Universality for one--dimensional hierarchical coalescence processes with double and triple merges
A. Faggionato
We consider one--dimensional hierarchical coalescence processes (in short HCP) where two or three neighbouring domains can merge. An HCP consists of an infinite sequence of stochastic coalescence processes: each process occurs in a different ``epoch" and evolves for an infinite time, while the evolutions in subsequent epochs are linked in such a way that the initial distribution of epoch $n+1$ coincides with the final distribution of epoch $n$. Inside each epoch a domain can incorporate one of its neighbouring domains or both of them if its % its left or its right or both its left and right %neighboring interval. Merging events occur with quite general rates %with the constraint that only intervals whose length belongs to a certain epoch-dependent finite range. % are active, \ie they can %C merge with their left or right neighboring interval with quite general rates. Inactive intervals are frozen. %incorporate other intervals. %Inactive intervals cannot incorporate their neighbours and can increase their length only if they % are incorporated by active neighbours. %The activity ranges are such that after a merging %step the newly produced interval %always becomes inactive for that epoch but active for some future %epoch. Assuming that the distribution at the beginning of the first epoch is described by a renewal simple point process, we prove limit theorems for the domain length and for the position of the leftmost point (if any). %These results show the existence of universality %classes which explain why very different models and very different %initial distribution lead to the same limiting behavior. Our analysis extends the results obtained in \cite{FMRT0} to a larger family of models, including relevant examples from the physics literature \cite{BDG}, \cite{SE}. It reveals the presence of a common abstract structure behind models which are apparently very different, thus leading to very similar limit theorems. Finally, we give here a full characterization of the infinitesimal generator for the dynamics inside each epoch, thus allowing to describe the time evolution of the expected value of regular observables in terms of an ordinary differential equation. \bigskip \noindent {\em Keywords}: coalescence process, simple point process, renewal process, universality, non-equilibrium dynamics. skip \noindent{\em AMS 2010 Subject Classification}: 60G55, % point processes 60B10. % convergence of probability measures %82C24 % Interface problems; diffusion-limited aggregation
1112.4630.tex
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\title[1D  Hierarchical Coalescence Processes]{Universality for one--dimensional hierarchical
coalescence processes with double and triple merges}
\author{A. Faggionato}
\address{Alessandra Faggionato. Dip. Matematica ``G. Castelnuovo", Univ. ``La
  Sapienza''. P.le Aldo Moro  2, 00185  Roma, Italy. e--mail:
  faggiona@mat.uniroma1.it}


 \author[C. Roberto]{C. Roberto}
% \email{cyril.roberto@univ-mlv.fr}
 \address{C. Roberto.
MODAL'X, Universit\'{e} Paris Ouest Nanterre La D\'{e}fense, 200
avenue de la R\'{e}publique 92000 Nanterre, France.
% Cyril Roberto. L.A.M.A., Univ. Marne-la-Vall\'ee, 5 bd Descartes 77454 Marne-la-Vall\'ee, France.
e--mail: croberto@math.cnrs.fr
 }

\author[C. Toninelli]{C. Toninelli}
\address{Cristina Toninelli. L.P.M.A. and
  CNRS-UMR 7599, Univ. Paris VI-VII 4, Pl. Jussieu 75252
  Paris, France. e--mail: cristina.toninelli@upmc.fr}

\thanks{Work supported by the European Research Council through the ``Advanced
Grant'' PTRELSS 228032}




\begin{document}

\begin{abstract}
We consider one--dimensional hierarchical  coalescence processes
(in short HCP) where two or three neighbouring domains can merge. An HCP
consists of an infinite sequence of stochastic coalescence
processes: each process occurs in a different ``epoch" and evolves
for an infinite time, while the evolutions in subsequent epochs are
 linked in such a way that the initial distribution of epoch $n+1$ coincides with
the final distribution of  epoch $n$. Inside each epoch a domain
 can incorporate one of   its neighbouring domains or both of them
 if its
% its left or its right or both its left and right
%neighboring interval. Merging events occur with quite general rates
%with the constraint that only intervals whose
length belongs to a certain  epoch-dependent finite range.
% are \emph{active}, \ie they can
%C merge with their left or right neighboring interval with quite general rates. Inactive intervals are frozen.
%incorporate other intervals.
%Inactive intervals cannot incorporate their neighbours and can increase their length only if they
% are incorporated by active neighbours.
%The activity ranges are such that after a merging
%step the newly produced interval
%always becomes inactive for that epoch but active for some future
%epoch.

Assuming that the distribution at the beginning of the first epoch
is described by a renewal simple point process,  we prove limit
theorems  for the domain length and for the position of the leftmost
point (if any). %These results show the existence of universality
%classes which explain why very different models and very different
%initial distribution lead to the same limiting behavior.
Our analysis extends the results obtained in \cite{FMRT0} to a
larger family of models, including relevant examples from the
physics literature \cite{BDG}, \cite{SE}. It reveals the presence of
a common abstract structure behind models which are apparently very
different, thus leading to very similar limit theorems. Finally,
 we give here a full characterization
of the infinitesimal generator for the dynamics inside each epoch,
thus allowing to describe the time evolution of the expected value
of regular observables in terms of an  ordinary differential
equation.
\bigskip


\noindent {\em Keywords}: coalescence process, simple point process,
renewal process, universality, non-equilibrium dynamics.


\smallskip

\noindent{\em AMS 2010 Subject Classification}:
60G55, % point processes
60B10. % convergence of probability measures
%82C24 % Interface problems; diffusion-limited aggregation
\end{abstract}

\maketitle

\section{Introduction}

%We consider a large class of one-dimensional hierarchical
%coalescence processes, HCP's.
  A one-dimensional hierarchical
coalescence process (HCP) consists of an infinite sequence of
one-dimensional coalescence processes:
%$\{\xi^{(n)}(\cdot)\}_{n\ge 1}$:
each process
occurs in a different {\sl epoch}
%(indexed by $n$)
and evolves for an
infinite time, while the evolution in subsequent epochs are
 linked in such a way that the initial distribution of epoch $n+1$
 %$\xi^{(n+1)}$
 coincides with
the final distribution of epoch $n$.
%Then the corresponding HCP is such that
%$\xi^{(n)}$. I
%**need informal descriotion**.
At a given time inside epoch $n$ the state of the process  is
described  by a simple point process on $\bbR$, {\it i.e.} by a
random  locally finite subset of $\bbR$,  such that the intervals
among consecutive points ({\sl domains}) are not smaller than
$d^{(n)}$, where $\{d^{(n)}\}_{n\geq 1}$ is an a priori  fixed
sequence of strictly increasing  and diverging positive numbers. The
evolution inside epoch $n$ can be informally described as follows.
% process evolves as follows.
%Inside each epoch the process is
%described by a suitable simple point process on $\bbR$ or on $\bbZ$ representing the
%boundaries between adjacent intervals (domains), evolves as follows.
Only
intervals whose length belongs to the
%epoch-dependent
 finite range $[d^{(n)},d^{(n+1)})$ are
\emph{active} \ie they can
%C merge with their left or right neighboring interval with quite general rates. Inactive intervals are frozen.
incorporate their left    neighbouring domain, their right
neighbouring domain or  both of them. Inactive domains cannot
incorporate their neighbours and can increase their length only if
they
 are incorporated by active neighbours.
The  rates of the merging events and the sequence $\{d^{(n)}\}_{n\geq 1}$ are quite general, with the important feature that the activity ranges  $[d^{(n)},d^{(n+1)})$  should be  such that after each merging step the newly produced interval always becomes inactive for that epoch but active for some future epoch.

We have introduced the concept of HCP in \cite{FMRT0}, considering
only left  or right merging of domains, {\it i.e.} a domain cannot
incorporate simultaneously both its neighbours. There
%, here instead we also allow triple merging:   We
we  proved that  if the initial distribution  is a renewal process,
such property is preserved at all times and epochs and the
distribution of certain rescaled variables -- the domain length and
the position of the leftmost point (if any) -- has a well defined
limiting behaviour corresponding to large universality classes (most
of the dynamical details disappear in the scaling limit).
% The analysis in
%\cite{FMRT0} is based on a robust combinatorial study of the
%coalescence  inside a given epoch, which becomes extremely hard in
%the present setting. Hence, here we have followed a different route.
Here  we extend these results to the more general HCP's defined
above which also allow triple merging and we determine the
corresponding limiting behavior and universality classes.

Besides the mathematical interest, our study has been motivated by
the fact that several HCP's have been implicitely introduced in
physics literature to model the non-equilibrium evolution of one
dimensional systems whose dynamics is dominated by the coalescence
of proper domains or droplets characterizing the experiments. We
refer to Section \ref{physics} for a review of some of these HCP's
and the corresponding  physical systems. A key common feature
emerges from the experiments on all these systems: an interesting
coarsening phenomena occurs which leads to  a scale-invariant
morphology for large times, namely the system is described by a
single (time-dependent) length and the distribution approaches a
scaling form. Several models, even very simple ones,
 have been proposed by physicists in order to capture and explain such intriguing behavior and in many cases these models turn out to be HCP's
 (see {\it e.g.}\ \cite{P},   \cite{DBG}, \cite{DGY1},  \cite{DGY2}, \cite{SE}, \cite{BDG}). Supported by computer simulations and under the key assumption
of a well defined limiting behavior under suitable rescaling,
physicists  have derived for these HCP's in the mean field
approximation some non trivial limiting distributions for the
relevant quantities and noticed that these distributions display a
certain degree of universality.
 The results we obtained in \cite{FMRT0} prove and generalize the findings of physicists.
 However the analysis in \cite{FMRT0} does not cover some cases of interests for physics which involve triple merging, {\it e.g.}
  the HCP which has introduced in \cite{BDG} to model Ising at zero temperature (see Section \ref{ising}).
These models are instead covered by the present study which explains
why the limiting distributions of several models, although
different, have a similar structure.







 The
 analysis in
\cite{FMRT0} is based on a robust combinatorial study of the
coalescence  inside a given epoch, which becomes extremely hard in
the present setting. Hence, here we have followed a different route
inspired by the approach of \cite{SE}. In particular, we start with
the infinitesimal generator of the one--epoch coalescence, giving a
complete characterization of its form and domain (Theorem
\ref{amico_marco_zero}). As well known, this allows to characterize
the time evolution of the expectation of regular observables  in
terms of an ordinary differential equation. Applied to the domain
length and the position of the leftmost point (if any), this method
leads to recursive equations between the Laplace transforms of the
involved quantities at the beginning and the end of each epoch, and
therefore at the beginning of two consecutive epochs (Theorems
\ref{differisco} and \ref{differiscobis}).

The study of the Markov generator for  stochastic processes whose
state at a give time is described by a simple point process on
$\bbR$ is rather heavy  \cite{Pr}.  Here, we have introduced a
lattice structure (which is somehow artificial from a geometric
point of view) that strongly simplifies the analysis of the Markov
generator, and in particular allows us to use the standard methods
described in  \cite{L}. However, such a discretization requires some
very special care, because of the use the vague topology on the the
space $\cN$ of locally finite subset of $\bbR$. Once obtained the
above mentioned system of recursive equations between Laplace
transforms, we have generalized the transformation introduced in
\cite[Section 5]{FMRT0} which in some sense linearizes the system
and allows to analyze the recursive identities and obtain the limit
behavior (Theorems \ref{teo2} and \ref{teo3}). The resulting
transformation is now a more abstract object and can therefore be
applied to a larger class of models.






Finally we stress that the heuristic technique developed by physicists (see \cite{BDG})
to derive the limiting distribution (under the assumption of the existence of a limiting behaviour)
is restricted to models with $d^{(n)}=n$ and it
becomes meaningless also at heuristic level  if the ratio
$d^{(n)}/ d^{(n+1)}$ does not converge to $1$ as $n$ goes to
$\infty$.
% In this class a particular interesting case is
%represented by the model in which (roughly) the smallest interval
%merges with its two neighbors.
Under the same hypothesis of  \cite{BDG}, namely $d^{(n)}=n$ and via
the mean field approximation, in \cite{CP} the authors proposed a
time evolution equation which should describe the domain size
distribution when the time variable $t$ is a continuous
approximation of the discrete label $n$ of the epochs and one
forgets how much time elapses between and during the merging events.
This equation has been rigorously analyzed in \cite{CP} and
\cite{GM} and in the latter work a limiting self-similar profile for
this equation has been proved. In this special case, a
transformation similar to one presented in more generality in
\cite{FMRT0}, and here, has been used.
% (see also
%\cite{Pe} for an interesting review) by means of non-linear analysis
%techniques.











\section{Model and results}\label{monello}


In this section we fix some notation and give our main results. We
first introduce the simple point processes we are interested in
(standard references are \cite{DV}, \cite{FKAS}). Then we define the
process called one--epoch coalescence process (in short OCP) and the
hierarchical coalescence process (HCP). Finally we provide some
examples of HCP's coming from the physics literature.
%\marginpar{prvoaprova}
%In this section, we present the hierarchical coalescence process \marginpar{da cambiare}
%(HCP) and our main results in the renewal case (generalizations are
%described in Appendix \ref{generale}). The
% HCP is given by a sequence of one--epoch
%coalescence processes (OCP) suitably linked, each one corresponding
%to  an annihilation dynamics on a simple point process (SPP).

\subsection{Simple point processes (SPP)}\label{sec_SPP}

We denote by $\cN$ the family   of locally finite subsets $\xi
\subset \bbR$.
 $\cN$ is a measurable space endowed with the
$\s$--algebra of measurable subsets  generated by
$$ \{ \xi \in \cN \,:\, |\xi \cap A_1|=n_1\,, \cdots , |\xi \cap A_k |=n_k
\}  ,$$   $A_1, \dots , A_k$ being bounded Borel sets in $\bbR$ and
$n_1, \dots, n_k \in \bbN$. We recall that any probability measure
on the measurable space $\cN$ defines a simple point process (SPP).


 We call {\sl domains} the intervals $[x,x']$
between nearest--neighbour points $x,x'$ in $\xi\cup \{-\infty,
+\infty\} $.
 Note that the existence of the domain $[-\infty,x']$ corresponds to
 the fact that $\xi$ is bounded from the left and its leftmost point is given by $x'$.
A similar consideration holds for $[x, \infty]$. Points of $\xi$ are
also called {\sl domain separation points}. Given a point $x\in
\bbR$, we define
 $$d_x^\ell:= \inf \{ t > 0 \,:\, x-t \in \xi \} \,,\qquad
 d_x ^r := \inf \{ t > 0 \,:\, x+t \in \xi \} ,
 $$
 with the convention that the infimum of the empty set is $\infty$.
 Note that if $x \in \xi$ then $d_x^\ell$ ($d_x^r$)  is simply the
 length of the domain to the left (right) of
 $x$.

\smallskip


  In what
follows $\bbN$ ($\bbN_+$)  will denote the set of non-negative
(positive) integers.
\begin{definition}\
  \begin{enumerate}[(i)]
  \item We say that a SPP  $\xi$ is  {\em left-bounded} if it has a leftmost point and
has infinite cardinality.
%In this case we enumerate the points of $\xi$ as $\{x_k; \ k \in \bbN\}$ in increasing order.
\item  We say that a SPP $\xi$ is {\em $\bbZ$--stationary} if  $\xi \subset \bbZ$  and its law $\cQ$ is
 invariant by $\bbZ$--translations, {\it i.e.}\ if for any $x \in \bbZ$ the
 random set $\xi-x$ has law $\cQ$.
\item We say that a SPP $\xi$  is {\em stationary} if its law $\cQ$ is
 invariant under $\bbR$-translations,
 {\it i.e.}\ if for any $x \in \bbR$ the random set $\xi-x$ has law $\cQ$.
  \end{enumerate}
\end{definition}
If $\xi$ is $\bbZ$--stationary or
 stationary, then   a.s.\ the following dichotomy holds \cite{FKAS}:  $\xi$ is unbounded from the left and from the
 right or $\xi$ is  empty. In the sequel we will always assume the  first alternative to hold  a.s.\ and we will write $ \xi=\{x_k: k \in
\bbZ\}$ with the rules: $x_0 \leq 0 < x_1$ and
 $x_k < x_{k+1}$ for all $k \in \bbZ$. In the case of a left--bounded SPP,
 we enumerate the points of $\xi$ as $\{x_k\,:\, \ k \in \bbN\}$ in increasing order.




We now describe the main classes of SPP's we are interested in.
\begin{definition}
Let $\nu$ and  $\mu$ be  probability measures   on $\bbR$ and
$(0,\infty)$, respectively. Let $\xi $ be a SPP with law $\cQ$.

\begin{itemize}

\item We say that $\xi$ is a \emph{renewal SPP containing the origin
and with interval law $\mu$}, and write $\cQ = \text{Ren}(\mu\tc
0)$, if
\begin{enumerate}[$(i)$]
\item $0 \in \xi$ ,
\item $\xi$ is unbounded from the left and from the right and,
labelling the points in increasing order with $x_0=0$, the random
variables $d_k= x_k-x_{k-1}$, $k \in \bbZ$,  are i.i.d.\ with common
law $\mu$.
\end{enumerate}

\item

We say that $\xi$ is
 a \emph{right renewal}  SPP with first point law $\nu$ and interval law
$\mu$, and write $\cQ= \text{Ren}(\nu, \mu)$,  if
\begin{itemize}
\item[$(i)$] $\xi= \{x_k \,,\, k \in \bbN\} $ is a left-bounded SPP,
\item[$(ii)$] the first point $x_0$ has law $\nu$,
\item[$(iii)$]  $ d_k = x_k -x_{k-1}$ ($k \in \bbN_+$)
has law $\mu$,
\item[$(iv)$]
the random variables $x_0, \{d_k\}_{k \in \bbN_+}$  are independent.
  \end{itemize}

\item


 If $\mu$ has finite mean, we
say that $\xi$ is a {\sl stationary renewal SPP with interval law
$\mu$}, and write $\cQ=\text{Ren}   (\mu)$, if
\begin{itemize}
\item[$(i)$] $\xi$ is a  stationary SPP with finite intensity and $\xi$ is non-empty a.s.,
\item[$(ii)$] the random
variables $d_k= x_k-x_{k-1}$, $k \in \bbZ$, are i.i.d. with common
law $\mu$ w.r.t.\ the Palm distribution associated to    $\cQ$.
\end{itemize}




\item If  $\mu$ has support  on $\bbN_+$ and has  finite mean, we
say that  $\xi$   is a {\sl $\bbZ$--stationary renewal SPP with
interval law $\mu$}, and write  $\cQ=\text{Ren} _{\bbZ} (\mu)$, if
\begin{itemize}
\item[$(i)$] $\xi$ is $\bbZ$--stationary and a.s. non-empty,
\item[$(ii)$]   w.r.t.\ the conditional probability $\cQ(\cdot |0
\in \xi)$ the random variables $d_k= x_k-x_{k-1}$, $k \in \bbZ$, are
i.i.d.\ with common law $\mu$.
\end{itemize}
  \end{itemize}
\end{definition}


We recall that the intensity $\l_{\cQ}$  of a stationary SPP with
law $\cQ$ is defined as the expectation  $\l_{\cQ}:= \bbE_{\cQ}
\left( |\xi \cap [0,1]|\right)$.  A ($\bbZ$--)stationary renewal SPP
with interval law $\mu$ having infinite mean cannot exist (see
Proposition 4.2.I in \cite{DV} and Appendix C in \cite{FMRT0}). As
discussed after Theorem 1.3.4 in \cite{FKAS}, $\cQ=\text{Ren} (\mu)$
if and only if the following holds: the random variables
$d_k=x_k-x_{k-1}$, $k\not=1$, are i.i.d. with law $\mu$ and are
independent from the random vector $(x_0,x_1)$, which satisfies
\begin{equation}
\cQ ( -x_0> u,\, x_1> v) = \l _\cQ \int _{u+v}^\infty \bigl( 1-
F(t)\bigr) dt \,, \quad F(t):= \mu\bigl((0,t]\bigr)\,, \qquad
u,v>0 .
\end{equation}




\subsection{ The one--epoch coalescence process
(OCP)}\label{sec_OCP}
This process  depends on two constants $0<d_{\rm min}<d_{\rm max}$
and on non-negative bounded continuous functions $\l_\ell, \l _r,\l_a
$ defined on $[d_{\rm min} , \infty]$ which, with $ \l (d):= \l_\ell
(d)+ \l_r (d)+\l_a (d)$, satisfy the following assumptions:
\begin{itemize}
\item[(A1)] $\l(d) >0$ if and only if $d \in [d_{\rm min}, d_{\rm max})$,
\item[(A2)] if $d,d' \geq d_{\rm min}$, then $d+d'\geq  d_{\rm max}$.
\end{itemize}
Trivially, (A2) is equivalent to the bound $2d_{\rm min}\geq d_{\rm
max}$.

\smallskip

The admissible  starting configurations for the OCP
belong to the subset $\cN(d_{\rm min} )$ given by  the
configurations $\xi\in \cN$ having only domains of length not
smaller than $d_{\rm min}$, {\it i.e.}\
\begin{equation}\label{scudetto} \cN(d_{\rm min})= \{\xi \in \cN \,:\,
d_x^{\,\ell} \geq d_{\rm min}\,,\; d_x^{\, r} \geq  d_{\rm min}\;
\forall x \in \xi\}  .
\end{equation}

 Then, the stochastic evolution  of the OCP  is given by a jump dynamics
with
  c\`{a}dl\`{a}g paths   $\{ \xi(t) \}_{t \geq 0}$ in the Skorohod space $D\bigl([0,\infty),
 \cN(d_{\rm min}) \bigr)$ (cf.\ \cite{B}). Roughly speaking, the dynamics is the following. Each domain $\D$ of length $d$ waits an exponential time with
parameter $\l(d)$, afterwards exactly one of the following
annihilations takes place: the  left extreme of $\D$ is erased with
probability $\l_\ell (d)/\l(d)$, the right extreme of $\D$ is erased
with probability $\l_r(d)/\l(d)$, both the extremes of $\D$ are
erased with probability $\l_a(d)/\l(d)$. We say that the domain $\D$
incorporates its left domain, its right domain, both its
neighbouring domains, respectively. In Section  \ref{universal} we
present a full construction of all OCPs, varying the initial
configuration, on the same probability space (\emph{universal
coupling}).

Note that the assumptions (A1) and (A2) on the coalescence rates
imply that any domain which has been generated by a coalescence
event is not active, {\it i.e.}\ it cannot incorporate other domains.
This assumption comes from  several models of physical interest (see
Section \ref{physics}) and plays a fundamental role in our analysis.
% will be used when we establish the
%recursive identities between the Laplace transforms of the interval
%law at the beginning and at the end of the process (see Theorem
%\ref{differisco}). In particular the two assumptions are at the
%origin of the simple formula \eqref{davide} which governs the
%evolution of the distribution of the active domains.

\begin{remark}
Note that $\l_\ell$ and $\l_r$ correspond to
$\l_r ^*$ and $\l_\ell^*$ in \cite{FMRT0}. The case $\l_a \equiv 0$
has been treated in \cite{FMRT0} without the additional assumption
that $\l_\ell$, $\l_r$ are continuous functions.
\end{remark}

Formally, the Markov generator of
 the OCP is given by
 \begin{align} \label{santi}
\cL f(\xi)  = \sum_{\substack{[x,x+d]\\\text{domain in }\xi}} &
\Big\{ \l_\ell (d) \bigl[ f (\xi\setminus\{x\} )- f(\xi) \bigr]+
 \l_r (d) \bigl[ f (\xi\setminus\{x+d\} )- f(\xi) \bigr] \nonumber \\
& \quad +
\l_a (d) \bigl[ f (\xi\setminus\{x, x+d\} )- f(\xi) \bigr] \Big\} .
\end{align}
A precise description of the Markov generator $\mathcal{L}$  is
given  %\club \club 
below while its full rigorous analysis is
postponed to Section  for clarity of exposition.



 We will write $\bbP_{\cQ}$ for the law on $D\bigl([0,\infty),
\cN(d_{\rm min}) \bigr)$ of the OCP  with initial law $\cQ$ on $\cN
(d_{\rm min})$ and  $\cQ_t$ for its marginal at time $t$.




Since the OCP is an annihilation process, points can only disappear.
Furthermore, Assumptions $(A1)$ and $(A2)$ guarantee that the process converges
to a limiting configuration.
One can easily prove the following lemma already stated in \cite{FMRT0} in a less general setting
(details are left to the reader).

%\smallskip
%  Lemma 2.12 and Theorem
%2.13 of \cite{FMRT0} remain valid here by slightly changing their
%proofs (note that in Section \ref{universal} we give a detailed
%description of the universal coupling which is at the base of the
%proof of Lemma 2.12 and Theorem 2.13 in \cite{FMRT0}). Hence, we
%recall these results omitting the proofs:
\begin{Lemma} \label{volarelontano}
For any given initial condition $\xi \in \cN(d_{\rm min})$ the
following holds:
\begin{itemize}
\item[$(i)$] $\xi(t)
\subset \xi(s)$ if $s \leq t$,
%\item[$(ii)$]
% the configuration $\xi(t) $ is constant on
%bounded intervals eventually in $t$,
\item[$(ii)$] there exists a unique element $\xi(\infty)$ in $\cN(d_{\rm max})$
  such that
$\xi(t)\cap I=\xi(\infty)\cap I$ for all large enough $t$ (depending
on $I$) and all bounded intervals $I$.
\end{itemize}
\end{Lemma}

The next result is a simple generalization of \cite[Theorem
2.13]{FMRT0} (its proof is based on the universal coupling described
in Section \ref{universal}, we omit details). It states that if the
process starts with some right renewal (respectively stationary,
$\mathbb{Z}$--stationary, etc.) simple point process $\xi$, then at
any later time $t$, the process $\xi(t)$ is still of the same type.



\begin{Lemma} \label{vecchioteo1}
Let $\nu, \mu$ be two probability measures on $\bbR$ and $[d_{\rm min},
\infty)$, respectively. Then, for all $t\in [0,\infty]$ there exist
probability measures $\nu_t,\mu_t$  on $\bbR$ and $[d_{\rm min},
\infty)$ respectively  such that $\nu_0=\nu$, $\mu_0=\mu$ and
\begin{enumerate}[(i)]
 \item if $\cQ=\text{Ren}(\nu,\mu)$ then
 $\cQ_t=\text{Ren} (\nu_t, \mu_t)$,
\item if  $\cQ= \text{Ren} (\mu)$ then
 $\cQ_t= \text{Ren} (\mu_t)$,
\item if $\cQ= \text{Ren}_\bbZ (\mu)$ then
 $\cQ_t=\text{Ren}_\bbZ (\mu_t)$,
\item if $\cQ= \text{Ren} (\d_0, \mu)$ then $\cQ_t(\cdot \tc 0\in
  \xi)= \text{Ren}(\d_0,\mu_t)$,
\item $\lim_{t\to \infty}\nu_t=\nu_\infty$ and $\lim_{t\to
    \infty}\mu_t=\mu_\infty$ weakly.
\end{enumerate}
\end{Lemma}

Thanks to the previous results $\xi(\infty)$, $\mu_\infty$ and
$\nu_\infty$ are well defined. In fact there exists a recursive
identity between the Laplace transform of the interval law, and of
the first point law, at time $t=0$ and at time $t=\infty$. These
identities, stated in Theorem \ref{differisco} and Theorem
\ref{differiscobis} below, will be the keystones of the analysis of
the asymptotic of the hierarchical coalescence process.


Given a probability measures  $\mu$ on  $[d_{\rm min}, \infty)$, let
$\mu_t$ be as in Lemma  \ref{vecchioteo1}. Then, for $s \in \bbR_+$,
define
$$  G_t (s) = \int e^{-s x} \mu_t (dx) \,,\qquad  H_t (s) = \int _{[d_{\rm min}, d_{\rm max} )} e^{-sx} \mu_t (dx) .
$$

\begin{Theorem}[Recursive identities for the interval law]\label{differisco}
For any $s \in \bbR_+$, the functions $[0,\infty)\ni t \mapsto
G_t(s), H_t (s)  $ are differentiable and satisfy
\begin{align}
& \partial_t H_t(s)= -\int  \mu_t (dx) \l(x) e^{-s x} , \label{davide}\\
& \partial _t \bigl[G_t(s)-H_t(s)\bigr] = %\nonumber \\
%& \qquad \qquad
G_t(s)\int \!\mu_t(dx) (\l_\ell+\l_r)(x)
 e^{-sx}
 + \!G_t(s)^2 \int \!\mu_t(dx)\l_a(x)  e^{-sx}
 .\label{orecchiettebis}
 \end{align}
In particular, it holds
\begin{enumerate}[(i)]
\item
If $\l_a\equiv 0$, then $\partial_t G_t (s)= \partial_t H_t(s) (1- G_t(s))$.
%\begin{equation}
% \partial_t\bigl[ G_t (s)-H_t(s)\bigr]= -\partial_t H_t(s) \cdot
%G_t(s) \,,\label{speranza1}
%\end{equation}
Hence,
\begin{align}
& 1-G_t(s) = (1-G_0(s)) e ^{H_0(s)  - H_t(s) } , \qquad t\in \bbR_+, \label{caviale} \\%\label{uovo}\\
 & 1-G_\infty(s)= (1-G_0(s)) e ^{H_0(s)   }  . \label{uovone}
\end{align}
\item
If $\l_\ell+\l_r \equiv\g \l_a$ for some $\g\geq 0$, then $\partial _t G_t (s)= \partial_t H_t(s) \left( 1 -
\frac{G_t(s)(\gamma + G_t(s))}{1+\gamma}\right)$.
%\begin{equation}
%\partial _t \bigl[G_t (s)-H_t(s)\bigr]=
% -\frac{\g}{1+\g} \partial_t H_t(s)\cdot G_t(s)-\frac{1}{1+\g} \partial _t H_t(s)\cdot G_t (s)^2\,,\label{speranza2}
%\end{equation}
Hence, for $s>0$ it holds
\begin{align}
&  e^{-\frac{\g+2}{\g+1} H_t(s)} \frac{\g+1+G_t(s) }{1-G_t(s)} =
e^{-\frac{\g+2}{\g+1} H_0(s)} \frac{\g+1+G_0(s) }{1-G_0(s)}
 ,\qquad t\in \bbR_+,  \label{salsa}
\\
&  \frac{\g+1+G_\infty(s) }{1-G_\infty(s)} = e^{-\frac{\g+2}{\g+1} H_0(s)} \frac{\g+1+G_0(s) }{1-G_0(s)}.  \label{salsona}
\end{align}
\end{enumerate}
\end{Theorem}
In the above theorem, as in the rest of the paper, differentiability
at $t=0$ for a function on $[0,\infty)$  means differentiability
from the right.
\begin{remark}\label{pasquetta}
The restriction to the above  cases $(i)$ and $(ii)$ is technical
and motivated by the following. Set
$$
a_t(s):= \int \mu_t(dx) (\l_\ell+\l_r)(x) e^{-sx} \quad \mbox{ and } \quad b_t(s):= \int \mu_t(dx) \l_a(x)
 e^{-sx}.
$$
 Thanks to \eqref{davide}, Equation \eqref{orecchiettebis} can be rewritten as
\begin{equation}\label{serioso}
\partial_t G_t (s)= -a_t(s) - b_t(s) + a_t (s) G_t (s)+ b_t(s) G_t (s)^2 .
\end{equation}
Fixing functions $A_t (s)$ and $B_t(s)$ such that $\partial_t
A_t(s)=a_t(s)$ and $\partial_t B_t(s)=b_t(s)$, \eqref{serioso} leads
to
\begin{equation}\label{fulmini}
\frac{ e^{A_t(s)+2 B_t(s)} }{1-G_t(s)}= \frac{  e^{A_0(s)+2B_0(s) }
}{1-G_0(s)} +\int _0 ^t b_u(s) e^{A_u(s)+2 B_u(s) }du  .
\end{equation}
In order to have a recursive identity between $(G_0,H_0)$ and
$G_\infty$, one needs to find an explicit expression of the integral
in the right hand side of \eqref{fulmini}. This can   be achieved in
cases (i) and (ii) of Theorem \ref{differisco} by taking
$B_t(s)=b_t(s)=0$ and $A_t(s)=-H_t(s)$ in case $(i)$,  and by taking
$A_t(s)=\g B_t(s)$ and $A_t(s)+B_t(s)=-H_t (s)$ in case $(ii)$.
%Trivially, one recovers \eqref{uovo} and \eqref{salsa}

Finally we point out that,  since $\emph{arctanh}(x)= \frac{1}{2} \ln
\frac{1+x}{1-x}$ for $x \in (-1,1)$,  \eqref{salsa} with $\g =0$ can
be rewritten in the more compact form
$$ -H_t + \emph{arctanh}G_t(s)= -H_0 + \emph{arctanh} G_0(s) .$$
\end{remark}




The next result is concerned with the evolution of  the first point
law $\nu_t$ when starting with a SPP having law ${\rm Ren}(\nu,
\mu)$ (recall Lemma \ref{vecchioteo1}). First, we observe that if
$\xi$ is a SPP with law ${\rm Ren}( \d_0, \mu)$ and $V$ is a random
variable with law $\nu$ independent from $\xi$, then the translated
random subset $\{x +V\,:\,x \in\xi\}\subset \bbR$ is a SPP with law
${\rm Ren} (\nu, \mu)$. This simple observation and the definition
of the OCP, whose dynamics depends only on the sequence of the
domain lengths and not on the specific location of the domains,
allow to conclude that $\nu_t$ is the convolution
\begin{equation}\label{convoluto}
 \nu_t= \bar \nu_t \ast \nu , \end{equation}  where $ \bar
\nu_t$ denotes the evolution at time $t$ of the first point law when
starting from a SPP having law ${\rm Ren}(\d_0, \mu)$. Hence,
without loss we can  restrict our analysis to this case.



\begin{Theorem}[Recursive identities for the first point  law]\label{differiscobis}
Assume that $\nu= \d_0$. Then, for any $s \in \bbR_+$  the Laplace
transform
$$[0,\infty)\ni t \mapsto
 L_t(s):=  \int  e^{-s x} \nu_t (x) \in (0,1] $$ is  differentiable and satisfies
\begin{equation}\label{bucatini}
\frac{\partial_t L_t(s)}{L_t(s)}
 =
- \int \!\mu_t(dy) \left(\lambda_\ell(y) + \lambda_a(y) \right) +  \int \!\mu_t(dy) \lambda_\ell(y)e^{-sy}  %\right. \\
+ G_t(s) \int \! \mu_t(dy) \lambda_a(y)e^{-sy}   .
\end{equation}
In particular, it holds:
\begin{enumerate}[(i)]
\item
If $\l_a\equiv 0$ and $\lambda_r \equiv \gamma \lambda_\ell $ for
some constant $\gamma \geq 0$, then it holds $\partial _t L_t (s)=
\frac{L_t(s)}{1+\gamma} \left( \partial _t H_t (0) -  \partial _t
H_t(s) \right)$.
%\begin{equation*} %\label{agognata1}
% \partial _t L_t (s)= \frac{L_t(s)}{1+\gamma} \left( \partial _t H_t (0) -  \partial _t H_t (s) \right) .
%\end{equation*}
Hence,
\begin{align}
& L_t (s)= L_0(s)\exp \left\{\frac{-H_t(s) + H_t(0) + H_0(s) - H_0(0)}{1+\gamma}\right\} \,,\qquad t\in \bbR_+ , \label{monte1}\\
& L_\infty(s)=  L_0(s) \exp\left\{\frac{H_0(s)-H_0(0)}{1+\gamma}\right\}
 . \label{monte2}
\end{align}
If   $\l_a \equiv 0$ and $ \l_\ell \equiv 0$,  then trivially $L_t(s)=L_0(s)$ for any $t \geq 0$. %and any $s \geq 0$.

\item
If $\l_\ell \equiv  0$ and $\l_r \equiv 0$, then $\partial _t L_t
(s)=L_t(s) (\partial_t H_t(0) - G_t(s) \partial_t H_t(s))$.
%\begin{equation*}\label{agognata2}
% \partial _t L_t (s)=L_t(s) G_t(s) \partial_t H_t(s) .
% \end{equation*}
Hence, for $s>0$ it holds
\begin{align}
    & L_t(s)=  L_0(s) \sqrt{ \frac{1-G_t^2(s)}{1-G_0^2(s)} }  e^{H_t(0)-H_0(0)} \,,\qquad t\in\bbR_+ ,\label{gransasso1} \\
& L_\infty(s)=  L_0(s)\sqrt{\frac{1-G_\infty^2(s)}{1-G_0^2(s)}} e^{-H_0(0)}
 . \label{gransasso2}
\end{align}

\end{enumerate}

\end{Theorem}
 We point out that cases $(i)$ and $(ii)$ of Theorem
\ref{differiscobis} are included into (but not equal to)  cases
$(i)$ and $(ii)$ of Theorem \ref{differisco}.


%In the above theorem,  if for $s \in [0,\infty)$ it holds
%$L_0(s)= \infty$ then we understand that $L_t (s) = \infty$ for all
%$t \geq 0$. If $L_0(s) < \infty$, then we understand that $L_t (s) <
%\infty$ for all $t\geq 0$, the map $t \mapsto L_t(s)$ is differentiable
%and satisfies the differential identities above.
%%\eqref{agognata1} and \eqref{agognata2} hold.
%Differentiability at $t=0$ means from the
%right.


\medskip   The previous results are based on our
analysis of the Markov generator $\cL$  of the OCP. In general, the
expected value at time $t$  of a regular observable evolves
according to an ordinary  differential equation that we describe
below. We first  fix some notation. Given $k \in \bbZ$ we set
\begin{equation} I_k:=
 \begin{cases}
 [k d_{\rm min}, (k+1) d_{\rm min} ) & \text{ if } k \geq 1 ,\\
(k d_{\rm min}, (k+1) d_{\rm min} ) & \text{ if } k=0 ,\\
(k d_{\rm min}, (k+1) d_{\rm min} ] & \text{ if } k\leq -1  .
\end{cases}
\end{equation}
% By definition of $\cN ( d_{\rm min} )$ (see \eqref{scudetto}), each configuration $\xi
% \in \cN ( d_{\rm min} )$ has at most one point in the interval $I_k$. Then,
 Given  $\xi \in \cN ( d_{\rm min} )$, we set for $k \in
\bbZ$ and $k<k'$ in $\bbZ$:
\begin{align*}
& \xi ^k := \xi \setminus I_k  \qquad \qquad \qquad \nabla_k f
(\xi):= f(
\xi^k)-f(\xi) , \\
&  \xi^{k, k'} := \xi\setminus (I_k \cup I_{k'})\, \qquad
 \nabla_{k,k'} f(\xi):= f(\xi^{k,k'} ) - f (\xi) .
\end{align*}
We define
\begin{equation*}
%\text{\club} 
\cR := \bbZ \cup \{ (k,k')\,:\,  k' \in
\{k+1,\dots,k+\lceil d_{\max}/d_{\rm min}\rceil\}, k,k'  \text{ in }
\bbZ \} ,
\end{equation*}
%\club 
$\lceil a \rceil$ being the smaller integer $n\geq a$. We
consider the space $\cN( d_{\rm min})$ endowed of the vague topology
(see Section \ref{dugundi}), making it a compact space.
 We write $\mathbb{B}$ for the Banach space of all continuous
functions $f:\cN(d_{\rm min}) \mapsto \mathbb{R}$ endowed with the
uniform norm that we denote by $\| \cdot \|$. Also, and for later
purpose, we let $\mathbb{B}_{\rm loc}$ be the set of functions $f
\in \mathbb{B}$ that are local, {\it i.e.}\ such that there exists a
bounded interval $I \subset \mathbb{R}$ with $f(\xi)=f(\xi \cap I)$
for all $\xi \in \cN(d_{\rm min} )$. Then, similarly to the analysis
of interacting particle systems \cite{L}, we define
$$
\D_f (r):= \sup _{ \xi \in \cN (d_{\rm min}) } \bigl| \nabla _r f
(\xi)\bigr| \,, \quad f \in \bbB, r \in \cR .
$$
%Note that $\D_f (r)$ is finite (since $\cN (d_{\rm min})$ is
%compact). Finally,
and we  introduce the  subset $\bbD$ of $ \bbB$ as
\begin{equation}\label{ddd}
 \bbD:= \bigl\{
f \in \bbB\,:\, |||f|||:= \sum _{r \in \cR } \D_f (r) < \infty
\bigr\} .
\end{equation}
Observe that $\mathbb{B}_{\rm loc} \subset \mathbb{D}$. The
following result characterize completely the Markov generator of the
OCP:

\begin{Theorem}\label{amico_marco_zero}
The subspaces $\bbB_{\rm loc}$ and $\bbD$ are a core of the Markov
generator $\cL$, \it{i.e.}\    $\cL$ is the closure of the operator
obtained by restriction to $\bbB_{\rm loc}$ or to $\bbD$. Moreover,
if $f \in
 \bbD$, $\cL f(\xi)$ equals the absolutely convergent series in the r.h.s.\ of \eqref{santi}.
\end{Theorem}
The proof is given in Section \ref{infinitesimi}. %\club \club
Although this analysis represents our starting point, we prefer to
postpone it to the end since rather technical.
 As a consequence of the above theorem and standard
theory of Markov generators, we get the following characterization
of the time evolution of expected observables:
\begin{Corollary}\label{corri}
Given  $f \in \bbD$, the the map $f(t,\xi):=\bbE_\xi\bigl[ f(\xi_t)
\bigr]$ (the expectation of $f$ for the OCP at time $t$ starting
from $\xi$) is differentiable in $t$ as function in $\bbB$ and
moreover $ \frac{d}{dt} f(t, \cdot) = \cL f$.
\end{Corollary}







\subsection{The hierarchical coalescence process}\label{sec_HCP}

We can now introduce the \emph{hierarchical coalescence process} (in short HCP).
The dynamics depends on a strictly increasing sequence of positive numbers
 $\{d^{(n)}\}_{n\ge 1}$ and a family of   bounded  continuous functions
$\l_\ell^{(n)}$, $\l_r^{(n)}$, $\l_a^{(n)}:[d^{(n)},
\infty]\rightarrow [0,A_n] \,,n \geq 1$. Without loss of generality,
at cost of a length  rescaling,  we may assume  \begin{equation}
d^{(1)}=1  .
\end{equation}
 We set
$\l^{(n)}:= \l^{(n)}_\ell + \l^{(n)}_r+ \l^{(n)}_a$ and we assume
\begin{itemize}
\item[(A1)] for any $n \in \bbN_+$, $\l^{(n)} (d) >0$ if and only if $d \in [d^{(n)},
d^{(n+1)})$,
\item[(A2)] for any $n \in \bbN_+$, if  $d,d'\geq d^{(n)}$, then   $d+d'\geq  d^{(n+1)}$ ({\it i.e.}\ $2d^{(n)} \geq
d^{(n+1)}$),
\item[(A3)] $\lim _{n\to \infty}d^{(n)}=\infty$.
\end{itemize}
For example one could take  $d^{(n)}=n$ or  $d^{(n)} =
a^{n-1}$ with $a \in (1,2]$.


\smallskip


The  HCP is then given by a sequence of one-epoch coalescence
processes, suitably linked. More precisely, at the beginning of the
first epoch one starts with a   SPP with support on $\cN( d^{(1)})=
\cN(1) $. Then
 the stochastic evolution
of the HCP is described by the sequence of random paths $\{ \xi^{(n)
} (\cdot)\}_{n\ge 1}$, where each $\xi ^{(n)}$  is the random
trajectory of the OCP with rates $\l^{(n)}_\ell$, $\l^{(n)}_r$,
$\l^{(n)}_a$, active domain lengths
 $d^{(n)}_{\rm min}=d^{(n)}, \ d^{(n)}_{\rm max}=d^{(n+1)}$ and initial condition  $\xi^{(n)}(0)=\xi ^{(n-1)}
 (\infty),\ n\geq 2$.
Informally we refer to $\xi ^{(n)}$ as describing the evolution in
the $n^{th}$-epoch. Note that, by Lemma  \ref{volarelontano}, one
can prove recursively that at the end of the $n^{th}$--epoch  the
random configuration $\xi^{(n)}(\infty)$ belongs to $\cN( d^{(n+1)}
)$, hence it is an admissible starting configuration for the OCP
associated to the  $(n+1)^{th}$--epoch.


\smallskip

Lemma \ref{vecchioteo1} gives us information on the evolution and
its asymptotics inside each epoch when the initial condition is a
SPP of the renewal type. If {\it e.g.}\ the initial distribution $\cQ$ for
the first epoch is ${\rm Ren}(\nu,\mu)$, where $\mu$ has support on
$[d^{(1)}, \infty)=[1, \infty)$,  we can use Lemma \ref{vecchioteo1}
together with the link $\xi^{(n+1)}(0)=\xi ^{(n)}
 (\infty)$ between two consecutive epochs to recursively define the measures
$\mu^{(n)},\nu^{(n)}$ by
\begin{align}
  \label{eq:1}
\mu^{(n+1)}&:=\mu_\infty^{(n)},\quad \mu^{(1)}:=\mu\,,\cr
\nu^{(n+1)}&:=\nu_\infty^{(n)},\quad \nu^{(1)}:=\nu.
\end{align}
%Note that each $\mu^{(n)} $ (the interval law at the beginning of
%the epoch $n$) has support in $[d^{(n)}, \infty)$, thus allowing to
%apply recursively Theorem \ref{vecchioteo1}.
 With this position it
is then natural to ask if, in some suitable sense, the measures
$\mu^{(n)},\nu^{(n)}$ have a well defined limiting behaviour as $n\to
\infty$. The affirmative answer is contained in the following
theorem, which is the core of the paper, for some specific choice of
transition rates. Before stating it we recall a useful result on the
Laplace transform of probability measures on $[1,\infty)$.

\begin{Lemma}[\cite{FMRT0}] \label{prelim:mainth}
Let $\mu$ be a probability measure on $[1, \infty)$ and let $g(s)$ be
its Laplace transform, \ie $g(s)=  \int e^{-s x}
 \mu(dx)\,, s \in \bbR_+$.
\begin{enumerate}[i)]
\item If
 \begin{equation}
\label{lim_arrosto}
 \lim_{s \downarrow 0} -\frac{s
g'(s)}{1-g(s)}=c_0,
\end{equation}
then necessarily $0\le c_0\le 1$.
\item The existence of the limit \eqref{lim_arrosto} holds if:
\\
a) $\mu
  $ has finite mean and then $c_0=1$ or
\\
b) for some $\a\in (0,1)$ $\mu$ belongs to the domain of attraction of an
$\a$--stable law or, more  generally,  $\mu\bigl((x,\infty)\bigr)=x^{-\a} L(x)$
where $L(x)$ is a slowly varying\footnote{A function $L$ is said to
be slowly varying at infinity, if for all $c>0$, $\lim\limits_{x \to
\infty} L(cx)/L(x)=1$.} function at $+\infty$, $\a\in [0,1]$, and in
this case $c_0=\alpha$.
\end{enumerate}
\end{Lemma}

The reader may find the proof in \cite[Appendix A]{FMRT0} together
with an example for which  the limit \eqref{lim_arrosto} does not
exist.

\begin{Theorem}\label{teo2}
 Let $\nu, \mu$ be probability measures on
$\bbR$ and $[1, \infty)$ respectively.
% and let $g(s)$ be the Laplace transform of $\mu$.
 Suppose that


\begin{itemize}

\item
  the law $\cQ$  of $\xi^{(1)}(0)$  is
either $\cQ=\text{Ren} (\nu, \mu)$ or $\cQ= \text{Ren} (\mu)$ or
$\cQ=\text{Ren}_\bbZ(\mu)$,

\item  it  holds  $(i)$
$\l_a^{(n)} \equiv 0$ for all $n\geq 1$, or $(ii)$ $\l_\ell^{(n)} +
\l_r^{(n)} \equiv \gamma \l_a^{(n)}$ for all $n\geq 1 $ and for some
$\g\geq 0$ independent from $n$,

\item the Laplace transform $g(s)$ of $\mu$ satisfies \eqref{lim_arrosto}.
\end{itemize}

For any $n\geq 1$ let $X^{(n)}$ be a random variable  with law
$\mu^{(n) }$  defined in \eqref{eq:1}
so that $g(s):=\bbE \bigl[ e^{-s X^{(1)}}\bigr]$. %and $Z^{(n)}:=X^{(n)}/d^{(n)}$.

\medskip

Then the following holds:


\begin{itemize}
\item If $c_0=0$,  then
 the rescaled variable $Z^{(n)}=X^{(n)}/d^{(n)}$ weakly
 converges to the random variable $Z^{(\infty)}_{0}= \infty$.

 \item If  $c_0 \in (0,1]$, then
the rescaled variable $Z^{(n)}=X^{(n)}/d^{(n)}$ weakly
 converges to the random variable $ Z^{(\infty)}_{\k }$ with
 values in $[1, \infty)$,
whose Laplace transform is given by
\begin{equation}\label{macedonia}
g^{(\infty)}_{\k}(s)= \cR \left(\k \int _1^\infty \frac{e^{-sx}}{x}
dx \right) \,, \qquad s >0 ,
\end{equation}
where
\begin{equation}\label{galattico}
\begin{cases}
\k :=c_0 \;\;\;\;\;\;\;\;\text{ and } \;\;\;  \cR(x) := 1 - e^{-x} & \text{ in case $(i)$} ,\\
\k := \frac{\g+1}{\g+2} c_0 \;\; \text{ and} \;\; \cR(x):=\frac{
\exp \left\{\frac{\g+2}{\g+1}x\right\} -1}{
\exp\left\{\frac{\g+2}{\g+1}x\right\}+ \frac{1}{\g+1} }
 & \text{ in case $(ii)$}  .
\end{cases}
\end{equation}
\end{itemize}
%,\qquad \mbox{and}\qquad
%\beta = \left\{
%\begin{array}{ll}
%1  &  \mbox{in case } (i),\\
%\frac{\gamma+1}{\gamma +2} &   \mbox{in case } (ii) .\\
%\end{array}
%\right.
\end{Theorem}
The proof of Theorem \ref{teo2} is given in Section
\ref{fegatograsso}. Case (i) has already been proved in \cite{FMRT0}
with a more combinatorial method, not suited for extensions.
\begin{remark}
In the above result the only reminiscence of the initial
distribution is through the constant $c_0$ which is ``universal''
for a large class of initial interval laws $\mu$ (see Lemma
\ref{prelim:mainth}).
%Hence the term \emph{universality} in the title.
%C spostato qui
In particular, starting with a stationary or $\bbZ$--stationary
renewal SPP (which necessarily corresponds to a law $\mu$ with
finite mean), the weak limit of $Z^{(n)}$ always exists and  is
universal ($c_0=1$), depending on the rates only through the
fulfilment of case (i) or case (ii), and not depending on
%not depending on the rates $\l^{(n)}_\ell $, $\l^{(n)}_r$, $\l^{(n)}_a$ and
 the sequence $\{d^{(n)}\}_{n\ge 1}$ which defines the active
intervals.

 We also underline that our results cover a slightly more general
class of HCP. Indeed, following exactly the same lines of our
proofs, we can also treat more general  triple merging allowing
{\it e.g.}\ an active domain to incorporate either
  its two neighbours to the left and/or its two neighbours
   to the right and/or its left and right neighbours (this last case is the
    only one considered in this paper). For this more general class of HCP both the above asymptotic result as well as the one in Theorem \ref{teo3}  are unchanged (instead the single epoch evolution expressed by the differential equation \eqref{basilea} has  to be properly changed by adding to $\lambda_a$ the rates of these new triple mergence events).
\end{remark}
%introduced by mistake
%\begin{remark}
%We point out that the asymptotic  Laplace distribution
%$g^{(\infty)}_{c_0}$ can be written also as $$g^{(\infty)}_{c_0}(s)= 1-
%\exp\Big \{ - c_0 \int _s^\infty \frac{e^{-x}}{x} dx \Big\} = 1-
%\exp\Big \{ - c_0 \, \text{Ei}(s) \Big\}\,,
%$$
%where $Ei(\cdot)$ denotes the exponential integral function \footnote{Note that the function that we denote by $\text{Ei}(s)$ (following the notation of \cite{SE} and our paper \cite{FMRT0}) is instead more frequently denoted in mathematics literature by  $\text{E}_1(s)$.}.  This
%is indeed the form appearing in \cite{DGY2} and \cite{SE} with $c_0
%=1$ (see previous Remark).
%%The remarkable fact of the above result is that the only reminiscence
%%of the initial distribution in the limiting law is through the
%%constant $c_0$ which, as
%%proved in Lemma \ref{prelim:mainth}, is ``universal'' for a large class
%%of initial laws $\mu$. Hence the term \emph{universality} in the title.
%%%C spostato qui
%%We also stress that, starting with a stationary or $\bbZ$--stationary
%%renewal SPP, the weak limit of $Z^{(n)}$ always exists
%%and it is universal ($c_0=1$), not depending on the rates
%%$\l^{(n)}_\ell $, $\l^{(n)}_r$.
%\end{remark}
\begin{remark}
The asymptotic  Laplace distribution $g^{(\infty)}_{\k}$ can be
written also as
$$
g^{(\infty)}_{\k}(s)= \cR \left(  \k \int _s^\infty \frac{e^{-x}}{x}
dx \right)  =  \cR \left(  \k \, \text{Ei}(s) \right)
$$
where $Ei(\cdot)$ denotes the exponential integral
function\footnote{Note that the function that we denote by
$\text{Ei}(s)$ (following the notation of \cite{SE} and our paper
\cite{FMRT0})  is instead more frequently denoted in mathematics
literature by  $\text{E}_1(s)$.}.  This is indeed the form appearing
in \cite{DBG} and \cite{SE}. Moreover, in case $(ii)$ with $\g=0$ in
the above theorem (as in \cite{DBG}), one simply has $\k= c_0/2$ and
$ g^{(\infty)}_{c_0/2}= \tanh \left( \frac{c_0}{2} \text{Ei}(s)
\right)$.
\end{remark}

Next we concentrate on the asymptotic behaviour of the first point law
when starting with a right renewal SPP. %\marginpar{teorema da scrivere}
\begin{Theorem}\label{teo3}
%Let $\mu$ be probability measures on
%$\bbR$  and consider the hierarchical coalescence
%process such that the
%initial law $\cQ$ of $\xi^{(1)}$ is $\text{Ren} (\delta_0, \mu)$.
 Let $\nu, \mu$ be probability measures on
$\bbR$ and $[1, \infty)$ respectively.
% and let $g(s)$ be the Laplace transform of $\mu$.
 Suppose that


\begin{itemize}

\item
  the law $\cQ$  of $\xi^{(1)}(0)$  is
 $\text{Ren} (\nu, \mu)$,

\item  it  holds
$(i)$  $\l_a^{(n)} \equiv 0$ and  $\l^{(n)}_r \equiv \gamma
\l_{\ell}^{(n)}$ for all $n\geq 1$ and for some $\g\geq 0$ independent
from $n$, or $(ii)$ $\l_\ell^{(n)}\equiv 0$  and $ \l_r^{(n)} \equiv
0$ for all $n\geq 1$.

\item the Laplace transform $g(s)$ of $\mu$ satisfies \eqref{lim_arrosto}.
\end{itemize}
For any $n\geq 1$ let  $X_0^{(n)}$ be
 the position of the  first
point  of the HCP at the beginning of the $n$--th epoch and let
$Y^{(n)}$ be the rescaled random variable $Y^{(n)}:=X_0^{(n)} /
d^{(n)}$.

\smallskip

 Then the following holds:
\begin{itemize}

\item  \cite{FMRT0} In case $(i)$, as $n\to \infty$, $Y^{(n)}$ weakly converges to the positive random variable
$Y^{(\infty)}_{c_0}$ with Laplace transform given by
\begin{equation}\label{thenero}
\bbE\bigl( e^{-s Y^{\infty)}_{c_0}} \bigr)=\exp \left\{-
\frac{c_0}{1+\g} \int _{(0,1)} \frac{1-e^{-sy} }{y} dy
\right\}\,,\qquad s\in \bbR_+ .
\end{equation}
%%%
%%%
\item In case $(ii)$,  supposing that $\int z \mu (dz)< \infty$ and  that
\begin{equation}\label{pato}
\lim _{z \to \infty} \frac{1}{z} \int _{[1,z]}   x^2 \mu( dx) = 0
 ,
\end{equation} as $n\to \infty$ the variable $Y^{(n)}$ weakly
converges
% $\infty$ if $\g>0$ and, for $\g=0$,
 to the  random variable $Y^{(\infty)}$ with values in $(0,\infty)$ and  Laplace
transform given by
\begin{equation}\label{duro}
 \bbE\bigl( e^{-s Y^{(\infty)}} \bigr)=
  \frac{e^{-\bar\gamma/2}}{2} \sqrt{ \frac{1-\tanh^2 ( {\rm Ei}(s)/2)}{s}}, \qquad s \in \mathbb{R}_+,
%\exp \left\{ - \frac{1}{ 2}\left( e^{-s} -1 +\int_{(0,1)} s e^{-s y}
%+ \int_{(0,1)} \frac{e^{-sy}-1}{y} dy \right)\right\}
%\sqrt{e/s} \left( 1 - \frac{1}{4} \tanh \left( \frac{1}{2}\int_s^\infty \frac{e^{-x}}{x} dx \right)  \right)^{1/2}
\end{equation}
where we let $\bar\gamma = - \int_0^\infty e^{-t}(\log t)  dt \simeq
0,577$ be the Euler-Mascheroni constant. Condition \eqref{pato} is
satisfied if $\int x^{1+\e} \mu (dx)<\infty$ for some
$\e>0$.\end{itemize}
\end{Theorem}
The proof is given in Section \ref{provateo3}.   Case $(i)$ in the
above theorem has been stated only for completeness. It has already
been obtained in \cite{FMRT0} (see Theorem 2.24 there) . Finally, we
point out that, due to Lemma \ref{prelim:mainth}, under condition
\eqref{pato} it must be $c_0=1 $ in the limit \eqref{lim_arrosto}.




\begin{remark}
Extensions  of the  results presented in this section to  OCP's and
HCP's starting from an exchangeable SPP can be easily achieved
following the arguments  reported in \cite[Appendix D]{FMRT0}.
\end{remark}





\subsection{Examples of HCP's}\label{physics}
We conclude this section by discussing some HCP's coming from the physics
literature.

\subsubsection{The HCP associated to the East model at low
temperature \cite{SE,FMRT1}} An interesting and highly non trivial
example of HCP has been devised in physics literature \cite{SE} to
model
 the high
density (or low temperature) non-equilibrium dynamics of the {\em
East model}  when a deep quench from a normal density state is performed.
% (see
%\cite{SE} for physics motivations and discussions and
%\cite{FMRT1} for a mathematical analysis).
%C
The East model \cite{SE,EJ} is  a well known example of kinetically
constrained stochastic particle system with site exclusion which
evolves according to a  Glauber dynamics submitted to the following
constraint: the $0/1$ occupancy variable at a given site  $x\in
\bbZ$ can change only if the site $x+1$ is empty ({\it i.e.}\ the
corresponding occupation variable equals zero). The change of the
occupation variable, when allowed by this constraint, occurs at rate
$q$ (respectively $1-q$) if it corresponds to a change towards an
empty (respectively occupied) site. Note that each configuration can
also be represented by a sequence of domains on $\bbZ$, where a
domain represents a maximal sequence of consecutive occupied sites
delimited by two empty sites. If the equilibrium vacancy density is
very low ({\it i.e.}\ in the limit  $q\to 0$) and the initial distribution
has a normal density ({\it e.g.} $q=1/2$) most of the non-equilibrium
evolution will try to remove the excess of vacancies of the initial
state and will thus be dominated by the coalescence of domains. In
this setting,  under a proper rescaling \cite{FMRT1}, the East
process can be well described by an HCP with the following
parameters: $d^{(1)}=1$, $d^{(n)}=2^{n-2}+1$ for $n\geq 2$,
$\l^{(n)}_r(d)=\l^{(n)}_a(d)=0$ for any value of the domain length
$d$, thus  $\lambda^{(n)}=\l^{(n)}_\ell$ where $\l^{(n)}_\ell$ is a
function expressed via a proper large deviation probability (see
\cite{FMRT1} for the precise form of this function).
%and it holds  $\l^{(n)}_\ell(d)>0$ for any $d$ which belongs to the active range $d\in[d^{(n)},^{(n+1)})$.
We provide here only a very short explanation to justify the above
choices of the parameters and refer the reader to  \cite{SE} for an
heuristic explanation of the connection of this HCP with East and to
Section 3 of \cite{FMRT1} for a rigorous description. The choice
$\l^{(n)}_a=0$ is due to the fact that the relevant event for East
corresponds to the disappearance of one zero at a time, namely to
the coalescence of two domains (triple domain merging is not
allowed). The asymmetry between the right and left coalescence is
due to the orientated character of the East constraints which
implies that  only the left domains can be incorporated. Finally,
the apparently weird choice of the active ranges $d^{(n)}$ is due to
the fact that in order to remove the vacancy sitting at the left
border of a domain of size $\ell\in[2^{n-1}+1,2^n]$ one needs to
create  at least $n$ additional vacancies inside the domain (again,
see \cite{SE,FMRT1} for details of the combinatorial argument
leading to this result). Thus energy barrier considerations imply
that  this event requires a typical time of order $1/q^n$ which in
turn means that  in the regime $q\to 0$ domains of sizes
$\ell,\ell'$ with  $\ell\in[2^{n-1}+1,2^n]$ ,
$\ell'\in[2^{m-1}+1,2^m]$ and $n\neq m$ are active (namely their
left border can disappear) on very well separated time scales.
\subsubsection{The  paste--all model \cite{DGY2}}
Another interesting HCP  has been  ``introduced" in \cite{DGY2} and
named {\em Paste-all-model}.
 The model  was intended to describe breath figures, namely the patterns formed by growing and  coalescing droplets when vapour condenses on a non wetting surface.
  A common feature of breath figure experiments is the occurrence of a scale-invariant regime with a stable distribution of the drop sizes.
  In \cite{DGY2} several simplified one dimensional models were proposed to understand this phenomenon, including  the HCP named Paste-all-model.
In   this case all the domains are subintervals of the integer
 lattice, a single length is active in each epoch and domains
 merge with their left/right neighbour with rate one, namely $d^{(n)}=n$,
$\l^{(n)}_\ell(n)=\l^{(n)}_r(n)=1$ and $\l^{(n)}_a(n)=0$ (drops can
coalesce either with their right or left neighbour and the smaller
droplets are the first that disappear).

\subsubsection{The HCP associated to the 1d Ising model \cite{BDG}}\label{ising}
Finally, we recall the HCP which has been ``introduced" in
\cite{BDG} to model the zero temperature Glauber dynamics of the one
dimensional Ising model evolved from a random initial condition. In
this case the domains correspond to the ordered spin regions, namely
the maximal sequence of consecutive sites with the same value of the
spin, either up or down. At late stages of the dynamics a
scale-invariant morphology develops: the structure at different
times is statistically similar apart from an overall change of
scale, {\it i.e.}\ the system is described by a single,
time-dependent length scale. Instead of considering the stochastic
Glauber dynamics the authors of \cite{BDG} start from the well known
simpler deterministic model which is expected to mimics this
dynamics, namely the time-dependent Ginzburg-Landau equation for a
scalar field in $d=1$, $\partial_t\phi=\partial_x^2\phi-dV/d\phi$
with $V(\phi)$ a symmetric double well potential with minima at
$\phi=\pm 1$ corresponding to the up and down phases for the Ising
model. If the model starts with a $\phi$ profile corresponding to a
random initial condition for the Ising model, then it evolves
rapidly to a phase of subsequent regions
 were $\phi$ is close to $\pm 1$ (corresponding to the ordered domains) and the dynamics is dominated by the events that bring together and annihilate the closest pair of domain walls. This in turn corresponds to the fact that  the smaller domains merge with the two neighbouring domains.
 % (suppose the smaller domain is in the up phase, then by definition the two neighbouring domains are in the down phase and the first event will typically correspond to the fact that the up phase regions becomes a down phase thus leading to the formation of a larger down phase domain corresponding to the region initially covered by three domains).
Consequently, the HCP which has been introduced in \cite{BDG}  to mimic this domain dynamics has parameters: $d^{(n)}:=n$ (only the smallest length is active at each epoch),
$\l^{(n)}_\ell(n)=\l^{(n)}_r(n)=0$ and $\l^{(n)}_a(n)=1$ (only triple merging occurs).
%and the sequence
%\subsection{Outline of the paper} In Section \ref{dugundi} we
% analyze the metric structure on the space $\cN (d_{\rm min})$. In
%Section \ref{universal} we present the universal coupling of the
%OCP's. Both Sections \ref{dugundi} and \ref{universal} are
%preliminary to the study of the Markov generator of the OCP,
%performed in Sections \ref{semigruppo_OCP} and \ref{infinitesimi},
%and leading the Theorem \ref{amico_marco}. In Section
%\ref{prova_differisco} we apply the results concerning the Markov
%generator of the OCP to prove Theorem  \ref{differisco}. In Section
%\ref{astrattismo} we prove Theorem \ref{astratto} \club finire


%\subsection{Outline of the paper}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



\section{Metric structure of $\cN ( d_{\rm min} )$}\label{dugundi}

%In this section we we collect some useful results on the vague topology and we explain how $\cN(d_{\min})$ can be metrized.

%\subsection{Metric structure of $\cN ( d_{\rm min} )$}
Let us write $\cM$ for the space of Radon measures on $\bbR$, {\it
i.e.}\ locally finite Borel non-negative measures. We consider this
space endowed of the vague topology, such that $\mu_n \to \mu$ in
$\cM$ if and only if $\mu_n(f) \to \mu(f)$ for all continuous
functions  $f$ on $\bbR$ with compact support (shortly, $f \in
C_0$). Then $\cM$ can be metrizied by a suitable  metric $m$ making it a Polish space (see \cite[Sec. A2.6]{DV} and observe that, since the Euclidean space $\bbR$ is Polish and locally
compact, the vague topology coincides with the $\hat w$--topology as
discussed before \cite[Cor. A2.6.V]{DV}). We recall the definition
of $m$ since useful below:
 $$
 m(\mu,\nu):=\int_0^\infty e^{-r}\frac{ d_r\bigl( \mu^{(r)},\nu^{(r)}
\bigr)}{1+ d_r\bigl( \mu^{(r)},\nu^{(r)} \bigr) } dr ,\qquad
\mu,\nu \in \cM,
$$
where $\mu^{(r)}$, $\nu^{(r)}$ denote the
restriction to $(-r,r)$ of $\mu,\nu$, while $d_r$ stands for the
Prohorov distance for measures on $(-r,r)$ (see \cite[Sec.
A2.5]{DV}).

 The space $\cN$ introduced in Section
\ref{sec_SPP} can be thought of as a subspace of $\cM$, identifying
the set $\xi \in \cN$ with  the measure $\sum_{ x \in \xi} \d_x$.
Then one gets that the $\s$--algebra of its Borel subsets coincides
with the $\s$--algebra of measurable subsets introduced in Section
\ref{sec_SPP} (see \cite[Ch.\ 7]{DV}, in particular Prop.7.1.III and
Cor.\ 7.1.VI there). Therefore, the same property holds for $\cN
(d_{\rm min})$ ({\it i.e.}\ Borel subsets and measurable subsets
coincide).




\begin{Lemma}\label{gnocchi}
The following holds:
\begin{enumerate}[$(i)$]
\item  %A sequence $(\xi_n)_{n \geq 1}$ in $\cN( d_{\rm min})$
%vaguely converges to $\xi$
 $\xi_n \to \xi$ in $\cN( d_{\rm min})$
 if and only if $|\xi_n \cap [a,b]| \to |\xi
\cap [a,b]|$ for each interval $[a,b]$ such that $\xi\cap
\{a,b\}=\emptyset$. The same criterion holds replacing closed
intervals $[a,b]$ by open intervals $(a,b)$.

\item Suppose that  $\xi_n \to \xi$ in $\cN( d_{\rm min})$. Fix
  $a<b$ with $\xi \cap \{a,b\} = \emptyset$. Then $\xi_n \cap(a,b) \to \xi \cap (a,b)$ in $\cN(d_{\rm min} )$.
Moreover,  for $n$ large enough   $\xi_n \cap (a,b)$ has the same
cardinality of $\xi \cap (a,b)$ and $\lim_{n\to \infty} x_i^{(n)}=
x_i$ for $1\leq i \leq k$, where $\xi_n\cap(a,b)=
\{x_1^{(n)}<x_2^{(n)} < \cdots <x_k^{(n)} \}$ and $\xi\cap(a,b)=
\{x_1<x_2 < \cdots <x_k\}$.



\item The space $\cN ( d_{\rm min})$ is a closed subset of $\cM$.
In particular,  it is a Polish space endowed of the metric
$m$.

\item The space  $\cN (d_{\rm min})$ is compact.
\end{enumerate}
\end{Lemma}


\begin{proof} Part $(i)$ with closed intervals follows from \cite[Prop. A2.6.II]{DV} (see also \cite[Th.1.1.16]{FKAS} with $P_n:=\delta_{\xi_n}$ and
$P:= \d_\xi$). The same criterion with open interval is a simple
derivation from the one with closed interval.

Let us consider Part $(ii)$. Applying the criterion in Part $(i)$ it
is trivial to check that $\xi_n \cap (a,b) \to \xi \cap (a,b)$. Take
now $\e >0$ small enough that all the intervals
$J_i=[x_i-\e,x_i+\e]$, $1\leq i \leq k$, are disjoint and intersect
$\xi$ only at $x_i$. Then, by item $(i)$ for $n$ large $\xi_n$ has
exactly one point in each $J_i$. Similarly, $\xi_n$ has exactly $k$
points in $(a,b)$ for $n$ large. By the arbitrariness of $\epsilon$
we can conclude.

To prove Part $(iii)$ call $\bar \cN$ the family of counting
measures in $\bbR$, {\it i.e.}\ $\xi \in \bar \cN$ if and only if
$\xi= \sum _i k_i \d_{x_i}$ with $k_i \in \bbN_+$ and $\{x_i\}$
being a locally finite countable subset of $\bbR$. By
\cite[Prop.7.1.III]{DV}, $\bar \cN$ is a closed subset of $\cM$.
Hence, if $\xi_n\in \cN (d_{\rm min}) $ and $\xi_n \to\xi$ with $\xi
$ in $\cM$, then $\xi \in \bar \cN$. We only need to show that $\xi
\in \cN( d_{\rm min})$. Suppose by contradiction that
$\xi(\{x\})\geq 2$ for some $x \in \bbR$. Take $I= [x-\epsilon,
x+\epsilon]$ such that $\xi ( \{x-\epsilon, x+\epsilon\})=0$ and $2
\e < d_{\rm min}$  (the existence of $\epsilon$ is guaranteed by the
fact that $\xi \in \bar \cN$). By Part $(i)$ it must be $\xi_n(I)
\geq 2$ for $n$ large enough, in contradiction with the fact that
$\xi_n$ can have at most one point in $I$.

Due to Part $(iii)$, Part $(iv)$ is a simple consequence of the
compactness criterion given in \cite[Cor. A2.6.V]{DV}.
\end{proof}



 Recall that  $\mathbb{B}$ denotes
the Banach space of all continuous  functions $f:\cN(d_{\rm min})
\to \mathbb{R}$ endowed with the uniform norm $\| \cdot \|$ and that
$\mathbb{B}_{\rm loc}$ denotes the set of local  functions $f \in
\mathbb{B}$.

\begin{Lemma}\label{lemino}
The set $\mathbb{B}_{\rm loc}$ is dense in $\mathbb{B}$. In
particular, given   $f \in \bbB$ and defining $f_N(\xi):= \int
_{N}^{N+1}  f(\xi \cap(-r,r) )dr$, it holds $f_N \in \bbB_{\rm loc}$
and $f_N \to f$ in $\bbB$.
\end{Lemma}

Note that the map $\bbR_+ \ni r \mapsto f(\xi \cap (-r,r))\in \bbR$
is stepwise, with a finite number of jumps in any finite interval.
Hence, the above function $f_N$ is well defined.

\begin{proof}
Take $ f\in \bbB$. Since $\cN(d_{\min})$ is compact, $f$ is
uniformly continuous. Hence, given $\epsilon >0$,  there exists
$\delta_0>0$ such that $m(\xi,\xi')<\delta_0$ implies
$|f(\xi)-f(\xi')|<\epsilon$. Take $N_0\in \bbN$ large enough that
$e^{-N_0} \leq \delta_0$. By the definition of $m$ we have  $m (\xi,
\xi \cap (-N,N) )\leq \int_{N}^\infty e^{-a}da \leq \d_0$ for any $N
\geq N_0$. This implies that $|f(\xi)-f(\xi\cap(-r,r)) |\leq
\epsilon$ for all $r \geq N_0$ and therefore $\|f- f_N\|\leq
\epsilon$. Trivially $f_N$ is a local function, it remains to prove
that $f_N$ is continuous. To this aim, fix $\xi \in \cN (d_{\rm
min})$. Then the set $R=\{r\in [N,N+1]\,:\,\xi \cap \{-r,r\}\not=
\emptyset\}$ is finite. In particular, by Lemma \ref{gnocchi}
$(ii)$, if $\xi_n \to \xi$ then $\xi_n \cap(-r,r)\to\xi \cap (-r,r)$
for all $r \in [N,N+1]\setminus R$. Since $f$ is continuous, we get
that
$$f(\xi_n\cap (-r,r) ) \to f(\xi \cap (-r,r) )\qquad \forall r\in
[N, N+1]\setminus R .
$$
We conclude applying now the dominated convergence theorem.
\end{proof}














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\section{OCP process: proof of Theorem \ref{differisco} and Theorem \ref{differiscobis}}\label{prova_differisco}

 In this section  we prove Theorem
\ref{differisco} and Theorem \ref{differiscobis} applying our
analysis of the Markov generator of the OCP (recall Corollary
\ref{corri}).






\subsection{Differential equation for $\mu_t$ and proof of Theorem \ref{differisco}}
 As application of Corollary \ref{corri}
  we
 can prove the following result:
 \begin{Proposition}\label{diuretico}
Let  $f:[0,\infty)\to \bbR$ be a  continuous   function such that
\begin{equation}\label{neve}
\sum _{k =0}^\infty \sup _{x \geq k} |f(x)|<\infty .
\end{equation}
Let $\mu$ be a probability measure on $[d_{\rm min},\infty)$ and
$\mu_t$ be as in Lemma \ref{vecchioteo1} with the choice $Q=
\text{Ren} (\mu)$. Then, the function $[0,\infty)\ni t\mapsto
\mu_t(f)\in \bbR$ is differentiable and
\begin{align}\label{basilea}
\frac{d}{dt}  \mu_t(f)
& =
-\int \mu_t (dx) \l(x) f(x) +\int \mu_t (dx) \int \mu_t (dy) \bigl( \l _r(x)+\l_\ell (y) \bigr)f(x+y)
\nonumber \\
& \quad +
\int \mu_t (dx) \int \mu_t (dy) \int \mu_t (dz) \l_a (y) f(x+y+z)  .
\end{align}
 \end{Proposition}
\begin{proof}
 Set   $\cQ= \text{Ren} (\d_0, \mu)$.
Note that $\bbP_\cQ$--a.s. $\xi(t)$ belongs to the set  $\cN_*$ of
configurations $\xi \in \cN(d_{\rm min})$ such that $\xi \subset [0,
\infty)$, $\xi\cap(0,d_{\rm min}/2]=\emptyset$ and $\xi$ is given by an  increasing sequence of points
diverging to
$\infty$. %Note that $\cN_*$ is left invariant by the annihilation of a finite set of points.
Points in $\xi \in \cN_*$ are labeled as $x_0(\xi),x_1(\xi),x_2
(\xi), \dots$ in increasing order.  Then,  by Lemma
\ref{vecchioteo1},
 $\mu_t$ equals the law of $x_1(\xi(t))$ under $\bbP_\cQ( \cdot |
0\in \xi (t) )$.
 Hence we can write $ \mu_t(f) = N_t/D_t $ where
\begin{align*}
&  N_t=  \bbE_\cQ\left[ f\bigl(x_1 (\xi(t)  )\bigr); 0\in \xi (t)
\right]\,, \qquad
 D_t =\bbP_\cQ(  0\in \xi (t) )  .
\end{align*}
Let $\rho: \bbR \to [0,1]$ be a continuous function such that
$\rho(x)=0$ for $x \not \in (-\frac{d_{\rm min}}{2}, \frac{d_{\rm
min}}{2})$ and $\rho(0)=1$. By definition of vague convergence (see
Section \ref{dugundi}), the function $\Phi:\cN (d_{\rm min})\mapsto
\bbR$ defined as $\Phi(\xi):= \sum _{x \in \xi} \rho(x)$ is a
continuous map. Since local it belongs to $\bbB_{\rm loc}$ and
moreover it satisfies $\Phi(\xi)= \mathds{1}_{ 0 \in \xi}$  for all
$\xi \in \cN_*$.  In Lemma \ref{tosse} below we exhibit a function
$\Psi \in \bbD$ that satisfies  $\Psi (\xi)=f\bigl(x_1 (\xi
)\bigr)\mathds{1}_{ 0 \in \xi}$ for all $\xi \in \cN_*$. Hence, we
can write
$$ N_t =\bbE_\cQ\left[\Psi (\xi(t) ) \right]\,, \qquad D_t=\bbE_\cQ\left[ \Phi (\xi(t) )
\right] .
$$
By standard properties of Markov generators, we
conclude that the maps $N_t, D_t$ are differentiable  and that
$$
N_t'=\bbE_\cQ\left[ \cL\Psi (\xi(t) ) \right]\,,
\qquad D'_t = \bbE_\cQ\left[\cL \Phi (\xi(t) )\right] .
$$
Since $\Psi, \Phi \in \bbD$, we can use Equation \eqref{santi} %apply Theorem \ref{amico_marco}
to compute $\cL \Psi $ and $\cL \Phi $. We need their value only on $\cN_*$.
%Since at the end we only
%need their value on $\cN_*$ and since the space $\cN_*$ is left
%invariant by annihilation of a finite number of points in $\xi \in
%\cN_*$, this computation is simple and does not depend on the value
%of $\Phi, \Psi$ outside $\cN_*$.  Since we are dealing with right
%renewal SPP's points are numbered in increasing order starting from
%$x_0$.
 Suppose that $\z, \xi \in \cN_*$ are such that $\z \subset \xi$ and  $0\in \xi$. %Recall that $d_i:= x_{i}- x_{i-1}$ and that
%points are enumerated such that $x_0 \leq 0 \leq x_1$.
Writing $x_i$ and $d_i$ instead of $x_i(\z)$ and $d_i (\z)= x_i
(\z)-x_{i-1}(\z)$, we get
$$
\begin{cases}
 \cL\Psi (\z) = \mathds{1}(0\in \z) G (\z) ,\\
  \cL \Phi (\z)= \mathds{1}(0 \in
 \z) H(\z)
\end{cases}
$$
 where $ H(\z):= - \l_\ell  (d_1 ) -\l_a( d_1 )$ and
\begin{align*}
  G(\z)& := -\bigl[\l_\ell (d_1)+ \l_a (d_1) \bigr] f(x_1) +
\bigl[ \l_r (d_1)+ \l_\ell (d_2) \bigr] \bigl[f(x_2)-f(x_1) \bigr] \\
& \quad +
\l_a (d_2) \bigl[ f(x_3)-f(x_1) \bigr] .
\end{align*}
Since
$N_t$ and $D_t$ are derivable, we get that $N_t/D_t$ is derivable
and that
$$
\frac{d}{dt} \mu_t (f) = \frac{d}{dt} \frac{N_t}{D_t} = \frac{N_t'}{D_t}- \frac{N_t}{D_t} \frac{D_t '}{D_t}.
$$
Writing $F(\xi)= f( x_1 (\xi) )$, the above identities imply that
\begin{equation}\label{malattia}
\frac{d}{dt} \mu_t (f)=
%\frac{d}{dt} \frac{N_t}{D_t}=
\bbE_\cQ\left( G(\xi(t) )\, | \, 0\in
\xi (t) \right)- \bbE_\cQ \left( F (\xi(t) )\, |\, 0\in \xi
(t)\right ) \bbE_\cQ\left( H(\xi(t) )\, | \, 0\in \xi (t) \right) .
\end{equation}
 By Lemma \ref{vecchioteo1} $(iv)$, we
can write (brackets should help to follow the computations)
\begin{align}\label{infettiva}
 & \bbE_\cQ\left( G(\xi(t) )\, | \, 0\in \xi (t) \right)
 =
- \left\{ \mu_t ( \l_\ell f) +\mu_t (\l_a f )\right\} \nonumber \\
& \qquad
+ \left\{ \int \mu_t(dx)\int  \mu_t (dy)\bigl[\l_r(x)+ \l_\ell(y) \bigr]f(x+y)
-\mu_t (\l_r f) -\mu_t (\l_\ell) \mu_t (f) \right\} \nonumber \\
& \qquad
+ \left\{ \int
\mu_t(dx) \int \mu_t (dy)\int\mu_t(dz) \l_a(y) f(x+y+z) - \mu_t
(\l_a) \mu_t (f) \right\}
\end{align}
and
\begin{equation}\label{preghiamo}
\bbE_\cQ \left( F (\xi(t) )\, |\, 0\in \xi (t)\right )
\bbE_\cQ\left( H(\xi(t) )\, | \, 0\in \xi (t) \right)
=
- \mu_t (f)\mu_t (\l_\ell) -\mu_t (f) \mu_t (\l_a) .
\end{equation}
Combining the above identities \eqref{malattia}, \eqref{infettiva}
and \eqref{preghiamo} we get the thesis.
\end{proof}

In the proof of Proposition \ref{diuretico} above, we used the following technical lemma.

\begin{Lemma}\label{tosse}
Let  $f$ be a real continuous  function on $[0,\infty)$ satisfying
\eqref{neve} and extend it to a continuous function on $\bbR$
constant on $(-\infty,0]$. Given $s \in \bbR$ define
$$ f_s (\xi) =
 \begin{cases}
0 & \text{ if } \bigl|\xi \cap (s, \infty)\bigr|\leq 1  \\
 f \left ( z\bigl(\xi\cap (s,\infty)\bigr) \right)  & \text{ otherwise}
\end{cases}
$$
where $ z\bigl(\xi\cap (s,\infty)\bigr)$ denotes the second point
from the left of $\xi \cap(s, \infty)$. Then the function
$$F:\cN(d_{\rm min} ) \ni \xi \mapsto \frac{1}{d_{\rm min} }\int
_{-d_{\rm min}}^{0} f_s(\xi ) ds \in \bbR$$ belongs to $\bbB$.
Moreover, the function $\Psi(\xi)= \Phi(\xi) F(\xi)$ belongs to
$\bbD$ and  $\Psi (\xi)=f\bigl(x_1 (\xi )\bigr)\mathds{1}_{ 0 \in
\xi}$ for all $\xi \in \cN_*$ (for the definition of $\Phi$ and
$\cN_*$ see the the proof of Proposition \ref{diuretico}).
\end{Lemma}
The integrand in the definition of $F$ is a stepwise function with a
finite number of jumps, hence it is integrable.
\begin{proof}
Let us  prove the continuity of  $F$. Take  $\xi_n \to \xi $ in
$\cN(d_{\rm min})$ and set $R:=\{s \in (-d_{\rm min},0)\,:\, s
\not\in \xi \}$. We claim that, fixed  $s \in R$, it holds $f_s
(\xi_n) \to f(\xi)$.
 Let us first suppose that $|\xi \cap
(s,\infty)|\geq 2$. Let $a<b$ be the first two points of $\xi
\cap(s,\infty)$ and take $c$ larger than $b$ such that $\xi$ has no
point in $(b,c]$. Then by Lemma \ref{gnocchi} $(ii)$ $\xi_n
\cap(s,c)$ has exactly two points $a^{(n)}<b^{(n)}$ eventually in
$n$, moreover $a^{(n)} \to a$ and $b^{(n)} \to b$. By the continuity
of $f$, we have $$f_s(\xi_n)= f(b^{(n)}) \to f(b) = f_s (\xi) .
$$
Let us now suppose that $|\xi \cap (s,\infty)|\leq  1$. Suppose
first that $\xi \cap (s, \infty)$ has only one point, denoted by
$x_*$. Given $\e>0$ fix $L>x_*$ such that $L \not \in \xi$ and
$|f(x)| \leq \e$ for $x \geq L$ ($L$ exists due to \eqref{neve}). By
Lemma \ref{gnocchi} $(i)$ for $n$ large $\xi_n$ has exactly one
point in $(s, L)$. This assures that $|f_s(\xi_n)|\leq \e$ for $n$
large and therefore that $\lim _{n\to \infty} f_s(\xi_n)=0=f_s
(\xi)$. A similar argument can be applied when $\xi $ has no point
in $(s, \infty)$. This concludes the proof of our claim.


 Combining our claim with the dominated convergence theorem and
with the fact that $R$ is a finite set, we get that $F(\xi_n) \to
F(\xi)$, thus proving the continuity of $F$.

If $\xi \in \cN_*$ it is simple to check that  $\Psi
(\xi)=f\bigl(x_1 (\xi )\bigr)\mathds{1}_{ 0 \in \xi}$.  It remains
to prove that $|||\Psi|||<\infty$. Suppose that $k \in \bbZ $ and
$\nabla _k \Psi (\xi)\not = 0$. Then $k \geq -1$ and $\xi$ has at
least two points in $(-d_{\min}, \infty)$, the first or the second
one (from the left) must lie in $I_k$. In particular, it must be
$|\nabla_k \Psi (\xi) |\leq 2 \sup_{x \geq k d_{\rm min} } |f(x) |$.
Take now %\club \club 
$k<k'\leq k+\lceil d_{\max}/d_{\min}\rceil $ in
$\bbZ$. Suppose that $\nabla_{(k,k')} \Psi (\xi)\not =0$. If $k<-1$
then $\nabla_{(k,k')} \Psi (\xi)=\nabla_{k'} \Psi (\xi)$ which can
be bounded as above. If $k,k'\geq -1$, then we conclude that $\xi$
has at least two points in $(-d_{\min}, \infty)$, the first or the
second one (from the left) must lie in $I_k\cup I_{k'}$. Hence,
$|\nabla_{(k,k')} \Psi (\xi) |\leq 2 \sup_{x \geq k d_{\rm min} }
|f(x) |$. The above bounds  and condition \eqref{neve} allow to
conclude.
\end{proof}



%Suppose that $\xi \subset \bbZ$ for $\cQ$--a.a. $\xi$. Then the
%function  $f(x)= \mathds{1}(x=d)$ satisfies the hypothesis of
%Proposition \ref{diuretico}, while $\mu_t(f)= \mu_t
%(\{d\})=:\mu_t(d)$. Hence, \eqref{basilea} reads
%\begin{multline}
%\frac{d}{dt}  \mu_t(d)= -\mu_t (d) \l(d) +\sum _{x,y\in
%\bbN_+:x+y=d}  \mu_t (x) \mu_t (dy) \bigl( \l _r(x)+\l_\ell (y)
%\bigr)
%\\+ \sum _{x,y,z\in
%\bbN_+:x+y+z=d}  \mu_t (x)\mu_t (y)\mu_t (z) \l_a (y)  \,.
%\end{multline}

We have now all the tools to prove Theorem \ref{differisco}.


\begin{proof}[Proof of Theorem \ref{differisco}]
 To prove \eqref{davide} and \eqref{orecchiettebis} we can restrict
to $s>0$. Indeed, writing these differential equations as integral
identities one can take the limit $s\downarrow 0$ and recover the
case $s=0$.

 We can apply  Proposition \ref{diuretico} to the function  $f(x)=
e^{-s x}$, $x \geq 0$, getting that  $G_t(s) = \mu_t (f)$ is
$t$--differentiable, with derivative given by \eqref{basilea}.

We can write $H_t(s) = \mu_t (\tilde f)$ where
 $\tilde f (x):=e^{-s x}\mathds{1}(x<d_{\rm max} )$. Obviously
 $\tilde f$ is not suited to Proposition \ref{diuretico} since not
 continuous. If $\mu$ had support on a lattice, {\it e.g.}\ $\bbN$,
 trivially $\tilde f $ could be replaced by a nice function. In the
 general case we need more care.
For $\e>0 $ small enough, we fix a continuous function $f_\e$ on
$[0, \infty)$  with values in $[0,1]$  such that   $f_\e (x)=\tilde
f (x) $ if $x \not \in \bigl(d_{\rm max} -\e,d_{\rm max}\bigr)$.
Applying Proposition \ref{diuretico} we get that the function
$[0,\infty) \ni t \mapsto \mu_t (f_\e)$ is differentiable with
derivative given by \eqref{basilea} (with $f$ replaced by $f_\e$).
Since $\mu_t$ has support in  $[d_{\rm min}, \infty)$ and since
$f_\e(x+y)=0$, $f_\e(x+y+z)=0$ if  $x,y, z \geq d_{\rm min}$ (recall
assumption (A2) in Section \ref{sec_OCP}), from \eqref{basilea} we
conclude that $ \mu_t' (f_\e)= - \mu_t ( \l f_\e)$.

Let $\a_t(\e):= \mu_t \bigl( (d_{\rm max}-\e, d_{\rm max}) \bigr)$.
Trivially, $\lim _{\e \downarrow 0} \a_t (\e)=0$ and $0\leq
\a_t(\e)\leq 1$.
 Since $ |\mu_t (\tilde f) - \mu_t (f_\e)| \leq \a_t( \e) $ and
 $$
 | \mu_t ' (f_\e)+ \mu_t (\l \tilde{f}) |
 =
| \mu_t  (\l f_\e)- \mu_t (\l \tilde{f}) |
 \leq
 \a_t( \e)\|\l\|_\infty,
 $$
 applying the Dominated Convergence Theorem we
 get
\begin{equation*}%\label{befana}
H_t (s)=\mu_t (\tilde f)= \lim _{\e \downarrow 0} \mu_t (f_\e) =
\lim _{\e \downarrow 0}\bigl[ \mu_0(f_\e) - \int _0 ^t\mu_u( \l
f_\e) du\bigr]=\mu_0(\tilde f) - \int _0 ^t\mu_u( \l \tilde f) du .
\end{equation*}
Since  $\l$ is a continuous function (extendable on $[0,\infty)$)
and is zero on  $[ d_{\rm max}, \infty)$, we can apply Proposition
\ref{diuretico} to the function  $\l \tilde f$ concluding that the
map $[0,\infty) \ni t \mapsto \mu_t (\l \tilde f)$ is differentiable
and therefore continuous. This observation together with the above
identity allows to conclude that $H_t(s)$ is $t$--differentiable and
its derivative satisfies \eqref{davide} (note that $\l \tilde{f}= \l
f $ by assumption (A1) in Section \ref{sec_OCP}). Knowing that
$\partial _t G_t(s)$ is given by \eqref{basilea} and using
\eqref{davide} we get \eqref{orecchiettebis}.
%we conclude that
%\begin{align}\label{orecchiette}
% \partial _t \bigl[G_t(s)-H_t(s)\bigr]
% & =
% \int \mu_t (dx) \int \mu_t (dy) \bigl( \l _r(x)+\l_\ell (y) \bigr)e^{-s(x+y)}  \\
% & \quad + \int \mu_t (dx) \int \mu_t (dy) \int \mu_t (dz) \l_a
%(y)e^{-s(x+y+z)} \,.. \nonumber
%\end{align}
%Since   $\mu_t$ has support on $[d_{\rm min},\infty)$, by assumption
%(A2) we can remove the characteristic functions in
%\eqref{orecchiette} getting  \eqref{orecchiettebis}.




We observe that in case $(i)$  it holds $ \l_\ell + \l_r=\l$, while
in case $(ii)$ it holds $\l_\ell+\l_r= \l\g/ (1+\g)$ and $\l_a = \l
/(1+\g)$. These identities allow to derive from \eqref{davide} and
\eqref{orecchiettebis} that $\partial_t G_t (s)= \partial_t H_t(s)
(1- G_t(s))$ in case $(i)$ and that $\partial _t G_t (s)=
\partial_t H_t(s) \left( 1 - \frac{G_t(s)(\gamma +
G_t(s))}{1+\gamma}\right)$ in case $(ii)$. The rest of the proof
follows by the computations outlined in Remark \ref{pasquetta} using
\eqref{davide}, \eqref{orecchiettebis} and the fact that $\lim
_{t\to \infty} G_t(s)= G_\infty (s)$, $\lim _{t \to \infty} H_t(s)
=H_\infty (s)=0$ (which   is due to Lemma \ref{volarelontano}
$(iii)$ and Lemma \ref{vecchioteo1} $(v)$ ). We only point out that
with the definition of $A_t(s), B_t(s)$ given in Remark
\ref{pasquetta} one gets $ b_u(s) e^{A_u(s) + 2 B_u(s) } =
 \frac{1}{\g+2}\partial _u e^{- H_u \frac{\g+2}{\g+1} }
 $   in
case $(ii)$.  \end{proof}




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\subsection{Differential equation for $\nu_t$ and proof of Theorem \ref{differiscobis}}
As in the case of the interval law $\mu_t$, in order to prove Theorem \ref{differiscobis}, we need first to establish a differential equation
 for the expectation $\nu_t(f)$ for nice functions $f$.
 %Recall that $\lfloor \cdot \rfloor$ denotes the entire part.


\begin{Proposition}\label{vongole}
%Consider a hierarchical coalescence process starting from a right renewal SPP $\mathcal{Q} = Ren(\nu,\mu)$ with.
%Then, for any continuous function  $f: [\inf \mbox{supp}(\nu),\infty) \to \bbR$ such that
%\begin{equation}\label{nuvole}
%\sum _{k =\lfloor \inf \mbox{supp}(\nu)\rfloor }^\infty \sup _{x \in [k, \infty)} |f(x)|<\infty\,,
%\end{equation}
Let $f: [0,\infty) \to \bbR$ be as in Proposition \ref{diuretico}.
Let $\mu$ be a probability measure on $[d_{\rm min},\infty)$ and
$\nu_t$ be as in Lemma  \ref{vecchioteo1} with the choice $Q=
\text{Ren} (\d_0,\mu)$. Then the map $[0,\infty)\ni t\mapsto
\nu_t(f) \in \bbR$ is differentiable and
\begin{align}\label{carbonara}
\frac{d}{dt}  \nu_t(f)
& =
- \! \int \!\nu_t(dx) \! \int \!\mu_t(dy) \left(\lambda_\ell(y) + \lambda_a(y) \right) f(x)
+ \int \!\nu_t (dx)\! \int \!\mu_t (dy)  \l_\ell (y) f(x+y)  \nonumber \\
& \quad
+
\int \nu_t (dx) \int \mu_t (dy) \int \mu_t (dz) \l_a (y) f(x+y+z)  .
\end{align}
\end{Proposition}
%In the above statement, if $\inf \mbox{supp}(\nu)=-\infty$, then we
%understand that $[\inf \mbox{supp}(\nu),\infty)=\mathbb{R}$.

\begin{proof}
We extend $f$ as continuous function to all $\bbR$, constant on
$(-\infty,0]$. Given $s \in \bbR$ define
$$ f_s (\xi) =
 \begin{cases}
0 & \text{ if }  \xi \cap (s, \infty)=0  \\
 f \left ( z\bigl(\xi\cap (s,\infty)\bigr) \right)  & \text{ otherwise}
\end{cases}
$$
where $ z\bigl(\xi\cap (s,\infty)\bigr)$ denotes the first point
from the left of $\xi \cap(s, \infty)$. Then the function
$$\Theta :\cN(d_{\rm min} ) \ni \xi \mapsto \frac{1}{d_{\rm min} }\int
_{-d_{\rm min}}^{0} f_s(\xi ) ds \in \bbR$$ belongs to $\bbD$ and
$\Theta (\xi) = f(x_0(\xi)) $ if $\xi\in \cN_*$. The proof is
similar to the one of Lemma \ref{tosse} and we omit the details.

Set   $\cQ= \text{Ren} (\d_0, \mu)$. Note that $\bbP_\cQ$--a.s.
$\xi(t)$ belongs to the set  $\cN_*$ of configurations $\xi \in
\cN(d_{\rm min})$ such that $\xi \subset [0, \infty)$, $\xi \cap (0,
d_{\rm min}/2]=\emptyset$ and $\xi$ is given by an increasing
sequence of points diverging to $\infty$. Points in $\xi \in \cN_*$
are labeled as $x_0(\xi),x_1(\xi),x_2 (\xi), \dots$ in increasing
order. Hence, we can write
$$
 \nu_t(f) = \mathbb{E}_{\cQ} \bigl[ f\bigl(x_0 (\xi(t))\bigr)  \bigr] =
\mathbb{E}_{\cQ} \bigl[  \Theta(\xi(t))\bigr]  .
$$
Using that $\Theta \in \bbD$ and therefore \eqref{santi},
 one concludes that the map $t\to \nu_t(f)$
is differentiable and that
\begin{align*}
\frac{d}{dt}  \nu_t(f) & =
\mathbb{E}_\cQ \bigl[\mathcal{L} \Theta(\xi(t))\bigr] \\
& =
\mathbb{E}_{\cQ} \left[  \,\lambda_\ell(x_1-x_0)[f(x_1)-f(x_0)] +\lambda_a(x_1-x_0)[f(x_2)-f(x_0)]  \,   \right] \\
& =
 \int   \nu_t(dx)  \int\mu_t(dy)\lambda_\ell(y)[f(x+y)-f(x)]  \\
& \quad +  \int   \nu_t(dx) \int  \mu_t(dy) \int\mu_t(dz)
\lambda_a(y)[f(x+y+z)-f (x)]
\end{align*}
where we used, for simplicity of notation, $x_0=x_0(\xi(t))$,
$x_1=x_1(\xi(t))$ and $x_2=x_2(\xi(t))$ and the fact that $x_0$ has
law $ \nu_t$, while $x_1-x_0$ and $x_2-x_1$ have law $\mu_t$.
\end{proof}


\begin{proof}[Proof of Theorem \ref{differiscobis}] As in the proof
of Theorem \ref{differisco} we can take $s>0$. Using Proposition
\ref{vongole} with the function $f : x \mapsto e^{-sx}$,
 we get that $t \mapsto L_t(s)$ is
differentiable and that
\begin{align*}
\frac{d}{dt} L_t(s) & =
- \int \!\nu_t(dx)  \!\int\! \mu_t(dy) \left(\lambda_\ell(y) + \lambda_a(y) \right) e^{-sx} +\int \!\nu_t (dx) \int \!\mu_t (dy)  \l_\ell (y) e^{-sx-sy}  \nonumber \\
& \quad
+
\int \nu_t (dx) \int \mu_t (dy) \int \mu_t (dz) \l_a (y) e^{-sx-sy-sz}  .
\end{align*}
The above equation corresponds to \eqref{bucatini}.

Consider the case $\lambda_a\equiv0$. Then if $\lambda_\ell\equiv
0$, trivially $L_t=L_0$ for any $t \geq  0$. While for
$\lambda_r\equiv\gamma \lambda_\ell$, one has
$\lambda_\ell\equiv\frac{1}{1+\gamma} \lambda$ so that the
differential equation \eqref{bucatini} satisfied by $L_t$ reads
$$
\partial_t L_t(s) = \frac{L_t(s)}{1+\gamma} \left[ -\int \!\mu_t(dy)
\lambda(y) + \int \! \mu_t(dy) \lambda(y)e^{-sy} \right] =
 \frac{L_t(s)}{1+\gamma} \left( \partial _t H_0(s)  - \partial_t H_t(s) \right)
$$
where we used \eqref{davide}. Integrating  and using that $\lim _{t
\to \infty} H_t (s) = 0$ leads to \eqref{monte1} and \eqref{monte2}.

\smallskip

 Now consider the case $\lambda_\ell \equiv 0$, $ \lambda_r \equiv0$.
Noticing that $\lambda_a \equiv \lambda$ and using \eqref{davide},
from \eqref{bucatini} we obtain that $
\partial_t \ln  L_t(s) = \partial _t H_t(0)- G_t (s) \partial_t
H_t(s) $. At this point we apply Point (ii) in Theorem
\ref{differisco} with $\g=0$ getting for $s>0$
$$ \partial_t \ln L_t(s)= \partial _t H_t(0) - \frac{G_t(s)\partial_t G_t(s) }{1-
G_t(s)^2} = \partial _t \left\{ H_t (0) +\frac{1}{2} \ln
(1-G_t(s)^2)\right\} .
$$
This leads to \eqref{gransasso1}, which implies \eqref{gransasso2}
after taking the limit $t\to \infty$.
\end{proof}





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%    HCP
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Abstract generalization
of the transformation introduced in \cite{FMRT0}}\label{trasformo}


% proof of Theorem \ref{teo2} and Theorem \ref{teo3}}
%In this section we prove Theorem \ref{teo2} and Theorem \ref{teo3}.
%This will be achieved using an abstract result that transforms the
%nonlinear identities on the Laplace transforms appearing in Theorem
%\ref{differisco} and Theorem \ref{differiscobis} into  linear
%identities involving Radon measures. \club {\bf Ale:} {\sl non sono
%completamente sicura che l'aggettivo "linear" sia  ragionevole}

%\subsection{Abstract result}

We extend here a transformation developed in \cite[Sec. 5]{FMRT0}
allowing to rephrase the non-linear identities on the
Laplace transforms appearing in Theorem \ref{differisco} and Theorem
\ref{differiscobis} into  linear identities involving Radon
measures.  This transformation will be crucial in our analysis of the limiting behaviour of the HCP process (see Section \ref{fegatograsso} and \ref{provateo3}).
%\club {\bf Ale:} {\sl non sono completamente sicura che
%l'aggettivo "linear" sia  ragionevole}
%revealing why the limit states for the interval law and the first
%point law of the HCP have a similar structure when comparing case
%(i), i.e.\ $\l_a\equiv 0$, and case (ii), i.e. $\l_\ell+\l_r \equiv
%\g\l_a$.

Consider the  OCP starting from a renewal SPP  with interval law $\mu$
 having
support on $[d_{\rm min}, \infty)$ ({\it i.e.}\ $\xi(0)$ has law ${\rm
Ren}(\nu, \mu) $ or ${\rm Ren} (\mu)$ or ${\rm Ren}_\bbZ(\mu)$). We
recall that $\mu_\infty$ denotes the interval law at the end of the
epoch (see Lemma \ref{vecchioteo1}) and  we call $X_0, X_\infty$
some generic random variables with law $\mu, \mu_\infty$,
respectively. Then we define the rescaled random variables
$$
Z_0=X_0/d_{\rm min} \;\; \text{ and }\;\; Z_\infty=X_\infty/d_{\rm max}
$$
 and we set, for $s \geq 0$,
\begin{equation*} %\label{ridono}
g_0(s) = \bbE \bigl( e^{-s Z_0}\bigr)\,, \;\; g_\infty (s)=\bbE
\bigl (e^{- s Z_\infty } \bigr) \,, \;\;  h_0 (s) = \bbE \bigl(
e^{-s Z_0 }; Z_0< a  \bigr) \,,\;\; a:= \frac{d_{\rm max}}{d_{\rm
min}}   .
\end{equation*}
   By definition and because of
Assumption (A2) and Lemma \ref{volarelontano} $(iii)$,  we have that
$Z_0\ge 1$, $\ Z_\infty\ge 1$ and $ a \in [1,2]$. In particular,
$g_0(s),g_\infty(s) \in (0,1)$ for $s>0$.

\smallskip


We observe that Equations \eqref{uovone} and \eqref{salsona} have
the following common structure:
$$
\cF\bigl( g_\infty (as ) \bigr ) = \cF \bigl( g_0(s) \bigr)
-h_0(s)\,, \qquad s>0 ,
$$
where
\begin{equation}
\mathcal{F}(x):= \begin{cases} -\ln (1-x) & \text{ for Equation
\eqref{uovone}} ,\\
 \frac{\g+1}{\g+2}\ln \frac{
1+x/(\g+1)}{1-x}  & \text{ for Equation \eqref{salsona} } .
\end{cases}
\label{richiamo2}
\end{equation}
 With these examples in mind, we introduce the following
definition.

\begin{definition}[Hypothesis (H)] \label{cavolo}
We say that a real function $\mathcal{F}$ satisfies Hypothesis (H)
if there  exists $\e>0$ such that $\cF$ is defined on $(-\e,1)$ and
\begin{itemize}
\item (H1)  $\cF$ is $C^1$,
\item (H2) the derivative $\cF'$ admits an  analytic expansion on $(0,1)$ of the form $\cF'(x)=\sum_{n=0}^\infty c_n x^n
$ with $c_n \geq 0$ for all $n \geq 0$,
\item (H3) $\cF(0)=0$,  $\cF$ is bijective from $(-\e, \e)$ to an open interval
$U$ containing $0$ such that $\cR:= \cF^{-1} : U \to (-\e, \e)$ is an
analytic function and $\cR'(0)=1$ ({\it i.e.}\ $\cF'(0)=1)$.
\end{itemize}
\end{definition}
By analytic expansion in  (H3) we mean that $\cR(x)= \sum
_{k=1}^\infty r_k x^k$ for all $x \in U$, where  the series in the
r.h.s. if absolutely convergent.

\medskip

One can easily verifies that both functions $\mathcal{F}$ defined in \eqref{richiamo2}
%two examples above
 satisfy
Hypothesis (H) since for $|x|<1$ we have the  analytic expansions $
-\ln(1-x)=x+x^2/2+x^3/3+\cdots $, while
\begin{equation}\label{zittizitti} \ln \frac{ 1+x/(\g+1)}{1-x} =
%\sum_{n=1}(-1)^{n+1} \frac{x^n}{n(1+\g)^n} + \sum _{n=1}^\infty
%\frac{x^n}{n}=
\sum_{n=1}^\infty \frac{x^{n}}{n} \left[1+(-1)^{n+1}
(\g+1)^{-n}\right].
\end{equation}
%and $ a_n:=1+(-1)^{n+1} (\g+1)^{-n}> 0$, $a_1 =(\g+2)/(\g+1)$.
Moreover, for $|x| <1$, it holds \begin{equation}\label{spaziale}
\cR(x)=
\begin{cases}
1- e^{-x} & \text{ if } \cF(x)= -\ln (1-x) ,\\
 \frac{ \exp \left\{\frac{\g+2}{\g+1}x\right\} -1}{ \exp\left\{\frac{\g+2}{\g+1}x\right\}+ \frac{1}{\g+1} }   & \text{  if } \cF(x)=
\frac{\g+1}{\g+2}\ln \frac{ 1+x/(\g+1)}{1-x}  .
\end{cases}
\end{equation}



\smallskip

 Finally,  we  introduce the following notation.
  Given
an increasing function $\phi:[0,\infty) \rightarrow [0,\infty)$ and
a Radon measure $\mathfrak{m}$ on $[0,\infty)$,  we denote by
$\mathfrak{m}\circ \phi$ the new Radon  measure on $[0,\infty)$
defined by
\begin{equation*}%\label{gianni}
\mathfrak{m} \circ \phi  (A)= \mathfrak{m}( \phi (A)) , \qquad
A\subset \bbR \text{ Borel} .
\end{equation*}
Note that $\mathfrak{m}\circ \phi$ is indeed a measure, due to the
injectivity of $\phi$. Moreover, it holds
\begin{equation}\label{regoletta}
\int _0^\infty f(x) \mathfrak{m} \circ \phi (dx)= \int
_{[\phi(0),\phi(\infty)]} f( \phi^{-1} (x) ) \mathfrak{m}(dx) .
\end{equation}
Above, and in what follows, we use the short notation
$\int_0^\infty$ for $\int_{[0, \infty)}$.


\begin{Theorem}\label{astratto}
Let $\cF$ be a function satisfying Hypothesis (H). Then there exist
unique Radon non-negative measures $t_0 (dx)$ and $t_\infty(dx)$ on
$[0,\infty)$ such that for all $s >0$ it holds
\begin{align}
& \cF \bigl( g_0(s)\bigr)= \int _0^\infty \frac{e^{-s(1+x) }}{1+x}
t_0
(dx) ,\label{sorriso1} \\
& \cF \bigl(g_\infty (s)\bigr)= \int _0^\infty \frac{e^{-s(1+x)
}}{1+x} t_\infty
(dx)  ,\label{sorriso2}\\
& h_0(s)=\int_{[0,a-1) } \frac{e^{-s(1+x)}}{1+x}
t_0(dx) .\label{sorriso3}
\end{align}
Moreover, the equation
\begin{equation}\label{fidanzato}
\cF\bigl( g_\infty (a s ) \bigr ) = \cF \bigl( g_0(s) \bigr) -h_0(s)
 , \qquad   s>0  , \end{equation}
 is equivalent to the
relation
\begin{equation}\label{bianchini}
  t_\infty = (1/a) \, t_0 \circ \phi
  \end{equation}
where  the linear function $\phi:[0,\infty) \rightarrow [0,\infty)$
is defined as $\phi(x)= a(1+x)-1$.
\end{Theorem}

\begin{remark}\label{codroipo}
Combining (H2) and (H3) in Definition \ref{cavolo}, it follows that
the map $\cF$ is strictly increasing  on $[0,1)$. In particular,
Equation \eqref{fidanzato} univocally determines $g_\infty$ knowing
$g_0$ and $h_0$ on $(0,\infty)$, and similarly Equations
\eqref{sorriso1} and \eqref{sorriso2} univocally determine
$g_\infty$ and $g_0$ knowing $t_\infty$ and $t_0$, respectively.
\end{remark}


We divide the proof of the above theorem in different steps.
\begin{Lemma}\label{gola}
Let $Z$ be a random variable such that $Z \geq 1$ and define
$g(s)=\bbE [ e^{-sZ} ]$, $s \geq 0$.  Let  $w:(0, \infty)
\rightarrow \bbR$ be the  unique function such that
\begin{equation}\label{pompiere0}
\cF \bigl( g(s)\bigr) = \int _s ^\infty du\, e^{-u} w(u)  ,\qquad
s>0 ,
\end{equation}
{\it i.e.}\
\begin{equation}\label{pompiere1}
 w (s):= -e^s \cF' (g(s) ) g'(s)  , \qquad s >0 .
\end{equation}
Then the function $w$
  is completely monotone\footnote{Recall that a function $f : (0,\infty) \to \mathbb{R}$ is said to be completely monotone if it is $\mathcal{C}^\infty$ and if for any integer $k$, $(-1)^k f^{(k)} \geq 0$.}.
 In particular,  there exists a  unique  Radon  measure $t(dx)$ on $[0, \infty)$ (not
necessarily of finite total mass) such that
\begin{equation}\label{cotoletta}
w(s)= \int _0 ^\infty e^{-sx} t(dx) , \qquad  s>0 ,
\end{equation}
and therefore
\begin{equation}\label{animainpena}
\cF \bigl(g(s)\bigr)= \int _0^\infty \frac{e^{-s(1+x) }}{1+x} t
(dx) , \qquad s>0   .
\end{equation}
%   Moreover,
%  \begin{equation}\label{fuoco}
%  \limsup _{s \downarrow 0}-\frac{s g'(s)}{1-g (s)}\in [0 ,1]\,.
%  \end{equation}
   Moreover, the above identity \eqref{animainpena} univocally
determines $t(dx)$.
\end{Lemma}

\begin{proof}  %\club \club 
The last statement    follows  from the
inversion formula of the Laplace transform.  For the rest, the proof
is similar to the proof of Lemma 5.1 in \cite{FMRT0}. The only
slight difference is  in the following argument.
By condition (H2) and since $g(s) \in (0,1)$
for $s>0$, we can write $w=f \sum _{k=0}^\infty c_k g^k$, $f =-e^s
g'(s)$.
 Since $c_k \geq 0$ for all $k \geq 0$ and since the product and the sum  of
 completely monotone functions is again  completely monotone
 (cf.\ \cite{F}) we get that $ \sum _{k=0}^\infty c_k g^k$ is completely
 monotone. The rest of the proof is as in \cite{FMRT0}.
%  On the other hand, $f$ is completely monotone. Indeed, by
% the Leibniz rule one has $D^n f (s)= e^s \bbE\left( e^{-s Z}
% Z(1-Z)^ n \right)$ (see the proof of Lemma 5.1 in \cite{FMRT0}). From this formula and the assumption $Z\geq 1$, we conclude that
%   $(-1)^n D^n f(s)
% \geq 0$ for $s >0$.
% We can apply Theorem 1a in Section XIII.4 of
%\cite{F} to get that there exists a  Radon measure $t(dx)$ on $[0,
%\infty)$ (not necessarily of finite total mass) satisfying
%\eqref{cotoletta}. Moreover, the above measure $t$ is univocally
%determined due to the \emph{inversion formula} given in Theorem 2,
%Section XIII.4 \cite{F}. Finally, we derive \eqref{animainpena} for
%$s
%>0$ from  \eqref{pompiere0}, \eqref{cotoletta} and the identity
%\begin{equation}\label{pizzette}
%\int_0 ^\infty \frac{e^{-s (1+x) }}{1+x} t (dx)= \int _s ^\infty du
%e^{-u} \int_0 ^\infty e^{-u x} t (dx) , \qquad s>0  .
%\end{equation}
\qedhere\end{proof}

\begin{Lemma}\label{limonata}
Let $Z$ be a random variable such that $Z \geq 1$ and let $g(s)$ be
its Laplace transform. Let $t$ be the unique Radon measure on
$[0,\infty)$ satisfying \eqref{animainpena} and call $m(dx)$ the
Radon  measure with support in $[1,\infty)$ such that
\begin{equation}\label{bilancia}
m (A)= \int_0 ^\infty \frac{\mathds{1}_{1+x \in A} }{1+x} t(dx) .
\end{equation}
For each $k \geq 1$,    consider the convolution measure $ m ^{(k)}$
with support in $[k,\infty)$ defined as
\begin{equation}
 m^{(k)} (A)= \int _1^\infty
m(dx_1)\int _1^\infty m(dx_2)\cdots \int _1^\infty m(dx_k)
\mathds{1}_{x_1 +x_2+ \cdots+ x_k\in A}  .
\end{equation}
Then the law  of $Z$ is  given by the measure
%\begin{equation}\label{barrocciaio}
$m_*:=\sum _{k=1}^\infty r_k  m^{(k)}$,
%\end{equation}
where the coefficients $r_k$ are determined by the series expansion
$\cR(x)= \sum _{k=1}^\infty r_k x^k$ of the function $\cR$ around
$0$ (recall condition (H3) in Definition \ref{cavolo}).
  In particular
\begin{equation} \label{brucogiallo}
\bbE\bigl[ e^{-sZ}; Z< a \bigr]= \int_{[0,a-1) } \frac{e^{-s(1+x)}}{1+x} t(dx)
 ,\qquad s \geq 0 .
\end{equation}
\end{Lemma}

We point out that, given a bounded Borel set $A$, since
$m^{(k)}$ has support in $[k,\infty)$, the series
$m_*(A)=\sum _{k=1}^\infty r_k  m^{(k)}(A) $ is a finite sum.
%The thesis includes that this
%sum is a nonnegative number and that the set--function $A\mapsto
%m_*(A)$, defined on bounded Borel sets, extend univocally to a Radon
%measure on all Borel sets.

\begin{proof}
The proof is a generalization of the proof of Lemma 5.2 in
\cite{FMRT0}. It reveals the fundamental structure behind the
transformation introduced in \cite{FMRT0}.


 By definition of $m(dx)$ and
by \eqref{animainpena} we can write
\begin{equation}\label{succhetto}
\cF \bigl( g(s) \bigr)= \int _0^\infty \frac{e^{-s(1+x) }}{1+x} t
(dx) = \int_0^\infty e^{-sx} m(dx) , \qquad s>0 .
\end{equation}
Since $\lim _{s \to \infty} g(s)=0$,  by (H3) we conclude that $\cF
(g(s) )$ goes to zero as $s$ goes to $\infty$. In particular, by (H3), for
$s$ large enough we can invert \eqref{succhetto} and use the
analytic expansion of $\cR$ getting
\begin{equation}\label{asiagoDOP}
g(s) = \cR\left(\int_0^\infty e^{-sx} m(dx)\right)= \sum
_{k=1}^\infty r_k \int _0 ^k e^{-s x} m^{(k)} (dx) , \qquad s
\text{ large} ,
\end{equation}
where, in the last equality, we used that $\Big(
\int_0^\infty e^{-sx} m(dx) \Big)^k = \int_0^\infty e^{-sx } m^{(k)}
(dx)$. From now  on $s$ has to be thought large. We can rewrite the
right hand side of \eqref{asiagoDOP} as
$
 \sum _{k=1}^\infty \Big(\sum_{j=k}^\infty a_{k,j}\Big) $, where
$a_{k,j} = r_k  \int_{I_j} e^{-s x} m^{(k)} (dx)$ and $ I_j=[j,j+1)
\text{ for } j \geq 1$. Due to the analytic expansion of $\cR(x)$
around $0$, we have that $ \sum _{k=1}^\infty |r_k x^k|<\infty$ for
$|x| $ small, hence we can write
\begin{align*}
\sum _{k=1}^\infty \sum _{j=1 } ^\infty |a_{j,k}|
 =
%\sum_{k=1}^\infty  |r_k|\left( \sum _{j=1}^\infty \int_{I_j} e^{-s x} m^{(k)} (dx)\right)=
 \sum _{k=1}^\infty |r_k| \int _0 ^\infty e^{-s x} m^{(k)} (dx)
  =
 \sum _{k=1}^\infty |r_k|\Big( \int_0^\infty e^{-sx} m(dx) \Big)^k<\infty ,
\end{align*}
thus implying that in the series  $\sum _{k=1}^\infty
\Big(\sum_{j=k}^\infty a_{k,j}\Big)$ we can indeed arrange the terms
as we prefer. In particular, we can invert $k$ and $j$ getting $$
g(s) =\sum _{j=1}^\infty \Big( \sum _{k=1}^j a_{k,j}\Big)= \sum
_{j=1}^\infty \int _{I_j} e^{-s x} m_*(dx)= \int _{[0,\infty)}
e^{-sx }m_*(dx)  .$$ Let us write $m_*=m_* ^{(+)}- m_*^{(-)}$ as
the sum of non-negative measures with disjoint supports   \cite{H}.
Writing $p_Z$ for the law of $Z$, we then have that the Laplace
transforms of $p_Z+ m_*^{(-)}$ and   $m_*^{(+)}$ are identical  for
$s$ large. By Theorem 1a in Section XIII.1 [F] we conclude that
$p_Z+ m_*^{(-)}=m_*^{(+)}$. Since $m_*^{(-)}$ and $m_*^{(+)}$ have
disjoint supports,   $m_*^{(-)}$ must be zero and therefore
$m_*=m_*^{(+)}$ is a non-negative measure.

To conclude the proof it remains  to check \eqref{brucogiallo}. It
is enough to prove the thesis for $s >0$, since the case $s=0$
follows by monotonicity. To this aim we observe that, since
$m^{(k)}$ has support contained in $[k, \infty)$ and since $r_1=
\cR'(0)=1$ by (H3) , the measure $m_*$ equals $m$ on $[1, 2)$. Since
$a\leq 2$ and using the definition of the measure $m$ given by
\eqref{bilancia}, we obtain that
\begin{equation*} \bbE\bigl[ e^{-sZ} ; Z< a\bigr]= \int _{ [1,a) }
e^{-sx } p_Z(dx)= \int _{ [1,a) } e^{-sx } m(dx)= \int _{[0,a-1)}
\frac{e^{-s(1+x)}}{1+x} t(dx) .
\end{equation*}
This concludes the proof of \eqref{brucogiallo}.
\end{proof}

We are now in position to prove Theorem \ref{astratto}.

\begin{proof}[Proof of Theorem \ref{astratto}]
Observe that Equations \eqref{sorriso1} and
\eqref{sorriso2} follow from Lemma \ref{gola}, and that Equation
\eqref{sorriso3} follows from \eqref{brucogiallo} in Lemma
\ref{limonata}.

To prove the last statement we  write
$\rho(dx) $ for the measure in the r.h.s.\ of \eqref{bianchini}.
Using that $a[\phi^{-1}(x)+1]= 1+ x $, we obtain for $s\geq 0$  that
\begin{equation*}
 \int_0^\infty   \frac{e^{-a s(1+x) }}{1+x}\rho (dx)
= a^{-1} \int_{[\phi(0), \infty)}\frac{e^{-s
(1+x)}}{a^{-1}(1+x)}t_0(dx) = \int_{[a-1,\infty )}\frac{e^{-s
(1+x)}}{1+x} t_0 (dx)  .
\end{equation*}
Using also \eqref{sorriso1}, \eqref{sorriso2}, \eqref{sorriso3} we
conclude that
 Equation \eqref{fidanzato} is equivalent to
\begin{equation}\label{petrolio}
\int _0^\infty   \frac{e^{-as(1+x) }}{1+x}t_{\infty}(dx) = \int
_0^\infty \frac{e^{-a s(1+x)}}{1+x}\rho(dx) , \qquad \forall s
> 0 .
\end{equation}
Thinking the above integrals as Laplace transforms of suitable
non-negative measures in the variables $as$, by Theorem 1a  in
Section XIII.1 in \cite{F} we conclude that \eqref{petrolio}  is
equivalent to the identity $t_\infty= \rho$.
\end{proof}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Asymptotic of the interval law for HCP: Proof of Theorem \ref{teo2}} \label{fegatograsso}
%We use  notation and definitions of Theorem \ref{teo2}.
% \club {\bf Ale:} {\sl prima venivano richiamate alcune defizioni
% del Teorema \ref{teo2} ma vi erano alcune imprecisioni. penso che
% la cosa naturale e'  che il lettore si rilegga il Th. \ref{teo2} il
% cui statement e' preciso e breve}
% and Theorem \ref{teo3}. In particular $\mu$ and $\nu$ are probability measures
%on $[1,\infty)$ and $\mathbb{R}$ respectively. We define here
%$X^{(n)}$, $n \in \bbN_+$,  as the length of the leftmost  domain
%inside $(0,\infty)$ at the beginning of the $n$--th epoch, i.e.
%$X^{(n)}= x^{(n)} _2 (0)- x^{(n)}_1 (0)$. Moreover we set
%$Z^{(n)}=X^{(n)}/d^{(n)}$. Note that $X^{(n)}$ has law $\mu ^{(n)}$.
% is a random variable with law $\mu^{(n)}$, $n\in \bbN_+$, and
% Also,
%starting indifferently from $\cQ=\text{Ren}(\nu,\mu)$,
%$\cQ=\text{Ren}(\mu)$ or $\cQ=\text{Ren}_{\mathbb{Z}}(\mu)$.
%In addition,

The key result of this section is Theorem \ref{astrattobis} which in
turn allows to prove easily Theorem  \ref{teo2}. Theorem
\ref{astrattobis} is proved by using the recursive identities for
the OCP process established in Section \ref{prova_differisco} and
our extension of the transformation of \cite{FMRT0}  derived in the
previous section.

 Let us start by recalling the notation of Theorem \ref{teo2} which will be used throughout this section and by giving a few more definitions. We let $\mu$ be a
probability measure on $[d^{(1)}, \infty)=[1,\infty)$ and
 consider the
HCP such that $\xi^{(1)}(0)$ has law  of the form ${\rm Ren} (\nu,
\mu)$, ${\rm Ren}(\mu)$ or $ {\rm Ren}_\bbZ(\mu)$ ($\nu$ being a
probability measure on $\bbR$). Call
 $\mu^{(n)}$  the interval law of $\xi ^{(n)}(0) $, {\it i.e.}\ at
the beginning of epoch $n$ and let
$X^{(n)}$ be
a generic random variable with law $\mu^{(n)}$ and $Z^{(n)}$ be the rescaled variable $Z^{(n)} =
X^{(n)} /d^{(n)}$. Finally, for any $n \geq 1 $ and  $s \geq 0$ set
\begin{equation}\label{sciopero}
g^{(n)}(s):=\mathbb{E}\bigl(e^{-sZ^{(n)}}\bigr) , \qquad
h^{(n)}(s):=\mathbb{E}\bigl(e^{-sZ^{(n)}} \mathds{1}_{1 \leq Z^{(n)}
<a_n}\bigr)
\end{equation}
with \begin{equation}\label{sciopero2}a_n:=d^{(n+1)}/d^{(n)}.\end{equation} Note that $\mu= \mu_1$ and
$g^{(1)}(s)= g(s): = \int e^{-sx} \mu(dx)$. The following holds

\begin{Theorem}\label{astrattobis}
Let $\cF$ be a function satisfying Hypothesis (H) (see Definition \ref{cavolo})
and  assume  that for some number $\k$ it holds
\begin{equation}\label{baseperarrosto}
\lim _{s\downarrow 0} -s \cF' \bigl( g(s) \bigr) g'(s) = \k
\end{equation}
%} $\lim_{x \to \infty} \cF^{-1}(x) = 1$, that $\lim_{x \to 1}
%\cF'(x)(1-x)$ exists,
and that
\begin{equation}\label{mennea}
\cF(g^{(n+1)}(a_n s)) = \cF(g^{(n)}( s)) - h^{(n)}(s) ,  \qquad   n
\geq 1 , \;  s >0   .
\end{equation}
Then  it must be $\k \geq 0$. Moreover, the rescaled variable
$Z^{(n)} $ weakly
 converges to the random variable $Z^{(\infty)}\equiv Z^{(\infty)}_{\k}$
whose Laplace transform $g^{(\infty)}_\k$ satisfies
\begin{equation}\label{radicchio}
\cF \bigl( g^{(\infty)}_{\k}(s) \bigr) =    \k  \int _1^\infty
\frac{e^{-sx}}{x} dx   , \qquad s >0 .
\end{equation}
If $\k=0$ then $Z^{(\infty)}_{\k }= \infty$, while if $\k >0$ then
$Z^{(\infty)}_\k $ takes value in $[1, \infty)$.
%where $\beta = \lim_{x \to 1} \cF'(x)(1-x)$.
\end{Theorem}
\begin{proof}
 We first apply  Theorem \ref{astratto} getting   that, for each $n
\geq 1$,  there exists a unique  measure $t^{(n)}$ on $[0,\infty)$
such that
\begin{equation}\label{asilobello}
\cF \bigl( g^{(n)}(s) \bigr) =  \int _0^\infty
\frac{e^{-s(1+x)}}{1+x} t^{(n)}(dx) , \qquad s>0 .
\end{equation}
Due to \eqref{mennea} and  Theorem \ref{astratto}  again,  for $n
\geq 2$ it holds $t^{(n)} = \frac{1}{a_{n-1}}t^{(n-1)} \circ
\phi_{n-1}$, with $\phi_{n-1}(x)=a_{n-1}(1+x)-1$. The recursive
identities relying the $t^{(n)}$'s can be explicitly solved, leading
to
\begin{equation} \label{brumsporcobis}
t^{(n)} = \frac{1}{d^{(n)}}\,    t^{(1)} \circ \psi_{n-1} \,, \qquad n \geq 2
\end{equation}
with $\psi_{n-1}(x)=d^{(n)} (1+x)-1$. %\club  
Defining $ U^{(n)}(x)=
 t^{(n)} ([0,x])  \mathds{1}(x \geq 0)$,
we get that %\club  
$dU^{(n)} = t^{(n)}$ and $U^{(n)}(x)=0$ for
$x<0$. By \eqref{brumsporcobis} it holds that
 \begin{equation}\label{pescefritto}
% \begin{split}
%\text{\club} 
U^{(n)}(x)
%& = \frac{1}{d^{(n)}}\Big[ U^{(1)}(\psi_{n-1}
%(x))-
%U^{(1)}(\psi_{n-1} (0)-)\Big] ,\\
 =\frac{1}{d^{(n)}} \Big[ U^{(1)} \big(
d^{(n)} (1+x)-1\big)- U^{(1)} \bigl((
d^{(n)}-1)-\bigr)\Big] , \qquad n\geq 1 .
%\end{split}
\end{equation}
Moreover, for each $n \geq 1$, integrating  by parts and using that
$U^{(n)}(0-)=0$, we can rewrite the integral in the r.h.s. of
\eqref{asilobello} as
 \begin{equation}\label{ananas}
%\text{\club} 
\int_0^\infty \frac{e^{-s(1+x) }}{1+x} t^{(n)}(dx)
 %& =
 %\int _0^\infty \frac{e^{-s(1+x) }}{1+x} dU^{(n)}(x) \nonumber \\
= \lim_{y \uparrow \infty} \frac{e^{-s(1+y) }}{1+y} U^{(n)}(y)- \int
_0^\infty \left( \frac{d}{dx} \left( \frac{e^{-s(1+x) }}{1+x}\right)
\right)
 U^{(n)}(x) dx .
\end{equation}
We now use the key additional hypothesis \eqref{baseperarrosto}.
Since $g^{(1)}(s)= g(s)$ because $d^{(1)}=1$, if $w^{(1)}$ denotes
the Laplace transform of $t^{(1)}$ ({\it i.e.}\ $w^{(1)}(s) =
\int_0^\infty e^{-s x} t^{(1)}(dx) $),  then \eqref{baseperarrosto}
together with \eqref{pompiere1} implies %\club 
that $ \lim
_{s\downarrow 0} s\, w^{(1)}(s)= \k$. The above limit and    the
Tauberian  Theorem 2 in Section XIII.5 of \cite{F} allow to conclude
that
\begin{equation}\label{marina}
\lim _{y\uparrow \infty} \frac{ U^{(1)}(y)}{y}= \k  .
\end{equation} The above limit together with \eqref{pescefritto}
implies that there exists a suitable constant $C>0$ such that
\begin{equation}\label{bound} U^{(n)}(x)
\leq C (1+x) , \qquad n\geq 1 ,\; x \geq 0 .
\end{equation} In
particular, the limit in the r.h.s. of \eqref{ananas} is zero and
\begin{equation}
\label{miserve}
\int_0^\infty \frac{e^{-s(1+x) }}{1+x} t^{(n)}(dx)=- \int _0^\infty
\left( \frac{d}{dx} \left( \frac{e^{-s(1+x) }}{1+x}\right) \right)
 U^{(n)}(x) dx ,\qquad n\geq 1  .
\end{equation}
By \eqref{pescefritto}, \eqref{marina} and the fact that $d^{(n)}
\to \infty$, we conclude that $\lim_{n\to \infty}U^{(n)}(x) = \k x$
for all $x \geq 0$. This limit together with \eqref{bound} allows us
to apply the Dominated Convergence Theorem,  getting that
\begin{equation}\label{monnalisa}
\lim_{n\to \infty} \int _0^\infty \frac{e^{-s(1+x) }}{1+x}
t^{(n)}(dx) = -\k \int_0 ^\infty \left( \frac{d}{dx} \left(
\frac{e^{-s(1+x) }}{1+x}\right) \right) x dx = \k \int_0 ^\infty
\frac{e^{-s (1+x) }}{1+x} \, dx
\end{equation}
(in the last identity we have simply integrated by parts).

Let us come back to \eqref{asilobello}.  We know   the  limit of the
r.h.s. as $n\to \infty$ by \eqref{monnalisa}. Let us analyze the
l.h.s.
% Since $Z_n \geq 1$ for any $n$, given $s >0$  it holds
%$e^{g^{(n)}(s)\leq e^{-s} \in (0,1)$ for any $n \geq 1$.
We claim that, given $s>0$,  the sequence $\{g^{(n)}(s)\}_{n\geq 1}$
converges to some number in $[0,e^{-s}]$. Indeed, since $Z_n \geq
1$, it holds $ g^{(n)}(s) \in (0, e^{-s}]$. If the sequence was not
convergent, by compactness we could find two subsequence
$\{n_k\}_{k\geq 1}$ and $\{n_r\}_{r\geq 1}$ such that $ \lim _{k \to
\infty}g^{(n_k)}(s)< \lim _{r \to \infty}g^{(n_r)}(s)$ and both
limits exist and belong to $[0, e^{-s}]$. On the other hand,  by
hypothesis $(H1)$ and Remark \ref{codroipo} the function $\cF$ is
continuous and strictly increasing on $[0,1)$. Hence,
%
 $$
\lim _{k \to \infty}\cF\bigl(g^{(n_k)}(s)\bigr)=
 \cF \bigl(\lim _{k \to \infty}g^{(n_k)}(s)\bigr)< \cF\bigl(\lim
_{r \to \infty}g^{(n_r)}(s)\bigr)=\lim _{r \to
\infty}\cF\bigl(g^{(n_r)}(s)\bigr)$$ in contradiction with the fact
that the first member and the last member  equal the r.h.s. of
\eqref{monnalisa}, by \eqref{asilobello} and \eqref{monnalisa}.

Since we have proved that for all $s >0$ the sequence
$\{g^{(n)}(s)\}_{n \geq 1}$ converges to some number
$g^{(\infty)}_{\k}(s)\in [0, e^{-s}]$, using  the continuity  of
$\cF$ on $[0,1)$, \eqref{asilobello} and \eqref{monnalisa}, we
conclude that $g^{(\infty)}_\k$ satisfies \eqref{radicchio}.



%Equation \eqref{asilobello} and the limit \eqref{monnalisa} imply
%that, on $(0,\infty)$,  $g^{(n)} $ converges pointwise to the a well
%defined function $g^{(\infty)}_{\k}(s)$, which satisfies
%\eqref{radicchio}.




Since by Hypothesis (H) the function $\cF'$ is positive on $[0,1)$,
the limit $\k$ in \eqref{baseperarrosto} must be non-negative. Let us
first consider the case $\k=0$. Then,  by \eqref{radicchio}, the
fact that $\cF$ is strictly increasing on $[0,1)$ and $\cF(0)=0$, we
conclude that $g^{(\infty)}(s)=0$ for all $s>0$. This implies that
the law  of the random variable $Z^{(n)}$ weakly converges to
$\d_\infty$.

We now consider the case $\k>0$.
 As  pointwise limit of decreasing functions, also
$g^{(\infty)}_\k$ is decreasing on $(0,\infty)$. In particular  the
limit $\lim_{s \downarrow 0} g^{(\infty)}_\k(s)$ exists and belongs
to $[0,1]$. Let us call $z$ this limit and prove that $z=1$. Suppose
by absurd that $z \in [0,1)$. Then, by the continuity of $\cF$ on
$[0,1)$ and Equation \eqref{radicchio}, we would have
$$  \cF (z)= \lim _{s \downarrow 0} \cF\bigl(g^{(\infty)}_\k(s)
\bigr) = \lim _{s \downarrow 0} \k \int _1 ^\infty
\frac{e^{-sx}}{1+x} dx = \infty  .
$$
Since $\cF$ takes finite value on $[0,1)$ it cannot be
$\cF(z)=\infty$, thus implying that $z =1$. In conclusion we have
proved that  $\lim _{s\downarrow 0 } g^{(\infty)}_{\k}(s)=1$. Then,
by Theorem 2 in Section XIII.1 of \cite{F}, we conclude that
$g^{(\infty)}_{\k}$ is the Laplace transform of some non-negative
(finite) random variable $Z _{\k}^{(\infty)}$ and that   $Z^{(n)}$
weakly converges to $Z _{\k}^{(\infty)}$. The fact that  $Z
_{\k}^{(\infty)}\geq 1$ a.s. follows from the fact $Z ^{(n)}\geq 1$
for all $n \geq 1$.
\end{proof}


\begin{proof}[Proof of Theorem \ref{teo2}]
Thanks to Theorem
\ref{differisco} and the discussion before Definition \ref{cavolo},
the Laplace transforms of the rescaled variables $Z^{(n)}$ satisfy
$$
\cF(g^{(n+1)}(a_n s)) = \cF(g^{(n)}( s)) - h^{(n)}(s) \qquad \forall
n \geq 1, \quad \forall s > 0 ,
$$
where
 $$\cF(x) =
 \begin{cases} - \ln (1-x) & \text{ in case $(i)$} ,\\
  \frac{\gamma+1}{\gamma+2} \ln \frac{1 +\frac{x}{\g+1}}{1-x} &
  \text{ in case $(ii)$} ,
\end{cases}
$$
respectively. We have already observed, that in both cases $\cF$
satisfies the Hypothesis (H). Computing $\cF'$ we get
% Since, for $x \in [0,1)$,
%$$ \cF'(x)=
%\begin{cases}
% \frac{1}{1-x} & \text{ in case $(i)$}\,,\\
%  \frac{\gamma+1}{\gamma+2} \left( \frac{1}{\g+1+x}+\frac{1}{1-x}
%  \right)&
%  \text{ in case $(ii)$}\,,
%\end{cases}
%$$
%using also that $\lim _{s \downarrow 0} g(s)=1$,  we have
\begin{equation*}%\label{mammina}
 \lim _{s \downarrow 0 } -s \cF'(g(s) ) g'(s)=
\begin{cases}
-\lim _{s \downarrow 0} \frac{s g'(s)}{1-g(s)} & \text{ in case
$(i)$} ,\\
 -\lim _{s \downarrow 0} \frac{\gamma+1}{\gamma+2} \left(
 \frac{s g'(s)}{\g+1+g(s)}+\frac{s g'(s)}{1-g(s)}
  \right) & \text{ in case $(ii)$} .
  \end{cases}
  \end{equation*}
Since we have assumed the limit  \eqref{lim_arrosto} and  since
$1-g(s)=o(1)$ for $s$ small,  it must be $\lim _{s \downarrow 0}
sg'(s)=0$. This last observation allows to conclude that
\begin{equation}\label{mammina}
 \lim _{s \downarrow 0 } -s
\cF'(g(s) ) g'(s)=
\begin{cases}
 c_0 & \text{ in case
$(i)$}\\
  \frac{\gamma+1}{\gamma+2} c_0 & \text{ in case
 $(ii)$} .
  \end{cases}
\end{equation}
At this point Theorem \ref{teo2} is an immediate consequence of
Theorem \ref{astrattobis} and the computation of $\cR= \cF^{-1}$
given in  \eqref{spaziale}.
\end{proof}


%\begin{remark}
%% We point out that in the general case where $d^{(1)}$ is not necessarly equal to $1$,
%It is useful to observe that if the initial scale $d^{(1)}$ was
%different from one than necessarily $g(s)\neq g^{(1)}(s)$. However,
%and that is the reason why we could fix $d^{(1)}=1$, the limit
%\eqref{lim_arrosto} is invariant under rescaling the variable $s$ by
%a constant \ie  \eqref{lim_arrosto} for $g$ implies the same limit
%for $g^{(1)}$. {\bf Ale}: {\sl questo remark va migliorato}
%\end{remark}


\section{Asymptotic of the first point law: proof of Theorem \ref{teo3}}\label{provateo3}

In this section we prove Theorem \ref{teo3}. While in the derivation
of Theorem \ref{teo2} we have tried to keep the discussion at a
general and abstract level in order to catch the fundamental
structure of the transformation introduced in \cite{FMRT0} and
therefore explain the similar asymptotics of very different HCP's,
we restrict here to the special cases mentioned  in Theorem
\ref{teo3}. Indeed, as the reader will see, the proof goes through
estimates which are very model--dependent.


\begin{proof}[Proof of  Theorem \ref{teo3}] Case $(i)$
  has been solved in \cite[Theorem 2.24]{FMRT0}.
Hence we focus on case $(ii)$. Without loss of generality we can
restrict to the case $\nu=\delta_0$, {\it i.e.}\ when the HCP starts with
$\xi^{(1)} (0)$ having law ${\rm Ren} (\d_0, \mu)$, $\mu$ being a
probability measure on $[d^{(1)}, \infty)=[1, \infty)$. Indeed, in
the general case $X^{(n)}_0$  can be expresses as $V+ \bar
X^{(n)}_0$, where $\bar X^{(n)}_0$ is the first point in
$\xi^{(n)}_0$ for the above HCP starting with distribution ${\rm
Ren}(\d_0, \mu)$, while $V$ is a random variable with law $\nu$
independent from $\bar X^{(n)}_0$. Since $d^{(n)} \to \infty$, when
taking the rescaled random variable $Y^{(n)}= X^{(n)}_0/ d^{(n)}$
the effect of the random translation $V$ disappears as $n \to
\infty$.
From Lemma \ref{vecchioteo1} we know that
 the configuration $\xi^{(n)}(0)$ at the beginning of epoch $n$ has
 law ${\rm Ren}\bigl( \nu^{(n)}, \mu^{(n)}\bigr)$. As in the previous section
 $X^{(n)}$ will be a random variable with law $\mu^{(n)}$ and $Z^{(n)}$ the rescaled random variable $Z^{(n)}= X^{(n)}/
 d^{(n)}$.
 Moreover, we write $X^{(n)}_0$ for a generic random
 variable with law $\nu^{(n)}$ and set
$$
\ell^{(n)}(s)= \mathbb{E}\Big( e^{ -s Y^{(n)}}\Big)  , \qquad s \in
\mathbb{R}_+  , \qquad Y^{(n)}= X^{(n)}_0 /d^{(n)} .
$$
Recalling the definitions of $g^{(n)}(s)$, $h^{(n)}(s)$ and  $a_n$ in equations \eqref{sciopero} and \eqref{sciopero2},  we use formula
\eqref{convoluto} and  Theorem \ref{differiscobis} (ii) to obtain the
recursive equations
$$
\ell^{(n+1)}(a_n s) = \ell^{(n)}(s)  \sqrt{\frac{1-g^{(n+1)}(a_n
s)^2}{1-g^{(n)}(s)^2}} e^{-h^{(n)}(0)} , \qquad n\geq 1  .
$$
By iteration, we get
\begin{equation}\label{pescedaprile}
\begin{split}
\ell^{(n)}(s)
& =
\ell^{(1)}(s/d^{(n)})\sqrt{
\frac{1-g^{(n)}(s)^2}{1-g^{(1)}(s/d^{(n)})^2} }
\exp \Big\{ - \sum_{j=1}^{n-1} h^{(j)}(0) \Big\} \\
& =
\ell^{(1)}(s/d^{(n)})\sqrt{ \frac{1-g^{(n)}(s)^2}{s}}  \sqrt{ \frac{s/d^{(n)}}{1-g^{(1)}(s/d^{(n)})^2}}
 \exp \Big\{  \frac{1}{2} \log d^{(n)} -  \sum_{j=1}^{n-1} h^{(j)}(0) \Big\}
\end{split}
\end{equation}
%where
%$$
%F^{(n)}(s):=  \sum_{j=1}^{n-1} h^{(j)}(sd^{(j)}/d^{(n)}) \,.
%$$
Since  $d^{(n)} \to \infty$,  we have $\lim_{n \to \infty}
\ell^{(1)}(s/d^{(n)}) =1$. By assumption $\mu$ has finite mean $\bar
\mu= -g'(0)$. Hence, $ \frac{s/d^{(n)}}{1-g^{(1)}(s/d^{(n)})^2}$
converges to $1/ 2\bar \mu$ as $n \to \infty$. Finally, invoking
Theorem \ref{teo2}, from \eqref{pescedaprile} we get
\begin{equation}\label{balena}
\lim _{n \to \infty} \ell^{(n)}(s)= \frac{1}{\sqrt{2 \bar \mu}}
\sqrt{ \frac{1-\tanh ^2\bigl(  \text{Ei}(s)/2 \bigr) }{s}} \lim
_{n\to \infty}\exp \Big\{ \frac{1}{2} \log d^{(n)} -
\sum_{j=1}^{n-1} h^{(j)}(0) \Big\} .
\end{equation}
In remains to study the last limit in \eqref{balena}. To this aim we
come back to the measures $t^{(n)}$. As already observed in the
proof of Theorem  \ref{astrattobis} and Theorem \ref{teo2}, applying
Theorem \ref{astratto} one gets that for each $n\geq 1$ there exists
a unique measure $t^{(n)}$ on $[0,\infty)$ satisfying
\eqref{asilobello} with $\cF(x)=\frac{1}{2} \ln \frac{1 +x}{1-x}$
%%%%%%%%%%%%%\frac{\gamma+1}{\gamma+2} \ln \frac{1 +\frac{x}{\g+1}}{1-x}
for $x \in [0,1)$. Moreover, by Equation \eqref{sorriso3} it holds
 $$
 h^{(n)}(s) = \int_{[0, a_n-1)} \frac{e^{-s (1+x)}}{1+x}
 t^{(n)}(dx)
$$
 and  by formula \eqref{brumsporcobis}  it holds
%for each $n \geq 2$  it holds
% $t^{(n)} = \frac{1}{a_{n-1}}t^{(n-1)}  \circ \Phi_{n-1}$, with $\Phi_{n-1}(x)=a_{n-1}(1+x)-1$,  leading to
%\begin{equation} \label{brumsporcter}
$t^{(n)} = \bigl(1/d^{(n)}\bigr)     t^{(1)} \circ \psi_{n-1}$ with
$\psi_{n-1}(x)=d^{(n)} (1+x)-1$, for all $n\geq 2$.
 Combining the last identities, from \eqref{regoletta} one gets
\begin{equation*}\label{nuvolettabis}
 h^{(n)}(s) =  \int _{[d^{(n)}-1,
\,d^{(n+1)}-1)} \frac{ e^{-s(1+x)/d^{(n)} }}{1+x} t^{(1)} (dx)  ,
\qquad  n \geq 1 .
\end{equation*}
The above  integral representation implies
$$
\sum_{j=1}^{n-1} h^{(j)}(0)
 =
\int _{[0, d^{(n)}-1)} \frac{1}{1+x}t^{(1)}
(dx)  .
$$
%
%\begin{equation*}\label{nuvolettater}
%h^{(j)}( s\, d^{(j)}/d^{(n)}) = \int _{[d^{(j)}-1, d^{(j+1)}-1)}
%\frac{ e^{-s(1+x)/d^{(n)} }    }{1+x}t^{(1)} (dx)  \,, \qquad j\geq
%1\,.
%\end{equation*}
%This allows to write
%\begin{equation}\label{secchio}
%F^{(n)}(s)= \sum _{j=1}^{n-1}
%  h^{(j)}(s d^{(j)}/d^{(n)})  =
%\int _{[0, d^{(n)}-1)} \frac{ e^{-s(1+x)/d^{(n)} }    }{1+x}t^{(1)}
%(dx) \,.
%\end{equation} Setting $U^{(1)}(x)= t^{(1)}( [0,x])$, we can use
%integration by parts and the change of variable $y=(1+x)/d^{(n)}$ to
%conclude that \begin{equation}\label{brodino} F^{(n)}(s)=
%\bigl[e^{-s}\bigr] \frac{U^{(1)}(d^{(n)}-1)}{d^{(n)}}+
%\int_{[1/d^{(n)},1) }e^{-sy }\left[ \frac{s}{y}+ \frac{1}{y^2}
%\right]  \frac{ U^{(1)}( d^{(n)} y -1 ) }{ d^{(n)}  }dy\,.
%\end{equation}
%As already observed, the above function $\cF$ satisfies all the
%hypothesis of  Theorem \ref{astrattobis}. In particular,
%\eqref{baseperarrosto} is valid with $\k= 1/2$ (recall Remark
%\ref{paradiso} and \eqref{mammina}). As pointed out in the proof of
%Theorem \ref{astrattobis}, \eqref{baseperarrosto} implies the
%Tauberian limit \eqref{marina}, i.e.\ $U^{(1)} (y)/y \to 1/2$ as $y
%\to \infty$. Using this limit and the Dominated Convergence Theorem,
%we conclude that
%\begin{equation}\label{sognando}
%\lim _{n \to \infty} \Big ( F^{(n)}(s)-
%F^{(n)}(0)\Bigr)=\frac{1}{2}\left( e^{-s} -1 +\int_{(0,1)} s e^{-s
%y} + \int_{(0,1)} \frac{e^{-sy}-1}{y} dy \right)\,.
%\end{equation}
%We point out that the above algebra follows closely the derivation
%of (6.19) in \cite{FMRT0}: this has been possible due to the
%abstractization of the transformation introduced in \cite{FMRT0}.
%
%\medskip
%
%Having \eqref{balena} and \eqref{sognando}, it remains to analyze
%the limit
%\begin{equation}\label{volando}
%\lim _{n \to \infty} \left( \frac{1}{2}\ln  d^{(n)}- F^{(n)}(0)
%\right)= \lim _{n \to \infty}\left( \frac{1}{2}\ln  d^{(n)}
% -\int _{[0, d^{(n)}-1)}
%\frac{1}{1+x}t^{(1)} (dx)\right)
% \,.
%\end{equation}
%For the identity we have used  \eqref{secchio}.  Here the arguments
%become very model--dependent and there is nothing similar in
%structure analyzed in  \cite{FMRT0}.
Equation \eqref{duro}  then follows from  Claim \ref{30gradi}.  From
this formula one can check that  $\lim_{s \to 0}
\mathbb{E}(e^{-sY^{(\infty)}})=1$ and $\lim_{s \to \infty}
\mathbb{E}(e^{-sY^{(\infty)}})=0$, thus implying
$Y^{(\infty)}\in(0,\infty)$. Indeed,   it is known that $Ei(x) = -
\bar \gamma - \log (x) - \sum_{n=1}^\infty \frac{(-x)^n}{n \cdot
n!}$ for $x >0$, which, after few computation leads to the limit
when $s \to 0$, while for $s \to \infty$, it is enough to observe
that $Ei(s) \to 0$ and thus $\tanh (Ei(s)/2) \to 0$.


Finally, we remark that condition \eqref{pato} is satisfied
if $\mu$ has finite $(1+\e)$--moment. Indeed, under
this hypothesis it holds   $\int _{[1,z]} x^2 \mu(dx) \leq z^{1-\e}
\int x^{1+\e} \mu(dx) \leq C z^{1-\e}$ for $\e \in (0,1)$ and  $\int
_{[1,\infty)} x^2 \mu(dx)<\infty$ if $\e \geq 1$.
\end{proof}

\begin{claim}\label{30gradi}
\begin{equation}
 \lim _{z \to
\infty}\left(  \int _{[0, z-1)}
\frac{1}{1+x}t^{(1)} (dx)- \frac{1}{2} \ln z  \right)=
 \frac{1}{2}\log 2 + \frac{\bar\gamma}{2}
-\frac{1}{2}\ln(\bar  \mu)
\end{equation}
where $\bar\gamma  \simeq 0,577$ is the Euler-Mascheroni constant.
\end{claim}
%for a suitable  positive constant $C_*$.  Assuming the above claim
% we conclude that
%$\sqrt{d^{(n)}} \exp \left\{ - F^{(n)}(s) \right\}$ goes to
%$e^{-C_*} \sqrt{\bar \mu}$. Combining this limit with \eqref{balena}
%and  \eqref{sognando} one gets immediately the thesis.

\begin{proof}[Proof of Claim \ref{30gradi}]

First of all we give an explicit formula for  the measure $m(dx)$
with support in $[1, \infty)$ such that
\begin{equation}\label{falchetto}
\int _A m (dx)= \int_{A-1} \frac{1}{1+x} t^{(1)}(dx) , \qquad A
\subset[1,\infty) \text{ Borel} .
\end{equation}

\begin{Lemma}
Let $m(dx)$ be the measure defined by \eqref{falchetto}. Let $
\otimes_k\mu$ be the convolution of $k$ copies of the interval law
$\mu$. Then
$$ m(A)= \sum _{k=1} ^\infty
\a_k [\otimes_k \mu]  (A)  , \qquad A \subset [1,\infty) \text{
Borel}
$$
where $\a_k :=\bigl(1+(-1)^{k+1} \bigr)/(2k)$.
\end{Lemma}
Note that, since $\mu$ has support in $[1, \infty)$, the probability
measure  $\otimes _k \mu$ has support in $[k,\infty)$.
\begin{proof}
We know that $t^{(1)}$ satisfies \eqref{asilobello} with
$\cF(x)={\mbox{arctanh}}(x)$. Since $g^{(1)}(s)= g(s):= \int e^{-sx}
\mu(dx)$, by \eqref{falchetto} the identity \eqref{asilobello}  can
be rewritten as
%\begin{equation}
$\cF( g(s) )  = \int _1 ^\infty e^{-sx} m(dx)$.
%\end{equation}
 For $s$ large $g(s)$ goes to zero, hence we can use
the analytic expansion of $\cF(x)$   around zero (recall that ${\mbox{arctanh}}(x)=1/2\ln\frac{1+x}{1-x}$ and use \eqref{zittizitti} with $\gamma=0$)
getting
\begin{equation*}
%\label{saretta}
 \sum_{k=1}^\infty \a_k  g(s)^k = \int _1 ^\infty
e^{-sx} m(dx)  , \qquad s \text{ large }.
\end{equation*}
Since $g(s)^k = \int e^{-sx} [\otimes _k \mu ](dx)$, the above
equation can be written as \begin{equation*} \sum_{k=1}^\infty \a_k
\int e^{-sx}   [\otimes_k \mu](dx)= \int _1 ^\infty e^{-sx} m(dx)
 , \qquad s \text{ large }.
\end{equation*}
The thesis then follows from Theorem 1a in \cite[Section XIII.1]{F}.
\end{proof}
Let  $W_1, W_2, \dots, W_k$ be i.i.d. random
variables with common law $\mu$.  Then, $\otimes _k \mu$ is the law of $W_1+W_2+
\cdots + W_k$.
Due to the above lemma and since $W_i \geq 1$ a.s., we can write
\begin{equation}\label{tamburo} \int _{[0,z-1)}
\frac{t^{(1)}(dx)}{1+x}= \int _{[1,z)} m(dx)= \sum _{k=1} ^{ \lfloor
z \rfloor}\a_k   \bbP( W_1+ \cdots +W_k \leq z)
\end{equation}
where $\lfloor z\rfloor$ denotes the integer part of $z$.
%Since the
%r.h.s. is a monotone function in $z$, for the proof of our claim it
%is enough to restrict to the case of integer $z$.

Recall that $\bar \mu := \int x \mu(dx) = \bbE( W_i)\geq 1$. If
$\bar \mu=1$ then $\mu= \d_1$ and, as the reader can check, the
arguments below become trivial. Hence, we assume that $\bar \mu>1$.

Given $z >1$ we define $ \tilde W_i := W_i \mathds{1} (W_i  \leq  z ) $
and $\bar \mu(z):= \bbE( \tilde W_i) =\bbE(W_i; W_i \leq  z)$. We
can estimate the variance of $\tilde W_i$ as
\begin{equation}\label{barbalalla} \Var( \tilde W_i) \leq \bbE( \tilde
W_i^ 2)= \bbE( W_i^2; W_i \leq z)  . \end{equation}
Fix $\e>0$. We deal separately with the case (i) $k\leq \frac{z}{\bar \mu} (1-\e)$ and (ii)
$k\geq \frac{z}{\bar \mu} (1+\e)$.  \\
$\bullet$ Case (i). Since $\lim _{ z \to \infty} \bar \mu (z) = \bar
\mu$, this implies that there exists $z(\e)$ large enough and
independent from $k$ such that for $z\geq z(\e)$ it holds
\begin{equation}k\label{equationpalla}
\bar \mu (z) <z \qquad \text{ and } \qquad
%$$\frac{\bar \mu (z) - \bar \mu}{ z/k -\bar \mu (z) }\leq 1$$
%which in turn implies
%\begin{equation}\label{palline}
\left(\frac{z- k \bar \mu }{ z-k \bar \mu(z) }\right)^2\leq 4.
\end{equation}Therefore for $z\geq z(\e)$ thanks to \eqref{equationpalla} we can use
the Markov inequality to obtain
\begin{equation}\label{ben10}
\begin{split}
 \bbP( W_1 + \cdots +W_k >z) & \leq \bbP( \exists i \leq k\,:\, W_i \not = \tilde W_i)
+\bbP(\tilde W_1 + \cdots +\tilde W_k >z)\\
& \leq k \bbP( W_1 >z) + \bbP\Big(\frac{\tilde W_1 + \cdots +\tilde
W_k}{k} -\bar \mu(z) > \frac{z- k \bar \mu(z)}{k}\Big)\\ & \leq k
\bbP(W_1>z) + \frac{k\Var (\tilde W_1)}{ (z-k \bar \mu (z) )^2} .
%%\\ & k \bbP(W_1>z) + \frac{k)}{ (z-k   \mu (z) )^2}  \,.
\end{split}
\end{equation}
Then, combining \eqref{barbalalla}, \eqref{equationpalla}  and
\eqref{ben10} we get for $z\geq z(\e)$
\begin{align}\label{ben110}
\a_k \bbP( W_1 + \cdots +W_k >z)
& \leq
\bbP( W_1>z) + \frac{ \bbE ( W_1^2; W_1 \leq z) }{ (z-k \bar \mu(z) )^2} \nonumber \\
& \leq
\bbP( W_1>z) + 4\frac{ \bbE ( W_1^2; W_1 \leq z) }{ (z-k \bar \mu )^2}.
\end{align}
$\bullet$ Case  (ii) By similar arguments one can prove that there
exists $\bar z(\e)$ such that for $z\geq\bar z(\e)$ it holds
\begin{equation}\label{ben111}
\a_k \bbP( W_1 + \cdots +W_k \leq z)
\leq
\bbP( W_1>z) + 4\frac{ \bbE ( W_1^2; W_1 \leq z) }{ (z-k \bar \mu
)^2}.
\end{equation}

% Using
%the Markov inequality,  if $k \bar \mu (z) <z$ we can bound
%\begin{equation}\label{ben10}
%\begin{split}
% \bbP( W_1 + \cdots +W_k >z) & \leq \bbP( \exists i \leq k\,:\, W_i \not = \tilde W_i)
%+\bbP(\tilde W_1 + \cdots +\tilde W_k >z)\\
%& \leq k \bbP( W_1 >z) + \bbP\Big(\frac{\tilde W_1 + \cdots +\tilde
%W_k}{k} -\bar \mu(z) > \frac{z- k \bar \mu(z)}{k}\Big)\\ & \leq k
%\bbP(W_1>z) + \frac{k\Var (\tilde W_1)}{ (z-k \bar \mu (z) )^2}\,.
%%\\ & k \bbP(W_1>z) + \frac{k)}{ (z-k   \mu (z) )^2}  \,.
%\end{split}
%\end{equation}
%Let us suppose that $k \leq   \frac{z}{\bar \mu} (1-\e)$. Since $
%\lim _{ z \to \infty} \bar \mu (z) = \bar \mu$, this implies that $k
%\bar \mu (z) <z$ for $z\geq z(\e)$ large enough independently from
%$k$. In particular, we are allowed to use \eqref{ben10}. In
%addition, for $z \geq z(\e)$ and $k \leq   \frac{z}{\bar \mu}
%(1-\e)$, it holds
%\begin{equation}\label{palline}\frac{z- k \bar \mu }{ z-k \bar \mu
%(z) }= 1+ \frac{\bar \mu (z) - \bar \mu}{ z/k -\bar \mu (z) }\leq 2
%\end{equation}
%
%Combining \eqref{barbalalla}, \eqref{ben10} and \eqref{palline}, for
%any  $\e>0$ and for $z\geq z(\e)$ we can bound
%$$  \a_k \bbP( W_1 + \cdots +W_k >z) \leq  \bbP( W_1>z) + 4\frac{ \bbE ( W_1^2; W_1 \leq z) }{ (z-k \bar \mu
%)^2} \,, \qquad \forall k \leq   \frac{z}{\bar \mu} (1-\e)\,.
%$$
%By similarly arguments, the same estimate holds also for any $\e>0$,
%$z \geq z(\e)$ and  $k \geq\frac{z}{\bar \mu} (1+\e)$.

At this point
we get
\begin{equation}\label{perpetuo}
\sum_{k=1}^{\lfloor \frac{z}{\bar \mu} (1-\e)\rfloor} \a_k+\cE_1\leq
 \sum _{k=1} ^{ \lfloor z \rfloor} \a_k \bbP( W_1+ \cdots +W_k \leq
 z)\leq
\sum_{k=1}^{\lfloor \frac{z}{\bar \mu} (1+\e)\rfloor } \a_k+ \cE_2
%+ \sum_{k=\frac{z}{\bar \mu} (1-\e)}^{\frac{z}{\bar \mu} (1+\e)}\a_k \bbP( W_1+ \cdots +W_k \leq z)+
\end{equation} where
%, using that $0\leq a_k k \leq 2$,
 the error $\cE_1$
  can be bounded  via \eqref{ben110} as
\begin{multline*}
 |\cE_1|  \leq \sum_{k=1}^{\lfloor \frac{z}{\bar \mu}
(1-\e)\rfloor }\Big (    \bbP( W_1>z) + 4\frac{ \bbE ( W_1^2; W_1
\leq z) }{ (z-k \bar \mu )^2}\Big) \\
 \leq z  \bbP( W_1>z) +C  \bbE ( W_1^2; W_1 \leq z)  \int_{\e z} ^z
\frac{1}{x^2} dx  \leq z  \bbP( W_1>z) + \frac{C'}{\e z} \bbE (
W_1^2; W_1 \leq z)  , %\label{prete}
\end{multline*}
 and similarly the error $\cE_2$ can be bounded via \eqref{ben111}
as
\begin{equation*}
  |\cE_2|
  %\sum _{k=\lfloor
%\frac{z}{\bar \mu} (1+\e)\rfloor }^z \Big (    \bbP( W_1>z) +
%4\frac{ \bbE ( W_1^2; W_1 \leq z) }{ (z-k \bar \mu )^2}\Big)
%\\ \leq z  \bbP( W_1>z) +C  \bbE ( W_1^2; W_1 \leq z)  \int_{\e z} ^z\frac{1}{x^2} dx
  \leq z  \bbP( W_1>z) + \frac{C'}{\e z} \bbE (
W_1^2; W_1 \leq z)  . %\label{pretebis}
\end{equation*}
The bound  $\bar \mu=\bbE(W_1)<\infty$ trivially implies that $\lim
_{z \to \infty} z \bbP( W_1 >z)=0$. This observation, together with
the hypothesis \eqref{pato}, assures that  for any fixed $\e>0$
it holds
\begin{equation}\label{richiamo}
\lim_{z\to  \infty}\cE_1=\lim_{z\to  \infty}\cE_2=0.
\end{equation} We point
out that the above estimates follow closely the arguments used to
prove the weak LLN. If $\mu$ has finite variance, exactly as in the
proof of the LLN, the truncation $\tilde W_i$ would be unnecessary
and a direct  application of the Markov inequality would allow to
estimate $\cE_1, \cE_2$.

It remains to study the behavior of the series $  \sum _{k=1} ^n
\a_k$ for $n$ integer. It is known that $ \sum_{k=1}^n \frac{1}{k} =
\log n + \bar\gamma + o(1)$, where $\bar\gamma$ is Euler-Mascheroni
constant. Assume that $n$ is even and  $n=2p$. Then,
$$
\phi(n)  = \sum_{\genfrac{}{}{0pt}{}{k=1}{k \text{ odd}}}^n
\frac{1}{k} = \sum_{k=1}^n \frac{1}{k} - \frac{1}{2}\sum_{k=1}^p
\frac{1}{k}
%= \log n + \alpha - \frac{1}{2}(\log p + \alpha) +o(1)
= \frac{1}{2} \log 2 + \frac{1}{2} \log n + \frac{\bar\gamma}{2} +
o(1).
$$
%\bigskip
%We first take $\g=0$.
% Let us restrict to $z$ integer and odd. Since
% $\frac{1}{k}= \frac{1}{2} \int _k^{k+2} \frac{1}{k} dx $, we can write
%$$ \sum _{k=1}^z \a_k = \sum _{k \text{ odd} , 1\leq k \leq z}
%\frac{ 1}{2} \int _k ^{k+2} \frac{1}{k} dx = \sum _{k \text{ odd} ,
%1\leq k \leq z}\frac{1}{2}  \int _k ^{k+2} \frac{1}{x} dx+
% \sum _{k \text{ odd} ,
%1\leq k \leq z}\frac{1}{2}  \int _k ^{k+2} \left(\frac{1}{k} -
%\frac{1}{x} \right) dx\,.
%$$
%Let us analyze the r.h.s.  The first sum in the r.h.s. equals
%$\frac{1}{2} \ln (z+2)$.
% We note that for $x \in [k, k+2]$ it
%holds $|x^{-1} - k^{-1}| \leq (x-1)^{-2}$. Hence   the last sum is
%absolutely   convergent as $z $ goes to $\infty$. Since $| \ln z -
%\ln (z+2)| $ goes to zero as $z $ goes to $\infty$ and singe $ \a_k
%$ goes to zero as $k $ goes to $\infty$ we conclude that there
%exists a finite constant $C_*$  such that
 For $n$ odd, one obtains a similar expression. Hence,we conclude
that
\begin{equation}\label{russia}
 \lim _{z \to \infty} \left(\sum
_{k=1}^{  \lfloor z  \rfloor} \a_k - \frac{1}{2} \ln z\right) =
\frac{1}{2} \log 2 + \frac{\bar\gamma}{2}  .
\end{equation}
%without any restriction on $z$.
% \centerline{ {\bf Ale:} la suddetta formula si dovrebbe estendere
% a tutti i $\g$. Nel seguito assumo che sia provata nel caso generale}
Collecting \eqref{tamburo}, \eqref{perpetuo}, \eqref{richiamo} and
\eqref{russia} we get %for $\g=0$
that
$$C_*  +\frac{1}{2} \ln \left(\frac{z(1-\e)}{\bar \mu}\right) -
o(1)  \leq  \int _{[0,z-1) } \frac{t^{(1)}(dx)}{1+x}\leq C_*
+\frac{1}{2} \ln \left(\frac{z(1+\e)}{\bar \mu} \right) + o(1)
$$ where $o(1)$ goes to zero as $z \to \infty$ (for any fixed $\e>0$)  and $C_* =
\frac{1}{2} \log 2 + \frac{\bar\gamma}{2} $. Hence
$$ \left|\int_{[0, z-1)} \frac{t^{(1)}(dx)}{1+x}- \frac{1}{2} \ln (z/\bar \mu)-C_*  \right|
\leq C (\e  +  o(1)) .
$$
 At this point take
first the limit $z \to\infty$ and then the limit $\e \downarrow
0$, thus concluding the proof of our claim.



\medskip

\end{proof}
%Recall that $\bar \mu := \int x \mu(dx) = \bbE( W_i)\geq 1$. Set
%$\bar
%W_i:= W_i - \bar \mu$.  We distinguish two special  cases:\\
%$\bullet$ {\bf Case $k \bar \mu < z$}. Then
%\begin{multline*}
% \bbP( W_1 + \cdots +W_k >z)=\\
% \bbP( \bar W_1 + \cdots +\bar W_k >z-k \bar
% \mu) \leq \frac{ \bbE\left[ ( \bar W_1+ \cdots +\bar W_k )
% ^2\right] }{ (z- k \bar \mu)^2 }= \frac{k Var(W_i) }{ (z- k \bar \mu)^2
% }\,.
%\end{multline*}
%$\bullet$ {\bf Case $k \bar \mu > z$}. By similar computations we
%get
%$$
%\bbP( W_1 + \cdots +W_k \leq z) \leq\frac{k \Var(W_i) }{ (z- k \bar
%\mu)^2 }\,.$$
%
%
%Due to the above two cases, taking $\e>0$ small , we get
%%
%\begin{equation}\label{perpetuo}
%\sum_{k=1}^{\lfloor \frac{z}{\bar \mu} (1-\e)\rfloor}
%\a_k+\cE_1(z)\leq
% \sum _{k=1} ^{ \lfloor z \rfloor} \a_k \bbP( W_1+ \cdots +W_k \leq
% z)\leq
%\sum_{k=1}^{\lfloor \frac{z}{\bar \mu} (1+\e)\rfloor } \a_k+
%\cE_2(z)
%%+ \sum_{k=\frac{z}{\bar \mu} (1-\e)}^{\frac{z}{\bar \mu} (1+\e)}\a_k \bbP( W_1+ \cdots +W_k \leq z)+
%\end{equation} where
%%, using that $0\leq a_k k \leq 2$,
% the errors $\cE_1(z)$
%and $\cE_2(z)$
%  can be estimated as
%\begin{align}
%& |\cE_1(z)| \leq \sum_{k=1}^{\lfloor \frac{z}{\bar \mu}
%(1-\e)\rfloor } \a_k \frac{k \Var(W_i) }{ (z- k \bar \mu)^2  } \leq
%C \int_{\e z} ^z
%\frac{1}{x^2} dx \leq   C' \frac{1}{\e z}\,, \label{prete}\\
%& |\cE_2(z)|\leq \sum _{k=\lfloor \frac{z}{\bar \mu} (1+\e)\rfloor
%}^z \a_k \frac{k \Var(W_i) }{ (z- k \bar \mu)^2  } \leq C \int_{\e
%z} ^z \frac{1}{x^2} dx \leq   C' \frac{1}{\e z}\,. \label{perpetua}
%\end{align}
%%{\bf Ale}:{ \sl nelle formule sopra bisogna mettere un po' di parti
%%intere. bisogna essere un po' piu' precisi e distingure $\bar mu=1$
%%e $\bar \mu>1$. comunque la ciccia e' la stessa. il caso $\bar
%%\mu=1$ avrebbe persino meno termini}. \\
%
%It remains to study the behavior of the series $ \sum _{k=1} ^n
%\a_k$ for $n$ integer. It is known that
%$$
%\sum_{k=1}^n \frac{1}{k} = \log n + \alpha + o(1) ,
%$$
%where $\alpha$ is Euler-Mascheroni constant. Assume that $n=2p$.
%Then,
%$$
%\sum_{k=1}^n \alpha_k = \sum_{\genfrac{}{}{0pt}{}{k=1}{k \text{ odd}}}^n \frac{1}{k} =  \sum_{k=1}^n \frac{1}{k} -  \frac{1}{2}\sum_{k=1}^p \frac{1}{k}
%%= \log n + \alpha - \frac{1}{2}(\log p + \alpha) +o(1)
%= \frac{1}{2} \log 2 + \frac{1}{2} \log (n) + \frac{\alpha}{2} + o(1).
%$$
%%\bigskip
%%We first take $\g=0$.
%% Let us restrict to $z$ integer and odd. Since
%% $\frac{1}{k}= \frac{1}{2} \int _k^{k+2} \frac{1}{k} dx $, we can write
%%$$ \sum _{k=1}^z \a_k = \sum _{k \text{ odd} , 1\leq k \leq z}
%%\frac{ 1}{2} \int _k ^{k+2} \frac{1}{k} dx = \sum _{k \text{ odd} ,
%%1\leq k \leq z}\frac{1}{2}  \int _k ^{k+2} \frac{1}{x} dx+
%% \sum _{k \text{ odd} ,
%%1\leq k \leq z}\frac{1}{2}  \int _k ^{k+2} \left(\frac{1}{k} -
%%\frac{1}{x} \right) dx\,.
%%$$
%%Let us analyze the r.h.s.  The first sum in the r.h.s. equals
%%$\frac{1}{2} \ln (z+2)$.
%% We note that for $x \in [k, k+2]$ it
%%holds $|x^{-1} - k^{-1}| \leq (x-1)^{-2}$. Hence   the last sum is
%%absolutely   convergent as $z $ goes to $\infty$. Since $| \ln z -
%%\ln (z+2)| $ goes to zero as $z $ goes to $\infty$ and singe $ \a_k
%%$ goes to zero as $k $ goes to $\infty$ we conclude that there
%%exists a finite constant $C_*$  such that
%In turn,
%\begin{equation}\label{russia}
% \lim _{z \to \infty} \left(\sum
%_{k=1}^z \a_k - \frac{1}{2} \ln z\right) = \frac{1}{2} \log 2 +  \frac{\alpha}{2} ,
%\end{equation}
%without any restriction on $z$.
%% \centerline{ {\bf Ale:} la suddetta formula si dovrebbe estendere
%% a tutti i $\g$. Nel seguito assumo che sia provata nel caso generale}
%
%Collecting \eqref{perpetuo}, \eqref{prete}, \eqref{perpetua} and
%\eqref{russia} we get %for $\g=0$
%that
%$$C_*  +\frac{1}{2} \ln \left(\frac{z(1-\e)}{\bar \mu}\right) -
%o(1) - \frac{C}{\e z} \leq  \int _{[0,z-1) }
%\frac{t^{(1)}(dx)}{1+x}\leq C_*  +\frac{1}{2} \ln \left(\frac{z(1+\e)}{\bar
%\mu} \right) + o(1) + \frac{C}{\e z}$$ where $o(1)$ goes to zero as
%$z \to \infty$ and $C_* = \frac{1}{2} \log 2 +  \frac{\alpha}{2} $. Hence
%$$ \left|\int_{[0, z-1)} \frac{t^{(1)}(dx)}{1+x}- \frac{1}{2} \ln (z/\bar \mu)-C_*  \right|
%\leq C (\e  + 1/ \e z+ o(1)) .
%$$
% At this point take
%first the limit $z \to \infty$ and then the limit $\e \downarrow
%0$, thus concluding the proof of our claim.
%\end{proof}
%
%
%
%
%
%Let us come back to the case $\g>0$. For $z$ integer  we can write
%$$ \sum _{k=1}^z \a_k = \sum _{k=1}^z \frac{1}{k} + \sum _{k=1}^z
%\frac{(-1)^{k+1}}{(1+\g)^k}\,.$$ Since $\g>0$ the second sum in the
%r.h.s. is convergent. Reasoning as in the proof of \eqref{russia} we
%have that  $ \lim _{z \to \infty} \left (  \sum _{k=1}^z
%\frac{1}{k}- \int _1 ^z \frac{1}{x} dx \right)$ exists finite. Since
%the last integral equals $\ln z$ we conclude that
% there
%exists a finite constant $C_*(\g)$  such that
%\begin{equation}\label{russiabis}
% \lim _{z \to \infty} \left(\sum
%_{k=1}^z \a_k - \ln z\right) = C_*(\g)\,, \qquad [\g>0]\,.
%\end{equation}
%As above, in the above limit we can drop the restriction of $z$
%integer. The proof of the claim follows again combining
%\eqref{russiabis} with \eqref{perpetuo}, \eqref{prete},
%\eqref{perpetua} and taking the limit $z \uparrow  \infty$ and
%afterwards $\e \downarrow 0$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%  UNIVERSAL COUPLING
%%%%%%%
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\section{Universal coupling: graphical construction of the dynamics}\label{universal}

In this section we describe the universal coupling for the OCP's.
The   construction is standard and very similar to the one presented
in   Section 3.1 of \cite{FMRT0}.  On the other hand, it will be
used in Section \ref{semigruppo_OCP} and is fundamental in order to
recover   results as Lemma \ref{hollanda} and the first part of
Proposition \ref{note}.

Given $ \xi \in  \cN (d_{\rm min})$, we  enumerate
its points in increasing order  with the rule that the smallest
positive one (if it exists) gets the label $1$, while the largest
non-positive one (if it exists) gets the label $0$. We write $N(x,
\xi)$ for the integer number labelling the point $x \in \xi $. This
allows to enumerate the domains of $\xi$ as follows: a domain
$[x,x']$ is said to be the $k^{th}$-domain if (i) $x$ is finite and
$N(x, \xi )=k$, or (ii) $x=-\infty $ and   $N(x', \xi )=k+1$. Recall
that if $x=-\infty $, then $\xi$  is unbounded from the left and
$x'$ is the smallest number in $\xi$.

We set $\|\l\| _\infty = \sup_{d \in[d_{\min},d_{\max} )}  \l (d)$ where we
recall that $\l=\l_r+\l_\ell+\l_a$.  We consider  a  probability
space $\bigl(\O, \cF,P\bigr)$ on which the following random objects
are defined and are all independent: the Poisson processes
$\cT^{(k)} = \{ T_m^{(k)}\,:\, m\in \bbN\}$, $\bar \cT^{(k)} =
\{\bar T_m^{(k)}\,:\, m\in \bbN\}$ and $\tilde{\cT}^{(k)} =
\{\tilde{T}_m^{(k)}\,:\, m\in \bbN\}$ of parameter $\| \l\|_\infty$,
indexed by $k \in \bbZ$, and the random variables $U^{(k)}_m$, $\bar
U^{(k)}_m$ and $\tilde{U}_m^{(k)}$, uniformly distributed in
$[0,1]$, indexed  by $k \in \bbZ$ and $m \in \bbN$. Above, the
Poisson processes are described in terms  of the jump times $T_m
^{(k)}$, $\bar T_m^{(k)}$, $\tilde{T}_m^{(k)}$.
  By discarding a set of $P$-probability $0$, we may assume that
\begin{equation} \label{disgarding}
\mbox{As $k_1,k_2,k_3$ vary in $\bbZ$, the sets } \cT^{(k_1)},\! \bar
\cT^{(k_2)} \mbox{ and } \tilde{\cT}^{(k_3)} \mbox{ are locally
finite and   disjoint.}
\end{equation}

Next, given $\z\in \cN (d_{\rm min})$ and $\o\in \O$, to each domain
$\Delta $ that belongs to  $\z$ we associate the Poisson process
$\cT^{(k)}$
 if $\Delta $ is the $k$--th domain in $\z$. In this case, we write $\cT^{(\D)}$
 instead of $\cT^{(k)}$. Similarly we define  $\bar \cT^{(\D)}$, $\tilde{\cT}^{(\D)}$, $U^{(\D)}_m$, $\bar
 U^{(\D)}_m$ and $\tilde{U}^{(\D)}_m$. The idea behind the
 construction of the universal coupling is the following: if for
  example $s=T^{(\D)}_m$ for some $m \in \bbN$ and if the domain
  $\D$ is present at time $s-$, then the left extreme of $\D$ has to
  be erased at time $s$ if and only if $U_m ^{(\D)}\leq \l _\ell
  (d)/\|\l\|_\infty$, $d$ being the length of the domain $\D$. Similarly,
  the right extreme (or both the extremes) of $\D$ can be erased at time $s=\bar
  T^{(\D)}_m$ (resp. $\tilde{T}^{(\D)}_m$). Working with infinite
  domains, to formalize the  above construction one needs some
  percolation argument as presented below.


 We define $\cW_t[\o,\z]$ as
the set of domains $\D$ in  $\z$ such that
\begin{align}\label{caldo-bestiale}
\Big\{ s\in[0,t] : s \in  \cT^{(\D)}\cup \bar \cT^{(\D)} \cup \tilde
{\cT}^{(\D)} ,  \text{ or } s \in
\cT^{(\D')}  & \cup \bar \cT^{(\D')} \cup \tilde{\cT}^{(\D')} \nonumber \\
 \text{ for some domain } & \D'
\text{neighbouring } \D \Big\} \neq \emptyset  .
\end{align}
On $\cW_t[\o,\z]$ we define a graph structure putting an edge
between domains $\D$ and $\D'$ if and only if they are neighbouring
in $\z$. Since the function $\l$ is bounded from above, we deduce
that the set
\begin{equation*}
\cB(\z):= \Big\{ \o \,:\,\cW_t[\o,\z] \text{ has all connected
components of finite cardinality}\; \forall t \geq 0 \Big\}
\end{equation*}
has $P$--probability equal to $1$. Note that the event $ \cB(\z)$
depends on $\z$ only through the infimum and the supremum of the set
$  \{ N(x,\z)\in \bbZ\,:\, x \in \z\}$. By a simple argument based
on countability, we conclude that
 $ P(\cB)=1$, where $\cB$ is defined as the family  of elements  $\o \in \O$ satisfying \eqref{disgarding}
 and belonging to
 $\cap _{\z \in \cN (d_{\rm min})} \cB (\z)$:
 %. and  such that all the  sets $\cT^{(k)}[\o]$, $\bar \cT^{(k)}[\o]$ and $\tilde{T}^{(k)}[\o]$,   $k \in \bbZ$, are
%disjoint:
\begin{equation} \label{spagna}
\cB = \bigcap _{\z \in \cN (d_{\rm min})} \cB (\z) \cap
\left\{\omega \in \Omega : \omega \mbox{ satisfies }
\eqref{disgarding}\right\} .
\end{equation}

\medskip

In order to define the path $\{\xi(s)\}_{s\ge 0}:= \{\xi^\z(s,\o)\}_{s\ge 0}$
associated to $\z \in \cN (d_{\rm min}
)$ and $\o \in \O$, we first fix a time $t>0$ and define the path up
to time $t$. If $\o \not \in \cB$, then we set $$ \xi(s)=\z ,
\qquad \forall s \in [0,t] .$$ If $\o \in \cB$, recall the
definition of the graph $\cW_t[\o,\z] $. Given a set of domains $ V$
we write $\bar V$ for the set of the associated extremes, {\it i.e.}\
$x \in \bar V$  if and only if there exists a domain in $V$ having
$x$ as left or right extreme. Moreover, we write $\cV_t[\o,\z] $ for
the set of all domains in $\z$ that do not belong to $\cW_t[\o,\z]$.
We require that
\begin{equation}\label{heidi}
\xi (s)\cap \overline{ \cV_t[\o,\z]}:= \overline{ \cV_t[\o,\z]} ,
\qquad \forall s \in [0,t] ,
\end{equation}
{\it i.e.}\ up to time $t$ all points in $\overline{ \cV_t[\o,\z] }$
survive.
% Since $\o \not \in \cB$, for all time $t \geq 0$
% this graphs has all clusters of finite cardinality. If the domain
% $[x,x']$ does not belong to $\cW_t[\o,\z] $, then we impose that
% $x,x' \in \xi  (t )[\z, \o]$.
 Let us now fix a    cluster $ \cC$  in
the graph $\cW_t[\o,\z]$.
% and   write $C$ for the set of points that
% are extremes of some domain in $\cC$.
  The path  $\bigl( \xi (s )\cap \bar \cC \,:\, s \in [0,t]\bigr)$ is implicitly defined  by
  the following rules (the definition is well posed since
   $\o  \in \cB$).
If  $s \in [0,t]$ equals $T ^{(\D)}_m$ with $\Delta=[x,x'] \in \cC$
and  $x ,x'\in \xi(s-)$, then the ring at time $T^{(\D)}_m$ is
called {\sl legal} if
\begin{equation} \label{eq:uniformbis}
U_m^{(\D)} \leq \frac{ \l_\ell ( x'-x)}{\|\l\|_\infty}
\end{equation}
and in this case we set $\xi(s)  \cap \bar  \cC :=(\xi(s-) \cap \bar
\cC) \setminus \{x\}$, otherwise we set  $\xi(s) \cap \bar \cC =
\xi(s-) \cap \bar\cC$. In the first case we say that $x$ is erased
and that the domain  $[x,x']$ has incorporated the domain on its
left. Similarly, if $s \in [0,t]$ equals $ \bar T ^{(\D)}_m$ with
$\Delta=[x,x'] \in  \cC$ and $x,x' \in \xi(s-)$, then the ring at
time $\bar T^{(\D)}_m$ is called {\sl legal} if
\begin{equation} \label{eq:uniform2bis}
\bar U_m^{(\D)} \leq \frac{ \l_r ( x'-x)}{\|\l\|_\infty}
\end{equation}
and in this case we  set $\xi(s) \cap \bar \cC :=(\xi(s-) \cap \bar
\cC) \setminus \{x'\}$, otherwise we set  $\xi(s) \cap \bar \cC =
\xi(s-) \cap \bar \cC$. Again, in  the first case we say  that $x'$
is erased and that the domain $[x,x']$ has incorporated the domain
on its right. Finally, if $s \in [0,t]$ equals $ \tilde{T}^{(\D)}_m$
with $\Delta=[x,x ' ] \in  \cC$ and $x,x' \in \xi(s-)$, then the
ring at time $\tilde{T}^{(\D)}_m$ is called {\sl legal} if
\begin{equation} \label{eq:uniform3bis}
\tilde{U}_m^{(\D)} \leq \frac{ \l_a ( x'-x)}{\|\l\|_\infty}
\end{equation}
and in this case we  set $\xi(s) \cap \bar \cC :=(\xi(s-) \cap \bar
\cC) \setminus \{x,x'\}$, otherwise we set  $\xi(s) \cap \bar \cC =
\xi(s-) \cap \bar \cC$. Again, in  the first case we say  that $x$
and $x'$ are erased and that the domain $[x,x']$ has incorporated
both the domain on its right on its left.

\smallskip

We point out that $\bar \cC \cap \bar \cC'= \emptyset$ if $ \cC$ and
$ \cC'$ are distinct clusters in $\cW_t[\o,\z]$. On the other hand,
it could be $\bar \cC \cap \overline{ \cV_t[\o,\z]}\not =
\emptyset$. Let $x $ a point in the intersection and suppose for
example that $[a,x] \in \cC$ while $[x,b]\in \cV_t[\o,\z] $. Then,
by definition of $\cW_t [\o, \z]$, one easily derives that the
Poisson processes associated to the domains  $[a,x] $ and $[x,b]$ do
not intersect $[0,t]$, while  at least one of the Poisson processes
associated to the domain on the left of $[a,x] $ intersects $[0,t]$.
In particular, $x \in \xi(s) \cap \bar \cC $ for all $s \in [0,t]$,
in agreement with \eqref{heidi}. The same conclusion is reached if
$[a,x]\in \cV_t[\o,\z] $ and $[x,b] \in \cC$. This allows to
conclude that the definition of
 the path $\{\xi(s)\}_{s\ge 0}$ up to time $t$ is well posed. We point out that this definition
 is $t$--dependent. The reader can easily check
that, increasing $t$, the resulting paths coincide on the
intersection of their time domains. Joining these paths together we
get $\{\xi(s)\}_{s\ge 0}$.

\smallskip

Given a configuration $\z \in \cN(d_{\rm min})$, the law of the
corresponding random  path $\{\xi(s)\}_{s\ge 0}$ is that of the OCP
with initial condition $\z$. The advantage of the above construction
is that all OCP's, obtained by varying the initial configuration,
can be realized on  the same probability space. Given a probability
measure $\cQ$ on $\cN (d_{\rm min})$, the OCP with initial
distribution $\cQ$ can be realized by the random path $\{\xi^{\cdot}
(s,\cdot )\}_{s\ge 0}$ defined on the product space $\O \times \cN
(d_{\rm min})$ endowed with the probability measure $P\times \cQ $.

\smallskip

The next  result (similar to  \cite[Lemma 2.2]{S}) is an immediate consequence of the above construction and of the metric
defined on $\cN(d_{\rm min})$. We omit the proof.

\begin{Lemma} \label{hollanda}
For any $(\zeta,\omega) \in \cN
(d_{\rm min}) \times \cB$, the function $[0,\infty) \ni s \mapsto
\xi^\zeta(s,\omega) \in \cN (d_{\rm min}) $ is c\`adl\`ag.
 In other words, $\{\xi^\zeta(s,\omega)\}_{s\ge 0}$ belongs to the Skohorod space
$D\bigl( [0,\infty), \cN (d_{\rm min})\bigr)$.
\end{Lemma}





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\section{Proof of Theorem \ref{amico_marco_zero}}\label{infinitesimi}


This section is dedicated to the construction and the analysis of
the Markov generator $\mathcal{L}$ %\club
 of the  OCP.
%Since such a construction is partially standard, we omit some details
%and focus on the relevant proofs.
We first introduce the Markov semigroup associated to the graphical
construction of Section \ref{universal} and then introduce the
pregenerator $\mathbb{L}$.


%\club \club 
 If points (domain extremes) belong always to a given
countable subset of $\bbR$ (for example points belong to $\bbZ$),
then one can directly apply the methods developed for interacting
particle systems on countable space \cite{L}, identifying each
domain extreme with a particle. In the general case,
 we have introduced a lattice
structure (see Section \ref{sec_OCP}) which strongly simplifies the
problem of the Markov generator from an analytic viewpoint, and
allows us to use again the methods described in \cite{L}. Endowing
the space $\cN$ of locally finite subset of $\bbR$ of the vague
topology, the map $\cN \ni \xi \mapsto \xi \cap [a,b]\in \cN$ is not
continuous, hence the above discretization requires  some  special
care.

\subsection{Markov semigroup and pregenerator}\label{semigruppo_OCP}


 Given an initial configuration $\xi \in \cN(d_{\rm min})$, define the path $\{\xi(s)\}_{s\ge 0}
= \{\xi^\xi(s)\}_{s\ge 0}$ as in Section  \ref{universal}, with
$\zeta=\xi$. Note that the dependence on the element $\o \in \O$ is
understood. In what follows we will alternatively use the notation
$\{\xi^\xi(s,\omega)\}_{s \geq 0}$, or $\{\xi^\xi(s)\}_{s \geq 0}$
or $\{\xi(s)\}_{s \geq 0}$, depending on the context.
% for the process starting from $\zeta=\xi$.

 Let $\mathbb{P}_\xi$ be the law
of the OCP starting from $\xi\in \cN(d_{\rm min}) $:
% {\it i.e.} the probability measure
%on $D([0,\infty), \cN(d_{\rm min}))$ defined by
$$
\mathbb{P}_\xi (A) = P\left( \left\{ \omega\in \O : \left\{ \xi(s)
\right\}_{s \geq 0} \in A \right\} \right)  \qquad \forall A \subset
D([0,\infty), \cN(d_{\rm min}))\;\; \text{Borel}  .
$$
We write $\mathbb{E}_\xi$ for the corresponding expectation. Then,
for any $f \in \mathbb{B}$, we set
$$
P_t f(\xi) = \mathbb{E}_\xi \left( f\left( \xi(t) \right) \right)
\qquad \forall t \geq 0 .
$$


Below we shall prove that $(P_t)_{t \geq 0}$ is a Markov semigroup
on $\mathbb{B}$ in the sense of the following definition \cite{L}:
\begin{definition}[Markov semigroup] \label{markov}
A family of linear operator $(S_t)_{t \geq 0}$ on $\mathbb{B}$ is
called a Markov semigroup
if it is Feller, {\it i.e.}\ $S_t f \in \bbB$ for all $f\in  \bbB$, and satisfies the following properties:\\
$(i)$ $S_0 = \mathds{1}_\mathbb{B}$, the identity operator on $\mathbb{B}$;\\
$(ii)$ for any $f \in \mathbb{B}$, $\lim_{t \to 0} \| S_t f - f \| =0$;\\
$(iii)$ for any $s,t \geq 0$, any $f \in \mathbb{B}$, $ S_{t+s} f = S_t ( S_s f)$;\\
$(iv)$ for any $t \geq 0$, $S_t \mathds{1} = \mathds{1}$;\\
$(v)$ for any $f \in \mathbb{B}$, $f \geq 0 \Rightarrow P_t f \geq
0$.
\end{definition}

 Before moving to the proof of the fact that $(P_t)_{t \geq 0}$ is
a Markov semigroup, we need to introduce some operators and to fix
some notation.

\smallskip


 Given $s>0$ we consider the operator $L_s$ on $\bbB$
defined as
 \begin{align} \label{santinumi}
L_s f(\xi)
 =
\!\!\!\!\!\! \sum_{\genfrac{}{}{0pt}{}{[x,x+d]\text{ domain}}{\text{in } \xi\cap (-s,s) } }
\!\!\! & \Big\{ \l_\ell (d) \bigl[ f (\xi\setminus\{x\} )- f(\xi) \bigr]
+  \l_r (d) \bigl[ f (\xi\setminus\{x+d\} )- f(\xi) \bigr] \nonumber \\
& \quad + \l_a (d) \bigl[ f (\xi\setminus\{x, x+d\} )- f(\xi) \bigr]
\Big\}  .
\end{align}
Since $\xi $ is locally finite, the r.h.s.\ is given by  a finite sum
and therefore is well defined. Given an integer $n\in \bbN_+$, we
define the operator $\bbL_n$ on $\bbB$ as \begin{equation}\label{ln}
\bbL_n f (\xi) := d_{\rm min}^{-1} \int _{n d_{\rm min} }^{(n+1)
d_{\rm min} } L_s f(\xi) ds  .
\end{equation}
Note that, given $\xi\in \cN(d_{\rm min} )$,  the integrand is a
bounded stepwise function with a finite family of jumps. Hence, it
is integrable.

Recall the notation at the
beginning  of Section \ref{prova_differisco}. Given $k \in \bbZ$ and
$\xi \in \cN(d_{\rm min})$, let
\begin{align*}
& c_k( \xi) := \mathds{1} \bigl( |\xi\cap I_k|=1\bigr)\bigl[  \l _r
( d_{z_k } ^\ell)+\l _\ell ( d_{z_k} ^r)\bigr] ,\\
& c_{k,k'} (\xi):=  \mathds{1} \bigl(|\xi\cap I_k|=1\,,\, |\xi \cap
I_{k'}|=1 \,,\, |\xi \cap I_r|=0 \; \forall r: k <r<k' \bigr) \l_a
\bigl( z_{k'}-z_k\bigr) ,
\end{align*}
where for any $\xi$ such that $ |\xi\cap I_k|=1$ we set $z_k:=
\xi\cap I_k$ (due to the definition of $\cN(d_{\rm min})$ each
interval $I_k$ contains at most one point of $\xi$).

Finally,  for any $f \in \bbD$ (recall \eqref{ddd}),  set
\begin{equation}\label{preistorico}
\bbL f (\xi): = \sum _{r \in \cR} c_r (\xi) \nabla _r f(\xi) , \qquad
\xi \in \cN(d_{\min}) .
\end{equation}
Since the rates $\l_\ell,\l_r,\l_a$  are bounded, for all $f \in
\bbD$  the series in the r.h.s.\ of \eqref{preistorico} is absolutely
convergent, hence $\bbL f (\xi)$ is well defined.

\begin{Lemma}\label{giostra}
The following holds:
\begin{enumerate}[(i)]
\item  For each $n \in \bbN_+$,  $\bbL _n$ is a bounded operator from
$\bbB$ to $\bbB$.





\item  For each $f \in \bbD$,   $\bbL f \in \bbB$ and $\bbL f = \lim _{n \to \infty}\bbL_n f$.  In particular, $\bbL$ is an
operator with domain $\cD(\bbL):=\bbD$ into $\bbB$.
\item $\bbB_{\rm loc} \subset \bbD$ ($\bbB_{\rm loc}$ being the set of local functions $f\in \bbB$).
\end{enumerate}
\end{Lemma}

\begin{remark}\label{linguini}
Observe that,  for any $f \in \bbB_{\rm loc}$ and any $\xi \in
\cN(d_{\min})$, $ \bbL f(\xi)$ equals the r.h.s.\ of
 \eqref{santi}.
On the other hand, we point out that the operator $\bbL: \bbB
\supset \bbD \to \bbB$ is a \emph{Markov pregenerator}  as defined
in  \cite[Ch.\ 1, Def.\ 2.1]{L}. Indeed, it holds (i) $\mathds{1}
\in \bbD$, (ii) $\bbD$ is dense in $\bbB$ since it contains the
subset $\bbB_{\rm loc}$ which we know by Lemma \ref{lemino} to be
dense and finally (iii) if $f \in \bbD$ and $f(\xi)= \min \bigl\{
f(\xi') \,:\, \xi ' \in \cN (d_{\rm min})\bigr\}$ then $\bbL f (\xi)
\geq 0$. Due to \cite[Ch.\ 1, Prop.\ 2.2]{L}, these conditions
ensure  that $\bbL$ is a Markov pregenerator.
\end{remark}




\begin{proof} Without loss of generality,  for simplicity of notation we take $d_{\rm
min}=1$.

We consider Part $(i)$. Let $\xi_k \to \xi$ in $\cN(d_{\rm min})$.
%Let us prove that $\bbL_s f (\xi_k)\to \bbL f(\xi)$. To this aim we
We set $R:=\{ s\in [n,n+1]\,:\, \xi \cap\{-s,s\} = \emptyset\}$.
%Then $R \cap [n,n+1]$ is finite.
We claim that $L_s f(\xi_k) \to L_s f(\xi)$ for $s \in R$. To this
aim we apply Lemma \ref{gnocchi} $(ii)$. For $k$ large, it holds that $\xi
\cap (-s,s)$ and $\xi_k\cap(-s,s)$ have the same finite cardinality
$N$. Writing $x_j$ and $x_j^{(k)}$ for their $j$--th point (from the
left), we can write
\begin{align*}
L_s f (\xi)
=
\sum_{j=1}^{N-1} & \Big\{ \l_\ell (x_{j+1}-x_j) \bigl[ f (\xi\setminus\{x_j\} )- f(\xi) \bigr]  \\
& \quad
+ \l_r (x_{j+1}-x_j) \bigl[ f (\xi\setminus\{x_{j+1} \} )- f(\xi) \bigr]\\
& \quad
+ \l_a (x_{j+1}-x_j) \bigl[ f (\xi\setminus\{x_j,x_{j+1} \} )- f(\xi) \bigr]\Big\}
\end{align*}
and a similar expression for $ L_s f (\xi_k)$. The thesis then
follows from (a)  the convergence $x_j \to x_j ^{(k)}$ as $k \to
\infty$ due to Lemma \ref{gnocchi} $(ii)$, (b)  the continuity of the
jump rates, (c) the convergence $\xi_k\setminus \{x^{(k)}_j\} \to\xi
\setminus \{x _j\}$, $\xi_k\setminus \{x^{(k)}_{j+1}\} \to\xi
\setminus \{x _{j+1}\}$ and $\xi_k\setminus \{x^{(k)}_j,
x^{(k)}_{j+1} \} \to\xi \setminus \{x_j, x_{j+1}\}$ as $k \to
\infty$ for $j: 1\leq j < N$, (d) the continuity of $f$.


We can now prove that $\bbL_n f$ belongs to $\bbB$. To this aim  it
is enough to apply the dominated convergence theorem together with
the above claim and the following  observations: (a) $R \setminus
[n,n+1]$ is finite, (b) due to the definition of $\cN(d_{\rm min})$
the function $L_s f $ has uniform norm bounded by $C s\|f\|$, $C$
being independent from $s$.

\smallskip


Let us now prove Part $(ii)$. Since we already now that $\bbL_n f
\in \bbB$, it is enough to show that $\sup _{\xi \in \cN(d_{\rm
min})} | \bbL f(\xi) -\bbL_n f(\xi)|$ converges to zero as $n \to
\infty$.  By the
 boundedness of the rates it holds
\begin{equation*}
| \bbL f(\xi) -\bbL_n f(\xi)|\leq  C \!\!\!\!\!\!\!\sum_{\genfrac{}{}{0pt}{}{r\in \cR}{{\rm supp}(r) \not \subset [-n,n]}} \!\!\!\!\!\! \D_f(r)
\end{equation*}
where the support of $r$ is defined as $ {\rm supp}(r)= k$ if $r=k $
and ${\rm supp}(r)= \{k,k'\}$ if $r=(k,k')$. The above estimate and
the fact that $f \in \bbD$ allow to conclude.


\smallskip
Part $(iii)$ is obvious.
\end{proof}


\begin{Proposition} \label{note}
The family of linear operators $(P_t)_{t \geq 0}$ is a  Markov
semigroup on $\bbB$  given by contraction maps ({\it i.e.}\  $\| P_t
f \| \leq \|f\|$
 for all $f \in \mathbb{B}$).
 Moreover, for any $f \in \bbB_{\rm loc}$,
 it holds
 % and any $\xi \in \cN(d_{\min})$, the function
%$t \mapsto P_tf(\xi)$ is differentiable at $t=0$ with derivative
%$\bb
%$$
%\frac{d}{dt} P_t f(\xi)_{|_{t=0}} = \bbL f(\xi) ,
%$$
%and the limit holds uniformly in $\xi$:
\begin{equation}\lim_{t \downarrow  0} \left\| \frac{P_t f -f}{t} - \bbL
f\right\| = 0 .
\end{equation}
\end{Proposition}



\begin{proof}
We focus on the only point that is not standard, namely the Feller
property. The rest is either a direct consequence of the graphical
construction, or can be easily derived using the arguments presented
in  \cite[Ch. 2]{S}. Details are left to the reader.

Let us prove the Feller property. Fix $f \in \mathbb{B}$ and $\e>0$.
%$\varepsilon >0$, $t >0$ and $\eta \in \cN (d_{\min})$. Our aim is
%to prove that there exists some $\delta>0$ such that for any $\eta'
%\in \cN(\d_{\min})$ with $m(\eta,\eta') < \delta$ (recall
%\eqref{vague-distance}), then $|P_tf(\eta)-P_tf(\eta')| <
%\varepsilon$.
 Thanks to Lemma \ref{lemino}, setting $f_N(\xi) =\int_N^{N+1}f(\xi
\cap (-r,r) dr$, we are guaranteed that
  $f _N \in \bbB_{\rm
loc}$ and $\lim _{N \to \infty}\|f-f_N\|=0$. Since $\|P_t f-P_t
f_N\|_\infty\leq \|f-f_N\|_\infty $ as functions on $\cN (d_{\rm
min} )$, approximating $f$ by $f_N$ we conclude that it is enough to
show that  $P_tf_N\in \bbB$, or equivalently that $P_t f \in \bbB$
for any $f \in\bbB_{\rm loc}$.


Let us fix $f \in\bbB_{\rm loc}$ and suppose that $f$ has support
inside $(-N,N)$ for some $N\geq 1$.
 For simplicity of notation we take $d_{\rm
min}\geq 1$ (the general case is completely similar). Since
$\cN(d_{\min})$ is compact, $f$ is uniformly continuous. Hence,
there exists $\delta_0>0$ such that
\begin{equation} \label{fuoriberlusconi}
m(\zeta,\eta)<\delta_0 \qquad \Longrightarrow \qquad
|f(\zeta)-f(\eta)| \leq \e .
\end{equation}
Recall the universal coupling discussed in Section
 \ref{universal} and the notation introduced therein. Depending on $\varepsilon$, we can fix $\gamma > 10 $
  large enough
 such that $P(\cC) \geq 1- \varepsilon$ where $\cC$ is the event given
 by the elements $\o \in \O$ for which there exist integers  $k,k'$
 with $ 10N \leq k,k'\leq \g N$ and
 $$
 [0,t]\cap\left( \cT ^{(j)} \cup \bar{\cT}^{(j)} \cup \tilde{\cT}^{(j)
 }\right) = \emptyset \qquad \forall j=k,k-1,-k',-(k'-1) .
 $$
 Given a generic configuration $\zeta \in \cN( d_{\rm
min})$, all the points $x$ of $\zeta \cap(-N,N)$ have index
$N(x,\zeta)$ belonging to  $[-N,N]$ due to our assumption $d_{\rm
min} \geq 1$.

 We claim that, if $\o \in \cC$, then the configuration
$\xi^{\zeta }(t)[\omega] $ inside $(-N,N)$ is univocally determined
knowing $ \cT ^{(j)}$, $ \bar{\cT}^{(j)}$, $ \tilde{\cT}^{(j)}$,
$(U^{(j)}_m)_{m \geq 0}$, $(\bar{ U}^{(j)}_m)_{m \geq 0}$,
$(\tilde{U}^{(j)}_m)_{m \geq 0}$ for $j \in [-\gamma N, \gamma N]$.
 In
order to explain this, suppose for example that $\zeta$ is unbounded
from the left and from the right. Then,   the Poisson processes
associated to the domains $[x,y]$ and $[y,z]$ do not have any time
inside $[0,t]$, where $N(x,\zeta)=k-1$, $N(y,\zeta)=k$. In
particular, both these domains can be incorporated but  cannot
incorporate other domains. This implies that the point $y$ survives
for all times in  $[0,t]$. Hence, whatever has happened on the right
of $y$ up to time $t$ has  not influenced the dynamics on the left
of $y$. The same argument holds observing the domains $[x',y']$ and
$[y',z']$ with $N(x',\zeta)=-k'$, $N(y',\zeta)=-(k'-1)$. If $\zeta$
is bounded from the left of from the right, the proof of our claim
is even simpler.

Due to the above claim, for each $\zeta$ it holds
$$
\xi^{\zeta }(t)[\omega] \cap (-N,N)=\xi^{\bar \zeta }(t)[\omega]
\cap (-N,N)
$$
 if $\o \in \cC$ and if $ \bar \zeta$ is the configuration obtained
 from $\zeta$ by removing all the points $x \in\zeta$ with $|N(x,
 \zeta)|> \gamma N$.
 Recalling that $f$ has support in $(-N,N)$, we have
\begin{align}\label{cantare}
  | P_t f (\zeta)-P_t f ( \bar \zeta) |
  & =
  \Big|\int P(d\o) f\Big(\xi^{\zeta
 }(t)[\omega]\cap (-N,N)\Big) \nonumber \\
 & \quad - \int P(d\o) f\Big(\xi^{\bar\zeta
 }(t)[\omega]\cap (-N,N)\Big)\Big| %\nonumber \\
  \leq 2 P(\cC^c) \|f\|\leq
 2  \|f\| \varepsilon.
 \end{align}
Fixed now $\zeta$. Let us suppose for simplicity that $\zeta $ is
unbounded from the left and from the right (the other cases can be
treated similarly). Then $\bar \zeta$ contains all the points $x\in
\zeta$ with index $N(x,\zeta)\in [-\g N,\g N]$. We have $\bar \zeta
= \zeta\cap(-a,b)$ for suitable $a,b>0$.  Due to Lemma
\ref{gnocchi},  one can prove  that there exists $\d>0$ (smaller
than $\delta_0$, defined in \eqref{fuoriberlusconi}) such that if
$\eta \in \cN (d_{\rm min} )$ and $m(\zeta,\eta) \leq \d$ then $\eta
\cap (-a,b) $ has the same cardinality of $\zeta \cap (-a, b)$. In
particular, $\eta \cap (-a,b)$ is given by all the points $x$ of
$\eta$ with index $N(x, \eta) \in [-\g N,\g N]$. This implies that
for all $\eta \in \cN ( d_{\rm min})$ such that $m( \zeta,\eta) \leq
\delta$ it holds $ \bar \eta= \eta \cap(-a,b) $. Fix $\delta_1
>0$. Taking $\delta$ smaller if necessary, we can assume that
if $m(\zeta,\eta) \leq \d$ then  any two  points  $x\in \zeta$ and
$x' \in \eta$ with $N(x,\zeta)=N(x',\eta)$ satisfy $|x-x'|\leq
\d_1$.

Since \eqref{cantare} has been obtained for any configuration in $
\cN (d _{\rm min} )$, we conclude that
\begin{equation}\label{incendio}
| P_t f (\zeta)-P_t f ( \eta ) |\leq 2 \|f\| \varepsilon + |P_t f (
\zeta \cap(-a,b) ) -P_t f(\eta \cap(-a,b) |\qquad \forall
\eta:m(\zeta, \eta ) \leq \d .
\end{equation}
Hence, in order to prove that $\zeta \mapsto P_t f (\zeta)$ is
continuous, it remains to prove that $|P_t f ( \zeta \cap(-a,b) )
-P_t f(\eta \cap(-a,b) |$ is small with $\e$. Fix an integer $L$
that will be chosen later and $\eta$ so that $m(\zeta, \eta) <
\delta$. Then we decompose the expectation according to the event
that the total (random) number $X$ of clock rings
 inside
$(-a,b)$, up to time $t$, is smaller or larger than $L$. Namely
\begin{align}\label{corrovia}
& |P_t f ( \zeta \cap(-a,b) ) -P_t f(\eta \cap(-a,b) | \nonumber \\
& \quad \leq
|\mathbb{E} (f(\xi^{\zeta \cap(-a,b)}(t)) \mathds{1}_{X \leq L}) -
\mathbb{E} (f(\xi^{\eta \cap(-a,b)}(t)) \mathds{1}_{X \leq L}) | + 2 \|f \| P(X \geq L)
\end{align}
where  $X$ is the cardinality of the set $$ \Big \{ s \in[0,t]\,:\,
s\in \cT^{(k)} \cup \bar \cT^{(k)} \cup \tilde\cT ^{(k)} \text{ for
some }  k \in \bbZ \cap [-\g N, \g N] \Big\}
$$
 Let $t_1< \dots<t_X $ be clock  rings in the above set.
 Consider the first ring $t_1$. Either
this ring is legal/not legal (see \eqref{eq:uniformbis},
\eqref{eq:uniform2bis}, \eqref{eq:uniform3bis}) for both processes
({\it i.e.}\ the dynamics starting from $\z \cap (-a,b)$ and  the dynamics
starting from $\eta\cap(-a,b)$), or it is legal for one process and
not legal for the other one. In the first case one easily sees that
$m(\xi^{\zeta \cap(-a,b)}(t_1) \cap (-N,N)),\xi^{\eta
\cap(-a,b)}(t_1) \cap (-N,N))<\delta$ (and thus $m(\xi^{\zeta
\cap(-a,b)}(s) \cap (-N,N)),\xi^{\eta \cap(-a,b)}(s) \cap
(-N,N))<\delta$ for any $s \in [0,t_2)$). The second case takes
place with probability bounded by
$$
c(\d_1):=\sup _{ i=a,\ell,r} \sup _{d,d'\geq 0: |d-d'| \leq 2\d_1}
\frac{ |\l_i(d)-\l_i(d') |}{\|\l \|}
$$
By assumption, the jump rates $\l_a, \l_\ell,\l_r$ are continuous functions with support in
$[0,d_{\rm max}]$, hence they are uniformly continuous and thus $\lim _{\d_1\downarrow 0} c(\d_1)=0$.
Iterating the above  argument, we end up with
\begin{align}\label{scotta}
& | \mathbb{E} (f(\xi^{\zeta \cap(-a,b)}(t)) \mathds{1}_{X \leq L})
-
\mathbb{E} (f(\xi^{\eta \cap(-a,b)}(t)) \mathds{1}_{X \leq L}) | \leq Lc( \delta_1) \nonumber \\
&  +   \mathbb{E}\left( |f(\xi^{\zeta \cap(-a,b)}(t) ) -
f_N(\xi^{\eta \cap(-a,b)}(t) )|
\mathds{1}_{m(\xi^{\zeta \cap(-a,b)}(t) \cap (-N,N)),\xi^{\eta \cap(-a,b)}(t) \cap (-N,N))<\delta} \right) \nonumber \\
& \phantom{AAAAAAAAAAAAAAAAAAAAAAAAAAAAai} \leq L c(\delta_1) + \varepsilon
\end{align}
where in the last line we used \eqref{fuoriberlusconi} (together
with the fact that $\delta < \delta_0$).

In remains to estimate the deviation $P(X \geq L) $ with $X$ a
Poisson variable of mean $3tM$, where $M$ is the cardinality of $
[-\g N,\g N] \cap \bbZ$. Since $E( e^{ X}) = \exp\bigl\{ (e-1)
3tM\bigr\}$,  setting $L= \k t M$ by Chebyshev inequality we get
\begin{equation} \label{sole}
 P(X \geq \k t M ) \leq \exp \bigl\{ 3tM (e-1)- \k t M
\bigr\} \leq e^{-\k t M/2 } \end{equation}  for $\k \geq\k_0$.
 Summing up the  above estimates (see \eqref{incendio}, \eqref{corrovia}, \eqref{scotta}, \eqref{sole})
 we finally
get the following.    Fixed $\d_1>0$ and $ \k>\k_0$, for $\delta$
small enough  the bound  $m(\zeta,\eta)<\delta$ implies
$$
|P_tf(\zeta)-P_tf(\eta)| \leq 2\|f\| \e +
 \k t M  c(\d_1)+ \e  +
\|f \| e^{-\k t M/2} .
$$
Choosing $\k$ large enough, and then $\delta_1$ small enough amounts
to the desired result.
\end{proof}







\subsection{Proof of Theorem \ref{amico_marco_zero}}

By definition, the Markov generator $\cL:
 \bbB \supset \cD(\cL) \to \bbB$, associated to the Markov semigroup
$\{ P_t: t \geq 0\}$ acting on the space $\bbB$, has domain
$\cD(\cL)$ given by
$$
\cD (\cL):= \left\{ f\in \bbB\,:\, \lim _{t\downarrow 0}\frac{ P_t f
- f}{t} \text{ exists in $\bbB$} \right\} .
$$
Moreover, given $f \in \cD (\cL)$, one sets $\cL f := \lim
_{t\downarrow 0}\frac{ P_t f - f}{t}$.
 We stress that the above limits are
thought w.r.t.\ the uniform norm. In addition, we recall that the
space $\bbB$ depends on the parameter $d_{\rm min}$, although
omitted. Note that, when speaking of Markov generators, we do not
follow the definition given in \cite[Ch.\ 1]{L} (even if, invoking
the Hille--Yosida Theorem, the two definitions coincide).

Our aim is to prove the following theorem, which corresponds to
Theorem \ref{amico_marco}:
\begin{Theorem}\label{amico_marco}
The subspaces $\bbB_{\rm loc}$ and $\bbD$ are a core of the Markov
generator $\cL$, {\it i.e.}\    $\cL$ is the closure of the operator
 $\bbL : \bbD \ni f \mapsto \bbL f\in \bbB$, and of its restriction  to $\bbB_{\rm loc}$. Moreover, if $f \in
 \bbD$, $\cL f(\xi)$ equals the absolutely convergent series in the r.h.s.\ of \eqref{santi}.
\end{Theorem}

We need some preparation.
 Our first target is to prove that the
image of $\mathds{1} -\l \bbL$ (where $\mathds{1}$ is the identity
operator) is dense in $\bbB$ for $\l$ sufficiently small.
% and that $P_t f \in \bbD$ for all $f \in \bbD$ and $t\geq 0$.
To this aim, we follow a strategy similar to the one adopted for
particle systems in \cite[Ch.\ 1]{L}.
%Indeed, we first start with
%some key estimate (compare with \cite[Ch.\ 1, Lemma 3.4]{L}:
Set $\| c\|_\infty :=\sup _{r \in \cR } \|c_r \|_\infty$ and note that, by boundedness of the rates, $\| c\|_\infty < \infty$.




\begin{Lemma}\label{pixar:deigeni}
 Suppose that $f \in \bbD$ and $f-\l \bbL f = g$ for some $\l \geq
0$. Then for any $r \in\cR$ it holds
\begin{equation}\label{wall-e}
\D _f (r) \leq \D_g (r) +\l  \sum _{ r'\in \cR,\, r' \not = r }
\g(r,r')  \D _f (r')
\end{equation}
where $\g(r,r') :=\sup_{\xi \in \cN(d_{\rm min})}
\bigl|c_{r'}(\xi^r)-c_{r'}(\xi)\bigr|$.
\end{Lemma}


\begin{proof}
  Fix  $\e>0$ and a finite subset $\hat \cR \subset \cR$. Take $\xi
\in \cN (d _{\rm min})$ such that $ \D _f (r)\leq \e+ |\nabla_r
f(\xi) |$.  Since  the map $\xi \mapsto f(\xi^r)$ can be
discontinuous we cannot avoid the error $\e$ (the setting in
\cite{L} is different due to continuity).
 We first consider the case that $|\nabla_r f (\xi)|=
\nabla _r f (\xi)$. Then, it holds
\begin{equation}\label{nanomech}
\D_f (r) \leq \e +\nabla_r f(\xi)=\e+ \nabla_r g(\xi )+ \l \bbL f
(\xi^r) -\l \bbL f (\xi) \leq \e + \D_g (r)+ \l \bbL f (\xi^r) -\l
\bbL f (\xi)  .
\end{equation}
Since $\xi^r = \xi \setminus I_r$ ($I_r:=I_k \cup I_{k'} $ if
$r=(k,k')$), $c_r(\xi^r)=0$ and $\nabla_r f(\xi) \geq 0$, we have
\begin{equation}\label{rum}
\begin{split}
\bbL f ( \xi^r)-\bbL f (\xi) & = \sum _{ r'\in \cR   } \left\{
c_{r'}(\xi^r) \nabla _{r'} f\bigl( \xi ^r \bigr)-
c_{r'} (\xi)  \nabla_{r'} f (\xi) \right\}  \\
& \leq  \sum _{ r'\in \cR,\, r' \not = r  } \left\{ c_{r'}(\xi^r)
\nabla _{r'} f\bigl( \xi ^r \bigr)- c_{r'} (\xi)  \nabla_{r'} f
(\xi) \right\}
% \leq 2 \sum _{r'\in \cR,\, r' \not = r  } \| c_{r'}\|_\infty \D_f (r')
 . \end{split}
\end{equation}
%Hence, in the case $ \D _f (r)\leq \e+ \nabla_r f(\xi) $ w
By our choice of $\xi$ we can write
$$ f\bigl( (\xi^{r'})^r \bigr) - f\bigl( \xi^{r'}\bigr) \leq \D_f(r) \leq \e+
f(\xi^r) -f(\xi) ,$$ thus implying  that $\nabla _{r'} f\bigl(
\xi^r \bigr) \leq \e +\nabla_{r'} f(\xi) $. In particular, it holds
\begin{align}\label{stima1}
c_{r'}(\xi^r) \nabla _{r'} f\bigl( \xi ^r \bigr)- c_{r'} (\xi)
\nabla_{r'} f (\xi)
& \leq
\bigl[ c_{r'}(\xi^r)  - c_{r'} (\xi) \bigr]\nabla_{r'} f (\xi)+ \e\|c\|_\infty \nonumber \\
& \leq \g(r,r')\Delta_f(r') + \e \|c\|_\infty .
\end{align}
On the other hand, we have the trivial bound
\begin{equation}\label{stima2}
c_{r'}(\xi^r) \nabla _{r'} f\bigl( \xi ^r \bigr)- c_{r'} (\xi)
\nabla_{r'} f (\xi)  \leq 2\|c\|_\infty \D_f(r') .
\end{equation}
Combining \eqref{nanomech}, \eqref{rum} and using  \eqref{stima1}
for $r'\in \hat \cR$ and \eqref{stima2} for $r' \in \cR \setminus
\hat \cR$, we get
\begin{equation}\label{eve}
\D_f (r) \leq \e+\D_g (r)+ \l \sum _{r' \in \hat \cR: r'\not= r}
 \g(r,r')\Delta_f(r')
+ \l\e \|c\|_\infty |\hat \cR|
 +2 \l  \| c\|_\infty  \sum _{ r'\in
\cR\setminus \hat \cR  } \D_f (r') .
\end{equation}
It is simple to check, by similar arguments, that the above bound
\eqref{eve} holds also in  the case $|\nabla_r f (\xi)|= -\nabla _r
f (\xi)$. Note moreover that, since $f \in \bbD$, the last series in
\eqref{eve} is finite and converges to zero as $\hat \cR \nearrow
\cR$.
 Taking first the limit $\e \downarrow 0$ and then the
limit $\hat \cR \nearrow \cR$ we get the thesis.
\end{proof}

We can finally prove our first  target:


\begin{Lemma}
The image $\{ f -\l \bbL f \,:\, f \in \bbD\}$ is dense in $\bbB$ for $\l \geq 0$ small enough.
\end{Lemma}


\begin{proof}
Part of the  proof is similar to the proof of \cite[Ch.\ 1, Thm.\
3.9]{L}. We give it for completeness.  Without loss of generality,
for simplicity of notation we take $d_{\min}=1$. Consider the
operator $ \bbL_n$ defined in \eqref{ln}. As already observed in
Lemma \ref{giostra}, $\bbL_n$ is a bounded operator $\bbL_n : \bbB
\to \bbB$. It is simple to check that $\bbL_n$ is a Markov
pregenerator (see the criterion in Remark \ref{linguini}). Being
$\bbL_n$ a bounded Markov pregenerator, the image of $\mathds{1} -
\l \bbL_n$ is the entire space $\bbB$ for each $\l \geq 0$ (see
\cite[Ch.\ 1,Prop. 2.8]{L}). Hence, fixed $g \in \bbD$ we can find
$f_n \in \bbB$ such that
$$ f_n - \l \bbL_n f_n =g .$$
Take $ s \in (n , n+1)$. Fix $r \in \cR$. If
$\nabla_r f_n(\xi)\geq 0$ we can bound
\begin{equation*}%\label{rumbis}
L_s  f_n ( \xi^r)-L_s f_n (\xi)  \leq   {\sum}_{\genfrac{}{}{0pt}{}{r'\in \cR: r' \not = r,}
{{\rm supp}(r')\subset[-n-1,n+1) } }
\cU\Big( c_{r'}(\xi^r) \nabla _{r'} f_n\bigl( \xi ^r \bigr)- c_{r'}
(\xi) \nabla_{r'} f_n (\xi) \Big)
\end{equation*}
where $\cU(x)=x \mathds{1}_{\{x \geq 0\}}$. Hence, averaging over $s$,
the same estimate holds for $\bbL_n$ instead of $L_s$. Using this
observation and the same arguments used in the proof of Lemma
\ref{pixar:deigeni}, we get
\begin{equation}\label{wall-eeee}
\D _{f_n} (r) \leq \D_g (r) + \l  {\sum}_{ \substack{
r'\in \cR: r' \not = r, \\
 {\rm supp}(r')\subset[-n-1,n+1)
 }} \g(r,r') \D _{f_n} (r') .
\end{equation}
 Introduce now the bounded operator  $\G: \ell_1 ( \cR) \to \ell_1 (\cR)$
 as
$$(\G \underline x)(r)= \sum _{ r'\in
\cR:    r' \not = r } \g(r,r') \underline x (r') , \qquad
\underline x \in \ell_1 (\cR) .
$$
The operator is bounded since $\g(r,r') $ is bounded by
$\|c\|_\infty$ and is zero if the supports of $r$ and $r'$ are at
distance larger than a suitable constant depending on $d_{\rm min }$
and $d_{\rm max}$ only (recall that that rates $\l_\ell,\l_r,\l_a$
are zero when evaluated at $d \geq d_{\rm max}$).
 Then, the bound \eqref{wall-eeee}
implies that $ [\mathds{1}-   \l
 \G] \D _{f_n} \leq  \D_{g}$. If $\l$ is small enough, the
 operator $\mathds{1}-  \l \G$ can be inverted and therefore
we get
 \begin{equation}\label{capitano}
\D_{f_n} \leq [\mathds{1}-  \l \G]^{-1} \D _g .
 \end{equation}
Let us define $g_n := f_n - \l \bbL f_n$. Then \begin{equation*}
\begin{split}\|g- g_n\| =
\l \|( \bbL - \bbL_n)f_n \| & \leq {\sum}_{\substack{ r \in \cR\,:\\
{\rm supp}(r) \not \subset (-n,n)}} \| c_r\|_\infty \D_{f_n} (r)
\\ &\leq \|c\|_\infty {\sum}_{\substack{ r \in \cR\,:\\ {\rm supp}( r)
\not \subset (-n,n)}} [\mathds{1}-  \l \G]^{-1} \D _g (r) .
\end{split}
\end{equation*}
Since $[\mathds{1}- \l \G]^{-1} \D _g \in \ell_1 (\cR)$, the above
bound implies that $\lim_{n \to \infty} \|g- g_n \|=0$. Recalling
that $g \in \bbD$ and that  $g_n$ belongs to the image of
$\mathds{1}- \l \bbL$,  we conclude that the image of this last
operator is dense in $\bbD$ and therefore  in $\bbB$.
\end{proof}



%



%We need some preparation.
% Our first target is to prove that the
%image\marginpar{\club ho cambiato "imagine" in "image"} of $\mathds{1} -\l \bbL$ (where $\mathds{1}$ is the identity
%operator) is dense in $\bbB$ for $\l$ sufficiently small.
%% and that $P_t f \in \bbD$ for all $f \in \bbD$ and $t\geq 0$.
%To this aim, we follow a strategy similar to the one adopted for
%particle systems in \cite[Ch.\ 1]{L}.
%%Indeed, we first start with
%%some key estimate (compare with \cite[Ch.\ 1, Lemma 3.4]{L}:
%Set $\| c\|_\infty :=\sup _{r \in \cR } \|c_r \|_\infty$ and note that, by boundedness of the rates, $\| c\|_\infty < \infty$.
%
%
%\begin{Lemma}\label{pixar:deigeni}
%Suppose that $f \in \bbD$ and $f-\l \bbL f = g$ for some $\l \geq
%0$. Then for any $r \in\cR$ it holds
%\begin{equation}\label{wall-e}
%\D _f (r) \leq \D_g (r) + 2\l \| c \|_\infty \sum _{ r'\in \cR,\, r'
%\not = r  } \D _f (r') \,.
%\end{equation}
%\end{Lemma}
%
%
%\begin{proof}
%Given $\e>0$, take $\xi \in \cN (d _{\rm min})$ such that $ \D _f
%(r)\leq \e+ |\nabla_r f(\xi) |$.  We first consider the case that
%$|\nabla_r f (\xi)|= \nabla _r f (\xi)$. Then, it holds
%$$
%\D_f (r) \leq \e +\nabla_r f(\xi)=\e+ \nabla_r g(\xi )+ \l \bbL f
%(\xi^r) -\l \bbL f (\xi) \leq \e + \D_g (r)+ \l \bbL f (\xi^r) -\l
%\bbL f (\xi) .
%$$
%Since $\xi^r = \xi \setminus I_r$, $c_r(\xi^r)=0$ and $\nabla_r
%f(\xi) \geq 0$, it holds
%\begin{align}\label{rum}
%\bbL f ( \xi^r)-\bbL f (\xi)
%& =
%\sum _{ r'\in \cR,\, r' \not = r  }
%\left\{ c_{r'}(\xi^r) \nabla _{r'} f\bigl( \xi ^r \bigr)-
%c_{r'} (\xi)  \nabla_{r'} f (\xi) \right\} - c_r (\xi) \nabla _r f (\xi) \nonumber \\
%& \leq
%2 \sum _{r'\in \cR,\, r' \not = r  } \| c_{r'}\|_\infty \D_f
%(r') .
%\end{align}
%%Hence, in the case $ \D _f (r)\leq \e+ \nabla_r f(\xi) $ w
%We conclude that
%\begin{equation}\label{eve}
%\D_f (r) \leq \e+\D_g (r)+ 2 \l  \| c\|_\infty  \sum _{ r'\in \cR,\,
%r' \not = r  }  \D_f (r')\,.
%\end{equation}
%It is simple to check, by similar arguments, that the above bound
%\eqref{eve} holds also in  the case $|\nabla_r f (\xi)|= -\nabla _r
%f (\xi)$. By the arbitrariness of $\e$ we get the thesis.
%\end{proof}
%
%
%We can finally prove our first  target:


%\begin{Lemma}
%The image $\{ f -\l \bbL f \,:\, f \in \bbD\}$ is dense in $\bbB$ for $\l \geq 0$ small enough.
%\end{Lemma}
%
%
%\begin{proof}
%The proof is similar to the one presented for particle systems in
%the proof of \cite[Ch.\ 1, Thm.\ 3.9]{L}. We give it for
%completeness.  Consider the operator $ \bbL_n$ defined in
%\eqref{ln}. As already observed in Lemma \ref{giostra}, $\bbL_n$ is
%a bounded operator $\bbL_n : \bbB \to \bbB$. It is simple to check
%that $\bbL_n$ is a Markov pregenerator (see the criterion in Remark
%\ref{linguini}). Being $\bbL_n$ a bounded Markov pregenerator, the
%image of $\mathds{1} - \l \bbL_n$ is the entire space $\bbB$ for
%each $\l \geq 0$ (see \cite[Ch.\ 1,Prop. 2.8]{L}). Hence, fixed $g
%\in \bbD$ we can find $f_n \in \bbB$ such that
%$$ f_n - \l \bbL_n f_n =g\,.$$
%Take $ s \in (n d_{\rm min}, (n+1)d_{\rm min})$. Fix $r \in \cR$.
%Given $\e>0$, take $\xi \in \cN (d _{\rm min})$ such that $ \D
%_{f_n} (r)\leq \e+ |\nabla_r f_n(\xi) |$.  We   consider the case
%that $|\nabla_r f_n (\xi)|= \nabla _r f_n (\xi)$ (the other case can
%be treated similarly).
% Since $\xi^r = \xi \setminus I_r$, $c_r(\xi^r)=0$ and $\nabla_r
%f(\xi) \geq 0$, we have
%\begin{equation}\label{rumbis}
%\begin{split}
%L_s  f_n ( \xi^r)-L_s f_n (\xi) & \leq   {\sum}_{ \substack{
%r'\in \cR: r' \not = r,\;r' \sim r\\
% {\rm supp}(r')\subset[-n-1,n+1)
% }}
%\left| c_{r'}(\xi^r) \nabla _{r'} f_n\bigl( \xi ^r \bigr)-
%c_{r'} (\xi)  \nabla_{r'} f_n (\xi) \right| - c_r (\xi) \nabla _r f _n(\xi) \\
%& \leq 2  {\sum}_{ \substack{
%r'\in \cR: r' \not = r,\;r' \sim r\\
% {\rm supp}(r')\subset[-n-1,n+1)
% }} \| c_{r'}\|_\infty \D_{f_n} (r') \,,
%\end{split}
%\end{equation}
%where we write $r' \sim r$ if the Euclidean distance between the
%supports of $r$ and $r'$ is bounded by $ 100 d_{\rm max}/d_{\rm min}
%$. This last constraint comes from the fact that $c_{r'}
%(\xi)=c_{r'}(\xi^r)$ if the supports of $r,r'$ are far enough  from
%each other since $\l_\ell, \l_r, \l_a$ are zero on $[d_{\rm max},
%\infty)$\marginpar{ho cambiato $[d_{\rm max},0)$ in $[d_{\rm max}, \infty)$}. Taking the uniform average over $ s \in (n d_{\rm min},
%(n+1)d_{\rm min})$ we conclude that the above bound \eqref{rumbis}
%holds also with $L_s$ replaced by $\bbL_n$.
%
% Repeating the arguments in the proof of Lemma
%\ref{pixar:deigeni}, we then get
%\begin{equation}\label{terra!!!}
%\D_{f_n} (r) \leq \D_{g} (r)+ 2\l \| c \|_\infty {\sum}_{ \substack{
%r'\in \cR: r' \not = r,\;r' \sim r\\
% {\rm supp}(r')\subset[-n-1,n+1)
% }} \D _{f_n} (r')\,, \qquad r
%\in \cR \,,
%\end{equation}
%%where we write $r' \sim r$ if the Euclidean distance between the
%%supports of $r$ and $r'$ is bounded by $ 100 d_{\rm max}/d_{\rm min}
%%$. The last constraint follows from the fact that $c_{r'}
%%(\xi)=c_{r'}(\xi^r)$ if the supports of $r,r'$ are far enough  from
%%each other since $\l_\ell, \l_r, \l_a$ are zero on $[d_{\rm max},
%%0)$.
%\marginpar{ho levato tre righe che erano proprio uguale a quelle sopra}
% Since $g\in \bbD$ and $f_n \in \bbB$, the above bound \eqref{terra!!!} implies that $f_n\in \bbD$.
%
% Introduce now the bounded operator  $\G: \ell_1 ( \cR) \to \ell_1 (\cR)$
% as
%$$(\G \underline x)(r)=
%  2 \| c \|_\infty \sum _{\substack{ r'\in
%\cR:\\   r' \not = r,\, r' \sim r}  } \underline x (r')\,, \qquad
%\underline x \in \ell_1 (\cR)\,.
%$$
%Then, the bound \eqref{terra!!!} implies that  $ [\mathds{1}- 2 \l
% \G] \D _{f_n} \leq  \D_{g}$. If $\l$ is small enough, the
% operator $\mathds{1}- 2 \l \G$ can be inverted and therefore
%we get
% \begin{equation}\label{capitano}
%\D_{f_n} \leq [\mathds{1}- 2 \l \G]^{-1} \D _g\,.
% \end{equation}
%Let us define $g_n := f_n - \l \bbL f_n$. Then \begin{equation*}
%\begin{split}\|g- g_n\| =
%\l \|( \bbL - \bbL_n)f_n \| & \leq \sum _{\substack{ r \in \cR\,:\\
%{\rm supp}(r) \not \subset [-n-1,n+1)}} \| c_r\|_\infty \D_{f_n} (r)
%\\ &\leq \|c\|_\infty \sum _{\substack{ r \in \cR\,:\\ {\rm supp}( r)
%\not \subset [-n-1,n+1)}} [\mathds{1}- 2 \l \G]^{-1} \D _g (r)\,.
%\end{split}
%\end{equation*}
%Since $[\mathds{1}- 2 \l \G]^{-1} \D _g \in \ell_1 (\cR)$, the above
%bound implies that $\lim_{n \to \infty} \|g- g_n \|=0$. Recalling
%that $g \in \bbD$ and that  $g_n$ belongs to the image of
%$\mathds{1}- \l \bbL$,  we conclude that the image of this last
%operator is dense in $\bbD$ and therefore  in $\bbB$.
%\end{proof}


As a consequence of the above result and Remark \ref{linguini},  we
get that the closure $\bar \bbL$  of $\bbL$ is a Markov generator in
the sense of \cite[Ch. 1, Def.  2.7]{L} (briefly, we will say that
$\bbL$  is a L--Markov generator).


\begin{Lemma}\label{cavallino}
If $f\in \bbD$, then there exists a sequence $f_n\in \bbB_{\rm loc}$
such that
% setting $f_n (\xi):= f(\xi \cap (-n d_{\rm min},n
%d_{\rm min}))$, then $f_n \in \bbB_{\rm loc}$,
$f_n \to f$  and
$\bbL f_n \to \bbL f$ in $\bbB$.
\end{Lemma}





\begin{proof}
  Given $n$ set $f_n(\xi):= \int_n^{n+1} f( \xi \cap (-s,s)) ds$.
Due to Lemma \ref{lemino}, we know that $\|f-f_n\|\to 0$ and $f_n
\in \bbB_{\rm loc}$. Let us prove that $\| \bbL f_n-\bbL f \|\to 0$.
To this aim, setting $\xi_s:= \xi \cap (-s,s)$ and observing that
$(\xi_s)^r= (\xi^r)_s$ for all $r \in \cR$,  for any integer $N$ we
can write
\begin{align}
\Big| \bbL  f(\xi)-\bbL f_n (\xi)\Big| &=
\Big| \int_n^{n+1} \sum _{r\in \cR} c_r(\xi) \bigl( \nabla_r f(\xi) - \nabla _r f (\xi_s)) ds \Big| \nonumber\\
&  \leq \Big| \int_n^{n+1} \!\!\!\!\!\!
\sum_{\genfrac{}{}{0pt}{}{r\in \cR: }{{\rm supp}(r)\not \subset
[-N,N]}} \!\!\!\!\!\! c_r(\xi) \bigl( \nabla_r f(\xi) - \nabla _r f
(\xi_s)\bigr) ds\Big|
\nonumber \\
& \quad  + \Big| \int_n^{n+1} \!\!\!\!\!\!
\sum_{\genfrac{}{}{0pt}{}{r\in \cR: }{{\rm supp}(r) \subset [-N,N]}}
\!\!\!\!\!\!  c_r(\xi)
\bigl( \nabla_r f(\xi) - \nabla _r f (\xi_s)\bigr)ds \Big|\nonumber \\
& \leq 2\|c\|_\infty \!\!\!\!\!\! \sum_{\genfrac{}{}{0pt}{}{r\in
\cR: }{{\rm supp}(r)\not \subset [-N,N]}}
\!\!\!\!\!\! \Delta_f(r) \label{mentana1} \\
& \quad + 2 \|c\|_\infty |\{r\in \cR :{\rm supp}(r) \subset
[-N,N]\}| \cdot\|f-f_n\|\,.\label{mentana2}
\end{align}
Given $\varepsilon>0$ we choose  $N$ large enough that
\eqref{mentana1}  is smaller than $\varepsilon$ (this is possible
since $f\in \bbD$).
%$\sum _{r\in \cR: {\rm supp}(r)\not \subset [-N,N]}\Delta_f(r)\leq
%\varepsilon$.
Afterwards, for $n$ large enough   \eqref{mentana2}  is smaller than
$\varepsilon$ (recall that $f_n \to f$ in $\bbB$). Then we conclude
that $\|\bbL f - \bbL f_n\|\leq 2 \varepsilon$ for $n$ large enough.
\end{proof}


 We can finally prove Theorem \ref{amico_marco}.

\begin{proof}[Proof of Theorem \ref{amico_marco}]
In Proposition  \ref{note} we have already  showed that $\cL f=\bbL
f $   if $f \in
 \bbB_{\rm loc}$. As observed after Lemma \ref{giostra}, in this
 case $\cL f$ must equal \eqref{santi}.
By Lemma \ref{cavallino}, $\bar \bbL$ is the closure of the
restriction of $\bbL$ to $\bbB_{\rm loc}$. Hence, $\bbB_{\rm loc}$
is a core of $\bar \bbL$. By Lemma \ref{giostra} $(i)$, given $f\in
\bbD$ the value $\bbL f (\xi)$ equals the r.h.s. of
 \eqref{santi} which is an absolutely convergent series.




It remains to prove that $\bar \bbL=\cL$. Since $\bbL f=\cL f$ for
all $f \in \bbB_{\rm loc}$, Lemma \ref{cavallino} and the closure of
$\cL$ implies that $f\in \cD(\cL)$ and $\bbL f=\cL f $ for all $f
\in \bbD$ (the fact that $\cL$ is close is a standard fact: combine
Def. 2.1 in \cite[Ch. 1]{L} with the Hille--Yosida Theorem as stated
in Thm. 2.9 in \cite[Ch. 1]{L} leading to the fact that $\cL$ is an
L--Markov generator, and therefore close). This observation implies
that $\cL$ is an extension of $\bar \bbL$. It is a general fact that
this implies that $\cL= \bar \bbL$ (cf. \cite[Prop. 3.13]{S}
together with  the Hille--Yosida Theorem as stated in Thm. 2.9 in
\cite[Ch. 1]{L}).


\end{proof}



%
%
%. It is a general fact that any Markov pregenerator is a
%closable operator, which is again a Markov pregenerator (cf.
%\cite[Ch.\ 1, Prop. 2.5]{L}). We write $\bar{\bbL}:\bbB\supset \cD
%(\bar \bbL)\to \bbB$ for the closure of $\bbL$.
%
%






\subsection*{Acknowledgements}
We thank F. Martinelli for useful discussions. We  thank   the
Laboratoire de Probabilit\'{e}s et Mod\`{e}les Al\'{e}atoires of the
University Paris VII and the Department  of Mathematics of the
University of Roma Tre for the support and the kind hospitality. C.
Toninelli acknowledges the support of the French Ministry of
Education through the  grant
 ANR-2010-BLAN-0108.





\begin{thebibliography}{2}

\bibitem[A]{A} D.\ J.\ Aldous, {\em
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%\bibitem[BCC]{BCC}
% L.\ Bertini, N.\ Cancrini, F.\ Cesi, {\em The spectral gap for a
%Glauber--type dynamics in a continuous gas}. Ann. Inst. H.
%Poincar\'{e} Probab. Statist. {\bf38} 91--108 (2002).

\bibitem[Be1]{Be1} J.\ Bertoin, {\em
Random fragmentation and coagulation processes}.  Cambridge studies
in advanced mathematics {\bf 102},
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\bibitem[Be2]{Be2} J.\ Bertoin, {\em Eternal additive coalescents and certain bridges
with exchangeable increments}.  Annals of Probability {\bf 29}, No
1, 344--360 (2001).



\bibitem[B]{B} P. Billingsley, {\sl Convergence of probability measures}.
 Wiley series in probability and mathematical
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\bibitem[BDG]{BDG} A.J.\ Bray, B.\ Derrida, C.\ Gordr\`{e}che, {\em
Non--trivial algebraic decay in a soluble model of coarsening}.
Europhys. Lett. {\bf 27 } (3), 175--180 (1994).

%\bibitem[CKLS]{CKLS} \club \club \club E. Caglioti, K.Kondratiev, J. L. Lebowitz, E. R. Speer, {\sl
%Point Processes with Specified Low Order Correlations},  Markov Process. Related Fields {\bf 12}, No. 2, 257-272 (2006).

\bibitem[CP]{CP} J. Carr, R. Pego, {\em Proc.R.Soc.Lond} A {\bf 436}, 569--583 (1992).

\bibitem[DBG]{DBG} B.\ Derrida, A.J.\ Bray, C.\ Godr\`{e}che, {\em Non-trivial exponents in the
zero temperature dynamics of the 1d Ising and Potts model}.  J.
Phys.  A {\bf 27}, L357-L361 (1994).


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 (1990).

 \bibitem[DGY2]{DGY2} B.\ Derrida, C.\ Godr\`{e}che, I.1 Yekutieli, {\em
 Scale invariant regime in the one dimensional models of growing and coalescing droplets}.  Phys. Rev. A {\bf 44},
  6241--6251 (1991).




%\bibitem[D]{D} R. Durrett, {\em Probability: theory and examples}.
%Second edition, Duxbury Press (1996).

% {\em Lecture notes on particle systems
%and percolation}, in {\em Ecole d'et\'{e} de probabilit\'{e}s de
%Saint-Flour XXIII, 1993}.     Lecture Notes in Mathematics 1608,
%Springer Verlag, Berlin  (1995).


\bibitem[DV]{DV} D.J.\ Daley, D.\ Vere--Jones, {\em An introduction to the theory
of point processes.} Springer Verlag, New York (1988).

\bibitem[EJ]{EJ} S.~Eisinger, J.~Jackle,  {\em A
   hierarchically constrained kinetic ising model.} Z. Phys. B {\bf
    84}, 115--124, (1991)

\bibitem[FMRT0]{FMRT0} A. Faggionato, F. Martinelli, C. Roberto, C.
Toninelli, {\em Universality in one
  dimensional hierarchical
coalescence processes}. Ann. Probab. To appear.


\bibitem[FMRT1]{FMRT1} A. Faggionato, F. Martinelli, C. Roberto, C.
Toninelli, {\em Aging through hierarchical coalescence in the East
Model}. Commun. Math. Phys. To appear.


\bibitem[F]{F}  W. Feller, {\sl
An  introduction to probability theory and its applications}. Vol.2.
Second edition.  Wiley series in probability and mathematical
statistics.  Wiley, New York (1971).

%\bibitem[H]{H} P.R. Halmos, {\sl Measure Theory}. Graduate Text in
%Mathematics, Springer Verlag, Berlin (1974).


\bibitem[FKAS]{FKAS} P.\ Franken, D.\  K\"{o}nig, U.\  Arndt, V.\ Schmidt, {\sl  Queues
and Point Processes}.  John Wiley and Sons, Chichester (1982).


\bibitem[GM]{GM} T.~Gallay, A.~Mielke,{\em Convergence results for a coarsening model using global
              linearization}. J. Nonlinear Sci., {\bf 13}, 311--346  (2003).


\bibitem[H]{H} P.R.\ Halmos, {\sl Measure theory}. New York,
Springer--Verlag (1974).


%\bibitem[K]{K} O.\ Kallenberg, {\sl Probabilistic symmetries and invariance
%principles}. New York, Springer--Verlag (2005).


\bibitem[L]{L} T.M.\ Liggett, {\sl Interacting particle systems}.
Grundlehren der mathematischen Wissenschaften {\bf 276}, Springer
Verlag, Berlin (2005).

%\bibitem[LS]{LS} Limic, V. and Sturm, A., \emph{The spatial $\Lambda$-coalescent}, Electr. J. Probab. {\bf 11}  (2006).

\bibitem[Pr]{Pr}  C. Preston, \emph{ Spatial birth-and-death processes}. Proceedings of the
40th Session of the International Statistical Institute (Warsaw,
1975), Bull. Inst. Internat. Statist. {\bf 46}  371391 (1975).

\bibitem[P]{P} V. Privman (editor), {\sl Non equilibrium statistical physics
in one dimension}. Cambridge University Press (1997).

\bibitem[S]{S} T.\ Sepp\"{a}l\"{a}inen, {\sl Translation Invariant Exclusion
Processes}. Available online.

\bibitem[SE]{SE} P. Sollich, M.R. Evans,  {\sl Glassy
dynamics in the asymmetrically constrained kinetic Ising chain}.
Phys. Rev. E {\bf 68}, 031504 (2003).


\end{thebibliography}



\end{document}



1112.4630.extracted_bib
\begin{thebibliography}{2}

\bibitem[A]{A} D.\ J.\ Aldous, {\em
Deterministic and stochastic models for coalescence (aggregation and
coagulation): a review of the mean-field theory for probabilists}.
Bernoulli {\bf 5} (1), 3--48 (1999).

%\bibitem[BCC]{BCC}
% L.\ Bertini, N.\ Cancrini, F.\ Cesi, {\em The spectral gap for a
%Glauber--type dynamics in a continuous gas}. Ann. Inst. H.
%Poincar\'{e} Probab. Statist. {\bf38} 91--108 (2002).

\bibitem[Be1]{Be1} J.\ Bertoin, {\em
Random fragmentation and coagulation processes}.  Cambridge studies
in advanced mathematics {\bf 102},
 Cambridge University Press (2006).

\bibitem[Be2]{Be2} J.\ Bertoin, {\em Eternal additive coalescents and certain bridges
with exchangeable increments}.  Annals of Probability {\bf 29}, No
1, 344--360 (2001).



\bibitem[B]{B} P. Billingsley, {\sl Convergence of probability measures}.
 Wiley series in probability and mathematical
statistics, J. Wiley, New York (1968).



\bibitem[BDG]{BDG} A.J.\ Bray, B.\ Derrida, C.\ Gordr\`{e}che, {\em
Non--trivial algebraic decay in a soluble model of coarsening}.
Europhys. Lett. {\bf 27 } (3), 175--180 (1994).

%\bibitem[CKLS]{CKLS} \club \club \club E. Caglioti, K.Kondratiev, J. L. Lebowitz, E. R. Speer, {\sl
%Point Processes with Specified Low Order Correlations},  Markov Process. Related Fields {\bf 12}, No. 2, 257-272 (2006).

\bibitem[CP]{CP} J. Carr, R. Pego, {\em Proc.R.Soc.Lond} A {\bf 436}, 569--583 (1992).

\bibitem[DBG]{DBG} B.\ Derrida, A.J.\ Bray, C.\ Godr\`{e}che, {\em Non-trivial exponents in the
zero temperature dynamics of the 1d Ising and Potts model}.  J.
Phys.  A {\bf 27}, L357-L361 (1994).


 \bibitem[DGY1]{DGY1} B.\ Derrida, C.\ Godr\`{e}che, I.\ Yekutieli, {\em
 Stable distributions of growing and coalescing droplets}.  Europhys. Lett. {\bf 12}, 385--390
 (1990).

 \bibitem[DGY2]{DGY2} B.\ Derrida, C.\ Godr\`{e}che, I.1 Yekutieli, {\em
 Scale invariant regime in the one dimensional models of growing and coalescing droplets}.  Phys. Rev. A {\bf 44},
  6241--6251 (1991).




%\bibitem[D]{D} R. Durrett, {\em Probability: theory and examples}.
%Second edition, Duxbury Press (1996).

% {\em Lecture notes on particle systems
%and percolation}, in {\em Ecole d'et\'{e} de probabilit\'{e}s de
%Saint-Flour XXIII, 1993}.     Lecture Notes in Mathematics 1608,
%Springer Verlag, Berlin  (1995).


\bibitem[DV]{DV} D.J.\ Daley, D.\ Vere--Jones, {\em An introduction to the theory
of point processes.} Springer Verlag, New York (1988).

\bibitem[EJ]{EJ} S.~Eisinger, J.~Jackle,  {\em A
   hierarchically constrained kinetic ising model.} Z. Phys. B {\bf
    84}, 115--124, (1991)

\bibitem[FMRT0]{FMRT0} A. Faggionato, F. Martinelli, C. Roberto, C.
Toninelli, {\em Universality in one
  dimensional hierarchical
coalescence processes}. Ann. Probab. To appear.


\bibitem[FMRT1]{FMRT1} A. Faggionato, F. Martinelli, C. Roberto, C.
Toninelli, {\em Aging through hierarchical coalescence in the East
Model}. Commun. Math. Phys. To appear.


\bibitem[F]{F}  W. Feller, {\sl
An  introduction to probability theory and its applications}. Vol.2.
Second edition.  Wiley series in probability and mathematical
statistics.  Wiley, New York (1971).

%\bibitem[H]{H} P.R. Halmos, {\sl Measure Theory}. Graduate Text in
%Mathematics, Springer Verlag, Berlin (1974).


\bibitem[FKAS]{FKAS} P.\ Franken, D.\  K\"{o}nig, U.\  Arndt, V.\ Schmidt, {\sl  Queues
and Point Processes}.  John Wiley and Sons, Chichester (1982).


\bibitem[GM]{GM} T.~Gallay, A.~Mielke,{\em Convergence results for a coarsening model using global
              linearization}. J. Nonlinear Sci., {\bf 13}, 311--346  (2003).


\bibitem[H]{H} P.R.\ Halmos, {\sl Measure theory}. New York,
Springer--Verlag (1974).


%\bibitem[K]{K} O.\ Kallenberg, {\sl Probabilistic symmetries and invariance
%principles}. New York, Springer--Verlag (2005).


\bibitem[L]{L} T.M.\ Liggett, {\sl Interacting particle systems}.
Grundlehren der mathematischen Wissenschaften {\bf 276}, Springer
Verlag, Berlin (2005).

%\bibitem[LS]{LS} Limic, V. and Sturm, A., \emph{The spatial $\Lambda$-coalescent}, Electr. J. Probab. {\bf 11}  (2006).

\bibitem[Pr]{Pr}  C. Preston, \emph{ Spatial birth-and-death processes}. Proceedings of the
40th Session of the International Statistical Institute (Warsaw,
1975), Bull. Inst. Internat. Statist. {\bf 46}  371391 (1975).

\bibitem[P]{P} V. Privman (editor), {\sl Non equilibrium statistical physics
in one dimension}. Cambridge University Press (1997).

\bibitem[S]{S} T.\ Sepp\"{a}l\"{a}inen, {\sl Translation Invariant Exclusion
Processes}. Available online.

\bibitem[SE]{SE} P. Sollich, M.R. Evans,  {\sl Glassy
dynamics in the asymmetrically constrained kinetic Ising chain}.
Phys. Rev. E {\bf 68}, 031504 (2003).


\end{thebibliography}