11footnotetext: Institut de Mathématiques, Université de Neuchâtel,
Switzerland.22footnotetext: Université de Lorraine, CNRS, Inria, IECL, F-54000
Nancy, France.

# Degenerate processes killed at the boundary of a domain

M Benaïm1, N Champagnat2, W Oçafrain2, D Villemonais2

###### Abstract

We investigate certain properties of degenerate Feller processes that are
killed when exiting a relatively compact set. Our main result provides general
conditions ensuring that such a process possesses a (possibly non unique)
quasi stationary distribution. Conditions ensuring uniqueness and exponential
convergence are discussed. The results are applied to nonelliptic and
hypoelliptic stochastic differential equations.

Keywords: Absorbed Markov processes, Quasi-Stationary distributions,
Exponential mixing, Hypoellipticity, Stochastic differential equations,
Potential theory.

2020 Mathematics Subject Classification: 60J25, 60J60, 60F, 60B10, 35H10,
37A25, 37A30, 47A35, 47A75.

##  1 Introduction

###  1.1 Context and objectives

This paper investigates certain properties of a Feller Markov process
$(X_{t})_{t\geq 0}$ living on some metric space $M,$ killed when it exits a
relatively compact set $D.$

We discuss conditions ensuring:

(a)

The existence of a quasi-stationary distribution (QSD). That is, a probability
$\mu$ on $D$ such that

$\mathbb{P}_{\mu}(X_{t}\in\cdot\mid\tau_{D}^{out}>t)=\mu(\cdot)$

where $\tau_{D}^{out}$ is the first exit time of $D;$

(b)

Uniqueness of such a QSD;

(c)

Convergence of the conditional laws
$\mathbb{P}_{\nu}(X_{t}\in\cdot\mid\tau_{D}^{out}>t)$ toward this QSD, in
total variation.

We refer the reader to the survey paper [36] and the monograph [14] for a
comprehensive introduction to the theory of quasi-stationarity.

Probabilistic criteria (based on Lyapunov functions and minorization
conditions) implying $(c)$ are given in [10, 11]. These apply to elliptic
diffusions, as shown in [9] and in [11, Section 4]. Quasi-stationarity for
elliptic diffusions has also been widely studied through spectral arguments,
see for example [40, 20, 8, 23, 30, 34, 35]. However, the treatment of quasi-
stationarity beyond elliptic diffusions is a more delicate problem. Spectral
arguments cannot be applied and the conditions in [10, 11] rely on certain
global Harnack type inequalities (condition (A2) in [10] or (E3) in [11]) that
may be difficult to verify in general, although some results are known in
particular cases [19]. For non elliptic, but strong Feller, diffusions, the
question of proving $(c)$ has been very recently considered in [21, 32]. As we
will see, the strong Feller assumption, although natural in certain
situations, is unnecessary. Also, we find interesting to understand under what
conditions the existence property $(a)$ (respectively the uniqueness property
$(b)$) holds while $(b)$ (respectively $(c)$) does not. A situation which is
typical of degenerate (and not strong Feller) processes. Briefly put, this is
the main objective of this paper.

The remainder of this introduction is devoted to the presentation of our main
results in the specific case where $(X_{t})$ is a degenerate (i.e non
elliptic) diffusion process. Then, Section 2 sets up the general framework and
addresses points (a) (Theorem 2.9), (b) (Theorem 2.13) and (c) (Theorem 2.16).
These results are applied in Section 3 to prove (among other things) the
results stated below for diffusions. Section 4 briefly discusses a simple
example that is not given by a diffusion, but by a piecewise deterministic
Markov process, which opens perspectives for other applications of our
results.

###  1.2 Description of the results for SDEs

Let $M={\mathbb{R}}^{n}$ and $D\subset M$ be an open connected set with
compact closure $K=\overline{D}$ and boundary $\partial D=K\setminus D.$

Consider a stochastic differential equation on $M$

$dX_{t}=S^{0}(X_{t})dt+\sum_{j=1}^{m}S^{j}(X_{t})\circ dB_{t}^{j},$ (1)

where $\circ$ refers to the Stratonovich stochastic integral,
$S^{0},S^{j},j=1,\ldots,m$ are smooth vector fields on $M$ and
$B^{1},\ldots,B^{m}$ $m$ independent Brownian motions.Equivalently, for the
reader familiar with Ito’s calculus,

$dX_{t}=[S^{0}(x)+\frac{1}{2}\sum_{j=1}^{m}DS^{j}(x)S^{j}(x)]dt+\sum_{j=1}^{m}S^{j}(X_{t})dB_{t}^{j},$

where $DS^{j}(x)$ stands for the Jacobian matrix of $DS^{j}$ at $x.$

As usual, the law of $(X_{t})_{t\geq 0}$ when $X_{0}=x$ is denoted
$\mathbb{P}_{x}.$ For $x,y\in M$ and $A\subset M,$ we write $\langle
x,y\rangle=\sum_{i=1}^{n}x_{i}y_{i},\|x\|=\sqrt{\langle x,x\rangle}$ and
$d(x,A)=\inf_{y\in A}\|x-y\|.$ If $A$ is Borel, we let $\tau_{A}=\inf\\{t\geq
0\>:X_{t}\in A\\}$ and $\tau_{A}^{out}=\tau_{M\setminus A}.$

Associated to (1) is the Stroock and Varadhan deterministic control system:

$\dot{y}(t)=S^{0}(y(t))+\sum_{j=1}^{m}u^{j}(t)S^{j}(y(t))$ (2)

where the control function
$u=(u^{1},\ldots,u^{m}):{\mathbb{R}}_{+}\mapsto{\mathbb{R}}^{m},$ can be
chosen to be piecewise continuous. Given such a control function, we let
$y(u,x,\cdot)$ denote the maximal solution to (2) starting from $x$ (i.e
$y(u,x,0)=x$).

###### Definition 1.1

An open set $U\subset{\mathbb{R}}^{n}$ is said to be accessible by
$\\{S^{0},(S^{j})\\}$ from $x\in{\mathbb{R}}^{n}$ if there exist a control $u$
and $t\geq 0$ such that $y(u,x,t)\in U.$ If in addition, $y(u,x,s)\in D$ for
all $0\leq s\leq t,$ (in which case $x\in D$ and $U\cap D\neq\emptyset$) we
say that $U$ is $D$-accessible by $\\{S^{0},(S^{j})\\}$ from $x.$

By abuse of language, we say that a point $y$ is accessible (respectively
$D$-accessible) by $\\{S^{0},(S^{j})\\}$ from $x,$ provided every neighborhood
$U$ of $y$ is accessible (respectively $D$-accessible) by
$\\{S^{0},(S^{j})\\}$ from $x.$

Throughout, we will always assume that the following hypothesis is satisfied
(or, like in Corollary 1.9, implied by other assumptions).

######  Hypothesis  (H1)

The set $M\setminus K$ is accessible by $\\{S^{0},(S^{j})\\}$ from all $x\in
K.$

It easily follows from Stroock and Varadhan support theorem and compactness of
$K$ (see Propositions 3.1 and 2.3) that, under (H1),

$\mathbb{P}_{x}(\tau^{out}_{K}<\infty)=1$

for all $x\in K.$

The following definition is classical (see e.g [3]).

###### Definition 1.2

A point $p\in\partial D$ is called regular for $M\setminus K$ with respect to
(1), provided $\mathbb{P}_{p}(\tau^{out}_{K}=0)=1.$

Our second standing hypothesis is the regularity of $\partial D.$

######  Hypothesis  (H2)

Every point $p\in\partial D$ is regular for $M\setminus K$ with respect to
(1).

For the usual Brownian motion in ${\mathbb{R}}^{n}$ (i.e $S^{0}=0,m=n,$ and
$(S^{1},\ldots,S^{n})$ is an orthonormal basis in ${\mathbb{R}}^{n}$), a
classical condition on $D$ ensuring (H2) is the so-called exterior cone
condition: For all $p\in\partial D,$ there exists an open truncated cone at
vertex $p$ contained in $M\setminus K$ (see e.g [3], Proposition 21.8). If the
diffusion is elliptic at $p$ (meaning that $S^{1}(p),\ldots S^{m}(p)$ span
${\mathbb{R}}^{n}$), the exterior cone condition also ensures that $p$ is
regular. In case the diffusion is in divergence form, this follows directly
from Aronson’s estimates [45, 33] which allow to bound from below the law of
$(X_{t})$ by (up to a constant) the law of a Brownian motion. The general case
follows by a straightforward application of Girsanov theorem.

For degenerate diffusions, a practical condition is the stronger hypothesis
(H2’) given below. It turns out that this assumption will also prove to be
useful for our results on hypoelliptic diffusions.

###### Definition 1.3

We say that $D$ satisfies the exterior sphere condition at $p\in\partial D,$
if there exists a vector $v\in{\mathbb{R}}^{n},$ called a unit outward normal
vector at $p,$ such that $\|v\|=1$ and $d(p+rv,K)=r$ for some $r>0.$

In other words, the open Euclidean ball with center $p+rv$ and radius $r$ is
contained in $M\setminus K$ and its closure touches $K$ at $p.$ If the
condition holds at every $p\in\partial D,$ we simply say that $D$ satisfies
the exterior sphere condition.

###### Remark 1.4

Examples of sets verifying the exterior sphere condition are sets with $C^{2}$
boundary and convex sets. In the first case every $p\in\partial D$ has a
unique unit outward normal vector. In the second case, a point $p\in\partial
D$ may have infinitely many unit outward normal vectors (think of a convex
polygon). If $D\subset{\mathbb{R}}^{2}$ is a nonconvex polygon and
$p\in\partial D$ is a vertex at which the interior angle is $>\pi$, the
exterior sphere condition is not satisfied at $p$ although the exterior cone
is.

The following condition (H2’) implies (H2). This will be proved in Section 3,
Proposition 3.12.

######  Hypothesis  (H2’)

$D$ satisfies the exterior sphere condition and for each $p\in\partial D,$
there exist an outward unit normal vector $v$ at $p,$ and
$i\in\\{1,\ldots,m\\}$ such that $\langle S^{i}(p),v\rangle\neq 0.$

The next result is an existence theorem. Its proof, in Sections 3.1 and 3.2,
shows that it only requires the Lipschitz continuity of $S^{0}$ and the
$C^{2}$ regularity of the $S^{j},j\geq 1.$

###### Theorem 1.5

Assume that:

(i)

Hypotheses (H1) and (H2) (or (H2’)) hold;

(ii)

For some $\varepsilon>0,$ the set $D_{\varepsilon}=\\{x\in D:\>d(x,\partial
D)>\varepsilon\\}$ is $D$-accessible by $\\{S^{0},(S^{j})\\}$ from all $x\in
D\setminus D_{\varepsilon}.$

Then, there exists a QSD for $(X_{t})$ on $D.$

Simple criteria ensuring (H1) and condition $(ii)$ of Theorem 1.5 will be
discussed in Section 3 (Propositions 3.3 and 3.8). In particular, Proposition
3.8 has the following useful consequence that, when $\partial D$ is $C^{2},$
(H2’) implies condition $(ii)$ of Theorem 1.5. Therefore,

###### Corollary 1.6

Assume that Hypothesis (H1) holds, $\partial D$ is $C^{2},$ and for each
$p\in\partial D$ one of the vectors $S^{j}(p),j=1,\ldots m,$ is transverse to
$\partial D$ (i.e $S^{j}(p)\not\in T_{p}\partial D$). Then, there exists a QSD
for $(X_{t})$ on $D.$

If the SDE (1) enjoys certain hypoellipticity properties, more can be said.
Given a family $\mathcal{S}$ of smooth vector fields on $M$ and
$k\in{\mathbb{N}},$ we let $[\mathcal{S}]_{k}$ denote the set of vector fields
recursively defined by $[\mathcal{S}]_{0}=\mathcal{S},$ and

$[\mathcal{S}]_{k+1}=[\mathcal{S}]_{k}\cup\\{[Y,Z]\>:Y,Z\in[\mathcal{S}]_{k}\\}$

where $[Y,Z]$ stands for the Lie bracket of $Y$ and $Z.$ Set
$[\mathcal{S}]=\cup_{k}[\mathcal{S}]_{k}$ and
$[\mathcal{S}](x)=\\{Y(x)\>:Y\in[\mathcal{S}]\\}.$

###### Definition 1.7

A point $x^{*}\in M$ is said to satisfy the weak Hörmander condition
(respectively the Hörmander condition, respectively the strong Hörmander
condition) if $[\\{S^{0},\ldots,S^{m}\\}](x^{*})$ (respectively

${\\{S^{1}(x^{*}),\ldots,S^{m}(x^{*})\\}\cup\\{[Y,Z](x^{*}):\>Y,Z\in[\\{S^{0},\ldots,S^{m}\\}]\\}},$

respectively

$[\\{S^{1},\ldots,S^{m}\\}](x^{*})\ \text{)}$

spans ${\mathbb{R}}^{n}.$

###### Theorem 1.8

Assume that:

(i)

Hypotheses (H1) and (H2’) hold;

(ii)

The weak Hörmander condition is satisfied at every point $x\in K;$

(iii)

Every $y\in D$ is $D$-accessible by $\\{S^{0},(S^{j})\\}$ from every $x\in D.$

Then the $QSD$ $\mu$ is unique. Its topological support equals $K$ and it has
a smooth density with respect to the Lebesgue measure.

Suppose furthermore that there exists a point $x^{*}\in D$ at which the weak
Hörmander condition is strengthened to the Hörmander condition. Then there
exist $\alpha>0$, $C\in]0,+\infty[$ and a continuous function
$h:D\mapsto]0,\infty[$ with $h(x)\rightarrow 0,$ as $x\rightarrow\partial D,$
satisfying $\mu(h)=1$, such that for all $\rho\in\mathcal{M}_{1}(D)$ (the set
of probability measures over $D$),

$\|\mathbb{P}_{\rho}(X_{t}\in\cdot\mid\tau_{D}^{out}>t)-\mu(\cdot)\|_{TV}\leq\frac{C}{\rho(h)}e^{-\alpha
t}$

where $\|\cdot\|_{TV}$ stands for the total variation distance and
$\rho(h):=\int hd\rho.$

Note that Condition (ii) of Theorem 1.5 readily follows from Condition (iii)
of Theorem 1.8, so that the assumptions of Theorem 1.8 imply the existence of
a QSD.

###### Corollary 1.9

Assume that Hypothesis (H2’) holds, and that the strong Hörmander condition is
satisfied at every point $x\in K.$ Then, the conclusions of Theorem 1.8 hold
and the density of $\mu$ is positive on $D.$

Proof:  Let, for $\varepsilon\geq 0,$ $y^{\varepsilon}(x,u,\cdot)$ be defined
like $y(x,u,\cdot)$ when $S^{0}$ is replaced by $\varepsilon S^{0}.$ By Chow’s
theorem (see e.g [27], Chapter 2, Theorem 3), the strong Hörmander condition
implies that for all $x,y\in D$ there exists a control $u$ piecewise
continuous with $u^{i}(s)\in\\{-1,0,1\\}$ and $t\geq 0$ such that
$y^{0}(x,u,s)\in D$ for all $0\leq s\leq t$ and $y^{0}(x,u,t)=y.$ By
continuity of solutions to an ODE, with respect to some parameter (or by a
simple application of Gronwall’s lemma),
$y^{\varepsilon}(x,u,\cdot)\rightarrow y^{0}(x,u,\cdot)$ uniformly on $[0,t]$
as $\varepsilon\rightarrow 0.$ Let
$u^{\varepsilon}(s)=\frac{u(\frac{s}{\varepsilon})}{\varepsilon}.$ Then
$y^{\varepsilon}(x,u,s)=y(x,u^{\varepsilon},\varepsilon s).$ This proves that
$y$ is $D$-accessible from $x.$ Because the strong Hörmander condition also
holds in a neighborhood of $K,$ the same proof also shows that, for $y$ in a
neighborhood of $K$ and $x\in K,$ $y$ is accessible from $x$, so (H1) is
satisfied. The conditions, hence the conclusions, of Theorem 1.8 are then
satisfied. Positivity of $\frac{d\mu}{dx}$ is proved in Lemma 3.16. $\Box$

###### Remark 1.10

The results above easily extend to the situation where $M$ is a
$n$-dimensional manifold, provided $\partial D$ is a $(n-1)$-dimensional
$C^{2}$ sub-manifold. This is illustrated in Examples 1.12 and 1.13.

###### Remark 1.11

The recent paper [21] considers QSDs and their properties under the assumption
that the underlying process is strong Feller. Quasi-stationarity for
degenerate strong Feller diffusion processes have also been the focus of
interest in the recent literature [21, 32, 41]. Observe that none of the
conditions of Theorem 1.5, neither the weak Hörmander condition assumed in
Theorem 1.8 imply that the semigroup induced by (1) is strong Feller (see
Example 1.12 below).

###### Example 1.12

Suppose $M$ is the cylinder, $M={\mathbb{R}}/{\mathbb{Z}}\times{\mathbb{R}}.$
Let $m=1,$

$S^{0}(x,y)=\partial_{x}\mbox{ and }S^{1}(x,y)=\partial_{y}.$

Let $D={\mathbb{R}}/{\mathbb{Z}}\times]0,1[.$ Here the conditions of Theorem
1.8, are easily seen to be satisfied. However, the process is not strong
Feller (the dynamics in the $x$-variable being deterministic). It is not hard
to check that the unique QSD is the measure

$\mu(dxdy)=2\frac{\sin(\pi y)}{\pi}{\mathbbm{1}}_{D}(x,y)dxdy.$ (3)

######  Example 1.13 (Example 1.12, continued)

Let $M$ and $D$ be like in Example 1.12, $m=1,$

$S^{0}(x,y)=a(y)\partial_{x}\mbox{ and }S^{1}(x,y)=\partial_{y},$

where $a$ is a smooth function $\geq 1.$ Like in example 1.12, the unique QSD
$\mu$ is given by (3). Suppose that $a^{\prime}(y^{*})\neq 0$ for some
$0<y^{*}<1.$ Then the Hörmander condition holds at $(x,y^{*})$, so that by
Theorem 1.8, $(\mathbb{P}_{x}(X_{t}\in\cdot|\tau_{D}^{out}>t))_{t\geq 0}$
converges at an exponential rate to $\mu.$

###### Remark 1.14

In absence of condition $(iii)$ in Theorem 1.8, there is no guarantee that the
QSD is unique, as shown by the next example. Still every QSD has a smooth
density (see Lemma 3.16).

###### Example 1.15

Let $M=\mathbb{R}$ and $D=]0,5[$. We consider smooth functions
$\varphi^{1},\varphi^{2},\psi^{1},\psi^{2}:D\to[0,1]$ such that
$\psi^{1}+\psi^{2}=1$ and

$\displaystyle\begin{cases}\varphi^{1}_{\rvert]0,1]}\equiv 1,\
\varphi^{1}_{\rvert[1,2]}\leq 1,\ \varphi^{1}_{\rvert[2,5[}\equiv 0,\\\
\varphi^{2}_{\rvert]0,3]}\equiv 0,\ \varphi^{2}_{\rvert[3,4]}\leq 1,\
\varphi^{2}_{\rvert[4,5[}\equiv 1,\\\ \psi^{1}_{\rvert]0,2]}\equiv 1,\
0<\psi^{1}_{\rvert]2,3[}\leq 1,\ \psi^{1}_{\rvert[3,5[}\equiv 0,\\\
\psi^{2}_{\rvert]0,2]}\equiv 0,\ 0<\psi^{2}_{\rvert]2,3[}\leq 1,\
\psi^{2}_{\rvert[3,5[}\equiv 1.\end{cases}$

For all $\alpha>0$, we consider the absorbed diffusion process $X^{\alpha}$
evolving according to the Itô SDE

$\displaystyle
dX^{\alpha}_{t}=\left(\varphi^{1}(X^{\alpha}_{t})+\sqrt{\alpha}\,\varphi^{2}(X^{\alpha}_{t})\right)\,dB_{t}\,+(\psi^{1}(X_{t})+\alpha\,\psi^{2}(X_{t}))\,dt.$

and absorbed when it reaches $\partial D=\\{0,5\\}$. While $X^{\alpha}$
satisfies the conditions (i) and (ii) of Theorem 1.8, it does not satisfy
condition (iii) and it admits either one or two QSDs, depending on the value
of $\alpha$. Indeed, as shown in [5], there exists $\alpha_{c}>0$ such that

  * •

for all $\alpha\in]0,\alpha_{c}]$, $X^{\alpha}$ admits a unique QSD supported
by $[3,5]$,

  * •

for all $\alpha\in]\alpha_{c},+\infty[$, $X^{\alpha}$ admits exactly two QSDs,
supported respectively by $[3,5]$ and $[0,5]$.

##  2 Killed processes

Throughout, we let $M$ denote a separable and locally compact metric space,
and $D\subset M$ a nonempty set with compact closure $K=\overline{D}$ in $M$,
such that $D$ is open relative to $K.$ That is $D={\cal O}\cap K$ for some
open set ${\cal O}\subset M.$

We let $(P_{t})_{t\geq 0}$ denote a Markov Feller semigroup on $M$. By this,
we mean (as usual) that $(P_{t})_{t\geq 0}$ is a semigroup of Markov operators
on $C_{0}(M)$ (the space of continuous functions on $M$ vanishing at infinity)
and that $P_{t}f(x)\rightarrow f(x)$ as $t\rightarrow 0$ for the uniform norm
for all $f\in C_{0}(M).$ Observe that since we are interested by the behavior
of the process killed outside $K$, the behavior of $(P_{t})_{t\geq 0}$ at
infinity is irrelevant (and the reader can think of $M$ as compact without
loss of generality). Feller processes, named after William Feller, forms a
large class of processes whose sample path properties have been studied, among
others, by Doob, Kinney, Dynkin, Blumenthal, Hunt and Ray. We refer the reader
to [12, 28] for properties and historical notes on these processes, see also
[31, 29, 42] for expository texts on Feller semi-groups and their sample path
properties.

By classical results (see e.g Le Gall [31], Theorem 6.15), there exist a
filtered space $(\Omega,{\cal F},({\cal F}_{t}))$ with $({\cal F}_{t})$ right
continuous and complete, a family of probabilities $(\mathbb{P}_{x})_{x\in M}$
on $(\Omega,{\cal F})$ and a continuous time adapted process $(X_{t})$ on
$(\Omega,{\cal F},({\cal F}_{t}))$ taking values in $M,$ such that:

(i)

$(X_{t})$ has cad-lag paths,

(ii)

$\mathbb{P}_{x}(X_{0}=x)=1$ and,

(iii)

$(X_{t})$ is a Markov process with semigroup $(P_{t}),$ meaning that

$\mathbb{E}_{x}(f(X_{t+s})|{\cal F}_{t})=P_{s}f(X_{t})$

for all $t,s\geq 0$ and $f$ measurable bounded (or $\geq 0$).

For any Borel set $A\subset M$ we let $\tau_{A}=\inf\\{t\geq 0:\>X_{t}\in
A\\}$ and $\tau^{out}_{A}=\tau_{M\setminus A}.$ The assumptions on $({\cal
F}_{t})$ (right continuous and complete) imply that $\tau_{A}$ and
$\tau^{out}_{A}$ are stopping times with respect to $({\cal F}_{t})$ (see e.g.
Bass [2]).

###### Remark 2.1

For $X_{0}=x\in D,$ $\tau^{out}_{D}\leq\tau^{out}_{K}$ but it is not true in
general that $\tau^{out}_{D}=\tau^{out}_{K}.$ Consider for example the ODE on
${\mathbb{R}}^{2}$ given by

$\left\\{\begin{array}[]{c}\dot{x}=1\\\ \dot{y}=0\end{array}\right.$

Let

$D=\\{(x,y)\in{\mathbb{R}}^{2}\>-1<x<1,-1<y<x^{2}\\}.$

For $-1<x<0$ and $y=0$ the trajectory $(x(t),y(t))=(x+t,0)$ starting at
$(x,0)$ leaves $D$ at time $-x$ and $K$ at time $-x+1.$

###### Definition 2.2

An open set $U\subset M$ is said to be accessible (by $(P_{t})$) from $x\in
M,$ if there exists $t\geq 0$ such that
$P_{t}(x,U)=P_{t}{\mathbbm{1}}_{U}(x)>0.$

Here also, by abuse of language, we say that $y\in M$ is accessible from $x\in
M,$ if every open neighborhood $U$ of $y$ is accessible from $x.$

We shall assume throughout all this section that the following assumption is
satisfied.

######  Hypothesis  (Standing Hypothesis)

The set $M\setminus K$ is accessible from all $x\in K.$

The next result is a basic, but very useful, consequence of this assumption.
It is interesting to point out that it only requires Feller continuity,
accessibility of $M\setminus K$ and compactness of $K$. Note that similar -
albeit different- results can be found in the literature (see e.g [3],
Proposition 21.2).

###### Proposition 2.3

There exist positive constants $C,\Lambda$ such that

$\mathbb{P}_{x}(\tau^{out}_{K}>t)\leq Ce^{-\Lambda t}$

for all $t\geq 0$ and $x\in K.$ In particular $\tau^{out}_{K}<\infty$, and
hence $\tau^{out}_{D}<\infty$, $\mathbb{P}_{x}$ almost surely for all $x\in
K.$

Proof:  Let $U=M\setminus K$. By Feller continuity, for all $t\geq 0,$ the set
$O_{t}=\\{x\in M:\>P_{t}(x,U)>0\\}$ is open (possibly empty). We detail the
proof since the argument will be used several times in the sequel. We notice
that the sequence of functions $f_{n}(x)=(nd(x,M\setminus U)\wedge 1)$ is non-
decreasing and converges to $\mathbbm{1}_{U}(x)$. Therefore, $x\in O_{t}$ if
and only if $x\in O_{t}^{n}=\\{x\in M:\>P_{t}f_{n}(x)>0\\}$ for some $n\geq
1$. Since $O^{n}_{t}$ is open by Feller continuity of the semi-group $P_{t}$,
we deduce that $O_{t}$ is open. By assumption, the family
$\\{O_{t}:\>t\in{\mathbb{R}}^{+}\\}$ covers $K,$ so that, by compactness,
there exist $t_{1},\ldots,t_{n}$ such that $K\subset\cup_{i=1}^{n}O_{t_{i}}.$
In particular, for some $\delta>0$ and $t=\max\\{t_{1},\ldots,t_{n}\\}$
$\mathbb{P}_{x}(\tau_{U}>t)\leq 1-\sup_{1\leq i\leq n}P_{t_{i}}(x,U)\leq
1-\delta$ for all $x\in K.$ Thus, by the Markov property,
$\mathbb{P}_{x}(\tau_{U}>kt)\leq(1-\delta)^{k}.$ This proves the result.
$\Box$

###  2.1 Green kernel and QSDs

Let $B(D)$ denote the set of bounded measurable functions
$f:D\mapsto{\mathbb{R}}.$ For all $f\in B(D),x\in D$ and $t\geq 0$ set

$P_{t}^{D}f(x)=\mathbb{E}_{x}(f(X_{t}){\mathbbm{1}}_{\tau^{out}_{D}>t}).$

By Proposition 2.3, $(P_{t}^{D})_{t\geq 0}$ is a well defined sub-Markovian
semigroup on $B(D).$ The semigroup property is a consequence of the Markov
property and the fact that $\tau^{out}_{D}$ is a stopping time.

The Green kernel $G^{D}$ is the bounded (by Proposition 2.3) operator defined
on $B(D)$ by

$G^{D}f(x)=\int_{0}^{\infty}P_{t}^{D}f(x)dt=\mathbb{E}_{x}(\int_{0}^{\tau^{out}_{D}}f(X_{t})dt).$

For all $x\in D$ and $A\subset M,$ a Borel set, we let

$G^{D}(x,A)=G^{D}{\mathbbm{1}}_{A\cap D}(x).$

#### Quasi-stationary Distributions

A Quasi-stationary Distribution (QSD) for $(P_{t}^{D})$ is a probability
measure $\mu$ on $D$ such that, for all $t\geq 0$,

$\mu P_{t}^{D}=e^{-\lambda t}\mu$ (4)

for some $\lambda>0.$ For further reference we call $\lambda$ the absorption
rate (or simply the rate) of $\mu.$ Equivalently,

$\frac{\mu P_{t}^{D}(\cdot)}{\mu
P_{t}^{D}{\mathbbm{1}}_{D}}=\mathbb{P}_{\mu}(X_{t}\in\cdot\mid\tau_{D}^{out}>t)=\mu(\cdot).$

###### Lemma 2.4

Equation (4) holds if and only if $\mu G^{D}=\frac{1}{\lambda}\mu.$

Proof:  Clearly, by definition of $G^{D}$, equation (4) implies that $\mu
G^{D}=\frac{1}{\lambda}\mu.$ Conversely, assume that $\mu
G^{D}=\frac{1}{\lambda}\mu.$ Then for every bounded nonnegative measurable map
$f:D\mapsto{\mathbb{R}},$ $\mu(G^{D}f)=\frac{1}{\lambda}\mu(f)$ and also

$\mu(G^{D}P_{t}^{D}f)=\frac{1}{\lambda}\mu(P_{t}^{D}f).$

That is

$\mu(\int_{t}^{\infty}P_{s}^{D}fds)=\frac{1}{\lambda}\mu(P^{D}_{t}f).$

Equivalently,

$\mu(G^{D}f-\int_{0}^{t}P_{s}^{D}fds)=\frac{1}{\lambda}\mu(P_{t}^{D}f).$

This shows that the bounded map $v(t)=\mu(P_{t}^{D}f)$ satisfies the integral
equation

$v(t)-v(0)=-\lambda\int_{0}^{t}v(s)ds,$

for all $t\geq 0$. It follows that $v(t)=v(0)e^{-\lambda t}.$ $\Box$
Let $C_{b}(D)\subset B(D)$ denote the set of bounded continuous functions on
$D,$ and $C_{0}(D)\subset C_{b}(D)$ the subset of functions $f$ such that
$f(x)\rightarrow 0$ when $x\rightarrow\partial_{K}D:=K\setminus D.$ We define
similarly $C_{b}(K)$ and $C_{0}(K).$

###### Remark 2.5

In the recent paper [18], the authors prove the existence and convergence to a
QSD under the condition that the sub-Markovian semigroup is strong Feller.
Observe that in our case, although $(P_{t})$ is Feller, there is no evidence
in general that $(P_{t}^{D})$ is strong Feller nor that it preserves
$C_{b}(D).$ On the other hand, under rather weak, reasonable conditions,
$G^{D}$ maps $C_{b}(D)$ into $C_{0}(D)$, as illustrated by the following
example.

###### Example 2.6

Consider the ODE on ${\mathbb{R}}$ given by $\dot{x}=-1.$ For $D=]0,1[$
$P_{t}^{D}f(x)=f(x-t){\mathbbm{1}}_{x>t}$ is not Feller,
$G^{D}f(x)=\int_{0}^{x}f(u)du$ is Feller (and even strong Feller). If now
$D=]0,1],$ then $G^{D}$ maps $C_{b}(D)$ into $C_{0}(D).$

The condition that $G^{D}(C_{b}(D))\subset C_{0}(D)$ plays a key role in the
next Theorem 2.9 and will be investigated in the subsequent sections.

###### Definition 2.7

An open set $U\subset M$ is said $D$-accessible (by $(P_{t})$) from $x\in D$
if $P_{t}^{D}(x,U)>0$ for some $t\geq 0.$ A point $y\in K$ is said
$D$-accessible from $x\in D$ if every open neighborhood of $y$ is
$D$-accessible from $x.$

###### Lemma 2.8

An open set $U\subset M$ is $D$-accessible from $x\in D$ if and only if
$G^{D}(x,U)>0.$ In particular, the set of $D$-accessible points from $x$
coincides with the topological support of $G^{D}(x,\cdot).$

Proof:  Suppose that $P^{D}_{t}(x,U)>0$ for some open set $U$ and $t\geq 0.$
Fatou’s Lemma and right continuity of paths imply that

$\liminf_{s\rightarrow
t,s>t}P^{D}_{s}(x,U)\geq\mathbb{E}_{x}(\liminf_{s\rightarrow
t,s>t}{\mathbbm{1}}_{U}(X_{s}){\mathbbm{1}}_{\tau_{D}^{out}>s)}\geq
P^{D}_{t}(x,U)>0.$

This proves that $s\rightarrow P^{D}_{s}(x,U)$ is positive on some interval
$[t,t+\varepsilon],$ hence $G^{D}(x,U)>0.$ The converse implication is
obvious. $\Box$

###### Theorem 2.9

Assume that $G^{D}(C_{b}(D))\subset C_{0}(D).$ Then, the conditions (i),(ii)
below are equivalent and imply the existence of a QSD.

(i)

There exists an open set $U\subset M,$ $D$-accessible from all $x\in D$ and
such that $\overline{U\cap K}\subset D.$

(ii)

There exists a function $\phi\in C_{0}(D),$ positive on $D$ and $\theta>0,$
such that $G^{D}\phi\geq\theta\phi.$

Proof:  We first show that $(i)$ implies $(ii).$ For $\varepsilon>0,$ let

$\overline{U}^{\varepsilon}_{K}=\\{x\in K:\>d(x,U\cap K)\leq\varepsilon\\}.$

Choose $\varepsilon>0$ small enough so that
$\overline{U}_{K}^{\varepsilon}\subset D$.

Let $\psi(x)=(1-\frac{d(x,U\cap K)}{\varepsilon})^{+}$ and $\phi=G^{D}\psi.$
Then, $\phi\in C_{0}(D)$ (because $G^{D}(C_{b}(D))\subset C_{0}(D)$) and for
all $x\in D,$

$\phi(x)\geq G^{D}{\mathbbm{1}}_{U\cap K}(x)=G^{D}{\mathbbm{1}}_{U}(x)>0,$

where the last inequality holds by Lemma 2.8. Thus, by compactness of
$\overline{U}^{\varepsilon}_{K},$
$\theta=\inf_{x\in\overline{U}_{K}^{\varepsilon}}\phi(x)>0.$ Since $\psi=0$ on
$K\setminus\overline{U}_{K}^{\varepsilon},$ it follows that for all $x\in D,$

$\phi(x)\geq\theta\mathbbm{1}_{\overline{U}^{\varepsilon}_{K}}\geq\theta\psi(x).$

Therefore, $G^{D}\phi(x)\geq\theta\phi(x).$

We now prove that $(ii)$ implies $(i)$. For all $\varepsilon\geq 0$ and $x\in
D,$ set

$V_{\varepsilon}=\\{y\in D:\>\phi(y)>\varepsilon\\},$

and

$\eta=\eta(x):=\inf\\{\varepsilon\geq 0:\>G^{D}(x,V_{\varepsilon})=0\\}.$

By monotone convergence,
$G^{D}(x,V_{\eta})=\lim_{n\rightarrow\infty}G^{D}(x,V_{\varepsilon+1/n})=0.$
In particular, $\eta$ is positive because, by (ii), $G^{D}(x,D)$ is.
Therefore, by definition of $\eta,$ $G^{D}(x,V_{\eta/2})>0,$ so that
$\mathbb{P}_{x}(X_{t}\in V_{\eta/2},\,t<\tau_{D}^{out})>0$ for some $t\geq 0.$
Furthermore, $G^{D}(X_{t},V_{\eta})=0$ $\mathbb{P}_{x}$-almost surely on the
event $\\{t<\tau_{D}^{out}\\}.$ Indeed,

$\mathbb{E}_{x}(G^{D}(X_{t},V_{\eta}){\mathbbm{1}}_{t<\tau_{D}^{out}})=P_{t}^{D}G^{D}({\mathbbm{1}}_{V_{\eta}})(x)=\int_{t}^{\infty}P_{s}^{D}(x,V_{\eta})ds\leq
G^{D}(x,V_{\eta})=0.$

In particular, $X_{t}\in D\setminus V_{\eta}$ for all $\tau_{D}^{out}>t\geq 0$
almost surely, by Lemma 2.8. This shows that there exists a point $y\in
V_{\eta/2}\setminus V_{\eta}$ with $G^{D}(y,V_{\eta})=0.$ Thus

$\theta\leq\frac{G^{D}\phi(y)}{\phi(y)}=\frac{G^{D}(\phi{\mathbbm{1}}_{D\setminus
V_{\eta}})(y)}{\phi(y)}\leq 2G^{D}{\mathbbm{1}}_{D}(y)\leq 2\sup_{x\in
D\setminus V_{\eta}}G^{D}{\mathbbm{1}}_{D}(x).$ (5)

Now, since $G^{D}{\mathbbm{1}}_{D}\in C_{0}(D)$, there exists $\alpha>0$ such
that

$\sup_{x\in D,\,d(x,\partial
D)\leq\alpha}G^{D}{\mathbbm{1}}_{D}(x)<\frac{\theta}{2}$

and, since $\phi$ is positive on $D$ and $\phi\in C_{0}(D)$, there exists
$\eta_{0}>0$ such that $D\setminus V_{\eta_{0}}\subset\\{x\in D,\,d(x,\partial
D)\leq\alpha\\}$. Therefore, (5) implies that the map $x\mapsto\eta(x)$ is
bounded from below by $\eta_{0}$. Since $G^{D}(x,V_{\eta_{0}/2})\geq
G^{D}(x,V_{\eta/2})>0$, Lemma 2.8 proves $(i)$ with $U=V_{\eta_{0}/2}.$

Our last goal is to prove the existence of a QSD under Condition(ii). Let
$B(K)$ (respectively $C_{b}(K)$) be the space of bounded (respectively
continuous) functions over $K.$ The operator $G^{D}$ extends to a bounded
operator $G$ on $B(K)$ defined as

$Gf(x)=\left\\{\begin{array}[]{ll}G^{D}(f|_{D})(x)&\mbox{ for }x\in D\\\
0&\mbox{ for }x\in\partial_{K}D\end{array}\right.$

The assumption that $G^{D}(C_{b}(D))\subset C_{0}(D)$ implies that $G$ leaves
$C_{b}(K)$ invariant. That is $G(C_{b}(K))\subset C_{b}(K).$

Let ${\cal M}_{1}(\phi)$ be the set of Borel finite nonnegative measures $\mu$
on $K$ such that $\mu(\phi)=1$ and let $T:{\cal M}_{1}(\phi)\mapsto{\cal
M}_{1}(\phi)$ be the map defined by

$T(\mu)=\frac{\mu G}{\mu G\phi}$

We first observe that $T$ is continuous for the weak* topology: if
$\mu_{n}\rightharpoonup\mu$ for the weak* topology in $\mathcal{M}_{1}(\phi)$,
then $\mu_{n}G\rightharpoonup\mu G$ and $\mu_{n}G\phi\rightarrow\mu G\phi$.
Since in addition $\mu_{n}G\phi\geq\theta\mu_{n}(\phi)=\theta>0$ by $(ii),$
$T(\mu_{n})\rightharpoonup T(\mu)$.

Choose an open neighborhood ${\cal N}$ of $\partial_{K}D$ such that
$G{\mathbbm{1}}_{K}\leq\theta/2$ on ${\cal N}\cap K$ and set $C=\sup_{x\in
K\setminus{\cal N}}\frac{(G{\mathbbm{1}}_{K})(x)}{\phi(x)}<\infty.$ Then, for
all $\mu\in{\cal M}_{1}(\phi)$

$T(\mu)(K)=\frac{\mu(G{\mathbbm{1}}_{K})}{\mu
G\phi}\leq\frac{\mu(G{\mathbbm{1}}_{K})}{\theta}\leq\frac{\theta/2\mu(K)+C\mu(\phi)}{\theta}=\frac{\mu(K)}{2}+\frac{C}{\theta}.$

It follows that for any $R\geq 2\frac{C}{\theta}$ the set

${\cal M}_{1}^{R}(\phi)=\\{\mu\in{\cal M}_{1}(\phi)\>:\mu(K)\leq R\\}$

is invariant by $T.$ Since ${\cal M}_{1}^{R}(\phi)$ is convex and compact (for
the weak* topology), $T$ admits a fixed point by Tychonoff Theorem. If $\mu$
is such a fixed point, $\mu(D)>0$ since $\mu(\phi)=1$ and $\phi(x)=0$ for all
$x\in K\setminus D$, so we can define the probability measure
$\bar{\mu}=\frac{\mu(\cdot\cap D)}{\mu(D)}$ on $D$. Since, for all $f\in
B(K)$, $\mu Gf=\mu({\mathbbm{1}}_{D}G^{D}f|_{D})$, we deduce that, for all
$f\in B(D)$, $\bar{\mu}G^{D}f=(\mu G\phi)\bar{\mu}(f)$ with $\mu G\phi>0$.
Then $\mu$ is a QSD for $(P_{t}^{D})$ by Lemma 2.4. $\Box$

###### Remark 2.10

The last part of the proof of Theorem 2.9 is reminiscent of the proof of the
existence Theorem 4.2 in [13].

###  2.2 Uniqueness and convergence criteria

#### Right eigenfunctions

We say that $h\in C_{0}(D)$ is a positive right eigenfunction for $G^{D}$ if
$h(x)>0$ for all $x\in D$ and

$G^{D}h=\frac{1}{\lambda}h$ (6)

for some $\lambda>0.$

###### Lemma 2.11

If $h$ is a positive right eigenfunction, the parameter $\lambda$ in (6)
necessarily equals the absorption rate of any QSD and

$P_{t}^{D}h=e^{-\lambda t}h$

for all $t\geq 0.$

Proof:  If $\mu$ is a QSD with rate $\lambda^{\prime}$, then $\mu
G^{D}h=\frac{1}{\lambda^{\prime}}\mu h=\frac{1}{\lambda}\mu h$ so that
$\lambda=\lambda^{\prime}.$ The proof of the second statement is similar to
the proof of Lemma 2.4 and left to the reader. $\Box$
Observe that there is no assumption in Lemma 2.11 that $G^{D}(C_{b}(D))\subset
C_{0}(D)$ but when this is the case, there always exists a QSD, by application
of Theorem 2.9.

The following result shows that a strengthening in the assumptions of Theorem
2.9 ensures the existence of a positive right eigenfunction for $G^{D}.$

###### Corollary 2.12

Assume that:

(i)

$G^{D}(C_{b}(D))\subset C_{0}(D)$ and $G^{D}$ is a compact operator on
$C_{0}(D);$

(ii)

For all $x,y\in D$ $y$ is $D$-accessible from $x.$

Then, there exists a positive right eigenfunction for $G^{D}.$

Proof:  Let $r=\lim_{n\rightarrow\infty}\|(G^{D})^{n}\|^{1/n}$ be the spectral
radius of $G^{D}$ on $C_{0}(D).$ Let $\mu$ be a QSD (whose existence is given
by Theorem 2.9) with rate $\lambda.$ For any $f\in C_{0}(D)$ such that $0\leq
f\leq 1$ and $\mu(f)\neq 0$
$\|(G^{D})^{n}\|\geq\mu((G^{D})^{n}f)=\frac{1}{\lambda^{n}}\mu(f).$ Hence
$r\geq\frac{1}{\lambda}>0.$

Let $C_{0}^{+}(D)=\\{f\in C_{0}(D)\>:f\geq 0\\}.$ It is readily seen that
$C_{0}^{+}(D)$ is a reproducing cone in $C_{0}(D),$ invariant by $G^{D},$
meaning that $C_{0}(D)^{+}$ is a cone, $C_{0}(D)=\\{u-v:\>u,v\in
C_{0}(D)^{+}\\}$ and $G^{D}(C_{0}(D)^{+})\subset C_{0}(D)^{+}.$ Therefore, by
Krein Rutman Theorem ([16], Theorem 19.2), there exists $h\in
C_{0}(D)^{+}\setminus\\{0\\}$ such that $G^{D}h=rh.$ Let $y\in D$ be such that
$h(y)>0.$ Then, $h\geq\frac{h(y)}{2}{\mathbbm{1}}_{U\cap D}$ for some
neighborhood $U$ of $y.$ Therefore, by $(ii),$ for all $x\in D,$

$rh(x)=(G^{D}h)(x)\geq\frac{h(y)}{2}(G^{D}{\mathbbm{1}}_{U\cap D})(x)>0.$

This concludes the proof. $\Box$

#### Uniqueness and convergence

We say that $(P_{t}^{D})$ is irreducible if there exists a nontrivial positive
measure $\xi$ on $D$ such that for all $x\in D$ and $A$ Borel,

$\xi(A)>0\Rightarrow G^{D}(x,A)>0.$

###### Theorem 2.13

Assume that:

(i)

$G^{D}(C_{b}(D))\subset C_{0}(D)$;

(ii)

There exists a positive right eigenfunction for $G^{D};$

(iii)

$(P_{t}^{D})$ is irreducible.

Then $(P_{t}^{D})$ has a unique QSD.

###### Remark 2.14

Conditions $(i)$ and $(ii)$ of Theorem 2.13 are implied by the assumptions of
Corollary 2.12.

Proof:  Let $h$ be a positive right eigenfunction and $\mu$ a QSD with rate
$\lambda.$ Let $Q$ and $\pi$ respectively denote the Markov kernel and the
probability on $D$ defined by

$Q(f)=\lambda\frac{G^{D}(fh)}{h},$

and

$\pi(f)=\frac{\mu(fh)}{\mu(h)}$

for all $f\in B(D).$ Then, $\pi$ is invariant by $Q.$ The assumption that
$(P_{t}^{D})$ is irreducible makes $Q$ irreducible, in the sense that
$\xi(A)>0\Rightarrow Q(x,A)>0$ for all $x\in D$ and $A$ Borel. Therefore, by a
standard result (see e.g [17] or [37]), $\pi$ is the unique invariant
probability of $Q.$ Assume now that $\nu$ is another QSD with rate $\alpha.$
Then $\nu(G^{D}h)=\frac{1}{\alpha}\nu(h)=\frac{1}{\lambda}\nu(h).$ Since $h$
is positive on $D$, we deduce that $\alpha=\lambda.$ It follows that the
probability $\pi^{\prime}$ defined like $\pi$ with $\nu$ in place of $\mu$ is
invariant by $Q.$ By uniqueness, $\pi=\pi^{\prime}$ and consequently
$\mu=\nu.$ $\Box$
A sufficient (and often more tractable than the definition) condition ensuring
irreducibility is given by the next lemma.

###### Lemma 2.15

Suppose that there exists an open set $U\subset M,$ $D$-accessible from all
$x\in D,$ and a non trivial measure $\xi$ such that for all $x\in U$
$G^{D}(x,\cdot)\geq\xi(\cdot).$ Then $(P_{t}^{D})$ is irreducible.

Proof:  For all $x\in D,$ there exists, by $D$-accessibility, $t\geq 0$ such
that $P_{t}^{D}(x,U)>0.$ For every Borel set $A\subset M,$

$G^{D}(x,A)\geq\int_{0}^{\infty}P_{t+s}^{D}(x,A)ds=\int_{0}^{\infty}\int_{M}P_{t}^{D}(x,dy)P_{s}^{D}(y,A)ds$
$\geq\int_{0}^{\infty}\int_{U}P_{t}^{D}(x,dy)P_{s}^{D}(y,A)ds\geq
P_{t}(x,U)\xi(A).$

$\Box$
If the local minorization $G^{D}(x,\cdot)\geq\xi(\cdot)$ appearing in Lemma
2.15 can be improved to local minorization involving $(P_{t}^{D}),$ we also
get the exponential convergence of the conditional semigroup toward $\mu.$
More precisely,

###### Theorem 2.16

Suppose that:

(i)

$G^{D}(C_{b}(D))\subset C_{0}(D)$;

(ii)

There exists a positive right eigenfunction $h\in C_{0}(D)$ for $G^{D}$;

(iii)

There exist an open set $U\subset M,$ $D$-accessible from all $x\in D,$ a non
trivial measure $\xi$ on $D,$ and $T>0$ such that for all $x\in U,$

$P_{T}^{D}(x,\cdot)\geq\xi(\cdot).$

Then there exist $C,\alpha>0$ such that, for all $\rho\in\mathcal{M}_{1}(D)$,

$\left\|\frac{\rho P_{t}^{D}}{\rho
P_{t}^{D}{\mathbbm{1}}_{D}}-\mu(\cdot)\right\|_{TV}\leq\frac{C}{\rho(h)}e^{-\alpha
t}$

where $\|\cdot\|_{TV}$ stands for the total variation distance.

Proof:  Let $h$ be a positive right eigenfunction and $\mu$ a QSD with rate
$\lambda.$ For all $f\in B(D)$ and $t\geq 0,$ let

$Q_{t}(f)=e^{\lambda t}\frac{P_{t}^{D}(fh)}{h}$

and

$\pi(f)=\frac{\mu(fh)}{\mu(h)}$

It is readily seen that $(Q_{t})$ is a Markov semigroup (usually called the
$Q$–process induced by $\mu$) having $\pi$ as invariant probability. In order
to prove the theorem we will show that:

Step 1 There exists a probability $\nu$ on $D$ and $T_{0}>0$ such that for
every compact set $\tilde{K}\subset D$ there is some integer $n$ and some
constant $c$ (both depending on $\tilde{K}$) such that

$Q_{nT_{0}}(x,\cdot)\geq c\nu(\cdot)$

for all $x\in\tilde{K}.$

Step 2 The function $V=1/h$ is a continuous and proper Lyapunov function for
$Q_{T_{0}}$, that is

$\lim_{x\rightarrow\partial D}V(x)=\infty,$

and

$Q_{T_{0}}V\leq\rho V+C$ (7)

for some $0\leq\rho<1$ and $C\geq 0.$

From step 2 we deduce that for all $n\geq 0$

$Q_{nT_{0}}V\leq\rho^{n}V+\frac{C}{1-\rho}.$ (8)

Choose $R>\frac{2C}{1-\rho}$ and set $\tilde{K}=\\{x\in D:\>V(x)\leq R\\}.$
Then $\tilde{K}$ is a compact subset of $D$ and, by step 1, there is some
$n\geq 1$ such that

$Q_{nT_{0}}(x,\cdot)\geq c\nu(\cdot)$ (9)

on $\tilde{K}.$ Now, relying on a version of Harris’s theorem proved by Hairer
and Mattingly in [22], (8) and (9) imply that for all $f:D\mapsto{\mathbb{R}}$
measurable, and $k\geq 0,$

$|Q_{knT_{0}}(f)(x)-\pi(f)|\leq C\gamma^{k}(1+V(x))\|f\|_{V}$

for all $x\in D,$ where $0\leq\gamma<1$ and $\|f\|_{V}=\sup_{x\in
D}\frac{|f(x)|}{1+V(x)}.$ Recalling that $h$ is bounded (it lies in
$C_{0}(D)$) and $V=1/h$, one has $\inf_{x\in D}V(x)>0.$ This entails the
existence of a constant $C>0$ such that, for all measurable function $f$ such
that $\|f\|_{V}\leq 1$, for all $k\geq 0$, for all $x\in D$,

$|Q_{knT_{0}}(f)(x)-\pi(f)|\leq CV(x)\gamma^{k}.$

Integrating this last inequality over $\rho(dx)$, for any probability measure
$\rho\in\mathcal{M}_{1}(D)$,

$|\rho Q_{knT_{0}}(f)-\pi(f)|\leq C\rho(V)\gamma^{k}.$

Hence, by the property of semigroup for $(Q_{t})$, for any
$k\in\mathbb{Z}_{+}$ and $s\in[0,n{T_{0}}]$, for all
$\rho\in\mathcal{M}_{1}(D)$,

$\displaystyle|\rho Q_{s+kn{T_{0}}}(f)-\pi(f)|$ $\displaystyle\leq C\rho
Q_{s}(V)\gamma^{k}.$

Now, for all $s\in[0,n{T_{0}}]$ and $\rho\in\mathcal{M}_{1}(D)$,

$\rho Q_{s}(V)=\rho Q_{s}(1/h)=\int_{D}\rho(dx)\frac{e^{\lambda
s}P_{s}^{D}(\mathbbm{1}_{D})(x)}{h(x)}\leq e^{\lambda n{T_{0}}}\rho(1/h).$

Thus, setting $\alpha=-\ln\gamma$, we deduce that, for any $t\geq 0$ and $f$
such that $\|f\|_{1/h}\leq 1$,

$|\rho Q_{t}f-\pi(f)|\leq Ce^{\lambda n{T_{0}}}\rho(1/h)\gamma^{-1}e^{-\alpha
t}.$

In particular, by definition of $\pi$, for any $f$ such that
$\|f\|_{\infty}\leq 1$ and $\rho\in\mathcal{M}_{1}(D)$ such that
$\rho(1/h)<+\infty$, up to a change in the constant $C>0$,

$\left|\rho Q_{t}\left[\frac{f}{h}\right]-\frac{\mu(f)}{\mu(h)}\right|\leq
C\rho(1/h)e^{-\alpha t}.$

For any $\rho\in\mathcal{M}_{1}(D)$, denote
$h\circ\rho(dx):=\frac{h(x)\rho(dx)}{\rho(h)}\in\mathcal{M}_{1}(D)$. Remark
then that, for all $\rho\in\mathcal{M}_{1}(D)$,
$h\circ\rho(\frac{1}{h})=\frac{1}{\rho(h)}<+\infty$. Then the last inequality
(applied to $h\circ\rho$ instead of $\rho$) entails that, for any $f$ such
that $\|f\|_{\infty}\leq 1$ and $\rho\in\mathcal{M}_{1}(D)$, for all $t\geq
0$,

$\frac{\mu(f)}{\mu(h)}-\frac{C}{\rho(h)}e^{-\alpha
t}\leq(h\circ\rho)Q_{t}\left[\frac{f}{h}\right]\leq\frac{\mu(f)}{\mu(h)}+\frac{C}{\rho(h)}e^{-\alpha
t}.$ (10)

Moreover, using that $P_{t}^{D}f(x)=e^{-\lambda t}h(x)Q_{t}[f/h](x)$,

$\frac{\rho P_{t}^{D}f}{\rho
P_{t}^{D}1}=\frac{(h\circ\rho)Q_{t}[f/h]}{(h\circ\rho)Q_{t}[\mathbbm{1}_{D}/h]}.$

Thus, fixing $\rho\in\mathcal{M}_{1}(D)$ and denoting
$t_{\rho}:=\frac{1}{\alpha}\log\left(\frac{C\mu(h)}{\rho(h)}\right)$, this
equality and (10) entail that, for any $t>t_{\rho}$,

$\frac{\frac{\mu(f)}{\mu(h)}-\frac{C}{\rho(h)}e^{-\alpha
t}}{\frac{1}{\mu(h)}+\frac{C}{\rho(h)}e^{-\alpha t}}\leq\frac{\rho
P_{t}^{D}f}{\rho
P_{t}^{D}1}\leq\frac{\frac{\mu(f)}{\mu(h)}+\frac{C}{\rho(h)}e^{-\alpha
t}}{\frac{1}{\mu(h)}-\frac{C}{\rho(h)}e^{-\alpha t}}.$ (11)

Therefore, since $|\mu(f)|\leq\|f\|_{\infty}\leq 1$, for all $t\geq
1+t_{\rho}$,

$\mu(f)-2\frac{C\mu(h)}{\rho(h)}e^{-\alpha t}\leq\frac{\rho P_{t}^{D}f}{\rho
P_{t}^{D}1}\leq\mu(f)+2\frac{\frac{C\mu(h)}{\rho(h)}e^{-\alpha
t}}{1-\frac{C\mu(h)}{\rho(h)}e^{-\alpha t}}\\\
\leq\mu(f)+2\frac{\frac{C\mu(h)}{\rho(h)}e^{-\alpha t}}{1-e^{-\alpha}}.$

Hence, we have proved that, for all $t>1+t_{\rho}$,

$\left\|\frac{\rho P_{t}^{D}}{\rho
P_{t}^{D}1}-\mu\right\|_{TV}\leq\frac{2C}{1-e^{-\alpha}}\frac{\mu(h)}{\rho(h)}e^{-\alpha
t}.$

If $t\leq 1+t_{\rho}$,

$\left\|\frac{\rho P_{t}^{D}}{\rho P_{t}^{D}1}-\mu\right\|_{TV}\leq 2\leq
2e^{\alpha(1+t_{\rho})}e^{-\alpha
t}=\frac{2e^{\alpha}C\mu(h)}{\rho(h)}e^{-\alpha t}.$

To sum up, there exists two constants $C,\alpha>0$ such that, for all
$\rho\in\mathcal{M}_{1}(D)$ and $t\geq 0$,

$\left\|\frac{\rho P_{t}^{D}}{\rho
P_{t}^{D}1}-\mu\right\|_{TV}\leq\frac{C\mu(h)}{\rho(h)}e^{-\alpha t}.$

Hence, it remains to prove steps 1 and 2 to conclude the proof.

We start with a preliminary step, ensuring that there exist a non trivial
measure $\zeta$ with $\zeta(U)>0,$ and positive numbers $T_{0}>\varepsilon>0$
such that for all $x\in U,T_{0}-\varepsilon\leq t\leq T_{0},$

$\displaystyle P_{t}^{D}(x,\cdot)\geq\zeta(\cdot).$ (12)

Preliminary step: Since $U$ is $D$-accessible from all $x\in D$, we deduce
that

$\xi G^{D}{\mathbbm{1}}_{U}(x)=\int_{0}^{\infty}\xi
P^{D}_{t}\mathbbm{1}_{U}(x)\,dt>0.$

Hence there exists $t_{\xi}>0$ such that $\xi
P^{D}_{t_{\xi}}\mathbbm{1}_{U}>0$. Setting $\zeta^{\prime}=\xi
P^{D}_{t_{\xi}}$ and $T^{\prime}=T+t_{\xi}$, we then obtain
$\zeta^{\prime}(U)>0$ and, for all $x\in U$,

$P^{D}_{T^{\prime}}(x,\cdot)=\delta_{x}P^{D}_{T}P^{D}_{t_{\xi}}\geq\xi
P^{D}_{t_{\xi}}=\zeta^{\prime}.$

Now, by Fatou’s Lemma and right continuity of paths (like in the proof of
Lemma 2.8), $\liminf_{t\to
0}\zeta^{\prime}P^{D}_{t}(U)\geq\zeta^{\prime}(U)>0.$ Then, there exist
$\varepsilon,\delta>0$ such that
$\zeta^{\prime}P^{D}_{t}\mathbbm{1}_{U}\geq\delta$ for all
$t\in[0,\varepsilon]$. Hence, for all $x\in U$ and all $t\in[0,\varepsilon]$,

$P^{D}_{2T^{\prime}+t}(x,\cdot)\geq\zeta^{\prime}P^{D}_{T^{\prime}+t}\geq\zeta^{\prime}P^{D}_{t}(\mathbbm{1}_{U}P^{D}_{T^{\prime}}(\cdot))\geq\delta\zeta^{\prime}.$

Setting $T_{0}=2T^{\prime}+\varepsilon$ and $\zeta=\delta\zeta^{\prime}$, we
conclude that (12) holds true.

Proof of Step 1: For $\delta>0$ let $O_{\delta}=\\{x\in
D:\>G^{D}(x,U)>\delta\\}.$ Using a similar argument as for Proposition 2.3, it
is not hard to see that $O_{\delta}$ is open in $D.$ By (iii), the family
$(O_{\delta})_{\delta>0}$ covers $D.$ Thus, for every compact set
$\tilde{K}\subset D$ there exists $\delta>0$ such that for all
$x\in\tilde{K},G^{D}(x,U)\geq\delta.$ Now, relying on Proposition 2.3, one can
choose $S>0$ large enough so that, for all $x\in\tilde{K},$

$\int_{0}^{S}P_{t}^{D}(x,U)dt>\frac{\delta}{2}.$

Consequently, for all $x\in\tilde{K}$ there is some $0\leq t_{x}\leq S$ such
that

$P_{t_{x}}^{D}(x,U)\geq\frac{\delta}{2S}=:\delta^{\prime}.$

On the other hand, $h$ is bounded below by a positive constant on $U,$ because
for all $x\in U$ $h(x)=e^{\lambda T_{0}}P_{T_{0}}^{D}h(x)\geq e^{\lambda
T_{0}}\zeta(h)>0.$ Hence,

$Q_{t_{x}}(x,U)\geq\delta^{\prime\prime}$

for some $\delta^{\prime\prime}>0$ and all $x\in\tilde{K}.$ By (12), there
exists $c^{\prime}>0$ such that

$Q_{t}(x,\cdot)\geq c^{\prime}\zeta^{\prime}(\cdot)$

for all $x\in U$ and $T_{0}-\varepsilon\leq t\leq T_{0}$, where
$\zeta^{\prime}(f)=\zeta(fh)$. Choose now $n$ sufficiently large so that
$\frac{S}{n}<\varepsilon.$ Then for all $x\in\tilde{K},nT_{0}=t_{x}+n\tau_{x}$
for some $\tau_{x}\in[T_{0}-\varepsilon,T_{0}].$ Thus

$Q_{nT_{0}}(x,\cdot)\geq\int_{U}Q_{t_{x}}(x,dy)Q_{n\tau_{x}}(y,\cdot)\geq\delta^{\prime\prime}(c^{\prime}\zeta^{\prime}(U))^{n-1}c^{\prime}\zeta^{\prime}(\cdot).$

Proof of Step 2: By Markov inequality,

$\mathbb{P}_{x}(\tau_{D}^{out}>T_{0})\leq\frac{\mathbb{E}_{x}(\tau^{out}_{D})}{T_{0}}=\frac{G^{D}(x,D)}{T_{0}}.$

Let $\theta>0$ be such that $\rho=e^{\lambda T_{0}}\theta<1.$ By the
assumption that $G^{D}(C_{b}(D))\subset C_{0}(D),$ there exists $\eta>0$ and
$C^{\prime}\geq 0$ such that

$\mathbb{P}_{x}(\tau_{D}^{out}>T_{0})\leq\theta+C^{\prime}{\mathbbm{1}}_{\\{x\in
D:\>d(x,\partial D)\geq\eta\\}}.$

Then

$Q_{T_{0}}(V)(x)=e^{\lambda
T_{0}}\frac{\mathbb{P}_{x}(\tau_{D}^{out}>T_{0})}{h(x)}\leq\rho
V(x)+\frac{C^{\prime}e^{\lambda T_{0}}}{\inf_{y\in D,\,d(y,\partial
D)\geq\eta}h(y)}.$

$\Box$

##  3 Application to SDEs

The main purpose of this section is to prove Theorems 1.5 and to provide
criteria allowing to check their assumptions. We assume throughout that
$M={\mathbb{R}}^{n}$ and $D\subset M$ is an open connected set with compact
closure $K.$ We let $(P_{t}),(P_{t}^{D})$ and $G^{D},$ respectively denote,
the Markov semigroup, sub-Markov semigroup and Green kernel, induced by the
SDE (1).

###  3.1 Accessibility

This section discusses the accessibility properties and provides simple
criteria ensuring that Hypothesis (H1) (accessibility of $M\setminus K$ from
$K$) and condition (ii) of Theorem 1.5 (accessibility of $D_{\varepsilon}$
from $D$) are satisfied.

Recall that $y(u,x,\cdot)$ denotes the maximal solution to the control system
(2) starting from $x$ (i.e $y(u,x,0)=x$).

The following proposition easily follows from the celebrated Stroock and
Varadhan’s support theorem [46] (see also Theorem 8.1, Chapter VI in [26]). It
justifies the terminology used in Definition 1.1 of Section 1.

###### Proposition 3.1

Assume that the $S^{j},j\geq 1,$ are bounded with bounded first and second
derivatives and $S^{0}$ is Lipschitz and bounded. Let $x\in M$ and $U\subset
M$ open. Then $U$ is accessible (respectively $D$-accessible) by $(P_{t})$
from $x$ (see Definitions 2.2 and 2.7) if it is accessible (respectively
$D$-accessible) by $\\{S^{0},(S^{j})\\}$ from $x$ (see Definition 1.1).

###### Remark 3.2

The boundedness assumption is free of charge here, since - by compactness of
$K$ \- we can always modify the $S^{j}$ outside $K$ so that they have compact
support.

As an illustration of this latter proposition, we provide an elementary proof
of the following result, originally due to Pinsky [39].

###### Proposition 3.3

Suppose there exists $\tilde{x}\in M\setminus K$ and $\delta>0$ such that for
all $x\in D$

$\sum_{j=1}^{m}\langle
S^{j}(x),x-\tilde{x}\rangle^{2}\geq\delta\|x-\tilde{x}\|^{2}.$ (13)

Then $M\setminus K$ is accessible by $\\{S^{0},(S^{j})\\}$ (hence by
$(P_{t})$) from all $x\in K.$

Proof:  Note that by continuity and compactness (of
$K\times\\{x\in{\mathbb{R}}^{n}:\>\|x\|=1\\}$), one can always assume that
(13) holds true on some larger open bounded domain $D^{\prime}$ with $K\subset
D^{\prime}.$ For $x\in D$ and $j\in\\{1,\ldots,m\\}$ set

$u^{j}(x)=-\frac{1}{2\delta\varepsilon}\langle S^{j}(x),x-\tilde{x}\rangle$

where $\varepsilon$ will be chosen later. Consider the ODE

$\dot{x}=S^{0}(x)+\sum_{j}u^{j}(x)S^{j}(x)$

and set $v(t)=\|x(t)-\tilde{x}\|^{2}.$ Then, as long as $x(t)\in D^{\prime},$

$\frac{dv(t)}{dt}\leq-\frac{1}{\varepsilon}v(t)+a$

where $a=\sup_{x\in D^{\prime}}2(\langle S^{0}(x),x-\tilde{x}\rangle).$ Thus

$v(t)=\|x(t)-\tilde{x}\|^{2}\leq
e^{-t/\varepsilon}(v_{0}-a\varepsilon)+a\varepsilon.$

One can choose $\varepsilon$ small enough so that $x(t)$ meets
$D^{\prime}\setminus K.$ $\Box$
The next result (Proposition 3.8) provides a natural condition ensuring that
condition $(ii)$ of Theorem 1.5 holds.

Suppose that $D$ satisfies the exterior sphere condition as defined in
Definition 1.3. Let ${\cal N}_{p}$ denote the set of unit outward normal
vectors at $p\in\partial D$ and let

${\cal N}=\\{(p,v):\>p\in\partial D,v\in{\cal N}_{p}\\}.$

###### Definition 3.4

We say that a vector $w\in{\mathbb{R}}^{n}$ points inward $D$ at $p\in\partial
D$ if

$\langle w,v\rangle\leq 0\mbox{ for all }v\in{\cal N}_{p},\mbox{ \bf and
}\langle w,v\rangle<0\mbox{ for at least one }v\in{\cal N}_{p}.$

We say that it points strictly inward $D$ at $p$ if $\langle w,v\rangle<0$ for
all $v\in{\cal N}_{p}.$ We say that a vector field $F$ points inward
(respectively strictly inward) $D$ if $F(p)$ points inward (strictly inward)
$D$ at $p$ for all $p\in\partial D.$

###### Lemma 3.5

Suppose that $D$ satisfies the exterior sphere condition. Let $F$ be a
Lipschitz vector field pointing inward $D$ and let $\Phi=\\{\Phi_{t}\\}$ be
its flow. Then

(i)

$\Phi_{t}(K)\subset D$ for all $t>0;$

(ii)

There exists a compact set $A\subset D$ invariant under $\\{\Phi_{t}\\}$ (i.e
$\Phi_{t}(A)=A$ for all $t\in{\mathbb{R}}$) such that for all $x\in D$
$\omega_{\Phi}(x)\subset A,$ where $\omega_{\Phi}(x)$ stands for the omega
limit set of $x$ for $\Phi.$

Proof:  We first show that $\Phi_{t}(K)\subset K$ for all $t\geq 0.$ Suppose
not. Then, for some $p\in\partial D$ and $\varepsilon>0,$ $d(\Phi_{t}(p),K)>0$
on $]0,\varepsilon].$ The function

$t\rightarrow V(t):=d(\Phi_{t}(p),K),$

being Lipschitz on $[0,\varepsilon]$ it is absolutely continuous, hence almost
everywhere derivable and $V(t)=\int_{0}^{t}\dot{V}(u)du.$ Let
$0<t_{0}\leq\varepsilon$ be a point at which it is derivable,
$x_{0}=\Phi_{t_{0}}(p)$ and $p_{0}\in\partial D$ be such that
$\|x_{0}-p_{0}\|=d(x_{0},K).$ In particular,
$\frac{x_{0}-p_{0}}{\|x_{0}-p_{0}\|}\in\mathcal{N}_{p_{0}}$.Then for all
$t>0,$

$\frac{V(t_{0}+t)-V(t_{0})}{t}=\frac{d(\Phi_{t}(x_{0}),K)-d(x_{0},K)}{t}\leq\frac{\|\Phi_{t}(x_{0})-p_{0}\|-\|x_{0}-p_{0}\|}{t}.$

Letting $t\rightarrow 0,$ using that $F(p_{0})$ points inward $D$ at $p_{0}$,
we get

$\displaystyle\dot{V}(t_{0})\leq\frac{\langle
F(x_{0}),x_{0}-p_{0}\rangle}{\|x_{0}-p_{0}\|}$ $\displaystyle\leq\frac{\langle
F(x_{0})-F(p_{0}),x_{0}-p_{0}\rangle}{\|x_{0}-p_{0}\|}$ $\displaystyle\leq
L\|x_{0}-p_{0}\|=LV(t_{0}),$

where $L$ is a Lipschitz constant for $F$. By Gronwall’s lemma we then get
that for all $0<s<t\leq\varepsilon,$ $V(t)\leq e^{L(t-s)}V(s).$ Since
$V(0)=0,$ $V$ cannot be positive. This proves the desired result. Note that
this first result doesn’t require that the exterior sphere condition holds
everywhere but only that, for all $p\in\partial D$ at which it holds, $\langle
F(p),v\rangle\leq 0$ for all $v\in{\cal N}_{p}$ (compare with Theorem 2.3 in
[7]).

We now show that $\Phi_{t}(K)\subset D$ for $t>0.$ This amounts to show that
for all $p\in\partial D$ and all $\varepsilon>0$ small enough,
$\Phi_{-\varepsilon}(p)\in M\setminus K.$ So let $p\in\partial D$ and choose
$v\in{\cal N}_{p}$ such that $\delta:=-\langle F(p),v\rangle>0.$ By assumption
$d(p+rv,K)=r$ for some $r>0.$ Thus for all $\varepsilon>0$ such that
$\varepsilon\|F(p)\|^{2}<r\delta,$

$\|(p-\varepsilon F(p))-(p+rv)\|^{2}<r^{2}-\varepsilon\delta r.$

Since $\|\Phi_{-\varepsilon}(p)-(p-\varepsilon F(p)\|=o(\varepsilon),$ this
shows that $\|\phi_{-\varepsilon}(p)-(p+rv)\|<r$ for $\varepsilon>0$ small
enough, so that $\Phi_{-\varepsilon}(p)\not\in K$.

Assertion $(ii)$ is a consequence of $(i).$ It suffices to set

$A=\cap_{t\geq 0}\Phi_{t}(K).$

$\Box$
For $(p,v)\in{\cal N},$ set $R(p,v)=\sup\\{r>0\>:d(p+rv,K)=r\\}\in(0,\infty],$
and let

$R_{\partial D}=\inf_{(p,v)\in{\cal N}}R(p,v).$

###### Definition 3.6

We say that $D$ satisfies the strong exterior sphere condition if it satisfies
the exterior sphere condition and $R_{\partial D}\neq 0.$

###### Remark 3.7

If $D$ is convex or $\partial D$ is $C^{2},$ then $D$ satisfies the strong
exterior sphere condition. However, the following example shows that the
exterior sphere condition and the strong exterior sphere condition are not
equivalent: For $1\leq\alpha\leq 2,$ let $D^{\alpha}\subset{\mathbb{R}}^{2}$
be defined as

$D^{\alpha}=\\{(x,y)\in{\mathbb{R}}^{2}\>:0<x<1,|y|<x^{\alpha}\\}.$

Then $D^{\alpha}$ satisfies the exterior sphere condition for all
$1\leq\alpha\leq 2$ but not the strong one for $1<\alpha<2.$ Indeed, for such
an $\alpha,$ ${\cal N}_{(0,0)}=\\{v\>:\|v\|=1,v_{1}<0\\}$ and

$\lim_{v\rightarrow(0,1),v\in{\cal N}_{(0,0)}}R((0,0),v)=0.$

###### Proposition 3.8

Assume that $D$ satisfies the strong exterior sphere condition. Recall that
${\cal N}_{p}$ denotes the set of unit outward normal vectors at $p\in\partial
D.$

(i)

For each $p\in\partial D,$ the two following conditions are equivalent:

(a)

${\cal N}_{p}\cap-{\cal N}_{p}=\emptyset$ and for each $v\in{\cal N}_{p}$

$\langle S^{0}(p),v\rangle<0\mbox{ or }\sum_{i=1}^{m}\langle
S^{i}(p),v\rangle^{2}\neq 0;$

(b)

There exists a vector $w\in\mathsf{Span}\\{S^{1}(p),\ldots,S^{m}(p)\\}$ such
that $S^{0}(p)+w$ points strictly inward $D$ at $p.$

(ii)

If for all $p\in\partial D$ condition $(i)-(a)$ (or $(i)-(b)$) holds, then

$D_{\varepsilon}=\\{x\in D:\>d(x,\partial D)>\varepsilon\\}$

is $D$-accessible by $\\{S^{0},(S^{j})\\}$ from all $x\in D,$ for some
$\varepsilon>0.$

Proof:  We first Point $(ii).$ Assume that $(i)-(b)$ holds at every
$p\in\partial D$ and prove that $D_{\varepsilon}$ is accessible for some
$\varepsilon>0.$ The assumption $R_{\partial D}>0$ makes ${\cal N}$ closed
(hence compact). Indeed, if $(p_{n},v_{n})\rightarrow(p,v)$ with
$(p_{n},v_{n})\in{\cal N},$ then, for any $0<r<R_{\partial D}$,
$d(p_{n}+rv_{n},K)=r.$ Thus $d(p+rv,K)=r.$

This has the consequence that, if a continuous vector field $F$ points
strictly inward $D$ at $p\in\partial D,$ it points strictly inward $D$ at
$q\in\partial D$ for all $q$ in a neighborhood of $p.$ Therefore, by
compactness, there exists a covering of $\partial D$ by open sets
$U_{1},\ldots,U_{k},$ and vector fields
$W_{1},\ldots,W_{k}\in\mathsf{Span}\\{S^{1},\ldots,S^{m}\\}$ such that for all
$p\in\partial D\cap U_{i}$ $F_{i}(p):=S^{0}(p)+W_{i}(p)$ points strictly
inward $D$ at $p.$ Set $U_{0}=M\setminus\partial D$ and let
$\\{\rho_{i}\\}_{i=0,\ldots,k}$ be a partition of unity subordinate to
$\\{U_{i}\\}_{i=0,\ldots,k}.$ That is $\rho_{i}\in C^{\infty}(M),\rho_{i}\geq
0,\sum_{i=0}^{k}\rho_{i}=1,$ and $supp(\rho_{i})\subset U_{i}.$ Define
$F=\rho_{0}S^{0}+\sum_{i=1}^{k}\rho_{i}F_{i}.$ Then, $F$ points strictly
inward $D$ and writes

$F=S^{0}+\sum_{i=1}^{m}u^{i}S^{i}$ (14)

with $u^{i}\in C^{\infty}(M).$ In view of (14), Lemma 3.5 proves the result.

Point $(i).$ We now prove that conditions $(i)-(a)$ and $(i)-(b)$ are
equivalent. The implication $(i)-(b)\Rightarrow(i)-(a)$ is straightforward. We
focus on the converse implication. Let

$Cone({\cal N}_{p})=\\{tv,t\geq 0,v\in{\cal N}_{p}\\}$

and $conv({\cal N}_{p})$ be the convex hull of ${\cal N}_{p}.$ We claim that

$conv({\cal N}_{p})\subset Cone({\cal N}_{p})$

and $0\not\in conv({\cal N}_{p}).$ To prove the first inclusion, it suffices
to show that $Cone({\cal N}_{p})$ is convex. To shorten notation, assume
(without loss of generality) that $p=0.$ Let $x,y\in Cone({\cal N}_{0})$ and
$0\leq t\leq 1.$ By definition of ${\cal N}_{0},$
$Cone(\mathcal{N}_{0})=\\{z\in\mathbb{R}^{n},\exists r>0\text{ s.t.\
}d(rz,K)=\|rz\|\\}$, so there exists $r>0$ such that $d(rx,K)=\|rx\|$ and
$d(ry,K)=\|ry\|.$ Thus for all $z\in K$

$\|rx-z\|^{2}-\|rx\|^{2}=\|z\|^{2}-2\langle rx,z\rangle\geq 0.$

Similarly $\|z\|^{2}-2\langle ry,z\rangle\geq 0.$ Thus

$\|r(tx+(1-t)y)-z\|^{2}-\|r(tx+(1-t)y)\|^{2}$ $=t(\|z\|^{2}-2\langle
rx,z\rangle)+(1-t)(\|z\|^{2}-2\langle ry,z\rangle)\geq 0.$

This proves that $tx+(1-t)y\in Cone({\cal N}_{0}),$ hence convexity of
$Cone({\cal N}_{0}).$

The fact that $0\not\in conv({\cal N}_{p})$ follows from the fact that ${\cal
N}_{p}\cap-{\cal N}_{p}=\emptyset.$ Indeed, suppose to the contrary that
$0=\sum_{i=1}^{k}t_{i}x_{i}$ with $k\geq 2,x_{i}\in{\cal N}_{p},t_{i}>0$ and
$\sum_{i=1}^{k}t_{i}=1.$ Then

$-\frac{t_{1}}{1-t_{1}}x_{1}\in conv(x_{2},\ldots,x_{k})\subset conv({\cal
N}_{p})\subset cone({\cal N}_{p}).$

Thus $-x_{1}\in{\cal N}_{p}.$ A contradiction.

We shall now deduce the implication $(i)-(a)\Rightarrow(i)-(b)$ from the
Minimax theorem (see e.g [44]). For all $j\in\\{1,\ldots,m\\}$ set
$S^{-j}=-S^{j}.$ Let $J=\\{-m,\ldots,0,\ldots m\\}$ and

$\Delta(J)=\\{\alpha\in{\mathbb{R}}^{J}:\>\alpha_{j}\geq 0,\sum_{j\in
J}\alpha_{j}=1\\}.$

By condition $(i)-(a)$ and compactness of ${\cal N}_{p}$ there exists
$\delta>0$ such that for all $v\in{\cal N}_{p}$

$\min_{j\in J}\langle S^{j}(p),v\rangle\leq-\delta.$

Thus for all $v\in Cone({\cal N}_{p})$

$\min_{j\in J}\langle S^{j}(p),v\rangle\leq-\delta\|v\|.$

The set ${\cal N}_{p}$ being compact (in finite dimension) its convex hull is
also compact by Carathéodory theorem. Thus, because $0\not\in conv({\cal
N}_{p}),$ $\|v\|\geq\frac{\delta^{\prime}}{\delta}$ for some
$\delta^{\prime}>0$ and all $v\in conv({\cal N}_{p}).$ It then follows that

$\sup_{v\in conv({\cal N}_{p})}\inf_{\alpha\in\Delta(J)}\langle\sum_{j\in
J}\alpha_{j}S^{j}(p),v\rangle\leq\sup_{v\in conv({\cal N}_{p})}\min_{j\in
J}\langle S^{j}(p),v\rangle\leq-\delta^{\prime}.$

By the Minimax theorem, the left hand side also equals

$\inf_{\alpha\in\Delta(J)}\sup_{v\in conv({\cal N}_{p})}\langle\sum_{j\in
J}\alpha_{j}S^{j}(p),v\rangle$

and this infimum is achieved for some $\beta\in\Delta(J).$ If $\beta_{0}\neq
0$ this implies that

$\sup_{v\in conv({\cal N}_{p})}\langle S^{0}(p)+\sum_{j\in J,j\neq
0}\frac{\beta_{j}}{\beta_{0}}S^{j}(p),v\rangle\leq-\frac{\delta^{\prime}}{\beta_{0}}<0.$

If $\beta_{0}=0,$ for $R>0$ sufficiently large

$\sup_{v\in conv({\cal N}_{p})}\langle S^{0}(p)+R\sum_{j\in
J}\beta_{j}S^{j}(p),v\rangle\leq-R\delta^{\prime}+\|S^{0}(p)\|<0.$

This concludes the proof. $\Box$

###### Remark 3.9

It follows from Proposition 3.8 that whenever $\partial D$ is $C^{2},$
Hypothesis (H2’) implies condition $(ii)$ of Theorem 1.5 because at each point
$p\in\partial D$ there is a unique outward unit normal. The following example
shows that this is not true in general. Let $D^{1}$ be as in Remark 3.7, with
$\alpha=1,$ and let $(X_{t})$ be solution to

$dX_{t}=S^{1}(X_{t})\circ dB_{t}$

where $S^{1}(x,y)=(1,2).$ At each point $p\in\partial D$ there is at least one
$v\in{\cal N}_{p}$ such that $\langle S^{1}(p),v\rangle\neq 0$ so that
condition $(ii)-(a)$ is satisfied. However, for
$0<\eta<\varepsilon,D^{1}_{\varepsilon}$ is not $D$-accessible from
$(\eta,0).$

Observe also that none of the conditions required in Proposition 3.8 is
necessary for $D_{\varepsilon}$ to be accessible. Let $D^{\alpha}$ be as in
Remark 3.7, with $1<\alpha\leq 2$ and let

$dX_{t}=e_{1}\circ dB_{t}^{1}+e_{2}\circ dB_{t}^{2},$

with $(e_{1},e_{2})$ the canonical basis of ${\mathbb{R}}^{2}.$ As shown in
Corollary 1.9 (and its proof), $D_{\varepsilon}^{\alpha}$ is accessible; while
for $1<\alpha<2,R_{\partial D}=0$ and for $\alpha=2,{\cal N}_{0,0}\cap-{\cal
N}_{0,0}\neq\emptyset.$

###  3.2 Proof of Theorem 1.5

In order to prove Theorem 1.5, it suffices to show that the two assumptions of
Theorem 2.9 are satisfied. Assumption (ii) of Theorem 2.9 is implied by
Condition $(ii)$ of Theorem 1.5 and Proposition 3.1. Recall that sufficient
conditions for this property are given by Proposition 3.8. We shall show here
that the property $G^{D}(C_{b}(D))\subset C_{0}(D)$ is implied by Hypotheses
(H1) and (H2); and that Hypothesis (H2’) implies (H2).

#### Feller properties

Let $C({\mathbb{R}}_{+},M)$ be the set of continuous paths
$\eta:{\mathbb{R}}_{+}\mapsto M$ equipped with the topology of uniform
convergence on compact intervals.

Let $(\eta_{n})_{n\geq 0}$ be a sequence converging to $\eta$ in
$C({\mathbb{R}}_{+},M).$ Set

$\tau^{n,out}_{D}=\inf\\{t\geq 0:\>\eta_{n}(t)\in M\setminus
D\\},\tau^{out}_{D}=\inf\\{t\geq 0:\>\eta(t)\in M\setminus D\\}$

Define $\tau^{n,out}_{K}$ and $\tau^{out}_{K}$ similarly.

###### Lemma 3.10

(i)

Suppose $\eta(0)\in D.$ Then for all $f\in C_{b}(D)$ such that $f\geq 0$ and
all $t\geq 0,$

$\liminf_{n\rightarrow\infty}f(\eta_{n}(t)){\mathbbm{1}}_{\\{\tau^{n,out}_{D}>t\\}}\geq
f(\eta(t)){\mathbbm{1}}_{\\{\tau^{out}_{D}>t\\}}.$

In particular

$\liminf_{n\rightarrow\infty}\tau^{n,out}_{D}\geq\tau^{out}_{D}.$

(ii)

Suppose $\eta(0)\in K.$ Then

$\limsup_{n\rightarrow\infty}\tau^{n,out}_{K}\leq\tau^{out}_{K}$

and for all $f\in C_{0}(K),f\geq 0$ and all $t\geq 0,$

$\limsup_{n\rightarrow\infty}f(\eta_{n}(t)){\mathbbm{1}}_{\\{\tau^{n,out}_{K}>t\\}}\leq
f(\eta(t)){\mathbbm{1}}_{\\{\tau^{out}_{K}>t\\}}.$

Proof:  $(i)$ If $\tau^{out}_{D}\leq t$ the statement is obvious. If
$\tau^{out}_{D}>t,$ then $\eta([0,t])\subset D$ so that, for $n$ large enough,
$\eta_{n}([0,t])\subset D.$ That is $\tau^{n,out}_{D}>t$ and the statement
follows. The assertion that
$\liminf_{n\rightarrow\infty}\tau^{n,out}_{D}\geq\tau^{out}_{D}$ follows by
choosing $f={\mathbbm{1}}_{D}.$

$(ii)$ Suppose to the contrary that
$\tau^{n,out}_{K}>\tau^{out}_{K}+\varepsilon$ for some $\varepsilon>0$ and
infinitely many $n.$ Then $\eta^{n}([0,\tau^{out}_{K}+\varepsilon])\subset K.$
Hence $\eta([0,\tau^{out}_{K}+\varepsilon])\subset K.$ A contradiction. The
last assertion directly follows for $t\neq\tau^{out}_{K}$ and $f\in C_{b}(K).$
If now $t=\tau^{out}_{K}$ and $f\in C_{0}(K),$ $f(\eta^{n}(t))\rightarrow
f(\eta(t))=0.$ $\Box$
The next lemma shows that under (H1) and (H2), $G^{D}(C_{b}(D))\subset
C_{0}(D).$ In this lemma, $P_{t}^{K}$ denotes the semigroup defined as
$P_{t}^{K}f(x)=\mathbb{E}_{x}(f(X_{t}){\mathbbm{1}}_{\tau_{K}^{out}>t})$ for
all $f$ bounded and measurable on $K$.

###### Lemma 3.11

Suppose that Hypothesis (H1) holds. Then,

(i)

For all $f\geq 0,f\in C_{b}(D),x\in D$ and $t\geq 0,$

$\liminf_{y\rightarrow x}P_{t}^{D}f(y)\geq P_{t}^{D}f(x),$

(ii)

For all $f\geq 0,f\in C_{0}(K),x\in K$ and $t\geq 0$

$\limsup_{y\rightarrow x,y\in K}P_{t}^{K}f(y)\leq P^{K}_{t}f(x)$

(iii)

Suppose, in addition, that Hypothesis (H2) holds. Then,

(a)

For all $x\in D,$

$\mathbb{P}_{x}(\tau_{D}^{out}=\tau_{K}^{out})=1;$

(b)

For all $f\in C_{0}(D)$ and $t\geq 0,$ $P_{t}^{D}f\in C_{b}(D);$

(c)

For all $f\in C_{b}(D)$ and $t\geq 0,$ $G^{D}f\in C_{0}(D).$

Proof:  Let $(X_{t}^{x})$ be the (strong) solution to (1) with initial
condition $X_{0}^{x}=x$. We can always assume that $(X_{t}^{x})_{t\geq
0,\,x\in M}$ is defined on the Wiener space space
$C({\mathbb{R}}_{+},{\mathbb{R}}^{m})$ equipped with its Borel sigma field and
the Wiener measure $\mathbb{P}$ (the law of $(B^{1},\ldots,B^{m})$). That is
$\mathbb{P}_{x}(\cdot)=\mathbb{P}(X^{x}\in\cdot).$ Also, for all $\omega\in
C({\mathbb{R}}_{+},{\mathbb{R}}^{m}),$ the map $x\in M\rightarrow
X^{x}(\omega)\in C({\mathbb{R}}_{+},M)$ is continuous (see for instance [31],
Theorem 8.5).

Assertions $(i)$ and $(ii)$ then follow from Lemma 3.10 and Fatou’s lemma.

We now pass to the proof of $(iii).$

$(a)$ follows from the strong Markov property, valid since $(P_{t})_{t\geq 0}$
is Feller (see e.g. [31], Theorem 6.17), as follows. For all $x\in D,$

$\mathbb{P}_{x}(\tau_{D}^{out}=\tau_{K}^{out})=\mathbb{E}_{x}(\mathbb{P}_{X_{\tau^{out}_{D}}}(\tau_{K}^{out}=0))=1.$

$(b)$ Let $x\in D.$ The property
$\mathbb{P}_{x}(\tau^{out}_{D}=\tau^{out}_{K})=1$ implies that
$P_{t}^{D}f(x)=P_{t}^{K}{\color[rgb]{0,0,1}\hat{f}}(x)$ for all $f\in
C_{b}(D),$ where $\hat{f}(x)=f(x)$ if $x\in D$ and $\hat{f}(x)=0$ if $x\in
K\setminus D.$ Hence, by $(i)$ and $(ii),$

$\lim_{y\rightarrow x}P_{t}^{D}f(y)=P_{t}^{D}f(x)$

for all $f\in C_{0}(D),f\geq 0.$ If now $f\in C_{0}(D)$ it suffices to write
$f=f^{+}-f^{-}$ with $f^{+}=\max(f,0)$ and $f^{-}=(-f)^{+}.$

$(c)$ Write $\tau^{x,out}_{D}$ for $\inf\\{t\geq 0:\>X_{t}^{x}\not\in D\\}.$
Again, the property $\mathbb{P}_{x}(\tau^{out}_{D}=\tau^{out}_{K})=1$ combined
with Lemma 3.10, imply that, almost surely, the maps $x\in
D\rightarrow\tau^{x,out}_{D},$ and $x\in
D\rightarrow\int_{0}^{\tau^{x,out}_{D}}f(X_{s}^{x})ds$ are continuous for all
$f\in C_{b}(D).$ Also,

$\sup_{x\in
D}\mathbb{E}\left[\left(\int_{0}^{\tau^{x,out}_{D}}f(X_{s}^{x})ds\right)^{2}\right]\leq\|f\|^{2}\sup_{x\in
D}\mathbb{E}_{x}[(\tau^{out}_{D})^{2}]<\infty$

where the last inequality follows from Proposition 2.3. This shows that the
family $(\int_{0}^{\tau^{x,out}_{D}}f(X_{s}^{x})ds)_{x\in D}$ is uniformly
integrable. Thus, $x\in
D\rightarrow\mathbb{E}(\int_{0}^{\tau^{x,out}_{D}}f(X_{s}^{x})ds)=G^{D}f(x)$
is continuous.

To conclude, observe that $|G^{D}f(x)|\leq\|f\|G^{D}{\mathbbm{1}}_{D}(x)$ and
that $G^{D}{\mathbbm{1}}_{D}(x)=\mathbb{E}(\tau_{D}^{x,out}).$ As
$x\rightarrow p\in\partial D$ this last quantity converges to
$\mathbb{E}(\tau_{D}^{p,out})=0.$ $\Box$
In the next result, we prove that (H2’) implies (H2).

###### Proposition 3.12

Let $p\in\partial D.$ Assume that there exist a unit outward vector $v$ at $p$
and $j\in\\{1,\ldots,m\\}$ such that $\langle v,S^{j}(p)\rangle\neq 0.$ Then,
$p$ is regular with respect to $M\setminus K$ for (1). In particular,
Hypothesis $(H2^{\prime})$ implies $(H2).$

Proof:  Without loss of generality we can assume that $j=1.$ By Definition
1.3, there is $r>0$ such that $d(p+rv,K)=r.$ Let
$\Psi:\>{\mathbb{R}}^{n}\mapsto{\mathbb{R}}$ be defined as

$\Psi(x)=r^{2}-\|x-(p+rv)\|^{2}.$

Then

$\Psi(x)>0\Rightarrow x\in M\setminus K,$

and $\langle\nabla\Psi(p),S^{1}(p)\rangle\neq 0.$ Hence, for some neighborhood
$U$ of $p,$

$(\langle\nabla\Psi(x),S^{1}(x)\rangle)^{2}\geq a>0$

on $\overline{U}.$

By Ito’s formula, since $\Psi(p)=0,$

$\Psi(X^{p}_{t\wedge\tau^{out}_{U}})=\int_{0}^{t\wedge\tau^{out}_{U}}L\Psi(X_{s}^{p})ds+M_{t\wedge\tau^{out}_{U}}$

where

$L\Psi=S^{0}(\Psi)+\frac{1}{2}\sum_{j=1}^{m}(S^{j})^{2}(\Psi),$
$M_{t}=\sum_{j=1}^{m}\int_{0}^{t}\sigma^{j}(X_{s}^{p})dB^{j}_{s},$

and $\sigma^{j}(x)$ is any bounded measurable function coinciding with
$\langle\nabla\Psi(x),S^{j}(x)\rangle$ on $\overline{U}.$ In the definition of
$L$ above, we used the standard notation for differential operators defined
from vector fields: $S^{i}(f)(x)=\langle S^{i}(x),\nabla f(x)\rangle$. For
convenience, we set $\sigma^{1}(x)=\sqrt{a}$ for $x\not\in\overline{U}.$
Therefore,

$\Psi(X^{p}_{t\wedge\tau^{out}_{U}})\geq
M_{t\wedge\tau^{out}_{U}}-b(t\wedge\tau^{out}_{U})$ (15)

with $b=\sup_{x\in\overline{U}}|L\Psi(x)|,$ and

$\langle
M\rangle_{t}=\sum_{j=1}^{m}\int_{0}^{t}\sigma^{j}(X_{s}^{p})^{2}ds\geq at.$

By Dubins-Schwarz Theorem (see e.g [31], Theorem 5.13) there exists a Brownian
motion $(\beta)$ such that $M_{t}=\beta_{\langle M\rangle_{t}}$ for all $t\geq
0.$ Thus, for all $\varepsilon>0$

$\sup_{0\leq t\leq\varepsilon}(M_{t}-bt)\geq\sup_{0\leq
t\leq\varepsilon}(\beta_{\langle M\rangle_{t}}-\frac{b}{a}\langle
M\rangle_{t})\geq\sup_{0\leq t\leq a\varepsilon}(\beta_{t}-\frac{b}{a}t).$

By the law of the iterated logarithm for the Brownian motion, the right hand
term is almost surely positive.

Now, using (15), it follows that $\tau^{out}_{K}\leq\varepsilon$ almost surely
on the event $\\{\tau^{out}_{U}>\varepsilon\\}.$ Thus
$\mathbb{P}_{p}(\tau^{out}_{K}=0)=1$ because
$\mathbb{P}_{p}(\tau^{out}_{U}>0)=1.$ $\Box$

###  3.3 Proof of Theorem 1.8

#### Consequences of Hörmander conditions

We assume throughout all this subsection, as in Theorem 1.8, that:

(i)

Hypotheses (H1) and (H2’) holds true;

(ii)

The weak Hörmander condition (see Definition 1.7) is satisfied at every $x\in
K.$

###### Lemma 3.13

The operator $G^{D}$ is a compact operator on $C_{0}(D).$

Proof:  By Lemma 3.11, (iii) (c) and Proposition 2.3, the operator $G^{D}$ is
a bounded operator on $C_{b}(D)$ whose image is in $C_{0}(D).$ It then defines
a bounded operator on $C_{0}(D).$ To prove compactness, we rely on the
following (easy) consequence of Ascoli’s theorem (see e.g [38], exercise 6,
chapter 7): A family ${\cal F}\subset C_{0}(D)$ is relatively compact if (and
only if) it is bounded, equicontinuous at every point $x\in D$ and vanishes
uniformly at $\partial D;$ That is for all $\varepsilon>0$ there exists a
compact set $W\subset D$ such that $|f(x)|\leq\varepsilon$ for all $f\in{\cal
F}$ and $x\in D\setminus W.$

Let ${\cal F}=\\{G^{D}(f):\>f\in C_{0}(D),\|f\|\leq 1\\}.$ Then ${\cal F}$ is
bounded and vanishes uniformly at $\partial D$ because for all $f$ in the unit
ball of $C_{0}(D)$ $|G^{D}(f)|\leq G^{D}{\mathbbm{1}}_{D}\in C_{0}(D).$

We now prove the equicontinuity property. Let $L$ be the differential operator
defined as $L=S^{0}+\frac{1}{2}\sum_{j=1}^{m}(S^{j})^{2}.$

We claim that for every $f\in C_{0}(D),$ the map $g=G^{D}f$ satisfies $Lg=-f$
on $D,$ in the sense of distributions. First, assume that the claim is proved.
Then, by a theorem due to Rothschild and Stein (Theorem 18 in [43]) there
exists $0<\alpha<1$ depending only on the family $S^{0},S^{1},\ldots,S^{m}$
such that $G^{D}(C_{0}(D))\subset\Lambda^{\alpha}(D).$ Here (see [43], Section
16), $\Lambda_{\alpha}(D)$ denotes the set of $f:D\mapsto{\mathbb{R}}$ such
that for every compact $W\subset D,$ $f|_{W}$ ($f$ restricted to $W$) is
$\alpha$-Hölder.

Let now $x_{0}\in D,$ $W\subset D$ be a compact neighborhood of $x_{0},$
$C(W)$ the space of continuous functions on $W$ equipped with the uniform
norm, $C^{\alpha}(W)$ the space of $\alpha$-Hölder functions on $W$ equipped
with the corresponding Hölder norm and $i_{W}:C_{0}(D)\mapsto C(W),f\mapsto
f|_{W}.$ The operator $i_{W}\circ G_{D}:C_{0}(D)\mapsto C(W)$ is bounded. Its
graph is then closed in $C_{0}(D)\times C(W)$ and consequently also in
$C_{0}(D)\times C^{\alpha}(W).$ Hence, by the closed graph theorem, it is a
bounded operator from $C_{0}(D)$ into $C^{\alpha}(W).$ In particular, for all
$f$ in the unit ball of $C_{0}(D)$ and $x,y\in W$
$|G^{D}(f)(x)-G^{D}f(y)|\leq\kappa\|x-y\|^{\alpha}$ for some $\kappa$
depending only on $W.$ This proves that ${\cal F}$ is equicontinuous at
$x_{0}.$

We now pass to the proof of the claim.

By a theorem of Bony ([7], Théorème 5.2), for any $a>0$ and $f\in C_{b}(K),$
the Dirichlet problem:

$\left\\{\begin{array}[]{l}Lg-ag=-f\mbox{ on }D\mbox{ (in the sense of
distributions)}\\\ g|_{\partial D}=0;\end{array}\right.$ (16)

has a unique solution, call it $g_{a},$ continuous on $K.$ Furthermore, if $f$
is smooth on $D$ so is $g_{a}.$ Note that the assumptions required for this
theorem are implied by condition $(ii)$ of Theorem 1.5 and the weak Hörmander
condition.

Suppose that $f$ is smooth on $D$ (so that $g_{a}$ is smooth on $D$). Then by
Ito’s formula

$\left(e^{-{at\wedge\tau^{out}_{D}}}g_{a}(X_{t\wedge\tau^{out}_{D}})+\int_{0}^{t\wedge\tau^{out}_{D}}e^{-as}f(X_{s})ds\right)_{t\geq
0}$

is a local martingale. Being bounded, it is a uniformly integrable martingale.
Thus, taking the expectation and letting $t\rightarrow\infty$, we get that

$\int_{0}^{\infty}e^{-as}P_{s}^{D}f(x)ds=\mathbb{E}_{x}\left(\int_{0}^{\tau_{D}^{out}}e^{-as}f(X_{s})ds\right)=g_{a}(x).$
(17)

In particular, $G^{D}f(x)=\lim_{a\rightarrow 0}g_{a}(x)$, where the
convergence is uniform by Proposition 2.3.

For every smooth test function $\Phi$ with compact support in $D$

$\langle g_{a},L^{*}\Phi\rangle-a\langle g_{a},\Phi\rangle=-\langle
f,\Phi\rangle$

where $\langle h,\Phi\rangle=\int h(x)\Phi(x)dx.$ Letting $a\rightarrow 0$, we
get that $\langle G^{D}f,L^{*}\Phi\rangle=-\langle f,\Phi\rangle$, that is

$LG^{D}(f)=-f,$

in the sense of distributions. This proves the claim whenever $f$ is smooth on
$D.$ If now $f$ is only continuous, let $(f_{n})$ be smooth (on a neighborhood
of $K$) with $f_{n}\rightarrow f$ uniformly on $K$. Then
$G^{D}(f_{n})\rightarrow G^{D}(f)$ uniformly; and, by the same argument as
above, $LG^{D}f=-f$ in the sense of distributions.

$\Box$
The proofs of the next two lemmas are similar to the proof of Corollary 5.4 in
[4]. For convenience we provide details.

###### Lemma 3.14

Let $p\in D$ be such that $S^{i}(p)\neq 0$ for some $i\in\\{1,\ldots,m\\}.$
Then, there exist disjoint open sets $U,V\subset D$ with $p\in U$ and a
nontrivial measure $\xi$ on $V$ (i.e. s.t. $\xi(V)>0,\xi(M\setminus V)=0$)
such that for all $x\in U,$

$G^{D}(x,\cdot)\geq\xi(\cdot).$

Proof:  We first assume that for all $x\in D$ there is some
$i\in\\{1,\ldots,m\\}$ such that $S^{i}(x)\neq 0.$ Then, by Theorem 6.1 in
[7], for all $a>0,$ there exists a map $K_{a}:K^{2}\mapsto{\mathbb{R}}_{+}$
smooth on $D^{2}\setminus\\{(x,x):\>x\in D\\}$ such that for all $f\in
C_{b}(K),$ and $x\in D$

$\int_{0}^{\infty}e^{-as}P_{s}^{D}f(x)ds=\int_{D}K_{a}(x,y)f(y)dy.$

To be more precise, Theorem 6.1 in [7] asserts that the solution to the
Dirichlet problem (16) can be written under the form given by the right hand
side of the last equality. On the other hand, we have shown in the proof of
Lemma 3.13, that this solution is given by the left hand side.

Given $p\in D,$ choose $q\neq p$ such that $K_{a}(p,q)>0.$ Such a $q$ exists,
for otherwise, we would have
$\int_{0}^{\infty}e^{-as}P_{s}^{D}{\mathbbm{1}}_{D}(p)ds=0.$ That is
$\tau^{out}_{D}=0,\mathbb{P}_{p}$ almost surely. By continuity of $K_{a}$ off
the diagonal, there exist disjoint neighborhoods $U,V$ of $p$ and $q$ and some
$c>0$ such that $K_{a}(x,y)\geq c$ for all $x\in U,y\in V.$ Thus, for all
$x\in U,$

$G^{D}(x,\cdot)\geq\int_{0}^{\infty}e^{-as}P_{s}^{D}(x,\cdot)\geq
cLeb(V\cap\cdot)$

where $Leb$ stands for the Lebesgue measure on ${\mathbb{R}}^{n}.$

We now pass to the proof of the Lemma. Using a local chart around $p$ we can
assume without loss of generality that $p=0,S^{1}(0)\neq 0$ and
$\frac{S^{1}(0)}{\|S^{1}(0)\|}=e_{1},$ where $(e_{1},\ldots,e_{n})$ stands for
the canonical basis on ${\mathbb{R}}^{n}.$ Let
$B_{\varepsilon}=\\{x\in{\mathbb{R}}^{n}:\>\sum_{i=1}^{n}|x_{i}|<\varepsilon\\}.$
For $\varepsilon>0$ small enough $S^{1}(x)\neq 0$ for all $x\in
B_{\varepsilon}$ and for all $x\in\overline{B_{\varepsilon}}$ there is a
vector $u$ normal to $\overline{B_{\varepsilon}}\setminus B_{\varepsilon}$
such that $\langle S^{1}(x),u\rangle\neq 0.$ The preceding reasoning can then
be applied with $D_{\varepsilon}$ in place of $D.$ Thus, for some disjoint
open sets $U,V\subset D_{\varepsilon}$ with $p\in U$ and for all $x\in U,$

$G^{D}(x,\cdot)\geq G^{D_{\varepsilon}}(x,\cdot)\geq cLeb(V\cap\cdot).$

$\Box$

###### Lemma 3.15

Let $p\in D$ be such that the Hörmander condition holds at $p.$ Then, there
exist a neighborhood $U$ of $p,$ a non trivial measure $\xi$ on $D$ and $T>0$
such that for all $x\in U,P_{T}^{D}(x,\cdot)\geq\xi(\cdot).$

Proof:  Let $V$ be a neighborhood of $p$ such that the Hörmander condition
holds at every $x\in V.$ By ([25], Theorem 3 and its proof), there exists a
nonnegative map $p_{t}^{V}(x,y)$ smooth in the variables
$(t,x,y)\in{\mathbb{R}}_{+}^{*}\times V\times V$ such that
$P_{t}^{V}(x,dy)=p_{t}^{V}(x,y)dy,$ where $P_{t}^{V}(x,\cdot)$ stands for the
law of $X_{t}^{x}$ killed at $D\setminus V.$ Choose $q\in V$ and $T>0$ such
that $p_{T}^{V}(p,q):=c>0.$ Then, by continuity, there exist neighborhoods
$U,W$ of $p$ and $q$ such that $p_{T}^{V}(x,y)\geq c/2>0$ for all $(x,y)\in
U\times W.$ Thus, for all $x\in U,$ $P_{T}^{D}(x,dy)\geq
P_{T}^{V}(x,dy)\geq\frac{c}{2}{\mathbbm{1}}_{W}(y)dy.$ $\Box$

###### Lemma 3.16

Let $\mu$ be a QSD for $(X_{t})$ on $D.$ Then $\mu$ has a smooth density with
respect to the Lebesgue measure on $D.$ If furthermore the strong Hörmander
conditions holds at every $x\in K,$ this density is positive.

Proof:  Let $\Phi$ be a smooth function with compact support in $D.$ By Ito’s
formula
$(\Phi(X_{t\wedge\tau^{out}_{D}})-\int_{0}^{t\wedge\tau^{out}_{D}}L\Phi(X_{s})ds)_{t\geq
0}$ is a bounded local martingale, hence a true martingale. Thus, taking the
expectation, letting $t\rightarrow\infty$ and using Proposition 2.3, it comes
that $G^{D}(L\Phi)(x)=-\Phi(x)$ for all $x\in D.$ Let $\mu$ be a QSD with rate
$\lambda.$ Then $-\mu(\Phi)=\mu G^{D}(L\Phi)=\frac{1}{\lambda}\mu(L\Phi).$
This shows that $L^{*}\mu+\lambda\mu=0$ on $D$ in the sense of distributions.
Now, for all distribution $f$ on $D$,

$L^{*}f=\tilde{S}^{0}f+\frac{1}{2}\sum_{j=1}^{m}(S^{j})^{2}f+Tf$

where $T$ is a smooth function and $S^{0}+\tilde{S}^{0}\in
Span(S^{1},\ldots,S^{m}).$ Therefore $L^{*}$ satisfies the weak Hörmander
property. By Hörmander Theorem [24], it is hypoelliptic. This implies that
$\mu$ has a smooth density.

For $a>0,$ set $L^{*}_{a}f=L^{*}f-af$ and choose $a$ sufficiently large so
that $L^{*}_{a}1=T-a<0.$ If the strong Hörmander condition is satisfied at
every point $x\in K,$ the same is true for $L^{*}_{a}.$ Now,
$L^{*}_{a}(-\mu)=(\lambda+a)\mu\geq 0.$ Therefore, by application of Bony’s
maximum principle ([7], Corollary 3.1), if the density of $-\mu$ vanishes at
some $x\in D,$ it has to be zero on $D$. This is impossible because $\mu$ is a
probability measure. $\Box$

#### Proof of Theorem 1.8

We now assume that Conditions $(i),(ii),(iii)$ of Theorem 1.8 hold. By Lemma
3.13 and Corollary 2.12, there exists a positive right eigenfunction for
$G^{D}.$ By Conditions $(i)$ and $(iii)$ of Theorem 1.8, there exists a point
$p\in D$ near $\partial D,$ $D$-accessible, at which $S^{j}(p)\neq 0$ for some
$j\geq 1.$ Then, by Lemma 3.14 and Lemma 2.15, $(P_{t}^{D})$ is irreducible.
Thus, according to Theorem 2.13, it has a unique QSD. Such a QSD has a smooth
density by Lemma 3.16. This prove the first part of Theorem 1.8. The last part
is similar and follows from Lemma 3.15 and Theorem 2.16.

##  4 Beyond diffusions

This section briefly discusses a simple example given by a piecewise
deterministic Markov process. This illustrates the applicability of our
results beyond diffusions and justifies some of the abstract conditions made
in Section 2. In particular, the facts that $M$ can be chosen to be a general
metric space rather than ${\mathbb{R}}^{n}$ or a manifold, and that $D$ can be
chosen to be open relative to $K=\overline{D}$ rather than open.

Let $M={\mathbb{R}}\times\\{0,1\\}$ endowed with the metric
$d((x,i),(y,j))=|i-j|+|x-y|$ and $(Z_{t})=((X_{t},I_{t}))$ be the Markov
process on $M$ defined by

$\dot{X_{t}}=2I_{t}-1,$

and $(I_{t})$ is a continuous time Markov chain on $\\{0,1\\}$ independent of
$(X_{t})$ having jump rates $\lambda_{0,1}=\lambda_{1,0}=\lambda>0.$ In words,
$(X_{t})$ moves at velocity $1$ either to the left or to the right and changes
direction at a constant rate $\lambda.$ It is easy to check that the semi-
group associated to $(Z_{t})$ is Feller.

Suppose now that $(Z_{t})$ is killed when $(X_{t})$ exits $]0,1[.$ At the exit
time, one clearly has $X_{t}=I_{t}$ almost surely, so that it is natural to
set

$D=]0,1]\times\\{0\\}\cup[0,1[\times\\{1\\}.$

Thus, $K=\overline{D}=[0,1]\times\\{0,1\\}$ and

$\partial_{K}D=\\{(0,0)\\}\cup\\{(1,1)\\}.$

It is not hard to show that $M\setminus K$ is accessible from all $z\in K$ (in
the sense of definition 2.2), and that $G^{D}(C_{b}(D))\subset C_{0}(D).$ The
verification is left to the reader.

For any function $f:M\mapsto{\mathbb{R}}$ and $i\in\\{0,1\\}$ we write
$f_{i}(x)=f(x,i).$

###### Lemma 4.1

Let $H:[0,1]\mapsto{\mathbb{R}}_{+}$ and $\omega>0$ be defined as follows:

  * •

If $\lambda>1,H(x)=\frac{\sin(\theta
x)}{\theta},\omega=\lambda(1+\cos(\theta)),$ where $0<\theta<\pi$ is the
unique solution to $\lambda\sin(\theta)=\theta;$

  * •

If $\lambda=1,H(x)=x,\omega=2$;

  * •

If $\lambda<1,H(x)=\frac{\sinh(\theta
x)}{\theta},\omega=\lambda(1+\cosh(\theta)),$ where $\theta>0$ is the unique
solution to $\lambda\sinh(\theta)=\theta.$

Then,

(i)

The map $h\in C_{0}(D)$ defined by

$h_{i}(x)=(1-i)H(x)+iH(1-x)$ (18)

is a positive right eigenfunction for $G^{D}$ with eigenvalue
$\frac{1}{\omega}.$

(ii)

The probability on $D$ defined by

$\mu=\frac{1}{2\int_{0}^{1}H(s)ds}(h_{1}(x)dx\otimes\delta_{0}+h_{0}(x)dx\otimes\delta_{1})$
(19)

is a QSD for $(Z_{t})$ on $D$ with absorption rate $\omega.$

Proof:  Let $L$ denote the infinitesimal generator of the process $(Z_{t})$
and ${\cal D}(L)$ its domain (defined in the usual sense for Feller processes,
see e.g. [31]; see also [15] more specifically on piecewise deterministic
Markov processes). It is easy to verify that if $f:M\mapsto{\mathbb{R}}$ is
$C^{1}$ in $x,$ with compact support, then $f\in{\cal D}(L)$ and

$Lf(x,i)=\left\\{\begin{array}[]{c}-f^{\prime}_{0}(x)+\lambda(f_{1}(x)-f_{0}(x))\mbox{
for }i=0\\\ f^{\prime}_{1}(x)+\lambda(f_{0}(x)-f_{1}(x))\mbox{ for
}i=1.\end{array}\right.$ (20)

Now, observe that the map $H$ satisfies the identity

$-H^{\prime}(x)+\lambda(H(1-x)-H(x))=-\omega H(x)$ (21)

for all $0\leq x\leq 1.$ Extending $H$ to a smooth function on ${\mathbb{R}}$
having compact support, formula (18) defines a map $h\in{\cal D}(L)$ such
that, for all $z\in D$, $Lh(z)=-\omega h(z).$ Therefore,
$((h(Z_{t\wedge\tau_{D}^{out}})-h(z)-\int_{0}^{t\wedge\tau_{D}^{out}}Lh(Z_{s})ds))_{t}$
is a $\mathbb{P}_{z}$ martingale (because $h\in{\cal D}(L)$) bounded in
$L^{1},$ and by the stopping time theorem

$-h(z)+\omega\mathbb{E}_{z}\left(\int_{0}^{\tau_{D}^{out}}h(Z_{s})ds\right)=\mathbb{E}_{z}\left(h(Z_{\tau_{D}^{out}})-h(z)-\int_{0}^{\tau_{D}^{out}}Lh(Z_{s})ds\right)=0.$

That is $G^{D}h(z)=\frac{1}{\omega}h(z).$ This prove $(i).$

$(ii).$ In order to prove that $\mu$ is a QSD with rate $\omega,$ it suffices
to show that for all $f\in C_{0}(D),$ $\mu G^{D}(f)=\frac{1}{\omega}\mu(f).$
Given such a $f,$ one can easily solve the problem $Lg=-f$ on $D$ with $g\in
C_{0}(D).$ Namely,

$g_{0}(x)=\int_{0}^{x}[f_{0}(u)-\lambda(x-u)(f_{0}(u)+f_{1}(u))]du+a\lambda x$

and

$g_{1}(x)=\int_{0}^{x}[-f_{1}(u)-\lambda(x-u)(f_{0}(u)+f_{1}(u))]du+a(\lambda
x+1),$

where $a$ is determined by $g_{1}(1)=0.$ These formulae show that
$g_{0},g_{1}$ can be extended to $C^{1}$ maps on ${\mathbb{R}}$ with compact
support. Reasoning exactly like in $(i)$ this shows that $G^{D}(f)=g$ on $D.$
The problem of showing that $\mu G^{D}(f)=\frac{1}{\omega}\mu(f)$ then reduces
to show that $\mu(Lg)=-\omega\mu(g).$ This latter identity can be easily
checked, using (19), (21) and integration by part. $\Box$

###### Remark 4.2

Observe that $(P_{t}^{D})$ is not a semigroup on $C_{b}(D)$ because for any
$0<t<1/2,\,P_{t}^{D}{\mathbbm{1}}_{D}$ is discontinuous. Indeed,
$P_{t}^{D}{\mathbbm{1}}_{D}(x,0)=1$ for $t<x<1/2,$ while
$P_{t}^{D}{\mathbbm{1}}_{D}(t,0)=1-e^{-\lambda t}.$ It is, however, a strongly
continuous semigroup on $C_{0}(D).$ Let $L^{D}$ be its generator and ${\cal
D}(L^{D})$ its domain. A by product of the proof above is that ${\cal
D}(L^{D})$ consists of the maps $f\in C_{0}^{1}(D)$ such that

$f_{0}(0)=f_{1}(1)=-f^{\prime}_{0}(0)+\lambda
f_{1}(0)=f^{\prime}_{1}(1)+\lambda f_{0}(1)=0,$

and $L^{D}f(x)=Lf(x)$ for all $x\in D$ and $f\in{\cal D}(L^{D}).$ Here $L$ is
the operator defined by (20) and $C_{0}^{1}(D)$ stands for the maps $f\in
C_{0}(D)$ such that $f_{0},f_{1}$ can be extended to $C^{1}$ maps on
${\mathbb{R}}.$

###### Remark 4.3

The formula defining $g=G^{D}(f)$ in the previous proof is also valid for
$f\in C_{0}(D)+{\mathbb{R}}{\mathbbm{1}}_{D}.$ In particular,

$G^{D}({\mathbbm{1}}_{D})(x,0)=\mathbb{E}_{x,0}(\tau_{D}^{out})=x(1+\lambda-\lambda
x).$

Observe that the map $x\rightarrow\mathbb{E}_{x,0}(\tau_{D}^{out})$ achieves
its maximum at $x=1$ for $\lambda\leq 1$ and at $x=\frac{1+\lambda}{2\lambda}$
for $\lambda\geq 1.$

###### Proposition 4.4

Let $\mu$ $h$ be as in Lemma 4.1. There exist $C,\alpha>0$ such that for all
probability $\rho,$ on $D$

$\left\|\frac{\rho P_{t}^{D}}{\rho
P_{t}^{D}{\mathbbm{1}}_{D}}-\mu(\cdot)\right\|_{TV}\leq\frac{C}{\rho(h)}e^{-\alpha
t}$

Proof:  We rely on Theorem 2.16. Assertions $(i)$ and $(ii)$ are satisfied.
For $(iii),$ fix $\varepsilon<1/10$ and let
$U=]\frac{1}{2}-\varepsilon,\frac{1}{2}+\varepsilon[\times\\{0\\}.$ Then $U$
is clearly $D$-accessible from all $z\in D.$

Set $T=\frac{1}{2}-2\varepsilon.$ For $z=(x,0)\in U,$ and $A$ a Borel subset
of $]\frac{1}{2}-\varepsilon,\frac{1}{2}+\varepsilon[,$

$P_{T}^{D}(z,A\times\\{0\\})\geq\mathbb{P}(x-\tau_{1}+(\tau_{2}-\tau_{1})-(T-\tau_{2})\in
A,\tau_{2}<T<\tau_{3}),$

where $0<\tau_{1}<\tau_{2}<\tau_{3}$ are the three first jump times of
$I_{t}.$ Thus, using the fact that, given $\tau_{2}<t<\tau_{3}$,
$(\tau_{1},\tau_{2})$ is distributed as the order statistics of
$(U_{1},U_{2})$, where $U_{1}$ and $U_{2}$ are i.i.d. uniform r.v. on $[0,t]$,

$\displaystyle P_{T}^{D}(z,A\times\\{0\\})$
$\displaystyle\geq{\mathsf{P}}(x-T+2(\tau_{2}-\tau_{1})\in
A|\tau_{2}<T<\tau_{3})\frac{(\lambda T)^{2}}{2}e^{-\lambda T}$
$\displaystyle=\frac{(\lambda T)^{2}}{2}e^{-\lambda
T}\int_{A}\frac{(x+T-u)}{2T^{2}}du\geq cLeb(A)$

for some $c>0.$ This shows condition $(iii)$ of Theorem 2.16 with
$d\xi=c{\mathbbm{1}}_{U}du\otimes\delta_{0}.$ $\Box$

###### Remark 4.5

The minorization of $P_{t}^{D}(z,\cdot)$ proved in Proposition 4.4 could also
be obtained for more general PDMPs under appropriate bracket conditions on the
vector fields defining the PDMP, along the lines of the general results proved
in [1] or [6]

### Acknowledgment

We thank Emmanuel Trelat for fruitful discussions and his suggestion to use
reference [43], in Lemma 3.13. We thank Josef Hofbauer for his valuable
suggestions on Lemma 3.5. We also thank 3 anonymous referees for their careful
reading and valuable suggestions to improve the first version of this paper.
This research is supported by the Swiss National Foundation grant 200020
196999.

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