In (A.26), we focus on

$$ \int_{S_N} [\eta(i, m) - \eta(i, m + 1)] f(\eta, \xi) d\nu_N(\eta, \xi) = \int_{S_N} \eta(i, m) f(\eta, \xi) d\nu_N(\eta, \xi) - \int_{S_N} \eta(i, m + 1) f(\eta, \xi) d\nu_N(\eta, \xi). \quad (A.27) $$

In the first integral in (A.27), we perform the change of variable

$$ \eta \mapsto \eta^{(i,m),(i,m+1)} := \eta^{\hat{k}_0, \hat{k}_1}. $$

This gives

$$ \begin{aligned} & \int_{S_N} [\eta(i, m) - \eta(i, m + 1)] f(\eta, \xi) d\nu_N(\eta, \xi) \\ &= \int_{S_N} \eta(\hat{k}_1) [f(\eta^{\hat{k}_0, \hat{k}_1}, \xi) - f(\eta, \xi)] d\nu_N(\eta, \xi) \\ &= \int_{S_N} \eta(\hat{k}_1) \underbrace{\left[ \sqrt{f(\eta^{\hat{k}_0, \hat{k}_1}, \xi)} - \sqrt{f(\eta, \xi)} \right] \left[ \sqrt{f(\eta^{\hat{k}_0, \hat{k}_1}, \xi)} + \sqrt{f(\eta, \xi)} \right]}_{\text{use (I3) at this point}} d\nu_N(\eta, \xi). \\ & \le \frac{B}{2} \int_{S_N} \eta(\hat{k}_1) \left[ \sqrt{f(\eta^{\hat{k}_0, \hat{k}_1}, \xi)} - \sqrt{f(\eta, \xi)} \right]^2 d\nu_N(\eta, \xi) \end{aligned} \quad (A.28) $$

$$ + \frac{1}{2B} \int_{S_N} \eta(\hat{k}_1) \left[ \sqrt{f(\eta^{\hat{k}_0, \hat{k}_1}, \xi)} + \sqrt{f(\eta, \xi)} \right]^2 d\nu_N(\eta, \xi), \quad (A.29) $$

where $B > 0$ is to be determined later in this proof. In (A.29) we now use (I4) and the fact that $f$ is a density with respect to $\nu_N$ to write

$$ \begin{aligned} & \int_{S_N} [\eta(i, m) - \eta(i, m + 1)] f(\eta, \xi) d\nu_N(\eta, \xi) \\ & \le \frac{B}{2} \int_{S_N} \eta(\hat{k}_1) \left[ \sqrt{f(\eta^{\hat{k}_0, \hat{k}_1}, \xi)} - \sqrt{f(\eta, \xi)} \right]^2 d\nu_N(\eta, \xi) + \frac{2}{B}. \end{aligned} \quad (A.30) $$

Incorporating (A.30) into (A.26) yields

$$ \begin{aligned} & \int_{S_N} V(s, \eta) f(\eta, \xi) d\nu_N(\eta, \xi) \\ & \le \sum_{k \in [-\epsilon N, \epsilon N]^{p-1}} \sum_{\ell=[N-1-\epsilon N]}^{N-1} \frac{c_{N,\epsilon}}{N^{p-1}} \sum_{i \in \mathbb{T}_N^{p-1}} G(s, \frac{i}{N}, \frac{N-1}{N}) \\ & \qquad \times \sum_{m=\ell}^{N-2} \left[ \frac{B}{2} \int_{S_N} \eta(\hat{k}_1) \left[ \sqrt{f(\eta^{\hat{k}_0, \hat{k}_1}, \xi)} - \sqrt{f(\eta, \xi)} \right]^2 d\nu_N(\eta, \xi) + \frac{2}{B} \right] \\ & \le \|G(s, \cdot)\|_{L^\infty(\Lambda)} \left[ \frac{B}{2N^{p-1}} \sum_{i \in \mathbb{T}_N^{p-1}} \sum_{m=0}^{N-2} \int_{S_N} \left[ \sqrt{f(\eta^{\hat{k}_0, \hat{k}_1}, \xi)} - \sqrt{f(\eta, \xi)} \right]^2 d\nu_N(\eta, \xi) + \frac{2\epsilon N}{B} \right] \end{aligned} \quad (A.31) $$

where we used that $\sum_k \sum_\ell c_{N,\epsilon} = 1$, $\frac{1}{N^{p-1}} \sum_i 1 = 1$, and

$$ N - 1 - \ell \le \epsilon N \quad \text{for all } \ell \in \llbracket N - 1 - \epsilon N \rbrack; N - 1 \rbrack. $$
In (A.26), we focus on

$$ \int_{S_N} [\eta(i, m) - \eta(i, m + 1)] f(\eta, \xi) d\nu_N(\eta, \xi) = \int_{S_N} \eta(i, m) f(\eta, \xi) d\nu_N(\eta, \xi) - \int_{S_N} \eta(i, m + 1) f(\eta, \xi) d\nu_N(\eta, \xi). \quad (A.27) $$

In the first integral in (A.27), we perform the change of variable

$$ \eta \mapsto \eta^{(i,m),(i,m+1)} := \eta^{\hat{k}_0, \hat{k}_1}. $$

This gives

$$ \begin{aligned} & \int_{S_N} [\eta(i, m) - \eta(i, m + 1)] f(\eta, \xi) d\nu_N(\eta, \xi) \\ &= \int_{S_N} \eta(\hat{k}_1) [f(\eta^{\hat{k}_0, \hat{k}_1}, \xi) - f(\eta, \xi)] d\nu_N(\eta, \xi) \\ &= \int_{S_N} \eta(\hat{k}_1) \underbrace{\left[ \sqrt{f(\eta^{\hat{k}_0, \hat{k}_1}, \xi)} - \sqrt{f(\eta, \xi)} \right] \left[ \sqrt{f(\eta^{\hat{k}_0, \hat{k}_1}, \xi)} + \sqrt{f(\eta, \xi)} \right]}_{\text{use (I3) at this point}} d\nu_N(\eta, \xi). \\ & \le \frac{B}{2} \int_{S_N} \eta(\hat{k}_1) \left[ \sqrt{f(\eta^{\hat{k}_0, \hat{k}_1}, \xi)} - \sqrt{f(\eta, \xi)} \right]^2 d\nu_N(\eta, \xi) \end{aligned} \quad (A.28) $$

$$ + \frac{1}{2B} \int_{S_N} \eta(\hat{k}_1) \left[ \sqrt{f(\eta^{\hat{k}_0, \hat{k}_1}, \xi)} + \sqrt{f(\eta, \xi)} \right]^2 d\nu_N(\eta, \xi), \quad (A.29) $$

where $B > 0$ is to be determined later in this proof. In (A.29) we now use (I4) and the fact that $f$ is a density with respect to $\nu_N$ to write

$$ \begin{aligned} & \int_{S_N} [\eta(i, m) - \eta(i, m + 1)] f(\eta, \xi) d\nu_N(\eta, \xi) \\ & \le \frac{B}{2} \int_{S_N} \eta(\hat{k}_1) \left[ \sqrt{f(\eta^{\hat{k}_0, \hat{k}_1}, \xi)} - \sqrt{f(\eta, \xi)} \right]^2 d\nu_N(\eta, \xi) + \frac{2}{B}. \end{aligned} \quad (A.30) $$

Incorporating (A.30) into (A.26) yields

$$ \begin{aligned} & \int_{S_N} V(s, \eta) f(\eta, \xi) d\nu_N(\eta, \xi) \\ & \le \sum_{k \in [-\epsilon N, \epsilon N]^{p-1}} \sum_{\ell=[N-1-\epsilon N]}^{N-1} \frac{c_{N,\epsilon}}{N^{p-1}} \sum_{i \in \mathbb{T}_N^{p-1}} G(s, \frac{i}{N}, \frac{N-1}{N}) \\ & \qquad \times \sum_{m=\ell}^{N-2} \left[ \frac{B}{2} \int_{S_N} \eta(\hat{k}_1) \left[ \sqrt{f(\eta^{\hat{k}_0, \hat{k}_1}, \xi)} - \sqrt{f(\eta, \xi)} \right]^2 d\nu_N(\eta, \xi) + \frac{2}{B} \right] \\ & \le \|G(s, \cdot)\|_{L^\infty(\Lambda)} \left[ \frac{B}{2N^{p-1}} \sum_{i \in \mathbb{T}_N^{p-1}} \sum_{m=0}^{N-2} \int_{S_N} \left[ \sqrt{f(\eta^{\hat{k}_0, \hat{k}_1}, \xi)} - \sqrt{f(\eta, \xi)} \right]^2 d\nu_N(\eta, \xi) + \frac{2\epsilon N}{B} \right] \end{aligned} \quad (A.31) $$

where we used that $\sum_k \sum_\ell c_{N,\epsilon} = 1$, $\frac{1}{N^{p-1}} \sum_i 1 = 1$, and

$$ N - 1 - \ell \le \epsilon N \quad \text{for all } \ell \in \llbracket N - 1 - \epsilon N \rbrack; N - 1 \rbrack. $$
Now observe that we can make the Dirichlet form $D_N^{\text{field}}$ appear in (A.31) thanks to (D1). This results in

$$ \int_{S_N} V(s, \eta) f(\eta, \xi) \, d\nu_N(\eta, \xi) \le \|G(s, \cdot)\|_{L^\infty(\Lambda)} \left[ \frac{2B}{dN^{p-1}} D_N^{\text{field}}(\sqrt{f}, \nu_N) + \frac{2\epsilon N}{B} \right]. \quad (A.32) $$

By gathering (A.23), (A.25) and (A.32), we obtain

$$ \begin{aligned} & \mathbb{E}_N^{\mu_N} \left[ \left| \int_0^t (\text{A.20}) \, ds \right| \right] \\ & \le \int_0^t \sup_{\substack{f \text{ density} \\ \text{with respect to } \nu_N}} \left\{ \|G(s, \cdot)\|_{L^\infty(\Lambda)} \left[ \frac{2B}{dN^{p-1}} D_N^{\text{field}}(\sqrt{f}, \nu_N) + \frac{2\epsilon N}{B} \right] + \frac{1}{aN^p} \langle \mathcal{L}_N \sqrt{f}, \sqrt{f} \rangle_{\nu_N} \right\} ds + \frac{C_0}{a} \\ & \le \int_0^t \sup_{\substack{f \text{ density} \\ \text{with respect to } \nu_N}} \left\{ \left[ \frac{2B \|G(s, \cdot)\|_{L^\infty(\Lambda)}}{dN^{p-1}} - \frac{1}{aN^{p-2}} \right] D_N^{\text{field}}(\sqrt{f}, \nu_N) + \frac{2\epsilon N \|G(s, \cdot)\|_{L^\infty(\Lambda)}}{B} \right] ds + \frac{C_0 + 2c(b)\alpha}{a}, \end{aligned} \quad (A.33) $$

where we used the whole Lemma A.1 (with $\gamma = b$) and the fact that $-\frac{1}{2}\mathbb{I}_N^{\text{road}}$, $-\frac{1}{2}I_N^{\text{up}}$, $-\frac{1}{2}I_N^{\text{reac}}$ and $-\frac{1}{2}I_N^{\text{Rob}}$ are nonpositive to provide the last inequality. At this point, we make the choice $B := dN/(2a\|G(s, \cdot)\|_{L^\infty(\Lambda)})$ to cancel the supremum in (A.33), so that we are left with

$$ \mathbb{E}_N^{\mu_N} \left[ \left| \int_0^t (\text{A.20}) \, ds \right| \right] \le \frac{4T\epsilon a}{d} \sup_{s \in [0,T]} \|G(s, \cdot)\|_{L^\infty(\Lambda)}^2 + \frac{C_0 + 2c(b)\alpha}{a}. \quad (A.34) $$

By letting $\epsilon \to 0$ and then $a \to \infty$, we finally get the vanishing of $\mathbb{E}_N^{\mu_N} [|\int_0^t (\text{A.20}) \, ds|]$ which, combined with those of $\mathbb{E}_N^{\mu_N} [|\int_0^t (\text{A.18}) \, ds|]$ and $\mathbb{E}_N^{\mu_N} [|\int_0^t (\text{A.19}) \, ds|]$ in (A.21), completes this proof. $\square$

## A.3 Proof of the energy estimate

**Proof of Lemma 4.3 (Energy estimate).** For $q \in [1; p]$, $G \in C_c^{0,2}([0, T] \times \bar{\Lambda})$ and $v \in L^2(0, T; L^2(\Lambda))$, let us write the quantity below the temporal integral in (4.2) as

$$ J_G(s) = J_G(s, v, q) := \langle v(s), \partial_{e_q} G(s) \rangle_{\Lambda} - \frac{1}{2} \|G(s)\|_{L^2(\Lambda)}^2. $$

Now consider an enumerate sequence $(G_n)_{n \in \mathbb{N}}$ dense in $C_c^{0,2}([0, T] \times \bar{\Lambda})$, and observe that it is sufficient to show that there is a constant $C > 0$ such that for any $n_0 \in \mathbb{N}$, we have

$$ \mathbb{E}_{\infty} \left[ \max_{0 \le n \le n_0} \left\{ \int_0^T J_{G_n}(s) \, ds \right\} \right] < C, \quad (A.35) $$

where $\mathbb{E}_{\infty}$ denotes the expectation with respect to $\mathbb{Q}_{\infty}$. Since the maps

$$ (\pi_N(t))_{t \in [0, T]} \mapsto \int_0^T \left( \langle \pi_N^{\text{field}}(s), \partial_{e_q} G_n(s) \rangle - \frac{1}{2} \|G_n(s)\|_{L^2(\Lambda)}^2 \right) ds $$

are continuous with respect to the Skorokhod topology, and since the probability $\mathbb{Q}_{\infty}$ is in the weak closure of $(\mathbb{Q}_N^{\mu_N})_{N \ge 2}$, then the expectation in (A.35) can be recast

$$ \lim_{N \to \infty} \mathbb{E}_N^{\mu_N} \left[ \max_{0 \le n \le n_0} \left\{ \int_0^T \left( \langle \pi_N^{\text{field}}(s), \partial_{e_q} G_n(s) \rangle - \frac{1}{2} \|G_n(s)\|_{L^2(\Lambda)}^2 \right) ds \right\} \right], $$
Now observe that we can make the Dirichlet form $D_N^{\text{field}}$ appear in (A.31) thanks to (D1). This results in

$$ \int_{S_N} V(s, \eta) f(\eta, \xi) \, d\nu_N(\eta, \xi) \le \|G(s, \cdot)\|_{L^\infty(\Lambda)} \left[ \frac{2B}{dN^{p-1}} D_N^{\text{field}}(\sqrt{f}, \nu_N) + \frac{2\epsilon N}{B} \right]. \quad (A.32) $$

By gathering (A.23), (A.25) and (A.32), we obtain

$$ \begin{aligned} & \mathbb{E}_N^{\mu_N} \left[ \left| \int_0^t (\text{A.20}) \, ds \right| \right] \\ & \le \int_0^t \sup_{\substack{f \text{ density} \\ \text{with respect to } \nu_N}} \left\{ \|G(s, \cdot)\|_{L^\infty(\Lambda)} \left[ \frac{2B}{dN^{p-1}} D_N^{\text{field}}(\sqrt{f}, \nu_N) + \frac{2\epsilon N}{B} \right] + \frac{1}{aN^p} \langle \mathcal{L}_N \sqrt{f}, \sqrt{f} \rangle_{\nu_N} \right\} ds + \frac{C_0}{a} \\ & \le \int_0^t \sup_{\substack{f \text{ density} \\ \text{with respect to } \nu_N}} \left\{ \left[ \frac{2B \|G(s, \cdot)\|_{L^\infty(\Lambda)}}{dN^{p-1}} - \frac{1}{aN^{p-2}} \right] D_N^{\text{field}}(\sqrt{f}, \nu_N) + \frac{2\epsilon N \|G(s, \cdot)\|_{L^\infty(\Lambda)}}{B} \right] ds + \frac{C_0 + 2c(b)\alpha}{a}, \end{aligned} \quad (A.33) $$

where we used the whole Lemma A.1 (with $\gamma = b$) and the fact that $-\frac{1}{2}\mathbb{I}_N^{\text{road}}$, $-\frac{1}{2}I_N^{\text{up}}$, $-\frac{1}{2}I_N^{\text{reac}}$ and $-\frac{1}{2}I_N^{\text{Rob}}$ are nonpositive to provide the last inequality. At this point, we make the choice $B := dN/(2a\|G(s, \cdot)\|_{L^\infty(\Lambda)})$ to cancel the supremum in (A.33), so that we are left with

$$ \mathbb{E}_N^{\mu_N} \left[ \left| \int_0^t (\text{A.20}) \, ds \right| \right] \le \frac{4T\epsilon a}{d} \sup_{s \in [0,T]} \|G(s, \cdot)\|_{L^\infty(\Lambda)}^2 + \frac{C_0 + 2c(b)\alpha}{a}. \quad (A.34) $$

By letting $\epsilon \to 0$ and then $a \to \infty$, we finally get the vanishing of $\mathbb{E}_N^{\mu_N} [|\int_0^t (\text{A.20}) \, ds|]$ which, combined with those of $\mathbb{E}_N^{\mu_N} [|\int_0^t (\text{A.18}) \, ds|]$ and $\mathbb{E}_N^{\mu_N} [|\int_0^t (\text{A.19}) \, ds|]$ in (A.21), completes this proof. $\square$

### A.3 Proof of the energy estimate

**Proof of Lemma 4.3 (Energy estimate).** For $q \in [1; p]$, $G \in C_c^{0,2}([0, T] \times \bar{\Lambda})$ and $v \in L^2(0, T; L^2(\Lambda))$, let us write the quantity below the temporal integral in (4.2) as

$$ J_G(s) = J_G(s, v, q) := \langle v(s), \partial_{e_q} G(s) \rangle_{\Lambda} - \frac{1}{2} \|G(s)\|_{L^2(\Lambda)}^2. $$

Now consider an enumerate sequence $(G_n)_{n \in \mathbb{N}}$ dense in $C_c^{0,2}([0, T] \times \bar{\Lambda})$, and observe that it is sufficient to show that there is a constant $C > 0$ such that for any $n_0 \in \mathbb{N}$, we have

$$ \mathbb{E}_{\infty} \left[ \max_{0 \le n \le n_0} \left\{ \int_0^T J_{G_n}(s) \, ds \right\} \right] < C, \quad (A.35) $$

where $\mathbb{E}_{\infty}$ denotes the expectation with respect to $\mathbb{Q}_{\infty}$. Since the maps

$$ (\pi_N(t))_{t \in [0,T]} \mapsto \int_0^T \left( \langle \pi_N^{\text{field}}(s), \partial_{e_q} G_n(s) \rangle - \frac{1}{2} \|G_n(s)\|_{L^2(\Lambda)}^2 \right) ds $$

are continuous with respect to the Skorokhod topology, and since the probability $\mathbb{Q}_{\infty}$ is in the weak closure of $(\mathbb{Q}_N^{\mu_N})_{N \ge 2}$, then the expectation in (A.35) can be recast

$$ \lim_{N \to \infty} \mathbb{E}_N^{\mu_N} \left[ \max_{0 \le n \le n_0} \left\{ \int_0^T \left( \langle \pi_N^{\text{field}}(s), \partial_{e_q} G_n(s) \rangle - \frac{1}{2} \|G_n(s)\|_{L^2(\Lambda)}^2 \right) ds \right\} \right], $$
that is, by explaining the empirical measure $\pi_N^{\text{field}}(s)$,

$$ \lim_{N \to \infty} \mathbb{E}_N^{\mu_N} \left[ \max_{0 \le n \le n_0} \left\{ \int_0^T \left( \frac{1}{N^p} \sum_{\hat{\iota} \in \Lambda_N} \partial_{e_q} G_n(s, \frac{\hat{\iota}}{N}) \times \eta_s(\hat{\iota}) - \frac{1}{2} \|G_n(s)\|_{L^2(\Lambda)}^2 \right) ds \right\} \right] =: J_{G_n}^N(s, \eta, q) = J_{G_n}^N(s). \quad (A.36) $$

For a fixed $a > 0$, we use the entropy inequality (see [34, Appendix 1, Section 8]) to bound the expectation in (A.36) with

$$ \mathbb{E}_N^{\mu_N} \left[ \max_{0 \le n \le n_0} \left\{ \int_0^T J_{G_n}^N(s) ds \right\} \right] \le \frac{1}{aN^p} \log \left( \mathbb{E}_N^{\nu_N} \left[ \exp \left( aN^p \times \max_{0 \le n \le n_0} \left\{ \int_0^T J_{G_n}^N(s) ds \right\} \right) \right] \right) + \frac{1}{aN^p} \mathcal{H}(\mu_N | \nu_N). \quad (A.37) $$

Thanks to the control we have on $\mathcal{H}(\mu_N|\nu_N)$, as outlined in (A.1), the second term in (A.37) is bounded by $C_0/a$ and does not pose a significant issue while $a$ remains far from 0. Now focusing on the first term in (A.37), notice that

$$ \mathbb{E}_N^{\nu_N} \left[ \exp \left( aN^p \times \max_{0 \le n \le n_0} \left\{ \int_0^T J_{G_n}^N(s) ds \right\} \right) \right] \le \mathbb{E}_N^{\nu_N} \left[ \sum_{n=0}^{n_0} \exp \left( aN^p \int_0^T J_{G_n}^N(s) ds \right) \right] \\ = \sum_{n=0}^{n_0} \mathbb{E}_N^{\nu_N} \left[ \exp \left( aN^p \int_0^T J_{G_n}^N(s) ds \right) \right]. $$

This control allows to bootstrap (I1) (cf. Lemma A.2), providing

$$ \limsup_{N \to \infty} \left\{ \frac{1}{aN^p} \log \left( \mathbb{E}_N^{\nu_N} \left[ \exp \left( aN^p \times \max_{0 \le n \le n_0} \left\{ \int_0^T J_{G_n}^N(s) ds \right\} \right) \right] \right) \right\} \\ \le \max_{0 \le n \le n_0} \limsup_{N \to \infty} \left\{ \frac{1}{aN^p} \log \left( \mathbb{E}_N^{\nu_N} \left[ \exp \left( aN^p \times \int_0^T J_{G_n}^N(s) ds \right) \right] \right) \right\}. \quad (A.38) $$

Working on the term between the bracket in (A.38), we use the Feynman-Kac's inequality — see [4] (Lemma 7.3 in Appendix) — with the operator $\mathcal{L}_N + aN^p J_{G_n}^N(s, \cdot, q)$. By using the variational formula (Rayleigh quotient) for the largest (principal) eigenvalue of this operator, we are led to

$$ \frac{1}{aN^p} \log \left( \mathbb{E}_N^{\nu_N} \left[ \exp \left( aN^p \times \int_0^T J_{G_n}^N(s) ds \right) \right] \right) \\ \le \int_0^T \sup_{\substack{f \text{ density} \\ \text{with respect to } \nu_N}} \left\{ \int_{S_N} \left( \frac{1}{N^p} \sum_{\hat{\iota} \in \Lambda_N} \eta(\hat{\iota}) \partial_{e_q} G_n(s, \frac{\hat{\iota}}{N}) \right) f(\eta, \xi) \, d\nu_N(\eta, \xi) + \frac{1}{aN^p} \langle \mathcal{L}_N \sqrt{f}, \sqrt{f} \rangle_{\nu_N} \right\} ds \quad (A.39) \\ - \frac{1}{2} \int_0^T \|G_n(s)\|_{L^2(\Lambda)}^2 ds. $$

Considering the integral term into the supremum in (A.39), the regularity of $G$ enables to write

$$ \frac{1}{N^p} \partial_{e_q} G_n(s, \frac{\hat{\iota}}{N}) = \frac{1}{N^{p-1}} \left( G_n(s, \frac{\hat{\iota}+e_q}{N}) - G_n(s, \frac{\hat{\iota}}{N}) + o(1/N) \right), $$
that is, by explaining the empirical measure $\pi_N^{\text{field}}(s)$,

$$ \lim_{N \to \infty} \mathbb{E}_N^{\mu_N} \left[ \max_{0 \le n \le n_0} \left\{ \int_0^T \left( \frac{1}{N^p} \sum_{\hat{\iota} \in \Lambda_N} \partial_{e_q} G_n(s, \frac{\hat{\iota}}{N}) \times \eta_s(\hat{\iota}) - \frac{1}{2} \|G_n(s)\|_{L^2(\Lambda)}^2 \right) ds \right\} \right] =: J_{G_n}^N(s, \eta, q) = J_{G_n}^N(s). \quad (A.36) $$

For a fixed $a > 0$, we use the entropy inequality (see [34, Appendix 1, Section 8]) to bound the expectation in (A.36) with

$$ \mathbb{E}_N^{\mu_N} \left[ \max_{0 \le n \le n_0} \left\{ \int_0^T J_{G_n}^N(s) ds \right\} \right] \le \frac{1}{aN^p} \log \left( \mathbb{E}_N^{\nu_N} \left[ \exp \left( aN^p \times \max_{0 \le n \le n_0} \left\{ \int_0^T J_{G_n}^N(s) ds \right\} \right) \right] \right) + \frac{1}{aN^p} \mathcal{H}(\mu_N | \nu_N). \quad (A.37) $$

Thanks to the control we have on $\mathcal{H}(\mu_N|\nu_N)$, as outlined in (A.1), the second term in (A.37) is bounded by $C_0/a$ and does not pose a significant issue while $a$ remains far from 0. Now focusing on the first term in (A.37), notice that

$$ \mathbb{E}_N^{\nu_N} \left[ \exp \left( aN^p \times \max_{0 \le n \le n_0} \left\{ \int_0^T J_{G_n}^N(s) ds \right\} \right) \right] \le \mathbb{E}_N^{\nu_N} \left[ \sum_{n=0}^{n_0} \exp \left( aN^p \int_0^T J_{G_n}^N(s) ds \right) \right] \\ = \sum_{n=0}^{n_0} \mathbb{E}_N^{\nu_N} \left[ \exp \left( aN^p \int_0^T J_{G_n}^N(s) ds \right) \right]. $$

This control allows to bootstrap (I1) (cf. Lemma A.2), providing

$$ \limsup_{N \to \infty} \left\{ \frac{1}{aN^p} \log \left( \mathbb{E}_N^{\nu_N} \left[ \exp \left( aN^p \times \max_{0 \le n \le n_0} \left\{ \int_0^T J_{G_n}^N(s) ds \right\} \right) \right] \right) \right\} \\ \le \max_{0 \le n \le n_0} \limsup_{N \to \infty} \left\{ \frac{1}{aN^p} \log \left( \mathbb{E}_N^{\nu_N} \left[ \exp \left( aN^p \times \int_0^T J_{G_n}^N(s) ds \right) \right] \right) \right\}. \quad (A.38) $$

Working on the term between the bracket in (A.38), we use the Feynman-Kac's inequality — see [4] (Lemma 7.3 in Appendix) — with the operator $\mathcal{L}_N + aN^p J_{G_n}^N(s, \cdot, q)$. By using the variational formula (Rayleigh quotient) for the largest (principal) eigenvalue of this operator, we are led to

$$ \frac{1}{aN^p} \log \left( \mathbb{E}_N^{\nu_N} \left[ \exp \left( aN^p \times \int_0^T J_{G_n}^N(s) ds \right) \right] \right) \\ \le \int_0^T \sup_{\substack{f \text{ density} \\ \text{with respect to } \nu_N}} \left\{ \int_{S_N} \left( \frac{1}{N^p} \sum_{\hat{\iota} \in \Lambda_N} \eta(\hat{\iota}) \partial_{e_q} G_n(s, \frac{\hat{\iota}}{N}) \right) f(\eta, \xi) \, d\nu_N(\eta, \xi) + \frac{1}{aN^p} \langle \mathcal{L}_N \sqrt{f}, \sqrt{f} \rangle_{\nu_N} \right\} ds \quad (A.39) \\ - \frac{1}{2} \int_0^T \|G_n(s)\|_{L^2(\Lambda)}^2 ds. $$

Considering the integral term into the supremum in (A.39), the regularity of $G$ enables to write

$$ \frac{1}{N^p} \partial_{e_q} G_n(s, \frac{\hat{\iota}}{N}) = \frac{1}{N^{p-1}} \left( G_n(s, \frac{\hat{\iota}+e_q}{N}) - G_n(s, \frac{\hat{\iota}}{N}) + o(1/N) \right), $$
so that

$$
\begin{align*}
& \int_{S_N} \left( \frac{1}{N^p} \sum_{\hat{\iota} \in \Lambda_N} \eta(\hat{\iota}) \partial_{e_q} G_n(s, \frac{\hat{\iota}}{N}) \right) f(\eta, \xi) \, d\nu_N(\eta, \xi) \\
&= \int_{S_N} \left( \frac{1}{N^{p-1}} \sum_{\hat{\iota} \in \Lambda_N} \eta(\hat{\iota}) \left( G_n(s, \frac{\hat{\iota}+e_q}{N}) - G_n(s, \frac{\hat{\iota}}{N}) \right) \right) f(\eta, \xi) \, d\nu_N(\eta, \xi) + o_N(1).
\end{align*}
$$

Using then a summation by part (mind the compact support of G) and the change of variable $\eta \mapsto \eta^{\hat{\iota}, \hat{\iota}+e_q}$, we obtain

$$
\begin{align*}
& \int_{S_N} \left( \frac{1}{N^p} \sum_{\hat{\iota} \in \Lambda_N} \eta(\hat{\iota}) \partial_{e_q} G_n(s, \frac{\hat{\iota}}{N}) \right) f(\eta, \xi) \, d\nu_N(\eta, \xi) - o_N(1) \\
&= \frac{1}{N^{p-1}} \sum_{\hat{\iota} \in \Lambda_N} \int_{S_N} \eta(\hat{\iota}) G_n(s, \frac{\hat{\iota}}{N}) \left( f(\eta, \xi) - f(\eta^{\hat{\iota}, \hat{\iota}+e_q}, \xi) \right) \, d\nu_N(\eta, \xi) \\
&= \frac{1}{N^{p-1}} \sum_{\hat{\iota} \in \Lambda_N} \int_{S_N} \eta(\hat{\iota}) \underbrace{G_n(s, \frac{\hat{\iota}}{N}) \left[ \sqrt{f(\eta^{\hat{\iota}, \hat{\iota}+e_q}, \xi)} - \sqrt{f(\eta, \xi)} \right] \left[ \sqrt{f(\eta^{\hat{\iota}, \hat{\iota}+e_q}, \xi)} + \sqrt{f(\eta, \xi)} \right]}_{\text{use (I3) at this point}} \, d\nu_N(\eta, \xi) \\
&\le \frac{1}{N^{p-1}} \sum_{\hat{\iota} \in \Lambda_N} \frac{B}{2} \int_{S_N} \eta(\hat{\iota}) \left[ \sqrt{f(\eta^{\hat{\iota}, \hat{\iota}+e_q}, \xi)} - \sqrt{f(\eta, \xi)} \right]^2 \, d\nu_N(\eta, \xi) \tag{A.40}
\end{align*}
$$

$$
+ \frac{1}{N^{p-1}} \sum_{\hat{\iota} \in \Lambda_N} \frac{1}{2B} \left( G_n(s, \frac{\hat{\iota}}{N}) \right)^2 \int_{S_N} \eta(\hat{\iota}) \left[ \sqrt{f(\eta^{\hat{\iota}, \hat{\iota}+e_q}, \xi)} + \sqrt{f(\eta, \xi)} \right]^2 d\nu_N(\eta, \xi), \quad (A.41)
$$

where $B > 0$ is to be determined later in this proof. In (A.41) we use now (I4) and the fact that $f$ is a density with respect to $\nu_N$ to write

$$
\begin{align*}
& \int_{S_N} \left( \frac{1}{N^p} \sum_{\hat{\iota} \in \Lambda_N} \eta(\hat{\iota}) \partial_{e_q} G_n(s, \frac{\hat{\iota}}{N}) \right) f(\eta, \xi) \, d\nu_N(\eta, \xi) - o_N(1) \\
&\le \frac{1}{N^{p-1}} \sum_{\hat{\iota} \in \Lambda_N} \frac{B}{2} \int_{S_N} \eta(\hat{\iota}) \left[ \sqrt{f(\eta^{\hat{\iota}, \hat{\iota}+e_q}, \xi)} - \sqrt{f(\eta, \xi)} \right]^2 d\nu_N(\eta, \xi) + \frac{1}{N^{p-1}} \sum_{\hat{\iota} \in \Lambda_N} \frac{2}{B} \left( G_n(s, \frac{\hat{\iota}}{N}) \right)^2 \\
&\le \frac{2B}{dN^{p-1}} \mathbb{D}_N^{\text{field}}(\sqrt{f}, \nu_N) + \frac{2}{BN^{p-1}} \sum_{\hat{\iota} \in \Lambda_N} \left( G_n(s, \frac{\hat{\iota}}{N}) \right)^2, \tag{A.42}
\end{align*}
$$

where we used (D1) to provide (A.42). By gathering (A.37), (A.39) and (A.42), we obtain

$$
\begin{align*}
& \mathbb{E}_N^{\mu_N} \left[ \max_{0 \le n \le n_0} \left\{ \int_0^T J_{G_n}^N(s) \, ds \right\} \right] \\
&\le \int_0^T \sup_{f \text{ density with respect to } \nu_N} \left\{ \frac{2B}{dN^{p-1}} \mathbb{D}_N^{\text{field}}(\sqrt{f}, \nu_N) + \frac{1}{aN^p} \langle \mathcal{L}_N \sqrt{f}, \sqrt{f} \rangle_{\nu_N} \right\} \, ds \\
&\quad + \int_0^T \left( \frac{2}{BN^{p-1}} \sum_{\hat{\iota} \in \Lambda_N} \left( G_n(s, \frac{\hat{\iota}}{N}) \right)^2 - \frac{1}{2} \| G_n(s) \|_{L^2(\Lambda)}^2 \right) \, ds + \frac{C_0}{a} + o_N(1) \\
&\le \int_0^T \sup_{f \text{ density with respect to } \nu_N} \left\{ \left[ \frac{2B}{dN^{p-1}} - \frac{1}{aN^{p-2}} \right] \mathbb{D}_N^{\text{field}}(\sqrt{f}, \nu_N) \right\} \, ds \tag{A.43} \\
&\quad + \int_0^T \left( \frac{2}{BN^{p-1}} \sum_{\hat{\iota} \in \Lambda_N} \left( G_n(s, \frac{\hat{\iota}}{N}) \right)^2 - \frac{1}{2} \| G_n(s) \|_{L^2(\Lambda)}^2 \right) \, ds + \frac{C_0 + c(b)\alpha}{a} + o_N(1),
\end{align*}
$$
so that

$$
\begin{align*}
& \int_{S_N} \left( \frac{1}{N^p} \sum_{\hat{\iota} \in \Lambda_N} \eta(\hat{\iota}) \partial_{e_q} G_n(s, \frac{\hat{\iota}}{N}) \right) f(\eta, \xi) \, d\nu_N(\eta, \xi) \\
&= \int_{S_N} \left( \frac{1}{N^{p-1}} \sum_{\hat{\iota} \in \Lambda_N} \eta(\hat{\iota}) \left( G_n(s, \frac{\hat{\iota}+e_q}{N}) - G_n(s, \frac{\hat{\iota}}{N}) \right) \right) f(\eta, \xi) \, d\nu_N(\eta, \xi) + o_N(1).
\end{align*}
$$

Using then a summation by part (mind the compact support of G) and the change of
variable $\eta \mapsto \eta^{\hat{\imath}, \hat{\imath}+e_q}$, we obtain

$$
\begin{align*}
& \int_{S_N} \left( \frac{1}{N^p} \sum_{\hat{\iota} \in \Lambda_N} \eta(\hat{\iota}) \partial_{e_q} G_n(s, \frac{\hat{\iota}}{N}) \right) f(\eta, \xi) \, d\nu_N(\eta, \xi) - o_N(1) \\
&= \frac{1}{N^{p-1}} \sum_{\hat{\iota} \in \Lambda_N} \int_{S_N} \eta(\hat{\iota}) G_n(s, \frac{\hat{\iota}}{N}) \left( f(\eta, \xi) - f(\eta^{\hat{\imath}, \hat{\imath}+e_q}, \xi) \right) \, d\nu_N(\eta, \xi) \\
&= \frac{1}{N^{p-1}} \sum_{\hat{\iota} \in \Lambda_N} \int_{S_N} \eta(\hat{\iota}) G_n(s, \frac{\hat{\iota}}{N}) \underbrace{\left[ \sqrt{f(\eta^{\hat{\imath}, \hat{\imath}+e_q}, \xi)} - \sqrt{f(\eta, \xi)} \right] \left[ \sqrt{f(\eta^{\hat{\imath}, \hat{\imath}+e_q}, \xi)} + \sqrt{f(\eta, \xi)} \right]}_{\text{use (I3) at this point}} \, d\nu_N(\eta, \xi) \\
&\le \frac{1}{N^{p-1}} \sum_{\hat{\iota} \in \Lambda_N} \frac{B}{2} \int_{S_N} \eta(\hat{\iota}) \left[ \sqrt{f(\eta^{\hat{\imath}, \hat{\imath}+e_q}, \xi)} - \sqrt{f(\eta, \xi)} \right]^2 \, d\nu_N(\eta, \xi) \tag{A.40}
\end{align*}
$$

$$
+ \frac{1}{N^{p-1}} \sum_{\hat{\iota} \in \Lambda_N} \frac{1}{2B} \left( G_n\left(s, \frac{\hat{\iota}}{N}\right) \right)^2 \int_{S_N} \eta(\hat{\iota}) \left[ \sqrt{f(\eta^{\hat{\iota}, \hat{\iota}+e_q}, \xi)} + \sqrt{f(\eta, \xi)} \right]^2 d\nu_N(\eta, \xi), \quad (A.41)
$$

where $B > 0$ is to be determined later in this proof. In (A.41) we use now (I4) and the
fact that $f$ is a density with respect to $\nu_N$ to write

$$
\begin{align*}
& \int_{S_N} \left( \frac{1}{N^p} \sum_{\hat{\iota} \in \Lambda_N} \eta(\hat{\iota}) \partial_{e_q} G_n(s, \frac{\hat{\iota}}{N}) \right) f(\eta, \xi) \, d\nu_N(\eta, \xi) - o_N(1) \\
&\le \frac{1}{N^{p-1}} \sum_{\hat{\iota} \in \Lambda_N} \frac{B}{2} \int_{S_N} \eta(\hat{\iota}) \left[ \sqrt{f(\eta^{\hat{\iota}, \hat{\iota}+e_q}, \xi)} - \sqrt{f(\eta, \xi)} \right]^2 d\nu_N(\eta, \xi) + \frac{1}{N^{p-1}} \sum_{\hat{\iota} \in \Lambda_N} \frac{2}{B} \left( G_n(s, \frac{\hat{\iota}}{N}) \right)^2 \\
&\le \frac{2B}{dN^{p-1}} \mathbb{D}_N^{\text{field}}(\sqrt{f}, \nu_N) + \frac{2}{BN^{p-1}} \sum_{\hat{\iota} \in \Lambda_N} \left( G_n(s, \frac{\hat{\iota}}{N}) \right)^2, \tag{A.42}
\end{align*}
$$

where we used (D1) to provide (A.42). By gathering (A.37), (A.39) and (A.42), we obtain

$$
\begin{align*}
& \mathbb{E}_N^{\mu_N} \left[ \max_{0 \le n \le n_0} \left\{ \int_0^T J_{G_n}^N(s) \, ds \right\} \right] \\
&\le \int_0^T \sup_{f \text{ density} \\ \text{with respect} \\ \text{to } \nu_N} \left\{ \frac{2B}{dN^{p-1}} \mathbb{D}_N^{\text{field}}(\sqrt{f}, \nu_N) + \frac{1}{aN^p} \langle \mathcal{L}_N \sqrt{f}, \sqrt{f} \rangle_{\nu_N} \right\} ds \\
&\quad + \int_0^T \left( \frac{2}{BN^{p-1}} \sum_{\hat{\iota} \in \Lambda_N} \left( G_n(s, \frac{\hat{\iota}}{N}) \right)^2 - \frac{1}{2} \|G_n(s)\|_{L^2(\Lambda)}^2 \right) ds + \frac{C_0}{a} + o_N(1) \\
&\le \int_0^T \sup_{f \text{ density} \\ \text{with respect} \\ \text{to } \nu_N} \left\{ \left[ \frac{2B}{dN^{p-1}} - \frac{1}{aN^{p-2}} \right] \mathbb{D}_N^{\text{field}}(\sqrt{f}, \nu_N) \right\} ds \tag{A.43} \\
&\quad + \int_0^T \left( \frac{2}{BN^{p-1}} \sum_{\hat{\iota} \in \Lambda_N} \left( G_n(s, \frac{\hat{\iota}}{N}) \right)^2 - \frac{1}{2} \|G_n(s)\|_{L^2(\Lambda)}^2 \right) ds + \frac{C_0 + c(b)\alpha}{a} + o_N(1),
\end{align*}
$$
where we used Lemma A.1 to provide the last inequality. At this point, we make the choice $B := dN/2a$ to cancel the supremum in (A.43), so that we are left with

$$ \mathbb{E}_N^{\mu_N} \left[ \max_{0 \le n \le n_0} \left\{ \int_0^T J_{G_n}^N(s) ds \right\} \right] \\ \le \int_0^T \left( \frac{2a}{dN^p} \sum_{\hat{\lambda} \in \Lambda_N} \left( G_n(s, \frac{\hat{\lambda}}{N}) \right)^2 - \frac{1}{2} \|G_n(s)\|_{L^2(\Lambda)}^2 \right) ds + \frac{C_0 + c(b)\alpha}{a} + o_N(1). \quad (A.44) $$

We take now $a := d/4$ to face the norm with the Riemann sum in (A.44), this results in

$$ \mathbb{E}_N^{\mu_N} \left[ \max_{0 \le n \le n_0} \left\{ \int_0^T J_{G_n}^N(s) ds \right\} \right] \le \frac{4C_0 + 4c(b)\alpha}{d} + o_N(1). $$

It remains to let $N \to \infty$ to eventually reach (A.35) with $C := (4C_0 + 4c(b)\alpha)/d$. This completes the proof. $\square$

Table of Notations

<table>
  <thead>
    <tr>
      <th>Notation</th>
      <th>Description</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td><i>p</i></td>
      <td>Dimension of the field (the road is (<i>p</i> &minus; 1)-dimensional)</td>
    </tr>
    <tr>
      <td>T</td>
      <td>One-dimensional torus R/Z</td>
    </tr>
    <tr>
      <td>T<sup><i>p</i>&minus;1</sup></td>
      <td>Macroscopic road</td>
    </tr>
    <tr>
      <td>&Lambda;</td>
      <td>Macroscopic field T<sup><i>p</i>&minus;1</sup> &times; (0, 1)</td>
    </tr>
    <tr>
      <td>&Lambda;&#773;</td>
      <td>Closure of &Lambda;</td>
    </tr>
    <tr>
      <td>&Gamma;</td>
      <td>Macroscopic frontier of the field &part;&Lambda; = T<sup><i>p</i>&minus;1</sup> &times; {0, 1}</td>
    </tr>
    <tr>
      <td>&Gamma;<sup>up</sup></td>
      <td>Macroscopic upper boundary of the field</td>
    </tr>
    <tr>
      <td>&Gamma;<sup>low</sup></td>
      <td>Macroscopic lower boundary of the field</td>
    </tr>
    <tr>
      <td><i>N</i></td>
      <td>Size of the microscopic particle system</td>
    </tr>
    <tr>
      <td>T<sub><i>N</i></sub></td>
      <td>One-dimensional discrete torus Z/NZ</td>
    </tr>
    <tr>
      <td>T<sup><i>p</i>&minus;1</sup><sub><i>N</i></sub></td>
      <td>Microscopic road</td>
    </tr>
    <tr>
      <td>&Lambda;<sub><i>N</i></sub></td>
      <td>Microscopic field T<sub><i>N</i></sub> &times; [[1; <i>N</i> &minus; 1]]</td>
    </tr>
    <tr>
      <td>&Gamma;<sub><i>N</i></sub></td>
      <td>Microscopic frontier of the field &part;&Lambda;<sub><i>N</i></sub> = T<sub><i>N</i></sub> &times; {0, 1}</td>
    </tr>
    <tr>
      <td>&Gamma;<sup>up</sup><sub><i>N</i></sub></td>
      <td>Microscopic upper boundary of the field</td>
    </tr>
    <tr>
      <td>&Gamma;<sup>low</sup><sub><i>N</i></sub></td>
      <td>Microscopic lower boundary of the field</td>
    </tr>
    <tr>
      <td>&#x0102;<sub><i>i</i>, <i>j</i></sub></td>
      <td>Microscopic point on &Lambda;<sub><i>N</i></sub> (<i>i</i> &isin; T<sub><i>N</i></sub> and <i>j</i> &isin; [[1; <i>N</i> &minus; 1]])</td>
    </tr>
    <tr>
      <td>&#x0103;<sub><i>k</i>, <i>l</i></sub></td>
      <td>Alternative microscopic point on &Lambda;<sub><i>N</i></sub> if needed</td>
    </tr>
    <tr>
      <td>&#x0104;<sub><i>x</i>, <i>y</i></sub></td>
      <td>Macroscopic point on &Lambda; (<i>x</i> &isin; T and <i>y</i> &isin; (0, 1))</td>
    </tr>
    <tr>
      <td>&#x0105;<sub><i>z</i>, <i>&omega;</i></sub></td>
      <td>Alternative macroscopic point on &Lambda; if needed</td>
    </tr>
  </tbody>
</table>
where we used Lemma A.1 to provide the last inequality. At this point, we make the choice $B := dN/2a$ to cancel the supremum in (A.43), so that we are left with

$$ \mathbb{E}_N^{\mu_N} \left[ \max_{0 \le n \le n_0} \left\{ \int_0^T J_{G_n}^N(s) ds \right\} \right] \\ \le \int_0^T \left( \frac{2a}{dN^p} \sum_{\hat{\lambda} \in \Lambda_N} \left( G_n(s, \frac{\hat{\lambda}}{N}) \right)^2 - \frac{1}{2} \|G_n(s)\|_{L^2(\Lambda)}^2 \right) ds + \frac{C_0 + c(b)\alpha}{a} + o_N(1). \quad (A.44) $$

We take now $a := d/4$ to face the norm with the Riemann sum in (A.44), this results in

$$ \mathbb{E}_N^{\mu_N} \left[ \max_{0 \le n \le n_0} \left\{ \int_0^T J_{G_n}^N(s) ds \right\} \right] \le \frac{4C_0 + 4c(b)\alpha}{d} + o_N(1). $$

It remains to let $N \to \infty$ to eventually reach (A.35) with $C := (4C_0 + 4c(b)\alpha)/d$. This completes the proof. □

Table of Notations

<table><thead><tr><th>Notation</th><th>Description</th></tr></thead><tbody><tr><td><i>p</i></td><td>Dimension of the field (the road is (<i>p</i> − 1)-dimensional)</td></tr><tr><td>T</td><td>One-dimensional torus R/Z</td></tr><tr><td>T<sup><i>p</i>−1</sup></td><td>Macroscopic road</td></tr><tr><td>Λ</td><td>Macroscopic field T<sup><i>p</i>−1</sup> × (0, 1)</td></tr><tr><td>Λ&#773;</td><td>Closure of Λ</td></tr><tr><td>Γ</td><td>Macroscopic frontier of the field ∂Λ = T<sup><i>p</i>−1</sup> × {0, 1}</td></tr><tr><td>Γ<sup>up</sup></td><td>Macroscopic upper boundary of the field</td></tr><tr><td>Γ<sup>low</sup></td><td>Macroscopic lower boundary of the field</td></tr><tr><td><i>N</i></td><td>Size of the microscopic particle system</td></tr><tr><td>T<sub><i>N</i></sub></td><td>One-dimensional discrete torus Z/NZ</td></tr><tr><td>T<sup><i>p</i>−1</sup><sub><i>N</i></sub></td><td>Microscopic road</td></tr><tr><td>Λ<sub><i>N</i></sub></td><td>Microscopic field T<sub><i>N</i></sub> × [[1; <i>N</i> − 1]]</td></tr><tr><td>Γ<sub><i>N</i></sub></td><td>Microscopic frontier of the field ∂Λ<sub><i>N</i></sub> = T<sub><i>N</i></sub> × {0, 1}</td></tr><tr><td>Γ<sup>up</sup><sub><i>N</i></sub></td><td>Microscopic upper boundary of the field</td></tr><tr><td>Γ<sup>low</sup><sub><i>N</i></sub></td><td>Microscopic lower boundary of the field</td></tr><tr><td>⟨<i>i</i>, <i>j</i>⟩</td><td>Microscopic point on Λ<sub><i>N</i></sub> (<i>i</i> ∈ T<sub><i>N</i></sub> and <i>j</i> ∈ [[1; <i>N</i> − 1]])</td></tr><tr><td>⟨<i>k</i>, <i>l</i>⟩</td><td>Alternative microscopic point on Λ<sub><i>N</i></sub> if needed</td></tr><tr><td>⟨<i>x</i>, <i>y</i>⟩</td><td>Macroscopic point on Λ (<i>x</i> ∈ T and <i>y</i> ∈ (0, 1))</td></tr><tr><td>⟨<i>z</i>, <i>ω</i>⟩</td><td>Alternative macroscopic point on Λ if needed</td></tr></tbody></table>
<table><tr><td>e<sub>q</sub></td><td>q<sup>th</sup> canonical vector of R<sup>p</sup> (1 &le; q &le; p)</td></tr><tr><td>(&eta;, &xi;)</td><td>State of the system (&eta; for the field and &xi; for the road)</td></tr><tr><td>S<sup>field</sup><sub>N</sub></td><td>State space for the field {0, 1}<sup>&Lambda;<sub>N</sub></sup> &ni; &eta;</td></tr><tr><td>S<sup>road</sup><sub>N</sub></td><td>State space for the road {0, 1}<sup>&Gamma;<sup>low</sup><sub>N</sub></sup> &ni; &xi;</td></tr><tr><td>S<sub>N</sub></td><td>Whole state space S<sup>field</sup><sub>N</sub> &times; S<sup>road</sup><sub>N</sub></td></tr><tr><td>&pi;<sup>field</sup><sub>N</sub>(&eta;)</td><td>Empirical measure on S<sup>field</sup><sub>N</sub> associated with &eta;</td></tr><tr><td>&pi;<sup>road</sup><sub>N</sub>(&xi;)</td><td>Empirical measure on S<sup>road</sup><sub>N</sub> associated with &xi;</td></tr><tr><td>&pi;<sub>N</sub>(&eta;, &xi;)</td><td>Empirical measure on S<sub>N</sub> associated with (&eta;, &xi;)</td></tr><tr><td>M<sup>field</sup></td><td>Set of positive measures on S<sup>field</sup><sub>N</sub> bounded by 1</td></tr><tr><td>M<sup>road</sup></td><td>Set of positive measures on S<sup>road</sup><sub>N</sub> bounded by 1</td></tr><tr><td>M</td><td>Cartesian product M<sup>field</sup> &times; M<sup>road</sup></td></tr><tr><td>T</td><td>Time horizon</td></tr><tr><td>d, D</td><td>Diffusion coefficients</td></tr><tr><td>&alpha;</td><td>Exchange coefficient</td></tr><tr><td>b</td><td>Birth rate at the upper boundary</td></tr><tr><td>&gamma;</td><td>Parameter of the Bernoulli product measures</td></tr><tr><td>P<sup>&mu;<sub>N</sub></sup><sub>N</sub></td><td>Probability measure on S<sub>N</sub> induced by &mu;<sub>N</sub> and (&eta;<sub>t</sub>, &xi;<sub>t</sub>)<sub>t&isin;[0,T]</sub></td></tr><tr><td>Q<sup>&mu;<sub>N</sub></sup><sub>N</sub></td><td>Probability measure on M induced by P<sup>&mu;<sub>N</sub></sup><sub>N</sub> and &pi;<sub>N</sub></td></tr><tr><td>Q<sub>&infin;</sub></td><td>A point in the closure of (Q<sup>&mu;<sub>N</sub></sup><sub>N</sub>)<sub>N&ge;2</sub></td></tr><tr><td>E<sup>&bull;</sup><sub>N</sub></td><td>Expectation with respect to P<sup>&bull;</sup><sub>N</sub></td></tr><tr><td>E<sup>&mu;<sub>N</sub></sup><sub>N</sub></td><td>Expectation with respect to P<sup>&mu;<sub>N</sub></sup><sub>N</sub></td></tr><tr><td>E<sup>&nu;<sub>N</sub></sup><sub>N</sub></td><td>Expectation with respect to P<sup>&nu;<sub>N</sub></sup><sub>N</sub></td></tr><tr><td>E<sub>&infin;</sub></td><td>Expectation with respect to Q<sub>&infin;</sub></td></tr><tr><td>&Delta;</td><td>Laplacian operator for the field</td></tr><tr><td>&Delta;<sub>x</sub></td><td>Laplacian operator for the road</td></tr><tr><td>&Delta;<sup>N</sup><sub>x</sub></td><td>Discrete Laplacian operator for the road</td></tr><tr><td>&part;<sup>N</sup><sub>yy</sub></td><td>Discrete Laplacian operator in the y-direction</td></tr><tr><td>&nabla;</td><td>Gradient operator for the field</td></tr><tr><td>&nabla;<sub>x</sub></td><td>Gradient operator for the road</td></tr><tr><td>Tr</td><td>Trace operator</td></tr><tr><td>G</td><td>Test functions for the field</td></tr><tr><td>H</td><td>Test functions for the road</td></tr><tr><td>I<sup>field</sup><sub>N</sub></td><td>Bulk field part of the generator</td></tr><tr><td>I<sup>road</sup><sub>N</sub></td><td>Bulk road part of the generator</td></tr><tr><td>L<sup>Rob</sup><sub>N</sub></td><td>Robin exchange part of the generator</td></tr></table>
<table><thead><tr><th></th><th></th></tr></thead><tbody><tr><td>e<sub>q</sub></td><td>q<sup>th</sup> canonical vector of ℝ<sup>p</sup> (1 ≤ q ≤ p)</td></tr><tr><td>(η, ξ)</td><td>State of the system (η for the field and ξ for the road)</td></tr><tr><td>S<sup>field</sup><sub>N</sub></td><td>State space for the field {0, 1}<sup>Λ<sub>N</sub></sup> ∋ η</td></tr><tr><td>S<sup>road</sup><sub>N</sub></td><td>State space for the road {0, 1}<sup>Γ<sup>low</sup><sub>N</sub></sup> ∋ ξ</td></tr><tr><td>S<sub>N</sub></td><td>Whole state space S<sup>field</sup><sub>N</sub> × S<sup>road</sup><sub>N</sub></td></tr><tr><td>π<sup>field</sup><sub>N</sub>(η)</td><td>Empirical measure on S<sup>field</sup><sub>N</sub> associated with η</td></tr><tr><td>π<sup>road</sup><sub>N</sub>(ξ)</td><td>Empirical measure on S<sup>road</sup><sub>N</sub> associated with ξ</td></tr><tr><td>π<sub>N</sub>(η, ξ)</td><td>Empirical measure on S<sub>N</sub> associated with (η, ξ)</td></tr><tr><td>M<sup>field</sup></td><td>Set of positive measures on S<sup>field</sup><sub>N</sub> bounded by 1</td></tr><tr><td>M<sup>road</sup></td><td>Set of positive measures on S<sup>road</sup><sub>N</sub> bounded by 1</td></tr><tr><td>M</td><td>Cartesian product M<sup>field</sup> × M<sup>road</sup></td></tr><tr><td>T</td><td>Time horizon</td></tr><tr><td>d, D</td><td>Diffusion coefficients</td></tr><tr><td>α</td><td>Exchange coefficient</td></tr><tr><td>b</td><td>Birth rate at the upper boundary</td></tr><tr><td>γ</td><td>Parameter of the Bernoulli product measures</td></tr><tr><td>P<sup>μ<sub>N</sub></sup><sub>N</sub></td><td>Probability measure on S<sub>N</sub> induced by μ<sub>N</sub> and (η<sub>t</sub>, ξ<sub>t</sub>)<sub>t∈[0,T]</sub></td></tr><tr><td>Q<sup>μ<sub>N</sub></sup><sub>N</sub></td><td>Probability measure on M<sub>N</sub> induced by P<sup>μ<sub>N</sub></sup><sub>N</sub> and π<sub>N</sub></td></tr><tr><td>Q<sub>∞</sub></td><td>A point in the closure of (Q<sup>μ<sub>N</sub></sup><sub>N</sub>)<sub>N≥2</sub></td></tr><tr><td>E<sup>•</sup><sub>N</sub></td><td>Expectation with respect to P<sup>•</sup><sub>N</sub></td></tr><tr><td>E<sup>μ<sub>N</sub></sup><sub>N</sub></td><td>Expectation with respect to P<sup>μ<sub>N</sub></sup><sub>N</sub></td></tr><tr><td>E<sup>ν<sub>N</sub></sup><sub>N</sub></td><td>Expectation with respect to P<sup>ν<sub>N</sub></sup><sub>N</sub></td></tr><tr><td>E<sub>∞</sub></td><td>Expectation with respect to Q<sub>∞</sub></td></tr><tr><td>Δ</td><td>Laplacian operator for the field</td></tr><tr><td>Δ<sub>x</sub></td><td>Laplacian operator for the road</td></tr><tr><td>Δ<sup>N</sup><sub>x</sub></td><td>Discrete Laplacian operator for the road</td></tr><tr><td>∂<sup>N</sup><sub>yy</sub></td><td>Discrete Laplacian operator in the y-direction</td></tr><tr><td>∇</td><td>Gradient operator for the field</td></tr><tr><td>∇<sub>x</sub></td><td>Gradient operator for the road</td></tr><tr><td>Tr</td><td>Trace operator</td></tr><tr><td>G</td><td>Test functions for the field</td></tr><tr><td>H</td><td>Test functions for the road</td></tr><tr><td>L<sup>field</sup><sub>N</sub></td><td>Bulk field part of the generator</td></tr><tr><td>L<sup>road</sup><sub>N</sub></td><td>Bulk road part of the generator</td></tr><tr><td>L<sup>Rob</sup><sub>N</sub></td><td>Robin exchange part of the generator</td></tr></tbody></table>
<table><tr><td>L<sup>reac</sup><sub>N</sub></td><td>Reaction exchange part of the generator</td></tr><tr><td>L<sup>up</sup><sub>N</sub></td><td>Upper reservoir part of the generator</td></tr><tr><td>D<sub>N</sub></td><td>Dirichlet form</td></tr><tr><td>D<sup>field</sup><sub>N</sub></td><td>Bulk field part of the Dirichlet form</td></tr><tr><td>D<sup>road</sup><sub>N</sub></td><td>Bulk road part of the Dirichlet form</td></tr><tr><td>D<sup>Rob</sup><sub>N</sub></td><td>Robin exchange part of the Dirichlet form</td></tr><tr><td>D<sup>reac</sup><sub>N</sub></td><td>Reaction exchange part of the Dirichlet form</td></tr><tr><td>D<sup>up</sup><sub>N</sub></td><td>Upper reservoir part of the Dirichlet form</td></tr><tr><td>H(μ|ν)</td><td>Relative entropy of μ with respect to ν</td></tr><tr><td>M<sub>N</sub>(t)</td><td>The Martingale</td></tr><tr><td>M<sup>field</sup><sub>N,G</sub>(t)</td><td>Field part of the Martingale M<sub>N</sub>(t)</td></tr><tr><td>M<sup>road</sup><sub>N,H</sub>(t)</td><td>Road part of the Martingale M<sub>N</sub>(t)</td></tr><tr><td>N<sup>field</sup><sub>N,G</sub>(t)</td><td>Quadratic variation of M<sup>field</sup><sub>N,G</sub>(t)</td></tr><tr><td>N<sup>road</sup><sub>N,H</sub>(t)</td><td>Quadratic variation of M<sup>road</sup><sub>N,H</sub>(t)</td></tr><tr><td>B<sup>field</sup><sub>N,G</sub>(t)</td><td>“B”-part of the martingale N<sup>field</sup><sub>N,G</sub>(t)</td></tr><tr><td>B<sup>road</sup><sub>N,H</sub>(t)</td><td>“B”-part of the martingale N<sup>road</sup><sub>N,H</sub>(t)</td></tr><tr><td>S</td><td>Set of measures whose densities satisfy (W1) and (W2)</td></tr><tr><td>S<sub>W1</sub></td><td>Set of measures whose densities satisfy (W1)</td></tr><tr><td>S<sub>W2</sub></td><td>Set of measures whose densities satisfy (W2)</td></tr><tr><td>W<sub>u,v</sub>(t)</td><td>Functional of the weak formulation W<sub>u,v</sub> = W<sup>field</sup><sub>v,G</sub> + W<sup>road</sup><sub>u,H</sub></td></tr><tr><td>W<sup>field</sup><sub>v,G</sub>(t)</td><td>Functional of the weak formulation in the field</td></tr><tr><td>W<sup>road</sup><sub>u,H</sub>(t)</td><td>Functional of the weak formulation on the road</td></tr><tr><td>U<sup>up</sup><sub>ε</sub></td><td>Upper unit approximation</td></tr><tr><td>U<sup>low</sup><sub>ε</sub></td><td>Upper unit approximation</td></tr></table>

## References

[1] E. AFFILI, *A Fisher-KPP model with a fast diffusion line in periodic media*, arXiv:2009.14760, (2020).

[2] M. ALFARO AND C. CHAINAIS-HILLAIRET, *Long time behavior of the field-road diffusion model: An entropy method and a finite volume scheme*, arXiv:2309.16242, (2023).

[3] M. ALFARO, R. DUCASSE, AND S. TRÉTON, *The field-road diffusion model: Fundamental solution and asymptotic behavior*, Journal of Differential Equations, 367 (2023), pp. 332–365.

[4] R. BALDASSO, O. MENEZES, A. NEUMANN, AND R. R. SOUZA, *Exclusion Process with Slow Boundary*, Journal of Statistical Physics, 167 (2017), pp. 1112–1142.
<table><tr><td>L<sup>reac</sup><sub>N</sub></td><td>Reaction exchange part of the generator</td></tr><tr><td>L<sup>up</sup><sub>N</sub></td><td>Upper reservoir part of the generator</td></tr><tr><td>D<sub>N</sub></td><td>Dirichlet form</td></tr><tr><td>D<sup>field</sup><sub>N</sub></td><td>Bulk field part of the Dirichlet form</td></tr><tr><td>D<sup>road</sup><sub>N</sub></td><td>Bulk road part of the Dirichlet form</td></tr><tr><td>D<sup>Rob</sup><sub>N</sub></td><td>Robin exchange part of the Dirichlet form</td></tr><tr><td>D<sup>reac</sup><sub>N</sub></td><td>Reaction exchange part of the Dirichlet form</td></tr><tr><td>D<sup>up</sup><sub>N</sub></td><td>Upper reservoir part of the Dirichlet form</td></tr><tr><td>H(μ|ν)</td><td>Relative entropy of μ with respect to ν</td></tr><tr><td>M<sub>N</sub>(t)</td><td>The Martingale</td></tr><tr><td>M<sup>field</sup><sub>N,G</sub>(t)</td><td>Field part of the Martingale M<sub>N</sub>(t)</td></tr><tr><td>M<sup>road</sup><sub>N,H</sub>(t)</td><td>Road part of the Martingale M<sub>N</sub>(t)</td></tr><tr><td>N<sup>field</sup><sub>N,G</sub>(t)</td><td>Quadratic variation of M<sup>field</sup><sub>N,G</sub>(t)</td></tr><tr><td>N<sup>road</sup><sub>N,H</sub>(t)</td><td>Quadratic variation of M<sup>road</sup><sub>N,H</sub>(t)</td></tr><tr><td>B<sup>field</sup><sub>N,G</sub>(t)</td><td>“B”-part of the martingale N<sup>field</sup><sub>N,G</sub>(t)</td></tr><tr><td>B<sup>road</sup><sub>N,H</sub>(t)</td><td>“B”-part of the martingale N<sup>road</sup><sub>N,H</sub>(t)</td></tr><tr><td>S</td><td>Set of measures whose densities satisfy (W1) and (W2)</td></tr><tr><td>S<sub>W1</sub></td><td>Set of measures whose densities satisfy (W1)</td></tr><tr><td>S<sub>W2</sub></td><td>Set of measures whose densities satisfy (W2)</td></tr><tr><td>W<sub>u,v</sub>(t)</td><td>Functional of the weak formulation W<sub>u,v</sub> = W<sup>field</sup><sub>v,G</sub> + W<sup>road</sup><sub>u,H</sub></td></tr><tr><td>W<sup>field</sup><sub>v,G</sub>(t)</td><td>Functional of the weak formulation in the field</td></tr><tr><td>W<sup>road</sup><sub>u,H</sub>(t)</td><td>Functional of the weak formulation on the road</td></tr><tr><td>U<sup>up</sup><sub>ε</sub></td><td>Upper unit approximation</td></tr><tr><td>U<sup>low</sup><sub>ε</sub></td><td>Upper unit approximation</td></tr></table>

## References

[1] E. AFFILI, *A Fisher-KPP model with a fast diffusion line in periodic media*, arXiv:2009.14760, (2020).

[2] M. ALFARO AND C. CHAINAIS-HILLAIRET, *Long time behavior of the field-road diffusion model: An entropy method and a finite volume scheme*, arXiv:2309.16242, (2023).

[3] M. ALFARO, R. DUCASSE, AND S. TRÉTON, *The field-road diffusion model: Fundamental solution and asymptotic behavior*, Journal of Differential Equations, 367 (2023), pp. 332–365.

[4] R. BALDASSO, O. MENEZES, A. NEUMANN, AND R. R. SOUZA, *Exclusion Process with Slow Boundary*, Journal of Statistical Physics, 167 (2017), pp. 1112–1142.
[5] H. BERESTYCKI, A.-C. COULON, J.-M. ROQUEJOFFRE, AND L. ROSSI, *Speed-up of reaction-diffusion fronts by a line of fast diffusion*, Séminaire Laurent Schwartz — EDP et applications, (2013/2014), pp. 1–25.

[6] —, *The effect of a line with nonlocal diffusion on Fisher-KPP propagation*, Mathematical Models and Methods in Applied Sciences, 25 (2015), pp. 2519–2562.

[7] H. BERESTYCKI, R. DUCASSE, AND L. ROSSI, *Generalized principal eigenvalues for heterogeneous road-field systems*, Communications in Contemporary Mathematics, 22 (2020), p. 1950013.

[8] —, *Influence of a road on a population in an ecological niche facing climate change*, Journal of Mathematical Biology, 81 (2020), pp. 1059–1097.

[9] H. BERESTYCKI, J.-M. ROQUEJOFFRE, AND L. ROSSI, *Fisher-KPP propagation in the presence of a line: Further effects*, Nonlinearity, 26 (2013), pp. 2623–2640.

[10] —, *The influence of a line with fast diffusion on Fisher-KPP propagation*, Journal of Mathematical Biology, 66 (2013), pp. 743–766.

[11] —, *The shape of expansion induced by a line with fast diffusion in Fisher-KPP equations*, Communications in Mathematical Physics, 343 (2016), pp. 207–232.

[12] —, *Travelling waves, spreading and extinction for Fisher-KPP propagation driven by a line with fast diffusion*, Nonlinear Analysis, 137 (2016), pp. 171–189.

[13] C. BERNARDIN, P. GONCALVES, AND B. O. JIMENEZ, *Slow to fast infinitely extended reservoirs for the symmetric exclusion process with long jumps*, Markov Processes and Related Fields, 25 (2019), pp. 217–274.

[14] P. BILLINGSLEY, *Convergence of Probability Measures*, Wiley Series in Probability and Statistics, John Wiley & Sons, Inc., New York, 2nd ed., 2013.

[15] T. BODINEAU, B. DERRIDA, AND J. L. LEBOWITZ, *A Diffusive System Driven by a Battery or by a Smoothly Varying Field*, Journal of Statistical Physics, 140 (2010), pp. 648–675.

[16] B. BOGOSEL, T. GILETTI, AND A. TELLINI, *Propagation for KPP bulk-surface systems in a general cylindrical domain*, Nonlinear Analysis, 213 (2021), p. 112528.

[17] H. BREZIS, *Analyse Fonctionnelle*, Masson, théorie et applications ed., 1983.

[18] D. CUSSEDDU, L. EDELSTEIN-KESHET, J. A. MACKENZIE, S. PORTET, AND A. MADZVAMUSE, *A coupled bulk-surface model for cell polarisation*, Journal of Theoretical Biology, 481 (2019), pp. 119–135.

[19] A. DE MASI, E. PRESUTTI, D. TSAGKAROGIANNIS, AND M. E. VARES, *Current Reservoirs in the Simple Exclusion Process*, Journal of Statistical Physics, 144 (2011), pp. 1151–1170.

[20] —, *Non-equilibrium Stationary States in the Symmetric Simple Exclusion with Births and Deaths*, Journal of Statistical Physics, 147 (2012), pp. 519–528.
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[6] —, *The effect of a line with nonlocal diffusion on Fisher-KPP propagation*, Mathematical Models and Methods in Applied Sciences, 25 (2015), pp. 2519–2562.

[7] H. BERESTYCKI, R. DUCASSE, AND L. ROSSI, *Generalized principal eigenvalues for heterogeneous road-field systems*, Communications in Contemporary Mathematics, 22 (2020), p. 1950013.

[8] —, *Influence of a road on a population in an ecological niche facing climate change*, Journal of Mathematical Biology, 81 (2020), pp. 1059–1097.

[9] H. BERESTYCKI, J.-M. ROQUEJOFFRE, AND L. ROSSI, *Fisher-KPP propagation in the presence of a line: Further effects*, Nonlinearity, 26 (2013), pp. 2623–2640.

[10] —, *The influence of a line with fast diffusion on Fisher-KPP propagation*, Journal of Mathematical Biology, 66 (2013), pp. 743–766.

[11] —, *The shape of expansion induced by a line with fast diffusion in Fisher-KPP equations*, Communications in Mathematical Physics, 343 (2016), pp. 207–232.

[12] —, *Travelling waves, spreading and extinction for Fisher-KPP propagation driven by a line with fast diffusion*, Nonlinear Analysis, 137 (2016), pp. 171–189.

[13] C. BERNARDIN, P. GONCALVES, AND B. O. JIMENEZ, *Slow to fast infinitely extended reservoirs for the symmetric exclusion process with long jumps*, Markov Processes and Related Fields, 25 (2019), pp. 217–274.

[14] P. BILLINGSLEY, *Convergence of Probability Measures*, Wiley Series in Probability and Statistics, John Wiley & Sons, Inc., New York, 2nd ed., 2013.

[15] T. BODINEAU, B. DERRIDA, AND J. L. LEBOWITZ, *A Diffusive System Driven by a Battery or by a Smoothly Varying Field*, Journal of Statistical Physics, 140 (2010), pp. 648–675.

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[18] D. CUSSEDU, L. EDELSTEIN-KESHET, J. A. MACKENZIE, S. PORTET, AND A. MADZVAMUSE, *A coupled bulk-surface model for cell polarisation*, Journal of Theoretical Biology, 481 (2019), pp. 119–135.

[19] A. DE MASI, E. PRESUTTI, D. TSAGKAROGIANNIS, AND M. E. VARES, *Current Reservoirs in the Simple Exclusion Process*, Journal of Statistical Physics, 144 (2011), pp. 1151–1170.

[20] —, *Non-equilibrium Stationary States in the Symmetric Simple Exclusion with Births and Deaths*, Journal of Statistical Physics, 147 (2012), pp. 519–528.
[21] —, *Truncated correlations in the stirring process with births and deaths*, Electronic Journal of Probability, 17 (2012), pp. 1–35.

[22] R. DUCASSE, *Influence of the geometry on a field-road model: The case of a conical field*, Journal of the London Mathematical Society, 97 (2018), pp. 441–469.

[23] H. EGGER, K. FELLNER, J. F. PIETSCHMANN, AND B. Q. TANG, *Analysis and numerical solution of coupled volume-surface reaction-diffusion systems with application to cell biology*, Applied Mathematics and Computation, 336 (2018), pp. 351–367.

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[27] T. FRANCO, P. GONÇALVES, AND A. NEUMANN, *Hydrodynamical behavior of symmetric exclusion with slow bonds*, Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, 49 (2013), pp. 402–427.

[28] —, *Phase transition in equilibrium fluctuations of symmetric slowed exclusion*, Stochastic Processes and their Applications, 123 (2013), pp. 4156–4185.

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