Scaling limits for Rudvalis card shuffles]Scaling limits for Rudvalis card shuffles

P. Gonçalves
[P. Gonçalves]Center for Mathematical Analysis, Geometry and Dynamical Systems, Instituto Superior Técnico, Universidade de Lisboa, 1049-001 Lisboa, Portugal

M. Jara
[M. Jara]Instituto de Matemática Pura e Aplicada, Estrada Dona Castorina 110, 22460-320
RIO DE JANEIRO, BRAZIL
<http://w3.impa.br/ monets/index.html>

R. Marinho
[R. Marinho]Universidade Federal de Santa Maria, Campus Cachoeira do Sul, Rod. Taufik Germano, 3013, 96503-205, Cachoeira do Sul, Brasil

D. Moreira
[D. Moreira]Center for Mathematical Analysis, Geometry and Dynamical Systems, Instituto Superior Técnico, Universidade de Lisboa, 1049-001 Lisboa, Portugal

We consider the Rudvalis card shuffle and some of its variations that were introduced by Diaconis and Saloff-Coste in [5], and we project them to some stochastic interacting particle system. For the latter, we derive the hydrodynamic limits  and we study the equilibrium fluctuations. Our results show that, for these shuffles, when we consider an asymmetric variation of the Rudvalis shuffle,  the hydrodynamic limit in Eulerian scale is given in terms of a transport equation with a constant that depends on the exchange rates of the system; while for  symmetric and weakly asymmetric variations of this shuffle, in diffusive time scale,  the evolution is given by the solution of a martingale problem.

§ INTRODUCTION

Fluid dynamics  are commonly modelled by stochastic interacting particle systems. With these models it may be possible to derive partial differential equations as the  limits, in some sense, of some observables of the system that evolve both in space and  time. Those are called scaling limits [13].

One of the most famous stochastic interacting  particle systems is the symmetric simple exclusion process (SSEP), where particles perform symmetric nearest-neighbor random walks on a graph with the exclusion rule, which forbids more than one particle at the same vertex. When the graph is the finite cycle $\bb Z_n=\bb Z/n \bb Z$, for instance, the total mass (number of particles) is conserved and one can recover the heat equation on the torus $\bb T=[0,1)$ as the limit, in probability, of the empirical measure, which is a measure on $\bb T$ obtained by re-scaling time  diffusively  as $n^2$, re-scaling space by $n^{-1}$, and assigning mass $n^{-1}$ to each particle. The limit of the empirical measure is known as the hydrodynamic limit and it can be seen as a Law of Large Numbers for the density of particles. A more challenging task is to prove a central limit theorem from which one can observe the equilibrium fluctuations of the process.

An interesting fact is that the SSEP can be obtained as a projection of a very famous card shuffle known as the interchange process. In order to see this projection, one just has to color some cards in black, the remaining ones in red, and to associate black cards with particles and red cards with empty sites. This projection and its scaling limits were used in [14] and in [17] to study the mixing times of both processes, i.e.  the time at which the total variation distance between the law of the process and the stationary measure is smaller than some threshold $\varepsilon>0$, both in the interval and in the cycle. Inspired by this, in this work we  color the cards of a deck with black and red colours, but we study the corresponding stochastic interacting particle system obtained when we perform a peculiar card shuffle, the Rudvalis shuffle. This card shuffle was introduced by Arunas Rudvalis in [4] and it can be explained as follows: remove the card on the top of a deck and insert it back, uniformly at random, either  at the bottom or at the second position from the bottom (see Figure <ref>).

Insertion of the top card at position $n-1$ or $n$.

Lower and upper bounds on the mixing time for the Rudvalis card shuffle were obtained in [18] and in [9]. Matching bounds are  still unknown. In the case of the interchange process, the knowledge about the hydrodynamic limit and the fluctuations was crucial to determine the lower and upper bounds of the mixing time. Motivated by this, in this work we investigate the hydrodynamic behavior and the equilibrium fluctuations of the Rudvalis card shuffle and some of its variations. In [5] for instance, Diaconis and Saloff-Coste introduced a symmetric version of the Rudvalis shuffle where one inserts the top card at the bottom of the deck  with probability $1/4$, or inserts the bottom card at the top  of the deck with probability $1/4$,  or transposes the cards at positions $1$ and $2$ with probability $1/4$,  or nothing is done with probability $1/4$. In order to obtain distinct hydrodynamic behaviours for the associated stochastic interacting particle systems,  we introduce a continuous-time random walk on the symmetric group from which not only  we can get the symmetric and asymmetric variations of the Rudvalis card shuffle, but also a weakly asymmetric one. We prove that, in the Eulerian  time scale, the hydrodynamics and the equilibrium fluctuations of the projection of the asymmetric Rudvalis shuffle are driven by a transport equation with periodic boundary conditions. Furthermore, we show that, in the diffusive time scale, hydrodynamics and fluctuations for the symmetric and weakly asymmetric variations are driven by a solution of a non-trivial martingale problem. In [7], Goel introduced a variation of the Rudvalis card shuffle where one puts the top card uniformly at random in the bottom-$k$ positions of the deck. For this reason, this card shuffle is known as the Top to bottom-$k$ shuffle. For the top-to-bottom $k$-shuffle, our methodology yields the  hydrodynamic limits when $k=o(n)$ and the equilibrium  fluctuations when $k=o(\sqrt{n})$.

This work is organized as follows: in Section <ref> we introduce the continuous-time card shuffle and we map it to a stochastic  interacting particle systems. Moreover, in that section we state our main results, and to do so, we introduce some notation and definitions as Sobolev spaces, the empirical measure, the density fluctuation field, some weak solutions of partial differential equations and some martingale problems. In Section <ref>  we prove the hydrodynamic limits in the Eulerian and diffusive time scales,  and in Section <ref>  we obtain the scaling limit of  the equilibrium fluctuations in both time scales.
In Section <ref>, we prove the uniqueness of the solution of the martingale problem obtained in diffusive time scale.

§ NOTATION AND RESULTS

§.§ The Generalized Rudvalis shuffle

Let $n\in\mathbb N$ be a scaling parameter and fix four sequences of real numbers $\{a_n\}_{n},\{b_n\}_n,\{c_n\}_n,\{d_n\}_n  \in [0,1]$ converging, as $n \to \infty$, respectively, to $a,b,c,d $.
Let $S_n$ denote the symmetric group. $S_n$ must be understood as the set of all possible permutations of a deck with $n$ cards. Given a permutation $\sigma \in S_n$, we denote the card at position $x$ by $\sigma(x)$. Given $\sigma\in S_n$ and $x<y$, if we denote $\sigma=(\sigma(1),\sigma(2),\cdots,\sigma(x),\cdots, \sigma(y),\cdots, \sigma(n))\in S_n$, we define $\sigma^{x \rightarrow y}$ as the permutation obtained from $\sigma$ after we remove the card at position $x$ from the deck and insert it at position $y$, i.e.
$$\sigma^{x \rightarrow y}=(\sigma(1),\sigma(2),\cdots, \sigma(x-1),\sigma(x+1),\cdots, \sigma(y),\sigma(x),\sigma(y+1),\cdots, \sigma(n)),$$

while for $x>y$ and $\sigma=(\sigma(1),\sigma(2),\cdots,\sigma(y),\cdots, \sigma(x),\cdots, \sigma(n))$ we define

$$\sigma^{x \rightarrow y}=(\sigma(1),\sigma(2),\cdots, \sigma(y-1),\sigma(y),\sigma(x),\sigma(y+1),\cdots, \sigma(x-1),\sigma(y+1),\cdots, \sigma(n)).$$

We also define  $\sigma^{x \leftrightarrow y}$ as  the permutation  obtained after we transpose the cards at positions $x$ and $y$, i.e.
for $\sigma=(\sigma(1),\sigma(2),\cdots,\sigma(x-1),\sigma(x),\sigma(x+1),\cdots, \sigma(y-1),\sigma(y),\sigma(y+1),\cdots, \sigma(n))$, we define

$$\sigma^{x \leftrightarrow y}=\sigma=(\sigma(1),\sigma(2),\cdots,\sigma(x-1),\sigma(y),\sigma(x+1),\cdots, \sigma(y-1),\sigma(x),\sigma(y+1),\cdots, \sigma(n)).$$

We  consider the continuous-time Markov chain $\{ \sigma_t \}_{t \geq 0 }$ with state space $S_n$ and generator $\mcb L_n^{\spadesuit} $ given on $f: S_n \to \bb R$ and $\sigma\in S_n$ by

L_n^ f(σ)= a_n ( f(σ^1 →n-1)- f(σ)) + b_n ( f(σ^1 →n)- f(σ)) + c_n ( f(σ^n →1)- f(σ)) +d_n ( f(σ^1 ↔2)- f(σ))

and we call $\{ \sigma_t\}_{ t \geq 0 }$ the generalized Rudvalis card shuffle.

We observe that for $a_n=b_n=1/2$ and $c_n=d_n=0$ we obtain the original Rudvalis shuffle introduced in [4]. Similarly, when $a_n=0$ and $b_n=c_n=d_n=1/4$ we obtain the symmetric version of the Rudvalis  shuffle introduced in [5]. In [18] it is also considered the case $a_n \in (0,1)$, $b_n=1-a_n$ and $c_n=d_n=0$.

Our aim is to project the generalized Rudvalis shuffle to a stochastic  interacting particle system, where the latter is defined as a projection of the colours of the cards. Let $\bb Z_n=\bb Z/ n \bb Z$ and let  us colour each card of the deck in either red or black. For each permutation $\sigma$ and each $x,y \in \bb Z_n$, we define the colouring  $\eta=\eta(\sigma)$ by

\eta(x)=\begin{cases}
1 &, \text{ if the card } \sigma(x) \text{ is black},\\
0 &, \text{ if the card } \sigma(x) \text{ is red}.
\end{cases}

Let $\Omega_n=\{0,1\}^{\bb Z_n}$. For each $x, y \in \bb Z_n$ define $\eta^{x \rightarrow y}=\eta(\sigma^{x \rightarrow y})$ and $\eta^{x \leftrightarrow y}=\eta(\sigma^{x \leftrightarrow y})$. The projection defined above allows mapping black cards to particles and red cards to holes. From  (<ref>), we see that  $\{\eta_t \}_{t \geq 0 }$ has an infinitesimal generator given on $f: \Omega_n \to \bb R$ and $\eta\in\Omega_n$ by

L_n f(η)= a_n ( f(η^1 →n-1)- f(η)) + b_n ( f(η^1 →n)- f(η)) + c_n ( f(η^n →1)- f(η)) +d_n ( f(η^1 ↔2)- f(η)).

We will consider the process speeded up in the time scale $tn^\beta$ with $\beta>0$ and the generator of $\{\eta_{tn^\beta}\}_{t\geq0}$ is simply given by $n^\beta\mcb L_n$. We will consider two cases, the hyperbolic time scale, corresponding to the choice $\beta=1$ and  the diffusive time scale, corresponding to the choice $\beta=2$.

§.§ Stationary measures

We assume that  $a_n,b_n,c_n,d_n$ are chosen in such a way that the  processes $\{\sigma_t\}_{  t \geq 0 }$ and $\{\eta_t\}_{  t \geq 0 }$ are irreducible and in those cases, the uniform measure on $S_n$ will be the stationary measure of the card shuffle $\{\sigma_t\}_{  t \geq 0 }$. Therefore,  if the initial number of particles $\ell$ is fixed, then the uniform measure on the hyperplane $\Omega_{n,\ell}:=\{ \eta \in \Omega_n ; \sum_{x \in \bb Z_n} \eta(x)=\ell \}$ is the stationary measure of the particle system.

For $\varrho\in [0,1]$, let $\nu_\varrho$ be the Bernoulli product measure with parameter $\varrho $, that is, for each $\eta \in \Omega_n$,

ν_ϱ(η)=∏_x ∈Z_n { ϱη(x) +(1-ϱ)(1-η(x)) }.

It is straightforward to check that $\nu_\varrho$ is stationary for any $\varrho \in [0,1]$.
In order to define the space where our functionals will act, we need to introduce the so-called Sobolev spaces.

§.§ Sobolev Spaces

Let $\bb T$ be the unit torus.  For $f,g:\bb T \to \bb C$, let  $\left<f,g \right> $ denote the inner product in $L^2(\bb T)$, which is given by
$$\left<f,g \right>=\int_{\bb T} f(u) \overline{g(u)} du.$$
For each $k \in \bb Z$ let $\psi_k:\bb T \to \bb C$ be defined by

√(2) sin(2 πk u)    ,  if  k<0,

1    ,  if  k=0,

√(2) cos(2 πk u)    ,  if  k>0.

for any $u \in \bb T$. The family $\{ \psi_k \}_{ k \in \bb Z }$ forms an orthonormal basis of $L^2(\bb T)$. Note that the functions $\psi_k$ are eigenfunctions of the positive symmetric linear operator $\mcb L=(1-\Delta)$, i.e.  for each $k \in \bb Z$,
\mcb L \psi_k=\gamma_k \psi_k,
with $\gamma_k=1+4 \pi^2 k^2$.
Now we are ready to define Sobolev spaces.  For each positive integer $m$, we denote by $\mcb H_{m}$ the Hilbert space obtained as the completion of $C^\infty(\bb T)$ endowed with the inner product $\left\< \cdot,\cdot\right\>_{m}$ defined by

\left\< f,g\right\>_{m}=\left\< f,\mcb L^m g \right\>.

It is easy to verify that $\mcb H_m$ is the subspace of $L^2(\bb T)$ consisting of all functions $f$ such that

\sum_{k \in \bb Z} |\left\< f, \psi_k \right\>|^2 \gamma_k^m<\infty.

For each positive integer $m$, we denote by $\mcb H_{-m}$ the dual of $\mcb H_m$ with respect to the inner product  $\langle\cdot, \cdot \rangle$. It  is easy to check  that $\mcb H_{-m}$ consists of all sequences $\{ \left\<f,\psi_k \right\>\}_{  k \in \bb Z }$ such that

\sum_{k \in \bb Z} |\left\< f, \psi_k \right\>|^2 \gamma_k^{-m}<\infty

and that the inner product $\left\< f,g\right\>_{-m}$ of two functions $f,g \in \mcb H_{-m}$ can be written as

\begin{equation}\label{eq:norm-}
\left\< f,g \right\>_{-m}= \sum_{k \in \bb Z} \left\< f, \psi_k \right>  \left\< g, \psi_k \right> \gamma_k^{-m}.
\end{equation}

If we denote $L^2(\bb T)$ by $\mcb H_0$, then
from the explicit characterization of $\mcb H_{-m}$, we obtain the following inclusion of spaces

\cdots \subset \mcb H_2 \subset \mcb  H_1 \subset \mcb H_0 \subset \mcb H_{-1} \subset \mcb H_{-2} \subset \cdots

§.§ Empirical measure

We note that the number of particles (or the numbers of black cards) is conserved  by the dynamics  of $\{\eta_t\}_{t\geq 0}$. Our first question is related to understanding how does  the distribution of particles evolve in space and time.   To properly answer this question we need to define the empirical measure associated to the conserved quantity, i.e.  the number of particles. To that end,
let $\bb T$ be the unit torus and  fix a time horizon $[0,T]$, with $T>0$. We denote by $\mcb  M$  the space of positive measures on $\bb T$ with total mass bounded by one, endowed with the weak topology.

For each $t \in [0,T]$, let $\pi_t^{n,\beta} \in \mcb M$ be the empirical measure  on $\bb T$ obtained by rescaling space by $n^{-1}$, and assigning mass $n^{-1}$ to each particle (black card), that is,

π_t^n,β(η,du):=1/n∑_x∈Z_n  η_tn^β(x)δ_xn  (du),

where $\delta_u$ is the Dirac measure concentrated on $u\in\mathbb T$. For convenience of notation,  we denote the integral of  $f:\bb T \to \mathbb R$ with respect to $\pi_t^{n,\beta}$ as
\langle \pi_t^{n,\beta},f\rangle$.
Let $\{\mu_n\}_{n\in\bb N}$ be a sequence of probability measures in $\Omega_n$. For  $n \in \bb N$, let $\mathbb{P}_{\mu_n}$ be the probability measure on the Skorokhod space $\mcb {D}([0,T],\Omega_n)$ induced by  $\{\eta_{tn^\beta}\}_{t \geq 0}$ and the initial measure $\mu_n$ and  let $\mathbb{E}_{\mu_n}$ be the expectation with respect to $\mathbb{P}_{\mu_n}$. Let $\{\bb {Q}_{n,\beta} \}_{n\in\bb N}$ be the sequence of probability measures on $\mcb{D} ([0,T],  \mcb {M})$ induced by $\{\pi_{t}^{n,\beta}\}_{t \geq 0}$ and $\mathbb{P}_{\mu_n}$.

Let $\mathfrak h: \mathbb{R}\rightarrow[0,1]$ be a measurable function and $\{\mu_n\}_{n\in\
bb N}$ a sequence of probability measures in $\Omega_n$. We say that $\{\mu_n\}_{n\in \bb N}$  is associated with $\mathfrak h$, if for any continuous function $G:\mathbb T\rightarrow \mathbb R$ and any $\delta > 0$,

\begin{equation*}
\lim _{n\to \infty } \mu_n \Big( \eta \in \Omega_n : \Big|  \frac{1}{n} \sum_{x\in \bb Z_n} G(\tfrac{x}{n}) \eta(x)   - \int_{\mathbb{R}} G(u)\mathfrak h(u)\,du \, \Big|    > \delta \Big)= 0.
\end{equation*}

§.§ Hydrodynamic limit

In order to state our first result, we introduce the notion of weak solution of a transport equation on $\bb T$:

Let $\mathfrak g :\bb T \to [0,1]$ be a measurable function. Let $\theta \in \bb R$. We say that $\varrho^\theta: [0,T] \times \bb T \to [0,1]$ is a weak solution of the transport equation

∂_t ϱ^θ(t,u) = θ∇ϱ^θ(t,u)    ,∀u ∈T;

ϱ(0,u)=𝔤(u)    , ∀u ∈T.

if for all $f \in C^{1}(\bb T)$ and all $t \in[0,T]$ we have

0=\left\< \varrho^\theta_t  - \mathfrak g , f\right\>  +\theta \int_0^t \langle  \varrho_s^\theta ,\nabla f\rangle \,ds.

Equation (<ref>) has a unique weak solution given by $\varrho_t^\theta(u)=\varrho_0^\theta(u+\theta t)$.
Now we state the  hydrodynamic limit in the hyperbolic time scale ($\beta=1)$:

Let $\beta=1$ and $\varrho_0: \bb T \to [0,1]$ be a measurable function.    Let $\{\mu_n\}_{n\ \geq 1}$ be a sequence of probability measures in $\Omega_n$ associated to the profile $\varrho_0$.
Assume that
\begin{equation}\label{def_kappa}
\lim_{n\to+\infty}|a_n+b_n-c_n|=\kappa:=a+b-c\neq 0.
\end{equation}

Then, for any $t \in [0,T]$, the sequence  $\{{\bb Q}_{n,1}\}_{ n \in \bb N}$ converges, as $n\to+\infty$, with respect to the weak topology of $\mcb D ([0,T],\mcb H_{-m})$, for $m>3/2$, to ${\bb Q}_{1}$, which is a Dirac measure on the trajectory $\pi^1_t(du)=\varrho^\kappa(t,u)du$, where $\varrho^\kappa$ is the unique weak solution of  (<ref>) with $\theta=\kappa$.

We observe that since the limiting equation is a transport equation, we could perform a Galilean transformation at the level of the empirical measure and, in the hyperbolic time scale, we would  have a trivial evolution (which also happens if $\kappa=0$). 	In that sense, we need to speed up the time scale in order to see a non-trivial limit. In fact, the trivial behavior goes up to $\beta<2$. Therefore, we consider now the diffusive time scale corresponding to $\beta=2$. In this regime the limit is not longer deterministic since the contribution of the martingale term is relevant and brings an additional term to the equation.  Let us now define properly the notion of solution to the equation that we obtain in this regime.

Let $\nu,\sigma>0$ and $\mcb v\in\mathbb R$. Fix $m >0$.  We say that a process $\{X_t\}_{ t \geq 0}$ with continuous trajectories in $\mcb  H_{ -m}$ is a solution of the martingale problem associated to

\begin{equation}
\label{arica}
\partial_t X_t = \nu \Delta X_t + \mcb v \nabla X_t + \sigma\nabla X_t d \mc B_t,
\end{equation}

where  $\{ \mc B_t\}_{t \geq 0}$ is  a standard Brownian motion,
if for every $f \in  C^\infty(\bb T)$ the process $\{\mcb M_t(f)\}_{ t \geq 0}$ given by

\[
\mcb M_t(f) := X_t(f) - X_0(f) - \int_0^t X_s(  \nu \Delta f - \mcb v \nabla f) ds
\]

for every $t \geq 0$, is a martingale with respect to the filtration $\mathcal F_t:=\sigma \{ X_s(g);\,\,s \leq t \text{ and } g \in C^\infty(\bb T)  \}$ whose quadratic variation is given by

\begin{equation}\label{QVofmp}
\< \mcb M(f) \>_t := \int_0^t  \sigma^2 X_s(\nabla f)^2 ds.
\end{equation}

Since our proofs combine tightness and the unique characterization of the limit point, the next result is crucial in what follows. We did not find in the literature the proof of this result, therefore we present it in Section <ref>.

Any two  solutions of the martingale problem  associated to (<ref>) starting from the same initial condition have the same law.

Now we state our second result.

Let $\beta=2$ and $\varrho_0: \bb T \to [0,1]$ be a measurable function.    Let $\{\mu_n\}_{n\in\bb N}$ be a sequence of probability measures in $\Omega_n$ associated to the profile $\varrho_0$.
Assume that $c>0$ and that
for some constant $\gamma \in \bb R$.  Then, the sequence $\{{\bb Q}_{n,2}\}_{ n \in \bb N}$  converges, as $n\to+\infty$, with respect to  the weak topology of $\mcb  D ([0,T],\mcb  H_{-m})$, for $m>5/2$,  to $\bb Q_2$  which is concentrated on the solution of the martingale problem associated to (<ref>) for $\nu=c$, $\mcb v=\gamma$ and $\sigma=\sqrt {2c}$ and with initial condition $\varrho_0$.

Now that we have established the hydrodynamic limit, which is a Law of Large Numbers for the empirical measure, we can analyse the fluctuations around that limit. That is the purpose of the next section, though we consider the process starting from the invariant state $\nu_\varrho$, since the non-equilibrium scenario is much more complicated and the equilibrium case already gives interesting results.

§.§ Equilibrium density fluctuations

In the last subsection we proved the hydrodynamic limit in two different time scales: in the hyperbolic time scale the limit is deterministic, but in the diffusive time scale, the limit is random. Now we analyse the
corresponding central limit theorems starting from the stationary measure $\nu_\varrho$. To that end, fix $\varrho \in [0,1]$. We define the density fluctuation field on functions $f\in \mcb H_{m}$, by

Y_t^n,β(f)=1/√(n) ∑_x ∈Z_n ( η_tn^β(x) - ϱ) f( xn ).

Now, we denote by $\widetilde {\bb Q}_{n,\beta} $ the probability measures on the Skorohod space  $\mcb D ([0,T],\mcb H_{-m})$ induced by the probability measure $\bb P_{\nu_\varrho}$ and the fluctuation field $\{\mcb Y_t^{n,\beta}\}_{t\geq 0}$. Let also $\mcb C ([0,T],\mcb H_{-m})$  be the space of  continuous trajectories with values in $\mcb H_{-m}.$  Recall the value of $\kappa$ given in (<ref>).

Let $\beta=1$. Assume that
$$\lim_{n\to+\infty}|a_n+b_n-c_n|=\kappa\neq 0$$
and that  we start the process from the Bernoulli product measure $\nu_\varrho$, with $\varrho \in [0,1]\,$.
For $m>3/2$, let $\widetilde{\bb Q}_{1}$ be the probability measure on
$\mcb C ([0,T],\mcb H_{-m})$ corresponding to a stationary Gaussian process with mean $0$ and covariance
given on $f,g\in\mcb H_{m}$ by

\begin{equation}\label{cov_field}
\bb E[\mcb Y_t(f)\mcb Y_s(g)] =\varrho(1-\varrho)\langle T^\kappa_{t-s}\,f,g\rangle,
\end{equation}
where $T_t f(u)=f(u+\kappa t)$ is the semigroup associated to the transport equation given in (<ref>) with $\theta=\kappa$.
Then the sequence $\{\widetilde {\bb Q}_{n,1}\}_{n\in\bb N} $ converges weakly, as $n\to+\infty$, to    $\widetilde{\bb Q}_{1}$.

Let $\beta=2$. Assume that $c>0$, that
for some constant $\gamma \in \bb R$ and  that the system  starts  from the Bernoulli product measure $\nu_\varrho$, with $\varrho \in [0,1]\,$.  Then, the sequence $\{\widetilde {\bb Q}_{n,2}\}_{ n \in \bb N}$  converges with respect to  the weak topology of $\mcb  D ([0,T],\mcb  H_{-m})$, as $n\to+\infty$, for $m>5/2$,  to $\widetilde{\bb Q}_2$  which is concentrated on the solution of the martingale problem associated to (<ref>) for $\nu=c$, $\mcb v=\gamma$ and $\sigma=\sqrt {2c}$ and with initial condition  $\mcb Y_0$,  such that for each $f \in C^\infty(\bb T)  $,  $\mcb Y_0(f)$ is a normal random variable with mean zero and variance $\varrho (1-\varrho)\|f\|_{L^2(\bb T)}^2$.

§ HYDRODYNAMIC LIMITS

In this section we prove Theorems <ref> and <ref>. Recall the definition of  ${\bb Q}_{n,\beta}$ given above Definition <ref>. We first show that $\{{\bb Q}_{n,\beta}\} _{ n \in \bb N}$ is tight with respect to the weak topology of $\mcb D([0,T],\mcb H_{-m})$ for any $m>3/2$ when $\beta=1$ and $m>5/2$ when $\beta=2$. Then we characterize the limit points.

§.§ Tightness

In order to prove tightness, we follow [8] and we invoke the following criterion.

The sequence of probability measures $\{\bb Q_{n,\beta}\}_{ n \in \bb N}$ on $\mcb D ([0,T], \mcb H_{-m})$ is tight if

* $\lim_{A\to \infty} \limsup_{n \to \infty} {\bb Q}_{n,\beta} \left(  \sup_{t \in [0,T]}
\|\pi_t^{n,\beta}\|_{-m}>A \right)=0$;
* $\lim_{\delta \to 0} \limsup_{n \to \infty} {\bb Q}_{n,\beta}\left( \sup_{\stackrel{|s-t|\leq \delta}{0\leq s, t \leq T}} \|\pi_t^{n,\beta} - \pi_s^{n,\beta}\|_{-m} \geq \varepsilon \right)=0$ for every $\varepsilon>0$.

In order to prove the validity of (1) and (2) above,  we use the martingale characterization of the empirical measure. We can recast (1) and (2) in terms of $\mathbb P_{\mu_n}$ instead  of  ${\bb Q}_{n,\beta} $.
From  Dynkin's formula (see for instance <cit.>), for  any $f \in C^{\infty}( \bb T)$,

M_t^n,β(f):=⟨π_t^n,β,f⟩-⟨π_0^n,β,f⟩- ∫_0^t ( ∂_s + n^β L_n ) ⟨π_s^n,β, f⟩ds

N_t^n,β(f):=(M_t^n,β(f))^2 - ∫_0^t n^βL_n (⟨π_s^n,β, f⟩)^2 -2 ⟨π_s^n,β, f⟩L_n ⟨π_s^n,β, f⟩ds

are mean-zero martingales with respect to the natural filtration of the process $\{\eta_t\}_{t\geq 0}$.
A simple computation shows that

\begin{equation}
\begin{split}
&\mcb L_n\eta(1)=(a_n+b_n+d_n)(\eta(2)-\eta(1))+c_n(\eta(n)-\eta(1))
\\&
\mcb L_n\eta(2)=(a_n+b_n)(\eta(3)-\eta(2))+(c_n+d_n)(\eta(1)-\eta(2))\\&
\mcb L_n\eta(n-1)=a_n(\eta(1)-\eta(n-1))++b_n(\eta(n)-\eta(n-1))+c_n(\eta(n-2)-\eta(n-1))\\&
\mcb L_n\eta(n)=b_n(\eta(1)-\eta(n))+c_n(\eta(n-1)-\eta(n))
\end{split}
\end{equation}
and $\mcb L_n\eta(x)=(a_n+b_n)(\eta(x+1)-\eta(x))+c_n(\eta(x-1)-\eta(x))$ for all $x\in\{3,4,\cdots, n-2\}$.
After long, but elementary computations, we get,

\begin{equation}\label{martingale01}
\begin{split}
n^{\beta} \mcb L_n \langle \pi_{s}^{n,\beta},f\rangle&=- (a_n+b_n)n^{\beta-2} \sum_{x \in \bb Z_n} \eta_{s n^{\beta}}(x) \nabla _n^- f(\tfrac{x}{n} )+ c_n n^{\beta-2} \sum_{x \in \bb Z_n} \eta_{s n^{\beta}}(x) \nabla_n^+ f(\tfrac{x}{n})\\
&-a_n n^{\beta-2} \left( \eta_{s n^{\beta}}(1)-\eta_{s n^{\beta}}(n) \right) \nabla_n^- f\left(0 \right) + d_n  n^{\beta-2} \left( \eta_{s n^{\beta}}(1)-\eta_{s n^{\beta}}(2) \right) \nabla_n^+ f(\tfrac{1}{n} ).
\end{split}
\end{equation}

Above,  $\nabla_n^-$, $\nabla_n^+$  are  discrete operators defined on  $f: \bb T \to \bb R$ and  $x \in \bb Z_n$ by

\nabla_n^-f(\tfrac{x}{n} )=n( f(\tfrac{x}{n}) -f(\tfrac{x-1}{n})) \quad \textrm{and} \quad \nabla_n^+f(\tfrac{x}{n})=\nabla_n^-f(\tfrac{x+1}{n}).$$

The quadratic variation of the martingale given in (<ref>) is equal to

M^n,β(f) _t   = ∫_0^t (a_n+b_n) n^β-2 ( π_sn^β^n(∇_n^- f) )^2 ds + ∫_0^t c_n n^β-2 ( π_sn^β^n(∇_n^+ f) )^2 ds

+∫_0^t 2 a_n n^β-3 (η_sn^β(1)-η_sn^β(n)) ∇_n^- f(0) π_sn^β^n(∇_n^- f)  ds

+∫_0^t a_n n^β-4 (η_sn^β(1)-η_sn^β(n))^2 ( ∇_n^- f(0) )^2 ds
+ ∫_0^t d_n n^β-4 (η_sn^β(1)-η_sn^β(2))^2 ( ∇_n^+ f(1n))^2 ds.
Now we prove the validity of (1). Recall  (<ref>) and (<ref>). From Markov's inequality,

P_μ_n (  sup_t ∈[0,T] π_t^n,β_-m>A ) ≤1/A^2    ∑_k ∈Z γ_k^-m E_μ_n [ sup_t ∈[0,T]| π_t^n,β, ψ_k |^2 ].

To estimate last identity we analyse (<ref>) for each time scale. Note that  for $\beta=1$ and from a Taylor's expansion, we get

⟨π_t^n,1,ψ_k⟩   =M_t^n,1(ψ_k)+⟨π_0^n,1,ψ_k⟩- (a_n+b_n-c_n)  ∫_0^t  ⟨π_s ^n,1,∇ψ_k⟩ds +  O (n^-1),

while for $\beta=2$, we get

⟨π_t^n,2,ψ_k⟩   =M_t^n,2(ψ_k)+⟨π_0^n,2,ψ_k⟩+ c_n   ∫_0^t ⟨π_s^n,2,Δ_n ψ_k⟩ds +(c_n- (a_n+b_n))∫_0^t ⟨π_s^n,2,∇_n^- ψ_k⟩ds

-a_n  ∫_0^t ( η_s n^2(1)-η_s n^2(n) ) ∇_n^- ψ_k(0) ds + d_n  ∫_0^t ( η_s n^2(1)-η_s n^2(2)) ∇_n^+ ψ_k(1n ) ds.
Above  the discrete Laplacian operator is defined on $f: \bb T \to \bb R$ and  $x \in \bb Z_n$ by
$$\Delta_nf(\tfrac{x}{n} )=n^2( f(\tfrac{x+1}{n}) +f(\tfrac{x-1}{n})-2f(\tfrac{x}{n})).$$

By the convexity inequality $|\sum_{k=1}^\ell x_k|^2 \leq \ell \sum_{k=1}^\ell |x_k|^2$ for $\ell \in \bb N$, we  are left to estimate the contribution of each term in the previous identities (<ref>) and (<ref>). We note that the initial term can be estimated by the Cauchy-Schwarz inequality, as

E_μ_n[  |⟨π_0^n,β,ψ_k⟩|^2  ]≤1/n∑_x ∈Z_n |ψ_k(x n)|^2.

Now we split the proof and we start with the hyperbolic case.

From Doob's inequality, (<ref>) for $\beta=1$ and noting that $\eta(x)\leq 1$ for all $x\in\mathbb Z_n$, we get

E_μ_n   [ sup_t ∈[0,T] |M_t^n,1(ψ_k)|^2  ] ≤4   E_μ_n[  |M_T^n,1(ψ_k)|^2  ] ≲n^-1.

Now, it remains to estimate the third term on the right hand-side of (<ref>).
By the definition of $\psi_k$ and using the Cauchy-Schwarz inequality twice, we get

E_μ_n[ sup_t ∈[0,T] |∫_0^t  ⟨π_s ^n,1,∇ψ_k ⟩ds|^2  ] ≤T   E_μ_n[  ∫_0^T| ⟨π_s ^n,1,∇ψ_k⟩|^2 ds  ] ≤T^2/n∑_x ∈Z_n |∇ψ_k(xn)|^2.

From the previous results and since $\frac{1}{n}\sum_{x \in \bb Z_n} |f(\tfrac xn)|^2$ converges to $\left\< f,f \right\>$ for any $f \in C^\infty (\bb T)$, we have

lim sup_n →∞ P_μ_n (  sup_t ∈[0,T] π_t^n,1_-m>A )
≲ (T^2+1) A^2    ∑_k ∈Z γ_k^-m {   ψ_k,ψ_k +  ∇ψ_k, ∇ψ_k }

≲ (T^2+1)  A^2(1+2 ∑_k=1^∞{  1+4 π^2 k^2 }^1-m).

Therefore, (1) follows from the fact that the series above converges for $m>3/2$.

Now we focus on the case $\beta=2$.
From Doob's inequality, (<ref>) for $\beta=2$, noting that $\eta(x)\leq 1$ for all $x\in\mathbb Z_n$, and by  the Cauchy-Schwarz inequality, we get

E_μ_n[ sup_t ∈[0,T] |M_t^n,2(ψ_k)|^2  ]    ≤4   E_μ_n[  |M_T^n,2(ψ_k)|^2  ]
= 4   ∫_0^T (a_n+b_n) E_μ_n[ ⟨π_s^n,2,∇_n^- ψ_k⟩^2]  ds +4 ∫_0^t c_n E_μ_n[  ⟨π_s^n,2,∇_n^+ ψ_k⟩^2] ds + O (n^-1)

≲ (a_n+b_n)T/n ∑_x ∈Z_n  |∇_n^- ψ_k (xn)|^2 +  c_n T/n ∑_x ∈Z_n  |∇_n^+ ψ_k (xn)|^2.
With the same arguments as used in  (<ref>) we see that

E_μ_n [ sup_t ∈[0,T]  |∫_0^t ⟨π_s^n,2,Δ_n ψ_k⟩ds |^2   ] ≤T^2/n∑_x ∈Z_n |Δ_n ψ_k(xn)|^2,

E_μ_n [ sup_t ∈[0,T]  |∫_0^t  ⟨π_s^n,2,∇_n^- ψ_k⟩ds |^2   ] ≤T^2/n∑_x ∈Z_n |∇_n^- ψ_k(xn)|^2.

Finally, by the Cauchy-Schwarz inequality we have

E_μ_n [ sup_t ∈[0,T]  |a_n  ∫_0^t ( η_s n^2(1)-η_s n^2(n) ) ∇_n^- ψ_k(0 ) ds  |^2   ] ≤|a_n|^2 T |∇_n^- ψ_k(0)|^2,

E_μ_n [ sup_t ∈[0,T]  |d_n  ∫_0^t ( η_s n^2(1)-η_s n^2(2) ) ∇_n^+ ψ_k(1n ) ds  |^2   ] ≤|d_n|^2 T | ∇_n^+ ψ_k(1n) |^2.

By (<ref>), (<ref>), (<ref>), (<ref>), (<ref>) and (<ref>), we get

lim sup_n →∞      P_μ_n (  sup_t ∈[0,T] π_tn^2^n_-m>A ) ≤1/A^2    ∑_k ∈Z γ_k^-m lim sup_n →∞ E_μ_n [ sup_t ∈[0,T]|π_tn^2^n(ψ_k)|^2  ]

≲1/A^2    ∑_k ∈Z γ_k^-m {   ψ_k,ψ_k +  ∇ψ_k, ∇ψ_k +  Δψ_k, Δψ_k } + (a^2+d^2)T/A^2    ∑_k ∈Z γ_k^-m |∇ψ_k (0)|^2

≲1 /A^2+ 1/A^2    ∑_k=1^∞{  1+4 π^2 k^2+ (2πk)^4 } {  1+ (2 πk)^2 }^-m +  π^2 (a^2+d^2)T/A^2    ∑_k =1^∞k^2{  1+  π^2 k^2 }^-m.

Observe that the series above converge if $m>5/2$. This concludes the proof of (1).

Now we prove (2). Observe that, arguing similarly, one can show that if $m>3/2$ for $\beta=1$ or $m>5/2$ for $\beta=2$, then

lim_N →∞ lim sup_n →∞ 𝔼_μ_n  [  sup_t ∈[0,T]   ∑_|k| ≥N | ⟨π_t^n,β,ψ_k⟩|^2 γ_k^-m   ]=0.

Thus, (2) follows if for any integer $N$ and every $\varepsilon>0$,

\lim_{\delta \to 0} \limsup_{n \to \infty} \bb P_{\mu_n}  \Bigg[   \sup_{\substack{|s-t|\leq \delta\\0\leq s, t \leq T}} \,\,\sum_{|k| \leq N} |\langle \pi_{t}^{n,\beta},\psi_k\rangle-\langle \pi_{s}^{n,\beta},\psi_k\rangle|^2 \gamma_k^{-m}\geq \varepsilon   \Bigg]=0.

Observe also that to prove the above limit it is sufficient to show that

\begin{equation}\label{end_tight}
\lim_{\delta \to 0} \limsup_{n \to \infty} \bb P_{\mu_n}  \Bigg[   \sup_{\stackrel{|s-t|\leq \delta}{0\leq s, t \leq T}}| \langle \pi_{t}^{n,\beta},\psi_k\rangle-\langle \pi_{s}^{n,\beta},\psi_k\rangle|^2 \geq \varepsilon    \Bigg]=0
\end{equation}

for every $k \in \bb Z$. 	Now we split the proof and we start with the hyperbolic regime.
Recall (<ref>) for $\beta=1$. Then  the previous limit is a consequence of the next two results

\begin{equation*}
\begin{split}
&(A)\quad \quad  \lim_{\delta \to 0} \limsup_{n \to \infty}  \bb P_{\mu_n} \Bigg[  \sup_{\substack{|s-t|\leq \delta\\0\leq s, t \leq T}} \left|\int_s^t \langle \pi_{r}^{n,1} ,\nabla \psi_k\rangle dr\right| > \varepsilon  \Bigg]=0.\\
&	(B)\quad \quad  \lim_{\delta \to 0} \limsup_{n \to \infty}  \bb P_{\mu_n} \Bigg[  \sup_{\substack{|s-t|\leq \delta\\0\leq s, t \leq T}} |\mcb M_t^{n,1}(\psi_k)-\mcb M_s^{n,1}(\psi_k)| > \varepsilon  \Bigg]=0.\end{split}
\end{equation*}

We start by proving (A). Note that from  Markov and Cauchy-Schwarz inequalities, and since $|t-s| \leq \delta$ and $s,t \leq T$, we get

P_μ_n [  sup_|s-t|≤δ
0≤s, t ≤T |∫_s^t ⟨π_r^n,1 ,∇ψ_k⟩dr| > ε]
≤δT/ ε^2 E_μ_n [  ∫_0^T |⟨π_r^n,1 ,∇ψ_k⟩|^2   dr ]≤δT^2/ ε^2 n ∑_x ∈Z_n |∇ψ_k (xn)|^2.

Now we prove (B).
For $\beta>0$, define

ω_δ(M^n,β(f))=inf_t_i max_i ∈[0,r) sup_t_i ≤s < t <t_i+1 |M_t^n,β(f)-M_s^n,β(f)|,

where the first infimum is taken over all partitions of $[0,T]$ such that
0=t_0 < t_1 < \cdots <t_r=T$ and $
t_{i+1}-t_i>\delta$ for $0 \leq i <r.$
$$\sup_{t\in[0,T]} \Big|\mcb M_t^{n,\beta}(f)-\mcb  M_{t-}^{n,\beta}(f)\Big|=\sup_{t\in[0,T]} \Big|\langle \pi_{t}^{n,\beta},\psi_k\rangle-\langle \pi_{t^-}^{n,\beta},\psi_k\rangle\Big|\leq \frac{2}{n} \|\nabla \psi_k \|_{\infty}$$

\sup_{\substack{|s-t|\leq \delta\\0\leq s, t \leq T}} \|\mcb M_t^{n,\beta}(\psi_k)-\mcb M_s^{n,\beta}(\psi_k)\|_{-m}  \leq 2 \omega_{\delta}(\mcb M^{n,\beta}(\psi_k)) + \sup_{t\in[0,T]} \Big|\mcb M_t^{n,\beta}(\psi_k)-\mcb M_{t-}^{n,1}(\psi_k)\Big|,

the proof ends (for $\beta=1$) if we show that

\lim_{\delta \to 0} \limsup_{n \to \infty} \bb P_{\mu_n} \Bigg[  \omega_{\delta}(\mcb M^{n,1}(\psi_k)) > \varepsilon \Bigg]=0

for every $\varepsilon>0$. By <cit.>, it is enough to check that for every $\varepsilon>0$

\lim_{\delta \to 0} \limsup_{n \to \infty} \sup_{\stackrel{\tau \in \mathcal T_T}{\theta \in [0,\delta]}} \bb P_{\mu_n} \Bigg[ \Big|\mcb M_{(\tau+\theta)\wedge T}^{n,1}(\psi_k)-\mcb M_{\tau}^{n,1}(\psi_k)\Big| >\varepsilon    \Bigg]=0,

where $\mathcal T_T$ stands for all stopping times bounded from above by $T$. From Markov's inequality, since $\mcb M_t^{n,1}(\psi_k)$ is a martingale and since $\tau$ is a bounded stopping time,

P_μ_n [ |M_τ+θ^n,1(f)-M_τ^n,1f)| >ε]    ≤1/ε^2 E_μ_n [ M^n,1(f)_τ+θ -M^n,1(f)_τ    ]≲Cn^-1.

And this ends the proof for the case $\beta=1$. For $\beta=2$, recall (<ref>).
Then (<ref>)
is a consequence of (A) (which also holds for $\beta=2$) and to prove (B), it is enough to note that

P_μ_n    [ |M_τ+θ^n,2(f)-M_τ^n,2(f)| >ε]
≤1/ε^2 E_μ_n[ ∫_τ^τ+θ (a_n+b_n)  (⟨π_s^n,2,∇_n^- ψ_k ⟩)^2 ds + ∫_τ^τ+θ c_n  (⟨π_s^n,2,∇_n^+ f⟩)^2 ds  ] + O (n^-1)

≤δ/ε^2[(a_n+b_n)∇_n^- f_∞^2 + c_n∇_n^+ f_∞^2 ] + O (n^-1)

and last display vanishes by taking $n\to+\infty$ and then $\delta\to 0.
This ends the proof of Lemma <ref> for $\beta\in\{1,2\}$.

From last results, we  know that $\{\bb Q_{n,\beta}\}_{ n \in \bb N }$ is tight in the weak topology of $\mcb D([0,T],\mcb H_{-m})$ for any $m>3/2$ when $\beta=1$ and $m>5/2$ for $\beta=2$.  Now we characterize the limit point.

§.§ Characterization of limit points

Now we complete the proof of Theorems <ref> and <ref>.
From the computations of last subsection, we know that there exists a limiting point  $\mathbb Q_{\beta}$ of  the sequence $\{\bb Q_{n,\beta}\}_{ n \in \bb N }$. Since in both cases of $\beta$ we have $\eta(x)\leq 1$, then  $\mathbb Q_{\beta}$  is supported on trajectories of measures that are absolutely continuous with respect to the Lebesgue measure:
$$\bb Q_\beta(\pi_\cdot: \forall t\in[0,T], \,\pi_t(du)=\varrho(t,u)du)=1.$$

Now we characterize the density.
We start with the hyperbolic case and we present an heuristic argument, from which one easily deduces the rigorous one.
Recall (<ref>) with $\beta=1$.
From the previous section we know that the martingale (<ref>) vanishes in $\mathbb L^2(\mathbb P_{\mu_n})$, as $n\to+\infty$.   Since the limit $\pi^1_t(du)$ satisfies $\pi^1_t(du)=\varrho^\kappa(t,u)du$ and $\mu_n $ is associated to $\varrho_0$,  we get

0=\langle \varrho^\kappa_t,f\rangle -\langle \varrho_0,f\rangle + \int_0^t  (a+b-c)\langle \varrho_s^\kappa,\nabla  f\rangle ds

which matches with the notion of weak solution given in  Definition <ref> with $\mathfrak g=\varrho_0$ and  $\kappa=a+b-c$.
Now we go to the diffusive case. Recall (<ref>) with $\beta=2$. By reorganizing terms and using the fact that $a_n+b_n-c_n=\tfrac{\gamma}{n}$, we can rewrite

M_t^n,2(f)   =⟨π_t^n,2,f ⟩-⟨π_0^n,2, f⟩- c_n   ∫_0^t ⟨π_s^n,2,Δ_n f⟩ds +γ∫_0^t ⟨π_s^n,2,∇_n^- f⟩ds

+a_n  ∫_0^t ( η_s n^2(1)-η_s n^2(n) ) ∇_n^- f(0) ds - d_n  ∫_0^t ( η_s n^2(1)-η_s n^2(2) ) ∇_n^+ f(1n ) ds.
In this case, the argument is similar to one used in  $\beta=1$, but now the martingale  $\mcb M_t^{n,2}(f)$ does not vanish, and showing that the terms in the last line of last display vanish is more complicated since we need to use the dynamics of the process. To overcome this problem we use the two lemmas below, which allows to conclude that in the limit, as $n\to+\infty$, we obtain

\begin{equation*}
\mcb M^2_t(f) := \langle \pi^2_t,f\rangle-\langle\pi^2_0, f\rangle -c\int_0^t \langle \pi^2_s,\Delta f\rangle ds +\gamma\int_0^t \langle \pi^2_s,\nabla f\rangle ds.
\end{equation*}

The next lemma is necessary to control the boundary terms in (<ref>).

Let $r\in\{2,n\}$ and assume that if $r=2$ then $d_n>0$ or if $r=n$ then $b_n>0$.  For any $t >0$ and $\delta\in(0,1)$

E_μ_n[ | ∫_0^t (η_sn^2(1)-η_sn^2(r)) ds | ] ≲tmax{1/n^δ,1/ℭ_r n^1-δ},

where $~\mathfrak C_r=d_n\textbf{1}_{r=2}+b_n \textbf{1}_{r=n}$.

The next lemma is necessary to control the martingale appearing in (<ref>), which, contrarily to the case $\beta=1$ does not vanish.

The sequence of martingales $\{ \mcb M_t^{n,2}(f); t \geq 0 \}_{n \in \bb N}$ introduced in (<ref>) with $\beta=2$, converges, in distribution as $n\to+\infty$, to a mean-zero martingale $\mcb M^2_t(f)$ with quadratic variation given by

\left\< \mcb M^2(f) \right\>_t:=2c\int_0^t \langle \pi^2_s,\nabla f\rangle ^2 ds.

First observe that $\mu_n$ is a general measure  on $\Omega_n$ and by the entropy inequality we can pay a price to replace it by the stationary measure, the Bernoulli product measure $\nu_\varrho$. A simple computation shows that  there exists a constant $C=C(\varrho)$ such that

H(μ|ν_ϱ)= ∑_η∈Ω_n μ(η) log( μ(η)/ν_ϱ(η) )≤C n.

This together with  Jensen's inequality, gives for any $A>0$ that

E_μ_n[ | ∫_0^t (η_sn^2(1)-η_sn^2(r)) ds | ] ≤C/A + 1/A n logE_ν_ϱ[ exp{ A n | ∫_0^t (η_sn^2(1)-η_sn^2(r)) ds |  } ].

Since $e^{|x|}\leq e^{x}+e^{-x}$ and

\limsup_{n \to \infty}  \log (B_n + C_n)=\max \left\{  \limsup_{n \to \infty}  \log (B_n)\,,\,\limsup_{n \to \infty} \log (C_n) \right\},

we can remove the absolute value on the right-hand side of (<ref>). From  Feynman-Kac's formula,  the  rightmost term  of (<ref>) is bounded from above by

∫_0^t   sup_f { ∫(η(1)- η(r)) f(η)   ν_ϱ (dη) - n/A⟨L_n√(f),√(f)⟩_ν_ϱ } ds,

where the supremum runs over all densities $f$ with respect to $\nu_\varrho$. Now we introduce
the quadratic operator  given on  $g:\Omega_n\to\bb R$ by

D_n (g;ν_ϱ)    = a_n/2 ∫( √( g(η^1 →n-1))-√(  g(η)))^2 dν_ϱ+  b_n/2 ∫(√( g(η^1 →n))-√( g(η)))^2 dν_ϱ
+  c_n/2 ∫(√( g(η^n →1))-√( g(η)))^2 dν_ϱ+ d_n/2∫(√( g(η^1 ↔2))-√( g(η)))^2 dν_ϱ.

A simple computation shows that
the  Dirichlet form associated with the generator $ \mcb  L_n$, i.e.  $\langle \mcb L_n g,g\rangle_{\nu_\varrho}$ is given by 	$\mf D_n\,(g;\nu_\varrho)$.
Now, we write the leftmost term in the supremum in (<ref>) as one-half plus one-half of itself, and in one of the parcels we make  the change of variables $\eta \mapsto \eta^{1 \rightarrow r}$ for $r=n$ or  $\eta \mapsto \eta^{1 \leftrightarrow r}$ for $r=2$, and note that the probability measure $\nu_\varrho$ is invariant for both transformations. Then, from the elementary identity $(x-y)=(\sqrt{x}-\sqrt{y})(\sqrt{x}+\sqrt{y})$ and  Young's  inequality, we can bound that term from above by

B/2 ∫η(1) ( √(f(η^1 →n))+√(f(η))   )^2  dν_ϱ+1/2B ∫( √(f(η^1 →n))-√(f(η))  )^2  dν_ϱ,

for $r=n$ and a similar expression for $r=2$. Above $B>0$.
Recalling (<ref>), using the fact that $f$ is a density with respect to $\nu_\varrho$ and choosing $	B=A/(b_n n)$, last display is bounded from above by

\frac{2A}{b_n n} + \frac{n}{A} \, \mf D_n \,(\sqrt{f};\nu_\varrho).

Taking $A=n^\delta$ the proof ends. For $r=2$, we repeat the same strategy as above and we take  instead $B=A/(d_n n)$.

In order to prove Lemma <ref>, we invoke <cit.>, which we state in our setting as:

Let $\{M_t^n \,; \,t \in [0,T]\}_{ n \in \bb N }$ be a sequence of martingales in $\mcb D ([0,T],\bb R)$ with quadratic variation $\left\< M^n\right\>_t$. Assume that

* For any $n \in \bb N$ the quadratic variation process $\left\< M^n\right\>_t$ has continuous trajectories almost surely;

* $\lim_{n \to \infty} \bb E \Bigg[ \sup_{t \in [0,T]} | M_t^n - M_{t^-}^n | \Bigg]=0$;

* $\left\< M^n\right\>_t$ converges to $\left\< M\right\>_t $, in distribution, as $n \to \infty$.

Then,  $\{M_t^n \,; \,t \in [0,T]\}_{ n \in \bb N }$ converges, in distribution, to  a mean zero martingale $\{M_t \}_{t \in [0,T] }$ with continuous trajectories and with quadratic variation  $\left\< M\right\>_t$.

We apply Theorem <ref> to the sequence $\{\mcb M_t^{n,2}(f);\, t \geq 0\}_{ n \in \bb N}$ of martingales given in (<ref>) with $\beta=2$.
From (<ref>) the condition (1) is trivially satisfied.   In order to verify (2), observe that at each time there is at most one type of change, either $\eta^{1 \rightarrow n-1}$, or $\eta^{1 \rightarrow n}$ or $\eta^{n \rightarrow 1}$ or $\eta^{1 \leftrightarrow 2}$.  Let us suppose that at time $t
$ it happens $\eta^{1 \rightarrow n}$, the others can be treated analogously. Note that in this case, we have $\eta_t(x-1)=\eta_{t^-}(x)$. Therefore, for  any function $f \in C^\infty(\bb T)$ we have that

|M_t^n,2(f)-M_t^-^n,2(f)|    = | π_t^n,2 (f) -  π_t^-^n,2 (f)|=| 1/n^2∑_x∈ℤ_n∇_n^+f(xn)η_tn^2(x)|≤ ∇f _∞/n,

which trivially implies (2).
Finally,  note that for $\beta=2$, the quadratic variation in (<ref>) is given by

M^n,2(f) _t   = ∫_0^t (a_n+b_n)  ⟨π_s^n,2,∇_n^- f⟩^2 ds + ∫_0^t c_n  ⟨π_s^n,2,∇_n^+ f⟩^2 ds

+∫_0^t 2 a_n n^-1 (η_sn^2(1)-η_sn^2(n)) ∇_n^- f(0) ⟨π_s^n,2,∇_n^- f⟩ds

+∫_0^t a_n n^-2 (η_sn^2(1)-η_sn^2(n))^2 ( ∇_n^- f(0) )^2 ds
+ ∫_0^t d_n n^-2 (η_sn^2(1)-η_sn^2(2))^2 ( ∇_n^+ f(1n) )^2 ds.
From last identity, condition  (3) follows from the fact that $\{\pi_t^2\}_{t\geq 0}$ is a limit point, in distribution, of $ \{\pi_{t}^{n,2}\}_{t\geq 0}$.

§ EQUILIBRIUM FLUCTUATIONS

In this section we prove Theorems <ref> and <ref>.

§.§ Tightness

We start by showing tightness of $\{\widetilde {\bb Q}_{n,\beta}\}_{n\in\bb N} $. As above we use Lemma <ref> and we start by showing (1). Note that  by Dynkin's formula, there exist two mean-zero martingales $\{\widetilde{\mcb M}_t^{n,\beta}(f)\}_{t\geq 0}$ and $\{\widetilde{\mcb N}_t^{n,\beta}(f)\}_{t\geq 0}$ as given in (<ref>) and (<ref>), respectively. We start the proof of (1) in Lemma <ref> for  the hyperbolic case, i.e.   $\beta=1$. Then, for $f \in C^{\infty}( \bb T)$ we obtain

M_t^n,1(f)   =Y_t^n,1(f)-Y_0^n,1(f)
+ (a_n+b_n)  ∫_0^t Y_s^n,1(∇_n^- f) ds -c_n  ∫_0^t Y_s^n,1(∇_n^+ f) ds
+a_n n^-1/2 ∫_0^t ( η_s n(1)-η_s n(n) ) ∇_n^- f(0 ) ds - d_n n^-1/2 ∫_0^t ( η_s n(1)-η_s n(2) ) ∇_n^+ f(1n )ds,
$\widetilde{\mcb N}_t^{n,1}(f):=\left(\widetilde{\mcb M}_t^{n,1}(f)\right)^2 - \left\<  \widetilde{\mcb M}^{n,1}(f)\right\>_t,

M^n,1(f)_t   =∫_0^t (a_n+b_n) n^-1 ( Y_s^n,1(∇_n^- f) )^2 ds + ∫_0^t c_n n^-1 ( Y_s^n,1(∇_n^+ f) )^2 ds

+ ∫_0^t 2 a_n n^-2 (η_sn(1)-η_sn(n)) ∇_n^- f(0) Y_s^n,1(∇_n^- f) ds

+ ∫_0^t a_n n^-2 (η_sn(1)-η_sn(n))^2 ( ∇_n^- f(0) )^2 ds
+  ∫_0^t d_n n^-2 (η_sn(1)-η_sn(2))^2 ( ∇_n^+ f(1n) )^2 ds.
Recall  $\psi_k$ defined in (<ref>). To prove  (1) we note that by Doob's inequality,  last identity and the fact that $\nu_\varrho$ is invariant, we get

E_ν_ϱ   [ sup_t ∈[0,T] |M_t^n,1(ψ_k)|^2  ] ≤4   E_ν_ϱ[  |M_T^n,1(ψ_k)|^2  ]
= 4 E_ν_ϱ[  ∫_0^T (a_n+b_n) n^-1 ( Y_s^n,1(∇_n^- ψ_k) )^2 ds + ∫_0^t c_n n^-1 (Y_s^n,1(∇_n^+ ψ_k) )^2 ds ] +(a_n+d_n)k^2 T/n^2=O(n^-1).

A simple computation shows that

E_ν_ϱ[  |Y_0^n,1(ψ_k) |^2  ]= ϱ(1-ϱ)/n∑_x ∈Z_n |ψ_k(xn)|^2.

E_ν_ϱ[ sup_t ∈[0,T] |∫_0^t  Y_s^n,1(∇ψ_k) ds|^2  ] ≤T   E_ν_ϱ[  ∫_0^T| Y_s^n,1(∇ψ_k)|^2 ds  ] ≤ϱ(1-ϱ)T^2/n∑_x ∈Z_n |∇ψ_k(xn )|^2.

From the previous results we can easily show that

lim sup_n →∞ P_ν_ϱ (  sup_t ∈[0,T] Y_t^n,1_-m>A )    ≤T^2+1  /A^2(1+ 2    ∑_k=1^∞{  1+4 π^2 k^2 }^1-m).

Note that the series above converges for $m>3/2$.

Now we prove (1) for $\beta=2$. Note that in this case

M_t^n,2(f)   =Y_t^n,2(f)-Y_0^n(f) - c_n   ∫_0^t Y_0^n,2(Δ_n f) ds -(c_n- (a_n+b_n))n ∫_0^t Y_s^n,2(∇_n^- f) ds

+a_n √(n) ∫_0^t ( η_s n^2(1)-η_s n^2(n) ) ∇_n^- f(0 ) ds - d_n √(n) ∫_0^t ( η_s n^2(1)-η_s n^2(2) ) ∇_n^+ f(1n )ds,
and the quadratic variation equals to

M^n,2(f) _t
=∫_0^t (a_n+b_n)  ( Y_s^n,2(∇_n^- f) )^2 ds + ∫_0^t c_n  ( Y_s^n,2(∇_n^+ f) )^2 ds

+ ∫_0^t 2 a_n n^-1 (η_sn^2(1)-η_sn^2(n)) ∇_n^- f(0) Y_s^n,2(∇_n^- f) ds

+ ∫_0^t a_n n^-1 (η_sn^2(1)-η_sn^2(n))^2 ( ∇_n^- f(0) )^2 ds
+  ∫_0^t d_n n^-1 (η_sn^2(1)-η_sn^2(2))^2 ( ∇_n^+ f(1n) )^2 ds.

From Doob's inequality, (<ref>) and the fact that $\nu_\varrho$ is invariant, we get  that

E_ν_ϱ   [ sup_t ∈[0,T] |M_t^n,2(ψ_k)|^2  ] ≲ (a_n+b_n)T/n ∑_x ∈Z_n  |∇_n^- ψ_k (xn)|^2 + c_n T/n ∑_x ∈Z_n  |∇_n^+ ψ_k (xn)|^2 +   (a_n+d_n) T k^2/n.

The initial condition can be estimated as in the previous case. Moreover, as above we have
𝔼_ν_ϱ[ sup_t ∈[0,T] |∫_0^t  Y_s^n,2(∇_n^- ψ_k) ds|^2  ] ≤ϱ(1-ϱ)T^2/n∑_x ∈Z_n |∇_n^- ψ_k(xn)|^2.
The contribution of the boundary terms in  (<ref>) is estimated in the next lemma whose proof we present at the end of the proof of (1).

Let $r\in\{2,n\}$ and assume that if $r=2$ then $d_n>0$ or if $r=n$ then $b_n>0$.  For any $t \in [0,T]$ we have

E_ν_ϱ [ sup_t ∈[0,T]|   √(n) ∫_0^t ( η_s n^2(1)-η_s n^2(r) )   ds |^2 ]≲T1/ ℭ_r n,

where we recall that $~\mathfrak C_r=d_n\textbf{1}_{r=2}+b_n \textbf{1}_{r=n}$.

From the previous arguments, we conclude that

lim sup_n →∞  P_ν_ϱ (  sup_t ∈[0,T]  Y_t^n,2_-m>A ) ≤C /A^2( 1+2    ∑_k=1^∞{  1+4π^2 k^2+ (2πk)^4 } {  1+ 4 π^2 k^2 }^-m).

The series above converges if $m>5/2$. This ends the proof of (1) in both cases $\beta\in\{1,2\}$. Now we prove Lemma <ref>.

By Kipnis-Varadhan inequality (see <cit.>), the expectation in the statement of the lemma
is bounded from above by

T sup_f ∈L^2(ν_ϱ) {  √(n) ∫(η(1)-η(r) )f(η) dν_ϱ- n^2 𝔇_n (f;ν_ϱ)  }.
Above we used the operator $\mathfrak D_n $ introduced in
As in the proof of Lemma <ref>, we can rewrite, for $r=n$,  the previous expression as

T sup_f ∈L^2(ν_ϱ) {  √(n)∫η(1)  (f(η) - f(η^1 →n))    dν_ϱ- n^2 𝔇_n (f;ν_ϱ)  }.

For $r=2$ we get exactly the previous  expression but with the transformation $\eta \mapsto \eta^{1 \leftrightarrow 2}$. From Young's inequality, the expression inside the supremum above is bounded by

A√(n)/2    ∫(f(η) - f(η^1 →n))^2   d ν_ϱ+ √(n)ϱ/2A    - n^2 𝔇_n (f;ν_ϱ),

and for the choice $A= b_n n^{3/2}$ the proof ends.  For $r=2$ we make the choice $A=d_n n^{3/2}$.

Now we prove (2) of Lemma <ref>. In both cases, it is enough to show that

\begin{equation}
\lim_{\delta \to 0} \limsup_{n \to \infty} \bb P_{\nu_\varrho} \Bigg[   \sup_{\substack{|s-t|\leq \delta\\0\leq s, t \leq T}}\Big| \mcb Y_{t}^{n,\beta}(\psi_k)-\mcb Y_{t}^{n,\beta}(\psi_k)\Big|^2 \geq \varepsilon   \Bigg]=0
\end{equation}

for every $k \in \bb Z$.
Now we start with the case $\beta=1$. Last limit is  a consequence of the next two results.
For any function $f \in C^\infty(\bb T)$ and any $\varepsilon>0$ we have

P_ν_ϱ [  sup_|s-t|≤δ
0≤s, t ≤T |∫_s^t Y_s^n,1 (∇f) dr| > ε]  ≤δ/ ε^2 E_ν_ϱ [  ∫_0^T |Y_s^n,1  (∇f)|^2   dr ]= ϱ(1-ϱ)δT/ ε^2 n ∑_x ∈Z_n |∇f (xn)|^2.

Moreover, for any function $f \in C^\infty(\bb T)$ and any $\varepsilon>0$ we have

\lim_{\delta \to 0} \limsup_{n \to \infty}  \bb P_{\nu_\varrho} \Bigg[  \sup_{\substack{|s-t|\leq \delta\\0\leq s, t \leq T}} \Big|\widetilde{\mcb M}_t^{n,1}(f)-\widetilde{\mcb M}_s^{n,1}(f)\Big| > \varepsilon  \Bigg]=0.

Last result is a consequence of the fact that

P_ν_ϱ [ |M_τ+θ^n,1(f)-M_τ^n,1(f)| >ε]    ≤O(n^-1).

This proves (2) for the case $\beta=1$. Now we analyse (2) for $\beta=2$, which is a consequence of the next results.
For any function $f \in C^\infty(\bb T)$ and any $\varepsilon>0$ we have

\lim_{\delta \to 0} \limsup_{n \to \infty}  \bb P_{\nu_\varrho} \Bigg[  \sup_{\substack{|s-t|\leq \delta\\0\leq s, t \leq T}} \Bigg|\int_s^t \mcb Y_{t}^{n,2} (\nabla_n^- f) dr\Bigg| > \varepsilon  \Bigg]=0,

And the same limit above holds for $\Delta_n f$ instead of $\nabla_n^- f$.
From a similar argument to the one used to prove Lemma <ref>
we get for $r\in\{2,n\}$ that

\lim_{\delta \to 0} \limsup_{n \to \infty}  \bb P_{\nu_\varrho} \Bigg[  \sup_{\substack{|s-t|\leq \delta\\0\leq s, t \leq T}} \Big|   \sqrt{n} \int_s^t \left( \eta_{r n^{2}}(1)-\eta_{r n^{2}}(r) \right) dr   \Big| > \varepsilon  \Bigg]=0.

Finally, as above it remains to check that for every $\varepsilon>0$

\lim_{\delta \to 0} \limsup_{n \to \infty} \sup_{\substack{\tau \in \mathcal T_T\\\theta \in [0,\delta]}} \bb P_{\nu_\varrho} \Bigg[\Big |\widetilde{\mcb M}_{(\tau+\theta)\wedge T}^{n,2}(f)-\widetilde{\mcb M}_{\tau}^{n,2}(f)\Big| >\varepsilon    \Bigg]=0,

and as in the hyperbolic case it is easy to check that

P_ν_ϱ    [|M_(τ+θ)∧T^n,2(f)-M_τ^n,2(f)| >ε] ≲δ/ε^2ϱ(1-ϱ)[(a_n+b_n)∇_n^- f_∞^2 + c_n∇_n^+ f_∞^2 ] +   (a_n+d_n)k^2 T/n.

And this ends the proof of tightness.

§.§ Characterization of the limit points

In order to conclude the proofs of Theorems <ref> and <ref>, we  need to show that the limit points are solutions of the martingale problem given in the statements of the results. The hyperbolic case is trivial and we leave the details to the reader. We just note that the limit field satisfies

0=\mcb Y^1_t(f)-\mcb Y^1_0(f) + \int_0^t  (a+b-c)\mcb Y^1_s(\nabla  f) ds

for any $f\in C^1(\bb T)$
and from this identity it follows that the covariance is given by (<ref>).

In the diffusive case, the result follows as a direct consequence of the expressions given in (<ref>) and (<ref>), Lemmas   <ref> and <ref>.

§ PROOF OF THEOREM <REF>

In order to prove the theorem, we   show the uniqueness of strong solutions of
(<ref>) and then, we prove that every solution of the martingale problem is a strong solution. Since all the strong solutions have the same, we are done.

We define a strong solution to (<ref>) in the following way.
We say that a process $\{X_t\}_{t\geq 0}$ in $ \mcb C([0,T], \mcb H_{-m})$ is a strong solution of (<ref>) if for all $f\in\mcb H_{m}$ it holds

\begin{equation}
\label{iquique}
d\langle X_t,f\rangle =\langle X_t, (\nu\Delta-\mcb v\nabla )f\rangle dt-\sigma \langle X_t,\nabla f\rangle d \mcb B_t.
\end{equation}

In order to prove uniqueness of strong solutions, first, recall that the functions $g_k(u)=e^{ 2\pi i k u }$, $u \in \bb T,k \in \bb Z$, form an orthonormal basis of $L^2(\bb T)$. For each $k \in \bb Z$, let $\widehat f(k)=\left\< f,g_k \right\>$ and let us rewrite $X_t$ as a linear combination of the aforementioned eigenfunctions:

X_t   = ∑_k ∈ZX_t(k) g_k.

Observe that for each $k$ the process $\widehat X_t(k)$ is solution of

d \widehat X_t(k)=-\lambda_k \widehat X_t(k) dt  +\theta i \widehat X_t(k) d {\mcb  B}_t,

where $\lambda_k=\nu(2 \pi k)^2-2 \pi \mcb v k i$ and $\theta_k=2\pi  k\sigma $. As a consequence, it is enough to show uniqueness of the strong solution of the previous equation.  Let  $Y_t(k):=\widehat X_t(k)e^{\lambda_k t}$. From the above identity, we obtain the Schrödinger equation

\begin{equation}\label{eq:schor}
dY_t(k)= i \theta_k Y_t(k) d {\mcb B}_t,
\end{equation}

whose solutions are of the form

Y_t(k)=Y_0(k)+\int_0^t i \theta_k Y_s(k) d{\mcb B}_t.

Let us assume that there exist two strong solutions $Y_t^1(k)$ and $Y_t^2(k)$ of the above equation starting from the same initial condition. Define $\Phi_t(k):=Y_t^1(k)-Y_t^2(k)$ and $\varphi_t(k)=\bb E\left[\|\Phi_t(k)\|^2\right]$. Thus,

φ_t(k)=E Φ_t(k)^2    =E[  ∫_0^t  i θ_k Φ_s(k) dℬ_s ^2]= θ_k^2 ∫_0^t  Φ_s(k)^2 ds=  θ_k^2 ∫_0^t  φ_s(k) ds.

Note that in the third identity above, we used the Itô's isometry.
By Grönwall's Lemma and since $\varphi_0(k)=Y_0^1(k)-Y_0^2(k)=0$, we conclude that $\varphi_t(k)=0$ for any $t \geq 0$ and any $k \in \bb Z$. Therefore, the strong solution of the Schrödinger equation (<ref>) is unique and thus the strong solution of (<ref>) is unique.

Now we prove that every solution of the martingale problem given in Definition <ref> is a strong solution.
To that end,  let $\{X_t\}_{t \geq 0}$ be a solution of that martingale problem.  Observe that $\{\mcb M_t(f)\}_{ t \geq 0}$ is a continuous martingale, so, in particular, the representation theorem and the Itô's formula hold for these martingales. Fix a countable  subset $\mcb D$ of $C^\infty(\bb T)$.
For each $f \in \mcb D$, let $\{B_t(f)\}_{ t \geq 0}$ be a Brownian motion such that

\[
\mcb M_t(f)  = \int_0^t \sigma X_s( \nabla f) dB_s(f)
\]

for every $t \geq 0$. By the representation theorem, the Brownian motion $\{B_t(f)\}_{ t \geq 0}$ exists in an extended probability space. The importance of  $\mcb D$ to be countable is to guarantee that  all the Brownian motions $\{B_t(f)\}_{ t \geq 0, f \in \mcb D}$ are defined on the same extended space. Our aim is to prove that we can choose $\{B_t(f)\}_{ t \geq 0}$  independently of $f$. For $f,g \in \mcb D$, from the polarization identity and  (<ref>),  the joint quadratic variation of $\{\mcb M_t(f)\}_{ t \geq 0}$ and $\{\mcb M_t(g)\}_{ t \geq 0}$ can be computed as follows

\begin{equation}
\label{coquimbo}\begin{split}
\langle \mcb M(f), \mcb M(g)\rangle_t =& \frac{1}{2}\Big(\langle \mcb M(f)+\mcb M(g)\rangle_t-\langle \mcb M(f)\rangle_t-\langle \mcb M(g)\rangle_t \Big)\\
=&\int_0^t \sigma^2\Big(X_s(\nabla f )+X_s (\nabla g)\Big)^2 ds- \int_0^t \sigma^2X_s(\nabla f )^2 ds-\int_0^t \sigma^2X_s (\nabla g)^2 ds\\=&
\int_0^t \sigma^2X_s( \nabla f) \,X_s(\nabla g) ds.\end{split}
\end{equation}

On the other hand,  from the representation theorem (see Theorem 3.4.2 of [12]), the joint quadratic variation is given by

\begin{equation}
\label{laserena}
\langle \mcb M(f), \mcb M(g)\rangle_t = \int_0^t \sigma^2 X_s(\nabla f) X_s(\nabla g) \,d \langle B(f), B(g)\rangle_s.
\end{equation}

Let us assume for a moment that $X_t(\sigma \nabla f) \,X_t(\sigma \nabla g) \neq 0$ for every $t \geq 0$. If this is the case, then (<ref>) and (<ref>) imply that

\[
\< B(f), B(g)\>_t = t.
\]

Another application of the polarization identity gives that $\<B(f) -B(g)\>_t=0$, from where we conclude that $B_t(f) = B_t(g)$ for every $t \geq 0$. However, we do not have any a priori reason to assume that $X_t(\sigma \nabla f) \,X_t(\sigma \nabla  g) \neq 0$ for every $t \geq 0$, nor we can derive this fact from the martingale property. Therefore, we will proceed more carefully. Let $\{f_i\}_{ i \in \bb N}$ be an enumeration of $\mcb D$ and let $\{A_i\}_{ i \in \bb N}$ be given on $i\in\mathbb N$ by

\[
A_i := \{t \geq 0; \, X_t(\sigma \nabla f_i) =0\}.
\]

The representation theorem states that $\{B_t(f)\}_{ t \geq 0}$ admits the representation

\[
B_t(f_i) = \int_0^t \Big( \mathbb{1}_{\{s \notin A_i\}} X_s(\sigma \nabla f_i)^{-1} d \mcb M_s(f_i) + \mathbb{1}_{\{s \in A_i\}}  d {B}^i_s \Big),
\]

where $\{B_t^i\}_{ t \geq0}$ is an auxiliary Brownian motion. According to the representation theorem, we have the freedom to define this auxiliary Brownian motion using a Brownian motion defined in our original filtered space, using a Brownian motion independent of our original filtered space, or using a combination of both procedures. We adopt the latter option, as we explain below.

Let $A_0:= [0,\infty)$, and define the sets $\{C_i\}_{ i \in \bb N}$ as

\[
C_i:= \bigcap_{j=0}^{i-1}A_j \setminus A_i,
\]

and let $C_\infty : = \cap_{i \in \bb N} A_i$. Observe that $C_1=A_0\setminus A_1$, $C_2=(A_0\cap A_1)\setminus A_2$ and so on.   Note that $\{C_i\}_{ i \in \bb N\cup \infty}$ is a partition of $A_0$.
Let $\{W_s\}_{s\geq 0}$ be a Brownian motion independent of $\{B_s^i\}_{s\geq 0}$ and $\{\mcb M_s(f_i)\}_{s\geq0}$, for all $i\geq 1$. Define

\[
\widetilde{B}_t(f_1) := \int_0^t \Big( \sum_{i=1}^\infty \mathbb{1}_{\{s \in C_i\}} X_s(\sigma \nabla f_i)^{-1} d \mcb M_s(f_i) + \mathbb{1}_{\{s \in C_\infty\}} d {W_s} \Big).
\]

From the representation theorem, we have that

\[
\mcb M_t(f_1) = \int_0^t X_s(\sigma \nabla f_1) d \widetilde{B}_s(f_1).
\]

Recall that by (<ref>) and (<ref>), $d B_t(f_1) = d B_t(f_2)$ for $t \notin A_1 \cup A_2$. We also have that

\[
d \widetilde{B}_t(f_1) = d {B}_t(f_1) \text{ for } t \notin A_1 \cup A_2,
\]

\[
d \widetilde{B}_t(f_1) = d B_t(f_2) \text{ for } t \in A_1 \setminus A_2,
\]

from where we conclude that $d \widetilde{B}_t(f_1) = d B_t(f_2)$ for $t \notin A_2$. Let us define

\[
\widetilde{B}_t(f_2) := \int_0^t \Big( \mathbb{1}_{\{ s \notin A_2\}} d B_s(f_2) + \mathbb{1}_{\{s \in A_2\}} d \widetilde{B}_s(f_1)\Big)
\]

and observe that

\[
\mcb M_t(f_2) = \int_0^t X_s(\sigma \nabla f_2) d \widetilde{B}_s(f_2)
\]

and $\widetilde{B}_t(f_1) = \widetilde{B}_t(f_2)$ for every $t \geq 0$.
Inductively, we can define

\[
\widetilde{B_t}(f_{\ell+1}) := \int_0^t \Big( \mathbb{1}_{\{s \notin A_{\ell+1}\}} d B_s(f_{\ell+1}) + \mathbb{1}_{\{s \in A_{\ell+1}\}} d \widetilde{B}_s(f_\ell)\Big)
\]

and we can check that $\widetilde{B}_t(f_{\ell+1}) = \widetilde{B}_t(f_\ell)$ for every $t \geq 0$. We conclude that $\widetilde{B}_t:= \widetilde{B}_t(f_\ell)$ does not depend on $\ell$ and

\[
\mcb M_t(f_\ell) = \int_0^t X_s(\sigma \nabla f_\ell) d \widetilde{B}_s(f_\ell)
\]

for every $\ell \in \bb N$. This shows that the solution of the martingale problem $\{X_t\}_{ t \geq 0}$ of (<ref>) satisfies

\[
X_t(f) = X_0(f) + \int_0^t X_s(\nu \Delta f - \gamma \nabla f) ds + \int_0^t X_s(\sigma \nabla f) d \widetilde{B}_s
\]

for every $f \in \mcb D$. Observe that in order to show uniqueness of strong solutions of (<ref>), we only need to verify (<ref>) for the trigonometric functions $\{g_k\}_{ k \in \bb Z}$.
Choosing $\mc D =: \{ \textrm{Re} (g_k), \textrm{Im} (g_k)\}_{ k \in \bb Z}$, we conclude that every martingale solution of (<ref>) is also a strong solution of (<ref>), from where uniqueness follows.

P.G. thanks  FCT/Portugal for financial support
through CAMGSD, IST-ID, projects UIDB/04459/2020 and UIDP/04459/2020.  R.M. thanks the National Council for Scientific and Technological Development (CNPq). M.J. has been funded by  CNPq grant 201384/2020-5 and FAPERJ grant E-26/201.031/2022.
This work was partially supported by the National Council for Scientific and Technological Development (CNPq),  by the Simons Foundation and by the Mathematisches Forschungsinstitut Oberwolfach.
This project has received funding from the European Research Council (ERC) under the European Union's Horizon $2020$ research and innovative programme (grant agreement No $715734$).

[2] Chang, C. C., Landim, C., and Olla, S. Equilibrium fluctuations of asymmetric simple exclusion processes in dimension $d \geq 3$. Probability Theory and Related Fields 119.3 (2001): 381–409.

[4] Diaconis, P. Group representations in probability and statistics. Lecture notes-monograph series 11 (1988): 1–192.

[5] Diaconis, P., and Saloff-Coste, L. Random walks on finite groups: a survey of analytic techniques. Probability measures on groups and related structures, XI (Oberwolfach, 1994) (1995): 44–75.

[7] Goel, S. Analysis of top to bottom-$k$ shuffles. The Annals of Applied Probability 16.1 (2006): 30–55.

[8] Holley, R. A., and Stroock, D. W. Generalized Ornstein-Uhlenbeck processes and infinite particle branching Brownian motions. Publications of the Research Institute for Mathematical Sciences 14.3 (1978): 741–788.

[9] Hildebrand, M. Rates of Convergence of Some Random Processes on Finite Groups. PhD thesis, Harvard University, 1990.

[10] Jacod, J., and A. N. Shiryaev. Limit Theorems for Stochastic Processes. Springer. (1987).

[11] Jara, M., and  Menezes, O. Non-equilibrium fluctuations of interacting particle systems. arXiv preprint arXiv:1810.09526 (2018).

[12]
Karatzas, I., and Shreve, S. E. Brownian motion and stochastic calculus. Second edition. Graduate Texts in Mathematics, 113. Springer-Verlag, New York, 1991.

[13] Kipnis, C., and Landim, C. Scaling Limits of Interacting Particle Systems. Vol. 320. Springer Science and Business Media, 1998.

[14] Lacoin, H. Mixing time and cutoff for the adjacent transposition shuffle and the simple exclusion. The Annals of Probability 44.2 (2016): 1426–1487.

[17] Wilson, D. B. Mixing times of lozenge tiling and card shuffling Markov chains. The Annals of Applied Probability 14.1 (2004): 274–325.

[18] Wilson, D. B. Mixing time of the Rudvalis shuffle. Electronic Communications in Probability 8 (2003): 77–85.


# Hallucination in Perceptual Metric-Driven
Speech Enhancement Networks
††thanks: This work was supported by the Centre for Doctoral Training in
Speech and Language Technologies (SLT) and their Applications funded by UK
Research and Innovation [grant number EP/S023062/1]. This work was also funded
in part by TOSHIBA Cambridge Research Laboratory.

George Close, Thomas Hain, and Stefan Goetze {glclose1, t.hain,
<EMAIL_ADDRESS>0000-0002-9478-5421 0000-0003-0939-3464
0000-0003-1044-7343 Speech and Hearing Group, Department of Computer Science,
University of Sheffield, Sheffield, United Kingdom

###### Abstract

Within the area of speech enhancement, there is an ongoing interest in the
creation of neural systems which explicitly aim to improve the perceptual
quality of the processed audio. In concert with this is the topic of non-
intrusive (i.e. without clean reference) speech quality prediction, for which
neural networks are trained to predict human-assigned quality labels directly
from distorted audio. When combined, these areas allow for the creation of
powerful new speech enhancement systems which can leverage large real-world
datasets of distorted audio, by taking inference of a pre-trained speech
quality predictor as the sole loss function of the speech enhancement system.
This paper aims to identify a potential pitfall with this approach, namely
hallucinations which are introduced by the enhancement system ‘tricking’ the
speech quality predictor.

###### Index Terms:

Speech enhancement, non-intrusive speech quality prediction, generative models
for signal enhancement

##  I Introduction

There has been an ongoing interest in the development of neural _non-
intrusive_ speech quality (SQ) predictors, for both, (i) replacing traditional
signal-processing-based _intrusive_ SQ metrics [1, 2] and (ii) prediction of
human mean opinion score (MOS) quality labels [3, 4, 5, 6]. Such neural-
network-based predictors have two major benefits: In the case of predictors of
traditional metrics, they allow for the creation of non-intrusive versions of
normally intrusive SQ metrics, such as e.g. the frequently-used Perceptual
Evaluation of Speech Quality (PESQ) [7]. While traditional metrics are often
non-differentiable, these neural-network-based metrics can then be used in a
loss function for training an speech enhancement (SE) system [8, 9]. Direct
MOS predictors estimating human signal quality assessment are also able to
leverage large datasets of audio with MOS labels as training data where the
model is trained to accurately predict the MOS of the input audio. During
inference, these models are able to mimic the effect of actually conducting
listening tests which is often a costly and time-consuming process. Inference
of MOS predictors can also be used as a training objective for SE systems,
potentially allowing for the leveraging of large amounts of _real_ noisy audio
(where no reference signal exists) as training data. Approaches combining
metric and MOS prediction [10] have also proved successful, especially when a
strong correlation between metric and human labels can be found.

In the recent CHiME7 challenge unsupervised domain adaptation speech
enhancement (UDASE) task [11, 12] it was shown that high metric scores from
non-intrusive neural SQ predictors do not always match with actual human MOS
evaluation. The evaluation of the SE system entries to the UDASE task had two
stages; first the entries were evaluated in terms of the scores from the Deep
Noise Suppression Mean Opinion Score (DNSMOS) [13] neural non-intrusive SQ
metric. Then, in the second evaluation stage, listening tests were conducted
and MOS scores for audio enhanced by the challenge entries were computed from
these listening tests. The best-performing system in the first evaluation
stage was [14], an SE system which utilises a non-intrusive MetricGAN [2]
framework to directly optimise towards the DNSMOS metric. However, this system
was scoring lowest of the entries going forward to the listening-test
evaluation stage; by optimising directly for high DNSMOS scores, the SE system
may learn to introduce specific distortions which result in high DNSMOS scores
but which negatively impact the actual perceptual quality of the enhanced
audio when assessed by humans.

In general, it was found in the UDASE task results [12] that quality ratings
from non-intrusive quality predictors such as DNSMOS and TorchAudio-SQUIM [1]
did not correlate strongly with the MOS ratings obtained in the second
evaluation stage by listening tests and that traditional intrusive signal
processing based metrics such as PESQ and Short-Time Objective Intelligibility
(STOI) showed significantly stronger correlation.

This work therefore has two major objectives. Firstly, to better understand
how SE systems like that in [14] learn to optimise their outputs towards
neural non-intrusive SQ metrics during training. Secondly, to identify why
neural non-intrusive SQ metrics fail to properly assess the human assessed
quality of the output of SE systems, even in the setting that the SE system
does not directly optimise the metric in training.

This paper is structured as follows. In Section II a novel non-intrusive SQ
predictor system is introduced. Then in Section III an SE system which uses
inference of the SQ predictor in its training loss function is detailed. In
Section IV, an experiment training the SE system with varying degrees of
influence of the SQ predictor is conducted, and the results analysed. In
Section V a small listening test experiment is carried out using the models
trained in the proceeding section, and the results analysed. Section VI
concludes the paper.

##  II Non-Intrusive Speech Quality Predictor

The non-intrusive SQ predictor $\mathcal{D}(\cdot)$ used in this work is based
on [15] and consists of a Transformer [16] block, followed by a feed-forward
attention block with a sigmoid activation on a single output neuron which
represents the predicted quality $\hat{q}^{\prime}$ of the input audio,
normalised between $0$ and $1$.

The proposed predictor differs from that in [15] as follows: Rather than an
input feature derived from the XLS-R representation, the input feature of
$\mathcal{D}(\cdot)$ is the output of the Transformer Encoder stage of a pre-
trained Whisper [17] automatic speech recognition (ASR) network. This
representation has been shown to be a useful feature representation for
similar non-intrusive prediction tasks [18, 19]. In this work, the whisper-
small111https://huggingface.co/openai/whisper-small model, trained on $680$k
hours of labelled speech data is used. The encoder stage of this model returns
a representation of fixed dimension $F_{\mathrm{Enc}}\times
T_{\mathrm{Enc}}=768\times 1500$. Note that the Whisper encoder block is used
solely as a feature extractor, and its parameters are not updated during the
training of $\mathcal{D}(\cdot)$.

The metric prediction network $\mathcal{D}(\cdot)$ is trained as follows: The
MOS label $q$ in most datasets is expressed as a value between $1$ and $5$,
higher being better. In the training and inference of $\mathcal{D}(\cdot)$,
this value is normalized between $0.2$ and $1$, which is denoted as
$q^{\prime}$. For a pair of audio and normalized MOS label
$\\{x[n],\,q^{\prime}\\}$, the model is trained with a loss between the output
of the model (i.e. the predicted quality of $x[n]$) and the true normalized
MOS label $q^{\prime}$:

$L_{\mathcal{D}}=(\mathcal{D}(x[n])-q^{\prime})^{2}$ (1)

The model is trained following a scheme similar to that proposed in [3] where
training halts if the validation performance does not improve after $20$
epochs.

The performance of the proposed non-intrusive metric prediction network
$\mathcal{D}(\cdot)$, trained and tested on the NISQA [3] dataset is shown in
Table I, compared to the NISQA baseline. The NISQA test set and (baseline)
model are widely used benchmarks for the SQ prediction task. The proposed
predictor network outperforms this baseline both in terms of spearman
correlation $r$ and root mean squared error (RMSE) across all three NISQA
testsets (P501, FOR and LIVETALK), and is comparable or better than state-of-
the-art systems [15, 20] on these testsets. In addition, a variant of
$\mathcal{D}(\cdot)$, denoted as $\mathcal{D_{B}}(\cdot)$ in Table I, is
trained based on additional datasets, i.e NISQA [3], Tencent [6] and PTSN [4]
speech quality datasets, which shows similar, in mean further increased
performance.

Table I: Proposed SQ Predictor compared with baseline NISQA model.

Testset | P501 | FOR | LIVETALK | MEAN
---|---|---|---|---
Model | r$\uparrow$ | RMSE$\downarrow$ | r$\uparrow$ | RMSE$\downarrow$ | r$\uparrow$ | RMSE$\downarrow$ | r$\uparrow$ | RMSE$\downarrow$
NISQA | 0.89 | 0.46 | 0.88 | 0.40 | 0.70 | 0.67 | 0.82 | 0.51
$\mathcal{D}(\cdot)$ | 0.94 | 0.35 | 0.93 | 0.32 | 0.81 | 0.54 | 0.89 | 0.40
$\mathcal{D}_{B}(\cdot)$ | 0.93 | 0.37 | 0.94 | 0.32 | 0.85 | 0.50 | 0.91 | 0.40

##  III Speech Enhancement System

The DPT-FSNet [21] single-channel speech enhancement architecture which is
based on the dual-path Transformer (DPT) architecture is used as the baseline
speech enhancement system denoted as $\mathcal{G}(\cdot)$ in this work. This
model has shown state-of-the-art performance in this task, despite a
relatively small parameter count. It takes as input the real and imaginary
short-time Fourier transform (STFT) components $\mathbf{X}_{r}$ and
$\mathbf{X}_{i}$ of the noisy time domain signal $x[n]$, and returns mask
matrices $\mathbf{M}_{r}$ and $\mathbf{M}_{i}$ which are multiplied with the
inputs to produce estimated of the clean complex signal spectra, i.e.
$\mathbf{\hat{S}}_{r}$ and $\mathbf{\hat{S}}_{i}$. These are then used as
inputs to an inverse short-time Fourier transform (ISTFT) operation to produce
the enhanced time domain audio $\hat{s}[n]$. For a detailed description of the
architecture see [21].

###  III-A Network Structure

The DPT-FSNet has three sequential stages. The first stage is an encoder
comprised of a $1$-D convolutional layer followed by a dilated-dense block
with $4$ dilated convolutional layers. The input to the encoder are the real
and imaginary STFT components of the noisy input signal $x[n]$ in the form
$2\times T\times F$, and the output is a higher dimensional representation
with dimensions $64\times T\times F$, i.e. $64$ time-frequency maps.

The second stage comprises $4$ dual path [22] Transformer blocks, which
operate over the $T$ and $F$ dimensions sequentially. The Transformer blocks
use a slight variation on the standard structure; i.e. a gated recurrent unit
(GRU) layer is incorporated into the feed-forward network following the multi-
head attention.

The final stage is a decoder structured as a mirror of the initial encoder
stage; it takes as input the output of the second stage and returns predicted
real and imaginary masks $\mathbf{M}_{r},\mathbf{M}_{i}$ with dimensions
$2\times T\times F$.

###  III-B Loss Function

To train the proposed adaptation of the DPT-FSNet network
$\mathcal{G}(\cdot)$, an extended loss function

$L=\alpha L_{\mathrm{Spec}}+(1-\alpha)L_{\mathrm{SQ}}$ (2)

is proposed which adds a loss term

$L_{\mathrm{SQ}}=(1-\mathcal{D}_{B}(\hat{s}[n]))^{2}$ (3)

based on inference of the non-intrusive pre-trained SQ predictor (cf. Section
II) of the enhanced audio $\hat{s}[n]$ to the loss used in the original DPT-
FSNet paper [21]

$\begin{split}L_{\mathrm{spec}}=\frac{1}{TF}\sum_{t=0}^{T-1}\sum_{f=0}^{F-1}\left|\left(|\mathbf{S}_{r}[t,f]|-|\mathbf{\hat{S}}_{r}[t,f]|\right)+\right.\\\
\left.\left(|\mathbf{S}_{i}[t,f]|-|\mathbf{\hat{S}}_{i}[t,f]|\right)\right|,\end{split}$
(4)

which is based on the distance of enhanced real and imaginary spectrogram
components $\mathbf{\hat{S}}_{r}$ and $\mathbf{\hat{S}}_{i}$ to the
spectrogram components $\mathbf{S}_{r}$ and $\mathbf{S}_{i}$ of the reference
audio $s[n]$. Note that the time domain loss term as outlined in [21] is not
utilised here.

Table II: Performance of Speech Enhancement for different $\alpha$ in (2) for
the VoiceBank-DEMAND testset.

Best performance denoted in bold font. Unprocessed data denoted in italic
font.  Composite DNSMOS Loss $\alpha$ in (2) PESQ STOI CSIG CBAK COVL SISDR
SIG BAK OVR $\mathcal{D}_{B}(\cdot)$ - clean 4.50 1.00 5.00 5.00 5.00 91.14
4.27 4.36 3.88 0.67 - noisy 1.97 0.92 3.34 2.44 2.63 8.44 4.24 3.32 3.36 0.58
$L_{\mathrm{spec}}$ only, (4) 1 2.99 0.95 4.09 3.57 3.55 19.82 4.14 4.42 3.86
0.65 0.9 2.93 0.94 3.96 3.52 3.44 19.70 4.12 4.46 3.95 0.68 0.8 2.63 0.93 3.59
3.33 3.09 19.62 4.05 4.41 3.91 0.70 0.7 2.72 0.94 3.78 3.38 3.24 19.39 4.08
4.30 3.78 0.71 0.6 2.63 0.93 3.45 3.25 3.00 19.25 4.00 4.33 3.79 0.79 0.5 2.65
0.93 3.57 3.29 3.07 19.43 4.04 4.36 3.84 0.77 0.4 2.66 0.93 3.67 3.31 3.14
18.92 4.06 4.28 3.75 0.76 0.3 2.68 0.93 3.79 3.34 3.22 18.98 4.11 4.38 3.83
0.77 0.2 2.58 0.93 3.47 3.25 3.00 18.51 3.92 4.24 3.70 0.75 $\leftarrow L$,
(2) $\rightarrow$ 0.1 2.37 0.91 3.29 3.10 2.79 17.72 4.02 4.29 3.75 0.76
$L_{SQ}$ only, (3) 0 1.43 0.41 1.00 1.03 1.02 -29.68 2.55 2.54 2.42 0.88

The hyperparameter $\alpha$ in (2) is a value between $0$ and $1$ which
controls the relative weight of the intrusive and non-intrusive terms which
will be analysed in the following.

##  IV Experiment 1 - Scaling the Quality Estimator’s Influence

In this experiment, the SE system $\mathcal{G}(\cdot)$ is trained for
different $\alpha$ in (2), i.e. for varying degrees of influence of the
quality estimator $\mathcal{D}_{B}(\cdot)$ in the loss function. In doing
this, it is possible to compare the performance of at one pole, a traditional
signal-processing-based intrusive loss function, i.e. $L_{\mathrm{spec}}$ in
(4) only, and at the other a purely non-intrusive SQ predictor loss, i.e.
$L_{SQ}$ in (3) only, as well as points between these poles, i.e. the combined
loss in (2) .

###  IV-A Experiment Setup

Each speech enhancement system model, i.e. for varying $\alpha$ is trained for
$200$ epochs on the VoiceBank-DEMAND [23] training set, a widely used dataset
for for training speech enhancement networks. It consists of clean English
read speech, artificially corrupted with environmental noise from the
DEMAND[24] noise dataset. The Adam [25] optimizer is used; following [21], a
dynamic strategy to adjust the learning rate is employed, where the learning
rate steadily increases during the first few model updates and then scales
down over the remaining training epochs.

###  IV-B Results

Table II shows the speech enhancement performance of the experiment described
in Section IV-A for the VoiceBank-DEMAND testset. The models are evaluated by
frequently-used signal-processing-based intrusive measures PESQ, STOI, the
three terms of the Composite measure [26] CSIG, CBAK and COVL and the scale
invariant signal-distortion ratio (SI-SDR) [27]. The models are also evaluated
using the non-intrusive neural SQ measure DNSMOS [13] as well as in terms of
the score assigned by $\mathcal{D}_{B}(\cdot)$ detailed above in Section II.

The best performing model in terms of the standard intrusive measures is the
model with $\alpha=1$ in (2), i.e. where no inference of
$\mathcal{D}_{B}(\cdot)$ is used, and the loss function consists solely of the
tempo-spectral distance in the loss term defined in (4). Generally, as the
value of $\alpha$ decreases, so do the scores. At $\alpha=0$ (i.e. solely
using inference of $\mathcal{D}_{B}$ as defined in (3) as the loss function),
the performance degrades significantly, being drastically worse than even the
input noisy data in all intrusive measures. The difference in performance
between an $\alpha=0$ and $\alpha=0.1$ is stark, suggesting that even a small
weighting of the intrusive loss term (4) is enough to greatly improve
performance.

Performance assessed by the non-intrusive measures in the right part of Table
II follows a somewhat different pattern. All models degrade the DNSMOS SIG
score in comparison to the noisy (as well as the clean) audio. This is
consistent with the findings in masking-based SE in general and for the UDASE
task in particular [12], where all enhancement systems show degraded DNSMOS
SIG with the exception of those systems which explicitly optimise towards it
in training. While the DNSMOS ratings generally decrease as $\alpha$ does, the
model for $\alpha=0.9$ outperforms the model with $\alpha=1$ in terms of the
BAK and OVR components. Furthermore, the model for $\alpha=0.9$ performs only
slightly worse than the model for $\alpha=1.0$ in terms of the intrusive
metrics, suggesting that a small weighting of (3) might be beneficial to the
overall audio quality. However, this DNSMOS performance should be interpreted
with some scrutiny; the results here show that some of the models outperform
even the clean reference audio in terms of DNSMOS, which might be surprising
in the first instance. However, later spectrogram analysis (cf. Figure 1)
shows that _clean_ signals sometimes contain noise (primarily breathing
sounds) which are removed by the SE system. Furthermore, all DNSMOS scores for
$\alpha=0$ are much too high given that this system completely destroys the
input signal. As noted in [14], this might be due to an extreme failure to
generalise in DNSMOS. As it is to be expected, the $\mathcal{D}_{B}(\cdot)$
score increases as $\alpha$ decreases and inference of
$\mathcal{D}_{B}(\cdot)$ is weighted more heavily in the loss function.

###  IV-C Spectrogram Comparison

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

Figure 1: (Magnitude) spectrogram comparison for differing values of $\alpha$.

Figure 1 exemplarily shows spectrograms for the VoiceBank-DEMAND testset file
p232_005.wav; clean reference $\mathbf{S}$ and noisy audio $\mathbf{X}$ (top
two panels) as well as enhanced signals $\mathbf{\hat{S}}$ for different
$\alpha$ in (2) are shown. With the exception of $\alpha=0$, all models
successfully remove the distortion tone present in the first $2$ seconds of
the input noisy signal at approx. $500$ Hz, and generally do enhance the noisy
input such that it resembles the clean reference.

As the value of $\alpha$ decreases however, a distortion in the first half
second of the audio becomes more prominent. This distortion is interesting for
a number of reasons. It does not resemble the noise in this region in the
noisy input signal and occurs consistently in appearance spectrally and in
audible sound across all audio enhanced by the models, indicating that it can
be best characterised as a hallucination of the enhancement model(s). This
hallucination is most prominent in the model where $\alpha=0$; other than the
hallucination the outputs of this model consist of seemingly meaningless
content which does not resemble the noisy input signal at all. Given that the
hallucination appears more strongly as the influence of the quality predictor-
based loss term (3) increases, it is likely caused by the speech enhancement
system learning to trick $\mathcal{D}_{B}(\cdot)$. The consistent form of the
distortion can also be explained as follows; during the training of
$\mathcal{D}_{B}$, it learned to assign a high-quality rating for input audio
which contained a sound like this hallucination. _Then during the training of
the SE models, the SE models learn to exploit this quirk of the training of
$\mathcal{D}_{B}$ by introducing the hallucination in order to minimise the
loss function._ The consistent temporal position of the hallucination can be
explained by the short non-speech region at the start of the audio file which
is often present across all audio in the VoiceBank-DEMAND and similar
datasets. The presence of this hallucination is likely the cause of the
decrease in performance in terms of intrusive signal processing metrics in
Table II while the non-intrusive neural SQ metrics change less uniformly; the
intrusive metrics all involve a direct comparison with the reference audio
which explicitly penalise the presence of the hallucination. The hallucination
has a _speech-like_ characteristic which is possibly the reason that the SQ
predictor models reward its presence.

##  V Experiment 2 - Listening Test

In order to better understand the performance of the trained SE models and the
human perception of the hallucination distortion, a small listening test
experiment was carried out.

###  V-A Setup

Noisy audio files from the VoiceBank-DEMAND testset and audio enhanced by
enhancement models with $\alpha$ values of $0$, $0.1$, $0.5$ and $0.9$ were
randomly selected for a total of $15$ files ($3$ files from each of the $4$
$\alpha$ values plus the noisy signal). The ITU-T P.835 [28] methodology was
used, inspired by [11]. $16$ participants sequentially rated each file in
terms of the naturalness of the speech signal, the intrusiveness of the
background noise distortion and overall quality on $5$ point Likert scales
(i.e SIG, BAK and OVRL), for a total of $48$ ratings per audio file. The
listening test audio is available
online222https://leto19.github.io/nisqa_se_demo.html.

###  V-B Results

The results of the listening test are shown in Table III. In terms of signal
quality SIG, the noisy input audio scores the highest; this is in line with
the MOS results reported in [11] and results in Table II. For BAK and OVRL,
the listening-test results follow those of the metrics in Table II with the
model for $\alpha=1.0$ being the best performing.

Interestingly, $\alpha=0.1$ significantly outperforms $\alpha=0.5$ in all
aspects in the listening tests, outperforming even $\alpha=0.9$ which showed
the best performance in Table II in terms of SIG. The low BAK and OVRL scores
of for $\alpha=0.5$ and $0.1$ suggest that the hallucination is perceptible,
but that the listeners considered it as an aspect of the background rather
than a distortion in the speech signal itself. This is important when
considering the disconnect between intrusive metric scores and human
perception MOS. An intrusive metric like PESQ is directly comparative such
that deviation in the test signal from the oracle reference signal always
results in a lower output score. On the other hand, human MOS is indirectly
comparative; the score is informed wholly by the listener’s preconceived
notion of speech quality, which varies not only between individuals but also
unconsciously over time during the listening test. Likewise, non-intrusive SQ
predictors are also indirectly comparative, with the output score informed by
the training data. This is exemplified clearly by comparing the Composite
measure CSIG score of noisy audio in Table II with the analogous DNSMOS SIG
score in the same table and the real MOS SIG average in Table III. The noisy
signals are generally dissimilar to the clean references, meaning that the
intrusive CSIG score suffers but this does not in reality mean that the human
perception (or a predictor of human perception) of the speech distortion
suffers drastically.

Table III: Listening Test Results

Best performance denoted in bold. Unprocessed data denoted in italic.

| SIG | BAK | OVRL
---|---|---|---
$\alpha$ | MEAN | STD | MEAN | STD | MEAN | STD
noisy | 4.54 | 0.65 | 2.92 | 0.82 | 3.67 | 0.88
1.0 | 4.50 | 0.62 | 4.67 | 0.66 | 4.42 | 0.74
0.9 | 4.31 | 0.72 | 4.54 | 0.65 | 4.25 | 0.73
0.5 | 4.06 | 0.86 | 3.58 | 1.03 | 3.63 | 0.96
0.1 | 4.35 | 0.70 | 3.83 | 0.69 | 3.94 | 0.81

##  VI Conclusion

SE models which are optimised using non-intrusive neural SQ predictors are
shown to produce hallucinatory artefacts in output audio. These hallucinations
do not represent meaningful content but are learned by the SE system in order
to optimise the audio towards maximising the score awarded by the SQ
predictor. Intrusive metrics like PESQ are sensitive to these hallucinations,
and they are shown to generally be perceptible in human listening tests.

## References

  * [1] A. Kumar, K. Tan, Z. Ni, P. Manocha, X. Zhang, E. Henderson, and B. Xu, “Torchaudio-Squim: Reference-Less Speech Quality and Intelligibility Measures in Torchaudio,” _ICASSP 2023_.
  * [2] S.-W. Fu, C. Yu, K.-H. Hung, M. Ravanelli, and Y. Tsao, “MetricGAN-U: Unsupervised Speech Enhancement/ Dereverberation Based Only on Noisy/ Reverberated Speech,” in _ICASSP 2022_.
  * [3] G. Mittag, B. Naderi, A. Chehadi, and S. Möller, “NISQA: A deep CNN-self-attention model for multidimensional speech quality prediction with crowdsourced datasets,” in _Interspeech 2021_.
  * [4] G. Mittag, R. Cutler, Y. Hosseinkashi, M. Revow, S. Srinivasan, N. Chande, and R. Aichner, “DNN No-Reference PSTN Speech Quality Prediction,” in _Interspeech 2020_.
  * [5] B. Cauchi, K. Siedenburg, J. F. Santos, T. H. Falk, S. Doclo, and S. Goetze, “Non-Intrusive Speech Quality Prediction Using Modulation Energies and LSTM-Network,” _IEEE/ACM Transactions on Audio, Speech, and Language Processing_ , 2019.
  * [6] G. Yi, W. Xiao, Y. Xiao, B. Naderi, S. Möller, W. Wardah, G. Mittag, R. Cutler, Z. Zhang, D. S. Williamson _et al._ , “ConferencingSpeech 2022 Challenge: Non-intrusive Objective Speech Quality Assessment (NISQA) Challenge for Online Conferencing Applications,” 2022.
  * [7] A. Rix, J. Beerends, M. Hollier, and A. Hekstra, “Perceptual evaluation of speech quality (PESQ)-a new method for speech quality assessment of telephone networks and codecs,” in _ICASSP_ , 2001.
  * [8] S.-W. Fu, C. Yu, T.-A. Hsieh, P. Plantinga, M. Ravanelli, X. Lu, and Y. Tsao, “MetricGAN+: An Improved Version of MetricGAN for Speech Enhancement,” in _Interspeech 2021_.
  * [9] G. Close, T. Hain, and S. Goetze, “MetricGAN+/-: Increasing Robustness of Noise Reduction on Unseen Data,” in _EUSIPCO 2022_ , Belgrade, Serbia, Aug. 2022.
  * [10] G. Close, S. Hollands, T. Hain, and S. Goetze, “Non-intrusive Speech Intelligibility Metric Prediction for Hearing Impaired Individuals,” in _Proc. Interspeech 2022_ , 2022, pp. 3483–3487.
  * [11] S. Leglaive, L. Borne, E. Tzinis, M. Sadeghi, M. Fraticelli, S. Wisdom, M. Pariente, D. Pressnitzer, and J. R. Hershey, “The CHiME-7 UDASE task: Unsupervised domain adaptation for conversational speech enhancement,” 2023.
  * [12] S. Leglaive, M. Fraticelli, H. ElGhazaly, L. Borne, M. Sadeghi, S. Wisdom, M. Pariente, J. R. Hershey, D. Pressnitzer, and J. P. Barker, “Objective and subjective evaluation of speech enhancement methods in the udase task of the 7th chime challenge,” 2024.
  * [13] C. K. A. Reddy, V. Gopal, and R. Cutler, “Dnsmos: A non-intrusive perceptual objective speech quality metric to evaluate noise suppressors,” in _ICASSP 2021 - 2021 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_ , 2021, pp. 6493–6497.
  * [14] G. Close, W. Ravenscroft, T. Hain, and S. Goetze, “CMGAN+/+: The University of Sheffield CHiME-7 UDASE Challenge Speech Enhancement System,” in _Proc. 7th Int. Workshop on Speech Processing in Everyday Environments (CHiME 2023)_ , Dublin, Ireland, Aug. 2023.
  * [15] B. Tamm, R. Vandenberghe, and H. Van hamme, “Analysis of xls-r for speech quality assessment,” 2023.
  * [16] A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez, L. Kaiser, and I. Polosukhin, “Attention is all you need,” 2017.
  * [17] A. Radford, J. W. Kim, T. Xu, G. Brockman, C. McLeavey, and I. Sutskever, “Robust Speech Recognition via Large-Scale Weak Supervision,” 2022.
  * [18] Santiago Cuervo, Ricard Marxer, “Temporal-hierarchical features from noise-robust speech foundation models for non-intrusive intelligibility prediction,” in _Clarity Workshop 2022_. [Online]. Available: https://claritychallenge.org/clarity2023-workshop/papers/CPC2_E011_report.pdf
  * [19] R. Mogridge, G. Close, R. Sutherland, T. Hain, J. Barker, S. Goetze, and A. Ragni, “Non-intrusive speech intelligibility prediction for hearing-impaired users using intermediate asr features and human memory models,” in _Proc. ICASSP 2024 (accepted)_ , 2024.
  * [20] K. Shen, D. Yan, and L. Dong, “Msqat: A multi-dimension non-intrusive speech quality assessment transformer utilizing self-supervised representations,” _Applied Acoustics_ , 2023.
  * [21] F. Dang, H. Chen, and P. Zhang, “DPT-FSNet: Dual-Path Transformer Based Full-Band and Sub-Band Fusion Network for Speech Enhancement,” in _ICASSP 2022_.
  * [22] Y. Luo, Z. Chen, and T. Yoshioka, “Dual-Path RNN: Efficient Long Sequence Modeling for Time-Domain Single-Channel Speech Separation,” in _ICASSP 2020_.
  * [23] C. Valentini-Botinhao, “Noisy speech database for training speech enhancement algorithms and tts models,” 2017. [Online]. Available: https://doi.org/10.7488/ds/2117
  * [24] J. Thiemann, N. Ito, and E. Vincent, “DEMAND: a collection of multi-channel recordings of acoustic noise in diverse environments,” Jun. 2013.
  * [25] D. P. Kingma and J. Ba, “Adam: A method for stochastic optimization,” _CoRR_ , 2014.
  * [26] Z. Lin, L. Zhou, and X. Qiu, “A composite objective measure on subjective evaluation of speech enhancement algorithms,” _Applied Acoustics_ , vol. 145, 2019.
  * [27] J. L. Roux, S. Wisdom, H. Erdogan, and J. R. Hershey, “Sdr – half-baked or well done?” _ICASSP 2019 - 2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_ , 2018.
  * [28] “Subjective test methodology for evaluating speech communication systems that include noise suppression algorithm.” International Telecommunication Union (ITU), Standard, 2003.

  *[SQ]: speech quality
  *[MOS]: mean opinion score
  *[PESQ]: Perceptual Evaluation of Speech Quality
  *[SE]: speech enhancement
  *[UDASE]: unsupervised domain adaptation speech enhancement
  *[DNSMOS]: Deep Noise Suppression Mean Opinion Score
  *[STOI]: Short-Time Objective Intelligibility
  *[ASR]: automatic speech recognition
  *[RMSE]: root mean squared error
  *[DPT]: dual-path Transformer
  *[STFT]: short-time Fourier transform
  *[ISTFT]: inverse short-time Fourier transform
  *[GRU]: gated recurrent unit
  *[SI-SDR]: scale invariant signal-distortion ratio
  *[ SE]: speech enhancement