Proof. We prove (2.6), (2.8) and (2.9) in this order. The proofs of (2.5), (2.7) are similar and thus omitted.

Proof of (2.6). Fix a positive measure $\mu$ on D (the result is trivial if $\mu=0$ ). For all $\varepsilon>0$ and for all $x\in D$ such that $j_S(\mu)+\varepsilon<j_S(x)$ , we have, by definition of $j_S(x)$ and the fact that $(j_S(\mu)+\varepsilon+j_S(x))/2<j_S(x)$ ,

\begin{align*} \liminf_{n\to+\infty} \theta_{0,S}^{-n}n^{-(j_S(\mu)+\varepsilon+j_S(x))/2}\delta_x S_n{\unicode{x1D7D9}}_D>0 \end{align*}

and hence, since $(j_S(\mu)+\varepsilon+j_S(x))/2>j_S(\mu)+\varepsilon$ ,

\begin{align*} \liminf_{n\to+\infty} \theta_{0,S}^{-n}n^{-j_S(\mu)-\varepsilon}\delta_x S_n{\unicode{x1D7D9}}_D=+\infty. \end{align*}

Using Fatou’s lemma, we obtain

\begin{align*} 0=\liminf_{n\to+\infty} \ \theta_{0,S}^{-n}n^{-j_S(\mu)-\varepsilon}\mu S_n{\unicode{x1D7D9}}_D &\geq \mu\left(\liminf_{n\to+\infty} \ \theta_{0,S}^{-n}n^{-j_S(\mu)-\varepsilon} S_n{\unicode{x1D7D9}}_D\right)\\[5pt] &\geq \mu\left(+\infty\, {\unicode{x1D7D9}}_{j_S(\cdot)>j_S(\mu)+\varepsilon}\right)\!. \end{align*}

This implies that, for all $\varepsilon>0$ , $\mu\{x,\ j_S(x)>j_S(\mu)+\varepsilon\}=0$ , and hence that $\mu\{x,\ j_S(x)>j_S(\mu)\}=0$ . In particular, any $\ell\geq 0$ such that $\mu\{x,j_S(x)\geq \ell\}>0$ satisfies $\ell\leq j_S(\mu)$ . We have thus proved that

(2.10) \begin{align} j_S(\mu)\geq \ell_\mu\;:\!=\;\sup\left\{\ell\geq 0,\ \mu\{x,\ j_S(x)\geq \ell\}>0\right\}. \end{align}

Proof of (2.8) and the fact that $j_S$ is lower semi-continuous. We assume that $\mu\in \mathcal M_+(W_S)$ is a positive measure and that Assumption (A) holds true, and we prove $j_S(\mu)\leq \ell_\mu$ , where $\ell_\mu$ is defined in (2.10). Fix $\ell>\ell_\mu$ , so $j_S(x)<\ell$ $\mu(\!\operatorname{d}\!x)$ -almost everywhere. Then, by Assumption (A),

for some constant $C>0$ . This also implies that $|\theta_{0,S}^{-n}n^{-\ell}\delta_xS_n{\unicode{x1D7D9}}_D|$ is bounded, up to a multiplicative constant, by the $\mu$ -integrable function $W_S$ , and hence, by the Lebesgue dominated convergence theorem, we obtain

\begin{align*} \lim_{n\to+\infty} \theta_{0,S}^{-n}n^{-\ell}\mu S_n{\unicode{x1D7D9}}_D=0. \end{align*}

This entails that $\ell\geq j_S(\mu)$ . Since $\ell>\ell_\mu$ was arbitrary, we deduce that $\ell_\mu\geq j_S(\mu)$ . This concludes the proof of (2.8).

Now let $(\mu_n)_{n\in\mathbb{N}}$ be a sequence of elements of $\mathcal M_+(W_S)$ converging towards $\mu$ in $\mathcal M_+(W_S)$ . Then, for all measurable subsets $A\subset D$ , we have $\mu_n(A)\to \mu(A)$ and hence, for all $\ell\geq 0$ such that $\mu\{x,\ j_S(x)\geq \ell\}>0$ ,

\[ \liminf_{n\geq +\infty} \mu_n\{x,\ j_S(x)\geq \ell\}>0, \]

so that

\[ \liminf_{n\geq +\infty} j_S(\mu_n)\geq \ell. \]

This holds true for all $\ell<j_S(\mu)$ , so

\[ \liminf_{n\geq +\infty} j_S(\mu_n)\geq j_S(\mu), \]

which concludes the proof of the fact that $j_S$ is lower semi-continuous.

Proof of (2.9) and that $j_S(X_n)$ is almost surely non-increasing. We continue to assume that $\mu\in \mathcal M_+(W_S)$ and that Assumption (A) holds true. Let us first prove that $j_S(\mu)=j_S(\mu S_1)$ . We have, for all $\ell\geq 0$ ,

\begin{align*} \theta_{0,S}^{-n} n^{-\ell}(\mu S_1)S_n&=\left(\frac{n}{n+1}\right)^\ell\theta_{0,S}\,\theta_{0,S}^{-(n+1)}(n+1)^{-\ell}\mu S_{n+1}\\[5pt] &\sim_{n\to+\infty} \theta_{0,S}\,\theta_{0,S}^{-(n+1)}(n+1)^{-\ell}\mu S_{n+1}. \end{align*}

This implies that the limit inferior of $\theta_0^{-n} n^{-\ell}(\mu S_1)S_n$ equals 0 if and only if the limit inferior of $\theta_{0,S}^{-n}\,n^{-\ell}\mu S_{n}$ equals 0, and hence that $j_S(\mu S_1)=j_S(\mu)$ .

We conclude by proving the last assertion of the proposition. We have

\[ j_S(\mu S_1)=\sup\left\{\ell\geq 0,\ \mu S_1\{x,\ j_S(x)\geq \ell\}>0\right\}\!, \]

hence

\[ \mu S_1\{x,\ j_S(x)>j_S(\mu S_1)\}=0. \]

Using the equality $j_S(\mu S_1)=j_S(\mu)$ and the fact that $\mu S_1={{\operatorname{P}}}_\mu(X_1\in\cdot,\ X_1\neq\partial)$ , we deduce that

\[ {{\operatorname{P}}}_\mu(\,j_S(X_1)>j_S(\mu))=0, \]

where we used $j_S(\partial)=0$ due to our notational convention about extension of functions. This, together with a straightforward application of the Markov property, concludes the proof of the proposition.

Proof. According to (2.3) we have, for all $x\in D$ ,

(2.11) \begin{align} \theta_{0,S}^{-n} n^{-j_S(x)} {{\operatorname{E}}}_x({\unicode{x1D7D9}}_D(X_n){\unicode{x1D7D9}}_{n< \tau_\partial})\xrightarrow[n\to+\infty]{} \eta_S(x) \end{align}

and

(2.12) \begin{align} \left|\theta_{0,S}^{-n} n^{-j_S(x)} {{\operatorname{E}}}_x({\unicode{x1D7D9}}_D(X_n){\unicode{x1D7D9}}_{n< \tau_\partial})\right|&\leq |\eta_S(x)|+\alpha_{S,n} |W_S(x)|\, \|{\unicode{x1D7D9}}_D\|_{W_S}\leq C W_S(x). \end{align}

Denote by $\theta_\nu$ the absorption parameter of $\nu$ . Assume that $\theta_\nu>\theta_{0,S}$ . Then, for any $\ell\geq 1$ such that $\nu\{\,j_S(\!\cdot\!)\leq \ell\}=1$ , and $n\geq 1$ large enough so that $\theta_\nu^{-n}\leq \theta_{0,S}^{-n}n^{-\ell-1}$ ,

\begin{align*} \nu(D) &=\theta_{\nu}^{-n} {{\operatorname{E}}}_\nu({\unicode{x1D7D9}}_D(X_n){\unicode{x1D7D9}}_{n< \tau_\partial})\\[5pt] &\leq n^{-1}\nu\left(\theta_{0,S}^{-n} n^{-j_S(\cdot)} {{\operatorname{E}}}_\cdot({\unicode{x1D7D9}}_D(X_n){\unicode{x1D7D9}}_{n< \tau_\partial})\right)\xrightarrow[n\to+\infty]{} 0, \end{align*}

using (2.12). This is a contradiction and hence $\theta_\nu\leq \theta_{0,S}$ .

Assume now that $\theta_\nu<\theta_{0,S}$ . Then Fatou’s lemma entails that

\begin{align*} 1&=\nu({\unicode{x1D7D9}}_D)= \theta_{\nu}^{-n} {{\operatorname{E}}}_\nu({\unicode{x1D7D9}}_D(X_n){\unicode{x1D7D9}}_{n< \tau_\partial})\\[5pt] &\geq\nu\left(\theta_{\nu}^{-n} n^{-j_S(\cdot)} {{\operatorname{E}}}_\cdot({\unicode{x1D7D9}}_D(X_n){\unicode{x1D7D9}}_{n< \tau_\partial})\right)\xrightarrow[n\to+\infty]{} +\infty \end{align*}

by (2.11) and since $\nu(\eta_S)>0$ . Hence, we have proved that $\theta_\nu=\theta_{0,S}$ .

Proof. The property $j_S(\nu)=0$ is immediate, while the second equality derives immediately from (2.6) in Proposition 2.1.

If in addition Assumption (A) holds true and $\nu\in \mathcal M_+(W_S)$ , then (2.11) and (2.12) are satisfied. Using the fact that $\nu(W_S)<+\infty$ , we deduce from Lebesgue’s dominated convergence theorem that

\begin{align*} 1&=\nu({\unicode{x1D7D9}}_D)= \theta_{0,S}^{-n} {{\operatorname{E}}}_\nu({\unicode{x1D7D9}}_D(X_n){\unicode{x1D7D9}}_{n< \tau_\partial})\\[5pt] &=\nu\left(\theta_{0,S}^{-n} n^{-j_S(\cdot)} {{\operatorname{E}}}_\cdot({\unicode{x1D7D9}}_D(X_n){\unicode{x1D7D9}}_{n< \tau_\partial})\right)\xrightarrow[n\to+\infty]{} \nu(\eta_S), \end{align*}

which shows that $\nu(\eta_S)=1$ .

Finally, integrating (2.3) with respect to $\nu$ and using the fact that $j_S(x)=0$ $\nu(\textrm dx)$ -almost surely, we deduce that, for all $f\in L^\infty(W_S)$ ,

\begin{align*} \Big|\theta_{0,S}^{-n} {{\operatorname{E}}}_\nu(f(X_n){\unicode{x1D7D9}}_{n< \tau_\partial})-\sum_{i\in I_S} \nu(\eta_{S,i})\nu_{S,i}(f)\Big|\leq \alpha_{S,n} \nu(W_S)\,\|f\|_{W_S}. \end{align*}

Since $\theta_{0,S}^{-n}\,{{\operatorname{E}}}_\nu(f(X_n){\unicode{x1D7D9}}_{n< \tau_\partial})=\nu(f)$ and $\nu(W_S)<+\infty$ , we deduce that $\nu=\sum_{i\in I_S}\nu(\eta_{S,i})\nu_{S,i}$ . This concludes the proof of the proposition.

Proof. Proposition 2.3 entails that any quasi-stationary distribution in $\mathcal M_+(W_S)$ with absorption parameter $\theta_{0,S}$ is a convex combination of the probability measures $\nu_{S,i}$ . Conversely, for any $x\in M_i$ , we have $\eta_S(x)=\eta_{S,i}(x)>0$ and hence, according to (2.3),

\[ \frac{1}{\eta_{S}(x)}\sum_{i\in I_S}\eta_{S,i}(x)\nu_{S,i}=\nu_{S,i} \]

is the limit when $n\to+\infty$ of the conditional distribution ${{\operatorname{P}}}_x(X_n\in\cdot\mid n<\tau_\partial)$ , hence it is a quasi-limiting distribution and thus a quasi-stationary distribution in $\mathcal M_+(W_S)$ . Moreover, $\nu_{S,i}$ satisfies $\nu_{S,i}(M_i)=1$ and, since $\eta_{S,i}>0$ on $M_i$ , $\nu_{S,i}(\eta_S)>0$ . Proposition 2.2 entails that the absorption parameter of $\nu_{S,i}$ is $\theta_{0,S}$ . In particular, for any convex combination $\nu=\sum_{i\in I_S}\lambda_i \nu_{S,i}$ , we have $\nu\in\mathcal M_+(W_S)$ and

\begin{align*} {{\operatorname{P}}}_\nu(X_n\in\cdot)&=\sum_{i\in I_S}\lambda_i {{\operatorname{P}}}_{\nu_{S,i}}(X_n\in\cdot)=\sum_{i\in I_S}\lambda_i \theta_{0,S}^n \nu_{S,i}= \theta_{0,S}^n \nu, \end{align*}

which implies that $\nu$ is a quasi-stationary distribution with absorption parameter $\theta_{0,S}$ .

To conclude the proof, we simply observe that any quasi-stationary distribution $\nu$ in $\mathcal M_+(W_S)$ such that $\nu(D\setminus N)>0$ satisfies $\nu(\eta_S)>0$ and hence, according to Proposition 2.2, its absorption parameter is $\theta_{0,S}$ .

Proof. We have $\sum_{i\in I_S} \eta_{S,i}\nu_{S,i}(W_S)\in L^\infty(W_S)$ by assumption. It follows from (2.3) with $n=1$ that $\mathcal{M}(W_S)$ is stable under $\mathfrak{S}$ . Therefore, the stability of $G_\ell$ under $\mathfrak{S}$ is a consequence of (2.9) in Proposition 2.1. Then, for all $\mu\in G_\ell$ and $f\in L^\infty(W_S)$ such that $\|f\|_{W_S}\leq 1$ , we have, using Assumption (A),

\begin{align*} \left|n^{-\ell}(\mathfrak{S}^n\mu)(f)-(E_{\mathfrak{S}} \mu)(f)\right| &=\bigg|n^{-\ell}\theta_{0,S}^{-n}\mu S_n f-\sum_{i\in I_S}\mu\left({\unicode{x1D7D9}}_{j_S(\cdot)=\ell}\eta_{S,i}\right)\nu_{S,i}(f)\bigg| \\[5pt] & \leq \bigg|\theta_{0,S}^{-n}\mu\left(n^{-j_S(\cdot)} {\unicode{x1D7D9}}_{j_S(\cdot)=\ell} S_n f\right)-\sum_{i\in I_S}\mu\left({\unicode{x1D7D9}}_{j_S(\cdot)=\ell}\eta_{S,i}\right)\nu_{S,i}(f)\bigg| \\[5pt] & \qquad+\frac{1}{n}\theta_{0,S}^{-n}|\mu|\left(n^{-j_S(\cdot)} {\unicode{x1D7D9}}_{j_S(\cdot)\leq \ell-1} S_n W_S\right) \\[5pt] & \leq\alpha_{S,n}|\mu|\left({\unicode{x1D7D9}}_{j_S(\cdot)=\ell}W_S\right)+\frac{1}{n}\bigg[\sum_{i\in I_S}|\mu|\left({\unicode{x1D7D9}}_{j_S(\cdot)\leq\ell-1}\eta_{S,i}\right)\nu_{S,i}(W_S)\bigg. \\[5pt] & \qquad \bigg. +\alpha_{S,n}|\mu|\left({\unicode{x1D7D9}}_{j_S(\cdot)\leq \ell-1}W_S\right)\bigg]\\[5pt] & \leq \alpha_{S,n} |\mu|(W_S)+\frac{1}{n}|\mu|\Bigg(\sum_{i\in I_S}\eta_{S,i}\nu_{S,i}(W_S)\Bigg){\unicode{x1D7D9}}_{\ell\geq 1}\\[5pt] &\leq \alpha_{S,n} |\mu|(W_S) + \frac{1}{n}\Bigg\|\sum_{i\in I_S}\eta_{S,i}\nu_{S,i}(W_S)\Bigg\|_{W_S}{\unicode{x1D7D9}}_{\ell\geq 1} |\mu|(W_S). \end{align*}

This implies (2.15). Finally, since $G_\ell$ is a closed subset of the Banach space $\mathcal{M}(W_S)$ , we deduce that $E_{\mathfrak{S}}\mu\in G_\ell$ for all $\mu\in G_\ell$ and the fact that $E_{\mathfrak{S}}$ is a bounded operator on $G_\ell$ follows from (2.15).

Proof. Note that, according to Lemma 2.1, inequality (2.15) holds true. Assertions (i) and (ii) are thus immediate. For assertion (iii), we observe that

\begin{align*} &\Pi {\mathfrak{S}}E_{\mathfrak{S}}-E_{\mathfrak{S}}\Pi \leq \Pi n^{-\ell}{\mathfrak{S}}^{n+1}-{\mathfrak{S}}E_{\mathfrak{S}}\Pi \\[5pt] &\quad +\Pi (n+1)^{-\ell}{\mathfrak{S}}^{n+1}-n^{-\ell}{\mathfrak{S}}^{n+1}\Pi+\Pi (n+1)^{-\ell} {\mathfrak{S}}^{n+1}-E_{\mathfrak{S}}\Pi \end{align*}

where

\begin{align*} \Pi n^{-\ell}{\mathfrak{S}}^{n+1}-{\mathfrak{S}}E_{\mathfrak{S}}\Pi &=\Pi {\mathfrak{S}}(n^{-\ell}{\mathfrak{S}}^n-E_{\mathfrak{S}})\Pi \leq \Pi {\mathfrak{S}}\Pi\,\alpha_{{\mathfrak{S}},n} \end{align*}

and

\begin{align*} \Pi (n+1)^{-\ell}{\mathfrak{S}}^{n+1}-n^{-\ell}{\mathfrak{S}}^{n+1}\Pi & \leq \left(\frac{(n+1)^{\ell}}{n^{\ell}}-1\right)\Pi (n+1)^{-\ell}\mathfrak{S}^{n+1}\Pi \\[5pt] & \leq \left(\frac{(n+1)^{\ell}}{n^{\ell}}-1\right) (\alpha_{\mathfrak{S},n}+\Pi E_{\mathfrak{S}} \Pi) \end{align*}

by (i), and

\begin{align*} \Pi (n+1)^{\ell} {\mathfrak{S}}^{n+1}-E_{\mathfrak{S}}\Pi \leq \alpha_{{\mathfrak{S}},n+1}, \end{align*}

so that $\Pi {\mathfrak{S}}E_{{\mathfrak{S}}}-E_{\mathfrak{S}} \Pi \to 0$ when $n\to+\infty$ , and hence

\[ {\mathfrak{S}}E_{\mathfrak{S}} =E_{\mathfrak{S}} . \]

Similarly, we have

\begin{align*} &\Pi E_{{\mathfrak{S}}}{\mathfrak{S}}-E_{\mathfrak{S}}\Pi \leq \Pi n^{-\ell}{\mathfrak{S}}^{n+1}-E_{\mathfrak{S}} {\mathfrak{S}}\Pi \\[5pt] &\quad +\Pi (n+1)^{-\ell}{\mathfrak{S}}^{n+1}-n^{-\ell}{\mathfrak{S}}^{n+1}\Pi +\Pi (n+1)^{\ell} {\mathfrak{S}}^{n+1}-E_{\mathfrak{S}}\Pi, \end{align*}

where

\begin{align*} \Pi n^{-\ell}{\mathfrak{S}}^{n+1}-E_{\mathfrak{S}} {\mathfrak{S}}\Pi \leq \Pi {\mathfrak{S}}\Pi \,\Pi n^{-\ell}{\mathfrak{S}}^n-E_{\mathfrak{S}}\Pi\leq \alpha_{{\mathfrak{S}},n}, \end{align*}

and the other terms go to 0 as in the previous case. Hence, we deduce that

\[ E_{\mathfrak{S}} {\mathfrak{S}}=E_{\mathfrak{S}} . \]

Proof of Proposition 2.4 . (i) Fix $x_*\in D$ such that $\eta_S(x_*)>0$ and let $\ell_*=j_S(x_*)$ . According to Lemma 2.2 with $\ell=\ell_*$ , the operator

\[ E_{\mathfrak{S}}\mu=\sum_{i\in I_S}\mu\left({\unicode{x1D7D9}}_{j_S(\cdot)=\ell_*}\eta_{S,i}\right)\nu_{S,i}\]

on $G_{\ell_*}$ satisfies

\[ (E_{\mathfrak{S}} \mathfrak{S})\delta_{x}=(\mathfrak{S} E_{\mathfrak{S}})\delta_{x}=E_{\mathfrak{S}}\delta_{x},\quad \text{for all $x\in D$ such that $j_S(x)\leq\ell_*$.} \]

This means that, for all $x\in D$ such that $j_S(x)\leq\ell_*$ ,

(2.16) \begin{align} \theta_{0,S}^{-1}\sum_{i\in I_S}\delta_{x} S_1\left({\unicode{x1D7D9}}_{j_S(\cdot)=\ell_*}\eta_{S,i}\right)\nu_{S,i} &={\unicode{x1D7D9}}_{j_S(x)=\ell_*}\theta_{0,S}^{-1}\sum_{i\in I_S} \eta_{S,i}(x)\nu_{S,i}S_1 \end{align}

(2.17) \begin{align} &={\unicode{x1D7D9}}_{j_S(x)=\ell_*}\sum_{i\in I_S} \eta_{S,i}(x)\nu_{S,i}. \end{align}

Since $j_S(x_*)=\ell_*$ , we deduce from the equality between the right-hand-side of (2.16) and (2.17) that

\[\nu\;:\!=\;\frac{1}{\eta_S(x_*)}\sum_{i\in I_S} \eta_{S,i}(x_*)\nu_{S,i}\in\mathcal M_+(W_S)\]

is a quasi-stationary distribution for $S_1$ with absorption parameter $\theta_{0,S}$ . In addition, according to Proposition 2.3, we have $\nu(\eta_S)=1$ and $\nu({\unicode{x1D7D9}}_{j_S(\cdot)=0})=1$ , so that $\nu({\unicode{x1D7D9}}_{j_S(\cdot)=0}\eta_S)=1$ . Therefore, there exists $x\in D$ such that $\eta_S(x)>0$ and $j_S(x)=0$ .

(ii) Fix $x\in D$ . Then, applying Lemma 2.2 as above with $\ell=j_S(x)$ instead of $\ell_*$ , we obtain (2.16) and (2.17) with $\ell_*$ replaced by $j_S(x)$ . Integrating the test function ${\unicode{x1D7D9}}_D$ on both sides, this implies that

\[ \eta_S(x)=\sum_{i\in I_S} \eta_{S,i}(x)= \theta_{0,S}^{-1}\sum_{i\in I_S}\delta_{x} S_1\left({\unicode{x1D7D9}}_{j_S(\cdot)=j_S(x)}\eta_{S,i}\right)=\theta_{0,S}^{-1}\delta_x S_1\left({\unicode{x1D7D9}}_{j_S(\cdot)=j_S(x)}\eta_S\right)\!. \]

Hence,

\begin{align*} {{\operatorname{E}}}_x\left[\eta_S(X_1){\unicode{x1D7D9}}_{j_S(X_1)=j_S(x)}\right]=\theta_{0,S}\eta_S(x). \end{align*}

We deduce that, for all $n\geq 1$ , ${{\operatorname{P}}}_x$ -almost surely,

\begin{align*} {{\operatorname{E}}}_{X_{n-1}}\left[\eta_S(X_1){\unicode{x1D7D9}}_{j_S(X_1)=j_S(x)}\right]{\unicode{x1D7D9}}_{j_S(X_{n-1})=j_S(x)}=\theta_{0,S}\eta_S(X_{n-1}){\unicode{x1D7D9}}_{j_S(X_{n-1})=j_S(x)}. \end{align*}

Taking the expectation and using the Markov property, we deduce that

\begin{align*} {{\operatorname{E}}}_x\left[\eta_S(X_{n}){\unicode{x1D7D9}}_{j_S(X_n)=j_S(x)}{\unicode{x1D7D9}}_{j_S(X_{n-1})=j_S(x)}\right]=\theta_{0,S}{{\operatorname{E}}}_x\left[\eta_S(X_{n-1}){\unicode{x1D7D9}}_{j_S(X_{n-1})=j_S(x)}\right]\!. \end{align*}

Because of the last assertion of Proposition 2.1, we deduce that $\{\,j_S(X_{n})=j_S(x)\}=\{\,j_S(X_{n})=j_S(X_{n-1})=\cdots=j_S(x)\}$ up to a ${{\operatorname{P}}}_x$ -negligible event, so that

\begin{align*} {{\operatorname{E}}}_x\left[\eta_S(X_{n}){\unicode{x1D7D9}}_{j_S(X_n)=j_S(x)}\right]=\theta_{0,S}{{\operatorname{E}}}_x\left[\eta_S(X_{n-1}){\unicode{x1D7D9}}_{j_S(X_{n-1})=j_S(x)}\right]\!. \end{align*}

Then property (ii) follows by induction.

(iii) Fix $n\geq 1$ and let $\mu\in\mathcal{M}_+(W_S)$ such that $\mu(\eta_S)>0$ and $\mu(n^{j_S(\cdot)}W_S)<+\infty$ . Integrating (2.3) with respect to the measure $n^{j_S(x)}\,\mu(\operatorname{d} x)$ , we obtain, for all $f\in L^\infty(W_S)$ such that $\|f\|_{W_S}\leq 1$ ,

(2.18) \begin{equation} \Bigg|\theta_{0,S}^{-n}{{\operatorname{E}}}_\mu(f(X_n){\unicode{x1D7D9}}_{n<\tau_\partial})-\sum_{i\in I_S}\mu\left(n^{j_S(\cdot)}\eta_{S,i}\right)\nu_{S,i}(f)\Bigg|\leq \alpha_{S,n}\,\mu(n^{j_S(\cdot)}W_S). \end{equation}

Note that $W_S\geq 1$ (by assumption) and hence this inequality also applies to $f\equiv 1$ , which will be used shortly. Then we have, for all measurable functions $f\;:\;D\to \mathbb R$ bounded by 1,

\begin{align*} & \left|\frac{{{\operatorname{E}}}_\mu f(X_n){\unicode{x1D7D9}}_{n<\tau_\partial}}{{{\operatorname{P}}}_\mu(n<\tau_\partial)}-\frac{\sum_{i\in I_S}\mu\left(n^{j_S(\cdot)}\eta_{S,i}\right)\nu_{S,i}(f)}{\sum_{i\in I_S}\mu\left(n^{j_S(\cdot)}\eta_{S,i}\right)}\right| \\[5pt] & \quad \leq\frac{{{\operatorname{E}}}_\mu |f(X_n)|{\unicode{x1D7D9}}_{n<\tau_\partial}}{{{\operatorname{P}}}_\mu(n<\tau_\partial)}\,\frac{|\sum_{i\in I_S}\mu\left(n^{j_S(\cdot)}\eta_{S,i}\right)-\theta_{0,S}^{-n}{{\operatorname{P}}}_\mu(n<\tau_\partial)|}{\sum_{i\in I_S}\mu\left(n^{j_S(\cdot)}\eta_{S,i}\right)} \\[5pt] &\qquad +\frac{|\theta_{0,S}^{-n}{{\operatorname{E}}}_\mu f(X_n){\unicode{x1D7D9}}_{n<\tau_\partial}-\sum_{i\in I_S}\mu\left(n^{j_S(\cdot)}\eta_{S,i}\right)\nu_{S,i}(f)|}{\sum_{i\in I_S}\mu\left(n^{j_S(\cdot)}\eta_{S,i}\right)} \\[5pt] &\quad \leq 2\alpha_{S,n}\frac{\mu\left(n^{j_S(\cdot)}W_S\right)}{\mu\left(n^{j_S(\cdot)}\eta_S\right)}, \end{align*}

which concludes the proof.

(iv) Let $\mu$ be a measure in $\mathcal M(W_S)$ . Property (iv) is an immediate consequence of Lemma 2.1 with $\ell=j_S(|\mu|)$ .

Proof of Theorem 4.1 . In what follows, we set, for all index values $\ell\in\{-1,0,1,\ldots,\bar\ell\}$ ,

\begin{align*} E_\emptyset^{(\ell)}\;:\!=\;\{x\in E_\emptyset,\text{ such that index}(x)=\ell\}.\end{align*}

The proof is based on an induction argument, based on a specific decomposition of the state space $D\cup\{\partial\}$ into an increasing sequence of closed subsets, as defined below. We call a subset $\bar{D}$ of $D\cup\{\partial\}$ a closed set if, for all $x\in D$ , ${{\operatorname{P}}}_x(\exists n\geq 0,\ X_n\not\in \bar D)=0$ . We first observe that $E_\emptyset^{(-1)}\cup\{\partial\}$ is closed and, by Assumption (B2) and the definition of $F_0$ ,

(4.8) \begin{equation} S_0\;:\!=\;\bigcup_{i\in F_0} E_i\cup E^{(-1)}_\emptyset\cup\{\partial\}\end{equation}

is also closed. Similarly, the sets

(4.9) \begin{equation} S_n\;:\!=\;\bigcup_{\ell=0}^{n}\bigcup_{i\in F_\ell} E_i \cup \bigcup_{k=-1}^{n-1} E_\emptyset^{(k)}\cup\{\partial\},\end{equation}

for all $n\in\{1,\ldots,\bar\ell\}$ , and

(4.10) \begin{equation} S'_n\;:\!=\;\bigcup_{\ell=0}^{n}\bigcup_{i\in F_\ell} E_i \cup \bigcup_{k=-1}^{n} E_\emptyset^{(k)}\cup\{\partial\},\end{equation}

for all $n\in\{0,\ldots,\bar\ell\}$ , are also closed. Below, we prove by induction that the following property is true on any of the previous sets. Given a closed subset $\bar{D}$ of $D\cup\{\partial\}$ , we say that property (P) is satisfied on $\bar D$ if the semi-group R of the process X restricted to $\bar D$ satisfies Assumption (A) with $\theta_{0,R}=\bar\theta$ , $I_{R}=F_0$ , $W_R$ the restriction of $W_S$ to $\bar D\setminus\{\partial\}$ , $j_R(x)=\operatorname{index}(x)\vee 0$ for all $x\in \bar D\setminus\{\partial\}$ , for all $i\in I_R$ ,

\begin{align*} \nu_{S,i}\propto \nu_i+\sum_{\ell\geq 0}\theta_{0,S}^{-\ell-1}{{\operatorname{P}}}_{\nu_i}(X_{1}\not\in E_i,\ X_{\ell+1}\in\cdot), \\[5pt] \eta_{S,i}(x)>0\quad\forall x\in \bar D\text{ with }{{\operatorname{P}}}_x(T_{E_i}<\infty)>0\end{align*}

and

\[\eta_{S,i}(x)=0\quad\forall x\in \bar D\text{ with }{{\operatorname{P}}}_x(T_{E_i}<\infty)=0.\]

This will prove Theorem 4.1 since, by definition of $\bar\ell$ , $D=S'_{\bar\ell}$ .

Step 1. Proof that (P) is satisfied on the set $S_0$ . Our aim is to apply Theorem 3.1 with

\[D_1=\cup_{i\in F_0} E_i\quad\text{and}\quad D_2=E_\emptyset^{(-1)}.\]

In what follows, we set $W_R=W{\unicode{x1D7D9}}_{D_2}$ and $W_P=C_W W{\unicode{x1D7D9}}_{D_1}$ . According to (4.4) in Assumption (B1), we have, for all $x\in D_1$ ,

\[{{\operatorname{E}}}_x(W_R(X_1))\leq{{\operatorname{E}}}_x(W(X_1))\leq C_W W(x)=W_P(x),\]

so that the first part of (3.10) holds true. In addition, since $D_2$ is closed, we have, for all $x\in D_2$ ,

\begin{align*} {{\operatorname{E}}}_x(W_R(X_n))={{\operatorname{E}}}_x(W_\emptyset(Y^{(\emptyset)}_n))\leq C\gamma^n W_\emptyset(x)=C\gamma^n W_R(x),\end{align*}

where we used (4.3) from Assumption (B3) for the last inequality. Hence, the second part of (3.10) holds true.

Finally, for all $i\in F_0$ and $x\in E_i$ , we have by Assumption (B1),

\begin{align*} \left|\bar\theta^{-n} {{\operatorname{E}}}_x(f(X_n){\unicode{x1D7D9}}_{n< T_{\cup_{j\neq i}E_j\cup\{\partial\}}})- \eta_{i}(x)\nu_{i}(f)\right|\leq \alpha_{i,n} W_i(x)\|f\|_{W}.\end{align*}

Since $D_1\cup D_2\cup\{\partial\}$ is closed, we deduce that this reduces to

\begin{align*} \left|\bar\theta^{-n} {{\operatorname{E}}}_x(f(X_n){\unicode{x1D7D9}}_{n< T_{\mathcal{A}}})- \eta_{i}(x)\nu_{i}(f)\right|\leq \alpha_{i,n} W_i(x)\|f\|_{W},\end{align*}

where

\[\mathcal{A}\;:\!=\;{\cup_{j\in F_0,j\neq i}E_j\cup D_2\cup \{\partial\}}.\]

Since, in addition, by definition of $F_0$ , $j\not\prec i$ for all $i\neq j\in F_0$ , we deduce that

\begin{align*} \left|\bar\theta^{-n} {{\operatorname{E}}}_x(f(X_n){\unicode{x1D7D9}}_{n< T_{D_2\cup \{\partial\}}})- \eta_{i}(x)\nu_{i}(f)\right|\leq \alpha_{i,n} W_i(x)\|f\|_{W}.\end{align*}

Summing over $i\in F_0$ , we conclude that, for all $x\in D_1$ ,

\begin{align*} \bigg|\bar\theta^{-n} {{\operatorname{E}}}_x(f(X_n){\unicode{x1D7D9}}_{n< T_{D_2\cup \{\partial\}}})- \sum_{i\in F_0} \eta_{i}(x)\nu_{i}(f)\bigg|\leq {\sum_{i\in F_0} \alpha_{i,n} W_i(x)\|f\|_{W}.}\end{align*}

In particular, the process restricted to $D_1$ satisfies Assumption (A) with $j_{0,S}=0$ . We conclude that Assumption (A1) holds true and hence, according to Theorem 3.1, that the process restricted to D satisfies Assumption (A) with $W_S=W{\unicode{x1D7D9}}_{D_2}+C_W W{\unicode{x1D7D9}}_{D_1}$ , $j_S\equiv 0$ , $I_S=F_0$ ,

\begin{align*} \nu_{S,i}\propto \nu_i+\sum_{\ell\geq 0}\theta_{0,S}^{-\ell-1}{{\operatorname{P}}}_{\nu_i}(X_{1}\not\in E_i,\ X_{\ell+1}\in\cdot),\end{align*}

and, for all $i\in I_S$ , $\eta_{S,i}(x)>0$ if and only if $x\in E_i$ with $i\in F_0$ . This proves that property (P) is satisfied on the set $S_0$ . Step 2. Proof that (P) is satisfied on a set of the form (4.10) assuming it is satisfied on a set of the form (4.9). Assume that property (P) is satisfied on the set $S_n$ for some $n\in\{0,1,\ldots,\bar\ell\}$ . Let us prove that it is satisfied on the set $S'_n$ . To do so, we aim to apply Theorem 3.2 with

\begin{align*} D_1=E_\emptyset^{(n)}\quad\text{and}\quad D_2=\bigcup_{\ell=0}^{n}\bigcup_{i\in F_\ell} E_i \cup \bigcup_{\ell=-1}^{n-1} E_\emptyset^{(\ell)}.\end{align*}

Our induction assumption applies to $D_2\cup\{\partial\}$ , so that the semi-group R of the process X restricted to $D_2\cup\{\partial\}$ satisfies Assumption (A) with

\begin{align*} \theta_{0,R}=\bar{\theta},\quad I_R={F}_0,\quad W_R={W_\emptyset+}\sum_{\ell=0}^{n}\sum_{i\in F_\ell}^k W_i, \\[5pt] j_R(x)=\operatorname{index}(x)\vee 0,\quad\forall x\in \bigcup_{\ell=0}^{n}\bigcup_{i\in F_\ell} E_i\cup \bigcup_{\ell=-1}^{n-1}E_\emptyset^{(\ell)},\end{align*}

and, for all $i\in I_R$ ,

\begin{align*} \nu_{S,i}\propto \nu_i+\sum_{\ell\geq 0}\theta_{0,S}^{-\ell-1}{{\operatorname{P}}}_{\nu_i}(X_{1}\not\in E_i,\ X_{\ell+1}\in\cdot), \\[5pt] \eta_{S,i}(x)>0,\quad\forall x\in \bar D\text{ with }{{\operatorname{P}}}_x(T_{E_i}<\infty)>0,\end{align*}

and

\[\eta_{R,i}(x)=0,\quad\forall x\in D_2\text{ with }{{\operatorname{P}}}_x(T_{E_i}<\infty)=0.\]

In addition, similarly to Step 1, one checks that the first and second parts of (3.14) hold true. In particular, for all $x\in D_1$ ,

\[{{\operatorname{E}}}_x(W_\emptyset(Y^{(\emptyset)}_k){\unicode{x1D7D9}}_{Y^{(\emptyset)}_k\in E^{(n)}_\emptyset})\leq {{\operatorname{E}}}_x(W_\emptyset(Y^{(\emptyset)}_k){\unicode{x1D7D9}}_{Y^{(\emptyset)}_k\in E_\emptyset})\leq C\gamma^k W_\emptyset(x).\]

We deduce that Assumption (A2) holds true and we can thus apply Theorem 3.2. Since, for all $x\in D_1$ , there exists $i\in F_{n}$ such that ${{\operatorname{P}}}_x(T_{E_i}<+\infty)>0$ , we deduce that, for all $x\in D_1$ ,

\[\max_{k\geq 0} j_R(\delta_x P_kQ)=n=\operatorname{index}(x).\]

This, together with Theorem 3.2, proves that property (P) is satisfied on the set $S'_n$ .

Step 3. Proof that (P) is satisfied on a set of the form (4.9) assuming it is satisfied on a set of the form (4.10). Assume that property (P) is satisfied on the set $S'_n$ for some $n\in\{1,\ldots,\bar\ell-1\}$ . Let us prove that is it satisfied on the set $S_{n+1}$ . In this case, we aim to apply Theorem 3.3 with

\begin{align*} D_1=\bigcup_{i\in F_{n+1}} E_i\quad\text{and}\quad D_2=\bigcup_{\ell=0}^{n}\bigcup_{i\in F_\ell} E_i \cup \bigcup_{\ell=-1}^{n} E_\emptyset^{(\ell)}.\end{align*}

Using our induction assumption, we deduce that Assumption (A) holds true for the process restricted to $D_2$ , with $j_{0,R}=n$ and $\theta_{0,R}=\bar\theta$ . As in Step 1, it is also clear that Assumption (A) holds true for the process restricted to $D_1$ , with $j_{0,P}=0$ and $\theta_{0,P}=\bar\theta$ . Also as in Step 1, we observe that (3.18) holds true with $W_R=W{\unicode{x1D7D9}}_{D_2}$ and $W_P=C_W W {\unicode{x1D7D9}}_{D_1}$ .

In order to check that Assumption (A3) holds true, it remains to prove (3.19) and (3.20), with $\ell^*=n$ . Since $j_{0,R}=n=\max_{x\in D_2} j_R(x)$ , equality (3.19) is immediate. For (3.20), we observe that, for all $x\in D_1$ , $\operatorname{index}(x)=n+1$ and hence that the process starting from x can reach a point of index n (otherwise, its index would be smaller than or equal to n by definition of the sets $F_i$ ), and hence that ${{\operatorname{P}}}_x(\operatorname{index}(X_{T_{D_2}})=n\text{ and }T_{D_2}<\infty)>0$ . Since $\operatorname{index}(y)=n$ implies that $j_R(y)=n$ and $\eta_R(y)>0$ (by the induction assumption), we deduce that (3.20) also holds true.

We deduce that Assumption (A3) holds true and we can thus apply Theorem 3.3, which concludes the proof.

Proof of Theorem 5.1. For all $x\in D$ , let $C_x$ be the communication class of x, and let $(x_i)_{i\in I}$ , where I is either $\mathbb{N}$ or $\{1,\ldots,n\}$ for some n, be such that D is the disjoint union of the sets $C_{x_i}$ , $i\in I$ . We take (without loss of generality) $x_1=x_0$ and write $C_i$ instead of $C_{x_i}$ .

Let $i\in I$ be such that ${{\operatorname{P}}}_{x_i}(\exists n\geq 0,\ X_n=x_i)>0$ . By assumption, this is the case for $i=1$ . Then the process X restricted to $C_{i}$ is irreducible and satisfies Assumption (E) in [Reference Champagnat and Villemonais11] (this is a direct adaptation to the discrete-time setting of the proof of Theorem 5.1 in [Reference Champagnat and Villemonais11]). By [Reference Champagnat and Villemonais11, Corollary 2.7], this implies that the process X restricted to $C_{i}$ satisfies Assumption (A) with $j_S\equiv 0$ , $\eta_S$ positive and $W_S=V_{|{C_i}}$ . We denote by $\theta_i$ the associated exponential convergence parameter. In particular, it follows from Assumption (A) that there exists a constant $A_i$ such that, for all $x\in C_i$ and all $n\geq 0$ ,

(5.2) \begin{equation} {{\operatorname{E}}}_x(V(X_n){\unicode{x1D7D9}}_{n<T_{\{\partial\}\cup D\setminus C_i}})\leq A_i\theta_i^n V(x).\end{equation}

Let $i\in I$ be such that ${{\operatorname{P}}}_{x_i}(\exists n\geq 0,\ X_n=x_i)=0$ . Then $C_{i}=\{x_i\}$ . Now define

\[J\;:\!=\;\left\{i\in I,\ \frac{{{\operatorname{E}}}_x(V(X_1){\unicode{x1D7D9}}_{1<\tau_\partial})}{V(x)}<\theta_1\quad\forall x\in C_i\right\}.\]

By assumption, there exist only finitely many points $x\in D$ such that $\frac{{{\operatorname{E}}}_x(V(X_1){\unicode{x1D7D9}}_{1<\tau_\partial})}{V(x)}\geq \theta_1/2$ , and hence there exists $\rho<\theta_1$ such that

\[J=\left\{i\in I,\ \frac{{{\operatorname{E}}}_x(V(X_1){\unicode{x1D7D9}}_{1<\tau_\partial})}{V(x)}\leq\rho\quad\forall x\in C_i\right\}\]

In particular, for all $x\in \cup_{j\in J} C_j$ ,

\begin{align*} \frac{{{\operatorname{E}}}_x( V(X_1){\unicode{x1D7D9}}_{X_1\in \cup_{j\in J} C_j})}{V(x)}\leq \rho,\end{align*}

so we deduce from Markov’s property that, for all $n\geq 1$ , using the notation $\tau_J\;:\!=\;T_{\{\partial\}\cup D\setminus \cup_{j\in J} C_j}$ ,

(5.3) \begin{equation} {{\operatorname{E}}}_x\left( V(X_n){\unicode{x1D7D9}}_{n<\tau_J}\right)\leq \rho^n V(x).\end{equation}

Note that, by assumption, $I\setminus J$ is finite. Recall that all $i\in I\setminus J$ are either such that ${{\operatorname{P}}}_{x_i}(\exists n\geq 0,\ X_n=x_i)=0$ and $C_i=\{x_i\}$ , or such that ${{\operatorname{P}}}_{x_i}(\exists n\geq 0,\ X_n=x_i)>0$ , which implies that the process restricted to $C_{i}$ satisfies Assumption (A) as above with exponential convergence parameter $\theta_i$ .

We then define

\[J'\;:\!=\;\left\{i \in I\setminus J\text{ such that }{{\operatorname{P}}}_{x_i}\left(\exists n\geq 0,\ X_n=x_i\right)=0\right\}\]

and, setting $\bar\theta =\max_{i\in I\setminus(J\cup J')}\theta_i$ ,

\[J''\;:\!=\;\left\{i\in I\setminus(J\cup J')\text{ such that }\theta_i<\bar\theta\right\}.\]

Since J ^″ is finite, $\hat\theta\;:\!=\;\sup_{j\in J''} \theta_i<\bar\theta$ .

We now set

\[E_\emptyset=\bigcup_{j\in J\cup J'\cup J''} C_j\]

and enumerate the $C_i$ , $i\in I\setminus (J\cup J'\cup J'')$ , as $E_1,\ldots,E_k$ . We shall apply Theorem 4.1 to the partition of D into the disjoint sets $E_\emptyset,E_1,\ldots,E_k$ . Note that Assumptions (B1) and (B2) are satisfied for all $E_i$ , $1\leq i\leq k$ , with $W_i=V_{\vert E_i}$ . Note also that, because of (5.1) and since $V\geq 1$ , Assumption (B4) is satisfied with $W=V$ , that is, for all $x\in D$ ,

(5.4) \begin{equation} {{\operatorname{E}}}_x (V(X_1){\unicode{x1D7D9}}_{1<\tau_\partial})\leq {A}V(x)\end{equation}

for some constant A.

Let us now check that Assumption (B3) is satisfied with $W_\emptyset=V_{\vert E_\emptyset}$ . We set $\gamma_\emptyset=\rho\vee \hat \theta<\bar\theta$ . Fix $n\geq 0$ . Given any path $(X_k,0\leq k\leq n)$ of X in $E_\emptyset$ , we introduce an auxiliary process $(J_k,0\leq k\leq n)$ defined as follows: we set $J_k=j\in J'\cup J''$ if $X_k\in C_j$ , and otherwise, we set $J_k=\aleph$ . This means that $J_k=\aleph$ whenever $X_k\in C_j$ for any $j\in J$ . Given any path $\textbf{j}=(j_0,\ldots,j_n)\in (\{\aleph\}\cup J'\cup J'')^{n+1}$ of $(J_k,0\leq k\leq n)$ , we denote by $n_{\operatorname{trans}}(\textbf{j})$ the number of transitions in the sequence $\textbf{j}$ , that is, the number of $k\in\{0,\ldots,n-1\}$ such that $j_k\neq j_{k+1}$ and by $n''(\textbf{j})$ the number of visits of J” in $\textbf{j}$ , that is the number of pairs $(k,\ell)\in\{0,\ldots,n\}$ such that $k<\ell$ , $j_k=j_{k+1}=\ldots=j_\ell\in J''$ , $j_{k-1}\neq j_k$ or $k=0$ , and $j_{\ell+1}\neq j_\ell$ or $\ell=n$ .

We shall prove by induction on $n_{\operatorname{trans}}(\textbf{j})$ that for all $\textbf{j}\in \bigcup_{n\geq 0}(\{\aleph\}\cup J'\cup J'')^{n+1}$ ,

(5.5) \begin{equation} {{\operatorname{E}}}_x\left(V(X_n){\unicode{x1D7D9}}_{(J_k,0\leq k\leq n)=\textbf{j}}\right)\leq A^{n_{\operatorname{trans}}(\textbf{j})}\left(\max_{j\in J''}A_j \right)^{n''(\textbf{j})}\gamma_\emptyset^{n-n_{\operatorname{trans}}(\textbf{j})}. \end{equation}

First, if $\textbf{j}\in(\{\aleph\}\cup J'\cup J'')^{n+1}$ is such that $n_{\operatorname{trans}}(\textbf{j})=0$ , this means that $J_0=J_1=\ldots=J_n$ . If $J_0\in J'$ , this means that $n=0$ , so (5.5) is clear. If $J_0\in J''$ , then $n''(\textbf{j})=1$ and (5.5) follows from (5.2). If $J_0=\aleph$ , (5.5) follows from (5.3).

Assume now that we have proved (5.5) for all $\textbf{j}$ such that $n_{\operatorname{trans}}(\textbf{j})=k\geq 0$ and let $\textbf{j}\in(\{\aleph\}\cup J'\cup J'')^{n+1}$ be such that $n_{\operatorname{trans}}(\textbf{j})= k+1$ . This means that $\textbf{j}=(\textbf{j}',j,j,\ldots,j)$ with $j\in \{\aleph\}\cup J'\cup J''$ repeated $\ell$ times for some $\ell\geq 1$ and $\textbf{j}'\in(\{\aleph\}\cup J'\cup J'')^{n-\ell+1}$ is such that $n_{\operatorname{trans}}(\textbf{j}')=n$ . If $j=\aleph$ , it follows from Markov’s property that

\begin{align*} {{\operatorname{E}}}_x\left(V(X_n){\unicode{x1D7D9}}_{(J_k,0\leq k\leq n)=\textbf{j}}\right) & ={{\operatorname{E}}}_x\left[{\unicode{x1D7D9}}_{(J_k,0\leq k\leq n-\ell+1) =(\textbf{j}',j)}{{\operatorname{E}}}_{X_{n-\ell+1}}\left(V(X_{\ell-1}){\unicode{x1D7D9}}_{J_p=j,\,0\leq p\leq \ell-1 }\right)\right] \\[5pt] & \leq \gamma_\emptyset^\ell {{\operatorname{E}}}_x\left[{\unicode{x1D7D9}}_{(J_k,0\leq k\leq n-\ell) =\textbf{j}'}{{\operatorname{E}}}_{X_{n-\ell}} \left(V(X_1)\right)\right] \\[5pt] &\leq A \gamma_\emptyset^\ell {{\operatorname{E}}}_x\left[{\unicode{x1D7D9}}_{(J_k,0\leq k\leq n-\ell) =\textbf{j}'} V(X_{n-\ell})\right] \end{align*}

where we used (5.3) in the second line and (5.4) in the last line. Observing that $n''(\textbf{j})=n''(\textbf{j}')$ , inequality (5.5) for $\textbf{j}$ follows from the induction assumption. We proceed similarly if $j\in J'$ using that $\ell=1$ and (5.4) and if $j\in J''$ using (5.2) and (5.4).

Let n be fixed and let $\mathcal{P}_n$ be the set of $\textbf{j}\in (\{\aleph\}\cup J'\cup J'')^{n+1}$ such that ${{\operatorname{P}}}_x((J_k,0\leq k\leq n)=\textbf{j})>0$ . For all $\textbf{j}\in\mathcal{P}_n$ , since the $C_j$ are communication classes for all $j\in J'\cup J''$ , they are visited at most once by $\textbf{j}$ , that is, there exist at most one $i\in\{0,\ldots,n-1\}$ such that $j_i=j$ and $j_{i+1}\neq j$ , and at most one $i'\in\{1,\ldots,n\}$ such that $j_{i'}\neq j$ and $j_{i'+1}=j$ , and in addition if $j\in J'$ , there exists at most one $i''\in\{0,\ldots,n\}$ such that $j_{i''}=j$ . This means that, for all $\textbf{j}\in\mathcal{P}_n$ , $n''(\textbf{j})\leq \# J''$ and $n_{\operatorname{trans}}(\textbf{j})\leq 2(\# J'+\# J'')$ , and thus it follows from (5.5) that

\[ {{\operatorname{E}}}_x\left(V(X_n){\unicode{x1D7D9}}_{(J_k,0\leq k\leq n)=\textbf{j}}\right)\leq C'\gamma_\emptyset^{n} \]

for a constant C’ independent of n.

Now

\[ \# \mathcal{P}_n\leq (\# J'+\# J'')!\,(n+1)^{2\# J'+2\# J''+1} \]

since, to construct a path $\textbf{j}\in\mathcal{P}_n$ , one must first choose an order of (possibly empty) visits of the classes $C_j$ for $j\in J'\cup J''$ and then one must choose the length of the (possibly empty) path in $\aleph$ before each of these visits, the length of this visit and the length of the (possibly empty) path in $\aleph$ after this visit, and they are all less than $n+1$ . Therefore, given any $\gamma'_\emptyset\in(\gamma_\emptyset,\bar{\theta})$ , there exists a constant C” independent of n such that

\[ {{\operatorname{E}}}_x\left(V(X_n){\unicode{x1D7D9}}_{n<T_{\{\partial\}\cup E_1\cup\ldots\cup E_k}}\right)=\sum_{\textbf{j}\in\mathcal{P}_n}{{\operatorname{E}}}_x\left(V(X_n){\unicode{x1D7D9}}_{(J_k,0\leq k\leq n)=\textbf{j}}\right) \leq C'' (\gamma'_\emptyset)^n. \]

Hence (B3) is proved and we deduce from Theorem 4.1 that X satisfies Assumption (A).

In order to prove the last statement of Theorem 5.1, we apply the above proof to the process X restricted to $D_{x_0}\cup\{\partial\}$ , where

\[D_{x_0}=\{x\in D\text{ such that }{{\operatorname{P}}}_{x_0}(\exists n\geq 0,\ X_n=x)>0\}.\]

We deduce from (4.6) in Theorem 4.1 that $\eta_S(x)>0$ in Assumption (A) for this process, so that Proposition 2.4(iii) entails the claim for X restricted to $D_{x_0}\cup\{\partial\}$ . But the definition of $D_{x_0}$ clearly implies that $T_{\{\partial\}\cup D\setminus D_{x_0}}=\tau_\partial$ ${{\operatorname{P}}}_{x_0}$ -almost surely, which concludes the proof.