#  Branching process and homogeneization for epidemics on spatial random
graphs

Vincent Bansaye  École Polytechnique CMAP, École Polytechnique, Route de
Saclay 91128 Palaiseau Cedex, FRANCE<EMAIL_ADDRESS>and
Michele Salvi  Dipartimento di Matematica, Università di Roma Tor Vergata, Via
della ricerca scientifica 1, 00133, Rome, Italy<EMAIL_ADDRESS>https://www.mat.uniroma2.it/~salvi/

###### Abstract.

Consider a graph where the sites are distributed in space according to a
Poisson point process on $\mathbb{R}^{n}$. We study a population evolving on
this network, with individuals jumping between sites with a rate which
decreases exponentially in the distance. Individuals give also birth
(infection) and die (recovery) at constant rate on each site. First, we
construct the process, showing that it is well-posed even when starting from
non-bounded initial conditions. Secondly, we prove hydrodynamic limits in a
diffusive scaling. The limiting process follows a deterministic reaction
diffusion equation. We use stochastic homogenization to characterize its
diffusion coefficient as the solution of a variational principle. The proof
involves in particular the extension of a classic Kipnis–Varadhan estimate to
cope with the non-reversiblity of the process, due to births and deaths. This
work is motivated by the approximation of epidemics on large networks and
results are extended to more complex graphs including percolation of edges.

_Key words:_ Epidemics, branching process, random graphs, stochastic
homogenization.

_MSC 2020:_ 92D25, 05C81, 35B27.

##  1\. Introduction and main results

Consider a graph $\mathcal{G}$ whose vertices $V$ are placed according to a
Poisson point process on $\mathbb{R}^{n}$ with $n\geq 2$ and with edge set $E$
drawn from some distribution. Attach to each unoriented edge $\\{x,y\\}\in E$
a rate $r(x,y)=r(y,x)={\rm e}^{-\|x-y\|}$, where $\|\cdot\|$ indicates the
Euclidean distance. Consider individuals that perform independent random walks
on $\mathcal{G}$ with jump rates $r(x,y)$. They also give birth to new
individuals at rate $b\geq 0$ and die at rate $d\geq 0$. The main goal of the
present work is to describe the limiting behaviour of this particle system
under a diffusive rescaling.

The motivation for studying this kind of process comes from the analysis of
real-world networks with agents moving on spatially inhomogeneous structures.
Metapopulation models (or metacommunity for several species) aim at describing
the habitat of a population as a collection of patches. Exchanges between two
patches can depend on several features, in particular the distance. From the
pioneering works of Levins [Levins(1969)], metapopulations have a long story
in biology and ecology. Issues come from conservation of species (see e.g.
[Bascompte and Sole(1996), Bansaye and Lambert(2013)]), evolution of
dispersion (see e.g. [Hastings(1983)]), impact of fragmentation of habitats
(see e.g. [Hiebeler(2000)]) and effect of heterogeneity of habitats (see e.g.
[Pulliam(1988)]). While for the sake of simplicity one would consider a small
number of patches, applications often ask for the study of large
metapopulations. As far as we know the literature, large metapopulations are
considered either in a mean field approximation (see [Levins(1969)]) or with a
spatially explicit large structure using cellular automates and simulations
([Bascompte and Sole(1996)]) or in a periodic environment. Random networks
provide a relevant mathematical framework to analyze models which do not fall
in the mean field approximation and that would make explode the parameters’
complexity if considered as large explicit graphs. Rigorous works which
combine motion and demography (birth, death, infections…) on large random
landscapes are rare for now. Our interest is understanding how an epidemics
would spread on such structures. As a driving example, one can consider the
spread of an infection among cattle on the French network of farms, see [Qi et
al.(2019)Qi, Beaunée, Arnoux, Dutta, Joly, Vergu, and Ezanno]. In this first
work we identify the diffusive behaviour of an epidemics in its first stages,
which corresponds to the classic branching process approximation for small
ratio of infected individuals. In this case $b$ represents the contamination
rate and $d$ is the recovery rate for the infected population. This
approximation is valid on a time window where the infected population remains
locally small compared to the population size, see e.g. [Ball and
Donnelly(1995), Barbour and Reinert(2013), Montagnon(2019)] for the classical
mixed SIR model.

From a mathematical point of view, the first challenge is represented by the
unboundedness of the jump rates: on the one hand a site in $V$ can have a huge
number of close-by neighbours, so that the jump rate of a single individual
can be arbitrarily large at that site; on the other hand there is no
restriction on the number of individuals that can occupy a given site. Proving
that such a process is well-posed is in itself not trivial: both classic and
more recent techniques for proving existence fail to apply to our framework.
The second challenge is represented by the irregularity of the support $V$
combined with the lack of reversibility of the system, due to births and
deaths. In order to study the limiting behaviour of the process, we need to
gather approaches coming from statistical mechanics and mathematical biology.
To be more precise, the way we can cope with the random geometry of the
underlying graph is through stochastic homogenization and in particular the
results of [Faggionato(2022a)]. The theory of homogenization, first started in
a deterministic context by analysts, describes how the microscopic
irregularities of a medium affect the macroscopic behavior of the system. It
is by now well understood how to use this technique to derive hydrodynamic
limits for reversible particles systems, see e.g. [Gonçalves and Jara(2008),
Faggionato(2022b)]. Yet, to our knowledge, one fundamental requirement for
obtaining these results has been the reversibility of the process. In our
context we need to adapt some tools to non–reversible population models, in
the vein of e.g. [Kurtz(1981), Bansaye and Méléard(2015)]. In particular, a
fundamental ingredient is the extension of an inequality for the supremum of a
particle process due to Kipnis and Varadhan. This estimate is required for the
proof of tightness, for the identification of the limit and to show that this
limit has a density with respect to the Lebesgue measure.

###  1.1. Model and main results

For some probability space $(\Omega,\mathbb{P},\mathcal{F})$ and
$\omega\in\Omega$, let $V=V(\omega)$ be the points of a Poisson point process
on $\mathbb{R}^{n}$ with $n\geq 2$ and intensity $\gamma>0$ under
$\mathbb{P}$. Let $E=E(\omega)=\\{\\{x,y\\},\,x,y\in V\\}$ be the set of
unoriented edges between the points of $V$. We will consider at first the
complete graph $\mathcal{G}=(V,E)$ as a support for our particle system, while
in Section 6 we will discuss how to extend our results by generalizing
$\mathcal{G}$ via bond percolation. Given $\omega$ and a configuration of
particles $\eta\in\mathbb{N}^{V}$ consider the transitions

$\displaystyle\eta\longrightarrow\begin{cases}\eta^{x,y}\quad&\mbox{with rate
}\eta(x)r(x,y)\\\ \eta^{x,+}\quad&\mbox{with rate }b\,\eta(x)\\\
\eta^{x,-}\quad&\mbox{with rate }d\,\eta(x)\end{cases}$ (1)

where $\eta^{x,y}=\eta-\mathbf{1}\\!_{\,x}+\mathbf{1}\\!_{\,y}$ is the
configuration obtained from $\eta$ by subtracting one particle in $x\in V$ and
adding one in $y\in V$, $\eta^{x,+}=\eta+\mathbf{1}\\!_{\,x}$ adds one
particle in $x\in V$ and $\eta^{x,-}=\eta-\mathbf{1}\\!_{\,x}$ has one
particle less in $x\in V$. The positive numbers

$r(x,y)=r(y,x)={\rm e}^{-\|x-y\|}$

are the jump rates for each particle to go from point $x\in V$ to point $y\in
V$, and vice-versa. For simplicity we set $r(x,x)=0$ for all $x\in V$. We let

$r(x):=\sum_{y\in V:\,\\{x,y\\}\in E}r(x,y)$

be the total jump rate of a particle at site $x\in V$. It is not hard to show
that, $\mathbb{P}$–almost surely, $r(x)$ is finite for every $x\in V$. The
parameters $b,d\geq 0$ are the individual rate of birth and death of the
particles, respectively.

For a given realization $\mathcal{G}(\omega)$ of the graph, we introduce a
probability space with measure $P^{\omega}$ under which we will construct our
particle process. $E^{\omega}$ indicates the associated expectation. Let
$\eta_{0}$ be the initial configuration of particles. Our first result
establishes that, for $\mathbb{P}$–almost every $\omega$, there exists a
Markov process with jump rates given by (1) as soon as $\eta_{0}$ has
uniformly bounded expectation on each site.

######  Theorem 1.1.

For $\mathbb{P}$-a.a. $\omega\in\Omega$ the following holds. Let $\eta_{0}$ be
a random variable on $\mathbb{N}^{V}$ such that, for some $M\in\mathbb{N}$,
one has $\mathbb{E}^{\omega}[\eta_{0}(x)]\leq M$ for all $x\in V$. Then, for
all $T>0$, there exists a Markov process $(\eta_{t})_{t\in[0,T]}$ with initial
value $\eta_{0}$ and paths in the Skohorod space
$\mathcal{D}([0,T],\mathbb{N}^{V})$ that satisfies the following: for
functions $f_{G}:\mathbb{N}^{V}\to\mathbb{R}$ of the form
$f_{G}(\eta)=\sum_{x\in V}G(x)\eta(x)$ with $G$ compactly supported on
$\mathbb{R}^{n}$, the generator $\mathcal{L}$ of $(\eta_{t})_{t\in[0,T]}$ is
given by

$\displaystyle\mathcal{L}f_{G}(\eta)=$ $\displaystyle\sum_{x,y\in
V}\eta(x)r(x,y)\big{(}G(y)-G(x)\big{)}+\sum_{x\in
V}\eta(x)\big{(}b-d\big{)}G(x)\,.$ (2)

Our second result establishes the hydrodynamic limit of the process. Let
$\mathcal{M}(\mathbb{R}^{n})$ be the Polish space of non-negative Radon
measures on $\mathbb{R}^{n}$ endowed with the vague topology. For
$\pi\in\mathcal{M}(\mathbb{R}^{n})$ and a continuous function
$G\in\mathcal{C}(\mathbb{R}^{n})$ we write
$\langle\pi,G\rangle=\int_{\mathbb{R}^{n}}G(y)\,\pi({\rm d}y)$. We consider a
scaling parameter $N\in\mathbb{N}$ and associate to each element
$\eta\in\mathbb{N}^{V}$ the empirical measure
$\pi^{N}=\pi^{N}(\eta)=N^{-n}\sum_{x\in
V}\eta(x)\delta_{x/N}\in\mathcal{M}(\mathbb{R}^{n})$, where $\delta_{y}$
represents a Dirac mass at $y\in\mathbb{R}^{n}$. Conversely, we can recover
$\eta$ from $\pi^{N}$ via
$\eta(\cdot)=\eta(\pi^{N})(\cdot)=N^{n}\pi^{N}(\cdot/N)$, so that for any
fixed $N$ we may use $\pi^{N}$ and $\eta$ indifferently. In this work, we are
interested in the regime where the motion is faster than births and deaths
(resp. of infection and recovery rates for epidemics). Thus, for a given $N$,
we introduce now the process $\eta^{N}$ with sped–up motion. For $G$ compactly
supported, its generator is given by (recall that $f_{G}(\eta)=\sum_{x\in
V}G(x)\eta(x)$)

$\displaystyle\mathcal{L}^{N}f_{G}(\eta)=$ $\displaystyle\sum_{x\in
V}\eta(x)L^{N}G(x/N)+\sum_{x\in V}\eta(x)\big{(}b-d\big{)}G(x/N)\,.$ (3)

Here

$\displaystyle L^{N}G(x/N)=\sum_{y\in
V}N^{2}r(x,y)\big{(}G(y/N)-G(x/N)\big{)}$ (4)

is the generator of the random walk on $V/N:=\\{x/N:\,x\in V(\omega)\\}$ with
transition rates $N^{2}r(\cdot,\cdot)$. The associated measure-valued process
is defined as

$\displaystyle\pi_{t}^{N}:=\frac{1}{N^{n}}\sum_{x\in
V}\eta_{t}^{N}(x)\delta_{x/N}\;.$ (5)

Theorem 1.1 guarantees that, for all $T>0$ and fixed $N\in\mathbb{N}$,
$(\pi^{N}_{t})_{t\in[0,T]}$ is a well-defined Markov process with values in
$\mathcal{D}([0,T],\mathcal{M}(\mathbb{R}^{n}))$, the space of measure-valued
càdlàg processes.

For the scaling limit, we need to consider initial conditions such that the
tails of $\eta_{0}$ are dominated by a product of (translated) Poisson
distributions indexed by $V$, as precisely defined here below. This allows in
particular to invoke the existence and characterization stated in Theorem 1.1.
For example, one can take configurations with a number of particles that is a
constant on each site or that is distributed as i.i.d. Poisson random
variables, or a sum of the two. We also need that the initial conditions
converge as $N$ goes to infinity. For a given realization of the graph
$\omega\in\Omega$, we make thus the following assumption.

######  Domination $\&$ Convergence Assumption.

The sequence of random configurations $(\eta_{0}^{N})_{N\in\mathbb{N}}$
satisfies the following:

  * (i)

There exists $M\in\mathbb{N}_{0}$ and $\rho>0$ such that for any
$N\in\mathbb{N}$, $x\in V$ and for any $A\subset V$ and $(n_{x})_{x\in
A}\in\mathbb{N}^{|A|}$,

$\displaystyle P^{\omega}\Big{(}\forall x\in A,\,\eta_{0}^{N}(x)\geq
M+n_{x}\Big{)}\leq\prod_{x\in
A}\Big{(}\sum_{j=n_{x}}^{\infty}\frac{\rho^{j}{\rm e}^{-\rho}}{j!}\Big{)}.$
(6)

  * (ii)

There exists a bounded Borel function $\rho_{0}:\,\mathbb{R}^{n}\to[0,\infty)$
such that, for any $\mathcal{C}^{\infty}$ function with compact support
$G\in\mathcal{C}^{\infty}_{c}(\mathbb{R}^{n})$,

$\displaystyle\lim_{N\to\infty}N^{-n}\sum_{x\in
V}\eta_{0}^{N}(x)G(x/N)=\int_{\mathbb{R}^{n}}G(x)\rho_{0}(x)\,{\rm d}x\,$ (7)

in $P^{\omega}$–probability.

Consider now $B(\Omega)$, the family of bounded Borel functions on $\Omega$,
and let $\sigma^{2}\geq 0$ be characterized by the variational formula

$\displaystyle\sigma^{2}:=\frac{1}{2}\inf_{\psi\in
B(\Omega)}\mathbb{E}_{0}\Big{[}\sum_{y\in
V}r(0,y)\big{(}y_{1}+\psi(\theta_{y}\omega)-\psi(\omega)\big{)}^{2}\Big{]}\,.$
(8)

Here $y_{1}$ denotes the first coordinate of $y\in\mathbb{R}^{n}$ and
$\theta_{y}\omega$ is the environment translated by $y$ (see Section 3.1 for
the precise meaning of this). The expectation $\mathbb{E}_{0}$ is taken with
respect to the Palm measure relative to the underlying Poisson point process,
which can be obtained by just adding to the configuration a point at the
origin (see [Daley and Vere-Jones(2008)] for a complete account of Palm
measures). Calling $I_{n}$ the $n$-dimensional identity matrix, we point out
that $2\sigma^{2}I_{n}$ is the diffusion matrix of the Brownian motion
obtained by rescaling diffusively the random walk on the Poisson point process
with transition rates $r(x,y)$, see e.g. [Faggionato(2022a)].

######  Theorem 1.2.

For $\mathbb{P}$-a.a. $\omega\in\Omega$ the following holds. Let
$(\eta_{0}^{N})_{N\in\mathbb{N}}$ be a sequence of random variables on
$\mathbb{N}^{V}$ which satisfies the Domination $\&$ Convergence Assumption
for some bounded Borel function $\rho_{0}:\,\mathbb{R}^{n}\to[0,\infty)$. Then
the sequence of processes $\\{(\pi^{N}_{t})_{t\in[0,T]}\\}_{N\in\mathbb{N}}$
with initial value $\pi^{N}_{0}=\pi^{N}(\eta_{0}^{N})$ converges in law in
$\mathcal{D}([0,T],\mathcal{M}(\mathbb{R}^{n}))$ to the deterministic
trajectory $(\rho(t,u)\,{\rm d}u)_{t\in[0,T]}$, where
$\rho(\cdot,\cdot):[0,T]\times\mathbb{R}^{n}\to\mathbb{R}$ is the unique weak
solution of the problem

$\displaystyle\begin{cases}\quad\partial_{t}\rho\,=\,\sigma\Delta\rho+(b-d)\rho\\\
\rho(0,\cdot)\,=\,\rho_{0}\,\end{cases}.$ (9)

Since the sequence of processes converges in distribution to a deterministic
process, we obtain immediately the following convergence in probability.

######  Corollary 1.3.

Under the hypothesis of Theorem 1.2 we have that, for all $t>0$, $G\in
C_{c}(\mathbb{R}^{n})$ and $\varepsilon>0$,

$\displaystyle\lim_{N\to\infty}P^{\omega}\Big{(}\Big{|}N^{-n}\sum_{x\in
V}G(x/N)\eta_{t}(x)-\int_{\mathbb{R}^{n}}G(x)\rho(x,t)\,{\rm
d}x\Big{|}\geq\varepsilon\Big{)}=0\,.$

###  1.2. State of the art, techniques and structure of the paper

The rest of the paper is substantially divided into two parts corresponding to
the proofs of the two main theorems.

Section 2:

Theorem 1.1 establishes the well-posedness of the process. As mentioned
before, our setting does not seem to be treated in the previous literature,
even if we set the rates of birth and death equal to 0. The case $b=d=0$
corresponds to an instance of the so called zero-range process on
$\mathcal{G}$. The existence of the zero-range process on an arbitrary
countable state space was proved in the classical work [Liggett(1973)] and
then under weaker assumptions in [Andjel(1982)]. A first requirement for those
constructions is that, in some sense, the rate of jump of each particle must
be uniformly bounded from above, a condition that fails in our setting due to
the irregularity of $V$. A second problem is that in [Liggett(1973)] and
[Andjel(1982)] one must impose a restriction on the initial configuration of
particles $\eta_{0}$. Namely, one accepts only $\eta_{0}$ satisfying
$\sum_{x\in V}\eta_{0}(x)\alpha(x)<\infty$ for some function $\alpha$ such
that $\sum_{y\in V}p(x,y)\alpha(y)\leq M\alpha(x)$, where $M>0$ is a given
constant and $p(x,y)$ indicates the probability to go from $x$ to $y$ when the
particle jumps. In our case, again because of the irregularity of $V$, this
condition would not allow us to consider, for example, initial conditions with
a constant number of particles on each site. Neither more recent approaches to
prove existence for general particle systems on random graphs, like [Ganguly
and Ramanan(2022)], cover our model, because of the unboundedness of the jump
rates.

We adopt a different approach which borrows from [Andjel(1982)] the idea of
ghost particles. In Section 2.1 we enlarge our space and consider a richer
measure-valued process where, roughly put, particles are labelled and leave a
“ghost” behind them every time they jump to a new site. To show existence of
the original process, we pass through the well-posedness of the stochastic
differential equation (2.1) associated to this richer measure-valued process.
In Section 2.2 we prove the existence when we restrict the dynamics to a
finite subgraph of $V$. We also pin down a key estimate of how many particles
have visited a given compact set up to time $T$ in mean, making use of the
ghosts (Lemma 2.1). In Section 2.3 we extend the existence of the process when
considering the whole infinite graph, but under the condition of having a
finite number of particles at time $0$. This is achieved by showing that the
range covered by the particles stays finite almost surely, see Proposition
2.2. Finally, in Section 2.4, we include in our construction also the case of
an infinite number of initial particles.

Section 3:

In this section we prepare some of the technical tools that are necessary for
the proof of Theorem 1.2. The operator $L^{N}$ can be thought of as a
discretization of the operator $\sigma^{2}\Delta$. For a given
$G\in\mathcal{C}_{c}^{\infty}(\mathbb{R}^{n})$, though, some difficulties
arise if one tries to prove directly the convergence of $L^{N}G$, due to the
possible lack of regularity of this last object. To overcome the problem, one
wants to substitute $G$ by a regularized version $G_{N}^{\lambda}$ for which
$L^{N}G^{\lambda}_{N}$ directly yields the expected limit $\sigma^{2}\Delta
G$. This procedure, introduced in [Jara and Landim(2008)] in the context of
hydrodynamic limits and further developed in [Gonçalves and Jara(2008)],
requires results from stochastic homogenization theory. First of all, in
Section 3.1, we prove that indeed we are allowed to use the homogenization
machinery elaborated in [Faggionato(2022a)]. In Section 3.2 we introduce
$G_{N}^{\lambda}$, prove some bounds in norm for this function and use the
homogenization results to show its convergence to $G$ in $L^{1}$ and in
$L^{2}$, see Lemma 3.3 and Lemma 3.4.

Section 4:

One of the main technical ingredients for proving the hydrodynamic limit of
the sequence of processes $\\{(\pi^{N}_{t})_{t\in[0,T]}\\}_{N\in\mathbb{N}}$
is the Kipnis-Varadhan estimate to control the supremum of the particle
process integrated against a test function, see [Kipnis and Varadhan(1986)].
The estimate in its classic form, though, is valid only for reversible
processes. In Lemma 4.1 we adapt the Kipnis-Varadhan estimate to our model
without births and deaths, which is reversible but presents some issues due to
the irregularity of $V$. In Lemma 4.3 we extend the estimate to the non-
reversible setting. The idea is to look separately at each branch of the
genealogical tree of the particles in the initial configuration. The process
that looks at particles of a given branch can then be dominated by another
(reversible) process, to which we can apply the original Kipnis-Varadhan type
of estimate. This dominating process is obtained via a percolation procedure
on the particles in the initial configuration.

Section 5:

The strategy to prove Theorem 1.2 follows a classical tightness and
identification procedure, which relies on the two previous sections. In
Section 5.1 we consider the martingale problem and show that the process
$M^{N}$ appearing in (74) is an $L^{2}$ martingale via a truncation argument.
We also prove that $M^{N}$ tends to $0$ in $L^{2}$ as $N$ tends to infinity.
This enables to easily conclude the proof of tightness by Aldous’ criterion.
Finally in Lemma 5.6 we prove that a limiting value $(\pi_{t})_{t\in[0,T]}$ of
the sequence $\\{(\pi^{N}_{t})_{t\in[0,T]}\\}_{N\in\mathbb{N}}$ must have a
density with respect to the Lebesgue measure and that it has to satisfy a
suitable differential equation that admits a unique weak solution, cfr. (71).
For simplicity of exposition the proof until this point has been elaborated
for the case without deaths, $d=0$, and we conclude the section by extending
the result to the general case $d>0$, see Section 5.4.

Section 6:

In the very last part of the paper we show that our two main theorems continue
to hold if we consider a percolation procedure on the edges of the complete
graph with nodes $V$. As special cases of interest for applications, we
analyze the long-range percolation and scale-free percolation random graphs.
We conclude in Section 6.2 with a discussion of open problems.

###  1.3. Notation

For a given realization of the graph $\mathcal{G}=\mathcal{G}(\omega)$, with
$\omega\in\Omega$, we recall that $P^{\omega}$ is the probability measure
under which we have built the process defined in Theorem 1.1 and $E^{\omega}$
is the relative expectation. We will make clear each time what initial
distribution of particles the process is starting from, but sometimes we will
further stress the initial condition with a subscript. For example, if the
initial distribution of particles on $\mathcal{G}(\omega)$ is $\mu$, then we
can write $P^{\omega}_{\mu}$.

######  Remark 1.4.

Through most of the proofs of the paper, we will talk directly about
$P^{\omega}$, without specifying each time that $\omega\in\Omega$ is a
realization of the underlying graph sampled according to measure $\mathbb{P}$.
All the processes that appear will evolve under $P^{\omega}$. All the claims
about these processes have to be intended to be true for $\mathbb{P}$–almost
all $\omega$, even when we do not mention it explicitly.

As mentioned before, $\mathcal{M}=\mathcal{M}(\mathbb{R}^{n})$ stands for the
Polish space of non–negative Radon measures on $\mathbb{R}^{n}$ endowed with
the vague topology (namely, a sequence of measures $\nu_{n}$ converges to a
measure $\nu$ in $\mathcal{M}$ if
$\langle\nu_{n},f\rangle\to\langle\nu,f\rangle$ for all
$f\in\mathcal{C}_{c}(\mathbb{R}^{n})$). Consequently,
$\mathcal{D}([0,T],\mathcal{M}(\mathbb{R}^{n}))$ indicates the space of
measure-valued càdlàg processes.

##  2\. Existence and characterization of the process

We will prove the existence of the process just with $d=0$. Indeed, a positive
rate of death of the particles cannot contribute to the explosion of the
process in finite time (if anything, it can help prevent it). So, if the
process is well-defined for $d=0$, a completely analogous construction proves
that it is also well-defined for any $d>0$.

###  2.1. Measure valued process

In order to prove Theorem 1.1 with $d=0$ we will have to consider an auxiliary
process that encodes more information than $(\eta_{t})_{t\in[0,T]}$ and that
lives in the space of measure-valued processes. Let
$\mathcal{I}:=\mathbb{N}\times\bigcup_{k\geq 0}\\{1,2\\}^{k}$. Under measure
$P^{\omega}$, let $(\mathcal{N}_{i}^{x,y})_{i\in\mathcal{I},x,y\in V}$ be a
collection of Poisson point measures on $\mathbb{R}_{+}$ with intensity
$r(x,y)\,{\rm d}t$ and recall that $r(x,x)=0$ for each $x\in V$. These Poisson
point measures are chosen independent for each ordered couple $(x,y)$ and they
are also independent of the initial state $\eta_{0}$. Also let
$(\mathcal{N}_{i}^{b})_{i\in\mathcal{I}}$ be a collection of independent
Poisson point measures on $\mathbb{R}_{+}$ with intensity $b\,{\rm d}t$ and
independent of the $\mathcal{N}_{i}^{x,y}$’s and $\eta_{0}$. The
interpretation is the following: $\mathcal{I}$ shall be thought of as the
space of labels attached to each single particle. Particles that are present
at time $0$ will be just labelled with the natural numbers. If particle $i$ is
present at time $t\geq 0$, we call $X_{t}^{i}\in V$ its position. Suppose a
particle with label $i=n\,i_{1}i_{2}\dots i_{k}$, with $n\in\mathbb{N}$ and
$k\in\mathbb{N}_{0}$ and $i_{j}\in\\{1,2\\}$ for $j=1,\dots,k$, is at position
$X_{t-}^{i}$ at time $t-$ and suppose that
$\mathcal{N}_{i}^{b}(t)-\mathcal{N}_{i}^{b}(t-)=1$. Then particle $i$
disappears at time $t$ and is replaced by two particles with labels
$n\,i_{1}i_{2}\dots i_{k}1$ and $n\,i_{1}i_{2}\dots i_{k}2$ on the same site.
If instead particle $i$ is at $X_{t-}^{i}=x$ at time $t-$ and
$\mathcal{N}_{i}^{x,y}(t)-\mathcal{N}_{i}^{x,y}(t-)=1$, then particle $i$
disappears and generates particle $i1$ at site $y$, that is, $X_{t}^{i1}=y$,
and it leaves behind a ghost particle on site $x$ labelled with $i$. This way
of labelling the particles is commonly known as the Ulam–Harris–Neveu
notation.

Let

$\widehat{\pi}_{t}=\sum_{i\in\mathcal{A}_{t}}\delta_{(i,X_{t}^{i},a)}+\sum_{i\in\mathcal{G}_{t}}\delta_{(i,X_{t}^{i},g)}$

be the measure on $\mathcal{I}\times V\times\\{a,g\\}$ keeping track of
position and state of each particle. For any $i\in\mathcal{I}$ and
$u\in\\{a,g\\}$, one has $\widehat{\pi}_{t}(\\{i\\}\times
V\times\\{u\\})\in\\{0,1\\}$. More precisely,

$\displaystyle\mathcal{A}_{t}:=\\{i\in\mathcal{I}:\widehat{\pi}_{t}(\\{i\\}\times
V\times\\{a\\})>0\\}$ (10)

is the set of particles that are present at time $t$ and that can jump or give
birth, also called alive particles, while

$\displaystyle\mathcal{G}_{t}:=\\{i\in\mathcal{I}:\widehat{\pi}_{t}(\\{i\\}\times
V\times\\{g\\})>0\\}$ (11)

is the set of ghost particles present at time $t$.

Our aim is to construct the process $(\widehat{\pi}_{t})_{t\geq 0}$ which
satisfies $P^{\omega}$–a.s., on every compact set,

$\displaystyle\widehat{\pi}_{t}=\widehat{\pi}_{0}$
$\displaystyle+\int_{0}^{t}\sum_{i\in\mathcal{A}_{s-},\,y\in
V}\left(\delta_{(i1,y,a)}+\delta_{(i,X_{s-}^{i},g)}-\delta_{(i,X_{s-}^{i},a)}\right)\mathcal{N}_{i}^{X_{s-}^{i},y}({\rm
d}s)$
$\displaystyle+\int_{0}^{t}\sum_{i\in\mathcal{A}_{s-}}\left(\delta_{(i1,X_{s-}^{i},a)}+\delta_{(i2,X_{s-}^{i},a)}-\delta_{(i,X_{s-}^{i},a)}\right)\mathcal{N}_{i}^{b}({\rm
d}s)$ (12)

where, for $i\in\mathcal{A}_{t}$, $X_{t}^{i}$ is the unique element $x\in V$
such that $\widehat{\pi}_{t}(\\{i\\}\times\\{x\\}\times\\{a\\})=1$. The
initial configuration $\widehat{\pi}_{0}$ might be random under $P^{\omega}$.

For a Borel set $A\subseteq\mathbb{R}^{n}$, we also let

$\displaystyle\pi_{t}(A):=\widehat{\pi}_{t}(\mathcal{I}\times(A\cap
V)\times\\{a\\})$ (13)

be the total number of alive particles in $A$ at time $t$. We will see that
$(\pi_{t})_{t\in[0,T]}$ corresponds to the measure-valued process
$(\pi_{t}^{N})_{t\in[0,T]}$ introduced in (5) with $N=1$.

We will show the existence of the process $(\eta_{t})_{t\in[0,T]}$ by
constructing the richer process $(\widehat{\pi}_{t})_{t\in[0,T]}$ in three
steps. First, in Section 2.2, we will show the existence of an analogous
process restricted to a finite graph. We will then build on this to extend the
existence of the process on the infinite graph, but only when the initial
configuration has a finite number of particles, see Section 2.3. Finally in
Section 2.4 we will conclude with the existence of
$(\widehat{\pi}_{t})_{t\in[0,T]}$ under the conditions of Theorem 1.1.

###  2.2. Existence of the process on a finite graph

In this section we deal with a version of the process
$(\widehat{\pi}_{t})_{t\in[0,T]}$ for which the underlying spatial point
process is restricted to a finite number of points. Fix a bounded set
$B\subset\mathbb{R}^{n}$. The process $(\widehat{\pi}^{B}_{t})_{t\in[0,T]}$ is
defined as the strong solution of a stochastic differential equation whose
jumps are represented by the Poisson point measures introduced in Section 2.1.
Analogously to (10) and (11), let
$\mathcal{A}_{t}^{B}:=\\{i\in\mathcal{I}:\widehat{\pi}^{B}_{t}(\\{i\\}\times
V\cap B\times\\{a\\})>0\\}$ and
$\mathcal{G}_{t}^{B}:=\\{i\in\mathcal{I}:\widehat{\pi}_{t}^{B}(\\{i\\}\times
V\cap B\times\\{g\\})>0\\}$. For $i\in\mathcal{A}_{t}^{B}$, we write
$X_{t}^{i}$ the location of particle $i$ at time $t$, that is the unique point
$x\in V$ such that $\widehat{\pi}_{t}^{B}(\\{i,x,a\\})=1$. Then
$(\widehat{\pi}^{B}_{t})_{t\in[0,T]}$ is defined via

$\displaystyle\widehat{\pi}_{t}^{B}=\widehat{\pi}_{0}^{B}$
$\displaystyle+\int_{0}^{t}\sum_{i\in\mathcal{A}_{s-}^{B},\,y\in V\cap
B\,}\left(\delta_{(i1,y,a)}+\delta_{(i,X_{s-}^{i},g)}-\delta_{(i,X_{s-}^{i},a)}\right)\mathcal{N}_{i}^{X_{s}^{i},y}({\rm
d}s)$
$\displaystyle+\int_{0}^{t}\sum_{i\in\mathcal{A}_{s-}^{B}}\left(\delta_{(i1,X_{s-}^{i},a)}+\delta_{(i2,X_{s-}^{i},a)}-\delta_{(i,X_{s-}^{i},a)}\right)\mathcal{N}_{i}^{b}({\rm
d}s)\,,$ (14)

where the initial configuration $\widehat{\pi}^{B}_{0}$ is a point measure on
$\mathcal{I}\times(V\cap B)\times\\{a,g\\}$, possibly random under
$P^{\omega}$. Notice that we do not impose any restriction on the initial
configuration at this stage.

Let us justify that there exists a unique solution to this stochastic
differential equation. Since we deal with a countable discrete state space,
the existence of such a solution can be shown just by constructing a
stochastic process with a classical inductive scheme, where the successive
jumps are given by the Poisson point measures. This is a strong Markov process
which is well defined until the potential accumulation point of the jumps (if
explosion occurs) and it is the solution of (2.2) until the time of explosion
by construction. We just need to prove that explosion does not occur almost
surely. For that purpose, let us introduce the projection of the process on
the last two coordinates

$Z_{t}^{B}(K,u):=\widehat{\pi}_{t}^{B}(\mathcal{I},V\cap K,u)\qquad
K\subseteq\mathbb{R}^{n},\,u\subseteq\\{a,g\\}\,.$

$Z_{t}^{B}(K,u)$ counts the alive or ghost particles in $K$ at time $t$. Under
$P^{\omega}$, $(Z_{t}^{B})_{t\in[0,T]}$ is a multi-type branching process with
a finite number of types (that is, $(V\cap B)\times\\{a,g\\}$) and bounded
reproduction mean: at rate $r(x,y)$ each particle of type $(x,a)$ is replaced
by two particles of types $(x,g)$ and $(y,a)$. At rate $b$ each particle of
type $(x,a)$ creates a new particle of type $(x,a)$. The particles of type
$(\cdot,g)$ do not evolve. Using classical first moment estimates of the
branching process $(Z_{t}^{B})_{t\in[0,T]}$ we obtain non-explosivity of the
process, so $(\widehat{\pi}^{B}_{t})_{t\in[0,T]}$ is well defined for any
positive time $T>0$, see e.g. [Bansaye and Méléard(2015)]. Actually, we will
need more quantitative estimates on the first moment of $Z_{t}^{B}$ for the
limiting procedure in the next section, in particular the dependance on the
transition rates. These estimates are given in the next lemma, using the
harmonic function of the branching process, which is here constant in space.

######  Lemma 2.1.

Consider two compact sets $K,B\subset\mathbb{R}^{n}$ with $K\subseteq B$. Take
a (possibly random) initial configuration $Z_{0}^{B}$ such that
$Z_{0}^{B}(B,\\{g\\})=0$ and, for some $M>0$,
$E^{\omega}[Z_{0}^{B}(x,\\{a\\})]\leq M$ for all $x\in B\cap V$. Then it
holds, for all $T>0$,

$\displaystyle E^{\omega}[Z_{T}^{B}(K,\\{a,g\\})]\leq C_{K}M{\rm e}^{bT},$
(15)

where $C_{K}=\sum_{x\in K\cap V}(b^{-1}r(x)+1)$. In particular $C_{K}$ does
not depend on $B$.

###### Proof.

We define the matrix $M_{t}^{B}$ with entries indexed by the types

$\displaystyle
M^{B}_{t}\big{(}(x,u),\,(y,v)\big{)}=E^{\omega}_{\delta_{(x,\\{u\\})}}\big{[}Z^{B}_{t}(y,\\{v\\})\big{]}\qquad
x,y\in B\cap V,\,u,v\in\\{a,g\\}\,,$

so that $M^{B}_{t}\big{(}(x,u),\,(y,v)\big{)}$ indicates the mean number of
particles of type $(y,v)$ present at time $t$ if we started with a unique
particle of type $(x,u)$ at time $0$. This matrix is the first moment
semigroup associated to a branching process and thus coincides with
$\exp(tA)$, where $A=A(B)$ is a finite matrix given by

$\displaystyle A=\begin{pmatrix}A_{1}&A_{2}\\\ 0&0\end{pmatrix}$

with the following blocks: $A_{1}$ is the submatrix accounting for the
evolution of $a$–particles, that is, for $x\neq y\in{B\cap}V$, we have
$A_{1}(x,x)=b-r^{B}(x)$ with $r^{B}(x):=\sum_{z\neq x,z\in B\cap V}r(x,z)$ and
$A_{1}(x,y)=r(x,y)$. $A_{2}$ accounts for the generation of $g$–particles from
$a$–particles, that is, $A_{2}(x,y)=r(x,y)$. The two lower blocks have all the
entries equal to $0$, since $g$–particles neither move nor generate other
$g$–particles. To see that, one may use Kolmogorov forward equation or apply
the differential equation (2.2) to $(\mathcal{I},y,v)$ with
$\widehat{\pi}_{0}^{B}=\delta_{\\{1,x,\\{u\\}\\}}$ and take expectation on
both sides to get
$M^{B}_{t}\big{(}(x,u),\,(y,v)\big{)}=\delta_{(x,u)=(y,v)}+\int_{0}^{t}M_{s}A\big{(}(x,u),\,(y,v)\big{)}\,{\rm
d}s.$

We compute now $\exp(At)$:

$\displaystyle A^{k}=\begin{pmatrix}A_{1}^{k}&A_{1}^{k-1}A_{2}\\\
0&0\end{pmatrix}\quad\Longrightarrow\quad{\rm e}^{At}=\begin{pmatrix}{\rm
e}^{A_{1}t}&\sum_{k\geq 1}\frac{A_{1}^{k-1}A_{2}}{k!}t^{k}\\\
0&Id\end{pmatrix}$

where $Id$ is the identity matrix. Call $z_{0}=(M,M,\dots,M\,;\,0,0,\dots,0)$
the configuration having $M$ alive particles on each site of $B\cap V$ and no
ghost particles. Then, for any initial configuration $Z_{0}^{B}$ with less
than $M$ alive particles per site in $B\cap V$ in average and no ghosts as in
the hypothesis of the lemma, it holds

$\displaystyle
E^{\omega}\big{[}Z_{T}^{B}(K,\\{a,g\\})\big{]}=E^{\omega}\big{[}Z_{0}^{B}{\rm
e}^{AT}(K,\\{a,g\\})\big{]}=E^{\omega}[Z_{0}^{B}]{\rm e}^{AT}(K,\\{a,g\\})\leq
z_{0}{\rm e}^{AT}(K,\\{a,g\\})\,.$

Since $\bar{1}=(1,1,\dots,1)$ is an eigenvector for the matrix $A_{1}$ with
eigenvalue $b$ (that is, $\bar{1}A_{1}=(b,b,\dots,b)$) and since

$\bar{1}\sum_{k\geq 1}\frac{A_{1}^{k-1}A_{2}}{k!}t^{k}=\sum_{k\geq
1}\frac{b^{k-1}(1,\dots,1)A_{2}}{k!}t^{k}=\sum_{k\geq
1}\frac{b^{k-1}}{k!}t^{k}(r^{B}(x))_{x\in B}=b^{-1}({\rm
e}^{bt}-1)(r^{B}(x))_{x\in B}\,,$

it holds

$\displaystyle z_{0}{\rm e}^{At}=M({\rm e}^{bt},\dots,{\rm
e}^{bt}\,;\,b^{-1}({\rm e}^{bt}-1)r(\cdot),\dots,b^{-1}({\rm
e}^{bt}-1)r(\cdot))$

and we can conclude that

$\displaystyle E^{\omega}[Z_{T}^{B}(K,\\{a,g\\})]\leq M{\rm e}^{bT}\\#\\{K\cap
V\\}+Mb^{-1}{\rm e}^{bT}\sum_{x\in K\cap V}r(x)\,,$

which implies (15). ∎

###  2.3. Existence of the process on the infinite graph with a finite number
of initial particles

In this section we want to show that the process $(\widehat{\pi}_{t})_{t\geq
0}$ described in (2.1) is well defined when we start with a configuration with
a finite number of particles. We will show that $(\widehat{\pi}_{t})_{t\geq
0}$ can be in fact obtained as the limit of the process
$(\widehat{\pi}_{t}^{B_{N}})_{t\geq 0}$ introduced in Section 2.2, where
$B_{N}$ is the $n$-dimensional box $[-N,N]^{n}$.

Consider the process $(\widehat{\pi}_{t}^{B_{N}})_{t\in[0,T]}$ introduced in
Section 2.2 up to time $T>0$. We want to show now that this process
“stabilizes” as $N$ tends to infinity. That is, suppose to use the same source
of randomness (i.e. the same realization of the Poisson processes
$\mathcal{N}_{i}^{x,y},\mathcal{N}_{i}^{b}$) to construct the process
$(\widehat{\pi}_{t}^{B_{N}})_{t\in[0,T]}$ for all different $N$’s. Then, with
$P^{\omega}$–probability $1$, there exists $N_{0}\in\mathbb{N}$ such that, for
all $N\geq N_{0}$,

$\displaystyle(\widehat{\pi}_{t}^{B_{N}})_{t\in[0,T]}\equiv(\widehat{\pi}_{t}^{B_{N_{0}}})_{t\in[0,T]}\,.$
(16)

To this end, we first of all prove that the progeny of a finite number of
particles remains into a finite region up to time $T>0$ with probability $1$
as $N\to\infty$.

For an initial configuration of alive and ghost particles $z_{0}$, define the
maximal displacement at time $T$ as

$\displaystyle
R_{N}(z_{0},T):=\sup_{t\in[0,T]}\sup_{i\in\mathcal{A}_{t}}\|X_{t}^{N,i}\|\,,$

where $X_{t}^{N,i}$ is the position of particle $i$ at time $t$ in the process
$(\widehat{\pi}_{t}^{B_{N}})_{t\in[0,T]}$.

######  Proposition 2.2.

Consider an initial configuration $z_{0}$ with a finite number of alive
particle. Then, $P^{\omega}$–almost surely, there exists $Q>0$ such that
$R_{N}(z_{0},T)\leq Q$ for all $N\in\mathbb{N}$.

###### Proof.

We first consider $z_{0}$ to be constituted of a unique alive particle,
labelled with $1$. Without loss of generality we can imagine particle $1$ to
start at the origin. Abbreviate $R=R_{N}(z_{0},T)$ and consider
$(\widehat{\pi}_{t}^{B_{N}})_{t\in[0,T]}$ for any $N$. For $M>0$, we can bound

$\displaystyle
P^{\omega}(R>M)=\sum_{\ell=0}^{\infty}P^{\omega}(R>M\,|\,E_{2\ell+1})P^{\omega}(E_{2\ell+1})$
(17)

with

$E_{\ell}=E_{\ell}(T):=\\{\mbox{particle 1 had $\ell-1$ descendants up to time
$T$}\\}\,.$

By descendant of particle $1$ we mean a particle with label starting by $1$
and that was generated via a birth event (so we do not count the particles
whose label start by $1$ that were generated with a change of label due to a
jump event). We are considering only odd integers $2\ell+1$ since each time a
particle disappears it generates two new particles. For $\ell\in\mathbb{N}$,
the quantity $P^{\omega}(E_{\ell})$ is clearly dominated by
$P^{\omega}(E_{\ell}^{+})$, with

$E_{\ell}^{+}:=\\{\mbox{particle 1 had at least $\ell-1$ descendants up to
time $T$}\\}\,.$

The number $Z_{t}^{B_{N}}(\mathbb{R}^{n},\\{a\\})$ of alive particles at time
$t$ follows a $\mathbb{N}$-valued Markov process starting in $1$ and that goes
from $k$ to $k+1$ with rate $kb$. Let $(e_{k})_{k\in\mathbb{N}}$ be
independent exponential random variables under $P^{\omega}$ with
$E^{\omega}[e_{k}]=(bk)^{-1}$ and let $S_{\ell}:=\sum_{k=1}^{\ell}e_{k}$. We
bound, for all $\theta>0$,

$\displaystyle P^{\omega}\big{(}S_{\ell}\leq T\big{)}\leq{\rm e}^{\theta
T}E^{\omega}[{\rm e}^{-\theta S_{\ell}}]={\rm e}^{\theta
T}\prod_{k=1}^{\ell}\Big{(}1-\frac{\theta}{kb+\theta}\Big{)}\leq{\rm
e}^{\theta T-\sum_{k=1}^{\ell}\frac{\theta}{kb+\theta}},$ (18)

where for the first inequality we have exploited the exponential Markov
inequality, while for the second passage we have used the independence of the
$e_{k}$’s and the formula for the moment generating function of the
exponential distribution.

Notice that if the total number of descendants of particle $1$ at time $T$ is
$2\ell$, then $Z_{t}^{B_{N}}(\mathbb{R}^{n},\\{a\\})=\ell+1$. A (non-
optimized) choice of $\theta=4b$ in (18) yields therefore

$\displaystyle
P^{\omega}(E_{2\ell+1}^{+})=P^{\omega}(Z_{t}^{B_{N}}(\mathbb{R}^{n},\\{a\\})\geq
n+1)=P^{\omega}\Big{(}S_{\ell+1}\leq T\Big{)}\leq{\rm
e}^{4bT-\sum_{k=1}^{\ell+1}\frac{4}{k+4}}\leq C\ell^{-4}$ (19)

for $\ell$ sufficiently large and some universal constant $C>0$.

We move to the analysis of the term $P(R>M\,|\,E_{\ell})$ in (17). Abandoning
for a moment the Ulam–Harris–Neveu notation, let us look at the descendants of
particle $1$ and just label them $2,3,4,...$ in chronological order of birth
(particles $2j$ and $2j+1$ are born in the same instant, for all $j$). Let
$x_{k}\in V$ and $t_{k}\in[0,T]$ be the site and the time where the $k$-th
particle was born and let $(X_{t}^{k})$ be its trajectory while alive. Let
$R_{k}$ be the maximal displacement of particle $k$, that is,
$R_{k}:=\sup_{t\in[t_{k},T]}\|X_{t}^{k}-x_{k}\|$. We observe that

$\displaystyle P^{\omega}(R>M\,|\,E_{\ell})$
$\displaystyle=P^{\omega}\big{(}\exists
k\in\\{1,\dots,\ell\\}:\,R_{k}>M/\ell\mbox{ and }R_{j}<M/\ell\mbox{ for all
$j<k$}\,|\,E_{\ell}\big{)}$
$\displaystyle\leq\sum_{k=1}^{\ell}P^{\omega}\big{(}R_{k}>M/\ell\,\big{|}\,A_{k,\ell}\big{)},$
(20)

with $A_{k,\ell}:=\\{R_{j}<M/\ell\mbox{ for all $j<k$}\\}\cap E_{\ell}$. We
have therefore to study the probability that $(X_{t}^{k})$ left the ball
$B_{M/\ell}(x_{k})$ before time $T$ knowing that the first $k-1$ particles had
a displacement smaller than $M/\ell$. We point out that, under $A_{k,\ell}$,
we have that $B_{M/\ell}(x_{k})$ is completely contained in $B_{M}$: it
follows that all $x\in B_{M/\ell}(x_{k})$ have $r(x)<C\log M$ by Lemma 2.3
(see here below) and that $B_{M/\ell}(x_{k})$ contains at most $CM^{n}\log
M/\ell^{n}$ points of $V$ (this follows from item (ii) in the proof of Lemma
2.3).

Let

$\displaystyle\tau_{k}:=\inf\big{\\{}t\in[t_{k},T]:\,X_{t}^{k}\not\in
B_{M/\ell}(x_{k})\big{\\}}\,.$

We decompose the event $\\{\tau_{k}\leq T\\}=\\{\tau_{k}^{\prime}<\tau_{k}\leq
T\\}\cup\\{\tau_{k}\leq T,\tau_{k}^{\prime}\\}$ with

$\tau_{k}^{\prime}:=\inf\big{\\{}t\in[t_{k},T]:\,\|X_{t}^{k}-X_{t-}^{k}\|\geq\sqrt{M/\ell}\big{\\}}$

the first time that particle $k$ makes a jump longer than $\sqrt{M/\ell}$.
Under $A_{k,\ell}$, the event that $(X_{t}^{k})$ makes a jump of length larger
than $\sqrt{M/\ell}$ inside $B_{M/\ell}(x_{k})$ has rate smaller than ${\rm
e}^{-\sqrt{M/\ell}}$ times the number of the points in $B_{M/\ell}(x_{k})$. It
follows that

$\displaystyle P^{\omega}(\tau_{k}^{\prime}<\tau_{k}\leq T\,|\,A_{k,\ell})\leq
P^{\omega}(\xi\leq T)\leq cT{\rm e}^{-\sqrt{M/\ell}}M^{n}\log M/\ell^{n}$ (21)

for some universal constant $c>0$, where $\xi$ is an exponential random
variable with parameter $C{\rm e}^{-\sqrt{M/\ell}}M^{n}\log M/\ell^{n}$. On
the other hand, the event $\\{\tau_{k}\leq T,\tau_{k}^{\prime}\\}$ implies
that $X^{k}$ has performed more than $\sqrt{M/\ell}$ jumps before time $T$.
Remember that under $A_{k,\ell}$ each jump has rate smaller than $C\log M$ by
Lemma 2.3. Hence, if $Y$ is a Poisson random variable of parameter $CT\log M$,

$\displaystyle P^{\omega}(\tau_{k}\leq T,\tau_{k}^{\prime}\,|\,A_{k,\ell})\leq
P^{\omega}(Y>\sqrt{M/\ell})\leq{\rm e}^{-CT\log M}\big{(}(M/\ell)^{-1/2}{\rm
e}CT\log M\big{)}^{\sqrt{M/\ell}}$ (22)

where the last bound holds for $M/\ell$ sufficiently large, for example when
$\ell\leq\sqrt{M}$, see e.g. [Vershynin(2018), Exercise 2.3.3]. Continuing
from (2.3), bounds (21) and (22) together yield, for $M$ sufficiently large
and $\ell\leq\sqrt{M}$,

$\displaystyle P^{\omega}(R>M\,|\,E_{\ell})$
$\displaystyle\leq\sum_{k=1}^{\ell}P^{\omega}(\tau_{k}\leq
T\,\big{|}\,A_{k,\ell})\leq c_{1}\ell{\rm e}^{-c_{2}\sqrt{M/\ell}}\,.$ (23)

Going back to (17) and using (54) and (23), we finally have

$\displaystyle P^{\omega}(R>M)$
$\displaystyle\leq\sum_{\ell=0}^{\sqrt{M}}P^{\omega}(R>M\,|\,E_{2\ell+1})+\sum_{\ell=\sqrt{M}}^{\infty}P^{\omega}(E_{2\ell+1}^{+})$
$\displaystyle\leq c_{1}M{\rm e}^{-c_{2}M^{1/4}}+CM^{-3/2}.$

This quantity is summable in $M$, which implies the claim by the Borel-
Cantelli lemma for a single initial particle. The argument can be easily
generalized to any finite number of initial particles. ∎

######  Lemma 2.3.

There exists $C>0$ such that, for $\mathbb{P}$-a.a. $\omega$, the following
holds: there exists $\bar{N}=\bar{N}(\omega)$ such that $\forall N\geq\bar{N}$
one has

$\max_{x\in B_{N}\cap V}r(x)<C\log N\,.$

###### Proof.

We will use the two following trivial facts about Poisson point processes.
Recall that $B_{N}=[-N,N]^{n}$. Let $(\mathcal{B}_{i})_{i=1,\dots,(2N)^{n}}$
be a collection of disjoint (up to their border) volume-$1$ cubes covering
$B_{N}$ and let $\mathcal{C}_{j}:=\\{x\in B_{j+1}\setminus B_{j}\\}$, for
$j\in\mathbb{N}$, be the $j$-th square-crown around the origin. Then there
exist constants $c_{1},c_{2}>0$ only depending on the dimension $n$ such that,
for $\mathbb{P}$-a.a. $\omega$,

  * (i)

there exists $N_{1}=N_{1}(\omega)$ such that for all $N\geq N_{1}$

$\\#\\{x\in\mathcal{C}_{N}\cap V\\}\leq c_{1}N^{n-1}\,;$

  * (ii)

there exists $\bar{N}_{2}=\bar{N}_{2}(\omega)$ such that for all
$N\geq\bar{N}_{2}$ and for all $i=1,\dots,(2N)^{n}$

$\\#\\{x\in\mathcal{B}_{i}\cap V\\}\leq c_{2}\log N\,.$

Both facts can be checked by using classic concentration inequalities for
Poisson random variables around their mean and then the Borel-Cantelli lemma.

Take $N\geq\max\\{N_{1},N_{2}\\}$ and write, for $x\in B_{N}$,

$\displaystyle r(x)$ $\displaystyle=\sum_{y\in B_{2N}\cap V}{\rm
e}^{-\|x-y\|}+\sum_{y\in B_{2N}^{c}\cap V}{\rm e}^{-\|x-y\|}\,.$ (24)

For the first sum we divide $B_{2N}$ into $\mathcal{B}_{i}$’s as for item (ii)
above, with $i=1,\dots,(2N)^{n}$. Notice that, for all $k\in\mathbb{N}$, there
are less than $c_{3}k^{n-1}$ such boxes at distance $k$ from $x$, for some
$c_{3}>0$ that only depends on the dimension $n$. Furthermore, in each of
these boxes there are at most $c_{2}\log(2N)$ vertices by (ii). Hence it holds

$\displaystyle\sum_{y\in B_{2N}\cap V}{\rm
e}^{-\|x-y\|}\leq\sum_{k=0}^{2N}c_{3}{\rm e}^{-k}k^{n-1}\cdot
c_{2}\log(2N)\leq c_{5}\log N\,.$ (25)

For the second sum in (24) we use item (i) and bound

$\displaystyle\sum_{y\in B_{2N}^{c}\cap V}{\rm
e}^{-\|x-y\|}=\sum_{k=N}^{\infty}\sum_{y\in\mathcal{C}_{k}\cap V}{\rm
e}^{-\|x-y\|}\leq\sum_{k=N+1}^{\infty}c_{1}k^{n-1}{\rm e}^{-(k-N)}\leq c_{6}$
(26)

for some $c_{6}>0$. Putting (25) and (26) into (24) gives the result. ∎

######  Corollary 2.4 (Corollary of Proposition 2.2).

Consider a compact set $Q\subset\mathbb{R}^{n}$. Let
$Z_{0}\in\mathbb{N}^{V}\times\mathbb{N}^{V}$ be an initial configuration such
that $Z_{0}(x,\\{a,g\\})=0$ for all $x\not\in Q\cap V$ and
$E^{\omega}[Z_{0}(x,a)]\leq M$ for all $x\in Q\cap V$. Then,
$P^{\omega}$–a.s., for every $I\in\mathcal{I}$, $K\subset\mathbb{R}^{n}$
compact, $U\subseteq\\{a,g\\}$ and $t>0$, the following limit exists:

$\widehat{\pi}_{t}(I,K,U):=\lim_{N\to\infty}\widehat{\pi}_{t}^{B_{N}}(I,K,U)\,.$

Furthermore, the following holds:

  * (i)

The measure $(\widehat{\pi}_{t})_{t\geq 0}$ verifies equation (2.1), where the
two sides are finite measures and coincide on $\mathbb{R}^{n}$.

  * (ii)

Defining for all compact sets $K\subset\mathbb{R}^{n},$ for all
$u\subseteq\\{a,g\\}$ and for all $t\geq 0$

$Z_{t}(K,u):=\widehat{\pi}_{t}(\mathcal{I},K,u)\,,$

one has, for all $T>0$,

$\displaystyle E^{\omega}[Z_{T}(K,\\{a,g\\})]\leq C_{K}M{\rm e}^{bT}$ (27)

where $C_{K}=\sum_{x\in K}(b^{-1}r(x)+1)$.

###### Proof.

The existence of $\widehat{\pi}_{t}$ follows immediately from Proposition 2.2,
since it implies that (16) holds $P^{\omega}$–a.s. for all $T>0$. In
particular, (16) and the fact that
$\widehat{\pi}_{T}^{B_{N}}(\mathcal{I},\mathbb{R}^{n},\\{a,g\\})<\infty$
almost surely (which follows by (15)) imply that
$\widehat{\pi}_{T}(\mathcal{I},\mathbb{R}^{n},\\{a,g\\})<\infty$ almost
surely. We let $\mathcal{A}_{s}:=\lim_{N\to\infty}\mathcal{A}_{s}^{B_{N}}$ for
almost every realization of the process.

For item (i), we first notice that $(\widehat{\pi}^{B_{N}}_{t})$ satisfies
(2.2) with $B=B_{N}$. $P^{\omega}$–a.s., for all bounded test functions $f$
with support on some set $C\subset\mathbb{R}^{n}$ and for all $t\in[0,T]$ we
have

$\displaystyle\int_{0}^{t}\sum_{i\in\mathcal{A}_{s-},\,y\in
V}\left|f{(i1,y,a)}+f{(i,X_{s-}^{i},g)}-f{(i,X_{s-}^{i},a)}\right|\mathcal{N}_{i}^{X_{s-}^{i},y}({\rm
d}s)$ $\displaystyle\qquad\qquad\leq
3\|f\|_{\infty}\Big{(}\int_{0}^{T}\sum_{i\in\mathcal{A}_{s-},\,X_{s-}^{i}\in
C,\,y\in V}\mathcal{N}_{i}^{X_{s-}^{i},y}({\rm
d}s)+\int_{0}^{T}\sum_{i\in\mathcal{A}_{s-},\,X_{s-}^{i}\not\in C,\,y\in
C}\mathcal{N}_{i}^{X_{s-}^{i},y}({\rm d}s)\Big{)}$
$\displaystyle\qquad\qquad\leq
3\|f\|_{\infty}\widehat{\pi}_{T}(\mathcal{I},C,\\{a,g\\})<\infty$ (28)

since $\widehat{\pi}_{T}(\mathcal{I},\mathbb{R}^{n},\\{a,g\\})<\infty$.
Similarly the number of births in $C$ is a.s. controlled by
$\widehat{\pi}_{T}(\mathcal{I},C,\\{a,g\\})$:

$\displaystyle\int_{0}^{t}\sum_{i\in\mathcal{A}_{s-}}\left|f{(i1,X_{s-}^{i},a)}+f{(i2,X_{s-}^{i},a)}-f{(i,X_{s-}^{i},a)}\right|\mathcal{N}_{i}^{b}({\rm
d}s)\leq 3\|f\|_{\infty}\widehat{\pi}_{T}(\mathcal{I},C,\\{a,g\\})\,.$ (29)

Indeed, for each newborn in $C$ there is either an active particle in $C$ or,
at least, a ghost. To sum up, the integrals appearing on the right hand side
of (2.1) are almost surely well defined on $\mathbb{R}^{n}$ and finite and (i)
follows from $\eqref{skype}$ by letting $N$ go to infinity.

We turn our attention to item (ii). By Fatou’s lemma and Proposition 2.1,

$\displaystyle
E^{\omega}[Z_{T}(K,\\{a,g\\})]\leq\liminf_{N\to\infty}E^{\omega}[Z_{T}^{B_{N}}(K,\\{a,g\\})]\leq
C_{K}M{\rm e}^{bT}\,.$

This ends the proof. ∎

###  2.4. Existence of the process on the infinite graph with infinitely many
initial particles

In the previous section we have shown that the process
$(\widehat{\pi}_{t})_{t\in[0,T]}$ is well defined as soon as the initial
condition involves only a finite number of particles. We want to show the
existence of $(\widehat{\pi}_{t})_{t\in[0,T]}$ also for initial configurations
where the average number of particles on each site is bounded.

Consider an initial configuration of particles
$Z_{0}\in\mathbb{N}^{V}\times\mathbb{N}^{V}$ of alive and ghost particles such
that $E^{\omega}[Z_{0}(x,a)]\leq M$. For $N\in\mathbb{N}$ the truncated
configuration $Z_{0,N}$ is obtained by considering only the particles in
$Z_{0}$ that are inside the ball $B_{N}$:

$Z_{0,N}(x,\cdot)=Z_{0}(x,\cdot)\mathbf{1}\\!_{\,x\in B_{N}}\,.$

A central observation is that we have monotonicity in $N$ of the process: take
$N_{1}<N_{2}$ and couple the processes started in $Z_{0,N_{1}}$ and
$Z_{0,N_{2}}$, call them $(\widehat{\pi}_{t,N_{j}})_{t\geq 0}$ for $j=1,2$.
Then we have $P^{\omega}$–a.s.

$\displaystyle\widehat{\pi}_{t,N_{1}}(i,x,u)\leq\widehat{\pi}_{t,N_{2}}(i,x,u)\qquad\forall
i\in\mathcal{I},\,x\in V,\,u\in\\{a,g\\}\,.$ (30)

As a consequence, we have the following proposition.

######  Proposition 2.5.

Consider a configuration of particles
$Z_{0}\in\mathbb{N}^{V}\times\mathbb{N}^{V}$ such that
$E^{\omega}[Z_{0}(x,a)]\leq M$ for all $x\in V$. Let
$(\widehat{\pi}_{t,N})_{t\geq 0}$ be the process on the infinite graph started
in $Z_{0,N}$. Then, $P^{\omega}$–a.s., for every $I\in\mathcal{I}$,
$K\subset\mathbb{R}^{n}$ compact, $U\subseteq\\{a,g\\}$ and $t\geq 0$, the
following limit exists and is finite:

$\widehat{\pi}_{t}(I,K,U):=\lim_{N\to\infty}\widehat{\pi}_{t,N}(I,K,U)\,.$

Furthermore one has, for all $T>0$,

$\displaystyle E^{\omega}[\widehat{\pi}_{T}(\mathcal{I},K,\\{a,g\\})]\leq
C_{K}M{\rm e}^{bT}$ (31)

where $C_{K}=\sum_{x\in K}(b^{-1}r(x)+1)$.

Notice that we have called the limiting process again
$(\widehat{\pi}_{t})_{t\geq 0}$, since we have extended the definition
appearing in Corollary 2.4 to a larger set of initial conditions.

###### Proof.

The existence of the limit follows by the monotonicity in (30). Fix any $T>0$.
We want to show now that, $P^{\omega}$–a.s.,
$\widehat{\pi}_{t}(\mathcal{I},K,\\{a,g\\})$ does not explode for any compact
$K\subset\mathbb{R}^{n}$ and $0\leq t\leq T$. Keeping in mind (30), we can use
monotone convergence in $N$ to see that

$\displaystyle
E^{\omega}\Big{[}\sup_{t\in[0,T]}\widehat{\pi}_{t}(\mathcal{I},K,\\{a,g\\})\Big{]}$
$\displaystyle=E^{\omega}[\widehat{\pi}_{T}(\mathcal{I},K,\\{a,g\\})]$
$\displaystyle=\lim_{N\to\infty}E^{\omega}[\widehat{\pi}_{T,N}(\mathcal{I},K,\\{a,g\\})]\stackrel{{\scriptstyle\eqref{sangiovanni}}}{{\leq}}C_{K}M{\rm
e}^{bT}<\infty\,,$ (32)

where for the first equality we have used the fact that
$\widehat{\pi}_{t}(\mathcal{I},K,\\{a,g\\})$ is also monotone in $t$, since
each new event does not decrease the total number of particles in $K$. ∎

It follows that $\sup_{t\in[0,T]}\widehat{\pi}_{t}(\mathcal{I},K,\\{a,g\\})$
is finite $P^{\omega}$–almost surely. Notice in particular that this implies
that $\widehat{\pi}_{t,N}(I,K,U)=\widehat{\pi}_{t,M}(I,K,U)$ for all $N,M$
large enough. If this was not the case, we would have an infinite sequence of
initial particles, coming from arbitrary far away, whose progeny would enter
$K$ before time $T$, thus making $\widehat{\pi}_{T}(\mathcal{I},K,\\{a,g\\})$
explode.

######  Corollary 2.6.

The following holds.

  * (i)

For any $t\geq 0$ and
$f:\mathcal{I}\times\mathbb{R}^{n}\times\\{a,g\\}\rightarrow\mathbb{R}$
measurable and compactly supported in the second coordinate:

$\displaystyle E^{\omega}\Big{[}\int_{0}^{t}\sum_{i\in\mathcal{A}_{s-},y\in
V}\left|f(i1,y,a)+f(i,X_{s-}^{i},g)-f(i,X_{s-}^{i},a)\right|\mathcal{N}_{i}^{X_{s-}^{i},y}({\rm
d}s)\Big{]}<\infty.$ $\displaystyle
E^{\omega}\big{[}\int_{0}^{t}\sum_{i\in\mathcal{A}_{s-}}\left|f(i1,X_{s-}^{i},a)+f(i2,X_{s-}^{i},a)-f(i,X_{s-}^{i},a)\right|\mathcal{N}_{i}^{b}({\rm
d}s)\Big{]}<\infty.$

  * (ii)

For such functions $f$ and $t\geq 0$, the following identity holds a.s.

$\displaystyle\langle\widehat{\pi}_{t},f\rangle=\langle\widehat{\pi}_{0},f\rangle$
$\displaystyle+\int_{0}^{t}\sum_{i\in\mathcal{A}_{s-},\,y\in
V}\left(f(i1,y,a)+f(i,X_{s-}^{i},g)-f(i,X_{s-}^{i},a)\right)\mathcal{N}_{i}^{X_{s-}^{i},y}({\rm
d}s)$
$\displaystyle+\int_{0}^{t}\sum_{i\in\mathcal{A}_{s-}}\left(f(i1,X_{s-}^{i},a)+f(i2,X_{s-}^{i},a)-f(i,X_{s-}^{i},a)\right)\mathcal{N}_{i}^{b}({\rm
d}s).$

###### Proof.

The first part is a consequence (2.3) and (29) for bounded functions $f$ with
support on some compact set $C\subset\mathbb{R}^{n}$, together with (2.4)
which guarantees finiteness. We are left to show that $(\widehat{\pi}_{t})$ is
a solution of equation (2.1) on any compact set, where now the initial
population can be non bounded. By choosing $N_{0}$ large, the terms involved
in (2.1) for $(\widehat{\pi}_{t,N})_{t\in[0,T]}$ are all constant for $N\geq
N_{0}$, which ends the proof. ∎

Recall that $\pi_{t}$ is the projection of $\widehat{\pi}_{t}$ on alive
particles, i.e. for $B\subset\mathbb{R}^{n}$ Borel set
$\pi_{t}(B):=\widehat{\pi}_{t}(\mathcal{I}\times(B\cap V)\times\\{a\\})$. For
every $f:\mathbb{R}^{n}\rightarrow\mathbb{R}$ with compact support, we get

$\displaystyle\langle\pi_{t},f\rangle=\langle\pi_{0},f\rangle+\int_{0}^{t}\sum_{i\in\mathcal{A}_{s-},y\in
V}\big{(}f(y)-f(X_{s-}^{i})\big{)}\,\mathcal{N}_{i}^{X_{s-}^{i},y}({\rm
d}s)+\int_{0}^{t}\sum_{i\in\mathcal{A}_{s-}}f(X_{s-}^{i})\,\mathcal{N}_{i}^{b}({\rm
d}s)\,.$ (33)

We can now justify that the generator of this process is given by (2) and end
the proof of Theorem 1.1. More precisely, let us check that for all $G$
compactly supported on $\mathbb{R}^{n}$,

$M^{G}_{t}=\langle\pi_{t},G\rangle-\langle\pi_{0},G\rangle-\int_{0}^{t}\mathcal{L}f_{G}(\eta_{s})\,{\rm
d}s$

is indeed a martingale, where we recall that $\eta_{t}(x)=\pi_{t}(\\{x\\})$
and for $f_{G}(\eta)=\sum_{x\in V}G(x)\eta(x)$

$\displaystyle\mathcal{L}f_{G}(\eta)=$ $\displaystyle\sum_{x,y\in
V}\eta(x)r(x,y)\big{(}G(y)-G(x)\big{)}+\sum_{x\in
V}\eta(x)\big{(}b-d\big{)}G(x).$ (34)

The fact that $\langle\pi_{t},G\rangle$ is integrable is due to (31). The fact
that $E^{\omega}[\int_{0}^{t}|\mathcal{L}f_{G}|(\eta_{s}){\rm d}s]$ is finite
is due to (2.3) and (29), which allows us to bound this term by
$E^{\omega}[\widehat{\pi}_{t}(\mathcal{I},C,\\{a,g\\})]$. Besides

$\displaystyle M^{G}_{t}=\langle\widehat{\pi}_{0},f\rangle$
$\displaystyle+\int_{0}^{t}\sum_{i\in\mathcal{A}_{s-},\,y\in
V}\left(f(i1,y,a)+f(i,X_{s-}^{i},g)-f(i,X_{s-}^{i},a)\right)\widetilde{\mathcal{N}}_{i}^{X_{s-}^{i},y}({\rm
d}s)$
$\displaystyle+\int_{0}^{t}\sum_{i\in\mathcal{A}_{s-}}\left(f(i1,X_{s-}^{i},a)+f(i2,X_{s-}^{i},a)-f(i,X_{s-}^{i},a)\right)\widetilde{\mathcal{N}}_{i}^{b}({\rm
d}s),$

where $\widetilde{\mathcal{N}}_{i}$ and $\widetilde{\mathcal{N}}_{i}^{b}$ are
the compensated Poisson point measures. Again, (2.3) and (29) provide the
integrability condition for stochastic $L^{1}$ martingale with jumps, see for
example [Ikeda and Watanabe(1989)]. Thus, $M^{G}$ inherits the martingale
property. This ensures that $\mathcal{L}$ provides the generator for functions
of the form $f_{G}$.

##  3\. Input from homogenization

In this section we set the homogenization tools that are needed to prove the
hydrodynamic limit in Theorem 1.2. Notice that the results we collect are
mainly inherent to the environment $\omega\in\Omega$: the specific particle
dynamics we are analyzing only enters in these results through the generator
of the simple random walk $L^{N}$.

###  3.1. Assumptions for homogenization on point processes

In [Faggionato(2022a)] Faggionato proves homogenization for a wide class of
random walks on purely atomic measures on $\mathbb{R}^{n}$ under some
regularity assumptions for the environment, called (A1),…,(A9). Our proof of
Theorem 1.2 relies on these homogenization results. We first state these
assumptions in a simplified way, adapted to our context. We check then that
they are indeed satisfied by our model.

Consider the Abelian group $\mathbb{G}=\mathbb{R}^{n}$ acting on a probability
space $(\Omega,\mathbb{P},\mathcal{F})$ in the following way (see
$(P_{1}),\dots,(P4)$ in [Faggionato(2022a)]): for $g\in\mathbb{G}$ we consider
the measurable map $\theta_{g}:\,\Omega\to\Omega$ such that $\theta_{0}$ is
the identity; $\theta_{g}\circ\theta_{g^{\prime}}=\theta_{g+g^{\prime}}$ for
all $g,g^{\prime}\in\mathbb{G}$; the map $(g,\omega)\to\theta_{g}\omega$ is
measurable; $\mathbb{P}\circ\theta_{g}^{-1}=\mathbb{P}$ for all
$g\in\mathbb{G}$. The group $\mathbb{G}$ acts also on the space
$\mathbb{R}^{n}$ as space-translations $(\tau_{g})_{g\in\mathbb{G}}$ such that
$\tau_{g}x=x+g$ for all $g\in\mathbb{G}$ and $x\in\mathbb{R}^{n}$. Suppose to
have a random purely atomic locally finite non-negative measure
$\mu_{\omega}\in\mathcal{M}(\mathbb{R}^{n})$

$\displaystyle\mu_{\omega}=\sum_{x\in\hat{\omega}}n_{x}(\omega)\delta_{x},\qquad
n_{x}(\omega):=\mu_{\omega}(\\{x\\}),\qquad\hat{\omega}:=\\{x\in\mathbb{R}^{n}:\,n_{x}(\omega)>0\\}\,.$

Let $\mathbb{P}_{0}$ be the Palm measure associated to $\mathbb{P}$ and
$\mathbb{E}_{0}$ the associated expectation (see for example
[Faggionato(2022a), equation (9)] for the precise definition or [Daley and
Vere-Jones(2008)] for a more complete account of Palm measures). Finally let
$r:\,(\omega,x,y)\to r(\omega,x,y)\in[0,\infty)$ be the jump rates with
$r(\omega,x,x)=0$ for all $x\in\mathbb{R}^{n}$ and $\omega\in\Omega$, and
$r(\omega,x,y)=0$ when $x$ or $y$ is not in $\hat{\omega}$. Then the nine
assumptions are the following, with $\Omega_{*}$ some measurable, translation
invariant subset of $\Omega$ with $\mathbb{P}(\Omega_{*})=1$:

  * (A1)

$\mathbb{P}$ is stationary and ergodic w.r.t. $(\theta_{g})_{g\in\mathbb{G}}$.
That is, $\mathbb{P}\circ\theta_{g}^{-1}=\mathbb{P}$ for all $g\in\mathbb{G}$
and, for each $A\subseteq\Omega$ such that $A=\theta_{g}A$ for all
$g\in\mathbb{G}$, one has $\mathbb{P}(A)\in\\{0,1\\}$;

  * (A2)

$0<\mathbb{E}[\mu_{\omega}([0,1)^{n})]<\infty$;

  * (A3)

for all $\omega\in\Omega_{*}$ and all $g\neq g^{\prime}$ it holds
$\theta_{g}\omega\neq\theta_{g^{\prime}}\omega$;

  * (A4)

for all $\omega\in\Omega_{*}$, $\mu_{\omega}$ is $\mathbb{G}$–stationary: for
all $x,y\in\mathbb{R}^{n}$ and for all $g\in\mathbb{G}$ it holds
$\mu_{\theta_{g}\omega}=\tau_{g}\mu_{\omega}$ and
$r(\theta_{g}\omega,x,y)=r(\omega,\tau_{g}x,\tau_{g}y)$;

  * (A5)

for all $\omega\in\Omega_{*}$ and for all $x,y\in\hat{\omega}$ it holds
$n_{x}(\omega)r(\omega,x,y)=n_{y}(\omega)r(\omega,y,x)$;

  * (A6)

for all $\omega\in\Omega_{*}$ and for all $x,y\in\hat{\omega}$ there exists a
path $x=x_{0},\,x_{1},\dots,x_{n-1},x_{n}=y$ such that
$r(\omega,x_{i},x_{i+1})>0$ for all $i=0,\dots,n-1$;

  * (A7)

$\mathbb{E}_{0}\big{[}\sum_{x\in\hat{\omega}}r(\omega,0,x)|x|^{k}\big{]}<\infty$
for $k=0,2$;

  * (A8)

$L^{2}(\mathbb{P}_{0})$ is separable;

  * (A9)

setting $N_{z}(\omega):=\mu_{\omega}(z+[0,1)^{n})$ for $z\in\mathbb{Z}^{n}$,
it holds $\mathbb{E}[N_{0}^{2}]<\infty$ and, for some $C\geq 0$,
$|Cov(N_{z},N_{z^{\prime}})|\leq C|z-z^{\prime}|^{-1}$.

We prove now these assumptions are satisfied for our model.

######  Lemma 3.1.

The complete graph $\mathcal{G}=(V,E)$ on a Poisson point process of parameter
$\gamma>0$ in $\mathbb{R}^{n}$ with transition rates $r(x,y)=r(y,x)={\rm
e}^{-\|x-y\|}$ (with the convention $r(x,x)=0$) satisfies assumptions
$(A1),\dots,(A9)$ of [Faggionato(2022a)].

###### Proof of Lemma 3.1.

In our case $\mu_{\omega}$ is the point measure associated to the Poisson
point process, so that we have almost surely $n_{x}(\omega)=1$ for each point
$x\in V$. We also notice that $\hat{\omega}$ coincides with $V$. (A1), (A2)
and (A3) clearly hold. (A4) and (A5) also come from the stationarity of the
Poisson point process and from our choice of the rates. (A6) is trivial since
we are considering the complete graph. For (A7) and (A8), we mention that the
Palm measure associated to the underlying Poisson point process can be
obtained by just adding an additional point to the configuration at the
origin. (A7) is easy to verify, while for (A8) see the comment at the end of
Section 2.4 in [Faggionato(2022a)]. Finally, $N_{z}$ is just the number of
points in the box $z+[0,1)^{n}$, so that
$\mathbb{E}[N_{0}^{2}]=\gamma^{2}+\gamma<\infty$ and the covariance appearing
in (A9) is just equal to 0. ∎

###  3.2. The Poisson equation

Fix $\omega\in\Omega$. Recall from (4) that $L^{N}$ is the generator of the
diffusively rescaled random walk on $V/N:=\\{x/N:\,x\in V(\omega)\\}$ with
transition rates $N^{2}r(\cdot,\cdot)$. We think of $L^{N}$ as acting on
functions in $L^{2}(\mu_{N})$, where $\mu_{N}=\mu_{N}(\omega)$ is the uniform
measure on $V/N$, that is

$\mu_{N}:=N^{-n}\sum_{x\in V}\delta_{x/N}\,.$

We write $(\cdot,\cdot)_{\mu_{N}}$ and $\|\cdot\|_{L^{2}(\mu_{N})}$ for,
respectively, the scalar product and the norm in $L^{2}(\mu_{N})$. Note that
$L^{N}$ is a negative-definite symmetric operator: for any $f,g\in
L^{2}(\mu_{N})$

$(f,L^{N}g)_{\mu_{N}}=(L^{N}f,g)_{\mu_{N}}\qquad\mbox{and}\qquad(f,-L^{N}f)_{\mu_{N}}\geq
0\,.$

The following definition is justified by the fact that $L^{N}$ should approach
in some sense the continuous operator $\sigma^{2}\Delta$.

######  Definition 3.2.

Given $\lambda>0$, $G\in C_{c}^{\infty}(\mathbb{R}^{n})$ and $N\in\mathbb{N}$,
we define $G^{\lambda}_{N}$ to be the unique element in $L^{2}(\mu_{N})$ such
that

$\displaystyle\lambda G_{N}^{\lambda}-L^{N}G^{\lambda}_{N}=H_{N}\,$ (35)

where $H_{N}$ is the restriction to $V/N$ of the function
$H=H(\lambda)=\lambda G-\sigma^{2}\Delta G\in C_{c}^{\infty}(\mathbb{R}^{n})$.

Notice that the introduction of $\lambda>0$ is just an artifice to make
$\lambda Id-L^{N}$ invertible, where $Id$ is the identity operator, and that
$\lambda$ will be fixed and play basically no role in what follows. The idea
for introducing $G_{N}^{\lambda}$ is that $L^{N}G_{N}^{\lambda}$ is more
regular than $L^{N}G$ (for example inequality (38) here below might fail for a
general $G$). This regularizing procedure is associated to the so-called
corrected empirical measure in the literature, see [Gonçalves and Jara(2008)]
for more comments on this.

The next result is where homogenization theory enters the game, and in
particular the results of [Faggionato(2022a)].

######  Lemma 3.3.

Fix $\lambda>0$. Then for $\mathbb{P}$-a.a. $\omega$ and for each $G\in
C_{c}^{\infty}(\mathbb{R}^{n})$ it holds

$\displaystyle(G_{N}^{\lambda},-L^{N}G_{N}^{\lambda})_{\mu_{N}}\leq
c(\lambda,G)$ (36)
$\displaystyle\|G_{N}^{\lambda}\|_{L^{1}(\mu_{N})}\,,\;\|G_{N}^{\lambda}\|_{L^{2}(\mu_{N})}\leq
c(\lambda,G)$ (37)
$\displaystyle\|L^{N}G_{N}^{\lambda}\|_{L^{1}(\mu_{N})}\,,\;\|L^{N}G_{N}^{\lambda}\|_{L^{2}(\mu_{N})}\leq
c(\lambda,G)\,,$ (38)

where $c(\lambda,G)>0$ is a constant not depending on $N$. Furthermore

$\displaystyle\lim_{N\to\infty}\|G_{N}^{\lambda}-G\|_{L^{1}(\mu_{N})}$
$\displaystyle=0$ (39)
$\displaystyle\lim_{N\to\infty}\|G_{N}^{\lambda}-G\|_{L^{2}(\mu_{N})}$
$\displaystyle=0\,.$ (40)

###### Proof.

We follow the proof of [Faggionato(2010), Lemma 3.1]. By taking the scalar
product with $G_{N}^{\lambda}$ in the left and right hand sides of (35) and
using Cauchy-Schwarz inequality we get

$\displaystyle\lambda\|G_{N}^{\lambda}\|^{2}_{L^{2}(\mu_{N})}+(-L^{N}G^{\lambda}_{N},G^{\lambda}_{N})_{\mu_{N}}=(H_{N},G^{\lambda}_{N})_{\mu_{N}}\leq\|G_{N}^{\lambda}\|_{L^{2}(\mu_{N})}\|H_{N}\|_{L^{2}(\mu_{N})}\,.$

Notice that $G_{N}^{\lambda}\in L^{2}(\mu_{N})$ and that
$\sup_{{N\in\mathbb{N}}}\|H_{N}^{\lambda}\|_{L^{2}(\mu_{N})}<\infty$, so that
we don’t have a trivial inequality. Recalling that
$(-L^{N}G^{\lambda}_{N},G^{\lambda}_{N})\geq 0$ by negativity of the operator
$L^{N}$ and noticing that $\|G_{N}^{\lambda}\|_{L^{2}(\mu_{N})}$ appears with
a square on the l.h.s., we obtain the $L^{2}$ bound in (37). As a consequence
we also get (36) and, since $L^{N}G_{N}^{\lambda}=\lambda
G_{N}^{\lambda}+H_{N}$, the $L^{2}$ bound in (38).

For the $L^{1}$ bounds we need the integral representation

$\displaystyle G_{N}^{\lambda}(x/N)=\sum_{y\in V}\int_{0}^{\infty}{\rm
e}^{-\lambda t}p_{t}^{N}(x/N,y/N)H_{N}(y/N)\,{\rm d}t\qquad x\in V$ (41)

where $p^{N}_{t}(x/N,y/N)$ indicates the probability that the random walk on
$V/N$ with generator $L^{N}$ started at $x/N$ is in $y/N$ at time $t\geq 0$,
for $x,y\in V$. Since by reversibility
$p^{N}_{t}(x/N,y/N)=p^{N}_{t}(y/N,x/N)$,

$\displaystyle\|G_{N}^{\lambda}\|_{L^{1}(\mu_{N})}\leq\frac{1}{N^{n}}\sum_{x,y\in
V}\int_{0}^{\infty}{\rm e}^{-\lambda t}p_{t}^{N}(y/N,x/N)|H_{N}(y/N)|\,{\rm
d}t$ (42)
$\displaystyle\qquad\qquad\qquad=\frac{1}{\lambda}\|H_{N}\|_{L^{1}(\mu_{N})}\xrightarrow{N\to\infty}\frac{1}{\lambda}\|H\|_{L^{1}(\mathbb{R}^{n})}<\infty,$
(43)

where $H=\lambda G-\sigma^{2}\Delta G$. This proves the first inequality of
(37), and the $L^{1}$ bound for $L^{N}G_{N}^{\lambda}$ in (38) follows as
before since
$\sup_{N\in\mathbb{N}}\|H_{N}^{\lambda}\|_{L^{1}(\mu_{N})}<\infty$.

We move to the homogenization results (39) and (40). By Lemma 3.1 we can apply
the results of [Faggionato(2022a)]. By [Faggionato(2022a), Theorem 1] we know
that $G_{N}^{\lambda}\to G$ in the strong sense described in
[Faggionato(2022a), Definition 3.11]. As pointed out in [Faggionato(2022a)],
see discussion after formula (167) therein, if condition (A9) is fulfilled as
in our case, then one also has the $L^{2}$ convergence in (40), since $G$ is
compactly supported.

Finally, we turn our attention to (39). For $\ell>0$, using Cauchy Schwarz
inequality, the quantity $\|G_{N}^{\lambda}-G\|_{L^{1}(\mu_{N})}$ can be upper
bounded by

$\displaystyle\|G_{N}^{\lambda}(\cdot)\mathbf{1}\\!_{\left\\{\|\,\cdot\,\|>\ell\right\\}}\|_{L^{1}(\mu_{N})}+\|G(\cdot)\mathbf{1}\\!_{\left\\{\|\,\cdot\,\|>\ell\right\\}}\|_{L^{1}(\mu_{N})}+\frac{\\#\\{x\in
B_{N\ell}\cap V\\}^{1/2}}{N^{n/2}}\|G_{N}^{\lambda}-G\|_{L^{2}(\mu_{N})}\,.$

The second term is null for $\ell$ large enough, since $G$ has compact
support. The third term goes to $0$ as $N\to\infty$ since the fraction is
$\mathbb{P}$–a.s. converging to a constant while
$\|G_{N}^{\lambda}-G\|_{L^{2}(\mu_{N})}$ goes to $0$ by (40). It remains to
show that

$\displaystyle\limsup_{\ell\to\infty}\limsup_{N\to\infty}\|G_{N}^{\lambda}(\cdot)\mathbf{1}\\!_{\left\\{\|\,\cdot\,\|>\ell\right\\}}\|_{L^{1}(\mu_{N})}=0\,.$
(44)

Since $H\in\mathcal{C}^{\infty}_{c}(\mathbb{R}^{n})$, we can find two non-
negative functions $H^{+},H^{-}\in\mathcal{C}^{\infty}_{c}(\mathbb{R}^{n})$
such that $H^{-}\leq H\leq H^{+}$ and so, for their restrictions
$H^{-}_{N},H^{+}_{N}$ to $V_{N}$, it holds $H_{N}^{-}\leq H_{N}\leq
H^{+}_{N}$. Call $f_{N}^{\lambda},F_{N}^{\lambda}$ the solutions in
$L^{2}(\mu_{N})$ of the equations

$\displaystyle\lambda
f_{N}^{\lambda}-L^{N}f_{N}^{\lambda}=H_{N}^{-}\qquad\lambda
F_{N}^{\lambda}-L^{N}F_{N}^{\lambda}=H^{+}_{N}\,.$

From (41) we derive that $f_{N}^{\lambda},F_{N}^{\lambda}$ are also non-
negative and that $-f_{N}^{\lambda}\leq G_{N}^{\lambda}\leq F_{N}^{\lambda}$
on $V_{N}$. In particular, in order to prove (44) we can just prove the same
equation with $f_{N}^{\lambda}$ and $F_{N}^{\lambda}$ instead of
$G_{N}^{\lambda}$. Therefore, without loss of generality, we can just assume
that $G_{N}^{\lambda}$ and $H$ are non-negative, the same proof working for
$G_{N}^{\lambda}$ and $H$ non-positive. By (35) and an integral representation
as in (41) we see that in this case the function $G$ such that $H=\lambda
G-\sigma^{2}\Delta G$ is non-negative, too. Cauchy Schwarz inequality yields

$\displaystyle\|G_{N}^{\lambda}(\cdot)$
$\displaystyle\mathbf{1}\\!_{\left\\{\|\,\cdot\,\|>\ell\right\\}}\|_{L^{1}(\mu_{N})}=\|G_{N}^{\lambda}(\cdot)\|_{L^{1}(\mu_{N})}-\|G_{N}^{\lambda}(\cdot)\mathbf{1}\\!_{\left\\{\|\,\cdot\,\|\leq\ell\right\\}}\|_{L^{1}(\mu_{N})}$
$\displaystyle\qquad\leq\|G_{N}^{\lambda}(\cdot)\|_{L^{1}(\mu_{N})}-\|G(\cdot)\mathbf{1}\\!_{\left\\{\|\,\cdot\,\|\leq\ell\right\\}}\|_{L^{1}(\mu_{N})}+\|(G_{N}^{\lambda}(\cdot)-G(\cdot))\mathbf{1}\\!_{\left\\{\|\,\cdot\,\|\leq\ell\right\\}}\|_{L^{1}(\mu_{N})}$
$\displaystyle\qquad\leq\|G_{N}^{\lambda}(\cdot)\|_{L^{1}(\mu_{N})}-\|G(\cdot)\mathbf{1}\\!_{\left\\{\|\,\cdot\,\|\leq\ell\right\\}}\|_{L^{1}(\mu_{N})}+\frac{\\#\\{x\in
B_{N\ell}\cap V\\}^{1/2}}{N^{d/2}}\|G_{N}^{\lambda}-G\|_{L^{2}(\mu_{N})}\,.$

The third summand goes to $0$ as $N\to\infty$ as seen before. To handle the
first summand, we notice that, since $G_{N}^{\lambda}$ and $H$ are non-
negative, the inequality in (42) is in fact an equality. Hence

$\displaystyle\limsup_{N\to\infty}\|G_{N}^{\lambda}(\cdot)\mathbf{1}\\!_{\left\\{\|\,\cdot\,\|>\ell\right\\}}\|_{L^{1}(\mu_{N})}=\|G(\cdot)\mathbf{1}\\!_{\left\\{\|\,\cdot\,\|>\ell\right\\}}\|_{L^{1}(\mu_{N})},$

which is null for $\ell$ large enough, since $G$ has compact support. ∎

As a result of having to deal with a non-conservative system, in order to
study the hydrodynamic limits we will also have to control the
$L^{2}(\mu_{N})$ norm of $L^{N}G$.

######  Lemma 3.4.

Let $G\in\mathcal{C}_{c}^{\infty}(\mathbb{R}^{n})$ and $n\geq 2$. Then,
$\mathbb{P}$–a.s.,

$\lim_{N\to\infty}N^{-n}\|L^{N}G\|_{L^{2}(\mu_{N})}=0\,.$

###### Proof.

First of all we bound the second moment of $\|L^{N}G\|_{L^{2}(\mu_{N})}$. Call
$S_{G}$ the support of $G$ and indicate with $NS_{G}$ the support blown by a
factor $N$. We have

$\displaystyle\mathbb{E}\big{[}\|L^{N}G\|_{L^{2}(\mu_{N})}^{2}\big{]}$
$\displaystyle=\mathbb{E}\Big{[}N^{-n}\sum_{x\in V}\Big{(}\sum_{y\in
V}N^{2}r(x,y)\big{(}G(y/N)-G(x/N)\big{)}\Big{)}^{2}\Big{]}$ (45)
$\displaystyle\leq 2N^{4-n}\big{(}(A)+(B)+(C)\big{)},$ (46)

where

$\displaystyle(A)$ $\displaystyle=\mathbb{E}\Big{[}\sum_{x\in NS_{G}\cap
V}\Big{(}\sum_{y\in B_{R}(x)\cap
V}r(x,y)\big{(}G(y/N)-G(x/N)\big{)}\Big{)}^{2}\Big{]}$ $\displaystyle(B)$
$\displaystyle=\mathbb{E}\Big{[}\sum_{x\in NS_{G}\cap V}\Big{(}\sum_{y\in
B_{R}^{c}(x)\cap V}r(x,y)\big{(}G(y/N)-G(x/N)\big{)}\Big{)}^{2}\Big{]}$
$\displaystyle(C)$ $\displaystyle=\mathbb{E}\Big{[}\sum_{x\in(NS_{G})^{c}\cap
V}\Big{(}\sum_{y\in NS_{G}\cap V}r(x,y)G(y/N)\Big{)}^{2}\Big{]},$

where $B_{R}(x)$ is a ball around $x$ of radius $R=\log N^{n}$. We proceed by
estimating separately the three parts. We can easily deal with part $(B)$
thanks to Slivnyak-Mecke theorem (see [Moller and Waagepetersen(2003), Theorem
13.3] or [Daley and Vere-Jones(2008), Chapter 13] for more general versions of
the theorem), which yields

$\displaystyle(B)$ $\displaystyle\leq\|G\|_{\infty}^{2}\int_{x\in
NS_{G}}\int_{y\notin B_{R}(x)}\Big{(}r(x,y)^{2}+r(x,y)\int_{z\notin
B_{R}(x)}r(x,z)\,{\rm d}z\Big{)}\,{\rm d}y\,{\rm d}x\leq cN^{n}{\rm
e}^{-R}\,,$ (47)

where the factor $N^{n}$ comes from the size of $NS_{G}$ and the factor ${\rm
e}^{-R}$ comes from the internal integrals.

Developing the square and using again Slivnyak-Mecke theorem, term $(C)$
becomes

$\displaystyle(C)=$ $\displaystyle\int_{x\notin NS_{G}}\int_{y\in
NS_{G}}r(x,y)^{2}G(y/N)^{2}\,{\rm d}y\,{\rm d}x$ $\displaystyle+\int_{x\notin
NS_{G}}\int_{y\in NS_{G}}\int_{z\in NS_{G}}r(x,y)r(x,z)G(y/N)G(z/N)\,{\rm
d}y\,{\rm d}z\,{\rm d}x.$ (48)

Since $G\in\mathcal{C}^{\infty}_{c}(\mathbb{R}^{n})$, $G$ must be Lipshitz
with Lipshitz constant, say, $K>0$. Call $d(x,A)$ the distance between
$x\in\mathbb{R}^{n}$ and the border of the set $A\subset\mathbb{R}^{n}$.
Noticing that $\|x-y\|\geq d(x,NS_{G})+d(y,NS_{G})$ if $x\not\in NS_{G}$ and
$y\in S_{G}$, we see that the first double integral on the r.h.s. of (48) is
smaller than

$\displaystyle\int_{x\notin NS_{G}}\int_{y\in NS_{G}}{\rm
e}^{-2(d(x,NS_{G})+d(y,NS_{G}))}$
$\displaystyle\Big{(}K\frac{d(y,NS_{G})}{N}\Big{)}^{2}\,{\rm d}y\,{\rm d}x$
$\displaystyle\leq c_{1}N^{-2}\int_{x\notin NS_{G}}{\rm
e}^{-2d(x,NS_{G})}N^{n-1}\,{\rm d}x\leq c_{2}N^{d-3}$

with $c_{1},c_{2}>0$ constants that depend on $G$. Regarding the triple
integral on the r.h.s. of (48) we can do something similar and bound it by

$\displaystyle\int_{x\notin NS_{G}}{\rm e}^{-2d(x,NS_{G})}\Big{(}\int_{y\in
NS_{G}}{\rm e}^{-2d(y,NS_{G})}\Big{(}K\frac{d(y,NS_{G})}{N}\Big{)}\,{\rm
d}y\Big{)}^{2}\,{\rm d}x\leq cN^{2n-4},$

with $c>0$ a constant depending on $G$. Plugging these two last bounds back
into (48) we got

$\displaystyle(C)\leq cN^{2n-4}.$ (49)

Finally we turn our attention to $(A)$. We use once more Slivnyak-Mecke
theorem and a first order Taylor approximation and obtain

$\displaystyle(A)$ $\displaystyle=\int_{x\in
NS_{G}}\mathbb{E}\Big{[}\Big{(}\sum_{y\in B_{R}(x)\cap
V}r(x,y)\Big{(}\sum_{i=1,\dots,n}\frac{y_{i}-x_{i}}{N}\tfrac{\rm d}{{\rm
d}x_{j}}G(x/N)+O\big{(}\|x-y\|^{2}/N^{2}\big{)}\Big{)}\Big{)}^{2}\Big{]}$
$\displaystyle\leq c_{1}N^{-2}\|\nabla G\|_{\infty}^{2}\int_{x\in
NS_{G}}\mathbb{E}\Big{[}\Big{(}\sum_{y\in B_{R}(x)\cap
V}r(x,y)\big{(}\|x-y\|+\|x-y\|^{2}/N\big{)}\Big{)}^{2}\Big{]}\,{\rm d}x$
$\displaystyle\leq c_{2}N^{-2}\|\nabla G\|_{\infty}^{2}\int_{x\in
NS_{G}}U(x,R)\,{\rm d}x$ $\displaystyle\leq c_{3}N^{n-2}$ (50)

where we have used the fact that

$\displaystyle U(x,R):=\int_{y\in
B_{R}(x)}\Big{(}r(x,y)^{2}\|x-y\|^{2}+\int_{z\in
B_{R}(x)}r(x,y)r(x,z)\|x-y\|\,\|x-z\|\,{\rm d}z\Big{)}{\rm d}y\leq c$

for some $c>0$.

We finally put (47), (49) and (50) back into (46) to obtain that

$\displaystyle\mathbb{E}[\|L^{N}G\|_{L^{2}(\mu_{N})}^{2}]\leq cN^{n}\,.$

By Markov inequality we obtain now that, for all $\varepsilon>0$,

$\displaystyle\mathbb{P}(N^{-n}\|L^{N}G\|_{L^{2}(\mu_{N})}>\varepsilon)\leq
c\,\varepsilon^{-2}N^{-n}\,,$

which tells us that the sequence $N^{-n}\|L^{N}G\|_{L^{2}(\mu_{N})}$ converges
almost completely to $0$ for $n\geq 2$ and hence almost surely. ∎

##  4\. A non-conservative Kipnis–Varadhan estimate

Recall the Domination $\&$ Convergence Assumption and in particular (6). For
constants $\rho>0$ and $M\in\mathbb{N}_{0}$, call
$\nu_{M,\rho}(\cdot)=\nu_{M,\rho}(\omega,\cdot)$ the measure that dominates
all initial conditions. That is, $\nu_{M,\rho}$ is the product measure on
$\mathbb{N}^{V}$ such that its restriction on each site $x\in V$ is a Poisson
random variable of parameter $\rho$ plus the constant $M\in\mathbb{N}$:

$\displaystyle\nu_{M,\rho}\Big{(}\prod_{x\in
A}[M+n_{x},\infty)\Big{)}=\prod_{x\in
A}\Big{(}\sum_{j=n_{x}}^{\infty}\frac{\rho^{j}{\rm
e}^{-\rho}}{j!}\Big{)}\qquad\forall A\subset V,\,(n_{x})_{x\in
A}\in\mathbb{N}^{|A|}\,.$ (51)

######  Lemma 4.1.

Consider an initial condition given by $\nu_{0,\rho}$, the product of Poisson
random variables of parameter $\rho>0$. Under $P_{\nu_{0,\rho}}^{\omega}$, let
each particle perform an independent random walk on $V/N$ with generator
$L^{N}$ (without births nor deaths) and call $(Y_{t})_{t\geq 0}$ the evolution
of their configuration, so that $Y_{t}(x)$ is the number of particles in $x\in
V$ at time $t$. Let $H$ be a nonnegative function on $V/N$ belonging to
$L^{1}(\mu_{N})\cap L^{2}(\mu_{N})$ and such that $L^{N}H$ belongs to
$L^{2}(\mu_{N})$. Then for any $T,A>0$ it holds

$\displaystyle P_{\nu_{0,\rho}}^{\omega}\Big{(}\sup_{0\leq t\leq
T}\frac{1}{N^{n}}\sum_{x\in V}Y_{t}(\tfrac{x}{N})H(\tfrac{x}{N})>A\Big{)}$
$\displaystyle\leq
c(\rho,T)A^{-1}{\left|\kern-1.07639pt\left|\kern-1.07639pt\left|H\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{N}$
(52) $\displaystyle P_{\nu_{0,\rho}}^{\omega}\Big{(}\sup_{0\leq t\leq
T}\frac{1}{N^{n}}\sum_{x\in V}Y_{t}(\tfrac{x}{N})^{2}H(\tfrac{x}{N})>A\Big{)}$
$\displaystyle\leq\tilde{c}(\rho,T)A^{-1}\sqrt{{\left|\kern-1.07639pt\left|\kern-1.07639pt\left|H\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{N}^{2}+N^{2-2n}\sum_{x\in
V}r(x)H(\tfrac{x}{N})}$ (53)

with $c(\rho,T)=(\rho^{2}+\rho+T\rho)^{1/2}$, $\tilde{c}(\rho,T)^{2}$ a
polynomial in $\rho$ and $T$ and

$\displaystyle{\left|\kern-1.07639pt\left|\kern-1.07639pt\left|H\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{N}^{2}:=\|H\|_{L^{1}(\mu_{N})}^{2}+N^{-n}\|H\|_{L^{2}(\mu_{N})}\,\|L^{N}H\|_{L^{2}(\mu_{N})}\,.$
(54)

######  Remark 4.2.

This sort of inequalities are typically carried out for all powers of the
number of particles $Y_{t}^{k}$ at once, at the only cost of a constant on the
r.h.s. varying with $k$, see for example [Faggionato(2010), Lemma 3.2]. In our
setting, though, we cannot hope for such a “clean” result for all values of
$k$, due to the irregularity of the support $V=V(\omega)$. In the rest of the
paper we only need $k=1$, but we bound here also the case $k=2$ for future
interest.

###### Proof of Lemma 4.1.

The particle dynamics without births or deaths is reversible with respect to
$\nu_{0,\rho}$. Hence, by Kipnis-Varadhan inequality ([Kipnis and
Varadhan(1986)], see also [Kipnis and Landim(1998), Theorem 11.1 in Appendix
1]) we know that, for $k\geq 1$,

$\displaystyle P_{\nu_{0,\rho}}^{\omega}\Big{(}\sup_{0\leq t\leq
T}\frac{1}{N^{n}}\sum_{x\in
V}Y_{t}(\tfrac{x}{N})^{k}H(\tfrac{x}{N})>A\Big{)}\leq\frac{\rm
e}{A}\sqrt{\langle g,g\rangle_{\nu_{0,\rho}}+T\,\langle
g,-{N^{2}\mathcal{L}_{*}}g\rangle_{\nu_{0,\rho}}}$ (55)

where ${N^{2}\mathcal{L}_{*}}$ is the generator of $(Y_{t})_{t\geq 0}$ and
$g:\mathbb{N}^{V}\to\mathbb{R}$ is given by

$g(\eta):=\frac{1}{N^{n}}\sum_{x\in V}g_{x}(\eta)H(x/N),\qquad
g_{x}(\eta):=\eta(x)^{k}.$

Notice that $\mathcal{L}_{*}$ corresponds to $\mathcal{L}$ appearing in (2)
with $b=d=0$. Now we calculate

$\displaystyle\langle g,g\rangle_{\nu_{0,\rho}}$
$\displaystyle=\frac{1}{N^{2n}}\sum_{x,y\in
V}H(x/N)H(y/N)\nu_{0,\rho}[g_{x}g_{y}]\leq
c_{0}(\rho,k)\|H\|_{L^{1}(\mu_{N})}^{2}$ (56)

where $c_{0}(\rho,k)=\mathbb{E}[\xi_{\rho}^{2k}]$ indicates the $2k$-th moment
of $\xi_{\rho}\sim$Poisson($\rho$). Moving to the second summand under the
root in (55), we write

$\displaystyle\langle g,-N^{2}\mathcal{L}_{*}g\rangle_{\nu_{0,\rho}}$
$\displaystyle=-N^{2-2n}\sum_{x,y\in
V}H(x/N)H(y/N)\nu_{0,\rho}[g_{x}\,\mathcal{L}_{*}g_{y}]\,.$ (57)

Besides, we have

$\displaystyle\mathcal{L}_{*}g_{y}(\eta)=\eta(y)r(y)\big{(}(\eta(y)-1)^{k}-\eta(y)^{k}\big{)}+\sum_{z\in
V}\eta(z)r(z,y)\big{(}(\eta(y)+1)^{k}-\eta(y)^{k}\big{)}.$

For $x=y$, we get

$\displaystyle\nu_{0,\rho}\big{[}g_{x}\,\mathcal{L}_{*}g_{x}\big{]}$
$\displaystyle=c_{1}(\rho,k)r(x)$

with

$c_{1}(\rho,k)=\mathbb{E}\big{[}\xi_{\rho}^{k+1}((\xi_{\rho}-1)^{k}-\xi_{\rho}^{k})\big{]}+\mathbb{E}[\xi_{\rho}]\mathbb{E}\big{[}\xi_{\rho}^{k}((\xi_{\rho}+1)^{k}-\xi_{\rho}^{k})\big{]}\leq
0\,.$

For $x\neq y$, using that
$\mathcal{L}_{*}g_{y}(\eta)-\eta(x)r(x,y)\big{(}(\eta(y)+1)^{k}-\eta(y)^{k})$
is independent of $\eta(x)$ under $\nu_{0,\rho}$ and that
$\nu_{0,\rho}[\mathcal{L}_{*}f]=0$ for all $f$,

$\displaystyle\nu_{0,\rho}\big{[}g_{x}\,\mathcal{L}_{*}g_{y}\big{]}$
$\displaystyle=\nu_{0,\rho}\Big{[}\eta^{k}(x)\cdot\eta(x)r(x,y)\big{(}(\eta(y)+1)^{k}-\eta(y)^{k}\big{)}\Big{]}$
$\displaystyle\quad+\nu_{0,\rho}[\eta^{k}]\nu_{0,\rho}\Big{[}\mathcal{L}_{*}(g_{y})-\eta(x)r(x,y)\big{(}(\eta(y)+1)^{k}-\eta(y)^{k}\big{)}\Big{]}$
$\displaystyle=c_{2}(\rho,k)r(x,y)\,.$

with

$c_{2}(\rho,k)=\big{(}\mathbb{E}[\xi_{\rho}^{k+1}]-\mathbb{E}[\xi_{\rho}^{k}]\mathbb{E}[\xi_{\rho}]\big{)}\mathbb{E}\big{[}(\xi_{\rho}+1)^{k}-\xi_{\rho}^{k}\big{]}\,.$

When $k=1$ we magically have $c_{2}(\rho,1)=-c_{1}(\rho,1)=\rho$, so that

$\displaystyle\langle g,-\mathcal{L}_{*}g\rangle_{\nu_{0,\rho}}$
$\displaystyle=\rho N^{2-2n}\Big{(}\sum_{x\in V}r(x)H(x/N)^{2}-\sum_{x\neq
y\in V}r(x,y)H(x/N)H(y/N)\Big{)}$ $\displaystyle=\rho N^{-2n}\sum_{x\in
V}H(x/N)\Big{(}-N^{2}\sum_{y\neq x}r(x,y)\big{(}H(y/N)-H(x/N)\big{)}\Big{)}$
$\displaystyle=\rho N^{-n}\langle H,L^{N}H\rangle_{\mu_{N}}$
$\displaystyle\leq\rho
N^{-n}\|H\|_{L^{2}(\mu_{N})}\,\|L^{N}H\|_{L^{2}(\mu_{N})}\,.$

Putting this and (56) back into (55) together with the fact that
$c_{0}(\rho,1)=\mathbb{E}[\xi_{\rho}^{2}]=\rho^{2}+\rho$ gives (52).

When $k=2$, explicit calculation yield
$c_{1}(\rho,2)=-\rho(4\rho^{2}+8\rho+1)$ and
$c_{2}(\rho,2)=4\rho^{3}+4\rho^{2}+\rho$. They do not cancel out as in the
case $k=1$ and as a consequence we have another term appearing from the term
$\langle g,-\mathcal{L}_{*}g\rangle_{\nu_{0,\rho}}$, that is

$\displaystyle\langle g,-\mathcal{L}_{*}g\rangle_{\nu_{0,\rho}}$
$\displaystyle\leq
c_{2}(\rho,k)N^{-n}\|H\|_{L^{2}(\mu_{N})}\,\|L^{N}H\|_{L^{2}(\mu_{N})}+\big{|}c_{1}(\rho,k)+c_{2}(\rho,k)\big{|}\,R$

with

$R=N^{2-2n}\sum_{x\in V}r(x)H(x/N)^{2}.$

Putting the pieces together as before we obtain (53). ∎

Let us turn to the non-conservative case.

######  Lemma 4.3.

Let $\eta_{0}^{N}$ be an initial distribution of particles whose law is
dominated by $\nu_{M,\rho}$ for some $M\in\mathbb{N}_{0}$ and $\rho\geq 0$ in
the sense of (6). Let $H$ be a nonnegative function on $V/N$ belonging to
$L^{1}(\mu_{N})$ and $L^{2}(\mu_{N})$. Then there exist a constant
$c_{1}=c(M,\rho,T)>0$ and an absolute constant $c_{2}>0$ such that

$\displaystyle P^{\omega}\Big{(}\sup_{0\leq t\leq T}\frac{1}{N^{n}}\sum_{x\in
V}\eta^{N}_{t}(x)H(x/N)>A\Big{)}\leq A^{-1}c_{1}{\rm
e}^{c_{2}bT}{\left|\kern-1.07639pt\left|\kern-1.07639pt\left|H\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{N}$
(58)

for all $A>0$, where
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|H\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{N}$
is defined in (54).

###### Proof.

The probability appearing in (58) can be clearly upper bounded by the
probability of the same event starting with a configuration sampled with
$\nu_{M,\rho}$. As a first step, we would like to further bound the initial
condition in order to have a pure product of a Poisson number of particles per
site, which will allow us to use the result of Lemma 4.1 in the following. To
this end we first focus on the case $M=1$, $\rho=0$. In this case we have that
the initial condition $\nu_{1,0}$ is given by a single particle on each site
of $V$. Take two random variables $X,Y\sim\text{Poisson}(\log 2)$ such that

$\displaystyle P(X+Y\geq 1)=1\,.$

We can dominate $\nu_{1,0}$ by the random initial condition $\bar{\nu}$ given
by the following: the number of particles on site $x\in V$ is given by
$X(x)+Y(x)$, with $X(x)\sim X$ and $Y(x)\sim Y$ and $(X(x),Y(x))_{x\in V}$
independent for different $x\in V$. Now we notice that if we want
$N^{-n}\sum_{x\in V}\eta^{N}_{t}(x)H(\tfrac{x}{N})$ to be greater than $A$, it
must be that $N^{-n}\sum_{x\in V}\eta^{N,X}_{t}(x)H(\tfrac{x}{N})$ is larger
than $A/2$, with $\eta_{t}^{N,X}$ are the particles descending from initial
particles “of type X”, or $N^{-n}\sum_{x\in
V}\eta^{N,Y}_{t}(x)H(\tfrac{x}{N})$ has to be greater than $A/2$. So with a
union bound we get

$\displaystyle P_{\nu_{1,0}}^{\omega}\Big{(}\sup_{0\leq t\leq
T}\frac{1}{N^{n}}\sum_{x\in V}\eta^{N}_{t}(x)H(\tfrac{x}{N})>A\Big{)}\leq
2P^{\omega}_{\nu_{0,\log 2}}\Big{(}\sup_{0\leq t\leq
T}\frac{1}{N^{n}}\sum_{x\in V}\eta^{N}_{t}(x)H(\tfrac{x}{N})>A/2\Big{)}\,.$

It is straightforward to generalize the previous argument to the case $M\geq
1$ and $\rho\geq 0$ which yields

$\displaystyle P_{\nu_{M,\rho}}^{\omega}\Big{(}\sup_{0\leq t\leq
T}\frac{1}{N^{n}}$ $\displaystyle\sum_{x\in
V}\eta^{N}_{t}(x)H(\tfrac{x}{N})>A\Big{)}$
$\displaystyle\leq(M+1)P^{\omega}_{\nu_{0,\rho\vee\log 2}}\Big{(}\sup_{0\leq
t\leq T}\frac{1}{N^{n}}\sum_{x\in
V}\eta^{N}_{t}(x)H(\tfrac{x}{N})>\tfrac{A}{M+1}\Big{)}\,.$

From this we see that, at the cost of a constant factor depending on $M$, we
can prove (58) with initial particle configuration $\nu_{0,\rho}$, where we
have replaced the original $\rho$ with $\rho\vee\log 2$.

We use a new labelling notation for the particles, not to be confused with the
one appearing in Section 2.1. The individuals at time $0$ are labelled by
$\mathbb{N}$. To label their descendants, we introduce the binary tree

$\mathcal{J}=\cup_{k\in\mathbb{N}_{0}}\\{1,2\\}^{k}.$

For $j=(j_{1},\dots,j_{k})\in\mathcal{J}$, $k\in\mathbb{N}_{0}$ and
$n\in\mathbb{N}$, we write $(n,j)=(n,j_{1},\dots,j_{k})$. In particular,
$(n,j)=(n,j_{1},...,j_{k})$ is an individual of generation $|j|=k$. When a
particle $(n,j)\in\mathbb{N}\times\\{1,2\\}^{k}$ reproduces, it disappears and
generates particles $(n,j_{1},...,j_{k},1)$ and $(n,j_{1},...,j_{k},2)$. For a
subset $A=I\times J$ with $I\subset\mathbb{N}$ and $J\subset\mathcal{J}$, we
write $(\eta_{t}^{N,A})$ for the process restricted to the subset of particles
labelled by elements of $A$, that is, at time $t$ we look at $\eta_{t}^{N}$
and ignore all the particles with labels not belonging to $A$. Since

$\eta_{t}^{N}=\sum_{j\in\mathcal{J}}\eta_{t}^{N,\mathbb{N}\times\\{j\\}}$

we have

$\Sigma_{T}^{N}:=\sup_{0\leq t\leq T}\frac{1}{N^{n}}\sum_{x\in
V}\eta^{N}_{t}(x)H(x/N)\leq\sum_{j\in\mathcal{J}}\Sigma_{T}^{N,j}$

where

$\Sigma_{T}^{N,j}:=\sup_{0\leq t\leq T}\frac{1}{N^{n}}\sum_{x\in
V}\eta^{N,\mathbb{N}\times\\{j\\}}_{t}(x)H(x/N)\,.$

Using that $\sum_{j\in\mathcal{J}}4^{-|j|}=\sum_{k\geq 0}2^{k}4^{-k}=2$, we
can bound

$\displaystyle P^{\omega}_{\nu_{0,\rho}}(\Sigma_{T}^{N}\geq A)\leq
P^{\omega}_{\nu_{0,\rho}}\left(\cup_{j\in\mathcal{J}}\\{\Sigma_{T}^{N,j}\geq
4^{-|j|}A/2\\}\right)\leq\sum_{j\in\mathcal{J}}P^{\omega}_{\nu_{0,\rho}}\left(\Sigma_{T}^{N,j}\geq
4^{-|j|}A/2\right)\,.$ (59)

The key point is now to see that the process
$(\eta_{t}^{N,\mathbb{N}\times\\{j\\}})$ can be dominated by another process
$(Y_{t}^{N,j})$, obtained by a percolation procedure on the initial
distribution of particles. More precisely, $Y_{0}^{N,j}$ is obtained from
$\eta_{0}^{N}$ as follows: for $\ell\in\mathbb{N}$, the particle with label
$\ell$ in $\eta_{0}^{N}$ is kept in $Y_{0}^{N,j}$ only if particle $(\ell,j)$
is born before time $T$ for the process $(\eta_{t}^{N})$. Notice that this
happens with probability

$\displaystyle p_{j}=\mathbb{P}(\text{Poisson}(bT)\geq|j|)={\rm
e}^{-bT}\sum_{k\geq|j|}\frac{(bT)^{k}}{k!}$ (60)

since, along each lineage, birth events follow a Poisson process with
intensity $b$. If present in $Y_{0}^{N,j}$, then, particle $\ell$ evolves in
the process $(Y_{t}^{N,j})$ by following the trajectory of $(\ell,j)$ and its
ancestors in $(\eta_{0}^{N})$; once $(\ell,j)$ has disappeared in
$(\eta_{0}^{N})$, the particle continues to evolve following the trajectory of
any lineage of descendants of $(\ell,j)$. From this coupling, it is clear
that, for all $t\in[0,T]$ and $j\in\mathcal{J}$,

$\displaystyle\eta_{t}^{N,\mathbb{N}\times\\{j\\}}\leq Y_{t}^{N,j}$

and we have obtained

$\displaystyle P^{\omega}_{\nu_{0,\rho}}\left(\Sigma_{T}^{N,j}\geq
4^{-|j|}A/2\right)\leq P^{\omega}_{\nu_{0,\rho}}\Big{(}\sup_{0\leq t\leq
T}\frac{1}{N^{n}}\sum_{x\in V}Y_{t}^{N,j}(x)H(x/N)>4^{-|j|}A/2\Big{)}\,.$

At this point, we can use Lemma 4.1 for the process
$(\tilde{Y}_{t}^{N,j})_{t\in[0,T]}$ on $V/N$, with
$\tilde{Y}_{t}^{N,j}(x/N)=Y_{t}^{N,j}(x)$ for all $x\in V$ and $t\in[0,T]$,
since for this process the present particles just perform independent random
walks on $V/N$ generated by $L^{N}$. We notice furthermore that the initial
particles of process $(Y_{t}^{N,j})$ have distribution $\nu_{0,\rho p_{j}}$,
cfr. (60). Hence

$\displaystyle P^{\omega}_{\nu_{0,\rho}}\Big{(}\sup_{0\leq t\leq
T}\frac{1}{N^{n}}\sum_{x\in V_{N}}Y_{t}^{N,j}(x)H(x/N)>4^{-|j|}A/2\Big{)}\leq
4^{|j|}2A^{-1}c(\rho
p_{j},T){\left|\kern-1.07639pt\left|\kern-1.07639pt\left|H\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{N}$

where the function $c(\cdot,\cdot)$ is the same appearing in Lemma 4.1. Going
back to (59) we have obtained

$\displaystyle P^{\omega}_{\nu_{0,\rho}}(\Sigma_{T}^{N}\geq
A)\leq\sum_{j\in\mathcal{J}}4^{|j|}2A^{-1}c(\rho
p_{j},T){\left|\kern-1.07639pt\left|\kern-1.07639pt\left|H\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{N}\leq\bar{c}(\rho,T)A^{-1}\sum_{j\in\mathcal{J}}4^{|j|}p_{j}^{1/2}.$
(61)

Recall that if $X\sim Poisson(\lambda)$ one has the bound $\mathbb{P}(X\geq
t)\leq{\rm e}^{-\lambda}({\rm e}\lambda/t)^{t}$ for all $t>\lambda$ (see for
example [Vershynin(2018), Exercise 2.3.3]). Using also that the number of
$j$’s of length $\ell$ is $2^{\ell}$, we set $\bar{\ell}=\lceil 81{\rm
e}bT\rceil$ and compute

$\displaystyle\sum_{j\in\mathcal{J}}4^{|j|}p_{j}^{1/2}\leq\sum_{\ell=0}^{\bar{\ell}}8^{\ell}+\sum_{\ell=\bar{\ell}+1}^{\infty}8^{\ell}\Big{(}\frac{{\rm
e}bT}{\ell}\Big{)}^{\ell/2}\leq c{\rm e}^{c_{2}bT}$

with $c,c_{2}>0$ absolute constants, which together with (61) yields the
result of the lemma. ∎

##  5\. Proof of Theorem 1.2

As in Section 2, through the whole section we fix some realization of the
underlying graph $\omega\in\Omega$ sampled according to measure $\mathbb{P}$.
All the processes in what follows will evolve under measure $P^{\omega}$, and
all the claims have to be intended to be true $\mathbb{P}$–almost surely.

###  5.1. An $L^{2}$ martingale

In this section we will pave the way for the proof of tightness of the
sequence of process
$\left((\langle\pi_{t}^{N},G\rangle)_{t\in[0,T]}\right)_{N}$ and
identification of the limit. For
$G\in\mathcal{C}_{c}^{\infty}(\mathbb{R}^{n})$ let us define the process

$\displaystyle
M_{t}^{N}=M_{t}^{N}(G_{N}^{\lambda}):=\langle\pi_{t}^{N},G_{N}^{\lambda}\rangle-\langle\pi_{0}^{N},G_{N}^{\lambda}\rangle-\int_{0}^{t}\langle\pi_{s}^{N},L^{N}G_{N}^{\lambda}+bG_{N}^{\lambda}\rangle{\rm
d}s\,.$ (62)

By Lemma 3.3 and Lemma 4.3, we know that $M^{N}$ is almost surely well defined
when starting from some $\eta_{0}^{N}$ satisfying (6). We aim at proving the
following result:

######  Lemma 5.1.

Consider a sequence of initial configurations
$(\eta_{0}^{N})_{N\in\mathbb{N}}$ satisfying the Domination $\&$ Convergence
Assumption. For all $\varepsilon>0$ and for all
$G\in\mathcal{C}_{c}^{\infty}(\mathbb{R}^{n})$ it holds

$\displaystyle\lim_{N\to\infty}P^{\omega}\Big{(}\sup_{0\leq t\leq
T}\big{|}M_{t}^{N}\big{|}\geq\varepsilon\Big{)}=0\,.$

In fact, we will not only show Lemma 5.1, but also that $M^{N}$ is a square
integrable martingale which converges in $L^{2}$ to $0$ and obtain a speed of
convergence, see next Lemma 5.2. To do so, we will use a truncation argument
and exploit the results already obtained in Section 2.3 while constructing the
process. More precisely, for any $a\in\mathbb{N}$, consider the process
$(\pi^{N,a}_{t})_{t\in[0,T]}$ (and the corresponding
$(\eta^{N,a}_{t})_{t\in[0,T]}$) where the initial configuration of particles
is truncated outside the box $[-a,a]^{n}$, that is, only the particles in the
finite set $V\cap[-a,a]^{n}$ are retained for the initial configuration and
all the others are deleted. By (16), we know that all the particles of the
process $\pi^{N,a}$ a.s. remain in a finite box during time interval $[0,T]$
(recall that $\widehat{\pi}_{t}^{B_{N}}$ was the process restricted to a box
of size $B_{N}$ and that for finite initial conditions $\pi^{N}_{t}$ was
obtained as the restriction to the second coordinate of the limit for
$N\to\infty$ of $\widehat{\pi}_{t}^{B_{N}}$, cfr. Corollary 2.4 and (13)). It
follows that the number of births and of jumps is a.s. finite in a finite time
interval and therefore the following equation holds for any locally bounded
function $H$:

$\displaystyle\langle\pi_{t}^{N,a},H\rangle=\langle\pi_{0}^{N,a},H\rangle$
$\displaystyle+\frac{1}{N^{n}}\int_{0}^{t}\int_{\mathbb{R}^{+}}\sum_{x,y\in
V}\mathbf{1}\\!_{\left\\{u\leq\eta_{s}^{N,a}(x)N^{2}r(x,y)\right\\}}\big{(}H(y/N)-H(x/N)\big{)}\mathcal{N}^{x,y}({\rm
d}s,{\rm d}u)$
$\displaystyle+\frac{1}{N^{n}}\int_{0}^{t}\int_{\mathbb{R}^{+}}\sum_{x\in
V}\mathbf{1}\\!_{\left\\{u\leq
b\eta_{s}^{N,a}(x)\right\\}}\,H(x/N)\,\mathcal{Q}^{x}({\rm d}s,{\rm d}u)\,.$
(63)

Notice that we have adopted here a slightly different description of the
process for convenience. The underlying Poisson point processes are indexed by
sites and not by individuals as before. That is, measures $\mathcal{N}^{x,y}$
and $\mathcal{Q}^{x}$ with intensity ${\rm d}s\,{\rm d}u$ on
$\mathbb{R}_{+}^{2}$ are replacing the previous
$\mathcal{N}_{i}^{X_{s}^{i},y}$ and $\mathcal{N}_{i}^{b}$. Equation (5.1) can
be rewritten as

$\displaystyle\langle\pi_{t}^{N,a},H\rangle$
$\displaystyle=\langle\pi_{0}^{N,a},H\rangle+\int_{0}^{t}\langle\pi_{s}^{N,a},L^{N}H+bH\rangle\,{\rm
d}s+M^{N,a}_{t}(H),$ (64)

where $M^{N,a}(H)$ is defined by

$\displaystyle M^{N,a}_{t}(H)=$
$\displaystyle\frac{1}{N^{n}}\int_{0}^{t}\int_{\mathbb{R}^{+}}\sum_{x,y\in
V}\mathbf{1}\\!_{\left\\{u\leq\eta_{s}^{N,a}(x)N^{2}r(x,y)\right\\}}\big{(}H(y/N)-H(x/N)\big{)}\widetilde{\mathcal{N}}^{x,y}({\rm
d}s,{\rm d}u)$
$\displaystyle+\frac{1}{N^{n}}\int_{0}^{t}\int_{\mathbb{R}^{+}}\sum_{x\in
V}\mathbf{1}\\!_{\left\\{u\leq
b\eta_{s}^{N,a}(x)\right\\}}\,H(x/N)\,\widetilde{{\mathcal{Q}}}^{x}({\rm
d}s,{\rm d}u),$

and $\widetilde{\mathcal{N}}^{x,y}$ and $\widetilde{\mathcal{Q}}^{x}$ are the
compensated measures of $\mathcal{N}^{x,y}$ and ${\mathcal{Q}}^{x}$.

We turn our attention to $H=G_{N}^{\lambda}$. On the one hand,
$\langle\pi_{t}^{N,a},G_{N}^{\lambda}\rangle$ increases a.s. as
$a\rightarrow\infty$ to $\langle\pi_{t}^{N},G_{N}^{\lambda}\rangle$, which is
a.s. finite (using for example (58) and Lemma 3.3). On the other hand, the
fact that $L^{N}G_{N}^{\lambda}=\lambda G_{N}^{\lambda}-H_{N}$ (cfr. (35)) and
(58) ensure that

$\int_{0}^{t}\langle\pi_{s}^{N},|L^{N}G_{N}^{\lambda}|+bG_{N}^{\lambda}\rangle\,ds<\infty\quad\text{a.s.}$

and it follows by bounded convergence that a.s.

$\lim_{a\rightarrow\infty}\int_{0}^{t}\langle\pi_{s}^{N,a},L^{N}G_{N}^{\lambda}+bG_{N}^{\lambda}\rangle\,{\rm
d}s=\int_{0}^{t}\langle\pi_{s}^{N},L^{N}G_{N}^{\lambda}+bG_{N}^{\lambda}\rangle\,{\rm
d}s\,.$

We obtain from (64) that for any $t\geq 0$,
$M^{N,a}_{t}=M^{N,a}_{t}(G_{N}^{\lambda})$ converges a.s. as $a\to\infty$ to
$M_{t}^{N}$, which is given by (74) and is a.s. finite.

To wrap up, we have defined a càdlàg process $(M_{t}^{N})_{t\in[0,T]}$
satisfying identity (74) and such that, for any $t$, $M^{N}_{t}$ is the a.s.
limit of $M^{N,a}_{t}$, defined as an integral against compensated jump
measures. Let us check now that these processes are also square integrable
martingale and that they tend to $0$ in $L^{2}$ and probability as
$N\to\infty$. This in particular implies Lemma 5.1.

######  Lemma 5.2.

For any $N\geq 1$ and $a>0$, $M^{N,a}$ and $M^{N}$ are càdlàg square
integrable martingales and, for any $T>0$,

$E^{\omega}\Big{[}\sup_{t\leq
T}\big{(}M^{N,a}_{t}-M^{N}_{t}\big{)}^{2}\Big{]}\stackrel{{\scriptstyle
a\rightarrow\infty}}{{\longrightarrow}}0\,.$

Furthermore, for any $a>0$ and $N\geq 1$,

$\displaystyle E^{\omega}\Big{[}\sup_{t\leq
T}(M^{N,a}_{t})^{2}\Big{]}+E^{\omega}\Big{[}\sup_{t\leq
T}(M^{N}_{t})^{2}\Big{]}\leq\frac{C_{T}}{N^{n}}$ (65)

for some constant $C_{T}$ which only depends on $T$.

###### Proof.

We first prove that $M^{N,a}=M^{N,a}(G_{N}^{\lambda})$ is a square integrable
martingale. Its quadratic variation is

$\displaystyle\langle M^{N,a}\rangle_{t}=$
$\displaystyle\frac{N^{2}}{N^{2n}}\int_{0}^{t}\sum_{x,y\in
V}\eta_{s}^{N,a}(x)r(x,y)\left(G_{N}^{\lambda}(y/N)-G_{N}^{\lambda}(x/N)\right)^{2}{\rm
d}s$ $\displaystyle+\frac{1}{N^{2n}}\int_{0}^{t}\sum_{x\in
V}b\eta_{s}^{N,a}(x)G_{N}^{\lambda}(x/N)^{2}\,{\rm d}s\,.$

Since $E^{\omega}[\eta_{s}^{N,a}(x)]\leq E^{\omega}[\eta_{s}^{N}(x)]\leq C{\rm
e}^{bs}$ (cfr. equation (31) and recall that $\eta_{s}$ is the projection on
alive $a$–particles for $Z_{s}$) we get

$\displaystyle E^{\omega}\left[\langle M^{N,a}\rangle_{t}\right]$
$\displaystyle\leq C^{\prime}e^{bt}\Big{(}\frac{N^{2}}{N^{2n}}\sum_{x,y\in
V}r(x,y)\left(G_{N}^{\lambda}(y/N)-G_{N}^{\lambda}(x/N)\right)^{2}+\frac{1}{N^{2n}}\sum_{x\in
V}bG_{N}^{\lambda}(x/N)^{2}\Big{)}\,.$ (66)

Rewriting

$\displaystyle
r(x,y)\left(G_{N}^{\lambda}(y/N)-G_{N}^{\lambda}(x/N)\right)^{2}$
$\displaystyle\quad=-r(x,y)G_{N}^{\lambda}(x/N)\left(G_{N}^{\lambda}(y/N)-G_{N}^{\lambda}(x/N)\right)-r(y,x)G_{N}^{\lambda}(y/N)\left(G_{N}^{\lambda}(x/N)-G_{N}^{\lambda}(y/N)\right)$

we obtain

$\displaystyle E^{\omega}\left[\langle M^{N,a}\rangle_{t}\right]$
$\displaystyle\leq\frac{C^{\prime\prime}{\rm
e}^{2bt}}{N^{n}}\big{(}(G_{N}^{\lambda},-L^{N}G_{N}^{\lambda})_{\mu_{N}}+\|G_{N}^{\lambda}\|_{L^{2}(\mu_{N})}^{2}\big{)}$
(67)

which is finite by (36) and (37). It follows that $M^{N,a}$ is a square
integrable martingale and using Doob’s inequality we also obtain the relative
$L^{2}$ bound appearing in (65).

We prove now by Cauchy criterion that $M^{N,a}$ converges to some right-
continuous square integrable martingale, since the space of $L^{2}$ right-
continuous martingales is complete (see e.g. [Ikeda and Watanabe(1989), Lemma
2.1]). By uniqueness, this limit will then have to be $M^{N}$. Notice that
(65) will automatically follow, since the $L^{2}$ bound for $M^{N}$ can be
derived from that of $M^{N,a}$ taking the limit. More precisely let
$a<a^{\prime}$. Then

$\displaystyle M^{N,a^{\prime}}_{t}-M^{N,a}_{t}=$
$\displaystyle\frac{1}{N^{n}}\int_{0}^{t}\int_{\mathbb{R}^{+}}\sum_{x,y\in
V}\mathbf{1}\\!_{\left\\{\eta_{s}^{N,a}<u\leq\eta_{s}^{N,a^{\prime}}(x)N^{2}r(x,y)\right\\}}\left(G(y/N)-G(x/N)\right)\widetilde{\mathcal{N}}^{x,y}({\rm
d}s,{\rm d}u)$
$\displaystyle+\frac{1}{N^{n}}\int_{0}^{t}\int_{\mathbb{R}^{+}}\sum_{x\in
V}\mathbf{1}\\!_{\left\\{b\eta_{s}^{N,a}(x)<u\leq
b\eta_{s}^{N,a^{\prime}}(x)\right\\}}\,G(x/N)\,\widetilde{{\mathcal{Q}}}^{x}({\rm
d}s,{\rm d}u),$

and

$\displaystyle\langle M^{N,a^{\prime}}-M^{N,a}\rangle_{t}=$
$\displaystyle\frac{N^{2}}{N^{2n}}\int_{0}^{t}\sum_{x,y\in
V}\big{(}\eta_{s}^{N,a^{\prime}}(x)-\eta_{s}^{N,a}(x)\big{)}\,r(x,y)\left(G_{N}^{\lambda}(y/N)-G_{N}^{\lambda}(x/N)\right)^{2}{\rm
d}s$ $\displaystyle+\frac{1}{N^{2n}}\int_{0}^{t}\sum_{x\in
V}b\big{(}\eta_{s}^{N,a^{\prime}}(x)-\eta_{s}^{N,a}(x)\big{)}G_{N}^{\lambda}(x/N)^{2}{\rm
d}s\,.$

As $M^{N,a^{\prime}}-M^{N,a}$ is a square integrable martingale and by
monotonicity with respect to $a^{\prime}$,

$\displaystyle E^{\omega}\Big{[}$ $\displaystyle\sup_{t\leq
T}(M^{N,a^{\prime}}_{t}-M^{N,a}_{t})^{2}\Big{]}$ $\displaystyle\leq$
$\displaystyle\frac{4N^{2}}{N^{2n}}\int_{0}^{t}\sum_{x,y\in
V}E^{\omega}\big{[}\eta_{s}^{N}(x)-\eta_{s}^{N,a}(x)\big{]}r(x,y)\left(G_{N}^{\lambda}(y/N)-G_{N}^{\lambda}(x/N)\right)^{2}{\rm
d}s$ $\displaystyle+\frac{4}{N^{2n}}\int_{0}^{t}\sum_{x\in
V}bE^{\omega}\big{[}\eta_{s}^{N}(x)-\eta_{s}^{N,a}(x)\big{]}G_{N}^{\lambda}(x/N)^{2}{\rm
d}s$

We can apply bounded convergence, see (66) and (67) to get that the right hand
side goes to $0$ as $a$ goes to infinity. ∎

###  5.2. Proof of tightness

Having proved in the previous section that the process $M^{N}$ defined in (74)
is an $L^{2}$ martingale makes now the proof of tightness quite straight-
forward by using Aldous criterion.

######  Lemma 5.3.

Consider a sequence of initial configurations
$(\eta_{0}^{N})_{N\in\mathbb{N}}$ satisfying the Domination $\&$ Convergence
Assumption. The sequence of processes
$\\{(\pi_{t}^{N})_{t\in[0,T]}\\}_{N\in\mathbb{N}}$ is tight in
$D([0,T],\mathcal{M})$.

Since $\mathcal{M}$ is separable and the vague topology in $\mathcal{M}$ is
metrizable, in order to prove tightness of $\pi^{N}_{t}$ in
$D([0,T],\mathcal{M})$ it is enough to show tightness of
$\langle\pi^{N}_{t},G\rangle$ in $D([0,T],\mathbb{R})$ for $G$ in a dense
subset of $\mathcal{C}(\mathbb{R}^{n})$, the set of continuous functions in
$\mathbb{R}^{n}$. We consider functions $G\in\mathcal{C}_{c}(\mathbb{R}^{n})$
that are continuous with compact support. The proof of this fact can be
obtained by following [Kipnis and Landim(1998), Proposition 1.7, Chapter IV]
or [Meleard and Roelly(1993)]. Furthermore, the tightness of
$\langle\pi^{N}_{t},G\rangle$ can be deducted by that of
$\langle\pi^{N}_{t},G_{N}^{\lambda}\rangle$:

######  Lemma 5.4.

For any $G\in\mathcal{C}^{\infty}_{c}(\mathbb{R}^{n})$ and for
$\mathbb{P}-a.a.$ $\omega$ the following holds. Consider a sequence of initial
configurations $(\eta_{0}^{N})_{N\in\mathbb{N}}$ satisfying the Domination
$\&$ Convergence Assumption. Then the process
$(\langle\pi_{t}^{N},G_{N}^{\lambda}\rangle)_{t\in[0,T]}$ is well defined for
all $N\in\mathbb{N}$. The tightness of
$\\{(\langle\pi_{t}^{N},G_{N}^{\lambda}\rangle)_{t\in[0,T]}\\}_{N\in\mathbb{N}}$
in $D([0,T],\mathbb{R})$ implies the tightness of
$\\{(\langle\pi_{t}^{N},G\rangle)_{t\in[0,T]}\\}_{N\in\mathbb{N}}$ in
$D([0,T],\mathbb{R})$ and therefore the tightness of
$\\{(\pi_{t}^{N})_{t\in[0,T]}\\}_{N\in\mathbb{N}}$ in $D([0,T],\mathcal{M})$ .

###### Proof.

Notice that the combination of Lemma 3.3 and of Lemma 4.3 guarantees that the
process $(\langle\pi_{t}^{N},G_{N}^{\lambda}\rangle)_{t\in[0,T]}$ is well
defined. Furthermore, for each $\varepsilon>0$,

$\displaystyle\lim_{N\to\infty}P^{\omega}\Big{(}\sup_{0\leq t\leq
T}\big{|}\langle\pi_{t}^{N},G_{N}^{\lambda}\rangle-\langle\pi_{t}^{N},G\rangle\big{|}\geq\varepsilon\Big{)}=0\,.$
(68)

This fact can be shown by using Lemma 4.3 to bound the probability in (68) by

$\displaystyle\varepsilon^{-1}c_{1}{\rm
e}^{c_{2}bT}\sqrt{\|G_{N}^{\lambda}-G\|^{2}_{L^{1}(\mu_{N})}+N^{-n}\|G_{N}^{\lambda}-G\|_{L^{2}(\mu_{N})}\|L^{N}(G_{N}^{\lambda}-G)\|_{L^{2}(\mu_{N})}}$

and then applying Lemma 3.3 and Lemma 3.4 to see that this converges to $0$ as
$N\to\infty$. ∎

###### Proof of Lemma 5.3.

By Lemma 5.4 we will just have to prove tightness of
$(\langle\pi_{t}^{N},G_{N}^{\lambda}\rangle)_{t\in[0,T]}$. We will use Aldous
criterion, see for example [Kipnis and Landim(1998), Section 4: Proposition
1.2 and Proposition 1.6]. Let us work with the set $\mathcal{T}_{N}(\theta)$
of couples $(\tau,h)$ such that $\tau$ is a stopping time for the process
$(\eta_{t}^{N})_{t\in[0,T]}$ and $h\in\mathbb{R}$ is such that $0\leq
h\leq\theta$ and $\tau+h\leq T$. Using identity (74) we have

$\displaystyle\langle\pi_{\tau+h}^{N}-\pi_{\tau}^{N},G_{N}^{\lambda}\rangle\ $
$\displaystyle=\int_{\tau}^{\tau+h}\langle\pi_{s}^{N},L^{N}G_{N}^{\lambda}+bG_{N}^{\lambda}\rangle{\rm
d}s+M_{\tau,\tau+h}^{N}(G),$ (69)

where $M_{\tau,\tau+h}^{N}(G)=M_{\tau+h}^{N}(G)-M_{\tau}^{N}(G)$.

Equation (3.2) allows us to write
$L^{N}G_{N}^{\lambda}=\lambda(G^{\lambda}_{N}-G)+\sigma^{2}\Delta G$, so that

$\displaystyle\left|\int_{\tau}^{\tau+h}\langle\pi_{s}^{N},L^{N}G_{N}^{\lambda}\rangle\,{\rm
d}s\right|\leq h\sup_{0\leq s\leq
T}\langle\pi_{s}^{N},\lambda|G^{\lambda}_{N}-G|+|\sigma^{2}\Delta G|\rangle.$

We want to use Lemma 4.3 to this quantity and then take the limit
$\theta\downarrow 0$. Using that $\|G_{N}^{\lambda}-G\|_{L^{1}(\mu_{N})}$,
$\|G_{N}^{\lambda}-G\|_{L^{2}(\mu_{N})}$ and
$\|L^{N}G_{N}^{\lambda}\|_{L^{2}(\mu_{N})}$ are bounded by Lemma 3.3, using
that $G$ is compactly supported and using also Lemma 3.4, we obtain

$\lim_{\theta\downarrow
0}\sup_{\begin{subarray}{c}(\tau,h)\in\mathcal{T}_{N}(\theta)\\\ N\geq
1\end{subarray}}P^{\omega}\left(\left|\int_{\tau}^{\tau+h}\langle\pi_{s}^{N},L^{N}G_{N}^{\lambda}\rangle\,{\rm
d}s\right|\geq\delta\right)=0$

for any $\delta>0$. Similarly we also see that

$\lim_{\theta\downarrow
0}\sup_{\begin{subarray}{c}(\tau,h)\in\mathcal{T}_{N}(\theta)\\\ N\geq
1\end{subarray}}P^{\omega}\left(\left|\int_{\tau}^{\tau+h}\langle\pi_{s}^{N},bG_{N}^{\lambda}\rangle\,{\rm
d}s\right|\geq\delta\right)=0\,.$

Finally we also know that the contribution of $M_{\tau,\tau+h}^{N}(G)$ is
negligible by Lemma 5.1. All in all equation (69) combined with these
estimates yields

$\lim_{\delta\downarrow
0}\limsup_{N\rightarrow\infty}\sup_{(\tau,h)\in\mathcal{T}_{N}(\theta)}P^{\omega}\left(|\langle\pi_{\tau+h}^{N},G_{N}^{\lambda}\rangle-\langle\pi_{\tau}^{N},G_{N}^{\lambda}\rangle|\geq
3\delta\right)=0$

for any $\delta>0$, which ends the proof. ∎

###  5.3. Proof of identification and convergence

We are finally ready to identify the limiting points of
$(\pi_{t}^{N})_{t\in[0,T]}$ as deterministic measure-valued processes with
density. We will show that this limit can be characterized as the weak
solution of (9).

######  Lemma 5.5.

For $\mathbb{P}-a.a.$ $\omega$ the following holds. Consider a sequence of
initial configurations $(\eta_{0}^{N})_{N\in\mathbb{N}}$ satisfying the
Domination $\&$ Convergence Assumption. For all $\varepsilon>0$ and
$G\in\mathcal{C}(\mathbb{R}^{n})$, it holds

$\displaystyle\lim_{N\to\infty}P^{\omega}\Big{(}\sup_{0\leq t\leq
T}\Big{|}\int_{0}^{t}\,\langle\pi_{s}^{N},G_{N}^{\lambda}-G\rangle\,{\rm
d}s\Big{|}>\varepsilon\Big{)}=0\,.$

###### Proof.

As a direct consequence of Lemma 4.3 it holds

$\displaystyle P^{\omega}\Big{(}\sup_{0\leq t\leq
T}\Big{|}\int_{0}^{t}\,\langle\pi_{s}^{N},G_{N}^{\lambda}-G\rangle\,{\rm
d}s\Big{|}>\varepsilon\Big{)}\leq\varepsilon^{-1}Tc_{1}e^{c_{2}bT}|\kern-1.07639pt|\kern-1.07639pt|G_{N}^{\lambda}-G|\kern-1.07639pt|\kern-1.07639pt|_{N}$

with
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\cdot\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}$
defined in (54). As in the proof of Lemma 5.4, one concludes by observing that
$|\kern-1.07639pt|\kern-1.07639pt|G_{N}^{\lambda}-G|\kern-1.07639pt|\kern-1.07639pt|_{N}$
goes to $0$ as $N\to\infty$. ∎

Thanks to Lemma 5.4, we can consider now a limiting value of of
$(\pi_{t}^{N})_{t\in[0,T]}$, call it $(\pi_{t})_{t\in[0,T]}$, in the space
$D([0,T],\mathcal{M})$, where $\mathcal{M}=\mathcal{M}(\mathbb{R}^{n})$ is as
usual endowed with the vague topology. Noticing that (35) is equivalent to

$\displaystyle
L^{N}G_{N}^{\lambda}=\lambda(G_{N}^{\lambda}-G)+\sigma^{2}\Delta G\,,$

we can use representation (74) and put together the results of Lemma 5.1 and
Lemma 5.5 and (68) to infer that for any $\varepsilon>0$

$\displaystyle\lim_{N\to\infty}P^{\omega}\Big{(}\sup_{t\in[0,T]}\Big{|}\langle\pi_{t}^{N},G\rangle-\langle\pi_{0}^{N},G\rangle-\int_{0}^{t}\langle\pi_{s}^{N},\sigma^{2}\Delta
G+bG\rangle\,{\rm d}s\Big{|}>\varepsilon\Big{)}=0\,.$ (70)

As $\langle\pi_{t}^{N},G\rangle$ converges to $\langle\pi_{t},G\rangle$ along
a subsequence, we get, for any $G\in\mathcal{C}_{c}^{\infty}(\mathbb{R}^{n})$,

$\langle\pi_{t},G\rangle=\langle\pi_{0},G\rangle+\int_{0}^{t}\langle\pi_{s},\sigma^{2}\Delta
G+bG\rangle\,{\rm d}s$ (71)

$\mathbb{P}$–almost surely. We use now classical techniques to check that the
solutions have a density and prove uniqueness of the limiting problem. The
result may be classical, even if the fact that our domain is non bounded or
our weak formulation make that we do not know the appropriate reference. For
convenience of the reader, we provide the proof.

######  Lemma 5.6.

i) For $\mathbb{P}$-a.a. $\omega$ the following holds. Consider a sequence of
initial configurations $(\eta_{0}^{N})_{N\in\mathbb{N}}$ satisfying the
Domination $\&$ Convergence Assumption for some $\rho_{0}$. Consider a
limiting point $\pi$ of $\pi^{N}$ in the space $D([0,T],\mathcal{M})$. Then
$\pi$ satisfies Equation (71), with all terms well defined, for any $G$ in

$\mathcal{G}=\\{G\in\mathcal{C}^{\infty}(\mathbb{R}^{n})\cap L^{1}({{\rm
d}x}):\sigma^{2}\Delta G\in L^{1}({{\rm d}x})\\}.$

ii) Equation (71) has a unique weak solution in the space cadlag positive
Radon measure valued function. Besides this solution $\pi$ has a bounded
density : $\pi_{t}({\rm d}x)=\rho_{t}(x){\rm d}x$ a.s., for a.e. $t\geq 0$,
and $\rho_{\cdot}\in L^{\infty}([0,T]\times\mathbb{R}^{n},\mathbb{R}_{+})$.

The second part of the statement ensures that the limiting processes $\pi$ are
deterministic. For the proof of the existence of a density, we exploit our
results coming from Kipnis Varadhan estimates, but one may also invoke results
focusing on the solution of (71).

###### Proof.

Let us prove $(i)$ and consider $G\in\mathcal{G}$ non-negative. Approximate
$G$ with non-negative functions
$G_{k}\in\mathcal{C}^{\infty}_{c}(\mathbb{R}^{n})$ such that $G_{k}(x)=G(x)$
for $x\in B_{k}$, with $B_{k}$ the ball of radius $k$ centered in the origin,
and $G_{k}(x)\leq G(x)$ for $x\notin B_{k}$. Then, as $k\to\infty$, dominated
convergence theorem yields that
$\langle\pi_{t},G_{k}\rangle\to\langle\pi_{t},G\rangle$,
$\langle\pi_{0},G_{k}\rangle\to\langle\pi_{0},G\rangle$ and
$\int_{0}^{t}\langle\pi_{s},bG_{k}\rangle\,{\rm
d}s\to\int_{0}^{t}\langle\pi_{s},bG\rangle\,{\rm d}s$. It follows that also
$G$ satisfies (71), the r.h.s. being well defined and integrable by part
$(i)$.

Let us move to $(ii)$. Let us move to $(ii)$. For the purpose, we consider the
subspace of signed Radon measures on $\mathbb{R}^{n}$ associated with the norm

$\|\mu\|_{TV}=\sup_{f\in\mathcal{G},\,\|f\|_{L^{1}({\rm d}x)}\leq
1}|\langle\mu,f\rangle|\,.$

We also introduce the space $\mathcal{G}_{T}$ of functions
$G:[0,T]\times\mathbb{R}^{n}\rightarrow\mathbb{R}$ that verify the following
properties: $G(t,.)\in\mathcal{G}$ for any $t\in[0,T]$;
$\sup_{t\in[0,T]}\left\\{\|G(t,.)\|_{L^{1}({\rm d}x)}+\|\sigma^{2}\Delta
G(t,.)\|_{L^{1}({\rm d}x)}\right\\}<\infty$; $s\rightarrow G(s,x)$ is
differentiable for any $x\in\mathbb{R}^{n}$ and
$\sup_{s\in[0,T]}|\partial_{s}G(s,.)|\in L^{1}({\rm d}x)$. By approximating
$G(\cdot,\cdot)\in\mathcal{G}_{T}$ with a function which is piece-wise linear
in time and proceeding as in [Gonçalves and Jara(2008), Section 3], we observe
that (71) can be extended to functions in $\mathcal{G}_{T}$. This is the weak
form of (9).

Finally, to get uniqueness, consider $G^{T}(t,x)=P_{T-t}\varphi(x)$ with
$\varphi\in\mathcal{G}$ where $P_{t}f(x)=\mathbb{E}[f(x+B_{t})]$ is the
semigroup of the Brownian motion with generator $\sigma^{2}\Delta$. Then
$\partial_{s}G^{T}(s,\cdot)=-\sigma^{2}\Delta G^{T}(s,\cdot)$ and
$G^{T}\in\mathcal{G}_{T}$ and

$\langle\pi_{t},G^{T}(t,.)\rangle=\langle\pi_{0},G(0,.)\rangle+\int_{0}^{t}\langle\pi_{s},bG^{T}(s,.)\rangle\,{\rm
d}s.$

This implies, for two solutions $\pi^{1}$ and $\pi^{2}$ of Equation (71) with
same initial values $\pi_{0}$,

$\langle\pi^{1}_{T}-\pi^{2}_{T},\varphi\rangle=\int_{0}^{T}\langle\pi^{1}_{s}-\pi^{2}_{s},bG^{T}(s,\cdot)\rangle\,{\rm
d}s$

Adding that $\|G^{T}(t,.)\|_{L^{1}({\rm d}x)}=\|\varphi\|_{L^{1}({\rm d}x)}$,
we obtain

$\|\pi^{1}_{T}-\pi^{2}_{T}\|_{TV}\leq
b\int_{0}^{T}\|\pi^{1}_{s}-\pi^{2}_{s}\|_{TV}\,{\rm d}s\,.$

Gronwall lemma yields $\pi^{1}=\pi^{2}$ and uniqueness of the solution
$\eqref{eqlimite}$ in the space of positive Radon measure is proved.

Finally we prove that this solution admits a density. We can exploit the first
part $i)$ and see this solution $\pi$ as the limit of our sequence of
processes. Take $H\in\mathcal{C}^{\infty}_{c}$ to be a nonnegative function on
$V/N$ belonging to $L^{1}(\mu_{N})$ and $L^{2}(\mu_{N})$. Letting $N\to\infty$
in equation (58), we invoke Portemanteaux theorem and the fact that the
supremum over $[0,T]$ of $\langle\,\cdot\,,H\rangle$ is a continuous
functional on $D([0,T],\mathcal{M}(\mathbb{R}^{n}))$ to obtain

$\displaystyle P^{\omega}_{\rho_{0}}\Big{(}\sup_{0\leq t\leq
T}\,\langle\pi_{t},H\rangle\,>A\Big{)}\leq A^{-1}c(T)\|H\|_{L^{1}({\rm d}x)}$
(72)

with $c(T)=c_{1}{\rm e}^{c_{2}bT}$. On the r.h.s. we have used the fact that
$N^{-n}\,\|L^{N}H\|_{L^{2}(\mu_{N})}$ converges a.s. to $0$ using Lemma 3.4
and that $\|H\|_{L^{1}(\mu_{N})}$ tends a.s. to $\|H\|_{L^{1}({\rm d}x)}$. For
a given $\omega$ the process $\pi_{t}$ must be deterministic, since it
verifies (71). Hence, by (72), we have obtained that

$\sup_{0\leq t\leq T}\,\langle\pi_{t},H\rangle\leq c(T)\|H\|_{L^{1}({\rm
d}x)}\,.$

for any $H\in\mathcal{C}^{\infty}_{c}$. By approximation, this identity can be
extended to any non-negative $H\in L^{1}({\rm d}x)$. As a consequence, the
process $\pi_{t}$ is concentrated on measures absolutely continuous with
respect to the Lebesgue measure and we call $\rho_{t}(x)$ its density at time
$t\in[0,T]$ and for any $t\in[0,T]$, $\rho_{t}\leq C(T)$ a.e. and $\rho\in
L^{\infty}([0,T]\times\mathbb{R}^{n},\mathbb{R}_{+})$.

It follows that also $G$ satisfies (71), the r.h.s. being well defined and
integrable by part $(i)$.

∎

###  5.4. Proof of Theorem 1.2 with mortality ($d>0$)

We want to adapt the proof of Theorem 1.2 to the case where particles die at
rate $d>0$.

As mentioned before, the construction of the process $(\eta_{t})_{t\in[0,T]}$
is still valid when $d>0$. By following the proof of Section 2 one can prove
that the corresponding measure-valued process satisfies an equivalent of (33)
that now reads, for $G:\mathbb{R}^{n}\rightarrow\mathbb{R}$ with compact
support,

$\displaystyle\langle\pi_{t},G\rangle=\langle\pi_{0},G\rangle$
$\displaystyle+\int_{0}^{t}\sum_{i\in\mathcal{A}_{s},y\in
V}\big{(}G(y)-G(X_{s}^{i})\big{)}\,\mathcal{N}_{i}^{X_{s}^{i},y}({\rm d}s)$
$\displaystyle+\int_{0}^{t}\sum_{i\in\mathcal{A}_{s}}G(X_{s}^{i})\,\mathcal{N}_{i}^{b}({\rm
d}s)-\int_{0}^{t}\sum_{i\in\mathcal{A}_{s}}G(X_{s}^{i})\,\mathcal{N}_{i}^{d}({\rm
d}s)$ (73)

where $(\mathcal{N}_{i}^{b})_{i\in\mathcal{I}}$ is another collection of
independent Poisson point measures on $\mathbb{R}_{+}$ with intensity $d\,{\rm
d}t$, independent from the $\mathcal{N}_{i}^{x,y}$’s and the
$\mathcal{N}_{i}^{b}$’s.

There is no problem in updating the technical tools presented in Section 3 and
Section 4 to the case $d>0$. The homogenization results do not depend on the
specific particle dynamics, so the results of Lemma 3.3 are needed as they
are. The non-reversible Kipnis-Varadhan estimate Lemma 4.3 holds when $d>0$,
too. To see that, one can couple the dynamics with $d=0$ and the one with
$d>0$ in a way that guarantees that the supremum in equation (58) always
decreases (for example, one can use the same Poisson processes for generating
jumps and births of the particles).

The results of Section 5.1 still hold for the process

$\displaystyle
M_{t}^{N}=\langle\pi_{t}^{N},G_{N}^{\lambda}\rangle-\langle\pi_{0}^{N},G_{N}^{\lambda}\rangle-\int_{0}^{t}\langle\pi_{s}^{N},L^{N}G_{N}^{\lambda}+(b-d)G_{N}^{\lambda}\rangle{\rm
d}s\,.$ (74)

In particular, $M^{N}$ is a square integrable martingale and it satisfies
Lemma 5.1. This can be again achieved by the same truncation argument. Finally
the proofs of tightness, identification and convergence (corresponding to
Section 5.2 and Section 5.3) for $d>0$ follow those of the case $d=0$ as they
are fundamentally based on the homogenization results and on the non-
reversible Kipnis-Varadhan estimates.

##  6\. Extension of the results and perspective

###  6.1. Extension to random graphs

For the sake of clarity we have made the choice to state and prove our main
results for the particle system evolving over the complete graph
$\mathcal{G}=(V,E)$, where $V$ are the points of an homogeneous Poisson point
process of intensity $\gamma>0$ on $\mathbb{R}^{n}$, with $n\geq 2$. In this
section we discuss how one can consider a broader class of graphs. To this
end, we will consider graphs
$\overline{\mathcal{G}}=(\overline{V},\overline{E})$ with
$\overline{V}\subseteq V$ and $\overline{E}\subseteq E$ obtained under measure
$\mathbb{P}$ (eventually enlarging the probability space $\Omega$) by
performing a random percolation procedure on the edges of $\mathcal{G}$. For a
graph $\overline{\mathcal{G}}$ we say that the particle system evolves on
$\overline{\mathcal{G}}$ when the particles move on the nodes $\overline{V}$
with transition rates $r(x,y)$ substituted by
$\overline{r}(x,y):=r(x,y)\mathbf{1}\\!_{\left\\{\\{x,y\\}\in\overline{E}\right\\}}$.

The next theorem is the generalization of Theorem 1.1. In this case one can
perform any percolation procedure on the bonds of the complete graph
$\mathcal{G}$. Notice that we include cases where the graph becomes
disconnected.

######  Theorem 1.1’.

For $\mathbb{P}$-a.a. realizations of the underlying Poisson point process,
the following holds. Let $\overline{\mathcal{G}}=(\overline{V},\overline{E})$
be any graph such that $\overline{V}=V$ and $\overline{E}\subseteq E$. Let
$\eta_{0}$ be a random variable on $\mathbb{N}^{\overline{V}}$ such that
$\mathbb{E}[\eta_{0}(x)]\leq M$ for all $x\in\overline{V}$, for some
$M\in\mathbb{N}$. Then, for all $T>0$, there exists a process
$(\eta_{t})_{t\in[0,T]}$ with initial value $\eta_{0}$ and paths in the
Skohorod space $\mathcal{D}([0,T],\mathbb{N}^{\overline{V}})$ and generator
$\mathcal{L}$ given by the following. For each function
$f_{G}:\mathbb{N}^{\overline{V}}\to\mathbb{R}$ of the form
$f_{G}(\eta)=\sum_{x\in\overline{V}}G(x)\eta(x)$ with $G$ compactly supported
on $\mathbb{R}^{n}$, it holds

$\displaystyle\mathcal{L}f_{G}(\eta)=$
$\displaystyle\sum_{x,y\in\overline{V}}\eta(x)\overline{r}(x,y)\big{(}G(y)-G(x)\big{)}+\sum_{x\in\overline{V}}\eta(x)\big{(}b-d\big{)}G(x)\,.$

###### Proof.

The existence of the process on the finite graph as in Section 2.2 goes easily
through. Also the quantitative estimates of Lemma 2.1 work with $r(x)$
substituted by the corresponding quantity
$\overline{r}(x)=\sum_{y\in\overline{V}}\overline{r}(x,y)$. For the case of a
finite number of initial particle, cfr. Section 2.3, we notice that Lemma 2.3
is true for the percolated graph since $\overline{r}(x)\leq r(x)$. This in
turn implies the equivalent of the key Proposition 2.2 and hence of the
equivalent of Corollary 2.4. The existence of the process with an infinite
number of initial particles can be checked then by following the proof of
Section 2.4. ∎

Generalizing the results of Theorem 1.2 is much more subtle. In this case one
is not authorized to freely percolate the edges of $\mathcal{G}$. Indeed, our
proof crucially relies on the homogenization results discussed in Section 3.1.
On the other hand, as long as the assumptions that imply homogenization are
fulfilled, our machinery continues to work and we obtain the following more
general version of Theorem 1.2.

######  Theorem 1.2’.

Consider under $\mathbb{P}$ a random graph
$\overline{\mathcal{G}}=(\overline{V},\overline{E})$, with
$\overline{V}\subseteq V$ and $\overline{E}\subseteq E$ verifying conditions
(A1),…,(A9) of Section 3.1. The results of Theorem 1.2 hold for the particle
system evolving on $\overline{\mathcal{G}}$. In this case, the matrix
$\sigma\Delta$ appearing in (9) has to be substituted by the matrix $D$ such
that, for any $a\in\mathbb{R}^{n}$,

$\displaystyle a\cdot Da=\frac{1}{2}\inf_{\psi\in
L^{\infty}(\mathbb{P}_{0})}\mathbb{E}_{0}\Big{[}\sum_{y\in\overline{V}}\bar{r}(0,y)\big{(}a\cdot
y+\psi(\theta_{y}\omega)-\psi(\omega)\big{)}^{2}\Big{]}$ (75)

where $\theta_{y}\omega$ is the shift defined in Section 3.1.

###### Proof.

The homogenization results of Lemma 3.3 are still valid exactly because
(A1),…, (A9) have been chosen so that they would work. Lemma 3.4 works since
$\overline{V}\subseteq V$ and since $\bar{r}(x,y)\leq r(x,y)$ for all $x,y\in
V$. The non-conservative Kipnis-Varadhan estimate of Lemma 4.3 carries through
to the edge-percolated case since the reversibility of the process without
births and deaths is maintained on $\overline{\mathcal{G}}$. Sections 5.1, 5.2
and 5.3 follow a general strategy that relies on the previous estimates and
remains substantially identical for the percolated graph. ∎

######  Remark 6.1.

It is possible, in principle, that the diffusion matrix described by (75) is
degenerate.

To give more substance to Theorems 1.1’ and 1.2’, we exhibit now two well-
known models that are obtained from the complete graph $\mathcal{G}=(V,E)$ via
a bond-percolation procedure. For both of them we check that assumptions
(A1),…,(A9) are in fact satisfied, at least in some range of the parameters.

Long-range percolation. Long-range percolation is a well studied random graph
model, usually defined on the $\mathbb{Z}^{n}$ lattice, see for example
[Berger(2002), Hutchcroft(2021)] and references therein. Extending its
definition to continuous space (as for example in [Penrose(1991)]), we
consider again the set of vertices $V$ given by a Poisson point process of
intensity $\gamma>0$ and the set of all possible edges $E$. Under
$\mathbb{P}$, independently for each $\\{x,y\\}\in E$ we retain the edge with
probability $1-{\rm e}^{-\beta\|x-y\|^{-\alpha}}$ and delete it otherwise,
with $\alpha,\beta>0$ two parameters of the model. Theorem 1.1’ guarantees
that our particle process on this structure is always well-defined.

To study the hydrodynamic limit as in Theorem 1.2’ we need some adjustment.
The main problem arising from the percolation procedure, in fact, is that the
resulting graph might not be connected and (A6), for example, would not stand.
To overcome this issue we must consider parameters $\alpha,\beta$ and $\gamma$
that guarantee the existence of a unique infinite giant component for the
graph with $\mathbb{P}$–probability $1$ (see [Penrose(1991)] for results on
the existence of an infinite component in continuous long-range percolation).
Then we define $\overline{V}$ to be the set of vertices in the infinite
component of the graph and $\overline{E}$ to be the set of retained edges
connecting points of $\overline{V}$. More precisely, to use the notation of
Section 3.1, we identify $\hat{\omega}$ with the points in the infinite
component and let $n_{x}(\omega)=1$ for $x\in\hat{\omega}$ and
$n_{x}(\omega)=0$ otherwise. We show now that (A1),…,(A8) hold and discuss
(A9) after. (A1) comes from the stationarity and ergodicity of the Poisson
point process combined with the independence of the percolation procedure.
Conditions (A2), (A3) and (A7) are inherited from the underlying Poisson point
process. (A4) and (A5) are straightforward. (A6) is clear since we have
restricted ourselves to the infinite component and every jump within it has
positive probability. (A8) can be also deducted from the analogous property
for the Poisson point process. The only point left to check is the decay of
correlation (A9). Clearly $\mathbb{E}[N_{0}^{2}]<\infty$. The fact that
$|Cov(N_{k},N_{k^{\prime}})|\leq C_{0}|k-k^{\prime}|^{-1}$ is more delicate,
since $N_{k}$ might depend on the structure of the configuration of the
infinite cluster in a far away box $k^{\prime}+[0,1)^{n}$. While we believe
the bound to be true, we could not find a proper reference in the literature
and its proof might be technically involved. One way to avoid this problem, is
to consider the parameters $\alpha$ and $\beta$ in the range where all the
points of $V$ belong to the infinite component. This happens, for example,
when $\alpha<n$. In this case (A9) is clearly verified.

Scale-free percolation. Scale-free percolation is an inhomogeneous version of
the long-range percolation model. It was originally introduced with nodes
placed on the lattice $\mathbb{Z}^{n}$ in [Deijfen et al.(2013)Deijfen, van
der Hofstad, and Hooghiemstra] and then also studied on a Poisson point
process in [Deprez and Wüthrich(2018)] and [Dalmau and Salvi(2021)]. Under
measure $\mathbb{P}$, we let $V$ be the realization of a Poisson point process
of parameter $\gamma>0$ and assign independently to each vertex $x\in V$ a
random weight $W_{x}\in[1,\infty)$ such that
$\mathbb{P}(W_{x}>w)=w^{-(\tau-1)}L(w)$ for some $\tau>1$ and with $L$ a
slowly varying function. Then, independently for each edge of the complete
associated graph $\\{x,y\\}\in E$, we retain the edge in $\overline{E}$ with
probability $1-{\rm e}^{-\beta W_{x}W_{y}\|x-y\|^{-\alpha}}$ for some
$\alpha,\beta>0$ and delete it otherwise. The particle system on the resulting
graph is again well defined by Theorem 1.1’.

For Theorem 1.2’ to work, we restrict to values of $\alpha,\beta,\gamma$ and
$\tau$ such that a unique infinite component $\overline{V}$ exists with
$\mathbb{P}$–probability $1$, see [Deprez and Wüthrich(2018)] for the precise
range of parameters. The random graph
$\overline{\mathcal{G}}=(\overline{V},\overline{E})$ induced on the infinite
component equipped with rates $\overline{r}(x,y)$ verifies (A1),…,(A8). This
can be deducted as for long-range percolation. To verify (A9) we have the same
problem as before with correlations. To be on the safe side, one can take
$\alpha\leq n$ or $\alpha(\tau-1)/n\leq 1$, which guarantees all the points of
$V$ to be in the infinite cluster, see [Dalmau and Salvi(2021)].

###  6.2. Open problems and perspectives

The methods we present in our paper can be easily used for a wider class of
models. First of all, we point out that the reason for choosing a Poisson
point process as a support is mainly due to the seek of clarity rather than to
technical obstacles of dealing with more general settings. It should be
possible to replace the Poisson point process with any sufficiently regular
simple point process satisfying the nine conditions of [Faggionato(2022a)],
see Section 3.1. Likewise, the transition rates $r(x,y)$ do not have to assume
the exact form ${\rm e}^{-\|x-y\|}$: any rates decaying sufficiently fast
(possibly depending on the dimension $n$ of the space) should work.

A more challenging task would be generalizing the birth and death mechanism of
the particles: the techniques being used in this paper rely on the linearity
of the birth rate $b$ and of the death rate $d$. By a domination argument, it
is possible to show that whenever one chooses a bounded birth rate, the
corresponding process is well defined. The methods for proving tightness can
be extended, too, but the identification of the limit becomes a more difficult
matter. A motivating and realistic generalization would be to consider a death
rate $d=d_{x}(\eta)$ that depends on the amount of particles present at each
given site and $b$ constant. A classical example is the local logistic model
with death rate $d_{x}(\eta)=d+c\eta(x)$, where $c>0$ is a competition factor.
In this case, when the population becomes large on each site, we expect the
limiting density $\rho_{t}(x)$ to satisfy a reaction-diffusion equation of the
kind $\partial_{t}\rho\,=\,\sigma\Delta\rho+(b-d-c\rho)\rho$. Otherwise, if
the population remains of order of magnitude $1$ on each site, we expect rapid
stirring (see e.g. [Durrett(1995)] or [Perrut(2000)]). In this case the
limiting reaction coefficient should come from an averaging procedure of local
population size in quasi-state stationary law. This follows the spirit of the
replacement lemma used for zero-range processes in statistical mechanics, see
[Kipnis and Landim(1998)], which, roughly put, allows one to compare the
particle density on microscopic boxes with the particle density on macroscopic
ones. Another interesting extension would be to consider births (or deaths)
that depend via a non-local kernel on the population size in a surrounding
region. This would mimic the inclination of individuals to reproduce less in
crowded communities and would give further importance to the spatial aspect of
the particle system. Again we believe that rapid stirring techniques can be
applied to tackle this problem.

Finally, in this paper we have worked with a scaling corresponding to a fast
motion, compared to the population dynamics or epidemics. In fact, the jump
rates carry an additional $N^{2}$ factor, while births and deaths do not. It
would be relevant for our motivations in epidemiology (ANR project Cadence)
and interesting for the branching process analysis to consider less separated
scales.

Acknowledgment The authors are very grateful to Jerome Coville, Alessandra
Faggionato and Chi Tran Viet for stimulating discussions and relevant
suggestions.