# General criteria for the study of quasi-stationarity

Nicolas Champagnat, Denis Villemonais

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Nicolas Champagnat, Denis Villemonais. General criteria for the study of quasi-stationarity. Electronic Journal of Probability, In press, 10.1214/22-EJP880. hal-01672979v3

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# General criteria for the study of quasi-stationarity

Nicolas Champagnat<fnref n="1" />, Denis Villemonais<fnref n="1" />

September 14, 2022

## Abstract

For Markov processes with absorption, we provide general criteria ensuring the existence and the exponential non-uniform convergence in weighted total variation norm to a quasi-stationary distribution. We also characterize a subset of its domain of attraction by an integrability condition, prove the existence of a right eigenvector for the semigroup of the process and the existence and exponential ergodicity of the Q-process. These results are applied to one-dimensional and multi-dimensional diffusion processes, to pure jump continuous time processes, to reducible processes with several communication classes, to perturbed dynamical systems and discrete time processes evolving in discrete state spaces.

**Keywords:** Markov processes with absorption; quasi-stationary distribution; Q-process; mixing property; diffusion processes; birth and death processes; reducible processes; perturbed dynamical systems; Galton-Watson processes.

*2010 Mathematics Subject Classification.* Primary: 37A25, 60B10, 60F99, 60J05, 60J10, 60J25, 60J27; Secondary: 60J60, 60J75, 60J80, 93E03.

## Contents

<table><tr><td><b>1 Introduction</b></td><td><b>3</b></td></tr><tr><td><b>2 Main Results</b></td><td><b>8</b></td></tr><tr><td><b>3 Other formulations and particular cases of Assumption (E)</b></td><td><b>12</b></td></tr><tr><td>3.1 General comments on the assumptions</td><td>13</td></tr><tr><td>3.1.1 Construction of Lyapunov functions satisfying (E2)</td><td>13</td></tr><tr><td>3.1.2 Checking (E1) and (E3) from comparisons between transition probabilities</td><td>14</td></tr></table>

<footnote n="1">¹Université de Lorraine, CNRS, Inria, IECL, F-54000 Nancy, France
E-mail: Nicolas.Champagnat@inria.fr, Denis.Villemonais@univ-lorraine.fr</footnote>
<table>
  <tbody>
    <tr>
      <td>3.1.3</td>
      <td>Optimal value of θ<sub>2</sub> in (E2)</td>
      <td>14</td>
    </tr>
    <tr>
      <td>3.2</td>
      <td>On continuous time</td>
      <td>14</td>
    </tr>
    <tr>
      <td>3.3</td>
      <td>The case of uniform exponential convergence</td>
      <td>17</td>
    </tr>
    <tr>
      <td><b>4</b></td>
      <td><b>Application to diffusion processes</b></td>
      <td><b>17</b></td>
    </tr>
    <tr>
      <td>4.1</td>
      <td>A general criterion in any dimension</td>
      <td>17</td>
    </tr>
    <tr>
      <td>4.2</td>
      <td>Application to uniformly elliptic diffusion processes</td>
      <td>19</td>
    </tr>
    <tr>
      <td>4.3</td>
      <td>Non-uniformly elliptic diffusions: the Feller diffusion with competition</td>
      <td>22</td>
    </tr>
    <tr>
      <td>4.4</td>
      <td>Diffusion processes with killing</td>
      <td>24</td>
    </tr>
    <tr>
      <td>4.5</td>
      <td>The case of one-dimensional diffusions</td>
      <td>25</td>
    </tr>
    <tr>
      <td><b>5</b></td>
      <td><b>Application to processes in discrete state space and continuous time</b></td>
      <td><b>31</b></td>
    </tr>
    <tr>
      <td><b>6</b></td>
      <td><b>On reducible examples</b></td>
      <td><b>36</b></td>
    </tr>
    <tr>
      <td>6.1</td>
      <td>Three successive sets</td>
      <td>36</td>
    </tr>
    <tr>
      <td>6.2</td>
      <td>Countably many communication classes</td>
      <td>42</td>
    </tr>
    <tr>
      <td><b>7</b></td>
      <td><b>Application to processes in continuous state space and discrete time</b></td>
      <td><b>45</b></td>
    </tr>
    <tr>
      <td>7.1</td>
      <td>Two sided estimates for processes with killing</td>
      <td>45</td>
    </tr>
    <tr>
      <td>7.2</td>
      <td>Perturbed dynamical systems</td>
      <td>48</td>
    </tr>
    <tr>
      <td>7.2.1</td>
      <td>The case of unbounded perturbation with bounded density</td>
      <td>48</td>
    </tr>
    <tr>
      <td>7.2.2</td>
      <td>An example with unbounded perturbation and singular density</td>
      <td>50</td>
    </tr>
    <tr>
      <td>7.2.3</td>
      <td>Two examples with bounded perturbation</td>
      <td>52</td>
    </tr>
    <tr>
      <td><b>8</b></td>
      <td><b>Irreducible processes in discrete state space and discrete time</b></td>
      <td><b>57</b></td>
    </tr>
    <tr>
      <td>8.1</td>
      <td>R-positive matrices</td>
      <td>58</td>
    </tr>
    <tr>
      <td>8.2</td>
      <td>Application to the extinction of biological populations dominated by Galton-Watson processes</td>
      <td>60</td>
    </tr>
    <tr>
      <td><b>9</b></td>
      <td><b>Proof of Theorem 2.1</b></td>
      <td><b>64</b></td>
    </tr>
    <tr>
      <td>9.1</td>
      <td>Main steps of the proof</td>
      <td>64</td>
    </tr>
    <tr>
      <td>9.2</td>
      <td>Preliminary results</td>
      <td>67</td>
    </tr>
    <tr>
      <td>9.3</td>
      <td>Proof of Proposition 9.1</td>
      <td>71</td>
    </tr>
    <tr>
      <td>9.4</td>
      <td>Proof of Proposition 9.2</td>
      <td>72</td>
    </tr>
    <tr>
      <td>9.5</td>
      <td>Proof of Proposition 9.3</td>
      <td>74</td>
    </tr>
    <tr>
      <td>9.6</td>
      <td>Proof of Lemma 9.4</td>
      <td>75</td>
    </tr>
    <tr>
      <td>9.7</td>
      <td>Conclusion of the proof of (2.1) for the norm ‖·‖<sub>TV(φ<sub>1</sub>)</sub></td>
      <td>76</td>
    </tr>
    <tr>
      <td>9.8</td>
      <td>The case where P<sub>x</sub>(n &lt; τ<sub>θ</sub>) = 0 for some x ∈ E and n ≥ 1</td>
      <td>78</td>
    </tr>
  </tbody>
</table>
<table><thead><tr><th></th><th></th></tr></thead><tbody><tr><td><b>10 Proof of the other results of Section 2</b></td><td><b>79</b></td></tr><tr><td>10.1 Proof of the existence of the eigenfunction η</td><td>79</td></tr><tr><td>10.2 Proof of Theorem 2.3</td><td>81</td></tr><tr><td>10.3 Proof of Corollary 2.4 and end of the proof of Theorem 2.2</td><td>83</td></tr><tr><td>10.4 Proof of Corollary 2.5</td><td>84</td></tr><tr><td>10.5 Proof of Corollary 2.6</td><td>85</td></tr><tr><td><b>11 Proof of the results of Section 3</b></td><td><b>85</b></td></tr><tr><td>11.1 Proof of Lemma 3.1</td><td>86</td></tr><tr><td>11.2 Proof of Lemma 3.2</td><td>86</td></tr><tr><td>11.3 Proof of Proposition 3.3</td><td>87</td></tr><tr><td>11.4 Proof of Lemma 3.4</td><td>88</td></tr><tr><td>11.5 Proof of Theorem 3.5</td><td>88</td></tr><tr><td>11.5.1 Proof of (E)</td><td>88</td></tr><tr><td>11.5.2 Existence of a quasi-stationary distribution for (X<sub>t</sub>)<sub>t∈I</sub></td><td>91</td></tr><tr><td>11.5.3 Convergence to η</td><td>93</td></tr><tr><td>11.6 Proof of Lemma 3.6</td><td>94</td></tr><tr><td>11.7 Proof of Proposition 3.7</td><td>94</td></tr><tr><td><b>12 Proof of the results of Section 4.1</b></td><td><b>96</b></td></tr><tr><td>12.1 Construction of the diffusion process X and Markov property</td><td>96</td></tr><tr><td>12.2 Harnack inequalities</td><td>101</td></tr><tr><td>12.3 Proof of Theorem 4.1</td><td>103</td></tr><tr><td>12.4 Proof of Corollary 4.2</td><td>105</td></tr></tbody></table>

# 1 Introduction

Let $(X_t, t \in I)$ be a Markov process in $E \cup \{\partial\}$ where $E$ is a measurable space and $\partial \notin E$, with set of time indices $I$ which might be $\mathbb{R}_+$ or $\frac{1}{k}\mathbb{Z}_+$ for some $k \in \mathbb{N} := \{1, 2, \dots\}$, where $\mathbb{Z}_+ := \{0, 1, \dots\}$. For all $x \in E \cup \{\partial\}$, we denote as usual by $\mathbb{P}_x$ the law of $X$ given $X_0 = x$ and for any probability measure $\mu$ on $E \cup \{\partial\}$, we define $\mathbb{P}_\mu = \int_{E \cup \{\partial\}} \mathbb{P}_x \mu(dx)$. We also denote by $\mathbb{E}_x$ and $\mathbb{E}_\mu$ the associated expectations. We assume that $\partial$ is absorbing, which means that $X_t = \partial$ for all $t \ge \tau_\partial$, $\mathbb{P}_x$-almost surely, where

$$ \tau_\partial = \inf\{t \in I, X_t = \partial\}. $$

Our goal is to study the existence of *quasi-limiting distributions* on $E$ for the process $X$, i.e. probability measures $\nu$ such that

$$ \lim_{t \in I, t \to +\infty} \mathbb{P}_\mu(X_t \in A | t < \tau_\partial) = \nu(A) $$
for some probability measure $\mu$ on $E$ and for all $A \subset E$ measurable. Such a measure $v$ is a *quasi-stationary distribution* for $X$, i.e. a probability measure such that $P_v(X_t \in \cdot | t < \tau_\partial) = v(\cdot)$ for all $t \in I$. We refer the reader to [34, 83, 104] for general introductions to quasi-stationary distributions. In particular, it is well-known that there exists a constant $\lambda_0 \ge 0$, called the decay parameter of the quasi-stationary distribution $v$, such that $P_v(t < \tau_\partial) = e^{-\lambda_0 t}$ for all $t \in I$ (for discrete time processes, i.e. $I = \mathbb{Z}_+$, the term refers to $\theta_0 = e^{-\lambda_0}$).

More precisely, our first goal is to give general criteria involving Lyapunov-type functions $\varphi_1 \ge 1$ and $\varphi_2 \le 1$ ensuring the existence of a quasi-stationary distribution $v_{QSD}$ such that

$$ \| \mathbb{P}_{\mu}(X_t \in \cdot | t < \tau_{\partial}) - v_{QSD} \|_{TV(\varphi_1)} \le C \alpha^t \frac{\mu(\varphi_1)}{\mu(\varphi_2)}, \quad \forall t \in I, \quad (1.1) $$

for some constants $C \in (0, +\infty)$ and $\alpha \in (0, 1)$ and for all probability measure $\mu$ on $E$ such that $\mu(\varphi_1) < +\infty$ and $\mu(\varphi_2) > 0$, where $\mu(\varphi) := \int_E \varphi(x) \mu(dx)$ and, for all probability measures $\mu_1$ and $\mu_2$,

$$ \| \mu_1 - \mu_2 \|_{TV(\varphi_1)} = \sup_{f:E \to \mathbb{R} \text{ measurable s.t. } |f| \le \varphi_1} |\mu_1(f) - \mu_2(f)|. $$

When $\varphi_1$ is bounded, we recover convergence for the usual total variation distance $\| \cdot \|_{TV(1)}$ since the norms $\| \cdot \|_{TV(1)}$ and $\| \cdot \|_{TV(\varphi_1)}$ are equivalent. The measure $v_{QSD}$ in (1.1) is the only quasi-stationary distribution $v$ such that $v(\varphi_1) < +\infty$ and $v(\varphi_2) > 0$.

Our second goal is to show how our criteria can be applied to a wide range of Markov processes, including several classes of processes for which even the existence of a quasi-stationary distribution was not known, such as diffusions in irregular domains or perturbed dynamical systems in unbounded domains.

General criteria ensuring that the convergence in (1.1) holds uniformly with respect to the initial distribution $\mu$ have been studied in [10, 20]. In this case, $v_{QSD}$ is the quasi-limiting distribution of any initial distributions. However, these results do not apply to processes admitting several quasi-stationary distributions, which is known to happen in a variety of specific cases, even for processes irreducible in $E$ (including branching processes [95, 2, 73, 76], one-dimensional birth and death processes [99, 47, 46, 108] and one-dimensional diffusion processes [75, 81]). In addition, as for non-absorbed processes, uniform convergence with respect to the initial distribution only happens for processes that come back quickly in compact sets [85, 20] or are killed fast [106]. The present paper provides general criteria generalizing those of [20] to cases of non-uniform convergence.
Given a quasi-stationary distribution $v$, its domain of attraction is defined as the set of probability measures $\mu$ on $E$ such that $P_\mu(X_t \in \cdot | t < \tau_\partial)$ converges in total variation norm to $v$. In the case where the domain of attraction of $v$ contains all Dirac masses, $v$ is called the *Yaglom limit*, or the *minimal quasi-stationary distribution*. In all the models admitting several quasi-stationary distributions cited above, it has been proved that the minimal quasi-stationary distribution exists. The convergence (1.1) implies in addition that the domain of attraction of the Yaglom limit $v_{QSD}$ actually contains all measures $\mu$ such that $\mu(\varphi_1) < \infty$ and $\mu(\varphi_2) > 0$.

We provide in Section 2 criteria ensuring (1.1) for all $t \in \mathbb{Z}_+$. We also obtain several consequences, including a large subset of the domain of attraction of $v_{QSD}$ and the geometric uniform convergence of $x \mapsto e^{\lambda_0 n} P_x(n < \tau_\partial)/\varphi_1(x)$ as $n \to +\infty$ to $\eta/\varphi_1$, where $\eta$ is a function which satisfies $E_x(\eta(X_n)1_{n<\tau_\partial}) = e^{-\lambda_0 n}\eta(x)$ for all $n \in \mathbb{Z}_+$ and $x \in E$. We also obtain the existence of the process $(X_n, n \in \mathbb{Z}_+)$ conditioned to never be absorbed (the so-called Q-process) and its geometric ergodicity. Links between ergodicity of the Q-processes and quasi-limiting properties were already studied in various context (see for instance [1, 53, 86, 98, 49, 89]). All these results are proved in Sections 9 and 10.

The criterion developed in Section 2 assumes that $(X_n, n \in \mathbb{Z}_+)$ is aperiodic but of course applies to 1-periodic processes $(X_t, t \in I)$. Under additional periodicity assumptions, we show in Section 3 how the previous results extend to general time indices $t \in I$ and provide practical versions of our criteria for continuous-time processes. We also provide alternative conditions allowing to check our criteria, that are easier to check in some cases. We also show that the known criteria for uniform convergence in (1.1) obtained in [20] can be recovered using this new approach. These results are proved in Section 11.

These results allow us to put in a unified framework a large body of works on quasi-stationary distributions as illustrated by the rest of the paper, which is devoted to the application of our abstract criteria. We start in Section 4 with diffusion processes in $\mathbb{R}^d$, $d \ge 1$, absorbed at the boundary of a domain $D$. Our analysis provides for example the following general result.

**Theorem 1.1.** Assume that $E = D$ is a bounded connected open subset of $\mathbb{R}^d$ and that $(X_t, t \in \mathbb{R}_+)$ is solution to

$$dX_t = b(X_t)dt + \sigma(X_t)dB_t$$

until its first exit time $\tau_\partial$ from $D$, where $B$ is a $r$-dimensional Brownian motion and $b: \mathbb{R}^d \to \mathbb{R}^d$ and $\sigma: \mathbb{R}^d \to \mathbb{R}^{d \times r}$ are Hölder functions, such that $\sigma$ is uniformly elliptic. Then, the process $X$ has a unique quasi-stationary distribution
$v_{QSD}$ which satisfies

$$ \| \mathbb{P}_{\mu}(X_t \in \cdot | t < \tau_{\partial}) - v_{QSD} \|_{TV} \le \frac{C}{\mu(\eta)} \alpha^t, \forall t \in [0, +\infty), $$

for some constants $C < +\infty$ and $\alpha \in (0, 1)$, where the function $\eta$ is $C^2(D)$ and satisfies

$$ \sum_{i=1}^{d} b_i(x) \frac{\partial \eta}{\partial x_i}(x) + \frac{1}{2} \sum_{i,j=1}^{d} \sum_{k=1}^{r} \sigma_{ik}(x) \sigma_{jk}(x) \frac{\partial^2 \eta}{\partial x_i \partial x_j}(x) = -\lambda_0 \eta(x), \quad \forall x \in D $$

and

$$ \eta(x) = \lim_{t \to +\infty} e^{\lambda_0 t} \mathbb{P}_x(t < \tau_{\partial}), \quad \forall x \in D, $$

where the convergence is uniform in $D$.

We emphasize that one of the main contributions of this result with respect to the existing literature (see for example [90, 53, 15, 70, 43, 17, 26]) is that it applies to any bounded domain $D$ without any regularity assumption, with possible applications to recent Monte-Carlo methods (see [92, 109]). Theorem 1.1 is in fact obtained in Section 4 as a particular case of a criterion for unbounded domains and coefficients $b$ and $\sigma$ only locally Hölder and locally uniformly elliptic in $D$. We also consider the case of diffusions with killing in Section 4.4. All these results are proved in Section 12.

Absorbed one-dimensional diffusions with or without killing have received a lot of attention (see for instance [78, 33, 75, 81, 96, 14, 74, 71, 61, 87, 23, 22]). We consider these models in Section 4.5. Our main contributions with respect to the literature are the characterization of a larger subset of the domain of attraction of the minimal quasi-stationary distribution, weaker regularity of the drift and diffusion coefficients and explicit general bounds on $\varphi_1$ and $\lambda_0$ allowing practical verification of our assumptions. Our criteria also provide alternative approaches to other classes of processes in continuous time and space, as those studied for example in [32, 6] using a spectral approach based on Tychonov's fixed point theorem, in [62, 49, 56, 13, 7] based on compactness or quasi-compactness properties, and in [80] for branching Markov processes using Lyapunov conditions on the conditioned semigroup.

The case of continuous-time Markov processes in discrete state spaces is considered in Section 5 with application to multitype birth and death processes absorbed at the exit of any connected $E \subset \mathbb{Z}_+^d$ (in the sense of the nearest neighbors structure of $\mathbb{Z}_+^d$). Note that the quasi-stationary behavior of finite state space processes [39] and of one-dimensional birth and death processes [67, 54, 16, 68, 99, 100] has been extensively studied using spectral methods that do not
generalize easily to the multi-dimensional countable state-space setting. The quasi-stationary behavior of multi-dimensional birth and death processes was studied in the case of uniform convergence in (1.1) in [21, 26, 30, 31].

All the previous examples assumed irreducibility of $X$ in $E$. In Section 6, we show that our criteria also apply to reducible cases, as those considered in [88] (for Galton-Watson processes), [55] (for discrete processes), [19] (for Feller diffusions) and [18, 104] (in the finite case). We first give a general criterion in Subsection 6.1 and we study in details an example with a countable infinity of communication classes in Subsection 6.2.

In Section 7, we consider general models in discrete time and continuous space, first extending the criteria of [10, 17] in order to cover the case of Euler schemes for stochastic differential equations absorbed at the boundary of a domain (as defined in [79, 51]) and penalized semigroups (as in [41, 42]; note that all our results naturally extend to penalized homogeneous semigroups, provided the penalization rate is bounded from above, see [24, 25]). We then study in details the case of perturbed dynamical systems, as those considered for example in [9, 5, 62], where the quasi-stationary behavior was studied using the criterion of [10]. As an illustration of our method, let us mention the following original result.

**Theorem 1.2.** Let $E = D$ be a measurable set of $\mathbb{R}^d$ with positive Lebesgue measure and let $\partial \notin D$. Assume that

$$X_{n+1} = \begin{cases} f(X_n) + \xi_n & \text{if } X_n \neq \partial \text{ and } f(X_n) + \xi_n \in D, \\ \partial & \text{otherwise,} \end{cases}$$

where $f: \mathbb{R}^d \to \mathbb{R}^d$ is a locally bounded measurable function such that

$$|x| - |f(x)| \xrightarrow[|x| \to +\infty]{} +\infty$$

and $(\xi_n)_{n \in \mathbb{N}}$ is an i.i.d. non-degenerate Gaussian sequence in $\mathbb{R}^d$. Then (1.1) is satisfied for $\varphi_1(x) = e^{|x|}$ and a positive measurable function $\varphi_2$ on $D$.

Finally, we study in Section 8 the case of processes in discrete time and discrete space. This is the most studied situation in the literature since it covers both the Galton-Watson processes [112, 59, 64, 2] and the general discrete case [38, 95, 47, 48, 46, 45, 55, 82]. We first show in Subsection 8.1 that our results allow to recover the general criterion of [45], based on the theory of R-positive matrices. We then consider general population processes dominated by population-dependent multi-type Galton-Watson processes in Subsection 8.2. The case of population-dependent Galton-Watson processes with a single type
was studied in [55] using quasi-compactness methods. We also obtain as a corollary several results on subcritical multi-type Galton-Watson processes. We do not recover the optimal $L \log L$ assumption on the offspring distribution [64, 60] for the existence of a minimal quasi-stationary distribution $v_{QSD}$ having finite first moment, but we obtain a stronger form of convergence in (1.1), a larger subset of its domain of attraction and stronger moments properties on $v_{QSD}$.

## 2 Main Results

Let $(X_t, t \in I)$ be a Markov process in $E \cup \{\partial\}$ where $E$ is a measurable space and $\partial \notin E$, with set of time indices $I$ which might be $\mathbb{Z}_+ = \{0, 1, \dots\}$, $\mathbb{R}_+$ or $\frac{1}{k}\mathbb{Z}_+$ for some $k \in \mathbb{N} = \{1, 2, \dots\}$. We define the absorption time $\tau_\partial$ as

$$ \tau_\partial = \inf\{t \in I, X_t = \partial\}. $$

In this section, we study the sub-Markovian transition semigroup of $X$ considered at integer times, $(P_n)_{n \in \mathbb{Z}_+}$, defined as

$$ P_n f(x) = \mathbb{E}_x (f(X_n) 1_{n < \tau_\partial}), \forall n \in \mathbb{Z}_+ $$

for all bounded or nonnegative measurable function $f$ on $E$ and all $x \in E$. We also define as usual the left-action of $P_n$ on measures as

$$ \mu P_n f = \mathbb{E}_\mu (f(X_n) 1_{n < \tau_\partial}) = \int_E P_n f(x) \mu(dx), $$

for all probability measure $\mu$ on $E$. We make the following assumption.

**Assumption (E).** There exist a positive integer $n_1$, positive real constants $\theta_1, \theta_2, c_1, c_2, c_3$, two functions $\varphi_1, \varphi_2: E \to \mathbb{R}_+$ and a probability measure $v$ on a measurable subset $K \subset E$ such that

(E1) (Local Dobrushin coefficient). $\forall x \in K$,

$$ \mathbb{P}_x(X_{n_1} \in \cdot) \geq c_1 v(\cdot \cap K). $$

(E2) (Global Lyapunov criterion). We have $\theta_1 < \theta_2$ and

$$ \inf_{x \in E} \varphi_1(x) \geq 1, \sup_{x \in K} \varphi_1(x) < \infty $$

$$ \inf_{x \in K} \varphi_2(x) > 0, \sup_{x \in E} \varphi_2(x) \leq 1, $$

$$ P_1 \varphi_1(x) \leq \theta_1 \varphi_1(x) + c_2 1_K(x), \forall x \in E $$

$$ P_1 \varphi_2(x) \geq \theta_2 \varphi_2(x), \forall x \in E. $$
(E3) (Local Harnack inequality). We have

$$ \sup_{n \in \mathbb{Z}_+} \frac{\sup_{y \in K} \mathbb{P}_y(n < \tau_\partial)}{\inf_{y \in K} \mathbb{P}_y(n < \tau_\partial)} \le c_3 $$

(E4) (Aperiodicity). For all $x \in K$, there exists $n_4(x)$ such that, for all $n \ge n_4(x)$,

$$ \mathbb{P}_x(X_n \in K) > 0. $$

Note that it follows from (E2) that $\theta_2 \le 1$ and thus $\theta_1 < 1$. We also emphasize that our assumptions neither require that $\tau_\partial < +\infty \mathbb{P}_x$-a.s., nor that $\mathbb{P}_x(n < \tau_\partial) > 0$ for all $t \ge 0$ and $x \in E$. Several examples of Markov processes satisfying Assumption (E) are provided in Sections 4 to 8.

Assumption (E) is an extension of the ergodicity criteria developed in [84]. Indeed, if we assume that $\tau_\partial = \infty \mathbb{P}_x$-almost surely for all $x \in E$, then Condition (E3) becomes void and one can take $\varphi_2 \equiv 1$ in (E2), so that $\theta_2 = \theta_0 = 1$. We recognize in (E1) the standard “small set” assumption of [84], in (E2) for $\varphi_1$ a standard Foster-Lyapunov criterion and in (E4) an aperiodicity condition. As such, it is well-known that alternative formulations of these conditions can be given. In the general case, we provide in Section 3.1 conditions ensuring the existence of Lyapunov functions satisfying (E2) in terms of exponential moment of hitting times for $\varphi_1$ and exponential decay of the probability to be in $K$ for $\varphi_2$, and conditions ensuring (E1) and (E3) based on comparisons between transition probabilities. Similarly as for the ergodicity criteria developed in [84], we extend our criterion to the continuous-time setting in Section 3.2.

In the rest of this section, we state the main general consequences of Assumption (E). We start with the exponential contraction in total variation of the conditional marginal distributions of the process given non-absorption. Its proof is given in Section 9.

**Theorem 2.1.** Assume that Condition (E) holds true. Then there exist a constant $C > 0$, a constant $\alpha \in (0, 1)$, a probability measure $v_{QSD}$ on $E$ such that $v_{QSD}(K) > 0$ and such that

$$ \left\| \frac{\mu P_n}{\mu P_n \mathbb{1}_E} - v_{QSD} \right\|_{TV(\varphi_1)} \le C \alpha^n \frac{\mu(\varphi_1)}{\mu(\varphi_2)}, \quad \forall n \ge 0, \qquad (2.1) $$

for all probability measure $\mu$ on $E$ such that $\mu(\varphi_1) < \infty$ and $\mu(\varphi_2) > 0$. In addition, $v_{QSD}$ is the unique quasi-stationary distribution satisfying $v_{QSD}(\varphi_2) > 0$ and $v_{QSD}(\varphi_1) < \infty$.
Remark 1. For all $p \ge 1$, Hölder's inequality entails

$$P_1(\varphi_1^{1/p}) \le (\theta_1 \varphi_1 + c_2 \mathbb{1}_K)^{1/p} \le \theta_1^{1/p} \varphi_1^{1/p} + c_2^{1/p} \mathbb{1}_K,$$

so that $(\varphi_1^{1/p}, \varphi_2)$ satisfies Assumption (E) for all $p < \log \theta_1 / \log \theta_2$. Therefore, the exponential convergence (2.1) actually holds true for the norm $\| \cdot \|_{TV(\varphi_1^{1/p})}$ and measures $\mu$ such that $\mu(\varphi_1^{1/p}) < +\infty$ for some $p < \log \theta_1 / \log \theta_2$.

In the following result, we show the existence of an eigenfunction $\eta$ of $P_1$ for the eigenvalue $\theta_0$, where $\theta_0 \in (0, 1]$ is such that

$$\mathbb{P}_{v_{QSD}}(n < \tau_{\partial}) = \theta_0^n, \quad \forall n \in \mathbb{N}.$$

We recall that the existence of the decay parameter $\theta_0$ is a classical general result for quasi-stationary distributions [83, 34]. The proof of the following result is initiated in Section 10.1 and concluded in Section 10.3. To state this result, we define for all positive function $\psi$ on $E$ the space $L^\infty(\psi)$ as the set of measurable real functions $f$ on $E$ such that $\|f\|_{L^\infty(\psi)} := \sup_{x \in E} f(x)/\psi(x) < \infty$. Note that $(L^\infty(\psi), \| \cdot \|_{L^\infty(\psi)})$ is a Banach space.

**Theorem 2.2.** Assume that Condition (E) holds true. Then there exists a function $\eta: E \to \mathbb{R}_+$ such that

$$\eta(x) = \lim_{n \to +\infty} \frac{\mathbb{P}_x(n < \tau_{\partial})}{\mathbb{P}_{v_{QSD}}(n < \tau_{\partial})} = \lim_{n \to +\infty} \theta_0^{-n} \mathbb{P}_x(n < \tau_{\partial}), \quad \forall x \in E, \quad (2.2)$$

where the convergence is geometric in $L^\infty(\varphi_1)$. In addition, we have $\inf_{y \in K} \eta(y) > 0$, $v_{QSD}(\eta) = 1$, $\eta \in L^\infty(\varphi_1^{\log(1/\theta_0)/\log(1/\theta_1)})$,

$$P_1 \eta = \theta_0 \eta \quad \text{and} \quad \theta_0 \ge \theta_2 > \theta_1.$$

Remark 2. In general, there is no simple relation between $\varphi_2$ and $\eta$, in particular $\varphi_2$ is not necessarily an element of $L^\infty(\eta)$. However, it is true that, for all $x \in E$, $P_k \varphi_2(x) > 0$ for some $k \ge 0$ if and only if $\eta(x) > 0$ (see Corollary 2.5 below).

Remark 3. Note that, when $\eta$ is bounded, the last result implies that one can actually take $\varphi_2 = \eta / \|\eta\|_\infty$ in Condition (E2). Results with unbounded $\varphi_2$ or $1/\varphi_1$ can also be obtained by taking the $\varphi_1$-transform of $(P_n)_{n \in \mathbb{Z}_+}$ (see [4, 25]).

We consider now the Q-process and its ergodicity properties under Condition (E). In the next result, proved in Section 10.2, $\Omega = E^{\mathbb{Z}_+}$ is the canonical state space of Markov chains on $E$ and $(\mathcal{F}_n)_{n \in \mathbb{Z}_+}$ is the associated canonical filtration. We emphasize that the constant $\alpha$ may differ from the one in Theorem 2.1. In the following result, we define

$$E' := \{x \in E, \eta(x) > 0\}.$$
**Theorem 2.3.** Condition (E) implies the following properties.

(i) Existence of the Q-process. There exists a family $(\mathbb{Q}_x)_{x \in E'}$ of probability measures on $\Omega$ defined by

$$ \lim_{n \to +\infty} \mathbb{P}_x(A | n < \tau_\partial) = \mathbb{Q}_x(A) $$

for all $x \in E'$, for all $\mathcal{F}_m$-measurable set $A$ and for all $m \ge 0$. The process $(\Omega, (\mathcal{F}_n)_{n \in \mathbb{Z}_+}, (X_n)_{n \in \mathbb{Z}_+}, (\mathbb{Q}_x)_{x \in E'}$ is an $E'$-valued homogeneous Markov chain.

(ii) Semigroup. The semigroup of the Markov process $X$ under $(\mathbb{Q}_x)_{x \in E'}$ is given for all bounded measurable function $\varphi$ on $E'$ and $n \ge 0$ by

$$ \tilde{P}_n \varphi(x) = \frac{\theta_0^{-n}}{\eta(x)} P_n(\eta \varphi)(x). \quad (2.3) $$

(iii) Exponential ergodicity. The probability measure $\beta$ on $E'$ defined by

$$ \beta(dx) = \eta(x) v_{QSD}(dx). $$

is the unique invariant distribution of the Markov process $X$ under $(\mathbb{Q}_x)_{x \in E'}$. Moreover, there exist constants $C > 0$ and $\alpha \in (0, 1)$ such that, for all initial distributions $\mu$ on $E'$ such that $\mu(\varphi_1/\eta) < \infty$ and

$$ \| \mu \tilde{P}_n - \beta(h) \|_{TV(\varphi_1/\eta)} \le C \alpha^n \mu(\varphi_1/\eta), \quad \forall n \ge 0, \quad (2.4) $$

where $\mathbb{Q}_\mu = \int_{E'} \mathbb{Q}_x \mu(dx)$. In addition, for all initial distributions $\mu$ on $E'$,

$$ \| \mu \tilde{P}_n - \beta \|_{TV} \xrightarrow[n \to \infty]{} 0. \quad (2.5) $$

We conclude this section with corollaries of the last theorem. The following result is proved in Section 10.3.

**Corollary 2.4.** Assume that Condition (E) holds true. Then there exist constants $C > 0$ and $\alpha \in (0, 1)$ such that, for all probability measure $\mu$ on $E$ such that $\mu(\varphi_1) < +\infty$,

$$ \| \theta_0^{-n} \mu P_n - \mu(\eta) v_{QSD} \|_{TV(\varphi_1)} \le C \alpha^n \mu(\varphi_1). \quad (2.6) $$

*Remark 4.* The proof of Theorem 2.3 makes use of [57, 58], which allows to derive explicit expressions for the constants $C$ and $\alpha$ (we refer the interested reader to Remark 22). In particular, using these estimates in the proof of Corollary 2.4 would also provide explicit constants in (2.6).
Remark 5. The formulation (2.6) for the convergence of the semigroup is natural in this setting, since a property of equivalence between (2.6) and Condition (E) is proved in [4, 25].

The last corollary has consequences on the attraction domain of $v_{QSD}$.

**Corollary 2.5.** Assume that Condition (E) holds true. Then

$$E' = \{x \in E : \exists k \ge 0, P_k \varphi_2(x) > 0\}$$

and the domain of attraction of $v_{QSD}$ for the total variation norm contains all probability measures on E such that $\mu(E') > 0$ and $\mu(\varphi_1^{1/p}) < +\infty$ for some $p < \log\theta_1 / \log\theta_2$. If in addition $\varphi_1$ is bounded, then the domain of attraction of $v_{QSD}$ is the set of probability measures on E such that $\mu(E') > 0$ and $v_{QSD}$ is the unique quasi-stationnary distribution giving positive mass to $E'$.

Convergence estimates can also be obtained for initial distributions on $E'$ satisfying $\mu(\eta) < +\infty$ but not necessarily $\mu(\varphi_1) < +\infty$. The following result is proved in 10.5.

**Corollary 2.6.** Assume that Condition (E) holds true. Then, for all probability measures $\mu$ on $E'$ such that $\mu(\eta) < +\infty$,

$$\|\theta_0^{-n} \mu P^n - \mu(\eta) v_{QSD}\|_{TV(\eta)} \xrightarrow{n \to +\infty} 0. \qquad (2.7)$$

In particular, if $\eta$ is positive on E, then $v_{QSD}$ is the unique quasi-stationnary distribution of X such that $v_{QSD}(\eta) < +\infty$. If in addition $\eta$ is lower bounded away from 0 on E, then for all probability measures $\mu$ on E such that $\mu(\eta) < +\infty$, we have

$$\|\mathbb{P}_{\mu}(X_n \in \cdot | n < \tau_{\partial}) - v_{QSD}\|_{TV(\eta)} \xrightarrow{n \to +\infty} 0. \qquad (2.8)$$

In particular, the domain of attraction of $v_{QSD}$ contains all probability measures $\mu$ on E such that $\mu(\eta) < +\infty$.

# 3 Other formulations and particular cases of Assumption (E)

In this section, we provide general comments on Assumption (E). Alternative formulations of our assumptions and simple criteria are gathered in Subsection 3.1. Subsection 3.2 focuses on criteria adapted to continuous time processes and we consider the case of uniform convergence in Theorem 2.1 in Subsection 3.3.
## 3.1 General comments on the assumptions

We propose here alternative formulations of Condition (E2) and criteria ensuring (E1) and (E3) when (E2) and (E4) are satisfied, that may be easier to check in some practical situations. In particular, we make strong use of these results in Sections 7 and 8.

### 3.1.1 Construction of Lyapunov functions satisfying (E2)

In order to prove the existence of functions $\varphi_1$ and $\varphi_2$ in Condition (E2), one may use probabilistic properties of the Markov process $X$, as stated in the following lemmas, proved in Sections 11.1 and 11.2. The first lemma shows a way to construct $\varphi_2$.

**Lemma 3.1.** Let $K$ be a measurable subset of $E$. If there exists $\theta_2 > 0$ such that

$$ \inf_{x \in K} \theta_2^{-n} \mathbb{P}_x(X_n \in K) \xrightarrow{n \to +\infty} +\infty, $$

then the function $\varphi_2: E \to [0, 1]$ defined by $\varphi_2(x) = \frac{\theta_2^{-1}-1}{\theta_2^{-\ell}-1} \sum_{k=0}^{\ell-1} \theta_2^{-k} \mathbb{P}_x(X_k \in K)$, for any $\ell$ is such that $\theta_2^{-\ell} \inf_{x \in K} \mathbb{P}_x(X_\ell \in K) \ge 1$, verifies $\inf_K \varphi_2 > 0$ and $P_1 \varphi_2(x) \ge \theta_2 \varphi_2(x)$. Moreover, (E4) is satisfied.

The second lemma shows how $\varphi_1$ can be constructed. This is a well-known result in the case without absorption [84], which can provide easier ways to check (E2) in some situations. We define

$$ T_K = \inf\{n \in \mathbb{Z}_+, X_n \in K\}. \qquad (3.1) $$

**Lemma 3.2.** Let $K$ be a measurable subset of $E$. If there exists a constant $\theta_1 > 0$ such that

$$ \mathbb{E}_x \left( \theta_1^{-T_K \wedge \tau_\delta} \right) < +\infty \quad \forall x \in E \text{ and } C := \sup_{y \in K} \mathbb{E}_y \left( \mathbb{E}_{X_1} \left( \theta_1^{-T_K \wedge \tau_\delta} \right) 1_{1<\tau_\delta} \right) < +\infty, $$

then the function $\varphi_1: E \to [1, +\infty)$ defined by $\varphi_1(x) = \mathbb{E}_x \left( \theta_1^{-T_K \wedge \tau_\delta} \right)$ satisfies

$$ \sup_K \varphi_1 < +\infty \quad \text{and} \quad P_1 \varphi_1 \le \theta_1 \varphi_1 + \frac{C}{\theta_1} 1_K. $$

Conversely, if there exist two constants $C > 0$, $\theta_1 > 0$ and a function $\varphi_1: E \to [1, +\infty)$ such that $\sup_K \varphi_1 < +\infty$ and $P_1 \varphi_1 \le \theta_1 \varphi_1 + C 1_K$, then, for all $\theta > \theta_1$, there exists a constant $C_\theta$ such that

$$ \mathbb{E}_x \left( \theta_1^{-T_K \wedge \tau_\delta} \right) \le C_\theta \varphi_1(x) \quad \forall x \in E \text{ and } \sup_{y \in K} \mathbb{E}_y \left( \mathbb{E}_{X_1} \left( \theta_1^{-T_K \wedge \tau_\delta} \right) 1_{1<\tau_\delta} \right) < +\infty. $$
Note that the hitting time $T_K$ is defined from the process $(X_n)_{n \in \mathbb{Z}_+}$. When $I \neq \mathbb{Z}_+$, it might be easier to use criteria based on the hitting time $\tau_K$ defined from the full process $(X_n)_{n \in I}$. We refer the reader to Lemma 3.6 below for that.

### 3.1.2 Checking (E1) and (E3) from comparisons between transition probabilities

Condition (E3) is a form of Harnack inequality, and one can indeed use general versions of these inequalities to check (E3) and (E1) (for example, our results on diffusions given in Section 4 use this idea, cf. Section 12.2). We propose below another criterion, based on comparison techniques on transition probabilities, to check that Conditions (E1) and (E3) hold true when Conditions (E2) and (E4) are satisfied. This result is proved in Subsection 11.3.

**Proposition 3.3.** Assume that Conditions (E2) and (E4) are satisfied and that there exist two constants $C > 0$ and $n_0 \le m_0 \in \mathbb{N}$ such that

$$ \mathbb{P}_x(X_{n_0} \in \cdot \cap K) \le C \mathbb{P}_y(X_{m_0} \in \cdot), \forall x \in E \text{ and } y \in K. \quad (3.2) $$

Then Condition (E) is satisfied.

### 3.1.3 Optimal value of $\theta_2$ in (E2)

As many results of Section 2 make use of the function $\phi_1^{1/p}$ with a parameter $p \in [1, \log\theta_1/\log\theta_2)$, it is important to characterize the largest possible value of $\theta_2$. This result is proved in Section 11.4.

**Lemma 3.4.** If Condition (E) is satisfied for some functions $\phi_1$ and $\phi_2$ with constants $\theta_1$ and $\theta_2$, then, for all $\theta'_2 \in (\theta_1, \theta_0)$ it is also satisfied for $\phi_1$ and some function $\phi'_2$ with constants $\theta_1$ and $\theta'_2$.

## 3.2 On continuous time

In Section 2, we only considered the conditional behavior of the process $X$ at integer times. In general, the results of Section 2 do not give information about the process at intermediate times. In this section, we derive a sufficient condition which is well suited for practical verification in the case of continuous time Markov processes or for aperiodic Markov processes, in particular because (F2) below is usually easier to check than (E2). We consider an absorbed Markov process $(X_t)_{t \in I}$ with time parameter in $I = \mathbb{Z}_+$ or $[0, +\infty)$.

**Assumption (F).** There exist positive real constants $\gamma_1, \gamma_2, c_1, c_2$ and $c_3$, $t_1, t_2 \in I$, a measurable function $\psi_1: E \to [1, +\infty)$, and a probability measure $\nu$ on a measurable subset $L \subset E$ such that
(F0) (A strong Markov property). Defining

$$ \tau_L := \inf\{t \in I : X_t \in L\}, \qquad (3.3) $$

assume that for all $x \in E$, $X_{\tau_L} \in L$, $\mathbb{P}_x$-almost surely on the event $\{\tau_L < \infty\}$ and for all $t \in I$ and all measurable $f: E \cup \{\partial\} \to \mathbb{R}_+$,

$$ \mathbb{E}_x[f(X_t)1_{\tau_L \le t < \tau_\partial}] = \mathbb{E}_x[1_{\tau_L \le t \wedge \tau_\partial} \mathbb{E}_{X_{\tau_L}}[f(X_{t-u})1_{t-u < \tau_\partial}]|_{u=\tau_L}]. $$

(F1) (Local Dobrushin coefficient). $\forall x \in L$,

$$ \mathbb{P}_x(X_{t_1} \in \cdot) \ge c_1 \nu(\cdot \cap L). $$

(F2) (Global Lyapunov criterion). We have $\gamma_1 < \gamma_2$ and

$$ \mathbb{E}_x(\psi_1(X_{t_2})1_{t_2 < \tau_L \wedge \tau_\partial}) \le \gamma_1^{t_2} \psi_1(x), \forall x \in E $$

$$ \mathbb{E}_x(\psi_1(X_t)1_{t < \tau_\partial}) \le c_2, \forall x \in L, \forall t \in [0, t_2] \cap I, $$

$$ \gamma_2^{-t} \mathbb{P}_x(X_t \in L) \xrightarrow{t \to +\infty} +\infty, \forall x \in L. $$

(F3) (Local Harnack inequality). We have

$$ \sup_{t \ge 0} \frac{\sup_{y \in L} \mathbb{P}_y(t < \tau_\partial)}{\inf_{y \in L} \mathbb{P}_y(t < \tau_\partial)} \le c_3 $$

Be careful that the definition of $\tau_L$ in (3.3) is different from that of $T_L$ in (3.1). Note also that, in (F2), the Lyapunov function $\varphi_2$ has been replaced by an alternative condition similar to Lemma 3.1. Both are actually equivalent thanks to (F0) (see the beginning of Section 11.5.1).

The following result is proved in Section 11.5.

**Theorem 3.5.** Under Assumption (F), $(X_t)_{t \in I}$ admits a quasi-stationary distribution $v_{QSD}$, which is the unique one satisfying $v_{QSD}(\psi_1) < \infty$ and $v_{QSD}(L) > 0$ for some $t \in I$. Moreover, there exist constants $\alpha \in (0, 1)$ and $C > 0$ such that, for all probability measures $\mu$ on E satisfying $\mu(\psi_1) < \infty$ and $\mu(\psi_2) > 0$,

$$ \| \mathbb{P}_{\mu}(X_t \in \cdot | t < \tau_{\partial}) - v_{QSD} \|_{TV}(\psi_1) \le C \alpha^t \frac{\mu(\psi_1)}{\mu(\psi_2)}, \forall t \in I, \qquad (3.4) $$

where $\psi_2(x) = \sum_{k=0}^{n_0} \gamma_2^{-kt_2} \mathbb{P}_x(X_{kt_2} \in L)$ for some $n_0 \ge 1$ large enough. In addition, there exists a constant $\lambda_0 \ge 0$ such that $\lambda_0 \le \log(1/\gamma_2) < \log(1/\gamma_1)$ and $\mathbb{P}_{v_{QSD}}(t < \tau_\partial) = e^{-\lambda_0 t}$ for all $t \ge 0$, and there exists a function $\eta$ such that

$$ \eta(x) = \lim_{t \to +\infty} e^{\lambda_0 t} \mathbb{P}_x(t < \tau_\partial), \quad \forall x \in E, \qquad (3.5) $$

where the convergence is exponential in $L^\infty(\psi_1^{1/p})$ for all $p \in [1, \log(1/\gamma_1)/\lambda_0)$, and $P_t \eta(x) = e^{-\lambda_0 t} \eta(x)$ for all $x \in E$ and $t \in I$.
A key point that guided our formulation of Condition (F) is that, for continuous-time Markov processes, usual practical conditions for the existence of $\psi_1$ are provided by Foster-Lyapunov inequalities (cf. [84]). They involve the extended infinitesimal generator $\mathcal{L}$ of the process $X$ (see e.g. [84, 26]) and take the form

$$ \mathcal{L}\psi_1(x) \leq -\lambda_1\psi_1(x) + C1_K(x), \quad \forall x \in E. \qquad (3.6) $$

This inequality does not imply, in general, that (E2) holds true for $\varphi_1 = \psi_1$. However, Equation (3.6) implies (formally, assuming one can apply Dynkin's formula) that $\mathbb{E}_x[1_{1\leq\tau_L\wedge\tau_\partial}\psi_1(X_1)] \leq e^{-\lambda_1}\psi_1(x)$ and $\mathbb{E}_x[\psi_1(X_t)1_{t<\tau_\partial}] \leq e^{ct}\psi_1(x)$. Hence the first two lines of (F2) can be deduced from classical Foster Lyapunov criteria. This will be used for diffusion processes in Section 4 or in discrete state space in Section 5.

Alternatively, one can use controls on the exponential moments for the return times in $L$. The following result, similar to Lemma 3.2, is proved in Section 11.6.

**Lemma 3.6.** Assume that there exist positive constants $\gamma_1 > 0$ and $t_2 \in I$ such that

$$ \mathbb{E}_x(\gamma_1^{-\tau_L \wedge \tau_\partial}) < \infty, \forall x \in E \quad \text{and} \quad \sup_{x \in L} \mathbb{E}_x(\mathbb{E}_{X_{t_2}}(\gamma_1^{-\tau_L \wedge \tau_\partial}) 1_{\{t_2 < \tau_\partial\}}) < +\infty, $$

then $\psi_1(x) = \mathbb{E}_x(\gamma_1^{-\tau_L \wedge \tau_\partial})$ satisfies

$$ \mathbb{E}_x(\psi_1(X_{t_2}) 1_{\{t_2 < \tau_L \wedge \tau_\partial\}}) \leq \gamma_1^{t_2} \psi_1(x), \forall x \in E $$

$$ \mathbb{E}_x(\psi_1(X_t) 1_{\{t < \tau_\partial\}}) \leq c_2, \forall x \in L, \forall t \in [0, t_2] \cap I, $$

for some constant $c_2 > 0$.

**Remark 6.** In the proof of Theorem 3.5, we will show that Assumption (F) implies that Assumption (E) is satisfied for the sub-Markovian semigroup $(P_n)_{n \geq 0}$ of the absorbed Markov process $(X_{nt_2})_{n \in \mathbb{Z}_+}$, with the functions $\varphi_1 = \psi_1$ and $\varphi_2 =$

$$ \frac{\gamma_2^{-t_2} - 1}{\gamma_2^{-(n_0+1)t_2} - 1} \psi_2, \text{ any } \theta_1 \in (\gamma_1^{t_2}, \gamma_2^{t_2}), \theta_2 = \gamma_2^{t_2} \text{ and the set} $$

$$ K = \{y \in E, \mathbb{P}_y(\tau_L \leq t_2) / \psi_1(y) \geq (\theta_1 - \gamma_1^{t_2}) / c_2\} \supset L. $$

In particular, all the consequences of (E) stated in Section 2 hold true. Moreover, it is also possible to obtain a continuous-time version of Theorem 2.3 about the Q-process by adapting the proof given in Section 10.2.

**Remark 7.** If $I = \mathbb{R}_+$, it follows from the fact that $P_t\eta = e^{-\lambda_0 t}\eta$ that, setting $\eta(\partial) = 0$, the function $\eta$ defined on $E \cup \{\partial\}$ belongs to the domain of the infinitesimal generator $\mathcal{L}$ of the semigroup of the Markov process $X$ on $E \cup \{\partial\}$, seen as acting on $L^\infty(\psi_1^{1/p})$ for $p \in [1, \log(1/\gamma_1)/\lambda_0)$, and $\mathcal{L}\eta = -\lambda_0\eta$.
### 3.3 The case of uniform exponential convergence

We now want to characterize the case of exponential convergence in total variation of the conditional distributions of $(X_n)$ to $v_{QSD}$, uniformly with respect to the initial distribution $\mu$. This question was already studied in [20]. The next result, proved in Section 11.7, gives a necessary and sufficient condition based on Condition (E).

**Proposition 3.7.** There exists constants $C$ and $\alpha < 1$ such that, for all probability measure $\mu$ on $E$ and all integer $n$,

$$ \| \mathbb{P}_{\mu}(X_n \in \cdot | n < \tau_{\partial}) - v_{QSD} \|_{TV} \le C\alpha^n, \qquad (3.7) $$

if and only if Condition (E) is satisfied with a bounded function $\varphi_1$ and there exists an integer $n'_4 > 0$ such that

$$ \underline{c} := \inf_{x \in E} \mathbb{P}_x(X_{n'_4} \in K | n'_4 < \tau_{\partial}) > 0. \qquad (3.8) $$

## 4 Application to diffusion processes

In this section, we apply the criteria (E) and (F) to diffusion processes absorbed at the boundary of a domain. We give a general criterion in Subsection 4.1 and apply it to uniformly elliptic diffusions in Subsection 4.2 and to an example with vanishing diffusion coefficient at the boundary of the domain in Subsection 4.3. Our criteria are extended to diffusions with killing in Subsection 4.4 and the particular case of one-dimensional diffusions is studied in Subsection 4.5.

### 4.1 A general criterion in any dimension

We consider a diffusion process $X$ on a connected, open domain $D \subset \mathbb{R}^d$ for some $d \ge 1$, solution to the SDE

$$ dX_t = b(X_t)dt + \sigma(X_t)dB_t, \qquad (4.1) $$

where $B$ is a standard, $r$-dimensional Brownian motion and $b: D \to \mathbb{R}^d$ and $\sigma: D \to \mathbb{R}^{d \times r}$ are locally Hölder functions, such that $\sigma$ is locally uniformly elliptic in $D$, i.e.

$$ \forall K \subset D \text{ compact, } \inf_{x \in K} \inf_{s \in \mathbb{R}^d \setminus \{0\}} \frac{s^* \sigma(x) \sigma^*(x) s}{|s|^2} > 0, $$

where $|\cdot|$ is the standard Euclidean norm on $\mathbb{R}^d$. We assume that the process is immediately absorbed at some cemetery point $\partial \notin D$ at its first exit time of
$D$, denoted $\tau_\partial$. The existence and basic properties of this process need some care since the coefficients $b$ and $\sigma$ are only defined in the open set $D$ without any assumption on the boundary of $D$, and so may not be possible to extend as continuous functions out of this set. Details are given in Subsection 12.1. For the moment, let us only observe that, for all $k \ge 1$, defining the compact set

$$K_k = \{x \in D : |x| \le k \text{ and } d(x, D^c) \ge 1/k\},$$

a weak solution to (4.1) can be constructed up to the first exit time $\tau_{K_k^c}$ of $K_k$ as defined in (3.3). The proper definition of the absorption time $\tau_\partial$ is

$$\tau_\partial = \sup_{k \ge 1} \tau_{K_k^c}. \qquad (4.2)$$

We introduce the differential operator associated to the SDE (4.1), related to the infinitesimal generator of the process $X$: for all $f \in \mathcal{C}^2(D)$, we define for all $x \in D$

$$\mathcal{L} f(x) := \sum_{i=1}^{d} b_i(x) \frac{\partial f}{\partial x_i}(x) + \frac{1}{2} \sum_{i,j=1}^{d} \sum_{k=1}^{r} \sigma_{ik}(x) \sigma_{jk}(x) \frac{\partial^2 f}{\partial x_i \partial x_j}(x). \quad (4.3)$$

We also define the constant

$$\lambda_0 := \inf \left\{ \lambda > 0, \text{ s.t. } \liminf_{t \to +\infty} e^{\lambda t} \mathbb{P}_x(X_t \in B) > 0 \right\} \quad (4.4)$$

for some $x \in D$ and some open ball $B$ such that $\bar{B} \subset D$. It is standard to prove using Harnack inequalities (proved in our case in Section 12.2) that, under the previous assumptions, $\lambda_0 < +\infty$ and its value is independent of the choice of $x \in D$ and of the non-empty, open ball $B$ such that $\bar{B} \subset D$.

The following result is proved in Sections 12.1 to 12.3.

**Theorem 4.1.** Assume that there exist some constants $C > 0$, $\lambda_1 > \lambda_0$, a $\mathcal{C}^2(D)$ function $\varphi: D \to [1, +\infty)$ and a subset $D_0 \subset D$ closed in $D$ such that $\sup_{x \in D_0} \varphi(x) < +\infty$ and

$$\mathcal{L}\varphi(x) \le -\lambda_1\varphi(x) + C \mathbb{1}_{x \in D_0}, \quad \forall x \in D. \qquad (4.5)$$

Assume also that there exists a time $s_1 > 0$ such that

$$\sup_{x \in D_0} \mathbb{P}_x(s_1 < \tau_{K_k} \wedge \tau_\partial) \xrightarrow{k \to \infty} 0. \qquad (4.6)$$

Then $X$ admits a quasi-stationary distribution $v_{QSD}$ which satisfies $v_{QSD}(\varphi^{1/p}) < +\infty$ for all $p > 1$. Moreover, for all $p \in (1, \lambda_1/\lambda_0)$, there exist a constant $\alpha_p \in (0, 1)$,
a constant $C_p$ and a function $\varphi_{2,p}: D \to (0, +\infty)$ uniformly bounded away from 0 on compact subsets of D such that, for all probability measures $\mu$ on E satisfying $\mu(\varphi^{1/p}) < \infty$,

$$ \| \mathbb{P}_{\mu}(X_t \in \cdot | t < \tau_{\partial}) - v_{QSD} \|_{TV(\varphi^{1/p})} \le C_p \alpha_p^t \frac{\mu(\varphi^{1/p})}{\mu(\varphi_{2,p})}, \quad \forall t \in [0, +\infty). $$

In particular, $v_{QSD}$ is the only quasi-stationary distribution of X which satisfies $v_{QSD}(\varphi^{1/p}) < +\infty$ for at least one value of $p \in (1, \lambda_1/\lambda_0)$.

**Remark 8.** Note that $\tau_{K_k} = 0$ $\mathbb{P}_x$-a.s. for all $x \in K_k$, thus

$$ \sup_{x \in D_0} \mathbb{P}_x(s_1 < \tau_{K_k} \wedge \tau_{\partial}) = \sup_{x \in D_0 \setminus K_k} \mathbb{P}_x(s_1 < \tau_{K_k} \wedge \tau_{\partial}). $$

Hence Condition (4.6) requires the process to be absorbed or return in $K_k$ fast starting in $D_0 \setminus K_k$.

**Remark 9.** We shall actually prove that, under the conditions of the previous theorem, Assumption (F) is satisfied with $L = K_k$ for some $k \ge 1$, and $\psi_1 = \varphi^{1/p}$, for any $p \in (1, \lambda_1/\lambda_0)$.

**Remark 10.** In general, the assumptions of Theorem 4.1 do not ensure the non-explosion of the Markov process X. In the case of an explosive Markov process, the definition of $\tau_{\partial}$ in (4.2) implies that, in the event of an explosion, the absorption time $\tau_{\partial}$ is defined as equal to the explosion time.

The last result has other consequences of interest, gathered in the next corollary, proved in Section 12.4.

**Corollary 4.2.** Under the assumptions of Theorem 4.1, the infimum defining the constant $\lambda_0$ in (4.4) is actually a minimum and it satisfies $\mathbb{P}_{v_{QSD}}(t < \tau_{\partial}) = e^{-\lambda_0 t}$ for all $t \ge 0$. In addition, the function $\eta$ of Theorem 3.5 satisfies $P_t\eta = e^{-\lambda_0 t}\eta$ for all $t \ge 0$. In particular, $\eta$ belongs to the domain of the infinitesimal generator of the semigroup of the process X defined as acting on the Banach space $L^{\infty}(\varphi_1)$, and it is an eigenfunction for the eigenvalue $-\lambda_0$. In addition, $\eta \in C^2(D)$ and $\mathcal{L}\eta(x) = -\lambda_0\eta(x)$ for all $x \in D$.

## 4.2 Application to uniformly elliptic diffusion processes

We consider the case where $\sigma$ can be extended to $\mathbb{R}^d$ as a locally uniformly elliptic matrix-valued function. In the following corollary, we give a general situation where (4.6) holds true. We emphasize that, contrary to previous results on existence of quasi-stationary distributions for diffusions in a domain (see e.g. [90, 53, 70, 43, 17]), no regularity on the boundary of D is required.
**Corollary 4.3.** Let D be an open connected subset of $\mathbb{R}^d$, $d \ge 1$. Let X be solution to the SDE

$$dX_t = b(X_t)dt + \sigma(X_t)dB_t, \ t < \tau_{\partial}, \qquad (4.7)$$

where $b: \mathbb{R}^d \to \mathbb{R}^d$ and $\sigma: \mathbb{R}^d \to \mathbb{R}^{d \times r}$ are locally Hölder continuous in $\mathbb{R}^d$ and $\sigma$ is locally uniformly elliptic on $\mathbb{R}^d$. Recall the definition (4.4) of $\lambda_0$ and assume that there exist constants $C > 0$, $\lambda_1 > \lambda_0$, a $\mathcal{C}^2(D)$ function $\varphi: D \to [1, +\infty)$ and a bounded subset $D_0 \subset D$ closed in D such that

$$\mathcal{L}\varphi(x) \le -\lambda_1\varphi(x) + C1_{x \in D_0}, \quad \forall x \in D. \qquad (4.8)$$

Then the process X absorbed at the boundary of D (in the sense of (4.2)) satisfies the assumptions of Theorem 4.1.

Note that we do not assume that $\varphi(x) \to +\infty$ when $|x| \to +\infty$, hence the process X may be explosive (see Remark 10).

*Proof.* Let us consider the diffusion process Y solution to (4.7) on $\mathbb{R}^d$. Due to our regularity assumptions on b and $\sigma$, this process is well-defined up to a possibly finite explosion time $\tau_{\text{expl}}$. The Harnack inequality (12.6) applied to Y on the compact set $\bar{D}_0$ ensures the existence of constants $\delta > 0$ and N such that, for all $f: \mathbb{R}^d \to [0, 1]$, for all $x \in \bar{D}_0$ and all $y \in B(x, \delta)$,

$$\mathbb{E}_x[1_{\delta+\delta^2<\tau_{\text{expl}}} f(Y_{\delta+\delta^2})] \le N \mathbb{E}_y[1_{\delta+2\delta^2<\tau_{\text{expl}}} f(Y_{\delta+2\delta^2})].$$

By compactness of $\bar{D}_0$, there exist a positive integer $n$ and $y_1, \dots, y_n \in D_0$ such that $\bar{D}_0 \subset \bigcup_{i=1}^n B(y_i, \delta)$. Setting $s_1 = \delta + \delta^2$, we deduce that, for all $k \ge 1$ and all $x \in D_0$,

$$\mathbb{P}_x(Y_{s_1} \in D \setminus K_k) \le N \max_{1 \le i \le n} \mathbb{P}_{y_i} (Y_{s_1+\delta^2} \in D \setminus K_k) \xrightarrow{k \to +\infty} 0.$$

Hence (4.6) is satisfied. This and Theorem 4.1 end the proof of Corollary 4.3. $\square$

We give three examples of application.

*Example 1.* Assume that D is bounded. Then, one can choose $D_0 = D$ and $\varphi = 1$ in Corollary 4.3. This implies that Assumption (F) is satisfied for $\psi_1 = \varphi^p$ bounded (see Remark 9), so that it follows from Theorem 3.5 that the convergence of $e^{\lambda_0 t} \mathbb{P}_X(t < \tau_{\partial})$ to $\eta$ is uniform and that $\eta$ is bounded. Theorem 3.5 also implies that Assumption (E) is satisfied for some bounded $\varphi_1$ and $\varphi_2$. Since
$P_1\eta = e^{-\lambda_0}\eta$ and $e^{-\lambda_0} \ge \theta_2 > \theta_1$, we deduce that (E) is still satisfied if $\varphi_2$ is replaced by $\eta/\|\eta\|_{\infty}$. Therefore,

$$ \| \mathbb{P}_{\mu}(X_n \in \cdot | n < \tau_{\partial}) - v_{QSD} \|_{TV} \le \frac{C}{\mu(\eta)} \alpha^n, \forall n \in \mathbb{N}. $$

The extension to any $t \in [0, +\infty)$ can be obtained using the same argument as in Section 11.5.2 replacing $\varphi_2$ and $\varphi'_2$ with $\eta$. This implies Theorem 1.1 of the introduction.

**Example 2.** Assume that $D \subset \mathbb{R}_+^d$ is open connected and that

$$ dX_t = b(X_t)dt + \sigma(X_t)dB_t $$

in $D$, where $b: \mathbb{R}^d \to \mathbb{R}^d$ and $\sigma: \mathbb{R}^d \to \mathbb{R}^{d \times r}$ are locally Hölder continuous in $\mathbb{R}^d$, $\sigma$ is locally uniformly elliptic on $\mathbb{R}^d$ and

$$ \frac{\langle b(x), 1 \rangle}{\langle x, 1 \rangle} \underset{|x| \to +\infty}{\longrightarrow} -\infty, $$

where $\langle \cdot, \cdot \rangle$ is the standard Euclidean product in $\mathbb{R}^d$ and $|\cdot|$ is the associated norm. Then (4.8) is satisfied for $\varphi(x) = 1+x_1+\dots+x_d$ and hence the process $X$ absorbed at the boundary of $D$ satisfies the assumptions of Theorem 4.1.

**Example 3.** Assume that $D \subset \mathbb{R}^d$ is open connected and that

$$ dX_t = b(X_t)dt + dB_t $$

in $D$, where $b: \mathbb{R}^d \to \mathbb{R}^d$ is locally Hölder continuous in $\mathbb{R}^d$ and

$$ \limsup_{|x| \to +\infty} \frac{\langle b(x), x \rangle}{|x|} < -\frac{3}{2} \sqrt{\lambda_0}, \quad (4.9) $$

where $\langle \cdot, \cdot \rangle$ is the standard Euclidean product in $\mathbb{R}^d$ and $\lambda_0$ is defined in (4.4). Then the process $X$ absorbed at the boundary of $D$ satisfies the assumptions of Theorem 4.1.

Indeed, let us check that (4.8) is satisfied for $\varphi(x) = \exp(\sqrt{\lambda_0}|x|)$. One has, for all $x \ne 0$,

$$ \begin{aligned} \mathcal{L}\varphi(x) &= \sum_{i=1}^{d} \frac{e^{\sqrt{\lambda_0}|x|}}{2} \left( \frac{\sqrt{\lambda_0}}{|x|} - \frac{\sqrt{\lambda_0}x_i^2}{|x|^3} + \frac{\lambda_0 x_i^2}{|x|^2} \right) + \sum_{i=1}^{d} e^{\sqrt{\lambda_0}|x|} \frac{\sqrt{\lambda_0}b_i(x)x_i}{|x|} \\ &\le \sqrt{\lambda_0}\varphi(x) \left( \frac{d-1}{2|x|} + \frac{\sqrt{\lambda_0}}{2} + \frac{\langle b(x), x \rangle}{|x|} \right) \\ &\le -(\lambda_0 + \varepsilon)\varphi(x) \end{aligned} $$
for some $\varepsilon > 0$ and for all $x$ such that $|x|$ is large enough. This implies (4.8).

To apply this criterion, it is necessary to obtain a priori bounds on $\lambda_0$. We will give some ideas about how to do so for one-dimensional diffusions in Section 4.5. In general, one can also use of course that (4.9) is implied by

$$ \lim_{|x| \to +\infty} \frac{\langle b(x), x \rangle}{|x|} = -\infty. $$

### 4.3 Non-uniformly elliptic diffusions: the Feller diffusion with competition

We provide an example where the diffusion matrix $\sigma$ cannot be extended out of $D$ as a locally uniformly elliptic matrix. This example deals with Feller diffusions with competition and is motivated by models of population dynamics with $d$ species in interaction, where absorption corresponds to the extinction of one of the populations [15, 26].

Assume that $D = (0, \infty)^d$ and

$$ dX_t^i = \sqrt{\gamma_i X_t^i} dB_t^i + X_t^i b_i(X_t) dt, $$

where $\gamma_i > 0$ for all $1 \le i \le d$, $B^1, \dots, B^d$ are independent standard Brownian motions and $b_i$ are locally Hölder in $(0, \infty)^d$ and locally bounded in $\mathbb{R}_+^d$.

**Proposition 4.4.** Assume that there exist constants $c_0, c_1 > 0$ such that

$$ \sum_{i=1}^{d} \frac{x_i b_i(x)}{\gamma_i} \le c_0 - c_1|x|, \quad \forall x \in (0, \infty)^d. $$

Then the process $X$ absorbed at the boundary of $D$ satisfies the assumptions of Theorem 4.1.

Compared to the existing literature on multi-dimensional Feller diffusions [15, 26], the main novelty of this result is that it covers cases where the process does not come down from infinity, e.g. $b_i(x) = r_i - \sum_{j=1}^{d} c_{ij} \frac{x_j}{1+x_j}$, for some positive constants $r_i$ and $c_{ij}$ such that $r_i < c_{ii}$ for all $1 \le i \le d$, and where $b$ does not derive from a potential (see for instance [15], based on a spectral theoretic approach). While our results on existence and convergence to quasi-stationary distributions are more general than those of [15], we do not recover finer results on the spectrum of the process, such as its discreteness.

*Proof.* Our aim is to prove that the assumptions of Theorem 4.1 hold true with $\varphi(x) = \exp(c(x_1/\gamma_1 + \dots + x_n/\gamma_n))$, where $c = c_1 \min_i \gamma_i / \sqrt{d}$.
We have, for all $x \in D$,

$$ \mathcal{L}\varphi(x) = \sum_{i=1}^{d} \left( \frac{x_i c^2}{2\gamma_i} + \frac{cx_i b_i(x)}{\gamma_i} \right) \varphi(x) \le \left( c_0 c - \frac{c_1 c|x|}{2} \right) \varphi(x). $$

Choosing $\lambda_1 = \lambda_0 + 1$ and $D_0 = \{x \in D, \text{s.t. } |x| \le (2c_0 + 2\lambda_1/c)/c_1\}$, one deduces that (4.5) holds true with $C = c_0c \max_{D_0} \varphi$.

Let us now prove that

$$ \mathbb{P}_x(1 < \tau_{\partial}) \xrightarrow{x \to \partial D, x \in D_0} 0, \qquad (4.10) $$

which implies that (4.6) holds true with $s_1 = 1$. Fix $\varepsilon > 0$ and define the set $F = \{x \in \mathbb{R}_+^d, \text{s.t. } \varphi(x) \ge e^C \sup_{y \in D_0} \varphi(y)/\varepsilon\}$. Using Itô's formula (see the proof of (12.9) in Section 12.3 for details), we deduce from (4.5) that, For all $x \in D_0$,

$$ \mathbb{P}_x(\tau_F \le 1) e^C \sup_{y \in D_0} \varphi(y)/\varepsilon \le \mathbb{E}_x \left( \varphi(X_{\tau_F \wedge 1}) 1_{\tau_F \wedge 1 < \tau_\partial} \right) \le e^C \varphi(x), $$

so that $\mathbb{P}_x(\tau_F \le 1) \le \varepsilon$ for all $x \in D_0$. Since $F^c$ is bounded, we have

$$ \beta := \sup_{x \in F^c, i \in \{1, \dots, d\}} |b_i(x)| < +\infty. $$

Let $(Z_t)_{t \in [0, +\infty)} := (Z_t^1, \dots, Z_t^d)_{t \in [0, +\infty)}$ be the solution of the system of SDEs

$$ dZ_t^i = \sqrt{\gamma_i Z_t^i} dB_t^i + Z_t^i \beta dt, \quad Z_0^i = X_0^i \in (0, +\infty), $$

with absorption at the boundary of $D$. Note that the components of $Z$ are independent one dimensional diffusion processes such that $0$ is reachable and hence that

$$ \mathbb{P}_x \left( \forall t \in [0, 1], \forall i \in \{1, \dots, d\}, Z_t^i > 0 \right) \xrightarrow{x \to \partial D} 0. $$

Standard comparison arguments show that $X_t^i \le Z_t^i$ for all $t < \tau_\partial \wedge \tau_F \wedge 1$ and all $i \in \{1, \dots, d\}$, so that

$$ \mathbb{P}_x \left( \forall t \in [0, 1], \forall i \in \{1, \dots, d\}, X_t^i > 0 \text{ and } 1 < \tau_F \right) \xrightarrow{x \to \partial D} 0. $$

But $\mathbb{P}_x(1 < \tau_F) \ge 1 - \varepsilon$, so that

$$ \limsup_{x \to \partial D} \mathbb{P}_x \left( \forall t \in [0, 1], \forall i \in \{1, \dots, d\}, X_t^i > 0 \right) \le \varepsilon. $$

Since this is true for all $\varepsilon > 0$ and since $\{\forall t \in [0, 1], \forall i \in \{1, \dots, d\}, X_t^i > 0\} = \{1 < \tau_\partial\}$, we deduce that (4.10) holds true, which concludes the proof or Proposition 4.4. $\square$
## 4.4 Diffusion processes with killing

This section is devoted to the study of diffusion processes with killing. More precisely, we consider as above a diffusion process X on a connected, open domain $D \subset \mathbb{R}^d$ for some $d \ge 1$, solution to the SDE

$$dX_t = b(X_t)dt + \sigma(X_t)dB_t \quad (4.11)$$

absorbed in $\partial$ at its first exit time $\tau_{\text{exit}}$ of $D$, as defined in (4.2), with the same assumptions as in Section 4.1. We also assume that the process is subject to an additional measurable killing rate $\kappa: D \to \mathbb{R}_+$ which is locally bounded: there exists an independent exponential random variable $\xi$ with parameter 1 such that the process is instantaneously sent to the cemetery point $\partial \notin D$ at time

$$\tau_{\partial} = \tau_{\text{exit}} \wedge \inf \left\{ t \ge 0, \int_{0}^{t} \kappa(X_s) ds > \xi \right\}.$$

Since $\kappa$ is assumed to be locally bounded, one easily checks that $\lambda_0$ in (4.4) is finite, and that it does not depend on $x \in D$ or on the open ball $B$ such that $\bar{B} \subset D$.

The following result is an extension to the multi-dimensional setting of [71, Theorem 4.3].

**Theorem 4.5.** Assume that there exist a subset $D_0 \subset D$ closed in $D$ such that

$$\inf_{x \in D \setminus D_0} \kappa(x) > \lambda_0, \quad (4.12)$$

and a time $s_1 > 0$ such that

$$\sup_{x \in D_0} \mathbb{P}_x(s_1 < \tau_{\partial} \wedge \tau_{K_k}) \xrightarrow{k \to +\infty} 0. \quad (4.13)$$

Then the process X absorbed at time $\tau_{\partial}$ admits a unique quasi-stationary distribution $v_{QSD}$ and there exist a positive function $\varphi_2$ on D (uniformly bounded away from 0 on compact subsets of D) and a positive constant C such that

$$\| \mathbb{P}_{\mu}(X_t \in \cdot | t < \tau_{\partial}) - v_{QSD} \|_{TV} \le \frac{C}{\mu(\varphi_2)} \alpha^t, \forall t \in [0, +\infty)$$

for all probability measures $\mu$ on E.

**Remark 11.** Let us make some comments on the assumptions of the above result.
1. If the process without killing rate satisfies (4.13), then the process with killing rate also satisfies this property. Hence the analysis provided in Section 4.2 can also be used to check the assumptions of the above theorem.

2. If $\inf_{x \in D \setminus K_k} \kappa(x) \to +\infty$ when $k \to +\infty$, then the assumptions of Theorem 4.7 are trivially satisfied.

3. In order to reach the conclusion of Theorem 4.1 in the setting of killed diffusion, it is also possible to use a Lyapunov type criterion: the assumption (4.5) can be simply replaced by the assumption that there exist $\lambda > \lambda_0$ and $C > 0$ such that

$$ \mathcal{L}\varphi(x) - \kappa(x)\varphi(x) \leq -\lambda\varphi(x) + C1_{\{x \in D_0\}}. $$

Note that (4.12) of course implies the last inequality for $\varphi \equiv 1$. This extension follows from a simple adaptation of the arguments of Theorem 4.1 observing that

$$ \mathbb{E}_x[f(X_t)1_{\{t < \tau_{\partial}\}}] = \mathbb{E}_x \left[ f(X_t^D)1_{\{t < \tau_{\text{exit}}\}} \exp\left(-\int_0^t \kappa(X_s^D)ds\right) \right], $$

where the process $X^D$ is the process solution to (4.11) without killing, absorbed at its first exit time of $D$, at time $\tau_{\text{exit}}$.

4. If in addition the killing rate $\kappa$ is locally Hölder in $D$, we can apply [50, Cor. 3.1] as in Section 12.4 to prove that $\eta$ is $\mathcal{C}^2(D)$ and $\mathcal{L}\eta(x) - \kappa(x)\eta(x) = -\lambda_0\eta(x)$ for all $x \in D$.

*Proof.* The proof follows the same lines as the proof of Theorem 4.1 in Section 12. We emphasize that the construction of the process in Section 12.1 is still valid. The same is true for the Harnack inequalities of Section 12.2 since they are based on Krylov's and Safonov's general result [72] which is obtained for diffusion processes with a bounded and measurable killing rate. The rest of the proof is exactly the same, replacing $\varphi_1 = \varphi$ by $\varphi_1 = 1$. $\square$

## 4.5 The case of one-dimensional diffusions

In this section, we consider the case of one-dimensional diffusion processes. Here, the Hölder regularity of the coefficients is not needed. Let $X$ be the solution in $D = (\alpha, \beta)$, where $-\infty \le \alpha < \beta \le +\infty$, to the SDE

$$ dX_t = \sigma(X_t) dB_t + b(X_t) dt, \quad X_0 \in D, $$
where $\sigma: D \to (0, +\infty)$ and $b: D \to \mathbb{R}$ are measurable functions such that $(1 + |b|)/\sigma^2$ is locally integrable on $D$. We assume that the process is sent to a cemetery point $\partial$ when it reaches the boundary of $D$ and that it is subject to an additional killing rate $\kappa: D \to \mathbb{R}_+$ which is measurable and locally integrable w.r.t. Lebesgue's measure. This assumption implies that the killed process is regular in the sense that, for all $x, y \in D$, $P_x(\tau_{\{y\}} < \infty) > 0$.

We define $\lambda_0$ as in (4.4). The fact that $\lambda_0$ does not depend on $x$ nor $B$ is a consequence of the regularity of the process.

Let $\delta: D \to \mathbb{R}_+$ and $s: D \to \mathbb{R}$ be defined by

$$ \delta(x) = \exp \left( -2 \int_{\alpha_0}^{x} \frac{b(u)}{\sigma(u)^2} du \right) \quad \text{and} \quad s(x) = \int_{\alpha_0}^{x} \delta(u) du, $$

for some arbitrary $\alpha_0 \in D$. We recall that $s$ is the scale function of $X$ (unique up to an affine transformation), meaning that $s(X_t)$ is a local martingale. We also recall that the boundary $\alpha$ (and similarly for $\beta$) is said to be reachable (for the process without killing) if $s(\alpha_+) > -\infty$ and

$$ \int_{\alpha}^{+} \frac{s(x) - s(\alpha_{+})}{\sigma(x)^2 \delta(x)} dx < +\infty. $$

**Theorem 4.6.** Assume that one among the following conditions (i), (ii) or (iii) holds true:

(i) $\alpha$ and $\beta$ are reachable boundaries;

(ii) $\alpha$ is reachable and there exist $\lambda_1 > \lambda_0$, a $\mathscr{C}^2(D)$ function $\varphi: D \to [1, +\infty)$ and $x_1 \in D$ such that, for all $x \ge x_1$,

$$ \frac{\sigma(x)^2}{2} \varphi''(x) + b(x)\varphi'(x) - \kappa(x)\varphi(x) \le -\lambda_1 \varphi(x); \quad (4.14) $$

(iii) there exist $\lambda_1 > \lambda_0$, a $\mathscr{C}^2(D)$ function $\varphi: D \to [1, +\infty)$ and $x_0 < x_1 \in D$ such that (4.14) holds true for all $x \in (\alpha, x_0) \cup (x_1, \beta)$.

Then the conclusions of Theorem 4.1 hold true.

**Remark 12.** We shall not detail the proof of this result since it is very close to the proof of Theorem 4.1 given in Section 12. We only explain the places that need to be modified. First, weak existence, weak uniqueness and the strong Markov property are well-known under the assumptions that $\sigma > 0$ and $(1 + |b|)/\sigma^2 \in L_{\text{loc}}^1(D)$ (weak existence and uniqueness in law are proved up to an explosion time in [66, Thm. 5.5.15], so we can construct a unique weak solution and prove
the strong Markov property as in Section 12.1). Second, in order to construct an appropriate function $\varphi$ on $D$, we choose $D_0 = (\alpha, x_1]$ in case (ii) and $D_0 = [x_0, x_1]$ in case (iii) and we can extend $\varphi$ on $D_0$ as a bounded $\mathcal{C}^2(D)$ function. In case (i), we can take $\varphi \equiv 1$ and $D_0 = D$. Third, (4.6) follows from the fact that the boundaries $\alpha$ and $\beta$ are reachable in case (i) and $\alpha$ is reachable in case (ii), since

$$ \sup_{x \in (\alpha, \alpha+1/k]} \mathbb{P}_x(s_1 < \tau_\partial) \le \mathbb{P}_{\alpha+1/k}(s_1 < \tau_{\{\alpha\}}) \xrightarrow{k \to +\infty} 0. $$

In case (iii), the limit is trivial since $D_0 \subset K_k$ for $k$ large enough. Finally, all the arguments using Harnack's inequality can be replaced by arguments using the regularity of the process and standard coupling arguments for one-dimensional diffusions (see [23, 22]).

In order to apply this result in practice, one needs to find computable estimates for $\lambda_0$ and candidates for $\varphi$. One may for instance use the bounds for the first eigenvalue of the (Dirichlet) infinitesimal generator of $(X_t, t \ge 0)$ obtained in a $L^2$ (symmetric) setting using Rayleigh-Ritz formula in [91, 110, 111], as observed in [71]. We propose here two different upper bounds for $\lambda_0$ which follow from the characterization (4.4) of the eigenvalue $\lambda_0$ and Dynkin's formula.

**Proposition 4.7.** For all $\alpha < a < b < \beta$, we have

$$ \lambda_0 \le \sup_{x \in [a, b]} \left\{ \frac{1}{2} \left( \frac{\pi \sigma(x)}{\int_a^b \exp\left(-2 \int_x^y \frac{b(z)}{\sigma^2(z)} dz\right) dy} \right)^2 + \kappa(x) \right\}. $$

If $x \mapsto b(x)/\sigma(x)^2$ is $\mathcal{C}^1([a, b])$, then

$$ \lambda_0 \le \sup_{x \in [a, b]} \frac{\pi^2 \sigma(x)^2}{2(b-a)^2} + \sigma(x)^2 \left(\frac{b}{2\sigma^2}\right)'(x) + \frac{b(x)^2}{2\sigma(x)^2} + \kappa(x). $$

*Proof.* For the proof of the first inequality, set

$$ f(x) = \sin \left( \pi \frac{s(x) - s(a)}{s(b) - s(a)} \right). $$

Then, for all $x \in (a, b)$,

$$ \begin{aligned} \frac{\sigma(x)^2}{2} f''(x) + b(x) f'(x) - \kappa(x) f(x) &= - \left( \frac{\pi^2 \sigma(x)^2 \delta(x)^2}{2(s(b) - s(a))^2} + \kappa(x) \right) f(x) \\ &= - \left( \frac{\pi^2 \sigma(x)^2}{2 \left( \int_a^b \exp\left(-2 \int_x^y \frac{b(z)}{\sigma^2(z)} dz \right) dy \right)^2} + \kappa(x) \right) f(x) \\ &\ge -C f(x), \end{aligned} $$
where

$$
C := \sup_{x \in [a, b]} \left\{ \frac{1}{2} \left( \frac{\pi \sigma(x)}{\int_a^b \exp\left(-2 \int_x^y \frac{b(z)}{\sigma^2(z)} dz \right) dy} \right)^2 + \kappa(x) \right\}.
$$

Since $f$ is $C^2$ and bounded, we deduce from Itô's formula that, for all $x \in (a, b)$,

$$
\mathbb{E}_x(f(X_t) \mathbb{1}_{t<\tau_{[a,b]}}) \geq e^{-Ct} f(x).
$$

Now, using the fact that $0 < f(x) \le 1$ for all $x \in (a, b)$, we deduce that

$$
\mathbb{P}_x(X_t \in (a, b)) \geq e^{-Ct} f(x), \forall x \in D.
$$

As a consequence, the definition of $\lambda_0$ entails $\lambda_0 \le C$.

The proof of the second inequality is the same, using instead the function

$$
f(x) := \exp\left(-\int_c^x \frac{b(u)}{\sigma(u)^2} du\right) \sin\left(\pi \frac{x-a}{b-a}\right)
$$

for some $c \in (a, b)$.

<figure><img src="image_2.png" /></figure>

The next result provides two candidates for $\varphi$. Its proof is a straightforward
computation.

**Proposition 4.8.** Let $\varphi : (0, +\infty) \text{ be any } \mathscr{C}^2(D) \text{ function such that, for some con-}$
**stants } \alpha_- < \alpha_0 < \alpha_+ \in D,$

$$
\varphi(x) = \begin{cases} \sqrt{s(x)} & \text{if } x \ge \alpha_+, \\ \sqrt{-s(x)} & \text{if } x \le \alpha_- \end{cases} \tag{4.15}
$$

Then, for all $x \in (\alpha, \alpha_-] \cup [\alpha_+, \beta)$

$$
\frac{\sigma(x)^2}{2} \varphi''(x) + b(x) \varphi'(x) - \kappa(x) \varphi(x) \le - \left( \frac{\sigma(x)^2 \delta(x)^2}{8s(x)^2} + \kappa(x) \right) \varphi(x).
$$

If $x \mapsto b(x)/\sigma(x)^2$ is $C^1(D)$, then

$$
\varphi(x) = \exp\left(-\int_{\alpha_0}^{x} \frac{b(u)}{\sigma^2(u)} du\right) \quad (4.16)
$$

satisfies

$$
\frac{\sigma(x)^2}{2} \varphi''(x) + b(x) \varphi'(x) - \kappa(x) \varphi(x) = - \left( \frac{b^2(x)}{2\sigma^2(x)} + \frac{\sigma^2(x)}{2} \left( \frac{b}{\sigma^2} \right)'(x) + \kappa(x) \right) \varphi(x).
$$
Remark 13. The first function $\varphi$ is always uniformly lower bounded on $(\alpha, \alpha_-] \cup [\alpha_+, \beta)$ by $\min\{\sqrt{s(\alpha_+)}, \sqrt{-s(\alpha_-)}\}$. To ensure that the second one is also uniformly lower bounded, one needs further assumptions on the behavior of $b/\sigma^2$ close to $\alpha$ and $\beta$.

The above results can be used as follows. In the case where $\alpha$ is reachable and $b \equiv 0$, Condition (ii) of Theorem 4.6 holds true if

$$ \liminf_{x \to \beta^{-}} \frac{\sigma^2(x)}{8(x - \alpha)^2} + \kappa(x) > \lambda_0, $$

choosing $\alpha_0 = \alpha$ and using the function $\varphi$ of (4.15). Similarly, in the case where $\alpha$ is reachable, $\sigma \equiv 1$ and $b$ is $C^1$, condition (ii) of Theorem 4.6 holds true if

$$ \liminf_{x \to \beta^{-}} \frac{b^2(x)}{2} + \frac{b'(x)}{2} + \kappa(x) > \lambda_0, $$

using the function $\varphi$ of (4.16).

We give below more precise examples.

**Example 4.** Assume that $D = (0, +\infty)$, $\kappa$ is locally bounded and that $X$ is solution to the SDE in $D$

$$ dX_t = \sqrt{X_t} dB_t - X_t dt. $$

Then $0$ is reachable for $X$ and, since

$$ \frac{\sigma(x)^2 \delta(x)^2}{8s(x)^2} \xrightarrow{x \to +\infty} +\infty, $$

we deduce from Proposition 4.8 and Theorem 4.6 that $X$ admits a quasi-stationary distribution $v_{QSD}$ and, for all $p \ge 1$, there exist positive constants $C_p, \gamma_p$ and a positive function $\varphi_{2,p}$ on $(0, +\infty)$ such that

$$ \| \mathbb{P}_{\mu}(X_t \in \cdot | t < \tau_{\partial}) - v_{QSD} \|_{TV(\exp(\cdot/p))} \le C_p \frac{\int_{(0, +\infty)} \exp(x/p) \mu(dx)}{\mu(\varphi_{2,p})} e^{-\gamma_p t}, $$

for all probability measure $\mu$ on $D$. In particular, one deduces that the domain of attraction $v_{QSD}$ contains any initial distribution $\mu$ admitting a finite exponential moment. Note that, in the case where $\kappa \equiv 0$, the process $X$ is a continuous state branching process (Feller diffusion), for which quasi-stationarity was already studied (see [73] and the references therein).
**Example 5.** Assume that $(\alpha, \beta) = \mathbb{R}$, that $b \equiv 0$ and $\sigma$ is bounded measurable on $\mathbb{R}$. Assume also that the absorption of $X$ is due to the killing rate $\kappa(x) = \kappa_0(1 - \frac{1}{1+|x|})$ for some constant $\kappa_0 > 0$. We deduce from the first inequality of Proposition 4.7 (taking $b > 0$ and $a = -b$) that

$$ \lambda_0 \le \frac{\pi^2 \|\sigma\|_{\infty}^2}{8b^2} + \kappa_0 \left(1 - \frac{1}{1+b}\right) \le \kappa_0 \left(1 - \frac{1}{1+2b}\right) $$

for $b$ large enough. Moreover, choosing $\varphi = 1$ and $x_0 = -3b$, $x_1 = 3b$, one deduces that, for all $x \notin [-x_1, x_1]$,

$$ \frac{\sigma(x)^2}{2} \varphi''(x) - \kappa(x)\varphi(x) \le -\kappa_0 \left(1 - \frac{1}{1+3b}\right) \varphi(x). $$

Hence Theorem 4.6 implies that there exists a unique quasi-stationary distribution $v_{QSD}$ for $X$ and that it attracts all probability measures $\mu$ on $D$.

**Example 6.** We consider the case $(\alpha, \beta) = (0, +\infty)$, $\sigma(x) = 1$, $b(x) = x \sin x$, and $\kappa(x) = \kappa_0(1 - \frac{1}{1+x})$ for some constant $\kappa_0 > \pi^2 + 3$. This corresponds to a SDE $dX_t = dB_t + \nabla U(X_t)dt$ where the potential $U(x) = \sin x - x \cos x$ has infinitely many wells with arbitrarily large depths, meaning that the process $X$ without killing has a tendency to be “trapped” away from zero for large initial conditions. Nevertheless, thanks to the killing, we are able to prove convergence to a unique quasi-stationary distribution. Indeed, using the second inequality of Proposition 4.7, we have

$$ \lambda_0 \le \sup_{x \in (0,1)} \frac{\pi^2}{2} + \frac{\sin x + x \cos x + x^2 \sin^2 x}{2} + \kappa_0 \left(1 - \frac{1}{1+x}\right) \le \frac{\pi^2}{2} + \frac{3}{2} + \kappa_0/2. $$

Moreover, 0 is a reachable boundary for $X$ and, taking $\varphi = 1$, one has, for all $x_1 > 0$ and all $x > x_1$,

$$ \frac{\sigma(x)^2}{2} \varphi''(x) + b(x)\varphi'(x) - \kappa(x)\varphi(x) \le -\kappa_0 \left(1 - \frac{1}{1+x_1}\right) \varphi(x) $$

Hence, since we assumed that $\kappa_0 > \pi^2 + 3$, one deduces that there exists a unique quasi-stationary distribution $v_{QSD}$ for $X$ and that it attracts all probability measures $\mu$ on $D$.

**Remark 14.** The case of general one-dimensional diffusion processes [65] can be handled using our framework, although using the infinitesimal generator is more tricky [63]. However, in the case of a regular diffusion process on $(0, +\infty)$
such that 0 is a reachable boundary and such that $+\infty$ is entrance, one easily shows (see for instance [23]) that, for all $\lambda > 0$, there exists $y > 0$ such that

$$ \sup_{x \in (0, +\infty)} \mathbb{E}_x \left( e^{\lambda \tau_{[0, y]}} \right) < +\infty. $$

Hence, using the same proof as in Theorem 4.1 and using Lemma 3.6, we deduce that there exists a unique quasi-stationary distribution $v_{QSD}$ for $X$ and that it satisfies

$$ \| \mathbb{P}_{\mu}(X_t \in \cdot | t < \tau_{\partial}) - v_{QSD} \|_{TV} \le \frac{1}{\mu(\varphi_2)} \alpha^t, \forall t \in [0, +\infty) $$

for some positive function $\varphi_2$ and some $\alpha < 1$. Whether the convergence to $v_{QSD}$ holds uniformly with respect to the initial distribution (as in Proposition 3.7) without further assumptions remains an open problem. It has been shown to be true for a wide range of cases in [23, 22].

# 5 Application to processes in discrete state space and continuous time

Let $X$ be a non-explosive<fnref n="1" /> Markov process in a countable state space $E \cup \{\partial\}$ absorbed in $\partial$, with jump rate $q_{x,y}$ from $x$ to $y \ne x$ such that $\sum_{y \in E \cup \{\partial\} \setminus \{x\}} q_{x,y} < \infty$ for all $x \in E$. The extended generator $\mathcal{L}$ acts on nonnegative real functions $f$ on $E \cup \{\partial\}$ such that $\sum_{y \in E \cup \{\partial\}} q_{x,y} f(y) < \infty$ for all $x \in E$ as

$$ \mathcal{L}f(x) = \sum_{y \ne x \in E \cup \{\partial\}} q_{x,y}(f(y) - f(x)), \quad \forall x \in E, \quad Lf(\partial) = 0. \quad (5.1) $$

**Theorem 5.1.** Assume that there exists a finite subset $D_0$ of $E$ such that $\mathbb{P}_x(X_1 = y) > 0$ for all $x, y \in D_0$, so that the constant

$$ \lambda_0 := \inf \left\{ \lambda > 0, \text{ s.t. } \liminf_{t \to +\infty} \inf e^{\lambda t} \mathbb{P}_x (X_t = x) > 0 \right\} $$

is finite and independent of $x \in D_0$. If in addition there exist constants $C > 0$, $\lambda_1 > \lambda_0$, a function $\varphi: E \cup \{\partial\} \to \mathbb{R}_+$ such that $\varphi|_E \ge 1$, $\varphi(\partial) = 0$, $\sum_{y \in E \setminus \{x\}} q_{x,y} \varphi(y) < \infty$ for all $x \in E$ and such that

$$ \mathcal{L}\varphi(x) \le -\lambda_1\varphi(x) + C1_{x \in D_0}, \quad \forall x \in E, \quad (5.2) $$

<footnote n="1">¹One could actually consider the case of explosive Markov processes as in Section 4 (see Remark 10), with $\tau_{\partial}$ defined as the infimum between the first hitting time of $\partial$ and the explosion time.</footnote>
then Assumption (F) is satisfied with $L = D_0$, $\gamma_1 = e^{-\lambda_1}$, any $\gamma_2 \in (e^{-\lambda_1}, e^{-\lambda_0})$ and $\psi_1 = \varphi|_E$. In addition, $\mathbb{P}_{v_{QSD}}(t < \tau_\partial) = e^{-\lambda_0 t}$ for all $t \ge 0$, the function $\eta$ of Theorem 2.2 satisfies $P_t\eta = e^{-\lambda_0 t}\eta$ for all $t \ge 0$ and $\sum_{y \in E \setminus \{x\}} q_{x,y}\eta(y) < \infty$ and $\mathcal{L}\eta(x) = -\lambda_0\eta(x)$ for all $x \in E$.

Remark 15. If in addition to the assumptions of Theorem 5.1 we assume that $\lambda_1 > \sup_{x \in E} q(x, \partial)$, it is possible to adapt the proof of Theorem 3.5 given in Section 11.5 to prove that the conclusion of Theorem 3.5 holds true with $\psi_2 \equiv 1$. Therefore, we obtain the improved convergence, for all $h \in L^\infty(\varphi)$,

$$|\mathbb{E}_{\mu}(h(X_t) | t < \tau_{\partial}) - v_{QSD}(h)| \le C \mu(\varphi) \alpha^t \|h/\varphi\|_{\infty}, \forall t \ge 0,$$

instead of (3.4). If moreover $\varphi$ is bounded over $E$, the convergence is uniform and there exists a unique quasi-stationary distribution.

Before turning to the proof of Theorem 5.1, we give an example of application.

**Example 7.** Assume that $X$ is a birth and death process with killing on $E = \mathbb{N}$ and $\partial = 0$. This means that there exist non-negative numbers $(b_x)_{x \in \mathbb{N}}$, $(d_x)_{x \in \mathbb{N}}$, $(\kappa_x)_{x \in \mathbb{N}}$ such that $b_x > 0$ for all $x \ge 1$, $d_x > 0$ for all $x \ge 2$, and $d_1 = 0$, and such that, for all $x \in E$,

$$q_{x,y} = \begin{cases} b_x & \text{if } y = x + 1, \\ d_x & \text{if } y = x - 1, \\ \kappa_x & \text{if } y = 0, \\ 0 & \text{otherwise.} \end{cases}$$

We set

$$S := \sum_{k \ge 1} \frac{1}{d_k \alpha_k} \sum_{l \ge k} \alpha_l, \qquad (5.3)$$

with $\alpha_k = (\prod_{i=1}^{k-1} b_i) / (\prod_{i=1}^k d_i)$. Recent advances on existence of quasi-stationary distribution of birth and death processes with killing were obtained in [35, 100, 101], see also the nice survey [104].

In this setting, Theorems 5.1 and 3.5 translate as follows: if there exists a function $\varphi: \mathbb{Z}_+ \to [1, +\infty)$ such that $\varphi(0) = 0$ and

$$\lambda_0 < \liminf_{x \to +\infty} - \frac{b_x(\varphi(x+1) - \varphi(x)) + d_x(\varphi(x-1) - \varphi(x))}{\varphi(x)} + \kappa_x, \quad (5.4)$$

then there exists a unique quasi-stationary distribution $v_{QSD}$ such that $v_{QSD}(\varphi) < +\infty$ which attracts exponentially fast all initial distributions integrating $\varphi$. To
check (5.4), one may use in practice the fact that $\lambda_0 \le \inf_{x \in \mathbb{N}} b_x + d_x + \kappa_x$, or
adapt the ideas of Section 4.5 to birth and death processes, or use the finer up-
per bounds for $\lambda_0$ proved in [105]. We consider now three situations where the
criterion (5.4) improves known results in the literature.

First, if $S < +\infty$, [35, Theorem 6.6] proves that there exists a unique quasi-stationary distribution for $X$ assuming that $(\kappa_x)_{x \in \mathbb{N}}$ has finite support. We extend this result to any killing rates $(\kappa_x)_{x \in \mathbb{N}}$ and also prove that the unique quasi-stationary distribution attracts all initial distributions exponentially fast. We can indeed check that (5.4) is satisfied by a bounded function $\varphi$ defined as follows: fix $\lambda_1 > \lambda_0$ and choose $x_0 \in \mathbb{N}$ large enough such that (see for instance [20, Equation (4.7)])

$$
\sup_{x \in \mathbb{N}} \mathbb{E}_x (e^{\lambda_1 \tau_{D_0} \wedge \tau_{\partial}}) < +\infty.
$$

where $D_0 = \{1, \dots, x_0\}$. Then we define $\varphi(0) = 0$ and

$$
\varphi(x) = \mathbb{E}_x \left( e^{\lambda_1 \tau_{D_0} \wedge \tau_{\partial}} \right), \quad \forall x \in \mathbb{N}.
$$

Using Markov's property at the first time of jump, one checks that (since $b_x + d_x \to +\infty$ when $x \to +\infty$, we assume w.l.o.g. that $b_x + d_x + \kappa_x > \lambda_1$ for all $x \ge x_0 + 1$),

$$
\varphi(x) = \frac{d_x}{b_x + d_x + \kappa_x - \lambda_1}\varphi(x-1) + \frac{b_x}{b_x + d_x + \kappa_x - \lambda_1}\varphi(x+1) \\
\qquad + \frac{\kappa_x}{b_x + d_x + \kappa_x - \lambda_1}, \quad \forall x \ge x_0 + 1.
$$

Hence $\lambda_1 = -[b_x(\varphi(x+1) - \varphi(x)) + d_x(\varphi(x-1) - \varphi(x))]/\varphi(x) + \kappa_x$ for $x \ge x_0 + 1$ and (5.4) is satisfied.

Second, if $S = +\infty$ and if $\lambda_0 < \liminf_{x \to +\infty} \kappa_x$, it was proved in [101, Theorem 4.3] that there exists a quasi-stationary distribution. The criterion (5.4) improves this result since it implies that the quasi-stationary distribution is unique and that it attract all initial distributions exponentially fast. Indeed, (5.4) is clearly satisfied for $\varphi \equiv 1$.

Last, we can also extend [101, Theorem 4.3] to processes that do not neces-
sarily admit a unique quasi-stationary distribution, and in particular that do not
come down from infinity. For example, assuming that, for some $\varepsilon > 0$,

$$
\liminf_{x \to +\infty} \kappa_x + \frac{\varepsilon}{1+\varepsilon} d_x - \varepsilon b_x > \lambda_0,
$$

Condition (5.4) is satisfied for $\varphi(x) = (1 + \varepsilon)^x$.

Note that, because of Corollary 2.4, our criteria imply the $\lambda_0$-positive recur-
rence of the process $X$ (cf. e.g. [104, Eq. (26)]). Therefore, it can only apply to
such situations. For results on birth and death processes with killing which are not $\lambda_0$-positive recurrent, we refer the reader to [101, Theorem 4.2].

*Example 8.* We consider general multitype birth and death processes in continuous time, taking values in a connected (in the sense of the nearest neighbors structure of $\mathbb{Z}^d$) subset $E$ of $\mathbb{Z}_+^d$ for some $d \ge 1$, with transition rates

$$ q_{x,y} = \begin{cases} b_i(x) & \text{if } y = x + e_i, \\ d_i(x) & \text{if } y = x - e_i, \\ 0 & \text{otherwise,} \end{cases} $$

with $e_i = (0, \dots, 0, 1, 0, \dots, 0)$ where the nonzero coordinate is the $i$-th one and with the convention that the process is sent instantaneously to $\partial$ when it jumps to a point $y \notin E$ according to the previous rates. To ensure irreducibility, it is sufficient (although not optimal) to assume that $b_i(x) > 0$ and $d_i(x) > 0$ for all $1 \le i \le d$ and $x \in E$.

We show below that Theorem 5.1 applies either under the assumption that

$$ \frac{1}{1+|x|} \sum_{i=1}^{d} (d_i(x) - b_i(x)) \xrightarrow{x \in E, |x| \to +\infty} +\infty. \quad (5.5) $$

or that there exists $\delta > 1$ such that

$$ \sum_{i=1}^{d} (d_i(x) - \delta b_i(x)) \xrightarrow{x \in E, |x| \to +\infty} +\infty. \quad (5.6) $$

This improves the general criteria obtained in [26] since this reference assumes (among other assumptions) that $E = \mathbb{Z}_+^d$ and that $\sum_{i=1}^d (d_i(x) - b_i(x)) \ge |x|^{1+\eta}$ for some $\eta > 0$ and $|x|$ large enough.

Let us first show that (5.5) implies that the assumptions of Theorem 5.1 are satisfied. In order to do so, we define $\varphi(x) = |x| + 1 = x_1 + \dots + x_d + 1$ and $\varphi(\partial) = 0$ and obtain

$$ \begin{aligned} \mathcal{L}\varphi(x) &= \sum_{i=1}^{d} (b_i(x) - d_i(x)) - \sum_{i=1}^{d} (b_i(x) \mathbb{1}_{x+e_i \notin E}\varphi(x+e_i) + d_i(x) \mathbb{1}_{x-e_i \notin E}\varphi(x-e_i)) \\ &\le -\varphi(x) \frac{\sum_{i=1}^{d} (d_i(x) - b_i(x))}{|x| + 1} \end{aligned} $$

The proof is concluded by setting $D_0 = \{ x \in E, \text{ s.t. } \frac{\sum_{i=1}^{d} (d_i(x) - b_i(x))}{|x|+1} \le \lambda_0 + 1 \}$.

Let us now show that (5.6) implies that the assumptions of Theorem 5.1 are satisfied. Setting $\varphi(x) = \exp\langle a, x \rangle$ for a given $a \in (0, \infty)^d$ and $\varphi(\partial) = 0$, we obtain

$$ \mathcal{L}\varphi(x) \le -\varphi(x) \left( \sum_{i=1}^{d} (1 - e^{-a_i}) d_i(x) + (1 - e^{a_i}) b_i(x) \right). $$
Choosing $a = (\epsilon, \dots, \epsilon)$ with $\epsilon$ small enough, we have

$$ \liminf_{x \in E, |x| \to +\infty} \sum_{i=1}^{d} (1 - e^{-a_i}) d_i(x) + (1 - e^{a_i}) b_i(x) = +\infty. $$

Taking $D_0 = \{x \in E, \text{s.t. } \sum_{i=1}^{d} (1 - e^{-a_i})d_i(x) + (1 - e^{a_i})b_i(x) \le \lambda_0 + 1\}$ allows us to conclude the proof.

**Proof of Theorem 5.1.** The fact that $\lambda_0$ is independent of $x$ is classical for irreducible processes (cf. e.g. [69]). We set $L = D_0$. Since $X$ is a non-explosive pure jump continuous time process, it satisfies the strong Markov property and the entrance times $\tau_L$ and $\tau_\partial$ are stopping times. This entails (F0).

For all $x, y \in L$, we have

$$ \mathbb{P}_x(X_2 \in \cdot) \ge \inf_{u, v \in L} \mathbb{P}_u(X_1 = v) \mathbb{P}_y(X_1 \in \cdot), $$

where $\inf_{u, v \in L} \mathbb{P}_u(X_1 = v) > 0$ by assumption, which implies (F1) and (F3).

We set $\psi_1 = \varphi$. For all $0 \le s \le 1$, using (5.2) and Dynkin's formula, one has that for all $x \in L$

$$ \mathbb{E}_x (\psi_1(X_s) \mathbb{1}_{s<\tau_\partial}) \le e^{Cs} \sup_{y \in L} \psi_1(y). $$

Similarly, setting $\gamma_1 = e^{-\lambda_1}$, for all $x \in E \setminus L$,

$$ \mathbb{E}_x (\psi_1(X_1) \mathbb{1}_{1<\tau_L \wedge \tau_\partial}) \le e^{-\lambda_1} \psi_1(x) = \gamma_1 \psi_1(x). $$

Choosing any $\gamma_2 \in (\gamma_1, e^{-\lambda_0})$, one obtains that (F2) is satisfied and the first part of Theorem 5.1 is proved.

The inequality $\sum_{y \in E \setminus \{x\}} q_{x,y}\eta(y) < \infty$ for all $x \in E$ follows from the fact that $\eta \in L^\infty(\psi_1)$ and the fact that $P_t\eta(x) = e^{-\lambda_0 t}\eta(x)$ was proved in Theorem 3.5. It then follows from Markov's property and the last equality that $(e^{\lambda_0 t}\eta(X_t), t \ge 0)$ is a martingale for the canonical filtration associated to $X$, with the convention that $\eta(\partial) = 0$. Now, it is standard to represent the Markov process $X$ as a solution to a stochastic differential equation driven by a Poisson point process: assume that the elements of the finite or countable set $E$ are labeled by distinct positive integers, that $\partial = 0$ and, for all $x, i \in \mathbb{Z}_+$, let $\kappa_i(x) = q_{x,0} + q_{x,1} + \dots + q_{x,i}$ with the convention that $q_{x,x} = 0$ and $q_{x,i} = 0$ for all $x$ or $i \notin E \cup \{\partial\}$ and set $q(x) = \sum_{i \in \mathbb{Z}_+} q_{x,i} < \infty$. Given a Poisson point measure $N(ds, d\theta)$ on $\mathbb{R}_+^2$ with intensity the Lebesgue measure on $\mathbb{R}_+^2$, the process $X$ solution

$$ X_t = X_0 + \int_0^t \int_0^t q^{(X_{s-})} \sum_{i=0}^{\infty} \mathbb{1}_{\theta \in [\kappa_{i+1}(X_{s-}), \kappa_i(X_{s-}))} (i - X_{s-}) N(ds, d\theta) $$
is well-defined for all time $t \ge 0$ almost surely and is a Markov process with matrix of jump rates $(q_{i,j})_{i,j \in \mathbb{Z}_+}$. Introducing the compensated Poisson measure $\tilde{N}(ds, d\theta) = N(ds, d\theta) - ds d\theta$, it follows from basic stochastic calculus for jump processes (cf. e.g. [93]) that

$$e^{\lambda_0 t} \eta(X_t) = X_0 + \int_0^t \int_0^{q(X_{s-})} e^{\lambda_0 s} \sum_{i=0}^{\infty} \mathbb{1}_{\theta \in [\kappa_{i+1}(X_{s-}), \kappa_i(X_{s-}))} (\eta(i) - \eta(X_{s-})) \tilde{N}(ds, d\theta) \\ + \int_0^t e^{\lambda_0 s} \left( \sum_{i=0}^{\infty} q_{X_s, i}(\eta(i) - \eta(X_s)) + \lambda_0 \eta(X_s) \right) ds.$$

Since $e^{\lambda_0 t} \eta(X_t)$ is a $\mathbb{P}_x$-martingale, the Doob-Meyer decomposition theorem entails that

$$\int_0^t e^{\lambda_0 s} \left( \sum_{i=0}^{\infty} q_{X_s, i}(\eta(i) - \eta(X_s)) + \lambda_0 \eta(X_s) \right) ds = 0$$

$\mathbb{P}_x$-almost surely for all $t \ge 0$ and all $x \in E$. Hence, if there exists $y \in E$ such that $\mathcal{L}\eta(y) \ne -\lambda_0\eta(y)$, by irreducibility, there exists an event with positive probability under $\mathbb{P}_x$ such that the previous integral is non-constant. We obtain a contradiction and hence $\mathcal{L}\eta(x) = -\lambda_0\eta(x)$ for all $x \in E$. $\square$

# 6 On reducible examples

The criteria and examples studied in the last two sections assume that the process $X$ is irreducible in $E$. However, the abstract results of Section 2 do not require the state space to be irreducible. Our goal in this section is to explain that our criteria are also well-suited to cases of reducible absorbed Markov processes, in the sense that the state space $E$ can be partitioned in a finite or countable family of communication classes. The study of quasi-stationary behavior for such processes has been up to now restricted to the case of finite state spaces or to particular classes of models [77, 38, 88, 55, 19, 102, 103, 18, 104, 8]. Our criteria provide new practical tools to tackle this problem, further exploited in [28].

In Subsection 6.1, we consider a general setting with three successive sets. In Subsection 6.2, we consider a birth and death process with a countable infinity of communication classes.

## 6.1 Three successive sets

In this section, we consider a discrete time Markov process $(X_n, n \in \mathbb{Z}_+)$ evolving in a measurable set $E \cup \{\partial\}$ with absorption at $\partial \notin E$. We assume that the transition probabilities of $X$ satisfy the structure displayed in Figure 1: one can find
<figure><img src="image_3.png" /><figcaption>Figure 1: Transition graph displaying the relation between the sets $D_1, D_2, D_3$ and $\partial$.</figcaption></figure>

a partition $\{D_1, D_2, D_3\}$ of E such that the process starting from $D_1$ can access $D_1 \cup D_2 \cup D_3 \cup \{\partial\}$, the process starting from $D_2$ can only access $D_2 \cup D_3 \cup \{\partial\}$, and the process starting from $D_3$ can only access $D_3 \cup \{\partial\}$. More formally, we assume that $\mathbb{P}_x(T_{D_3} \wedge \tau_\partial < T_{D_1}) = 1$ for all $x \in D_2$ and that $\mathbb{P}_x(\tau_\partial < T_{D_1 \cup D_2}) = 1$ for all $x \in D_3$, where we recall that, for any measurable set $A \subset E$, $T_A = \inf\{n \in \mathbb{Z}_+, X_n \in A\}$.

Our aim is to provide sufficient conditions ensuring that X satisfies Assumption (E). In order to do so, we assume that Assumption (E) is satisfied by the process X before exiting $D_2$. This corresponds to the following assumption.

**Assumption (H1).** The absorbed Markov process Y evolving in $D_2 \cup \{\partial\}$, defined by

$$Y_n = \begin{cases} X_n & \text{if } n < T_{D_1 \cup D_3 \cup \{\partial\}}, \\ \partial & \text{if } n \ge T_{D_1 \cup D_3 \cup \{\partial\}}, \end{cases}$$

satisfies Assumption (E). In what follows, we denote the objects related to Y with a superscript Y, for instance, the constants of Assumption (E) for Y are denoted by $\theta_1^Y > 0, \theta_2^Y > 0$.

We also assume that the exit times from $D_1$ and $D_3$ for the process X admit exponential moments of sufficiently high order, as stated by the following assumption.

**Assumption (H2).** There exists a positive constant $\gamma < \theta_0^Y$ such that, for all $x \in D_1$,

$$\mathbb{E}_x(\gamma^{-T_{D_2}} \varphi_1^Y(X_{T_{D_2}}) \mathbb{1}_{T_{D_2} < T_{D_3} \wedge \tau_\partial}) < +\infty, \quad \mathbb{E}_x(\gamma^{-T_{D_3} \wedge \tau_\partial} \mathbb{1}_{T_{D_3} \wedge \tau_\partial < T_{D_2}}) < +\infty,$$
and such that

$$ \sup_{x \in D_3} \mathbb{E}_x (\gamma^{-\tau_\partial}) < +\infty. $$

We are now able to state the main result of this section.

**Theorem 6.1.** Under Assumptions (H1) and (H2), the process X satisfies Assumption (E) with $K = K^Y$,

$$ \varphi_1(x) = \mathbb{E}_x (\gamma^{-T_K \wedge \tau_\partial}) \quad \text{and} \quad \varphi_2(x) \ge c \mathbb{1}_{\{x \in K\}}, \forall x \in E, \quad (6.1) $$

for some constant $c > 0$. In particular, it admits a unique quasi-stationary distribution $v_{QSD}$ such that $v_{QSD}(\varphi_1) < \infty$ and $v_{QSD}(\varphi_2) > 0$. Moreover, there exist two constants $C > 0$ and $\alpha \in (0, 1)$ such that, for all probability measure $\mu$ on E such that $\mu(\varphi_1) < \infty$ and $\mu(\varphi_2) > 0$,

$$ \| \mathbb{P}_{\mu}(X_n \in \cdot | n < \tau_{\partial}) - v_{QSD} \|_{TV}(\varphi_1) \le C \alpha^n \frac{\mu(\varphi_1)}{\mu(\varphi_2)}. \quad (6.2) $$

Finally, $\theta_0 = \theta_0^Y$, $v_{QSD}(D_1) = 0$ and the function $\eta$ of Theorem 2.2 vanishes on $D_3$.

Before turning to the proof of this result, let us make some remarks.

**Remark 16.** 1. The fact that there are three different sets $D_1$, $D_2$ and $D_3$ in the decomposition of E is not restrictive on the number of communication classes. Indeed, the three sets can contain several communication classes.

2. A similar result can be obtained for continuous time processes, based on Assumption (F) instead of (E), with the additional technical assumption that the strong Markov property can be applied at the exit times of $D_1$ and $D_2$.

3. We emphasize that, besides the exponential moment assumption, there is no additional requirement on the behavior of the Markov process in $D_1$ and $D_3$. In these sets, the process might be for instance periodic or deterministic and could satisfy that $\mathbb{P}_x(n < \tau_\partial) = 0$ for some $n \in \mathbb{N}$.

4. One easily checks from the proof that the function $\varphi_1$ in (6.1) is bounded (up to a multiplicative positive constant) from above by

$$ \mathbb{E}_x \left( \gamma^{-T_{D_2}} \varphi_1^Y \left( X_{T_{D_2}} \mathbb{1}_{T_{D_2} < T_{D_3} \wedge \tau_\partial} \right) + \mathbb{E}_x \left( \gamma^{-T_{D_3} \wedge \tau_\partial} \mathbb{1}_{T_{D_3} \wedge \tau_\partial < T_{D_2}} \right) \right) $$

on $D_1$, by $\varphi_1^Y$ on $D_2$ and by a constant on $D_3$.
5. In particular, if $\varphi_1^Y$ is uniformly bounded and if the first statement in Assumption (H2) is replaced by

$$ \sup_{x \in D_1} \mathbb{E}_x (\gamma^{-T_{D_2 \cup D_3} \wedge \tau_\partial}) < +\infty, $$

then one can also choose a bounded function $\varphi_1$ in Assumption (E) for $X$.

*Remark 17.* In general, processes on reducible state spaces may not satisfy Assumption (E). For example the convergence in (6.2) may not be exponential, or quasi-stationary distributions may not be unique, even if the process $X$ restricted to $D_1$, $D_2$ or $D_3$ satisfy condition (E). We refer the reader to [28] for a more general discussion on quasi-stationary distributions and quasi-limiting behavior for general processes on reducible state spaces.

*Proof of Theorem 6.1.* Let us prove that Assumption (E) is satisfied by the process $X$. Note that, because of Lemma 3.4, one can assume without loss of generality that $\gamma < \theta_2^Y$.

*Step 1. Assumption (E1).*

We set $K = K^Y$, $n_1 = n_1^Y$, $c_1 = c_1^Y$ and $\nu = \nu^Y$. Assumption (E1) for $X$ is an immediate consequence of Assumption (E1) for $Y$.

*Step 2. Assumption (E2).*

We set $\theta_2 = \theta_2^Y$ and

$$ \varphi_2(x) = \begin{cases} \varphi_2^Y(x) & \text{if } x \in D_2 \\ 0 & \text{if } x \in D_1 \cup D_3. \end{cases} $$

Then the second and fourth lines of Assumption (E) for $X$ are direct consequences of the same lines of Assumption (E) for $Y$.

Without loss of generality, we assume (increasing $\gamma$ if necessary, which does not change the fact that Assumptions (H2) is true) that $\gamma \in (\theta_1^Y, \theta_2^Y)$. We define

$$ \varphi_1(x) = \mathbb{E}_x (\gamma^{-T_K \wedge \tau_\partial}), \quad \forall x \in E \cup \{\partial\}. $$

Let us first check that $\varphi_1$ is finite on $E$. For all $x \in D_3$, using that $\mathbb{P}_x(\tau_\partial < T_{D_1 \cup D_2}) = 1$ and that $K \subset D_2$, one deduces that

$$ \varphi_1(x) = \mathbb{E}_x (\gamma^{-\tau_\partial}) \le A := \sup_{x \in D_3} \mathbb{E}_x (\gamma^{-\tau_\partial}) < +\infty. $$
For all $x \in D_2$, using the strong Markov property and inequality (9.7) for the process $Y$, one deduces that

$$ 
\begin{aligned}
\varphi_1(x) &= \mathbb{E}_x \left( \gamma^{-T_K \wedge T_{D_2^c}} \mathbb{1}_{T_K < T_{D_2^c}} \right) + \mathbb{E}_x \left( \gamma^{-\tau_\partial} \mathbb{1}_{T_{D_2^c} < T_K} \right) \\
&= \mathbb{E}_x \left( \gamma^{-T_K \wedge T_{D_2^c}} \mathbb{1}_{T_K < T_{D_2^c}} \right) + \mathbb{E}_x \left( \gamma^{-T_K \wedge T_{D_2^c}} \mathbb{1}_{T_{D_2^c} < T_K} \mathbb{E}_{X_{T_{D_2^c}}} \left( \gamma^{-\tau_\partial} \right) \right) \\
&\le A \mathbb{E}_x \left( \gamma^{-T_K \wedge T_{D_2^c}} \right) \le \frac{A}{1 - \theta_1^Y / \gamma} \varphi_1^Y(x). \qquad (6.3)
\end{aligned}
$$

For all $x \in D_1$, one has, using the Markov property and the above inequalities,

$$ 
\begin{aligned}
\mathbb{E}_x (\gamma^{-T_K \wedge \tau_\partial}) &= \mathbb{E}_x \left( \gamma^{-T_{D_2 \cup D_3} \wedge \tau_\partial} \varphi_1 (X_{T_{D_2 \cup D_3} \wedge \tau_\partial}) \right) \\
&\le \frac{A}{1 - \theta_1^Y / \gamma} \left[ \mathbb{E}_x \left( \gamma^{-T_{D_2}} \varphi_1^Y (X_{T_{D_2}}) \mathbb{1}_{T_{D_2} < T_{D_3} \wedge \tau_\partial} \right) + \mathbb{E}_x \left( \gamma^{-T_{D_3} \wedge \tau_\partial} \mathbb{1}_{T_{D_3} \wedge \tau_\partial < T_{D_2}} \right) \right],
\end{aligned}
\qquad (6.4) $$

which is finite by Assumption (H2).

The definition of $\varphi_1$ immediately implies that $\inf_E \varphi_1 \ge 1$ and, since $\varphi_1^Y$ is uniformly bounded over $K \subset D_2$, (6.3) implies that $\sup_K \varphi_1 < +\infty$. Hence the first line of Assumption (E2) is satisfied. Moreover, for all $x \in K$,

$$ 
\begin{aligned}
P_1 \varphi_1(x) &= \mathbb{E}_x \left( \mathbb{1}_{X_1 \in D_2} \mathbb{E}_{X_1} \left( \gamma^{-T_K \wedge \tau_\partial} \right) \right) + \mathbb{E}_x \left( \mathbb{1}_{X_1 \in D_3} \mathbb{E}_{X_1} \left( \gamma^{-\tau_\partial} \right) \right) \\
&\le \mathbb{E}_x \left( \mathbb{1}_{X_1 \in D_2} \frac{A}{1 - \theta_1^Y / \gamma} \varphi_1^Y (X_1) \right) + A \\
&= \frac{A}{1 - \theta_1^Y / \gamma} P_1^Y \varphi_1^Y (x) + A \le \frac{A}{1 - \theta_1^Y / \gamma} \left( \theta_1^Y \sup_K \varphi_1^Y + c_2^Y \right) + A.
\end{aligned}
$$

Hence, the third line of (E2) for $X$ with $\theta_1 = \gamma$ follows from Lemma 3.2.

Step 3. Assumption (E3).

For all $x \in K$, we have, for all $n \ge 1$,

$$ \mathbb{P}_x(n < \tau_\partial) \le \mathbb{P}_x(n < \tau_\partial \wedge T_{D_3}) + \mathbb{P}_x(T_{D_3} \le n < \tau_\partial). \qquad (6.5) $$

On the one hand, by Lemma 9.9, there exists a constant $C > 0$ such that

$$ \mathbb{P}_x(n < \tau_\partial \wedge T_{D_3}) \le \frac{C \varphi_1^Y(x)}{1 - \theta_1^Y / \theta_2^Y} \inf_{y \in K} \mathbb{P}_y(n < T_{D_2^c}) \le \frac{C \sup_K \varphi_1^Y}{1 - \theta_1^Y / \theta_2^Y} \inf_{y \in K} \mathbb{P}_y(n < T_{D_2^c}). $$

On the other hand, using Markov's property and Markov's inequality,

$$ 
\begin{aligned}
\mathbb{P}_x(T_{D_3} \le n < \tau_\partial) &= \mathbb{E}_x \left( \mathbb{1}_{T_{D_3} \le n} \mathbb{P}_{X_{T_{D_3}}} (n - u < \tau_\partial) \Big|_{u=T_{D_3}} \right) \\
&\le \mathbb{E}_x \left( \mathbb{1}_{T_{D_3} \le n} \varphi_1(X_{T_{D_3}}) \gamma^{n-T_{D_3}} \right) \le A \mathbb{E}_x \left( \mathbb{1}_{T_{D_2^c} \le n} \gamma^{n-T_{D_2^c}} \right),
\end{aligned}
$$
since {$T_{D_3} \le n$} $\subset$ {$T_{D_2^c} = T_{D_3}$}. Now, using Theorem 2.2 and the fact that $\eta^Y$ is uniformly bounded from above and away from 0 on $K$, we deduce that there exist constants $C, C' > 0$ such that

$$
\begin{align*}
\mathbb{E}_x \left( \mathbb{1}_{T_{D_2^c} \le n} \gamma^{n-T_{D_2^c}} \right) &= \sum_{k=1}^{n} \mathbb{P}_x (T_{D_2^c} = k) \gamma^{n-k} \le \sum_{k=1}^{n} \mathbb{P}_x (T_{D_2^c} > k-1) \gamma^{n-k} \\
&\le C \sum_{k=1}^{n} (\theta_0^Y)^{k-1} \gamma^{n-k} \le C (\theta_0^Y)^{n-1} \frac{1}{1-\gamma/\theta_0^Y} \\
&\le C C' \frac{(\theta_0^Y)^{-1}}{1-\gamma/\theta_0^Y} \inf_{y \in K} \mathbb{P}_y (n < T_{D_2^c}).
\end{align*}
$$

Finally, we obtain from (6.5) that there exists a constant $C'' > 0$ such that, for all $x \in K$,

$$
\mathbb{P}_x(n < \tau_\partial) \le C'' \inf_{y \in K} \mathbb{P}_y(n < T_{D_2^c}) \le C'' \inf_{y \in K} \mathbb{P}_y(n < \tau_\partial). \quad (6.6)
$$

This concludes Step 3.

Step 4. Conclusion.

Assumption (E4) for the process X is an immediate consequence of Assump-
tion (E4) for the process Y, and hence we have checked that X satisfies Assump-
tion (E). The convergence result of Theorem 6.1 is exactly the convergence result
obtained in Theorem 2.1.

Note that (6.6) entails that, for any $x \in K$,

$$
\limsup_{n \to +\infty} (\theta_0^Y)^{-n} \mathbb{P}_x(n < T_{D_2^c}) &\le \limsup_{n \to +\infty} (\theta_0^Y)^{-n} \mathbb{P}_x(n < \tau_\partial) \\
&\le C'' \limsup_{n \to +\infty} (\theta_0^Y)^{-n} \mathbb{P}_x(n < T_{D_2^c})
$$

and that Theorem 2.2 applied to Y entails

$$
\limsup_{n \to +\infty} (\theta_0^Y)^{-n} \mathbb{P}_x (n < T_{D_2^c}) = \eta^Y (x) < +\infty.
$$

Since it follows from Theorem 2.2 applied to $X$ that $\lim_{n \to +\infty} \theta_0^{-n} \mathbb{P}_x(n < \tau_\partial) > 0$, we deduce that $\theta_0 = \theta_0^Y$.

Finally, for all $x \in K$, the structure of the transition graph of $X$ implies that

$$
0 = \mathbb{P}_x(X_n \in D_1 \mid n < \tau_\partial) \xrightarrow[n \to +\infty]{n \to +\infty} \nu_{QSD}(D_1),
$$

so that $v_{QSD}(D_1) = 0$. Moreover, for all $x \in D_3$, Markov's inequality and Assump-
tion (H2) yield the inequality $\mathbb{P}_x(n < \tau_\partial) \le A\gamma^n$, for all $x \in K$ and all $n \ge 1$. Since
$\theta_0 = \theta_0^Y > \gamma$ by assumption, we deduce that, for all $x \in K$, $\lim_{n \to +\infty} \theta_0^{-n} \mathbb{P}_x(n <
\tau_\partial) = 0$, which means that $\eta(x) = 0$.

This concludes the proof of Theorem 6.1.

<figure><img src="image_4.png" /></figure>
## 6.2 Countably many communication classes

In this section, we study a particular case of a continuous time càdlàg Markov process $(X_t)_{t \in [0, +\infty)}$ with a countable infinity of communication classes and we show that the process admits a quasi-stationary distribution.

More precisely, we assume that $X$ evolves in the state space $\mathbb{N} \times \mathbb{Z}_+$ and, denoting $N_t \in \mathbb{N}$ and $Y_t \in \mathbb{Z}_+$ the two components of $X_t$ for all $t \in [0, +\infty)$, that there exist three positive functions $b, d, f: \mathbb{N} \to (0, +\infty)$ such that

* $N$ is a Poisson process with intensity 1,

* $Y$ is a process such that, at time $t$,

$$ Y \text{ jumps from } Y_t \text{ to } y \in \mathbb{Z}_+ \text{ with rate } \begin{cases} f(N_t) b(Y_t) & \text{if } y = Y_t + 1 \text{ and } Y_t \ge 1, \\ f(N_t) d(Y_t) & \text{if } y = Y_t - 1 \text{ and } Y_t \ge 1, \\ 0 & \text{otherwise.} \end{cases} $$

The set $\mathbb{N} \times \{0\}$ is absorbing for $X$ and we are interested in the quasi-stationary behavior of $X$ conditioned to not hit this set. Note that, in this case, each set $\{n\} \times \mathbb{N}$ is a communication class.

*Remark 18.* This process can be used to model the survival of an individual (for example a bacterium) whose metabolic efficiency (for example its ability to consume resources) changes with time, due to aging [96]. Here $Y$ is the vitality of the individual, who dies when its vitality hits 0, $f(N)$ is the metabolic rate of the individual, which may for example decrease in the early life of the individual up to age $n_0$ and then accelerates progressively.

This can also model the accumulation of deleterious mutations in a population under the assumption that mutations do not overlap, i.e. that when a mutant succeeds to invade the population (either because they are advantaged or due to genetic drift for deleterious mutations), other types of mutants disappear rapidly. Here $Y$ represents the size of the population and $N$ the number of mutations. It is typical to assume that the first $n_0$ mutations that invade are advantageous (which corresponds to adaptation), and afterwards that deleterious mutations start to accumulate, hence accelerating the extinction of the species (extinction vortex [37, 36]).

In both cases, it is relevant to assume that $f$ is decreasing on $\{1, 2, \dots, n_0\}$ and increasing on $\{n_0, n_0 + 1, \dots\}$.

We assume that $(d(y)-b(y))/y \to +\infty$ when $y \to +\infty$ or that there exists $\delta > 1$ such that $d(y) - \delta b(y) \to +\infty$. Hence the birth and death process $Z$ evolving in $\mathbb{N}$, with birth rates $(b(z))_{z \in \mathbb{N}}$ and death rates $(d(z))_{z \in \mathbb{N}}$, satisfies Assumption (F)
by Theorem 5.1 (see Example 8). In particular, there exist an eigenvalue $\lambda_0^Z > 0$ and eigenfunction $\eta^Z : \mathbb{N} \to (0, +\infty)$ such that, for all $z \in \mathbb{N}$, $\mathcal{L}^Z \eta^Z = -\lambda_0^Z \eta^Z$, where the operator $\mathcal{L}^Z$ is defined as the operator $\mathcal{L}$ in (5.1).

**Theorem 6.2.** Assume also that there exists a unique $n_0 \in \mathbb{N}$ such that $f(n_0) = \min_{n \in \mathbb{N}} f(n)$ and that $\liminf_{n \to +\infty} f(n) > f(n_0) + \frac{1}{\lambda_0^Z}$. Then the process $X$ satisfies Assumption (F) and admits a quasi-stationary distribution $v_{QSD}$ whose domain of attraction contains all Dirac measures $\delta_{n,y}$, with $n \le n_0$ and $y \in \mathbb{N}$.

Of course, all the consequences of Theorem 3.5 also apply here, taking the functions $\psi_1$ and $\psi_2$ as described in the proof.

*Proof.* The proof makes use of the special structure of the process $Y$, which can be constructed as

$$Y_t = Z_{\int_0^t f(N_s)ds}^T, \quad \forall t \ge 0.$$

In general, we shall denote the objects related to $Z$ with a superscript $Z$, for example $\psi_1^Z$ is the functions involved in (F2) and $L^Z$ is the set involved in (F) for $Z$. We can assume without loss of generality as in Theorem 5.1 that $L^Z = D_0^Z$, i.e.

$$\mathcal{L}^Z \psi_1^Z \le -\lambda_1^Z \psi_1^Z + \bar{C} \mathbb{1}_{L^Z} \quad (6.7)$$

with $\psi_1^Z(0) = 0$ and $\lambda_1^Z > \lambda_0^Z$.

Our goal is to apply Theorem 5.1 to the process $X = (N, Y)$. We define the finite set $D_0 = \{n_0\} \times L^Z$, so that $\mathbb{P}_{(n_0,x)}(X_1 = (n_0, y)) > 0$ for all $(n_0, x)$ and $(n_0, y)$ in $D_0$, and check that $\lambda_0 \le f(n_0)\lambda_0^Z + 1$. Indeed, for all $y \in L^Z$,

$$e^{t(f(n_0)\lambda_0^Z + 1)} \mathbb{P}_{(n_0, y)}((N_t, Y_t) = (n_0, y)) \ge e^{tf(n_0)\lambda_0^Z} \mathbb{P}_y^Z(Z_{f(n_0)t} = y) \\ \xrightarrow{t \to +\infty} \eta^Z(y) v_{QSD}^Z(\{y\}) > 0.$$

We fix $\lambda_1$ such that

$$f(n_0)\lambda_0^Z + 1 < \lambda_1 < \left( \lambda_0^Z \inf_{n \ne n_0} f(n) + 1 \right) \wedge \left( \lambda_0^Z \liminf_{n \to +\infty} f(n) \right) \wedge \left( \lambda_1^Z f(n_0) + 1 \right)$$

and we choose

* $n_1 > n_0$ such that, for all $n \ge n_1$, $\lambda_1 < \lambda_0^Z f(n)$;

* $c > 0$ small enough so that $\psi_1^Z(x) \ge c\eta^Z(x)$ for all $x \ge 1$ (such a constant exists thanks to Theorem 2.2);

* $a > 0$ large enough so that $\lambda_1 < \lambda_1^Z f(n_0) + 1 - e^{-a}$;
• $\varepsilon > 0$ small enough so that $\lambda_1 < (\lambda_0^Z - \varepsilon) \inf_{n \neq n_0} f(n) + 1$;

• $b > a$ large enough so that $\lambda_1 < (\lambda_0^Z - \varepsilon) \inf_{n \neq n_0} f(n) + 1 - e^{-b}$ and $\bar{C}e^{a-b} < \varepsilon \inf_{y \in L^Z} \eta^Z(y)$, where the constant $\bar{C}$ is the one of (6.7).

We can now define

$$ \psi_1(n, y) = \begin{cases} \psi_1^Z(y) & \text{if } n = n_0, \\ e^{a(n_0-n)} \psi_1^Z(y) + e^{b(n_0-n)} \eta^Z(y) & \text{if } n < n_0, \\ c e^{-a(n-n_0)} \eta^Z(y) & \text{if } n_0 < n < n_1, \\ c e^{-a(n_1-n_0)} \eta^Z(y) & \text{if } n_1 \le n. \end{cases} $$

In the case where $n < n_0$, it follows from (6.7) that

$$ \begin{aligned} \mathcal{L}\psi_1(n, y) &\le -(\lambda_1^Z f(n) + 1 - e^{-a}) e^{a(n_0-n)} \psi_1^Z(y) \\ &\quad - (\lambda_0^Z f(n) + 1 - e^{-b} \mathbb{1}_{n<n_0-1}) e^{b(n_0-n)} \eta^Z(y) \\ &\quad + \frac{\bar{C}}{\inf_{z \in L^Z} \eta^Z(z)} f(n) e^{a(n_0-n)} \eta^Z(y) \\ &\le -\lambda_1 e^{a(n_0-n)} \psi_1^Z(y) - [(\lambda_0^Z - \varepsilon) f(n) + 1 - e^{-b} \mathbb{1}_{n<n_0-1}] e^{b(n_0-n)} \eta^Z(y) \\ &\quad + \varepsilon f(n) e^{a(n_0-n)} (e^{b-a} - e^{(b-a)(n_0-n)}) \eta^Z(y) \\ &\le -\lambda_1 \psi_1(n, y). \end{aligned} $$

When $n = n_0$, we have

$$ \begin{aligned} \mathcal{L}\psi_1(n_0, y) &\le -\lambda_1^Z f(n_0) \psi_1^Z(y) + \bar{C} \mathbb{1}_{L^Z}(y) f(n_0) + c e^{-a} \eta^Z(y) - \psi_1^Z(y) \\ &\le -\lambda_1 \psi_1(n_0, y) + \bar{C} f(n_0) \mathbb{1}_{D_0}(n_0, y). \end{aligned} $$

When $n_0 < n < n_1$, we have

$$ \begin{aligned} \mathcal{L}\psi_1(n, y) &\le -\lambda_0^Z f(n) c e^{-a(n-n_0)} \eta^Z(y) + c e^{-a(n-n_0)} \eta^Z(y) (e^{-a} - 1) \\ &\le -\lambda_1 \psi_1(n, y). \end{aligned} $$

When $n_1 \le n$, we have

$$ \mathcal{L}\psi_1(n, y) \le -\lambda_0^Z f(n) \eta^Z(y) \le -\lambda_1 \psi_1(n, y). $$

Finally we have proved that $\mathcal{L}\psi_1(n, y) \le -\lambda_1 \psi_1(n, y) + \bar{C}f(n)\mathbb{1}_{D_0}(n, y)$, where $\lambda_1 > \lambda_0$. Now, note that, since $Z$ is a birth-death process, basic comparison arguments imply that $\eta^Z(k) \ge \eta^Z(1) > 0$ for all $k \ge 1$. Therefore, the function $\psi_1$ is uniformly lower bounded, so that it satisfies the assumptions of Theorem 5.1 up to a multiplicative constant.

Hence, Theorem 5.1 allows us to conclude the proof. The fact that all Dirac masses $\delta_{(n,y)}$ with $n \le n_0$ belong to the domain of attraction follows from Corollary 2.5. $\square$
# 7 Application to processes in continuous state space and discrete time

Discrete time Markov models in continuous state space and with absorption naturally arise in many applications. Examples of such processes are given by perturbed dynamical systems, cf. e.g. [44, 9, 5, 62], or piecewise deterministic Markov processes when one looks at the process at jump times only (see e.g. [3]). We provide in Section 7.1 a general criterion applying to such processes with arbitrarily close to 1, state-dependent killing probability, and we give applications to Euler schemes for diffusions absorbed at the boundary of a domain. In Section 7.2, we consider perturbed dynamical systems in finite dimension. We first consider the case of unbounded domains with unbounded perturbation. Subsection 7.2.1 assumes that the perturbation has bounded density with respect to Lebesgue's measure and Subsection 7.2.2 provides examples with perturbations with unbounded density. Finally, the case of bounded perturbations is studied in Subsection 7.2.3. Theorem 1.2 of the Introduction is obtained as an application of the results of Section 7.2.1.

## 7.1 Two sided estimates for processes with killing

Let $(Y_n, n \in \mathbb{Z}_+)$ be a Markov process evolving on a measurable state space $E \cup \{\partial\}$ with transition kernel $(Q(y, \cdot)_{y \in E \cup \{\partial\}})$ such that $\partial \notin E$ is absorbing (i.e. $Q(\partial, \{\partial\}) = 1$) satisfying a two-sided estimate (see for instance [10, 40, 17]), which means that there exist a probability measure $\zeta$ on $E$, a positive function $g: E \to (0, +\infty)$ and a constant $C > 1$ such that, for all $y \in E$ and all measurable sets $A \subset E$,

$$g(y)\zeta(A) \leq Q(y, A) \leq Cg(y)\zeta(A). \quad (7.1)$$

Condition (7.1) is known to be satisfied for various models (see e.g. [9] or the references in [17]). It is also well known (see [10, 17]) that this implies that $Y$ admits a unique quasi-stationary distribution $v_{QSD}^Y$ for which the convergence in (2.1) holds true for the total variation distance with geometric speed uniform with respect to the initial distribution $\mu$ on $E$. Our aim is to generalize this result to processes obtained from $Y$ with additional killing (or penalization).

More precisely, let $p: E \times E \to (0, 1]$ be measurable and consider the Markov process $X$ evolving in $E \cup \{\partial\}$ with transition kernel $P(x, \cdot)_{x \in E \cup \{\partial\}}$ defined by

$$P(x, dy) = \begin{cases} p(x, y)Q(x, dy) + (1 - p(x, y))\delta_{\partial}(dy) & \text{if } x \in E \\ \delta_{\partial}(dy) & \text{if } x = \partial. \end{cases}$$
Observe that Condition (7.1) may not be satisfied by the kernel $P$ in cases where $\inf_{x,y \in E} p(x, y) = 0$. We also emphasize that the kernel $P$ generates a penalized semigroup of $(Y_n)_{n \in \mathbb{Z}_+}$, in the sense that, for any function $f: E \to \mathbb{R}_+$, all $x \in E$ and all $n \ge 1$, one has

$$ \mathbb{E}_x(f(X_n)1_{n<\tau_\partial}) = \mathbb{E}_x(p(x, Y_1) \cdots p(Y_{n-1}, Y_n) f(Y_n)1_{n<\tau_\partial^Y}), $$

where $\tau_\partial$, resp. $\tau_\partial^Y$, is the absorption time for $X$, resp. $Y$, in $\partial$.

**Theorem 7.1.** Assume that there exists an increasing sequence $(L_k)_{k \ge 1}$ of measurable subsets of $E$ such that $E = \cup_{k=1}^{+\infty} L_k$ and $\inf_{x,y \in L_k} p(x, y) > 0$ for all $k \ge 1$. Then $X$ satisfies Assumption (E) with $\varphi_1 = 1$ and $\varphi_2$ positive on $E$. In particular, $X$ admits a unique quasi-stationary distribution whose domain of attraction contains all probability measures on $E$.

*Example 9.* Typical examples of discrete-time Markov processes in continuous state space are given by Euler schemes for stochastic differential equations. We consider the SDE $dY_t = b(Y_t)dt + \sigma(Y_t)dB_t$ in $\mathbb{R}^d$, with $b$ and $\sigma$ bounded measurable on $\mathbb{R}^d$ and $\sigma$ uniformly elliptic on $\mathbb{R}^d$. Its standard Euler scheme with time-step $\delta$ is the Markov chain $(X_n, n \ge 0)$ defined as

$$ X_{n+1} = b(X_n)\delta + \sqrt{\delta}\sigma(X_n)G_n, \qquad (7.2) $$

where $(G_n, n \ge 0)$ is an i.i.d. sequence of $\mathcal{N}(0, 1)$ Gaussian variables in $\mathbb{R}^d$. In the case of a SDE absorbed at its first exit time of a bounded open connected domain $D \subset \mathbb{R}^d$, the “naive” Euler scheme, constructed as above with the additional rule that $X_n$ is immediately sent to $\partial$ when $X_n \notin D$, is not good in terms of weak error. Indeed, when $X_n$ is close to the boundary of $D$ and $X_{n+1}$ remains in $D$, the path of the SDE $Y$ in the time interval $[n\delta, (n+1)\delta]$ might have exited $D$. In this case, it is more efficient to construct the Brownian path that links 0 to $G_n$ on the time interval $[n\delta, (n+1)\delta]$ as a Brownian bridge $(\tilde{G}_t, t \in [n\delta, (n+1)\delta])$ such that $\tilde{G}_{n\delta} = 0$ and $\tilde{G}_{(n+1)\delta} = G_n$, so that one can approximate the path of the diffusion on this time interval as

$$ \tilde{X}_t = b(X_n)(t - n\delta) + \sqrt{\delta}\sigma(X_n)\tilde{G}_t, \quad \forall t \in [n\delta, (n+1)\delta], $$

and approximate the absorption event as $\{\exists t \in [n\delta, (n+1)\delta] : \tilde{X}_t \notin D\}$. The corresponding Euler scheme is thus obtained as the Markov chain $X$ as defined in (7.2) with the penalization $p(X_n, X_{n+1}) = \mathbb{P}(\exists t \in [n\delta, (n+1)\delta] : \tilde{X}_t \notin D)$. For a detailed presentation and study of this kind of modified Euler schemes, we refer the reader to [79, 51, 52, 11].
Using Theorem 7.1, we obtain the existence and convergence to a unique quasi-stationary distribution for this modified Euler scheme. Indeed, (7.1) is satisfied for the naive Euler scheme with $\zeta$ equal to the restriction of Lebesgue's measure to $D$ and a constant function $g$, thanks to the boundedness of the domain $D$, the uniform ellipticity of $\sigma$ and the boundedness of $b$ and $\sigma$. In addition, it follows from the connectedness of the domain $D$, the uniform ellipticity of $\sigma$ and the boundedness of $b$ and $\sigma$ that $\inf_{x,y \in K} p(x, y) > 0$ for any compact subset $K$ of $D$.

*Proof of Theorem 7.1.* For all $k \ge 1$, we define the set $K_k = \{x \in L_k \text{ s.t. } g(x) \ge 1/k\}$. Let $k_0$ be large enough so that $\zeta(K_{k_0}) > 0$. Then one has, for all $k \ge k_0$, all $x \in K_k$ and all measurable set $A \subset E$,

$$ \mathbb{P}_x(X_1 \in A \cap K_{k_0}) \ge g(x) \int_{A \cap K_{k_0}} p(x, y) \zeta(dy) \ge \frac{\zeta(K_{k_0}) \inf_{u,v \in L_k} p(u, v)}{k} v(A \cap K_{k_0}), \quad (7.3) $$

where $v$ is the probability measure on $K_{k_0}$ defined by $v(A) = \zeta(A)/\zeta(K_{k_0})$. We fix $k \ge k_0$ large enough so that $C/k < \frac{\zeta(K_{k_0}) \inf_{u,v \in L_k} p(u, v)}{k_0}$, where the constant $C$ is the one of (7.1), and set $K = K_k$.

Let us now check that Condition (E) is satisfied with the above choices of $K$ and $v$ (extended by $0$ to $K_k \setminus K_{k_0}$), and with $\theta_1 = C/k$ and $\theta_1 < \theta_2 < \frac{\zeta(K_{k_0}) \inf_{u,v \in L_k} p(u, v)}{k_0}$. Setting $\varphi_1 = 1$, one has

$$ P_1 \varphi_1(x) \le 1, \quad \forall x \in K, $$

$$ P_1 \varphi_1(x) \le C g(x) \le \theta_1 = \theta_1 \varphi_1(x), \quad \forall x \in E \setminus K, $$

so that the first and third lines of Condition (E2) are satisfied. Using Markov's property, one deduces from (7.3) that $\theta_2^{-n} \inf_{x \in K} \mathbb{P}_x(X_n \in K_{k_0}) \to +\infty$ when $n \to +\infty$. Hence Lemma 3.1 implies that the second and fourth lines of Condition (E2) are satisfied. It also implies that Condition (E4) is satisfied. Note also that the function $\varphi_2$ provided by Lemma 3.1 is positive on $E$ since $g$ is positive in (7.1).

Moreover, for all $x \in E$, all $y \in K$ and all measurable set $A \subset E$,

$$ \begin{aligned} \mathbb{P}_x(X_1 \in A \cap K) &\le C g(x) \zeta(A \cap K) \le \frac{C g(x) k g(y)}{\inf_{K \times K} p} \int_{A \cap K} p(y, z) \zeta(dz) \\ &\le \frac{C \|g\|_{\infty} k}{\inf_{K \times K} p} \mathbb{P}_y(X_1 \in A \cap K). \end{aligned} $$

We deduce from Proposition 3.3 with $n_0 = m_0 = 1$ that Conditions (E1) and (E3) are satisfied, which concludes the proof of Theorem 7.1. $\square$
## 7.2 Perturbed dynamical systems

We consider the following perturbed dynamical system

$$X_{n+1} = f(X_n) + \xi_n,$$

where $f: \mathbb{R}^d \to \mathbb{R}^d$ is a measurable function and $(\xi_n)_{n \in \mathbb{N}}$ is an i.i.d. sequence in $\mathbb{R}^d$. We assume that the process evolves in a measurable set $D$ of $\mathbb{R}^d$ with positive Lebesgue measure, meaning that it is immediately sent to $\partial \not\in \mathbb{R}^d$ as soon as $X_n \not\in D$. We shall consider two situations below, where the random variables $\xi_n$ are unbounded or almost surely bounded. In the unbounded case, different methods must be used depending on whether $\xi_n$ has a bounded density with respect to Lebesgue's measure or not.

The same arguments would also work if $X_{n+1} = f(X_n) + \xi_n(X_n)$, where the sequence of random maps $(x \mapsto \xi_n(x))_{n \ge 0}$ are i.i.d. We leave the appropriate extensions of our assumptions and arguments to the reader.

### 7.2.1 The case of unbounded perturbation with bounded density

We consider here the case where the random variables $\xi_n$ have support $\mathbb{R}^d$.

**Proposition 7.2.** Assume that $f$ is locally bounded, that the law of $\xi_n$ has a bounded density $g(x)$ with respect to Lebesgue's measure such that

$$\inf_{|x| \le R} g(x) > 0, \quad \forall R > 0,$$

and that there exists a locally bounded function $\varphi: \mathbb{R}^d \to [1, +\infty)$ such that $x \mapsto \mathbb{E}(\varphi(x + \xi_1))$ is locally bounded on $\mathbb{R}^d$ and such that

$$\limsup_{|x| \to +\infty, x \in D} \frac{\mathbb{E}(\varphi(f(x) + \xi_1))}{\varphi(x)} = 0. \qquad (7.4)$$

Then Condition (E) is satisfied with $\varphi_1 = \varphi$ and $\varphi_2$ positive on $D$.

Note that, if $D$ is bounded, the last result is already a consequence of the classical criterion based on (7.1). Before proving this result, let us give three applications.

**Example 10.** If there exists $\alpha > 0$ such that $\mathbb{E}e^{\alpha|\xi_1|} < +\infty$ and if $|x| - |f(x)| \to +\infty$ when $|x| \to +\infty$, then Proposition 7.2 applies. Indeed, choosing $\varphi(x) = \exp(\alpha|x|)$, we have

$$\frac{\mathbb{E}\varphi(|f(x) + \xi_1|)}{\varphi(x)} \le e^{\alpha(|f(x)| - |x|)} \mathbb{E}e^{\alpha|\xi_1|} \underset{|x| \to +\infty}{\longrightarrow} 0.$$

For instance, this covers the case of Gaussian perturbations, as stated in Theorem 1.2 in the introduction.
**Example 11.** If there exists $p > 0$ such that $\mathbb{E}(\xi_1^p) < +\infty$ and if $|f(x)| = o(|x|)$ when $|x| \to +\infty$, then Proposition 7.2 applies. Indeed, choosing $\varphi(x) = (1 + |x|)^p$, we have

$$ \frac{\mathbb{E}\varphi(|f(x) + \xi_1|)}{\varphi(x)} \le \frac{(1 + |f(x)|)^p}{(1 + |x|)^p} \mathbb{E}[(1 + |\xi_1|)^p] \underset{|x| \to +\infty}{\longrightarrow} 0. $$

**Example 12.** If $\mathbb{E}\log(1 + |\xi_1|) < \infty$ and $|f(x)| \le C|x|^{\varepsilon(x)}$ for some $C > 0$ and some $\varepsilon(x) \to 0$ when $|x| \to +\infty$, then Proposition 7.2 applies. Indeed, choosing $\varphi(x) = \log(e + |x|)$, we have

$$ \frac{\mathbb{E}\varphi(|f(x) + \xi_1|)}{\varphi(x)} \le \frac{\log(e + C) + \varepsilon(x) \log(e + |x|)}{\log(e + |x|)} + \frac{\mathbb{E}\log(1 + |\xi_1|)}{\log(e + |x|)}. $$

**Proof of Proposition 7.2.** We first prove Conditions (E2) and (E4) and conclude the proof with Proposition 3.3.

**Step 1.** Conditions (E2) and (E4) are satisfied.

Let $K_1 \subset D$ be a bounded measurable set with positive Lebesgue measure. Then, for all $x \in K_1$, denoting by $\lambda_d$ the Lebesgue measure on $\mathbb{R}^d$,

$$ \mathbb{P}_x(X_1 \in K_1) = \mathbb{P}(f(x) + \xi_1 \in K_1) \ge \lambda_d(K_1) \inf_{u \in K_1 + B(0, \sup_{K_1} |f|)} g(u) > 0. $$

Fix $\theta_2 \in (0, \lambda_d(K_1) \inf_{u \in K_1 + B(0, \sup_{K_1} |f|)} g(u))$, we deduce that, for all $x \in K_1$,

$$ \theta_2^{-n} \inf_{x \in K_1} \mathbb{P}_x(X_n \in K_1) \ge \theta_2^{-n} \inf_{x \in K_1} \mathbb{P}_x(X_1 \in K_1, \dots, X_n \in K_1) \xrightarrow{n \to +\infty} +\infty. $$

Fix $0 < \theta_1 < \theta_2$, and, using (7.4), consider a bounded subset $K \subset D$ containing $K_1$ and such that, for all $x \in D \setminus K$, $P_1\varphi(x) \le \theta_1\varphi(x)$. Since $K$ is bounded, one has

$$ \inf_{x \in K} \mathbb{P}_x(X_1 \in K_1) \ge \lambda_d(K_1) \inf_{u \in K_1 + B(0, \sup_K |f|)} g(u) > 0, $$

so that

$$ \theta_2^{-n} \inf_{x \in K} \mathbb{P}_x(X_n \in K) \ge \theta_2^{-n} \lambda_d(K_1) \inf_{u \in K_1 + B(0, \sup_K |f|)} g(u) \inf_{x \in K_1} \mathbb{P}_x(X_{n-1} \in K_1) $$

and thus $\theta_2^{-n} \inf_{x \in K} \mathbb{P}_x(X_n \in K)$ converges to $+\infty$ when $n \to +\infty$. Lemma 3.1 then entail that Condition (E4) is satisfied and that there exists a function $\varphi_2 : D \to [0, 1]$ such that $P_1\varphi_2(x) \ge \theta_2\varphi_2(x)$ for all $x \in D$ and such that $\inf_K \varphi_2 > 0$. In addition, for all $x \in D$, $\mathbb{P}_x(X_1 \in K) \ge \lambda_d(K) \inf_{u \in K-f(x)} g(u) > 0$, so that $P_1 \mathbb{1}_K(x) > 0$. Hence, the function $\varphi_2$ of Lemma 3.1 also satisfies that $\varphi_2(x) > 0$ for all $x \in E$.

Setting $\varphi_1 = \varphi$, we deduce that Conditions (E2) and (E4) are satisfied for the set $K$.
### Step 2. Comparison of transition probabilities.

Let us prove that Proposition 3.3 applies with $n_0 = m_0 = 1$. For all $x \in D$, we have

$$ \mathbb{P}_x(X_1 \in \cdot \cap K) \leq \sup_{u \in \mathbb{R}^d} g(u) \lambda_d(\cdot \cap K). $$

Moreover, for all $y \in K$,

$$ \begin{aligned} \mathbb{P}_y(X_1 \in \cdot) &\geq \mathbb{P}(f(y) + \xi_1 \in \cdot \cap K) \\ &\geq \inf_{u \in K+B(0,\sup_K|f|)} g(u) \lambda_d(\cdot \cap K). \end{aligned} $$

Hence, for all $x \in E$ and all $y \in K$,

$$ \mathbb{P}_x(X_1 \in \cdot \cap K) \leq \frac{\sup_{\mathbb{R}^d} g}{\inf_{K+B(0,\sup_K|f|)} g} \mathbb{P}_y(X_1 \in \cdot). $$

We deduce from Step 1 and Proposition 3.3 that Condition (E) is satisfied with the functions $\varphi_1$ and $\varphi_2$, which concludes the proof. □

### 7.2.2 An example with unbounded perturbation and singular density

The last result made strong use of the boundedness of $g$. Actually, our criteria also apply to perturbations with singular density. We consider here the following example: assume that $f(x) = Ax + B$, where $A$ is an invertible $d \times d$ matrix and $B \in \mathbb{R}^d$, and that there exists $a > 0$ such that the density $g$ of $\xi_n$ satisfies for some constant $C_g$

$$ g(x) \leq C_g \left( \frac{1}{|x|^{d-a}} \vee 1 \right) \quad \forall x \in \mathbb{R}^d. \qquad (7.5) $$

We have the following result.

**Proposition 7.3.** Let $\| \cdot \|$ be a norm on $\mathbb{R}^d$ and assume that

$$ \sup_{x \in \mathbb{R}^d \setminus \{0\}} \frac{\|Ax\|}{\|x\|} < 1. \qquad (7.6) $$

Assume also that $\mathbb{E}e^{\alpha|\xi_1|} < \infty$ for some $\alpha > 0$ and that

$$ \inf_{|x| \le R} g(x) > 0, \quad \forall R > 0. $$

Then Condition (E) is satisfied with $\varphi_1 = \varphi$ and $\varphi_2$ positive on $D$.

The proof of Proposition 7.2 made use of Proposition 3.3 with $n_0 = m_0 = 1$. The proof of Proposition 7.3 requires to apply Proposition 3.3 with $n_0 \ge 2$.
*Proof.* The first step of the proof of Proposition 7.2 remains valid taking $\varphi(x) = e^{\alpha\|x\|}$ for $\alpha > 0$ small enough and using (7.6) and the equivalence of the norms $|\cdot|$ and $\|\cdot\|$ (the computation is similar to the one of Example 10). So we only have to prove that (3.2) is satisfy and apply Proposition 3.3.

We define $n_0 = \lceil d/a \rceil$ and we assume without loss of generality (reducing slightly $a$ if needed) that $n_0 a > d$. We observe that

$$X_{n_0} = A^{n_0} x + A^{n_0-1}(B + \xi_1) + \dots + B + \xi_{n_0}.$$

Using (7.5) and the fact that $\sup_{x \neq 0} \frac{|Ax|}{|x|} \le C_{\|\cdot\|}^2$ where the constant $C_{\|\cdot\|}$ is such that $C_{\|\cdot\|}^{-1}|\cdot| \le \|\cdot\| \le C_{\|\cdot\|}|\cdot|$, the density $g_2$ of $A\xi_1 + \xi_2$ satisfies

$$
\begin{align}
g_2(x) &= \frac{1}{|\det A|} \int_{\mathbb{R}^d} g(x-y)g(A^{-1}y)dy \nonumber \\
&\le \frac{C_g^2}{|\det A|} \int_{\{y:|A^{-1}y|\le 1\}\cap B(x,1)} \frac{1}{|x-y|^{d-a}} \frac{1}{|A^{-1}y|^{d-a}} dy + C_g \left(1 + \frac{1}{|\det A|}\right) \nonumber \\
&\le \frac{C_g^2 C_{\|\cdot\|}^{2(d-a)}}{|\det A|} \int_{B(0,C_{\|\cdot\|}^2)} \frac{1}{|x-y|^{d-a}} \frac{1}{|y|^{d-a}} dy + C_g \left(1 + \frac{1}{|\det A|}\right) \nonumber \\
&= \frac{C_g^2 C_{\|\cdot\|}^{2(d-a)}}{|\det A|} \frac{1}{|x|^{d-2a}} \int_{B(0,C_{\|\cdot\|}^2/|x|)} \frac{1}{|\frac{x}{|x|}-u|^{d-a}} \frac{1}{|u|^{d-a}} du + C_g \left(1 + \frac{1}{|\det A|}\right), \tag{7.7}
\end{align}
$$

where we made the change of variable $u = y/|x|$.

If $2a > d$ (i.e. if $n_0 = 2$), we can bound the integral in the right-hand side as follows:

$$
\begin{align*}
\int_B \left(0, \frac{c_{\|\cdot\|}^2}{|x|}\right) \frac{1}{\left|\frac{x}{|x|}-u\right|^{d-a}} \frac{1}{|u|^{d-a}} du &\le C + 2^d \int_B \left(0, \frac{c_{\|\cdot\|}^2}{|x|}\right) \wedge B(0,2) \frac{1}{|u|^{2d-2a}} du \\
&\le C + \frac{C}{2a-d} \frac{1}{|x|^{2a-d}},
\end{align*}
$$

where the constant $C$ may change from line to line. Therefore, $g_2$ is bounded if $2a > d$.

Otherwise, if $2a < d$, the integral in the right-hand side of (7.7) can be bounded by the same integral over $\mathbb{R}^d$ and thus it is uniformly bounded with respect to $x$, so $g_2$ is bounded by $C(1 \vee 1/|x|^{d-2a})$. In this case, we can proceed similarly to bound the density $g_3$ of $A^2\xi_1 + A\xi_2 + \xi_3$, and prove by induction that the density $g_{n_0}$ of $A^{n_0-1}\xi_1 + \dots + \xi_{n_0}$ is bounded.
We deduce that

$$ \mathbb{P}_x(X_{n_0} \in \cdot \cap K) \le \sup_{u \in \mathbb{R}^d} g_{n_0}(u) \lambda_d(\cdot \cap K). $$

The end of the proof is the same as for Proposition 7.2, using Proposition 3.3 with $m_0 = n_0$. □

### 7.2.3 Two examples with bounded perturbation

The case where $\xi_1$ is a bounded random variable is more involved. To avoid complications, we will focus on the case where $\xi_n$ is a uniform random variable on the unit ball $B(0, 1)$ of $\mathbb{R}^d$. Extensions to different distributions are possible.

We start with the simpler case of bounded domain $D$ and contracting dynamical system $f$.

**Proposition 7.4.** Assume that $D$ is a bounded, connected open set of $\mathbb{R}^d$, that $f$ is continuous and satisfies $|f(x) - x| < 1$ for all $x \in D$. Then Condition (E) is satisfied.

*Proof.* Again, the proof makes use of the criterion of Proposition 3.3.

*Step 1. Construction and properties of the sets $K_\varepsilon$, $\varepsilon > 0$.*

For all $\varepsilon > 0$, let $K'_\varepsilon$ be the connected component of $\{x \in D : d(x, \partial D) \ge 2\varepsilon\}$ with larger Lebesgue measure and let

$$ K_\varepsilon := \bigcup_{x \in K'_\varepsilon} \overline{B(x, \varepsilon)}, $$

which is also a connected compact subset of $D$ with distance to $D^c$ larger than $\varepsilon$. For all $\delta > 0$ and all $x, y \in K_\varepsilon$, we call a sequence $(x_0, x_1, \dots, x_n) \in K_\varepsilon^{n+1}$ for some $n \in \mathbb{N}$ a $\delta$-path linking $x$ to $y$ in $K_\varepsilon$ if $x_0 = x, x_n = y$ and $|x_k - x_{k-1}| < \delta$ for all $1 \le k \le n$. By construction, the set $K_\varepsilon$ satisfies that, for all $\delta > 0$ and all $x, y \in K_\varepsilon$, there exists a $\delta$-path linking $x$ to $y$ in $K_\varepsilon$. In addition, since $K_\varepsilon$ is compact, there exists an integer $n_{\varepsilon, \delta}$ depending only on $\varepsilon$ and $\delta$ such that, for all $x, y \in K_\varepsilon$, there exists a $\delta$-path in $K_\varepsilon$ linking $x$ to $y$ with length less than $n_{\varepsilon, \delta}$. For all $x \in K_\varepsilon$ and all $k \in \{1, \dots, n_{\varepsilon, \delta}\}$ let us define

$$ K_{\varepsilon, \delta}^{(k)}(x) = \left\{ y \in \mathbb{R}^d : \exists x_1, \dots, x_{k-1} \in K_{\varepsilon}, |x_{\ell} - x_{\ell-1}| < \delta \text{ for all } 1 \le \ell \le k \text{ with } x_0 = x \text{ and } x_k = y \right\}. $$

Note that in general, $K_{\varepsilon, \delta}^{(k)}$ is not included in $K_\varepsilon$, but it is included in $D$ if $\delta < \varepsilon$. It follows from above that $K_{\varepsilon, \delta}^{(n_{\varepsilon, \delta})}(x) \supset K_\varepsilon$ for all $x \in K_\varepsilon$.
Let us also prove that $\cup_{\varepsilon>0} K_{\varepsilon} = D$. Let $(x_n)_{n \ge 1}$ be a dense sequence in $D$ and for all $n \ge 1$, let $r_n = d(x_n, \partial D)/2$. Since $D = \cup_{n \ge 1} B(x_n, r_n)$, there exists $n_0 \ge 1$ such that $\cup_{1 \le n \le n_0} B(x_n, r_n)$ has Lebesgue measure larger than $\lambda_d(D)/2$. Since $D$ is connected, there exists a continuous path in $D$ linking $x_i$ to $x_j$ for all $1 \le i, j \le n_0$. Since the distance between this path and $\partial D$ is positive, there exists $\varepsilon > 0$ small enough such that all the points $x_1, \dots, x_{n_0}$ belong to the same connected component of $\{x \in D : d(x, \partial D) \ge 2\varepsilon\}$. We can assume without loss of generality that $\varepsilon < r_n/2$ for all $1 \le n \le n_0$, so that this connected component actually contains $\cup_{1 \le n \le n_0} B(x_n, r_n)$ and hence has the largest Lebesgue measure among all the connected components of $\{x \in D : d(x, \partial D) \ge 2\varepsilon\}$. In particular, $K_{\varepsilon}$ contains $B(x_1, r_1)$ for all $\varepsilon$ small enough. Now, given any $x \in D$, there exists a path linking $x$ to $x_1$ in $D$. Since the distance between this path and $\partial D$ is positive, $x$ belongs to $K_{\varepsilon}$ for all $\varepsilon > 0$ small enough. Hence, we have proved that $\cup_{\varepsilon>0} K_{\varepsilon} = D$ and that the family $(K_{\varepsilon})_{\varepsilon>0}$ is non-increasing with respect to $\varepsilon > 0$ when $\varepsilon$ is small enough.

**Step 2.** *Proof of Condition (3.2) of Proposition 3.3.*

For all $\varepsilon > 0$, since $f$ is continuous,

$$ \delta_{\varepsilon} := \left( 1 - \sup_{x \in K_{\varepsilon}} |f(x) - x| \right) \wedge \varepsilon > 0. $$

Hence, for all $x \in K_{\varepsilon}$,

$$ \mathbb{P}_x(X_1 \in \cdot \cap B(x, \delta_{\varepsilon})) \ge c_d \lambda_d(\cdot \cap B(x, \delta_{\varepsilon})), \quad (7.8) $$

for a positive constant $c_d$ only depending on the dimension of the space. In other words, for all $x \in K_{\varepsilon}$,

$$ \mathbb{P}_x(X_1 \in \cdot) \ge c_d \mathbb{P}(x + U \in \cdot) $$

where $U$ is a uniform random variable on $B(0, \delta_{\varepsilon})$. Hence, defining the Markov chain $Y_n = Y_0 + U_1 + \dots + U_n$ where $U_i$ are i.i.d. uniform random variable on $B(0, \delta_{\varepsilon})$, we deduce that

$$ \mathbb{P}_x(X_k \in \cdot) \ge c_d^k \mathbb{P}_x(Y_1, \dots, Y_{k-1} \in K_{\varepsilon} \text{ and } Y_k \in \cdot), \quad \forall x \in K_{\varepsilon}, \forall k \in \mathbb{N}. \quad (7.9) $$

In view of Step 1, the following Lemma 7.5 about the process $Y$ implies that there exists a constant $c' > 0$ such that

$$ \mathbb{P}_x(X_{n_{\varepsilon, \delta_{\varepsilon}/3}} \in \cdot) \ge c' \lambda_d(\cdot \cap K_{\varepsilon}), \quad \forall x \in K_{\varepsilon}. \quad (7.10) $$

Since the law of $X_1$ is dominated by the Lebesgue measure independently of $X_0$, we have proved that, for all $\varepsilon > 0$, (3.2) is satisfied for $K = K_{\varepsilon}$, $n_0 = 1$ and $m_0 = n_{\varepsilon, \delta_{\varepsilon}/3}$. This concludes Step 2 of the proof.
**Lemma 7.5.** For all $1 \le k \le n_{\varepsilon, \delta_{\varepsilon}/3}$, there exists a constant $c'_k > 0$ such that, for all $x \in K_{\varepsilon}$,

$$ \mathbb{P}_x(Y_1, \dots, Y_{k-1} \in K_{\varepsilon} \text{ and } Y_k \in \cdot) \ge c'_k \lambda_d \left( \cdot \cap K_{\varepsilon, \delta_{\varepsilon}/3}^{(k)}(x) \right), \quad (7.11) $$

where $\lambda_d$ is Lebesgue's measure on $\mathbb{R}^d$.

**Step 3.** *Proof of (E2) and (E4).*

Fix $\varepsilon_0 > 0$ such that $K_{\varepsilon_0}$ is non-empty and $(K_{\varepsilon})_{\varepsilon \in (0, \varepsilon_0]}$ is non-increasing. It follows from the definition of $K_{\varepsilon}$ that $\inf_{x \in K_{\varepsilon_0}} \lambda_d(K_{\varepsilon_0} \cap B(x, \delta_{\varepsilon_0})) > 0$. Fixing

$$ \theta_2 < 4 \wedge \left\{ c_d \inf_{x \in K_{\varepsilon_0}} \lambda_d(K_{\varepsilon_0} \cap B(x, \delta_{\varepsilon_0})) \right\}, $$

we deduce from (7.8) that

$$ \lim_{n \to +\infty} \theta_2^{-n} \inf_{x \in K_{\varepsilon_0}} \mathbb{P}_x(X_n \in K_{\varepsilon_0}) = +\infty. \quad (7.12) $$

Since the law of $X_1$ is dominated by the Lebesgue measure and $D = \cup_{0<\varepsilon \le \varepsilon_0} K_{\varepsilon}$, there exists $\varepsilon_1 \in (0, \varepsilon_0]$ small enough such that

$$ \sup_{x \in D} \mathbb{P}_x(X_1 \in D \setminus K_{\varepsilon_1}) \le \theta_2/4. $$

Hence, the function

$$ \varphi_1 : x \in D \to \begin{cases} 1 & \text{if } x \in K_{\varepsilon}, \\ 4/\theta_2 & \text{if } x \in D \setminus K_{\varepsilon_1}, \end{cases} $$

satisfies $P_1\varphi_1(x) \le 2 \le (\theta_2/2)\varphi_1(x)$ for all $x \in D \setminus K_{\varepsilon_1}$. Hence the first and third lines of Condition (E2) are satisfied with $\theta_1 = \theta_2/2$ and $K = K_{\varepsilon_1}$.

We also deduce from (7.10), (7.12), the fact that $K_{\varepsilon_0} \subset K_{\varepsilon_1}$ and Markov's property that

$$ \lim_{n \to +\infty} \theta_2^{-n} \inf_{x \in K_{\varepsilon_1}} \mathbb{P}_x(X_n \in K_{\varepsilon_1}) = +\infty. $$

Hence, it follows from Lemma 3.1 that (E4) is satisfied with $K = K_{\varepsilon_1}$ and that there exists a function $\varphi_2$ satisfying the conditions of (E2) with $\theta_2$ defined above and $K = K_{\varepsilon_1}$.

Therefore, the result follows from Step 2 and Proposition 3.3 with $K = K_{\varepsilon_1}$, $n_0 = 1$ and $m_0 = n_{\varepsilon_1, \delta_{\varepsilon_1}/3}$. $\square$

**Proof of Lemma 7.5.** We prove this result by induction over $k$. Since $Y_1 = x + U_1$ is uniform in $B(x, \delta_{\varepsilon})$, the case $k = 1$ is clear since $K_{\varepsilon, \delta_{\varepsilon}/3}^{(1)} = B(x, \delta_{\varepsilon}/3) \subset B(x, \delta_{\varepsilon})$.
So assume that (7.11) is satisfied for some $1 \le k \le n_{\varepsilon, \delta_{\varepsilon}/3} - 1$ and let us prove it for $k + 1$. Let $A \subset \mathbb{R}^d$ be measurable. Using (7.11) for $k$ and the fact that $Y_{k+1}$ is uniform in $B(Y_k, \delta_{\varepsilon})$ conditionally on $Y_k$, we have

$$
\begin{align*}
& \mathbb{P}_x(Y_1, \dots, Y_k \in K_\varepsilon, Y_{k+1} \in A) \\
& \ge \mathbb{P}_x \left( Y_1, \dots, Y_{k-1} \in K_\varepsilon, Y_k \in K_{\varepsilon, \delta_\varepsilon/3}^{(k)}(x) \cap K_\varepsilon, Y_{k+1} \in A \cap B(Y_k, \delta_\varepsilon) \right) \\
& \ge \frac{c_k'}{\lambda_d(B(0, \delta_\varepsilon))} \int_{K_{\varepsilon, \delta_\varepsilon/3}^{(k)}(x) \cap K_\varepsilon} \mathrm{d}y \int_{A \cap B(y, \delta_\varepsilon)} \mathrm{d}z \\
& = \frac{c_k'}{\lambda_d(B(0, \delta_\varepsilon))} \int_A \lambda_d \left\{ K_{\varepsilon, \delta_\varepsilon/3}^{(k)}(x) \cap K_\varepsilon \cap B(z, \delta_\varepsilon) \right\} \mathrm{d}z \\
& \ge \frac{c_k'}{\lambda_d(B(0, \delta_\varepsilon))} \int_{A \cap K_{\varepsilon, \delta_\varepsilon/3}^{(k+1)}(x)} \lambda_d \left\{ K_{\varepsilon, \delta_\varepsilon/3}^{(k)}(x) \cap K_\varepsilon \cap B(z, \delta_\varepsilon) \right\} \mathrm{d}z,
\end{align*}
$$

where the third equality follows from Fubini's theorem.

Now, for all $z \in K_{\varepsilon, \delta_{\varepsilon}/3}^{(k+1)}(x)$, there exists a path $x_0 = x, x_1, \dots, x_k \in K_{\varepsilon}$ such that $|x_{\ell} - x_{\ell-1}| < \delta_{\varepsilon}/3$ for all $1 \le \ell \le k$ and $|x_k - z| < \delta_{\varepsilon}/3$. By definition of $K_{\varepsilon}$, there exists $y \in K_{\varepsilon}$ such that $x_{k-1} \in B(y, \varepsilon) \subset K_{\varepsilon}$. Let $y'$ be the unique point such that $|y' - x_{k-1}| = \delta_{\varepsilon}/6$ of the half-line with initial point $x_{k-1}$ and containing $y$. Then $B(y', \delta_{\varepsilon}/6) \subset K_{\varepsilon}$. Since $|x_k - z| < \delta_{\varepsilon}/3$ and $|x_{k-1} - x_k| < \delta_{\varepsilon}/3$, we also have $B(y', \delta_{\varepsilon}/6) \subset B(z, \delta_{\varepsilon})$. In addition, for all $y'' \in B(y', \delta_{\varepsilon}/6)$, the path $x_0 = x, x_1, \dots, x_{k-1}, y''$ lies in $K_{\varepsilon}$ and has distance between consecutive point smaller than $\delta_{\varepsilon}/3$. Therefore, $B(y', \delta_{\varepsilon}/6) \subset K_{\varepsilon, \delta_{\varepsilon}/3}^{(k)}(x)$. We conclude that, for all $z \in K_{\varepsilon, \delta_{\varepsilon}/3}^{(k+1)}(x)$,

$$
\lambda_d \left\{ K_{\varepsilon, \delta_{\varepsilon}/3}^{(k)}(x) \cap K_{\varepsilon} \cap B(z, \delta_{\varepsilon}) \right\} \geq \lambda_d(B(0, \delta_{\varepsilon}/6)).
$$

Hence

$$
\mathbb{P}_x(Y_1, \dots, Y_k \in K_\varepsilon, Y_{k+1} \in A) \ge c'_{k+1} \lambda_d \left( A \cap K_{\varepsilon, \delta_\varepsilon/3}^{(k+1)}(x) \right)
$$

for a positive constant $c'_{k+1}$.

<figure><img src="image_5.png" /></figure>

The general case of dynamical systems with bounded perturbations raises
several additional difficulties. We illustrate two of them with the next example
in dimension 1. We consider the Markov process in $D = (0, +\infty)$ defined as

$$
X_0 \in (0, +\infty), \quad X_{n+1} = \alpha X_n - \frac{1}{1+X_n} + \xi_n, \quad \forall n \ge 0
$$
where $\alpha \in (0, 1)$ and $\xi_n$ are i.i.d. with uniform distribution on $[-1, 1]$ and the process is immediately sent to the cemetery point $\partial$ when it leaves $D$. The first difficulty comes from the fact that

$$ \mathbb{P}_x(X_1 > 0) = \left[ 1 - \left( \frac{1}{1+x} - \alpha x \right) \right] \vee 0 \xrightarrow{n \to +\infty} 0, $$

which means that the probability of immediate absorption converges to 1 when $x$ approaches the boundary of $D$. The second difficulty comes from the fact that $|f(x) - x|$ is unbounded on $D$ (in contrast with Proposition 7.4). This example is covered by the following general result.

**Proposition 7.6.** Assume that $X_{n+1} = f(X_n) + \xi_n$ with $D = (0, +\infty)$, $\xi_n$ i.i.d. uniform on $[-1, 1]$, $f$ continuous and there exists $x^* \in D$ such that

$$ (0, x^*) = \{x \in D : |f(x) - x| < 1\} \quad \text{and} \quad [x^*, +\infty) = \{x \in D : f(x) + 1 \le x\}. $$

Then Condition (E) is satisfied.

*Proof.* Fix $K_0 \subset (0, x^*)$ a closed interval with non-empty interior. As in the proof of Proposition 7.4, using in particular (7.9) and (7.11), there exists $n_0 \ge 1$ and $c_0 > 0$ such that, for all $x \in K_0$,

$$ \mathbb{P}_x(X_{n_0} \in \cdot) \ge c_0 \lambda_1(\cdot \cap K_0). $$

Hence there exists a constant $\theta_2 \in (0, 1)$ such that

$$ \theta_2^{-n} \inf_{x \in K_0} \mathbb{P}_x(X_n \in K_0) \xrightarrow{n \to +\infty} +\infty. \quad (7.13) $$

Fix now $\theta_1 < \theta_2$ and $K \subset (0, x^*)$ a closed interval such that $K_0 \subset K$ and

$$ \lambda_1 \{(0, x^*) \setminus K\} \le \frac{\theta_1}{M}, $$

where

$$ M := \frac{2(1 + e^{(x^*+2)/\theta_1})}{\theta_1}. $$

As above, there exists $n_1 \ge 1$ and $c_1 > 0$ such that, for all $x \in K$,

$$ \mathbb{P}_x(X_{n_1} \in \cdot) \ge c_1 \lambda_1(\cdot \cap K). $$

In particular, $\inf_{x \in K} \mathbb{P}_x(X_{n_1} \in K_0) > 0$, so that, using Markov property and (7.13), we deduce that

$$ \theta_2^{-n} \inf_{x \in K} \mathbb{P}_x(X_n \in K) \xrightarrow{n \to +\infty} +\infty. $$
Using Lemma 3.1, we deduce that there exists a function $\varphi_2$ satisfying the conditions of (E2) and that (E4) is satisfied. For all $x \in D$, let

$$ \varphi_1(x) = \begin{cases} 1 & \text{if } x \in K, \\ M & \text{if } x \in (0, x^*) \setminus K, \\ e^{x/\theta_1} & \text{if } x \ge x^*. \end{cases} $$

For $x \ge x^*$, using the fact that the density of $X_1$ on $D$ with respect to Lebesgue measure is bounded by $\frac{1}{2} \mathbb{1}_D$ for all value of $X_0$, we have

$$ \begin{aligned} P_1 \varphi_1(x) &\le \mathbb{E}_x(e^{X_1/\theta_1} \mathbb{1}_{X_1 \ge x^*}) + \mathbb{P}_x(X_1 \in K) + M \mathbb{P}_x(X_1 \in (0, x^*) \setminus K) \\ &\le \mathbb{E}_x(e^{X_1/\theta_1}) + \frac{M}{2} \lambda_1 \{(0, x^*) \setminus K\} \\ &\le \varphi_1(x) e^{(f(x)-x)/\theta_1} \mathbb{E}_x e^{\xi_1/\theta_1} + \frac{\theta_1}{2} \\ &\le \varphi_1(x) e^{-\theta_1^{-1}} \frac{e^{\theta_1^{-1}} - e^{-\theta_1^{-1}}}{2\theta_1^{-1}} + \frac{\theta_1}{2} \varphi_1(x) \le \theta_1 \varphi_1(x). \end{aligned} $$

For $x \in (0, x^*) \setminus K$, since $f(x) + \xi_1 \le x + 2 \le x^* + 2$,

$$ \begin{aligned} P_1 \varphi_1(x) &\le \mathbb{P}_x(X_1 \in K) + e^{(x^*+2)/\theta_1} \mathbb{P}_x(X_1 \ge x^*) + M \mathbb{P}_x(X_1 \in (0, x^*) \setminus K) \\ &\le 1 + e^{(x^*+2)/\theta_1} + \frac{M}{2} \lambda_1 \{(0, x^*) \setminus K\} \\ &\le M \left( \frac{\theta_1}{2} + \frac{\theta_1}{2M} \right) \le \theta_1 \varphi_1(x). \end{aligned} $$

Since $P_1 \varphi_1(x)$ is clearly bounded for $x \le x^*$, we have proved (E2).

To conclude, it remains to observe that (3.2) can be deduced for $n_0 = 1$ and $m_0$ large enough exactly as in the proof of Proposition 7.4. Hence the result follows from Proposition 3.3. $\square$

# 8 Irreducible processes in discrete state space and discrete time

The theory of R-positive matrices is a powerful tool to study absorbed Markov processes in discrete time and space [45]. The goal of Section 8.1 is to show that our criteria allow to recover the results on convergence to quasi-stationarity of this theory. We then study in Section 8.2 a class of discrete Markov chains in discrete time to which criteria based on R-positive matrices do not apply easily.
## 8.1 R-positive matrices

We consider a Markov chain $(X_n, n \in \mathbb{Z}_+)$ in a countable state space $E \cup \{\partial\}$ with $\partial \notin E$ an absorbing point and with irreducible transition probabilities in $E$, i.e. such that for all $x, y \in E$, there exists $n = n(x, y) \ge 1$ such that $\mathbb{P}_x(X_n = y) > 0$. In this case, the most general criterion for existence and convergence to a quasi-stationary distribution is provided in [45]. In this paper, the authors obtain a convergence result similar to the one of Theorem 2.1 restricted to Dirac initial distributions, and the pointwise convergence to $\eta$ as in Theorem 2.2, using the theory of R-positive matrices. In this section, we show how our criterion allows to recover these results, providing in addition the several refinements of Section 2 (including the characterization of a non-trivial subset of the domain of attraction, the convergence of (2.1) for unbounded functions $f$ and a stronger convergence to $\eta$).

We denote by $P$ the transition matrix of the chain $(X_n, n \in \mathbb{Z}_+)$ and we assume that the absorption time $\tau_\partial$ is almost surely finite. Without loss of generality, we will assume that the process is aperiodic, meaning that $\mathbb{P}_x(X_n = y) > 0$ for all $x, y \in E$ provided $n$ is large enough; the extension to general periodic processes is routine, as observed in [45] (see also [27] on this topic in our general setting).

**Proposition 8.1.** *The assumptions of [45, Theorem 1] imply Assumption (E).*

*Proof.* Since $E$ is finite or countable and because of the irreducibility assumption, it is known [107] that the limit

$$ \frac{1}{R} := \lim_{n \to +\infty} \mathbb{P}_x(X_n = y)^{1/n} \qquad (8.1) $$

exists with $1 \le R < \infty$, and is independent of $x, y \in E$. Using [45, Lemma 1], the assumptions of [45, Theorem 1] can be stated as follows: there exist a non-empty set $K \subset E$ and $x_0 \in K$ such that

(a) there exist $\varepsilon_0 > 0$ and a constant $C_1$ such that, for all $x \in K$ and all $n \ge 0$,

$$ \mathbb{P}_x(n < \sigma_K \wedge \tau_\partial) \le C_1 (R + \varepsilon_0)^{-n}, $$

where $\sigma_K$ is the first return time in $K$

$$ \sigma_K := \inf\{n \ge 1, X_n \in K\}. $$

(b) there exists a constant $C_2$ such that, for all $x \in K$ and $n \ge 0$,

$$ \mathbb{P}_x(n < \tau_\partial) \le C_2 \mathbb{P}_{x_0}(n < \tau_\partial); $$
(c) there exist $n_0 \ge 0$ and a constant $C_3 > 0$ such that, for all $x \in K$,

$$ \mathbb{P}_x(T_{\{x_0\}} \le n_0) \ge C_3, $$

where we recall that $T_L := \inf\{n \in \mathbb{Z}_+ : X_n \in L\}$ for all $L \subset E$.

Let us first prove (E1). By aperiodicity and irreducibility, there exists $m_1 \ge 1$ such that, for all $n \ge m_1$, $\mathbb{P}_{x_0}(X_n = x_0) > 0$. Combining this with (c), the Markov Property entails that, for all $x \in K$,

$$ \mathbb{P}_x(X_{n_0+m_1} = x_0) \ge C_3 \min_{m_1 \le k \le n_0+m_1} \mathbb{P}_{x_0}(X_k = x_0). $$

This is (E1) with $v = \delta_{x_0}$ and $n_1 = n_0 + m_1$.

We now prove (E2) and (E4). Condition (a) implies that

$$ \left(R + \frac{\varepsilon_0}{2}\right) \sup_{y \in K} \mathbb{E}_y \left[ 1_{1 < \tau_\partial} \mathbb{E}_{X_1} \left( \left(R + \frac{\varepsilon_0}{2}\right)^{T_K \wedge \tau_\partial} \right) \right] = \sup_{y \in K} \mathbb{E}_y \left[ \left(R + \frac{\varepsilon_0}{2}\right)^{\sigma_K \wedge \tau_\partial} \right] < \infty. $$

For all $x \in E \setminus K$, the irreducibility assumption implies that there exist $y \in K$ and $n = n(x, y) \ge 1$ such that $\mathbb{P}_y(X_n = x$ and $n < \sigma_K) > 0$. By Markov's property,

$$ \mathbb{E}_y \left[ \left(R + \frac{\varepsilon_0}{2}\right)^{\sigma_K \wedge \tau_\partial} \right] \ge \mathbb{P}_y(X_n = x \text{ and } n < \sigma_K) \mathbb{E}_x \left[ \left(R + \frac{\varepsilon_0}{2}\right)^{\sigma_K \wedge \tau_\partial} \right]. $$

Since $\sigma_K = T_K$ almost surely under $\mathbb{P}_x$ for $x \in E \setminus K$, Lemma 3.2 provides a function $\varphi_1$ satisfying the conditions of (E2), with $\theta_1 := (R + \frac{\varepsilon_0}{2})^{-1}$. According to [45, (1.16)], which holds true under their assumption by [45, Theorem 1], and setting $\theta_2 = (R + \frac{\varepsilon_0}{3})^{-1}$, one has

$$ \lim_{n \to +\infty} \theta_2^{-n} \mathbb{P}_{x_0}(X_n = x_0) = +\infty. $$

Using Markov's property, Condition (c) immediately entails that

$$ \lim_{n \to +\infty} \theta_2^{-n} \inf_{x \in K} \mathbb{P}_x(X_n \in K) = +\infty. $$

Using Lemma 3.1, we deduce that there exists a function $\varphi_2 : E \to [0, 1]$ satisfying the conditions of (E2) and that (E4) holds true. This concludes the proof of (E2) and (E4).

To conclude, Conditions (b) and (E1) imply, for all $n \ge 0$,

$$ \inf_{y \in K} \mathbb{P}_y(n < \tau_\partial) \ge \inf_{y \in K} \mathbb{P}_y(n + n_1 < \tau_\partial) \ge c_1 \mathbb{P}_{x_0}(n < \tau_\partial) \ge \frac{c_1}{C_2} \sup_{y \in K} \mathbb{P}_y(n < \tau_\partial). $$

This proves (E3) and concludes the proof of Proposition 8.1. $\square$

<figure><img src="image_6.png" /></figure>
Remark 19. One can actually prove that, in the particular case of a discrete state space E and aperiodic and irreducible transition probability on E, Assumption (E) is equivalent to the Conditions (a), (b) and (c) of [45]. Besides the additional properties provided in Section 2, one of our main contribution in this particular setting is to provide a more tractable criterion. Indeed, the use of Lyapunov type functions has the advantage to be quite flexible.

## 8.2 Application to the extinction of biological populations dominated by Galton-Watson processes

In this section, we show how our criteria can be applied to general population processes dominated by population-dependent Galton-Watson processes. In particular, we refine existing results for the classical multi-type Galton-Watson process.

More precisely, we consider an aperiodic and irreducible Markov population process $(Z_n)_{n \in \mathbb{N}}$ on $\mathbb{Z}_+^d = E \cup \{\partial\}$ absorbed at $\partial = 0$ such that, for all $n \ge 0$,

$$ \|Z_{n+1}\| \le \sum_{i=1}^{|Z_n|} \xi_{i,n}^{(Z_n)}, \qquad (8.2) $$

where $\| \cdot \|$ is a norm on $\mathbb{R}^d$ and $|z| = z_1 + \dots + z_d$ for all $z \in \mathbb{Z}_+^d$ and, for all $n \ge 0$, the nonnegative random variables $\xi_{1,n}^{(Z_n)}$, \dots, $\xi_{|Z_n|,n}^{(Z_n)}$ are assumed independent (but not necessarily identically distributed) given $Z_n$ and the families $(\xi_{i,n}^{(z)}, z \in \mathbb{Z}_+^d, 1 \le i \le |z|)$ are i.i.d. for $n \in \mathbb{Z}_+$.

We assume that

$$ \mathbb{E} \left( \sum_{i=1}^{|z|} \xi_{i,n}^{(z)} \right) \le m \|z\|, \quad \forall z \in \mathbb{Z}_+^d \text{ such that } |z| \ge n_0, \qquad (8.3) $$

for some $m < 1$ and $n_0 \in \mathbb{N}$. This means that the population size has a tendency to decrease (in mean) when it is too large. This also implies that $\tau_{\partial} < \infty$ a.s.

In the following theorem, $R > 0$ is the limiting value defined in (8.1).

**Theorem 8.2.** Assume that $(Z_n, n \in \mathbb{Z}_+)$ is aperiodic irreducible, that it satisfies the assumptions (8.2) and (8.3) and that, for some $q_0 > \frac{\log R}{\log(1/m)} \vee 1$,

$$ \sup_{z \in \mathbb{Z}_+^d, 1 \le i \le |z|} \mathbb{E}[(\xi_{i,1}^{(z)})^{q_0}] < \infty, $$

Then Condition (E) holds true with $\varphi_1(x) = \|x\|^q$, for all $q \in (\frac{\log R}{\log(1/m)} \vee 1, q_0]$.
**Remark 20.** This result easily applies if $\sup_{z \in \mathbb{Z}_+^d, 1 \le i \le |z|} \mathbb{E}[(\xi_{i,n}^{(z)})^q] < \infty$ for all $q > 0$. In other cases, we need an upper bound for $R > 0$ to check the assumptions of Theorem 8.2. For instance, one may use the fact that $R \le 1/ \sup_{z \in \mathbb{Z}_+^d} \mathbb{P}_z(Z_1 = z)$. One may also use Lyapunov techniques, in the same spirit as in Section 4.7 for diffusion processes.

**Remark 21.** A particular case of application of the above theorem is when $Z$ is obtained from a Galton-Watson multi-type process (see below for a more precise definition) with additional population-dependent death rates. For example, one can assume that additional death events may affect a fraction of the population, modelling global death events. In this case, compared to the Galton-Watson case, the independence between the progeny of individuals breaks down. Another situation covered by the above result is the case where the domain of absorption of $Z$ is a larger set than 0, for example the process may be absorbed when it reaches one edge of $\mathbb{Z}_+^d$ (i.e. when one type disappears). Another typical application of Theorem 8.2 is the case of population-dependent Galton-Watson processes, i.e. of processes such that, given $Z_n, Z_{n+1}$ is the sum of $|Z_n|$ i.i.d. random variables whose law may depend on $Z_n$. In this situation, Theorem 8.2 and its consequences stated in Section 2 generalize the results of [55] to the multi-type situation and provides finer results on the domain of attraction of the minimal quasi-stationary distribution. The reducible cases considered in [55] can also be recovered using the criterion of Theorem 6.1 in Section 6.1 or the criteria of [28]. Of course, the above cases may be combined.

Let us now consider the case of multi-type Galton-Watson processes. A Markov process $(Z_n, n \in \mathbb{Z}_+)$ evolving in $\mathbb{Z}_+^d = E \cup \{\partial\}$ absorbed at $\partial = 0$ is called a Galton-Watson process with $d$ types if, for all $n \ge 0$ and all $i \in \{1, \dots, d\}$,

$$Z_{n+1}^i = \sum_{k=1}^d \sum_{\ell=1}^{Z_n^k} \zeta_{k,i}^{(n,\ell)}, \quad (8.4)$$

where the random variables $(\zeta_{k,1}^{(n,\ell)}, \dots, \zeta_{k,d}^{(n,\ell)})_{n,\ell,k}$ in $\mathbb{Z}_+$ are assumed independent and such that, for all $k \in \{1, \dots, d\}$, $(\zeta_{k,1}^{(n,\ell)}, \dots, \zeta_{k,d}^{(n,\ell)})_{n,\ell}$ is an i.i.d. family. We define the matrix $M = (M_{k,i})_{1 \le k, i \le d}$ of mean offspring as

$$M_{k,i} = \mathbb{E}(\zeta_{k,i}^{(n,\ell)}), \quad \forall k, i \in \{1, \dots, d\},$$

and assume that $M_{k,i} < +\infty$ and that there exists $n \ge 1$ such that $[M^n]_{k,i} > 0$ for all $k, i \in \{1, \dots, d\}$.

Using the classical formalism of [59], we consider a positive right eigenvector $v$ of the matrix $M$ of mean offspring and we denote by $\rho(M)$ its spectral
radius. The sub-critical case corresponds to $\rho(M) < 1$. It is well-known [64] (see also [60, 2]) that this implies the existence of a quasi-stationary distribution whose domain of attraction contains all Dirac measures (a so-called *Yaglom limit* or *minimal quasi-stationary distribution*). The authors also prove that $v_{QSD}(|\cdot|) < \infty$ if and only if $\mathbb{E}[|Z_1|\log(|Z_1|) | Z_0 = (1, \dots, 1)] < \infty$. While the following result makes the stronger assumption that $\mathbb{E}[|Z_1|^{q_0} | Z_0 = (1, \dots, 1)] < \infty$ for some $q_0 > 1$, we obtain the finer results of Section 2, including a stronger form of convergence (in total variation norm with exponential speed), a non-trivial subset of the domain of attraction of the minimal quasi-stationary distribution and stronger moment properties for this quasi-stationary distribution.

**Corollary 8.3.** If $(Z_n, n \ge 0)$ is a d-type irreducible, aperiodic sub-critical Galton-Watson process, and if, for some $q_0 > 1$,

$$ \mathbb{E}[|Z_1|^{q_0} | Z_0 = (1, \dots, 1)] < \infty, $$

then Condition (E) holds true with $\phi_1(z) = |z|^q$ for any $q \in (1, q_0]$. In particular, the domain of attraction of $v_{QSD}$ contains all the probability measures such that $\mu(|\cdot|^q) < \infty$ for some $q > 1$.

This corollary easily derives from Theorem 8.2. Indeed, setting $\|z\| = \langle v, z \rangle$ and $\xi_{i,n}^{(Z_n)} = \sum_{j=1}^d v_j \zeta_{k,j}^{(n,l)}$ (assuming that $i$ is the $l$-th individual of type $k$ in the population), one obtains

$$ \|Z_{n+1}\| = \sum_{i=1}^{|Z_n|} \xi_{i,n}^{(Z_n)} $$

and

$$ \mathbb{E} \left( \left| \sum_{i=1}^{|Z_n|} \xi_{i,n}^{(Z_n)} \right| Z_n = z \right) = \sum_{k=1}^d \sum_{\ell=1}^d \sum_{j=1}^d v_j \mathbb{E} \left( \zeta_{k,j}^{(n,\ell)} \right) = \rho(M) \|z\|, $$

for all $z \in \mathbb{Z}_+^d$. Since, in the case of multi-type Galton-Watson process, one has $R = 1/\rho(M)$ (see for instance Theorems 2 and 3 of [64]), Theorem 8.2 applies with $m = \rho(M)$.

To prove Theorem 8.2, we use the following lemma.

**Lemma 8.4.** For all $q \in (\frac{\log R}{\log(1/m)} \vee 1, q_0]$, there exists a constant $C_q$ such that, for all $z \in \mathbb{Z}_+^d$,

$$ \mathbb{E} \left[ \left( \sum_{i=1}^{|z|} \xi_{i,n}^{(z)} - \mathbb{E}(\xi_{i,n}^{(z)}) \right)^q \right] \le C_q |z|^{1/(q/2)}. $$
*Proof.* If $q \in (1,2]$, this is exactly Lemma 1 of [29]. If $q \ge 2$, Burkholder's inequality [12] implies that there exists a constant $c_q$ such that

$$
\begin{align*}
\mathbb{E}\left[\left(\sum_{i=1}^{|z|} \xi_{i,n}^{(z)} - \mathbb{E}(\xi_{i,n}^{(z)})\right)^q\right] &\le c_q \mathbb{E}\left[\left(\sum_{i=1}^{|z|} \left\{\xi_{i,n}^{(z)} - \mathbb{E}(\xi_{i,n}^{(z)})\right\}^2\right)^{q/2}\right] \\
&= c_q |z|^{q/2} \mathbb{E}\left[\left(\frac{1}{|z|} \sum_{i=1}^{|z|} \left\{\xi_{i,n}^{(z)} - \mathbb{E}(\xi_{i,n}^{(z)})\right\}^2\right)^{q/2}\right] \\
&\le c_q |z|^{q/2} \mathbb{E}\left[\frac{1}{|z|} \sum_{i=1}^{|z|} \left|\xi_{i,n}^{(z)} - \mathbb{E}(\xi_{i,n}^{(z)})\right|^q\right] \\
&\le c_q |z|^{q/2} \mathbb{E}\left[\frac{1}{|z|} \sum_{i=1}^{|z|} \left|\xi_{i,n}^{(z)}\right|^q + \mathbb{E}(\xi_{i,n}^{(z)})^q\right] \\
&\le 2c_q |z|^{q/2} \sup_{z \in \mathbb{Z}_+^d, 1 \le i \le |z|} \mathbb{E}[(\xi_{i,n}^{(z)})^q],
\end{align*}
$$

where we used Jensen's inequality in the third line, that the r.v. $\xi_{i,n}^{(z)}$ are nonnegative in the fourth line and Hölder's inequality in the last inequality. $\square$

*Proof of Theorem 8.2.* We introduce an increasing sequence $(K_k, k \ge 0)$ of finite subsets of $\mathbb{Z}_+^d \setminus \{\partial\}$, where $K_k$ is the smallest set containing $\{z \in \mathbb{Z}_+^d : 1 \le |z| \le k\}$ such that the process $Z$ restricted to $K_k$ is irreducible and aperiodic. The existence of this set follows from the irreducibility assumption and the fact that $\mathbb{Z}_+^d$ is countable. We shall choose $K = K_k$ for an appropriate value of $k \ge 0$.

Fix $q \in (\frac{\log R}{\log(1/m)} \vee 1, q_0]$, $\theta_1 \in (m^q, 1/R)$, $\theta_2 \in (\theta_1, 1/R)$ and $\varphi_1(z) = \|z\|^q$. Using Minkowski's inequality in the first inequality, Lemma 8.4 in the third line and the equivalence between norms on $\mathbb{R}_+^d$,

$$
\begin{align}
P_1 \varphi_1(z) = \mathbb{E} \left( \left| \sum_{i=1}^{|z|} \xi_{i,n}^{(z)} \right|^q \right) &\le \left[ \mathbb{E} \left( \left| \sum_{i=1}^{|z|} \xi_{i,n}^{(z)} - \mathbb{E}(\xi_{i,n}^{(z)}) \right|^q \right)^{1/q} + \sum_{i=1}^{|z|} \mathbb{E}(\xi_{i,n}^{(z)}) \right]^q \nonumber \\
&\le \left[ (C_q |z|^{1/(q/2)})^{1/q} + m \|z\|^q \right]^q \nonumber \\
&= m^q \|z\|^q \left( 1 + C'_q |z|^{1/(q\wedge2)-1} \right)^q \nonumber \\
&\le m^q \|z\|^q + C''_q |z|^{q-1+1/(q\wedge2)}, \tag{8.5}
\end{align}
$$

for constants $C'_q$ and $C''_q$ only depending on $q$ and $m$. Since $q-1+1/(q\wedge2) < q$, there exists $k_1 \ge 0$ such that, for all $z \not\in K_{k_1}$,

$$ P_1 \varphi_1(z) \le \theta_1 \varphi_1(z). \quad (8.6) $$
We also deduce that, for all $z \in K_{k_1}$,

$$P_1\varphi_1(z) \leq \max_{x \in K_{k_1}} m^q \|x\|^q + C_q'' |x|^{q-1+1/(q\wedge 2)} < +\infty.$$

Setting $K = K_{k_1}$, we deduce that the first and third lines of Condition (E2) are satisfied.

By definition of $R$, we have $\lim_{n \to \infty} \theta_2^{-n} \inf_{z \in K} \mathbb{P}_z(X_n \in K) = +\infty$ and hence, using Lemma 3.1, there exists a function $\varphi_2: E \to [0, 1]$ such that the second and fourth lines of Condition (E2) are satisfied. It also implies that Condition (E4) holds true.

Since the process is irreducible and aperiodic and $K$ is finite, (3.2) is clearly satisfied for $n_0 = 1$ and $m_0$ large enough, so that Theorem 8.2 follows from Proposition 3.3. $\square$

# 9 Proof of Theorem 2.1

In all the proof, the constants $C$ are all positive and finite and may change from line to line. We first assume from Subsections 9.1 to 9.6 that for all $n \ge 0$ and all $x \in E$, $\mathbb{P}_x(n < \tau_\partial) > 0$. The general case will be handled in Subsection 9.8.

## 9.1 Main steps of the proof

The proof is based on a careful study of the semigroup of the process conditioned to not be absorbed before time $T$. In this section, we give the main ideas and steps of the proof of (2.1) for the total variation norm $\| \cdot \|_{TV} := \| \cdot \|_{TV(1)}$ in place of $\| \cdot \|_{TV(\varphi_1)}$, and leave the details for the following subsections, where preliminary results and the following Propositions 9.1, 9.2, 9.3 and Lemma 9.4 are proved. The general case of the $\| \cdot \|_{TV(\varphi_1)}$ norm is handled in Subsection 9.7.

For any $T \in \mathbb{Z}_+$, we consider the law of the process $X$ conditioned to not be absorbed before time $T$. We introduce the linear operators $(S_{m,n}^T)_{0 \le m \le n \le T}$ defined by

$$S_{m,n}^T f(x) = \mathbb{E}(f(X_n) | X_m = x, T < \tau_\partial) = \frac{P_{n-m}(f P_{T-n} \mathbb{1}_E)(x)}{P_{T-m} \mathbb{1}_E(x)}.$$

It is well-known that $(S_{m,n}^T)_{0 \le m \le n \le T}$ forms a time-inhomogeneous semigroup (i.e. $S_{m,n}^T S_{n,p}^T = S_{m,p}^T$ for all $m \le n \le p \le T$) and that the process $(X_n, 0 \le n \le T)$ under $\mathbb{P}_x^{S_0^T}$ is a (time-inhomogeneous) Markov process, where we denote by $\mathbb{P}_x^{S_0^T}$ the law of the process $(X_n, 0 \le n \le T)$ conditionally on $T < \tau_\partial$ and $X_0 = x$.
Fix $\theta \in (\theta_1/\theta_2, 1)$. For any $T \ge 0$, we set, for $x \in E$,

$$ \psi_T(x) = \mathbb{E}_x(\theta^{-T_K \wedge T} | T < \tau_\partial) = \mathbb{E}_x^{S_0^T} (\theta^{-T_K \wedge T}), $$

where

$$ T_K := \inf\{n \in \mathbb{Z}_+ : X_n \in K\} $$

is the first hitting time of $K$ by the process $(X_n, n \in \mathbb{Z}_+)$. Be careful that $T_K$ is not the first hitting time of $K$ by the full process $(X_t, t \in I)$, unless $I = \mathbb{Z}_+$.

The following proposition provides a Lyapunov-type property for the inhomogeneous semigroup $S$.

**Proposition 9.1.** There exists a constant $\bar{C} > 0$ such that, for all $0 \le m < T$ and $1 \le k \le T - m$,

$$ S_{m,m+k}^T \psi_{T-(m+k)}(x) \le \theta^k \psi_{T-m}(x) + \bar{C}, \quad \forall x \in E. \quad (9.1) $$

The next proposition provides a Dobrushin coefficient-type property for the inhomogeneous semigroup $S$.

**Proposition 9.2.** There exists a constant $\alpha_0 \in (0, 1)$ such that, for all $R > 0$, there exists $k_R \ge 1$ such that, for all $T \ge k_R$ and all $x, y \in E$ such that $\psi_T(x) + \psi_T(y) \le R$, we have

$$ \| \delta_x S_{0,k_R}^T - \delta_y S_{0,k_R}^T \|_{TV} \le 2(1 - \alpha_0). $$

The following property is a consequence of the two previous ones.

**Proposition 9.3.** There exist constants $n_0 \ge 1$, $C > 0$ and $\alpha \in (0, 1)$ such that, $\forall n \ge 1$ and all $x, y \in E$,

$$ \| \delta_x S_{0,n_0 n}^{n_0 n} - \delta_y S_{0,n_0 n}^{n_0 n} \|_{TV} \le C \alpha^n (2 + \psi_{n_0 n}(x) + \psi_{n_0 n}(y)). $$

Let us now deduce (2.1) with the total variation norm in place of $\| \cdot \|_{TV(\phi_1)}$, from the last proposition. We have, for all $x, y \in E$,

$$ \begin{aligned} \| \delta_x P_{nn_0} - \delta_x P_{nn_0} \mathbb{1}_E \delta_y S_{0,n_0 n}^{n_0 n} \|_{TV} & \le C \alpha^n (2 \delta_x P_{nn_0} \mathbb{1}_E + \mathbb{E}_x (\theta^{-T_K \wedge n n_0} \mathbb{1}_{n n_0 < \tau_\partial}) + \psi_{n_0 n}(y) \delta_x P_{nn_0} \mathbb{1}_E). \end{aligned} $$

Hence, for any probability measure $\mu$ on $E$, integrating the above inequality over $\mu(dx)$ leads to

$$ \begin{aligned} \| \mu P_{nn_0} - \mu P_{nn_0} \mathbb{1}_E \delta_y S_{0,n_0 n}^{n_0 n} \|_{TV} & \le C \alpha^n (2 \mu P_{nn_0} \mathbb{1}_E + \mathbb{E}_\mu (\theta^{-T_K \wedge n n_0} \mathbb{1}_{n n_0 < \tau_\partial}) + \psi_{n_0 n}(y) \mu P_{nn_0} \mathbb{1}_E). \end{aligned} $$

We make use of the following lemma.
**Lemma 9.4.** For all $\theta \in (\theta_1/\theta_2, 1)$, there exists a constant $C$ such that, for all $0 \le m \le T$ and all probability measure $\mu$ over $E$ such that $\mu(\varphi_2) > 0$,

$$ \mathbb{E}_{\mu} (\theta^{-T_K \wedge T} 1_{T < \tau_{\partial}}) \le C \frac{\mu(\varphi_1)}{\mu(\varphi_2)} \mathbb{P}_{\mu}(T < \tau_{\partial}). $$

This implies that, for all $\mu$ such that $\mu(\varphi_2) > 0$,

$$ \begin{aligned} \left\| \mu P_{nn_0} - \delta_y S_{0,n_0n}^{n_0n} \mu P_{nn_0} 1_E \right\|_{TV} & \le C \alpha^n \left( 2\mu P_{nn_0} 1_E + \frac{\mu(\varphi_1)}{\mu(\varphi_2)} \mu P_{nn_0} 1_E + \psi_{n_0n}(y) \mu P_{nn_0} 1_E \right). \end{aligned} $$

Hence

$$ \left\| \frac{\mu P_{nn_0}}{\mu P_{nn_0} 1_E} - \delta_y S_{0,n_0n}^{n_0n} \right\|_{TV} \le C \alpha^n \left( 2 + \frac{\mu(\varphi_1)}{\mu(\varphi_2)} + \psi_{n_0n}(y) \right). $$

Using the same procedure w.r.t. $y$, we deduce that, for any probability measures $\mu_1$ and $\mu_2$ on $E$ such that $\mu_1(\varphi_2) > 0$ and $\mu_2(\varphi_2) > 0$,

$$ \left\| \frac{\mu_1 P_{nn_0}}{\mu_1 P_{nn_0} 1_E} - \frac{\mu_2 P_{nn_0}}{\mu_2 P_{nn_0} 1_E} \right\|_{TV} \le C \alpha^n \left( \frac{\mu_1(\varphi_1)}{\mu_1(\varphi_2)} + \frac{\mu_2(\varphi_1)}{\mu_2(\varphi_2)} \right), $$

where we used the fact that $\mu(\varphi_1)/\mu(\varphi_2) \ge 1$ for all probability measure $\mu$ on $E$ such that $\mu(\varphi_2) > 0$.

Because of Lemma 9.6 below, we deduce that, for some constant $D_1 > 0$ and for all $0 \le k < n_0$,

$$ \begin{aligned} \left\| \frac{\mu_1 P_{nn_0+k}}{\mu_1 P_{nn_0+k} 1_E} - \frac{\mu_2 P_{nn_0+k}}{\mu_2 P_{nn_0+k} 1_E} \right\|_{TV} & \le C \alpha^n \left( \frac{\mu_1 P_k \varphi_1}{\mu_1 P_k \varphi_2} + \frac{\mu_2 P_k \varphi_1}{\mu_2 P_k \varphi_2} \right) \\ & \le C \alpha^n \left( \frac{\mu_1(\varphi_1)}{\mu_1(\varphi_2)} \vee D_1 + \frac{\mu_2(\varphi_1)}{\mu_2(\varphi_2)} \vee D_1 \right). \end{aligned} $$

Therefore, up to a change in the constant $C$ and replacing $\alpha$ by $\alpha^{1/n_0}$, we deduce that, for all probability measures $\mu_1$ and $\mu_2$ on $E$ such that $\mu_1(\varphi_2) > 0$ and $\mu_2(\varphi_2) > 0$ and for all $n \ge 0$,

$$ \left\| \frac{\mu_1 P_n}{\mu_1 P_n 1_E} - \frac{\mu_2 P_n}{\mu_2 P_n 1_E} \right\|_{TV} \le C \alpha^n \left( \frac{\mu_1(\varphi_1)}{\mu_1(\varphi_2)} + \frac{\mu_2(\varphi_1)}{\mu_2(\varphi_2)} \right). \quad (9.2) $$

Fix $x_0 \in K$. We set $\mu_1 = \delta_{x_0}$ and $\mu_2 = \frac{\mu_1 P_1}{\mu_1 P_1 1_E}$ in (9.2). Since $\frac{\mu_1 \varphi_1}{\mu_1 \varphi_2} < \infty$ and because of Lemma 9.6 below, we have $\frac{\mu_2 \varphi_1}{\mu_2 \varphi_2} < \infty$. We deduce that, for some constant $C > 0$,

$$ \left\| \frac{\delta_{x_0} P_{n+1}}{\delta_{x_0} P_{n+1} 1_E} - \frac{\delta_{x_0} P_n}{\delta_{x_0} P_n 1_E} \right\|_{TV} \le C \alpha^n, $$
and hence, using the completeness of the space of probability measures on E for the total variation norm, we deduce that there exists a quasi-limiting measure $v_{QSD}$ (which is hence a quasi-stationary distribution) such that

$$ \left\| \frac{\delta_{x_0} P_n}{\delta_{x_0} P_n \mathbb{1}_E} - v_{QSD} \right\|_{TV} \le \frac{2C}{1-\alpha} \alpha^n. $$

In particular, it follows from Lemma 9.8 below that $v_{QSD}(K) > 0$ and hence that $v_{QSD}(\varphi_2) > 0$. Since Lemma 9.6 implies that $\frac{P_n \varphi_1(x_0)}{P_n \mathbb{1}_E(x_0)}$ is uniformly bounded in $n \ge 0$, we deduce that $v_{QSD}(\varphi_1 \wedge M)$ is bounded uniformly in $M > 0$ and hence $v_{QSD}(\varphi_1) < \infty$.

Using (9.2) again (up to another change of the constant C), we obtain that, for all probability measure $\mu$ on E such that $\frac{\mu(\varphi_1)}{\mu(\varphi_2)} < \infty$,

$$ \left\| \frac{\mu P_n}{\mu P_n \mathbb{1}_E} - v_{QSD} \right\|_{TV} \le C \alpha^n \frac{\mu(\varphi_1)}{\mu(\varphi_2)}. $$

This also entails that there exists a unique quasi-stationary distribution such that $v_{QSD}(\varphi_1)/v_{QSD}(\varphi_2) < \infty$.

This ends the proof of (2.1) for the total variation norm. The general case with the norm $\| \cdot \|_{TV}(\varphi_1)$ is proved in Subsection 9.7.

## 9.2 Preliminary results

We start by proving two basic inequalities which are direct consequences of (E2).

**Lemma 9.5.** For all $x \in E \setminus K$ and all $n \ge 0$,

$$ \mathbb{P}_x(n < T_K \wedge \tau_\partial) \le \mathbb{E}_x[\varphi_1(X_n) \mathbb{1}_{n < T_K \wedge \tau_\partial}] \le \theta_1^n \varphi_1(x). $$

For all $x \in E$ and $n \ge 0$,

$$ \mathbb{P}_x(n < \tau_\partial) \ge \mathbb{E}_x[\varphi_2(X_n) \mathbb{1}_{n < \tau_\partial}] \ge \theta_2^n \varphi_2(x). $$

*Proof of Lemma 9.5.* These two properties follow easily by induction from (E2). For example, the first one makes use of the following relation: for all $n \ge 1$ and $x \in E$,

$$ \mathbb{E}_x[\varphi_1(X_n) \mathbb{1}_{n < T_K \wedge \tau_\partial}] = \mathbb{1}_{x \in E \setminus K} P_1 \left[ \mathbb{E}_x \left( \varphi_1(X_{n-1}) \mathbb{1}_{n-1 < T_K \wedge \tau_\partial} \right) \right] (x). $$

This and (E2) entail the property at time $n = 1$ and, by induction, at any time $n \ge 1$. □
The next lemma states that the expectation of $\varphi_1(X_n)$ is controlled by the expectation of $\varphi_2(X_n)$ uniformly in time.

**Lemma 9.6.** For all $\theta \in (\theta_1/\theta_2, 1]$, there exists a finite constant $D_\theta > 0$ such that, for all probability measure $\mu$ on E such that $\mu(\varphi_1)/\mu(\varphi_2) < \infty$, for all $T \in \mathbb{Z}_+$ and all $x \in E$,

$$ \frac{\mu P_T \varphi_1}{\mu P_T \varphi_2} \le \left( \theta^T \frac{\mu(\varphi_1)}{\mu(\varphi_2)} \right) \vee D_\theta. \qquad (9.3) $$

*Proof of Lemma 9.6.* It follows from (E2) that

$$ \mu P_{T+1} \varphi_1 \le \theta_1 \mu P_T \varphi_1 + C \mu P_T \mathbb{1}_K $$

and

$$ \mu P_{T+1} \varphi_2 \ge \theta_2 \mu P_T \varphi_2. $$

Hence

$$ \begin{aligned} \frac{\mu P_{T+1} \varphi_1}{\mu P_{T+1} \varphi_2} &\le \frac{\theta_1 \mu P_T \varphi_1 + C \mu P_T \mathbb{1}_K(x)}{\theta_2 \mu P_T \varphi_2} \\ &\le \frac{\theta_1}{\theta_2} \frac{\mu P_T \varphi_1}{\mu P_T \varphi_2} + \frac{C}{\theta_2 \inf_{y \in K} \varphi_2(y)}. \end{aligned} $$

Since $\theta_1/\theta_2 < \theta$, these arithmetico-geometric inequalities entail (9.3). $\square$

<figure><img src="image_7.png" /></figure>

We now give an irreducibility inequality.

**Lemma 9.7.** For all $C \ge 1$, there exists a time $n_5(C) \in \mathbb{N}$ such that

$$ a_5(C) := \inf_{\mu \in \mathcal{M}_1(E) \text{ s.t. } \mu(\varphi_1) \le C\mu(\varphi_2)} \mathbb{P}_{\mu}(X_{n_5(C)} \in K) > 0. \qquad (9.4) $$

*Proof of Lemma 9.7.* It follows from (E4) that there exists a time $n_v \in \mathbb{N}$ such that, for all $n \ge n_v$, $\mathbb{P}_v(X_n \in K) > 0$, and, using (E1), that for all $n \ge n_v + n_1$,

$$ \inf_{x \in K} \mathbb{P}_x(X_n \in K) \ge c_1 \mathbb{P}_v(X_{n-n_1} \in K) > 0. $$

Let $C \ge 1$ and $\mu$ be such that $\mu(\varphi_1) \le C\mu(\varphi_2)$. It follows from Lemma 9.5 that, for all $n \ge 1$,

$$ \mathbb{P}_{\mu}(T_K \wedge \tau_{\partial} > n) \le \mathbb{E}_{\mu}[\varphi_1(X_n) \mathbb{1}_{T_K \wedge \tau_{\partial} > n}] \le \theta_1^n \mu(\varphi_1) \le C \theta_1^n \mu(\varphi_2). $$

and

$$ \mathbb{P}_{\mu}(n < \tau_{\partial}) \ge \mathbb{E}_{\mu}[\varphi_2(X_n)] \ge \theta_2^n \mu(\varphi_2). $$
Therefore,

$$ \mathbb{P}_{\mu}(T_K \le n < \tau_{\partial}) \ge (\theta_2^n - C\theta_1^n) \mu(\varphi_2). $$

Choosing $n(C) = \lceil 2C / \log(\theta_2 / \theta_1) \rceil$, we deduce that

$$ \mathbb{P}_{\mu}(T_K \le n(C) < \tau_{\partial}) \ge \frac{\theta_2^{n(C)}}{2} \mu(\varphi_2) \ge \frac{\theta_2^{n(C)}}{2C}. $$

Therefore,

$$ \begin{aligned} \mathbb{P}_{\mu}(X_{n(C)+n_v+n_1} \in K) &\ge \mathbb{E}_{\mu} \left[ \mathbb{1}_{T_K \le n(C)} \mathbb{P}_{X_{T_K}} (X_{n(C)+n_v+n_1-k} \in K) \Big|_{k=T_K} \right] \\ &\ge \min_{n_v+n_1 \le k \le n_v+n_1+n(C)} \inf_{x \in K} \mathbb{P}_x (X_k \in K) \frac{\theta_2^{n(C)}}{2C}. \end{aligned} $$

Hence we have proved Lemma 9.7 with $n_5(C) = n_v + n_1 + n(C)$. $\square$

<figure><img src="image_8.png" /></figure>

The next lemma shows that conditional distributions with initial conditions in $K$ give to $K$ a mass uniformly bounded from below.

**Lemma 9.8.** There exists a time $n_6 \in \mathbb{N}$ such that

$$ \inf_{T \ge n_6} \inf_{x \in K} \mathbb{P}_x(X_T \in K \mid T < \tau_{\partial}) > 0. $$

*Proof of Lemma 9.8.* Since $\varphi_1 / \varphi_2$ is bounded over $K$, we deduce from Lemma 9.6 that, setting $C := D_1 + \sup_{x \in K} \frac{\varphi_1(x)}{\varphi_2(x)}$, we have for all $x \in K$ and all $T \ge n_5(C)$,

$$ \frac{P_{T-n_5(C)}\varphi_1(x)}{P_{T-n_5(C)}\varphi_2(x)} \le C. \qquad (9.5) $$

Using Lemma 9.7 applied to $\mu = \frac{\delta_x P_{T-n_5(C)}}{\delta_x P_{T-n_5(C)} \mathbb{1}_E}$, we deduce that, for all $x \in K$ and $T \ge n_5(C)$,

$$ \mathbb{P}_x(X_T \in K \mid T < \tau_{\partial}) = \frac{\mu P_{n_5(C)} \mathbb{1}_K}{\mu P_{n_5(C)} \mathbb{1}_E} \ge \mu P_{n_5(C)} \mathbb{1}_K \ge a_5(C). \quad \square $$

<figure><img src="image_9.png" /></figure>

The next lemma shows that survival probabilities are controlled by the function $\varphi_1$.

**Lemma 9.9.** There exists a constant $C > 0$ such that, for all $p \in [1, \log\theta_1 / \log\theta_2)$, $x \in E$ and $n \ge 1$,

$$ \mathbb{P}_x(n < \tau_{\partial}) \le C \frac{\varphi_1(x)^{1/p}}{1 - \theta_1^{1/p} / \theta_2} \inf_{y \in K} \mathbb{P}_y(n < \tau_{\partial}). \qquad (9.6) $$
Proof of Lemma 9.9. It follows from Lemma 9.5 that, for all $p \ge 1$, $x \in E \setminus K$ and $n \ge 1$,

$$ \mathbb{P}_x(n < T_K \wedge \tau_\partial) \le \theta_1^{n/p} \varphi_1(x)^{1/p}. \quad (9.7) $$

Note that this inequality is trivial for $x \in K$. In particular, for $p \ge 1$ such that $\theta_1^{1/p} < \theta_2$, for all $x \in K$,

$$ \mathbb{E}_x(\theta_2^{-T_K \wedge \tau_\partial}) \le \frac{\varphi_1(x)^{1/p}}{1 - \theta_1^{1/p} / \theta_2}. \quad (9.8) $$

Fix $p \in [1, \log\theta_1 / \log\theta_2)$. Using (9.7), the second inequality of Lemma 9.5 and (E3), we have for all $x \in E$

$$ \begin{aligned} \mathbb{P}_x(n < \tau_\partial) &= \mathbb{P}_x(n < T_K \wedge \tau_\partial) + \mathbb{P}_x(T_K \wedge \tau_\partial \le n < \tau_\partial) \\ &\le \theta_2^n \varphi_1(x)^{1/p} + \sum_{k=0}^n \mathbb{P}_x(T_K \wedge \tau_\partial = k) \sup_{y \in K} \mathbb{P}_y(n-k < \tau_\partial) \\ &\le \frac{\inf_{z \in K} \mathbb{P}_z(n < \tau_\partial)}{\inf_{z \in K} \varphi_2(z)} \varphi_1(x)^{1/p} + c_3 \sum_{k=0}^n \mathbb{P}_x(T_K \wedge \tau_\partial = k) \inf_{y \in K} \mathbb{P}_y(n-k < \tau_\partial) \\ &\le C \inf_{z \in K} \mathbb{P}_z(n < \tau_\partial) \varphi_1(x)^{1/p} + C \inf_{z \in K} \mathbb{P}_z(n < \tau_\partial) \sum_{k=0}^n \mathbb{P}_x(T_K \wedge \tau_\partial = k) \theta_2^{-k}, \end{aligned} \quad (9.9) $$

where we used the fact that, for some constant $C > 0$, for all $n \ge k \ge 0$ and all $z \in K$,

$$ \mathbb{P}_z(n < \tau_\partial) \ge C \theta_2^k \inf_{y \in K} \mathbb{P}_y(n-k < \tau_\partial). \quad (9.10) $$

This is proved using the three following equations. For all $n \ge k \ge n_6$ and all $z \in K$, by Lemmata 9.8 and 9.5,

$$ \begin{aligned} \mathbb{P}_z(n < \tau_\partial) &\ge \mathbb{P}_z(X_k \in K | k < \tau_\partial) \mathbb{P}_z(k < \tau_\partial) \inf_{y \in K} \mathbb{P}_y(n-k < \tau_\partial) \\ &\ge C \theta_2^k \varphi_2(z) \inf_{y \in K} \mathbb{P}_y(n-k < \tau_\partial) \\ &\ge C \theta_2^k \inf_{y \in K} \mathbb{P}_y(n-k < \tau_\partial), \end{aligned} $$

since $\inf_{z \in K} \varphi_2(z) > 0$. Also, for all $n \ge n_6 \ge k$, using the last inequality,

$$ \begin{aligned} \mathbb{P}_z(n < \tau_\partial) &\ge C \theta_2^{n_6} \inf_{y \in K} \mathbb{P}_y(n - n_6 < \tau_\partial) \\ &\ge C \theta_2^{n_6} \inf_{y \in K} \mathbb{P}_y(n - k < \tau_\partial) \\ &\ge (C \theta_2^{n_6}) \theta_2^k \inf_{y \in K} \mathbb{P}_y(n - k < \tau_\partial). \end{aligned} $$
Finally, for all $k \le n < n_6$,

$$ \mathbb{P}_z(n < \tau_\partial) \ge \mathbb{P}_z(n_6 < \tau_\partial) \ge C\theta_2^{n_6} \ge (C\theta_2^{n_6}) \theta_2^k \inf_{y \in K} \mathbb{P}_y(n-k < \tau_\partial), $$

so (9.10) is proved.

Combining (9.8) and (9.9) ends the proof of Lemma 9.9. $\square$

## 9.3 Proof of Proposition 9.1

Markov's property implies that, for all $x \in E \setminus K$ and $T, m \ge 1$,

$$ S_{0,1}^T \psi_{T-1}(x) = S_{m,m+1}^{T+m} \psi_{T-1}(x) = \theta \psi_T(x). \quad (9.11) $$

Indeed,

$$ \begin{aligned} \theta \psi_T(x) &= \frac{\mathbb{E}_x(\theta^{1-T_K \wedge T} \mathbb{1}_{T < \tau_\partial})}{\mathbb{P}_x(T < \tau_\partial)} \\ &= \frac{\mathbb{E}_x[\mathbb{1}_{1 < \tau_\partial} \mathbb{E}_{X_1}(\theta^{-T_K \wedge (T-1)} | T-1 < \tau_\partial) \mathbb{P}_{X_1}(T-1 < \tau_\partial)]}{\mathbb{P}_x(T < \tau_\partial)} = S_{0,1}^T \psi_{T-1}(x). \end{aligned} $$

Similarly, for all $x \in K$,

$$ S_{0,1}^T \psi_{T-1}(x) = S_{m,m+1}^{T+m} \psi_{T-1}(x) = \theta \mathbb{E}_x^{S_{0,1}^T} (\theta^{-\sigma_K \wedge T}), \quad (9.12) $$

where

$$ \sigma_K := \min\{n \ge 1, X_n \in K\} $$

is the first return time in $K$. Setting

$$ C := \sup_{T \ge 0} \sup_{x \in K} \mathbb{E}_x^{S_{0,1}^T} (\theta^{-\sigma_K \wedge T}), $$

which is finite (see Lemma 9.10), we can apply recursively (9.11) and (9.12) to obtain

$$ \begin{aligned} S_{m,m+k}^T \psi_{T-(m+k)} &= S_{m,m+k-1}^T (\mathbb{1}_{E \setminus K} S_{m+k-1,m+k}^T (\psi_{T-(m+k)})) \\ &\quad + S_{m,m+k-1}^T (\mathbb{1}_K S_{m+k-1,m+k}^T (\psi_{T-(m+k)})) \\ &\le \theta S_{m,m+k-1}^T \psi_{T-(m+k-1)} + C\theta \\ &\le \dots \le \theta^k \psi_{T-m}(x) + C \sum_{\ell=1}^k \theta^\ell. \end{aligned} $$

Hence Proposition 9.1 follows from the next lemma.
**Lemma 9.10.** For all $\theta \in (\theta_1/\theta_2, 1)$,

$$ \sup_{T \ge 0} \sup_{x \in K} \mathbb{E}_x^{S_0^T} (\theta^{-\sigma_K \wedge T}) < \infty. $$

*Proof of Lemma 9.10.* Fix $x \in K$. On the one hand, by Lemma 9.9 (with $p = 1$), we have for any $1 \le n < T$,

$$ \begin{aligned} \mathbb{P}_x(n < \sigma_K \text{ and } T < \tau_\partial) &= \mathbb{E}_x(\mathbb{1}_{n < \sigma_K \wedge \tau_\partial} \mathbb{P}_{X_n}(T - n < \tau_\partial)) \\ &\le C \inf_{y \in K} \mathbb{P}_y(T - n < \tau_\partial) \mathbb{E}_x(\mathbb{1}_{n < \sigma_K \wedge \tau_\partial} \varphi_1(X_n)). \end{aligned} $$

Using (E2) and Markov's property as in the proof of Lemma 9.5, we deduce

$$ \mathbb{P}_x(n < \sigma_K \text{ and } T < \tau_\partial) \le C \inf_{y \in K} \mathbb{P}_y(T - n < \tau_\partial) \theta_1^{n-1} P_1 \varphi_1(x) \quad (9.13) $$

$$ \le C \inf_{y \in K} \mathbb{P}_y(T - n < \tau_\partial) \theta_1^n. \quad (9.14) $$

On the other hand, Lemma 9.8 implies the existence of a constant $C > 0$ such that, for all $x \in K$ and all $n \ge n_6$,

$$ \mathbb{P}_x(X_n \in K) \ge C \mathbb{P}_x(n < \tau_\partial). $$

We deduce from Markov's property and Lemma 9.5 that, for all $T \ge n \ge n_6$,

$$ \begin{aligned} \mathbb{P}_x(T < \tau_\partial) &\ge \mathbb{P}_x(X_n \in K) \inf_{y \in K} \mathbb{P}_y(T - n < \tau_\partial) \\ &\ge C \mathbb{P}_x(n < \tau_\partial) \inf_{y \in K} \mathbb{P}_y(T - n < \tau_\partial) \\ &\ge C \theta_2^n \inf_{y \in K} \mathbb{P}_y(T - n < \tau_\partial). \end{aligned} $$

Combining this with (9.13), we finally deduce that there exists a constant $C > 0$ such that, for all $x \in K$ and all $T \ge n \ge n_6$,

$$ \mathbb{P}_x(n < \sigma_K | T < \tau_\partial) \le C \left( \frac{\theta_1}{\theta_2} \right)^n. \quad (9.15) $$

The extension to any $T \ge n$ is trivial, so the conclusion follows. $\square$

## 9.4 Proof of Proposition 9.2

We start by stating a lemma proved at the end of this subsection.
**Lemma 9.11.** For all $x \in K$ and $n_1 + n_6 \le n \le T$,

$$ \mathbb{P}_x(X_n \in \cdot | T < \tau_\partial) \ge c_1' \nu, \qquad (9.16) $$

where the measure $\nu$ and the integer $n_1$ are the one of Condition (E1), the integer $n_6$ is from Lemma 9.7 and $c_1' > 0$ is independent of $x, n$ and $T$.

Fix $\theta \in (\theta_1/\theta_2, 1)$ and set $k_R = \lceil \log(2R)/\log(1/\theta) \rceil + n_1 + n_6$ and fix $T \ge k_R$. For all $x \in E$ such that $\psi_T(x) \le R$, Markov's inequality implies that

$$ \mathbb{P}_x(T_K > k_R - n_1 - n_6 | T < \tau_\partial) = \mathbb{P}_x^{S_{0,\cdot}^T} (T_K > k_R - n_1 - n_6) \le \frac{R}{\theta^{-k_R+n_1+n_6}} \le \frac{1}{2}. $$

It follows from Lemma 9.11 that, for all measurable $A \subset E$,

$$ \begin{aligned} \mathbb{P}_x^{S_{0,\cdot}^T} (X_{k_R} \in A) &\ge \frac{\mathbb{E}_x \left[ \sum_{k=1}^{k_R-n_1-n_6} \mathbb{1}_{T_K=k} \mathbb{P}_{X_k} (X_{k_R-k} \in A, T-k < \tau_\partial) \right]}{\mathbb{P}_x(T < \tau_\partial)} \\ &\ge c_1' \nu(A) \frac{\mathbb{E}_x \left[ \sum_{k=1}^{k_R-n_1-n_6} \mathbb{1}_{T_K=k} \mathbb{P}_{X_k} (T-k < \tau_\partial) \right]}{\mathbb{P}_x(T < \tau_\partial)} \\ &= c_1' \nu(A) \mathbb{P}_x (T_K \le k_R - n_1 - n_6 | T < \tau_\partial) \\ &\ge \frac{1}{2} c_1' \nu(A). \end{aligned} $$

This concludes the proof of Proposition 9.2 with $\alpha_0 = c_1'/2$.

**Proof of Lemma 9.11.** For all measurable set $A \subset K$, we deduce from Markov's property that, for all $x \in K$ and all $T \ge n \ge n_1 + n_6$,

$$ \begin{aligned} \mathbb{P}_x(X_n \in A, T < \tau_\partial) &\ge \mathbb{E}_x \left[ \mathbb{1}_{X_{n-n_1} \in K} \mathbb{E}_{X_{n-n_1}} \left( \mathbb{1}_{X_{n_1} \in A} \mathbb{P}_{X_{n_1}} (T-n < \tau_\partial) \right) \right] \\ &\ge \mathbb{E}_x \left[ \mathbb{1}_{X_{n-n_1} \in K} \mathbb{P}_{X_{n-n_1}} (X_{n_1} \in A) \right] \inf_{y \in K} \mathbb{P}_y (T-n < \tau_\partial) \\ &\ge c_1 \nu(A) \mathbb{P}_x (X_{n-n_1} \in K) \inf_{y \in K} \mathbb{P}_y (T-n < \tau_\partial), \end{aligned} \qquad (9.17) $$

where we used (E1). Now, using Lemma 9.9, we deduce that there exists a constant $c > 0$ such that

$$ \begin{aligned} \mathbb{P}_x(T < \tau_\partial) &\le \mathbb{P}_x(T - n_1 < \tau_\partial) = \mathbb{E}_x \left( \mathbb{1}_{n-n_1 < \tau_\partial} \mathbb{P}_{X_{n-n_1}} (T - n < \tau_\partial) \right) \\ &\le c \mathbb{E}_x \left( \mathbb{1}_{n-n_1 < \tau_\partial} \varphi_1(X_{n-n_1}) \right) \inf_{y \in K} \mathbb{P}_y (T - n < \tau_\partial). \end{aligned} $$

Since $\varphi_1(x)/\varphi_2(x)$ is uniformly bounded over $x \in K$, Lemma 9.6 implies that there exists a constant $c' > 0$ such that, for all $x \in K$,

$$ \mathbb{E}_x [ \mathbb{1}_{n-n_1 < \tau_\partial} \varphi_1(X_{n-n_1}) ] \le c' \mathbb{E}_x [ \mathbb{1}_{n-n_1 < \tau_\partial} \varphi_2(X_{n-n_1}) ] \le c' \mathbb{P}_x (n-n_1 < \tau_\partial). $$
But $n - n_1 \ge n_6$, hence Lemma 9.8 entails that there exists a constant $c'' > 0$ such that, for all $x \in K$,

$$ \mathbb{P}_x(n - n_1 < \tau_\partial) \le c'' \mathbb{P}_x(X_{n-n_1} \in K). $$

Hence we obtain

$$ \mathbb{P}_x(T < \tau_\partial) \le c c' c'' \mathbb{P}_x(X_{n-n_1} \in K) \inf_{y \in K} \mathbb{P}_y(T - n < \tau_\partial). $$

Combining this with (9.17), we obtain

$$ \mathbb{P}_x(X_n \in A | T < \tau_\partial) \ge \frac{c_1}{cc'c''} \nu(A). $$

This ends the proof of Lemma 9.11.

<figure><img src="image_10.png" /></figure>

## 9.5 Proof of Proposition 9.3

We transpose the ideas of [57] (see also [58]) to the time-inhomogeneous setting. We fix the constants $R = 4\bar{C}/(1 - \theta)$ and $\beta = \alpha_0/2\bar{C}$, where $\bar{C}$ is the constant of Proposition 9.1. For all $T \ge 0$ and all $\varphi: E \to \mathbb{R}$, we set

$$ \|\varphi\|_T = \sup_{x,y \in E} \frac{|\varphi(x) - \varphi(y)|}{2 + \beta\psi_T(x) + \beta\psi_T(y)}. $$

Fix $n$ and $T \ge 0$ such that $(n+1)k_R \le T$ and let $\varphi$ be such that $\|\varphi\|_{T-(n+1)k_R} \le 1$. Then, replacing $\varphi$ by $\varphi + c$ for some appropriate constant $c$, one has $|\varphi| \le 1 + \beta\psi_{T-(n+1)k_R}$ (see Lemma 3.8 p.14 in [57]).

If $\psi_{T-nk_R}(x) + \psi_{T-nk_R}(y) > R$, then, using Proposition 9.1,

$$ \begin{aligned} \left| S_{nk_R,(n+1)k_R}^T \varphi(x) - S_{nk_R,(n+1)k_R}^T \varphi(y) \right| & \le 2 + \theta \beta \psi_{T-nk_R}(x) + \theta \beta \psi_{T-nk_R}(y) + 2\beta \bar{C} \\ & \le 2 + (\theta + (1-\theta)/2) (\beta \psi_{T-nk_R}(x) + \beta \psi_{T-nk_R}(y)) \\ & \quad - (R\beta)(1-\theta)/2 + 2\beta \bar{C} \\ & \le (1-\alpha_1)(2 + \beta \psi_{T-nk_R}(x) + \beta \psi_{T-nk_R}(x)), \end{aligned} $$

where $\alpha_1 \in (0, 1)$ is such that $2 + (\theta + (1-\theta)/2)y \le (1-\alpha_1)(2+y)$ for all $y \ge \beta R$.

If $\psi_{T-nk_R}(x) + \psi_{T-nk_R}(y) \le R$, then, considering

$$ \varphi = \varphi' + \varphi'', $$
with $|\varphi'| \le 1$ and $|\varphi''| \le \beta\psi_{T-(n+1)k_R}$, Propositions 9.1 and 9.2 entail

$$ \left| S_{nk_R,(n+1)k_R}^T \varphi(x) - S_{nk_R,(n+1)k_R}^T \varphi(y) \right| \le 2(1-\alpha_0) + \beta\theta\psi_{T-nk_R}(x) + \beta\theta\psi_{T-nk_R}(y) + 2\beta\bar{C}. $$

Our choice $\beta = \alpha_0/2\bar{C}$ implies that

$$ \left| S_{nk_R,(n+1)k_R}^T \varphi(x) - S_{nk_R,(n+1)k_R}^T \varphi(y) \right| \le (1-\alpha_2)(2 + \beta\psi_{T-nk_R}(x) + \beta\psi_{T-nk_R}(y)). $$

for the constant $\alpha_2 = \frac{\alpha_0}{2} \wedge (1-\theta) > 0$.

Hence, we obtained

$$ \left\| S_{nk_R,(n+1)k_R}^T \varphi \right\|_{T-nk_R} \le (1-\alpha_1 \wedge \alpha_2) \left\| \varphi \right\|_{T-(n+1)k_R}, $$

which implies by iteration that

$$ \left\| S_{0,nk_R}^{nk_R} \varphi \right\|_{nk_R} \le (1 - \alpha_1 \wedge \alpha_2)^n \left\| \varphi \right\|_0 \le (1 - \alpha_1 \wedge \alpha_2)^n \|\varphi\|_{\infty} / (1 + \beta). $$

This concludes the proof of Proposition 9.3.

## 9.6 Proof of Lemma 9.4

This lemma in a generalization of Lemma 9.10. Its proof is based on similar computations. We give the details for sake of completeness.

For all probability measure $\mu$ on $E$, for any $0 \le n < T$, using Lemma 9.9 for the second inequality and Lemma 9.5 for the third inequality, we have

$$ \begin{aligned} \mathbb{P}_{\mu}(n < T_K \text{ and } T < \tau_{\partial}) &\le \mathbb{E}_{\mu}(\mathbb{1}_{n < T_K} \mathbb{P}_{X_n}(T - n < \tau_{\partial})) \\ &\le C \inf_{y \in K} \mathbb{P}_y(T - n < \tau_{\partial}) \mathbb{E}_{\mu}(\mathbb{1}_{n < T_K} \varphi_1(X_n)) \\ &\le C \inf_{y \in K} \mathbb{P}_y(T - n < \tau_{\partial}) \theta_1^n \mu(\varphi_1). \end{aligned} \quad (9.18) $$

For all integer $n \ge n_{\mu}$, where

$$ n_{\mu} := \left\lfloor n_5(D_{\theta}) + \frac{\log \frac{\mu(\varphi_1)}{D_{\theta}\mu(\varphi_2)}}{\log(1/\theta)} \right\rfloor, $$

it follows from Lemma 9.6 that

$$ \frac{\mu P_{n-n_5(D_\theta)} \varphi_1}{\mu P_{n-n_5(D_\theta)} \varphi_2} \le D_\theta \lor \left( \theta^{n-n_5(D_\theta)} \frac{\mu(\varphi_1)}{\mu(\varphi_2)} \right) \le D_\theta $$
and from Lemma 9.7 that

$$ \frac{\mu P_n 1_K}{\mu P_n 1_E} \geq a_5(D_\theta) > 0. $$

Therefore, we obtain from the Markov property and Lemma 9.5 that

$$ \begin{aligned} \mathbb{P}_{\mu}(T < \tau_{\partial}) &\geq \mathbb{P}_{\mu}(X_n \in K) \inf_{y \in K} \mathbb{P}_y(T - n < \tau_{\partial}) \\ &\geq a_5(D_{\theta}) \mathbb{P}_{\mu}(n < \tau_{\partial}) \inf_{y \in K} \mathbb{P}_y(T - n < \tau_{\partial}) \\ &\geq a_5(D_{\theta}) \theta_2^n \mu(\varphi_2) \inf_{y \in K} \mathbb{P}_y(T - n < \tau_{\partial}). \end{aligned} $$

Combining this with (9.18), we obtain that, for all $n \geq n_{\mu}$,

$$ \mathbb{P}_{\mu}(n < T_K \text{ and } T < \tau_{\partial}) \leq \frac{C}{a_5(D_{\theta})} \left(\frac{\theta_1}{\theta_2}\right)^n \frac{\mu(\varphi_1)}{\mu(\varphi_2)} \mathbb{P}_{\mu}(T < \tau_{\partial}). $$

Hence

$$ \mathbb{E}_{\mu} \left( \theta^{-T_K \wedge T} 1_{T_K \ge n_{\mu}, T < \tau_{\partial}} \right) \le C \frac{\mu(\varphi_1)}{\mu(\varphi_2)} \mathbb{P}_{\mu} (T < \tau_{\partial}). $$

We deduce that

$$ \mathbb{E}_{\mu} \left( \theta^{-T_K \wedge T} 1_{T < \tau_{\partial}} \right) \le \left( C \frac{\mu(\varphi_1)}{\mu(\varphi_2)} + \theta^{-n_{\mu}} \right) \mathbb{P}_{\mu} (T < \tau_{\partial}). $$

Since $\theta^{-n_{\mu}} \le \frac{\theta^{-(n_5(D_{\theta})+1)} \mu(\varphi_1)}{D_{\theta}\mu(\varphi_2)}$, we have proved Lemma 9.4.

## 9.7 Conclusion of the proof of (2.1) for the norm $\| \cdot \|_{TV}(\varphi_1)$

For all $n \geq 1$, we introduce the linear operator on $L^{\infty}(\varphi_1)$, defined for all $h \in L^{\infty}(\varphi_1)$ as

$$ R_n h(x) = \mathbb{E}_x \left( h(X_n) 1_{T_K \le n < \tau_{\partial}} \right), \quad \forall x \in E. \qquad (9.19) $$

Note that this operator is well-defined since $|R_n h(x)| \le \|h\|_{L^{\infty}(\varphi_1)} P_n \varphi_1(x) < \infty$. We first give some properties of $R_n$, which can be seen as a bounded approximation of $P_n$ in $L^{\infty}(\varphi_1)$.

**Lemma 9.12.** We have

$$ \bar{R} := \sup_{n \ge 1} \sup_{x \in E} R_n \varphi_1(x) < \infty, $$

and for all $n \ge 1$ and $x \in E$,

$$ 0 \le P_n \varphi_1(x) - R_n \varphi_1(x) \le \theta_1^n \varphi_1(x). $$
Proof. Using Markov's property,

$$
\begin{align*}
R_n \varphi_1(x) &= \sum_{k \le n} \mathbb{E}_x [1_{T_k=k} P_{n-k} \varphi_1(X_k)] \\
&\le \sup_{y \in K, k \ge 0} P_k \varphi_1(y) \mathbb{P}_x(T_K \le n) \\
&\le \sup_{y \in K, k \ge 0} \frac{P_k \varphi_1(y)}{P_k \varphi_2(y)} \le D_1 \vee \sup_{y \in K} \frac{\varphi_1(y)}{\varphi_2(y)} < +\infty
\end{align*}
$$

by Lemma 9.6. This proves the first inequality. For the second one, we observe
that for all $x \in E$,

$$
P_n \varphi_1(x) - R_n \varphi_1(x) = \mathbb{E}_x (\varphi_1(X_n) 1_{\mathbb{N}_{<T_K}}) \le \theta_1^n \varphi_1(x)
$$

by Lemma 9.5.

<figure><img src="image_11.png" /></figure>

We fix $1 \le k \le n$, $h$ such that $|h| \le \varphi_1$ and $\mu$ such that $\mu(\varphi_1)/\mu(\varphi_2) \le D_\theta$, where $\theta = \frac{1+\theta_1/\theta_2}{2}$ and $D_\theta$ is from Lemma 9.6. The inequality (2.1) with $\|\cdot\|_{TV}$ in place of $\|\cdot\|_{TV}(\varphi_1)$ and Lemma 9.12 entail

$$
\left| \frac{\mu P_{n-k} R_k h}{\mu P_{n-k} \mathbb{1}_E} - v_{QSD}(R_k h) \right| \le C \alpha^{n-k} \frac{\mu(\varphi_1)}{\mu(\varphi_2)} \sup_{x \in E} |R_k h(x)| \le C D_\theta \bar{R} \alpha^{n-k}.
$$

The second inequality of Lemma 9.12 implies

$$
|v_{QSD}[(P_k - R_k)h]| \le \theta_1^k v_{QSD}(\varphi_1)
$$

and, by Lemma 9.6,

$$
\frac{\mu P_{n-k} (P_k - R_k) h}{\mu P_{n-k} \mathbb{1}_E} \le \theta_1^k \frac{\mu P_{n-k} \varphi_1}{\mu P_{n-k} \varphi_2} \le \theta_1^k \left( D_\theta \vee \frac{\mu(\varphi_1)}{\mu(\varphi_2)} \right) = \theta_1^k D_\theta.
$$

Combining the last three inequalities and recalling that $v_{QSD}P_k h = \theta_0^k v_{QSD}(h)$,
we obtain that, for some constant $C > 0$,

$$
\left| \frac{\mu P_n h}{\theta_0^k \mu P_{n-k} 1_E} - v_{QSD}(h) \right| \le C \left( \alpha^{n-k} \theta_0^{-k} + (\theta_1 / \theta_0)^k \right).
$$

Applying the last inequality to $h = 1_E$, we obtain

$$
\left| \frac{1}{\theta_0^k \mu P_{n-k} 1_E} - \frac{1}{\mu P_n 1_E} \right| \le \frac{C(\alpha^{n-k} \theta_0^{-k} + (\theta_1/\theta_0)^k)}{\mu P_n 1_E}
$$
so that, using Lemma 9.6,

$$ \left| \frac{\mu P_n h}{\theta_0^k \mu P_{n-k} 1_E} - \frac{\mu P_n h}{\mu P_n 1_E} \right| \le C(\alpha^{n-k} \theta_0^{-k} + (\theta_1/\theta_0)^k) \frac{\mu P_n \varphi_1}{\mu P_n 1_E} \le C D_\theta (\alpha^{n-k} \theta_0^{-k} + (\theta_1/\theta_0)^k). $$

Hence, for some $\bar{\alpha} < 1$, for all $n \ge 0$,

$$ \left| \frac{\mu P_n h}{\mu P_n 1_E} - v_{QSD}(h) \right| \le C(\alpha^{n-k} \theta_0^{-k} + (\theta_1/\theta_0)^k) \le C \bar{\alpha}^n. $$

Finally, if $\mu(\varphi_1)/\mu(\varphi_2) > D_\theta$, then let $T = \left\lfloor \frac{\ln(D_\theta \mu(\varphi_1)/\mu(\varphi_2))}{-\ln\theta} \right\rfloor$, so that $\mu P_T \varphi_1 / \mu P_T \varphi_2 \le D_\theta$ according to Lemma 9.6. We deduce from the previous inequality applied to $\mu P_T / \mu P_T 1_E$ that, for all $n \ge 0$,

$$ \left| \frac{\mu P_{T+n} h}{\mu P_{T+n} 1_E} - v_{QSD}(h) \right| \le C \bar{\alpha}^n \le C \bar{\alpha}^n \theta^{T-1} \frac{\mu(\varphi_1)}{\mu(\varphi_2)} $$

while, using again Lemma 9.6, we obtain, for all $n \in \{0, T-1\}$,

$$ \left| \frac{\mu P_n h}{\mu P_n 1_E} - v_{QSD}(h) \right| \le D_\theta \vee \left( \theta^n \frac{\mu(\varphi_1)}{\mu(\varphi_2)} \right) + v_{QSD}(\varphi_1) \le C \theta^n \frac{\mu(\varphi_1)}{\mu(\varphi_2)}. $$

The last two inequalities conclude the proof of (2.1) with $\alpha = \bar{\alpha} \vee \theta$ and hence of Theorem 2.1.

## 9.8 The case where $P_x(n < \tau_\partial) = 0$ for some $x \in E$ and $n \ge 1$

In this section, we assume that $X$ satisfies assumption (E), but we do not assume anymore that $P_x(n < \tau_\partial) > 0$ for all $x \in E$ and all $n \ge 1$. We introduce $\tilde{E} = \{x \in E, P_x(n < \tau_\partial) > 0 \ \forall n \ge 0\}$ and $\tilde{E} = E \setminus \tilde{E}$. One immediately deduces from (E2) that, for all $x \in \tilde{E}$ and all $n \ge 0$, $\varphi_2(x) = 0$ and $P_x(X_n \in K) = 0$, and hence that $\delta_x P_n \varphi_1 \le \theta_1^n \varphi_1(x)$ by Lemma 9.5. In addition, one easily checks that the semi-group $P$ restricted to $\tilde{E} \cup \{\partial\}$ still satisfies assumption (E), and in particular (2.1) applies.

Let $\mu$ be a probability measure on $E$ such that $\mu(\varphi_2) > 0$ and $\mu(\varphi_1) < +\infty$. Then, for all $n \ge 0$ and all $|h| \le \varphi_1$,

$$ \left| \mu P_n h - v_{QSD}(h) \mu P_n 1_E \right| \le \left| \mu P_n h - \mu_{|\tilde{E}} P_n h \right| + \left| \mu_{|\tilde{E}} P_n h - v_{QSD}(h) \mu_{|\tilde{E}} P_n 1_E \right| + v_{QSD}(\varphi_1) \left| \mu_{|\tilde{E}} P_n 1_E - \mu P_n 1_E \right|. $$

Each term can be bounded as follows:

$$ |\mu P_n h - \mu_{|\tilde{E}} P_n h| \le \mu_{|\tilde{E}} P_n \varphi_1 \le \theta_1^n \mu \varphi_1, $$

$$ |\mu_{|\tilde{E}} P_n h - v_{QSD}(h) \mu_{|\tilde{E}} P_n 1_E| \le C \alpha^n \frac{\mu_{|\tilde{E}}(\varphi_1)}{\mu_{|\tilde{E}}(\varphi_2)} \mu_{|\tilde{E}} P_n 1_E \le C \alpha^n \frac{\mu(\varphi_1)}{\mu(\varphi_2)} \mu P_n 1_E, $$

$$ v_{QSD}(\varphi_1) |\mu_{|\tilde{E}} P_n 1_E - \mu P_n 1_E| \le v_{QSD}(\varphi_1) \mu_{|\tilde{E}} P_n \varphi_1 \le v_{QSD}(\varphi_1) \theta_1^n \mu \varphi_1. $$
Since $\mu P_n 1_E \ge \theta_2^n \mu(\varphi_2)$, we deduce that

$$ \left| \frac{\mu P_n h}{\mu P_n 1_E} - v_{QSD}(h) \right| \le \left( (\theta_1/\theta_2)^n + v_{QSD}(\varphi_1)(\theta_1/\theta_2)^n + C\alpha^n \right) \frac{\mu(\varphi_1)}{\mu(\varphi_2)}. $$

This concludes the proof of (2.1) in the general case.

# 10 Proof of the other results of Section 2

The previous section ensures the existence of a quasi-stationary distribution $v_{QSD}$ such that $v_{QSD}(\varphi_1) < +\infty$ and $v_{QSD}(K) > 0$. Denoting by $\theta_0$ its associated decay parameter, we observe that $\theta_2 \le \theta_0$, since Lemma 9.5 entails that, for all $n \ge 1$,

$$ \theta_0^n = \mathbb{P}_{v_{QSD}}(n < \tau_\partial) \ge v_{QSD}(K) \inf_{y \in K} \mathbb{P}_y(n < \tau_\partial) \ge v_{QSD}(K) \theta_2^n \inf_{y \in K} \varphi_2(y). $$

We begin to prove Theorem 2.2 in Section 10.1, except for the exponential convergence in $L^\infty(\varphi_1)$. We then prove Theorem 2.3 in Section 10.2. In Section 10.3, we conclude the proof of Theorem 2.2 and prove Corollary 2.4. We prove Corollary 2.6 in Subsection 10.5.

## 10.1 Proof of the existence of the eigenfunction $\eta$

In this section, we show that the limit (2.2) is well defined pointwise, $v_{QSD}(\eta) = 1$, $P_1\eta = \theta_0\eta$, $\eta$ is lower bounded away from 0 on $K$ and $\eta \in L^\infty(\varphi_1^{\log(1/\theta_0)/\log(1/\theta_1)})$. For all $n \ge 0$ and $x \in E \cup \{\partial\}$, let us denote

$$ \eta_n(x) = \theta_0^{-n} \mathbb{P}_x(n < \tau_\partial) = \frac{\mathbb{P}_x(n < \tau_\partial)}{\mathbb{P}_{v_{QSD}}(n < \tau_\partial)}. $$

By Lemma 9.9, for all $x \in E$,

$$ \begin{aligned} \eta_n(x) &\le C \theta_0^{-n} \inf_{y \in K} \mathbb{P}_y(n < \tau_\partial) \varphi_1(x) \\ &\le \frac{C}{v_{QSD}(K)} \theta_0^{-n} \mathbb{P}_{v_{QSD}}(n < \tau_\partial) \varphi_1(x) = \frac{C \varphi_1(x)}{v_{QSD}(K)}. \end{aligned} \quad (10.1) $$

This implies that the sequence $(\eta_n)_{n \ge 0}$ is uniformly bounded in $L^\infty(\varphi_1)$.

For all probability measure $\mu$ on $E$ and for all $n, m \ge 0$, by Markov's property,

$$ \mu(\eta_{n+m}) = \mu(\eta_n) \mathbb{E}_\mu \left[ \theta_0^{-m} \mathbb{P}_{X_n}(m < \tau_\partial) \mid n < \tau_\partial \right]. $$