# Bridging bulk and surface: An interacting particle system towards the field-road diffusion model

Matthieu Alfaro, Mustapha Mourragui, Samuel Tréton

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Matthieu Alfaro, Mustapha Mourragui, Samuel Tréton. Bridging bulk and surface: An interacting particle system towards the field-road diffusion model. 2024. hal-04618109

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# Bridging bulk and surface: An interacting particle system towards the field-road diffusion model

Matthieu Alfaro, Mustapha Mourragui, Samuel Tréton

► To cite this version:

Matthieu Alfaro, Mustapha Mourragui, Samuel Tréton. Bridging bulk and surface: An interacting particle system towards the field-road diffusion model. 2024. hal-04618109

HAL Id: hal-04618109

https://hal.univ-lorraine.fr/hal-04618109v1

Preprint submitted on 20 Jun 2024

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

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

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

Distributed under a Creative Commons Attribution 4.0 International License
# Bridging bulk and surface: An interacting particle system towards the field-road diffusion model

Matthieu ALFARO<fnref n="1" />, Mustapha MOURRAGUI² and Samuel TRÉTON³

## Abstract

We recover the so-called field-road diffusion model as the hydrodynamic limit of an interacting particle system. The former consists of two parabolic PDEs posed on two sets of different dimensions (a “field” and a “road” in a population dynamics context), and coupled through exchange terms between the field’s boundary and the road. The latter stands as a Symmetric Simple Exclusion Process (SSEP): particles evolve on two microscopic lattices following a Markov jump process, with the constraint that each site cannot host more than one particle at the same time. The system is in contact with reservoirs that allow to create or remove particles at the boundary sites. The dynamics of these reservoirs are slowed down compared to the diffusive dynamics, to reach the reactions and the boundary conditions awaited at the macroscopic scale. This issue of bridging two spaces of different dimensions is, as far as we know, new in the hydrodynamic limit context, and raises perspectives towards future related works.

**Key words:** hydrodynamic limit, interacting particle system, simple exclusion processes, slow boundary reservoirs, field-road diffusion model.

**MSC2020:** 60J27, 82C22, 35K57, 37N25.

## Contents

<table><tr><td>1 Introduction</td><td>2</td></tr><tr><td>2 Notations and main result</td><td>4</td></tr><tr><td>3 Martingales and tightness</td><td>12</td></tr><tr><td>4 Characterization of the limit points of (Q<sub>N</sub><sup>μ<sub>N</sub></sup>)<sub>N≥2</sub></td><td>16</td></tr><tr><td>5 Uniqueness of the solution</td><td>22</td></tr><tr><td>Appendix</td><td>23</td></tr><tr><td>Table of notations</td><td>35</td></tr><tr><td>References</td><td>37</td></tr></table>

<footnote n="1">¹,²,³ Univ. Rouen Normandie, CNRS, Normandie Univ., LMRS UMR 6085, F-76000 Rouen, France.</footnote>
# Bridging bulk and surface: An interacting particle system towards the field-road diffusion model

Matthieu ALFARO<fnref n="1" />, Mustapha MOURRAGUI² and Samuel TRÉTON³

## Abstract

We recover the so-called field-road diffusion model as the hydrodynamic limit of an interacting particle system. The former consists of two parabolic PDEs posed on two sets of different dimensions (a “field” and a “road” in a population dynamics context), and coupled through exchange terms between the field’s boundary and the road. The latter stands as a Symmetric Simple Exclusion Process (SSEP): particles evolve on two microscopic lattices following a Markov jump process, with the constraint that each site cannot host more than one particle at the same time. The system is in contact with reservoirs that allow to create or remove particles at the boundary sites. The dynamics of these reservoirs are slowed down compared to the diffusive dynamics, to reach the reactions and the boundary conditions awaited at the macroscopic scale. This issue of bridging two spaces of different dimensions is, as far as we know, new in the hydrodynamic limit context, and raises perspectives towards future related works.

**Key words:** hydrodynamic limit, interacting particle system, simple exclusion processes, slow boundary reservoirs, field-road diffusion model.

**MSC2020:** 60J27, 82C22, 35K57, 37N25.

## Contents

<table><tr><td>1 Introduction</td><td>2</td></tr><tr><td>2 Notations and main result</td><td>4</td></tr><tr><td>3 Martingales and tightness</td><td>12</td></tr><tr><td>4 Characterization of the limit points of (Q<sub>N</sub><sup>μ<sub>N</sub></sup>)<sub>N≥2</sub></td><td>16</td></tr><tr><td>5 Uniqueness of the solution</td><td>22</td></tr><tr><td>Appendix</td><td>23</td></tr><tr><td>Table of notations</td><td>35</td></tr><tr><td>References</td><td>37</td></tr></table>

<footnote n="1">¹,²,³ Univ. Rouen Normandie, CNRS, Normandie Univ., LMRS UMR 6085, F-76000 Rouen, France.</footnote>
# 1 Introduction

The goal of the present work is to derive the field-road diffusion model as the hydrodynamic limit of an interacting particle system. The former was introduced by Berestycki, Roquejoffre and Rossi [10] in order to describe spread of diseases or invasive species in presence of networks with accelerated propagation. It consists of two parabolic PDEs posed on two sets of different dimensions (a field and a road in a population dynamics context), and coupled through exchange terms between the field's boundary and the road — see subsection 1.1 for details. To asymptotically retrieve this deterministic model from a stochastic interacting particle system, we consider a Symmetric Simple Exclusion Process (SSEP) which evolves both on a finite discrete cylinder (the field) and its lower boundary (the road). Characterizing the SSEP, the microscopic dynamics is tied with a simple exclusion rule that forces each site to host at most one particle at the same time. To manage in particular the coupling between the field and the road, the system is in contact with reservoirs that allow to create or remove particles at the boundary sites of the cylinder. The activity of these reservoirs is slowed down compared to the diffusive dynamics, in order to align with the exchange terms awaited at the macroscopic scale. The originality of our analysis stands in the coupling between two domains of different dimensions, an issue that, as far as we know, has never been considered when recovering diffusive PDEs as the hydrodynamic limit of exclusion processes — see subsection 1.2.

## 1.1 The field-road model for fast diffusion channels

Recently, there has been a growing recognition of the importance of fast diffusion channels on biological invasions: for instance, an accidental transportation via human activities of some individuals towards northern and eastern France may be the cause of accelerated propagation of the pine processionary moth [46]. In Canada, some GPS data revealed that wolves travel faster along seismic lines (i.e. narrow strips cleared for energy exploration), thus increasing their chances to meet a prey [41]. It is also acknowledged that fast diffusion channels (roads, airlines, etc.) play a central role in the propagation of epidemics. As is well known, the spread of the black plague, which killed about a third of the European population in the 14th century, was favoured by the trade routes, especially the Silk Road, see [48]. More recently, some evidences of the radiation of the COVID epidemic along highways and transportation infrastructures were found [31].

In this context, the field-road model introduced by Berestycki, Roquejoffre and Rossi [10] writes as

$$ \left\{ \begin{array}{ll} \partial_t v = d\Delta v + f(v), & t > 0, \ x \in \mathbb{R}^{p-1}, \ y > 0, \\ -d\partial_y v|_{y=0} = \alpha u - \beta v|_{y=0}, & t > 0, \ x \in \mathbb{R}^{p-1}, \\ \partial_t u = D\Delta u + \beta v|_{y=0} - \alpha u, & t > 0, \ x \in \mathbb{R}^{p-1}. \end{array} \right. \quad (1.1) $$

The mathematical problem then amounts to describing survival and propagation in a non-standard physical space: the geographical domain consists in the half-space (the “field”) $x \in \mathbb{R}^{p-1}$, $y > 0$, bordered by the hyperplane (the “road”) $x \in \mathbb{R}^{p-1}$, $y = 0$. In the field, individuals diffuse with coefficient $d > 0$ and their density is given by $v = v(t, x, y)$. In particular $\Delta v$ has to be understood as $\Delta_x v + \partial_{yy} v$. On the road, individuals typically diffuse faster ($D > d$) and their density is given by $u = u(t, x)$. In particular $\Delta u$ has to be understood as $\Delta_x u$. The exchanges of population between the road and the field
# 1 Introduction

The goal of the present work is to derive the field-road diffusion model as the hydrodynamic limit of an interacting particle system. The former was introduced by Berestycki, Roquejoffre and Rossi [10] in order to describe spread of diseases or invasive species in presence of networks with accelerated propagation. It consists of two parabolic PDEs posed on two sets of different dimensions (a field and a road in a population dynamics context), and coupled through exchange terms between the field's boundary and the road — see subsection 1.1 for details. To asymptotically retrieve this deterministic model from a stochastic interacting particle system, we consider a Symmetric Simple Exclusion Process (SSEP) which evolves both on a finite discrete cylinder (the field) and its lower boundary (the road). Characterizing the SSEP, the microscopic dynamics is tied with a simple exclusion rule that forces each site to host at most one particle at the same time. To manage in particular the coupling between the field and the road, the system is in contact with reservoirs that allow to create or remove particles at the boundary sites of the cylinder. The activity of these reservoirs is slowed down compared to the diffusive dynamics, in order to align with the exchange terms awaited at the macroscopic scale. The originality of our analysis stands in the coupling between two domains of different dimensions, an issue that, as far as we know, has never been considered when recovering diffusive PDEs as the hydrodynamic limit of exclusion processes — see subsection 1.2.

## 1.1 The field-road model for fast diffusion channels

Recently, there has been a growing recognition of the importance of fast diffusion channels on biological invasions: for instance, an accidental transportation via human activities of some individuals towards northern and eastern France may be the cause of accelerated propagation of the pine processionary moth [46]. In Canada, some GPS data revealed that wolves travel faster along seismic lines (i.e. narrow strips cleared for energy exploration), thus increasing their chances to meet a prey [41]. It is also acknowledged that fast diffusion channels (roads, airlines, etc.) play a central role in the propagation of epidemics. As is well known, the spread of the black plague, which killed about a third of the European population in the 14th century, was favoured by the trade routes, especially the Silk Road, see [48]. More recently, some evidences of the radiation of the COVID epidemic along highways and transportation infrastructures were found [31].

In this context, the field-road model introduced by Berestycki, Roquejoffre and Rossi [10] writes as

$$ \left\{ \begin{array}{ll} \partial_t v = d\Delta v + f(v), & t > 0, \quad x \in \mathbb{R}^{p-1}, \quad y > 0, \\ -d\partial_y v|_{y=0} = \alpha u - \beta v|_{y=0}, & t > 0, \quad x \in \mathbb{R}^{p-1}, \\ \partial_t u = D\Delta u + \beta v|_{y=0} - \alpha u, & t > 0, \quad x \in \mathbb{R}^{p-1}. \end{array} \right. \quad (1.1) $$

The mathematical problem then amounts to describing survival and propagation in a non-standard physical space: the geographical domain consists in the half-space (the “field”) $x \in \mathbb{R}^{p-1}$, $y > 0$, bordered by the hyperplane (the “road”) $x \in \mathbb{R}^{p-1}$, $y = 0$. In the field, individuals diffuse with coefficient $d > 0$ and their density is given by $v = v(t, x, y)$. In particular $\Delta v$ has to be understood as $\Delta_x v + \partial_{yy} v$. On the road, individuals typically diffuse faster ($D > d$) and their density is given by $u = u(t, x)$. In particular $\Delta u$ has to be understood as $\Delta_x u$. The exchanges of population between the road and the field
are described by the second equation in system (1.1), where $\alpha > 0$ and $\beta > 0$. These boundary conditions, and the zeroth-order term on the road, link the field and the road equations and are the core of the model (see also the volume-surface systems [18], [26], [23] in the context of chemical processes or asymmetric stem cell division).

In a series of works [10, 9, 11, 12], Berestycki, Roquejoffre and Rossi studied the field-road system with $p = 2$ and $f$ a Fisher-KPP nonlinearity. They shed light on an acceleration phenomenon: when $D > 2d$, the road enhances the global diffusion and the spreading speed exceeds the standard Fisher-KPP invasion speed. This new feature has stimulated many works and, since then, many related problems taking into account heterogeneities, more complex geometries, nonlocal diffusions, etc. have been studied [5, 6], [33], [44, 45, 43], [51], [47], [22], [7, 8], [1, 52], [16].

Very recently, the purely diffusive field-road system — obtained by letting $f \equiv 0$ in (1.1) — has attracted some attention. Hence, an explicit expression for both the fundamental solution and the solution to the associated Cauchy problem, and a sharp (possibly up to a logarithmic term) decay rate of the $L^\infty$ norm of the solution were obtained in [3]. In a bounded domain, the long time convergence was studied [2] through entropy methods, in both the continuous and the discrete (finite volume scheme) settings.

From now on, we thus consider the purely diffusive field-road model. By using the rescaling

$$ \tilde{v}(t, x, y) = v\left(\frac{t}{\lambda^2}, \frac{x}{\lambda}, \frac{y}{\lambda}\right), \quad \tilde{u}(t, x) = \lambda u\left(\frac{t}{\lambda^2}, \frac{x}{\lambda}\right), \quad \lambda = \frac{\alpha}{\beta}, $$

we see that it is enough to consider the case $\alpha = \beta$. Also, for $p \ge 2$, we work in the $p$-dimensional open finite cylinder

$$ \Lambda := \mathbb{T}^{p-1} \times (0, 1), $$

where $\mathbb{T}$ is the one-dimensional torus $\mathbb{R}/\mathbb{Z}$. For $v$, we impose the zero Neumann boundary conditions on the upper boundary $\mathbb{T}^{p-1} \times \{y = 1\}$. This insures the conservation of the total mass, namely $\int_{\mathbb{T}^{p-1} \times (0,1)} v(t, x, y)dxdy + \int_{\mathbb{T}^{p-1}} u(t, x)dx$, therefore modeling a purely diffusive process within a closed environment. Denoting $n$ the unit outward normal vector to $\partial\Lambda$, the considered system is thus

$$ \left\{ \begin{array}{lll} \partial_t v = d\Delta v, & t > 0, \quad x \in \mathbb{T}^{p-1}, \quad y \in (0, 1), \\ -d\partial_y v|_{y=0} = \alpha u - \alpha v|_{y=0}, & t > 0, \quad x \in \mathbb{T}^{p-1}, \quad y = 0, \\ \partial_t u = D\Delta u + \alpha v|_{y=0} - \alpha u, & t > 0, \quad x \in \mathbb{T}^{p-1}, \\ \frac{\partial v}{\partial n} = 0, & t > 0, \quad x \in \mathbb{T}^{p-1}, \quad y = 1, \end{array} \right. \quad (1.2) $$

supplemented with an initial condition

$$ \left\{ \begin{array}{ll} v|_{t=0} = v_0 \in L^\infty(\Lambda) \cap [0, 1]^\Lambda, & x \in \mathbb{T}^{p-1}, \quad y \in (0, 1), \\ u|_{t=0} = u_0 \in L^\infty(\mathbb{T}^{p-1}) \cap [0, 1]^{\mathbb{T}^{p-1}}, & x \in \mathbb{T}^{p-1}. \end{array} \right. \quad (1.3) $$

Note that, given the linear nature of the system (1.2), the use of initial data bounded by 1 is a simplification that does not compromise the generality of our analysis. This choice is actually imposed by the exclusion rule, see Remark 2.3.
are described by the second equation in system (1.1), where $\alpha > 0$ and $\beta > 0$. These boundary conditions, and the zeroth-order term on the road, link the field and the road equations and are the core of the model (see also the volume-surface systems [18], [26], [23] in the context of chemical processes or asymmetric stem cell division).

In a series of works [10, 9, 11, 12], Berestycki, Roquejoffre and Rossi studied the field-road system with $p = 2$ and $f$ a Fisher-KPP nonlinearity. They shed light on an acceleration phenomenon: when $D > 2d$, the road enhances the global diffusion and the spreading speed exceeds the standard Fisher-KPP invasion speed. This new feature has stimulated many works and, since then, many related problems taking into account heterogeneities, more complex geometries, nonlocal diffusions, etc. have been studied [5, 6], [33], [44, 45, 43], [51], [47], [22], [7, 8], [1, 52], [16].

Very recently, the purely diffusive field-road system — obtained by letting $f \equiv 0$ in (1.1) — has attracted some attention. Hence, an explicit expression for both the fundamental solution and the solution to the associated Cauchy problem, and a sharp (possibly up to a logarithmic term) decay rate of the $L^\infty$ norm of the solution were obtained in [3]. In a bounded domain, the long time convergence was studied [2] through entropy methods, in both the continuous and the discrete (finite volume scheme) settings.

From now on, we thus consider the purely diffusive field-road model. By using the rescaling

$$ \tilde{v}(t, x, y) = v\left(\frac{t}{\lambda^2}, \frac{x}{\lambda}, \frac{y}{\lambda}\right), \quad \tilde{u}(t, x) = \lambda u\left(\frac{t}{\lambda^2}, \frac{x}{\lambda}\right), \quad \lambda = \frac{\alpha}{\beta}, $$

we see that it is enough to consider the case $\alpha = \beta$. Also, for $p \ge 2$, we work in the $p$-dimensional open finite cylinder

$$ \Lambda := \mathbb{T}^{p-1} \times (0, 1), $$

where $\mathbb{T}$ is the one-dimensional torus $\mathbb{R}/\mathbb{Z}$. For $v$, we impose the zero Neumann boundary conditions on the upper boundary $\mathbb{T}^{p-1} \times \{y = 1\}$. This insures the conservation of the total mass, namely $\int_{\mathbb{T}^{p-1} \times (0,1)} v(t, x, y)dxdy + \int_{\mathbb{T}^{p-1}} u(t, x)dx$, therefore modeling a purely diffusive process within a closed environment. Denoting $n$ the unit outward normal vector to $\partial\Lambda$, the considered system is thus

$$ \left\{ \begin{array}{lll} \partial_t v = d\Delta v, & t > 0, \quad x \in \mathbb{T}^{p-1}, \quad y \in (0, 1), \\ -d\partial_y v|_{y=0} = \alpha u - \alpha v|_{y=0}, & t > 0, \quad x \in \mathbb{T}^{p-1}, \quad y = 0, \\ \partial_t u = D\Delta u + \alpha v|_{y=0} - \alpha u, & t > 0, \quad x \in \mathbb{T}^{p-1}, \\ \frac{\partial v}{\partial n} = 0, & t > 0, \quad x \in \mathbb{T}^{p-1}, \quad y = 1, \end{array} \right. \quad (1.2) $$

supplemented with an initial condition

$$ \left\{ \begin{array}{lll} v|_{t=0} = v_0 \in L^\infty(\Lambda) \cap [0, 1]^\Lambda, & x \in \mathbb{T}^{p-1}, \quad y \in (0, 1), \\ u|_{t=0} = u_0 \in L^\infty(\mathbb{T}^{p-1}) \cap [0, 1]^{\mathbb{T}^{p-1}}, & x \in \mathbb{T}^{p-1}. \end{array} \right. \quad (1.3) $$

Note that, given the linear nature of the system (1.2), the use of initial data bounded by 1 is a simplification that does not compromise the generality of our analysis. This choice is actually imposed by the exclusion rule, see Remark 2.3.
## 1.2 Interacting Particle Systems and Simple Exclusion Processes

The field of interacting particle systems is a branch of probability theory that emerged in the early 1970s, focusing on Markov processes inspired by models from statistical physics and biology. Analysis occurs at both the microscopic level of particle dynamics and by scaling from microscopic to macroscopic levels. This involves space and time renormalization procedures to derive hydrodynamic limits, represented by PDEs that describe key model quantities such as particle densities.

Introduced by Frank Spitzer in [49], the exclusion process are interacting particle systems from which can be recovered a large variety of diffusive systems driven out of equilibrium, see the pioneering works [50], [40, 39]. We refer to the seminal book [34] for the complete derivation of the Heat equation on a torus from a nearest-neighbor exclusion process which consists in a collection of continuous-time random walks evolving on a lattice (see below for details).

When boundaries are considered, the system is in contact with some so-called reservoirs, see [36], [19, 21, 20] where Dirichlet boundary conditions are recovered. Recently, a lot of effort has been put in understanding the case of exclusion process whose dynamics is perturbed by the presence of a slow bond [15], [27], or by slow boundary effects [27, 28, 29], [4], [35], [37], [13].

Let us comment more precisely on some of the outcomes obtained in [4]. The authors specifically examine the hydrodynamic behavior of a symmetric simple exclusion process with slow boundary. This means that, at the boundary sites, particles can be born or die at slower rates (depending on the scaling parameter $N$) than events occurring in the bulk. The hydrodynamic limit is then the Heat equation, supplemented with Dirichlet, Robin, or Neumann boundary conditions, depending on the scaling of the boundary rates.

The present work stands at the crossroads of this framework, the reaction-diffusion issues and epidemiology/population dynamics modeling. With that respect, let us mention the very recent work [42] where a reaction-diffusion system modeling the sterile insect technique is retrieved. The very originality of our work stands in the fact that the considered system is posed on sets of different dimensions a case which, as far as we know, is considered for the first time in the interacting particle system literature.

# 2 Notations and main result

All the notations used in this paper are gathered in the Table of Notations at the end of this document.

## 2.1 Sets and related notations

As announced above, in the macroscopic setting, we work in the $p$-dimensional open finite cylinder

$$ \Lambda := \mathbb{T}^{p-1} \times (0, 1), $$

where $\mathbb{T}$ designates the one-dimensional torus $\mathbb{R}/\mathbb{Z}$. The boundary of the domain is denoted

$$ \Gamma := \partial\Lambda = \{(x, y) \in \bar{\Lambda} \mid y = 0 \text{ or } y = 1\} = \mathbb{T}^{p-1} \times \{0, 1\}, $$
## 1.2 Interacting Particle Systems and Simple Exclusion Processes

The field of interacting particle systems is a branch of probability theory that emerged in the early 1970s, focusing on Markov processes inspired by models from statistical physics and biology. Analysis occurs at both the microscopic level of particle dynamics and by scaling from microscopic to macroscopic levels. This involves space and time renormalization procedures to derive hydrodynamic limits, represented by PDEs that describe key model quantities such as particle densities.

Introduced by Frank Spitzer in [49], the exclusion process are interacting particle systems from which can be recovered a large variety of diffusive systems driven out of equilibrium, see the pioneering works [50], [40, 39]. We refer to the seminal book [34] for the complete derivation of the Heat equation on a torus from a nearest-neighbor exclusion process which consists in a collection of continuous-time random walks evolving on a lattice (see below for details).

When boundaries are considered, the system is in contact with some so-called reservoirs, see [36], [19, 21, 20] where Dirichlet boundary conditions are recovered. Recently, a lot of effort has been put in understanding the case of exclusion process whose dynamics is perturbed by the presence of a slow bond [15], [27], or by slow boundary effects [27, 28, 29], [4], [35], [37], [13].

Let us comment more precisely on some of the outcomes obtained in [4]. The authors specifically examine the hydrodynamic behavior of a symmetric simple exclusion process with slow boundary. This means that, at the boundary sites, particles can be born or die at slower rates (depending on the scaling parameter $N$) than events occurring in the bulk. The hydrodynamic limit is then the Heat equation, supplemented with Dirichlet, Robin, or Neumann boundary conditions, depending on the scaling of the boundary rates.

The present work stands at the crossroads of this framework, the reaction-diffusion issues and epidemiology/population dynamics modeling. With that respect, let us mention the very recent work [42] where a reaction-diffusion system modeling the sterile insect technique is retrieved. The very originality of our work stands in the fact that the considered system is posed on sets of different dimensions a case which, as far as we know, is considered for the first time in the interacting particle system literature.

# 2 Notations and main result

All the notations used in this paper are gathered in the Table of Notations at the end of this document.

## 2.1 Sets and related notations

As announced above, in the macroscopic setting, we work in the $p$-dimensional open finite cylinder

$$ \Lambda := \mathbb{T}^{p-1} \times (0, 1), $$

where $\mathbb{T}$ designates the one-dimensional torus $\mathbb{R}/\mathbb{Z}$. The boundary of the domain is denoted

$$ \Gamma := \partial\Lambda = \{(x, y) \in \bar{\Lambda} \mid y = 0 \text{ or } y = 1\} = \mathbb{T}^{p-1} \times \{0, 1\}, $$
with $\bar{\Lambda}$ the closure of $\Lambda$. We partition $\Gamma$ into two parts representing the lower and upper boundaries of the cylinder:

$$ \Gamma^{\text{low}} := \mathbb{T}^{p-1} \times \{0\} \quad \text{and} \quad \Gamma^{\text{up}} := \mathbb{T}^{p-1} \times \{1\}. $$

At the microscopic level, given an integer $N \ge 2$, we define $\Lambda_N$ and $\Gamma_N$ as the corresponding discrete microscopic sets. Specifically, by letting $\mathbb{T}_N := \mathbb{Z}/N\mathbb{Z}$ the discrete one-dimensional torus of length $N$, and using the notation $\llbracket a; b \rrbracket := [a, b] \cap \mathbb{Z}$ for any $a, b \in \mathbb{R}$,

$$ \Lambda_N := \mathbb{T}_N^{p-1} \times \llbracket 1; N-1 \rrbracket $$

represents the cylinder in $\mathbb{Z}^p$ of height $N-1$ and basis $\mathbb{T}_N^{p-1}$, its boundary being

$$ \Gamma_N := \{(i, j) \in \Lambda_N \mid j = 1 \text{ or } N-1\} = \mathbb{T}_N^{p-1} \times \{1, N-1\}. $$

Similarly, $\Gamma_N = \Gamma_N^{\text{low}} \cup \Gamma_N^{\text{up}}$ with

$$ \Gamma_N^{\text{low}} := \mathbb{T}_N^{p-1} \times \{1\} \quad \text{and} \quad \Gamma_N^{\text{up}} := \mathbb{T}_N^{p-1} \times \{N-1\}. $$

The elements of $\bar{\Lambda}$ are represented by

$$ \hat{x} = (x, y) \quad \text{and} \quad \hat{z} = (z, \omega), $$

with $x, z \in \mathbb{T}^{p-1}$ and $y, \omega \in (0, 1)$, while those of $\Lambda_N$ are symbolized by the letters

$$ \hat{i} = (i, j) \quad \text{and} \quad \hat{k} = (k, \ell), $$

with $i, k \in \mathbb{T}_N^{p-1}$ and $j, \ell \in \llbracket 1; N-1 \rrbracket$.

## 2.2 Description of the microscopic model

We consider the evolution of two kinds of interacting particles on the lattices $\Lambda_N$ (the microscopic field) and $\Gamma_N^{\text{low}}$ (the microscopic road). The associated stochastic dynamics is described by the temporal evolution of a Markov process denoted by $(\eta_t, \xi_t)_{t \in [0, T]}$, where $T > 0$ is a given temporal horizon. Particles tied to the dynamics of $\eta$ (the “field-particles”) evolve in the whole microscopic field $\Lambda_N$, while the particles corresponding to the dynamics of $\xi$ (the “road-particles”) evolve solely on the microscopic road $\Gamma_N^{\text{low}}$, that stands as the lower frontier of the microscopic field<fnref n="1" />. Both types of particles follow an exclusion rule in its respective environment: each site $\hat{i} = (i, j) \in \Lambda_N$ can host at most one field-particle, and similarly, each site $i \in \Gamma_N^{\text{low}}$ can host at most one road-particle. Note in particular that at a site $i \in \Gamma_N^{\text{low}}$, there may be a field-particle and a road-particle.

The overall dynamics emerge from the superposition of several independent ones, which are individually specified below and collectively depicted in Figure I:

<footnote n="1">¹It is important to note that the microscopic road $\Gamma_N^{\text{low}}$ is actually embedded in $\mathbb{Z}^{p-1}$. For the purpose of simplifying notations and facilitating understanding, we often make an identification between $(i, 1)$ and $i$, establishing a one-to-one correspondence between the lower boundary of the microscopic field and the $(p-1)$-dimensional torus. In line with this simplification, we will use $\xi(i)$ and $\eta(i)$ to denote what are, in fact, $\xi(i, 1)$ and $\eta(i, 1)$. This is a deliberate choice to streamline the expressions without compromising the accuracy of the mathematical representations involved. Similarly, on the upper microscopic boundary $\Gamma_N^{\text{up}}$, we may write $\eta(i)$ to represent $\eta(i, N-1)$, provided this does not lead to any ambiguity.</footnote>
with $\bar{\Lambda}$ the closure of $\Lambda$. We partition $\Gamma$ into two parts representing the lower and upper boundaries of the cylinder:

$$ \Gamma^{\text{low}} := \mathbb{T}^{p-1} \times \{0\} \quad \text{and} \quad \Gamma^{\text{up}} := \mathbb{T}^{p-1} \times \{1\}. $$

At the microscopic level, given an integer $N \ge 2$, we define $\Lambda_N$ and $\Gamma_N$ as the corresponding discrete microscopic sets. Specifically, by letting $\mathbb{T}_N := \mathbb{Z}/N\mathbb{Z}$ the discrete one-dimensional torus of length $N$, and using the notation $\llbracket a; b \rrbracket := [a, b] \cap \mathbb{Z}$ for any $a, b \in \mathbb{R}$,

$$ \Lambda_N := \mathbb{T}_N^{p-1} \times \llbracket 1; N-1 \rrbracket $$

represents the cylinder in $\mathbb{Z}^p$ of height $N-1$ and basis $\mathbb{T}_N^{p-1}$, its boundary being

$$ \Gamma_N := \{(i, j) \in \Lambda_N \mid j = 1 \text{ or } N-1\} = \mathbb{T}_N^{p-1} \times \{1, N-1\}. $$

Similarly, $\Gamma_N = \Gamma_N^{\text{low}} \cup \Gamma_N^{\text{up}}$ with

$$ \Gamma_N^{\text{low}} := \mathbb{T}_N^{p-1} \times \{1\} \quad \text{and} \quad \Gamma_N^{\text{up}} := \mathbb{T}_N^{p-1} \times \{N-1\}. $$

The elements of $\bar{\Lambda}$ are represented by

$$ \hat{x} = (x, y) \quad \text{and} \quad \hat{z} = (z, \omega), $$

with $x, z \in \mathbb{T}^{p-1}$ and $y, \omega \in (0, 1)$, while those of $\Lambda_N$ are symbolized by the letters

$$ \hat{i} = (i, j) \quad \text{and} \quad \hat{k} = (k, \ell), $$

with $i, k \in \mathbb{T}_N^{p-1}$ and $j, \ell \in \llbracket 1; N-1 \rrbracket$.

## 2.2 Description of the microscopic model

We consider the evolution of two kinds of interacting particles on the lattices $\Lambda_N$ (the microscopic field) and $\Gamma_N^{\text{low}}$ (the microscopic road). The associated stochastic dynamics is described by the temporal evolution of a Markov process denoted by $(\eta_t, \xi_t)_{t \in [0, T]}$, where $T > 0$ is a given temporal horizon. Particles tied to the dynamics of $\eta$ (the “field-particles”) evolve in the whole microscopic field $\Lambda_N$, while the particles corresponding to the dynamics of $\xi$ (the “road-particles”) evolve solely on the microscopic road $\Gamma_N^{\text{low}}$, that stands as the lower frontier of the microscopic field<fnref n="1" />. Both types of particles follow an exclusion rule in its respective environment: each site $\hat{i} = (i, j) \in \Lambda_N$ can host at most one field-particle, and similarly, each site $i \in \Gamma_N^{\text{low}}$ can host at most one road-particle. Note in particular that at a site $i \in \Gamma_N^{\text{low}}$, there may be a field-particle and a road-particle.

The overall dynamics emerge from the superposition of several independent ones, which are individually specified below and collectively depicted in Figure I:

<footnote n="1">¹It is important to note that the microscopic road $\Gamma_N^{\text{low}}$ is actually embedded in $\mathbb{Z}^{p-1}$. For the purpose of simplifying notations and facilitating understanding, we often make an identification between $(i, 1)$ and $i$, establishing a one-to-one correspondence between the lower boundary of the microscopic field and the $(p-1)$-dimensional torus. In line with this simplification, we will use $\xi(i)$ and $\eta(i)$ to denote what are, in fact, $\xi(i, 1)$ and $\eta(i, 1)$. This is a deliberate choice to streamline the expressions without compromising the accuracy of the mathematical representations involved. Similarly, on the upper microscopic boundary $\Gamma_N^{\text{up}}$, we may write $\eta(i)$ to represent $\eta(i, N-1)$, provided this does not lead to any ambiguity.</footnote>
• *Diffusion in the field.* Within $\Lambda_N$, the field-particles follow a simple exclusion process and jump at exponential times. The dynamics of this process is as follows: a particle located at site $\hat{i}$ awaits an exponential time after which it jumps to a neighboring site $\hat{k}$ with speeded rate $N^2d$, for some fixed $d > 0$. However, if the site $\hat{k}$ is already occupied, the jump is prevented in accordance with the exclusion rule.

• *Diffusion on the road.* Similarly, the road-particles follow a simple exclusion process on $\mathbb{T}_N^{p-1}$: a particle positioned at site $i$ awaits an exponential time after which it jumps to a neighboring site $k$ with speeded rate $N^2D$, for some fixed $D > 0$. However, if the site $k$ is already occupied, the jump is inhibited.

• *Reservoir at the upper field’s boundary.* The dynamics defined on the upper boundary $\Gamma_N^{up}$ act as reservoirs for the field-particles that are much slower compared to the rate of jumps in the bulk. Fix a constant $0 \le b \le 1$, for each site $\hat{i} \in \Gamma_N^{up}$, the following events occur, according to exponential times that are independent of all others:

- In the absence of a particle, a new one is generated with rate $b$.

- Conversely, if a particle is present, it is eliminated with rate $1 - b$.

•• *Exchange dynamics between the lower field’s boundary and the road.* We now describe the interacting behavior between the road-particles and the field-particles at the lower boundary of the microscopic field $\Gamma_N^{low}$. Fix $\alpha > 0$, for each site $\hat{i} = (i, 1) \in \Gamma_N^{low}$, according to exponential times, the following scenarios may occur:

- If a road-particle is present and no field-particle exists, then a field-particle is generated at site $\hat{i}$ with speeded rate $N\alpha$ (♥). Independently, the road-particle is eliminated with rate $\alpha$ (♣).

- Conversely, if a field-particle is present without a road-particle, then the field-particle is eliminated with speeded rate $N\alpha$ (♦). Independently, a road-particle is generated with rate $\alpha$ (♠).

The configuration space is given by

$$S_N := \underbrace{\{0, 1\}^{\Lambda_N}}_{=: S_N^{\text{field}}} \times \underbrace{\{0, 1\}^{\Gamma_N^{\text{low}}}}_{=: S_N^{\text{road}}}$$

which we endow with the product topology. The elements of $S_N$, referred to as configurations, are denoted by $(\eta, \xi)$. The first marginal $\eta$ represents a configuration within the state space $S_N^{\text{field}}$. To be more specific, in a given configuration $\eta$, for any $\hat{i}$ in $\Lambda_N$, $\eta(\hat{i}) = 1$ means the site $\hat{i}$ is occupied. Conversely, $\eta(\hat{i}) = 0$ signifies that the site $\hat{i}$ is empty. Similarly, the second marginal $\xi$ stands for a configuration within the state space $S_N^{\text{road}}$, and for any $i$ in $\Gamma_N^{\text{low}}$, the value $\xi(i)$ indicates the occupancy status of particle at site $i$ in a given configuration $\xi$.

For any configuration $\eta$ in $S_N^{\text{field}}$ (resp. $\xi$ in $S_N^{\text{road}}$), and any sites $\hat{i}, \hat{k}$ in $\Lambda_N$ (resp. $i, k$ in $\Gamma_N^{\text{low}}$), let $\eta^{\hat{i}, \hat{k}}$ (resp. $\xi^{i, k}$) be the configuration obtained from $\eta$ (resp. $\xi$) by switching the values at $\hat{i}$ and $\hat{k}$ (resp. $i$ and $k$), namely

$$(\eta^{\hat{i}, \hat{k}})(\hat{m}) = \begin{cases} \eta(\hat{i}) & \text{if } \hat{m} = \hat{k}, \\ \eta(\hat{k}) & \text{if } \hat{m} = \hat{i}, \\ \eta(\hat{m}) & \text{otherwise,} \end{cases} \quad \text{resp. } (\xi^{i,k})(m) = \begin{cases} \xi(i) & \text{if } m = k, \\ \xi(k) & \text{if } m = i, \\ \xi(m) & \text{otherwise} \end{cases}.$$
• *Diffusion in the field.* Within $\Lambda_N$, the field-particles follow a simple exclusion process and jump at exponential times. The dynamics of this process is as follows: a particle located at site $\hat{i}$ awaits an exponential time after which it jumps to a neighboring site $\hat{k}$ with speeded rate $N^2d$, for some fixed $d > 0$. However, if the site $\hat{k}$ is already occupied, the jump is prevented in accordance with the exclusion rule.

• *Diffusion on the road.* Similarly, the road-particles follow a simple exclusion process on $\mathbb{T}_N^{p-1}$: a particle positioned at site $i$ awaits an exponential time after which it jumps to a neighboring site $k$ with speeded rate $N^2D$, for some fixed $D > 0$. However, if the site $k$ is already occupied, the jump is inhibited.

• *Reservoir at the upper field’s boundary.* The dynamics defined on the upper boundary $\Gamma_N^{up}$ act as reservoirs for the field-particles that are much slower compared to the rate of jumps in the bulk. Fix a constant $0 \le b \le 1$, for each site $\hat{i} \in \Gamma_N^{up}$, the following events occur, according to exponential times that are independent of all others:

- In the absence of a particle, a new one is generated with rate $b$.

- Conversely, if a particle is present, it is eliminated with rate $1-b$.

•• *Exchange dynamics between the lower field’s boundary and the road.* We now describe the interacting behavior between the road-particles and the field-particles at the lower boundary of the microscopic field $\Gamma_N^{low}$. Fix $\alpha > 0$, for each site $\hat{i} = (i, 1) \in \Gamma_N^{low}$, according to exponential times, the following scenarios may occur:

- If a road-particle is present and no field-particle exists, then a field-particle is generated at site $\hat{i}$ with speeded rate $N\alpha$ (♥). Independently, the road-particle is eliminated with rate $\alpha$ (♣).

- Conversely, if a field-particle is present without a road-particle, then the field-particle is eliminated with speeded rate $N\alpha$ (♦). Independently, a road-particle is generated with rate $\alpha$ (♠).

The configuration space is given by

$$S_N := \underbrace{\{0, 1\}^{\Lambda_N}}_{=: S_N^{\text{field}}} \times \underbrace{\{0, 1\}^{\Gamma_N^{\text{low}}}}_{=: S_N^{\text{road}}}$$

which we endow with the product topology. The elements of $S_N$, referred to as configurations, are denoted by $(\eta, \xi)$. The first marginal $\eta$ represents a configuration within the state space $S_N^{\text{field}}$. To be more specific, in a given configuration $\eta$, for any $\hat{i}$ in $\Lambda_N$, $\eta(\hat{i}) = 1$ means the site $\hat{i}$ is occupied. Conversely, $\eta(\hat{i}) = 0$ signifies that the site $\hat{i}$ is empty. Similarly, the second marginal $\xi$ stands for a configuration within the state space $S_N^{\text{road}}$, and for any $i$ in $\Gamma_N^{\text{low}}$, the value $\xi(i)$ indicates the occupancy status of particle at site $i$ in a given configuration $\xi$.

For any configuration $\eta$ in $S_N^{\text{field}}$ (resp. $\xi$ in $S_N^{\text{road}}$), and any sites $\hat{i}, \hat{k}$ in $\Lambda_N$ (resp. $i, k$ in $\Gamma_N^{\text{low}}$), let $\eta^{\hat{i}, \hat{k}}$ (resp. $\xi^{i, k}$) be the configuration obtained from $\eta$ (resp. $\xi$) by switching the values at $\hat{i}$ and $\hat{k}$ (resp. $i$ and $k$), namely

$$(\eta^{\hat{i}, \hat{k}})(\hat{m}) = \begin{cases} \eta(\hat{i}) & \text{if } \hat{m} = \hat{k}, \\ \eta(\hat{k}) & \text{if } \hat{m} = \hat{i}, \\ \eta(\hat{m}) & \text{otherwise,} \end{cases} \quad \text{resp. } (\xi^{i,k})(m) = \begin{cases} \xi(i) & \text{if } m = k, \\ \xi(k) & \text{if } m = i, \\ \xi(m) & \text{otherwise} \end{cases}.$$
For the sites $\hat{i} \in \Gamma_N$ (resp. $i \in \Gamma_N^{\text{low}}$), let $\eta^{\hat{i}}$ (resp. $\xi^i$) be the configuration obtained from $\eta$ (resp. $\xi$) by flipping the occupation number at site $\hat{i}$ (resp. $i$), namely

$$ (\eta^{\hat{i}})(\hat{m}) = \begin{cases} 1 - \eta(\hat{m}) & \text{if } \hat{m} = \hat{i}, \\ \eta(\hat{m}) & \text{otherwise,} \end{cases} \quad \text{(resp. } (\xi^i)(m) = \begin{cases} 1 - \xi(m) & \text{if } (m) = i, \\ \xi(m) & \text{otherwise} \end{cases}). $$

<figure><img src="image_2.png" /><figcaption>**Figure I** — The microscopic dynamics in dimension $p=2$. In the field, particles are represented by green dots ● and jump towards one of their adjacent sites at exponential times with mean frequency $N^2d$ — “with rate $N^2d$” for short. Moves to already occupied sites are prohibited by the exclusion rule, and are indicated by the symbol ⊙. Similarly, particles on the road are depicted by the blue dots ● and jump to their neighboring unoccupied sites with rate $N^2D$. At the upper boundary of the field $\Gamma_N^{\text{up}}$, particles emerge in empty sites with rate $b$ and are removed with rate $1-b$. The interactions at the lower boundary of the field $\Gamma_N^{\text{low}}$ allow the coupling between the field and the road and play a central role in the model. These are detailed in the “Exchange dynamics” panel on the right-hand-side of the figure. For clarity and to facilitate understanding, $\Gamma_N^{\text{low}}$ is represented twice to distinguish between the particles at the lower boundary of the field and those on the road. Notice also that not all possible jumps are represented.</figcaption></figure>

Fix $\alpha, d, D \in (0, \infty)$, and $0 \le b \le 1$. The generator of the microscopic dynamics, $\mathcal{L}_N: \mathbb{R}^{S_N} \to \mathbb{R}^{S_N}$, is split as follows:

$$ \mathcal{L}_N = \underbrace{N^2 \mathbb{L}_N^{\text{field}}}_{\text{field diffusion}} + \underbrace{N^2 \mathbb{L}_N^{\text{road}}}_{\text{lower Robin condition}} + \underbrace{N \mathbb{L}_N^{\text{Rob}}}_{\text{upper reservoir}} + \underbrace{\mathbb{L}_N^{\text{reac}}}_{\text{road reaction}} + \underbrace{\mathbb{L}_N^{\text{up}}}_{\text{upper reservoir}}, \quad (2.1) $$
For the sites $\hat{i} \in \Gamma_N$ (resp. $i \in \Gamma_N^{\text{low}}$), let $\eta^{\hat{i}}$ (resp. $\xi^i$) be the configuration obtained from $\eta$ (resp. $\xi$) by flipping the occupation number at site $\hat{i}$ (resp. $i$), namely

$$ (\eta^{\hat{i}})(\hat{m}) = \begin{cases} 1 - \eta(\hat{m}) & \text{if } \hat{m} = \hat{i}, \\ \eta(\hat{m}) & \text{otherwise,} \end{cases} \quad \text{(resp. } (\xi^i)(m) = \begin{cases} 1 - \xi(m) & \text{if } (m) = i, \\ \xi(m) & \text{otherwise} \end{cases}). $$

<figure><img src="image_2.png" /><figcaption>**Figure I** — The microscopic dynamics in dimension $p=2$. In the field, particles are represented by green dots ● and jump towards one of their adjacent sites at exponential times with mean frequency $N^2d$ — “with rate $N^2d$” for short. Moves to already occupied sites are prohibited by the exclusion rule, and are indicated by the symbol ⊘. Similarly, particles on the road are depicted by the blue dots ● and jump to their neighboring unoccupied sites with rate $N^2D$. At the upper boundary of the field $\Gamma_N^{\text{up}}$, particles emerge in empty sites with rate $b$ and are removed with rate $1-b$. The interactions at the lower boundary of the field $\Gamma_N^{\text{low}}$ allow the coupling between the field and the road and play a central role in the model. These are detailed in the “Exchange dynamics” panel on the right-hand-side of the figure. For clarity and to facilitate understanding, $\Gamma_N^{\text{low}}$ is represented twice to distinguish between the particles at the lower boundary of the field and those on the road. Notice also that not all possible jumps are represented.</figcaption></figure>

Fix $\alpha, d, D \in (0, \infty)$, and $0 \le b \le 1$. The generator of the microscopic dynamics, $\mathcal{L}_N: \mathbb{R}^{S_N} \to \mathbb{R}^{S_N}$, is split as follows:

$$ \mathcal{L}_N = \underbrace{N^2 \mathbb{L}_N^{\text{field}}}_{\text{field diffusion}} + \underbrace{N^2 \mathbb{L}_N^{\text{road}}}_{\text{lower Robin condition}} + \underbrace{N \mathbb{L}_N^{\text{Rob}}}_{\text{upper reservoir}} + \underbrace{\mathbb{L}_N^{\text{react}}}_{\text{road reaction}} + \underbrace{\mathbb{L}_N^{\text{up}}}_{\text{upper reservoir}}, \quad (2.1) $$
where, for any $f: S_N \to \mathbb{R}$ and any $(\eta, \xi) \in S_N$,

$$ (\mathbb{L}_N^{\text{field}} f)(\eta, \xi) = \frac{d}{2} \sum_{\substack{\hat{i}, \hat{k} \in \Lambda_N \\ |\hat{i} - \hat{k}| = 1}} [f(\eta^{\hat{i}}, \hat{k}), \xi) - f(\eta, \xi)], \quad (2.2) $$

$$ (\mathbb{L}_N^{\text{road}} f)(\eta, \xi) = \frac{D}{2} \sum_{\substack{i,k \in \Gamma_N^{\text{low}} \\ |i-k|=1}} [f(\eta, \xi^{i,k}) - f(\eta, \xi)], \quad (2.3) $$

$$ (L_N^{\text{Rob}} f)(\eta, \xi) = \alpha \sum_{i \in \Gamma_N^{\text{low}}} (\eta(i) - \xi(i))^2 [f(\eta^i, \xi) - f(\eta, \xi)], \quad (2.4) $$

$$ (L_N^{\text{reac}} f)(\eta, \xi) = \alpha \sum_{i \in \Gamma_N^{\text{low}}} (\eta(i) - \xi(i))^2 [f(\eta, \xi^i) - f(\eta, \xi)], \quad (2.5) $$

$$ (L_N^{\text{up}} f)(\eta, \xi) = \sum_{i \in \Gamma_N^{\text{up}}} (b(1 - \eta(i)) + (1 - b)\eta(i)) [f(\eta^i, \xi) - f(\eta, \xi)]. \quad (2.6) $$

In (2.2) and (2.3), we use $|\cdot|$ to denote the infinity norm in $\mathbb{R}^p$, that is

$$ |\hat{i}| = \max(|i_1|, \dots, |i_{p-1}|, |j|), \quad \forall \hat{i} \in \Lambda_N. $$

Also, we highlight that the flip rate $\alpha(\eta(i) - \xi(i))^2$ in (2.4) and (2.5) arises from equality

$$ (1 - \eta(i)) \xi(i) + \eta(i) (1 - \xi(i)) = (\eta(i) - \xi(i))^2, $$

which holds since both $\eta(i)$ and $\xi(i)$ belong to $\{0, 1\}$.

The parts (2.2), (2.3) and (2.6) are rather classical, see [34], [4], [42] for instance. On the other hand, the parts (2.4) and (2.5) are original, their role being to catch the exchange condition in the field-road model. We refer to [28] and [30] for related issues.

For a given time horizon $T > 0$, we denote $(\eta_t, \xi_t)_{t \in [0,T]}$ the Markov process with state space $S_N$ associated to the generator $\mathcal{L}_N$. We define $D([0,T]; S_N)$ as the path space for càdlàg time trajectories valued in $S_N$. Given a measure $\mu_N$ on $S_N$, we denote by $\mathbb{P}_N^{\mu_N}$ the probability measure on $D([0,T]; S_N)$ induced by $\mu_N$ and $(\eta_t, \xi_t)_{t \in [0,T]}$, and we write $\mathbb{E}_N^{\mu_N}$ the expectation with respect to $\mathbb{P}_N^{\mu_N}$. Moreover, the notation $\langle \cdot, \cdot \rangle_{\mu_N}$ refers to the scalar product on $L_N^2(S_N)$.

## 2.3 Functional spaces and macroscopic equations

For any integers $n$ and $m$, we define the functional spaces

$$ \mathcal{C}^{n,m}([0, T] \times \bar{\Lambda}) \quad \text{and} \quad \mathcal{C}^{n,m}([0, T] \times \mathbb{T}^{p-1}) $$

which respectively consist of functions

$$ G = G(t, \hat{x}) : [0, T] \times \bar{\Lambda} \to \mathbb{R} \quad \text{and} \quad H = H(t, x) : [0, T] \times \mathbb{T}^{p-1} \to \mathbb{R}, $$

that possess $n$ continuous derivatives with respect to the time variable on $[0, T]$, and $m$ continuous derivatives with respect to the spatial variable on $\bar{\Lambda}$ and $\mathbb{T}^{p-1}$ respectively. We also introduce the subset $\mathcal{C}_c^{n,m}([0, T] \times \bar{\Lambda})$ of functions with compact support in $[0, T] \times \Lambda$
where, for any $f: S_N \to \mathbb{R}$ and any $(\eta, \xi) \in S_N$,

$$ (\mathbb{L}_N^{\text{field}} f)(\eta, \xi) = \frac{d}{2} \sum_{\substack{\hat{i}, \hat{k} \in \Lambda_N \\ |\hat{i} - \hat{k}| = 1}} [f(\eta^{\hat{i}, \hat{k}}, \xi) - f(\eta, \xi)], \quad (2.2) $$

$$ (\mathbb{L}_N^{\text{road}} f)(\eta, \xi) = \frac{D}{2} \sum_{\substack{i,k \in \Gamma_N^{\text{low}} \\ |i-k|=1}} [f(\eta, \xi^{i,k}) - f(\eta, \xi)], \quad (2.3) $$

$$ (L_N^{\text{Rob}} f)(\eta, \xi) = \alpha \sum_{i \in \Gamma_N^{\text{low}}} (\eta(i) - \xi(i))^2 [f(\eta^i, \xi) - f(\eta, \xi)], \quad (2.4) $$

$$ (L_N^{\text{reac}} f)(\eta, \xi) = \alpha \sum_{i \in \Gamma_N^{\text{low}}} (\eta(i) - \xi(i))^2 [f(\eta, \xi^i) - f(\eta, \xi)], \quad (2.5) $$

$$ (L_N^{\text{up}} f)(\eta, \xi) = \sum_{i \in \Gamma_N^{\text{up}}} (b(1 - \eta(i)) + (1 - b)\eta(i)) [f(\eta^i, \xi) - f(\eta, \xi)]. \quad (2.6) $$

In (2.2) and (2.3), we use $|\cdot|$ to denote the infinity norm in $\mathbb{R}^p$, that is

$$ |\hat{i}| = \max(|i_1|, \dots, |i_{p-1}|, |j|), \quad \forall \hat{i} \in \Lambda_N. $$

Also, we highlight that the flip rate $\alpha(\eta(i) - \xi(i))^2$ in (2.4) and (2.5) arises from equality

$$ (1 - \eta(i)) \xi(i) + \eta(i) (1 - \xi(i)) = (\eta(i) - \xi(i))^2, $$

which holds since both $\eta(i)$ and $\xi(i)$ belong to $\{0, 1\}$.

The parts (2.2), (2.3) and (2.6) are rather classical, see [34], [4], [42] for instance. On the other hand, the parts (2.4) and (2.5) are original, their role being to catch the exchange condition in the field-road model. We refer to [28] and [30] for related issues.

For a given time horizon $T > 0$, we denote $(\eta_t, \xi_t)_{t \in [0, T]}$ the Markov process with state space $S_N$ associated to the generator $\mathcal{L}_N$. We define $D([0, T]; S_N)$ as the path space for càdlàg time trajectories valued in $S_N$. Given a measure $\mu_N$ on $S_N$, we denote by $\mathbb{P}_N^{\mu_N}$ the probability measure on $D([0, T]; S_N)$ induced by $\mu_N$ and $(\eta_t, \xi_t)_{t \in [0, T]}$, and we write $\mathbb{E}_N^{\mu_N}$ the expectation with respect to $\mathbb{P}_N^{\mu_N}$. Moreover, the notation $\langle \cdot, \cdot \rangle_{\mu_N}$ refers to the scalar product on $L_N^2(S_N)$.

## 2.3 Functional spaces and macroscopic equations

For any integers $n$ and $m$, we define the functional spaces

$$ \mathcal{C}^{n,m}([0, T] \times \bar{\Lambda}) \quad \text{and} \quad \mathcal{C}^{n,m}([0, T] \times \mathbb{T}^{p-1}) $$

which respectively consist of functions

$$ G = G(t, \hat{x}) : [0, T] \times \bar{\Lambda} \to \mathbb{R} \quad \text{and} \quad H = H(t, x) : [0, T] \times \mathbb{T}^{p-1} \to \mathbb{R}, $$

that possess $n$ continuous derivatives with respect to the time variable on $[0, T]$, and $m$ continuous derivatives with respect to the spatial variable on $\bar{\Lambda}$ and $\mathbb{T}^{p-1}$ respectively. We also introduce the subset $\mathcal{C}_c^{n,m}([0, T] \times \bar{\Lambda})$ of functions with compact support in $[0, T] \times \Lambda$
within $C^{n,m}([0, T] \times \bar{\Lambda})$. Similarly, we denote $C^m(\bar{\Lambda})$ and $C^m(\mathbb{T}^{p-1})$ the sets of functions with $m$ continuous derivatives on $\bar{\Lambda}$ and $\mathbb{T}^{p-1}$ respectively.

In the whole document, if $\varphi$ is a function that depends both on the time and the spatial variables, the abbreviation $\varphi(t)$ obviously stands for $\varphi(t, \cdot)$.

In the sequel, the notations $\langle \cdot, \cdot \rangle_{\Lambda}$ and $\langle \cdot, \cdot \rangle_{\mathbb{T}^{p-1}}$ respectively represent the $L^2(\Lambda)$ and $L^2(\mathbb{T}^{p-1})$ inner products.

We consider the Sobolev space $\mathcal{H}^1(\Lambda)$ as the set of functions $g$ in $L^2(\Lambda)$ such that for every $q$ ranging from 1 to $p$, there is an element $\partial_{e_q} g$ in $L^2(\Lambda)$, for which we have

$$ \langle \partial_{e_q} G, g \rangle_{\Lambda} = - \langle G, \partial_{e_q} g \rangle_{\Lambda}, \quad \forall G \in \mathcal{C}_{c}^{\infty}(\Lambda), $$

where $\partial_{e_q} \varphi$ denotes the derivative with respect to the $q$th canonical vector of $\mathbb{R}^p$. We then define the norms on the Sobolev space $\mathcal{H}^1(\Lambda)$ by

$$ \|g\|_{\mathcal{H}^1(\Lambda)} = \left( \|g\|_{L^2(\Lambda)}^2 + \sum_{q=1}^{p} \|\partial_{e_q} g\|_{L^2(\Lambda)}^2 \right)^{1/2}. $$

With respect to a given Banach space $B$ (most of the time, $B = \mathcal{H}^1(\Lambda)$ or $B = L^2(\mathbb{T}^{p-1})$), we define $L^2(0, T; B)$ as the function space composed of maps $\varphi : [0, T] \to B$ satisfying

$$ \int_{0}^{T} \|\varphi(t)\|_{B}^{2} dt < +\infty. $$

In order to define the value of an element $g$ in $\mathcal{H}^1(\Lambda)$ at the boundaries $\Gamma = \Gamma^{\text{up}} \cup \Gamma^{\text{low}}$, we need to introduce the notion of trace. The trace operator on the space $\mathcal{H}^1(\Lambda)$ can be defined as the bounded linear operator

$$ \text{Tr} : \mathcal{H}^1(\Lambda) \to L^2(\Gamma) $$

such that Tr extends the classical trace, that is

$$ \text{Tr}(g) = g|_{\Gamma}, \quad \forall g \in \mathcal{H}^1(\Lambda) \cap \mathcal{C}^0(\bar{\Lambda}). $$

We refer to [24, Part II, Section 5] for a detailed survey of the trace operator. In the sequel, for any point $\hat{x} = (x, y)$ in $\Gamma$ and any function $g$ belonging to the space $L^2(0, T; \mathcal{H}^1(\Lambda))$, all the expressions $g(t, \hat{x})$, $g(t, x)$ or $g(t, \cdot)|_{\Gamma}(x)$ represent the trace operator applied to $g(t, \cdot)$ at position $\hat{x} \in \Gamma$. Additionally, observe that $\partial_y g(t, \cdot)$ (resp. $-\partial_y g(t, \cdot)$) stands for the normal derivative of the function $g(t, \cdot)$ on the boundary $\Gamma^{\text{up}}$ (resp. $\Gamma^{\text{low}}$).

We are now in a position to introduce our notion of weak solution for the field-road problem (1.2)-(1.3).

**Definition 2.1** (Solving the field-road system) Fix a time horizon $T > 0$. For any measurable initial data $v_0 : \Lambda \to [0, 1]$ and $u_0 : \mathbb{T}^{p-1} \to [0, 1]$, a couple of functions $(v, u)$ is said to be a weak solution to the initial value problem (1.2)-(1.3) as soon as the following two conditions (W1)-(W2) hold true:

$$ (\textbf{W1}) \quad v \in L^2(0, T; \mathcal{H}^1(\Lambda)) \text{ and } u \in L^2(0, T; L^2(\mathbb{T}^{p-1})). $$
within $C^{n,m}([0, T] \times \bar{\Lambda})$. Similarly, we denote $C^m(\bar{\Lambda})$ and $C^m(\mathbb{T}^{p-1})$ the sets of functions with $m$ continuous derivatives on $\bar{\Lambda}$ and $\mathbb{T}^{p-1}$ respectively.

In the whole document, if $\varphi$ is a function that depends both on the time and the spatial variables, the abbreviation $\varphi(t)$ obviously stands for $\varphi(t, \cdot)$.

In the sequel, the notations $\langle \cdot, \cdot \rangle_{\Lambda}$ and $\langle \cdot, \cdot \rangle_{\mathbb{T}^{p-1}}$ respectively represent the $L^2(\Lambda)$ and $L^2(\mathbb{T}^{p-1})$ inner products.

We consider the Sobolev space $\mathcal{H}^1(\Lambda)$ as the set of functions $g$ in $L^2(\Lambda)$ such that for every $q$ ranging from 1 to $p$, there is an element $\partial_{e_q} g$ in $L^2(\Lambda)$, for which we have

$$ \langle \partial_{e_q} G, g \rangle_{\Lambda} = - \langle G, \partial_{e_q} g \rangle_{\Lambda}, \quad \forall G \in \mathcal{C}_{c}^{\infty}(\Lambda), $$

where $\partial_{e_q} \varphi$ denotes the derivative with respect to the $q$th canonical vector of $\mathbb{R}^p$. We then define the norms on the Sobolev space $\mathcal{H}^1(\Lambda)$ by

$$ \|g\|_{\mathcal{H}^1(\Lambda)} = \left( \|g\|_{L^2(\Lambda)}^2 + \sum_{q=1}^{p} \|\partial_{e_q} g\|_{L^2(\Lambda)}^2 \right)^{1/2}. $$

With respect to a given Banach space $B$ (most of the time, $B = \mathcal{H}^1(\Lambda)$ or $B = L^2(\mathbb{T}^{p-1})$), we define $L^2(0, T; B)$ as the function space composed of maps $\varphi : [0, T] \to B$ satisfying

$$ \int_{0}^{T} \|\varphi(t)\|_{B}^{2} dt < +\infty. $$

In order to define the value of an element $g$ in $\mathcal{H}^1(\Lambda)$ at the boundaries $\Gamma = \Gamma^{\text{up}} \cup \Gamma^{\text{low}}$, we need to introduce the notion of trace. The trace operator on the space $\mathcal{H}^1(\Lambda)$ can be defined as the bounded linear operator

$$ \text{Tr} : \mathcal{H}^1(\Lambda) \to L^2(\Gamma) $$

such that Tr extends the classical trace, that is

$$ \text{Tr}(g) = g|_{\Gamma}, \quad \forall g \in \mathcal{H}^1(\Lambda) \cap \mathcal{C}^0(\bar{\Lambda}). $$

We refer to [24, Part II, Section 5] for a detailed survey of the trace operator. In the sequel, for any point $\hat{x} = (x, y)$ in $\Gamma$ and any function $g$ belonging to the space $L^2(0, T; \mathcal{H}^1(\Lambda))$, all the expressions $g(t, \hat{x})$, $g(t, x)$ or $g(t, \cdot)|_{\Gamma}(x)$ represent the trace operator applied to $g(t, \cdot)$ at position $\hat{x} \in \Gamma$. Additionally, observe that $\partial_y g(t, \cdot)$ (resp. $-\partial_y g(t, \cdot)$) stands for the normal derivative of the function $g(t, \cdot)$ on the boundary $\Gamma^{\text{up}}$ (resp. $\Gamma^{\text{low}}$).

We are now in a position to introduce our notion of weak solution for the field-road problem (1.2)-(1.3).

**Definition 2.1** (Solving the field-road system) Fix a time horizon $T > 0$. For any measurable initial data $v_0 : \Lambda \to [0, 1]$ and $u_0 : \mathbb{T}^{p-1} \to [0, 1]$, a couple of functions $(v, u)$ is said to be a weak solution to the initial value problem (1.2)-(1.3) as soon as the following two conditions (W1)-(W2) hold true:

$$ (\textbf{W1}) \quad v \in L^2(0, T; \mathcal{H}^1(\Lambda)) \text{ and } u \in L^2(0, T; L^2(\mathbb{T}^{p-1})). $$
(W2) For any $G \in \mathcal{C}^{1,2}([0, T] \times \bar{\Lambda})$, any $H \in \mathcal{C}^{1,2}([0, T] \times \mathbb{T}^{p-1})$, any $t \in [0, T]$, there holds

$$ \begin{aligned} \langle v(t), G(t) \rangle_{\Lambda} - \langle v_0, G(0) \rangle_{\Lambda} = & \int_{0}^{t} \langle v(s), \partial_{s} G(s) \rangle_{\Lambda} ds + \int_{0}^{t} \langle v(s), d\Delta G(s) \rangle_{\Lambda} ds \\ & - \int_{0}^{t} \langle v|_{y=1}(s), d\partial_{y} G|_{y=1}(s) \rangle_{\mathbb{T}^{p-1}} ds + \int_{0}^{t} \langle v|_{y=0}(s), d\partial_{y} G|_{y=0}(s) \rangle_{\mathbb{T}^{p-1}} ds \\ & + \int_{0}^{t} \alpha \langle u(s) - v|_{y=0}(s), G|_{y=0}(s) \rangle_{\mathbb{T}^{p-1}} ds, \end{aligned} \quad (2.7) $$

together with

$$ \begin{aligned} \langle u(t), H(t) \rangle_{\mathbb{T}^{p-1}} - \langle u_0, H(0) \rangle_{\mathbb{T}^{p-1}} = & \int_{0}^{t} \langle u(s), \partial_{s} H(s) \rangle_{\mathbb{T}^{p-1}} ds + \int_{0}^{t} \langle u(s), D\Delta_{x} H(s) \rangle_{\mathbb{T}^{p-1}} ds \\ & + \int_{0}^{t} \alpha \langle v|_{y=0}(s) - u(s), H(s) \rangle_{\mathbb{T}^{p-1}} ds. \end{aligned} \quad (2.8) $$

## 2.4 Main Results

We use the notations $\mathcal{M}^{\text{field}}$ and $\mathcal{M}^{\text{road}}$ to denote the sets of positive measures on $\Lambda$ and $\Gamma^{\text{low}}$ whose total mass is bounded by 1, and we define the space $\mathcal{M}$ as the Cartesian product $\mathcal{M}^{\text{field}} \times \mathcal{M}^{\text{road}}$. We denote the integrals of functions against measures indifferently for $\Lambda$ or $\Gamma^{\text{low}}$, namely, for any measures $\mu \in \mathcal{M}^{\text{field}}$ and $\nu \in \mathcal{M}^{\text{road}}$, and any functions $G \in L_{\mu}^{1}(\Lambda)$ and $H \in L_{\mu}^{1}(\mathbb{T}^{p-1})$,

$$ \langle \mu, G \rangle = \int_{\Lambda} G(\hat{x}) \mu(d\hat{x}) \quad \text{and} \quad \langle \mu, H \rangle = \int_{\mathbb{T}^{p-1}} H(x) \nu(dx). $$

We endow $\mathcal{M}$, $\mathcal{M}^{\text{field}}$ and $\mathcal{M}^{\text{road}}$ with a topology induced by the weak convergence of measures. It is worth mentioning that all these spaces are compact and Polish.

The empirical measure of a configuration $(\eta, \xi) \in S_N$ is defined as $\pi_N(\eta, \xi)$, where the map $\pi_N: S_N \to \mathcal{M}$ is given by

$$ \pi_N(\eta, \xi) := \underbrace{\left( \frac{1}{N^p} \sum_{\hat{i} \in \Lambda_N} \eta(\hat{i}) \delta_{\hat{i}/N} \right)}_{=:\pi_N^{\text{field}}(\eta)}, \quad \underbrace{\left( \frac{1}{N^{p-1}} \sum_{i \in \Gamma_N^{\text{low}}} \xi(i) \delta_{i/N} \right)}_{=:\pi_N^{\text{road}}(\xi)}. \quad (2.9) $$

In (2.9), the notation $\delta_{\hat{i}/N}$ (resp. $\delta_{i/N}$) stands for the Dirac mass at place $\hat{i}/N$ (resp. $i/N$). For any configuration $(\eta, \xi) \in S_N$, any $G \in \mathcal{C}^0(\bar{\Lambda})$ and any $H \in \mathcal{C}^0(\mathbb{T}^{p-1})$, we denote

$$ \langle \pi_N(\eta, \xi), [G, H] \rangle := \langle \pi_N^{\text{field}}(\eta), G \rangle + \langle \pi_N^{\text{road}}(\xi), H \rangle. $$

We introduce $(\pi_N(t))_{t \in [0, T]} := (\pi_N(\eta_t, \xi_t))_{t \in [0, T]}$, the Markov process on the state space $\mathcal{M}$ induced from $(\eta_t, \xi_t)_{t \in [0, T]}$. The trajectories of this process occupy $D([0, T]; \mathcal{M})$, the designated path space for càdlàg time trajectories valued in $\mathcal{M}$. We endow $D([0, T]; \mathcal{M})$ with the Skorokhod topology. For further details regarding this topology, we refer the reader to [14], which provides an extensive survey on this subject. For $G \in \mathcal{C}^{0,0}([0, T] \times \bar{\Lambda})$, $H \in \mathcal{C}^{0,0}([0, T] \times \mathbb{T}^{p-1})$, and $t \in [0, T]$, we denote

$$ \begin{aligned} \langle \pi_N(t), [G(t), H(t)] \rangle &:= \langle \pi_N^{\text{field}}(t), G(t) \rangle + \langle \pi_N^{\text{road}}(t), H(t) \rangle \\ &= \langle \pi_N^{\text{field}}(\eta_t), G(t) \rangle + \langle \pi_N^{\text{road}}(\xi_t), H(t) \rangle. \end{aligned} $$
(W2) For any $G \in \mathcal{C}^{1,2}([0, T] \times \bar{\Lambda})$, any $H \in \mathcal{C}^{1,2}([0, T] \times \mathbb{T}^{p-1})$, any $t \in [0, T]$, there holds

$$ \begin{aligned} \langle v(t), G(t) \rangle_{\Lambda} - \langle v_0, G(0) \rangle_{\Lambda} = & \int_{0}^{t} \langle v(s), \partial_{s} G(s) \rangle_{\Lambda} ds + \int_{0}^{t} \langle v(s), d\Delta G(s) \rangle_{\Lambda} ds \\ & - \int_{0}^{t} \langle v|_{y=1}(s), d\partial_{y} G|_{y=1}(s) \rangle_{\mathbb{T}^{p-1}} ds + \int_{0}^{t} \langle v|_{y=0}(s), d\partial_{y} G|_{y=0}(s) \rangle_{\mathbb{T}^{p-1}} ds \\ & + \int_{0}^{t} \alpha \langle u(s) - v|_{y=0}(s), G|_{y=0}(s) \rangle_{\mathbb{T}^{p-1}} ds, \end{aligned} \quad (2.7) $$

together with

$$ \begin{aligned} \langle u(t), H(t) \rangle_{\mathbb{T}^{p-1}} - \langle u_0, H(0) \rangle_{\mathbb{T}^{p-1}} = & \int_{0}^{t} \langle u(s), \partial_s H(s) \rangle_{\mathbb{T}^{p-1}} ds + \int_{0}^{t} \langle u(s), D\Delta_x H(s) \rangle_{\mathbb{T}^{p-1}} ds \\ & + \int_{0}^{t} \alpha \langle v|_{y=0}(s) - u(s), H(s) \rangle_{\mathbb{T}^{p-1}} ds. \end{aligned} \quad (2.8) $$

## 2.4 Main Results

We use the notations $\mathcal{M}^{\text{field}}$ and $\mathcal{M}^{\text{road}}$ to denote the sets of positive measures on $\Lambda$ and $\Gamma^{\text{low}}$ whose total mass is bounded by 1, and we define the space $\mathcal{M}$ as the Cartesian product $\mathcal{M}^{\text{field}} \times \mathcal{M}^{\text{road}}$. We denote the integrals of functions against measures indifferently for $\Lambda$ or $\Gamma^{\text{low}}$, namely, for any measures $\mu \in \mathcal{M}^{\text{field}}$ and $\nu \in \mathcal{M}^{\text{road}}$, and any functions $G \in L_{\mu}^{1}(\Lambda)$ and $H \in L_{\mu}^{1}(\mathbb{T}^{p-1})$,

$$ \langle \mu, G \rangle = \int_{\Lambda} G(\hat{x}) \mu(d\hat{x}) \quad \text{and} \quad \langle \mu, H \rangle = \int_{\mathbb{T}^{p-1}} H(x) \nu(dx). $$

We endow $\mathcal{M}$, $\mathcal{M}^{\text{field}}$ and $\mathcal{M}^{\text{road}}$ with a topology induced by the weak convergence of measures. It is worth mentioning that all these spaces are compact and Polish.

The empirical measure of a configuration $(\eta, \xi) \in S_N$ is defined as $\pi_N(\eta, \xi)$, where the map $\pi_N : S_N \to \mathcal{M}$ is given by

$$ \pi_N(\eta, \xi) := \underbrace{\left( \frac{1}{N^p} \sum_{\hat{i} \in \Lambda_N} \eta(\hat{i}) \delta_{\hat{i}/N} \right)}_{=:\pi_N^{\text{field}}(\eta)} , \quad \underbrace{\left( \frac{1}{N^{p-1}} \sum_{i \in \Gamma_N^{\text{low}}} \xi(i) \delta_{i/N} \right)}_{=:\pi_N^{\text{road}}(\xi)} . \quad (2.9) $$

In (2.9), the notation $\delta_{\hat{i}/N}$ (resp. $\delta_{i/N}$) stands for the Dirac mass at place $\hat{i}/N$ (resp. $i/N$). For any configuration $(\eta, \xi) \in S_N$, any $G \in \mathcal{C}^0(\bar{\Lambda})$ and any $H \in \mathcal{C}^0(\mathbb{T}^{p-1})$, we denote

$$ \langle \pi_N(\eta, \xi), [G, H] \rangle := \langle \pi_N^{\text{field}}(\eta), G \rangle + \langle \pi_N^{\text{road}}(\xi), H \rangle. $$

We introduce $(\pi_N(t))_{t \in [0, T]} := (\pi_N(\eta_t, \xi_t))_{t \in [0, T]}$, the Markov process on the state space $\mathcal{M}$ induced from $(\eta_t, \xi_t)_{t \in [0, T]}$. The trajectories of this process occupy $D([0, T] ; \mathcal{M})$, the designated path space for càdlàg time trajectories valued in $\mathcal{M}$. We endow $D([0, T] ; \mathcal{M})$ with the Skorokhod topology. For further details regarding this topology, we refer the reader to [14], which provides an extensive survey on this subject. For $G \in \mathcal{C}^{0,0}([0, T] \times \bar{\Lambda})$, $H \in \mathcal{C}^{0,0}([0, T] \times \mathbb{T}^{p-1})$, and $t \in [0, T]$, we denote

$$ \begin{aligned} \langle \pi_N(t), [G(t), H(t)] \rangle &:= \langle \pi_N^{\text{field}}(t), G(t) \rangle + \langle \pi_N^{\text{road}}(t), H(t) \rangle \\ &= \langle \pi_N^{\text{field}}(\eta_t), G(t) \rangle + \langle \pi_N^{\text{road}}(\xi_t), H(t) \rangle. \end{aligned} $$
Given an initial probability measure $\mu_N$ on $S_N$, we define, for $N \ge 2$, the probability measure $\mathbb{Q}_N^{\mu_N} := \mathbb{P}_N^{\mu_N}(\pi_N^{-1})$ on the set of measures $\mathcal{M}$, as the law of the Markov process

$$ (\pi_N(t))_{t \in [0,T]} = (\pi_N(\eta_t, \xi_t))_{t \in [0,T]}. $$

Essentially, $\mathbb{Q}_N^{\mu_N}$ allows to provide a description through measures on the macroscopic space of the state distribution of the process when initiated from the measure $\mu_N$.

**Definition 2.2** (Sequence of measures associated with the initial data) Let

$$ v_0 : \Lambda \to [0, 1] \quad \text{and} \quad u_0 : \mathbb{T}^{p-1} \to [0, 1] $$

be two measurable functions. We say that a sequence of probability measures

$$ (\mu_N)_{N \ge 2} = (\mu_N^{\text{field}}, \mu_N^{\text{road}})_{N \ge 2} $$

on $S_N = S_N^{\text{field}} \times S_N^{\text{road}}$ is associated with $(v_0, u_0)$ if

$$ \lim_{N \to \infty} \mu_N^{\text{field}} \left[ \eta \in S_N^{\text{field}} : \left| \langle \pi_N^{\text{field}}(\eta), G \rangle - \langle v_0, G \rangle_{\Lambda} \right| \ge \delta \right] = 0, \quad (2.10) $$

and

$$ \lim_{N \to \infty} \mu_N^{\text{road}} \left[ \xi \in S_N^{\text{road}} : \left| \langle \pi_N^{\text{road}}(\xi), H \rangle - \langle u_0, H \rangle_{\Gamma^{\text{low}}} \right| \ge \delta \right] = 0, \quad (2.11) $$

for any $\delta > 0$ and any $G \in \mathcal{C}^0(\bar{\Lambda})$ and $H \in \mathcal{C}^0(\mathbb{T}^{p-1})$.

**Remark 2.3** Observe in Definition 2.2 that we ask $v_0$ and $u_0$ to be valued in $[0, 1]$. This condition cannot be relaxed because of the exclusion rule, that enforces the sites to host at most one particle. To be convinced with this, consider the case $u_0 \equiv 2$, and take $H \equiv 1$. We then have $\langle u_0, H \rangle_{\Gamma^{\text{low}}} = 2$, and

$$ \langle \pi_N^{\text{road}}(\xi), H \rangle = \frac{1}{N^{p-1}} \sum_{i \in \Gamma_N^{\text{low}}} \xi(i) \le \frac{1}{N^{p-1}} \sum_{i \in \Gamma_N^{\text{low}}} 1 = 1, $$

so that no configuration allows the limit (2.11) to hold, and the data $u_0 \equiv 2$ is unreachable.

Here is the main contribution of the present work.

**Theorem 2.4** (Hydrodynamic limit) Fix a time horizon $T > 0$. Let $v_0 : \Lambda \to [0, 1]$ and $u_0 : \mathbb{T}^{p-1} \to [0, 1]$ be two measurable functions, and $(\mu_N)_{N \ge 2}$ a sequence of initial probability measures on $S_N$ associated with $(v_0, u_0)$ in the sense of Definition 2.2. Then the sequence of probability measures $(\mathbb{Q}_N^{\mu_N})_{N \ge 2}$ converges weakly towards some $\mathbb{Q}_\infty$ which gives mass 1 to the path

$$ \left( \pi^{\text{field}}(t, dxdy), \pi^{\text{road}}(t, dx) \right)_{t \in [0,T]} = \left( v(t, x, y)dxdy, u(t, x)dx \right)_{t \in [0,T]}, $$

where $(v, u)$ is the unique weak solution to the Cauchy problem (1.2)-(1.3) in the sense of Definition 2.1. In particular, for any $t \in [0, T]$, any $\delta > 0$, and any test functions

$$ G \in \mathcal{C}^{1,2}([0, T] \times \bar{\Lambda}) \quad \text{and} \quad H \in \mathcal{C}^{1,2}([0, T] \times \Gamma^{\text{low}}), $$

we have

$$ \lim_{N \to \infty} \mathbb{P}_N^{\mu_N} \left[ \eta_t \in S_N^{\text{field}} : \left| \langle \pi_N^{\text{field}}(t), G(t) \rangle - \langle v(t), G(t) \rangle_{\Lambda} \right| > \delta \right] = 0, $$

and

$$ \lim_{N \to \infty} \mathbb{P}_N^{\mu_N} \left[ \xi_t \in S_N^{\text{road}} : \left| \langle \pi_N^{\text{road}}(t), H(t) \rangle - \langle u(t), H(t) \rangle_{\Gamma^{\text{low}}} \right| > \delta \right] = 0. $$
Given an initial probability measure $\mu_N$ on $S_N$, we define, for $N \ge 2$, the probability measure $\mathbb{Q}_N^{\mu_N} := \mathbb{P}_N^{\mu_N}(\pi_N^{-1})$ on the set of measures $\mathcal{M}$, as the law of the Markov process

$$ (\pi_N(t))_{t \in [0,T]} = (\pi_N(\eta_t, \xi_t))_{t \in [0,T]}. $$

Essentially, $\mathbb{Q}_N^{\mu_N}$ allows to provide a description through measures on the macroscopic space of the state distribution of the process when initiated from the measure $\mu_N$.

**Definition 2.2** (Sequence of measures associated with the initial data) Let

$$ v_0 : \Lambda \to [0, 1] \quad \text{and} \quad u_0 : \mathbb{T}^{p-1} \to [0, 1] $$

be two measurable functions. We say that a sequence of probability measures

$$ (\mu_N)_{N \ge 2} = (\mu_N^{\text{field}}, \mu_N^{\text{road}})_{N \ge 2} $$

on $S_N = S_N^{\text{field}} \times S_N^{\text{road}}$ is associated with $(v_0, u_0)$ if

$$ \lim_{N \to \infty} \mu_N^{\text{field}} \left[ \eta \in S_N^{\text{field}} : \left| \langle \pi_N^{\text{field}}(\eta), G \rangle - \langle v_0, G \rangle_{\Lambda} \right| \ge \delta \right] = 0, \quad (2.10) $$

and

$$ \lim_{N \to \infty} \mu_N^{\text{road}} \left[ \xi \in S_N^{\text{road}} : \left| \langle \pi_N^{\text{road}}(\xi), H \rangle - \langle u_0, H \rangle_{\Gamma^{\text{low}}} \right| \ge \delta \right] = 0, \quad (2.11) $$

for any $\delta > 0$ and any $G \in \mathcal{C}^0(\bar{\Lambda})$ and $H \in \mathcal{C}^0(\mathbb{T}^{p-1})$.

**Remark 2.3** Observe in Definition 2.2 that we ask $v_0$ and $u_0$ to be valued in $[0, 1]$. This condition cannot be relaxed because of the exclusion rule, that enforces the sites to host at most one particle. To be convinced with this, consider the case $u_0 \equiv 2$, and take $H \equiv 1$. We then have $\langle u_0, H \rangle_{\Gamma^{\text{low}}} = 2$, and

$$ \langle \pi_N^{\text{road}}(\xi), H \rangle = \frac{1}{N^{p-1}} \sum_{i \in \Gamma_N^{\text{low}}} \xi(i) \le \frac{1}{N^{p-1}} \sum_{i \in \Gamma_N^{\text{low}}} 1 = 1, $$

so that no configuration allows the limit (2.11) to hold, and the data $u_0 \equiv 2$ is unreachable.

Here is the main contribution of the present work.

**Theorem 2.4** (Hydrodynamic limit) Fix a time horizon $T > 0$. Let $v_0 : \Lambda \to [0, 1]$ and $u_0 : \mathbb{T}^{p-1} \to [0, 1]$ be two measurable functions, and $(\mu_N)_{N \ge 2}$ a sequence of initial probability measures on $S_N$ associated with $(v_0, u_0)$ in the sense of Definition 2.2. Then the sequence of probability measures $(\mathbb{Q}_N^{\mu_N})_{N \ge 2}$ converges weakly towards some $\mathbb{Q}_\infty$ which gives mass 1 to the path

$$ \left( \pi^{\text{field}}(t, dxdy), \pi^{\text{road}}(t, dx) \right)_{t \in [0,T]} = \left( v(t, x, y)dxdy, u(t, x)dx \right)_{t \in [0,T]}, $$

where $(v, u)$ is the unique weak solution to the Cauchy problem (1.2)-(1.3) in the sense of Definition 2.1. In particular, for any $t \in [0, T]$, any $\delta > 0$, and any test functions

$$ G \in \mathcal{C}^{1,2}([0, T] \times \bar{\Lambda}) \quad \text{and} \quad H \in \mathcal{C}^{1,2}([0, T] \times \Gamma^{\text{low}}), $$

we have

$$ \lim_{N \to \infty} \mathbb{P}_N^{\mu_N} \left[ \eta_t \in S_N^{\text{field}} : \left| \langle \pi_N^{\text{field}}(t), G(t) \rangle - \langle v(t), G(t) \rangle_{\Lambda} \right| > \delta \right] = 0, $$

and

$$ \lim_{N \to \infty} \mathbb{P}_N^{\mu_N} \left[ \xi_t \in S_N^{\text{road}} : \left| \langle \pi_N^{\text{road}}(t), H(t) \rangle - \langle u(t), H(t) \rangle_{\Gamma^{\text{low}}} \right| > \delta \right] = 0. $$
## 2.5 The steps to prove Theorem 2.4 and organisation of the paper

The proof of the hydrodynamic limit, as outlined in Theorem 2.4, is inspired by works such as those by Kipnis and Landim [34] or Baldasso et al. [4], and unfolds in three distinct steps. The first one is to prove the tightness of the probability measures $(Q_N^{\mu_N})_{N \ge 2}$ within the Skorokhod topology. This point, established in Section 3, is crucial to ensure the existence of accumulation points for this sequence. Following this, we characterize, in Section 4, the limit points $Q_\infty$ of the sequence $(Q_N^{\mu_N})_{N \ge 2}$. Specifically, it is demonstrated that every $Q_\infty$ concentrates on measure processes with density relative to the Lebesgue measure at every moment (subsection 4.1), and that the corresponding densities satisfy conditions (W1)-(W2) (subsections 4.2 and 4.4) that characterize our notion of solution. This proves that the density of the measures loaded by $Q_\infty$ are weak solutions to the Cauchy problem (1.2)-(1.3). The third and final step, performed in Section 5, consists in showing — thanks to very adequate test functions — that the Cauchy problem (1.2)-(1.3) admits a unique solution.

# 3 Martingales and tightness

## 3.1 Martingales

We now proceed to explain the martingales associated to our system. Fix a couple of functions $(G, H)$ with $G \in \mathcal{C}^{1,2}([0, T] \times \bar{\Lambda})$ and $H \in \mathcal{C}^{1,2}([0, T] \times \Gamma^{\text{low}})$, and consider the martingale $\mathcal{M}_N = \mathcal{M}_{N,G,H}$ with respect to the natural filtration $\sigma((\eta_s, \xi_s)_{0 \le s \le t})$ given for any $t \in [0, T]$ by the Dynkin's formula

$$ \mathcal{M}_N(t) := \langle \pi_N(t), [G(t), H(t)] \rangle - \langle \pi_N(0), [G(0), H(0)] \rangle - \int_0^t (\partial_s + \mathcal{L}_N) (\langle \pi_N(s), [G(s), H(s)] \rangle) ds. \quad (3.1) $$

Expanding the empirical measure $\pi_N$ with (2.9) in (3.1), we see that $\mathcal{M}_N$ can be split into $\mathcal{M}_{N,G}^{\text{field}} + \mathcal{M}_{N,H}^{\text{road}}$ where

$$ \begin{aligned} \mathcal{M}_{N,G}^{\text{field}}(t) &= \langle \pi_N^{\text{field}}(t), G(t) \rangle - \langle \pi_N^{\text{field}}(0), G(0) \rangle - \int_0^t [\partial_s + \mathcal{L}_N] [\langle \pi_N^{\text{field}}(s), G(s) \rangle] ds, \\ \mathcal{M}_{N,H}^{\text{road}}(t) &= \langle \pi_N^{\text{road}}(t), H(t) \rangle - \langle \pi_N^{\text{road}}(0), H(0) \rangle - \int_0^t [\partial_s + \mathcal{L}_N] [\langle \pi_N^{\text{road}}(s), H(s) \rangle] ds. \end{aligned} \quad (3.2) $$

The quadratic variation of $\mathcal{M}_N$ is given by the martingale $\mathcal{N}_N = \mathcal{N}_{N,G,H}$ defined for any $t \in [0, T]$ as

$$ \mathcal{N}_N(t) := [\mathcal{M}_N(t)]^2 - \int_0^t \mathcal{B}_N(s) ds, \quad (3.3) $$

where

$$ \mathcal{B}_N(s) := \mathcal{L}_N [\langle \pi_N(s), [G(s), H(s)] \rangle^2] - 2 \langle \pi_N(s), [G(s), H(s)] \rangle \times \mathcal{L}_N [\langle \pi_N(s), [G(s), H(s)] \rangle]. \quad (3.4) $$

A proof that $\mathcal{M}_N$ and $\mathcal{N}_N$ are martingales with respect to the natural filtration is classical and follows the very same lines as [34, Appendix 1, Section 5].

In the integral terms of $\mathcal{M}_{N,G}^{\text{field}}$ and $\mathcal{M}_{N,H}^{\text{road}}$ in (3.2) we expand the expressions of $\pi_N^{\text{field}}$, $\pi_N^{\text{road}}$ (2.9), and $\mathcal{L}_N$ (2.1), and use at some point two discrete summations by parts. After
## 2.5 The steps to prove Theorem 2.4 and organisation of the paper

The proof of the hydrodynamic limit, as outlined in Theorem 2.4, is inspired by works such as those by Kipnis and Landim [34] or Baldasso et al. [4], and unfolds in three distinct steps. The first one is to prove the tightness of the probability measures $(Q_N^{\mu_N})_{N \ge 2}$ within the Skorokhod topology. This point, established in Section 3, is crucial to ensure the existence of accumulation points for this sequence. Following this, we characterize, in Section 4, the limit points $Q_\infty$ of the sequence $(Q_N^{\mu_N})_{N \ge 2}$. Specifically, it is demonstrated that every $Q_\infty$ concentrates on measure processes with density relative to the Lebesgue measure at every moment (subsection 4.1), and that the corresponding densities satisfy conditions (W1)-(W2) (subsections 4.2 and 4.4) that characterize our notion of solution. This proves that the density of the measures loaded by $Q_\infty$ are weak solutions to the Cauchy problem (1.2)-(1.3). The third and final step, performed in Section 5, consists in showing — thanks to very adequate test functions — that the Cauchy problem (1.2)-(1.3) admits a unique solution.

# 3 Martingales and tightness

## 3.1 Martingales

We now proceed to explain the martingales associated to our system. Fix a couple of functions $(G, H)$ with $G \in \mathcal{C}^{1,2}([0, T] \times \bar{\Lambda})$ and $H \in \mathcal{C}^{1,2}([0, T] \times \Gamma^{\text{low}})$, and consider the martingale $\mathcal{M}_N = \mathcal{M}_{N,G,H}$ with respect to the natural filtration $\sigma((\eta_s, \xi_s)_{0 \le s \le t})$ given for any $t \in [0, T]$ by the Dynkin's formula

$$ \mathcal{M}_N(t) := \langle \pi_N(t), [G(t), H(t)] \rangle - \langle \pi_N(0), [G(0), H(0)] \rangle - \int_0^t (\partial_s + \mathcal{L}_N) (\langle \pi_N(s), [G(s), H(s)] \rangle) ds. \quad (3.1) $$

Expanding the empirical measure $\pi_N$ with (2.9) in (3.1), we see that $\mathcal{M}_N$ can be split into $\mathcal{M}_{N,G}^{\text{field}} + \mathcal{M}_{N,H}^{\text{road}}$ where

$$ \begin{aligned} \mathcal{M}_{N,G}^{\text{field}}(t) &= \langle \pi_N^{\text{field}}(t), G(t) \rangle - \langle \pi_N^{\text{field}}(0), G(0) \rangle - \int_0^t [\partial_s + \mathcal{L}_N] [\langle \pi_N^{\text{field}}(s), G(s) \rangle] ds, \\ \mathcal{M}_{N,H}^{\text{road}}(t) &= \langle \pi_N^{\text{road}}(t), H(t) \rangle - \langle \pi_N^{\text{road}}(0), H(0) \rangle - \int_0^t [\partial_s + \mathcal{L}_N] [\langle \pi_N^{\text{road}}(s), H(s) \rangle] ds. \end{aligned} \quad (3.2) $$

The quadratic variation of $\mathcal{M}_N$ is given by the martingale $\mathcal{N}_N = \mathcal{N}_{N,G,H}$ defined for any $t \in [0, T]$ as

$$ \mathcal{N}_N(t) := [\mathcal{M}_N(t)]^2 - \int_0^t \mathcal{B}_N(s) ds, \quad (3.3) $$

where

$$ \mathcal{B}_N(s) := \mathcal{L}_N [\langle \pi_N(s), [G(s), H(s)] \rangle]^2 - 2 \langle \pi_N(s), [G(s), H(s)] \rangle \times \mathcal{L}_N [\langle \pi_N(s), [G(s), H(s)] \rangle]. \quad (3.4) $$

A proof that $\mathcal{M}_N$ and $\mathcal{N}_N$ are martingales with respect to the natural filtration is classical and follows the very same lines as [34, Appendix 1, Section 5].

In the integral terms of $\mathcal{M}_{N,G}^{\text{field}}$ and $\mathcal{M}_{N,H}^{\text{road}}$ in (3.2) we expand the expressions of $\pi_N^{\text{field}}$, $\pi_N^{\text{road}}$ (2.9), and $\mathcal{L}_N$ (2.1), and use at some point two discrete summations by parts. After
some tedious but straightforward computations — using moves such as $(\eta(i) - \xi(i))^2(1 - 2\eta(i)) = \xi(i) - \eta(i)$ or $(b(1 - \eta(i)) + (1 - b)\eta(i))(1 - 2\eta(i)) = b - \eta(i)$ — we reach

$$ \begin{aligned} \mathcal{M}_{N,G}^{\text{field}}(t) = & \langle \pi_N^{\text{field}}(t), G(t) \rangle - \langle \pi_N^{\text{field}}(0), G(0) \rangle - \int_0^t \langle \pi_N^{\text{field}}(s), \partial_s G(s) \rangle ds \\ & - \frac{1}{N^p} \int_0^t \sum_{\hat{i} \in \Lambda_N} d\Delta_x^N G(s, \frac{\hat{i}}{N}) \times \eta_s(\hat{i}) ds - \frac{1}{N^p} \int_0^t \sum_{\hat{i} \in \Lambda_N \setminus \Gamma_N} d\partial_{yy}^N G(s, \frac{\hat{i}}{N}) \times \eta_s(\hat{i}) ds \\ & + \frac{1}{N^{p-1}} \int_0^t \sum_{i \in \Gamma_N^{\text{up}}} d\partial_y^N G(s, \frac{i}{N}) \times \eta_s(i) ds - \frac{1}{N^{p-1}} \int_0^t \sum_{i \in \Gamma_N^{\text{low}}} d\partial_y^N G(s, \frac{i}{N}) \times \eta_s(i) ds \\ & - \frac{\alpha}{N^{p-1}} \int_0^t \sum_{i \in \Gamma_N^{\text{low}}} G(s, \frac{i}{N}) \times (\xi_s(i) - \eta_s(i)) ds - \frac{1}{N^p} \int_0^t \sum_{i \in \Gamma_N^{\text{up}}} G(s, \frac{i}{N}) \times (b - \eta_s(i)) ds, \end{aligned} \quad (3.5) $$

and

$$ \begin{aligned} \mathcal{M}_{N,H}^{\text{road}}(t) = & \langle \pi_N^{\text{road}}(t), H(t) \rangle - \langle \pi_N^{\text{road}}(0), H(0) \rangle - \int_0^t \langle \pi_N^{\text{road}}(s), \partial_s H(s) \rangle ds \\ & - \frac{1}{N^{p-1}} \int_0^t \sum_{i \in \Gamma_N^{\text{low}}} D \Delta_x^N H(s, \frac{i}{N}) \times \xi_s(i) ds - \frac{\alpha}{N^{p-1}} \int_0^t \sum_{i \in \Gamma_N^{\text{low}}} H(s, \frac{i}{N}) \times (\eta_s(i) - \xi_s(i)) ds. \end{aligned} \quad (3.6) $$

In (3.5) and (3.6), the discrete Laplacians $\Delta_x^N$ and $\partial_{yy}^N$ are defined by (time dependency is locally drop for the sake of clarity)

$$ \begin{aligned} \Delta_x^N H(\frac{i}{N}) &:= N^2 \sum_{q=1}^{p-1} \left[ H(\frac{i+e_q}{N}) - 2H(\frac{i}{N}) + H(\frac{i-e_q}{N}) \right], & \forall i \in \mathbb{T}_N^{p-1}, \\ \Delta_x^N G(\frac{i}{N}, \frac{j}{N}) &:= N^2 \sum_{q=1}^{p-1} \left[ G(\frac{i+e_q}{N}, \frac{j}{N}) - 2G(\frac{i}{N}, \frac{j}{N}) + G(\frac{i-e_q}{N}, \frac{j}{N}) \right], & \forall (i, j) \in \Lambda_N, \end{aligned} \quad (3.7) $$

$$ \partial_{yy}^N G(\frac{i}{N}, \frac{j}{N}) := N^2 \left[ G(\frac{i}{N}, \frac{j+1}{N}) - 2G(\frac{i}{N}, \frac{j}{N}) + G(\frac{i}{N}, \frac{j-1}{N}) \right], \quad \forall (i, j) \in \Lambda_N \setminus \Gamma_N, $$

and the discrete derivative $\partial_y^N$ at the boundary of the microscopic field by (time dependency is locally drop for the sake of clarity)

$$ \partial_y^N G(\frac{i}{N}, \frac{j}{N}) := \begin{cases} N \left[ G(\frac{i}{N}, \frac{j+1}{N}) - G(\frac{i}{N}, \frac{j}{N}) \right] & \text{if } j=1, \\ N \left[ G(\frac{i}{N}, \frac{j}{N}) - G(\frac{i}{N}, \frac{j-1}{N}) \right] & \text{if } j=N-1. \end{cases} \quad (3.8) $$

Similarly, we also develop $\mathcal{B}_N$ from (3.4) with the expression of $\mathcal{L}_N$ in (2.1). The computations show that $\mathcal{B}_N$ can be split into $\mathcal{B}_{N,G}^{\text{field}} + \mathcal{B}_{N,H}^{\text{road}}$ with

$$ \begin{aligned} \mathcal{B}_{N,G}^{\text{field}}(s) = & \frac{d}{N^{2p}} \sum_{\substack{\hat{i}, \hat{k} \in \Lambda_N \\ |\hat{i} - \hat{k}| = 1}} \left[ \eta_s(\hat{i}) - \eta_s(\hat{k}) \right]^2 \times \left[ N(G(s, \frac{\hat{i}}{N}) - G(s, \frac{\hat{k}}{N})) \right]^2 \\ & + \frac{\alpha}{N^{2p-1}} \sum_{i \in \Gamma_N^{\text{low}}} \left[ \eta_s(i) - \xi_s(i) \right]^2 \times \left[ G(s, \frac{i}{N}) \right]^2 \\ & + \frac{1}{N^{2p}} \sum_{i \in \Gamma_N^{\text{up}}} \left[ b(1 - \eta_s(i)) + (1 - b)\eta_s(i) \right] \times \left[ G(s, \frac{i}{N}) \right]^2 \end{aligned} \quad (3.9) $$
some tedious but straightforward computations — using moves such as $(\eta(i) - \xi(i))^2(1 - 2\eta(i)) = \xi(i) - \eta(i)$ or $(b(1 - \eta(i)) + (1 - b)\eta(i))(1 - 2\eta(i)) = b - \eta(i)$ — we reach

$$ \begin{aligned} \mathcal{M}_{N,G}^{\text{field}}(t) = & \langle \pi_N^{\text{field}}(t), G(t) \rangle - \langle \pi_N^{\text{field}}(0), G(0) \rangle - \int_0^t \langle \pi_N^{\text{field}}(s), \partial_s G(s) \rangle ds \\ & - \frac{1}{N^p} \int_0^t \sum_{\hat{i} \in \Lambda_N} d\Delta_x^N G(s, \frac{\hat{i}}{N}) \times \eta_s(\hat{i}) ds - \frac{1}{N^p} \int_0^t \sum_{\hat{i} \in \Lambda_N \setminus \Gamma_N} d\partial_{yy}^N G(s, \frac{\hat{i}}{N}) \times \eta_s(\hat{i}) ds \\ & + \frac{1}{N^{p-1}} \int_0^t \sum_{i \in \Gamma_N^{\text{up}}} d\partial_y^N G(s, \frac{i}{N}) \times \eta_s(i) ds - \frac{1}{N^{p-1}} \int_0^t \sum_{i \in \Gamma_N^{\text{low}}} d\partial_y^N G(s, \frac{i}{N}) \times \eta_s(i) ds \\ & - \frac{\alpha}{N^{p-1}} \int_0^t \sum_{i \in \Gamma_N^{\text{low}}} G(s, \frac{i}{N}) \times (\xi_s(i) - \eta_s(i)) ds - \frac{1}{N^p} \int_0^t \sum_{i \in \Gamma_N^{\text{up}}} G(s, \frac{i}{N}) \times (b - \eta_s(i)) ds, \end{aligned} \quad (3.5) $$

and

$$ \begin{aligned} \mathcal{M}_{N,H}^{\text{road}}(t) = & \langle \pi_N^{\text{road}}(t), H(t) \rangle - \langle \pi_N^{\text{road}}(0), H(0) \rangle - \int_0^t \langle \pi_N^{\text{road}}(s), \partial_s H(s) \rangle ds \\ & - \frac{1}{N^{p-1}} \int_0^t \sum_{i \in \Gamma_N^{\text{low}}} D \Delta_x^N H(s, \frac{i}{N}) \times \xi_s(i) ds - \frac{\alpha}{N^{p-1}} \int_0^t \sum_{i \in \Gamma_N^{\text{low}}} H(s, \frac{i}{N}) \times (\eta_s(i) - \xi_s(i)) ds. \end{aligned} \quad (3.6) $$

In (3.5) and (3.6), the discrete Laplacians $\Delta_x^N$ and $\partial_{yy}^N$ are defined by (time dependency is locally drop for the sake of clarity)

$$ \begin{aligned} \Delta_x^N H(\frac{i}{N}) &:= N^2 \sum_{q=1}^{p-1} \left[ H(\frac{i+e_q}{N}) - 2H(\frac{i}{N}) + H(\frac{i-e_q}{N}) \right], & \forall i \in \mathbb{T}_N^{p-1}, \\ \Delta_x^N G(\frac{i}{N}, \frac{j}{N}) &:= N^2 \sum_{q=1}^{p-1} \left[ G(\frac{i+e_q}{N}, \frac{j}{N}) - 2G(\frac{i}{N}, \frac{j}{N}) + G(\frac{i-e_q}{N}, \frac{j}{N}) \right], & \forall (i, j) \in \Lambda_N, \end{aligned} \quad (3.7) $$

$$ \partial_{yy}^N G(\frac{i}{N}, \frac{j}{N}) := N^2 \left[ G(\frac{i}{N}, \frac{j+1}{N}) - 2G(\frac{i}{N}, \frac{j}{N}) + G(\frac{i}{N}, \frac{j-1}{N}) \right], \quad \forall (i, j) \in \Lambda_N \setminus \Gamma_N, $$

and the discrete derivative $\partial_y^N$ at the boundary of the microscopic field by (time dependency is locally drop for the sake of clarity)

$$ \partial_y^N G(\frac{i}{N}, \frac{j}{N}) := \begin{cases} N \left[ G(\frac{i}{N}, \frac{j+1}{N}) - G(\frac{i}{N}, \frac{j}{N}) \right] & \text{if } j=1, \\ N \left[ G(\frac{i}{N}, \frac{j}{N}) - G(\frac{i}{N}, \frac{j-1}{N}) \right] & \text{if } j=N-1. \end{cases} \quad (3.8) $$

Similarly, we also develop $\mathcal{B}_N$ from (3.4) with the expression of $\mathcal{L}_N$ in (2.1). The computations show that $\mathcal{B}_N$ can be split into $\mathcal{B}_{N,G}^{\text{field}} + \mathcal{B}_{N,H}^{\text{road}}$ with

$$ \begin{aligned} \mathcal{B}_{N,G}^{\text{field}}(s) = & \frac{d}{N^{2p}} \sum_{\substack{\hat{i}, \hat{k} \in \Lambda_N \\ |\hat{i} - \hat{k}| = 1}} \left[ \eta_s(\hat{i}) - \eta_s(\hat{k}) \right]^2 \times \left[ N \left( G(s, \frac{\hat{i}}{N}) - G(s, \frac{\hat{k}}{N}) \right) \right]^2 \\ & + \frac{\alpha}{N^{2p-1}} \sum_{i \in \Gamma_N^{\text{low}}} \left[ \eta_s(i) - \xi_s(i) \right]^2 \times \left[ G(s, \frac{i}{N}) \right]^2 \\ & + \frac{1}{N^{2p}} \sum_{i \in \Gamma_N^{\text{up}}} \left[ b(1 - \eta_s(i)) + (1 - b)\eta_s(i) \right] \times \left[ G(s, \frac{i}{N}) \right]^2 \end{aligned} \quad (3.9) $$
and

$$ \mathcal{B}_{N,H}^{\text{road}}(s) = \frac{D/2}{N^{2(p-1)}} \sum_{\substack{i,k \in \Gamma_N^{\text{low}} \\ |i-k|=1}} \left[ \xi_s(i) - \xi_s(k) \right]^2 \times \left[ N(H(s, \frac{i}{N}) - H(s, \frac{k}{N})) \right]^2 + \frac{\alpha}{N^{2(p-1)}} \sum_{i \in \Gamma_N^{\text{low}}} \left[ \eta_s(i) - \xi_s(i) \right]^2 \times \left[ H(s, \frac{i}{N}) \right]^2. \quad (3.10) $$

## 3.2 Tightness

We are now in position to prove the tightness of $(\mathbb{Q}_N^{\mu_N})_{N \ge 2}$.

**Proposition 3.1 (Tightness)** For any sequence of initial measures $(\mu_N)_{N \ge 2}$ on $S_N$, the sequence $(\mathbb{Q}_N^{\mu_N})_{N \ge 2}$ is tight in the Skorokhod topology of $D([0, T]; \mathcal{M})$.

**Proof of Proposition 3.1.** In accordance with the method presented in [34, Chapter 4, Section 1], we must verify the following two statements for any $G \in C^2(\bar{\Lambda})$ and any $H \in C^2(\Gamma^{\text{low}})$:

(T1) For all $t \in [0, T]$ and all $\varepsilon > 0$, there is $M > 0$ such that

$$ \sup_{N \ge 2} \left\{ \mathbb{Q}_N^{\mu_N} \left( \left| \langle \pi_N(t), [G, H] \rangle \right| \ge M \right) \right\} \le \varepsilon. \quad (3.11) $$

(T2) For all $\varepsilon > 0$, we have

$$ \lim_{\delta \to 0} \limsup_{N \to \infty} \sup_{t \in [\delta, T-\delta]} \sup_{|\theta| \le \delta} \left\{ \mathbb{Q}_N^{\mu_N} \left( \left| \langle \pi_N(t+\theta), [G, H] \rangle - \langle \pi_N(t), [G, H] \rangle \right| \ge \varepsilon \right) \right\} = 0. \quad (3.12) $$

Note that, within this proof, we require the test functions $G$ and $H$ to depend solely on the spatial variable.

• *Proof of (T1).* Since the empirical measures $\pi_N^{\text{field}}$ and $\pi_N^{\text{road}}$ are both bounded by 1, we almost surely have

$$ \left| \langle \pi_N^{\text{field}}(t), G \rangle \right| \le \|G\|_{L^\infty(\Lambda)} \quad \text{and} \quad \left| \langle \pi_N^{\text{road}}(t), H \rangle \right| \le \|H\|_{L^\infty(\Gamma^{\text{low}})}. $$

As a result, (3.11) is achieved with $M = \varepsilon/(\|G\|_{L^\infty(\Lambda)} + \|H\|_{L^\infty(\Gamma^{\text{low}})})$.

• *Proof of (T2).* By expanding the terms $\langle \pi_N(t+\theta), [G, H] \rangle$ and $\langle \pi_N(t), [G, H] \rangle$ with those of the Dynkin's formula in (3.1) and using the Markov and the triangular inequalities, we can see that (3.12) holds if we prove the following limits:

$$ \lim_{\delta \to 0} \limsup_{N \to \infty} \sup_{t \in [\delta, T-\delta]} \sup_{|\theta| \le \delta} \left\{ \mathbb{E}_N^{\mu_N} \left( \left| \mathcal{M}_N(t+\theta) - \mathcal{M}_N(t) \right| \right) \right\} = 0, \quad (3.13) $$

$$ \lim_{\delta \to 0} \limsup_{N \to \infty} \sup_{t \in [\delta, T-\delta]} \sup_{|\theta| \le \delta} \left\{ \mathbb{E}_N^{\mu_N} \left( \left| \int_t^{t+\theta} \mathcal{L}_N \left[ \langle \pi_N^{\text{field}}(s), G \rangle \right] ds \right| \right) \right\} = 0, \quad (3.14) $$
and

$$ \mathcal{B}_{N,H}^{\text{road}}(s) = \frac{D/2}{N^{2(p-1)}} \sum_{\substack{i,k \in \Gamma_N^{\text{low}} \\ |i-k|=1}} \left[ \xi_s(i) - \xi_s(k) \right]^2 \times \left[ N(H(s, \frac{i}{N}) - H(s, \frac{k}{N})) \right]^2 + \frac{\alpha}{N^{2(p-1)}} \sum_{i \in \Gamma_N^{\text{low}}} \left[ \eta_s(i) - \xi_s(i) \right]^2 \times \left[ H(s, \frac{i}{N}) \right]^2. \quad (3.10) $$

## 3.2 Tightness

We are now in position to prove the tightness of $(\mathbb{Q}_N^{\mu_N})_{N \ge 2}$.

**Proposition 3.1 (Tightness)** For any sequence of initial measures $(\mu_N)_{N \ge 2}$ on $S_N$, the sequence $(\mathbb{Q}_N^{\mu_N})_{N \ge 2}$ is tight in the Skorokhod topology of $D([0, T]; \mathcal{M})$.

**Proof of Proposition 3.1.** In accordance with the method presented in [34, Chapter 4, Section 1], we must verify the following two statements for any $G \in C^2(\bar{\Lambda})$ and any $H \in C^2(\Gamma^{\text{low}})$:

(T1) For all $t \in [0, T]$ and all $\varepsilon > 0$, there is $M > 0$ such that

$$ \sup_{N \ge 2} \left\{ \mathbb{Q}_N^{\mu_N} \left( \left| \langle \pi_N(t), [G, H] \rangle \right| \ge M \right) \right\} \le \varepsilon. \quad (3.11) $$

(T2) For all $\varepsilon > 0$, we have

$$ \lim_{\delta \to 0} \limsup_{N \to \infty} \sup_{t \in [\delta, T-\delta]} \sup_{|\theta| \le \delta} \left\{ \mathbb{Q}_N^{\mu_N} \left( \left| \langle \pi_N(t+\theta), [G, H] \rangle - \langle \pi_N(t), [G, H] \rangle \right| \ge \varepsilon \right) \right\} = 0. \quad (3.12) $$

Note that, within this proof, we require the test functions $G$ and $H$ to depend solely on the spatial variable.

• *Proof of* (T1). Since the empirical measures $\pi_N^{\text{field}}$ and $\pi_N^{\text{road}}$ are both bounded by 1, we almost surely have

$$ \left| \langle \pi_N^{\text{field}}(t), G \rangle \right| \le \|G\|_{L^\infty(\Lambda)} \quad \text{and} \quad \left| \langle \pi_N^{\text{road}}(t), H \rangle \right| \le \|H\|_{L^\infty(\Gamma^{\text{low}})}. $$

As a result, (3.11) is achieved with $M = \varepsilon / (\|G\|_{L^\infty(\Lambda)} + \|H\|_{L^\infty(\Gamma^{\text{low}})})$.

• *Proof of* (T2). By expanding the terms $\langle \pi_N(t + \theta), [G, H] \rangle$ and $\langle \pi_N(t), [G, H] \rangle$ with those of the Dynkin's formula in (3.1) and using the Markov and the triangular inequalities, we can see that (3.12) holds if we prove the following limits:

$$ \lim_{\delta \to 0} \limsup_{N \to \infty} \sup_{t \in [\delta, T-\delta]} \sup_{|\theta| \le \delta} \left\{ \mathbb{E}_N^{\mu_N} \left( \left| \mathcal{M}_N(t+\theta) - \mathcal{M}_N(t) \right| \right) \right\} = 0, \quad (3.13) $$

$$ \lim_{\delta \to 0} \limsup_{N \to \infty} \sup_{t \in [\delta, T-\delta]} \sup_{|\theta| \le \delta} \left\{ \mathbb{E}_N^{\mu_N} \left( \left| \int_t^{t+\theta} \mathcal{L}_N \left[ \langle \pi_N^{\text{field}}(s), G \rangle \right] ds \right| \right) \right\} = 0, \quad (3.14) $$
$$ \lim_{\delta \to 0} \limsup_{N \to \infty} \sup_{t \in [\delta, T-\delta]} \sup_{|\theta| \le \delta} \left\{ \mathbb{E}_N^{\mu_N} \left( \left| \int_t^{t+\theta} \mathcal{L}_N \left[ \langle \pi_N^{\text{road}}(s), H \rangle \right] ds \right| \right) \right\} = 0. \quad (3.15) $$

We start to show (3.13). Focusing on the terms between the brackets, the Cauchy-Schwarz inequality yields

$$ \begin{aligned} \mathbb{E}_N^{\mu_N} \left( \left| \mathcal{M}_N(t+\theta) - \mathcal{M}_N(t) \right| \right) &\le \sqrt{\mathbb{E}_N^{\mu_N} \left( \left[ \mathcal{M}_N(t+\theta) - \mathcal{M}_N(t) \right]^2 \right)} \\ &= \sqrt{\mathbb{E}_N^{\mu_N} \left( \left[ \mathcal{M}_N(t+\theta) \right]^2 \right) - \mathbb{E}_N^{\mu_N} \left( \left[ \mathcal{M}_N(t) \right]^2 \right)}, \end{aligned} \quad (3.16) $$

where the second line arises from the martingale property. We use then the fact that the quadratic variation $N$ defined in (3.3) is a zero-mean martingale to simplify (3.16) into

$$ \mathbb{E}_N^{\mu_N} \left( \left| \mathcal{M}_N(t+\theta) - \mathcal{M}_N(t) \right| \right) \le \sqrt{\mathbb{E}_N^{\mu_N} \left( \int_t^{t+\theta} |\mathcal{B}_N(s)| ds \right)}. \quad (3.17) $$

Now observe from (3.9) and (3.10) that, for any $s \in (t - \delta, t + \delta)$,

$$ |\mathcal{B}_{N,G}^{\text{field}}(s)| \le \frac{d/2}{N^{2p}} \|\nabla G\|_{L^{\infty}(\Lambda)}^2 \left( \sum_{\substack{\hat{i}, \hat{k} \in \Lambda_N \\ |\hat{i} - \hat{k}| = 1}} 1 \right) + \frac{\alpha}{N^{2p-1}} \|G\|_{L^{\infty}(\Lambda)}^2 \left( \sum_{i \in \Gamma_N^{\text{low}}} 1 \right) + \frac{1}{N^{2p}} \|G\|_{L^{\infty}(\Lambda)}^2 \left( \sum_{i \in \Gamma_N^{\text{low}}} 1 \right) $$

and

$$ |\mathcal{B}_{N,H}^{\text{road}}(s)| \le \frac{D/2}{N^{2(p-1)}} \|\nabla_x H\|_{L^{\infty}(\mathbb{T}^{p-1})}^2 \left( \sum_{\substack{i,k \in \Gamma_N^{\text{low}} \\ |i-k|=1}} 1 \right) + \frac{\alpha}{N^{2(p-1)}} \|H\|_{L^{\infty}(\mathbb{T}^{p-1})}^2 \left( \sum_{i \in \Gamma_N^{\text{low}}} 1 \right). $$

As a result, we have $|\mathcal{B}_{N,G}^{\text{field}}(s)| = O(1/N^p)$ and $|\mathcal{B}_{N,H}^{\text{road}}(s)| = O(1/N^{p-1})$, and then $|\mathcal{B}_N(s)| = O(1/N^{p-1})$. Combining this control with (3.17), the proof of the limit (3.13) is then completed.

We move now on the proofs of (3.14) and (3.15). By similar computations as those used to develop $\mathcal{M}_{N,G}^{\text{field}}$ and $\mathcal{M}_{N,H}^{\text{road}}$ in (3.5) and (3.6), we express the terms under the integrals in (3.14) and (3.15):

$$ \begin{aligned} \mathcal{L}_N \left[ \langle \pi_N^{\text{field}}(s), G \rangle \right] &= \frac{1}{N^p} \sum_{\hat{i} \in \Lambda_N} d \Delta_x^N G(\frac{\hat{i}}{N}) \times \eta_s(\hat{i}) + \frac{1}{N^p} \sum_{\hat{i} \in \Lambda_N \setminus \Gamma_N} d \partial_{yy}^N G(\frac{\hat{i}}{N}) \times \eta_s(\hat{i}) \\ &\quad - \frac{1}{N^{p-1}} \sum_{i \in \Gamma_N^{\text{up}}} d \partial_y^N G(\frac{i}{N}) \times \eta_s(i) + \frac{1}{N^{p-1}} \sum_{i \in \Gamma_N^{\text{low}}} d \partial_y^N G(\frac{i}{N}) \times \eta_s(i) \\ &\quad + \frac{\alpha}{N^{p-1}} \sum_{i \in \Gamma_N^{\text{low}}} G(\frac{i}{N}) \times (\xi_s(i) - \eta_s(i)) + \frac{1}{N^p} \sum_{i \in \Gamma_N^{\text{up}}} G(\frac{i}{N}) \times (b - \eta_s(i)) \end{aligned} \quad (3.18) $$

and

$$ \mathcal{L}_N \left[ \langle \pi_N^{\text{road}}(s), H \rangle \right] = \frac{1}{N^{p-1}} \sum_{i \in \Gamma_N^{\text{low}}} D \Delta_x^N H(\frac{i}{N}) \times \xi_s(i) + \frac{\alpha}{N^{p-1}} \sum_{i \in \Gamma_N^{\text{low}}} H(\frac{i}{N}) \times (\eta_s(i) - \xi_s(i)), \quad (3.19) $$

where we recall that the discrete operators $\Delta_x^N$, $\partial_{yy}^N$, and $\partial_y^N$ are defined in (3.7) and (3.8). Using then some Taylor expansions to control the discrete derivatives in (3.18) and (3.19),
$$ \lim_{\delta \to 0} \limsup_{N \to \infty} \sup_{t \in [\delta, T-\delta]} \sup_{|\theta| \le \delta} \left\{ \mathbb{E}_N^{\mu_N} \left( \left| \int_t^{t+\theta} \mathcal{L}_N \left[ \langle \pi_N^{\text{road}}(s), H \rangle \right] ds \right| \right) \right\} = 0. \quad (3.15) $$

We start to show (3.13). Focusing on the terms between the brackets, the Cauchy-Schwarz inequality yields

$$ \begin{aligned} \mathbb{E}_N^{\mu_N} \left( \left| \mathcal{M}_N(t+\theta) - \mathcal{M}_N(t) \right| \right) &\le \sqrt{\mathbb{E}_N^{\mu_N} \left( \left[ \mathcal{M}_N(t+\theta) - \mathcal{M}_N(t) \right]^2 \right)} \\ &= \sqrt{\mathbb{E}_N^{\mu_N} \left( \left[ \mathcal{M}_N(t+\theta) \right]^2 \right) - \mathbb{E}_N^{\mu_N} \left( \left[ \mathcal{M}_N(t) \right]^2 \right)}, \end{aligned} \quad (3.16) $$

where the second line arises from the martingale property. We use then the fact that the quadratic variation $N$ defined in (3.3) is a zero-mean martingale to simplify (3.16) into

$$ \mathbb{E}_N^{\mu_N} \left( \left| \mathcal{M}_N(t+\theta) - \mathcal{M}_N(t) \right| \right) \le \sqrt{\mathbb{E}_N^{\mu_N} \left( \int_t^{t+\theta} |\mathcal{B}_N(s)| ds \right)}. \quad (3.17) $$

Now observe from (3.9) and (3.10) that, for any $s \in (t - \delta, t + \delta)$,

$$ |\mathcal{B}_{N,G}^{\text{field}}(s)| \le \frac{d/2}{N^{2p}} \|\nabla G\|_{L^{\infty}(\Lambda)}^2 \left( \sum_{\substack{\hat{i}, \hat{k} \in \Lambda_N \\ |\hat{i} - \hat{k}| = 1}} 1 \right) + \frac{\alpha}{N^{2p-1}} \|G\|_{L^{\infty}(\Lambda)}^2 \left( \sum_{i \in \Gamma_N^{\text{low}}} 1 \right) + \frac{1}{N^{2p}} \|G\|_{L^{\infty}(\Lambda)}^2 \left( \sum_{i \in \Gamma_N^{\text{low}}} 1 \right) $$

and

$$ |\mathcal{B}_{N,H}^{\text{road}}(s)| \le \frac{D/2}{N^{2(p-1)}} \|\nabla_x H\|_{L^{\infty}(\mathbb{T}^{p-1})}^2 \left( \sum_{\substack{i,k \in \Gamma_N^{\text{low}} \\ |i-k|=1}} 1 \right) + \frac{\alpha}{N^{2(p-1)}} \|H\|_{L^{\infty}(\mathbb{T}^{p-1})}^2 \left( \sum_{i \in \Gamma_N^{\text{low}}} 1 \right). $$

As a result, we have $|\mathcal{B}_{N,G}^{\text{field}}(s)| = O(1/N^p)$ and $|\mathcal{B}_{N,H}^{\text{road}}(s)| = O(1/N^{p-1})$, and then $|\mathcal{B}_N(s)| = O(1/N^{p-1})$. Combining this control with (3.17), the proof of the limit (3.13) is then completed.

We move now on the proofs of (3.14) and (3.15). By similar computations as those used to develop $\mathcal{M}_{N,G}^{\text{field}}$ and $\mathcal{M}_{N,H}^{\text{road}}$ in (3.5) and (3.6), we express the terms under the integrals in (3.14) and (3.15):

$$ \begin{aligned} \mathcal{L}_N \left[ \langle \pi_N^{\text{field}}(s), G \rangle \right] &= \frac{1}{N^p} \sum_{\hat{i} \in \Lambda_N} d \Delta_x^N G(\frac{\hat{i}}{N}) \times \eta_s(\hat{i}) + \frac{1}{N^p} \sum_{\hat{i} \in \Lambda_N \setminus \Gamma_N} d \partial_{yy}^N G(\frac{\hat{i}}{N}) \times \eta_s(\hat{i}) \\ &\quad - \frac{1}{N^{p-1}} \sum_{i \in \Gamma_N^{\text{up}}} d \partial_y^N G(\frac{i}{N}) \times \eta_s(i) + \frac{1}{N^{p-1}} \sum_{i \in \Gamma_N^{\text{low}}} d \partial_y^N G(\frac{i}{N}) \times \eta_s(i) \\ &\quad + \frac{\alpha}{N^{p-1}} \sum_{i \in \Gamma_N^{\text{low}}} G(\frac{i}{N}) \times (\xi_s(i) - \eta_s(i)) + \frac{1}{N^p} \sum_{i \in \Gamma_N^{\text{up}}} G(\frac{i}{N}) \times (b - \eta_s(i)) \end{aligned} \quad (3.18) $$

and

$$ \mathcal{L}_N \left[ \langle \pi_N^{\text{road}}(s), H \rangle \right] = \frac{1}{N^{p-1}} \sum_{i \in \Gamma_N^{\text{low}}} D \Delta_x^N H(\frac{i}{N}) \times \xi_s(i) + \frac{\alpha}{N^{p-1}} \sum_{i \in \Gamma_N^{\text{low}}} H(\frac{i}{N}) \times (\eta_s(i) - \xi_s(i)), \quad (3.19) $$

where we recall that the discrete operators $\Delta_x^N$, $\partial_{yy}^N$, and $\partial_y^N$ are defined in (3.7) and (3.8). Using then some Taylor expansions to control the discrete derivatives in (3.18) and (3.19),
we reach

$$
\begin{aligned}
|\mathcal{L}_N [\langle \pi_N^{\text{field}}(s), G \rangle]| & \le \frac{d}{N^p} \left( \sup_{\Lambda} |\Delta_x G| + O(1/N) \right) \left( \sum_{\hat{\iota} \in \Lambda_N} 1 \right) + \frac{d}{N^p} \left( \sup_{\Lambda} |\partial_{yy} G| + O(1/N) \right) \left( \sum_{\hat{\iota} \in \Lambda_N \setminus \Gamma_N} 1 \right) \\
& \quad + \frac{d}{N^{p-1}} \left( \sup_{\Gamma^{\text{up}}} |\partial_y G| + O(1/N) \right) \left( \sum_{i \in \Gamma_N^{\text{up}}} 1 \right) + \frac{d}{N^{p-1}} \left( \sup_{\Gamma^{\text{low}}} |\partial_y G| + O(1/N) \right) \left( \sum_{i \in \Gamma_N^{\text{low}}} 1 \right) \\
& \quad + \frac{\alpha}{N^{p-1}} \left( \sup_{\Gamma^{\text{low}}} |G| \right) \left( \sum_{i \in \Gamma_N^{\text{low}}} 1 \right) + \left( \sup_{\Gamma^{\text{up}}} |G| \right) \left( \sum_{i \in \Gamma_N^{\text{up}}} 1 \right)
\end{aligned}
$$

and

$$
|\mathcal{L}_N [\langle \pi_N^{\text{road}}(s), H \rangle]| \le \frac{D}{N^{p-1}} \left( \sup_{\mathbb{T}^{p-1}} |\Delta_x H| + O(1/N) \right) \left( \sum_{i \in \Gamma_N^{\text{low}}} 1 \right) + \frac{\alpha}{N^{p-1}} \left( \sup_{\mathbb{T}^{p-1}} |H| \right) \left( \sum_{i \in \Gamma_N^{\text{low}}} 1 \right).
$$

From this, it follows that $|\mathcal{L}_N [\langle \pi_N^{\text{field}}(s), G \rangle]|$ and $|\mathcal{L}_N [\langle \pi_N^{\text{road}}(s), H \rangle]|$ remain bounded.
Therefore, for some constant $C > 0$, we have

$$
\mathbb{E}_N^{\mu_N} \left( \left| \int_t^{t+\theta} \mathcal{L}_N \left[ \langle \pi_N^{\text{field}}(s), G \rangle \right] ds \right| \right) \le C\delta
$$

and

$$
\mathbb{E}_N^{\mu_N} \left( \left| \int_t^{t+\theta} \mathcal{L}_N \left[ \langle \pi_N^{\text{road}}(s), H \rangle \right] ds \right| \right) \le C\delta,
$$

whom the limits (3.14) and (3.15) are the straight consequence.

Having verified both conditions (T1) and (T2), we can conclude that the sequence
$(\mathbb{Q}_N^{\mu_N})_{N \ge 2}$ is tight with respect to the Skorokhod topology on $D([0, T]; \mathcal{M})$. This com-
pletes the proof.
□

4 Characterization of the limit points of (Q<sub>N</sub><sup>μ<sub>N</sub></sup>)<sub>N≥2</sub>

4.1 The limit points of (Q<sub>N</sub><sup>μ<sub>N</sub></sup>)<sub>N≥2</sub> load paths with density

**Proposition 4.1** (Loading Lebesgue continuous measure processes) *Any limit point of the sequence (Q<sub>N</sub><sup>μ<sub>N</sub></sup>)<sub>N≥2</sub>, referred to as Q<sub>∞</sub>, is concentrated on the set D<sup>0</sup>([0, T]; M) of couple of measure processes that are, at any time, absolutely continuous with respect to the Lebesgue measure on Λ and T<sup>p-1</sup> respectively. More precisely, for any Q<sub>∞</sub> in the closure of (Q<sub>N</sub><sup>μ<sub>N</sub></sup>)<sub>N≥2</sub>, we have*

$$
\mathbb{Q}_{\infty} (D^0([0, T]; \mathcal{M})) = 1,
$$

where

$$
D^0([0, T]; \mathcal{M}) := \left\{
\begin{array}{@{}l@{}}
  (\pi^{\text{field}}(t), \pi^{\text{road}}(t))_{t \in [0, T]} \in D([0, T]; \mathcal{M}) \text{ such that,} \\
  \quad \text{for any } t \in [0, T], (\pi^{\text{field}}(t), \pi^{\text{road}}(t)) \text{ is absolutely} \\
  \quad \text{continuous with respect to the Lebesgue measure on} \\
  \quad (\Lambda \times \mathbb{T}^{p-1}), \text{ with density } (v(t, \cdot), u(t, \cdot)) \in [0, 1]^\Lambda \times [0, 1]^{\mathbb{T}^{p-1}}
\end{array}
\right\}.
$$

Indeed, the simple exclusion rule provides the controls

$$
|\langle \pi^{\text{field}}(t), G \rangle| \le \|G\|_{L^1(\Lambda)} \quad \text{and} \quad |\langle \pi^{\text{road}}(t), H \rangle| \le \|H\|_{L^1(\mathbb{T}^{p-1})},
$$

for any $t \in [0, T]$, $G \in C(\bar{\Lambda})$ and $H \in C(\mathbb{T}^{p-1})$. The conclusion of the proposition follows
from this and the Lusin theorem.
we reach

$$
\begin{aligned}
|\mathcal{L}_N [\langle \pi_N^{\text{field}}(s), G \rangle]| &\le \frac{d}{N^p} \left( \sup_{\Lambda} |\Delta_x G| + O(1/N) \right) \left( \sum_{\hat{\iota} \in \Lambda_N} 1 \right) + \frac{d}{N^p} \left( \sup_{\Lambda} |\partial_{yy} G| + O(1/N) \right) \left( \sum_{\hat{\iota} \in \Lambda_N \setminus \Gamma_N} 1 \right) \\
&\quad + \frac{d}{N^{p-1}} \left( \sup_{\Gamma^{\text{up}}} |\partial_y G| + O(1/N) \right) \left( \sum_{i \in \Gamma_N^{\text{up}}} 1 \right) + \frac{d}{N^{p-1}} \left( \sup_{\Gamma^{\text{low}}} |\partial_y G| + O(1/N) \right) \left( \sum_{i \in \Gamma_N^{\text{low}}} 1 \right) \\
&\quad + \frac{\alpha}{N^{p-1}} \left( \sup_{\Gamma^{\text{low}}} |G| \right) \left( \sum_{i \in \Gamma_N^{\text{low}}} 1 \right) + \left( \sup_{\Gamma^{\text{up}}} |G| \right) \left( \sum_{i \in \Gamma_N^{\text{up}}} 1 \right)
\end{aligned}
$$

and

$$
|\mathcal{L}_N [\langle \pi_N^{\text{road}}(s), H \rangle]| \le \frac{D}{N^{p-1}} \left( \sup_{\mathbb{T}^{p-1}} |\Delta_x H| + O(1/N) \right) \left( \sum_{i \in \Gamma_N^{\text{low}}} 1 \right) + \frac{\alpha}{N^{p-1}} \left( \sup_{\mathbb{T}^{p-1}} |H| \right) \left( \sum_{i \in \Gamma_N^{\text{low}}} 1 \right).
$$

From this, it follows that $|\mathcal{L}_N [\langle \pi_N^{\text{field}}(s), G \rangle]|$ and $|\mathcal{L}_N [\langle \pi_N^{\text{road}}(s), H \rangle]|$ remain bounded.
Therefore, for some constant $C > 0$, we have

$$
\mathbb{E}_N^{\mu_N} \left( \left| \int_t^{t+\theta} \mathcal{L}_N \left[ \langle \pi_N^{\text{field}}(s), G \rangle \right] ds \right| \right) \le C\delta
$$

and

$$
\mathbb{E}_N^{\mu_N} \left( \left| \int_t^{t+\theta} \mathcal{L}_N \left[ \langle \pi_N^{\text{road}}(s), H \rangle \right] ds \right| \right) \le C\delta,
$$

whom the limits (3.14) and (3.15) are the straight consequence.

Having verified both conditions (T1) and (T2), we can conclude that the sequence
$(\mathbb{Q}_N^{\mu_N})_{N \ge 2}$ is tight with respect to the Skorokhod topology on $D([0, T]; \mathcal{M})$. This com-
pletes the proof.
□

4 Characterization of the limit points of (Q<sub>N</sub><sup>μ<sub>N</sub></sup>)<sub>N≥2</sub>

4.1 The limit points of (Q<sub>N</sub><sup>μ<sub>N</sub></sup>)<sub>N≥2</sub> load paths with density

**Proposition 4.1** (Loading Lebesgue continuous measure processes) *Any limit point of the sequence (Q<sub>N</sub><sup>μ<sub>N</sub></sup>)<sub>N≥2</sub>, referred to as Q<sub>∞</sub>, is concentrated on the set D<sup>0</sup>([0, T]; M) of couple of measure processes that are, at any time, absolutely continuous with respect to the Lebesgue measure on Λ and T<sup>p-1</sup> respectively. More precisely, for any Q<sub>∞</sub> in the closure of (Q<sub>N</sub><sup>μ<sub>N</sub></sup>)<sub>N≥2</sub>, we have*

$$
\mathbb{Q}_{\infty} (D^0([0, T]; \mathcal{M})) = 1,
$$

where

$$
D^0([0, T]; \mathcal{M}) := \left\{ (\pi^{\text{field}}(t), \pi^{\text{road}}(t))_{t \in [0, T]} \in D([0, T]; \mathcal{M}) \text{ such that,} \right. \\
\qquad \left. \text{for any } t \in [0, T], (\pi^{\text{field}}(t), \pi^{\text{road}}(t)) \text{ is absolutely} \right. \\
\qquad \left. \text{continuous with respect to the Lebesgue measure on} \right. \\
\qquad \left. (\Lambda \times \mathbb{T}^{p-1}), \text{ with density } (v(t, \cdot), u(t, \cdot)) \in [0, 1]^\Lambda \times [0, 1]^{\mathbb{T}^{p-1}} \right\}.
$$

Indeed, the simple exclusion rule provides the controls

$$
|\langle \pi^{\text{field}}(t), G \rangle| \le \|G\|_{L^1(\Lambda)} \quad \text{and} \quad |\langle \pi^{\text{road}}(t), H \rangle| \le \|H\|_{L^1(\mathbb{T}^{p-1})},
$$

for any $t \in [0, T]$, $G \in C(\bar{\Lambda})$ and $H \in C(\mathbb{T}^{p-1})$. The conclusion of the proposition follows
from this and the Lusin theorem.
## 4.2 The limit points of $(\mathbb{Q}_N^{\mu_N})_{N \ge 2}$ load paths whose densities satisfy (W1)

The next step consists in showing that the limit trajectories own the regularity claimed in (W1). Let us define the set $\mathcal{S}_{W1}$ by

$$ \mathcal{S}_{W1} := \left\{ \left( \pi^{\text{field}}(t), \pi^{\text{road}}(t) \right)_{t \in [0,T]} = \left( v(t, \hat{x}) d\hat{x}, u(t, x) dx \right)_{t \in [0,T]} \mid (v, u) \text{ satisfies condition (W1)} \right\}. $$

Then we have the following proposition.

**Proposition 4.2 (Identification of the limit sets)** Let $\mathbb{Q}_\infty$ be a limit point of the sequence $(\mathbb{Q}_N^{\mu_N})_{N \ge 2}$. Then,

$$ \mathbb{Q}_\infty(\mathcal{S}_{W1}) = 1. \qquad (4.1) $$

The function $u(t, \cdot)$ being in $L^2(\mathbb{T}^{p-1})$ directly follows from the simple exclusion rule which forces $u$ to be positive and bounded by 1. On the other hand, $v(t, \cdot)$ being in $\mathcal{H}^1(\Lambda)$ comes from the Riesz representation theorem combined with the following energy estimate.

**Lemma 4.3 (Energy estimate)** Given $v \in L^2(0, T; L^2(\Lambda))$ and $1 \le q \le p$, consider the (potentially infinite) quantity

$$ E_q(v) := \sup_{G \in \mathcal{C}_c^{0,2}([0,T] \times \bar{\Lambda})} \left\{ \int_0^T \langle v(s), \partial_{e_q} G(s) \rangle_\Lambda ds - \frac{1}{2} \int_0^T \|G(s)\|_{L^2(\Lambda)}^2 ds \right\}. \quad (4.2) $$

Then, for any $q \in [1; p]$,

$$ \mathbb{Q}_\infty \left( \left( \pi^{\text{field}}(t), \pi^{\text{road}}(t) \right)_{t \in [0,T]} = \left( v(t, \hat{x}) d\hat{x}, u(t, x) dx \right)_{t \in [0,T]} \mid E_q(v) < +\infty \right) = 1. $$

A proof of Lemma 4.3 is given in Appendix A.3.

## 4.3 Replacement lemmas

Before to proceed with the limit equations satisfied by the densities loaded by the limit points of $(\mathbb{Q}_N^{\mu_N})_{N \ge 2}$, we need to state two Replacement lemmas to correctly ensure the convergence of the boundary terms of the martingale $\mathcal{M}_N$ towards those of the weak formulation in (W2) — see subsection (4.4) for details. The essence of these lemmas lies in comparing the occupancy status of $\eta$ at the boundary sites to the average number of particles in their immediate vicinity. To state such a result, we must define these “substitute objects”. For fixed $\hat{\iota} \in \Gamma_N$ and $\varepsilon > 0$, let

$$ \Lambda_{\hat{\iota}}^{\varepsilon N} := \{ \hat{k} \in \Lambda_N : |\hat{\iota} - \hat{k}| \le \varepsilon N \} = \Lambda_N \cap \{ \hat{\iota} + [-\varepsilon N, \varepsilon N]^p \}, \quad (4.3) $$

and define for $\eta \in S_N^{\text{field}}$ the average number of particle of $\eta$ inside the box $\Lambda_{\hat{\iota}}^{\varepsilon N}$, that is

$$ \eta^{\varepsilon N}(\hat{\iota}) := c_{N,\varepsilon} \sum_{\hat{k} \in \Lambda_{\hat{\iota}}^{\varepsilon N}} \eta(\hat{k}), \qquad (4.4) $$

where $c_{N,\varepsilon}$ is the number of sites inside $\Lambda_{\hat{\iota}}^{\varepsilon N}$, namely,

$$ c_{N,\varepsilon} := \frac{1}{|\Lambda_{\hat{\iota}}^{\varepsilon N}|} = \left[ (2\lfloor \varepsilon N \rfloor + 1)^{p-1} \times (\lfloor \varepsilon N \rfloor + 1) \right]^{-1}. \quad (4.5) $$

The upper Replacement lemma is the following.
## 4.2 The limit points of $(\mathbb{Q}_N^{\mu_N})_{N \ge 2}$ load paths whose densities satisfy (W1)

The next step consists in showing that the limit trajectories own the regularity claimed in (W1). Let us define the set $\mathcal{S}_{W1}$ by

$$ \mathcal{S}_{W1} := \left\{ \left( \pi^{\text{field}}(t), \pi^{\text{road}}(t) \right)_{t \in [0,T]} = \left( v(t, \hat{x}) d\hat{x}, u(t, x) dx \right)_{t \in [0,T]} \mid (v, u) \text{ satisfies condition (W1)} \right\}. $$

Then we have the following proposition.

**Proposition 4.2 (Identification of the limit sets)** Let $\mathbb{Q}_\infty$ be a limit point of the sequence $(\mathbb{Q}_N^{\mu_N})_{N \ge 2}$. Then,

$$ \mathbb{Q}_\infty(\mathcal{S}_{W1}) = 1. \quad (4.1) $$

The function $u(t, \cdot)$ being in $L^2(\mathbb{T}^{p-1})$ directly follows from the simple exclusion rule which forces $u$ to be positive and bounded by 1. On the other hand, $v(t, \cdot)$ being in $\mathcal{H}^1(\Lambda)$ comes from the Riesz representation theorem combined with the following energy estimate.

**Lemma 4.3 (Energy estimate)** Given $v \in L^2(0, T; L^2(\Lambda))$ and $1 \le q \le p$, consider the (potentially infinite) quantity

$$ E_q(v) := \sup_{G \in \mathcal{C}_c^{0,2}([0,T] \times \bar{\Lambda})} \left\{ \int_0^T \langle v(s), \partial_{e_q} G(s) \rangle_\Lambda ds - \frac{1}{2} \int_0^T \|G(s)\|_{L^2(\Lambda)}^2 ds \right\}. \quad (4.2) $$

Then, for any $q \in [1; p]$,

$$ \mathbb{Q}_\infty \left( \left( \pi^{\text{field}}(t), \pi^{\text{road}}(t) \right)_{t \in [0,T]} = \left( v(t, \hat{x}) d\hat{x}, u(t, x) dx \right)_{t \in [0,T]} \mid E_q(v) < +\infty \right) = 1. $$

A proof of Lemma 4.3 is given in Appendix A.3.

## 4.3 Replacement lemmas

Before to proceed with the limit equations satisfied by the densities loaded by the limit points of $(\mathbb{Q}_N^{\mu_N})_{N \ge 2}$, we need to state two Replacement lemmas to correctly ensure the convergence of the boundary terms of the martingale $\mathcal{M}_N$ towards those of the weak formulation in (W2) — see subsection (4.4) for details. The essence of these lemmas lies in comparing the occupancy status of $\eta$ at the boundary sites to the average number of particles in their immediate vicinity. To state such a result, we must define these “substitute objects”. For fixed $\hat{\iota} \in \Gamma_N$ and $\varepsilon > 0$, let

$$ \Lambda_{\hat{\iota}}^{\varepsilon N} := \{ \hat{k} \in \Lambda_N : |\hat{\iota} - \hat{k}| \le \varepsilon N \} = \Lambda_N \cap \{ \hat{\iota} + [-\varepsilon N, \varepsilon N]^p \}, \quad (4.3) $$

and define for $\eta \in S_N^{\text{field}}$ the average number of particle of $\eta$ inside the box $\Lambda_{\hat{\iota}}^{\varepsilon N}$, that is

$$ \eta^{\varepsilon N}(\hat{\iota}) := c_{N,\varepsilon} \sum_{\hat{k} \in \Lambda_{\hat{\iota}}^{\varepsilon N}} \eta(\hat{k}), \quad (4.4) $$

where $c_{N,\varepsilon}$ is the number of sites inside $\Lambda_{\hat{\iota}}^{\varepsilon N}$, namely,

$$ c_{N,\varepsilon} := \frac{1}{|\Lambda_{\hat{\iota}}^{\varepsilon N}|} = \left[ (2\lfloor \varepsilon N \rfloor + 1)^{p-1} \times (\lfloor \varepsilon N \rfloor + 1) \right]^{-1}. \quad (4.5) $$

The upper Replacement lemma is the following.
**Lemma 4.4** (Replacement at the upper boundary) Let $(\mu_N)_{N \ge 2}$ be a sequence of initial measures on $S_N$. For any $t \in [0, T]$, and any test functions $G \in C^{1,2}([0, T] \times \bar{\Lambda})$, we have

$$ \limsup_{\varepsilon \to 0} \limsup_{N \to \infty} \mathbb{E}_N^{\mu_N} \left[ \left| \int_0^t \frac{1}{N^{p-1}} \sum_{\hat{\iota} \in \Gamma_N^{\text{up}}} G(s, \frac{\hat{\iota}}{N}) [\eta_s^{\varepsilon N}(\hat{\iota}) - \eta_s(\hat{\iota})] ds \right| \right] = 0. \quad (4.6) $$

<figure><img src="image_3.png" /><figcaption>**Figure II — The upper Replacement lemma.** For $\hat{\iota} = (i, j) \in \Gamma_N^{\text{up}}$, we define $\Lambda_{\hat{\iota}}^{\varepsilon N}$ as the intersection between $\Lambda_N$ and the $p$-dimensional box of range $2\varepsilon N$ centred on $\hat{\iota}$. Notice that for fixed $\varepsilon > 0$, the size of the box remains constant at the macroscopic scale. Lemma 4.4 establishes that, as $N \to \infty$ followed by $\varepsilon \to 0$, the value of $\eta(\hat{\iota})$ can be replaced by $\eta^{\varepsilon N}(\hat{\iota})$ which is the mean value of $\eta$ inside $\Lambda_{\hat{\iota}}^{\varepsilon N}$.</figcaption></figure>

Similarly, the lower Replacement lemma is the following.

**Lemma 4.5** (Replacement at the lower boundary) Let $(\mu_N)_{N \ge 2}$ be a sequence of initial measures on $S_N$. For any $t \in [0, T]$, and any test functions $G \in C^{1,2}([0, T] \times \bar{\Lambda})$, we have

$$ \limsup_{\varepsilon \to 0} \limsup_{N \to \infty} \mathbb{E}_N^{\mu_N} \left[ \left| \int_0^t \frac{1}{N^{p-1}} \sum_{\hat{\iota} \in \Gamma_N^{\text{low}}} G(s, \frac{\hat{\iota}}{N}) [\eta_s^{\varepsilon N}(\hat{\iota}) - \eta_s(\hat{\iota})] ds \right| \right] = 0. \quad (4.7) $$

The proof of the Lemma 4.4 is given in Appendix A.2, while that of Lemma 4.5 is omitted since it follows essentially identical arguments.

## 4.4 The limit points of $(\mathbb{Q}_N^{\mu_N})_{N \ge 2}$ load paths whose densities satisfy (W2)

We claim now that all the limit trajectories are supported on the set $\mathcal{S}_{W2}$ of measures with density $(v, u)$ that satisfy the weak formulation associated with (1.2)-(1.3), namely

$$ \mathcal{S}_{W2} := \left\{ \left( \pi^{\text{field}}(t), \pi^{\text{road}}(t) \right)_{t \in [0, T]} = (v(t, \hat{x})d\hat{x}, u(t, x)dx)_{t \in [0, T]} \mid (v, u) \text{ satisfies condition (W2)} \right\}. $$
**Lemma 4.4** (Replacement at the upper boundary) Let $(\mu_N)_{N \ge 2}$ be a sequence of initial measures on $S_N$. For any $t \in [0, T]$, and any test functions $G \in C^{1,2}([0, T] \times \bar{\Lambda})$, we have

$$ \limsup_{\varepsilon \to 0} \limsup_{N \to \infty} \mathbb{E}_N^{\mu_N} \left[ \left| \int_0^t \frac{1}{N^{p-1}} \sum_{\hat{\iota} \in \Gamma_N^{\text{up}}} G(s, \frac{\hat{\iota}}{N}) [\eta_s^{\varepsilon N}(\hat{\iota}) - \eta_s(\hat{\iota})] ds \right| \right] = 0. \quad (4.6) $$

<figure><img src="image_3.png" /><figcaption>**Figure II — The upper Replacement lemma.** For $\hat{\iota} = (i, j) \in \Gamma_N^{\text{up}}$, we define $\Lambda_{\hat{\iota}}^{\varepsilon N}$ as the intersection between $\Lambda_N$ and the $p$-dimensional box of range $2\varepsilon N$ centred on $\hat{\iota}$. Notice that for fixed $\varepsilon > 0$, the size of the box remains constant at the macroscopic scale. Lemma 4.4 establishes that, as $N \to \infty$ followed by $\varepsilon \to 0$, the value of $\eta(\hat{\iota})$ can be replaced by $\eta^{\varepsilon N}(\hat{\iota})$ which is the mean value of $\eta$ inside $\Lambda_{\hat{\iota}}^{\varepsilon N}$.</figcaption></figure>

Similarly, the lower Replacement lemma is the following.

**Lemma 4.5** (Replacement at the lower boundary) Let $(\mu_N)_{N \ge 2}$ be a sequence of initial measures on $S_N$. For any $t \in [0, T]$, and any test functions $G \in C^{1,2}([0, T] \times \bar{\Lambda})$, we have

$$ \limsup_{\varepsilon \to 0} \limsup_{N \to \infty} \mathbb{E}_N^{\mu_N} \left[ \left| \int_0^t \frac{1}{N^{p-1}} \sum_{\hat{\iota} \in \Gamma_N^{\text{low}}} G(s, \frac{\hat{\iota}}{N}) [\eta_s^{\varepsilon N}(\hat{\iota}) - \eta_s(\hat{\iota})] ds \right| \right] = 0. \quad (4.7) $$

The proof of the Lemma 4.4 is given in Appendix A.2, while that of Lemma 4.5 is omitted since it follows essentially identical arguments.

## 4.4 The limit points of $(\mathbb{Q}_N^{\mu_N})_{N \ge 2}$ load paths whose densities satisfy (W2)

We claim now that all the limit trajectories are supported on the set $\mathcal{S}_{W2}$ of measures with density $(v, u)$ that satisfy the weak formulation associated with (1.2)-(1.3), namely

$$ \mathcal{S}_{W2} := \left\{ (\pi^{\text{field}}(t), \pi^{\text{road}}(t))_{t \in [0, T]} = (v(t, \hat{x})d\hat{x}, u(t, x)dx)_{t \in [0, T]} \mid (v, u) \text{ satisfies condition (W2)} \right\}. $$
**Proposition 4.6** (Identification of the limit equations) Let $Q_\infty$ be a limit point of the sequence $(Q_N^{\mu_N})_{N \ge 2}$. Then,

$$Q_\infty(\mathcal{S}_{W_2}) = 1. \qquad (4.8)$$

To establish Proposition 4.6, we need to demonstrate that each component of the martingale $M_N = M_{N,G}^{\text{field}} + M_{N,H}^{\text{road}}$, as detailed in (3.5)-(3.6), converges towards its respective counterpart in the weak formulation (2.7)-(2.8), and then that the martingale itself vanishes as $N \to \infty$. This requires the two Replacement lemmas (Lemma 4.4 and Lemma 4.5), which allow the substitution of terms not expressed through the empirical measure $\pi_N$ in the martingale $M_N$, and the identification of the limit sets (Proposition 4.2), that is essential to define the trace of the function $v(t, \cdot)$ at the lower boundary.

**Proof of Proposition 4.6.** Let $G \in \mathcal{C}^{1,2}([0, T] \times \bar{\Lambda})$ and $H \in \mathcal{C}^{1,2}([0, T] \times \mathbb{T}^{p-1})$. For practical reasons, let us define here the functional $\mathcal{W}_{u,v}(t) := \mathcal{W}_{u,v,G}^{\text{field}}(t) + \mathcal{W}_{u,v,H}^{\text{road}}(t)$ associated with the weak formulation (2.7)-(2.8) in (W2):

$$ \begin{aligned} \mathcal{W}_{u,v,G}^{\text{field}}(t) := & \langle v(t), G(t) \rangle_{\Lambda} - \langle v_0, G(0) \rangle_{\Lambda} - \int_0^t \langle v(s), \partial_s G(s) \rangle_{\Lambda} ds - \int_0^t \langle v(s), d\Delta G(s) \rangle_{\Lambda} ds \\ & + \int_0^t \langle v|_{y=1}(s), d\partial_y G|_{y=1}(s) \rangle_{\mathbb{T}^{p-1}} ds - \int_0^t \langle v|_{y=0}(s), d\partial_y G|_{y=0}(s) \rangle_{\mathbb{T}^{p-1}} ds \quad (4.9) \\ & - \int_0^t \alpha \langle u(s) - v|_{y=0}(s), G|_{y=0}(s) \rangle_{\mathbb{T}^{p-1}} ds, \end{aligned} $$

and

$$ \begin{aligned} \mathcal{W}_{u,v,H}^{\text{road}}(t) := & \langle u(t), H(t) \rangle_{\mathbb{T}^{p-1}} - \langle u_0, H(0) \rangle_{\mathbb{T}^{p-1}} - \int_0^t \langle u(s), \partial_s H(s) \rangle_{\mathbb{T}^{p-1}} ds \\ & - \int_0^t \langle u(s), D\Delta_x H(s) \rangle_{\mathbb{T}^{p-1}} ds - \int_0^t \alpha \langle v|_{y=0}(s) - u(s), H(s) \rangle_{\mathbb{T}^{p-1}} ds. \end{aligned} \qquad (4.10) $$

Given $Q_\infty$ in the closure of the sequence $(Q_N^{\mu_N})_{N \ge 2}$, the statement of Proposition 4.6 can then be reformulated as

$$ Q_\infty\left(\left(\pi^{\text{field}}(t), \pi^{\text{road}}(t)\right)_{t \in [0, T]} = \left(v(t, \hat{x})d\hat{x}, u(t, x)dx\right)_{t \in [0, T]} \mid \mathcal{W}_{u,v,G}^{\text{field}}(t) = \mathcal{W}_{u,v,H}^{\text{road}}(t) = 0\right) = 1. $$

To prove this, it is sufficient to establish that of any $t \in [0, T]$ and $\delta > 0$ there holds

$$ Q_\infty\left(\left|\mathcal{W}_{u,v}(t)\right| > \delta\right) = 0. \qquad (4.11) $$

At this point, to work with the probability measures $Q_N^{\mu_N}$ (that do not load paths with densities) instead of $Q_\infty$, we need to substitute the quantity $\mathcal{W}_{u,v}(t)$ with another one which only depends on measures $\pi^{\text{field}}$ and $\pi^{\text{road}}$ instead of $v$ and $u$. This substitution cannot be made directly because of the boundary terms in (4.9) and (4.10). To overcome this issue, let us introduce the two families of unit approximations indexed by $\varepsilon > 0$,

$$ \mathbb{U}_\varepsilon^{\text{up}} := \frac{1}{\varepsilon(2\varepsilon)^{p-1}} \mathbb{1}_{[-\varepsilon, \varepsilon]^{p-1} \times [0, \varepsilon]} \quad \text{and} \quad \mathbb{U}_\varepsilon^{\text{low}} := \frac{1}{\varepsilon(2\varepsilon)^{p-1}} \mathbb{1}_{[-\varepsilon, \varepsilon]^{p-1} \times [-\varepsilon, 0]}, $$
**Proposition 4.6** (Identification of the limit equations) Let $Q_\infty$ be a limit point of the sequence $(Q_N^{\mu_N})_{N \ge 2}$. Then,

$$Q_\infty(\mathcal{S}_{W_2}) = 1. \qquad (4.8)$$

To establish Proposition 4.6, we need to demonstrate that each component of the martingale $M_N = M_{N,G}^{\text{field}} + M_{N,H}^{\text{road}}$, as detailed in (3.5)-(3.6), converges towards its respective counterpart in the weak formulation (2.7)-(2.8), and then that the martingale itself vanishes as $N \to \infty$. This requires the two Replacement lemmas (Lemma 4.4 and Lemma 4.5), which allow the substitution of terms not expressed through the empirical measure $\pi_N$ in the martingale $M_N$, and the identification of the limit sets (Proposition 4.2), that is essential to define the trace of the function $v(t, \cdot)$ at the lower boundary.

**Proof of Proposition 4.6.** Let $G \in \mathcal{C}^{1,2}([0, T] \times \bar{\Lambda})$ and $H \in \mathcal{C}^{1,2}([0, T] \times \mathbb{T}^{p-1})$. For practical reasons, let us define here the functional $\mathcal{W}_{u,v}(t) := \mathcal{W}_{u,v,G}^{\text{field}}(t) + \mathcal{W}_{u,v,H}^{\text{road}}(t)$ associated with the weak formulation (2.7)-(2.8) in (W2):

$$ \begin{aligned} \mathcal{W}_{u,v,G}^{\text{field}}(t) := & \langle v(t), G(t) \rangle_{\Lambda} - \langle v_0, G(0) \rangle_{\Lambda} - \int_0^t \langle v(s), \partial_s G(s) \rangle_{\Lambda} ds - \int_0^t \langle v(s), d\Delta G(s) \rangle_{\Lambda} ds \\ & + \int_0^t \langle v|_{y=1}(s), d\partial_y G|_{y=1}(s) \rangle_{\mathbb{T}^{p-1}} ds - \int_0^t \langle v|_{y=0}(s), d\partial_y G|_{y=0}(s) \rangle_{\mathbb{T}^{p-1}} ds \quad (4.9) \\ & - \int_0^t \alpha \langle u(s) - v|_{y=0}(s), G|_{y=0}(s) \rangle_{\mathbb{T}^{p-1}} ds, \end{aligned} $$

and

$$ \begin{aligned} \mathcal{W}_{u,v,H}^{\text{road}}(t) := & \langle u(t), H(t) \rangle_{\mathbb{T}^{p-1}} - \langle u_0, H(0) \rangle_{\mathbb{T}^{p-1}} - \int_0^t \langle u(s), \partial_s H(s) \rangle_{\mathbb{T}^{p-1}} ds \\ & - \int_0^t \langle u(s), D\Delta_x H(s) \rangle_{\mathbb{T}^{p-1}} ds - \int_0^t \alpha \langle v|_{y=0}(s) - u(s), H(s) \rangle_{\mathbb{T}^{p-1}} ds. \end{aligned} \qquad (4.10) $$

Given $Q_\infty$ in the closure of the sequence $(Q_N^{\mu_N})_{N \ge 2}$, the statement of Proposition 4.6 can then be reformulated as

$$ Q_\infty\left(\left(\pi^{\text{field}}(t), \pi^{\text{road}}(t)\right)_{t \in [0, T]} = \left(v(t, \hat{x})d\hat{x}, u(t, x)dx\right)_{t \in [0, T]} \middle| \mathcal{W}_{u,v,G}^{\text{field}}(t) = \mathcal{W}_{u,v,H}^{\text{road}}(t) = 0\right) = 1. $$

To prove this, it is sufficient to establish that of any $t \in [0, T]$ and $\delta > 0$ there holds

$$ Q_\infty\left(\left|\mathcal{W}_{u,v}(t)\right| > \delta\right) = 0. \qquad (4.11) $$

At this point, to work with the probability measures $Q_N^{\mu_N}$ (that do not load paths with densities) instead of $Q_\infty$, we need to substitute the quantity $\mathcal{W}_{u,v}(t)$ with another one which only depends on measures $\pi^{\text{field}}$ and $\pi^{\text{road}}$ instead of $v$ and $u$. This substitution cannot be made directly because of the boundary terms in (4.9) and (4.10). To overcome this issue, let us introduce the two families of unit approximations indexed by $\varepsilon > 0$,

$$ \mathbb{U}_\varepsilon^{\text{up}} := \frac{1}{\varepsilon(2\varepsilon)^{p-1}} \mathbb{1}_{[-\varepsilon, \varepsilon]^{p-1} \times [0, \varepsilon]} \quad \text{and} \quad \mathbb{U}_\varepsilon^{\text{low}} := \frac{1}{\varepsilon(2\varepsilon)^{p-1}} \mathbb{1}_{[-\varepsilon, \varepsilon]^{p-1} \times [-\varepsilon, 0]}, $$
defined for all $x \in \mathbb{T}^{p-1}$ and $y \in \mathbb{R}$. Observe that $\eta^{\varepsilon N}(\hat{\iota})$ defined in (4.4) can then be rewritten for any $\hat{\iota} \in \Gamma_N = \Gamma_N^{\text{up}} \cup \Gamma_N^{\text{low}}$, $\varepsilon > 0$ and $N \ge 2$ as

$$ \eta^{\varepsilon N}(\hat{\iota}) = \tilde{c}_{N,\varepsilon} [\pi_N^{\text{field}} * \mathbb{U}_{\varepsilon}^{\text{up}}](\hat{\iota}/N), \quad \text{if } \hat{\iota} \in \Gamma_N^{\text{up}}, \qquad (4.12) $$

and

$$ \eta^{\varepsilon N}(\hat{\iota}) = \tilde{c}_{N,\varepsilon} [\pi_N^{\text{field}} * \mathbb{U}_{\varepsilon}^{\text{low}}](\hat{\iota}/N), \quad \text{if } \hat{\iota} \in \Gamma_N^{\text{low}}, \qquad (4.13) $$

where

$$ \tilde{c}_{N,\varepsilon} = \frac{(2\varepsilon N)^{p-1}(\varepsilon N)}{(2[\varepsilon N]+1)^{p-1}([\varepsilon N]+1)} \quad \text{for any } \varepsilon > 0 \text{ and any } N \ge 2. $$

Now for $\varepsilon > 0$, $\pi = (\pi^{\text{field}}, \pi^{\text{road}}) \in \mathcal{M}$, and $t \in [0, T]$, we let $\mathcal{W}_{\pi}^{\varepsilon}(t) := \mathcal{W}_{\pi, G, \varepsilon}^{\text{field}}(t) + \mathcal{W}_{\pi, H, \varepsilon}^{\text{road}}(t)$ with

$$ \begin{aligned} \mathcal{W}_{\pi, G, \varepsilon}^{\text{field}}(t) := & \langle \pi^{\text{field}}(t), G(t) \rangle - \langle \pi^{\text{field}}(0), G(0) \rangle - \int_0^t \langle \pi^{\text{field}}(s), \partial_s G(s) \rangle ds - \int_0^t \langle \pi^{\text{field}}(s), d\Delta G(s) \rangle ds \\ & + \int_0^t \langle [\pi^{\text{field}}(s) * \mathbb{U}_{\varepsilon}^{\text{up}}]|_{y=1}, d\partial_y G|_{y=1}(s) \rangle_{\mathbb{T}^{p-1}} ds - \int_0^t \langle [\pi^{\text{field}}(s) * \mathbb{U}_{\varepsilon}^{\text{low}}]|_{y=0}, d\partial_y G|_{y=0}(s) \rangle_{\mathbb{T}^{p-1}} ds \quad (4.14) \\ & - \int_0^t \alpha \langle \pi^{\text{road}}(s), G|_{y=0}(s) \rangle + \int_0^t \alpha \langle [\pi^{\text{field}}(s) * \mathbb{U}_{\varepsilon}^{\text{low}}]|_{y=0}, G|_{y=0}(s) \rangle_{\mathbb{T}^{p-1}} ds, \end{aligned} $$

and

$$ \begin{aligned} \mathcal{W}_{\pi, H, \varepsilon}^{\text{road}}(t) := & \langle \pi^{\text{road}}(t), H(t) \rangle - \langle \pi^{\text{road}}(0), H(0) \rangle - \int_0^t \langle \pi^{\text{road}}(s), \partial_s H(s) \rangle ds \\ & - \int_0^t \langle \pi^{\text{road}}(s), D\Delta_x H(s) \rangle ds - \int_0^t \alpha \langle [\pi^{\text{field}}(s) * \mathbb{U}_{\varepsilon}^{\text{low}}]|_{y=0}, H(s) \rangle_{\mathbb{T}^{p-1}} + \int_0^t \alpha \langle \pi^{\text{road}}(s), H(s) \rangle ds. \end{aligned} \qquad (4.15) $$

Thanks to Proposition 4.2, we know that $\mathbb{Q}_{\infty}$ loads paths with densities $(v(t, \cdot))_{t \in [0,T]}$ in $L^2(0, T; \mathcal{H}^1(\Lambda))$. As a consequence, the trace of $v(t, \cdot)$ at the boundaries $\Gamma^{\text{up}}$ and $\Gamma^{\text{low}}$ is well-defined and we have (see [25, Section 5.3])

$$ \lim_{\varepsilon \to 0} [\pi^{\text{field}}(s) * \mathbb{U}_{\varepsilon}^{\text{up}}]|_{\hat{\iota} \in \Gamma} = \text{Tr}(v), \quad \mathbb{Q}_{\infty}\text{-almost-surely in } \Gamma. $$

As a result of this

$$ \lim_{\varepsilon \to 0} |\mathcal{W}_{u,v}(t) - \mathcal{W}_{\pi}^{\varepsilon}(t)| = 0, \quad \mathbb{Q}_{\infty}\text{-almost-surely in } \Gamma. \qquad (4.16) $$

We can now bound the probability in (4.11) as follows

$$ \begin{aligned} \mathbb{Q}_{\infty} \left( \sup_{0 \le t \le T} |\mathcal{W}_{u,v}(t)| > \delta \right) &= \mathbb{Q}_{\infty} \left( \sup_{0 \le t \le T} |\mathcal{W}_{u,v}(t) - \mathcal{W}_{\pi}^{\varepsilon}(t) + \mathcal{W}_{\pi}^{\varepsilon}(t)| > \delta \right) \\ &\le \mathbb{Q}_{\infty} \left( \sup_{0 \le t \le T} |\mathcal{W}_{u,v}(t) - \mathcal{W}_{\pi}^{\varepsilon}(t)| > \delta/2 \right) + \mathbb{Q}_{\infty} \left( \sup_{0 \le t \le T} |\mathcal{W}_{\pi}^{\varepsilon}(t)| > \delta/2 \right) \\ &\le \mathbb{Q}_{\infty} \left( \sup_{0 \le t \le T} |\mathcal{W}_{u,v}(t) - \mathcal{W}_{\pi}^{\varepsilon}(t)| > \delta/2 \right) + \liminf_{N \to \infty} \mathbb{Q}_{N}^{\mu_N} \left( \sup_{0 \le t \le T} |\mathcal{W}_{\pi_N}^{\varepsilon}(t)| > \delta/2 \right), \quad (4.17) \end{aligned} $$

where we used the Portmanteau Theorem to write the last inequality. While the vanishing of the first term in (4.17) as $\varepsilon$ goes to zero is a straight consequence of (4.16), the second
defined for all $x \in \mathbb{T}^{p-1}$ and $y \in \mathbb{R}$. Observe that $\eta^{\varepsilon N}(\hat{\iota})$ defined in (4.4) can then be rewritten for any $\hat{\iota} \in \Gamma_N = \Gamma_N^{\text{up}} \cup \Gamma_N^{\text{low}}$, $\varepsilon > 0$ and $N \ge 2$ as

$$ \eta^{\varepsilon N}(\hat{\iota}) = \tilde{c}_{N,\varepsilon} [\pi_N^{\text{field}} * \mathbb{U}_{\varepsilon}^{\text{up}}](\hat{\iota}/N), \quad \text{if } \hat{\iota} \in \Gamma_N^{\text{up}}, \qquad (4.12) $$

and

$$ \eta^{\varepsilon N}(\hat{\iota}) = \tilde{c}_{N,\varepsilon} [\pi_N^{\text{field}} * \mathbb{U}_{\varepsilon}^{\text{low}}](\hat{\iota}/N), \quad \text{if } \hat{\iota} \in \Gamma_N^{\text{low}}, \qquad (4.13) $$

where

$$ \tilde{c}_{N,\varepsilon} = \frac{(2\varepsilon N)^{p-1}(\varepsilon N)}{(2[\varepsilon N]+1)^{p-1}([\varepsilon N]+1)} \quad \text{for any } \varepsilon > 0 \text{ and any } N \ge 2. $$

Now for $\varepsilon > 0$, $\pi = (\pi^{\text{field}}, \pi^{\text{road}}) \in \mathcal{M}$, and $t \in [0, T]$, we let $\mathcal{W}_{\pi}^{\varepsilon}(t) := \mathcal{W}_{\pi, G, \varepsilon}^{\text{field}}(t) + \mathcal{W}_{\pi, H, \varepsilon}^{\text{road}}(t)$ with

$$ \begin{aligned} \mathcal{W}_{\pi, G, \varepsilon}^{\text{field}}(t) := & \langle \pi^{\text{field}}(t), G(t) \rangle - \langle \pi^{\text{field}}(0), G(0) \rangle - \int_0^t \langle \pi^{\text{field}}(s), \partial_s G(s) \rangle ds - \int_0^t \langle \pi^{\text{field}}(s), d\Delta G(s) \rangle ds \\ & + \int_0^t \langle [\pi^{\text{field}}(s) * \mathbb{U}_{\varepsilon}^{\text{up}}]|_{y=1}, d\partial_y G|_{y=1}(s) \rangle_{\mathbb{T}^{p-1}} ds - \int_0^t \langle [\pi^{\text{field}}(s) * \mathbb{U}_{\varepsilon}^{\text{low}}]|_{y=0}, d\partial_y G|_{y=0}(s) \rangle_{\mathbb{T}^{p-1}} ds \quad (4.14) \\ & - \int_0^t \alpha \langle \pi^{\text{road}}(s), G|_{y=0}(s) \rangle + \int_0^t \alpha \langle [\pi^{\text{field}}(s) * \mathbb{U}_{\varepsilon}^{\text{low}}]|_{y=0}, G|_{y=0}(s) \rangle_{\mathbb{T}^{p-1}} ds, \end{aligned} $$

and

$$ \begin{aligned} \mathcal{W}_{\pi, H, \varepsilon}^{\text{road}}(t) := & \langle \pi^{\text{road}}(t), H(t) \rangle - \langle \pi^{\text{road}}(0), H(0) \rangle - \int_0^t \langle \pi^{\text{road}}(s), \partial_s H(s) \rangle ds \\ & - \int_0^t \langle \pi^{\text{road}}(s), D\Delta_x H(s) \rangle ds - \int_0^t \alpha \langle [\pi^{\text{field}}(s) * \mathbb{U}_{\varepsilon}^{\text{low}}]|_{y=0}, H(s) \rangle_{\mathbb{T}^{p-1}} + \int_0^t \alpha \langle \pi^{\text{road}}(s), H(s) \rangle ds. \end{aligned} \qquad (4.15) $$

Thanks to Proposition 4.2, we know that $\mathbb{Q}_{\infty}$ loads paths with densities $(v(t, \cdot))_{t \in [0,T]}$ in $L^2(0, T; \mathcal{H}^1(\Lambda))$. As a consequence, the trace of $v(t, \cdot)$ at the boundaries $\Gamma^{\text{up}}$ and $\Gamma^{\text{low}}$ is well-defined and we have (see [25, Section 5.3])

$$ \lim_{\varepsilon \to 0} [\pi^{\text{field}}(s) * \mathbb{U}_{\varepsilon}^{\text{up}}]|_{\hat{\iota} \in \Gamma} = \text{Tr}(v), \quad \mathbb{Q}_{\infty}\text{-almost-surely in } \Gamma. $$

As a result of this

$$ \lim_{\varepsilon \to 0} |\mathcal{W}_{u,v}(t) - \mathcal{W}_{\pi}^{\varepsilon}(t)| = 0, \quad \mathbb{Q}_{\infty}\text{-almost-surely in } \Gamma. \qquad (4.16) $$

We can now bound the probability in (4.11) as follows

$$ \begin{aligned} \mathbb{Q}_{\infty} \left( \sup_{0 \le t \le T} |\mathcal{W}_{u,v}(t)| > \delta \right) &= \mathbb{Q}_{\infty} \left( \sup_{0 \le t \le T} |\mathcal{W}_{u,v}(t) - \mathcal{W}_{\pi}^{\varepsilon}(t) + \mathcal{W}_{\pi}^{\varepsilon}(t)| > \delta \right) \\ &\le \mathbb{Q}_{\infty} \left( \sup_{0 \le t \le T} |\mathcal{W}_{u,v}(t) - \mathcal{W}_{\pi}^{\varepsilon}(t)| > \delta/2 \right) + \mathbb{Q}_{\infty} \left( \sup_{0 \le t \le T} |\mathcal{W}_{\pi}^{\varepsilon}(t)| > \delta/2 \right) \\ &\le \mathbb{Q}_{\infty} \left( \sup_{0 \le t \le T} |\mathcal{W}_{u,v}(t) - \mathcal{W}_{\pi}^{\varepsilon}(t)| > \delta/2 \right) + \liminf_{N \to \infty} \mathbb{Q}_{N}^{\mu_N} \left( \sup_{0 \le t \le T} |\mathcal{W}_{\pi_N}^{\varepsilon}(t)| > \delta/2 \right), \quad (4.17) \end{aligned} $$

where we used the Portmanteau Theorem to write the last inequality. While the vanishing of the first term in (4.17) as $\varepsilon$ goes to zero is a straight consequence of (4.16), the second
one requires further attention. By considering the probability under the limit in this second term, we control it by this way:

$$
\begin{align}
\mathbb{Q}_N^{\mu_N} \left( \sup_{0 \le t \le T} |\mathcal{W}_{\pi_N}^{\varepsilon}(t)| > \delta/2 \right) &= \mathbb{Q}_N^{\mu_N} \left( \sup_{0 \le t \le T} |\mathcal{W}_{\pi_N}^{\varepsilon}(t) - \mathcal{M}_N(t) + \mathcal{M}_N(t)| > \delta/2 \right) \nonumber \\
&\le \mathbb{Q}_N^{\mu_N} \left( \sup_{0 \le t \le T} |\mathcal{W}_{\pi_N}^{\varepsilon}(t) - \mathcal{M}_N(t)| > \delta/4 \right) + \mathbb{Q}_N^{\mu_N} \left( \sup_{0 \le t \le T} |\mathcal{M}_N(t)| > \delta/4 \right), \nonumber \\
&\le \frac{4}{\delta} \mathbb{E}_N^{\mu_N} \left( \sup_{0 \le t \le T} |\mathcal{W}_{\pi_N}^{\varepsilon}(t) - \mathcal{M}_N(t)| \right) + \mathbb{Q}_N^{\mu_N} \left( \sup_{0 \le t \le T} |\mathcal{M}_N(t)| > \delta/4 \right), \tag{4.18}
\end{align}
$$

where we recall that the martingale $\mathcal{M}_N(t)$ is defined in (3.1). In (4.18), the vanishing of $\mathbb{Q}_N^{\mu_N}(|\mathcal{M}_N(t)| > \delta/4)$ can be shown by using the Doob's inequality:

$$
\begin{align*}
\mathbb{Q}_N^{\mu_N} \left( \sup_{0 \le t \le T} |\mathcal{M}_N(t)| > \delta/4 \right) &\le \frac{16}{\delta^2} \mathbb{E}_N^{\mu_N} \left( \left[ \mathcal{M}_N(t) \right]^2 \right) \stackrel{(3.3)}{=} \frac{16}{\delta^2} \left[ \mathbb{E}_N^{\mu_N} \left( \mathcal{N}_N(t) + \int_0^t \mathcal{B}_N(s) ds \right) \right] \\
&= \frac{16}{\delta^2} \left[ \mathbb{E}_N^{\mu_N} \left( \int_0^t \mathcal{B}_N(s) ds \right) \right],
\end{align*}
$$

that goes to zero as $N \to \infty$ — see the control of $\mathcal{B}_N(s)$ below (3.17). Now we focus on the remaining expectation in (4.18), and note that the quantity $|\mathcal{W}_{\pi_N}^{\varepsilon}(t) - \mathcal{M}_N(t)|$ can be bounded by the sum of the following terms

$$
\sup_{0 \le t \le T} \left| \int_0^t \langle \pi_N^{\text{field}}(s), d(\Delta - \Delta_x^N + \partial_{yy}^N)G(s) \rangle + O(1/N) \, ds \right|, \quad (4.19)
$$

$$
\sup_{0 \le t \le T} \left| \int_0^t \left\langle \left[ \pi_N^{\text{field}}(s) * \mathbb{U}_\varepsilon^{\text{up}} \right] \right|_{y=1} d\partial_y G \left|_{y=1}(s) \right\rangle \right|_{\mathbb{T}^{p-1}} - \frac{1}{N^{p-1}} \sum_{i \in \Gamma_N^{\text{up}}} d\partial_y^N G(s, \frac{i}{N}) \times \eta_s(i) \, ds \right|, \quad (4.20)
$$

$$
\sup_{0 \le t \le T} \left| \int_0^t \left\langle \left[ \pi_N^{\text{field}}(s) * \mathbb{U}_\varepsilon^{\text{low}} \right] \right|_{y=0} d\partial_y G \left|_{y=0}(s) \right\rangle \right|_{\mathbb{T}^{p-1}} - \frac{1}{N^{p-1}} \sum_{i \in \Gamma_N^{\text{low}}} d\partial_y^N G(s, \frac{i}{N}) \times \eta_s(i) \, ds \right|, \quad (4.21)
$$

$$
\sup_{0 \le t \le T} \left| \int_0^t \alpha \left\langle \left[ \pi_N^{\text{field}}(s) * \mathbb{U}_\varepsilon^{\text{low}} \right] \right|_{y=0} G \left|_{y=0}(s) \right\rangle \right|_{\mathbb{T}^{p-1}} - \frac{\alpha}{N^{p-1}} \sum_{i \in \Gamma_N^{\text{low}}} G(s, \frac{i}{N}) \times \eta_s(i) \, ds \right|, \quad (4.22)
$$

$$
\sup_{0 \le t \le T} \left| \int_0^t \alpha \langle \pi_N^{\text{road}}(s), G|_{y=0}(s) - G|_{y=\frac{1}{N}}(s) \rangle ds \right|, \quad (4.23)
$$

$$
\sup_{0 \le t \le T} \left| \int_0^t \frac{1}{N^p} \sum_{i \in \Gamma_N^{\text{up}}} G(s, \frac{i}{N}) \times (b - \eta_s(i)) ds \right|, \quad (4.24)
$$

for the “field parts”, and

$$
\sup_{0 \le t \le T} \left| \int_0^t \langle \pi_N^{\text{road}}(s), D(\Delta_x - \Delta_x^N) H(s) \rangle ds \right|, \quad (4.25)
$$

$$
\sup_{0 \le t \le T} \left| \int_0^t \alpha \left\langle \left[ \pi_N^{\text{field}}(s) * \mathbb{U}_\varepsilon^{\text{low}} \right] \right|_{y=0} H(s) \right\rangle_{\mathbb{T}^{p-1}} - \frac{\alpha}{N^{p-1}} \sum_{i \in \Gamma_N^{\text{low}}} H(s, \frac{i}{N}) \times \eta_s(i) \, ds \right|, \quad (4.26)
$$
one requires further attention. By considering the probability under the limit in this second term, we control it by this way:

$$
\begin{align}
\mathbb{Q}_N^{\mu_N} \left( \sup_{0 \le t \le T} |\mathcal{W}_{\pi_N}^{\varepsilon}(t)| > \delta/2 \right) &= \mathbb{Q}_N^{\mu_N} \left( \sup_{0 \le t \le T} |\mathcal{W}_{\pi_N}^{\varepsilon}(t) - \mathcal{M}_N(t) + \mathcal{M}_N(t)| > \delta/2 \right) \nonumber \\
&\le \mathbb{Q}_N^{\mu_N} \left( \sup_{0 \le t \le T} |\mathcal{W}_{\pi_N}^{\varepsilon}(t) - \mathcal{M}_N(t)| > \delta/4 \right) + \mathbb{Q}_N^{\mu_N} \left( \sup_{0 \le t \le T} |\mathcal{M}_N(t)| > \delta/4 \right), \nonumber \\
&\le \frac{4}{\delta} \mathbb{E}_N^{\mu_N} \left( \sup_{0 \le t \le T} |\mathcal{W}_{\pi_N}^{\varepsilon}(t) - \mathcal{M}_N(t)| \right) + \mathbb{Q}_N^{\mu_N} \left( \sup_{0 \le t \le T} |\mathcal{M}_N(t)| > \delta/4 \right), \tag{4.18}
\end{align}
$$

where we recall that the martingale $\mathcal{M}_N(t)$ is defined in (3.1). In (4.18), the vanishing of $\mathbb{Q}_N^{\mu_N}(|\mathcal{M}_N(t)| > \delta/4)$ can be shown by using the Doob's inequality:

$$
\begin{align*}
\mathbb{Q}_N^{\mu_N} \left( \sup_{0 \le t \le T} |\mathcal{M}_N(t)| > \delta/4 \right) &\le \frac{16}{\delta^2} \mathbb{E}_N^{\mu_N} \left( \left[ \mathcal{M}_N(t) \right]^2 \right) \stackrel{(3.3)}{=} \frac{16}{\delta^2} \left[ \mathbb{E}_N^{\mu_N} \left( \mathcal{N}_N(t) + \int_0^t \mathcal{B}_N(s) ds \right) \right] \\
&= \frac{16}{\delta^2} \left[ \mathbb{E}_N^{\mu_N} \left( \int_0^t \mathcal{B}_N(s) ds \right) \right],
\end{align*}
$$

that goes to zero as $N \to \infty$ — see the control of $\mathcal{B}_N(s)$ below (3.17). Now we focus on the remaining expectation in (4.18), and note that the quantity $|\mathcal{W}_{\pi_N}^{\varepsilon}(t) - \mathcal{M}_N(t)|$ can be bounded by the sum of the following terms

$$
\sup_{0 \le t \le T} \left| \int_0^t \langle \pi_N^{\text{field}}(s), d(\Delta - \Delta_x^N + \partial_{yy}^N)G(s) \rangle + O(1/N) \, ds \right|, \quad (4.19)
$$

$$
\sup_{0 \le t \le T} \left| \int_0^t \left\langle \left[ \pi_N^{\text{field}}(s) * \mathbb{U}_\varepsilon^{\text{up}} \right]_{y=1} \right\rangle_{\mathbb{T}^{p-1}} - \frac{1}{N^{p-1}} \sum_{i \in \Gamma_N^{\text{up}}} d\partial_y^N G(s, \frac{i}{N}) \times \eta_s(i) \, ds \right|, \quad (4.20)
$$

$$
\sup_{0 \le t \le T} \left| \int_0^t \left\langle \left[ \pi_N^{\text{field}}(s) * \mathbb{U}_\varepsilon^{\text{low}} \right]_{y=0} \right\rangle_{\mathbb{T}^{p-1}} - \frac{1}{N^{p-1}} \sum_{i \in \Gamma_N^{\text{low}}} d\partial_y^N G(s, \frac{i}{N}) \times \eta_s(i) \, ds \right|, \quad (4.21)
$$

$$
\sup_{0 \le t \le T} \left| \int_0^t \alpha \left\langle \left[ \pi_N^{\text{field}}(s) * \mathbb{U}_\varepsilon^{\text{low}} \right]_{y=0} \right\rangle_{\mathbb{T}^{p-1}} - \frac{\alpha}{N^{p-1}} \sum_{i \in \Gamma_N^{\text{low}}} G(s, \frac{i}{N}) \times \eta_s(i) \, ds \right|, \quad (4.22)
$$

$$
\sup_{0 \le t \le T} \left| \int_0^t \alpha \langle \pi_N^{\text{road}}(s), G|_{y=0}(s) - G|_{y=\frac{1}{N}}(s) \rangle ds \right|, \quad (4.23)
$$

$$
\sup_{0 \le t \le T} \left| \int_0^t \frac{1}{N^p} \sum_{i \in \Gamma_N^{\text{up}}} G(s, \frac{i}{N}) \times (b - \eta_s(i)) ds \right|, \quad (4.24)
$$

for the “field parts”, and

$$
\sup_{0 \le t \le T} \left| \int_0^t \langle \pi_N^{\text{road}}(s), D(\Delta_x - \Delta_x^N) H(s) \rangle ds \right|, \quad (4.25)
$$

$$
\sup_{0 \le t \le T} \left| \int_0^t \alpha \left\langle \left[ \pi_N^{\text{field}}(s) * \mathbb{U}_\varepsilon^{\text{low}} \right]_{y=0} \right\rangle_{\mathbb{T}^{p-1}} - \frac{\alpha}{N^{p-1}} \sum_{i \in \Gamma_N^{\text{low}}} H(s, \frac{i}{N}) \times \eta_s(i) \, ds \right|, \quad (4.26)
$$
for the “road parts”. To conclude this proof, it remains to argue that all these terms (4.19)-(4.26) vanish in the limit $N \to \infty$ then $\varepsilon \to 0$. First of all, due to the regularity of the test functions $G$ and $H$, it is clear that (4.19), (4.23) and (4.25) go to zero as $N \to \infty$. Notice then that (4.24) associated to the upper spawn/kill dynamics is a $O(1/N)$. Lastly, the vanishing of (4.20)-(4.21)-(4.22)-(4.26) arises from the Replacement lemmas (Lemma 4.4 and Lemma 4.5), the approximation of $\eta_s^{\varepsilon N}(i)$ with the convolution products (4.12) and (4.13), and the regularity of the test functions $G$ and $H$. $\square$

# 5 Uniqueness of the solution

In this last section, we establish the uniqueness of the weak solutions to the Cauchy problem (1.2)-(1.3) in the sense (W1)-(W2). Our proof relies on testing the weak formulation (W2) against the solutions $(G, H)$ to a “dual problem” related to (1.2).

**Proposition 5.1 (Uniqueness)** *There exists at most one solution $(v, u)$ to the Cauchy problem (1.2)-(1.3) in the sense (W1)-(W2).*

**Proof of Proposition 5.1 (Uniqueness).** By linearity it is enough to consider the case $(v_0, u_0) = (0, 0)$. Given any $\varphi \in C_c^\infty([0, T] \times \Lambda)$ and any $\psi \in C_c^\infty([0, T] \times \mathbb{T}^{p-1})$, we consider the problem

$$ egin{cases} 
-\partial_t G - d\Delta G = \varphi, & t \in (0, T), \quad x \in \mathbb{T}^{p-1}, \quad y \in (0, 1), \\
-d\partial_y G|_{y=0} = \alpha(H - G|_{y=0}), & t \in (0, T), \quad x \in \mathbb{T}^{p-1}, \quad y = 0, \\
-\partial_t H - D\Delta_x H - \alpha(G|_{y=0} - H) = \psi, & t \in (0, T), \quad x \in \mathbb{T}^{p-1}, \\
\partial_y G|_{y=0} = 0, & t \in (0, T), \quad x \in \mathbb{T}^{p-1}, \quad y = 1, 
\end{cases} 
\quad (5.1) $$

supplemented with final condition

$$ egin{cases} 
G|_{t=T} = 0, & x \in \mathbb{T}^{p-1}, \quad y \in (0, 1), \\
H|_{t=T} = 0, & x \in \mathbb{T}^{p-1}. 
\end{cases} 
\quad (5.2) $$

By letting $(\tilde{G}(t), \tilde{H}(t)) = (G(T-t), H(T-t))$, we can notice that the problem (5.1)-(5.2) is actually a reversed-time field-road system with sources, namely

$$ egin{cases} 
\partial_t \tilde{G} = d\Delta \tilde{G} + \varphi, & t \in (0, T), \quad x \in \mathbb{T}^{p-1}, \quad y \in (0, 1), \\
-d\partial_y \tilde{G}|_{y=0} = \alpha(\tilde{H} - \tilde{G}|_{y=0}), & t \in (0, T), \quad x \in \mathbb{T}^{p-1}, \quad y = 0, \\
\partial_t \tilde{H} = D\Delta_x \tilde{H} + \alpha(\tilde{G}|_{y=0} - \tilde{H}) + \psi, & t \in (0, T), \quad x \in \mathbb{T}^{p-1}, \\
\partial_y \tilde{G}|_{y=0} = 0, & t \in (0, T), \quad x \in \mathbb{T}^{p-1}, \quad y = 1, 
\end{cases} 
\quad (5.3) $$

provided with trivial initial data. In absence of sources, the solution is explicitly known through the “field-road heat kernel”, that could be computed as in [3]. By combining this with the Duhamel principle, see [32, Chapter 4, Section 3] for instance, we obtain the classical solution to the above problem with sources. As a result, we own $G \in C^{1,2}([0, T] \times \bar{\Lambda})$ and $H \in C^{1,2}([0, T] \times \mathbb{T}^{p-1})$ that satisfy (5.1)-(5.2), and that are sufficiently smooth to be tested in the weak formulation (W2). By plugging $(G, H)$ into (2.7) and (2.8), and then summing the two obtained results, we are left with

$$ \int_0^T \langle v(s), \varphi(s) \rangle_{\Lambda} ds + \int_0^T \langle u(s), \psi(s) \rangle_{\mathbb{T}^{p-1}} ds = 0, $$
for the “road parts”. To conclude this proof, it remains to argue that all these terms (4.19)-(4.26) vanish in the limit $N \to \infty$ then $\varepsilon \to 0$. First of all, due to the regularity of the test functions $G$ and $H$, it is clear that (4.19), (4.23) and (4.25) go to zero as $N \to \infty$. Notice then that (4.24) associated to the upper spawn/kill dynamics is a $O(1/N)$. Lastly, the vanishing of (4.20)-(4.21)-(4.22)-(4.26) arises from the Replacement lemmas (Lemma 4.4 and Lemma 4.5), the approximation of $\eta_s^{\varepsilon N}(i)$ with the convolution products (4.12) and (4.13), and the regularity of the test functions $G$ and $H$. $\square$

# 5 Uniqueness of the solution

In this last section, we establish the uniqueness of the weak solutions to the Cauchy problem (1.2)-(1.3) in the sense (W1)-(W2). Our proof relies on testing the weak formulation (W2) against the solutions $(G, H)$ to a “dual problem” related to (1.2).

**Proposition 5.1 (Uniqueness)** *There exists at most one solution $(v, u)$ to the Cauchy problem (1.2)-(1.3) in the sense (W1)-(W2).*

**Proof of Proposition 5.1 (Uniqueness).** By linearity it is enough to consider the case $(v_0, u_0) = (0, 0)$. Given any $\varphi \in C_c^\infty([0, T] \times \Lambda)$ and any $\psi \in C_c^\infty([0, T] \times \mathbb{T}^{p-1})$, we consider the problem

$$ egin{cases} 
-\partial_t G - d\Delta G = \varphi, & t \in (0, T), \quad x \in \mathbb{T}^{p-1}, \quad y \in (0, 1), \\
-d\partial_y G|_{y=0} = \alpha(H - G|_{y=0}), & t \in (0, T), \quad x \in \mathbb{T}^{p-1}, \quad y = 0, \\
-\partial_t H - D\Delta_x H - \alpha(G|_{y=0} - H) = \psi, & t \in (0, T), \quad x \in \mathbb{T}^{p-1}, \\
\partial_y G|_{y=0} = 0, & t \in (0, T), \quad x \in \mathbb{T}^{p-1}, \quad y = 1, 
\end{cases} 
\quad (5.1) $$

supplemented with final condition

$$ egin{cases} 
G|_{t=T} = 0, & x \in \mathbb{T}^{p-1}, \quad y \in (0, 1), \\
H|_{t=T} = 0, & x \in \mathbb{T}^{p-1}. 
\end{cases} 
\quad (5.2) $$

By letting $(\tilde{G}(t), \tilde{H}(t)) = (G(T-t), H(T-t))$, we can notice that the problem (5.1)-(5.2) is actually a reversed-time field-road system with sources, namely

$$ egin{cases} 
\partial_t \tilde{G} = d\Delta \tilde{G} + \varphi, & t \in (0, T), \quad x \in \mathbb{T}^{p-1}, \quad y \in (0, 1), \\
-d\partial_y \tilde{G}|_{y=0} = \alpha(\tilde{H} - \tilde{G}|_{y=0}), & t \in (0, T), \quad x \in \mathbb{T}^{p-1}, \quad y = 0, \\
\partial_t \tilde{H} = D\Delta_x \tilde{H} + \alpha(\tilde{G}|_{y=0} - \tilde{H}) + \psi, & t \in (0, T), \quad x \in \mathbb{T}^{p-1}, \\
\partial_y \tilde{G}|_{y=0} = 0, & t \in (0, T), \quad x \in \mathbb{T}^{p-1}, \quad y = 1, 
\end{cases} 
\quad (5.3) $$

provided with trivial initial data. In absence of sources, the solution is explicitly known through the “field-road heat kernel”, that could be computed as in [3]. By combining this with the Duhamel principle, see [32, Chapter 4, Section 3] for instance, we obtain the classical solution to the above problem with sources. As a result, we own $G \in C^{1,2}([0, T] \times \bar{\Lambda})$ and $H \in C^{1,2}([0, T] \times \mathbb{T}^{p-1})$ that satisfy (5.1)-(5.2), and that are sufficiently smooth to be tested in the weak formulation (W2). By plugging $(G, H)$ into (2.7) and (2.8), and then summing the two obtained results, we are left with

$$ \int_0^T \langle v(s), \varphi(s) \rangle_{\Lambda} ds + \int_0^T \langle u(s), \psi(s) \rangle_{\mathbb{T}^{p-1}} ds = 0, $$
that holds for any $\varphi \in C_c^\infty([0, T] \times \Lambda)$ and any $\psi \in C_c^\infty([0, T] \times \mathbb{T}^{p-1})$. In particular we have

$$ \left\{ \begin{array}{ll} \displaystyle \int_0^T \langle v(s), \varphi(s) \rangle_{\Lambda} ds = 0, & \forall \varphi \in C_c^\infty([0, T] \times \Lambda), \\ \\ \displaystyle \int_0^T \langle u(s), \psi(s) \rangle_{\mathbb{T}^{p-1}} ds = 0, & \forall \psi \in C_c^\infty([0, T] \times \mathbb{T}^{p-1}). \end{array} \right. \qquad (5.4) $$

From (5.4), we can deduce that $v$ and $u$ are both identically zero, see [17, Lemma IV.2], and the proof is therefore completed. □

**Acknowledgement.** Matthieu Alfaro is supported by the *région Normandie* project BIOMA-NORMAN 21E04343 and the ANR project DEEV ANR-20-CE40-0011-01. Samuel Tréton would like to acknowledge the *région Normandie* for the financial support of his PhD.

# Appendix

## A.1 Some tools and basic estimates

To prove the upper Replacement lemma (Lemma 4.4) and the energy estimate (Lemma 4.3), we first need to introduce the relative entropy $\mathcal{H}$ and the Dirichlet form $\mathcal{D}_N$. These notions are rather classical and can be found in [34, Appendix 1, Sections 7-8-9-10] for instance.

For $\gamma \in (0, 1)$, we denote by $\nu_N = \nu_{N,\gamma}^{\text{field}} \otimes \nu_{N,\gamma}^{\text{road}}$ the Bernoulli product measure on $S_N$ whose marginals are given by

$$ \nu_{N,\gamma}^{\text{field}}[\eta(\hat{\iota}) = \cdot] = \begin{cases} \gamma & \text{if } \cdot = 1, \\ 1-\gamma & \text{if } \cdot = 0, \end{cases} \quad \text{for any } \hat{\iota} \in \Lambda_N, \text{ and} $$

$$ \nu_{N,\gamma}^{\text{road}}[\xi(i) = \cdot] = \begin{cases} \gamma & \text{if } \cdot = 1, \\ 1-\gamma & \text{if } \cdot = 0, \end{cases} \quad \text{for any } i \in \Gamma_N^{\text{low}}. $$

The probability measure $\nu_N$ on $S_N$ offers interesting properties to work with the relative entropy and the Dirichlet form. Indeed, since $\nu_N$ gives a positive probability to each configuration $(\eta, \xi) \in S_N$, any probability measure $\mu_N$ on $S_N$ is absolutely continuous with respect to $\nu_N$. Moreover, the changes of variable of type “flip” or “switch” as done below are very simple to express when integrating with respect to $\nu_N$.

For a probability measure $\mu_N$ on $S_N$, the entropy of $\mu_N$ with respect to $\nu_N$ is defined as the positive value

$$ \mathcal{H}(\mu_N|\nu_N) := \sup_{f \in \mathbb{R}^{S_N}} \left\{ \int_{S_N} f(\eta, \xi) \, d\mu_N(\eta, \xi) - \log \left[ \int_{S_N} e^{f(\eta, \xi)} \, d\nu_N(\eta, \xi) \right] \right\}. $$

Since $\mu_N$ is absolutely continuous with respect to $\nu_N$, the entropy can be explicitly written as [34, Appendix 1, Sections 8, Theorem 8.3]

$$ \mathcal{H}(\mu_N|\nu_N) = \int_{S_N} \log \left[ \frac{d\mu_N}{d\nu_N}(\eta, \xi) \right] d\mu_N(\eta, \xi) = \sum_{(\eta, \xi) \in S_N} \mu_N(\eta, \xi) \log \left[ \frac{\mu_N(\eta, \xi)}{\nu_N(\eta, \xi)} \right], $$
that holds for any $\varphi \in C_c^\infty([0, T] \times \Lambda)$ and any $\psi \in C_c^\infty([0, T] \times \mathbb{T}^{p-1})$. In particular we have

$$ \left\{ \begin{array}{ll} \displaystyle \int_0^T \langle v(s), \varphi(s) \rangle_\Lambda ds = 0, & \forall \varphi \in C_c^\infty([0, T] \times \Lambda), \\ \\ \displaystyle \int_0^T \langle u(s), \psi(s) \rangle_{\mathbb{T}^{p-1}} ds = 0, & \forall \psi \in C_c^\infty([0, T] \times \mathbb{T}^{p-1}). \end{array} \right. \qquad (5.4) $$

From (5.4), we can deduce that $v$ and $u$ are both identically zero, see [17, Lemma IV.2], and the proof is therefore completed. □

**Acknowledgement.** Matthieu Alfaro is supported by the *région Normandie* project BIOMA-NORMAN 21E04343 and the ANR project DEEV ANR-20-CE40-0011-01. Samuel Tréton would like to acknowledge the *région Normandie* for the financial support of his PhD.

# Appendix

## A.1 Some tools and basic estimates

To prove the upper Replacement lemma (Lemma 4.4) and the energy estimate (Lemma 4.3), we first need to introduce the relative entropy $\mathcal{H}$ and the Dirichlet form $\mathcal{D}_N$. These notions are rather classical and can be found in [34, Appendix 1, Sections 7-8-9-10] for instance.

For $\gamma \in (0, 1)$, we denote by $\nu_N = \nu_{N,\gamma}^{\text{field}} \otimes \nu_{N,\gamma}^{\text{road}}$ the Bernoulli product measure on $S_N$ whose marginals are given by

$$ \nu_{N,\gamma}^{\text{field}}[\eta(\hat{\iota}) = \cdot] = \begin{cases} \gamma & \text{if } \cdot = 1, \\ 1-\gamma & \text{if } \cdot = 0, \end{cases} \quad \text{for any } \hat{\iota} \in \Lambda_N, \text{ and} $$

$$ \nu_{N,\gamma}^{\text{road}}[\xi(i) = \cdot] = \begin{cases} \gamma & \text{if } \cdot = 1, \\ 1-\gamma & \text{if } \cdot = 0, \end{cases} \quad \text{for any } i \in \Gamma_N^{\text{low}}. $$

The probability measure $\nu_N$ on $S_N$ offers interesting properties to work with the relative entropy and the Dirichlet form. Indeed, since $\nu_N$ gives a positive probability to each configuration $(\eta, \xi) \in S_N$, any probability measure $\mu_N$ on $S_N$ is absolutely continuous with respect to $\nu_N$. Moreover, the changes of variable of type “flip” or “switch” as done below are very simple to express when integrating with respect to $\nu_N$.

For a probability measure $\mu_N$ on $S_N$, the entropy of $\mu_N$ with respect to $\nu_N$ is defined as the positive value

$$ \mathcal{H}(\mu_N|\nu_N) := \sup_{f \in \mathbb{R}^{S_N}} \left\{ \int_{S_N} f(\eta, \xi) d\mu_N(\eta, \xi) - \log \left[ \int_{S_N} e^{f(\eta, \xi)} d\nu_N(\eta, \xi) \right] \right\}. $$

Since $\mu_N$ is absolutely continuous with respect to $\nu_N$, the entropy can be explicitly written as [34, Appendix 1, Sections 8, Theorem 8.3]

$$ \mathcal{H}(\mu_N|\nu_N) = \int_{S_N} \log \left[ \frac{d\mu_N}{d\nu_N}(\eta, \xi) \right] d\mu_N(\eta, \xi) = \sum_{(\eta, \xi) \in S_N} \mu_N(\eta, \xi) \log \left[ \frac{\mu_N(\eta, \xi)}{\nu_N(\eta, \xi)} \right], $$
where $d\mu_N/d\nu_N$ denotes the Radon-Nikodym derivative of $\mu_N$ with respect to $\nu_N$ and the last equality is a consequence of the finiteness of $S_N$. By decomposing $\mu_N$ as a convex combination of Dirac masses and using the convexity of the entropy, we can show that

$$ \mathcal{H}(\mu_N|\nu_N) \leq C_0 N^p, \qquad (A.1) $$

for $C_0 := -\log(\min(\gamma, 1-\gamma)) > 0$.

Given a function $f: S_N \to \mathbb{R}$, the Dirichlet form associated to the dynamics writes

$$ \mathcal{D}_N(f, \nu_N) := -\langle \mathcal{L}_N f, f \rangle_{\nu_N} \qquad (A.2) $$

and may be split following the different parts of $\mathcal{L}_N$ — see (2.1) — into

$$ \mathcal{D}_N = N^2 \mathbb{D}_N^{\text{field}} + N^2 \mathbb{D}_N^{\text{road}} + N \mathcal{D}_N^{\text{Rob}} + \mathcal{D}_N^{\text{reac}} + \mathcal{D}_N^{\text{up}}. \qquad (A.3) $$

At some points in the proofs of upper Replacement lemma (Lemma 4.4) and the energy estimate (Lemma 4.3), it becomes essential to control each component of $\mathcal{D}_N$. These controls are encapsulated in Lemma A.1 below, and require to introduce the functional

$$ \mathcal{I}_N = \mathcal{I}_N(f, \nu_N) := N^2 \mathbb{I}_N^{\text{field}} + N^2 \mathbb{I}_N^{\text{road}} + N \mathcal{I}_N^{\text{Rob}} + \mathcal{I}_N^{\text{reac}} + \mathcal{I}_N^{\text{up}}, \qquad (A.4) $$

where, for any $f: S_N \to \mathbb{R}$,

$$ \mathbb{I}_N^{\text{field}}(f, \nu_N) := \frac{d}{2} \sum_{\substack{\hat{i}, \hat{k} \in \Lambda_N \\ |\hat{i} - \hat{k}| = 1}} \int_{S_N} \left[ \sqrt{f(\eta^{\hat{i}, \hat{k}}, \xi)} - \sqrt{f(\eta, \xi)} \right]^2 d\nu_N(\eta, \xi), $$

$$ \mathbb{I}_N^{\text{road}}(f, \nu_N) := \frac{D}{2} \sum_{\substack{i,k \in \Gamma_N^{\text{low}} \\ |i-k|=1}} \int_{S_N} \left[ \sqrt{f(\eta, \xi^{i,k})} - \sqrt{f(\eta, \xi)} \right]^2 d\nu_N(\eta, \xi), $$

$$ I_N^{\text{Rob}}(f, \nu_N) := \alpha \sum_{i \in \Gamma_N^{\text{low}}} \int_{S_N} (\eta(i) - \xi(i))^2 \left[ \sqrt{f(\eta^i, \xi)} - \sqrt{f(\eta, \xi)} \right]^2 d\nu_N(\eta, \xi), $$

$$ I_N^{\text{reac}}(f, \nu_N) := \alpha \sum_{i \in \Gamma_N^{\text{low}}} \int_{S_N} (\eta(i) - \xi(i))^2 \left[ \sqrt{f(\eta, \xi^i)} - \sqrt{f(\eta, \xi)} \right]^2 d\nu_N(\eta, \xi), $$

$$ I_N^{\text{up}}(f, \nu_N) := \sum_{i \in \Gamma_N^{\text{up}}} \int_{S_N} \left( b(1 - \eta(i)) + (1 - b)\eta(i) \right) \left[ \sqrt{f(\eta^i, \xi)} - \sqrt{f(\eta, \xi)} \right]^2 d\nu_N(\eta, \xi), $$

are all nonnegative.

**Lemma A.1 (Control of the Dirichlet forms)** For any density function $f: S_N \to \mathbb{R}$ with respect to the Bernoulli product measure $\nu_N$ on $S_N$ with parameter $\gamma$, we have

(D1) For any $\gamma \in (0, 1)$,

$$ \langle \mathbb{I}_N^{\text{field}} \sqrt{f}, \sqrt{f} \rangle_{\nu_N} = - \mathbb{D}_N^{\text{field}}(\sqrt{f}, \nu_N) = - \frac{1}{2} \mathbb{I}_N^{\text{field}}(f, \nu_N). \qquad (A.5) $$
where $d\mu_N/d\nu_N$ denotes the Radon-Nikodym derivative of $\mu_N$ with respect to $\nu_N$ and the last equality is a consequence of the finiteness of $S_N$. By decomposing $\mu_N$ as a convex combination of Dirac masses and using the convexity of the entropy, we can show that

$$ \mathcal{H}(\mu_N|\nu_N) \leq C_0 N^p, \qquad (A.1) $$

for $C_0 := -\log(\min(\gamma, 1-\gamma)) > 0$.

Given a function $f: S_N \to \mathbb{R}$, the Dirichlet form associated to the dynamics writes

$$ \mathcal{D}_N(f, \nu_N) := -\langle \mathcal{L}_N f, f \rangle_{\nu_N} \qquad (A.2) $$

and may be split following the different parts of $\mathcal{L}_N$ — see (2.1) — into

$$ \mathcal{D}_N = N^2 \mathbb{D}_N^{\text{field}} + N^2 \mathbb{D}_N^{\text{road}} + N \mathcal{D}_N^{\text{Rob}} + \mathcal{D}_N^{\text{reac}} + \mathcal{D}_N^{\text{up}}. \qquad (A.3) $$

At some points in the proofs of upper Replacement lemma (Lemma 4.4) and the energy estimate (Lemma 4.3), it becomes essential to control each component of $\mathcal{D}_N$. These controls are encapsulated in Lemma A.1 below, and require to introduce the functional

$$ \mathcal{I}_N = \mathcal{I}_N(f, \nu_N) := N^2 \mathbb{I}_N^{\text{field}} + N^2 \mathbb{I}_N^{\text{road}} + N \mathcal{I}_N^{\text{Rob}} + \mathcal{I}_N^{\text{reac}} + \mathcal{I}_N^{\text{up}}, \qquad (A.4) $$

where, for any $f: S_N \to \mathbb{R}$,

$$ \mathbb{I}_N^{\text{field}}(f, \nu_N) := \frac{d}{2} \sum_{\substack{\hat{i}, \hat{k} \in \Lambda_N \\ |\hat{i} - \hat{k}| = 1}} \int_{S_N} \left[ \sqrt{f(\eta^{\hat{i}, \hat{k}}, \xi)} - \sqrt{f(\eta, \xi)} \right]^2 d\nu_N(\eta, \xi), $$

$$ \mathbb{I}_N^{\text{road}}(f, \nu_N) := \frac{D}{2} \sum_{\substack{i,k \in \Gamma_N^{\text{low}} \\ |i-k|=1}} \int_{S_N} \left[ \sqrt{f(\eta, \xi^{i,k})} - \sqrt{f(\eta, \xi)} \right]^2 d\nu_N(\eta, \xi), $$

$$ I_N^{\text{Rob}}(f, \nu_N) := \alpha \sum_{i \in \Gamma_N^{\text{low}}} \int_{S_N} (\eta(i) - \xi(i))^2 \left[ \sqrt{f(\eta^i, \xi)} - \sqrt{f(\eta, \xi)} \right]^2 d\nu_N(\eta, \xi), $$

$$ I_N^{\text{reac}}(f, \nu_N) := \alpha \sum_{i \in \Gamma_N^{\text{low}}} \int_{S_N} (\eta(i) - \xi(i))^2 \left[ \sqrt{f(\eta, \xi^i)} - \sqrt{f(\eta, \xi)} \right]^2 d\nu_N(\eta, \xi), $$

$$ I_N^{\text{up}}(f, \nu_N) := \sum_{i \in \Gamma_N^{\text{up}}} \int_{S_N} \left( b(1 - \eta(i)) + (1 - b)\eta(i) \right) \left[ \sqrt{f(\eta^i, \xi)} - \sqrt{f(\eta, \xi)} \right]^2 d\nu_N(\eta, \xi), $$

are all nonnegative.

**Lemma A.1 (Control of the Dirichlet forms)** For any density function $f: S_N \to \mathbb{R}$ with respect to the Bernoulli product measure $\nu_N$ on $S_N$ with parameter $\gamma$, we have

(D1) For any $\gamma \in (0, 1)$,

$$ \langle \mathbb{I}_N^{\text{field}} \sqrt{f}, \sqrt{f} \rangle_{\nu_N} = - \mathbb{D}_N^{\text{field}}(\sqrt{f}, \nu_N) = - \frac{1}{2} \mathbb{I}_N^{\text{field}}(f, \nu_N). \qquad (A.5) $$
(D2) For any $\gamma \in (0, 1)$,

$$ \langle \mathbb{I}_N^{\text{road}} \sqrt{f}, \sqrt{f} \rangle_{\nu_N} = -\mathbb{D}_N^{\text{road}}(\sqrt{f}, \nu_N) = -\frac{1}{2} \mathbb{I}_N^{\text{road}}(f, \nu_N). \quad (A.6) $$

(D3) For any $\gamma \in (0, 1)$,

$$ \langle L_N^{\text{Rob}} \sqrt{f}, \sqrt{f} \rangle_{\nu_N} = -D_N^{\text{Rob}}(\sqrt{f}, \nu_N) = -\frac{1}{2} I_N^{\text{Rob}}(f, \nu_N) + \varepsilon_N(\alpha, \gamma, f). \quad (A.7) $$

(D4) For any $\gamma \in (0, 1)$,

$$ \langle L_N^{\text{reac}} \sqrt{f}, \sqrt{f} \rangle_{\nu_N} = -D_N^{\text{reac}}(\sqrt{f}, \nu_N) = -\frac{1}{2} I_N^{\text{reac}}(f, \nu_N) + \varepsilon_N(\alpha, \gamma, f). \quad (A.8) $$

(D5) For $\gamma = b$,

$$ \langle L_N^{\text{up}} \sqrt{f}, \sqrt{f} \rangle_{\nu_N} = -D_N^{\text{up}}(\sqrt{f}, \nu_N) = -\frac{1}{2} I_N^{\text{up}}(f, \nu_N). \quad (A.9) $$

In (D3) and (D4), $\varepsilon_N(\alpha, \gamma, f)$ is twice the same quantity, and such that

$$ |\varepsilon_N(\alpha, \gamma, f)| \leq c(\gamma) \alpha N^{p-1}. \quad (A.10) $$

**Proof of Lemma A.1, (D1).** By writing $\mathbb{D}_N^{\text{field}}$ from (A.2) and (2.2), we get

$$ -\mathbb{D}_N^{\text{field}}(\sqrt{f}, \nu_N) = \frac{d}{4} \sum_{\substack{\hat{\gamma}, \hat{\kappa} \in \Lambda_N \\ |\hat{\gamma} - \hat{\kappa}| = 1}} \int_{S_N} \left[ \sqrt{f(\eta^{\hat{\gamma}, \hat{\kappa}}, \xi)} - \sqrt{f(\eta, \xi)} \right] \sqrt{f(\eta, \xi)} \, d\nu_N(\eta, \xi) \\ + \frac{d}{4} \sum_{\substack{\hat{\gamma}, \hat{\kappa} \in \Lambda_N \\ |\hat{\gamma} - \hat{\kappa}| = 1}} \int_{S_N} \left[ \sqrt{f(\eta^{\hat{\gamma}, \hat{\kappa}}, \xi)} - \sqrt{f(\eta, \xi)} \right] \sqrt{f(\eta, \xi)} \, d\nu_N(\eta, \xi). $$

We use then the change of variable $\tilde{\eta} := \eta^{\hat{\gamma}, \hat{\kappa}}$ in the second integral. Since the measure $\nu_N$ is a Bernoulli product measure with constant parameter $\gamma$, this change of variable remains transparent for $\nu_N$. We thus have

$$ -\mathbb{D}_N^{\text{field}}(\sqrt{f}, \nu_N) = \frac{d}{4} \sum_{\substack{\hat{\gamma}, \hat{\kappa} \in \Lambda_N \\ |\hat{\gamma} - \hat{\kappa}| = 1}} \int_{S_N} \left[ \sqrt{f(\eta^{\hat{\gamma}, \hat{\kappa}}, \xi)} - \sqrt{f(\eta, \xi)} \right] \sqrt{f(\eta, \xi)} \, d\nu_N(\eta, \xi) \\ + \frac{d}{4} \sum_{\substack{\hat{\gamma}, \hat{\kappa} \in \Lambda_N \\ |\hat{\gamma} - \hat{\kappa}| = 1}} \int_{S_N} \left[ \sqrt{f(\eta, \xi)} - \sqrt{f(\eta^{\hat{\gamma}, \hat{\kappa}}, \xi)} \right] \sqrt{f(\eta^{\hat{\gamma}, \hat{\kappa}}, \xi)} \, d\nu_N(\eta, \xi) $$

from which follows (A.5).

□

**Proof of Lemma A.1, (D2).** Follow the same method as in the proof of (D1).

□
(D2) For any $\gamma \in (0, 1)$,

$$ \langle \mathbb{I}_N^{\text{road}} \sqrt{f}, \sqrt{f} \rangle_{\nu_N} = -\mathbb{D}_N^{\text{road}}(\sqrt{f}, \nu_N) = -\frac{1}{2} \mathbb{I}_N^{\text{road}}(f, \nu_N). \quad (A.6) $$

(D3) For any $\gamma \in (0, 1)$,

$$ \langle L_N^{\text{Rob}} \sqrt{f}, \sqrt{f} \rangle_{\nu_N} = -D_N^{\text{Rob}}(\sqrt{f}, \nu_N) = -\frac{1}{2} I_N^{\text{Rob}}(f, \nu_N) + \varepsilon_N(\alpha, \gamma, f). \quad (A.7) $$

(D4) For any $\gamma \in (0, 1)$,

$$ \langle L_N^{\text{reac}} \sqrt{f}, \sqrt{f} \rangle_{\nu_N} = -D_N^{\text{reac}}(\sqrt{f}, \nu_N) = -\frac{1}{2} I_N^{\text{reac}}(f, \nu_N) + \varepsilon_N(\alpha, \gamma, f). \quad (A.8) $$

(D5) For $\gamma = b$,

$$ \langle L_N^{\text{up}} \sqrt{f}, \sqrt{f} \rangle_{\nu_N} = -D_N^{\text{up}}(\sqrt{f}, \nu_N) = -\frac{1}{2} I_N^{\text{up}}(f, \nu_N). \quad (A.9) $$

In (D3) and (D4), $\varepsilon_N(\alpha, \gamma, f)$ is twice the same quantity, and such that

$$ |\varepsilon_N(\alpha, \gamma, f)| \le c(\gamma) \alpha N^{p-1}. \quad (A.10) $$

**Proof of Lemma A.1, (D1).** By writing $\mathbb{D}_N^{\text{field}}$ from (A.2) and (2.2), we get

$$ -\mathbb{D}_N^{\text{field}}(\sqrt{f}, \nu_N) = \frac{d}{4} \sum_{\substack{\hat{\gamma}, \hat{\kappa} \in \Lambda_N \\ |\hat{\gamma} - \hat{\kappa}| = 1}} \int_{S_N} \left[ \sqrt{f(\eta^{\hat{\gamma}, \hat{\kappa}}, \xi)} - \sqrt{f(\eta, \xi)} \right] \sqrt{f(\eta, \xi)} \, d\nu_N(\eta, \xi) \\ + \frac{d}{4} \sum_{\substack{\hat{\gamma}, \hat{\kappa} \in \Lambda_N \\ |\hat{\gamma} - \hat{\kappa}| = 1}} \int_{S_N} \left[ \sqrt{f(\eta^{\hat{\gamma}, \hat{\kappa}}, \xi)} - \sqrt{f(\eta, \xi)} \right] \sqrt{f(\eta, \xi)} \, d\nu_N(\eta, \xi). $$

We use then the change of variable $\tilde{\eta} := \eta^{\hat{\gamma}, \hat{\kappa}}$ in the second integral. Since the measure $\nu_N$ is a Bernoulli product measure with constant parameter $\gamma$, this change of variable remains transparent for $\nu_N$. We thus have

$$ -\mathbb{D}_N^{\text{field}}(\sqrt{f}, \nu_N) = \frac{d}{4} \sum_{\substack{\hat{\gamma}, \hat{\kappa} \in \Lambda_N \\ |\hat{\gamma} - \hat{\kappa}| = 1}} \int_{S_N} \left[ \sqrt{f(\eta^{\hat{\gamma}, \hat{\kappa}}, \xi)} - \sqrt{f(\eta, \xi)} \right] \sqrt{f(\eta, \xi)} \, d\nu_N(\eta, \xi) \\ + \frac{d}{4} \sum_{\substack{\hat{\gamma}, \hat{\kappa} \in \Lambda_N \\ |\hat{\gamma} - \hat{\kappa}| = 1}} \int_{S_N} \left[ \sqrt{f(\eta, \xi)} - \sqrt{f(\eta^{\hat{\gamma}, \hat{\kappa}}, \xi)} \right] \sqrt{f(\eta^{\hat{\gamma}, \hat{\kappa}}, \xi)} \, d\nu_N(\eta, \xi) $$

from which follows (A.5).

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

**Proof of Lemma A.1, (D2).** Follow the same method as in the proof of (D1).

<figure><img src="image_5.png" /></figure>
**Proof of Lemma A.1, (D3).** For clarity, let us introduce the notations

$$F := \sqrt{f(\eta, \xi)} \quad \text{and} \quad F^i := \sqrt{f(\eta^i, \xi)}. \qquad (A.11)$$

By writing $\mathbb{D}_N^{\text{field}}$ from (A.2) and (2.4), we get

$$-D_N^{\text{Rob}}(\sqrt{f}, \nu_N) = \frac{\alpha}{2} \sum_{i \in \Gamma_N^{\text{low}}} \int_{S_N} (\eta(i) - \xi(i))^2 (F^i - F) F \, d\nu_N(\eta, \xi) \\ + \frac{\alpha}{2} \sum_{i \in \Gamma_N^{\text{low}}} \int_{\eta(i)=0} (\eta(i) - \xi(i))^2 (F^i - F) F \, d\nu_N(\eta, \xi) \\ + \frac{\alpha}{2} \sum_{i \in \Gamma_N^{\text{low}}} \int_{\eta(i)=1} (\eta(i) - \xi(i))^2 (F^i - F) F \, d\nu_N(\eta, \xi).$$

We then use the change of variable $\tilde{\eta} := \eta^i$ in the integral of the second and third line so that we have

$$\nu_N(\tilde{\eta}, \xi) = \nu_N(\eta, \xi) \times \begin{cases} \gamma/(1-\gamma) & \text{if } \eta(i) = 0, \\ (1-\gamma)/\gamma & \text{if } \eta(i) = 1, \end{cases} = \nu_N(\eta, \xi) \times \begin{cases} 1 + \frac{2\gamma-1}{1-\gamma} & \text{if } \eta(i) = 0, \\ 1 + \frac{1-2\gamma}{\gamma} & \text{if } \eta(i) = 1. \end{cases}$$

This results in

$$-D_N^{\text{Rob}}(\sqrt{f}, \nu_N) = \frac{\alpha}{2} \sum_{i \in \Gamma_N^{\text{low}}} \int_{S_N} (\eta(i) - \xi(i))^2 (F^i - F) F \, d\nu_N(\eta, \xi) \\ + \frac{\alpha}{2} \sum_{i \in \Gamma_N^{\text{low}}} \int_{S_N} (1 - \eta(i) - \xi(i))^2 (F - F^i) F^i \, d\nu_N(\eta, \xi) \\ + \frac{\alpha}{2} \sum_{i \in \Gamma_N^{\text{low}}} \frac{1-\gamma}{2\gamma-1} \int_{\tilde{\eta}(i)=1} (1 - \tilde{\eta}(i) - \xi(i))^2 (F - F^i) F^i \, d\nu_N(\tilde{\eta}, \xi) \\ + \frac{\alpha}{2} \sum_{i \in \Gamma_N^{\text{low}}} \frac{\gamma}{1-2\gamma} \int_{\tilde{\eta}(i)=0} (1 - \tilde{\eta}(i) - \xi(i))^2 (F - F^i) F^i \, d\nu_N(\tilde{\eta}, \xi).$$

Now observe that $(1 - \eta(i) - \xi(i))^2 = (\eta(i) - \xi(i))^2 + (1 - 2\eta(i))(1 - 2\xi(i))$ and plug this into the second line above. This directly yields (A.7) with

$$\begin{aligned} \varepsilon_N(\alpha, \gamma, f) = \frac{\alpha}{2} \sum_{i \in \Gamma_N^{\text{low}}} \int_{S_N} (1 - 2\eta(i))(1 - 2\xi(i))(F - F^i) F^i \, d\nu_N(\eta, \xi) \\ + \frac{\alpha}{2} \sum_{i \in \Gamma_N^{\text{low}}} \frac{1-\gamma}{2\gamma-1} \int_{\eta(i)=1} (1 - \eta(i) - \xi(i))^2 (F - F^i) F^i \, d\nu_N(\eta, \xi) \\ + \frac{\alpha}{2} \sum_{i \in \Gamma_N^{\text{low}}} \frac{\gamma}{1-2\gamma} \int_{\eta(i)=0} (1 - \eta(i) - \xi(i))^2 (F - F^i) F^i \, d\nu_N(\eta, \xi), \end{aligned}$$

and the control of $\varepsilon_N$ (A.10) arises from the Cauchy-Schwarz inequality and the fact that $f$ is a density with respect to the Bernoulli product measure $\nu_N$. $\square$

**Proof of Lemma A.1, (D4).** Follow the same method as in the proof of (D3). $\square$
**Proof of Lemma A.1, (D3).** For clarity, let us introduce the notations

$$F := \sqrt{f(\eta, \xi)} \quad \text{and} \quad F^i := \sqrt{f(\eta^i, \xi)}. \qquad (A.11)$$

By writing $\mathbb{D}_N^{\text{field}}$ from (A.2) and (2.4), we get

$$-D_N^{\text{Rob}}(\sqrt{f}, \nu_N) = \frac{\alpha}{2} \sum_{i \in \Gamma_N^{\text{low}}} \int_{S_N} (\eta(i) - \xi(i))^2 (F^i - F) F \, d\nu_N(\eta, \xi) \\ + \frac{\alpha}{2} \sum_{i \in \Gamma_N^{\text{low}}} \int_{\eta(i)=0} (\eta(i) - \xi(i))^2 (F^i - F) F \, d\nu_N(\eta, \xi) \\ + \frac{\alpha}{2} \sum_{i \in \Gamma_N^{\text{low}}} \int_{\eta(i)=1} (\eta(i) - \xi(i))^2 (F^i - F) F \, d\nu_N(\eta, \xi).$$

We then use the change of variable $\tilde{\eta} := \eta^i$ in the integral of the second and third line so that we have

$$\nu_N(\tilde{\eta}, \xi) = \nu_N(\eta, \xi) \times \begin{cases} \gamma/(1-\gamma) & \text{if } \eta(i) = 0, \\ (1-\gamma)/\gamma & \text{if } \eta(i) = 1, \end{cases} = \nu_N(\eta, \xi) \times \begin{cases} 1 + \frac{2\gamma-1}{1-\gamma} & \text{if } \eta(i) = 0, \\ 1 + \frac{1-2\gamma}{\gamma} & \text{if } \eta(i) = 1. \end{cases}$$

This results in

$$-D_N^{\text{Rob}}(\sqrt{f}, \nu_N) = \frac{\alpha}{2} \sum_{i \in \Gamma_N^{\text{low}}} \int_{S_N} (\eta(i) - \xi(i))^2 (F^i - F) F \, d\nu_N(\eta, \xi) \\ + \frac{\alpha}{2} \sum_{i \in \Gamma_N^{\text{low}}} \int_{S_N} (1 - \eta(i) - \xi(i))^2 (F - F^i) F^i \, d\nu_N(\eta, \xi) \\ + \frac{\alpha}{2} \sum_{i \in \Gamma_N^{\text{low}}} \frac{1-\gamma}{2\gamma-1} \int_{\tilde{\eta}(i)=1} (1 - \tilde{\eta}(i) - \xi(i))^2 (F - F^i) F^i \, d\nu_N(\tilde{\eta}, \xi) \\ + \frac{\alpha}{2} \sum_{i \in \Gamma_N^{\text{low}}} \frac{\gamma}{1-2\gamma} \int_{\tilde{\eta}(i)=0} (1 - \tilde{\eta}(i) - \xi(i))^2 (F - F^i) F^i \, d\nu_N(\tilde{\eta}, \xi).$$

Now observe that $(1 - \eta(i) - \xi(i))^2 = (\eta(i) - \xi(i))^2 + (1 - 2\eta(i))(1 - 2\xi(i))$ and plug this into the second line above. This directly yields (A.7) with

$$\begin{aligned} \varepsilon_N(\alpha, \gamma, f) = \frac{\alpha}{2} \sum_{i \in \Gamma_N^{\text{low}}} \int_{S_N} (1 - 2\eta(i))(1 - 2\xi(i))(F - F^i) F^i \, d\nu_N(\eta, \xi) \\ + \frac{\alpha}{2} \sum_{i \in \Gamma_N^{\text{low}}} \frac{1-\gamma}{2\gamma-1} \int_{\eta(i)=1} (1 - \eta(i) - \xi(i))^2 (F - F^i) F^i \, d\nu_N(\eta, \xi) \\ + \frac{\alpha}{2} \sum_{i \in \Gamma_N^{\text{low}}} \frac{\gamma}{1-2\gamma} \int_{\eta(i)=0} (1 - \eta(i) - \xi(i))^2 (F - F^i) F^i \, d\nu_N(\eta, \xi), \end{aligned}$$

and the control of $\varepsilon_N$ (A.10) arises from the Cauchy-Schwarz inequality and the fact that $f$ is a density with respect to the Bernoulli product measure $\nu_N$. $\square$

**Proof of Lemma A.1, (D4).** Follow the same method as in the proof of (D3). $\square$
**Proof of Lemma A.1, (D5).** As in the proof of (D3), we employ the notations $F$ and $F^i$ defined in (A.11). By writing $D_N^{\text{up}}$ from (A.2) and (2.6), we get

$$ -D_N^{\text{up}}(\sqrt{f}, \nu_N) = b \sum_{i \in \Gamma_N^{\text{up}}} \int_{\eta(i)=0} S_N (F^i F - \frac{1}{2} F^2) d\nu_N(\eta, \xi) - b \sum_{i \in \Gamma_N^{\text{up}}} \int_{\eta(i)=0} \frac{1}{2} F^2 d\nu_N(\eta, \xi) \\ + (1-b) \sum_{i \in \Gamma_N^{\text{up}}} \int_{\eta(i)=1} S_N (F^i F - \frac{1}{2} F^2) d\nu_N(\eta, \xi) - (1-b) \sum_{i \in \Gamma_N^{\text{up}}} \int_{\eta(i)=1} \frac{1}{2} F^2 d\nu_N(\eta, \xi). $$

We then use the change of variable $\tilde{\eta} := \eta^i$ in the second and fourth integrals above so that we have

$$ \nu_N(\tilde{\eta}, \xi) = \nu_N(\eta, \xi) \times \begin{cases} \gamma/(1-\gamma) & \text{if } \eta(i) = 0, \\ (1-\gamma)/\gamma & \text{if } \eta(i) = 1, \end{cases} $$

resulting in

$$ -D_N^{\text{up}}(\sqrt{f}, \nu_N) = b \sum_{i \in \Gamma_N^{\text{up}}} \int_{\eta(i)=0} S_N (F^i F - \frac{1}{2} F^2) d\nu_N(\eta, \xi) - \frac{(1-\gamma)b}{\gamma} \sum_{i \in \Gamma_N^{\text{up}}} \int_{\tilde{\eta}(i)=1} S_N \frac{1}{2} (F^i)^2 d\nu_N(\tilde{\eta}, \xi) \\ + (1-b) \sum_{i \in \Gamma_N^{\text{up}}} \int_{\eta(i)=1} S_N (F^i F - \frac{1}{2} F^2) d\nu_N(\eta, \xi) - \frac{\gamma(1-b)}{1-\gamma} \sum_{i \in \Gamma_N^{\text{up}}} \int_{\tilde{\eta}(i)=0} S_N \frac{1}{2} (F^i)^2 d\nu_N(\tilde{\eta}, \xi). $$

By choosing $\gamma = b$, we get

$$ -D_N^{\text{up}}(\sqrt{f}, \nu_N) = -\frac{b}{2} \sum_{i \in \Gamma_N^{\text{up}}} \int_{\eta(i)=0} S_N (F^2 - 2F^i F + (F^i)^2) d\nu_N(\eta, \xi) \\ - \frac{1-b}{2} \sum_{i \in \Gamma_N^{\text{up}}} \int_{\eta(i)=1} S_N (F^2 - 2F^i F + (F^i)^2) d\nu_N(\eta, \xi) \\ = -\frac{1}{2} \sum_{i \in \Gamma_N^{\text{up}}} \int_{S_N} (b(1-\eta(i)) + (1-b)\eta(i)) (F^i - F)^2 d\nu_N(\eta, \xi), $$

that is (A.9).

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

We also need some useful inequalities that are gathered in the following lemma.

**Lemma A.2 (Useful inequalities)**

(I1) For any sequence of positive numbers $(a_N)$ and $(b_N)$, we have

$$ \limsup_{N \to \infty} \frac{1}{N} \log(a_N + b_N) \le \max \left( \limsup_{N \to \infty} \frac{1}{N} \log(a_N), \limsup_{N \to \infty} \frac{1}{N} \log(b_N) \right). $$

(I2) For any $z \in \mathbb{R}$ we have $e^{|z|} \le e^z + e^{-z}$.

(I3) For any $X, Y \in \mathbb{R}$ and any $B > 0$, we have $XY \le \frac{1}{2B}X^2 + \frac{B}{2}Y^2$.

(I4) For any $X, Y \in \mathbb{R}$, we have $\frac{(X+Y)^2}{2} \le X^2 + Y^2$.
**Proof of Lemma A.1, (D5).** As in the proof of (D3), we employ the notations $F$ and $F^i$ defined in (A.11). By writing $D_N^{\text{up}}$ from (A.2) and (2.6), we get

$$-D_N^{\text{up}}(\sqrt{f}, \nu_N) = b \sum_{i \in \Gamma_N^{\text{up}}} \int_{\eta(i)=0} S_N (F^i F - \frac{1}{2} F^2) d\nu_N(\eta, \xi) - b \sum_{i \in \Gamma_N^{\text{up}}} \int_{\eta(i)=0} \frac{1}{2} F^2 d\nu_N(\eta, \xi) \\ + (1-b) \sum_{i \in \Gamma_N^{\text{up}}} \int_{\eta(i)=1} S_N (F^i F - \frac{1}{2} F^2) d\nu_N(\eta, \xi) - (1-b) \sum_{i \in \Gamma_N^{\text{up}}} \int_{\eta(i)=1} \frac{1}{2} F^2 d\nu_N(\eta, \xi).$$

We then use the change of variable $\tilde{\eta} := \eta^i$ in the second and fourth integrals above so that we have

$$\nu_N(\tilde{\eta}, \xi) = \nu_N(\eta, \xi) \times \begin{cases} \gamma/(1-\gamma) & \text{if } \eta(i) = 0, \\ (1-\gamma)/\gamma & \text{if } \eta(i) = 1, \end{cases}$$

resulting in

$$-D_N^{\text{up}}(\sqrt{f}, \nu_N) = b \sum_{i \in \Gamma_N^{\text{up}}} \int_{\eta(i)=0} S_N (F^i F - \frac{1}{2} F^2) d\nu_N(\eta, \xi) - \frac{(1-\gamma)b}{\gamma} \sum_{i \in \Gamma_N^{\text{up}}} \int_{\tilde{\eta}(i)=1} S_N \frac{1}{2} (F^i)^2 d\nu_N(\tilde{\eta}, \xi) \\ + (1-b) \sum_{i \in \Gamma_N^{\text{up}}} \int_{\eta(i)=1} S_N (F^i F - \frac{1}{2} F^2) d\nu_N(\eta, \xi) - \frac{\gamma(1-b)}{1-\gamma} \sum_{i \in \Gamma_N^{\text{up}}} \int_{\tilde{\eta}(i)=0} S_N \frac{1}{2} (F^i)^2 d\nu_N(\tilde{\eta}, \xi).$$

By choosing $\gamma = b$, we get

$$-D_N^{\text{up}}(\sqrt{f}, \nu_N) = -\frac{b}{2} \sum_{i \in \Gamma_N^{\text{up}}} \int_{\eta(i)=0} S_N (F^2 - 2F^i F + (F^i)^2) d\nu_N(\eta, \xi) \\ -\frac{1-b}{2} \sum_{i \in \Gamma_N^{\text{up}}} \int_{\eta(i)=1} S_N (F^2 - 2F^i F + (F^i)^2) d\nu_N(\eta, \xi) \\ = -\frac{1}{2} \sum_{i \in \Gamma_N^{\text{up}}} \int_{S_N} (b(1-\eta(i)) + (1-b)\eta(i)) (F^i - F)^2 d\nu_N(\eta, \xi),$$

that is (A.9).

<figure><img src="image_6.png" /></figure>

We also need some useful inequalities that are gathered in the following lemma.

**Lemma A.2 (Useful inequalities)**

(11) For any sequence of positive numbers $(a_N)$ and $(b_N)$, we have

$$\limsup_{N \to \infty} \frac{1}{N} \log(a_N + b_N) \le \max \left( \limsup_{N \to \infty} \frac{1}{N} \log(a_N), \limsup_{N \to \infty} \frac{1}{N} \log(b_N) \right).$$

(12) For any $z \in \mathbb{R}$ we have $e^{|z|} \le e^z + e^{-z}$.

(13) For any $X, Y \in \mathbb{R}$ and any $B > 0$, we have $XY \le \frac{1}{2B}X^2 + \frac{B}{2}Y^2$.

(14) For any $X, Y \in \mathbb{R}$, we have $\frac{(X+Y)^2}{2} \le X^2 + Y^2$.
## A.2 Proof of the upper Replacement lemma

**Proof of Lemma 4.4 (Replacement at the upper boundary).** Consider the term under the integral in (4.6), namely

$$A_{N,\varepsilon}(G(s, \frac{\hat{l}}{N}), \eta_s) := \frac{1}{N^{p-1}} \sum_{\hat{l} \in \Gamma_N^{\text{up}}} G(s, \frac{\hat{l}}{N}) \left[ \eta_s^{\varepsilon N}(\hat{l}) - \eta_s(\hat{l}) \right]. \quad (A.12)$$

We develop $\eta_s^{\varepsilon N}(\hat{l})$ in (A.12) with (4.4) and express that $\hat{l} + \hat{k}$, with $\hat{k} = (k, \ell)$, browses $\Lambda_{\hat{l}}^{\varepsilon N}$. This yields

$$A_{N,\varepsilon}(G(s, \frac{\hat{l}}{N}), \eta_s) = \sum_{k \in [-\varepsilon N], [\varepsilon N]}^{N-1} \sum_{\ell=[N-1-\varepsilon N]}^{N-1} \frac{c_{N,\varepsilon}}{N^{p-1}} \sum_{\hat{l} \in \Gamma_N^{\text{up}}} G(s, \frac{\hat{l}}{N}) \left[ \eta_s(\hat{l} + \hat{k}) - \eta_s(\hat{l}) \right]. \quad (A.13)$$

In (A.13), focus on

$$\sum_{\hat{l} \in \Gamma_N^{\text{up}}} G(s, \frac{\hat{l}}{N}) \eta_s(\hat{l} + \hat{k}) = \sum_{\hat{l} \in \Gamma_N^{\text{up}}} \left[ G(s, \frac{\hat{l}}{N}) - G(s, \frac{\hat{l}+\hat{k}}{N}) \right] \eta_s(\hat{l} + \hat{k}) + \sum_{\hat{l} \in \Gamma_N^{\text{up}}} G(s, \frac{\hat{l}+\hat{k}}{N}) \eta_s(\hat{l} + \hat{k}). \quad (A.14)$$

Since $G \in C^{1,2}([0, T] \times \bar{\Lambda})$, the Mean Value Theorem allows to control, in the first sum of (A.14),

$$|G(s, \frac{\hat{l}}{N}) - G(s, \frac{\hat{l}+\hat{k}}{N})| \le p\varepsilon \sup_{s \in [0,T]} \|\nabla G(s, \cdot)\|_{L^{\infty}(\Lambda)}. \quad (A.15)$$

For the second sum in (A.14), notice that we have, thanks to the periodicity of the torus $\mathbb{T}_N$ and the Mean Value Theorem,

$$\begin{aligned} \sum_{\hat{l} \in \Gamma_N^{\text{up}}} G(s, \frac{\hat{l}+\hat{k}}{N}) \eta_s(\hat{l} + \hat{k}) &= \sum_{i \in \mathbb{T}_N^{p-1}} G(s, \frac{i+k}{N}, \frac{\ell}{N}) \eta_s(i+k, \ell) \\ &= \sum_{i \in \mathbb{T}_N^{p-1}} G(s, \frac{i}{N}, \frac{\ell}{N}) \eta_s(i, \ell) \\ &= \sum_{i \in \mathbb{T}_N^{p-1}} \left[ G(s, \frac{i}{N}, \frac{N-1}{N}) + \frac{N-1-\ell}{N} C_1(G, N) \right] \eta_s(i, \ell), \quad (A.16) \end{aligned}$$

where there is $y \in (0, 1)$ such that

$$|C_1(G, N)| = |\partial_y G(s, \frac{i}{N}, y)| \le \sup_{s \in [0,T]} \|\nabla G(s, \cdot)\|_{L^{\infty}(\Lambda)}. \quad (A.17)$$

Now using (A.14) and (A.16) into the expression of $A_{N,\varepsilon}$ in (A.13), we obtain

$$A_{N,\varepsilon}(G(s, \frac{\hat{l}}{N}), \eta_s) = \sum_{k \in [-\varepsilon N], [\varepsilon N]} \sum_{\ell=[N-1-\varepsilon N]}^{N-1} \frac{c_{N,\varepsilon}}{N^{p-1}} \sum_{\hat{l} \in \Gamma_N^{\text{up}}} \left[ G(s, \frac{\hat{l}}{N}) - G(s, \frac{\hat{l}+\hat{k}}{N}) \right] \eta_s(\hat{l} + \hat{k}) \quad (A.18)$$

$$+ \sum_{k \in [-\varepsilon N], [\varepsilon N]} \sum_{\ell=[N-1-\varepsilon N]}^{N-1} \frac{c_{N,\varepsilon}}{N^{p-1}} \sum_{i \in \mathbb{T}_N^{p-1}} \frac{N-1-\ell}{N} C_1(G, N) \eta_s(i, \ell) \quad (A.19)$$

$$+ \sum_{k \in [-\varepsilon N], [\varepsilon N]} \sum_{\ell=[N-1-\varepsilon N]}^{N-1} \frac{c_{N,\varepsilon}}{N^{p-1}} \sum_{i \in \mathbb{T}_N^{p-1}} G(s, \frac{i}{N}, \frac{N-1}{N}) [\eta_s(i, \ell) - \eta_s(i, N-1)]. \quad (A.20)$$
## A.2 Proof of the upper Replacement lemma

**Proof of Lemma 4.4 (Replacement at the upper boundary).** Consider the term under the integral in (4.6), namely

$$A_{N,\varepsilon}(G(s, \frac{\hat{l}}{N}), \eta_s) := \frac{1}{N^{p-1}} \sum_{\hat{l} \in \Gamma_N^{\text{up}}} G(s, \frac{\hat{l}}{N}) \left[ \eta_s^{\varepsilon N}(\hat{l}) - \eta_s(\hat{l}) \right]. \quad (A.12)$$

We develop $\eta_s^{\varepsilon N}(\hat{l})$ in (A.12) with (4.4) and express that $\hat{l} + \hat{k}$, with $\hat{k} = (k, \ell)$, browses $\Lambda_{\hat{l}}^{\varepsilon N}$. This yields

$$A_{N,\varepsilon}(G(s, \frac{\hat{l}}{N}), \eta_s) = \sum_{k \in [-\varepsilon N], [\varepsilon N]}^{N-1} \sum_{\ell=[N-1-\varepsilon N]}^{N-1} \frac{c_{N,\varepsilon}}{N^{p-1}} \sum_{\hat{l} \in \Gamma_N^{\text{up}}} G(s, \frac{\hat{l}}{N}) \left[ \eta_s(\hat{l} + \hat{k}) - \eta_s(\hat{l}) \right]. \quad (A.13)$$

In (A.13), focus on

$$\sum_{\hat{l} \in \Gamma_N^{\text{up}}} G(s, \frac{\hat{l}}{N}) \eta_s(\hat{l} + \hat{k}) = \sum_{\hat{l} \in \Gamma_N^{\text{up}}} \left[ G(s, \frac{\hat{l}}{N}) - G(s, \frac{\hat{l}+\hat{k}}{N}) \right] \eta_s(\hat{l} + \hat{k}) + \sum_{\hat{l} \in \Gamma_N^{\text{up}}} G(s, \frac{\hat{l}+\hat{k}}{N}) \eta_s(\hat{l} + \hat{k}). \quad (A.14)$$

Since $G \in C^{1,2}([0, T] \times \bar{\Lambda})$, the Mean Value Theorem allows to control, in the first sum of (A.14),

$$|G(s, \frac{\hat{l}}{N}) - G(s, \frac{\hat{l}+\hat{k}}{N})| \le p\varepsilon \sup_{s \in [0,T]} \|\nabla G(s, \cdot)\|_{L^{\infty}(\Lambda)}. \quad (A.15)$$

For the second sum in (A.14), notice that we have, thanks to the periodicity of the torus $\mathbb{T}_N$ and the Mean Value Theorem,

$$\begin{aligned} \sum_{\hat{l} \in \Gamma_N^{\text{up}}} G(s, \frac{\hat{l}+\hat{k}}{N}) \eta_s(\hat{l}+\hat{k}) &= \sum_{i \in \mathbb{T}_N^{p-1}} G(s, \frac{i+k}{N}, \frac{\ell}{N}) \eta_s(i+k, \ell) \\ &= \sum_{i \in \mathbb{T}_N^{p-1}} G(s, \frac{i}{N}, \frac{\ell}{N}) \eta_s(i, \ell) \\ &= \sum_{i \in \mathbb{T}_N^{p-1}} \left[ G(s, \frac{i}{N}, \frac{N-1}{N}) + \frac{N-1-\ell}{N} C_1(G, N) \right] \eta_s(i, \ell), \quad (A.16) \end{aligned}$$

where there is $y \in (0, 1)$ such that

$$|C_1(G, N)| = |\partial_y G(s, \frac{i}{N}, y)| \le \sup_{s \in [0, T]} \|\nabla G(s, \cdot)\|_{L^{\infty}(\Lambda)}. \quad (A.17)$$

Now using (A.14) and (A.16) into the expression of $A_{N,\varepsilon}$ in (A.13), we obtain

$$A_{N,\varepsilon}(G(s, \frac{\hat{l}}{N}), \eta_s) = \sum_{k \in [-\varepsilon N], [\varepsilon N]} \sum_{\ell=[N-1-\varepsilon N]}^{N-1} \frac{c_{N,\varepsilon}}{N^{p-1}} \sum_{\hat{l} \in \Gamma_N^{\text{up}}} \left[ G(s, \frac{\hat{l}}{N}) - G(s, \frac{\hat{l}+\hat{k}}{N}) \right] \eta_s(\hat{l} + \hat{k}) \quad (A.18)$$

$$+ \sum_{k \in [-\varepsilon N], [\varepsilon N]} \sum_{\ell=[N-1-\varepsilon N]}^{N-1} \frac{c_{N,\varepsilon}}{N^{p-1}} \sum_{i \in \mathbb{T}_N^{p-1}} \frac{N-1-\ell}{N} C_1(G, N) \eta_s(i, \ell) \quad (A.19)$$

$$+ \sum_{k \in [-\varepsilon N], [\varepsilon N]} \sum_{\ell=[N-1-\varepsilon N]}^{N-1} \frac{c_{N,\varepsilon}}{N^{p-1}} \sum_{i \in \mathbb{T}_N^{p-1}} G(s, \frac{i}{N}, \frac{N-1}{N}) [\eta_s(i, \ell) - \eta_s(i, N-1)]. \quad (A.20)$$
Therefore we have

$$ \mathbb{E}_N^{\mu_N} \left[ \left| \int_0^t \frac{1}{N^{p-1}} \sum_{\hat{\iota} \in \Gamma_N^{\text{up}}} G(s, \frac{\hat{\iota}}{N}) [\eta_s^{\varepsilon N}(\hat{\iota}) - \eta_s(\hat{\iota})] ds \right| \right] \le \\ \mathbb{E}_N^{\mu_N} \left[ \left| \int_0^t (\text{A.18}) ds \right| \right] + \mathbb{E}_N^{\mu_N} \left[ \left| \int_0^t (\text{A.19}) ds \right| \right] + \mathbb{E}_N^{\mu_N} \left[ \left| \int_0^t (\text{A.20}) ds \right| \right]. \quad (\text{A.21}) $$

In (A.21), the vanishing of the two first expectations is a straight consequence of the controls (A.15) and (A.17). Indeed, for the first expectation in (A.21), we have

$$ \begin{aligned} & \mathbb{E}_N^{\mu_N} \left[ \left| \int_0^t (\text{A.18}) ds \right| \right] \\ &= \mathbb{E}_N^{\mu_N} \left[ \left| \int_0^t \sum_{k \in [-\varepsilon N], [\varepsilon N]}^{N-1} \sum_{\ell=[N-1-\varepsilon N]}^{N-1} \frac{c_{N,\varepsilon}}{N^{p-1}} \sum_{\hat{\iota} \in \Gamma_N^{\text{up}}} \left[ G(s, \frac{\hat{\iota}}{N}) - G(s, \frac{\hat{\iota}+\hat{\kappa}}{N}) \right] \eta_s(\hat{\iota} + \hat{\kappa}) ds \right| \right] \\ &\le \int_0^t \sum_{k \in [-\varepsilon N], [\varepsilon N]}^{N-1} \sum_{\ell=[N-1-\varepsilon N]}^{N-1} \frac{c_{N,\varepsilon}}{N^{p-1}} \sum_{\hat{\iota} \in \Gamma_N^{\text{up}}} \underbrace{\left| G(s, \frac{\hat{\iota}}{N}) - G(s, \frac{\hat{\iota}+\hat{\kappa}}{N}) \right|}_{\text{control this with (A.15)}} ds \\ &\le T p \varepsilon \sup_{s \in [0,T]} \|\nabla G(s, \cdot)\|_{L^{\infty}(\Lambda)} \times \underbrace{\sum_{k \in [-\varepsilon N], [\varepsilon N]}^{N-1} \sum_{\ell=[N-1-\varepsilon N]}^{N-1} c_{N,\varepsilon}}_{=1} \times \underbrace{\frac{1}{N^{p-1}} \sum_{\hat{\iota} \in \Gamma_N^{\text{up}}} 1}_{=1} \\ &= T p \varepsilon \sup_{s \in [0,T]} \|\nabla G(s, \cdot)\|_{L^{\infty}(\Lambda)} \end{aligned} $$

that vanishes as $\varepsilon \to 0$. Similarly,

$$ \begin{aligned} & \mathbb{E}_N^{\mu_N} \left[ \left| \int_0^t (\text{A.19}) ds \right| \right] \\ &= \mathbb{E}_N^{\mu_N} \left[ \left| \int_0^t \sum_{k \in [-\varepsilon N], [\varepsilon N]}^{N-1} \sum_{\ell=[N-1-\varepsilon N]}^{N-1} \frac{c_{N,\varepsilon}}{N^{p-1}} \sum_{i \in \mathbb{T}_N^{p-1}} \underbrace{\frac{N-1-\ell}{N}}_{\le \varepsilon} C_1(G, N) \eta_s(i, \ell) ds \right| \right] \\ &\le T \varepsilon \sum_{k \in [-\varepsilon N], [\varepsilon N]}^{N-1} \sum_{\ell=[N-1-\varepsilon N]}^{N-1} \frac{c_{N,\varepsilon}}{N^{p-1}} \sum_{\hat{\iota} \in \Gamma_N^{\text{up}}} \underbrace{|C_1(G, N)|}_{\text{control this with (A.17)}} \\ &\le T \varepsilon \sup_{s \in [0,T]} \|\nabla G(s, \cdot)\|_{L^{\infty}(\Lambda)} \end{aligned} $$

that also vanishes as $\varepsilon \to 0$. The last expectation outlined in (A.21), namely

$$ \mathbb{E}_N^{\mu_N} \left[ \left| \int_0^t (\text{A.20}) ds \right| \right] = \mathbb{E}_N^{\mu_N} \left[ \left| \int_0^t \sum_{k \in [-\varepsilon N], [\varepsilon N]}^{N-1} \sum_{\ell=[N-1-\varepsilon N]}^{N-1} \frac{c_{N,\varepsilon}}{N^{p-1}} \sum_{i \in \mathbb{T}_N^{p-1}} G(s, \frac{i}{N}, \frac{N-1}{N}) [\eta_s(i, \ell) - \eta_s(i, N-1)] ds \right| \right] \quad (\text{A.22}) $$

captures key information about the y-direction of $\eta_s$ in the region $\Lambda_{\hat{\iota}}^{\varepsilon N}$. The vanishing of this expectation is actually the core of this proof.
Therefore we have

$$
\begin{aligned}
& \mathbb{E}_N^{\mu_N} \left[ \left| \int_0^t \frac{1}{N^{p-1}} \sum_{\hat{\iota} \in \Gamma_N^{\text{up}}} G(s, \frac{\hat{\iota}}{N}) [\eta_s^{\varepsilon N}(\hat{\iota}) - \eta_s(\hat{\iota})] ds \right| \right] \le \\
& \qquad \mathbb{E}_N^{\mu_N} \left[ \left| \int_0^t (\text{A.18}) ds \right| \right] + \mathbb{E}_N^{\mu_N} \left[ \left| \int_0^t (\text{A.19}) ds \right| \right] + \mathbb{E}_N^{\mu_N} \left[ \left| \int_0^t (\text{A.20}) ds \right| \right]. \quad (\text{A.21})
\end{aligned}
$$

In (A.21), the vanishing of the two first expectations is a straight consequence of the controls (A.15) and (A.17). Indeed, for the first expectation in (A.21), we have

$$
\begin{aligned}
& \mathbb{E}_N^{\mu_N} \left[ \left| \int_0^t (\text{A.18}) \, ds \right| \right] \\
&= \mathbb{E}_N^{\mu_N} \left[ \left| \int_0^t \sum_{k \in [-\epsilon N], [\epsilon N]}^{p-1} \sum_{\ell=[N-1-\epsilon N]}^{N-1} \frac{c_{N,\epsilon}}{N^{p-1}} \sum_{\hat{\iota} \in \Gamma_N^{\text{up}}} \left[ G(s, \frac{\hat{\iota}}{N}) - G(s, \frac{\hat{\iota}+\hat{\kappa}}{N}) \right] \eta_s(\hat{\iota} + \hat{\kappa}) \, ds \right| \right] \\
&\le \int_0^t \sum_{k \in [-\epsilon N], [\epsilon N]}^{p-1} \sum_{\ell=[N-1-\epsilon N]}^{N-1} \frac{c_{N,\epsilon}}{N^{p-1}} \sum_{\hat{\iota} \in \Gamma_N^{\text{up}}} \underbrace{\left| G(s, \frac{\hat{\iota}}{N}) - G(s, \frac{\hat{\iota}+\hat{\kappa}}{N}) \right|}_{\text{control this with (A.15)}} \, ds \\
&\le T p \epsilon \sup_{s \in [0, T]} \|\nabla G(s, \cdot)\|_{L^{\infty}(\Lambda)} \times \underbrace{\sum_{k \in [-\epsilon N], [\epsilon N]}^{N-1} \sum_{\ell=[N-1-\epsilon N]}^{N-1} c_{N,\epsilon}}_{=1} \times \underbrace{\frac{1}{N^{p-1}} \sum_{\hat{\iota} \in \Gamma_N^{\text{up}}} 1}_{=1} \\
&= T p \epsilon \sup_{s \in [0, T]} \|\nabla G(s, \cdot)\|_{L^{\infty}(\Lambda)}
\end{aligned}
$$

that vanishes as $\epsilon \to 0$. Similarly,

$$
\begin{aligned}
& \mathbb{E}_N^{\mu_N} \left[ \left| \int_0^t (\text{A.19}) \, ds \right| \right] \\
&= \mathbb{E}_N^{\mu_N} \left[ \left| \int_0^t \sum_{k \in [-\epsilon N], [\epsilon N]}^{p-1} \sum_{\ell=[N-1-\epsilon N]}^{N-1} \frac{c_{N,\epsilon}}{N^{p-1}} \sum_{i \in \mathbb{T}_N^{p-1}} \overbrace{\frac{N-1-\ell}{N}}^{\le \epsilon} C_1(G, N) \eta_s(i, \ell) \, ds \right| \right] \\
&\le T \epsilon \sum_{k \in [-\epsilon N], [\epsilon N]}^{p-1} \sum_{\ell=[N-1-\epsilon N]}^{N-1} \frac{c_{N,\epsilon}}{N^{p-1}} \sum_{\hat{\iota} \in \Gamma_N^{\text{up}}} \underbrace{|C_1(G, N)|}_{\text{control this with (A.17)}} \\
&\le T \epsilon \sup_{s \in [0, T]} \|\nabla G(s, \cdot)\|_{L^{\infty}(\Lambda)}
\end{aligned}
$$

that also vanishes as $\epsilon \to 0$. The last expectation outlined in (A.21), namely

$$
\begin{aligned}
& \mathbb{E}_N^{\mu_N} \left[ \left| \int_0^t (\text{A.20}) \, ds \right| \right] \\
&= \mathbb{E}_N^{\mu_N} \left[ \left| \int_0^t \sum_{k \in [-\epsilon N], [\epsilon N]}^{p-1} \sum_{\ell=[N-1-\epsilon N]}^{N-1} \frac{c_{N,\epsilon}}{N^{p-1}} \sum_{i \in \mathbb{T}_N^{p-1}} G(s, \frac{i}{N}, \frac{N-1}{N}) [\eta_s(i, \ell) - \eta_s(i, N-1)] \, ds \right| \right] \quad (\text{A.22})
\end{aligned}
$$

captures key information about the y-direction of $\eta_s$ in the region $\Lambda_{\hat{\iota}}^{\epsilon N}$. The vanishing of this expectation is actually the core of this proof.
Given any fixed $a > 0$ (which will eventually be increased to $+\infty$), we use the entropy inequality (see [34, Appendix 1, Section 8]) on (A.22). This yields

$$ \mathbb{E}_{N}^{\mu_N} \left[ \left| \int_{0}^{t} (\text{A.20}) \, ds \right| \right] \leq \frac{1}{aN^p} \log \left[ \mathbb{E}_{N}^{\nu_N} \left[ \exp \left( aN^p \left| \int_{0}^{t} (\text{A.20}) \, ds \right| \right) \right] \right] + \frac{1}{aN^p} \mathcal{H}(\mu_N | \nu_N). \quad (A.23) $$

Thanks to the control we have on $\mathcal{H}(\mu_N|\nu_N)$, as outlined in (A.1), the second term in (A.23) is bounded by $C_0/a$ that vanishes when $a \to \infty$. Therefore, to conclude the proof, it is enough to show that, for any $a > 0$,

$$ \lim_{\epsilon \to 0} \lim_{N \to \infty} \frac{1}{aN^p} \log \left[ \mathbb{E}_{N}^{\nu_N} \left[ \exp \left( aN^p \left| \int_{0}^{t} (\text{A.20}) \, ds \right| \right) \right] \right] = 0. $$

Now, we remark that we can get rid of the absolute value into the exponential of this last expression. Indeed, by combining (I2) and (I1) in Lemma A.2 with

$$ z = aN^p \int_0^t (\text{A.20}) \, ds, \quad a_n = \mathbb{E}_N^{\nu_N}(e^z), \quad \text{and} \quad b_n = \mathbb{E}_N^{\nu_N}(e^{-z}), $$

we have

$$ \limsup_{N \to \infty} \frac{1}{aN^p} \log \left( \mathbb{E}_N^{\nu_N}(e^{|z|}) \right) \leq \max \left[ \limsup_{N \to \infty} \frac{1}{aN^p} \log \left( \mathbb{E}_N^{\nu_N}(e^z) \right), \limsup_{N \to \infty} \frac{1}{aN^p} \log \left( \mathbb{E}_N^{\nu_N}(e^{-z}) \right) \right], $$

and therefore (up to take $-G$ instead of $G$) we only have to prove the vanishing of

$$ \frac{1}{aN^p} \log \left( \mathbb{E}_N^{\nu_N}(e^z) \right) = \frac{1}{aN^p} \log \left[ \mathbb{E}_N^{\nu_N} \left[ \exp \left( aN^p \int_0^t (\text{A.20}) \, ds \right) \right] \right]. $$

We use now the Feynman-Kac's inequality — see [4, Lemma 7.3 in Appendix] — with the operator $\mathcal{L}_N + aN^p V(s, \cdot)$ where

$$ V(s, \cdot) : \eta \mapsto \sum_{k \in [-\epsilon N], [\epsilon N]}^{N-1} \sum_{\ell=[N-1-\epsilon N]}^{N-1} \frac{c_{N,\epsilon}}{N^{p-1}} \sum_{i \in \mathbb{T}_N^{p-1}} G(s, \frac{i}{N}, \frac{N-1}{N}) [\eta(i, \ell) - \eta(i, N-1)]. \quad (A.24) $$

With the variational formula (Rayleigh quotient) for the largest (principal) eigenvalue of the operator $\mathcal{L}_N + aN^p V(s, \cdot)$, we are led to

$$ \frac{1}{aN^p} \log \left[ \mathbb{E}_N^{\nu_N} \left[ \exp \left( aN^p \int_0^t (\text{A.20}) \, ds \right) \right] \right] \leq \\ \int_0^t \sup_{\substack{f \text{ density} \\ \text{with respect} \\ \text{to } \nu_N}} \left\{ \int_{S_N} V(s, \eta) f(\eta, \xi) \, d\nu_N(\eta, \xi) + \frac{1}{aN^p} \langle \mathcal{L}_N \sqrt{f}, \sqrt{f} \rangle_{\nu_N} \right\} ds. \quad (A.25) $$

Now focus on the integral term into the supremum in (A.25). By using a telescopic sum to write the differences $\eta(i, \ell) - \eta(i, N-1)$ in $V(s, \eta)$ as defined in (A.24), we get

$$ \int_{S_N} V(s, \eta) f(\eta, \xi) \, d\nu_N(\eta, \xi) = \sum_{k \in [-\epsilon N], [\epsilon N]}^{N-1} \sum_{\ell=[N-1-\epsilon N]}^{N-1} \frac{c_{N,\epsilon}}{N^{p-1}} \sum_{i \in \mathbb{T}_N^{p-1}} G(s, \frac{i}{N}, \frac{N-1}{N}) \\ \times \sum_{m=\ell}^{N-2} \int_{S_N} [\eta(i, m) - \eta(i, m+1)] f(\eta, \xi) \, d\nu_N(\eta, \xi). \quad (A.26) $$
Given any fixed $a > 0$ (which will eventually be increased to $+\infty$), we use the entropy inequality (see [34, Appendix 1, Section 8]) on (A.22). This yields

$$ \mathbb{E}_{N}^{\mu_N} \left[ \left| \int_{0}^{t} (\text{A.20}) \, ds \right| \right] \leq \frac{1}{aN^p} \log \left[ \mathbb{E}_{N}^{\nu_N} \left[ \exp \left( aN^p \left| \int_{0}^{t} (\text{A.20}) \, ds \right| \right) \right] \right] + \frac{1}{aN^p} \mathcal{H}(\mu_N | \nu_N). \quad (A.23) $$

Thanks to the control we have on $\mathcal{H}(\mu_N|\nu_N)$, as outlined in (A.1), the second term in (A.23) is bounded by $C_0/a$ that vanishes when $a \to \infty$. Therefore, to conclude the proof, it is enough to show that, for any $a > 0$,

$$ \lim_{\epsilon \to 0} \lim_{N \to \infty} \frac{1}{aN^p} \log \left[ \mathbb{E}_{N}^{\nu_N} \left[ \exp \left( aN^p \left| \int_{0}^{t} (\text{A.20}) \, ds \right| \right) \right] \right] = 0. $$

Now, we remark that we can get rid of the absolute value into the exponential of this last expression. Indeed, by combining (I2) and (I1) in Lemma A.2 with

$$ z = aN^p \int_0^t (\text{A.20}) \, ds, \quad a_n = \mathbb{E}_N^{\nu_N}(e^z), \quad \text{and} \quad b_n = \mathbb{E}_N^{\nu_N}(e^{-z}), $$

we have

$$ \limsup_{N \to \infty} \frac{1}{aN^p} \log \left( \mathbb{E}_N^{\nu_N}(e^{|z|}) \right) \leq \max \left[ \limsup_{N \to \infty} \frac{1}{aN^p} \log \left( \mathbb{E}_N^{\nu_N}(e^z) \right), \limsup_{N \to \infty} \frac{1}{aN^p} \log \left( \mathbb{E}_N^{\nu_N}(e^{-z}) \right) \right], $$

and therefore (up to take $-G$ instead of $G$) we only have to prove the vanishing of

$$ \frac{1}{aN^p} \log \left( \mathbb{E}_N^{\nu_N}(e^z) \right) = \frac{1}{aN^p} \log \left[ \mathbb{E}_N^{\nu_N} \left[ \exp \left( aN^p \int_0^t (\text{A.20}) \, ds \right) \right] \right]. $$

We use now the Feynman-Kac's inequality — see [4, Lemma 7.3 in Appendix] — with the operator $\mathcal{L}_N + aN^p V(s, \cdot)$ where

$$ V(s, \cdot) : \eta \mapsto \sum_{k \in [-\epsilon N], [\epsilon N]}^{N-1} \sum_{\ell=[N-1-\epsilon N]}^{N-1} \frac{c_{N,\epsilon}}{N^{p-1}} \sum_{i \in \mathbb{T}_N^{p-1}} G(s, \frac{i}{N}, \frac{N-1}{N}) [\eta(i, \ell) - \eta(i, N-1)]. \quad (A.24) $$

With the variational formula (Rayleigh quotient) for the largest (principal) eigenvalue of the operator $\mathcal{L}_N + aN^p V(s, \cdot)$, we are led to

$$ \frac{1}{aN^p} \log \left[ \mathbb{E}_N^{\nu_N} \left[ \exp \left( aN^p \int_0^t (\text{A.20}) \, ds \right) \right] \right] \leq \\ \int_0^t \sup_{\substack{f \text{ density} \\ \text{with respect} \\ \text{to } \nu_N}} \left\{ \int_{S_N} V(s, \eta) f(\eta, \xi) \, d\nu_N(\eta, \xi) + \frac{1}{aN^p} \langle \mathcal{L}_N \sqrt{f}, \sqrt{f} \rangle_{\nu_N} \right\} ds. \quad (A.25) $$

Now focus on the integral term into the supremum in (A.25). By using a telescopic sum to write the differences $\eta(i, \ell) - \eta(i, N-1)$ in $V(s, \eta)$ as defined in (A.24), we get

$$ \int_{S_N} V(s, \eta) f(\eta, \xi) \, d\nu_N(\eta, \xi) = \sum_{k \in [-\epsilon N], [\epsilon N]}^{N-1} \sum_{\ell=[N-1-\epsilon N]}^{N-1} \frac{c_{N,\epsilon}}{N^{p-1}} \sum_{i \in \mathbb{T}_N^{p-1}} G(s, \frac{i}{N}, \frac{N-1}{N}) \\ \times \sum_{m=\ell}^{N-2} \int_{S_N} [\eta(i, m) - \eta(i, m+1)] f(\eta, \xi) \, d\nu_N(\eta, \xi). \quad (A.26) $$