1. Notation

Throughout, we shall use the following notation:

• $\mathbb{N} = \{1,2, \ldots, \}$ and $\mathbb{Z}_+ = \{0\} \cup \mathbb{N}$ .

• $\mathcal{M}_1(E)$ denotes the space of the probability measures whose support is included in E.

• $\mathcal{B}(E)$ denotes the set of the measurable bounded functions defined on E.

• $\mathcal{B}_1(E)$ denotes the set of the measurable functions f defined on E such that $\|f\|_\infty \leq 1$ .

• For all $\mu \in \mathcal{M}_1(E)$ and $p \in \mathbb{N}$ , $\mathbb{L}^p(\mu)$ denotes the set of the measurable functions $f \;:\; E \mapsto \mathbb{R}$ such that $\int_E |f(x)|^p \mu(dx) < + \infty$ .

• For any $\mu \in \mathcal{M}_1(E)$ and $f \in \mathbb{L}^1(\mu)$ , we define

  \begin{equation*}\mu(f) \;:\!=\; \int_E f(x) \mu(dx).\end{equation*}

• For any positive function $\psi$ ,

  \begin{equation*}\mathcal{M}_1(\psi) \;:\!=\; \{\mu \in \mathcal{M}_1(E) \;:\; \mu(\psi) < + \infty\}.\end{equation*}

• Id denotes the identity operator.

3. Ergodic theorem for asymptotically periodic time-inhomogeneous semigroup.

Asymptotic periodicity: the definition. Let $(E,{\mathcal E})$ be a measurable space. Consider $\{(E_t, {\mathcal E}_t)_{t \geq 0}, (P_{s,t})_{s \leq t}\}$ a Markovian time-inhomogeneous semigroup, giving a family of measurable subspaces of $(E, {\mathcal E})$ , denoted by $(E_t, {\mathcal E}_t)_{t \geq 0}$ , and a family of linear operator $(P_{s,t})_{s \leq t}$ , with $P_{s,t} \;:\; \mathcal{B}(E_t) \to \mathcal{B}(E_s)$ , satisfying for any $r \leq s \leq t$ ,

\begin{equation*}P_{s,s} = \text{Id},\;\;\;\;\;\;\;\;P_{s,t}\mathbb{1}_{E_t} = \mathbb{1}_{E_s},\;\;\;\;\;\;\;\;P_{r,s}P_{s,t} = P_{r,t}.\end{equation*}

In particular, associated to $\{(E_t, {\mathcal E}_t)_{t \geq 0}, (P_{s,t})_{s \leq t}\}$ is a Markov process $(X_t)_{t \geq 0}$ and a family of probability measures $(\mathbb{P}_{s,x})_{s \geq 0, x \in E_s}$ such that, for any $s \leq t$ , $x \in E_s$ , and $A \in {\mathcal E}_t$ ,

\begin{equation*}\mathbb{P}_{s,x}[X_t \in A] = P_{s,t}\mathbb{1}_{A}(x).\end{equation*}

We denote by $\mathbb{P}_{s,\mu} \;:\!=\; \int_{E_s} \mathbb{P}_{s,x} \mu(dx)$ any probability measure $\mu$ supported on $E_s$ . We also denote by $\mathbb{E}_{s,x}$ and $\mathbb{E}_{s,\mu}$ the expectations associated to $\mathbb{P}_{s,x}$ and $\mathbb{P}_{s,\mu}$ respectively. Finally, the following notation will be used for $\mu \in \mathcal{M}_1(E_s)$ , $s \leq t$ , and $f \in \mathcal{B}(E_t)$ :

\begin{equation*}\mu P_{s,t} f \;:\!=\; \mathbb{E}_{s,\mu}[f(X_t)],\;\;\;\;\;\;\mu P_{s,t} \;:\!=\; \mathbb{P}_{s,\mu}[X_t \in \cdot].\end{equation*}

The periodicity of a time-inhomogeneous semigroup is defined as follows. We say a semigroup $\{(F_t, {\mathcal F}_t)_{t \geq 0},(Q_{s,t})_{s \leq t}\}$ is $\gamma$ -periodic (for $\gamma > 0$ ) if, for any $s \leq t$ ,

\begin{equation*}(F_t, {\mathcal F}_t) = (F_{t+k\gamma}, {\mathcal F}_{t+k\gamma}),\;\;\;\;Q_{s,t} = Q_{s+k \gamma,t+k\gamma},\;\;\;\;\forall k \in \mathbb{Z}_+.\end{equation*}

It is now possible to define an asymptotically periodic semigroup.

When $\psi_s = \tilde{\psi}_s = \mathbb{1}$ for all $s \geq 0$ , we say that the semigroup $(P_{s,t})_{s \leq t}$ is asymptotically periodic in total variation. By extension, we will say that the process $(X_t)_{t \geq 0}$ is asymptotically periodic (in total variation) if the associated semigroup $\{(E_t, {\mathcal E}_t)_{t \geq 0},(P_{s,t})_{s \leq t}\}$ is asymptotically periodic (in total variation).

In what follows, the functions $(\psi_s)_{s \geq 0}$ and $(\tilde{\psi}_s)_{s \in [0,\gamma)}$ will play the role of Lyapunov functions (that is to say, satisfying Assumption 1(ii) below) for the semigroups $(P_{s,t})_{s \leq t}$ and $(Q_{s,t})_{s \leq t}$ , respectively. The introduction of these functions in the definition of asymptotically periodic semigroups will allow us to establish an ergodic theorem for processes satisfying the Lyapunov/minorization conditions stated below.

Lyapunov/minorization conditions. The main assumption of Theorem 1, which will be provided later, will be that the asymptotically periodic Markov process satisfies the following assumption.

When a semigroup $(P_{s,t})_{s \leq t}$ satisfies Assumption 1 as stated above, we will say that the functions $(\psi_s)_{s \geq 0}$ are Lyapunov functions for the semigroup $(P_{s,t})_{s \leq t}$ . In particular, under (ii) and (iii), it is easy to prove that for any $s \leq t$ ,

(7) \begin{equation} P_{s,t} \psi_t \leq C \bigg(1 + \frac{C}{1-\theta}\bigg) \psi_s. \end{equation}

We remark in particular that Assumption 1 implies an exponential weak ergodicity in $\psi_t$ -distance; that is, we have the existence of two constants $C' > 0$ and $\kappa > 0$ such that, for all $s \leq t$ and for all probability measures $\mu_1, \mu_2 \in \mathcal{M}_1(E_s)$ ,

(8) \begin{equation} \| \mu_1 P_{s,t} - \mu_2 P_{s,t} \|_{\psi_t} \leq C' [\mu_1(\psi_s) + \mu_2(\psi_s)] e^{-\kappa (t-s)}, \end{equation}

where, for a given function $\psi$ , $\| \mu - \nu \|_\psi$ is the $\psi$ -distance, defined to be

\begin{equation*}\| \mu - \nu \|_{\psi} \;:\!=\; \sup_{|f| \leq \psi} \left| \mu(f) - \nu(f) \right|,\;\;\;\;\forall \mu, \nu \in \mathcal{M}_1(\psi).\end{equation*}

In particular, when $\psi = \mathbb{1}$ for all $t \geq 0$ , the $\psi$ -distance is the total variation distance. If we have weak ergodicity (8) in the time-homogeneous setting (see in particular [Reference Hairer and Mattingly15]), the proof of [Reference Hairer and Mattingly15, Theorem 1.3] can be adapted to a general time-inhomogeneous framework (see for example [Reference Champagnat and Villemonais6, Subsection 9.5]).

The main theorem and proof. The main result of this paper is the following.

Theorem 1. Let $\{ (E_t, {\mathcal E}_t)_{t \geq 0},(P_{s,t})_{s \leq t}, (X_t)_{t \geq 0}, (\mathbb{P}_{s,x})_{s \geq 0, x \in E_s}\}$ be an asymptotically $\gamma$ -periodic time-inhomogeneous Markov process, with $\gamma > 0$ , and denote by $\{(F_t, {\mathcal F}_t)_{t \geq 0},(Q_{s,t})_{s \leq t}\}$ its periodic auxiliary semigroup. Also, denote by $(\psi_s)_{s \geq 0}$ and $(\tilde{\psi}_s)_{s \geq 0}$ the two families of functions as defined in Definition 1. Assume moreover the following:

1. 1. The semigroups $(P_{s,t})_{s \leq t}$ and $(Q_{s,t})_{s \leq t}$ satisfy Assumption 1, with $(\psi_s)_{s \geq 0}$ and $(\tilde{\psi}_s)_{s \geq 0}$ respectively as Lyapunov functions.

2. 2. For any $s \in [0,\gamma)$ , $(\psi_{s+n \gamma})_{n \in \mathbb{Z}_+}$ converges pointwise to $\tilde{\psi}_s$ .

Then, for any $\mu \in \mathcal{M}_1(E_0)$ such that $\mu(\psi_0) < + \infty$ ,

(9) \begin{equation} \Bigg\|\frac{1}{t} \int_0^t \mu P_{0,s}[\psi_s \times \cdot] ds - \frac{1}{\gamma} \int_0^\gamma \beta_{\gamma} Q_{0,s}\big[\tilde{\psi}_s \times \cdot\big] ds \Bigg\|_{TV} \underset{t \to \infty}{{\longrightarrow}} 0, \end{equation}

where $\beta_\gamma \in \mathcal{M}_1(F_0)$ is the unique invariant probability measure of the skeleton semigroup $(Q_{0,n \gamma})_{n \in \mathbb{Z}_+}$ satisfying $\beta_\gamma(\tilde{\psi}_0) < + \infty$ . Moreover, for any $f \in \mathcal{B}(E)$ we have the following:

1. 1. For any $\mu \in \mathcal{M}_1(E_0)$ ,

   (10) \begin{equation} \mathbb{E}_{0,\mu}\!\left[\Bigg|\frac{1}{t} \int_0^t f(X_s)ds - \frac{1}{\gamma} \int_0^\gamma \beta_\gamma Q_{0,s}f ds\Bigg|^2\right] \underset{t \to \infty}{\longrightarrow} 0.\end{equation}

2. 2. If moreover $\mu(\psi_0) < + \infty$ , then

   (11) \begin{equation}\frac{1}{t} \int_0^t f(X_s) ds \underset{t \to \infty}{\longrightarrow} \frac{1}{\gamma} \int_0^\gamma \beta_\gamma Q_{0,s}f ds,\;\;\;\;\mathbb{P}_{0,\mu}\textit{-almost surely.}\end{equation}

Doeblin condition. There exist $t_0 \geq 0$ , $c > 0$ , and a family of probability measures $(\nu_t)_{t \geq 0}$ on $(E_t)_{t \geq 0}$ such that, for any $s \geq 0$ and $x \in E_s$ ,

(12) \begin{equation} \delta_x P_{s,s+t_0} \geq c \nu_{s+t_0}.\end{equation}

In fact, if we assume that Assumption 1(i) holds for $K_s = E_s$ , the Doeblin condition holds if we set $t_0 \;:\!=\; n_0 t_1$ . Conversely, the Doeblin condition implies the conditions (i), (ii), and (iii) with $K_s = E_s$ and $\psi_s = \mathbb{1}_{E_s}$ for all $s \geq 0$ , so that these conditions are equivalent. In fact, (ii) and (iii) straightforwardly hold true for $(K_s)_{s \geq 0} = (E_s)_{s \geq 0}$ , $(\psi_s)_{s \geq 0} = (\mathbb{1}_{E_s})_{s \geq 0}$ , $C = 1$ , any $\theta \in (0,1)$ , and any $t_1 \geq 0$ . If we set $t_1 = t_0$ and $n_0 = 1$ , the Doeblin condition implies that, for any $s \in [0,t_1)$ ,

\begin{equation*}\delta_x P_{s,s+t_1} \geq c \nu_{s+t_1},\;\;\;\;\forall x \in E_s.\end{equation*}

Integrating this inequality over $\mu \in \mathcal{M}_1(E_s)$ , one obtains

\begin{equation*}\mu P_{s,s+t_1} \geq c \nu_{s+t_1},\;\;\;\;\forall s \in [0,t_1), \ \forall \mu \in \mathcal{M}_1(E_s).\end{equation*}

Then, by the Markov property, for all $s \in [0,t_1)$ , $x \in E_s$ , and $n \in \mathbb{N}$ , we have

\begin{equation*}\delta_x P_{s,s+nt_1} = (\delta_x P_{s,s+(n-1)t_1})P_{s + (n-1)t_1, s + nt_1} \geq c \nu_{s+nt_1},\end{equation*}

which is (i).

Theorem 1 then implies the following corollary.

Corollary 1. Let $(X_t)_{t \geq 0}$ be asymptotically $\gamma$ -periodic in total variation distance. If $(X_t)_{t \geq 0}$ and its auxiliary semigroup satisfy a Doeblin condition, then the convergence (10) is improved to

\begin{equation*} \sup_{\mu \in \mathcal{M}_1(E_0)} \sup_{f \in \mathcal{B}_1(E)} \mathbb{E}_{0,\mu}\left[\Bigg|\frac{1}{t} \int_0^t f(X_s)ds - \frac{1}{\gamma} \int_0^\gamma \beta_\gamma Q_{0,s}f ds\Bigg|^2\right] \underset{t \to \infty}{\longrightarrow} 0.\end{equation*}

Moreover, the almost sure convergence (11) holds for any initial measure $\mu$ .

Remark 2. We also note that, if the convergence (6) holds for all

\begin{equation*}x \in \bigcup_{k=0}^\infty \bigcap_{l \geq k} E_{s+l \gamma} \cap F_s,\end{equation*}

then this implies (6) and therefore the pointwise convergence of $(\psi_{s+n \gamma})_{n \in \mathbb{Z}_+}$ to $\tilde{\psi}_s$ (by taking $n = 0$ in (6)).

First step. Since the auxiliary semigroup $(Q_{s,t})_{s \leq t}$ satisfies Assumption 1 with $(\tilde{\psi}_s)_{s \geq 0}$ as Lyapunov functions, the time-homogeneous semigroup $(Q_{0,n\gamma})_{n \in \mathbb{Z}_+}$ satisfies Assumptions 1 and 2 of [Reference Hairer and Mattingly15], which we now recall (using our notation).

Assumption 2. ([Reference Hairer and Mattingly15, Assumption 1].) There exist $V \;:\; F_0 \to [0,+\infty)$ , $n_1 \in \mathbb{N}$ , and constants $K \geq 0$ and $\kappa \in (0,1)$ such that

\begin{equation*}Q_{0,n_1\gamma} V \leq \kappa V + K.\end{equation*}

Assumption 3. ([Reference Hairer and Mattingly15, Assumption 2].) There exist a constant $\alpha \in (0,1)$ and a probability measure $\nu$ such that

\begin{equation*}\inf_{x \in \mathcal{C}_R} \delta_x Q_{0,n_1\gamma} \geq \alpha \nu(\cdot),\end{equation*}

with $\mathcal{C}_R \;:\!=\; \{x \in F_0 \;:\; V(x) \leq R \}$ for some $R > 2 K/(1-\kappa)$ , where $n_1$ , K, and $\kappa$ are the constants from Assumption 2.

In fact, since $(Q_{s,t})_{s \leq t}$ satisfies (ii) and (iii) of Assumption 1, there exist $C > 0$ , $\theta \in (0,1)$ , $t_1 \geq 0$ , and $(K_s)_{s \geq 0}$ such that

(13) \begin{equation}Q_{s,s+t_1} \tilde{\psi}_{s+t_1} \leq \theta \tilde{\psi}_s + C \mathbb{1}_{K_s},\;\;\;\;\forall s \geq 0,\end{equation}

and

\begin{equation*}Q_{s,s+t}\tilde{\psi}_{s+t} \leq C \tilde{\psi}_s,\;\;\;\;\forall s \geq 0, \forall t \in [0,t_1).\end{equation*}

We let $n_2 \in \mathbb{N}$ be such that $\theta^{n_2} C \bigl(1 + \frac{C}{1-\theta}\bigl) < 1$ . By (13) and recalling that $\tilde{\psi}_t = \tilde{\psi}_{t+\gamma}$ for all $t \geq 0$ , one has for any $s \geq 0$ and $n \in \mathbb{N}$ ,

(14) \begin{equation}Q_{s,s+nt_1}\tilde{\psi}_{s+nt_1} \leq \theta^n \tilde{\psi}_s + \frac{C}{1 - \theta}.\end{equation}

Thus, for all $n_1 \geq \lceil \frac{n_2t_1}{\gamma} \rceil$ ,

\begin{align*}Q_{0,n_1\gamma}\tilde{\psi}_0 &= Q_{0,n_1 \gamma - n_2t_1} Q_{n_1 \gamma - n_2t_1, n_1 \gamma} \tilde{\psi}_{n_1 \gamma}\\[5pt] &\leq \theta^{n_2} Q_{0,n_1 \gamma -n_2t_1} \tilde{\psi}_{n_1\gamma-n_2t_1} + \frac{C}{1-\theta}\\[5pt] &\leq \theta^{n_2}C\bigg(1 + \frac{C}{1-\theta}\bigg) \tilde{\psi}_0 + \frac{C}{1 - \theta},\end{align*}

where we successively used the semigroup property of $(Q_{s,t})_{s \leq t}$ , (14), and (7) applied to $(Q_{s,t})_{s \leq t}$ . Hence one has Assumption 2 by setting $V = \tilde{\psi}_0$ , $\kappa \;:\!=\; \theta^{n_2}C\bigl(1 + \frac{C}{1-\theta}\bigl)$ , and $K \;:\!=\; \frac{C}{1 - \theta}$ .

We now prove Assumption 3. To this end, we introduce a Markov process $(Y_t)_{t \geq 0}$ and a family of probability measures $(\hat{\mathbb{P}}_{s,x})_{s \geq 0, x \in F_s}$ such that

\begin{equation*}\hat{\mathbb{P}}_{s,x}(Y_t \in A) = Q_{s,t}\mathbb{1}_A(x),\;\;\;\;\forall s \leq t,\ x \in F_s,\ A \in {\mathcal F}_t.\end{equation*}

In what follows, for all $s \geq 0$ and $x \in F_s$ , we will use the notation $\hat{\mathbb{E}}_{s,x}$ for the expectation associated to $\hat{\mathbb{P}}_{s,x}$ . Moreover, we define

\begin{equation*}T_K \;:\!=\; \inf\!\big\{n \in \mathbb{Z}_+ \;:\; Y_{n t_1} \in K_{nt_1}\big\}.\end{equation*}

Then, using (13) recursively, for all $k \in \mathbb{N}$ , $R > 0$ , and $x \in \mathcal{C}_R$ (recalling that $\mathcal{C}_R$ is defined in the statement of Assumption 3), we have

\begin{align*}\hat{\mathbb{E}}_{0,x}\big[\tilde{\psi}_{kt_1}(Y_{kt_1})\mathbb{1}_{T_K > k}\big] &= \hat{\mathbb{E}}_{0,x}\big[\mathbb{1}_{T_K > k-1} \hat{\mathbb{E}}_{(k-1)t_1,Y_{(k-1)t_1}}\big(\tilde{\psi}_{kt_1}(Y_{kt_1})\mathbb{1}_{T_K > k}\big)\big] \\[5pt] &\leq \theta \hat{\mathbb{E}}_{0,x}\big[\tilde{\psi}_{(k-1)t_1}(Y_{(k-1)t_1})\mathbb{1}_{T_K > k-1}\big] \leq \theta^{k} \tilde{\psi}_0(x) \leq R \theta^k.\end{align*}

Since $\tilde{\psi}_{kt_1} \geq 1$ for all $k \in \mathbb{Z}_+$ , we have that for all $x \in \mathcal{C}_R$ , for all $k \in \mathbb{Z}_+$ ,

\begin{equation*}\hat{\mathbb{P}}_{0,x}(T_K > k) \leq R \theta^k.\end{equation*}

In particular, there exists $k_0 \geq n_0$ such that, for all $k \geq k_0 - n_0$ ,

\begin{equation*}\hat{\mathbb{P}}_{0,x}(T_K > k) \leq \frac{1}{2}.\end{equation*}

Hence, for all $x \in \mathcal{C}_R$ ,

\begin{align*}\delta_x Q_{0,k_0t_1} = \hat{\mathbb{P}}_{0,x}\big(Y_{k_0t_1} \in \cdot\big) \geq &\sum_{i=0}^{k_0-n_0} \hat{\mathbb{E}}_{0,x}\big(\mathbb{1}_{T_K = i} \hat{\mathbb{P}}_{it_1,Y_{it_1}}\big(Y_{k_0t_1} \in \cdot\big)\big) \\[5pt] &\geq c \sum_{i=0}^{k_0-n_0} \hat{\mathbb{E}}_{0,x}\big(\mathbb{1}_{T_K = i}\big) \times \nu_{k_0t_1} \\[5pt] &= c \hat{\mathbb{P}}_{0,x}\big(T_K \leq k_0 - n_0\big) \nu_{k_0t_1} \\[5pt] &\geq \frac{c}{2} \nu_{k_0t_1}.\end{align*}

Hence, for all $n_1 \geq \bigl\lceil \frac{k_0t_1}{\gamma} \bigl\rceil$ , for all $x \in \mathcal{C}_R$ ,

\begin{equation*}\delta_x Q_{0,k_0t_1}Q_{k_0t_1,n_1\gamma} \geq \frac{c}{2} \nu_{k_0t_1} Q_{k_0t_1, n_1 \gamma}.\end{equation*}

Thus, Assumption 3 is satisfied if we take $n_1 \;:\!=\; \bigl\lceil \frac{n_2 t_1}{\gamma}\bigl\rceil \lor \bigl\lceil \frac{k_0t_1}{\gamma} \bigl\rceil$ , $\alpha \;:\!=\; \frac{c}{2}$ , and $\nu(\cdot) \;:\!=\; \nu_{k_0t_1} Q_{k_0t_1,n_1\gamma}$ .

Then, by [Reference Hairer and Mattingly15, Theorem 1.2], Assumptions 2 and 3 imply that $Q_{0,n_1 \gamma}$ admits a unique invariant probability measure $\beta_\gamma$ . Furthermore, there exist constants $C > 0$ and $\delta \in (0,1)$ such that, for all $\mu \in \mathcal{M}_1(F_0)$ ,

(15) \begin{equation}\| \mu Q_{0,nn_1\gamma} - \beta_\gamma\|_{\tilde{\psi}_0} \leq C \mu(\tilde{\psi}_0) \delta^n.\end{equation}

Since $\beta_\gamma$ is the unique invariant probability measure of $Q_{0,n_1 \gamma}$ , and noting that $\beta_\gamma Q_{0,\gamma}$ is invariant for $Q_{0,n_1 \gamma}$ , we deduce that $\beta_\gamma$ is the unique invariant probability measure for $Q_{0,\gamma}$ , and by (15), for all $\mu$ such that $\mu(\tilde{\psi}_0) < + \infty$ ,

\begin{equation*}\|\mu Q_{0,n\gamma} - \beta_\gamma\|_{\tilde{\psi}_0} \underset{n \to \infty}{\longrightarrow} 0.\end{equation*}

Now, for any $s \geq 0$ , note that $\delta_x Q_{s,\lceil \frac{s}{\gamma} \rceil \gamma} \tilde{\psi}_0 < + \infty$ for all $x \in F_s$ (this is a consequence of (7) applied to the semigroup $(Q_{s,t})_{s \leq t}$ ), and therefore, taking $\mu = \delta_x Q_{s,\lceil \frac{s}{\gamma} \rceil \gamma}$ in the above convergence,

\begin{equation*}\|\delta_x Q_{s,n\gamma} - \beta_\gamma\|_{\tilde{\psi}_0} \underset{n \to \infty}{\longrightarrow} 0\end{equation*}

for all $x \in F_s$ . Hence, since $Q_{n\gamma, n \gamma +s}\tilde{\psi}_s \leq C \bigl(1 + \frac{C}{1-\theta}\bigl) \tilde{\psi}_{n \gamma}$ by (7), we conclude from the above convergence that

(16) \begin{equation} \| \delta_{x} Q_{s,s+n\gamma} - \beta_\gamma Q_{0,s} \|_{\tilde{\psi}_s} \leq C \bigg(1 + \frac{C}{1-\theta}\bigg) \|\delta_x Q_{s,n\gamma} - \beta_\gamma\|_{\tilde{\psi}_0} \underset{n \to \infty}{\longrightarrow} 0.\end{equation}

Moreover, $\beta_\gamma(\tilde{\psi}_0) < + \infty$ .

Second step. The first part of this step (up to the equality (20)) is inspired by the proof of [Reference Bansaye, Cloez and Gabriel1, Theorem 3.11].

We fix $s \in [0,\gamma]$ . Without loss of generality, we assume that $\bigcap_{l \geq 0} E_{s+l \gamma} \cap F_s \ne \emptyset$ . Then, by Definition 1, there exists $x_s \in \bigcap_{l \geq 0} E_{s+l \gamma} \cap F_s$ such that for any $n \geq 0$ ,

\begin{equation*} \big\|\delta_{x_s} P_{s+k \gamma, s+(k+n)\gamma}[\psi_{s+(k+n)\gamma} \times \cdot] - \delta_{x_s} Q_{s, s + n \gamma}\big[\tilde{\psi}_{s} \times \cdot\big]\big\|_{TV} \underset{k \to \infty}{\longrightarrow} 0,\end{equation*}

which implies by (16) that

(17) \begin{equation} \lim_{n \to \infty} \lim_{k \to \infty} \big\|\delta_{x_s} P_{s+k \gamma, s+(k+n)\gamma}[\psi_{s+(k+n)\gamma} \times \cdot] - \beta_\gamma Q_{0,s}\big[\tilde{\psi}_s \times \cdot\big] \big\|_{TV} = 0.\end{equation}

Then, by the Markov property, (8), and (7), one obtains that, for any $k,n \in \mathbb{N}$ and $x \in \bigcap_{l \geq 0} E_{s+l \gamma}$ ,

(18) \begin{align} \|\delta_{x} & P_{s,s+(k+n)\gamma} - \delta_{x} P_{s+k\gamma, s+(k+n)\gamma} \|_{\psi_{s+(k+n)\gamma}} \notag \\[5pt] & \qquad = \|\left(\delta_{x} P_{s,s+k \gamma} \right) P_{s+k\gamma,s+(k+n)\gamma} - \delta_{x} P_{s+k\gamma, s+(k+n)\gamma} \|_{\psi_{s+(k+n)\gamma}} \notag \\[5pt] & \qquad \leq C' [P_{s,s+k \gamma} \psi_{s+k \gamma}(x) + \psi_{s+k \gamma}(x)] e^{- \kappa \gamma n} \notag \\[5pt] & \qquad \leq C''[\psi_s(x) + \psi_{s+k \gamma}(x)] e^{-\kappa \gamma n},\end{align}

where $C'' \;:\!=\; C' \bigl(C \bigl(1 + \frac{C}{1-\theta}\bigl) \lor 1\bigl)$ . Then, for any $k,n \in \mathbb{N}$ ,

(19) \begin{align} \| \delta_{x_s} P_{s,s+(k+n)\gamma}[\psi_{s+(k+n)\gamma} \times \cdot] - \beta_\gamma Q_{0,s} [\tilde{\psi}_{s} \times \cdot] \|_{TV} \end{align}

\begin{align*} \leq C''[\psi_s(x) + \psi_{s+k \gamma}(x)] e^{-\kappa \gamma n} +\big\|\delta_{x_s} P_{s+k \gamma, s+(k+n)\gamma}[\psi_{s+(k+n)\gamma} \times \cdot] - \beta_\gamma Q_{0,s}\big[\tilde{\psi}_s \times \cdot\big] \big\|_{TV}, \end{align*}

which by (17) and the pointwise convergence of $(\psi_{s+k \gamma})_{k \in \mathbb{Z}_+}$ implies that

(20) \begin{align} \lim_{n \to \infty} &\big\| \delta_{x_s} P_{s,s+n\gamma}[\psi_{s+n\gamma} \times \cdot] - \beta_\gamma Q_{0,s}\big[\tilde{\psi}_s \times \cdot\big] \big\|_{TV} \notag \\[5pt] & = \lim_{n \to \infty} \limsup_{k \to \infty} \| \delta_{x_s} P_{s,s+(k+n)\gamma}[\psi_{s+(k+n)\gamma} \times \cdot] - \beta_\gamma Q_{0,s} \big[\tilde{\psi}_{s} \times \cdot\big] \big\|_{TV} \notag \\[5pt] & = 0.\end{align}

The weak ergodicity (8) implies therefore that the previous convergence actually holds for any initial distribution $\mu \in \mathcal{M}_1(E_0)$ satisfying $\mu(\psi_0) < + \infty$ , so that

(21) \begin{equation} \big\| \mu P_{0,s+n \gamma}[\psi_{s+n \gamma} \times \cdot] - \beta_\gamma Q_{0,s}\big[\tilde{\psi}_s \times \cdot\big] \big\|_{TV} \underset{n \to \infty}{\longrightarrow} 0.\end{equation}

Since

\begin{equation*}\big\|\mu P_{0,s+n\gamma}[\psi_{s+n\gamma} \times \cdot] - \beta_\gamma Q_{0,s}\big[\tilde{\psi}_s \times \cdot\big]\big\|_{TV} \leq 2\end{equation*}

for all $\mu \in \mathcal{M}_1(E_0)$ , $s \geq 0$ , and $n \in \mathbb{Z}_+$ , (21) and Lebesgue’s dominated convergence theorem imply that

\begin{equation*}\frac{1}{\gamma} \int_0^\gamma \big\|\mu P_{0,s+n\gamma}[\psi_{s+n\gamma} \times \cdot] - \beta_\gamma Q_{0,s}\big[\tilde{\psi}_s \times \cdot\big]\big\|_{TV}ds \underset{n \to \infty}{\longrightarrow} 0,\end{equation*}

which implies that

\begin{equation*}\bigg\|\frac{1}{\gamma}\int_0^\gamma \mu P_{0,s+n\gamma}[\psi_{s+n \gamma} \times \cdot]ds - \frac{1}{\gamma} \int_0^\gamma \beta_\gamma Q_{0,s}\big[\tilde{\psi}_{s} \times \cdot\big] ds \bigg\|_{TV} \underset{n \to \infty}{\longrightarrow} 0.\end{equation*}

By Cesaro’s lemma, this allows us to conclude that, for any $\mu \in \mathcal{M}_1(E_0)$ such that $\mu(\psi_0) < + \infty$ ,

\begin{align*} \bigg\|\frac{1}{t} \int_0^t &\mu P_{0,s} [\psi_{s} \times \cdot] ds - \frac{1}{\gamma} \int_0^\gamma \beta_\gamma Q_{0,s} \big[\tilde{\psi}_{s} \times \cdot\big] ds \bigg\|_{TV} \\[5pt] & \leq \frac{1}{\lfloor \frac{t}{\gamma} \rfloor} \sum_{k=0}^{\lfloor \frac{t}{\gamma} \rfloor}\bigg\|\frac{1}{\gamma}\int_0^\gamma \mu P_{0,s+k\gamma}[\psi_{s+k \gamma} \times \cdot]ds - \frac{1}{\gamma} \int_0^\gamma \beta_\gamma Q_{0,s}\big[\tilde{\psi}_{s} \times \cdot\big] ds \bigg\|_{TV} \\[5pt] & \quad\qquad\qquad\qquad\qquad\qquad\qquad\qquad + \Bigg\|\frac{1}{t} \int_{\lfloor \frac{t}{\gamma} \rfloor \gamma}^t \mu P_{0,s}[\psi_{s} \times \cdot] ds\Bigg\|_{TV}\underset{t \to \infty}{\longrightarrow} 0,\end{align*}

which concludes the proof of (9) .

Third step. In the same manner, we now prove that, for any $\mu \in \mathcal{M}_1(E_0)$ such that $\mu(\psi_0) < + \infty$ ,

(22) \begin{equation} \bigg\|\frac{1}{t} \int_0^t \mu P_{0,s} ds - \frac{1}{\gamma} \int_0^\gamma \beta_\gamma Q_{0,s} ds \bigg\|_{TV} \underset{t \to \infty}{\longrightarrow} 0.\end{equation}

In fact, for any function f bounded by 1 and $\mu \in \mathcal{M}_1(E_0)$ such that $\mu(\psi_0) < + \infty$ ,

\begin{align*}&\bigg| \mu P_{0,s+n \gamma}\bigg[\psi_{s+n\gamma} \times \frac{f}{\psi_{s+n \gamma}}\bigg] - \beta_\gamma Q_{0,s}\bigg[\tilde{\psi}_s \times \frac{f}{\tilde{\psi}_s}\bigg]\bigg| \\[5pt] & \leq \bigg| \mu P_{0,s+n \gamma}\bigg[\psi_{s+n\gamma} \times \frac{f}{\psi_{s+n \gamma}}\bigg] - \beta_\gamma Q_{0,s}\bigg[\tilde{\psi}_s \times \frac{f}{{\psi}_{s+n\gamma}}\bigg]\bigg| + \bigg| \beta_\gamma Q_{0,s}\bigg[\tilde{\psi}_s \times \frac{f}{{\psi}_{s+n\gamma}}\bigg] \\[5pt] & \quad - \beta_\gamma Q_{0,s}\bigg[\tilde{\psi}_s \times \frac{f}{\tilde{\psi}_s}\bigg]\bigg| \\[5pt] & \leq \bigg\| \mu P_{0,s+n \gamma}[\psi_{s+n \gamma} \times \cdot] - \beta_\gamma Q_{0,s}\big[\tilde{\psi}_s \times \cdot\big] \bigg\|_{TV} + \bigg| \beta_\gamma Q_{0,s}\bigg[\tilde{\psi}_s \times \frac{f}{{\psi}_{s+n\gamma}}\bigg] \\[5pt] & \quad - \beta_\gamma Q_{0,s}\bigg[\tilde{\psi}_s \times \frac{f}{\tilde{\psi}_s}\bigg]\bigg|.\end{align*}

We now remark that, since $\psi_{s+n \gamma} \geq 1$ for any s and $n \in \mathbb{Z}_+$ , one has that

\begin{equation*}\bigg| \frac{\tilde{\psi}_s}{\psi_{s+n \gamma}} - 1 \bigg| \leq 1 + \tilde{\psi}_s.\end{equation*}

Since $(\psi_{s + n \gamma})_{n \in \mathbb{Z}_+}$ converges pointwise towards $\tilde{\psi}_s$ and $\beta_\gamma Q_{0,s} \tilde{\psi}_s < + \infty$ , Lebesgue’s dominated convergence theorem implies

\begin{equation*}\sup_{f \in \mathcal{B}_1(E)} \bigg| \beta_\gamma Q_{0,s}\bigg[\tilde{\psi}_s \times \frac{f}{{\psi}_{s+n\gamma}}\bigg] - \beta_\gamma Q_{0,s}\bigg[\tilde{\psi}_s \times \frac{f}{\tilde{\psi}_s}\bigg]\bigg| \underset{n \to \infty}{\longrightarrow} 0.\end{equation*}

Then, using (21), one has

\begin{equation*} \left\| \mu P_{0,s+n \gamma} - \beta_\gamma Q_{0,s} \right\|_{TV} \underset{n \to \infty}{\longrightarrow} 0,\end{equation*}

which allows us to conclude (22), using the same argument as in the first step.

Fourth step. In order to show the $\mathbb{L}^2$ -ergodic theorem, we let $f \in \mathcal{B}(E)$ . For any $x \in E_0$ and $t \geq 0$ ,

\begin{align*} \mathbb{E}_{0,x}&\Bigg[\bigg|\frac{1}{t} \int_0^t f(X_s)ds - \mathbb{E}_{0,x}\bigg[\frac{1}{t} \int_0^t f(X_s)ds\bigg] \bigg|^2\Bigg] \\[5pt] & = \frac{2}{t^2} \int_0^t \int_s^t (\mathbb{E}_{0,x}[f(X_s)f(X_u)] - \mathbb{E}_{0,x}[f(X_s)]\mathbb{E}_{0,x}[f(X_u)])du\,ds\\[5pt] & = \frac{2}{t^2} \int_0^t \int_s^t \mathbb{E}_{0,x}[f(X_s)(f(X_u) - \mathbb{E}_{0,x}[f(X_u)])]du\,ds\\[5pt] & = \frac{2}{t^2} \int_0^t \int_s^t \mathbb{E}_{0,x}[f(X_s)(\mathbb{E}_{s,X_s}[f(X_u)] - \mathbb{E}_{s,\delta_x P_{0,s}}[f(X_u)])]du\,ds,\end{align*}

where the Markov property was used in the last line. By (8) (weak ergodicity) and (7), one obtains for any $s \leq t$

(23) \begin{equation} \big|\mathbb{E}_{s,X_s}[f(X_t)] - \mathbb{E}_{s,\delta_x P_{0,s}}[f(X_t)]\big| \leq C'' \|f\|_\infty [\psi_s(X_s) + \psi_0(x)] e^{- \kappa (t-s)},\;\;\;\;\mathbb{P}_{0,x}\text{-almost surely,}\end{equation}

where C ′ was defined in the first part. As a result, for any $x \in E_0$ and $t \geq 0$ ,

\begin{align*} \mathbb{E}_{0,x} &\Bigg[\bigg|\frac{1}{t} \int_0^t f(X_s)ds - \mathbb{E}_{0,x}\bigg[\frac{1}{t} \int_0^t f(X_s)ds\bigg] \bigg|^2\Bigg] \\[5pt] &\leq \frac{2C''\|f\|_\infty}{t^2} \int_0^t \int_s^t \mathbb{E}_{0,x}[|f(X_s)|(\psi_s(X_s)+\psi_0(x))] e^{-\kappa(u-s)}du\,ds \\[5pt] &= \frac{2C''\|f\|_\infty}{t^2} \int_0^t \mathbb{E}_{0,x}[|f(X_s)|(\psi_s(X_s)+\psi_0(x))] e^{\kappa s} \frac{e^{-\kappa s} - e^{- \kappa t}}{\kappa}ds \\[5pt] &= \frac{2C''\|f\|_\infty}{\kappa t} \times \mathbb{E}_{0,x}\bigg[\frac{1}{t} \int_0^t |f(X_s)|(\psi_s(X_s) + \psi_0(x))ds\bigg] \\[5pt] & \quad - \frac{2C''\|f\|_\infty e^{-\kappa t}}{\kappa t^2} \int_0^t e^{\kappa s} \mathbb{E}_{0,x}[|f(X_s)|(\psi_s(X_s)+\psi_0(x))]ds.\end{align*}

Then, by (9), there exists a constant $\tilde{C} > 0$ such that, for any $x \in E_0$ , when $t \to \infty$ ,

(24) \begin{align} & \mathbb{E}_{0,x}\Bigg[\bigg|\frac{1}{t} \int_0^t f(X_s)ds - \mathbb{E}_{0,x}\bigg[\frac{1}{t} \int_0^t f(X_s)ds\bigg] \bigg|^2\Bigg] \leq \frac{\tilde{C} \|f\|_{\infty} \psi_0(x)}{t} \nonumber \\[5pt] & \quad \times \frac{1}{\gamma} \int_0^\gamma \beta_\gamma Q_{0,s}\big[|f|\tilde{\psi}_s\big] ds + o\bigg(\frac{1}{t}\bigg).\end{align}

Since $f \in \mathcal{B}(E)$ and by definition of the total variation distance, (22) implies that, for all $x \in E_0$ ,

\begin{equation*}\bigg| \frac{1}{t} \int_0^t P_{0,s}f(x) - \frac{1}{\gamma} \int_0^\gamma \beta_\gamma Q_{0,s} f ds \bigg| \leq \|f\|_\infty \bigg\|\frac{1}{t} \int_0^t \delta_x P_{0,s} ds - \frac{1}{\gamma} \int_0^\gamma \beta_\gamma Q_{0,s} ds \bigg\|_{TV} \underset{t \to \infty}{\longrightarrow} 0.\end{equation*}

Then, using (22), one deduces that for any $x \in E_0$ and bounded function f,

\begin{align*}& \mathbb{E}_{0,x}\Bigg[\bigg|\frac{1}{t} \int_0^t f(X_s)ds - \frac{1}{\gamma} \int_0^\gamma \beta_\gamma Q_{0,s} f ds \bigg|^2\Bigg] \\[5pt] & \leq 2 \Bigg(\mathbb{E}_{0,x}\Bigg[\!\bigg(\frac{1}{t} \int_0^t f(X_s)ds - \frac{1}{t} \int_0^t P_{0,s}f(x)\!\bigg)^2\Bigg] + \left| \frac{1}{t} \int_0^t P_{0,s}f(x) - \frac{1}{\gamma} \int_0^\gamma \beta_\gamma Q_{0,s} f ds \right|^2\Bigg) \!\underset{t \to \infty}{\longrightarrow} \!0.\end{align*}

The convergence for any probability measure $\mu \in \mathcal{M}_1(E_0)$ comes from Lebesgue’s dominated convergence theorem.

Fifth step. We now fix nonnegative $f \in \mathcal{B}(E)$ , and $\mu \in \mathcal{M}_1(E_0)$ satisfying $\mu(\psi_0) < + \infty$ . The following proof is inspired by the proof of [Reference Vassiliou26, Theorem 12].

Since $\mu(\psi_0) < + \infty$ , the inequality (24) implies that there exists a finite constant $C_{f,\mu} \in (0,\infty)$ such that, for t large enough,

\begin{equation*}\mathbb{E}_{0,\mu}\Bigg[\bigg|\frac{1}{t} \int_0^t f(X_s)ds - \mathbb{E}_{0,\mu}\bigg[\frac{1}{t} \int_0^t f(X_s)ds\bigg] \bigg|^2\Bigg] \leq \frac{C_{f,\mu}}{t}.\end{equation*}

Then, for n large enough,

\begin{equation*}\mathbb{E}_{0,\mu}\Bigg[\Bigg|\frac{1}{n^2} \int_0^{n^2} f(X_s)ds - \mathbb{E}_{0,\mu}\Bigg[\frac{1}{n^2} \int_0^{n^2} f(X_s)ds\Bigg] \Bigg|^2\Bigg] \leq \frac{C_{f,\mu}}{n^2}.\end{equation*}

Then, by Chebyshev’s inequality and the Borel–Cantelli lemma, this last inequality implies that

\begin{equation*}\Bigg|\frac{1}{n^2} \int_0^{n^2} f(X_s)ds - \mathbb{E}_{0,\mu}\Bigg[\frac{1}{n^2} \int_0^{n^2} f(X_s)ds\Bigg] \Bigg| \underset{n \to \infty}{{\longrightarrow}} 0,\;\;\;\;\mathbb{P}_{0,\mu}\text{-almost surely.}\end{equation*}

One thereby obtains by the convergence (22) that

(25) \begin{equation}\frac{1}{n^2} \int_0^{n^2} f(X_s)ds \underset{n \to \infty}{{\longrightarrow}} \frac{1}{\gamma} \int_0^\gamma \beta_\gamma Q_{0,s} f ds,\;\;\;\;\mathbb{P}_{0,\mu}\text{-almost surely.}\end{equation}

Since the nonnegativity of f is assumed, this implies that for any $t > 0$ we have

\begin{equation*}\int_0^{\lfloor \sqrt{t} \rfloor^2} f(X_s)ds \leq \int_0^t f(X_s)ds \leq \int_0^{\lceil \sqrt{t} \rceil^2} f(X_s)ds.\end{equation*}

These inequalities and (25) then give that

\begin{equation*}\frac{1}{t} \int_0^t f(X_s)ds \;\underset{t \to \infty}{{\longrightarrow}}\; \frac{1}{\gamma} \int_0^\gamma \beta_\gamma Q_{0,s} f ds,\;\;\;\;\mathbb{P}_{0,\mu}\text{-almost surely.}\end{equation*}

In order to conclude that the result holds for any bounded measurable function f, it is enough to decompose $f = f_+ - f_-$ with $f_+ \;:\!=\; f \lor 0$ and $f_- = (\!-\!f) \lor 0$ and apply the above convergence to $f_+$ and $f_-$ . This concludes the proof of Theorem 1.

Remark 3. The proof of Theorem 1, as written above, does not allow us to deal with semigroups satisfying a Doeblin condition with time-dependent constant $c_s$ , that is, such that there exist $t_0 \geq 0$ and a family of probability measure $(\nu_t)_{t \geq 0}$ on $(E_t)_{t \geq 0}$ such that, for all $s \geq 0$ and $x \in E_s$ ,

\begin{equation*}\delta_x P_{s,s+t_0} \geq c_{s+t_0} \nu_{s+t_0}.\end{equation*}

In fact, under the condition written above, we can show (see for example the proof of the formula (2.7) of [Reference Champagnat and Villemonais9, Theorem 2.1]) that, for all $s \leq t$ and $\mu_1, \mu_2 \in \mathcal{M}_1(E_s)$ ,

\begin{equation*}\| \mu_1 P_{s,t} - \mu_2 P_{s,t} \|_{TV} \leq 2 \prod_{k=0}^{\left \lfloor \frac{t-s}{t_0} \right \rfloor-1} (1-c_{t-k t_0}).\end{equation*}

Hence, by this last inequality with $\mu_1 = \delta_x P_{s,s+k\gamma}$ , $\mu_2 = \delta_x$ , replacing s by $s+k\gamma$ and t by $s+(k+n)\gamma$ , one obtains

\begin{equation*}\|\delta_x P_{s,s+(k+n)\gamma} - \delta_x P_{s+k\gamma,s+(k+n)\gamma}\|_{TV} \leq 2 \prod_{l=0}^{\big\lfloor \frac{n\gamma}{t_0}\big\rfloor-1} (1-c_{s+(k+n)\gamma-lt_0}),\end{equation*}

which replaces the inequality (18) in the proof of Theorem 1. Plugging this last inequality into the formula (19), one obtains

\begin{equation*}\| \delta_{x} P_{s,s+(k+n)\gamma} - {\beta}_{\gamma} Q_{0,s} \|_{TV} \leq 2 \prod_{l=0}^{\left \lfloor \frac{n \gamma}{t_0} \right \rfloor-1} (1-c_{s + (k+n)\gamma-l t_0}) + \| \delta_{x} P_{s+k\gamma, s+(k+n)\gamma} - \beta_\gamma Q_{0,s} \|_{TV}.\end{equation*}

Hence, we see that we cannot conclude a similar result when $c_s \longrightarrow 0$ as $s \to + \infty$ , since, for n fixed,

\begin{equation*}\limsup_{k \to \infty} \prod_{l=0}^{\left \lfloor \frac{n \gamma}{t_0} \right \rfloor-1} (1-c_{s+(k+n)\gamma-l t_0}) = 1.\end{equation*}

4. Application to quasi-stationarity with moving boundaries

In this section, $(X_t)_{t \geq 0}$ is assumed to be a time-homogeneous Markov process. We consider a family of measurable subsets $(A_t)_{t \geq 0}$ of E, and define the hitting time

\begin{equation*}\tau_A \;:\!=\; \inf\{t \geq 0 \;:\; X_t \in A_t\}.\end{equation*}

For all $s \leq t$ , denote by ${\mathcal F}_{s,t}$ the $\sigma$ -field generated by the family $(X_u)_{s \leq u \leq t}$ , with ${\mathcal F}_t \;:\!=\; {\mathcal F}_{0,t}$ . Assume that $\tau_A$ is a stopping time with respect to the filtration $({\mathcal F}_{t})_{t \geq 0}$ . Assume also that for any $x \not \in A_0$ ,

\begin{equation*}\mathbb{P}_{0,x}[\tau_A < + \infty] = 1 \;\;\;\;\;\;\text{ and }\;\;\;\;\;\; \mathbb{P}_{0,x}[\tau_A > t] > 0,\;\;\forall t \geq 0.\end{equation*}

We will be interested in a notion of quasi-stationarity with moving boundaries, which studies the asymptotic behavior of the Markov process $(X_t)_{t \geq 0}$ conditioned not to hit $(A_t)_{t \geq 0}$ up to the time t. For non-moving boundaries ( $A_t = A_0$ for any $t \geq 0$ ), the quasi-limiting distribution is defined as a probability measure $\alpha$ such that, for at least one initial measure $\mu$ and for all measurable subsets $\mathcal{A} \subset E$ ,

\begin{equation*}\mathbb{P}_{0,\mu}[X_t \in \mathcal{A} | \tau_A > t] \underset{t \to \infty}{\longrightarrow} \alpha(\mathcal{A}).\end{equation*}

Such a definition is equivalent (still in the non-moving framework) to the notion of quasi-stationary distribution, defined as a probability measure $\alpha$ such that, for any $t \geq 0$ ,

(26) \begin{equation}\mathbb{P}_{0,\alpha}[X_t \in \cdot | \tau_A > t] = \alpha.\end{equation}

If quasi-limiting and quasi-stationary distributions are in general well-defined for time-homogeneous Markov processes and non-moving boundaries (see [Reference Collet, Martínez and San Martín11, Reference Méléard and Villemonais23] for a general overview of the theory of quasi-stationarity), these notions are nevertheless not well-defined for time-inhomogeneous Markov processes or moving boundaries, for which they are no longer equivalent. In particular, under reasonable assumptions on irreducibility, it was shown in [Reference Oçafrain24] that the notion of quasi-stationary distribution as defined by (26) is not well-defined for time-homogeneous Markov processes absorbed by moving boundaries.

Another asymptotic notion to study is the quasi-ergodic distribution, related to a conditional version of the ergodic theorem and usually defined as follows.

Definition 2. A probability measure $\beta$ is a quasi-ergodic distribution if, for some initial measure $\mu \in \mathcal{M}_1(E \setminus A_0)$ and for any bounded continuous function f,

\begin{equation*}\mathbb{E}_{0,\mu}\bigg[\frac{1}{t} \int_0^t f(X_s)ds \bigg| \tau_A > t\bigg] \underset{t \to \infty}{\longrightarrow} \beta(f).\end{equation*}

In the time-homogeneous setting (in particular for non-moving boundaries), this notion has been extensively studied (see for example [Reference Breyer and Roberts2, Reference Champagnat and Villemonais8, Reference Chen and Jian10, Reference Colonius and Rasmussen12, Reference Darroch and Seneta13, Reference He16–Reference He, Zhang and Zhu18, Reference Oçafrain24]). In the ‘moving boundaries’ framework, the existence of quasi-ergodic distributions has been dealt with in [Reference Oçafrain24] for Markov chains on finite state spaces absorbed by periodic boundaries, and in [Reference Oçafrain25] for processes satisfying a Champagnat–Villemonais condition (see Assumption (A′) below) absorbed by converging or periodic boundaries. In this last paper, the existence of the quasi-ergodic distribution is dealt with through the following inequality (see [Reference Oçafrain25, Theorem 1]), which holds for any initial state x, $s \leq t$ , and for some constants $C, \gamma > 0$ independent of x, s, and t:

\begin{equation*}\|\mathbb{P}_{0,x}(X_s \in \cdot | \tau_A > t) - \mathbb{Q}_{0,x}(X_s \in \cdot)\|_{TV} \leq C e^{-\gamma (t-s)},\end{equation*}

where the family of probability measures $(\mathbb{Q}_{s,x})_{s \geq 0, x \in E_s}$ is defined by

\begin{equation*}\mathbb{Q}_{s,x}[\Gamma] \;:\!=\; \lim_{T \to \infty} \mathbb{P}_{s,x}[\Gamma | \tau_A > T],\;\;\;\;\forall s \leq t,\ x \in E \setminus A_s,\ \Gamma \in {\mathcal F}_{s,t}.\end{equation*}

Moreover, by [Reference Champagnat and Villemonais9, Proposition 3.1], there exists a family of positive bounded functions $(\eta_t)_{t \geq 0}$ defined in such a way that, for all $s \leq t$ and $x \in E_s$ ,

\begin{equation*}\mathbb{E}_{s,x}(\eta_t(X_t)\mathbb{1}_{\tau_A > t}) = \eta_s(x).\end{equation*}

Then we can show (this is actually shown in [Reference Champagnat and Villemonais9]) that

\begin{equation*}\mathbb{Q}_{s,x}(\Gamma) = \mathbb{E}_{s,x}\bigg(\mathbb{1}_{\Gamma, \tau_A > t} \frac{\eta_t(X_t)}{\eta_s(x)}\bigg)\end{equation*}

and that, for all $\mu \in \mathcal{M}_1(E_0)$ ,

\begin{equation*}\|\mathbb{P}_{0,\mu}(X_s \in \cdot | \tau_A > t) - \mathbb{Q}_{0,\eta_0*\mu}(X_s \in \cdot)\|_{TV} \leq C e^{-\gamma (t-s)},\end{equation*}

where

\begin{equation*}\eta_0 * \mu(dx) \;:\!=\; \frac{\eta_0(x)\mu(dx)}{\mu(\eta_0)}.\end{equation*}

By the triangle inequality, one has

(27) \begin{equation}\bigg\|\frac{1}{t} \int_0^t \mathbb{P}_{0,\mu}[X_s \in \cdot | \tau_A > t]ds - \frac{1}{t} \int_0^t \mathbb{Q}_{0,\eta_0 * \mu}[X_s \in \cdot]ds \bigg\|_{TV} \leq \frac{C}{\gamma t},\;\;\;\;\forall t > 0.\end{equation}

In particular, the inequality (27) implies that there exists a quasi-ergodic distribution $\beta$ for the process $(X_t)_{t \geq 0}$ absorbed by $(A_t)_{t \geq 0}$ if and only if there exist some probability measures $\mu \in \mathcal{M}_1(E_0)$ such that $\frac{1}{t} \int_0^t \mathbb{Q}_{0,\eta_0 * \mu}[X_s \in \cdot]ds$ converges weakly to $\beta$ , when t goes to infinity. In other words, under Assumption (A′), the existence of a quasi-ergodic distribution for the absorbed process is equivalent to the law of large numbers for its Q-process.

We now state Assumption (A′).

Assumption 4. There exists a family of probability measures $(\nu_t)_{t \geq 0}$ , defined on $E \setminus A_t$ for each t, such that the following hold:

(A′1) There exist $t_0 \geq 0$ and $c_1 > 0$ such that

\begin{equation*}\mathbb{P}_{s,x}[X_{s+t_0} \in \cdot | \tau_A > s + t_0] \geq c_1 \nu_{s+t_0},\;\;\;\;\forall s \geq 0,\ \forall x \in E \setminus A_s.\end{equation*}

(A′2) There exists $c_2 > 0$ such that

\begin{equation*}\mathbb{P}_{s,\nu_s}[\tau_A > t] \geq c_2 \mathbb{P}_{s,x}[\tau_A > t],\;\;\;\;\forall s \leq t,\ \forall x \in E \setminus A_s.\end{equation*}

In what follows, we say that the pair $\{(X_t)_{t \geq 0}, (A_t)_{t \geq 0}\}$ satisfies Assumption (A′) when the assumption holds for the Markov process $(X_t)_{t \geq 0}$ considered as absorbed by the moving boundary $(A_t)_{t \geq 0}$ .

The condition (A′1) is a conditional version of the Doeblin condition (12), and (A′2) is a Harnack-like inequality on the probabilities of surviving, necessary to deal with the conditioning. They are equivalent to the set of conditions presented in [Reference Bansaye, Cloez and Gabriel1, Definition 2.2], when the non-conservative semigroup is sub-Markovian. In the time-homogeneous framework, we obtain the Champagnat–Villemonais condition defined in [Reference Champagnat and Villemonais5] (see Assumption (A)), shown as being equivalent to the exponential uniform convergence to quasi-stationarity in total variation.

In [Reference Oçafrain25], the existence of a unique quasi-ergodic distribution is proved only for converging or periodic boundaries. However, we can expect such a result on existence (and uniqueness) for other kinds of movement for the boundary. Hence, the aim of this section is to extend the results on the existence of quasi-ergodic distributions obtained in [Reference Oçafrain25] to Markov processes absorbed by asymptotically periodic moving boundaries.

Now let us state the following theorem.

Theorem 2. Assume that there exists a $\gamma$ -periodic sequence of subsets $(B_t)_{t \geq 0}$ such that, for any $s \in [0,\gamma)$ ,

\begin{equation*}E'_{\!\!s} \;:\!=\; E \setminus \bigcap_{k \in \mathbb{Z}_+} \bigcup_{l \geq k} A_{s+l \gamma} \cup B_s \ne \emptyset,\end{equation*}

and there exists $x_s \in E_s$ such that, for any $n \leq N$ ,

(28) \begin{equation} \|\mathbb{P}_{s+k \gamma, x_s}[X_{s+(k+n)\gamma} \in \cdot, \tau_A > s+(k+N)\gamma] - \mathbb{P}_{s, x_s}[X_{s+n\gamma} \in \cdot, \tau_B > s+N\gamma]\|_{TV} \underset{k \to \infty}{\longrightarrow} 0.\end{equation}

Assume also that Assumption (A′) is satisfied by the pairs $\{(X_t)_{t \geq 0}, (A_t)_{t \geq 0}\}$ and $\{(X_t)_{t \geq 0}, (B_t)_{t \geq 0}\}$ .

Then there exists a probability measure $\beta \in \mathcal{M}_1(E)$ such that

(29) \begin{equation}\sup_{\mu \in \mathcal{M}_1(E \setminus A_0)} \sup_{f \in \mathcal{B}_1(E)} \mathbb{E}_{0,\mu}\Bigg[ \bigg| \frac{1}{t} \int_0^t f(X_s)ds - \beta(f) \bigg|^2 \Bigg| \tau_A > t\Bigg] \underset{t \to \infty}{\longrightarrow} 0.\end{equation}

Remark 4. Observe that the condition (28) implies that, for any $n \in \mathbb{Z}_+$ ,

\begin{equation*}\mathbb{P}_{s+k \gamma,x_s}[\tau_A > s +(k+n) \gamma] \underset{k \to \infty}{\longrightarrow} \mathbb{P}_{s,x_s}[\tau_B > s + n \gamma].\end{equation*}

Under the additional condition $B_t \subset A_t$ for all $t \geq 0$ , these two conditions are equivalent, since for all $n \leq N$ ,

\begin{align*} & \|\mathbb{P}_{s+k \gamma, x_s}[X_{s+(k+n)\gamma} \in \cdot, \tau_A > s+(k+N)\gamma] - \mathbb{P}_{s, x_s}[X_{s+n\gamma} \in \cdot, \tau_B > s+N\gamma]\|_{TV} \\[5pt] & \qquad\qquad = \|\mathbb{P}_{s+k\gamma,x_s}[X_{s+(k+n)\gamma} \in \cdot, \tau_B \leq s+(k+N) \gamma < \tau_A] \|_{TV} \\[5pt] & \qquad\qquad\qquad\qquad \leq \mathbb{P}_{s+k\gamma,x_s}[\tau_B \leq s+(k+N) \gamma< \tau_A] \\[5pt] & \qquad\qquad\qquad\qquad\qquad = |\mathbb{P}_{s+k \gamma, x_s}[\tau_A > s+(k+N)\gamma] - \mathbb{P}_{s, x_s}[\tau_B > s+N\gamma]|,\end{align*}

where we used the periodicity of $(B_t)_{t \geq 0}$ , writing

\begin{equation*}\mathbb{P}_{s, x_s}[X_{s+n\gamma} \in \cdot, \tau_B > s+N\gamma] = \mathbb{P}_{s+k\gamma, x_s}[X_{s+(k+n)\gamma} \in \cdot, \tau_B > s+(k+N)\gamma]\end{equation*}

for all $k \in \mathbb{Z}_+$ . This implies the following corollary.

Corollary 2. Assume that there exists a $\gamma$ -periodic sequence of subsets $(B_t)_{t \geq 0}$ , with $B_t \subset A_t$ for all $t \geq 0$ , such that, for any $s \in [0,\gamma)$ , there exists $x_s \in E'_{\!\!s}$ such that, for any $n \leq N$ ,

\begin{equation*}\mathbb{P}_{s+k \gamma,x_s}[\tau_A > s +(k+n) \gamma] \underset{k \to \infty}{\longrightarrow} \mathbb{P}_{s,x_s}[\tau_B > s + n \gamma].\end{equation*}

Assume also that Assumption (A′) is satisfied by $\{(X_t)_{t \geq 0}, (A_t)_{t \geq 0}\}$ and $\{(X_t)_{t \geq 0}, (B_t)_{t \geq 0}\}$ .

Then there exists $\beta \in \mathcal{M}_1(E)$ such that (29) holds.

The main idea of this proof is to apply Corollary 1. Since $\{(X_t)_{t \geq 0}, (A_t)_{t \geq 0}\}$ and $\{(X_t)_{t \geq 0}, (B_t)_{t \geq 0}\}$ satisfy Assumption (A′), [Reference Oçafrain25, Theorem 1] implies that there exist two families of probability measures $\big(\mathbb{Q}^A_{s,x}\big)_{s \geq 0, x \in E \setminus A_s}$ and $\big(\mathbb{Q}^B_{s,x}\big)_{s \geq 0, x \in E \setminus B_s}$ such that, for any $s \leq t$ , $x \in E \setminus A_s$ , $y \in E \setminus B_s$ , and $\Gamma \in {\mathcal F}_{s,t}$ ,

\begin{equation*}\mathbb{Q}_{s,x}^A[\Gamma] = \lim_{T \to \infty} \mathbb{P}_{s,x}[\Gamma | \tau_A > T] \;\;\text{ and } \;\;\mathbb{Q}_{s,y}^B[\Gamma] = \lim_{T \to \infty} \mathbb{P}_{s,y}[\Gamma | \tau_B > T].\end{equation*}

In particular, the quasi-ergodic distribution $\beta$ is the limit of $t \mapsto \frac{1}{t} \int_0^t \mathbb{Q}^B_{0, \mu}[X_s \in \cdot]ds$ , when t goes to infinity (see [Reference Oçafrain25, Theorem 5]). Also, by [Reference Oçafrain25, Theorem 1], there exist constants $C > 0$ and $\kappa > 0$ such that, for any $s \leq t \leq T$ , for any $x \in E \setminus A_{s}$ ,

\begin{equation*}\big\| \mathbb{Q}^A_{s,x}[X_t \in \cdot] - \mathbb{P}_{s, x}[X_{t} \in \cdot | \tau_A > T] \big\|_{TV} \leq C e^{-\kappa (T-t)},\end{equation*}

and for any $x \in E \setminus B_s$ ,

\begin{equation*}\big\| \mathbb{Q}^B_{s,x}[X_{t} \in \cdot] - \mathbb{P}_{s, x}[X_{t} \in \cdot | \tau_B > T] \big\|_{TV} \leq C e^{-\kappa (T-t)}.\end{equation*}

Moreover, for any $s \leq t \leq T$ and $x \in E'_{\!\!s}$ ,

(30) \begin{align}\| \mathbb{P}_{s,x}&[X_{t} \in \cdot | \tau_A > T] - \mathbb{P}_{s,x}[X_{t} \in \cdot | \tau_B > T] \|_{TV} \notag \\[5pt] &=\bigg\| \frac{\mathbb{P}_{s,x}[X_{t} \in \cdot, \tau_A > T]}{\mathbb{P}_{s,x}[\tau_A > T]} - \frac{\mathbb{P}_{s,x}[X_{t} \in \cdot, \tau_B > T]}{\mathbb{P}_{s,x}[\tau_B > T]} \bigg\|_{TV} \notag \\[5pt] &= \bigg\| \frac{\mathbb{P}_{s,x}(\tau_B > T)}{\mathbb{P}_{s,x}(\tau_A > T)}\frac{\mathbb{P}_{s,x}[X_{t} \in \cdot, \tau_A > T]}{\mathbb{P}_{s,x}[\tau_B > T]} - \frac{\mathbb{P}_{s,x}[X_{t} \in \cdot, \tau_B > T]}{\mathbb{P}_{s,x}[\tau_B > T]} \bigg\|_{TV} \notag \\[5pt] &\leq \bigg\| \frac{\mathbb{P}_{s,x}(\tau_B > T)}{\mathbb{P}_{s,x}(\tau_A > T)}\frac{\mathbb{P}_{s,x}[X_{t} \in \cdot, \tau_A > T]}{\mathbb{P}_{s,x}[\tau_B > T]} - \frac{\mathbb{P}_{s,x}[X_{t} \in \cdot, \tau_A > T]}{\mathbb{P}_{s,x}[\tau_B > T]} \bigg\|_{TV} \notag \\[5pt] &\;\;\;\;\;\;+ \bigg\| \frac{\mathbb{P}_{s,x}[X_{t} \in \cdot, \tau_A > T]}{\mathbb{P}_{s,x}[\tau_B > T]} - \frac{\mathbb{P}_{s,x}[X_{t} \in \cdot, \tau_B > T]}{\mathbb{P}_{s,x}[\tau_B > T]} \bigg\|_{TV} \notag \\[5pt] &\leq \frac{|\mathbb{P}_{s,x}(\tau_B > T) - \mathbb{P}_{s,x}(\tau_A > T)|}{\mathbb{P}_{s,x}(\tau_B > T)} + \frac{\|\mathbb{P}_{s,x}[X_t \in \cdot, \tau_A > T] - \mathbb{P}_{s,x}[X_t \in \cdot, \tau_B > T]\|_{TV}}{\mathbb{P}_{s,x}[\tau_B > T]} \notag \\[5pt] &\leq 2 \frac{\|\mathbb{P}_{s,x}[X_t \in \cdot, \tau_A > T] - \mathbb{P}_{s,x}[X_t \in \cdot, \tau_B > T]\|_{TV}}{\mathbb{P}_{s,x}[\tau_B > T]},\end{align}

since

\begin{equation*}|\mathbb{P}_{s,x}(\tau_B > T) - \mathbb{P}_{s,x}(\tau_A > T)| \leq \|\mathbb{P}_{s,x}[X_t \in \cdot, \tau_A > T] - \mathbb{P}_{s,x}[X_t \in \cdot, \tau_B > T]\|_{TV}.\end{equation*}

Then we obtain, for any $s \leq t \leq T$ and $x \in E'_{\!\!s}$ ,

(31) \begin{align} \bigl\|\mathbb{Q}_{s,x}^A[X_{t} \in \cdot] &- \mathbb{Q}_{s,x}^B[X_{t} \in \cdot] \bigl\|_{TV} \nonumber \\[5pt] &\leq 2 C e^{-\kappa (T - t)} + 2 \frac{\|\mathbb{P}_{s,x}[X_t \in \cdot, \tau_A > T] - \mathbb{P}_{s,x}[X_t \in \cdot, \tau_B > T]\|_{TV}}{\mathbb{P}_{s,x}[\tau_B > T]}.\end{align}

The condition (28) implies the existence of $x_s \in E_s$ such that, for any $n \leq N$ , for all $k \in \mathbb{Z}_+$ ,

\begin{equation*}\lim_{k \to \infty} \|\mathbb{P}_{s+k \gamma, x_s}[X_{s+(k+n)\gamma} \in \cdot, \tau_A > s+(k+N)\gamma] - \mathbb{P}_{s, x_s}[X_{s+n\gamma} \in \cdot, \tau_B > s+N\gamma]\|_{TV} = 0,\end{equation*}

which implies by (31) that, for any $n \leq N$ ,

\begin{equation*}\limsup_{k \to \infty} \bigl\|\mathbb{Q}_{s+k\gamma,x_s}^A[X_{s+(k+n)\gamma} \in \cdot] - \mathbb{Q}_{s+k\gamma,x_s}^B[X_{s+(k+n)\gamma} \in \cdot] \bigl\|_{TV} \leq 2Ce^{-\kappa \gamma (N-n)}.\end{equation*}

Now, letting $N \to \infty$ , for any $n \in \mathbb{Z}_+$ we have

\begin{align*}\lim_{k \to \infty} & \bigl\|\mathbb{Q}_{s+k\gamma,x_s}^A[X_{s+(k+n)\gamma} \in \cdot] - \mathbb{Q}_{s+k\gamma,x_s}^B[X_{s+(k+n)\gamma} \in \cdot] \bigl\|_{TV}\\[5pt] &= \lim_{k \to \infty} \bigl\| \mathbb{Q}_{s+k\gamma,x_s}^A(X_{s+(k+n)\gamma}\in \cdot)-\mathbb{Q}_{s,x_s}^B(X_{s+n\gamma}\in \cdot)\bigl\|_{TV}\\[5pt] &= 0.\end{align*}

In other words, the semigroup $\big(Q^A_{s,t}\big)_{s \leq t}$ defined by

\begin{equation*}Q_{s,t}^Af(x) \;:\!=\; \mathbb{E}_{s,x}^{\mathbb{Q}^A}(f(X_t)),\;\;\;\;\forall s \leq t,\ \forall f \in \mathcal{B}(E \setminus A_t),\ \forall x \in E \setminus A_s,\end{equation*}

is asymptotically periodic (according to Definition 1, with $\psi_s = \tilde{\psi}_s = 1$ for all $s \geq 0$ ), associated to the auxiliary semigroup $\big(Q^B_{s,t}\big)_{s \leq t}$ defined by

\begin{equation*}Q_{s,t}^Bf(x) \;:\!=\; \mathbb{E}_{s,x}^{\mathbb{Q}^B}(f(X_t)),\;\;\;\;\forall s \leq t,\ \forall f \in \mathcal{B}(E \setminus B_t),\ \forall x \in E \setminus B_s.\end{equation*}

Moreover, since Assumption (A′) is satisfied for $\{(X_t)_{t \geq 0}, (A_t)_{t \geq 0}\}$ and $\{(X_t)_{t \geq 0}, (B_t)_{t \geq 0}\}$ , the Doeblin condition holds for these two Q-processes. As a matter of fact, by the Markov property, for all $s \leq t \leq T$ and $x \in E \setminus A_s$ ,

(32) \begin{align} \mathbb{P}_{s,x}(X_t \in \cdot | \tau_A > T) &= \mathbb{E}_{s,x}\bigg[\mathbb{1}_{X_t \in \cdot, \tau_A > t} \frac{\mathbb{P}_{t,X_t}(\tau_A > T)}{\mathbb{P}_{s,x}(\tau_A > T)}\bigg] \notag \\[5pt] &= \mathbb{E}_{s,x}\bigg[\frac{\mathbb{1}_{X_t \in \cdot, \tau_A > t}}{\mathbb{P}_{s,x}(\tau_A > t)} \frac{\mathbb{P}_{t,X_t}(\tau_A > T)}{\mathbb{P}_{t,\phi_{t,s}(\delta_x)}(\tau_A > T)}\bigg] \notag \\[5pt] &= \mathbb{E}_{s,x}\bigg[\mathbb{1}_{X_t \in \cdot} \frac{\mathbb{P}_{t,X_t}(\tau_A > T)}{\mathbb{P}_{t,\phi_{t,s}(\delta_x)}(\tau_A > T)} \bigg| \tau_A > t\bigg], \end{align}

where, for all $s \leq t$ and $\mu \in \mathcal{M}_1(E_s)$ , $\phi_{t,s}(\mu) \;:\!=\; \mathbb{P}_{s,\mu}(X_t \in \cdot | \tau_A > t)$ . By (A′1), for any $s \geq 0$ , $T \geq s+t_0$ , $x \in E \setminus A_s$ , and measurable set $\mathcal{A}$ ,

\begin{equation*}\mathbb{E}_{s,x}\bigg[\mathbb{1}_{X_{s+t_0} \in \mathcal{A}} \frac{\mathbb{P}_{s+t_0,X_{s+t_0}}(\tau_A > T)}{\mathbb{P}_{s+t_0,\phi_{s+t_0,s}(\delta_x)}(\tau_A > T)} \bigg| \tau_A > s+t_0\bigg] \geq c_1 \int_{\mathcal{A}} \nu_{s+t_0}(dy) \frac{\mathbb{P}_{s+t_0,y}(\tau_A > T)}{\mathbb{P}_{s+t_0,\phi_{s+t_0,s}(\delta_x)}(\tau_A > T)};\end{equation*}

that is, by (32),

\begin{equation*}\mathbb{P}_{s,x}(X_{s+t_0} \in \mathcal{A} | \tau_A > T) \geq c_1 \int_{\mathcal{A}} \nu_{s+t_0}(dy) \frac{\mathbb{P}_{s+t_0,y}(\tau_A > T)}{\mathbb{P}_{s+t_0,\phi_{s+t_0,s}(\delta_x)}(\tau_A > T)}.\end{equation*}

Letting $T \to \infty$ in this last inequality and using [Reference Champagnat and Villemonais9, Proposition 3.1], for any $s \geq 0$ , $x \in E \setminus A_s$ , and measurable set $\mathcal{A}$ ,

\begin{equation*}\mathbb{Q}_{s,x}^A(X_{s+t_0} \in \mathcal{A}) \geq c_1 \int_\mathcal{A} \nu_{s+t_0}(dy) \frac{\eta_{s+t_0}(y)}{\phi_{s+t_0,s}(\delta_x)(\eta_{s+t_0})}.\end{equation*}

The measure

\begin{equation*}\mathcal{A} \mapsto \int_\mathcal{A} \nu_{s+t_0}(dy) \frac{\eta_{s+t_0}(y)}{\phi_{s+t_0,s}(\delta_x)(\eta_{s+t_0})}\end{equation*}

is then a positive measure whose mass is bounded below by $c_2$ , by (A′2), since for all $s \geq 0$ and $T \geq s + t_0$ ,

\begin{equation*}\int_{E \setminus A_{s+t_0}} \nu_{s+t_0}(dy) \frac{\mathbb{P}_{t,y}(\tau_A > T)}{\mathbb{P}_{t,\phi_{t,s}(\delta_x)}(\tau_A > T)} \geq c_2.\end{equation*}

This proves a Doeblin condition for the semigroup $\big(Q_{s,t}^A\big)_{s \leq t}$ . The same reasoning also applies to prove a Doeblin condition for the semigroup $\big(Q_{s,t}^B\big)_{s \leq t}$ . Then, using (27) followed by Corollary 1, we have

\begin{align*}\lim_{t \to \infty} \frac{1}{t} \int_0^t \mathbb{P}_{0,\mu}[X_s \in \cdot | \tau_A > t]ds &= \lim_{t \to \infty} \frac{1}{t} \int_0^t \mathbb{Q}_{0,\eta_0*\mu}^A(X_s \in \cdot)ds\\[5pt] &= \lim_{t \to \infty} \frac{1}{t} \int_0^t \mathbb{Q}^B_{0,\eta_0*\mu}[X_s \in \cdot]ds = \beta,\end{align*}

where the limits refer to convergence in total variation and hold uniformly in the initial measure.

For any $\mu \in \mathcal{M}_1(E \setminus A_0)$ , $f \in \mathcal{B}_1(E)$ , and $t \geq 0$ ,

\begin{equation*}\mathbb{E}_{0,\mu}\Bigg[\bigg| \frac{1}{t} \int_0^t f(X_s)ds \bigg|^2\Bigg| \tau_A > t\Bigg] = \frac{2}{t^2} \int_0^t \int_s^t \mathbb{E}_{0,\mu}[f(X_s)f(X_u) | \tau_A > t]du\,ds.\end{equation*}

Then, by [Reference Oçafrain25, Theorem 1], for any $s \leq u \leq t$ , for any $\mu \in \mathcal{M}_1(E \setminus A_0)$ and $f \in \mathcal{B}(E)$ ,

\begin{equation*}\Big| \mathbb{E}_{0,\mu}[f(X_s)f(X_u) | \tau_A > t] - \mathbb{E}^{\mathbb{Q}^A}_{0,\eta_0*\mu}[f(X_s)f(X_u)]\Big| \leq C \|f\|_\infty e^{-\kappa (t-u)},\end{equation*}

where the expectation $\mathbb{E}^{\mathbb{Q}^A}_{0,\eta_0*\mu}$ is associated to the probability measure $\mathbb{Q}_{0,\eta_0*\mu}^A$ . Hence, for any $\mu \in \mathcal{M}_1(E \setminus A_0)$ , $f \in \mathcal{B}_1(E)$ , and $t > 0$ ,

\begin{align*} & \Bigg| \mathbb{E}_{0,\mu}\Bigg[\bigg| \frac{1}{t} \int_0^t f(X_s)ds - \beta(f) \bigg|^2\Bigg| \tau_A > t\Bigg] - \mathbb{E}_{0,\eta_0*\mu}^{\mathbb{Q}^A}\Bigg[\bigg| \frac{1}{t} \int_0^t f(X_s)ds - \beta(f) \bigg|^2\Bigg] \Bigg| \\[5pt] & \qquad \leq \frac{4C}{t^2} \int_0^t \int_s^t e^{-\kappa(t-u)}du\,ds \\[5pt] & \qquad \leq \frac{4C}{\kappa t} - \frac{4C (1 - e^{-\kappa t})}{\kappa^2 t^2}.\end{align*}

Moreover, since $\big(Q^A_{s,t}\big)_{s \leq t}$ is asymptotically periodic in total variation and satisfies the Doeblin condition, like $\big(Q^B_{s,t}\big)_{s \leq t}$ , Corollary 1 implies that

\begin{equation*}\sup_{\mu \in \mathcal{M}_1(E \setminus A_0)} \sup_{f \in \mathcal{B}_1(E)} \mathbb{E}_{0,\eta_0*\mu}^{\mathbb{Q}^A}\Bigg[\bigg| \frac{1}{t} \int_0^t f(X_s)ds - \beta(f) \bigg|^2\Bigg] \underset{t \to \infty}{\longrightarrow} 0.\end{equation*}

Then

\begin{equation*}\sup_{\mu \in \mathcal{M}_1(E \setminus A_0)} \sup_{f \in \mathcal{B}_1(E)} \mathbb{E}_{0,\mu}\Bigg[\bigg| \frac{1}{t} \int_0^t f(X_s)ds - \beta(f) \bigg|^2\Bigg| \tau_A > t\Bigg] \underset{t \to \infty}{\longrightarrow} 0.\end{equation*}

5. Examples

5.1. Asymptotically periodic Ornstein–Uhlenbeck processes

Let $(X_t)_{t \geq 0}$ be a time-inhomogeneous diffusion process on $\mathbb{R}$ satisfying the stochastic differential equation

\begin{equation*}dX_t = dW_t - \lambda(t) X_t dt,\end{equation*}

where $(W_t)_{t \geq 0}$ is a one-dimensional Brownian motion and $\lambda \;:\; [0, \infty) \to [0, \infty)$ is a function such that

\begin{equation*}\sup_{t \geq 0} |\lambda(t)| < + \infty\end{equation*}

and such that there exists $\gamma > 0$ such that

\begin{equation*}\inf_{s \geq 0} \int_s^{s+\gamma} \lambda(u)du > 0.\end{equation*}

By Itô’s lemma, for any $s \leq t$ ,

\begin{equation*}X_t = e^{-\int_s^t \lambda(u)du}\bigg[X_s + \int_s^t e^{\int_s^u \lambda(v)dv} dW_u\bigg].\end{equation*}

In particular, denoting by $(P_{s,t})_{s \leq t}$ the semigroup associated to $(X_t)_{t \geq 0}$ , for any $f \in \mathcal{B}(\mathbb{R})$ , $t \geq 0$ , and $x \in \mathbb{R}$ ,

\begin{equation*}P_{s,t}f(x) = \mathbb{E}\!\left[f\!\left(e^{-\int_s^t \lambda(u)du}x + e^{-\int_s^t \lambda(u)du} \sqrt{\int_s^t e^{2\int_s^u \lambda(v)dv}du} \times \mathcal{N}(0,1)\right)\right],\end{equation*}

where $\mathcal{N}(0,1)$ denotes a standard Gaussian variable.

Theorem 3. Assume that there exists a $\gamma$ -periodic function g, bounded on $\mathbb{R}$ , such that $\lambda \sim_{t \to \infty} g$ . Then the assumptions of Theorem 1 hold.

Proof. In our case, the auxiliary semigroup $(Q_{s,t})_{s \leq t}$ of Definition 1 will be defined as follows: for any $f \in \mathcal{B}(\mathbb{R})$ , $t \geq 0$ , and $x \in \mathbb{R}$ ,

\begin{equation*}Q_{s,t}f(x) = \mathbb{E}\left[f\left(e^{-\int_s^t g(u)du}x + e^{-\int_s^t g(u)du} \sqrt{\int_s^t e^{2\int_s^u g(v)dv}du} \times \mathcal{N}(0,1)\right)\right].\end{equation*}

In particular, the semigroup $(Q_{s,t})_{s \leq t}$ is associated to the process $(Y_t)_{t \geq 0}$ following

\begin{equation*}dY_t = dW_t - g(t) Y_t dt.\end{equation*}

We first remark that the function $\psi \;:\; x \mapsto 1 + x^2$ is a Lyapunov function for $(P_{s,t})_{s \leq t}$ and $(Q_{s,t})_{s \leq t}$ . In fact, for any $s \geq 0$ and $x \in \mathbb{R}$ ,

\begin{align*} P_{s,s+\gamma} \psi(x) &= 1 + e^{-2 \int_s^{s+\gamma} \lambda(u)du}x^2 + e^{-2\int_s^{s+\gamma} \lambda(u)du} \int_s^{s+\gamma} e^{2\int_s^u \lambda(v)dv}du \\[5pt] &= e^{-2\int_s^{s+\gamma} \lambda(u)du} \psi(x) + 1 - e^{-2\int_s^{s+\gamma} \lambda(u)du} + e^{-2\int_s^{s+\gamma} \lambda(u)du} \int_s^{s+\gamma} e^{2\int_s^u \lambda(v)dv}du \\[5pt] &\leq e^{-2 \gamma c_{\inf}} \psi(x) + C,\end{align*}

where $C \in (0,+\infty)$ and $c_{\inf} \;:\!=\; \inf_{t \geq 0} \frac{1}{\gamma} \int_t^{t+\gamma}\lambda(u)du > 0$ . Taking $\theta \in (e^{-2 \gamma c_{\inf}},1)$ , there exists a compact set K such that, for any $s \geq 0$ ,

\begin{equation*}P_{s,s+\gamma} \psi(x) \leq \theta \psi(x) + C \mathbb{1}_K(x).\end{equation*}

Moreover, for any $s \geq 0$ and $t \in [0, \gamma)$ , the function $P_{s,s+t}\psi/\psi$ is upper-bounded uniformly in s and t. It remains therefore to prove Assumption 1(i) for $(P_{s,t})_{s \leq t}$ , which is a consequence of the following lemma.

Lemma 1. For any $a,b_{-},b_{+} > 0$ , define the subset $\mathcal{C}(a,b_{-},b_{+}) \subset \mathcal{M}_1(\mathbb{R})$ as

\begin{equation*}\mathcal{C}(a,b_{-},b_{+}) \;:\!=\; \{ \mathcal{N}(m,\sigma) \;:\; m \in [\!-\!a,a], \sigma \in [b_{-},b_{+}]\}.\end{equation*}

Then, for any $a,b_{-},b_{+} > 0$ , there exist a probability measure $\nu$ and a constant $c > 0$ such that, for any $\mu \in \mathcal{C}(a,b_{-},b_{+})$ ,

\begin{equation*}\mu \geq c \nu.\end{equation*}

The proof of this lemma is postponed until after the end of this proof.

Since $\lambda \sim_{t \to \infty} g$ and these two functions are bounded on $\mathbb{R}_+$ , Lebesgue’s dominated convergence theorem implies that, for all $s \leq t$ ,

\begin{equation*}\bigg| \int_{s+k\gamma}^{t+k\gamma} \lambda(u)du - \int_s^{t} g(u)du\bigg| \underset{k \to \infty}{\longrightarrow} 0.\end{equation*}

In the same way, for all $s \leq t$ ,

\begin{equation*}\int_{s+k\gamma}^{t+k\gamma} e^{2\int_{s+k\gamma}^u \lambda(v)dv}du \underset{k \to \infty}{\longrightarrow} \int_s^{t} e^{2\int_s^u g(v)dv}du.\end{equation*}

Hence, for any $s \leq t$ ,

\begin{equation*}e^{-\int_{s+k\gamma}^{t+k\gamma} \lambda(u)du} \underset{k \to \infty}{\longrightarrow} e^{-\int_s^{t} g(u)du},\end{equation*}

and

\begin{equation*}e^{-\int_{s+k \gamma}^{t+k\gamma} \lambda(u)du} \sqrt{\int_{s+k\gamma}^{t+k\gamma} e^{2\int_{s+k\gamma}^u \lambda(v)dv}du} \underset{k \to \infty}{\longrightarrow} e^{-\int_s^{t} g(u)du} \sqrt{\int_s^{t} e^{2\int_s^u g(v)dv}du}.\end{equation*}

Using [Reference Devroye, Mehrabian and Reddad14, Theorem 1.3], for any $x \in \mathbb{R}$ ,

(33) \begin{equation}\| \delta_x P_{s+k\gamma, t+k\gamma} - \delta_x Q_{s+k\gamma, t+k\gamma} \|_{TV} \underset{k \to \infty}{\longrightarrow} 0.\end{equation}

To deduce the convergence in $\psi$ -distance, we will draw inspiration from the proof of [Reference Hening and Nguyen19, Lemma 3.1]. Since the variances are uniformly bounded in k (for $s \leq t$ fixed), there exists $H > 0$ such that, for any $k \in \mathbb{N}$ and $s \leq t$ ,

(34) \begin{equation}\delta_x P_{s+k\gamma, t+k\gamma}\big[\psi^2\big] \leq H\;\;\text{ and }\;\;\delta_x Q_{s, t}\big[\psi^2\big] \leq H.\end{equation}

Since $\lim_{|x| \to \infty} \frac{\psi(x)}{\psi^2(x)} = 0$ , for any $\epsilon > 0$ there exists $l_\epsilon > 0$ such that, for any function f such that $|f| \leq \psi$ and for any $|x| \geq l_\epsilon$ ,

\begin{equation*}|f(x)| \leq \frac{\epsilon \psi(x)^2}{H}.\end{equation*}

Combining this with (34), and letting $K_\epsilon \;:\!=\; [\!-\!l_\epsilon, l_\epsilon]$ , we find that for any $k \in \mathbb{Z}_+$ , f such that $|f| \leq \psi$ , and $x \in \mathbb{R}$ ,

\begin{equation*}\delta_x P_{s+k \gamma, t+k\gamma}[f \mathbb{1}_{K_\epsilon^c}] \leq \epsilon\;\;\text{ and }\;\;\delta_x Q_{s, t}[f \mathbb{1}_{K_\epsilon^c}] \leq \epsilon.\end{equation*}

Then, for any $k \in \mathbb{Z}_+$ and f such that $|f| \leq \psi$ ,

(35) \begin{align} | \delta_x P_{s+k \gamma, t+k\gamma}f - \delta_x Q_{s, t}f| &\leq 2 \epsilon + | \delta_x P_{s+k \gamma, t+k\gamma}[f \mathbb{1}_{K_\epsilon}] - \delta_x Q_{s, t}[f \mathbb{1}_{K_\epsilon}]| \end{align}

(36) \begin{align} & \qquad\qquad\qquad\qquad\qquad \leq 2 \epsilon + (1+l_\epsilon^2) \| \delta_x P_{s+k\gamma, t+k\gamma} - \delta_x Q_{s, t} \|_{TV}. \end{align}

Hence, (33) implies that, for k large enough, for any f bounded by $\psi$ ,

(37) \begin{align} | \delta_x P_{s+k \gamma, t+k\gamma}f - \delta_x Q_{s, t}f| &\leq 3 \epsilon,\end{align}

implying that

\begin{equation*}\| \delta_x P_{s+k \gamma, t+k\gamma} - \delta_x Q_{s, t} \|_{\psi} \underset{k \to \infty}{\longrightarrow} 0.\end{equation*}

We now prove Lemma 1.

Proof of Lemma 1. Defining

\begin{equation*}f_\nu(x) \;:\!=\; e^{-\frac{(x-a)^2}{2 {b_-}^2}} \land e^{-\frac{(x+a)^2}{2 {b_-}^2}},\end{equation*}

we conclude easily that, for any $m \in [\!-\!a,a]$ and $\sigma \geq b_-$ , for any $x \in \mathbb{R}$ ,

\begin{equation*}e^{-\frac{(x-m)^2}{2 \sigma^2}} \geq f_\nu(x).\end{equation*}

Imposing moreover that $\sigma \leq b_+$ , one has

\begin{equation*}\frac{1}{\sqrt{2 \pi} \sigma} e^{-\frac{(x-m)^2}{2 \sigma^2}} \geq \frac{1}{\sqrt{2 \pi} b_+} f_\nu(x),\end{equation*}

which concludes the proof.

5.2. Quasi-ergodic distribution for Brownian motion absorbed by an asymptotically periodic moving boundary

Let $(W_t)_{t \geq 0}$ be a one-dimensional Brownian motion, and let h be a $\mathcal{C}^1$ -function such that

\begin{equation*}h_{\min} \;:\!=\; \inf_{t \geq 0} h(t) > 0\;\;\;\;\text{ and }\;\;\;\;h_{\max} \;:\!=\; \sup_{t \geq 0} h(t) < + \infty.\end{equation*}

We assume also that

\begin{equation*}-\infty < \inf_{t \geq 0} h'(t) \leq \sup_{t \geq 0} h'(t) < + \infty.\end{equation*}

Define

\begin{equation*}\tau_h \;:\!=\; \inf\{t \geq 0 \;:\; |W_t| \geq h(t)\}.\end{equation*}

Since h is continuous, the hitting time $\tau_h$ is a stopping time with respect to the natural filtration of $(W_t)_{t \geq 0}$ . Moreover, since $\sup_{t \geq 0} h(t) < + \infty$ and $\inf_{t \geq 0} h(t) > 0$ ,

\begin{equation*}\mathbb{P}_{s,x}[\tau_h < + \infty] = 1 \;\;\text{ and }\;\; \mathbb{P}_{s,x}[\tau_h > t] > 0,\;\;\;\;\forall s \leq t,\ \forall x \in [\!-\!h(s),h(s)].\end{equation*}

The main assumption on the function h is the existence of a $\gamma$ -periodic function g such that $h(t) \leq g(t)$ , for any $t \geq 0$ , and such that

\begin{equation*}h \sim_{t \to \infty} g \;\;\text{and}\;\;h' \sim_{t \to \infty} g'.\end{equation*}

Similarly to $\tau_h$ , define

\begin{equation*}\tau_g \;:\!=\; \inf\{t \geq 0 \;:\; |W_t| = g(t)\}.\end{equation*}

Finally, let us assume that there exists $n_0 \in \mathbb{N}$ such that, for any $s \geq 0$ ,

(38) \begin{equation} \inf \{ u \geq s \;:\; h(u) = \inf_{t \geq s} h(t)\} - s \leq n_0 \gamma.\end{equation}

This condition says that there exists $n_0 \in \mathbb{N}$ such that, for any time $s \geq 0$ , the infimum of the function h on the domain $[s, + \infty)$ is reached on the subset $[s, s + n_0 \gamma]$ .

We first prove the following proposition.

Proposition 1. The Markov process $(W_{t})_{t \geq 0}$ , considered as absorbed by h or by g, satisfies Assumption (A′).

Proof. In what follows, we will prove Assumption (A′) with respect to the absorbing function h. The proof can easily be adapted for the function g.

• Proof of (A′1). Define $\mathcal{T} \;:\!=\; \{s \geq 0 \;:\; h(s) = \inf_{t \geq s} h(t)\}$ . The condition (38) implies that this set contains an infinity of times.

In what follows, the following notation is needed: for any $z \in \mathbb{R}$ , define $\tau_z$ as

\begin{equation*}\tau_z \;:\!=\; \inf\{t \geq 0 \;:\; |W_t| = z\}.\end{equation*}

Also, let us state that, since the Brownian motion absorbed at $\{-1,1\}$ satisfies Assumption (A) of [Reference Champagnat and Villemonais5] at any time (see [Reference Champagnat and Villemonais7]), it follows that, for a given $t_0 > 0$ , there exist $c > 0$ and $\nu \in \mathcal{M}_1((\!-\!1,1))$ such that, for any $x \in (\!-\!1,1)$ ,

(39) \begin{equation} \mathbb{P}_{0,x}\bigg[W_{\frac{t_0}{h_{\max}^2} \land t_0} \in \cdot \bigg| \tau_1 > \frac{t_0}{h_{\max}^2} \land t_0\bigg] \geq c \nu.\end{equation}

Moreover, in relation to the proof of [Reference Champagnat and Villemonais7, Section 5.1], the probability measure $\nu$ can be expressed as

(40) \begin{equation} \nu = \frac{1}{2}\left(\mathbb{P}_{0,1-\epsilon}[W_{t_2} \in \cdot | \tau_1 > t_2] + \mathbb{P}_{0,-1+\epsilon}[W_{t_2} \in \cdot | \tau_1 > t_2]\right),\end{equation}

for some $0 < t_2 < \frac{t_0}{h_{\max}^2} \land t_0$ and $\epsilon \in (0,1)$ .

The following lemma is very important for the next part of the argument.

Lemma 2. For all $z \in [h_{\min},h_{max}]$ ,

\begin{equation*}\mathbb{P}_{0,x}[W_{u} \in \cdot | \tau_z > u] \geq c \nu_z,\;\;\;\;\forall x \in (\!-\!z,z),\ \forall u \geq t_0,\end{equation*}

where $t_0$ is as previously mentioned, $c > 0$ is the same constant as in (39), and

\begin{equation*}\nu_z(f) = \int_{(\!-\!1,1)} f(z x) \nu(dx),\end{equation*}

with $\nu \in \mathcal{M}_1((\!-\!1,1))$ defined in (40).

The proof of this lemma is postponed until after the current proof.

Let $s \in \mathcal{T}$ . Then, for all $x \in (\!-\!h(s),h(s))$ and $t \geq 0$ ,

\begin{equation*}\mathbb{P}_{s,x}[W_{s+t} \in \cdot | \tau_{h} > s+t] \geq \frac{ \mathbb{P}_{s,x}[\tau_{h(s)} > s+t]}{\mathbb{P}_{s,x}[\tau_h > s+t]} \mathbb{P}_{s,x}[W_{s+t} \in \cdot | \tau_{h(s)} > s+t].\end{equation*}

By Lemma 2, for all $x \in (\!-\!h(s),h(s))$ and $t \geq t_0$ ,

\begin{equation*}\mathbb{P}_{s,x}[W_{s+t} \in \cdot | \tau_{h(s)} > s+t] \geq c \nu_{h(s)},\end{equation*}

which implies that, for any $t \in [t_0, t_0 + n_0 \gamma]$ ,

(41) \begin{align} \mathbb{P}_{s,x}[W_{s+t} \in \cdot | \tau_h > s+t] &\geq \frac{ \mathbb{P}_{s,x}[\tau_{h(s)} > s+t]}{\mathbb{P}_{s,x}[\tau_h > s+t]} c \nu_{h(s)} \notag \\[5pt] &\geq \frac{ \mathbb{P}_{s,x}[\tau_{h(s)} > s+t_0+n_0 \gamma]}{\mathbb{P}_{s,x}[\tau_h > s+t_0]} c \nu_{h(s)}.\end{align}

Let us introduce the process $X^h$ defined by, for all $t \geq 0$ ,

\begin{equation*}X^h_t \;:\!=\; \frac{W_t}{h(t)}.\end{equation*}

By Itô’s formula, for any $t \geq 0$ ,

\begin{equation*}X^h_t = X^h_0 + \int_0^t \frac{dW_s}{h(s)} - \int_0^t \frac{h'(s)}{h(s)} X^h_s ds.\end{equation*}

Define

\begin{equation*}\big(M^h_t\big)_{t \geq 0} \;:\!=\; \bigg(\int_0^t \frac{1}{h(s)}dW_s\bigg)_{t \geq 0}.\end{equation*}

By the Dubins–Schwarz theorem, it is well known that the process $M^h$ has the same law as

\begin{equation*}\bigg(W_{\int_0^t \frac{1}{h^2(s)}ds}\bigg)_{t \geq 0}.\end{equation*}

Then, defining

\begin{equation*}I^h(s) \;:\!=\; \int_0^s \frac{1}{h^2(u)}du\end{equation*}

and, for any $s \leq t$ and for any trajectory w,

(42) \begin{align} {\mathcal E}_{s,t}^h(w) \;:\!=\; \sqrt{\frac{h(t)}{h(s)}} \exp\bigg(& -\frac{1}{2} \bigg[h'(t)h(t) w_{I^h(t)}^2 - h'(s)h(s) w_{I^h(s)}^2 \end{align}

(43) \begin{align} & \qquad\qquad\qquad\qquad\qquad + \int_{s}^{t} w_{I^h(u)}^2[(h'(u))^2 - [h(u)h'(u)]']du\bigg]\bigg), \end{align}

Girsanov’s theorem implies that, for all $x \in (\!-\!h(s),h(s))$ ,

(44) \begin{equation}\mathbb{P}_{s,x}[\tau_h > s+t_0] = \mathbb{E}_{I^h(s),\frac{x}{h(s)}}\bigg[{\mathcal E}_{s,s+t_0}^h(W) \mathbb{1}_{\tau_1 > \int_0^{s+t_0} \frac{1}{h^2(u)} du}\bigg].\end{equation}

On the event

\begin{equation*}\bigg\{\tau_1 > \int_0^{s+t_0} \frac{1}{h^2(u)} du\bigg\},\end{equation*}

and since h and h ′ are bounded on $\mathbb{R}_+$ , the random variable ${\mathcal E}_{s,s+t_0}^h(W)$ is almost surely bounded by a constant $C > 0$ , uniformly in s, such that for all $x \in (\!-\!h(s),h(s))$ ,

(45) \begin{equation}\mathbb{E}_{I^h(s),\frac{x}{h(s)}}\bigg[{\mathcal E}_{s,s+t_0}^h(W) \mathbb{1}_{\tau_1 > \int_0^{s+t_0} \frac{1}{h^2(u)} du}\bigg] \leq C \mathbb{P}_{0,\frac{x}{h(s)}}\bigg[\tau_1 > \int_s^{s+t_0} \frac{1}{h^2(u)} du\bigg].\end{equation}

Since $h(t) \geq h(s)$ for all $t \geq s$ (since $s \in \mathcal{T}$ ),

\begin{equation*}I^h(s+t_0) - I^h(s) \leq \frac{t_0}{h(s)^2}.\end{equation*}

By the scaling property of the Brownian motion and by the Markov property, one has for all $x \in (\!-\!h(s),h(s))$

\begin{align*}\mathbb{P}_{s,x}[\tau_{h(s)} > s+t_0] &= \mathbb{P}_{0,x}[\tau_{h(s)} > t_0] \\[5pt] &= \mathbb{P}_{0,\frac{x}{h(s)}}\bigg[\tau_1 > \frac{t_0}{h^2(s)}\bigg] \\[5pt] &= \mathbb{E}_{0,\frac{x}{h(s)}}\bigg[\mathbb{1}_{\tau_1 > \int_s^{s+t_0} \frac{1}{h^2(u)}du} \mathbb{P}_{0,W_{ \int_s^{s+t_0} \frac{1}{h^2(u)}du}}\bigg[\tau_1 > \frac{t_0}{h^2(s)} - \int_s^{s+t_0} \frac{1}{h^2(s)}ds\bigg]\bigg] \\[5pt] &= \mathbb{P}_{0,\frac{x}{h(s)}}\bigg[\tau_1 > \int_s^{s+t_0} \frac{1}{h^2(u)} du\bigg] \\[5pt] & \quad \mathbb{P}_{0,\phi_{I^h(s+t_0) - I^h(s)}(\delta_x)}\bigg[\tau_1 > \frac{t_0}{h^2(s)} - \int_s^{s+t_0} \frac{1}{h^2(u)}du\bigg],\end{align*}

where, for any initial distribution $\mu$ and any $t \geq 0$ ,

\begin{equation*}\phi_{t}(\mu) \;:\!=\; \mathbb{P}_{0,\mu}[W_t \in \cdot | \tau_1 > t].\end{equation*}

The family $(\phi_t)_{t \geq 0}$ satisfies the equality $\phi_t \circ \phi_s = \phi_{t+s}$ for all $s,t \geq 0$ . By this property, and using that

\begin{equation*}I^h(s+t_0) - I^h(s) \geq \frac{t_0}{h_{\max}^2}\end{equation*}

for any $s \geq 0$ , the minorization (39) implies that, for all $s \geq 0$ and $x \in (\!-\!1,1)$ ,

\begin{equation*}\phi_{I^h(s+t_0) - I^h(s)}(\delta_x) \geq c \nu.\end{equation*}

Hence, by this minorization, and using that h is upper-bounded and lower-bounded positively on $\mathbb{R}_+$ , one has for all $x \in (\!-\!1,1)$

\begin{align*}\mathbb{P}_{0,\phi_{I^h(s+t_0)-I^h(s)}(\delta_x)} &\bigg[\tau_1 > \frac{t_0}{h^2(s)} - \int_s^{s+t_0} \frac{1}{h^2(u)}du\bigg]\\[5pt] &\geq c \mathbb{P}_{0,\nu}\bigg[\tau_1 > \inf_{s \geq 0} \bigg\{\frac{t_0}{h^2(s)} - \int_s^{s+t_0} \frac{1}{h^2(u)}du\bigg\}\bigg];\end{align*}

that is to say,

\begin{equation*}\frac{\mathbb{P}_{s,x}[\tau_{h(s)} > s+t_0]}{\mathbb{P}_{0,\frac{x}{h(s)}}\bigl[\tau_1 > \int_s^{s+t_0} \frac{1}{h^2(u)} du\bigl]} \geq c \mathbb{P}_{0,\nu}\bigg[\tau_1 > \inf_{s \geq 0} \bigg\{\frac{\gamma}{h^2(s)} - \int_s^{s+t_0} \frac{1}{h^2(u)}du\bigg\}\bigg].\end{equation*}

In other words, we have just shown that, for all $x \in (\!-\!h(s),h(s))$ ,

(46) \begin{equation} \frac{ \mathbb{P}_{s,x}[\tau_{h(s)} > s+t_0]}{\mathbb{P}_{s,x}[\tau_h > s+t_0]} \geq \frac{c}{C} \mathbb{P}_{0,\nu}\bigg[\tau_1 > \inf_{s \geq 0} \bigg\{\frac{t_0}{h^2(s)} - \int_s^{s+t_0} \frac{1}{h^2(u)}du\bigg\}\bigg] > 0.\end{equation}

Moreover, by Lemma 2 and the scaling property of the Brownian motion, for all $x \in (\!-\!h(s),h(s))$ ,

(47) \begin{align}\frac{\mathbb{P}_{s,x}[\tau_{h(s)} > s + t_0 + n_0 \gamma]}{\mathbb{P}_{s,x}[\tau_{h(s)} > s+ t_0]} &= \mathbb{P}_{0,\mathbb{P}_{0,x}[W_{t_0} \in \cdot | \tau_{h(s)} > t_0]}[\tau_{h(s)} > n_0 \gamma] \notag \\[5pt] &\geq c \mathbb{P}_{0,\nu_{h(s)}}[\tau_{h(s)} > n_0 \gamma] \notag \\[5pt] &= c \int_{(\!-\!1,1)} \nu(dy) \mathbb{P}_{h(s)y}[\tau_{h(s)} > n_0 \gamma] \notag\\[5pt] &\geq c \mathbb{P}_{0,\nu}\Bigg[\tau_1 > \frac{n_0 \gamma}{h_{\min}^2}\Bigg] > 0.\end{align}

Thus, combining (41), (46), and (47), for any $x \in (\!-\!h(s),h(s))$ and any $t \in [t_0, t_0 + n_0 \gamma]$ ,

(48) \begin{equation}\mathbb{P}_{s,x}[W_{s+t} \in \cdot | \tau_h > s+t] \geq c_1 \nu_{h(s)},\end{equation}

where

\begin{equation*}c_1 \;:\!=\; c \mathbb{P}_{0,\nu}\bigg[\tau_1 > \frac{n_0 \gamma}{h_{\max}^2}\bigg] \times \frac{c}{C} \mathbb{P}_{0,\nu}\bigg[\tau_1 > \inf_{s \geq 0} \bigg\{\frac{\gamma}{h^2(s)} - \int_s^{s+\gamma} \frac{1}{h^2(u)}du\bigg\}\bigg] c.\end{equation*}

We recall that the Doeblin condition (48) has, for now, been obtained only for $s \in \mathcal{T}$ . Consider now $s \not \in \mathcal{T}$ . Then, by the condition (38), there exists $s_1 \in \mathcal{T}$ such that $s < s_1 \leq s + n_0 \gamma$ . The Markov property and (48) therefore imply that, for any $x \in (\!-\!h(s),h(s))$ ,

\begin{equation*}\mathbb{P}_{s,x}[W_{s+t_0 + n_0 \gamma} \in \cdot | \tau_h > s+t_0 + n_0 \gamma] = \mathbb{P}_{s_1,\phi_{s_1,s}}[W_{s+t_0 + n_0 \gamma} \in \cdot | \tau_h > s+t_0 + n_0\gamma] \geq c_1 \nu_{h(s_1)},\end{equation*}

where, for all $s \leq t$ and $\mu \in \mathcal{M}_1((\!-\!h(s),h(s)))$ ,

\begin{equation*}\phi_{t,s}(\mu) \;:\!=\; \mathbb{P}_{s,\mu}[W_t \in \cdot | \tau_h > t].\end{equation*}

This concludes the proof of (A′1).

• Proof of (A′2). Since $(W_t)_{t \geq 0}$ is a Brownian motion, note that for any $s \leq t$ ,

  \begin{equation*}\sup_{x \in (\!-\!1,1)} \mathbb{P}_{s,x}[\tau_h > t] = \mathbb{P}_{s,0}[\tau_h > t].\end{equation*}
  Also, for any $a \in (0,h(s))$ ,
  \begin{equation*}\inf_{[\!-\!a,a]}\mathbb{P}_{s,x}[\tau_h > t] = \mathbb{P}_{s,a}[\tau_h > t].\end{equation*}
  Thus, by the Markov property, and using that the function $s \mapsto \mathbb{P}_{s,0}[\tau_g > t]$ is non-decreasing on [0, t] (for all $t \geq 0$ ), one has, for any $s \leq t$ ,
  (49) \begin{align} \mathbb{P}_{s,a}[\tau_h > t] \geq \mathbb{E}_{s,a}[\mathbb{1}_{\tau_0 < s+\gamma < \tau_h} \mathbb{P}_{\tau_0,0}[\tau_h > t]] \geq \mathbb{P}_{s,a}[\tau_0 < s+\gamma < \tau_h] \mathbb{P}_{s,0}[\tau_h > t].\end{align}
  Defining $a \;:\!=\; \frac{h_{\min}}{h_{\max}}$ , by Lemma 2 and taking $s_1 \;:\!=\; \inf\{u \geq s \;:\; u \in \mathcal{T}\}$ , one obtains that, for all $s \leq t$ ,
  \begin{align*}\mathbb{P}_{s,\nu_{h(s_1)}}[\tau_h > t] &= \int_{(\!-\!1,1)} \nu(dx) \mathbb{P}_{s,h(s_1)x}[\tau_h > t] \\[5pt] &\geq \nu([\!-\!a,a]) \mathbb{P}_{s,h(s_1)a}[\tau_h > t] \\[5pt] & \geq \nu([\!-\!a,a]) \mathbb{P}_{0,h_{\min}}[\tau_0 < \gamma < \tau_h] \sup_{x \in (\!-\!h(s),h(s))} \mathbb{P}_{s,x}[\tau_h > t].\end{align*}
  This concludes the proof, since, using (40), one has $\nu([\!-\!a,a]) > 0$ .

We now prove Lemma 2.

Proof of Lemma 2. This result comes from the scaling property of a Brownian motion. In fact, for any $z \in [h_{\min},h_{\max}]$ , $x \in (\!-\!z,z)$ , and $t \geq 0$ , and for any measurable bounded function f,

\begin{align*} \mathbb{E}_{0,x}[f(W_t) | \tau_z > t] &= \mathbb{E}_{0,x}\bigg[f\bigg(z \times \frac{1}{z}W_{z^2 \frac{t}{z^2}}\bigg) \bigg| \tau_z > t\bigg] \\[5pt] &= \mathbb{E}_{0,\frac{x}{z}}\bigg[f\bigg(z \times W_{\frac{t}{z^2}}\bigg) \bigg| \tau_1 > \frac{t}{z^2}\bigg].\end{align*}

Then the minorization (39) implies that for any $x \in (\!-\!1,1)$ ,

\begin{equation*}\mathbb{P}_{0,x}\bigg[W_{\frac{t_0}{h_{\max}^2}} \in \cdot \bigg| \tau_1 > \frac{t_0}{h_{\max}^2}\bigg] \geq c \nu.\end{equation*}

This inequality holds for any time greater than $\frac{t_0}{h_{\max}^2}$ . In particular, for any $z \in [h_{\min},h_{\max}]$ and $x \in (\!-\!1,1)$ ,

\begin{equation*}\mathbb{P}_{0,x}\bigg[W_{\frac{t_0}{z^2}} \in \cdot \bigg| \tau_1 > \frac{t_0}{z^2}\bigg] \geq c \nu.\end{equation*}

Then, for any $z \in [a,b]$ , f positive and measurable, and $x \in (\!-\!z,z)$ ,

\begin{equation*}\mathbb{E}_{0,x}[f(W_{t_0}) | \tau_z > t_0] \geq c \nu_z\left(f\right),\end{equation*}

where $\nu_z(f) \;:\!=\; \int_E f(z \times x)\nu(dx)$ . This completes the proof of Lemma 2.

We now conclude the section by stating and proving the following result.

Theorem 4. For any $s \leq t$ , $n \in \mathbb{N}$ , and any $x \in \mathbb{R}$ ,

\begin{equation*}\mathbb{P}_{s+k \gamma, x}[\tau_h \leq t+k \gamma < \tau_g] \underset{k \to \infty}{\longrightarrow} 0.\end{equation*}

In particular, Corollary 2 holds for $(W_t)_{t \geq 0}$ absorbed by h.

Proof. Recalling (43), by the Markov property for the Brownian motion, one has, for any $k,n \in \mathbb{N}$ and any $x \in \mathbb{R}$ ,

\begin{align*} \mathbb{P}_{s+k\gamma,x}[\tau_h > t+k\gamma] &= \sqrt{\frac{h(t+k\gamma)}{h(s+k \gamma)}} \mathbb{E}_{0,x}\bigg[\exp\bigg(-\frac{1}{2} \mathcal{A}^h_{s,t,k}(W) \bigg) \mathbb{1}_{\tau_1 > I^h(t+k\gamma) - I^h(s+k \gamma)}\bigg], \end{align*}

where, for any trajectory $w = (w_u)_{u \geq 0}$ ,

\begin{align*} \mathcal{A}_{s,t,k}^h(w) =h'(t+k\gamma)h(t+k\gamma) w_{I^h(t+k\gamma) - I^h(s+k \gamma)}^2 - h'(s+k \gamma)h(s+k \gamma) w_0^2 \\[5pt] + \int_{0}^{t-s} w_{I^h(u+s+k\gamma) - I^h(s+k \gamma)}^2[(h'(u+s+k\gamma))^2 - [h(u+s+k \gamma)h'(u+s+k\gamma)]']du.\end{align*}

Since $h \sim_{t \to \infty} g$ , one has for any $s,t \in [0, \gamma]$

\begin{equation*}\sqrt{\frac{h(t+k\gamma)}{h(s+k \gamma)}} \underset{k \to \infty}{\longrightarrow} \sqrt{\frac{g(t)}{g(s)}}.\end{equation*}

For the same reasons, and using that the function h is bounded on $[s+k \gamma, t+k\gamma]$ for all $s \leq t$ , Lebesgue’s dominated convergence theorem implies that

\begin{equation*}I^h(t+k\gamma) - I^h(s+k\gamma) \underset{k \to \infty}{\longrightarrow} I^g(t) - I^g(s)\end{equation*}

for all $s \leq t \in [0,\gamma]$ . Moreover, since $h \sim_{t \to \infty} g$ and $h' \sim_{t \to \infty} g'$ , one has for all trajectories $w = (w_u)_{u \geq 0}$ and $s \leq t \in [0,\gamma]$

\begin{equation*} \mathcal{A}_{s,t,k}^h(w) \underset{k \to \infty}{\longrightarrow} g'(t)g(t) w_{I^g(t) - I^g(s)}^2 - g'(s)g(s)w_0^2 + \int_{s}^{t} w_{I^g(u)}^2[(g'(u))^2 - [g(u)g'(u)]']du.\end{equation*}

Since the random variable

\begin{equation*}\exp\bigg(-\frac{1}{2} \mathcal{A}^h_{s,t,k}(W) \bigg) \mathbb{1}_{\tau_1 > I^h(t+k\gamma) - I^h(s+k \gamma)}\end{equation*}

is bounded almost surely, Lebesgue’s dominated convergence theorem implies that

\begin{equation*}\mathbb{P}_{s+k \gamma, x}[\tau_h > t+k \gamma] \underset{k \to \infty}{\longrightarrow} \mathbb{P}_{s,x}[\tau_g > t],\end{equation*}

which concludes the proof.