11footnotetext: Université de Lorraine, IECL, UMR 7502, Campus Scientifique,
B.P. 70239, Vandœuvre-lès-Nancy Cedex, F-54506, France22footnotetext: CNRS,
IECL, UMR 7502, Vandœuvre-lès-Nancy, F-54506, France33footnotetext: Inria,
TOSCA team, Villers-lès-Nancy, F-54600, France.
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
# Lyapunov criteria for uniform convergence of conditional distributions of
absorbed Markov processes

Nicolas Champagnat1,2,3, Denis Villemonais1,2,3

###### Abstract

We study the quasi-stationary behavior of multidimensional processes absorbed
when one of the coordinates vanishes. Our results cover competitive or weakly
cooperative Lotka-Volterra birth and death processes and Feller diffusions
with competitive Lotka-Volterra interaction. To this aim, we develop original
non-linear Lyapunov criteria involving two Lyapunov functions, which apply to
general Markov processes.

Keywords: stochastic Lotka-Volterra systems; multitype population dynamics;
multidimensional birth and death process; multidimensional Feller diffusions;
process absorbed on the boundary; quasi-stationary distribution; uniform
exponential mixing property; Lyapunov function

2010 Mathematics Subject Classification. Primary: 60J27; 37A25; 60B10.
Secondary: 92D25; 92D40.

##  1 Introduction

We consider a Markov process $(X_{t},t\geq 0)$ on a topological state space
$E\cup\partial$, where $\partial\cap E=\emptyset$ and $\partial$ is absorbing,
meaning that $X_{t}\in\partial$ for all $t\geq\tau_{\partial}$, where
$\tau_{\partial}=\inf\\{t\geq 0:X_{t}\in\partial\\}$. We denote by
$\mathbb{P}_{x}$ its law given $X_{0}=x$ and we assume that the process $X$ is
absorbed in $\mathbb{P}_{x}$-a.s. finite time $\tau_{\partial}$ in $\partial$
for all $x\in E$. A quasi-stationary distribution for $X$ is a probability
measure $\nu_{QSD}$ on $E$ such that

$\mathbb{P}_{\nu_{QSD}}(X_{t}\in\cdot\mid
t<\tau_{\partial})=\nu_{QSD},\quad\forall t\geq 0,$

where $\mathbb{P}_{\nu}=\int_{E}\mathbb{P}_{x}\nu(dx)$ for all probability
measure $\nu$ on $E$.

Our goal is to prove the existence and uniqueness of a quasi-stationary
distribution for two models which are very standard in ecology and evolution
(cf. e.g. [5, 6, 3] and the references therein). In fact, we prove the uniform
convergence in total variation of the law of $X_{t}$ given
$X_{t}\not\in\partial$ when $t\rightarrow+\infty$ to a unique quasi-stationary
distribution. The first model is the multidimensional Lotka-Volterra birth and
death process, and the second one the multidimensional competitive Lotka-
Volterra (or logistic) Feller diffusion. These two models have received a lot
of attention in the one-dimensional case, and a lot is known about their
quasi-stationary behaviour [19, 2, 8], so we focus here on the
multidimensional case, where the processes evolve on the state spaces
$E\cup\partial=\mathbb{Z}_{+}^{d}$ for birth and death processes (with
$\mathbb{Z}_{+}=\\{0,1,2,\ldots\\}$) and $E\cup\partial=\mathbb{R}_{+}^{d}$
for diffusion processes, where $d\geq 2$. The case where
$\partial=\\{(0,\ldots,0)\\}$ can be handled combining the results of the
present paper and those known in the one-dimensional case [9, 12, 8] following
the methods of [3, Thm. 1.1]. So we focus here on the case where absorption
corresponds to the extinction of a single population, instead of the
extinction of the whole population. This case corresponds to
$\partial=\mathbb{Z}^{d}_{+}\setminus\mathbb{N}^{d}$ and $E=\mathbb{N}^{d}$
(where $\mathbb{N}=\\{1,2,\ldots\\}$) for multidimensional birth and death
processes and $\partial=\mathbb{R}_{+}^{d}\setminus(0,+\infty)^{d}$ and
$E=(0,+\infty)^{d}$ for multidimensional diffusions.

A general Lotka-Volterra birth and death process in dimension $d\geq 2$ is a
Markov process $(X_{t},t\geq 0)$ on $\mathbb{Z}_{+}^{d}$ with transition rates
$q_{n,m}$ from $n=(n_{1},\ldots,n_{d})\in\mathbb{Z}_{+}^{d}$ to $m\neq n$ in
$\mathbb{Z}_{+}^{d}$ given by

$q_{n,m}=\begin{cases}n_{i}(\lambda_{i}+\sum_{j=1}^{d}\gamma_{ij}n_{j})&\text{if
}m=n+e_{i},\text{ for some $i\in\\{1,\ldots,d\\}$}\\\
n_{i}(\mu_{i}+\sum_{j=1}^{d}c_{ij}n_{j})&\text{if }m=n-e_{i},\text{ for some
$i\in\\{1,\ldots,d\\}$}\\\ 0&\text{otherwise,}\end{cases}$

where $e_{i}=(0,\ldots,0,1,0,\ldots,0)$ where the 1 is at the $i$-th
coordinate. Note that $q_{n,n-e_{i}}=0$ if $n_{i}=0$, so that the process
remains in the state space $\mathbb{Z}_{+}^{d}$. Since in addition $q_{n,m}=0$
for all $n$ such that $n_{i}=0$ and $m$ such that $m_{i}\geq 1$, the set
$\partial=\mathbb{Z}^{d}_{+}\setminus\mathbb{N}^{d}$ is absorbing for the
process. We make the usual convention that

$q_{n,n}:=-q_{n}:=-\sum_{m\neq n}q_{n,m}.$

From the biological point of view, the constant $\lambda_{i}\geq 0$ is the
birth rate per individual of type $i\in\\{1,\ldots,d\\}$, the constant
$\mu_{i}\geq 0$ is the death rate per individual of type $i$, $c_{ij}\geq 0$
is the rate of death of an individual of type $i$ from competition with an
individual of type $j$, and $\gamma_{ij}\geq 0$ is the rate of birth of an
individual of type $i$ from cooperation with (or predation of) an individual
of type $j$. In general, a Lotka-Volterra process could be explosive if some
of the $\gamma_{ij}$ are positive, but the assumptions of the next theorem
ensure that it is not the case and that the process is almost surely absorbed
in finite time.

######  Theorem 1.1.

Consider a competitive Lotka-Volterra birth and death process $(X_{t},t\geq
0)$ in $\mathbb{Z}_{+}^{d}$ as above. Assume that the matrix
$(c_{ij}-\gamma_{ij})_{1\leq i,j\leq d}$ defines a positive operator on
$\mathbb{R}_{+}^{d}$ in the sense that, for all
$(x_{1},\ldots,x_{d})\in\mathbb{R}_{+}^{d}\setminus\\{0\\}$,
$\sum_{ij}x_{i}(c_{ij}-\gamma_{ij})x_{j}>0$. Then the process has a unique
quasi-stationary distribution $\nu_{QSD}$ and there exist constants
$C,\gamma>0$ such that, for all probability measures $\mu$ on
$E=\mathbb{N}^{d}$,

$\left\|\mathbb{P}_{\mu}(X_{t}\in\cdot\mid
t<\tau_{\partial})-\nu_{QSD}\right\|_{TV}\leq Ce^{-\lambda t},\quad\forall
t\geq 0,$ (1.1)

where $\|\cdot\|_{TV}$ is the usual total variation distance defined by
$\|\mu\|_{TV}=\sup_{f\in L^{\infty}(E),\ \|f\|_{\infty}\leq 1}|\mu(f)|$.

This result was previously known only in the competitive case (i.e. when
$\gamma_{ij}=0$ for all $1\leq i,j\leq d$) and when the constants $c_{ij}$
satisfy some bounds (see [10]). As explained in the last reference, an
important difficulty of this problem is the fact that the absorption rate
(i.e. the rate of jump from a state in $E$ to a state in $\partial$) is not
bounded with respect to the initial distribution. The existence of a quasi-
stationary distribution for this kind of multi-dimensional birth and death
processes can also be obtained using the theory of positive matrices, as
exposed in [14], but the uniform exponential convergence (1.1) also implies
uniqueness and several other properties, as recalled below.

A competitive Lotka-Volterra Feller diffusion process in dimension $d\geq 2$
is a Markov process $(X_{t},t\geq 0)$ on $\mathbb{R}_{+}^{d}$, where
$X_{t}=(X^{1}_{t},\ldots,X^{d}_{t})$, is a solution of the stochastic
differential equation

$dX^{i}_{t}=\sqrt{\gamma_{i}X^{i}_{t}}dB^{i}_{t}+X^{i}_{t}\left(r_{i}-\sum_{j=1}^{d}c_{ij}X^{j}_{t}\right)dt,\quad\forall
i\in\\{1,\ldots,d\\},$ (1.2)

where $(B^{1}_{t},t\geq 0),\ldots,(B^{d}_{t},t\geq 0)$ are independent
standard Brownian motions. The Brownian terms and the linear drift terms
correspond to classical Feller diffusions, and the quadratic drift terms
correspond to Lotka-Volterra interactions between coordinates of the process.
The variances per individual $\gamma_{i}$ is a positive number, and the growth
rates per individual $r_{i}$ can be any real number, for all $1\leq i\leq d$.
The competition parameters $c_{ij}$ are assumed nonnegative for all $1\leq
i,j\leq d$, which corresponds to competitive Lotka-Volterra interaction. It is
well known that, in this case, there is global strong existence and pathwise
uniqueness for the SDE (1.2), and that it is almost surely absorbed in finite
time in $\partial=\mathbb{R}_{+}^{d}\setminus(0,+\infty)^{d}$ if $c_{ii}>0$
for all $i\in\\{1,\ldots,d\\}$ (see [3] and Section 4).

######  Theorem 1.2.

Consider a competitive Lotka-Volterra Feller diffusion $(X_{t},t\geq 0)$ in
$\mathbb{R}_{+}^{d}$ as above. Assume that $c_{ii}>0$ for all
$i\in\\{1,\ldots,d\\}$. Then the process has a unique quasi-stationary
distribution $\nu_{QSD}$ and there exist constants $C,\gamma>0$ such that, for
all probability measures $\mu$ on $E=(0,\infty)^{d}$,

$\left\|\mathbb{P}_{\mu}(X_{t}\in\cdot\mid
t<\tau_{\partial})-\nu_{QSD}\right\|_{TV}\leq Ce^{-\lambda t},\quad\forall
t\geq 0.$

This results was previously known in dimension 2 when the constants $c_{ij}$
and $\gamma_{i}$ satisfy $c_{12}\gamma_{1}=c_{21}\gamma_{2}$, which implies
that the process (after some transformations) is a Kolmogorov diffusion (i.e.
of the form $dY_{t}=dW_{t}-\nabla V(Y_{t})dt$ for some Brownian motion $W$ and
some $C^{2}$ function $V$, see [3]). Our result is valid in any dimension and
has no restriction on the coefficients. One can also expect to extend our
result to cooperative cases (e.g. with $c_{21}<0$ and $c_{12}<0$ as in [3]) by
using our abstract Lyapunov criterion with Lyapunov functions combining those
used to prove Theorems 1.1 and 1.2.

To prove these two results, we make use of necessary and sufficient conditions
for the convergence (1.1) obtained in [9]. These conditions are hard to check
in practice (see e.g. [8, 4, 10]), so we first obtain in Section 2 several
criteria based on generalized Lyapunov functions to check these conditions.
These criteria apply to general Markov processes, not necessarily of the form
of competitive Lotka-Volterra birth and death or diffusion processes. Our
simplest Lyapunov criterion involves two bounded nonnegative functions $V$ and
$\gamma$ such that $V(x)/\varphi(x)\rightarrow+\infty$ out of compact subsets
of $E$ such that

$-L\varphi\leq C\mathbbm{1}_{K}$

for some compact subset $K$ of $E$ and

$LV+a\frac{V^{1+\varepsilon}}{\varphi^{\varepsilon}}\leq b\varphi$

for some $\varepsilon>0$ and some constants $a>0$ and $b\geq 0$, where $L$
denotes (an extension of) the infinitesimal generator of the Markov process
$X$. Together with additional conditions, the existence of such Lyapunov
functions $(V,\varphi)$ implies the convergence (1.1).

Many other properties can be deduced from (1.1). First, it implies the
existence of a so-called _mortality/extinction plateau_ : there exists
$\lambda_{0}>0$ limit of the extinction rate of the population (see [20]). The
constant $-\lambda_{0}$ is actually the largest non-trivial eigenvalue of the
generator $L$ and satisfies

$\mathbb{P}_{\alpha}(t<\tau_{\partial})=e^{-\lambda_{0}t},\quad\forall t\geq
0.$

In addition, (1.1) implies that $x\mapsto
e^{\lambda_{0}t}\mathbb{P}_{x}(t<\tau_{\partial})$ converges uniformly to a
function $\eta:E\rightarrow(0,+\infty)$ when $t\rightarrow+\infty$ [11,
Theorem 2.1]. Moreover, $\eta$ is the eigenfunction of $L$ corresponding to
the eigenvalue $-\lambda_{0}$ [7, Prop. 2.3]. It also implies the existence
and the exponential ergodicity of the associated $Q$-process, defined as the
process $X$ conditionned to never be extinct (see [7, Thm. 3.1] for a precise
definition). The convergence of the conditional laws of $X$ to the law of the
$Q$-process holds also uniformly in total variation norm [11, Theorem 2.1],
which entails conditional ergodic properties [11, Corollary 2.2].

Sections 3 and 4 are devoted to the study of (extensions of) competitive
Lotka-Volterra birth and death processes and competitive Lotka-Volterra Feller
diffusions and the proofs of Theorems 1.1 and 1.2.

##  2 General Lyapunov citeria for exponential convergence of conditional
distributions

In order to obtain results as general as possible, we consider a general
framework inspired from [21]. We first define this framework in Subsection
2.1, and we then state and prove a first version of our Lyapunov criterion in
Subsection 2.2 and a second (simpler) version in Subsection 2.3.

###  2.1 Definitions and notations

We consider a right-continuous time-homogeneous Markov process
$(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\geq 0},(X_{t},t\geq
0),(\mathbb{P}_{x})_{x\in E\cup\\{\partial\\}})$ (see [22]) with state space
$E\cup\partial$ such that $E\cap\partial=\emptyset$ where $\partial$ and $E$
are topological spaces equiped with their Borel $\sigma$-fields. We assume
that $\partial$ is an absorbing set for the process, which means that
$X_{t}\in\partial$ for all $t\geq\tau_{\partial}$, where

$\tau_{\partial}=\inf\\{t\geq 0:X_{t}\in\partial\\}$

is such that, for all $x\in E$ and $t\geq 0$,
$\mathbb{P}_{x}(\tau_{\partial}<+\infty)=1$ and
$\mathbb{P}_{x}(t<\tau_{\partial})>0$.

Throughout the paper, we let $\\{O_{n},n\in\mathbb{N}\\}$ and
$\\{U_{n},n\in\mathbb{N}\\}$ denote two fixed increasing families of subsets
of $E$ such that

$E=\bigcup_{n\in\mathbb{N}}O_{n}\quad\text{and}\quad\text{cl}(O_{n})\subset\text{int}(U_{n+1})\subset\text{cl}(U_{n+1})\subset\text{int}(O_{n+1}),\
\forall n\in\mathbb{N},$

where $\text{cl}(A)$ and $\text{int}(A)$ denote respectively the closure and
the interior in $E$ of $A\subset E$. We assume that, for all $n\in\mathbb{N}$,

$T_{n}:=\inf\\{t\geq 0:X_{t}\not\in O_{n}\\}$

is a $(\mathcal{F}_{t})_{t\geq 0}$-stopping time and

$\tau_{U_{n}}:=\inf\\{t\geq 0:X_{t}\in U_{n}\\}$

is a $(\mathcal{F}_{t})_{t\geq 0}$-stopping time to which the strong Markov
property applies for $(X_{t})_{t\geq 0}$. For example, the fact that $T_{n}$
and $\tau_{U_{n}}$ are stopping times holds true when $X$ is continuous,
$(\mathcal{F}_{t})_{t\geq 0}$ is the natural filtration, $O_{n}$ are open sets
and $U_{n}$ are closed sets. This is also true if $X$ is only right-
continuous, $(\mathcal{F}_{t})_{t\geq 0}$ is right-continuous, $O_{n}$ are
closed sets and $U_{n}$ are open subsets. However, in the last case, the
strong Markov property at time $\tau_{U_{n}}$ is harder to check. This is true
for Feller processes, but for general stochastic differential equations, the
well-posedness of the martingale problem only implies the strong Markov
property on the natural filtration (see e.g. the books of Le Gall [17] or
Rogers and Williams [22, 23]).

We assume that the process $(X_{t},t\geq 0)$ is _regularly absorbed_ in the
sense that

$\tau_{\partial}=\lim_{n\rightarrow+\infty}T_{n},\quad\mbox{a.s.\ under
}\mathbb{P}_{x},\ \forall x\in E.$ (2.1)

Thus, $\tau_{\partial}$ is a $(\mathcal{F}_{t})_{t\geq 0}$-stopping time.

We also assume that the paths of $(X_{t},t\geq 0)$ are càdlàg (right-
continuous and admitting left limits) up to time $T_{n}$ for all
$n\in\mathbb{N}$. Note that the process $X$ needs not be càdlàg since it may
not admit a left limit at time $\tau_{\partial}=\lim_{n}T_{n}$.

We shall make use of the following weakened notion of generator for $X$.

######  Definition 2.1.

We say that a measurable function $V:E\cup\partial\rightarrow\mathbb{R}$
belongs to the domain $\mathcal{D}(L)$ of the weakened generator $L$ of $X$ if
there exists a measurable function $W:E\rightarrow\mathbb{R}$ such that, for
all $n\in\mathbb{N}$, $t\geq 0$ and $x\in E$,

$\mathbb{E}_{x}V(X_{t\wedge T_{n}})=V(x)+\mathbb{E}_{x}\left[\int_{0}^{t\wedge
T_{n}}W(X_{s})ds\right]\mbox{ and }\mathbb{E}_{x}\left|\int_{0}^{t\wedge
T_{n}}W(X_{s})ds\right|<\infty,$ (2.2)

and we define $LV=W$ on $E$. We also define $LV(x)=0$ for all $x\in\partial$.

So far, the operator $L$ may be multi-valued, but it is enough for us to
define $LV$ as any $W$ satisfying the above property. Of course, this
definition extends the usual definition of the extended generator of $X$, and
therefore of the usual weak and infinitesimal generators (cf. [13, 21]).

We introduce the following notation.

######  Definition 2.2.

We say that a function $f:E\rightarrow\mathbb{R}$ converges to a limit $\ell$
out of $(O_{n})_{n\geq 1}$ and we write

$\lim_{x\not\in O_{n},\ n\rightarrow\infty}f(x)=\ell$

if, for all sequence $(x_{k})_{k\geq 1}$ in $E$ such that
$\\{k\in\mathbb{N}:x_{k}\in O_{n}\\}$ is finite for all $n\geq 1$,

$\lim_{k\rightarrow\infty}f(x_{k})=\ell.$

Equivalently, this means that

$\lim_{n\rightarrow+\infty}\sup_{x\in E\setminus O_{n}}|f(x)-\ell|=0.$

We also define the set of admissible functions to which our Lyapunov criteria
apply.

######  Definition 2.3.

We say that a couple $(V,\varphi)$ of functions $V$ and $\varphi$ measurable
from $E\cup\partial$ to $\mathbb{R}$ is an _admissible couple of functions_ if

(i)

$V$ and $\varphi$ are bounded nonnegative on $E\cup\partial$ and positive on
$E$, $V(x)=\varphi(x)=0$ for all $x\in\partial$ and

$\inf_{x\in E}\frac{V(x)}{\varphi(x)}>0.$ (2.3)

(ii)

We have the convergences

$\lim_{x\not\in O_{n},\ n\rightarrow\infty}\frac{V(x)}{\varphi(x)}=+\infty$
(2.4)

and

$\lim_{n\rightarrow+\infty}V(X_{T_{n}})=0\quad\text{a.s.}$ (2.5)

(iii)

$V$ and $\varphi$ belong to the domain of the weakened generator $L$ of $X$,
$LV$ is bounded from above and $L\varphi$ is bounded from below.

Note that since $V(X_{t})=V(X_{t})\mathbbm{1}_{t<\tau_{\partial}}$, (2.5) is
actually equivalent to

$\lim_{n\rightarrow+\infty}V(X_{t\wedge T_{n}})=V(X_{t}),\quad\forall t\geq
0,\mbox{ a.s.}$ (2.6)

Moreover, since $V$ is bounded, (2.4) implies that

$\lim_{x\not\in O_{n},\ n\rightarrow\infty}\varphi(x)=0.$

Therefore, using the fact that $X_{T_{n+1}}\not\in O_{n}$ for all $n\geq 1$
since $X$ is right continuous, an admissible couple of functions $(V,\varphi)$
also satisfies

$\lim_{n\rightarrow+\infty}\varphi(X_{t\wedge
T_{n}})=\varphi(X_{t}),\quad\forall t\geq 0,\ \mbox{a.s.}$ (2.7)

###  2.2 A non-linear Lyapunov criterion

Given a couple of admissible functions $(V,\varphi)$, our first Lyapunov
criterion is based on the following assumption.

######  Assumption 1.

The couple of admissible functions $(V,\varphi)$ satisfies that there exist
constants $\varepsilon,A,B>0$ such that, for all probability measure $\mu$ on
$E$,

$\mu(LV)-\mu(V)\frac{\mu(L\varphi)}{\mu(\varphi)}\leq
A\mu(\varphi)-B\frac{\mu(V)^{1+\varepsilon}}{\mu(\varphi)^{\varepsilon}}$
(2.8)

and there exist constants $r_{0},p_{0}>0$ such that, for $n$ large enough,

$\mathbb{P}_{x}(r_{0}<\tau_{\partial})\leq p_{0}V(x),\quad\forall x\in
E\setminus O_{n}.$ (2.9)

Assumption (2.8) can be seen as a nonlinear Lyapunov criterion. We also need
to assume a local Doeblin property.

######  Assumption 2.

There exist $k_{0}\in\mathbb{N}$, $\theta_{0},\theta_{1},a_{1}>0$ and a
probability measure $\nu$ on $E$ such that, for all $x\in O_{k_{0}}$ and all
$s\in[\theta_{1},\theta_{1}+\theta_{0}]$,

$\mathbb{P}_{x}(X_{s}\in\cdot)\geq a_{1}\nu.$

In addition, we assume the following weak irreducibility: for all $n\geq
k_{0}$, there exists $s_{n}\geq 0$ such that

$D_{n}:=\inf_{x\in O_{n}}\mathbb{P}_{x}(X_{s_{n}}\in O_{k_{0}})>0.$

The following assumption is about the uniform boundedness of exponential
moments of the hitting time of the sets $U_{n}\cup\partial$.

######  Assumption 3.

For all $\lambda>0$, there exists $n\geq 1$ such that

$\sup_{x\in
E}\mathbb{E}_{x}(e^{\lambda(\tau_{U_{n}}\wedge\tau_{\partial})})<\infty.$
(2.10)

Our last assumption is a form of local Harnack inequality, uniform in time.

######  Assumption 4.

For all $n\geq 0$, there exists a constant $C_{n}$ such that, for all $t\geq
0$,

$\sup_{x\in O_{n}}\mathbb{P}_{x}(t<\tau_{\partial})\leq C_{n}\inf_{x\in
O_{n}}\mathbb{P}_{x}(t<\tau_{\partial}).$ (2.11)

We are now able to state our first result, which will be applied in the
subsequent sections.

######  Theorem 2.4.

Assume that the process $(X_{t},t\geq 0)$ is regularly absorbed and that there
exists a couple of admissible functions $(V,\varphi)$ satisfying Assumption 1.
Assume also that Assumptions 2, 3 and 4 are satisfied. Then the process $X$
admits a unique quasi-stationary distribution $\nu_{QSD}$ and there exist
constants $C,\gamma>0$ such that for all probability measure $\mu$ on $E$,

$\|\mathbb{P}_{\mu}(X_{t}\in\cdot\mid t<\tau_{\partial})-\nu_{QSD}\|_{TV}\leq
Ce^{-\gamma t},\quad\forall t\geq 0.$ (2.12)

The proof of Theorem 2.4 is based on the next result.

######  Proposition 2.5.

Assume that the process $(X_{t},t\geq 0)$ is regularly absorbed and that there
exists a couple of admissible functions $(V,\varphi)$ satisfying Assumption 1.
Then, for all $x\in E$ and $t\geq 0$,

$\frac{\mathbb{E}_{x}[V(X_{t})]}{\mathbb{E}_{x}[\varphi(X_{t})]}=\frac{V(x)}{\varphi(x)}+\int_{0}^{t}\left\\{\frac{\mathbb{E}_{x}[LV(X_{s})]}{\mathbb{E}_{x}[\varphi(X_{s})]}-\frac{\mathbb{E}_{x}[V(X_{s})]}{\mathbb{E}_{x}[\varphi(X_{s})]}\,\frac{\mathbb{E}_{x}[L\varphi(X_{s})]}{\mathbb{E}_{x}[\varphi(X_{s})]}\right\\}\,ds,$
(2.13)

where the value of the integral in the r.h.s. is well-defined since, for all
$t\geq 0$,

$\frac{\mathbb{E}_{x}[LV(X_{s})]}{\mathbb{E}_{x}[\varphi(X_{s})]}-\frac{\mathbb{E}_{x}[V(X_{s})]}{\mathbb{E}_{x}[\varphi(X_{s})]}\,\frac{\mathbb{E}_{x}[L\varphi(X_{s})]}{\mathbb{E}_{x}[\varphi(X_{s})]}\in
L^{1}([0,t]).$

###### Proof.

Using the Definition 2.1 of the weakened infinitesimal generator, we have for
all $n\geq 1$

$\mathbb{E}_{x}V(X_{t\wedge T_{n}})=V(x)+\mathbb{E}_{x}\int_{0}^{t\wedge
T_{n}}LV(X_{s})ds=V(x)+\mathbb{E}_{x}\int_{0}^{t}LV(X_{s})\mathbbm{1}_{s<T_{n}}ds.$
(2.14)

Applying (2.6) and Lebesgue’s theorem to the left-hand side and Fatou’s lemma
to the right-hand side (using that $LV$ is bounded from above), we obtain

$\mathbb{E}_{x}V(X_{t})\leq
V(x)+\mathbb{E}_{x}\left[\int_{0}^{t\wedge\tau_{\partial}}LV(X_{s})\right]\,ds=V(x)+\int_{0}^{t}\mathbb{E}_{x}\left[LV(X_{s})\right]\,ds.$

Since $V\geq 0$ and $LV$ is bounded from above, we deduce that
$\mathbb{E}_{x}LV(X_{s})\in L^{1}([0,t])$ and $LV(X_{s})\in
L^{1}(\Omega\times[0,t])$. Therefore, we can actually apply Lebesgue’s Theorem
to the right-hand side of (2.14) and hence

$\mathbb{E}_{x}V(X_{t})=V(x)+\int_{0}^{t}\mathbb{E}_{x}\left[LV(X_{s})\right]\,ds.$

The same argument applies to $-\varphi$ using (2.7):

$\mathbb{E}_{x}\varphi(X_{t})=\varphi(x)+\int_{0}^{t}\mathbb{E}_{x}\left[L\varphi(X_{s})\right]\,ds.$
(2.15)

Therefore (cf. e.g. [1, Thm. VIII.2 and Lem. VIII.2]), for all $T>0$,
$t\mapsto(\mathbb{E}_{x}V(X_{t}),\mathbb{E}_{x}\varphi(X_{t}))$ belongs to the
Sobolev space $W^{1,1}([0,T],\mathbb{R}^{2})$ (the set of functions from
$[0,T]$ to $\mathbb{R}^{2}$ in $L^{1}$ admitting a derivative in the sense of
distributions in $L^{1}$).

In particular, $t\mapsto\mathbb{E}_{x}\varphi(X_{t})$ is continuous on
$[0,T]$. Since $\mathbb{P}_{x}(t<\tau_{\partial})>0$ and hence
$\mathbb{E}_{x}\varphi(X_{t})>0$ for all $t\in[0,T]$, we deduce that
$\inf_{t\in[0,T]}\mathbb{E}_{x}\varphi(X_{t})>0$. Therefore, we deduce from
standard properties of $W^{1,1}$ functions [1, Cor. VIII.9 and Cor. VIII.10]
that $t\mapsto\mathbb{E}_{x}V(X_{t})/\mathbb{E}_{x}\varphi(X_{t})$ also
belongs to $W^{1,1}$ and admits as derivative

$t\mapsto\frac{\mathbb{E}_{x}LV(X_{t})}{\mathbb{E}_{x}\varphi(X_{t})}-\frac{\mathbb{E}_{x}V(X_{t})\,\mathbb{E}_{x}L\varphi(X_{t})}{\mathbb{E}_{x}[\varphi(X_{t})]^{2}}\in
L^{1}([0,T]).$

Hence we have proved (2.13). ∎

Before proving Theorem 2.4, let us prove that Assumption 2 is equivalent to a
seemingly stronger assumption.

######  Lemma 2.6.

Assumption 2 implies that, for all $n_{0}\geq k_{0}$ and for all $t_{0}>0$,
there exist $t_{1},c_{1}>0$ such that, for all $s\in[t_{1},t_{1}+t_{0}]$ and
$x\in O_{n_{0}}$,

$\mathbb{P}_{x}(X_{s}\in\cdot)\geq c_{1}\nu.$ (2.16)

In addition, for all $n\geq n_{0}\geq k_{0}$,

$\inf_{x\in O_{n}}\mathbb{P}_{x}(X_{s_{n}}\in O_{n_{0}})\geq D_{n}>0.$ (2.17)

###### Proof.

Equation (2.17) is an immediate consequence of the definition of $D_{n}$ and
the fact that $O_{k_{0}}\subset O_{n_{0}}$.

To prove (2.16), let $m_{0}\geq k_{0}$ be such that $\nu(O_{m_{0}})>0$. Then,
for all $x\in O_{m_{0}}$ and all
$s\in[s_{m_{0}}+\theta_{1},s_{m_{0}}+\theta_{1}+\theta_{0}]$,

$\displaystyle\mathbb{P}_{x}(X_{s}\in\cdot)\geq D_{m_{0}}a_{1}\nu(\cdot).$

Hence, for all $\ell\in\mathbb{N}$, $x\in O_{m_{0}}$ and
$s\in[\ell(s_{m_{0}}+\theta_{1}),\ell(s_{m_{0}}+\theta_{1}+\theta_{0})]$,

$\displaystyle\mathbb{P}_{x}(X_{s}\in\cdot)\geq\left(D_{m_{0}}a_{1}\right)^{\ell}\nu(O_{m_{0}})^{\ell-1}\nu(\cdot).$

Therefore, for all $n_{0}\geq k_{0}$, $x\in O_{n_{0}}$ and
$s\in[s_{n_{0}}+\ell(s_{m_{0}}+\theta_{1}),s_{n_{0}}+\ell(s_{m_{0}}+\theta_{1}+\theta_{0})]$,

$\displaystyle\mathbb{P}_{x}(X_{s}\in\cdot)\geq
D_{n_{0}}\left(D_{m_{0}}a_{1}\right)^{\ell}\nu(O_{m_{0}})^{\ell-1}\nu(\cdot).$

This ends the proof of Lemma 2.6. ∎

###### Proof of Theorem 2.4.

The convergence (2.12) can be proved using the necessary and sufficient
criterion obtained in [7] for general Markov processes. This criterion is
given by the two conditions (A1) and (A2) below.

There exists a probability measure $\nu$ on $E$ such that

  * (A1)

there exist $t^{\prime}_{0},c^{\prime}_{1}>0$ such that for all $x\in E$,

$\mathbb{P}_{x}(X_{t^{\prime}_{0}}\in\cdot\mid
t^{\prime}_{0}<\tau_{\partial})\geq c^{\prime}_{1}\nu(\cdot);$

  * (A2)

there exists $c^{\prime}_{2}>0$ such that for all $x\in E$ and $t\geq 0$,

$\mathbb{P}_{\nu}(t<\tau_{\partial})\geq
c^{\prime}_{2}\mathbb{P}_{x}(t<\tau_{\partial}).$

The proof is divided in two steps, where (A1) and (A2) are proved
respectively.

Step 1: Proof of (A1).
Assumption (2.8) and Proposition 2.5 imply that

$\displaystyle\frac{\mathbb{E}_{x}V(X_{t})}{\mathbb{E}_{x}\varphi(X_{t})}$
$\displaystyle\leq\frac{V(x)}{\varphi(x)}+At-B\int_{0}^{t}\left(\frac{\mathbb{E}_{x}V(X_{s})}{\mathbb{E}_{x}\varphi(X_{s})}\right)^{1+\varepsilon}\,ds.$

Fix $t>0$ and assume that for all $s\in[0,t]$,
$\mathbb{E}_{x}V(X_{s})/\mathbb{E}_{x}\varphi(X_{s})\geq a$, where
$a=(2A/B)^{1/(1+\varepsilon)}$. Then

$\frac{\mathbb{E}_{x}V(X_{t})}{\mathbb{E}_{x}\varphi(X_{t})}\leq\frac{V(x)}{\varphi(x)}-\frac{B}{2}\int_{0}^{t}\left(\frac{\mathbb{E}_{x}V(X_{s})}{\mathbb{E}_{x}\varphi(X_{s})}\right)^{1+\varepsilon}\,ds.$

Integrating this differential inequality entails

$\displaystyle\frac{\mathbb{E}_{x}V(X_{t})}{\mathbb{E}_{x}\varphi(X_{t})}$
$\displaystyle\leq\left[\left(\frac{\varphi(x)}{V(x)}\right)^{\varepsilon}+\frac{\varepsilon
Bt}{2}\right]^{-1/\varepsilon}\leq\left(\frac{2}{\varepsilon
Bt}\right)^{1/\varepsilon}.$

Choosing $t=t_{0}$ such that $\left(\frac{2}{\varepsilon
Bt_{0}}\right)^{1/\varepsilon}<a$ gives a contradiction, hence there exists
$u_{x}\in[0,t_{0}]$ such that
$\mathbb{E}_{x}V(X_{u_{x}})/\mathbb{E}_{x}\varphi(X_{u_{x}})<a$. Note that the
value of $u_{x}$ may depend on $x\in E$ but the value of $t_{0}$ is
independent of $x\in E$. We also define $n_{0}$ as the first integer $n$ such
that

$\inf_{y\in E\setminus O_{n}}\frac{V(y)}{\varphi(y)}\geq 2a,$

whose existence is ensured by Assumption (2.4).

Now,

$\displaystyle a$
$\displaystyle\geq\frac{\mathbb{E}_{x}V(X_{u_{x}})}{\mathbb{E}_{x}\varphi(X_{u_{x}})}$
$\displaystyle\geq\frac{\mathbb{E}_{x}[V(X_{u_{x}})\mathbbm{1}_{X_{u_{x}}\in
E\setminus O_{n_{0}}}]+\mathbb{E}_{x}[V(X_{u_{x}})\mathbbm{1}_{X_{u_{x}}\in
O_{n_{0}}}]}{\sup_{y\in E\setminus
O_{n_{0}}}\frac{\varphi(y)}{V(y)}\mathbb{E}_{x}[V(X_{u_{x}})\mathbbm{1}_{X_{u_{x}}\in
E\setminus O_{n_{0}}}]+\sup_{y\in
E}\frac{\varphi(y)}{V(y)}\mathbb{E}_{x}[V(X_{u_{x}})\mathbbm{1}_{X_{u_{x}}\in
O_{n_{0}}}]}$
$\displaystyle\geq\frac{\mathbb{E}_{x}[V(X_{u_{x}})\mathbbm{1}_{X_{u_{x}}\in
E\setminus O_{n_{0}}}]+\mathbb{E}_{x}[V(X_{u_{x}})\mathbbm{1}_{X_{u_{x}}\in
O_{n_{0}}}]}{\frac{1}{2a}\mathbb{E}_{x}[V(X_{u_{x}})\mathbbm{1}_{X_{u_{x}}\in
E\setminus O_{n_{0}}}]+\sup_{y\in
E}\frac{\varphi(y)}{V(y)}\mathbb{E}_{x}[V(X_{u_{x}})\mathbbm{1}_{X_{u_{x}}\in
O_{n_{0}}}]}.$

Since $a>\mathbb{E}_{x}V(X_{u_{x}})/\mathbb{E}_{x}\varphi(X_{u_{x}})\geq
1/\sup_{y\in E}(\varphi(y)/V(y))$, we deduce that

$\displaystyle\mathbb{E}_{x}[V(X_{u_{x}})\mathbbm{1}_{X_{u_{x}}\in
O_{n_{0}}}]$ $\displaystyle\geq\frac{1}{2(a\sup_{y\in
E}\frac{\varphi(y)}{V(y)}-1)}\mathbb{E}_{x}[V(X_{u_{x}})\mathbbm{1}_{X_{u_{x}}\in
E\setminus O_{n_{0}}}].$

On the one hand, this last inequality, Assumption (2.9) and Markov’s property
entails (where $s_{n_{0}}$ is taken from (2.17) of Lemma 2.6)

$\mathbb{P}_{x}\left(u_{x}+s_{n_{0}}\left\lceil\frac{r_{0}}{s_{n_{0}}}\right\rceil<\tau_{\partial}\right)\leq\mathbb{P}_{x}(u_{x}+r_{0}<\tau_{\partial})\\\
\begin{aligned} &\leq\mathbb{P}_{x}(X_{u_{x}}\in
O_{n_{0}})+\mathbb{E}_{x}[\mathbb{P}_{X_{u_{x}}}(r_{0}<\tau_{\partial})\mathbbm{1}_{X_{u_{x}}\in
E\setminus O_{n_{0}}}]\\\ &\leq\mathbb{P}_{x}(X_{u_{x}}\in
O_{n_{0}})+p_{0}\mathbb{E}_{x}[V(X_{u_{x}})\mathbbm{1}_{X_{u_{x}}\in
E\setminus O_{n_{0}}}]\\\ &\leq\mathbb{P}_{x}(X_{u_{x}}\in
O_{n_{0}})+2p_{0}\left(a\sup_{y\in
E}\frac{\varphi(y)}{V(y)}-1\right)\mathbb{E}_{x}[V(X_{u_{x}})\mathbbm{1}_{X_{u_{x}}\in
O_{n_{0}}}]\\\ &\leq\mathbb{P}_{x}(X_{u_{x}}\in
O_{n_{0}})\left[1+2p_{0}\|V\|_{\infty}\left(a\sup_{y\in
E}\frac{\varphi(y)}{V(y)}-1\right)\right].\end{aligned}$

On the other hand, using (2.17) and Markov’s property, for all $x\in E$, and
all $k\geq 1$,

$\displaystyle\mathbb{P}_{x}(X_{u_{x}+ks_{n_{0}}}\in O_{n_{0}})$
$\displaystyle\geq\mathbb{P}_{x}(X_{u_{x}}\in O_{n_{0}})\inf_{y\in
O_{n_{0}}}\mathbb{P}_{y}(X_{ks_{n_{0}}}\in O_{n_{0}})$ $\displaystyle\geq
D^{k}_{n_{0}}\mathbb{P}_{x}(X_{u_{x}}\in O_{n_{0}}).$

Therefore, choosing $k=\lceil\frac{r_{0}}{s_{n_{0}}}\rceil$, we have proved
that

$\mathbb{P}_{x}(X_{u_{x}+ks_{n_{0}}}\in O_{n_{0}}\mid
u_{x}+ks_{n_{0}}<\tau_{\partial})\\\
\geq\frac{D^{k}_{n_{0}}}{1+2p_{0}\|V\|_{\infty}\left(a\sup_{y\in
E}\varphi(y)/V(y)-1\right)}=:q.$

Now, since $u_{x}\in[0,t_{0}]$, Equation (2.16) of Lemma 2.6 entails
$\mathbb{P}_{y}(X_{t_{1}+t_{0}-u_{x}}\in\Gamma)\geq c_{1}\nu(\Gamma)$ for all
$y\in O_{n_{0}}$ and all $\Gamma\subset E$ measurable. Hence, denoting
$v_{x}=u_{x}+ks_{n_{0}}$,

$\displaystyle\mathbb{P}_{x}(X_{t_{1}+t_{0}+ks_{n_{0}}}\in\Gamma\mid
t_{1}+t_{0}+k$ $\displaystyle s_{n_{0}}<\tau_{\partial})$
$\displaystyle=\frac{\mathbb{E}_{x}\left[\mathbbm{1}_{v_{x}<\tau_{\partial}}\mathbb{P}_{X_{v_{x}}}(X_{t_{1}+t_{0}-u_{x}}\in\Gamma)\right]}{\mathbb{P}_{x}(t_{1}+t_{0}+ks_{n_{0}}<\tau_{\partial})}$
$\displaystyle\geq c_{1}\nu(\Gamma)\,\frac{\mathbb{P}_{x}(X_{v_{x}}\in
O_{n_{0}})}{\mathbb{P}_{x}(t_{1}+t_{0}+ks_{n_{0}}<\tau_{\partial})}$
$\displaystyle\geq c_{1}\nu(\Gamma)\mathbb{P}_{x}(X_{v_{x}}\in O_{n_{0}}\mid
v_{x}<\tau_{\partial})$ $\displaystyle\geq c_{1}q\nu(\Gamma).$

This concludes the proof of (A1), with $c^{\prime}_{1}=c_{1}q$ and
$t^{\prime}_{1}=t_{1}+t_{0}+ks_{n_{0}}$.

Step 2: Proof of (A2).
Since $\cup_{n}O_{n}=E$, we can fix $n_{1}\geq n_{0}$ such that
$\nu(O_{n_{1}})\geq 1/2$. Let $m\geq n_{0}$ and $0\leq s\leq t$, and define
$\ell$ the smallest integer such that $s\leq
s_{m}+\ell(t^{\prime}_{1}+s_{n_{1}})$. Using (2.17), we obtain, for all $x\in
O_{m}$,

$\displaystyle\mathbb{P}_{x}(t<\tau_{\partial})$
$\displaystyle\geq\mathbb{P}_{x}(X_{s_{m}}\in
O_{n_{0}},\,X_{s_{m}+t^{\prime}_{1}}\in
O_{n_{1}},\,X_{s_{m}+t^{\prime}_{1}+s_{n_{1}}}\in
O_{n_{0}},\,X_{s_{m}+2t^{\prime}_{1}+s_{n_{1}}}\in O_{n_{1}},$
$\displaystyle\qquad\ldots,X_{s_{m}+\ell(t^{\prime}_{1}+s_{n_{1}})}\in
O_{n_{0}},\,t-s+s_{m}+\ell(t^{\prime}_{1}+s_{n_{1}})<\tau_{\partial})$
$\displaystyle\geq
D_{m}\left(\frac{c^{\prime}_{1}}{2}D_{n_{1}}\right)^{\ell}\inf_{x\in
O_{n_{0}}}\mathbb{P}_{x}(t-s<\tau_{\partial})$ $\displaystyle\geq
D_{m}\left(\frac{c^{\prime}_{1}}{2}D_{n_{1}}\right)^{1+s/(t^{\prime}_{1}+s_{n_{1}})}\inf_{x\in
O_{m}}\mathbb{P}_{x}(t-s<\tau_{\partial})$
$\displaystyle\geq\frac{c^{\prime}_{1}D_{m}D_{n_{1}}}{2}\inf_{x\in
O_{m}}\mathbb{P}_{x}(t-s<\tau_{\partial})\exp(-\lambda s),$

where

$\lambda:=\frac{1}{t^{\prime}_{1}+s_{n_{1}}}\log\left(\frac{2}{c^{\prime}_{1}D_{n_{1}}}\right)>0$

is independent of $m\geq n_{0}$ and $x\in O_{m}$. Using Assumption 4, we
obtain for all $0\leq s\leq t$

$\inf_{x\in
O_{m}}\mathbb{P}_{x}(t<\tau_{\partial})\geq\frac{c_{1}D_{m}D_{n_{1}}}{2C_{m}}\sup_{x\in
O_{m}}\mathbb{P}_{x}(t-s<\tau_{\partial})\exp(-\lambda s).$ (2.18)

Now, we apply Assumption 3 for $\lambda$ as defined above: there exists $m\geq
n_{0}\vee n_{1}$ such that

$M:=\sup_{x\in
E}\mathbb{E}_{x}[\exp(\lambda(\tau_{U_{m}}\wedge\tau_{\partial}))]<\infty.$
(2.19)

Recall that we assumed that the strong Markov property applies at the stopping
time $\tau_{U_{m}}$. Note also that $X_{\tau_{U_{m}}}\in O_{m}$ on the event
$\\{\tau_{U_{m}}<\infty\\}$. Hence, using (2.19) and the strong Markov
property,

$\displaystyle\mathbb{P}_{x}(t<\tau_{\partial})$
$\displaystyle=\mathbb{P}_{x}(t<\tau_{U_{m}}\wedge\tau_{\partial})+\mathbb{P}_{x}(\tau_{U_{m}}\leq
t<\tau_{\partial})$ $\displaystyle\leq Me^{-\lambda t}+\int_{0}^{t}\sup_{y\in
O_{m}}\mathbb{P}_{y}(t-s<\tau_{\partial})\mathbb{P}_{y}(\tau_{U_{m}}\in ds).$

Now, for all $s\leq t$, $\mathbb{P}_{y}(\tau_{U_{m}}\in
ds)\leq\mathbb{P}_{y}(\tau_{U_{m}}\wedge\tau_{\partial}\in ds)$. Hence, using
(2.18) twice and the fact that $\nu(O_{n_{1}})\geq 1/2$, we have for all $x\in
E$

$\displaystyle\mathbb{P}_{x}(t<\tau_{\partial})$ $\displaystyle\leq
Me^{-\lambda t}+\int_{0}^{t}\sup_{y\in
O_{m}}\mathbb{P}_{y}(t-s<\tau_{\partial})\mathbb{P}_{y}(\tau_{U_{m}}\wedge\tau_{\partial}\in
ds)$ $\displaystyle\leq\frac{2C_{m}}{c_{1}D_{m}D_{n_{1}}}\left[M\inf_{y\in
O_{m}}\mathbb{P}_{y}(t<\tau_{\partial})\right.$
$\displaystyle\qquad\qquad\qquad\left.+\inf_{y\in
O_{m}}\mathbb{P}_{y}(t<\tau_{\partial})\int_{0}^{t}e^{\lambda
s}\mathbb{P}_{x}(\tau_{U_{m}}\wedge\tau_{\partial}\in ds)\right]$
$\displaystyle\leq\frac{4MC_{m}}{c_{1}D_{m}D_{n_{1}}}\inf_{y\in
O_{n_{1}}}\mathbb{P}_{y}(t<\tau_{\partial})$
$\displaystyle\leq\frac{8MC_{m}}{c_{1}D_{m}D_{n_{1}}}\mathbb{P}_{\nu}(t<\tau_{\partial}).$

This is (A2). ∎

###  2.3 A simpler Lyapunov criterion

Condition (2.8) may be hard to check in practice, so we give a stronger
condition, easier to check.

######  Proposition 2.7.

Assume that $(V,\varphi)$ is a couple of admissible functions such that
$\inf_{x\in O_{m}}\varphi(x)>0$ for all integer $m$. Then Condition (2.8) is
implied by the following two conditions:

(a)

There exist an integer $n$ and a constant $C\geq 0$ such that

$-L\varphi\leq C\mathbbm{1}_{O_{n}}.$ (2.20)

(b)

There exist constants $C^{\prime}>0$ and $C^{\prime\prime}\geq 0$ such that

$LV+C^{\prime}\frac{V^{1+\varepsilon}}{\varphi^{\varepsilon}}\leq
C^{\prime\prime}\varphi.$ (2.21)

Note that Conditions (a) and (b) imply that $LV$ is bounded from above and
$L\varphi$ is bounded from below (which are parts of the conditions for a
couple of functions to be admissible). This result immediately entails the
following corollary, which will be used in the applications provided in
Sections 3 and 4.

######  Corollary 2.8.

Assume that the process $(X_{t},t\geq 0)$ is regularly absorbed and that there
exists a couple of admissible functions $(V,\varphi)$ satisfying (2.9) and
Conditions (a) and (b) of Proposition 2.7 and such that $\inf_{x\in
O_{m}}\varphi(x)>0$ for all integer $m$. Assume also that Assumptions 2, 3 and
4 are satisfied. Then the process $X$ admits a unique quasi-stationary
distribution $\nu_{QSD}$ and there exist constants $C,\gamma>0$ such that, for
all probability measure $\mu$ on $E$,

$\|\mathbb{P}_{\mu}(X_{t}\in\cdot\mid t<\tau_{\partial})-\nu_{QSD}\|_{TV}\leq
Ce^{-\gamma t},\quad\forall t\geq 0.$

###### Proof of Proposition 2.7.

Let $\mu$ be any probability measure on $E$. On the one hand, it follows from
(2.20) that

$\displaystyle-\frac{\mu(L\varphi)}{\mu(\varphi)}$
$\displaystyle\leq\frac{C}{\inf_{x\in O_{n}}\varphi(x)}.$

On the other hand, Hölder’s inequality with $p=(1+\varepsilon)/\varepsilon$
and $q=1+\varepsilon$ yields

$\displaystyle\mu(V)^{1+\varepsilon}$
$\displaystyle\leq\mu\left(\frac{V^{q}}{\varphi^{\varepsilon
q/(1+\varepsilon)}}\right)^{(1+\varepsilon)/q}\mu(\varphi^{\varepsilon
p/(1+\varepsilon)})^{(1+\varepsilon)/p}$
$\displaystyle=\mu\left(\frac{V^{1+\varepsilon}}{\varphi^{\varepsilon}}\right)\,\mu(\varphi)^{\varepsilon}.$

Hence we deduce from (2.21) that

$\mu(LV)-\frac{\mu(L\varphi)}{\mu(\varphi)}\,\mu(V)\\\ \begin{aligned} &\leq
C^{\prime\prime}\mu(\varphi)-C^{\prime}\mu\left(\frac{V^{1+\varepsilon}}{\varphi^{\varepsilon}}\right)+\frac{C}{\inf_{x\in
O_{n}}\varphi(x)}\,\mu(V)\\\ &\leq
C^{\prime\prime}\mu(\varphi)+\mu\left[V\left(\frac{C}{\inf_{x\in
O_{n}}\varphi(x)}-\frac{C^{\prime}V^{\varepsilon}}{2\varphi^{\varepsilon}}\right)\right]-\frac{C^{\prime}}{2}\,\frac{\mu(V)^{1+\varepsilon}}{\mu(\varphi)^{\varepsilon}}.\end{aligned}$

Now, because of Assumption (2.4), there exists $m$ large enough such that, for
all $x\in E\setminus O_{m}$,

$\frac{C^{\prime}V^{\varepsilon}(x)}{2\varphi^{\varepsilon}(x)}\geq\frac{C}{\inf_{y\in
O_{n}}\varphi(y)}.$

Therefore,

$\mu(LV)-\frac{\mu(L\varphi)}{\mu(\varphi)}\,\mu(V)\\\ \begin{aligned} &\leq
C^{\prime\prime}\mu(\varphi)+\frac{C}{\inf_{x\in
O_{n}}\varphi(x)}\mu(V\mathbbm{1}_{O_{m}})-\frac{C^{\prime}}{2}\,\frac{\mu(V)^{1+\varepsilon}}{2\mu(\varphi)^{\varepsilon}}\\\
&\leq\left(C^{\prime\prime}+\frac{C\sup_{x\in E}V(x)}{\inf_{x\in
O_{n}}\varphi(x)\,\inf_{x\in
O_{m}}\varphi(x)}\right)\mu(\varphi)-\frac{C^{\prime}}{2}\,\frac{\mu(V)^{1+\varepsilon}}{2\mu(\varphi)^{\varepsilon}}.\end{aligned}$

This is (2.8). ∎

##  3 Application to multidimensional birth and death processes absorbed when
one of the coordinates hits 0

We consider general multitype birth and death processes in continuous time,
taking values in $\mathbb{Z}_{+}^{d}$ for some $d\geq 2$. Let $(X_{t},t\geq
0)$ be a Markov process on $\mathbb{Z}_{+}^{d}$ with transition rates

$\text{from }n=(n_{1},\ldots,n_{d})\text{ to }\begin{cases}n+e_{j}&\text{with
rate }n_{j}b_{j}(n),\\\ n-e_{j}&\text{with rate }n_{j}d_{j}(n)\end{cases}$

for all $1\leq j\leq d$, with $e_{j}=(0,\ldots,0,1,0,\ldots,0)$, where the
nonzero coordinate is the $j$-th one, $b(n)=(b_{1}(n),\ldots,b_{r}(n))$ and
$d(n)=(d_{1}(n),\ldots,d_{r}(n))$ are functions from $\mathbb{Z}_{+}^{d}$ to
$\mathbb{R}_{+}^{d}$. This model represents a density-dependent population
dynamics with $d$ types of individuals (say $d$ species), where $b_{i}(n)$
(resp. $d_{i}(n)$) represents the reproduction rate (resp. death rate) per
individuals of species $i$ when the population is in state $n$.

Note that the forms of the birth and death rates imply that, once a coordinate
$X^{j}_{t}$ of the process hits 0, it remains equal to 0. This corresponds to
the extinction of the population of type $j$. Hence, the set
$\partial:=\mathbb{Z}_{+}^{r}\setminus\mathbb{N}^{r}$ is absorbing for the
process $X$ and we denote as usual by $\tau_{\partial}$ its absorption time.

We define for all $1\leq k\leq d$

$\displaystyle\bar{d}(k)$ $\displaystyle=\sup_{n\in\mathbb{N}^{d},\
|n|=k}|n|\sum_{i=1}^{d}\mathbbm{1}_{n_{i}=1}d_{i}(n),$
$\displaystyle\text{and}\quad\underline{d}(k)$
$\displaystyle=\inf_{n\in\mathbb{N}^{d},\
|n|=k}\sum_{i=1}^{d}n_{i}\left[\mathbbm{1}_{n_{i}\neq
1}d_{i}(n)-b_{i}(n)\right],$

where $|n|:=n_{1}+\ldots+n_{d}$. We shall assume

######  Assumption 5.

There exists $\eta>0$ small enough such that, for all $k\in\mathbb{N}$ large
enough,

$\underline{d}(k)\geq\eta\bar{d}(k),$ (3.1)

and

$\frac{\underline{d}(k)}{k^{1+\eta}}\xrightarrow[k\rightarrow+\infty]{}+\infty.$
(3.2)

Note that, assuming that the set $O_{n}$ is finite for all $n$, any function
$f:\mathbb{Z}_{+}^{d}\rightarrow\mathbb{R}$ is in the domain of the weakened
infinitesimal generator of $X$ and, for all $n\in\mathbb{N}_{+}^{d}$,

$Lf(n)=\sum_{j=1}^{d}[f(n+e_{j})-f(n)]n_{j}b_{j}(n)+\sum_{j=1}^{d}[f(n-e_{j})-f(n)]n_{j}d_{j}(n).$

Under Assumption (3.2), setting $W(n)=|n|$, we have

$\displaystyle\frac{LW(n)}{W(n)^{1+\eta}}\rightarrow-\infty.$

This classically entails that

$\sup_{n\in\mathbb{N}^{r}}\mathbb{E}_{n}(|X_{1}|)<+\infty.$ (3.3)

(The argument is very similar to the one used for Theorem 2.4.) In particular,
the process is non-explosive and $\tau_{\partial}$ is finite almost surely.
Therefore, if we define for all $m\geq 1$

$O_{m}=U_{m}=\\{x\in\mathbb{N}^{d}:|x|\leq m\\},$

it is clear that the process $X$ satisfies the conditions of Section 2.1 and
is regularly absorbed. We can now state the main result of the section.

######  Theorem 3.1.

Under Assumption 5, the multi-dimensional competitive birth and death process
$(X_{t},t\geq 0)$ absorbed when one of its coordinates hits 0 admits a unique
quasi-stationnary distribution $\nu_{QSD}$ and there exist constants
$C,\gamma>0$ such that, for all probability measure $\mu$ on $\mathbb{N}^{d}$,

$\|\mathbb{P}_{\mu}(X_{t}\in\cdot\mid t<\tau_{\partial})-\nu_{QSD}\|_{TV}\leq
Ce^{-\gamma t},\quad\forall t\geq 0.$

It is easy to check that Assumption 5 is satisfied in the general Lotka-
Volterra birth and death process of the introduction. Indeed, we clearly have
$\bar{d}(k)\leq Ck^{2}$ and

$\displaystyle\underline{d}(k)$ $\displaystyle\geq\inf_{n\in\mathbb{N}^{d},\
|n|=k}\sum_{i=1}^{d}(d_{i}(n)-b_{i}(n))-\sup_{n\in\mathbb{N}^{d},\
|n|=k}\sum_{i=1}^{n}d_{i}(n)\mathbbm{1}_{n_{i}=1}$ $\displaystyle\geq-
Ck+\inf_{n\in\mathbb{N}^{d},\
|n|=k}\sum_{i,j=1}^{d}n_{i}(c_{ij}-\gamma_{ij})n_{j}$

for $C=\sup_{i}(b_{i}+d_{i})+\sup_{ij}c_{ij}$. Under the assumptions of
Theorem 1.1, there exists $C^{\prime}>0$ such that, for all
$n\in\mathbb{N}^{d}$,

$\sum_{i,j=1}^{d}n_{i}(c_{ij}-\gamma_{ij})n_{j}\geq C^{\prime}|n|^{2}.$

This entails Assumption 5.

###### Proof.

We are going to use the criterion of Corollary 2.8. Using (3.3) and copying
the arguments of [9, Sec. 4.1.1 and Thm. 4.1], one deduces that Assumption 2
is satisfied with $\nu=\delta_{(1,\ldots,1)}$ and Assumptions 3 and 4 are also
satisfied.

Hence we only have to find a couple of admissible functions $(V,\varphi)$
satisfying Conditions (a) and (b) of Proposition 2.7 and satisfying (2.9)
(note that the fact that $\varphi$ is positive on $\mathbb{N}^{d}$ entails
immediately that $\inf_{x\in O_{m}}\varphi(x)>0$). This couple of functions is
given for all $n\in\mathbb{Z}_{+}^{d}$ by

$V(n)=\begin{cases}\sum_{k=1}^{|n|}\frac{1}{k^{\alpha}}&\text{if
}n\in\mathbb{N}^{d},\\\ 0&\text{if }n\in\partial\end{cases}$

and

$\varphi(n)=\begin{cases}\sum_{k=|n|+1}^{+\infty}\frac{1}{k^{\beta}}&\text{if
}n\in\mathbb{N}^{d},\\\ 0&\text{if }n\in\partial,\end{cases}$

for appropriate choices of $\alpha,\beta>1$. Note that the two functions are
bounded, nonnegative and positive of $\mathbb{N}^{d}$. Note also that, since
$\inf_{n\in\mathbb{N}^{d}}V(n)>0$, Conditions (2.3) and (2.9) are obviously
satisfied. Note also that (2.5) is clear for non-explosive Markov processes on
discrete state spaces, and that (2.4) is true from the definition of $V$ and
$\varphi$.

Hence, we only have to check Conditions (a) and (b) since they imply the last
condition needed for the couple of functions $(V,\varphi)$ to be admissible:
the fact that $LV$ is bounded from above and $L\varphi$ bounded from below. So
we compute

$\displaystyle L\varphi(n)$
$\displaystyle=-\sum_{i=1}^{d}\frac{n_{i}b_{i}(n)}{(1+|n|)^{\beta}}+\sum_{i=1}^{d}\frac{n_{i}\mathbbm{1}_{n_{i}\neq
1}d_{i}(n)}{|n|^{\beta}}-\sum_{i=1}^{d}\mathbbm{1}_{n_{i}=1}d_{i}(n)\sum_{k=|n|+1}^{+\infty}\frac{1}{k^{\beta}}$
$\displaystyle\geq\frac{1}{|n|^{\beta}}\left[\underline{d}(|n|)-\frac{\bar{d}(|n|)}{\beta-1}\right],$

where we used the fact that

$\sum_{k=x+1}^{+\infty}\frac{1}{k^{\beta}}\leq\int_{x}^{+\infty}\frac{dy}{y^{\beta}}=\frac{1}{(\beta-1)x^{\beta-1}}.$

Hence it follows from Assumption (3.1) that there exists $\beta>1$ large
enough such that $L\varphi(n)\geq 0$ for all $|n|$ large enough. This entails
Condition (a).

We fix such a value of $\beta$. Using that

$\sup_{n\in\mathbb{N}^{d}}V(n)=\sum_{k=1}^{+\infty}\frac{1}{k^{\alpha}}\leq
1+\int_{1}^{\infty}\frac{dx}{x^{\alpha}}=\frac{\alpha}{\alpha-1}$

and

$\varphi(n)\geq\int_{|n|+1}^{\infty}\frac{dx}{x^{\beta}}=\frac{(1+|n|)^{1-\beta}}{\beta-1}\geq\frac{|n|^{1-\beta}}{2(\beta-1)}$

for $|n|$ large enough, we compute for such $n$

$\displaystyle LV(n)+\frac{V^{1+\varepsilon}(n)}{\varphi^{\varepsilon}(n)}$
$\displaystyle\leq\sum_{i=1}^{d}\frac{n_{i}b_{i}(n)}{(|n|+1)^{\alpha}}-\sum_{i=1}^{d}\frac{n_{i}d_{i}(n)\mathbbm{1}_{n_{i}\neq
1}}{|n|^{\alpha}}+C|n|^{\varepsilon(\beta-1)}$
$\displaystyle\leq-\frac{\underline{d}(|n|)}{|n|^{\alpha}}+C|n|^{\varepsilon(\beta-1)},$

where $C=[\alpha/(\alpha-1)]^{1+\varepsilon}[2(\beta-1)]^{\varepsilon}$.
Choosing $\alpha=1+\eta/2$ and $\varepsilon=\eta/[2(\beta-1)]$, Assumption
(3.2) implies that
$LV(n)+\frac{V^{1+\varepsilon}(n)}{\varphi^{\varepsilon}(n)}\leq 0$ for
$n\not\in O_{m}$ with $m$ large enough. Since $\inf_{n\in O_{m}}\varphi(n)>0$,
we have proved Condition (b). ∎

##  4 Application to multidimensional Feller diffusions absorbed when one of
the coordinates hits 0

We consider a general multitype Feller diffusion $(X_{t},t\geq 0)$ in
$\mathbb{R}_{+}^{d}$, solution to the stochastic differential equation

$dX^{i}_{t}=\sqrt{\gamma_{i}X^{i}_{t}}dB^{i}_{t}+X^{i}_{t}\,r_{i}(X_{t})dt,\quad
1\leq i\leq d,$ (4.1)

where $(B^{i}_{t},t\geq 0)$ are independent standard Brownian motions,
$\gamma_{i}$ are positive constants and $r_{i}$ are measurable maps from
$\mathbb{R}_{+}^{d}$ to $\mathbb{R}$. From the biological point of view,
$r_{i}(x)$ represents the growth rate per individual of species $i$ in a
population of size vector $x\in\mathbb{R}_{+}^{d}$. We shall make the
following assumption.

######  Assumption 6.

Assume that, for all $i\in\\{1,\ldots,d\\}$, $r_{i}$ is locally Hölder on
$\mathbb{R}_{+}^{d}$ and that there exist $a>0$ and $0<\eta<1$ such that

$\displaystyle r_{i}(x)\leq a^{\eta}-x_{i}^{\eta},$ (4.2)

and there exist constants $B_{a}>a$, $C_{a}>0$ and $D_{a}>0$ such that

$\sum_{i=1}^{d}\mathbbm{1}_{x_{i}\geq B_{a}}r_{i}(x)\leq
C_{a}\left(\sum_{i=1}^{d}\mathbbm{1}_{x_{i}\leq
a}r_{i}(x)+D_{a}\right),\quad\forall x\in\mathbb{R}_{+}^{d}.$ (4.3)

Assumptions (4.2) and (4.3) correspond to some form of competitivity for the
system of interacting Feller diffusions. Moreover, Assumption (4.2) entail
that each coordinate of the process can be upper bounded by the (strong)
solution of the one-dimensional Feller diffusion

$d\hat{X}^{i}_{t}=\sqrt{\gamma_{i}\hat{X}^{i}_{t}}dB^{i}_{t}+a^{\eta}\hat{X}^{i}_{t}dt,\quad
1\leq i\leq d,$ (4.4)

with initial value $\hat{X}^{i}_{0}=X^{i}_{0}$. To prove this, one can apply
the transformation $Y^{i}_{t}=2\sqrt{X^{i}_{t}/\gamma_{i}}$ to obtain a SDE of
the form

$dY^{i}_{t}=dB^{i}_{t}+\hat{r}_{i}(Y_{t})dt,\quad 1\leq i\leq d,$

with $\hat{r}_{i}$ measurable, locally Hölder, hence locally bounded in
$(0,+\infty)^{d}$, and (possibly) singular at
$\partial:=\mathbb{R}_{+}^{d}\setminus(0,+\infty)^{d}$, and apply classical
comparison results to solutions of SDEs with constant diffusion coefficient.
Since the SDE (4.4) is non-explosive ($+\infty$ is a natural boundary and $0$
is an exit boundary for this diffusion by classical criteria, cf. e.g. [15]),
the SDE (4.1) is non explosive. Similarly, Assumption (4.2) implies that
$X^{i}_{t}\leq\bar{X}^{i}_{t}$ where

$d\bar{X}^{i}_{t}=\sqrt{\gamma_{i}\bar{X}^{i}_{t}}dB^{i}_{t}+\bar{X}^{i}_{t}\left[a^{\eta}-\left(\bar{X}^{i}_{t}\right)^{\eta}\right]dt,\quad
1\leq i\leq d,$ (4.5)

for some constant $B$ and with initial value $\bar{X}^{i}_{0}=X^{i}_{0}$.
Since $\bar{X}^{i}$ is a diffusion on $\mathbb{R}_{+}$ for which $+\infty$ is
an entrance boundary and $0$ an exit boundary, we deduce that
$\tau_{\partial}<+\infty$ almost surely.

We also deduce from the fact that $r_{i}$ is locally bounded and from (4.2)
that there is strong existence and pathwise uniqueness for (4.1). The ideas to
prove this seem quite standard, but we couldn’t find a proof of this fact in
our situation, so we explain the main steps below. Assume that $X^{i}_{0}>0$
for all $1\leq i\leq d$ (otherwise, we will see below what to do). Since the
drifts $\hat{r}_{i}$ of $Y^{i}$ are locally bounded in $(0,+\infty)^{d}$, we
can use the well-known result of Veretennikov [25] to prove local strong
existence and pathwise uniqueness until possible explosion or the first exit
time $\tau_{\text{exit}}$ from all compact subsets of $(0,+\infty)^{d}$. As
explained above, the process is non-explosive. In addition, on the event
$\\{\tau_{\text{exit}}<\infty\\}$, the process admits a left limit at the time
$\tau_{\text{exit}}$ since its drift and diffusion coefficients are bounded on
each compact subset of $\mathbb{R}_{+}^{d}$. Hence, we can define
$X_{\tau_{\text{exit}}}$ as the right limit of $X_{t}$ at this time, which has
at least one zero coordinate, i.e. $\tau_{\partial}=\tau_{\text{exit}}$. We
then set $X^{j}_{t}=0$ for all $t\geq\tau_{\partial}$ and all $j$ such that
$X^{j}_{\tau_{\partial}}=0$. We can then proceed as above for the set of
coordinates such that $X^{i}_{\tau_{\partial}}>0$. This gives the existence of
a global strong solution.

For pathwise uniqueness, we know that it holds locally between the times of
absorption of a coordinate, but only for the process formed by the nonzero
coordinates, so we need to prove that no solution of (4.1) starting with one
zero coordinate, say $X^{i}_{0}=0$, can have $X^{i}_{t}>0$ for some positive
time $t$. This can be proved using again the comparison between $X^{i}$ and
the Feller diffusions (4.4), since 0 is absorbing (since it is an exit
boundary) for this diffusion.

Strong existence and pathwise uniqueness imply well-posedness of the
martingale problem, hence the strong Markov property hold on the canonical
space with respect to the natural filtration (see e.g. [23]). Since the paths
of $X$ are continuous, the hitting times of closed subsets of
$\mathbb{R}_{+}^{d}$ are stopping times for this filtration. Hence, defining
$O_{n}=\\{x\in(0,+\infty)^{d}:1/(2n)<x_{i}<2n,\ \forall 1\leq i\leq d\\}$ and
$U_{n}=\\{x\in(0,+\infty)^{d}:1/(2n-1)\leq x_{i}\leq 2n-1,\ \forall 1\leq
i\leq d\\}$ for all $n\geq 1$, it is then clear that the process $X$ satisfies
the conditions of Section 2.1 and, in view of the previous construction of the
process, it is regularly absorbed. In addition, it follows from Itô’s formula
and the local boundedness of the coefficients of the SDE that any measurable
function $f:\mathbb{R}_{+}^{d}\rightarrow\mathbb{R}$ twice continuously
differentiable on $(0,+\infty)$ belongs to the domain of the weakened
generator of $X$ and

$Lf(x)=\sum_{i=1}^{d}\frac{\gamma_{i}x_{i}b_{i}(x)}{2}\frac{\partial
f}{\partial x_{i}}(x)+\sum_{i=1}^{d}x_{i}\frac{\partial^{2}f}{\partial
x_{i}^{2}}(x),\quad\forall x\in(0,+\infty)^{d}.$

We can now state the main result of the section.

######  Theorem 4.1.

Under Assumption 6, the multi-dimensional Feller diffusion process
$(X_{t},t\geq 0)$ absorbed when one of its coordinates hits 0 admits a unique
quasi-stationnary distribution $\nu_{QSD}$ and there exist constants
$C,\gamma>0$ such that, for all probability measure $\mu$ on $\mathbb{N}^{d}$,

$\|\mathbb{P}_{\mu}(X_{t}\in\cdot\mid t<\tau_{\partial})-\nu_{QSD}\|_{TV}\leq
Ce^{-\gamma t},\quad\forall t\geq 0.$

It is straightforward to check that Assumption 6 is satisfied in the
competitive Lotka-Volterra case, that is when
$r_{i}(x)=r_{i}-\sum_{i=1}^{d}c_{ij}x_{j}$ with $c_{ij}\geq 0$ and $c_{ii}>0$
for all $1\leq i,j\leq d$. Hence Theorem 1.2 is an immediate corollary of
Theorem 4.1. Assumption 6 allows for other biologically relevant models. For
instance, one can consider ecosystems where the competition among individuals
only begins when the population size reaches a level $K>0$, which leads for
instance to the SDE

$\displaystyle
dX^{i}_{t}=\sqrt{\gamma_{i}X^{i}_{t}}\,dB^{i}_{t}+X^{i}_{t}\left(r-\sum_{j=1}^{d}c_{ij}\,(X^{j}_{t}-K)_{+}\right)\,dt,\
X^{i}_{0}>0,$

where $c_{ij}$ are non-negative constants and $c_{ii}>0$ for all
$i,j\in\\{1,\ldots,d\\}$. Note also that a similar approach (i.e. using
Theorem 2.4 with Lyapunov functions of the form $\prod_{i=1}^{d}h(x_{i})$) can
also be used to handle diffusion processes evolving in bounded boxes.

###### Proof.

Up to a linear scaling of the coordinates, we can assume without loss of
generality that $\gamma_{i}=2$ for all $1\leq i\leq d$, so we will only
consider this case from now on. Note that Assumption 6 is not modified by the
rescaling.

We shall make use of Corollary 2.8. We divide the proof into five steps,
respectively devoted to the construction of a function $\varphi$ satisfying
Conditions (a), of a function $V$ satisfying Condition (b), to the proof of
(2.9), to the proof of a local Harnack inequality, and to check Assumption 2,
Assumption 3 and Assumption 4.

_Step 1: construction of a function $\varphi$ satisfying Condition (a)._

Recall the definition of the constants $a>0$ and $B_{a}>a$ from Assumption 6.
We use the following lemma, proved at the end of this section.

######  Lemma 4.2.

There exists $M>0$ such that, for all $\beta\geq M$, there exists a function
$h_{\beta}:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$ twice continuously
differentiable on $(0,+\infty)$ such that

$h_{\beta}(x)=\begin{cases}4x^{2}/a^{2}&\text{if }x\in[0,a/2],\\\
B_{a}^{\beta}(2x)^{-\beta}&\text{if }x\geq B_{a},\end{cases}$

$h_{\beta}(x)\geq 1$ for all $x\in[a/2,a]$, $h_{\beta}$ is nonincreasing and
convex on $[a,+\infty)$,

$M^{\prime}:=\sup_{\beta\geq
M}\sup_{x\in[a/2,a]}|h^{\prime}_{\beta}(x)|<+\infty\quad\text{and}\quad
M^{\prime\prime}:=\sup_{\beta\geq
M}\sup_{x\in[a/2,a]}|h^{\prime\prime}_{\beta}(x)|<+\infty.$

We set $\beta=M+(2\vee aM^{\prime})/C_{a}+1$,

$\varphi(x)=\prod_{i=1}^{d}h_{\beta}(x_{i}),\quad\forall
x\in\mathbb{R}_{+}^{d}$

and we shall prove that $\varphi$ satisfies Condition (a). We have

$\displaystyle\frac{L\varphi(x)}{\varphi(x)}$
$\displaystyle=\sum_{i=1}^{d}\frac{x_{i}h^{\prime}_{\beta}(x_{i})r_{i}(x)+x_{i}h^{\prime\prime}_{\beta}(x)}{h_{\beta}(x_{i})}.$

Now, it follows from the properties of $h_{\beta}$ and Assumptions 6 that, for
all $x\in\mathbb{R}_{+}$ and all $1\leq i\leq d$,

$\frac{x_{i}h^{\prime}_{\beta}(x_{i})r_{i}(x)+x_{i}h^{\prime\prime}_{\beta}(x)}{h_{\beta}(x_{i})}\geq\begin{cases}0&\text{if
}x_{i}\geq a,\\\ -\beta r_{i}(x)&\text{if }x_{i}\geq B_{a},\\\
2r_{i}(x)+\frac{2}{x_{i}}&\text{if }x_{i}\leq a/2,\\\
aM^{\prime}(r_{i}(x)-a^{\eta})-a^{1+\eta}M^{\prime}-aM^{\prime\prime}&\text{if
}a/2\leq x_{i}\leq a,\end{cases}$

where we used in the last inequality the fact that $r_{i}(x)-a^{\eta}\leq 0$
for all $x$. Using once again this property, we deduce that, for some constant
$B$ independent of $\beta\geq M$,

$\frac{x_{i}h^{\prime}_{\beta}(x_{i})r_{i}(x)+x_{i}h^{\prime\prime}_{\beta}(x)}{h_{\beta}(x_{i})}\geq\begin{cases}0&\text{if
}x_{i}\geq a,\\\ -\beta r_{i}(x)&\text{if }x_{i}\geq B_{a},\\\ (2\vee
aM^{\prime})r_{i}(x)+\frac{2}{x_{i}}-B&\text{if }x_{i}\leq a.\end{cases}$
(4.6)

Hence, for all $x\in\mathbb{R}_{+}^{d}$,

$\displaystyle\frac{L\varphi(x)}{\varphi(x)}$
$\displaystyle\geq-\beta\sum_{i=1}^{d}\mathbbm{1}_{x_{i}\geq
B_{a}}r_{i}(x)+\sum_{i=1}^{d}\mathbbm{1}_{x_{i}\leq a}\left((2\vee
aM^{\prime})r_{i}(x)+\frac{2}{x_{i}}\right)-dB.$

This and Assumption (4.3) imply that

$\displaystyle\frac{L\varphi(x)}{\varphi(x)}$
$\displaystyle\geq-\sum_{i=1}^{d}\mathbbm{1}_{x_{i}\geq
B_{a}}r_{i}(x)+2\sum_{i=1}^{d}\mathbbm{1}_{x_{i}\leq
a}\frac{1}{x_{i}}-dB-(2\vee aM^{\prime})D_{a}$
$\displaystyle\geq\sum_{i=1}^{d}\left(x_{i}^{\eta}+\frac{2}{x_{i}}\right)-B^{\prime},$

for some constant $B^{\prime}$, where we used Assumption (4.2) in the last
inequality.

Hence, there exists $n\geq 1$ and a constant $C>0$ such that

$\displaystyle L\varphi(x)\geq-C\mathbbm{1}_{x\in O_{n}}.$

This ends the proof that $\varphi$ satisfies Condition (a).

_Step 2: construction of a function $V$ satisfying Condition (b) and
verification that $(V,\varphi)$ is a couple of admissible functions._

For $V$, we define

$V(x)=\prod_{i=1}^{d}g(x_{i}),\quad\forall x\in\mathbb{R}_{+}^{d},$

where the function $g:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$ is twice
continuously differentiable on $(0,+\infty)$, increasing concave and such that

$g(x)=\begin{cases}x^{\gamma}&\text{if }x\leq 1\\\ \delta-x^{-\eta/2}&\text{if
}x\geq 2,\end{cases}$

for some constants $\gamma<1$ and $\delta>0$ and where $\eta$ is defined in
Assumption (4.2). Since $g^{\prime}(1)=\gamma$ and $g^{\prime}(2)=\eta
2^{-2-\eta/2}$, it is possible to find $\delta>0$ such that such a function
$g$ exists as soon as $\eta 2^{-2-\eta/2}<\gamma$. Hence, we shall assume that
$\gamma$ belongs to the non-empty interval $(\eta 2^{-2-\eta/2},1)$. We have

$\displaystyle\frac{LV(x)}{V(x)}$
$\displaystyle=\sum_{i=1}^{d}\frac{x_{i}g^{\prime}(x_{i})r_{i}(x)+x_{i}g^{\prime\prime}(x_{i})}{g(x_{i})}$

and

$\displaystyle\frac{x_{i}g^{\prime}(x_{i})r_{i}(x)+x_{i}g^{\prime\prime}(x_{i})}{g(x_{i})}\leq\begin{cases}\displaystyle\gamma
r_{i}(x)-\frac{\gamma(1-\gamma)}{x_{i}}&\text{if }x_{i}\leq 1,\\\
\displaystyle 2a^{\eta}\sup_{1\leq x\leq 2}g^{\prime}(x)&\text{if }1\leq
x_{i}\leq 2,\\\ \displaystyle\frac{\eta r_{i}(x)x_{i}^{-\eta/2}}{2(\delta-
x_{i}^{-\eta/2})}&\text{if }x_{i}\geq 2.\end{cases}$

We deduce from Assumptions (4.2) that there exist constants
$B^{\prime},B^{\prime\prime}>0$ such that

$\displaystyle\frac{x_{i}g^{\prime}(x_{i})r_{i}(x)+x_{i}g^{\prime\prime}(x)}{g(x_{i})}\leq
B^{\prime}-\begin{cases}\frac{\gamma(1-\gamma)}{x_{i}}&\text{if }x_{i}\leq
1,\\\ 0&\text{if }1\leq x_{i}\leq 2,\\\
B^{\prime\prime}x_{i}^{\eta/2}&\text{if }x_{i}\geq 2.\end{cases}$ (4.7)

Thus, since $h_{\beta}(x_{i})\geq D_{\beta}\left(x_{i}^{2}\wedge
x_{i}^{-\beta}\right)$ for some constant $D_{\beta}>0$ and since
$g(x_{i})\leq\delta$,

$\displaystyle\frac{LV(x)}{V(x)}+\frac{V(x)^{\varepsilon}}{\varphi(x)^{\varepsilon}}$
$\displaystyle\leq
B^{\prime}d-\gamma(1-\gamma)\sum_{i=1}^{d}\frac{\mathbbm{1}_{x_{i}\leq
1}}{x_{i}}-B^{\prime\prime}\sum_{i=1}^{d}\mathbbm{1}_{x_{i}\geq
2}x_{i}^{\eta/2}$
$\displaystyle+\left(\frac{\delta}{D_{\beta}}\right)^{d\varepsilon}\prod_{i=1}^{d}\left(x_{i}^{\varepsilon\beta}\vee
x_{i}^{-2\varepsilon}\right)$ $\displaystyle\leq
B^{\prime}d+\gamma(1-\gamma)+B^{\prime\prime}2^{\eta/2}-\gamma(1-\gamma)\left(\inf_{i}x_{i}\right)^{-1}-B^{\prime\prime}\left(\sup_{i}x_{i}\right)^{\eta/2}$
$\displaystyle+\left(\frac{\delta}{D_{\beta}}\right)^{d\varepsilon}\left[\left(\sup_{i}x_{i}\right)^{\beta
d\varepsilon}+\left(\inf_{i}x_{i}\right)^{-2d\varepsilon}\right].$

Therefore, choosing $\varepsilon>0$ such that $d(1+\alpha)\varepsilon<1$ and
$\beta d\varepsilon<\eta/2$,
$LV(x)+V(x)^{1+\varepsilon}/\varphi(x)^{\varepsilon}\leq 0$ for all
$x\in\mathbb{R}_{+}^{d}$ such that $\inf_{i}x_{i}$ is small enough or
$\sup_{i}x_{i}$ is big enough. Since $LV$ is bounded from above by (4.7) and
since $V$ and $\varphi$ are positive continuous on any compact subset of
$(0,+\infty)^{d}$, we have proved Condition (b).

We can now check that $(V,\varphi)$ is a couple of admissible functions.
First, $V$ and $\varphi$ are both bounded, positive on $(0,+\infty)$ and
vanishing on $\partial$. They both belong to the domain of the weakened
infinitesimal generator of $X$. Since the function $g/h_{\beta}$ is positive
continuous on $(0,+\infty)$ and

$\frac{g(x)}{h_{\beta}(x)}=\begin{cases}x^{\gamma-2}&\text{if }x\leq 1\wedge
a/2,\\\ (\delta-x^{-\eta/2})(2x)^{\beta}/B_{a}^{\beta}&\text{if }x\geq
B_{a}\vee 2,\end{cases}$

we deduce that $\inf_{x\in(0,+\infty)}g(x)/h_{\beta}(x)>0$, hence (2.3) holds
true, and (2.4) is also clear. Since $V$ is continuous on $\mathbb{R}_{+}^{d}$
and $X_{T_{n}}\rightarrow X_{\tau_{\partial}}$ almost surely, (2.5) also holds
true. Finally, since $LV$ is bounded from above and $L\varphi$ is bounded from
below, we have proved that $(V,\varphi)$ is a couple of admissible functions.

_Step 3: proof of ( 2.9)._

Using the upper bound $X^{i}_{t}\leq\bar{X}^{i}_{t}$ for all $t\geq 0$ and
$1\leq i\leq d$, where $\bar{X}^{i}$ is solution to the SDE (4.5), and noting
that the processes $(\bar{X}^{i})_{1\leq i\leq d}$ are independent, we have
for all $x\in\mathbb{R}_{+}^{d}$ and all $t_{2}>0$,

$\displaystyle\mathbb{P}_{x}(t_{2}<\tau_{\partial})\leq\prod_{i=1}^{d}\mathbb{P}_{x_{i}}(\bar{X}^{i}_{t_{2}}>0).$

Now, there exist constants $D$ and $D^{\prime}$ such that

$\mathbb{P}_{x_{i}}(\bar{X}^{i}_{t_{2}}>0)\leq(Dx_{i})\wedge 1\leq
D^{\prime}g(x_{i})\quad\text{for all }x_{i}>0.$ (4.8)

To prove this, we can consider a scale function $s$ of the diffusion
$\bar{X}^{i}$ such that $s(0)=0$. Using the expression of the scale function
and the speed measure (see e.g. [23, V.52]), one easily checks that
$s(x_{i})\sim x_{i}$ when $x_{i}\rightarrow 0$ and that Proposition 4.9 of [8]
is satisfied, so that $\mathbb{P}_{x_{i}}(\bar{X}^{i}_{t_{2}}>0)\leq
Ms(x_{i})$ for some $M>0$. Since $s(x_{i})\sim x_{i}$ when $x_{i}\rightarrow
0$, (4.8) is proved and hence (2.9) holds true.

_Step 4: Harnack inequality for $u$._ Consider a bounded measurable function
$f$ and define the application $u:(t,x)\in\mathbb{R}_{+}\times
E\mapsto\mathbb{E}_{x}(f(X_{t}))$. Our aim is to prove that, for all $m\geq
1$, there exist two constants $N_{m}>0$ and $\delta_{m}>0$, which do not
depend on $f$, such that

$\displaystyle u(\delta_{m}+\delta_{m}^{2},x)\leq
N_{m}u(\delta_{m}+2\delta_{m}^{2},y),\text{ for all $x,y\in O_{m}$ such that
}|x-y|\leq\delta_{m}.$ (4.9)

First, the function $u$ is continuous on $\mathbb{R}_{+}^{*}\times E$. Indeed,
since the SDE (4.1) has Hölder coefficients in $O_{m}$, using the same
approach as in the proof of [24, Thm 7.2.4], one deduces that, for all $x\in
E$ and for all $\varepsilon>0$, there exists $\delta>0$ such that, for all
$y\in B(x,\delta)$, all $s,t\in[\delta,2\delta]$ and all bounded measurable
function $\varphi:E\rightarrow\mathbb{R}$,

$\displaystyle\left|\mathbb{E}_{x}\left(\varphi(X_{s})\right)-\mathbb{E}_{y}\left(\varphi(X_{t})\right)\right|\leq\varepsilon\|\varphi\|_{\infty}.$

This result applied to the function
$\varphi:x\mapsto\mathbb{E}_{x}\left(f(X_{t-\delta})\right)$ for $t\geq\delta$
and the Markov property entails the continuity of $u$ on
$\mathbb{R}_{+}^{*}\times E$.

Second, fix $m\geq 1$ and let $K$ be a compact set with $C^{\infty}$ boundary
such that $O_{m}\subset K\subset E$ and such that $d(O_{m},\partial K)>0$. We
set $\delta=d(O_{m},\partial K)/3$. Let $(h_{n})_{n\geq 1}$ be a sequence of
bounded $C^{\infty}$ functions from $([\delta,\infty[\times\partial
K)\cup(\\{\delta\\}\times K)$ to $\mathbb{R}$ such that $(h_{n})_{n\geq 1}$
converges uniformly locally toward $u$ restricted to
$([\delta,\infty[\times\partial K)\cup(\\{\delta\\}\times K)$ (such a sequence
exists because of the continuity of $u$). For all $n\geq 1$, we let
$u_{n}:[\delta,+\infty[\times K\rightarrow\mathbb{R}$ be the solution to the
linear parabolic equation

$\displaystyle\begin{cases}\displaystyle\partial_{t}u_{n}=\sum_{i=1}^{d}x_{i}\frac{\partial^{2}u_{n}}{\partial
x_{i}^{2}}+x_{i}r_{i}(x)\frac{\partial u_{n}}{\partial x_{i}},\\\
\displaystyle u_{n}(t,x)=h_{n}(t,x),\
\forall(t,x)\in([\delta,\infty[\times\partial K)\cup(\\{\delta\\}\times
K).\end{cases}$

By [18, Thm 5.1.15], for all $n\geq 1$, $u_{n}$ is of regularity $C^{1,2}$.
Moreover, using the Harnack inequality provided by [16, Theorem 1.1] (with
$\theta=2$ and $R=\delta$), we deduce that there exists a constant $N>0$ which
does not depend on $f$ nor on $n$ such that

$\displaystyle u_{n}(\delta+\delta^{2},x)\leq
Nu_{n}(\delta+2\delta^{2},y),\text{ for all $x,y\in O_{m}$ such that
}|x-y|\leq\delta.$ (4.10)

In particular, applying Itô’s formula to $s\mapsto u_{n}(\delta+t-s,X_{s})$ at
time $\tau_{K}\wedge t$ and taking the expectation, one deduces that, for all
$t\geq 0$,

$\displaystyle u_{n}(\delta+t,x)$
$\displaystyle=\mathbb{E}_{x}\left[u_{n}(\delta+t-\tau_{K}\wedge
t,X_{\tau_{K}\wedge t})\right]$
$\displaystyle=\mathbb{E}_{x}\left(\mathbbm{1}_{\tau_{K}\leq
t}u_{n}(t+\delta-\tau_{K},X_{\tau_{K}})\right)+\mathbb{E}_{x}\left(\mathbbm{1}_{\tau_{K}>t}u_{n}(\delta,X_{t})\right)$
$\displaystyle=\mathbb{E}_{x}\left(\mathbbm{1}_{\tau_{K}\leq
t}h_{n}(t+\delta-\tau_{K},X_{\tau_{K}})\right)+\mathbb{E}_{x}\left(\mathbbm{1}_{\tau_{K}>t}h_{n}(\delta,X_{t})\right).$

By Lebesgue’s theorem, the last quantity converges when $n\rightarrow+\infty$
to

$\mathbb{E}_{x}\left(\mathbbm{1}_{\tau_{K}\leq
t}u(t+\delta-\tau_{K},X_{\tau_{K}})\right)+\mathbb{E}_{x}\left(\mathbbm{1}_{\tau_{K}>t}u(\delta,X_{t})\right)\\\
\begin{aligned} &=\mathbb{E}_{x}\left(\mathbbm{1}_{\tau_{K}\leq
t}\mathbb{E}_{X_{\tau_{K}}}(f(X_{t+\delta-\tau_{K}})\right)+\mathbb{E}_{x}\left(\mathbbm{1}_{\tau_{K}>t}\mathbb{E}_{X_{t}}(f(X_{\delta}))\right)\\\
&=\mathbb{E}_{x}(f(X_{\delta+t}))=u(\delta+t,x),\end{aligned}$

where we used the strong Markov property at time $\tau_{K}$.

We deduce from this convergence and the Harnack inequalities (4.10) that there
exist two constants $N_{m}>0$ and $\delta_{m}>0$, which do not depend on $f$,
such that (4.9) holds true.

_Step 5 : proof that Assumptions 2, 3 and 4 are satisfied._ Fix $x_{1}\in
O_{1}$ and let $\nu$ denote the conditional law
$\mathbb{P}_{x_{1}}(X_{\delta_{1}+\delta_{1}^{2}}\in\cdot\mid\delta_{1}+\delta_{1}^{2}<\tau_{\partial})$.
Then the Harnack inequality (4.9) entails that, for all $x\in O_{1}$ such that
$|x-x_{1}|\leq\delta_{1}$,

$\displaystyle\mathbb{P}_{x}(X_{\delta_{1}+2\delta_{1}^{2}}\in\cdot)\geq\frac{\mathbb{P}_{x_{1}}(\delta_{1}+\delta_{1}^{2}<\tau_{\partial})}{N_{1}}\,\nu.$

Replacing $O_{1}$ by its intersection with the open ball with center $x_{1}$
and radius $\delta_{1}$, this implies that Assumption 2 is satisfied for
$k_{0}=1$ (the second part of this assumption is a classical property for
locally elliptic diffusion processes).

Assumption 3 is a direct consequence of the domination by solutions to (4.5),
since these solutions come down from infinity and hit $0$ in finite time
almost surely (cf. e.g. [2]).

Finally, Assumption 4 is a consequence of (4.9). Indeed, for any fixed $m$ and
for all $t\geq\delta_{m}+2\delta_{m}^{2}$, this equation applied to
$f(x)=\mathbb{P}_{x}(t-\delta_{m}-2\delta_{m}^{2}<\tau_{\partial})$ and the
Markov property entails that

$\displaystyle\mathbb{P}_{x}(t-\delta_{m}^{2}<\tau_{\partial})\leq
N_{m}\mathbb{P}_{y}(t<\tau_{\partial}),\text{ for all $x,y\in O_{m}$ such that
}|x-y|\leq\delta_{m}.$

Since $s\mapsto\mathbb{P}_{x}(s<\tau_{\partial})$ is non-increasing, we deduce
that

$\displaystyle\mathbb{P}_{x}(t<\tau_{\partial})\leq
N_{m}\mathbb{P}_{y}(t<\tau_{\partial}),\text{ for all $x,y\in O_{m}$ such that
}|x-y|\leq\delta_{m}.$

Since $O_{m}$ has a finite diameter and is connected, we deduce that there
exists $N^{\prime}_{m}$ such that, for all $t\geq\delta_{m}+2\delta_{m}^{2}$,

$\displaystyle\mathbb{P}_{x}(t<\tau_{\partial})\leq
N^{\prime}_{m}\mathbb{P}_{y}(t<\tau_{\partial}),\text{ for all $x,y\in
O_{m}$.}$

Now, for $t\leq\delta_{m}+2\delta_{m}^{2}$, we simply use the fact that
$x\mapsto\mathbb{P}_{x}(\delta_{m}+2\delta_{m}^{2}<\tau_{\partial})$ is
uniformly bounded from below on $O_{m}$ by a constant
$1/N^{\prime\prime}_{m}>0$. In particular,

$\displaystyle\mathbb{P}_{x}(t<\tau_{\partial})\leq 1\leq
N^{\prime\prime}_{m}\mathbb{P}_{y}(\delta_{m}+2\delta_{m}^{2}<\tau_{\partial})\leq
N^{\prime\prime}_{m}\mathbb{P}_{y}(t<\tau_{\partial}),\text{ for all $x,y\in
O_{m}$.}$

As a consequence, Assumption 4 is satisfied.

Finally, we deduce from Steps 1, 2, 3 and 5 that all the assumptions of
Theorem 2.8 are satisfied. This concludes the proof of Theorem 1.2. ∎

###### Proof of Lemma 4.2.

We consider a $C^{\infty}$ function $\varphi:\mathbb{R}\rightarrow[0,1]$ such
that $\varphi(x)=0$ for all $x\leq a/2$ and $\varphi(x)=1$ for all $x\geq a$.
For any $\beta>0$, we define the function $h_{\beta}$ as

$\displaystyle h_{\beta}(x)=\begin{cases}4x^{2}/a^{2}&\text{if
}x\in[0,a/2],\\\ P_{1}(x)&\text{if }x\in[a/2,a],\\\ P_{2}(x)&\text{if
}x\in[a,B_{a}],\\\ B_{a}^{\beta}(2x)^{-\beta}&\text{if }x\geq
B_{a},\end{cases}$

where

$\displaystyle P_{1}(x)=4x^{2}/a^{2}(1-\varphi(x))+\varphi(x)P_{2}(x)$

and

$\displaystyle P_{2}(x)=2^{-\beta}-\frac{\beta
2^{-\beta}}{B_{a}}(x-B_{a})+\frac{\beta(\beta+1)2^{-\beta-1}}{B_{a}^{2}}(x-B_{a})^{2}+C_{\beta}(x-B_{a})^{4},$

with

$\displaystyle C_{\beta}=\frac{-2^{-\beta}+\beta
2^{-\beta}(a/B_{a}-1)-\beta(\beta+1)2^{-\beta-1}(a/B_{a}-1)^{2}+1}{(a-B_{a})^{4}}.$

Note that the coefficients of the polynomial $P-2$ have been chosen so that
$h_{\beta}$ is $C^{2}$ at $B_{a}$ and $P_{2}(a)=1$.

There exists $M>0$ such that $C_{\beta}>0$ for all $\beta\geq M$, so that the
first and second derivative of $P_{2}$ satisfy, for all $\beta\geq M$,

$\displaystyle P^{\prime}_{2}(x)$ $\displaystyle=-\frac{\beta
2^{-\beta}}{B_{a}}+\frac{\beta(\beta+1)2^{-\beta}}{B_{a}^{2}}(x-B_{a})+4C_{\beta}(x-B_{a})^{3}\leq
0,\ \forall x\leq B_{a},$ $\displaystyle P^{\prime\prime}_{2}(x)$
$\displaystyle=\frac{\beta(\beta+1)2^{-\beta}}{B_{a}^{2}}+12C_{\beta}(x-B_{a})^{2}\geq
0,\ \forall x\in\mathbb{R}.$

In particular, $P_{2}$ is decreasing and convex on $(-\infty,B_{a}]$.
Moreover, since $P_{2}(a)=1$, we deduce that $P_{2}(x)\geq 1$ for all $x\leq
a$.

Finally, one easily deduces from the above definitions and properties that,
for all $\beta\geq M$, $h_{\beta}$ is of regularity $C^{2}$, that
$h_{\beta}(x)\geq 1$ for all $x\in[a/2,a]$, that $h_{\beta}$ is decreasing and
convex on $[a,+\infty)$. The finiteness of $M^{\prime}$ and $M^{\prime\prime}$
are simple consequences of the fact that the coefficients of $P_{2}$ are
uniformly bounded in $\beta\geq M$. This concludes the proof of Lemma 4.2. ∎

## References

  * [1] H. Brezis.  Analyse fonctionnelle.  Collection Mathématiques Appliquées pour la Maîtrise. [Collection of Applied Mathematics for the Master’s Degree]. Masson, Paris, 1983\.  Théorie et applications. [Theory and applications].
  * [2] P. Cattiaux, P. Collet, A. Lambert, S. Martínez, S. Méléard, and J. San Martín.  Quasi-stationary distributions and diffusion models in population dynamics.  Ann. Probab., 37(5):1926–1969, 2009.
  * [3] P. Cattiaux and S. Méléard.  Competitive or weak cooperative stochastic Lotka-Volterra systems conditioned on non-extinction.  J. Math. Biol., 60(6):797–829, 2010.
  * [4] N. Champagnat, K. Coulibaly-Pasquier, and D. Villemonais.  Quasi-stationary distribution for diffusion processes on manifolds.  Work in progress, 2015.
  * [5] N. Champagnat, R. Ferrière, and S. Méléard.  Unifying evolutionary dynamics: From individual stochastic processes to macroscopic evolution.  Theor. Pop. Biol., 69:297–321, 2006.
  * [6] N. Champagnat, R. Ferrière, and S. Méléard.  Individual-based probabilistic models of adaptive evolution and various scaling approximations.  In Seminar on Stochastic Analysis, Random Fields and Applications V, volume 59 of Progr. Probab., pages 75–113. Birkhäuser, Basel, 2008.
  * [7] N. Champagnat and D. Villemonais.  Exponential convergence to quasi-stationary distribution and Q-process.  Probability Theory and Related Fields, pages 1–41, 2015.
  * [8] N. Champagnat and D. Villemonais.  Exponential convergence to quasi-stationary distribution for one-dimensional diffusions.  ArXiv e-prints, June 2015.
  * [9] N. Champagnat and D. Villemonais.  Exponential convergence to quasi-stationary distribution and Q-process.  Probability Theory and Related Fields, 164(1):243–283, 2016.
  * [10] N. Champagnat and D. Villemonais.  Population processes with unbounded extinction rate conditioned to non-extinction.  ArXiv e-prints, Nov. 2016.
  * [11] N. Champagnat and D. Villemonais.  Uniform convergence to the Q-process.  ArXiv e-prints, Nov. 2016.
  * [12] J.-R. Chazottes, P. Collet, and S. Méléard.  On time scales and quasi-stationary distributions for multitype birth-and-death processes.  ArXiv e-prints, Feb. 2017.
  * [13] E. B. Dynkin.  Markov processes. Vols. I, II, volume 122 of Translated with the authorization and assistance of the author by J. Fabius, V. Greenberg, A. Maitra, G. Majone. Die Grundlehren der Mathematischen Wissenschaften, Bände 121.  Academic Press Inc., Publishers, New York, 1965.
  * [14] P. A. Ferrari, H. Kesten, and S. Martínez.  $R$-positivity, quasi-stationary distributions and ratio limit theorems for a class of probabilistic automata.  Ann. Appl. Probab., 6(2):577–616, 1996.
  * [15] K. Itô and H. P. McKean, Jr.  Diffusion processes and their sample paths, volume 125 of Die Grundlehren der mathematischen Wissenschaften.  Springer-Verlag, Berlin, 1974.  Second printing, corrected.
  * [16] N. V. Krylov and M. V. Safonov.  A property of the solutions of parabolic equations with measurable coefficients.  Izv. Akad. Nauk SSSR Ser. Mat., 44(1):161–175, 239, 1980.
  * [17] J.-F. Le Gall.  Brownian motion, martingales, and stochastic calculus, volume 274 of Graduate Texts in Mathematics.  Springer, [Cham], french edition, 2016.
  * [18] A. Lunardi.  Analytic semigroups and optimal regularity in parabolic problems, volume 16 of Progress in Nonlinear Differential Equations and their Applications.  Birkhäuser Verlag, Basel, 1995.
  * [19] S. Martinez, J. San Martin, and D. Villemonais.  Existence and uniqueness of a quasi-stationary distribution for Markov processes with fast return from infinity.  J. Appl. Probab., 2013.  to appear.
  * [20] S. Méléard and D. Villemonais.  Quasi-stationary distributions and population processes.  Probab. Surv., 9:340–410, 2012.
  * [21] S. P. Meyn and R. L. Tweedie.  Stability of Markovian processes. III. Foster-Lyapunov criteria for continuous-time processes.  Adv. in Appl. Probab., 25(3):518–548, 1993.
  * [22] L. C. G. Rogers and D. Williams.  Diffusions, Markov processes, and martingales. Vol. 1.  Cambridge Mathematical Library. Cambridge University Press, Cambridge, 2000.  Foundations, Reprint of the second (1994) edition.
  * [23] L. C. G. Rogers and D. Williams.  Diffusions, Markov processes, and martingales. Vol. 2.  Cambridge Mathematical Library. Cambridge University Press, Cambridge, 2000.  Itô calculus, Reprint of the second (1994) edition.
  * [24] D. W. Stroock and S. R. S. Varadhan.  Multidimensional diffusion processes.  Classics in Mathematics. Springer-Verlag, Berlin, 2006.  Reprint of the 1997 edition.
  * [25] A. Veretennikov.  On strong solutions and explicit formulas for solutions of stochastic integral equations.  Math USSR Sbornik, 39(3):387–403, 1981.