Strong local survival of branching random walks \\ is not monotone
Daniela Bertacchi
The aim of this paper is the study of the strong local survival property for discrete-time and continuous-time branching random walks. We study this property by means of an infinite dimensional generating function $G$ and a maximum principle which, we prove, is satisfied by every fixed point of $G$. We give results about the existence of a strong local survival regime and we prove that, unlike local and global survival, in continuous time, strong local survival is not a monotone property in the general case (though it is monotone if the branching random walk is quasi transitive). We provide an example of an irreducible branching random walk where the strong local property depends on the starting site of the process. By means of other counterexamples we show that the existence of a pure global phase is not equivalent to nonamenability of the process, and that even an irreducible branching random walk with the same branching law at each site may exhibit non-strong local survival. Finally we show that the generating function of a irreducible BRW can have more than two fixed points; this disproves a previously known result. % in % \cite{cf:Spataru89}.
SLS.tex
\documentclass[10pt,reqno]{amsart}
\usepackage{amsmath, latexsym, amsfonts, amssymb,amsthm, amscd,epsfig,enumerate}
%\usepackage{showkeys}
\usepackage{subfigure,color}

\pagestyle{plain}

\advance\hoffset -.75cm

\oddsidemargin=.4cm
\evensidemargin=.4cm
\textwidth=16.5cm
\textheight=22cm


\usepackage[usesnames]{xcolor}
%\usepackage[color]{showkeys}

\definecolor{refkey}{gray}{.75}
\definecolor{labelkey}{gray}{.75}

\def\theequation{\thesection.\arabic{equation}}

%\pagewidth{35pc}

\newcommand{\Q}{\mathbb Q}
\newcommand{\C}{\mathbb C}
\newcommand{\R}{\mathbb R}
\newcommand{\Z}{\mathbb Z}
\newcommand{\N}{\mathbb N}
\newcommand{\Prob}{\mathbb P}
\newcommand{\E}{\mathbb E}
\newcommand{\Rd}{{\mathbb R}^d}
\newcommand{\Zd}{{{\mathbb Z}^d}}
\newcommand{\bPr}{\mathbf P}
\newcommand{\bE}{\mathbf E}
\newcommand{\diff}{\mathrm{d}}

\newcommand{\I}{\mathbf 1}

\newcommand{\eps}{\varepsilon}
\newcommand{\s}{\varphi}
\newcommand{\La}{\Lambda}
\newcommand{\sg}{\sigma}

\newcommand{\dd}{{\mathrm d}}

\newcommand{\cE}{\mathcal{E}}
\newcommand{\cG}{\mathcal{G}}
\newcommand{\cF}{\mathcal{F}}
\newcommand{\cS}{\mathcal{S}}
\newcommand{\cH}{\mathcal{H}}
\newcommand{\cP}{\mathcal{P}}
\newcommand{\cB}{\mathcal{B}}
\newcommand{\cK}{\mathcal{K}}
\newcommand{\cL}{\mathcal{L}}
\newcommand{\cO}{\mathcal{O}}
\newcommand{\cA}{\mathcal{A}}

\newcommand{\Pone}{\widetilde{\Prob}_N }
\newcommand{\Ptwo}{\Prob }
\newcommand{\Pthree}{\Prob _N}
\newcommand{\Eone}{\widetilde{\E}_N }
\newcommand{\Etwo}{\E }
\newcommand{\Ethree}{\E _N}
\newcommand{\pr}{\mathbb P}
\newcommand{\T}{\mathbb T}

\newcommand{\sbs}{\subset}
\newcommand{\ovl}{\overline}
\newcommand{\unl}{\underline}
\newcommand{\second}{{\prime\prime}}
\newcommand{\ra}{\rightarrow}
\newcommand{\lra}{\longrightarrow}
\newcommand{\ident}{{\mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l}
{\rm 1\mskip-4.5mu l} {\rm 1\mskip-5mu l}}}
\newcommand{\undertilde}[1]{\underset{\widetilde{}}{#1}}


% \renewcommand{\thesubfigure}{\arabic{subfigure}}
% \makeatletter
% \renewcommand{\@thesubfigure}{\tiny Figure \thesubfigure: \space}
% \renewcommand{\p@subfigure}{}
% \makeatother
% %\newcommand{\deg}{{\rm deg}}


\newtheorem{teo}{Theorem}[section]
\newtheorem{lem}[teo]{Lemma}
\newtheorem{cor}[teo]{Corollary}
\newtheorem{rem}[teo]{Remark}
\newtheorem{pro}[teo]{Proposition}
\newtheorem{defn}[teo]{Definition}
\newtheorem{exmp}[teo]{Example}
\newtheorem{nota}[teo]{Notation}
\newtheorem{assump}[teo]{Assumption}
\newtheorem{step}{Step}


\setlength{\topmargin}{-0.35in}
\setlength{\textheight}{8.5in}   % 11.0 - 1.125 - 0.875
\setlength{\textwidth}{6.0in}    %  8.5 - 1.375 - 1.125
\setlength{\oddsidemargin}{0.5in}
\setlength{\evensidemargin}{0.5in}
\setlength{\headheight}{26pt}
\setlength{\headsep}{8pt}


\title%[Recent results on branching random walks]
{Strong local survival of branching random walks \\ is not monotone}
% {Non-monotonicity of the strong local survival property of branching random walks}


\author[D.~Bertacchi]{Daniela Bertacchi}
\address{D.~Bertacchi, Dipartimento di Matematica e Applicazioni,
Universit\`a di Milano--Bicocca,
via Cozzi 53, 20125 Milano, Italy.}
\email{daniela.bertacchi\@@unimib.it}


\author[F.~Zucca]{Fabio Zucca}
\address{F.~Zucca, Dipartimento di Matematica,
Politecnico di Milano,
Piazza Leonardo da Vinci 32, 20133 Milano, Italy.}
\email{fabio.zucca\@@polimi.it}

\date{}

\begin{document}

\begin{abstract}
The aim of this paper is the study of the  strong local survival property for
 discrete-time and continuous-time branching random walks. We study this property
by means of an infinite dimensional generating function $G$ and a 
maximum principle which, we prove, is satisfied
by every fixed point of $G$. 
We give results about the existence of a strong local survival regime and 
we prove that, unlike local and global survival, in continuous time, strong local survival is not a monotone property
in the general case (though it is monotone if the branching random walk is quasi transitive). 
We provide an example of an irreducible branching random walk
where the strong local property depends on the starting site of the process.
By means of other counterexamples we show that the existence of a pure global phase is not equivalent
to nonamenability of the process, and that even an irreducible branching random walk with the same branching law
at each site may exhibit non-strong local survival. Finally we show that the generating function
of a irreducible BRW can have more than two fixed points; this disproves a previously known result.
%  in 
% \cite{cf:Spataru89}.
\end{abstract}

%\tableofcontents

\maketitle
\noindent {\bf Keywords}: branching random walk, branching process, strong local survival, recurrence, generating function,
maximum principle.

\noindent {\bf AMS subject classification}: 60J05, 60J80.


\section{Introduction}
\label{sec:intro}

% 
% Branching random walks can be considered as processes which simultaneously generalize
% the concepts of branching process and of random walk.
A branching process is a very simple population model (introduced in \cite{cf:GW1875})
where particles breed and die (independently of each other) according to some random law.
At any time, this process is completely characterized by the total number of particles alive.
Branching random walks (in short, BRWs) add space to this picture: particles live in a spatially structured environment
and the reproduction law, which may depend on the location, not only tells how many children the particle
has, but also where it places them. The state of the process, at any time, is thus described by the
collection of the numbers of particles alive at $x$, where $x$ varies among the possible sites.
% Although particles do not actually move, an observer would see a random movement of the population.
% Moreover if we identify every particle with one of its children (if there are any), then we may view
% the BRW as a system of random walkers which may disappear (i.e.~the corresponding
% particle has no children) or split into two or more independent walkers (i.e.~the corresponding
% particle has two or more children).
% 
In the literature one can find BRWs both in continuous and discrete time.
The continuous-time setting has been studied by many authors
(see \cite{cf:HuLalley, cf:Ligg1, cf:Ligg2, cf:MadrasSchi, cf:MountSchi} just to name a few) along with some
variants of the process (see \cite{cf:BL, cf:BBZ, cf:BLZ, cf:BPZ, cf:BZ3}).
% As we see in Section~\ref{subsec:continuous},
% in this case one studies a family of BRWs which depends on the choice
% of a parameter $\lambda$.
% There are two (possibly coinciding) parameters of interest:
%  $\lambda_w\le\lambda_s$. If $\lambda<\lambda_w$ there is almost sure extinction,
% if $\lambda_w<\lambda\le \lambda_s$ there is global but not local survival and
% if $\lambda_s<\lambda$ there is local and global survival
%  (see for instance \cite{cf:PemStac1, cf:Stacey03, cf:BZ, cf:BZ2}).
The discrete-time case has been initially considered as a natural
generalization of branching processes
(see \cite{cf:AthNey, cf:Big1977, cf:Big1978, cf:BigKypr97, cf:BigRah05, cf:Harris63}).
The definition of discrete-time BRW that we give in Section~\ref{subsec:discrete}
is sufficiently general to include the discrete-time counterpart that 
every continuous-time BRW admits.
Since every continuous-time BRW and its discrete-time counterpart have the same asymptotic
behavior, it suffices to provide results for the discrete-time case. On the other hand,
continuous-time examples naturally yield discrete-time ones.
Our definition also includes as particular cases: BRWs with independent diffusion
(where particles are first generated and then dispersed independently according to
a diffusion matrix $P$, see Section~\ref{subsec:discrete} and equation~\eqref{eq:particular1});
BRWs with no death (where each particle has null probability of having no children);
BRWs whose total number of particles behaves as a branching process (where the law of the number
of offspring does not depend on the site, we call these BRWs locally isomorphic to a branching process,
see Section~\ref{subsec:FBRWs}).


The basic question which arises studying the BRW is whether it survives with positive 
probability and, in this case, if it visits a given  site infinitely many times. 
% ; while the classical question for random walks
% is whether the walker returns (with positive probability or, equivalently, with probability one)
% infinitely many times to some fixed site. Transposed into BRW theory, 
The first question
asks whether there is global survival, that is, with
positive probability at any time there is someone alive \textit{somewhere}); while the second question
deals with local survival, that is, whether with
positive probability the process returns infinitely many times to some fixed sites.
It is clear that the probability of global survival is larger or equal to the probability of
local survival. If the probability of global survival is strictly larger than the one of local
survival, then the latter may be positive or null. In the first case, we say that there is non-strong local survival, in the second case there is pure global survival.
When on the contrary, the probabilities of global and local survival
are equal and strictly positive, we say that the BRW has strong local survival.
Hence, strong local survival means that the events of local and global survival coincide
(but for a null probability set) and have positive probability.

The interest on the strong local behavior is fairly recent
(see for instance \cite{cf:GMPV09, cf:Muller08-2}).
The aim of this paper is to study some properties of the strong local survival, comparing
them with the corresponding ones of local and global survival.

As in the case of branching processes, the main tool is that probabilities of extinction are fixed points of 
an infinite-dimensional generating function $G$ (see Sections~\ref{subsec:genfun} for the definition
and \ref{subsec:survivalprob} for its link to the extinction probabilities).
It is worth noting that, unlike the branching process case, it is not true that $G$ has at most
two fixed points, even in the irreducible case (where with positive probability a particle
at site $x$ can have a progenies at site $y$, for all $x$ and $y$). 
Indeed, we prove this indirectly by providing examples of irreducible BRWs which survive locally
but with a smaller probability than the one of global survival (hence non-strong locally, see 
Examples~\ref{ex:nonstronglocalFBRW} and \ref{ex:nonstronglocalFBRW2}) and directly by an explicit construction
of three fixed points for the $G$ of a certain BRW (Remark~\ref{rem:Spataru}).
By Corollary~\ref{cor:finiteBRW} we have that in the irreducible case a
sufficient condition for the existence of at most two fixed points for $G$ is the finiteness of the
set of vertices.

In the particular case where there is no branching,
one gets a random walk and the role of $G$ and its fixed points is played 
by the transition matrix and the harmonic functions, respectively. It is thus natural to look
for a maximum principle in the context of branching random walks as well (see
Proposition~\ref{pro:maximumprinciple}).
As an application, we have that in the irreducible case, pure global survival is independent
of the starting vertex. This is also true for local and global survival, but it does not hold
for strong local survival, unless that the probability of having zero children is positive for all
sites or if the BRW is quasi transitive  (see Sections~\ref{subsec:FBRWs}
and \ref{subsec:survivalprob} and 
Corollary~\ref{pro:qtransitive}). Example~\ref{exm:irreduciblenotstrongeverywhere} shows that
we may have strong local survival starting from some vertices and non-strong local survival
starting from others.
% By means of the generating function and its properties we obtain conditions for strong local
% survival (Theorems~\ref{rem:strongconditioned}, \ref{th:Fgraph}, Corollary~\ref{pro:qtransitive})
% and we provide various counterexamples.



The speed of reproduction of a continuous-time BRW is proportional to a positive parameter
$\lambda$ (see Section~\ref{subsec:continuous}). It is easily seen that the probability of local and 
global survival are nondecreasing functions with respect to $\lambda$; thus local and global
survival are monotone properties (meaning that if one of them holds for some $\lambda_0$
then it holds for all $\lambda \ge \lambda_0$) and it is possible to define the local
and global critical parameters $\lambda_s$ and $\lambda_w$ (see Section~\ref{subsec:continuous}). 
We show that monotonicity in $\lambda$ does not hold for strong local
survival and it is thus impossible in general to define a strong local critical parameter: 
for the irreducible BRW in Example~\ref{rem:nonstrongandstrong}
if $\lambda$ is small enough or large enough there is
strong local survival but in a intermediate interval for $\lambda$ there is global and local
survival with different probabilities. 
% We also show that, unlike local and global survival, even in the case of
% an irreducible BRW with null probability of having no children for a particle living at 
% $x$ (for some $x \in X$), one may have
% strong local
% survival or non-strong local survival depending on the starting vertex (see
%  Example~\ref{exm:irreduciblenotstrongeverywhere}).

Here is the outline of the paper.
 In Section~\ref{sec:basic} we give the necessary definitions and some basic facts about
discrete-time BRWs (Section~\ref{subsec:discrete}), 
continuous-time BRWs (Section~\ref{subsec:continuous}), the infinite-dimensional
generating function $G$, defined on $[0,1]^X$ associated to a BRW (Section~\ref{subsec:genfun}) and the
special class of $\mathcal F$-BRWs (Section~\ref{subsec:FBRWs}). This class contains
properly the class of BRWs on quasi-transitive graphs (which were studied in \cite{cf:Stacey03}).
We also exhibit the explicit expression of $G$ in a particular case of BRW with independent
diffusion (equation~\eqref{eq:Gcontinuous}) and of BRW with no death constructed from a BRW with death
disregarding all particles with finite progenies (equation~\eqref{eq:G-one}).
Moreover in Section~\ref{subsec:genfun} we state a maximum principle
for the solutions of the equation $G(v) \ge v$, including all fixed points of $G$
 (Proposition~\ref{pro:maximumprinciple}).


Section~\ref{sec:survival} is devoted to the study of all the types of survival.
We first recall, in Section~\ref{subsec:local}, results on local and global survival
(Theorems~\ref{th:discretesurv} and \ref{th:continuoussurv}).
In Section~\ref{subsec:survivalprob} extinction probabilities
are seen as fixed points of the generating function $G$. 
Theorem~\ref{rem:strongconditioned} gives equivalent conditions for strong
local survival, in terms of extinction probabilities, which are useful to 
prove that strong local survival is not
monotone. % (Example~\ref{rem:nonstrongandstrong}). 

From the maximum principle %(Proposition~\ref{pro:maximumprinciple}) 
we derive
Theorem~\ref{th:Fgraph}, which describes some properties of fixed points of $G$ for $\mathcal F$-BRWs,
and Corollaries~\ref{cor:finiteBRW} and \ref{pro:qtransitive}.
Corollary~\ref{pro:qtransitive} shows that for an irreducible, 
quasi-transitive BRW, there are only three possible behaviours
(independently of the starting vertex):
global extinction, pure global survival or strong local survival.
% (while Examples~\ref{ex:nonstronglocalFBRW} and
% \ref{ex:nonstronglocalFBRW2} show that this does not hold for a generic irreducible $\mathcal F$-BRW).
Thus, in this case % for a quasi-transitive, irreducible, continuous-time BRW, 
strong local survival is monotone
and the critical parameter is $\lambda_s$.
A characterization of strong local survival in terms of the existence of a solution of some inequalities
involving the generating function $G$ is given by Theorem~\ref{th:MenshikovVolkov}.

Section~\ref{sec:examples} is devoted to examples and counterexamples.
For a continuous-time irreducible $\mathcal F$-BRW,  the existence of a pure global phase
is equivalent to nonamenability (see Section~\ref{subsec:discrete} and 
Section~\ref{subsec:stronglocal}). 
% In our construction of a continuous-time BRW where the strong local survival is not
% monotone there is a deep relation with the existence of a pure global phase.
% This last property, in the case of an irreducible $\mathcal F$-BRW is equivalent
% to nonamenability. 
Nevertheless in general nonamenability neither implies nor is implied by the existence of
a pure global phase (Example~\ref{exm:amenable}). 
% Example~\ref{rem:nonstrongandstrong} shows that strong local survival is not monotone;
% Example~\ref{exm:irreduciblenotstrongeverywhere} that it may depend on the starting vertex.
% Example~\ref{exm:irreduciblenotstrongeverywhere} shows
% that even for an irreducible BRW with null probability of having no children for a particle living at 
% $x$ 
% (for some $x \in X$), it is possible to have strong local survival starting from some vertices and
% non-strong local survival starting from others. 
Finally we show (Examples~\ref{ex:nonstronglocalFBRW} and
\ref{ex:nonstronglocalFBRW2}) that even fairly simple BRWs (such as irreducible 
BRWs with independent diffusion
and with offspring distribution independent of the site) may have non-strong local survival. 
This implies that, even in the irreducible case, the generating function $G$ may have more than two fixed points
in $[0,1]^X$ and disproves a result in \cite{cf:Spataru89} (see Remark~\ref{rem:Spataru}).




\section{Basic definitions and preliminaries}
\label{sec:basic}

\subsection{Discrete-time Branching Random Walks}
\label{subsec:discrete}

We start with the construction of a generic discrete-time BRW $\{\eta_n\}_{n \in \N}$
(see also \cite{cf:BZ2} where it is
called \textit{infinite-type branching process}) on a set $X$ which is
at most countable; $\eta_n(x)$ represents the number of particles alive
at $x \in X$ at time $n$. To this aim we
consider a family $\mu=\{\mu_x\}_{x \in X}$
of probability measures % , with the condition
%%(which we will use only in the points 2. of the two following theorems) that
%$\mathrm{supp}(\mu_x)\subseteq S_X$, where
on the (countable) measurable space $(S_X,2^{S_X})$
where $S_X:=\{f:X \to \N\colon \sum_yf(y)<\infty\}$.
To obtain generation $n+1$ from generation $n$ we proceed as follows:
a particle at site $x\in X$ lives one unit of time,
then a function $f \in S_X$ is chosen at random according to the law $\mu_x$
and the original particle is replaced by $f(y)$ particles at
$y$, for all $y \in X$; this is done independently for all particles of
generation $n$ (a similar construction in random environment can be found
in \cite{cf:GMPV09}).
Note that the choice of $f$ assigns simultaneously the total number of children
and the location where they will live.
We denote the BRW by the couple $(X,\mu)$.

Equivalently we could introduce the BRW by choosing first the number of children and afterwards
their location.
Indeed define $\cH:S_X \rightarrow \N$ as $\cH(f):=\sum_{y \in X} f(y)$
which represents the total number of children associated to $f$.
Denote by $\rho_x$ the measure on $\N$ defined by
$\rho_x(\cdot):=\mu_x(\cH^{-1}(\cdot))$; this is the law of the random number of children
of a particle living at $x$. For each particle, independently, we pick a number $n$ at random,
according to the law $\rho_x$, then we choose a function $f \in \cH^{-1}(n)$ with probability
$\mu_x(f)/\rho_x(n)\equiv\mu_x(f)/\sum_{g \in \cH^{-1}(n)}\mu_x(g)$ and
we replace the particle at $x$ with $f(y)$ particles at $y$ (for all $y \in X$).


% To be precise, to construct the process,
% pick a family $\{ f_{i,n,x}\}_{i,n \in \N , x \in X}$
% of independent $S_X$-valued random variables such that, for every $x \in X$, $\{f_{i,n,x}\}_{i,n \in \N}$
% have the common law $\mu_x$, and an initial state $\eta_0$ such that $\sum_{x \in X} \eta_0(x) <+\infty$.
% The discrete-time BRW $\{\eta_n\}_{n \in \N}$ is defined
% iteratively as follows
% \begin{equation}\label{eq:evolBRW}
% \eta_{n+1}(x)=\sum_{y \in X} \sum_{i=1}^{\eta_n(y)} f_{i,n,y}(x)=
% \sum_{y \in X} \sum_{j=0}^\infty \ident_{\{\eta_n(y)=j\}} \sum_{i=1}^{j} f_{i,n,y}(x).
% \end{equation}
% Even though in this chapter the initial state  will always be deterministic,
% considering a random initial distribution $\sum_{x \in X} \eta_0(x) <+\infty$ a.s., would not
% be a significant generalization.

%One can show that this frame includes every \textsl{infinite-type} BRW as well. Indeed,
%consider a process where each particle on $X$ is labeled with some
%$i \in I$ (where $I$ is an at-most-countable set) and it can possibly have
%offsprings carrying different labels: this process can be seen as a discrete-time BRW
%on $X \times I$.   Moreover, our model is time-homogeneous, nevertheless any
%\textsl{time-inhomogeneous} BRW on $X$ can be seen as a particular case of a discrete-time
%BRW on $X \times \N$ (simply by coupling time and space).

In BRW theory  a fundamental role is played by the \textit{first-moment matrix}
$M=(m_{xy})_{x,y \in X}$,
where
$m_{xy}:=\sum_{f\in S_X} f(y)\mu_x(f)$ is
 the expected number of particles from $x$ to $y$
(that is, the expected number of children that a particle living
at $x$ sends to $y$).
 We
%\textcolor{magenta}{
suppose that $\sup_{x \in X} \sum_{y \in X} m_{xy}<+\infty$; most of the results
of this paper still hold without this hypothesis, nevertheless
it allows us to avoid dealing with an infinite expected number of offsprings.
%}
The expected number of children generated by a particle living at $x$
is $\sum_{y \in X} m_{xy} = \sum_{n \ge 0} n \rho_x(n)=:\bar \rho_x$.
Given a function $f$ defined on $X$ we denote by $Mf$ the function $Mf(x):=\sum_{y \in X} m_{xy}f(y)$
whenever the right-hand side converges absolutely for all $x$.
We denote by
 $m^{(n)}_{xy}$
%its iterates (that is,
the entries of the $n$th power matrix $M^n$ and we define
\begin{equation}\label{eq:generalgeomparam}
M_s(x,y) := \limsup_{n \to \infty} \sqrt[n]{m_{xy}^{(n)}}, \quad
M_w(x) := \liminf_{n \to \infty} \sqrt[n]{\sum_{y \in X} m_{xy}^{(n)}}, \qquad \forall x,y \in X.
\end{equation}

Explicit computations of  $M_s(x,y)$ and $M_w(x)$ are possible in some cases (see \cite{cf:BZ, cf:BZ2}):
in particular $M_s(x,x)$ can be obtained by means of a generating function (see  \cite[Section 3.2]{cf:Z1}).
In this paper we do not need to compute explicitly $M_s$ and $M_w$ except for some specific examples
where justifications will be provided. 

For a generic BRW, we call \textit{diffusion matrix} the matrix $P$ with entries $p(x,y)=m_{xy}/\bar \rho_x$.
% In this case
%  the expected number of children at $y$ of a particle
% living at $x$ is
% %\begin{equation}\label{eq:meanparticular}
%  $m_{xy}=p(x,y) \bar \rho_x$.
% %\end{equation}
In particular if $\bar \rho_x$ does not depend on $x \in X$, we have that $M_w(x)=\bar \rho$ for all $x \in X$ and
$M_s(x,y)=\bar \rho \cdot \limsup_{n \to \infty} \sqrt[n]{p^{(n)}(x,y)}$ (where the $\limsup$ defines
the \textit{spectral radius} of $P$ according to \cite[Chapter I, Section 1.B]{cf:Woess}). 

Note that, in the general case, the locations of the offsprings are not chosen independently
(they are assigned by the chosen $f\in S_X$).
When
the offsprings are dispersed independently according to $P$ we call the process a
\textit{BRWs with independent diffusion}:
in this case
\begin{equation}\label{eq:particular1}
\mu_x(f)=\rho_x \left (\sum_y f(y) \right )\frac{(\sum_y f(y))!}{\prod_y f(y)!} \prod_y p(x,y)^{f(y)},
\quad \forall f \in S_X.
\end{equation}


% 
% Some special cases are discussed at the end of this section, after 
% Definition~\ref{def:locallyisomorphic} and at the beginning of Section~\ref{subsec:local}.

% From equation~\eqref{eq:evolBRW},
% it is straightforward to prove that the expected number of particles,
% starting from an initial state $\eta_0$,
% satisfies the recurrence equation $\E^{\eta_0}(\eta_{n+1}(x))
% =\sum_{y \in X}m_{yx} \E^{\eta_0}(\eta_n(y))$
% %independence between f_{i,n,x} and \eta_n
% hence
% \[ %begin{equation}\label{eq:evolexpectednumber}
% \E^{\eta_0}(\eta_n(x)) = \sum_{y \in X} m^{(n)}_{yx} \eta_0(y).
% \] %end{equation}


% \begin{rem}\label{rem:BRWasRW}
% Note that a BRW can be seen as a random walk on $\N^X$ (to be precise, on $S_X \subseteq \N^X$),
% where $\mathbf{0}$ is an absorbing state.
% If $\rho_x(0)>0$ for all $x \in X$ then every state in $S_X \setminus \{\mathbf{0}\}$ is transient. Basically this is
% due to the fact that the probability of going into the state $\mathbf{0}$ starting from a state
% $\eta \in S_X$ in one step is $\prod_{x \in X}\rho_x(0)^{\eta(x)}>0$; hence the probability of visiting
% infinitely often the state $\eta$ without ending in the trap state $\mathbf{0}$ is $0$
% (for a formal proof in the case of a branching process see \cite[Theorem 6.2]{cf:Harris63}).
% \end{rem}

% We introduce here some terminology borrowed from random walk and graph theory.
% In general our definitions extend the classical ones which apply to graphs in
% the following way: a discrete-time counterpart of an edge-breeding continuous-time BRW
% (see Section~\ref{subsec:continuous}) has the %\textsl{extended}
% property
% $\mathcal{P}$ if and only if the underlying graph has the usual
% property $\mathcal{P}$.



To  a generic discrete-time BRW we associate a graph $(X,E_\mu)$ where $(x,y) \in E_\mu$  
if and only if $m_{xy}>0$.
% a particle living at $x$ can
% send a child at $y$ with positive probability (from now on \textit{wpp}). Let $E_\mu$
% be the set of edges. 
We denote by $\mathrm{deg}(x)$
the degree of a vertex $x$, that
is, the cardinality of the set $ \mathcal{N}_x:=\{y\in X\colon  (x,y) \in E_\mu%\mu(x,y)>0
\}$.
%We denote by $d(x,y)$ the ``distance'' from $x$ to $y$ (not necessarily symmetric)
%by $d(x,y)=\inf\{n \in \N: x \to y \}$ (the distance $d(A,B)$ between two subsets of $X$
%is defined, as usual, as $\inf_{x \in A, y \in B} d(x,y) \equiv \min_{x \in A, y \in B} d(x,y)$).
We say that there is a path from $x$ to $y$, and we write $x \to y$, if it is
possible to find a finite sequence $\{x_i\}_{i=0}^n$ (where $n \in \N$)
such that $x_0=x$, $x_n=y$ and $(x_i,x_{i+1}) \in E_\mu$
for all $i=0, \ldots, n-1$. If $x \to y$ and $y \to x$ we write $x \rightleftharpoons y$.
Observe that there is always a path of length $0$ from $x$ to itself. The equivalence 
class $[x]$ of $x$ with respect to $\rightleftharpoons$ is called \textit{irreducible class of $x$}.
It is easy to show that if $x \rightleftharpoons x^\prime$ and $y \rightleftharpoons y^\prime$ then
$M_s(x,y)=M_s(x^\prime,y^\prime)$ and $M_w(x)=M_w(x^\prime)$. Moreover, 
$m^{(n)}_{xx}$ and $M_s(x,x)$ depend only on the entries $(m_{ww^\prime})_{w,w^\prime \in [x]}$. 
We call the matrix $M=(m_{xy})_{x,y \in X}$ \textit{irreducible} if and only if
the graph $(X,E_\mu)$ is \textit{connected} (that is, there is only one irreducible class), 
otherwise we call it \textit{reducible}.
From the BRW point of view, the irreducibility of $M$ means that the progeny of any particle can spread to any site of the 
graph. For an irreducible BRW, $M_s(x,y)=M_s$ and $M_w(x)=M_w$ for all $x,y \in X$.


The BRW $(X,\mu)$ is called \textit{non-oriented} or \textit{symmetric} if $m_{xy}=m_{yx}$ for every $x,y \in X$.
Note that if $(X,\mu)$ is non-oriented then the graph $(X,E_\mu)$ is non-oriented (that is, $(x,y) \in E_\mu$
if and only if $(y,x) \in E_\mu$).
$(X,\mu)$ is called \textit{nonamenable} if and only if
\[ %\begin{equation}%\label{eq:amenable}
\inf
\left \{
\frac{\sum_{x \in S, y \in S^\complement} m_{xy}}{|S|}\colon  S \subseteq X, |S| < \infty
\right \}%=:\iota_{(X,\mu)}
>0,
\] %\end{equation}
and it is called \textit{amenable} otherwise.

The idea behind the definition of nonamenability is that the expected number of children placed outside every finite subset of 
$X$ is always comparable with the size of the subset itself. This suggests, in principle, that it should be possible 
for the BRW to survive and, at the same time, to escape from every finite set.
This is true for a subclass of BRWs but not in general, see Example~\ref{exm:amenable} and the preceding discussion.
We note that, if $m_{xy}\in\{0, \lambda\}$ (for some fixed $\lambda>0$) then the BRW is nonamenable if and only if the graph $(X, E_\mu)$ is  
nonamenable according to the usual definition for graphs (see \cite[Chapter II, Section 12.B]{cf:Woess}).
% 
% 
% 
%  (for a comparison between this definition
% and the usual one on graphs see the end of Section~\ref{subsec:continuous}).
% The value $\iota_{(X,\mu)}$ is called \textit{isoperimetric constant} since in the
% case of a continuous-time, edge-breeding BRW (see the end of Section~\ref{subsec:continuous})
% this is the usual isoperimetric constant of the underlying multigraph and, in that case,
% the nonamenability of the BRW is equivalent to the nonamenability of the multigraph
% (see \cite[Section 3.3]{cf:BZ} for the definition).

% For a generic discrete-time BRW, the set $X$ is not \textit{a priori} a graph;
% nevertheless, the family of probability measures, $\{\mu_x\}_x$ induces in a natural way
% a graph structure on $X$ that
% we denote by $(X,E_\mu)$ where $E_\mu:=\{(x,y):m_{xy}>0\}\equiv\{(x,y): \exists f \in S_X, \mu_x(f)>0, f(y)>0\}$.


Depending on the initial configuration, the process can survive in different ways.
We consider initial configurations with only one particle placed at a fixed site $x$:
let $\pr^{\delta_{x}}$ be the law of this process. Throughout this paper \textit{wpp}
is shorthand for ``with positive probability''.

\begin{defn}\label{def:survival} $\ $
\begin{enumerate}
 \item 
The
process \textsl{survives locally wpp} in $A \subseteq X$ starting from $x \in X$
if
%\[
$
{\mathbf{q}}(x,A):=1-\pr^{\delta_x}(\limsup_{n \to \infty} \sum_{y \in A} \eta_n(y)>0)<1.
$
%\]
\item
The process \textsl{survives globally wpp} starting from $x$ if
$
\bar {\mathbf{q}}(x):={\mathbf{q}}(x,X) 
%\equiv 1- \pr^{\delta_x} \Big (\sum_{w \in X} \eta_n(w)>0, \forall n \in N \Big )
<1.
$
% (or, equivalently, $\pr^{\delta_x}(\limsup_{n \to \infty} \sum_{w \in X}\eta_n(w)>0$).
\item
%Let us define the probabilities of extinction
% ${\mathbf{q}}(x,y):=1-\pr^{\delta_x}(\limsup_{n \to \infty}  \eta_n(y)>0)$  and
% $\bar {\mathbf{q}}(x):= 1-\pr^{\delta_x} \Big (\sum_{w \in X} \eta_n(w)>0, \forall n \in N \Big )$.
There is \textsl{strong local survival wpp} in $A \subseteq X$ starting from $x \in X$
if
$ %\[
%\pr^{\delta_x}(\limsup_{n \to \infty} \eta_n(y)>0)=
%\pr^{\delta_x} \Big (\sum_{w \in X} \eta_n(w)>0, \forall n \in N \Big )>0.
{\mathbf{q}}(x,A)=\bar {\mathbf{q}}(x)<1
$ %\]
and \textsl{non-strong local survival wpp} in $A$ if $\bar {\mathbf{q}}(x)<{\mathbf{q}}(x,A)<1$.
\item
The BRW is in a \textit{pure global survival phase} starting from $x$ if
$ %\[
\bar {\mathbf{q}}(x)<{\mathbf{q}}(x,x)=1
$ %\]
(where we write ${\mathbf{q}}(x,y)$ instead of ${\mathbf{q}}(x, \{y\})$ for all $x,y \in X$).
\end{enumerate}
\end{defn}
\noindent
From now on, when we talk about survival, ``wpp'' will be tacitly understood.
Often we will say simply that local survival occurs ``starting from $x$'' or ``at $x$'':
in this case we mean that $x=y$. When there is no survival wpp, we say that there is extinction
and the fact that extinction occurs with probability one will be tacitly understood.

Note that ${\mathbf{q}}(x,A)$ are the probabilities of extinction in $A$ starting from $x$.
Roughly speaking, 
% there is global survival if there are particles alive somewhere at all times wpp and
% there is local survival at $y$ if there are particles alive at $y$ at arbitrarily large times wpp.
% Strong local survival at $y$ starting from $x$ requires that the probability of local survival at $y$
% equals the probability of global survival starting from $x$ and that they are both positive.
% Equivalently, 
there is strong survival at $y$ starting from $x$ if and only if the probability
of local survival at $y$ starting from $x$ conditioned on global survival starting from $x$ is $1$.
% One can show
% that strong local survival implies 
Thus, strong local survival means that for almost all realizations the process either survives locally
(hence globally) or it goes globally extinct. 
% The typical case (but not the only one)
% where there is global but no strong local survival is being in a pure global survival phase.
There are many relations between $\bar {\mathbf{q}}(x)$ and ${\mathbf{q}}(x,y)$ and between ${\mathbf{q}}(w,x)$ and
${\mathbf{q}}(w,y)$ where $x,y, w \in X$ (see for instance Section~\ref{subsec:survivalprob} or \cite{cf:BZ4, cf:Z1}).
% ;
% we observe that $\bar {\mathbf{q}}(x) \le {\mathbf{q}}(x,A)$ for all $x \in X$, $A \subseteq X$.

In order to avoid trivial situations where particles have one offspring almost surely, we assume
henceforth the following.
\begin{assump}\label{assump:1}
For all $x \in X$ there is a vertex $y \rightleftharpoons x$ such that
$\mu_y(f\colon  \sum_{w\colon w \rightleftharpoons y} f(w)=1)<1$,
 that is, in every equivalence class (with respect to $\rightleftharpoons$)
there is at least one vertex where a particle
can have inside the class a number of children different from one wpp.
\end{assump}

% \begin{rem}\label{rem:assump1}
% The previous assumption guarantees that the restriction of the BRW
% to an equivalence class is \textit{nonsingular} (see \cite[Definition II.6.2]{cf:Harris63}).
% %%%%or, in other words, that there are no \textit{final classes} for the BRW (see \cite[Definition 10.1]{cf:Harris63}).
% %We made the previous assumption in order to avoid trivial situations.
% %Indeed, in many cases, one can easily delete from $X$ a vertex $x$ such that
% %$\mu_x(\cH^{-1}(1))=1$. Suppose that $\mu_x(\delta_x)=0$, $\mu_x(\cH^{-1}(1))=1$ and consider this new dynamics which does
% %not involve $x$: whenever a particle at a vertex $y$ gives birth to a certain number of particles
% %at $x$, replace each one of them immediately with another particle
% %placed accordingly to the distribution $\mu_x$; these new particles
% %are now considered as the offsprings of the particle $y$ and the vertex $x$ in not involved
% %in the dynamics of this new BRW (the case $\mu_x(\delta_x)\in (0,1)$ can be treated
% %analogously by means of a limit procedure and the case $\mu_x(\delta_x)=1$ clearly implies local
% %survival at $x$ starting from $x$ with probability 1).
% %Clearly the new BRW has the same global behavior and the same local behavior (except
% %for at $x$ which in not an admissible state for the new BRW) than the original one.
% %
% There is a technical reason behind the previous assumption.
% The classical Galton--Watson branching process is a particular BRW where $X:=\{x\}$ is a singleton
% and $S_X$ and $\mu_x$ can be identified with $\N$ and a probability measure on $\mathbb N$
% respectively. It is well-known that
% \begin{itemize}
%  \item if $\mu_x(1)=1$ then $m_{xx}=1$ and there is survival with probability $1$;
% 
% \item if $\mu_x(1)<1$ then there is survival wpp if and
% only if $m_{xx}>1$,
% \end{itemize}
% (see also Example~\ref{ex:branchingprocess}).
% Hence the condition  $m_{xx}>1$ is equivalent to survival under
% Assumption~\ref{assump:1}.
% %This will be used implicitly in Theorem~\ref{th:equiv1}(1) and (5),
% %but it is not needed, for instance, in Theorem~\ref{th:equiv1}(2), (3) and (4).
% Remark~\ref{rem:nodeath} gives better insight of the meaning of Assumption~\ref{assump:1}.
% \end{rem}



%\begin{rem}
% \label{rem:comparison}
%Let $\{\eta_n\}_{n \in \N}$ and $\{\eta^m_n\}_{n \in \N}$ be defined by
%equations~\eqref{eq:evolBRW} and \eqref{eq:evolBRWm} respectively. If
%$\eta_0 \ge \eta_0^m$ a.s.~then, by induction, $\eta_n \ge \eta^m_n$ a.s.~
%for all $n \in \N$.
%\end{rem}





\subsection{Continuous-time Branching Random Walks}
\label{subsec:continuous}


% Continuous-time BRWs have been studied extensively by many authors;
% in this section we make use of a natural correspondence between continuous-time BRWs and
% discrete-time BRWs which preserves both local and global behaviors.

In continuous time each particle has an exponentially distributed
random lifetime with parameter 1. The breeding mechanisms can
be regulated by means of a nonnegative matrix $K=(k_{xy})_{x,y \in X}$ in such a way that
for each particle alive at $x$,
there is a clock with $Exp(\lambda k_{xy})$-distributed intervals (where $\lambda>0$),
each time the clock
rings the particle places one son at $y$. We say that the BRW has a death rate 1 and a
reproduction rate $\lambda k_{xy}$ from $x$ to $y$.
We observe (see Remark~\ref{rem:deathrate}) that the assumption
of a nonconstant death rate does not represent a
significant generalization.
We denote by  $(X,K)$ a family of continuous-time BRWs (depending on the parameter $\lambda>0$),
while we use the notation $(X,\mu)$ for
a discrete-time BRW.

% Equivalently, one can associate to each particle at $x$
% a clock with $Exp(\lambda k(x))$-distributed intervals ($k(x)=\sum_y k_{xy}$):
% each time the clock rings the particle breeds and the offspring is placed
% at random according to a stochastic matrix $P$
% (where $p(x,y)=k_{xy}/k(x)$).
% %We assume that $\sup_{x \in X} k(x)<\infty$.

To a continuous-time BRW
one can associate a discrete-time counterpart which takes into account all the offsprings 
of a particle before it dies;
% here is the construction.
% The initial particles represent the generation $0$ of the discrete-time BRW;
% the generation $n+1$ (for all $n \ge 0$) is obtained by
% considering the children of all particles of generation $n$
% (along with their positions).
% Clearly the progenies of the original continuous-time BRW and of its discrete-time counterpart
% are both finite (or both infinite) at the same time.
% Moreover, almost surely, the two processes have the same local and global behavior.
in this sense the theory of continuous-time BRWs, as long as %we are interested in 
as it concerns the probabilities of survival (local, strong local and global),
is a particular case of the theory of discrete-time BRWs.
%The construction proceeds as follows:
Elementary calculations show that $\mu_x$ satisfies %is given by
equation~\eqref{eq:particular1}, where
\begin{equation}\label{eq:counterpart}
\rho_x(i)=\frac{1}{1+\lambda k(x)} \left ( \frac{\lambda k(x)}{1+\lambda k(x)} \right )^i, \qquad
p(x,y)=\frac{k_{xy}}{k(x)},
\end{equation}
($k(x):=\sum_{y \in X} k_{xy}$).
Note that the discrete-time
counterpart of a continuous-time BRW is a BRW with independent diffusion and that
$\rho_x$ depends only on $\lambda k(x)$.
It is straightforward to show that $m_{xy}=\lambda k_{xy}$ and $\bar \rho_x=\lambda k(x)$.
%(look at the sentence after equation~\eqref{eq:particular1}).
Moreover equation~\eqref{eq:counterpart} shows that
% Note that,
% for a continuous-time BRW the first-moment matrix $M$ equals $\lambda K$.
% From equation~\eqref{eq:counterpart} we have that, for any $\lambda>0$, 
the discrete-time
counterpart satisfies Assumption~\ref{assump:1}.
% 
% %, $f(y)$ in site $y$, for all $y$, with
% % probability
% %\[
% %\mu_x(f)=\frac{(\sum_y f(y))!\prod_y k_{xy}^{f(y)}}
% %{(1+\sum_y k_{xy})^{1+\sum_y f(y)} \prod_y {f(y)!}}.
% %\]
All the definitions given in the discrete-time case extend to continuous-time BRWs:
a continuous-time BRW has some property if and only if its discrete-time
counterpart has it.

\begin{rem}\label{rem:deathrate}
The same construction applies to continuous-time BRWs with a death rate $d(x)>0$ dependent on $x \in X$. In this
case the discrete-time counterpart satisfies equation~\eqref{eq:particular1} where
\[ %begin{equation}\label{eq:counterpart}
\rho_x(i)=\frac{d(x)}{d(x)+\lambda k(x)} \left ( \frac{\lambda k(x)}{d(x)+\lambda k(x)} \right )^i, \qquad
p(x,y)=\frac{k_{xy}}{k(x)}.
\] %end{equation}
Hence, from the point of view of local and global survival, this process is equivalent to a continuous-time BRW with death rate
$1$ and reproduction rate $\lambda k_{xy}/d(x)$ from $x$ to $y$.
\end{rem}

% 
% All the definitions given in the previous section extend to the continuous-time case:
% a continuous-time BRW has a certain property if and only if its discrete-time counterpart
% has it. In particular, we observe that, for a continuous-time BRW, the isoperimetric
% constant defined in equation~\eqref{eq:amenable} equals to $\lambda \iota_{(X,K)}$
% where $\iota_{(X,K)}$ is the constant when $\lambda=1$. Hence either the BRW
% is nonamenable for all $\lambda >0$ or it is amenable for all $\lambda >0$.
% 
% \smallskip

Given $x \in X$, two critical parameters are associated to the
continuous-time BRW: the \textit{global} 
\textit{survival critical parameter} $\lambda_w(x)$ and the  \textit{local} 
 \textit{survival critical parameter} $\lambda_s(x)$.
They are defined as
\[ %begin{equation}\label{eq:criticalparameters}
\begin{split}
\lambda_w(x)&:=\inf \Big \{\lambda>0\colon \,
\pr^{\delta_{x}}\Big (\sum_{w \in X} \eta_t(w)>0, \forall t\Big) >0 \Big \},\\
%\pr^{\delta_{x}}\left(\exists t\colon \eta_t=\mathbf{0}\right)<1\}\\
\lambda_s(x)&:=
\inf\{\lambda>0\colon \,
\pr^{\delta_{x}} \big(\limsup_{t \to \infty} \eta_t(x)>0 \big) >0
\}.
%\pr^{\delta_{x}}\left(\exists \bar t\colon \eta_t(x)=0,\,\forall t\ge\bar t\right)<1\},
  \end{split}
\] %end{equation}
% If the graph $(X, E_{\mu})$
% is connected (that is, the BRW is irreducible) then these values do not depend on the initial
% configuration, provided that this configuration is finite (that
% is, it has only a finite number of individuals), nor on the choice
% of $x$. 
These values are constant in every irreducible class; in particular they do not depend
on $x$ if the BRW is irreducible.
The process is called
\textit{globally supercritical}, \textit{critical} or \textit{subcritical}
if $\lambda>\lambda_w$, $\lambda=\lambda_w$ or $\lambda<\lambda_w$;
an analogous definition is given for the local behavior using $\lambda_s$ instead of $\lambda_w$.
% If we have $(X,K)$ and $(Y,\bar K)$ such that $Y\subseteq X$
% and $k_{xy}\ge \bar k_{xy}$ for all $x,y\in Y$ then for all $x\in Y$
% we have $\lambda_s^X(x)\le \lambda_s^Y(x)$ and  $\lambda_w^X(x)\le \lambda_w^Y(x)$.
% %See \cite{cf:BZ} and \cite{cf:BZ2} for a more detailed
% %discussion on the values of $\lambda_w(x)$ and $\lambda_s(x)$.
In particular we say that there exists a \textit{pure global survival phase} starting from $x$ if the
interval $(\lambda_w(x),\lambda_s(x))$ is not empty; clearly, if $\lambda  \in (\lambda_w(x),\lambda_s(x))$
then the BRW is in a pure global survival phase according to Definition~\ref{def:survival}.
% We prove in this paper that it is not possible to define a \textit{strong local survival critical parameter}
% analogously (see Example~\ref{rem:nonstrongandstrong}).

Given a continuous-time BRW $(X,K)$ we define %the analogs of $M_s(x,y)$ and $M_w(x)$
\[ %\begin{equation}%\label{eq:geomparam}
K_s(x,y) := \frac{M_s(x,y)}\lambda\equiv\limsup_{n \to \infty} \sqrt[n]{k_{xy}^{(n)}}, \quad
K_w(x) := \frac{M_w(x)}\lambda\equiv\liminf_{n \to \infty} \sqrt[n]{\sum_{y \in X} k_{xy}^{(n)}}, \qquad \forall x,y \in X,
\] %\end{equation}
where $M_s(x,y)$ and $M_w(x)$ are the corresponding parameters of the discrete-time counterpart.
$K_s(x,y)$ and $K_w(x)$ depend only on the equivalence classes of $x$ and $y$, hence
if the BRW is irreducible, then they do not depend on $x,y \in X$.
% Note that $K_s$ and $K_w$ were
% introduced in \cite{cf:BZ, cf:BZ2} where they were called $M_s$ and $M_w$.
% Note that supermultiplicative arguments imply that
% $K_s(x,x) =\lim_{n} (k^{(d(x)n)}_{xx})^{1/d(x)n}$ where
% $d(x):=\textrm{gcd}\{n >0 \colon  m^{(n)}_{xx}>0\}$ is the period of $x \in X$
% (see \cite[Definition 2.19]{cf:Woess09};
% hence,
% for all $x \in X$, we have that
% $K_s(x,x) \le K_w(x)$.
% If the BRW is irreducible then $K_s(x,y)$ and $K_w(x)$ do not depend on $x,y \in X$.
% 
% 
% 
% Two special cases are particularly interesting in the continuous-time setting:
% \textit{site-breeding BRWs} and \textit{edge-breeding BRWs}.

We say that a BRW is \textit{site-breeding} if $k(x)$ does not depend on $x \in X$.
%(cfr.~Definition~\ref{def:locallyisomorphic}). 
%; on one hand the number of children of a particle
% is independent of the site, on the other hand, clearly, the diffusion matrix $P=(p(x,y))_{x,y \in X}$
% can be inhomogeneous.
We say that a BRW is \textit{edge-breeding} if $k_{xy} \in \N$. The typical edge-breeding BRW can be
constructed from a multigraph with set of vertices $X$ by defining $k_{xy}$ as
the number of edges from $x$ to $y$; in this case to each edge there corresponds a constant
reproduction rate $\lambda$.
% Thus the total reproduction rate for a particle living at $x$ is $\lambda \cdot \mathrm{deg}(x)$
% where $\mathrm{deg}(x)$ is the number of edges \textit{from} $x$. The diffusion matrix $P$ in this
% case is the transition matrix of the simple random walk on the underlying multigraph.
% Note that if the multigraph is \textit{regular} (i.e.~$\mathrm{deg}(x)$ does not depend on $x$) then
% the edge-breeding BRW is site breeding.
If the multigraph is a graph, then it coincides with the graph $(X,E_\mu)$ associated 
with the discrete-time counterpart of the edge-breeding BRW.


% 
% 
% \subsection{Other dynamics}
% \label{subsec:otherdynamics}
% 
% We describe some slightly different dynamics which, in fact,
% have a natural discrete-time counterpart which has the
% same local and global behavior.
% 
% 
% Multitype BRWs has been studied by some authors (see for instance \cite{cf:MachadoMenshikovPopov}).
% In these processes there is a metapopulation which consists of individuals
% carrying a certain characteristic chosen among a family $I$ of types.
% A particle of type $i\in I$ living
% at $x \in X$ generates a random number of children of any type which are randomly
% placed in $X$. More precisely, this can be seen as a discrete-time BRW where
% the space is $X \times I$ and the family of probability measures $\{\mu_{x,i}\}_{x \in X, i \in I}$
% are defined on $S_{X \times I}$. Thus, a particle of type $i$ living
% at $x$ at the end of its lifetime dies and generates $f(y,j)$ children of
% type $j$ at $y$ (for all $y \in X$ and $j \in I$) with probability $\mu_{x,i}(f)$.
% For the multitype BRW the global survival of the metapopulation (resp.~the local
% survival of a fixed type) is equivalent to the global (resp.~local) survival
% of the single-type BRW on $X \times I$. On the other hand, the global survival
% of a fixed type $i_0$ is equivalent to the survival in the subset $X \times \{i_0\}$,
% while local survival at $x$ of the metapopulation is equivalent to the
% survival on $\{x\} \times I$.
% 
% Time-dependent BRWs can be described similarly, if we have a time-dependent
% family of laws $\{\bar \mu_{x,n}\}_{x \in X, n \in \N}$ we can construct a BRW
% on $X\times \N$ where, at the end of its lifetime, a particle
% at $(x, n)$ dies and generates $f(y)$ children at $(y, n+1)$, for all $y \in X$, with probability
% $\mu_{x,n}(f)$.
% 
% 
% We could define a continuous-time BRW where a particle living at $x$
% is endowed with a Poisson clock of parameter $\lambda k(x)$; when this
% clock rings it picks a function $f \in S_X$ with probability $\mu_{x}(f)$
% and reproduces accordingly. All particles reproduce a random number of times
% during their exponentially distributed (with mean $1$) lifetime.
% This process has a discrete-time counterpart which can be constructed as in
% Section~\ref{subsec:continuous}. In this case the discrete-time counterpart
% in general does not satisfy equation~\eqref{eq:particular1}.
% 
% Another definition of BRW, which is used by some authors (see for instance \cite{cf:KestenSidorav} or
% \cite{cf:Zaehle}) is
% the following. Each particle moves according to a random walk on $X$. After a random
% number of steps it dies and is replaced by a random number of children, whose law may depend
% on the final position. Again this process has a natural counterpart:
% if in the original BRW a particle starts its life at $x$ and dies at $y$ giving
% birth to $n$ children then in the discrete-time counterpart this particle does not move
% and, after one unit of time, it generates $n$ children at $y$.   Note that
% in terms of local and global survival these two processes are equivalent.
% 
% 
% 
% \subsection{Reproduction trails}\label{subsec:trails}
% 
% A fundamental tool which allows us to give an alternative construction of the BRW is
% the reproduction trail (see \cite{cf:PemStac1}).
% We fix an injective map $\phi:X\times X \times \Z \times \N \rightarrow \N$.
% Let the family $\{ f_{i,n,x}\}_{i \in \Z,n \ge 0 , x \in X}$ be as in Section~\ref{subsec:discrete} and let
% $\eta_0$ be the initial value. For any fixed realization of the process we call \textit{reproduction trail}
% to $(x,n) \in X \times \N$ a sequence
% \begin{equation}\label{eq:trail}
% (x_0,i_0,1),(x_1,i_1,j_1), \ldots ,(x_{n},i_{n}, j_{n})
% \end{equation}
% such that $x=x_n$, $-\eta_0(x_0) \le i_0 <0$, $0 < j_l \le f_{i_{l-1}, l-1, x_{l-1}}(x_l)$
% and $\phi(x_{l-1}, x_l, i_{l-1}, j_l)=i_l$, where $0 <l \le n$.
% The interpretation  is the following:
% $i_n$ is the identification number of the particle, which
% lives at $x_n$ at time $n$ and is the $j_n$-th offspring
% of its parent. The sequence $\{x_0, x_1, \ldots, x_n\}$ is
% the path induced by the trail (sometimes, we say
% that the trail is based on this path).
% Given any element $(x_l, i_l, j_l)$ of the trail~\eqref{eq:trail}, we say that the particle identified
% by $i_n$ is a descendant of generation $n-l$ of the particle identified by $i_l$ and the trail
% joining them is $(x_l,i_l,j_l), \ldots ,(x_{n},i_{n}, j_{n})$. We also say that the trail
% of the particle $i_n$ is a prolongation of the trail of the particle $i_l$.
% 
% Roughly speaking the trail represents the past history
% of each single particle back to its original ancestor, that is, the one living at time $0$; we note that from the couple
% $(n, i_n)$, since the map $\phi$ is injective, we can trace back the entire genealogy of the particle.
% The random variable $\eta_n(x)$ can be alternatively defined as the number of reproduction trails to
% $(x,n)$.
% This construction does not coincide with the one
% %Clearly the construction of the process given by the reproduction trails is not the same as the one
% induced by the equation~\eqref{eq:evolBRW} but the resulting processes have the same laws.
% 
% 



\subsection{Infinite-dimensional generating function}\label{subsec:genfun}

% %In the following sections we make use of various
% Later on we will need some generating functions, both
% %some of them are
% 1-dimensional and
% %some are
% infinite dimensional.
% Define $T^n_x:=\sum_{y \in X}m^{(n)}_{xy}$ and
% $\varphi^{(n)}_{xy}:=\sum_{x_1,\ldots,x_{n-1} \in X \setminus\{y\}} m_{x x_1} m_{x_1 x_2} \cdots m_{x_{n-1} y}$
% (by definition $\varphi^{(0)}_{xy}:=0$ for all $x,y \in X$).
% $T^n_x$ is the expected number of particles alive at time $n$ when the initial state is a single
% particle at $x$.
% Roughly speaking, $\varphi^{(n)}_{xy}$ is
% the expected number of particles alive at $y$ at time $n$
% when the initial state is just one particle at $x$ and the
% process behaves like a BRW except that every particle reaching
% $y$ at any time $i <n$ is immediately killed (before breeding).
% % In other words $\varphi^{(n)}_{xy}$ is the expected number of particles alive at $y$ at time $n$
% % whose trail did not hit any $(y,i,k)$ with $k <n$.
% 
% Let us consider the following family of 1-dimensional generating functions
% (depending on $x,y \in X$), where $\lambda \in \C$:
% \[
% \begin{split}
% \Gamma(x,y|\lambda)&:=\sum_{n =0}^\infty m^{(n)}_{xy} \lambda^n,
% \qquad \Phi(x,y|\lambda):=\sum_{n =1}^\infty \varphi_{xy}^{(n)} \lambda^n.
% \end{split}
% \]
% % To compare with random walk theory, $\Gamma$ is the analog of the
% % \textit{Green function} (cfr.~\cite[Section 1.C]{cf:Woess09}) and
% % $\Phi$ is the analog of the generating function of
% % the first-return probabilities (cfr.~the function $U$ of \cite[Section 1.C]{cf:Woess09}).
% % %note that the radii of convergence of $\Gamma(x,y|\lambda)$ and $\Theta(x|\lambda)$ are
% % %$1/K_s(x,y)$ and $1/K_w(x)$ respectively.
% % %
% % %The following relation holds
% % %\begin{equation}\label{eq:HTheta}
% % %\Gamma(x,y|\lambda)=\Phi(x,y|\lambda)\Gamma(y,y|\lambda)+\delta_{xy}, \quad \forall
% % %\lambda: |\lambda|< \left (\limsup_{n \in \N} \sqrt[n]{m^{(n)}_{xy}} \right )^{-1}.
% % %\end{equation}
% % %Since
% It is easy to prove that $\Gamma(x,x|\lambda)= \sum_{i \in \N} \Phi(x,x|\lambda)^i$ for all $\lambda>0$, hence
% \[ %begin{equation}\label{eq:genfun1}
% \Gamma(x,x|\lambda)= \frac{1}{1-\Phi(x,x|\lambda)},
% \qquad \forall \lambda \in \C: |\lambda|< %\left (\limsup_{n \in \N} \sqrt[n]{m^{(n)}_{xx}} \right )
% M_s(x,x)^{-1},
% \] %end{equation}
% and we have that $M_s(x,x)%(\limsup_{n \in \N} \sqrt[n]{m^{(n)}_{xx}} )
% ^{-1}=\max\{ \lambda %\geq 0
% \in {\mathbb R}:\Phi(x,x|\lambda)\leq 1\}$
% for all $x \in X$. % (see \cite[Section 2.2]{cf:BZ} for details).
% In particular $\Phi(x,x|1) \le 1$ if and only if
% $%\limsup_{n \in \N} \sqrt[n]{m^{(n)}_{xx}}
% M_s(x,x) \le 1$.
% % The interpretation of $\Gamma(x,y|1)$ is the expected value of the total number of descendants at $y$ of
% % a common ancestor living at $x$. On the other hand, $\Phi(x,y|1)$ is the expected number of descendants at $y$, of
% % a common ancestor living at $x$, whose trails start from $x$ and arrive at $y$ for the first time.
% % 
% % The classical approach to branching processes (see for instance \cite{cf:Harris63}) makes use of the one-dimensional
% % generating function of the offspring distribution $\rho$: $\widetilde G(z)=\sum_{n \in \N} \rho(n) z^n$, whose
% % minimal fixed point is the probability of extinction (see also Example~\ref{ex:branchingprocess}). Inspired by
% % this approach, 

We associate a generating function $G:[0,1]^X \to [0,1]^X$
to the family $\{\mu_x\}_{x \in X}$
which can be considered as an infinite dimensional power series. % (see also \cite[Section 3]{cf:BZ2}). 
More precisely,
for all ${\mathbf{z}} \in [0,1]^X$, $G({\mathbf{z}}) \in [0,1]^X$ is defined as the following weighted sum of (finite) products
\[ %\begin{equation}
%\label{eq:genfun}
G({\mathbf{z}}|x):= \sum_{f \in S_X} \mu_x(f) \prod_{y \in X} {\mathbf{z}}(y)^{f(y)},
\] %\end{equation}
where $G({\mathbf{z}}|x)$ is the $x$ coordinate of $G({\mathbf{z}})$.
The family $\{\mu_x\}_{x \in X}$ is uniquely determined by $G$. Indeed fix a finite $X_0 \subseteq X$ and $x \in X$.
For every $\mathbf{z}$ with support in $X_0$, we have  
$G({\mathbf{z}}|x)= \sum_{f \in S_{X_0}} \mu_x(f) \prod_{y \in X_0} {\mathbf{z}}(y)^{f(y)}$
which can be identified with a power series with several variables (defined on $[0,1]^{X_0}$).
Suppose now we have another generating function $ \overline G$ (associated to  $\{\overline \mu_x\}_{x \in X}$)
such that $G=\overline G$. In particular $G({\mathbf{z}}|x)=\overline G({\mathbf{z}}|x)$ for every 
$\mathbf{z}$ with support in $X_0$. Thus $\mu_x(f)=\overline \mu_x(f)$ for all $f \in S_{X_0}$.
Since $S_X =\bigcup_{\{X_0 \subseteq X \colon X_0 \textrm{ finite}\}}S_{X_0}$ we have that
$\mu_x(f)=\overline \mu_x(f)$ for
all $f \in S_X$. 


Note that $G$ is continuous with respect to the \textit{pointwise convergence topology} of $[0,1]^X$  and nondecreasing
with respect to the usual partial order of $[0,1]^X$ (see \cite[Sections 2 and 3]{cf:BZ2} for further details).
Moreover, $G$ represents the 1-step reproductions; we denote by $G^{(n)}$ the generating function
associated to the $n$-step reproductions, which is inductively defined as $G^{(n+1)}({\mathbf{z}})=G^{(n)}(G({\mathbf{z}}))$,
where $G^{(0)}$ is the identity.
% It is worth mentioning that the function $G$ arises naturally when computing the extinction probabilities
% of the process, as shown in Section~\ref{subsec:survivalprob} (see also \cite{cf:Harris63} 
% for the use of generating
% functions in the study of branching processes); indeed 
Extinction probabilities are fixed points
of $G$ and the smallest one is $\bar {\mathbf{q}}$ (see Section~\ref{subsec:survivalprob} for details). 

% 
% The generating function $G$ can be explicitly computed in some cases: for instance,
% for a BRW with independent diffusion (i.e.~if equation~\eqref{eq:particular1} holds).
% Indeed in this case it is straightforward
% to show that
% $G({\mathbf{z}}|x)=F_x(P{\mathbf{z}}(x))$
% where $F_x(y)=\sum_{n=0}^\infty \rho_x(n)y^n$ is the generating function of the number of children
% and $P{\mathbf{z}}(x)=\sum_{y \in X} p(x,y){\mathbf{z}}(y)$ is the transition operator of the corresponding random walk.
% In particular if $\rho_x(n)=\frac{1}{1+\bar \rho_x} (\frac{\bar \rho_x}{1+\bar \rho_x} )^n$
% (for instance if we are dealing with the discrete-time counterpart of a continuous-time BRW, see
% equation~\eqref{eq:counterpart}), we have
% $G({\mathbf{z}}|x)=\frac{1}{1+\bar \rho_x(1-P{\mathbf{z}}(x))}$ (see \cite[Section 3.1]{cf:BZ2}),
% that is,
% \begin{equation}\label{eq:Gcontinuous}
% G({\mathbf{z}})= \frac{\mathbf{1}}{\mathbf{1}+M(\mathbf{1}-{\mathbf{z}})}
% \end{equation}
% where $\mathbf{1}(x):=1$ for all $x \in X$, the ratio is to be intended as coordinatewise and
% $Mv$ was defined in Section~\ref{subsec:discrete}; in this case
% $m_{xy}$ is given by equation~\eqref{eq:meanparticular}, thus $M=\lambda K$.
%
% \smallskip
An example where the function $G$ can be explicitly computed is a BRW 
with independent diffusion: in this case it is not difficult to see that
$G({\mathbf{z}}|x)=\sum_{n \in \N} \rho_x(n) (P{\mathbf{z}}(x))^n$
where $P{\mathbf{z}}(x)=\sum_{y \in X} p(x,y){\mathbf{z}}(y)$.
If, in particular, $\rho_x(n)=\frac{1}{1+\bar \rho_x} (\frac{\bar \rho_x}{1+\bar \rho_x} )^n$ 
(as in the discrete-time counterpart of a continuous-time BRW) then
the previous expression becomes $G({\mathbf{z}}|x)=(1+\bar \rho_x P(\mathbf{1}-\mathbf{z})(x))^{-1}$.
The previous equality can be written in a more compact way as
\begin{equation}\label{eq:Gcontinuous}
 G({\mathbf{z}})= \frac{\mathbf{1}}{\mathbf{1}+M(\mathbf{1}-{\mathbf{z}})}
\end{equation}
where $M$ is the first-moment matrix and $M \mathbf{v}(x)=\bar \rho_x P\mathbf{v}(x)$ 
(by definition of $P$).
In equation~\eqref{eq:Gcontinuous} and hereafter, whenever ${\mathbf{z}}, {\mathbf{v}} \in [0,1]^X$ the ratio ${\mathbf{z}}/{\mathbf{v}}$ will be taken coordinatewise,
that is, $({\mathbf{z}}/{\mathbf{v}})(x):={\mathbf{z}}(x)/{\mathbf{v}}(x)$ for all $x$ such that ${\mathbf{v}}(x)>0$ 
(the value of $({\mathbf{z}}/{\mathbf{v}})(x)$
if ${\mathbf{v}}(x)=0$, if any, will be explicitly  defined when needed). 

When one is interested in the question whether a global surviving BRW survives strong locally,
it may be useful to condition the process on global survival. 
Given a generic discrete-time BRW such that $\bar {\mathbf{q}}(x)<1$ for all $x \in X$, 
by conditioning on global survival, we associate a BRW with no death
(that is, a BRW such that $\rho_x(0)=0$).
Let $\{\eta_n\}_{n \in \N}$ be the original BRW. Consider the event $\Omega_\infty=
\{\sum_{x \in X} \eta_n(x)>0, \, \forall n \in \N\}$ and define
the process $\{\widehat \eta_n\}_{n \in \N}$ as follows:
$\widehat \eta_n(x, \omega)$ equals the number of particles in $\eta_n(x, \omega)$ with at least one infinite line of descent
when $\omega \in \Omega_\infty$ and it equals  $0$ when $\omega \not \in \Omega_\infty$.
Roughly speaking, $\{\widehat \eta_n\}_{n \in \N}$ is obtained
by $\{\eta_n\}_{n \in \N}$ by removing all the particles with finite progeny, which
are clearly irrelevant in view of the survival due to the fact that $\bar {\mathbf{q}}(x)<1$ for all $x \in X$.
%
% Counterexample: X:=\{1,2\} s.t. from 1 to 1 there is a supercritical BP
% and every particle at 1 send one particle to 2. Every particle at 2 dies outside% without breeding.
% In this case there is local survival at 2 starting from 1 for \eta but not for \widehat \eta.
%$ When $\bar {\mathbf{q}}(x)<1$ for all $x\ in X$, local survival of \eta implies, by Borel-Cantelli,
% that an infinite number of particles at every point have an infinite progeny.
%
Hence, we have that the probability of local survival of
$\{\widehat \eta_n\}_{n \in \N}$ in $A$ (for all $A \subseteq X$),
starting from $x$ is equal to the same probability for $\{\eta_n\}_{n \in \N}$,
that is, ${\mathbf{q}}(x,A)$.
It can be shown that this process, restricted to $\Omega_\infty$ is a BRW 
that we call the \textit{no-death BRW associated to $\{\eta_n\}_{n \in \N}$}
(we still denote it by
$\{\widehat \eta_n\}_{n \in \N}$). Its
generating function is
\begin{equation}\label{eq:G-one}
\widehat G({\mathbf{z}}|x) = \frac{G(v({\mathbf{z}})|x)-\bar {\mathbf{q}}(x)}{1-\bar {\mathbf{q}}(x)},
\end{equation}
where %$\bar {\mathbf{q}}(x)$ is the probability of global extinction starting from $x \in X$
%(see Sections~\ref{subsec:discrete} and \ref{subsec:survivalprob}),
$G$ is the generating function of the original BRW and $v:[0,1]^X \rightarrow [0,1]^X$ is
defined as $v({\mathbf{z}}|x):=\bar {\mathbf{q}}(x)+{\mathbf{z}}(x)(1-\bar {\mathbf{q}}(x))$.
In a more compact way equation~\eqref{eq:G-one} can be written as $\widehat G=T^{-1}_{\bar {\mathbf{q}}} \circ G \circ T_{\bar {\mathbf{q}}}$ where
$T_{\mathbf{w}}:[0,1]^X \to \{{\mathbf{z}} \in [0,1]^X\colon  {\mathbf{w}} \le {\mathbf{z}}\}$ is defined as $T_{\mathbf{w}}{\mathbf{z}}(x):={\mathbf{z}}(x)(1-{\mathbf{w}}(x))+{\mathbf{w}}(x)$; note that
$T_{\mathbf{w}}$ is nondecreasing and, if ${\mathbf{w}}(x)  <{1}$ for all $x \in X$, bijective.
In particular if $\bar {\mathbf{q}} < \mathbf{1}$ then $T_{\bar {\mathbf{q}}}$ is a bijective map from the set
of fixed points of $\widehat G$ to the set of fixed points of $G$.

Clearly, for all $A \subseteq X$, the probability of local survival in $A$ of the associated no-death BRW
 starting from $x$ is %equal to the same probability for the original BRW
%that is, $q(x,A)$. In particular the probability of local survival at
%$A$ starting from $x$ conditioned on global survival starting from $x$ is 
the probability of local survival in $A$ of the original BRW  conditioned on 
global survival (starting from $x$), that is, 
$1-(T_{\bar {\mathbf{q}}}^{-1} {\mathbf{q}}(\cdot,A))(x)= (1-{\mathbf{q}}(x,A))/(1-\bar {\mathbf{q}}(x))$.



The following proposition is a sort of \textit{maximum principle} for
the function $({\mathbf{z}}-\bar {\mathbf{q}})/(\mathbf{1}-\bar {\mathbf{q}})$ where ${\mathbf{z}}$ is such that $G({\mathbf{z}}) \ge {\mathbf{z}}$
(note that we are not assuming that $\bar {\mathbf{q}}(x)<1$ for all $x \in X$).

\begin{pro}\label{pro:maximumprinciple} 
Given ${\mathbf{z}} \in [0,1]^X$ such that ${\mathbf{z}} \ge \bar {\mathbf{q}}$ is 
a solution of the inequality $G({\mathbf{z}}) \ge {\mathbf{z}}$,
%and define for every $x \in X$ the set $\mathcal{N}_x:=\{y\colon  x \to y\}$.
we define
%$\widehat {\mathbf{z}}(x):=({\mathbf{z}}(x)-\bar {\mathbf{q}}(x))/(\mathbf{1}-\bar {\mathbf{q}}(x))$ if $\bar {\mathbf{q}}(x) < 1$ and
$\widehat {\mathbf{z}}:=({\mathbf{z}}-\bar {\mathbf{q}})/(\mathbf{1}-\bar {\mathbf{q}})$ where
 $\widehat {\mathbf{z}}(x):=1$ for all
$x$ such that $\bar {\mathbf{q}}(x)=1$. %\textcolor{red}{-forse qui bisogna mettere 0 al posto di 1})
Then for all $x \in X$ such that the set $\mathcal{N}_x=\{y\colon (x,y) \in E_\mu\}$
is not empty,
either $\widehat {\mathbf{z}}(y) = \widehat {\mathbf{z}}(x)$ for all $y \in \mathcal{N}_x$ or there exists
$y \in  \mathcal{N}_x$ such that $\widehat {\mathbf{z}}(y)> \widehat {\mathbf{z}}(x)$. In particular
if $\widehat {\mathbf{z}}(x)=1$ then for all $y \in \mathcal{N}_x$ we have $\widehat {\mathbf{z}}(y)=1$.
The same results hold if we take the set $\{y \in X \colon  x \to y\}$ instead of $\mathcal{N}_x$.
\end{pro}

% The proof, which makes use of some arguments of Section~\ref{subsec:nodeath},
% can be found in Section~\ref{sec:proofs}.
As an application, in a finite, final irreducible class
(for instance if the BRW is irreducible and the set $X$ is finite) if $\mathbf{z}$ is as in Proposition~\ref{pro:maximumprinciple}, then
$\widehat {\mathbf{z}}$ is a
constant vector.
% % Since the probabilities of extinction
% % $\{q(\cdot,A)\}_{A \subseteq X}$ %(see Section~\ref{subsec:survivalprob} for the definition)
% % are solutions of $G({\mathbf{z}})={\mathbf{z}}$,
% Proposition~\ref{pro:maximumprinciple} applies, for instance,  to ${\mathbf{z}}(\cdot)={\mathbf{q}}(\cdot,A)$ %for all $A \subseteq X$
% (see Section~\ref{subsec:survivalprob}).
% % Note that if $\mu_x(\mathbf{0})=0$ for %some
% % all $x \in X$ (see Section~\ref{subsec:nodeath})
% % then $\bar q= \mathbf{0}$ and $\widehat {\mathbf{z}}={\mathbf{z}}$.
% 
% % As an application,
% % if we have an irreducible BRW on $\N$ where $m_{xy}>0$ implies $|x-y| \le 1$ we get the following behavior of
% % the extinction probabilities: for all $A \subseteq X$,
% % there exists $x_0 \in \N \cup \{+\infty\}$ such that $q(x,A)=q(0,A)$ for all $x \le x_0$ and
% % $q(x,A)<q(x+1,A)$ for all $x \ge x_0$.
% 
% % Finally it is worth mentioning that for a BRW with independent
% % diffusion the following generating function is very
% % useful:
% % \[
% %  L({\mathbf{z}}|x):={\mathbf{z}}(x)\frac{M{\mathbf{z}}(x)}{\mathbf{1}+ \bar \rho_x}+\frac{\mathbf{1}}{\mathbf{1}+\bar \rho_x}.
% % \]
% % While $G$ is obtained by conditioning on the number
% % of total children of a particle at the end of its life, in continuous time $L$
% % can be obtained by conditioning on the first event (birth or death of
% % a particle).  By algebraic manipulation we have that,
% % if $G$ satisfies equation~\eqref{eq:Gcontinuous}, then
% % ${\mathbf{z}}=G({\mathbf{z}})$ if and only if ${\mathbf{z}}=L({\mathbf{z}})$.
% % In particular, using the same arguments as in Section~\ref{subsec:survivalprob},
% % we see that by means of $L$ one can compute iteratively the probability of extinction
% % before the $n$th event (either birth or death).
% % %Clearly,
% % %\[
% % % \begin{cases}
% % %e_{n+1}&:= H(e_n)\\
% % %e_0&= \mathbf{0}\\
% % % \end{cases}
% % %\]
% % %where $e_n(x)$ is the probability of extinction before the $n$th
% % %event (birth or death of a particle).
% % 
% % % \subsection{Derived generating functions}\label{subsec:dergenfun}
% % % 
% % % In this section we show how it is possible to construct the generating functions of
% % % some particular subprocesses of the BRW.
% % % 
% % % First of all it is easy to show that if $\{\eta_n\}_{n \in \N}$ and $\{\xi_t\}_{t \ge 0}$ are a
% % % discrete-time BRW and a continuous time BRW respectively, then 
% % % $\{\eta_{kn}\}_{n \in \N}$ and $\{\xi_{\alpha n}\}_{n \in \N}$ (where $k \in \N\setminus\{0\}$ and
% % % $\alpha >0$) are two discrete-time BRWs.
% % % In particular the generating function of $\{\eta_{kn}\}_{n \in \N}$ can be computed
% % % easily from the generating function $G$ of $\{\eta_{n}\}_{n \in \N}$ as
% % % $G^{(k)}$ (as defined in Section~\ref{subsec:genfun} where $G^{(0)}$ is the
% % % identity in $[0,1]^X$).
% % % 
% % % From now on we consider the BRW $\{\eta_{n}\}_{n \in \N}$.
% % % Another interesting function is the generating fucntion of the number of
% % % children at $y$ of a particle at $x$; this is easily computed from $G$ as 
% % % \[
% % %  \mathcal V_{x\, y}(t):=G(\mathbf 1 - (1-t) \delta_y |x)= \sum_{n=0}^\infty \mu_x(f:f(y)=n) t^n
% % % \]
% % % where $\mathbf 1(x)=1$ for all $x \in X$ and $\delta_y$ is the usual Dirac function.
% % % 
% % % From the previous function we can obtain the generating function of the number
% % % of children of a particle along a path $\gamma=\{x_i\}_{i=0}^n$. Indeed
% % % this is clearly 
% % % \[
% % %  \mathcal V_\gamma:=\mathcal V_{x_0 \, x_1} \circ \cdots \circ \mathcal V_{x_{n-1} \, x_n}.
% % % \]
% % % 
% % %  
% % 



%\section{Special processes - TENERE SOLO DEFINIZIONI ESSENZIALI}\label{sec:specialprocesses}

\subsection{$\mathcal F$-BRWs}\label{subsec:FBRWs}

Some results can be achieved if the BRW has some regularity; to this aim
we introduce the concept of $\mathcal F$-BRW (see also \cite[Definition 4.2]{cf:Z1}),
which extends the concept of quasi-transitivity (see below).


\begin{defn}
\label{def:locallyisomorphic} 
$\ $
\begin{enumerate}
 \item 
A BRW $(X, \mu)$ is locally isomorphic to a BRW $(Y,\nu)$ if there exists a
surjective map $g:X\to Y$ such that
$
%\begin{equation}
%\label{eq:fgraph}
%\nu_{g(x)}(f)=\mu_x\left(h\colon \forall y\in Y, f(y)=\sum_{z\in g^{-1}(y)}h(z)\right),
%\quad \forall f\in S_Y.
\nu_{g(x)}(\cdot)=
\mu_x\left(\pi_g^{-1}(\cdot)\right)
$, %\end{equation}
where $\pi_g:S_X \rightarrow S_Y$ is defined as $\pi_g(f)(y)=\sum_{z\in g^{-1}(y)}f(z)$ for all $f\in S_X$, $y \in Y$.
\item
$(X, \mu)$ is a $\mathcal F$-BRW if it is locally isomorphic to some
BRW $(Y,\nu)$ on a finite set $Y$.
\end{enumerate}
\end{defn}
Clearly, if $(X,\mu)$ is locally isomorphic to $(Y, \nu)$ then
\begin{equation} \label{eq:Gfunctions}
G_X({\mathbf{z}} \circ g|x)=G_Y({\mathbf{z}}|g(x))
\end{equation}
for all ${\mathbf{z}} \in [0,1]^Y$ and $x \in X$. 
% Indeed $\pi_g$ is surjective and
% \[
% \begin{split}
%  G_X({\mathbf{z}} \circ g|x)&=\sum_{f \in S_X} \mu_x(f) \prod_{w \in X} {\mathbf{z}}(g(w))^{f(w)}
% =  \sum_{h \in S_Y}  \sum_{f \in \pi_g^{-1}(h) } \mu_x(f) \prod_{w \in X} {\mathbf{z}}(g(w))^{f(w)}\\
% &=  \sum_{h \in S_Y}  \sum_{f \in \pi_g^{-1}(h) } \mu_x(f) \prod_{v \in Y} \prod_{w \in g^{-1}(v)} {\mathbf{z}}(v)^{f(w)}\\
% &=  \sum_{h \in S_Y}  \mu_x(\pi_g^{-1}(h)) \prod_{v \in Y} {\mathbf{z}}(v)^{h(v)} \\
% &= \sum_{h \in S_Y}  \nu_{g(x)}(h) \prod_{v  \in y} {\mathbf{z}}(v)^{h(v)}=G_Y ({\mathbf{z}} | g(x)).
% \end{split}
% \]
% %Hence, $G_X({\mathbf{z}} \circ g)=G_Y({\mathbf{z}})\circ g$ and b
% By induction on $n$
% \[
% G^{(n+1)}_X({\mathbf{z}} \circ g)=G_X(G_X^{(n)}({\mathbf{z}} \circ g))=G_X(G^{(n)}_Y({\mathbf{z}})\circ g)=G_Y(G^{(n)}_Y({\mathbf{z}}))\circ g=
% G^{(n+1)}_Y({\mathbf{z}})\circ g,
% \]
% whence
% \begin{equation}\label{eq:G-FBRWs}
%  G^{(n)}_X({\mathbf{z}} \circ g)=G^{(n)}_Y({\mathbf{z}})\circ g
% \end{equation}
% for all $n \in \N$.
% %The equality $G_X({\mathbf{z}} \circ g)=G_Y({\mathbf{z}})\circ g$ implies that $\bar q^Y_{n}(g(x))=\bar q^X_{n}(x)$ for all $n \in \N$. By induction on $n$; when $n=0$ it is straightforward.
% %If $\bar q^Y_{n}(g(x))=\bar q^X_{n}(x)$ then
% %\[
% % \begin{split}
% % \bar q^Y_{n+1}(g(x))&= G_Y ( \bar q^Y_{n}| g(x))=G_X( \bar q^Y_{n} \circ g|x)=G_X( \bar q^X_{n} |x)=\bar q^X_{n+1}(x).
% % \end{split}
% %\]
% %
% %Finally, as $n \to \infty$, we have that the minimal fixed points
% %of these generating functions satisfy $\bar q^Y(g(x))=\bar q^X(x)$.
% %\item If $(X, \mu)$ is locally isomorphic to $(Y,\nu)$ then
We note that, since $\mu$ is uniquely determined by $G$,
 equation~\eqref{eq:Gfunctions} holds  if and only if $(X,\mu)$ is locally
isomorphic to $(Y,\nu)$ and $g$ is the map in Definition~\ref{def:locallyisomorphic}.
% To see the ``only if'' part, define $\widehat \nu$ by using equation~\eqref{eq:fgraph} (substitute $\nu$ with $\widehat \nu$),
% then equation~\eqref{eq:Gfunctions} holds with
% $\widehat G$ instead of $G_Y$; thus $\widehat G=G_Y$ and this implies that equation~\eqref{eq:fgraph}
% holds for $\nu$.
If $\{\eta_n\}_{n \in \N}$ is a realization of the BRW $(X, \mu)$ then $\{\pi_g(\eta_n)\}_{n \in \N}$ is a realization
of the BRW $(Y, \nu)$.

Using equation~\eqref{eq:Gfunctions} and the fact that $\bar {\mathbf{q}}= \lim_{n \to \infty} G^{(n)}(\mathbf 0)$
(see equation~\eqref{eq:extprobab} with $A=X$), it is possible to prove that
there is global survival for $(X,\mu)$ starting from $x$ if and only if
there is global survival for $(Y,\nu)$ starting from $g(x)$  (see \cite[Theorem 4.3]{cf:Z1}).
% In particular note that the total offspring generating functions $G(t \mathbf 1|x)$ satisfy
% $G_X(t\mathbf 1_X|x)= G_X(t\mathbf 1_Y \circ g|x)=G_Y(t\mathbf 1_Y | g(x))$.
% , hence the offspring distribution
% of a particle at $x$ behaving according to $(X,\mu)$ is the same of the offspring distribution of a particle
% at $g(x)$ behaving according to $(Y,\nu)$.
% 
% If $(X,\mu)$ is locally isomorphic to a BRW $(Y, \nu)$ where $Y$ is a singleton (equivalently,
% if the law of the total number of children $\rho_x$ does
% not depend on $x \in X$) then we say that the BRW is \textit{locally isomorphic to a branching process}
% (this case is discussed in details in Section~\ref{subsec:locisomBP}).
% 
% It is easy to prove that $\sum_{w \in X} m^X_{xw}v(x)= \frac{\diff}{\diff t} G_X(\mathbf 1 -(1-t)v|x)|_{t=1}$ for all $x \in X$ and
% $v \in [0,1]^X$, hence, using equation~\eqref{eq:G-FBRWs}, we have
% %\[
% %\begin{split}
%   $\sum_{w \in X} m^X_{xw}{\mathbf{z}}(g(x))=
% %\frac{\diff}{\diff t} G^X((\mathbf 1 -(1-t){\mathbf{z}}) \circ g|x)|_{t=1} \\
% %&=
%  %\frac{\diff}{\diff t} G^Y(\mathbf 1 -(1-t){\mathbf{z}}|g(x))|_{t=1}=
% \sum_{y \in Y} m^Y_{g(x)y}{\mathbf{z}}(y)$,
% %\end{split}
% %\]
% for all $x \in X$ and ${\mathbf{z}} \in [0,1]^Y$. This, in turn, implies that
% $\sum_{w \in X} m^{X,(n)}_{xw}{\mathbf{z}}(g(w))=\sum_{y \in Y} m^{Y,(n)}_{g(x)y}{\mathbf{z}}(y)$ for all $n \in \N$.
% In particular, when $n=1$ and ${\mathbf{z}} = \mathbf 1$ we have $\bar \rho^X_{x}=\bar \rho^Y_{g(x)}$.
It is not difficult to prove (the details can be found in \cite{cf:Z1} before Theorem 4.3) that, for all $x \in X$, $y \in Y$ and
$n \in \N$, 
$\widetilde m^{(n)}_{g(x)y} = \sum_{z \in g^{-1}(y)} m^{(n)}_{xz}$ where $\widetilde M$ is the first-moment matrix of the BRW $(Y,\nu)$.
This implies $M^{X}_w(x)=M^{Y}_w(g(x))$ for all $x \in X$.

In continuous time (see \cite{cf:BZ2})
one can prove that $(X,K)$ is \textit{locally isomorphic} to $(Y,\widetilde K)$ if and only if
there exists a surjective map $g:X \to Y$ such that
$\sum_{z \in g^{-1}(y)} k_{xz}=\widetilde k_{g(x)y}$
for all
$x \in X$ and $y \in Y$, whence 
% We observe that
% $(X,K)$ is locally isomorphic to $(Y,\widetilde K)$
% if and only if the discrete-time counterparts satisfy
% Definition~\ref{def:locallyisomorphic}; this can easily be checked
% by proving that the equation~\eqref{eq:Gfunctions} holds when
% $G_X$ and $G_Y$ satisfy equation~\eqref{eq:Gcontinuous}.
% Note that a continuous-time BRW is site-breeding if and only if it
% is locally isomorphic to a branching process.
% On the other hand a continuous-time, edge-breeding BRW is locally isomorphic
% to another edge-breeding BRW if and only if the underlying multigraphs
% satisfy \cite[Definition 3.1]{cf:BZ}.
%
% \begin{defn}\label{def:quasitransitive}
$K^{X}_w(x)=K^{Y}_w(g(x))$ for all $x \in X$.
In other words, the total rate at which particles at $x$ generate children placing them in the set
of vertices with ``label'' $y$, depends only on $y$ and on $g(x)$.

Roughly speaking, an $\mathcal F$-BRW is a BRW where the vertices of $X$ can be labelled by means of
a finite alphabet $Y$ in such a way that the law of the labels of the positions of the children of 
a particle depends only on the label of the position of the father.  
As an example, consider a graph $(X,E(X))$ such that $\sup_{x \in X} \mathrm{deg}(x)<+\infty$ and where 
$\mathrm{deg}(x)=\mathrm{deg}(y)$ implies $\#\{z \in \mathcal{N}_x \colon \mathrm{deg}(z)=j\}=
\#\{z \in \mathcal{N}_y \colon \mathrm{deg}(z)=j\}$ (for all $j$); an example 
of such a graph is a tree with two alternating degrees.
In this case a BRW on $X$ with independent diffusion where $\rho_x$ depends only on $\mathrm{deg}(x)$  
is an $\mathcal{F}$-BRW and the label
of $x$ is $\mathrm{deg}(x)$. 


It is worth mentioning a particular subclass of $\mathcal F$-BRWs:
a BRW is \textit{locally isomorphic to a branching process}
if and only if  the laws of the offspring number $\rho_x=\rho$ is independent
of $x \in X$. % (see Definition~\ref{def:locallyisomorphic}).
In this case the BRW is locally isomorphic to a BRW on a singleton $Y:=\{y\}$ where 
the law of the number of children of each particle is $\rho$ and $g(x):=y$ for all $x \in X$.
%Hence $M_w(x)=\bar \rho$ for all $x \in X$. 
The explicit computations of $M_w$ and $M_s$ in this case can be found after equation~\eqref{eq:generalgeomparam}.
In particular a continuous-time BRW is 
% In this case the branching process can be constructed as the BRW
% $(\{0\}, \nu)$, where $\nu_0:=\rho$.
% All the results of this section also apply to continuous-time BRWs
% where $\rho_x$ is independent of $x \in X$, since %his implies that the
% their discrete-time
% counterpart is  locally isomorphic to a branching process. In particular,
% for a continuous-time BRW $(X,K)$, $\rho_x$ is uniquely determined by $k(x) \equiv \sum_{y \in X} k_{xy}$
% (and by $\lambda$),
% hence $(X;K)$ is 
locally isomorphic to a branching process if and only if $k(x)=k$ for all $x \in X$ 
(that is, if and only if it is an site-breeding BRW). In this case $K_w(x)=k$
and $K_s(x,y)= k\cdot \limsup_{n \to \infty} \sqrt[n]{p^{(n)}(x,y)}$. 

% Among the BRWs which are locally isomorphic to a branching process there are those
% with independent diffusion. %: 
% % %each particle has $n$ children with probability $\rho(n)$ 
% % %and every child is placed independently according to a transition matrix $P=(p(x,y))_{x,y \in X}$.
% % it is a simple exercise to prove that, in this case, $G({\mathbf{z}}|x)= \sum_{n \in \N} \rho(n) (P{\mathbf{z}}(x))^n$
% % where $P{\mathbf{z}}(x):=\sum_{y \in X} p(x,y){\mathbf{z}}(y)$ (see Examples~\ref{ex:nonstronglocalFBRW} and \ref{ex:nonstronglocalFBRW2}).

Let $\gamma:X \to X$ be an injective map. %Does it need to be bijective?
We say that $\mu=\{\mu_x\}_{x \in X}$ is $\gamma$-invariant if
for all $x%,y 
\in X$ and $f \in S_X$ we have
$\mu_x(f)=\mu_{\gamma(x)}( f \circ \gamma^{-1})$ (where $f \circ \gamma^{-1}$ is extended to a function
on $X$ by setting $0$ outside $\gamma(X)$). In particular, a BRW with independent diffusion is $\gamma$-invariant
if and only if $\rho_x=\rho_{\gamma(x)}$ and $p(x,y)=p(\gamma(x),\gamma(y))$ for all $x,y \in X$.

Moreover $(X,\mu)$ is \textit{quasi transitive} if and only if there exists
a finite subset $X_0 \subseteq X$ such that for all $x \in X$ there exists a bijective
map $\gamma:X\to X$ and $x_0 \in X_0$ satisfying $\gamma(x_0)=x$ and $\mu$ is $\gamma$-invariant.
An edge-breeding BRW on a graph $(X,E)$ is quasi transitive if and only if $(X,E)$ is a quasi-transitive graph. 
% \end{defn}

% The previous definition generalizes the usual one (which applies to graphs) in the following way:
% a discrete-time counterpart of an edge-breeding continuous-time BRW is quasi transitive if and only
% if the underlying graph is quasi transitive (that is, the action of the group of automorphisms
% has only finitely many orbits).

We note that every quasi-transitive BRW is an $\cF$-BRW (see \cite[Section 6.2]{cf:Z1}). 
% Indeed, consider
% the equivalence relation $x \sim y$ if and only if there exists a bijective map
% $\gamma:X \to X$ such that $\gamma(x)=y$. Clearly if $Y:=X/_\sim$ then $\# Y \le \# X_0$.
% Let $g$ be the usual projection from $X$ onto $X/_\sim$ and $\nu_{g(x)}(\cdot):=\mu_x(\pi_g^{-1}(\cdot))$.
% We have to show that the last definition is well-posed. Note that if $\mu$ is $\gamma$ invariant
% then $g=g \circ \gamma$ which implies $\pi_g(f)=\pi_{g \circ \gamma}(f)=\pi_g(f \circ \gamma^{-1})$; indeed
% \[
%  \pi_g(f)(y)=\sum_{z \in g^{-1}(y)} f(z)= \sum_{z \in (g \circ \gamma)^{-1}(y)} f(z)=
% \sum_{z \in g^{-1}(y)} f(\gamma^{-1}(z))
% \]
% whence
% for all $h \in S_Y$
% \[
% \begin{split}
%  \mu_x(\pi_g^{-1}(h))&=\mu_{\gamma(x)}(\{f \circ \gamma^{-1}: \pi_g(f)=h\})\\
% &=
% \mu_{y}(\{f \circ \gamma^{-1}: \pi_g(f \circ \gamma^{-1})=h\})=
% \mu_y(\pi_g^{-1}(h)).
% \end{split}
% \]
The class of $\mathcal{F}$-BRWs is strictly larger than the class of quasi-transitive BRWs.
For instance consider the BRW described in Example~\ref{ex:nonstronglocalFBRW}.
Indeed, in this case %it is not difficult to see that 
the BRW is $\gamma$-invariant if and only if, for all $i,j \in \N$,
$p(\gamma(i),\gamma(j))=p(i,j)$. This implies that $\gamma(0)=0$ and, by induction, $\gamma(i)=i$ for all $i \in \N$.
Thus, the only invariant map $\gamma$ is the identity on $\N$, whence there is no finite $X_0$ as described in the definition
of quasi transitivity. Nevertheless, the BRW is locally isomorphic to a branching process, thus it is an $\mathcal{F}$-BRW.
%Other examples of $\mathcal F$-BRWs which are not quasi transitive are \cite[Examples 3.1 and 3.2]{cf:BZ}).
Other examples are the edge-breeding BRWs associated to the following graphs: take a square and attach to each
vertex an infinite branch of a homogeneous tree $\mathbb{T}_3$ of degree 3 (see Figure~\ref{fig:square}); now attach to each vertex of the new graph
an edge with a new endpoint (see Figure~\ref{fig:square-end}). They are both $\mathcal{F}$-BRWs which are not quasi-transitive; moreover while
the first graph is regular (it has constant degree $3$), the second one is not since it has vertices with degree $4$ and vertices with degree
$1$.
\begin{figure}
    \begin{center}
    \subfigure[\tiny Regular graph with degree $3$.]{\includegraphics[height=5cm]{square.eps}
    \label{fig:square}}
   \hspace{0.5cm}
    \subfigure[\tiny Irregular graph.]{\includegraphics[height=5cm]{square-end.eps}
    \label{fig:square-end}}
    \end{center}
\caption{}
\end{figure}

% 
% \begin{exmp}\textbf{\cite[Example 3.1]{cf:BZ}}\label{rem:pureFBRW}
% Take a square and attach to every vertex a branch of a homogeneous tree of degree $3$, obtaining a regular graph (of degree $3$)
% which is not quasi transitive (see Figure~\ref{fig:square}).
% If we attach now to each vertex a new edge with a new endpoint we obtain a
% non-oriented, nonamenable
% $\cF$-graph $(X,E(X))$ (see \cite[Definition 3.1]{cf:BZ}) which is neither regular nor quasi transitive. It is easily seen
% (see \cite[Lemma 3.2]{cf:BZ}) to be locally isomorphic to a multigraph with
% adjacency matrix
% \[
% N=
% \begin{pmatrix}
% 3 & 1\\
% 1 & 0 \\
% \end{pmatrix}.
% \]
% The corresponding continuous-time, edge-breeding BRW is an $\mathcal{F}$-BRW which is defined on a
% (nonamenable) multigraph which is neither regular nor quasi transitive (thus, the process is not quasi transitive).
% %According to Theorem \ref{th:nonam}, the BRW on this graph has a pure weak phase.
% \end{exmp}
% 
% 
% 
% 
% \begin{figure}
%     \begin{center}
%     \subfigure[\tiny The regular graph of Example~\ref{rem:pureFBRW}.]{\includegraphics[height=5cm]{square.eps}
%     \label{fig:square}}
%    \hspace{0.5cm}
%     \subfigure[\tiny The graph $X$ of Example~\ref{exm:amenable}.]{\includegraphics[height=3.5cm]{t3n.eps}
%     \label{fig:amenable}}
%     \end{center}
% \caption{}
% \end{figure}

% \subsection{BRWs with and without death: a comparison}
% \label{subsec:nodeath}
% 
% 
% 
% 
% 
% Some authors (see for instance \cite{cf:CMP98, cf:GMPV09, cf:MachadoMenshikovPopov, cf:M08, cf:Muller08-2} and some
% results of \cite{cf:MenshikovVolkov}) have extensively studied the special case
% where $\mu_x(\mathbf 0)=0$ for all $x \in X$
% or, that is the same, $\rho_x(0)=0$ for all $x \in X$.  We call this kind of process a
% \textit{BRW with no death}; to be honest, in the usual interpretation each particle still dies but
% it has at least one descendant almost surely.
% On the other hand one can think that particles never die (in this case, $\rho_x(n)$ must be interpreted
% as the probability of having $n-1$ children). We stick with the first interpretation.
% 
% Being the global survival trivial in this case,
% it is interesting to explore the local behavior of the BRW. This situation is
% the closest one to the random walk theory.
% In particular,
% the process can become extinct locally at $x$ with probability one (some authors call the
% process \textit{transient} in this case) or survive locally with positive probability.
% While in the general situation the colony cannot always survive with probability one (if
% $\rho_x( 0)>0$ the process starting from $x$ might die at the first step),
% a BRW with no death can survive locally at $x$ either with probability one
% (strong local survival or \textit{recurrent BRW}) or with a strictly positive probability
% different from one (\textit{weakly recurrent BRW}).
% 
% In this section we want to discuss how we may interpret results on
% BRWs with no death in the general case. The first idea, which was introduced by Harris
% in the case of a branching process (see for instance \cite{cf:Harris48}
% or \cite[Chapter I.12]{cf:AthNey}), is to condition on global survival.
% It is clear that a generic BRW,
% such that $\bar q(x)<1$ for all $x \in X$,  %(where $\rho_x(0) \not = 0$
% conditioned on global survival is not a BRW with no death
% (it is not even a BRW). Indeed, for the conditioned process,
% the behaviors of different particles of the same generation
% are not independent: the probability of dying without breeding is strictly positive
% but the probability that all particles in the same generation die out
% without breeding is $0$. Nevertheless it is possible to associate to a generic irreducible BRW, with a fixed
% starting configuration,
% a BRW with no death.
% Given a generic irreducible BRW $\{\eta_n\}_{n \in \N}$ consider the event $\Omega_\infty=
% \{\sum_{x \in X} \eta_n(x)>0, \, \forall n \in \N\}$ and define
% the process $\{\widehat \eta_n\}_{n \in \N}$ as follows:
% $\widehat \eta_n(x, \omega)$ equals the number of particles in $\eta_n(x, \omega)$ with an infinite line of descent
% when $\omega \in \Omega_\infty$ and it equals  $0$ when $\omega \not \in \Omega_\infty$.
% It can be shown that this process, restricted to $\Omega_\infty$ is a BRW (that we call again
% $\{\widehat \eta_n\}_{n \in \N}$) and its generating function is
% \begin{equation}\label{eq:G-one}
% \widehat G({\mathbf{z}}|x) = \frac{G(v({\mathbf{z}})|x)-\bar q(x)}{1-\bar q(x)}
% \end{equation}
% where $\bar q(x)$ is the probability of global extinction starting from $x \in X$
% (see Sections~\ref{subsec:discrete} and
% \ref{subsec:survivalprob}),
% $G$ is the generating function of the original BRW and $v:[0,1]^X \rightarrow [0,1]^X$ is
% defined as $v({\mathbf{z}}|x):=\bar q(x)+{\mathbf{z}}(x)(1-\bar q(x))$.
% In a more compact way equation~\eqref{eq:G-one} can be written as $\widehat G=T^{-1}_{\bar q} \circ G \circ T_{\bar q}$ where
% $T_w:[0,1]^X \to \{{\mathbf{z}} \in [0,1]^X: w \le {\mathbf{z}}\}$ is defined as $T_w{\mathbf{z}}(x):={\mathbf{z}}(x)(1-w(x))+w(x)$; note that
% $T_w$ is nondecreasing and, if $w(x)  <{1}$ for all $x \in X$, bijective.
% More explicitly
% \[
% \begin{split}
%  \widehat G({\mathbf{z}}|x)&=\sum_{f \in S_X} \widehat \mu_x(f)
% \prod_{y \in X} {\mathbf{z}}(y)^{f(y)},\\
% \textrm{where }
% \widehat \mu_x(f)&=
% \begin{cases}
%  \displaystyle \frac{
% \sum_{g \in S_X:g \ge f} \mu_x(g) \prod_{y \in X} \binom{g(y)}{f(y)} \bar q(y)^{g(y)-f(y)}(1-\bar q(y))^{f(y)}}{1-\bar q(x)} &
% \ \textrm{if }f \not = \mathbf{0}\\
% \displaystyle 0 &\ \textrm{if }  f = \mathbf{0}.
% \end{cases}
% \end{split}
% \]
% Indeed $\{\widehat \eta_n\}_{n \in \N}$ is a Markov process and if we consider the offsprings
% of a particle living at $x$ at some time $n$, then its reproduction law conditioned
% on the global survival of the process and on having an infinite line of descent is equal to
% the reproduction law of an initial particle at $y$ conditioned on $\mathcal{A}_x$, that is
% % The reproduction law of a particle at $y$ in the $n$-th generation
% % conditioned on the global survival and on having an infinite line of descent is equal to $\pr(\widehat \eta_1=f|\mathcal{A}_y)$.
% \[
%  \pr(\widehat \eta_1=f|\mathcal{A}_x)=
% \begin{cases}
%  \displaystyle\frac{\pr(\widehat \eta_1=f)}{1-\bar q(x)}
% & f > \mathbf{0}\\
% 0 & f=\mathbf{0}
% \end{cases}
% \]
% where $\mathcal{A}_x$ is the event that there is global survival (i.e.~the first particle has an infinite line of descent) starting
% from one particle at $x \in X$ (clearly $\pr(\mathcal{A}_x)=1-\bar q(x)$).
% Hence, if $f > \mathbf{0}$,
% \[
%  \pr(\widehat \eta_1=f|\mathcal{A}_x)=
% \frac{1}{1-\bar q(x)} \sum_{g \in S_X: g \ge f} \mu_x(g) \prod_{y \in Y} \binom{g(y)}{f(y)} (1-\bar q(y))^{f(y)}
% \bar q(y)^{g(y)-f(y)}.
% \]
% This implies that
% \[
% \begin{split}
% \widehat G({\mathbf{z}}|x)&=
%  \sum_{f \in S_X} \pr(\widehat \eta=f|\mathcal{A}_x) \prod_{y \in Y} {\mathbf{z}}(y)^{f(y)} \\
% &= \sum_{g \in S_X: g > \mathbf{0}}\frac{\mu_x(g)}{1-\bar q(x)} \sum_{f \in S_X: \mathbf{0} <f \le g}
% \prod_{y \in Y} \binom{g(y)}{f(y)} ({\mathbf{z}}(y)(1-\bar q(y)))^{f(y)}
% \bar q(y)^{g(y)-f(y)}\\
% &=\sum_{g \in S_X: g > \mathbf{0}}\frac{\mu_x(g)}{1-\bar q(x)} \left [\prod_{y \in Y}
% (T_{\bar q}{\mathbf{z}}(y))^{g(y)}- \prod_{y \in Y} \bar q(y)^{g(y)}
% \right ]\\
% &=\frac{1}{1-\bar q(x)} \left [G(T_{\bar q}{\mathbf{z}}|x)-G(\bar q|x)\right ]
% =\frac{1}{1-\bar q(x)} \left [G(T_{\bar q}{\mathbf{z}}|x)-\bar q(x)\right ].\\
% \end{split}
% \]
% 
% It can be shown, following \cite{cf:AthNey}, that many results about survival
% are true for $\{\eta_n\}_{n \in \N}$ if and only if
% they are true for $\{\widehat \eta_n\}_{n \in \N}$. Proving this equivalence
% in details goes beyond the purpose of this chapter.
% Nevertheless we observe that, if $\bar q < \mathbf{1}$ then $T_{\bar q}$ is a bijective map from the set
% of fixed points of $\widehat G$ to the set of fixed points of $G$. Moreover,
% since $\{\widehat \eta_n\}_{n \in \N}$ is obtained
% by $\{\eta_n\}_{n \in \N}$ by removing all the particles with finite progeny, which
% are clearly irrelevant in view of the survival due to the fact that $\bar q(x)<1$ for all $x \in X$,
% %
% % Counterexample: X:=\{1,2\} s.t. from 1 to 1 there is a supercritical BP
% % and every particle at 1 send one particle to 2. Every particle at 2 dies outside% without breeding.
% % In this case there is local survival at 2 starting from 1 for \eta but not for \widehat \eta.
% %$ When $\bar q(x)<1$ for all $x\ in X$, local survival of \eta implies, by Borel-Cantelli,
% % that an infinite number of particles at every point have an infinite progeny.
% %
% we have immediately that the probability of local survival of
% $\{\widehat \eta_n\}_{n \in \N}$ in $A$ (for all $A \subseteq X$),
% starting from $x$ is equal to the same probability for $\{\eta_n\}_{n \in \N}$,
% that is, $q(x,A)$. In particular the probability of local survival at
% $A$ starting from $x$ conditioned on $\mathcal{A}_x$
% is $1-(T_{\bar q}^{-1} q(\cdot,A))(x)= (1-q(x,A))/(1-\bar q(x))$.
% 
% We call the process $\{\widehat \eta_n\}_{n \in \N}$ conditioned on $\mathcal{A}_x$ the \textit{no-death BRW associated to}
% $\{\eta_n\}_{n \in \N}$ starting from $x \in X$.
% 
% 


\section{Conditions for survival and extinction}
\label{sec:survival}




\subsection{Local and global survival}
\label{subsec:local}

The following theorems summarize the main results about local and global survival for discrete-time BRWs and
continuous-time BRWs respectively (see \cite[Theorems 4.1, 4.7, 4.8 and Proposition 4.5]{cf:BZ2}, \cite[Theorems 4.1 and 4.3]{cf:Z1}).  
In particular Theorem~\ref{th:discretesurv}(4) is a straightforward generalization of \cite[Theorem 3.6]{cf:BZ} (we omit the proof). 

\begin{teo}\label{th:discretesurv}
Let $(X,\mu)$ be a discrete-time BRW.
\begin{enumerate}
 \item 
There is
local survival starting from $x$ if and only if $M_s(x,x) %\limsup_{n\to\infty}\sqrt[n]{m^{(n)}_{xx}}
>1$.
\item
There is global survival starting from $x$ if and only if there exists
${\mathbf{z}}\in [0,1]^X$, ${\mathbf{z}}(x)<1$
such that $G({\mathbf{z}}|y) = {\mathbf{z}}(y)$, for all $y \in X$
(equivalently, such that $G({\mathbf{z}}|y) \le {\mathbf{z}}(y)$, for all $y \in X$).
\item
If $(X,\mu)$ is an $\mathcal F$-BRW then
there is global survival starting from $x$ if and only if $M_w(x)>1$.
\item If $(X,\mu)$ is an irreducible, non-oriented BRW then $M_s < M_w$ if and only if 
$(X,\mu)$ is nonamenable. 
\end{enumerate}
\end{teo}
Note that the fact that there is local survival or not, depends only on the first-moment matrix $M$.
% 
% 
% \begin{teo}\label{th:equiv1local} - TENERE PRIMA PARTE
% Let $(X,\mu)$ be a BRW.
% \begin{enumerate}[(1)]
%  \item
% \item
% If $\sup_{w \in X: x \to w \to y} M_s(w,w) %\limsup_{n\to\infty}\sqrt[n]{m^{(n)}_{ww}}
% >1$ then
% there is local survival at $y$ starting from $x$. Moreover if the cardinality of
% ${w \in X: x \to w \to y}$ is finite (for instance if $X$ is finite) the converse is true.
% \end{enumerate}
% \end{teo}
%that is, if and only if 
% 
% We recall that a useful characterization $M_s(x,x)=%\limsup_{n\to\infty} \sqrt[n]{m^{(n)}_{xx}}=
% \left ( \max \{\lambda \in \mathbb{R}: \Phi(x,x|\lambda) \le 1\}\right )^{-1}$ was
% given in Section~\ref{subsec:genfun}. %(see also Section~\ref{subsec:locisomBP}).
% Clearly, since $t \mapsto\Phi(x,x|t)$ is left continuous and strictly increasing,
% $M_s(x,x)>1$ if and only if $\Phi(x,x|1)>1$.
% , which is another condition equivalent to local survival at $x$.
% Note that $M_s(x,x)$ depends only on the values  $\{m_{wy}\}_{w,y \rightleftharpoons x}$.
In particular the BRW survives locally at $x$ if and only if it does so when restricted to the irreducibility
class $[x]$ of $x$.
% If the BRW is irreducible then
% $M_s(x,x) %\limsup_{n\to\infty}\sqrt[n]{m^{(n)}_{xx}}
% =M_s(w,y)$ %\limsup_{n\to\infty}\sqrt[n]{m^{(n)}_{wy}}$
% for all $x,w,y \in X$; hence there is local survival at $y$ starting from $x$ if and only if
% $M_s(w,w) %\limsup_{n\to\infty}\sqrt[n]{m^{(n)}_{ww}}
% >1$ for some $w \in X$ (equivalently, for all $w \in X$).
% 
% 
% To compare with random walk theory,
% the reader might recall the definition of spectral radius of a random walk on $X$ with transition matrix $P$
% as $1/\limsup_{n\to\infty}\sqrt[n]{p^{(n)}{(x,x)}}$
% (see the discussion after Proposition~\ref{pro:F-BRW1} and \cite[Section 2.C]{cf:Woess09}).
% If $\limsup_{n\to\infty}\sqrt[n]{p^{(n)}{(x,x)}}<1$
% then the random walk is
% transient, i.e.~it returns to $x$ a finite number of times almost surely
% (where $p^{(n)}(x,x)$ are the $n$-step return probabilities); on
% the other hand if $\limsup_{n\to\infty}\sqrt[n]{p^{(n)}{(x,x)}}=1$ then the process
% may be either recurrent or transient. We observe that, while for a random walk
% the probability of returning to a site infinitely many times obeys a 0-1 law,
% a BRW can survive locally with any probability in $[0,1]$. Of course,
% survival with probability one is possible only in the no-death case.
% 
% We show that the sufficient condition $\sup_{w \in X: x \to w \to y} M_s(w,w)%\limsup_{n\to\infty}\sqrt[n]{m^{(n)}_{ww}}
% >1$
% stated in Theorem~\ref{th:equiv1local} is not necessary in the reducible case when the cardinality of $X$
% is infinite.
% 
% \begin{exmp}\label{exm:reducible}
%  Let $X:=\N \times \{0,1\}$ (see Figure~\ref{fig:stripn}), fix $p>1/2$
%  and consider the BRW with the following reproduction rules:
% \begin{enumerate}[(a)]
%  \item every particle at $(i,0)$ has 2 children at $(i+1,0)$ and 1 child at $(i,1)$ with probability $p$
% and no children with probability $1-p$ (for all $i \in \N$);
% \item every particle at $(i,1)$ has 2 children at $(i-1,1)$ with probability $p$
% and no children with probability $1-p$ (for all $i \ge 1$);
% \item every particle at $(0,1)$ has 1 child  at $(0,1)$ with probability $p$
% and no children with probability $1-p$.
% \end{enumerate}
% Clearly $\limsup_{n\to\infty}\sqrt[n]{m^{(n)}_{ww}}=0$ for all $x \in X$. Nevertheless,
% there is local survival at $(i,1)$ for all $i \in \N$ starting from $(j,0)$ for any fixed $j \in \N$
% (note that there is no local survival at any $(i,0)$ for every starting point).
% \end{exmp}
% 
% \begin{figure}
%     \begin{center}
%     \subfigure[\tiny The graph $\N \times \{0,1\}$ of Example~\ref{exm:reducible}.]{\includegraphics[height=3cm]{stripn.eps}
%     \label{fig:stripn}}
%        \hspace{0.5cm}
%      \subfigure[\tiny The graph of Example~\ref{exm:nonamenable} (the circles are loops).]{\includegraphics[height=4.5cm]{t3complete.eps}
%     \label{fig:t3complete}}
%     \end{center}
%     \caption{}
% \end{figure}
% 
% 
% %\begin{figure}
% %     \begin{center}
% %\includegraphics[height=3.5cm]{stripn.eps}
% %    \caption{The graph $\N \times \{0,1\}$ of Example~\ref{exm:reducible}.}
% %    \end{center}\label{fig:stripn}
% %\end{figure}
% 
It is worth noting that if $[x]$
% , the irreducible
% class of $x \in X$, 
is finite, then $M_s(x,x)$
is the Perron-Frobenius eigenvalue
of the submatrix $M^\prime:=(m_{yz})_{y,z \in [x]}$. In this case
there is local survival at $x$ if and only if $\max\{t >0 \colon 
\exists v \not = \mathbf{0}, \, M^\prime v =t v\}>1$.
% 
% \begin{rem}\label{rem:nodeath}
% By looking at the proof of \cite[Theorem 4.1]{cf:Z1} we note that
% it is crucial to assume that each particle in the Galton-Watson branching process
% (with a different time-scale and living at $x$), costructed in the proof itself, has
% a number of children different from $1$ with positive probability. Roughly speaking
% this means that either it has a positive probability of dying without breeding or
% with positive probability it can give birth to more than one particle. 
% Indeed if this Galton-Watson branching process breeds with probability 1 then the original 
% BRW survives locally at $x$ with probability 1.
% 
% This holds under Assumption~\ref{assump:1}; indeed if the event A:=``the number of children 
% at $X$ from a particle living at $x$ for the Galton-Watson process is 1'' had probability 1 then it would
% be clearly independent of all events. Now consider the vertex $y$ given by Assumption~\ref{assump:1}:
% since $y$ is in the class of $x$ then $\mu_y(\mathbf{0})\not =1$. If $\mu_y(\mathbf{0})>0$ then the
% probability of $A$ conditioned to the events ``there are no offsprings from any particle at $y$'' and
% ``there is at least one offspring from at least one particle at $y$'' (where the particle at $y$ is
% a descendent of the original particle at $x$) have different probabilities and this is a contradiction.
% On the other hand, if there exists $n>1$ such that $\mu_y(f:\sum_{w:w \rightleftharpoons y}f(w)=n)>0$
% then clearly there exists a positive probability (by the definition of equivalence class) that at least 
% one descendent from each one among these $n$ particles reaches $x$. 
% 
% Finally, if Assumption~\ref{assump:1} does not hold then $m_{xx}=1$ and local survival
% occurs with probability one, which is in contrast with Theorem~\ref{th:equiv1local}. 
% 
% % Observe that, from Section~\ref{subsec:dergenfun}, the probability
% % of having no children along a path $\gamma=\{x_i\}_{i=0}^n$ is $\mathcal V_\gamma(0)$
% % and this is $0$ if and only if $\mathcal V_{x_i\, x_{i+1}}(0)=0$ for all $i=0, \ldots, n-1$,
% % that is, if and only if $\mu_{x_i}(f:f(x_{i+1})=0)=0$ for all $i=0, \ldots, n-1$.
% % Hence the probability of dying without breeding for that particular Galton-Watson branching process
% % is $0$ if and only if there exists a closed path $\gamma$ starting from $x$ 
% % such that $\mathcal V_\gamma(0)=0$.
% % If we consider the graph $(X,E_G)$ of  where $(x,y ) \in E_G$ if and only if $\mu_{x}(f:f(y)=0)=0$
% % (it is clearly a subgraph of $(X,E_\mu)$),
% % then the probability of dying without breeding for the Galton-Watson branching process in the
% % proof of \cite[Theorem 4.1]{cf:Z1} is $0$ if and only if there is a loop from $x$ to $x$.
% % This implies immediately that the proof  of \cite[Theorem 4.1]{cf:Z1} holds
% % if (and only if) we assume that  the graph $(X,E_G)$ is a (possibly oriented and disconnected)
% % tree.
% % 
% % For a BRW with independent diffusion (i.e.~if equation~\eqref{eq:particular1} holds) then
% % $\mathcal V_\gamma(0)=0$ if and only if $\rho_x{i}(0)=0$ and $p(x_i,x_i+1)>0$ for all $i=0, \ldots, n-1$
% % Remember that in this case $(x,y) \in E_\mu$ if and only if $p(x,y)>0$).
% % Hence the probability of dying without breeding for the Galton-Watson branching process in the
% % proof of \cite[Theorem 4.1]{cf:Z1} is $0$ if and only if there exists a path from $x$ to $x$ in
% % $(X,E_\mu)$ visiting all vertices in the set $\{y \in X: \rho_y(0)=0\}$.
% % 
% % % 
% % % 
% % %    
% % % The probability of this event equals
% % % \begin{equation}\label{eq:assumpnodeath}
% % % 1-\sum_{\gamma \in \Gamma_x} \prod_{i=0}^{l_\gamma-1} \mu_{\gamma_i}(f: f(\gamma_{i+1})>0)
% % % \end{equation}
% % % where $\Gamma_x$ is the set of all closed paths $\gamma=\{\gamma_i\}_{i=0}^{l_\gamma}$ starting from $x$ and ending at $x$
% % % after $l_\gamma \ge 1$ steps and
% % % never visiting x in-between (that is, $\gamma_0=\gamma_{l_\gamma}=x$ and $\gamma_i \not= x$ for all $i=1,\ldots,l_\gamma-1$).
% % % 
% % % Clearly if the probability in equation~\eqref{eq:assumpnodeath} is $0$ then the BRW survives
% % % locally at $x$ with probability $1$. Assumption~\ref{assump:1}, on the other hand, implies 
% % % that this probability is strictly positive.
% % % 
% % % We note that for a BRW with independent diffusion (i.e.~if equation~\eqref{eq:particular1} holds)
% % % then the probability in equation~\eqref{eq:assumpnodeath} is 
% % % \[
% % % 1-\sum_{\gamma \in \Gamma_x} \prod_{i=0}^{l_\gamma-1} \rho_{\gamma_i}([1,+\infty))
% % % p(\gamma_i,\gamma_{i+1})
% % % \]
% % % which, in turn, becomes
% % % \[
% % % 1-F(x,x|\rho([1,+\infty)))
% % % \]
% % % if $\rho_x=\rho$ for all $x$
% % % (where $F$ the generating function of the hitting probabilities,
% % % cfr.~the function $U$ of \cite[Section 1.C]{cf:Woess09}).
% \end{rem}
% 
% 
% In the case of continuous-time BRWs with
% rates $\{\lambda k_{xy}\}_{x,y \in X}$ one is also interested in the characterization
% of the local critical parameter $\lambda_s(x)$. Moreover one may also wonder
% whether at $\lambda= \lambda_s(x)$ there is survival or extinction.
% We already observed that the behavior of the continuous-time BRW is equivalent  to
% the behavior of
% its discrete-time counterpart (that is, the BRW with independent diffusion
% where $\{\mu_x\}_{x \in X}$ is
% given by equations \eqref{eq:particular1} and \eqref{eq:counterpart}).
% If we apply Theorem~\ref{th:equiv1local} then
% we obtain the following corollary, which gives the strong critical value and states
% that at the local critical value there is local extinction a.s.
% 
% \begin{cor}\textbf{\cite[Theorems 4.1 and 4.7]{cf:BZ2}}\label{cor:pemantleimproved}
In general, the global behavior does not depend only on $M$ 
(see \cite[Example 4.4]{cf:Z1}) unless
there is a
one-to-one correspondence
between first moment matrices and processes.
This is true for instance in the class of
BRWs with independent diffusion %satisfying %equations~\eqref{eq:counterpart} and equation~\eqref{eq:particular1} 
such that  $\rho_x(n)=\frac{1}{1+\bar \rho_x} (\frac{\bar \rho_x}{1+\bar \rho_x} )^n$
(hence for a continuous-time BRW). Indeed 
%(see \cite{cf:Z1,cf:BZ2}) 
in that case an equivalent condition
for global survival starting from $x \in X$ is the existence of 
${\mathbf{v}} \in [0,1]^X$, ${\mathbf{v}}(x)>0$ such that
\[
 M{\mathbf{v}} \ge {\mathbf{v}} /(\mathbf{1}-{\mathbf{v}}), \qquad \text{(equivalently, }M{\mathbf{v}} = {\mathbf{v}} /(\mathbf{1}-{\mathbf{v}}) \text{)}
\]
which comes from Theorem~\ref{th:discretesurv}(2) given $\mathbf{z}:=\mathbf{1}-{\mathbf{v}}$ and
the explicit expression \eqref{eq:Gcontinuous} of $G$.
In particular, for a BRW with independent diffusion, the local survival probability $\mathbf{v}_A:=\mathbf{1}-\mathbf{q}(\cdot,A)$
satisfies $M \mathbf{v}_A=\mathbf{v}_A/(\mathbf{1}-\mathbf{v}_A)$, which becomes
$\lambda K  \mathbf{v}_A=\mathbf{v}_A/(\mathbf{1}-\mathbf{v}_A)$ for a continuous-time BRW.

\begin{teo}\label{th:continuoussurv}
Let $(X,K)$ be a continuous-time BRW.
\begin{enumerate}
 \item 
$\lambda_s(x)=1/K_s(x,x)$ and if 
$\lambda=\lambda_s(x)$ then there is local extinction at $x$.
\item
$
 \lambda_w(x) \ge 1/K_w(x).
$
\item If $(X,K)$ is an $\mathcal F$-BRWs then  
$\lambda_w(x)=1/K_w(x)$ and when $\lambda=\lambda_w(x)$ there
is global extinction starting from $x$.
\item If $(X,K)$ is an irreducible, non-oriented $\mathcal{F}$-BRW then $\lambda_s > \lambda_w$ if and only if 
$(X,\mu)$ is nonamenable.
\end{enumerate}
\end{teo}




% \end{cor}

% 
% It is worth mentioning that, for a edge-breeding, irreducible BRW the critical value
% $\lambda_s$ (not depending on $x$) was already identified in \cite[Lemma 3.1]{cf:PemStac1}, even though
% the critical behavior was not known.
% 
% 
% Another result about local survival, in the context of irreducible BRWs with no death, is the following which relies on
% the existence of positive superharmonic functions; this one has
% a well-known counterpart in the random walk theory (see \cite[Theorem 6.21]{cf:Woess09}).
% 
% 
% \begin{teo}\textbf{\cite[Theorem 2.1]{cf:M08}}
% \label{th:Muller08}
% Let $(X,\mu)$ be an irreducible BRW such that $\rho_x(0)=0$ for all $x \in X$. There is local extinction if and only if
% there exists a strictly positive function $f$ on $X$ such that $Mf \le f$.
% \end{teo}
% 
% The proof of the previous theorem is inspired by the proof of
% Theorem~\ref{th:MenshikovVolkov}.
% Note that in the irreducible case with no death, one can obtain  Theorem~\ref{th:equiv1local}
% directly from Theorem~\ref{th:Muller08}
% Indeed, see \cite[Section 7.A]{cf:Woess},
% %\[ %begin{equation}\label{eq:superharmonic}
% $ M_s %\limsup_{n\to\infty}\sqrt[n]{m^{(n)}_{xy}}
% = \min \{t>0: \exists f:X \rightarrow (0,+\infty), \, Mf \le tf \}$
% %\] %end{equation}
% (remember that in the irreducible case $M_s(x,y)$ does not depend on $x,y \in X$).
% 
% 
% \subsection{Global survival - TENERE IL RISULTATO PRINCIPALE - UNIFICARE CON SEZIONE PRECEDENTE}
% \label{subsec:global}
% 
% The classical approach to estimate the probability of extinction of a branching process
% uses the fact that this probability is the minimal fixed point in $[0,1]$ of the generating
% function of the law of the number of children. Theorem~\ref{th:equiv1global} extends this approach to
% BRWs; indeed
% in this case the vector of global extinction probability $\bar q \in [0,1]^X$ is
% the smallest fixed point of the infinite dimensional generating function $G$
% (see Section~\ref{subsec:survivalprob}). Global survival is equivalent to the existence
% of a fixed point of $G$ strictly smaller than $\mathbf{1}$.
% We remark here that given $v,w \in [0,1]^X$
% by $v < w$ we mean, as usual, that $v \le w$ and $v \not = w$, that is
% $v(x) \le w(x)$ for all $x \in X$ and $v(x_0)<w(x_0)$ for some $x_0 \in X$.
% Theorem~\ref{th:equiv1global} gives an equivalent condition for global survival and a necessary one.
% 
% \begin{teo}\textbf{\cite[Theorem 4.1]{cf:Z1}}\label{th:equiv1global}
% Let $(X,\mu)$ be a discrete-time BRW.
% \begin{enumerate}[(1)]%[1.]
% \item There is
%global survival starting from $x$ if and only if there exists 
% % \item
% % If there is global survival starting from $x$, then
% % there exists $v\in[0,1]^X$, $v(x)>0$, such that
% %  $Mv \ge v$. Moreover, for all $y$, $Mv(y)=v(y)$ if and only if   %\Longleftrightarrow
% % $G(\mathbf 1 -(1-t) v|y)=1-(1-t)v(y), \forall t \in [0,1]$.
% % \end{enumerate}
% % \end{teo}
% It is easy to show (see for instance \cite[Section 3]{cf:BZ2}) that the first condition
% implies that $\bar q \le z < \mathbf{1}$, that is, $z$ is an upper bound for the
% probabilities of global extinction.
% %By taking the derivative at $t=0$ of $G$ evaluated on the line $\mathbf{1}-tv$ (where $v=\mathbf{1}-\bar q$) one
% As for the second condition, one has that if  $\bar q < \mathbf{1}$ then taking
%  $v=\mathbf{1}-\bar q$ (the probabilities of global survival) one obtains the inequality $Mv \ge v$.
% This is the analog for a BRW of the well-known result which states that a
% branching process survives if and only if the expected number of children is strictly larger than one.
% Nevertheless, for a BRW, even if we prove that $Mv(y)>v(y)$ for all $y \in X$
% and some $v \not = \mathbf{0}$ this does not suffice for
% global survival: a counterexample is given by Example~\ref{exm:noext}.
For a generic BRW when $\lambda=\lambda_w(x)$ there might be global survival
(see \cite[Example 3]{cf:BZ2}).
% On the other hand, for a generic BRW,
% according to \cite[Example 4.4]{cf:Z1}
% the global survival does not depend exclusively on $M$.
% 
% \begin{exmp}\textbf{}
% 
% In particular cases we can characterize global survival in terms of $M$:
% the following corollary follows easily from Theorem~\ref{th:equiv1global}(1) and
% equation~\eqref{eq:Gcontinuous} by taking $z=\mathbf{1}-v$.
% 
% \begin{cor}\label{cor:forgotten}
%  Suppose that the generating function of $(X,\mu)$ satisfies equation~\eqref{eq:Gcontinuous}
% (for instance, if $(X,\mu)$ is a BRW with independent diffusion %satisfies equation~\eqref{eq:particular1}
% where
% $\rho_x(n)=\frac{1}{1+\bar \rho_x} (\frac{\bar \rho_x}{1+\bar \rho_x} )^n$) then
% there is global survival starting from $x \in X$ if and only if there exists
% $v \in [0,1]^X$, $v(x)>0$ such that
% \[
%  Mv \ge v /(\mathbf{1}-v), \qquad \text{(equivalently, }Mv = v /(\mathbf{1}-v) \text{)}
% \]
% (where the ratio is taken coordinatewise).
% \end{cor}
% Observe that the solution $v$ in the previous corollary provides a lower bound
% for the probabilities of global survival $\mathbf{1}-\bar q$;
% moreover, among all the solutions of either equations, the largest one is $\mathbf{1}-\bar q$.
% 
% 
% Now we notice that in order to characterize local survival we studied the behavior
% of the expected number of $n$-th generation offsprings coming back to $x$, that is
% $m^{(n)}_{xx}$, see Theorem~\ref{th:equiv1local}.
% For the global survival problem, we are naturally lead to investigate the behavior of
% the expected number of $n$-th generation offsprings whose ancestor is a single particle at $x$,
% that is $\sum_y m^{(n)}_{xy}$.
% First of all, observe that it is easy to show, by using supermultiplicative arguments,
% that  %$\liminf_{n\to\infty} \sqrt[n]{\sum_y m^{(n)}_{xy}}
% $M_w(x) \ge M_s(x,x)$. %\limsup_{n\to\infty} \sqrt[n]{m^{(n)}_{xx}}$.
% % indeed $\sum_y m^{(n)}_{xy} \ge m^{(n-i_0)}_{xx} m^{i_0}_{xy}$ and $d_x|n-i_0$
% % where $d_x$ is the period of $x$. Supermultiplicative arguments imply that
% % $\limsup_{n\to\infty} \sqrt[n]{m^{(n)}_{xx}}= \lim_{n\to\infty} \sqrt[nd_x]{m^{(nd_x)}_{xx}}$
% %
% %Theorem~\ref{th:equiv1} extends \cite[Theorems 4.1, 4.3 and 4.7]{cf:BZ2}.
% %Indeed in terms of survival, studying a continuous-time BRW with
% %rates $\{\lambda k_{xy}\}_{x,y \in X}$ is equivalent  to studying
% %its discrete-time counterpart (that is, a BRW where $\{\mu_x\}_{x \in X}$ is
% %given by equations \eqref{eq:particular1} and \eqref{eq:counterpart}).
% %
% moreover, for an irreducible BRW,
% $M_w(x)$ %\liminf_{n\to\infty} \sqrt[n]{\sum_y m^{(n)}_{xy}}$
% does not depend on $x \in X$.
% 
% A complete description of global survival in terms of $M_w(x)$ %\liminf_{n \in \N} \sqrt[n]{\sum_{y \in X} m^{(n)}_{xy}}$
% is not possible
% in general (we have a necessary condition); indeed Example~\ref{exm:noext} shows that
% global survival does not depend on $M$ alone.
% Nevertheless, a characterization of global survival by means of $M_w(x)$
% %\liminf_{n \in \N} \sqrt[n]{\sum_{y \in X} m^{(n)}_{xy}}$
% holds for the
% class of $\mathcal F$-BRWs.
% 
% \begin{teo}\textbf{\cite[Theorems 4.1 and 4.3]{cf:Z1}}\label{th:equiv1global3}
% Let $(X,\mu)$ be a discrete-time BRW.
% \begin{enumerate}[(1)]%[1.]
% \item If there is global survival starting from $x$, then $M_w(x) %\liminf_{n \in \N} \sqrt[n]{\sum_{y \in X} m^{(n)}_{xy}}
% \ge 1$.
% \item If $(X,\mu)$ is an $\mathcal{F}$-BRW then there is global survival for $(X,\mu)$ starting from $x$ if and only if
% $M_w(x) %\liminf_{n \in \N} \sqrt[n]{\sum_{y \in X} m^{(n)}_{xy}}
% > 1$.
% \end{enumerate}
% \end{teo}
% We already observed that if the BRW is irreducible then $\bar q<\mathbf{1}$ implies $\bar q(x) <1$ for all
% $x \in X$; Example~\ref{exm:reducible} shows that in the reducible case it might happen that
% $\bar q< \mathbf{1}$ and $\bar q(x)=1$ for some $x \in X$.
% 
% 
% 
% In the case of continuous-time BRWs, one is interested in identifying $\lambda_w(x)$ and the
% behavior of the process when $\lambda=\lambda_w(x)$.
% The following result is a corollary of Theorems~\ref{th:equiv1global} and \ref{th:equiv1global3}.
% It characterizes $\lambda_w(x)$ in the case of $\mathcal F$-BRWs and gives a lower bound
% in the general case.
% 
% \begin{cor}\textbf{\cite[Theorems 4.3 and 4.8, Proposition 4.5]{cf:BZ2}}\label{cor:globalsurvivalcontinuous}
% Consider a continuous-time BRW $(X,K)$.
% \begin{enumerate}[(1)]
%  \item
% %If there is global survival starting from $x$, then
% $\lambda_w(x) \ge 1/K_w(x)=1/\liminf_{n \in \N} \sqrt[n]{\sum_{y \in X} k^{(n)}_{xy}}$.
% \item
% If $(X,\mu)$ is an $\mathcal{F}$-BRW then $\lambda_w(x)=1/K_w(x)$. Moreover, if $\lambda=\lambda_w(x)$ then
% there is global extinction starting from $x$.
% \item
% Suppose that for all $y \in X$ there exists $x \in X$ such that $x \to y$. If there is global survival starting from $x$, then
% there exists $v\in[0,1]^X$, $v(x)>0$, such that $\lambda Kv > v$.
% \end{enumerate}
% \end{cor}
% Note the difference between Theorem~\ref{th:equiv1global}(2) and Corollary~\ref{cor:globalsurvivalcontinuous}(3):
% in the second case we have the strict inequality $Mv=\lambda Kv>v$.
% As a consequence of the previous corollary we can compute the global critical value for two frequently used
% classes of continuous-time BRWs which are locally isomorphic to a branching process: for a site-breeding BRW
% $\lambda_w(x)=1/k$ for all $x \in X$ (where $k(x)=k$ for all $x \in X$),
% while for an edge-breeding BRW on a regular graph of degree $d$ we have that $\lambda_w(x)=1/d$ for all $x \in X$.
% % The last result follows from the fact that G is strictly convex.
% 
% \begin{rem}\label{rem:finiteclasses}
%  Consider a BRW $(X,\mu)$ where $X$ is finite. Following \cite[Remark 4.4]{cf:BZ2} one can prove that
% $M_w(x) %\liminf_{n \in \N} \sqrt[n]{\sum_{y \in X} m^{(n)}_{xy}}
% =\max_{w:x \to w}
% M_s(w,w)$; %\limsup_{n \in \N} \sqrt[n]{m^{(n)}_{ww}}$;
% remember that global survival starting from $x$ is equivalent to global (hence local)
% survival in some class (hence at some point $w$) since $X$ is finite.
% For a continuous-time BRW, this means that $K_w(x)=\max_{w:x \to w} K_s(w,w)$ and hence
% $\lambda_w(x)=\min_{w:x \to w} \lambda_s(w)$.
% If the BRW is finite and irreducible then there is only one class of irreducibility and the previous
% results hold without $\max$ and $\min$.
% \end{rem}
% 
% 
% %Clearly the previous result states that $\lambda_w(x) \ge  1/K_w(x)$ and the equality holds,
% %for instance, for $\mathcal{F}$-BRWs.
% Still in the case of continuous-time BRWs, we have a
% %The following theorem
% characterization of $\lambda_w(x)$, which makes use of the existence of a solution of certain
% systems of inequalities.
A characterization of $\lambda_w(x)$ has been given in \cite[Theorem 4.2]{cf:BZ2}
by means of the so-called
 \textit{lower
Collatz-Wielandt number}.
%  $v$ (see \cite{cf:FN1}, \cite{cf:FN2} and \cite{cf:Marek1}).
% In particular, according to Theorem~\ref{th:equiv1global2}(4), we have that
% for a continuous-time BRW
% \[
% \begin{split}
%  \lambda_w(x)&=\inf \{\lambda > 0 : \exists v \in l^\infty_+(X), v(x)> 0 \text{ such that } \lambda K v \ge  v \}\\
% &=\inf \{\lambda > 0 : \exists v \in l^\infty_+(X), v(x)> 0 \text{ such that } \lambda K v =  v \}
% \end{split}
% \]
% while, according to Theorem~\ref{th:Muller08}, if the BRW is irreducible,
% \[
%  \lambda_s:=\max \{\lambda > 0 : \exists v > 0 \text{ such that } \lambda K v \le  v \}.
% \]

% Finally, we know that a continuous-time BRW dies out locally at $x$ when $\lambda=\lambda_s(x)$ (see Theorem~\ref{th:equiv1local}).
% %It also true that, if the process is an $\mathcal{F}$-BRW, when $\lambda=\lambda_w$ there is global extinction.
% Theorem~\ref{th:equiv1global2}(2) states that the vector of probabilities of survival $v$ of a generic BRW at the critical point $\lambda_w(x)$
% if it is not equal to $\mathbf{0}$ it satisfies $\inf_{y \in X} v(y)=0$. This proves immediately that an irreducible $\mathcal{F}$-BRW
% dies out globally when $\lambda=\lambda_w$ (which is independent of $x \in X$).
% % in the irreducible case $v>\mathbf{0}$ iff $v(x)>0$ for all $x \in X$ and for an $\cF$-BRW $\inf_{x \in X} v(x)=\min_{x \in X} v(x)$
% Theorem~\ref{th:equiv1global2}(2) is the most reasonable result we can expect in full generality; indeed, here is an example of an irreducible BRW which
% survives globally when $\lambda=\lambda_w$.
% 
% \begin{exmp}\textbf{\cite[Example 3]{cf:BZ2}}
% \label{exm:4}
% Let $X:=\N$ and $K$ be defined by
% $k_{0\, 1}:=2$, $k_{n\, n+1}:=(1+1/n)^2$, $k_{n\, n-1}:= 1/3^{n}$
% (for all $n \ge 1$) and $0$ otherwise.
% Note that the corresponding continuous-time BRW is irreducible.
% In order to show that $\lambda_w=1$ we look for solutions of the inequality
%  $\lambda Kv \ge v/(\mathbf{1}-v)$. The system becomes
% \[
% \begin{cases}
% 2 \lambda v(1) \ge v(0)/(1-v(0)) & \\
% \lambda(v(n+1) (1+1/n)^2+v(n-1)/3^n) \ge v(n)/(1-v(n))& \text{ for all }n\ge1.\\
% \end{cases}
% \]
% Clearly, for all $\lambda \ge 1$,
% $v(0)=1/2$ and $v(n):=1/(n+1)$ (for all $n \ge 1$) is a solution; this implies, according to Theorem~\ref{th:equiv1global2}(1), that there is global survival
% for $\lambda \ge 1$, thus
% $\lambda_w \le 1$.
% If $\lambda <1$ then there are no solutions in $l^\infty_+(X)$.
% Indeed one can prove by induction that any solution must satisfy
% $v(n+1)/v(n) \ge \frac1\lambda\left(\frac{n}{n+1}\right)^2\left(
% 1-\frac{1}{2^n}\right)$ for all $n \ge 2$. Thus
% $v(n+1)/v(n)$ is eventually larger than $1+\eps$ for some $\eps>0$,
% hence either $v=\mathbf 0$ or $\lim_n v(n)= +\infty$.
% This implies that $\lambda_w=1$ and there is global survival if $\lambda=\lambda_w$.
% \end{exmp}
% 
% 
% Another result, which applies to edge-breeding, irreducible, continuous-time BRWs
% and which deals with the relation between $K_w$ and $\lambda_w$ is the following.
% 
% 
% \begin{teo}\textbf{\cite[Theorem 3.3]{cf:BZ}}\label{th:Tnk}
% Let $(X,K)$ be an edge-breeding, irreducible, continuous-time BRW on a multigraph $X$;
% let us suppose that there exists $x_0 \in X$, $Y \subseteq X$ and $n_0 \in \N$
% such that
% \begin{enumerate}[(1)]
% \item for all $x \in X$ we have that $B^+(x,n_0) \cap Y\not = \emptyset$;
% \item for all $y \in Y$ there exists an injective map $\varphi_y:X \to X$, such that $\varphi_y(x_0)=y$ and
% $k_{\varphi_y(x)\varphi_y(z)} \geq k_{xz}$ for all $x,z \in X$,
% \end{enumerate}
% where $B^+(x,n_0)$ is the set of all vertices which can be reached from $x$ in at most
% $n_0$ steps.
% Then $\lambda_w=1/K_w$.
% \end{teo}
% 
% %-------- controllare se serve nel teorema precedente che il grafo sia nonorientato; in tal caso scrivere $B(x,n)$ al posto
% %di $B^+(x,n)$ --------
% 
% The previous theorem is based on the following result (see \cite[Theorem 3.1]{cf:BZ})
% which gives an interesting sufficient condition for the equality
%  $\lambda_w=1/K_w$.
% If the multigraph satisfies this geometrical condition:
% \[
% \forall \varepsilon>0\, \exists\bar n=\bar n(\varepsilon):
% \sup_{n\le \bar n}
% \sqrt[n]{T_x^{n}}\ge K_w-\varepsilon,\, \forall x\in X
% \]
% then $\lambda_w=1/K_w$.
% Note that, by definition of $K_w$, for all fixed $\varepsilon>0$ and $x \in X$, there exists
% $n_x$ such that $\sqrt[n_x]{T_x^{n_x}}\ge K_w-\varepsilon$.
% The above condition is a request of uniformity in $x$.
% An application of Theorem~\ref{th:Tnk} is given in
% the following example.
% 
% \begin{exmp}\textbf{\cite[Example 3.3]{cf:BZ}}\label{exmp:0}
% Given a sequence of positive natural numbers $\{m_k\}_{k\ge 1}$
% we construct a non-oriented, rooted tree $\T$ (with root $o$) such that if $x\in \T$ satisfies $d(o,x)=k$
% then it has $m_{k+1}$ neighbors $y$ such that $d(o,y)=k+1$ (where $d$ is the natural distance of
% a non-oriented graph). We call this radial graph $T_{\{m_k\}}$-tree.
% If the sequence is periodic of period $b$, then Theorem~\ref{th:Tnk} applies
% with $x_0=o$, $n_0=b$, $Y:=\cup_{n\in\N} S(o,nb)$ (where $S(o,nb)$ is the sphere with center $o$ and radius
% $nb$ with respect to the distance $d$) and  $\varphi_y$
% (where $y \in Y$) maps isomorphically
% the tree $\T$ onto the subtree branching from $y$; in this case
% the global critical parameter $\lambda_w$ of the
% (irreducible) edge-breeding, continuous-time BRW on the tree
% equals $1/K_w$.
% Note that for every periodic sequence the BRW is
% not an $\cF$-BRW, hence Corollary~\ref{cor:globalsurvivalcontinuous}(2) does not apply.
% %We call $T_i$
% %the $T_{\{m^\prime_k\}}$-tree obtained by means of this construction where $m^\prime_k:=m_{k+i-1}$.
% %Roughly speaking we construct $T_1, \ldots, T_{b}$,
% %by using cyclic permutations of the sequence $\{m_1, \ldots, m_{b}\}$. Obviously $\T=T_1$.
% %Since $T_i$ may be mapped into $T_j$ for all $i,j=1, \ldots, b$ (in the sense of Remark~\ref{rem:finitegraph})
% %then $\lambda_w^{T_i}$, $\lambda_s^{T_i}$, $K_w^{T_i}$, and $K_s^{T_i}$ do not depend on $i$.
% %
% %Let us consider the finite cyclic graph $\widetilde Y:=\{y_1, \ldots, y_b\}$ where $n_{y_i y_{i+1}}=n_{y_{i+1} y_i}=1$
% %for all $i=1,\ldots b$ (with the identification $y_{b+1} = y_1$). To each vertex $y_i$ we attach
% %$m_i-1$ copies of $T_{i+1}$ (again with the identification $T_{b+1}=T_1$), each of them by using a two-way edge.
% %We denote this  connected, non-oriented %$\cF$-
% %graph  by $(X,E(X))$; indeed it may be mapped
% %onto the finite multigraph $Y^\prime$ where $Y^\prime=\widetilde Y$ and $n^\prime_{y_i y_{i+1}}=m_i$, $n^\prime_{y_{i+1} y_i}=1$
% %for all $i=1,\ldots b$. Note that $X$ is neither quasi transitive nor regular, unless $m_i = 1$ for all $i$
% %(see Remark~\ref{rem:exmp0} for further properties on the edge-breeding, continuous-time BRWs on these graphs).
% \end{exmp}
% 
% 
% 
% 
% We already observed that local survival depends only on the first moment matrix $M$.
% It is clear that if we investigate the global survival in a class of BRWs where there is a
% one-to-one correspondence
% between first moment matrices and processes (as in the case of continuous-time BRWs), then also
% the global survival depends only
% on $M$. This is also true, by Corollary~\ref{cor:forgotten},
% for a BRW with independent diffusion satisfying %equations~\eqref{eq:particular1} and
% equation~\eqref{eq:counterpart}.
% On the other hand, for a generic BRW,
% according to the following example,
% the global survival does not depend exclusively on $M$; in
% particular, even $M_w(x) %\liminf_{n\to\infty} \sqrt[n]{\sum_y m^{(n)}_{x_0y}}
% >1$ does not imply global survival starting from $x$.
% 
% \begin{exmp}\textbf{\cite[Example 4.4]{cf:Z1}}\label{exm:noext}
% Let $X=\mathbb N$ and consider the family of BRWs $(\mathbb N,\mu)$ with
% $\mu_i=p_i \delta_{n_i \ident_{\{i+1\}}+\ident_{\{i-1\}}} +(1-p_i)\delta_{\bf 0}$ (for all $i\ge 1$) and
% $\mu_0=p_0 \delta_{n_0 \ident_{\{1\}}} +(1-p_0)\delta_{\bf 0}$.
% Roughly speaking, each particle at $i \ge 1$ has $n_i$ children at $i+1$ and $1$ at $i-1$ with probability $p_i$ and
% no children at all with probability $1-p_i$; each particle at $0$ has $n_0$ children at $1$ with probability $p_0$
% and no children at all otherwise.
% 
% According to Theorem~\ref{th:equiv1global}(1) global survival starting from $0$ is equivalent
% to the existence of $z\in [0,1]^{\mathbb N}$, $z(0)<1$,
% such that $G(z|i)\le z(i)$, for all $i$ where
% \[
% G(z|i)=
%  \begin{cases}
% p_i z(i+1)^{n_{i}}z(i-1) +1-p_i & i \ge 1\\
% p_0 z(1)^{n_{0}} +1-p_0 & i =0.
%  \end{cases}
% \]
% The trick is to choose the sequences $\{p_i\}_{i\in \N}$ and $\{n_i\}_{i\in \N}$ such that
% $p_i \to 0$ fast enough and $p_i n_i=2$ for all $i \in \N$; this way, the unique solutions
% of $G(z)\le z$ is $z= \mathbf 1$.
% All the details can be found in \cite[Example 4.4]{cf:Z1}.
% 
% On the other hand, if the BRW is given by
% $\mu_i=1/2 \, \delta_{4 \ident_{\{i+1\}}}+p_i \delta_{\ident_{\{i-1\}}} +(1/2-p_i)\delta_{\bf 0}$ (for all $i\ge 1$) and
% $\mu_0=1/2 \delta_{4 \ident_{\{1\}}} +1/2 \delta_{\bf 0}$ (where $p_i$ is the same as before) then
% the first-moment matrix $M$ is the same as before, but in this case the process
% survives globally
% (the total number of particles dominates a branching process with $\bar \rho=2$).
% Moreover, $M_w(x_0) %\liminf_{n\to\infty} \sqrt[n]{\sum_y m^{(n)}_{x_0y}}
% \ge 2$ since at each step the expected number
% of children $\bar \rho_x$ is at least $2$ for all $x \in X$.
% %Thus, even for irreducible BRWs, global survival does not depend only
% %on the first-moment matrix $M$ and
% % $\liminf_{n\to\infty} \sqrt[n]{\sum_y m^{(n)}_{x_0y}}>1$
% %does not imply, in general, global survival.%, even if the BRW is irreducible.
% \end{exmp}
% 
% Another legitimate question arises from Theorem~\ref{th:equiv1global3}: is it true that $\sum_{y \in X} m_{xy}<1$
% for all $x \in X$ implies global extinction? According to the following example
% (see also \cite[Example 1]{cf:BZ2}), the answer is negative.
% 
% \begin{exmp}\label{exm:4.5}
% We start by considering a reducible BRW.
% Let $X=\N$, $\{p_n\}_{n\in\N}$ be a sequence in $(0,1]$ and suppose that a particle at $n$
% has one child at %of type
% $n+1$ with
% probability $p_n$ and no children with probability $1-p_n$.
% The generating function of this process is
% $\widetilde G(z|n)=1-p_n + p_n z(n+1) $.
% The probability of extinction of this process, starting with one particle at $n$, equals
% $z(n)=1-\prod_{i=n}^{\infty} p_i$ ($z$ is the smallest solution of $\widetilde G(z)=z$); hence it survives wpp,
% if and only if $\sum_{i=1}^\infty (1-p_i) < +\infty$.
% 
% This process is stochastically dominated by the irreducible BRW where
% each particle at  $n\ge1$ has one child at $n+1$ with
% probability $p_n$, one child at $n-1$ with probability $(1-p_n)/2$
% (if $n=0$ then it has one child at $0$ with probability $(1-p_0)/2$)
% and no children at all with probability $(1-p_n)/2$.
% The generating function $G$ can be explicitly computed
% \[
% G(z|n)=
% \begin{cases}
% \frac{1-p_n}{2} + \frac{1-p_n}{2} z(n-1)+ p_n z(n+1) & n \ge 1 \\
% \frac{1-p_0}{2} + \frac{1-p_0}{2} z(0) + p_0 z(1) & n=0.\\
% \end{cases}
% \]
% By coupling this process with the previous one (see \cite[Section 3.3]{cf:Z1})
% or, simply, by applying Theorem~\ref{th:equiv1global}(1)
% ($z(n)=1-\prod_{i=n}^{\infty} p_i$ is a solution of $G(z) \le z$) one can prove that
% $\sum_{i=1}^\infty (1-p_i) < +\infty$ implies global survival. Note that here
% $\sum_{j \in \N} m_{ij} = (1+p_i)/2 <1$; clearly, $M_w(i) %\liminf_{n \to \infty} \sqrt[n]{\sum_{j \in \N} m^{(n)}_{ij}}
% =1$.
% \end{exmp}
% 
% The following example shows how to apply the results of this section
% to the study of a couple of interesting branching processes.
% The first one appears for instance in the proof of \cite[Proposition 3.6]{cf:Muller08-2}
% (see also Corollary~\ref{cor:Mueller08}).
% 
% \begin{exmp}\label{exm:BP1-2}
% Let $\rho$ be a measure on $\N$ with generating function
% $\phi(z):=\sum_{n\in \N} \rho(n) z^n$ and denote by $\bar \rho=\frac{\diff}{\diff z}\phi(z)|_{z=1}$
% (suppose that $\rho(0)<1$).
% Consider the following Galton-Watson branching processes. BP$_1$ is the process
% where each particle gives birth to $n$ children with probability $\rho(n)$ and each
% newborn particle is killed (independently) with probability $1-p$. BP$_2$ is the process
% where each particle is killed (independently) before breeding with probability $1-p$, otherwise it
% gives birth to $n$ children with probability $\rho(n)$. We suppose that $p \in(0,1)$ to avoid trivial
% situations.
% 
% In order to study these two branching processes simultaneously, consider the BRW
% $\{\eta_n\}_{n \in \N}$ defined by $X:=\{1,2\}$ and
% \[
%  \mu_1(a,b):=
% \begin{cases}
%  0 & \textrm{if } a\not = 0 \\
%  \rho(b) & \textrm{if } a = 0 \\
% \end{cases}
% \qquad
%  \mu_2(a,b):=
% \begin{cases}
%  0 & \textrm{if } b\not = 0  \textrm{ or } a\ge 2\\
%  1-p & \textrm{if } a =b= 0 \\
%  p & \textrm{if } a =1,\, b= 0, \\
% \end{cases}
% \]
% where $\mu_j(a,b)\equiv\mu_j(f:f(1)=a,f(2)=b)$.
% Clearly
% \[
%  G(z_1,z_2)=\left ( \phi(z_2), pz_1+1-p
% \right ), \qquad \forall z_1,z_2 \in [0,1].
% \]
% Note that $\{\eta_{2n}(1)\}_{n \in \N}$ and $\{\eta_{2n}(2)\}_{n \in \N}$ are
% realizations of BP$_1$ and BP$_2$ respectively.
% Indeed the 2-step generating function of $\{\eta_n\}_{n \in \N}$ is
% given by $G^{(2)}(z_1,z_2|1)=\phi(pz_1+1-p)$ and $G^{(2)}(z_1,z_2|2)=p\phi(z_2)+1-p$;
% $z \mapsto \phi(pz+1-p)$ is the generating function of BP$_1$ and $z \mapsto p\phi(z)+1-p$
% is the generating function of BP$_2$.
% Note that there is no distinction between local and global survival for the BRW since it is irreducible and
% $X$ is finite. Moreover the survivals of the BRW, the BP$_1$ and the BP$_2$ are all equivalent and, in turn, they are
% equivalent, for instance, to $1 < \frac{\diff}{\diff z} \phi(pz+1-p)|_{z=1}=p \bar \rho$
% (see Example~\ref{ex:branchingprocess}).
% The vector of extinction probabilities
% satisfies $\bar q=G(\bar q)$ that is
% \[
%  \begin{cases}
% \phi(\bar q(2))=\bar q(1) \hfill &\\
% p\bar q(1) +1-p=\bar q(2). \hfill &\\
%  \end{cases}
% \]
% Note that $\bar q(1)$ (resp.~$\bar q(2)$) is also the extinction probability of BP$_1$ (resp.~BP$_2$)
% since $\bar q=(\bar q(1), \bar q(2))$ is also the smallest fixed point of
% $G^{(2)}$ (see Section~\ref{subsec:survivalprob}).
% % \{G^{(n \cdot n_o)}(\mathbf{0})\}_{n \in \N}$ converges to $\bar q$.
% Clearly $\bar q(1)<1$ if and only if $\bar q(2)<1$ and, in this case, $p(\bar q(2)-\bar q(1))= (1-p)(1-\bar q(2))>0$.
% This
% implies that if there is survival then the probability of survival of BP$_1$ is strictly larger
% than the probability of survival of BP$_2$.
% The same result can be obtained by convexity and by the fact that $\phi(1)=1$ which implies
% the following order relation between the generating functions of BP$_1$ and BP$_2$:
% $\phi(pz+1-p) < p\phi(z)+1-p$ whenever $z \not = 1$ and $p\in (0,1)$, whence $\bar q(1)<\bar q(2)$.
% % TOGLIERE nella versione finale
% Finally if we denote by $\bar \alpha$ the probability of extinction of BP$_1$ when $p=1$, that is,
% the smallest solution in $[0,1]$ of $\phi(z)=z$, from the inequality $\phi(\bar q(2))=\bar q(1)< \bar q(2)$ we have
% also $\phi(\bar q(1)) <\phi(\bar q(2)) =\bar(q(1))$. Hence $\bar \alpha < \min(\bar q(1), \bar q(2))$.
% % fine - TOGLIERE nella versione finale
% \end{exmp}
% 
% 
% \subsection{Strong local survival}
% \label{subsec:stronglocal}

% % 
% The main result of this section is the following proposition.
%\smallskip



\subsection{Probabilities of extinction and strong local survival}
\label{subsec:survivalprob}


Define ${\mathbf{q}}_n(x,A)$ as the probability of extinction in $A$ no later than the $n$-th
generation starting with one particle at $x$, namely
${\mathbf{q}}_n(x,A)=\Prob(\eta_k(x)=0,\, \forall k\ge n,\,\forall x\in A)$. It is
clear that $\{{\mathbf{q}}_n(x,A)\}_{n \in \N}$ is a nondecreasing sequence satisfying
\begin{equation}\label{eq:extprobab}
 \begin{cases}
 {\mathbf{q}}_{n}(\cdot,A)=G({\mathbf{q}}_{n-1}(\cdot, A)),& \quad \forall n \ge 1\\
 {\mathbf{q}}_0(x,A)=0, &\quad \forall x \in A,
\end{cases}
\end{equation}
hence there is a limit ${\mathbf{q}}(x,A)=\lim_{n \to \infty} {\mathbf{q}}_n(x, A) \in
[0,1]^X$ which is the probability of local extinction in $A$
starting with one particle at $x$ (see
Definition~\ref{def:survival}). Note that equation~\eqref{eq:extprobab}
defines completely the sequence $\{{\mathbf{q}}_n(\cdot,A)\}_{n \in \N}$ only when
$A=X$ (otherwise one needs the values ${\mathbf{q}}_0(x,A)$ for $x \not \in A$).
%
Since $G$ is continuous we have that ${\mathbf{q}}(\cdot,A)=G({\mathbf{q}}(\cdot,
A))$, hence these extinction probabilities are
fixed points of $G$.


Note that ${\mathbf{q}}(\cdot, \emptyset)= \mathbf 1$.
% and $q(\cdot, \{y\})=q(\cdot, y)$ (see Definition~\ref{def:survival}).
%Define $q_n(x,y)$ as the probability of local extinction, before
%generation $n+1$, at $y$ starting with one particle at $x$. The
%sequence $\{q_n(\cdot, y)\}_{n \in \mathbb N}$ is non decreasing
%in $[0,1]^X$ for all $y \in X$. Moreover it is easy to show that
%$q_{n+1}(x,y)=G(q_n(\cdot,y);x)$, thus the probability $q(x,y)$ of
%local extinction at $y$ starting with one particle at $x$
%satisfies $q(x,y)=\lim_{n \to \infty}q_n(x,y)$ which implies that
%$q(\cdot,y)=G(q(\cdot,y))$. Hence $q(\cdot,y)$ is another fixed
%point for $G$ for every $y \in X$. Clearly
%
Since $\bar {\mathbf{q}}=\lim_{n \to \infty} G^{(n)}(\mathbf{0})$ we have that
$\bar {\mathbf{q}}$ is the smallest fixed point of $G$ in $[0,1]^X$ (see \cite[Corollary 2.2]{cf:BZ2}).
Using the same arguments, one can prove that $\bar {\mathbf{q}}$ is the smallest
fixed point of $G^{(m)}$ for all $m \in \N$.

Note that $A \subseteq B \subseteq X$ implies ${\mathbf{q}}(\cdot,A)
\ge {\mathbf{q}}(\cdot,B)
\ge \bar {\mathbf{q}}$. Since for all finite $A\subseteq X$ we have ${\mathbf{q}}(x,A) \ge 1-\sum_{y \in A} (1-{\mathbf{q}}(x,y))$
then, for any given finite $A \subseteq X$, ${\mathbf{q}}(x,A)=1$ if and only if ${\mathbf{q}}(x,y)=1$ for all $y \in A$.
% 
% Moreover, given a BRW $(X,\mu)$ and $Y\subseteq X$,
% consider $(Y,\nu)$ obtained by killing all particles outside $Y$; in this case
% $q^X(x,A)\le q^Y(x,A)$ for all $x\in Y$, $A\subseteq Y$.

If $x \to x^\prime$ and $A \subseteq X$ then ${\mathbf{q}}(x^\prime,A)<1$ implies
${\mathbf{q}}(x,A)<1$; as a consequence,
if $x \rightleftharpoons x^\prime$ then
%given $A,A^\prime \subseteq X$ such that $y \in A$, $y^\prime \in A^\prime$ we have
%$q(x,A)<1$ if and only if $q(x^\prime,A^\prime)<1$; in particular
%$q(x,y)<1$ if and only if $q(x^\prime,y^\prime)<1$ and one can easily show that
${\mathbf{q}}(x,A)<1$ if and only if ${\mathbf{q}}(x^\prime,A)<1$. Moreover if  $y \rightleftharpoons y^\prime$ 
we have ${\mathbf{q}}(x,y)={\mathbf{q}}(x,y^\prime)$ for all $x \in X$.
% In the irreducible case $q(x,A)<1$  for some $x \in X$ if and only if
% $q(w,A)<1$  for all $w \in X$; in particular
% $\bar q(x)<1$ for some $x \in X$ if and only if $\bar q(w)<1$ for all $w \in X$.
% Moreover $q(x,A)<1$ for some $x \in X$ and a finite $A \subseteq X$ if and only if
% $q(w,B)<1$ for all $w \in X$ and all finite $B \subseteq X$. Indeed, in the irreducible case,
% one can prove that $q(x,A)=q(x,x)$ for all $x \in X$ and every finite $A \subseteq X$:
% since surviving in a finite subset $A$ is equivalent to surviving in at least
% one of its points, then it is enough to prove it in the case $A:=\{y\}$ for $y \in X$;
% in this case the conclusion follows from a Borel-Cantelli argument.

In the irreducible case, if $\rho_x(0)>0$ for all $x \in X$, we have that $\bar {\mathbf{q}}(x)={\mathbf{q}}(x,A)$ for
some $x \in X$ and a finite subset $A \subseteq X$ if and only if $\bar {\mathbf{q}}(y)={\mathbf{q}}(y,B)$ for
all $y \in X$ and all finite subsets $B \subseteq X$ (hence, strong local survival is a common property
of all subsets and all starting vertices).
% Indeed, if $\bar q(x)=1$ then $q(y,B)=1$ for all $y \in X$ and $B \subseteq X$ and there is nothing to prove.
% Suppose that $\bar q(x)=q(x,A)<1$ and $\bar q(y)<q(y,B)$ for some $x,y \in X$ and $A,B \subseteq X$
% finite. By irreducibility $q(x,A)=q(x,x)=q(x,B)$ hence we can assume that $A=B$.
% We know that there is a positive probability that the process, starting from $x$
% has at least one descendant at $y$. There is also a positive probability that all the particles (except one at $y$)
% die out and the progeny of the surviving particle survives globally but not locally at $A$. Thus,
% there is a positive probability, starting from $x$, of surviving globally but not locally at $A$ and this is a contradiction.
Clearly, this may not be true in the reducible case. 
Besides, if we drop the assumption $\rho_x(0)>0$ for all $x \in X$, we might actually
have $\bar {\mathbf{q}}(x)={\mathbf{q}}(x,A)<1$ and $\bar {\mathbf{q}}(y)<{\mathbf{q}}(y,A)$ for some $x,y \in X$ and a finite $A \subseteq X$
 even when the BRW is irreducible (see Example~\ref{exm:irreduciblenotstrongeverywhere}).
Hence, in general, even for irreducible BRWs, 
strong local survival is not a common property of all vertices
 as local and global survival are.

The following theorem, in the case of global survival, gives equivalent conditions for strong local survival
in terms of extinction probabilities.


\begin{teo}\label{rem:strongconditioned}
We observe that the following assertions are equivalent for every nonempty subset $A \subseteq X$.
\begin{enumerate}[(1)]
\item  ${\mathbf{q}}(x, A) = \bar {\mathbf{q}}(x)$, for all $x \in X$;
\item ${\mathbf{q}}_0(x,A) \le \bar {\mathbf{q}}(x)$, for all $x \in X$;
\item  the probability of visiting $A$ at least once starting from $x$ is larger than or equal to the probability of
global survival starting from $x$, for all $x \in X$:
\item for all $x \in X$, either $\bar {\mathbf{q}}(x)=1$ or
the probability of visiting $A$ at least once starting from $x$ conditioned on global survival starting from $x$ is $1$;
\item  for all $x \in X$, either $\bar {\mathbf{q}}(x)=1$ or
the probability of local survival in $A$ starting from $x$ conditioned on global survival starting from $x$ is $1$
(strong local survival in $A$ starting from $x$).
\end{enumerate}
\end{teo}


From this theorem we have that 
if there exists $x \in X$ such that ${\mathbf{q}}(x,A)>\bar {\mathbf{q}}(x)$ (that is, there is a positive probability of global survival and
nonlocal survival in $A$ starting from $x$) then there exists $y \in X$ such that
${\mathbf{q}}_0(y,A)>\bar {\mathbf{q}}(y)$ (that is, there is a positive probability that the colony survives globally starting from $y$ without
ever visiting $A$). Note that, ${\mathbf{q}}_0(x,A)>\bar {\mathbf{q}}(x)$ implies ${\mathbf{q}}(x,A)>\bar {\mathbf{q}}(x)$ but 
the converse is not true.
% Take a BRW on $\N$ with a drift to the right a.s. If $A:=\{x_0\}$ then ${\mathbf{q}}(x,A)=1$ for all $x$ but ${\mathbf{q}}_0(x,A)=0$ for $x \le x_0$.
In particular for a BRW with no death there is strong local survival in $A$ starting from $x$ for all $x \in X$ if and only if
the probability of visiting $A$ is $1$ starting from every vertex.
% This is the BRW counterpart of an analogous result in random walk theory; a vertex $x$
% is transient
% if and only if there exists a vertex $y$ such that with positive probability the walker never visits $x$ starting
%  from $y$.

We note that, \textit{a priori}, there is no order relation between the events
``visiting $A$ at least once starting from $x$'' and ``global survival starting from $x$''. Nevertheless
if, for all $x \in X$,  the probability of ``visiting $A$ at least once starting from $x$'' is larger than or equal to
the probability of ``global survival starting from $x$'' then, by Theorem~\ref{rem:strongconditioned}
we have that the probability of ``global survival starting from $x$ never visiting $A$'' is $0$
and this implies, whenever $\bar {\mathbf{q}}(x)<1$, that there is strong local survival in $A$
starting from $x$.


% An application of the previous theorem to the construction of a BRW which survives globally and
% locally, but not strong locally, is given in Example~\ref{rem:nonstrongandstrong}.
% 
% 
% If $G$ has only one fixed point ${\mathbf{z}} < \mathbf{1}$ then
% $q(\cdot, y)=\mathbf{1}$ or $q(\cdot, y)=\bar q(\cdot)$.
% More precisely, if one can prove that $q(\cdot, y)< \mathbf{1}$ then $q(\cdot,y)=q(\cdot, A)=\bar q(\cdot)$ for all $A \ni y$.
% In this case, global survival starting from $x$ (i.e.~$\bar q(x)<1$) is equivalent to local survival at $y$ starting from $x$ and
% it implies strong local survival at $y$ starting from $x$.
% If for some $y \in X$ we have $q(\cdot,y)= \bar q$ then the global survival starting from $x$ implies
% the strong local survival at $y$ starting from $x$.
% %Also note that $q(\cdot,y)= \bar q$
% %if and only if $q_0(\cdot,y) \le \bar q$.
% 
% 
% For a BRW with independent diffusion such that $\rho_x(n)=\frac{1}{1+\bar \rho_x} (\frac{\bar \rho_x}{1+\bar \rho_x} )^n$,
% we have that the survival probability in $A$, ${\mathbf{v}}_A:=\mathbf{1}-{\mathbf{q}}(\cdot, A)$, satisfies the equality
% $M {\mathbf{v}}_A={\mathbf{v}}_A/(\mathbf{1}-{\mathbf{v}}_A)$
% which is equivalent to $G(\mathbf{1}-{\mathbf{v}}_A)=\mathbf{1}-{\mathbf{v}}_A$ when $G$ is given by equation~\eqref{eq:Gcontinuous}. 
% %  (see equation~\eqref{eq:Gcontinuous}).
% In particular in the continuous-time case we have $\lambda K {\mathbf{v}}_A={\mathbf{v}}_A/(\mathbf{1}-{\mathbf{v}}_A)$.

% \begin{exmp}
% \label{ex:branchingprocess}
% In the case of a Galton--Watson branching process, the generating function is
% $G({\mathbf{z}})=\sum_{n \in \N} \mu(n) {\mathbf{z}}^n$ and its smallest fixed point $\bar q \in [0,1]$
% satisfies $\bar q<1$ if and only if $1< \frac{\diff}{\diff {\mathbf{z}}}G({\mathbf{z}})|_{z=1}=\sum_{n \in \N} n \mu(n)= m_{xx}$
% (cfr.~Assumption~\ref{assump:1} and Remark~\ref{rem:assump1}).
% 
% Moreover denote by $H(t)=\sum_{i=1}^\infty r_i t^i$ the generating function
% of the total number of descendants, that is, $r_i$ is the probability 
% that the total number of descendants (including the original particle) is $i$ for all $i \in \N$. Note that clearly $r_0=0$. 
% It is easy to show that $H(t)=t \cdot G(H(t))$. Clearly $G(1)=1$ and 
% $H(1) \leq 1$ (since there might be a positive probability of an infinite progeny, for instance
% if $G^\prime(1)>1$. More precisely, $H(1)=\sum_{i \in \N} r_i=\bar q$, hence it is the smallest fixed point of $G$.
% The expected number of descendants conditioned on extinction (if $\bar q \not = 0$) is $H^\prime(1)/\bar q$ 
% where $H^\prime(1)$ satisfies
% $H^\prime(1)=G(H(1))+G^\prime(H(1)) H^\prime(1)=G(\bar q)+ G^\prime(\bar q) H^\prime(1)$. 
% Note that, under Assumption~\ref{assump:1}, $G^\prime(\bar q)=1$ if and only if $G^\prime(1)=1$ and this
% implies that $\bar q=1$.
% Hence, %the expected da cui
% %$H^\prime(1)(1-G^\prime(H(1)))=G(H(1))=H(1)=\bar q$.
% %Se il numero medio di figli
% %$m=G^\prime(1) \leq 1$ (che implica $1=\bar q =H(1)$) allora
% %$H^\prime(1)(1-m)=1$. Questo \`e equivalente a
% \[
% H^\prime(1)=
% \begin{cases}
% 1/(1-G^\prime(\bar q)) & \text{if } G^\prime(1) \not = 1 \\
% +\infty & \text{se } G^\prime(1)=1.
% \end{cases}
% \]
% In particular, $H^\prime(1)$ is finite if and only if $G^\prime(1) \not = 1$.
% % then 
% %Se invece $m >1$ allora si ha $H^\prime(1)=\bar q/(1-G^\prime(\bar q)))$;
% %in questo caso $H^\prime(1)/\bar q$, che equivale a $1/(1-G^\prime(\bar q)))$,
% %rappresenta il numero medio di discendenti condizionato all'estinzione.
% 
% If we are dealing with a continuous-time branching process with reproduction rate
% $\lambda k_{xx}$ then $G(z)=1/(1+\lambda k_{xx}(1-z))$ and $\bar q= \min(1,1/\lambda k_{xx})$.
% We see that, according to the general case, $\bar q <1$ if and only if $\lambda k_{xx}>1$.
% \end{exmp}
% 




In the case of an $\mathcal F$-BRW the fixed-points of $G$
have an interesting property stated in the following theorem.
\begin{teo}\label{th:Fgraph}
 Let $(X, \mu)$ be an $\mathcal F$-BRW.
\begin{enumerate} 
 \item 
There exists at most one fixed point ${\mathbf{z}}$ for $G$ such that $\sup_{x \in X} {\mathbf{z}}(x)<1$,
namely ${\mathbf{z}}=\bar {\mathbf{q}}$.
\item
%Suppose that
%for all $y \in Y$ $\mu_y(\mathbf 0)>0$ and, there exist
%$\bar y \in X$, $f \in S_Y$ such that $\nu_{\bar y}(f)>0$ and $\mathcal H (f) \ge 2$.
For all $x \in X$, either
${\mathbf{q}}(\cdot,x) = \bar {\mathbf{q}}(\cdot)$ or $\sup_{w \in X} {\mathbf{q}}(w,x)= 1$. In particular when $(X,\mu)$ is irreducible then
it is either ${\mathbf{q}}(x,x)=\bar {\mathbf{q}}(x)$ for all $x \in X$ or $\sup_{x \in X} {\mathbf{q}}(x,x)= 1$.
\end{enumerate}
\end{teo}
 It is worth noting that, unlike the 
branching process, for a generic irreducible $\mathcal F$-BRW, when $\bar {\mathbf{q}} < \mathbf{1}$,
there might be other fixed points for $G$ (see Examples~\ref{ex:nonstronglocalFBRW}, \ref{ex:nonstronglocalFBRW2} and
Remark~\ref{rem:Spataru}). Nevertheless this cannot happen when $X$ is finite.

\begin{cor}\label{cor:finiteBRW}
If $X$ is finite and the BRW is irreducible then
there are at most two solutions of  $G(\mathbf{z}) \ge \mathbf{z}$ when $\mathbf{z} \ge \bar {\mathbf{q}}$, that is, $\bar {\mathbf{q}}$ and $\mathbf{1}$.
\end{cor}


Using Theorem~\ref{th:Fgraph} we can describe the case when $X$ is finite
(not necessarily irreducible).
Clearly in this case $\bar {\mathbf{q}}(w)={\mathbf{q}}(w, A_w)$ where $A_w:=\{x \in X \colon w \to x\}$.
% $\min_{x \in X\colon  w \to x} {\mathbf{q}}(w,x)$, hence for all $w$ such that $\bar {\mathbf{q}}(w)<1$
% there exists $x$ such that ${\mathbf{q}}(w,x)=\bar {\mathbf{q}}(w)$.
Moreover, %using Theorem~\ref{th:Fgraph}, 
for all $x \in X$ we have that
it is either ${\mathbf{q}}(\cdot, x)=\bar {\mathbf{q}}(\cdot)$ or there exists $w \in X$ such that ${\mathbf{q}}(w,x)=1$. If the BRW is
irreducible (and $X$ is finite) then it is %either
$\bar {\mathbf{q}}(w)={\mathbf{q}}(w,w)$ for all $w \in X$ or ${\mathbf{q}}(w,x)=1$ for all $w,x \in X$.


\begin{cor}\label{pro:qtransitive}
 Let $(X, \mu)$ be an irreducible and quasi-transitive BRW.
%Suppose that
%for all $x \in X$ $\mu_x(\mathbf 0)>0$ and, there exist
%$\bar x \in X$, $f \in S_X$ such that $\mu_{\bar x}(f)>0$ and $\mathcal H (f) \ge 2$.
Then the existence of $x \in X$ such that there is local survival at $x$ (i.e.~${\mathbf{q}}(x,x) < 1$)
implies that there is strong local survival at $y$ starting from $w$ for
every $w,y \in X$ (i.e~${\mathbf{q}}(w,y)=\bar {\mathbf{q}}(w)$).
\end{cor}

Hence for a quasi-transitive, irreducible BRW, whenever there is local
survival, it is a strong local survival; in continuous-time this implies that
there is global and local extinction if $\lambda \in [0, \lambda_w]$,
pure global survival if $\lambda \in (\lambda_w, \lambda_s]$ and
strong local survival if $\lambda \in (\lambda_s, +\infty)$
% (the global and local critical behaviors have been proved in
% \cite{cf:BZ, cf:BZ2}).
(see also Theorem~\ref{th:continuoussurv}).

In the particular case of a quasi-transitive, irreducible BRW with no death
and with independent diffusion, %satisfying equation~\eqref{eq:particular1}
Corollary~\ref{pro:qtransitive} was proved in~\cite[Theorem 3.7]{cf:Muller08-2}.
The proof we give in Section~\ref{sec:proofs} is of a different nature.
Unlike Theorem~\ref{th:Fgraph}, Corollary~\ref{pro:qtransitive} does not hold for every
$\mathcal F$-BRW; indeed, as Examples~\ref{ex:nonstronglocalFBRW} and \ref{ex:nonstronglocalFBRW2}
show, for an irreducible $\mathcal F$-BRW there might be non-strong local survival.
% 
% 
% The proof of this theorem, which can be found in Section~\ref{sec:proofs}, relies on
% Lemmas~\ref{lem:convexity}~and~\ref{lem:generationn} which guarantee the strict convexity of
% the function $G$ evaluated on a line in $[0,1]^X$.
% The existence of an example of an irreducible $\mathcal F$-BRW where $\bar q(x) < q(x,x)<1$ for all $ x \in X$ is given
% in Example~\ref{ex:nonstronglocalFBRW}.
% 
% 
% In particular we can describe the case when $X$ is finite
% (not necessarily irreducible).
% Clearly in this case $\bar q(w)=\min_{x \in X: w \to x} q(w,x)$, hence for all $w$ such that $\bar q(w)<1$
% there exists $x$ such that $q(w,x)=\bar q(w)$.
% Moreover, using Theorem~\ref{th:Fgraph}, for all $x \in X$ we have that
% it is either $q(\cdot, x)=\bar q(\cdot)$ or there exists $w \in X$ such that $q(w,x)=1$. If the BRW is
% irreducible (and $X$ is finite) then it is %either
% $\bar q(w)=q(w,w)$ for all $w \in X$ or $q(w,x)=1$ for all $w,x \in X$.
% %We observe that the previous theorem generalizes Theorem~\ref{th:indep}; moreover, Theorem~\ref{th:qtransitive}
% %can be obtained as a Corollary of Theorem~\ref{th:Fgraph} since for a quasi-transitive BRW $q(\cdot, x)< \mathbf 1$
% %implies $\sup_{w \in X} q(w, x) = \max_{w \in X} q(w, x) < 1$.
% %We recall that there is strong local survival at $A \subseteq X$ starting from $x \in X$ if and
% %only if $\bar q(x)=q(x,A)<1$.
% Recall that, in the irreducible case,  if $\rho_x(0)>0$ for
% all $x \in X$, then strong local survival is a common property of all vertices
% as local and global survival are.


% If we are dealing with a continuous-time BRW, it might happen that if $\lambda$ is small enough or large enough there is
% strong local survival but in a intermediate interval for $\lambda$ there might be global and local
% survival with different probabilities.
% You can find this behavior in the BRW of Example~\ref{rem:nonstrongandstrong}
% which is inspired by Theorem~\ref{rem:strongconditioned}.
% In particular this shows that, unlike local and global survival, strong local survival
% is not \textit{monotonic}.



The following result follows by applying \cite[Theorem 3.1]{cf:MenshikovVolkov} to
the no-death BRW associated to a generic BRW as described in
Section~\ref{subsec:genfun} (hence we omit the proof).
The original result \cite[Theorem 3.1]{cf:MenshikovVolkov} can be recovered from this one by assuming
$\rho_x(0)=0$ for all $x \in X$ which implies that $\bar {\mathbf{q}}={\mathbf{0}}$ and  
$T^{-1}_{\bar {\mathbf{q}}}$ is equal to the identity.

\begin{teo}
\label{th:MenshikovVolkov} 
Let $(X,\mu)$ be an irreducible, globally surviving BRW. %satisfying equation~\eqref{eq:particular1}
%where $\rho_x(0)=0$ for all $x \in X$.
Then there is no strong local survival
if and only if  there exists a finite, nonempty set $A \subseteq X$ and a function ${\mathbf{v}} \in [0,1]^X$ such that
$\bar {\mathbf{q}} \le {\mathbf{v}}$ and
\[ %\begin{equation}%\label{eq:MenshikovVolkov}
 \begin{cases}
  G({\mathbf{v}}|x) \ge {\mathbf{v}}(x), & \forall x \in A^\complement,\\
  (T^{-1}_{\bar {\mathbf{q}}}{\mathbf{v}})(x_0) > \max_{x \in A} (T^{-1}_{\bar {\mathbf{q}}}{\mathbf{v}})(x) & \textrm{for some } x_0 
  \in A^\complement,
 \end{cases}
\] %\end{equation}
where $T^{-1}_{\bar {\mathbf{q}}}{\mathbf{v}}=({\mathbf{v}}-\bar {\mathbf{q}})/(\mathbf{1}-\bar {\mathbf{q}})$.
\end{teo}

% It is worth mentioning at least one result for irreducible BRWs with no death.
% The following proposition gives a general criterion for the strong local survival
% of a BRW with no death and with independent diffusion. %which satisfies equation~\eqref{eq:particular1}.
% 
% 
% \begin{pro}\textbf{\cite[Lemma 3.4]{cf:Muller08-2}}\label{pro:Mueller08}
%  Let $(X,\mu)$ be an irreducible BRW with independent diffusion %satisfying equation~\eqref{eq:particular1}
% where $\rho_x(0)=0$ for all $x \in X$. If for some $c>0$ the set $C:=\{x:q(x,x) \le 1-c\}$
% is visited infinitely often by the BRW, then there is strong local survival.
% \end{pro}
% 
% As a corollary one can prove Proposition~\ref{cor:Mueller08} concerning BRWs which are locally isomorphic
% to branching processes. Note that, since in the previous proposition we deal with BRWs with
% independent diffusion, it is possible to substitute the hypothesis that ``the BRW visits infinitely often
% the set $C$'' with ``the set $C$ is recurrent for the random walk $(X,P)$''.
% 
% In view of the discussion of Section~\ref{subsec:nodeath}, a reasonable hypothesis for a generalization
% of the previous result to a BRW where $\rho_x(0) \neq 0$ for some $x \in X$, could be
% the fact that the BRW visits infinitely often the set $C:=\{x:(1-q(x,x))/(1-\bar q(x)) \ge c\}$
% for some $c>0$.

% 
% It is not possible, in general, to define a critical parameter for the strong local survival since this
% is not a monotone property. More precisely,
% in Example~\ref{rem:nonstrongandstrong} we construct a
% continuous-time BRW, where if $\lambda$ is small enough or large enough there is
% strong local survival but in a intermediate interval for $\lambda$ there is global and local
% survival with different probabilities. This construction relies on
% Theorem~\ref{rem:strongconditioned}.


\subsection{Pure global survival}
\label{subsec:stronglocal}


% Our construction of an example of a continuous-time BRW where there is no
% monotonicity for the strong local behavior relies on the existence of a
% \textit{pure global phase}.
% 
% \subsection{Pure global survival - TENERE CIò CHE SERVE PER DISCUTERE STRONG LOCAL}
% \label{subsec:pureweak}
% 
The idea of \textit{pure global survival} (see Definition~\ref{def:survival}(4)) 
has been first introduced in continuous-time
BRW theory (and, more generally, in interacting
particle theory) to define the situation where $\lambda_s(x)>\lambda_w(x)$. In this case
for every $\lambda \in (\lambda_w(x), \lambda_s(x)]$ there is a positive probability
of global survival starting from $x$ but the colony dies out locally at $x$ almost surely.
% We know that when $\lambda=\lambda_s(x)$ there is local extinction at $x$ due to
% Corollary~\ref{cor:pemantleimproved}, while if
%   $\lambda=\lambda_w(x)$ both global extinction or global survival starting from $x$
% are possible (due to Example~\ref{exm:4} and Corollary~\ref{cor:globalsurvivalcontinuous}(2)
% respectively).
% Hence it is conceivable that when $\lambda=\lambda_w(x)=\lambda_s(x)$
% it might happen that the process dies out locally but survives globally (we do not know
% of any example though). From now on we agree with many authors by %and we stick with
% defining the phase of \textit{pure global survival} at $x$ when $\lambda_w(x)<\lambda_s(x)$.
A necessary condition for the existence of a pure global survival phase starting from $x$
is $K_s(x,x)<K_w(x)$ (see Theorem~\ref{th:continuoussurv}).
According to Theorem~\ref{th:continuoussurv}(3), for an $\mathcal{F}$-BRW this condition is also sufficient.

Clearly for an irreducible, continuous-time BRW, the existence of pure global survival does not depend
 on the starting vertex since
$\lambda_w(x)=\lambda_w$ and $\lambda_s(x)=\lambda_s$ for all $x \in X$.
This is still true for an irreducible discrete-time BRW as a consequence of
Proposition~\ref{pro:maximumprinciple}. Indeed, if ${\mathbf{z}}(\cdot):={\mathbf{q}}(\cdot,A)$
we have that $\widehat {\mathbf{z}}(x)$ can be interpreted as the probability of
local extinction in $A$ conditioned on global survival (starting from $x$).
Thus, according to Proposition~\ref{pro:maximumprinciple}, 
if the BRW is irreducible, then this conditional probability is one everywhere, provided it
is one somewhere. This means that if there is pure global survival starting from some $x$
then there is pure global survival starting from every $x$. 

Theorem~\ref{th:continuoussurv}(4) tells us that
an irreducible, continuous-time $\mathcal{F}$-BRW has a pure global survival phase 
if and only if it is nonamenable. This is not true if the process is not an $\mathcal{F}$-BRW
as shown by Example~\ref{exm:amenable}.
% 
% We observe that in the irreducible case in the pure global survival phase, the colony
% survives globally by clearing all finite subsets $A\subseteq X$. We might think of it
% as a sort of drifting towards some sort of boundary of the graph (we do not want to
% give more details on this).
% 
% 
% \begin{exmp}\label{exmp:pureglobalsurvival}
% The classical examples of amenable and nonamenable graphs are $\Z^d$ (for all $d \in \N$)
% and $\mathbb{T}_d$ (with $d \ge 3$) respectively. These properties
% can be verified by computing explicitly $\lambda_s$ and $\lambda_w$ for the edge-breeding
% continuous-time BRWs on these graphs.
% 
% Let us consider the Euclidean lattice $X=\Z^d$ and the edge-breeding BRW on $X$.
% In this case Corollaries~\ref{cor:pemantleimproved} and \ref{cor:globalsurvivalcontinuous}
% apply and $\lambda_s=\lambda_w=1/2d$ since $\bar \rho=2 \lambda d$.
% Indeed $\Z^d$ is an amenable graph.
% 
% If we consider an edge-breeding BRW on the homogeneous tree $\mathbb{T}_d$, where
% each edge has $d \ge 3$ neighbors, the situation is different. %The graph is nonamenable and
% Easy computations show that $\lambda_w=1/d$ (since the graph is regular) and
% $\lambda_s=1/2\sqrt{d-1}$. Indeed, observe that, in this case, %$\bar \rho=\lambda d$
% $\lambda_s=r/d$
% where
% $r=\max\{t \in \mathbb{R} : F(x,x|t) \le 1\}$ (as explained in Section~\ref{subsec:locisomBP}
% after Proposition~\ref{pro:F-BRW1}). In this case it is easy to find the explicit
% expression of the function $F(x,x|t)= (d-\sqrt{d^2-4(d-1)t^2})/(2(d-1)t)$
% (see, for instance, the proof of \cite[Lemma 1.24]{cf:Woess}), whence
% $r=d/2\sqrt{d-1}$. We have verified by direct computation that $\lambda_w < \lambda_s$.
% \end{exmp}
% 
% 
% we show in the following remark.
% 
% \begin{rem}\textbf{\cite[Remark 3.2]{cf:BZ}}\label{rem:finitegraph}
% For simplicity, in this remark we consider only edge-breeding BRWs
% on multigraphs, that is, continuous-time BRWs on $X$ where $k_{xy}$ is the number of edges
% from $x$ to $Y$ for all $x,y \in X$.
% %If $(X,E(X))$, $(Y,E(Y))$ are two multigraphs and $\psi:Y \to X$
% %is an injective map such that $n_{\psi(x)\psi(y)} \ge \widetilde n_{xy}$ for
% %all $x,y \in Y$ (where $\widetilde n$ refers to $Y$) then
% %$\lambda_w^X \le \lambda_w^Y$,
% %$\lambda_s^X \le \lambda_s^Y$,
% %$K_w^X \ge K_w^Y$, $K_s^X \ge K_s^Y$.
% Note that if $(Y,E(Y))$ is a submultigraph of $(X,E(X))$ then
% $\lambda_w^X \le \lambda_w^Y$,
% $\lambda_s^X \le \lambda_s^Y$,
% $K_w^X \ge K_w^Y$, $K_s^X \ge K_s^Y$.
% 
% Suppose that there exists a finite subset $S \subseteq X$
% such that $X \setminus S$ is the disjoint union of
% a finite number of connected multigraphs $X_1, \ldots, X_n$
% then, the existence of a pure global survival phase on
% $X$, implies the existence of a pure global survival phase on some $X_i$.
% Indeed
% for every
% $\lambda \in (\lambda_w(X),\lambda_s(X))$ the $\lambda$-BRW leaves
% eventually a.s.~the subset $S$.
% Hence it survives (globally but not locally) at least on one connected component; this means that,
% although
% $\lambda_s(X_i) \ge \lambda_s(X)$, $\lambda_w(X_i) \ge \lambda_w(X)$ for all $i=1, \ldots,n$,
% there exists $i_0$ such that $\lambda_w(X_{i_0}) = \lambda_w(X)$. The existence of a pure
% global survival phase on $X_{i_0}$ follows from
% $\lambda_s(X_{i_0}) \geq \lambda_s(X)>\lambda_w(X)=\lambda_w(X_{i_0})$.
% 
% Moreover if there exists a subset $S$ as above such that $\lambda_w(X_i) > \lambda_w(X)$ for all $i$,
% then there is no pure global survival for the BRW on $X$.
% Take for instance
% a graph $(X^\prime,E(X^\prime))$ and $k \in \N$ such that
% $1/k < \lambda_w(X^\prime)$. Attach a \textsl{complete graph} of degree $k$ to a vertex of
% $X^\prime$ by an edge (a complete graph is a finite set where every couple $(x,y)$ is an edge),
% we obtain a new graph $X$ such that $\lambda_s(X)=\lambda_w(X)
% \leq 1/k < \lambda_w(X^\prime)$; hence even if the BRW on $X^\prime$ has a pure global survival phase,
% the BRW on $X$ has none.
% Nevertheless, using the same arguments as in Example~\ref{rem:nonstrongandstrong}, if the original BRW has
% a pure global survival phase, the new one has non-strong local survival.
% \end{rem}
The same example shows that pure global survival is a fragile property of a BRW. Indeed, finite modifications,
such as for an edge-breeding BRW attaching a complete finite graph to a vertex
or removing a set of
vertices and/or edges,
 can create it or destroy it.

\section{Examples}\label{sec:examples}

 The first example shows that there are irreducible amenable BRWs with pure global survival and
irreducible nonamenable BRWs with no
pure global survival (see also \cite{cf:PemStac1}).
% Recall that, for an edge-breeding BRW on a graph (or a multigraph), nonamenability is equivalent
% to the usual nonamenability of the graph.

\begin{exmp}\label{exm:amenable}
In this example we use many times the following argument (which is an adaptation from \cite[Remark 3.2]{cf:BZ}).
Consider a continuous-time BRW adapted to a connected graph $X$, in the sense that
 $k_{xy}>0$ if and only if $(x,y)$ is an edge.
In some cases it is easy to show that  the existence of a pure global survival of a BRW
implies the existence of a pure global survival of the BRW restricted to some subgraph
(where all the rates $k_{xy}$ are turned to $0$ if $x$ or $y$ do not belong to the subgraph). 
Indeed
if $Y$ is a finite subset of $X$ such that $X \setminus Y$ is divided into
a finite number of connected graphs $X_1, \ldots, X_n$ (which is certainly true if
$k_{xy}>0$ is equivalent to $k_{yx} >0$ for all $x,y \in X \setminus Y$), then for every
$\lambda \in (\lambda_w^{X},\lambda_s^{X})$ the BRW on $X$ leaves eventually a.s.~the subset $Y$.
Hence it survives (globally but not locally) at least on one connected component; this means that,
although $\lambda_s^{X_i} \ge \lambda_s^{X}$, $\lambda_w^{X_i} \ge \lambda_w^{X}$ for all $i=1, \ldots,n$
(since $X_i \subseteq X$),
there exists $i_0$ such that $\lambda_w^{X_{i_0}} = \lambda_w^{X}$. The existence of a pure
global survival on $X_{i_0}$ follows from
$\lambda_s^{X_{i_0}} \geq \lambda_s^{X}>\lambda_w^{X}=\lambda_w^{X_{i_0}}$.
Hence if there exists a subset $Y$ as above such that $\lambda_w^{X_i} > \lambda_w^{X}$ for all $i$,
then there is no pure global survival for the BRW on $X$.
% 
% Take for instance
% a graph $(X^\prime,E(X^\prime))$ and $k \in \N$ such that
% $1/k < \lambda_w(X^\prime)$. Attach a complete graph of degree $k$ to a vertex of
% $X^\prime$, we obtain a new graph $X$ such that $\lambda_s^{X}=\lambda_w^{X}
% \leq 1/k < \lambda_w(X^\prime)$; hence even if the BRW on $X^\prime$ has a pure weak phase,
% the BRW on $X$ has none.

Consider an irreducible, edge-breeding continuous-time BRW on the (non-oriented) graph $X$ obtained by attaching to a copy of
 $\N$ one branch $T$ of the homogeneous tree $\mathbb T_3$. % (see Figure~\ref{fig:amenable}).
 The BRW is amenable by the presence of the copy of $\N$.
We claim that $\lambda_s^X=\lambda_s^{\mathbb T_3}$ and $\lambda_w^X = \lambda_w^{\mathbb T_3}$.
Indeed $T\subset X\subset \mathbb T_3$, hence
 $\lambda_s^T\ge \lambda_s^X\ge\lambda_s^{\mathbb T_3}$ and  $\lambda_w^T\ge
 \lambda_w^X \ge \lambda_w^{\mathbb T_3}$.  But by approximation, $\lambda_s^T=\lambda_s^{\mathbb T_3}$.
 Indeed $\lambda_s^T\ge \lambda_s^{\mathbb T_3}$ and does not depend on the starting vertex; moreover
 $T$ contains arbitrarily large balls isomorphic to balls of $\mathbb T_3$, hence by 
\cite[Theorem 5.2]{cf:Z1}\footnote{In \cite[Section 5.1]{cf:Z1}
the hypotheses that $M$ is a nonnegative matrix is missing, even though it is implicitly used.
Moreover \cite[Theorems 5.1 and 5.2]{cf:Z1} hold without the irreducibility hypothesis: the key is to note that,
given a sequence $\{X_n\}_{n \in \N}$ of subsets of $X$ such that $\liminf_n X_n=X$ and defined the sequence of matrices
$\{_nM\}_{n \in \N}$ as $_nM:=(m_{x,y})_{x,y \in X_n}$, one can
prove that, for all $x_0 \in X$, we have $(_nM)_s(x_0,x_0) \rightarrow M_s(x_0,x_0)$ (as defined
in equation~\eqref{eq:generalgeomparam}).}
or \cite[Theorem 3.1]{cf:BZ3}
 their critical local parameters coincide.
Explicit computations show that $\lambda_w^{\mathbb T_3}=1/3<1/2\sqrt{2}=\lambda_s^{\mathbb T_3}$ (there is pure global survival
on $\T_3$).
 Since $\T_3$ can be obtained by attaching three copies of $T$ to a root, the above discussion about surviving on a subgraph, implies
that $\lambda_w^T=\lambda_w^{\mathbb T_3}$.
 % Using Example~\ref{exmp:pureglobalsurvival} 
Then we have
 $\lambda_w^X = \lambda_w^T \le \lambda_w^{\mathbb T_3}<\lambda_s^{\mathbb T_3}=\lambda_s^X$ and there is pure global survival on $X$.
% 
% It is worth mentioning an alternative proof of $\lambda_w^T=\lambda_w^{\mathbb T_3}$ which makes use
% of Theorem~\ref{th:equiv1global2}(4). Indeed $\lambda_w^{\mathbb T_3}=1/3$ and %for $\lambda=1/3$
% it is clear that
% the function $v \in l^\infty_+(\mathbb T_3)$ defined by
% \[
%  v(x):=
% \begin{cases}
%  1 & x=o \\
%  5 - 2^{2-n} & d(x,o)=n\\
% \end{cases}
% \]
% (where $o$ is the root of $T$ and $d(x,o)$ denotes the natural distance on the graph between $x$ and $o$)
% is a solution of $\frac{1}{3} K v =v$; this implies $\lambda_w^{T}\le 1/3$.


On the other hand, consider a nonamenable graph $X^\prime$ such that the corresponding edge-breeding continuous-time
BRW has a pure global survival phase (take for instance $X^\prime:=\mathbb{T}_3$ the homogeneous tree
with degree $3$). %Following \cite[Remark 3.2]{cf:BZ}, a
Attach
to a vertex of $X^\prime$ a complete graph $Y$ with degree $k>1/\lambda_w^{X^\prime}$ by an edge.
% (see Figure~\ref{fig:t3complete}).
It is easy to show that the resulting
graph $X$ is still nonamenable, nevertheless %, according to \cite[Remark 3.2]{cf:BZ}, 
there
is no pure global survival for the corresponding edge-breeding BRW. Indeed, by the above discussion, if there were
pure global survival on $X$ then one of the connected components of $X \setminus Y$ should have the same global critical value
of $X$; but $X\setminus Y=X^\prime$ and $\lambda_w^X \le 1/k<\lambda_w^{X^\prime}$.
Roughly speaking, %since $\lambda_w^X \le 1/k<\lambda_w^{X^\prime}$, %1/3=\lambda_w^{\mathbb T_3}$, 
it happens that for every
$\lambda \in (\lambda_w^X,\lambda_w^{X^\prime})$ the process cannot survive globally in
$X^\prime$ %:=\mathbb{T}_3$ 
hence it hits infinitely often with positive probability the complete
graph, thus $\lambda_s^X=\lambda_w^X$.
%
%Another example can be found in \cite[Section 4]{cf:PemStac1}; here is a sketch
%of the construction. Let $(X,K)$ be an irreducible, edge-breeding continuous-time
%BRW on the (non-oriented) tree $X$ where
%each vertex has degree $m+1$ or $d+1$ with $d>m\ge 2$ (which implies nonamenability).
%Suppose that $m(1+m/d) < 1/2 \sqrt{d}$.
%Let us fix a root $o \in X$ and let us call a vertex $y$ a \textit{child} of a vertex $x$ if
%and only if $x$ and $y$ are neighbors and $d(y,o)=d(x,o)+1$. Using this language, we say that
%each vertex (except for the root $o$) has $d$ or $m$ children. We note that $X$ is a subset of
%a homogeneous tree of degree $d+1$ and
%it contains an homogeneous
%tree of degree $m+1$ as a subset, hence
%$1/2 \sqrt{d} \le \lambda_s \le 1/2\sqrt{m}$ and $1/(d+1) \le \lambda_w \le 1/(m+1)$. Suppose that for all $k$ there is a vertex
%whose descendants for $k$ generations have $d$ children. In this case, according to Theorem~\ref{th:spatial}
%(using similar arguments as in Example~\ref{exm:amenable}), $\lambda_s \le 1/2 \sqrt{d}$; thus
%$\lambda_s = 1/2 \sqrt{d}$. Nevertheless if the vertices with $d$ children are sufficiently rare in the tree
%(in the sense of~\cite[Theorem 2.1(ii)]{cf:PemStac1}) then also $\lambda_w=1/2\sqrt{d}$.
\end{exmp}

% The following result gives a useful sufficient condition for the absence of pure global
% survival for a continuous-time BRW $(X,K)$
% which is based only on the geometry of the graph $(X,E_K)$ generated by the BRW
% (where $E_K:=\{(x,y) \in X^2: k_{xy}>0\}$). It is a slight generalization of \cite[Proposition 2.1]{cf:BZ}.
% 
% \begin{teo}\label{th:subexponentialgrowth}
%  Let $(X,K)$ be a continuous-time non-oriented BRW and let $x_0 \in X$.
% Suppose that there exists $\kappa \in (0,1]^X$ and $\{c_n\}_{n \in \N}$  such that, for all $n \in \N$
% \[
% \begin{cases}
%  \kappa(y)/\kappa(x_0)\le c_n & \forall y \in B(x_0,n) \\
% \kappa(x)k_{xy}=\kappa(y)k_{yx} & \forall x,y \in X, \\
% \end{cases}
% \]
% where $B(x,n)$ is the ball of center $x$ and radius $n$
% w.r.~to the natural distance of the graph $(X,E_K)$.
% If $\sqrt[n]{c_n} \to 1$ and $\sqrt[n]{|B(x_0,n)|} \to 1$ as $n \to 1$
% then $K_s(x_0,x_0)=K_w(x_0)$ and there is no pure global survival starting from $x_0$.
% \end{teo}
% 
% The condition $\sqrt[n]{|B(x,n)|} \to 1$ as $n \to 1$ is usually called \textit{subexponential growth}.
% The previous result applies, for instance to BRWs  on
% $\Z^d$  or $d$-dimensional combs (see \cite{cf:BZ03} for the definition).
% This result extends easily to discrete-time non-oriented BRWs using
% $M_w(x)$ %\liminf_n\sqrt[n]{\sum_ym^{(n)}_{xy}}$
% and $M_s(x,x)$ %$\limsup_n\sqrt[n]{m^{(n)}_{xx}}$
% instead of $K_w(x)$
% and $K_s(x,x)$ respectively.
% 
% \begin{rem}\label{rem:exmp0}
% We can apply the previous arguments to the family of rooted trees in Example~\ref{exmp:0}.
% $X$ is nonamenable if and only if $T_1$ is nonamenable, that is, if and only if there exists $i$ such that
% $m_i\ge 2$. In this case, according to Theorem \ref{th:nonam}, $\lambda_w^{X}< \lambda_s^X$, hence by Remark~\ref{rem:finitegraph}
% (considering $X \setminus \widetilde Y$) there exists
% $i$ such that $\lambda_w^{T_i}< \lambda_s^{T_i}$.
% This means that for all $i= 1, \ldots, b$ we have $\lambda_w^{T_i}<  \lambda_s^{T_i}$ and there is pure
% global survival on $T_i$. On the other hand, if $m_i\equiv 1$ for all $i=1,\ldots,b$ the graph has
% subexponential growth, then there is no
% pure global survival.
% \end{rem}

 
The following example shows that the strong local survival is not monotone.
The counterexample is obtained by modifying the edge-breeding
BRW on a particular graph, namely the homogeneous
tree $\mathbb{T}_d$. The crucial property that we need here is the existence of a pure global survival
phase, thus the procedure applies to every BRW with such a phase.

\begin{exmp}\label{rem:nonstrongandstrong}
Consider the edge-breeding continuous-time BRW on the homogeneous
tree $\mathbb{T}_d$ with degree $d\ge 3$. 
Since the graph has constant degree $d$, the BRW can be seen also as a site-breeding process
where $k(x)=d$ for all $x \in \mathbb{T}_d$.
Hence it is locally isomorphic to a branching process which implies that $\lambda_w(x)=1/K_w(x)=1/d$ for all $x \in \mathbb{T}_d$
and if $\lambda \le 1/d$ then the probabilities of survival are $0$ (see Theorem~\ref{th:continuoussurv}(3)).
%Section~\ref{subsec:local} for details).
Similarly, according to Theorem~\ref{th:continuoussurv}(1), $\lambda_s(x)=1/K_s(x,x)$ which does not depend on $x$. 
By the definition of $P$ and the discussion after equation~\eqref{eq:counterpart}, 
we have that $K_s(x,x) = d \cdot \limsup_{n \to \infty} \sqrt[n]{p^{(n)}(x,y)}$
where $P$ is the diffusion matrix of the simple random walk on $\mathbb{T}_d$. 
Using \cite[Lemma 1.24]{cf:Woess}, we obtain $K_s(x,x)=2 \sqrt{d-1}$ which implies $\lambda_s(x)=1/2\sqrt{d-1}$ for all 
$x \in \mathbb{T}_d$ (and there is global extinction when $\lambda=\lambda_w$).
Hence, 
if $\lambda > 1/2\sqrt{d-1}$ there is strong local survival (see Corollary~\ref{pro:qtransitive})
while 
if $\lambda \in (1/d ,1/2 \sqrt{d-1}]$ the probability
of global survival is positive and
independent of the starting point and the probability of local survival at any finite $A \subseteq X$
is $0$.

Fix $\lambda \in (1/d, 1/2 \sqrt{d-1}]$ and a finite  $A \subseteq X$. According to Theorem~\ref{rem:strongconditioned},
there exists  $x \in X$  such that
there is a positive probability of global survival starting from $x$
without ever visiting $A$ (clearly $x\not\in A$). In this case, any modification of the rates in the subset $A$ provides 
a new BRW such that %, again,
there is still a positive probability of global survival starting from $x$
without ever visiting $A$ (since, the original BRW and the new one coincide until the first hitting time on $A$).
On the other hand, if there is $y \in A$ such that $x \to y$ and we add a loop in $y$
and a rate $k_{yy}>1/\lambda$ then $\bar {\mathbf{q}}(x)<{\mathbf{q}}(x,y)<1$;
the first inequality holds by the above discussion on local modifications
and the second one holds since $\lambda k_{yy}>1$ implies local survival at $y$
(then irreducibility implies local survival at $y$ starting from $x$).
This means that, for this fixed value of $\lambda$, we obtained a locally and globally
(but not strong-locally) surviving BRW at $y$ starting from $x$.

Suppose now that $k_{yy}>d$;
then, as in Example~\ref{exm:amenable},
we have a new BRW such that $\lambda_w^\prime=\lambda_s^\prime \le 1/k_{yy}$. In this case, when $\lambda \le \lambda_w^\prime$
there is global extinction.
When $\lambda > 1/2\sqrt{d-1}$ there is  strong local survival for the new BRW since 
%(by Corollary~\ref{pro:qtransitive}) 
there is strong local survival for the original one
(the probability of hitting $x$ conditioned on global survival is $1$ for both
processes and Theorem~\ref{rem:strongconditioned} applies).
If $\lambda \in (\lambda_w^\prime, 1/d]$ there is local and global survival
with the same probability since in order to survive globally,
the process must visit $x$ infinitely many times (it cannot survive globally in the branches of $\mathbb{T}_d$).
If $\lambda \in (1/d, 1/2\sqrt{d-1}]$ then $k_{yy}>1/\lambda$ and, according to the previous discussion,
% there is a positive probability
%of global survival without ever hitting $x$ for the original BRW and again,
%using Remark~\ref{rem:strongconditioned},
there is non-strong local survival for the new BRW.
\end{exmp}


We show that even in the irreducible case, if $\rho_x(0)=0$ for some $x \in X$, we might have
strong local survival starting from some vertices and not from others.


\begin{exmp}\label{exm:irreduciblenotstrongeverywhere}
 Let us consider a modification of the discrete-time counterpart of the edge-breeding BRW on
$\mathbb{T}_d$ with degree $d\ge 3$ and $\lambda \in (1/d, 1/2\sqrt{d-1}]$ . Let us fix
a vertex $y$; in this modified version we add, with probability one,
one child at $y$ for every particle at $y$. In this case $\bar {\mathbf{q}}(y)={\mathbf{q}}(y,A)=0$ for all $A \subseteq X$.
On the other hand as  in Example~\ref{rem:nonstrongandstrong},
there is a vertex $x$ such that $\bar {\mathbf{q}}(x)<{\mathbf{q}}(x,y)$.
\end{exmp}
\smallskip

In the last few examples we make use of the subclass of BRWs which are
locally isomorphic to a branching process (which are particular
$\mathcal F$-BRWs, see Section~\ref{subsec:FBRWs}). 
By using Theorems~\ref{th:discretesurv} and \ref{th:continuoussurv} and the explicit
computations for $M_s$ and $M_w$ given after equation~\eqref{eq:generalgeomparam},
it is easy to show that for such a process: (1) there is global survival if and only if $\bar \rho>1$;
(2) there is local survival at $x$
if and only if $\bar \rho > 1/\limsup_{n \to \infty} \sqrt[n]{p^{(n)}(x,x)}=:r(x,x)$.
% (that is, 
% if and only if either $F(x,x|\bar \rho) >1$ or $F(x,y| \bar \rho)$ diverges).
% 
% 
% The following result characterizes local and global survival for this class.
% Remember the definition of the diffusion matrix given in Section~\ref{subsec:discrete}
% as $p(x,y):=m_{xy}/\bar \rho_x$; note that, for a BRW with independent diffusion, %has satisfies equation~\eqref{eq:particular1}
% the diffusion matrix coincides with the transition probability matrix $P$.
% In this proposition we denote by
% $F$ the generating function of the hitting probabilities,
% (cfr.~the function $U$ of \cite[Section 1.C]{cf:Woess09}).
% %in particular $F(x,y|t)=+\infty$
% %for all $t>0$ such that the series does not converge and, as usual, $+\infty > a$ for all $a \in \mathbb{R}$.
% 
% \begin{pro}
%  \label{pro:F-BRW1}
% Let the BRW be locally isomorphic to a branching process %(i.e.~$\mathcal F$-BRW where $Y$ is a singleton)
% and denote
% by $\rho$ the common offspring law.
% Then
% \begin{enumerate}[(1)]
%  \item there is global survival if and only if $\bar \rho>1$;
%  \item there is local survival at $x$
% if and only if $\bar \rho > 1/\limsup_{n \to \infty} \sqrt[n]{p^{(n)}(x,x)}$;
%  \item there is local survival at $x$
% if and only if either $F(x,x|\bar \rho) >1$ or $F(x,y| \bar \rho)$ diverges.
% \end{enumerate}
% \end{pro}
% 
% A sketch of the proof can be found in Section~\ref{sec:proofs}.
% We note that there is local survival at $x$ if and only if
% \[
%  \bar \rho > \max\{t \in \mathbb{R}: F(x,x| t) \le 1\} \equiv r(x,x)
% \]
% where $r(x,x)=1/\limsup_{n \to \infty} \sqrt[n]{p^{(n)}(x,x)}$ is the spectral radius %of convergence of the Green function
% of the random walk $P$ %corresponding to the couple of vertices $(x,x)$
% (see \cite[Section 2.C]{cf:Woess09}).
Hence, 
in the irreducible case, there is pure global survival if and only if
$1< \bar \rho \le r$ (where $r=r(x,x)$ in this case does not depend on $x \in X$ due to irreducibility).
This is possible if and only if $r>1$ which is equivalent to nonamenability % (see Theorem~\ref{th:nonam})
since in this case
$%\limsup_{n \to \infty} \sqrt[n]{m^{(n)}_{xy}}
M_s(x,y)= \bar \rho / r$ and
$M_w(x) %\liminf_{n \to \infty} \sqrt[n]{\sum_{y \in X} m^{(n)}_{xy}}
=\bar \rho$.
% Notice that if there is pure global survival then $P$ defines a transient random walk but the converse is not true:
% if $P$ is the simple random walk on $\Z^d$ there is no pure global survival for any $\bar \rho$ and $d$.
% 
It is clear that, given a
continuous-time BRW which is locally isomorphic to a branching process,
$\lambda_w=1/k$ and $\lambda_s(x)=r(x,x)/k$ (where $k=k(x)$ for all $x \in X$).

In general there may be non-strong local survival, even if the BRW is irreducible, 
locally isomorphic to a branching process
and it has independent diffusion %satisfies equation~\eqref{eq:particular1}
as Examples~\ref{ex:nonstronglocalFBRW} and \ref{ex:nonstronglocalFBRW2} show.
This disproves \cite[Theorem 3 and Corollary 4]{cf:Spataru89} (see also Remark~\ref{rem:Spataru}) since
$\bar{\mathbf{q}} < \mathbf{q}(\cdot,0) < \mathbf{1}$ are three distinct fixed points of $G$.
% Before discussing the examples we need an easy lemma, whose proof can be found in Section~\ref{sec:proofs}.
% 
% \begin{lem}
% \label{lem:test0}
%  Let $\{\alpha_i\}_{i \in \N}$ and $\{k_i\}_{i\in \N}$ be such that $\alpha_i \in (-\infty, 1)$ and
% $k_i \ge 0$ for all $i \in \N$. Then
% \[
%         \sum_{i \in \N} k_i \alpha_i < +\infty \Longleftarrow \prod_{i \in \N} (1-\alpha_i)^{k_i}>0.
%        \]
% Moreover if $\alpha_i \in [0, 1)$ and $k_i \ge 1$ eventually as $i \to \infty$ then
% \[
%         \sum_{i \in \N} k_i \alpha_i < +\infty \Longleftrightarrow \prod_{i \in \N} (1-\alpha_i)^{k_i}>0.
%        \]
% \end{lem}

\begin{exmp}\label{ex:nonstronglocalFBRW}
Fix $X:=\N$ and consider a BRW with the following reproduction probabilities.
Every particle has two children with probability $3/4$ and no children with probability $1/4$. Each newborn
particle is dispersed independently according to a nearest neighbor matrix $P$ on $\N$.
More precisely
\[
 p(i,j):=
\begin{cases}
 p_i & \textrm{ if } j=i+1\\
1-p_i  & \textrm{ if } j=i-1,\\
\end{cases}
\]
and $p_0=1$.
%Every particle at $x \in X=\N$, $x \not = 0$ has $2$ children at $x+1$ with probability $3p_x/4$, $2$ children at $x-1$
%with probability $3(1-p_x)/4$ and no children with probability $1/4$; a particle at $0$ has $2$ children at $1$ with
%probability $3/4$ and no children with probability $1/4$.
The process described above is
an irreducible $\mathcal F$-BRW for every choice of the set $\{p_i\}_{i \in \N \setminus \{0\}}$
such that $p_i \in (0,1)$ for all $i>0$.
%Hence ${\mathbf{q}}(x,y)={\mathbf{q}}(x,x)$ for all $x \in \N$ and $\bar {\mathbf{q}}(x) \not = 1$ (resp.~${\mathbf{q}}(x,x) \not = 1$)
%for some $x \in X$ if and only if $\bar {\mathbf{q}}(x) \not = 1$ (resp.~${\mathbf{q}}(x,x) \not = 1$)
%for all $x \in X$.
The generating function of the total number of children is $z \mapsto 3 z^2/4+1/4$ and its minimal
fixed point is $1/3=\bar {\mathbf{q}}(x)$ (for all $x \in \N$).

Choose $p_1 < 5/9$; it is easy to show that the process confined to $\{0,1\}$ (that is, every particle sent outside $\{0,1\}$ is
killed) survives,
since the expected number of children at $0$ every two generations (starting from $0$)
is $ (3/2)^2 (1-p_1)>1$. Since the confined process is stochastically dominated by the original one,
we have local survival, for instance, at $x=0$.
By irreducibility this implies that ${\mathbf{q}}(x,y)<1$ and $\bar {\mathbf{q}}(x)<1$ for all $x,y \in \N$.

Choose the $p_i$s such that $\prod_{i=1}^\infty p_i^{2^i}>0$ (or, equivalently, 
 $\sum_{i=1}^\infty 2^i (1-p_i)< +\infty$).
%Remember the construction of a BRW as a
%markov chain indexed by trees (see for instance \cite[Section 5.C]{cf:Woess09});
%the above inequality implies that conditioned on global survival (that is, conditioned on
%an infinite Galton-Watson tree in the construction) with probability at least $\alpha$
%. Roughly speaking, every realization
Consider the branching process $N_n$ representing the total number of particles alive at time $n$:
for all $n$,  $N_n \le 2^n$ almost surely.
The probability, conditioned on global survival, that every particle places its children (if any) to its right,
is the conditional expected value of $\prod_{i=1}^\infty p_i^{N_i}$.
But
$\prod_{i=1}^\infty p_i^{N_i} \ge \prod_{i=1}^\infty p_i^{2^i}>0$ almost surely.
 Hence, conditioning on global survival
there is a positive probability of non-local survival. This implies ${\mathbf{q}}(\cdot, y) \not = \bar {\mathbf{q}}$
for every $y \in \N$. Note that, according to Theorem~\ref{th:Fgraph}, $\sup_{x \in \N} {\mathbf{q}}(x,x)=1$.
This proves that, even in the irreducible case, the generating function $G$ can have more than two
fixed points (see also Remark~\ref{rem:Spataru}).
% 
% Claim: if $\overline z<\mathbf{1}$ is a non decreasing fixed point of the above $G$ 
% \[
%  G(z|n)=
% \begin{cases}
%  3 \big (p_n z(n+1)+(1-p_n)z(n-1) \big )^2/4+1/4 & n \ge 1 \\
%  3 z(1)^2/4+1/4 & n=0,
% \end{cases}
% \]
% then for every, $z(0) \in (1/3, \overline z(0))$, the recursion
% \[
% z(n+1)=
%  \begin{cases}
%   \sqrt{\frac{4z(n)-1}{3}} \frac{4^n}{4^n-1} -\frac{z(n-1)}{4^n-1} & n \ge 2 \\
% \sqrt{\frac{4z(0)-1}{3}} & n=1 \\
%  \end{cases}
% \]
% defines iteratively a fixed point $z \in [0,1]^\N$ of $G$; thus there are infinitely (uncountably) many
% fixed points (cfr.~Spataru ---INSERIRE COMMENTO E CITAZIONE). 
% 
% Note that the constant vectors 1/3 and 1 are always fixed points of $G$ regardless the choice of $\{p_n\}$.
% An explicit construction of \{p_n\}$ and $\{z(n)\}$ can be done as follows: take $z(0) \in (1/3, 1)$ and $p_1<5/9$, $p_0=1$. 
%  Since $1>\sqrt{({4x-1})/{3}}> x$ for all $x \in (1/3,1)$ then $1>z(1)>z(0)>1/3$. Suppose that $1>z(n) > z(n-1) > \cdots >z(0)>1/3$
% for some $p_0, p_1, \ldots, p_{n-1}$. The equation $x y+(1-x)z(n-2) \uparrow y$ as $x\to 1^-$, hence for every $p$ sufficiently 
% close to 1, $p<1$, we have $3 \big (p z(n+1)+(1-p)z(n-1) \big )^2/4+1/4=z(n)$ implies z(n+1) \in (z(n),1)$; fix $p_n$ like this.
% By induction we have a new fixed point $z$ of this $G$ (associated to the sequence $\{p_n\}$) such that $1/3<z(n)<z(n+1)<1$ for all $n$
% and $\lim_{n \to \infty} p_n=\lim_{n \to \infty} z(n)=1^-$. 
% 
% 
% This construction provides an example of a generating function $G$, which in this case is
% \[
%  G(z|n)=
% \begin{cases}
%  3 \big (p_n z(n+1)+(1-p_n)z(n-1) \big )^2/4+1/4 & n \ge 1 \\
%  3 z(1)^2/4+1/4 & n=0,
% \end{cases}
% \]
% of an irreducible BRW with more
% than 2 fixed points. An explicit example is given by $p_1:=1/2$ and $p_n:=1-4^{-n}$ for all $n \ge 2$.
% In this case, for every $z(0) \in [1/3,1]$ the equation
% \[
% z(n+1)=
%  \begin{cases}
%   \sqrt{\frac{4z(n)-1}{3}} \frac{4^n}{4^n-1} -\frac{z(n-1)}{4^n-1} & n \ge 2 \\
% \sqrt{\frac{4z(0)-1}{3}} & n=1 \\
%  \end{cases}
% \]
% defines iteratively a fixed point $z \in [0,1]^\N$ of $G$; thus there are infinitely (uncountably) many
% fixed points (cfr.~Spataru ---INSERIRE COMMENTO E CITAZIONE). 
% Indeed, for every $z(0) \in (1/3,1)$, the sequence $\{z(n)\}_{n \in \N}$ is strictly increasing and $z(n) \in (1/3,1)$
% for all $n \in \N$
% (while if $z(0) \in \{1/3,1\}$ the sequence is constant. It is easy to show that $1>\sqrt{({4x-1})/{3}}> x$ for all $x \in (1/3,1)$.
% By induction, suppose that $1>z(n)>z(n-1)> 1/3$ (it holds if $n=1$). Then,
% $z(n+1)=\sqrt{({4z(n)-1})/{3}} +   (\sqrt{({4z(n)-1})/{3}}-z(n-1) )/({4^n-1})>z(n)$. On the other hand,
% \textcolor{red}{$1-z(n+1)= 4^n \big (1-\sqrt{1-{4(1-z(n))}/{3}} \big )/(4^n-1)-(1-z(n-1))/(4^n-1)>0$}. Hence $1>z(n+1)>z(n)>z(n-1)> 1/3$.
\end{exmp}

The key in the previous example is that the total number of particles alive at time $n$ is bounded.
This is not an essential assumption. The following example shows that, given any law $\rho$ of a
surviving
branching process (that is, $\bar \rho=\sum_{n \in \N} n \rho(n)>1$), it is possible to construct
an irreducible BRW which is locally isomorphic to a branching process with non-strong local survival.
% (which means that if there is local survival then it is non-strong local survival).

\begin{exmp}\label{ex:nonstronglocalFBRW2}
Let $X=\N$ and $\rho_x:=\rho$ for all $x \in \N$; $\rho$ being the law of a surviving branching process.
We know that $\bar {\mathbf{q}}(x) \equiv \bar q$ for all $x \in \N$
where $\bar q<1$ is the smallest fixed point of $z \mapsto \sum_{n\ in \N} \rho(n) z^n$.
Pick a sequence of natural numbers
$\{N_i\}_{i \in \N}$ satisfying
\begin{equation} \label{eq:defineNi}
\prod_{i \in \N} \rho([0,N_{i+1}])^{\prod_{j=0}^i N_j} >\bar q,
\end{equation}
where $N_0:=1$.
Note that the probability of the event $\mathcal{A}$=``every particle alive at time $i$ has at most $N_{i+1}$ children
for all $i \in \N$'' is bounded from below by the LHS of equation~\eqref{eq:defineNi}.
Thus, from equation~\eqref{eq:defineNi}, with a probability larger than
$\prod_{i \in \N} \rho([0,N_{i+1}])^{\prod_{j=0}^i N_j} -\bar q>0$ the colony survives globally and
the total size of the population at time $n$ is not larger than $\prod_{j=0}^n N_j$
(i.e.~the intersection between $\mathcal{A}$ and global survival has positive probability).

We define a BRW with independent diffusion %satisfying equation~\eqref{eq:particular1}
where $P$ is as follows
\[
 p(i,j)
:=
\begin{cases}
 p_i &  j=i+1,\, i \ge 0\\
1-p_i & j=i-1, \, i \ge 1\\
1-p_0 & i=j=0.\\
%0 & \textrm{otherwise}.
\end{cases}
\]
Let $p_0$ such that $(1-p_0)\bar \rho>1$; this implies local survival.
We choose the sequence $\{p_i\}_{i \in \N}$, where $p_i \in (0,1)$ in such a way that
\begin{equation} \label{eq:definepi}
\prod_{i \in \N} p_i^{\prod_{j=0}^i N_j} >0
\end{equation}
(or, equivalently, 
$\sum_{i \in \N} (1-p_i){\prod_{j=0}^i N_j}< \infty$).
Using equation~\eqref{eq:definepi}, if we condition on $\mathcal A$,
the probability that, every particle places its children (if any) to its right is bounded from
below by $\prod_{i \in \N} p_i^{\prod_{j=0}^i N_j}$.
This implies that there is a positive probability of global, non-local survival.


The choice of the sequences $\{N_i\}_{i \in \N}$ and $\{p_i\}_{i \in \N}$ satisfying equations~\eqref{eq:defineNi} and
\eqref{eq:definepi} respectively can be done as follows.
Choose a sequence $\{\alpha_i\}_{i \in \N}$ such that $\alpha_i \in (0,1)$ for all $i \in \N$ and
$\prod_{i \in \N} \alpha_i >1-\bar q$. Then, iteratively, if we fixed $N_0, \ldots, N_k$, since $\lim_{x \to \infty}\rho([0,x])=1$
there exists $N_{k+1} \in \N$ such that $\rho([0,N_{k+1}])>\alpha_{k+1}^{1/\prod_{j=0}^k N_j}$.
Let us take, for instance, $p_i > 1/(i \cdot {\prod_{j=0}^i N_j})$.
%This is equivalent, in the construction of the branching random walk as a random walk indexed by trees,
%to the existence of a set (of positive probability) $\mathcal{A}$ of Galton-Watson trees where the number of descendant of a
%vertex at distance $k$ from the root is at most $N_{k+1}$.

We note that the class constructed in this example includes
discrete-time counterparts of continuous-time BRWs where
$\rho$ can be chosen as in equation~\eqref{eq:counterpart} where $k(x) \equiv k$ does not depend on $x$,
$k_{xy}:=k \cdot p(x,y)$ (where $P$ is defined as before) and $\lambda>\lambda_s$ is fixed.
Finally we observe that this example extends naturally to an example
of a site-breeding BRW on a radial tree where the number of branches of a vertex
at distance $k$ from the root is at least $1/p(k,k+1)$.
\end{exmp}

Even though the local extinction probability ${\mathbf{q}}(\cdot,y)$ (for any fixed $y \in \mathbb{N}$)
of Examples~\ref{ex:nonstronglocalFBRW} and \ref{ex:nonstronglocalFBRW2} provides
a fixed point which is different from both $\overline {\mathbf{q}}$ and $\mathbf{1}$ for the function $G$
of an irreducible BRW, in the following example we give a more explicit construction of such a fixed point.
\begin{rem}\label{rem:Spataru}
Consider a generating function 
\begin{equation}\label{eq:explicit}
 G({\mathbf{z}}|n)=
\begin{cases}
 3 \big (p_n {\mathbf{z}}(n+1)+(1-p_n){\mathbf{z}}(n-1) \big )^2/4+1/4 & n \ge 1 \\
 3 {\mathbf{z}}(1)^2/4+1/4 & n=0,
\end{cases}
\end{equation}
on $[0,1]^\mathbb{N}$ where $p_n \in (0,1)$ for all $n \in \mathbb{N}$.
This is the generating function of an irreducible BRW and
the constant vectors $\overline {\mathbf{q}} \equiv 1/3$ and $\mathbf{1}$ 
are always fixed points of $G$ regardless of the choice of $\{p_n\}$.
An explicit construction of $\{p_n\}$ and of a third fixed point ${\mathbf{z}} \in (0,1)^X$ can be 
carried out recursively as follows. Take ${\mathbf{z}}(0) \in (1/3, 1)$,  $p_0=1$ and $p_1<5/9$.
The explicit expression of the equation $G({\mathbf{z}})={\mathbf{z}}$ is easily derived from equation~\eqref{eq:explicit}.
  Since $1>\sqrt{({4x-1})/{3}}> x$ for all $x \in (1/3,1)$ then $1>{\mathbf{z}}(1)>{\mathbf{z}}(0)>1/3$. 
Suppose that $1>{\mathbf{z}}(n) > {\mathbf{z}}(n-1) > \cdots >{\mathbf{z}}(0)>1/3$
for some $p_0, p_1, \ldots, p_{n-1} \in (0,1)$.
Choose ${\mathbf{z}}(n+1)<1$ such that $3 {\mathbf{z}}(n+1)^2/4+1/4 >{\mathbf{z}}(n)$. 
% Since $p {\mathbf{z}}(n+1)+(1-p){\mathbf{z}}(n-1) \to {\mathbf{z}}(n+1)$ as $p\to 1^-$, 
By continuity, there exists $p_n<1$ such
that $3 \big (p_n {\mathbf{z}}(n+1)+(1-p_n){\mathbf{z}}(n-1) \big )^2/4+1/4={\mathbf{z}}(n)$.
By induction we have a new fixed point ${\mathbf{z}}$ of this $G$ (associated to the sequence 
$\{p_n\}$) such that $1/3<{\mathbf{z}}(n)<{\mathbf{z}}(n+1)<1$ for all $n$
and $\lim_{n \to \infty} p_n=\lim_{n \to \infty} {\mathbf{z}}(n)=1^-$. 

This disproves \cite[Theorem 3 and Corollary 4]{cf:Spataru89}. Indeed there is a
gap in the proof of 
\cite[Theorem 3]{cf:Spataru89}: in the line 8 of the proof, the sentence ``Clearly 
$B^\prime \not = \emptyset$'' is incorrect when the set $X$ is infinite as the following example shows.
%
% First of all $C$ is a compact set with empty interior if $X$ is infinite. Hence
% the topological boundary of $C$ is $C$ itself.
%
Using the notation of \cite{cf:Spataru89} take ${\mathbf{x}}, {\mathbf{t}} \in [0,1]^X$ such that
${\mathbf{x}}$ is constant, say ${\mathbf{x}}(i)=a<1$ for all $i \in X$ and $1>{\mathbf{t}}(i)>a$ for all $i \in X$. Suppose that
$\sup_{i \in X} {\mathbf{t}}(i)=1$. Then the half-line $\{{\mathbf{x}} + \theta({\mathbf{t}}-{\mathbf{x}})\colon \theta \ge 0\}$ exits from
the set $[0,1]^X$ at $\theta=1$, that is, at the  point ${\mathbf{t}}$. 
Indeed if $\theta >1$ then $\sup_{i \in X} ({\mathbf{x}}(i)+\theta({\mathbf{t}}(i)-{\mathbf{x}}(i)))
=\theta \sup_{i \in X} {\mathbf{t}}(i) -a(\theta-1)=\theta -a(\theta-1)=1+(\theta-1)(1-a)>1$.
But $B^\prime:=\{ i \in X \colon {\mathbf{t}}(i)=1\}=\emptyset$; roughly speaking, in this case 
there is not a smallest
value for $\theta \ge 0$ such that some coordinates of the point ${\mathbf{x}}+\theta({\mathbf{t}}-{\mathbf{x}})$ are $1$.  
\end{rem}

% The following easy theorem gives another sufficient condition for the strong local survival
% of a BRW which is locally isomorphic to a branching process. The proof can be found in Section~\ref{sec:proofs}.
% 
% \begin{teo}\label{th:recurrent} - DA TENERE
% Suppose that $(X,\mu)$ is an irreducible BRW with independent diffusion
% %satisfying equation~\eqref{eq:particular1} %(2.3) of \cite{cf:Zu}
% such that
% $P$ is the transition matrix of a recurrent random walk
% and $\rho_x=\rho$ for all $x \in X$. %There are no fixed points for $G$ outside $\{\bar q, \mathbf 1\}$.
% %Moreover  % $q(x,y)$ does not depend on $x$ and it is a fixed point of $F(t)=\sum_{n=0}^n \rho(n) t^n$;
% % hence if
% %$q(\cdot,y)<\mathbf 1$ then $q(\cdot,y)=\bar q$.
% %$q(\cdot,y)=\bar q$.
% Then, global survival starting from some $x \in X$ implies strong local survival at $y$ starting from $w$
% for  all $w,y \in X$.
% \end{teo}
% 
% 
% 
% In case of a BRW with no death and with independent diffusion %satisfying equation~\eqref{eq:particular1}
% one can prove the following proposition which makes use of Proposition~\ref{pro:Mueller08}.
% By $U(x,y|z)$ we mean the usual generating function of the first-return probabilities of the
% random walk $P$ as defined in \cite[equation (1.26)]{cf:Woess09}; in particular
% $U(x,y)$ is the probability of visiting $y$ \textit{after} starting from $x$.
% 
% \begin{pro}\textbf{\cite[Proposition 3.6]{cf:Muller08-2}}\label{cor:Mueller08}
%  Let $(X,\mu)$ be an irreducible BRW with independent diffusion %satisfying equation~\eqref{eq:particular1}
% where $\rho_x=\rho$ for all $x \in X$ and $\rho(0)=0$.
% If $\bar \rho > \sup_{x \in X} 1/U(x,x)$ then there is strong local survival.
% \end{pro}
% 
% A result like this one could be proved without the no-death hypothesis
% using the comparison described in
% Section~\ref{subsec:nodeath}; in this case a reasonable hypothesis should be
% $\bar \rho > \sup_{x \in X} 1/U(x,x|1-\rho(0))$.
% 
% 
% \begin{rem}
%  \label{rem:dominatingBP}
% In order to extend some results, as Proposition~\ref{cor:Mueller08} for instance, from
% the homogeneous case ($\rho_x=\rho$ for all $x \in X$) to the inhomogeneous case,
% we have to find sufficient conditions such that the infimum of the probabilities of
% survival of the branching processes with laws $\{\rho_x\}_{x \in X}$ is strictly larger than 0.
% %To this aim,
% %remember the usual definition of \textit{stochastic order}: $X \succeq Y$ if and
% %only if $\pr(X \ge t) \ge \pr(Y \ge t)$ for all $t \in \mathbb R$.
% %Equivalently, $X \succeq Y$ if and
% %only if for all nondecreasing, positive,
% %%
% %%(on one side it suffices to take $f:=\ident_{[t, +\infty)},$ on the other hand
% %%notice that any measurable, positive, nondecreasing $f$ is the limit of
% %%simple functions $\sum_i a_i \ident_{[t_i, \infty)}$).
% Observe that if $\rho \succeq \widehat \rho$ then for all nondecreasing, positive,
% measurable function $f$ we have $\sum_{n \in \N} f(n) \rho(n) \ge
% \sum_{n \in \N} f(n) \widehat \rho(n)$ which, in turn,
% implies $\sum_{n \in \N} z^n \rho(n)
% \le \sum_{n \in \N} z^n \widehat \rho(n)$ for all $z \in [0,1]$.
% Hence if $\sum_{n \in \N} z^n \widehat \rho(n) \le z$ then $\sum_{n \in \N} z^n \rho(n)\le z$.
% Thus if  $\rho_x \succeq \widehat \rho$ for all $x \in X$
% then the probabilities of extinction of the branching processes associated to
% the $\rho_x$s are all dominated by the probability
% of extinction of the branching process associated to $\widehat \rho$.
% Hence one possibility would be to assume that for all $x \in X$,
% $\widehat \rho_x$ dominates some law of a surviving branching process $\widehat \rho$. This way the results
% for inhomogeneous BRWs are simply corollaries of the homogeneous case. Clearly this is not a
% significant improvement.
% 
% One might guess that if the expected number of offsprings $\bar \rho_x$ is sufficiently large then
%  the supremum of the probabilities of
% extinction of the branching processes with laws $\{\rho_x\}_{x \in X}$ is strictly smaller than 1.
% But clearly, if $\sup_{x \in X} \rho_x(0)=1$ then the supremum of the probabilities of
% extinction is $1$. Even bounding $\rho_x(0), \ldots, \rho_x(k)$ (for all $x \in X$) is not sufficient.
% Indeed consider the set $\mathcal A$ of all probability generating functions
% and a subset $\mathcal A_{a_0,a_1; \ldots, a_k,m}$ defined as
% \[
%  \begin{split}
%   \mathcal A&:=\{\phi(z)=\sum_i \rho(i) z^i: \rho(i) \ge 0 \ \forall i, \ \sum_i \rho(i)=1\}\\
% \mathcal A_{a_0,a_1, \ldots, a_k,m}&:=\{\phi\in \mathcal A:\phi(z)=\sum_i \rho(i) z^i, \  \rho(i) \le a_i \
% \forall i \le k, \ \bar \rho  %=\sum_i i \rho(i)
% \ge m\}\\
%  \end{split}
% \]
% (where $a_i \in [0, 1]$ and $m \in \mathbb R$, $m>1$).
% %To each $\phi \in \mathcal A$ there corresponds
% %a branching process whose generating function is $\phi$
% %(and the law is $\rho$, where $\{\rho(n)\}_{n \in \N}$ are the coefficients of $\phi$).
% %One could conjecture that if $m$ and $k$ are large enough and $\{a_i\}_{i \le k}$ are small enough
% %then the supremum of the probabilities of extinction might be strictly smaller than $1$.
% %Unfortunately,
% In this case, either there is a surviving branching process
% (which does not necessarily belong to $\mathcal A_{a_0,a_1, \ldots, a_k,m}$)
% whose law is stochastically dominated by all
% $\rho \in \mathcal A_{a_0,a_1, \ldots, a_k,m}$
% or the supremum of the probabilities of extinction is $1$.
% 
% %---Inizio della proof che si pu\`o togliere dalla versione finale----
% 
% In order to prove this claim,
% let $k_0:= \max \{i=0, \ldots, k: \sum_{j=0}^k a_i < 1\}$
% and %Note that $\mathcal A_{a_0,a_1, \ldots, a_k,m}=\mathcal A_{a_0,a_1, \ldots, a_{k_0},m}$
% %since, in this case, the remaining constraints are dependent. Define
% define $\widehat \phi(z):= \sum_{i=0}^{k_0} a_i z^i+ (1-\sum_{i=0}^{k_0} a_i)z^{k_0+1}$.
% Clearly the branching process corresponding to $\widehat \phi$ (which might not belong
% to $\mathcal A_{a_0,a_1, \ldots, a_k,m}$) is dominated
% by every branching process in $\mathcal A_{a_0,a_1, \ldots, a_k,m}$
% and $\widehat \phi \le \phi$ in $[0,1]$ for every $\phi \in \mathcal A_{a_0,a_1, \ldots, a_k,m}$.
% This implies that the minimal fixed point, $c \le 1$, of $\widehat \phi$ is an upper bound
% of the minimal fixed point of
% $\phi \in \mathcal A_{a_0,a_1, \ldots, a_k,m}$.
% If $c<1$, that is, $\sum_{i=0}^{k_0} ia_i + (1-\sum_{i=0}^{k_0} a_i)(k_0+1)>1$
% (i.e.~the branching process survives) there is nothing to prove.
% On the other hand, if $c=1$ then consider
% \[
% \phi_N=\sum_{i=0}^{k_0} a_i z^i+ w_N z^{k_0+1} +(1-w_N-\sum_{i=0}^{k_0} a_i)z^N
% \]
% where
% $w_N=(\sum_{i=0}^{k_0} i a_i - m +(1-\sum_{i=0}^{k_0} a_i)N)/(N-(k_0+1)) \in [0, 1-\sum_{i=0}^{k_0} a_i]$ eventually
% as $N \to \infty$
% (remember that $c=1$ if and only if $1 \ge \frac{\diff}{\diff z} {\widehat \phi}(z)|_{z=1}
% \equiv \sum_{i=0}^{k_0} ia_i + (1-\sum_{i=0}^{k_0} a_i)(k_0+1)$).
% Then $\frac{\diff}{\diff z} \phi_N(z)|_{z=1}=m$, $\phi_N \in \mathcal A_{a_0,a_1, \ldots, a_k,m}$
% and $\lim_{N \to \infty} \phi_N \equiv \widehat \phi$.
% It is straightforward to show that the minimal fixed point $c_N$ of $\phi_N$ converges
% to $c=1$ as $N \to \infty$.
% 
% %---Fine della proof che si pu\`o togliere dalla versione finale----
% 
% \end{rem}

% \section{Approximation}
% \label{sec:approx}
% 
% 
% 
% \subsection{Spatial approximation}
% \label{subsec:spatial}
% 
% The first kind of approximation is based upon a result on approximation of nonnegative matrices
% which is interesting in itself.
% Recall the usual classification of indices of a matrix $M=(m_{xy})_{x,y \in X}$ (which is supposed to be
% nonnegative throughout this section)
% as described in \cite[Chapter 1]{cf:Sen}.
% For any index $x$ we denote by $[x]$ its \textit{class}, that is, the set of indices
% which communicate with $x$.
% We define the convergence parameters $R(x,y):=M_s(x,y)%\left (\limsup_{n \in \N} \sqrt[n]{m^{(n)}_{xy}} \right )
% ^{-1}$
% and $R:= \inf_{x,y \in X} R(x,y)$.
% %(where $m^{(n)}_{ij}$ are the entries of the matrix $M^n$);
% It is straightforward to show that $M_s(x,y)=M_s(x_1,y_1)$
% %$\limsup_{n \in \N} \sqrt[n]{m^{(n)}_{xy}}=\limsup_{n \in \N} \sqrt[n]{m^{(n)}_{x_1y_1}}$
% if
% $[x]=[x_1]$ and $[y]=[y_1]$; %does not depend on the choice of $i,j$ inside a class,
% this implies that for irreducible matrices, $R(x,y)$ is independent of $x,y \in X$.
% 
% %Suppose $I=\N$ and
% Let $\{X_n\}_{n \in \N}$  be a %nondecreasing
% sequence of subsets of $X$ %such that $\bigcup_{n \in \N} I_n = \N$
% and denote by $_nR$ the convergence parameter of  $M_n=(m_{xy})_{x,y \in X_n}$;
% clearly, if the sequence $\{X_n\}_{n \in \N}$ is nondecreasing, we have that $_nR \geq {_{n+1}R}$.
% The following theorem  generalizes \cite[Theorem 6.8]{cf:Sen}
% (note that the submatrices $\{M_n\}_{n \in \N}$ are not necessarily irreducible); it
% is the key to prove our main
% result about spatial approximation (Theorem~\ref{th:spatial}).
% 
% 
% \begin{teo}\textbf{\cite[Theorem 5.1]{cf:Z1}}\footnote{We observe that in \cite[Section 5.1]{cf:Z1}
% the hypotheses that $M$ is a nonnegative matrix is missing, even though it is implicitly used.}\label{th:genseneta}
%  Let $\{X_n\}_{n \in \N}$  be a general sequence of subsets of $X$ such that
% $\liminf_{n \to \infty} X_n =X$ and suppose that $M=(m_{xy})_{x,y \in X}$ is a nonnegative matrix.
% Then for all $x_0 \in X$ we have $_nR(x_0,x_0) \rightarrow R(x_0,x_0)$.
% Moreover if $M$ is irreducible and $M_n=(m_{xy})_{x,y \in X_n}$ then $_nR \rightarrow R$ as $n \to \infty$ and, in particular,
% for all $x_0 \in X$ we have $_nR(x_0,x_0) \rightarrow R$.
% \end{teo}
% 
% 
% Note that in the previous theorem the subsets $\{X_n\}_{n \in \N}$ can be chosen
% arbitrarily; in particular they may be finite proper subsets. Moreover
% the result extends easily to the case of a sequence of nonnegative matrices
% $M_n=( {m(n)}_{xy})_{x,y \in X_n}$ where $\liminf_{n \to \infty} X_n =X$,
% $0 \le {m(n)}_{xy} \le m_{xy}$ for all $x,y \in X_n$ and
% $\lim_{n \to \infty} {m(n)}_{xy} =m_{xy}$ for all $x,y \in X$
% (note that $m(n)_{xy}$ is eventually well defined for all $x,y \in X$ as $n \to \infty$).
% The idea of the proof is essentially contained in \cite[Theorem 5.2]{cf:Z1}.
% 
% 
% 
% %We stated the results of the previous section considering
% %matrices with natural indices in order to keep the same notation
% %as in \cite{cf:Sen}. %The same results still hold if
% %Here we consider a generic (at most countable) set $X$ instead of $\N$.
% 
% %Description of the order relation between BRWs and notation
% %for the following theorem.
% 
% Given a sequence of BRWs $\{(X_n,\mu_n)\}_{n \in \N}$ such that $\liminf_{n \to \infty} X_n=X$,
% %where the subsets $\{X_n\}_{n \in \N}$ satisfy $\liminf_{n \to \infty} X_n =X$.
% %The BRWs $\{(X_n,\mu_n)\}_{n \in \N}$
% we define $m(n)_{xy}:=\sum_{f \in S_{X_n}} f(y) \mu_{n,x}(f)$ and
% the corresponding sequence of matrices $\{M_n\}_{n \in \N}$. % of $M$ where $M_n=(m(n)_{xy})_{x,y \in X_n}$.
% Note that in the following result we are not assuming that the BRW is irreducible.
% 
% \begin{teo}\textbf{\cite[Theorem 5.2]{cf:Z1}}\label{th:spatial}
% Let us %$(X,E_\mu)$ be connected and
% fix a vertex $x_0 \in X$. If $\liminf_{n \to \infty} X_n=X$ and
% $m(n)_{xy} \le m_{xy}$ for all $x,y \in X_n$, $n \in \N$ and
%  $m(n)_{xy} \to m_{xy}$ as $n \to \infty$ then
% \begin{enumerate}[(1)]
% \item
% $(X,\mu)$ dies out locally (resp.~globally) a.s.~starting from $x_0$ $\Longrightarrow$ $(X_n,\mu_n)$ dies out locally (resp.~globally)
% a.s~starting from $x_0$  for all $n \in \N$;
% \item
% $(X,\mu)$ survives locally starting from $x_0$  $\Longrightarrow$ $(X_n,\mu_n)$ survives locally
% starting from $x_0$ eventually as $n \to\infty$.
% \end{enumerate}
% \end{teo}
% 
% \noindent Clearly the discrete-time counterpart of a spatially
% confined BRW in continuous-time
% is obtained by spatially confining the discrete-time counterpart
% of the continuous-time BRW.
% Hence, Theorem~\ref{th:spatial} yields an analogous result for BRWs in continuous time.
% 
% \begin{cor}\textbf{\cite[Theorem 3.1]{cf:BZ2}}\label{cor:sen2}
% Let $(X,K)$ be a continuous-time BRW and
% let us consider a sequence of continuous-time BRWs $\{(X_n, {K_n})\}_{n\in \N}$
% such that $\limsup_{n \to \infty} X_n = X$.
% Let us suppose that
% $k_{xy}(n)  \le k_{xy}$ for all $n\in\N$, $x,y\in X_n$
% and $k_{xy}(n) \to k_{xy}$ as $n \to \infty$ for all $x,y\in X$.
% %\hfill\break
% Then $\lambda_s(X_n, {K_n})\ge \lambda_s(X,K)$ and
% $\lambda_s(X_n, {K_n})
% \to \lambda_s(X,K)$ as $n\to \infty$.
% \end{cor}
% 
% 
% 
% Among all possible choices of the sequence
% $\{(X_n,\mu_n)\}_{n \in \N}$ there is one which is
% \textit{induced} by $(X,\mu)$ on the subsets $\{X_n\}_{n\in \N}$;
% more precisely, one can take $\mu_n(g):=\sum_{f \in S_X:f|_{X_n}=g} \mu_x(f)$
% for all $x \in X_n$ and $g \in S_{X_n}$.
% Roughly speaking, this choice means that all reproductions outside $X_n$ are suppressed.
% In this case it is simply
% $m(n)_{xy}=m_{xy}$ for all $x,y \in X_n$.
% 
% 
% Since
% Theorem~\ref{th:spatial} deals with local survival, one can wonder what
% can be said about global survival. First of all, if the process $(X,\mu)$ survives %both
% globally and locally then eventually $(X_n,\mu_n)$ survives locally and thus globally.
% The question is nontrivial when $(X,\mu)$ survives globally but not locally, which we assume henceforth in this brief discussion.
% In this last case,
% if, for instance, $X_n$ is finite for every $n \in \N$ and the graph $(X_n, E_{\mu_n})$ is connected
% then, by Theorem~\ref{th:spatial}(1),  $(X_n,\mu_n)$ dies out (locally and globally) a.s.~for all values of $n \in \N$.
% On the other hand, the case where $X_n$ is finite for every $n \in \N$ and the graph $(X_n, E_{\mu_n})$ is not connected
% is more complicated and can be treated as in \cite[Remark 4.4]{cf:BZ2}.
% When $X_n$ is infinite for infinitely many values of $n$,
% it is possible that there is no global survival for infinitely many values of $n$.
% An example in the discrete-time case can be found in \cite[Remark 5.3]{cf:Z1} while
% an example in the continuous-time case can be constructed using \cite[Remark 3.2]{cf:BZ}.
% Taking the couple $(X_n,\mu_n)$ random, some results can be achieved as an
% application of Theorem~\ref{th:spatial} (see for instance
% \cite[Theorem 7.1]{cf:BZ3} or \cite[Theorem 2.4]{cf:GMPV09}).
% 
% 
% \begin{exmp}\label{exmp:applicationspatial}
% Consider the edge-breeding (continuous-time) BRW on $\Z^d$.
% We saw in Example~\ref{exmp:pureglobalsurvival} that if
% $\lambda>\lambda_s=1/2d$ then there is local survival.
% Suppose that $d>1$ and that $\lambda \le 1/2$ and
% consider the infinite cylinder $X_n:=\{x \in \Z^d : |x(i)| \le n, \, \forall i=2, \ldots, d\}$.
% It is clear that there is no local survival for the BRW restricted to $X_0=\Z \subseteq \Z^d$,
% nevertheless according to Corollary~\ref{cor:sen2} there exists $n_0$ such that
% there is local survival on $X_n$ for all $n \ge n_0$.
% This shows a difference between random walks and BRWs: the simple random walk on $X_n$ is recurrent for all $n\in \N$
% (as the simple random walk on $\Z$); on the other hand, while the BRW restricted to $X_0=\Z$ dies out locally, it
% survives when restricted to
% $X_n$ if $n$ is sufficiently large (in some sense the BRW on $X_n$ approaches the  BRW on $\Z^d$ as $n$ tends to infinity).
% 
% Another consequence of Theorem~\ref{th:spatial} is the following
% (see also \cite[Section 5]{cf:Z1}):
% consider the edge-breeding continuous-time BRW on $\Z^d$
% (but the argument extends easily to any translation invariant BRW in discrete
% and continuous time). Let us choose a connected subset
% $Y \subset \Z^d$ such that every finite
% subset $A \subset \Z^d$ is a subset of a suitable translation of $Y$ in $\Z^d$.
% Then the strong critical parameter $\lambda_s^\prime$ of the BRW restricted
% to $Y$ is equal to $\lambda_s=1/2d$.
% A possible choice is $Y:=\{y \in \Z^d :\langle y, y_0 \rangle \geq \alpha \|y\|\cdot\|y_0\| \}$
% for some fixed nontrivial  $y_0 \in \Z^d$
% and $\alpha <1$ (where $\langle \cdot, \cdot \rangle$ and $\|\cdot\|$ represent the
% usual scalar product and norm of $\Z^d$ respectively).
% \end{exmp}
% 
% 
% 
% 
% \subsection{Approximation by truncated BRWs}
% \label{subsec:truncated}
% 
% The family of discrete-time BRWs can be extended to the more general class of \textit{truncated BRWs} where
% a maximum of $m \in \N \cup \{\infty\}$ particles per site is allowed. We denote
% this process as a BRW$_m$.
% The dynamics is described by the following
% recursive relation
% \[ %begin{equation}\label{eq:evolBRWm}
% \eta^m_{n+1}(x)=m \wedge \sum_{y \in X} \sum_{i=1}^{\eta^m_n(y)} f_{i,n,y}(x)=
% m \wedge \sum_{y \in X} \sum_{j=0}^\infty \ident_{\{\eta^m_n(y)=j\}} \sum_{i=1}^{j} f_{i,n,y}(x).
% \] %end{equation}
% Clearly the BRW$_\infty$ is the usual BRW and the BRW$_1$ is the well-known \textit{contact process}.
% Note that while for the BRW the reproductions of two particles are independent, this is not true for the BRW$_m$
% which is a truly interacting particle system.
% 
% There is an analogous class of continuous-time processes that we still call truncated BRWs
% which are subject to the same constraint (see \cite{cf:BZ3}).
% 
% \begin{rem} \label{rem:restrained}
% Even though the BRW$_m$ is the only generalization that we need here,
% there is a more general class of continuous-time processes, called
% \textsl{restrained} BRWs, which is worth mentioning and
% which has been introduced and studied in \cite{cf:BPZ}
% (an analogous construction in discrete-time is straightforward and we omit it).
% Consider an infinite connected graph $X$ %as the environment where the population lives
% and let $\eta(x)$ be the number of individuals living at the site $x\in X$.
% The lifespan of each individual is an exponential random variable of mean 1.
% During its lifetime each individual tries to reproduce following a Poisson
% process of intensity $\lambda$.
% Every time the clock associated to the Poisson process rings, the individual tries to send an offspring to a
% randomly chosen target neighboring site.
% The target neighboring site is chosen using the transition matrix $P=(p(x,y))_{x,y\in X}$
% of a nearest neighbor random walk on $X$.
% Call the target site $y$.
% The reproduction on $y$ is effective only with probability $c(\eta(y))/\lambda$, where $c:\N\to\R^+$ is a
% non-increasing and nonnegative function with $c(0)=\lambda$. In this case the population
% living at $y$ increases by one individual, otherwise nothing happens.\break \indent
% Observe that the restrained BRW is a Markov process and
% the continuous-time truncated BRW$_m$ are
% special cases ($c\equiv \lambda$ if $m=\infty$ and $c=\lambda \ident_{\{0,1,\ldots, m-1\}})$ otherwise).
% Results about survival and stationary measures of this process can be found in \cite{cf:BPZ}; it
% is worth noting that this process may have an \textit{ecological equilibrium}, that is,
% a phase of local survival where the expected number of individuals per site is bounded from above. This is
% not possible for the BRW where local survival implies almost surely an unbounded population
% (see the proof of \cite[theorem 4.1(1)]{cf:Z1} and \cite[Theorem 6.2]{cf:Harris63}).
% \end{rem}
% 
% We observe that the discrete-time counterpart of a
% continuous-time truncated BRW is not a discrete-time truncated BRW. Indeed, in order to construct
% the discrete-time counterparts we lose the original time scale: on one hand, particles which are in the same generation
% in the discrete-time process might have disjoint lifespan intervals in the continuous-time process and,
% on the other hand, particles living at the same time in the continuous-time process might belong
% to disjoint generations in the discrete-time counterpart.
% Hence the results about approximation of continuous-time BRWs by means of continuous-time truncated
% BRWs (see \cite{cf:BZ3}) cannot be considered as particular cases of the analogous results
% for discrete-time processes (see \cite{cf:Z1}). Nevertheless, the techniques used are very
% similar and the results essentially coincides.
% 
% The goal of this section is to study the approximation of a BRW $\{\eta_n\}_{n \in \N}$
% by means of the sequence of truncated BRWs $\{\{\eta_n^m\}_{n \in \N}\}_{m \in \N}$.
% It is clear, by stochastic domination (see \cite[Section 3.3]{cf:Z1}), that
% if the BRW dies out locally (resp.~globally)
% a.s.~then any truncated BRW dies out locally (resp.~globally).
% We are going to prove here a result similar to Theorem~\ref{th:spatial}
% as $m$ tends to infinity.
% For discrete-time BRWs this has been done in \cite{cf:Z1} while the results
% for continuous-time BRWs can be found in
% \cite{cf:BZ3}.
% 
% From now on we make some assumptions on $(X,\mu)$.
% First, we assume that $X$ is countable. Indeed, the finite case is uninteresting
% since the the truncated BRW $\{\eta_n^m\}_{n \in \N}$ do not survive
% for any $m \in \N$ in discrete and continuous time (by standard Markov chain arguments,
% being $\mathbf{0}$ an absorbing state).
% Second, we require that the graph $(X,E_{\mu})$ has finite geometry, that is,
% $\sup_{x \in X} \mathrm{deg}(x)<+\infty$ and that
% the matrix $M$ is irreducible- We denote its
% convergence parameter by $R_\mu$.
% We observe that, using this notation,
% according to Theorem~\ref{th:equiv1local}, local survival
% is equivalent to $R_\mu<1$.
% For a continuous-time BRW $(X,K)$ we denote the convergence parameter of the matrix
% $K$ by $R_K$ and we observe that according to Corollary~\ref{cor:pemantleimproved}, local survival
% is equivalent to $1/R_K<\lambda$.
% 
% Finally, we suppose that
% \begin{equation}\label{eq:suprho}
%  \sup_{x \in X} \rho_x([n,+\infty)) \to 0, \qquad \text{as } n \to +\infty.
% \end{equation}
% This assumption allows to use the measure $\rho$ defined as
% \[%begin{equation}\label{eq:chooserho}
% \rho(n)=\sup_{x \in X} \rho_x([n,+\infty))-\sup_{x \in X} \rho_x([n+1,+\infty)),
% \]%end{equation}
% to stochastically dominate all the laws $\{\rho_x\}_{x \in X}$.
% Indeed equation~\eqref{eq:suprho} is equivalent to the existence of a measure
% which dominates the $\rho_x$s.
% The measure $\rho$ has finite second (hence first) moment if and only if
% $\int_0^\infty \sup_{x \in X} \rho_x([\sqrt{t},+\infty)) \diff t<+\infty$
% (that we assume henceforth).
% For a continuous-time BRW $(X,K)$ we simply assume that $\sup_{x \in X} k(x) <+\infty$
% and this implies immediately the stochastic domination by means of a continuous-time branching process
% with parameter $\sup_{x \in X} k(x)$ (see \cite{cf:BZ3} for details).
% \smallskip
% 
% We start with the result on the approximation of the local behavior of a BRW by means of
% the sequence of truncated BRWs $\{\{\eta_n^m\}_{n \in \N}\}_{m \in \N}$.
% 
% \begin{teo}\textbf{\cite[Theorem 6.3]{cf:Z1}} \label{th:mainlocal}$ $\\
% Suppose that at least one of the following conditions holds
% \begin{enumerate}[(1)]
% \item  $(X,\mu)$ is %(non-oriented)
% % l'ipotesi non-oriented si puo` togliere perche'
% % consideriamo p^{(m)} e il grafo e` irriducibile,
% % in tal caso si prende come indici X con in piu` gli
% % edge "inversi"
% %an $\mathcal F$-BRW;
%  quasi transitive and irreducible;
% \item $(X,\mu)$ is irreducible and there exists
% $\gamma$
% bijection on $X$ such that
% \begin{enumerate}[(a)]
% \item
% $\mu$ is $\gamma$-invariant;
% \item for some $x_0\in X$ we have $x_0=\gamma^nx_0$ if and only if $n=0$.
% \end{enumerate}
% \end{enumerate}
% If %we have the following.
% %\begin{enumerate}[(i)]
%  %\item
% %  If
% if $\{\eta_n\}_{n \in \N}$ survives locally (starting from $x_0$) then $\{\eta^m_n\}_{n \in \N}$
% survives locally (starting from $x_0$) eventually as $m\to +\infty$.
% % \item
% %  If $\liminf_{n\to\infty} \sqrt[n]{\sum_y m^{(n)}_{xy}}= \limsup_{n\to\infty} \sqrt[n]{m^{(n)}_{xx}}$
% %and  $\{\eta_n\}_{n \in \N}$ survives globally then $\{\eta^m_n\}_{n \in \N}$ survives globally eventually as $m\to +\infty$.
% %\end{enumerate}
% \end{teo}
% 
% In the continuous-time case there is an analogous result which
% gives an approximation of $\lambda_s$ by $\lambda_s^m$, where $\lambda_s^m$ is
% the local critical parameter of the truncated BRW with (at most) $m$ particles per site.
% 
% \begin{teo}\textbf{\cite[Theorem 5.1]{cf:BZ3}} \label{th:main}$ $\\
% Let $(X,K)$ be a continuous-time BRW and suppose that at least one of the following conditions holds
% \begin{enumerate}[(1)]
% \item  $(X,K)$ is %(non-oriented)
% % l'ipotesi non-oriented si puo` togliere perche'
% % consideriamo p^{(m)} e il grafo e` irriducibile,
% % in tal caso si prende come indici X con in piu` gli
% % edge "inversi"
% quasi transitive;
% \item $(X,K)$ is irreducible and there exists
% $\gamma$
% bijection on $X$ such that
% \begin{enumerate}[(a)]
% \item
% $\mu$ is $\gamma$-invariant;
% \item for some $x_0\in X$ we have $x_0=\gamma^nx_0$ if and only if $n=0$.
% \end{enumerate}
% \end{enumerate}
% Then
% \[
% \lim_{m\to+\infty}\lambda^m_s=\lambda_s
% %=R_\mu
% \ge\lim_{m\to+\infty}\lambda^m_w\ge\lambda_w.
% \]
% Moreover if $\lambda_s=\lambda_w$ then
% $\lambda^m_w\downarrow_{m\to+\infty} \lambda_w$.
% \end{teo}
% \medskip
% 
% Let us consider now the global behavior; as before, we start
% with a discrete-time process.
% We take $(X,\mu)$ with $X=\Z \times Y$ (for some set $Y$) and we denote by $g:X \to \Z$ the usual
% projection from $X$ onto $\Z$, namely $g(n,y):=n$.
% In the following we use the same notation as in Section~\ref{subsec:FBRWs}.
% We suppose that $\nu=\mu\circ g^{-1}$ is translation invariant %(that is, $\nu_i=\nu_0$ for all $i\in\Z$)
% (that is, $\gamma$-invariant according to Definition~\ref{def:quasitransitive} for every
% translation operator $\gamma$ on $\Z$)
% and
% we denote the common distribution and the expected number of offsprings
% of the BRW
% by $\rho$ and $\bar \rho=\sum_{y \in X} m_{xy}$; observe that they.
% %We note that $\rho$ and $\bar \rho$
% do not depend on $x \in X$ or $i \in \Z$
% since $\nu$ is translation invariant.
% 
% 
% \begin{teo}\textbf{\cite[Theorem 6.5]{cf:Z1}} \label{th:zdriftglobal}
% Let $X=\Z \times Y$ and suppose that the BRW $(X, \mu)$ is locally isomorphic to $(\Z, \nu)$ where  $\nu$ is
% translation invariant.
% If $m_{xy}=0$ whenever $|g(x)-g(y)|>1$ then
% \begin{enumerate}[(1)]
% \item
% the BRW survives globally starting from $x$ if and only if $\bar \rho=\sum_{y \in \Z} m_{xy} > 1$;
% \item
% if the BRW survives globally (starting from $x$) then
% then the BRW$_m$ survives globally (starting from $x$) for every sufficiently
% large $m$.
% \end{enumerate}
% \end{teo}
% Note that the hypotheses that we made in the previous theorem implies that the BRW $(\Z,\nu)$ is
% ``nearest neighbor'' in the sense that reproductions are possible only in the same site
% or towards neighboring sites (in the usual graph $\Z$).
% Theorem~\ref{th:zdriftglobal} applies to translation invariant BRWs on two particular graphs:
% $\Z^d$ and the homogeneous tree $\mathbb{T}_{r}$ with degree $r$.
% In particular the application to $\mathbb{T}_{r}$ is possible since the product
% $\Z \times Y$ is meant as a set product and not a graph product; indeed
% the set of vertices of $\mathbb{T}_{r}$
% can be seen as $\Z^2$ and the projection $g$ as the horocyclic map
% (see \cite[Section 12.13]{cf:Woess}).
% 
% 
% \begin{cor}\textbf{\cite[Corollary 6.6]{cf:Z1}} \label{cor:zd}
% If the BRW $(\Z^d,\mu)$ is translation invariant and there exists a projection $g$ on one of the coordinates
% such that $m_{xy}=0$ whenever $|g(x)-g(y)|>1$, then
% \begin{enumerate}[(1)]
% \item
% the BRW survives globally (starting from $x$) if and only if $\bar \rho=\sum_{y \in \Z} m_{xy} > 1$;
% \item
% if the BRW survives globally (starting from $x$)
% then the BRW$_m$ survives globally (starting from $x$) for every sufficiently
% large $m$.
% \end{enumerate}
% \end{cor}
% 
% 
% 
% 
% \begin{cor}\textbf{\cite[Corollary 6.7]{cf:Z1}} \label{th:tree}
% Let $\mathbb{T}_r$ be a homogeneous tree and suppose that the BRW $(\mathbb{T}_r,\mu)$ is
% $\gamma$-invariant for every automorphism $\gamma$ of
%  $\mathbb{T}_r$. %of degree $r$
% If
% $\mu_x(f)\not = 0$ implies $\mathrm{supp}(f) \subseteq B(x,1)$ (where $B(x,1)$
% is the usual ball of radius 1 and center $x$ of the graph $\mathbb{T}_r$)
% then
% \begin{enumerate}[(1)]
% \item
% the BRW survives globally (starting from $x$) if and only if $\bar \rho=\sum_{y \in \Z} m_{xy} > 1$;
% \item
% if the BRW survives globally (starting from $x$) then
% the BRW$_m$ survives globally (starting from $x$) for every sufficiently
% large $m$.
% \end{enumerate}
% \end{cor}
% 
% Let us consider now the continuous-time case; there are analogous results  which
% give an approximation of $\lambda_w$ by $\lambda_w^m$, where $\lambda_w^m$ is
% the global critical parameter of the truncated BRW with (at most) $m$ particles per site.
% From now on we deal with a site-breeding BRW; thus, $k(x)=k$ for all $x \in X$, that is,
% we set $k_{xy}=k p(x,y)$ where $P$ is a transition matrix of a random walk.
% %All our results extend trivially to a generic site-breeding BRW by rescaling
% %the total reproduction rate.
% We
% stress that in this case $\lambda_w=1/k$.
% We are concerned with the question whether
% $\lambda_w^m\downarrow\lambda_w=1/k$ or not. Under the hypotheses of
% Theorem~\ref{th:main}, this is the case when the BRW has no pure
% global survival phase (i.e.~$r=1$ where $r$ is the spectral radius of the random walk $P$).
% The interesting case is $r>1$. Most natural examples are drifting random walks on $\Z^d$
% and the simple random walk on homogeneous trees: as for discrete-time processes,
% in both cases we show that $\lambda^m_w\stackrel{m\to\infty}{\lra}\lambda_w$.
% 
% \begin{teo}\textbf{\cite[Theorem 6.1]{cf:BZ3}}\label{th:zdrift}
% Let $P$ be a transition matrix of a nearest-neighbor, translation invariant random walk on $\Z$.
% Then $\lim_{m\to+\infty}\lambda^m_w=1/k=\lambda_w$.
% \end{teo}
% 
% This result immediately extends to the case of a class of
% more general spaces (including multidimensional lattices $\Z^d$) in
% the following way.
% 
% \begin{cor}\textbf{\cite[Corollary 6.1]{cf:BZ3}}\label{cor:zdcontinuous}
% Let us consider the BRW $(Y\times\Z, K_\alpha)$
% where $K_\alpha=\alpha ({\mathbb I}^Y\times P) +(1-\alpha) (Q\times {\mathbb
% I}^\Z)$ and $Q$ and $P$ are transition matrices of a random walk on $Y$ and of a translation invariant random walk on
% $\Z^d$ (as in
% Theorem~\ref{th:zdrift}) respectively.
% Then $\lim_{k\to+\infty}\lambda^k_w=1/k=\lambda_w$.
% \end{cor}
% 
% For a homogeneous tree the following result holds.
% 
% \begin{teo}\textbf{\cite[Theorem 6.2]{cf:BZ3}}\label{th:treecontinuous}
% If $X=\mathbb{T}_d$ is the homogeneous tree of
% degree $d$ and $P$ is the simple random walk on $X$
% then $\lim_{m\to+\infty}\lambda^m_w=1/k=\lambda_w$.
% \end{teo}
% 
% Observe that Theorem~\ref{th:treecontinuous} can be immediately extended to the edge-breeding
% BRW on $\mathbb{T}_d$ (on regular graphs, the edge-breeding BRW is a particular site-breeding BRW).
% %is just a matter of rescaling the reproduction rates).
% In the edge-breeding case we have that
% $\lim_{m\to+\infty}\lambda^m_w=1/d=\lambda_w$; on the other hand, according to Theorem~\ref{th:main},
% $\lim_{m\to+\infty}\lambda^m_s=1/2\sqrt{d-1}=\lambda_s$. Thus $\lambda_w^m<\lambda_s^m$ eventually as
% $m \to \infty$ (pure global survival phase for truncated BRWs).
% In particular in \cite{cf:Pem} it was shown that $\lambda_w^1< \lambda_s^1$, hence we conjecture that
% $\lambda_w^m<\lambda_s^m$ for all $m \in \N$.
% 
% Discussing the details of the proofs goes beyond the purpose of this chapter. We just observe
% that they rely on a comparison between the processes and a suitable oriented percolation.
% Such a strategy has been introduced in \cite{cf:BD88} and widely used since then. The difference
% in our case is that the percolation is not even one-dependent and this brings some additional
% difficulties from a technical point of view (see \cite[Section 4]{cf:BZ3} and \cite[Section 6.1]{cf:Z1}).
% Some applications and a slight generalization of these results of approximation can be found in
% \cite[Theorem 3.4]{cf:BBZ} and \cite[Theorem 1]{cf:BLZ}.
% 


\section{Proofs}
\label{sec:proofs}

\begin{proof}[Proof of Proposition~\ref{pro:maximumprinciple}]
If $\bar {\mathbf{q}}=\mathbf{1}$ there is nothing to prove. Suppose that $\bar {\mathbf{q}}<\mathbf{1}$.
Without loss of generality we can suppose that $\bar {\mathbf{q}}(x)<1$ for all $x \in X$. Indeed,
given $x_0$ such that $\bar {\mathbf{q}}(x_0)=1$ then for all $x \in \mathcal{N}_{x_0}$ we have
$\bar {\mathbf{q}}(x)=1$. Since we defined $\widehat {\mathbf{z}}(x):=1$ whenever $\bar {\mathbf{q}}(x)=1$ we can remove
these vertices obtaining a new set $X^\prime \subseteq X$.
Consider the restricted BRW on $X^\prime$ (obtained by killing all the particles going outside $X^\prime$).
It is clear that
${\mathbf{q}}^X(x,A)\le {\mathbf{q}}^{X^\prime}(x,A)$ for all $x\in X^\prime$, $A\subseteq X^\prime$.
The generating function $G^\prime$ of the new BRW satisfies $G^\prime(({\mathbf{z}}_{|{X^\prime}})|x) \ge G({\mathbf{z}}|x)$ for all
$x \in X^\prime$, hence $G({\mathbf{z}}) \ge {\mathbf{z}}$ implies $G^\prime({\mathbf{z}}_{|{X^\prime}}) 
\ge {\mathbf{z}}_{|{X^\prime}}$ (where ${\mathbf{z}}_{|{X^\prime}}$ is ${\mathbf{z}}$ restricted to $X^\prime$).
Moreover $\widehat {\mathbf{z}}$ satisfies the conclusions of the proposition if and only 
if $\widehat{{\mathbf{z}}_{|{X^\prime}}}\equiv
 \widehat {\mathbf{z}}_{|{X^\prime}}$
does. Thus, it is enough to prove the result for the BRW restricted to $X^\prime$.

%We use the notation of Section~\ref{subsec:nodeath}.
Note that $\widehat {\mathbf{z}}:=T_{\bar {\mathbf{q}}}^{-1}({\mathbf{z}})$, %({\mathbf{z}}-\bar {\mathbf{q}})/(\mathbf{1}-\bar {\mathbf{q}})$
thus $G({\mathbf{z}}) \ge {\mathbf{z}}$  is equivalent to $\widehat G(\widehat {\mathbf{z}}) \ge \widehat {\mathbf{z}}$.
%\textcolor{red}{Attenzione a definire bene $T_{\bar {\mathbf{q}}}^{-1}$ nei punti in cui
%$\bar {\mathbf{q}}(x)=1$!!! C'\`e dimostrazione senza utilizzare $\widetilde G$?}
Hence it is enough to prove the proposition when $\mu_x(\mathbf{0})=0$ for all $x \in X$
which implies $\bar {\mathbf{q}}=\mathbf{0}$ and $\widehat {\mathbf{z}}= {\mathbf{z}}$.
Suppose that $\mathcal{N}_x$ is nonempty, ${\mathbf{z}}(y) \le {\mathbf{z}}(x)$ for all $y \in  \mathcal{N}_x$
and ${\mathbf{z}}(y_0)< {\mathbf{z}}(x)$ for some $y_0 \in  \mathcal{N}_x$.
Then, using the fact that ${\mathbf{z}} \le \mathbf{1}$ and that $\prod_{y \in X} {\mathbf{z}}(y)^{f(y)} \le {\mathbf{z}}(x)$ if $\mathcal{H}(f) \ge 1$,
we have that
${\mathbf{z}}(x) \le G({\mathbf{z}}|x) \le \sum_{f \in S_X\colon f(y_0)=0} \mu_x(f) {\mathbf{z}}(x) +\sum_{f \in S_X\colon f(y_0)>0} \mu_x(f) {\mathbf{z}}(y_0)<  {\mathbf{z}}(x)$
which is a contradiction.
As for the second part, since ${\mathbf{z}}(y) \le 1= {\mathbf{z}}(x)$ for all $y \in X$ then we have ${\mathbf{z}}(y)=1$ for all $y \in X$.
Finally, by induction we obtain the result for the set $\{y \in X\colon x \to y\}$.
\end{proof}
% 
% 
% \begin{proof}[Proof of Theorem~\ref{th:equiv1local}]
%  The first part of the theorem is \cite[Theorem 4.1]{cf:Z1}. The sufficient condition
% in the second part follows easily from the first part. Clearly, it is equivalent to study the BRW restricted to
% $Y:=\{w:x \to w \to y\}$ which is finite. In this case
% $q(w,w)=1$ for all $w$ in a finite irreducible class implies a.s.~extinction in the class; if the number
% of classes is finite then $q(x,y)=1$.
% \end{proof}

\begin{proof}[Proof of Theorem~\ref{rem:strongconditioned}]
 Indeed,
since $\{{\mathbf{q}}_n(\cdot,A)\}_{n \in\N}$ is non decreasing, ${\mathbf{q}}_{n}(\cdot,A)=G({\mathbf{q}}_{n-1}(\cdot, A))$ and
$\bar {\mathbf{q}}$ is the smallest fixed point of $G$, we have immediately that
\begin{equation}\label{eq:strongconditioned}
{\mathbf{q}}(\cdot, A) = \bar {\mathbf{q}}(\cdot) \Longleftrightarrow {\mathbf{q}}_0(\cdot,A) \le \bar {\mathbf{q}}(\cdot),
\end{equation}
that is, (1)$\Longleftrightarrow$(2).
Moreover
the event ``local survival in $A$ starting from $x$'' implies both ``global survival starting from $x$''
and ``visiting $A$ at least once starting from $x$'', hence
${\mathbf{q}}(x, A) = \bar {\mathbf{q}}(x)<1$ if and only if
the probability of visiting $A$ infinitely many times starting from $x$ conditioned on global survival
is $1$ and (1)$\Longleftrightarrow$(5)$\Longrightarrow$(4). Trivially (2)$\Longleftrightarrow$(3)
and (4)$\Longrightarrow$(3). This proves the equivalence.
\end{proof}


Before proving
Corollary~\ref{pro:qtransitive} and Theorem~\ref{th:Fgraph}
we need two technical lemmas.

\begin{lem}\label{lem:convexity}
 Let $(X,\mu)$ be a BRW and fix ${\mathbf{z}},v \in [0,1]^X$ such that ${\mathbf{z}}+\varepsilon v \in [0,1]^X$ for some
$\varepsilon>0$.
Then the function $t \mapsto G({\mathbf{z}}+vt|x)$ is strictly convex if and only if
\begin{equation}\label{eq:supp}
 \exists f \colon  \mu_x(f)>0 ,\, \sum_{y \in \mathrm{supp}(v)} f(y) \ge 2, \,
\mathrm{supp}({\mathbf{z}}) \cup \mathrm{supp}(v)\supseteq \mathrm{supp}(f).
\end{equation}
\end{lem}

%We note that, if $\mu_x(\mathbf 0)>0$ for all $x \in X$ and ${\mathbf{z}} \ge \bar q$ then
%$\mathrm{supp}({\mathbf{z}}) \cup \mathrm{supp}(v)\supseteq \mathrm{supp}(f)$ for all $f\in S_X$ and $v \in [0,1]^X$.


\begin{proof}[Proof of Lemma~\ref{lem:convexity}]
Let us evaluate the function $G$ on the line $t \mapsto {\mathbf{z}}+tv$
where $t \in [0, T)$ and $T:=\sup\{s>0:{\mathbf{z}}+sv \in [0,1]^X\}$.

\[ %begin{equation}\label{eq:Gline}
\begin{split}
G({\mathbf{z}}+tv|x)&=
\sum_{f \in S_X} \mu_x(f) \prod_{y \in X} \sum_{i=0}^{f(y)} \binom{f(y)}{i} {\mathbf{z}}(y)^{f(y)-i}v(y)^i t^i\\
&=
\sum_{f \in S_X} \mu_x(f)  \sum_{g \in S_X\colon g \le f} \prod_{y \in X}\binom{f(y)}{g(y)} {\mathbf{z}}(y)^{f(y)-g(y)}v(y)^{g(y)} t^{g(y)}\\
&=
\sum_{f \in S_X} \mu_x(f)  \sum_{g \in S_X\colon g \le f} t^{\mathcal H(g)} \prod_{y \in X}\binom{f(y)}{g(y)} {\mathbf{z}}(y)^{f(y)-g(y)}v(y)^{g(y)}  \\
&=
\sum_{f \in S_X} \mu_x(f)  \sum_{i=0}^{\infty} \sum_{g \in S_X\colon \mathcal H (g)=i, g \le f}
t^{i} \prod_{y \in X}\binom{f(y)}{g(y)} {\mathbf{z}}(y)^{f(y)-g(y)}v(y)^{g(y)}  \\
&=
\sum_{i=0}^\infty t^{i} \left ( \sum_{f,g \in S_X\colon \mathcal H (g)=i, g \le f}
\mu_x(f)  \prod_{y \in X}\binom{f(y)}{g(y)} {\mathbf{z}}(y)^{f(y)-g(y)}v(y)^{g(y)} \right ) \\
\end{split}
\] %end{equation}
The strict convexity of a power series in $t$ with nonnegative coefficients is equivalent to the strict positivity
of at least one coefficient corresponding to $t^i$ with $i \ge 2$.
Hence it is easy to show that each of the following assertions is equivalent to the next one and
that they are all equivalent to the strict
convexity of $t \mapsto G({\mathbf{z}}+vt|x)$
\begin{enumerate}
 \item $\exists f,g \colon  \mathcal H(g) \ge 2, \, f \ge g, \, \mu_x(f)>0 , \mathrm{supp}(v) \supseteq \mathrm{supp}(g), \,
\mathrm{supp}({\mathbf{z}}) \supseteq \mathrm{supp}(f-g)$;
 \item $ \exists f,g \colon  \mathcal H(g) \ge 2, \, f \ge g, \, \mu_x(f)>0 , g=f \ident_{\mathrm{supp}(v)}, \,
\mathrm{supp}({\mathbf{z}}) \supseteq \mathrm{supp}(f) \setminus \mathrm{supp}(v)$;
 \item $\exists f \colon  \mu_x(f)>0 , \sum_{y \in \mathrm{supp}(v)} f(y) \ge 2, \,
\mathrm{supp}({\mathbf{z}}) \supseteq \mathrm{supp}(f) \setminus \mathrm{supp}(v)$;
 \item $\exists f \colon  \mu_x(f)>0 , \sum_{y \in \mathrm{supp}(v)} f(y) \ge 2, \,
\mathrm{supp}({\mathbf{z}}) \cup \mathrm{supp}(v)\supseteq \mathrm{supp}(f)$;
%  \item $ \exists f \colon  \mu_x(f)>0 , \sum_{y \in \mathrm{supp}(v)} f(y) \ge 2, \,
% \mathrm{supp}({\mathbf{z}}+v)\supseteq \mathrm{supp}(f)$.
\end{enumerate}
%Clearly if $\#X=1$ we do not need to assume $\mu_x(\mathbf 0)>0$, since it is enough that ${\mathbf{z}}+v \not =0$ to have
%strict convexity (cfr.~the statements of Theorems~\ref{th:indep} and~\ref{th:Fgraph}).
\end{proof}


\begin{lem}\label{lem:generationn}
 Let $(X, \mu)$ be a BRW and fix $x_0 \in X$. Suppose that for some
$\bar x$ in the same irreducible class of $x_0$ and $f \in S_X$ we have that $\mu_{\bar x}(f)>0$,
$\sum_{w\colon w \rightleftharpoons x_0} f(w) \ge 2$.
We can fix $\bar n \in\N$ such that
if the process starts with one particle at $x_0 \in X$ then we have at least 2 particles at
$x_0$ in the generation $\bar n$ wpp.
\end{lem}

\begin{proof}[Proof of Lemma~\ref{lem:generationn}]
Consider a path $x_0, x_1, \ldots, x_m=\bar x$ and let $f \in S_X$ be
such that $\mu_{\bar x}(f)>0$ and $\sum_{w\colon w \rightleftharpoons x_0} f(w) \ge 2$.
We can have two cases.
%\begin{enumerate}[(a)]
% \item

\smallskip

\noindent \textbf{(a)}.~There exists $x_{m+1} \in X$ such that $x_{m+1} \rightleftharpoons x_0$ and $f(x_{m+1}) \ge 2$; in this case consider the closed path
$x_0, x_{1}, x_{2}, \ldots, x_m, x_{m+1}, \ldots, x_n=x_0$ and take $\bar n:=n$. Since any particle at $x_i$
has at least one child at $x_{i+1}$ wpp and a particle at $\bar x$ has at least 2 children at $x_{m+1}$ wpp,
then any particle at $x_0$ has at least 2 descendants
at $x_0$ in the $\bar n$th generation.
Indeed, denote by $f_i \in S_X$ such that $\mu_{x_i}(f_i)>0$, $f_i(x_{i+1}) \ge 1$ for all $i =0, \ldots \bar n-1$
($f_m$ being $f$), then the probability  that a particle at $x_0$ has at least 2 particles at $x_0$
in the $\bar n$th generation is bounded from below by $\prod_{i=0}^m \mu_i(f_i) \prod_{j=m+1}^{\bar n-1} \mu_j(f_i)^2$.
%\item
\smallskip

\noindent \textbf{(b)}.~There exists a couple of different vertices
$x_{m+1}, y_{m+1}$ %$ in the same class of irriducibility of $x_0$
%\in X$
such that $x_{m+1}, y_{m+1}\rightleftharpoons x_0$ and \break
%$x_{m+1} \not = y_{m+1}$,  $x_{m+1}, y_{m+1} and
$f(x_{m+1}), f(y_{m+1}) \ge 1$; in
this case consider the paths $x_0, x_1, \ldots x_m, x_{m+1},
\ldots, x_{n_1}=x_0$ and $x_0, x_1, \ldots x_m, y_{m+1}, \ldots,
y_{n_2}=x_0$ and take $\bar n:= GCD(n_1,n_2)$ (the conclusion is
similar as before).
%\end{enumerate}
\end{proof}


\begin{proof}[Proof of Theorem~\ref{th:Fgraph}]
$(1)$. 
For every fixed point ${\mathbf{z}}$ of $G$, we know that ${\mathbf{z}} \ge \bar {\mathbf{q}}$ and ${\mathbf{z}} \le \mathbf 1_X$; this implies
that if $\sup_{x \in X} {\mathbf{z}}(x)<1$ for some fixed point then necessarily $\sup_{x \in X} \bar {\mathbf{q}}(x)<1$.
Hence, if $\bar {\mathbf{q}}=\mathbf 1$ there is nothing to prove. Otherwise,
we show that if $G({\mathbf{z}})={\mathbf{z}}$ and ${\mathbf{z}} \not = \bar {\mathbf{q}}$ then $\sup_{w \in X} {\mathbf{z}}(w)=1$.
Suppose that  the BRW is locally isomorphic to $(Y,\nu)$ through the map $g$
and define ${\mathbf{h}}(y):=\sup_{w \in g^{-1}(y)} {\mathbf{z}}(w)$.
Clearly ${\mathbf{h}} \in [0,1]^Y$ and ${\mathbf{h}} \circ g \ge {\mathbf{z}}$ which implies that
$G_Y({\mathbf{h}}) \ge {\mathbf{h}}$.
Indeed
\[
\begin{split}
G_Y({\mathbf{h}}|y)&= \sup_{x \in g^{-1}(y)}G_Y({\mathbf{h}}|g(x))=\sup_{x \in g^{-1}(y)}G({\mathbf{h}} \circ g|x) \\
&\ge
\sup_{x \in g^{-1}(y)}G({\mathbf{z}}|x)= \sup_{x \in g^{-1}(y)} {\mathbf{z}}(x) = {\mathbf{h}}(y).
\end{split}
\]
If $Y$ finite then we can choose $\widetilde y \in Y$ which minimizes
\[
 t(y):= \frac{1-\bar {\mathbf{q}}^Y(y)}{{\mathbf{h}}(y)-\bar {\mathbf{q}}^Y(y)}
\]
(where $t(y):=+\infty$ if ${\mathbf{h}}(y)=\bar {\mathbf{q}}^Y(y)$);
note that $t(y) \ge 1$ for all $y \in Y$ and $t(\widetilde y)<+\infty$.
By applying the maximum principle (Proposition~\ref{pro:maximumprinciple})
to the function $1/t(y)$ (where $y$ is ranging in the set $\{w\colon \bar {\mathbf{q}}^Y(w)<1\}$)
we have that it is constant on $\{y\colon \widetilde y \to y\}$.
Since $\bar {\mathbf{q}}^Y (\widetilde y)<1$ and $Y$ is finite,
then there exists $y_0$ such that $\widetilde y \to y_0$ and
there is local survival at $y_0$ starting from $y_0$.
Since $(Y,\nu)$ satisfies Assumption~\ref{assump:1}
then there exists $\bar y \rightleftharpoons y_0$ such that
a particle living at $\bar y$ wpp has at least 2 children in the
irreducible class of $y_0$.
Then by taking $y_0$ instead of $x_0$ in
Lemma~\ref{lem:generationn} we have that we can
find $\bar n \in \N$ such that the function
\[
\phi(t):=G_Y^{(\bar n)}(\bar {\mathbf{q}}^Y+t ({\mathbf{h}}-\bar {\mathbf{q}}^Y)|y_0)-\bar {\mathbf{q}}^Y(y_0)-t ({\mathbf{h}}(y_0)-\bar {\mathbf{q}}^Y(y_0))
\]
is strictly convex by Lemma~\ref{lem:convexity}.
Indeed $G_Y^{(\bar n)}$ is the generating function of the BRW constructed by considering
the $n$-th generations of the original BRW where $\bar n | n$ and, under our hypotheses, it satisfies
equation~\eqref{eq:supp}. %It is straightforward to prove that the generating
%function of this new BRW is $G^{(n)}$ (the $n$th iteration of $G$), hence

Note that $\phi$ is well defined in $[0,t(y_0)]$ since
\[
\mathbf{r}_t(y):=\bar {\mathbf{q}}^Y(y)+t ({\mathbf{h}}(y)-\bar {\mathbf{q}}^Y(y)) \le \bar {\mathbf{q}}^Y(y)+t(y_0) ({\mathbf{h}}(y)-\bar {\mathbf{q}}^Y(y))\le 1
\]
hence $\mathbf{r}_t \in [0,1]^Y$ for all $t \in [0,t(y_0)]$.

Clearly every fixed point of $G_Y$ is a fixed point of $G_Y^{(\bar n)}$;
in particular, $G^{(\bar n)}({\mathbf{z}})={\mathbf{z}}$ and $G^{(\bar n)}_Y(\bar {\mathbf{q}}^Y)=\bar {\mathbf{q}}^Y$, whence
$\phi(0)=0$ and $\phi(1)=G^{(\bar n)}_Y({\mathbf{h}}|y_0)-{\mathbf{h}}(y_0)$.
Now, using equation~\eqref{eq:Gfunctions}, $G^{(\bar n)}_Y({\mathbf{h}}) \ge {\mathbf{h}}$
%\[
%G^{(\bar n)}_Y(h) \circ g=G^{(\bar n)}(h\circ g) \ge G^{(\bar n)}({\mathbf{z}})={\mathbf{z}}
%\]
%which implies
%\[
%G^{(\bar n)}_Y(h|y_0) \ge \sup_{w \in g^{-1}(y_0)} {\mathbf{z}}(w)=h(y_0)
%\]
and this, in turn, implies $\phi(1) \ge 0$. 
Since $\phi$ is strictly convex we have that $\phi(t)>0$ for all $t \in (1, t(y_0)]$.
If $t(y_0)>1$ then $0 < \phi(t(y_0))=G^{(\bar n)}_Y({\mathbf{r}}_{t(y_0)}|y_0)-1$ but this is a contradiction
since ${\mathbf{r}}_{t(y_0)} \in [0,1]^Y$ and $G^{(\bar n)}_Y({\mathbf{r}}_{t(y_0)}) \in [0,1]^Y$.
In the end $t(y_0)=1$, thus $1={\mathbf{h}}(y_0)=\sup_{w \in X} {\mathbf{z}}(w)$.


$(2)$
This applies to $\mathbf{z}(\cdot)=\mathbf{q}(\cdot,x)$ for any fixed $x \in X$. If the BRW is irreducible 
$\mathbf{q}(w,x)= \mathbf{q}(w,y)$ for all $x,y,w \in X$. 
Thus, there exists $x \in X$ such that $\mathbf{q}(\cdot,x)=\bar{\mathbf{q}(\cdot)}$
if and only if  $\mathbf{q}(w,w)=\bar{\mathbf{q}(w)}$ for all $w \in X$; analogously,
there exists $x \in X$ such that $\sup_{w \in X} \mathbf{q}(w,x)=1$
if and only if  $\sup_{w \in X} \mathbf{q}(w,w)=1$.

\end{proof}
Note that, from the first part of the previous proof, % from $h \circ g \ge {\mathbf{z}}$, we have that $G_Y(h) \ge h$
%hence,
if the BRW on $Y$ is irreducible then by the maximum principle %(see Proposition~\ref{pro:maximumprinciple})
we have
that $(\mathbf{h}-\bar {\mathbf{q}}^Y)/(\mathbf{1}-\bar {\mathbf{q}}^Y)$ is a constant function, thus %, the previous proof implies that
$\mathbf{h}(y)=\sup_{w \in g^{-1}(y)} {\mathbf{z}}(w)=1$ for all $y \in Y$.

\begin{proof}[Proof of Corollary~\ref{cor:finiteBRW}]
 
When $X$ is finite, $(X, \mu)$ is clearly an $\mathcal{F}$-BRW.
If $\bar {\mathbf{q}}=\mathbf{1}$ 
there is nothing to prove. Suppose that $\bar {\mathbf{q}}<\mathbf{1}$,  since the BRW is irreducible
we have that $\bar {\mathbf{q}}(x)<{1}$ for all $x \in X$. 
Let $\bar{\mathbf{q}}<\mathbf{z}<\mathbf{1}$ be a solution of $G(\mathbf{z}) \ge \mathbf{z}$.
Since $X$ is finite and $\bar{\mathbf{q}}<\mathbf{z}$ from Theorem~\ref{th:Fgraph}(1) we have 
that $\mathbf{z}(x)=1$ for some $x \in X$. By Proposition~\ref{pro:maximumprinciple}, using irreducibility, 
$\widehat {\mathbf{z}}= \mathbf{1}$ which contradicts $\mathbf{z}<\mathbf{1}$.

\end{proof}


\begin{proof}[Proof of Corollary~\ref{pro:qtransitive}]
Since  $(X, E_\mu)$ is irreducible we have that ${\mathbf{q}}(x,y)={\mathbf{q}}(x,x)$ for all $x,y \in X$ and
if $\bar {\mathbf{q}}<\mathbf 1$ (resp.~${\mathbf{q}}(\cdot,y)<\mathbf 1$)
then $\bar {\mathbf{q}}(x) <1$ (resp.~${\mathbf{q}}(x,y)<1$) for all $x \in X$.
Moreover, quasi transitivity implies that if ${\mathbf{q}}(\cdot,y)<\mathbf 1$ then
$\sup_{x \in X} {\mathbf{q}}(x,y) <1$.
% which, in turn, implies
%$\sup_{x \in X} \bar q(x)  <1$.
Thus, according to Theorem~\ref{th:Fgraph}, ${\mathbf{q}}(\cdot,y) \not = \mathbf 1$
implies ${\mathbf{q}}(\cdot,y)=\bar {\mathbf{q}}$.
\end{proof}

% 
% \begin{proof}[Proof of Theorem~\ref{th:MenshikovVolkov}]
% According to Section~\ref{subsec:nodeath}, there is
% strong local survival at $y$ starting from $x$ for the BRW
% $\{\eta_n\}_{n \in \N}$
% if and only if there is a.s.~local survival at $y$ for the associated BRW with no death,
% that is, $\{\widehat \eta_n\}_{n \in \N}$
% conditioned on $\mathcal{A}_x$ (global survival starting from $x$).
% Moreover $v$ satisfies equation~\eqref{eq:MenshikovVolkov}
% if and only if $v_1:=T_{\bar q}^{-1}v$ satisfies
% \[%begin{equation}\label{eq:MenshikovVolkov}
%  \begin{cases}
%   \widehat G(v_1|x) \ge v_1(x), & \forall x \in A^\complement,\\
%   v_1(x_0) > \max_{x \in A} v_1(x) & \textrm{for some } x_0 \in A^\complement,
%  \end{cases}
% \] %end{equation}
% which is equation~\eqref{eq:MenshikovVolkov} in the case of the associated BRW with no death.
% Hence it is enough to prove the result for the case $\rho_x(0)=0$ for all $x \in X$.
% 
% For completeness, we sketch the proof of \cite[Theorem 3.1]{cf:MenshikovVolkov}.
% Suppose that there exist a function $v$ and a set $A$ as in the statement of
% the theorem.
% Recall the definition of $G$ given by equation~\eqref{eq:genfun}, define
% $\widetilde Q_n:=\prod_{x \in X} v(x)^{\eta_n(x)}$ and $\sigma:=\min\{n \in \N: \sum_{x \in A} \eta_n(x)>0\}$,
% where $\{\eta_n\}_{n \in \N}$ is a
% realization of the BRW. As usual $\min \emptyset :=+\infty$. Let $Q_n:=\widetilde Q_{n \wedge \sigma}$.
% If $\bar v(x):=\mathbb E [ \widetilde Q_{n+1} | \eta_n=\delta_x]$ then it is easy to show that
% $\bar v%(x)
% = G(v)$. %|x)$.
% Using the same arguments as in  \cite[Theorem 3.1]{cf:MenshikovVolkov}
% we can show that $\{Q_n\}_{n\ \in \N}$ is a nonnegative supermartingale, hence there exists
% $Q_\infty:=\lim_{n \to \infty} Q_n$ in $L^1$ and almost surely. Clearly $\mathbb E[Q_\infty] \ge \mathbb E[Q_0]$.
% If $\eta_0:=\delta_{x_0}$ where $x_0 \not \in A$ satisfies the hypotheses of Theorem~\ref{th:MenshikovVolkov}
% and if there were strong local survival then at least one particle would hit $A$ a.s., thus
% $Q_\infty \le \max_{x \in A} v(x) < v(x_0)$ which is a contradiction. This yields the first part
% of the proof.
% 
% Assume now that there is no strong local survival. Fix $\bar x \in X$ and $A:=\{\bar x\}$. Define
% $v(x):=q_0(x,A)$, the probability of never hitting $A$ starting from $x$. Since the BRW is irreducible,
% then there is no strong local survival if and only if $v(x)>0$ for some $x$. Clearly $v(\bar x)=0<v(x_0)$
% for some $x_0 \not \in A$ and
% \[
%  v(x)= \sum_{g \in S_X: g(\bar x)=0} \mu_x(g) \prod_{y \in X} v(y)^{g(y)} \le G(v|x),
% \quad \forall x \not = \bar x
% \]
% and the theorem is proved.
% \end{proof}
% 
% \begin{proof}[Proof of Theorem~\ref{th:nonam}]
%  The proof is essentially the same as in \cite[Section 3.3]{cf:BZ}. We just sketch
% the main steps. %Define $M_w:=\liminf_{n \to \infty} \sqrt[n]{\sum_{y \in X} m^{(n)}_{xy}}$
% %and $M_s:=\limsup_{n \to \infty} \sqrt[n]{m^{(n)}_{xy}}$.
% Let $Y$ be the finite set onto which $X$ can be mapped by definition of $\mathcal F$-BRW.
% Instead of the operators $N$ and $\widetilde N$ we use
% $Mf(x):=\sum_{w \in X} m_{xw}f(w)$ (for all $x \in X$)
% and  $\widetilde Mf(y):=\sum_{w \in Y} \widetilde m_{yw}f(w)$ (for all $y \in Y$)
% where $m_{xw}=m^X_{xw}$ and $\widetilde m_{yw}=m^Y_{yw}$. These are well-defined,
% bounded, linear operators from $l^2(X)$ and $l^2(Y)$ into itself respectively.
% One can prove that $\|M\|=\rho(M)=M_s$ where $\rho(M)$ is the spectral radius of the operator
% (this can be done as in \cite[Lemma 3.3]{cf:BZ} and \cite[Lemma 2.2]{cf:Stacey03}).
% Similarly $\|\widetilde M\|=\rho(\widetilde M)=\widetilde M_s=\widetilde M_w=M_w$
% (here we use the finiteness of $Y$).
% 
% Moreover, it can be shown that
% for nonamenable BRWs
% there exists $c>0$ such that, for all $f\in l^2(X)$,
% \[
% \| f\|_{D(2)}\ge c\| f\|_2,
% \]
% where the Dirichlet norm is defined as
% \[
% \| f\|_{D(2)} = \left( \sum_{x,y\in X} m_{xy}|f(x)-f(y)|^2
% \right)^{1/2}.
% \]
% The proof of this inequality is analogous to the one of
% \cite[Theorem 2.6]{cf:Stacey03}, the only difference being
% the presence of $m_{xy}$ which can be easily dealt with.
% 
% The rest of the proof is the same as \cite[Theorem 3.6]{cf:BZ} using the term
% $\sum_{x \in S, y \in S^\complement} m_{xy}$ instead of $|\partial_E S|$ and using the
% graph $G^2$ induced by
% $M^2$. %$m^{(2)}$.
% \end{proof}
% 
% 
% \begin{proof}[Proof of Theorem~\ref{th:subexponentialgrowth}]
% Note that $\kappa(x)k^{(n)}_{xy}=\kappa(y)k^{(n)}_{yx}$ for all $x,y \in X$,
% $n \in \N$.
% Moreover, by the Cauchy-Schwartz inequality, for all $n \in \N$,
% \[
% K_s(x_0,x_0)^{2n}\ge k^{(2n)}_{x_0x_0}=\sum_{y \in X}k^{(n)}_{x_0y}k^{(n)}_{yx_0}=
% \sum_{y \in B(x_0,n)} (k^{(n)}_{x_0y})^2\frac{\kappa(x_0)}{\kappa(y)}\ge
% \frac{\left(\sum_yk^{(n)}_{x_0y}\right)^2}{c_n|B(x_0,n)|}.
% \]
% Hence
% \[
% K_w(x_0)=\liminf_n\sqrt[n]{\sum_yk^{(n)}_{x_0y}}=
% \liminf_n\sqrt[2n]{\frac{\left(\sum_yk^{(n)}_{x_0y}\right)^2}{c_n|B(x_0,n)|}} \le K_s(x_0,x_0).
% \]
% \end{proof}
% 
% 
% 
% \begin{proof}[Proof of Proposition~\ref{pro:F-BRW1}]
% \textit{(1)} and \textit{(2)} follow easily from Theorems~\ref{th:equiv1local}(1) and \ref{th:equiv1global3}(2).
% As for \textit{(3)},
%  we note that $m^{(n)}(x,y)= \bar \rho^n p^{(n)}(x,y)$ and that the generating
% function $\Phi$ defined in Section~\ref{subsec:genfun} satisfies $\Phi(x,y|t)=F(x,y|t \bar \rho)$.
% Thus, $\Phi(x,x|1)>1$, which is equivalent to local survival at $x$, is equivalent to
% $F(x,x|\bar \rho) >1$.
% \end{proof}
% 
% \begin{proof}[Proof of Lemma~\ref{lem:test0}]
% Clearly $\prod_{i \in \N} (1-\alpha_i)^{k_i}>0$ if and only if $\sum_{i \in \N} k_i \log(1-\alpha_i)>-\infty$.
% Observe that $\log(1-x) \le -x$ for all $x < 1$ hence
% \[ %begin{equation}\label{eq:test0}
%   \sum_{i \in \N} k_i \alpha_i \le -\sum_{i \in \N} k_i \log(1-\alpha_i)<\infty.
% \] %end{equation}
% 
% If $\alpha_i \in [0, 1)$ and $k_i \ge 1$ eventually as $i \to \infty$ then
% there is no loss of generality by assuming that $\alpha_i \in [0, 1)$ and $k_i \ge 1$ for all $i$.
% In this case, since $k_i \ge 1$ both sides imply $\alpha_i \to 0$. Thus $\log(1-\alpha_i) \sim -\alpha_i$ and
% \[
%  \sum_{i \in \N} k_i \log(1-\alpha_i) > -\infty \Longleftrightarrow \sum_{i \in \N} k_i \alpha_i < \infty.
% \]
% \end{proof}
% 
% \begin{proof}[Proof of Theorem~\ref{th:recurrent}]
% Since $\mu$ satisfies equation~\eqref{eq:particular1} %(2.3) of \cite{cf:Zu}
% then
% $G(z| x) = \sum_{n=0}^\infty \rho_x(n) (Pz(x))^n$ where
% $Pz(x)=\sum_{y \in X} p(x,y)z(y)$.
% On the other hand, $\rho_x=\rho$ for all $x \in X$,
% thus  $\bar q(x)=\bar q$ for all $x \in X$ where $\bar q$ is
% the smallest fixed point in
% $[0,1]$ of $t \mapsto G(t \mathbf{1}| x) \equiv \sum_{n=0}^\infty \rho(n) t^n=:F(t)$.
% %Since $q(\cdot, y)$ is a fixed point (for all $y \in X$) of $G$, we look for
% %all solutions in $[0,1]^X$ of the equation $G(z)=z$.
% Clearly, any fixed point $z$ of $G$ must satisfy the inequality $z(x) \ge \bar q(x)=\bar q$.
% Since $F(t) < t$ for all $t \in (\bar q,1)$ then
% \[
% z(x)=G(z|x)= F(Pz(x)) \le Pz(x),
% \]
% hence $z$ is a bounded subharmonic function. It is well known that the existence
% of non-constant subharmonic functions which are bounded from above is equivalent to transience,
% thus, in the recurrent case we have necessarily $z=t \mathbf 1$ which implies that $t=F(t)$
% and $t \in \{\bar q, 1\}$.
% Suppose that $\bar q<1$, since the random walk is recurrent,
% then $q_0(\cdot, A) \le \bar q \mathbf{1}$ (for all $A \subseteq X$),
% hence by Remark~\ref{rem:strongconditioned} $q(\cdot, A) = \bar q \mathbf{1}$
% which is equivalent to strong local survival in $A$.
% %To show that $t=\bar q$
% %we make use of the construction
% %of a BRW as a
% %markov chain $\{Z_\nu\}$ indexed by trees (see \cite[Section 5.C]{cf:Woess09}). If we fix an infinite,
% %self-avoiding path $\{\gamma_n\}_{n \in \N}$ starting from the root of each infinite
% %Galton-Watson tree, then the process $\{Z_{\gamma_n}\}_{n \in \N}$ is a (recurrent) random walk
% %with transition matrix $P$. Remember that in this construction, the set of global survival is equal
% %to the set of infinite Galton-Watson trees, hence the probability of local survival conditioned on global
% %survival is $1$.
% \end{proof}
% 
% 

\section*{Acknowledgments}
The authors are grateful to the anonymous referee for carefully reading the manuscript and for useful suggestions
which helped to improve the paper.

\begin{thebibliography}{99}



\bibitem{cf:AthNey}
K.B.~Athreya,  P.E.~Ney,
Branching processes,
Die Grundlehren der mathematischen Wissenschaften, \textbf{196},
Springer-Verlag, 1972.

\bibitem{cf:BL} L.~Belhadji, N.~Lanchier, Individual versus cluster recoveries within a
spatially structured population,  Ann.~Appl.~Probab.~\textbf{16}  (2006),  no.1, 403--422.

% 
\bibitem{cf:BBZ}
 L.~Belhadji, D.~Bertacchi, F.~Zucca,
A self-regulating and patch subdivided population,
Adv.~Appl.~Probab.~\textbf{42} n.3 (2010), 899--912.

\bibitem{cf:BLZ}
D.~Bertacchi, N.~Lanchier, F.~Zucca,
Contact and voter processes on the infinite
percolation cluster as models of host-symbiont interactions,
Ann.~Appl.~Probab.~\textbf{21} n.~4 (2011), 1215--1252.

\bibitem{cf:BPZ}
D.~Bertacchi, G.~Posta, F.~Zucca,
        Ecological equilibrium for restrained random walks,
        Ann.~Appl.~Probab.~\textbf{17} n.~4 (2007), 1117--1137.

% \bibitem{cf:BZ03}
% D.~Bertacchi, F.Zucca,
% Uniform asymptotic estimates of transition probabilities on combs,
% J. Aust. Math. Soc.~\textbf{75} n.~3 (2003), 325--353.

\bibitem{cf:BZ}
D.~Bertacchi, F.~Zucca,
Critical behaviors and critical values of branching random walks
on multigraphs, J.~Appl.~Probab.~\textbf{45} (2008), 481--497.

\bibitem{cf:BZ2}
D.~Bertacchi, F.~Zucca,
Characterization of the critical values of branching random walks on
weighted graphs through infinite-type branching processes,
J.~Stat.~Phys.~\textbf{134} n.~1 (2009), 53--65.

\bibitem{cf:BZ3}
D.~Bertacchi, F.~Zucca,
Approximating critical parameters of branching random walks,
J.~Appl.~Probab.~\textbf{46} (2009), 463--478.

\bibitem{cf:BZ4}
D.~Bertacchi, F.~Zucca,
Recent results on branching random walks, 
Statistical Mechanics and Random Walks: Principles, Processes and Applications, 
Nova Science Publishers (2012), 289-340.

% 
% \bibitem{cf:BD88}
% M.~Bramson, R.~Durrett,
% A simple proof of the stability criterion of Gray
% and Griffeath,
%   Probab.~Theory Related Fields \textbf{80}  (1988),  no. 2, 293--298.
% 
\bibitem{cf:Big1977}
J.D.~Biggins,
Martingale convergence in the branching random walk,
J.~Appl.~Probab.~\textbf{14}  n.~1 (1977), 25--37.

\bibitem{cf:Big1978}
J.D.~Biggins,
The asymptotic shape of the branching random walk,
Adv.~Appl.~Probab.~\textbf{10} n.~1 (1978), 62--84.


\bibitem{cf:BigKypr97}
J.D.~Biggins, A.E.~Kyprianou,
Seneta-Heyde norming in the branching random walk,
Ann.~Probab.  \textbf{25} n.~1 (1997), 337--360.

\bibitem{cf:BigRah05}
J.D.~Biggins, A.~Rahimzadeh Sani,
Convergence results on multitype, multivariate branching random walks,
Adv.~Appl.~Probab.  \textbf{37} n.~3 (2005), 681--705.

% 
% \bibitem{cf:CMP98}
% F.~Comets, M.V.~Menshikov, S.Yu.~Popov,
% One-dimensional branching random walk in random environment: A classification,
% Markov Process.~Related Fields~\textbf{4} (1998), 465--477.
% 
% \bibitem{cf:FN1}
% K.~H.~F\"orster, B.~Nagy,
% On the Collatz-Wielandt numbers and the local spectral radius of a nonnegative operator,
% Proceedings of the Fourth Haifa Matrix Theory Conference (Haifa, 1988),
% Linear Algebra Appl.~\textbf{120} (1989), 193--205.
% 
% \bibitem{cf:FN2}
% K.~H.~F\"orster, B.~Nagy,
% Local spectral radii and Collatz-Wielandt numbers of monic operator polynomials with nonnegative coefficients,
% Linear Algebra Appl.~\textbf{268} (1998), 41--57.

\bibitem{cf:GW1875}
F.~Galton, H.W.~Watson, On the probability of the extinction of
families, Journal of the Anthropological Institute of Great Britain and Ireland \textbf{4} (1875), 138--144.

\bibitem{cf:GMPV09}
N.~Gantert, S.~M\"uller, S.Yu.~Popov, M.~Vachkovskaia,
Survival of branching random walks in random environment,
J.~Theoret.~Probab.~\textbf{23} (2010), no.~4, 1002--1014. 
% 
% \bibitem{cf:Harris48}
% T.E.~Harris,
% Branching processes,
% Ann.~Math.~Statistics \textbf{19} (1948). 474--494.

\bibitem{cf:Harris63}
T.E.~Harris, The theory of branching processes, Springer-Verlag, Berlin, 1963.

% \bibitem{cf:DHMP99}
% F.~den Hollander, M.V.~Menshikov, S.Yu.~Popov,
% A note on transience versus recurrence for a branching random walk in random environment,
% J.~Stat.~Phys.~\textbf{95}  (1999), 587--614.

\bibitem{cf:HuLalley}
I.~Hueter, S.P.~Lalley, {Anisotropic branching random walks
on homogeneous trees},  Probab.~Theory Related Fields  \textbf{116},
(2000),  n.1, 57--88.

% 
% \bibitem{cf:KestenSidorav}
% H.~Kesten, V.~Sidoravicius,
% Branching random walk with catalysts,
% Electron.~J.~Probab.~\textbf{8} (2003), no. 5.

\bibitem{cf:Ligg1}
T.M.~Liggett,
{Branching random walks and contact processes on homogeneous trees},
Probab.~Theory Related Fields  \textbf{106},  (1996),  n.4, 495--519.

\bibitem{cf:Ligg2}
T.M.~Liggett,
{Branching random walks on finite trees},
Perplexing problems in probability,  315--330, Progr.~Probab.,
\textbf{44}, Birkh\"auser Boston, Boston, MA, 1999.

% 
% 
% \bibitem{cf:MachadoMenshikovPopov}
% F.P.~Machado,M.~V.~Menshikov, S.Yu.~Popov,
% Recurrence and transience of multitype
% branching random walks,
% Stoch.~Proc.~Appl.~\textbf{91} (2001), 21--37.
% 
% \bibitem{cf:MP00}
% F.P.~Machado, S.Yu.~Popov,
% One-dimensional branching random walk in a Markovian random environment,
% J.~Appl.~Probab.~\textbf{37} (2000), 1157--1163.
% 
% \bibitem{cf:MP03}
% F.P.~Machado, S.Yu.~Popov,
% Branching random walk in random environment on trees,
% Stoch.~Proc.~Appl.~\textbf{106} (2003), 95--106.

\bibitem{cf:MadrasSchi}
N.~Madras, R.~Schinazi, {Branching random walks on trees},
Stoch.~Proc.~Appl.~\textbf{42},  (1992),  n.2, 255--267.
% 
% \bibitem{cf:Marek1}
% I.~Marek,
% Collatz-Wielandt numbers in general partially ordered spaces,
% Linear Algebra Appl.~\textbf{173}, (1992), 165--180.

\bibitem{cf:MenshikovVolkov}
M.~V.~Menshikov, S.~E.~Volkov, Branching Markov chains: Qualitative characteristics,
Markov Proc.~and~rel.~Fields.~\textbf{3} (1997), 225–241.

\bibitem{cf:MountSchi}
T.~Mountford, R.~Schinazi, A note on branching random walks on
finite sets, J.~Appl.~Probab.~\textbf{42} (2005),  287--294.

% \bibitem{cf:M08}
% S.~M\"uller,
% A criterion for transience of multidimensional branching random walk
% in random environment,
% Electron.~J.~Probab.~\textbf{13} (2008).
% 

\bibitem{cf:Muller08-2}
S.~M\"uller, Recurrence for branching Markov chains,
Electron.~Commun.~Probab.~\textbf{13} (2008), 576--605.

% 
% \bibitem{cf:Pem}
% R.~Pemantle, {The contact process on trees}, Ann.~Prob.
% \textbf{20}, (1992), 2089--2116.
% 
\bibitem{cf:PemStac1}
R.~Pemantle, A.M.~Stacey, {The branching random walk and
contact process on Galton--Watson and nonhomogeneous trees},
Ann.~Prob.~\textbf{29}, (2001),
 n.4, 1563--1590.
% 
% 
% \bibitem{cf:Sen}
%         E.~Seneta, {Non-negative matrices and Markov chains},
%         Springer Series in Statistics, Springer, New York, 2006.
% 
\bibitem{cf:Spataru89}
A.~Spataru,
Properties of branching processes with denumerable many types,


\bibitem{cf:Stacey03}
A.M.~Stacey, {Branching random walks on quasi-transitive graphs},
Combin.~Probab.~Comput.~\textbf{12}, (2003), n.3
  345--358.

\bibitem{cf:Woess}
        W.~Woess, {Random walks on infinite graphs and groups},
        Cambridge Tracts in Mathematics, {\textbf 138},
    Cambridge Univ.~Press, 2000.
% 
% \bibitem{cf:Woess09}
%         W.~Woess,
% Denumerable Markov chains,
% Generating functions, boundary theory, random walks on trees.
% EMS Textbooks in Mathematics,
% European Mathematical Society (EMS), 2009.
% 
% \bibitem{cf:Zaehle}
% I.~Z\"ahle, Renormalizations of branching random walks in equilibrium,
% Electron.~J.~Probab.~\textbf{7} (2002), no. 7.

\bibitem{cf:Z1}
F.~Zucca,
Survival, extinction and approximation of discrete-time branching random walks,
J.~Stat.~Phys., \textbf{142} n.4 (2011), 726--753.


\end{thebibliography}

\end{document}


1208.4010.extracted_bib
\begin{thebibliography}{99}



\bibitem{cf:AthNey}
K.B.~Athreya,  P.E.~Ney,
Branching processes,
Die Grundlehren der mathematischen Wissenschaften, \textbf{196},
Springer-Verlag, 1972.

\bibitem{cf:BL} L.~Belhadji, N.~Lanchier, Individual versus cluster recoveries within a
spatially structured population,  Ann.~Appl.~Probab.~\textbf{16}  (2006),  no.1, 403--422.

% 
\bibitem{cf:BBZ}
 L.~Belhadji, D.~Bertacchi, F.~Zucca,
A self-regulating and patch subdivided population,
Adv.~Appl.~Probab.~\textbf{42} n.3 (2010), 899--912.

\bibitem{cf:BLZ}
D.~Bertacchi, N.~Lanchier, F.~Zucca,
Contact and voter processes on the infinite
percolation cluster as models of host-symbiont interactions,
Ann.~Appl.~Probab.~\textbf{21} n.~4 (2011), 1215--1252.

\bibitem{cf:BPZ}
D.~Bertacchi, G.~Posta, F.~Zucca,
        Ecological equilibrium for restrained random walks,
        Ann.~Appl.~Probab.~\textbf{17} n.~4 (2007), 1117--1137.

% \bibitem{cf:BZ03}
% D.~Bertacchi, F.Zucca,
% Uniform asymptotic estimates of transition probabilities on combs,
% J. Aust. Math. Soc.~\textbf{75} n.~3 (2003), 325--353.

\bibitem{cf:BZ}
D.~Bertacchi, F.~Zucca,
Critical behaviors and critical values of branching random walks
on multigraphs, J.~Appl.~Probab.~\textbf{45} (2008), 481--497.

\bibitem{cf:BZ2}
D.~Bertacchi, F.~Zucca,
Characterization of the critical values of branching random walks on
weighted graphs through infinite-type branching processes,
J.~Stat.~Phys.~\textbf{134} n.~1 (2009), 53--65.

\bibitem{cf:BZ3}
D.~Bertacchi, F.~Zucca,
Approximating critical parameters of branching random walks,
J.~Appl.~Probab.~\textbf{46} (2009), 463--478.

\bibitem{cf:BZ4}
D.~Bertacchi, F.~Zucca,
Recent results on branching random walks, 
Statistical Mechanics and Random Walks: Principles, Processes and Applications, 
Nova Science Publishers (2012), 289-340.

% 
% \bibitem{cf:BD88}
% M.~Bramson, R.~Durrett,
% A simple proof of the stability criterion of Gray
% and Griffeath,
%   Probab.~Theory Related Fields \textbf{80}  (1988),  no. 2, 293--298.
% 
\bibitem{cf:Big1977}
J.D.~Biggins,
Martingale convergence in the branching random walk,
J.~Appl.~Probab.~\textbf{14}  n.~1 (1977), 25--37.

\bibitem{cf:Big1978}
J.D.~Biggins,
The asymptotic shape of the branching random walk,
Adv.~Appl.~Probab.~\textbf{10} n.~1 (1978), 62--84.


\bibitem{cf:BigKypr97}
J.D.~Biggins, A.E.~Kyprianou,
Seneta-Heyde norming in the branching random walk,
Ann.~Probab.  \textbf{25} n.~1 (1997), 337--360.

\bibitem{cf:BigRah05}
J.D.~Biggins, A.~Rahimzadeh Sani,
Convergence results on multitype, multivariate branching random walks,
Adv.~Appl.~Probab.  \textbf{37} n.~3 (2005), 681--705.

% 
% \bibitem{cf:CMP98}
% F.~Comets, M.V.~Menshikov, S.Yu.~Popov,
% One-dimensional branching random walk in random environment: A classification,
% Markov Process.~Related Fields~\textbf{4} (1998), 465--477.
% 
% \bibitem{cf:FN1}
% K.~H.~F\"orster, B.~Nagy,
% On the Collatz-Wielandt numbers and the local spectral radius of a nonnegative operator,
% Proceedings of the Fourth Haifa Matrix Theory Conference (Haifa, 1988),
% Linear Algebra Appl.~\textbf{120} (1989), 193--205.
% 
% \bibitem{cf:FN2}
% K.~H.~F\"orster, B.~Nagy,
% Local spectral radii and Collatz-Wielandt numbers of monic operator polynomials with nonnegative coefficients,
% Linear Algebra Appl.~\textbf{268} (1998), 41--57.

\bibitem{cf:GW1875}
F.~Galton, H.W.~Watson, On the probability of the extinction of
families, Journal of the Anthropological Institute of Great Britain and Ireland \textbf{4} (1875), 138--144.

\bibitem{cf:GMPV09}
N.~Gantert, S.~M\"uller, S.Yu.~Popov, M.~Vachkovskaia,
Survival of branching random walks in random environment,
J.~Theoret.~Probab.~\textbf{23} (2010), no.~4, 1002--1014. 
% 
% \bibitem{cf:Harris48}
% T.E.~Harris,
% Branching processes,
% Ann.~Math.~Statistics \textbf{19} (1948). 474--494.

\bibitem{cf:Harris63}
T.E.~Harris, The theory of branching processes, Springer-Verlag, Berlin, 1963.

% \bibitem{cf:DHMP99}
% F.~den Hollander, M.V.~Menshikov, S.Yu.~Popov,
% A note on transience versus recurrence for a branching random walk in random environment,
% J.~Stat.~Phys.~\textbf{95}  (1999), 587--614.

\bibitem{cf:HuLalley}
I.~Hueter, S.P.~Lalley, {Anisotropic branching random walks
on homogeneous trees},  Probab.~Theory Related Fields  \textbf{116},
(2000),  n.1, 57--88.

% 
% \bibitem{cf:KestenSidorav}
% H.~Kesten, V.~Sidoravicius,
% Branching random walk with catalysts,
% Electron.~J.~Probab.~\textbf{8} (2003), no. 5.

\bibitem{cf:Ligg1}
T.M.~Liggett,
{Branching random walks and contact processes on homogeneous trees},
Probab.~Theory Related Fields  \textbf{106},  (1996),  n.4, 495--519.

\bibitem{cf:Ligg2}
T.M.~Liggett,
{Branching random walks on finite trees},
Perplexing problems in probability,  315--330, Progr.~Probab.,
\textbf{44}, Birkh\"auser Boston, Boston, MA, 1999.

% 
% 
% \bibitem{cf:MachadoMenshikovPopov}
% F.P.~Machado,M.~V.~Menshikov, S.Yu.~Popov,
% Recurrence and transience of multitype
% branching random walks,
% Stoch.~Proc.~Appl.~\textbf{91} (2001), 21--37.
% 
% \bibitem{cf:MP00}
% F.P.~Machado, S.Yu.~Popov,
% One-dimensional branching random walk in a Markovian random environment,
% J.~Appl.~Probab.~\textbf{37} (2000), 1157--1163.
% 
% \bibitem{cf:MP03}
% F.P.~Machado, S.Yu.~Popov,
% Branching random walk in random environment on trees,
% Stoch.~Proc.~Appl.~\textbf{106} (2003), 95--106.

\bibitem{cf:MadrasSchi}
N.~Madras, R.~Schinazi, {Branching random walks on trees},
Stoch.~Proc.~Appl.~\textbf{42},  (1992),  n.2, 255--267.
% 
% \bibitem{cf:Marek1}
% I.~Marek,
% Collatz-Wielandt numbers in general partially ordered spaces,
% Linear Algebra Appl.~\textbf{173}, (1992), 165--180.

\bibitem{cf:MenshikovVolkov}
M.~V.~Menshikov, S.~E.~Volkov, Branching Markov chains: Qualitative characteristics,
Markov Proc.~and~rel.~Fields.~\textbf{3} (1997), 225–241.

\bibitem{cf:MountSchi}
T.~Mountford, R.~Schinazi, A note on branching random walks on
finite sets, J.~Appl.~Probab.~\textbf{42} (2005),  287--294.

% \bibitem{cf:M08}
% S.~M\"uller,
% A criterion for transience of multidimensional branching random walk
% in random environment,
% Electron.~J.~Probab.~\textbf{13} (2008).
% 

\bibitem{cf:Muller08-2}
S.~M\"uller, Recurrence for branching Markov chains,
Electron.~Commun.~Probab.~\textbf{13} (2008), 576--605.

% 
% \bibitem{cf:Pem}
% R.~Pemantle, {The contact process on trees}, Ann.~Prob.
% \textbf{20}, (1992), 2089--2116.
% 
\bibitem{cf:PemStac1}
R.~Pemantle, A.M.~Stacey, {The branching random walk and
contact process on Galton--Watson and nonhomogeneous trees},
Ann.~Prob.~\textbf{29}, (2001),
 n.4, 1563--1590.
% 
% 
% \bibitem{cf:Sen}
%         E.~Seneta, {Non-negative matrices and Markov chains},
%         Springer Series in Statistics, Springer, New York, 2006.
% 
\bibitem{cf:Spataru89}
A.~Spataru,
Properties of branching processes with denumerable many types,


\bibitem{cf:Stacey03}
A.M.~Stacey, {Branching random walks on quasi-transitive graphs},
Combin.~Probab.~Comput.~\textbf{12}, (2003), n.3
  345--358.

\bibitem{cf:Woess}
        W.~Woess, {Random walks on infinite graphs and groups},
        Cambridge Tracts in Mathematics, {\textbf 138},
    Cambridge Univ.~Press, 2000.
% 
% \bibitem{cf:Woess09}
%         W.~Woess,
% Denumerable Markov chains,
% Generating functions, boundary theory, random walks on trees.
% EMS Textbooks in Mathematics,
% European Mathematical Society (EMS), 2009.
% 
% \bibitem{cf:Zaehle}
% I.~Z\"ahle, Renormalizations of branching random walks in equilibrium,
% Electron.~J.~Probab.~\textbf{7} (2002), no. 7.

\bibitem{cf:Z1}
F.~Zucca,
Survival, extinction and approximation of discrete-time branching random walks,
J.~Stat.~Phys., \textbf{142} n.4 (2011), 726--753.


\end{thebibliography}