# Stationary fluctuations of run-and-tumble particles

Frank Redig111Delft Institute of Applied Mathematics, TU Delft, Delft, The
Netherlands<EMAIL_ADDRESS>Hidde van Wiechen222Delft Institute of
Applied Mathematics, TU Delft, Delft, The Netherlands<EMAIL_ADDRESS>
## Abstract

  * We study the stationary fluctuations of independent run-and-tumble particles. We prove that the joint densities of particles with given internal state converges to an infinite dimensional Ornstein-Uhlenbeck process. We also consider an interacting case, where the particles are subjected to exclusion. We then study the fluctuations of the total density, which is a non-Markovian Gaussian process, and obtain its covariance in closed form. By considering small noise limits of this non-Markovian Gaussian process, we obtain in a concrete example a large deviation rate function containing memory terms.

##  1 Introduction

In this paper we consider a system of independent run-and-tumble particles on
${\mathbb{Z}}$ and study the stationary fluctuations of its empirical
distribution. Because particles have positions and internal states (which
determine the direction in which they move and/or their rate of hopping over
lattice edges), the hydrodynamic limit is a system of linear reaction-
diffusion equations, describing the macroscopic joint evolution of the
densities of particles with a given internal state. In this sense, the paper
can be viewed as a study of macroscopic properties of the multi-layer particle
systems which we studied in [12]. The study of hydrodynamic limits and
fluctuations around the hydrodynamic limit for particles with internal states,
or alternatively, multi-layer systems is quite recent, and to our knowledge at
present only a limited set of results is known: see [5], [6], [7], [13].

Our interest in multi-layer systems is motivated from the study of active
particles (see e.g. [3]), the study of double diffusivity models (see e.g. [6]
and references therein), and finally the study of particle systems described
macrosopically by equations containing memory terms. In this paper we consider
multi-layer systems in which duality can be applied. Duality is a powerful
tool which reduces the study of the hydrodynamic limit to the scaling limit of
a single (dual) particle, and as we show in this paper (see Section 3.1 below)
also determines uniquely the covariance of the stationary fluctuations of the
empirical density of particles. Provided one can show that the stationary
fluctuations converge to a Gaussian limiting (distribution-valued) process,
this limiting covariance uniquely determines the limiting stationary Gaussian
process.

In our paper we prove that the fluctuation fields of the densities of
particles with given internal state converge to a system of stochastic partial
differential equations. In these limiting equations, the drift is determined
by the hydrodynamic limit, whereas the noise has both a conservative part
coming from the transport of particles with a given internal state as well as
a non-conservative part coming from the flipping of internal states. We first
deal with a system of independent particles, which has a simple dual
consisting of independent particles with reversed velocities. Next we indicate
how to deal with interacting particles such as layered exclusion processes,
where still duality can be used.

One of our motivations of studying fluctuation fields of particles with
internal states is to understand fluctuation properties of the total density,
i.e., disregarding the internal states of the particles. The configuration
which gives at each site the total number of particles is one of the simplest
examples of a non-Markovian interacting particle system. The study of the
hydrodynamic limit, fluctuations and large deviations around the hydrodynamic
limit for non-Markovian particle systems is largely terra incognita.
Therefore, we believe that simple examples in which one can have some grip on
the explicit form of fluctuations and large deviations are important to
obtain.

In our setting, we prove that the fluctuations of the total density of
particles converges to a Gaussian distribution-valued process which satisfies
a non-Markovian SPDE. We provide a concrete example where we can explicitly
characterize the large deviations of the limiting SPDE in the small noise
limit. These large deviations give an indication of the large deviations of
the total density of particles. The latter can of course also be obtained via
a contraction principle from the large deviations of the joint densities of
particles with a given internal state. However, the large deviation rate
function obtained via this contraction principle is very implicit, and
therefore in this paper we preferred not to follow this road in order to
obtain an explicit form of the memory terms of the rate function.

The rest of our paper is organized as follows. In Section 2 we introduce the
run-and-tumble particle model and state preliminary results on ergodic
measures, duality and hydrodynamic limit, the latter of which will be proven
in the appendix A. In Section 3 we state the main result on stationary
fluctuations for independent particles, Theorem 3.1, provide a direct proof of
the limiting covariance in Section 3.1, and consider an interacting case,
namely a multi-layer version of the symmetric exclusion process, in Section
3.2. In Section 4 we study the hydrodynamic limit and the fluctuations of the
total density of particles, and prove a large deviations result for the
limiting fluctuation process in a particular case. In Section 5 we prove the
Theorem 3.1.

##  2 Basic notations and definitions

In this paper we will look at the run-and-tumble particle process, which is a
process designed to model active particles. Let $V:=\mathbb{Z}\times S$, with
$S\subset\mathbb{Z}$ a finite set. The set $V$ is the state space of a single
run-and-tumble particle. We see elements $v=(x,\sigma)\in V$ as particles with
position $x\in\mathbb{Z}$ and internal state $\sigma\in S$. The dynamics of a
single run-and-tumble particle are now as follows

  * i.

At rate $\kappa N^{2}$ the particle performs a nearest neighbor jump, i.e.,
$(x,\sigma)\to(x\pm 1,\sigma)$

  * ii.

At rate $\lambda N$ the particle performs an active jump in the direction of
its internal state, i.e., $(x,\sigma)\to(x+\sigma,\sigma)$.

  * iii.

At rate $c(\sigma,\sigma^{\prime})$ the particle changes its internal state
from $\sigma$ to $\sigma^{\prime}$, i.e. $(x,\sigma)\to(x,\sigma^{\prime})$.
Here we assume that the rates
$\left\\{c(\sigma,\sigma^{\prime}):\sigma,\sigma^{\prime}\in S\right\\}$ are
irreducible and symmetric, i.e.,
$c(\sigma,\sigma^{\prime})=c(\sigma^{\prime},\sigma)$.

The run-and-tumble particle process is the process of configurations
consisting of independent run-and-tumble particles. More precisely it is a
Markov process $\\{\eta_{t}:t\geq 0\\}$ on the state space
$\Omega:=\mathbb{N}^{V}$ consisting of independent random walkers on $V$ where
every particle has the dynamics as described above.

From the dynamics we can write down the following generator $L_{N}$ acting on
local functions, i.e., functions $f:\Omega\to\mathbb{R}$ which only depend on
a finite number of sites in $V$.

$\begin{split}L_{N}f(\eta)&=\kappa N^{2}\sum_{(x,\sigma)\in
V}\eta(x,\sigma)\left(f\big{(}\eta^{(x,\sigma)\to(x+1,\sigma)}\big{)}+f\big{(}\eta^{(x,\sigma)\to(x-1,\sigma)}\big{)}-2f(\eta)\right)\\\
&\ \ \ \ +\lambda N\sum_{(x,\sigma)\in
V}\eta(x,\sigma)\left(f\big{(}\eta^{(x,\sigma)\to(x+\sigma,\sigma)}\big{)}-f(\eta)\right)\\\
&\ \ \ \ +\sum_{(x,\sigma)\in V}\sum_{\sigma^{\prime}\in
S}\eta(x,\sigma)c(\sigma,\sigma^{\prime})\left(f\big{(}\eta^{(x,\sigma)\to(x,\sigma^{\prime})}\big{)}-f(\eta)\right).\end{split}$
(1)

Here $\eta(x,\sigma)$ denotes the number of particles at site $(x,\sigma)\in
V$ in the configuration $\eta$, and $\eta^{(x,\sigma)\to(y,\sigma^{\prime})}$
denotes the configuration $\eta$ where a single particle has moved from
$(x,\sigma)$ to $(y,\sigma^{\prime})$.

With this choice of scaling, in the macroscopic limit, the densities of
particles with a given internal state satisfy a system of linear reaction-
diffusion equations (see section 5.1 below for the explicit form).
Equivalently, one can view the choice of scaling as a diffusive time scale
($t\to N^{2}t$), a weak asymmetry (active jumps in the direction of the
velocity occur at rate $N=N^{-1}N^{2}$), and a slow reaction term (changes of
internal state happen at rate $1=N^{-2}N^{2}$. The scaling is also such that
the motion of a single particle converges to a multi-layer Brownian motion
with layer-dependent drift (cf. section 2.1 below).

###  2.1 Scaling limit of the single particle dynamics

We will denote by $\mathscr{L}_{N}$ the Markov generator of a single run-and-
tumble particle (rescaled in space), more precisely, the generator of the
process $(\tfrac{X_{t}}{N},\sigma_{t})$ where $X_{t}$ denotes the position and
$\sigma_{t}$ the internal state of the particle.

This generator acts on a core consisting of test functions on the space
$\mathbb{R}\times S$, which we denote by $C_{c,S}^{\infty}$, and which is
defined via

$C_{c,S}^{\infty}:=\left\\{\phi:\mathbb{R}\times
S\to\mathbb{R}:\phi(\cdot,\sigma)\in C_{c}^{\infty}(\mathbb{R})\text{ for all
$\sigma\in S$}\right\\}.$

The generator $\mathscr{L}_{N}$ then reads as follows:

$\displaystyle\mathscr{L}_{N}\phi(x,\sigma)$ $\displaystyle=\kappa
N^{2}(\phi(x+\tfrac{1}{N},\sigma)+\phi(x-\tfrac{1}{N},\sigma)-2\phi(x,\sigma))+\lambda
N(\phi(x+\tfrac{\sigma}{N},\sigma)-\phi(x,\sigma))$ $\displaystyle\ \ \ \
+\sum_{\sigma^{\prime}\in
S}c(\sigma,\sigma^{\prime})(\phi(x,\sigma^{\prime})-\phi(x,\sigma)).$

Corresponding to this generator we have the corresponding Markov semigroup
which we denote by $S^{N}_{t}$. Via Taylor approximation we obtain that
$\mathscr{L}_{N}\phi\to A\phi$ uniformly as $N\to\infty$, where $A$ is the
differential operator given by

$A\phi(x,\sigma)=\left(\tfrac{\kappa}{2}\partial_{xx}+\sigma\lambda\partial_{x}\right)\phi(x,\sigma)+\sum_{\sigma^{\prime}\in
S}c(\sigma,\sigma^{\prime})\big{(}\phi(x,\sigma^{\prime})-\phi(x,\sigma)\big{)}.$
(2)

Because $A$ generates a Markov semigroup as well, as a consequence of the
convergence of the generators we can also obtain $S^{N}_{t}\phi\to e^{tA}\phi$
uniformly for all $\phi\in C_{0,S}$, i.e., the functions space consisting of
functions $\phi:\mathbb{R}\times S\to\mathbb{R}$ such that
$\phi(\cdot,\sigma)\in C_{0}(\mathbb{R})$ for all $\sigma\in S$.

The operator $A$ above is also an operator on (a subset of) the Hilbert space
$L^{2}(\mathop{}\\!\mathrm{d}x\times|\cdot|_{S})$, where $|\cdot|_{S}$ is the
counting measure over $S$. The inner product on this Hilbert space, denoted by
$\left<\left<\cdot,\cdot\right>\right>$, is the following

$\left<\left<\phi,\psi\right>\right>:=\sum_{\sigma\in
S}\int_{\mathbb{R}}\phi(x,\sigma)\psi(x,\sigma)\mathop{}\\!\mathrm{d}x.$ (3)

Later on we will need the adjoint of the operator $A$ with respect to this
inner product, which acts on $\phi\in C_{c,S}^{\infty}$ as follows:

$A^{*}\phi(x,\sigma)=\left(\tfrac{\kappa}{2}\partial_{xx}-\sigma\lambda\partial_{x}\right)\phi(x,\sigma)+\sum_{\sigma^{\prime}\in
S}c(\sigma,\sigma^{\prime})\big{(}\phi(x,\sigma^{\prime})-\phi(x,\sigma)\big{)}.$
(4)

###  2.2 Basic properties of independent run-and-tumble particles

Before we state the theorem of the stationary fluctuations, we first review a
few known results on run-and-tumble particles which we need.

####  2.2.1 Stationary ergodic product measures

We define the measures $\mu_{\rho}$, with $\rho\in[0,\infty)$, as the product
Poisson measure with density $\rho$, i.e.

$\mu_{\rho}:=\bigotimes_{(x,\sigma)\in V}\mathrm{Pois}(\rho).$

In [12] it is proved that these measures are stationary and ergodic with
respect for run-and-tumble particle process $\\{\eta_{t}:t\geq 0\\}$. For this
reason, when we study the stationary fluctuations of the densities of
particles with given internal state, we will start the process
$\\{\eta_{t}:t\geq 0\\}$ from the measure $\mu_{\rho}$.

####  2.2.2 Duality

######  Definition 2.1.

We say that two Markov processes $\\{\eta_{t}:t\geq 0\\}$ and
$\\{\xi_{t}:t\geq 0\\}$, on the state spaces $\Omega$ and $\Omega^{\prime}$
respectively, are dual to one another with respect to a duality function
$D:\Omega\times\Omega^{\prime}\to\mathbb{R}$ if

$\mathbb{E}_{\eta}\left[D(\xi,\eta_{t})\right]=\widehat{\mathbb{E}}_{\xi}\left[D(\xi_{t},\eta)\right]<\infty,$
(5)

where $\mathbb{E}_{\eta}$ denotes the expectation in $\\{\eta_{t}:t\geq 0\\}$
starting from $\eta$ and $\widehat{\mathbb{E}}_{\xi}$ the expectation in the
dual process $\\{\xi_{t}:t\geq 0\\}$ starting from $\xi$.

In [12] it is proved that the run-and-tumble particle process is dual to its
time-reversed process where the active jumps are in the reverse direction,
i.e., the process corresponding to the following generator

$\begin{split}\widehat{L}_{N}f(\eta)&=\kappa N^{2}\sum_{(x,\sigma)\in
V}\eta(x,\sigma)\left(f\big{(}\eta^{(x,\sigma)\to(x+1,\sigma)}\big{)}+f\big{(}\eta^{(x,\sigma)\to(x-1,\sigma)}\big{)}-2f(\eta)\right)\\\
&\ \ \ \ +\lambda N\sum_{(x,\sigma)\in
V}\eta(x,\sigma)\left(f\big{(}\eta^{(x,\sigma)\to(x-\sigma,\sigma)}\big{)}-f(\eta)\right)\\\
&\ \ \ \ +\sum_{(x,\sigma)\in V}\sum_{\sigma^{\prime}\in
S}\eta(x,\sigma)c(\sigma,\sigma^{\prime})\left(f\big{(}\eta^{(x,\sigma)\to(x,\sigma^{\prime})}\big{)}-f(\eta)\right).\end{split}$

The duality function is then given by

$D(\xi,\eta)=\prod_{(x,\sigma)\in
V}\frac{\eta(x,\sigma)!}{\xi(x,\sigma)!(\eta(x,\sigma)-\xi(x,\sigma))!}\cdot
I\big{(}\xi(x,\sigma)\leq\eta(x,\sigma)\big{)},$

where $I$ denotes the indicator function, and where $\xi$ is assumed to be a
finite configuration, i.e.,

$\sum_{(x,\sigma)}\xi(x,\sigma)<\infty$

In our paper we will mostly need this duality relation in the form of duality
with a single dual particle, i.e.,

$\mathbb{E}_{\eta}[\eta_{t}(x,\sigma)]=\widehat{\mathbb{E}}_{(x,\sigma)}[\eta(\widehat{X}_{t},\widehat{\sigma}_{t})],$

where $(\tfrac{\widehat{X}_{t}}{N},\widehat{\sigma}_{t})$ is the process
corresponding to the (time-reversed) generator $\widehat{\mathscr{L}}_{N}$
given by

$\displaystyle\widehat{\mathscr{L}}_{N}\phi(x,\sigma)$ $\displaystyle=\kappa
N^{2}(\phi(x+\tfrac{1}{N},\sigma)+\phi(x-\tfrac{1}{N},\sigma)-2\phi(x,\sigma))+\lambda
N(\phi(x-\tfrac{\sigma}{N},\sigma)-\phi(x,\sigma))$ $\displaystyle\ \ \ \
+\sum_{\sigma^{\prime}\in
S}c(\sigma,\sigma^{\prime})(\phi(x,\sigma^{\prime})-\phi(x,\sigma)).$

We denote the corresponding Markov semigroup of this process as
$\widehat{S}_{t}^{N}$. By a Taylor expansion, we obtain that
$\widehat{\mathscr{L}}_{N}\phi\to A^{*}\phi$, with $A^{*}$ defined as in (4),
uniformly in $N$ for all $\phi\in C_{c,S}^{\infty}$, and therefore we are able
to write for all $\phi\in C_{0,S}$ that $\widehat{S}^{N}_{t}\phi\to
e^{tA^{*}}\phi$ uniformly.

###  2.3 Hydrodynamic limit

In this section we will briefly mention the hydrodynamic limit of the run-and-
tumble particle process. For the proof, which follows standard methodology, we
refer to the appendix.

Given a function $\rho:\mathbb{R}\times S\to\mathbb{R}$ such that
$\rho(\cdot,\sigma)\in C_{b}^{2}(\mathbb{R})$ for all $\sigma\in S$, we start
by defining the product Poisson measures $\mu_{\rho}^{N}$ for every
$N\in\mathbb{N}$ as follows

$\mu_{\rho}^{N}:=\bigotimes_{(x,\sigma)\in
V}\mathrm{Pois}\big{(}\rho(\tfrac{x}{N},\sigma)\big{)}.$ (6)

This is the local equilibrium distribution corresponding to the macroscopic
profile $\rho$.

Furthermore, for every $N\in\mathbb{N}$, the process $\\{\eta_{t}^{N}:t\geq
0\\}$ is the run-and-tumble particle process started from
$\eta_{0}^{N}\sim\mu_{\rho}^{N}$. We can now define the empirical measures of
the process, denoted by $\pi^{N}=\left\\{\pi_{t}^{N}:t\geq 0\right\\}$, as
follows

$\pi^{N}_{t}:=\frac{1}{N}\sum_{(x,\sigma)\in
V}\eta^{N}_{t}(x,\sigma)\delta_{(\tfrac{x}{N},\sigma)},$ (7)

where $\delta$ is the dirac measure. We think of $\pi^{N}_{t}$ as the
macroscopic profile corresponding to the microscopic configuration $\eta_{t}$.
In the rhs of (7) every particle of type $\sigma$ contributes a mass $1/N$ at
the “macro spatial location” $x/N$.

For every $t\geq 0$, $\pi^{N}_{t}$ is a positive measure on $\mathbb{R}\times
S$ such that when paired with a test function $\phi\in C_{c,S}^{\infty}$ we
obtain

$\pi^{N}_{t}(\phi):=\left<\phi,\pi^{N}_{t}\right>=\frac{1}{N}\sum_{(x,\sigma)\in
V}\eta^{N}_{t}(x,\sigma)\phi(\tfrac{x}{N},\sigma).$

By the choice of the initial distribution, we have at time $t=0$ zero that

$\pi^{N}_{0}(\phi)\to\int\rho(x,\sigma)\phi(x,\sigma)dx$

We then have the following result for the hydrodynamic limit.

######  Theorem 2.1.

For every $t\geq 0$, $\varepsilon>0$ and $\phi\in C_{c,S}^{\infty}$, we have
that

$\lim_{N\to\infty}\mathbb{P}\left(\left|\pi_{t}^{N}(\phi)-\sum_{\sigma\in
S}\int\rho_{t}(x,\sigma)\phi(x,\sigma)dx\right|>\varepsilon\right)=0,$

where $\rho_{t}(x,\sigma)$ solves the PDE $\dot{\rho}_{t}=A^{*}\rho_{t}$ with
initial condition $\rho_{0}(x,\sigma)=\rho(x,\sigma)$.

This results is actually a corollary of an stronger theorem which shows
convergence of the trajectories $\pi^{N}$ in the path space
$D([0,T];\textbf{M})$ equipped with the Skorokhod topology, where M is the
space of Radon measures on $\mathbb{R}\times S$. Let $\pi=\\{\pi_{t}:t\geq
0\\}$ denote the trajectory of measures on $\mathbb{R}\times S$ such that for
all $t\geq 0,\phi\in C_{c,S}^{\infty}$ we have that
$\left<\phi,\pi_{t}\right>=\left<\left<\phi,\rho_{t}\right>\right>$, where
$\rho_{t}$ solves the PDE in the above theorem. The trajectory $\pi$ is then
the unique continuous path in $D([0,T];\textbf{M})$ such that for all $\phi\in
C_{c,S}^{\infty}$

${\mathscr{M}}^{\phi}_{t}(\pi)=\pi_{t}(\phi)-\pi_{0}(\phi)-\int_{0}^{t}\pi_{s}(A\phi)\mathop{}\\!\mathrm{d}s=0.$
(8)

######  Theorem 2.2.

For any $N\in\mathbb{N}$, let $P^{N}$ be the law of the process $\pi^{N}$.
Then $P^{N}\to\delta_{\pi}$ weakly in $D([0,T];\emph{{M}})$ for any $T>0$,
with $\pi$ the unique continuous path solving (8).

For the sake of self-containedness, the proof of Theorem 2.2 is provided in
the appendix. The method of proof is standard and it follows Seppäläinen, in
[14, Chapter 8].

###  2.4 Fluctuation fields

For every $N\in\mathbb{N}$, we define the fluctuation field
$Y^{N}:=\\{Y_{t}^{N}:t\geq 0\\}$ as

$Y^{N}_{t}=\frac{1}{\sqrt{N}}\sum_{x\in\mathbb{Z}}\big{(}\eta_{t}(x,\sigma)-\rho\big{)}\delta_{(\tfrac{x}{N},\sigma)}.$
(9)

This process takes values in the space of distributions on $\mathbb{R}\times
S$, denoted by $(C_{c,S}^{\infty})^{*}$. We expect the fluctuation field
$Y^{N}$ to converge weakly to a generalized stationary Ornstein-Uhlenbeck
process. Before we can state the result we first recall some basic definitions
of space-time white noise (see e.g. [8] for a detailed account).

######  Definition 2.2.

A random distribution $\mathscr{W}$ is called a _white noise_ on
$\mathbb{R}\times S$ if $\\{\left<\phi,\mathscr{W}\right>:\phi\in
C^{\infty}_{c,s}\\}$ is jointly centered Gaussian with covariance

$\mathbb{E}[\left<\phi,\mathscr{W}\right>\left<\psi,\mathscr{W}\right>]=\left<\left<\phi,\psi\right>\right>.$

where $\langle\langle,\rangle\rangle$ denotes the inner product defined in
(3). We denote by $\mathop{}\\!\mathrm{d}\mathscr{W}_{t}$ the time-
differential of space-time white noise. This object is such that when paired
with a test function $\phi\in C_{c,S}^{\infty}$ and integrated over time gives
a Brownian motion, i.e.,

$\int_{0}^{t}\left<\phi,\mathop{}\\!\mathrm{d}\mathscr{W}_{s}\right>=B(\left<\left<\phi,\phi\right>\right>t),$

where $B(\cdot)$ is a standard Brownian motion on $\mathbb{R}$. We denote by
$\frac{\mathop{}\\!\mathrm{d}\mathscr{W}_{t}}{\mathop{}\\!\mathrm{d}t}$ the
corresponding space-time white noise. This random space-time distribution is
such that for all $\phi:[0,T]\times\mathbb{R}\times S\to\mathbb{R}$, with
$\phi(t,\cdot)$ a test function
$\langle\phi,\frac{\mathop{}\\!\mathrm{d}\mathscr{W}_{t}}{\mathop{}\\!\mathrm{d}t}\rangle$
is jointly Gaussian with covariance

$\mathbb{E}\left[\left\langle\phi,\frac{\mathop{}\\!\mathrm{d}\mathscr{W}_{t}}{\mathop{}\\!\mathrm{d}t}\right\rangle\left\langle\psi,\frac{\mathop{}\\!\mathrm{d}\mathscr{W}_{t}}{\mathop{}\\!\mathrm{d}t}\right\rangle\right]=\int_{0}^{T}\left\langle\left\langle\phi(t,\cdot),\psi(t,\cdot)\right\rangle\right\rangle\mathop{}\\!\mathrm{d}t.$

######  Remark 2.1.

Informally speaking, a white noise on $\mathbb{R}\times S$ is a Gaussian field
$W(x,\sigma)$ with covariance $\delta(x-y)\delta_{\sigma,\sigma^{\prime}}$,
and a space-time white noise on $\mathbb{R}\times S$ is a Gaussian field
$W(t,x,\sigma)$ with covariance
$\delta(t^{\prime}-t)\delta(x-y)\delta_{\sigma,\sigma^{\prime}}$.

##  3 Stationary fluctuations

We are now ready to state our result on stationary fluctuations. We start with
the case of independent particles; in Section 3.2 below we will consider an
interacting case.

######  Theorem 3.1.

Assume that $\eta_{0}$ is distributed according to the Poisson product measure
$\mu_{\rho}$. For every $N\in\mathbb{N}$, let $Q^{N}$ denote the law of the
process $Y^{N}$ defined in (9). Then $Q^{N}\to Q$ weakly in
$D([0,T];(C_{c,S}^{\infty})^{*})$ for any $T>0$, where $Q$ is the law of the
stationary Gaussian process $Y$ satisfying the following SPDE

$\mathop{}\\!\mathrm{d}Y_{t}=A^{*}Y_{t}\mathop{}\\!\mathrm{d}t+\sqrt{2\kappa\rho}\partial_{x}\mathop{}\\!\mathrm{d}\mathscr{W}_{t}+\sqrt{2\rho\Sigma}\mathop{}\\!\mathrm{d}\mathscr{\tilde{W}}_{t}.$
(10)

Here $\mathop{}\\!\mathrm{d}\mathscr{W}_{t}$ and
$\mathop{}\\!\mathrm{d}\mathscr{\tilde{W}}_{t}$ are two independent space-time
white noises on the space $\mathbb{R}\times S$, and $\Sigma$ is the operator
working on test functions $\phi\in C_{c,S}^{\infty}$ as

$(\Sigma\phi)(x,\sigma)=-\sum_{\sigma^{\prime}\in
S}c(\sigma,\sigma^{\prime})\big{(}\phi(x,\sigma^{\prime})-\phi(x,\sigma)\big{)}.$
(11)

By the assumed symmetry of the rates $c(\sigma,\sigma^{\prime})$, for
$\phi,\psi\in C_{c,S}^{\infty}$ we have
$\left<\left<\Sigma\phi,\psi\right>\right>=\left<\left<\phi,\Sigma\psi\right>\right>$,
and moreover $\left<\left<\Sigma\phi,\psi\right>\right>\geq 0$. Hence the
operator is bounded, self-adjoint and non-negative and therefore its square
root $\sqrt{\Sigma}$ is well-defined. The process
$\partial_{x}\mathop{}\\!\mathrm{d}\mathscr{W}_{t}$ is defined as the process
of distributions such that for all $\phi\in C_{c,S}^{\infty}$

$\left<\phi,\partial_{x}\mathop{}\\!\mathrm{d}\mathscr{W}_{t}\right>=-\left<\partial_{x}\phi,\mathop{}\\!\mathrm{d}\mathscr{W}_{t}\right>.$

The rigorous meaning of the SPDE in (10) is defined in terms of a martingale
problem as in [9]. More precisely, the map $\phi\mapsto Y_{t}(\phi)$ is the
solution of the following martingale problem: for every $\phi\in
C_{c,S}^{\infty}$, the following two processes

$\begin{split}\mathscr{M}^{\phi}_{t}(Y)&=Y_{t}(\phi)-Y_{0}(\phi)-\bigintsss_{0}^{t}Y_{s}(A\phi)ds,\\\
\mathscr{N}^{\phi}_{t}(Y)&=\mathscr{M}^{\phi}_{t}(Y)^{2}-2t\kappa\rho\left<\left<\partial_{x}\phi,\partial_{x}\phi\right>\right>-2t\rho\left<\left<\phi,\Sigma\phi\right>\right>\end{split}$
(12)

are martingales with respect to the natural filtration
$\mathscr{F}_{t}=\sigma(Y_{s}:0\leq s\leq t)$.

###  3.1 Stationary covariance of the fluctuation fields via duality

We will first compare the covariance structure of the limiting process of
$Y^{N}$ with the covariance structure of the process solving the SPDE in (10).
This covariance uniquely characterizes the process. More precisely, if we can
prove that $Y^{N}_{t}\to Y_{t}$ where $Y_{t}$ is a distribution-valued
stationary Gaussian process, then the covariance
$\mathbb{E}(Y_{t}(\phi)Y_{0}(\psi))$ uniquely determines this process. In that
sense, the computation of the covariance already determines the only possible
candidate limit $Y_{t}$. As we show below, the covariance is in turn
completely determined by the scaling limit of a single dual particle. This
shows that for systems with duality, both the hydrodynamic limit and the
stationary fluctuations are uniquely determined by the scaling limit of a
single dual particle.

######  Proposition 3.2.

For all $\phi,\psi\in C_{c}^{\infty}(\mathbb{R}\times S)$

$\lim_{N\to\infty}\mathbb{E}[Y_{t}^{N}(\phi)Y_{0}^{N}(\psi)]=\mathbb{E}[Y_{t}(\phi)Y_{0}(\psi)]=\rho\cdot\left<\left<e^{tA}\phi,\psi\right>\right>.$

Here $\mathbb{E}$ denotes the stationary expectation starting from the initial
configuration distributed according to $\eta_{0}\sim\mu_{\rho}$.

###### Proof.

If $Y$ is a solution to the SPDE in (10), then we can write

$Y_{t}(\phi)=\mathscr{M}_{t}^{\phi}(Y)+Y_{0}(\phi)+\int_{0}^{t}Y_{s}(A\phi)\mathop{}\\!\mathrm{d}s,$

where $\mathscr{M}_{t}^{\phi}(Y)$ is a martingale with respect to the
filtration $\mathscr{F}_{t}=\sigma\left(Y_{s}:0\leq s\leq t\right)$ such that
$\mathscr{M}_{0}^{\phi}(Y)=0$. By the martingale property we have that

$\mathbb{E}[\mathscr{M}_{t}^{\phi}(Y)Y_{0}(\psi)]=\mathbb{E}\big{[}\mathbb{E}[\mathscr{M}_{t}^{\phi}(Y)Y_{0}(\psi)|\mathscr{F}_{0}]\big{]}=\mathbb{E}\big{[}Y_{0}(\psi)\mathbb{E}[\mathscr{M}_{t}^{\phi}(Y)|\mathscr{F}_{0}]\big{]}=0,$

and so

$\displaystyle\mathbb{E}[Y_{t}(\phi)Y_{0}(\psi)]=\mathbb{E}[Y_{0}(\phi)Y_{0}(\psi)]+\int_{0}^{t}\mathbb{E}[Y_{s}(A\phi)Y_{0}(\psi)]\mathop{}\\!\mathrm{d}s.$

Therefore, using that
$\mathbb{E}[Y_{0}(\phi)Y_{0}(\psi)]=\left<\left<\phi,\psi\right>\right>$ we
obtain that if $Y$ is a solution of (10), then we have

$\mathbb{E}[Y_{t}(\phi)Y_{0}(\psi)]=\mathbb{E}[Y_{0}(e^{tA}\phi)Y_{0}(\psi)]=\rho\cdot\left<\left<e^{tA}\phi,\psi\right>\right>.$

On the other hand, for any $N\in\mathbb{N}$ we have that

$\displaystyle\mathbb{E}\left[Y^{N}_{t}(\phi)Y^{N}_{0}(\psi)\right]$
$\displaystyle=\frac{1}{N}\sum_{(x,\sigma)\in V}\sum_{(y,\sigma^{\prime})\in
V}\phi(\tfrac{x}{N},\sigma)\psi(\tfrac{y}{N},\sigma^{\prime})\int\mathbb{E}_{\eta}\left[(\eta_{t}(x,\sigma)-\rho)(\eta(y,\sigma^{\prime})-\rho)\right]\mathop{}\\!\mathrm{d}\mu_{\rho}(\eta)$
$\displaystyle=\frac{1}{N}\sum_{(x,\sigma)\in V}\sum_{(y,\sigma^{\prime})\in
V}\phi(\tfrac{x}{N},\sigma)\psi(\tfrac{y}{N},\sigma^{\prime})\int\widehat{\mathbb{E}}_{(x,\sigma)}\left[(\eta(\widehat{X}_{t},\widehat{\sigma}_{t})-\rho)(\eta(y,\sigma^{\prime})-\rho)\right]\mathop{}\\!\mathrm{d}\mu_{\rho}(\eta)$
$\displaystyle=\frac{1}{N}\sum_{(x,\sigma)\in V}\sum_{(y,\sigma^{\prime})\in
V}\phi(\tfrac{x}{N},\sigma)\psi(\tfrac{y}{N},\sigma^{\prime})\widehat{\mathbb{E}}_{(x,\sigma)}\left[\mathrm{Cov}_{\mu_{\rho}}\left(\eta(\widehat{X}_{t},\widehat{\sigma}_{t}),\eta(y,\sigma^{\prime})\right)\right],$
(13)

where we used duality for the second equality and Fubini for the last
equality. Now note that, because $\mu_{\rho}$ is a product of Poisson
measures, the covariance term is equal to $\rho$ if and only if
$(\widehat{X}_{t},\widehat{\sigma}_{t})=(y,\sigma^{\prime})$ and zero
otherwise. Therefore

$\displaystyle\sum_{(y,\sigma^{\prime})\in
V}\psi(\tfrac{y}{N},\sigma^{\prime})\widehat{\mathbb{E}}_{(x,\sigma)}$
$\displaystyle\left[\mathrm{Cov}_{\mu_{\rho}}\left(\eta(\widehat{X}_{t},\widehat{\sigma}_{t}),\eta(y,\sigma^{\prime})\right)\right]$
$\displaystyle=\rho\sum_{(y,\sigma^{\prime})\in
V}\psi(\tfrac{y}{N},\sigma^{\prime})\widehat{\mathbb{E}}_{(x,\sigma)}\left[I\left((\widehat{X}_{t},\widehat{\sigma}_{t})=(y,\sigma^{\prime})\right)\right]$
$\displaystyle=\rho\cdot(\widehat{S}_{t}^{N}\psi)(\tfrac{x}{N},\sigma).$ (14)

Here $\widehat{S}_{t}^{N}$ is the semigroup of the Markov process
$(\tfrac{\widehat{X}_{t}}{N},\widehat{\sigma}_{t})$, for which we have the
following uniform convergence $\widehat{S}_{t}^{N}\psi\to e^{tA^{*}}\psi$ (see
Section 2.2.2) . By now combining (3.1) and (3.1), we find that

$\mathbb{E}\left[Y^{N}_{t}(\phi)Y^{N}_{0}(\psi)\right]=\rho\cdot\frac{1}{N}\sum_{(x,\sigma)\in
V}\sum_{(y,\sigma^{\prime})\in
V}\phi(\tfrac{x}{N},\sigma)(\widehat{S}_{t}^{N}\psi)(\tfrac{x}{N},\sigma)\to\rho\cdot\left<\left<\phi,e^{tA^{*}}\psi\right>\right>=\rho\cdot\left<\left<e^{tA}\phi,\psi\right>\right>,$

which concludes the proof. ∎

######  Remark 3.1.

In Proposition 3.2, the only place where the independence of the particles is
manifest is in the pre-factor $\rho$ which corresponds to the limiting
variance of the fluctuation field at time zero, because $\eta_{0}$ is
distributed as $\mu_{\rho}$. When considering any other system which satisfies
duality, when $A$ is the scaling limit of the single particle generator, and
$\chi(\rho)$ is the limiting variance of the fluctuation field at time zero,
we find that the limiting covariance is given by

$\mathbb{E}[Y_{t}(\phi)Y_{0}(\psi)]=\chi(\rho)\left<\left<e^{tA}\phi.\psi\right>\right>.$

E.g. for the exclusion process studied in the section below,
$\chi(\rho)=\rho(\alpha-\rho)$.

###  3.2 Interacting case: the multi-layer SEP

The multi-layer symmetric exclusion process, or multi-layer SEP, is a
generalization of the symmetric exclusion process on $\mathbb{Z}$ to the
multi-layered setting on $\mathbb{Z}\times S$. For this process we look at
configurations $\eta\in\\{0,1,...,\alpha\\}^{V}$ with $\alpha\in\mathbb{N}$,
i.e., there are at most $\alpha$ particles per site $v\in V$. Instead of
having an active component on every layer $\sigma\in S$ like the run-and-
tumble particle system, multi-layer SEP switches to a different diffusion
coëfficient, denoted by $\kappa_{\sigma}$, between the layers. The generator
of this process is then as follows

$\begin{split}L^{SEP}_{N}f(\eta)&=N^{2}\sum_{(x,\sigma)\in
V}\kappa_{\sigma}\sum_{|x-y|=1}\eta(x,\sigma)\left(\alpha-\eta(y,\sigma)\right)\left(f\big{(}\eta^{(x,\sigma)\rightarrow(y,\sigma)}\big{)}-f(\eta)\right)\\\
&\ \ \ \ \ \ \ \ +\sum_{(x,\sigma)\in V}\sum_{\sigma^{\prime}\in
S}c(\sigma,\sigma^{\prime})\eta(x,\sigma)(\alpha-\eta(x,\sigma^{\prime}))\left(f\big{(}\eta^{(x,\sigma)\rightarrow(x,\sigma^{\prime})}\big{)}-f(\eta)\right).\end{split}$

In [12] it is proved that this process is self-dual and has ergodic measures
given by product Binomial measures $\nu_{\rho}=\bigotimes_{v\in
V}\mathrm{Bin}(\alpha,\rho)$ where $\rho\in(0,1)$ is constant.

The corresponding single-particle generator is then given by

$\mathscr{L}_{N}^{SEP}\phi(x,\sigma)=\alpha\kappa_{\sigma}\big{(}(\phi(x+\tfrac{1}{N},\sigma)+\phi(x-\tfrac{1}{N},\sigma)-2\phi(x,\sigma)\big{)}+\sum_{\sigma^{\prime}\in
S}c(\sigma,\sigma^{\prime})\big{(}\phi(x,\sigma^{\prime})-\phi(x,\sigma)\big{)},$

and $\mathscr{L}_{N}^{SEP}\phi\to B\phi$ uniformly, where

$(B\phi)(x,\sigma)=\frac{\alpha\kappa_{\sigma}}{2}\partial_{xx}\phi(x,\sigma)+\sum_{\sigma^{\prime}\in
S}\alpha
c(\sigma,\sigma^{\prime})\big{(}\phi(x,\sigma^{\prime})-\phi(x,\sigma)\big{)}.$

Since we took the rates $c(\sigma,\sigma^{\prime})$ symmetric, this operator
is self-adjoint in the Hilbert space
$L^{2}(\mathop{}\\!\mathrm{d}x\times|\cdot|_{S})$.

Using the same line of proof as in Section 5 below, we obtain the following
SPDE for the stationary fluctuation field,

$\mathop{}\\!\mathrm{d}Y_{t}=BY_{t}\mathop{}\\!\mathrm{d}t+\sqrt{2\rho(\alpha-\rho)K}\partial_{x}\mathop{}\\!\mathrm{d}\mathscr{W}_{t}+\sqrt{2\rho(\alpha-\rho)\Sigma}\mathop{}\\!\mathrm{d}\mathscr{\tilde{W}}_{t}.$
(15)

Here $K$ is the operator given by
$(K\phi)(x,\sigma)=\kappa_{\sigma}\phi(x,\sigma)$. Note in the noise terms the
appearance of the terms $\rho(\alpha-\rho)$ instead of $\rho$ as in (10). This
comes from the fact that for $(x,\sigma)\neq(y,\sigma^{\prime})$

$\mathbb{E}_{\nu_{\rho}}[\eta_{s}(x,\sigma)(\alpha-\eta_{s}(y,\sigma^{\prime}))]=\rho(\alpha-\rho),$

which plays a role in the calculation of the expectation of the Carré du champ
operator.

##  4 Scaling limits of the total density

If we sum over the layers, i.e., over the $\sigma$-variables, then the
resulting configuration which gives the total number of particles at each site
is no longer a Markov process. Therefore, both in the hydrodynamic limit as
well as in the fluctuations we expect memory terms to appear in the form of
higher order time derivatives in the limiting equations. The stationary
fluctuations of the empirical distribution of the total number of particles
will then become a non-Markovian Gaussian process which we can identify
explicitly.

Next, we consider the small-noise limit of these fluctuations. We then obtain
a large deviation principle via large deviations of Schilder’s type for
Gaussian processes (i.e., small variance limit of Gaussian processes, see e.g.
[4] p. 88, and also [10]), and we have memory terms in the corresponding large
deviation rate function. We will make these memory effects explicit in the
simplest possible setting where $\kappa=0$ in (1). To our knowledge, this is
the first example of an explicit expression for a large deviation rate
function of the empirical distribution of particles in a non-Markovian
context. In general such rate functions can be obtained from the contraction
principle of the Markovian multi-layer system, but this expression in the form
of an infimum is implicit, can rarely be made explicit, and therefore does not
make manifest the effect of memory terms.

In the whole of this section, for notational simplicity, we further restrict
to $S=\\{-1,1\\}$ (two layers) and put $c(1,-1)=c(-1,1)=:\gamma$. The aim is
then to study the fluctuations of the total density of particles, where we sum
up the particles in both layers. This produces an empirical measure and
fluctuation field on $\mathbb{R}$ given by

$\zeta^{N}_{t}=\frac{1}{N}\sum_{(x,\sigma)\in
V}\eta^{N}_{t}(x,\sigma)\delta_{\tfrac{x}{N}},\ \ \ \ \ \
Z^{N}_{t}=\frac{1}{\sqrt{N}}\sum_{(x,\sigma)\in
V}(\eta_{t}(x,\sigma)-\rho)\delta_{\tfrac{x}{N}}.$

###  4.1 Hydrodynamic equation for the total density

From Theorem 2.1 we can deduce that $\zeta_{t}^{N}$ converges in probability
to $\varrho_{t}(x)\mathop{}\\!\mathrm{d}x$, where the density $\varrho_{t}(x)$
is the sum of the densities on both layers, i.e.,
$\varrho_{t}(x)=\rho_{t}(x,1)+\rho_{t}(x,-1)$ with $\rho_{t}(x,\sigma)$ the
solution to the hydrodynamic equation $\dot{\rho}_{t}=A^{*}\rho_{t}$. We can
rewrite this equation as a coupled system of linear PDE’s given by

$\begin{cases}\dot{\rho}_{t}(x,1)=\left(\frac{\kappa}{2}\partial_{xx}-\lambda\partial_{x}\right)\rho_{t}(x,1)+\gamma(\rho_{t}(x,-1)-\rho_{t}(x,1)),\\\
\\\
\dot{\rho}_{t}(x,-1)=\left(\frac{\kappa}{2}\partial_{xx}+\lambda\partial_{x}\right)\rho_{t}(x,-1)+\gamma(\rho_{t}(x,1)-\rho_{t}(x,-1)).\end{cases}$

Summing up both equations gives us a PDE for the total density
$\varrho_{t}(x)$. This PDE also depends on the difference of the densities,
which we will denote by $\Delta_{t}(x):=\rho_{t}(x,1)-\rho_{t}(x,-1)$, and
therefore we get a new system of PDE’s,

$\begin{cases}\dot{\varrho}_{t}(x)=\frac{\kappa}{2}\partial_{xx}\varrho_{t}(x)-\lambda\partial_{x}\Delta_{t}(x),\\\
\\\
\dot{\Delta}_{t}(x)=\frac{\kappa}{2}\partial_{xx}\Delta_{t}(x)-\lambda\partial_{x}\varrho_{t}(x)-2\gamma\Delta_{t}(x).\end{cases}$
(16)

From this system we can actually find a closed equation for $\varrho(x)$.
Namely, by first taking a second derivative in time of the upper equation we
find that

$\displaystyle\ddot{\varrho}_{t}(x)=\frac{\kappa}{2}\partial_{xx}\dot{\varrho}_{t}(x)-\lambda\partial_{x}\dot{\Delta}_{t}(x)=\frac{\kappa}{2}\partial_{xx}\dot{\varrho}_{t}(x)-\lambda\partial_{x}\left(\frac{\kappa}{2}\partial_{xx}\Delta_{t}(x)-\lambda\partial_{x}\varrho_{t}(x)-2\gamma\Delta_{t}(x)\right).$
(17)

Now we use that from the first equation in (16) we have
$-\lambda\partial_{x}\Delta_{t}(x)=\dot{\varrho}_{t}(x)-\frac{\kappa}{2}\partial_{xx}\varrho_{t}(x)$.
Substituting this in (17), we find the following closed equation for the total
density

$\ddot{\varrho}_{t}(x)-(\kappa\partial_{xx}+2\gamma)\dot{\varrho}_{t}(x)=\left((\lambda^{2}-\gamma\kappa)\partial_{xx}-\frac{\kappa^{2}}{4}(\partial_{x})^{4}\right)\varrho_{t}(x).$

###  4.2 Fluctuations of the total density

For the analysis of the fluctuation field of the total density we first define
the fluctuation fields of each layer, and then by taking higher order
derivatives as in the previous subsection, we obtain a second order SPDE for
the fluctuations of the total density (cf. (4.4) below). We first set up a
framework where we can rigorously deal with the various distributions coming
from the SPDE given in (10) corresponding to both layers. We start by defining
a fluctuation field for each layer individually.

$Y_{t,\sigma}^{N}=\frac{1}{\sqrt{N}}\sum_{x\in\mathbb{Z}}(\eta_{t}(x,\sigma)-\rho)\delta_{\tfrac{x}{N}},\
\sigma\in\\{-1,1\\}$

The relation between these fluctuation fields and $Z^{N}_{t}$ is as follows:
for every $\phi\in C_{c}^{\infty}$ we have that

$\left<\phi,Z^{N}_{t}\right>=\left<\phi,Y_{t,1}^{N}\right>+\left<\phi,Y_{t,-1}^{N}\right>.$
(18)

However, there is also a direct relation between the fluctuation fields on
both layers and the fluctuation field $Y^{N}_{t}$ on $\mathbb{R}\times S$
defined in (9): for every $\phi\in C_{c,S}^{\infty}$ the following holds

$\left<\phi,Y_{t}^{N}\right>=\left<\phi(\cdot,1),Y_{t,1}^{N}\right>+\left<\phi(\cdot,-1),Y_{t,-1}^{N}\right>.$
(19)

In this way $Y_{t}^{N}$, but more importantly its limiting process $Y_{t}$,
can be interpreted as a column vector of distributions,
$Y_{t}=\left(\begin{matrix}Y_{t,1}&Y_{t,-1}\end{matrix}\right)^{T}$, working
on a row vector of functions,
$\phi=\left(\begin{matrix}\phi(\cdot,1)&\phi(\cdot,-1)\end{matrix}\right)$.
With this in mind, we can look at the vector representation of the measure
$A^{*}Y_{t}$. We have that

$\displaystyle\left<\phi,A^{*}Y_{t}\right>=\left<A\phi,Y_{t}\right>$
$\displaystyle=\left<(\tfrac{\kappa}{2}\partial_{xx}+\lambda\partial_{x})\phi(\cdot,1),Y_{t,1}\right>+\left<\phi(\cdot,1),\gamma(Y_{t,-1}-Y_{t,1})\right>$
$\displaystyle\ \ \ \ \
+\left<(\tfrac{\kappa}{2}\partial_{xx}-\lambda\partial_{x})\phi(\cdot,-1),Y_{t,-1}\right>+\left<\phi(\cdot,-1),\gamma(Y_{t,1}-Y_{t,-1})\right>$
$\displaystyle=\left<\phi(\cdot,1),(\tfrac{\kappa}{2}\partial_{xx}-\lambda\partial_{x})Y_{t,1}+\gamma(Y_{t,-1}-Y_{t,1})\right>$
$\displaystyle\ \ \ \ \
+\left<\phi(\cdot,-1),(\tfrac{\kappa}{2}\partial_{xx}+\lambda\partial_{x})Y_{t,-1}+\gamma(Y_{t,1}-Y_{t,-1})\right>.$

Therefore $A^{*}Y_{t}$ corresponds to the following vector of distributions

$A^{*}Y_{t}=\left(\begin{matrix}(\tfrac{\kappa}{2}\partial_{xx}-\lambda\partial_{x})Y_{t,1}+\gamma(Y_{t,-1}-Y_{t,1})\\\
\\\
(\tfrac{\kappa}{2}\partial_{xx}+\lambda\partial_{x})Y_{t,-1}+\gamma(Y_{t,1}-Y_{t,-1})\end{matrix}\right).$

In a similar way we can find a vector representation of the noise part in the
SPDE (10), namely

$\displaystyle\sqrt{2\kappa\rho}\partial_{x}\mathop{}\\!\mathrm{d}\mathscr{W}_{t}+\sqrt{2\rho\Sigma}\mathop{}\\!\mathrm{d}\mathscr{\tilde{W}}_{t}$
$\displaystyle=\sqrt{2\kappa\rho}\partial_{x}\left(\begin{matrix}\mathop{}\\!\mathrm{d}W_{t,1}\\\
\mathop{}\\!\mathrm{d}W_{t,-1}\end{matrix}\right)+\sqrt{2\rho\Sigma}\left(\begin{matrix}\mathop{}\\!\mathrm{d}\tilde{W}_{t,1}\\\
\mathop{}\\!\mathrm{d}\tilde{W}_{t,-1}\end{matrix}\right)$
$\displaystyle=\left(\begin{matrix}\sqrt{2\kappa\rho}\partial_{x}\mathop{}\\!\mathrm{d}W_{t,1}+\sqrt{\gamma\rho}\left(\mathop{}\\!\mathrm{d}\tilde{W}_{t,-1}-\mathop{}\\!\mathrm{d}\tilde{W}_{t,1}\right)\\\
\sqrt{2\kappa\rho}\partial_{x}\mathop{}\\!\mathrm{d}W_{t,-1}+\sqrt{\gamma\rho}\left(\mathop{}\\!\mathrm{d}\tilde{W}_{t,1}-\mathop{}\\!\mathrm{d}\tilde{W}_{t,-1}\right)\end{matrix}\right),$

where all the
$\mathop{}\\!\mathrm{d}W_{t,i},\mathop{}\\!\mathrm{d}\tilde{W}_{t,i}$ are
independent space-time white noises on $\mathbb{R}$. In this notation, the
SPDE in (10) actually gives us a system of SPDE’s given by

$\begin{cases}\mathop{}\\!\mathrm{d}Y_{t,1}=\left[\frac{\kappa}{2}\partial_{xx}Y_{t,1}-\lambda\partial_{x}Y_{t,1}+\gamma\left(Y_{t,-1}-Y_{t,1}\right)\right]\mathop{}\\!\mathrm{d}t+\sqrt{2\kappa\rho}\partial_{x}\mathop{}\\!\mathrm{d}W_{t,1}+\sqrt{\gamma\rho}\left(\mathop{}\\!\mathrm{d}\tilde{W}_{t,-1}-\mathop{}\\!\mathrm{d}\tilde{W}_{t,1}\right),\\\
\\\
\mathop{}\\!\mathrm{d}Y_{t,-1}=\left[\frac{\kappa}{2}\partial_{xx}Y_{t,-1}+\lambda\partial_{x}Y_{t,-1}+\gamma\left(Y_{t,1}-Y_{t,-1}\right)\right]\mathop{}\\!\mathrm{d}t+\sqrt{2\kappa\rho}\partial_{x}\mathop{}\\!\mathrm{d}W_{t,-1}+\sqrt{\gamma\rho}\left(\mathop{}\\!\mathrm{d}\tilde{W}_{t,1}-\mathop{}\\!\mathrm{d}\tilde{W}_{t,-1}\right).\end{cases}$

Now we are able to sum up these equations to get an SPDE for the fluctuation
process of the total density $Z_{t}$. Just like in the hydrodynamic limit,
this will again depend on the difference of the two processes above, denoted
by $R_{t}:=Y_{t,1}-Y_{t,-1}$. This gives us the following system of coupled
SPDE’s

$\begin{cases}\mathop{}\\!\mathrm{d}Z_{t}=\left[\frac{\kappa}{2}\partial_{xx}Z_{t}-\lambda\partial_{x}R_{t}\right]\mathop{}\\!\mathrm{d}t+2\sqrt{\kappa\rho}\partial_{x}\mathop{}\\!\mathrm{d}W_{t,Z},\\\
\\\
\mathop{}\\!\mathrm{d}R_{t}=\left[\frac{\kappa}{2}\partial_{xx}R_{t}-\lambda\partial_{x}Z_{t}-2\gamma
R_{t}\right]\mathop{}\\!\mathrm{d}t+2\sqrt{\kappa\rho}\partial_{x}\mathop{}\\!\mathrm{d}W_{t,R}+2\sqrt{2\gamma\rho}\mathop{}\\!\mathrm{d}\tilde{W}_{t},\end{cases}$
(20)

where

$W_{t,Z}=\frac{1}{\sqrt{2}}\left(W_{t,1}+W_{t,-1}\right),\ \ \ \
W_{t,R}=\frac{1}{\sqrt{2}}\left(W_{t,1}-W_{t,-1}\right),\ \ \ \
\tilde{W}_{t}=\frac{1}{\sqrt{2}}\left(\tilde{W}_{t,1}-\mathop{}\\!\mathrm{d}\tilde{W}_{t,-1}\right),$

which are all independent space-time white noises on $\mathbb{R}$.

###  4.3 Covariance of the total density

The process $Z_{t}$ introduced as in (20) is a (non-Markovian) stationary
Gaussian processes. Therefore, we can characterize $Z_{t}$ through its
covariances. Using (18) and (19), we can actually relate this covariance to
the covariance structure of $Y_{t}$, which we have already calculated in
Proposition 3.2. In order to do so, for a given $\phi,\psi\in C_{c}^{\infty}$
we define the functions $\bar{\phi},\bar{\psi}\in C_{c,S}^{\infty}$ via
$\bar{\phi}(x,\sigma)=\phi(x)$ and $\bar{\psi}(x,\sigma)=\psi(x)$. The
covariance can then be computed as follows

$\displaystyle\mathbb{E}[\left<\phi,Z_{t}\right>\left<\psi,Z_{0}\right>]$
$\displaystyle=\mathbb{E}[\left(\left<\phi,Y_{t,1}\right>+\left<\phi,Y_{t,-1}\right>\right)\left(\left<\psi,Y_{0,1}\right>+\left<\psi,Y_{0,-1}\right>\right)]$
$\displaystyle=\mathbb{E}[\left(\left<\bar{\phi}(\cdot,1),Y_{t,1}\right>+\left<\bar{\phi}(\cdot,-1),Y_{t,-1}\right>\right)\left(\left<\bar{\psi}(\cdot,1),Y_{0,1}\right>+\left<\bar{\psi}(\cdot,-1),Y_{0,-1}\right>\right)]$
$\displaystyle=\mathbb{E}[\left<\bar{\phi},Y_{t}\right>\left<\bar{\psi},Y_{0}\right>]$
$\displaystyle=\rho\cdot\left<\left<e^{tA}\bar{\phi},\bar{\psi}\right>\right>.$
(21)

This covariance strongly resembles the covariance of a stationary Ornstein-
Uhlenbeck process, but notice that the semigroup $e^{tA}$ works on the
“extended” functions $\bar{\phi},\bar{\psi}$, which corresponds to the non-
Markovianity of the process $\\{Z_{t},t\geq 0\\}$.

Notice that the formula for the covariance obtained in (4.3) is solely based
on duality, and is therefore valid as long as we have duality for the multi-
layer system, i.e., beyond the case of two internal states and including also
interacting cases such as the multi-layer SEP.

###  4.4 Closed form equation and large deviations for the case $\kappa=0$

In the case of $\kappa=0$ the noise term vanishes in the upper equation of
(20) and therefore we we can solve the system explicitly. Namely, we then find
that

$\begin{cases}\mathop{}\\!\mathrm{d}Z_{t}=-\lambda\partial_{x}R_{t}\mathop{}\\!\mathrm{d}t,\\\
\\\ \mathop{}\\!\mathrm{d}R_{t}=-\left[\lambda\partial_{x}Z_{t}+2\gamma
R_{t}\right]\mathop{}\\!\mathrm{d}t+2\sqrt{\gamma\rho}\mathop{}\\!\mathrm{d}\tilde{W}_{t}.\end{cases}$

Just like for the hydrodynamic limit of the total density, by now taking a
second derivative in time in the first equation we find that
$\mathop{}\\!\mathrm{d}^{2}Z_{t}=-\lambda\partial_{x}\mathop{}\\!\mathrm{d}R_{t}\mathop{}\\!\mathrm{d}t$.
By now filling in $\mathop{}\\!\mathrm{d}R_{t}$ from the lower equation, we
have that

$\displaystyle\frac{\mathop{}\\!\mathrm{d}^{2}Z_{t}}{\mathop{}\\!\mathrm{d}t^{2}}$
$\displaystyle=\lambda^{2}\partial_{xx}Z_{t}-2\gamma\lambda\partial_{x}R_{t}+2\lambda\sqrt{\gamma\rho}\partial_{x}\frac{\mathop{}\\!\mathrm{d}\tilde{W}_{t}}{\mathop{}\\!\mathrm{d}t}$
$\displaystyle=\lambda^{2}\partial_{xx}Z_{t}+2\gamma\frac{\mathop{}\\!\mathrm{d}Z_{t}}{\mathop{}\\!\mathrm{d}t}+2\lambda\sqrt{\gamma\rho}\partial_{x}\frac{\mathop{}\\!\mathrm{d}\tilde{W}_{t}}{\mathop{}\\!\mathrm{d}t}.$
(22)

From the expression above we are also able to obtain the rate function for the
large deviations of $Z_{t}^{(\varepsilon)}$ in the small noise regime, i.e.,
where we add a factor $\varepsilon$ before the noise $\tilde{W}_{t}$ which we
will send to zero. I.e., we are interested in the large deviations of Schilder
type (see [4], [10]) for the family of Gaussian process given by

$\frac{\mathop{}\\!\mathrm{d}^{2}Z_{t}^{(\varepsilon)}}{\mathop{}\\!\mathrm{d}t^{2}}=\lambda^{2}\partial_{xx}Z_{t}^{(\varepsilon)}+2\gamma\frac{\mathop{}\\!\mathrm{d}Z_{t}^{(\varepsilon)}}{\mathop{}\\!\mathrm{d}t}+\varepsilon
2\lambda\sqrt{\gamma\rho}\partial_{x}\frac{\mathop{}\\!\mathrm{d}\tilde{W}_{t}}{\mathop{}\\!\mathrm{d}t}.$
(23)

We use that

$\mathbb{P}\left(\varepsilon\partial_{x}\frac{\mathop{}\\!\mathrm{d}\tilde{W}_{t}}{\mathop{}\\!\mathrm{d}t}\asymp\Gamma(t,x)\right)\asymp\exp\left(-\varepsilon^{-2}\frac{1}{2}\int_{0}^{T}||\Gamma(t,\cdot)||^{2}_{H_{-1}}\mathop{}\\!\mathrm{d}t\right),$

which has to be interpreted in the sense of the large deviation principle in
the space of space-time distributions. The rate function in the above equation
can be derived from the log-moment-generating function of a space-time white
noise on $\mathbb{R}$, which for a test function $\phi\in
C_{c}^{\infty}([0,T]\times\mathbb{R})$ is equal to

$\Lambda(\phi)=\lim_{\varepsilon\to
0}\varepsilon^{2}\log\left(\mathbb{E}[e^{\varepsilon^{-1}\left<\phi,\partial_{x}\frac{\mathop{}\\!\mathrm{d}W_{t}}{\mathop{}\\!\mathrm{d}t}\right>}]\right)=\frac{1}{2}\left<\partial_{x}\phi,\partial_{x}\phi\right>_{L^{2}(\mathbb{R}\times[0,T])}.$

The Legendre transform of $\Lambda$ then yields the rate function,

$\begin{split}\Lambda^{*}(\Gamma(t,x))&=\sup_{\phi\in
C_{c}^{\infty}([0,T]\times\mathbb{R})}\left\\{\left<\phi,\Gamma\right>_{L^{2}([0,T]\times\mathbb{R})}-\frac{1}{2}\left<\partial_{x}\phi,\partial_{x}\phi\right>_{L^{2}([0,T]\times\mathbb{R})}\right\\}\\\
&=\frac{1}{2}\int_{0}^{T}\left|\left|\Gamma(t,\cdot)\right|\right|^{2}_{H^{-1}}\mathop{}\\!\mathrm{d}t.\end{split}$

As a consequence, we obtain the large deviation principle for the random
space-time distribution $Z_{t}^{(\varepsilon)}$, namely from $\eqref{Zte}$ it
follows that

$\begin{split}\mathbb{P}\left(Z_{t}^{(\varepsilon)}\asymp\Gamma(t,x)\right)&=\mathbb{P}\left(\varepsilon\partial_{x}\frac{\mathop{}\\!\mathrm{d}\tilde{W_{t}}}{\mathop{}\\!\mathrm{d}t}\asymp\frac{1}{2\lambda\sqrt{\gamma\rho}}\left(\ddot{\Gamma}(t,x)-2\gamma\dot{\Gamma}(t,x)-\lambda^{2}\partial_{xx}\Gamma(t,x)\right)\right)\\\
&\asymp\exp\left(-\varepsilon^{-2}\frac{1}{4\lambda\sqrt{\gamma\rho}}\int_{0}^{T}\left|\left|\ddot{\Gamma}(t,\cdot)-2\gamma\dot{\Gamma}(t,\cdot)-\lambda^{2}\partial_{xx}\Gamma(t,\cdot)\right|\right|^{2}_{H_{-1}}\mathop{}\\!\mathrm{d}t\right).\end{split}$
(24)

##  5 Proof of Theorem 3.1

In this section we prove Theorem 3.1, following the line of proof of Van
Ginkel and Redig in [15]. For the readers convenience we sketch the main
steps.

We start by introducing the Dynkin martingales of $Y_{t}^{N}(\phi)$. For every
$\phi\in C_{c,S}^{\infty}$ and $N\in\mathbb{N}$, let
$\\{\mathscr{F}^{N}_{t}:t\geq 0\\}$ be the filtration generated by
$\\{Y^{N}_{t}:t\geq 0\\}$. Because the configuration process
$\\{\eta_{t}:t\geq 0\\}$ is a Markov processes the following processes

$\begin{split}\mathscr{M}^{N,\phi}_{t}(Y^{N})&=Y^{N}_{t}(\phi)-Y^{N}_{0}(\phi)-\int_{0}^{t}L_{N}Y^{N}_{s}(\phi)\mathop{}\\!\mathrm{d}s,\\\
\mathscr{N}^{N,\phi}_{t}(Y^{N})&=\mathscr{M}^{N,\phi}_{t}(Y^{N})^{2}-\int_{0}^{t}\Gamma_{s}^{N,\phi}(Y^{N})\mathop{}\\!\mathrm{d}s,\end{split}$
(25)

are $\mathscr{F}^{N}_{t}$-martingales, where $\Gamma_{s}^{N,\phi}$ is the so-
called Carré du champ operator given by

$\Gamma_{s}^{N,\phi}(Y^{N}):=L_{N}\big{(}Y_{s}^{N}(\phi)^{2}\big{)}-2Y_{s}^{N}(\phi)L_{N}Y_{s}^{N}(\phi).$
(26)

The aim is then to prove that as $N\to\infty$, the martingales in (25)
converge to the martingales from (12). This fact, complemented with a proof of
tightness and the fact that the martingale problem (12) has a unique solution,
then completes the proof. In Section 5.1 we prove the convergence of the
martingales, in Section 5.2 we prove the tightness, and in Section 5.3 we
prove the uniqueness of the solution of the martingale problem (12).

###  5.1 Substituting the martingales

Our goal for this section is to show that in the limit as $N\to\infty$, we can
substitute $\mathscr{M}^{\phi}_{t}(Y^{N})$ and $\mathscr{N}^{\phi}_{t}(Y^{N})$
(with $\mathscr{M}^{\phi}_{t}$ and $\mathscr{N}^{\phi}_{t}$ defined as in
(12)) for ${\mathscr{M}}^{N,\phi}_{t}(Y^{N})$ and
${\mathscr{N}}^{N,\phi}_{t}(Y^{N})$ respectively. We do so in the Propositions
5.1 and 5.4. We recall the reader that the expectation $\mathbb{E}$ stands for
the stationary expectation starting from the initial configuration distributed
according to $\eta_{0}\sim\mu_{\rho}$.

######  Proposition 5.1.

For all $\phi\in C_{c,S}^{\infty}$ we have

$\displaystyle\lim_{N\to\infty}\mathbb{E}\left[\left|{\mathscr{M}}^{N,\phi}_{t}(Y^{N})-\mathscr{M}^{\phi}_{t}(Y^{N})\right|^{2}\right]=0.$

###### Proof.

First of all, note that by definition

$\mathbb{E}\left[\left|{\mathscr{M}}^{N,\phi}_{t}(Y^{N})-\mathscr{M}^{\phi}_{t}(Y^{N})\right|^{2}\right]=\mathbb{E}\left[\left|\int_{0}^{t}L_{N}Y_{s}^{N}(\phi)\mathop{}\\!\mathrm{d}s-\int_{0}^{t}Y^{N}_{s}(A\phi)\mathop{}\\!\mathrm{d}s\right|^{2}\right].$

For a given $(x,\sigma)\in V$ we have that

$\begin{split}L_{N}\eta(x,\sigma)&=\kappa
N^{2}[\eta(x+1,\sigma)+\eta(x-1,\sigma)-2\eta(x,\sigma)]\\\ &\ \ \ +\lambda
N[\eta(x-\sigma,\sigma)-\eta(x,\sigma)]\\\ &\ \ \ +\sum_{\sigma^{\prime}\in
S}c(\sigma,\sigma^{\prime})[\eta(x,\sigma^{\prime})-\eta(x,\sigma)],\end{split}$

and so in particular we find that

$\begin{split}L_{N}Y_{s}^{N}(\phi)&=\frac{1}{\sqrt{N}}\sum_{(x,\sigma)\in
V}\big{(}L_{N}\eta_{s}(x,\sigma)\big{)}\phi(\tfrac{x}{N},\sigma)=\frac{1}{\sqrt{N}}\sum_{(x,\sigma)\in
V}\eta_{s}(x,\sigma)\cdot(\mathscr{L}_{N}\phi)(\tfrac{x}{N},\sigma),\end{split}$

where we remind the reader that $\mathscr{L}_{N}$ is the generator of a single
run-and-tumble particle on the rescaled space $\frac{1}{N}\mathbb{Z}\times S$.
Now, using that for any $\phi\in C_{c,S}^{\infty}$ we have that

$\sum_{(x,\sigma)\in V}\rho\cdot(\mathscr{L}_{N}\phi)(\tfrac{x}{N},\sigma)=0,$

we are able to write

$L_{N}Y_{s}^{N}(\phi)=\frac{1}{\sqrt{N}}\sum_{(x,\sigma)\in
V}(\eta_{s}(x,\sigma)-\rho)\cdot(\mathscr{L}_{N}\phi)(\tfrac{x}{N},\sigma).$

Since $\mathscr{L}_{N}\phi\to A\phi$ uniformly, where $A$ is defined in (2),
we have that

$\begin{split}L_{N}Y_{s}^{N}(\phi)&=\frac{1}{\sqrt{N}}\sum_{(x,\sigma)\in
V}(\eta_{s}(x,\sigma)-\rho)\cdot(A\phi)(\tfrac{x}{N},\sigma)+R_{1}(\phi,N,s),\end{split}$
(27)

where $R_{1}(\phi,N,s)$ is an error term produced by the Taylor
approximations. Since $\phi$ is compactly supported, if we define
$V_{\phi}^{N}:=\\{(x,\sigma)\in V,\phi(\tfrac{x}{N},\sigma)\neq 0\\}$ then
$|V_{\phi}^{N}|=\mathcal{O}(N)$. Furthermore, the error term is bounded in the
following way

$|R_{1}(\phi,N,s)|\leq\frac{1}{N^{3/2}}\sum_{(x,\sigma)\in
V_{\phi}^{N}}(\eta_{s}(x,\sigma)-\rho)(\kappa||\partial_{xxx}\phi||_{\infty}+\lambda\sigma^{2}||\partial_{xx}\phi||_{\infty}).$
(28)

Therefore we find that for every $\phi\in C_{c,S}^{\infty}$ and $t\geq 0$,

$\begin{split}\mathbb{E}\left[R_{1}(\phi,N,s)^{2}\right]&\leq\frac{1}{N^{3}}\mathbb{E}\left[\sum_{(x,\sigma),(y,\sigma^{\prime})\in
V_{\phi}^{N}}(\eta_{s}(x,\sigma)-\rho)(\eta_{s}(y,\sigma^{\prime})-\rho)(\kappa||\partial_{xxx}\phi||_{\infty}+\lambda\sigma^{2}||\partial_{xx}\phi||_{\infty})^{2}\right]\\\
&=\frac{1}{N^{3}}\sum_{(x,\sigma),(y,\sigma^{\prime})\in
V_{\phi}^{N}}\mathrm{Cov}\big{(}\eta_{s}(x,\sigma),\eta_{s}(y,\sigma^{\prime})\big{)}(\kappa||\partial_{xxx}\phi||_{\infty}+\lambda\sigma^{2}||\partial_{xx}\phi||_{\infty})^{2}.\\\
\end{split}$ (29)

Since we are starting the process $\eta_{t}$ from the invariant product
measure $\mu_{\rho}$, we have that

$\mathrm{Cov}\big{(}\eta_{s}(x,\sigma),\eta_{s}(y,\sigma^{\prime})\big{)}=\rho\cdot
I\big{(}(x,\sigma)=(y,\sigma^{\prime})\big{)}.$ (30)

Therefore,

$\mathbb{E}\left[R_{1}(\phi,N,s)^{2}\right]\leq\frac{1}{N^{3}}|V_{\phi}^{N}|\rho(\kappa||\partial_{xxx}\phi||_{\infty}+\lambda\sigma^{2}||\partial_{xx}\phi||_{\infty})^{2}\to
0,$

where we used the fact that $|V_{\phi}^{N}|=\mathcal{O}(N)$. Note that the
above convergence is uniform in $s$, and therefore by dominated convergence we
find that

$\displaystyle\lim_{N\to\infty}\mathbb{E}\left[\left|{\mathscr{M}}^{N,\phi}_{t}(Y^{N})-\mathscr{M}^{\phi}_{t}(Y^{N})\right|^{2}\right]=\lim_{N\to\infty}\int_{0}^{t}\mathbb{E}\left[R_{1}(\phi,N,s)^{2}\right]\mathop{}\\!\mathrm{d}s=0,$

which concludes the proof. ∎

The substitution of ${\mathscr{N}}_{t}^{\phi}(Y^{N})$ is a bit more work and
requires a fourth moment estimate. We start by proving two lemmas. The proof
of the substitution result in Proposition 5.4 immediately follows from these
lemmas.

######  Lemma 5.2.

For all $\phi\in C_{c,S}^{\infty}$ we have the following

$\lim_{k\to\infty}\mathbb{E}\left[\left({\mathscr{M}}_{t}^{N,\phi}(Y^{N})^{2}-{\mathscr{M}}_{t}^{\phi}(Y^{N})^{2}\right)^{2}\right]=0.$
(31)

###### Proof.

We start with the following application of Hölder’s inequality

$\begin{split}\mathbb{E}\left[\left({\mathscr{M}}_{t}^{N,\phi}(Y^{N})^{2}-{\mathscr{M}}_{t}^{\phi}(Y^{N})^{2}\right)^{2}\right]&=\mathbb{E}\left[\left({\mathscr{M}}_{t}^{N,\phi}(Y^{N})-{\mathscr{M}}_{t}^{\phi}(Y^{N})\right)^{2}\left({\mathscr{M}}_{t}^{N,\phi}(Y^{N})+{\mathscr{M}}_{t}^{\phi}(Y^{N})\right)^{2}\right]\\\
&\leq\left(\mathbb{E}\left[\left({\mathscr{M}}_{t}^{N,\phi}(Y^{N})-{\mathscr{M}}_{t}^{\phi}(Y^{N})\right)^{4}\right]\cdot\mathbb{E}\left[\left({\mathscr{M}}_{t}^{N,\phi}(Y^{N})+{\mathscr{M}}_{t}^{\phi}(Y^{N})\right)^{4}\right]\right)^{\tfrac{1}{2}}.\end{split}$
(32)

We will first show that the first expectation in the last line vanishes as
$N\to\infty$, and afterwards we will show that the second expectation is
uniformly bounded in $N$. Note that by (27)

$\displaystyle\mathbb{E}\left[\left({\mathscr{M}}_{t}^{N,\phi}(Y^{N})-{\mathscr{M}}_{t}^{\phi}(Y^{N})\right)^{4}\right]=\mathbb{E}\left[\left(\int_{0}^{t}\left[R_{1}(\phi,N,s)\right]\mathop{}\\!\mathrm{d}s\right)^{4}\right]\leq
t^{3}\int_{0}^{T}\mathbb{E}\left[R_{1}(\phi,N,s)^{4}\right]\mathop{}\\!\mathrm{d}s.$

Using the bound in $\eqref{R1}$ we find that

$\mathbb{E}\left[R_{1}(\phi,N,s)^{4}\right]\leq\frac{1}{N^{6}}\sum_{\begin{subarray}{c}(x_{i},\sigma_{i})\in
V_{\phi}^{N}\\\ 1\leq i\leq
4\end{subarray}}\mathbb{E}\left[\prod_{i=1}^{4}(\eta_{s}(x_{i},\sigma_{i})-\rho)\right](\kappa||\partial_{xxx}\phi||_{\infty}+\lambda\sigma^{2}||\partial_{xx}\phi||_{\infty})^{4}.$

Since we start from the product Poisson measure $\mu_{\rho}$, we only get non-
zero contributions in the expectation on the right-hand side when all
$(x_{i},\sigma_{i})$ are equal or when we have two distinct pairs, given by

$\mathbb{E}\left[(\eta_{s}(x,\sigma)-\rho)^{4}\right]=3\rho^{2}+\rho,\ \ \ \
\mathbb{E}\left[(\eta_{s}(x,\sigma)-\rho)^{2}(\eta_{s}(y,\sigma^{\prime})-\rho)^{2}\right]=\rho^{2}.$

Therefore, it follows that

$\mathbb{E}\left[R_{1}(\phi,N,s)^{4}\right]\leq\frac{1}{N^{6}}\left(|V_{\phi}^{N}|(3\rho^{2}+\rho)+|V_{\phi}^{N}|^{2}\rho^{2}\right)(\kappa||\partial_{xxx}\phi||_{\infty}+\lambda\sigma^{2}||\partial_{xx}\phi||_{\infty})^{4},$
(33)

and so $R_{1}(\phi,N,s,\sigma)\xrightarrow{L^{4}}0$ uniformly in $s$. From
this we can conclude that

$\mathbb{E}\left[\left({\mathscr{M}}_{t}^{N,\phi}(Y^{N})-{\mathscr{M}}_{t}^{\phi}(Y^{N})\right)^{4}\right]\leq
t^{3}\int_{0}^{t}\mathbb{E}\left[R_{1}(\phi,N,s)^{4}\right]\mathop{}\\!\mathrm{d}s\to
0.$

To now show that the second expectation in the last line of (32) is uniformly
bounded in $N$, note that

$\mathbb{E}\left[\left({\mathscr{M}}_{t}^{N,\phi}(Y^{N})+{\mathscr{M}}_{t}^{\phi}(Y^{N})\right)^{4}\right]\leq
8\left(\mathbb{E}\left[\left({\mathscr{M}}_{t}^{N,\phi}(Y^{N})\right)^{4}\right]+\mathbb{E}\left[\left({\mathscr{M}}_{t}^{\phi}(Y^{N})\right)^{4}\right]\right),$
(34)

and similarly

$\begin{split}\mathbb{E}\left[\left({\mathscr{M}}_{t}^{\phi}(Y^{N})\right)^{4}\right]\leq
27\left(\mathbb{E}\left[Y_{t}^{N}(\phi)^{4}\right]+\mathbb{E}\left[Y_{0}^{N}(\phi)^{4}\right]+\mathbb{E}\left[\left(\int_{0}^{t}Y_{s}^{N}(A\phi)\mathop{}\\!\mathrm{d}s\right)^{4}\right]\right).\end{split}$
(35)

Now we need to show that three expectations on the right-hand-side are
uniformly bounded. For the first expectation, we find that

$\begin{split}\mathbb{E}\left[Y_{t}^{N}(\phi)^{4}\right]&\leq\frac{1}{N^{2}}\cdot\sum_{(x_{1},\sigma_{1})\in
V_{\phi}^{N}}\cdots\sum_{(x_{4},\sigma_{4})\in
V_{\phi}^{N}}\mathbb{E}\left[\prod_{i=1}^{4}(\eta_{t}(x_{i},\sigma_{i})-\rho)\right]||\phi||_{\infty}.\end{split}$

Similarly as in (33), we find that

$\mathbb{E}\left[Y_{t}^{N}(\phi)^{4}\right]\leq\frac{1}{N^{2}}\left(|V_{\phi}^{N}|(3\rho^{2}+\rho)+|V_{\phi}^{N}|^{2}\rho^{2}\right)||\phi||_{\infty}=\mathcal{O}(1),$
(36)

hence it is uniformly bounded, and similar approaches can be used for
$\mathbb{E}\left[Y_{0}^{N}(\phi)^{4}\right]$ and
$\mathbb{E}\left[Y_{s}^{N}(A\phi)^{4}\right]$. The fact that the last
expectation in (35) is uniformly bounded now follows from an application of
Hölder’s inequality, namely

$\mathbb{E}\left[\left(\int_{0}^{t}Y_{s}^{N}(A\phi)\mathop{}\\!\mathrm{d}s\right)^{4}\right]\leq
t^{3}\int_{0}^{T}\mathbb{E}\left[\left(Y_{s}^{N}(A\phi)\right)^{4}\right]\mathop{}\\!\mathrm{d}s.$

Therefore we know that
$\mathbb{E}\left[\big{(}{\mathscr{M}}_{t}^{\phi}(Y^{N})\big{)}^{4}\right]$ is
uniformly bounded. The proof for
$\mathbb{E}\left[\big{(}{\mathscr{M}}_{t}^{N,\phi}(Y^{N})\big{)}^{4}\right]$
works the same way if we use that

$\begin{split}\mathbb{E}\left[\left(L_{N}Y_{s}^{N}(\phi)\right)^{4}\right]=8\left(\mathbb{E}\left[\left(Y_{s}^{N}(A\phi)\right)^{4}\right]+\mathbb{E}\left[R_{1}(\phi,N,t,\sigma)^{4}\right]\right),\end{split}$

where by (33) we already know that
$\mathbb{E}\left[R_{1}(\phi,N,t,\sigma)^{4}\right]$ is uniformly bounded.
Hence we can conclude that (31) holds. ∎

######  Lemma 5.3.

For all $\phi\in C_{c,S}^{\infty}$ we have the following

$\lim_{N\to\infty}\mathbb{E}\left[\left(\int_{0}^{t}\Gamma_{s}^{N,\phi}(Y^{N})\mathop{}\\!\mathrm{d}s-2t\kappa\rho\left<\left<\partial_{x}\phi,\partial_{x}\phi\right>\right>-2t\rho\left<\left<\phi,\Sigma\phi\right>\right>\right)^{2}\right]=0,$

with $\Sigma$ defined as in (11).

###### Proof.

First we recall that for a Markov process with generator $L$ determined by the
transition rates $r(\eta,\eta^{\prime})$ the carré du champ operator is
computed as follows.

$\begin{split}Lf^{2}(\eta)-2f(\eta)\cdot
Lf(\eta)&=\sum_{\eta^{\prime}\in\Omega}r(\eta,\eta^{\prime})\Big{(}\big{(}f^{2}(\eta^{\prime})-f^{2}(\eta)\big{)}-2\big{(}f(\eta)f(\eta^{\prime})-f^{2}(\eta)\big{)}\Big{)}\\\
&=\sum_{\eta^{\prime}\in\Omega}r(\eta,\eta^{\prime})\big{(}f(\eta^{\prime})-f(\eta)\big{)}^{2},\end{split}$
(37)

Translating this to our setting with $L=L_{N}$ and $f=Y^{N}$ we obtain

$\begin{split}\Gamma_{s}^{N,\phi}(Y^{N})&=\kappa N\sum_{(x,\sigma)\in
V}\eta_{s}(x,\sigma)\left(\phi(\tfrac{x+1}{N},\sigma)-\phi(\tfrac{x}{N},\sigma))^{2}+(\phi(\tfrac{x-1}{N},\sigma)-\phi(\tfrac{x}{N},\sigma))^{2}\right)\\\
&\ \ \ +\lambda\sum_{(x,\sigma)\in
V}\eta_{s}(x,\sigma)\left(\phi(\tfrac{x+\sigma}{N},\sigma)-\phi(\tfrac{x}{N},\sigma)\right)^{2}\\\
&\ \ \ +\frac{1}{N}\sum_{(x,\sigma)\in V}\sum_{\sigma^{\prime}\in
S}c(\sigma,\sigma^{\prime})\eta_{s}(x,\sigma)\big{(}\phi(x,\sigma^{\prime})-\phi(x,\sigma))^{2}.\end{split}$

Using Taylor expansion with rest term, we can write

$\begin{split}\Gamma_{s}^{N,\phi}(Y^{N})&=\frac{2\kappa}{N}\sum_{(x,\sigma)\in
V}\eta_{s}(x,\sigma)\left(\partial_{x}\phi(\tfrac{x}{N},\sigma)\right)^{2}+\frac{1}{N}\sum_{(x,\sigma)\in
V}\sum_{\sigma^{\prime}\in
S}c(\sigma,\sigma^{\prime})\eta_{s}(x,\sigma)(\phi(\tfrac{x}{N},\sigma^{\prime})-\phi(\tfrac{x}{N},\sigma)^{2}\\\
&\ \ \ \ \ \ \ \ +R_{2}(\phi,s,N),\end{split}$ (38)

with $R_{2}(\phi,s,N)$ the error term, which is bounded as follows

$|R_{2}(\phi,s,N)|\leq\kappa\frac{1}{N^{3}}\sum_{(x,\sigma)\in
V_{\phi}^{N}}\eta_{s}(x,\sigma)\kappa||\partial_{xx}\phi||_{\infty}+\frac{1}{N^{2}}\sum_{(x,\sigma)\in
V_{\phi}^{N}}\eta_{s}(x,\sigma)\lambda\sigma||\phi^{\prime}||_{\infty}.$

Following the line of thought leading to (29), we obtain that
$R_{2}(\phi,s,N)\xrightarrow{L^{2}}0$. Therefore, for the expectation we find
that

$\begin{split}\mathbb{E}\left[\Gamma_{s}^{N,\phi}(Y^{N})\right]&=\frac{2\kappa\rho}{N}\sum_{(x,\sigma)\in
V}(\partial_{x}\phi(\tfrac{x}{N},\sigma))^{2}+\frac{2\rho}{N}\sum_{\sigma^{\prime}\in
S}c(\sigma,\sigma^{\prime})(\phi(\tfrac{x}{N},\sigma^{\prime})-\phi(\tfrac{x}{N},\sigma))^{2}+\mathbb{E}\left[R_{2}(\phi,s,N)\right]\\\
&\to
2\kappa\rho\left<\left<\partial_{x}\phi,\partial_{x}\phi\right>\right>+2\rho\left<\left<\phi,\Sigma\phi\right>\right>,\end{split}$
(39)

and for the variance

$\begin{split}\mathrm{Var}\left[\Gamma_{s}^{N,\phi}(Y^{N})\right]&\leq\frac{C(\phi,s)}{N^{2}}\sum_{(x,\sigma),(y,\sigma^{\prime})\in
V_{\phi}^{N}}\mathrm{Cov}\big{(}\eta_{s}(x,\sigma),\eta_{s}(y,\sigma^{\prime})\big{)}\\\
&=\frac{C(\phi,s)}{N^{2}}|V_{\phi}^{N}|\rho\to 0,\end{split}$ (40)

with $C(\phi,s)$ some constant and where we have used (30) for the equality.
Since the variance converges to zero, this means that
$\Gamma_{s}^{N,\phi}(Y^{N})$ converges to its mean in $L^{2}$. Therefore

$\displaystyle\lim_{N\to\infty}$
$\displaystyle\mathbb{E}\left[\left(\int_{0}^{t}\Gamma_{s}^{N,\phi}(Y^{N})\mathop{}\\!\mathrm{d}s-2t\kappa\rho\left<\left<\partial_{x}\phi,\partial_{x}\phi\right>\right>-2t\rho\left<\left<\phi,\Sigma\phi\right>\right>\right)^{2}\right]$
$\displaystyle\leq\lim_{N\to\infty}\int_{0}^{t}\mathbb{E}\left[\left(\Gamma_{s}^{N,\phi}(Y^{N})-2\kappa\rho\left<\left<\partial_{x}\phi,\partial_{x}\phi\right>\right>-2\rho\left<\left<\phi,\Sigma\phi\right>\right>\right)^{2}\right]\mathop{}\\!\mathrm{d}s$
$\displaystyle=0,$

where we used dominated convergence for the last equality. ∎

######  Proposition 5.4.

For all $\phi\in C_{c,S}^{\infty}$

$\lim_{N\to\infty}\mathbb{E}\left[\left|{\mathscr{N}}^{N,\phi}_{t}(Y^{N})-{\mathscr{N}}^{\phi}_{t}(Y^{N})\right|^{2}\right]=0.$

###### Proof.

We have that

$\displaystyle\mathbb{E}\left[\left|{\mathscr{N}}^{N,\phi}_{t}(Y^{N})-{\mathscr{N}}^{\phi}_{t}(Y^{N})\right|^{2}\right]$
$\displaystyle\leq
2\mathbb{E}\left[\left({\mathscr{M}}_{t}^{N,\phi}(Y^{N})^{2}-{\mathscr{M}}_{t}^{\phi}(Y^{N})^{2}\right)^{2}\right]$
$\displaystyle\ \ \ \
+2\mathbb{E}\left[\left(\int_{0}^{t}\Gamma_{s}^{N,\phi}(Y^{N})\mathop{}\\!\mathrm{d}s-2t\kappa\rho\left<\left<\partial_{x}\phi,\partial_{x}\phi\right>\right>-2t\rho\left<\left<\phi,\Sigma\phi\right>\right>\right)\right].$

The proof now follows from Lemma 5.2 and 5.3. ∎

###  5.2 Tightness

In this section we will show the tightness of the collection
$\\{Y^{N}:N\in\mathbb{N}\\}$.

######  Proposition 5.5.

$\\{Y^{N}:N\in\mathbb{N}\\}$ is tight in $D([0,T];(C_{c,S}^{\infty})^{*})$.

###### Proof.

Since $C_{c,S}^{\infty}$ is a nuclear space, by Mitoma [11, Theorem 4.1] it
suffices to prove that for a fixed $\phi\in C_{c,S}^{\infty}$ we have that
$\\{Y^{N}(\phi):N\in\mathbb{N}\\}$ is tight in the path space
$D([0,T];\mathbb{R})$. Aldous’ criterion, as stated in [1, Theorem 1], tells
us that it suffices to show the following two things:

  1. A.1

For all $t\in[0,T]$ and $\varepsilon>0$ there exists a compact
$K(t,\varepsilon)\in\mathbb{R}$ such that

$\sup_{N\in\mathbb{N}}\mathbb{P}\big{(}Y_{t}^{N}(\phi)\notin
K(t,\varepsilon)\big{)}\leq\varepsilon.$

  2. A.2

For all $\varepsilon>0$

$\lim_{\delta\to
0}\limsup_{N\to\infty}\sup_{\begin{subarray}{c}\uptau\in\mathscr{T}_{T}\\\
\theta\leq\delta\end{subarray}}\mathbb{P}\big{(}|Y_{\uptau}^{N}(\phi)-Y_{\uptau+\theta}^{N}(\phi)|>\varepsilon\big{)}=0,$

with $\mathscr{T}_{T}$ the set of all stopping times bounded by $T$.

Fix $t\in[0,T]$ and $\phi\in C_{c,S}^{\infty}$. Then, for every $\sigma\in S$
we have that

$\begin{split}&\mathbb{E}[Y_{t}^{N}(\phi)]=\frac{1}{\sqrt{N}}\sum_{(x,\sigma)\in
V}\mathbb{E}\left[\eta_{t}(x,\sigma)-\rho\right]\phi(\tfrac{x}{N},\sigma)=0,\\\
&\mathrm{Var}[Y_{t}^{N}(\phi)]=\frac{1}{\sqrt{N}}\sum_{(x,\sigma)\in
V}\mathrm{Var}\left[\eta_{t}(x,\sigma)-\rho\right]\phi(\tfrac{x}{N},\sigma)=\frac{1}{N}\rho\sum_{(x,\sigma)\in
V}\phi^{2}(\tfrac{x}{N},\sigma).\end{split}$

By the central limit theorem, we therefore see that every $Y_{t}^{N}(\phi)$
converges in distribution to the normal distribution
$\mathcal{N}\big{(}0,\rho\left<\left<\phi,\phi\right>\right>\big{)}$. This
implies the tightness of the real-valued random variables
$\\{Y_{t}^{N}(\phi):N\in\mathbb{N}\\}$, and therefore also A.1.

To prove A.2, we note that for every bounded stopping time
$\uptau\in\mathscr{T}_{T}$ we have that

$Y_{\uptau}^{N}(\phi)={\mathscr{M}}^{N,\phi}_{\uptau}(Y^{N})+Y_{0}^{N}(\phi)+\int_{0}^{\uptau}L_{N}Y_{s}^{N}(\phi)ds,$

with ${\mathscr{M}}^{N,\phi}_{\uptau}(Y^{N})$ the Dynkin martingale of
$Y_{\uptau}^{N}(\phi)$. Using the Markov inequality, we can then deduce that

$\begin{split}\mathbb{P}\big{(}|Y_{\uptau}^{N}(\phi)-Y_{\uptau+\theta}^{N}(\phi)|>\varepsilon\big{)}&\leq\frac{1}{\varepsilon^{2}}\mathbb{E}\left[\left(Y_{\uptau}^{N}(\phi)-Y_{\uptau+\theta,\sigma}^{N}(\phi)\right)^{2}\right]\\\
&\leq\frac{2}{\varepsilon^{2}}\left(\mathbb{E}\left[\left({\mathscr{M}}^{N,\phi}_{\uptau}(Y^{N})-{\mathscr{M}}_{\uptau+\theta}^{N,\phi}(Y^{N})\right)^{2}\right]+\mathbb{E}\left[\left(\int_{\uptau}^{\uptau+\theta}L_{N}Y_{s}^{N}(\phi)\mathop{}\\!\mathrm{d}s\right)^{2}\right]\right).\end{split}$
(41)

For the integral term, note that by the Cauchy-Schwarz inequality and Fubini
we have that

$\begin{split}\mathbb{E}\left[\left(\int_{\uptau}^{\uptau+\theta}L_{N}Y_{s}^{N}(\phi)dr\right)^{2}\right]&\leq\sqrt{\theta}\cdot\left(\mathbb{E}\left[\int_{0}^{T+\theta}\left(L_{N}Y_{s}^{N}(\phi)\right)^{2}\mathop{}\\!\mathrm{d}s\right]\right)^{\frac{1}{2}}\\\
&=\sqrt{\theta}\cdot\left(\int_{0}^{T+\theta}\mathbb{E}\left[\left(L_{N}Y_{s}^{N}(\phi)\right)^{2}\right]\mathop{}\\!\mathrm{d}s\right)^{\frac{1}{2}}.\\\
\end{split}$ (42)

In the proof of Lemma 5.2 we have shown that
$\\{L_{N}Y_{s}^{N}(\phi):N\in\mathbb{N}\\}$ is uniformly bounded in $L^{4}$,
hence it is also uniformly bounded in $L^{2}$, i.e.

$C:=\sup_{N\in\mathbb{N}}\mathbb{E}\left[\left(L_{N}Y_{s}^{N}(\phi)\right)^{2}\right]<\infty.$
(43)

Combining (42) and (43), we find that

$\lim_{\delta\to
0}\limsup_{N\to\infty}\sup_{\begin{subarray}{c}\uptau\in\mathscr{T}_{T}\\\
\theta\leq\delta\end{subarray}}\mathbb{E}\left[\left(\int_{\uptau}^{\uptau+\theta}L_{N}Y_{s}^{N}(\phi)dr\right)^{2}\right]\leq\lim_{\delta\to
0}\sqrt{\delta CT}=0.$ (44)

For the martingale, by the martingale property we have that

$\mathbb{E}\left[{\mathscr{M}}_{\uptau}^{N,\phi}(Y^{N}){\mathscr{M}}_{\uptau+\theta}^{N,\phi}(Y^{N})\right]=\mathbb{E}\left[\left({\mathscr{M}}_{\uptau}^{N,\phi}(Y^{N})\right)^{2}\right],$

hence we see that

$\begin{split}\mathbb{E}\left[\left({\mathscr{M}}^{N,\phi}_{\uptau}(Y^{N})-{\mathscr{M}}_{\uptau+\theta}^{N,\phi}(Y^{N})\right)^{2}\right]=\mathbb{E}\left[\left({\mathscr{M}}_{\uptau+\theta}^{N,\phi}(Y^{N})\right)^{2}-\Big{(}{\mathscr{M}}_{\uptau}^{N,\phi}(Y^{N})\Big{)}^{2}\right].\end{split}$

Since $\mathbb{E}\left[{\mathscr{M}}_{0}^{N,\phi}(Y^{N})\right]=0$, we can use
that

$\mathbb{E}\left[\left({\mathscr{M}}_{t}^{N,\phi}(Y^{N})\right)^{2}\right]=\mathbb{E}\left[\int_{0}^{t}\Gamma_{s}^{N,\phi}(Y^{N})\right]\mathop{}\\!\mathrm{d}s,$

because $\int_{0}^{t}\Gamma_{s}^{N,\phi}(Y^{N})\mathop{}\\!\mathrm{d}s$ is the
quadratic variation of the process ${\mathscr{M}}_{t}^{N,\phi}(Y^{N})$.
Furthermore,
$\mathbb{E}\left[\left(\Gamma_{s}^{N,\phi}(Y^{N})\right)^{2}\right]$ is
uniformly bounded since $\Gamma_{s}^{N,\phi}(Y^{N})$ converges in $L^{2}$,
hence

$\begin{split}\sup_{N\in\mathbb{N}}\mathbb{E}\left[\left({\mathscr{M}}^{N,\phi}_{\uptau}(Y^{N})-{\mathscr{M}}_{\uptau+\theta}^{N,\phi}(Y^{N})\right)^{2}\right]&=\sup_{N\in\mathbb{N}}\mathbb{E}\left[\int_{\uptau}^{\uptau+\theta}\Gamma_{s}^{N,\phi}(Y^{N})\right]\mathop{}\\!\mathrm{d}s,\\\
&\leq\sqrt{\theta}\cdot\left(\int_{0}^{T+\theta}\sup_{N\in\mathbb{N}}\mathbb{E}\left[\left(\Gamma_{s}^{N,\phi}(Y^{N})\right)^{2}\right]\mathop{}\\!\mathrm{d}s\right)^{\frac{1}{2}}<\infty,\end{split}$

where we used Cauchy Schwarz in the second line. From this we can again
conclude that

$\lim_{\delta\to
0}\limsup_{N\to\infty}\sup_{\begin{subarray}{c}\uptau\in\mathscr{T}_{T}\\\
\theta\leq\delta\end{subarray}}\mathbb{E}\left[\left({\mathscr{M}}^{N,\phi}_{\uptau}(Y^{N})-{\mathscr{M}}_{\uptau+\theta}^{N,\phi}(Y^{N})\right)^{2}\right]=0.$
(45)

Combining (44) and (45) with (41), we indeed find that (A.2) holds. ∎

###  5.3 Uniqueness of limits

By the tightness, there exists a subsequence $N_{k}$ and a process $Y\in
D([0,T];(C_{c,S}^{\infty})^{*})$ such that $Y^{N_{k}}\to Y$ in distribution.

######  Lemma 5.6.

For each $\phi\in C_{c,S}^{\infty}$ we have that $t\mapsto Y_{t}(\phi)$ is
a.s. continuous.

###### Proof.

We define the following functions

$w_{\delta}(X)=\sup_{|t-s|<\delta}|X_{t}-X_{s}|,\ \ \ \ \
w_{\delta}^{\prime}(X)=\inf_{\begin{subarray}{c}0=t_{0}<t_{1}<...<t_{r}=1\\\
t_{i}-t_{i-1}<\delta\end{subarray}}\ \max_{1\leq i\leq r}\ \sup_{t_{i-1}\leq
s<t\leq t_{i}}|X_{t}-X_{s}|,$

then we have the following inequality

$w_{\delta}(X)\leq 2w_{\delta}^{\prime}(X)+\sup_{t}|X_{t}-X_{t^{-}}|.$ (46)

From A.2 it follows for all $\varepsilon>0$ and all $\sigma\in S$ we have that

$\lim_{\delta\to
0}\limsup_{N\to\infty}\mathbb{P}(w^{\prime}_{\delta}(Y^{N}(\phi))\geq\varepsilon)=0.$
(47)

Now note that

$\sup_{t}\left|Y_{t}^{N}(\phi)-Y_{t^{-}}^{N}(\phi)\right|\leq\sup_{t}\frac{1}{\sqrt{N}}\sum_{v\in
V}\left|(\eta_{t}(v)-\eta_{t^{-}}(v))\phi(v)\right|\leq\frac{1}{\sqrt{N}}||\phi||_{\infty}\to
0,$ (48)

where we used that there can be at most one jump between the times $t$ and
$t^{-}$ for the second inequality. Therefore, by combining (47) and (48) with
(46) we can conclude that

$\lim_{\delta\to
0}\limsup_{N\to\infty}\mathbb{P}(w_{\delta}(Y^{N}(\phi))\geq\varepsilon)=0.$

Therefore we find that $t\mapsto Y_{t}(\phi)$ is a.s. continuous. ∎

Finally we show that $Y$ solves the martingale problem in (12).

######  Proposition 5.7.

For every $\phi\in C_{c,S}^{\infty}$ the processes
${\mathscr{M}}_{t}^{\phi}(Y)$ and ${\mathscr{N}}_{t}^{\phi}(Y)$ defined in
(12) are martingales with respect to the filtration $\\{\mathscr{F}_{t}:t\geq
0\\}$ generated by $Y$.

###### Proof.

Fix arbitrary $n\in\mathbb{N}$, $s\geq 0$, $0\leq s_{1}\leq...\leq s_{n}\leq
s$, $\psi_{1},...,\psi_{n}\in C_{c,S}^{\infty}$ and $\Psi\in
C_{b}(\mathbb{R}^{n})$, and define the function
$\mathcal{I}:D([0,T];(C_{c,S}^{\infty})^{*})\to\mathbb{R}$ as

${\mathcal{I}}(X):=\Psi\left(X_{s_{1}}(\psi_{1}),...,X_{s_{n}}(\psi_{n})\right).$

To show that ${\mathscr{M}}_{t}^{\phi}(Y)$ and ${\mathscr{N}}_{t}^{\phi}(Y)$
are $\mathscr{F}_{t}$-martingales, it suffices to show that

$\lim_{k\to\infty}\mathbb{E}\left[{\mathscr{M}}_{t}^{N_{k},\phi}(Y^{N_{k}}){\mathcal{I}}(Y^{N_{k}})\right]=\mathbb{E}\left[{\mathscr{M}}_{t}^{\phi}(Y){\mathcal{I}}(Y)\right],\
\ \ \
\lim_{k\to\infty}\mathbb{E}\left[{\mathscr{N}}_{t}^{N_{k},\phi}(Y^{N_{k}}){\mathcal{I}}(Y^{N_{k}})\right]=\mathbb{E}\left[{\mathscr{N}}_{t}^{\phi}(Y){\mathcal{I}}(Y)\right],$

with ${\mathscr{M}}_{t}^{N,\phi}$ and ${\mathscr{N}}_{t}^{N,\phi}$ the Dynkin
martingales defined in (25). Namely, by the martingale property we then have
that

$\mathbb{E}\left[{\mathscr{M}}_{t}^{\phi}(Y){\mathcal{I}}(Y)\right]=\lim_{k\to\infty}\mathbb{E}\left[{\mathscr{M}}_{t}^{N_{k},\phi}(Y^{N_{k}}){\mathcal{I}}(Y^{N_{k}})\right]=\lim_{k\to\infty}\mathbb{E}\left[{\mathscr{M}}_{s}^{N_{k},\phi}(Y^{N_{k}}){\mathcal{I}}(Y^{N_{k}})\right]=\mathbb{E}\left[{\mathscr{M}}_{s}^{\phi}(Y){\mathcal{I}}(Y)\right],$

and analogous for ${\mathscr{N}}_{t}^{\phi}(Y)$.

We start by proving ${\mathscr{M}}_{t}^{\phi}(Y)$ is a martingale. First of
all, note that from Proposition 5.1 we can conclude

$\lim_{k\to\infty}\mathbb{E}\left[{\mathscr{M}}_{t}^{N_{k},\phi}(Y^{N_{k}}){\mathcal{I}}(Y^{N_{k}})\right]=\lim_{k\to\infty}\mathbb{E}\left[{\mathscr{M}}_{t}^{\phi}(Y^{N_{k}}){\mathcal{I}}(Y^{N_{k}})\right].$

Furthermore, in Lemma 5.2 we have shown that the process
${\mathscr{M}}_{t}^{\phi}(Y^{N})$ is uniformly bounded in $L^{4}$, hence it is
also uniformly bounded in $L^{2}$, therefore

$\sup_{k\in\mathbb{N}}\mathbb{E}\left[\left|{\mathscr{M}}_{t}^{\phi}(Y^{N_{k}}){\mathcal{I}}(Y^{N_{k}})\right|^{2}\right]\leq||\Psi||_{\infty}^{2}\sup_{k\in\mathbb{N}}\sum_{\sigma\in
S}\mathbb{E}\left[\left({\mathscr{M}}_{t}^{\phi}(Y^{N_{k}})\right)^{2}\right]<\infty.$

This implies that we have uniform integrability of
${\mathscr{M}}_{t}^{\phi}(Y^{N_{k}}){\mathcal{I}}(Y^{N_{k}})$. It now suffices
to show that ${\mathscr{M}}_{t}^{\phi}(Y^{N_{k}}){\mathcal{I}}(Y^{N_{k}})$
converges to ${\mathscr{M}}_{t}^{\phi}(Y){\mathcal{I}}(Y)$ in distribution.
One usually concludes this using the Portmanteau theorem, but because the path
space $D([0,T];(C_{c,S}^{\infty})^{*}_{S})$ is not metrizable, we cannot
directly use this. Instead, using the exact same method as introduced in [15,
Proposition 5.2], one can work around the problem of non-metrizability via the
continuity of $t\mapsto Y_{t}(\phi)$.

The proof that ${\mathscr{N}}_{t}^{\phi}(Y)$ is a martingale works in the same
way. First we note that by Proposition 5.4 we have that

$\lim_{k\to\infty}\mathbb{E}\left[{\mathscr{N}}_{t}^{N_{k},\phi}(Y^{N_{k}}){\mathcal{I}}(Y^{N_{k}})\right]=\lim_{k\to\infty}\mathbb{E}\left[{\mathscr{N}}_{t}^{\phi}(Y^{N_{k}}){\mathcal{I}}(Y^{N_{k}})\right].$

Therefore we only need to show that

$\sup_{k\in\mathbb{N}}\mathbb{E}\left[\left|{\mathscr{N}}_{t}^{\phi}(Y^{N_{k}}){\mathcal{I}}(Y^{N_{k}})\right|^{2}\right]<\infty.$
(49)

Afterwards the convergence of
${\mathscr{N}}_{t}^{\phi}(Y^{N_{k}}){\mathcal{I}}(Y^{N_{k}})$ to
${\mathscr{N}}_{t}^{\phi}(Y){\mathcal{I}}(Y)$ in distribution follows from the
same arguments as above.

To see that (49) holds, note that

$\displaystyle\mathbb{E}\left[\left({\mathscr{N}}_{t}^{\phi}(Y^{N_{k}})\right)^{2}\right]\leq
2\mathbb{E}\left[\left({\mathscr{M}}_{t}^{\phi}(Y^{N_{k}})\right)^{4}\right]+8t^{2}\rho^{2}\left(\kappa\left<\left<\partial_{x}\phi,\partial_{x}\phi\right>\right>+\left<\left<\phi,\Sigma\phi\right>\right>\right)^{2}.$

In the proof of Lemma 5.2, we have already shown that
$\mathbb{E}\left[\big{(}{\mathscr{M}}_{t}^{\phi}(Y^{N})\big{)}^{4}\right]$ is
uniformly bounded in $N$, hence the result follows. ∎

Acknowledgement  The authors would like to thank Christian Maes for helpful
discussions.

## References

  * [1] D. Aldous.  Stopping Times and Tightness.  The Annals of Probability, 6(2):335 – 340, 1978.
  * [2] Francesco Casini, Cristian Giardinà, and Cecilia Vernia.  Uphill in reaction-diffusion multi-species interacting particles systems.  Journal of Statistical Physics, 190(8):132, 2023.
  * [3] T. Demaerel and C. Maes.  Active processes in one dimension.  Physical review. E, 97 3-1:032604, 2018.
  * [4] J.D. Deuschel and D.W. Stroock.  Large Deviations.  AMS Chelsea Publishing Series. American Mathematical Society, 2001\.
  * [5] C. Erignoux.  Hydrodynamic limit for an active exclusion process.  Mémoires de la Société Mathématique de France, 169, May 2021.
  * [6] S. Floreani, C. Giardinà, F. den Hollander, S. Nandan, and F. Redig.  Switching Interacting Particle Systems: Scaling Limits, Uphill Diffusion and Boundary Layer.  Journal of Statistical Physics, 186(3):33, January 2022.
  * [7] A. Gidi, C. Bahadoran, O. Busani, and E. Saada.  Invariant measures for multilane exclusion process.  arXiv preprint arXiv:2105.12974, 2021.
  * [8] T. Hida.  Brownian Motion.  Stochastic Modelling and Applied Probability. Springer New York, 2012.
  * [9] C. Kipnis and C. Landim.  Scaling Limits of Interacting Particle Systems.  Springer, 1999.
  * [10] M. Lifshits.  Lectures on Gaussian Processes.  SpringerBriefs in Mathematics. Springer Berlin Heidelberg, 2012.
  * [11] I. Mitoma.  Tightness of Probabilities On $C([0,1];\mathscr{Y}^{\prime})$ and $D([0,1];\mathscr{Y}^{\prime})$.  The Annals of Probability, 11(4):989 – 999, 1983.
  * [12] F. Redig and H. van Wiechen.  Ergodic theory of multi-layer interacting particle systems.  Journal of Statistical Physics, 190, 04 2023.
  * [13] M. Sasada.  Hydrodynamic limit for two-species exclusion processes.  Stochastic processes and their applications, 120(4):494–521, 2010\.
  * [14] T. Seppäläinen.  Translation invariant exclusion process (book in progress).  Department of Mathematics, University of Wisconsin, 2008.
  * [15] B. van Ginkel and F. Redig.  Equilibrium fluctuations for the symmetric exclusion process on a compact riemannian manifold.  Markov Processes and Related Fields, 28(1):29–52, 2022.

##  Appendix A Hydrodynamic limit

In this section we give the proof of the hydrodynamic limit, i.e., of Theorem
2.2. We follow the standard methodology of [14].

###  A.1 Preliminary results

Before we start the proof of Theorem 2.2, we first show the following lemma
which, using duality, provides uniform upper bounds for the first and second
moment of the expected particle number when starting from the local
equilibrium distribution (6).

######  Lemma A.1.

For all $N\in\mathbb{N}$, $t\geq 0$ and $(x,\sigma)\in V$ we have that

$\mathbb{E}_{\mu_{\rho}^{N}}\left[\eta_{t}^{N}(x,\sigma)\right]\leq||\rho||_{\infty},$
(50)
$\mathbb{E}_{\mu_{\rho}^{N}}\left[\eta_{t}^{N}(x,\sigma)^{2}\right]\leq||\rho||_{\infty}^{2}+||\rho||_{\infty}.$
(51)

###### Proof.

For the first inequality, note that by duality we have that

$\displaystyle\mathbb{E}_{\mu_{\rho}^{N}}\left[\eta_{t}^{N}(x,\sigma)\right]$
$\displaystyle=\int\mathbb{E}_{\eta^{N}}\left[D(\delta_{(x,\sigma)},\eta_{t}^{N})\right]\mathop{}\\!\mathrm{d}\mu_{\rho}^{N}(\eta^{N})$
$\displaystyle=\int\widehat{\mathbb{E}}_{(x,\sigma)}\left[D(\delta_{(\widehat{X}_{t},\widehat{\sigma}_{t})},\eta^{N})\right]\mathop{}\\!\mathrm{d}\mu_{\rho}^{N}(\eta^{N})$
$\displaystyle=\widehat{\mathbb{E}}_{(x,\sigma)}\left[\rho(\tfrac{\widehat{X}_{t}}{N},\widehat{\sigma}_{t})\right]\leq||\rho||_{\infty}.$

Similarly for the second inequality, we have that

$\displaystyle\mathbb{E}_{\mu_{\rho}^{N}}\left[\eta_{t}^{N}(x,\sigma)^{2}\right]$
$\displaystyle=\int\mathbb{E}_{\eta^{N}}\left[D(2\delta_{(x,\sigma)},\eta_{t}^{N})+D(\delta_{(x,\sigma)},\eta_{t}^{N})\right]\mathop{}\\!\mathrm{d}\mu_{\rho}^{N}(\eta^{N})$
$\displaystyle=\int\widehat{\mathbb{E}}_{(x,\sigma),(x,\sigma)}\left[D(\delta_{\widehat{X}^{(1)}_{t},\widehat{\sigma}^{(1)}_{t})}+\delta_{(\widehat{X}^{(2)}_{t},\widehat{\sigma}^{(2)}_{t})},\eta^{N})+D(\delta_{(\widehat{X}^{(1)}_{t},\widehat{\sigma}^{(1)}_{t})},\eta^{N})\right]\mathop{}\\!\mathrm{d}\mu_{\rho}^{N}(\eta^{N})$
$\displaystyle=\widehat{\mathbb{E}}_{(x,\sigma),(x,\sigma)}\left[\rho(\tfrac{\widehat{X}^{(1)}_{t}}{N},\widehat{\sigma}^{(1)}_{t})\rho(\tfrac{\widehat{X}^{(2)}_{t}}{N},\widehat{\sigma}^{(2)}_{t})+\rho(\tfrac{\widehat{X}^{(1)}_{t}}{N},\widehat{\sigma}^{(1)}_{t})\right]\leq||\rho||_{\infty}^{2}+||\rho||_{\infty}.$

∎

Now we will define the processes ${\mathscr{M}}^{\phi}_{t}(\pi^{N})$ and
${\mathscr{M}}^{N,\phi}_{t}(\pi^{N})$ the same way as in (12) and (25)
respectively. We will again show that we can exchange these processes in the
limit.

######  Proposition A.2.

For all $t\geq 0$ and $\phi\in C_{c,S}^{\infty}$, we have that

$\lim_{N\to\infty}\mathbb{E}\left[\left|{\mathscr{M}}^{\phi}_{t}(\pi^{N})-{\mathscr{M}}^{N,\phi}_{t}(\pi^{N})\right|\right]=0.$

###### Proof.

Through similar calculations as in the proof of Proposition 5.1, we find that

$L_{N}\pi_{s}^{N}(\phi)=\pi_{s}^{N}(A\phi)+R_{3}(\phi,N,s).$ (52)

Here $R_{3}(\phi,N,s)$ is the error term of the Taylor approximations, which
is bounded as follows

$\left|R_{3}(\phi,N,s)\right|\leq\frac{1}{N^{2}}\sum_{(x,\sigma)\in
V_{N}}\eta_{s}^{N}(x,\sigma)\big{(}\kappa||\phi_{xxx}||_{\infty}+\lambda\sigma^{2}||\phi_{xx}||_{\infty}\big{)},$
(53)

and so by (50)

$\displaystyle\lim_{N\to\infty}\mathbb{E}\left[\left|{\mathscr{M}}^{\phi}_{t}(\pi^{N})-{\mathscr{M}}^{N,\phi}_{t}(\pi^{N})\right|\right]$
$\displaystyle=\lim_{N\to\infty}\int_{0}^{t}\mathbb{E}\big{[}\left|R_{3}(\phi,N,s)\right|\big{]}\mathop{}\\!\mathrm{d}s$
$\displaystyle\leq\lim_{N\to\infty}\frac{1}{N^{2}}t|V_{N}|\cdot||\rho||_{\infty}\big{(}\kappa||\phi_{xxx}||_{\infty}+\lambda\sigma^{2}||\phi_{xx}||_{\infty}\big{)}$
$\displaystyle=0,$

which concludes the proof. ∎

Lastly we will prove that the martingale ${\mathscr{M}}^{N,\phi}_{t}(\pi^{N})$
actually vanishes in the limit.

######  Lemma A.3.

For any $\phi\in C_{c,S}^{\infty}$ we have that

$\lim_{N\to\infty}\mathbb{E}\left[\sup_{t\in[0,T]}\left|{\mathscr{M}}_{t}^{N,\phi}(\pi^{N})\right|^{2}\right]=0.$

###### Proof.

First of all, by Doob’s maximal inequality, we have that

$\mathbb{E}\left[\sup_{t\in[0,T]}\left|{\mathscr{M}}_{t}^{N,\phi}(\pi^{N})\right|^{2}\right]\leq
4\mathbb{E}\left[\left({\mathscr{M}}_{T}^{N,\phi}(\pi^{N})\right)^{2}\right].$

Since ${\mathscr{M}}_{t}^{N,\phi}(\pi^{N})$ is a mean-zero martingale, this
expectation is equal to the expectation of the quadratic variation of
${\mathscr{M}}_{t}^{N,\phi}(\pi^{N})$, i.e.,

$\mathbb{E}\left[\left({\mathscr{M}}_{T}^{N,\phi}(\pi^{N})\right)^{2}\right]=\mathbb{E}\left[\int_{0}^{T}\Gamma_{s}^{N,\phi}(\pi^{N})\mathop{}\\!\mathrm{d}s\right]=\int_{0}^{T}\mathbb{E}\left[\Gamma_{s}^{N,\phi}(\pi^{N})\right]\mathop{}\\!\mathrm{d}s,$

where $\Gamma_{s}^{N,\phi}$ is as defined in (26). By using the same
calculations to get (38) we find that

$\displaystyle\Gamma_{s}^{N,\phi}(\pi^{N})$
$\displaystyle=\frac{2\kappa}{N^{2}}\sum_{(x,\sigma)\in
V}\eta^{N}_{s}(x,\sigma)(\partial_{x}\phi(\tfrac{x}{N},\sigma))^{2}+\frac{1}{N^{2}}\sum_{(x,\sigma)\in
V}\sum_{\sigma^{\prime}\in
S}c(\sigma,\sigma^{\prime})\eta^{N}_{s}(x,\sigma)(\phi(\tfrac{x}{N},\sigma^{\prime})-\phi(\tfrac{x}{N},\sigma))^{2}$
$\displaystyle\ \ \ \ \ \ \ \ +R_{4}(\phi,s,N,\sigma),$

with $R_{4}(\phi,s,N)$ bounded as follows

$|R_{4}(\phi,s,N)|\leq\kappa\frac{1}{N^{4}}\sum_{(x,\sigma)\in
V_{N}}\eta^{N}_{s}(x,\sigma)(\kappa||\phi_{xx}||_{\infty}+\lambda\sigma||\phi_{x}||_{\infty}).$

By dominated convergence and (50) we can then conclude that

$\lim_{N\to\infty}\mathbb{E}\left[\left({\mathscr{M}}_{T}^{N,\phi}(\pi^{N})\right)^{2}\right]=\lim_{N\to\infty}\int_{0}^{T}\mathbb{E}\left[\Gamma_{s}^{N,\phi}(\pi^{N})\right]\mathop{}\\!\mathrm{d}s=0,$

and the result follows. ∎

###  A.2 Tightness

We now prove the tightness result for the sequence
$\\{\pi^{N}:N\in\mathbb{N}\\}$.

######  Proposition A.4.

$\\{\pi^{N}:N\in\mathbb{N}\\}$ is tight in $D([0,T];\textbf{\emph{M}})$.

###### Proof.

In the space $D([0,T];\textbf{M})$ we can prove tightness by showing that the
following two assertions hold.

  1. B.1

For all $t\in[0,T]$ and $\varepsilon>0$ there exists a compact
$K(t,\varepsilon)\subset\textbf{M}$ such that

$\sup_{N\in\mathbb{N}}\mathbb{P}\big{(}\pi_{t}^{N}\notin
K(t,\varepsilon)\big{)}\leq\varepsilon.$

  2. B.2

For all $\varepsilon>0$

$\lim_{\delta\to
0}\limsup_{N\to\infty}\mathbb{P}\big{(}\omega(\pi^{N},\delta)\geq\varepsilon\big{)}=0,$

where

$\omega(\alpha,\delta)=\sup\\{d\big{(}\alpha_{s},\alpha_{t}\big{)}:s,t\in[0,T],|t-s|<\delta\\},$

and $d$ is the metric on M given by

$d\big{(}\alpha,\beta\big{)}=\sum_{j=1}^{\infty}2^{-j}\left(1\wedge\left|\alpha(\phi_{j})-\beta(\phi_{j})\right|\right)$

for some specific choice of test functions $\phi_{j}\in C_{c,S}^{\infty}$.

We start by proving B.1. Fix $\varepsilon>0$ and $t\geq 0$, and for some $C>0$
let $K_{C}$ be the following set

$K_{C}=\left\\{\mu\in\textbf{M}:\mu([-k,k]\times S)\leq C(2k+1)k^{2}\text{ for
all $k\in\mathbb{N}$}\right\\}.$

By[14, Proposition A.25], this is a compact set in M, and by Markov’s
inequality we now have that

$\displaystyle\mathbb{P}(\pi_{t}^{N}([-k,k]\times S)\geq C(2k+1)k^{2})$
$\displaystyle\leq\frac{1}{C(2k+1)k^{2}}\mathbb{E}\left[\pi_{t}^{N}([-k,k]\times
S)\right]$
$\displaystyle=\frac{1}{C(2k+1)k^{2}N}\sum_{(x,\sigma)\in[-kN,kN]\times
S}\mathbb{E}\left[\eta^{N}_{t}(x,\sigma)\right]$
$\displaystyle\leq\frac{1}{C(2k+1)k^{2}N}(2k+1)N|S|\cdot||\rho||_{\infty}$
$\displaystyle=\frac{1}{Ck^{2}}|S|\cdot||\rho||_{\infty}.$

Here we have used the inequality in (50). Therefore

$\displaystyle\mathbb{P}(\pi_{t}^{N}\notin
K_{C})\leq\sum_{k=1}^{\infty}\mathbb{P}(\pi_{t}^{N}([-k,k])\geq
C(2k+1)k)\leq\frac{1}{C}|S|\cdot||\rho||_{\infty}\sum_{k=1}^{\infty}\frac{1}{k^{2}}<\infty.$

By now taking $C$ big enough, we then have that for all $N\in\mathbb{N}$ that
$\mathbb{P}(\pi_{t}^{N}\notin K_{C})\leq\varepsilon$, which finishes the proof
of B.1.

In order to prove that B.2 holds, note first that

$\begin{split}\omega(\pi^{N},\delta)&=\sup_{\begin{subarray}{c}s,t\in[0,T]\\\
|t-s|<\delta\end{subarray}}\sum_{j=1}^{\infty}2^{-j}\left(1\wedge\left|\pi^{N}_{t}(\phi_{j})-\pi^{N}_{s}(\phi_{j})\right|\right)\\\
&\leq 2^{-m}+\sum_{j=1}^{m}\sup_{\begin{subarray}{c}s,t\in[0,T]\\\
|t-s|<\delta\end{subarray}}2^{-j}\left(1\wedge\left|\pi^{N}_{t}(\phi_{j})-\pi^{N}_{s}(\phi_{j})\right|\right)\\\
&\leq 2^{-m}+\sum_{j=1}^{m}\sup_{\begin{subarray}{c}s,t\in[0,T]\\\
|t-s|<\delta\end{subarray}}\left|\pi^{N}_{t}(\phi_{j})-\pi^{N}_{s}(\phi_{j})\right|.\end{split}$
(54)

Here we have taken $m$ arbitrarily, so the first term can be made as small as
we want. We now want to show that the expecation of the sum vanishes as we let
$N\to\infty$ and $\delta\downarrow 0$. Afterwards, the claim can be shown by
using the Markov inequality.

Note first that we have the following,

$\begin{split}\mathbb{E}\left[\sup_{\begin{subarray}{c}s,t\in[0,T]\\\
|t-s|<\delta\end{subarray}}\left|\pi^{N}_{t}(\phi_{j})-\pi^{N}_{s}(\phi_{j})\right|^{2}\right]&=\mathbb{E}\left[\sup_{\begin{subarray}{c}s,t\in[0,T]\\\
|t-s|<\delta\end{subarray}}\left|{\mathscr{M}}_{t}^{N,\phi_{j}}(\pi^{N})-{\mathscr{M}}_{s}^{N,\phi_{j}}(\pi^{N})-\int_{s}^{t}L_{N}\pi_{r}^{N}(\phi_{j})\mathop{}\\!\mathrm{d}r\right|^{2}\right]\\\
&\leq
4\mathbb{E}\left[\sup_{t\in[0,T]}\left({\mathscr{M}}_{t}^{N,\phi_{j}}(\pi^{N})\right)^{2}\right]+2\mathbb{E}\left[\sup_{\begin{subarray}{c}s,t\in[0,T]\\\
|t-s|<\delta\end{subarray}}\left|\int_{s}^{t}L_{N}\pi_{r}^{N}(\phi_{j})\mathop{}\\!\mathrm{d}r\right|^{2}\right].\\\
\end{split}$ (55)

By Lemma A.3, the first term goes to zero as $N\to\infty$. For the second
term, by filling in (52) we find that

$\begin{split}\left|\int_{s}^{t}L_{N}\pi_{r}^{N}(\phi_{j})\mathop{}\\!\mathrm{d}r\right|^{2}&=\left(\int_{s}^{t}\left(\pi_{r}^{N}(A\phi_{j})+R_{3}(\phi_{j},N,r)\right)\mathop{}\\!\mathrm{d}r\right)^{2}\\\
&\leq
2\left(\int_{s}^{t}\pi_{r}^{N}(A\phi_{j})\mathop{}\\!\mathrm{d}r\right)^{2}+2\left(\int_{s}^{t}R_{3}(\phi_{j},N,r,\sigma)\mathop{}\\!\mathrm{d}r\right)^{2}.\end{split}$
(56)

By the upper bound on $R_{3}(\phi_{j},N,r)$ in (53) and by (50), we can see
that the last term vanishes in expectation when $N\to\infty$. For the other
term we have that

$\displaystyle\left(\int_{s}^{t}\pi_{r}^{N}(A\phi_{j})\mathop{}\\!\mathrm{d}r\right)^{2}=\frac{1}{N^{2}}\left[\int_{s}^{t}\sum_{(x,\sigma)\in
V}\eta_{r}^{N}(x,\sigma)\cdot(A\phi_{j})(\tfrac{x}{N},\sigma)\mathop{}\\!\mathrm{d}r\right]^{2}.$

Using that $|t-s|<\delta$ and applying Hölder a number of times, we find that

$\displaystyle\left(\int_{s}^{t}\pi_{r}^{N}(A\phi_{j})\mathop{}\\!\mathrm{d}r\right)^{2}$
$\displaystyle\leq\frac{1}{N^{2}}|V_{\phi_{j}}^{N}|\delta\cdot||A\phi||_{\infty}\sum_{(x,\sigma)\in
V_{\phi_{j}}^{N}}\int_{0}^{T}\left(\eta_{r}^{N}(x,\sigma)\right)^{2}\mathop{}\\!\mathrm{d}r.$

Using the inequality in (51), we find that

$\displaystyle\mathbb{E}\left[\sup_{\begin{subarray}{c}s,t\in[0,T]\\\
|t-s|<\delta\end{subarray}}\left(\int_{s}^{t}\pi_{r}^{N}(A\phi_{j})\mathop{}\\!\mathrm{d}r\right)^{2}\right]\leq\frac{1}{N^{2}}|V_{\phi_{j}}^{N}|^{2}\delta
T\cdot||A\phi||_{\infty}(||\rho||_{\infty}^{2}+||\rho||_{\infty})=\mathcal{O}(\delta).$

Therefore

$\lim_{\delta\downarrow
0}\limsup_{N\to\infty}\mathbb{E}\left[\sup_{\begin{subarray}{c}s,t\in[0,T]\\\
|t-s|<\delta\end{subarray}}\left|\pi^{N}_{t}(\phi_{j})-\pi^{N}_{s}(\phi_{j})\right|^{2}\right]=\lim_{\delta\downarrow
0}\limsup_{N\to\infty}\mathbb{E}\left[\sup_{\begin{subarray}{c}s,t\in[0,T]\\\
|t-s|<\delta\end{subarray}}\left(\int_{s}^{t}\pi_{r}^{N}(A\phi_{j})\mathop{}\\!\mathrm{d}r\right)^{2}\right]=0.$
(57)

So, by going back to (54) and using the Markov inequality, we get the
following:

$\displaystyle\mathbb{P}(\omega(\pi^{N},\delta)\geq\varepsilon)\leq\frac{1}{\varepsilon}\left(2^{-m}+\sum_{j=1}^{m}\mathbb{E}\left[\sup_{\begin{subarray}{c}s,t\in[0,T]\\\
|t-s|<\delta\end{subarray}}\left|\pi^{N}_{t}(\phi_{j})-\pi^{N}_{s}(\phi_{j})\right|\right]\right).$

By now taking $m$ such that $2^{-m}<\varepsilon^{2}$ and using (57) we see
that

$\displaystyle\lim_{\delta\downarrow
0}\limsup_{N\to\infty}\mathbb{P}(\omega(\pi^{N},\delta)\geq\varepsilon)<\varepsilon,$

which ultimately proves the tightness result. ∎

###  A.3 Proof of hydrodynamic limit

Now we have everything needed to prove the result.

###### Proof of Theorem 2.2.

From the tightness of the sequence $\\{P^{N}:N\in\mathbb{N}\\}$ we know that
there exists a subsequence $\\{P^{N_{k}}:k\in\mathbb{N}\\}$ that converges
weakly in the Skorokhod topology, i.e., $P^{N_{k}}\xrightarrow{w}P$ for some
probability measure $P$ on $D([0,T];\textbf{M})$. If we can show that every
convergent subsequence converges to the dirac measure $P=\delta_{\pi}$ with
$\pi$ the unique continuous path solving (8), then the result follows.

First of all, by B.2, we immediately know that $P$ is concentrated on
continuous paths in $D([0,T];\textbf{M})$. Now define for $\phi\in
C_{c,S}^{\infty}$, $\varepsilon>0$ and $T>0$ the following set

$H(\phi,\varepsilon):=\left\\{\alpha\in
D([0,T];\textbf{M}):\sup_{t\in[0,T]}\left|\alpha_{t}(\phi)-\alpha_{0}(\phi)-\int_{0}^{t}\alpha_{s}(A\phi)\mathop{}\\!\mathrm{d}s\right|\leq\varepsilon\right\\}.$

Analogously as in [14, Lemma 8.7] one can prove that this set is closed in the
Skorokhod topology. Since the set $H(\phi,\varepsilon)$ is closed, we can
apply the Portmanteau Theorem to see that

$\displaystyle P\big{(}H(\phi,\varepsilon)\big{)}$
$\displaystyle\geq\limsup_{k\to\infty}P^{N_{k}}\big{(}H(\phi,\varepsilon)\big{)}$
$\displaystyle=\limsup_{k\to\infty}\mathbb{P}\left(\sup_{t\in[0,T]}\left|\pi^{N_{k}}_{t}(\phi)-\pi^{N_{k}}_{0}(\phi)-\int_{0}^{t}\pi^{N_{k}}_{s}(A\phi)\mathop{}\\!\mathrm{d}s\right|\leq\varepsilon\right)$
$\displaystyle=\limsup_{k\to\infty}\mathbb{P}\left(\sup_{t\in[0,T]}\left|{\mathscr{M}}_{t}^{\phi}(\pi^{N_{k}})\right|\leq\varepsilon\right)$
$\displaystyle=\limsup_{k\to\infty}\mathbb{P}\left(\sup_{t\in[0,T]}\left|{\mathscr{M}}_{t}^{N_{k},\phi}(\pi^{N_{k}})\right|\leq\varepsilon\right).$

Here we have used Proposition A.2 for the last equality. By Lemma A.3 and the
Markov inequality we then have that

$\mathbb{P}\left(\sup_{t\in[0,T]}\left|{\mathscr{M}}_{t}^{N_{k},\phi}(\pi^{N_{k}})\right|>\varepsilon\right)\leq\frac{1}{\varepsilon^{2}}\mathbb{E}\left[\sup_{t\in[0,T]}\left|{\mathscr{M}}_{t}^{N_{k},\phi}(\pi^{N_{k}})\right|^{2}\right]\to
0,$

so $P\big{(}H(\phi,\varepsilon)\big{)}=1$. Since we took $\varepsilon>0$
arbitrarily, we indeed find that $P=\delta_{\pi}$. ∎


In this paper, we extend Manin and Schechtman's higher Bruhat orders for $S_n$ to higher Bruhat orders for non-longest words $w$ in $S_n$. We prove that the higher Bruhat orders of non-longest words are ranked posets with unique minimal and maximal elements. As in Manin and Schechtman's original paper, the $k$-th Bruhat order for $w$ is created out of equivalence classes of maximal chains in its $(k-1)$-st Bruhat order. We also define the second and third Bruhat orders for arbitrary realizable $k$-sets, and prove that the second Bruhat order has a unique minimal and maximal element. Lastly, we also outline how this extension may guide future research into developing higher Bruhat orders for affine type A Weyl groups.

[1]
Ben Elias.
A diagrammatic category for generalized Bott-Samelson bimodules
and a diagrammatic categorification of induced trivial modules for Hecke
Proceedings of the London Mathematical Society, page 924–978,

[2]
Ben Elias.
A Diamond lemma for Hecke-type algebras, Apr 2021.

[3]
Mitra Koley.
Thicker Soergel calculus for B3.
Journal of Algebra and Its Applications, 19(12):2050230, 2019.

[4]
Nicolas Libedinsky.
New bases of some Hecke algebras via Soergel bimodules.
Advances in Mathematics, 228(2):1043–1067, 2011.

[5]
Yu. I. Manin and V. V. Schechtman.
Arrangements of Hyperplanes, Higher Braid Groups and higher
Bruhat orders.
Algebraic Number Theory — in honor of K. Iwasawa, page
289–308, 1989.

[6]
Seth Shelley-Abrahamson and Suhas Vijaykumar.
Higher Bruhat orders in type B.
The Electronic Journal of Combinatorics, 23(3), 2016.

[7]
Günter M. Ziegler.
Higher Bruhat orders and cyclic hyperplane arrangements.
Topology, 32(2):259–279, 1993.