$. By (2.19), we have

$$ \mathcal{L}^N \left( \frac{1}{N} \sum_{i=1}^N \tilde{H}(Z_t^{i,N}) \right) \le \tilde{B} - \frac{\lambda}{4} \left( \frac{1}{N} \sum_{i=1}^N \tilde{H}(Z_t^{i,N}) \right). $$
• $R_0, \phi_{min}$: intermediate constants (see Assumption 2),

• $C_{init,2}$: uniform in time bound on the second moment of the processes (see Lemma 2.3),

• $C_1, C_2, C_z$: constants used to quantify the control our modify distance has over the usual $L^1$ and $L^2$ distance (see Lemma 2.6 for the control and Assumption 2 for explicit values),

• $\phi_{rc}, \phi_{sc}, \xi$: $\phi_{rc}$ and $\phi_{sc}$ are two Lipschitz continuous functions used to define the coupling method, and their definitions involve a parameter $\xi$ which converges to 0 in the end (see the beginning of Section 3),

• $C_{r,H}$: used to explicit the control of the Lyapunov function $H$ over the distance $r$ (see Lemma 2.1),

## References

[BFFT12] Javier Baladron, Diego Fasoli, Olivier Faugeras, and Jonathan Touboul. Mean-field description and propagation of chaos in networks of hodgkin-huxley and fitzhugh-nagumo neurons. *The Journal of Mathematical Neuroscience*, 2(1):10, May 2012.

[BFO19] Mireille Bossy, Joaquin Fontbona, and Hector Olivero. Synchronization of stochastic mean field networks of hodgkin-huxley neurons with noisy channels. *Journal of Mathematical Biology*, 78(6):1771–1820, May 2019. arXiv: 1803.04379.

[BGM10] François Bolley, Arnaud Guillin, and Florent Malrieu. Trend to equilibrium and particle approximation for a weakly selfconsistent Vlasov-Fokker-Planck equation. *ESAIM, Math. Model. Numer. Anal.*, 44(5):867–884, 2010.

[BHV21] Şeyma Bayrak, Philipp Hövel, and Vesna Vuksanović. Modeling functional connectivity on empirical and randomized structural brain networks. *Differential Equations and Dynamical Systems*, 29(4):789–805, 2021.

[BL12] Nils Berglund and Damien Landon. Mixed-mode oscillations and interspike interval statistics in the stochastic fitzhugh-nagumo model. *Nonlinearity*, 25(8):2303–2335, 2012.

[DEGZ20] Alain Durmus, Andreas Eberle, Arnaud Guillin, and Raphael Zimmer. An elementary approach to uniform in time propagation of chaos. *Proc. Amer. Math. Soc.*, 148(12):5387–5398, 2020.

[Ebe16] Andreas Eberle. Reflection couplings and contraction rates for diffusions. *Probab. Theory Relat. Fields*, 166(3-4):851–886, 2016.

[GLM21] Arnaud Guillin, Pierre Le Bris, and Pierre Monmarché. Convergence rates for the vlasov-fokker-planck equation and uniform in time propagation of chaos in non convex cases, 2021. arXiv: 2105.09070.

[HH52] Alan L. Hodgkin and Andrew F. Huxley. A quantitative description of membrane current and its application to conduction and excitation in nerve. *The Journal of Physiology*, 117(4):500–544, Aug 1952.

[IW89] Nobuyuki Ikeda and Shinzo Watanabe. *Stochastic Differential Equations and Diffusion Processes*. Number 24 in North-Holland Mathematical Library. North-Holland [u.a.], 2. ed edition, 1989.

[Kac56] Mark Kac. Foundations of kinetic theory. In *Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, Volume 3: Contributions to Astronomy and Physics*, pages 171–197, Berkeley, Calif., 1956. University of California Press.
In particular,

$$ \frac{d}{dt} \left[ E \left( \frac{1}{N} \sum_{i=1}^{N} \tilde{H}(Z_t^{i,N}) \right) \right] \leq E \left[ \mathcal{L}^N \left( \frac{1}{N} \sum_{i=1}^{N} \tilde{H}(Z_t^{i,N}) \right) \right] \leq \tilde{B} - \frac{\lambda}{4} E \left( \frac{1}{N} \sum_{i=1}^{N} \tilde{H}(Z_t^{i,N}) \right), $$

and we can use the same method as above.

Finally, we have proved that for each $j \le N$, $E(\tilde{H}(Z_t^{j,N}))$ and $E(\tilde{H}(\bar{Z}_t^j))$ are bounded uniformly in time. Thus, $E(G_t^i)$ is bounded uniformly in time (and in $N$).

To bound the second moment of $G_t^i$, we have to bound each type of the following expectations $E[\tilde{H}(Z_t^{j_1,N})\tilde{H}(Z_t^{j_2,N})]$, $E[\tilde{H}(Z_t^{j_1,N})\tilde{H}(\bar{Z}_t^{j_2})]$, $E[\tilde{H}(\bar{Z}_t^{j_1})\tilde{H}(\bar{Z}_t^{j_2})]$, $E[\tilde{H}(Z_t^{j,N})^2]$ and $E[\tilde{H}(\bar{Z}_t^j)^2]$. By Cauchy-Schwarz inequality, it is in fact enough to bound $E[\tilde{H}(Z_t^{j,N})^2]$ and $E[\tilde{H}(\bar{Z}_t^j)^2]$.

First, by the definition of $\tilde{H}$ in (2.8),

$$ \begin{aligned} \tilde{H}(z)^2 &= \left( \frac{2}{a^2} \exp \left( a\sqrt{H(z)} \right) \left( a\sqrt{H(z)} - 1 \right) + \frac{2}{a^2} \right)^2 \\ &\le 2 \frac{2^2}{a^4} \exp \left( 2a\sqrt{H(z)} \right) \left( a\sqrt{H(z)} - 1 \right)^2 + 2 \frac{2^2}{a^4} \\ &\le \frac{8}{a^4} \exp \left( 2a\sqrt{H(z)} \right) \left( 2a^2 H(z) + 2 \right) + \frac{8}{a^4}. \end{aligned} $$

As for the first moment, the study of $Z_t^{j,N}$ is very similar to the one of $\bar{Z}_t^j$. Here, we only focus on the second one.

Using Cauchy-Schwarz inequality, bounds on $E[H(\bar{Z}_t^j)^2]$ and $E[\exp(4a\sqrt{H(\bar{Z}_t^j)})]$ are sufficient to bound $E[\tilde{H}(\bar{Z}_t^j)^2]$. The latter has already been bounded uniformly in time, and the former can be obtained by the same calculations as previously, replacing $a$ by $4a$ (and thus assuming $\tilde{a} \ge 4\sqrt{2a} \max(\sqrt{\gamma}, 1)$, which we do).

Finally, we deduce $E((G_t^i)^2)$ is bounded uniformly in time. □

**Proof of Lemma 3.7.** Using $\partial_X H(z) = \gamma x + \beta$, we have

$$ \begin{aligned} & \left| \partial_X \tilde{H}(Z_t^{i,N}) - \partial_X \tilde{H}(\bar{Z}_t^i) \right| \\ &= \left| \left( \gamma X_t^{i,N} + \beta \right) \exp \left( a\sqrt{H(Z_t^{i,N})} \right) - \left( \gamma \bar{X}_t^i + \beta \right) \exp \left( a\sqrt{H(\bar{Z}_t^i)} \right) \right| \\ &\le \left| \gamma X_t^{i,N} - \gamma \bar{X}_t^i \right| \left( \exp \left( a\sqrt{H(Z_t^{i,N})} \right) + \exp \left( a\sqrt{H(\bar{Z}_t^i)} \right) \right) \\ &\quad + \left| \gamma \bar{X}_t^i + \beta \right| \left| \exp \left( a\sqrt{H(Z_t^{i,N})} \right) - \exp \left( a\sqrt{H(\bar{Z}_t^i)} \right) \right|. \end{aligned} $$

Since $|X_t^{i,N} - \bar{X}_t^i| \le r_t^i$,

$$ \begin{aligned} & \left| \gamma X_t^{i,N} - \gamma \bar{X}_t^i \right| \left( \exp \left( a\sqrt{H(Z_t^{i,N})} \right) + \exp \left( a\sqrt{H(\bar{Z}_t^i)} \right) \right) \\ &\le \gamma r_t^i \left( \exp \left( a\sqrt{H(Z_t^{i,N})} \right) + \exp \left( a\sqrt{H(\bar{Z}_t^i)} \right) \right) \end{aligned} $$

By Lemma 2.1 (ii), we have $H(z) \ge \frac{1}{2} \min(\frac{1}{\gamma}, 1) (\gamma x + \beta)^2$. By the mean value theorem, for all $y_1 \le y_2$ in $\mathbb{R}$, there exists $y_3 \in [y_1, y_2]$ such that $e^{ay_1} - e^{ay_2} = a(y_1 - y_2)e^{ay_3}$. In
[KNRS20] Chaman Kumar, Neelima, Christoph Reisinger, and Wolfgang Stockinger. Well-posedness and tamed schemes for mkean-vlasov equations with common noise. arXiv:2006.00463 [cs, math], May 2020. arXiv: 2006.00463.

[Li21] Fuzhi Li. Dynamics for stochastic fitzhugh-nagumo systems with general multiplicative noise on thin domains. *Mathematical Methods in the Applied Sciences*, 44(6):5050–5078, 2021.

[LL19] Yangrong Li and Fuzhi Li. Limiting dynamics for stochastic fitzhugh-nagumo equations on large domains. *Stochastics and Dynamics*, 19(5):25, 2019.

[LP21] Eric Luçon and Christophe Poquet. Periodicity induced by noise and interaction in the kinetic mean-field fitzhugh-nagumo model. *The Annals of Applied Probability*, 31(2):561–593, Apr 2021.

[LR86] Torgny Lindvall and L. C. G. Rogers. Coupling of multidimensional diffusions by reflection. *The Annals of Probability*, 14(3):860–872, Jul 1986.

[LS14] Eric Luçon and Wilhelm Stannat. Mean field limit for disordered diffusions with singular interactions. *The Annals of Applied Probability*, 24(5), 2014.

[LS18] José R. León and Adeline Samson. Hypoelliptic stochastic fitzhugh-nagumo neuronal model: mixing, up-crossing and estimation of the spike rate. *The Annals of Applied Probability*, 28(4):2243–2274, 2018.

[LW10] Yan Lv and Wei Wang. Limit dynamics for the stochastic fitzhugh-nagumo system. *Nonlinear Analysis. Real World Applications*, 11(4):3091–3105, 2010.

[LX21] Fuzhi Li and Dongmei Xu. Regular dynamics for stochastic fitzhugh-nagumo systems with additive noise on thin domains. *Discrete and Continuous Dynamical Systems. Series B*, 26(7):3517–3542, 2021.

[MQT16] Stéphane Mischler, Cristóbal Quiñinao, and Jonathan Touboul. On a kinetic fitzhugh-nagumo model of neuronal network. *Communications in Mathematical Physics*, 342(3):1001–1042, Mar 2016. arXiv: 1503.00492.

[RS22] Christoph Reisinger and Wolfgang Stockinger. An adaptive euler-maruyama scheme for mkean-vlasov sdes with super-linear growth and application to the mean-field fitzhugh-nagumo model. *Journal of Computational and Applied Mathematics*, 400:23, 2022.

[RST19] Goncalo Dos Reis, William Salkeld, and Julian Tugaut. Freidlin-wentzell ldp in path space for mkean-vlasov equations and the functional iterated logarithm law. *The Annals of Applied Probability*, 29(3), Jun 2019. arXiv: 1708.04961.

[Sch22] Katharina Schuh. Global contractivity for langevin dynamics with distribution-dependent forces and uniform in time propagation of chaos, 2022. arXiv:2206.03082.

[Szn91] Alain-Sol Sznitman. Topics in propagation of chaos. In Paul-Louis Hennequin, editor, *Ecole d'Eté de Probabilités de Saint-Flour XIX — 1989*, pages 165–251, Berlin, Heidelberg, 1991. Springer Berlin Heidelberg.

[TBSST13] D. Tatchim Bemmo, M. Siewe Siewe, and C. Tchawoua. Combined effects of correlated bounded noises and weak periodic signal input in the modified fitzhugh-nagumo neural model. *Communications in Nonlinear Science and Numerical Simulation*, 18(5):1275–1287, May 2013.

[Thi13] Michèle Thieullen. *Deterministic and Stochastic FitzHugh-Nagumo Systems*, volume 2058 of *Lecture Notes in Mathematics*, page 175–186. Springer Berlin Heidelberg, 2013.
particular, we have the following control $|e^{ay_1} - e^{ay_2}| \le a|y_1 - y_2|(e^{ay_1} + e^{ay_2})$. Thus

$$ 
\begin{aligned}
& |\gamma \bar{X}_t^i + \beta| \left| \exp \left( a \sqrt{H(Z_t^{i,N})} \right) - \exp \left( a \sqrt{H(\bar{Z}_t^i)} \right) \right| \\
& \le a \sqrt{\frac{2H(\bar{Z}_t^i)}{\min\left(\frac{1}{\gamma}, 1\right)}} \left| \sqrt{H(Z_t^{i,N})} - \sqrt{H(\bar{Z}_t^i)} \right| \left( \exp \left( a \sqrt{H(Z_t^{i,N})} \right) + \exp \left( a \sqrt{H(\bar{Z}_t^i)} \right) \right) \\
& \le a \sqrt{2 \max(\gamma, 1)} \left| H(Z_t^{i,N}) - H(\bar{Z}_t^i) \right| \left( \exp \left( a \sqrt{H(Z_t^{i,N})} \right) + \exp \left( a \sqrt{H(\bar{Z}_t^i)} \right) \right).
\end{aligned}
$$

Then by the definition of $H$ we get

$$ 
\begin{aligned}
& |H(Z_t^{i,N}) - H(\bar{Z}_t^i)| \\
& = \left| \frac{1}{2}\gamma \left( (X_t^{i,N})^2 - (\bar{X}_t^i)^2 \right) + \beta(X_t^{i,N} - \bar{X}_t^i) + \frac{1}{2} \left( (C_t^{i,N})^2 - (\bar{C}_t^i)^2 \right) + \alpha(C_t^{i,N} - \bar{C}_t^i) \right| \\
& \le \frac{1}{2}\gamma \left| X_t^{i,N} - \bar{X}_t^i \right| \left| X_t^{i,N} + \bar{X}_t^i \right| + \beta \left| X_t^{i,N} - \bar{X}_t^i \right| + \frac{1}{2} \left| C_t^{i,N} - \bar{C}_t^i \right| \left| C_t^{i,N} + \bar{C}_t^i \right| \\
& \qquad + \alpha \left| C_t^{i,N} - \bar{C}_t^i \right|.
\end{aligned}
$$

Now, by Lemma 2.1 (i), we have $H(z) \ge \frac{\gamma}{4}x^2 + \frac{1}{4}c^2$ and since $|X_t^{i,N} - \bar{X}_t^i| \le r_t^i$ and $|C_t^{i,N} - \bar{C}_t^i| \le r_t^i/\delta$, we get

$$ |X_t^{i,N} - \bar{X}_t^i| \left( \frac{1}{2}\gamma |X_t^{i,N} + \bar{X}_t^i| + \beta \right) \le r_t^i \left( \sqrt{\gamma} \left( \sqrt{H(Z_t^{i,N})} + \sqrt{H(\bar{Z}_t^i)} \right) + \beta \right) $$

and

$$ |C_t^{i,N} - \bar{C}_t^i| \left( \frac{1}{2} |C_t^{i,N} + \bar{C}_t^i| + \alpha \right) \le \frac{r_t^i}{\delta} \left( \sqrt{H(Z_t^{i,N})} + \sqrt{H(\bar{Z}_t^i)} + \alpha \right). $$

Thus

$$ |H(Z_t^{i,N}) - H(\bar{Z}_t^i)| \le \left(\beta + \frac{\alpha}{\delta}\right) r_t^i + \left(\sqrt{\gamma} + \frac{1}{\delta}\right) r_t^i \left(\sqrt{H(Z_t^{i,N})} + \sqrt{H(\bar{Z}_t^i)}\right). $$

Finally,

$$ 
\begin{aligned}
& |\partial_X \tilde{H}(Z_t^{i,N}) - \partial_X \tilde{H}(\bar{Z}_t^i)| \\
& \le \gamma r_t^i \left( \exp \left( a \sqrt{H(Z_t^{i,N})} \right) + \exp \left( a \sqrt{H(\bar{Z}_t^i)} \right) \right) \\
& \qquad + a \sqrt{2 \max(\gamma, 1)} \left( \beta + \frac{\alpha}{\delta} \right) r_t^i \left( \exp \left( a \sqrt{H(Z_t^{i,N})} \right) + \exp \left( a \sqrt{H(\bar{Z}_t^i)} \right) \right) \\
& \qquad + a \sqrt{2 \max(\gamma, 1)} \left( \sqrt{\gamma} + \frac{1}{\delta} \right) r_t^i \left( \sqrt{H(Z_t^{i,N})} + \sqrt{H(\bar{Z}_t^i)} \right) \\
& \qquad \qquad \times \left( \exp \left( a \sqrt{H(Z_t^{i,N})} \right) + \exp \left( a \sqrt{H(\bar{Z}_t^i)} \right) \right) \\
& \le r_t^i \left( \gamma + a \sqrt{2 \max(\gamma, 1)} \left( \beta + \frac{\alpha}{\delta} \right) \right) \left( \exp \left( a \sqrt{H(Z_t^{i,N})} \right) + \exp \left( a \sqrt{H(\bar{Z}_t^i)} \right) \right) \\
& \qquad + a r_t^i \sqrt{2 \max(\gamma, 1)} \left( \sqrt{\gamma} + \frac{1}{\delta} \right) \\
& \qquad \qquad \times \left( 2 \sqrt{H(Z_t^{i,N})} \exp \left( a \sqrt{H(Z_t^{i,N})} \right) + 2 \sqrt{H(\bar{Z}_t^i)} \exp \left( a \sqrt{H(\bar{Z}_t^i)} \right) \right).
\end{aligned}
$$
[TRW03] Henry C. Tuckwell, Roger Rodriguez, and Frederic Y. M. Wan. Determination of firing times for the stochastic fitzhugh-nagumo neuronal model. *Neural Computation*, 15(1):143–159, 2003.

[Uda19] Kenneth Uda. Ergodicity and spike rate for stochastic fitzhugh-nagumo neural model with periodic forcing. *Chaos, Solitons and Fractals*, 123:383–399, 2019.
Now, we can finally use Lemma 2.5, and more precisely (2.9) and (2.10), we obtain

$$
\begin{aligned}
& |\partial_X \tilde{H}(Z_t^{i,N}) - \partial_X \tilde{H}(\bar{Z}_t^i)| \\
& \leq r_t^i \left( \gamma + a\sqrt{2\max(\gamma, 1)} \left(\beta + \frac{\alpha}{\delta}\right) \right) \left( \tilde{H}(Z_t^{i,N}) + \tilde{H}(\bar{Z}_t^i) + \frac{4}{a^2} \left(e^{a^2/2} - 1\right) \right) \\
& \quad + ar_t^i \sqrt{2\max(\gamma, 1)} \left(\sqrt{\gamma} + \frac{1}{\delta}\right) \left( 2a\tilde{H}(Z_t^{i,N}) + \frac{2}{a}(e-2) + 2a\tilde{H}(\bar{Z}_t^i) + \frac{2}{a}(e-2) \right) \\
& \leq r_t^i \left( \tilde{H}(Z_t^{i,N}) + \tilde{H}(\bar{Z}_t^i) \right) \left[ \gamma + a\sqrt{2\max(\gamma, 1)} \left(\beta + \frac{\alpha}{\delta}\right) + 2a^2 \sqrt{2\max(\gamma, 1)} \left(\sqrt{\gamma} + \frac{1}{\delta}\right) \right] \\
& \quad + r_t^i \left[ \left( \gamma + a\sqrt{2\max(\gamma, 1)} \left(\beta + \frac{\alpha}{\delta}\right) \right) \frac{4}{a^2} \left(e^{a^2/2} - 1\right) \right. \\
& \qquad \left. + 4\sqrt{2\max(\gamma, 1)} \left(\sqrt{\gamma} + \frac{1}{\delta}\right) (e-2) \right].
\end{aligned}
$$

We denote by $C_{f,1}$ and $C_{f,2}$ (given in Lemma 2.6) the following constants

$$
\begin{aligned}
C_{f,1} &= 4 \left[ \left( \gamma + a \sqrt{2 \max(\gamma, 1)} \left( \beta + \frac{\alpha}{\delta} \right) \right) \frac{4}{a^2} \left( e^{a^2/2} - 1 \right) \right. \\
& \qquad \left. + 4 \sqrt{2 \max(\gamma, 1)} \left( \sqrt{\gamma} + \frac{1}{\delta} \right) (e-2) \right] \\
C_{f,2} &= 4 \left[ \gamma + a \sqrt{2 \max(\gamma, 1)} \left( \beta + \frac{\alpha}{\delta} \right) + 2a^2 \sqrt{2 \max(\gamma, 1)} \left( \sqrt{\gamma} + \frac{1}{\delta} \right) \right]
\end{aligned}
$$

By the definition of $G_t^i$ and since $G_t^i \ge 1$, we obtain

$$
|\partial_X \tilde{H}(Z_t^{i,N}) - \partial_X \tilde{H}(\bar{Z}_t^i)| \le r_t^i \frac{G_t^i C_{f,2}}{\epsilon} + r_t^i G_t^i \frac{C_{f,1}}{4},
$$

and eventually

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

<figure><img src="image_4.png" /></figure>

**B Proof of Theorem 1.4 in the case $\sigma_X = 0$ and $\sigma_C > 0$**

We quickly explain in this section how we may also deal with the case $\sigma_X = 0$ and $\sigma_C > 0$. Recall how the choice of the coupling method was motivated by the observation in (1.9) that the difference of potentials $|C_t^{i,N} - \bar{C}_t^i|$ was naturally contracting when $|X_t^{i,N} - \bar{X}_t^i|$ was close to 0. This lead us to use a reflection coupling on the Brownian motions acting on the potential $X$, to bring the difference close to 0, and it was thus necessary for $\sigma_X$ to be positive ($\sigma_C$ however did not matter). In the case $\sigma_X = 0$, we then have to assume $\sigma_C > 0$, and we do a change of variable, motivated by the following
observation. We have, when $\sigma_X = 0$

$$ 
\begin{aligned}
d(X_t^{i,N} - \bar{X}_t^i) &= ((X_t^{i,N} - \bar{X}_t^i) - ((X_t^{i,N})^3 - (\bar{X}_t^i)^3) - (C_t^{i,N} - \bar{C}_t^i)) dt \\
&\quad + \left( \frac{1}{N} \sum_{j=1}^N K_X(Z_t^{i,N} - Z_t^{j,N}) - K_X * \bar{\rho}(\bar{Z}_t^i) \right) dt \\
&= (2(X_t^{i,N} - \bar{X}_t^i) - (C_t^{i,N} - \bar{C}_t^i) - (X_t^{i,N} - \bar{X}_t^i) - ((X_t^{i,N})^3 - (\bar{X}_t^i)^3)) dt \\
&\quad + \left( \frac{1}{N} \sum_{j=1}^N K_X(Z_t^{i,N} - Z_t^{j,N}) - K_X * \bar{\rho}(\bar{Z}_t^i) \right) dt.
\end{aligned}
$$

Thus

$$ 
\begin{aligned}
d|X_t^{i,N} - \bar{X}_t^i| = &\operatorname{sign}(X_t^{i,N} - \bar{X}_t^i) (2(X_t^{i,N} - \bar{X}_t^i) - (C_t^{i,N} - \bar{C}_t^i)) dt \\
& - (|(X_t^{i,N})^3 - (\bar{X}_t^i)^3| - |X_t^{i,N} - \bar{X}_t^i|) dt \\
& + \operatorname{sign}(X_t^{i,N} - \bar{X}_t^i) \left( \frac{1}{N} \sum_{j=1}^N K_X(Z_t^{i,N} - Z_t^{j,N}) - K_X * \bar{\rho}(\bar{Z}_t^i) \right) dt.
\end{aligned}
$$

The quantity $|X_t^{i,N} - \bar{X}_t^i|$ is therefore naturally contracting when $|2(X_t^{i,N} - \bar{X}_t^i) - (C_t^{i,N} - \bar{C}_t^i)|$ is close to 0. Thanks to the presence of a Brownian motion in the stochastic differential equations defining the potential $C$, we can now use a reflection coupling to have $|2(X_t^{i,N} - \bar{X}_t^i) - (C_t^{i,N} - \bar{C}_t^i)|$ go to 0. Consider the following coupling

$$ 
\left\{
\begin{aligned}
dX_t^{i,N} &= (X_t^{i,N} - (X_t^{i,N})^3 - C_t^{i,N} - \alpha)dt + \frac{1}{N} \sum_{j=1}^N K_X(Z_t^{i,N} - Z_t^{j,N})dt \\
dC_t^{i,N} &= (\gamma X_t^{i,N} - C_t^{i,N} + \beta)dt + \frac{1}{N} \sum_{j=1}^N K_C(Z_t^{i,N} - Z_t^{j,N})dt \\
&\quad + \sigma_C \varphi_{\text{sc}} \left( |2(X_t^{i,N} - \bar{X}_t^i) - (C_t^{i,N} - \bar{C}_t^i)| \right) dB_t^{i,\text{sc},C} \\
&\quad + \sigma_C \varphi_{\text{rc}} \left( |2(X_t^{i,N} - \bar{X}_t^i) - (C_t^{i,N} - \bar{C}_t^i)| \right) dB_t^{i,\text{rc},C},
\end{aligned}
\right.
\quad (B.1) $$

and

$$ 
\left\{
\begin{aligned}
d\bar{X}_t^i &= (\bar{X}_t^i - (\bar{X}_t^i)^3 - \bar{C}_t^i - \alpha)dt + K_X * \bar{\rho}(\bar{Z}_t^i)dt \\
d\bar{C}_t^i &= (\gamma\bar{X}_t^i - \bar{C}_t^i + \beta)dt + K_C * \bar{\rho}(\bar{Z}_t^i)dt \\
&\quad + \sigma_C \varphi_{\text{sc}} \left( |2(X_t^{i,N} - \bar{X}_t^i) - (C_t^{i,N} - \bar{C}_t^i)| \right) dB_t^{i,\text{sc},C} \\
&\quad - \sigma_C \varphi_{\text{rc}} \left( |2(X_t^{i,N} - \bar{X}_t^i) - (C_t^{i,N} - \bar{C}_t^i)| \right) dB_t^{i,\text{rc},C} \\
\bar{\rho} &= \mathcal{L}((\bar{X}_t^1, \bar{C}_t^1)),
\end{aligned}
\right.
\quad (B.2) $$

and for $\delta > 0$, the following modified distance

$$ r_t^i = \delta|X_t^{i,N} - \bar{X}_t^i| + |2(X_t^{i,N} - \bar{X}_t^i) - (C_t^{i,N} - \bar{C}_t^i)|. $$

Like previously, we consider a modified semi-metric of the form $\frac{1}{N} \sum f(r_t^i) G_t^i$ and similar calculations yield

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

where $M_t^i$ is a continuous local martingale and

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

$$ \tilde{K}_t^i = G_t^i \left[ 2cf(r_t^i) + 2f''(r_t^i) \sigma_C^2 \varphi_{\text{rc}} \left( |2(X_t^{i,N} - \bar{X}_t^i) - (C_t^{i,N} - \bar{C}_t^i)| \right)^2 \right. \\ \qquad + f'(r_t^i) \left( |2(X_t^{i,N} - \bar{X}_t^i) - (C_t^{i,N} - \bar{C}_t^i)| (\delta + 1) - |(X_t^{i,N})^3 - (\bar{X}_t^i)^3| (\delta - 2) \right. \\ \qquad + |X_t^{i,N} - \bar{X}_t^i| (-\delta + \gamma + L_X(\delta + 2) + L_C) + |C_t^{i,N} - \bar{C}_t^i|(L_X(\delta + 2) + L_C) \\ \qquad \left. + \sigma_C^2 \varphi_{\text{rc}} \left( |2(X_t^{i,N} - \bar{X}_t^i) - (C_t^{i,N} - \bar{C}_t^i)| \right)^2 (\epsilon C_{f,1} + C_{f,2}) r_t^i \right] \\ \qquad \left. + \epsilon f(r_t^i) \left( 4\tilde{B} - \frac{\lambda}{8}\tilde{H}(\bar{Z}_t^i) - \frac{\lambda}{8}\tilde{H}(Z_t^{i,N}) - \frac{\lambda}{8N} \sum_{j=1}^N \tilde{H}(\bar{Z}_t^j) - \frac{\lambda}{32N} \sum_{j=1}^N \tilde{H}(Z_t^{j,N}) \right), $$

$$ I_t^{1,i} = G_t^i f'(r_t^i) \left[ (\delta + 2) \left( \left| \frac{1}{N} \sum_{j=1}^N K_X(\bar{Z}_t^i - \bar{Z}_t^j) - K_X * \bar{\mu}_t(\bar{Z}_t^i) \right| \right) \right. \\ \qquad \left. + \left( \left| \frac{1}{N} \sum_{j=1}^N K_C(\bar{Z}_t^i - \bar{Z}_t^j) - K_C * \bar{\mu}_t(\bar{Z}_t^i) \right| \right) \right], $$

$$ I_t^{2,i} = G_t^i f'(r_t^i) \left[ (\delta + 2) \left( \frac{L_X}{N} \left( \sum_{j=1}^N |X_t^{j,N} - \bar{X}_t^j| + |C_t^{j,N} - \bar{C}_t^j| \right) \right) \right. \\ \qquad \left. + \left( \frac{L_C}{N} \left( \sum_{j=1}^N |X_t^{j,N} - \bar{X}_t^j| + |C_t^{j,N} - \bar{C}_t^j| \right) \right) \right] \\ \qquad - cf(r_t^i)G_t^i - \epsilon f(r_t^i) \left[ \frac{\lambda}{32} H(\bar{Z}_t^i) \exp \left( a\sqrt{H(\bar{Z}_t^i)} \right) + \frac{\lambda}{32} H(Z_t^{i,N}) \exp \left( a\sqrt{H(Z_t^{i,N})} \right) \right] \\ \qquad - \epsilon f(r_t^i) \left[ \frac{\lambda}{32N} \sum_{j=1}^N H(\bar{Z}_t^j) \exp \left( a\sqrt{H(\bar{Z}_t^j)} \right) \right. \\ \qquad \left. + \frac{\lambda}{32N} \sum_{j=1}^N H(Z_t^{j,N}) \exp \left( a\sqrt{H(Z_t^{j,N})} \right) \right], $$

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

We then have the additional constraint of $\delta > 2$ (so that the coefficient appearing in front of $|(X_t^{i,N})^3 - (\bar{X}_t^i)^3|$ in the expression of $\tilde{K}_t^i$ is non-positive). Otherwise, we deal with the various terms exactly as previously, through the choice of a sufficiently concave
function f and a law of large numbers, and by considering the regions of space

$$ \text{Reg}_1^i = \{ (\bar{Z}_t^i, Z_t^{i,N}) \text{ s.t. } |2(X_t^{i,N} - \bar{X}_t^i) - (C_t^{i,N} - \bar{C}_t^i)| \ge \xi \text{ and } r_t^i \le R \}, $$

$$ \text{Reg}_2^i = \{ (\bar{Z}_t^i, Z_t^{i,N}) \text{ s.t. } |2(X_t^{i,N} - \bar{X}_t^i) - (C_t^{i,N} - \bar{C}_t^i)| < \xi \text{ and } r_t^i \le R_1 \}, $$

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

Index

Throughout this article, we define many parameters and constants. For the sake of clarity, we list the main ones here so as to give the reader an index to refer to.

• X, C, Z: X and C are the processes we consider (see (1.1) and (1.2)) and we often refer to $Z = (X, C)$,

• $\bar{\mu}_t = \text{Law}(\bar{Z}_t)$: the density of the non-linear limit (see (1.2)),

• $\alpha, \beta, \gamma, \sigma_X, \sigma_C$: parameters of the problem (see (1.1)),

• $K_X, K_C, L_X, L_C, L_{X,max}, L_{C,max}$: $K_X$ (resp. $K_C$) is a Lipschitz continuous interaction kernel, with Lipschitz constant $L_X \in [0, L_{X,max}]$ (resp. $L_C \in [0, L_{C,max}]$), as given in Assumption 1.1. In the case of uniform in time propagation of chaos, the inequalities $L_X$ and $L_C$ must satisfy are listed in Subsection 2.4,

• $W_p$: the usual Wasserstein distance associated to the $L^p$ distance (see (1.3)),

• $a, \tilde{a}, C_{\text{init,exp}}$: constants used to give an exponential initial moment to the problem (see the assumptions of Theorem 1.4 and Section 2.3),

• $\lambda, B, \tilde{B}, H, \tilde{H}, \alpha_X, \alpha_C, \beta_X, \beta_C$: $H$ (resp. $\tilde{H}$) is a Lyapunov functions given in (2.1) (resp. (2.8)). Its main property involves parameters $\lambda$ and $B$ (resp. $\lambda$ and $\tilde{B}$), as can for instance be seen in (2.3) (resp. (2.14)). $\alpha_X, \alpha_C, \beta_X$ and $\beta_C$ are intermediate constants given in Lemma 2.2,

• c: a contraction rate (see Subsection 2.4),

• $r, f, g, \phi, \Phi, G, \rho, \delta, R, \epsilon, C_{f,1}, C_{f,2}$: $f$ (see (2.20)) is a concave function, the definition of which involves $g, \phi, \Phi$ (see Subsection 2.4). Function $G$ (see (2.26)) is then used to define $\rho$ (see (2.25)), the semi-metric we consider in the end. All these notations thus refer to the modified distance we consider. These functions will be applied to a modification $r$ of the usual $L^1$ distance (see equation (2.24)). Then, parameters $\delta, R, \epsilon, C_{f,1}$, and $C_{f,2}$ are used to define such functions (see Subsection 2.4 for some explicit values),

• $R_0, \phi_{\min}$: intermediate constants (see Subsection 2.4),

• $C_{\text{init},2}$: uniform in time bound on the second moment of the processes (see Lemma 2.4),

• $C_1, C_2, C_3$: constants used to quantify the control our modified distance has over the usual $L^1$ and $L^2$ distance (see Lemma 2.7 for the control and Subsection 2.4 for explicit values),

• $\phi_{rc}, \phi_{sc}, \xi$: $\phi_{rc}$ and $\phi_{sc}$ are two Lipschitz continuous functions used to define the coupling method, and their definitions involve a parameter $\xi$ which converges to 0 in the end (see the beginning of Section 3),

• $C_{r,H}$: used to explicit the control of the Lyapunov function $H$ over the distance $r$ (see Lemma 2.1),
## References

[1] Javier Baladron, Diego Fasoli, Olivier Faugeras, and Jonathan Touboul, Mean-field description and propagation of chaos in networks of Hodgkin-Huxley and FitzHugh-Nagumo neurons, Journal of Mathematical Neuroscience 2 (2012), Art. 10, 50, doi. MR-2974499

[2] Şeyma Bayrak, Philipp Hövel, and Vesna Vuksanović, Modeling functional connectivity on empirical and randomized structural brain networks, Differential Equations and Dynamical Systems 29 (2021), no. 4, 789–805, doi.

[3] Nils Berglund and Damien Landon, Mixed-mode oscillations and interspike interval statistics in the stochastic FitzHugh-Nagumo model, Nonlinearity 25 (2012), no. 8, 2303–2335, doi. MR-2946187

[4] François Bolley, Arnaud Guillin, and Florent Malrieu, Trend to equilibrium and particle approximation for a weakly selfconsistent Vlasov-Fokker-Planck equation, M2AN. Mathematical Modelling and Numerical Analysis 44 (2010), no. 5, 867–884, doi. MR-2731396

[5] Mireille Bossy, Olivier Faugeras, and Denis Talay, Clarification and complement to "Mean-field description and propagation of chaos in networks of Hodgkin-Huxley and FitzHugh-Nagumo neurons", Journal of Mathematical Neuroscience 5 (2015), Art. 19, 23, doi. MR-3392551

[6] Mireille Bossy, Joaquin Fontbona, and Héctor Olivero, Synchronization of stochastic mean field networks of Hodgkin-Huxley neurons with noisy channels, Journal of Mathematical Biology 78 (2019), no. 6, 1771–1820, doi. MR-3968981

[7] Gonçalo dos Reis, William Salkeld, and Julian Tugaut, Freidlin-Wentzell LDP in path space for McKean-Vlasov equations and the functional iterated logarithm law, The Annals of Applied Probability 29 (2019), no. 3, 1487–1540, doi. MR-3914550

[8] Alain Durmus, Andreas Eberle, Arnaud Guillin, and Raphael Zimmer, An elementary approach to uniform in time propagation of chaos, Proceedings of the American Mathematical Society 148 (2020), no. 12, 5387–5398, doi. MR-4163850

[9] Andreas Eberle, Reflection couplings and contraction rates for diffusions, Probability Theory and Related Fields 166 (2016), no. 3-4, 851–886, doi. MR-3568041

[10] Arnaud Guillin, Pierre Le Bris, and Pierre Monmarché, Convergence rates for the Vlasov-Fokker-Planck equation and uniform in time propagation of chaos in non convex cases, Electronic Journal of Probability 27 (2022), Paper No. 124, 44, doi. MR-4489825

[11] Alan L. Hodgkin and Andrew F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, The Journal of Physiology 117 (1952), no. 4, 500–544, doi.

[12] Nobuyuki Ikeda and Shinzo Watanabe, Stochastic differential equations and diffusion processes, second ed., North-Holland Mathematical Library, vol. 24, North-Holland Publishing Co., Amsterdam; Kodansha, Ltd., Tokyo, 1989. MR-1011252

[13] Mark Kac, Foundations of kinetic theory, Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954–1955, Vol. III, University of California Press, Berkeley-Los Angeles, Calif., 1956, pp. 171–197. MR-0084985

[14] Chaman Kumar, Neelima, Christoph Reisinger, and Wolfgang Stockinger, Well-posedness and tamed schemes for McKean-Vlasov equations with common noise, The Annals of Applied Probability 32 (2022), no. 5, 3283–3330, doi. MR-4497846

[15] José R. León and Adeline Samson, Hypoelliptic stochastic FitzHugh-Nagumo neuronal model: Mixing, up-crossing and estimation of the spike rate, The Annals of Applied Probability 28 (2018), no. 4, 2243–2274, doi. MR-3843828

[16] Fuzhi Li, Dynamics for stochastic Fitzhugh-Nagumo systems with general multiplicative noise on thin domains, Mathematical Methods in the Applied Sciences 44 (2021), no. 6, 5050–5078, doi. MR-4235550

[17] Fuzhi Li and Dongmei Xu, Regular dynamics for stochastic Fitzhugh-Nagumo systems with additive noise on thin domains, Discrete and Continuous Dynamical Systems. Series B. A Journal Bridging Mathematics and Sciences 26 (2021), no. 7, 3517–3542, doi. MR-4251845
[18] Yangrong Li and Fuzhi Li, *Limiting dynamics for stochastic FitzHugh-Nagumo equations on large domains*, Stochastics and Dynamics **19** (2019), no. 5, 1950037, 25, doi. MR-3994161

[19] Torgny Lindvall and L. Chris G. Rogers, *Coupling of multidimensional diffusions by reflection*, The Annals of Probability **14** (1986), no. 3, 860–872. MR-841588

[20] Eric Luçon and Christophe Poquet, *Periodicity induced by noise and interaction in the kinetic mean-field FitzHugh-Nagumo model*, The Annals of Applied Probability **31** (2021), no. 2, 561–593, doi. MR-4254489

[21] Eric Luçon and Wilhelm Stannat, *Mean field limit for disordered diffusions with singular interactions*, The Annals of Applied Probability **24** (2014), no. 5, 1946–1993, doi. MR-3226169

[22] Yan Lv and Wei Wang, *Limit dynamics for the stochastic FitzHugh-Nagumo system*, Nonlinear Analysis. Real World Applications. An International Multidisciplinary Journal **11** (2010), no. 4, 3091–3105, doi. MR-2661971

[23] Sima Mehri, Michael Scheutzow, Wilhelm Stannat, and Bian Z. Zangeneh, *Propagation of chaos for stochastic spatially structured neuronal networks with delay driven by jump diffusions*, The Annals of Applied Probability **30** (2020), no. 1, 175–207, doi. MR-4068309

[24] Stéphane Mischler, Cristóbal Quiñinao, and Jonathan Touboul, *On a kinetic Fitzhugh-Nagumo model of neuronal network*, Communications in Mathematical Physics **342** (2016), no. 3, 1001–1042, doi. MR-3465438

[25] Christoph Reisinger and Wolfgang Stockinger, *An adaptive Euler-Maruyama scheme for McKean-Vlasov SDEs with super-linear growth and application to the mean-field FitzHugh-Nagumo model*, Journal of Computational and Applied Mathematics **400** (2022), Paper No. 113725, 23, doi. MR-4293705

[26] Daniel Revuz and Marc Yor, *Continuous martingales and Brownian motion*, third ed., Grundlehren Der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 293, Springer-Verlag, Berlin, 1999, doi. MR-1725357

[27] Katharina Schuh, *Global contractivity for Langevin dynamics with distribution-dependent forces and uniform in time propagation of chaos*, 2022, arXiv:2206.03082, doi.

[28] Alain-Sol Sznitman, *Topics in propagation of chaos*, École d'Été de Probabilités de Saint-Flour XIX—1989, Lecture Notes in Math., vol. 1464, Springer, Berlin, 1991, pp. 165–251, doi. MR-1108185

[29] D. Tatchim Bemmo, Martin Siewe Siewe, and Clément Tchawoua, *Combined effects of correlated bounded noises and weak periodic signal input in the modified FitzHugh-Nagumo neural model*, Communications in Nonlinear Science and Numerical Simulation **18** (2013), no. 5, 1275–1287, doi. MR-2998587

[30] Michèle Thieullen, *Deterministic and stochastic FitzHugh-Nagumo systems*, Stochastic Biomathematical Models, Lecture Notes in Math., vol. 2058, Springer, Heidelberg, 2013, pp. 175–186, doi. MR-3051033

[31] Henry C. Tuckwell, Roger Rodriguez, and Frederic Y. M. Wan, *Determination of firing times for the stochastic FitzHugh-Nagumo neuronal model*, Neural Computation **15** (2003), no. 1, 143–159, doi.

[32] Kenneth Uda, *Ergodicity and spike rate for stochastic FitzHugh-Nagumo neural model with periodic forcing*, Chaos, Solitons & Fractals **123** (2019), 383–399, doi. MR-3942215

**Acknowledgments.** During this study, Laetitia Colombani was a PhD student under the supervision of Patrick Cattiaux and Manon Costa, and Pierre Le Bris a PhD student under the supervision of Arnaud Guillin and Pierre Monmarché. The authors would like to thank them, as well as Samir Salem, for their help throughout the redaction of the present article. This work has been (partially) supported by the Project EFI ANR-17-CE40-0030 of the French National Research Agency.