# Condensation, boundary conditions, and effects of slow sites in zero-range
systems

Sunder Sethuraman  Department of Mathematics, University of Arizona, Tucson,
AZ 85721, USA<EMAIL_ADDRESS>and  Jianfei Xue  Department of
Mathematics, University of Missouri, Columbia, MO 65211,USA<EMAIL_ADDRESS>
###### Abstract.

We consider the space-time scaling limit of the particle mass in zero-range
particle systems on a $1$D discrete torus $\mathbb{Z}/N\mathbb{Z}$ with a
finite number of defects. We focus on two classes of increasing jump rates
$g$, when $g(n)\sim n^{\alpha}$, for $0<\alpha\leq 1$, and when $g$ is a
bounded function. In such a model, a particle at a regular site $k$ jumps
equally likely to a neighbor with rate $g(n)$, depending only on the number of
particles $n$ at $k$. At a defect site $k_{j,N}$, however, the jump rate is
slowed down to $\lambda_{j}^{-1}N^{-\beta_{j}}g(n)$ when $g(n)\sim
n^{\alpha}$, and to $\lambda_{j}^{-1}g(n)$ when $g$ is bounded. Here, $N$ is a
scaling parameter where the grid spacing is seen as $1/N$ and time is speeded
up by $N^{2}$.

Starting from initial measures with $O(N)$ relative entropy with respect to an
invariant measure, we show the hydrodynamic limit and characterize boundary
behaviors at the macroscopic defect sites
$x_{j}=\lim_{N\uparrow\infty}k_{j,N}/N$, for all defect strengths. For rates
$g(n)\sim n^{\alpha}$, at critical or super-critical slow sites
($\beta_{j}=\alpha$ or $\beta_{j}>\alpha$), associated Dirichlet boundary
conditions arise as a result of interactions with evolving atom masses or
condensation at the defects. Differently, when $g$ is bounded, at any slow
site ($\lambda_{j}>1$), we find the hydrodynamic density must be bounded above
by a threshold value reflecting the strength of the defect. Moreover, due to
interactions with masses of atoms stored at the slow sites, the associated
boundary conditions bounce between being periodic and Dirichlet.

###### Key words and phrases:

interacting particle system, zero-range, hydrodynamic, boundary condition,
defect, inhomogeneity, condensation

###### 2020 Mathematics Subject Classification:

60K35

##  1\. Introduction

The purpose of this article is to understand the macroscopic boundary
conditions which arise in hydrodynamic scaling limits for the space-time
‘bulk’ mass evolution of zero-range processes with a finite number of
‘defects’, and also related effects of ‘condensation’ at these defect
locations. Such an aim more broadly fits into the larger study of how
macroscopic boundary conditions emerge from inhomogeneous microscopic
interactions.

In this view, there has been much interesting work on one-dimensional ($1$D)
exclusion models where a site or bond is ‘slowed’ down. In [12], [13], [14],
in computing the hydrodynamic limit in symmetric systems, different boundary
conditions from Dirichlet to Neumann, and also Robin have been derived; see
also [10]. There are however only a few works with respect to different
interactions, in particular zero-range systems, which do not limit the
particle numbers at a location. Among these, [17] studies the hydrodynamic
limit for a system of $1$D symmetric independent particles moving in a
(random) trap environment. In [19], with respect to a $1$D totally asymmetric
zero-range process, with bounded, increasing rate function $g(\cdot)$, effects
of a slow site and a slow particle are found when starting from a ‘flat’
initial measure. In [4] (see also [5]), with respect to a class of such $1$D
totally asymmetric zero-range systems, however with a nontrivial density of
site disorders, hydrodynamics is shown with respect to an effective flux
function, constant at supercritical densities; see also the conjectures in
Section 3.3 in [4] about slow sites in asymmetric models.

We also mention that $1$D systems with ‘reservoir’ boundaries have been
studied; see [9] for a discussion of ‘hydrostatics’, and related references,
with respect to exclusion models. In zero-range processes, ‘static’ reservoir
effects are studied and dynamical conjectures are discussed in [15].

In this context, the general goal of our work is to consider the effect of a
finite number of ‘slow’ sites in a class of $1$D symmetric zero-range systems
in a torus $\mathbb{Z}/N\mathbb{Z}$. In such a model, particles would
‘condense’ on a defect site if jump rates from it are ‘slow’ enough. One would
expect that for the particle continuum mass, different boundary conditions
would result depending on how ‘slow’ the defect is. Our main results describe
corresponding hydrodynamic limits, in terms of a nonlinear parabolic PDE and
evolving point masses, with specified boundary conditions at defects
reflecting different types of condensation. To our knowledge, these are the
first results for a general class of zero-range interactions, with bounded and
unbounded rate functions $g$. In particular, the behaviors found differ
depending on the structure of the rate $g$. The proof method, in the scheme of
the ‘entropy’ method, specifies local ‘replacements’ which may be useful in
other problems.

By specifying the locations of the ‘slow’ defects in the system, we fix the
macroscopically separated points where ‘condensation’ can occur. There seems
to be little work on the dynamical structure in such systems. This is in
contrast to the well-developed study of ‘condensation’, which ‘spontaneously’
forms at a random location by introducing more particles in a zero-range
system with a bounded decreasing rate function than is allowed to
equilibriate. See [2] for a discussion of both mechanisms with respect to
canonical and grand canonical invariant measures of a bounded rate zero-range
process with a single defect.

In passing, we mention, among the recent literature on the ‘spontaneous’
formation of ‘condensation’, [3] considers, with respect to a thermodynamic
limit in $1$D symmetric zero-range models, the evolution of the random
‘condensate’ in a certain time-scale. In [20], with respect to $1$D asymmetric
dynamics in a set of $L$ fixed sites, motion of the ‘condensate’ is described.
In [6], a ‘martingale problem’ approach for the condensate dynamics is
developed. In [23], some partial results on a hydrodynamic limit is given. See
also references therein in these papers for a more complete history of the
subject. For related notions of ‘metastability’, see books and surveys [7],
[21], [24].

###  1.1. Sketch of results

To describe our results, we consider zero-range processes on the torus
${\mathbb{T}}_{N}:=\\{0,1,\ldots,N-1\\}$ where the site $N$ (the right
neighbor of $N-1$) is identified as $0$. The jump rate functions
$g:\mathbb{N}_{0}\mapsto[0,\infty)$ focused upon satisfy
$\lim_{n\uparrow\infty}g(n)/n^{\alpha}=1$ (to fix a time-scale) and
$\alpha\in[0,1]$. Moreover, we classify the $g$’s into two families: (1)
$g(n)\sim n^{\alpha}$ when $0<\alpha\leq 1$, and (2) $g$ is bounded. In both
settings, we will assume also that $g$ is a Lipschitz, increasing function,
which rules out the case $\alpha>1$. Such an assumption is convenient to bound
the rate $g$ in estimates, and also allows ‘attractive’ process couplings; see
Section 3.2 for a discussion of the use of ‘attractive’ coupling.

In this process, at a regular site $k\in\mathbb{T}_{N}$, if there are $n$
particles there, one of them leaves with rate $g(n)$ to a neighbor, jumping
either to $k-1$ or $k+1$, with equal probability. However, at a defect site
$k$, the departure rate is altered as follows: Let $\lambda>0$ and
$\beta\in\mathbb{R}$. If there are $n$ particles at the defect $k$, one of
them leaves at rate $\tfrac{1}{\lambda N^{\beta}}g(n)$. We will say the rate
is ‘slow’ when $\beta>0$, or $\beta=0$ and $\lambda>1$.

The zero-range system tracks the evolution of the unlabeled particles on
$\mathbb{T}_{N}$. We denote by $\xi_{t}=\\{\xi_{t}(k):k\in\mathbb{T}_{N}\\}$
the configuration of the process, where $\xi_{t}(k)$ is the number of
particles at site $k$ at time $t\geq 0$. Given the symmetric transitions, it
will be useful to define also the speeded-up process $\eta_{t}=\xi_{N^{2}t}$.
For this Markov system, there is a family of product (reversible) invariant
measures $\mathscr{R}^{N}_{c}$, indexed by an interval of ‘density’ parameters
$c$; see Section 2.1.

For the system with rate $g(n)\sim n^{\alpha}$, we will start the process from
measures $\mu^{N}$, associated to an initial macroscopic measure $\pi_{0}$ on
the unit torus $\mathbb{T}=[0,1)$, with $O(N)$ relative entropy with respect
to an invariant distribution $\mathscr{R}^{N}_{c_{0}}$, and stochastically
bounded by another invariant distribution $\mathscr{R}^{N}_{c^{\prime}}$. With
respect to bounded rates $g$, $\mu^{N}$ satisfies a similar but slightly
different criteria, since when $\|g\|_{\infty}=g_{\infty}<\infty$, there will
be a finite effective critical density above which the invariant measure is
not defined; see Condition 3.1.

Here, in the setting $g(n)\sim n^{\alpha}$, the initial profile $\pi_{0}$ will
be in form

$\pi_{0}(dx)=\rho_{0}(x)dx+\sum_{j:\beta_{j}=\alpha}{\mathfrak{m}}_{0,j}\delta_{x_{j}}(dx),$

with a similar formulation in the $g$ bounded setting. Examples of suitable
initial measures $\mu^{N}$ are given by local equilibrium product measures;
see Section 3.3.

####  1.1.1.  Rates $g(n)\sim n^{\alpha}$

To describe the main result in the setting $g(n)\sim n^{\alpha}$, it will be
helpful to get a sense of the ‘condensation’ of particles, under an invariant
measure $\mathscr{R}^{N}_{c}$. Typically at a slow site, (1) the number of
particles will be $O(N^{\beta/\alpha})$ when $\beta>\alpha$, referred to as
‘super-slow’ or ‘super-critical’, (2) order $O(N)$ when $\beta=\alpha$, called
‘critical’, and (3) $o(N)$ when $\beta<\alpha$, named as ‘sub-critical’. So,
with a finite number of slow sites, if one of the $k_{j,N}$’s is ‘super-slow’,
there will be a superlinear $O(N^{\beta_{j}/\alpha})$ number of particles at
that location. Whereas, when all the $\beta_{j}\leq\alpha$, there will be
$O(N)$ particles on $\mathbb{T}_{N}$. See Section 2.2 for precise statements.

Let ${\mathfrak{D}}_{s,N}$ be the set of super-slow sites $k_{j,N}$, and let
${\mathfrak{D}}_{s}$ be the corresponding set of continuum points $x_{j}\sim
k_{j,N}/N$. Since, at these locations, the particle numbers are superlinear,
we omit them in the definition of the empirical measure with respect to the
diffusively scaled system:

$\pi^{N}_{t}=\frac{1}{N}\sum_{k\in\mathbb{T}_{N}\setminus{\mathfrak{D}}_{s,N}}\eta_{t}(k)\delta_{k/N}.$

By a hydrodynamic limit, we mean, with respect to test functions $G\in
C(\mathbb{T})$, that $\langle G,\pi^{N}_{t}\rangle$ converges in probability
to $\langle G,\pi_{t}\rangle$, where $\pi_{t}$ is a measure-valued weak
solution of a specified macroscopic evolution.

Our first result (Theorem 4.1) is that three different behaviors and boundary
conditions, at the macroscopic defect sites $x_{j}$, arise in the hydrodynamic
limit depending on when $\beta_{j}>\alpha$, $\beta_{j}=\alpha$ and
$\beta_{j}<\alpha$. As might be suspected, when $\beta_{j}<\alpha$, the defect
is not ‘slow’ enough to be seen in the continuum limit. However, defects where
$\beta_{j}\geq\alpha$ on the other hand do register. To describe the bulk mass
macroscopic flow, segregate the unit torus $\mathbb{T}$ into intervals with
endpoints $x_{j}$ corresponding to $\beta_{j}\geq\alpha$. In the interior of
each interval, the hydrodynamic limit is described by a nonlinear heat
equation

$\partial_{t}\rho=\partial_{xx}\Phi(\rho).$ (1.1)

Here, the ‘fugacity’ function $\Phi$ is a continuum homogenization of the
microscopic rate $g$ (cf. (2.4)). We note this equation also arises in the
context of the zero-range system without disorder. (cf. Chapter 4 [18]).

However, at a defect point $x_{j}$ where $\beta_{j}=\alpha$, an atom evolves
with a mass given by
$\mathfrak{m}_{j}(t)=\big{\\{}\lambda_{j}\Phi(\rho(t,x_{j}))\big{\\}}^{1/\alpha}$,
in terms of $\rho$. Such a statement is natural, given that under $\mu^{N}$,
not far from $\mathscr{R}^{N}_{c_{0}}$, there are $O(N)$ particles at
$k_{j,N}$. Differently, when $\beta_{j}>\alpha$, we will observe initially a
superlinear $O(N^{\beta_{j}/\alpha})$ number of particles at $k_{j,N}$ due to
the relative entropy bound assumption. This level of condensation will not
evolve much for $t>0$ and, as a result, leads to a macroscopic boundary
condition $\rho(t,x_{j})=c_{0}$ where $c_{0}$ is the parameter of the
reference measure $\mathscr{R}_{c_{0}}^{N}$, reflecting a level of initial
condensation at $x_{j}$. The derivation of such Dirichlet boundary condition
at super-slow sites ($\beta_{j}>\alpha$), to our knowledge, is novel.

Taken together, the hydrodynamic limit $\pi_{t}$ on $\mathbb{T}$ will be in
form

$\pi_{t}=\rho(t,x)dx+\sum_{j:\beta_{j}=\alpha}\mathfrak{m}_{j}(t)\delta_{x_{j}}(dx),\quad\text{for
$t>0$.}$

At time $t=0$, $\pi_{t}$ reduces to $\pi_{0}$ mentioned earlier. In general,
the PDE (1.1) for the bulk mass $\rho$ will not be closed without
${\mathfrak{m}}_{j}(t)$ being specified. In fact, the limit $\pi_{t}$ is
characterized as the unique weak solution to a system in terms of
$(\rho(t,x),\\{\mathfrak{m}_{j}(t)\\}_{j})$:

$\partial_{t}\pi_{t}=\partial_{xx}\Phi(\rho)$ (1.2)

with boundary behaviors of $\rho(t,x)$ at slow sites $x_{j}$ given earlier;
see Definition 4.4 and uniqueness Theorem 11.1. See Example 4.7, for a
specific discussion when there is only one defect in the system. We mention in
[17], when $g(n)\equiv n$ ($\alpha=1$), that is for independent particles, a
case of the above hydrodynamic limit may be inferred when
$\beta_{j}\equiv\alpha=1$, with $\Phi(u)\equiv u$.

####  1.1.2. Bounded rates $g$

Our second result (Theorem 4.2) concerns the process with a rate function
$g(n)$, bounded and increasing say to level $g_{\infty}=1$ in the limit as
$n\uparrow\infty$. Slow sites $k_{j,N}$ that we consider will be those where
the jump rate is
$\tfrac{1}{\lambda_{j}N^{\beta_{j}}}g(n)=\tfrac{1}{\lambda_{j}}g(n)$, when
$\beta_{j}=0$ and $\lambda_{j}>1$, when there are $n$ particles at $k$. As
before, with a finite number of slow sites $k_{j,N}$, there is a family of
product invariant measures $\\{\mathscr{R}^{N}_{c}\\}$, but now with densities
$c$ limited to $0\leq\max_{j}\\{\lambda_{j}N^{\beta_{j}}\vee 1\\}\Phi(c)\leq
g_{\infty}$. There is no non-trivial product invariant measure for $c$ above
this level. With this understanding, it is natural to focus on $\beta_{j}=0$,
as if we would slow down with $\beta_{j}>0$, there would be no such non-
trivial invariant measure, as the fugacity $\lambda_{j}N^{\beta_{j}}\Phi(c)$
would exceed $g_{\infty}$ for a finite $N$.

Phenomenologically, the slow site effect is different here than when $g(n)\sim
n^{\alpha}$ in that the system and in particular the slow sites under
$\mathscr{R}^{N}_{c}$ have at most $O(N)$ particles on them. As before,
between macroscopic defects $\\{x_{j}\sim k_{j,N}/N\\}$, the hydrodynamic
limit satisfies (1.1). An atom of mass $\mathfrak{m}_{j}(t)$ may also form
though at macroscopic site $x_{j}\sim k_{j,N}/N$ where $\beta_{j}=0$ and
$\lambda_{j}>1$.

However, unlike in the case $g(n)\sim n^{\alpha}$, this atomic mass may be
evanescent: We show that, at such a defect $x_{j}$, the hydrodynamic density
$\rho(t,x_{j})$ satisfies a bound $\rho(t,x_{j})\leq c_{j,\max}$ where
$c_{j,\max}:=\Phi^{-1}(g_{\infty}/\lambda_{j})$ for all $t>0$. When this
inequality is strict, it holds necessarily that $\mathfrak{m}_{j}(t)=0$. In
particular, the boundary condition near $x_{j}$ may bounce between being
periodic and Dirichlet.

For an informal description, suppose that $\rho(t,x_{j})$ is strictly less
than $c_{j,\max}$ at $t=t_{0}$ and ${\mathfrak{m}}_{j}(t_{0})=0$. Here, the
macroscopic flow will evolve as if the defect at $x_{j}$ were not present,
namely, we will observe a periodic boundary condition at $x=x_{j}$. This will
last until the first moment $t=t_{1}$ when the density $\rho(t,x_{j})$ tends
toward exceeding the threshold value $c_{j,\max}$. The defect is then ‘set
off’ to keep $\rho(t,x_{j})$ at the value $c_{j,\max}$, with any excess mass
stored at the defect to form a delta mass. In other words, we start to observe
${\mathfrak{m}}_{j}(t)>0$ as well as a Dirichlet boundary condition
$\rho(t,x_{j})=c_{j,\max}$ after $t=t_{1}$. Besides being a place of storage,
the slow site $x_{j}$ also serves as a source of mass to maintain the
mentioned Dirichlet boundary when needed as long as ${\mathfrak{m}}_{j}(t)>0$.
When the stored mass is used up (when ${\mathfrak{m}}_{j}(t)$ becomes $0$) and
the density $\rho(t,x_{j})$ tends to drop below $c_{j,\max}$, say at
$t=t_{3}$, the boundary condition switches to the periodic one again after
$t=t_{3}$. Such a ‘bouncing’ phenomenon between boundary conditions takes
place at all slow sites. Finally, characterization of the limit

$\pi_{t}=\rho(t,x)dx+\sum_{j:\beta_{j}=0,\lambda_{j}>0}\mathfrak{m}_{j}(t)\delta_{x_{j}}(dx)$

is given through a weak formulation (1.2) with boundary conditions described
earlier, shown to have a unique solution; see Definition 4.5 and uniqueness
Theorem 11.2.

We remark that a sufficient condition to not feel the defects $\\{x_{j}\\}$ in
the limit would be to start the process from initial (local equilibrium)
measures $\mu^{N}$ associated to $\pi_{0}(dx)=\rho_{0}(x)dx$, where
$\|\rho_{0}\|_{\infty}<\Phi^{-1}\big{(}[\max_{j}\\{\lambda_{j}\\}]^{-1}\big{)}$,
so that by say attractiveness $\rho(t,x)$ also satisfies this bound. On the
other hand, a sufficient condition so that atom masses form is that the
initial density
$\rho_{0}(x)>\Phi^{-1}\big{(}[\max_{j}\\{\lambda_{j}\\}]^{-1}\big{)}$ on
$\mathbb{T}$: Indeed, if no atoms are formed, by the maximum principle,
$\rho(t,x)>\Phi^{-1}\big{(}[\max_{j}\\{\lambda_{j}\\}]^{-1}\big{)}$ on
$\mathbb{T}$, a contradiction. We believe the results in Theorem 4.2 are the
first for a general class of bounded symmetric zero-range processes with
defects.

###  1.2.  Proof ideas for Theorems 4.1 and 4.2

We now discuss ideas in the proofs. We use the general scheme of the ‘entropy’
method, discussed in [18], although there are several departures since the
dynamics is not translation invariant. In particular, mixing in time estimates
are used to make microscopic to macroscopic homogenizations in the ‘bulk’.
However, to capture the boundary effects and correspondences with macroscopic
mass of atoms, we need to estimate the local behavior near defects, for which
we give a more refined argument.

Let $G$ be a test function on $\mathbb{T}$ when $g$ is bounded, and with
compact support away from super-slow sites $x_{j}\in{\mathfrak{D}}_{s}$ when
$g\sim n^{\alpha}$. Consider the stochastic differential

$d\langle
G,\pi^{N}_{t}\rangle=\frac{1}{N}\sum_{k\in\mathbb{T}_{N}}\Delta_{N}G\big{(}k/N\big{)}g_{k}(\eta_{t}(k))dt+dM^{N}(t).$

Here, $g_{k}=\tfrac{1}{\lambda_{j}N^{\beta_{j}}}g$ when $k=k_{j,N}$ and
$g_{k}=g$ otherwise. Also, $M^{N}(t)$ is a martingale, which will vanish as
$N\uparrow\infty$. We state a ‘bulk’ replacement in Lemma 8.4. The proof that
we give makes use of local replacements, as these will be useful to deduce
boundary behaviors at the defects.

In particular, we may replace the local function $g(\eta_{t}(k))$ at regular
sites $k$ by $\Phi\big{(}\eta^{\theta N}_{t}\big{)}$ where $\eta^{\ell}$ is
the $\ell$-window average $\frac{1}{2\ell+1}\sum_{|y-k|\leq\ell}\eta(y)$, by
use of an entropy inequality and a ‘Rayleigh’ estimate. The ‘Rayleigh’
estimate controls the difference between expectations of a local function
under non-equilibrium and equilibrium measures in terms of a Dirichlet form
and a term written in terms of the spectral gap of the process localized to an
interval. No special bound is required–this gap needs only to be positive, or
equivalently that the localized dynamics is ergodic. We remark, as a technical
device, attractiveness is used to handle large densities, difficult to analyze
otherwise, in the proofs of these local replacement, ‘1-block’ Lemma 8.1 and
‘2-block’ Lemma 8.2. As a consequence of these estimates, the hydrodynamic
equation (1.1) may be derived in the ‘bulk’. We now discuss derivation of the
boundary conditions in the two settings.

####  1.2.1. Boundary derivations for $g(n)\sim n^{\alpha}$

In the setting where $g(n)\sim n^{\alpha}$ at a defect $k_{j,N}$, when
$0<\beta_{j}<\alpha$, under $\mu^{N}$, the number of particles
$\eta_{t}(k_{j,N})=O(N^{\beta_{j}/\alpha})$ is sublinear in $N$. Hence, with
respect to the empirical measure, the term

$\langle
G,N^{-1}\delta_{k_{j,N}/N}\rangle=N^{-1}\eta_{t}(k_{j,N})G(k_{j,N}/N)\leq\|G\|_{\infty}N^{-1}\eta_{t}(k_{j,N})=O(N^{-1+\beta_{j}/\alpha})$

vanishes as $N\uparrow\infty$. In effect, we can ignore such defects in the
continuum limit.

However, when $\beta_{j}\geq\alpha$, to deduce a non-trivial boundary
condition, we consider the rate
$g_{k_{j,N}}(\eta_{t}(k_{j,N}))=\tfrac{1}{\lambda_{j}N^{\beta_{j}}}g(\eta_{t}(k_{j,N})$
at the slow site $k_{j,N}$. The following discussion leading to (1.3) will
hold actually for both $g$ of $n^{\alpha}$ type and $g$ bounded settings. To
fix ideas, let us say $k_{j,N}=0$, the origin. Under $\mu^{N}$, not far from
the reversible $\mathscr{R}^{N}_{c_{0}}$ in terms of relative entropy, we show
that this rate is close to the neighboring rate $g(\eta_{t}(1))$ say; see
Lemma 9.1. By the local ‘1-block’ estimate, we show that $g(\eta_{t}(1))$ is
close to $\Phi(\eta^{\ell,+}_{t}(0))$, where
$\eta_{t}^{\ell,+}(0)=\frac{1}{\ell}\sum_{y=1}^{\ell}\eta_{t}(y)$ is the
average in the $\ell$-block to the right of the defect site $0$; see Lemma
9.2. Now, by the local ‘2-block’ bound, we have $\Phi(\eta^{\ell,+}_{t}(0))$
is close to the macroscopic quantity $\Phi((\eta^{\theta N,+}_{t}(0))$ for
$\theta>0$ small. Together, we conclude the fundamental relation,

$\Phi(\eta^{\theta N,+}_{t}(0))\sim g_{0}(\eta_{t}(0)).$ (1.3)

After establishment of tightness of the empirical measure trajectories, and
absolute continuity estimates in Lemmas 6.1 and 7.1, we may consider a
subsequential limit $\pi$ on $\mathbb{T}\setminus{\mathfrak{D}}_{s}$ in the
form $\pi=\rho(t,x)dx+\sum_{j:\beta_{j}=\alpha}\mathfrak{m}_{j}(t)$. From
(1.3), one has for this limit point that

$\Phi(\rho(t,0))\sim g_{0}(\eta_{t}(0)).$

When $\beta_{j}=\alpha$, we have

$\Phi(\rho(t,0))\sim
g_{0}(\eta_{t}(0))\sim\big{(}N^{-1}\eta_{t}(0)\big{)}^{\alpha}\lambda_{j}^{-1}\sim\big{(}{\mathfrak{m}}_{j}(t)\big{)}^{\alpha}\lambda_{j}^{-1},$

which is the boundary condition relating the atom strength to the local
density in Theorem 4.1; see Lemma 7.4.

When $\beta_{j}>\alpha$, we may take the same passage to obtain

$\Phi(\rho(t,0))\sim
g_{0}(\eta_{t}(0))\sim\big{(}N^{-\beta_{j}/\alpha}\eta_{t}(0)\big{)}^{\alpha}\lambda_{j}^{-1}.$

But, in our time-scale, we show $N^{-\beta_{j}/\alpha}\eta_{t}(0)$ does not
vary much from the time $t=0$ quantity $N^{-\beta_{j}/\alpha}\eta_{0}(0)$. For
instance, if $k_{j,N}=0$ were the only defect in the system, by mass
conservation,

$|\eta_{t}(0)-\eta_{0}(0)|=\big{|}\sum_{k\neq
0}\big{(}\eta_{t}(k)-\eta_{0}(k)\big{)}\big{|}.$

Starting under $\mu^{N}$, by use of the entropy inequality, $\sum_{k\neq
0}\eta_{t}(k)=O(N)$ for all $t\geq 0$. Hence,

$N^{-\beta_{j}/\alpha}\big{|}\eta_{t}(0)-\eta_{0}(0)\big{|}=O(N^{-\beta_{j}/\alpha+1})$

vanishes as $N\uparrow\infty$. Moreover, by a calculation with respect to the
initial measure $\mu^{N}$, we may show that

$\big{(}N^{-\beta_{j}/\alpha}\eta_{0}(0)\big{)}^{\alpha}\sim\lambda_{j}\Phi(c_{0}).$

As a consequence, we have

$\Phi(\rho(t,0))\sim\big{(}N^{-\beta_{j}/\alpha}\eta_{0}(0)\big{)}^{\alpha}\lambda_{j}^{-1}\sim\Phi(c_{0}),$

yielding the boundary behavior in Theorem 4.1; see Lemma 7.2 which addresses
the multiple defect setting.

####  1.2.2. Boundary derivations for bounded rate $g$

We now discuss the case of bounded rate function $g$:
$\|g\|_{\infty}=g_{\infty}<\infty$. There are subtleties here with respect to
boundary estimates, different than in the $n^{\alpha}$ rate setting, although
the bulk hydrodynamic limit between defects $x_{j}\sim k_{j,N}/N$ follows the
same procedure as above to derive the equation (1.1).

At a slow site $k_{j,N}$, say again at the origin, slowed down by
$\lambda_{j}^{-1}$ (and $\beta_{j}=0$), we deduce

$\Phi(\rho(t,0))\sim\frac{1}{\lambda_{j}}g(\eta_{t}(0))\leq\frac{g_{\infty}}{\lambda_{j}}.$

Hence, the density near the slow site $x_{j}=0$ is restricted,

$\rho(t,0)\leq\Phi^{-1}(g_{\infty}/\lambda_{j}).$

When the restriction is strict, the intuition is that the particle mass is not
high enough to impact much the flow of particle numbers through the slow site
$0$. However, when equality in the above relation holds, an atom at the slow
site $0$ may form to store excess mass while maintaining the boundary
condition. More precisely, we have ${\mathfrak{m}}_{j}(t)$, the atomic mass
accumulated by $x_{j}=0$, satisfies
${\mathfrak{m}}_{j}(t)={\mathfrak{m}}_{j}(t)\mathbbm{1}_{\rho(t,0)=\Phi^{-1}(g_{\infty}/\lambda_{j})}$;
this is established by showing a microscopic version of the following key
relation

${\mathfrak{m}}_{j}(t)\big{(}\Phi^{-1}(g_{\infty}/\lambda_{j})-\rho(t,0)\big{)}=0;$

see Lemma 7.5. These prescriptions give the boundary conditions specified in
Theorem 4.2.

###  1.3. Paper outline

The plan of the paper is as follows: In Section 2 and 3, we introduce
carefully the zero-range model with defects, their invariant measures, and the
initial measures considered. In Section 4, we state results; see Section 4.7
for a discussion of the example when there is only one defect in the system.
In Section 5, we give the proof outline of the main results, Theorems 4.1 and
4.2, referring to estimates in the sequel. In Section 6, we discuss tightness
of the empirical measures. In Section 7, we discuss properties of limit
points, including importantly their boundary behaviors near macroscopic
defects. In Section 8, we show local ‘1-block’ and ‘2-block’ replacements, and
as a consequence ‘bulk’ replacement. In Section 9, we discuss replacements
near the boundaries of defects needed to derive the macroscopic boundary
conditions. In Sections 10 and 11, we derive energy estimates and prove
uniqueness theorems for the weak formulations.

##  2\. Model description

We will consider symmetric zero-range processes on the discrete torus
$\mathbb{T}_{N}:=\mathbb{Z}/N\mathbb{Z}=\\{0,1,\ldots,N-1\\}$ with a finite
number $n_{0}\geq 0$ of defects located in $\mathbb{T}_{N}$. We will always
assume that $N>n_{0}$ so that there is enough space in $\mathbb{T}_{N}$ to
record the defects.

More carefully, the structure of the defects is the following: Let $J$ be the
index set $\\{1,2,\ldots,n_{0}\\}$. For each $j\in J$, fix
$(x_{j},\beta_{j},\lambda_{j})$ such that $x_{j}\in\mathbb{T}$,
$\beta_{j}\in\mathbb{R}$ and $\lambda_{j}\in(0,\infty)$. Here, $\mathbb{T}$
stands for $[0,1)$ viewed as the unit torus. We will assume that all $x_{j}$’s
are different, that is macroscopically separated. For each $j$, the point
$x_{j}$ denotes the macroscopic location of a defect and
$(\beta_{j},\lambda_{j})$ characterizes its strength. Let
${\mathfrak{D}}:=\\{x_{j}\\}_{j\in J}$ be the set of all macroscopic defect
locations. For each $j\in J$ and $N\in\mathbb{N}$, we now define
$k_{j,N}=\lfloor x_{j}N\rfloor$ be the integer part of $x_{j}N$. Then, the set
${\mathfrak{D}}_{N}:=\\{k_{j,N}\\}_{j\in J}$ stands for the set of microscopic
locations of the defects.

Let now $\mathbb{N}_{0}:=\\{0\\}\cup\mathbb{N}$. For each $N$, let
$\Omega_{N}:=\mathbb{N}_{0}^{\mathbb{T}_{N}}$ be the space of all particle
configurations on $\mathbb{T}_{N}$. With respect to $\xi\in\Omega_{N}$, at a
normal site $k\in\mathbb{T}_{N}\setminus{\mathfrak{D}}_{N}$, a particle jumps
to neighboring sites $k\pm 1$ equally likely at rate $g(\xi(k))/\xi(k)$ where
$\xi(k)$ is the number of particles at site $k$, and
$g:\mathbb{N}_{0}\mapsto[0,\infty)$ is a jump rate function. At a defect site
$k=k_{j,N}$, the jump rate is
$(\lambda_{j}N^{\beta_{j}})^{-1}g(\xi(k))/\xi(k)$. In particular, the site
$k_{j,N}$ is a slow site if $\lambda_{j}N^{\beta_{j}}>1$ and a fast or normal
site if $\lambda_{j}N^{\beta_{j}}\leq 1$ for $N$ large. Let

$g_{k,N}(\cdot)=\begin{cases}g(\cdot)&k\in\mathbb{T}_{N}\setminus{\mathfrak{D}}_{N},\\\
\dfrac{g(\cdot)}{\lambda_{j}N^{\beta_{j}}}&k=k_{j,N},\ j\in J.\end{cases}$

The zero-range process is a Markov process $\xi_{t}$ with the generator

$\begin{split}L_{N}f(\xi)=\sum_{k\in\mathbb{T}_{N}}\Big{\\{}g_{k,N}(\xi(k))\big{(}f(\xi^{k,k+1})-f(\xi)\big{)}+g_{k,N}(\xi(k))\big{(}f(\xi^{k,k-1})-f(\xi)\big{)}\Big{\\}}\end{split}$
(2.1)

where

$\xi^{x,y}(k)=\begin{cases}\xi(x)-1&k=x,\\\ \xi(y)+1&k=y,\\\ \xi(k)&k\neq
x,y.\end{cases}$ (2.2)

So that the process is irreducible, we will assume that the jump rate function
$g$ is such that $g(n)=0$ exactly when $n=0$. We will also assume the
following condition.

######  Condition 2.1.

The jump rate function $g(\cdot)$ satisfies

  1. (1)

Lipschitz: there exists $g^{*}>0$ such that $|g(n+1)-g(n)|\leq g^{*}$ for all
$n\in\mathbb{N}_{0}$.

  2. (2)

Power Interaction: there exists $\alpha\in[0,1]$ such that $g(n)\sim
n^{\alpha}$, that is

$\lim_{n\to\infty}\dfrac{g(n)}{n^{\alpha}}=1.$

  3. (3)

Monotonicity: $g(n)\leq g(n+1)$ for all $n\in\mathbb{N}_{0}$.

When the constant $\alpha=0$, in the power interaction condition, we note the
function $g(n)$ increases to $g_{\infty}:=1$ as $n\to\infty$. We will refer to
this case as ‘$g$ is of bounded type’ or simply, $g$ is bounded. On the other
hand, when $\alpha\in(0,1]$, we will say ‘$g$ is of $n^{\alpha}$ type’.

We remark that, when $g(n)\equiv n$, the dynamics is non-interactive, a
superposition of independent random walks. The Lipschitz condition, which
rules out the case $\alpha>1$, is convenient to bound $g(n)$ by $g^{*}n$, and
in using the entropy inequality in places. Moreover, the assumption that $g$
is increasing, an ‘attractive’ dynamics assumption, allows use of the ‘basic
coupling’ in our proofs; for more discussion of this point; see Section 3.2.

Also, as discussed in the introduction, we will suppose in the $g$ bounded
setting that, at a defect site $k_{j,N}$, the parameter $\beta_{j}\leq 0$.
Such a condition ensures existence of a non-trivial family of invariant
measures; see Section 2.1.

######  Condition 2.2.

In the $g$ bounded setting, we will assume $\beta_{j}\leq 0$ for a defect site
$k=k_{j,N}$.

Our goal in this work is to study hydrodynamic limits of the dynamics
generated by $L_{N}$. In particular, in the limit macroscopic flow, there will
be different behaviors at the defect locations depending on the associated
strength parameters. To prepare for these statements, it will be helpful to
introduce a partition on the index set $J=J_{s}\cup J_{c}\cup J_{b}$. Here,
the subscripts $s$, $c$, and $b$ stand for super-critical or ‘super-slow’,
critical, and sub-critical respectively.

There will be different partitions of $J$ depending on whether $g$ is of
$n^{\alpha}$ type or bounded. When $g$ is of $n^{\alpha}$ type, we let

  * •

$J_{s}:=\\{j\in J:\beta_{j}>\alpha\\}$,

  * •

$J_{c}:=\\{j\in J:\beta_{j}=\alpha\\}$, and

  * •

$J_{b}:=\\{j\in J:\beta_{j}<\alpha\\}$.

When $g$ is bounded, we take

  * •

$J_{s}:=\\{j\in J:\beta_{j}>0\\}$ (= $\emptyset$),

  * •

$J_{c}:=\\{j\in J:\beta_{j}=0,\ \lambda_{j}>1\\}$, and

  * •

$J_{b}:=\\{j\in J:\beta_{j}<0\text{ or }\beta_{j}=0\text{ with
}\lambda_{j}<1\\}$.

In terms of the partition of $J$, the sets ${\mathfrak{D}}$ and
${\mathfrak{D}}_{N}$, the macroscopic and microscopic locations of the
defects, are then divided into corresponding subsets. For example, we have
${\mathfrak{D}}_{s}:=\\{x_{j}\in{\mathfrak{D}}:j\in J_{s}\\}$ and
${\mathfrak{D}}_{s,N}:=\\{k_{j,N}\in{\mathfrak{D}}_{N}:j\in J_{s}\\}$. The
sets ${\mathfrak{D}}_{c}$, ${\mathfrak{D}}_{b}$, ${\mathfrak{D}}_{c,N}$, and
${\mathfrak{D}}_{b,N}$ are defined in the same fashion.

###  2.1. Invariant measures

The construction of invariant measures under $L_{N}$ is based on
$\\{\pazocal{P}_{\phi}\\}$, a family of Poisson-like distributions indexed by
‘fugacities’ $\phi$. In order to define $\pazocal{P}_{\phi}$, we first
introduce the partition function:

$Z(\phi):=\sum_{n=0}^{\infty}\dfrac{\phi^{n}}{g(n)!}.$

Let $r_{g}$ be the convergence radius of $Z(\cdot)$. It holds that
$r_{g}=\lim_{n\to\infty}g(n)$. In particular, when $g$ is of $n^{\alpha}$
type, we have $r_{g}=\infty$, namely, the “FEM” condition (cf. p. 69, [18]) is
satisfied. When $g$ is bounded, $r_{g}=g_{\infty}=1$. In either case, it holds
that $\lim_{\phi\uparrow r_{g}}Z(\phi)=\infty$.

For each $\phi\in[0,r_{g})$, define $\pazocal{P}_{\phi}$ by

$\pazocal{P}_{\phi}(n)=\dfrac{1}{Z(\phi)}\dfrac{\phi^{n}}{g(n)!},\quad\text{for
}n\geq 0.$ (2.3)

Here, $g(0)!:=1$ and $g(n)!:=\prod_{k=1}^{n}g(k)$ for $n\geq 1$. Let
$R(\phi)=E_{\pazocal{P}_{\phi}}[X]$, where $X(n)=n$, be the mean of the
distribution $\pazocal{P}_{\phi}$. A direct computation yields that $R(0)=0$
and $\lim_{\phi\uparrow r_{g}}R(\phi)=\infty$. Moreover, it holds
$R^{\prime}(\phi)=\sigma^{2}(\phi)/\phi$ and $\lim_{\phi\downarrow
0}R^{\prime}(\phi)=1/g(1)$ where $\sigma^{2}(\phi)={\rm
Var}_{\pazocal{P}_{\phi}}[X]$ is the variance of $X$ under
$\pazocal{P}_{\phi}$.

Since $R$ is strictly increasing, it has an inverse, denoted by
$\Phi:[0,\infty)\mapsto[0,r_{g})$. We may parametrize the family of
distributions $\pazocal{P}_{\phi}$ by their means: For $\rho\geq 0$, let
$\pazocal{Q}_{\rho}=\pazocal{P}_{\Phi(\rho)}$, so that
$E_{\pazocal{Q}_{\rho}}[X]=E_{\pazocal{P}_{\Phi(\rho)}}[X]=R(\Phi(\rho))=\rho$.
A straightforward computation yields that $E_{\pazocal{P}_{\phi}}[g(X)]=\phi$
for $\phi\geq 0$. Thus,

$\Phi(\rho)=E_{\pazocal{P}_{\Phi(\rho)}}[g(X)]=E_{\pazocal{Q}_{\rho}}[g(X)],\quad\rho\,\geq\,0.$
(2.4)

As $g(n)\leq g^{*}n$, we have that $\Phi(\rho)\leq g^{*}\rho$. As
$\Phi(\cdot)=R^{-1}(\cdot)$, we have
$\Phi^{\prime}(\rho)=\Phi(\rho)/\sigma^{2}(\Phi(\rho))$. Under our assumptions
on $g$, in fact, it holds that $0\leq\Phi^{\prime}(\rho)\leq g^{*}$ for all
$\rho\geq 0$ (cf. p. 33, [18]). In particular, $\Phi\in C^{1}[0,\infty)$ is an
increasing function with a uniformly bounded derivative. We note, in the case
$g(n)\equiv n$, that $\Phi(\rho)\equiv\rho$ and $\pazocal{P}_{\phi}$ is a
Poisson measure with mean $\phi$.

We now introduce the invariant measures. For each $N$, let

$q_{N}=\max\\{1,\lambda_{j}N^{\beta_{j}}|j\in J\\}.$

For $c$ so that $\Phi(c)\in[0,r_{g}/q_{N})$, denote by $\mathscr{R}_{c}^{N}$
the product measure on $\Omega_{N}$ whose marginals are given by

$\mathscr{R}_{c}^{N}(\xi(k)=n)=\begin{cases}\pazocal{P}_{\Phi(c)}(n)&\text{
for }k\in\mathbb{T}_{N}\setminus{\mathfrak{D}}_{N}\text{ and }n\geq 0,\\\
\pazocal{P}_{\lambda_{j}N^{\beta_{j}}\Phi(c)}(n)&\text{ for }k=k_{j,N},\ j\in
J\text{, and }n\geq 0.\end{cases}$ (2.5)

Notice that the condition $\Phi(c)\in[0,r_{g}/q_{N})$ is needed since the
distributions $\\{\pazocal{P}_{\phi}\\}$ are defined for $\phi\in[0,r_{g})$.
When $g$ is of $n^{\alpha}$ type, as $r_{g}=\infty$, we have that
$\\{\mathscr{R}_{c}^{N}\\}$ are defined for all $c\in[0,\infty)$. However,
when $g$ is bounded, we have $r_{g}=g_{\infty}=1$, and thus the measures
$\\{\mathscr{R}_{c}^{N}\\}$ are defined only for $c\in[0,R(1/q_{N}))$.
Moreover, in the $g$ bounded setting, we have that $r_{g}/q_{N}$ is bounded
away from $0$ for all large $N$ exactly when $J_{s}=\emptyset$, that is
$\beta_{j}\leq 0$, the reason for the assumption $J_{s}=\emptyset$ in this
case (cf. Condition 2.2).

With $\mathscr{R}_{c}^{N}$ defined, it is straightforward (cf. [1], [11]) to
check the following lemma.

######  Lemma 2.3.

For $c$ so that $\Phi(c)\in[0,r_{g}/q_{N})$, $\mathscr{R}_{c}^{N}$ is
invariant and reversible with respect to the generator $L_{N}$ in (2.1).

###  2.2. Static limit

Before studying the hydrodynamic limits, it will be useful to understand the
particle mass behavior under an invariant measure $\mathscr{R}^{N}_{c}$. For a
configuration $\xi\in\Omega_{N}$, define the associated scaled mass empirical
measure:

$\widehat{\pi}^{N}(dx):=\dfrac{1}{N}\sum_{k\in\mathbb{T}_{N}}\xi(k)\delta_{k/N}(dx).$
(2.6)

In this formulation, each particle has mass $N^{-1}$. Here and in the sequel,
$\delta_{z}$ refers to a delta point mass at $z$.

For a test function $G\in C(\mathbb{T})$, let

$\langle
G,\widehat{\pi}^{N}\rangle:=N^{-1}\sum_{k\in\mathbb{T}_{N}}\xi(k)G(k/N).$

More generally, we will use the notation $\langle G,\mu\rangle:=\int Gd\mu$.
We now compute the limit of $\langle G,\widehat{\pi}^{N}\rangle$ with respect
to a sequence of invariant measures $\mathscr{R}_{c}^{N}$ with $c$ fixed as
$N\to\infty$.

We assume first $g$ is of $n^{\alpha}$ type and $c>0$. Because of the product
structure of $\mathscr{R}_{c}^{N}$, $\\{\xi(k)\\}_{k\in\mathbb{T}_{N}}$ are
independent and have a commmon marginal $\pazocal{P}_{\Phi(c)}$ for all
$k\neq{\mathfrak{D}}_{N}$. As $\pazocal{P}_{\Phi(c)}$ has expectation $c$ and
finite variance, we have $N^{-1}\sum_{k\notin{\mathfrak{D}}_{N}}\xi(k)G(k/N)$
converges in probability to $\int_{\mathbb{T}}G(x)c\,dx$ as $N\to\infty$.

It remains to investigate the behavior of $N^{-1}\xi(k)$ for
$k=k_{j,N}\in{\mathfrak{D}}_{N}$. As $\xi(k_{j,N})$ has distribution
$\pazocal{P}_{\lambda_{j}N^{\beta_{j}}\Phi(c)}$, by the later Lemma 2.6, for
all $\beta_{j}>0$,

$E_{\mathscr{R}_{c}^{N}}[\xi(k_{j,N})]\sim(\lambda_{j}\Phi(c))^{1/\alpha}N^{\beta_{j}/\alpha},\quad\text{and}\quad{\rm
Var}_{\mathscr{R}_{c}^{N}}[\xi(k_{j,N})]=o(N^{2\beta_{j}/\alpha}).$

Then, according to the value of $\beta_{j}$, there are three different types
of behaviors at $k\in{\mathfrak{D}}_{N}$:

  1. (1)

if $\beta_{j}<\alpha$, $N^{-1}\xi(k_{j,N})\to 0$ in probability;

  2. (2)

if $\beta_{j}=\alpha$,
$N^{-1}\xi(k_{j,N})\to\left(\lambda_{j}\Phi(c)\right)^{1/\alpha}$ in
probability;

  3. (3)

if $\beta_{j}>\alpha$,
$N^{-\beta_{j}/\alpha}\xi(k_{j,N})\to\left(\lambda_{j}\Phi(c)\right)^{1/\alpha}$
in probability.

In other words, the defect site $k_{j,N}$ becomes macroscopically invisible
when $\beta_{j}<\alpha$ as typically it contains $o(N)$ number of particles
and each particle has mass $N^{-1}$. In the case $\beta_{j}=\alpha$, typically
the number of particles at $k_{j,N}$ is of order $N$ and a delta mass of
magnitude $\left(\lambda_{j}\Phi(c)\right)^{1/\alpha}$ emerges. When the site
is super-slow, that is $\beta_{j}>\alpha$, the particle number at $k_{j,N}$ is
of order $N^{\beta_{j}/\alpha}$, corresponding to an infinite macroscopic
mass. Recall, the partition $J=J_{b}\cup J_{c}\cup J_{s}$ in Section 2 matches
with this classification of the $\beta_{j}$’s.

As the macroscopical mass at $x_{j}\in{\mathfrak{D}}_{s}$ explodes, to
consider the remaining mass, we define microscopic empirical measures which
exclude the super-critical defect set ${\mathfrak{D}}_{s,N}$:

$\pi^{N}(dx):=\dfrac{1}{N}\sum_{k\in\mathbb{T}_{N}\setminus{\mathfrak{D}}_{s,N}}\xi(k)\delta_{k/N}(dx).$
(2.7)

Then, we may summarize the above discussion as follows.

######  Proposition 2.4.

Assume $g$ is of $n^{\alpha}$ type. Then, for any $G\in C(\mathbb{T})$, $c\geq
0$, and $\delta>0$:

$\lim_{N\to\infty}\mathscr{R}_{c}^{N}\big{[}\,\big{|}\langle
G,\pi^{N}\rangle-\langle G,\pi\rangle\big{|}>\delta\,\big{]}=0$

where $\pi(dx)=c\,dx+\sum_{j\in
J_{c}}\left(\lambda_{j}\Phi(c)\right)^{1/\alpha}\delta_{x_{j}}(dx)$. Moreover,
for all $j\in{\mathfrak{D}}_{s}$, we have

$\lim_{N\to\infty}\mathscr{R}_{c}^{N}\big{[}\,\big{|}N^{-\beta_{j}/\alpha}\xi(k_{j,N})-\left(\lambda_{j}\Phi(c)\right)^{1/\alpha}\big{|}>\delta\,\big{]}=0.$

Now we turn to the case when $g$ is bounded. Recall, in this case,
$\mathscr{R}_{c}^{N}$ is defined for $c$ such that $\Phi(c)\in[0,1/q_{N})$
where $q_{N}=\max\\{1,\lambda_{j}N^{\beta_{j}}|j\in J\\}$. Recall also, in
this setting, that $\beta_{j}\leq 0$, that is $J_{s}=\emptyset$. When
$J_{c}\neq\emptyset$, we will define $\lambda_{\max}:=\max_{j\in
J_{c}}\\{\lambda_{j}\\}$. Otherwise, when $J_{c}=\emptyset$, we will take
$\lambda_{\max}=1$. Hence, the domain for $c$ so that $\mathscr{R}_{c}^{N}$ is
defined for all $N$ is $[0,R(1/\lambda_{\max})$.

For $c\in[0,R(1/\lambda_{\max}))$, we also have
$N^{-1}\sum_{k\notin{\mathfrak{D}}_{N}}\xi(k)G(k/N)$ converges in probability
to $\int_{\mathbb{T}}G(x)c\,dx$ as $N\to\infty$. Given $\xi(k)$ has finite
$\mathscr{R}_{c}^{N}$-expectation and variance for all
$k\in{\mathfrak{D}}_{N}$, the following is easily obtained.

######  Proposition 2.5.

Assume $g$ is bounded and $J_{s}=\emptyset$. Then, for
$c\in[0,R(1/\lambda_{\max}))$, for any $G\in C(\mathbb{T})$ and $\delta>0$, we
have

$\lim_{N\to\infty}\mathscr{R}_{c}^{N}\big{[}\,\big{|}\langle
G,\pi^{N}\rangle-\int_{\mathbb{T}}G(x)c\,dx\big{|}>\delta\,\big{]}=0.$

We finish this section with a technical lemma used in Proposition 2.4.

######  Lemma 2.6.

Assume $g(k)\sim k^{\alpha}$ for some $\alpha\in(0,1]$. For each $\varphi>0$,
let $X$ be a random variable with distribution $\pazocal{P}_{\varphi}$ (cf.
(2.3)). Then, for each $n\in\mathbb{N}$, $E[X^{n}]\sim\varphi^{n/\alpha}$ as
$\varphi\to\infty$. As a result, $\ln Z(\varphi)\sim\alpha\varphi^{1/\alpha}$
and ${\rm Var}[X]=o\big{(}\varphi^{2/\alpha}\big{)}$ as $\varphi\to\infty$.

###### Proof.

We first show $E[X^{n}]\sim\varphi^{n/\alpha}$ for all $n\in\mathbb{N}$. To
this end, let us assume for now

$\sum_{k=1}^{\infty}\dfrac{k^{n\alpha}\varphi^{k}}{g(k)!}\sim\sum_{j=n}^{\infty}\dfrac{\varphi^{k}}{g(j-n)!}.$
(2.8)

Let $Y=X^{\alpha}$. Then,

$E[Y^{n}]=\dfrac{1}{Z(\varphi)}\sum_{k=1}^{\infty}\dfrac{k^{n\alpha}\varphi^{k}}{g(k)!}\sim\dfrac{1}{Z(\varphi)}\sum_{k=n}^{\infty}\dfrac{\varphi^{k}}{g(k-n)!}=\varphi^{n}.$

As $\alpha\in(0,1]$, we may find $p\in[0,1)$ and $l\in\mathbb{N}$ such that
$\alpha^{-1}=p+(1-p)l$. Also, since $E[X^{n}]=E[Y^{n/\alpha}]$, by Jensen’s
and Hölder’s inequalities,

$E[Y^{n}]^{1/\alpha}\leq E[X^{n}]\leq E[Y^{n}]^{p}E[Y^{nl}]^{1-p}.$

Since $E[Y^{n}]\sim\varphi^{n}$ and $E[Y^{nl}]\sim\varphi^{nl}$, we obtain
$E[X^{n}]\sim\varphi^{n/\alpha}$.

For the limit behavior of $E[X^{n}]$, it remains to show the claim (2.8). As
$g(k)\sim k^{\alpha}$, for any $A>0$, we may find $\lambda_{1}=\lambda_{1}(A)$
and $\lambda_{2}=\lambda_{2}(A)$, such that $\lambda_{1}k^{\alpha}\leq
g(k)\leq\lambda_{2}k^{\alpha}$ for all $k\geq A$ and
$\lim_{A\to\infty}\lambda_{1}=\lim_{A\to\infty}\lambda_{2}=1$. Then, for all
$k\geq A+n$,

$\lambda_{2}^{-n}\leq\dfrac{k^{n\alpha}}{\prod_{k-n<k^{\prime}\leq
k}g(k^{\prime})}\leq\Big{(}\dfrac{A+n}{A}\Big{)}^{n}\lambda_{1}^{-n}.$

Therefore,

$\lambda_{2}^{-n}\sum_{k=A+n}^{\infty}\dfrac{\varphi^{k}}{g(k-n)!}\leq\sum_{k=A+n}^{\infty}\dfrac{k^{n\alpha}\varphi^{k}}{g(k)!}\leq\Big{(}\dfrac{A+n}{A}\Big{)}^{n}\lambda_{1}^{-n}\sum_{k=A+n}^{\infty}\dfrac{\varphi^{k}}{g(k-n)!}$

Notice that, if $\varphi$ is sent to infinity in the above display, we may
replace $\sum_{k\geq A+n}$ by either $\sum_{k\geq 0}$ or $\sum_{k\geq n}$.
Then the claim (2.8) follows from taking $\varphi\to\infty$ and then
$A\to\infty$.

We have shown $E[X^{n}]\sim\varphi^{n/\alpha}$ for all $n\in\mathbb{N}$. Then,
it follows that ${\rm Var}[X]=o\big{(}\varphi^{2/\alpha}\big{)}$ as
$E[X^{2}]\sim{E[X]}^{2}\sim\varphi^{2/\alpha}$. To prove $\ln
Z(\varphi)\sim\alpha\varphi^{1/\alpha}$, notice $\dfrac{d}{d\varphi}\ln
Z(\varphi)=\varphi^{-1}E[X]\sim\varphi^{1/\alpha-1}$ and then apply
L’Hospital’s rule, to finish the argument. ∎

##  3\. Initial measures

In this section, we specify the assumptions on the initial measures
$\\{\mu^{N}\\}$ we use to start our dynamics. Roughly speaking,
$\\{\mu^{N}\\}$ should be associated with a macroscopic profile which gives
the initial condition for the hydrodynamic limit. We will also require
$\mu^{N}$ to possess certain relative entropy estimates and to be
stochastically bounded with respect to invariant measures.

To specify these conditions, recall, for $\mu$, $\nu$, two probability
measures on $\Omega_{N}$, we say that $\mu\leq\nu$, that is $\mu$ is
stochastically bounded by $\nu$, if for all $f:\Omega_{N}\mapsto\mathbb{R}$
coordinately increasing, we have $E_{\mu}(f)\leq E_{\nu}(f)$. Fix also $\pi$,
a nonnegative measure on $\mathbb{T}$, such that

$\pi(dx)=\rho_{0}(x)dx+\sum_{j\in
J_{c}}{\mathfrak{m}}_{0,j}\,\delta_{x_{j}}(dx)$ (3.1)

where $\rho_{0}(x)\in L^{1}(\mathbb{T})$ and ${\mathfrak{m}}_{j,0}\geq 0$ for
$j\in J_{c}$. Recall also, the empirical measure $\pi^{N}$ defined in (2.7).

Throughout this work, we will assume the following on the sequence of initial
measures $\\{\mu^{N}\\}_{N\in\mathbb{N}}$ on $\Omega_{N}$.

######  Condition 3.1.

The following hold:

  1. (1)

$\\{\mu^{N}\\}_{N\in\mathbb{N}}$ has macroscopic profile $\pi$ on
$\mathbb{T}\setminus{\mathfrak{D}}_{s}$, i.e. for all $G(x)\in C(\mathbb{T})$
and $\delta>0$,

$\lim_{N\to\infty}\mu^{N}\Big{[}\,\Big{|}\langle G,\pi^{N}\rangle-\langle
G,\pi\rangle\Big{|}>\delta\,\Big{]}=0.$

  2. (2)

There exists $c_{0}>0$ such that the relative entropy of $\mu^{N}$ with
respect to $\mathscr{R}_{c_{0}}^{N}$ is of order $N$: Let
$f_{N}:=d\mu^{N}/d\mathscr{R}_{c_{0}}^{N}$. Then,
$H(\mu^{N}|\mathscr{R}_{c_{0}}^{N}):=\int f_{N}\ln
f_{N}d\mathscr{R}_{c_{0}}^{N}=O(N)$.

  3. (3)

When $g$ is of $n^{\alpha}$ type, there exists $c^{\prime}$ such that
$\mu^{N}$ is stochastically bounded by $\mathscr{R}_{c^{\prime}}^{N}$ for all
$N$.

  4. (3′)

When $g$ is bounded, there exists $c^{\prime}$ such that, when restricted to
$\mathbb{T}_{N}\setminus{\mathfrak{D}}_{c,N}$, $\mu^{N}$ is stochastically
bounded by $\kappa_{c^{\prime}}^{N}$, that is $E_{\mu^{N}}[f]\leq
E_{\kappa_{c^{\prime}}^{N}}[f]$ for coordinate increasing functions supported
on $\\{\eta(k):k\in\mathbb{T}_{N}\setminus{\mathfrak{D}}_{c,N}\\}$. Here,

$\kappa_{c^{\prime}}^{N}=\prod_{k\in\mathbb{T}_{N}\setminus{\mathfrak{D}}_{b,N}}\pazocal{P}_{\Phi(c^{\prime})}\times\prod_{k=k_{j,N}\in{\mathfrak{D}}_{b,N}}\pazocal{P}_{\lambda_{j}N^{\beta_{j}}\Phi(c^{\prime})}.$

We comment that item (3) is sufficiently general in the $n^{\alpha}$ setting
to allow $\\{\mu^{N}\\}$ to be associated with profiles of the form
$\pi_{0}(dx)=\rho_{0}(x)dx+\sum_{j\in
J_{c}}\mathfrak{m}_{0,j}\delta_{x_{j}}(dx)$, for densities $\rho_{0}$ and
masses $\\{\mathfrak{m}_{0,j}\\}$, as demonstrated in Section 3.3. In the
bounded $g$ case, however, if $\\{\mu^{N}\\}$ satisfied item (3), then $O(N)$
accumulations at points ${\mathfrak{D}}_{c,N}$ would not be allowed. In this
case, the only profiles allowed would be of form $\pi_{0}(dx)=\rho_{0}(x)dx$,
where $\|\Phi(\rho_{0})\|_{\infty}\leq\min_{j\in J_{c}}\tfrac{1}{\lambda_{j}}$
(cf. Section 3.2). As mentioned in the introduction, hydrodynamic evolution
from such profiles would not see any defects.

In this context, item (3′) is formulated so that, with respect to
$\\{\mu^{N}\\}$, $O(N)$ accumulations are possible on ${\mathfrak{D}}_{c,N}$
as well as later point masses on ${\mathfrak{D}}_{c}$ in the hydrodynamic
evolution.

In fact, conditions $(3)$ and $(3^{\prime})$ can be made to accommodate a
larger class of initial measures $\\{\mu^{N}\\}$. For example, one may remove
stochastic bounded assumptions on coordinates in ${\mathfrak{D}}_{b,N}$. We
have however chosen to state the $(3)$ and $(3^{\prime})$ in the forms given
to streamline arguments and avoid more piecemeal calculations.

Define now $\mu_{t}^{N}$ as the distribution of the process $\xi_{t}$, with
initial measure $\mu^{N}$.

In the rest of this section, we make remarks on the consequences of the
relative entropy bound in terms of $\mu^{N}_{t}$-particle numbers, discuss the
use of attractiveness, and also provide a large class of examples of $\mu^{N}$
which satisfy Condition 3.1.

###  3.1. $\mu^{N}_{t}$-particle numbers and relative entropy

We first comment on the $\mu_{t}^{N}$-particle numbers in the system. Denote
$H_{N}:=H(\mu^{N}|\mathscr{R}_{c_{0}}^{N})=O(N)$. Notice that the entropy does
not increase in $t$, that is $H(\mu_{t}^{N}|\mathscr{R}_{c_{0}}^{N})\leq
H_{N}$ for $t\geq 0$ (cf. pp 340, [18]).

Assume now that $g$ is of $n^{\alpha}$ type. By the entropy inequality
$E_{\mu}[f]\leq H(\mu|\nu)+\ln E_{\nu}[e^{f}]$ (cf. p.338 [18]),

$E_{\mu_{t}^{N}}\Big{[}\sum_{k\notin{\mathfrak{D}}_{s,N}}\xi(k)\Big{]}\leq
H(\mu_{t}^{N}|\mathscr{R}_{c_{0}}^{N})+\ln
E_{\mathscr{R}_{c_{0}}^{N}}\big{[}e^{\sum_{k\notin{\mathfrak{D}}_{s,N}}\xi(k)}\big{]}.$

As $\mathscr{R}_{c_{0}}^{N}$ is product measure, the $\ln$ term is written as
$\sum_{k\notin{\mathfrak{D}}_{s,N}}\ln
E_{\mathscr{R}_{c_{0}}^{N}}[e^{\xi(k)}]$ which is $O(N)$ by Lemma 2.6. Thus,
the condition $H_{N}=O(N)$ specifies $O(N)$ particles on the sites
$\mathbb{T}_{N}\setminus{\mathfrak{D}}_{s,N}$ for times $t\geq 0$.

On the other hand, at a super-slow site $k_{j,N}\in{\mathfrak{D}}_{s,N}$, the
$\mathscr{R}_{c_{0}}^{N}$-particle number is typically of order
$O(N^{\beta_{j}/\alpha})$, as discussed in Section 2.2. Then, because
$H_{N}=O(N)$, as consequence of the entropy inequality, we may conclude, with
respect to $\mu_{t}^{N}$, that $N^{-\beta_{j}/\alpha}\xi(k_{j,N})$ converges
in probability to $\left(\lambda_{j}\Phi(c_{0})\right)^{1/\alpha}$; see near
(7.6) for a proof.

Therefore, the net exchange of particle numbers between super-slow sites
${\mathfrak{D}}_{s,N}$ and the rest of the system is of order $N$ and the
total particle number on $\mathbb{T}_{N}\setminus{\mathfrak{D}}_{s,N}$ remains
$O(N)$ for all times $t\geq 0$.

We now turn to case when $g$ is bounded. Since in this setting
$J_{s}=\emptyset$, that is ${\mathfrak{D}}_{s}=\emptyset$ (Condition 2.2), by
the previous entropy inequality discussion, it follows that the total number
of particles in $\mathbb{T}_{N}$ is of order $N$ for all times $t\geq 0$.

###  3.2. Basic coupling

We discuss the use of the stochastic boundedness assumption (3) and (3′) in
Condition 3.1. Since $g(n)$ is an increasing function in $n$, the dynamics
generated by $L_{N}$ is ‘attractive’: if initially $\xi_{0}$ is distributed
according to measures $\mu\leq\nu$, then we have $\mu_{t}\leq\nu_{t}$ for all
$t\geq 0$ where $\mu_{t}$ and $\nu_{t}$ are distributions of $\xi_{t}$ (cf.
[1], Chapter II in [22]).

In the case when $g$ of $n^{\alpha}$ type, as $\mathscr{R}_{c^{\prime}}^{N}$
is invariant, the assumption $\mu^{N}\leq\mathscr{R}_{c^{\prime}}^{N}$ implies
that $\mu_{t}^{N}\leq\mathscr{R}_{c^{\prime}}^{N}$ for all $t\geq 0$. However,
when $g$ is bounded, the domain for $c$ in $\mathscr{R}_{c}^{N}$ is
$c<R(1/\lambda_{\max})$ (cf. Proposition 2.5). Then, an assumption
$\mu^{N}\leq\mathscr{R}_{c^{\prime}}^{N}$ would imply that $\pi$, the
macroscopic profile associated to $\mu^{N}$, is $\pi(dx)=\rho_{0}(x)dx$ with
$\|\rho_{0}\|_{\infty}<R(1/\lambda_{\max})<\infty$. To accommodate more
initial profiles $\pi$ (and observe more involved limit evolutions), we have
assumed $(3^{\prime})$ in Condition 3.1 instead, that is, for all coordinately
increasing $f:\Omega_{N}\mapsto\mathbb{R}$ depending only on
$\\{\xi(k)\\}_{k\notin{\mathfrak{D}}_{c,N}}$, we have
$E_{\mu_{t}^{N}}[f(\xi)]\leq E_{\kappa_{c^{\prime}}^{N}}[f(\xi)]$.

We now illustrate how we will apply attractiveness under assumption
$(3^{\prime})$. Instead of an evolution with respect to particle numbers in
$\mathbb{N}_{0}$ and configurations in $\Omega_{N}^{\mathbb{T}_{N}}$, we may
consider a dynamics corresponding to
$\overline{\mathbb{N}}_{0}:=\mathbb{N}_{0}\cup\\{\infty\\}$ and
$\overline{\Omega}_{N}:=\overline{\mathbb{N}}_{0}^{\mathbb{T}_{N}}$. Recall
the constant $c^{\prime}$ in $(3^{\prime})$. Define
$\overline{g}_{k,N}(\cdot)$ by $\overline{g}_{k,N}(n)=g_{k,N}(n)$ for
$n\in\mathbb{N}_{0}$ and $\overline{g}_{k,N}(\infty)=\Phi(c^{\prime})$.
Consider now the following generator on $\overline{\Omega}_{N}$

$\overline{L}_{N}f(\xi)=\sum_{k\in\mathbb{T}_{N}}\Big{\\{}\overline{g}_{k,N}(\xi(k))\big{(}f(\xi^{k,k+1})-f(\xi)\big{)}+\overline{g}_{k,N}(\xi(k))\big{(}f(\xi^{k,k-1})-f(\xi)\big{)}\Big{\\}}.$

Here, $\xi^{x,y}$ is defined as in (2.2) by using the convention $\infty\pm
1=\infty$. When starting with configurations in $\Omega_{N}$,
$\overline{L}_{N}$ coincides with $L_{N}$. Once a site starts with $\infty$
particles, it will serve as a ‘reservoir’ which pumps particles into its
neighbors at the rate of $\Phi(c^{\prime})$. Let $\delta_{\infty}$ be the
Dirac measure on the extended number $\infty$. Define
$\overline{\kappa}_{c^{\prime}}^{N}$ as the product measure on
$\overline{\Omega}_{N}$ that coincides with $\kappa_{c^{\prime}}^{N}$ on
$k\notin{\mathfrak{D}}_{c,N}$ and has marginal $\delta_{\infty}$ for
$k\in{\mathfrak{D}}_{c,N}$. It is straightforward to check that
$\overline{\kappa}_{c^{\prime}}^{N}$ is invariant under $\overline{L}_{N}$.
Also, we have that $\mu^{N}\leq\overline{\kappa}_{c^{\prime}}^{N}$ by
$(3^{\prime})$ in Condition 3.1. As attractiveness and the basic coupling are
still in effect for $\overline{L}_{N}$, we obtain that the later time $t$
distribution satisfies $\mu_{t}^{N}\leq\overline{\kappa}_{c^{\prime}}^{N}$ for
any $t\geq 0$. In particular, for a coordinate-increasing function $f(\xi)$
which does not depend on $\\{\xi(k)\\}_{k\in\mathscr{D}_{c,N}}$, we have
$E_{\mu^{t}_{N}}[f]\leq
E_{\overline{\kappa}^{N}_{c^{\prime}}}[f]=E_{\kappa^{N}_{c^{\prime}}}[f]$.

Finally, as a general remark, we make technical use of ‘attractiveness’ most
essentially in the cutoff of large densities in the local replacements (Lemma
8.1 and Lemma 8.2), and in proving the limit mass profile has $L^{2}$
absolutely continuous part (Lemma 7.1). The latter property is needed in the
proof of uniqueness of weak solutions (Theorem 11.1 and 11.2).

###  3.3. Local equilibria

We now give explicit examples of initial measures that satisfy the Condition
3.1. These examples will be denoted by $\mu^{N}_{\text{le}}$ as they are
related with the usual ‘local equilibrium’ measures in setting without defects
(cf. [18]).

Let $\pi$ be as in (3.1) with $\rho_{0}(x)\in L^{\infty}(\mathbb{T})$ and
${\mathfrak{m}}_{0,j}\geq 0$ for each $j\in J_{c}$. For each
$k\in\mathbb{T}_{N}$, define

$\rho_{k,N}=N\int_{(k-1)/N}^{k/N}\rho_{0}(x)dx.$

We will construct $\\{\mu^{N}\\}$ separately for $g$ of $n^{\alpha}$ and
bounded types. Consider first $g$ of $n^{\alpha}$ type. Fix $c_{0}>0$ and
define

$\varphi_{k,N}=\begin{cases}\Phi(\rho_{k,N})&k\in\mathbb{T}_{N}\setminus{\mathfrak{D}}_{N}\\\
0&k=k_{j,N}\in{\mathfrak{D}}_{b,N}\\\
\left(N{{\mathfrak{m}}_{0,j}}\right)^{\alpha}&k=k_{j,N}\in{\mathfrak{D}}_{c,N}\\\
\lambda_{j}N^{\beta_{j}}\Phi(c_{0})&k=k_{j,N}\in{\mathfrak{D}}_{s,N}.\end{cases}$

For each $N\in\mathbb{N}$, let $\mu^{N}_{\text{le}}$ be the product measure on
$\Omega_{N}$ with marginals given by

$\mu^{N}_{\text{le}}(\xi(k)=n)\;=\;\pazocal{P}_{\varphi_{k,N}}(n),\quad\text{for
}k\in\mathbb{T}_{N}\,,\,n\geq 0.$

######  Lemma 3.2.

Suppose $g$ is of $n^{\alpha}$ type. Then, $\\{\mu^{N}=\mu^{N}_{\text{le}}\\}$
satisfies Condition 3.1.

###### Proof.

Let $c^{\prime}$ be such that
$\Phi(c^{\prime})=\max\\{\Phi\left(\|\rho_{0}\|_{\infty}\right),({\mathfrak{m}}_{0,j})^{\alpha},\Phi(c_{0})\\}_{j\in
J_{c}}$. As $\pazocal{P}_{\phi_{1}}\leq\pazocal{P}_{\phi_{2}}$ if
$\phi_{1}\leq\phi_{2}$ (cf. [22], pp. 32), we have that the product measure
$\mu^{N}_{\text{le}}$ is stochastically bounded by
$\mathscr{R}_{c^{\prime}}^{N}$ for all $N$. As $G$ is uniformly continuous,
that $\mu^{N}_{\text{le}}$ is associated with the given macroscopic profile
$\pi$ holds straightforwardly from Chebychev’s inequality and Lemma 2.6.

It remains to check the desired entropy bound
$H(\mu^{N}_{\text{lc}}|\mathscr{R}_{c_{0}}^{N})=O(N)$. Let
$\varphi=\Phi(c_{0})$. We compute

$H(\mu^{N}_{\text{le}}|\mathscr{R}_{c_{0}}^{N})=\sum_{j\in
J}H(\pazocal{P}_{\varphi_{k_{j,N},N}}|\pazocal{P}_{\lambda_{j}N^{\beta_{j}}\varphi})+\sum_{k\in\mathbb{T}_{N}\setminus{\mathfrak{D}}_{N}}H(\pazocal{P}_{\varphi_{k,N}}|\pazocal{P}_{\varphi}).$

Note that, for any $\phi_{1}\geq 0$ and $\phi_{2}>0$,

$H(\pazocal{P}_{\phi_{1}}|\pazocal{P}_{\phi_{2}})=\ln\dfrac{\phi_{1}}{\phi_{2}}E_{\pazocal{P}_{\phi_{1}}}[X]+\ln\dfrac{Z(\phi_{2})}{Z(\phi_{1})}$

where $X$ is defined as $X(n)=n$. We also adopt $0\ln 0=0$ in the case
$\phi_{1}=0$. By Lemma 2.6, we conclude the desired relative entropy bound
$H(\mu^{N}_{\text{le}}|\mathscr{R}_{c_{0}}^{N})=O(N)$. ∎

We now consider the $g$ bounded case. Note by assumption (Condition 2.2) that
$J_{s}=\emptyset$ in this setting. Let
$\tilde{\varphi}_{k,N}=\Phi(\rho_{k,N})$ for
$k\in\mathbb{T}_{N}\setminus{\mathfrak{D}}_{N}$ and $\tilde{\varphi}_{k,N}=0$
for $k\in{\mathfrak{D}}_{b,N}$. Also, let $\pazocal{P}^{\prime}_{\lambda}$
denote the Poisson distribution with mean $\lambda$.

We define $\tilde{\mu}_{\text{le}}^{N}$ as the product measure with marginals
$\pazocal{P}_{\tilde{\varphi}_{k,N}}$ on sites $k\notin{\mathfrak{D}}_{c,N}$
and $\pazocal{P}^{\prime}_{{\mathfrak{m}}_{0,j}N}$ for
$k=k_{j,N}\in{\mathfrak{D}}_{c,N}$. It is straightforward that $(1)$ and
$(3^{\prime})$ in Condition 3.1 hold with
$\mu^{N}=\tilde{\mu}_{\text{le}}^{N}$. The choice of Poisson distributions at
sites ${\mathfrak{D}}_{c,N}$ allows for some explicit computation.

Fix any $c_{0}\in(0,R(1/\lambda_{\max}))$, we now argue that
$H(\tilde{\mu}_{\text{le}}^{N}|\mathscr{R}_{c_{0}}^{N})=O(N)$. It suffices to
check that $H(\pazocal{P}^{\prime}_{aN}|\pazocal{P}_{\varphi})=O(N)$ for any
fixed $a\geq 0$ and $\varphi\in(0,1)$, where $g_{\infty}=1$. To see this, let
$f_{N}(n)=\pazocal{P}^{\prime}_{aN}(X=n)$ and
$f(n)=\pazocal{P}_{\varphi}(X=n)$ where $X(n)=n$. Then
$H(\pazocal{P}^{\prime}_{aN}|\pazocal{P}_{\varphi})=E_{\pazocal{P}^{\prime}_{aN}}[\ln
f_{N}(X)]-E_{\pazocal{P}^{\prime}_{aN}}[\ln f(X)]$. The term
$E_{\pazocal{P}^{\prime}_{aN}}[\ln f_{N}]$ is computed as $aN\ln(aN)-aN-
E_{\pazocal{P}^{\prime}_{aN}}[\ln X!]$. By Stirling’s formula,
$n!\geq\sqrt{2\pi n}e^{-n}n^{n}\geq e^{-n}n^{n}$, and Jensen’s inequality, we
have $E_{\pazocal{P}^{\prime}_{aN}}[\ln X!]\geq
E_{\pazocal{P}^{\prime}_{aN}}[X\ln X-X]\geq aN\ln(aN)-aN$. Therefore, we have
$E_{\pazocal{P}^{\prime}_{aN}}[\ln f_{N}]$ is $O(N)$. For the term
$E_{\pazocal{P}^{\prime}_{aN}}[\ln f]$, we may write it as $aN\ln\varphi-\ln
Z(\varphi)-E_{\pazocal{P}^{\prime}_{aN}}[\ln g(X)!]$. Note that $g(X)\leq 1$,
so that $E_{\pazocal{P}^{\prime}_{aN}}[\ln f]=O(N)$. Hence,
$H(\tilde{\mu}_{\text{le}}^{N}|\mathscr{R}_{c_{0}}^{N})=O(N)$.

We now summarize the above calculations.

######  Lemma 3.3.

Suppose $g$ is bounded. Then $\mu^{N}=\tilde{\mu}^{N}_{\text{le}}$ satisfies
Condition 3.1.

##  4\. Results

Suppose that $\\{\mu^{N}\\}$ satisfies Condition 3.1. Consequently, $\mu^{N}$
has macroscopic profile

$\pi(dx)=\rho_{0}(x)dx+\sum_{j\in
J_{c}}{\mathfrak{m}}_{0,j}\,\delta_{x_{j}}(dx)$

on $\mathbb{T}\setminus{\mathfrak{D}}_{s}$ and we have
$H(\mu^{N}|\mathscr{R}_{c_{0}}^{N})=O(N)$ for some $c_{0}\geq 0$.

On $\mathbb{T}_{N}$, we will observe the evolution of the zero-range process
speeded up by $N^{2}$, in diffusive scale. Denote the process
$\eta_{t}:=\xi_{N^{2}t}$, generated by $N^{2}L_{N}$ (cf. (2.1)), for times
$0\leq t\leq T$. Here, $T>0$ refers to a fixed time horizon. We will access
the space-time structure of the process through the scaled mass empirical
measure:

$\pi_{t}^{N}(dx):=\dfrac{1}{N}\sum_{k\in\mathbb{T}_{N}\setminus{\mathfrak{D}}_{s,N}}\eta_{t}(k)\delta_{k/N}(dx).$
(4.1)

Throughout, we will view each $\pi^{N}_{t}$ as a member of $\pazocal{M}$, the
space of finite nonnegative measures on
$\mathbb{T}\setminus{\mathfrak{D}}_{s}$. We will place a metric
$d(\cdot,\cdot)$ on $\pazocal{M}$ which realizes the vague convergence on
$\mathbb{T}\setminus{\mathfrak{D}}_{s}$, (see Section 6 for a definitive
choice). Here, the trajectories $\\{\pi^{N}_{t}:0\leq t\leq T\\}$ are elements
of the Skorokhod space $D([0,T],\pazocal{M})$, endowed with the associated
Skorokhod topology.

In the following, the process measure and associated expectation governing
$\eta_{\cdot}$ starting from $\mu$ will be denoted by $\mathbb{P}_{\mu}$ and
$\mathbb{E}_{\mu}$. When the process starts from
$\\{\mu^{N}\\}_{N\in\mathbb{N}}$, in the class satisfying Condition 3.1, we
will denote by $\mathbb{P}_{N}:=\mathbb{P}_{\mu^{N}}$ and
$\mathbb{E}_{N}:=\mathbb{E}_{\mu^{N}}$, the associated process measure and
expectation.

###  4.1. Hydrodynamic limits

We now state our main results. Recall the initial profile $\pi$ and the
constant $c_{0}$ from the beginning of this section.

######  Theorem 4.1.

Asssume $g$ is of $n^{\alpha}$ type. Then, for any $t>0$, test function $G\in
C(\mathbb{T})$, and $\delta>0$, we have

$\lim_{N\to\infty}\mathbb{P}_{N}\Big{[}\Big{|}\langle
G,\pi^{N}_{t}\rangle-\langle G,\pi_{t}\rangle\Big{|}>\delta\Big{]}=0,$ (4.2)

where $\pi_{t}(dx)=\rho(t,x)dx+\sum_{j\in
J_{c}}{\mathfrak{m}}_{j}(t)\,\delta_{x_{j}}(dx)$ is the unique weak solution
to

$\begin{cases}\partial_{t}\pi_{t}=\partial_{xx}\Phi(\rho(t,x)),&x\in\mathbb{T}\setminus{\mathfrak{D}}_{s},\
t\in(0,T),\\\
\pi_{t}|_{t=0}=\pi,\quad\Phi(\rho(t,x_{j}))=\Phi(c_{0}),&t\in(0,T),\ j\in
J_{s},\\\
{\mathfrak{m}}_{j}(t)=\left(\lambda_{j}\Phi(\rho(t,x_{j}))\right)^{1/\alpha},&t\in(0,T),\,j\in
J_{c}.\end{cases}$ (4.3)

######  Theorem 4.2.

Assume $g$ is bounded. Then, for any $t>0$, test function $G\in
C(\mathbb{T})$, and $\delta>0$, we have

$\lim_{N\to\infty}\mathbb{P}_{N}\Big{[}\Big{|}\langle
G,\pi^{N}_{t}\rangle-\langle G,\pi_{t}\rangle\Big{|}>\delta\Big{]}=0,$ (4.4)

where $\pi_{t}(dx)=\rho(t,x)dx+\sum_{j\in
J_{c}}{\mathfrak{m}}_{j}(t)\,\delta_{x_{j}}(dx)$ is the unique weak solution
to

$\begin{cases}\partial_{t}\pi_{t}=\partial_{xx}\Phi(\rho(t,x)),&x\in\mathbb{T},\
t\in(0,T),\\\ \pi_{t}|_{t=0}=\pi,\quad\Phi(\rho(t,x_{j}))\leq
1/\lambda_{j},&t\in(0,T),\,j\in J_{c},\\\
{\mathfrak{m}}_{j}(t)={\mathfrak{m}}_{j}(t)\mathbbm{1}_{\Phi(\rho(t,x_{j}))=1/\lambda_{j}},&t\in(0,T),\,j\in
J_{c}.\end{cases}$ (4.5)

We now define the weak solutions to the limit PDE (4.3) and (4.5).

######  Definition 4.3.

Let $f(t,x)$ and $g(t,x)$ be in $L_{loc}^{1}([0,T]\times D)$ where $D$ is a
domain of $x$. We say $f$ is weakly differentialble with respect to $x\in D$
if for all $G(t,x)\in C_{c}^{0,1}([0,T]\times D)$ that

$\int_{0}^{T}\int_{\mathbb{T}}\partial_{x}G(t,x)f(t,x)dxdt=-\int_{0}^{T}\int_{\mathbb{T}}G(t,x)g(t,x)dxdt;$

The weak derivative will be denoted by $\partial_{x}f(t,x)$ and
$\partial_{x}f(t,x):=g(t,x)$.

We comment that if $f(t,x)$ is weakly differentiable with respect to $x\in D$
as defined above, then for a.e. $t\in[0,T]$, $f(t,\cdot)$ is absolutely
continuous and $f(t,b)-f(t,a)=\int_{a}^{b}\partial_{x}f(t,x)\,dx$ for all
connected $a,b\in D$. In particular, the evaluation of
$f(t,x^{\prime})=f(t,x)|_{x=x^{\prime}}$ at a given value $x^{\prime}\in D$ is
well defined.

######  Definition 4.4.

We say $\pi_{t}(dx)=\rho(t,x)dx+\sum_{j\in
J_{c}}{\mathfrak{m}}_{j}(t)\,\delta_{x_{j}}(dx)$ is a weak solution to the
system (4.3) if

  1. (1)

$\rho(t,x)$ is in $L^{2}([0,T]\times\mathbb{T})$ and $\Phi(\rho(t,x))$ is
weakly differentiable with respect to $x\in\mathbb{T}$ with
$\partial_{x}\Phi(\rho(t,x))\in L^{2}([0,T]\times\mathbb{T})$;

  2. (2)

$\Phi(\rho(t,x_{j}))=\Phi(c_{0})$, for almost all $t\in(0,T)$ and all
$x_{j}\in{\mathfrak{D}}_{s}$;

  3. (3)

${\mathfrak{m}}_{j}(t)=\left(\lambda_{j}\Phi(\rho(t,x_{j}))\right)^{1/\alpha}$
for almost all $t\in(0,T)$ and $j\in J_{c}$;

  4. (4)

for all $G(t,x)\in
C_{c}^{\infty}\left([0,T)\times(\mathbb{T}\setminus{\mathfrak{D}}_{s})\right)$

$\begin{split}\int_{0}^{T}\int_{\mathbb{T}}\partial_{t}G(t,x)\pi_{t}(dx)dt+\int_{\mathbb{T}}G(0,x)\pi(dx)+\int_{0}^{T}\int_{\mathbb{T}}\partial_{xx}G(t,x)\Phi(\rho(t,x))dxdt=0.\end{split}$
(4.6)

######  Definition 4.5.

We say $\pi_{t}(dx)=\rho(t,x)dx+\sum_{j\in
J_{c}}{\mathfrak{m}}_{j}(t)\,\delta_{x_{j}}(dx)$ is a weak solution to the
system (4.5) if

  1. (1)

$\rho(t,x)$ is in $L^{2}([0,T]\times\mathbb{T})$ and $\Phi(\rho(t,x))$ is
weakly differentiable with respect to $x\in\mathbb{T}$ with
$\partial_{x}\Phi(\rho(t,x))\in L^{2}([0,T]\times\mathbb{T})$;

  2. (2)

$\Phi(\rho(t,x_{j}))\leq 1/\lambda_{j}$, for almost all $t\in(0,T)$ and all
$x_{j}\in{\mathfrak{D}}_{c}$;

  3. (3)

${\mathfrak{m}}_{j}(t)={\mathfrak{m}}_{j}(t)\mathbbm{1}_{\Phi(\rho(t,x_{j}))=1/\lambda_{j}}$
for almost all $t\in(0,T)$ and $j\in J_{c}$;

  4. (4)

the weak formulation (4.6) holds for all $G(t,x)\in
C_{c}^{\infty}\left([0,T)\times\mathbb{T}\right)$.

######  Remark 4.6.

In Definitions 4.4 and 4.5, we have assumed $\Phi(\rho(t,x))$ to be weakly
differentialbe with respect to $x\in\mathbb{T}$. Note that $\Phi(\cdot)$ is
invertible and the inverse function $\Phi^{-1}(\cdot)$ is a continuous mapping
from $[0,r_{g})$ to $[0,\infty)$ where $r_{g}=\infty$ for $g$ of $n^{\alpha}$
type and $r_{g}=1$ for $g$ bounded (cf. Section 2.1). Then
$\rho(t,x)=\Phi^{-1}\left(\Phi(\rho(t,x))\right)$ is continuous in $x$ for
a.e. $t\in[0,T]$ (viewed as $[0,\infty]$ valued function for $g$ bounded). In
particular, the evaluation of $\Phi(\rho(t,x))$ at $x=x_{j}$ may be written in
terms $\rho(t,x_{j})$ instead.

Moreover, notice that, under the settings of Theorems 4.1 and 4.2, it holds
$\|\rho\|_{\infty}<\infty$; see Lemma 7.1. As $\Phi^{-1}(\cdot)$ is Lipschitz
on $[0,\phi]$ where $\phi=\Phi(\|\rho\|_{\infty})$ (cf. Section 2.1), we have
that $\rho(t,x)$ is also weakly differentiable with respect to
$x\in\mathbb{T}$ and $\partial_{x}\rho(t,x)\in L^{2}([0,T]\times\mathbb{T})$.

To illustrate the relation between boundary conditions and effects on defect
sites, we consider the case with only a single defect site.

######  Example 4.7 (Effects of a single slow site).

As an example to illustrate how defects affect the macroscopic bulk evolution,
we look at systems with a single defect site. Without loss of generality, we
may assume the defect location is at $0$ and the altered jump rate is
$(\lambda N^{\beta})^{-1}g(\xi(0))$.

Consider first $g$ of $n^{\alpha}$ type. By Theorem 4.1, the hydrodynamic
limit $\pi_{t}$ is governed by the PDE (4.3). As the defect site is at $x=0$,
we have $\pi_{t}(dx)=\rho(t,x)dx$ when restricted to open interval $(0,1)$.
Indeed, (4.3) is now reduced to a nonlinear heat equation regarding only
$\rho(t,x)$ with different boundary conditions at $x=0$ and $x=1$ depending on
the value of $\beta$. Precisely, we have the following.

  1. (1)

When $\beta<\alpha$, the defect site is invisible in the limit and (4.3)
becomes

$\begin{cases}\partial_{t}\rho(t,x)=\partial_{xx}\Phi(\rho(t,x)),\quad
x\in\mathbb{T},\ t\in(0,T),\\\ \rho(0,x)=\rho_{0}(x),\end{cases}$ (4.7)

that is $\rho(t,x)$ satisfies periodic boundary conditions.

  2. (2)

When $\beta=\alpha$, we have
$\pi_{t}(dx)=\rho(t,x)dx+{\mathfrak{m}}(t)\delta_{0}(dx)$ with
${\mathfrak{m}}(t)=(\lambda\Phi(\rho(t,0)))^{1/\alpha}$. The atomic mass
${\mathfrak{m}}(t)$ can also be expressed in terms of the its initial value
${\mathfrak{m}}_{0}$ and net change of the bulk mass as the total mass is
conserved:

${\mathfrak{m}}_{0}+\int_{0}^{1}\rho_{0}(x)dx={\mathfrak{m}}(t)+\int_{0}^{1}\rho(t,x)dx,\quad\text{for
all }t>0.$

Therefore, we have
$\Phi(\rho(t,0))=\lambda^{-1}\big{[}{\mathfrak{m}}_{0}+\int_{0}^{1}(\rho_{0}(x)-\rho(t,x))dx\big{]}^{\alpha}$.
Noticing that $x=0$ and $x=1$ coincide on $\mathbb{T}$, we obtain

$\begin{cases}\partial_{t}\rho(t,x)=\partial_{xx}\Phi(\rho(t,x)),\quad
x\in(0,1),\ t\in(0,T),\\\ \rho(0,x)=\rho_{0}(x),\\\
\Phi(\rho(t,0))=\Phi(\rho(t,1))=\lambda^{-1}\big{[}{\mathfrak{m}}_{0}+\int_{0}^{1}\left(\rho_{0}(x)-\rho(t,x)\right)dx\big{]}^{\alpha}.\end{cases}$

We comment that, in a system with more than one critical slow sites
($\beta=\alpha$), conservative of mass is not sufficient to determine the
atomic masses individually. For a closed equation, one needs to stay with a
form such as (4.3).

  3. (3)

When $\beta>\alpha$, as ${\mathfrak{D}}={\mathfrak{D}}_{s}=\\{0\\}$, the PDE
(4.3) is

$\begin{cases}\partial_{t}\rho(t,x)=\partial_{xx}\Phi(\rho(t,x)),\quad
x\in(0,1),\ t\in(0,T),\\\ \rho(0,x)=\rho_{0}(x),\\\
\rho(t,0)=\rho(t,1)=c_{0}.\end{cases}$

We now turn to case when $g$ is bounded. By Theorem 4.2, the macroscopic flow
of $\pi_{t}$ is described by (4.4). Notice that here we have $\beta\leq 0$.
When the defect is a fast site, that is $\beta<0$ or $\beta=0$ with
$\lambda<1$, the defect site is invisible macroscopically and the limit
evolution is the usual nonlinear heat equation (4.7) with periodic boundary
condition. When the defect is a slow site, that is $\beta=0$ and $\lambda>1$,
we can write $\mathfrak{m}(t)$, the atomic mass at the slow site, in term of
‘mass conservation’ as in part (2) above to form a closed equation of
$\rho(t,x)$:

$\begin{cases}\partial_{t}\rho(t,x)=\partial_{xx}\Phi(\rho(t,x)),\quad
x\in\mathbb{T},\ t\in(0,T),\\\
\rho(0,x)=\rho_{0}(x),\quad\rho(t,0)=\rho(t,1)\leq\Phi^{-1}(1/\lambda),\\\
{\mathfrak{m}}(t)={\mathfrak{m}}(t)\mathbbm{1}_{\rho(t,0)=\Phi^{-1}(1/\lambda)},\quad{\mathfrak{m}}(t)={\mathfrak{m}}_{0}+\int_{0}^{1}\left(\rho_{0}(x)-\rho(t,x)\right)dx.\end{cases}$

Informally, during the evolution, we observe periodic boundary conditions when
${\mathfrak{m}}(t)=0$ and Dirichlet boundary conditions
$\rho(t,0)=\rho(t,1)=\Phi^{-1}(1/\lambda)$ when ${\mathfrak{m}}(t)>0$.

###  4.2. Remarks on general defects

In our model, we have assumed that the jumping rate at a defect site $k_{j,N}$
is slowed down by a factor of $\lambda_{j}N^{\beta_{j}}$. However, the form of
$\lambda_{j}N^{\beta_{j}}$ is only a choice for convenience. In fact, Theorems
4.1 and 4.2 hold for more general choices of slow down factors, as can be
perused.

We may assume the jumping rate at a defect $k_{j,N}$ is
$g(\cdot)/\beta_{j,N}$. (In terms of the current setting
$\beta_{j,N}=\lambda_{j}N^{\beta_{j}}$.) Assume that $\beta_{j,N}>0$ and
$\lim_{N\to\infty}\beta_{j,N}/g(N)$ exists (might be $\infty$ though). To
extend Theorems 4.1 and 4.2 to the new setting, we need to clarify the sets of
$J_{s}$, $J_{c}$ and $\lambda_{j}$ for $j\in J_{c}$. When $g$ is of
$n^{\alpha}$ type, $J_{c}=\\{j\in J:\beta_{j,N}\sim N^{\alpha}\\}$ and when
$g$ is bounded, $J_{c}=\\{j\in J:\lim_{N\to\infty}\beta_{j,N}>1\\}$. In either
case of $g$, the super-critical index set $J_{s}$ consists of $j$’s such that
$\beta_{j,N}\gg g(N)$ and $J_{b}=J\setminus\\{J_{s}\cup J_{c}\\}$. The
constant $\lambda_{j}$ for $j\in J_{c}$ can be defined as
$\lambda_{j}:=\lim_{N\to\infty}\beta_{j,N}/g(N)$.

##  5\. Proof outline

In this section, we prove Theorems 4.1 and 4.2. We begin with analyzing
$\langle G,\pi^{N}_{t}\rangle$ by computing its stochastic differential in
terms of certain martingales.

###  5.1. Stochastic differentials

Let $G$ be a smooth function on $[0,T]\times\mathbb{T}$ and we write
$G_{t}(x):=G(t,x)$. For $t\geq 0$, consider $\widehat{\pi}_{t}^{N}$, the
emprical measure associated with $\eta_{t}$ on $\mathbb{T}_{N}$:

$\widehat{\pi}_{t}^{N}(dx):=\dfrac{1}{N}\sum_{k\in\mathbb{T}_{N}}\eta_{t}(k)\delta_{k/N}(dx).$
(5.1)

Note that if $G$ has compact support on
$[0,T]\times\mathbb{T}\setminus{\mathfrak{D}}_{s}$, it holds $\langle
G_{t},\widehat{\pi}_{t}^{N}\rangle=\langle G_{t},\pi_{t}^{N}\rangle$ for all
$N$ large (cf. (4.1)).

We have

$M^{N,G}_{t}=\left\langle
G_{t},\widehat{\pi}^{N}_{t}\right\rangle-\left\langle
G_{0},\widehat{\pi}^{N}_{0}\right\rangle-\int_{0}^{t}\Big{\\{}\left\langle\partial_{s}G_{s},\widehat{\pi}_{s}^{N}\right\rangle+N^{2}L_{N}\left\langle
G_{s},\widehat{\pi}^{N}_{s}\right\rangle\Big{\\}}ds$

is a mean zero martingale. Denote the discrete Laplacian $\Delta_{N}$ by

$\Delta_{N}G\Big{(}\frac{k}{N}\Big{)}:=N^{2}\Big{(}G\Big{(}\frac{k+1}{N}\Big{)}+G\Big{(}\frac{k-1}{N}\Big{)}-2G\Big{(}\frac{k}{N}\Big{)}\Big{)}.$

Then, for $N$ large, we compute

$N^{2}L_{N}\left\langle
G,\widehat{\pi}^{N}_{s}\right\rangle=\dfrac{1}{N}\sum_{k\in\mathbb{T}_{N}}\Delta_{N}G_{s}\Big{(}\frac{k}{N}\Big{)}g_{k,N}(\eta_{s}(k)).$
(5.2)

The quadratic variation of $M^{N,G}_{t}$ is given by

$\langle M^{N,G}\rangle_{t}=\int_{0}^{t}\left\\{N^{2}L_{N}\left(\left\langle
G_{s},\widehat{\pi}^{N}_{s}\right\rangle^{2}\right)-2\left\langle
G_{s},\widehat{\pi}^{N}_{s}\right\rangle N^{2}L_{N}\left\langle
G_{s},\widehat{\pi}^{N}_{s}\right\rangle\right\\}ds$

which by standard calculation equals

$\int_{0}^{t}\sum_{k\in\mathbb{T}_{N}}g_{k,N}(\eta_{s}(k))\left\\{\Big{(}G_{s}\big{(}\frac{k+1}{N}\big{)}-G_{s}\big{(}\frac{k}{N}\big{)}\Big{)}^{2}+\Big{(}G_{s}\big{(}\frac{k-1}{N}\big{)}-G_{s}\big{(}\frac{k}{N}\big{)}\Big{)}^{2}\right\\}ds.$

This variation may be bounded as follows.

######  Lemma 5.1.

For any test function $G(x)\in C^{\infty}(\mathbb{T})$, there is a constant
$C$ independent of $N$ such that, for all $N$ large,

$\sup_{0\leq t\leq T}\mathbb{E}_{N}\langle M^{N,G}\rangle_{t}\leq CN^{-1}.$

###### Proof.

Since $G$ is smooth, we obtain, for $N$ large

$\mathbb{E}_{N}\langle M^{N,G}\rangle_{t}\leq
2(\|\partial_{x}G\|_{\infty})^{2}N^{-1}\mathbb{E}_{N}\Big{[}\displaystyle\int_{0}^{t}\dfrac{1}{N}\sum_{k\in\mathbb{T}_{N}}g_{k,N}(\eta_{s}(k))ds\Big{]}.$

Then, the result follows from (5.6) in the next Lemma 5.2. ∎

It will be useful to bound local particle numbers and rates. Although there
are different ways to prove parts of the following statements, say using the
entropy inequality and $g(n)/n^{\alpha}<C$ for $0\leq\alpha\leq 1$, we will
use attractiveness to facilitate a short proof.

######  Lemma 5.2.

We have the following:

  1. (1)

The expectation of total mass at all but super-critical sites is uniformly
bounded:

$\sup_{N\in\mathbb{N}}\sup_{t\geq
0}\mathbb{E}_{N}\Big{[}\dfrac{1}{N}\sum_{k\in\mathbb{T}_{N}\setminus{\mathfrak{D}}_{s,N}}\eta_{t}(k)\Big{]}<\infty;$
(5.3)

  2. (2)

The expected particle number at each regular site $k\notin{\mathfrak{D}}_{N}$
is uniformly bounded:

$\sup_{N\in\mathbb{N}}\sup_{t\geq
0}\sup_{k\notin{\mathfrak{D}}_{N}}\mathbb{E}_{N}\Big{[}\eta_{t}(k)\Big{]}<\infty;$
(5.4)

  3. (3)

The expectation of weighted jump rate $N^{-1}g_{k,N}$ vanishes uniformly as
$N\to\infty$:

$\lim_{N\to\infty}\sup_{t\geq
0}\sup_{k\in\mathbb{T}_{N}}\mathbb{E}_{N}\Big{[}\dfrac{1}{N}g_{k,N}\left(\eta_{t}(k)\right)\Big{]}=0;$
(5.5)

  4. (4)

The expectation of total weighted jump rate is uniformly bounded:

$\sup_{N\in\mathbb{N}}\sup_{t\geq
0}\mathbb{E}_{N}\Big{[}\dfrac{1}{N}\sum_{k\in\mathbb{T}_{N}}g_{k,N}(\eta_{t}(k))\Big{]}<\infty.$
(5.6)

###### Proof.

First, in both rate $g$ settings, the total mass estimate (5.3) follow
directly from the initial bound of the entropy
$H(\mu^{N}|\mathscr{R}_{c_{0}}^{N})=O(N)$ and the entropy inequality; see
Section 3.1 for an explicit proof.

Moreover, (5.4) and (5.5) follow by straightforwardly applying attractiveness,
$\mu_{t}^{N}\leq\mathscr{R}_{c^{\prime}}^{N}$ and
$\mu_{t}^{N}\leq\overline{\kappa}_{c^{\prime}}^{N}$ for $g$ of $n^{\alpha}$
type and bounded type respectively (see Section 3.2). Indeed, in the $g(n)\sim
n^{\alpha}$ case,, we have, for all $k\not\in{\mathfrak{D}}_{N}$, that
$\mathbb{E}_{N}[\eta_{t}(k)]\leq
E_{\mathscr{R}_{c^{\prime}}^{N}}[\xi(k)]=c^{\prime}$. Also, as $g$ is
increasing, $\mathbb{E}_{N}[g_{k,N}(\eta_{t}(k))]\leq
E_{\mathscr{R}_{c^{\prime}}^{N}}[g_{k,N}(\xi(k))]=\Phi(c^{\prime})$ (cf.
(2.4)). When $g$ is bounded, we have $\mathbb{E}_{N}[\eta_{t}(k)]\leq
E_{\overline{\kappa}_{c^{\prime}}^{N}}[\xi(k)]=c^{\prime}$ for regular sites
$k\notin{\mathfrak{D}}_{N}$. For the bound of
$\mathbb{E}_{N}[g_{k,N}(\eta_{t}(k))]$, it holds trivially that
$\mathbb{E}_{N}[g_{k,N}(\eta_{t}(k))]\leq g_{\infty}=1$ when
$k\notin{\mathfrak{D}}_{N}$ and $\mathbb{E}_{N}[g_{k,N}(\eta_{t}(k))]\leq
1/\lambda_{j}$ when $k=k_{j,N}$ with $\beta_{j}=0$. When $k=k_{j,N}$ with
$\beta_{j}<0$, we use attractiveness again
$\mathbb{E}_{N}[g_{k,N}(\eta_{t}(k))]\leq
E_{\overline{\kappa}_{c^{\prime}}^{N}}[g_{k,N}(\xi(k))]=\Phi(c^{\prime})$.

Finally, to show (5.6), we separate the sum of $k$ into two sums consisting of
the defect sites ${\mathfrak{D}}_{N}$ and the regular sites respectively: The
sum over ${\mathfrak{D}}_{N}$ is bounded via (5.5) and that
${\mathfrak{D}}_{N}$ has finite cardinality $n_{0}$ independent of $N$. Recall
that $g_{k,N}(\cdot)=g(\cdot)$ for $k\notin{\mathfrak{D}}_{N}$. Now, bounding
$g(n)\leq g^{*}n$, the sum over the regular sites is bounded due to (5.3). ∎

###  5.2. Proof outline of Theorems 4.1 and 4.2

We sketch the proofs of Theorems 4.1 and 4.2. Let $Q^{N}$ be the probability
measure on the trajectory space $D([0,T],\pazocal{M})$ governing
$\pi_{\cdot}^{N}$ when the process starts from $\mu^{N}$. By Lemma 6.1, the
family of measures $\left\\{Q^{N}\right\\}_{N\in\mathbb{N}}$ is tight with
respect to the uniform topology, stronger than the Skorokhod topology. Let now
$Q$ be any limit measure. For both cases of $g$, we first show that $Q$ is
supported on a class of weak solutions to the associated nonlinear PDE (4.3)
or (4.5). Then, by uniqueness results from Section 11, we prove the desired
results (4.2) and (4.4).

Step 1. Let $G(t,x)$ be a smooth function with compact support in
$[0,T)\times(\mathbb{T}\setminus{\mathfrak{D}}_{s})$. Notice that $\langle
G_{t},\pi_{t}^{N}\rangle=\langle G_{t},\widehat{\pi}_{t}^{N}\rangle$ and
recall the martingale $M^{N,G}_{t}$ and its quadratic variation $\langle
M^{N,G}\rangle_{t}$ in the last section. By Lemma 5.1, we have
$\mathbb{E}_{N}\big{(}M^{N,G}_{T}\big{)}^{2}=\mathbb{E}_{N}\langle
M^{N,G}\rangle_{T}$ vanishes as $N\to\infty$. For each $\delta>0$, by
Chebychev’s inequality,

$\displaystyle\mathbb{P}_{N}\Big{[}\big{|}\big{\langle}G_{T},\pi^{N}_{T}\big{\rangle}-\big{\langle}G_{0},\pi^{N}_{0}\big{\rangle}-\int_{0}^{T}\big{(}\big{\langle}\partial_{s}G_{s},\pi^{N}_{s}\big{\rangle}+N^{2}L_{N}\big{\langle}G_{s},\pi^{N}_{s}\big{\rangle}\big{)}ds\big{|}>\delta\Big{]}$
$\displaystyle=\mathbb{P}_{N}\Big{[}\big{|}M_{T}^{N,G}\big{|}>\delta\Big{]}\leq\dfrac{1}{\delta^{2}}\mathbb{E}_{N}\big{\langle}M^{N,G}\big{\rangle}_{T}\
\rightarrow\ 0\ \ {\rm as\ }N\to\infty.$

Note that $G_{T}(x)=0$ and recall the evaluation of $N^{2}L_{N}\left\langle
G_{s},\pi^{N}_{s}\right\rangle$ in (5.2). Then,

$\displaystyle\lim_{N\to\infty}\mathbb{P}_{N}\Big{[}\Big{|}\left\langle
G_{0},\pi^{N}_{0}\right\rangle$ (5.7) $\displaystyle\ \ \ \ \ \ \ \ \
+\int_{0}^{T}\Big{\\{}\left\langle\partial_{s}G_{s},\pi^{N}_{s}\right\rangle+\dfrac{1}{N}\sum_{k\in\mathbb{T}_{N}}\Delta_{N}G_{s}\big{(}\frac{k}{N}\big{)}g_{k,N}(\eta_{s}(k))\Big{\\}}ds\Big{|}>\delta\Big{]}=0.$

As
$\mathbb{E}_{N}\big{[}N^{-1}\sum_{k\in\mathbb{T}_{N}}g_{k,N}(\eta_{t}(k))\big{]}$
is uniformly bounded by (5.6), we may replace the discrete Laplacian
$\Delta_{N}G_{s}(\cdot)$ in (5.7) by $\partial_{xx}G_{s}(\cdot)$.

Let ${\mathfrak{D}}^{\varepsilon}=\cup_{j\in
J}(x_{j}-\varepsilon,x_{j}+\varepsilon)$ and
$F_{\varepsilon}(s,x)=\mathbbm{1}_{\mathbb{T}\setminus{\mathfrak{D}}^{\varepsilon}}(x)\partial_{ss}G(s,x)$.
To prepare for the ‘bulk’ replacement in the next step, we will further
replace $\partial_{xx}G_{s}(\cdot)$ by its approximation
$F_{\varepsilon}(s,\cdot)$. It suffices to show

$\lim_{\varepsilon\to 0}\sup_{N\in\mathbb{N}}\sup_{t\geq
0}\mathbb{E}_{N}\big{[}N^{-1}\sum_{k/N\in{\mathfrak{D}}^{\varepsilon}}g_{k,N}(\eta_{t}(k))\big{]}=0.$

As the sum of $k$ is over a set of cardinality at most $(2\varepsilon
N+1)n_{0}$, the desired limit follows from separating $k$’s into defects and
regular ones, and applying (5.5) and (5.4). Therefore, we obtain

$\displaystyle\lim_{\varepsilon\to
0}\lim_{N\to\infty}\mathbb{P}_{N}\Big{[}\Big{|}\left\langle
G_{0},\pi^{N}_{0}\right\rangle$ (5.8) $\displaystyle\ \ \ \ \ \ \ \ \
+\int_{0}^{T}\Big{\\{}\left\langle\partial_{s}G_{s},\pi^{N}_{s}\right\rangle+\dfrac{1}{N}\sum_{k\in\mathbb{T}_{N}}F_{\varepsilon}\big{(}\frac{k}{N}\big{)}g_{k,N}(\eta_{s}(k))\Big{\\}}ds\Big{|}>\delta\Big{]}=0.$

Step 2. We now replace the nonlinear term $g_{k,N}(\eta_{s}(k))$ by a function
of the empirical density of particles. To be precise, let
$\eta^{l}(x)=\dfrac{1}{2l+1}\sum_{|y-x|\leq l}\eta(y)$, that is the average
density of particles in the box centered at $x$ with length $2l+1$. Therefore,
using the Bulk Replacement Lemma (Lemma 8.4), we obtain from (5.8),

$\begin{split}&\lim_{\varepsilon\to 0}\limsup_{\theta\to
0}\limsup_{N\to\infty}\mathbb{P}_{N}\Big{[}\Big{|}\left\langle
G_{0},\pi^{N}_{0}\right\rangle+\int_{0}^{T}\Big{\\{}\left\langle\partial_{s}G_{s},\pi^{N}_{s}\right\rangle\\\
&\qquad\qquad\qquad+\dfrac{1}{N}\sum_{k\in\mathbb{T}_{N}}F_{\varepsilon}\big{(}s,\frac{k}{N}\big{)}\Phi\left(\eta_{t}^{\theta
N}(k)\right)\Big{\\}}ds\Big{|}>\delta\Big{]}=0.\end{split}$ (5.9)

Step 3. For each $\theta>0$, take
$\iota_{\theta}=(2\theta)^{-1}\mathbbm{1}_{[-\theta,\theta]}$. The average
density $\eta_{t}^{\theta N}(k)$ is written as a function of the empirical
measure $\pi^{N}_{t}$

$\eta_{t}^{\theta N}(k)=\dfrac{2\theta N}{2\theta
N+1}\langle\iota_{\theta}(\cdot-k/N),\pi^{N}_{t}\rangle.$

Since $\Phi$ is Lipschitz continuous and the total number of particles has
expectation of order $N$ on the bulk (cf. Lemma 5.2), we may replace
$\eta_{t}^{\theta N}(k)$ by
$\langle\iota_{\theta}(\cdot-k/N),\pi^{N}_{t}\rangle$. Hence, we get from
(5.9) in terms of the induced distribution $Q^{N}$ that

$\begin{split}&\lim_{\varepsilon\to 0}\limsup_{\theta\to
0}\limsup_{N\to\infty}Q^{N}\Big{[}\Big{|}\left\langle
G_{0},\pi^{N}_{0}\right\rangle+\int_{0}^{T}\Big{\\{}\left\langle\partial_{s}G_{s},\pi^{N}_{s}\right\rangle\\\
&\qquad\qquad\qquad+\int_{\mathbb{T}}F_{\varepsilon}(s,x)\Phi\big{(}\langle\iota_{\theta}(\cdot-x),\pi^{N}_{s}\rangle\big{)}dx\Big{\\}}ds\Big{|}>\delta\Big{]}=0.\end{split}$
(5.10)

Notice that the discrete sum on $k$ is also replaced by the corresponding
integral.

As the absolute value term in (5.10) is continuous (with respect to the
uniform topology) on $D([0,T],\pazocal{M})$, the set of trajectories in (5.10)
is open, Taking $N\to\infty$, we obtain

$\begin{split}&\lim_{\varepsilon\to 0}\limsup_{\theta\to
0}Q\Big{[}\Big{|}\left\langle
G_{0},\pi_{0}\right\rangle+\int_{0}^{T}\Big{\\{}\left\langle\partial_{s}G_{s},\pi_{s}\right\rangle\\\
&\qquad\qquad\qquad+\int_{\mathbb{T}}F_{\varepsilon}(s,x)\Phi\big{(}\langle\iota_{\theta}(\cdot-x),\pi_{s}\rangle\big{)}dx\Big{\\}}ds\Big{|}>\delta\Big{]}=0.\end{split}$
(5.11)

Step 4. We show in Lemma 7.1 that $Q$ is supported on trajectories

$\pi_{s}(dx)=\rho(s,x)dx+\sum_{j\in
J_{c}}{\mathfrak{m}}_{j}(s)\delta_{x_{j}}(dx).$

Then, for $x\notin{\mathfrak{D}}^{\varepsilon}$ and $\theta<\varepsilon$,
$\langle\iota_{\theta}(\cdot-x),\pi_{s}\rangle=(2\theta)^{-1}\int_{x-\theta}^{x+\theta}\rho(s,u)du$.
Note that $\Phi$ is Lipschitz and $\rho$ is integrable on
$[0,T]\times\mathbb{T}$. Hence, for all $\varepsilon$ small, we have,
$Q$-almost surely

$\lim_{\theta\to
0}\int_{0}^{T}\int_{\mathbb{T}}F_{\varepsilon}(s,x)\Phi(\langle\iota_{\theta}(\cdot-x),\pi_{s}\rangle)dxds=\int_{0}^{T}\int_{\mathbb{T}}F_{\varepsilon}(s,x)\Phi(\rho(s,x))dxds.$

Since almost sure convergence impiies convergence in probability, we obtain
from (5.11) that, for all $\delta>0$

$\begin{split}&\lim_{\varepsilon\to 0}Q\Big{[}\Big{|}\left\langle
G_{0},\pi_{0}\right\rangle+\int_{0}^{T}\Big{\\{}\left\langle\partial_{s}G_{s},\pi_{s}\right\rangle+\int_{\mathbb{T}}F_{\varepsilon}(s,x)\Phi(\rho(s,x))dx\Big{\\}}ds\Big{|}>\delta\Big{]}=0.\end{split}$

Taking $\varepsilon\to 0$ we may also replace $F_{\varepsilon}$ by
$\partial_{xx}G$. As $\delta$ is arbitrary, we have

$\begin{split}Q\Big{[}\left\langle
G_{0},\pi_{0}\right\rangle+\int_{0}^{T}\Big{\\{}\left\langle\partial_{s}G_{s},\pi_{s}\right\rangle+\int_{\mathbb{T}}\partial_{xx}G_{s}\left(x\right)\Phi(\rho(s,x))dx\Big{\\}}ds=0\Big{]}=1.\end{split}$

By Condition 3.1, the initial condition $\pi_{0}=\pi$ holds. Thus, we conclude
the limit measure $Q$ is concentrated on trajectories $\pi_{\cdot}$ that
satisfies the weak formulation (4.6).

Step 5. That $\rho(s,x)\in L^{2}([0,T]\times\mathbb{T})$ follows from Lemma
7.1. The weak spatial differentiability of $\Phi(\rho(t,x))$ and the $L^{2}$
integrability of the weak derivative are addressed in Proposition 10.1 and
Remark 10.2. When $g$ is of $n^{\alpha}$ type, by Lemma 7.2 and Lemma 7.4, we
obtain the boundary conditions
${\mathfrak{m}}_{j}(t)=\left(\lambda_{j}\Phi(\rho(t,x_{j}))\right)^{1/\alpha}$
for all $j\in J_{c}$ and $\Phi(\rho(t,x_{j}))=\Phi(c_{0})$ for all $j\in
J_{s}$. When $g$ is bounded, by Lemma 7.5, it holds that
$\Phi(\rho(t,x_{j}))\leq 1/\lambda_{j}$ and
${\mathfrak{m}}_{j}(t)={\mathfrak{m}}_{j}(t)\mathbbm{1}_{\Phi(\rho(t,x_{j}))=1/\lambda_{j}}$
for all $j\in J_{c}$. Therefore, $\pi_{\cdot}$ is a weak solution to (4.3) or
(4.5) when $g$ is of $n^{\alpha}$ type or bounded respectively (cf.
Definitions 4.4 and 4.5).

In Section 11, we show that there is at most one weak solution $\pi_{\cdot}$
to (4.3) or (4.5). We conclude then that the sequence of $Q^{N}$ converges
weakly to the Dirac measure on $\pi_{\cdot}$. Finally, as $Q^{N}$ converges to
$Q$ with respect to the uniform topology, we have weak convergence of
$\pi^{N}_{t}$ for each $0\leq t\leq T$. That is, for all $G$ in
$C_{c}(\mathbb{T}\setminus{\mathfrak{D}}_{s})$, $\langle G,\pi_{t}^{N}\rangle$
weakly converges to the constant $\langle G,\pi_{t}\rangle$, and therefore
convergence in probability as stated in (4.2) and (4.4). To finish the proof,
it remains to extend the test function from $G\in
C_{c}(\mathbb{T}\setminus{\mathfrak{D}}_{s})$ to $G\in C(\mathbb{T})$. When
$g$ is bounded, it holds that
$C_{c}(\mathbb{T}\setminus{\mathfrak{D}}_{s})=C(\mathbb{T})$ as
${\mathfrak{D}}_{s}=\emptyset$. When $g$ is of $n^{\alpha}$ type, notice that
$\rho(t,\cdot)$ is in $L^{1}(\mathbb{T})$ for all $t\geq 0$ (cf. Lemma 7.1).
Then, the desired extension follows from standard approximations and (5.4) of
Lemma 5.2.

##  6\. Tightness

In this section, we address the tightness of the probability measures
associated with the trajectories of empirical measures $\pi_{\cdot}^{N}$.
Recall that ${\mathfrak{D}}_{s}$ is the macroscopic locations of the super-
slow sites; in particular, ${\mathfrak{D}}_{s}=\emptyset$ when $g$ is bounded.
To unify treatmentss, in both rate $g$ settings, we let $\pazocal{M}$ be the
space of locally finite nonnegative measures on
$\mathbb{T}\setminus{\mathfrak{D}}_{s}$. Let
$C_{c}(\mathbb{T}\setminus{\mathfrak{D}}_{s})$ be the space of continuous
functions with compact support on $\mathbb{T}\setminus{\mathfrak{D}}_{s}$. Let
$\\{f_{k}\\}_{k\in\mathbb{N}}$ be a countable dense set in
$C_{c}(\mathbb{T}\setminus{\mathfrak{D}}_{s})$ in the sense that for all $f\in
C_{c}(\mathbb{T}\setminus{\mathfrak{D}}_{s})$ there exists a subsequence
$\\{f_{n_{k}}\\}$ such that ${\rm supp\,}f_{n_{k}}\subset{\rm supp\,}f$ for
all $k$ and $f_{n_{k}}$ converges uniformly to $f$. Equipped with the distance

$d(\mu,\nu)=\sum_{k=1}^{\infty}2^{-k}\frac{\left|\int_{0}^{1}f_{k}(d\mu-d\nu)\right|}{1+\left|\int_{0}^{1}f_{k}(d\mu-d\nu)\right|}.$

the space $(\pazocal{M},d(\cdot,\cdot))$ is a complete separable metric space.
The metric $d(\cdot,\cdot)$ realizes the vague topology, that is,
$\lim_{n\to\infty}d(\mu_{n},\mu)=0$ if and only if $\lim_{n\to\infty}\int
fd\mu_{n}=\int fd\mu$ for all $f\in
C_{c}(\mathbb{T}\setminus{\mathfrak{D}}_{s})$.

Let $Q^{N}$ be the probability measure on the trajectory space
$D([0,T],\pazocal{M})$ governing $\pi_{\cdot}^{N}$ when the process starts
from $\mu^{N}$. We show that $\left\\{Q^{N}\right\\}$ is tight with respect to
the uniform topology, stronger than the Skorokhod topology on
$D([0,T],\pazocal{M})$.

######  Lemma 6.1.

$\left\\{Q^{N}\right\\}_{N\in\mathbb{N}}$ is relatively compact with respect
to the uniform topology. As a consequence, all limit points $Q$ are supported
on trajectories $\\{\pi_{t}\\}_{t\in[0,T]}$ vaguely continuous on
$\mathbb{T}\setminus{\mathfrak{D}}_{s}$, that is, for all $G\in
C_{c}(\mathbb{T}\setminus{\mathfrak{D}}_{s})$, the mapping
$t\in[0,T]\mapsto\langle G,\pi_{t}\rangle$ is continuous.

###### Proof.

To deduce that $\\{Q^{N}\\}$ is relatively compact with respect to uniform
topology, we show the following items (cf. Theorem 15.5 in [8]).

  1. (1)

For each $t\in[0,T]$, $\epsilon>0$, there exists a compact set
$K_{t,\epsilon}\subset\pazocal{M}$ such that

$\sup_{N}Q^{N}\left[\pi^{N}_{\cdot}:\pi^{N}_{t}\notin
K_{t,\epsilon}\right]\leq\epsilon.$ (6.1)

  2. (2)

For every $\epsilon>0$,

$\limsup_{r\to
0}\limsup_{N\to\infty}Q^{N}\Big{[}\pi^{N}_{\cdot}:\sup_{|t-s|<r}d(\pi^{N}_{t},\pi^{N}_{s})>\epsilon\Big{]}=0.$
(6.2)

Step 1. We first consider (6.1). Notice that, for any $A>0$, the set
$\left\\{\mu\in\pazocal{M}:\langle 1,\mu\rangle\leq A\right\\}$ is compact in
$\pazocal{M}$. Also, by (5.3), we have
$\mathbb{E}_{N}\Big{[}N^{-1}\sum_{k\notin{\mathfrak{D}}_{s,N}}\eta_{t}(k)\Big{]}\leq
C$ for some constant $C<\infty$ independent of $N$. As $Q^{N}\big{[}\langle
1,\pi^{N}_{t}\rangle>A\big{]}\leq
A^{-1}\mathbb{E}_{N}\big{[}N^{-1}\sum_{k\notin{\mathfrak{D}}_{s,N}}\eta_{t}(k)\big{]}$,
the first condition (6.1) is checked by taking $A$ large.

Step 2. To verify the second condition (6.2), it is enough to show a
counterpart of the condition for the distributions of $\langle
G,\pi^{N}_{\cdot}\rangle$ where $G$ is any smooth test function compactly
supported on $\mathbb{T}\setminus{\mathfrak{D}}_{s}$ (cf. p. 54, [18]). In
other words, we need to show, for every $\epsilon>0$,

$\limsup_{r\to
0}\limsup_{N\to\infty}Q^{N}\Big{[}\pi^{N}_{\cdot}:\sup_{|t-s|<r}\Big{|}\langle
G,\pi^{N}_{t}\rangle-\langle G,\pi^{N}_{s}\rangle\Big{|}>\epsilon\Big{]}=0.$
(6.3)

Note that $\left\langle G,\pi^{N}_{t}\right\rangle=\left\langle
G,\pi^{N}_{0}\right\rangle+\int_{0}^{t}N^{2}L_{N}\left\langle
G,\pi^{N}_{s}\right\rangle ds+M^{N,G}_{t}$. Then, we only need to consider the
oscillations of $\int_{0}^{t}N^{2}L_{N}\left\langle G,\pi^{N}_{s}\right\rangle
ds$ and $M^{N,G}_{t}$ respectively.

Step 3. For the oscillations of the martingale $M^{N,G}_{t}$, by
$\big{|}M^{N,G}_{t}-M^{N,G}_{s}\big{|}\leq\big{|}M^{N,G}_{t}\big{|}+\big{|}M^{N,G}_{s}\big{|}$,
we have
$\mathbb{P}_{N}\big{[}\sup_{|t-s|<r}\big{|}M^{N,G}_{t}-M^{N,G}_{s}\big{|}>\epsilon\big{]}\leq
2\mathbb{P}_{N}\big{[}\sup_{0\leq t\leq
T}\big{|}M^{N,G}_{t}\big{|}>\epsilon/2\big{]}$. Using Chebyshev and Doob’s
inequality, we further bound it by

$\dfrac{8}{\epsilon^{2}}\mathbb{E}_{N}\Big{[}\Big{(}\sup_{0\leq t\leq
T}\big{|}M^{N,G}_{t}\big{|}\Big{)}^{2}\Big{]}\leq\dfrac{32}{\epsilon^{2}}\mathbb{E}_{N}\Big{[}\big{(}M^{N,G}_{T}\big{)}^{2}\Big{]}=\dfrac{32}{\epsilon^{2}}\mathbb{E}_{N}\langle
M^{N,G}\rangle_{T}.$

By Lemma 5.1, $\mathbb{E}_{N}\langle M^{N,G}\rangle_{T}=O(N^{-1})$. Then, we
conclude

$\lim_{r\to
0}\lim_{N\to\infty}\mathbb{P}_{N}\Big{[}\sup_{|t-s|<r}\left|M^{N,G}_{t}-M^{N,G}_{s}\right|>\epsilon\Big{]}=0.$

Step 4. To conclude (6.3), it suffices to show

$\limsup_{r\to
0}\limsup_{N\to\infty}Q^{N}\Big{[}\sup_{|t-s|<r}\int_{s}^{t}\big{|}N^{2}L_{N}\left\langle
G,\pi^{N}_{\tau}\right\rangle\big{|}d\tau>\epsilon\Big{]}=0.$

The absolute value term is bounded above by $\|\Delta
G\|_{\infty}\dfrac{1}{N}\sum_{k\notin{\mathfrak{D}}_{s,N}}g_{k,N}(\eta_{\tau}(k))$
(cf. (5.2) applied to $\pi^{N}_{\cdot}$). Then, by Markov’s inequality, it
remains to show

$\limsup_{r\to
0}\limsup_{N\to\infty}\mathbb{E}_{N}\Big{[}\sup_{|t-s|<r}\int_{s}^{t}\dfrac{1}{N}\sum_{k\in\mathbb{T}_{N}\setminus{\mathfrak{D}}_{s,N}}g_{k,N}(\eta_{\tau}(k))d\tau\Big{]}=0.$

By Lemma 5.2,
$\mathbb{E}_{N}\big{[}\int_{0}^{T}N^{-1}g_{k,N}(\eta_{t}(k))dt\big{]}$
vanishes as $N\to\infty$ for defect sites $k\in{\mathfrak{D}}_{c,N}$ or
${\mathfrak{D}}_{b,N}$. Therefore, we may restrict the summation term in the
previous display over $k\in\mathbb{T}_{N}\setminus{\mathfrak{D}}_{N}$. Note
that for each such $k$ we have $g_{k,N}(\cdot)=g(\cdot)$. When $g$ is bounded,
as
$N^{-1}\sum_{k\notin{\mathfrak{D}}_{N}}g(\eta_{\tau}(k))\leq\|g\|_{\infty}$,
the lemma is proved for this case.

The rest of this proof focuses on the case when $g$ is of $n^{\alpha}$ type.
Since $g(\cdot)$ grows at most linearly, we are left to show

$\lim_{r\to
0}\lim_{N\to\infty}\mathbb{E}_{N}\Big{[}\sup_{|t-s|<r}\Big{|}\int_{s}^{t}\dfrac{1}{N}\sum_{k\in\mathbb{T}_{N}\setminus{\mathfrak{D}}_{N}}\eta_{\tau}(k)d\tau\Big{|}\Big{]}=0.$
(6.4)

Step 5. To show (6.4). we introduce a truncation $\mathbbm{1}_{\eta(k)\leq A}$
with $A>0$. Notice that

$\mathbb{E}_{N}\Big{[}\sup_{|t-s|<r}\Big{|}\int_{s}^{t}\dfrac{1}{N}\sum_{k\notin{\mathfrak{D}}_{N}}\eta_{\tau}(k)\mathbbm{1}_{\eta_{\tau}(k)\leq
A}d\tau\Big{|}\Big{]}\leq\mathbb{E}_{N}\Big{[}\sup_{|t-s|<r}\Big{|}\int_{s}^{t}Ad\tau\Big{|}\Big{]}\leq
rA$

which vanishes when taking $N\to\infty$, $r\to 0$, and $A\to\infty$ in order.
It remains to show the error term with $\mathbbm{1}_{\eta(k)>A}$ also vanishes
in the limit. Note that the error term is estimated by

$\begin{split}\mathbb{E}_{N}\Big{[}\sup_{|t-s|<r}\Big{|}\int_{s}^{t}\dfrac{1}{N}\sum_{k\notin{\mathfrak{D}}_{N}}\eta_{\tau}(k)\mathbbm{1}_{\eta_{\tau}(k)>A}d\tau\Big{|}\Big{]}\leq\
&\mathbb{E}_{N}\Big{[}\int_{0}^{T}\dfrac{1}{N}\sum_{k\notin{\mathfrak{D}}_{N}}\eta_{\tau}(k)\mathbbm{1}_{\eta_{\tau}(k)>A}d\tau\Big{]}\\\
=&\int_{0}^{T}\mathbb{E}_{N}\Big{[}\dfrac{1}{N}\sum_{k\notin{\mathfrak{D}}_{N}}\eta_{\tau}(k)\mathbbm{1}_{\eta_{\tau}(k)>A}\Big{]}d\tau.\end{split}$

Recall the initial entropy bound $H(\mu^{N}|\mathscr{R}_{c_{0}}^{N})<CN$. By
entropy inequality, for any $\tau\in[0,T]$ and $B>0$, the term
$\mathbb{E}_{N}\Big{[}N^{-1}\sum\eta_{\tau}(k)\mathbbm{1}_{\eta_{\tau}(k)>A}\Big{]}$
is further bounded above by

$\dfrac{C}{B}+\dfrac{1}{BN}\ln
E_{\mathscr{R}_{c_{0}}^{N}}\Big{[}\exp\Big{\\{}B\sum_{k\notin{\mathfrak{D}}_{N}}\eta(k)\mathbbm{1}_{\eta(k)>A}\Big{\\}}\Big{]}.$
(6.5)

Since, with respect to $\mathscr{R}_{c_{0}}^{N}$, the $\eta(k)$’s in (6.5) are
i.i.d. with common distribution $\pazocal{P}_{\varphi}$ for
$\varphi=\Phi(c_{0})$, we have (6.5) is equal to

$\dfrac{C}{B}+\dfrac{N-n_{0}}{BN}\ln
E_{\pazocal{P}_{\varphi}}\left[e^{BX\mathbbm{1}_{X>A}}\right]\leq\dfrac{C}{B}+B^{-1}\ln
E_{\pazocal{P}_{\varphi}}\left[e^{BX\mathbbm{1}_{X>A}}\right].$

Here, $n_{0}$ is the number of defect sites and the distribution of $X$ is
$\pazocal{P}_{\varphi}$. As $E_{\pazocal{P}_{\varphi}}[e^{BX}]<\infty$ for all
$B>0$, we have $B^{-1}\ln
E_{\pazocal{P}_{\varphi}}\big{[}e^{BX\mathbbm{1}_{X>A}}\big{]}\to 0$ when
taking $A\to\infty$ and then $B\to\infty$. Hence, we obtain

$\lim_{A\to\infty}\lim_{r\to
0}\lim_{N\to\infty}\mathbb{E}_{N}\Big{[}\sup_{|t-s|<r}\Big{|}\int_{s}^{t}\dfrac{1}{N}\sum_{k\notin{\mathfrak{D}}_{N}}\eta_{\tau}(k)\mathbbm{1}_{\eta_{\tau}(k)>A}d\tau\Big{|}\Big{]}=0$

which completes the proof. ∎

##  7\. Properties of limit measures

By Lemma 6.1, the sequence $\left\\{Q^{N}\right\\}$ is relatively compact with
respect to the uniform topology. Consider any convergent subsequence of
$Q^{N}$ and relabel so that $Q^{N}\Rightarrow Q$. We now consider absolute
continuity and boundary behaviors at defect sites for trajectories under $Q$.

###  7.1. Absolute continuity

We now address absolute continuity.

######  Lemma 7.1.

We have $Q$ is supported on trajectories whose restriction on
$\mathbb{T}\setminus{\mathfrak{D}}_{c}$ is absolutely continuous and the
density is in $L^{\infty}$ uniformly in $t$: there exists $c\geq 0$ such that

$Q\Big{[}\pi_{\cdot}:\pi_{t}(dx)=\rho(t,x)dx+\sum_{j\in
J_{c}}{\mathfrak{m}}_{j}(t)\delta_{x_{j}}(dx)\text{ with
}\|\rho(t,\cdot)\|_{\infty}<c\text{ for all }0\leq t\leq T\Big{]}=1.$

###### Proof.

Let $D:=\mathbb{T}\setminus\\{{\mathfrak{D}}_{s}\cup{\mathfrak{D}}_{c}\\}$ and
$C_{c}^{+}(D)$ be the space of nonnegative continuous functions with compact
support on $D$, equipped with topology of uniform convergence on compact sets.
Take $\left\\{G_{n}\right\\}_{n\in\mathbb{N}}$ be a dense sequence of
$C_{c}^{+}(D)$. The lemma will follow if we have, for some $c\geq 0$,

$Q\Big{[}\pi_{\cdot}:\langle G_{n},\pi_{t}\rangle\leq\langle
G_{n},c\rangle\text{ for all }0\leq t\leq T\text{ and
}n\in\mathbb{N}\Big{]}=1.$

Here $\langle G,c\rangle$ is shorthand for $c\int_{\mathbb{T}}G(x)dx$.

To this end, recall $\kappa_{c^{\prime}}^{N}$ defined in Condition 3.1. Let
$\nu^{N}$ denote $\mathscr{R}_{c^{\prime}}^{N}$ when $g$ is of $n^{\alpha}$
type and $\kappa_{c^{\prime}}^{N}$ when $g$ is bounded. By the product
structure of $\nu^{N}$ and Chebyshev inequality, for each $\delta>0$, we have
$\nu^{N}[|\langle G_{n},\pi^{N}\rangle-\langle
G_{n},c^{\prime}\rangle|>\delta]\to 0$ as $N\to\infty$.

Fix $\varepsilon>0$ and $t\in[0,T]$. By attractiveness (cf. Section 3.2),
$Q^{N}\big{[}\langle G_{n},\pi^{N}_{t}\rangle\leq\langle
G_{n},c^{\prime}\rangle+\varepsilon\big{]}$ is bounded from below by
$\nu^{N}\big{[}\langle G_{n},\pi^{N}\rangle\leq\langle
G_{n},c^{\prime}\rangle+\varepsilon\big{]}.$ Then, we have, for all $t\geq 0$
and $\varepsilon>0$

$\lim_{N\to\infty}Q^{N}\left[\langle G_{n},\pi_{t}^{N}\rangle\leq\langle
G_{n},c^{\prime}\rangle+\varepsilon\right]=1.$

As compactness of $\\{Q^{N}\\}$ was shown in the uniform topology in Lemma
6.1, the distribution of $\langle G_{n},\pi^{N}_{t}\rangle$ under $Q^{N}$
converges weakly to $\langle G_{n},\pi_{t}\rangle$ under $Q$. Hence, for any
$\varepsilon>0$ and $t\in[0,T]$, we have

$Q\Big{[}\langle G_{n},\pi_{t}\rangle\leq\langle
G_{n},c^{\prime}\rangle+\varepsilon\Big{]}\geq\limsup_{N\to\infty}Q^{N}\Big{[}\langle
G_{n},\pi_{t}^{N}\rangle\leq\langle
G_{n},c^{\prime}\rangle+\varepsilon\Big{]}=1.$

Since $Q$ is supported on vaguely continuous trajectories by Lemma 6.1, we
obtain for all $\varepsilon>0$ that $Q\Big{[}\langle
G_{n},\pi_{t}\rangle\leq\langle G_{n},c^{\prime}\rangle+\varepsilon\text{ for
all }0\leq t\leq T,\,n\in\mathbb{N}\Big{]}=1.$ Then, we conclude the lemma by
taking $\varepsilon\to 0$ and choosing $c=c^{\prime}$. ∎

###  7.2. Boundary behavior

We proceed by showing the behavior of the limit density $\rho(t,x)$ around the
defects $x_{j}$ in ${\mathfrak{D}}_{s}$ or ${\mathfrak{D}}_{c}$. Notice that,
part of the results presented here are in terms of $\rho(t,x)$ as $x\to
x_{j}+$. The corresponding results for $x\to x_{j}-$ follow from similar
arguments. In the following, we will make use of a ‘local’ replacement lemma
Lemma 9.2 proved in Section 9.

For each $\theta\in(0,1)$, let $\hat{\iota}_{\theta}:(0,\theta)\mapsto[0,1]$
be a compactly supported continuous function such that
$\hat{\iota}_{\theta}(x)=1$ for $x\in$ $(\theta^{2},\theta-\theta^{2})$. Let
$\|\hat{\iota}_{\theta}\|$ denote the $L^{1}$ norm
$\int_{0}^{\theta}\hat{\iota}_{\theta}(x)dx$ and
$\|\hat{\iota}_{\theta}\|^{-1}:=(\|\hat{\iota}_{\theta}\|)^{-1}$. Notice that
$\big{|}\|\hat{\iota}_{\theta}\|^{-1}-\theta^{-1}\big{|}$ is bounded from
above for all $\theta$ small since
$\big{|}\|\hat{\iota}_{\theta}\|^{-1}-\theta^{-1}\big{|}$ is less than
$\big{|}(\theta-2\theta^{2})^{-1}-\theta^{-1}\big{|}$ which approaches $2$ as
$\theta\to 0$.

We first describe behavior near ‘super-slow’ sites in the $n^{\alpha}$
setting.

######  Lemma 7.2.

Let $g$ be of $n^{\alpha}$ type. Then, for any $j\in J_{s}$, $\delta>0$, and
$G\in C[0,T]$, we have

$\lim_{\theta\to
0}Q\Big{[}\Big{|}\int_{0}^{T}G(s)\big{(}\Phi\big{(}\rho^{+,\theta}_{x_{j}}(s)\big{)}-\Phi(c_{0})\big{)}ds\Big{|}>\delta\Big{]}=0$
(7.1)

where
$\rho^{+,\theta}_{x_{j}}(s):=\|\hat{\iota}_{\theta}\|^{-1}\langle\hat{\iota}_{\theta}(\cdot-
x_{j}),\rho\rangle=\|\hat{\iota}_{\theta}\|^{-1}\int_{\mathbb{T}}\hat{\iota}_{\theta}(x-x_{j})\rho(s,x)dx$.
Consequently,

$Q\big{[}\Phi(\rho(t,x_{j}))=\Phi(c_{0})\text{ for a.e.~{}$t\in[0,T]$ and all
$j\in J_{s}$}\big{]}=1.$ (7.2)

###### Proof.

The proof of is split into steps. We show (7.1) in the first four steps and
the last step derives (7.2) from (7.1).

Step 1. Fix $j\in J_{s}$ and also a test function $G\in C[0,T]$ for later use.
We first show that

$\lim_{N\to\infty}\sup_{s\geq
0}\mathbb{E}_{N}\big{[}\big{|}N^{-\beta_{j}}g(\eta_{s}(k_{j,N}))-N^{-\beta_{j}}(\eta_{s}(k_{j,N}))^{\alpha}\big{|}\big{]}=0.$
(7.3)

Note that $g(n)\sim n^{\alpha}$. For each $\epsilon>0$, let $A=A(\epsilon)$ be
such that $|n^{\alpha}/g(n)-1|<\epsilon$ for all $n>A$. Then, the expectation
term in (7.3) is bounded above by $E_{1}+E_{2}$ where $E_{1}$ and $E_{2}$ are
the same expectation as in (7.3) with the integrand multiplied by indicators
$\mathbbm{1}_{\eta_{s}(k_{j,N})\leq A}$ and
$\mathbbm{1}_{\eta_{s}(k_{j,N})>A}$ respectively. For each $A$, the term
$E_{1}$ vanishes as $N\to\infty$. By the definition of $A$ and Lemma 5.2, the
term $E_{2}$ is further bounded above by $\epsilon\sup_{s\geq
0}\mathbb{E}_{N}\big{[}N^{-\beta_{j}}g(\eta_{s}(k_{j,N}))\big{]}=O(\epsilon)$.
Letting $\epsilon\to 0$, we conclude (7.3).

Step 2. We now argue that $N^{-\beta_{j}}(\eta_{s}(k_{j,N}))^{\alpha}$ may be
replaced by $N^{-\beta_{j}}(\eta_{0}(k_{j,N}))^{\alpha}$. Let
$r_{N,s}=N^{-\beta_{j}/\alpha}(\eta_{s}(k_{j,N})-\eta_{0}(k_{j,N}))$. Let
$\tau>0$ be a constant such that there are no other defects within a
$\tau$-neighborhood of $x_{j}$. Take a test function
$F:\mathbb{T}\mapsto[0,1]$ such that $F$ has compact support in
$(x_{j}-\tau,x_{j}+\tau)$ and $F(x)=1$ for $|x-x_{j}|\leq\tau/2$. Recall
$\widehat{\pi}_{t}^{N}$ from (5.1) and notice that

$|r_{N,s}|\leq N^{1-\beta_{j}/\alpha}\big{|}\langle
F,\widehat{\pi}_{s}^{N}\rangle-\langle
F,\widehat{\pi}_{0}^{N}\rangle\big{|}+N^{-\beta_{j}/\alpha}{\sum_{k}}^{\circ}(\eta_{s}(k)+\eta_{0}(k)):=I_{1}+I_{2}.$

where the $\sum_{k}^{\circ}$ term is summation over $k\in\mathbb{T}_{N}$ such
that $|k/N-x_{j}|\leq\tau$ and $k\neq k_{j,N}$.

As $\beta_{j}>\alpha$, (5.3) of Lemma 5.2 asserts that $\sup_{s\geq
0}\mathbb{E}_{N}[I_{2}]$ vanishes as $N\to\infty$. To show
$\mathbb{E}_{N}[I_{1}]$ vanishes, notice that $\left\langle
F,\widehat{\pi}^{N}_{s}\right\rangle-\left\langle
F,\widehat{\pi}^{N}_{0}\right\rangle=\int_{0}^{s}N^{2}L_{N}\left\langle
F,\widehat{\pi}^{N}_{t}\right\rangle dt+M^{N,F}_{s}$. Using the generator
computation (5.2) and the jump rate bound (5.6), we have

$\sup_{N\in\mathbb{N}}\sup_{t\geq
0}\mathbb{E}_{N}\big{[}N^{2}L_{N}\left\langle
F,\widehat{\pi}^{N}_{t}\right\rangle\big{]}\leq\sup_{N\in\mathbb{N}}\sup_{t\geq
0}\|\Delta_{N}F(\cdot)\|_{\infty}\mathbb{E}_{N}\Big{[}\dfrac{1}{N}\sum_{k\in\mathbb{T}_{N}}g_{k,N}(\eta_{t}(k))\Big{]}<\infty.$

Also, by Lemma 5.1, the martingale term vanishes as its variance $\sup_{0\leq
s\leq T}\mathbb{E}_{N}\langle M^{N,G}\rangle_{s}=o(1)$. Thus, as
$\beta_{j}>\alpha$, we have $\mathbb{E}_{N}[I_{1}]$ and, therefore,
$\mathbb{E}_{N}[|r_{N,s}|]$ vanishes as $N\to\infty$ uniformly for all
$s\in[0,T]$.

As $\alpha\in(0,1]$, by the elementary inequality
$(|x|+|y|)^{\alpha}\leq|x|^{\alpha}+|y|^{\alpha}$, we have

$N^{-\beta_{j}}|(\eta_{s}(k_{j,N}))^{\alpha}-(\eta_{0}(k_{j,N}))^{\alpha}|\leq|r_{N,s}|^{\alpha}.$

Therefore, we conclude

$\lim_{N\to\infty}\sup_{0\leq s\leq
T}\mathbb{E}_{N}\Big{[}N^{-\beta_{j}}\big{|}(\eta_{s}(k_{j,N}))^{\alpha}-(\eta_{0}(k_{j,N}))^{\alpha}\big{|}\Big{]}=0.$
(7.4)

In particular, it follows from (7.3) and (7.4) that, for any $\delta>0$,

$\lim_{N\to\infty}\mathbb{P}_{N}\Big{[}\Big{|}\int_{0}^{T}G(s)\big{(}N^{-\beta_{j}}g(\eta_{s}(k_{j,N}))-N^{-\beta_{j}}(\eta_{0}(k_{j,N}))^{\alpha}\big{)}ds\Big{|}>\delta\Big{]}=0.$
(7.5)

Step 3. In this step, we show that
$N^{-\beta_{j}}(\eta_{0}(k_{j,N}))^{\alpha}$ may be replaced by
$\varphi=\lambda_{j}\Phi(c_{0})$. Precisely, as $\eta_{0}$ has distribution
$\mu^{N}$, we show that

$\lim_{N\to\infty}\mu^{N}\big{[}\big{|}N^{-\beta_{j}/\alpha}\xi(k_{j,N})-\varphi^{1/\alpha}\big{|}>\delta\big{]}=0.$
(7.6)

Indeed, let
$\pazocal{U}_{N}:=\big{\\{}\big{|}N^{-\beta_{j}/\alpha}\xi(k_{j,N})-\varphi^{1/\alpha}\big{|}\geq\delta\big{\\}}$.
Fix $r_{0}\in(1,\beta_{j}/\alpha)$. By the entropy inequality,

$\mu^{N}\left[\pazocal{U}_{N}\right]=E_{\mu^{N}}\left[\mathbbm{1}_{\pazocal{U}_{N}}\right]\leq
N^{-r_{0}}H(\mu^{N}|\mathscr{R}_{c_{0}}^{N})+N^{-r_{0}}\ln
E_{\mathscr{R}_{c_{0}}^{N}}\big{[}e^{N^{r_{0}}\mathbbm{1}_{\pazocal{U}_{N}}}\big{]}.$

Since $H(\mu^{N}|\mathscr{R}_{c_{0}}^{N})=O(N)$ (cf. Condition 3.1) and
$E_{\mathscr{R}_{c_{0}}^{N}}\big{[}e^{N^{r_{0}}\mathbbm{1}_{\pazocal{U}_{N}}}\big{]}\leq
1+e^{N^{r_{0}}}\mathscr{R}_{c_{0}}^{N}\left[\pazocal{U}_{N}\right]$, we have

$\lim_{N\to\infty}\mu^{N}\left[\pazocal{U}_{N}\right]\leq\max\big{\\{}0,\lim_{N\to\infty}N^{-r_{0}}\ln\big{(}e^{N^{r_{0}}}\mathscr{R}_{c_{0}}^{N}\left[\pazocal{U}_{N}\right]\big{)}\big{\\}}.$

By the following Lemma 7.3, we have
$\lim_{N\to\infty}N^{-r_{0}}\ln\mathscr{R}_{c_{0}}^{N}\left[\pazocal{U}_{N}\right]=-\infty$,
and therefore, (7.6) holds.

Step 4. Putting together (7.5) and (7.6), we have

$\lim_{N\to\infty}\mathbb{P}_{N}\Big{[}\Big{|}\int_{0}^{T}G(s)\big{(}N^{-\beta_{j}}g(\eta_{s}(k_{j,N}))-\lambda_{j}\Phi(c_{9})\big{)}ds\Big{|}>\delta\Big{]}=0.$

For each $\theta>0$, let $\eta_{s}^{\theta N,+}(k)=(\theta
N)^{-1}\sum_{0<j-k\leq\theta N}\eta_{s}(k)$. By Lemma 9.2, we may further
replace $N^{-\beta_{j}}g(\eta_{s}(k_{j,N}))$ by
$\lambda_{j}\Phi(\eta_{s}^{\theta N,+}(k_{j,N}))$ and obtain

$\lim_{\theta\to
0}\limsup_{N\to\infty}\mathbb{P}_{N}\Big{[}\Big{|}\int_{0}^{T}G(s)\big{(}\Phi(\eta_{s}^{\theta
N,+}(k_{j,N}))-\Phi(c_{0})\big{)}ds\Big{|}>\delta\Big{]}=0.$ (7.7)

Notice that $\big{|}\eta_{s}^{\theta
N,+}(k_{j,N})-\|\hat{\iota}_{\theta}\|^{-1}\langle\hat{\iota}_{\theta}(\cdot-
x_{j}),\pi_{s}^{N}\rangle\big{|}$ is bounded above by

$\big{|}\|\hat{\iota}_{\theta}\|^{-1}-\theta^{-1}\big{|}N^{-1}\sum_{0<k/N-x_{j}\leq\theta}\eta_{s}(k)+(\theta
N)^{-1}\sum_{(k/N-x_{j})\in(0,\theta^{2})\cup(\theta-\theta^{2},\theta]}\eta_{s}(k).$

Using the particle numbers bound (5.4) and that
$\big{|}\|\hat{\iota}_{\theta}\|^{-1}-\theta^{-1}\big{|}$ is bounded for all
$\theta$ small, the $\mathbb{E}_{N}$ expectation of the previous display
vanishes as $N\to\infty$. Since $\Phi(\cdot)$ is Lipschitz, we may replace
$\eta_{s}^{\theta N,+}(k_{j,N})$ in (7.7) by
$\|\hat{\iota}_{\theta}\|^{-1}\langle\hat{\iota}_{\theta}(\cdot-
x_{j}),\pi_{s}^{N}\rangle$ to obtain

$\lim_{\theta\to
0}\limsup_{N\to\infty}Q^{N}\Big{[}\Big{|}\int_{0}^{T}G(s)\big{(}\Phi\big{(}\|\hat{\iota}_{\theta}\|^{-1}\langle\hat{\iota}_{\theta}(\cdot-
x_{j}),\pi_{s}^{N}\rangle\big{)}-\Phi(c_{0})\big{)}ds\Big{|}>\delta\Big{]}=0.$

Since the absolute value term in the previous display is continuous (in the
uniform topology) when viewed as a function on the trajectory space
$D([0,T],\pazocal{M})$, we obtain

$\lim_{\theta\to
0}Q\Big{[}\Big{|}\int_{0}^{T}G(s)\big{(}\Phi\big{(}\|\hat{\iota}_{\theta}\|^{-1}\langle\hat{\iota}_{\theta}(\cdot-
x_{j}),\pi_{s}\rangle\big{)}-\Phi(c_{0})\big{)}ds\Big{|}>\delta\Big{]}=0.$

By Lemma 7.1, $\|\hat{\iota}_{\theta}\|^{-1}\langle\hat{\iota}_{\theta}(\cdot-
x_{j}),\pi_{s}\rangle$ is recognized as $\rho^{+,\theta}_{x_{j}}(s)$;
therefore (7.1) follows.

Step 5. We now conclude the proof of the lemma by showing (7.2) from (7.1). By
Lemma 7.1 and Proposition 10.1 (see also Remarks 4.6 and 10.2), it holds
almost surely under $Q$ that $\|\rho(\cdot,\cdot)\|_{\infty}<\infty$ and
$\rho(t,\cdot)\in C(\mathbb{T})$ for almost all $t\in[0,T]$. Then, we have,
$Q$-almost surely,
$\int_{0}^{T}G(s)\Phi\big{(}\rho^{+,\theta}_{x_{j}}(s)\big{)}ds$ converges to
$\int_{0}^{T}G(s)\Phi(\rho(s,x_{j}))ds$. As we have shown in (7.1) that
$\int_{0}^{T}G(s)\Phi\big{(}\rho^{+,\theta}_{x_{j}}(s)\big{)}ds$ also
converges to $\int_{0}^{T}G(s)\Phi(c_{0})ds$ in probability, we obtain, for
all $G\in C[0,T]$ and $j\in J_{s}$

$Q\Big{[}\int_{0}^{T}G(s)\big{(}\Phi\big{(}\rho(s,x_{j})\big{)}-\Phi(c_{0})\big{)}ds=0\Big{]}=1.$

As the continuous function space $C[0,T]$ is separable with respect to the
unform norm and $J_{s}$ is a finite set, we further obtain

$Q\Big{[}\int_{0}^{T}G(s)\big{(}\Phi\big{(}\rho(s,x_{j})\big{)}-\Phi(c_{0})\big{)}ds=0\text{
for all $G\in C[0,T]$ and $j\in J_{s}$}\Big{]}=1$

which implies (7.2), finishing the proof. ∎

We now show the estimate used in Step 3 of the above argument.

######  Lemma 7.3.

Let $g$ be of $n^{\alpha}$ type. Let $X_{N}$ be distributed according to
$\pazocal{P}_{N^{\beta}\varphi}$ for some $\beta>\alpha$ and $\varphi\geq 0$.
Then, for any $r_{0}<\beta/\alpha$ and $\delta>0$

$\lim_{N\to\infty}N^{-r_{0}}\ln
P\big{[}\big{|}N^{-\beta/\alpha}X_{N}-\varphi^{1/\alpha}\big{|}\geq\delta\big{]}=-\infty.$

###### Proof.

Let $\phi=\varphi^{1/\alpha}$. To prove the lemma, it suffices to show that

$\lim_{N\to\infty}N^{-r_{0}}\ln
P\big{[}X_{N}\geq(\phi+\delta)N^{\beta/\alpha}\big{]}=\lim_{N\to\infty}N^{-r_{0}}\ln
P\big{[}X_{N}\leq(\phi-\delta)N^{\beta/\alpha}\big{]}=-\infty.$

To this end, we notice that, for any $x>0$,

$\begin{split}\ln
P\big{[}X_{N}\geq(\phi+\delta)N^{\beta/\alpha}\big{]}\leq&\ln\big{(}e^{-(\phi+\delta)N^{\beta/\alpha}x}E[e^{xX_{N}}]\big{)}\\\
=&-(\phi+\delta)N^{\beta/\alpha}x+\ln\dfrac{Z(e^{x}N^{\beta}\varphi)}{Z(N^{\beta}\varphi)}.\end{split}$

By Lemma 2.6,
$\ln\dfrac{Z(e^{x}N^{\beta}\varphi)}{Z(N^{\beta}\varphi)}\sim\alpha\phi
N^{\beta/\alpha}(e^{x/\alpha}-1)$. Then, for

$K_{\delta,\phi,\alpha}:=\max_{x>0}\big{\\{}(\phi+\delta)x-\alpha\phi(e^{x/\alpha}-1)\big{\\}}$

we have

$\limsup_{N\to\infty}N^{-\beta/\alpha}\ln
P\big{[}X\geq(\phi+\delta)N^{\beta/\alpha}\big{]}\leq-
K_{\delta,\phi,\alpha}<0.$

Since $r_{0}<\beta/\alpha$, we obtain

$\lim_{N\to\infty}N^{-r_{0}}\ln
P\big{[}X_{N}\geq(\phi+\delta)N^{\beta/\alpha}\big{]}=-\infty.$ (7.8)

On the other hand, for all $x>0$, we also have

$\ln
P\big{[}X_{N}\leq(\phi-\delta)N^{\beta/\alpha}\big{]}\leq\ln\big{(}e^{(\phi-\delta)N^{\beta/\alpha}x}E[e^{-xX_{N}}]\big{)}.$

By a similar argument employed to prove (7.8), we have

$\lim_{N\to\infty}N^{-r_{0}}\ln
P\big{[}X_{N}\leq(\phi-\delta)N^{\beta/\alpha}\big{]}=-\infty.$

The lemma is now proved. ∎

We now consider the behavior near ‘critical’ slow sites in the $n^{\alpha}$
setting used in Step 5.

######  Lemma 7.4.

Let $g$ be of $n^{\alpha}$ type. Then, for any $j\in J_{c}$, $\delta>0$, and
$G\in C[0,T]$, we have

$\lim_{\theta\to
0}Q\Big{[}\Big{|}\int_{0}^{T}G(s)\Big{(}\lambda_{j}\Phi\big{(}\rho^{+,\theta}_{x_{j}}(s)\big{)}-({\mathfrak{m}}_{j}(s))^{\alpha}\Big{)}ds\Big{|}>\delta\Big{]}=0,$
(7.9)

where
$\rho^{+,\theta}_{x_{j}}(s)=\|\hat{\iota}_{\theta}\|^{-1}\langle\hat{\iota}_{\theta}(\cdot-
x_{j}),\rho\rangle$. Consequently,

$Q\big{[}{\mathfrak{m}}_{j}(t)=\big{(}\lambda_{j}\Phi\big{(}\rho(t,x_{j})\big{)}\big{)}^{1/\alpha}\text{
for a.e.~{}$t\in[0,T]$ and all $j\in J_{c}$}\big{]}=1.$ (7.10)

###### Proof.

Fix $j\in J_{c}$ and $G\in C[0,T]$. As $g(n)\sim n^{\alpha}$, arguments as in
Step $1$ of the proof of Lemma 7.2 show that $N^{-\alpha}g(\eta_{s}(k_{j,N}))$
may be replaced by $N^{-\alpha}(\eta_{s}(k_{j,N}))^{\alpha}$. Hence, we have

$\lim_{N\to\infty}\mathbb{P}_{N}\Big{[}\Big{|}\int_{0}^{T}G(s)\Big{(}N^{-\alpha}g(\eta_{s}(k_{j,N}))-(N^{-1}\eta_{s}(k_{j,N}))^{\alpha}\Big{)}ds\Big{|}>\delta\Big{]}=0.$
(7.11)

Let $\\{F_{\theta}(x)\\}$ be a sequence of nonnegative smooth functions such
that $F_{\theta}$’s are supported on $(x_{j}-\theta,x_{j}+\theta)$,
$\|F_{\theta}\|_{\infty}\leq 1$, and $F_{\theta}(x)=1$ for
$|x-x_{j}|\leq\theta/2$. Then,

$|\langle F_{\theta},\pi^{N}_{s}\rangle-N^{-1}\eta_{s}(k_{j,N})|\leq
B_{\theta,N}(\eta_{s})$

where
$B_{\theta,N}(\eta)=N^{-1}\sum_{k\notin{\mathfrak{D}}_{N}}\mathbbm{1}_{(-\theta,\theta)}(k/N-x_{j})\,\eta(k)$
for $\theta$ small and $N$ large. As $0<\alpha\leq 1$, we have

$\mathbb{E}_{N}\big{[}\big{|}\langle
F_{\theta},\pi^{N}_{s}\rangle^{\alpha}-(N^{-1}\eta_{s}(k_{j,N}))^{\alpha}\big{|}\big{]}\leq\mathbb{E}_{N}\big{[}(B_{\theta,N})^{\alpha}\big{]}\leq\mathbb{E}_{N}\big{[}B_{\theta,N}\big{]}^{\alpha},$

which vanishes as $N\to\infty$ and $\theta\to 0$ by (5.4) in Lemma 5.2.
Therefore, we may replace $N^{-1}\eta_{s}(k_{j,N})$ in (7.11) to obtain

$\lim_{\theta\to
0}\limsup_{N\to\infty}\mathbb{P}_{N}\Big{[}\Big{|}\int_{0}^{T}G(s)\Big{(}N^{-\alpha}g(\eta_{s}(k_{j,N}))-\langle
F_{\theta},\pi^{N}_{s}\rangle^{\alpha}\Big{)}ds\Big{|}>\delta\Big{]}=0.$

By Lemma 9.2, $N^{-\alpha}g(\eta_{s}(k_{j,N}))$ may be replaced by
$\lambda_{j}\Phi(\eta_{s}^{\theta N,+}(k_{j,N}))$. Moreover, following Step
$4$ in the proof of Lemma 7.2 to replace $\eta_{s}^{\theta N,+}(k_{j,N})$ with
$\rho^{+,\theta}_{x_{j}}(s)$, we obtain

$\lim_{\theta\to
0}Q\Big{[}\Big{|}\int_{0}^{T}G(s)\Big{(}\lambda_{j}\Phi\big{(}\rho^{+,\theta}_{x_{j}}(s)\big{)}-\langle
F_{\theta},\pi_{s}\rangle^{\alpha}\Big{)}ds\Big{|}>\delta\Big{]}=0.$

Moreover, by Lemma 7.1, $\int_{0}^{T}G(s)\langle
F_{\theta},\pi_{s}\rangle^{\alpha}ds$ converges to

$\int_{0}^{T}G(s)(\pi_{s}(\\{x_{j}\\}))^{\alpha}ds=\int_{0}^{T}G(s)({\mathfrak{m}}_{j}(s))^{\alpha}ds$

almost surely with respect to $Q$. Hence, with these observations, (7.9) in
the lemma holds. Following the argument in Step $5$ of Lemma 7.2, we obtain
(7.10) from (7.9), finishing the proof. ∎

We now turn to the behavior near ‘critical’ slow sites in the $g$ bounded
setting.

######  Lemma 7.5.

Let $g$ be bounded. Then, for any $j\in J_{c}$, $\delta>0$, and $G\in C[0,T]$,
we have

$\lim_{\theta\to
0}Q\Big{[}\int_{0}^{T}|G(s)|\big{(}\Phi\big{(}\rho^{+,\theta}_{x_{j}}(s)\big{)}-\phi_{j}\big{)}ds>\delta\Big{]}=0$
(7.12)

and

$\lim_{\theta\to 0}Q\Big{[}\Big{|}\int_{0}^{T}G(s)({\mathfrak{m}}_{j}(s)\wedge
1)\big{(}\phi_{j}-\Phi\big{(}\rho^{+,\theta}_{x_{j}}(s)\big{)}\big{)}ds\Big{|}>\delta\Big{]}=0.$
(7.13)

where
$\rho^{+,\theta}_{x_{j}}(s):=\|\hat{\iota}_{\theta}\|^{-1}\langle\hat{\iota}_{\theta}(\cdot-
x_{j}),\rho\rangle$, $\phi_{j}=\lambda^{-1}_{j}$, and $a\wedge
b=\min\\{a,b\\}$. Consequently,

$\displaystyle
Q\big{[}\Phi(\rho(t,x_{j}))\leq\lambda_{j}^{-1},\,{\mathfrak{m}}_{j}(t)={\mathfrak{m}}_{j}(t)\mathbbm{1}_{\Phi(\rho(t,x_{j}))=\lambda_{j}^{-1}}.$
(7.14) $\displaystyle\ \ \ \ \ \ \ \ \ \text{ for a.e.~{}$t\in[0,T]$ and all
$j\in J_{c}$}\big{]}=1.$

###### Proof.

Fix $j\in J_{c}$ and $G\in C[0,T]$. We first address (7.12). As $g(\cdot)\leq
1$, it is trivial that, for all $N$

$\mathbb{P}_{N}\Big{[}\int_{0}^{T}|G(s)|\left(\lambda_{j}^{-1}g(\eta_{s}(k_{j,N}))-\phi_{j}\right)ds>\delta\Big{]}=0.$

We now use the local ‘replacement’ Lemma 9.2 to replace
$\lambda_{j}^{-1}g(\eta_{s}(k_{j,N}))$ by $\Phi(\eta_{s}^{\theta
N,+}(k_{j,N}))$, to obtain

$\lim_{\theta\to
0}\limsup_{N\to\infty}\mathbb{P}_{N}\Big{[}\int_{0}^{T}|G(s)|\left(\Phi(\eta_{s}^{\theta
N,+}(k_{j,N}))-\phi_{j}\right)ds>\delta\Big{]}=0.$

Then, (7.12) follows by taking $N\to\infty$, cf. Step $4$ in Lemma 7.2.

To show (7.13), we take the same sequence $\\{F_{\theta}(x)\\}$ from the proof
of Lemma 7.4. We now observe a key microscopic boundary relation:

$\lim_{\theta\to
0}\limsup_{N\to\infty}\mathbb{E}_{N}\Big{[}\int_{0}^{T}\big{|}\left(\langle
F_{\theta},\pi^{N}_{s}\rangle\wedge
1\right)\big{(}\phi_{j}-\lambda_{j}^{-1}g(\eta_{s}(k_{j,N}))\big{)}\big{|}ds\Big{]}=0.$
(7.15)

Indeed, as $\lim_{n\to\infty}g(n)=1$, let $A=A(\varepsilon)$ be such that
$|g(n)-1|<\varepsilon$ for all $n\geq A$. Then, in the above display, on the
one hand, the absolute value is bounded above by $\varepsilon$ when
$\eta_{s}(k_{j,N})\geq A$. On the other hand, $\langle
F_{\theta},\pi^{N}_{s}\rangle$ is less than
$N^{-1}(A+\sum_{k}^{\circ,\theta}\eta_{s}(k))$ when $\eta_{s}(k_{j,N})<A$.
Here, the sum $\sum_{k}^{\circ,\theta}$ is over all $k$ such that
$|k/N-x_{j}|\leq\theta$ and $k\neq k_{j,N}$. In considering these cases, when
$\eta_{s}(k_{j,N})<A$ and $\eta_{s}(k_{j,N})\geq A$ respectively, the estimate
(7.15) follows from (5.4) of the ‘particle numbers’ Lemma 5.2.

Now, as $|\langle F_{\theta},\pi^{N}_{s}\rangle\wedge 1|$ is bounded, we
observe that the local ‘replacement’ Lemma 9.2 remains effective when the test
function is taken in form $G(s)\left(\langle
F_{\theta},\pi^{N}_{s}\rangle\wedge 1\right)$. Thus, we may replace
$\lambda_{j}^{-1}g(\eta_{s}(k_{j,N}))$ by $\Phi(\eta_{s}^{\theta
N,+}(k_{j,N}))$ and obtain

$\lim_{\theta\to
0}\limsup_{N\to\infty}\mathbb{P}_{N}\Big{[}\Big{|}\int_{0}^{T}G(s)\left(\langle
F_{\theta},\pi^{N}_{s}\rangle\wedge
1\right)\big{(}\phi_{j}-\Phi(\eta_{s}^{\theta
N,+}(k_{j,N}))\big{)}ds\Big{|}>\delta\Big{]}=0.$

Again, by following Step $4$ in Lemma 7.2, we have

$\lim_{\theta\to 0}Q\Big{[}\Big{|}\int_{0}^{T}G(s)\left(\langle
F_{\theta},\pi_{s}\rangle\wedge
1\right)\big{(}\phi_{j}-\Phi\big{(}\rho^{+,\theta}_{x_{j}}(s)\big{)}\big{)}ds\Big{|}>\delta\Big{]}=0.$

Also, by Lemma 7.1, it holds that $\lim_{\theta\to
0}\int_{0}^{T}\left|\left(\langle F_{\theta},\pi_{s}\rangle\wedge
1\right)-\left({\mathfrak{m}}_{j}(s)\wedge 1\right)\right|ds=0$ almost surely
with respect to $Q$. As $\phi_{j}$ and $\Phi(\cdot)$ are bounded, we conclude
the proof of (7.13).

It remains to show (7.14). Following Step $5$ in Lemma 7.2, we obtain from
(7.12) that, $Q$-almost surely, $\Phi(\rho(s,x_{j}))\leq\phi_{j}$ for a.e.
$t\in[0,T]$ and all $j\in J_{c}$. To finish the proof, it suffices to show
that ${\mathfrak{m}}_{j}(s)=0$ whenever $\Phi(\rho(s,x_{j}))<\phi_{j}$. In
fact, by (7.13), we have $({\mathfrak{m}}_{j}(t)\wedge
1)\big{(}\phi_{j}-\Phi\big{(}\rho(s,x_{j})\big{)}\big{)}=0$. As we have
$\Phi(\rho(s,x_{j}))\leq\phi_{j}$, we have that ${\mathfrak{m}}_{j}(s)\wedge
1=0$ (therefore ${\mathfrak{m}}_{j}(s)=0$) when
$\Phi(\rho(s,x_{j}))<\phi_{j}$, completing the proof. ∎

##  8\. Local $1$ and $2$-block estimates of bulk sites

In this section we address the $1$ and $2$-block estimates for non-defect
sites $\mathbb{T}_{N}\setminus{\mathfrak{D}}_{N}$. These estimates are
obtained through a Rayleigh-type estimation of a variational eigenvalue
expression derived from a Feynman-Kac bound. These bounds will hold in either
$g(n)\sim n^{\alpha}$ or $g$ bounded settings.

###  8.1. Local $1$-block estimate

We start from recalling the concept of spectral gap which is used to prove our
local $1$-block estimate. For $k\in\mathbb{T}_{N}$ and $l\geq 1$, define the
set
$\Lambda_{k,l}=\left\\{k-l,k-l+1,\ldots,k+l\right\\}\subset\mathbb{T}_{N}$.
Let $\Omega_{k,l}=\mathbb{N}_{0}^{\Lambda_{k,l}}$ be the state space of
configurations restricted on sites $\Lambda_{k,l}$. Define the state space of
configurations with exactly $j$ particles on the sites $\Lambda_{k,l}$:

$\Omega_{k,l,j}=\Big{\\{}\eta\in\Omega_{k,l}:\sum_{x\in\Lambda_{k,l}}\eta(x)=j\Big{\\}}.$

Consider the generator $L_{k,l}$ on $\Omega_{k,l}$ given by

$\begin{split}L_{k,l}f(\eta)=&\sum_{x,x+1\in\Lambda_{k,l}}\Big{\\{}g(\eta(x))\left[f\left(\eta^{x,x+1}\right)-f(\eta)\right]+g(\eta(x))\left[f\left(\eta^{x+1,x}\right)-f(\eta)\right]\Big{\\}}.\end{split}$

Recall the generator $L_{N}$ from (2.1). Notice that, for each $k,l$ such that
$\Lambda_{k,l}\cap{\mathfrak{D}}_{N}=\emptyset$, the generator $L_{k,l}$
coincides with $L_{N}$ localized on $\Lambda_{k,l}$.

For any $\rho>0$, let $\nu_{\rho}$ be the product measure on
$\Omega=\mathbb{N}_{0}^{\mathbb{T}_{N}}$ with common marginal
$\pazocal{P}_{\Phi(\rho)}$ on each site $k\in\mathbb{T}_{N}$, and let
$\nu^{\rho}_{k,l}$ be its restriction to $\Omega_{k,l}$. Let $\nu_{k,l,j}$ be
$\nu_{k,l}^{\rho}$ conditioned on total number of particles on $\Lambda_{k,l}$
being $j$. Notice that $\nu_{k,l,j}$ does not depend on $\rho$. It is well-
known that both $\nu^{\rho}_{k,l}$ and $\nu_{k,l,j}$ are invariant measures
with respect to the localized generator $L_{k,l}$ (cf. [1]). For
$\kappa=\nu_{k,l}^{\rho}$ or $\nu_{k,l,j}$, the corresponding Dirichlet form
is given by

$\begin{split}E_{\kappa}\left[f(-L_{k,l}f)\right]=&\sum_{x,x+1\in\Lambda_{k,l}}E_{\kappa}\left[g(\eta(x))\left(f\left(\eta^{x,x+1}\right)-f(\eta)\right)^{2}\right].\end{split}$
(8.1)

For $j\geq 1$, let $b_{l,j}$ be the spectral gap of $-L_{k,l}$ on
$\Omega_{k,l,j}$ (cf. p. 374, [18]):

$b_{l,j}:=\inf_{f}\dfrac{E_{\nu_{k,l,j}}[f(-L_{k,l}f)]}{{\rm
Var}_{\nu_{k,l,j}}(f)}$ (8.2)

where the infimum is taken over all $L^{2}(\nu_{k,l,j})$ functions $f$ from
$\Omega_{k,l,j}$ to $\mathbb{R}$. For all $l,j\geq 1$, as $\Omega_{k,l,j}$ is
a finite space and the localized process is irreducible, we have $b_{l,j}>0$.
As a consequence, we have the following Poincaré inequality: for all $f\in
L^{2}(\nu_{k,l,j})$

${\rm Var}_{\nu_{k,l,j}}(f)\leq
C_{l,j}E_{\nu{k,l,j}}\left[f(-L_{k,l}f)\right]$ (8.3)

where $C_{l,j}:=b_{l,j}^{-1}<\infty$ for $j\geq 1$ and $C_{l,0}=0$. We remark
that even though, for a large class of $g(\cdot)$’s, sharp estimates of
$b_{l,j}$ are available in the literature, we will only need that $b_{l,j}$ is
strictly positive for all $l,j\geq 1$.

We now prove the local $1$-block estimate for regular sites:

######  Lemma 8.1 (Local $1$-block estimate).

For any bounded function $G$ on $[0,T]\times\mathbb{T}$, we have

$\limsup_{l\to\infty}\limsup_{N\to\infty}\sup_{k,k^{\prime}}\mathbb{E}_{N}\Big{[}\Big{|}\int_{0}^{T}G(s,k^{\prime}/N)\left(g(\eta_{s}(k^{\prime}))-\Phi(\eta_{s}^{l}(k))\right)ds\Big{|}\Big{]}=0$

where the $\sup$ is taken over all $k$ and $k^{\prime}$ such that
$k^{\prime}\in\Lambda_{k,l}$ and
$\Lambda_{k,l}\cap{\mathfrak{D}}_{N}=\emptyset$.

###### Proof.

We separate the argument into $5$ steps.

Step 1. We first introduce a cutoff of large densities. Let

$V_{k,k^{\prime},l}(s,\eta):=G(s,k^{\prime}/N)\left(g(\eta(k^{\prime}))-\Phi(\eta^{l}(k))\right).$

As $g(n)\leq g^{*}n$ and $\Phi(x)\leq g^{*}x$, we have

$\mathbb{E}_{N}\Big{[}\Big{|}\int_{0}^{T}V_{k,k^{\prime},l}(s,\eta_{s})\mathbbm{1}_{\eta_{s}^{l}(k)>A}ds\Big{|}\Big{]}\leq
g^{*}\|G\|_{\infty}\int_{0}^{T}\mathbb{E}_{N}\left[\left(\eta_{s}(k^{\prime})+\eta_{s}^{l}(k)\right)\mathbbm{1}_{\eta_{s}^{l}(k)>A}\right]ds.$

By attractiveness (cf. Section 3.2), in both $g(n)\sim n^{\alpha}$ and bounded
settings (as $\Lambda_{k,l}\cap{\mathfrak{D}}_{N}=\emptyset$, the stochastic
bound $\kappa^{N}_{c^{\prime}}$ and $\mathscr{R}^{N}_{c^{\prime}}$ agree with
$\nu_{c^{\prime}}$ on $\Lambda_{k,l}$), the last expectation is bounded above
by

$\begin{split}E_{\mathscr{R}_{c^{\prime}}^{N}}\left[\left(\eta(k^{\prime})+\eta^{l}(k)\right)\mathbbm{1}_{\eta^{l}(k)>A}\right]\leq&A^{-1}E_{\nu_{c^{\prime}}}\left[\left(\eta(k^{\prime})\eta^{l}(k)\right)+\left(\eta^{l}(k)\right)^{2}\right]\\\
\leq&A^{-1}E_{\nu_{c^{\prime}}}\left[\left(\eta(k^{\prime})\right)^{2}+2\left(\eta^{l}(k)\right)^{2}\right].\end{split}$

Notice that
$\left(\eta^{l}(k)\right)^{2}\leq(2l+1)^{-1}\sum_{j\in\Lambda_{k,l}}\left(\eta(j)\right)^{2}$
and, under $\nu_{c^{\prime}}$, $\left\\{\eta(j)\right\\}_{j\in\Lambda_{k,l}}$
has common distribution $\pazocal{P}_{\Phi(c^{\prime})}$. We obtain
$\mathbb{E}_{N}\big{[}\big{|}\int_{0}^{T}V_{k,k^{\prime},l}(s,\eta_{s})\mathbbm{1}_{\eta_{s}^{l}(k)>A}ds\big{|}\big{]}\to
0$ as $N$, $l$, and $A$ approach $\infty$ in this order.

Therefore, to prove the lemma, it will be enough to show, for all $A>0$, that

$\limsup_{l\to\infty}\limsup_{N\to\infty}\sup_{k,k^{\prime}}\mathbb{E}_{N}\Big{[}\Big{|}\int_{0}^{T}V_{k,k^{\prime},l,A}(s,\eta_{s})ds\Big{|}\Big{]}=0$

where
$V_{k,k^{\prime},l,A}(s,\eta):=V_{k,k^{\prime},l}(s,\eta)\mathbbm{1}_{\\{\eta^{l}(k)\leq
A\\}}$.

Step 2. As $H(\mu^{N}|\mathscr{R}_{c_{0}}^{N})\leq CN$ for some $C<\infty$, it
follows from the entropy inequality

$\begin{split}\mathbb{E}_{N}\Big{[}\Big{|}\int_{0}^{T}V_{k,k^{\prime},l,A}(s,\eta_{s})ds\Big{|}\Big{]}\leq\dfrac{C}{\gamma}+\dfrac{1}{\gamma
N}\ln\mathbb{E}_{\mathscr{R}_{c_{0}}^{N}}\Big{[}\exp\Big{\\{}\gamma
N\Big{|}\int_{0}^{T}V_{k,k^{\prime},l,A}(s,\eta_{s})ds\Big{|}\Big{\\}}\Big{]}.\end{split}$

The absolute value in the right hand side of last inequality can be dropped by
using $e^{|x|}\leq e^{x}+e^{-x}$. By Feynman-Kac formula (cf. p.336, [18]),

$\begin{split}\dfrac{1}{\gamma
N}\ln\mathbb{E}_{\mathscr{R}_{c_{0}}^{N}}\Big{[}\exp\Big{\\{}\gamma
N\int_{0}^{T}V_{k,k^{\prime},l,A}(s,\eta_{s})ds\Big{\\}}\Big{]}\leq\dfrac{1}{\gamma
N}\int_{0}^{T}\lambda_{N,l}(s)ds\end{split}$

where $\lambda_{N,l}(s)$ is the largest eigenvalue of $N^{2}L_{N}+\gamma
NV_{k,k^{\prime},l,A}(s,\eta)$.

Step 3. Fix $s\in[0,T]$; we will omit the argument $s$ in $\lambda_{N,l}(s)$
to simplify notation, that is $\lambda_{N,l}=\lambda_{N,l}(s)$. Note the
variational formula for $\lambda_{N,l}$:

$(\gamma
N)^{-1}\lambda_{N,l}=\sup_{f}\big{\\{}E_{\mathscr{R}_{c_{0}}^{N}}\big{[}V_{k,k^{\prime},l,A}f\big{]}-\gamma^{-1}NE_{\mathscr{R}_{c_{0}}^{N}}\big{[}\sqrt{f}(-L_{N}\sqrt{f})\big{]}\big{\\}},$

where the supremum is over all $f$ which are densities with respect to
$\mathscr{R}_{c_{0}}^{N}$ (cf. [18], p. 377).

Let $f_{k,l}=E_{\mathscr{R}_{c_{0}}^{N}}\big{[}f|\Omega_{k,l}\big{]}$, be the
conditional expectation of $f$ given the variables on $\Lambda_{k,l}$. For
$\mu=\mathscr{R}_{c_{0}}^{N}$ let $\mu_{k,l}$ be $\mu$ restricted on
$\Lambda_{k,l}$. Clearly, $\mu_{k,l}=\nu_{k,l}^{c_{0}}$. Since the Dirichlet
form $E_{\mathscr{R}_{c_{0}}^{N}}\left[\sqrt{f}(-L_{N}\sqrt{f})\right]$ is
convex, we have

$(\gamma
N)^{-1}\lambda_{N,l}\leq\sup_{f_{k,l}}\left\\{E_{\mu_{k,l}}\left[V_{k,k^{\prime},l,A}f_{k,l}\right]-\gamma^{-1}NE_{\mu_{k,l}}\left[\sqrt{f_{k,l}}(-L_{k,l}\sqrt{f_{k,l}})\right]\right\\}.$

Step 4. We now decompose $f_{k,l}d\mu_{k,l}$ with respect to sets
$\Omega_{k,l,j}$ of configurations with total particle number $j$ on
$\Lambda_{k,l}$:

$E_{\mu_{k,l}}\left[V_{k,k^{\prime},l,A}f_{k,l}\right]=\sum_{j\geq
0}c_{k,l,j}(f)\int V_{k,k^{\prime},l,A}f_{k,l,j}d\mu_{k,l,j},$ (8.4)

where $c_{k,l,j}(f)=\int_{\Omega_{k,l,j}}f_{k,l}d\mu_{k,l}$ and
$f_{k,l,j}=c_{k,l,j}(f)^{-1}\mu_{k,l}\left(\Omega_{k,l,j}\right)f_{k,l}$.
Here, $\sum_{j\geq 0}c_{k,l,j}=1$ and $f_{k,l,j}$ is a density with respect to
$\mu_{k,l,j}$.

Straightforwardly, on $\Omega_{k,l,j}$, we have

$\begin{split}\dfrac{L_{k,l}\sqrt{f_{k,l}}}{\sqrt{f_{k,l}}}=\dfrac{L_{k,l}\sqrt{f_{k,l,j}}}{\sqrt{f_{k,l,j}}}.\end{split}$

Using (8.4), we write

$E_{\mu_{k,l}}\Big{[}\sqrt{f_{k,l}}(-L_{k,l}\sqrt{f_{k,l}})\Big{]}=\sum_{j\geq
0}c_{k,l,j}(f)E_{\mu_{k,l,j}}\Big{[}\sqrt{f_{k,l,j}}(-L_{k,l}\sqrt{f_{k,l,j}})\Big{]}.$

Then, we get

$(\gamma N)^{-1}\lambda_{N,l}\leq\sup_{0\leq j\leq
A(2l+1)}\sup_{f}\left\\{E_{\mu_{k,l,j}}\left[V_{k,k^{\prime},l,A}f\right]-\gamma^{-1}NE_{\mu_{k,l,j}}\left[\sqrt{f}(-L_{k,l}\sqrt{f})\right]\right\\},$

where the second supremum is on all densities $f$ with respect to
$\mu_{k,l,j}$.

Step 5. Let

$\widehat{V}_{k,k^{\prime},l,A}=V_{k,k^{\prime},l,A}-E_{\mu_{k,l,j}}\left[V_{k,k^{\prime},l,A}\right].$

Let $C_{l,A,G}$ be such that $\|\widehat{V}_{k,k^{\prime},l,A}\|_{\infty}\leq
C_{l,A,G}$. Recall $C_{l,j}$ the inverse spectral gap of $L_{k,l}$ (cf.
(8.3)). We now use the Rayleigh expansion (cf. [18], pp. 375–376, Appendix 3,
Theorem 1.1)

$\begin{split}&E_{\mu_{k,l,j}}\left[\widehat{V}_{k,k^{\prime},l,A}f\right]-\gamma^{-1}NE_{\mu_{k,l,j}}\big{[}\sqrt{f}(-L_{k,l}\sqrt{f})\big{]}\\\
\leq&\dfrac{\gamma N^{-1}}{1-2C_{l,A,G}C_{l,j}\,\gamma
N^{-1}}E_{\mu_{k,l,j}}\left[\widehat{V}_{k,k^{\prime},l,A}(-L_{k,l})^{-1}\widehat{V}_{k,l,A}\right].\end{split}$
(8.5)

The spectral gap of $L_{k,l}$ also implies that $\|L_{k,l}^{-1}\|_{2}$, the
$L^{2}(\mu_{k,l,j})$ norm of the operator $L_{k,l}^{-1}$ on mean zero
functions, is less than or equal to $C_{l,j}$. Now, by Cauchy-Schwarz and the
estimate of $\|L_{k,l}^{-1}\|_{2}$, we have

$\begin{split}E_{\mu_{k,l,j}}\left[\widehat{V}_{k,k^{\prime},l,A}(-L_{k,l})^{-1}\widehat{V}_{k,k^{\prime},l,A}\right]\leq
C_{l,j}E_{\mu_{k,l,j}}\left[\widehat{V}_{k,k^{\prime},l,A}^{2}\right]\leq
C_{l,j}C^{2}_{l,A,G}.\end{split}$

Accordingly, retracing our steps, noting (8.5), we have that
$\mathbb{E}_{N}\big{[}\big{|}\int_{0}^{T}V_{k,k^{\prime},l,A}(\eta_{s})ds\big{|}\big{]}$
is less than or equal to

$\begin{split}\dfrac{C_{0}}{\gamma}+\sup_{0\leq j\leq A(2l+1)}\dfrac{T\gamma
N^{-1}C_{l,j}C_{l,A,G}^{2}}{1-2C_{l,A,G}C_{l,j}\,\gamma N^{-1}}+T\sup_{0\leq
j\leq A(2l+1)}E_{\mu_{k,l,j}}\left[V_{k,k^{\prime},l,A}\right].\end{split}$

Taking $N\to\infty$, first $\sup$ term vanishes. Notice that the expression
$\sup E_{\mu_{k,l,j}}\left[V_{k,k^{\prime},l,A}\right]$ is independent of $N$
and vanishes as $l\to\infty$. In fact, as $\mu_{k,l,j}=\nu_{k,l,j}$ is
translation-invariant