# Extinction times of multitype continuous-state branching processes

Loïc Chaumont  and  Marine Marolleau  Loïc Chaumont – LAREMA – UMR CNRS 6093,
Université d’Angers, 2 bd Lavoisier, 49045 Angers cedex 01
<EMAIL_ADDRESS>Marine Marolleau – LAREMA – UMR CNRS 6093,
Université d’Angers, 2 bd Lavoisier, 49045 Angers cedex 01
<EMAIL_ADDRESS>
###### Abstract.

A multitype continuous-state branching process (MCSBP) ${\rm Z}=({\rm
Z}_{t})_{t\geq 0}$, is a Markov process with values in $[0,\infty)^{d}$ that
satisfies the branching property. Its distribution is characterised by its
branching mechanism, that is the data of $d$ Laplace exponents of
$\mathbb{R}^{d}$-valued spectrally positive Lévy processes, each one having
$d-1$ increasing components. We give an expression of the probability for a
MCSBP to tend to 0 at infinity in term of its branching mechanism. Then we
prove that this extinction holds at a finite time if and only if some
condition bearing on the branching mechanism holds. This condition extends
Grey’s condition that is well known for $d=1$. Our arguments bear on elements
of fluctuation theory for spectrally positive additive Lévy fields recently
obtained in [7] and an extension of the Lamperti representation in higher
dimension proved in [5].

###### Key words and phrases:

Multitype continuous-state branching process, extinction time, spectrally
positive additive Lévy field, Lamperti representation.

###### 2010 Mathematics Subject Classification:

60J80

##  1\. Introduction

A multitype continuous state branching process (MCSBP), ${\rm Z}$, with family
of probability measures $\mathbb{P}_{\rm r}$, ${\rm r}\in\mathbb{R}^{d}_{+}$,
$d\geq 1$, is a $[0,\infty)^{d}$-valued Markov process, satisfying the
_branching property_ :

$\mathbb{E}_{{\rm r}_{1}+{\rm r}_{2}}(e^{-\langle\lambda,{\rm
Z}_{t}\rangle})=\mathbb{E}_{{\rm r}_{1}}(e^{-\langle\lambda,{\rm
Z}_{t}\rangle})\mathbb{E}_{{\rm r}_{2}}(e^{-\langle\lambda,{\rm
Z}_{t}\rangle}),\;\;\lambda,{\rm r}_{1},{\rm r}_{2}\in\mathbb{R}^{d}_{+}.$

MCSBP’s were first introduced in the late 60’s by Watanabe [14] and there has
been renewed interest in such processes in more recent years with the works of
Duffie, Filipović and Schachermayer [8], Barczy, Li and Pap [1], Caballero,
Pérez Garmendia and Uribe Bravo [5], Kyprianou and Palau [11],…

Asymptotic behavior and extinction times are among the most studied questions
in recent times. In the articles [11] and [12], Kyprianou, Palau and Ren
provide a counterpart to the well known case of multitype Galton Watson
processes by proving that the asymptotic behaviour of the MCSBP ${\rm Z}$ is
characterized by the value of the Perron-Frobenius eigenvalue of the mean
matrix denoted here by $\rho$. More specifically, extinction occurs if and
only if $\rho\leq 0$. In this case, ${\rm Z}$ is said to be critical
($\rho=0$) or sub-critical ($\rho<0$). When $\rho>0$, the process is said to
be super-critical and ${\rm Z}$ either becomes extinct with positive
probability or has exponential growth under a log-condition. We will see here
that, as in dimension one, extinction in continuous time differs from the
discrete case. Indeed, in addition to the possibility of becoming extinct in a
finite time, that is there exists $t\geq 0$ such that for all $i\in[d]$,
$Z^{(i)}_{t}=0$, the process can be extinguished at infinity, that is for all
$t\geq 0$, there exists $i\in[d]$ such that $Z^{(i)}_{t}>0$ and
$\lim\limits_{t\rightarrow+\infty}{\rm Z}_{t}=0$. When $d=1$, Grey’s
condition, see [10], gives a necessary and sufficient condition for the
continuous state branching process $Z$ to become extinct at a finite time in
terms of its branching mechanism, $\varphi$. More precisely, $Z$ becomes
extinct at a finite time if and only if

(1.1) $\int^{\infty}\frac{ds}{\varphi(s)}<\infty\,.$

However, when $d>1$, we do not know how to distinguish these two ’types’ of
extinction. The aim of this article is to provide necessary and sufficient
conditions bearing on the mechanism of ${\rm Z}$ for extinction to take place
at a finite time, in order to extend Grey’s condition (1.1) to the multitype
case.

One of the major results used for our proof is the extension to the multitype
case of the Lamperti representation, recently obtained in [5]. This result
asserts that ${\rm Z}$ can be represented as the unique solution of the
following equation,

$(Z^{(1)}_{t},...,Z^{(d)}_{t})={\rm
r}+\left(\sum\limits_{j=1}^{d}X^{1,j}_{\int\limits_{0}^{t}Z^{(j)}_{s}ds},...,\sum\limits_{j=1}^{d}X^{d,j}_{\int\limits_{0}^{t}Z^{(j)}_{s}ds}\right),\;\;t\geq
0,$

where ${\rm X}^{(j)}={}^{t}(X^{1,j},\dots,X^{d,j})$, $j\in[d]$ are $d$
independent Lévy processes such that for all $j\in[d]$, $X^{j,j}$ is a
spectrally positive Lévy process, that is with no negative jumps, and for all
$i\neq j$, $X^{i,j}$ is a subordinator (here ${}^{t}{\rm u}$ means the
transpose of the vector ${\rm u}\in\mathbb{R}^{d}$). The right hand side of
the above equation can be re-written as ${\rm
r}+\mathbf{X}\left(\int\limits_{0}^{t}{\rm Z}_{s}ds\right)$, where
$\\{\mathbf{X}_{\rm t},{\rm t}\in\mathbb{R}^{d}_{+}\\}$ is the spectrally
positive additive Lévy field (spaLf) defined as the sum of the $d$ independent
Lévy processes ${\rm X}^{(j)}$, that is

$\mathbf{X}_{\rm t}:={\rm X}^{(1)}_{t_{1}}+\dots+{\rm
X}^{(d)}_{t_{d}},\;\;{\rm t}\in\mathbb{R}^{d}_{+}\,.\ $

Note that spaLf’s have been defined and studied in a previous paper under the
aspect of fluctuation theory, see [7]. When $d=1$, this result is due to
Lamperti who proved, in the $1970$’s, that any time continuous branching
process can be represented as a spectrally positive Lévy process time changed
by the inverse of some integral functional. Note that when $d=1$, the Lamperti
representation is constructive, that is the branching process can be made
explicit from the leading Lévy process, whereas this is not the case in higher
dimension.

Our main result asserts that if the Laplace exponent $\tilde{\varphi}_{i}$ of
each diagonal Lévy process $X^{i,i}$ in the above Lamperti representation
satisfies Grey’s condition, that is if

(1.2) $\int^{\infty}\frac{ds}{\tilde{\varphi}_{i}(s)}<\infty,\;\;\;\mbox{for
all $i\in[d]$}\,,$

then extinction can only occur at a finite time. If (1.2) is not satisfied for
some $i\in[d]$, then ${\rm Z}$ can only be extinguished at infinity. Moreover,
when starting from ${\rm r}\in\mathbb{R}_{+}^{d}$, the probability that ${\rm
Z}_{t}$ tends to $\bf 0$, when $t$ tends to $\infty$ is $e^{-\langle{\rm
r},\phi(\mathbf{0})\rangle}$, where $\phi$ is the inverse of the Laplace
exponent of the spaLf ${\bf X}$. In particular, ${\rm Z}$ becomes extinct
almost surely if and only if it is critical or sub-critical.

This result indicates in particular that the off-diagonal subordinators
$X^{i,j}$, $i\neq j$ have no influence on the nature of the extinction. It may
appear rather counter-intuitive. However, thinking of the neutral case helps
us to understand this phenomenon. Indeed, in the neutral case, that is when
the law of $\sum_{i=1}^{d}X^{i,j}$ does not depend on $j$, the MCSBP ${\rm Z}$
behaves like a single type continuous state branching process. More
specifically, $Z^{(1)}+\dots+Z^{(d)}$ is a continuous state branching process
whose branching mechanism is $d\hat{\varphi}$, where $\hat{\varphi}$ is the
Laplace exponent of $\sum_{i=1}^{d}X^{i,j}$. But provided the $X^{i,i}$’s are
not subordinators, which is excluded here,
$\int^{\infty}\frac{ds}{\hat{\varphi}(s)}<\infty$ if and only if (1.2) holds
for each $i$, see the discussion after Theorem 3.3. Therefore, from Grey’s
condition (1.1), ${\rm Z}$ becomes extinct at a finite time with positive
probability if and only if (1.2) is statisfied.

The next section is devoted to some reminders and preliminary results about
MCSBP’s and spaLf’s. Then in Section 3 we state our main results and in
Sections 4, 5 and the Appendix we give proofs of these results.

##  2\. Preliminary results on MCSBP’s and spaLf’s

We use the notation $\mathbb{R}_{+}=[0,\infty)$ and $[d]=\\{1,\dots,d\\}$,
where $d\geq 1$ is an integer. Vectors of $\mathbb{R}^{d}$ will be written in
roman characters and their coordinates in italic, as follows: ${\rm
x}=(x_{1},\dots,x_{d})$ and ${}^{t}{\rm x}={}^{t}(x_{1},\dots,x_{d})$ will
denote the transpose of ${\rm x}$. The $i$-th unit vector of
$\mathbb{R}_{+}^{d}$ will be denoted ${\rm e}_{i}=(0,\dots,0,1,0,\dots,0)$. We
set ${\bf 0}=(0,0,\dots,0)$ and ${\bf 1}=(1,1,\dots,1)$ respectively for the
zero vector of $\mathbb{R}_{+}^{d}$ and the vector of $\mathbb{R}_{+}^{d}$
whose all coordinates are equal to 1. For ${\rm s}=(s_{1},\dots,s_{d})$ and
${\rm t}=(t_{1},\dots,t_{d})\in\mathbb{R}_{+}^{d}$, we write ${\rm s}\leq{\rm
t}$ if $s_{i}\leq t_{i}$ for all $i\in[d]$ and we write ${\rm s}<{\rm t}$ if
${\rm s}\leq{\rm t}$ and there exists $i\in[d]$ such that $s_{i}<t_{i}$. We
will denote by $\langle{\rm x},{\rm y}\rangle$, ${\rm x},{\rm
y}\in\mathbb{R}^{d}$ the usual inner product on $\mathbb{R}^{d}$ and by $|{\rm
x}|$ the Euclidian norm of ${\rm x}$.

###  2.1. Basics on MCSBP’s

Let $\delta$ be an element external to $\mathbb{R}_{+}^{d}$ and
$R=\mathbb{R}_{+}^{d}\cup\\{\delta\\}$ be the one point Alexandrov
compactification of $\mathbb{R}_{+}^{d}$. An $R$-valued strong Markov process
${\rm Z}=\\{({\rm Z}_{t})_{t\geq 0},(\mathbb{P}_{{\rm r}})_{{\rm
r}\in\mathbb{R}^{d}_{+}}\\}$ with $\delta$ as a trap is called a multitype (or
a $d$-type) continuous state branching process (MCSBP) if it satisfies,

(2.3) $\mathbb{E}_{{\rm r}_{1}+{\rm r}_{2}}[e^{-\langle\lambda,{\rm
Z}_{t}\rangle}]=\mathbb{E}_{{\rm r}_{1}}[e^{-\langle\lambda,{\rm
Z}_{t}\rangle}]\mathbb{E}_{{\rm r}_{2}}[e^{-\langle\lambda,{\rm
Z}_{t}\rangle}],\;t\geq 0,\;{\rm r}_{1},{\rm
r}_{2},\lambda\in\mathbb{R}^{d}_{+}\,,$

where by convention $e^{-\langle\lambda,\delta\rangle}=0$, for all
$\lambda\in\mathbb{R}^{d}_{+}$. The property (2.3) is called the branching
property of ${\rm Z}$. We recall that $\mathbf{0}$ is an absorbing state for
${\rm Z}$ and that it is the only absorbing state other than $\delta$. The
branching property of ${\rm Z}$ implies directly the existence of a mapping
$(t,\lambda)\mapsto{\rm
u}_{t}(\lambda)=(u^{(1)}_{t}(\lambda),\dots,u^{(d)}_{t}(\lambda))$ satisfying

(2.4) $\mathbb{E}_{\rm r}[e^{-\langle\lambda,{\rm
Z}_{t}\rangle}]=e^{-\langle{\rm r},{\rm u}_{t}(\lambda)\rangle},\;\;t\geq
0,\;\lambda,{\rm r}\in\mathbb{R}^{d}_{+}.$

From the Markov property of ${\rm Z}$, we derive the semigroup property of
${\rm u}_{t}(\lambda)$,

${\rm u}_{t+s}(\lambda)={\rm u}_{t}({\rm u}_{s}(\lambda)),\;\;\;\mbox{for all
$s,t\geq 0$ and $\lambda\in\mathbb{R}^{d}_{+}$.}$

Moreover for each fixed $t$, ${\rm u}_{t}(\lambda)$ is differentiable in
$\lambda$. It was first proved in [14], see also Proposition 6.1 in [8], that
for each $\lambda\in\mathbb{R}_{+}^{d}$, $t\mapsto{\rm u}_{t}(\lambda)$ is
differentiable in $t$ and is the unique solution to the following differential
system,

$(S_{\varphi})\qquad\qquad\;\left\\{\begin{array}[]{ll}\dfrac{\partial}{\partial
t}u^{(i)}_{t}(\lambda)=-\varphi_{i}({\rm u}_{t}(\lambda))\\\\[6.0pt]
u^{(i)}_{0}(\lambda)=\lambda_{i}\end{array}\right.,\;\;\;i\in[d],\,t\geq 0,$

where each $\varphi_{i}$ is the Laplace exponent of a possibly killed
$d$-dimensional Lévy process ${\rm X}^{(i)}={}^{t}(X^{1,i},\dots,X^{d,i})$.
The coordinate $X^{i,i}$ is a (possibly killed) spectrally positive Lévy
process and $X^{i,j}$, $j\neq i$ are (possibly killed) subordinators. More
specifically, we define a probability measure $\mathbb{P}$ under which the
processes ${\rm X}^{(i)}$, $i\in[d]$ are $d$ independent possibly killed
$d$-dimensional Lévy processes. Then for each $i\in[d]$, $\varphi_{i}$ is
defined by

$\mathbb{E}[e^{-\langle\lambda,{\rm
X}^{(i)}_{t}\rangle}]=e^{t\varphi_{i}(\lambda)}\,,\;\;\;t\geq
0\,,\;\;\;{\bf\lambda}=(\lambda_{1},\dots,\lambda_{d})\in\mathbb{R}_{+}^{d}\,.$

Its Lévy-Khintchine decomposition is written as,

(2.5)
$\varphi_{i}(\lambda)=-\alpha_{i}-\sum_{j=1}^{d}a_{j,i}\lambda_{j}+\frac{1}{2}q_{i}\lambda_{i}^{2}-\int_{\mathbb{R}_{+}^{d}}(1-e^{-\langle\lambda,{\rm
x}\rangle}-\langle\lambda,{\rm x}\rangle 1_{\\{|{\rm x}|<1\\}})\,\pi_{i}(d{\rm
x})\,,\;\;\lambda\in\mathbb{R}^{d}_{+},$

where $\alpha_{i}\geq 0$, $(a_{j,i})_{i,j\in[d]}$ is an essentially
nonnegative matrix i.e. $a_{j,i}\geq 0$ for $i\neq j$, $q_{i}\geq 0$ and
$\pi_{i}$ is a measure on $\mathbb{R}_{+}^{d}$ such that
$\pi_{i}(\\{\mathbf{0}\\})=0$ and

$\int_{\mathbb{R}_{+}^{d}}\left[(1\wedge|{\rm x}|^{2})+\sum\limits_{j\neq
i}(1\wedge x_{j})\right]\pi_{j}(d{\rm x})<\infty\,.$

The mapping $\varphi=(\varphi_{1},\dots,\varphi_{d})$ is called the _branching
mechanism_ of the MCSBP ${\rm Z}$. We emphasize that for each $i\in[d]$,
$\varphi_{i}$ is a convex function. Moreover, for all $j\neq i$ and
$\lambda_{1},\dots,\lambda_{j-1}$, $\lambda_{j+1},\dots,\lambda_{d}$ the
function $\lambda_{j}\mapsto\varphi_{i}(\lambda)$ is non increasing and
whenever $X^{i,i}$ is not a subordinator, the function
$\lambda_{i}\mapsto\varphi_{i}(\lambda)$ tends to $\infty$ as
$\lambda_{i}\rightarrow\infty$. Conversely, still according to [14], for each
mapping such as $\varphi$ in (2.5) there exists a unique (in law) MCSBP ${\rm
Z}=\\{({\rm Z}_{t})_{t\geq 0},(\mathbb{P}_{{\rm r}})_{{\rm
r}\in\mathbb{R}^{d}_{+}}\\}$ with branching mechanism $\varphi$. Note that
MCSBP’s belong to a more general class of processes called affine processes.
Existence and uniqueness of the solution of equation $(S_{\varphi})$ is proved
in this general setting in [8].

A MCSBP ${\rm Z}$ is said to be _conservative_ if for all ${\rm
r}\in\mathbb{R}_{+}^{d}$ and $t>0$, $\mathbb{P}_{\rm r}({\rm
Z}_{t}\in\mathbb{R}_{+}^{d})=1$. From (2.4) this can be expressed in terms of
the Laplace exponent of ${\rm Z}$ as $e^{\langle{\rm r},{\rm
u}_{t}(\mathbf{0})\rangle}=1$, for all ${\rm r}\in\mathbb{R}_{+}^{d}$ and
$t>0$, so that ${\rm Z}$ is conservative if and only if

(2.6) $u_{t}^{(i)}(\mathbf{0})=0,\;\;\mbox{for all $t>0$ and $i\in[d]$.}$

When $d=1$, it is proved in [10] that ${\rm Z}$ is conservative if and only if
its mechanism $\varphi$ satisfies

$\int_{0+}\dfrac{d\alpha}{|\varphi(\alpha)|}=\infty\,.$

When $d>1$, finding necessary and sufficient conditions on the branching
mechanism for ${\rm Z}$ to be conservative is an open question which is not
our purpose here. Note however that sufficient conditions have been given in
Lemma 9.2 of [8]. It is also possible to construct examples from the neutral
case defined in the introduction, that is when the law of
$\sum_{i=1}^{d}X^{i,j}$ does not depend on $j$, since in this case, the
branching mechanism of ${\rm Z}$ is $d\hat{\varphi}$, where $\hat{\varphi}$ is
the Laplace exponent of $\sum_{i=1}^{d}X^{i,j}$. Let us also note that if for
some $i\in[d]$, $u^{(i)}_{t}(\mathbf{0})=0$, for all $t\geq 0$, then from
$(S_{\varphi})$, for all $t\geq 0$,

$\displaystyle\frac{\partial}{\partial t}u_{t}^{(i)}(\mathbf{0})$
$\displaystyle=$
$\displaystyle-\varphi_{i}(u_{t}^{(1)}(\mathbf{0}),\dots,u_{t}^{(i-1)}(\mathbf{0}),0,u_{t}^{(i+1)}(\mathbf{0}),\dots,u_{t}^{(d)}(\mathbf{0}))$
$\displaystyle=$ $\displaystyle 0.$

But $\varphi_{i}$ is non increasing on all coordinates $j\neq i$, so that

$-\varphi_{i}(u_{t}^{(1)}(\mathbf{0}),\dots,u_{t}^{(i-1)}(\mathbf{0}),0,u_{t}^{(i+1)}(\mathbf{0}),\dots,u_{t}^{(d)}(\mathbf{0}))\geq\alpha_{i}$

and hence $\alpha_{i}=0$, see also Proposition 9.1 in [8]. Therefore, if ${\rm
Z}$ is conservative, then for all $i\in[d]$, $\alpha_{i}=0$. We will assume
throughout this paper that ${\rm Z}$ is conservative.

There also exists a pathwise connection between MCSBP’s and their branching
mechanism which will be of great use for our results. For $d=1$, this has been
known for a long time as the Lamperti representation, see [13] and [4]. This
representation has recently been extended to the multitype case in [9] and
[5], see also [6] for discrete valued processes. More specifically, let ${\rm
r}\in\mathbb{R}_{+}^{d}$ and ${\rm X}^{(i)}$, $i\in[d]$ be $d$ independent
Lévy processes whose Laplace exponents have the form (2.5). Then Theorem 1 in
[5] asserts that there exists a unique solution to the equation,

(2.7)
$Z^{(i)}_{t}=r_{i}+\sum\limits_{j=1}^{d}X^{i,j}\left(\int\limits_{0}^{t}Z^{(j)}_{s}ds\right),\;\;t\geq
0,\;\;i\in[d].$

Moreover, if $\mathbb{P}_{\rm r}$ denotes the law of the solution ${\rm
Z}_{t}=(Z^{(1)}_{t},\dots,Z^{(d)}_{t})$, $t\geq 0$, then ${\rm Z}=\\{({\rm
Z}_{t})_{t\geq 0},(\mathbb{P}_{{\rm r}})_{{\rm r}\in\mathbb{R}^{d}_{+}}\\}$ is
a MCSBP with branching mechanism $\varphi$. Conversely, every MCSBP can be
obtained from this construction. Note that the paper [5] actually treats the
more general setting of affine processes.

###  2.2. Some reminders on spaLf’s

Whether characterized analytically through the differential system
$(S_{\varphi})$ or path by path through equations $(\ref{Lamperti})$, MCSBP’s
require a good knowledge of the underlying Lévy processes ${\rm X}^{(j)}$,
$j\in[d]$ in order to be properly studied. Note that in equation (2.7), these
Lévy processes live in different time scales and that the leading process is
actually the multivariate stochastic field

$\mathbf{X}_{\rm t}:={\rm X}_{t_{1}}^{(1)}+\dots+{\rm
X}^{(d)}_{t_{d}}=\left(\sum_{j=1}^{d}X^{i,j}_{t_{j}}\right)_{i\in[d]}\,,\;\;\;\mbox{for}\;\;\;{\rm
t}=(t_{1},\dots,t_{d})\in\mathbb{R}_{+}^{d}\,.$

This process is called a _spectrally positive additive Lévy field_ (spaLf) and
has been studied in [7] from the aspect of fluctuation theory. The mapping
$\varphi$ defined above is actually the Laplace exponent of this spaLf, that
is

$\mathbb{E}[e^{-\langle\lambda,\mathbf{X}_{\rm t}\rangle}]=e^{\langle{\rm
t},\varphi(\lambda)\rangle}\,,\;\;\;{\rm t},\lambda\in\mathbb{R}_{+}^{d}\,.$

Similarly to the case $d=1$, the construction of the solution of (2.7) only
requires the paths of $\mathbf{X}$ up to its first hitting time at level
$-{\rm r}$. This notion of first hitting times for spaLf’s has been defined in
[7] as the smallest solution ${\rm s}=(s_{1},\dots,s_{d})$ to the system

(2.8) $\sum_{j=1}^{d}X^{i,j}_{s_{j}}=-r_{i},\;\;\;i\in[d]_{\rm s}\,,$

where ’smallest’ is to be understood in the usual partial order of
$\mathbb{R}^{d}$ and $[d]_{\rm s}=\\{i\in[d]:s_{i}<\infty\\}$, see the
appendix. We will denote by $\mathbf{T}_{\rm r}=(T_{\rm r}^{(1)},\dots,T_{\rm
r}^{(d)})$ this solution and we will refer to it as _the $($multivariate$)$
first hitting time of level $-{\rm r}$ by the spaLf
$\mathbf{X}=\\{\mathbf{X}_{\rm t},{\rm t}\in\mathbb{R}^{d}_{+}\\}$_. In a
purely formal way we may set,

(2.9) $\mathbf{T}_{\rm r}=\inf\\{{\rm t}:\mathbf{X}_{{\rm t}}=-{\rm r}\\}\,.$

Note that from [7], $\mathbb{P}(\mathbf{T}_{\rm
r}\in\mathbb{R}_{+}^{d})+\mathbb{P}(T_{\rm r}^{(i)}=\infty,i\in[d])=1$, that
is, with probability one, either all coordinates of $\mathbf{T}_{\rm r}$ are
finite or all of them are infinite.

Let us now state two important hypothesis which will be in force throughout
this paper. The first one is,

$(H_{\varphi})\qquad\qquad\;D_{\varphi}=\\{\lambda\in\mathbb{R}^{d}_{+}:\varphi_{j}(\lambda)>0,j\in[d]\\}\;\text{is
not empty.}$

Assuming $(H_{\varphi})$ simply allows us to exclude the spaLf’s which do not
hit any level, almost surely, see Theorem 2.1 below. When $d=1$ this boils
down to exclude subordinators. Our second assumption regards the mean matrix
$M=\left(m_{ij}\right)_{i,j\in[d]}$, where
$m_{ij}:=\mathbb{E}(X_{1}^{i,j})=-\lim\limits_{\lambda\rightarrow
0}\dfrac{\partial}{\partial\lambda_{i}}\varphi_{j}(\lambda)$. We say that $M$
is irreducible if for all $i,j\in[d]$, there exist $n\geq 1$ and some indices
$i=i_{1},\dots,i_{n}=j$ such that for all $k\in[n-1]$, $m_{i_{k}i_{k+1}}\neq
0$. Note that for all $i,j\in[d]$, $m_{ij}\in(-\infty,\infty]$. A MCSBP ${\rm
Z}$ is said to be irreducible if the mean matrix $M$ of the underlying spaLf
${\bf X}$ in the Lamperti representation is irreducible. We will also
sometimes say that the spaLf ${\bf X}$ is irreducible. When ${\bf X}$ is
integrable, that is when $m_{ij}<\infty$, for all $i,j\in[d]$, Perron-
Frobenius theory asserts that $M$ has a real eigenvalue $\rho$ with
multiplicity equal to 1 and such that the real part of any other eigenvalue is
less than $\rho$. In this case the MCSBP ${\rm Z}$ will be said that it is
subcritical, critical or super critical according as $\rho<0$, $\rho=0$ or
$\rho>0$.

The following result is proved in [7], see Proposition 3.1, Theorems 3.3 and
3.4 therein.

######  Theorem 2.1.

Let $\mathbf{X}$ be a spaLf such that $\alpha_{i}=0$, for all $i\in[d]$. For
${\rm r}\in\mathbb{R}_{+}^{d}$, let $\mathbf{T}_{\rm r}$ be its first hitting
time of level $-{\rm r}$ as defined above. Then,

  * $1.$

$\mathbb{P}(\mathbf{T}_{\rm r}\in\mathbb{R}^{d}_{+})>0$ for some $($and hence
for all$)$ ${\rm r}\in\mathbb{R}^{d}_{+}$ if and only if $(H_{\varphi})$
holds.

  * $2.$

If $(H_{\varphi})$ holds then there is a mapping
$\phi=(\phi_{1},\dots,\phi_{d}):\mathbb{R}_{+}^{d}\rightarrow[0,\infty)^{d}$
such that

(2.10) $\mathbb{E}[e^{-\langle\lambda,\mathbf{T}_{\rm
r}\rangle}]=e^{-\langle{\rm
r},\phi(\lambda)\rangle},\;\;\;\lambda\in\mathbb{R}_{+}^{d}\,.$

Moreover, for all $\lambda\in(0,\infty)^{d}$, $\phi(\lambda)\in D_{\varphi}$
and the mapping $\phi:(0,\infty)^{d}\rightarrow D_{\varphi}$ is a
diffeormorphism whose inverse is
$\varphi:D_{\varphi}\rightarrow(0,\infty)^{d}$, that is

$\varphi(\phi(\lambda))=\lambda,\;\;\;\lambda\in(0,\infty)^{d}.$

  * $3.$

If ${\bf X}$ is irreducible and if $(H_{\varphi})$ holds, then the values
$\mathbf{0}$ and $\phi(\mathbf{0})$ are the only solutions of the equation
$\varphi(\lambda)=\mathbf{0}$, $\lambda\in\mathbb{R}_{+}^{d}$. Moreover,
either $\phi(\mathbf{0})=\mathbf{0}$ or $\phi(\mathbf{0})\in(0,\infty)^{d}$.

An immediate consequence of Theorem 2.1 is the following expression of the
probability for the first hitting time to be finite on each coordinate,

(2.11) $\mathbb{P}(\mathbf{T}_{\rm r}\in\mathbb{R}^{d}_{+})=e^{-\langle{\rm
r},\phi(\mathbf{0})\rangle}\,.$

When $d=1$ we derive directly from the Lamperti representation (2.7) the
almost sure equality $\mathbf{T}_{\rm r}=\int_{0}^{\infty}{\rm Z}_{s}\,ds$, on
the set $\\{{\rm Z}_{t}\in\mathbb{R}_{+},\,t\geq 0\\}$. The general case is
much more delicate to deal with. Indeed, the underlying spaLf being multi-
indexed, it may reach the level $-{\rm r}$ through infinitely many different
paths. However, the representation of $\mathbf{T}_{\rm r}$ for $d\geq 1$ is as
expected and the following result will be proved in the Appendix.

######  Proposition 2.1.

Let ${\rm Z}$ be a MCSBP and let ${\bf X}$ be the underlying spaLf in the
Lamperti representation $(\ref{Lamperti})$. Then for all $i\in[d]$, and ${\rm
r}\in\mathbb{R}_{+}^{d}$,

$T_{\rm r}^{(i)}=\int_{0}^{\infty}Z_{s}^{(i)}\,ds\,,\;\;\mbox{$\mathbb{P}_{\rm
r}$-a.s.~{}on the set $\\{{\rm Z}_{t}\in\mathbb{R}^{d}_{+},\,t\geq 0\\}$}\,.$

Let us finally specify that since the results of this paper are only concerned
with distributional properties of MCSBP’s, when refereing to such a process
${\rm Z}$, there will be no ambiguity in mentioning the underlying spaLf ${\bf
X}$ in the Lamperti representation (2.7). Then $\mathbb{P}$ will be a
reference probability measure under which ${\bf X}$ is a spaLf issued from 0
and ${\rm Z}$ is a MCSBP with family of probability measures
$(\mathbb{P}_{{\rm r}})_{{\rm r}\in\mathbb{R}^{d}_{+}}$.

##  3\. Main results

We start by fixing some natural conditions for the study of extinction times.
We have already mentioned that we assume conservativeness. When ${\rm Z}$ is
not irreducible, several subsets of types (that is sub-vectors of ${\rm Z}$)
can have an asymptotic behaviour that does not depend on other types. In such
cases, the study of extinction of ${\rm Z}$ is reduced to the study of partial
extinction of its irreducible classes. Therefore in order to study the
extinction property, it is natural to assume that ${\rm Z}$ is irreducible. On
the other hand, recall that from Theorem 2.1 if $(H_{\varphi})$ does not hold,
then the underlying spaLf satisfies $\mathbb{P}({\bf T}_{\rm
r}\in\mathbb{R}_{+}^{d})=0$, for all ${\rm r}\in\mathbb{R}_{+}^{d}$ and we can
derive from the Lamperti representation (2.7), that $\mathbb{P}_{\rm
r}\left(\lim\limits_{t\rightarrow\infty}{\rm Z}_{t}=\mathbf{0}\right)=0$, that
is, with probability one, extinction does not hold. Therefore, throughout the
rest of this paper, we will assume that

${\rm Z}$ is conservative, irreducible and $(H_{\varphi})$ holds.

In particular, these assumptions will not be recalled in the statements. Then,
recall from Theorem 2.1 that the branching mechanism
$\varphi:D_{\varphi}\rightarrow(0,\infty)^{d}$ of ${\rm Z}$ admits an inverse
denoted by $\phi$.

We first state the expression of the probability of extinction in terms of the
(inverse of) the branching mechanism.

######  Theorem 3.1.

Let ${\rm Z}$ be a MCSBP, then for all ${\rm r}\in\mathbb{R}_{+}^{d}$,

(3.12) $\mathbb{P}_{\rm r}\left(\lim\limits_{t\rightarrow\infty}{\rm
Z}_{t}=\mathbf{0}\right)=e^{-\langle{\rm r},\phi(\mathbf{0})\rangle}.$

Recall that when the underlying spaLf is integrable, that is
$\mathbb{E}(X_{1}^{i,j})<\infty$, for all $i,j\in[d]$, then from Theorem 3.4
in [7], $\phi(0)>0$ if and only if ${\rm Z}$ is supercritical. Hence our
result is consistent with Theorem 2 and the subsequent remark in [11] where it
is claimed that if ${\rm Z}$ is critical or subcritical then extinction (at a
finite time or at infinity) holds almost surely.

We define the probability of extinction at a finite time under
$\mathbb{P}_{\rm r}$, ${\rm r}\in\mathbb{R}_{+}^{d}$ by,

(3.13) $q_{\rm r}:=\mathbb{P}_{\rm r}(\mbox{${\rm Z}_{t}=\mathbf{0}$ for some
$t>0$}).$

Since the family of events $\\{{\rm Z}_{t}=\mathbf{0}\\}$ is non decreasing in
$t$ and converges to the event $\\{\mbox{${\rm Z}_{t}=\mathbf{0}$ for some
$t>0$\\}}$, the probability $q_{\rm r}$ can be written as

(3.14) $q_{\rm r}=\lim\limits_{t\rightarrow+\infty}\mathbb{P}_{\rm r}({\rm
Z}_{t}=\mathbf{0})\,.$

Note also that from (2.4), for all $t\geq 0$, the mapping $\lambda\mapsto{\rm
u}_{t}(\lambda)$ statisfies ${\rm u}_{t}(\lambda)\leq{\rm
u}_{t}(\lambda^{\prime})$, whenever $\lambda\leq\lambda^{\prime}$. Moreover,
the limit ${\rm u}_{t}(\infty)$ of ${\rm u}_{t}(\lambda)$ when all coordinates
$\lambda_{1},\dots,\lambda_{d}$ of $\lambda$ tend to $\infty$ does not depend
on the relative speeds at which the $\lambda_{i}$’s tend to $\infty$ and it
satisfies,

(3.15) $\mathbb{P}_{\rm r}({\rm Z}_{t}=\mathbf{0})=e^{-\langle{\rm r},{\rm
u}_{t}(\infty)\rangle}\,.$

The probability of extinction $q_{\rm r}$ depends on the finiteness of ${\rm
u}_{t}(\infty)$ as follows.

######  Theorem 3.2.

Let ${\rm Z}$ be a MCSBP. Assume that none of the processes $X^{i,j}$ for
$i,j\in[d]$ such that $i\neq j$ is a compound Poisson process. Then we have
the following dichotomy$:$

  * $(i)$

either for all $t\geq 0$ and $i\in[d]$, $u^{(i)}_{t}(\infty)<\infty$ and then
for all $t>0$ and ${\rm r}\in\mathbb{R}_{+}^{d}$, $\mathbb{P}_{\rm r}({\rm
Z}_{t}=\mathbf{0})>0$ and in this case,

$q_{\rm r}=e^{-\langle{\rm r},\phi(\mathbf{0})\rangle},\;\;\;\mbox{for all
${\rm r}\in\mathbb{R}_{+}^{d}$, }$

  * $(ii)$

or for all $t\geq 0$ and $i\in[d]$, $u^{(i)}_{t}(\infty)=\infty$ and then for
all ${\rm r}\in\mathbb{R}^{d}_{+}\setminus\\{\mathbf{0}\\}$, $q_{\rm r}=0$.

Our goal is to distinguish between two types of extinction defined by the two
exhaustive events

$\left\\{\lim\limits_{t\rightarrow\infty}{\rm Z}_{t}=\mathbf{0}\,\text{ and
}\,{\rm Z}_{t}>\mathbf{0},\;\mbox{for all
$t>0$}\right\\}\;\;\;\mbox{and}\;\;\;\left\\{\mbox{${\rm Z}_{t}=\mathbf{0}$
for some $t>0$}\right\\}.$

In the first case, we will say that extinction occurs at infinity and that the
MCSBP is extinguished and in the second case we will say that extinction
occurs at a finite time and that the MCSBP becomes extinct. Then, we wish to
find conditions bearing on the mechanism $\varphi$ allowing us to determine if
extinction holds at a finite time or not. The probability for ${\rm Z}$ to be
extinguished at infinity under $\mathbb{P}_{\rm r}$, ${\rm
r}\in\mathbb{R}_{+}^{d}$ will be denoted as follows:

(3.16) $\bar{q}_{\rm r}:=\mathbb{P}_{\rm
r}\left(\lim\limits_{t\rightarrow\infty}{\rm Z}_{t}=\mathbf{0}\,\text{ and
}\,{\rm Z}_{t}>\mathbf{0},\;\mbox{for all $t>0$}\right).$

Let us briefly recall the case $d=1$, from [10]. Let $\varphi$ be the
branching mechanism of the continuous state branching process $Z$. If the
following integral condition, called Grey’s condition, is satisfied,

$(G_{\varphi})\qquad\int^{\infty}\frac{ds}{\varphi(s)}<\infty,$

then extinction can only occur at a finite time. Moreover, under
$(G_{\varphi})$, if $Z$ starts from $r\geq 0$, then the probability of
extinction is $e^{-r\phi(0)}$, where $\phi$ is the inverse of $\varphi$. If
$(G_{\varphi})$ is not satisfied, then extinction of $Z$ can only occur at
infinity.

Let us now consider the general case $d\geq 1$. To this aim, we denote by
$\tilde{\varphi}_{i}$ the Laplace exponent of the spectrally positive Lévy
process $X^{i,i}$, that is

(3.17) $\tilde{\varphi}_{i}(s)=\varphi_{i}(s{\rm e}_{i}),\quad s\geq 0.$

Note that since $(H_{\varphi})$ is satisfied, none of the processes $X^{i,i}$
is a subordinator, so that
$\lim\limits_{s\rightarrow\infty}\tilde{\varphi}_{i}(s)=\infty$, for all
$i\in[d]$. Then let us introduce the following condition bearing on this
Laplace exponent,

$(G^{(i)}_{\varphi})\qquad\qquad\int^{\infty}\dfrac{ds}{\tilde{\varphi}_{i}(s)}<\infty\,.\
$

We are now able to state our main result. It provides an extension for MCSBP
of Grey’s condition. Recall the definition of $q_{\rm r}$ and $\bar{q}_{\rm
r}$ in (3.13) and (3.16).

######  Theorem 3.3.

Let ${\rm Z}$ be a MCSBP such that none of the processes $X^{i,j}$ for
$i,j\in[d]$, $i\neq j$ is a compound Poisson process.

  * $1.$

Assume that $(G_{\varphi}^{(i)})$ is satisfied for all $i\in[d]$. Then ${\rm
Z}$ can only become extinct at a finite time, that is $\bar{q}_{\rm r}=0$ for
all ${\rm r}\in\mathbb{R}^{d}_{+}$. Moreover,

(3.18) $q_{\rm r}=e^{-\langle{\rm r},\phi(\mathbf{0})\rangle},$

for all ${\rm r}\in\mathbb{R}^{d}_{+}$, and in this case, $q_{\rm
r}=\mathbb{P}_{\rm r}\left(\lim\limits_{t\rightarrow\infty}{\rm
Z}_{t}=\mathbf{0}\right)$.

  * $2.$

Assume that $(G_{\varphi}^{(i)})$ is not satisfied for some $i\in[d]$. Then,
${\rm Z}$ cannot become extinct at a finite time i.e. $q_{\rm r}=0$ for all
${\rm r}\in\mathbb{R}_{+}^{d}\setminus\\{\mathbf{0}\\}$. Moreover,

(3.19) $\bar{q}_{\rm r}=e^{-\langle{\rm r},\phi(\mathbf{0})\rangle},$

for all ${\rm r}\in\mathbb{R}^{d}_{+}\setminus\\{\mathbf{0}\\}$, and in this
case, $\bar{q}_{\rm r}=\mathbb{P}_{\rm
r}\left(\lim\limits_{t\rightarrow\infty}{\rm Z}_{t}=\mathbf{0}\right)$.

Note that as far as we know, part 2. of Theorem 3.3 when $d=1$ is never
mentioned in the literature.

As mentioned in the introduction, the fact that extinction at infinity is only
determined by the law of the diagonal Lévy processes $X^{i,i}$ may seem quite
surprising at first glance. The case of neutral MCSBP’s for which part 1. of
Theorem 3.3 can be verified directly, gives us a better intuition of this
result. Recall that a MCSBP is said to be neutral if the law of
$\sum_{i=1}^{d}X^{i,j}$ does not depend on $j$. It is plain that in this case,
$Z^{(1)}+\dots+Z^{(d)}$ is a continuous state branching process whose
branching mechanism is $d\hat{\varphi}$, where $\hat{\varphi}$ is the Laplace
exponent of $\sum_{i=1}^{d}X^{i,j}$. Therefore, from Grey’s result, ${\rm Z}$
becomes extinct at a finite time with positive probability if and only if

(3.20) $\int^{\infty}\frac{ds}{\hat{\varphi}(s)}<\infty.$

Let us denote by $\varphi_{ij}$ the Laplace exponent of the Lévy process
$X^{i,j}$, that is

(3.21) $\varphi_{ij}(s)=\varphi_{j}(s{\rm e}_{i}),\quad s\geq 0,$

and note that $\tilde{\varphi}_{i}=\varphi_{ii}$ according to our previous
notation. Note also that $\hat{\varphi}(s)=\varphi_{j}(s,s,\dots,s)$, for all
$s\geq 0$ and $j\in[d]$. Then from Lemma 5.3 and inequality (5.33),

(3.22)
$\sum_{i=1}^{d}\varphi_{ij}(s)\leq\hat{\varphi}(s)\leq\tilde{\varphi}_{j}(s)\,,\;\;\;j\in[d],\;s\geq
0.$

Therefore if $(\ref{6566})$ holds, then $(G_{\varphi}^{(i)})$ holds for all
$j\in[d]$. Now assume that $(G_{\varphi}^{(i)})$ holds for all $j\in[d]$. Then
from the asymptotic behaviour of Laplace exponents of spectrally positive Lévy
processes, see Chap. VII and in [3],
$\lim_{s\rightarrow\infty}\tilde{\varphi}_{j}(s)/s=\infty$ and since
$(\sum_{i\neq j}\varphi_{ij}(s))/s$ is bounded by a constant when $s$ is
large, $\sum_{i=1}^{d}\varphi_{ij}(s)$ is equivalent to
$\tilde{\varphi}_{j}(s)$ when $s$ tends to infinity. This means that
$\int^{\infty}\frac{ds}{\sum_{i=1}^{d}\varphi_{ij}(s)}<\infty$, so that
$(\ref{6566})$ holds. Our proof in the general case is based on similar
arguments. In order to prove part 1. of Theorem 3.3, we will actually
construct a function $\hat{\varphi}$ (no necessarily being a Laplace exponent)
which minimizes all Laplace exponents $\tilde{\varphi}_{j}$ and such that
condition (3.20) implies part $(i)$ in Theorem 3.2. The general idea is that
the distinction between extinction at a finite time and extinction at infinity
depends only on the asymptotic behaviour of $\varphi_{i}(\lambda)$, $i\in[d]$,
when $\lambda\rightarrow\infty$ and this is the same as the asymptotic
behaviour of $\tilde{\varphi}_{i}(\lambda_{i})$, $i\in[d]$, when
$\lambda_{i}\rightarrow\infty$. Let us emphasize that conditions for
conservativeness certainly depends on the law of the subordinators $X^{i,j}$.
Again, one can easily convince ourself of this fact by looking at the neutral
case.

We end this section by saying a few words about the exclusion of compound
Poisson processes. If for instance, type $1$ is such that for all $j\neq 1$,
$X^{1,j}$ is either identically equal to 0 or is a compound Poisson process,
then there is an a.s. positive random time $\tau$ such that $X^{1,j}_{t}=0$,
for all $j\neq 1$ and $t\in[0,\tau]$. Hence we derive from (2.7) that if ${\rm
r}$ is such that $r_{1}=0$, then $Z^{(1)}_{t}=0$, for all $t\in[0,\gamma]$ for
some a.s. positive random time $\gamma$. If moreover conditions
$(G_{\varphi}^{(j)})$ are satisfied for all $j\neq 1$, then from part $(ii)$
of the theorem, the process ${\rm Z}$ restricted to types $2,3,\dots,d$ can
reach ${\bf 0}$ with positive probability. Moreover, it is reasonable to think
that this hitting time of ${\bf 0}$ can occur in the time interval
$[0,\gamma]$ with positive probability. Therefore, the probability of
extinction at a finite time of ${\rm Z}$ is positive in this case. Actually,
whenever there exist processes $X^{i,j}$, $i\neq j$ that are compound Poisson
processes, it is possible that more than one condition $(G_{\varphi}^{(i)})$
is required for the process not to become extinct at a finite time. A general
result including compound Poisson processes could be proved, but we do not
think this case is much relevant.

##  4\. Proof of Theorems 3.1 and 3.2

Proof of Theorem 3.1: The Lamperti representation (2.7) yields the following
path by path relationship between ${\rm Z}$ starting at ${\rm Z}_{0}={\rm r}$
and the first passage times of its underlying spaLf,

(4.23) $\left\\{\lim\limits_{t\rightarrow\infty}{\rm
Z}_{t}=\mathbf{0}\right\\}\subset\left\\{\mathbf{T}_{{\rm
r}^{\prime}}\in\mathbb{R}_{+}^{d}\right\\},\;\;\;\mbox{for all ${\rm
r}^{\prime}$ such that $0\leq r_{i}^{\prime}<r_{i}$, for all $i\in[d]$.}$

Indeed, it follows directly from equation (2.7) that for ${\rm Z}$ to be as
close to $\mathbf{0}$ as possible, the spaLf ${\bf X}$ has to reach all
possible levels ${\rm r}^{\prime}$ such that $0\leq r_{i}^{\prime}<r_{i}$, for
all $i\in[d]$. Together with (2.11) inclusion (4.23) implies that

$\mathbb{P}_{\rm r}\left(\lim\limits_{t\rightarrow\infty}{\rm
Z}_{t}=\mathbf{0}\right)\leq e^{-\langle{\rm
r}^{\prime},\phi(\mathbf{0})\rangle},\;\;\;\mbox{for all ${\rm r}^{\prime}$
such that $0\leq r_{i}^{\prime}<r_{i}$, for all $i\in[d]$,}$

which gives one inequality in (3.12) by continuity.

In order to prove the other inequality, let us set $\int_{0}^{\infty}{\rm
Z}_{s}\,ds=\left(\int_{0}^{\infty}Z_{s}^{(i)}\,ds\right)_{i\in[d]}$ and note
that from (2.7), on the set $\left\\{\int_{0}^{\infty}{\rm
Z}_{s}\,ds\in\mathbb{R}_{+}^{d}\right\\}$, the process ${\rm Z}$ converges
$\mathbb{P}_{\rm r}$-almost surely and its limit is $\mathbf{0}$. This implies
the inclusion,

$\left\\{\int_{0}^{\infty}{\rm
Z}_{s}\,ds\in\mathbb{R}_{+}^{d}\right\\}\subset\left\\{\lim_{t\rightarrow\infty}{\rm
Z}_{t}=\mathbf{0}\right\\},\;\;\;\mbox{$\mathbb{P}_{\rm r}$-a.s.}$

Then recall from Proposition 2.1 that, ${\bf T}_{\rm r}=\int_{0}^{\infty}{\rm
Z}_{s}\,ds$, $\mathbb{P}_{\rm r}$-a.s. on the set $\\{{\rm
Z}_{t}\in\mathbb{R}^{d}_{+},\,t\geq 0\\}$ (which is supposed to be of
probability 1 here), for all ${\rm r}\in\mathbb{R}_{+}^{d}$, so that from
(2.11) and the above inclusion,

$\mathbb{P}({\bf T}_{\rm r}\in\mathbb{R}_{+}^{d})=e^{-\langle{\rm
r},\phi(\mathbf{0})\rangle}\leq\mathbb{P}_{\rm
r}\left(\lim_{t\rightarrow\infty}{\rm Z}_{t}=\mathbf{0}\right),$

which achieves our proof. $\quad\Box$

The next two lemmas will be needed for the proof of Theorem 3.2.

######  Lemma 4.1.

For all $\lambda\in D_{\varphi}$ and $t\geq 0$, ${\rm u}_{t}(\lambda)\in
D_{\varphi}$ and for all $i\in[d]$, the mapping $t\mapsto
u^{(i)}_{t}(\lambda)$ is decreasing. Moreover for all $i\in[d]$, the mapping
$t\mapsto u^{(i)}_{t}(\infty)$ is decreasing.

###### Proof.

Let $\lambda\in D_{\varphi}$. Then for all $i\in[d]$,
$\dfrac{\partial}{\partial
t}u^{(i)}_{t}(\lambda)|_{t=0}=-\varphi_{i}(\lambda)<0$ so that for all $h$
small enough, $u_{h}^{(i)}(\lambda)<\lambda_{i}=u_{0}^{(i)}(\lambda)$. That is
$[{\rm
u}_{h}(\lambda),\lambda]:=\prod_{i\in[d]}[u_{h}^{(i)}(\lambda),\lambda_{i}]\neq\emptyset$.
Now let us fix $i\in[d]$. Then thanks to the theorem of finite increments, for
all $h\geq 0$ small enough, there exists $\lambda^{\prime}\in[{\rm
u}_{h}(\lambda),\lambda]$ such that

$u^{(i)}_{t}({\rm
u}_{h}(\lambda))-u^{(i)}_{t}(\lambda)=u_{t+h}^{(i)}(\lambda)-u^{(i)}_{t}(\lambda)=\sum\limits_{j=1}^{d}\left(u^{(j)}_{h}(\lambda)-\lambda_{j}\right)\dfrac{\partial}{\partial\lambda_{j}}u^{(i)}_{t}(\lambda^{\prime})\,,\
$

where we have used the semigroup property of ${\rm u}_{t}(\lambda)$ in the
first equality. Then making $h$ tend to zero, we obtain

$\displaystyle\lim\limits_{h\rightarrow
0}\dfrac{u_{t+h}^{(i)}(\lambda)-u^{(i)}_{t}(\lambda)}{u^{(i)}_{h}(\lambda)-\lambda_{i}}$
$\displaystyle=$
$\displaystyle\dfrac{\partial}{\partial\lambda_{i}}u^{(i)}_{t}(\lambda)+$
$\displaystyle\sum\limits_{j\neq i}\dfrac{\partial}{\partial
t}u^{(j)}_{t}(\lambda)|_{t=0}\left(\dfrac{\partial}{\partial
t}u^{(i)}_{t}(\lambda)|_{t=0}\right)^{-1}\dfrac{\partial}{\partial\lambda_{j}}u^{(i)}_{t}(\lambda)$
$\displaystyle=$
$\displaystyle\dfrac{\partial}{\partial\lambda_{i}}u^{(i)}_{t}(\lambda)+\sum\limits_{j\neq
i}\dfrac{\varphi_{j}(\lambda)}{\varphi_{i}(\lambda)}\dfrac{\partial}{\partial\lambda_{j}}u^{(i)}_{t}(\lambda).$

This last quantity is positive. Indeed, it follows from Lemma 3.4 in [1] and
part $(ii)$ of Lemma A.1 in [2], that $\mathbb{E}_{e_{i}}(Z_{t}^{(j)})>0$, for
all $t\geq 0$ and all $i,j\in[d]$. (In [1] it is actually assumed that
$X^{i,j}$, $i,j\in[d]$ are integrable but our result can easily be extended to
the general case.) Therefore, for every $i,j\in[d]$, $\lambda_{j}\mapsto
u_{t}^{(i)}(\lambda)$ is increasing. Moreover, since $\lambda\in D_{\varphi}$,
$\varphi_{j}(\lambda)>0$, for all $j\in[d]$. This yields,

$\displaystyle\lim\limits_{h\rightarrow
0}\dfrac{u^{(i)}_{t+h}(\lambda)-u^{(i)}_{t}(\lambda)}{u^{(i)}_{h}(\lambda)-\lambda_{i}}$
$\displaystyle=\dfrac{\partial}{\partial
t}u^{(i)}_{t}(\lambda)\left(\dfrac{\partial}{\partial
t}u^{(i)}_{t}(\lambda)|_{t=0}\right)^{-1}>0.$

Thus $\dfrac{\partial}{\partial t}u^{(i)}_{t}(\lambda)$ has the same sign as
$\dfrac{\partial}{\partial
t}u^{(i)}_{t}(\lambda)|_{t=0}=-\varphi_{i}(\lambda)<0$ since $\lambda\in
D_{\varphi}$. In conclusion, it follows from the differential system
$(S_{\varphi})$ satisfied by ${\rm u}$ and $\varphi$, that for all $\lambda\in
D_{\varphi}$, $t\geq 0$, and $i\in[d]$

$\dfrac{\partial}{\partial t}u^{(i)}_{t}(\lambda)=-\varphi_{i}({\rm
u}_{t}(\lambda))<0,$

that is ${\rm u}_{t}(\lambda)\in D_{\varphi}$ and for all $i\in[d]$, $t\mapsto
u^{(i)}_{t}(\lambda)$ is decreasing.

The last assertion is a consequence of the previous one, but it is also
straightforward from (3.15).
∎

######  Lemma 4.2.

Assume that none of the processes $X^{i,j}$ for $i,j\in[d]$ such that $i\neq
j$ is a compound Poisson process. Then either for all $t>0$ and $i\in[d]$,
$u^{(i)}_{t}(\infty)=\infty$, or for all $t>0$ and $i\in[d]$,
$u^{(i)}_{t}(\infty)<\infty$.

###### Proof.

Assume that there exists $t>0$ such that $u^{(i)}_{t}(\infty)=\infty$ for some
$i\in[d]$. Let $j$ such that $j\neq i$ and $X^{i,j}$ is not identically equal
to 0. Such an index exists since ${\bf X}$ is irreducible. Assume moreover
that $u^{(j)}_{t}(\infty)<\infty$. Since $X^{i,j}$ is not a compound Poisson
process, $\varphi_{j}$ tends to $-\infty$ on the coordinate $i$. Therefore,
since $u^{(i)}_{t}(\infty)=\infty$, we have $\varphi_{j}({\rm
u}_{t}(\infty))=-\infty$ and this contradicts the fact that ${\rm
u}_{t}(\infty)\in\overline{D}_{\varphi}$ following from Lemma 4.1. In other
words, for all $j\in[d]$ such that $j\neq i$ and $X^{i,j}$ is not identically
equal to 0, $u^{(j)}_{t}(\infty)=\infty$. By irreducibility, we deduce that
for all $j\in[d]$, $u^{(j)}_{t}(\infty)=\infty$. We have proved that, if there
exists $t>0$ and $i\in[d]$ such that $u^{(i)}_{t}(\infty)<\infty$, then for
all $j\in[d]$, $u^{(j)}_{t}(\infty)<\infty$. Moreover from Lemma 4.1, for all
$s\geq t$ and $j\in[d]$, $u^{(j)}_{s}(\infty)<\infty$.

Now let $\tau^{(i)}=\inf\\{s\geq 0:u^{(i)}_{s}(\infty)<\infty\\}$. Thanks to
previous arguments, $\tau=\tau^{(i)}$ does not depend on $i\in[d]$. Assume
$\tau\in(0,\infty)$. For $0<s<\tau$ and $\tau-s<t<\tau$, for all $i\in[d]$ and
$\lambda\in D_{\varphi}$,

$u^{(i)}_{t+s}(\lambda)=u^{(i)}_{t}({\rm
u}_{s}(\lambda))\underset{\lambda\rightarrow+\infty}{\rightarrow}u^{(i)}_{t}(\infty)=\infty\,.\
$

We obtain a contradiction since $u^{(i)}_{t+s}(\infty)<\infty$. ∎

Proof of Theorem 3.2: It has already been proved in Lemma 4.2 that either for
all $t\geq 0$ and $i\in[d]$, $u^{(i)}_{t}(\infty)=\infty$ or for all $t>0$ and
$i\in[d]$, $u^{(i)}_{t}(\infty)<\infty$. Now recall from (3.15) that
$\mathbb{P}_{\rm r}({\rm Z}_{t}=\mathbf{0})=e^{-\langle{\rm r},{\rm
u}_{t}(\infty)\rangle}$. Hence if for all $t\geq 0$ and $i\in[d]$,
$u^{(i)}_{t}(\infty)=\infty$, then for all $t\geq 0$ and ${\rm
r}\in\mathbb{R}^{d}_{+}\setminus\\{\mathbf{0}\\}$, $\mathbb{P}_{\rm r}({\rm
Z}_{t}=\mathbf{0})=0$, so that for all ${\rm
r}\in\mathbb{R}^{d}_{+}\setminus\\{\mathbf{0}\\}$, $q_{\rm
r}=\lim\limits_{t\rightarrow\infty}\mathbb{P}_{\rm r}({\rm
Z}_{t}=\mathbf{0})=0$.

Again from (3.15), if for all $t\geq 0$ and $i\in[d]$,
$u^{(i)}_{t}(\infty)<\infty$, then for all $t\geq 0$ and ${\rm r}$,
$\mathbb{P}_{\rm r}({\rm Z}_{t}=\mathbf{0})=e^{-\langle{\rm r},{\rm
u}_{t}(\infty)\rangle}>0$ and since all coordinates of $t\mapsto{\rm
u}_{t}(\infty)$ are decreasing, $q_{\rm r}\in(0,1]$. Let us prove that in this
case, $q_{\rm r}=e^{-\langle{\rm r},\phi({\bf 0})\rangle}$. From (3.14) and
(3.15) this equality is equivalent to

(4.24) ${\rm k}=(k^{(1)},\dots,k^{(d)}):=\lim_{s\rightarrow\infty}{\rm
u}_{s}(\infty)=\phi({\bf 0}).$

In order to prove (4.24), we will use the semi-group property. That is for all
$i\in[d]$, $s,t\geq 0$ and $\lambda\in\mathbb{R}^{d}_{+}$,

(4.25) $u_{t+s}^{(i)}(\lambda)=u_{t}^{(i)}({\rm u}_{s}(\lambda)).$

Recall that for all $s>0$, ${\rm u}_{s}(\infty)\in\mathbb{R}^{d}_{+}$. Then
using (4.25) and making $\lambda$ tend to infinity, we obtain by the
continuity of $u_{t}^{(i)}$,

(4.26) $u_{t+s}^{(i)}(\infty)=u_{t}^{(i)}({\rm u}_{s}(\infty))\,.$

Recall from (4.24) the definition of ${\rm k}$. Since the mappings $s\mapsto
u^{(i)}_{s}(\infty)$, $i\in[d]$ are decreasing, see (3.15), we have ${\rm
k}\in\mathbb{R}_{+}^{d}$, so that from (4.26) and by continuity of
$u_{t}^{(i)}$ again, we obtain when $s$ tends to $\infty$ that for all
$i\in[d]$,

(4.27) $k^{(i)}=u_{t}^{(i)}({\rm k})\,.$

Therefore $t\mapsto{\rm u}_{t}({\rm k})$ is constant, so that from
$(S_{\varphi})$, $\varphi({\rm k})={\bf 0}$ and hence from part 3. of Theorem
2.1, ${\rm k}\in\\{\mathbf{0},\phi(\mathbf{0})\\}$. Moreover, thanks to Lemma
4.1, for all $\lambda\in D_{\varphi}$ and $t\geq 0$, ${\rm u}_{t}(\lambda)\in
D_{\varphi}$, thus ${\rm k}\in\overline{D}_{\varphi}$. Then it is proved at
the beginning of the proof of Theorem 3.4 in [7] that $\phi({\bf 0})$ is the
only solution of the equation $\varphi(\lambda)={\bf 0}$ in
$\overline{D}_{\varphi}$. This, shows that ${\rm k}=\phi(\mathbf{0})$ and we
conclude that $q_{\rm r}=e^{-\langle{\rm r},\phi({\bf 0})\rangle}$.
$\quad\Box$

##  5\. Proof of Theorem 3.3

For $i\in[d]$, let us call _the $i^{th}$ diagonal branching process_, the
continuous state branching process ${\tilde{Z}^{(i)}}$ which is solution of
the one-dimensional Lamperti representation,

(5.28)
$\tilde{Z}^{(i)}_{t}=r_{i}+X^{i,i}\left(\int\limits_{0}^{t}\tilde{Z}^{(i)}_{s}ds\right),\;\;\;t\geq
0.$

Its branching mechanism corresponds to the Laplace exponent
$\tilde{\varphi}_{i}$ of $X^{i,i}$ whose definition has been given in (3.17),
see also (3.21) below. Then the Laplace exponent $\tilde{u}^{(i)}$ of
$\tilde{Z}^{(i)}$ satisfies for all $\alpha\geq 0$,

$\left\\{\begin{array}[]{ll}\dfrac{\partial}{\partial
t}\tilde{u}^{(i)}_{t}(\alpha)&=-\tilde{\varphi}_{i}(\tilde{u}^{(i)}_{t}(\alpha))\\\
\tilde{u}^{(i)}_{0}(\alpha)&=\alpha.\end{array}\right.$

Extinction at a finite time of the $i^{th}$ diagonal branching process is
characterized through Grey’s condition for $\tilde{\varphi}_{i}$, see
$(G_{\varphi}^{(i)})$ right after (3.17).

Let us start with the proof of part 1. of Theorem 3.3. Let $\varphi$ be the
Laplace exponent of some spaLf satisfying $(G^{(i)}_{\varphi})$, for all
$i\in[d]$. We will show that $u^{(i)}_{t}(\infty)<\infty$, for all $t\geq 0$
and $i\in[d]$ and use Theorem 3.2. For this purpose, we need the following
series of Lemmas.

We first show that in the case $d=1$, for some functions that are more general
than Laplace exponents of spectrally positive Lévy processes, the system
$(S_{\varphi})$ defined in Subsection 2.1 still admits a unique solution with
nice properties.

######  Lemma 5.1.

Let $s_{0}\geq 0$ and $f:[s_{0},\infty)\rightarrow[0,\infty)$ be some
continuous, increasing function such that $f(s_{0})=0$ and
$\int_{s_{0}}ds/f(s)=\infty$. Fix $\delta>s_{0}$ and let

$F(x):=\int_{\delta}^{x}\frac{ds}{f(s)}\,,\;\;\;x\in(s_{0},\infty).$

Set $F(\infty):=\lim\limits_{x\rightarrow\infty}F(x)$. Then
$F:(s_{0},\infty)\rightarrow(-\infty,F(\infty))$ is a bijection. Let us denote
by $F^{-1}:(-\infty,F(\infty))\rightarrow(s_{0},\infty)$ its inverse. Then for
all $\lambda>s_{0}$, $v_{t}(\lambda)=F^{-1}(F(\lambda)-t)$, $t\geq 0$ is the
unique solution of the differential equation

(5.29)
$\left\\{\begin{array}[]{ll}&\frac{\partial}{\partial_{t}}v_{t}(\lambda)=-f(v_{t}(\lambda))\,,\;\;\;t>0,\\\
&u_{0}(\lambda)=\lambda\,.\end{array}\right.$

For every $\lambda>s_{0}$, the function $t\mapsto v_{t}(\lambda)$ is
decreasing and $\lim\limits_{t\rightarrow\infty}v_{t}(\lambda)=s_{0}$. For
every $t\geq 0$, the function $\lambda\mapsto v_{t}(\lambda)$ is increasing.
Moreover,
$v_{t}(\infty):=\lim\limits_{\lambda\rightarrow\infty}v_{t}(\lambda)<\infty$,
for all $t>0$ if and only if $\int_{\delta}^{\infty}ds/f(s)<\infty$.

###### Proof.

It is plain that $F:(s_{0},\infty)\rightarrow(-\infty,F(\infty))$ is a
bijection. Let $\lambda>s_{0}$. Then the function $t\mapsto
F^{-1}(F(\lambda)-t)$ is clearly differentiable and we readily check that it
satisfies (5.29). Uniqueness of the solution is given by integrating the
function
$t\mapsto-\frac{\partial}{\partial_{t}}v_{t}(\lambda)/f(v_{t}(\lambda))$
through an obvious change of variables allowing us to write

(5.30) $\int_{v_{t}(\lambda)}^{\lambda}\frac{ds}{f(s)}=t\,,$

and the expression of $v_{t}$ follows. The facts that the function $t\mapsto
v_{t}(\lambda)$ is decreasing, that
$\lim\limits_{t\rightarrow\infty}v_{t}(\lambda)=s_{0}$ and that the function
$\lambda\mapsto v_{t}(\lambda)$ is increasing are straightforward. Finally
(5.30) implies that for every $t>0$, $v_{t}(\infty)<\infty$ if and only if
$\int_{\delta}^{\infty}ds/f(s)<\infty$. ∎

Recall from (3.21) that $\varphi_{ij}$ is the Laplace exponent of the Lévy
process $X^{i,j}$ and that $\varphi_{ii}=\tilde{\varphi}_{i}$ according to our
notation. Then let us consider the following mappings:

$\displaystyle\bar{\varphi}:=\min\\{\tilde{\varphi}_{i},\,i\in[d]\\}\,,\;\;\bar{\bar{\varphi}}:=\sum_{k\neq
l}\varphi_{kl}\,,\;\;\varphi^{*}_{i}(\lambda):=\bar{\varphi}(\lambda_{i})+\sum_{j\neq
i}\bar{\bar{\varphi}}(\lambda_{j}).$

Let us define the diagonal on $\mathbb{R}_{+}^{d}\setminus\\{\mathbf{0}\\}$ by
$\Delta=\\{\lambda\in\mathbb{R}_{+}^{d}\setminus\\{\mathbf{0}\\}:\lambda_{1}=\lambda_{2}=\dots=\lambda_{d}\\}$.

######  Lemma 5.2.

Recall that the assumptions $(G^{(i)}_{\varphi})$, $i\in[d]$ are in force. Let
$s_{0}$ be the largest solution of the equation
$\bar{\varphi}(x)+(d-1)\bar{\bar{\varphi}}(x)=0$. Then for all
$\lambda\in\Delta$ with $\lambda_{1}>s_{0}$, the differential system

$(S_{\varphi^{*}})\;\;\qquad\;\left\\{\begin{array}[]{ll}\dfrac{\partial}{\partial
t}v^{(i)}_{t}(\lambda)&=-\varphi^{*}_{i}({\rm v}_{t}(\lambda))\\\\[8.5359pt]
v^{(i)}_{0}(\lambda)&=\lambda_{i}\end{array}\;\;\;,\;\;i\in[d]\,,\;\;t\geq
0\right.$

admits as a solution the mapping $t\mapsto{\rm v}_{t}(\lambda)$ given by

$v_{t}^{(1)}(\lambda)=\dots=v_{t}^{(d)}(\lambda)=F^{-1}(F(\lambda)-t),$

for all $t\geq 0$, where $F$ is given as in Lemma 5.1, with
$f(x):=\bar{\varphi}(x)+(d-1)\bar{\bar{\varphi}}(x)$, $x\geq s_{0}$.

###### Proof.

Let us check that $f$ satisfies the conditions of Lemma 5.1. First note that
$\bar{\varphi}+(d-1)\bar{\bar{\varphi}}$ is clearly continuous on
$[0,\infty)$. Set
$f_{i}(x):=\tilde{\varphi}_{i}(x)+(d-1)\bar{\bar{\varphi}}(x)$, $x\geq 0$.
Then $f_{i}$ is the characteristic exponent of a spectrally positive Lévy
process which is not a subordinator under our assumptions. Therefore, $f_{i}$
is convex with $\lim_{x\rightarrow\infty}f_{i}(x)=\infty$, and the equation
$f_{i}(x)=0$ has at most two roots. Let $s_{i}$ be the largest of these roots.
Then $f_{i}$ is increasing on $[s_{i},\infty)$. Moreover since
$\bar{\varphi}+(d-1)\bar{\bar{\varphi}}=\min\\{f_{i},i\in[d]\\}$, the point
$s_{0}=\max\\{s_{i},i\in[d]\\}$ is the largest root of the equation
$\bar{\varphi}(x)+(d-1)\bar{\bar{\varphi}}(x)=0$. Therefore, the function
$f:[s_{0},\infty)\rightarrow[0,\infty)$ defined by
$f(x):=\bar{\varphi}(x)+(d-1)\bar{\bar{\varphi}}(x)$, $x\geq s_{0}$ is
continuous, increasing and satisfies $f(s_{0})=0$.

Now let $I\subset[d]$ be the set of indices such that $s_{0}=s_{i}$, for all
$i\in I$. Then there is $\varepsilon>0$ such that $f(x)=\min\\{f_{i}(x),i\in
I\\}$ for all $x\in[s_{0},s_{0}+\varepsilon]$. Moreover, for all $i\in I$,

$\infty=\int_{s_{0}}^{s_{0}+\varepsilon}\frac{dx}{f_{i}(x)}\leq\int_{s_{0}}^{s_{0}+\varepsilon}\frac{dx}{f(x)}\,,$

where the equality follows from the fact that $0\leq
f_{i}^{\prime}(s_{0})<\infty$.

The result is then a consequence of Lemma 5.1 and the fact that for all
$\lambda\in\Delta$ such that $\lambda_{1}>s_{0}$ and for all $i,j\in[d]$,
$\varphi^{*}_{i}(\lambda)=\varphi^{*}_{j}(\lambda)=f(\lambda_{1})$. (Note also
that $t\mapsto{\rm v}_{t}(\lambda)$ is the only solution such that
$v_{t}^{(1)}(\lambda)=\dots=v_{t}^{(d)}(\lambda)$.) ∎

######  Lemma 5.3.

Assume that $d\geq 2$. Then for all $j\in[d]$ and
$\lambda\in\mathbb{R}_{+}^{d}$,

$\sum_{i=1}^{d}\varphi_{ij}(\lambda_{i})\leq\varphi_{j}(\lambda).$

Moreover, for all $j\in[d]$ there exists $i\neq j$ such that $\varphi_{ij}\neq
0$. As a consequence, for all $j\in[d]$ and
$\lambda\in(\mathbb{R}_{+}\setminus\\{0\\})^{d}$,

$\varphi_{j}^{*}(\lambda)<\varphi_{j}(\lambda).$

###### Proof.

By definition,

$\sum_{i=1}^{d}\varphi_{ij}(\lambda_{i})=-\sum_{i=1}^{d}a_{i,j}\lambda_{i}+\frac{1}{2}q_{j}\lambda_{j}^{2}+\int_{\mathbb{R}_{+}^{d}}\langle\lambda,{\rm
x}\rangle 1_{\\{|{\rm
x}|<1\\}}+\sum_{i=1}^{d}(e^{-\lambda_{i}x_{i}}-1)\,\pi_{j}(d{\rm x})$

and the first inequality is a consequence of the following one,
$\sum\limits_{i=1}^{d}(e^{-\lambda_{i}x_{i}}-1)\leq
e^{-\sum\limits_{i=1}^{d}\lambda_{i}x_{i}}-1$ which is valid for all
$\lambda,{\rm x}\in\mathbb{R}_{+}^{d}$. The second assertion follows from
irreducibility. The last assertion is a consequence of the first inequality
and the inequality
$\varphi_{j}^{*}(\lambda)<\sum\limits_{i=1}^{d}\varphi_{ij}(\lambda_{i})$, for
all $j\in[d]$ and $\lambda\in(\mathbb{R}_{+}\setminus\\{0\\})^{d}$. Indeed,

$\displaystyle\varphi_{j}^{*}(\lambda)$ $\displaystyle=$
$\displaystyle\bar{\varphi}(\lambda_{j})+\sum_{i\neq j}\sum_{k\neq
l}\varphi_{kl}(\lambda_{i})$ $\displaystyle=$
$\displaystyle\bar{\varphi}(\lambda_{j})+\sum_{i\neq
j}\varphi_{ij}(\lambda_{i})+\sum_{i\neq j}\sum_{k\neq
l,(k,l)\neq(i,j)}\varphi_{kl}(\lambda_{i})$ $\displaystyle\leq$
$\displaystyle\sum_{i=1}^{d}\varphi_{ij}(\lambda_{i})+\sum_{i\neq
j}\sum_{k\neq l,(k,l)\neq(i,j)}\varphi_{kl}(\lambda_{i}).$

Moreover, for the second term, we have

$\sum_{i\neq j}\sum_{k\neq
l,(k,l)\neq(i,j)}\varphi_{kl}(\lambda_{i})\leq\sum_{i\neq j}\sum_{k\neq
i}\varphi_{ki}(\lambda_{i})<0,$

where the last inequality follows from the fact that for all $i\in[d]$ there
exists $k\neq i$ such that $\varphi_{ki}\neq 0$. ∎

######  Corollary 5.1.

Assume that $d\geq 2$. Then for all $\lambda\in\Delta$ and $t\geq 0$, ${\rm
u}_{t}(\lambda)\leq{\rm v}_{t}(\lambda)$.

###### Proof.

Fix $\lambda\in\Delta$ and recall that for all $t\geq 0$, ${\rm
v}_{t}(\lambda)\in\Delta$. Then from Lemma 5.3,

$\displaystyle{\rm u}_{0}(\lambda)$ $\displaystyle={\rm
v}_{0}(\lambda)=\lambda$ $\displaystyle\text{ and
}\;\;\;\dfrac{\partial}{\partial
t}u^{(i)}_{t}(\lambda)|_{t=0}=-\varphi_{i}(\lambda)$
$\displaystyle<-\varphi^{*}_{i}(\lambda)=\dfrac{\partial}{\partial
t}v^{(i)}_{t}(\lambda)|_{t=0},\;i\in[d].$

Thanks to Taylor’s formula, for $t>0$ small enough,
$u^{(i)}_{t}(\lambda)<v^{(i)}_{t}(\lambda)$ for all $i\in[d]$.

Let $\tau^{(i)}=\inf\\{t>0:u^{(i)}_{t}(\lambda)>v^{(i)}_{t}(\lambda)\\}>0$ and
$\tau=\min\limits_{i\in[d]}\tau^{(i)}>0$. Assume $\tau<\infty$. By definition
of $\tau$ and by continuity, there exists at least one $i\in[d]$ such that
$u^{(i)}_{\tau}(\lambda)=v^{(i)}_{\tau}(\lambda)$ and for all $j\neq i$,
$u^{(j)}_{\tau}(\lambda)\leq v^{(j)}_{\tau}(\lambda)$. Recall from Lemma 5.3
that for all $\lambda\in(\mathbb{R}_{+}\setminus\\{0\\})^{d}$ and $j\in[d]$,
$\varphi^{*}_{j}(\lambda)<\varphi_{j}(\lambda)$ and also that if $k\neq j$,
then $\lambda_{k}\mapsto\varphi^{*}_{j}(\lambda)$ is non increasing. Note also
that, as justified in the proof of Lemma 4.1, $\mathbb{E}_{\rm
r}(Z^{(i)}_{\tau})>0$, for all ${\rm r}\in\mathbb{R}^{d}_{+}\setminus\\{{\bf
0}\\}$ and $i\in[d]$ and hence $u^{(i)}_{\tau}(\lambda)>0$, for all $i\in[d]$.
Let $I=\\{i\in[d]:u^{(i)}_{\tau}(\lambda)=v^{(i)}_{\tau}(\lambda)\\}$, then
for all $i\in I$,

$\dfrac{\partial}{\partial t}u^{(i)}_{t}(\lambda)|_{t=\tau}=-\varphi_{i}({\rm
u}_{\tau}(\lambda))<-\varphi^{*}_{i}({\rm
u}_{\tau}(\lambda))\leq-\varphi^{*}_{i}({\rm
v}_{\tau}(\lambda))=\dfrac{\partial}{\partial
t}v^{(i)}_{t}(\lambda)|_{t=\tau}\,.\ $

This implies that for $t>0$ small enough,
$u^{(i)}_{\tau+t}(\lambda)<v^{(i)}_{\tau+t}(\lambda)$ for all $i\in I$ and
this refutes the assumption $\tau<\infty$. We conclude that for all
$\lambda\in\Delta$, $t\geq 0$ and $i\in[d]$, $u^{(i)}_{t}(\lambda)\leq
v^{(i)}_{t}(\lambda)$. ∎

We are now able to end the proof of part 1. of Theorem 3.3. Let us first
assume that $d\geq 2$. Since $(G^{(i)}_{\varphi})$ are satisfied for all
$i\in[d]$, we can check that

(5.31)
$\int^{\infty}\frac{ds}{\bar{\varphi}(s)+(d-1)\bar{\bar{\varphi}}(s)}<\infty.$

Indeed, $\lim\limits_{s\rightarrow\infty}|\bar{\bar{\varphi}}(s)|/s$ is
bounded by a constant. Moreover, since $(G^{(i)}_{\varphi})$ are satisfied and
from the behaviour of Laplace exponents of spectrally positive Lévy processes,
see Chap. VII and in [3], we have
$\lim\limits_{s\rightarrow\infty}\tilde{\varphi}_{i}(s)/s=\infty$, for all
$i\in[d]$. This yields that
$\lim\limits_{s\rightarrow\infty}\bar{\varphi}(s)/s=\infty$. Hence the above
integral is finite if and only if
$\int^{\infty}\frac{ds}{\bar{\varphi}(s)}<\infty$. Then note the inequality
$\frac{1}{\bar{\varphi}(s)}\leq\max_{i\in[d]}\frac{1}{\tilde{\varphi}_{i}(s)}\leq\sum_{i\in[d]}\frac{1}{\tilde{\varphi}_{i}(s)}$,
which is valid whenever $s>0$ is such that $\tilde{\varphi}_{i}(s)>0$ for all
$i\in[d]$. This yields,

$\int^{\infty}\frac{ds}{\bar{\varphi}(s)}<\sum_{i\in[d]}\int^{\infty}\frac{ds}{\tilde{\varphi}_{i}(s)}<\infty\,.$

Now recall the definition of $s_{0}$ from Lemma 5.2 and for $s\geq s_{0}$, let
$t\mapsto{\rm v}_{t}(s{\bf 1})$, be the solution of $(S_{\varphi^{*}})$ in
this lemma. Then still according to this lemma, $t\mapsto v_{t}^{(1)}(s{\bf
1})=\dots=v_{t}^{(d)}(s{\bf 1})$ is the solution of the differential equation
(5.29) in Lemma 5.1, with $f(s)=\bar{\varphi}(s)+(d-1)\bar{\bar{\varphi}}(s)$.
Hence from Lemma 5.1 and (5.31),
$\lim\limits_{s\rightarrow\infty}v^{(i)}_{t}(s{\bf 1})<\infty$ for all $t>0$
and $i\in[d]$. Then as proved in Lemma 5.3,
$\varphi^{*}(\lambda)<\varphi(\lambda)$, for all
$\lambda\in(\mathbb{R}_{+}\setminus\\{0\\})^{d}$. Hence from Corollary 5.1,
for all $\lambda\in\Delta$ and $i\in[d]$, $u^{(i)}_{t}(\lambda)\leq
v_{t}^{(i)}(\lambda)$, so that $u^{(i)}_{t}(\infty)<\infty$, for all $t>0$ and
$i\in[d]$. Part $(ii)$ of Theorem 3.2 implies that $q_{\rm r}=e^{-\langle{\rm
r},\phi(\mathbf{0})\rangle}$. But since

(5.32) $\left\\{\mbox{${\rm Z}_{t}=\mathbf{0}$ for some
$t>0$}\right\\}\cup\left\\{\lim\limits_{t\rightarrow\infty}{\rm
Z}_{t}=\mathbf{0}\,\text{ and }\,{\rm Z}_{t}>\mathbf{0},\;\mbox{for all
$t>0$}\right\\}=\left\\{\lim\limits_{t\rightarrow\infty}{\rm
Z}_{t}=\mathbf{0}\right\\},$

we conclude from Theorem 3.1 that ${\rm Z}$ can only become extinct at a
finite time, that is $\bar{q}_{\rm r}=0$ for all ${\rm
r}\in\mathbb{R}^{d}_{+}$, and $q_{\rm r}:=\mathbb{P}_{\rm r}(\mbox{${\rm
Z}_{t}=\mathbf{0}$ for some $t>0$})=\mathbb{P}_{\rm
r}(\lim\limits_{t\rightarrow\infty}{\rm Z}_{t}=\mathbf{0})$. If $d=1$, then we
know directly from Grey’s result that $u_{t}(\infty)<\infty$, for all $t\geq
0$ and the same conclusion follows.

Let us now prove part 2. of Theorem 3.3. We will show that if one of the
diagonal branching processes $\tilde{Z}^{(i)}$ defined at the beginning of
this subsection is extinguished at infinity, that is if $(G_{\varphi}^{(i)})$
does not hold for some $i\in[d]$, then ${\rm Z}$ also is extinguished at
infinity. Let $i\in[d]$ and first note that since $\varphi_{i}$ is non
increasing in the variables $\lambda_{j}$, for $j\neq i$, for all
$\lambda\in(0,\infty)^{d}$,

(5.33) $\varphi_{i}(\lambda)\leq\varphi_{i}(\lambda_{i}{\rm
e}_{i})=\tilde{\varphi}_{i}(\lambda_{i})\,.$

Take $\lambda\in D_{\varphi}$ so that from Lemma 4.1, ${\rm u}_{s}(\lambda)\in
D_{\varphi}$ and in particular, $\varphi_{i}({\rm u}_{s}(\lambda))>0$, for all
$s\geq 0$. Then we derive from the differential system $(S_{\varphi})$ that
for all $s\geq 0$, $\dfrac{\frac{\partial}{\partial
s}u^{(i)}_{s}(\lambda)}{-\varphi_{i}({\rm u}_{s}(\lambda))}=1$, so that
integrating over $[0,t]$ and using the above inequality, we obtain for all
$t\geq 0$,

$\displaystyle t=\int_{0}^{t}\dfrac{\frac{\partial}{\partial
s}u^{(i)}_{s}(\lambda)}{-\varphi_{i}({\rm
u}_{s}(\lambda))}ds\geq\int_{0}^{t}\dfrac{\frac{\partial}{\partial
s}u^{(i)}_{s}(\lambda)}{-\tilde{\varphi_{i}}(u^{(i)}_{s}(\lambda))}ds=\int_{u^{(i)}_{t}(\lambda)}^{\lambda_{i}}\dfrac{d\alpha}{\tilde{\varphi}_{i}(\alpha)}.$

Since none of the Lévy processes $X^{i,i}$ is a subordinator,
$\lim\limits_{s\rightarrow\infty}\varphi_{i}(s{\bf 1})=\infty$, for all
$i\in[d]$, so that there is $A>0$ such that for all $s>A$, $s{\bf 1}\in
D_{\varphi}$. Fix $t>0$ and make $\lambda=s{\bf 1}$ tend to infinity. If
$u_{t}^{(i)}(\infty)<\infty$ and if $(G_{\varphi}^{(i)})$ is not satisfied,
then the right hand side of the above inequality tends to $\infty$, which is a
contradiction. Therefore $u^{(i)}_{t}(\infty)=\infty$ and we derive from Lemma
4.2 that for all $t\geq 0$ and $i\in[d]$, $u^{(i)}_{t}(\infty)=\infty$. In
particular, for all ${\rm r}\in\mathbb{R}_{+}^{d}\setminus\\{\mathbf{0}\\}$,
$\mathbb{P}_{\rm r}({\rm Z}_{t}=\mathbf{0})=e^{-\langle{\rm r},{\rm
u}_{t}(\infty)\rangle}=0$. Thus $q_{\rm
r}=\lim\limits_{t\rightarrow\infty}\mathbb{P}_{\rm r}({\rm
Z}_{t}=\mathbf{0})=0$ and the MCSBP ${\rm Z}$ can only be extinguished at
infinity. Finally, it follows from Theorem 3.1 and (5.32) that

$\bar{q}_{\rm r}:=\mathbb{P}_{\rm r}\left(\lim\limits_{t\rightarrow\infty}{\rm
Z}_{t}=\mathbf{0}\;\;\text{and}\;\;{\rm Z}_{t}>\mathbf{0},\;\mbox{for all
$t>0$}\right)=\mathbb{P}_{\rm r}(\lim\limits_{t\rightarrow\infty}{\rm
Z}_{t}=\mathbf{0})=e^{-\langle{\rm r},\phi(\mathbf{0})\rangle},$

which ends the proof of Theorem 3.3. $\qquad\Box$

##  Appendix A Proof of Proposition 2.1

Recall that a function $x:\mathbb{R}_{+}\rightarrow\mathbb{R}^{n}$, $n\geq 1$
is said to be _càdlàg_ , if it is right continuous on $\mathbb{R}_{+}$ and has
left limits on $(0,\infty)$. Such a function is said to be _downward skip
free_ if for all $s>0$, $x(s)-x(s-)\geq 0$. We will use the notation $x_{t}$
or $x(t)$ indifferently.

######  Definition A.1.

We call $\mathcal{E}_{d}$, the set of matrix valued functions
$\textsc{x}=\\{(x^{i,j}_{t_{j}})_{i,j\in[d]},\,{\rm
t}=(t_{1},\dots,t_{d})\in\mathbb{R}^{d}_{+}\\}$ such that for all $i,j$,
$x^{i,j}$ is a càdlàg function and

  * $(i)$

$x^{i,j}_{0}=0$, for all $i,j\in[d]$,

  * $(ii)$

for all $i\in[d]$, $x^{i,i}$ is downward skip free,

  * $(iii)$

for all $i,j\in[d]$ such that $i\neq j$, $x^{i,j}$ is non-decreasing.

For ${\rm s}\in\overline{\mathbb{R}}_{+}^{d}$, we denote by $[d]_{\rm s}$ the
set of indices of finite coordinates of ${\rm s}$, that is $[d]_{\rm
s}=\\{i\in[d]:s_{i}<\infty\\}$. For $i\neq j$, we set
$x^{i,j}(\infty)=x^{i,j}(\infty-)=\lim\limits_{s\rightarrow\infty}x^{i,j}(s)$.

######  Definition A.2.

Let $\textsc{x}\in\mathcal{E}_{d}$ and ${\rm
r}=(r_{1},\dots,r_{d})\in\mathbb{R}_{+}^{d}$. Then ${\rm
s}\in\overline{\mathbb{R}}_{+}^{d}$ is called a solution of the system $({\rm
r},\textsc{x})$ if it satisfies

$({\rm r},\textsc{x})\qquad\qquad
r_{i}+\sum_{j=1}^{d}x^{i,j}(s_{j}-)=0\,,\;\;\;i\in[d]_{\rm s}\,.$

$($In particular, ${\rm s}=(\infty,\infty,\dots,\infty)$ is always a solution
of the system $({\rm r},\textsc{x})$.$)$

Note that in $({\rm r},\textsc{x})$ it is implicit that
$\sum\limits_{j\in[d]\setminus[d]_{\rm s}}x^{i,j}(s_{j}-)<\infty$, for all
$i\in[d]_{\rm s}$, although by definition $s_{j}=\infty$, for
$j\in[d]\setminus[d]_{\rm s}$. Let us state Lemma $2.3$ in [7] which gives the
existence and uniqueness of a smallest solution to the system $({\rm
r},\textsc{x})$.

######  Lemma A.1.

Let $\textsc{x}\in\mathcal{E}_{d}$ and ${\rm
r}=(r_{1},\dots,r_{d})\in\mathbb{R}_{+}^{d}$. Then there exists a solution
${\rm s}=(s_{1},\dots,s_{d})\in\overline{\mathbb{R}}_{+}^{d}$ of the system
$({\rm r},\textsc{x})$ such that any other solution ${\rm t}$ of $({\rm
r},\textsc{x})$ satisfies ${\rm t}\geq{\rm s}$. The solution ${\rm s}$ will be
called the smallest solution of the system $({\rm r},\textsc{x})$.

For ${\rm r}=(r_{1},\dots,r_{d})\in\mathbb{R}^{d}_{+}$ and
$\textsc{x}\in\mathcal{E}_{d}$, let us now consider the functional equation,

(A.34)
$z^{(i)}_{t}=r_{i}+\sum\limits_{j=1}^{d}x^{i,j}\left(a^{(j)}_{t}\right),\;\;t\geq
0,i\in[d]\,,$

where $a^{(j)}_{t}:=\int_{0}^{t}z^{(j)}_{s}\,ds$. Existence and uniqueness of
solutions of this equation are studied in Section 3 of [5]. According to a
definition in [5], will say that a solution $z$ to (A.34) has no spontaneous
generation if whenever $z_{t}^{(i)}=0$ for some $t\geq 0$ and all $i$ in some
subset $I\subset[d]$, the function $s\mapsto a_{s}^{(i)}$ strictly increases
at $t$ for some $i\in I$ only if there exists $j\notin I$ such that
$x^{i,j}_{a^{(j)}_{t}}$ increases strictly to the right of $t$. A solution $z$
to (A.34) is said to be non-negative if all its coordinates $z^{(i)}$,
$i\in[d]$ are non-negative. Moreover, a non-negative solution $z$ to (A.34) is
said to be conservative if $z^{(i)}_{t}<\infty$ for all $i\in[d]$ and $t\geq
0$.

######  Lemma A.2.

Let ${\rm r}=(r_{1},\dots,r_{d})\in\mathbb{R}^{d}_{+}$ and
$\textsc{x}\in\mathcal{E}_{d}$. Denote by ${\rm t}_{\rm r}^{\textsc{x}}$ the
smallest solution of the system $({\rm r},\textsc{x})$. Assume that x is
continuous at time ${\rm t}_{\rm r}^{\textsc{x}}$ $($i.e. $x^{i,j}_{{\rm
t}_{\rm r}^{\textsc{x},j}-}=x^{i,j}_{{\rm t}_{\rm r}^{\textsc{x},j}}$,
$i,j\in[d]$$)$ and that there is a non-negative and conservative solution $z$
to the equation $(\ref{RL})$ which has no spontaneous generation. Then

(A.35) ${\rm t}^{\textsc{x}}_{\rm
r}=\lim_{t\rightarrow+\infty}(a^{(1)}_{t},\dots,a^{(d)}_{t}).$

###### Proof.

Let us first note that $\lim\limits_{t\rightarrow+\infty}{\rm
a}_{t}=\lim\limits_{t\rightarrow\infty}(a^{(1)}_{t},\dots,a^{(d)}_{t})$ is a
solution of the system $({\rm r},\textsc{x})$. Indeed, if
$a^{(i)}_{\infty}:=\lim\limits_{t\rightarrow\infty}a^{(i)}_{t}=\infty$ for all
$i\in[d]$, then $\lim\limits_{t\rightarrow+\infty}{\rm a}_{t}$ is a solution
by Definition A.2. On the other hand, assume that there is $i\in[d]$ such that
$a^{(i)}_{\infty}=\int_{0}^{\infty}z^{(i)}_{s}\,ds<\infty$. Since for all
$i\neq j$, $x^{i,j}$ is non-decreasing, according to the functional equation
(A.34), $z^{(i)}_{\infty}:=\lim\limits_{t\rightarrow\infty}z^{(i)}_{t}$ exists
and $z^{(i)}_{\infty}\in[0,\infty]$. But
$\int_{0}^{\infty}z^{(i)}_{s}\,ds<\infty$, so that $z^{(i)}_{\infty}=0$. From
(A.34) and Definition A.2, $\lim\limits_{t\rightarrow+\infty}{\rm a}_{t}$ is a
solution of the system $({\rm r},\textsc{x})$. In particular, from Lemma A.1,
$\lim\limits_{t\rightarrow+\infty}{\rm a}_{t}\geq{\rm t}^{\textsc{x}}_{\rm
r}$.

Now observe that the functions $s\mapsto a^{(i)}_{s}$ are continuous and non
decreasing and assume that $t$ is such that
$t:=\inf\\{s:a^{(i)}_{s}=t^{\textsc{x},i}_{\rm r}\\}<\infty$ for some indices
$i$ (in particular $t^{\textsc{x},i}_{\rm r}<\infty$) and
$a^{(j)}_{t}<t^{\textsc{x},j}_{\rm r}$ for all other indices. With no loss of
generality, we can assume that $d=2$,
$t:=\inf\\{s:a^{(1)}_{s}=t^{\textsc{x},1}_{\rm r}\\}<\infty$ and
$a^{(2)}_{t}<t^{\textsc{x},2}_{\rm r}$. Since $x^{1,2}$ is non decreasing,
$x^{1,2}_{a^{(2)}_{t}-}\leq x^{1,2}_{t^{\textsc{x},2}_{\rm r}-}$ and
$x^{1,2}_{a^{(2)}_{t}}\leq x^{1,2}_{t^{\textsc{x},2}_{\rm r}-}$. Moreover
since by definition $r_{1}+x^{1,1}_{t^{\textsc{x},1}_{\rm
r}-}+x^{1,2}_{t^{\textsc{x},2}_{\rm r}-}=0$, it follows, by the assumption of
continuity of $x^{1,1}$ at $t_{\rm r}^{\textsc{x},1}=a^{(1)}_{t}$, that

(A.36) $r_{1}+x^{1,1}_{a^{(1)}_{t}}+x^{1,2}_{a^{(2)}_{t}}\leq 0.$

If $x^{1,2}_{a^{(2)}_{t}}<x^{1,2}_{t^{\textsc{x},2}_{\rm r}-}$, then it
follows from above that $z^{(1)}_{t}<0$, which is a contradiction. Therefore
$x^{1,2}_{a^{(2)}_{t}}=x^{1,2}_{t^{\textsc{x},2}_{\rm r}-}$, that is $x^{1,2}$
is constant on the interval $[a^{(2)}_{t},t^{\textsc{x},2}_{\rm r})$ and
$z^{(1)}_{t}=0$.

Since by assumption ${\rm z}$ has no spontaneous generation, it implies that
$z^{(1)}$ is absorbed in 0 at time $t$, that is $z^{(1)}_{s}=0$, for
$s\in[t,t^{\prime}]$, where $t<t^{\prime}\leq\infty$ is such that
$t^{\prime}=\inf\\{s:a^{(2)}_{s}=t^{\textsc{x},2}_{\rm r}\\}$. Since $s\mapsto
a^{(1)}_{s}$ is constant on $[t,t^{\prime}]$, it follows that
$a^{(1)}_{t^{\prime}}=t^{\textsc{x},1}_{\rm r}$, so that
$(a^{(1)}_{t^{\prime}},a^{(2)}_{t^{\prime}})={\rm t}^{\textsc{x}}_{\rm r}$.
This implies, by the assumption of continuity of x at time ${\rm t}_{\rm
r}^{\textsc{x}}$, that $z_{t^{\prime}}=0$ and since $z$ has no spontaneous
generation, $z_{s}=0$, for all $s\geq t^{\prime}$ so that
$(a^{(1)}_{t^{\prime}},a^{(2)}_{t^{\prime}})=\lim_{s\rightarrow\infty}{\rm
a}_{s}$.
∎

Proof of Proposition 2.1: The proof is a direct application of Lemma A.2 to
the Lamperti representation $(\ref{Lamperti})$. As already mentioned, from
Theorem 1 in [5], for any ${\rm r}\in\mathbb{R}_{+}^{d}$ and any family of
Lévy processes $X^{(i)}$, $i\in[d]$ defined as in Subsection 2.2 and
satisfying $\alpha_{i}=0$, $i\in[d]$, there is a unique solution ${\rm Z}$ to
the equation $(\ref{Lamperti})$. Moreover, as a MCSBP, ${\rm Z}$ is non-
negative and its paths have no spontaneous generation in the sense defined
before Lemma A.2. Moreover, from Proposition 3.1 of [7], for every $i\in[d]$,
${\rm X}^{(i)}$ is a.s. continuous at time $T^{(i)}_{\rm r}$. Then on the set
$\\{{\rm Z}_{t}\in\mathbb{R}^{d}_{+},\,t\geq 0\\}$, which is supposed to be of
probability 1 here, the paths of the solution ${\rm Z}$ are conservative, in
the sense given above. Finally Proposition 2.1 follows from Lemma A.2.
$\;\;\;\Box$.

## References

  * [1] M. Barczy, Z. Li and G. Pap: Stochastic differential equation with jumps for multi-type continuous state and continuous time branching processes with immigration. ALEA Lat. Am. J. Probab. Math. Stat. 12, no. 1, 129–169, (2015).
  * [2] M. Barczy and G. Pap: Asymptotic behavior of critical, irreducible multi-type continuous state and continuous time branching processes with immigration. Stoch. Dyn. 16, no. 4, (2016).
  * [3] J. Bertoin: Lévy processes. Cambridge University Press, Cambridge, 1996.
  * [4] M.E. Caballero, A. Lambert, G. Uribe Bravo: Proof(s) of the Lamperti representation of continuous-state branching processes. Probab. Surv., 6, 62–89, (2009).
  * [5] M.E. Caballero, J.L. Pérez Garmendia, G. Uribe Bravo: Affine processes on $\mathbb{R}_{+}^{n}\times\mathbb{R}^{n}$ and multiparameter time changes. Ann. Inst. Henri Poincaré Probab. Stat., 53, 3, 1280–1304, (2017).
  * [6] L. Chaumont: Breadth first search coding of multitype forests with application to Lamperti representation. Séminaire de Probabilités XLVII, In Memoriam Marc Yor, (2015).
  * [7] L. Chaumont and M. Marolleau: Fluctuation theory for spectrally positive additive Lévy fields. Electronic Journal of Probability, Vol. 25, Issue none, 1-26 (2020).
  * [8] D. Duffie, D. Filipović and W. Schachermayer: Affine processes and applications in finance. Ann. Appl. Probab.13, no. 3, 984–1053, (2003).
  * [9] N. Gabrielli and J. Teichmann: Pathwise construction of affine processes. Preprint arXiv:1412.7837, (2014).
  * [10] D.R. Grey: Asymptotic behaviour of continuous time, continuous state-space branching processes. J. Appl. Probability , 11, 669–677, (1974).
  * [11] A.E. Kyprianou, S. Palau : Extinction properties of multi-type continuous-state branching processes. Stochastic Processes and their Applications, (2017).
  * [12] A.E. Kyprianou, S. Palau, Y.-X. Ren: Almost sure growth of supercritical multi-type continuous state branching process. ALEA, Lat. Am. J. Probab. Math. Stat., 15, 409–428, (2018).
  * [13] J. Lamperti: Continuous state branching processes. Bull. Amer. Math. Soc., Volume 73, Number 3, 382–386, 1967.
  * [14] S. Watanabe: On two dimensional Markov processes with branching property, Trans. Amer. Math. Soc., 136, 447–466, (1969).