Exponential quasi-ergodicity

for processes with discontinuous trajectories

Aurélien Velleret
[Aix-Marseille Université, CNRS, Centrale Marseille, I2M,  Marseille, France,
email<EMAIL_ADDRESS>
§.§ Abstract :

This paper establishes new results
which allow to deduce
the exponential convergence to a
unique quasi-stationary distribution
in the total variation norm
for quite general strong Markov processes.
we can treat
non-reversible processes with discontinuous trajectories,
which seems to be a substantial breakthrough.
Considering jumps driven by Poisson Point Processes
in four different applications,
we intend to illustrate the potential of these results
and motivate a quite technical criterion.
Such criterion is expected to be much easier to verify
than an implied property which is crucial in our proof,
namely a comparison of the asymptotic extinction rate
between different initial conditions.

Keywords :
continuous-time and continuous-space Markov process jumps quasi-stationary distribution survival capacity Q-process Harris recurrence

§ INTRODUCTION

§.§ Presentation

This work is concerned
with the long time behavior of
quite general strong Markov processes,
conditionally upon the fact that this process
has not been absorbed in some "cemetery state"

–is not "extinct".
The focus is here
on continuous-time and continuous-space processes
with discontinuous trajectories
(discontinuous processes in short).
Even for simple examples,
most usual criteria seem inapplicable
or too abstract to be easily verified.

In order to obtain quantitative results
for the speed of convergence
and uniqueness,
the approach we follow
appears to be the only one currently available.
It extends the principles of Harris' recurrence,
with the additional condition
on the asymptotic extinction rate :

\begin{align*}
\frl{x}\frlq{t\ge u}
\PR_x(t<\ext) \le c\, \PR_{\alpha_c}(t<\ext),
\EQn{BdSv}{}
\end{align*}

where $\alpha_c$ is a reference measure related to small sets
and the constants $c, u>0$ are uniform
over the initial conditions $x$.
This inequality is crucial
in any of the current proofs
extending the results presented in <cit.>.
It is not only implied by the conditions
–except in <cit.> where a local estimate can be sufficient–
but also a major step in the proofs
(be it for <cit.>, <cit.>, <cit.>, <cit.>
<cit.>,  <cit.>, <cit.> or <cit.>).
Methods for verifying inequality BdSv
have been so far developped in the case of discrete state space
on one side,
and exploiting Harnack's inequality
in the case of diffusion processes
on the other side.
An approach
for treating the case of jump diffusion processes
was still lacking.
It is the goal of this paper to propose
a very flexible approach in this direction.
It allows us to treat various kinds of Markov processes,
including jump Markov processes
and jump diffusion processes.

Although it may seem natural to focus on estimating directly
$\PR_x(t<\ext)$ in order to ensure BdSv,
this will certainly be inefficient for such an asymptotic result (in time).
in the situations we are interested in,
the results of this paper show that
there exists some unique asymptotic extinction rate,
identical for any initial condition.
This rate has a priori no chance to be obtained explicitly,
because it implies the whole set of admissible trajectories.
the unavoidable error in estimating lower- versus upper-bounds
on finite time-intervals
cannot be compensated.
Adapting the argument of <cit.>,
we rather intend to show that the trajectories starting
from the reference measure
can largely absorb (that is integrate itself into)
the ones starting from this test initial condition
–potentially with a time-shift.
In <cit.> and <cit.>
is already presented a simplified version
where we can neglect
initial conditions $x$
in some "transitory domain"
(whose exit time
has an exponential moment of a certain order).
With our new assumption,
we generalize in some sense
this idea that some components of the trajectories
(started from a test initial condition)
may be considered transitory.
Our assumption states a criteria on the success probability
of an "attempt of absorption"
to ensure that the remaining component (not absorbed)
does not play any substantial role
in the asymptotic survival
(starting from the test initial condition).

In addition,
we could relax the condition that jumps are locally bounded,

as long as the rate of the "large jumps" is locally bounded.
The bound introduced in <cit.>
is exploited to deduce a lower-bound
of the survival probability
in the long run
starting at some exit time.

Such exit times serve as reference
to initiate the comparison (the "competition")
between excursions far away for a long time
and much more conservative trajectories.
The idea we propose here
is in the choice of the competitors to
excursions initiated by these large jumps :

given a trajectory $(X_t)$ with such large jump,
the "competitor" $(\hat{X}_t)$ coincides with $(X_t)$
until the jump time $T_j$, but does not jump :

\begin{align*}
\frlq{t< T_j} \hat{X}_t = X_t\pv
\quad \hat{X}_{T_j}
= \textstyle \lim_{t\rightarrow T_j} X_t
\neq X_{T_j}
\end{align*}

and then behaves in a conservative way
(comes back to the core and stays there).
To make sure the competitors are well associated,
the conservative trajectories
begin the competition
according to some exponential rate.
This rate is chosen small enough
because the same trajectory may be called
against several competitors
(in which case only the last call is considered).

Finally, the last result of conditional ergodicity is added
to the results presented in <cit.>,
relying on the criteria of <cit.>.
It would hold also under the assumptions given in <cit.>.

As a result,
we precise a new set of key practical conditions on the process
such that the marginal
of the process –at time $t$–
conditioned on not being extinct –also at time $t$– (the MCNE)
converges as $t\ifty$ to a unique distribution $\alpha$,
called the quasi-stationary distribution (the QSD)
–cf Subsections <ref> and <ref>,
or chapter 2 in <cit.> for more details on this notion.
In the aftermath of <cit.>,
we obtain not only
the existence and uniqueness of the QSD,
but also the exponential convergence in total variation norm
–cf Theorem <ref>.

Since our proof follows essentially
the path presented in <cit.>,

we deduce likewise
the existence of a specific eigenfunction $\eta$ of the infinitesimal generator,
with the same eigenvalue as the QSD.
It is approached by finite time estimates of the relative extinction
$\eta_t(x)$ (at time $t$ with $x$ as starting point)
–cf. Theorem <ref>.
This convergence motivates the name survival capacity
that we give to $\eta$
–sometimes described as the "reproductive value" in ecological models.
Again, the convergence is exponential
–and still not uniform over the state space.
Moreover, we deduce
the existence  of the Q-process.
Its marginal at time $t$ is given by
the limit –as $T\ifty$–
of the marginal of the original process at time $t$
conditioned on not being extinct at time $T$,
cf. Theorem <ref>.
Thus, it is often described
as the process conditioned to never being absorbed.
Finally, we deduce
the existence and uniqueness of its stationary distribution $\beta$
together with some exponential convergence
of the marginals of the Q-process
and a result of exponential ergodicity of the conditional history.

General surveys
like in <cit.>, in <cit.>
or more specifically for population dynamics in
give some overview on the large literature dedicated to QSD
(for which Pollett has collected quite an impressive bibliography,
cf. <cit.>).
We focus here on the general context
of continuous-time and continuous-space Markov processes,
without any assumption of reversibility
(we mean without any help from the spectral theory).
the most elementary argument to ensure the existence of the QSD
has to do with the compactness of the semi-group,
like in the Krein-Rutman theorem.
In <cit.>,
the authors manage to characterize the QSD
as the fixed point of the normalized semi-group
(on a well-designed subspace).
Another classical technique,
called renewal,
aims at approximating some QSD by stationary distributions
of a sequence of processes without extinction
(specific jumps replace the extinction events).

Yet, these techniques do not give
any insight on the speed of convergence
or on the uniqueness (for cases where it holds true).

Up to now, the only existing strategy
for obtaining such results is the one
initiated in <cit.>
and notably extended in <cit.> and <cit.>.
The aim of such techniques is specifically to establish
the exponential convergence in total variation
to the QSD.
Note that in <cit.>,
the authors deduce the convergence
from any initial law sufficiently light-tailed
to a specific QSD even if is not unique.

These techniques seem to generalize naturally
to time-inhomogeneous processes,
cf. <cit.>, <cit.>, <cit.>.

the reversibility of the process
is ensured by an  easy to check criteria(detailed balance).
it may be useful to design reversible processes with jumps,
but this criteria is very stringent
and clearly not generic.
Given the constraints highlighted
in the Appendix A of the very recent <cit.>,
multidimensional processes with jumps
are rather expected not to be reversible.
And as explained in <cit.>,
these processes are probably not the fastest to converge either.

On the other hand,
concerning our specific interest in discontinuous processes,
the main issue in this case is that
Harnack's inequality is not anymore granted,
while it is the crucial ingredient
in <cit.>-<cit.>, <cit.>

(for continuous-time and continuous-space
Markov processes).
it seems hardly treated,
let alone in <cit.>,
which applies to semi-Markov processes
(the kind of pure jump processes
where the waiting time is not necessarily exponential).
And even in this article,
the aim seems rather to relate the QSD
to some other property of invariance
(involving the operator of jump
rather than the semi-group in time)
and it does not provide practical criteria to ensure
the existence of the QSD.
The techniques of this paper
are thus a breakthrough
in the case of jump diffusion Markov processes,
as well as pure jump Markov processes.
Of course such discontinuities
are rather an additional difficulty
and are not required.

One could wonder if our new assumption
could also replace assumption $(F3)$ in  <cit.>.
For our argument to hold,
we need however to consider only confined trajectories of the process
(our conditions involving $T_{\cD_m}$),
be it for our survival estimate $(A5)$
or our mixing assumption $(A2)$.
While any constructive proof of $(F1)$ in <cit.>
certainly ensures in fact $(A2)$,
it is clearly not as clear for the condition
on $\gamma_2>0$ in their $(F2)$.

But still,
our criterion may also be improved
for cases where the QSD is not unique,
in which case we need more precision
on the cost of "failed attempts".
And to estimate such cost,
the best way would probably be to express it
in term of a Lyapunov function
like the $\psi_1$ given in their $(F2)$.

The remainder of the paper is organized as follows.
Subsection <ref> describes our general notations ;
Subsection <ref> presents our specific setup
of a Markov process with extinction ;
and Subsection <ref>
the decomposition of the state space
on which we base our assumptions.

Subsection <ref> presents
the main set of conditions which we show to be sufficient
for the exponential convergence to the QSD.
Subsection <ref>
states the three main theorems of this paper,
that extend those of <cit.>,
dealing respectively with the QSD,
the survival capacity and the Q-process.

In Section <ref>,
we present four applications of our general theorems.
Theses results are rather generic,
although in a simple setting,
and completely new
for such behaviors with jumps.
We hope that they will help the reader
to get insight
on our criterion and its adaptability.
The application of our theorems
to more detailed biological models
is intended to be done in future work,
We refer notably to <cit.>,
where we couple
a diffusive process specifying the population size
to a piecewise deterministic process
for the adaptation of this population.
But already,
the last three applications of this paper
are somewhat related
to the solution of specific non-local
reaction-diffusion equations
at low density, see Subsection <ref>.
In fact,
it would also be true
for a simplified version of the first example.
in Section <ref>,
we detail our proofs.

§.§ Elementary notations

In this paper, $k\ge 1$ means $k\in \N$
while $\tp \ge 0$ –resp. $c>0$–
should be understood as $\tp \in \R_+:= [0, \infty)$
–resp. $c\in \R_+^* $ $:= (0, \infty)$.
We denote classical sets of integers by :
$\quad \Z_+~:= \Lbr 0,1,2...\Rbr$,$\; \N~:= \Lbr 1,2, 3...\Rbr$,
$\; [\![m, n ]\!]~:= \Lbr m,\, m+1, ..., n-1,\, n\Rbr$ (for $m\le n$),
where the notation $:=$ makes explicit that we define some notation by this equality.

For maxima and minima, we usually denote :
$s \vee t := \max\{s, t\}$,
$s \wedge t := \min\{s, t\}.$
Accordingly, for a function $\varphi$, $\varphi^{\wedge}$
–resp. $\varphi^{\vee}$–  will usually be used
for a lower-bound –resp. for an upper-bound– of $\varphi$.

Let $\big(\Omega; (\F_\tp)_{\tp\ge0}; (X_\tp)_{\tp\ge0}; (P_\tp)_{\tp\ge0}; (\PR_x)_{x\in \cX\cup {\partial}} \big)$ be a time homogeneous strong Markov
process with cadlag paths on some Polish space   [see <cit.>, Definition III.1.1], where $(\cX; \B)$ is a measurable space and .
We also assume that the filtration $(\F_\tp)_{\tp\ge0}$ is right-continuous and complete.
We recall that $\PR_x(X_0 = x) = 1$, $P_\tp$ is the transition function of the process satisfying the usual measurability assumptions
and the Chapman-Kolmogorov equation.
The hitting time (resp. the exit time out) of $\cD$,
for some domain $\cD \subset \cX$,
will generally be denoted by $\tau_{\cD}$ (resp. by $T_{\cD}$).

While dealing with the Markov property between different stopping times,
we wish to   clearly indicate with our notation
that we introduce a copy of $X$
–ie with the same semigroup $(P_\tp)$–

independent of $X$ given its initial condition.
This copy –and the associated stopping times–
is then denoted with a tilde
–$\wtd{X},\, \wtd{\ext},\, \wtd{T}_\cD$ etc.
In the notation
$\PR_{X(\UDc)} (t- \UDc < \wtd{\ext})$ for instance,
$\UDc$ and $X(\UDc)$ refer to the initial process $X$
while $\wtd{\ext}$ refers to the

§.§ The stochastic process with absorption

We consider a strong
Markov processes absorbed at $\partial$ : the cemetery.
More precisely, we assume that
$X_s = \partial$ implies $X_\tp = \partial$ for all $\tp \ge s$.
This implies that the extinction time :
$\quad \ext~:= \inf\Lbr \tp \ge 0 \pv X_\tp = \partial \Rbr\quad$
is a stopping time.
Thus, the family $(P_\tp)_{\tp \ge0}$
defines a non-conservative semigroup of operators on the set
–resp.  –
of positive –resp. bounded–
$(\cX,\B)$-measurable real-valued functions.
For any probability measure $\mu$ on $\cX$,
that is $\mu \in \Mone$, and
–or – we use the notations :

$\quad\PR_\mu (.)~:= \int_{\cX} P_x(.) \; \mu(dx), \quad
\langle \mu \bv f\rangle~:= \int_{\cX} f(x) \; \mu(dx)$.

We denote by $\E_x$ (resp. $\E_\mu$)
the expectation according to $\PR_x$ (resp. $\PR_\mu$).

\begin{align*}
\mu P_\tp(dy)~:= \PR_\mu(X_\tp \in dy),\qquad
&\langle \mu P_\tp\bv f\rangle \,=\, \langle \mu \bv P_\tp f\rangle
\,= \E_\mu[f(X_\tp)],
\\
\mu A_\tp(dy)~:= \PR_\mu(X_\tp \in dy \bv \tp < \ext), \qquad
&\langle\mu A_\tp\bv f\rangle \,= \E_\mu[f(X_\tp)\bv \tp< \ext],
\end{align*}

$\mu A_\tp$ is what we called the MCNE
at time $\tp$, with initial distribution $\mu$.

In this setting,
the family $(P_\tp)_{\tp\ge0}$
–resp. $(A_\tp)_{\tp\ge0}$–
defines a linear but non-conservative semigroup
–resp. a conservative but non-linear semigroup–
of operators on $\Mone$
endowed with the total variation norm :

$ \quad
\|\mu \|_{TV}~:= \sup\Lbr |\mu(A)|\pv A \in \B \Rbr
\text{ for } \mu \in \M(\cX).
\quad $

A probability measure $\alpha$ is said to be the quasi-limiting distribution of an initial condition $\mu$ if :

\frl{B \in \B}\quad
\limInf{\tp} \PR_\mu(X_\tp \in B\bv \tp< \ext) := \limInf{\tp} \mu A_\tp(B) = \alpha(B).
\quad $

It is now classical –cf e.g. Proposition 1 in <cit.>–
that $\alpha$ is then a quasi-stationary distribution or QSD, in the sense that :
$\frl{\tp\ge 0}\quad
%\PR_\alpha(X_\tp \in dy\bv \tp< \ext) :=
\alpha A_\tp(dy)
= \alpha(dy)\quad$
and that for any such QSD,
there exists an extinction rate $\laZ$
such that : $\frl{t \ge 0} \Pr_\alpha(t< \ext) = \exp[-\laZ\, t]$.

Our first purpose will be to prove that
the assumptions in Subsection <ref>
provide sufficient conditions
for the existence of a unique quasi-limiting distribution
$\alpha$, independent of the initial condition.

§.§ Specification on the state space

In the following Theorems, we will always assume :

$(A0)$ :
"Exhaustion of $\cX$"

There exists a sequence $(\cD_n)_{n\ge 1}$ of closed subsets of $\cX$ such that :

\begin{align*}
\frl{n\ge 1} \cD_n \subset \overset{\circ}{\cD}_{n+1} \quad\text{ and } \quad
\medcup_{n\ge 1} \cD_n= \cX.
\tag*{\RPA}
\end{align*}

This sequence will serve as a reference for the following statements.
In particular,
we will have control on the process
through the fact that the initial distribution
belongs to some set of the form :

\begin{align*}
\Mnx~:= \Lbr \mu \in \Mone\pv
\mu\lp \cD_n \rp \ge \xi\Rbr,
\hspace{1cm} \text{ with } \xi \in (0,1].
\EQn{mnx}{\Mnx}
\\&\hcm{1} \text{Note that :} \qquad
\frl{\xi \in (0,1]}, \quad \Mone = \medcup_{n\ge 1} \Mnx.\\
\text{Let also~:\,} &
%\Dp~:= \Lbr \cD_n \pv n\ge 1\Rbr, \quad
\Dps~:= \Lbr \cD\pv \cD \text{ is compact and }
\Ex{n\ge 1} \cD \subset \cD_n\Rbr.
\EQn{Dps}{\Dps}
\end{align*}

§ EXPONENTIAL CONVERGENCE TO THE $QSD$

§.§ Hypotheses

Like in <cit.>,
our results rely on a set $\mathbf{(AF)}$ of assumptions
which is actually implied by a much stronger
yet simpler set of assumption $\mathbf{(AF')}$.
The label of the assumption is chosen
to be consistent with the ones in <cit.>.
We detail first each basic assumption
and then define $\mathbf{(AF)}$ and $\mathbf{(AF')}$
in terms of those.

§.§.§ The assumptions common with <cit.>

Our first assumption is a relaxed version
of assumption $(A1)$ in <cit.> :

$(A1_G)$ : For any $\np \ge 1$,
there exists $\mpr \ge \np$ such that

the rate of jumps leading outside of $\cD_{\mpr}$
from any point inside $\cD_{\np}$
is bounded.

The following assumptions share common parameters
(we consider here the probability measure $\alc$ as a parameter).
Those appear between square brackets after the label of each assumption.
We will only keep this precision for the statements of $\mathbf{(AF)}$ and $\mathbf{(AF')}$,
and skip it for the proof.
We recall also that for any set $\cD$,
we defined its exit and its hitting times as :

$$T_{\cD}~:= \inf\Lbr  \tp \ge 0 \pv X_\tp \notin \cD \Rbr
\mVg\quad
\tau_\cD~:= \inf\Lbr \tp \ge 0\pv X_\tp \in \cD \Rbr.$$

$(A2)$ : "Mixing property"
[subscript : "$m x$"]

There exists a probability measure $\alc \in  \Mone$ such that,
for any $n\ge 1$,
there exists $m>n$ and  $c, t>0$ such that :

\begin{align*}
&  \frl{x \in \cD_{\np}}
\hspace{.5cm}
\PR_x \lc {X}_{\tp}\in dx \pv
\tp < \ext \wedge T_{\cD_m} \rc
\ge \cp\; \alc(dx).
\end{align*}

$(A4)$ : "Escape from the Transitory domain"
[sb : "$e T$"]

For a given $\rho >0$ and $\cD_c \in \Dps$ :

\begin{align*}
\Tsup{x\in \cX} \;\E_{x} \lp
\exp\lc\rho\, (\ext\wedge \UDc) \rc \rp < \infty.
\end{align*}

$(A5)$ : "Survival estimate"
[sb: "$s v$"]

There exists $(\Rsv, \cD_s) \in (0, \infty)\ltm \Dps$,

$\cp > 0$, $\mpr \ge 1$
such that $\cD_s \subset \cD_{\mpr}$ and :

\begin{align*}
&\frl{x\in \cD_s}
\frlq{\tp \ge 0}
& \PR_{x} (\tp < \ext\wedge T_{\cD_m})
\ge \cp \; \exp(-  \Rsv \, \tp)
\end{align*}

§.§.§  a coupling approach including failures to ensure BdSv

We now state the new conditions that replace $(A3)$.
As can be seen from our proof,
its purpose is to imply $(A3)$,
see Subsection <ref>.
We also postponed the remarks in Subsection <ref>.

$(A3_F)$ : "Absorption with failures"

Given $\alc \in \Mone$, $\rho > 0$
$\cD_c \in \Dps$,
for any $\fl\in (0,\, 1)$,
we can find $\tp, \cp >0$
such that for any $x \in \cD_c$
there exists a stopping time $\Uza$
such that

\begin{align*}
&  \Lbr\ext \wedge \tp \le \Uza \Rbr
= \Lbr\Uza = \infty\Rbr
% \text{ on } \Lbr\tp<   \Tfl \Rbr
% = \Lbr \Tfl = \infty \Rbr
% T^1_{f \ell} \le \tp \text{ on } \Lbr \Nstep \ge 1 \Rbr
\quad  \text{ and }\quad
\PR_{x} (\Uza = \infty, \,  t< \ext)
\le \fl\, \exp(-\rho\, \tp),
\end{align*}

while for some stopping time $\UCa$ :

\begin{align*}
&\PR_{x} \big(X(\Uza) \in dx' \pv \Uza < \ext \big)
\le \cp \,\PR_{\alc} \big(X(\UCa) \in dx'
\pv \UCa < \ext\big).
%  \tag*{\RAF[$\alc,\rho_{AF},\cD_c$]}
\end{align*}

We further require
that some random time $U_{Ab}$ extending $\Uza$
is indeed a stopping time :

$\bullet$   $U_{Ab} := \Uza$
on the event  $\Lbr \ext\wedge \Uza < \Tdc^t\Rbr$,
where $\Tdc^t := \inf\{s\ge t :
X_s \in \cD_c\}$.

$\bullet$ On the event
$\Lbr \Tdc^t \le \ext \wedge  \Uza \Rbr$
–which by construction equals
$\Lbr \Tdc^t < \ext\Rbr \cap
\Lbr \Uza = \infty \Rbr$–
and conditionally on $\F_{\Tdc^t}$,
the law of $U_{Ab} - \Tdc^t$ coincides
with the one of $\wtd{U}_{Ab}$
for a copy $\wtd{X}$ of the Markov process
with initial condition $X(\Tdc^t)$
and independent of $X$ conditionally on $X(\Tdc^t)$.
(that is a strong Markov property at time $\Tdc^t$
for the law of $\wtd{U}_{Ab}$).

§.§.§ The general sets of Assumptions

We say that assumption $\mathbf{(AF)}$ holds, whenever :

Assumptions ,
$(A1_G)$  hold as well as
–for some  –
for some $\Rsv >0$ and $\cD_s \in \Dps$ satisfying $\alc(\cD_s) >0$.
Moreover, there exist $\rho  > \Rsv$
and $\cD_c \in \Dps$ such that
and   hold."

Given stronger versions of assumptions   and ,
we no longer need any "survival estimate",
so that we can simplify the previous assumption :

We say that the (stronger) assumption $\mathbf{(AF')}$ holds, whenever :

"Assumptions , $(A1_G)$   and  hold,
with some .
Moreover, for any $\rho > 0$, there exists some set
$\cD_c \in \Dps$, such that

Remark :  Almost sure extinction is not at all needed for our proof (which would in fact include the case where there is no extinction, or only in some "transitory domain").

§.§ Main Theorems

Assume that $\mathbf{(AF)}$ holds.

Then, there exists a unique QSD $\alpha$.
Moreover, we have exponential convergence to $\alpha$ of the MCNE's.
More precisely,
there exists $\zeta >0$
and to any pair $\np \ge 0$ and $\xi \in (0,1]$,
we can associate a constant  $C(\np,\, \xi)$ such that :

\begin{align*}
\frl{\tp > 0}
\frlq{\mu \in \Mnx}
\|\, \PR_\mu \lc\, X_{\tp} \in dx \;
| \; \tp < \ext \rc  - \alpha(dx) \, \|_{TV}
\le C(\np,\, \xi) \; e^{-\zeta \; \tp}.
\EQn{ECvAl}{\textup{ECv:}\,\alpha}
\end{align*}

We recall that $\alpha$
is associated to some extinction rate $\laZ$ :

\begin{align*}
\frl{\tp\ge 0} \PR_\alpha (\tp< \ext) = e^{-\laZ\, \tp}
\quad \text{meaning that as a QSD, $\alpha$ satifies }
\alpha P_\tp = e^{-\laZ\, \tp}\, \alpha
\Eqn{LBz}{\laZ}
\end{align*}

- cf e.g.
Theorem 2.2 in <cit.>.
This motivates the definition :

\begin{align*}
\eta_\tp (x)~:= e^{\laZ\, \tp}  \PR_x (\tp< \ext)
=  \PR_x (\tp< \ext)/ \PR_\alpha (\tp< \ext).
\end{align*}

Assume again that $\mathbf{(AF)}$ holds.

we have uniform exponential convergence of $(\langle\mu\bv\eta_\tp\rangle)_{\tp\ge 0}$ to $\langle\mu\bv\eta\rangle$
(with the same rate $\zeta$ as in inequality ECvAl).
More precisely, there exists $C>0$ such that :

\begin{align*}
&\frl{\tp > 0}
\frlq{\mu \in \Mone}
\bv \langle\mu \bv \eta_\tp\rangle - \langle\mu\bv\eta\rangle  \bv
\le C \; e^{-\zeta \; \tp }.
\EQn{ECvEta}{\textup{ECv:}\,\eta}
\\&\text{In particular, } \hspace{2cm}
\frl{x\in \cX} \quad \eta(x)~:= \textstyle{\lim_{\tp\ifty }} \eta_\tp (x)
\text{ exists},
\end{align*}

where the function $\eta$,
which describes the "survival capacity"
of the initial condition,
is positive,
bounded on $\cX$
and vanishes on $\partial$.
It also belongs to the domain of the infinitesimal generator $\cL$, associated with the semi-group $(P_\tp)_{\tp \ge0}$ on $(B(\cX\cup\{\partial\} ); \Ninf{.})$,
and satisfies :

\begin{align*}
\cL\, \eta = -\laZ\, \eta, \qquad
so\quad \frl{\tp \ge 0}
P_\tp \, \eta = e^{-\laZ\, \tp }  \eta.
\EQn{EFeta}{}
\end{align*}

Remark :
As in <cit.>,
it is not difficult to show that there is no eigenvalue of $\cL$
between $0$ and $-\laZ$,
and that $\eta$ is the unique eigenvector
associated to $-\laZ$.

Under again the same assumptions $\mathbf{(AF)}$, we have :

There exists a family $(\Q_x)_{x \in \cX}$ of probability
measures on $\Omega$ defined by :

\begin{align*}
\limInf{\tp }
\PR_x(\Lambda_\spr \bv \tp  < \ext) = \Q_x(\Lambda_\spr),
\Eqn{eqd}{\Q_x}
\end{align*}

for all $\F_\spr$-measurable set $\Lambda_\spr$.
The process $(\Omega;(\F_\tp )_{\tp \ge 0};
(X_\tp )_{\tp \ge 0};(\Q_x)_{x \in \cX})$
is an $\cX$-valued homogeneous strong Markov process.

(ii) Transition kernel :

The transition kernel of the Markov process $X$ under
is given by :qt

\begin{align*}
&q(x; \tp ; dy) = e^{\laZ\, \tp } \, \eta(y)
\; p(x; \tp ; dy) / \eta(x),
\Eqn{qt}{q(x; \tp ; dy)}
\end{align*}

where $p(x; \tp ; dy)$ is the transition kernel of the Markov process $X$ under

(iii) Exponential ergodicity of the Q-process :

The measure $\quad \beta(dx) := \eta(x)\, \alpha(dx)$
is the unique invariant probability measure under $\Q$.

Moreover :

\begin{align*}
\frl{\tp  > 0}
\frlq{\mu \in \Mnx}
\NTV{ \Q_{\EtEt\mu_1} \lc\, X_{\tp } \in dx\rc
- \beta(dx) }
\le C(\np,\, \xi) \; e^{-\zeta \; \tp },
\EQn{ECvBeta}{\textup{ECv:}\,\beta}
\\ &\where
\Q_\mu (dw)~:= \textstyle{\int_\cX }\mu(dx) \, \Q_x (dw),
\qquad \eta_{*} \mu(dx)
:= \eta(x)\, \mu(dx)\, / \langle\mu\bv \eta\rangle.
\end{align*}

(iv) $\beta$ is the unique quasi-ergodic distribution :

For any bounded measurable function $f$, $\mu \in \Mone$
and $\eps >0$ :

\begin{align*}
\limInf{t} \PR_\mu\lc
|\tfrac{1}{t}\intO{t} f(X_s)\, ds - \langle\beta\bv f \rangle| > \eps
\bv t < \ext\rc
= 0.
\EQn{Qed}{}
\end{align*}

Remark :
$\bullet$ We refer to <cit.> for other equivalent statements on the transition kernel.

$\bullet$ To understand inequality ECvBeta,
it is worth noticing that,
considering some general initial condition
in the left-hand side of inequality ,
we obtain for the Q-process a biased initial condition, i.e. :

\begin{align*}
\frl{\mu \in \Mone}\quad
\textstyle{\lim_{\tp \ifty}}
\PR_\mu(\Lambda_\spr \bv \tp  < \ext) = \Q_{\eta_{*}\mu}(\Lambda_\spr).
\EQn{etaEt}{\Q_{\eta_{*}\mu}}
\end{align*}

For simplicity,
we will say that  the exponential quasi-ergocidicty holds
when the results of these three theorems hold
(see <cit.> for more details on this notion of quasi-ergodic measure).

§.§ Remarks on our new assumptions

Remark on $(A1_G)$:

The assumption $(A1)$ in <cit.>
that we are able to relax into $(A1_G)$
means actually forbidding those large jumps,
instead of having an upper-bound on their rates.
Note that the jumps can still be singular
(happening when reaching some specific subset of the state space)
as long as they do not lead the process too far away
and the other assumptions can be verified.
A further refinement is not excluded,
but would probably be much more specific to some model of interest.

Remarks on $(A3_F)$:

$\trr$ With the above  notations :
$ \Lbr \Uza < \ext \Rbr
=  \Lbr \Uza < t \Rbr
= \Lbr \Uza = \infty \Rbr^c$.

$\trr$ The r.v. $\Uza$ and $\UCa$
are expected to depend upon $\fl$ and $x$,
and to be related to $\tp$ and $\cp$,
while these two constants must be uniform in $x$.

$\trr$ The control on the failures
is an upper-bound on their probability via $\fl$
that takes into account
the time we waited hoping for a success.
In fact,
we will exploit this assumption for a given value of $\fl>0$,
which is explicitly
but not so simply related
to the other parameters
–cf Subs. <ref>.

$\trr$ It comes a priori from a coupling
that generalizes the approach of <cit.>
in the context of a time-periodic diffusion.
Yet, our condition is a priori
not uniform over the whole space.
Also, by including failures,
we can not a priori restrict ourselves
to the initial conditions $x$
optimizing the survival at some (large) time $t$.

$\trr$ The condition of existence of such $U_{Ab}$
comes from the fact that we want to apply inductively
(on the  number of failed attempts)
It has mainly to do with the regularity of the definition of $U_{ab}$
with respect to the initial condition $x$.
Like in our applications,
the definition of $U_{Ab}$ shall come quite naturally
from the conditions required by $U_{ab}$
and the proof that both are stopping times should be similar.

Although the law of $U_{Ab}$ is defined uniquely
(which is what we need),
it is a priori unclear how to define it generally.

If $\Uza$ is defined
through strong Markov processes with independent increments
like Brownian Motions,
Poisson Point Processes
or other Levy processes,
$U_{Ab}$ can be expressed through these increments
in the time-intervals
$[\Tdc^i, \Tdc^{i+1}]$, for $i\in [\![0, \infty[\![$,
where recursively:

\begin{align*}
\Tdc^{i+1} := \inf\{s\ge \Tdc^i +t : X_s \in \cD_c\}
\wedge \ext,
\AND
\Tdc^0 = 0.
\end{align*}

The Markov property on the incremental process
shall then imply the condition on

§.§ Refined versions
for the coupling approach including failures

Because our last assumption may appear too technical,
let us give some insight on where it comes from.
Recall that the last assumption needed in <cit.> is
the relaxed version of BdSv :

$(A3)$ : There exists $c, u>0$  such that :

\begin{align*}
\frl{x\in \cD_c}\frlq{t\ge u}
\PR_x(t<\ext) \le c\, \PR_{\alpha_c}(t<\ext),
\end{align*}

where  $\alc$ is the measure involved in $(A2)$.

Notably, rather than the uniformity on $\cX$
like in inequality BdSv,
the uniformity on $\cD_c$ is sufficient
thanks to assumption $(A4)$.

One way to verify $(A3)$ is to get the process
with initial condition $\alc$
to come close enough to any test initial condition $x$
with a given time-shift (and a given efficiency).
any set of long-time surviving trajectories of the process
with initial condition $x$
can in some sense be coupled
to behave like some of the long-time surviving trajectories
of the process with initial condition $\alc$.
Such containment property is what we call "absorption" in the following.
Generally, it means that,
for an absorption in one step to be deduced
(with efficiency parameter $1/\cp$),
we need to associate to any $x$
two stopping times $\Uza$ and $\UCa $ such that :

\begin{align*}
& \PR_{x} \big(X_{\Uza} \in dx\pv \Uza < \ext \big)
\le \cp \;\PR_{\alc} \big(X_{\UCa} \in dx\pv \UCa < \ext \big).
\EQn{Abs}{}
\end{align*}

Originally for continuous-time and continuous-space Markov processes
(see <cit.> or <cit.>),
this property is deduced from the Harnack inequality.
Yet, otherwise,
it may be very difficult to ensure in practice
such a strong property.
We might need for instance
in some multidimensional model
to wait for a jump on a specific dimension to happen,
while there may be positive probability for
very singular behavior to happen meanwhile
on the other dimensions.
This issue is in some sense avoided
in the simple application of Subsection <ref>,
thanks to our extension $(A3_G)$.
In general, the purpose of such an extension
is to get rid of singularities,
that is problematic events
which are rare in probability.
For instance,
if the process is already diffusive between jumps,
it may be interesting to consider the first jump
as the time of the first failure,
and exploit the Harnack inequality
in a time-scale so short
that jumps happen at that time-scale
with a small enough probability.

In the proof
–cf. Subsection <ref>–
we use a refinement of assumption $(A3_F)$
where $\Uza$ is more loosely bounded.
Indeed, it gives more insight on the role
of the constants involved.
Yet, it also requires to precise the times at which
failures are stated,
since there is no more reason for each step
to end before time $t$.
The statement is thus much more technical.
And in practice,
by reducing $\rho$
(while keeping $\rho' > \Rsv$),
one can always truncate the value of $\Uza$
and deal with the event that $\Uza$ takes large value
as exceptional.

For this statement, we first need the following definitions,
that we exploit for the proof in Subsection <ref>.
Assume that for any initial condition $x$,
we have defined some stopping times $\Uza$ and $\Tfl$.
We then say that the definitions
generalize to a path towards absorption
$(U_{Ab} \pv T^j_{f \ell}, \Tdc^j\mVg j \ge 0\pv \Nstep)$
if the following holds :

* $\Tfl^1 := \Tfl$ describes the time of the first "failed attempt".

* $U_{Ab}$ is a stopping time
that describes the time of the "absorption success"
–like in Abs.
It is infinite if extinction happens first
and $U_{Ab} = \Uza$ on $\Lbr \Tfl^1 = \infty\Rbr$.

* $\Nstep$ is a  $\Z_+$-valued $\F_{\Uza}$-measurable r.v.
that describes the number of needed absorption steps.
$\Nstep = 0$ corresponds to no failure,
that is $\Tfl^1 = \infty$.
We assume that  ,
yet it is a consequence of the following assumptions,
i.e. the Markov property and the uniform lower-bound on
$\PR_x(\Tfl^1 = \infty)$.

* there exists a double sequence of stopping times :

$$0< T^1_{f \ell} \le \Tdc^1
< T^2_{f \ell}\le \Tdc^2
< ...  < T^\Nstep_{f \ell}< U_{Ab}\wedge \ext
\quad \text{ such that~:}$$

$$\text{either } T^\Nstep_{f \ell} \le \Tdc^\Nstep < U_{Ab}\wedge \ext,\hcm{1.5}
\text{or }   T^\Nstep_{f \ell} < \ext < \infty = \Tdc^\Nstep = U_{Ab}.

* $T^j_{f \ell}$, for $1\le j\le \Nstep$,
shall be the time at which the $j$-th failure is stated
and $\Tdc^j$ is the first hitting time of $\cD_c$ after $T^j_{f \ell}$.

($X[ T^j_{f \ell}] \in \cD_c$ is not excluded,
in which case $T^j_{f \ell} = \Tdc^j$.)

* $T^j_{f \ell}= \Tdc^j = \infty$ for $j>\Nstep$.

* Compatibility with the Markov property :

Conditionally upon $\F_{\Tdc^1}$ on
$\Lbr \Tdc^1 < \ext\Rbr\subset\Lbr \Tfl^1 < \infty\Rbr$ :

with initial condition $x$  has the same law as
$\quad (\widetilde{U}_{Ab}
\pv \wTfl^j, \wTdc^j\mVg j \in [\![1, \infty[\![
\pv \wtd{\Nstep}\pv  \wtd{\ext}),\quad$
in term of the process $\wtd{X}$
with initial condition $X[\Tdc^1]$,

which is independent of $X$ conditionally on $X[\Tdc^1]$.

Assumption $(A3_{G})$ :
For any , there exists
such that
we can associate to any $x \in \cD_c$
two stopping times $\Uza$ and  $\Tfl^1$
for which

\begin{align*}
\Uza \vee \Tfl = \infty,
\quad \ext \le \Uza \wedge \Tfl \text{ implies } \Uza = \Tfl = \infty,
\\&
\E_{x} \lc \exp(\rho \, [\Uza\wedge \ext])
\pv \Tfl = \infty \rc
\le e_0,
%\tag*{\RAFb[$\alc, \cD_c, \rho$]}
\quad
\E_{x} \lc \exp(\rho \, T_{f \ell})
\pv \Tfl < \infty\rc
\le \fl,
\end{align*}

in such a way that these definitions
generalize to a path towards absorption

while for some stopping time  $\UCa$
and  any bounded measurable function

which is increasing in $u$ :

\begin{align*}
\E_{x} \lc g(X(\Uza),\, \Uza) \pv \Uza < \ext \rc
\le \cp \;\E_{\alc} \lc g(X(\UCa),\, \UCa + \tp)
\pv \UCa < \ext \rc.
\end{align*}

Remarks :
Given $\Uza$ and $\Tfl$ (for any $x$),
the condition of "Compatibility with the Markov property"
imposes the definition of the path to absorption (at least in law).
In particular,
on $\{ \Tdc^j< \infty\}$,
$\Tfl^{j+1} - \Tdc^j$ has the law of $\wTfl^1$
with initial condition $X(\Tdc^j)$on $\{ \Tfl^j< \infty\}$,
$\Tdc^{j} - \Tfl^j$ has the law of $\wTdc$
with initial condition $X(\Tfl^j)$.

The condition that the definitions of $\Uza$ and $\Tfl^1$
generalize to a path towards absorption
is in fact related to the regularity in $x\in \cX$
of these definitions.
Notably, it is a priori unclear
that the above construction
indeed produces a sequence of stopping times.
As one can see in our applications,
it relies on very classical results on stopping times.

§ FOUR MODELS TO WHICH OUR RESULTS APPLY

Jumps as a perturbation of the diffusion

§.§.§ Description of the process :

In  this illustration,
we generalize the process
considered in Subsection 3.2 of  <cit.>
so that it may additionnally jump.
As long as the jumps
happen with a locally bounded rate,
the exponential quasi-ergodicity still holds.

We consider a simple coupled process $(N, X)$ describing
the eco-evolutive dynamics of a population.
We model the population size by a logistic Feller diffusion $(N_t)_{t\ge0}$
where the growth rate $(r(X_t))_{t\ge0}$
is changing randomly.
We may also include catastrophes occurring randomly
that eliminate a substantial part of the population.
The adaptation of the population
and the change of the environment
are assumed to
effect the population through
a hidden process $(X_t)$ in $\R^d$,
from which the growth rate is deduced.
We will assume here that $X_t$ evolves as a càdlàg Markov process
driven by some Brownian Motion $B^X$, a drift
and a Poisson Point Process $M$ for the jumps.

For very low values of $r(X_t)$,
it is expected that the population shall vanish very quickly.
Yet, we want our result to be independent
of any bounds of the adaptation
and say that this large probability of extinction
is sufficient in itself
to bound the mal-adaptation,
keeping the idea that the initial condition indeed matters here.

In a general setting,
the process can be described as :

\begin{align*}
% \left\{
&N_t = n + \intO{t}(r(X_t) - c\ N_t)\ N_t\ dt
+ \intO{t}\sigma_N \ \sqrt{N_t}\ dB^N_t
\\&\hcm{2}
- \int_{[0, t] \times (0, 1] \times \R_+} p \, N_{s-}
\idc{u \le k_c(N_{s-}, X_{s-}, p)}
M^N(ds,\, dp,\, du),
\EQn{SN}{}
\\&
X_t =  x + \intO{t} b(X_t, N_t)\ dt +  \intO{t} \sigma_X(X_t, N_t) \ dB^X_t
\\&\hcm{2}
+ \int_{[0, t] \times \R^d\times \R_+} w \, \idc{u \le k_m(N_{s-}, X_{s-}, w)}
M^X(ds,\, dw,\, du).
\EQn{SX}{}
\end{align*}

with initial conditions $(n, x)$,
$B^N$ and $B^X$ two independent Brownian Motions,
$M^N$ and $M^X$ two independent Poisson Point Process  (PPP)
of respectively intensities  $ds\, \nu_P(dp)\, du$ and $ds\, \nu_W(dw)\, du$
(with $\nu_P(dp)$ and $\nu_W(dw)$ some reference measures).

$c, \sigma_N >0$, $r, b, \sigma_X$ are some measurable functions,
as well as $k_c$ and $k_m$.
We assume that the jump rates are locally bounded,
i.e.  for any compact $K$,
there exists $k^\vee< \infty$ such that
for any $y =(n, x) \in K$,

\begin{align*}
\intO{1} k_c(y, p) \, \nu_P(dp) + \int_{\R^d}\, k_m(y, w)\, \nu_W(dw) \le k^\vee.
\EQn{kv}{}
\end{align*}

We assume that this set of parameters
$(r, b, \sigma_X, k_c, k_m)$
is such that there exists a strong solution to the equation
Sufficient  conditions
guaranteeing  (strong)  existence and  uniqueness
are locally Lipschitz  and  linear growth conditions;
see e.g. Theorem 3, Chapter II in [15] or  [17].
Besides, the independence between $(B^N, B^X)$
and $(M^N, M^X)$ is automatic
according to Theorem 6.3 in Chapter II of [17].

We also add catastrophes,
arising at rate $\rho_c(x, n)$,
leading to the complete extinction of the population,
so that the extinction time is given by:

\begin{align*}
\ext := T_0 \wedge T_c
\EQn{AIIE}{}
\\&
\where
T_0 := \inf\{t\ge 0\pv N_t = 0 \}
\AND T_c := \inf\{t\ge 0\pv
\intO{t} \intO{\rho_c(X_s, N_s)} M^E(ds, du) \ge 1 \},
\end{align*}

where $M^E$ is again a PPP, with intensity $ds\ du$, independent of the other processes above.

§.§.§ Harnack inequalities for   and

In the following,
given  a Markov process $(Y_t)$ on $\Y$
with generator $\cL$
(including possibly an extinction rate $\rho_c$)
we say that
it  satisfies Assumption $(H)$ if :

For any compact sets $K, K'\subset \Y$
with $C^2$ boundaries such that $K\subset int(K')$,
$0<t_1<t_2$ and positive $C^\infty$
boundary condition :
$u_{\partial K'} : (\{0\} \times K') \cup ([0, t_2] \times \partial K')
\rightarrow \R_+$,

there exists a unique positive solution $u(t,x)$ to the Cauchy problem :

\begin{align*}
\partial_t u (t, y) = \cL u (t, y) \text{ on } [0, t_2] \times K'\pv
\\&%\hcm{1}
%u(0, x) = u_0(x) \text{ on } K'\pv
u(t, y) = u_{\partial K'}(y)
{\small \text{ on } (\{0\}\times K') \cup ([0, t_2] \times \partial K')},
\end{align*}

and it satisfies,
for some $C = C(t_1, t_2, K, K')>0$ independent of $u_{\partial K'}$ :

\begin{align*}
{\textstyle \inf_{y\in K} u(t_2, x)
\ge C\, \sup_{y\in K} u(t_1, y).}
\end{align*}

For the following theorem,
we consider a process $\hat{Y} = (\hat{N}, \hat{X})$
that corresponds to the diffusive part of $(N, X)$,
i.e. with generator $\cL$ given on any $f\in \cC_b^2(\R_+\times \R^d)$
by :

\begin{align*}
\cL f(n, x)
:= (r(x) - c\, n)\, n\, \partial_n f (n, x)
+ \tfrac{(\sigma_N)^2\, n}{2} \partial^2_n f (n, x)
+ b(x, n)\, \partial_x f (n, x)
+ \tfrac{\sigma_X(x, n))^2}{2} \partial^2_x f (n, x).
\end{align*}

Suppose that Assumption $(H)$ holds
for the above generator $\cL$.
Assume that $\limsup_{\|x\| \ifty} r(x) = -\infty$.
Then, with $\cD_\ell := [1/\ell, \ell]\ltm \bar{B}(0, \ell)$,
for $\ell \ge 1$,
$\mathbf{(AF')}$ holds, so that
the process is exponentially quasi-ergodic
(i.e. all the results of Subsection <ref> hold).

§.§.§ Proof of Theorem <ref>

Assumption $(A0)$ is clear.
The rate of jumps from any compact set is assumed to be bounded,
so that assumption $(A1_G)$ holds.
The proof of assumption   for any given $\rho$
can be easily adapted from <cit.>,
in the case where there is no jump.
this proof in <cit.>
only relies on some upper-bounds of $(N_t)$
–on different subsets–
through some uniform couplings.
The core of the argument
is that the population size
decreases sufficiently quickly,
or makes large increase very exceptionally,
while the extinction occurs with a large probability
when it gets too small.
Since the jumps of the population size
are assumed
to always reduce this population size,
these couplings still produce upper-bounds
on the process $N$ defined here with jumps.
the proof of assumption   can be taken almost
exactly from the one in Subsection 3.2.4 of <cit.>.

From <cit.>,
we know that for the process $(\hat{N}, \hat{X})$ without the jumps,
assumption   holds,
say with time $t_{mx}$
and trajectories restricted to stay in $\cD_m$.
Assuming that the jump rates are locally bounded,
we easily obtain a uniform lower-bound on the probability
of some event $\mathcal{P}_J$ that prevents jumps from occurring
before time $t_{mix}\wedge T_{\cD_m}$.
In practice,
consider some $k^\vee$ such that for any $y \in \cD_m$,

\begin{align*}
\intO{1} k_c(y, p) \, \nu_P(dp) + \int_{\R^d}\, k_m(y, w)\, \nu_W(dw) \le k^\vee.
\end{align*}

Then :

\begin{align*}
&\PR_{(n, x)} \lc (N_{t_{mx}}, X_{t_{mx}})\in (dn', dx') \pv
\tp < \ext \wedge T_{\cD_m}
\pv \mathcal{P}_J \rc
\\&\hcm{1}
\ge \PR(\mathcal{P}_J)\times
\PR_{(n, x)} \lc
(\hat{N}_{t_{mx}}, \hat{X}_{t_{mx}})\in (dn', dx') \pv
\tp < \ext \wedge T_{\cD_m} \rc
\\&\hcm{1}
\ge \exp[-k^\vee\, t_{mx}]\,
\cp\; \alc(dx).
%The associated cost
\end{align*}

Remark :
We refer to Subsection 3.2.3 in <cit.>
for a discussion on the way to obtain
a not too rough survival estimate.
Be it with jumps or without,
the main idea remains the same :
keep close to the most stable areas.

$\hcm{2}$ Proof of assumption

In the proof below,
we exploit again Harnack's inequality
to compare the long-term survival,
given that no jumps has still occurred
at the "coupling interval" $[0, \tZa]$.
Such a jump can be considered exceptional enough
by choosing $\tZa$ sufficiently small.
This criteria of exceptionality
is given by assumption
and here essentially linked to $\fl$.

Remark :
In practice,
if one compares the marginals after a very short time
starting from initial conditions far away from each other,
as we deduce here from the Harnack inequality,
the constant is very far from optimal.
For rather precise estimate,
it seems quite mandatory to exploit a result similar to assumption $\Rmix$
to first bring a part of the distribution starting from $\alc$
to a very close vicinity of the test initial condition $y$.
it may happen that the jumps actually help
for the "absorption",
so that rejecting all of them might be quite inefficient.

Thanks to the previous results on assumption $\RET$,
for any given $\rho>0$ and $\alc$,
we only need to prove assumption
for initial conditions in some $\cD_c = \cD_n\in \Dp$.
We can apply our Assumption $(H)$
for $K := \cD_n$ and $K' := \cD_{n+1}$.
Since the jump rate is assumed to be locally bounded :

\begin{align*}
k^{\vee} := \sup_{y\in K'}\{ %\rho_c(y)+
\intO{1} k_m(y, p)\, \nu_P(dp)
+ \int_{\R^d} k_c(y, w)\, \nu_W(dw)\}
\end{align*}

We define $T_{jp}$ as the first jump time of $(N, X)$.
We couple $(N, X)$
to a Markov process  $\hat{Y} := (\hat{N}, \hat{X})$
with generator $\cL$,
solution of :

\begin{align*}
\left\{
\begin{aligned}
&\hat{N}_t = n + \intO{t}(r(\hat{X}_t) - c\ \hat{N}_t)\ \hat{N}_t\ dt
+ \intO{t}\sigma_N \ \sqrt{\hat{N}_t}\ dB^N_t
\\&
\hat{X}_t =  x + \intO{t} b(\hat{X}_t, \hat{N}_t)\ dt
+  \intO{t} \sigma_X(\hat{X}_t, \hat{N}_t) \ dB^X_t
\end{aligned}
\right.
\qquad(\hat{S})
\end{align*}

By definition,
$(N, X)$ and $(\hat{N}, \hat{X})$ coincide until $T_{jp}$.

Suppose that assumption $(H)$ holds for $\cL$,
$K$ and $K'$ are compact subsets such that $K \subset \overset{\circ}{K'}$
and $\alc(K)>0$.
Then, there exists $c_H >0$
such that for any $y\in K$:

\begin{align*}
\PR_{y} \big(\hat{Y}(\tZa) \in dy \pv
\tZa < \hat{T}_{K'} \big)
\le c_H\, \PR_{\alc} \big(\hat{Y}(2\ \tZa) \in dy \pv
2\ \tZa < \hat{T}_{K'} \big).
\EQn{AcH}{}
\end{align*}

The proof of this lemma is easily adapted
from the one presented in <cit.>, Section 4, step 4,
so we will only sketch the proof.
The idea is first to
approximate the function :

$$u(\tp, y)~:= \E_{y}\lp  f(Y_{\tp}) \pv \tp < \ext \rp,
\with \; \tp  \ge \tza,\; y \in K'$$

defined for some $f \in C^\infty(\Y)$.
Although we can prove
(as they do, refering to Thm 5.1.15 in [18])
that $u$ is continuous,
it is a priori not regular enough
to apply the Harnack inequality directly.
Thus, we approximate it
on the parabolic boundary

by the family $(U_k)_{k\ge 1}$ of smooth –$\cC_+^\infty$ w.l.o.g.– functions.
We then deduce approximations of $u$ in $[t_\cR,\, \infty) \times \cR$
by (smooth) solutions of :

\begin{align*}
\partial_\tp u_k (\tp, y) - \cL u_k (\tp, y)= 0,\qquad
\tp \ge t_\cR,\; y \in \cR^\circ
\\& u_k(\tp, y) = U_k(\tp, y),\quad \tp \ge t_\cR,\; y \in \partial\cR, \quad
\text{ or } \tp = t_\cR,\, y \in \cR.
\end{align*}

By Assumption $(H)$,
the constant involved in the Harnack inequality
does not depend on the values on the boundary.

Thus, it applies with the same constant
for the whole family of approximations $u_k$.

We refer to the proof in <cit.>, Section 4, step 4,
to state that
the Harnack inequality then extends to the approximated function $u$,
concluding the proof of Lemma <ref>.

For any $\fl \in (0,1),$
we then choose $\tZa$, sufficiently small, such that,
$\hat{T}_{K'}$ denoting the exit time out of $K'$
for the process $\hat{Y}$ :

\begin{align*}
(1- \exp[-k^{\vee}\, \tZa])\;
\Tsup{y\in K} \PR_y(\hat{T}_{K'} \le \tZa)
\le \fl \, \exp[-\rho\, \tZa].
\EQn{tZaC}{\tZa}
\end{align*}

There exits such a $\tZa$, since the left-hand side
converges to 0 when $\tZa \rightarrow 0$,
while the right-hand side converges to $\fl$.
For $\Tfl := T_{jp} \wedge T_{K'}
= T_{jp} \wedge \hat{T}_{K'}$,
define :

$\Uza := \tZa$

on the event $\Lbr \tZa < \Tfl \wedge \ext \Rbr\;$;
$\quad \Uza := \infty$

on the event $\Lbr \Tfl \wedge \ext \le  \tZa \Rbr$.

We condition the jump processes
independently of $B^X$, $B^N$ and $M^E$
so as to ensure
that no jump can occur
before $\tZa \wedge \hat{T}_{K'}$
and deduce from $\Req{tZaC}$ :

\begin{align*}
\frlq{y\in \cD_n}
\PR_y(\Uza = \infty, \tZa < \ext)
\le \PR_y(\Tfl \le \tZa)
\le \fl \, \exp[-\rho\, \tZa].
\EQn{ATfl}{}
\end{align*}

By definition,
$(N, X)$ and $(\hat{N}, \hat{X})$ coincide until $T_{jp}$,
so that :

\begin{align*}
\PR_{y} \big(Y(\tZa) \in dy \pv \tZa < \Tfl \big)
\le \PR_{y} \big(\hat{Y}(\tZa) \in dy \pv
\tZa < \hat{T}_{K'} \big).
\EQn{Anj}{}
\end{align*}

On the other hand,
since the rate of jumps and extinction

is upper-bounded by $k^\vee$ on $K'$ :

\begin{align*}
& \PR_{\alc} \big(Y(2\ \tZa) \in dy \pv
2\ \tZa < \ext \big)
\ge  \PR_{\alc} \big(Y(2\ \tZa) \in dy \pv
2\ \tZa <  T_{K'} \wedge T_{jp}\wedge T_c \big)
\\&\hcm{1}
\ge \exp[-2\, k^\vee\, \tZa]\;
\PR_{\alc} \big(\hat{Y}(2\ \tZa) \in dy \pv
2\ \tZa <  \hat{T}_{K'} \big).
\EQn{ANJ}{}
\end{align*}

Combining inequalities Anj and ANJ with Lemma <ref>
with our above definition of $\Uza$,
$\UCaY := 2\ \tZa$
and $c_{abs}~:= c_H\,  \exp[2\, k^\vee\, \tZa]$,
we obtain finally :

\begin{align*}
& \PR_{y} \big(Y(\Uza) \in dy \pv \Uza < \ext \big)
\le c_{abs}\; \PR_{\alc} \big(Y(\UCaY) \in dy \pv
\UCaY < \ext \big).
\end{align*}

With inequality ATfl,
this concludes the proof of assumption ,
given the following lemma.

We can define a stopping time $U_{AB}$
extending the above definition of $\Uza$
as described in Assumption $(A3_F)$.

In Appendix UAB,
we show how to deduce a similar result
in a somewhat more difficult case
and let the reader infer the proof of Lemma <ref>.

Mutations compensating a drift leading to extinction

§.§.§ Description of the process :

We assume here that $X$ is governed
by a Piecewise-deterministic Markov process,
so that the mixing assumption
is a consequence of the jumps.
For simplicity, the drift is assumed to be constant,
which leads us to
the following system :

\begin{equation*}
= x - v\, t\, e_1
+ \int_{[0,t] \times \R^d \times \R_+ }
w \; \idc{u\le h(X_{s^-},w )}\, M(ds,dw,du)
\quad
\text{ as long as } t < \tau_d,
\EQn{eqS}{(S)}
\end{equation*}

$M$ is again a PPP over $\R_+ \times \R \times \R_+ $,
with intensity
$\pi (ds,dw,du) = ds \; dw \; du  $,
while $h(x, w)$ describes the rate of invasion of a new mutation of size $w$ in a population with type $x$,
and the change in type is given by the constant  speed $v>0$.
Moreover, the extinction occurs
at time $\ext$
with the rate $\rho_e (x), x\in\R^d$, i.e. :

\begin{align*}
\ext
:= \inf\{t\ge 0\pv \intO{t} \intO{\rho_c(X_s, N_s)} M^E(ds, du) \ge 1 \},
\end{align*}

where $M^E$ is again a PPP, with intensity $ds\ du$, independent of $M$.

* A3:

$\rho_e$ is locally bounded and
$\TlimInf{\|x\|}\rho_e(x)= +\infty$.
Also, explosion implies extinction :

$\ext \le \Tsup{n\ge 1} T_{\cD_n}$.

$h$ is measurable and for any compact $K\subset \R^d$, there exists :

an upper-bound $\roV$ on the jump-rate :
$\frlq{x\in K}
0<\rho_J(x) := \int_{\R^d} h(x, w) \, dw \le  \roV$.

a lower-bound $h_\wedge>0$ on jumps able to compensate the drift,
with some $0<\dS<S$ :

$\frl{x\in K} \frlq{w\in B(S\, e_1, \dS)}
h(x, w)\ge  h_\wedge$.

a tightness estimate for the jumps :

for any $\eps >0$, there exists $n_w$ such that :
$\frlq{x\in K}
\int_{\R^d} h(x, w)/\rho_J(x) \,\idc{\|w\|\ge n_w}\, dw \le  \eps$.

an upper-bound $h_\vee$ on the density for each jump :
$\frl{x\in K} \frlq{w\in\R^d} h(x,w) \le h_\vee\, \rho_J(x)$.

Consider the process $X$ given by equation eqS
under  A3.
with $\cD_n := \bar{B}(0, n), n\ge 1$,
$\mathbf{(AF')}$ holds,
thus the exponential quasi-ergodicity also.

Remark :
We do not need to assume any comparison
between $v$ and $h$.
The extinction rate of the QSD can thus be very large,
notably for large values of $v$.

§.§.§ Connection with reaction-diffusion equations

This quasi-ergodicity can be related to the behavior
of the solution $(u(t, x))_{t\ge 0, x\in \R}$
at low densities ($u \approx 0$)
to reaction-diffusion equations of the form :

\begin{align*}
&\partial_t u(t, x)
:= v\, \partial_x u(t, x)
+ \int_\R\,  h(y, x-y)\, u(t, y)\, dy
- \lp \int_\R\,  h(x, w)\,dw\rp\,  u(t, x)
\\&\hcm{1}
+ r(x, u(t, x))\, u(t, x).
\EQn{RDeq}{}
\end{align*}

At low density, the growth rate $r(x, u)$
can be approximated by $r_0(x) := r(x, 0)$,
so as to linearize RDeq.
The invasion at low density
is associated
to the principal eigenvalue of the operator  :

\begin{align*}
\cL^\star f (x)
:=  v\,f'(x)
+ \int_\R\,  h(y, x-y)\, f(y)\, dy
- \lp \int_\R\,  h(x, w)\,dw\rp\,  f(x)
+ r_0(x)\, f(x)
\end{align*}

(see for instance [5], [10]).

If we consider $r_0(x)$ rather as some upper-bound of $r(x)$,
the solution $\bar{u}$
to $\partial_t \bar{u}(t, x) = \cL^\star \bar{u}(t,x)$
shall provide an upper-bound of $u$ by maximum principle approaches.
If the eigenvalue $\lambda^\star$ of $\cL^\star$ is negative,
the solution $u$ is expected to asymptotically decline
at least quicker than at rate $-\lambda^\star$,
provided that the initial condition is sufficiently confined
(we may expect weaker conditions than compactly supported).
Thus, it has also implications as a criteria for non-persistence,
like in [3], [4], [5] , [6], [7], [8].

Assuming that $r_0$ is bounded by $R$,
we can then define $\rho(x) := R - r_0(x)$.
From A3,
A3 imply the existence
of a unique $\laZ$
and QSD $\alpha(dx)$
which can be seen as the eigenvector of $\cL^\star$ associated to $R-\laZ$
in the following sense :

For any functions $f, g$ with compact support and bounded derivatives :
\LAg \alpha \cL \bv f\RAg = - \laZ\, \LAg \alpha \bv f\RAg,\qquad$

noting that, by an immediate integration by parts :

\begin{align*}
&\LAg(g(x)\, dx)\cL\bv f \RAg
= \int_\R dx\, g(x) \, [-v\, f'(x) + \int_\R h(x, w)\, (f(x+w) - f(x))\, dw - \rho(x)\, f(x)]
\\&\hcm{2}
= \int_\R f(x) \cL^\star g(x)\, dx
- R\, \int_\R f(x)\, g(x)\, dx.
\end{align*}

More generally,
we expect that,
as long as $r(x, u)$ is well-approximated by $r_0$,
the solution $u(t,x)$ should stabilize
at an exponential rate quicker than $\zeta$
towards the profile :

\begin{align*}
u(t,x)\, dx
\sim \exp[(R-\laZ)\, t]\,
\lc \int_\R u(0, y)\, \eta(y)\, dy\rc \alpha(dx).
\end{align*}

This corresponds to the view that the individuals
do not interact at the limit of low density,
while they are sufficiently numerous
for $u$ to grow (or shrink)
driven by the law of a typical individual
(taking reproduction and death events into account).

Note that it is not crucial
for the proof
to assume  that $\rho(x) \cvifty{x} \infty$,
i.e. $r_0(x) \cvifty{x} - \infty$.
It suffices that $r_0$ is strictly smaller
than some reference growth rate
(the natural extension of $\rho_{sv}$ from (A5))
outside of some compact set
(or an extension of (A3) for less regular confinements).

Many authors are looking
at characterizing such eigenvalue
when there is possibly no regular eigenvector
(see e.g. [9], [12], [14],
[16], [11]).
Regarding other definitions presented for instance in [11],
we have an upper-bound on any $\lambda$ such that for
some $C^1$ non-negative function $\varphi$ :

\begin{align*}
\cL^\star \varphi  - \lambda\, \varphi \ge 0,\qquad
0< \int_\R \varphi(x)\, \eta(x)\, dx < \infty,
\end{align*}

where the last condition holds if $\varphi\in L^1(\R)$ and is non-degenerate
because $\eta$ is uniformly bounded, and positive.

Indeed, defining the initial condition $\mu(dx) = \varphi(x)\, dx$ :

\begin{align*}
&-\laZ\, \int_\R \varphi(x)\, \eta(x)\, dx
= \LAg \mu \bv \cL \eta \RAg
\\&
= \int_\R \, \eta(x)\,\cL^\star\varphi(x)\, dx
- R \int_\R \varphi(x)\, \eta(x)\, dx
\ge (\lambda - R) \int_\R \varphi(x)\, \eta(x)\, dx,
\end{align*}

so that necessarily :
$\lambda \le R -\laZ$.

Conversely and similarly,
we have the lower-bound $R -\laZ$
on any $\lambda$ such that for
some $C^1$ non-negative function$~\varphi$ :

\begin{align*}
\cL^\star \varphi  - \lambda\, \varphi \le 0,
\qquad \int_\R \varphi(x)\, \eta(x)\, dx < \infty.
\end{align*}

From the dual point of view,
we have the upper-bound $R -\laZ$
on any $\lambda$ such that for
some $C^1$ non-negative function$~\varphi$ :

\begin{align*}
\cL \varphi + (R - \lambda)\, \varphi \le 0,
\qquad 0<\LAg \alpha\bv \varphi\RAg < \infty,
\end{align*}

where this last condition is clearly true for bounded
and non-degenerate $\varphi$,
exploiting the fact that the following Lemma <ref>
implies  that $\alpha$ has a lower-bounded density on any compact set.
Similarly, we have indeed

\begin{align*}
&-\laZ\, \LAg \alpha\bv \varphi\RAg
= \LAg \alpha\, \cL \bv \varphi \RAg
\le (\lambda - R)\, \LAg \alpha\bv \varphi\RAg.
\end{align*}

Conversely and similarly,
we have the lower-bound $R -\laZ$
on any $\lambda$ such that for
some $C^1$ non-negative function$~\varphi$ :

\begin{align*}
\cL \varphi + (R - \lambda)\, \varphi \ge 0,
\qquad 0<\LAg \alpha\bv \varphi\RAg < \infty.
\end{align*}

Much more can certainly be obtained in this direction,
notably regarding the regularity of the QSD
and the survival capacity
or the conditions on the above-mentioned $\varphi$
(looking at the tails of $\alpha$ and $\eta$).
This shall be the purpose of a future work,
also extending the analysis to more general drift terms.

The process $(X_t)$ given in Example 1
can also be approximated by some autonomous process,
whose parameters are averaged
in order to take  into account
the fluctuations
of population size.
It can be quite a reasonable estimation
to assume  a time-scale separation
between the change of $r(X_t)$
and the fluctuations
of the population size process $(N_t)_{t\ge 0}$
around some quasi-equilibrium
(depending on $r(X_t)$).
The probability
that the extinction occurs
in the time-interval $[t, t+\delta t]$ for small $\delta t$
shall then be close to some value $\rho(X_t)\, \delta t$.
Here, the value $\rho(x)$ for $x\in \cX$
is deduced from the process $N^x$
defined as the population size process
with constant phentotypic lag $x$,
i.e. according to equation SN
with $X_t$ replaced by $x$.
A natural estimation of $\rho(x)$
is the extinction rate of of the QSD of $N^x$.

On the other hand,
the natural way to average the parameters
is to define :
$\bar{b}(x) := \int b(x, n)\, \mu_x(dn)$,
$\bar{\sigma}_X^2(x) := \int \sigma_X^2(x, n)\, \mu_x(dn)$
and $\bar{k}_m(x, w) :=  \int k_m(n, x, w)\, \mu_x(dn)$,
where the law of the fluctuations of $N^x$
are given by $\mu_x$.
We believe that
the best approximation is obtained
by taking $\mu_x$
as the stationary distribution of the Q-process
(for $N^x$)
rather than for instance the QSD.
Denoting $x := X_t$,
$\int k_m(n, x, w)\, \mu_x(dn)\, \delta t$
with the former choice of $\mu_x$
is notably expected to well-approximate
the probability that at least one jump of $X$
takes place in the time-interval $[t, t+\delta t]$,
as we condition upon the fact that $N^x$ is not extinct.

For such simplified model,
there is a similar connection to reaction-diffusion
equations of the form :

\begin{align*}
&\partial_t u(t, x)
:= \Delta_x[\bar{\sigma}^2_X(x)\, u(t, x)]
-\nabla_x [\bar{b}(x)\,  u(t, x)]
+ \int_{\R^d}\,  \bar{k}_m(y, x-y)\, u(t, y)\, dy
\\&\hcm{1}
- \lp \int_{\R^d}\,  \bar{k}_m(x, w)\,dw\rp\,  u(t, x)
+ r(x, u(t, x))\, u(t, x),
\end{align*}

where $\rho(x) = R - r_0(x)$ for some $R>0$.

§.§.§ Proof of A3

$(A0)$ is clear.
By , the rate of jumps from any compact set is bounded,
so that $(A1_G)$ holds.
Since the extinction rate outside of $\cD_n$
tends to infinity while $n\rightarrow \infty$,
for any $\rho>0$, we can find some $\cD_n$
for which assumption $(A4)$ is clear
(cf. Subsection 3.1.2 in <cit.>).
We defer to the end of this subsection the proof of :

Under $(A_{\ref{th.A3}}^{1,2,3})$, for any $n\ge 1$,
with $m := n+1$,
there exists $c, t>0$ such that :

\begin{align*}
&  \frl{x \in \cD_{\np}}
\hspace{.5cm}
\PR_x \lc {X}_{\tp}\in dx \pv
\tp < \ext \wedge T_{\cD_m} \rc
\ge \cp\; \idg{\cD_n}(dx).
\end{align*}

In particular,
Assumption $(A2)$ holds with $\alc$
uniform over $\cD_1$.

$\REM$ 1) The main idea is that we are able
to compensate the drift with the jumps
and introduce a bit of fluctuations at each step,
and thus we are able to ensure a dispersion
to any part of the space
(while keeping a bound on the norm).

Besides $(A2)$,
we need a lower-bounded density on any $\cD_n$
in order to prove assumption
(with a reaching time and efficiency of course dependent on this set).

In a following paper <cit.>,
we extend this proof to a coupled process
like in our first application.

Proof of Assumption $(A3_F)$

Consider a compact set $\cD_c = \bar{B}(0, n_c)$
and a test initial condition
$x\in \cD_c$.
We also assume that  $\fl, \Rsv>0$ are given.
W.l.o.g. we assume that $\frlq{y\notin \cD_c}$

$ \rho_e(y) \ge \rhoeT > \Rsv.\quad $

We first define $\tZa$ by the relation :

\begin{align*}
\exp[\Rsv\ltm (2\ n_c / v) -  (\rhoeT-\Rsv)\ltm(\tZa - 2\ n_c / v)]
= \fl /2.
\end{align*}

The l.h.s. is decreasing and converges to 0
when $\tZa \rightarrow \infty$, so that
$\tZa$ is well-defined.
Let $T_{jp}$ be the first jump time of $X$.
On the event $\Lbr \tZa < T_{jp}\Rbr$,
we set $\Uza = \infty$.
The choice of $\tZa$ is done to ensure
that the probability associated to the failure
is indeed exceptional enough
(with threshold $\fl/2$ and time-penalty $\Rsv$).
Any jump occurring before $\tZa$ thus occur from a position
$X(T_{jp}-) \in \bar{B}(0, n_c + v\, \tZa) := K$.
By A3,
we can then define $n_w$ such that  :

\begin{align*}
\frlq{x\in K}
\textstyle \int_{\R^d} h(x, w) \,\idc{\|w\|\ge n_w}\, dw
\le  \fl/2\ltm \exp[\Rsv\, \tZa].
\end{align*}

A jump size larger than $n_w$ is then the other criterion of failure.

On the event  $\Lbr  T_{jp} \le \tZa\Rbr
\cap \Lbr T_{jp} < \ext\Rbr
\cap \Lbr \|W\| \le n_w\Rbr$,
where  $W$ is the size of the first jump
(at time $T_{jp}$),
we set $\Uza := T_{jp} \le \tZa$.
Otherwise $\Uza := \infty$.

In particular, $ \Lbr\ext \wedge \tp \le \Uza \Rbr
= \Lbr\Uza = \infty\Rbr$ is clear.

We prove next that the failures are indeed exceptional enough :

\begin{align*}
\PR_x (\Uza = \infty,\; \tZa < \ext)
\le \PR_x ( \tZa \le T_{jp}\wedge\ext)
+ \PR_x ( T_{jp}<\ext\wedge\tZa\mVg \|W\| > n_w).
\EQn{Ad3.fl}{}
\end{align*}

By the definition of $n_w$, we deal with the second term :

\begin{align*}
\PR_x ( T_{jp}<\ext\wedge\tZa\mVg \|W\| > n_w)
\le \PR_x (\|W\| > n_w \bv T_{jp}<\ext\wedge\tZa)
\le  \fl/2\ltm \exp[\Rsv\, \tZa].
\EQn{Nw}{n_w}
\end{align*}

On the event $\Lbr \tZa \le T_{jp}\wedge\ext \Rbr$ :
$\frlq{x\le \tZa} X_t = x - v\, t\, e_1$.
Thus, $X$ is outside of $\cD_c$ in the time-interval
$[2\ n_c/v\mVg \tZa]$, with an extinction rate at least $\rhoeT$.
By the definition of $\tZa$ :

\begin{align*}
&\PR_x ( \tZa \le T_{jp}\wedge\ext)
\le \exp[-  \rhoeT\ (\tZa - 2\ n_c / v)]
\le \fl/2\ltm \exp[\Rsv\, \tZa].
\EQn{tVee}{\tZa}
\\&\quad
\text{This concludes : }
\qquad
\PR_x (\Uza = \infty,\; \tZa < \ext) \le \fl \ltm \exp[\Rsv\, \tZa].
\EQn{Ad3.N}{}
\end{align*}

On the other hand,
since the density of jumps and the jump rate
are bounded on $K$ :

\begin{align*}
\PR_x ( X(\Uza) \in dx'
\bv T_{jp}< \tZa\wedge \ext\mVg \|W\| \le n_w)
\le h_\vee\, \idc{x'\in \bar{B}(0, n_c + v\, \tZa+n_w)} dx'.
\end{align*}

We know also from Ad3.2 that there exists $t_{mix}, c_{mix}>0$
such that :

\begin{align*}
\PR_{\alc} \big(X(t_{mix}) \in dx\pv t_{mix} < \ext\big)
\ge c_{mix}\, \idc{x'\in \bar{B}(0, n_c + v\, \tZa+n_w)} dx'.
\end{align*}

With $\Uza = t_{mix}$ and $c_{abs} = h_\vee/ c_{mix}$
and inequality Ad3.N, this concludes the proof of $(A3_F)$,
thus of A3 (given Ad3.2).

Proof of Ad3.2 :

We wish to decompose the proof into elementary steps.
The first idea is that we cut $\cD_n$ into very small pieces,
both for the initial value (to ensure the uniformity),
and the final marginal (to ensure a density),
around some reference values, resp. $x_i$ and $x_f$.
We also cut the path from
the neighborhood of the initial value $x_i$
to the neighborhood of the final value $x_f$
into elementary steps with only one jump in each.
We see here that imposing one jump of size in $B(S\, e_1,\, \dS)$
in a time-interval $t_{el} := S/v$
gives to the marginal at time $t_{el}$
a bit of dispersion
in a neighborhood of $x_f$.

Let us formalize the above explanation.
Changing the scale of $X$,
we may assume w.l.o.g. that $S = 1$.
We also consider a characteristic length
of dispersion by $r := \dS/4$.

Given some $n\ge 1$, $x_i\in \cD_n$, $m := n+2$ and $c>0$,
we define :

\begin{align*}
&\cR^{(m)}(c) := \Lbr (t, x_f)\in \R_+\times \R^d \bv
\frl{x_0 \in B(x_i, r)}
\right.
\\&\hcm{2.5}
\left.
\PR_{x_0} \lc X_t\in dx
\pv t < T_{m} \rc
\ge c\; \idg{B(x_f, r)}(x)\,  dx\Rbr,
\EQn{cR}{\cR^{(m)}}
\end{align*}

where $T_m$ stands for the exit time $T_{\cD_m}$.

The proof then relies on the following three elementary lemmas :

Given any $(n , m)$ with $n\ge 1$, $m = n+2$, $x_i \in \cD_n$,
$t, c, c'>0$ and $x\in B(x_i, r)$ :

\begin{align*}
(t,x) \in \cR^{(m)}(c)
\imp \{(t+s, y)\bv (s,y)\in \cR^{(m)}(c')\} \subset \cR^{(m)}(c\times c').
\end{align*}

Given any $m = n+2$, $x_i \in \cD_n$,
there exists
$c_0, t_0, \delta t >0$, such that :

$$[t_0, t_0 + \delta t] \times \{x_i\} \subset \cR^{(m)}(c_0).$$

For any $x\in B(0, 2\, m -2)$, there exists $t', c'>0$ such that :

\begin{align*}
(t, x) \in \cR^{(m)}(c)\quad
(\text{for } t, c>0)
\quad \imp \quad
\{t+t'\} \times  B(x, r)\subset \cR^{(m)}(c\times c').
\end{align*}

Lemma <ref> is just an application of the Markov property.
The proofs of Lemmas <ref> and <ref>
are postponed to AP.A2 Appendix AP.A2.
Next we show how to deduce Ad3.2 from them.

Proof of Ad3.2

Step 1 : from the vicinity of any $x_i\in \cD_n$
to the vicinity of any $x_f\in \cD_n$.

Let $K > \Lfl \|x_f - x_i\| /r\Rfl$.
For $0\le k\le K$, let
Exploiting  Lemma <ref>,
we choose $(t_0, c_0)$ such that
$(t_0, x_i)  \in \cR^{(m)}(c_0)$.
By  Lemma <ref>,
and since $x_{k+1} \in B(x_k, r)$ for each $k$,
and by induction for any $k\le K$,
there exists of $t_k, c_k >0$ such that :
$(t_k, x_k) \in \cR^{(m)}(c_k)$.
In particular with $k=K$,
we get some $t_f, c_f>0$ such that $(t_f, x_f) \in \cR^{(m)}(c_f).$

$\hcm{1}$ Step 2 : a uniform time and efficiency.

By compactness, there exists $(x^\ell)_{\ell \le L}$ such that :
$\cD_n \subset \medcup_{\ell \le L} B(x^\ell, r)$.
Let $t_{max}$ be the larger time $t_f$ –as in step 1–
needed to reach
the vicinity of any $x_f \in \{x^\ell\}$
from any $x_i \in \{x^{\ell'}\}$.
To adjust the arrival time,
we make the process stay some time around $x_i$.
Let $t_{st} := t_0 \ltm \Lfl 1+ t_0/\delta t\Rfl$.
Then, for any $t \ge t_{st}$, we can find some $c>0$ for which
$(t, x_i) \in \cR^{(m)}(c)$.
We let the reader deduce this corollary from
Lemmas <ref> and  <ref>.

Combining this corollary with Lemma <ref>
ensures, with $t_{mix} := t_{max} + t_{st}$,
a global lower-bound $c_{mix}>0$ on the efficiency  :

\begin{align*}
\frl{\ell, \ell' \le L}
\frlq{x_0 \in B(x^{\ell'}_i, r)}
\PR_{x_0} \lc X(t_{mix})\in dx
\pv t_{mix}  < T_{m} \rc
\ge c_{mix}\; \idg{B(x^\ell_f, r)}(x)\,  dx.
\end{align*}

Since  $\cD_n \subset \medcup_{\ell \le L} B(x^\ell, r)$,
this completes the proof of Ad3.2.

Pure jump processes, the simplest case

§.§.§ Description of the process :

In the case of a pure jump process,
we have a new difficulty :
the process might survive better by doing no jump
and staying where it starts from.
By proving assumption $\RAF$, we ensure that this does not happen.

Let $(X_\tp)_{\tp\ge 0}$ be a pure jump process
on $\cX~:= \R^d$ defined by :

\begin{align*}
X_\tp~:= x + \int_{[0,\tp] \times \R^d\times \R_+}
w \,	\idc{ u \le h(X_{s^-}, w)}
M(ds,\, dw,\,du)
\EQn{X}{X_\tp}
\end{align*}

where $M$ is a PPP
with intensity  $ds\,dw\, du$.
It is again associated
to an extinction rate $\rho_e$.

$(X_\tp)$ is a Markov Process with piecewise constant trajectories.
Conditionally upon $X_\tp =x$,
the waiting-time and size of the next jump are independent,
the law of the waiting-time is exponential with rate
$\rho_J(x):=\int_{\R^d} h(x, w)\ dw<\infty$
and the law of the jump size is $h(x, w) / \rho_J(x)\, dw$.

* Ad4:
$h$ is measurable and for any compact $K\subset \R^d$, we have :

an upper-bound $\roV$ on the jump-rate :
$\frlq{x\in K}
0<\rho_J(x) := \int_{\R^d} h(x, w) \, dw \le  \roV$.

a lower-bound $h_\wedge>0$ on the jumps of size smaller than $r$ :

$\frl{x\in K} \frlq{w\in B(0, r)}
h(x, w)\ge  h_\wedge$.

a tightness estimate for the jumps :

for any $\eps >0$, there exists $n_w$ such that :
$\sup_{\{x\in K\}}
\int_{\R^d} h(x, w)\,\idc{\|w\|\ge n_w}\, dw /\rho_J(x) \le  \eps$.

an upper-bound $h_\vee$ on the density of jumps :
$%\frl{x\in K} \frlq{w\in\R^d}
\quad h(x,w) \le h_\vee\, \rho_J(x)\quad$
on $K\ltm \R^d$.

Assumption   holds for some $\Rsv>0$ such that
$\rho_e$ is lower-bounded by $\Rsv$ outside some compact set.
Moreover,  $\rho_e$ is locally bounded
and explosion implies extinction :
$\ext \le \Tsup{n\ge 1} T_{\cD_n}$.

No stable position :
$\Rsv$ from satisfies :

\begin{align*}
\Rsv < \rho_{sb}~:= \inf_{x\in \R^d} \Lbr \rho_J(x)+\rho_e(x) \Rbr.
\end{align*}

For this example, we simply choose $\qquad \cD_n = \bar{B}(0, n)$,
for $n\ge 1$.

Consider the process $X$ given by equation X
under  Ad4.

$\mathbf{(AF')}$ holds,
hence $X$ is exponentially quasi-ergodic.

In this model,
the speed of convergence
need not be not uniform :

There exists $h$ and $\rho_e$ such that
Ad4 is satisfied and such that :

For any $(t, \eps) \in (0,\infty)\ltm(0,1)$,
there exists $x\in \R^d$ such that :

\begin{align*}
\NTV{ \PR_x\lp X_\tp \in dy\Bv \tp < \ext \rp
- \PR_{0}\lp X_\tp \in dy\Bv \tp < \ext \rp}
\ge  1- \eps.
\Eqn{PjNU}{NU}
\end{align*}

The proof of this Proposition
is postponed
to Subsection <ref>.

Remarks :
Assumption   enables us to prove
that there will be no conservation
of the Dirac mass at the initial position

(in case this value would be
around the minimum of $\rho_J + \rho_e$).
In practice,
it seems not so clear to give not too stringent conditions
ensuring this.
Similar issues is described in [2]
in the context of a linear reaction-diffusion equation,
and our hope is to precise the connection in some future work.

But to get a more explicit toy-model,
we can see that Assumption   is implied
by the quite restrictive condition :

There exists $(x_{opt}, r_{opt}) \in \R^d\ltm(0, \infty)$
such that :

\begin{align*}
\rho_J(x)+\rho_e(x) \equiv
\inf \Lbr \rho_J(y)+\rho_e(y)\Bv y\in \R^d \Rbr,
\quad x \in B(x_{opt}, r_{opt}).
\Eqn{Pjrho}{\rho_J + \rho_e}
\end{align*}

Assumption   is then obtained by trajectories that stay in $B(x_{opt}, r_{opt})$ –the jumps inside this set improves a bit the rate of survival,
see <ref> below.

No precise regularity condition is needed for $h$ and $\rho_e$.

§.§.§ Connection with reaction diffusion-equations

This example is also specifically related
to a form of reaction-diffusion equation,
here :

\begin{align*}
\partial_t u(t, x)
:= v\, \partial_x u(t, x)
+ \int_{\R^d}\,  h(y, x-y)\, u(t, y)\, dy
- \lp \int_{\R^d}\,  h(x, w)\,dw\rp\,  u(t, x)
+ r(x, u(t, x))\, u(t, x).
\end{align*}

What we discussed in Subsection <ref> easily adapts
to the operators :

\begin{align*}
& \cL^\star f (x)
:= \int_{\R^d}\,  h(y, x-y)\, f(y)\, dy
- \lp \int_{\R^d}\,  h(x, w)\,dw\rp\,  f(x)
+ r_0(x)\, f(x);
\\&
\cL f(x) :=
\int_{\R^d} h(x, w)\, (f(x+w) - f(x))\, dw - \rho(x)\, f(x),
\end{align*}

where we defined $\rho(x) = R - r_0(x)$
for some upper-bound $R$ of $r_0$.

§.§.§ Proof of Ad4 :

Assumption   is clearly satisfied,
as well as $(A1_G)$ since the jump rate is locally bounded.

Assumptions $(A_{\ref{th.Ad4}}^{1,2, 5})$ imply assumption ,
with $\alc$ the uniform distribution over $\cD_1$.
More generally,
for any $n\ge 1$,
we can define $m>n$ and $\tm, \cm>0$ such that :

\begin{align*}
\frlq{x\in \cD_n}
\PR_x \lc X(\tm)\in dy\pv
\tm < \ext \mVg
\frl{\spr \le \tm} X_\spr \in \cD_{\mpr} \rc
\ge \cm\, \idc{y \in \cD_{\np}} \, dy
\end{align*}

For the proof of Lemma <ref>,
we can rely on exactly the same lemmas
as in the proof of Ad3.2.
This is even simpler here since,
with such pure jump process,
there is no constraint on time
(it is obvious
that for any $t>0$, there exists $c>0$ s. t. $(x_t, c)\in \cR^{(m)}(c)$,
and in Lemma <ref>,
$t'$ can take any value).
The proof is thus left to the reader.

Assumption  ensures that
there exists $\cD_n=: \cD_c$, $\Rsv$ and $\rho$ with $0<\Rsv < \rho$,
such that both assumptions $\RET$ and $\RSv$ are satisfied
(see Subsection 3.1.2 in <cit.>).
The proof of Ad4 is thus completed with  :

Ad4 implies that for any
$\cD_c = \cD_n$, assumption    holds with the above value of $\rho$.

Proof of Lemma <ref>
We consider here two types of "failed attempt",
in fact the same
as for our second application.
Either the process has not jumped for a very long time,
or the jump is too large.

Let $f_\ell >0$.
By assumption , we can find $\tZa>0$ such that :

\begin{align*}
& \hspace{5cm}
\exp[- ( \rho_{sb} - \Rsv) \,\tZa]  \le f_\ell/2,
\\&
\text{ so that with } x\in \cD_n,\,
T_{jp} \text{ the time of the first jump for the process starting from } x~:
\\&\hspace{1cm}
\exp[\Rsv\,\tZa] \,\PR_x(\tZa< T_{jp}\wedge \ext)
= \exp[- ( \rho_J(x)+\rho_e(x)-\Rsv) \,\tZa]
\le f_\ell /2.
\Eqn{eqd}{\tZa}
\end{align*}

We can then declare a failure
when the process starting from $x$
has not jumped at time $\tZa$,
i.e. $\Uza := \infty$ on the event $\Lbr \tZa < T_{jp}\Rbr$.
This clearly implies
$\Uza \le \tZa$
on the event $\Lbr\Uza < \infty \Rbr$.
$X(\Uza-) \in \cD_c$,
so that, with assumption ,
we can define some $n_w$ such that
$\Delta X ( \Uza) \ge n_w$
with probability less than $\fl\, \exp[-\Rsv\, \tZa] /2$.
The rest of the proof is exactly the same as for our second application
and is thus left to the reader.

§.§.§ Proof of Proposition <ref>

We consider the models where the jump rate is uniform
and the jump sizes are uniformly distributed on a given ball :

\begin{align*}
&R >0,\quad
h(x, w)~:= h_J\; \idc{w\in B(0, R) },
\with \rho_J(x) = \rho_J = h_J\ltm Leb\{B(0, R)\}
\Eqn{defRhoJ}{\rho_J}
\end{align*}

Thus,  Assumptions $(A_{\ref{th.Ad4}}^{1-4})$ clearly hold.
In order to ensure  Assumptions $(A_{\ref{th.Ad4}}^{5,6})$,
we impose assumption

\begin{align*}
&\text{with~: }\frl{x \in B(0, R)} \rho_e(x)~:= \rho_e^1 > 0,
\Eqn{defRhoE}{\rho_e}
%\\ & \hcm{1}
\quad
\frl{x \notin B(0, R)} \rho_e(x)~:= \rho_e^2 \ge \rho_e^1 + \rho_J.
\end{align*}

Inequalities   and
imply Ad4.

Before we prove this lemma,
we prove that the just defined process
satisfies inequality $\REQ{\thePjNU}$ :

Fix some $t>0$.
The MCNE of $X_t$ starting from 0
has most of its mass in some large enough ball $B_0$.

Outside of $B(0, R)$, the process behaves homogeneously. Thus, we can find some large enough ball $B(x, L)$, which does not intersect $B_0$, such that, conditionally on survival at time $t$, the process $(X_\spr)_{\spr\le \tp}$ starting from $x$
will not escape $B(x, L)$ with high probability.
escaping $B(x, L)$ before time $t$ cannot bring
a profit in terms of extinction
larger than $\exp([\rho_e^1 - \rho_e^2]\, t)$,
while the associated probability of escape
(while not taking the extinction into account)
goes to 0 as $L$ goes to infinity.
$x$ is then chosen sufficiently large to ensure distinct supports.
the MCNEs from $0$ and from $x$ at time $t$
have distinct support –with high probability–
proving inequality .

Proof of Lemma <ref> :

To deduce $(A_{\ref{th.Ad4}}^{5,6})$,
it suffices to prove
that assumption   holds with some $0<\Rsv < \rho_e^1 + \rho_J$.

In practice, we show that we can find $\tp >0$ and
$0<\Rsv < \rho_e^1 + \rho_J$ such that :

\begin{align*}
&\frl{x\in B(0, R)}
\hspace{0.5cm}
\PR_{x} \lc \frl{\spr\le \tp} X_\spr \in B(0, R)\pv
\tp< \ext\rc
\ge \exp(-\Rsv\,\tp).
\Eqn{RG}{R_g}
\end{align*}

(here we consider the Euclidean open balls).
Thanks to Lemma 2.5.2 in <cit.>,
this is sufficient to prove assumption   with $\cD_s =  B(0, R)$,
$\cD_{\mpr}~:= \bar{B}(0, \mpr)$,
for some $\mpr \ge R$
and this value of $\Rsv$.

In fact, we can choose $\tp >0$ arbitrarily
(the smaller it is,
the better should be our estimate
on $\Rsv$).
By definitions   and  :

\begin{align*}
&\frl{x\in B(0, R)}
\hspace{0.5cm}
\PR_{x} \lc \frl{\spr\le \tp} X_\spr = x \pv
\tp< \ext\rc
= \exp(-[\rho_e^1 + \rho_J] \,\tp).
\Eqn{NJ}{NJ}
\end{align*}

\begin{align*}
&\text{Let~: }x \in B(0, R),\quad  u~:= x/ \|x\|,\quad
z:= \lp \|x\| - R/2 \rp \, u  \in B\lp 0, R/2\rp
\\& \hcm{1}
\text{so that } B\lp z, R/2 \rp \subset B(0, R) \cap B(x, R),\quad
x \notin B\lp z, R/2 \rp
\Eq{eqt}
%\Eqn{Z}{B\lp z, R/2 \rp}
\\&% \hcm{1}
J_p~:= [0, \tp]\times B(z-x, R/2)\times [0,h_J],
\quad
\cN J~:= [0, \tp]\times B(0, R)\times [0,h_J]
\setminus J_p
\end{align*}

By equation X,
a.s. on $\Lbr M(Jp) = 1 \mVg M(\cN J) = 0\Rbr$,
there exists $T_{jp} \in (0, \tp)$
and $Z\in B(z, R/2)$ such that :

\begin{align*}
&\frl{0\le \spr< T_{jp}} \quad X_\spr = x,\qquad
\frl{T_{jp}\le \spr\le \tp} \quad X_\spr = Z
\end{align*}

Thus with inequalities   and  :

\begin{align*}
\PR_{x} \lc \frl{\spr\le \tp} X_\spr \in B(0, R)\pv
\tp< \ext\rc
\\&\hcm{1}
\ge \exp(-[\rho_e^1 + \rho_J] \,\tp)
+ \exp(-\rho_J \,\tp)\, \PR\lp M(Jp) = 1 \mVg M(\cN J) = 0\rp.
\Eq{equ}
\end{align*}

\begin{align*}
\text{Now, }&\PR\lp M(Jp) = 1 \mVg M(\cN J) = 0\rp
= (h_J\, \tp\, Leb\{B(z-x, R/2)\})\;
exp\lp - \rho_J\, \tp \rp
\\&\hcm{2}
:= rg >0 \qquad -\text{independent of } x.
\Eq{eqd}
\end{align*}

Inequalities  and  clearly imply inequality ,
thus Ad4.

Pure jump processes, a coordinate at a time

§.§.§ Description of the process :

Another difficulty occurs when each jump affects only
one component of the process at a time.
For the comparison assumption $\RAF$,
we thus need to wait at least until there has been a jump
for each of these components.

Let $(X_\tp)_{\tp\ge 0}$ be a pure jump process
on $\cX~:= \R^d$ defined by :

\begin{align*}
X_\tp~:= x + \sum_{i\le d} \int_{[0,\tp] \times \R\times \R_+}
w \,\mathbf{e_i}\, 	\idc{ u \le h_i(X_{s^-}, w)}
M_i(ds,\, dw,\,du)
\EQn{Xdim}{X_\tp}
\end{align*}

where $M_i$ are mutually independent PPP
with intensities  $ds\,dw\, du$.
The process is still associated
to an extinction rate $\rho_e$.

$(X_\tp)$ is a Markov Process with piecewise constant trajectories.
Conditionally upon $X_\tp =x$,
the waiting-time and size of the next jump are independent,
the law of the waiting-time is exponential of rate
$\rho_J(x):= \sum_{i\le d} \rho_J^i(x),$
where $ \rho_J^i(x) := \int_{\R^d} h_i(x, w)\ dw<\infty$.
The jump occurs on the $i$-th coordinate
with probability $\rho_J^i(x) / \rho_J(x)$,
then with size given by $h_i(x, w) / \rho_J^i(x)\, dw$.

* Ad5:
For any $i\le d$, $h_i$ is measurable
and for any compact $K\subset \R^d$, we have :

An upper-bound $\roV$ on the jump-rate :
$%\frlq{x\in K}
\sup_{\{x\in K\}} \rho_J(x) \le  \roV$.

A lower-bound $h_\wedge>0$ on the jumps of size lower than $r$ :

$\frl{x\in K} \frl{i\le d} \frlq{w\in B(0, r)}
h_i(x, w)\ge  h_\wedge$.

A tightness estimate for the jumps :

for any $\eps >0$, there exists $n_w$ such that :
$\quad \inf_{\{x\in K,\, i\le d\}}
\Lbr \int_{\R} h_i(x, w) \,\idc{\|w\|\ge n_w}\, dw /\rho_J(x)  \Rbr
%\frl{x\in K} \frlq{i\le d}
\le  \eps$.

An upper-bound $h_\vee$ on the density for each jump :

$\frl{x\in K} \frl{i\le d} \frlq{w\in\R}
h_i(x,w)/ \rho_J(x)  \le h_\vee$.

There is also a global lower-bound $p_\wedge$ on the probability that each direction gets involved in the jump :
$\quad \sup_{\{x\in \R^d,\, i\le d\}}
\Lbr \rho_J^i(x) / \rho_J(x)  \Rbr
\ge p_\wedge.$

Assumption   holds for some $\Rsv>0$ such that
$\rho_e$ is lower-bounded by $\Rsv$ outside some compact set.
Moreover,  $\rho_e$ is locally bounded
and explosion implies extinction :
$\ext \le \Tsup{n\ge 1} T_{\cD_n}$.

No stable subset :
With the value of $\Rsv$ given previously :

\begin{align*}
\Rsv < \inf_{\{x\in \R^d,\, i\le d\}}
\Lbr \rho_J^i(x)+\rho_e(x) \Rbr
~:= \rho_{sb}.
\end{align*}

For this example,
we can still consider $\; \cD_n = \bar{B}(0, n)$,
for $n\ge 1$
with rather here the infinite norm $\|.\|_\infty$.

Consider the process $X$
given by equation Xdim
under  Ad5.

$\mathbf{(AF')}$ holds,
thus also the exponential quasi-ergodicity.

$\REM$ The purpose of assumption
is to bound
the time $T_c$ at which
either extinction occurs or all of the coordinates have changed.
Assumption   indeed ensures
an exponential moment with parameter $\Rsv$.
Since there is a finite number $d$ of coordinates to be reassigned,
we can decompose the interval $[0, T_c]$ into at most $d$
intervals whose law is simpler.
The transition from one sub-interval to the next
occurs when there is a jump
affecting a coordinate not previously reassigned,
as long as the extinction has still not occurred.
The lengths of these intervals
can be easily upper-bounded by some exponential laws.
The main constraint on $\Rsv$
is then given by the worst condition on these steps,
namely where one waits for the last coordinate to be reassigned.

We also refer to Subsections <ref> and <ref>
for the connection with reaction-diffusion equations,
which holds in the same way,
this time for a non-local dispersion operator of the form :

\begin{align*}
\cM f (x)
:=  \sum_{i\le d}
\lc\int_{\R}\,  h_i(x - w_i\, e_i,\, w_i)
\, f(x - w_i\, e_i)\, dw_i
- \lp \int_{\R}\,  h(x, w_i)\,dw_i\rp\,  f(x)\rc
\end{align*}

§.§.§ Proof of Ad5 :

Assumption   is clearly satisfied,
as well as $(A1_G)$ since the jump rate is locally bounded.
Since the extinction rate outside of $\cD_n$
tends to infinity,
we can find for any $\rho>0$
some $\cD_n$ for which assumption $(A4)$ is clear.
Next is :

Assumptions $(A_{\ref{th.Ad5}}^{1,2, 5})$ imply assumption ,
with $\alc$ the uniform distribution over $\cD_1$.
More generally,
for any $n\ge 1$,
we can find $m>n$ and $\tm, \cm>0$ such that :

\begin{align*}
\frlq{x\in \cD_n}
\PR_x \lc X(\tm)\in dy\pv
\tm < \ext \mVg
\frl{\spr \le \tm} X_\spr \in \cD_{\mpr} \rc
\ge \cm\, \idc{y \in \cD_{\np}} \, dy
\end{align*}

For the proof of Lemma <ref>,
we can rely on exactly the same lemmas
as in the proof of Ad3.2
to prove the mixing on any coordinate.
It is then not difficult to combine the estimates
on the different coordinates.
The proof is again left to the reader.

The proof of Ad5 is thus achieved with  :

Ad5 implies that for any $\cD_c\in \Dp$, assumption    holds.

Proof of Ad5
We consider here three types of "failed attempts".
Either the process has not done all of its required jumps
despite a very long time of observation,
or there are too many of these jumps,
or at least one of these jumps is too large.

Definition of the stopping times and time of observation

For $k\le d$, let $T^k_J$ the first time
at which (at least) $k$ jumps have occurred in different coordinates.
On the event $\Lbr T^k_J < \ext\Rbr$ (for $0\le k\le d-1),$
and conditionally on $\F_{T^k_J}$,
we know from assumption
that $T^{k+1}_J\wedge \ext - T^k_J$
is upper-bounded by an exponential variable
with rate parameter $\rho_{sb} > \Rsv$.
Thus, with $\Rsv' := (\rho_{sb} + \Rsv) / 2$ :

\begin{align*}
&\frlq{x\in \R^d}
\E_x \exp[ \Rsv'\, (T^d_J\wedge \ext) ]
\le \lc 2\, \rho_{sb} / (\rho_{sb} - \Rsv) \rc^d := e_f < \infty,
\\&
\PR_x[T^d_J\wedge \ext > t]\, \exp[ \Rsv\,t]
\le e_f\, \exp[- (\rho_{sb} - \Rsv)\,t/2]
\cvifty{t} 0.
\EQn{EF}{}
\end{align*}

Let $f_\ell >0$.
By inequality EF, we can find $\tZa>0$ such that :

\begin{align*}
\exp[\Rsv\,\tZa] \,\PR_x(\tZa< T^d_J\wedge \ext)
\le f_\ell /3.
\EQn{tza4}{\tZa}
\end{align*}

On the event $\Lbr \tZa < T_J^d\Rbr$,
we set $\Uza := \infty$.
This clearly implies that
$\Uza \le \tZa$ on the event $\Lbr \tZa< \Tfl \Rbr$.

Upper-bound on the number of jumps

Thanks to assumption ,
at each new jump,
conditionally on the past until the previous jump,
there is a lower-bounded probability
that a new coordinate gets modified.
The number $N_J$ of jumps before $T^d_J$
(on $T^d_J<\ext$)
is thus upper-bounded
by a sum of $d$ mutually independent geometric
random variables.
Therefore, we can define $n_J^\vee \ge 1$ such that :

\begin{align*}
\frlq{x\in \R^d}
\PR_x(n_J^\vee \le N_J \mVg T^d_J<\ext)
\le f_\ell\, \exp[-\Rsv\,\tZa] /3.
\EQn{nJ4}{n_J^\vee}
\end{align*}

We thus declare a failure if
the $n_J^\vee$-th jump occurs
while $T^d_J$ still is not reached.

Upper-bound on the size of the jumps

Exploiting assumption  (recursively for each jump) in the same way
as in the proof of Lemma <ref>,
we can deduce a value $n_W$ such that for any $x\in \cD_c$ :

\begin{align*}
&\PR_x(\text{Some jump before the } n_J^\vee\text{-th one
and before } T^d_J\wedge \ext
\text{  has a size larger than } w^\vee )
\\&\hcm{5}
\le f_\ell\, \exp[-\Rsv\,\tZa]  /3.
\EQn{nW4}{w^\vee}
\end{align*}

We thus declare a failure if a jump larger than $w^\vee$ occurs.
From this, we deduce that the process has stayed in some
$\cD_L := B(0, n_c + n_J^\vee\ltm w^\vee)$
on the event $\{ \Tfl^1 =  \infty \}$.

On the event that at time $T_J^d < \ext$,
none of the three following conditions have been violated:

$\quad(i)$ $T_J^d$ still has not occurred at time $\tZa$,
$\quad $
or $(ii)$ the $n_J^\vee$-th jump has occurred,

$\qquad $ or $(iii)$ a jump of size larger than $n_W$
has occurred
(before time $T_J^d $),

we set $\Uza := T_J^d$.
Otherwise $\Uza := \infty$.

Given this construction
and inequalities tza4, nJ4 and nW4,
it is clear
that :

\begin{align*}
&  \Lbr\ext \wedge \tp \le \Uza \Rbr
= \Lbr\Uza = \infty\Rbr
\quad  \text{ and }\quad
\PR_{x} (\Uza = \infty, \,  t< \ext)
\le \fl\, \exp(-\rho\, \tp),
\end{align*}

The technical details are left to the reader.

As previously,
the proof is then completed with Lemma <ref> and :

Assume that $(A_{\ref{th.Ad5}}^{3-7})$ hold,
with the preceding notations.
Then, there exists $c>0$ such that :

\begin{align*}
\PR_x ( X(\Uza) \in dx'
\pv \Uza < \ext)
\le c\, \idc{x'\in \cD_L} dx'.
\end{align*}

The proof, which is technical,
is presented in detail in  Appendix APUza.

§ PROOFS OF THEOREMS <REF>-3

§.§ Relax Assumption   into Assumption $(A1_G)$

§.§.§ Stabilization of the marginals

In practice,
Assumption   is needed only
for the proof of Theorem 4.1. in <cit.>.
We thus only need to establish
the following extension of Theorem 4.1. in <cit.> :

Assume that $\mathbf{(AF)}$ holds {assumption   plays no role here}.
Then, there exists  $\xi_{\#} \in (0,1)$
and  $n_{\#} \ge m_{sv}$ (from assumption $(A5)$),
such that,
to any pair $(n, \xi)$,
with $\np \ge 1$ and $ \xi \in (0,1]$,
we can associate a time
$t_{\#} = t_{\#}(n, \xi)>0$ such that :

\begin{align*}
&\frl{\mu \in \Mnx}
\frl{\tp \ge \tXt}\qquad
\mu A_\tp \in \M_{\#} := \M_{n_{\#},\, \xi_{\#}}.
\EQn{Mxt}{\M_{\#}}
\end{align*}

§.§.§ The sequence of excursions

By $\mathbf{(AF)}$, there exists $\Rsv$, $\rhoeT$,
$\cD_s$ and  $\cD_{bk} := \cD_{n_{bk}}$ such that
$0< \Rsv<\rhoeT$, $\cD_s \subset \cD_{bk}$,
$(A5)$ holds (with $\Rsv$ and $\cD_s$)
as well as :

\begin{align*}
\Tsup{x} \E_{x} \lp \exp\lc \rhoeT\, (\tau_{bk}^1\wedge \ext) \rc \rp
< \infty,
\EQn{eqDbk}{\cD_{bk}}
\end{align*}

where $\tau_{bk}^1$ denotes the first hitting time of $\cD_{bk}$.
Let also $\cD_{out} := \cD_{n_{bk} +1}$.

Let $\Rsv'  \in (\Rsv, \rhoeT)$
and $\rho_S = \Rsv' - \Rsv$
and consider a family of i.i.d. r.v. $(T_S^i, i\ge 1)$
whose law is exponential with parameter $\rho_S$.

We then define by induction over $i\in \N$:

\begin{align*}
:= \inf\Lbr \tp \ge \tau_{bk}^i \pv
X_\tp \notin \cD_{out} \Rbr
\wedge (\tau_{bk}^{i}+T_S^i),\;
T^0_{out}:= 0,\\
:= \inf\Lbr \tp \ge T^i_{out}
\pv X_\tp \in \cD_{bk} \Rbr.
\end{align*}

Arguing similarly as in <cit.>,
the proof relies on the following three Lemmas,
whose statements are the same in this new extension,
except for the definitions of $T_{out}^i, i\ge 0$
(involving an additional stopping at exponential rate)
and consequently also those of $\tau_{bk}^i, i\ge 0$.

"First arrival in $\cD_{bk}$" :

Assume that eqDbk, and assumptions   and   hold.
Then, for any $n\ge 1$ and $\xi \in (0,1)$,
there exists $C_e = C_e(n,\, \xi) > 0$ such that :

\begin{align*}
&\frl{t_e > 0}
\frl{\mu\in  \Mnx}\quad
\PR_\mu ( t_e \le \tau_{bk}^1 \bv   t_e < \ext )
\le  C_e(n,\, \xi) \, e^{-(\rhoeT - \Rsv)\, t_e}.
\end{align*}

"Containment of the process after $T^i_{out}$" :

Suppose that assumptions   and   hold.

Then, for any $ n_{\ell j}\ge 1$,
there exist $n_{\#} > n_{\ell j}$,
$\Dxt~:= \cD_{n_{\#}}$
and $c_{\#} > 0$ such that,
with $T_{\#}~:= \inf\Lbr s> 0\pv X_s \notin \Dxt \Rbr $ :

\begin{align*}
&\frl{x\in \cD_{\ell j}}
\frl{t > 0}
\hspace{0.5cm}
\PR_{x} \lp
\tp < T_{out}^1\wedge T_{\#}\wedge \ext \rp
\ge c_{\#}\, \exp[- \Rsv'\, t].
\end{align*}

: Last exit from $\cD_{out}$

Suppose that assumptions ,  $(A1_G)$, , ,
and  eqDbk  hold,
with $\cD_s \subset \cD_{m} \subset \cD_{bk}
\subset \cD_{out} \subset \cD_{\ell j}$
and $\rhoeT > \Rsv$.

Then, there exists $C_v > 0$
such that for any $\mu\in  \Mone$ and any $t_e > t_m>0$,
the following holds :

\begin{align*}
\PR_\mu \lp T_{out}^{I(t_e)} \le t_e - t_m\mVg
t_e \le \tau_{bk}^{I(t_e)+1}\mVg
\tau_{bk}^1 \le t_e  \Bv   t_e < \ext \rp
\le  C_v \, e^{-(\rhoeT - \Rsv')\, t_m},\\
&\hspace{2.5cm}\text{ with }
I(t_e)~:= \max\Lbr i\ge 0\mVg T^i_{out} \le t_e\Rbr < \infty.
\end{align*}

The proof of the Lemma <ref> does not involve the sequence $T^i_{out}$
and is thus the same as its equivalent in <cit.>.
Of course, it implies the result where $\Rsv$ is replaced by $\Rsv'$.
For Lemma <ref>,
the stopping due to the exponential variable
adds artificially
a death rate $\rho_S$ to the original process,
and the proof is then almost the same
as its equivalent in <cit.>.
Only the proofs of Lemma <ref>
and the way these lemmas imply Theorem <ref>
are somewhat affected by these changes.
While these two proofs are very similar
to the ones in <cit.>,
they are based on the following lemma,
whose proof is deferred to
APfuj Appendix APfuj.

Let $U^i_{j}$ be the first jump time of $X$
from $\cD_{out}$ to outside of $\Dlj$ after $\tau^i_{bk}$.
Define $T_e^i := U^i_{j} \wedge (\tau^i_{bk}+T_S^i)$.
On the event
$\Lbr T_e^i = T^i_{out}<\ext \Rbr
\cap \Lbr t_e - t_m < T_e^i \Rbr$,
there exists a  $\sigma$-algebra $\F_{T_e^i}^*$
(specifying the past of the process until $T_e^i$),
such that the process satisfies
$X(T_{out}^i) = X(T_e^i) = X(T_e^i -)\in \cD_{out}$
with probability at least $p_S := \rho_S / (\rho_S + k^\vee)$.
Moreover $X(T_e^i -)$ and $T_e^i$
are $\F_{T_e^i}^*$-measurable.

Remark :
The role of the $T_S^i$
is to act as reference marks for us
to assign some set of trajectories
where $T^i_{out} = \tau_{bk}^{i}+T_S^i$
as competitors (in terms of their survival ability)
to the sets trajectories
where $X(T^i_{out})\notin \Dlj$,
in such a way that both coincide
until the common value of $T^i_{out}$.

§.§.§ Proof that Lemmas <ref>, <ref> and <ref>
imply Theorem <ref>

With Lemma <ref> and <ref>, we obtain a deterministic upper-bound –with probability close to 1– on how much time the process may have spent outside $\cD_{out}$.

Thus, we can associate most trajectories ending outside $\cD_{out}$
to others ending inside $\cD_{out}$. From this association, we deduce a lower-bound on the probability to see the process in $\cD_{out}$.

We first choose $n_{\#}$ according to Lemma <ref>.
Thanks to Lemma <ref>, we choose some $t_m> 0$ sufficiently large
to ensure : $ \frl{t_e > t_m}
\frl{\mu \in \Mone}$

\begin{align*}
& \PR_\mu \lp T_{out}^{I(t_e)} \le t_e - t_m\mVg
t_e \le \tau_{bk}^{I(t_e)+1}\mVg
\tau_{bk}^1 \le t_e  \Bv   t_e < \ext \rp
\le \Ndiv{4}.
\Eqn{equ}{Le}
\end{align*}

Let $n\ge 1,\, \xi \in (0,1]$.
Thanks to Lemma <ref>, we know that for some

\begin{align*}
\frl{t_e \ge \tXt}
\frl{\mu\in  \Mnx}\quad
\PR_\mu \lp t_e \le \tau_{bk}^1 \Bv   t_e < \ext \rp
\le  \Ndiv{4}.
\Eqn{eqd}{Fa}
\end{align*}

Let $\mu \in \Mnx$.

Because on the event
$\Lbr \tau_{bk}^{I(t_e)+1}\le t_e < \ext\Rbr$,
the process stays in $\cD_{out}\subset \cD_{\#}$
in the time-interval $[\tau_{bk}^{I(t_e)+1}, t_e]$ :

\begin{align*}
\PR_\mu \lp  \tau_{bk}^{I(t_e)+1}\le t_e
\Bv   t_e < \ext \rp
\ge \Ndiv{4}
\quad \text{ implies }\quad
\mu A_t(\Dxt) \ge 1/4.
\EQn{TBKe}{}
\end{align*}

In the following, we then assume :

\begin{align*}
\PR_\mu \lp  \tau_{bk}^{I(t_e)+1}\le t_e
\Bv   t_e < \ext \rp
< \Ndiv{4},
\\& \text{which implies with \REqu\, and \REqd~: } %\quad\\
\PR_\mu \lp t_e - t_m < T_{out}^{I(t_e)}\mVg
\tau_{bk}^1 \le t_e  \Bv   t_e < \ext \rp
\ge \Ndiv{4}.
\Eq{eqt}%{ t_e - t_m < T_{out}^{I(t_e)}}
\end{align*}

We wish to bound,
on the event $\Lbr t_e - t_m < T_{out}^i \Rbr$,
the probability :

$\PR_{X(T_{out}^i)} \lp
t_e - T_{out}^i < \wtd{T_{\#}}\wedge \wtd{T_{out}^{1}} \wedge \wtd{\ext}  \rp.\;$
This is easily done if $X(T_{out}^i)$ is still in $\Dlj$
thanks to Lemma <ref>,
but cannot generally be done otherwise.
if $X(T_{out}^i)$ is not in $\Dlj$,
it means that there is a jump from $\cD_{out}$ out of $\Dlj$
at time $T_{out}^i$.
By $(A1_G)$,
such jumps occur with a rate upper-bounded by some $k^\vee >0$.
On the other hand,
stopping $X$ at time $\tau^i_{bk}+T_S^i$
means stopping it with rate $\rho_S$,
at a time where a.s. no jump occurs.

By Lemma <ref>,
we obtain the lower-bound :

\begin{align*}
&\E_{\mu} \lc \PR_{X(T_{out}^i)} \lp
t_e - T_{out}^i < \wtd{T_{\#}}\wedge \wtd{T_{out}^{1}} \wedge \wtd{\ext}  \rp
\Bv \F_{ T_e^i}^* \rc
\ge p_S\;
\PR_{X(T_e^i-)} \lp
t_e - T_e^i < \wtd{T_{\#}}\wedge \wtd{T_{out}^{1}} \wedge \wtd{\ext}  \rp
\EQn{pm}{}
\\&\hcm{3}
\ge p_S\ c_{\#}\, \exp[-\Rsv'\, t_m].
\end{align*}

Thus, since $a.s.\; I(t_e) < \infty$, with Lemma <ref> :

\begin{align*}
&\PR_\mu \lp t_e - t_m < T_{out}^{I(t_e)}\mVg
\tau_{bk}^1 \le t_e\mVg   t_e < \ext \rp
\\&\hspace{1cm}
\le  \Tsum{i\ge 1}
\PR_{\mu} \lc t_e - t_m < T_{out}^i \rc
\\&\hspace{.5cm}
\le [e^{\Rsv'\, t_m}\, /(p_S \,c_{\#})]\ltm
\Tsum{i\ge 1}
\E_\mu \lc \PR_{X(T_{out}^i)} \lp
t_e - T_{out}^i < \wtd{T_{\#}}\wedge \wtd{T_{out}^{1}} \wedge \wtd{\ext}  \rp
\pv t_e - t_m < T_{out}^i \rc,
\\&\hspace{.7cm}
\le  [e^{\Rsv'\, t_m}\, /(p_S \,c_{\#})]\ltm
\PR_\mu \lc X_{t_e} \in \Dxt\mVg  t_e < \ext \rc.
\Eq{eqq}%{X_{t_e} \in \Dxt}
\end{align*}

\begin{align*}
& \text{By %\REqc,
\REqt\, and \REqq, with }
\xi_{\#}~:= p_S\, c_{\#}\, e^{-\Rsv\, t_m} /4 :\quad
\mu A_{t_e} \lp \Dxt\rp \ge \xi_{\#}.
\Eq{eqq}
\end{align*}

Now, recalling inequality $\Req{TBKe}$,
with $\M_{\#}~:= \Lbr \mu\in \Mone\pv \mu(\Dxt) \ge \xi_{\#}\Rbr$
($\xi_{\#}$ given by the previous formula
does not depend on $n$, $\xi$ or $\mu$ and is larger than $1/4$),
we indeed deduce inequality Mxt.

§.§.§ Proof of Lemma <ref>

We argue similarly as in Subsection <ref>
in order to adapt the proof of Lemma 4.1.3 from <cit.>.
Again, we only sketch the proof.
More details can be found in <cit.>.
Let :

\begin{align*}
\PR_\mu \lp T_{out}^{I(t_e)} \le t_e - t_m  \mVg
t_e \le \tau_{bk}^{I(t_e)+1}\wedge \ext	\mVg
\tau_{bk}^1 \le t_e  \rp
\\&\hspace{1.5cm}
\le  e_\T \,  \exp[- (\rhoeT- \Rsv) \, t_m]\;
\times \Tsum{i\ge 1}
\E_\mu \lc
\exp[-\Rsv\, ( t_e - \Tjo)]
\pv T_{out}^i \le (t_e - t_m)\wedge \ext \rc.
\end{align*}

On the event
$\Lbr T_{out}^i \le (t_e - t_m)\wedge \ext\Rbr
\cap \Lbr \Tjo < T_e^i\Rbr$,
the process cannot have exited $\cD_{out}$
by a jump leading outside of $\Dlj$,
which means $X(\Tjo) \in \Dlj$.
We hence deduce from Lemma <ref> that :

$a.s. \qquad \exp[-\Rsv'\, ( t_e - \Tjo)]
\le (1/c_{\#}) \times \PR_{X(\Tjo) } \lp
t_e - \Tjo < \wtd{T_{out}^1}\wedge \wtd{\ext} \rp.$

On the event
$\Lbr T_{out}^i \le (t_e - t_m)\wedge \ext\Rbr
\cap \Lbr \Tjo = T_e^i\Rbr\in \F^*_{T_e^i}$,
since $X(\Tjo) = X(T_e^i -) \in \cD_{out}\subset \Dlj$
with probability larger than $p_S$
conditionally on $\F^*_{T_e^i}$
(like in pm),
and from Lemma <ref>

\begin{align*}
&\E_\mu \lc
\PR_{X(\Tjo) } \lp
t_e - \Tjo < \wtd{T_{out}^1}\wedge \wtd{\ext} \rp
\bv \F^*_{T_J^i}  \rc
\ge p_S\times  \PR_{X(T_e^i -) } \lp
t_e - \Tjo < \wtd{T_{out}^1}\wedge \wtd{\ext} \rp
\\&\hcm{3}
\ge p_S\, c_{\#} \times \exp[-\Rsv'\, ( t_e - \Tjo)].
\end{align*}

Thus, combining both cases :

\begin{align*}
\le  [e_\T\, /(p_S \,c_{\#})]
\times  e^{- (\rhoeT- \Rsv') \, t_m}\;
\\&\hcm{2}
\times  \Tsum{i\ge 1}
\E_\mu \lc
\PR_{X(\Tjo) } \lp
t_e - \Tjo < \wtd{T_{out}^1}\wedge \wtd{\ext} \rp
\pv T_{out}^i \le (t_e - t_m)\wedge \ext \rc
\\&\hcm{1}
\le  [e_\T\, /(p_S \,c_{\#})]
\ltm e^{- (\rhoeT- \Rsv') \, t_m} \ltm
\PR_\mu \lc  t_e <  \ext \rc,
\end{align*}

which concludes the proof of Lemma <ref> with
$C_v~:= e_\T\, /(p_S \,c_{\#})$.

§.§ Absorption with failures

The fact that $\mathbf{(AF')}$ implies $\mathbf{(AF)}$
is due to the fact that a survival estimate like $(A5)$
can be deduced from $(A2)$ –cf Subsection 2.5.1 in <cit.>.

The rest of the proof is easily adapted
from the ones of Theorems 2.1-3 in <cit.>
thanks to the following theorem.
It extends in fact Theorem 4.2 in <cit.>
and their proofs  share many similarities,
as can be inferred from the frequent use of :

\begin{align*}
&\frl{ u\ge 0}
\frl {\tp \ge u + \tSb}
\hspace{0.5cm}
P_{\alc}(\tp - u < \ext)
\le \cSb\; e^{\Rsv\, u}\;  \PR_{\alc}\lp  \tp < \ext \rp,
\EQn{eqSb}{Sb}
\end{align*}

deduced from Corollary 4.2.1
in <cit.>.
This Corollary mainly relies on our preceding Theorem <ref>
with once more assumptions $(A2)$, $(A5)$
to ensure survival from $\cD_{\#}$,
so that   is of course not exploited to prove this result.

Assume that there exist $\rhoeT \ge \Rsv >0$, $e_\T>0$, $\cD_c \in \Dps$ and $\alc\in~\Mone$ such that
inequality eqSb,
and     hold.
Then, for some $c, u>0$ :

\begin{align*}
\frl{x \in \cX}\frlq{t\ge u}
\PR_x(t<\ext) \le c\, \PR_{\alpha_c}(t<\ext).
\tag*{\Req{BdSv}}
\end{align*}

Remark :

The following proof just ensures $(A3)$,
and we refer to Theorem 4.2 in <cit.>
to see how it implies inequality BdSv.

$\bullet$ We recall (cf. Subsection <ref>)
that we could have replaced in the statement of Theorem <ref>
assumption    by the more general assumption .

§.§.§   implies   :

We prove that
the choices of $t$, $c$, $\Uza$
from assumption   indeed satisfy the conditions of
assumption ,
where $\Tfl := t$ on the event $\Lbr t < \Uza \wedge \ext\Rbr$,
and $\Tfl := \infty$ otherwise.

$\Uza \vee \Tfl = \infty,$
$\quad \ext \le \Uza \wedge \Tfl$ implies $\Uza = \Tfl = \infty$
simply by assumption.

Since $\Uza \wedge \Tfl \wedge \ext \le t$,
for any $x\in \cX$ :

\begin{align*}
\E_{x} \lc \exp(\rho \,[\Uza \wedge \ext])
\pv \Tfl = \infty\rc
\le \exp(\rho \,t) := e_0.
\end{align*}

Since moreover  $\Tfl \le t$ on $\{\Tfl < \infty\}$ :

\begin{align*}
\E_{x} \lc \exp(\rho \,T_{f \ell}) \pv \Tfl \le \tZa\rc
\le \exp(\rho\, \tZa) \PR_x(\Tfl \le \tZa)
\le f_\ell.
\end{align*}

for any bounded measurable function

which is increasing in $u$ :

\begin{align*}
a.s.,\; \frl{y\in \cX}\quad g(y,\, \Uza)
\le g(y,\, \tZa) \le g(y,\, \UCa + \tZa),
\text{ thus~:}\\
& \E_{x} \lc g(X(\Uza),\, \Uza) \pv \Uza < \ext \rc
\le \E_{x} \lc g(X(\Uza),\, \tZa) \pv \Uza < \ext \rc \\
\le \cabs \; \E_{\alc} \lc g(X(\UCa),\, \tZa) \pv \UCa < \ext \rc
\by assumption \RAF\\
\le \cabs \; \E_{\alc} \lc g(X(\UCa),\, \UCa + \tZa) \pv \UCa < \ext \rc.
\end{align*}

This concludes the proof of $(A3)$.

§.§.§ Proof of Theorem <ref> with assumption

In the following,
every time we will apply assumption ,
we will use the parameters :

$	 \fl~:= (2\, \cSb\, e_\T )^{-1}\mVg
\tDc~:= \tZa + \tSb,
\quad$
from which we will easily deduce :

Assume that inequality eqSb  and assumption hold
with the above parameters.
Then :

\begin{align*}
\Ex{C_0 > 0}
\frl{x\in \cD_c}
\frl{\tp \ge \tDc}\quad
\PR_{x} (\Nstep = 0 \mVg \tp < \ext)
\le C_0 \; \PR_{\alc} (\tp < \ext)
\end{align*}

For $\fl$ sufficiently small, we expect this quantity to be the leading part of
$\PR_{x} ( \tp < \ext)$. We will prove indeed that the extension of survival during the failed coupling procedure –ie before $\Tab$– then outside $\cD_c$ –before $\Tdc^1$– is not sufficient to compensate the cost of such a failure.

Our idea is to distinguish the events according to the number of failures, and treat them inductively by replacing $x$ by $X(\UDc^1)$ and $t$ by $\tp - \UDc^1$.
Therefore, the induction is done not exactly depending on $\Nstep$, but more precisely on the value of the r.v. : Kt

\begin{align*}
\Nstep(t)~:= \sup\Lbr j \ge 0,\; \UDc^j < (\tp - \tDc)\wedge \ext\Rbr,
\Eqn{Kt}{\Nstep(t)}
\end{align*}

where we define  $\UDc^0~:= 0$.
To end the initialization –$\Nstep(t) =0$– we thus prove :

Assume that assumption $(A4)$,
inequalities eqSb  and assumption   hold
with $\rho> \Rsv$.
Then :

\begin{align*}
&\Ex{F_\ell > 0}
\frl{x\in \cD_c}
\frl{\tp \ge \tDc}\quad
\\&\hcm{.5}
\PR_{x} (\Nstep \ge 1 \mVg \tp - \tDc \le \UDc^1\mVg \tp < \ext)
\le F_\ell \; \PR_{\alc} (\tp < \ext)
\end{align*}

For any $\tp \ge 0$, $\Nstep(t) \le \Nstep < \infty$ –since $\UDc^j = \infty$ for $j > \Nstep$– so that the induction is achieved thanks to the :

Assume that assumption $(A4)$,
inequalities eqSb  and assumption   hold
with $\rho> \Rsv$.
If for some $j \ge 0$ and $C_j >0$, we have :

\begin{align*}
\frl{x \in \cD_c}
\frl{\tp \ge \tDc}
\PR_{x} (\tp < \ext \mVg \Nstep(t) = j) \le C_j \; \PR_{\alc} (\tp < \ext),
\Eqn{eqt}{C_k}
\\ & \text{Then }\hspace{1cm}
\frl{x \in \cD_c}
\frl{\tp \ge \tDc}
\PR_{x} (\tp < \ext \mVg \Nstep(t) = j+1)
\le  \tfrac{C_j}{2} \; \PR_{\alc} (\tp < \ext)
\end{align*}

Lemmas <ref>-3 imply BdSv :

With $\cDc~:= 2\, (C_0 + F_\ell)$, by Lemmas <ref>-3, we deduce that $  \frl{x \in \cD_c}
\frl{\tp \ge \tDc}$:

\begin{align*}
&\frl{j\ge 0}
\quad    \PR_{x} (\tp < \ext \mVg \Nstep(t) = j)
\le  2^{-j-1} \; \cDc \; \PR_{\alc} (\tp < \ext)
\\ &\hspace{1cm}
\PR_{x} (\tp < \ext)
= \Tsum{j\ge 0} \PR_{x} (\tp < \ext \mVg \Nstep(t) = j)
%  \\ &\hspace{2cm}
\le  \cDc \; \PR_{\alc} (\tp < \ext),
\end{align*}

which means that assumption $(A3)$ in <cit.> holds.
We refer to the proof of Theorem 4.2 in <cit.>
(inspired by <cit.>, cf. p.13)
to see how it implies inequality BdSv
with again inequality eqSb  and assumption $(A4)$.

Proof of Lemma <ref> :

\begin{align*}
&\PR_{x} (\Uza < \tp < \ext  \mVg   \Nstep = 0)
= \E_{x} \left[
\PR_{X\lc \Uza \rc } (\tp - \Uza< \wtd{\ext})\pv
\Uza < \ext\wedge t  \mVg   \Nstep =0 \right]\\
\le \cabs \; \E_{\alc} \left[
\PR_{X \lc \UCa \rc } (\tp - \UCa - \tZa< \wtd{\ext})\pv
\UCa < \ext \right]\\
\le   \cabs \; \PR_{\alc} \left[ \tp - \tZa  < \ext \right],
\\ &\hspace{.5cm}
\text{where we exploited  assumption } \RAFb \text{ with } g(x, u)~:= \PR_x\lp t-u < \ext\rp \text{ for }u < t,\, 1 \text{ otherwise.}
\\ &\hspace{1.5cm}
\le \cabs\;\cSb\;  e^{\rho\, \tZa}\;
\PR_{\alc} \left[ \tp < \ext \right]
\Eq{equ}%{\Uza < \tp}
\\ & \hcm{2.5} \text{by inequality \Req{eqSb}, with }
\tp \ge \tZa + \tSb.
\end{align*}

By inequalities   and again eqSb  :

\begin{align*}
&\PR_{x} (\tp < \ext\wedge \Uza \mVg   \Nstep = 0)
\le e^{-\rho\, \tp} \, \E_{x} \lc \exp(\rho_{AF} \, [\Uza\wedge \ext])
\pv \Nstep = 0\rc
\\&\hcm{2}
\le    \dfrac{ \cSb\;e_0}{e^{\rho\, \tSb}\; \PR_{\alc}(\tSb< \ext)}
\times \PR_{\alc}(\tp < \ext)
\Eq{eqd}%{\tp < \ext\wedge \Uza}
\end{align*}

and   imply Lemma <ref> with :
$\quad C_0~:= \cSb \; \lp \cabs\;e^{\rho\, \tZa}
+ \dfrac{e_0}{e^{\rho\, \tSb}\; \PR_{\alc}(\tSb< \ext)}\rp.$

Proof of Lemma <ref> :

We begin with the case $\Tab < t$ (recall that $\UDc^1$ is defined as the hitting time of $\cD_c$ after $\Tab$) :

\begin{align*}
&\PR_{x} (\tp - \tDc \le \UDc^1,\, \Tab < \tp < \ext,\, \Nstep \ge 1)
\\&\hspace{.5cm}
= \E_{x} \left[
\PR_{X\lc \Tab \rc }
(\tp - \tDc - \Tab \le \wTdc
\pv \tp - \Tab < \wtd{\ext})\pv
\Tab < \ext\wedge t\mVg
\Nstep \ge 1 \right]
\\&\hspace{1cm}
\le  e_\T \; \E_{x} \left[
\exp\lp - \rho [\tp - \tDc - \Tab] \rp
\pv \Tab < \ext\wedge t \mVg \Nstep \ge 1 \right],
\end{align*}

where we exploited the Markov inequality and assumption   to state that,

a.s. on $\Lbr \Tab < \ext\wedge t\mVg \Nstep \ge 1\Rbr~:$

$$\E_{X\lc \Tab \rc }
\lc \exp\lp \rho [\wTdc \wedge \wtd{\ext}] \rp \rc
\le e_\T.$$

Next :

\begin{align*}
&\PR_{x} \Big(\tp - \tDc \le \UDc^1\mVg
\Tab < \tp < \ext\mVg \Nstep \ge 1\Big)
\\&\hspace{2cm}
\le e_\T \; \exp\lp  \rho \tDc\rp \; \exp[-\rho\, t]\;
\E_{x} \lc \exp(\rho T^1_{f \ell}) \pv \Nstep \ge 1\rc  ,
\\ &\hspace{2cm}
\le e_\T \; \exp\lp \rho \tDc\rp \; \cSb\;
\dfrac{\PR_{\alc}(\tp < \ext)}{e^{\rho\, \tSb}\; \PR_{\alc}(\tSb< \ext)}  \times \fl,
\Eq{equ}
\end{align*}

where we have exploited inequalities   and eqSb
for the second inequality.

The case $t \le \Tab$ is treated in the same way as $t\le \Uza\wedge \ext$ to obtain :

\begin{align*}
&\PR_{x} (\tp < \Tab \mVg   \Nstep \ge 1)
\le    \dfrac{ \cSb\;\fl}{e^{\rho\, \tSb}\; \PR_{\alc}(\tSb< \ext)}
\times \PR_{\alc}(\tp < \ext).
\end{align*}

Combining the last inequality with ,
we deduce Lemma <ref> where :

\begin{align*}
\dfrac{ e_\T \; \exp\lp \rho \tDc\rp  + 1 }
{e^{\rho\, \tSb}\;   \PR_{\alc}(\tSb< \ext)}  \times \cSb\;  \fl.
\SQ
\end{align*}

Proof of Lemma <ref> :
By the Compatibility with the Markov property
–see assumption – and definition  :

\begin{align*}
&\PR_{x} (\tp < \ext\mVg \Nstep(t) = j+1)
\\ &\hspace{1cm}
= \E_{x} \left[
\PR_{X\lc \UDc^1 \rc } (\tp - \UDc^1< \wtd{\ext}\mVg
\wtd{\Nstep}(\tp - \UDc^1) = j)\pv
\Nstep(t) \ge 1%\mVg \UDc^1 < (\tp - \tDc)\wedge \ext
\right]
\\ &\hspace{1cm}
\le C_j\; \E_{x} \left[
\PR_{\alc } (\tp - \UDc^1< \wtd{\ext})\pv
\Nstep(t) \ge 1%\mVg \UDc^1 < (\tp - \tDc)\wedge \ext
\right],\;
\end{align*}

where we exploited    and the fact that $ \Nstep(t) \ge 1 \imp \tp - \UDc^1 \ge \tDc$.
Then, by inequality eqSb :

\begin{align*}
&\PR_{x} (\tp < \ext\mVg \Nstep(t) = j+1)
\le C_j\; \cSb\; \PR_{\alc } (\tp < \ext) \times
\E_{x} \left[
\exp(\rho \, \UDc^1);\;
\Nstep(t) \ge 1%\mVg \UDc^1 < (\tp - \tDc)\wedge \ext
\right]\;   %(Sb)
%  \cabs \; \PR_{\alc} \left[ \tp - \tDc  < \ext \right],
\\ &\hspace{2cm}
\le C_j\, \cSb\,  \PR_{\alc } (\tp < \ext)\;
%\\ &\hspace{1.5cm} \times
\E_{x} \big[
\E_{X\lc T^1_{f \ell} \rc} \lp
\exp(\rho \, \wTdc )\pv
\wTdc < (\tp - T^1_{f \ell}- \tDc) \wedge \wtd{\ext} \rp
\\ &\hspace{3.8cm}
\times \exp(\rho T^1_{f \ell})
\pv    \Nstep \ge 1\mVg T^1_{f \ell} < \ext\wedge t  \big],
\end{align*}

where we decomposed $\UDc^1$ into $T^1_{f \ell} + \wTdc$.

Now, by assumption   we have $a.s.$ on $\Lbr  \Nstep \ge 1\mVg T^1_{f \ell} < \ext\wedge t  \Rbr$ :

\begin{align*}
&\E_{X\lc T^1_{f \ell} \rc} \lp
\exp(\rho \, \wTdc)\pv
\wTdc < (\tp - T^1_{f \ell}- \tDc) \wedge \wtd{\ext} \rp
% \\ &\hspace{.5cm}
%  \le \sup_{y \in \cX}
%  \E_{y} \lp
%        \exp(\rho \, \wTdc\wedge \wtd{\ext}  ) \rp
\le e_\T; %\text{ since } \rhoeT > \rho
\\ &\hspace{.5cm}
\E_{x} \lc \exp(\rho T^1_{f \ell}) \pv \Nstep \ge 1\rc
\le \fl = \dfrac{1}{2\, \cSb\, e_\T}. \hspace{1cm}
\text{ by assumption \RAFb}
\end{align*}

\begin{align*}
\text{ Thus, }
\frl{x \in \cD_c}
\frl{\tp \ge \tDc}\quad
&\PR_{x} ( \tp < \ext \mVg \Nstep(t) = j+1)
%\\ &\hspace{1cm}
\le \tfrac{C_j}{2}\; \PR_{\alc } (\tp < \ext),
\end{align*}

which yields the result.

§.§ The quasi-ergodic distribution

Thanks to the proofs from <cit.>,
only the fact that the convergence of $\LAg \mu|\eta_t\RAg$
is uniform in $\mu$ and
point $(iv)$ in Theorem <ref>
remain to be proved.

Since there exists $n, \xi$ such that
for any $\mu \in \Mone$,
$\mu' := (\mu + \alpha)/2 \in \Mnx$,
while $\LAg \alpha|\eta_t\RAg = \LAg \alpha|\eta\RAg = 1$ :

\begin{align*}
|\LAg \mu|\eta_t\RAg - \LAg \mu|\eta\RAg|
= 2 |\LAg \mu'|\eta_t\RAg - \LAg \mu'|\eta\RAg|
\le 2\, C(n,\xi)\, \exp[-\zeta\, t].
\end{align*}

Exploiting the criteria given in Theorem 2.5 in <cit.>,

in order to establish $(iv)$
it is sufficient
to prove that,
for any $0<p< q<1$, $\mu \in \Mone$
and $f, g$ bounded measurable :

\begin{align*}
\limInf{t} \E_\mu[f(X_{p t})\, g(X_{q t}) \bv t < \ext]
= \langle\beta\bv f\rangle\ltm \langle\beta\bv g\rangle.
\EQn{QED}{}
\end{align*}

So we compute :

\begin{align*}
&\E_\mu[f(X_{p t})\, g(X_{q t}) \bv t < \ext]
= \dfrac{\langle\mu\bv \eta_{p t}\rangle}{\langle\mu\bv \eta_{t}\rangle}
\int_{\cX} \mu A_{p t}(dx)\, f(x)\,
\E_x\lc g(X_{(q-p)\, t})\, \exp[\laZ\, t\, (1-p)]\pv (1-p)\, t <\ext\rc
\\&\hcm{1}
= \dfrac{\langle\mu\bv \eta_{p t}\rangle}{\langle\mu\bv \eta_{t}\rangle}
\int_{\cX} \mu A_{p t}(dx)\, f(x)\, \eta_{(q-p)\, t}(x)\,
\int_{\cX} \delta_x A_{(q-p)\, t}(dy)\, g(y)\, \eta_{(1-q)\, t}(y).
\EQn{expFG}{}
\end{align*}

The difficulty arises from the fact
that the convergences in inequalities ECvAl and ECvEta
are not uniform.
Yet, they are when restricted to $\cD_n$ :

\begin{align*}
&\int_{\cD_n} \mu A_{p t}(dx)\, f(x)\, \eta_{(q-p)\, t}(x)\,
\int_{\cD_n} \delta_x A_{(q-p)\, t}(dy)\, g(y)\, \eta_{(1-q)\, t}(y)
\\&\hcm{2}
\cvifty{t} \int_{\cD_n} \alpha(dx)\, f(x)\, \eta(x)\,
\int_{\cD_n} \alpha(dy)\, g(y)\, \eta(y).
\end{align*}

In the following,
we prove that the error
of restricting values on $\cD_n$
becomes negligible for large $n$.
We recall that $\mu A_s$
is a probability measure for any $\mu$ and $s$,
and that we have a uniform bound $\Ninf{\eta_\bullet}$ on $\Ninf{\eta_t}$
–cf. (73) in Subsection 4.4 of <cit.>. So :

\begin{align*}
&\bv \int_{\cX \setminus \cD_n} \mu A_{p t}(dx)\, f(x)\, \eta_{(q-p)\, t}(x)
\,\int_{\cX} \delta_x A_{(q-p)\, t}(dy)\, g(y)\,
\eta_{(1-q)\, t}(y)\bv
\\&\hcm{2}
\le \Ninf{f}\, \Ninf{g}\, \Ninf{\eta_\bullet}^2\,
\mu A_{p\, t}(\cX\setminus \cD_n)
\cvifty{t} \Ninf{f}\, \Ninf{g}\, \Ninf{\eta_\bullet}^2\,
\alpha(\cX\setminus \cD_n).
\end{align*}

Likewise, since our convergence in inequality ECvAl
is uniform
w.r.t. the initial condition $x\in \cD_n$ :

\begin{align*}
&\bv \int_{\cD_n} \mu A_{p t}(dx)\, f(x)\, \eta_{(q-p)\, t}(x)
\,\int_{\cX \setminus \cD_n} \delta_x A_{(q-p)\, t}(dy)\, g(y)\,
\eta_{(1-q)\, t}(y)\bv
\\&\hcm{2}
\le \Ninf{f}\, \Ninf{g}\, \Ninf{\eta_\bullet}^2\,
\sup_{x\in \cD_n} \delta_x A_{(q-p)\, t}(\cX\setminus \cD_n)
\cvifty{t} \Ninf{f}\, \Ninf{g}\, \Ninf{\eta_\bullet}^2\,
\alpha(\cX\setminus \cD_n).
\end{align*}

The same upper-bounds apply of course to the limiting versions,
so that $n \ge1$ chosen sufficiently large ensures
that $\alpha(\cX\setminus \cD_n)$ is sufficiently small to neglect
the error of considering only $\cD_n$.
Finally, with the fact that
$\langle\mu\bv \eta_{p t}\rangle / \langle\mu\bv \eta_{t}\rangle $
converges to 1,
we have indeed proved equation QED,
which ensures equation Qed.

§ APPENDIX UAB : EXTENSION OF THE STOPPING TIME

Recall (with simplified notations)
that considering the process $(X, Y)$
with initial condition $(x, y)$, we define
for some $t>0$ :
$\quad \Uza := t$ on the event $\{t < T_{\delta y}\wedge U_j\}$,
$\quad \Uza := \infty$ otherwise,

\begin{align*}
\where
& T_{\dy} := \inf\Lbr s\pv |Y_s - y| \ge 2\,\dy \Rbr
< \ext
&\text{ for some  $\delta y>0$, }
\\&
U_j  := \inf\Lbr s\pv
M([0, s] \times \J) \ge 1 \Rbr,
\quad &\J := \R^d \times [0, f_\wedge]\times [0, g_\wedge]
&\text{ for some } f_\wedge, g_\wedge >0.
\end{align*}

Recursively, we also define :

\begin{align*}
\Tdc^{i+1} := \inf\{s\ge \Tdc^i +t : X_s \in \cD_c\}
\wedge \ext,
\AND
\Tdc^0 = 0,
\end{align*}

and on the event $\Lbr \Tdc^{i}  < \ext\Rbr$,
for any $i$, we set :

\begin{align*}
:= \inf\Lbr s\ge \Tdc^{i} \pv |Y_s - Y(\Tdc^{i})| \ge 2\,\dy \Rbr
\\&
U^i_j  := \inf\Lbr s\pv
M([\Tdc^{i}, \Tdc^{i} + s] \times \J) \ge 1 \Rbr,
\\&
:= \inf\{\Tdc^{i} + t\; :
\Tdc^{i} < \infty \mVg
\Tdc^{i} + t < T^i_{\dy}  \wedge U^i_j\},
\end{align*}

where in this notation,
the infimum equals $\infty$ if the set is empty,
$T^i_{\dy} := \infty$ and $U^i_j =\infty$
on the event
$\Lbr \ext \le \Tdc^{i}\Rbr$.
The proof that all these random times
define stopping times is very classical
and left to the reader.

It is obvious that $U_{AB}$ coincide with $\Uza$
on the event $\Lbr \Uza \wedge \ext \le \Tdc^1 \Rbr$.
the Markov property at time $\Tdc^1$
and the way $U_{AB}$ is defined
entails that on the event
$\Lbr  \Tdc^1 < \Uza \wedge \ext \Rbr$,
$U_{AB} - \Tdc^1$ has indeed the same law
as $\wtd{U}_{AB}$ associated the process
$(\wtd{N}, \wtd{X})$ solution of the system SN-SX,
see Subsection <ref>,
with initial condition $(N(\Tdc^1),\, X(\Tdc^1)\,)$
and driven by $\wtd{B}^N,\, \wtd{M}^N,\, \wtd{B}^X,\, \wtd{M}^X$
independent of the $B^N,\, M^N,\, B^X,\, M^X$
driving $(N, X)$.

§ APPENDIX AP.A2 : PROOFS OF LEMMAS <REF>
AND <REF>

§.§ Proof of Lemma <ref> :

With $t_0 := S/v$,  $x_0 \in B(x_i, r)$,
consider the first time of jump $U^j$ for $X$
and a positive and measurable test function $f$.
Concerning the constraint $t_0< T_m$,
note that :

\begin{align*}
\Lbr x_0 - v\,s\ e_1 \bv s \le t_0\Rbr
\cup \Lbr x_0 - v\,s\ e_1  +w
\bv s \le t_0\mVg w \in B(S\, e_1,\  4r)\Rbr
\subset \cD_m.
\end{align*}

Thus, making only such jumps will keep the process inside of $\cD_m$.

By Palm's formula
(see e.g. [13] Proposition 13.1.VII),
conditionally on $M$ having a dirac on $(s, w, u)$,
the law of $M$
is simply the one of $\wtd{M} + \delta_{(s, w, u)}$,
where $\wtd{M}$ is an independent copy of $M$.
By this way,
we can deal with this constraint $t_0< T_m$,
referring to the process $\hat{X}$
driven by $\wtd{M} + \delta_{(s, w, u)}$ :

\begin{align*}
&\E_{x_0}\lp f[X(t_0)] \pv t_0< T_m \rp
= f[x_0 - v\ t_0\ e_1]
\\&\hcm{1}
+ \E_{x_0}\Big( \int_{[0,t] \times \R^d \times \R_+ }
(f[X_{s^-} +w - v(t_0-s)e_1] - f[X_{s^-} - v(t_0-s)e_1])
\\&\hcm{6}
\; \idc{u\le h(X_{s^-},w )}\, \idc{t_0< \hat{T}_m}
\, M(ds,dw,du)\Big)
\\&%\hcm{1}
= f[x_0 - v\ t_0\ e_1]
+ \int_{[0,t] \times \R^d} \E_{x_0}\big(
(f[X_{s^-}+ w - v\ (t_0-s)\ e_1] - f[X_{s^-} - v\ (t_0-s)\ e_1])
\\&\hcm{6}
\;  h(X_{s^-},w )\, \idc{t_0< \hat{T}_m}  \big)\, ds\, dw.
\end{align*}

Here we specify by the hat (for $t_0< \hat{T}_m$)
that we consider the above-mentioned process
with $u = 0$,

remarking that $\hat{X}$ (with $(s, w, u)$)
equals  $\hat{X}_0$ (with $(s, w, 0)$)
on the event $\Lbr u\le h(X_{s^-},w )\Rbr$.
Note that $\hat{X}_{s-} = X_{s-}$.
We now define :

\begin{align*}
H(x_0) := \int_{[0,t_0] \times \R^d} h(x-v\,s\,e_1,w )\, ds\, dw
\\&
\AND
p(x_0) := \PR_{x_0} (t_0< U^j)
% \\&\hcm{1}
= \exp\lc- H(x_0)\rc
\ge \exp[- \roV\ t_0],
\EQn{nJ}{\roV}
\end{align*}

by assumption .
With a similar approach as above,
we can describe with Palm's formula

$\E_{x_0}\lp f[X(t_0)]
\pv  t_0< T_m
\mVg X \text{ has done two jumps before } t_0\rp\quad $
so as to obtain :

\begin{align*}
&\E_{x_0}\lp f[X(t_0)] \pv t_0< T_m \rp
=  p(x_0)\, f[x_0 - v\ t_0\ e_1]
+ p(x_0)\,H(x_0)\, f[x_0 - v\ t_0\ e_1]
\\&\hcm{1}
+ \int_{[0,t_0] \times \R^d}
p(x_0)\,(f[x_0 - v t_0 e_1 + w] - f[x_0 - v\ t_0\ e_1])
\;  h(x_0 - v\ s\ e_1,w ) \, ds\, dw
\\&\hcm{1}
+\E_{x_0}\lp f[X(t_0)]
\pv t_0< T_m \mVg
X \text{ has done two jumps before } t_0\rp
\\&
\text{ thus }\qquad
\E_{x_0}\lp f[X(t_0)]\pv t_0< T_m \rp
\ge p(x_0)\, \hw\ t_0\
\int_{B(S\, e_1,\  4r)} f[x_0 - v\ t_0\ e_1 + w]\,  \ dw,
\end{align*}

where we exploited assumption assumption   and the fact that $f$ is positive
for this lower-bound.
Now, with inequality nJ,
the facts  that $x_0\in B(x_i, r)$,
and that $f$ can be any positive and measurable function :

\begin{align*}
\PR_{x_0}[X(t_0) \in dx \pv t_0 < T_m\wedge \ext]
\ge \exp[- \roV\, t_0]\ \hw\ t_0\ \idc{x \in B(x_i, r)}\ dx.
\end{align*}

This concludes the proof of Lemma <ref>
with $c_0 := \exp[- \roV\ t_0]\ \hw\ t_0$.

§.§ Proof of Lemma <ref>

This proof follows similar lines
as that of Lemma <ref>,
so we just mention the adjustments.
Of course, the result relies on the Markov property
combined with a uniform estimate on the transitions
starting from $x_I$ with $x_I\in B(x, r)$.
For any $x_f \in B(x,\, r)$,
since $|x_I- x_f| \le 2r$,
we ensure that $X_{t_{0}}$
has a lower-bounded density on $B(x_f, r)$.
As previously,
still with the fact that

\begin{align*}
\Lbr x_I - v\,s\ e_1 \bv s \le t_0\Rbr
\cup \Lbr x_I - v\,s\ e_1  +w
\bv s \le t_0\mVg w \in B(S\, e_1,\  4r)\Rbr
\subset \cD_m,
\end{align*}

we deduce :
$\quad \PR_{x_I}\lp X(t_I)\in dy \pv t_0< T_m \rp
\ge c_0\, \idc{y \in B(x_I,\  4r)}\, dy
\ge c_0\, \idc{y \in B(x_f,\  r)}\, dy.$

By the Markov property,
if $(t, x) \in \cR^{(m)}(c)$,
then $(t+t_0, x_f)  \in \cR^{(m)}(c\ltm c_0)$.

§ APPENDIX APUZA : PROOF OF LEMMA <REF>
(DIFFUSION IN EACH OF THE COORDINATES)

The proof is based on an induction on the coordinates
affected by jumps in the time-interval $[0, \tZa]$.
We recall that,
thanks to our criterion of exceptionality,
we can restrict ourselves to trajectories
where any coordinate is affected
by at least one jump
in the time-interval $[0, \tZa]$,
while at most $n_J^\vee$ jumps have occurred in this time-interval.
Given any possible sequence of directions,
we shall also restrict ourselves to the event
such that the sequence of realized jump directions
is precisely the specified one.
Since the number of such possible sequences is finite,
the upper-bound on the density of $X(U_{ab})$
presented in Lemma <ref>
is implied by the restricted versions
for any such possible sequence of directions.

So let $U_J^k$ be the $k$-th jump of $X$.
Let also $i(k)\in [\![1, d]\!]$ for $k\le n_J \le n_J^\vee$
be a sequence of directions
such that, at $k = n_J$,
all the directions have been listed.
Let also $I(k)\in [\![1, d]\!]$ for $k\le n_J^\vee$,
be the sequence of random directions
that the successive jumps of $X$ follow.
Since in our model,
all directions are defined in a similar way,
we can simplify a bit our notations without loss of generality
by relabeling some of the directions.
Since we will go backwards
to progressively forget about the conditioning,
we order the coordinates by the time they appear
for the last time in $(i(k))_{k\le n_J}$.
It means that $i(n_J) = d$
and $\Lbr i(k)\pv K\le k\le n_J\Rbr = [\![j(K), d]\!]$.
Let then $ K(j)$ be the largest integer for which $j(k) \ge j$.

Remark :  In our case, $n_J$
is naturally chosen as the first integer
for which all the directions have been listed.
Yet, our induction argument
is more clearly stated
if we do not assume this condition on $n_J$.

Then, for any positive and measurable functions $(f_j)_{j\le d}$
and  any $x\in \cD_c$ :

\begin{align*}
E^d:= \E_x\lc \prod_{j\le d} f_j[X^j(U_J^{n_J})]
\pv U_J^{n_J} < \ext\wedge \tZa
\mVg \frl{k\le n_J} I(k) = i(k)
\mVg \|\Delta X(U_J^k)\| \le w^\vee \rc
\\&\hcm{.4}
\le \E_x\Bigg[ \prod_{j\le d-1} f_j[X^j(U_J^{n_J-1})]
%\ltm \PR_x\big[ I(n_J) = d \bv \F^*_{U_J^{n_J}}\big]
\ltm \E_x \big[ f_d[X^d(U_J^{n_J})]
\pv |\Delta X^d(U_J^{n_J})|\le w^\vee
\bv \F^*_{U_J^{n_J}} \big]  \pv
\\&\hcm{2.1}
U_J^{n_J} < \ext\wedge \tZa
\mVg I(n_J) = d
\mVg \frlq{k\le n_J-1} I(k) = i(k)
\mVg \|\Delta X(U_J^k)\| \le w^\vee \Big],
\EQn{APU1}{}
\\&
\where \F^*_{U_J^{n_J}} := \sigma(\F_{U_J^{n_J-1}}, \{ I(n_J) = d \}\cap \{U_J^{n_J} < \ext\wedge \tZa \})
\end{align*}

and $X(U_J^{n_J}-) = X(U_J^{n_J-1})$ is $\F_{U_J^{n_J-1}}$-measurable,
since we consider a pure jump process.
By the Markov property,
the law of the next jump only depends on $x' =  X(U_J^{n_J-1})$
through the functions $(w \mapsto h_j(x', w))_{j\le d}$.
With the $\sigma$-algebra $\F^*_{U_J^{n_J}}$,
we include the knowledge
of the direction of the jump at time $U_J^{n_J}$,
so that only the size of this jump (possibly negative)
remains random.
Noting that :
$\quad \Ninf{ X^d(U_J^{n_J}-)}\vee \Ninf{X^d(U_J^{n_J})}
\le n_c + n_J^\vee\ltm w^\vee := L\quad$
(independent of $n_J$ or the the particular choice of $i(k)\ $),
we deduce from assumption
that on the event
$\{ I(n_J) = d \}
\cap \{U_J^{n_J} < \ext\wedge \tZa \}$ :

\begin{align*}
\E_x \big[ f_d[X^d(U_J^{n_J})]
\pv |\Delta X^d(U_J^{n_J})|\le w^\vee
\bv  \F^*_{U_J^{n_J}}  \big]
\le h_\vee  \int_{[-L, L]} f_d(x^d)\, dx^d.
\EQn{APU3}{h_\vee}
\end{align*}

In the following,
we upper-bound by 1 the probability
of the event $\{ I(n_J) = d \}
\cap \{U_J^{n_J} < \ext\wedge \tZa \}$.
Combining inequalities APU1, APU3,
and our ordering with the definition of $K(j)$,
we deduce :

\begin{align*}
\le h_\vee  \int_{[-L, L]} f_d(x^d)\, dx^d
\times \E_x\Bigg[ \prod_{j\le d-1} f_j[X^j(U_J^{n_J-1})]
\pv U_J^{n_J-1} < \ext\wedge \tza \mVg
\\&\hcm{5.5}
\frl{k\le n_J-1} I(k) = i(k)
\mVg \|\Delta X(U_J^k)\| \le w^\vee \Big]
\\&\hcm{.5}
\le h_\vee  \int_{[-L, L]} f_d(x^d)\, dx^d
\times \E_x\Bigg[ \prod_{j\le d-1} f_j[X^j(U_J^{K(d-1)})]
\pv U_J^{K(d-1)} < \ext\wedge \tza \mVg
\\&\hcm{5.5}
\frl{k\le K(d-1)} I(k) = i(k)
\mVg \|\Delta X(U_J^k)\| \le w^\vee \Big],
\end{align*}

where in particular $i(\, K(d-1)\, ) = d-1$,
$K(d-1) \le n_J^\vee$.

We see that quite an immediate recursion ensures :

\begin{align*}
= \E_x\Bigg[ \prod_{j\le d-1} f_j[X^j(U_J^{K(d-1)})]
\pv U_J^{K(d-1)} < \ext\wedge \tza \mVg
\\&\hcm{4.5}
\frl{k\le K(d-1)} I(k) = i(k)
\mVg \|\Delta X(U_J^k)\| \le w^\vee \Big]
\\&\hcm{.5}
\le h_\vee  \int_{[-L, L]} f_{d-1}(x^{d-1})\, dx^{d-1}
\times \E_x\Bigg[ \prod_{j\le d-2} f_j[X^j(U_J^{K(d-2)})]
\pv U_J^{K(d-2)} < \ext\wedge \tza \mVg
\\&\hcm{5.5}
\frl{k\le K(d-2)} I(k) = i(k)
\mVg \|\Delta X(U_J^k)\| \le w^\vee \Big],
\end{align*}

and so on until finally :

\begin{align*}
\le (h_\vee)^d
\ltm \prod_{i\le d} \lp \int_{[-L, L]} f_{i}(x)\, dx \rp.
\end{align*}

We then sum over all possible sequences $i(k)$ for $k\le n_J\le n_J^\vee$
where $n_J$ is the first integer for which all the $d$ coordinates have been visited.
There are clearly less than $d^{n_J^\vee}$ possibilities
(surjection from the set of all sequences of length $n_J^\vee$).
Since for any positive
and measurable functions $(f_j)_{j\le d}$,
we have for any $x\in \cD_c$ :

\begin{align*}
\E_x\lc \prod_{j=1}^d f_j[X(\Uza)] \pv \Uza < \ext\rc
\le d^{n_J^\vee}\ltm (h_\vee)^d \;
\int_{\bar{B}(0, L)} \prod_{j=1}^d f_j(x_j) dx_1...dx_d,
\end{align*}

it is classical that it implies :

\begin{align*}
\frlq{x\in \cD_c}
\PR_x\lc X(\Uza) \in dx \pv \Uza < \ext\rc
\le d^{n_J^\vee}\ltm (h_\vee)^d \;
\idc{x\in \bar{B}(0, L)}\; dx.
\SQ
\end{align*}

Remark : It is quite conceivable that
the distinction between all the possible choices of directions
introduces quite artificially the combinatorial factor $d^{n_J^\vee}$,
which is certainly very rough.
This distinction is however very efficient
in order to simplify the proof.

§ APPENDIX APFUJ : PROOF OF LEMMA <REF>
(A SPECIFIC $\SIGMA$-ALGEBRA FOR JUMPS)

We recall that we define by induction over $i\in \N$:

\begin{align*}
:= \inf\Lbr \tp \ge \tau_{bk}^i \pv
X_\tp \notin \cD_{out} \Rbr
\wedge (\tau_{bk}^{i}+T_S^i),\;
T^0_{out}:= 0,\\
:= \inf\Lbr \tp \ge T^i_{out}
\pv X_\tp \in \cD_{bk} \Rbr,
\end{align*}

where $(T_S^i, i\ge 1)$
denotes a family of i.i.d. r.v.
whose law is exponential with parameter $\rho_S$.

Rather than the original $T_e^i$,
we consider $\hat{T}^i_e$ defined as :

$\bullet$ $\hat{T}^i_e
:= T^i_e
=U^i_{j} \wedge (\tau^i_{bk}+T_S^i)$
on the event $\Lbr T_e^i = T_{out}^i < \ext \Rbr$,
where we recall that $U^i_{j}$ is the first jump time of $X$
from $\cD_{out}$ to outside of $\Dlj$  after $\tau^i_{bk}$.

$\bullet$ $\hat{T}^i_e
:= \tau^i_{bk}+T_S^i \ge T_{out}^i$
on the event $\Lbr  T_{out}^i\wedge \ext < T_e^i\Rbr$.

It is clear that $\hat{T}^i_e$
still defines a stopping time.
Since our bound is needed solely
on the event $\Lbr T_e^i = T_{out}^i < \ext \Rbr$,
replacing $T^i_e$ by $\hat{T}^i_e$
poses no problem.

To alleviate notations since there is no difference in the proof,
we consider in the following
$T_{out} = T^1_{out}$, $T_e = \hat{T}_e^1$, $T_S = T_S^1$,
for initial condition $x\in \cD_{out}$ (so that $\tau^1_{bk} = 0$).
Then :

\begin{align*}
:= \sigma\big( \Lbr A_s \cap\Lbr s< T_e \Rbr\mVg s >0,\,
A_s \in \F_s \Rbr \big)
\end{align*}

Properties of $\F^*_{T_e}$ :
If $Z_s$ is $\F_s$-measurable and $s<t\in (0, \infty]$,
$Z_s\, \idc{s < T_e \le t}$ is  $\F^*_{T_e}$-measurable.

Lemma (APfuj1).
For any left-continuous and adapted process $Z$, $Z_{T_e}$ is $\F^*_{T_e}$-measurable.
$\F^*_{T_e}$ is in fact the smallest $\sigma$-algebra
generated by these random variables.

In particular, for any stopping time $T$, $\Lbr T_e \le T\Rbr \in \F^*_{T_e}$.

Notably, $\Lbr T_e = T_{out}<\ext\Rbr
= \Lbr T_e \le T_{out}\Rbr\cap\Lbr T_e<\ext\Rbr$
is indeed $\F^*_{T_e}$-measurable,
as well as $X(T_e-)$.

Lemma (APfuj2).
Conditionally on  $\F^*_{T_e}$
in particular on the event $\Lbr T_e = T_{out}<\infty\Rbr$,
the process satisfies $X(T_e)  = X(T_e-)$
with probability larger than $\rho_S / (k^\vee +\rho_S)$.

Conditioning by this $sigma$-algebra,
we thus have the knowledge of the past before $T_e$,
while keeping the randomness in the potential jump
at time $T^i_e$.

§.§ Proof of Lemma (APfuj1) :

For any left-continuous and adapted process $Z$,

\begin{align*}
Z_{T_e} = \limInf{n} \sum_{k \le n^2} Z_{\frac{k-1}{n}} \;
\idc{\frac{k-1}{n}< T_e \le \frac{k}{n}},
\end{align*}

where by previous property and the fact that $Z$ is adapted :

$$\frl{k, n} \quad Z_{\frac{k-1}{n}} \;
\idc{\frac{k-1}{n}< T_e \le \frac{k}{n}}
\text{ is $\F^*_{T_e}$-measurable}.$$

Reciprocally, for any $s>0$ and $A_s \in \F_s$ :

\begin{align*}
\idg{A_s\cup\{s< T_e\} }
= \lim_{n\ge 1} Z^n_{T_e}
\where Z^n_t := \{1\wedge [n\, (t-s)_+]\}\ltm \idg{A_s}
\end{align*}

Now, for any stopping time $T$,

\begin{align*}
& \frl{t \ge 0} \quad
\Lbr t \le T \Rbr \in \F_t,\qquad
\Lbr t \le T \Rbr
= \underset{s<t}{\cap} \Lbr s \le T  \Rbr\\
\imp \Lbr T_e \le T \Rbr\cap\Lbr T_e < \infty\Rbr \in \F^*_{T_e},\\
\Lbr T_e = T = \infty \Rbr
= \underset{s>0}{\cap} \Lbr s < T \Rbr \cap \Lbr s< T_e \le \infty\Rbr \in \F^*_{T_e}.
\SQ
\end{align*}

§.§ Proof of Lemma (APfuj2) :

For this proof,
we represent $T_S$
as the first point
of some PPP, $M^S$,
with intensity $\rho_S\, dt$ on $\R_+$.
Remark then :

\begin{align*}
&\idc{X(T_e) = X(T_e-)}
= \textstyle{ \int_{\R_+}}\, \idc{t \le T_e}\, M^S(dt),
\end{align*}

which holds also true when $\ext < T_e$, in which case $T_e= T_S$
and $X(T_e) = \partial = X(T_e-)$.
In particular, by the Markov property,
for any $s>0$
and on the event $\Lbr s < T_e\Rbr$ :

\begin{align*}
& \E_x\lc \idc{X(T_e) = X(T_e-)} \bv \F_s\rc
= \E_{X_s}\lp
\textstyle{ \int_{\R_+}}
\, \idc{t \le \wtd{T}_e}
\, \wtd{M^S}(dt)\rp
\\&\hcm{1}
=  \E_{X_s}\lp
\textstyle{ \int_{\R_+}}  \, \idc{t \le \wtd{T}_e}  \,\rho_S\  dt \rp,
\end{align*}

where the random time $\wtd{T}_e$
refer to the process $\wtd{X}$
with initial condition $X_s$,
as described in Subsection <ref>.

Then, by the Fubini theorem :

\begin{align*}
& \idc{ s < T_e}
\E_x\lc \idc{X(T_e) = X(T_e-)} \bv \F_s\rc
= \rho_S\ \idc{ s < T_e}
\textstyle{ \int_{\R_+}}
\PR_{X_s}\lp t \le \wtd{T}_e  \rp \,  dt.
\end{align*}

Now, by $(A1_G)$,
the rate at which $\wtd{U}_j$ may occur
is always upper-bounded by $\roV$,
so that $\wtd{T}_e$ is upper-bounded
by an exponential law of intensity $\rho_S  + \roV$.

$\frlq{t\ge 0}
\PR_{X_s}\lp t \le \wtd{T}_e \rp
\le \exp[- (\rho_S  + \roV)\ t]\quad$ and we deduce :

\begin{align*}
& \idc{ s < T_e}
\E_x\lc \idc{X(T_e) = X(T_e-)} \bv \F_s\rc
\ge \rho_S/(\rho_S  + \roV)\, \idc{ s < T_e}.
\end{align*}

That is, for any test r.v.
$Z_s\; \idc{s < T_e}$,
where $s>0$ and $Z_s$ is $\F_s$-measurable,

\begin{align*}
\E_x\lc \idc{X(T_e) = X(T_e-)}\, Z_s
\pv s < T_e \rp
\ge \E_x\lc  Z_s\, \rho_S/(\rho_S  + \roV)
\pv s < T_e\rc.
\end{align*}

By definition of $\F^*_{T_e}$,
this concludes the proof of the Lemma.

§ AKNOWLEDGMENT

I am very grateful to Etienne Pardoux, my PhD supervisor, for his great support all along the preparation of this article.

I would like finally to thank the very inspiring meetings and discussions brought about by the Consortium “Modélisation Mathématique et Biodiversité” of Veolia-Ecole Polytechnique-Museum National d'Histoire Naturelle.

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