# Convergence of cluster coagulation dynamics

L. Andreis111Politecnico di Milano, Dipartimento di matematica, Piazza
Leonardo da Vinci, 32, 20133 Milano., T. Iyer222Weierstrass Institute for
Applied Analysis and Stochastics, Mohrenstrasse 39, 10117 Berlin, Germany., E.
Magnanini22footnotemark: 2

###### Abstract

We study hydrodynamic limits of the cluster coagulation model; a coagulation
model introduced by Norris [Comm. Math. Phys., 209(2):407-435 (2000)]. In this
process, pairs of particles $x,y$ in a measure space $E$, merge to form a
single new particle $z$ according to a transition kernel $K(x,y,\mathrm{d}z)$,
in such a manner that a quantity, one may regard as the total _mass_ of the
system, is conserved. This model is general enough to incorporate various
inhomogeneities in the evolution of clusters, for example, their shape, or
their location in space. We derive sufficient criteria for trajectories
associated with this process to concentrate among solutions of a
generalisation of the _Flory equation_ , and, in some special cases, by means
of a uniqueness result for solutions of this equation, prove a weak law of
large numbers. This multi-type Flory equation is associated with _conserved
quantities_ associated with the process, which may encode different
information to conservation of mass (for example, conservation of centre of
mass in spatial models). We also apply criteria for _gelation_ in this process
to derive sufficient criteria for this equation to exhibit _gelling_
solutions. When this occurs, this multi-type Flory equation encodes, via the
associated conserved property, the interaction between the _gel_ and the
finite size _sol_ particles.

Keywords: Cluster coagulation, Marcus-Luschnikov process, Smoluchowski
equation, Flory equation, gelation.

AMS Subject Classification 2010: 60K35, 35Q70, 82C22.

##  1 Introduction

Coagulation models are prevalent across numerous disciplines, spanning from
physical chemistry (to describe polymer formation), to astrophysics, where
they simulate galaxy formation. A natural coagulation model involves particles
moving in space, with larger particles exhibiting slower movement compared to
smaller ones. A classical ‘mean field’ description of this process via
Markovian dynamics is known as the _Marcus-Lushnikov model_ [14, 8, 13]. Here,
pairs of particles of masses $x$ and $y$ merge at a rate $\bar{K}(x,y)$ (for
an appropriate function $\bar{K}(x,y)$). The re-scaled limiting behaviour of
the trajectories of empirical measures encoding particle masses in this model,
is generally expected to follow a system of measure-valued differential
equations known as the _Smoluchowski_ (or _Flory_) equations. This re-scaling,
which involves re-scaling time with the system size (to ‘slow time down’), and
looking at a fixed _volume_ (the whole space $E$), is known as taking a
_hydrodynamic limit_.

A popular question with regards to coagulation processes, concerns whether or
not one sees the occurrence of _macroscopic_ or _giant_ particles by some time
$t>0$; a phenomenon known as _gelation_. _Gelation_ is generally defined in
terms of whether or not a solution of the Smoluchowski (or Flory) equation
fails to ‘conserve mass’, which means, intuitively, that mass is lost to
‘infinite’ particles. Meanwhile, the term stochastic gelation, generally
refers to the formation of ‘large particles’ in a bounded amount of time in
the Marcus–Lushnikov process. When the trajectories of the Marcus–Lushnikov
process are concentrated on solutions of the Smoluchowski equation, Jeon
showed that these two notions of gelation are equivalent [12, Theorem 5].
However, in certain regimes, it may be the case, that, once gelation occurs,
the ‘gel’ or macroscopic particles, interact with the microscopic ones or
‘sol’, in such a way that the trajectories of the Marcus–Lushnikov process
only concentrate around solutions of the Smoluchowski equation for a short
time before gelation. In such regimes, a correction term encoding this ‘sol-
gel’ interaction is required in the Smoluchowkski equation, this is known as
the Flory equation. A weak law of large numbers of the Marcus–Lushnikov
process to the Flory equation in a particular case was first proved by Norris
[17], and later, concentration of trajectories around solutions of this
equation were proved in [7, 20].

A limitation of the classical Marcus–Lushnikov model is that the dynamics only
include information of the masses of particles, and not, for example, other
inhomogeneities such as their location in space. In [17], Norris introduced
the _cluster coagulation model_ , which allows one to incorporate these
features rather generally, (see also [15] for a variant where clusters move
diffusively). In a particular, limited regime of this model, Norris proved a
weak law of large numbers for the dynamics of this model, with exponential
rates of convergence [17, Theorem 4.2 & 4.3]. It should be noted that the
cluster coagulation model is general enough to encompass models such as those
studied by Jacquot [10] and later work of Heydecker and Patterson [9]. In
[10], Jacquot proves a weak law of large numbers for the ‘historical trees’
encoding histories of clusters in the Marcus-Lushnikov process is proved;
whilst in [9], Heydecker and Patterson prove a weak law of large numbers in
the special case of ‘bilinear’ kernels. For more examples of cluster
coagulation processes we refer the reader to [2, Section 3.1]. Also note that
more recently, from the perspective of analysis, a strain of research has
focused on the study of multi-component generalisations of the Smoluchowski
coagulation equations, where the mass variable is substituted by a variable in
a $d$-dimensional Euclidean space, and there is an additional source term
corresponding to the system not being in equilibrium (see for example, [6, 5,
21] and reference therein). In general, without the source term, these
equations are particular instances of the Smoluchowski equation associated
with the cluster coagulation model.

In recent work [2], the authors have proved sufficient criteria for stochastic
gelation to occur in the cluster coagulation process. In this paper, we derive
criteria for concentration of trajectories associated with the cluster
coagulation model around solutions of a multi-type Flory equation,
generalising the classical Flory equation. We also provide criteria for
solutions of this equation to be unique, hence for sufficient condition for a
weak law of large numbers associated with these trajectories. As a result of
the aforementioned device due to Jeon [12, Theorem 5] (reformulated in [2,
Theorem 1.1]) this concentration also leads to sufficient criteria for the
associated multi-type Flory equation to exhibit gelling solutions.

###  1.1 Overview on our contribution

The main novelty of this paper involves the idea of a _conserved quantity_
appearing in a generalisation of the Flory equation; corresponding to an
invariant associated with the cluster coagulation process. From the
perspective of applications, this quantity may correspond to the total mass of
the system, or, depending on the setting, for example, the ‘centre of mass’ or
‘momentum’.

  1. 1.

In Definition 2.1 we define a _multi-type Flory equation_ , expected to encode
the limiting behaviour of the cluster coagulation process, when there are
‘conserved quantities’. Then, in Theorem 2.3 we provide sufficient criteria
for tightness of trajectories associated with the cluster coagulation process,
and criteria for limit points associated with the process to concentrate on
solutions of this equation, proving, in particular, the existence of
solutions.

  2. 2.

In Theorem 2.4 we obtain a uniqueness result for this multi-type Flory
equation in the case when the kernel is ‘eventually conservative’ (see
Definition 2.2). Such a uniqueness result implies a _law of large numbers_ for
the paths of the stochastic cluster coagulation process in Corollary 2.5.

  3. 3.

In Corollaries 2.6 and 2.7, we state, without proof, conditions for solutions
of the multi-type Flory equation to have gelling solutions. These results rely
on previous results concerning stochastic gelation from the aforementioned
companion paper [2].

The notion of conserved quantities allows us to go beyond the setting of [16],
and to prove existence of solutions for the limiting equation under weaker
assumptions. The ‘eventually conservative’ property is a natural extension of
the _eventual multiplicativity_ property introduced in [17]. The approach we
use, using weak-compactness, and martingale techniques, whilst well-
established, allow for relatively few assumptions on the underlying space. We
only assume that the clusters take values in a $\sigma$-compact metric space
$E$. This means that our results encapsulate existing generalisations of the
Marcus-Lushnikov model in the literature, including those of [9, 10].

###  1.2 Structure

The rest of the paper is structured as follows.

  1. 1.

In Section 1.3 we introduce the model, terminology and global assumptions in
Assumption 1.1.

  2. 2.

In Section 2.1 we include our main definitions, including the definition of a
generalised Flory equation with a _conserved quantity_ in Definition 2.1. This
section also includes our main result, Theorem 2.3, concerning the
hydrodynamic limit of the process.

  3. 3.

In Section 2.2 we state a uniqueness result (Theorem 2.4) for the solutions of
the multi-type Flory equation in the case of _eventually conservative_
kernels.

  4. 4.

In Section 2.3 we draw connections with the related paper [2] on gelation for
these models to state results concerning existence of gelling solutions for
the multi-type Flory equation. As these results are an immediate consequence
of results appearing in [2], we omit explicit proofs.

  5. 5.

Section 3 is dedicated to proofs: the proof of Theorem 2.3 appearing in
Section 3.1, and the proof of Theorem 2.4 appearing in Section 2.2.

###  1.3 The model, terminology and global assumptions

####  1.3.1 Definition of the cluster coagulation process

Recall that in the cluster coagulation process [17], one begins with a
configuration of _clusters_ in a measurable space $(E,\mathcal{B})$.
Associated with a cluster $x\in E$ is a _mass function_
$m:E\rightarrow(0,\infty)$. Another important quantity associated with the
process is a _coagulation kernel_ $K:E\times
E\times\mathcal{B}\rightarrow[0,\infty)$, which satisfies the following:

  1. 1.

For all $A\in\mathcal{B}$ $(x,y)\mapsto K(x,y,A)$ is measurable,

  2. 2.

For all $x,y\in E$ $K(x,y,\cdot)$ is a measure on $E$,

  3. 3.

_symmetric:_ for all $A\in\mathcal{B},x,y\in E$ $K(x,y,A)=K(y,x,A)$,

  4. 4.

_finite:_ for all $x,y\in E$ $\bar{K}(x,y):=K(x,y,E)<\infty$

  5. 5.

_preserves mass:_ for all $x,y\in E$, $m(z)=m(x)+m(y)$ for $K(x,y,\cdot)$-a.a.
$z\in E$.

As a continuous time Markov chain, we associate the following dynamics.
Suppose that we begin with a configuration of clusters. Then,

  * •

to each pair of clusters $x,y\in E$, we associate an exponential random
variable with parameter $\bar{K}(x,y)$;

  * •

upon the elapsure of the next exponential random variable in the process,
corresponding to the pair $x$ and $y$, say, the clusters $x$ and $y$ are
removed and replaced by a new cluster $z\in E$, sampled according to the
probability measure

$\frac{K(x,y,\cdot)}{\bar{K}(x,y)}.$ (1)

####  1.3.2 Notation and preliminaries

In this paper, we consider the process as depending on a parameter
$N\in\mathbb{N}$, which one may consider as (up to random fluctuations) the
total _initial mass_ of the system, and analyse the process as in the limiting
regime as $N\to\infty$. We consider the configuration of clusters at time $t$
as being encoded by a _random point measure_ $\mathbf{L}^{(N)}_{t}$ on $E$, so
that, for any set $A\subseteq E$, $a\in(0,\infty)$,
$\mathbf{L}^{(N)}_{t}(A\cap m^{-1}([a,\infty)))$ denotes the random number of
clusters of mass at least $a$ belonging to $A$. We denote by
$\mathcal{M}_{+}(E)$ the set of finite, positive measures on $E$ (also
defining $\mathcal{M}_{+}(E\times E)$ in a similar manner). We may then
consider the process as a measure-valued Markov process, whose infinitesimal
generator $\mathcal{A}$ is defined as follows: for any bounded measurable test
function $F:\mathcal{M}_{+}(E)\rightarrow\mathbb{R}$, we have

$\mathcal{A}F(\mathbf{L}_{t})=\frac{1}{2}\int_{E\times E\times
E}{\mathbf{L}_{t}}(\mathrm{d}x)\left({\mathbf{L}_{t}}-\delta_{x}\right)(\mathrm{d}y)K(x,y,\mathrm{d}z)\left(F({\mathbf{L}}^{(x,y)\to
z}_{t})-F({\mathbf{L}_{t}})\right),$ (2)

where ${\mathbf{L}}^{(x,y)\to
z}:={\mathbf{L}}+\left(\delta_{z}-\delta_{x}-\delta_{y}\right)$. Note that, as
$\mathbf{L}^{(N)}_{t}$ is assumed to be a point measure, the above integral is
always with respect to a positive measure. The measure ${\mathbf{L}}^{(x,y)}$
describes the configuration of the system after a coagulation involving
clusters the two clusters $x,y\in E$ and ending with one cluster $z$ with
$m(z)=m(x)+m(y)$, for $K(x,y,\cdot)$-a.a. $z\in E$. Note the factor
$\frac{1}{2}$ in front of the generator, present to ensure that the total rate
at with clusters $x$ and $y$ interact is $\bar{K}(x,y)$ (and not
$2\bar{K}(x,y)$).

In this paper, we assume $E$ is a $\sigma$-compact metric space with metric
$d$. Given another space $F$, denote by $C_{b}(E;F)$, or resp. $C_{c}(E;F)$,
the spaces of continuous functions $E\rightarrow F$ which are bounded, or
resp., have compact support. In general, we write $C_{b}(E)$ (resp.
$C_{c}(E)$) as a shorthand for $C_{b}(E;\mathbb{R})$ (resp.
$C_{c}(E;\mathbb{R})$). We equip $\mathcal{M}_{+}(E)$ with a metric $d$ that
induces the vague topology on $\mathcal{M}_{+}(E)$. 333We recall that the
_vague_ (respectively _weak_) topologies on $\mathcal{M}_{+}(E)$ are the
smallest topologies that make the maps
$\mu\rightarrow\int_{E}f(x)\mu(\mathrm{d}x)$ continuous for all $f\in
C_{c}(E)$ (respectively $C_{b}(E)$). Note that, since $E$ is separable and
complete, $(\mathcal{M}_{+}(E),d)$ is a separable and complete space. For any
$n\in\mathbb{N}$, we denote by $\mathcal{E}_{n}$ the space

$\mathcal{E}_{n}:=\left\\{\mathbf{u}\in\mathcal{M}_{+}(E):\int_{E}m(x)\,\mathbf{u}(\mathrm{d}x)\leq
n\right\\}$

and $\mathcal{E}=\bigcup_{n\in\mathbb{N}}\mathcal{E}_{n}$. Note that on
$\mathcal{E}_{n}$ _weak_ and _vague_ topology coincides, while this is not the
case on $\mathcal{E}$. We generally will regard $\mathcal{E}$ as the state
space of the process, taking values in $D([0,\infty);\mathcal{E})$, the
_Skorokhod space_ of right-continuous functions
$f:[0,\infty)\rightarrow\mathcal{E}$ with _Skorokhod metric_ $d_{S}$ induced
by $d$.

Given a measure $\mu\in\mathcal{M}_{+}(E)$ and a measurable function
$f:E\rightarrow\mathbb{R}$, we denote by

$\|\mu\|:=\int_{E}\mu(\mathrm{d}x)\quad\text{and}\quad\langle
f,\mu\rangle:=\int_{E}f(x)\mu(\mathrm{d}x).$

At the level of the stochastic process, we denote by
$\mathbb{P}_{N}\left(\cdot\right)$ and $\mathbb{E}_{N}\left[\cdot\right]$
probability distributions and expectations with regards to the trajectories of
the process with generator $\mathcal{A}$ and (possibly random) initial
condition $\bar{\mathbf{L}}^{(N)}_{0}$.

####  1.3.3 Normalisation, global assumptions and the multi-type Flory
equation

We expect, _a priori_ that limiting equation, called the _multi-type Flory
equation_ (extending the Smoluchowski equation from [17]) encodes the
behaviour of the process $(\mathbf{L}^{(N)}_{t/N}/N)_{t\geq 0}$, for $N$
‘large’. The re-scaling of time is required to counter-balance the increase in
the number of interactions as the initial mass of clusters grows with $N$.
Thus, in general, we set

$\bar{\mathbf{L}}^{(N)}_{t}:=\mathbf{L}^{(N)}_{t/N}/N,$

and generally (by abuse of notation, since such a limit may not be unique) use
$(\bar{\mathbf{L}}^{*}_{t})_{t\geq 0}$ to denote a weak limit (a limit along a
subsequence) of the process.

Additionally, the initial condition of the process must meet certain natural
convergence assumptions. Here, we outline the primary assumptions required.

######  Assumption 1.1.

In this paper, we assume throughout that

  1. 1.

$E$ is a $\sigma$-compact metric space,

  2. 2.

we have $\bar{\mathbf{L}}^{(N)}_{0}=\sum_{i\in I}\frac{c_{i}\delta_{i}}{N}$
for some finite set $I\subseteq E$, $c_{i}\in\mathbb{N}$, and there exists
$c^{\prime}>0$ such that $\sum_{i\in I}\frac{c_{i}}{N}\leq c^{\prime}$ almost
surely,

  3. 3.

there exists $\mu\in\mathcal{E}$ such that

$\bar{\mathbf{L}}^{(N)}_{0}\rightarrow\mu$ (3)

weakly, in probability, and $\langle m,\mu\rangle>0$.

An informal definition of the multi-type Flory equation is as follows (see
Definitions 2.1 and 2.2 on the next page for a more formal treatment).
Included in this equation, is a function $\phi$, which one may regard as a
_conserved quantity_ of the system. Then, $(\mathbf{u}_{t},)_{t\geq 0}$,
taking values in $\mathcal{M}_{+}(E)$ is a solution of the _multi-type Flory
equation_ with _conserved quantity_ $\phi$, and initial condition
$\mathbf{u}_{0}$ if:

$\mathbf{u}_{t}-\mathbf{u}_{0}=\int_{0}^{t}\left[Q^{+}(\mathbf{u}_{s})-Q^{-}(\mathbf{u}_{s})\right]\mathrm{d}s;$
(4)

where $Q^{+}(\mathbf{u}_{s})$ and $Q^{-}(\mathbf{u}_{s})$ are measures defined
such that, for appropriate test functions ${J\in C_{c}(E;\mathbb{R})}$,

$\displaystyle\int_{E}J(y)Q^{+}(\mathbf{u}_{s})(\mathrm{d}y):=\frac{1}{2}\int_{E\times
E\times
E}J(z)K(x,y,\mathrm{d}z)\mathbf{u}_{s}(\mathrm{d}x)\mathbf{u}_{s}(\mathrm{d}y),$
(5)

and

$\displaystyle\int_{E}J(y)Q^{-}(\mathbf{u}_{s})(\mathrm{d}x):=\int_{E\times
E}J(y)\bar{K}(x,y)\mathbf{u}_{s}(\mathrm{d}x)\mathbf{u}_{s}(\mathrm{d}y)+\int_{E}J(y)g_{\infty}(y)\mathbf{u}_{s}(\mathrm{d}y);$
(6)

with $g_{\infty}$ defined such that

$g_{\infty}(y):=\int_{E}\phi(x,y)\mathbf{u}_{0}(\mathrm{d}x)-\int_{E}\phi(x,y)\mathbf{u}_{s}(\mathrm{d}x).$
(7)

##  2 Statements of main results

###  2.1 Concentration of trajectories on solutions of multi-type Flory
equations

We adapt the definition from [16, 17] of _solutions_ for the generalised Flory
equation to this setting.

######  Definition 2.1.

Given a function $\phi:E\times E\rightarrow\mathbb{R}$, we say a map
$t\mapsto\mathbf{u}_{t}\in\mathcal{M}_{+}(E),$ is a _solution of the multi-
type Flory equation_ with _conserved quantity_ $\phi$ if the following are
satisfied:

  1. 1.

for all Borel sets $A\subseteq E$ the map
$t\mapsto\mathbf{u}_{t}(A)\colon[0,\infty)\to[0,\infty]$ is measurable;

  2. 2.

for all $f\in C_{c}(E)$, and $t\geq 0$, we have $\langle
f,\mathbf{u}_{0}\rangle<\infty$,

$\displaystyle\int_{0}^{t}\int_{E\times
E}f(y)\bar{K}(x,y)\mathbf{u}_{s}(\mathrm{d}x)\mathbf{u}_{s}(\mathrm{d}y)\mathrm{d}s<\infty;$
(8) $\displaystyle\hskip 85.35826pt\text{and}\quad\int_{0}^{t}\int_{E\times
E}f(y)\phi(x,y)\mathbf{u}_{0}(\mathrm{d}x)\mathbf{u}_{s}(\mathrm{d}y)\mathrm{d}s<\infty;$
(9)

  3. 3.

for all $f\in C_{c}(E)$ and $t\geq 0$, with $Q^{+}$ and $Q^{-}$ as defined in
(5) and (6),

$\langle f,\mathbf{u}_{t}\rangle=\langle
f,\mathbf{u}_{0}\rangle+\int_{0}^{t}\langle
f,Q^{+}(\mathbf{u}_{s})-Q^{-}(\mathbf{u}_{s})\rangle\,\mathrm{d}s.$ (10)

  4. 4.

For each $x\in E$, $t\geq 0$ we have

$\int_{E}\phi(x,u)\mu_{t}(\mathrm{d}u)\leq\int_{E}\phi(x,u)\mu_{0}(\mathrm{d}u).$
(11)

Note that Item 1 ensures that the equation in Item 2 is well-defined (without
terms of the form $\infty-\infty$).

As alluded in the Definition 2.1, an important feature of a multi-type Flory
equation is a _conserved quantity_ $\phi$. This as a function $\phi:E\times
E\rightarrow\mathbb{R}$ such that, for any $x$, the quantity
$\langle\phi(\cdot,x),\bar{\mathbf{L}}^{(N)}_{t}\rangle$ is fixed for each
$t>0$. Perhaps the most natural conserved quantity in a coagulation process is
_mass_ , which corresponds to the choice of function $\phi(x,y)=m(x)$; as
masses add upon coagulation, this is fixed along trajectories of the process.
However, one may imagine, in models encoding more information about clusters,
that there are other quantities conserved, reflecting, for example, the
_centre of mass_ of clusters in space, or the _momentum_ of the system.

######  Definition 2.2 (Conserved or sub-conserved quantities).

A function $\phi:E\times E\rightarrow\mathbb{R}$ is said to be _conservative_
or a _conserved quantity_ if for all $x,y,q\in E$, for $K(x,y,\cdot)$ a.a.
$z$,

$\phi\left(z,q\right)=\phi(x,q)+\phi(y,q).$ (12)

It is, similarly, said to be _sub-conservative_ if for all $x,y,q\in E$, for
$K(x,y,\cdot)$ a.a. $z$,

$\phi\left(z,q\right)\leq\phi(x,q)+\phi(y,q).$ (13)

It is said to be _doubly conservative_ (similarly _doubly sub-conservative_),
if it is conservative in the second argument in addition to the first, so that
in addition for all $x,y,q\in E$, for $K(x,y,\cdot)$ a.a. $z$,

$\phi\left(q,z\right)=\phi(q,x)+\phi(q,y).$ (14)

Finally, if a function $\xi:E\rightarrow\mathbb{R}$ is such that
$\phi(x,y)=\xi(x)$ is conservative (resp. sub-conservative), we also say $\xi$
is conservative (resp. sub-conservative).

######  Examples 2.1.

Two natural examples of conserved quantities are $\phi\equiv 0$, and
$\phi(x,y)=m(x)\ell(y)$ for some measurable function
$\ell:E\rightarrow\mathbb{R}_{+}$. In the prior case, the associated multi-
type Flory equation corresponds to the generalised Smoluchowski equation
introduced in [17, Section 2], whilst the latter case corresponds to the Flory
equation (called the ‘modified Smoluchowski equation’ in [17]).

######  Examples 2.2.

If $\phi$ is symmetric and conservative, it is also doubly conservative. Thus,
some examples of doubly conservative functions include $\phi(x,y)=m(x)m(y)$.
Another example comes from the _bilinear coagulation process_ , studied in
[9], where $E=[0,\infty)^{d}$, $A\in[0,\infty)^{d\times d}$ is a symmetric
matrix with non-negative entries and
$K(x,y,\mathrm{d}z)=(x^{T}Ay)\delta_{x+y}$. In this case, for any matrix $M$
the function $\phi(x,y)=x^{T}My$ is a doubly conservative function. This means
that $\bar{K}=x^{T}Ay$ is itself is a doubly conservative function.

######  Remark 2.3.

Note that, if $\xi^{\prime}:[0,\infty)\rightarrow[0,\infty)$ is continuous and
sub-additive, the function
$\phi^{\prime}(x,y)=(\xi^{\prime}(m(x)))(\xi^{\prime}(m(y)))$ is doubly sub-
conservative. This is the analogue of ‘sublinear’ function used by Norris in
[17].

Our main assumptions are as follows. The first, Assumption 2.1 ensures that
the probability measures $(\mathbb{P}_{N})_{N\in\mathbb{N}}$ on the space
$D([0,\infty);\mathcal{E})$ induced by the processes
$(\bar{\mathbf{L}}^{(N)}_{t})_{t\in[0,\infty)}$ are tight; and thus, by
Prokhorov’s theorem, the collection of random trajectories
$\left\\{(\bar{\mathbf{L}}^{(N)}_{t})_{t\in[0,\infty)},N\in\mathbb{N}\right\\}$
contains weakly convergent subsequences.

######  Assumption 2.1 (Conditions for tightness).

Assume the following:

  1. 1.

There exists a doubly sub-conservative $\phi^{\prime}$ such that
$\bar{K}\leq\phi^{\prime}$ pointwise.

  2. 2.

We have

$\displaystyle\limsup_{N\to\infty}\mathbb{E}_{N}\left[\int_{E\times
E}\bar{\mathbf{L}}^{(N)}_{0}(\mathrm{d}x)\left(\bar{\mathbf{L}}^{(N)}_{0}(\mathrm{d}y)-\frac{\delta_{x}}{N}\right)\phi^{\prime}(x,y)\right]<\infty.$
(15)

  3. 3.

There exists a doubly sub-conservative function $\phi^{\prime\prime}:E\times
E\rightarrow[0,\infty)$, such that for all $n\in\mathbb{N}$ the set

$\mathcal{E}^{*}_{n}:=\left\\{\mathbf{u}\in\mathcal{M}_{+}(E\times
E):\int_{E}\mathbf{u}(\mathrm{d}x\times\mathrm{d}y)\phi^{\prime\prime}(x,y)\leq
n\right\\}$ (16)

is compact, and $\phi^{\prime\prime}$ satisfies (15).

Our next assumption, Assumption 2.2 (which contains Assumption 2.1), are
criteria under which limits of sub-sequences of the collection
$\left\\{(\bar{\mathbf{L}}^{(N)}_{t})_{t\in[0,\infty)},N\in\mathbb{N}\right\\}$
are concentrated on trajectories that solve the multi-type Flory equation. In
particular, Item 2 from Assumption 2.2 intuitively states that the coagulation
kernel $\bar{K}$ is more and more comparable to a conservative function $\phi$
outside larger and larger compact sets.

######  Assumption 2.2 (Conditions ensuring concentration of trajectories).

Assume that the following hold.

  1. 1.

Assumption 2.1 is satisfied and the functions $\bar{K}$ and $\phi^{\prime}$
are continuous.

  2. 2.

There exists a continuous, conservative function $\phi$ satisfying (15), such
that one of the following hold:

    1. (a)

For an increasing collection of sets $(C_{k})_{k\in\mathbb{N}}\subseteq E$,
with $\bigcap_{k\in\mathbb{N}}\overline{C^{c}_{k}}=\varnothing$ we have, for
any compact $C^{\prime}\subseteq E$

$\displaystyle\limsup_{k\to\infty}\sup_{x\in C^{c}_{k},\,y\in
C^{\prime}}\left|\bar{K}(x,y)-\phi(x,y)\right|<\infty.$ (17)

    2. (b)

There exists a continuous doubly sub-conservative function $\phi^{*}$
satisfying Equation (15), such that, for a collection of sets
$(C_{k})_{k\in\mathbb{N}}\subseteq E$, we have, for any compact
$C^{\prime}\subseteq E$

$\displaystyle\lim_{k\to\infty}\sup_{x\in C^{c}_{k},\,y\in
C^{\prime}}\frac{\left|\bar{K}(x,y)-\phi(x,y)\right|}{\phi^{*}(x,y)}=0.$ (18)

  3. 3.

The limiting initial condition $\mu$ from (3) is such that for any compact set
$C^{\prime}\subseteq E$

$\displaystyle\lim_{N\to\infty}\sup_{y\in
C^{\prime}}\bigg{|}\int_{E}\bar{\mathbf{L}}^{(N)}_{0}(\mathrm{d}x)\phi(x,y)-\int_{E}\mu(\mathrm{d}x)\phi(x,y)\bigg{|}=0\quad\text{almost
surely.}$ (19)

The main result of this section is the following.

######  Theorem 2.3.

Let
$\left\\{(\bar{\mathbf{L}}^{(N)}_{t})_{t\in[0,\infty)},N\in\mathbb{N}\right\\}$
be a sequence of cluster coagulation processes. Then, the following holds:

  1. 1.

If Assumption 2.1 is satisfied, the sequence
$\left\\{(\bar{\mathbf{L}}^{(N)}_{t})_{t\in[0,\infty)},N\in\mathbb{N}\right\\}$
contains converging sub-sequences;

  2. 2.

If Assumption 2.2 is satisfied, the limit $\bar{\mathbf{L}}^{*}$ of any weakly
convergent subsequence is, almost surely, a solution of the multi-type Flory
equation with conserved quantity $\phi$ and initial condition $\mu$ (according
to Definition 2.1).

######  Remark 2.4.

If we have uniqueness of the solution of the associated Flory equation, then
Theorem 2.3 also implies a weak law of large numbers for the cluster
coagulation process, (since weak convergence to a constant implies convergence
in probability). This has been shown by Norris [17] in the cases that either
$\bar{K}(x,y)\leq m(x)+m(y)$,
$\bar{K}(x,y)=\xi^{\prime}(m(x))\xi^{\prime}(m(y))$ where
$\xi^{\prime}:[0,\infty)\rightarrow[0,\infty)$ is continuous, sub-linear and
$(\xi^{\prime})^{2}$ is sub-linear, or if the kernel is ‘eventually
multiplicative’ (meaning that, for some $R>0$, $\bar{K}(x,y)=m(x)m(y)$ on
$(m^{-1}([0,R])\times m^{-1}([0,R]))^{c}$; see [17, Theorem 2.3]). Notably,
Norris also shows an exponential rate of convergence of the coagulation
process to the limit, in a the restricted case of ‘polymer models’ when the
limit is unique (see [17, Theorem 4.2 and Theorem 4.3]). We note, however,
that the setting we consider here is more general, and there are a wide range
of kernels satisfying Equation (18) which are not eventually multiplicative.
The natural extension of this result is to consider kernels that are, in some
sense, “eventually conservative” (c.f. Theorem 2.4).

######  Example 2.5.

Some particular instances where Theorem 2.3 applies, are listed in the
following.

  1. 1.

We can take $\phi^{\prime}=\phi^{\prime\prime}=\phi^{*}$ in Theorem 2.3 and
Lemma 3.2, with $\phi^{\prime}$ satisfying (15). In this case, if there exist
a nested sequence of compact sets $(C_{k})_{k\in\mathbb{N}}$ such that
$\bigcup_{k\in\mathbb{N}}C_{k}=E$ and $\phi^{\prime}(x,y)>k$ on $(C_{k}\times
C_{k})^{c}$, one readily verifies the compactness in (16), by Prokhorov’s
theorem and Markov’s inequality. Often, these sets $(C_{k})_{k\in\mathbb{N}}$
can also be used in (17) or (18).

    1. (a)

For example, assuming continuity of $\bar{K}$ we can take

$\phi^{\prime}(x,y)=\xi^{\prime}(m(x))\xi^{\prime}(m(y))$ (20)

for some continuous, sub-linear, positive function $\xi^{\prime}$, with
$\lim_{k\to\infty}\xi^{\prime}(k)=\infty$, and, assuming $m^{-1}([0,k])$ is
compact, choose $C_{k}=m^{-1}([0,k])$. This is a common assumption in the
setting of hydrodynamic limits converging to the Smoluchowski equation for
example in [17, Equation (2.4)].

    2. (b)

In a similar setting, we can choose $\phi^{\prime}(x,y)=m(x)\wedge m(y)$, in
which case (17) is satisfied with $\phi\equiv 0$. It may the case that

$\int_{E\times E}m(x)\wedge m(y)\mu(\mathrm{d}x)\mu(\mathrm{d}y)<\infty,$

but $\left\langle m,\mu\right\rangle=\infty$; and, as far as we are aware,
this case is not covered by previous results appearing in the literature. This
offers an alternative to the assumption [17, Equation (2.4)] in proving
convergence to solutions of the Smoluchowski equation. In a similar vein, we
can choose $\phi^{\prime}(x,y)=\xi^{\prime}(m(x)\wedge m(y))$, for a
continuous, sub-linear, positive function $\xi^{\prime}$.

    3. (c)

In the setting of the _bilinear coagulation process_ (see Examples 2.2), we
can take $\phi=\phi^{\prime}=\bar{K}(x,y)=x^{T}Ay$. If, for example, the
matrix $A$ has non-zero entries, the sets $\\{x,y\in[0,\infty)^{d}:x^{T}Ay\leq
k\\}$ are compact, and so Equation (16) is satisfied.

  2. 2.

Alternatively, we may choose $\phi(x,y)=m(x)\ell(y)$ for some measurable
function $\ell:E\to\mathbb{R}_{+}$, $\phi^{*}(x,y)=m(x)$, and
$\phi^{\prime}(x,y)=\xi^{\prime}(m(x))\xi^{\prime}(m(y))$, and
$\lim_{N\to\infty}\langle m,\bar{\mathbf{L}}^{(N)}_{0}\rangle=\langle
m,\mu\rangle$ almost surely (which may be used to show (19)). In the setting
of the classical Marcus-Lushnikov process, this example includes a mild
strengthening of [7, Theorem 2.3], showing concentration of trajectories
around the classical Flory equation. However, we allow for random initial
conditions $(\bar{\mathbf{L}}^{(N)}_{0})_{N\in\mathbb{N}}$, and do not require
$\xi^{\prime}(x)\geq 1$.

######  Example 2.6.

In [2, Example 3.3], a number of toy spatial coagulation models are
introduced. Following the notation introduced there, suppose that

$\lim_{n\to\infty}\frac{\bar{K}((p,n),(s,o))}{n}=\ell(s,o)<\infty,$ (21)

where $\ell$ is a continuous function that does not depend on the position $p$
of the mass going to infinity. Then, the function $\phi:=n\ell(s,o)$ satisfies
the properties in (18) and (12) for any choice of the location of the newly
formed particle, which we indicate by $X((p,m),(s,n))$. Indeed (21) implies
condition (18), and we can also check that also (12) is satisfied:

$\displaystyle\phi\left(\left(X((p,m),(s,n)),m+n\right),(v,j)\right)$
$\displaystyle=(m+n)\ell(v,j)=\phi((p,m)),(v,j))+\phi((s,n),(v,j)).$

Thus, in this context, we can apply Theorem 2.3 to deduce a spatial analogue
of the convergence to the Flory equation in [7, Theorem 2.3].

######  Remark 2.7.

If one considers the initial condition
$\bar{\mathbf{L}}^{(N)}_{0}:=\frac{\sum_{i=1}^{N}\delta_{X_{i}}}{N}$, where
$X_{i}$ are i.i.d samples from the limiting measure $\mu$, if we choose
$\phi^{\prime}$ according to (20), by applying the strong law of large
numbers, one can readily verify that Equations (15) is satisfied when
$\langle\xi^{\prime}\circ m,\mu\rangle<\infty$. In this case, the term
$\delta_{x}/N$ appearing in Equation (15) is crucial for this argument to
work, since it may be the case, for example, that $\langle\xi^{\prime}\circ
m\rangle<\infty$, but $\langle(\xi^{\prime}\circ m)^{2}\rangle=\infty$.

###  2.2 Sufficient condition for uniqueness: eventually conservative kernels

Uniqueness of solutions of a multi-type Flory equation is a rather delicate
issue even for the particular case of the classical Smoluchowski equation (see
[16] for an example of kernel which gives non-unique solutions). In the
following theorem we focus on certain kernels $\bar{K}(x,y)$ which coincide
with a conservative function outside a compact set. In analogy with [17,
Theorem 2.3] we call these kernels ‘eventually conservative’ and we prove that
this condition is sufficient to ensure uniqueness of solutions.

######  Theorem 2.4.

Suppose that $(\mu_{t})_{t\geq 0}$ is a solution to the multi-type Flory
equation (according to Definition 2.1) with conserved quantity $\phi$ and
initial condition $\mu_{0}$ such that $\|\mu_{0}\|<\infty$. Also, assume that,
for each $k\in\mathbb{N}$ sufficiently large, the set

$D_{k}:=\left\\{x\in E:\int_{E}\phi(x,y)\mu_{0}(\mathrm{d}y)\leq k\right\\}$
(22)

is compact, with $\bigcup_{k\in\mathbb{N}}D_{k}=E$; we have $\bar{K}(x,y)\leq
c^{\prime}\phi(x,y)$ for some $c^{\prime}>0$ and for some $R$ sufficiently
large, $\phi(x,y)=\bar{K}(x,y)$ on $(D_{R}\times D_{R})^{c}$. Then, the
solution $(\mu_{t})_{t\geq 0}$ is unique.

######  Corollary 2.5.

Under the hypotheses of Theorem 2.3 and Theorem 2.4; if $(\mu_{t})_{t\geq 0}$
denotes the associated unique solution to the multi-type Flory equation, we
have

$(\bar{\mathbf{L}}^{(N)})_{t\geq 0}\longrightarrow(\mu_{t})_{t\geq
0}\quad\text{ in probability, in $D([0,\infty);\mathcal{E})$.}$

######  Remark 2.8.

Note that, unlike in the context of Theorem 2.3, the condition that
$\phi(x,y)=\bar{K}(x,y)$ on $(D_{R}\times D_{R})^{c}$, combined with the
symmetry of $\bar{K}$ implies that $\phi(x,y)$ is symmetric.

######  Remark 2.9.

If we take $\phi(x,y)=m(x)m(y)$, one readily verifies that $\bar{K}(x,y)$ is
eventually conservative in the sense of Theorem 2.4 if the sets
$m^{-1}([0,R])$ are compact.

######  Remark 2.10.

Note that, in the setting of the _bilinear coagulation process_ (see Remark
2.2), the kernel $\bar{K}(x,y)=x^{T}Ay$ is a symmetric conservative function,
hence the uniqueness result, and weak law of large numbers from Theorem 2.4
and Corollary 2.5 extends the results of [9].

######  Remark 2.11.

It is possible to also apply these results to more general spaces $E$; as long
as $E$ is a $\sigma$-compact metric space. This means that we can take $E$ to
be, for example, an appropriate (restricted) space of functions and the kernel
to be of the form

$K(f,g,\mathrm{d}h)=\delta_{\frac{m(f)f+m(g)g}{m(f)+m(g)}}\bar{K}(f,g).$

With any kernel of this form, a symmetric, bilinear form gives rise to a
symmetric, conservative function $\phi$; for example, if $E=C([0,1];[0,1])$ we
see that $\phi(f,g)=\int f(x)g(x)\mathrm{d}x$ is symmetric and conservative.

###  2.3 Existence of gelling solutions to the multi-type Flory equation

In the companion paper [2], we focus on gelation for the cluster coagulation
model. In particular, we derive a general, sufficient criteria for _stochastic
gelation_ (see [2] for the different definitions of gelation) of the sequence
of cluster coagulation processes
$\left\\{(\bar{\mathbf{L}}^{(N)}_{t})_{t\in[0,\infty)},N\in\mathbb{N}\right\\}$.
The following is an immediate implication of Theorem 2.3 combined with [2,
Theorem 1.1 and Theorem 2.3], hence we omit providing an explicit proof.

######  Corollary 2.6.

Suppose a cluster coagulation process $(\bar{\mathbf{L}}^{(N)}_{t})_{t\geq 0}$
satisfies Assumptions 2.1 & 2.2 from [2] and the conditions of Theorem 2.3,
and that $m$ is continuous. Then, there exists a gelling solution to the
multi-type Flory equation (Definition 2.1), with conserved quantity $\phi$ and
initial condition $\mu$.

######  Remark 2.12.

Note that it may be the case that Assumptions 2.1 & 2.2 in [2] are satisfied,
but $\langle m,\mu\rangle=\infty$. In this case, if the conditions of Theorem
2.3 apply, trajectories of the cluster coagulation process still concentrate
around solutions of a Smoluchowski equation; and thus it is natural to use the
notion of _stochastic gelation_ to define gelling solutions of such equations.

One of the novelties of Corollary 2.6 comes from the criterion for gelation of
classical coagulation processes under the conditions of [2, Corollary 2.4]. In
particular, this confirms that a large class of homogeneous kernels, with
exponent $\gamma>1$ have gelling solutions, a well-known conjecture from
scientific modelling literature [1, 3]. Previously, Wagner showed that the
_mass flow process_ associated with such models is _explosive_ , a property
conjectured to hold for all coagulation kernels with gelling solutions [3,
22]. Since we believe this is an interesting result on its own, we state it
separately in the following corollary.

######  Corollary 2.7.

Suppose we are in the setting of the classical Marcus-Lushnikov process, that
is $E=(0,\infty)$, $K(x,y,\mathrm{d}z)=\bar{K}(x,y)\delta_{x+y}$, for a
continuous symmetric function $\bar{K}(x,y)$, and the mass function
$m(x)\equiv x$. Let $\left\langle m\mathbf{1}_{m\geq 1},\mu\right\rangle>0$
and the conditions of Theorem 2.3 hold. Suppose one of the following
conditions hold:

  1. 1.

we have $\inf_{i\in[1,2]}\bar{K}(1,i)>0$ and for all $x,y$ sufficiently large

$\bar{K}(cx,cy)=c^{\gamma}\bar{K}(x,y),\text{ with }\gamma>1$

or, alternatively,

  1. 2.

there exists $\epsilon>0$ such that, for all $x,y$ sufficiently large

$\bar{K}(x,y)\geq(x\wedge y)\log{(x\wedge y)}^{3+\epsilon}.$

Then, there exists a gelling solution to the multi-type Flory equation
(Definition 2.1), with conserved quantity $\phi$ and initial condition $\mu$.

##  3 Proofs of main results

In Section 3.1 we prove the main result of Theorem 2.3, whilst in Section 2.2
we prove Theorem 2.4. Note that the proof of Theorem 2.4 requires considerably
less technicalities, and may be read independently of Section 3.1.

###  3.1 Proof of Theorem 2.3

The normalised cluster coagulation process
$(\bar{\mathbf{L}}^{(N)}_{t})_{t\geq 0}$ induces a probability measure
$\mathbb{P}_{N}$ on the space $D([0,\infty);\mathcal{E})$. To prove Item 1 of
Theorem 2.3, in Section 3.1.1, we show that under Assumption 2.1 the family of
probability measures $(\mathbb{P}_{N})_{N\in\mathbb{N}}$ is tight, which
implies by Prokhorov’s theorem [18, Theorem IV.29, page 82] that
$(\mathbb{P}_{N})_{N\in\mathbb{N}}$ has a weakly convergent subsequence. We
then show in Section 3.1.2, that under Assumption 2.2 any such subsequence
concentrates on solutions of the _multi-type Flory equation_.

Throughout this section, it will be beneficial to have associated “conserved”
quantities, preserved by the dynamics of the process. The following lemma will
be useful throughout:

######  Lemma 3.1.

For any cluster coagulation process $(\bar{\mathbf{L}}^{(N)}_{t})_{t\geq 0}$,
given any doubly sub-conservative $\phi^{{}^{\prime}}:E\times
E\rightarrow\mathbb{R}$, almost surely for all $t\geq 0$ we have, for each
$y\in E$

$\int_{E}\left(\bar{\mathbf{L}}^{(N)}_{t}(\mathrm{d}x)\right)\phi^{\prime}(x,y)\leq\int_{E}\left(\bar{\mathbf{L}}^{(N)}_{0}(\mathrm{d}x)\right)\phi^{\prime}(x,y)$
(23)

and

$\displaystyle\int_{E\times
E}\bar{\mathbf{L}}^{(N)}_{t}(\mathrm{d}x)\left(\bar{\mathbf{L}}^{(N)}_{t}(\mathrm{d}y)-\frac{\delta_{x}}{N}\right)$
$\displaystyle\phi^{\prime}(x,y)\leq\int_{E\times
E}\bar{\mathbf{L}}^{(N)}_{0}(\mathrm{d}x)\left(\bar{\mathbf{L}}^{(N)}_{0}(\mathrm{d}y)-\frac{\delta_{x}}{N}\right)\phi^{\prime}(x,y).$
(24)

###### Proof.

For Equation (23) if $\tau_{1}<\tau_{2}$ denote times of two consecutive
coagulation events, with $\tau_{2}$ involving the coagulation of clusters
$x^{\prime}$ and $y^{\prime}$ to a new cluster $z$, we have

$\displaystyle\int_{E}\left(\bar{\mathbf{L}}^{(N)}_{\tau_{2}}(\mathrm{d}x)-\bar{\mathbf{L}}^{(N)}_{\tau_{1}}(\mathrm{d}x)\right)\phi^{\prime}(x,y)=\left(\phi^{\prime}(z,y)-\phi^{\prime}(x^{\prime},y)-\phi^{\prime}(y^{\prime},y)\right).$

The right-hand side is $0$ for $K(x^{\prime},y^{\prime},\mathrm{d}z)-$a.a.
$z$, hence almost surely. Now, for Equation (24), note that integrals of
$\phi^{\prime}(x,y)$ with respect to the product measure

$\bar{\mathbf{L}}^{(N)}_{s}(\mathrm{d}x)\left(\bar{\mathbf{L}}^{(N)}_{s}(\mathrm{d}y)-\frac{\delta_{x}}{N}\right)$

are nothing but sums of $\phi^{\prime}(x,y)$ across all the distinct pairs of
clusters $x,y$ in the process at time $s$. Thus, if $\tau_{1}<\tau_{2}$ denote
times of two consecutive coagulation events, with $\tau_{2}$ involving the
coagulation of clusters $x^{\prime}$ and $y^{\prime}$ to a new cluster $z$, we
have

$\displaystyle\int_{E\times
E}\bar{\mathbf{L}}^{(N)}_{\tau_{2}}(\mathrm{d}x)\bigg{(}\bar{\mathbf{L}}^{(N)}_{\tau_{2}}(\mathrm{d}y)-\frac{\delta_{x}}{N}\bigg{)}\phi^{\prime}(x,y)-\int_{E\times
E}\bar{\mathbf{L}}^{(N)}_{\tau_{1}}(\mathrm{d}x)\left(\bar{\mathbf{L}}^{(N)}_{\tau_{1}}(\mathrm{d}y)-\frac{\delta_{x}}{N}\right)\phi^{\prime}(x,y)$
$\displaystyle\hskip
28.45274pt=-\frac{\phi(x^{\prime},y^{\prime})+\phi(y^{\prime},x^{\prime})}{N^{2}}+\int_{E}\left(\bar{\mathbf{L}}^{(N)}_{\tau_{1}}-\frac{\delta_{x^{\prime}}}{N}-\frac{\delta_{y^{\prime}}}{N}\right)(\mathrm{d}u)\frac{\left(\phi^{\prime}(z,u)-\phi^{\prime}(x^{\prime},u)-\phi^{\prime}(y^{\prime},u)\right)}{N^{2}}$
$\displaystyle\hskip
56.9055pt+\int_{E}\left(\bar{\mathbf{L}}^{(N)}_{\tau_{1}}-\frac{\delta_{x^{\prime}}}{N}-\frac{\delta_{y^{\prime}}}{N}\right)(\mathrm{d}u)\frac{\left(\phi^{\prime}(u,z)-\phi^{\prime}(u,x^{\prime})-\phi^{\prime}(u,y^{\prime})\right)}{N^{2}}\leq
0,$

for $K(x^{\prime},y^{\prime},\mathrm{d}z)-$ a.a. $z$. Note that the first term
in the second line comes from the contribution to the integral from the pair
$x^{\prime},y^{\prime}$ involved with the coagulation, and the other integrals
in the second and third line, comes from the difference in the contributions
to the integrals from pairs $(v,u)$ where $v\in\\{x^{\prime},y^{\prime},z\\}$,
and $u$ is a cluster not involved in the coagulation event. The result follows
by iterating over the jumps in the process. ∎

####  3.1.1 Tightness: proof of Item 1 of Theorem 2.3

Item 1 of Theorem 2.3 is a consequence of tightness of the sequence of
measures $(\mathbb{P}_{N})_{N\in\mathbb{N}}$, which we prove in the following
lemma.

######  Lemma 3.2.

Assume that Assumption 2.1 is satisfied. Then the sequence of probability
measures $(\mathbb{P}_{N})_{N\in\mathbb{N}}$ is a tight family of probability
measures on the Skorokhod space $D([0,\infty);\mathcal{E})$.

###### Proof of Item 1 of Theorem 2.3.

This is an immediate consequence of Lemma 3.2 and Prokhorov’s theorem [18,
Theorem IV.29, page 82]. ∎

In order to prove Lemma 3.2, we apply some well-established tightness
criterion, stated in Appendix A.

###### Proof of Lemma 3.2.

We apply Lemma A.2. For the first compact containment criterion, first recall
that by (16), the set

$\mathcal{E}^{*}_{n}:=\left\\{\mathbf{u}\in\mathcal{M}_{+}(E\times
E):\int_{E}\mathbf{u}(\mathrm{d}x\times\mathrm{d}y)\phi^{\prime\prime}(x,y)\leq
n\right\\}$

is compact. Now, note that, by Lemma 3.1, if we have

$\bar{\mathbf{L}}^{(N)}_{0}(\mathrm{d}x)\left(\bar{\mathbf{L}}^{(N)}_{0}(\mathrm{d}y)-\frac{\delta_{x}}{N}\right)\in\mathcal{E}^{*}_{n}$

then, for all $s\geq 0$,
$\bar{\mathbf{L}}^{(N)}_{s}(\mathrm{d}x)\left(\bar{\mathbf{L}}^{(N)}_{s}(\mathrm{d}y)-\frac{\delta_{x}}{N}\right)\in\mathcal{E}^{*}_{n}.$
Suppose that we denote by $\mathcal{D}_{N}$ the set

$\mathcal{D}_{N}:=\left\\{\mathbf{u}\in\mathcal{E}:\mathbf{u}=\sum_{i\in
I}\frac{c_{i}\delta_{i}}{N},\>c_{i}\in\mathbb{N},\>I\subseteq E\right\\},$

and $\iota_{N}:\mathcal{D}_{N}\rightarrow\mathcal{M}_{+}(E\times E)$ denotes
the map
$\mathbf{u}\mapsto\mathbf{u}(\mathrm{d}x)\left(\mathbf{u}(\mathrm{d}y)-\frac{\delta_{x}}{N}\right)$;
and extend this map to a map
$\iota:\bigcup_{N\in\mathbb{N}}\mathcal{D}_{N}\rightarrow\mathcal{M}_{+}(E\times
E)$ such that $\iota\equiv\iota_{N}$ on $\mathcal{D}_{N}$. We now note that
for any $n\in\mathbb{N}$ the set

$B_{n}:=\left\\{\mathbf{u}\in\bigcup_{N\in\mathbb{N}}\mathcal{D}_{N}:\iota(\mathbf{u})\in\mathcal{E}^{*}_{n}\right\\}$

is relatively compact. Indeed, by the compactness of $\mathcal{E}^{*}_{n}$,
any sequence $(\iota(\mathbf{u}_{i}))_{i\in\mathbb{N}}$ has a convergent
subsequence $(\iota(\mathbf{u}_{i_{k}}))_{k\in\mathbb{N}}$. Suppose $\nu$
denotes a limit of this subsequence. There, are two cases: we can either find
a further subsequence (which we also denote
$(\iota(\mathbf{u}_{i_{k}}))_{k\in\mathbb{N}}$), such that, for some
$N^{\prime}\in\mathbb{N}$ we have
$(\iota(\mathbf{u}_{i_{k}}))_{k\in\mathbb{N}}=(\iota_{N^{\prime}}(\mathbf{u}_{i_{k}}))_{k\in\mathbb{N}}$,
or it is the case that for any $N^{\prime}\in\mathbb{N}$ there exists
$k\in\mathbb{N}$ such that
$\iota(\mathbf{u}_{i_{k}})=\iota_{j}(\mathbf{u}_{i_{k}})$ for some $j\geq
N^{\prime}$. In the latter case, (since the co-efficient of the $\delta_{x}$
term tends to $0$), we readily verify that

$\mathbf{u}_{i_{k}}\otimes\mathbf{u}_{i_{k}}\rightarrow\nu,$

hence $(\mathbf{u}_{i_{k}})_{k\in\mathbb{N}}$ also converges weakly. We may
similarly deduce the result in the first case, when

$\mathbf{u}_{i_{k}}(\mathrm{d}x)\left(\mathbf{u}_{i_{k}}(\mathrm{d}y)-\frac{\delta_{x}}{N^{\prime}}\right)\rightarrow\nu(\mathrm{d}x\times\mathrm{d}y).$

Now, since $\phi^{\prime\prime}$ satisfies (15) we know
$\mathbb{E}_{N}\left[\int_{E\times
E}\bar{\mathbf{L}}^{(N)}_{0}(\mathrm{d}x)\left(\bar{\mathbf{L}}^{(N)}_{0}(\mathrm{d}y)-\frac{\delta_{x}}{N}\right)\phi^{\prime\prime}(x,y)\right]<c_{0}$,
for some $c_{0}\in\mathbb{N}$. Therefore, by Markov’s inequality, for any
$c_{1}\in\mathbb{N}$,

$\liminf_{N\to\infty}\mathbb{P}_{N}\left(\forall t\geq
0\;\bar{\mathbf{L}}^{(N)}_{t}\in
B_{c_{1}}\right)=\liminf_{N\to\infty}\mathbb{P}_{N}\left(\left\langle\phi^{\prime\prime},\iota(\bar{\mathbf{L}}^{(N)}_{0})\right\rangle\leq
c_{1}\right)\geq 1-\frac{c_{0}}{c_{1}}.$

Thus, by fixing $c_{1}>c_{0}/\varepsilon$ and choosing the closure of
$B_{c_{1}}\subseteq\mathcal{E}$ as the required compact set, we have the
required compact containment condition (122).

For the second criterion, we define an appropriate family of test functions
$\mathbb{F}$, then apply Lemma A.1. In particular, we choose the family of
functions $\mathbb{F}$ from $\mathcal{E}$ to $\mathbb{R}$ such that

$\mathbb{F}:=\left\\{\tilde{J}:\>\tilde{J}(\mathbf{u})=\int_{E}J(x)\mathbf{u}(\mathrm{d}x);\>J\in
C_{c}(E;\mathbb{R})\right\\},$

where $C_{c}(E;\mathbb{R})$ denotes the set of continuous functions on $E$
with compact support. By the definition of the weak topology, this family
consists of continuous functions and it is straightforward to see that it is
closed under addition. Moreover, since $E$ is $\sigma$-compact, a measure
$\mu$ is uniquely determined by the values of $\langle f,\mu\rangle$, where
$f\in C_{c}(E;\mathbb{R})$, thus this family separates points. Now, let
$\tilde{J}\in\mathbb{F}$ be given, with associated function
$J:E\mapsto\mathbb{R}$, so that

$\tilde{J}(\bar{\mathbf{L}}_{t})=\langle
J,\bar{\mathbf{L}}_{t}\rangle=\int_{E}J(x)\bar{\mathbf{L}}_{t}(\mathrm{d}x).$
(25)

We seek to apply Lemma A.1 to the family of pushforward measures

$\left\\{\tilde{J}_{*}\mathbb{P}_{N}:\;N\in\mathbb{N}\right\\}.$

Note that these are measures on the space $D([0,\infty),\mathbb{R})$ which a
separable, and complete metric space, hence Lemma A.1 applies. Also note, that
as the continuous image of a compact set is compact, we can take
$\tilde{J}(\mathcal{E}_{c_{1}})$ as the compact set for the first condition,
and thus we need only verify the second condition of Lemma A.1.

We note that, for fixed $T$, and $\eta=\eta(T)$ sufficiently small, we can
find an integer $K\in\mathbb{N}$ such that $\eta<T/K=:\eta^{\prime}\leq
2\eta$. Therefore, we can define a partition $\\{t_{i}\\}$ of $[0,T]$ such
that $t_{i+1}-t_{i}=\eta^{\prime}>\eta$, so that

$\displaystyle\tilde{J}_{*}(\mathbb{P}_{N})\left(\left\\{f:w^{\prime}(f,\eta,T)\geq\varepsilon\right\\}\right)$
$\displaystyle=\mathbb{P}_{N}\left(w^{\prime}((\tilde{J}(\bar{\mathbf{L}}_{t}))_{t\in[0,T]},\eta,T)\geq\varepsilon\right)$
(26)
$\displaystyle\leq\mathbb{P}_{N}\left(\sup_{s,t\in[0,T],|s-t|\leq\eta^{\prime}}\left|\tilde{J}(\bar{\mathbf{L}}_{s})-\tilde{J}(\bar{\mathbf{L}}_{t})\right|\geq\varepsilon\right),$

where $w^{\prime}$ denotes the modulus of continuity defined in (121). We now
have the following claim:

######  Claim 3.2.1.

For some constant $C=C(T)$, we have

$\limsup_{N\to\infty}\mathbb{E}_{N}\left[\sup_{s,t\in[0,T],|s-t|\leq\eta}\left|\tilde{J}(\bar{\mathbf{L}}_{t})-\tilde{J}(\bar{\mathbf{L}}_{s})\right|\right]<C\eta,$
(27)

To complete the proof of Lemma 3.2 using Claim 3.2.1, observe that by Markov’s
inequality, for all $n\geq N$, we have

$\lim_{\eta\to
0}\limsup_{N\to\infty}\mathbb{P}_{N}\left(\sup_{s,t\in[0,T],|s-t|\leq\eta}\left|\tilde{J}(\bar{\mathbf{L}}_{t})-\tilde{J}(\bar{\mathbf{L}}_{s})\right|\geq\varepsilon\right)<\lim_{\eta\to
0}\frac{C\eta}{\varepsilon}=0,$

implying (123). ∎

###### Proof of Claim 3.2.1.

We first note some relevant facts: since $\bar{\mathbf{L}}_{t}$ is a pure jump
Markov process, and $\tilde{J}$ is bounded and measurable, it is well-known
that

$M_{N}(t):=\tilde{J}(\bar{\mathbf{L}}_{t})-\tilde{J}(\bar{\mathbf{L}}_{0})-\int_{0}^{t}\mathcal{A}_{N}\tilde{J}(\bar{\mathbf{L}}_{s})\mathrm{d}s$
(28)

and its quadratic variation

$Q_{N}(t):=M_{N}(t)^{2}-\int_{0}^{t}(\mathcal{A}_{N}\tilde{J}^{2}-2\tilde{J}\mathcal{A}_{N}\tilde{J})(\bar{\mathbf{L}}_{s})\mathrm{d}s$
(29)

are both martingales under $\mathbb{P}_{N}$ (see, for example, the proofs of
[19, Proposition 7.1.6, and Proposition 8.3.3]). From Equation (28), the
triangle inequality, and sub-additivity of taking suprema, it follows that

$\displaystyle\mathbb{E}_{N}\left[\sup_{s,t\in[0,T],|s-t|\leq\eta}\left|\tilde{J}[\bar{\mathbf{L}}(t)]-\tilde{J}[\bar{\mathbf{L}}_{s}]\right|\right]$
(30)
$\displaystyle\leq\mathbb{E}_{N}\left[\sup_{s,t\in[0,T],|s-t|\leq\eta}\left|M_{N}(t)-M_{N}(s)\right|\right]+\mathbb{E}_{N}\left[\sup_{s,t\in[0,T],|s-t|\leq\eta}\left|\int_{s}^{t}\mathcal{A}_{N}\tilde{J}(\bar{\mathbf{L}}_{\theta})\mathrm{d}\theta\right|\right]$
(31) $\displaystyle\leq 2\mathbb{E}_{N}\left[\sup_{0\leq t\leq
T}\left|M_{N}(t)\right|\right]+\mathbb{E}_{N}\left[\sup_{s,t\in[0,T],t-s\leq\eta}\left|\int_{s}^{t}\mathcal{A}_{N}\tilde{J}(\bar{\mathbf{L}}_{\theta})\mathrm{d}\theta\right|\right].$
(32)

To complete the proof, we bound the two terms on the right side of Equation
(32). For the first, observe that by Doob’s maximal inequality,

$\mathbb{E}_{N}\left[\left(\sup_{0\leq t\leq
T}|M(t)|\right)^{p}\right]\leq\left(\frac{p}{p-1}\right)^{p}\mathbb{E}_{N}\left[|M(_{N}T)|^{p}\right],$

so that, by setting $p=2$, and recalling that
$\mathbb{E}_{N}\left[Q(t)\right]=0$, we deduce from Equation (29) that

$\displaystyle\mathbb{E}_{N}\left[\sup_{0\leq t\leq T}|M_{N}(t)|\right]^{2}$
$\displaystyle\leq\mathbb{E}_{N}\left[\left(\sup_{0\leq t\leq
T}|M_{N}(t)|\right)^{2}\right]$ (33) $\displaystyle\leq
4\mathbb{E}_{N}\left[M_{N}(T)^{2}\right]=4\mathbb{E}_{N}\left[\int_{0}^{T}(\mathcal{A}_{N}\tilde{J}^{2}-2\tilde{J}\mathcal{A}_{N}\tilde{J})(\bar{\mathbf{L}}_{s})\mathrm{d}s\right].$
(34)

Now, we recall that the generator $\mathcal{A}_{N}$ of the normalised process
$(\bar{\mathbf{L}}^{(N)}_{t})_{t\geq 0}$ may be written as follows: for
bounded measurable test functions $F$, we have

$\displaystyle\mathcal{A}_{N}F(\bar{\mathbf{L}}^{(N)}_{t})$
$\displaystyle=\frac{N}{2}\int_{E\times E\times
E}(\bar{\mathbf{L}}^{(N)}_{t}(\mathrm{d}x)\left(\bar{\mathbf{L}}^{(N)}_{t}-\frac{\delta_{x}}{N}\right)(\mathrm{d}y)K(x,y,\mathrm{d}z)$
$\displaystyle\mspace{300.0mu}\times\left(F\left(\bar{\mathbf{L}}^{(N)}_{t}+\frac{\left(\delta_{z}-\delta_{x}-\delta_{y}\right)}{N}\right)-F\left(\bar{\mathbf{L}}^{(N)}_{t}\right)\right).$

For simplicity, for the remainder of this section, whenever unambiguous, we
drop the superscript (or subscript) $(N)$ when referring to
$\bar{\mathbf{L}}^{(N)}_{t}$, $M_{N}(t)$ and $\mathcal{A}_{N}$. Abusing
notation, for each $t$ we denote by $\bar{{\mathbf{L}}}^{(x,y)\to
z}_{t}:={\bar{\mathbf{L}}_{t}}+\left(\delta_{z}-\delta_{x}-\delta_{y}\right)/N$

Thus,

$\displaystyle(\mathcal{A}\tilde{J}^{2}-2\tilde{J}\mathcal{A}\tilde{J})(\bar{\mathbf{L}}_{s})$
$\displaystyle=\frac{N}{2}\int_{E\times E\times
E}\bar{\mathbf{L}}_{s}(\mathrm{d}x)\left(\bar{\mathbf{L}}_{s}-\frac{\delta_{x}}{N}\right)(\mathrm{d}y)K(x,y,\mathrm{d}z)\left(\tilde{J}(\bar{{\mathbf{L}}}^{(x,y)\to
z}_{s})^{2}-\tilde{J}(\bar{\mathbf{L}})^{2}\right)$
$\displaystyle\mspace{50.0mu}-N\tilde{J}(\bar{\mathbf{L}}_{s})\int_{E\times
E\times
E}\bar{\mathbf{L}}_{s}(\mathrm{d}x)\left(\bar{\mathbf{L}}_{s}-\frac{\delta_{x}}{N}\right)(\mathrm{d}y)K(x,y,\mathrm{d}z)\left(\tilde{J}(\bar{{\mathbf{L}}}^{(x,y)\to
z}_{s})-\tilde{J}(\bar{\mathbf{L}})\right)$
$\displaystyle=\frac{N}{2}\int_{E\times E\times
E}\bar{\mathbf{L}}_{s}(\mathrm{d}x)\left(\bar{\mathbf{L}}_{s}-\frac{\delta_{x}}{N}\right)(\mathrm{d}y)K(x,y,\mathrm{d}z)\left(\tilde{J}(\bar{{\mathbf{L}}_{s}}^{(x,y)\to
z})^{2}-\tilde{J}(\bar{\mathbf{L}}_{s})^{2}\right)$
$\displaystyle\mspace{30.0mu}-N\int_{E\times E\times
E}\bar{\mathbf{L}}_{s}(\mathrm{d}x)\left(\bar{\mathbf{L}}_{s}-\frac{\delta_{x}}{N}\right)(\mathrm{d}y)K(x,y,\mathrm{d}z)\left(\tilde{J}(\bar{\mathbf{L}}_{s})\tilde{J}(\bar{{\mathbf{L}}}^{(x,y)\to
z}_{s})-\tilde{J}(\bar{\mathbf{L}}_{s})^{2}\right)$
$\displaystyle=\frac{N}{2}\int_{E\times E\times
E}\bar{\mathbf{L}}_{s}(\mathrm{d}x)\left(\bar{\mathbf{L}}_{s}-\frac{\delta_{x}}{N}\right)(\mathrm{d}y)K(x,y,\mathrm{d}z)\left(\tilde{J}(\bar{{\mathbf{L}}}^{(x,y)\to
z}_{s})-\tilde{J}(\bar{\mathbf{L}}_{s})\right)^{2},$ (35)

where, by definition (25) of $\tilde{J}$, we have

$\tilde{J}(\bar{{\mathbf{L}}}^{(x,y)\to
z}_{s})-\tilde{J}(\bar{\mathbf{L}}_{s})=\left(J(z)-J(y)-J(x)\right)/N.$ (36)

Combining this with Equations (35) and (33), we get

$\displaystyle\mathbb{E}_{N}\left[\sup_{0\leq t\leq T}|M(t)|\right]^{2}$ (37)
$\displaystyle\leq\mathbb{E}_{N}\bigg{[}\frac{2}{N}\int_{0}^{T}\mathrm{d}s\int_{E\times
E\times
E}\bar{\mathbf{L}}_{s}(\mathrm{d}x)\left(\bar{\mathbf{L}}_{s}-\frac{\delta_{x}}{N}\right)(\mathrm{d}y)K(x,y,\mathrm{d}z)\left(J(z)-J(y)-J(x)\right)^{2}\bigg{]}.$
(38)

Moreover, recalling that $J$ is continuous with compact support, by the
extreme value theorem, it is bounded. Therefore, bounding
$(J(z)-J(y)-J(x))^{2}$ by a constant $c_{J}$, and recalling that, by
assumption, $\bar{K}\leq\phi^{\prime}$ pointwise, we have

$\displaystyle\mathbb{E}_{N}\left[\sup_{0\leq t\leq T}|M(t)|\right]^{2}$
$\displaystyle\leq\mathbb{E}_{N}\left[\frac{2c_{J}}{N}\int_{0}^{T}\mathrm{d}s\int_{E\times
E}\bar{\mathbf{L}}_{s}(\mathrm{d}x)\left(\bar{\mathbf{L}}_{s}(\mathrm{d}y)-\frac{\delta_{x}}{N}\right)\bar{K}(x,y)\right]$
(39)
$\displaystyle\leq\frac{2c_{J}}{N}\mathbb{E}_{N}\left[\int_{0}^{T}\mathrm{d}s\int_{E\times
E}\bar{\mathbf{L}}_{s}(\mathrm{d}x)\left(\bar{\mathbf{L}}_{s}(\mathrm{d}y)-\frac{\delta_{x}}{N}\right)\phi^{\prime}(x,y)\right]$
(40) $\displaystyle=\frac{2c_{J}T}{N}\mathbb{E}_{N}\left[\int_{E\times
E}\bar{\mathbf{L}}_{0}(\mathrm{d}x)\left(\bar{\mathbf{L}}_{0}(\mathrm{d}y)-\frac{\delta_{x}}{N}\right)\phi^{\prime}(x,y)\right].$
(41)

The last step is possible since we now observe, that, as $\phi^{\prime}$ is
doubly sub-conservative, for each $s\in[0,\infty)$ we have

$\displaystyle\int_{E\times
E}\bar{\mathbf{L}}^{(N)}_{s}(\mathrm{d}x)\left(\bar{\mathbf{L}}^{(N)}_{s}(\mathrm{d}y)-\frac{\delta_{x}}{N}\right)\phi^{\prime}(x,y)\leq\int_{E\times
E}\bar{\mathbf{L}}^{(N)}_{0}(\mathrm{d}x)\left(\bar{\mathbf{L}}^{(N)}_{0}(\mathrm{d}y)-\frac{\delta_{x}}{N}\right)\phi^{\prime}(x,y),$
(42)

almost surely. Thus, by Equations (41) and (42), we have

$\mathbb{E}_{N}\left[\sup_{0\leq t\leq
T}\left|M(t)\right|\right]\leq\sqrt{\frac{2c_{J}T}{N}\mathbb{E}_{N}\left[\int_{E\times
E}\bar{\mathbf{L}}_{0}(\mathrm{d}x)\left(\bar{\mathbf{L}}_{0}(\mathrm{d}y)-\frac{\delta_{x}}{N}\right)\phi^{\prime}(x,y)\right]}.$
(43)

In order to bound the second term on the right-side of (32), we apply a
similar argument:

$\displaystyle\left|\int_{s}^{t}\mathcal{A}\tilde{J}(\bar{\mathbf{L}}_{\theta})\mathrm{d}\theta\right|$
(44) $\displaystyle=\left|\frac{N}{2}\int_{s}^{t}\mathrm{d}\theta\int_{E\times
E\times
E}\bar{\mathbf{L}}_{t}(\mathrm{d}x)\left(\bar{\mathbf{L}}^{(N)}_{t}-\frac{\delta_{x}}{N}\right)(\mathrm{d}y)K(x,y,\mathrm{d}z)\tilde{J}(\bar{{\mathbf{L}}}^{(x,y)\to
z}_{\theta})-\tilde{J}(\bar{\mathbf{L}}_{\theta})\right|$
$\displaystyle\leq\frac{N}{2}\int_{s}^{t}\mathrm{d}\theta\int_{E\times E\times
E}\bar{\mathbf{L}}_{t}(\mathrm{d}x)\left(\bar{\mathbf{L}}_{t}-\frac{\delta_{x}}{N}\right)(\mathrm{d}y)K(x,y,\mathrm{d}z)\left|\tilde{J}(\bar{{\mathbf{L}}}^{(x,y)\to
z}_{\theta})-\tilde{J}(\bar{\mathbf{L}}_{\theta})\right|$

As before, using Equation (36), and the fact that $|x|=\sqrt{x^{2}}$, we may
bound the previous by

$\displaystyle\frac{\sqrt{c_{J}}}{2}\int_{s}^{t}\mathrm{d}\theta\int_{E\times
E}\bar{\mathbf{L}}_{s}(\mathrm{d}x)\left(\bar{\mathbf{L}}_{s}(\mathrm{d}y)-\frac{\delta_{x}}{N}\right)\bar{K}(x,y)$
(45) $\displaystyle\hskip
28.45274pt\leq\frac{\sqrt{c_{J}}(t-s)}{2}\int_{E\times
E}\bar{\mathbf{L}}_{0}(\mathrm{d}x)\left(\bar{\mathbf{L}}_{0}(\mathrm{d}y)-\frac{\delta_{x}}{N}\right)\phi^{\prime}(x,y),$
(46)

where the final inequality follows from (42). Thus, we have obtained the upper
bound

$\displaystyle\mathbb{E}_{N}\left[\sup_{s,t\in[0,T],|s-t|\leq\eta}\left|\int_{s}^{t}\mathcal{A}\tilde{J}(\bar{\mathbf{L}}_{\theta})\mathrm{d}\theta\right|\right]\leq\frac{\sqrt{c_{J}}}{2}\eta\mathbb{E}_{N}\left[\int_{E\times
E}\bar{\mathbf{L}}_{0}(\mathrm{d}x)\left(\bar{\mathbf{L}}_{0}(\mathrm{d}y)-\frac{\delta_{x}}{N}\right)\phi^{\prime}(x,y)\right].$
(47)

Thus, combining Equation (32) with Equations (43) and (47), and passing to the
limit as $N\to\infty$,

$\displaystyle\limsup_{N\to\infty}\mathbb{E}_{N}\left[\sup_{s,t\in[0,T],|s-t|\leq\eta}\left|\tilde{J}(\bar{\mathbf{L}}_{t})-\tilde{J}(\bar{\mathbf{L}}_{s})\right|\right]$
(48) $\displaystyle\hskip
56.9055pt\leq\frac{\sqrt{c_{J}}}{2}\eta\limsup_{N\to\infty}\mathbb{E}_{N}\left[\int_{E\times
E}\bar{\mathbf{L}}_{0}(\mathrm{d}x)\left(\bar{\mathbf{L}}_{0}(\mathrm{d}y)-\frac{\delta_{x}}{N}\right)\phi^{\prime}(x,y)\right]$
(49)

the latter bound being finite by (15). Setting

$C:=\frac{\sqrt{c_{J}}}{2}\limsup_{N\to\infty}\mathbb{E}_{N}\left[\int_{E\times
E}\bar{\mathbf{L}}_{0}(\mathrm{d}x)\left(\bar{\mathbf{L}}_{0}(\mathrm{d}y)-\frac{\delta_{x}}{N}\right)\phi^{\prime}(x,y)\right]$

concludes the proof of (27). ∎

####  3.1.2 Concentration of trajectories: proof of Item 2 of Theorem 2.3

Assume, now, that Assumption 2.2 is satisfied. This implies that Assumption
2.1 is satisfied, so that, by Item 1 of Theorem 2.3, the sequence of measures
$(\mathbb{P}_{N})_{N\in\mathbb{N}}$ is tight.

Let $\mathbb{P}^{*}$ denote an accumulation point of
$(\mathbb{P}_{N})_{N\in\mathbb{N}}$, and assume, by passing to a subsequence,
and re-indexing, that $\mathbb{P}_{N}\rightarrow\mathbb{P}^{*}$ with respect
to the weak topology on the space of measures on $D([0,\infty);\mathcal{E})$.
Following our previous notation, we denote by $\bar{\mathbf{L}}^{(N)}$ and
$\bar{\mathbf{L}}^{*}$ random trajectories sampled from these distributions.
Mostly out of convenience of notation, applying the Skorokhod representation
theorem [18, Theorem IV.13, page 71],444Noting that as a tight probability
measure on a metric space, $\mathbb{P}^{*}$ concentrates on a separable set.
we assume that $(\bar{\mathbf{L}}^{(N)})_{N\in\mathbb{N}}$ converges to
$\bar{\mathbf{L}}^{*}$ pointwise for all $\omega\in\Omega$ with respect to the
Skorokhod topology on $D([0,\infty);\mathcal{E})$ on some enlarged probability
space $(\Omega,\mathscr{F},\mathbb{P}(\cdot))$. For the rest of the section,
we use the notation $\mathbb{E}\left[\cdot\right]$ to denote expectations with
respect to this enlarged probability space. This allows us draw conclusions
about the limiting trajectory $\bar{\mathbf{L}}^{*}$, and thus the limiting
measure $\mathbb{P}_{*}$, more easily. We have the following proposition:

######  Proposition 3.3.

For any $t\in[0,\infty)$ we have
$\bar{\mathbf{L}}^{(N)}_{t}\rightarrow\bar{\mathbf{L}}^{*}_{t}$ almost surely
in the weak topology. In addition,
$\bar{\mathbf{L}}^{(N)}_{t}\otimes\bar{\mathbf{L}}^{(N)}_{t}\rightarrow\bar{\mathbf{L}}^{*}_{t}\otimes\bar{\mathbf{L}}^{*}_{t}$
almost surely in the weak topology, where the symbol $\otimes$ denotes the
product measure on the space $(\Omega,\mathscr{F},\mathbb{P}(\cdot))$.

###### Proof of Proposition 3.3.

The proof is the result of the following observations:

  1. (I)

First note that, for any $J\in C_{c}(E;\mathbb{R})$, the operator
$\tilde{J}:D([0,\infty);\mathcal{E})\rightarrow D([0,\infty);\mathbb{R})$ is
continuous, as the function $\tilde{J}:\mathcal{E}\rightarrow\mathbb{R}$
defined by $\tilde{J}(u)=\langle J,u\rangle$ is continuous (see for example,
[11, Theorem 4.3]). This implies that, if $\tilde{J}(\bar{\mathbf{L}}^{(N)})$
denotes the map $t\mapsto\tilde{J}(\bar{\mathbf{L}}^{(N)}_{t})$, for any $J\in
C_{c}(E;\mathbb{R})$, we have
$\tilde{J}(\bar{\mathbf{L}}^{(N)})\rightarrow\tilde{J}(\bar{\mathbf{L}}^{*})$
almost surely in $D([0,\infty);\mathbb{R})$.

  2. (II)

Applying (27), and observing that
$\sup_{s,t\in[0,T],|s-t|\leq\eta}\left|\tilde{J}(\bar{\mathbf{L}}_{t})-\tilde{J}(\bar{\mathbf{L}}_{s})\right|$
is monotone decreasing in $\eta$, we have

$\lim_{N\to\infty}\mathbb{E}\left[\lim_{\eta\to
0}\sup_{s,t\in[0,T],|s-t|\leq\eta}\left|\tilde{J}(\bar{\mathbf{L}}_{t})-\tilde{J}(\bar{\mathbf{L}}_{s})\right|\right]=0.$
(50)

In addition, one may readily verify that, for any $T\in[0,\infty)$ the
functional

$x\mapsto\lim_{\eta\to
0}\sup_{s,t\in[0,T],|s-t|\leq\eta}\left\|x(t)-x(s)\right\|$ (51)

is a continuous functional with respect to the Skorokhod topology.
Consequentially, by bounded convergence,

$\displaystyle 0$
$\displaystyle=\mathbb{E}\left[\lim_{N\to\infty}\lim_{\eta\to
0}\sup_{s,t\in[0,T],|s-t|\leq\eta}\left|\tilde{J}(\bar{\mathbf{L}}^{(N)}_{t})-\tilde{J}(\bar{\mathbf{L}}^{(N)}_{s})\right|\right]$
$\displaystyle\stackrel{{\scriptstyle\eqref{cont_fct}}}{{=}}\mathbb{E}\left[\lim_{\eta\to
0}\sup_{s,t\in[0,T],|s-t|\leq\eta}\left|\tilde{J}(\bar{\mathbf{L}}^{*}_{t})-\tilde{J}(\bar{\mathbf{L}}^{*}_{s})\right|\right]$

for any $J\in C_{c}(E;\mathbb{R})$, where in the final equality we have used
the continuity of $\eqref{cont_fct}$. Therefore, the function
$\tilde{J}(\mathbf{\bar{L}}^{*}):[0,\infty)\rightarrow\mathbb{R}$ such that
$t\mapsto\tilde{J}(\mathbf{\bar{L}}^{*}_{t})$ is almost surely continuous
(i.e., $\tilde{J}(\mathbf{\bar{L}}^{*})\leavevmode\nobreak\
\in\leavevmode\nobreak\ C([0,\infty),\mathbb{R})$ almost surely).

  3. (III)

This continuity implies that for any sequence $(t_{n})_{n\in\mathbb{N}}$ such
that $t_{n}\rightarrow t$, for any $J\in C_{c}(E;\mathbb{R})$ we have

$\tilde{J}(\mathbf{\bar{L}}^{*}_{t_{n}})=\int_{E}J(x)\mathbf{\bar{L}}^{*}_{t_{n}}(\mathrm{d}x)\rightarrow\int_{E}J(x)\mathbf{\bar{L}}^{*}_{t}(\mathrm{d}x)\quad\text{almost
surely}.$

But, since, by assumption on the initial condition we have
$\bar{\mathbf{L}}^{*}_{0}=\mu$, where $\mu$ denotes the limiting measure from
(3), and the dynamics of the process ensure that
$\|\bar{\mathbf{L}}^{(N)}_{t}\|$ is non-increasing for each $N$, we readily
verify that each for each $t\in[0,\infty)$ we have
$\mathbf{\bar{L}}^{*}_{t}(E)\leq\|\mu\|$. Thus, by approximating any $F\in
C_{b}(E;\mathbb{R})$ by compactly supported functions, for any $F\in
C_{b}(E;\mathbb{R})$ we have

$\int_{E}F(x)\mathbf{\bar{L}}^{*}_{t_{n}}(\mathrm{d}x)\rightarrow\int_{E}F(x)\mathbf{\bar{L}}^{*}_{t}(\mathrm{d}x)\quad\text{almost
surely}.$

This implies that $\bar{\mathbf{L}}^{*}$ is, almost surely, a continuous
trajectory of measures, i.e., $\bar{\mathbf{L}}^{*}\in
C([0,\infty);\mathcal{E})$.

  4. (IV)

It is well-known that in a Skorokhod space the projection map
$\pi_{t}:D([0,\infty);E)\rightarrow E$ is a continuous functional at any
trajectory $x\in D([0,\infty);E)$ for which $t$ is a continuity point. Since
every $t\in[0,\infty)$ is a continuity point of $\bar{\mathbf{L}}^{*}_{t}$,
this implies that for any $t\in[0,\infty)$ we have
$\bar{\mathbf{L}}^{(N)}_{t}\rightarrow\bar{\mathbf{L}}^{*}_{t}$ almost surely
in the weak topology, as required. Now, by a similar approach to the proof of
Lemma 3.2, the family of measures
$\left\\{\bar{\mathbf{L}}^{(N)}_{t}\otimes\bar{\mathbf{L}}^{(N)}_{t},N\in\mathbb{N}\right\\}$,
is (almost surely) tight, and by assumption uniformly bounded in norm, thus
almost surely relatively compact by [18, Theorem IV.29, page 82]; and any
accumulation point must be
$\bar{\mathbf{L}}^{*}_{t}\otimes\bar{\mathbf{L}}^{*}_{t}$. Thus,
$\bar{\mathbf{L}}^{(N)}_{t}\otimes\bar{\mathbf{L}}^{(N)}_{t}\rightarrow\bar{\mathbf{L}}^{*}_{t}\otimes\bar{\mathbf{L}}^{*}_{t}$
almost surely.

∎

Now we are ready to prove Item 2 of Theorem 2.3. Parts of the proof rely on
equations from the proof of Lemma 3.2, hence we recommend that the reader be
acquainted with this proof first.

###### Proof of Item 2 of Theorem 2.3.

In order to simplify some expressions, we make some shorthands. Recall that
for any compactly supported $J\in C_{c}(E;\mathbb{R})$, we denote by
$\tilde{J}$ the functional such that

$\tilde{J}(\bar{\mathbf{L}}^{(N)}_{s})=\langle
J,\bar{\mathbf{L}}^{(N)}_{s}\rangle=\int_{E}\bar{\mathbf{L}}^{(N)}_{s}(\mathrm{d}x)J(x).$

We also define the following functionals:

$\displaystyle G^{+}(\bar{\mathbf{L}}^{(N)}_{s},J):=\frac{1}{2}\int_{E\times
E\times
E}\bar{\mathbf{L}}^{(N)}_{s}(\mathrm{d}x)\bar{\mathbf{L}}^{(N)}_{s}(\mathrm{d}y)K(x,y,\mathrm{d}z)J(z).$
(52)

We recall that, by assumption, there exists a continuous function
$\phi:E\times E\mapsto\mathbb{R}^{+}$ that satisfies Equations (12), (18) and
(15). We then define $\hat{G}$ by

$\displaystyle\hat{G}(\bar{\mathbf{L}}^{(N)}_{s},J):=\int_{E\times
E}\bar{\mathbf{L}}^{(N)}_{s}(\mathrm{d}x)\bar{\mathbf{L}}^{(N)}_{s}(\mathrm{d}y)\left[\bar{K}(x,y)-\phi(x,y)\right]J(y).$
(53)

Finally, we define the functional

$\displaystyle H(\bar{\mathbf{L}}^{(N)}_{s},J)$
$\displaystyle:=\int_{E}\bar{\mathbf{L}}^{(N)}_{s}(\mathrm{d}y)\left\langle\phi(\cdot,y),\bar{\mathbf{L}}^{(N)}_{0}\right\rangle
J(y)=\int_{E\times
E}\bar{\mathbf{L}}^{(N)}_{s}(\mathrm{d}y)\bar{\mathbf{L}}^{(N)}_{0}(\mathrm{d}x)\phi(x,y)J(y).$
(54)

We now have the following claim:

######  Claim 3.3.1.

Almost surely, for any $t\in[0,\infty)$, $J\in C_{c}(E;\mathbb{R})$ we have

$\tilde{J}(\bar{\mathbf{L}}^{*}_{t})-\tilde{J}(\mu)=\int_{0}^{t}G^{+}(\bar{\mathbf{L}}^{*}_{s},J)-\hat{G}(\bar{\mathbf{L}}^{*}_{s},J)-H(\bar{\mathbf{L}}^{*}_{s},J)\,\mathrm{d}s.$
(55)

Note that the truth of Equation (55) for any $J\in C_{c}(E;\mathbb{R})$
implies that, almost surely, $(\bar{\mathbf{L}}^{*}_{t})_{t\geq 0}$ satisfies
(10) in Definition 2.1. Now, note that, as an application of Proposition 3.3,
for any $t\geq 0$ we have

$\bar{\mathbf{L}}^{(N)}_{t}(\mathrm{d}x)\left(\bar{\mathbf{L}}^{(N)}_{t}(\mathrm{d}y)-\frac{\delta_{x}}{N}\right)\rightarrow\bar{\mathbf{L}}^{*}_{t}\otimes\bar{\mathbf{L}}^{*}_{t}$
(56)

almost surely in the weak topology. Recall that we have
$\bar{K}\leq\phi^{\prime}$, where $\phi^{\prime}$ is doubly sub-conservative
and continuous by the first assumption of Theorem 2.3, and $\phi$ (which is
also continuous, and $\phi^{\prime}$ both satisfy (15). Thus, exploiting weak
convergence, and Lemma 3.1, we deduce that $(\bar{\mathbf{L}}^{*}_{t})_{t\geq
0}$ also satisfies Equations (8) and (11), thus is a solution of the multi-
type Flory equation in the sense given by Definition 2.1. Hence the claim
completes the proof of the theorem. ∎

It thus suffices to prove the claim.

###### Proof of Claim 3.3.1.

First note that by recalling the martingale from Equation (28), together with
the bound from (43) in the proof of Lemma 3.2, we obtain

$\displaystyle\lim_{N\to\infty}\mathbb{E}\Big{[}\Big{|}\tilde{J}(\bar{\mathbf{L}}^{(N)}_{t})-\tilde{J}(\bar{\mathbf{L}}^{(N)}_{0})$
(57) $\displaystyle-\frac{1}{2}\int_{0}^{t}\mathrm{d}s\int_{E\times E\times
E}\bar{\mathbf{L}}^{(N)}_{s}(\mathrm{d}x)\left(\bar{\mathbf{L}}^{(N)}_{s}(\mathrm{d}y)-\frac{\delta_{x}}{N}\right)K(x,y,\mathrm{d}z)\times\left(J(z)-J(y)-J(x)\right)\Big{|}\Big{]}$
(58) $\displaystyle=0.$ (59)

Now, we define

$\displaystyle G_{N}^{-}(\bar{\mathbf{L}}^{(N)}_{s},J)$
$\displaystyle:=\frac{1}{2}\int_{E\times
E}\bar{\mathbf{L}}^{(N)}_{s}(\mathrm{d}x)\left(\bar{\mathbf{L}}^{(N)}_{s}(\mathrm{d}y)-\frac{\delta_{x}}{N}\right)\bar{K}((x,y)(J(x)+J(y)),$
(60) $\displaystyle G^{+}_{N}(\bar{\mathbf{L}}^{(N)}_{s},J)$
$\displaystyle:=\frac{1}{2}\int_{E\times E\times
E}\bar{\mathbf{L}}^{(N)}_{s}(\mathrm{d}x)\left(\bar{\mathbf{L}}^{(N)}_{s}(\mathrm{d}y)-\frac{\delta_{x}}{N}\right)K(x,y,\mathrm{d}z)J(z),$
(61) $\displaystyle\hat{G}_{N}(\bar{\mathbf{L}}^{(N)}_{s},J)$
$\displaystyle:=\int_{E\times
E}\bar{\mathbf{L}}^{(N)}_{s}(\mathrm{d}x)\left(\bar{\mathbf{L}}^{(N)}_{s}(\mathrm{d}y)-\frac{\delta_{x}}{N}\right)\left[\bar{K}(x,y)-\phi(x,y)\right]J(y).$
(62)

We may thus re-write the inner integral appearing in Equation (57) as
$G_{N}^{+}(\bar{\mathbf{L}}^{(N)}_{s},J)-G_{N}^{-}(\bar{\mathbf{L}}^{(N)}_{s},J)$,
so that

$\lim_{N\to\infty}\mathbb{E}\left[\left|\tilde{J}(\bar{\mathbf{L}}^{(N)}_{t})-\tilde{J}(\bar{\mathbf{L}}^{(N)}_{0})-\int_{0}^{t}G_{N}^{+}(\bar{\mathbf{L}}^{(N)}_{s},J)-G^{-}_{N}(\bar{\mathbf{L}}^{(N)}_{s},J)\,\mathrm{d}s\right|\right]=0.$
(63)

Now, we seek to exploit the convergence of $\bar{\mathbf{L}}^{(N)}_{s}$ to
$\bar{\mathbf{L}}^{*}_{s}$, but note that as the integrand appearing in
Equation (60) is in general unbounded, and $G^{-}$ is, in general, not
continuous. However, it is possible to show that this functional coincides
with a continuous functional on the trajectories
$s\mapsto\bar{\mathbf{L}}^{(N)}_{s}$. Indeed, since $\phi$ is conservative, by
Lemma 3.1 the quantity
$\left\langle\phi(\cdot,y),\bar{\mathbf{L}}^{(N)}_{s}\right\rangle$ is fixed,
by adding and subtracting the term corresponding to $\int_{E\times
E}\bar{\mathbf{L}}^{(N)}_{s}(\mathrm{d}y)\bar{\mathbf{L}}^{(N)}_{0}(\mathrm{d}x)\phi(x,y)J(y)$,
we have

$G_{N}^{-}(\bar{\mathbf{L}}^{(N)}_{s},J)=\hat{G}_{N}(\bar{\mathbf{L}}^{(N)}_{s},J)+H(\bar{\mathbf{L}}^{(N)}_{s},J)-\mathcal{E}_{N}(\bar{\mathbf{L}}^{(N)}_{s},J),$

with

$\mathcal{E}_{N}(\bar{\mathbf{L}}^{(N)}_{s},J):=\frac{1}{N}\int_{E}\phi(x,x)J(x)\bar{\mathbf{L}}^{(N)}_{s}(\mathrm{d}x).$

Thus, re-writing Equation (63), we get

$\displaystyle\lim_{N\to\infty}\mathbb{E}\bigg{[}\bigg{|}\tilde{J}(\bar{\mathbf{L}}^{(N)}_{t})-\tilde{J}(\bar{\mathbf{L}}^{(N)}_{0})-\int_{0}^{t}G^{+}_{N}(\bar{\mathbf{L}}^{(N)}_{s},J)-\hat{G}_{N}(\bar{\mathbf{L}}^{(N)}_{s},J)-H(\bar{\mathbf{L}}^{(N)}_{s},J)+\mathcal{E}_{N}(\bar{\mathbf{L}}^{(N)}_{s},J)\,\mathrm{d}s\bigg{|}\bigg{]}=0.$
(64)

Now, in order to complete the proof of Equation (55), we need to argue that we
can pass the limit inside the expectation, and exploit weak convergence to
replace the terms corresponding to $\bar{\mathbf{L}}^{(N)}$ with
$\bar{\mathbf{L}}^{*}$.We can pass the limit inside if the term
$\hat{G}(\bar{\mathbf{L}}^{(N)}_{s},J)$ was bounded, and then, need to argue
continuity of the operators $\hat{G}$ and $H$. Consequentially, we first
approximate the functional $\hat{G}$ by truncations
$(\hat{G}^{(k)})_{k\in\mathbb{N}}$, such that, with compact sets as defined in
Equation (18)

$\displaystyle\hat{G}^{(k)}(\bar{\mathbf{L}}^{(N)}_{s},J):=\int_{C_{k}}\bar{\mathbf{L}}^{(N)}_{s}(\mathrm{d}x)\int_{E}\bar{\mathbf{L}}^{(N)}_{s}(\mathrm{d}y)\left(\bar{K}(x,y)-\phi(x,y)\right)J(y).$
(65)

We finish the proof with another claim.

######  Claim 3.3.2.

Almost surely, for $s,t\in[0,\infty)$, $J\in C_{c}(E)$ we have

$\lim_{k\to\infty}\limsup_{N\to\infty}\mathbb{E}\left[\left|\int_{0}^{t}\hat{G}_{N}(\bar{\mathbf{L}}^{(N)}_{s},J)-\hat{G}^{(k)}(\bar{\mathbf{L}}^{(N)}_{s},J)\,\mathrm{d}s\right|\right]=0,$
(66)
$\limsup_{N\to\infty}\left(\mathbb{E}\left[\left|\int_{0}^{t}G^{+}_{N}(\bar{\mathbf{L}}^{(N)}_{s},J)-{G}^{+}(\bar{\mathbf{L}}^{(N)}_{s},J)\,\mathrm{d}s\right|\right]+\mathbb{E}\left[\left|\int_{0}^{t}\mathcal{E}_{N}(\bar{\mathbf{L}}^{(N)}_{s},J)\,\mathrm{d}s\right|\right]\right)=0,$
(67)
$\lim_{k\to\infty}\mathbb{E}\left[\left|\int_{0}^{t}\hat{G}(\bar{\mathbf{L}}^{*}_{s},J)-\hat{G}^{(k)}(\bar{\mathbf{L}}^{*}_{s},J)\,\mathrm{d}s\right|\right]=0,$
(68)

and

$\lim_{N\to\infty}H(\bar{\mathbf{L}}^{(N)}_{s},J)=H(\bar{\mathbf{L}}^{*}_{s},J)\quad\text{almost
surely.}$ (69)

Indeed, if Equations (66), (67), (68) and (69) are satisfied, by approximating
$\hat{G}$ by $\hat{G}^{(k)}$ (using the triangle inequality) in the second
equality, and using bounded convergence for the third, we have

$\displaystyle 0$
$\displaystyle=\lim_{N\to\infty}\mathbb{E}\bigg{[}\bigg{|}\tilde{J}(\bar{\mathbf{L}}^{(N)}_{t})-\tilde{J}(\bar{\mathbf{L}}^{(N)}_{0})-\int_{0}^{t}G_{N}^{+}(\bar{\mathbf{L}}^{(N)}_{s},J)-\hat{G}_{N}(\bar{\mathbf{L}}^{(N)}_{s},J)-H_{N}(\bar{\mathbf{L}}^{(N)}_{s},J)+\mathcal{E}_{N}(\bar{\mathbf{L}}^{(N)}_{s},J)\,\mathrm{d}s\bigg{|}\bigg{]}$
(70) $\displaystyle\stackrel{{\scriptstyle\eqref{eq:goal-trunc-approx-
flory1},\eqref{eq:approx_G_+_N}}}{{=}}\lim_{k\to\infty}\lim_{N\to\infty}\mathbb{E}\bigg{[}\bigg{|}\tilde{J}(\bar{\mathbf{L}}^{(N)}_{t})-\tilde{J}(\bar{\mathbf{L}}^{(N)}_{0})-\int_{0}^{t}G^{+}(\bar{\mathbf{L}}^{(N)}_{s},J)-\hat{G}^{(k)}(\bar{\mathbf{L}}^{(N)}_{s},J)-H(\bar{\mathbf{L}}^{(N)}_{s},J)\,\mathrm{d}s\bigg{|}\bigg{]}$
(71)
$\displaystyle=\lim_{k\to\infty}\mathbb{E}\bigg{[}\lim_{N\to\infty}\bigg{|}\tilde{J}(\bar{\mathbf{L}}^{(N)}_{t})-\tilde{J}(\bar{\mathbf{L}}^{(N)}_{0})-\int_{0}^{t}G^{+}(\bar{\mathbf{L}}^{(N)}_{s},J)-\hat{G}^{(k)}(\bar{\mathbf{L}}^{(N)}_{s},J)-H(\bar{\mathbf{L}}^{(N)}_{s},J)\,\mathrm{d}s\bigg{|}\bigg{]}$
(72)
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:conv-H}}}{{=}}\lim_{k\to\infty}\mathbb{E}\bigg{[}\bigg{|}\tilde{J}(\bar{\mathbf{L}}^{*}_{t})-\tilde{J}(\mu)-\int_{0}^{t}G^{+}(\bar{\mathbf{L}}^{*}_{s},J)-\hat{G}^{(k)}(\bar{\mathbf{L}}^{*}_{s},J)-H(\bar{\mathbf{L}}^{*}_{s},J)\,\mathrm{d}s\bigg{|}\bigg{]}$
(73) $\displaystyle\stackrel{{\scriptstyle\eqref{eq:goal-trunc-approx-
flory2}}}{{=}}\mathbb{E}\left[\left|\tilde{J}(\bar{\mathbf{L}}^{*}_{t})-\tilde{J}(\mu)-\int_{0}^{t}G^{+}(\bar{\mathbf{L}}^{*}_{s},J)-\hat{G}(\bar{\mathbf{L}}^{*}_{s},J)-H(\bar{\mathbf{L}}^{*}_{s},J)\,\mathrm{d}s\right|\right].$
(74)

∎

Finally, we finish the proof of Claim 3.3.2.

###### Proof of Claim 3.3.2.

Note that we have

$\displaystyle\mathbb{E}\left[\left|\int_{0}^{t}\hat{G}_{N}(\bar{\mathbf{L}}^{(N)}_{s},J)-\hat{G}^{(k)}(\bar{\mathbf{L}}^{(N)}_{s},J)\,\mathrm{d}s\right|\right]$
(75)
$\displaystyle\leq\frac{1}{N}\mathbb{E}\left[\left|\int_{0}^{t}\int_{C_{k}}\bar{\mathbf{L}}^{(N)}_{s}(\mathrm{d}x)|\bar{K}(x,x)-\phi(x,x)||J(x)|\,\mathrm{d}s\right|\right]$
(76)
$\displaystyle\qquad+\mathbb{E}\left[\int_{0}^{t}\int_{C_{k}^{c}}\bar{\mathbf{L}}^{(N)}_{s}(\mathrm{d}x)\int_{E}\left(\bar{\mathbf{L}}^{(N)}_{s}(\mathrm{d}y)-\frac{\delta_{x}}{N}\right)\left|\bar{K}(x,y)-\phi(x,y)\right|\left|J(y)\right|\mathrm{d}s\right],$
(77)

where we immediately see that

$\limsup_{N\to\infty}\frac{1}{N}\mathbb{E}\left[\left|\int_{0}^{t}\int_{C_{k}}\bar{\mathbf{L}}^{(N)}_{s}(\mathrm{d}x)|\bar{K}(x,x)-\phi(x,x)||J(x)|\,\mathrm{d}s\right|\right]=0,$
(78)

since $\bar{K}$, $\phi$ and $J$ are continuous, they are bounded on the
support of $J$ and thus so is the expectation. Now, if Equation (17) applies,
then the integrand of the second term in (75) is bounded by some constant
$c^{\prime}>0$, thus by (56) and the Portmanteau theorem, we have

$\displaystyle\limsup_{N\to\infty}\mathbb{E}\left[\int_{0}^{t}\int_{C_{k}^{c}}\bar{\mathbf{L}}^{(N)}_{s}(\mathrm{d}x)\int_{E}\left(\bar{\mathbf{L}}^{(N)}_{s}(\mathrm{d}y)-\frac{\delta_{x}}{N}\right)\left|\bar{K}(x,y)-\phi(x,y)\right|\left|J(y)\right|\mathrm{d}s\right]$
(79) $\displaystyle\hskip 85.35826pt\leq
c^{\prime}\mathbb{E}\left[\int_{0}^{t}\bar{\mathbf{L}}^{*}_{s}\left(\overline{C_{k}^{c}}\right)\bar{\mathbf{L}}^{*}_{s}(E)\mathrm{d}s\right];$
(80)

and applying bounded convergence, using the fact that
$\bigcap_{k\in\mathbb{N}}\overline{C^{c}_{k}}=\varnothing$, we deduce (66).
Otherwise, in the case that Equation (18) applies, we bound the second term in
(75) as follows:

$\displaystyle\mathbb{E}\left[\int_{0}^{t}\int_{C_{k}^{c}}\bar{\mathbf{L}}^{(N)}_{s}(\mathrm{d}x)\int_{E}\left(\bar{\mathbf{L}}^{(N)}_{s}(\mathrm{d}y)-\frac{\delta_{x}}{N}\right)\left|\bar{K}(x,y)-\phi(x,y)\right|\left|J(y)\right|\mathrm{d}s\right],$
(81)
$\displaystyle\leq\mathbb{E}\left[\int_{0}^{t}\int_{C_{k}^{c}}\bar{\mathbf{L}}^{(N)}_{s}(\mathrm{d}x)\int_{E}\left(\bar{\mathbf{L}}^{(N)}_{s}(\mathrm{d}y)-\frac{\delta_{x}}{N}\right)\phi^{*}(x,y)\,\mathrm{d}s\right]\left\|J\right\|_{\infty}\sup_{x\in
C_{k}^{c},\,y\in{\rm
Supp}{(J)}}\left|\frac{\bar{K}(x,y)-\phi(x,y)}{\phi^{*}(x,y)}\right|,$ (82)

Since $\phi^{*}$ is doubly sub-conservative, and satisfies Equation (15), we
have

$\displaystyle\limsup_{N\to\infty}\mathbb{E}\left[\int_{0}^{t}\int_{C_{k}^{c}\times
E}\bar{\mathbf{L}}^{(N)}_{s}(\mathrm{d}x)\left(\bar{\mathbf{L}}^{(N)}_{s}(\mathrm{d}y)-\frac{\delta_{x}}{N}\right)\phi^{*}(x,y)\,\mathrm{d}s\right]$
(83)
$\displaystyle\leq\limsup_{N\to\infty}\,\mathbb{E}\left[\int_{0}^{t}\int_{E\times
E}\bar{\mathbf{L}}^{(N)}_{s}(\mathrm{d}x)\left(\bar{\mathbf{L}}^{(N)}_{s}(\mathrm{d}y)-\frac{\delta_{x}}{N}\right)\phi^{*}(x,y)\,\mathrm{d}s\right]$
(84) $\displaystyle\leq\limsup_{N\to\infty}t\,\mathbb{E}\left[\int_{E\times
E}\bar{\mathbf{L}}^{(N)}_{0}(\mathrm{d}x)\left(\bar{\mathbf{L}}^{(N)}_{0}(\mathrm{d}y)-\frac{\delta_{x}}{N}\right)\phi^{*}(x,y)\right]\stackrel{{\scriptstyle\eqref{eq:conv-
cons-prob}}}{{<}}\infty.$ (85)

Combining Equation (83), with (75), (78) and (81), we deduce Equation (66).

Equation (67) is proved in an analogous manner to (78), exploiting the compact
support of $J$. For (68), by the monotone convergence theorem, (56) and
Fatou’s lemma we have

$\displaystyle\mathbb{E}\left[\int_{0}^{t}\int_{E\times
E}\bar{\mathbf{L}}^{*}_{s}(\mathrm{d}x)\bar{\mathbf{L}}^{*}_{s}(\mathrm{d}y)\phi^{*}(x,y)\,\mathrm{d}s\right]=\mathbb{E}\left[\int_{0}^{t}\lim_{j\to\infty}\int_{E\times
E}\bar{\mathbf{L}}^{*}_{s}(\mathrm{d}x)\bar{\mathbf{L}}^{*}_{s}(\mathrm{d}y)\left(\phi^{*}(x,y)\wedge
j\right)\,\mathrm{d}s\right]$ (86) $\displaystyle\hskip
28.45274pt=\mathbb{E}\left[\int_{0}^{t}\lim_{j\to\infty}\lim_{N\to\infty}\int_{E\times
E}\bar{\mathbf{L}}^{(N)}_{s}(\mathrm{d}x)\left(\bar{\mathbf{L}}^{(N)}_{s}(\mathrm{d}y)-\frac{\delta_{x}}{N}\right)\left(\phi^{*}(x,y)\wedge
j\right)\,\mathrm{d}s\right]$ (87) $\displaystyle\hskip
28.45274pt\leq\limsup_{N\to\infty}\mathbb{E}\left[\int_{0}^{t}\int_{E\times
E}\bar{\mathbf{L}}^{(N)}_{s}(\mathrm{d}x)\left(\bar{\mathbf{L}}^{(N)}_{s}(\mathrm{d}y)-\frac{\delta_{x}}{N}\right)\phi^{*}(x,y)\,\mathrm{d}s\right]\stackrel{{\scriptstyle\eqref{eq:limsup-
bounded}}}{{<}}\infty.$ (88)

Applying Equation (86) we deduce Equation (68) in a similar manner to Equation
(66). Finally, recalling the definition of the functional $H$ from Equation
(54), we have

$\lim_{N\to\infty}H(\bar{\mathbf{L}}^{(N)}_{s},J)=\lim_{N\to\infty}\int_{E\times
E}\bar{\mathbf{L}}^{(N)}_{s}(\mathrm{d}y)\bar{\mathbf{L}}^{(N)}_{0}(\mathrm{d}x)\phi(x,y)J(y).$

Thus,

$\displaystyle\limsup_{N\to\infty}\left|H(\bar{\mathbf{L}}^{(N)}_{s},J)-H(\bar{\mathbf{L}}^{*}_{s},J)\right|$
(89) $\displaystyle\hskip 7.11317pt=\limsup_{N\to\infty}\left|\int_{E\times
E}\bar{\mathbf{L}}^{(N)}_{s}(\mathrm{d}y)\bar{\mathbf{L}}^{(N)}_{0}(\mathrm{d}x)\phi(x,y)J(y)-\int_{E\times
E}\bar{\mathbf{L}}^{*}_{s}(\mathrm{d}y)\mu(\mathrm{d}x)\phi(x,y)J(y)\right|$
(90) $\displaystyle\hskip 7.11317pt\leq\limsup_{N\to\infty}\left|\int_{E\times
E}\bar{\mathbf{L}}^{(N)}_{s}(\mathrm{d}y)\bar{\mathbf{L}}^{(N)}_{0}(\mathrm{d}x)\phi(x,y)J(y)-\int_{E\times
E}\bar{\mathbf{L}}^{(N)}_{s}(\mathrm{d}y)\mu(\mathrm{d}x)\phi(x,y)J(y)\right|$
(91) $\displaystyle\hskip 28.45274pt+\limsup_{N\to\infty}\left|\int_{E\times
E}\bar{\mathbf{L}}^{(N)}_{s}(\mathrm{d}y)\mu(\mathrm{d}x)\phi(x,y)J(y)-\int_{E\times
E}\bar{\mathbf{L}}^{*}_{s}(\mathrm{d}y)\mu(\mathrm{d}x)\phi(x,y)J(y)\right|.$
(92)

The second term in the upper bound of (89) is $0$, since the map
$y\mapsto\int_{E}\mu(\mathrm{d}x)\phi(x,y)J(y)$ is bounded and continuous
(because $J$ has compact support and $\phi$ is continuous), and
$\bar{\mathbf{L}}^{(N)}_{s}\rightarrow\bar{\mathbf{L}}^{*}_{s}$. On the other
hand, by applying Equation (19), with the compact set $C^{\prime}$ chosen to
be the support of $J$, for any $\varepsilon>0$, there exists $N_{0}$ such
that, for all $N\geq N_{0}$ we have

$\displaystyle\left|\int_{E\times
E}\bar{\mathbf{L}}^{(N)}_{s}(\mathrm{d}y)\bar{\mathbf{L}}^{(N)}_{0}(\mathrm{d}x)\phi(x,y)J(y)-\int_{E\times
E}\bar{\mathbf{L}}^{(N)}_{s}(\mathrm{d}y)\mu(\mathrm{d}x)\phi(x,y)J(y)\right|$
$\displaystyle\hskip
128.0374pt\leq\varepsilon\left|\int_{E}\bar{\mathbf{L}}^{(N)}_{s}(\mathrm{d}y)J(y)\right|,$

and thus, taking limits superior of both sides, since $J$ is bounded and
continuous,

$\displaystyle\limsup_{N\to\infty}\left|\int_{E\times
E}\bar{\mathbf{L}}^{(N)}_{s}(\mathrm{d}y)\bar{\mathbf{L}}^{(N)}_{0}(\mathrm{d}x)\phi(x,y)J(y)-\int_{E\times
E}\bar{\mathbf{L}}^{(N)}_{s}(\mathrm{d}y)\mu(\mathrm{d}x)\phi(x,y)J(y)\right|$
$\displaystyle\hskip
142.26378pt\leq\varepsilon\left|\int_{E}\bar{\mathbf{L}}^{*}_{s}(\mathrm{d}y)J(y)\right|.$

Sending $\varepsilon\to 0$, we deduce that the first term in the upper bound
of (89) is also $0$, hence conclude the proof of (69). ∎

###  3.2 Uniqueness: proof of Theorem 2.4

###### Proof of Theorem 2.4.

Suppose that $(\mu_{s})_{s\geq 0}$ and $(\hat{\mu}_{s})_{s\geq 0}$ denote two
solutions to the Flory equation, with a given initial condition $\mu_{0}$,
with $\|\mu_{0}\|<\infty$. Suppose that
$\left(\mu_{s}-\hat{\mu}_{s}\right)|_{D_{R}}$ denotes the measure
$\left(\mu_{s}-\hat{\mu}_{s}\right)$ restricted to $D_{R}$. By a well-known
property of the total variation distance, we may write

$\left\|\left(\mu_{s}-\hat{\mu}_{s}\right)|_{D_{R}}\right\|=\sup_{f:\|f\|_{\infty}=1}\left\langle
f\mathbf{1}_{D_{R}},\left(\mu_{s}-\hat{\mu}_{s})\right)\right\rangle.$

Note that, by a straightforward approximation argument (approximating a
measurable function pointwise by continuous functions), if $(\mu_{t})_{t\geq
0}$ is a solution to the multi-type Flory equation as in Definition 2.1,
Equation (10) is satisfied for all bounded measurable functions supported on
the compact set $D_{k}$, for $k\in\mathbb{N}$. Thus, for $f$ such that
$\|f\|_{\infty}=1$, we have

$\displaystyle\left\langle
f\mathbf{1}_{D_{R}},\left(\mu_{s}-\hat{\mu}_{s})\right)\right\rangle$ (93)
$\displaystyle\hskip 28.45274pt=\frac{1}{2}\int_{0}^{s}\int_{E\times E\times
E}f(z)\mathbf{1}_{D_{R}}(z)K(u,v,\mathrm{d}z)\left(\mu_{r}(\mathrm{d}u)\mu_{r}(\mathrm{d}v)-\hat{\mu}_{r}(\mathrm{d}u)\hat{\mu}_{r}(\mathrm{d}v)\right)\mathrm{d}r$
(94) $\displaystyle\hskip 56.9055pt-\int_{0}^{s}\int_{E\times
E}f(u)\mathbf{1}_{D_{R}}(u)\bar{K}(u,v)\left(\mu_{r}(\mathrm{d}u)\mu_{r}(\mathrm{d}v)-\hat{\mu}_{r}(\mathrm{d}u)\hat{\mu}_{r}(\mathrm{d}v)\right)\mathrm{d}r$
(95) $\displaystyle\hskip 56.9055pt-\int_{0}^{s}\int_{E\times
E}f(u)\mathbf{1}_{D_{R}}(u)\phi(u,v)\left(\mu_{r}(\mathrm{d}u)\mu_{0}(\mathrm{d}v)-\hat{\mu}_{r}(\mathrm{d}u)\mu_{0}(\mathrm{d}v)\right)\mathrm{d}r$
(96) $\displaystyle\hskip 56.9055pt+\int_{0}^{s}\int_{E\times
E}f(u)\mathbf{1}_{D_{R}}(u)\phi(u,v)\left(\mu_{r}(\mathrm{d}u)\mu_{r}(\mathrm{d}v)-\hat{\mu}_{r}(\mathrm{d}u)\hat{\mu}_{r}(\mathrm{d}v)\right)\mathrm{d}r.$
(97)

We now bound the values of each of the terms in the above display. For the
first, since $\phi$ is conservative, recalling the definition of $D_{R}$ in
(22), we have
$\mathbf{1}_{D_{R}}(z)\leq\mathbf{1}_{D_{R}}(u)\mathbf{1}_{D_{R}}(v)$ for
$K(u,v,\mathrm{d}z)$-a.a. $z$. Moreover, bounding $f(z)$ above by $1$, we
obtain

$\displaystyle\frac{1}{2}\int_{0}^{s}\int_{E\times E\times
E}f(z)\mathbf{1}_{D_{R}}(z)K(u,v,\mathrm{d}z)\left(\mu_{r}(\mathrm{d}u)\mu_{r}(\mathrm{d}v)-\hat{\mu}_{r}(\mathrm{d}u)\hat{\mu}_{r}(\mathrm{d}v)\right)\mathrm{d}r$
$\displaystyle\leq\frac{1}{2}\int_{0}^{s}\int_{E\times
E}\mathbf{1}_{D_{R}}(u)\mathbf{1}_{D_{R}}(v)\bar{K}(u,v)\left|\mu_{r}(\mathrm{d}u)\mu_{r}(\mathrm{d}v)-\hat{\mu}_{r}(\mathrm{d}u)\hat{\mu}_{r}(\mathrm{d}v)\right|\mathrm{d}r.$
(98)

We now have the following claim:

######  Claim 3.3.3.

We have

$\displaystyle\int_{E\times
E}\mathbf{1}_{D_{R}}(u)\mathbf{1}_{D_{R}}(v)\phi(u,v)\left|\mu_{r}(\mathrm{d}u)\mu_{r}(\mathrm{d}v)-\hat{\mu}_{r}(\mathrm{d}u)\hat{\mu}_{r}(\mathrm{d}v)\right|$
(99) $\displaystyle\hskip 85.35826pt\leq
2R\int_{E}\mathbf{1}_{D_{R}}(v)\left|\mu_{r}(\mathrm{d}v)-\hat{\mu}_{r}(\mathrm{d}v)\right|.$
(100)

By applying Claim 3.3.3, and bounding $\bar{K}(x,y)\leq c^{\prime}\phi(x,y)$,
we may now bound the right-side of (3.2):

$\displaystyle\frac{1}{2}\int_{0}^{s}\int_{E\times
E}\mathbf{1}_{D_{R}}(u)\mathbf{1}_{D_{R}}(v)\bar{K}(u,v)\left|\mu_{r}(\mathrm{d}u)\mu_{r}(\mathrm{d}v)-\hat{\mu}_{r}(\mathrm{d}u)\hat{\mu}_{r}(\mathrm{d}v)\right|\mathrm{d}r$
(101) $\displaystyle\hskip
28.45274pt\leq\frac{c^{\prime}}{2}\int_{0}^{s}\int_{E\times
E}\mathbf{1}_{D_{R}}(u)\mathbf{1}_{D_{R}}(v)\phi(u,v)\left|\mu_{r}(\mathrm{d}u)\mu_{r}(\mathrm{d}v)-\hat{\mu}_{r}(\mathrm{d}u)\hat{\mu}_{r}(\mathrm{d}v)\right|\mathrm{d}r$
(102) $\displaystyle\hskip 28.45274pt\leq
c^{\prime}\int_{0}^{s}R\int_{E}\mathbf{1}_{D_{R}}(v)\left|\mu_{r}(\mathrm{d}v)-\hat{\mu}_{r}(\mathrm{d}v)\right|\mathrm{d}r\leq
c^{\prime}R\int_{0}^{s}\left\|\left(\mu_{r}-\hat{\mu}_{r}\right)|_{D_{R}}\right\|\mathrm{d}r.$
(103)

Next, re-writing, and combining the second and fourth terms in (93), recalling
that $\phi(x,y)$ coincides with $\bar{K}(x,y)$ on $(D_{R}\times D_{R})^{c}$
(so in particular $D_{R}\times D_{R}^{c}$) we get

$\displaystyle\int_{0}^{s}\int_{E\times
E}f(u)\mathbf{1}_{D_{R}}(u)\left(\phi(u,v)-\bar{K}(u,v)\right)\left(\mu_{r}(\mathrm{d}u)\mu_{r}(\mathrm{d}v)-\hat{\mu}_{r}(\mathrm{d}u)\hat{\mu}_{r}(\mathrm{d}v)\right)\mathrm{d}r$
(104) $\displaystyle=\int_{0}^{s}\int_{E\times
E}f(u)\mathbf{1}_{D_{R}}(u)\mathbf{1}_{D_{R}}(v)\left(\phi(u,v)-\bar{K}(u,v)\right)$
(105) $\displaystyle\hskip
199.16928pt\times\left(\mu_{r}(\mathrm{d}u)\mu_{r}(\mathrm{d}v)-\hat{\mu}_{r}(\mathrm{d}u)\hat{\mu}_{r}(\mathrm{d}v)\right)\mathrm{d}r$
(106) $\displaystyle\hskip
28.45274pt\leq(c^{\prime}+1)\int_{0}^{s}\int_{E\times
E}\phi(u,v)\mathbf{1}_{D_{R}}(u)\mathbf{1}_{D_{R}}(v)\left|\mu_{r}(\mathrm{d}u)\mu_{r}(\mathrm{d}v)-\hat{\mu}_{r}(\mathrm{d}u)\hat{\mu}_{r}(\mathrm{d}v)\right|\mathrm{d}r$
(107) $\displaystyle\hskip 28.45274pt\stackrel{{\scriptstyle\eqref{eq:clm-
cross-
terms}}}{{\leq}}2R(c^{\prime}+1)\int_{0}^{s}\int_{E}\mathbf{1}_{D_{R}}(v)\left|\mu_{r}(\mathrm{d}v)-\hat{\mu}_{r}(\mathrm{d}v)\right|\mathrm{d}r$
(108) $\displaystyle\hskip
28.45274pt=2R(c^{\prime}+1)\int_{0}^{s}\left\|\left(\mu_{r}-\hat{\mu}_{r}\right)|_{D_{R}}\right\|\mathrm{d}r,$
(109)

where in the second to last inequality, we use the bound $\bar{K}(u,v)\leq
c^{\prime}\phi(u,v)$. Finally, for the third term in (93), observing that
$\mu_{0}$ is a positive measure, we make a similar computation:

$\displaystyle-\int_{0}^{s}\int_{E\times
E}f(u)\mathbf{1}_{D_{R}}(u)\phi(u,v)\left(\mu_{r}(\mathrm{d}u)\mu_{0}(\mathrm{d}v)-\hat{\mu}_{r}(\mathrm{d}u)\mu_{0}(\mathrm{d}v)\right)\mathrm{d}r$
(110) $\displaystyle\hskip 85.35826pt\leq\int_{0}^{s}\int_{E\times
E}\mathbf{1}_{D_{R}}(u)\phi(u,v)\left|\mu_{r}(\mathrm{d}u)\mu_{0}(\mathrm{d}v)-\hat{\mu}_{r}(\mathrm{d}u)\mu_{0}(\mathrm{d}v)\right|\mathrm{d}r$
(111) $\displaystyle\hskip
85.35826pt=\int_{0}^{s}\int_{E}\mathbf{1}_{D_{R}}(u)\int_{E}\phi(u,v)\mu_{0}(\mathrm{d}v)\left|\mu_{r}(\mathrm{d}u)-\hat{\mu}_{r}(\mathrm{d}u)\right|\mathrm{d}r$
(112) $\displaystyle\hskip 85.35826pt\leq
R\int_{0}^{s}\left\|\left(\mu_{r}-\hat{\mu}_{r}\right)|_{D_{R}}\right\|\mathrm{d}r$
(113)

Combining Equations (101), (109) and (113), to bound (93) we deduce that

$\displaystyle\left\|\left(\mu_{s}-\hat{\mu}_{s}\right)|_{D_{R}}\right\|=\sup_{f:\|f\|_{\infty}=1}\left\langle
f\mathbf{1}_{D_{R}},\left(\mu_{s}-\hat{\mu}_{s})\right)\right\rangle\leq
3R(c^{\prime}+1)\int_{0}^{s}\left\|\left(\mu_{r}-\hat{\mu}_{r}\right)|_{D_{R}}\right\|\mathrm{d}r.$

######  Claim 3.3.4.

Suppose that $(\mu_{t})_{t\geq 0}$ is a solution to the multi-type Flory
equation. Then, if $\xi:E\rightarrow\mathbb{R}_{+}$ is a sub-conservative
function, for each $t\geq 0$

$\int_{E}\xi(x)\mu_{t}(\mathrm{d}x)\leq\int_{E}\xi(x)\mu_{0}(\mathrm{d}x).$
(114)

By Claim 3.3.4 applied to the function $\xi(x)\equiv 1$, we know that for each
$s\geq 0$, we have

$\left\|\left(\mu_{s}-\hat{\mu}_{s}\right)|_{D_{R}}\right\|\leq 2\|\mu_{0}\|.$

We can thus apply Gronwall’s lemma to deduce that
$\left\|\left(\mu_{s}-\hat{\mu}_{s}\right)|_{D_{R}}\right\|=0$. As
$\bigcup_{k\in\mathbb{N}}D_{k}=E$, it must be the case that
$\left\|\mu_{s}-\hat{\mu}_{s}\right\|=0$, showing uniqueness. ∎

We finish with the proofs of Claim 3.3.3 and Claim 3.3.4:

###### Proof of Claim 3.3.3.

We bound

$\displaystyle\int_{E\times
E}\mathbf{1}_{D_{R}}(u)\mathbf{1}_{D_{R}}(v)\phi(u,v)\left|\mu_{r}(\mathrm{d}u)\mu_{r}(\mathrm{d}v)-\hat{\mu}_{r}(\mathrm{d}u)\hat{\mu}_{r}(\mathrm{d}v)\right|$
(115) $\displaystyle\hskip 42.67912pt\leq\int_{E\times
E}\mathbf{1}_{D_{R}}(u)\mathbf{1}_{D_{R}}(v)\phi(u,v)\left|\mu_{r}(\mathrm{d}u)\mu_{r}(\mathrm{d}v)-\mu_{r}(\mathrm{d}u)\hat{\mu}_{r}(\mathrm{d}v)\right|$
(116) $\displaystyle\hskip 85.35826pt+\int_{E\times
E}\mathbf{1}_{D_{R}}(u)\mathbf{1}_{D_{R}}(v)\phi(u,v)\left|\mu_{r}(\mathrm{d}u)\hat{\mu}_{r}(\mathrm{d}v)-\hat{\mu}_{r}(\mathrm{d}u)\hat{\mu}_{r}(\mathrm{d}v)\right|.$
(117)

For the first term on the right-hand side of (115), we integrate the variable
$u$, applying (11)

$\displaystyle\int_{E\times
E}\mathbf{1}_{D_{R}}(u)\mathbf{1}_{D_{R}}(v)\phi(u,v)\left|\mu_{r}(\mathrm{d}u)\mu_{r}(\mathrm{d}v)-\mu_{r}(\mathrm{d}u)\hat{\mu}_{r}(\mathrm{d}v)\right|\mathrm{d}r$
$\displaystyle\hskip
28.45274pt\leq\int_{E}\mathbf{1}_{D_{R}}(v)\int_{E}\phi(u,v)\mu_{r}(\mathrm{d}u)\left|\mu_{r}(\mathrm{d}v)-\hat{\mu}_{r}(\mathrm{d}v)\right|\mathrm{d}r$
$\displaystyle\hskip
28.45274pt\leq\int_{E}\mathbf{1}_{D_{R}}(v)\int_{E}\phi(u,v)\mu_{0}(\mathrm{d}u)\left|\mu_{r}(\mathrm{d}v)-\hat{\mu}_{r}(\mathrm{d}v)\right|\mathrm{d}r$
$\displaystyle\hskip 28.45274pt\leq
R\int_{E}\mathbf{1}_{D_{R}}(v)\left|\mu_{r}(\mathrm{d}v)-\hat{\mu}_{r}(\mathrm{d}v)\right|\mathrm{d}r;$

and applying a similar argument for the second term in (115), we deduce the
result. ∎

###### Proof of Claim 3.3.4.

Suppose first that $\xi$ is bounded. Then, applying (10) to the function
$\xi\mathbf{1}_{D_{k}}$, we have

$\displaystyle\left\langle\xi\mathbf{1}_{D_{k}},\mu_{s}-\mu_{0}\right\rangle$
(118) $\displaystyle=\frac{1}{2}\int_{0}^{s}\int_{E\times E\times
E}\left(\xi(z)\mathbf{1}_{D_{k}}(z)-\xi(u)\mathbf{1}_{D_{k}}(u)-\xi(v)\mathbf{1}_{D_{k}}(v)\right)K(u,v,\mathrm{d}z)\mu_{r}(\mathrm{d}u)\mu_{r}(\mathrm{d}v)\mathrm{d}r$
(119) $\displaystyle\hskip 85.35826pt+\int_{0}^{s}\int_{E\times
E}\xi(u)\mathbf{1}_{D_{k}}(u)\phi(u,v)\left(\mu_{r}(\mathrm{d}u)(\mu_{r}(\mathrm{d}v)-\mu_{0}(\mathrm{d}v))\right)\mathrm{d}r.$
(120)

By (11), since $\xi(u)\geq 0$, and $\mu_{r}$ is a positive measure, we deduce
that the second term on the right-side above is non-positive. In addition,
since $\phi$ is conservative, so is the function
$x\mapsto\left\langle\phi(x,\cdot),\mu_{0}\right\rangle$, and we deduce that
$\mathbf{1}_{D_{R}}(z)\leq\mathbf{1}_{D_{R}}(u)\mathbf{1}_{D_{R}}(v)$ for
$K(u,v,\mathrm{d}z)$-a.a. $z$. Finally, since $\xi$ is sub-conservative,

$\displaystyle\int_{E\times E\times
E}\left(\xi(z)\mathbf{1}_{D_{k}}(z)\right)K(u,v,\mathrm{d}z)\mu_{r}(\mathrm{d}u)\mu_{r}(\mathrm{d}v)$
$\displaystyle\hskip 56.9055pt\leq\int_{E\times E\times
E}\left(\xi(u)+\xi(v)\right)\mathbf{1}_{D_{k}}(u)\mathbf{1}_{D_{k}}(u)K(u,v,\mathrm{d}z)\mu_{r}(\mathrm{d}u)\mu_{r}(\mathrm{d}v)$
$\displaystyle\hskip 56.9055pt\leq\int_{E\times E\times
E}\left(\xi(u)\mathbf{1}_{D_{k}}(u)+\xi(v)\mathbf{1}_{D_{k}}(v)\right)K(u,v,\mathrm{d}z)\mu_{r}(\mathrm{d}u)\mu_{r}(\mathrm{d}v).$

Thus,
$\left\langle\xi\mathbf{1}_{D_{k}},\mu_{s}\right\rangle\leq\left\langle\xi\mathbf{1}_{D_{k}},\mu_{0}\right\rangle$,
and we deduce the result from monotone convergence. Finally, we can extend the
result to unbounded $\xi$, again from monotone convergence (approximating
$\xi$ from below by the sub-conservative functions $\xi\wedge j$, for
$j\in\mathbb{N}$). ∎

##  Appendix A General criteria for relative compactness

Recall that for each $N$, the cluster coagulation process
$(\bar{\mathbf{L}}^{(N)}_{t})_{t\in[0,\infty)}$ is defined as taking values in
the space

$\mathcal{E}=\bigcup_{n\in\mathbb{N}}\left\\{\mathbf{u}\in\mathcal{M}_{+}(\mathcal{E}):\langle
m,\mathbf{u}\rangle\leq n\right\\}$

Recall also that equip $\mathcal{E}$ with the Prokhorov metric, which metrises
the topology of weak convergence. We may interpret
$(\bar{\mathbf{L}}_{t})_{t\in[0,\infty)}$ as a trajectory in
$D([0,\infty);\mathcal{E})$, the space of right-continuous functions
$f:[0,\infty)\rightarrow\mathcal{E}$ with left-limits. We equip
$D([0,\infty);\mathcal{E})$ with the _Skorokhod metric_ $d$. Recall that for a
separable, complete metric space $(\mathcal{E},\delta)$ with $q:=\delta\wedge
1$, the Skorokhod metric on $D([0,\infty);\mathcal{E})$ is defined as follows:
Let $\Lambda$ denotes the set of all strictly increasing functions mapping
$[0,\infty)$ onto $[0,\infty)$, and $\Lambda^{\prime}\subseteq\Lambda$ the
subset of Lipschitz functions. Then, for $\lambda\in\Lambda^{\prime}$, define

$\gamma(\lambda):=\sup_{s>t\geq
0}\left|\log{\frac{\lambda(s)-\lambda(t)}{s-t}}\right|<\infty$

Then, for $f,g\in D([0,\infty);\mathcal{E})$, we define

$d(f,g):=\inf_{\lambda\in\Lambda}\left(\gamma(\lambda)\vee\int_{0}^{\infty}e^{-tu}\left(\sup_{t\geq
0}q(f(t\wedge u),g(t\wedge u))\right)\mathrm{d}u\right).$ (121)

It is well-established that $D([0,\infty);\mathbb{R})$ is a separable and
complete metric space see, for example, [4, Theorem 5.6]. In this paper, we
use the following, well-known criterion for tightness in Skorokhod spaces. The
first, from [4], has been slightly reformulated for out purposes. First, we
define the following modulus of continuity: for $f\in
D([0,\infty);\mathbb{R})$, $\eta>0$, $T\in[0,\infty)$, we define

$w^{\prime}(f,\eta,T):=\inf_{\\{t_{i}\\}}\max_{i}\sup_{s,t\in[t_{i-1},t_{i})}\left|f(s)-f(t)\right|;$

where $\\{t_{i}\\}$ ranges over all partitions of $[0,T]$, such that
$0=t_{0}<t_{1}<\cdots<t_{n}=T$, with $t_{i+1}-t_{i}>\eta$ and $n\geq 1$.

######  Lemma A.1 ([4, Corollary 7.4, page 129]).

A collection of probability measures
$\left\\{\mu_{n}\right\\}_{n\in\mathbb{N}}$ on the metric space
$D([0,\infty);\mathbb{R})$ is tight if and only if the following criteria are
satisfied:

  1. 1.

For all $t\in[0,\infty)\cap\mathbb{Q}$ and $\varepsilon>0$, there exists a
compact set $K(t,\varepsilon)\subseteq\mathcal{E}$ such that, for all
$n\in\mathbb{N}$

$\liminf_{n\to\infty}\mu_{n}\left(\left\\{f:f(t)\in
K(t,\varepsilon)\right\\}\right)\geq 1-\varepsilon,$ (122)

  2. 2.

For any $T\in[0,\infty)$, there exists $\eta>0$ such that, for all
$n\in\mathbb{N}$

$\lim_{\eta\to
0}\limsup_{n\to\infty}\mu_{n}\left(\left\\{f:w^{\prime}(f,\eta,T)\geq\varepsilon\right\\}\right)=0.$
(123)

$\blacksquare$

In literature surrounding stochastic processes, the first condition is often
known as _compact containment_. The following well-known tightness criterion
due to Jakubowski applies more generally to $D([0,\infty);F)$, where $F$ is a
completely regular Hausdorff spaces with metrisable compacts. Since we assume
$\mathcal{E}$ is a metric space, it applies to $D([0,\infty);\mathcal{E})$:

######  Lemma A.2 ([11, Theorem 4.6]).

A collection of probability measures $\left\\{\mu_{i}\right\\}_{i\in I}$ on
$D([0,\infty);\mathcal{E})$ is tight if and only if the following criteria are
satisfied:

  1. 1.

For any $t>0$ and $\varepsilon>0$ there is a compact set
$K(t,\varepsilon)\subseteq\mathcal{E}$ such that, for all $i\in I$,

$\mu_{i}\left(\left\\{f:\forall\,s\in[0,t]\,f(s)\in
K(t,\varepsilon)\right\\}\right)\geq 1-\varepsilon.$

  2. 2.

There exists a family of continuous functions $\mathbb{F}$ from $\mathcal{E}$
to $\mathbb{R}$ such that

    1. (a)

The family $\mathbb{F}$ separates points, i.e., for any $x,y\in\mathcal{E}$
there exists $f\in\mathbb{F}$ such that $f(x)\neq f(y)$.

    2. (b)

The family $\mathbb{F}$ is closed under addition, i.e., if $f,g\in\mathbb{F}$
then $f+g\in\mathbb{F}$.

    3. (c)

Let, for $f\in\mathbb{F}$, $\tilde{f}:D([0,\infty);\mathcal{E})\rightarrow
D([0,\infty);\mathbb{R})$ denote the map such that $\tilde{f}(x)=f\circ x$,
for $x\in\leavevmode\nobreak\ D([0,\infty);\mathcal{E})$. Then, for each
$f\in\mathbb{F}$ the family of pushforward measures
$\left\\{\tilde{f}_{*}(\mu_{i})\right\\}_{i\in I}$ is a tight family on
$D([0,\infty);\mathbb{R})$.

$\blacksquare$

Acknowledgements. We would like to thank Robert Patterson and Wolfgang König
for some helpful discussions. This research has been partially funded by the
Deutsche Forschungsgemeinschaft (DFG) (project number 443759178) through grant
SPP2265 “Random Geometric Systems”, Project P01: ‘Spatial Coagulation and
Gelation’. The authors acknowledge the support from Dipartimento di Matematica
e Informatica “Ulisse Dini”, University of Florence. LA acknowledges partial
support from “Indam-GNAMPA” Project CUP_E53C23001670001.

## References

  * [1] D. J. Aldous.  Deterministic and stochastic models for coalescence (aggregation and coagulation): a review of the mean-field theory for probabilists.  Bernoulli, 5(1):3–48, 1999.
  * [2] L. Andreis, T. Iyer, and E. Magnanini.  Gelation in cluster coagulation processes.  arXiv preprint arXiv:2308.10232v2, 2024.
  * [3] A. Eibeck and W. Wagner.  Stochastic particle approximations for Smoluchoski’s coagulation equation.  Ann. Appl. Probab., 11(4):1137–1165, 2001.
  * [4] S. N. Ethier and T. G. Kurtz.  Markov processes.  Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. John Wiley & Sons, Inc., New York, 1986.  Characterization and convergence.
  * [5] M. Ferreira, J. Lukkarinen, A. Nota, and J. J. L. Velàzquez.  Asymptotic localization in multicomponent mass conserving coagulation equations.  arXiv preprint arXiv:2203.08076, 2022.
  * [6] M. A. Ferreira, J. Lukkarinen, A. Nota, and J. J. L. Velázquez.  Localization in stationary non-equilibrium solutions for multicomponent coagulation systems.  Communications in Mathematical Physics, 388(1):479–506, 2021.
  * [7] N. Fournier and J.-S. Giet.  Convergence of the Marcus-Lushnikov process.  Methodol. Comput. Appl. Probab., 6(2):219–231, 2004.
  * [8] D. T. Gillespie.  The stochastic coalescence model for cloud droplet growth.  J. Atmos. Sci., 29:1496–1510, 1972.
  * [9] D. Heydecker and R. I. A. Patterson.  Bilinear coagulation equations.  arXiv preprint arXiv:1902.07686, 2019.
  * [10] S. Jacquot.  A historical law of large numbers for the Marcus-Lushnikov process.  Electron. J. Probab., 15:no. 19, 605–635, 2010.
  * [11] A. Jakubowski.  On the Skorokhod topology.  Ann. Inst. H. Poincaré Probab. Statist., 22(3):263–285, 1986.
  * [12] I. Jeon.  Existence of gelling solutions for coagulation-fragmentation equations.  Comm. Math. Phys., 194(3):541–567, 1998.
  * [13] A. Lushnikov.  Some new aspects of coagulation theory.  Izv. Acad. Sci. USSR, Ser. Phys. Atmos. Oceans, 14, 01 1978.
  * [14] A. H. Marcus.  Stochastic coalescence.  Technometrics, 10:133–143, 1968.
  * [15] J. Norris.  Measure solutions for the Smoluchowski coagulation-diffusion equation.  arXiv preprint arXiv:1408.5228, 2014.
  * [16] J. R. Norris.  Smoluchowski’s coagulation equation: uniqueness, nonuniqueness and a hydrodynamic limit for the stochastic coalescent.  The Annals of Applied Probability, 9(1):78 – 109, 1999.
  * [17] J. R. Norris.  Cluster coagulation.  Comm. Math. Phys., 209(2):407–435, 2000.
  * [18] D. Pollard.  Convergence of stochastic processes.  Springer Series in Statistics. Springer-Verlag, New York, 1984.
  * [19] D. Revuz and M. Yor.  Continuous martingales and Brownian motion, volume 293 of Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences].  Springer-Verlag, Berlin, third edition, 1999.
  * [20] F. Rezakhanlou.  Gelation for Marcus-Lushnikov process.  Ann. Probab., 41(3B):1806–1830, 2013.
  * [21] S. Throm.  Uniqueness of measure solutions for multi-component coagulation equations.  arXiv preprint arXiv:2303.00775, 2023.
  * [22] W. Wagner.  Explosion phenomena in stochastic coagulation-fragmentation models.  Ann. Appl. Probab., 15(3):2081–2112, 2005.