$\displaystyle\geq\frac{1}{4}\Big{(}\int_{0}^{\sigma}\mathcal{L}W(X_{s}^{x})\mathbbm{1}_{W(X_{s}^{x})>N}ds\Big{)}^{2}-\frac{1}{2}\Big{(}\int_{0}^{\sigma}\mathcal{L}W(X_{s}^{x})\mathbbm{1}_{W(X_{s}^{x})\leq
N}ds\Big{)}^{2}$ $\displaystyle\qquad-(W(X_{\sigma}^{x})-W(x))^{2}.$

Observe $W(X_{s}^{x})\geq m>N$ for all $s\in(\tau_{x,T},\sigma)$, and let
$A\coloneqq\sup_{W(x)\leq N}|\mathcal{L}W(x)|$, which is finite since
$\mathcal{L}W$ is continuous (by 2.9) and $W$ is proper (by 3). Since
$\tau_{x,T}\leq T$,

$\Big{(}\int_{0}^{\sigma}\mathcal{L}W(X_{s}^{x})1_{W(X_{s}^{x})\leq
N}ds\Big{)}^{2}=\Big{(}\int_{0}^{\tau_{x,T}}\mathcal{L}W(X_{s}^{x})1_{W(X_{s}^{x})\leq
N}ds\Big{)}^{2}\leq A^{2}T^{2}.$

Similarly, since $W(x)>N$ implies $\mathcal{L}W(x)\leq-1$ (recall 6.12), we
obtain

$\frac{1}{4}\Big{(}\int_{0}^{\sigma}\mathcal{L}W(X_{s}^{x})1_{W(X_{s}^{x})>N}ds\Big{)}^{2}\geq\frac{1}{4}(\sigma-\tau_{x,T})^{2}\geq\frac{1}{8}\sigma^{2}-\frac{1}{4}T^{2}.$

Finally, since we assumed $W(x)\leq m$, then it follows from 6.7 that

$(W(X_{\sigma}^{x})-W(x))^{2}\leq 4m^{2}\,.$

Overall,

$(M_{\sigma}^{W}(x))^{2}\geq\frac{1}{8}\sigma^{2}-A^{\prime}$

where $A^{\prime}\coloneqq\frac{1}{4}T^{2}+\frac{1}{2}A^{2}T^{2}+4m^{2}$ is a
positive constant. A substitution into (6.9) gives us

$0\geq\mathbb{E}[M_{\sigma}]\geq\mathbb{E}\Big{[}\frac{1}{8}\sigma^{2}-A^{\prime}-K^{2}\sigma-
KU(x)\Big{]}.$

Rearranging and using (6.8) gives

$\mathbb{E}[\sigma^{2}]\leq 8A^{\prime}+8K^{2}(W(x)+(K+1)T)+8KU(x),$

which proves the claim since the right hand side is uniformly bounded for
$x\in\mathcal{M}_{+}\cap\\{W\leq m\\}$. ∎

######  Lemma 6.15.

For every $T>0$, $m>N$ (see 6.12)

$\sup_{x\in\mathcal{M}_{+}\cap\\{W\leq
m\\}}\mathbb{E}[(V(X_{\sigma_{x,T,m}}^{x})-V(x))^{2}]<\infty\,.$

###### Proof.

Fix $T>0$, $m>N$, and $x\in\mathcal{M}_{+}\cap\\{W\leq m\\}$. First note that
for $M^{V}_{t}$ as in (2.1) and $H$ as in 4 (iii) (so $H$ agrees with
$\mathcal{L}V$ on $\mathcal{M}_{+}$) we have

$\mathbb{E}[(V(X_{\sigma_{x,T,m}}^{x})-V(x))^{2}]\leq
2\mathbb{E}[(M_{\sigma_{x,T,m}}^{V}(x))^{2}]+2\mathbb{E}\Big{[}\Big{(}\int_{0}^{\sigma_{x,T,m}}H(X_{s}^{x})ds\Big{)}^{2}\Big{]}\,.$

Since $H$ vanishes over $W^{\prime}$ (see 2.19, 2.18), there is $A>0$ such
that $|H|\leq A+W^{\prime}$. Thus,

$\mathbb{E}\Big{[}\Big{(}\int_{0}^{\sigma_{x,T,m}}H(X_{s}^{x})ds\Big{)}^{2}\Big{]}\leq
2A^{2}\mathbb{E}[\sigma_{x,T,m}^{2}]+2\mathbb{E}\Big{[}\Big{(}\int_{0}^{\sigma_{x,T,m}}W^{\prime}(X_{s}^{x})ds\Big{)}^{2}\Big{]}.$

Since $0\leq W^{\prime}\leq K-\mathcal{L}W$ (see 3),

$\mathbb{E}\Big{[}\Big{(}\int_{0}^{\sigma_{x,T,m}}W^{\prime}(X_{s}^{x})ds\Big{)}^{2}\Big{]}\leq
2K^{2}\mathbb{E}[\sigma_{x,T,m}^{2}]+2\mathbb{E}\Big{[}\Big{(}\int_{0}^{\sigma_{x,T,m}}\mathcal{L}W(X_{s}^{x})ds\Big{)}^{2}\Big{]}.$

Furthermore, by (2.1)

$\mathbb{E}\Big{[}\Big{(}\int_{0}^{\sigma_{x,T,m}}\mathcal{L}W(X_{s}^{x})ds\Big{)}^{2}\Big{]}\leq
2\mathbb{E}[(M_{\sigma_{x,T,m}}^{W}(x))^{2}]+2\mathbb{E}[(W(X_{\sigma_{x,T,m}}^{x})-W(x))^{2}]\,.$

By 6.7 and our assumption that $W(x)\leq m$,

$\mathbb{E}[(W(X_{\sigma_{x,T,m}}^{x})-W(x))^{2}]\leq 4m^{2}\,.$

Combining all of these inequalities,

$\mathbb{E}[(V(X_{\sigma_{x,T,m}}^{x})-V(x))^{2}]\\\ \leq
2\mathbb{E}[(M_{\sigma_{x,T,m}}^{V}(x))^{2}]+(4A^{2}+8K^{2})\mathbb{E}[\sigma_{x,T,m}^{2}]+16\mathbb{E}[(M_{\sigma_{x,T,m}}^{W}(x))^{2}]+64m^{2}.$

We finish the proof by noting that by 6.14 it holds that

$\sup_{x\in\mathcal{M}_{+}\cap\\{W\leq
m\\}}\mathbb{E}[\sigma_{x,T,m}^{2}]<\infty$

and by 4.2

$\mathbb{E}[(M_{\sigma_{x,T,m}}^{V}(x))^{2}]+\mathbb{E}[(M_{\sigma_{x,T,m}}^{W}(x))^{2}]\leq
2K(U(x)+K\mathbb{E}[\sigma_{x,T,m}])\,,$

where the right hand side is uniformly bounded for
$x\in\mathcal{M}_{+}\cap\\{W\leq m\\}$ due to 6.14 and the continuity of $U$
on the compact set $\\{W\leq m\\}$. ∎

###  6.4. Proof of 6.1

In this section we prove 6.1 and we start with an auxiliary lemma.

######  Lemma 6.16.

For $x\in\mathcal{M}_{+}$, $T>0$, and $m>0$,

$\inf_{0\leq
t<\sigma_{x,T,m}}(V(X_{t}^{x})-V(x))\mathbbm{1}_{W(X_{t}^{x})<m}\geq-\alpha
T\text{ a.s.}$

###### Proof.

By 6.4

$\inf_{0\leq t<\tau_{x,T}}(V(X_{t}^{x})-V(x))\geq-\alpha T\text{ a.s.}$

The claim follows since $W(X_{t}^{x})\geq m$ for $\tau_{x,T}\leq
t<\sigma_{x,T,m}$ (see 6.6). ∎

Next, we prove 6.1:

###### Proof.

Let $N$ be as in 6.12 and fix $\delta>0,M>N$. By 6.13 there is
$\epsilon\in\big{(}0,\frac{1}{M}\big{)}$, $n\in\mathbb{N}$, and
$T_{1},\dots,T_{n}>0$ such that for all

(6.10) $z\in\mathcal{N}_{\epsilon,M}:=\mathcal{M}_{+}\cap\\{W\leq M\\}\cap
N_{\epsilon}(\mathcal{M}_{0})$

(see (6.6)) there is $1\leq i(z)\leq n$ such that

(6.11) $\mathbb{E}[V(X_{\sigma_{z,T_{i(z)},M}}^{z})-V(z)]\geq\frac{\alpha
T_{i(z)}}{8}\,.$

To simplify the notation, we set $\sigma_{z}:=\sigma_{z,T_{i(z)},M}$ for
$z\in\mathcal{N}_{\epsilon,M}$ and $\sigma_{z}=1$ for
$z\notin\mathcal{N}_{\epsilon,M}$.

For $y\in\mathcal{M}_{+}$, we define an increasing sequence of stopping times
$\tau_{n}$ as $\tau_{0}=0$, $\tau_{n+1}=\tau_{n}+\sigma_{X_{\tau_{n}}^{y}}$.
(To interpret $\tau_{n+1}=\tau_{n}+\sigma_{X_{\tau_{n}}^{y}}$ rigorously, we
note that there is a measurable function
$\psi:\mathbb{D}_{[0,\infty)}(\mathcal{M})\to[0,\infty)$ such that
$\sigma_{z}=\psi(X^{z}_{\cdot})$ for all $z\in\mathcal{M}$, and we set
$\tau_{n+1}=\tau_{n}+\psi(X^{y}_{\tau_{n}+\cdot})$.) It follows from the
strong Markov property for $X_{t}^{y}$ (5.2) that the discrete-time process
$Y_{n}^{y}\coloneqq X_{\tau_{n}}^{y}$ is a Markov chain on $\mathcal{M}_{+}$
with respect to the filtration
$\mathcal{G}_{n}\coloneqq\mathcal{F}_{\tau_{n}}$.

Next, we claim that $\tau_{n}\to\infty$ a.s. as $n\to\infty$. Indeed, let
$T^{\ast}$ be the random variable such that $\tau_{n}\uparrow T^{\ast}$.
Suppose $T^{\ast}<\infty$ with positive probability, and we restrict to such
event. To simplify the notation, we set $\tau_{z}\coloneqq\tau_{z,T_{i(z)}}$
for $z\in\mathcal{N}_{\epsilon,M}$ and $\tau_{z}=1$ for
$z\notin\mathcal{N}_{\epsilon,M}$. Then
$\sigma_{Y^{y}_{n}}=\tau_{n+1}-\tau_{n}\to 0$, and since
$\sigma_{z}\geq\tau_{z}$ (see 6.6), we obtain $\tau_{Y^{y}_{n}}\to 0$. Note
that a.s. $t\mapsto\int_{0}^{t}H(X_{s}^{y})ds$ is continuous (recall that
$t\mapsto X_{t}^{y}$ is cadlag a.s.) and $t\mapsto M^{V}_{t}(y)$ is cadlag
(see 2.10), so $t\mapsto V(X_{t}^{y})$ is also cadlag. In particular, we have
that the limit as $t\uparrow T^{\ast}$ of the aforementioned functions exist,
so $\Big{|}\int_{\tau_{n}}^{\tau_{n}+\tau_{Y^{y}_{n}}}H(X_{s}^{y})ds\Big{|}$
and $|V(X_{\tau_{n}+\tau_{Y^{y}_{n}}}^{y})-V(X_{\tau_{n}}^{y})|$ both $\to 0$
as $n\to\infty$. By 6.4, this implies that $\tau_{Y^{y}_{n}}$ is eventually
equal to $T_{i(Y^{y}_{n})}$, contradicting $\tau_{Y^{y}_{n}}\to 0$, so
$T^{\ast}=\infty$ a.s.

We also define the discrete-time process

$Z_{n}^{y}\coloneqq V(Y_{\tau^{\prime}\wedge
n}^{y})+b(n-\tau^{\prime})_{+}-V(y)\,,$

where $b\coloneqq\frac{\alpha}{8}\min_{1\leq i\leq n}T_{i}$, $a_{+}:=a\vee 0$
for any $a\in\mathbb{R}$, and

$\tau^{\prime}:=\tau^{\prime}_{y}\coloneqq\inf\\{n\geq 0\mid
Y_{n}^{y}\notin\mathcal{N}_{\epsilon,M}\\}\,,$

which is a $\mathcal{G}_{n}$-stopping time. Next, we show that $Z_{n}^{y}$
satisfies the assumptions of 4.12. By (6.11),

(6.12)
$\inf_{z\in\mathcal{N}_{\epsilon,M}}\mathbb{E}[V(X_{\sigma_{z}}^{z})-V(z)]\geq
b>0$

and by 6.15, there is some $B>0$ such that

(6.13)
$\sup_{z\in\mathcal{N}_{\epsilon,M}}\mathbb{E}[(V(X_{\sigma_{z}}^{z})-V(z))^{2}]\leq
B.$

Furthermore, by the definition of $Z_{n}$,

(6.14)
$Z_{n}^{y}-Z_{n-1}^{y}=(V(Y_{n}^{y})-V(Y_{n-1}^{y}))\mathbbm{1}_{\tau^{\prime}>n-1}+b\mathbbm{1}_{\tau^{\prime}\leq
n-1}\,.$

Let $f(y)=\mathbb{E}[(V(Y_{1}^{y})-V(y))^{2}]$. Using (6.14), recalling that
$\tau^{\prime}$ is a stopping time with respect to $\mathcal{G}_{n}$, and
using the Markov property of $Y_{n}^{y}$, we have

$\mathbb{E}[(Z_{n}^{y}-Z_{n-1}^{y})^{2}\mid\mathcal{G}_{n-1}]=\mathbbm{1}_{\tau^{\prime}\leq
n-1}b^{2}+\mathbbm{1}_{\tau^{\prime}>n-1}f(Y_{n-1}^{y})\,.$

On the event $\tau^{\prime}>n-1$, $Y_{n-1}^{y}\in\mathcal{N}_{\epsilon,M}$ and
(6.13) implies

$\mathbbm{1}_{\tau^{\prime}>n-1}f(Y_{n-1}^{y})\leq\mathbbm{1}_{\tau^{\prime}>n-1}\sup_{z\in\mathcal{N}_{\epsilon,M}}\mathbb{E}[(V(X_{\sigma_{z}}^{z})-V(z))^{2}]\leq\mathbbm{1}_{\tau^{\prime}>n-1}B\,.$

Consequently,

$\mathbb{E}[(Z_{n}^{y}-Z_{n-1}^{y})^{2}\mid\mathcal{G}_{n-1}]\leq b^{2}+B.$

Similarly, by (6.12)

$\displaystyle\mathbb{E}[Z_{n}^{y}-Z_{n-1}^{y}\mid\mathcal{G}_{n-1}]$
$\displaystyle=\mathbbm{1}_{\tau^{\prime}\leq
n-1}b+\mathbbm{1}_{\tau^{\prime}>n-1}\mathbb{E}[V(Y_{n}^{y})-V(Y_{n-1}^{y})\mid\mathcal{G}_{n-1}]$
$\displaystyle\geq\mathbbm{1}_{\tau^{\prime}\leq
n-1}b+\mathbbm{1}_{\tau^{\prime}>n-1}\inf_{z\in\mathcal{N}_{\epsilon,M}}\mathbb{E}[V(X_{\sigma_{z}}^{z})-V(z)]$
$\displaystyle\geq b\,.$

Then by 4.12, there is some $C>0$ such that for all $y\in\mathcal{M}_{+}$

(6.15) $\mathbb{P}\Big{(}Z_{n}^{y}\geq\frac{b}{2}n-C\text{ for all }n\geq
0\Big{)}>1-\delta\,,$

where we stress that $C$ is independent of $y$. Also, by 4 (i), there is $A>0$
such that

(6.16) $V(x)\geq A\qquad\textrm{implies }\quad x\in
N_{\epsilon}(\mathcal{M}_{0})\,.$

We set $D\coloneqq A+C+\alpha\max_{1\leq i\leq n}T_{i}$ and fix
$y\in\mathcal{M}_{+}\cap\\{V>D\\}\cap\\{W\leq M\\}$. Denote $\Omega^{*}$ the
event that $Z_{n}^{y}\geq\frac{b}{2}n-C$ for all $n\geq 0$. Then (6.15)
implies that $\mathbb{P}(\Omega^{*})>1-\delta$, so we conclude the proof by
verifying the following claims:

(6.17) $\displaystyle\limsup_{t\to\infty,W(X_{t}^{y})\leq
M}V(X_{t}^{y})=\infty\textrm{ a.s. on }\Omega^{*}$ (6.18)
$\displaystyle\sup_{t\geq
0}d(X_{t}^{y},\mathcal{M}_{0})\mathbbm{1}_{W(X_{t}^{y})<M}\leq\frac{1}{M}\textrm{
a.s. on }\Omega^{*}$

First, we note that by 6.7 and the definition of $\mathcal{N}_{\epsilon,M}$ in
(6.10) we have that

$\tau^{\prime}=\inf\\{n\geq 0\mid Y_{n}^{y}\notin
N_{\epsilon}(\mathcal{M}_{0})\\}\,,$

so by (6.16) we obtain $V(Y_{\tau^{\prime}}^{y})<A$ a.s. on the event
$\\{\tau^{\prime}<\infty\\}$. In particular, on the event
$\\{\tau^{\prime}<\infty\\}$ we have a.s. that

$Z_{\tau^{\prime}}^{y}=V(Y_{\tau^{\prime}}^{y})-V(y)\leq
A-D=-C-\alpha\max_{1\leq i\leq n}T_{i}<-C\,.$

Since $Z_{n}^{y}\geq\frac{b}{2}n-C\geq-C$ for all $n\geq 0$ on $\Omega^{*}$,

(6.19) $\tau^{\prime}=\infty\quad{and}\quad
Z_{n}^{y}=V(Y_{n}^{y})-V(y)\textrm{ for all }n\geq 0\text{ a.s. on the event
}\Omega^{*}\,.$

Consequently, on $\Omega^{*}$

(6.20)
$V(Y_{n}^{y})=Z_{n}^{y}+V(y)>\frac{b}{2}n-C+D=\frac{b}{2}n+A+\alpha\max_{1\leq
i\leq n}T_{i}$

so that with $\mathcal{U}\coloneqq\mathcal{M}_{+}\cap\\{W\leq
M\\}\cap\\{V>A+\alpha\max_{1\leq i\leq
n}T_{i}\\}\subset\mathcal{N}_{\epsilon,M}$ we have

(6.21) $Y_{n}^{y}\in\mathcal{U}\textrm{ for all }n\geq 0\text{ a.s. on the
event }\Omega^{*}\,.$

By passing $n\to\infty$ in (6.20), noting that $W(Y_{n}^{y})\leq M$ by (6.21),
and recalling that $\tau_{n}\to\infty$ a.s., we obtain (6.17).

By 6.16, if $z\in\mathcal{U}$, then almost surely

$\inf_{0\leq
t<\sigma_{z}}(V(X_{t}^{z})-A)\mathbbm{1}_{W(X_{t}^{z})<M}\geq\inf_{0\leq
t<\sigma_{z}}(V(X_{t}^{z})-V(z)+\alpha\max_{1\leq i\leq
n}T_{i})\mathbbm{1}_{W(X_{t}^{z})<M}\geq 0$

so that by (6.16) we have

$\sup_{0\leq
t<\sigma_{z}}d(X_{t}^{z},\mathcal{M}_{0})\mathbbm{1}_{W(X_{t}^{z})<M}\leq\epsilon<\frac{1}{M}\text{
a.s.}$

In other words, defining $\tilde{\phi}:\mathcal{M}\to[0,1]$ by

$\tilde{\phi}(z)\coloneqq\mathbb{P}\Big{(}\sup_{0\leq
t<\sigma_{z}}d(X_{t}^{z},\mathcal{M}_{0})\mathbbm{1}_{W(X_{t}^{z})<M}<\frac{1}{M}\Big{)}\,,$

we have $\tilde{\phi}(z)=1$ for all $z\in\mathcal{U}$. Recall that
$Y_{n}^{y}=X_{\tau_{n}}^{y}$, so applying the strong Markov property (5.2) we
have

$\displaystyle\mathbb{P}\Big{(}\sup_{\tau_{n}\leq
t\leq\tau_{n+1}}d(X_{t}^{y},\mathcal{M}_{0})\mathbbm{1}_{W(X_{t}^{y})<M}<\frac{1}{M}\text{
and }Y_{n}^{y}\in\mathcal{U}\Big{)}$
$\displaystyle=\mathbb{E}[\tilde{\phi}(Y_{n}^{y})\mathbbm{1}_{Y_{n}^{y}\in\mathcal{U}}]$
$\displaystyle=\mathbb{P}(Y_{n}^{y}\in\mathcal{U})\,,$

or in other words

(6.22) $Y_{n}^{y}\in\mathcal{U}\qquad\textrm{implies }\quad\sup_{\tau_{n}\leq
t\leq\tau_{n+1}}d(X_{t}^{y},\mathcal{M}_{0})\mathbbm{1}_{W(X_{t}^{y})<M}<\frac{1}{M}\text{
a.s.}$

Thus, (6.18) follows from (6.21) and (6.22) by recalling that
$\tau_{n}\to\infty$ a.s. and writing

$\sup_{t\geq
0}d(X_{t}^{y},\mathcal{M}_{0})\mathbbm{1}_{W(X_{t}^{y})<M}=\sup_{n\geq
0}\sup_{\tau_{n}\leq
t\leq\tau_{n+1}}d(X_{t}^{y},\mathcal{M}_{0})\mathbbm{1}_{W(X_{t}^{y})<M}\,.$

∎

###  6.5. Proof of 3.2

In this section we provide the proof of 3.2:

###### Proof.

Since by enlarging $M$ we make the statement stronger, we can without loss of
generality assume $M>N$, where $N$ is as in 6.1. By 6.1, for all $n\geq 1$
there is a $D_{n}>0$ such that for all $y\in\mathcal{M}_{+}\cap\\{V\geq
D_{n}\\}\cap\\{W\leq M+n\\}$,

(6.23) $\mathbb{P}\Big{(}\sup_{t\geq
0}d(X_{t}^{y},\mathcal{M}_{0})\mathbbm{1}_{W(X_{t}^{y})<M+n}\leq\frac{1}{M+n}\\\
\text{ and }\limsup_{t\to\infty,W(X_{t}^{y})\leq
M+n}V(X_{t}^{y})=\infty\Big{)}\geq 1-\frac{\delta}{n},$

and we may without loss of generality assume that $D_{n}\uparrow\infty$. Fix
$y\in\mathcal{M}_{+}\cap\\{V\geq D_{1}\\}\cap\\{W\leq M\\}$ and define

$\tau_{n}:=\inf\\{t\geq 0\mid V(X_{t}^{y})>D_{n+1},W(X_{t}^{y})<M+2\\},$

and we claim that a.s. $\tau_{n}\uparrow\infty$. Indeed, by 2.10 $t\mapsto
M^{V}_{t}(y)$ is cadlag and so by the continuity of
$t\mapsto\int_{0}^{t}H(X_{s}^{y})ds$ it follows that $t\mapsto V(X_{t}^{y})$
is cadlag. However, on the event $\tau_{n}\to T^{*}<\infty$, have
$V(X_{\tau_{n}}^{y})\geq D_{n+1}\to\infty$, and therefore such event has zero
probability.

Next, let

$A_{n}=\\{\tau_{n}<\infty\\}\cap\Big{\\{}\limsup_{t\to\infty}d(X_{t}^{y},\mathcal{M}_{0})\mathbbm{1}_{W(X_{t}^{y})<M+n+1}\leq\frac{1}{M+n+1}\Big{\\}},$

a decreasing sequence of events. Define
$\tilde{\phi}:\mathcal{M}_{+}\to[0,\infty)$ by

$\tilde{\phi}(z)=\mathbb{P}\Big{(}\limsup_{t\to\infty}d(X_{t}^{z},\mathcal{M}_{0})\mathbbm{1}_{W(X_{t}^{z})<M+n+1}\leq\frac{1}{M+n+1}\Big{)}$

and by (6.23), $\tilde{\phi}(z)\geq 1-\frac{\delta}{n+1}$ for
$z\in\mathcal{M}_{+}\cap\\{V\geq D_{n+1}\\}\cap\\{W\leq M+n+1\\}$. In
particular, by the definition of $\tau_{n}$,

$\mathbbm{1}_{\tau_{n}<\infty}\tilde{\phi}(X_{\tau_{n}}^{y})\geq\mathbbm{1}_{\tau_{n}<\infty}\Big{(}1-\frac{\delta}{n+1}\Big{)}\,,$

where we used that $X_{\tau_{n}}^{y}\in\mathcal{M}_{+}\cap\\{V\geq
D_{n+1}\\}\cap\\{W\leq M+2\\}$ a.s. by the right continuity of $t\mapsto
X_{t}^{y}$ and the invariance of $\mathcal{M}_{+}$ (see 1). By the strong
Markov property (5.2),

$\mathbb{P}(A_{n})=\mathbb{E}[\mathbbm{1}_{\tau_{n}<\infty}\tilde{\phi}(X_{\tau_{n}}^{y})]\geq\Big{(}1-\frac{\delta}{n+1}\Big{)}\mathbb{P}(\tau_{n}<\infty).$

Since $\tau_{n}<\infty$ on the event that
$\limsup_{t\to\infty,W(X_{t}^{y})\leq M+1}V(X_{t}^{y})=\infty$, then for each
$n\geq 1$ we have by (6.23) that

$\mathbb{P}(\tau_{n}<\infty)\geq 1-\delta\,.$

Thus,

$\mathbb{P}\Big{(}\bigcap_{n=1}^{\infty}A_{n}\Big{)}=\lim_{n\to\infty}\mathbb{P}(A_{n})\geq
1-\delta.$

Since

$\bigcap_{n=1}^{\infty}A_{n}\subset\big{\\{}\lim_{M\to\infty}\limsup_{t\to\infty}d(X_{t}^{y},\mathcal{M}_{0})\mathbbm{1}_{W(X_{t}^{y})<M}=0\big{\\}},$

by 4.10

$\mathbb{P}\Big{(}\liminf_{t\to\infty}\frac{V(X_{t}^{y})}{t}\geq\alpha\Big{)}\geq
1-\delta\,,$

proving the claim. ∎

##  7\. Global Extinction

In this section we prove technical facts related to accessibility (see 3.4)
and give a proof of 3.5. In the entire section we fix a Markov quadruple
$(\mathcal{M},\mathcal{M}_{0},\mathcal{M}_{+},\\{X_{t}^{x}\\}_{x\in\mathcal{M},t\geq
0})$ satisfying 1–5.

######  Lemma 7.1.

For $x\in\mathcal{M}$ and $\mathcal{U}\subset\mathcal{M}$ open the following
are equivalent:

  1. (1)

$\int_{0}^{\infty}e^{-t}\mathbb{P}(X_{t}^{x}\in\mathcal{U})dt>0$.

  2. (2)

There exists $t\geq 0$ such that $\mathbb{P}(X_{t}^{x}\in\mathcal{U})>0$.

  3. (3)

$\mathbb{P}(\exists t\geq 0\text{ such that }X_{t}^{x}\in\mathcal{U})>0$.

###### Proof.

The statements (1) implies (2) and (2) implies (3) are straightforward. To
prove that (3) implies (1), denote $f(t)=\mathbbm{1}_{\mathcal{U}}(X_{t}^{x})$
so that (3) is equivalent to $\mathbb{P}(\exists t\geq 0\text{ such that
}f(t)>0)>0$. By the right continuity of $t\mapsto X_{t}^{x}$ and openness of
$\mathcal{U}$, this is equivalent to the existence of random $t_{2}>t_{1}\geq
0$ such that $f(t)>0$ for $t\in[t_{1},t_{2})$. In particular, the condition
(3) implies $\mathbb{P}(\int_{0}^{\infty}e^{-t}f(t)dt>0)>0$, which by
$f(t)\geq 0$, Tonelli’s theorem, and
$\mathbb{E}[f(t)]=\mathbb{P}(X_{t}^{x}\in\mathcal{U})$ is equivalent to (1). ∎

######  Lemma 7.2.

Suppose every point $x\in\mathcal{M}_{+}$ satisfies (3.1). Then there is some
$N>0$ such that for all $x\in\mathcal{M}_{+},D>0$ it holds that
$\\{V>D\\}\cap\\{W<N\\}$ is accessible from $x$.

###### Proof.

Let $N$ be as in 6.8 and fix $x\in\mathcal{M}_{+}$. Let $M$ satisfy (3.1) and
let $D>0$ be arbitrary. Set $T=3M$ and choose any $y\in\\{W\leq
M\\}\cap\mathcal{M}_{+}$. Using the notation of 6.8 we have

$\mathbb{P}(\eta_{N}(y)>T)\leq\frac{\mathbb{E}[\eta_{N}(y)]}{T}\leq\frac{W(y)}{T}\leq\frac{1}{3}\,.$

With $M^{V}_{t}(y)$ as defined in (2.1) and $H$ as in 4 so that $H$ agrees
with $\mathcal{L}V$ on $\mathcal{M}_{+}$, we have by Doob’s inequality, 4.2,
and (4.4) that

$\displaystyle\mathbb{E}[\sup_{t\leq T}|V(X_{t}^{y})-V(y)|]$
$\displaystyle\leq\mathbb{E}\Big{[}\sup_{t\leq
T}|M^{V}_{t}(y)|\Big{]}+\mathbb{E}\Big{[}\int_{0}^{T}|H(X_{s}^{y})|ds\Big{]}$
$\displaystyle\leq
1+4\mathbb{E}\Big{[}|M^{V}_{T}(y)|^{2}\Big{]}+\mathbb{E}\Big{[}\int_{0}^{T}|H(X_{s}^{y})|ds\Big{]}$
$\displaystyle\leq 1+4K(U(y)+KT)+bW(y)+(A+bK)T$ $\displaystyle\leq C\,,$

where $A,b$ are as in the proof of 4.3 and $C>0$ is a constant depending only
on $M$. Thus, for any $y\in\\{W\leq M\\}\cap\mathcal{M}_{+}$ we have by
Chebyshev inequality that

$\mathbb{P}(\exists t\leq T\text{ such that }|V(X_{t}^{y})-V(y)|\leq 3C\text{
and }W(X_{t}^{y})\leq N)\\\ \geq\mathbb{P}(\sup_{t\leq
T}|V(X_{t}^{y})-V(y)|\leq 3C\text{ and }\eta_{N}(y)\leq T)\geq\frac{1}{3}\,.$

Consequently, the strong Markov property (5.2), applied to the stopping time
$\tau=\inf\\{t:V(X_{t}^{x})>D+3C\text{ and }W(X_{t}^{x})<M\\}$ implies

$\mathbb{P}(\exists t\geq 0\text{ such that }V(X_{t}^{x})>D\text{ and
}W(X_{t}^{x})<N+1)\\\ \geq\frac{1}{3}\mathbb{P}(\exists t\geq 0\text{ such
that }V(X_{t}^{x})>D+3C\text{ and }W(X_{t}^{x})<M)\,,$

which is strictly positive by (3.1) (using condition 3 of 3.4). Since
$x\in\mathcal{M}_{+},D>0$ were arbitrary, the claim is proven. ∎

Now we prove 3.5:

###### Proof.

Let $N$ be as in 7.2 and fix $x\in\mathcal{M}_{+}$. By 3.2, there is $D>0$
such that for all $y\in\Psi_{D,N}:=\mathcal{M}_{+}\cap\\{V>D\\}\cap\\{W<N\\}$
we have

(7.1)
$\mathbb{P}\Big{(}\liminf_{t\to\infty}\frac{V(X_{t}^{y})}{t}\geq\alpha\Big{)}\geq\frac{1}{2}\,.$

Let $\tau_{0}^{\prime}=0$,
$\tau_{n}^{\prime}=\inf\\{t\geq\tau_{n-1}^{\prime}+1\mid
X^{x}_{t}\in\Psi_{D,N}\\}$, and
$\Omega^{*}\coloneqq\Big{\\{}\liminf_{t\to\infty}\frac{V(X_{t}^{x})}{t}\geq\alpha\Big{\\}}$.
Then the strong Markov property (5.2) and (7.1) imply that
$\mathbb{P}(\Omega^{*}\mid\mathcal{F}_{\tau_{n}^{\prime}})\geq 1/2$ on the
event $\\{\tau_{n}^{\prime}<\infty\\}$, where $n\geq 1$. By Levy’s 0-1 law,
$\lim_{n\to\infty}\mathbb{P}(\Omega^{*}\mid\mathcal{F}_{\tau_{n}^{\prime}})=\mathbbm{1}_{\Omega^{*}}$
a.s. Thus, (up to a set of measure zero) if $\omega\in\Omega$ is such that
$\tau_{n}^{\prime}(\omega)<\infty$ for all $n\geq 1$, then
$\mathbb{P}(\Omega^{*}\mid\mathcal{F}_{\tau_{n}^{\prime}})(\omega)\geq 1/2$
for all $n\geq 1$, which implies that $\mathbbm{1}_{\Omega^{*}}(\omega)=1$. To
summarize, almost surely we have that

(7.2) $\\{X_{t}^{x}\in\Psi_{D,N}\text{
i.o.}\\}\subset\Big{\\{}\liminf_{t\to\infty}\frac{V(X_{t}^{x})}{t}\geq\alpha\Big{\\}}=\Omega^{*}\,,$

where by “i.o.” we mean there are times $t_{n}\uparrow\infty$ such that
$X^{x}_{t_{n}}\in\Psi_{D,N}$. Thus, it suffices to show that almost surely

(7.3) $(\Omega^{*})^{c}\subset\\{X_{t}^{x}\in\Psi_{D,N}\text{ i.o.}\\}\,,$

which combined with (7.2) implies
$\mathbb{P}((\Omega^{*})^{c})=\mathbb{P}((\Omega^{*})^{c}\cap\Omega^{*})=0$.

In the rest of the proof we show (7.3). Let $K_{n}$ be as in 2.14 applied to
$A=\mathcal{M}_{+}$. Let $\tau_{n}:=\sum_{i=1}^{n}\sigma_{i}$, where
$\sigma_{i}$ are iid exponentially distributed (with parameter 1) random
variables independent of $X_{t}^{x}$, and $Y_{n}\coloneqq X_{\tau_{n}}^{x}$.
We claim that almost surely

(7.4) $(\Omega^{*})^{c}\subset\bigcup_{m=1}^{\infty}\\{Y_{n}\in K_{m}\text{
i.o.}\\}\,.$

By 4.9 and 4.8, almost surely on $(\Omega^{*})^{c}$ we may assume there are
$t_{n}\to\infty,\mu\in P(\mathcal{M})$ such that $\mu_{t_{n}}^{x}\to\mu$ and
$\mu(\mathcal{M}_{0})<1$ (where $\mu_{t}^{x}$ is as in 4.4). Then, by
Prohorov’s Theorem and $\mathcal{M}=\mathcal{M}_{+}\cup\mathcal{M}_{0}$, it
follows that there is some $m$ such that $\inf_{n}\mu_{t_{n}}^{x}(K_{m})>0$.
With the notation $B=\\{t\in[0,\infty)\mid X_{t}^{x}\in K_{m}\\}$, the
previous statement implies that

$\liminf_{n\to\infty}\frac{1}{t_{n}}\int_{0}^{t_{n}}\mathbbm{1}_{B}(t)\,dt=\liminf_{n\to\infty}\mu_{t_{n}}^{x}(K_{m})>0\,.$

In particular, $m(B)=\infty$, where $m(\cdot)$ denotes the Lebesgue measure on
$[0,\infty)$. By the definition of Poisson process we have for any $t>0$ that
$|\\{n:\tau_{n}\in B\cap[0,t]\\}|$ is a Poisson random variable with parameter
$m(B\cap[0,t])$, where $|S|$ denotes the cardinality of the set $S$. Thus, for
any $N>0$ we have $\lim_{t\to\infty}\mathbb{P}[|\\{n:\tau_{n}\in
B\cap[0,t]\\}|\leq N]=0$. It follows that $\mathbb{P}(|\\{n:\tau_{n}\in
B\\}|=\infty)=1$, and since $\tau_{n}\in B$ is equivalent to $Y_{n}\in K_{m}$,
this shows (7.4) ($Y_{n}\in K_{m}$ i.o.)

Finally, we prove that (7.4) implies (7.3). Fix $m$ and note that by Fatou’s
Lemma, 2, and Portmanteau’s Theorem, the function
$F:\mathcal{M}_{+}\to\mathbb{R}$ defined as

$F(y)\coloneqq\int_{0}^{\infty}e^{-t}\mathbb{P}(V(X_{t}^{y})>D\text{ and
}W(X_{t}^{y})<N)\,dt=\int_{0}^{\infty}e^{-t}\mathbb{P}(X_{t}^{y}\in\Psi_{D,N})\,dt$

is lower semicontinuous, where the equality follows from the invariance of
$\mathcal{M}_{+}$ (see 1). By 7.2 (using condition (1) of 3.4), we have
$F(y)>0$ for all $y\in K_{m}$ and thus $\inf_{y\in K_{m}}F(y)>0$. By strong
Markov property (5.2) and exponential distribution of $\sigma_{n+1}$ (see
definition of $\tau_{n}$), it follows that for each $n$

$\mathbb{P}(Y_{n+1}\in\Psi_{D,N}|Y_{n})=F(Y_{n})$

and so, similarly to the above, we have by Levy’s 0-1 law that almost surely

$\\{Y_{n}\in K_{m}\text{ i.o.}\\}\subset\\{Y_{n}\in\Psi_{D,N}\text{
i.o.}\\}\,,$

which by (7.4) shows (7.3), completing the proof. ∎

##  8\. Generator Details

In this section, we first expand upon the exposition given in Section 2.2
regarding the generator $\mathcal{L}$, the Carre Du Champ $\Gamma$, and the
domain $\mathcal{D}(\mathcal{M})$. Then we show that if an unbounded function
can be approximated in a certain sense by functions in
$\mathcal{D}(\mathcal{M})$ (resp. $\mathcal{D}_{2}(\mathcal{M})$), then it
lies in the extended domain $\mathcal{D}^{ext}_{+}(\mathcal{M})$ (resp.
$\mathcal{D}^{ext}_{2}(\mathcal{M})$). Finally, under general assumptions on
$\mathcal{M}$ and the action of the generator $\mathcal{L}$ on the domain
$\mathcal{D}(\mathcal{M})$, we show that any smooth enough function that does
not grow too quickly belongs to the extended domain. The proofs are mostly
routine and can be skipped on the first reading.

###  8.1. Generator for Bounded Functions

In this subsection we fix a Feller process
$\\{X_{t}^{x}\\}_{x\in\mathcal{M},t\geq 0}$ (see 2) on a locally compact
Polish space $\mathcal{M}$ with the Markov semigroup $\mathcal{P}_{s}$ (see •
‣ Section 2). Recall 2.7 and note that since $f\mapsto\mathcal{P}_{t}f$ is
linear, it is standard to check that $\mathcal{D}(\mathcal{M})$ is a subspace
of $C_{b}(\mathcal{M})$ and $\mathcal{L}:\mathcal{D}(\mathcal{M})\to
C_{b}(\mathcal{M})$ is a linear map.

######  Lemma 8.1.

For $f\in\mathcal{D}(\mathcal{M})$ and $t\geq 0$,
$\mathcal{P}_{t}f\in\mathcal{D}(\mathcal{M})$ and
$\mathcal{L}\mathcal{P}_{t}f=\mathcal{P}_{t}\mathcal{L}f$.

###### Proof.

By 2.7 (i), the definition of $\mathcal{P}_{t}$ (see • ‣ Section 2), 2.7 (iii)
and the Dominated convergence theorem, we have for any $t\geq 0$ that

$\mathcal{L}\mathcal{P}_{t}f=\lim_{s\downarrow
0}\frac{\mathcal{P}_{s}\mathcal{P}_{t}f-\mathcal{P}_{t}f}{s}=\lim_{s\downarrow
0}\mathcal{P}_{t}\Big{(}\frac{\mathcal{P}_{s}f-f}{s}\Big{)}=\mathcal{P}_{t}\Big{(}\lim_{s\downarrow
0}\frac{\mathcal{P}_{s}f-f}{s}\Big{)}=\mathcal{P}_{t}\mathcal{L}f,$

where all limits are pointwise. Thus, 2.7 (i) holds with $f$ replaced by
$\mathcal{P}_{t}f$. Next, since $\mathcal{L}f\in C_{b}(\mathcal{M})$ by 2.7
(ii), then $\mathcal{L}\mathcal{P}_{t}f=\mathcal{P}_{t}\mathcal{L}f\in
C_{b}(\mathcal{M})$ by 2 and so 2.7 (ii) holds with $f$ replaced by
$\mathcal{P}_{t}f$. Finally, since $\mathcal{P}_{t}$ is a contraction
($\|\mathcal{P}_{t}f\|\leq\|f\|$), then 2.7 (iii) implies

$\sup_{s>0}\Big{\|}\frac{\mathcal{P}_{s}\mathcal{P}_{t}f-\mathcal{P}_{t}f}{s}\Big{\|}=\sup_{s>0}\Big{\|}\mathcal{P}_{t}\Big{(}\frac{\mathcal{P}_{s}f-f}{s}\Big{)}\Big{\|}\leq\sup_{s>0}\Big{\|}\frac{\mathcal{P}_{s}f-f}{s}\Big{\|}<\infty$

and 2.7 (iii) with $f$ replaced by $\mathcal{P}_{t}f$ follows. ∎

Before proceeding we need a preliminary lemma, called quasi-left continuity,
which allows us to deduce that $t\mapsto P_{t}f(x)$ is continuous in $t$ (we
only assumed it was right-continuous in 2):

######  Lemma 8.2.

If $x\in\mathcal{M}$ and $\tau$ is a predictable stopping time, meaning there
are stopping times $\tau_{n}$ such that $\tau_{n}\uparrow\tau$ and
$\tau_{n}<\tau$ almost surely on $\\{\tau>0\\}$ for all $n$, then almost
surely on $\\{\tau<\infty\\}$ we have
$X_{\tau}^{x}=X_{\tau-}^{x}\coloneqq\lim_{t\uparrow\tau}X_{t}^{x}$. In
particular, if $f\in C_{b}(\mathcal{M})$ then $t\mapsto P_{t}f(x)$ is
continuous in $t$.

###### Proof.

We follow [Kal21, Proposition 17.29]. By [Kal21, Lemma 10.1v], $\tau\wedge n$
is predictable for all $n>0$, and since $\\{\tau<\infty\\}=\cup_{n\geq
1}\\{\tau\leq n\\}$ we may without loss of generality assume $\tau$ is finite.
Since $\mathcal{M}$ is Polish, there is a countable collection of bounded
continuous functions which separate the points, so it is enough to show that
for all $f\in C_{b}(\mathcal{M})$,
$\mathbb{E}[(f(X_{\tau-}^{x})-f(X_{\tau}^{x}))^{2}]=0$. Since $f$ is
continuous and $X_{t}^{x}$ is cadlag, then

$(f(X_{\tau-}^{x})-f(X_{\tau}^{x}))^{2}=\lim_{h\downarrow
0}\lim_{n\to\infty}(f(X_{\tau_{n}}^{x})-f(X_{\tau_{n}+h}^{x}))^{2}\text{
a.s.}$

Thus, (5.1) and the Dominated Convergence Theorem combined with 2 imply

$\displaystyle\mathbb{E}[(f(X_{\tau-}^{x})$
$\displaystyle-f(X_{\tau}^{x}))^{2}]=\lim_{h\downarrow
0}\lim_{n\to\infty}\mathbb{E}\big{[}\mathbb{E}[(f(X_{\tau_{n}}^{x})-f(X_{\tau_{n}+h}^{x}))^{2}\mid\mathcal{F}_{\tau_{n}}]\big{]}$
$\displaystyle=\lim_{h\downarrow
0}\lim_{n\to\infty}\mathbb{E}[f^{2}(X_{\tau_{n}}^{x})+P_{h}f^{2}(X_{\tau_{n}}^{x})-2f(X_{\tau_{n}}^{x})P_{h}f(X_{\tau_{n}}^{x})]$
$\displaystyle=\lim_{h\downarrow
0}\mathbb{E}[f^{2}(X_{\tau-}^{x})+P_{h}f^{2}(X_{\tau-}^{x})-2f(X_{\tau-}^{x})P_{h}f(X_{\tau-}^{x})]$
$\displaystyle=0\,,$

as desired. The remaining claim follows by taking $\tau=t$ and applying
Dominated Convergence Theorem to deduce left-continuity of $t\mapsto
P_{t}f(x)$, and then noting that we assumed right-continuity in 2. ∎

######  Corollary 8.3.

For all $f\in\mathcal{D}(\mathcal{M})$, $x\in\mathcal{M}$ the process
$M_{t}^{f}(x)$ given by (2.1) is a martingale.

###### Proof.

For any $t\geq 0$, $M_{t}^{f}(x)$ is $\mathcal{F}_{t}$ adapted, and in
addition $M_{t}^{f}(x)$ is integrable ($\mathbb{E}[|M_{t}^{f}(x)|]<\infty$)
since $f,\mathcal{L}f\in C_{b}(M)$. By Fubini’s theorem for conditional
expectation and the Markov property, for $t,s\geq 0$

$\displaystyle\mathbb{E}[M_{t+s}^{f}(x)-M_{s}^{f}(x)\mid\mathcal{F}_{s}]$
$\displaystyle=\mathbb{E}\Big{[}f(X^{x}_{t+s})-f(X_{s}^{x})-\int_{s}^{t+s}\mathcal{L}f(X_{a}^{x})da\Big{|}\mathcal{F}_{s}\Big{]}.$
$\displaystyle=\mathcal{P}_{t}f(X_{s}^{x})-f(X_{s}^{x})-\int_{0}^{t}\mathcal{P}_{a}\mathcal{L}f(X_{s}^{x})da.$

Thus, to show that $M_{t}^{f}(x)$ is a martingale it suffices to show that
$\mathcal{P}_{t}f=f+\int_{0}^{t}\mathcal{P}_{s}\mathcal{L}fds$ for all $t>0$.
By 8.1, for fixed $x\in M$ the function $g(t)=\mathcal{P}_{t}f(x)$ has right
derivative equal to $t\mapsto\mathcal{P}_{t}\mathcal{L}f(x)$, and both $g$ and
its right derivative are continuous by 8.2. Thus, by [Bru78, Theorem 1.3] $g$
is differentiable and $g^{\prime}(t)=\mathcal{P}_{t}\mathcal{L}f(x)$. By
fundamental theorem of calculus, $g(t)=g(0)+\int_{0}^{t}g^{\prime}(s)ds$,
which proves the claim. ∎

Next, we focus on the quadratic variation of $M_{t}^{f}(x)$ defined as
follows:

######  Definition 8.4.

Given a square integrable martingale $M_{t}$, we say that $\langle
M\rangle_{t}$ is the (predictable) quadratic variation of $M_{t}$ if $\langle
M\rangle_{t}$ is the unique integrable, predictable, increasing process such
that $M_{t}^{2}-\langle M\rangle_{t}$ is a martingale. The existence and
uniqueness of $\langle M\rangle_{t}$ is guaranteed by Doob-Meyer decomposition
theorem (see [Kal21, Theorem 10.5]).

######  Remark 8.5.

Note that for $f\in\mathcal{D}(\mathcal{M})$ (see 2.7) the martingale
$M_{t}^{f}(x)$ defined in (2.1) is bounded and thus square integrable, so 8.4
applies.

We also use the following definition.

######  Definition 8.6.

Let $\mathcal{D}_{2}(\mathcal{M})$ be the set of all
$f\in\mathcal{D}(\mathcal{M})$ such that $f^{2}\in\mathcal{D}(\mathcal{M})$.
For $f\in\mathcal{D}_{2}(\mathcal{M})$, let $\Gamma
f\coloneqq\mathcal{L}(f^{2})-2f\mathcal{L}f$.

######  Remark 8.7.

Note that $\Gamma f\geq 0$ since $\Gamma f=\lim_{s\downarrow
0}\frac{\mathcal{P}_{s}f^{2}-(\mathcal{P}_{s}f)^{2}}{s}$. Indeed,

$\displaystyle\mathcal{L}(f^{2})-2f\mathcal{L}f$
$\displaystyle=\lim_{s\downarrow
0}\frac{\mathcal{P}_{s}f^{2}-f^{2}}{s}-2f\frac{\mathcal{P}_{s}f-f}{s}=\lim_{s\downarrow
0}\frac{\mathcal{P}_{s}f^{2}-2f\mathcal{P}_{s}f+f^{2}}{s}$
$\displaystyle=\lim_{s\downarrow
0}\frac{\mathcal{P}_{s}f^{2}-(\mathcal{P}_{s}f)^{2}}{s}+\lim_{s\downarrow
0}\frac{(\mathcal{P}_{s}f-f)^{2}}{s}$ $\displaystyle=\lim_{s\downarrow
0}\frac{\mathcal{P}_{s}f^{2}-(\mathcal{P}_{s}f)^{2}}{s}+0\times(\mathcal{L}f)^{2}\,.$

######  Lemma 8.8.

For $f\in\mathcal{D}_{2}(\mathcal{M})$, $\langle
M^{f}(x)\rangle_{t}=\int_{0}^{t}\Gamma f(X_{s}^{x})ds$.

###### Proof.

This is proved in [Ben23, Lemma 9.1]. ∎

###  8.2. Extending the Domain of the Generator

In this subsection we fix a Feller process
$\\{X_{t}^{x}\\}_{x\in\mathcal{M},t\geq 0}$ (see 2) on a locally compact
Polish space $\mathcal{M}$ with the Markov semigroup $\mathcal{P}_{s}$ (see •
‣ Section 2).

In lieu of 8.3 and 8.8, we have shown that
$\mathcal{D}_{2}(\mathcal{M})\subset\mathcal{D}^{ext}_{2}(\mathcal{M})$ as
defined in 2.10 and that the two notions of $\mathcal{L}$ and $\Gamma$ agree.
Similarly, $\mathcal{D}(\mathcal{M})\cap\\{f\in C_{b}(\mathcal{M})\mid f\geq
0\\}\subset\mathcal{D}^{ext}_{+}(\mathcal{M})$ as defined in 2.9. Next we show
that $\mathcal{D}^{ext}_{2}(\mathcal{M}),\mathcal{D}^{ext}_{+}(\mathcal{M})$
are in some sense the closures of
$\mathcal{D}_{2}(\mathcal{M}),\mathcal{D}(\mathcal{M})\cap\\{f\in
C_{b}(\mathcal{M})\mid f\geq 0\\}$ respectively.

In order to verify $f\in\mathcal{D}^{ext}_{+}(A)$, we construct a localizing
sequence of stopping times for $M_{t}^{f}(x)$ based on 2.14. We expand upon a
similar proof given in [Ben23, Lemma 9.3].

######  Lemma 8.9.

Suppose $A\subset\mathcal{M}$ is an open invariant set (see 2.2) with $K_{n}$
as in 2.14. Then for any $x\in A$, the sequence

$\tau_{n}\coloneqq\inf\\{t\geq 0\mid X_{t}^{x}\notin K_{n}\\}\wedge n$

is an increasing sequence of stopping times such that
$\lim_{n\to\infty}\tau_{n}=\infty$ a.s.

###### Proof.

Since $K_{n}^{c}$ is open, $X_{t}^{x}$ is cadlag, and the filtration is right-
continuous, then $(\tau_{n})_{n\geq 1}$ is a non-decreasing sequence of
stopping times (see [Kal21, Lemma 9.6iii]) and the limit
$\tau=\lim_{n\to\infty}\tau_{n}$ is almost surely well defined.

We claim that $\tau_{n}<\tau$ almost surely. Indeed, for any $m\geq n\geq 1$
denote

$\Omega_{n,m}:=\\{\omega\in\Omega\mid\tau_{n}=\tau<\infty,X_{\tau}\in
K_{m}\\}\,.$

Then on $\Omega_{n,m}$, by right the continuity there is a random $h>0$ such
that $X_{t}^{x}\in K_{m+1}^{\circ}$ for all $\tau\leq t\leq\tau+h$, and since
$\tau_{n}=\tau$ we have that $X_{t}^{x}\in K_{n}\subset K_{m+1}^{\circ}$ for
all $t<\tau$. Thus, $X_{t}^{x}\in K_{m+1}$ for all $t\leq\tau+h$ and so
$\infty>\tau\geq\tau_{m+1}\geq\tau+h>\tau$. Since
$\mathbb{P}(\infty>\tau>\tau)=0$, $\Omega_{n,m}$ has zero probability.

Hence, by the invariance of $A=\cup_{m\geq n}K_{m}$ for each $n\geq 1$ we have

$\\{\tau_{n}=\tau<\infty\\}\subset\bigcup_{m\geq n}\Omega_{n,m}$

and the claim follows. In particular, we showed that $\tau$ is a predictable
stopping time.

We finish the proof by showing that $\mathbb{P}(\tau<\infty)=0$. For a
contradiction assume that there is $N>1$ such that $\mathbb{P}(\tau<N)>0$. On
$\\{\tau<N\\}$, for any $n>N$ it holds that $\tau_{n}<n$ and thus
$X_{\tau_{n}}^{x}\in\overline{K_{n}^{c}}$. Since $K_{n-1}\subset
K_{n}^{\circ}$, then

$\overline{K_{n}^{c}}\subset\overline{(K_{n}^{\circ})^{c}}=(K_{n}^{\circ})^{c}\subset
K_{n-1}^{c}\,,$

and therefore $X_{\tau_{n}}^{x}\in K_{n-1}^{c}$ for any $n>N$. By 8.2,
$\lim_{n\to\infty}X_{\tau_{n}}^{x}=X_{\tau}^{x}$ almost surely on the event
$\\{\tau<N\\}$, and thus $X_{\tau}^{x}\in\cap_{n}K_{n}^{c}=A^{c}$ which by 2.2
has probability $0$, a contradiction to $\mathbb{P}(\tau<N)>0$. ∎

######  Remark 8.10.

Note that if $f:A\to\mathbb{R}$ is a continuous function and $t\mapsto
X_{t}^{x}$ is a cadlag process, then $t\mapsto f(X_{t}^{x})$ is almost surely
cadlag, so all of the martingales defined in this section are cadlag (without
any modification).

######  Lemma 8.11.

Let $A\subset\mathcal{M}$ be an open invariant set (see 2.2) and assume
$f:A\to[0,\infty)$ is such that:

  1. (1)

There exists $f_{n}\in\mathcal{D}(\mathcal{M})$ such that $f_{n}\to f$
uniformly on compact subsets of $A$ and $f\geq f_{n}\geq 0$.

  2. (2)

$\mathcal{L}f_{n}$ converges uniformly on compact subsets of $A$ to a
continuous function from $A$ to $\mathbb{R}$, which we denote by
$\mathcal{L}f$.

Then for all $x\in A$, $M_{t}\coloneqq M_{t}^{f}(x)$ defined in (2.1) is a
cadlag local martingale with the localizing sequence $(\tau_{n})_{n\geq 1}$ as
defined in 8.9. In particular, $f\in\mathcal{D}^{ext}_{+}(A)$ (see 2.9).

###### Proof.

By 8.3, for any $n\geq 1$

(8.1) $M_{t}^{n}\coloneqq
f_{n}(X_{t}^{x})-f_{n}(x)-\int_{0}^{t}\mathcal{L}f_{n}(X_{s}^{x})ds$

is a martingale. Since for any $m\geq 1$ $(\mathcal{L}f_{n})_{n\in\mathbb{N}}$
converges uniformly on the compact set $K_{m}$ and $X^{x}_{t}\in K_{m}$ for
any $t<\tau_{m}\leq m$, then a.s.
$\int_{0}^{t\wedge\tau_{m}}\mathcal{L}f_{n}(X_{s}^{x})ds\to\int_{0}^{t\wedge\tau_{m}}\mathcal{L}f(X_{s}^{x})ds$.
Consequently, for any $m\geq 1$ we have that $M_{t\wedge\tau_{m}}^{n}\to
M_{t\wedge\tau_{m}}$ almost surely as $n\to\infty$. Thus, to show that $M$ is
a local martingale it suffices to show that $M_{\tau_{m}}^{n}\to M_{\tau_{m}}$
in $L^{1}$, or equivalently that $\\{M_{\tau_{m}}^{n}\\}_{n\in\mathbb{N}}$ is
uniformly integrable. By the uniform convergence
$\mathcal{L}f_{n}\to\mathcal{L}f$ on $K_{m}$ again, we obtain that
$\int_{0}^{\tau_{m}}\mathcal{L}f_{n}(X_{s}^{x})ds$ is uniformly bounded. Since
$f_{n}\leq f$, by (8.1) we just need to show that $f(X_{\tau_{m}}^{x})$ is
integrable, which follows from Fatou’s lemma and $M_{t}^{n}$ being a
martingale:

(8.2) $\displaystyle\mathbb{E}[f(X_{\tau_{m}}^{x})]$
$\displaystyle\leq\liminf_{n\to\infty}\mathbb{E}[f_{n}(X_{\tau_{m}}^{x})]$
$\displaystyle=\liminf_{n\to\infty}\mathbb{E}\Big{[}f_{n}(x)+\int_{0}^{\tau_{m}}\mathcal{L}f_{n}(X_{s}^{x})ds\Big{]}<\infty.$

∎

We can prove a similar claim for $\mathcal{D}_{2}(\mathcal{M})$ which can be
viewed as a generalization of 8.8:

######  Lemma 8.12.

Let $A\subset\mathcal{M}$ be an open invariant set (see 2.2) and
$f:A\to\mathbb{R}$ be such that:

  1. (i)

There exists $f_{n}\in\mathcal{D}_{2}(\mathcal{M})$ such that $f_{n}\to f$
uniformly on compact subsets of $A$ and $|f|\geq|f_{n}|$.

  2. (ii)

$\mathcal{L}f_{n}$ converges uniformly on compact subsets of $A$ to a
continuous function from $A$ to $\mathbb{R}$, which we denote by
$\mathcal{L}f$.

  3. (iii)

$\Gamma f_{n}$ converges uniformly on compact subsets of $A$ to a continuous
function from $A$ to $[0,\infty)$, which we denote by $\Gamma f$.

  4. (iv)

$\Gamma f\leq KU^{\prime}$ for some $U,U^{\prime}:\mathcal{M}\to[0,\infty)$
and $K>0$ satisfying 5 (i), (ii).

Then for all $x\in A$, $M_{t}\coloneqq M_{t}^{f}(x)$ from (2.1) is a square-
integrable cadlag martingale with $\langle M\rangle_{t}=\int_{0}^{t}\Gamma
f(X_{s}^{x})ds$ (see 8.4). In particular, $f\in\mathcal{D}^{ext}_{2}(A)$ (see
2.10).

###### Proof.

Let $(\tau_{n})_{n\geq 1}$ be as defined in 8.9. Recall from 8.3 and 8.8 that
$M_{t}^{n}$ defined in (8.1) is a square-integrable martingale with $\langle
M^{n}\rangle_{t}=\int_{0}^{t}\Gamma f_{n}(X_{s}^{x})ds$, and from the proof of
8.11 that $M_{t\wedge\tau_{m}}^{n}\to M_{t\wedge\tau_{m}}$ a.s. as
$n\to\infty$. Consequently the Monotone Convergence Theorem, 8.9, Doob’s
Inequality, and Fatou’s Lemma yield

$\displaystyle\mathbb{E}[\sup_{0\leq s\leq t}M_{s}^{2}]$
$\displaystyle=\lim_{m\to\infty}\mathbb{E}[\sup_{0\leq s\leq
t\wedge\tau_{m}}M_{s}^{2}]\leq\liminf_{m\to\infty}4\mathbb{E}[M_{t\wedge\tau_{m}}^{2}]$
$\displaystyle\leq\liminf_{m\to\infty}\liminf_{n\to\infty}4\mathbb{E}[(M_{t\wedge\tau_{m}}^{n})^{2}]\,,$

and then 8.8, Domintated convergence theorem, Monotone convergence theorem,
8.9, and Tonelli’s theorem imply

(8.3) $\displaystyle\mathbb{E}[\sup_{0\leq s\leq t}M_{s}^{2}]$
$\displaystyle\leq\liminf_{m\to\infty}\liminf_{n\to\infty}4\mathbb{E}\Big{[}\int_{0}^{t\wedge\tau_{m}}\Gamma
f_{n}(X_{s}^{x})ds\Big{]}$
$\displaystyle=\liminf_{m\to\infty}4\mathbb{E}\Big{[}\int_{0}^{t\wedge\tau_{m}}\Gamma
f(X_{s}^{x})ds\Big{]}=4\mathbb{E}\Big{[}\int_{0}^{t}\Gamma
f(X_{s}^{x})ds\Big{]}$ $\displaystyle=4\int_{0}^{t}\mathcal{P}_{s}\Gamma
f(x)ds<\infty\,,$

where in the last inequality we used our assumption (iv) and 4.1. Thus, $M$ is
a square-integrable martingale, so to finish the proof we show that
$M_{t}^{2}-\int_{0}^{t}\Gamma f(X_{s}^{x})ds$ is a martingale (see 8.4). Since
$\Gamma f\geq 0$ and (8.3) imply

$\mathbb{E}\Big{[}\sup_{0\leq s\leq t}\int_{0}^{s}\Gamma
f(X_{u}^{x})du\Big{]}=\mathbb{E}\Big{[}\int_{0}^{t}\Gamma
f(X_{s}^{x})ds\Big{]}<\infty\,,$

it is enough to establish that $M_{t}^{2}-\int_{0}^{t}\Gamma f(X_{s}^{x})ds$
is a local martingale. Similarly to the proof of 8.11, we obtain that that
almost surely for any $m\geq 1$

$(M_{t\wedge\tau_{m}}^{n})^{2}-\int_{0}^{t\wedge\tau_{m}}\Gamma
f_{n}(X_{s}^{x})ds\to(M_{t\wedge\tau_{m}})^{2}-\int_{0}^{t\wedge\tau_{m}}\Gamma
f(X_{s}^{x})ds\qquad\textrm{as}\quad n\to\infty\,,$

and consequently it suffices to show that
$\\{(M_{\tau_{m}}^{n})^{2}-\int_{0}^{\tau_{m}}\Gamma
f_{n}(X_{s}^{x})ds\\}_{n\in\mathbb{N}}$ is uniformly integrable. Since
$X_{s}^{x}\in K_{m}$ for any $s<\tau_{m}$ and $\Gamma f_{n}\to\Gamma f$
uniformly on $K_{m}$, then $\sup_{n}\int_{0}^{\tau_{m}}\Gamma
f_{n}(X_{s}^{x})ds<\infty$. We finish the proof by showing that
$\\{(M_{\tau_{m}}^{n})^{2}\\}_{n\in\mathbb{N}}$ is uniformly integrable. By
QM-AM inequality we have

$\displaystyle(M_{\tau_{m}}^{n})^{2}$
$\displaystyle=\Big{(}f_{n}(X_{\tau_{m}}^{x})-f_{n}(x)-\int_{0}^{\tau_{m}}\mathcal{L}f_{n}(X_{s}^{x})ds\Big{)}^{2}$
$\displaystyle\leq
3\Big{[}f_{n}(X_{\tau_{m}}^{x})^{2}+f_{n}(x)^{2}+\Big{(}\int_{0}^{\tau_{m}}\mathcal{L}f_{n}(X_{s}^{x})ds\Big{)}^{2}\Big{]},$

and as above, since $f_{n}\to f$ and $\mathcal{L}f_{n}\to\mathcal{L}f$
uniformly on $K_{m}$, the last two terms on the right hand side are bounded
uniformly in $n$. Finally, since $f_{n}(X_{\tau_{m}}^{x})^{2}\leq
f(X_{\tau_{m}}^{x})^{2}$ it suffices to show that
$\mathbb{E}[f(X_{\tau_{m}}^{x})^{2}]<\infty$, and this follows similarly to
the proof of 8.11:

$\displaystyle\mathbb{E}[f(X_{\tau_{m}}^{x})^{2}]$
$\displaystyle\leq\liminf_{n\to\infty}\mathbb{E}[f_{n}(X_{\tau_{m}}^{x})^{2}]$
$\displaystyle\leq\lim_{n\to\infty}3\mathbb{E}\Big{[}(M^{n}_{\tau_{m}})^{2}+f_{n}(x)^{2}+\Big{(}\int_{0}^{\tau_{m}}\mathcal{L}f_{n}(X_{s}^{x})ds\Big{)}^{2}\Big{]}$
$\displaystyle=\lim_{n\to\infty}3\mathbb{E}\Big{[}\int_{0}^{\tau_{m}}\Gamma
f_{n}(X_{s}^{x})ds\Big{]}+3f_{n}(x)^{2}+3\mathbb{E}\Big{[}\Big{(}\int_{0}^{\tau_{m}}\mathcal{L}f_{n}(X_{s}^{x})ds\Big{)}^{2}\Big{]}$
$\displaystyle=3\mathbb{E}\Big{[}\int_{0}^{\tau_{m}}\Gamma
f(X_{s}^{x})ds\Big{]}+3f(x)^{2}+3\mathbb{E}\Big{[}\Big{(}\int_{0}^{\tau_{m}}\mathcal{L}f(X_{s}^{x})ds\Big{)}^{2}\Big{]}$
$\displaystyle<\infty\,,$

as desired. ∎

###  8.3. Special Generator

In this subsection we assume that the generator $\mathcal{L}$ can be written
as $L_{1}+L_{2}$, where $L_{1}$ is a second order differential operator
representing the “smooth” part of the process and $L_{2}$ is the generator for
the “jump” part of the process. Below in applications $L_{1}$ arises when
Stochastic Differential Equations (SDEs) driven by Brownian motion are
involved, whereas $L_{2}$ appears in the context of random switching and other
jump processes. Our goal is to show that the assumptions of 8.11 and 8.12 are
satisfied if $L_{1}f,L_{2}f$ are well defined and continuous. We being with
some preliminary definitions and a technical lemma.

######  Definition 8.13.

For a Polish space $\mathcal{M}$, let $\mathscr{P}(\mathcal{M})$ denote the
set of all finite positive Borel measures on $\mathcal{M}$. We endow
$\mathscr{P}(\mathcal{M})$ with topology of weak convergence, that is,
$\mu_{n}\to\mu$ if for all $f\in C_{b}(\mathcal{M})$, $\mu_{n}f\to\mu f$.

######  Remark 8.14.

Note that the Portmanteau theorem is valid for $\mathscr{P}(\mathcal{M})$ as
well. Indeed, if $\mu_{n}\in\mathscr{P}(\mathcal{M})$ and $\mu_{n}\to\mu$
weakly, then for $f=1$ we have $\mu_{n}(\mathcal{M})\to\mu(\mathcal{M})$.
Thus, if $\mu(\mathcal{M})\neq 0$, then $\mu_{n}\to\mu$ if and only if
$\mu_{n}/\mu_{n}(\mathcal{M})\to\mu/\mu(\mathcal{M})$. Since
$\mu_{n}/\mu_{n}(\mathcal{M})$ is a probability measure, Portmanteau theorem
applies to $\mu_{n}\to\mu$. If $\mu(\mathcal{M})=0$, then the Portmanteau
theorem theorem is straightforward to prove since all relevant limits vanish.

######  Lemma 8.15.

Let $\mathcal{M}$ be a Polish space, let
$\mu:\mathcal{M}\to\mathscr{P}(\mathcal{M})$ and
$f:\mathcal{M}\times\mathcal{M}\to\mathbb{R}$ be continuous maps, and fix
$\epsilon>0$. Then the function $x\mapsto\int f(x,y)d\mu_{x}(y)$ is continuous
if $x\mapsto\int|f(x,y)|^{1+\epsilon}d\mu_{x}(y)$ is bounded above by a
continuous function.

###### Proof.

First we show that the function from $\mathcal{M}$ to
$\mathscr{P}(\mathcal{M}\times\mathcal{M})$ given by
$x\mapsto\delta_{x}\otimes\mu_{x}$ is continuous, where $\delta_{x}$ is dirac
delta measure and $\otimes$ denotes the product of measures. Indeed, for any
bounded Lipschitz function $f:\mathcal{M}\times\mathcal{M}\to\mathbb{R}$ with
the Lipschitz constant $C$ and any $x_{n}\to x$ we have

$\displaystyle|\delta_{x_{n}}\otimes\mu_{x_{n}}(f)-\delta_{x}\otimes\mu_{x}(f)|$
$\displaystyle=|\mu_{x_{n}}f(x_{n},\cdot)-\mu_{x}f(x,\cdot)|$
$\displaystyle\leq
C\mu_{x_{n}}(\mathcal{M})d(x_{n},x)+|\mu_{x_{n}}f(x,\cdot)-\mu_{x}f(x,\cdot)|\to
0\,,$

where in the limit $n\to\infty$ we used 8.14, continuity of $x\mapsto\mu_{x}$,
and $x_{n}\to x$.

Consequently, the function from $\mathcal{M}$ to $\mathscr{P}(\mathbb{R})$
given by $x\mapsto f^{*}(\delta_{x}\otimes\mu_{x})$ is continuous, where
$f^{*}$ denotes the pushforward by $f$. Since $a\mapsto a$ is a continuous
function from $\mathbb{R}\to\mathbb{R}$ which vanishes over the proper
function $a\mapsto|a|^{1+\epsilon}$, the result follows by the same argument
as in the proof of 4.7. ∎

######  Definition 8.16.

Let $\mathcal{X}$ be a locally compact Polish space, $n\geq 0$, and
$A\subset\mathbb{R}^{n}\times\mathcal{X}$. We define $\mathcal{C}^{2}(A)$ to
be the set of all continuous functions $f:A\to\mathbb{R}$ such that there is
an open set $\mathcal{U}\subset\mathbb{R}^{n}\times\mathcal{X}$ with
$A\subset\mathcal{U}$ such that $f$ extends to a continuous function
$\tilde{f}$ on $\mathcal{U}$ which is twice continuously differentiable with
respect to $b\in\mathbb{R}^{n}$. For $(b,x)\in\mathcal{U}$, we denote
$\partial_{i}f(b,x)$ the partial derivative of $\tilde{f}$ with respect to the
$i$th coordinate of $b$. We define $\mathcal{C}_{c}^{2}(A)$ to be the space of
compactly supported functions in $\mathcal{C}^{2}(A)$. If $n=0$, that is
$\mathbb{R}^{n}=\\{0\\}$, we abbreviate $\mathcal{C}^{2}(A)$ as
$\mathcal{C}(A)$, the set of all continuous functions $f:A\to\mathbb{R}$.

For the rest of the section, we fix a Feller quadruple
$(\mathcal{M},\mathcal{M}_{0},\mathcal{M}_{+},\\{X_{t}^{x}\\}_{x\in\mathcal{M},t\geq
0})$ (see 2.6) such that $\mathcal{M}=B\times\mathcal{X}$ where $B$ is a
closed subset of $\mathbb{R}^{n}$ for some $n\geq 0$ and $\mathcal{X}$ is a
locally compact Polish space. We also fix a continuous function
$\mu:B\times\mathcal{X}\to\mathscr{P}(B\times\mathcal{X})$, and we use
$\mu_{(b,x)}$ to denote $\mu(b,x)$.

Let $\Sigma_{ij},F_{i}:B\times\mathcal{X}\to\mathbb{R}$ be continuous. We
define operators $L_{1},L_{2},L,\Gamma$ for suitable functions $f$ by

$\displaystyle L_{1}f(b,x)$
$\displaystyle=\sum_{i,j=1}^{n}\frac{1}{2}\Sigma_{ij}(b,x)\partial_{i}\partial_{j}f(b,x)+\sum_{i=1}^{n}F_{i}(b,x)\partial_{i}f(b,x)$
$\displaystyle L_{2}f(b,x)$
$\displaystyle=\int_{A\times\mathcal{X}}f(c,y)-f(b,x)d\mu_{(b,x)}(c,y)$
$\displaystyle Lf(b,x)$ $\displaystyle=L_{1}f(b,x)+L_{2}f(b,x)$
$\displaystyle\Gamma f(b,x)$
$\displaystyle=\sum_{i,j=1}^{n}\Sigma_{ij}(b,x)\partial_{i}f(b,x)\partial_{j}f(b,x)$
$\displaystyle+\int_{A\times\mathcal{X}}(f(c,y)-f(b,x))^{2}d\mu_{(b,x)}(c,y)\,.$

Note that if $f\in\mathcal{C}_{c}^{2}(\mathcal{M})$, then $Lf,\Gamma f$ are
well defined and continuous.

######  Assumption 6.

Assume $\mathcal{C}_{c}^{2}(\mathcal{M})\subset\mathcal{D}(\mathcal{M})$ (see
2.7) and for all $f\in\mathcal{C}_{c}^{2}(\mathcal{M})$ the equality
$\mathcal{L}f=Lf$ holds.

######  Lemma 8.17.

Suppose the Feller quadruple
$(\mathcal{M},\mathcal{M}_{0},\mathcal{M}_{+},\\{X_{t}^{x}\\}_{x\in\mathcal{M},t\geq
0})$ satisfies 6 and let $A=\mathcal{M}$ or $\mathcal{M}_{+}$. Then for any
$f:A\to\mathbb{R}$:

  1. (1)

If $f\in\mathcal{C}^{2}(A)$, $f\geq 0$, and $L_{2}f$ is finite and continuous,
then $f\in\mathcal{D}^{ext}_{+}(A)$ with $\mathcal{L}f=Lf$ (see 2.9).

  2. (2)

If $f\in\mathcal{C}^{2}(A)$, $\Gamma f$ is finite and continuous, and 8.12
(iv) holds then $f\in\mathcal{D}^{ext}_{2}(A)$ with $\mathcal{L}f=Lf$ and
$\Gamma f$ as above (see 2.10).

###### Proof.

We only prove the second claim, since the first one follows similarly, using
8.11 instead of 8.12.

Let $\mathcal{U},\tilde{f}$ be as in 8.16. Let $K_{n}\subset\mathcal{U}$,
$n\geq 1$ be compact such that $K_{n}\subset K_{n+1}^{\circ}$, where the
interior is with respect to the subspace topology on $\mathcal{U}$ inherited
from $\mathbb{R}^{n}\times\mathcal{X}$, and $\cup_{n}K_{n}=\mathcal{U}$.

Next, we construct a continuous cut-off function which is smooth in
$b\in\mathbb{R}^{n}$. Our argument is standard, but since we were not able to
locate the exact statement in the literature, we provide details.

For each $n\geq 1$, by Urysohn’s lemma there is a continuous function
$g_{n}:\mathcal{U}\to[0,1]$ such that $K_{n}\subset\\{g_{n}=1\\}$ and
$\\{g_{n}\neq 0\\}\subset K_{n+1}$. By convolving $g_{n}(b,x)$ with a positive
mollifier $\phi_{n}(b)$, we may obtain a continuous function
$h_{n}:\mathcal{U}\to[0,1]$ such that $h_{n}$ is twice continuously
differentiable in $b$, $K_{n-1}\subset\\{h_{n}=1\\}$, and $\\{h_{n}\neq
0\\}\subset K_{n+2}$. Define $f_{n}=h_{n}\tilde{f}$ on $K_{n+2}$ and $f_{n}=0$
on $(\mathbb{R}^{n}\times\mathcal{X})\setminus K_{n+2}$, and therefore $f_{n}$
is a bounded continuous function on $\mathbb{R}^{n}\times\mathcal{X}$.
Restricting the domain of $f_{n}$ to $\mathcal{M}$, we have $\\{f_{n}\neq
0\\}\subset K_{n+2}\cap\mathcal{M}$. Since $\tilde{f}(b,x),h_{n}(b,x)$ are
twice continuously differentiable in $b$, then
$f_{n}\in\mathcal{C}_{c}^{2}(\mathcal{M})$. Thus, by 6,
$f_{n}\in\mathcal{D}(\mathcal{M})$ and $\mathcal{L}f_{n}=Lf_{n}$. Applying the
same argument to $f_{n}^{2}=h_{n}^{2}f^{2}$ and noting that $h_{n}^{2}$ shares
the same properties as $h_{n}$ shows that
$f_{n}^{2}\in\mathcal{C}_{c}(\mathcal{M})\subset\mathcal{D}(\mathcal{M})$, and
thus $f_{n}\in\mathcal{D}_{2}(\mathcal{M})$ and
$\mathcal{L}f_{n}^{2}=Lf_{n}^{2}$.

Since $\mathcal{M}\setminus\mathcal{M}_{+}$ is closed in $\mathcal{M}$, it is
closed in $\mathbb{R}^{n}\times\mathcal{X}$. Thus, if $A=\mathcal{M}_{+}$ we
may without loss of generality assume that $\mathcal{U}$ is disjoint from
$\mathcal{M}\setminus\mathcal{M}_{+}$ (otherwise intersect $\mathcal{U}$ with
the open set
$(\mathbb{R}^{n}\times\mathcal{X})\setminus(\mathcal{M}\setminus\mathcal{M}_{+})$),
or equivalently that
$\mathcal{U}\cap\mathcal{M}=\mathcal{U}\cap\mathcal{M}_{+}$. This ensures that
$|f|\geq|f_{n}|\geq 0$ on $\mathcal{M}$, since if $(b,x)\in\mathcal{M}$ then
either $(b,x)\in\mathcal{U}\cap\mathcal{M}=\mathcal{U}\cap\mathcal{M}_{+}$ in
which case $f_{n}(b,x)=h_{n}(b,x)f(b,x)$ or
$(b,x)\notin\mathcal{U}\cap\mathcal{M}$ in which case $f_{n}=0$. Similarly,
$f_{n}\geq 0$ on $\mathcal{M}$ if $f\geq 0$ on $\mathcal{M}$. Also note that
$(f_{n}^{+},f_{n}^{-})\to(f^{+},f^{-})$ uniformly on compact subsets of $A$
since $K_{n-1}\subset\\{h_{n}=1\\}$, so $f_{n}=f_{n}^{+}-f_{n}^{-}\to
f^{+}-f^{-}=f$ uniformly on compact subsets of $A$ as well. Thus, we have
verified 8.12 (i).

Next, we show that 8.12 (ii) holds true, that is, $\mathcal{L}f_{n}$ converges
uniformly on compact subsets of $A$ to $Lf$. Observe that $Lf=L_{1}f+L_{2}f$
is continuous since $f\in\mathcal{C}^{2}(A)$ implies that $L_{1}f$ is
continuous, and $\Gamma f$ being continuous with 8.15 yields that $L_{2}f$ is
continuous. Next, since $L_{1}f_{n}(b,x)=L_{1}f(b,x)$ for $(b,x)\in
K_{n-1}^{\circ}\cap A$, then $L_{1}f_{n}$ converges uniformly on compact
subsets of $A$ to $L_{1}f$. To treat $L_{2}f_{n}$, by the continuity of
$L_{2}f$ and $\Gamma f$ we have that $(b,x)\mapsto\mu_{(b,x)}f^{2}$ is
continuous. Consequently, by 8.15 the maps $(b,x)\mapsto\mu_{(b,x)}f^{+}$ and
$(b,x)\mapsto\mu_{(b,x)}f_{n}^{-}$ are continuous. Since
$(b,x)\mapsto\mu_{(b,x)}f_{n}^{+}$ is an increasing sequence of nonnegative
continuous functions and by Monotone Convergence Theorem it converges
pointwise to the continuous function $\mu_{(b,x)}f^{+}$, by Dini’s Theorem the
convergence is uniform on compact subsets of $A$. Similarly,
$\mu_{(b,x)}f_{n}^{-}\to\mu_{(b,x)}f^{-}$ uniformly on compact subsets of $A$.
This shows that $L_{2}f_{n}\to L_{2}f$ uniformly on compact subsets of $A$ and
thus $Lf_{n}=\mathcal{L}f_{n}\to Lf$ uniformly on compact subsets of $A$.

Applying the same argument to $f_{n}^{2}=h_{n}^{2}f^{2}$ shows that
$\mathcal{L}f_{n}^{2}\to Lf^{2}$ uniformly on compact subsets of $A$. Standard
computations show that $\Gamma f=Lf^{2}-2fLf$ and thus $\Gamma
f_{n}=Lf_{n}^{2}-2f_{n}Lf_{n}\to Lf^{2}-2fLf=\Gamma f$ uniformly on compact
subsets of $A$, verifying 8.12 (iii). Thus, by 8.12 we conclude that
$f\in\mathcal{D}^{ext}_{2}(A)$, as desired. ∎

##  9\. Some Important Classes of Examples

In this section we introduce model examples of Markov processes to which our
theory can be applied: switching diffusions, SDEs driven by Brownian motion,
and discrete-time Markov chains. In each case, under general assumptions, we
provide results that verify some or all of 1 – 5.

###  9.1. Switching Diffusions

In this section we consider a class of Markov processes given by solutions to
switching diffusions, which are a generalization of SDEs driven by Brownian
motion. In particular, we investigate the model from [YZ10, equations (2.2)
and (2.3)]:

(9.1) $\displaystyle dX^{(x,i)}(t)$
$\displaystyle=F(X^{(x,i)}(t),\alpha^{(x,i)}(t))dt+\sigma(X^{(x,i)}(t),\alpha^{(x,i)}(t))dw(t)$
$\displaystyle\mathbb{P}(\alpha^{(x,i)}(t)=j)$
$\displaystyle=q_{ij}(x)t+o(t)\quad\text{ for }i\neq j$

where $n,m,d\in\mathbb{N}$, $\mathcal{X}=\\{1,\dots,m\\}$,
$(x,i)\in\mathbb{R}^{n}\times\mathcal{X}$ is the initial condition,
$F:\mathbb{R}^{n}\times\mathcal{X}\to\mathbb{R}^{n}$ and
$\sigma:\mathbb{R}^{n}\times\mathcal{X}\to\mathbb{R}^{n\times d}$ are locally
Lipschitz, $w$ is an $\mathbb{R}^{d}$ Brownian motion, and
$Q:\mathbb{R}^{n}\times\mathcal{X}\times\mathcal{X}\to\mathbb{R}$ is
continuous, uniformly bounded, and such that $q_{ij}(x)\coloneqq Q(x,i,j)$
satisfies $q_{ij}(x)\geq 0$ if $i\neq j$ and $\sum_{j=1}^{m}q_{ij}=0$.

In other words, $\alpha$ is a pure jump process on the finite state
$\mathcal{X}$ with $x$-dependent generator $q(x)$. We define the following
operators as in [YZ10, (2,4)] for suitable functions $f$:

(9.2) $\displaystyle L_{1}f(x,\alpha)$
$\displaystyle=\sum_{i,j=1}^{n}\frac{1}{2}\Sigma_{ij}(x,\alpha)\partial_{i}\partial_{j}f(x,\alpha)+\sum_{i=1}^{n}F_{i}(x,\alpha)\partial_{i}f(x,\alpha)$
$\displaystyle L_{2}f(x,\alpha)$
$\displaystyle=\sum_{\beta\neq\alpha}q_{\alpha\beta}(x)(f(x,\beta)-f(x,\alpha))$
$\displaystyle Lf(x,\alpha)$ $\displaystyle=L_{1}f(x,\alpha)+L_{2}f(x,\alpha)$
$\displaystyle\Gamma f(x,\alpha)$
$\displaystyle=\sum_{i,j=1}^{n}\Sigma_{ij}(x,\alpha)\partial_{i}f(x,\alpha)\partial_{j}f(x,\alpha)$
$\displaystyle+\sum_{\beta\neq\alpha}q_{\alpha\beta}(x)(f(x,\beta)-f(x,\alpha))^{2}\,.$

where $\Sigma(x,\alpha)=\sigma(x,\alpha)\sigma(x,\alpha)^{T}$ (here T denotes
the transpose).

######  Lemma 9.1.

If there is a proper $W:\mathbb{R}^{n}\times\mathcal{X}\to[0,\infty)$ which is
twice continuously differentiable with respect to $x\in\mathbb{R}^{n}$ and
$LW\leq\gamma_{0}W$ for some constant $\gamma_{0}>0$, then there is a (unique)
solution $(X_{t}^{y},\alpha_{t}^{y})$ to (9.1) for any initial condition
$y=(x,i)$ and $(X_{t}^{y},\alpha_{t}^{y})$ satisfies 2 (the Feller property).

###### Proof.

The existence and uniqueness of solution $(X_{t}^{y},\alpha_{t}^{y})$ to (2.2)
and (2.3) for any initial condition $y=(x,i)$ follows from [YZ10, Proposition
2.20]. To show 2, define the truncated process $X_{t}^{N,y}$ as in [YZ10,
Proposition 2.20], and note that [YZ10, Proposition 2.20] implies that

$\mathbb{P}(\sup_{0\leq t<\beta_{N}^{x}}|X_{t}^{N,y}-X_{t}^{y}|>0)=0\,,$

where $\beta^{x}_{N}$ is the first exit time from a ball of radius $N$. If we
denote $W_{N}\coloneqq\inf_{|x|\geq N,i\in\mathcal{X}}W(x,i)$, then by [YZ10,
proof of Theorem 2.7], for any $T>0$ we have

$W(y)e^{\gamma_{0}T}=W(x,i)e^{\gamma_{0}T}\geq
W_{N}\mathbb{P}(\beta_{N}^{y}\leq T)\,,$

and since $W_{N}\to\infty$ as $N\to\infty$ ($W$ is proper) we obtain
$\mathbb{P}(\beta_{N}^{y}\leq T)\to 0$ as $N\to\infty$ uniformly on compact
subsets of $\mathbb{R}^{n}\times\mathcal{X}$ (for fixed $T$). Consequently, if
$g\in C_{b}(\mathbb{R}^{n}\times\mathcal{X})$, then the functions
$u_{N}(x,\alpha)=\mathbb{E}[g(X_{t}^{N,(x,\alpha)},\alpha_{t}^{(x,\alpha)})]$
converge to
$u(x,\alpha)=\mathbb{E}[g(X_{t}^{(x,\alpha)},\alpha_{t}^{(x,\alpha)})]$
uniformly on compact sets for any $t>0$. By [YZ10, Theorem 2.18], $u_{N}$ is
continuous, and so $u$ is also continuous and thus 2 holds. ∎

######  Lemma 9.2.

Supposed 2 holds for (9.1). Then if $A\subset\mathbb{R}^{n}\times\mathcal{X}$
is invariant (see 2.2) for $(X_{t}^{y},\alpha_{t}^{y})$ and
$f\in\mathcal{C}^{2}(A)$ (see 8.16), then

  1. (1)

If $f\geq 0$, then $f\in\mathcal{D}^{ext}_{+}(A)$ and $\mathcal{L}f=Lf$ (see
2.9).

  2. (2)

If 8.12 (iv) holds then $f\in\mathcal{D}^{ext}_{2}(A)$ (see 2.10),
$\mathcal{L}f=Lf$, and $\Gamma f$ is as above.

###### Proof.

To prove (1) and (2), by 8.17, it suffices to verify 6 since $L_{1}f$ and the
continuous part of $\Gamma$ are clearly continuous for
$f\in\mathcal{C}^{2}(A)$ and $L_{2}f$ and the jump part of $\Gamma f$ are
continuous for continuous $f$ since $q_{ij}$ are continuous. To verify 6, we
use the generalized Itô formula from [YZ10, (2.8)] that imply that $M^{f}_{t}$
in (2.1) is a martingale with $\mathcal{L}f=Lf$. Then 6 follows since the
converse of 8.3 is true: if $f,g$ are bounded continuous functions such that
(2.1) with $\mathcal{L}f$ replaced with $g$ is a martingale for all $x$, then
$f$ is in the domain of the generator and $\mathcal{L}f=g$ (see 2.7). ∎

###  9.2. SDEs

As a special case of Section 9, we suppose $\mathcal{X}$ is a singleton set so
that (9.1) is just a SDE driven by Brownian motion and $L_{2}=0$ in (9.2).
Then we have the following sufficient conditions for verifying the bulk of our
assumptions.

######  Lemma 9.3.

Suppose
$(\mathcal{M},\mathcal{M}_{0},\mathcal{M}_{+},\\{X_{t}^{x}\\}_{x\in\mathcal{M},t\geq
0})$ is a Feller quadruple, where $X_{t}^{x}$ is a solution to a SDE driven by
Brownian motion ((9.1) with $\alpha$ constant). Let
$V\in\mathcal{C}^{2}(\mathcal{M}_{+})$. Then 3, 4 (ii), and 5 are satisfied if
there is a proper map $\bar{U}:\mathcal{M}\to[1,\infty)$ in
$\mathcal{C}^{2}(\mathcal{M})$ and constants $K,c>0$ such that $L\bar{U}\leq
K-c\bar{U}$ and $|\mathcal{L}V|+\Gamma V\lesssim
2K-\frac{L\bar{U}}{\bar{U}}+\frac{\Gamma\bar{U}}{\bar{U}^{2}}$.

######  Remark 9.4.

In what follows, $f\lesssim g$ means that there is a constant $C>0$ depending
only on the parameters of the problem such that $f\leq Cg$.

######  Remark 9.5.

9.3 also applies in the case where $\mathcal{X}$ is not a singleton set but
$U$ only depends on $x\in\mathbb{R}^{n}$ so that $L_{2}U=0$.

###### Proof.

Without loss of generality we can assume $c<1<K$. We show that 3, 4 (ii), and
5 are satisfied for $U=\bar{U}^{1/2}$, $W=\bar{U}^{1/4}$,
$U^{\prime}=\frac{c}{8K}\bar{U}^{1/2}\phi$ and
$W^{\prime}=\frac{c}{16K}\bar{U}^{1/4}\phi$, where

(9.3) $\phi\coloneqq
2K-\frac{L\bar{U}}{\bar{U}}+\frac{\Gamma\bar{U}}{\bar{U}^{2}}=K+\Big{(}K-\frac{L\bar{U}}{\bar{U}}\Big{)}+\frac{\Gamma\bar{U}}{\bar{U}^{2}}\geq
K\,.$

Since $\bar{U}\geq 1$ and $\bar{U}\in\mathcal{C}^{2}(\mathcal{M})$, then
$U,W\in\mathcal{C}^{2}(\mathcal{M})$. By 9.2, we obtain
$U\in\mathcal{D}^{ext}_{+}(\mathcal{M})$ (5 (i)) and $\mathcal{L}U=LU$. By
calculus and (9.2),

$\mathcal{L}U=L\bar{U}^{1/2}=\frac{1}{2\bar{U}^{1/2}}L\bar{U}-\frac{1}{8\bar{U}^{3/2}}\Gamma{\bar{U}}=\bar{U}^{1/2}\Big{(}\frac{1}{2}\frac{L\bar{U}}{\bar{U}}-\frac{1}{8}\frac{\Gamma\bar{U}}{\bar{U}^{2}}\Big{)}\,.$

Since $\bar{U}$ is proper and satisfies $L\bar{U}\leq K-c\bar{U}$, then
outside of a compact set we have $\frac{L\bar{U}}{\bar{U}}\leq-\frac{3c}{4}$,
and so outside of the compact set we have
$\frac{1}{2}\frac{L\bar{U}}{\bar{U}}\leq\frac{c}{8K}(\frac{L\bar{U}}{\bar{U}}-2K)$,
where we used that for all $0<\gamma<\frac{1}{8}$ and
$x\leq-\frac{3c}{4}\leq-\frac{2c}{3}$ it holds that $\frac{1}{2}x\leq\gamma
x-\frac{c}{4}$. Then for large enough $K^{\prime}>\frac{K}{4}$, by
$\Gamma\bar{U}\geq 0$ and $c\leq 1\leq K$ we have

$\displaystyle\mathcal{L}U$ $\displaystyle\leq
K^{\prime}+\bar{U}^{1/2}\Big{(}\frac{c}{8K}\Big{(}\frac{L\bar{U}}{\bar{U}}-2K-\frac{\Gamma\bar{U}}{\bar{U}^{2}}\Big{)}+\Big{(}\frac{c}{8K}-\frac{1}{8}\Big{)}\frac{\Gamma\bar{U}}{\bar{U}^{2}}\Big{)}$
$\displaystyle\leq
K^{\prime}-\frac{c}{8K}\bar{U}^{1/2}\phi=K^{\prime}-U^{\prime}\,,$

and thus 5 (ii) holds.

Next, we focus on $W$:

$LW=L(\bar{U})^{\frac{1}{4}}=\frac{1}{4\bar{U}^{\frac{3}{4}}}L\bar{U}-\frac{3}{32\bar{U}^{\frac{7}{4}}}\Gamma{\bar{U}}=(\bar{U})^{\frac{1}{4}}\Big{(}\frac{1}{4}\frac{L\bar{U}}{\bar{U}}-\frac{3}{32}\frac{\Gamma\bar{U}}{\bar{U}^{2}}\Big{)}$

and by reusing the estimate
$\frac{1}{2}\frac{L\bar{U}}{\bar{U}}\leq\frac{c}{8K}(\frac{L\bar{U}}{\bar{U}}-2K)$
(which holds outside of a compact set) we obtain for sufficiently large
$K^{\prime\prime}>0$ that

(9.4) $\displaystyle LW$ $\displaystyle\leq
K^{\prime\prime}+(\bar{U})^{\frac{1}{4}}\Big{(}\frac{c}{16K}\Big{(}\frac{L\bar{U}}{\bar{U}}-2K-\frac{\Gamma\bar{U}}{\bar{U}^{2}}\Big{)}+\Big{(}\frac{c}{16K}-\frac{3}{32}\Big{)}\frac{\Gamma\bar{U}}{\bar{U}^{2}}\Big{)}$
$\displaystyle\leq
K^{\prime\prime}-\frac{c}{16K}(\bar{U})^{\frac{1}{4}}\phi=K^{\prime\prime}-W^{\prime}\,,$

where in the last inequality we used $\Gamma\bar{U}\geq 0$ and $c\leq 1\leq
K$. Furthermore, since $\phi\geq\frac{\Gamma\bar{U}}{\bar{U}^{2}}$, by
calculus

(9.5) $\Gamma
W=\Gamma(\bar{U}^{\frac{1}{4}})=\frac{\Gamma\bar{U}}{16\bar{U}^{\frac{3}{2}}}=\frac{1}{16}\bar{U}^{1/2}\frac{\Gamma\bar{U}}{\bar{U}^{2}}\leq\frac{1}{16}\bar{U}^{1/2}\phi=\frac{1}{16}U^{\prime}.$

Thus, $W$ satisfies 8.12(iv), and therefore 9.2 implies
$W\in\mathcal{D}^{ext}_{2}(\mathcal{M})$ and $\mathcal{L}W=LW$. By (9.4) and
(9.5), $W$ satisfies 3 and 5(iii), where we note that $W$ and $W^{\prime}$ are
proper on account of $\bar{U}$ being proper and (9.3).

Since $\bar{U}\geq 1$, then $\Gamma V\lesssim\phi\lesssim U^{\prime}$. Thus,
$V$ satisfies 8.12(iv), and therefore 9.2 implies
$V\in\mathcal{D}^{ext}_{2}(\mathcal{M}_{+})$ and 5(iv) holds true. Finally,
4(ii) is a consequence of $|LV|\lesssim\phi\lesssim
U^{\prime}\lesssim(W^{\prime})^{2}$. ∎

###  9.3. Discrete Time

In this section, we investigate discrete time processes. In Section 8.3 we
defined $\mathcal{M}=\mathbb{R}^{n}\times\mathcal{X}$, but since our process
is discrete, we assume $n=0$, that is $\mathbb{R}^{n}=\\{0\\}$. In the rest of
the section we suppress any dependence on the “continuous” variable
$b\in\mathbb{R}^{n}$ and simply write $\mathcal{M}$ instead of $\mathcal{X}$.

Suppose $\\{X_{n}^{x}\\}_{x\in\mathcal{M},n\in\mathbb{N}}$ is a discrete-time
Markov chain, which means that there is a filtered probability space
$(\Omega,\mathcal{F},\\{\mathcal{F}_{n}\\}_{n\in\mathbb{N}},\mathbb{P})$ and a
family of $\mathcal{M}-$valued random variables
$\\{X_{n}^{x}\\}_{x\in\mathcal{M},n\in\mathbb{N}}$ such that:

  * •

$X_{0}^{x}=x$ a.s.

  * •

$X^{x}_{\cdot}$ is adapted to $\\{\mathcal{F}_{n}\\}_{n\in\mathbb{N}}$,
meaning $X_{n}^{x}$ is $\mathcal{F}_{n}$ measurable for each $n\in\mathbb{N}$.

  * •

For all bounded measurable functions $f:\mathcal{M}\to\mathbb{R}$, the map

$\mathcal{M}\ni x\mapsto\mathcal{P}f(x)\coloneqq\mathbb{E}[f(X_{1}^{x})]$

is measurable and for any $n\geq 0$ we assume (homogeneity) that

$\mathcal{P}f(X_{n}^{x})=\mathbb{E}[f(X_{n+1}^{x})|\mathcal{F}_{n}]\,.$

We also define the operators

(9.6) $\displaystyle Lf(x)$
$\displaystyle\coloneqq\mathcal{P}f(x)-f(x)=\mathbb{E}[f(X_{1}^{x})-f(x)]$
$\displaystyle\Gamma f(x)$ $\displaystyle\coloneqq
Lf^{2}(x)-2f(x)Lf(x)=\mathbb{E}[(f(X_{1}^{x})-f(x))^{2}]$

Again we let $\mathcal{M}_{0}\subset\mathcal{M}$ be closed and
$\mathcal{M}_{+}\subset\mathcal{M}_{0}^{c}$ be open and dense. Let
$\\{N_{t}\\}_{t\geq 0}$ be a Poisson process with rate $1$ independent of
$X_{n}^{x}$ and define $Y_{t}^{x}\coloneqq X_{N_{t}}^{x}$. Then it is standard
to show that $Y_{t}^{x}$ is a continuous-time Markov process on $\mathcal{M}$
with cadlag sample paths and semigroup
$\mathcal{P}_{t}=e^{-t}e^{t\mathcal{P}}$. Indeed, heuristically we have for
any set $A$ that

$\displaystyle\mathcal{P}_{t}\mathbbm{1}_{A}$
$\displaystyle=\mathbb{P}(Y_{t}^{x}\in
A)=\sum_{n=0}^{\infty}\mathbb{P}(X_{n}^{x}\in A\cap
N_{t}=n)=\sum_{n=0}^{\infty}e^{-t}\frac{t^{n}}{n!}\mathbb{P}(X_{n}^{x}\in A)$
$\displaystyle=\sum_{n=0}^{\infty}e^{-t}\frac{t^{n}}{n!}\mathcal{P}^{n}\mathbbm{1}_{A}(x)=e^{-t}e^{t\mathcal{P}}\mathbbm{1}_{A}\,.$

The following lemma helps us translate between conditions on the discrete-time
Markov chain $X^{x}_{n}$ and the continuous-time Markov process $Y_{t}^{x}$.

######  Lemma 9.6.

Let $\\{X_{n}^{x}\\}_{x\in\mathcal{M},n\in\mathbb{N}}$ and
$\\{Y_{t}^{x}\\}_{x\in\mathcal{M},t\geq 0}$ be as above and $L,\Gamma$ be as
in (9.6). Then

  1. (i)

A probability measure $\mu$ is invariant for $Y_{t}$ if and only if $\mu$ is
invariant for $X_{t}$. Specifically, $\mu\mathcal{P}f=\mu f$ for each $f\in
C_{b}(\mathcal{M})$ if and only if $\mu\mathcal{P}_{t}f=\mu f$ for each $t\geq
0$ and $f\in C_{b}(\mathcal{M})$.

  2. (ii)

$\mathcal{U}$ is accessible from $x$ in the sense of 3.4 applied to $Y_{t}$ if
and only if one of the following equivalent conditions holds:

    1. (a)

$\sum_{n=0}^{\infty}e^{-n}\mathbb{P}(X_{n}^{x}\in\mathcal{U})>0$.

    2. (b)

There exists $n\in\mathbb{N}$ such that
$\mathbb{P}(X_{n}^{x}\in\mathcal{U})>0$.

    3. (c)

$\mathbb{P}(\exists n\in\mathbb{N}\text{ such that
}X_{n}^{x}\in\mathcal{U})>0$.

  3. (iii)

If for $A=\mathcal{M}_{0}$ and $A=\mathcal{M}_{+}$ it holds that for all $x\in
A,$ $\mathcal{P}\mathbbm{1}_{A}(x)=1$, then 1 holds for $Y_{t}^{x}$.

  4. (iv)

If $\mathcal{P}f\in C_{b}(\mathcal{M})$ for all $f\in C_{b}(\mathcal{M})$,
then 2 holds for $Y_{t}^{x}$.

In particular, if (iii) and (iv) hold then
$(\mathcal{M},\mathcal{M}_{0},\mathcal{M}_{+},\\{Y_{t}^{x}\\}_{x\in\mathcal{M},t\geq
0})$ is a Feller quadruple (see 2.6), which is assumed for the remaining
items.

  1. (v)

If $f\in C_{b}(\mathcal{M})$, then $f\in\mathcal{D}(\mathcal{M})$ and
$\mathcal{L}f=Lf$ (see 2.7).

  2. (vi)

Suppose $\Upsilon:\mathcal{M}\to[1,\infty)$ is proper (see 2.13) and
$V:\mathcal{M}_{+}\to\mathbb{R}$ is continuous. Suppose also that
$\mathbb{E}[|V(X_{1}^{x})-V(x)|^{2+\epsilon}]\lesssim\sqrt{\Upsilon(x)}$ (see
9.4) and $\mathcal{P}\Upsilon\leq\rho^{2}\Upsilon+C^{2}$ for some constants
$\epsilon>0,\rho\in(0,1),C>0$. Then 3, 4 (ii), and 5 hold for $Y_{t}^{x}$ with
$U=\sqrt{\Upsilon},U^{\prime}=(1-\rho)U,W=\Upsilon^{1/4},\text{ and
}W^{\prime}=(1-\sqrt{\rho})W$.

  3. (vii)

If 3–5 hold for
$(\mathcal{M},\mathcal{M}_{0},\mathcal{M}_{+},\\{Y_{t}^{x}\\}_{x\in\mathcal{M},t\geq
0})$, then 3.2 (and its corollaries 3.1, 3.5, and 3.9) hold for $X_{n}^{x}$
(with $t$ replaced with $n$).

###### Proof.

The statements (i)–(iv) are standard, so we omit the proofs.

For (v), fix $x\in\mathcal{M}$ and compute

$\displaystyle\lim_{s\downarrow 0}\frac{\mathcal{P}_{s}f(x)-f(x)}{s}$
$\displaystyle=\lim_{s\downarrow 0}\frac{e^{-s}e^{s\mathcal{P}}f(x)-f(x)}{s}$
$\displaystyle=\lim_{s\downarrow
0}\frac{se^{-s}(\sum_{n=0}^{\infty}s^{n}\frac{\mathcal{P}^{n+1}f(x)}{(n+1)!})-(1-e^{-s})f(x)}{s}$
$\displaystyle=\mathcal{P}f(x)-f(x)\,,$

where the last equality is justified since $\mathcal{P}$ is a contraction and

$e^{-s}\Big{|}\sum_{n=1}^{\infty}s^{n}\frac{\mathcal{P}^{n+1}f(x)}{(n+1)!}\Big{|}\leq
e^{-s}\Big{(}\frac{e^{s}-1-s}{s}\Big{)}\|f\|\to 0\qquad\textrm{as}\quad
s\downarrow 0\,.$

Similarly to the above, we obtain

$\displaystyle\sup_{s>0}\Big{\|}\frac{\mathcal{P}_{s}f-f}{s}\Big{\|}$
$\displaystyle=\sup_{s>0}\Big{\|}e^{-s}\sum_{n=0}^{\infty}s^{n}\frac{\mathcal{P}^{n+1}f}{(n+1)!}-\frac{(1-e^{-s})}{s}f\Big{\|}$
$\displaystyle\leq\sup_{s>0}\Big{|}\frac{e^{-s}(e^{s}-1)}{s}\Big{|}\|f\|+\sup_{s>0}\Big{|}\frac{(1-e^{-s})}{s}\Big{|}\|f\|$
$\displaystyle=2\sup_{s>0}\Big{|}\frac{(1-e^{-s})}{s}\Big{|}\|f\|<\infty\,.$

Thus, (v) is verified, which shows that 6 is satisfied with $L_{1}=0$ and
$\mu_{x}(A)=\mathcal{P}\mathbbm{1}_{A}(x)$.

To prove (vi) we first note that if $\mu_{x}$ is the law of $X_{1}^{x}$, then
by the Feller property, $x\mapsto\mu_{x}$ is continuous. Hence, by
$\mathbb{E}[|V(X_{1}^{x})-V(x)|^{2+\epsilon}]\lesssim U$ and 8.15, the
function $\Gamma V$ defined in (9.6) is finite and continuous. In addition, by
our assumption in (vi)

$\mathbb{E}[|U(X^{x}_{1})-U(x)|^{2}]\leq 2\mathcal{P}U^{2}(x)+2U^{2}(x)\leq
2\rho^{2}\Upsilon(x)+2C^{2}+2U^{2}(x)$

and so the continuity of $x\mapsto LU(x)$ follows from (9.6), the continuity
of $U$ and $x\mapsto\mu_{x}$, and 8.15. Analogously, we establish that $\Gamma
W$ is continuous. Thus, 8.17 implies $U\in\mathcal{D}^{ext}_{+}(\mathcal{M})$
and $V,W\in\mathcal{D}^{ext}_{2}(\mathcal{M})$, and consequently 3(i) and 5(i)
are satisfied. Next, by Jensen’s inequality and
$\sqrt{a+b}\leq\sqrt{a}+\sqrt{b}$ we have

$\mathcal{P}U=\mathcal{P}\sqrt{\Upsilon}\leq\sqrt{\mathcal{P}\Upsilon}\leq\rho
U+C\,,$

and therefore

(9.7) $LU=\mathcal{P}U-U\leq(\rho-1)U+C=C-U^{\prime}\,$

and 5(ii) follows. Similarly we obtain $LW\leq(\sqrt{\rho}-1)W+\sqrt{C}$,
which proves 3(ii). Also, since $W=\sqrt{U}\geq 0$ and $\mathcal{P}W\geq 0$,
using (9.7) we have

$\Gamma W=LU-2WLW=LU-2W\mathcal{P}W+2U\leq(\rho+1)U+C\lesssim U$

and 5(iii) holds true. Furthermore, by Jensen’s inequality

$|LV|\leq(\mathbb{E}[|V(X_{1}^{x})-V(x)|^{2+\epsilon}])^{\frac{1}{2+\epsilon}}\lesssim(W^{\prime}(x))^{\frac{2}{2+\epsilon}}$

and since $\epsilon>0$ and $W^{\prime}$ is proper, $LV$ vanishes over
$W^{\prime}$, establishing 4(ii). Finally, by Jensen’s inequality and $U\geq
1$ we have

$\Gamma
V\leq(\mathbb{E}[|V(X_{1}^{x})-V(x)|^{2+\epsilon}])^{\frac{2}{2+\epsilon}}\lesssim
U^{\frac{2}{2+\epsilon}}\lesssim U\,,$

which implies 5(iv) and the proof of (vi) is finished.

For (vii), recall that 1-2 are met since
$(\mathcal{M},\mathcal{M}_{0},\mathcal{M}_{+},\\{Y_{t}^{x}\\}_{x\in\mathcal{M},t\geq
0})$ is a Feller quadruple. Then by the strong law of large numbers we have
that almost surely $\lim_{t\to\infty}\frac{t}{N_{t}}=1$ and thus (almost
surely)

$\liminf_{n\to\infty}\frac{V(X_{n}^{x})}{n}=\liminf_{t\to\infty}\frac{V(Y_{t}^{x})}{N_{t}}=\liminf_{t\to\infty}\frac{V(Y_{t}^{x})}{t}\frac{t}{N_{t}}=\liminf_{t\to\infty}\frac{V(Y_{t}^{x})}{t}\,.$

∎

##  10\. Examples

In this section we illustrate how our theory can be applied to some
characteristic examples. We recommend that the reader is familiar with the
notation and definitions from Section 2 and the results in Section 3. The
first example is a model of an epidemic using a switching diffusion, the
second one is a SDE obtained by adding noise to the Lorenz system (a
prototypical model for turbulence), the third one is a very general class of
discrete-time ecological models, and the fourth is a very general class of
continuous-time ecological models based on Stochastic Kolmogorov equations. We
use the setup and results from Section 9.1 for the first example, from Section
9.2 for the second and fourth examples, and from Section 9.3 for the third
example.

###  10.1. Stochastic SIS epidemic model on network with Markovian switching

In this example, analyzed in [BO23], the dynamics of $(x(t),s(t))$ with
initial condition $(x_{0},s_{0})$ are given by the following switching
diffusion:

$\displaystyle dx_{i}$
$\displaystyle=[\beta(s)b_{i}(x,s)(1-x_{i})-\delta(s)x_{i}]dt+\sigma_{i}(x_{i},s)b_{i}(x,s)(1-x_{i})dW_{i}(t)$
$\displaystyle b_{i}(x,s)$
$\displaystyle\coloneqq\sum_{j=1}^{N}a_{ij}(s)x_{j}$
$\displaystyle\mathbb{P}(s(t)=s^{\prime})$
$\displaystyle=q_{s_{0}s^{\prime}}t+o(t)\quad\text{ for }s_{0}\neq
s^{\prime}\,.$

Here, we set $\mathcal{M}=[0,1]^{N}\times S=[0,1]^{N}\times\\{1,\dots,m\\}$
for some $N,m\in\mathbb{N}$, $\mathcal{M}_{0}=\\{0\\}\times S$, and
$\mathcal{M}_{+}=\mathcal{M}_{0}^{c}$. For $x\in[0,1]^{N}$ we write
$(x_{1},\dots,x_{n})$ for the components of $x$ and use $(x,s)$ to denote an
element of $\mathcal{M}$ (where $s\in S$). Also, $x_{i}$ represents the
probability that the $i$th node is infected and $s$ represents a different
regime/environment. The functions $\beta,\delta:S\to(0,\infty)$ represent the
infection and recovery rates and the parameters $a_{ij}:S\to\\{0,1\\}$ for
$1\leq i,j\leq N$ characterize the connections in the network so that the
matrix $A(s)$ with entries $a_{ij}(s)$ is the adjacency matrix of the network
when the environment is in state $s$. The intensity of the noise is given by
continuously differentiable functions $\sigma_{i}:[0,1]\times S\to[0,\infty)$
for $1\leq i\leq N$. We assume that $\sigma_{i}(x_{i},s)>0$ for
$x_{i}\in(0,1)$ in order to guarantee that $\mathcal{M}_{0}$ is accessible,
and also that $\sigma_{i}(0,s)=0$ so that $x_{i}$ remains nonnegative for all
times. The driving noise is an $N$-dimensional Brownian motion
$(W_{1}(t),\dots,W_{N}(t))$.

$s(t)$ is an irreducible Markov process on $S$ which is independent of the
Brownian motion and $q_{ss^{\prime}}$ denotes the transition rate from state
$s\in S$ to $s^{\prime}\in S$, meaning that $q_{ss^{\prime}}\geq 0$ if $s\neq
s^{\prime}$ and $\sum_{s^{\prime}=1}^{m}q_{ss^{\prime}}=0$. Since $s(t)$ is
irreducible, it has a unique invariant probability measure which we denote as
$\rho\in[0,1]^{m}$ satisfying $\sum_{s=1}^{m}\rho_{s}q_{ss^{\prime}}=0$ and
$\sum_{s=1}^{m}\rho_{s}=1$.

In [BO23, Theorem 1 and Remark 1] it is shown that the solution $(x(t),s(t))$
exists and is unique, and furthermore $(x(t),s(t))$ remains in $\mathcal{M}$
(resp. $\mathcal{M}_{+}$, $\mathcal{M}_{0}$) when the initial condition is in
$\mathcal{M}$ (resp. $\mathcal{M}_{+},\mathcal{M}_{0}$) so that 1 is
satisfied. Since $\mathcal{M}$ is compact, it follows from 9.1 that the Feller
property (2) holds. Thus,
$(\mathcal{M},\mathcal{M}_{0},\mathcal{M}_{+},\\{(x(t),s(t))\\}_{t\geq 0})$ is
a Feller quadruple (we suppress the superscript for the initial condition).

In the notation of (9.1) and (9.2) we have

(10.1) $\displaystyle F_{i}(x,s)$
$\displaystyle=\beta(s)b_{i}(x,s)(1-x_{i})-\delta(s)x_{i}$
$\displaystyle\Sigma_{ij}(x,s)$
$\displaystyle=\begin{cases}[\sigma_{i}(x_{i},s)b_{i}(x,s)(1-x_{i})]^{2}&\text{if
}i=j\\\ 0&\text{otherwise}\end{cases}\,.$

Our strategy is to use 3.9 and 3.13 to conclude that 3.5 applies to the
problem. For $(x,s)\in\mathcal{M}_{+}$, we set
$V(x,s)=-\frac{1}{2}\log{\|x\|^{2}}$, where
$\|x\|\coloneqq\sqrt{\sum_{i=1}^{N}x_{i}^{2}}$ is the Euclidean norm, and
observe that $V$ satisfies 4(i). Since $V$ is independent of $s$, from (9.2)
and (10.1) follows

(10.2) $\displaystyle LV(x,s)$
$\displaystyle=\sum_{i=1}^{n}\frac{1}{2}[\sigma_{i}(x_{i},s)b_{i}(x,s)(1-x_{i})]^{2}\frac{-\|x\|^{2}+2x_{i}^{2}}{\|x\|^{4}}$
$\displaystyle\qquad+\sum_{i=1}^{n}[\beta(s)b_{i}(x,s)(1-x_{i})-\delta(s)x_{i}]\frac{-x_{i}}{\|x\|^{2}}$
$\displaystyle=\delta(s)+\sum_{i=1}^{n}\frac{1}{2}[\sigma_{i}(x_{i},s)b_{i}(x,s)(1-x_{i})]^{2}\frac{-\|x\|^{2}+2x_{i}^{2}}{\|x\|^{4}}$
$\displaystyle\qquad+\sum_{i=1}^{n}[\beta(s)b_{i}(x,s)(1-x_{i})]\frac{-x_{i}}{\|x\|^{2}}$
$\displaystyle\Gamma V(x,s)$
$\displaystyle=\sum_{i=1}^{n}[\sigma_{i}(x_{i},s)b_{i}(x,s)(1-x_{i})]^{2}\Big{(}\frac{-x_{i}}{\|x\|^{2}}\Big{)}^{2}\,.$

Since $b_{i}(x,s)$ is a linear function of $x$, it satisfies
$|b_{i}(x,s)|\lesssim\|x\|$ and thus $\Gamma V$ is uniformly bounded on
$\mathcal{M}_{+}$. It follows from 9.2 that
$V\in\mathcal{D}^{ext}_{2}(\mathcal{M}_{+})$ with $\mathcal{L}V=LV$ and
$\Gamma V$ as above. Since

$\lim_{x\to
0}\sum_{i=1}^{n}[\beta(s)b_{i}(x,s)(1-x_{i})]\frac{-x_{i}}{\|x\|^{2}}$

does not exist (it depends on the direction $x$ approaches $0$), we need to
enlarge the state space. Let $\pi:S^{N-1}\times[0,\infty)\to\mathbb{R}^{N}$
denote the polar coordinate map, that is, $\pi(v,r)=rv$ for $v\in
S^{N-1}\coloneqq\\{v\in\mathbb{R}^{N}\mid\|v\|=1\\}$, $r\in[0,\infty)$. Let
$\mathcal{N}=\pi^{-1}([0,1]^{N})\times S$,
$\mathcal{N}_{0}=\pi^{-1}(\\{0\\})\times S$,
$\mathcal{N}_{+}=\mathcal{N}\setminus\mathcal{N}_{0}$.

It is routine to show that there exists a Markov process
$\\{(v(t),r(t),s(t))\\}_{t\geq 0}$ on $\mathcal{N}$ such that
$(x(t),s(t))=(\pi(v(t),r(t)),s(t))$, but we provide some details anyway. For
$x=\pi(v,r)$ we have $r=\|x\|$, $v=\frac{x}{\|x\|}$, and then Itô’s Formula
(note that $dx_{i}dx_{j}=0$) gives us

(10.3) $\displaystyle dr$
$\displaystyle=\sum_{i=1}^{n}\frac{x_{i}dx_{i}}{r}+\frac{1}{2}\sum_{i=1}^{n}\frac{r-x_{i}^{2}/r}{r^{2}}(dx_{i})^{2}$
$\displaystyle=r\Big{[}\sum_{i=1}^{n}v_{i}\frac{dx_{i}}{r}+\frac{1}{2}\sum_{i=1}^{n}(1-v_{i}^{2})\Big{(}\frac{dx_{i}}{r}\Big{)}^{2}\Big{]}$

and

(10.4) $\displaystyle dv_{i}$
$\displaystyle=x_{i}dr^{-1}+r^{-1}dx_{i}+dx_{i}dr^{-1}$
$\displaystyle=v_{i}\Big{(}\frac{-dr}{r}+\Big{(}\frac{dr}{r}\Big{)}^{2}\Big{)}+\frac{dx_{i}}{r}-\frac{dx_{i}}{r}\frac{dr}{r}\,.$

Since

$\frac{dx_{i}}{r}=[\beta(s)b_{i}(v,s)(1-rv_{i})-\delta(s)v_{i}]dt+\sigma_{i}(rv_{i},s)b_{i}(v,s)(1-rv_{i})dW_{i}(t)$

we obtain that (10.3) and (10.4) define a Markov process
$\\{(v(t),r(t),s(t))\\}_{t\geq 0}$ (again suppressing the superscript for the
initial condition) on $\mathcal{N}$. It is standard to verify that
$(\mathcal{N},\mathcal{N}_{0},\mathcal{N}_{+},\\{(v(t),r(t),s(t))\\}_{t\geq
0})$ is a Feller quadruple and $\tilde{\pi}\coloneqq(v,r,s)\mapsto(rv,s)$ is a
quadruple map from
$(\mathcal{N},\mathcal{N}_{0},\mathcal{N}_{+},\\{(v(t),r(t),s(t))\\}_{t\geq
0})$ to $(\mathcal{M},\mathcal{M}_{0},\mathcal{M}_{+},\\{(x(t),s(t))\\}_{t\geq
0})$ (see 3.7).

Substituting $x=rv$ into the formula for $LV(x,s)$ (10.2), we define the
continuous extension $H:\mathcal{N}\to\mathbb{R}$ of
$\mathcal{L}V\circ\tilde{\pi}$ by

$\displaystyle H(v,r,s)$
$\displaystyle\coloneqq\delta(s)+\sum_{i=1}^{n}\frac{1}{2}[\sigma_{i}(rv_{i},s)b_{i}(rv,s)(1-rv_{i})]^{2}\frac{-r^{2}+2v_{i}^{2}r^{2}}{r^{4}}$
$\displaystyle\qquad+\sum_{i=1}^{n}[\beta(s)b_{i}(rv,s)(1-rv_{i})]\frac{-rv_{i}}{r^{2}}$
$\displaystyle=\delta(s)+\sum_{i=1}^{n}\frac{1}{2}[\sigma_{i}(rv_{i},s)b_{i}(v,s)(1-rv_{i})]^{2}(-1+2v_{i}^{2})$
$\displaystyle\qquad-\sum_{i=1}^{n}[\beta(s)b_{i}(v,s)(1-rv_{i})]v_{i}\,,$

where the equality is by the linearity of $b$ in $x$. Since $\sigma(0,s)=0$
and $b(v,x)=A(s)v$, for $(v,0,s)\in\mathcal{N}_{0}$ we have

$\displaystyle H(v,0,s)$
$\displaystyle=\delta(s)-\beta(s)\sum_{i=1}^{n}b_{i}(v,s)v_{i}=\delta(s)-\beta(s)v^{T}A(s)v\,.$

Let $\lambda_{1}(s)$ denote the largest eigenvalue of $A(s)$. For any $\mu\in
P_{inv}(\mathcal{N}_{0})$, by using $\|v\|=1$, we have

$\mu
H\geq\mu(\delta(s)-\beta(s)\lambda_{1}(s))=\sum_{s=1}^{m}\rho_{s}(\delta(s)-\beta(s)\lambda_{1}(s))\,,$

where recall that $(\rho_{1},\dots,\rho_{m})$ are components of the unique
invariant measure for $s(t)$. Thus, if
$\sum_{s=1}^{m}\rho_{s}(\delta(s)-\beta(s)\lambda_{1}(s))>0$ then there is a
constant $\alpha>0$ such that $\mu H\geq\alpha$ for all $\mu\in
P_{inv}(\mathcal{N}_{0})$, and so (3.2) is satisfied.

Recall that $\sigma_{i}(x_{i},s)>0$ for $x_{i}\in(0,1)$ and $\sigma_{i}$ is
continuously differentiable in $x_{i}$, and consequently by the Stroock-
Varadhan support theorem (see [SV72]), $(0,s)$ is accessible from $(x,s)$ in
the sense of 3.6, and thus (3.1) is satisfied.

Furthermore, by 3.13 we have that 3, 4(ii), and 5 are satisfied for the Feller
quadruple
$(\mathcal{M},\mathcal{M}_{0},\mathcal{M}_{+},\\{X_{t}^{x}\\}_{x\in\mathcal{M},t\geq
0})$.

Thus, 3.9 and 3.5 provide the following result.

######  Theorem 10.1.

If $\sum_{s=1}^{m}\rho_{s}(-\delta(s)+\beta(s)\lambda_{1}(s))<0$ then for any
initial condition almost surely

$\limsup_{t\to\infty}\frac{\log{\|x\|}}{t}\leq\sum_{s=1}^{m}\rho_{s}(-\delta(s)+\beta(s)\lambda_{1}(s))<0\,,$

and in particular $x\to 0$ exponentially fast.

######  Remark 10.2.

Observe that [BO23, Theorem 2] shows the exponential convergence under the
stronger assumption that

$\sum_{s=1}^{m}\rho_{s}\Big{(}-\delta(s)+\beta(s)\lambda_{1}(s)+\frac{M(s)^{2}\lambda_{1}(s)^{2}}{32}\Big{)}<0\,,$

where $M(s)>0$ is a global bound on the strength of the noise
$\frac{\sigma_{i}(x_{i},s)}{x_{i}}$ for $x\in(0,1)^{N}$. Our 10.1 shows that
the extra term $\frac{M(s)^{2}\lambda_{1}(s)^{2}}{32}$ is unnecessary.
Heuristically, the term $\frac{M(s)^{2}\lambda_{1}(s)^{2}}{32}$ should not be
needed, since it depends on the behavior of the process everywhere on
$\mathcal{M}$, even at points which are far from $\mathcal{M}_{0}$, which
should not be affecting the rate of convergence to $\mathcal{M}_{0}$, as long
as the accessibility of $\mathcal{M}_{0}$ holds true.

######  Remark 10.3.

If the adjacency matrix of the network $A(s)=A$ is independent of $s$ and the
network is connected, then using the persistence theorem in [Ben23] and
observing that the only invariant measure of $(v(t),s(t))$ on
$\mathcal{N}_{0}$ is the one where $s(t)$ is distributed according to $\rho$
and $v(t)$ is constantly equal to the (unique) positive eigenvector of $A$
with norm 1, one obtains that the condition
$\sum_{s=1}^{m}\rho_{s}(-\delta(s)+\beta(s)\lambda_{1})>0$ gives persistence.
In the general case, one could also take $V(x,s)=-\log{\sum_{i=1}^{N}x_{i}}$
to obtain that the condition
$\sum_{s=1}^{m}\rho_{s}(-\delta(s)+\beta(s)d_{min}(s))>0$ gives persistence,
where $d_{\min}(s)$ is the minimum degree of nodes of the network with
adjacency matrix $A(s)$, an improvement over [BO23, Theorem 6].

###  10.2. Lorenz System

In this example taken from [CZH21], $\mathcal{M}=\mathbb{R}^{3}$,
$\mathcal{M}_{0}=\\{(x,y,z)\in\mathbb{R}^{3}\mid x=y=0\\}$ (the $z$-axis), and
$\mathcal{M}_{+}=\mathcal{M}_{0}^{c}$. The Markov process
$(X_{t},Y_{t},Z_{t})$ is given by the Lorenz system with constant noise in the
$Z$ component. Specifically, there are constants
$\sigma,\rho,\beta,\hat{\alpha_{0}}>0$ and a (one-dimensional) Brownian motion
$W_{t}$ such that $(X_{t},Y_{t},Z_{t})$ satisfies the SDE

$\displaystyle dX$ $\displaystyle=\sigma(Y-X)dt$ $\displaystyle dY$
$\displaystyle=[X(\rho-Z)-Y]dt$ $\displaystyle dZ$ $\displaystyle=[-\beta
Z+XY]dt+\hat{\alpha_{0}}dW$

(We use $\hat{\alpha_{0}}$ instead of $\hat{\alpha}$ in order to avoid
confusion with the $\alpha$ in 4.)

To consolidate the constants, [CZH21, (2.1)] makes a linear change of
variables $(x_{t},y_{t},z_{t})=(c_{1}X_{\chi t},c_{2}(Y_{\chi t}-X_{\chi
t}),z_{*}-c_{3}Z_{\chi t})$, where $c_{1},c_{2},c_{3},z_{*},\chi>0$ are
constants so that the system is rewritten as in [CZH21, (2.2)]:

$\displaystyle dx$ $\displaystyle=ydt$ $\displaystyle dy$
$\displaystyle=[x(z-2)-2y]dt$ $\displaystyle dz$
$\displaystyle=-[\gamma(z-z_{*})+x(x+\eta y)]dt+\alpha_{0}dW\,,$

where $\gamma,\eta>0$ are constants and $\alpha_{0}=c_{4}\hat{\alpha_{0}}$
($c_{4}>0$ is also a constant).

The operator $L$ in (9.2) is given by [CZH21, (2.5a)]:

(10.5) $L=y\partial_{x}+[x(z-2)-2y]\partial_{y}-[\gamma(z-z_{*})+x(x+\eta
y)]\partial_{z}+\frac{\alpha_{0}^{2}}{2}\partial_{z}^{2}\,.$

As in [CZH21, (4.11)], we define

$U(x,y,z):=\tilde{U}(X,Y,Z):=\exp(\frac{\beta}{2\hat{\alpha_{0}}^{2}}(X^{2}+Y^{2}+(Z-\sigma-\rho)^{2}))$

and we remark that the notation $V_{1},\tilde{V_{1}}$ was used instead of
$U,\tilde{U}$ in [CZH21]. By straightforward calculations (see [CZH21,
(4.14)]) we have $LU\leq K-c(1+x^{2}+y^{2}+z^{2})U$ for some constants
$K,c>0$.

######  Remark 10.4.

In fact, using $\tilde{U}$ with $\frac{\beta}{2\hat{\alpha_{0}}^{2}}$ replaced
by a constant $A$ shows that for fixed $\sigma,\beta$ there are constants
$A,K,c,p>0$ for which $L\tilde{U}\leq K-c(1+x^{2}+y^{2}+z^{2})\tilde{U}$ as
long as $\rho<1$ and $\alpha_{0}\leq p$.

By standard arguments $\mathcal{M}_{+},\mathcal{M}_{0}$ are invariant, and
consequently 9.1 with $W=\tilde{U}$ implies that
$(\mathcal{M},\mathcal{M}_{0},\mathcal{M}_{+},\\{(x_{t},y_{t},z_{t})\\}_{t\geq
0})$ is a Feller quadruple (see 2.1). In particular, 1–2 are satisfied.

Also, [CZH21, (2.3)] introduces a “cylindrical” change of variables via the
map $\pi:S^{1}\times[0,\infty)\times\mathbb{R}\to\mathbb{R}^{3}$ given by
$\pi(\theta,R,z)=(R\sin\theta,R(\cos\theta-\sin\theta),z)$, where $S^{1}$ is
viewed here as $\mathbb{R}/2\pi\mathbb{Z}$. With
$\mathcal{N}=S^{1}\times[0,\infty)\times\mathbb{R}$,
$\mathcal{N}_{0}=S^{1}\times\\{0\\}\times\mathbb{R}$, and
$\mathcal{N}_{+}=\mathcal{N}\setminus\mathcal{N}_{0}$ we see that $\pi$ is a
quadruple map (see 3.7) between the Feller quadruples
$(\mathcal{N},\mathcal{N}_{0},\mathcal{N}_{+},\\{(\theta_{t},R_{t},z_{t})\\}_{t\geq
0})$ and
$(\mathcal{M},\mathcal{M}_{0},\mathcal{M}_{+},\\{(x_{t},y_{t},z_{t})\\}_{t\geq
0})$ (we suppress superscripts for the initial condition), where the dynamics
of $(\theta_{t},R_{t},z_{t})$ are governed by the SDE

(10.6) $\displaystyle d\theta$ $\displaystyle=[1-z\sin^{2}\theta]dt$
$\displaystyle dR$ $\displaystyle=R[-1+\frac{z}{2}\sin(2\theta)]dt$
$\displaystyle dz$
$\displaystyle=-[\gamma(z-z_{*})+R^{2}\sin\theta(\sin\theta+\eta(\cos\theta-\sin\theta))]dt+\alpha_{0}dW$

We define $V:\mathcal{M}_{+}\to\mathbb{R}$ by
$V(x,y,z)=-\frac{1}{2}\log(x^{2}+(x+y)^{2})=-\log{R}$ and note that such $V$
satisfies 4(i). Since $LV\circ\pi=-\frac{dR}{R}=1-\frac{z}{2}\sin(2\theta)$
and $\Gamma V=0$, then 3, 4 (ii), and 5 follow from 9.3 with our fixed $V$ and
$\bar{U}=U$. Also, $\mathcal{L}V\circ\pi$ extends continuously to the function
$H(\theta,R,z)=1-\frac{z}{2}\sin(2\theta)$ on $\mathcal{N}$.

######  Remark 10.5.

In fact, by 10.4 and the definitions of $W,W^{\prime},K,U,U^{\prime}$ in the
proof of 9.3, it follows that 3 and 5 (i)-(iii) are satisfied using the same
$W,W^{\prime},K,U,U^{\prime}$ (independent of $\alpha_{0}$) as long as
$\alpha_{0}$ is small enough.

Concerning the accessibility, (3.1) is satisfied (see 3.6), by [CZH21,
Proposition 3.3] and the Stroock-Varhadan support theorem [SV72].

######  Remark 10.6.

It is not hard to show that (3.1) is satisfied, even without using [CZH21,
Proposition 3.3]. In the notation of [CZH21, Proposition 3.3], we only need
$|(x(T),y(T))|<\epsilon$ and $|z(T)|\leq M$ for some $M,T$ that may depend on
$(x_{0},y_{0},z_{0})$ but not on $\epsilon>0$. Finding a control $h$ so that
such a condition holds is not difficult since we can choose a fixed $z$ so
that the eigenvalues of $\begin{bmatrix}0&1\\\ z-2&-2\end{bmatrix}$ have
negative real parts and then have $h$ force $z(t)$ to eventually be equal to
$z$.

Finally, we focus on the assumption (3.2) from 3.9 and recall that
$H(\theta,R,z)=1-\frac{z}{2}\sin(2\theta)$. Since for $\alpha_{0}>0$ the set
$P_{inv}(\mathcal{N}_{0})$ consists of a single measure $\mu_{\alpha_{0}}$
(see the discussion preceding [CZH21, Theorem 4.1]), the assumption $\mu
H\geq\alpha>0$ for all $\mu\in P_{inv}(\mathcal{N}_{0})$ is equivalent to

(10.7)
$-\lambda_{\alpha_{0}}\coloneqq\int_{S^{1}\times\mathbb{R}}1-\frac{z}{2}\sin(2\theta)d\mu_{\alpha_{0}}(\theta,z)>0\,.$

Note that this is precisely the condition considered in [CZH21, Theorem 4.1].

Thus, assuming (10.7), 3.9 and 3.5 imply

$\mathbb{P}\Big{(}\limsup_{t\to\infty}\frac{\frac{1}{2}\log(x_{t}^{2}+(x_{t}+y_{t})^{2})}{t}\leq\lambda_{\alpha_{0}}<0\Big{)}=1$

for any initial condition in $\mathcal{M}_{+}$.

If $\alpha_{0}=0$ then all invariant measures $\mu_{0}$ on $\mathcal{N}_{0}$
have $z$ constantly equal to $z_{*}$ and, analyzing the eigenvalues of
$\begin{bmatrix}0&1\\\ z-2&-2\\\ \end{bmatrix}$,

$\lambda_{0}\coloneqq-\inf_{\mu_{0}\in
P_{inv}(\mathcal{N}_{0})}\mu_{0}H=\begin{cases}\sqrt{z_{*}-1}-1&\text{if
}z_{*}>1\\\ -1&\text{otherwise}\end{cases}\,.$

Fixing $R_{t}\equiv 0$ and $\alpha_{t}\equiv\alpha_{0}$ and noting that the
coefficients of (10.6) are locally Lipschitz in $(\theta,z,\alpha_{0})$, we
may use standard arguments similar to those in 9.1 to verify the Feller
property (2) for the Markov process $(\theta_{t},0,z_{t},\alpha_{t})$. Then it
follows from 10.5 that 3.11 applies to the Markov Processes
$(\theta_{t},0,z_{t})$ on $\mathcal{N}_{0}$ for
$\alpha_{0}\in\Theta\coloneqq[0,p]$ (where $p$ is as in 10.4), and we conclude
that for $\rho<1$ (which implies $z_{*}<2$, see [CZH21, page 5]) we have
$\limsup_{\alpha_{0}\downarrow 0}\lambda_{\alpha_{0}}\leq\lambda_{0}<0$, and
therefore (10.7) is satisfied for all sufficiently small $\alpha_{0}$. In
particular we recover the following half of [CZH21, Theorem 1.1].

######  Theorem 10.7.

If $\rho<1$ then there is a constant $\alpha_{*}>0$ (depending on
$\sigma,\beta$) such that if $0\leq\hat{\alpha_{0}}<\alpha_{*}$ then,
regardless of initial condition, $(x_{t},y_{t})\to 0$ exponentially fast
almost surely.

######  Remark 10.8.

Using the exact same average Lyapunov function $V$, the second half of [CZH21,
Theorem 1.1] (persistence) follows from [Ben23], [CZH21, Theorem 3.1], and
[CZH21, Theorem 5.2]. This shows that the only the construction of $V_{1}$ in
[CZH21, Section 4] is necessary for proving their main theorems. In
particular, there is no need to construct $V_{0}$.

###  10.3. Discrete Time Ecological Models

In this section we consider a general discrete-time ecological model discussed
in [HNC21, (2.1)]

(10.8) $\displaystyle X_{i}(t+1)$
$\displaystyle=X_{i}(t)F_{i}(Z(t),\xi(t)),\quad i=1,\dots,n$ $\displaystyle
Y(t+1)$ $\displaystyle=G(Z(t),\xi(t))\,.$

Here $\mathcal{M}=[0,\infty)^{n}\times\mathbb{R}^{\kappa_{0}}$, where each
coordinate of $X=(X_{1},\cdots,X_{n})\in[0,\infty)^{n}$ represents the
population of some species and $Y\in\mathbb{R}^{\kappa_{0}}$ represents
auxiliary variables such as “eco-environmental feedbacks, forcing, the
structure of each species or other factors” [HNC21]. The stochasticity is
represented by iid random variables $\\{\xi(t)\\}_{t\in\mathbb{N}}$ taking
values in a Polish space $\Xi$ which models the state of the environment on
the time interval $[t,t+1)$. The functions
$F:\mathcal{M}\times\Xi\to(0,\infty)^{n}$ and
$G:\mathcal{M}\times\Xi\to\mathbb{R}^{\kappa_{0}}$ are measurable in
$(z,\xi)\in\mathcal{M}\times\Xi$ and continuous in $z$ for every fixed $\xi$.

We also assume the existence of a proper function
$\Upsilon:\mathcal{M}\to[1,\infty)$ and constants
$\epsilon>0,\rho\in(0,1),C>0$ such that
$\mathcal{P}\Upsilon\leq\rho^{2}\Upsilon+C^{2}$ and

(10.9) $\sum_{i=1}^{n}\mathbb{E}[|\log
F_{i}(z,\xi(0))|^{2+\epsilon}]\lesssim\Upsilon(z)^{\frac{1}{4}}\leq\Upsilon(z)^{\frac{1}{2}}\,,$

where the last inequality follows since $\Upsilon\geq 1$.

######  Remark 10.9.

Compared to [HNC21], we do not need to assume [HNC21, Assumption 2.2]. Also,
the existence of our $\Upsilon$ is weaker than [HNC21, Assumption 2.1 A3]
since for any fixed $\gamma>0$, $|\log
x|^{4}\lesssim(x\vee\frac{1}{x})^{\gamma}$ for all $x>0$.

######  Remark 10.10.

By using more delicate arguments in 9.6 (vi), it suffices to assume
$\mathbb{E}[|\log F_{i}(z,\xi(0))|^{2+\epsilon}]\lesssim\Upsilon(z)^{\gamma}$
for some $\gamma<\frac{1}{2}$, but we choose $\gamma=\frac{1}{4}$ for ease and
concreteness. Also, for 10.12 it is enough to take $\gamma<1$.

For $I\subset\\{1,\dots,n\\}$ denote

$\displaystyle\mathcal{M}^{I}$
$\displaystyle\coloneqq\\{(x,y)\in\mathcal{M}\mid x_{i}=0\quad\forall i\notin
I\\}$ $\displaystyle\mathcal{M}_{0}^{I}$
$\displaystyle\coloneqq\\{(x,y)\in\mathcal{M}^{I}\mid\exists i\in I\text{ such
that }x_{i}=0\\}$ $\displaystyle\mathcal{M}_{+}^{I}$
$\displaystyle\coloneqq\mathcal{M}^{I}\setminus\mathcal{M}_{0}^{I}$

When $I=\\{1,\dots,n\\}$ we suppress the superscript and write $\mathcal{M}$
and $\mathcal{M}_{0}$ instead of $\mathcal{M}^{I}$ and $\mathcal{M}^{I}_{0}$.
Note that $\mathcal{M}_{0}$ is then the set of states where at least one
species is extinct. It is standard to verify (continuity of $F_{i}$ and the
dominated convergence theorem) that 9.6 (iii) and (iv) are satisfied for
$(\mathcal{M},\mathcal{M}_{0},\mathcal{M}_{+},\\{Z(t)\\}_{t\in\mathbb{N}})$ so
that $(\mathcal{M},\mathcal{M}_{0},\mathcal{M}_{+},\\{Z_{t}\\}_{t\geq 0})$ is
a Feller quadruple (see 2.6).

######  Remark 10.11.

We slightly abuse notation by using notation $\\{Z_{t}\\}_{t\geq 0}$ also for
the continuous-time Markov process $Z_{t}\coloneqq Z(N(t))$, where
$N:[0,\infty)\to\mathbb{N}$ is a Poisson process. We keep this notation for
the rest of this section, using $\cdot(t)$ for discrete-time Markov chains and
$\cdot_{t}\coloneqq\cdot(N(t))$ for the continuous-time counterpart.

For $i\in\\{1,\dots,n\\}$ we consider the functions
$V_{i}:\mathcal{M}_{+}\to\mathbb{R}$ defined by $V_{i}(x,y)=-\log{x_{i}}$. For
any $(x,y)=z\in\mathcal{M}_{+}$ we have

(10.10)
$V_{i}(Z(1))-V_{i}(z)=-\log{x_{i}F_{i}(z,\xi(0))}+\log{x_{i}}=-\log{F_{i}(z,\xi(0))}$

and then by the definition of $\mathcal{L}$ in (9.6) and our assumption
(10.9), it holds that

(10.11) $\displaystyle\mathcal{L}V_{i}(z)$ $\displaystyle=-\mathbb{E}[\log
F_{i}(z,\xi(0))]$
$\displaystyle\mathbb{E}[|V_{i}(Z(1))-V_{i}(z)|^{2+\epsilon}]$
$\displaystyle\lesssim\Upsilon(z)^{1/4}$

Recall that $F_{i}>0$, and therefore $\log(F_{i}(z,\xi(0)))$ is well defined.
Also, $\mathcal{L}V_{i}$ extends to the continuous function
$H_{i}(z)\coloneqq-\mathbb{E}[\log F_{i}(z,\xi(0))]$ on $\mathcal{M}$, where
the continuity follows from (10.9), 8.15, and 2 (the Feller property requires
continuity of $z\mapsto\mu^{z}_{Z}$, where $\mu^{z}_{Z}$ is the law of
$Z_{1}^{z}$). Similarly, we define
$\tilde{V}_{i}:\mathcal{M}_{+}\to[0,\infty)$ by $\tilde{V}_{i}\coloneqq
V_{i}\vee 0$. Since $a\mapsto a\vee 0$ is 1-Lipschitz, by (10.11)

(10.12)
$\mathbb{E}[|\tilde{V}_{i}(Z(1))-\tilde{V}_{i}(z)|^{2+\epsilon}]\lesssim\Upsilon(z)^{1/4}$

and, as above, $\mathcal{L}\tilde{V}_{i}$ extends to a continuous function
$\tilde{H}_{i}$ on $\mathcal{M}$. Explicitly, since
$\mathcal{L}\tilde{V_{i}}(z)$ is given by $-\mathbb{E}[\log
F_{i}(z,\xi(0))]+\mathbb{E}[(-\log x_{i}F_{i}(z,\xi(0)))\wedge
0]-\mathbb{E}[(-\log x_{i})\wedge 0]$, $\tilde{H}_{i}(z)$ can be defined by
formally setting $(-\log 0)\wedge 0=0$, so in particular $\tilde{H}_{i}$
agrees with $H_{i}$ on $\\{x_{i}=0\\}$.

Let us first focus on the case $I=\\{1,\cdots,n\\}$. If $p_{i}>0$ are
constants and $V\coloneqq\sum_{i=1}^{n}p_{i}V_{i}$, then by (10.11) and 9.6
(vi) we have that 3, 4 (ii), and 5 hold for
$(\mathcal{M},\mathcal{M}_{0},\mathcal{M}_{+},\\{Z_{t}\\}_{t\geq 0})$. In
addition, 4(i) follows from the definition of $V$.

Next, we provide sufficient condition for 4(iii) to hold. For ergodic $\mu\in
P_{inv}(\mathcal{M})$, [HNC21, (2.3)] defines the “expected per-capita growth
rate of species i”

(10.13) $r_{i}(\mu)\coloneqq-\mu H_{i}=-\mu\tilde{H}_{i}\,,$

which is well defined by 2.22. The second equality follows from [HNC21,
Proposition 2.1] and the fact that $\tilde{H}_{i}$ agrees with $H_{i}$ on
$\\{x_{i}=0\\}$. The following theorem ([HNC21, Theorem 2.4]) is an immediate
consequence of 9.6 (vii) and 3.5:

######  Theorem 10.12.

Suppose there exist $p_{i}>0$ such that for all ergodic $\mu\in
P_{inv}(\mathcal{M}_{0})$

$\sum_{i=1}^{n}p_{i}r_{i}(\mu)<0\,,$

so by 2.22 $V$ satisfies 4(iii). If $\mathcal{M}_{0}$ is accessible in the
sense of (3.1) (see 9.6 (ii)), then there exists an $\alpha>0$ such that for
all initial conditions $z\in\mathcal{M}_{+}$

$\mathbb{P}\Big{(}\limsup_{t\to\infty}\frac{\log\min_{1\leq i\leq
n}X_{i}^{z}(t)}{t}\leq-\alpha\Big{)}=1\,.$

######  Remark 10.13.

Note that the definition of accessibility in [HNC21, pages 11-12] implies
(3.1) as detailed in 3.6.

Next we focus on the case $I\subsetneq\\{1,\dots,n\\}$ and prove the
assertions of [HNC21, Theorem 2.5]. Note that as remarked in [HNC21], by using
[HN18] it is possible to derive the (nontrivial) corollaries [HNC21, Theorem
2.6, Theorem 2.7].

######  Theorem 10.14.

Suppose $I\subsetneq\\{1,\dots,n\\}$ is such that

  1. (i)

$P_{inv}(\mathcal{M}_{+}^{I})$ is nonempty and all ergodic measures $\mu\in
P_{inv}(\mathcal{M}_{+}^{I})$ satisfy $\max_{i\notin I}r_{i}(\mu)<0$.

  2. (ii)

For all $\nu\in P_{inv}(\mathcal{M}_{0}^{I})$ (not necessarily ergodic),
$\max_{i\in I}r_{i}(\nu)>0$.

Then there exists $\alpha_{I}>0$ such that for any compact set
$\mathcal{K}_{I}\subset\mathcal{M}_{+}^{I}$,

$\lim_{z\to\mathcal{K}_{I},z\in\mathcal{M}_{+}}\mathbb{P}\Big{(}\limsup_{t\to\infty}\frac{\log\max_{i\notin
I}X_{i}^{z}(t)}{t}\leq-\alpha_{I}\Big{)}=1\,.$

######  Remark 10.15.

Recall that $\mathcal{M}_{+}^{I}$ is the set where the species in $I$ all have
positive population while all other species are extinct. As explained in
[HNC21], the assumption (i) asserts that when all species in $I$ are alive and
a potential invading species not in $I$ has small population, on average the
invading species will die out. The assumption (ii) is equivalent to saying
that, in the absence of other species, the species in $I$ will coexist
(persist). Thus, heuristically, if the initial populations are close to
$\mathcal{M}_{+}^{I}$, then with high probability, the species not in $I$
disappear exponentially fast.

###### Proof.

As above, it is standard to verify that 9.6 (iii) and (iv) are satisfied for
$(\mathcal{M},\mathcal{M}^{I},\mathcal{M}_{+},\\{Z(t)\\}_{t\in\mathbb{N}})$,
so $(\mathcal{M},\mathcal{M}^{I},\mathcal{M}_{+},\\{Z_{t}\\}_{t\geq 0})$ is a
Feller quadruple.

First note that by 2.22 the sets $\\{(r_{i}(\nu))_{i\in I}\mid\nu\in
P_{inv}(\mathcal{M}_{0}^{I})\\}\subset\mathbb{R}^{|I|}$ and
$\\{(r_{i}(\nu))_{i\notin I}\mid\nu\in P_{inv}(\mathcal{M}_{0}^{I})\\}$ are
compact and convex. Thus, by Hahn-Banach separation theorem and (ii) there are
$p_{i}>0$ such that

(10.14) $\inf_{\nu\in P_{inv}(\mathcal{M}_{0}^{I})}\sum_{i\in
I}p_{i}r_{i}(\nu)>0\,,\qquad\sup_{\nu\in
P_{inv}(\mathcal{M}_{0}^{I})}\max_{i\notin I}|r_{i}(\nu)|<\infty\,.$

Similarly, $\\{(r_{i}(\mu))_{i\notin I}\mid\mu\in
P_{inv}(\mathcal{M}_{+}^{I})\\}$ is compact and so [HNC21, Proposition 2.1]
and (i) imply

(10.15) $\sup_{\mu\in P_{inv}(\mathcal{M}_{+}^{I})}\max_{i\notin
I}r_{i}(\mu)<0\,,\qquad r_{i}(\mu)=0\quad\textrm{for any}\quad i\in I,\mu\in
P_{inv}(\mathcal{M}_{+}^{I})\,.$

Since $\mathcal{M}^{I}=\mathcal{M}_{+}^{I}\cup\mathcal{M}^{I}_{0}$, then any
$\mu\in P_{inv}(\mathcal{M}^{I})$ can be decomposed as a convex combination of
invariant measures on $\mathcal{M}_{+}^{I}$ and $\mathcal{M}^{I}_{0}$
respectively. Thus, by (10.14) and (10.15) there is $\hat{p}>0$ such that

(10.16) $\inf_{\mu\in P_{inv}(\mathcal{M}^{I})}\sum_{i\in
I}p_{i}r_{i}(\mu)-\hat{p}\max_{i\notin I}r_{i}(\mu)>0\,.$

This motivates the choice of average Lyapunov function

(10.17) $V_{p}\coloneqq-\sum_{i\in
I}p_{i}\tilde{V}_{i}-\frac{\hat{p}}{p}\log\sum_{i\notin I}x_{i}^{p}$

for some $1>p>0$ to be determined later. Since $\tilde{V}_{i}\geq 0$, then 4
(i) holds and

$\limsup_{t\to\infty}-\frac{V_{p}(X_{i}^{z}(t))}{t}\geq\limsup_{t\to\infty}\frac{\hat{p}}{p}\log\sum_{i\notin
I}(X_{i}^{z}(t))^{p}\geq\hat{p}\limsup_{t\to\infty}\frac{\log\max_{i\notin
I}X_{i}^{z}(t)}{t}\,,$

so the proof will be finished once we show that 3.2 applies.

We will use 3.9 with a suitable change of coordinates for the variables
$x_{i}$ where $i\notin I$. Specifically, we introduce the Feller quadruple
$(\mathcal{N},\mathcal{N}_{0},\mathcal{N}_{+},$
$\\{(v_{p}(t),r_{p}(t),w_{p}(t),y_{p}(t))\\}_{t\geq 0})$ and the quadruple map
$\pi_{p}$ as follows.

Without loss of generality, otherwise relabel the variables, we assume that
$I$ are the last $|I|$ coordinates of $[0,\infty)^{n}$. Let
$\mathcal{N}=\bigtriangleup^{n-|I|-1}\times[0,\infty)\times[0,\infty)^{|I|}\times\mathbb{R}^{\kappa_{0}}$,
where

$\bigtriangleup^{n-|I|-1}=\Big{\\{}v\in[0,1]^{n-|I|}\mid\sum_{i=1}^{n-|I|}v_{i}=1\Big{\\}}$

is a simplex. For $v\in\bigtriangleup^{n-|I|-1}$, $r\in[0,\infty)$,
$w\in[0,\infty)^{|I|}$, $y\in\mathbb{R}^{\kappa_{0}}$ define
$\pi_{p}:\mathcal{N}\to\mathcal{M}$ by

$\pi_{p}(v,r,w,y)=(x,y)\text{ where }x_{I}=w\text{ and
}x_{i}=rv_{i}^{1/p}\text{ for }i\notin I\,,$

where $x_{I}$ is the element of $[0,\infty)^{|I|}$ with coordinates equal to
the last $|I|$ coordinates of $x$. With
$\mathcal{N}_{0}\coloneqq\pi_{p}^{-1}(\mathcal{M}^{I})=\\{r=0\\}$ and
$\mathcal{N}_{+}\coloneqq\pi_{p}^{-1}(\mathcal{M}_{+})$, we obtain a Feller
quadruple
$(\mathcal{N},\mathcal{N}_{0},\mathcal{N}_{+},\\{(v_{p,t},r_{p,t},w_{p,t},y_{p,t})\\}_{t\geq
0})$, where $(v_{p,t},r_{p,t},w_{p,t},y_{p,t})$ is the Markov process
corresponding to the Markov chain $(v_{p}(t),r_{p}(t),w_{p}(t),y_{p}(t))$ (see
10.11) given by setting
$(X(t),Y(t))=\pi_{p}(v_{p}(t),r_{p}(t),w_{p}(t),y_{p}(t))$ in (10.8):

(10.18) $\displaystyle v_{i}(t+1)$
$\displaystyle=v_{i}(t)\frac{\tilde{F}_{i}(v(t),r(t),w(t),y(t),\xi(t))^{p}}{\sum_{i\not\in
I}\tilde{F}_{i}(v(t),r(t),w(t),y(t),\xi(t))^{p}}\quad i\notin I$
$\displaystyle r(t+1)$ $\displaystyle=r(t)\Big{(}\sum_{i\not\in
I}v_{i}(t)\tilde{F}_{i}(v(t),r(t),w(t),y(t),\xi(t))^{p}\Big{)}^{1/p}$
$\displaystyle w(t)$ $\displaystyle=X_{I}(t)$ $\displaystyle y(t)$
$\displaystyle=Y(t)$

where we omit the subscript $p$ and use $\tilde{F}_{i}$ to denote
$F_{i}\circ\pi_{p}$. Then clearly $\pi_{p}$ is a quadruple map (see 3.7) from
$(\mathcal{N},\mathcal{N}_{0},\mathcal{N}_{+},\\{(v_{p,t},r_{p,t},w_{p,t},y_{p,t})\\}_{t\geq
0})$ to $(\mathcal{M},\mathcal{M}^{I},\mathcal{M}_{+},\\{Z_{t}\\}_{t\geq 0})$.

Next we compute $\mathcal{L}V_{p}$ in the new and old coordinates, where
recall $V_{p}$ was defined in (10.17). Since we have already discussed
$\mathcal{L}\tilde{V}_{i}$ above, we provide details for the
$\frac{\hat{p}}{p}\log\sum_{i\notin I}x_{i}^{p}$ term. Note that for the
initial condition $z=(x,y)=\pi_{p}(\vec{a})\in\mathcal{M}_{+}$ where
$\vec{a}=(v,r,w,y)\in\mathcal{N}_{+}$ we have

(10.19) $\displaystyle\log\sum_{i\notin I}X_{i}^{z}(1)^{p}-\log\sum_{i\notin
I}X_{i}^{z}(0)^{p}$ $\displaystyle=\log\sum_{i\notin
I}x_{i}^{p}F_{i}(x,y,\xi(0))^{p}-\log\sum_{i\notin I}x_{i}^{p}$
$\displaystyle=\log\frac{\sum_{i\notin
I}x_{i}^{p}F_{i}(x,y,\xi(0))^{p}}{\sum_{i\notin I}x_{i}^{p}}$
$\displaystyle=\log\sum_{i\notin
I}v_{i}^{p}\tilde{F_{i}}(\vec{a},\xi(0))^{p}\,.$

Thus, we have

$\displaystyle\mathcal{L}V_{p}(x,y)$ $\displaystyle=-\sum_{i\in
I}p_{i}\tilde{H}_{i}(x,y)-\frac{\hat{p}}{p}\mathbb{E}[\log\Big{(}\frac{\sum_{i\notin
I}x_{i}^{p}F_{i}(x,y,\xi(0))^{p}}{\sum_{i\notin I}x_{i}^{p}}\Big{)}]$
$\displaystyle\mathcal{L}V_{p}\circ\pi_{p}(\vec{a})$
$\displaystyle=-\sum_{i\in
I}p_{i}\tilde{H}_{i}\circ\pi_{p}(\vec{a})-\frac{\hat{p}}{p}\mathbb{E}\Big{[}\log\sum_{i\notin
I}v_{i}\tilde{F}_{i}(\vec{a},\xi(0))^{p}\Big{]}$

Then for any $v\in\bigtriangleup^{n-|I|-1}$ and $0<p<1$, since $|\log|$ is
quasiconvex,

(10.20) $\Big{|}\log\sum_{i\notin
I}v_{i}\tilde{F}_{i}(\vec{a},\xi(0))^{p}\Big{|}\leq\max_{i\notin
I}|\log\tilde{F}_{i}(\vec{a},\xi(0))^{p}|\leq p\sum_{i\notin
I}|\log\tilde{F}_{i}(\vec{a},\xi(0))|\,.$

In particular, (10.9) and 8.15 imply that $\mathcal{L}V_{p}\circ\pi_{p}$
extends to a continuous function $H_{p}$ on $\mathcal{N}$. By combining the
definition of $V_{p}$ in (10.17) with (10.19), (10.20), (10.9), and (10.12) we
obtain

$\mathbb{E}[|V_{p}(Z(1))-V_{p}(z)|^{2+\epsilon}]\lesssim\Upsilon(z)^{1/4}\,,$

and therefore by 9.6 (vi) we have that 3, 4 (ii), and 5 hold for
$(\mathcal{M},\mathcal{M}^{I},\mathcal{M}_{+},\\{Z_{t}\\}_{t\geq 0})$.

It remains to choose $1>p>0$ so that the assumption (3.2) in 3.9 holds, that
is, so that there is a constant $\alpha>0$ such that $\mu H_{p}\geq\alpha$ for
all $\mu\in P_{inv}(\mathcal{N}_{0})$. By (10.16) it suffices to show that

(10.21) $\lim_{p\downarrow 0}\inf_{\mu\in P_{inv}^{p}(\mathcal{N}_{0})}\mu
H_{p}\geq\inf_{\mu\in P_{inv}(\mathcal{M}^{I})}\sum_{i\in
I}p_{i}r_{i}(\mu)-\hat{p}\max_{i\notin I}r_{i}(\mu)\,,$

where $P_{inv}^{p}(\mathcal{N}_{0})$ is the set of all invariant measures
supported on $\mathcal{N}_{0}$ for (10.18) (see 9.6 (i)).

To show (10.21), we will define a function $H_{0}$ and a Markov Process which
are (in some sense) the limits of $H_{p}$ and (10.18) as $p\downarrow 0$.
Then, we verify (10.21) with $\lim_{p\downarrow 0}$ removed and $H_{p}$
replaced by $H_{0}$. Finally, by 3.11 we will conclude (10.21). Next we
provide details.

In the remainder of the argument we only deal with Markov processes on
$\mathcal{N}_{0}=\\{r=0\\}$. Since $\pi_{p}(v,0,w,y)$,
$\tilde{F}_{i}(v,0,w,y,\xi(0))$, and $\tilde{G}(v,0,w,y,\xi(0))$ are
independent of $v$ and $p$, we write $\pi(w,y)$, $\tilde{F}_{i}(w,y,\xi(0))$,
and $\tilde{G}(w,y,\xi(0))$ instead. Similarly, we write
$\tilde{\Upsilon}(w,y)$ for $\Upsilon\circ\pi_{p}(v,0,w,y)$. Additionally, we
drop $r$ and use $(v,w,y)$ to denote $(v,0,w,y)\in\mathcal{N}_{0}$.

We define the function $H_{0}$ on $\mathcal{N}_{0}$ by

$H_{0}(v,w,y)=-\sum_{i\in
I}p_{i}\tilde{H}_{i}\circ\pi(w,y)+\hat{p}\sum_{i\notin
I}v_{i}H_{i}\circ\pi(w,y)$

and the Markov chain $\\{(v_{0}(t),w_{0}(t),y_{0}(t))\\}_{t\in\mathbb{N}}$ on
$\mathcal{N}_{0}$ by

(10.22) $\displaystyle v_{0}(t+1)$ $\displaystyle=v_{0}(t)$ $\displaystyle
w_{0}(t)$ $\displaystyle=X_{I}(t)$ $\displaystyle y_{0}(t)$
$\displaystyle=Y(t)\,.$

Overall, we defined a collection of Markov processes
$\\{\\{(v_{p,t},w_{p,t},y_{p,t})\\}_{t\geq 0}\\}_{p\in[0,1/2]}$ on
$\mathcal{N}_{0}$ corresponding to the Markov chains given by (omitting the
subscript $p$)

(10.23) $\displaystyle v_{i}(t+1)$
$\displaystyle=v_{i}(t)\frac{\tilde{F}_{i}(w(t),y(t),\xi(t))^{p}}{\sum_{i\notin
I}v_{i}(t)\tilde{F}_{i}(w(t),y(t),\xi(t))^{p}}\quad i\notin I$ $\displaystyle
w_{i-n+|I|}(t+1)$
$\displaystyle=\tilde{F}_{i}(w(t),y(t),\xi(t))w_{i-n+|I|}(t)\quad i\in I$
$\displaystyle y(t+1)$ $\displaystyle=\tilde{G}(w(t),y(t),\xi(t))$

and a function $H:[0,1/2]\times\mathcal{N}_{0}\to\mathbb{R}$ given by

$H(p,v,w,y)=\begin{cases}-\sum_{i\in
I}p_{i}\tilde{H}_{i}\circ\pi(w,y)-\frac{\hat{p}}{p}\mathbb{E}[\log\sum_{i\notin
I}v_{i}\tilde{F}_{i}(w,y,\xi(0))^{p}]&\text{if }p>0\\\ -\sum_{i\in
I}p_{i}\tilde{H}_{i}\circ\pi(w,y)+\hat{p}\sum_{i\notin
I}v_{i}H_{i}\circ\pi(w,y)&\text{if }p=0\end{cases}$

To apply 3.11, we note that 9.6 (iv) implies 2 for the Markov process
$(p_{t},v_{p,t},w_{p,t},y_{p,t})$ on $[0,1/2]\times\mathcal{N}_{0}$, where
$p_{t}\equiv p$ (simply look at (10.23) and note that the right hand side is
continuous in $(p,v,w,y)$). Also, since we have already noted that by 9.6 (vi)
that $Z_{t}$ satisfies 3 and 5 (i)-(iii) with
$U=\sqrt{\Upsilon},U^{\prime}=(1-\rho)U,W=\Upsilon^{1/4},W^{\prime}=(1-\sqrt{\rho})W$,
and some $K>0$, it follows easily from
$\pi(w_{p,t}^{\vec{b}},y_{p,t}^{\vec{b}})=Z_{t}^{z}$ for all initial
conditions $\vec{b}=(v,w,y)\in\mathcal{N}_{0}$, $z=\pi(w,y)\in M^{I}$ that
each $(v_{p,t},w_{p,t},y_{p,t})$ satisfies 3 and 5 (i)-(iii) with
$U=\sqrt{\tilde{\Upsilon}},U^{\prime}=(1-\rho)\sqrt{\tilde{\Upsilon}},W=\tilde{\Upsilon}^{1/4},\text{
and }W^{\prime}=(1-\sqrt{\rho})\tilde{\Upsilon}^{1/4}$, and $K$ the same as
for $Z_{t}$. In particular, $W,W^{\prime},K,U,U^{\prime}$ are independent of
$p$.

To satisfy the rest of the assumptions of 3.11, we claim that $H$ is a
continuous function which vanishes over
$(p,v,w,y)\mapsto\tilde{\Upsilon}^{1/4}(w,y)$. The continuity on
$(0,1/2)\times\mathcal{N}_{0}$ follows from (10.9), (10.20), and 8.15. Also,
$H_{0}$ is clearly continuous. Thus, for continuity of $H$ it suffices to show
that $H_{p}\to H_{0}$ uniformly on compact sets.

As a preliminary calculation, for fixed $(v,w,y)\in\mathcal{N}_{0}$ and
$p\in(0,1/2)$, we set $\tilde{F}_{i}:=\tilde{F}_{i}(w,y,\xi(0))$ and estimate
by the mean value theorem that

(10.24) $\displaystyle\Big{|}\frac{v_{i}\tilde{F}_{i}^{p}}{\sum_{i\not\in
I}v_{i}\tilde{F}_{i}^{p}}-v_{i}\Big{|}$ $\displaystyle\leq
p\sup_{0<p*<p}\Big{|}\frac{v_{i}\tilde{F}_{i}^{p*}\log\tilde{F}_{i}}{\sum_{i\not\in
I}v_{i}\tilde{F}_{i}^{p*}}\Big{|}+\Big{|}\frac{v_{i}\tilde{F}_{i}^{p*}\sum_{i\not\in
I}v_{i}\tilde{F}_{i}^{p*}\log\tilde{F}_{i}}{(\sum_{i\not\in
I}v_{i}\tilde{F}_{i}^{p*})^{2}}\Big{|}$ $\displaystyle\leq
p|\log\tilde{F}_{i}|+p\max_{i\not\in I}|\log\tilde{F}_{i}|\leq
2p\sum_{i\not\in I}|\log\tilde{F}_{i}|$

Applying the mean value theorem to $p\mapsto\log\Big{(}\sum_{i\notin
I}v_{i}\tilde{F}_{i}^{p}\Big{)}$ and recalling that $\sum_{i\not\in I}v_{i}=1$
and $H_{i}\circ\pi=-\mathbb{E}[\log\tilde{F_{i}}]$, we obtain

$\displaystyle|H_{p}(v,w,y)-H_{0}(v,w,y)|$
$\displaystyle=\hat{p}\Big{|}\mathbb{E}\Big{[}\frac{1}{p}\Big{[}\log\Big{(}\sum_{i\notin
I}v_{i}\tilde{F}_{i}^{p}\Big{)}-\log\Big{(}\sum_{i\notin
I}v_{i}\Big{)}\Big{]}-\sum_{i\notin I}v_{i}\log\tilde{F_{i}}\Big{]}\Big{|}$
$\displaystyle\leq\sup_{0<p*<p}\hat{p}\mathbb{E}\Big{[}\Big{|}\sum_{i\notin
I}\Big{(}\frac{v_{i}\tilde{F}_{i}^{p*}}{\sum_{i\not\in
I}v_{i}\tilde{F}_{i}^{p*}}-v_{i}\Big{)}\log\tilde{F_{i}}\Big{|}\Big{]}$
$\displaystyle\lesssim\hat{p}p\mathbb{E}\Big{[}\Big{(}\sum_{i\not\in
I}|\log\tilde{F}_{i}|\Big{)}^{2}\Big{]}\lesssim\hat{p}p(\tilde{\Upsilon}^{1/4}(w,y))^{2/(2+\epsilon)}\,,$

where the first $\lesssim$ is by (10.24) and the second is by (10.9). This
shows that $H_{p}\to H_{0}$ uniformly on compacts, and therefore $H$ is
continuous. Additionally, it shows that $H$ vanishes over
$(p,v,w,y)\mapsto\tilde{\Upsilon}^{1/4}(w,y)$ if and only if $H_{0}$ vanishes
over $\tilde{\Upsilon}^{1/4}$. The latter is a consequence of the fact that
$H_{i}$ and $\tilde{H_{i}}$ vanish over $\Upsilon^{1/4}$ (in fact, over
$\Upsilon^{1/2}$), which is shown above (see (10.11) and (10.12)).

Thus, the assumptions of 3.11 are satisfied and so we conclude that

(10.25) $\lim_{p\downarrow 0}\inf_{\mu\in P_{inv}^{p}(\mathcal{N}_{0})}\mu
H_{p}\geq\inf_{\nu\in P_{inv}^{0}(\mathcal{N}_{0})}\nu H_{0}\,,$

where $P_{inv}^{0}(\mathcal{N}_{0})$ is the set of all invariant measures for
(10.22) (see 9.6 (i)). By 2.22, the right hand side of (10.25) is unchanged if
we restrict $\nu\in P_{inv}^{0}(\mathcal{N}_{0})$ to be ergodic, in which case
$\nu$ is supported on $\\{v=v_{0}\\}$ for some
$v_{0}\in\bigtriangleup^{n-|I|-1}$ (recall from (10.22) that
$v_{0}(t+1)=v_{0}(t)$). In particular, there is $\mu\in
P_{inv}(\mathcal{M}^{I})$ (equal to the pushforward of $\nu$ by $\pi$) such
that

$\nu H_{0}=-\sum_{i\in I}p_{i}\mu\tilde{H}_{i}+\hat{p}\sum_{i\notin
I}(v_{0})_{i}\mu H_{i}\geq\inf_{\mu\in P_{inv}(\mathcal{M}^{I})}\sum_{i\in
I}p_{i}r_{i}(\mu)-\hat{p}\max_{i\notin I}r_{i}(\mu)\,.$

Thus, (10.25) implies (10.21), which finishes the proof. ∎

###  10.4. Stochastic Kolmogorov Systems

In this subsection we mention how our results can be applied to Stochastic
Kolmogorov Equations, that is, SDEs of the form

(10.26) $dX_{i}=X_{i}f_{i}(X)dt+X_{i}g_{i}(X)dE_{i}\,,$

where $X_{t}\in[0,\infty)^{n}$, $f,g:[0,\infty)^{n}\to\mathbb{R}^{n}$ are
locally Lipschitz functions, and $(E_{1}(t),\dots,E_{n}(t))^{T}=A^{T}W(t)$,
where $A$ is a $d$ by $n$ matrix and $W(t)$ is a $d$-dimensional Brownian
motion. We denote $A^{T}A$ by $\Sigma=(\Sigma_{ij})_{1\leq i,j\leq n}$.
Similarly to the previous example, we could have also included auxiliary
variables $Y_{t}$, but we omit these for brevity. Our technical assumption is
that there exists a proper $U:[0,\infty)^{n}\to[1,\infty)$ in
$\mathcal{C}^{2}([0,\infty)^{n})$ and $K,c>0$ such that $\mathcal{L}U\leq
K-cU$ and $\sum_{i=1}^{n}|f_{i}(x)|+g_{i}(x)^{2}\lesssim
2K-\frac{\mathcal{L}U}{U}+\frac{\Gamma U}{U^{2}}$, where

$\displaystyle\mathcal{L}U(x)$
$\displaystyle=\sum_{i,j=1}^{n}\frac{1}{2}\Sigma_{ij}x_{i}x_{j}g_{i}(x)g_{j}(x)\partial_{i}\partial_{j}U(x)+\sum_{i=1}^{n}x_{i}f_{i}(x)\partial_{i}U(x)$
$\displaystyle\Gamma U(x)$
$\displaystyle=\sum_{i,j=1}^{n}\Sigma_{ij}x_{i}x_{j}g_{i}(x)g_{j}(x)\partial_{i}U(x)\partial_{j}U(x)\,.$

######  Remark 10.16.

If [HN18, Assumption 1.1 (3)] is satisfied then our technical assumption holds
with $U=(1+c^{T}x)^{\theta}$ for some small enough $1>\theta>0$. Similarly to
[HNC21, Assumption 3.1], if there exists a proper
$U:[0,\infty)^{n}\to[1,\infty)$ in $\mathcal{C}^{2}([0,\infty)^{n})$ such that
$\mathcal{L}U\leq K-cU\Big{(}1+\sum_{i=1}^{n}|f_{i}(x)|+g_{i}(x)^{2}\Big{)}$,
then our technical assumption is satisfied. We have no need of [HN18,
Assumption 1.1 (1)] (nondegeneracy), [HN18, Assumption 1.4], or [HNC21,
Assumption 3.2], although nondegeneracy is useful for deducing uniqueness of
invariant measures on certain sets and for deducing accessibility.

With $H_{i}(x)=\frac{1}{2}\Sigma_{ii}g_{i}(x)^{2}-f_{i}(x)$ and $r_{i}(\mu)$
defined as in (10.13), using 9.3 in place of 9.6 (vi) one can show that the
analogues of 10.12 and 10.14 hold for (10.26). We omit the proofs since they
are exactly the same as those given in Section 10.3 except that the details
are easier (Itô’s formula gives a closed-form expression for $H_{p}$ so we can
directly compute $H_{p}-H_{0}$ as opposed to using the mean value theorem
twice to bound $|H_{p}-H_{0}|$) and also instead of defining $\tilde{V_{i}}$
as $V_{i}\vee 0$ one should choose some smooth function
$v:\mathbb{R}\to[0,\infty)$ with bounded first and second derivatives with
$v(t)=t$ for $t\geq 1$ and instead set $\tilde{V_{i}}\coloneqq v\circ V_{i}$
(as in the proof of [Ben23, Theorem 5.1ii]).

Finally, we remark that the results in [HNS22] (under the weaker assumptions
described above) are corollaries of our theory. For example, [HNS22, Theorem
6.2] can be proved by considering
$V(x_{1},x_{2},x_{3})=-p_{1}\ln{x_{1}}-p_{2}\ln{x_{2}}-p_{3}\ln{x_{3}}-\frac{p_{0}}{p}v(-\ln({x_{1}^{p}+x_{2}^{p}+x_{3}^{p}}))$
for small enough $p>0$ and suitable $p_{0},p_{1},p_{2},p_{3}>0$ (see [HNS22,
Lemma 6.1]).

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