# A growth-fragmentation-isolation process on random recursive trees and
contact tracing

Vincent<EMAIL_ADDRESS>[ Chenlin
<EMAIL_ADDRESS>[ Linglong
<EMAIL_ADDRESS>[ CMAP, Ecole polytechnique, IPP,
Mathematics Department, NYU Shanghai & NYU-ECNU Institute of Mathematical
Sciences,  Yau Mathematical Sciences Center, Tsinghua University,  Department
of Mathematical Sciences, University of Liverpool,

###### Abstract

We consider a random process on recursive trees, with three types of events.
Vertices give birth at a constant rate (growth), each edge may be removed
independently (fragmentation of the tree) and clusters (or trees) are frozen
with a rate proportional to their sizes (isolation of connected component). A
phase transition occurs when the isolation is able to stop the growth
fragmentation process and cause extinction. When the process survives, the
number of clusters increases exponentially and we prove that the normalized
empirical measure of clusters a.s. converges to a limit law on recursive
trees. We exploit the branching structure associated to the size of clusters,
which is inherited from the splitting property of random recursive trees. This
work is motivated by the control of epidemics and contact tracing where
clusters correspond to trees of infected individuals that can be identified
and isolated. We complement this work by providing results on the Malthusian
exponent to describe the effect of control policies on epidemics.

60J27, 60J85, 60J80,

branching process, many-to-two formula, Kesten–Stigum theorem, non-
conservative semigroup, random recursive tree, strong law of large numbers,
epidemic, contact tracing,

###### keywords:

[class=MSC]

###### keywords:

, and

##  1 Introduction

The evolution of random trees is motivated by various fields: algorithms,
queuing systems, population modeling, etc. The random deletion of edges of a
tree has been studied in particular by [31, 10]. Initially, Meir and Moon [31]
were interested in the number of steps needed to isolate a distinguished
vertex in a random recursive tree, when every deleted edge is chosen
uniformly. Bertoin [11] and Marzouk [30] have then studied processes where
sets of vertices can be burnt. More precisely, a connected component of the
graph is removed (i.e. isolated) at each step. This component is determined by
a uniform choice among the vertices. Such dynamics combine the fragmentation
of the tree (when an edge is deleted) and the isolation of connected
components of the tree (when a vertex provokes a fire).

In this work, we are interested in the long-time behavior of similar dynamics
when the random recursive tree grows, following a binary branching process.
Our original motivation is to study the effects of control policies of an
epidemic. The growth of the infected population is modeled by a Yule process
in this work, i.e. a binary Markov branching process. The discrete structure
of the Yule process is a random recursive tree. The connected new vertices are
the new infected individuals. The fragmentation occurs when one edge is
removed, interpreted as the loss of infector-infectee information: tracing of
this contact becomes impossible. Various reasons may explain the loss or
absence of information on contacts, including memory and storage of
information or the fact that the contact is not accessible. We restrict
ourselves here to a simple model with a single parameter accounting for a rate
at which contacts (edges) get lost (removed) independently. Growth and
fragmentation will generate connected components, called clusters. Each
cluster is a set of connected infected individuals that can be isolated all
together as soon as one individual in the cluster is detected. An isolated
cluster is frozen, in the sense that no more event will happen to it. It is
interpreted as that all the individuals in the cluster are either under
treatment or self-isolating. This is the contact tracing strategy which has
been for decades the central public health response to control infectious
disease outbreaks.

Numerous research papers highlight the importance of contact-tracing for
controlling epidemics, using either simulations or real data, and we mention
[25, 19, 17] for Covid-19. For mathematical modeling, we refer to [28, 1, 9,
13] for related studies which are motivated by contact tracing and exploit a
branching structure. Let us describe the main differences. In [28], the author
provides explosion criterion (threshold for $R_{0}$, the reproduction number),
for different levels of tracing. The work [1] is more general regarding the
characteristics of epidemics but does not allow for backward tracing (i.e.
tracing and isolation of ancestors when a descendant is identified and
isolated). In [9], a discrete-time model is studied and a Malthusian behavior
is exhibited, with different methods and results. Here we exploit random
recursive trees and the growth fragmentation structure. Our model allows to
study the effect of tracing along time, in a Markovian dynamical way. It
allows to keep a simple description of the probabilistic structure at any
time, to achieve fine asymptotic analysis and to make emerge tractable key
quantities for epidemics (by theory or simulations). In particular, we will
exhibit a Malthusian coefficient which drives the main features of the
epidemics, more precisely the speed of explosion or extinction (depending on
its sign). We will see how it changes with the three parameters of the model,
which allows to see the respective effect of social distancing measures
(increase of infection rate), detection effort (increase of $\theta$) and
tracing effort (decrease of $\gamma$). The role of $\beta$ happens to be quite
surprising for us and somewhat reinforce the interest of “isolation-tracing”
strategy compared to social distancing. Finally, in [13] a model related to
ours is considered, where the contact information can be lost when infection
occurs (and not later). In this model, fragmentation disappears and more
explicit eigenelements and criterion for explosion are given. Combining both
effects (instantaneous or continuous loss of contact) would be interesting.

Figure 1: An illustration of the growth-fragmentation-isolation process with
62 active vertices (in red) and 77 inactive vertices (in blue). A cluster (or
a connected component, a tree) is formed by solid lines, while the dashed
lines split clusters.

Our growth-fragmentation-isolation model is thus a branching process to study
the effect of identification-tracing-isolation strategy in the context of
simple epidemics, with loss of contact information along time. The second
author also considered a similar model with numerical simulations in [20] to
study the propagation of Covid-19. The approximation of the outbreak by a
branching process is classical in the early stages of an epidemic, when the
whole population is large and the proportion of susceptible individuals is
close to one, see for instance [5]. We also obtain in this work estimations on
the speed of convergence of infected profiles, which allow to control in which
time window the Malthusian exponential growth indeed captures the dynamics of
the epidemics. There are more epidemiological and control features that could
be incorporated to reflect epidemics like Covid-19. In particular, no recovery
happens in our setting, and we consider only large homogeneously mixed
populations. We expect some extensions of our results on the long-time
behavior of infected population which would take into account these features,
even if the probabilistic structure of clusters becomes more complex. We give
a more detailed discussion in Section 8.3.

Random recursive trees (RRT) have a nice splitting property that allows us to
characterize a cluster by its size. More precisely, considering the collection
of active (non-isolated) clusters, the size process turns out to be a
branching process with a countable set of types (i.e. type being the size). At
fixed time, conditionally on the sizes of the clusters, the collection of
clusters are independent RRTs. We then study the ergodic properties of the
first moment semigroup of this branching process and obtain a phase transition
depending on the sign of the maximal eigenvalue (Malthusian growth rate). We
describe the a.s. behavior of the process when the active clusters survive,
proving a strong law of large numbers for the empirical measure of sizes and a
Kesten–Stigum type result for the growth of the population. With the knowledge
on the size process, we can deduce the a.s. behavior of the process of active
clusters conditioned on survival. We can also characterize the a.s. behavior
of the process of isolated clusters when the active clusters survive, which is
by itself non-Markovian.

The asymptotic analysis of the mean behavior, the weak convergence and the
estimation of the speed of convergence uses now well developed techniques for
branching processes with possibly infinitely many types. We refer to [12, 14,
7, 29, 34, 23] and references therein for related works on the asymptotic
analysis of growth fragmentation processes. Roughly, it relies on the ergodic
properties of the size of a typical cluster, and the fact that the common
ancestor of two clusters uniformly chosen at large times is found at small
times, see for instance Theorem 2 in [3], see also [22]. More precisely, we
exploit the fact that large clusters fragment fast, and with high probability
give one small cluster and one large cluster at fragmentation. Together with
isolation, it allows one to control the size of a typical cluster and the
eigenelements of the first moment semigroup. In particular, we prove that the
harmonic function is bounded and large clusters have no major impact on the
growth of epidemics. Indeed, large clusters are isolated before creating too
many small clusters, since isolation occurs at the same scale as
fragmentation. Once active clusters are well described, we can treat the
isolated clusters using an additive functional.

Besides, the fact that the number of types is infinite and the loss of Markov
property for the isolated clusters raise some technical difficulties to get
a.s. limits (strong convergence). We refer to [4, 2, 18] for classical
references on strong law of large numbers of some classes of multi-type
branching processes. We adapt here the argument of [4] for strong convergence.
In the context of growth fragmentation, let us mention respectively [15] and
[21, 23] for $L^{1}$ and strong convergence.

Let us describe the model more formally. We introduce a stochastic process on
a dynamic tree $G_{t}=(V_{t},E_{t})$ with two functions
$\Psi_{t}:V_{t}\to\\{0,1\\},\,\eta_{t}:E_{t}\to\\{0,1\\}.$

  * •

We identify the vertex set $V_{t}$ as the set of patients (individuals
infected up to time $t$), and label them with the infection time
$v\in[0,\infty)$. The function $\Psi_{t}$ tells us the state of a vertex at
time $t$, where vertex $v$ is active if $\Psi_{t}(v)=1$ and $v$ is inactive if
$\Psi_{t}(v)=0$. Only active vertices can infect new ones.

  * •

We identify the edge set $E_{t}$ as the set of (direct) infection links
between patients, and the function $\eta_{t}$ tells us the state of an edge at
time $t$. For an edge $e$, we say $e$ is open if $\eta_{t}(e)=1$ which means
the infection link can still be retrieved (i.e. it can be found who infected
the infectee), otherwise $\eta_{t}(e)=0$ and it is closed (i.e. it is not
possible to know who infected the infectee). A set of vertices connected by
open edges is called a cluster.

The growth-fragmentation-isolation model (GFI) process
$(G_{t},\Psi_{t},\eta_{t})_{t\geq 0}$ is a Markov jump process, starting from
an active vertex as patient zero $V_{0}=\\{0\\},\Psi_{0}(0)=1$, governed by
three positive parameters
$(\beta,\theta,\gamma)\in\mathbb{R}_{+}^{3}=(0,\infty)^{3}$ representing three
types of events, with the notation $\mathbb{P}$ for the probability and
$\mathbb{E}$ for the associated expectation.

  * •

_Infection_ (growth): every active vertex $v$ independently attaches a new
vertex in an exponential waiting time with parameter $\beta$. When a new
vertex $u$ is created and attached to $v$, it is active (i.e. $\Psi_{t}(u)=1$)
and the edge $\\{u,v\\}$ is open (i.e. $\eta_{t}(\\{u,v\\})=1$).

  * •

_Information decay_ (fragmentation): every open edge $e$ independently becomes
closed in an exponential waiting time with parameter $\gamma$.

  * •

_Confirmation and contact tracing_ (isolation): every active vertex
independently gets “confirmed” in an exponential waiting time with parameter
$\theta$, and once a vertex is confirmed, the associated cluster is isolated
and every vertex in this cluster becomes inactive.

See Figure 2 for an illustration of this model. If $\gamma=\theta=0$, this is
the well-known Yule tree process. As vertices are indexed by infection times,
every cluster is a labeled recursive tree (see Section 3.2 for a rigorous
definition).

(a) Growth: starting from vertex $0$, the vertices are attached one by one,
and form a recursive tree.

(b) Fragmentation: links get lost over time; for example the grey links
$\\{0,6\\},\\{1,4\\},\\{2,8\\}$ in the tree are lost.

(c) Isolation: when the vertex $2$ is confirmed, all the vertices in the same
cluster are isolated. These are the vertices in blue $\\{0,1,2,3,5,7\\}$.

(d) The isolated vertices are no longer active, while the active vertices
continue to attach new vertices.

Figure 2: An illustration of GFI process.

Because the vertices in a cluster have the same state, it is very natural to
decompose the dynamical tree $G_{t}$ into clusters of individuals connected by
open edges: for an isolated cluster, we call it inactive cluster; otherwise,
it is an active cluster. In this paper, we use isolated and inactive
interchangeably for clusters and also for vertices/patients/infected
individuals. We denote by $(\mathscr{X}_{t},\mathscr{Y}_{t})_{t\geq 0}$ the
associated cluster process, where $\mathscr{X}_{t}$ is the set of active
clusters and $\mathscr{Y}_{t}$ is the set of inactive clusters:

$\begin{split}\mathscr{X}_{t}&=\\{\mathcal{C}\,|\,\,\mathcal{C}\text{ is a
cluster in }G_{t};\,\forall v\in\mathcal{C},\Psi_{t}(v)=1\\},\\\
\mathscr{Y}_{t}&=\\{\mathcal{C}\,|\,\,\mathcal{C}\text{ is a cluster in
}G_{t};\,\forall v\in\mathcal{C},\Psi_{t}(v)=0\\}.\end{split}$

The process $(G_{t},\Psi_{t},\eta_{t})_{t\geq 0}$ stops evolving when
$\mathscr{X}_{t}$ is empty. We denote by $\tau$ the corresponding stopping
time:

$\displaystyle\tau:=\inf\\{t\geq 0\,|\,\mathscr{X}_{t}=\emptyset\\},$ (1)

and the event $\\{\tau<\infty\\}$ is called extinction, while
$\\{\tau=\infty\\}$ is called survival.

In this paper, we are motivated by the following questions. For what values of
the parameters $(\beta,\theta,\gamma)$ does the epidemic reach extinction
almost surely ? If the epidemic survives (with positive probability), what is
the long-time behavior of the population of active and inactive clusters ? We
give some answers to these questions by first determining the asymptotic
behavior of the first moment semigroup associated to the active clusters. It
depends on the maximal eigenvalue of this semigroup, which is called
Malthusian exponent. When this exponent is negative (subcritical case) or zero
(critical case), the population of active clusters reaches extinction almost
surely. It corresponds to the fact that the isolation process is strong enough
to stop the epidemic. When this exponent is positive (supercritical case),
survival occurs with positive probability and on this event, the growth of the
population is exponential with rate given by this Malthusian exponent. In that
case, we also shed some light on the genealogical structure of clusters and
describe the asymptotic behavior of the empirical distribution. We prove that
a.s. we get a collection of recursive trees whose sizes are distributed
following the left eigenvector associated to the maximal eigenvalue of the
semigroup, i.e. the Malthusian exponent. We also show similar asymptotic
behaviors for the inactive clusters on the survival event.

We give the main results and the outline of the paper in the next section. The
rest of the paper is dedicated to proofs and some additional results and
comments.

##  2 Main results

We first introduce the Malthusian exponent $\lambda$ which describes the
(mean) exponential growth (or decay) of the number of clusters. This growth
rate coincides for the active and inactive clusters, whose numbers at time $t$
are respectively denoted by $|\mathscr{X}_{t}|$ and $|\mathscr{Y}_{t}|$. As
expected in branching structures, its sign gives the global extinction and
survival criterion, leading to the classification of subcritical, critical and
supercritical phases.

######  Theorem 2.1 (Malthusian exponent).

The following limits exist and coincide and are finite

$\displaystyle\lambda:=\lim_{t\rightarrow\infty}\frac{1}{t}\log(\mathbb{E}[|\mathscr{X}_{t}|])=\lim_{t\rightarrow\infty}\frac{1}{t}\log(\mathbb{E}[|\mathscr{Y}_{t}|])\in(-\infty,\infty).$

If $\lambda\leq 0$, then extinction occurs a.s., i.e.
$\mathbb{P}[\tau<\infty]=1$. Otherwise, survival occurs with positive
probability, i.e. $\mathbb{P}[\tau=\infty]>0$.

The Malthusian exponent $\lambda$ corresponds to the maximal eigenvalue of the
first moment semigroup of $(X_{t})_{t\geq 0}$ and is also called Perron’s
root. The fact that the cluster size can be any positive integer leads us to
use techniques for ergodic behavior in infinite dimension, where the control
of large sizes is crucial. As usual, irreducibility on the state of sizes
ensures that the value $\lambda$ does not depend on the initial condition
(although the model starts with a single patient, i.e. a cluster of size $1$,
the setting of initial condition with a single random recursive tree of size
$n\geq 1$ will be used later, especially in the size processes). The diagram
of these different phases is illustrated in Figure 3, for fixed $\beta>0$.
Note that $\theta\geq\min(\beta,\gamma)$ implies a.s. extinction since in the
case $\theta\geq\beta$, individuals are isolated faster than they contaminate
and in the case $\theta\geq\gamma$, isolation is faster than fragmentation.

We want to emphasize that Theorem 2.1 and the forthcoming results do not
depend on the initial condition (random or deterministic), see the discussion
in Section 8.2. We also refer to Section 8.3 for other possible
generalizations with additional epidemic features.

Figure 3: An illustration of different phases.

To go further in the analysis of the model, we introduce the following size
process $(X_{t},Y_{t})_{t\geq 0}$, where the two empirical measures count the
clusters of different sizes

$X_{t}=\sum_{\mathcal{C}\in\mathscr{X}_{t}}\delta_{|\mathcal{C}|},\qquad
Y_{t}=\sum_{\mathcal{C}\in\mathscr{Y}_{t}}\delta_{|\mathcal{C}|}.$ (2)

Here $|\mathcal{C}|$ is the number of vertices in the cluster $\mathcal{C}$,
and we call it the size of cluster. The process $(X_{t})_{t\geq 0}$ is still a
branching Markov process with respect to its natural filtration. This comes
from the splitting property of RRT, which allows to preserve RRT clusters when
fragmentation occurs, see the forthcoming Proposition 3.1. Moreover, for any
fixed time, conditionally on cluster sizes, all (active and inactive) clusters
are independent, see Proposition 3.2. We can thus reduce the study of our GFI
process $(G_{t},\Psi_{t},\eta_{t})_{t\geq 0}$ to that of the size process
$(X_{t},Y_{t})_{t\geq 0}$; see Figure 4.

We prove the following strong law of large numbers in the supercritical case.
This provides the asymptotic behavior of $\langle
X_{t},f\rangle=\sum_{\mathcal{C}\in\mathscr{X}_{t}}f(|\mathcal{C}|)$, where
$f$ has at most polynomial growth, i.e. there exists $p>0$ such that
${\sup_{n\geq 1}|f(n)|/n^{p}<\infty}$. In particular, $f={\bf 1}_{m}$ yields
the number of active clusters of size $m$, while the identity function
provides the number of active individuals.

######  Theorem 2.2 (Law of large numbers for $(X_{t})_{t\geq 0}$).

Assume that $\lambda>0$. Then there exists a probability distribution $\pi$ on
$\mathbb{N}_{+}=\\{1,2,3,\cdots\\}$ and a random variable $W\geq 0$, such that
for any function $f:\mathbb{N}_{+}\to\mathbb{R}$ of at most polynomial growth,
we have $\left\langle{\pi,|f|}\right\rangle<\infty$ and

$\displaystyle e^{-\lambda t}\langle
X_{t},f\rangle\xrightarrow{t\to\infty}W\langle\pi,f\rangle,\qquad\text{ a.s.
and in }L^{2}.$ (3)

Besides, $\\{\tau=\infty\\}=\\{W>0\\}$ a.s. and on this event

$\displaystyle\frac{\langle X_{t},f\rangle}{\langle
X_{t},\mathbf{1}\rangle}\xrightarrow{t\to\infty}\langle\pi,f\rangle\quad\text{a.s.}.$
(4)

Recall that $\lambda$ is defined in Theorem 2.1. It gives the a.s. exponential
growth of the number of active clusters and active individuals. The
probability distribution $\pi$ gives the distribution of the size of clusters
for large times. This distribution is the (positive normalized) left
eigenvector of the first moment semigroup $M=(M_{t})_{t\geq 0}$ associated to
the size process $(X_{t})_{t\geq 0}$. Equivalently, it can be characterized as
the positive normalized left eigenvector of the generator $\mathcal{L}$ of
$M$, i.e. $\pi\mathcal{L}=\lambda\pi$, where

$\mathcal{L}f(n)=\beta n(f(n+1)-f(n))-\theta
nf(n)+\sum_{j=1}^{n-1}\frac{\gamma n}{j(j+1)}\left(f(j)+f(n-j)-f(n)\right),$

for real valued functions $f$ and $n\geq 0$. Thus $(\pi(n))_{n\geq 0}$
satisfies a linear system given by the dual operator of $\mathcal{L}$. We
refer to Section 4.1 for rigorous statements and details on the semigroup and
generator. The fact that $\left\langle{\pi,|f|}\right\rangle<\infty$ will also
be given in Proposition 4.1.

The random variable $W$ in the statement is the limit of the Malthusian
martingale $e^{-\lambda t}\left\langle{X_{t},h}\right\rangle$, where $h$ is
the right eigenvector of the semigroup: $M_{t}h=e^{\lambda t}h$, for any
$t\geq 0$; again see Proposition 4.1. Because the cluster size can be
arbitrarily large, the classical finite-dimensional Perron–Frobenius theorem
does not apply and we need a more precise analysis on the semigroup and its
generator to ensure the properties of eigenvectors; see Section 4.

Note that the above results hold for functions of at most polynomial growth.
This is inherited from our Lyapunov functions which are polynomial; see Lemma
4.2 and Proposition 4.1.

A similar result about the inactive clusters can be derived.

######  Corollary 1 (Law of large numbers for $(Y_{t})_{t\geq 0}$).

For any function $f:\mathbb{N}_{+}\to\mathbb{R}$ of at most polynomial growth,
we have that

$e^{-\lambda t}\langle
Y_{t},f\rangle\xrightarrow{t\to\infty}\left(\frac{\theta}{\lambda}\right)\,W\,\sum_{n=1}^{\infty}n\pi(n)f(n)\quad\text{
a.s. and in }L^{2},$

and

$\frac{\langle Y_{t},f\rangle}{\langle
Y_{t},\mathbf{1}\rangle}\xrightarrow{t\to\infty}\langle\widetilde{\pi},f\rangle,\qquad\,\,\,\,\,\,\quad\text{
a.s. on }\\{\tau=\infty\\},$

where $\widetilde{\pi}$ is a probability law on
$\mathbb{N}_{+}=\\{1,2,3,\cdots\\}$ given by

$\widetilde{\pi}(n):=\frac{\pi(n)n}{\sum_{j=1}^{\infty}\pi(j)j}.$ (5)

We observe that interestingly, $\widetilde{\pi}$ is a size-biased version of
$\pi$. It means that the isolated clusters are of larger size than the active
clusters. This phenomenon is also recorded in the recent work of Jean Bertoin
[13, Corollary 4.3]. The reason for this phenomenon is that every active
cluster gets isolated at a rate proportional to its size.

Figure 4: Illustration of the reduction of the study of GFI process
$(G_{t},\Psi_{t},\eta)_{t\geq 0}$. The first simplification is to decompose
the graph into clusters and study the cluster process
$(\mathscr{X}_{t},\mathscr{Y}_{t})_{t\geq 0}$, where every cluster is a RRT
whose law only depends on its size. The second simplification is to study size
process $(X_{t},Y_{t})_{t\geq 0}$, which provides the key information for the
law of $(\mathscr{X}_{t},\mathscr{Y}_{t})_{t\geq 0}$.

Recall that every vertex $v\in\mathbb{R}$ is labeled by its infection time and
every cluster is a tree. We consider these clusters up to an equivalence
relation, which consists in keeping the order between vertices but forgetting
their infection times, see Section 3.2 for a rigorous definition. We denote by
$\mathscr{T}$ the space of equivalence classes of trees of all sizes. We
denote by $T_{\pi}$ the random tree whose size is distributed as $\pi$, and
conditionally on its size the tree is a RRT. With a slight abuse of notation,
for any recursive tree $\mathrm{t}$, we write $\mathbf{t}\in\mathscr{T}$ for
its equivalence class and for any function $f:\mathscr{T}\mapsto\mathbb{R},$
we let $f(\mathrm{t})=f(\mathbf{t})$. We also write $|\mathbf{t}|$ for the
number of vertices in the equivalence class $\mathbf{t}.$

Using the splitting property, we derive the long-time behavior of the
empirical measures on (active and inactive) clusters from the results on the
size processes obtained above.

######  Theorem 2.3 (Limit of the empirical measure of clusters).

Consider any $f:\mathscr{T}\to\mathbb{R}$ such that there exists $p>0$
satisfying

$\sup_{\mathbf{t}\in\mathscr{T}}\frac{|f(\mathbf{t})|}{|\mathbf{t}|^{p}}<\infty.$

Then on the event $\\{\tau=\infty\\}$,

$\displaystyle\frac{1}{|\mathscr{X}_{t}|}\sum_{\mathcal{C}\in\mathscr{X}_{t}}f(\mathcal{C})\stackrel{{\scriptstyle
t\to\infty}}{{\longrightarrow}}\mathbb{E}[f(T_{\pi})],\quad\frac{1}{|\mathscr{Y}_{t}|}\sum_{\mathcal{C}\in\mathscr{Y}_{t}}f(\mathcal{C})\stackrel{{\scriptstyle
t\to\infty}}{{\longrightarrow}}\mathbb{E}[f(T_{\widetilde{\pi}})]\quad\text{a.s.}.$

The Malthusian exponent $\lambda$ is the key characteristic of the model and
is fully determined by the model parameters $(\beta,\gamma,\theta)$. For both
mathematical and practical purposes, it is highly important to understand how
$\lambda$ depends on $(\beta,\theta,\gamma)$ The main results are presented
below.

######  Theorem 2.4.

The mapping $(\beta,\theta,\gamma)\mapsto\lambda(\beta,\theta,\gamma)$ is
continuous, and the sets of parameters resulting in respectively
$\lambda>0,\lambda=0,\lambda<0$ are non-empty. The Malthusian exponent depends
on the three parameters monotonically that

  1. 1.

$\theta\mapsto\lambda(\beta,\theta,\gamma)$ is decreasing;

  2. 2.

$\gamma\mapsto\lambda(\beta,\theta,\gamma)$ is increasing;

  3. 3.

$\beta\mapsto\lambda(\beta,\theta,\gamma)$ is increasing if $\gamma>\theta$,
constant if $\gamma=\theta$, and decreasing if $\gamma<\theta$.

In the above statements, the continuity, the existence of phases, and also the
monotonicity with respect to $\theta$ and $\gamma$, are not surprising.
However, the change of monotonicity of
$\beta\mapsto\lambda(\beta,\theta,\gamma)$ seems not obvious. Indeed, the
increment of $\beta$ makes the clusters grow faster, thus the clusters will
split (fragmentation) or be detected (isolation) faster. Faster fragmentation
makes the number of clusters increase while faster detection has the opposite
effect. The complex competition between the two forces leads to the simple
Statement 3 in Theorem thm:Regularity surprisingly. At the end, the increment
of the infection rate $\beta$ will not necessarily speed up the epidemics, but
does make the epidemics weaker when $\gamma<\theta$. In practical terms, in
this regime, if the contact tracing policy is effective so that the population
is in the subcritical regime, the emergence of a more contagious variant will
make the epidemic decay faster, even if $\theta$ and $\gamma$ are unchanged
(it means we use the same contact tracing practice as before).

The rest of the paper is organized as follows. In Section 3 we introduce
notations and we recall the key splitting property of the random recursive
trees and derive the size process. The study of the first moment semigroup of
$(X_{t})_{t\geq 0}$ and the associated martingale and $L^{2}$ estimates is
carried out in Section 4, using in particular Lyapunov functions. We can then
prove Theorem 2.1. Section 5 is devoted to the strong convergences and we
prove Theorem 2.2, Corollary 1 and Theorem 2.3. Then we prove Theorem 2.4 in
Section 6 and present related simulations in Section 7. Finally we discuss
possible extensions and generalizations of the model in Section 8.

##  3 Preliminaries

In this section, we introduce notations and explain why the size process
captures the essential information of the cluster process.

Figure 5: All the recursive trees (as representatives of equivalence classes)
in $\mathscr{T}_{4}$.

###  3.1 Ulam-Harris-Neveu labeling of clusters

We introduce the notation for the genealogical tree of clusters (active and
inactive). Recall that every vertex is labeled by its infection time. For any
cluster $\mathcal{C}$ (active or inactive), we call the vertex with the
minimum label the root of $\mathcal{C}$, and denote it by
$\mathrm{root}(\mathcal{C})$. We label every cluster using the Ulam-Harris-
Neveu (UHN) notation:

$\displaystyle\mathcal{U}:=\bigcup_{n\geq 0}\\{1,2\\}^{n},$

where an element $u\in\mathcal{U}$ is called a label or word. For the initial
cluster, we use the label $\emptyset$ as convention. Then by induction, for
any cluster $\mathcal{C}$ labeled by a word $u\in\mathcal{U}$: this label is
unchanged during the growth of the cluster (infection); this label becomes
inactive when the cluster is isolated; this label is replaced by two labels
$u1$ and $u2$ when fragmentation occurs. By convention, $u1$ is the label of
the subcluster containing $\mathrm{root}(\mathcal{C})$ and is called the first
child, while $u2$ is for the other subcluster that we call the second child.

Every cluster except the initial one has a unique parent. There exists a
partial order $\preccurlyeq$ on $\mathcal{U}$ defined by the genealogy, i.e.
for two words $u$ and $uv$ with $v\neq\emptyset$, the former is an ancestor of
the latter, while the latter is a descendant of the former, and we denote
$u\preccurlyeq uv$. For two words $u,v\in\mathcal{U}$, we denote by $u\wedge
v$ the most recent common ancestor of $u$ and $v$.

Figure 6: An illustration of the genealogy of clusters.

We denote by $\mathcal{U}_{t}$ the collection of labels of active clusters at
time $t$, while $\mathcal{U}^{\dagger}_{t}$ gathers the labels of inactive
clusters at this time. For $u\in\mathcal{U}$, we use $\mathcal{U}(u)$ to
represent the genealogical tree rooted at $u$

$\displaystyle\mathcal{U}(u):=\left\\{w\in\mathcal{U}\,|w\succcurlyeq u\text{
or }w=u\right\\}.$

We also use $\mathcal{U}_{t}(u)$ (resp. $\mathcal{U}^{\dagger}_{t}(u)$) to
denote the set of labels of clusters in $\mathcal{U}(u)$ which are active
(resp. inactive) at time $t$. One should notice that $\mathcal{U}_{t}(u)$ can
be either the set of labels of active descendants of $u$ at time $t$, or $u$
itself if it is still alive at that moment, or empty if it encounters
isolation first at a time no later than $t$, so we consider
$\mathcal{U}_{t}(u)$ as the “set of labels of active clusters at $t$ issued
from the cluster of the label $u$”. Finally, if $u\in\mathcal{U}_{t}$ (resp.
$u\in\mathcal{U}^{\dagger}_{t}$), we denote by $\mathscr{X}_{t}^{u}$ (resp.
$\mathscr{Y}_{t}^{u}$) for its associated cluster, and $X_{t}^{u}$ (resp.
$Y_{t}^{u}$) its size, i.e. $X_{t}^{u}=|\mathscr{X}_{t}^{u}|$ (resp.
$Y_{t}^{u}=|\mathscr{Y}_{t}^{u}|$); we write $X_{t}^{u}=0$ for
$u\notin\mathcal{U}_{t}$. We also use $X_{t}(n)$ (resp. $Y_{t}(n)$) for the
number of active clusters (resp. inactive clusters) of size $n$ at time $t$.
With these notations, we have
$X_{t}=\sum_{u\in\mathcal{U}_{t}}\delta_{X_{t}^{u}}$ and for any function $f$
from $\mathbb{N}_{+}$ to $\mathbb{R}$:

$\displaystyle\left\langle{X_{t},f}\right\rangle=\sum_{\mathcal{C}\in\mathscr{X}_{t}}f(|\mathcal{C}|)=\sum_{u\in\mathcal{U}_{t}}f(X_{t}^{u})=\sum_{n=1}^{\infty}X_{t}(n)f(n).$

###  3.2 Random recursive trees

The genealogy in a cluster is given by a recursive tree and here we define
some notations related to the latter. Given
${V=\\{a_{1},\cdots,a_{n}\\}\subset\mathbb{R}}$ with increasing order
$a_{1}<a_{2}<\cdots<a_{n}$, a recursive tree $\mathrm{t}$ on $V$ is a rooted
tree with $V$ as the set of vertices, such that for any $a_{i},2\leq i\leq n$,
the path from $a_{1}$ to $a_{i}$ is increasing. Thus, a descendant has a
larger label than that of the parent. The minimal element $a_{1}$ is called
the root of $\mathrm{t}$. The collection of all the recursive trees on $V$ has
cardinality $(|V|-1)!$.

We define the equivalence relation $\sim$ between recursive trees on different
ordering sets. Denoting by $\mathrm{t}_{1}$ a recursive tree on $V_{1}$ and
$\mathrm{t}_{2}$ a recursive tree on $V_{2}$, then
$\mathrm{t}_{1}\sim\mathrm{t}_{2}$ if and only if there exists an order-
preserving function $\psi:V_{1}\to V_{2}$, which induces a bijection between
the trees $\mathrm{t}_{1}$ and $\mathrm{t}_{2}$. We denote by
$\mathscr{T}_{n}$ the set of recursive trees of size $n$ up to the equivalence
relation $\sim$, and use the recursive trees defined on $\\{1,\cdots,n\\}$ to
represent equivalence classes (canonical representation); see Figure 5 for an
example of $\mathscr{T}_{4}$. Finally, we define the space of finite recursive
trees up to equivalence relation $\sim$

$\displaystyle\mathscr{T}:=\bigcup_{n=1}^{\infty}\mathscr{T}_{n}.$ (6)

A (uniform) random recursive tree (RRT) of size $n$ is a random element chosen
uniformly in $\mathscr{T}_{n}$. We denote by $T_{n}$ this random equivalence
class. With a slight abuse, RRT can refer both to the equivalence class or a
specific labeling (for instance with the first integers). Since $\mathscr{T}$
defined in (6) contains only countably many elements, the space $\mathscr{T}$
is a Polish space under the trivial distance. We can construct more general
probability measures than uniform random recursive tree on $\mathscr{T}$. For
example, for any $\nu$ a probability measure on $\mathbb{N}_{+}$, we use the
notation $T_{\nu}$ to represent a random variable on $\mathscr{T}$, such that
we sample first the size by $\nu$, then sample an equivalence class uniformly
given its size, i.e. for any bounded function $f$ on $\mathscr{T}$,

$\begin{split}\mathbb{E}[f(T_{\nu})]=\sum_{n=1}^{\infty}\nu(n)\mathbb{E}[f(T_{n})]=\sum_{n=1}^{\infty}\nu(n)\left(\frac{1}{(n-1)!}\sum_{\mathbf{t}\in\mathscr{T}_{n}}f(\mathbf{t})\right).\end{split}$
(7)

This gives the rigorous definition for $\mathbb{E}[f(T_{\pi})]$ and
$\mathbb{E}[f(T_{\widetilde{\pi}})]$ in Theorem 2.3.

There are many ways to construct $T_{n}$. One classical construction is the
recursive approach: let $T_{1}$ be the tree with the single vertex $1$, and
construct $T_{k+1}$ by attaching the vertex labeled $(k+1)$ uniformly onto a
vertex of $T_{k}$. This construction explains why our infection process (Yule
process), conditioned on its size, is a RRT. Indeed each individual
contaminates a new individual with the same rate, which amounts to attaching a
new vertex to a uniformly chosen vertex of the tree, independent of the
history of construction given the current state.

The key property of RRT that we need is the splitting property. We state this
property below and its proof can be found in [31] and [10]. We will explain
what role this property plays in our model in the next section.

######  Proposition 3.1 (Splitting property).

Let $n\geq 2$ and $T_{n}$ the canonical random recursive tree of size $n$. We
choose uniformly one edge in $T_{n}$ and remove it. Then $T_{n}$ is split into
two subtrees $T_{n}^{0}$ and $T_{n}^{*}$, corresponding to two connected
components, where $T_{n}^{0}$ contains the root of $T_{n}$ and $T_{n}^{*}$
does not. Then

$\mathbb{P}\left[|T^{*}_{n}|=j\right]=\frac{n}{n-1}\frac{1}{j(j+1)},\qquad
j=1,2,\cdots,n-1.$ (8)

Furthermore, conditionally on $|T^{*}_{n}|=j$, $T_{n}^{0}$ and $T_{n}^{*}$ are
two independent RRTs of size respectively $(n-j)$ and $j$.

###  3.3 Reduction to the size process

Let us explain more explicitly how the study of
$(\mathscr{X}_{t},\mathscr{Y}_{t})_{t\geq 0}$ can be reduced to that of the
size process $(X_{t},Y_{t})_{t\geq 0}$, with the help of the splitting
property. We denote by $\mathcal{M}$ the finite point measures on
$\mathbb{N}_{+}$ and endow it with the weak topology and corresponding Borel
algebra. Then $(X_{t},Y_{t})_{t\geq 0}$ is a $\mathcal{M}^{2}$-valued process
and we denote by $(\mathcal{F}_{t})_{t\geq 0}$ its natural filtration. In
fact, $(\mathcal{F}_{t})_{t\geq 0}$ is also the natural filtration of
$(X_{t})_{t\geq 0}$ since $Y_{t}$ is a deterministic function of
$(X_{s})_{0\leq s\leq t}$ for any $t\geq 0$.

######  Proposition 3.2.

Fix arbitrary $t\geq 0$. Conditionally on $(X^{u}_{t})_{u\in\mathcal{U}_{t}}$
and $(Y^{u}_{t})_{u\in\mathcal{U}^{\dagger}_{t}}$, the clusters in
${\mathscr{X}_{t}\cup\mathscr{Y}_{t}}$ are independent RRTs whose sizes are
given by $(X^{u}_{t})_{u\in\mathcal{U}_{t}}$ and
$(Y^{u}_{t})_{u\in\mathcal{U}^{\dagger}_{t}}$. Moreover, $(X_{t},Y_{t})_{t\geq
0}$ is a measure-valued Markov branching process in
$(\mathcal{M}^{2},(\mathcal{F}_{t})_{t\geq 0},\mathbb{P})$.

###### Proof.

The RRT-distribution of clusters and conditional independence is obvious at
the initial time when there is only one single vertex. Let us check that this
property remains valid along time and at the same time that the size process
satisfies the Markov property. The branching property of the size process is a
direct consequence of the branching property of the cluster process.

We need to consider three events and their corresponding rates. First, a
cluster is isolated with a rate depending only on its size, and then becomes
inactive. Second, the growth rate of a cluster also only depends on its size,
and the new vertex is added independently of the state of other clusters.
Thus, after a growth, the new cluster remains independent from the other ones
(conditionally on the sizes). Moreover, by the construction of RRT by adding a
vertex uniformly at random, we know that the new cluster is also distributed
as a RRT. Third, for fragmentation, we invoke the splitting property
(Proposition 3.1), which guarantees that the two new clusters are independent
RRTs conditionally on their sizes. This also ensures the Markov property
thanks to the absence of memory for each event. ∎

Let us sum up the dynamics of active clusters and the transition for the size
process. Independently, each active cluster of size $n$

  * i)

becomes an inactive cluster of size $n$ at rate $\theta n$
(death of the active cluster and birth of an inactive cluster);

  * ii)

becomes a RRT of size $(n+1)$ at rate $\beta n$ (cluster size increases by 1);

  * iii)

splits into two RRTs of sizes $(n-j,j)$ at rate $\gamma n\frac{1}{j(j+1)}$,
for ${n\geq 2,1\leq j\leq n-1}$
(fragmentation). Here $(n-j)$ is the size of the first child (containing the
root) and $j$ is the size of the second child.

Thus, each active cluster of size $n$ lives an exponential time of parameter
$(\beta+\theta+\gamma)n-\gamma$. We introduce now the infinitesimal generator
$\mathcal{A}$ of the Markov process $(X_{t},Y_{t})_{t\geq 0}$ for sizes. It is
defined on a suitable subspace of measurable bounded functions on
$\mathcal{M}^{2}$. Consider two functions
$f,g:\mathbb{N}_{+}\rightarrow\mathbb{R}_{+}$ and
${F:\mathbb{R}^{2}\to\mathbb{R}}$ a bounded Borel function. We set

${F_{f,g}:(\mu,\nu)\in\mathcal{M}^{2}\rightarrow
F(\left\langle{\mu,f}\right\rangle,\left\langle{\nu,g}\right\rangle)\in\mathbb{R}},$

and define

$\begin{split}\mathcal{A}F_{f,g}(\mu,\nu)&=\sum_{n=1}^{\infty}\mu(\\{n\\})\beta
n\left(F(\left\langle{\mu+\delta_{n+1}-\delta_{n},f}\right\rangle,\left\langle{\nu,g}\right\rangle)-F(\left\langle{\mu,f}\right\rangle,\left\langle{\nu,g}\right\rangle)\right)\\\
&\quad+\sum_{n=1}^{\infty}\mu(\\{n\\})\theta
n\left(F(\left\langle{\mu-\delta_{n},f}\right\rangle,\left\langle{\nu+\delta_{n},g}\right\rangle)-F(\left\langle{\mu,f}\right\rangle,\left\langle{\nu,g}\right\rangle)\right)\\\
&\quad+\sum_{n=1}^{\infty}\mu(\\{n\\})\gamma
n\sum_{j=1}^{n-1}\left(\frac{1}{j(j+1)}\right)\\\
&\qquad\qquad\times\left(F(\left\langle{\mu+\delta_{j}+\delta_{n-j}-\delta_{n},f}\right\rangle,\left\langle{\nu,g}\right\rangle)-F(\left\langle{\mu,f}\right\rangle,\left\langle{\nu,g}\right\rangle)\right).\end{split}$
(9)

Although we start from one single active individual to define our model for
convenience (see Section 8.2 for a discussion on the initial condition), we
remark that the size process can be defined from any finite initial state
since it is a measure-valued process on integers.

##  4 First moment semigroup of $(X_{t})_{t\geq 0}$ and Perron’s root

In this part, we study the first moment semigroup associated to the process
$(X_{t})_{t\geq 0}$. We will establish the existence of Perron’s eigenelements
and speed of convergence and prove Theorem 2.1.

###  4.1 Semigroup and generator

Thanks to Section 3.2, the study of the model is reduced to the long-time
behavior of the measure-valued Markov branching process $(X_{t},Y_{t})_{t\geq
0}$. We denote the first moment semigroup associated to $(X_{t})_{t\geq 0}$ by
$M=(M_{t})_{t\geq 0}$. It is defined for any non-negative function $f$ by

$\displaystyle M_{t}f(n):=\mathbb{E}_{\delta_{n}}[\langle
X_{t},f\rangle],\quad\forall t\geq 0,n\geq 1,$ (10)

where $\mathbb{P}_{\delta_{n}}$ stands for the size process with initial
condition $(X_{0},Y_{0})=(\delta_{n},0)$ and $\mathbb{E}_{\delta_{n}}$ is its
associated expectation. In particular we consider for any
$n,m\in\mathbb{N}_{+}$,

$\displaystyle
M_{t}(n,m):=M_{t}\mathbf{1}_{m}(n)=\mathbb{E}_{\delta_{n}}[\langle
X_{t},\mathbf{1}_{m}\rangle]=\mathbb{E}_{\delta_{n}}\Big{[}\\#\\{\mathcal{C}\in\mathscr{X}_{t}\,:\,|\mathcal{C}|=m\\}\Big{]},$

which is the mean number of clusters of size $m$ at time $t$ descending from
one single RRT of size $n$ at time $0$.

We use the notation $[x^{p}]$ for the polynomial function such that
$[x^{p}](n)=n^{p},n\geq 1$. When $p=1$ it is just the identity function
$[x](n)=n$. We introduce $\mathcal{B}$ the set of functions from
$\mathbb{N}_{+}$ to $\mathbb{R}$ with at most polynomial growth:

$\displaystyle\mathcal{B}:=\left\\{f:\mathbb{N}_{+}\rightarrow\mathbb{R},\,\exists
p>0\text{ such that }\sup_{n\geq 1}|f(n)|/n^{p}<\infty\right\\}.$ (11)

Let $\mathcal{B}_{p}$ be the set of functions with $p>0$ fixed

$\displaystyle\mathcal{B}_{p}:=\left\\{f:\mathbb{N}_{+}\rightarrow\mathbb{R},\,\sup_{n\geq
1}|f(n)|/n^{p}<\infty\right\\}.$ (12)

Notice that $\mathcal{B}_{p}\subset\mathcal{B}_{p^{\prime}}\subset\mathcal{B}$
for $0<p\leq p^{\prime}$, and $\mathcal{B}=\bigcup_{p>0}\mathcal{B}_{p}.$ We
then extend now the first moment semigroup to the space $\mathcal{B}$.

######  Lemma 4.1.

  1. (i)

For any $p\geq 1,t\geq 0,n\geq 1$, we have

$\displaystyle M_{t}([x^{p}])(n)\leq e^{(2^{p-1}p\beta-\theta)t}n^{p}.$

  2. (ii)

For any $f\in\mathcal{B}$, setting $f_{+}$ (resp. $f_{-}$) to be the positive
part (resp. negative part) of $f$, the functions $t\in[0,\infty)\rightarrow
M_{t}f_{+}$ and $t\in[0,\infty)\rightarrow M_{t}f_{-}$ are well defined and
finite. We can thus set for any $t\geq 0$ and $n\in\mathbb{N}_{+}$,

$M_{t}f(n)=\mathbb{E}_{\delta_{n}}\big{[}\langle
X_{t},f\rangle\big{]}:=M_{t}f_{+}(n)-M_{t}f_{-}(n).$

  3. (iii)

$(M_{t})_{t\geq 0}$ is a positive semigroup on $\mathcal{B}$ and for any
$f\in\mathcal{B}$, we have for $t\geq 0$ and $n\geq 1$

$\displaystyle\frac{{\mathrm{d}}}{{\mathrm{d}}t}M_{t}f(n)=M_{t}(\mathcal{L}f)(n),$
(13)

where the linear operator $\mathcal{L}:\mathcal{B}\rightarrow\mathcal{B}$ is
defined for any $n\geq 1$ by

$\mathcal{L}f(n)=\underbrace{\beta
n(f(n+1)-f(n))}_{\text{``growth''}}\underbrace{-\theta
nf(n)}_{\text{``isolation''}}\\\ +\underbrace{\sum_{j=1}^{n-1}\frac{\gamma
n}{j(j+1)}\left(f(j)+f(n-j)-f(n)\right)}_{\text{``fragmentation''}}.$ (14)

###### Proof.

Linear operator $\mathcal{L}$ yields the generator of the first moment
semigroup of the size process. We focus on $(i)$ and study
$\mathcal{L}([x^{p}])$ at first . Notice that for $p\geq 1$ and any $x,y>0$,
$(x+y)^{p}\geq x^{p}+y^{p}$, so the contribution of the fragmentation term in
$\mathcal{L}([x^{p}])$ is negative. Thus we have

$\displaystyle\mathcal{L}([x^{p}])(n)\leq\beta n((n+1)^{p}-n^{p})-\theta
n^{p+1}.$

We then apply convexity for $n\in\mathbb{N}_{+}$

$\displaystyle(n+1)^{p}-n^{p}\leq p(n+1)^{p-1}\leq p2^{p-1}n^{p-1}.$

It gives us

$\displaystyle\mathcal{L}([x^{p}])\leq(2^{p-1}p\beta-\theta[x])[x^{p}]\leq(2^{p-1}p\beta-\theta)[x^{p}].$
(15)

Here we use simply the fact that $[x](n)=n\geq 1$ for all $n$ for the
isolation term.

The rest of the proof follows classical arguments of localization, see e.g.
Theorem 1 in [32], and we give here the main lines only. We assume that
$X_{0}=\delta_{n}$ for any given $n\geq 1.$ We consider the stopped process
$(X^{m}_{t},Y^{m}_{t})_{t\geq 0}$ defined by
$X_{t}^{m}=X_{t\wedge\tau_{m}},Y_{t}^{m}=Y_{t\wedge\tau_{m}}$, where

$\tau_{m}:=\inf\\{t\geq 0\,:\,\left\langle{X_{t},[x]}\right\rangle\geq m\\}.$

Notice that on the event $\\{\tau_{m}\geq t\\}$, we have
$\left\langle{X_{t},[x]}\right\rangle\leq m$ and
$\left\langle{X_{t},[x^{p}]}\right\rangle\leq m^{p}.$ The process
$(X^{m}_{t},Y^{m}_{t})_{t\geq 0}$ lives on a finite state space and has
bounded transition rates. Consider three functions
${f,g:\mathbb{N}_{+}\rightarrow\mathbb{R}_{+}}$ and
${F:\mathbb{R}^{2}\to\mathbb{R}}$ a bounded Borel function and recall that
$F_{f,g}(\mu,\nu)=F(\left\langle{\mu,f}\right\rangle,\left\langle{\nu,g}\right\rangle)$.
We get by Dynkin’s formula

$\mathbb{E}_{\delta_{n}}[F_{f,g}(X_{t}^{m},Y_{t}^{m})]=\mathbb{E}_{\delta_{n}}\left[F_{f,g}(X_{0}^{m},Y_{0}^{m})\right]+\mathbb{E}_{\delta_{n}}\left[\int_{0}^{t\wedge\tau_{m}}\mathcal{A}F_{f,g}(X_{s},Y_{s})\,{\mathrm{d}}s\right],$

where $\mathcal{A}$ is defined in (9). We apply this equation with
$F(x,y)=x\wedge m^{p}$ and $f=[x^{p}]$ and $g=0$ to obtain

$\mathbb{E}_{\delta_{n}}[\langle
X_{t}^{m},[x^{p}]\rangle]=\mathbb{E}_{\delta_{n}}[\langle
X_{0}^{m},[x^{p}]\rangle]+\mathbb{E}_{\delta_{n}}\left[\int_{0}^{t\wedge\tau_{m}}\langle
X_{s},\mathcal{L}[x^{p}]\rangle\,{\mathrm{d}}s\right].$ (16)

Using the above display and $\eqref{eq:lpBound}$ yields

$\mathbb{E}_{\delta_{n}}[\langle
X_{t}^{m},[x^{p}]\rangle]\leq\mathbb{E}_{\delta_{n}}[\langle
X_{0}^{m},[x^{p}]\rangle]+(2^{p-1}p\beta-\theta)\mathbb{E}_{\delta_{n}}\left[\int_{0}^{t\wedge\tau_{m}}\langle
X_{s},[x^{p}]\rangle\,{\mathrm{d}}s\right].$

Since the process $(X,Y)$ is non explosive, $\tau_{m}$ tends a.s. to infinity
as $m$ tends to infinity. Moreover, $\tau_{m}$ is increasing in $m$. Applying
Fatou’s lemma on the left hand side and monotone convergence on the right hand
side, the above inequality yields

$M_{t}[x^{p}](n)\leq
M_{0}[x^{p}](n)+(2^{p-1}p\beta-\theta)\int_{0}^{t}M_{s}[x^{p}](n)\,{\mathrm{d}}s.$

Grönwall’s lemma then ensures that $(i)$ and $(ii)$ are immediate
consequences.

For $(iii)$, thanks to the fact
$\mathcal{B}_{p}\subset\mathcal{B}_{p^{\prime}}\subset\mathcal{B}=\cup_{p>0}\mathcal{B}_{p}$
for any $p^{\prime}\geq p>0$, we only need to show that for any
$f\in\mathcal{B}_{p}$ with $p\geq 1$, we have

$\mathbb{E}_{\delta_{n}}[\langle
X_{t},f\rangle]=\mathbb{E}_{\delta_{n}}[\langle
X_{0},f\rangle]+\mathbb{E}_{\delta_{n}}\left[\int_{0}^{t}\langle
X_{s},\mathcal{L}f\rangle\,{\mathrm{d}}s\right].$ (17)

Let $C>0$ be such that $|f(n)|\leq Cn^{p},|\mathcal{L}f(n)|\leq Cn^{p+1}$ for
all $n$. Similarly to (16), we have

$\mathbb{E}_{\delta_{n}}[\langle
X_{t}^{m},f\rangle]=\mathbb{E}_{\delta_{n}}[\langle
X_{0}^{m},f\rangle]+\mathbb{E}_{\delta_{n}}\left[\int_{0}^{t\wedge\tau_{m}}\langle
X_{s},\mathcal{L}f\rangle\,{\mathrm{d}}s\right].$ (18)

We will show the limits of the left and right terms in above display. We first
study the left term which is the sum of the two terms below

$\mathbb{E}_{\delta_{n}}[\langle
X_{t}^{m},f\rangle\mathbf{1}_{\left\\{t<\tau_{m}\right\\}}]\,\,\text{ and
}\,\,\mathbb{E}_{\delta_{n}}[\langle
X_{t}^{m},f\rangle\mathbf{1}_{\left\\{t\geq\tau_{m}\right\\}}].$ (19)

For the first term in (19), we have

$\langle
X_{t}^{m},f\rangle\mathbf{1}_{\left\\{t<\tau_{m}\right\\}}\xrightarrow{m\to\infty}\langle
X_{t},f\rangle,\quad\text{ almost surely, }$

and

$|\langle
X_{t}^{m},f\rangle\mathbf{1}_{\left\\{t<\tau_{m}\right\\}}|\leq\langle
X_{t},|f|\rangle\leq C\langle X_{t},[x^{p}]\rangle.$

By $(i)$, the last term in the above display has a finite mean. Using
dominated convergence theorem, we obtain

$\lim_{m\to\infty}\mathbb{E}_{\delta_{n}}[\langle
X_{t}^{m},f\rangle\mathbf{1}_{\left\\{t<\tau_{m}\right\\}}]=\mathbb{E}_{\delta_{n}}[\langle
X_{t},f\rangle].$

We prove now that the second term in (19) vanishes. We use a coupling argument
below

$\mathbb{E}_{\delta_{n}}[|\langle
X_{t}^{m},f\rangle\mathbf{1}_{\left\\{t\geq\tau_{m}\right\\}}|]\leq
m^{p}\mathbb{P}_{\delta_{n}}[\tau_{m}\leq t]\leq
m^{p}\mathbb{P}_{\delta_{n}}[\langle\widetilde{X}_{t},[x]\rangle\geq m],$

where $(\widetilde{X}_{t})_{t\geq 0}$ is the size process with only growth
term (i.e. $\beta>0,\theta=\gamma=0$) and $\widetilde{X}_{0}=\delta_{n}$. Then
$(\langle\widetilde{X}_{t},[x]\rangle)_{t\geq 0}$ is a Yule process with
initial value $n$. Thus for fixed $t$, $\langle\widetilde{X}_{t},[x]\rangle$
follows the negative binomial distribution with parameters $n,1-e^{-\lambda}$.
Using the above display, we get

$\mathbb{E}_{\delta_{n}}[|\langle
X_{t}^{m},f\rangle\mathbf{1}_{\left\\{t\geq\tau_{m}\right\\}}|]\leq
m^{p}\frac{\mathbb{E}_{\delta_{n}}[(\langle\widetilde{X}_{t},[x]\rangle)^{2p}]}{m^{2p}}\xrightarrow{m\to\infty}0.$

Combining the analysis of the two terms in (19), the term on the left hand
side in (18) converges to $\mathbb{E}_{\delta_{n}}[\langle X_{t},f\rangle]$ as
$m\to\infty.$

Now we turn to the right hand side of (18). Note that

$\left|\int_{0}^{t\wedge\tau_{m}}\langle
X_{s},\mathcal{L}f\rangle\,{\mathrm{d}}s\right|\leq\int_{0}^{t}\langle
X_{s},|\mathcal{L}f|\rangle\,{\mathrm{d}}s\leq C\int_{0}^{t}\langle
X_{s},[x^{p+1}]\rangle\,{\mathrm{d}}s,\quad\forall m\in\mathbb{N}_{+}.$

Due to $(i)$, the last term has finite mean. Moreover
${\int_{0}^{t\wedge\tau_{m}}\langle
X_{s},\mathcal{L}f\rangle\,{\mathrm{d}}s\longrightarrow\int_{0}^{t}\langle
X_{s},\mathcal{L}f\rangle\,{\mathrm{d}}s},$ almost surely as $m\to\infty$.
Applying dominated convergence theorem yields

$\mathbb{E}_{\delta_{n}}\left[\int_{0}^{t\wedge\tau_{m}}\langle
X_{s},\mathcal{L}f\rangle\,{\mathrm{d}}s\right]\xrightarrow{m\to\infty}\mathbb{E}_{\delta_{n}}\left[\int_{0}^{t}\langle
X_{s},\mathcal{L}f\rangle\,{\mathrm{d}}s\right].$

Letting $m\to\infty$ in (18) gives (17) and ends the proof. ∎

###  4.2 Perron’s root and eigenvectors

In this part, we study the asymptotic behavior of the first moment semigroup
$(M_{t})_{t\geq 0}$ of $(X_{t})_{t\geq 0}.$ Under general assumptions
extending the Perron–Frobenius theory in finite dimension, the ergodic
behavior of the positive semigroup is given by the unique triplet of
eigenelements corresponding to the maximal eigenvalue. We refer in particular
to [7, 33] and references therein for general statements and applications to
growth fragmentation. In this work, we apply a general statement of [7] on the
ergodic behavior of positive semigroups. It allows us to exploit practical
sufficient conditions which are satisfied by our process: irreducibility
properties of the dynamic of the cluster sizes (22), and the fast splitting or
isolation of large clusters which provides a Lyapunov function for a typical
cluster (21). The fact that splitting is very asymmetric (such that one child
cluster is close to the parent in size) and the fact that a typical active
cluster at a given time has avoided isolation make the proof of the required
lower bound delicate. It involves a subtle compensation of fragmentation and
isolation terms. Moreover, we will show the exponential speed of convergence
of the semigroup. This will be useful in particular for the proof of the a.s.
convergences in the next section.

In what follows, notation “$f\leq g$” means the point-wise comparison for
functions (including constants). In this case, we say $f$ is upper bounded by
$g$ or $g$ is lower bounded by $f$. We also use $f\sim g$ to denote the fact
that $\lim_{n\to\infty}\frac{f(n)}{g(n)}=1$. We prove at first Lemma 4.2, the
key technical ingredient for Proposition 4.1, the latter is the main result of
this subsection. The following space of sublinear functions is useful to
control the harmonic function of the semigroup:

$\begin{split}\mathcal{S}:=\Big{\\{}f:\mathbb{N}_{+}\to&[1,\infty),\text{ such
that }\\\ &\,\,a)\,f\text{ is increasing and }\lim_{n\to\infty}f(n)=\infty\\\
&\,\,b)\,f\text{ is sublinear }\frac{f(n+1)}{n+1}\leq\frac{f(n)}{n},\text{ and
}C_{f}:=\sum_{j=1}^{\infty}\frac{f(j)}{j(j+1)}<\infty\Big{\\}}.\end{split}$
(20)

######  Lemma 4.2.

There exists a positive function $\psi$ defined on $\mathbb{N}_{+}$, such that

$\displaystyle{0<\inf_{\mathbb{N}_{+}}\psi<\sup_{\mathbb{N}_{+}}\psi\leq 1}$

and for every ${V\in\mathcal{S}\cup\\{[x^{p}]:p\geq 1\\}}$:

  1. (i)

There exist real constants $a<b$ and $\zeta>0$ such that

$\displaystyle\mathcal{L}V\leq aV+\zeta\psi,\qquad
b\psi\leq\mathcal{L}\psi\leq\xi\psi.$ (21)

  2. (ii)

For $R$ large enough, the set $K=\\{x\in\mathbb{N}_{+}:\psi(x)\geq V(x)/R\\}$
is a non-empty finite set and for any $x,y\in K$ and $t_{0}>0$,

$\displaystyle M_{t_{0}}(x,y)>0.$ (22)

###### Proof.

To find the Lyapunov-type functions in $(i)$, the main difficulty is to find
the lower bound of $\mathcal{L}\psi$ in (21). As we can see in (14), the
isolation term $-\theta nf(n)$ cannot be bounded from below uniformly in $n$
by $f$ times a constant. The strategy is to use the growth term and
fragmentation term to compensate the isolation term.

Step 1: Construction of $\psi$ \- setup. We set

$\displaystyle\psi(n):=A-(A-B)q^{n-1},$ (23)

with $A,B\in(0,\infty)$ and $q\in(0,1)$ to be chosen later. Then $\psi$ is
bounded between $A$ and $B$ and $\lim_{n\to\infty}\psi(n)=A$. We decompose
$\mathcal{L}\psi$ as follows

$\mathcal{L}\psi(n)=\underbrace{\beta
n(\psi(n+1)-\psi(n))}_{\mathbf{I}}-\underbrace{(\theta+\gamma)n\psi(n)}_{\mathbf{II}}+\gamma\psi(n)\\\
+\underbrace{\gamma
n\sum_{j=1}^{n-1}\frac{1}{j(j+1)}\left(\psi(j)+\psi(n-j)\right)}_{\mathbf{III}}.$
(24)

Firstly

$|\mathbf{I}|=|\beta(A-B)n(q^{n-1}-q^{n})|\leq C_{1}\psi(n),$ (25)

for some $C_{1}>0$, since $\psi\geq\min\\{A,B\\}>0$. Secondly, we observe that

$\displaystyle\lim_{n\to\infty}\frac{\sum_{j=1}^{n-1}\left(\frac{1}{j(j+1)}\psi(j)\right)}{\psi(n)}$
$\displaystyle=\frac{\sum_{j=1}^{\infty}\frac{1}{j(j+1)}\left(A-(A-B)q^{j-1}\right)}{A}$
$\displaystyle=1-\left(1-\frac{B}{A}\right)q^{-1}\Big{(}1+(q^{-1}-1)\ln(1-q)\Big{)}=:C_{q}^{A,B}.$

Here we used $\sum_{j=1}^{\infty}\frac{1}{j(j+1)}=1$ and
$\sum_{j=1}^{\infty}\frac{q^{j-1}}{j(j+1)}=q^{-1}\Big{(}1+(q^{-1}-1)\ln(1-q)\Big{)}$.
Moreover,

$\displaystyle\lim_{n\to\infty}\frac{\sum_{j=1}^{n-1}\frac{1}{j(j+1)}\psi(n-j)}{\psi(n)}$
$\displaystyle=\frac{\lim_{n\to\infty}\sum_{j=1}^{n-1}\frac{1}{j(j+1)}\left(A-(A-B)q^{n-j-1}\right)}{A}=1,$

since $\sum_{j=1}^{n-1}\frac{1}{j(j+1)}q^{n-j-1}$ goes to $0$ as
$n\rightarrow\infty$. Combining the above two displays, we obtain

$\mathbf{III}\sim(1+C_{q}^{A,B})\gamma n\psi(n),\quad\text{ as }n\to\infty.$

Step 2: Construction of $\psi$ \- choice of parameters. We add and subtract
the term ${(1+C_{q}^{A,B})\gamma n\psi(n)}$ and reformulate (24) as

$\mathcal{L}\psi(n)=\gamma\psi(n)+\underbrace{(C_{q}^{A,B}\gamma-\theta)n\psi(n)}_{\mathbf{II^{\prime}}}+\underbrace{R_{1}(n,q)+R_{2}(n,q)+\beta
n(\psi(n+1)-\psi(n))}_{\mathbf{III^{\prime}}},$

where the term $\mathbf{III^{\prime}}$ is the remainder term and

$\begin{split}R_{1}(n,q)&:=\gamma
n\left(\sum_{j=1}^{n-1}\left(\frac{1}{j(j+1)}\psi(j)\right)-C_{q}^{A,B}\psi(n)\right),\\\
R_{2}(n,q)&:=\gamma
n\left(\sum_{j=1}^{n-1}\left(\frac{1}{j(j+1)}\psi(n-j)\right)-\psi(n)\right).\end{split}$
(26)

We choose $q,A,B$ such that the term $\mathbf{II^{\prime}}$ is $0$ (i.e.
$C_{q}^{A,B}=\theta/\gamma$) and $0<A,B\leq 1$ (then
$0<\inf_{\mathbb{N}_{+}}\psi<\sup_{\mathbb{N}_{+}}\psi\leq 1$). More
precisely, we distinguish three cases:

  * •

If $\gamma=\theta$, we can choose $A=B=1$.

  * •

If $\gamma>\theta$, we can choose $q$ close to $1$ such that
${q^{-1}\Big{(}1+(q^{-1}-1)\ln(1-q)\Big{)}\in\left(1-\frac{\theta}{\gamma},1\right)}$
and then choose $0<B<A\leq 1$ such that $C_{q}^{A,B}=\theta/\gamma$.

  * •

If $\gamma<\theta$, it suffices to fix some $q\in(0,1)$ and then choose
$0<A<B\leq 1$ such that $C_{q}^{A,B}=\theta/\gamma$.

Besides, the convergences in Step 1 ensure that there exists
$C_{2}\in(0,\infty)$ such that

$\displaystyle\sup_{n\in\mathbb{N}_{+}}\left\\{\left|\frac{R_{1}(n,q)}{\psi(n)}\right|+\left|\frac{R_{2}(n,q)}{\psi(n)}\right|\right\\}\leq
C_{2}.$ (27)

Together with (25), we obtain that

$\displaystyle(\gamma-
C_{1}-C_{2})\psi\leq\mathcal{L}\psi\leq(\gamma+C_{1}+C_{2})\psi.$

This guarantees that the last two inequalities of (21) hold with the following
choice of parameters:

$\displaystyle b:=\gamma-C_{1}-C_{2},\qquad\xi:=\gamma+C_{1}+C_{2}.$

Step 3: Find $a,\zeta$. For $V=[x^{p}]$ with $p\geq 1$, we pick a real number
$a$ such that

$a<\min\\{2^{p-1}p\beta-\theta,b\\}.$

Using the first inequality in (15) and distinguishing if $2^{p-1}p\beta-\theta
n$ is larger than $a$ or not, we can write

$\displaystyle\mathcal{L}[x^{p}](n)$ $\displaystyle\leq
an^{p}+(2^{p-1}p\beta-\theta n-a)n^{p}\mathbf{1}_{\left\\{2^{p-1}p\beta-\theta
n\geq a\right\\}}$ $\displaystyle\leq a[x^{p}](n)+\zeta\psi(n),$

with $\zeta\in(0,\infty)$. The above result holds because $\psi$ is bounded
and there exist only finitely many $n$ satisfying ${2^{p-1}p\beta-\theta n\geq
a}$. This ends the proof of $(i)$ for $p\geq 1$. Besides, for any large $R$,
the set $K$ is finite and non-empty. The combination of growth, fragmentation
and isolation ensures the irreducibility of $(X_{t})_{t\geq 0}$ which allows
one to end the proof of $(ii)$. So both $(i)$ and $(ii)$ are proved for
$V=[x^{p}]$ with $p\geq 1.$

Now we treat the case $V\in\mathcal{S}$ and verify the condition $(i)$ and
$(ii)$ with $\psi$ given in Step 1. For condition $(ii)$, we only have to show
that $K$ is non-empty and finite, which is straightforward to see since $V$ is
increasing to infinity while $\psi$ is a bounded function. For condition
$(i)$, we only have to show the first half of (21). We calculate
$\mathcal{L}V$ and use the decomposition in (14). Note that
$\frac{V(n+1)}{n+1}\leq\frac{V(n)}{n}$ implies for the growth term that

$\displaystyle\mathbf{I}=\beta n(V(n+1)-V(n))\leq\beta
n\left(\frac{n+1}{n}V(n)-V(n)\right)\leq\beta V(n).$

Then the fact that $V$ increases and $C_{V}$ in (20) is finite yields for the
fragmentation term $\mathbf{III}$:

$\displaystyle\gamma(n-1)\sum_{j=1}^{n-1}\frac{n}{n-1}\frac{1}{j(j+1)}\left(V(j)+V(n-j)-V(n)\right)\leq\gamma
n\sum_{j=1}^{n-1}\frac{V(j)}{j(j+1)}<C_{V}\gamma n.$

The above two displays entail that

$\displaystyle\mathcal{L}V(n)\leq(\beta-\theta n)V(n)+C_{V}\gamma n.$ (28)

Now we pick a real number $a$ such that $a<\min\\{\beta-\theta,b\\}.$ Using
$\lim_{n\to\infty}V(n)=\infty$, we notice that
$E:=\\{n\in\mathbb{N}_{+}:(\beta-\theta n)V(n)+C_{V}\gamma n>aV(n)\\}$ is a
non-empty finite set. Then distinguishing the cases whether $n$ belongs to $E$
or not in (28) yields

$\displaystyle\mathcal{L}V(n)$ $\displaystyle\leq
aV(n)\mathbf{1}_{\left\\{n\in E^{c}\right\\}}+((\beta-\theta
n)V(n)+C_{V}\gamma n)\mathbf{1}_{\left\\{n\in E\right\\}}$
$\displaystyle=aV(n)+((\beta-a-\theta n)V(n)+C_{V}\gamma
n)\mathbf{1}_{\left\\{n\in E\right\\}}$ $\displaystyle\leq
aV(n)+\zeta\psi(n).$

Here the constant $\zeta$ is defined by

$\displaystyle\zeta:=\max_{n\in E}\frac{(\beta-a-\theta n)V(n)+C_{V}\gamma
n}{\psi(n)}\in(0,\infty).$

We conclude that both conditions $(i)$ and $(ii)$ are verified. The whole
proof is complete and finished. ∎

Now we come to the main result of this subsection which gives the existence of
eigenelements and asymptotic behavior of the semigroup, based on Theorem 2.1
in [7].

######  Proposition 4.1.

There exists a unique triplet $(\lambda,\pi,h)$ where $\lambda\in\mathbb{R}$
and ${\pi=(\pi(n))_{n\in\mathbb{N}_{+}}}$ is a positive vector of probability
distribution and $h:\mathbb{N}_{+}\rightarrow(0,\infty)$ is a positive
function, such that for all $t\geq 0$,

$\pi M_{t}=e^{\lambda t}\pi,\qquad M_{t}h=e^{\lambda t}h,$

and $0<\inf_{n\geq 1}h(n)\leq\sup_{n\geq 1}h(n)<\infty$ and $\sum_{n\geq
1}\pi(n)=\sum_{n\geq 1}\pi(n)h(n)=1.$

Besides, for every $p>0$ there exists $C,\omega>0$ such that for any $n,m\geq
1$, $t\geq 0$,

$\displaystyle\big{|}e^{-\lambda t}M_{t}(n,m)-h(n)\pi(m)\big{|}$
$\displaystyle\leq Cn^{p}m^{-p}e^{-\omega t},\qquad\sum_{n\geq
1}\pi(n)n^{p}<\infty.$ (29)

###### Proof.

Lemma 4.2 together with Lemma 4.1 (iii) ensures that the semigroup $M$
satisfies the drift and irreducibility conditions given in [7] (Propositions
2.2 and 2.3 therein). More precisely, these conditions are met with
$V=[x^{p}]$ for $p>0$ and $\varphi=\psi$ (defined in (23)), while $\psi\leq V$
is guaranteed by the fact that $\psi\leq 1$. Using these conditions, we can
apply Theorem 2.1 in [7], which yields the proposition, except the boundedness
of $h$. In particular, (29) is obtained by specifying the initial condition
$\mu=\delta_{n}$ and using the test function $1_{m}$.

We prove at first that $h$ is upper bounded. Lemma 3.4 in [7] ensures that $h$
is upper bounded by $V$ times a constant (i.e. $h\lesssim V$ in their
notation). Adding that Lemma 4.2 guarantees that we can pick a
$V\in\mathcal{S}$ that increases arbitrarily slowly, we obtain that $h$ is
upper bounded. Finally, we justify that $h$ is lower bounded. Indeed
$h(n)=e^{\lambda}M_{1}h(n)\geq ch(1)$ for all $n\geq 1$, where $c>0$. This is
because, due to (8), the probability that a cluster of size $n$ produces (by
fragmentation) a cluster of size one before unit time $1$ and that this latter
stays unchanged in the remaining time within the unit time is lower bounded by
a positive constant independent of its size $n$. ∎

Equation (29) ensures that for any $f\in\mathcal{B}_{p},$

$\displaystyle\big{|}e^{-\lambda
t}M_{t}f(n)-h(n)\left\langle{\pi,f}\right\rangle\big{|}$ $\displaystyle\leq
Cn^{p+2}\parallel f\parallel_{p}e^{-\omega t},$ (30)

where ${\parallel f\parallel_{p}:={\sum_{m\geq 1}m^{-(p+2)}|f(m)|<\infty}}$.
This result will be useful later.

The fact that the eigenvector $h$ is lower and upper bounded in $(0,\infty)$
ensures that the impact of the size of the initial cluster on the first order
approximation of the profile remains bounded. Furthermore, we expect that
restricting the set of test functions to bounded functions, the current result
(29) can be enhanced as uniform ergodic convergence. Indeed, we may apply
Theorem 3.5 in [6] with $\nu=\delta_{1}$, using again that large clusters
produce with high probability clusters of small sizes at fragmentation and are
fast isolated. The remaining difficulty lies in controlling uniformly
$M_{t}{\bf 1}(n)/M_{t}{\bf 1}(1)$ in time and size.

At this point one may want to apply [2] to prove strong convergence using the
asymptotic behavior of the first moment semigroup $(M_{t})_{t\geq 0}$. But [2]
requires stronger assumptions than what is obtained in (29), in particular in
terms of the stationary distribution $\pi$. Besides, we are interested in
finer and more quantitative estimates, with motivations coming from inference
and epidemiology. We thus follow another approach via $L^{2}$ estimates and
control of fluctuations.

###  4.3 $L^{2}$ martingale

Using the first moment semigroup, we can compute the second moment of $\langle
X_{t},f\rangle$ for $f\in\mathcal{B}$, which consists in the so-called formula
for forks or many-to-two formula, see e.g. [8, 29] and references therein. The
idea is to use the most recent common ancestor of two individuals to decouple
their values.

######  Lemma 4.3.

For any $x\in\mathbb{N}_{+}$ and $f\in\mathcal{B}$, we have

$\mathbb{E}_{\delta_{x}}\left[\langle X_{t},f\rangle^{2}\right]\\\
=M_{t}(f^{2})(x)+2\int_{0}^{t}\sum_{n\geq 1}M_{s}(x,n)\left(\sum_{1\leq j\leq
n-1}\kappa(n,j)M_{t-s}f(j)M_{t-s}f(n-j)\right)\,{\mathrm{d}}s,$

where $\kappa(n,j)=\frac{\gamma n}{j(j+1)}$ is the rate at which a cluster of
size $n$ breaks into two clusters of sizes $(n-j)$ (first child) and $j$
(second child).

######  Remark 1.

Combining this identity with the estimates on the semigroup $M$ obtained in
the previous subsection gives $L^{2}$ convergence of the empirical measure of
clusters, with exponential speed and size dependency; see next section.

###### Proof.

Recalling notations in Section 3.1, we have

$\langle X_{t},f\rangle=\sum_{u\in\mathcal{U}_{t}}f(X_{t}^{u}).$

Recall also that, for any $u,v\in\mathcal{U}$, $u\wedge v$ is the label of the
most recent common ancestor of $u$ and $v$, and $\mathcal{U}_{t}(u)$ as the
active clusters at $t$ issued from $u$. We first notice that

$\begin{split}\langle
X_{t},f\rangle^{2}=\sum_{u,v\in\mathcal{U}_{t}}f(X_{t}^{u})f(X_{t}^{v})&=\sum_{u\in\mathcal{U}_{t}}f^{2}(X_{t}^{u})+\sum_{\begin{subarray}{c}w\in\mathcal{U}\end{subarray}}\sum_{\begin{subarray}{c}u,v\in\mathcal{U}_{t},\\\
u\neq v,u\wedge v=w\end{subarray}}f(X_{t}^{u})f(X_{t}^{v})\\\
&=\sum_{u\in\mathcal{U}_{t}}f^{2}(X_{t}^{u})+\sum_{\begin{subarray}{c}w\in\mathcal{U}\end{subarray}}\mathbf{1}_{\left\\{b(w)<t\right\\}}I_{t}(w),\end{split}$
(31)

where for any $w\in\mathcal{U}$, $b(w)$ is the time at which the cluster
labeled by $w$ branches (i.e. the time when it splits into two clusters,
labeled $w1$ and $w2$; potentially infinite if that does not happen due to
isolation) and

$\displaystyle I_{t}(w)=\sum_{\begin{subarray}{c}i,j\in\\{1,2\\},\,i\neq j\\\
u\in\mathcal{U}_{t}(wi),\,v\in\mathcal{U}_{t}(wj)\end{subarray}}f(X_{t}^{u})f(X_{t}^{v})=2\left(\sum_{u\in\mathcal{U}_{t}(w1)}f(X_{t}^{u})\,\times\,\sum_{v\in\mathcal{U}_{t}(w2)}f(X_{t}^{v})\right).$

Thus $I_{t}(w)$ is the cross term between the active clusters on the two
genealogical subtrees rooted at $w1$ and $w2$.

Concerning the equation (31), we have firstly

$\displaystyle\mathbb{E}_{\delta_{x}}\left[\sum_{u\in\mathcal{U}_{t}}f^{2}(X_{t}^{u})\right]=M_{t}(f^{2})(x).$

Secondly, we deal with
$\mathbb{E}_{\delta_{x}}\left[\sum_{w\in\mathcal{U}}\mathbf{1}_{\left\\{b(w)<t\right\\}}\,I_{t}(w)\right]$.
For any $w\in\mathcal{U}$ and for any $i\in\\{1,2\\}$, we use strong Markov
property to get

$\displaystyle\mathbf{1}_{\left\\{b(w)<t\right\\}}\,\mathbb{E}_{\delta_{x}}\left[\sum_{u\in\mathcal{U}_{t}(wi)}f(X_{t}^{u})\,\Big{|}\,b(w),X_{b(w)}^{wi}\right]=\mathbf{1}_{\left\\{b(w)<t\right\\}}\,M_{t-b(w)}f(X_{b(w)}^{wi}).$

For any $w\in\mathcal{U}$, the branching property then yields

$\displaystyle\mathbf{1}_{\left\\{b(w)<t\right\\}}\mathbb{E}_{\delta_{x}}\left[I_{t}(w)\,\big{|}\mathcal{F}_{b(w)},b(w)\right]=2\mathbf{1}_{\left\\{b(w)<t\right\\}}\,M_{t-b(w)}f(X_{b(w)}^{w1})M_{t-b(w)}f(X_{b(w)}^{w2}).$

Combining these identities, we obtain

$\displaystyle\mathbb{E}_{\delta_{x}}\left[\sum_{w\in\mathcal{U}}\mathbf{1}_{\left\\{b(w)<t\right\\}}\,I_{t}(w)\right]$
$\displaystyle=2\mathbb{E}_{\delta_{x}}\left[\sum_{w\in\mathcal{U}}\mathbf{1}_{\left\\{b(w)<t\right\\}}\,M_{t-b(w)}f(X_{b(w)}^{w1})M_{t-b(w)}f(X_{b(w)}^{w2})\right]$
$\displaystyle=2\mathbb{E}_{\delta_{x}}\left[\sum_{w\in\mathcal{U}}\mathbf{1}_{\left\\{b(w)<t\right\\}}\,g(X^{w}_{b(w)-},b(w))\right],$

where we introduce

$\displaystyle
g(X^{w}_{b(w)-},b(w)):=\mathbb{E}_{\delta_{x}}\left[M_{t-b(w)}f(X_{b(w)}^{w1})M_{t-b(w)}f(X_{b(w)}^{w2})\,\big{|}X^{w}_{b(w)-},b(w)\right].$

This function involves the fragmentation event and can be written explicitly
by recalling that, when a cluster of size $n$ splits, the probability that the
size of the first child is $(n-j)$ and the second child is $j$ is
$n/((n-1)\cdot j\cdot(j+1))$. That is, we can obtain

$\displaystyle g(n,s)$ $\displaystyle=\sum_{1\leq j\leq
n-1}\frac{n}{n-1}\frac{1}{j(j+1)}M_{t-s}f(j)M_{t-s}f(n-j).\ $ (32)

Adding that the branching rate of a cluster of size $n$ is $\gamma(n-1)$, we
obtain

$\displaystyle\mathbb{E}_{\delta_{x}}\left[\sum_{w\in\mathcal{U}}\mathbf{1}_{\left\\{b(w)<t\right\\}}\,g(X^{w}_{b(w)-},b(w))\right]$
$\displaystyle=\int_{0}^{t}\sum_{w\in\mathcal{U},n\geq
1}g(n,s)\mathbb{P}_{\delta_{x}}[w\in\mathcal{U}_{s-},\,X^{w}_{b(w)-}=n,\,b(w)\in{\mathrm{d}}s]$
$\displaystyle=\int_{0}^{t}\sum_{w\in\mathcal{U},n\geq
1}g(n,s)\mathbb{P}_{\delta_{x}}[w\in\mathcal{U}_{s-},X^{w}_{b(w)-}=n]\,\gamma(n-1){\mathrm{d}}s$
$\displaystyle=\int_{0}^{t}\sum_{n\geq
1}g(n,s)\gamma(n-1)M_{s}(x,n)\,{\mathrm{d}}s.$

This equation and (32) give us the expression of $\kappa$. The proof is thus
completed. ∎

With the help of this $L^{2}$ expression, we can deal with the martingale
associated to the harmonic function $h$.

######  Proposition 4.2.

The process $(\mathscr{M}_{t})_{t\geq 0}$ defined as

$\displaystyle\mathscr{M}_{t}=e^{-\lambda
t}\left\langle{X_{t},h}\right\rangle,$ (33)

is a non-negative martingale, which converges almost surely to a non-negative
finite random variable $W$ as $t$ tends to infinity. Moreover, if $\lambda>0$,
$(\mathscr{M}_{t})_{t\geq 0}$ converges in the $L^{2}$ norm to $W$.

###### Proof.

The martingale property is classical and the proof is given for the sake of
completeness. Recall the notation $\mathcal{U}_{t}$ and $X_{t}^{u}$ introduced
in Section 3.1. For any $u\in\mathcal{U}_{t},$ recall that
$\mathcal{U}_{t+s}(u)$ is the set of labels of all the clusters active at time
$(t+s)$ issued from the cluster labeled by $u$ active at time $t$. Then we
have

$\displaystyle\mathbb{E}\left[\mathscr{M}_{t+s}\,|\mathcal{F}_{t}\right]$
$\displaystyle=e^{-\lambda(t+s)}\mathbb{E}\left[\sum_{u\in\mathcal{U}_{t+s}}h(X^{u}_{t+s})\,\Big{|}\mathcal{F}_{t}\right]$
$\displaystyle=e^{-\lambda(t+s)}\sum_{u\in\mathcal{U}_{t}}\mathbb{E}_{\delta_{X^{u}_{t}}}\left[\sum_{v\in\mathcal{U}_{t+s}(u)}h(X^{v}_{t+s})\,\Big{|}\mathcal{F}_{t}\right]$
$\displaystyle=e^{-\lambda(t+s)}\sum_{u\in\mathcal{U}_{t}}M_{s}h(X^{u}_{t})=\mathscr{M}_{t},$

since $M_{s}h=e^{\lambda s}h$. As $\mathscr{M}$ is non-negative, it converges
almost surely to a finite random variable.

Let us now prove the $L^{2}$ convergence. We apply Lemma 4.3 with $x=1$ and
obtain

$\displaystyle\mathbb{E}\left[\langle X_{t},h\rangle^{2}\right]$
$\displaystyle=M_{t}(h^{2})(1)+2\int_{0}^{t}\sum_{n\geq
1}M_{s}(1,n)\left(\sum_{1\leq j\leq
n-1}\kappa(n,j)M_{t-s}h(j)M_{t-s}h(n-j)\right)\,{\mathrm{d}}s$
$\displaystyle=M_{t}(h^{2})(1)+2e^{2\lambda t}J_{t},$

where

$\displaystyle J_{t}=\int_{0}^{t}\sum_{n\geq 1}e^{-2\lambda
s}M_{s}(1,n)\left(\sum_{1\leq j\leq
n-1}\kappa(n,j)h(j)h(n-j)\right)\,{\mathrm{d}}s.$

Using that $\kappa(n,j)=\gamma n/(j(j+1))$ for all $n\geq 1$ and $1\leq j\leq
n-1$ and that $h$ is bounded given in Proposition 4.1, we get that
$\sum_{1\leq j\leq n-1}\kappa(n,j)h(j)h(n-j)$ grows at most linearly with $n$.
Moreover we can apply (30) to control the gap between $e^{-\lambda
s}M_{s}(1,n)$ and $h(1)\pi(n)$. Combining these estimates ensures that for any
$p>2$, there exists $C>0$ such that

$\displaystyle 0\leq J_{t}\leq C\int_{0}^{t}e^{-\lambda s}\sum_{n\geq
1}n\gamma(h(1)\pi(n)+n^{-p}){\mathrm{d}}s,\quad\forall n\geq 1,t\geq 0,$

which is uniformly upper bounded for all $n\geq 1,t\geq 0$. Adding that
$\pi(n)$ decreases to $0$ faster than $n^{-3}$ ensures that $\sup_{t\geq
0}J_{t}<\infty.$ Finally

$\mathbb{E}\left[(\mathscr{M}_{t})^{2}\right]=e^{-2\lambda
t}\mathbb{E}\left[\langle X_{t},h\rangle^{2}\right]=e^{-2\lambda
t}M_{t}(h^{2})(1)+J_{t},$

and we apply (30) to conclude that $\sup_{t\geq
0}\mathbb{E}\left[(\mathscr{M}_{t})^{2}\right]<\infty.$ Then by the martingale
convergence theorem, we obtain that $\mathbb{E}[W^{2}]<\infty$ and
$(\mathscr{M}_{t})_{t\geq 0}$ converges in the $L^{2}$ norm to $W$. ∎

######  Remark 2.

In fact, Proposition 4.2 holds for $(X_{t})_{t\geq 0}$ under
$\mathbb{P}_{\delta_{n}}$ for any $n\geq 1$, with the limit $W$ depending on
$n$. The proof is essentially the same as for $n=1$. We state here the result
under $\mathbb{P}=\mathbb{P}_{\delta_{1}}$ so that the limit $W$ is consistent
with Theorem 2.2 and Corollary 1.

###  4.4 Proof of Theorem 2.1

With the help of Proposition 4.1 and Proposition 4.2, we are now ready to
prove our Theorem 2.1.

###### Proof of Theorem 2.1.

We use the classical notation $\varliminf$ and $\varlimsup$ respectively for
the limit inferior and limit superior of a sequence defined on discrete or
continuous time. We notice that
$\mathbb{E}[|\mathscr{X}_{t}|]=\mathbb{E}[\left\langle{X_{t},1}\right\rangle]=\sum_{j=1}^{\infty}M_{t}(1,j)$
and we apply (30) with $f\equiv\mathbf{1}$ (constant function) and $n=p=1$.
This ensures that
${\lim_{t\rightarrow\infty}\log(\mathbb{E}[|\mathscr{X}_{t}|])/t=\lambda}$.

To study the limit of $\log(\mathbb{E}[|\mathscr{Y}_{t}|])/t$, we use
Kolmogorov’s equation. More precisely, following the localization argument in
the proof of Lemma 4.1 $(ii)-(iii)$, we check that
${F_{f,g}(\mu,\nu)=\left\langle{\nu,1}\right\rangle}$ belongs to the domain of
the extended generator of $(X_{t},Y_{t})_{t\geq 0}$ (see (9)). Then we get

$\displaystyle\mathbb{E}[|\mathscr{Y}_{t}|]=\mathbb{E}[\left\langle{Y_{t},1}\right\rangle]=\int_{0}^{t}\mathbb{E}[\left\langle{X_{s},\theta[x]}\right\rangle]\,{\mathrm{d}}s=\int_{0}^{t}M_{s}(\theta[x])(1)\,{\mathrm{d}}s,$

and we conclude using (30).

Lastly, we study the the survival probability $\mathbb{P}[\tau=\infty]$.

  * •

In the subcritical phase $\lambda<0$, (30) and the classical first moment
estimate prove that extinction is almost sure.

  * •

In the supercritical phase $\lambda>0$, we use the $L^{2}$ martingale of
Proposition 4.2 and the optional stopping theorem to get

$\displaystyle
h(1)=\mathbb{E}\left[\lim_{t\to\infty}e^{-\lambda(t\wedge\tau)}\left\langle{X_{t\wedge\tau},h}\right\rangle\right]=\mathbb{E}\left[W\mathbf{1}_{\left\\{\tau=\infty\right\\}}\right].$

Adding that $h>0$ from Proposition 4.1 implies that
$\mathbb{P}[\tau=\infty]>0$ and $\mathbb{P}[W>0]>0$.

  * •

In the critical phase $\lambda=0$, we first observe that the probability of
extinction starting from one cluster, within a unit time, is greater than a
positive constant (uniformly with respect to the cluster size that we started
with). Besides, $\varliminf_{t\rightarrow\infty}|\mathscr{X}_{t}|<\infty$ a.s.
since Fatou’s lemma ensures that

$\mathbb{E}[\varliminf_{t\rightarrow\infty}|\mathscr{X}_{t}|]=\mathbb{E}[\varliminf_{t\rightarrow\infty}\left\langle{X_{t},\mathbf{1}}\right\rangle]\leq\lim_{t\rightarrow\infty}\mathbb{E}[\left\langle{X_{t},h}\right\rangle]/\inf_{n\geq
1}h(n)=h(1)/\inf_{n\geq 1}h(n)<\infty.$

This ensures that extinction occurs a.s. in finite time by a classical
argument for Markov processes with accessible absorbing points. Indeed, for
any $K\geq 1$, on the event
${\varliminf_{t\rightarrow\infty}|\mathscr{X}_{t}|\leq K}$, extinction occurs
a.s. since we can construct an infinite sequence of stopping times $T_{n}$
(separated at least by a unit time) such that $|\mathscr{X}_{T_{n}}|\leq K$
and for each $n$, extinction occurs with a positive (lower bounded)
probability during $[T_{n},T_{n}+1]$.

∎

##  5 Strong convergences

The Perron’s root $\lambda\in\mathbb{R}$ and associated eigenelements have
been characterized in Proposition 4.1. The sign of $\lambda$ determines if the
first moment semigroup goes to $0$ or infinity. We turn now to trajectorial
results and find first an equivalent statement of being extinct
(Kesten–Stigum). Then we focus on the supercritical regime $\lambda>0$ and
prove strong law of large numbers for the distribution of clusters.

###  5.1 Kesten–Stigum limit theorem

A fundamental and classical question is whether $\\{W>0\\}$ coincides with
survival event $\\{\tau=\infty\\}$ or not. This is one part of the
Kesten–Stigum theorem in branching processes, see e.g. [26, 27]. In our case,
the $L^{2}$ computation ensures that $\mathbb{P}[W>0]>0$ and we will get a
positive answer to the question posed.

######  Proposition 5.1.

Assume $\lambda>0$. Then $\mathbb{P}[W>0]>0$, and
$\\{W>0\\}=\\{\tau=\infty\\}\,\text{a.s.}$.

###### Proof.

The fact that $\mathbb{P}[W>0]>0$ comes from the $L^{2}$ martingale
convergence in Proposition 4.2. Besides, it is straightforward to see that
$\\{W>0\\}\subset\\{\tau=\infty\\}$. Thus if
$\mathbb{P}[W>0]=\mathbb{P}[\tau=\infty]$, the proof is done. The lines of the
proof are classical, even though the sizes of clusters being unbounded
requires some specific arguments.

First, we use the fact that any cluster can be isolated (before any other
event happening to it) during a unit time interval, with the isolation
probability lower bounded by a positive value for all sizes. As a result, the
number of clusters has to tend to infinity to survive:

$\\{\tau=\infty\\}=\\{\lim_{t\to\infty}|\mathscr{X}_{t}|=\infty\\}\quad\text{a.s.}.$

Second, we deduce from the above result that the number of clusters of size
$1$ tends to infinity on the survival event. Indeed, during a unit time
interval, clusters of size one have a positive probability to stay unchanged
and other clusters have a positive probability to create (by fragmentation)
one cluster of size one, and the latter probability is lower bounded uniformly
with respect to the size $n\geq 2$. By independence of clusters and Markov
inequality, this ensures that

$\displaystyle\varlimsup_{t\rightarrow\infty}X_{t}(1)=\infty\quad\text{a.s.}\text{
on }\\{\tau=\infty\\}.$

On the event $\\{\tau=\infty\\}$, we can thus define a sequence of stopping
times for $N\geq 1$

$\displaystyle\tau_{N}:=\inf\\{t:X_{t}(1)\geq N\\}.$

We obtain for $t\geq\tau_{N}$

$\displaystyle e^{-\lambda t}\left\langle{X_{t},h}\right\rangle$
$\displaystyle\geq
e^{-\lambda\tau_{N}}\sum_{u\in\mathcal{A}_{N}}e^{-\lambda(t-\tau_{N})}\sum_{v\in\mathcal{U}_{t}(u)}h(X^{v}_{t}),$

where $\mathcal{A}_{N}:=\\{u\in\mathcal{U}_{\tau_{N}}:X^{u}_{\tau_{N}}=1\\}.$
By Proposition 4.2,
${e^{-\lambda(t-\tau_{N})}\sum_{v\in\mathcal{U}_{t}(u)}h(X^{v}_{t})}$
converges to a non-negative random variable denoted by $W(u)$ which is equal
in law to $W$. Besides, $\\{W(u)\\}_{u\in\mathcal{A}_{N}}$ are i.i.d. random
variables. Thus we have

$\displaystyle\mathbb{P}[W=0,\tau=\infty]$
$\displaystyle\leq\mathbb{P}[W=0,\tau_{N}<\infty]$
$\displaystyle\leq\mathbb{P}[\\{\tau_{N}<\infty\\}\cap\\{W(u)=0,\forall
u\in\mathcal{A}_{N}\\}]\leq(\mathbb{P}[W=0])^{N}.$

As a result, $\mathbb{P}[W=0]=1$ or $\mathbb{P}[W=0,\tau=\infty]=0$ (by
letting $N\to\infty$). Only the latter is possible since we know that
$\mathbb{P}[W>0]>0.$ The proof is thus finished. ∎

###  5.2 Strong law of large numbers for the size process of active clusters

In this part, we prove Theorem 2.2 using the estimates of the first moment
semigroup, the $L^{2}$ estimates and the martingale associated to the harmonic
function. The $L^{2}$ estimates ensure weak convergence, and the convergence
speed obtained entails strong convergence of subsequences. The strategy is
then to control fluctuations to prove the strong convergence along
$t\in\mathbb{R}_{+}$. To that purpose, we follow the idea from [4]. We divide
the proof into three steps.

###### Proof of Theorem 2.2.

Step 1: $L^{2}$ convergence. We prove first the $L^{2}$ convergence of
$e^{-\lambda t}\left\langle{X_{t},f}\right\rangle$ to
$W\left\langle{\pi,f}\right\rangle$ for any $f\in\mathcal{B}_{p}$ with $p>0$.
We develop the difference as follows

$e^{-\lambda
t}\left\langle{X_{t},f}\right\rangle-W\left\langle{\pi,f}\right\rangle\\\
=\underbrace{e^{-\lambda t}\left\langle{X_{t},f}\right\rangle-e^{-\lambda
t}\left\langle{X_{t},h}\right\rangle\left\langle{\pi,f}\right\rangle}_{{\mathbf{I}}}+\underbrace{e^{-\lambda
t}\left\langle{X_{t},h}\right\rangle\left\langle{\pi,f}\right\rangle-W\left\langle{\pi,f}\right\rangle}_{{\mathbf{II}}}.$
(34)

The second term $\mathbf{II}$ is nothing but
$(\mathscr{M}_{t}-W)\left\langle{\pi,f}\right\rangle$, which converges in
$L^{2}$ to $0$ by Proposition 4.2. We only have to prove the $L^{2}$
convergence of the term $\mathbf{I}$ to $0$. Denoting
$g:=f-\left\langle{\pi,f}\right\rangle h$, Lemma 4.3 yields

$\displaystyle e^{2\lambda t}\mathbb{E}[|\text{$\mathbf{I}$}|^{2}]$
$\displaystyle=\mathbb{E}\left[\left\langle{X_{t},g}\right\rangle^{2}\right]=M_{t}(g^{2})(1)+J_{t},$
(35)

where

$\displaystyle J_{t}:=2\int_{0}^{t}\sum_{n\geq 1}M_{s}(1,n)\left(\sum_{1\leq
j\leq n-1}\kappa(n,j)M_{t-s}g(j)M_{t-s}g(n-j)\right)\,{\mathrm{d}}s.$

Recall that $\parallel f\parallel_{p}=\sum_{m\geq
1}|f(m)|m^{-(p+2)}\in(-\infty,\infty)$. Observe that $g\in\mathcal{B}_{p}$ and
let ${p^{\prime}>2p+8}$. By (30), there exists $C^{\prime}>0$ such that for
any $n\in\mathbb{N}_{+}$ and $s,t\geq 0,$

$\displaystyle|e^{-\lambda t}M_{t}g(n)-h(n)\left\langle{\pi,g}\right\rangle|$
$\displaystyle\leq C^{\prime}n^{p+2}\parallel g\parallel_{p}e^{-wt},$
$\displaystyle|e^{-\lambda s}M_{s}(1,n)-h(1)\pi(n)|$ $\displaystyle\leq
C^{\prime}n^{-p^{\prime}}e^{-ws}.$

Since $\left\langle{\pi,g}\right\rangle=0$ and $\kappa(n,j)\leq\gamma n$,
using the above two displays, there exists $C_{1}>0$ such that

$\displaystyle|J_{t}|$ $\displaystyle\leq C_{1}\parallel
g\parallel_{p}e^{2(\lambda-\omega)t}\int_{0}^{t}e^{(2\omega-\lambda)s}\sum_{n\geq
1}n^{2p+6}(h(1)\pi(n)+n^{-p^{\prime}})\,{\mathrm{d}}s.$

Using the second statement in (29) and $p^{\prime}>2p+8$, the sum $\sum_{n\geq
1}n^{2p+6}(h(1)\pi(n)+n^{-p^{\prime}})$ in the above display is finite and
there exists $C_{2}>0$ such that

$|J_{t}|\leq C_{2}\parallel
g\parallel_{p}e^{2(\lambda-\omega)t}\int_{0}^{t}e^{(2\omega-\lambda)s}{\mathrm{d}}s,\quad\forall
n\geq 1,t\geq 0.$

Moreover, by (30), $e^{-2\lambda
t}M_{t}(g^{2})(1)\leq\left\langle{\pi,g^{2}}\right\rangle h(1)e^{-\lambda
t}+C\parallel g^{2}\parallel_{p}e^{-(\lambda+\omega)t},$ for any $t\geq 0.$
Plugging in these estimates to (35), we see that there exists $C_{3}>0$ such
that

$\mathbb{E}[|\mathbf{I}|^{2}]\leq
C_{3}\left(\left\langle{\pi,g^{2}}\right\rangle+\parallel
g^{2}\parallel_{p}+\parallel g\parallel_{p}\right)te^{-(\lambda\wedge
2\omega)t},\quad\forall t\geq 0.$ (36)

Note that $C_{1},C_{2},C_{3}$ do not depend on $f$ and $t$. Then step 1 is
finished.

######  Remark 3.

A byproduct of (36) and Proposition 4.2 is that, for the case $\lambda>0$
there exists a constant $C_{0}>0$ and an exponent $\sigma\in(0,\lambda)$, such
that for any $f\in\mathcal{B}_{p}$,

$\displaystyle\mathbb{E}\left[\left\langle{X_{t},f}\right\rangle^{2}\right]\leq
C_{0}e^{2\lambda
t}\Big{(}|\left\langle{\pi,f}\right\rangle|^{2}+\left(\left\langle{\pi,g^{2}}\right\rangle+\parallel
g^{2}\parallel_{p}+\parallel g\parallel_{p}\right)e^{-\sigma
t}\Big{)},\quad\forall t\geq 0,$ (37)

with the notation $g=f-\left\langle{\pi,f}\right\rangle h$.

Step 2: Almost sure convergence for one type. The main idea is to extend an
elegant argument from [4] to our countable-type branching process.

First, we establish an almost sure convergence for a discrete scheme, using
the speed of convergence obtained from the $L^{2}$ estimates. We can pick a
step size $\Delta>0$ and apply the decomposition (34). Then the martingale
part $\mathbf{II}$ converges to $0$ almost surely, and for the term
$\mathbf{I}$, (36) yields

$\displaystyle\mathbb{E}\left[|e^{-\lambda
k\Delta}\left\langle{X_{k\Delta},f}\right\rangle-e^{-\lambda
k\Delta}\left\langle{X_{k\Delta},h}\right\rangle\left\langle{\pi,f}\right\rangle|^{2}\right]\leq
Ck\Delta e^{-(\lambda\wedge 2\omega)k\Delta},\quad\forall k\geq 0.$ (38)

By Borel–Cantelli lemma, we get

${e^{-\lambda
k\Delta}\left\langle{X_{k\Delta},f}\right\rangle\xrightarrow{k\to\infty}W\left\langle{\pi,f}\right\rangle},\qquad\text{
almost surely. }$ (39)

Recall that on the event $\\{W=0\\}$, by Proposition 5.1, the extinction
occurs a.s.. So we focus on the event $\\{W>0\\}$. Let us first prove that

$\displaystyle e^{-\lambda
t}X_{t}(n)\xrightarrow{t\to\infty}W\pi(n),\qquad\text{almost surely}.$ (40)

Given the almost sure convergence in discrete times, we need to control the
fluctuations in the intervals $[k\Delta,(k+1)\Delta)$. A nice observation in
[4] is that we only need to prove the following sufficient (and necessary)
condition

$\displaystyle\varliminf\limits_{t\to\infty}e^{-\lambda t}X_{t}(n)\geq
W\pi(n),\qquad\text{almost surely for all }n\geq 1.$ (41)

We first show that (41) implies (40) using that the martingale convergence
controls the dissipation of mass. To this purpose, for any fixed $n\geq 1$,
consider any sequence of (random) times $(t_{k})_{k\in\mathbb{N}_{+}}$ such
that $\varlimsup\limits_{t\to\infty}e^{-\lambda
t}X_{t}(n)=\lim_{k\to\infty}e^{-\lambda t_{k}}X_{t_{k}}(n)$. Proposition 4.2
and Fatou’s lemma and (41) ensure

$\begin{split}\varlimsup\limits_{t\to\infty}e^{-\lambda
t}X_{t}(n)h(n)&=\lim_{k\rightarrow\infty}\left(\sum_{i\geq 1}e^{-\lambda
t_{k}}X_{t_{k}}(i)h(i)-\sum_{i\geq 1,i\neq n}e^{-\lambda
t_{k}}X_{t_{k}}(i)h(i)\right)\\\ &\leq W-\sum_{i\geq 1,i\neq
n}\varliminf\limits_{k\to\infty}e^{-\lambda t_{k}}X_{t_{k}}(i)h(i)\\\ &\leq
W-\sum_{i\geq 1,i\neq n}W\pi(i)h(i)=W\pi(n)h(n).\end{split}$ (42)

Then together with (41) we obtain (40).

We need now to prove (41) following the argument in [4]. Recall $\Delta>0$ is
the time step size. The proof relies on the following lower bound:

$\displaystyle\forall t\in[k\Delta,(k+1)\Delta),\qquad X_{t}(n)\geq
X_{k\Delta}(n)-N_{k,\Delta}(n),$ (43)

where $N_{k,\Delta}(n)$ is the number of active clusters of size $n$ at time
$k\Delta$ that will encounter at least one event within
$(k\Delta,(k+1)\Delta)$. Indeed, to prove (41), we can find a lower bound for
$\varliminf_{t\to\infty}e^{-\lambda t}X_{t}(n)$ using the above display. More
precisely,

$\displaystyle\varliminf\limits_{t\to\infty}e^{-\lambda
t}X_{t}(n)\geq\varliminf\limits_{k\to\infty}e^{-\lambda(k+1)\Delta}X_{k\Delta}(n)-\varlimsup\limits_{k\to\infty}e^{-\lambda
k\Delta}N_{k,\Delta}(n).$

Using (39) for the first term on the right hand side, we obtain

$\varliminf\limits_{t\to\infty}e^{-\lambda t}X_{t}(n)\geq
e^{-\lambda\Delta}\pi(n)W-\varlimsup\limits_{k\to\infty}e^{-\lambda
k\Delta}N_{k,\Delta}(n).$

It suffices to prove that $\lim_{k\to\infty}e^{-\lambda
k\Delta}N_{k,\Delta}(n)=0$ a.s. and then let $\Delta$ go to $0$. We introduce

$\displaystyle
D_{k}=D_{\Delta,n,k,\varepsilon}:=\left\\{N_{k,\Delta}(n)>\varepsilon
X_{k\Delta}(n),\quad X_{k\Delta}(n)>k\right\\},\quad k\geq 1.$

By branching property, we know that

$\displaystyle
N_{k,\Delta}(n)\stackrel{{\scriptstyle\text{(d)}}}{{=}}\sum_{i=1}^{X_{k\Delta}(n)}\xi_{i},$

where $\\{\xi_{i}\\}_{i\geq 1}$ are i.i.d. Bernoulli random variables,
independent of $X_{k\Delta}(n)$ and

${\mathbb{P}[\xi_{i}=0]=1-\mathbb{P}[\xi_{i}=1]=\exp(-r_{n}\Delta)},\qquad{r_{n}=(\beta+\theta+\gamma)n-\gamma}.$

Indeed, $r_{n}$ is the total jump rate of an active cluster of size $n$.
Choose $\Delta$ small such that ${\mathbb{P}[\xi_{i}=1]<\varepsilon.}$ Then,

$\displaystyle\sum_{k\geq 1}\mathbb{P}[D_{k}]\leq\sum_{k\geq
1}\mathbb{P}\left[N_{k,\Delta}(n)>\varepsilon
X_{k\Delta}(n)\,\,|\,\,X_{k\Delta}(n)>k\right]<\infty,$

using that $\mathbb{P}[\sum_{i=1}^{k}\xi_{i}>\varepsilon k]$ decreases
exponentially as $k$ grows thanks to Hoeffding inequality. Borel–Cantelli
lemma then ensures that a.s. $D_{k}$ happens a finite number of times.

Recalling now from (39) that $X_{k\Delta}(n)$ grows exponentially on the event
$\\{W>0\\}$, so $\\{X_{k\Delta}\leq k\\}$ also happens a.s. a finite number of
times. As a result, a.s. on the event $\\{W>0\\}$, we have
$N_{k,\Delta}(n)\leq\varepsilon X_{k\Delta}(n)$ for $k$ large enough. We
conclude that $\lim_{k\to\infty}e^{-\lambda k\Delta}N_{k,\Delta}=0$ a.s. on
the event $\\{W>0\\}$ since $\varepsilon$ can be arbitrarily small. This ends
the proof of (41) and we obtain (40). We have thus also proved Theorem 2.2 for
functions with bounded support.

Step 3: Almost surely convergence - general test function. It suffices to
prove the results for functions in $\mathcal{B}_{p}$ with $p\geq 1$, thanks to
the fact $\mathcal{B}=\cup_{p>0}\mathcal{B}_{p}$. For convenience, we consider
instead

$\displaystyle\overline{\mathcal{B}}_{p}:=\left\\{f:\mathbb{N}_{+}\rightarrow\mathbb{R},\,\sup_{n\geq
1}|f(n)|/n^{p}<1\right\\},\quad p\geq 1.$ (44)

Indeed, $\overline{\mathcal{B}}_{p}\subset\mathcal{B}_{p}$ and for any
$f\in\mathcal{B}_{p}$ there exists $g\in\overline{\mathcal{B}}_{p}$ and
$c\in\mathbb{R}$ such that $f=cg.$

We define the cutoff operator at some level $K\in\mathbb{N}_{+}$

$\displaystyle f_{\leq K}(n):=f(n)\mathbf{1}_{\left\\{n\leq K\right\\}},\qquad
f_{>K}(n):=f(n)\mathbf{1}_{\left\\{n>K\right\\}}.$ (45)

First, using (40), we obtain

$\displaystyle\sup_{f\in\overline{\mathcal{B}}_{p}}|e^{-\lambda
t}\left\langle{X_{t},f_{\leq K}}\right\rangle-W\left\langle{\pi,f_{\leq
K}}\right\rangle|$ $\displaystyle\leq K^{p}\sum_{n=1}^{K}|e^{-\lambda
t}X_{t}(n)-W\pi(n)|\xrightarrow{t\to\infty}0,\quad a.s..$

Second, recall the definition of $[x^{p}]$ and $[x]$ introduced above (11).
Then

$\displaystyle\sup_{f\in\overline{\mathcal{B}}_{p}}|e^{-\lambda
t}\left\langle{X_{t},f_{>K}}\right\rangle|\leq e^{-\lambda
t}\left\langle{X_{t},[x^{p}]_{>K}}\right\rangle,\qquad\sup_{f\in\overline{\mathcal{B}}_{p}}|W\left\langle{\pi,f_{>K}}\right\rangle|\leq
W\left\langle{\pi,[x^{p}]_{>K}}\right\rangle.$

Combining these estimates and

$|e^{-\lambda
t}\left\langle{X_{t},f}\right\rangle-W\left\langle{\pi,f}\right\rangle|\\\
\leq|e^{-\lambda t}\left\langle{X_{t},f_{>K}}\right\rangle|+|e^{-\lambda
t}\left\langle{X_{t},f_{\leq K}}\right\rangle-W\left\langle{\pi,f_{\leq
K}}\right\rangle|+|W\left\langle{\pi,[x^{p}]_{>K}}\right\rangle|,$

it yields

$\displaystyle\varlimsup_{t\rightarrow\infty}\sup_{f\in\overline{\mathcal{B}}_{p}}|e^{-\lambda
t}\left\langle{X_{t},f}\right\rangle-W\left\langle{\pi,f}\right\rangle|\leq\varlimsup\limits_{t\to\infty}e^{-\lambda
t}\left\langle{X_{t},[x^{p}]_{>K}}\right\rangle+W\left\langle{\pi,f_{>K}}\right\rangle,$

for any $K\geq 1$. To show that the right hand side in the above display goes
to $0$ as $K$ goes to infinity, it suffices to prove:

$\displaystyle\lim_{K\to\infty}\varlimsup\limits_{t\to\infty}e^{-\lambda
t}\left\langle{X_{t},[x^{p}]_{>K}}\right\rangle=0.$ (46)

Since (39) ensures

$\displaystyle\lim_{k\to\infty}e^{-\lambda
k\Delta}\left\langle{X_{k\Delta},[x^{p}]_{>K}}\right\rangle=W\left\langle{\pi,[x^{p}]_{>K}}\right\rangle,$
(47)

we just need to control what happens on the time intervals
$[k\Delta,(k+1)\Delta)$.

For that purpose, we use a coupling argument. On every interval
$[k\Delta,(k+1)\Delta)$, we consider a size process $\widetilde{X}_{t}$
starting at time $k\Delta$ with the same value
$\widetilde{X}_{k\Delta}:=X_{k\Delta}$; we let the rates of fragmentation and
isolation be zero in $\widetilde{X}_{t}$, while for any cluster at $k\Delta$,
the growth events occurring to it on $(k\Delta,(k+1)\Delta)$ are constructed
by the common exponential clocks in $\widetilde{X}_{t}$ and $X_{t}$, until it
gets isolated or fragmented in the latter. Notice that the isolation events
make negative contribution to $\mathcal{L}([x^{p}]),p\geq 1$, so are the
fragmentation events because ${(a+b)^{p}\geq a^{p}+b^{p}}$ for all
$a,b>0,p\geq 1$. Therefore, we obtain

$\displaystyle\sup_{t\in[k\Delta,(k+1)\Delta)}\left\langle{X_{t},[x^{p}]_{>K}}\right\rangle\leq\sup_{t\in[k\Delta,(k+1)\Delta)}\left\langle{\widetilde{X}_{t},[x^{p}]_{>K}}\right\rangle.$

The term on the right hand side is monotone in $t$ and we get

$\displaystyle\sup_{t\in[k\Delta,(k+1)\Delta)}\left\langle{X_{t},[x^{p}]_{>K}}\right\rangle\leq\left\langle{\widetilde{X}_{(k+1)\Delta-},[x^{p}]_{>K}}\right\rangle.$

As a result, setting

$B_{k}=B_{\Delta,n,k}^{K}:=\left\\{\left\langle{\widetilde{X}_{(k+1)\Delta-},[x^{p}]_{>K}}\right\rangle>2\left\langle{X_{k\Delta},[x^{p}]_{>K}}\right\rangle\right\\},$

it suffices to prove

$\mathbb{P}[\\{\text{i.o.\ }B_{k}\\}\cap\\{W>0\\}]=0,$ (48)

to get that $\varlimsup\limits_{t\to\infty}e^{-\lambda
t}\left\langle{X_{t},[x^{p}]_{>K}}\right\rangle\leq
2\lim_{k\to\infty}e^{-\lambda
k\Delta}\left\langle{X_{k\Delta},[x^{p}]_{>K}}\right\rangle$ a.s.. Together
with (47), we can conclude that (46) holds.

To this purpose, we use a truncation technique. Define

$C_{k}=C^{K}_{\Delta,n,k,\varepsilon}:=\left\\{e^{-\lambda
k\Delta}\left\langle{X_{k\Delta},[x^{p}]_{>K}}\right\rangle\geq\varepsilon\right\\}\cap\left\\{e^{-\lambda
k\Delta}\left\langle{X_{k\Delta},[x^{2p}]}\right\rangle\leq
1/\varepsilon\right\\},$

for $\varepsilon>0$. We do a split

$\displaystyle\mathbb{P}[\\{\text{i.o.\
}B_{k}\\}\cap\\{W>0\\}]\leq\mathbb{P}[\\{\text{i.o.\ }B_{k}\cap
C_{k}\\}]+\mathbb{P}[\\{\text{i.o.\ }B_{k}\cap(C_{k})^{c}\\}\cap\\{W>0\\}].$
(49)

The second term on the right hand side has the following upper bound thanks to
(39) and dominated convergence theorem

$\mathbb{P}\Big{[}W\left\langle{\pi,[x^{p}]_{>K}}\right\rangle\in(0,2\varepsilon)\Big{]}+\mathbb{P}\Big{[}W\left\langle{\pi,[x^{2p}]}\right\rangle>1/(2\varepsilon)\Big{]},$

which converges to $0$ as $\varepsilon\to 0.$

We now deal with the first term on the right hand side in (49). Defining

$Z_{k,p,K}:=\left\langle{\widetilde{X}_{(k+1)\Delta-},[x^{p}]_{>K}}\right\rangle-\left\langle{\widetilde{X}_{k\Delta},[x^{p}]_{>K}}\right\rangle,$

and using Markov inequality,

$\begin{split}\mathbb{P}[B_{k}\,\,|\,\,\mathcal{F}_{k\Delta}]&=\mathbb{P}\left[\left\langle{\widetilde{X}_{(k+1)\Delta-},[x^{p}]_{>K}}\right\rangle-\left\langle{\widetilde{X}_{k\Delta},[x^{p}]_{>K}}\right\rangle>\left\langle{\widetilde{X}_{k\Delta},[x^{p}]_{>K}}\right\rangle\,\,\Big{|}\,\,\mathcal{F}_{k\Delta}\right]\\\
&\leq\frac{\operatorname{var}[Z_{k,p,K}|\,\,\mathcal{F}_{k\Delta}]}{\left(\left\langle{\widetilde{X}_{k\Delta},[x^{p}]_{>K}}\right\rangle-\mathbb{E}[Z_{k,p,K}|\mathcal{F}_{k\Delta}]\right)^{2}}.\end{split}$
(50)

We need now to evaluate the conditional expectation and variance. We will
apply the following lemma whose proof will be provided in Appendix A.

######  Lemma 5.1.

For any $k,K\in\mathbb{N}_{+}$ and $p\geq 1$, we have

$\displaystyle\mathbb{E}[Z_{k,p,K}\,\,|\,\,\mathcal{F}_{k\Delta}]$
$\displaystyle\leq C_{\Delta}\left\langle{X_{k\Delta},[x^{p}]}\right\rangle,$
(51)

where $C_{\Delta}=e^{2^{p-1}p\beta\Delta}-1+(1-e^{-\beta\Delta K})K^{p}$ and

$\displaystyle\operatorname{var}\left[Z_{k,p,K}\,\,|\,\,\mathcal{F}_{k\Delta}\right]\leq
2\beta\Delta(4^{p}p+K^{2p+1})\left\langle{X_{k\Delta},[x^{2p}]}\right\rangle.$
(52)

Plugging in the estimates (51) and (52) in Appendix to (50), we obtain

$\displaystyle 1_{C_{k}}\mathbb{P}[B_{k}\,\,|\,\,\mathcal{F}_{k\Delta}]$
$\displaystyle\leq\frac{2\beta\Delta(4^{p}p+K^{2p+1})\times\varepsilon^{-1}e^{\lambda
k\Delta}}{\Big{(}\varepsilon e^{\lambda
k\Delta}-\beta\Delta(2^{p}p+K^{p+1})\times\varepsilon^{-1}e^{\lambda
k\Delta}\Big{)}^{2}}.$

We pick $\Delta=\varepsilon^{2}$ with $\varepsilon$ small enough. Then we
obtain $\mathbb{P}[B_{k}\,\,|\,\,\mathcal{F}_{k\Delta}]\leq Ce^{-\lambda
k\Delta}$ conditional on $C_{k}$. Adding that
$\mathbb{P}[B_{k}\,\,|\,\,C_{k}]=\mathbb{E}\Big{[}\mathbb{P}[B_{k}\,\,|\,\,\mathcal{F}_{k\Delta}]\,\,\Big{|}\,\,C_{k}\Big{]}$,
we obtain $\sum_{k\geq 1}\mathbb{P}[B_{k}\,\,|\,\,C_{k}]<\infty$. By
Borel–Cantelli lemma, $\mathbb{P}[\text{i.o.\ }B_{k}\cap C_{k}]=0$ for
$\varepsilon$ small enough. Therefore we have proved the term on the right
hand side in (49) is equal to $0$. This implies (48) and the proof for general
test functions is finished. ∎

###  5.3 Strong law of large numbers for the size process of inactive clusters

We prove Corollary 1 in this part. A heuristic argument to obtain the
asymptotic limit is to use the generator (9) and the convergence of $X_{t}$ in
Theorem 2.2 :

$\displaystyle\lim_{s\searrow t}\frac{\mathbb{E}[\langle
Y_{s},f\rangle-\langle Y_{t},f\rangle|\mathcal{F}_{t}]}{s-t}$
$\displaystyle=\theta\langle X_{t},[x]f\rangle\sim_{t\rightarrow\infty}\theta
e^{\lambda t}W\langle\pi,[x]\rangle\langle\widetilde{\pi},f\rangle,$

with $\widetilde{\pi}$ defined in (5). In the sequel, we prove the result,
with a suitable set of test functions using in particular martingale analysis.

###### Proof of Corollary 1.

Let $f\in\mathcal{B}_{p}$ for some fixed $p>0$ throughout the proof. The proof
can be divided into 3 steps. In Step 1, we control the value
$\left\langle{Y_{t},f}\right\rangle$. In Step 2 we prove the result with a
specific function $f=h/[x]$ which gives us a martingale. In Step 3, we
generalize this result to general $f\in\mathcal{B}_{p}$. Let $C_{0}$ be a
constant, independent of $f$, which may change from line to line.

Step 1: $L^{2}$ estimate. We will use again the estimation (37). We can check
that $F_{g,f}(\mu,\nu)=\left\langle{\nu,f}\right\rangle$ belongs to the domain
of extended generator of $(X_{t},Y_{t})_{t\geq 0}$ (see (9)) using the same
localization argument as in the proof of Lemma 4.1. So we get

$\displaystyle\frac{{\mathrm{d}}}{{\mathrm{d}}t}\mathbb{E}\left[\left\langle{Y_{t},f}\right\rangle^{2}\right]$
$\displaystyle=2\theta\mathbb{E}\left[\left\langle{Y_{t},f}\right\rangle\left\langle{X_{t},[x]f}\right\rangle\right]+\theta\mathbb{E}\left[\left\langle{X_{t},[x]f^{2}}\right\rangle\right],\quad\forall
t\geq 0.$

Using Young’s inequality with $\alpha>0$ (to be chosen later),

$\theta\mathbb{E}\left[\left\langle{Y_{t},f}\right\rangle\left\langle{X_{t},[x]f}\right\rangle\right]\leq\alpha\mathbb{E}\left[\left\langle{Y_{t},f}\right\rangle^{2}\right]+\left(\frac{\theta^{2}}{\alpha}\right)\mathbb{E}\left[\left\langle{X_{t},[x]f}\right\rangle^{2}\right].$

Then we use Grönwall’s lemma to get

$\displaystyle\mathbb{E}\left[\left\langle{Y_{t},f}\right\rangle^{2}\right]\leq\int_{0}^{t}e^{\alpha(t-s)}\left(\left(\frac{\theta^{2}}{\alpha}\right)\mathbb{E}\left[\left\langle{X_{s},[x]f}\right\rangle^{2}\right]+\theta\mathbb{E}\left[\left\langle{X_{s},[x]f^{2}}\right\rangle\right]\right)\,{\mathrm{d}}s.$

Combining the $L^{2}$ estimate of $\left\langle{X_{s},[x]f}\right\rangle$
obtained in (37) and the $L^{1}$ estimate of
$\left\langle{X_{t},[x]f^{2}}\right\rangle$ in (30), we get

$\mathbb{E}\left[\left\langle{Y_{t},f}\right\rangle^{2}\right]\leq
C_{0}\int_{0}^{t}e^{\alpha(t-s)}\left(\frac{\theta^{2}}{\alpha}\right)\left(\left\langle{\pi,[x]f}\right\rangle^{2}e^{2\lambda
s}+\parallel f\parallel_{p}e^{(2\lambda-\sigma)s}\right)\,{\mathrm{d}}s\\\
+\int_{0}^{t}e^{\alpha(t-s)}\theta\left(\left\langle{\pi,[x]f^{2}}\right\rangle
e^{\lambda s}+C\parallel
f\parallel_{p}e^{(\lambda-w)s}\right)\,{\mathrm{d}}s.$

We choose $\alpha\in(0,\lambda-\max(\sigma/2,w))$ and conclude that there
exists $C^{\prime}>0$ such that

$\displaystyle\mathbb{E}[\left\langle{Y_{t},f}\right\rangle^{2}]\leq
C^{\prime}\left(\left\langle{\pi,[x]f}\right\rangle^{2}e^{2\lambda
t}+\left\langle{\pi,[x]f^{2}}\right\rangle e^{\lambda t}+\parallel
f\parallel_{p}\left(e^{(2\lambda-\sigma)t}+e^{(\lambda-w)t}\right)\right).$
(53)

Step 2: A martingale for $Y$ which tends to $0$. We introduce the function

$F_{h,h/[x]}(\mu,\nu)=\left\langle{\mu,h}\right\rangle-\left(\frac{\lambda}{\theta}\right)\left\langle{\nu,h/[x]}\right\rangle.$

We can again check that it belongs to the domain of the extended generator of
$(X_{t},Y_{t})_{t\geq 0}$. It turns out to be a harmonic function:
$\mathcal{A}F_{h,g}=0$. Then we obtain that

$\displaystyle
H_{t}:=\left\langle{X_{t},h}\right\rangle-\left(\frac{\lambda}{\theta}\right)\left\langle{Y_{t},h/[x]}\right\rangle.$
(54)

is a martingale with respect to $(\mathscr{F}_{t})_{t\geq 0}$. Let us prove
that $e^{-\lambda t}H_{t}$ converges to $0$ as $t\rightarrow\infty$ and thus
Corollary 1 holds for the specific test function $h/[x]$. This vanishing
property is due to the fact that the two parts in $H$ compensate each other.
We prove first $L^{2}$ convergence using (9):

$\displaystyle\frac{{\mathrm{d}}}{{\mathrm{d}}t}\mathbb{E}[|H_{t}|^{2}]$
$\displaystyle=\mathbb{E}\left[\sum_{n=1}^{\infty}X_{t}(n)\beta
n\left(|H_{t}+h(n+1)-h(n)|^{2}-|H_{t}|^{2}\right)\right]$
$\displaystyle\quad+\mathbb{E}\left[\sum_{n=1}^{\infty}X_{t}(n)\theta
n\left(\left|H_{t}-h(n)-\left(\frac{\lambda}{\theta}\right)h(n)/n\right|^{2}-|H_{t}|^{2}\right)\right]$
$\displaystyle\quad+\mathbb{E}\left[\sum_{n=1}^{\infty}\left(X_{t}(n)\gamma
n\sum_{j=1}^{n-1}\left(\frac{1}{j(j+1)}\right)\left(|H_{t}+h(j)+h(n-j)-h(n)|^{2}-|H_{t}|^{2}\right)\right)\right].$

We develop this equation and recognize the generator $\mathcal{L}$ defined in
(14). As $\mathcal{L}h=\lambda h$, we get

$\displaystyle\frac{{\mathrm{d}}}{{\mathrm{d}}t}\mathbb{E}[|H_{t}|^{2}]$
$\displaystyle=\mathbb{E}\left[\sum_{n=1}^{\infty}X_{t}(n)\left(\beta
n|h(n+1)-h(n)|^{2}+\theta
n\left|h(n)+\left(\frac{\lambda}{\theta}\right)h(n)/n\right|^{2}\right)\right]$
$\displaystyle\qquad+\mathbb{E}\left[\sum_{n=1}^{\infty}X_{t}(n)\left(\gamma
n\sum_{j=1}^{n-1}\left(\frac{1}{j(j+1)}\right)\left|h(j)+h(n-j)-h(n)\right|^{2}\right)\right].$

Since $h$ is bounded (see Proposition 4.1), we obtain

$\displaystyle\mathbb{E}[|H_{t}|^{2}]\leq
C_{0}\int_{0}^{t}\mathbb{E}\left[\left\langle{X_{s},[x]}\right\rangle\right]\,{\mathrm{d}}s\leq
C_{0}e^{\lambda t}.$ (55)

This implies the $L^{2}$ convergence of $e^{-\lambda t}H_{t}$ to $0$ as
$t\rightarrow\infty$. For the almost sure convergence, we set the step size
$\Delta>0$. As $(H_{t})_{t\geq 0}$ is a martingale, for any $\varepsilon>0$,
we combine Markov’s inequality, Doob’s inequality for $H_{t}$ and the estimate
(55) to obtain that

$\displaystyle\mathbb{P}\left[\sup_{t\in[k\Delta,(k+1)\Delta)}|e^{-\lambda
t}H_{t}|>\varepsilon\right]$ $\displaystyle\leq\mathbb{P}\left[e^{-\lambda
k\Delta}\sup_{t\in[k\Delta,(k+1)\Delta)}|H_{t}|>\varepsilon\right]$
$\displaystyle\leq\varepsilon^{-2}e^{-2\lambda
k\Delta}\mathbb{E}\left[\left(\sup_{t\in[k\Delta,(k+1)\Delta)}|H_{t}|\right)^{2}\right]$
$\displaystyle\leq 4\varepsilon^{-2}e^{-2\lambda
k\Delta}\mathbb{E}\left[|H_{(k+1)\Delta}|^{2}\right]$ $\displaystyle\leq
C_{0}\varepsilon^{-2}e^{-\lambda k\Delta}.$

By Borel–Cantelli lemma, we obtain the a.s. convergence of $e^{-\lambda
t}H_{t}$ to $0$.

Step 3: Convergence for general test functions. Now we need to obtain the
result for a general test function $f\in\mathcal{B}_{p}$. The idea is similar:
we define

$\displaystyle
H^{f}_{t}:=\left\langle{X_{t},f}\right\rangle-\left(\frac{\lambda}{\theta}\right)\left\langle{Y_{t},f/[x]}\right\rangle=\left\langle{\pi,f}\right\rangle
H_{t}+A_{t}+B_{t},$ (56)

where $A_{t}=\left\langle{X_{t},f-\left\langle{\pi,f}\right\rangle
h}\right\rangle$ and
$B_{t}=\left(\frac{\lambda}{\theta}\right)\left\langle{Y_{t},(f-\left\langle{\pi,f}\right\rangle
h)/[x]}\right\rangle$. We apply $L^{2}$ estimates from (37) and (53) to ensure
that

$\displaystyle e^{-2\lambda t}\mathbb{E}\left[A_{t}^{2}+B_{t}^{2}\right]\leq
C_{0}\left(1+\sup_{n\geq 1}\frac{|f(n)|}{n^{p}}\right)^{2}\left(e^{-\lambda
t}+e^{-\sigma t}+e^{-(\lambda+\sigma/2)t}\right).$

Here $C_{0}$ may depend on $p$ but not on (the specific choice of) $f.$ By
Borel–Cantelli lemma, this implies the convergence of $e^{-\lambda
t}H^{f}_{t}$ along a subsequence $\\{k\Delta\\}_{k\geq 1}$ with $\Delta>0$

$\displaystyle e^{-\lambda
k\Delta}H^{f}_{k\Delta}\xrightarrow{k\to\infty}0,\qquad\text{ in }L^{2}\text{
and almost surely}.$

We get

$\displaystyle\lim_{k\to\infty}e^{-\lambda
k\Delta}\left\langle{Y_{k\Delta},f/[x]}\right\rangle=\lim_{k\to\infty}\left(\frac{\theta}{\lambda}\right)e^{-\lambda
k\Delta}\left\langle{X_{k\Delta},f}\right\rangle=\left(\frac{\theta}{\lambda}\right)\left\langle{\pi,f}\right\rangle
W,\quad\text{ in }L^{2}\text{ and a.s.}.$

Finally, to obtain the convergence along $t\in\mathbb{R}_{+}$, we decompose
$f$ as the difference of two positive functions $f=f^{+}-f^{-}$ and use that
$Y_{t}$ is increasing in $t$:

$\displaystyle\forall t\in[k\Delta,(k+1)\Delta),\qquad
e^{-\lambda(k+1)\Delta}\left\langle{Y_{k\Delta},f^{+}}\right\rangle\leq
e^{-\lambda t}\left\langle{Y_{t},f^{+}}\right\rangle\leq e^{-\lambda
k\Delta}\left\langle{Y_{(k+1)\Delta},f^{+}}\right\rangle.$

We obtain then

$e^{-\lambda\Delta}\left(\frac{\theta}{\lambda}\right)\left\langle{\pi,[x]f^{+}}\right\rangle
W\leq\lim_{k\to\infty}e^{-\lambda(k+1)\Delta}\left\langle{Y_{k\Delta},f^{+}}\right\rangle\leq\varliminf_{t\to\infty}e^{-\lambda
t}\left\langle{Y_{t},f^{+}}\right\rangle\\\
\leq\varlimsup_{t\to\infty}e^{-\lambda
t}\left\langle{Y_{t},f^{+}}\right\rangle\leq\lim_{k\to\infty}e^{-\lambda
k\Delta}\left\langle{Y_{(k+1)\Delta},f^{+}}\right\rangle=e^{\lambda\Delta}\left(\frac{\theta}{\lambda}\right)\left\langle{\pi,[x]f^{+}}\right\rangle
W.$

We take $\Delta\searrow 0$ to get

$\displaystyle\lim_{t\to\infty}e^{-\lambda
t}\left\langle{Y_{t},f^{+}}\right\rangle=\left(\frac{\theta}{\lambda}\right)\left\langle{\pi,[x]f^{+}}\right\rangle
W,\quad a.s.$

A similar argument also works for $f^{-}$. We combine these two terms and use
$\widetilde{\pi}$ to prove the almost sure convergence in Corollary 1. The
$L^{2}$ convergence can be done similarly and we skip the details. ∎

###  5.4 Limit on recursive trees

In this part, we prove the convergence of the empirical measure on clusters.

###### Proof of Theorem 2.3.

We will prove only the convergence on active clusters $\mathscr{X}_{t}$, as
the proof for $\mathscr{Y}_{t}$ follows in the same manner. The main idea is
similar to the size process $(X_{t},Y_{t})_{t\geq 0}$, which involves one type
convergence and the cut-off argument. Without loss of generality, we suppose
that for any $\mathbf{t}\in\mathscr{T}$, $|f(\mathbf{t})|\leq|\mathbf{t}|^{p}$
for some $p>0$.

Step 1: Cut-off argument. We do the following decomposition on the event
$\mathscr{X}_{t}$ being not empty.

$\begin{split}&\frac{1}{|\mathscr{X}_{t}|}\sum_{\mathcal{C}\in\mathscr{X}_{t}}f(\mathcal{C})-\mathbb{E}[f(T_{\pi})]=\underbrace{\sum_{n=1}^{\infty}A_{t}(n)}_{\mathbf{I}}+\underbrace{\sum_{n=1}^{\infty}B_{t}(n)}_{\mathbf{II}},\\\
\end{split}$ (57)

where, if we write $g(n)=\mathbb{E}[f(T_{n})]$,

$A_{t}(n)=\frac{X_{t}(n)}{\langle
X_{t},1\rangle}\left(\frac{1}{X_{t}(n)}\sum_{\mathcal{C}\in\mathscr{X}_{t},|\mathcal{C}|=n}f(\mathcal{C})-g(n)\right),\quad
B_{t}(n)=\left(\frac{X_{t}(n)}{\langle X_{t},1\rangle}-\pi(n)\right)g(n).$

Theorem 2.2 implies the a.s. convergence of the second term $\mathbf{II}:$

$\displaystyle\lim_{t\to\infty}\sum_{n=1}^{\infty}B_{t}(n)=\lim_{t\to\infty}\left(\frac{\langle
X_{t},g\rangle}{\langle X_{t},1\rangle}-\langle\pi,g\rangle\right)=0,\text{ on
}\\{\tau=\infty\\}.$

For the first term $\mathbf{I}$, we use a cut-off argument and
$|f(\mathcal{C})|\leq|\mathcal{C}|^{p}$:

$\left|\sum_{n=1}^{\infty}A_{t}(n)\right|\leq\left|\sum_{n=1}^{K}A_{t}(n)\right|+2\left|\sum_{n=K+1}^{\infty}\frac{X_{t}(n)n^{p}}{\langle
X_{t},1\rangle}\right|.$ (58)

For the first term on the right hand side in (58), we admit the following
convergence (to be proved in Step 2)

$\displaystyle\forall
n\in\mathbb{N}_{+},\quad\lim_{t\to\infty}\frac{1}{X_{t}(n)}\sum_{\mathcal{C}\in\mathscr{X}_{t},|\mathcal{C}|=n}f(\mathcal{C})=\mathbb{E}[f(T_{n})],\qquad\text{
almost surely on }\\{\tau=\infty\\},$ (59)

which can also be seen as a generalized law of large numbers. By Theorem 2.2,
for any $n\in\mathbb{N}_{+}$, $\frac{X_{t}(n)}{\langle X_{t},1\rangle}$
converges a.s. as $t\to\infty$. Then the above display ensures that
${\lim_{t\to\infty}\left|\sum_{n=1}^{K}A_{t}(n)\right|=0}$, almost surely on
$\\{\tau=\infty\\}$. The second term on the right hand side in (58) also
converges a.s. due to Theorem 2.2:

$\displaystyle\lim_{t\to\infty}\left|\sum_{n=K+1}^{\infty}\frac{X_{t}(n)n^{p}}{\langle
X_{t},1\rangle}\right|=\langle\pi,[x^{p}]_{>K}\rangle,\qquad\text{ almost
surely on }\\{\tau=\infty\\},$

where $[x^{p}]_{>K}(n)=n^{p}\mathbf{1}_{\left\\{n>K\right\\}}$. We put these
results back in to (57) and obtain that

$\displaystyle\lim_{t\to\infty}\left|\frac{1}{|\mathscr{X}_{t}|}\sum_{\mathcal{C}\in\mathscr{X}_{t}}f(\mathcal{C})-\mathbb{E}[f(T_{\pi})]\right|\leq
2\langle\pi,[x^{p}]_{>K}\rangle,\qquad\text{ almost surely on
}\\{\tau=\infty\\}.$

Then we let $K\to\infty$ and prove Theorem 2.3.

Step 2: Convergence for one type. It remains to prove (59). We follow the same
spirit as in Step 2 in the proof of Theorem 2.2, with some minor technical
differences. We recall $\mathscr{T}_{n}$ the space of equivalence classes of
RRT of size $n$, and denote by

$\displaystyle\forall\mathbf{t}\in\mathscr{T}_{n},\qquad
X_{t}(\mathbf{t}):=\sum_{\mathcal{C}\in\mathscr{X}_{t}}\mathbf{1}_{\left\\{\mathcal{C}\sim\mathbf{t}\right\\}},$

the number of active clusters of type $\mathbf{t}$. Because the space
$\mathscr{T}_{n}$ is finite ($|\mathscr{T}_{n}|=(n-1)!$), it suffices to prove
that

$\forall
n\in\mathbb{N}_{+},\forall\mathbf{t}\in\mathscr{T}_{n},\qquad\lim_{t\to\infty}e^{-\lambda
t}X_{t}(\mathbf{t})=W\frac{\pi(n)}{(n-1)!},\quad a.s.,$ (60)

and this can be dealt with using the same technique in the proof of Theorem
2.2: we only need to prove the following

$\displaystyle\forall
n\in\mathbb{N}_{+},\forall\mathbf{t}\in\mathscr{T}_{n},\qquad\varliminf\limits_{t\to\infty}e^{-\lambda
t}X_{t}(\mathbf{t})\geq W\frac{\pi(n)}{(n-1)!}.$ (61)

This is similar to proving (42). Let $(t_{k})_{k\in\mathbb{N}_{+}}$ be any
(random) time sequence such that ${\varlimsup\limits_{t\to\infty}e^{-\lambda
t}X_{t}(\mathbf{t})=\lim_{k\to\infty}e^{-\lambda
t_{k}}X_{t_{k}}(\mathbf{t})}$. Then we have

$\begin{split}\varlimsup\limits_{t\to\infty}e^{-\lambda
t}X_{t}(\mathbf{t})&=\lim_{k\rightarrow\infty}\left(\sum_{\mathbf{t}^{\prime}\in\mathscr{T}_{n}}e^{-\lambda
t_{k}}X_{t_{k}}(\mathbf{t}^{\prime})-\sum_{\mathbf{t}^{\prime}\in\mathscr{T}_{n},\mathbf{t}^{\prime}\neq\mathbf{t}}e^{-\lambda
t_{k}}X_{t_{k}}(\mathbf{t}^{\prime})\right)\\\ &\leq
W\pi(n)-\sum_{\mathbf{t}^{\prime}\in\mathscr{T}_{n},\mathbf{t}^{\prime}\neq\mathbf{t}}\varliminf\limits_{k\to\infty}e^{-\lambda
t_{k}}X_{t_{k}}(\mathbf{t}^{\prime})\\\ &\leq
W\pi(n)-\sum_{\mathbf{t}^{\prime}\in\mathscr{T}_{n},\mathbf{t}^{\prime}\neq\mathbf{t}}W\frac{\pi(n)}{(n-1)!}=W\frac{\pi(n)}{(n-1)!}.\end{split}$

Here from the first line to the second line, we use Fatou’s lemma, and from
the second line to the third line we use (61) . This equation controls the
upper bound of $\varlimsup\limits_{t\to\infty}e^{-\lambda
t}X_{t}(\mathbf{t})$. Using also (61), we conclude that (60) is true.

Finally, we prove (61), which requires convergence along discrete time
sequences and the control of fluctuation. We calculate the $L^{2}$ moment as
follows

$\displaystyle\mathbb{E}\left[\left(e^{-\lambda
t}\sum_{\mathcal{C}\in\mathscr{X}_{t},|\mathcal{C}|=n}\left(\mathbf{1}_{\left\\{\mathcal{C}\sim\mathbf{t}\right\\}}-\frac{1}{(n-1)!}\right)\right)^{2}\right]$
$\displaystyle=e^{-2\lambda
t}\mathbb{E}\left[\sum_{\mathcal{C}\in\mathscr{X}_{t},|\mathcal{C}|=n}\mathbb{E}\left[\left(\mathbf{1}_{\left\\{\mathcal{C}\sim\mathbf{t}\right\\}}-\frac{1}{(n-1)!}\right)^{2}\,\,\Big{|}\,\,\mathcal{F}_{t}\right]\right]$
$\displaystyle\leq e^{-2\lambda
t}\mathbb{E}\left[X_{t}(n)\right]=O(e^{-\lambda t})\xrightarrow{t\to\infty}0.$

In the first line, we use the fact that the clusters are i.i.d. RRTs of size
$n$, see Proposition 3.2. In the second line, we use (37). Notice that the
$L^{2}$ moment decreases at least exponentially, we can thus take a discrete
time sequence $\\{k\Delta\\}_{k\geq 1}$ with $\Delta>0$ and use Borel–Cantelli
lemma to obtain that for any $\Delta>0$

$\displaystyle\forall
n\in\mathbb{N}_{+},\forall\mathbf{t}\in\mathscr{T}_{n},\qquad\lim_{k\to\infty}e^{-\lambda
k\Delta}X_{k\Delta}(\mathbf{t})=W\frac{\pi(n)}{(n-1)!}.$

Let us now control fluctuations and let $N_{k,\Delta}(\mathbf{t})$ be the
number of active clusters of type $\mathbf{t}$ at time $k\Delta$, on which
occurs some event during $(k\Delta,(k+1)\Delta)$. Then like (43), we have

$\displaystyle\forall t\in[k\Delta,(k+1)\Delta),\qquad X_{t}(\mathbf{t})\geq
X_{k\Delta}(\mathbf{t})-N_{k,\Delta}(\mathbf{t}),$

and it suffices to prove $\lim_{k\to\infty}e^{-\lambda
k\Delta}N_{k,\Delta}(\mathbf{t})=0$ to conclude (61). We skip the details as
it follows exactly the same lines starting from (43) in Step 2 in the proof of
Theorem 2.2. ∎

##  6 Characterization of phases and regularity

Recall the three parameters $\beta,\theta,\gamma$ introduced when describing
the GFI process in Section 1. The Malthusian exponent
$\lambda=\lambda(\beta,\theta,\gamma)$ allows us to define the phases in terms
of the sign of $\lambda$ (i.e. $\lambda>0,=0,<0$). So far, we have not shown
how many phases exist and how they are related. The main objective of this
section is to justify the phase diagram shown in Figure 3 and prove Theorem
2.4. The proof strategy is organized as follows: in Section 6.1, we prove some
regularity properties about the mapping
$(\beta,\theta,\gamma)\mapsto\lambda(\beta,\theta,\gamma)$. In Section 6.2, we
justify Figure 3. In Section 6.3, we give a brief proof for the change of
monotonicity of $\beta\mapsto\lambda(\beta.\theta,\gamma)$ and finally prove
Theorem 2.4.

###  6.1 Regularity of Perron’s root

The occurrence of different phases relies on the parameters
$(\beta,\theta,\gamma).$ We will study how the Perron’s root
$\lambda(\beta,\theta,\gamma)$ depends on the three parameters in the
following proposition. Recall that
$(\beta,\theta,\gamma)\in\mathbb{R}_{+}^{3}.$

######  Proposition 6.1 (Regularity of Perron’s root).

For the Perron’s root $\lambda$ associated to $\mathcal{L}$ in (14), the
mapping $(\beta,\theta,\gamma)\mapsto\lambda(\beta,\theta,\gamma)$ satisfies
the following properties:

  1. 1.

Monotonicity: $\lambda(\beta,\theta,\gamma)$ is increasing on $\gamma$ and
decreasing on $\theta$.

  2. 2.

Regime:

     * •

if $\theta\geq\beta$, or $\theta\in(0,\beta)$ and
$0<\gamma<\theta\left(2^{1-\frac{\theta}{\beta}}-1\right)^{-1}$, then
$\lambda<0$;

     * •

if $\theta\in(0,\beta)$ and
$\gamma>\max\left\\{2\theta\left(\frac{1+\frac{\theta}{\beta}}{1-\frac{\theta}{\beta}}\right),\frac{3}{2}\theta\left(1+\frac{\theta}{\beta}\right)\right\\}$,
then $\lambda>0$.

  3. 3.

Homogeneity: for any $\rho>0$, we have
$\lambda(\rho\beta,\rho\theta,\rho\gamma)=\rho\lambda(\beta,\theta,\gamma)$.

  4. 4.

Continuity: $\lambda(\beta,\theta,\gamma)$ is continuous with respect to the
three parameters.

###### Proof.

Monotonicity. If the clusters split faster, then the number of active clusters
will increase. If the clusters become inactive faster, then the number of
active clusters will decrease. This intuition is easy to prove by providing a
coupling argument for each of the parameters. Then we have the monotonicity
property.

Regime. To find a regime where $\lambda<0$, we make a test with $f=[x^{p}]$
with ${p\in(0,1)}$. Jensen’s inequality implies that

$\displaystyle\frac{x^{p}+y^{p}}{2}\leq\left(\frac{x+y}{2}\right)^{p},\quad\forall
x,y\geq 0.$

This entails that

$\displaystyle(j)^{p}+(n-j)^{p}-n^{p}\leq\left(2^{1-p}-1\right)n^{p}.$

We apply the above inequality into $\mathcal{L}f$ and get that

$\begin{split}\mathcal{L}f(n)&\leq\beta n((n+1)^{p}-n^{p})-\theta
n^{p+1}+\gamma\left(2^{1-p}-1\right)(n-1)n^{p}\\\
&\leq\left(p\beta-\left(2^{1-p}-1\right)\gamma\right)f(n)+\left(\left(2^{1-p}-1\right)\gamma-\theta\right)nf(n)\\\
&=\left(p\beta-\theta+\gamma(2^{1-p}-1-\theta)(n-1)\right)f(n).\end{split}$
(62)

Now, we distinguish two situations. If $\theta\in(0,\beta)$, the condition
$\gamma<\theta/\left(2^{1-\frac{\theta}{\beta}}-1\right)$ allows us to find
some $p<\theta/\beta$ such that $\mathcal{L}f(n)\leq(p\beta-\theta)f(n)$ for
all $n$. Consequently $M_{t}f\leq\exp((p\beta-\theta)t)f$. Then Proposition
4.1 entails $\lambda<0$. If $\theta\geq\beta$, it suffices to choose some $p$
close to $1$ such that $\gamma(2^{1-p}-1)-\theta<0$ and we get also
$\lambda<0$.

To find a regime where $\lambda>0$, we consider the test function

$\displaystyle
f(n)=\mathbf{1}_{\left\\{n=1\right\\}}+\kappa\mathbf{1}_{\left\\{n\geq
2\right\\}},$

with constant $\kappa\in(0,2)$ to be fixed. Recalling (14), it is clear that

$\displaystyle n=1,\quad\mathcal{L}f(1)$
$\displaystyle=\beta(\kappa-1)-\theta,$ $\displaystyle
n=2,\quad\mathcal{L}f(2)$ $\displaystyle=-2\theta\kappa+\gamma(2-\kappa),$
$\displaystyle n\geq 3,\quad\mathcal{L}f(n)$
$\displaystyle\geq-n\theta\kappa+\gamma(n-1).$

For

$\gamma>\max\left\\{2\theta\left(\frac{1+\frac{\theta}{\beta}}{1-\frac{\theta}{\beta}}\right),\frac{3}{2}\theta\left(1+\frac{\theta}{\beta}\right)\right\\},$

we get $\mathcal{L}f(n)>0$ for any $n\geq 1$. Taking
$\kappa=\frac{1}{2}\left(\left(1+\frac{\theta}{\beta}\right)+\left(\frac{2\gamma}{3\theta}\right)\right)$,
we obtain ${\mathcal{L}f(n)\geq Cf(n)}$ for all $n$, with $C>0$. It implies
$\lambda>0$.

Homogeneity. This is due to the fact that the generator $\mathcal{L}$ is
linear with respect to the three parameters. Let $\widetilde{\mathcal{L}}$ and
$(\widetilde{M}_{t})_{t\geq 0}$ be respectively the generator and semigroup
associated to $(\rho\beta,\rho\theta,\rho\gamma)$, while we denote by
$\lambda,h,\pi$ the eigenelements associated to $\mathcal{L}$. Then by the
definition of $(X_{t})_{t\geq 0}$, it is clear that
$\widetilde{\mathcal{L}}=\rho\mathcal{L}$, so $\widetilde{M}_{t}=M_{\rho t}$.
This also implies

$\displaystyle\pi\widetilde{\mathcal{L}}=\rho\pi\mathcal{L}=\rho\lambda\pi,\qquad\widetilde{\mathcal{L}}h=\rho\mathcal{L}h=\rho\lambda
h,$

so $\rho\lambda,h,\pi$ are eigenelements associated to
$\widetilde{\mathcal{L}}$. Then by Proposition 4.1, $\rho\lambda$ is Perron’s
root for $\widetilde{\mathcal{L}}=\rho\mathcal{L}$.

Continuity. We will use three steps to prove it.

Step 1: $\lambda(\beta,\theta,\gamma)$ is locally bounded. For any
$\varepsilon>0$ such that

$\frac{1}{1+\varepsilon}<\frac{\beta_{0}}{\beta}<1+\varepsilon,\quad\frac{1}{1+\varepsilon}<\frac{\theta_{0}}{\theta}<1+\varepsilon,\quad\frac{1}{1+\varepsilon}<\frac{\gamma_{0}}{\gamma}<1+\varepsilon,$

we have

$\displaystyle|\lambda(\beta_{0},\theta_{0},\gamma_{0})|\leq$
$\displaystyle\quad\quad\quad\quad(1+\varepsilon)\max\left\\{\big{|}\lambda(\beta,(1+\varepsilon)^{2}\theta,(1+\varepsilon)^{-2}\gamma)\big{|},\big{|}\lambda(\beta,(1+\varepsilon)^{-2}\theta,(1+\varepsilon)^{2}\gamma)\big{|}\right\\}$
.

The above result is due to the homogeneity of $\lambda$ on all parameters and
the monotonicity of $\lambda$ on $\theta$ and $\gamma$. Then the local
boundedness of $\lambda$ is proved.

Step 2: $\left\langle{\pi(\beta,\theta,\gamma),[x]}\right\rangle$ is locally
bounded. We prove it by contradiction. Assume that
$(\beta_{n},\theta_{n},\gamma_{n})$ converges to
$(\beta_{0},\theta_{0},\gamma_{0})$. Let
$(\beta,\theta,\gamma)=(\beta_{0},\theta_{0}/2,2\gamma_{0})$. Let
$M_{t},\pi,h,\lambda$ be associated to $(\beta,\theta,\gamma)$ and
$\widetilde{M}_{t},\widetilde{\pi},\widetilde{h},\widetilde{\lambda}$ to
$(\beta_{n},\theta_{n},\gamma_{n})$. We assume that
$\left\langle{\widetilde{\pi},[x]}\right\rangle$ converges to infinity as
$n\to\infty$. Due to homogeneity of $\lambda$ on its parameters, we can assume
$\beta_{n}=\beta_{0}$ without loss of generality.

By the assumption above, for any $K>0,$ let $n_{K}>0$ such that for any $n\geq
n_{K}$, we have $\left\langle{\widetilde{\pi},[x]}\right\rangle>K$ and
$\theta_{n}>3\theta_{0}/4$ and $\gamma_{n}<7\gamma_{0}/4$. By Proposition 4.1,
there exist $0<c<C$ such that $c\leq h(n)\leq C$ for any $n\geq 1$. Then using
(iii) in Lemma 4.1, we have

$\begin{split}\frac{{\mathrm{d}}}{{\mathrm{d}}t}\widetilde{M}_{t}h&=\widetilde{M}_{t}(\widetilde{\mathcal{L}}h)\\\
&\leq\widetilde{M}_{t}(Lh)-c(\theta_{n}-\theta_{0}/2)\widetilde{M}_{t}([x])+2c(\gamma_{n}-2\gamma_{0})\widetilde{M}_{t}([x])\\\
&=\lambda\widetilde{M}_{t}h-c(\theta_{n}-\theta_{0}/2)\widetilde{M}_{t}([x])+2c(\gamma_{n}-2\gamma_{0})\widetilde{M}_{t}([x]).\end{split}$
(63)

Using again Proposition 4.1, as $t\to\infty$,

$\widetilde{M}_{t}h(n)\sim\widetilde{h}\left\langle{\widetilde{\pi},h}\right\rangle
e^{\widetilde{\lambda}t},\quad\widetilde{M}_{t}([x])(n)\sim\widetilde{h}\left\langle{\widetilde{\pi},[x]}\right\rangle
e^{\widetilde{\lambda}t}.$ (64)

Then for any fixed $n\geq n_{K}$, and $t$ large enough, we have

$\widetilde{M}_{t}([x])(n)\geq\frac{K}{2C}\widetilde{M}_{t}h(n).$

Then we obtain that

$\begin{split}\frac{{\mathrm{d}}}{{\mathrm{d}}t}\widetilde{M}_{t}h(n)&\leq\lambda\widetilde{M}_{t}h(n)-c(\theta_{n}-\theta_{0}/2)\widetilde{M}_{t}([x])(n)+2c(\gamma_{n}-2\gamma_{0})\widetilde{M}_{t}([x])(n)\\\
&\leq\left(\lambda-
cK\frac{\theta_{0}+2\gamma_{0}}{8C}\right)\widetilde{M}_{t}h(n).\end{split}$
(65)

The above display implies that

$\widetilde{\lambda}\leq\lambda-cK\frac{\theta_{0}+2\gamma_{0}}{8C}.$

As $K$ can be arbitrarily large, we obtain that
$\widetilde{\lambda}\to-\infty$ as $n\to\infty$, which is contradictory to the
fact that $\lambda$ is locally bounded as proved in Step 1. Therefore we
conclude that $\left\langle{\pi(\beta,\theta,\gamma),[x]}\right\rangle$ is
locally bounded.

Step 3: $\lambda$ is continuous with respect to the three parameters. The
approach is very similar to that in Step 2. Assume that
$(\beta_{n},\theta_{n},\gamma_{n})$ converges to $(\beta,\theta,\gamma)$. Let
$M_{t},\pi,h,\lambda$ be associated to $(\beta,\theta,\gamma)$ and
$\widetilde{M}_{t},\widetilde{\pi},\widetilde{h},\widetilde{\lambda}$ to
$(\beta_{n},\theta_{n},\gamma_{n})$. We use again that there exist $0<c<C$
such that $c\leq h(n)\leq C$ for any $n\geq 1$. Due to homogeneity, we can
assume that $\gamma_{n}=\gamma$. We prove next
$\lim_{n\to\infty}\widetilde{\lambda}=\lambda.$

By the assumption above, for any $K>0,$ let $n_{K}>0$ such that for any $n\geq
n_{K}$, we have

$\displaystyle|\beta_{n}-\beta|\leq\beta/K,\quad|\theta_{n}-\theta|\leq\theta/K.$

Since $\left\langle{\widetilde{\pi},[x]}\right\rangle$ is locally bounded,
using (64) and $c\leq h\leq C$, there exists $l=l(\beta,\theta,\gamma)>1$ such
that for fixed $n\geq n_{K}$ and $t$ large enough, we have

$\widetilde{M}_{t}h(n)/l\leq\widetilde{M}_{t}([x])(n)\leq
l\widetilde{M}_{t}h(n).$

Then similarly to (65), we have

$\begin{split}\frac{{\mathrm{d}}}{{\mathrm{d}}t}\widetilde{M}_{t}(h)(n)&\leq\left(\lambda+Cl\frac{\beta+\theta}{K}\right)\widetilde{M}_{t}(h)(n).\end{split}$
(66)

Since $K$ can be arbitrarily large, we obtain that
$\varlimsup_{n\to\infty}\widetilde{\lambda}\leq\lambda.$ The other direction
can be proved similarly. Then the proof for continuity is finished. ∎

###  6.2 Existence of phases

The main objective of this part is to prove the following result, which
describes the phases and justifies Figure 3.

######  Proposition 6.2 (Classification of phases).

The regime of the three phases of the GFI process depends on the parameters
$(\beta,\theta,\gamma)$ in the following way.

  1. 1.

Critical phase: the regime is a critical surface defined as

$\displaystyle\\{\lambda(\beta,\theta,\gamma)=0\\}=\left\\{(\beta,\theta,\gamma)\in\mathbb{R}^{3}_{+}\,|0<\theta<\beta,\gamma=\gamma_{c}(\beta,\theta)\right\\}.$

Here $\gamma_{c}(\beta,\theta)$ is a function, such that for fixed $\beta$,
the mapping $\theta\mapsto\gamma_{c}(\beta,\theta)$ is strictly increasing,
continuous, and satisfies

$\displaystyle\lim_{\theta\searrow
0}\gamma_{c}(\beta,\theta)=0,\qquad\lim_{\theta\nearrow\beta}\gamma_{c}(\beta,\theta)=\infty.$

  2. 2.

Subcritical phase: the regime stays below the critical surface

$\displaystyle\\{\lambda(\beta,\theta,\gamma)<0\\}$
$\displaystyle=\left\\{(\beta,\theta,\gamma)\in\mathbb{R}^{3}_{+}\,|\theta\geq\beta\right\\}$
$\displaystyle\qquad\qquad\cup\left\\{(\beta,\theta,\gamma)\in\mathbb{R}^{3}_{+}\,|0<\theta<\beta,0<\gamma<\gamma_{c}(\beta,\theta)\right\\}$

  3. 3.

Supercritical phase: the regimes stays above the critical surface

$\displaystyle\\{\lambda(\beta,\theta,\gamma)>0\\}=\left\\{(\beta,\theta,\gamma)\in\mathbb{R}^{3}_{+}\,|0<\theta<\beta,\gamma>\gamma_{c}(\beta,\theta)\right\\}.$

The key step in the proof of Proposition 6.2 relies on some strict increment
estimates which are proved in the following two lemmas.

######  Lemma 6.1.

Perron’s root $\lambda$ associated to the generator $\mathcal{L}$ in (14)
satisfies the following estimates

$\begin{split}\forall\delta>0,\quad\lambda(\beta,\theta+\delta,\gamma)\leq\lambda(\beta,\theta,\gamma)-\delta,\\\
\forall\delta\in(0,\theta),\quad\lambda(\beta,\theta-\delta,\gamma)\geq\lambda(\beta,\theta,\gamma)+\delta.\end{split}$
(67)

###### Proof.

Let $\widetilde{\mathcal{L}}$ and $(\widetilde{M}_{t})_{t\geq 0}$ be the
generator and semigroup associated to $(\beta,\theta+\delta,\gamma)$, let $h$
be the eigenvector associated to $\mathcal{L}$ of parameters
$(\beta,\theta,\gamma)$. Using a similar development like (63) and using also
$[x]\geq 1$, we obtain

$\displaystyle\frac{{\mathrm{d}}}{{\mathrm{d}}t}\widetilde{M}_{t}h$
$\displaystyle=\widetilde{M}_{t}(\mathcal{L}h)-\delta\widetilde{M}_{t}([x]h)=\lambda\widetilde{M}_{t}h-\delta\widetilde{M}_{t}([x]h)\leq(\lambda-\delta)\widetilde{M}_{t}h.$

Using Grönwall’s inequality, the above display implies that
$\widetilde{M}_{t}h\leq e^{(\lambda-\delta)t}h$, which proves the first
statement in (67). For the second, the proof is similar and we skip it. ∎

######  Lemma 6.2.

Let $h$ be the eigenvector of Perron’s root associated to the generator
$\mathcal{L}$ in (14). Then in the critical phase $\lambda=0$, $h$ is
subadditive, i.e.

$\displaystyle\forall n,m\in\mathbb{N}_{+},\quad h(m+n)\leq h(m)+h(n).$ (68)

###### Proof.

The proof relies on the probabilistic representation of $h$. Let $N_{t}^{n}$
be the number of clusters at time $t$ with the initial state being a single
cluster of size $n$. Then we have

$\displaystyle\mathbb{E}[N_{t}^{n}]=\left\langle{\delta_{n}M_{t},1}\right\rangle.$

Using (30), in the case $\lambda=0$, we have

$\displaystyle\lim_{t\to\infty}\mathbb{E}[N_{t}^{n}]=\lim_{t\to\infty}\left\langle{\delta_{n}M_{t},1}\right\rangle=h(n)\left\langle{\pi,1}\right\rangle.$
(69)

Then we think of a coupling as follows: consider two processes of
$(\mathscr{X}_{t})_{t\geq 0}$, one with initial state being a single RRT of
size $(m+n)$, and the other one with the same initial state but has a
uniformly selected edge removed which results in two smaller clusters of sizes
respectively $m$ and $n$. By Proposition 3.1, these two clusters are
independent RRTs with sizes respectively $m,n$. Assume that the two processes
have the same random events in their life times, except that a splitting may
happen in the particular edge that exists in the first process and does not
exist in the second process. In this coupling, it is easy to see that the
number of clusters at any time $t>0$ in the first process will not be larger
than the number in the second process. Since the clusters evolve
independently, we have

$\displaystyle\mathbb{E}[N_{t}^{n+m}]\leq\mathbb{E}[N_{t}^{n}]+\mathbb{E}[N_{t}^{m}].$
(70)

Combining (69) and (70), we obtain the desired result (68). ∎

######  Remark 4.

In the critical case, since $\mathcal{L}h=0h=0,$ we solve $\mathcal{L}h(1)=0$
to obtain that $h(2)=\left(1+\frac{\theta}{\beta}\right)h(1)$. The criticality
also implies $\beta>\theta$. Therefore

$\displaystyle h(1)+h(1)-h(2)=\left(1-\frac{\theta}{\beta}\right)h(1)>0.$

So the $``="$ in (68) is not always established.

###### Proof of Proposition 6.2.

We divide the proof into 4 steps. Steps 1-3 are about the critical phase, and
the Step 4 proves the regime of the other phases.

Step 1: Unique critical point. Fix $\beta>0$ and $0<\theta<\beta$. Using the
regimes discovered for subcritical and critical phases in Proposition 6.1 and
also the continuity and monotonicity property of $\lambda$ therein, we know
that $\\{\gamma:\lambda(\beta,\theta,\gamma)=0\\}$ is a non-empty closed
interval. We aim to prove that this interval is a single point by