Lyapunov criteria for uniform convergence of conditional distributions of absorbed Markov processes

TL;DR

This paper develops Lyapunov criteria to analyze the uniform convergence of conditional distributions in absorbed Markov processes, including Lotka-Volterra and Feller diffusions, advancing understanding of their quasi-stationary behavior.

Contribution

It introduces novel non-linear Lyapunov criteria involving two functions, applicable to a broad class of Markov processes with absorption.

Findings

Criteria apply to Lotka-Volterra birth and death processes

Criteria extend to Feller diffusions with interactions

Results establish conditions for uniform convergence of conditional distributions

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.

Equations

$${\mathbb{P}}_{{\nu }_{QSD}}(X_t\in \cdot \mid t<{\tau }_{\partial })={\nu }_{QSD},\forall t\ge 0,$$

$$\Vert {\mathbb{P}}_{\mu }(X_t\in \cdot \mid t<{\tau }_{\partial })-{\nu }_{QSD}{\Vert }_{TV}\le C{e}^{-\gamma t},\forall t\ge 0,$$

$$\Vert {\mathbb{P}}_{\mu }(X_t\in \cdot \mid t<{\tau }_{\partial })-{\nu }_{QSD}{\Vert }_{TV}\le C{e}^{-\gamma t}\frac{\mu ({\phi }_1)}{\mu ({\phi }_2)},\forall t\ge 0,$$

$${\mathbb{P}}_x(X_{t_0^{\prime }}\in \cdot \mid t_0^{\prime }<{\tau }_{\partial })\ge c_1^{\prime }\nu (\cdot );$$

$${\mathbb{P}}_{\nu }(t<{\tau }_{\partial })\ge c_2^{\prime }{\mathbb{P}}_x(t<{\tau }_{\partial }).$$

$$-L\phi \le C_1\text{mathbbm}{1}_K$$

$$LV+C_2\frac{V^{1+\epsilon }}{{\phi }^{\epsilon }}\le C_3\phi$$

$$q_{n,m}=\{\begin{array}{ll}{n}_i({\lambda }_i+{\sum }_{j=1}^d{\gamma }_{ij}{n}_j)& \text{if }m=n+e_i,\text{ for some i∈{1,…,d}}\\ n_i({\mu }_i+{\sum }_{j=1}^d{c}_{ij}{n}_j)& \text{if }m=n-e_i,\text{ for some i∈{1,…,d}}\\ 0& \text{otherwise,}\end{array}$$

$$q_{n,n}:=-q_n:=-\sum _{m\ne n}{q}_{n,m}.$$

$${\Vert {\mathbb{P}}_{\mu }(X_t\in \cdot \mid t<{\tau }_{\partial })-{\nu }_{QSD}\Vert }_{TV}\le C{e}^{-\gamma t},\forall t\ge 0.$$

$$d{X}_t^i=\sqrt{{\gamma }_i{X}_t^i}d{B}_t^i+X_t^i\left(r_i-\sum _{j=1}^d{c}_{ij}{X}_t^j\right)dt,\forall i\in \{1,\dots ,d\},$$

$${\Vert {\mathbb{P}}_{\mu }(X_t\in \cdot \mid t<{\tau }_{\partial })-{\nu }_{QSD}\Vert }_{TV}\le C{e}^{-\gamma t},\forall t\ge 0.$$

$$O_n=\{x\in E:d(x,\partial )>1/(n+1)\text{ and }d(x,o)<n+1\},$$

$${\tau }_{\partial }=\inf \{t\ge 0,X_t\in \partial \}.$$

$${\tau }_{\partial }:=\underset{n\to +\infty }{\lim }{T}_n,\text{ where }{T}_n:=\inf \{t\ge 0,X_t\notin O_n\}$$

$${\mathbb{E}}_x{\int }_0^{t\wedge T_n}\left|W(X_s)\right|ds<\infty \text{ and }{\mathbb{E}}_{x}V(X_{t\wedge T_n})=V(x)+{\mathbb{E}}_x\left[{\int }_0^{t\wedge T_n}W(X_s)ds\right],$$

$$\underset{n\to +\infty }{\lim }\underset{x\in E\setminus O_n}{\inf }\frac{V(x)}{\phi (x)}=+\infty$$

$$\underset{n\to +\infty }{\lim }V(X_{T_n})=0\text{a.s.}$$

$$\frac{{\mathbb{E}}_x[LV(X_s)]}{{\mathbb{E}}_x[\phi (X_s)]}-\frac{{\mathbb{E}}_x[V(X_s)]}{{\mathbb{E}}_x[\phi (X_s)]}\frac{{\mathbb{E}}_x[L\phi (X_s)]}{{\mathbb{E}}_x[\phi (X_s)]}\in L^1([0,t])$$

$$\frac{{\mathbb{E}}_x[V(X_t)]}{{\mathbb{E}}_x[\phi (X_t)]}=\frac{V(x)}{\phi (x)}+{\int }_0^t\left\{\frac{{\mathbb{E}}_x[LV(X_s)]}{{\mathbb{E}}_x[\phi (X_s)]}-\frac{{\mathbb{E}}_x[V(X_s)]}{{\mathbb{E}}_x[\phi (X_s)]}\frac{{\mathbb{E}}_x[L\phi (X_s)]}{{\mathbb{E}}_x[\phi (X_s)]}\right\}ds.$$

$$-L\phi \le C_1\text{mathbbm}{1}_{O_{k_0}}\text{ and }LV+C_2\frac{V^{1+\epsilon }}{{\phi }^{\epsilon }}\le C_3\phi .$$

$${\mathbb{P}}_x(r_0<{\tau }_{\partial })\le p_{0}V(x),\forall x\in E\setminus O_{{\ell }_0}.$$

$${\mathbb{P}}_x(X_{{\theta }_n}\in \cdot )\ge a_n\nu ,\text{for all x∈On.}$$

$$\underset{x\in O_n}{\sup }{\mathbb{P}}_x(t<{\tau }_{\partial })\le D_n\underset{x\in O_n}{\inf }{\mathbb{P}}_x(t<{\tau }_{\partial }).$$

$$\underset{x\in E}{\sup }{\mathbb{E}}_x(e^{\lambda (S_n\wedge {\tau }_{\partial })})<\infty .$$

$$\Vert {\mathbb{P}}_{\mu }(X_t\in \cdot \mid t<{\tau }_{\partial })-{\nu }_{QSD}{\Vert }_{TV}\le C{e}^{-\gamma t},\forall t\ge 0.$$

$${\mathbb{E}}_{x}V(X_{t\wedge T_n})=V(x)+{\mathbb{E}}_x{\int }_0^{t\wedge T_n}LV(X_s)ds=V(x)+{\mathbb{E}}_x{\int }_0^{t}LV(X_s)\text{mathbbm}{1}_{s<T_n}ds.$$

$${\mathbb{E}}_{x}V(X_{t\wedge T_n})={\mathbb{E}}_x\left(\text{mathbbm}{1}_{t<T_n}V(X_t)\right)+{\mathbb{E}}_x\left(\text{mathbbm}{1}_{T_n\le t}V(X_{T_n})\right).$$

$$\underset{n\to +\infty }{\lim }{\mathbb{E}}_{x}V(X_{t\wedge T_n})={\mathbb{E}}_x\left(\text{mathbbm}{1}_{t<{\tau }_{\partial }}V(X_t)\right)={\mathbb{E}}_x(V(X_t)).$$

$${\mathbb{E}}_{x}V(X_t)\le V(x)+{\mathbb{E}}_x\left[{\int }_0^{t\wedge {\tau }_{\partial }}LV(X_s)ds\right]=V(x)+{\int }_0^t{\mathbb{E}}_x\left[LV(X_s)\right]ds,$$

Full text

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 $(X_{t},t\geq 0)$ evolving in a state space $E\cup\partial$, where $\partial\cap E=\emptyset$ and $\partial$ is absorbing. A quasi-stationary distribution for $X$ is a probability measure $\nu_{QSD}$ on $E$ such that

[TABLE]

where $\tau_{\partial}$ is the first hitting time of $\partial$ and, for any probability measure $\nu$ on $E$, $\mathbb{P}_{\nu}$ is the law of $X$ with initial distribution $\nu$.

Our goal is to provide a computational method, taking the form of a nonlinear Lyapunov type condition (sometimes also referred to as drift condition) ensuring the existence and uniqueness of a quasi-stationary distribution and the uniform convergence in total variation of the law of $X_{t}$ given $X_{t}\not\in\partial$ when $t\rightarrow+\infty$ to this quasi-stationary distribution, which means that there exist two constants $\gamma,C>0$ such that

[TABLE]

for all initial distribution $\mu$ on $E$, where $\|\cdot\|_{TV}$ is the usual total variation distance on the set of finite, signed measures on $E$, defined by $\|\mu\|_{TV}=\sup_{f\in L^{\infty}(E),\ \|f\|_{\infty}\leq 1}|\mu(f)|$. We apply this result to two standard models in ecology and evolution, called Lotka-Volterra (or logistic) birth-death or diffusion processes [27, 6, 7], which have attracted a lot of attention in the past and for which the question of uniform convergence toward a quasi-stationary distribution remains largely open in the multi-dimensional case.

Practical (linear) Lyapunov type criteria for convergence to quasi-stationary distributions were also developed in [12]. However, these results usually only entail non-uniform convergence with respect to the initial condition. In the applications we consider here, the results of Sections 4 and 5 in [12] would ensure the existence of two positive functions $\varphi_{1}\geq 1$ and $\varphi_{2}\leq 1$ on $E$ such that

[TABLE]

for all initial distribution $\mu$ on $E$. In the cases considered below, the function $\varphi_{1}$ may be taken bounded, but $\varphi_{2}$ is usually not bounded away from zero close to the boundary of $E$, which leads to a non-uniform convergence result.

As may be expected, the stronger convergence result (1.1) requires a finer control of the behavior of the process near the boundary, uniformly in $E$. As a consequence, our Lyapunov criteria are more involved than those of [12] and the techniques used here differ: in the present article, we use the control of the derivative of the continuous-time semi-group of the process to localize the conditioned process when it starts close to the boundary (see Proposition 2.3), while [12] makes use of the time-discretisation of the semi-group for this purpose. Additional arguments are also required to control the behavior of the conditioned process close to infinity. Note however that, in the end, our approach relies on the discrete time criterion of [9] and it is tempting to think that one may use a combination of the present approach and of the discrete time criterion of [12] in order to prove (1.2) with $\varphi_{2}=1$. Although this would lead to more complicated criteria than the one presented below, such an approach could also apply to processes that do not come down from infinity, such as Orstein-Uhlenbeck processes, with uniform convergence among initial distributions in bounded subsets. We leave this question for future research.

The uniform convergence (1.1) is also transferred to other properties of the process. Indeed, it provides uniformity in the time to the so-called mortality/extinction plateau (see [30]), uniform convergence of the process conditioned to late survival to the so-called $Q$-process, uniform exponential ergodicity of the $Q$-process (see [13]), and uniform convergence of $x\mapsto e^{\lambda_{0}t}\mathbb{P}_{x}(t<\tau_{\partial})$ to an eigenfunction $\eta:E\rightarrow(0,+\infty)$ for the semigroup, for some positive constant $\lambda_{0}>0$, when $t\rightarrow+\infty$ (see [9, Theorem 2.5]).

One of the main tools of our proofs is [9], where we showed that the uniform exponential convergence of conditional distributions in total variation to a unique quasi-stationary distribution is equivalent to the following conditions: there exists a probability measure $\nu$ on $E$ such that

• (A1)

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

[TABLE]

• (A2)

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

[TABLE]

Although it has the merit of generality, this criterion appears to be hard to check in practice [13, 5, 14, 11, 16, 23]. In particular, it lacks computational methods for verification. This is one of the purposes of the new criterion we present. It involves two bounded nonnegative functions $V$ and $\varphi$ such that $V(x)/\varphi(x)\rightarrow+\infty$ when $x$ converges to the boundary of $E$ or to $\infty$, satisfying

[TABLE]

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

[TABLE]

for some $\varepsilon>0$ and some constants $C_{1},C_{2},C_{3}>0$, where $L$ denotes (an extension of) the infinitesimal generator of the Markov process $X$.

We apply this criterion to Lotka-Volterra birth and death processes, and to competitive Lotka-Volterra Feller diffusion processes. The quasi-stationary behavior of (extensions of) these two models have received a lot of attention in the one-dimensional case [34, 33, 25, 29, 3, 14, 15, 22]. 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, with $d\geq 2$, and where absorption corresponds to the extinction of a single population. This means that $\partial=\mathbb{Z}^{d}_{+}\setminus\mathbb{N}^{d}$ 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. Non-uniform exponential convergence to quasi-stationary distributions for such processes can be obtained using [12], [35] or [20].

Remark 1*.*

The case where absorption corresponds to the extinction of the whole population, i.e. $\partial=\{(0,\ldots,0)\}$, can be handled combining the results of the present paper and those known in the one-dimensional case [13, 14] following the methods of [4, Thm. 1.1]. This case was also considered in [16].

A 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

[TABLE]

where $e_{i}=(0,\ldots,0,1,0,\ldots,0)$ where the 1 is at the $i$-th coordinate. Note that the set $\partial=\mathbb{Z}^{d}_{+}\setminus\mathbb{N}^{d}$ is absorbing for the process. We make the usual convention that

[TABLE]

From the biological point of view, the constant $\lambda_{i}>0$ is the birth rate per individual of type $i\in\{1,\ldots,d\}$, the constant $\mu_{i}>0$ 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 $E=\mathbb{N}^{d}$,

[TABLE]

An important difficulty here 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. Birth and death processes with bounded absorption rates are much easier to study, cf. e.g. [9]. 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 $R$-positive matrices, as exposed in [19], but without the uniform exponential convergence (1.1).

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

[TABLE]

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 $\gamma_{i}$ are positive numbers, and the growth rates per individual $r_{i}$ can be any real number, for all $1\leq i\leq d$. The 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 (1.3), 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 [4] and Section 5).

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 $E=(0,\infty)^{d}$,

[TABLE]

This results was previously known in dimension 2 when the constants $c_{ij}$ and $\gamma_{ij}$ 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 [4]). 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 [4]) by using our abstract Lyapunov criterion with functions combining those used to prove Theorems 1.1 and 1.2. Another motivations of our study comes from [1], where the coming down from infinity of Lotka-Volterra Feller diffusions is studied. It appears that such processes may go extinct far from compact sets for very large initial conditions. Theorem 1.2 proves that this does not prevent the conditioned process to come back to compact sets fast.

Our general non-linear Lyapunov criterion, Theorem 2.4, is stated in Section 2 and proved in Section 3. Sections 4 and 5 are devoted to the study of (extensions of) competitive Lotka-Volterra birth and death processes and competitive Lotka-Volterra Feller diffusions and to the proofs of Theorems 1.1 and 1.2.

Since the publication of the first version of this preprint on Arxiv, the criteria presented here have been applied to other models, including diffusion models of deadlocks in distributed algorithms [8] and stochastic reaction networks [21], with a weakened condition for birth and death processes.

2 A general Lyapunov criterion for uniform exponential convergence of conditional distributions

Our general framework is inspired from [31].

2.1 Definitions and notations

We consider a càdlàg (right continuous with left limits) time-homogeneous strong Markov process $(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\geq 0},(X_{t},t\geq 0),(\mathbb{P}_{x})_{x\in E\cup\{\partial\}})$ with state space $E\cup\partial$, which is assumed to be a metric space with distance function $d$, equipped with its Borel $\sigma$-fields and such that $E$ is measurable and $E\cap\partial=\emptyset$. In what follows, $o$ is a fixed arbitrary point in $E$ and we define the open sets

[TABLE]

for all $n\geq 1$.

We assume that $E\cap\partial=\emptyset$, that $\partial$ is an absorbing set for the process and we introduce

[TABLE]

We assume that $\tau_{\partial}<\infty$ a.s. and that the process is regularly absorbed, in the sense that

[TABLE]

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

We also assume that, for any closed set $C$, the entry time in $C$ defined by $\tau_{C}=\inf\{t\geq 0:X_{t}\in C\}$, are $({\cal F}_{t})_{t\geq 0}$-stopping times.111One can easily adapt our proofs to cases where entry times in other sets (e.g. open) are strong Markov times for the process. In particular, since $(E\cup\partial)\setminus O_{n}$ is closed, $T_{n}$ and thus $\tau_{\partial}$ are $(\mathcal{F}_{t})_{t\geq 0}$-stopping times.

We shall make use of the following weakened notion of generator for $X$, inspired from [31] and which extends the usual weak infinitesimal generator [18].

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 $x\in E$,

[TABLE]

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

Due to the local form of the above Dynkin formula, it is much easier to check that $V$ belongs to $\mathcal{D}(L)$ than to the usual domain of the weak infinitesimal generator.

We also define the set of admissible functions to which our Lyapunov criterion applies. We will say that a measurable function on $E$ or $E\cup\partial$ is locally bounded if it is bounded on $O_{n}$ for all $n\geq 1$.

Definition 2.2**.**

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, identically [math] on $\partial$, positive on $E$, and $1/V$ and $1/\varphi$ are locally bounded on $E$. *

(ii)

We have the convergences

[TABLE]

and

[TABLE]

(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.*

The main point of this definition of admissible functions is the following result, whose technical proof is postponed to Section 3.

Proposition 2.3**.**

Assume $(V,\varphi)$ is a couple of admissible functions. Then, for all $x\in E$ and $t\geq 0$,

[TABLE]

and

[TABLE]

2.2 A non-linear Lyapunov criterion

In the following assumption, we need a pair of functions, while usual drift conditions of Foster-Lyapunov criteria only use one (see for instance [31]). Roughly speaking, the function $V$ is used to control return time in compact sets from neighborhood of the boundary, while the second one, $\varphi$, is used to control the absorption rate. One of the main difficulty when checking the following assumption, is that $V$ needs also to be related to the absorption probability, via the inequality (2.7).

Assumption 1**.**

There exist constants $k_{0}\in\mathbb{N}$, $C_{1},C_{2},C_{3},\varepsilon>0$ such that

[TABLE]

and there exist constants $r_{0},p_{0}>0$ and $\ell_{0}\in\mathbb{N}$ such that

[TABLE]

The main role of the first part in the above assumption is to bound the derivative computed in Proposition 2.3, which will allow us to show that the quantity $\frac{\mathbb{E}_{x}[V(X_{t})]}{\mathbb{E}_{x}[\varphi(X_{t})]}$ is uniformly bounded in $x\in E$ for $t$ large enough. The second part of this assumption is needed to check that the boundedness of $\frac{\mathbb{E}_{x}[V(X_{t})]}{\mathbb{E}_{x}[\varphi(X_{t})]}$ implies that, conditionally on $t<\tau_{\partial}$, $X_{t}\in O_{n}$ with high probability for $n$ large. This implies that the problem can be localized in this set $O_{n}$, so that, in order to check Conditions (A1–A2), it is enough to assume the following local versions of (A1–A2), which are much easier to check.

Assumption 2**.**

There exists a probability measure $\nu$ on $E$ such that, for all $n\geq 1$, there exist $a_{n}>0$ and $\theta_{n}>0$ satisfying

[TABLE]

In addition, for all $n\geq 0$, there exists a constant $D_{n}$ such that, for all $t\geq 0$,

[TABLE]

The Lyapunov criterion of Assumption 1 will be useful to check that the conditioned process comes back quickly in bounded subsets of $E$ from any neighborhood of the boundary. However, it does not imply this property for initial distribution in a neighborhood of infinity. This is the purpose of the next assumption. Since it only concerns the unconditioned process, it may be proved using usual drift conditions or probabilistic arguments, as we illustrate in the two examples of Sections 4 and 5.

Assumption 3**.**

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

[TABLE]

where $S_{n}=\inf\{t\geq 0:X_{t}\in\overline{O_{n}}\}$.

We can now state our main general result. Its proof is given in Section 3.

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 and 3 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 $E$,

[TABLE]

3 Proof of the results of Section 2

We first give the proof of Proposition 2.3 in Subsection 3.1 and the proof of Theorem 2.4 in Subsection 3.2. The proofs of two technical lemmas are given in Subsections 3.3 and 3.4.

3.1 Proof of Proposition 2.3

In what follows, we use the classical definition of the integral of a signed function $f$ with respect to a positive measure $\mu$ by $\mu(f_{+})-\mu(f_{-})\in[-\infty,+\infty]$ (where $f_{+}$ and $f_{-}$ denote respectively the positive and negative parts of $f$), which is well defined as soon as at least one of the two terms is finite. Classical results (as Lebesgue’s theorem, Fatou’s lemma and Fubini’s theorem) still hold in this case.

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

[TABLE]

Note that

[TABLE]

Using (2.1), the Assumption (2.4) and that $V(X_{T_{n}})$ is uniformly bounded, Lebesgue’s theorem implies that

[TABLE]

Now Fatou’s lemma applied to the right-hand side of (3.1) (using that $LV$ is bounded from above) gives

[TABLE]

where we have used Fubini’s theorem for the last inequality. 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 that $V(X_{t})\in L^{1}(\Omega)$. Therefore, we can actually apply Lebesgue’s Theorem to the right-hand side of (3.1) and hence

[TABLE]

The same argument applies to $-\varphi$ (note that (2.3) and (2.4) imply that $\lim\varphi(X_{T_{n}})=0$ a.s.):

[TABLE]

Therefore $\mathbb{E}_{x}V(X_{t})$ and $\mathbb{E}_{x}\varphi(X_{t})$ are continuous with respect to $t$ and (cf. e.g. [2, 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, 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 from the continuity of $t\mapsto\mathbb{E}_{x}\varphi(X_{t})$ that $\inf_{t\in[0,T]}\mathbb{E}_{x}\varphi(X_{t})>0$. Therefore, we deduce from standard properties of $W^{1,1}$ functions [2, 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 $W^{1,1}([0,T],\mathbb{R})$ and admits as derivative

[TABLE]

Hence we have proved (2.5).

3.2 Proof of Theorem 2.4

The proof is based on two lemmas. The first one combines Proposition 2.3 and Assumption (2.6) to give uniform (in $x$) controls on $\frac{\mathbb{E}_{x}V(X_{t})}{\mathbb{E}_{x}\varphi(X_{t})}$ for $t$ large enough. Its proof is given in Subsection 3.3.

Lemma 3.1**.**

There exists two positive constants $A$ and $B$ such that, for all $x\in E$ and all $s\geq 0$,

[TABLE]

In particular, there exists $t_{0}>0$ such that, for all $x\in E$ and all $t\geq t_{0}$,

[TABLE]

The second lemma makes use of Assumption (2.7) to deduce from Lemma 3.1 the following inequality. Its proof is given in Subsection 3.4

Lemma 3.2**.**

There exist $n_{0}\geq 1$ and a constant $D>0$ such that, for all $t\geq t_{0}$ and all $x\in E$,

[TABLE]

We prove Theorem 2.4 by checking that the two conditions (A1) and (A2) in the introduction are satisfied (cf. [13]).

*Step 1: Proof of (A1).

*We first remark that there exists $m_{0}\geq 0$ such that $\nu(O_{m_{0}})>0$ (where $\nu$ and $\theta_{n}$ below are from Assumption 2) and hence such that, for all $n\geq 1$, all $x\in O_{n}$ and all $k\geq 1$,

[TABLE]

where we used Markov’s property and an induction procedure over $k$. Hence we can assume without loss of generality that, for all $n\geq 1$, $\theta_{n}\geq r_{0}$ (where $r_{0}$ is from Assumption 1).

As a consequence, Assumption (2.7), Inequality (3.6) and Markov’s property entail that, for all $t\geq t_{0}$,

[TABLE]

On the other hand,

[TABLE]

Setting $t^{\prime}_{0}=t_{0}+\theta_{n_{0}}$, the two above equations imply that, for all $x\in E$,

[TABLE]

and hence that Assumption (A1) holds true.

*Step 2: Proof of (A2).

*Using (3.7), for all $n\geq m_{0}$, all $x\in O_{n}$ and all $k\geq 1$, we obtain that, for all $t\in[\theta_{n}+k\theta_{m_{0}},\theta_{n}+(k+1)\theta_{m_{0}})$,

[TABLE]

Setting $\lambda:=-\ln(a_{m_{0}}\nu(O_{m_{0}}))/\theta_{m_{0}}$ and using inequality (2.8) of Assumption 2, we deduce that, for all $n\geq 1$ and all $s,t\geq 0$,

[TABLE]

Now, we apply Assumption 3 for $\lambda$ as defined above: there exists $n\geq m_{0}$ such that

[TABLE]

Note that, since $X_{t}$ is càdlàg, $X_{S_{n-1}}\in\overline{O}_{n-1}\subset O_{n}$ on the event $\{S_{n-1}<\infty\}$. Hence, using (3.9) and the strong Markov property at time $S_{n-1}$ (which is a stopping time since it is the entry time in a closed set), for all $x\in E$,

[TABLE]

Note that, if $S_{n-1}<\infty$, then $S_{n-1}<\tau_{\partial}$ and $S_{n-1}=S_{n}\wedge\tau_{\partial})$. Thus, for all $s\leq t$, $\mathbb{P}_{y}(S_{n-1}\in ds)=\mathbb{P}_{y}(S_{n-1}\wedge\tau_{\partial}\in ds,\,S_{n-1}<\infty)\leq\mathbb{P}_{y}(S_{n-1}\wedge\tau_{\partial}\in ds)$. Hence, using (3.8) twice, we have for all $x\in E$

[TABLE]

Since $O_{n}\supset O_{m_{0}}$, we have $\nu(O_{n})\geq\nu(O_{m_{0}})>0$, so the last inequality implies (A2).

3.3 Proof of Lemma 3.1

*Step 1: Proof of (3.4).

*Fix $x\in E$ and $s\geq 0$. On the one hand, it follows from (2.6) that

[TABLE]

and hence

[TABLE]

On the other hand, we deduce from (2.6) that

[TABLE]

where we used Hölder’s inequality to deduce that

[TABLE]

Now, because of Assumption (2.3), there exists $m$ large enough such that, for all $y\in E\setminus O_{m}$,

[TABLE]

Therefore, for such a value of $m$,

[TABLE]

Finally, we obtain from the last inequality, (3.10) and (3.11) that there exists two positive constants $A,B>0$ such that

[TABLE]

*Step 2: Proof of (3.5).

*We define $a=(2A/B)^{1/(1+\varepsilon)}$ and $t_{0}=\frac{4}{\varepsilon Ba^{\varepsilon}}$. Propositions 2.3 and (3.4) imply that, for all $t\geq 0$ and all $x\in E$,

[TABLE]

Since $\varepsilon>0$, this implies that, for all $x\in E$, there exists $u_{x}\in[0,t_{0}]$ such that $\frac{\mathbb{E}_{x}V(X_{u_{x}})}{\mathbb{E}_{x}\varphi(X_{u_{x}})}<a$ for any $x\in E$. We prove this by contradiction: assume on the contrary that for all $s\in[0,t_{0}]$, $\frac{\mathbb{E}_{x}V(X_{s})}{\mathbb{E}_{x}\varphi(X_{s})}\geq a$. Then, for all $t\in[0,t_{0}]$,

[TABLE]

Integrating this differential inequality up to time $t_{0}$ entails

[TABLE]

which gives a contradiction.

Hence, using Proposition 2.3 and (3.4) again, we deduce that, for all $t\in[t_{0},2t_{0}]$ and all $x\in E$,

[TABLE]

Using the same argument repetitively between time $kt_{0}$ (instead of [math]) and $(k+2)t_{0}$ (instead of $2t_{0}$) gives the result for all time $t\geq t_{0}$.

3.4 Proof of Lemma 3.2

Set $a^{\prime}=\left(\frac{2A}{B}\right)^{1/(1+\varepsilon)}+3At_{0}$. Equation (2.3) allows us to fix $n_{0}\geq\ell_{0}$ such that

[TABLE]

Lemma 3.1 implies that, for all $t\geq t_{0}$,

[TABLE]

Therefore,

[TABLE]

Since $a^{\prime}>\mathbb{E}_{x}V(X_{t})/\mathbb{E}_{x}\varphi(X_{t})\geq 1/\sup_{y\in E}(\varphi(y)/V(y))$, we deduce that there exists a constant $D>0$ such that

[TABLE]

4 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

[TABLE]

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, $b(n)=(b_{1}(n),\ldots,b_{d}(n))$ and $d(n)=(d_{1}(n),\ldots,d_{d}(n))$ are functions from $\mathbb{Z}_{+}^{d}$ to $(0,+\infty)^{d}$. 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 $\partial:=\mathbb{Z}_{+}^{d}\setminus\mathbb{N}^{d}$ is absorbing for the process $X$.

We define for all $k\geq 1$

[TABLE]

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

Assumption 4**.**

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

[TABLE]

and

[TABLE]

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

[TABLE]

Under Assumption (4.2), setting $W(n)=|n|$, we have

[TABLE]

This classically entails that

[TABLE]

(The argument is very similar to the one used for Lemma 3.1.) In particular, the process is non-explosive and $\tau_{\partial}$ is finite almost surely. Therefore, the process $X$ is regularly absorbed, as defined in Section 2.1. We can now state the main result of the section.

Theorem 4.1**.**

Under Assumption 4, 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-stationary distribution $\nu_{QSD}$ and there exist constants $C,\gamma>0$ such that, for all probability measure $\mu$ on $\mathbb{N}^{d}$,

[TABLE]

As will appear in the proof below, to check our conditions, it is sufficient to take functions $V$ and $\varphi$ of the form $f(|n|)\mathbbm{1}_{\mathbb{N}^{d}}(n)$ for some $f:\mathbb{N}\rightarrow\mathbb{R}_{+}$. More precisely, the first part of Condition (2.6) can be checked for $\varphi(n)=f(|n|)\mathbbm{1}_{\mathbb{N}^{d}}(n)$ with decreasing $f$ and the second part for $V(n)=g(|n|)\mathbbm{1}_{\mathbb{N}^{d}}(n)$ with increasing $g$, to take advantage of the drift of the birth and death process towards 0 for large $|n|$. However, let us emphasize that, although the choice $\varphi(n)=\mathbbm{1}_{\mathbb{N}^{d}}$ (i.e. $f\equiv 1$) is natural, it cannot satisfy the first part of Condition (2.6), since in this case $-L\varphi(n)=\sum_{i=1}^{d}\mathbbm{1}_{n_{i}=1}d_{i}(n)$ (the absorption rate) is unbounded. The direct study of $\mathbb{E}(V(X_{t})\mid t<\tau_{\partial})=\mathbb{E}[V(X_{t})]/\mathbb{E}[\mathbbm{1}_{n\in\mathbb{N}^{d}}(X_{t})]$ is possible, but only allows to recover particular cases of Theorem 4.1 (cf. [10, Section 4]). Our criterion is more flexible and better adapted to multidimensional birth and death models of interacting populations.

It is easy to check that Assumption 4 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

[TABLE]

for $C=\max_{i}\mu_{i}+\max_{i}\lambda_{i}+\max_{i,j}c_{ij}$. Under the assumptions of Theorem 1.1, there exists $C^{\prime}>0$ such that, for all $n\in\mathbb{N}^{d}$,

[TABLE]

This entails Assumption 4.

Proof of Theorem 4.1.

Using (4.3) and copying the arguments of [13, Sec. 4.1.1 and Thm. 4.1], one deduces that Assumption 2 is satisfied with $\nu=\delta_{(1,\ldots,1)}$ and that Assumption 3 is also satisfied.

Hence we only have to find a couple of admissible functions $(V,\varphi)$ satisfying Assumption 1. This couple of functions is given for all $n\in\mathbb{Z}_{+}^{d}$ by

[TABLE]

and

[TABLE]

for appropriate choices of $\alpha,\beta>1$. Note that the two functions are bounded, nonnegative and positive on $\mathbb{N}^{d}$. Note also that, since $O_{n}=\{n\in\mathbb{N}^{d},|n|\leq n+d\}$ (taking $o=(1,\ldots,1)$), so Conditions (i-ii) of Definition 2.2 are clearly satisfied (in this discrete state space case, the condition (2.4) is trivial since, almost surely, $T_{n}=\tau_{\partial}$ for all $n$ large enough). Note also that, since $\inf_{n\in\mathbb{N}^{d}}V(n)>0$, Condition (2.7) is also obviously satisfied.

Hence, we only have to check (2.6) since this of course implies that $V$ and $\varphi$ satisfy Point (iii) of Definition 2.2. So we compute

[TABLE]

where we used the fact that

[TABLE]

Hence it follows from Assumption (4.1) that there exists $\beta>1$ large enough such that $L\varphi(n)\geq 0$ for all $|n|$ large enough. This entails the first inequality in (2.6).

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

[TABLE]

and

[TABLE]

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

[TABLE]

where $C=[\alpha/(\alpha-1)]^{1+\varepsilon}[2(\beta-1)]^{\varepsilon}$. Choosing $\alpha=1+\eta/2$ and $\varepsilon=\eta/[2(\beta-1)]$, Assumption (4.2) 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 the second inequality in (2.6). ∎

5 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

[TABLE]

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 5**.**

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

[TABLE]

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

[TABLE]

This assumption implies in particular the non-explosion, strong existence and pathwise uniqueness for (5.1). Indeed, since $r_{i}$ is locally Hölder, standard arguments entail the strong existence and pathwise uniqueness for (5.1) up to the explosion time. Now, Assumption (5.2) and standard comparison results for one-dimensional diffusion processes (see e.g. Theorem 1.1 in [24, Chapter VI]) entail that each coordinate of the process can be upper bounded by the solution of the one-dimensional Feller diffusion

[TABLE]

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 [math] an exit boundary, we deduce that each coordinate is non-explosive and hence that the unique solution to (5.1) is non-explosive. Moreover, the subset $\partial=\mathbb{R}_{+}^{d}\setminus(0,+\infty)^{d}$ is an absorbing boundary for the diffusion process (5.1), so the process $X_{t}$ is regularly absorbed in the sense of Section 2.2.

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. [32]). Since the paths of $X$ are continuous, the hitting times of closed subsets of $\mathbb{R}_{+}^{d}$ are stopping times for this filtration. 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

[TABLE]

We can now state the main result of the section.

Theorem 5.1**.**

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

[TABLE]

It is straightforward to check that Assumption 5 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 5.1. Assumption 5 allows for other biologically relevant models. For instance, one can consider ecosystems where the competition among individuals only acts when the population size reaches a level $K>0$, which leads for instance to the SDE

[TABLE]

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.

Before giving the proof of Theorem 5.1, let us give some intuition about the choice of functions $V$ and $\varphi$. The behavior of the process when one of the coordinates is close to 0 is similar to the behavior of a one-dimensional diffusion absorbed at 0 started close to 0. This suggests to look for Lyapunov functions of the form $V(x)=\prod_{i=1}^{d}f(x_{i})$ and $\varphi(x)=\prod_{i=1}^{d}g(x_{i})$ where the functions $f$ and $g$ satisfy our criterion for one-dimensional diffusions like (5.4), say

[TABLE]

with generator $Af(x)=xf^{\prime\prime}(x)+x(r-x^{\eta})f^{\prime}(x)$. In the one dimensional case, Conditions (2.6) are restrictive only close to 0 and close to $+\infty$. When $x\rightarrow 0$, assuming $f(x)=x^{\alpha}$ and $g(x)=x^{\beta}$ when $x$ is close to 0, we obtain

[TABLE]

In particular, to satisfy the first part of (2.6) close to 0, we need $\beta>1$, and to satisfy the second part of (2.6), we need that

[TABLE]

in the neighborhood of 0. This holds true if $0<\alpha<1$ and $\varepsilon(\beta-\alpha)<1$. Similarly, assuming $f(x)=a-x^{-\gamma}$ and $g(x)=x^{-\delta}$ close to $+\infty$, with $a,\gamma,\delta>0$, we obtain

[TABLE]

For such functions $f$ and $g$ close to $+\infty$, the first part of (2.6) is always satisfied and the second part of (2.6) requires

[TABLE]

when $x\rightarrow+\infty$. This holds true if $\eta-\gamma>\delta\varepsilon$.

This gives the intuition to check our criterion in dimension 1222Note that more efficient criteria exist for one-dimensional diffusions (see [14]). The interest of our criterion comes from the fact that it applies to multi-dimensional processes.. The multi-dimensional case of Theorem 5.1 requires a more careful study, since we need to consider cases where some of the coordinates of the process are close to 0 or $+\infty$, and the others belong to some compact set.

Proof of Theorem 5.1.

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 5 is not modified by the rescaling (up to appropriate changes of the constants $a$, $B_{a}$, $C_{a}$ and $D_{a}$).

We divide the proof into five steps, respectively devoted to the construction of a function $\varphi$ satisfying the first inequality in (2.6), of a function $V$ satisfying the second inequality in (2.6), to the proof of (2.7), to the proof of a local Harnack inequality (needed to check Assumption 2), and the proofs of Assumption 2 and of Assumption 3.

Step 1: construction of a function $\varphi$ satisfying the first inequality in (2.6).

Recall the definition of the constants $a>0$ and $B_{a}>a$ from Assumption 5. We use the following lemma, whose proof is left to the reader (see Figure 1 for a typical graph of $h_{\beta}$).

Lemma 5.2**.**

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

[TABLE]

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

[TABLE]

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

[TABLE]

We have

[TABLE]

Now, it follows from the properties of $h_{\beta}$ and Assumptions 5 that, for all $x\in\mathbb{R}_{+}$ and all $1\leq i\leq d$,

[TABLE]

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 $\beta\geq M$,

[TABLE]

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

[TABLE]

This and Assumption (5.3) imply that

[TABLE]

for some constant $B^{\prime}$, where we used Assumption (5.2) in the last inequality.

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

[TABLE]

since $O_{n}=\{x\in[\frac{1}{n+1},+\infty)^{d},\,\sum x_{i}\leq n+1\}$ (we equip $\mathbb{R}_{+}^{d}$ with the $L^{1}$ distance). This ends the proof that $\varphi$ satisfies the first inequality in (2.6).

Step 2: construction of a function $V$ satisfying (2.6) and verification that $(V,\varphi)$ is a couple of admissible functions.

For $V$, we define

[TABLE]

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

[TABLE]

for some constants $\gamma<1$ and $\delta>0$ and where $\eta$ is defined in Assumption (5.2). 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

[TABLE]

and

[TABLE]

We deduce from Assumptions (5.2) that there exist constants $B^{\prime},B^{\prime\prime}>0$ such that

[TABLE]

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 $g(x_{i})\leq\delta$,

[TABLE]

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 (5.6) and since $V$ and $\varphi$ are positive continuous on any compact subset of $(0,+\infty)^{d}$, we have proved Condition (2.6).

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

[TABLE]

we deduce that (2.3) holds true. Condition (2.4) is also clear since $X_{T_{n}}\rightarrow X_{\tau_{\partial}}$ almost surely. 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 (2.7).

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 (5.4), 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 $t_{2}>0$,

[TABLE]

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

[TABLE]

To prove this, we can consider a scale function $s$ of the diffusion $\bar{X}^{i}$ such that $s(0)=0$ and $s(x)>0$ for $x>0$. Using the expression of the scale function and the speed measure (see e.g. [32, V.52]), one easily checks that $s(x_{i})\sim\alpha x_{i}$ when $x_{i}\rightarrow 0$ for some $\alpha\neq 0$ and that Proposition 4.9 of [14] is satisfied, so that $\mathbb{P}_{x_{i}}(\bar{X}^{i}_{t_{2}}>0)\leq Ms(x_{i})$ for some $M>0$. Since $s(x_{i})\sim\alpha x_{i}$ when $x_{i}\rightarrow 0$, (5.7) is proved and hence (2.7) holds true.

Step 4: Harnack inequality for $u$. Consider a bounded nonnegative measurable function $f$ and define the application $u:(t,x)\in\mathbb{R}_{+}\times(0,+\infty)^{d}\mapsto\mathbb{E}_{x}(f(X_{t})\mathbbm{1}_{t<\tau_{\partial}})$. 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

[TABLE]

We fix $m\geq 1$ and we will omit the indices $m$ for the constants $\delta$ and $N$ for the rest of Step 4. Let $K$ be a compact set with $C^{\infty}$ boundary such that $O_{m}\subset K\subset(0,+\infty)^{d}$ and such that $d(O_{m},\partial K)>0$. We set $\delta=d(O_{m},\partial K)/3$ and $\tau=\inf\{t\geq 0:X_{t}\in\partial K\}$. Let $x$ and $y$ be fixed in $K$ such that $|x-y|\leq\delta/2$. We define $\mu_{x}$ and $\mu_{y}$ as the joint law of $(\delta^{2}-\tau\wedge\delta^{2},X_{\tau\wedge\delta^{2}})$ starting from $X_{0}=x$ and $(2\delta^{2}-\tau\wedge(2\delta^{2}),X_{\tau\wedge(2\delta^{2})})$ starting from $X_{0}=y$, respectively. It follows from Lusin’s theorem (see e.g. [17, Thm. 7.5.2]) that there exists a sequence $(h_{n})_{n\geq 1}$ of bounded $C^{\infty}$ functions from $([0,\infty[\times\partial K)\cup(\{0\}\times K)$ to $\mathbb{R}_{+}$ such that $(h_{n})_{n\geq 1}$ converges bounded pointwisely toward $u$, $(\mu_{x}+\mu_{y})$-almost everywhere. For all $n\geq 1$, we let $u_{n}:[0,+\infty[\times K\rightarrow\mathbb{R}$ be the solution to the linear parabolic equation

[TABLE]

By [28, Thm 5.1.15], for all $n\geq 1$, $u_{n}$ is of regularity $C^{1,2}$. In particular, applying Itô’s formula to $s\mapsto u_{n}(\delta^{2}-s,X_{s})$ at time $\tau\wedge\delta^{2}$ and taking the expectation, one deduces that,

[TABLE]

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

[TABLE]

where we used the strong Markov property at time $\tau$. Similarly, $u_{n}(2\delta^{2},y)$ converges to $u(2\delta^{2},y)$.

Using the Harnack inequality provided by [26, Theorem 1.1] (with $\theta=2$ and $R=\delta$), we deduce that there exists a constant $N>0$ which does not depend on $f$, $x,y\in K$ such that $|x-y|\leq\delta/2$ nor on $n$ such that

[TABLE]

Hence, we deduce that

[TABLE]

and (5.8) is proved.

Step 5 : proof that Assumptions 2 and 3 are satisfied. Fix $x_{1}\in O_{1}$ and let $\nu$ denote the conditional law $\mathbb{P}_{x_{1}}(X_{\delta_{1}^{2}}\in\cdot\mid\delta_{1}^{2}<\tau_{\partial})$. Then the Harnack inequality (5.8) entails that, for all $x\in O_{1}$ such that $|x-x_{1}|\leq\delta_{1}/2$ and all measurable nonnegative bounded $f$ on $(0,+\infty)^{d}$,

[TABLE]

This means that

[TABLE]

Now, let $m\geq 1$. Since $O_{m}$ is bounded, connected and at a positive distance of $\partial$, $\mathbb{P}_{x}(X_{1}\in O_{1}\cap B(x_{1},\delta_{1}/2))$ is uniformly bounded from below in $O_{m}$ by a positive constant $M_{m}$. Therefore, Markov’s property implies that, for all $x\in O_{m}$,

[TABLE]

This is the first part of Assumption 2.

The second part of Assumption 2 is also a consequence of (5.8). Indeed, for any fixed $m$ and for all $t\geq 2\delta_{m}^{2}$, this equation applied to $f(x)=\mathbb{P}_{x}(t-2\delta_{m}^{2}<\tau_{\partial})$ and the Markov property entail that

[TABLE]

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

[TABLE]

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

[TABLE]

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

[TABLE]

As a consequence, the second part of Assumption 2 is satisfied.

Assumption 3 is a direct consequence of the domination by solutions to (5.4), since these solutions come down from infinity and hit [math] in finite time almost surely (cf. e.g. [3]).

Finally, we deduce from Steps 1, 2, 3 and 5 that all the assumptions of Theorem 2.4 are satisfied. This concludes the proof of Theorem 5.1. ∎