# The method of stochastic characteristics for linear second-order
hypoelliptic equations

Juraj Földes, David P. Herzog
email<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We study hypoelliptic stochastic differential equations (SDEs) and their
connection to degenerate-elliptic boundary value problems on bounded or
unbounded domains. In particular, we provide probabilistic conditions that
guarantee that the formal stochastic representation of a solution is smooth on
the interior of the domain and continuously approaches the prescribed boundary
data at a given boundary point. The main general results are proved using fine
properties of the process stopped at the boundary of the domain combined with
hypoellipticity of the operators associated to the SDE. The main general
results are then applied to deduce properties of the associated Green’s
functions and to obtain a generalization of Bony’s Harnack inequality. We
moreover revisit the transience and recurrence dichotomy for hypoelliptic
diffusions and its relationship to invariant measures.

##  1\. Introduction

###  1.1. Overview

Let $\mathfrak{m}\in\mathbf{N}$ and $U\subset\mathbf{R}^{\mathfrak{m}}$ be
nonempty, open set with nonempty boundary $\partial U$. Let
$\mathcal{X}\subset\mathbf{R}^{\mathfrak{m}}$ be an open set containing
$\overline{U}=U\cup\partial U$ and let $M_{\mathfrak{m}\times r}$ denote the
set of $\mathfrak{m}\times r$ matrices with entries in $\mathbf{R}$. Consider
a linear second-order differential operator of the form

$\displaystyle L=\sum_{i=1}^{\mathfrak{m}}(b(x))_{i}\frac{\partial}{\partial
x_{i}}+\frac{1}{2}\sum_{i,j=1}^{\mathfrak{m}}(\sigma(x)\sigma(x)^{T})_{i,j}\frac{\partial^{2}}{\partial
x_{i}\partial x_{j}}\,,$ (1.1)

where $b\in C^{\infty}(\mathcal{X};\mathbf{R}^{\mathfrak{m}})$ and $\sigma\in
C^{\infty}(\mathcal{X};M_{\mathfrak{m}\times r})$. In this paper, we study the
formal stochastic representation $u_{\text{stoc}}$ (see (1.4) below)
corresponding to the combined Dirichlet and Poisson problems for $L$ on $U$:

$\displaystyle\begin{cases}Lu=-f&\text{ on }\,\,U\,,\\\ u(x)\rightarrow
g(x_{*})&\text{ as }x\rightarrow x_{*}\in\partial U,\,x\in U\,,\end{cases}$
(1.2)

where $f\in C^{\infty}(U)$ and $g$ is continuous in a neighborhood of
$\partial U$. Importantly, our assumptions allow for $U$ to be unbounded and
for $L$ to be _hypoelliptic_ 111The operator $L$ is called _hypoelliptic on_
$U$ if for all distributions $v$ on $U$ with $Lv\in C^{\infty}(V)$ for some
open $V\subset U$, we have $v\in C^{\infty}(V)$.  on $U$.

Formally, the method of stochastic characteristics (see Section 1.2 below)
produces the formula $u_{\text{stoc}}$ as a candidate expression for a
classical solution of (1.2). However, in this setting, $u_{\text{stoc}}$ may
not be defined let alone be a classical solution of (1.2). The goal of this
paper is to provide practical, probabilistic conditions under which
$u_{\text{stoc}}$ is well defined and satisfies the properties required by the
problem (1.2):

  * •

$u_{\text{stoc}}\in C^{2}(U)$ and $Lu_{\text{stoc}}=f$ on $U$ in the classical
sense;

  * •

$u(x)\rightarrow g(x_{*})$ as $x\rightarrow x_{*}\in\partial U,\,x\in U$.

Results for classical well-posedness of (1.2) in the setting of a bounded
domain $U$ are also obtained.

Although we employ tools from analysis, the manuscript primarily focuses on
probabilistic methods. One may compare our work with the classical work of
Oksendal [Oks13, Chapter 9] adapted to our setting, but we aim to be more
self-contained on this particular topic. Specifically, we start from a
probabilistic construction of fundamental solutions and build up to a solution
theory for the boundary-value problem (1.2). As a consequence, one may also
compare this work to a portion of the classical elliptic theory text of
Gilbarg and Trudinger [GT15], but we use probabilistic techniques in the
possibly unbounded $U$, hypoelliptic $L$ setting.

###  1.2. The method of stochastic characteristics

In order to employ probabilistic techniques, we associate to $L$ a diffusion
process $x_{t}$ on $\mathcal{X}$ with infinitesimal generator $L$. That is, we
suppose that $x_{t}$ satisfies an Itô stochastic differential equation (SDE)
of the form

$\displaystyle dx_{t}$
$\displaystyle=b(x_{t})\,dt+\sigma(x_{t})\,d\mathbf{W}_{t}\,,$ (1.3)

where $\mathbf{W}_{t}=(W_{t}^{1},W_{t}^{2},\ldots,W_{t}^{r})^{T}$ is a
standard, $r$-dimensional Brownian motion defined on, and adapted to, a
filtered probability space
$(\Omega,\mathcal{F},\mathcal{F}_{t},\mathbf{P},\mathbf{E})$. The _formal
stochastic representation_ $u_{\text{stoc}}$ corresponding to the problem
(1.2) then has the form

$\displaystyle
u_{\text{stoc}}(x)=\mathbf{E}_{x}\int_{0}^{\tau}f(x_{s})\,ds+\mathbf{E}_{x}g(x_{\tau}),\qquad\,x\in
U,$ (1.4)

where

$\displaystyle\tau=\inf\\{t>0\,:\,x_{t}\notin U\\}$ (1.5)

is the first positive exit time from $U$. Note that in (1.4), the symbol
$\mathbf{E}_{x}$ means the expected value for the law $\mathbf{P}_{x}$ of the
process $x_{t}$ with $x_{0}=x$. The formula (1.4) is formally derived using
Dynkin’s/Itô’s formula (2.8) applied to a sufficiently _nice_ classical
solution of (1.2), which may or may not exist in this setting. Specifically,
in Dynkin’s formula (2.8) below, formally set $\phi(t,x)=u(x)$ and
$\sigma=\tau$ to arrive at (1.4). Compared with the usual method of
characteristics for first-order partial differential equations, this is the
analogous method for linear second-order equations, but this time the
characteristic trajectories $x_{t}$ are random.

Aside from it being unclear its precise relationship to (1.2), observe that
the expression $u_{\text{stoc}}$ in (1.4) is itself formal in two ways. First,
depending on the set $U$ and the coefficients $b,\sigma$, both the process
$x_{t}$ and the expected values in (1.4) may not be defined, so
$u_{\text{stoc}}$ in turn is not defined. Second, even if $u_{\text{stoc}}$ is
defined on $U$, it may not satisfy the equation (1.2) in the classical sense.
However, employing this formal argument via Dynkin’s formula (2.8),
$u_{\text{stoc}}$ is a _best guess_ at a solution of (1.2), and so it is
natural to study $u_{\text{stoc}}$ in relation to (1.2).

If the operator $L$ is hypoelliptic on $U$, then it has a smoothing effect
reminiscent of second-order uniformly elliptic operators on $U$ with smooth
coefficients. However, even if $U$ is a bounded domain with boundary $\partial
U$ satisfying the _exterior cone condition_ 222$\partial U$ satisfies the
_exterior cone condition_ if for all $x_{*}\in\partial U$ there exists a basis
$x_{1},x_{2},\ldots,x_{\mathfrak{m}}$ of $\mathbf{R}^{\mathfrak{m}}$ and a
parameter $\lambda_{*}>0$ such that
$\text{Cone}(x_{*},\lambda_{*}):=\\{x_{*}+\textstyle{\sum}_{i=1}^{\mathfrak{m}}\lambda_{i}x_{i}\,:\,\lambda_{i}\in(0,\lambda_{*})\\}\subset\overline{U}^{c}$.,
there are many examples where the problem (1.2) is ill-posed in the classical
sense (see 3.8 below). This is different compared to equation (1.2) with is
uniformly elliptic operator $L$ of the form (1.1). From a probabilistic
perspective, this difference can be explained intuitively, especially as it
relates to satisfaction of the boundary condition in equation (1.2). Indeed,
if $L$ is uniformly elliptic, then the noise in (1.3) is present in each
direction of the equation, and therefore for short times the particle visits
all points in a small ball. See [CFH21] for a rigorous formulation of this
statement. This is, however, not necessarily the case for a general
hypoelliptic diffusion. In particular, even if the process $x_{t}$ initiated
at any point $x\in U$ is defined, it may not hit certain portions of the
boundary with positive probability, and consequently the values of $g$ on
these portions do not influence $u_{\text{stoc}}$ in (1.4). Moreover, when
started on the boundary $\partial U$, the process may have a positive
probability of re-entering $U$ prior to exiting, so that $u_{\text{stoc}}$ may
not satisfy the boundary condition.

###  1.3. Previous results and layout of the paper

Despite these issues in the hypoelliptic setting, understanding when the
equation or boundary conditions in (1.2) are satisfied in the classical sense
by $u_{\text{stoc}}$ is of notable importance, as hypoelliptic operators, and
their corresponding boundary-value problems like (1.2), play a key role in a
number of problems in science and engineering. See, for example, hypoelliptic
diffusions arising in finite-dimensional models of turbulence [BCCM05, BCH07,
BL20, BHW12, EM01, FGHH21, GHW11, HN21, Rom04] or in statistical mechanics and
machine learning [CHSG21, CEHRB18, GOS+19, HM19, LSS20, LSR10]. Direct
applications also occur in the ergodic theory of SDEs, where functionals of
return times to compact sets can be seen as the formal stochastic
representations corresponding to equations of the form (1.2). Such functionals
are essential to understand precise rates of convergence to equilibrium [HM11,
RB06, EGZ19].

Historically, various aspects of the problem (1.2) in the hypoelliptic setting
have drawn interest from researchers dating back to Kolmogorov [Kol34], who
gave the first known example of a hypoelliptic diffusion that is not uniformly
elliptic. Later, the seminal work of Hörmander [Hör67] provided an efficient
tool to determine hypoellipticity of $L$ by locally calculating the Lie
algebra of vector fields that create the operator $L$ in the form (1.1), see
Section 2.4 for further information. For classical well-posedness of the
problem (1.2), the pioneering work of Bony [Bon69] is fundamental. There, Bony
gives conditions on a bounded open set $U$ and a hypoelliptic operator $M$ of
the form $M=L-a$, where $a\in C^{\infty}(U)$ is positive and bounded away from
zero on $U$, so that the problem (1.2) with $M$ replacing $L$ is well-posed in
the classical sense. Apart from the presence of the positive function $a$,
which in particular aides in the existence part of the problem (1.2), a
critical assumption guaranteeing continuity on $\overline{U}$ is that at every
point on $\partial U$, the noise in the equation points in the normal
direction. Such an assumption is often not satisfied for many hypoelliptic
operators of interest, for example if the noise is additive (that is,
spatially constant), see [BL20, BHW12, CHSG21, CEHRB18, EM01, FGHH21, GHW11,
HM19, HN21, LSS20, Rom04]. Thus, it is natural to investigate if classical
well-posedness holds under weaker assumptions.

More recently, the hypoelliptic Dirichlet problem ($f\equiv 0$ in (1.2)) in a
bounded domain $U$ was revisited in [Ram97], where Bony’s result was extended
to operators $L$ of the form (1.1); that is, with $a\equiv 0$, assuming $L$
satisfies a maximum principle and again that at every point on $\partial U$
there is noise pointing in the normal direction. Similar to Bony’s work, the
results in [Ram97] fail to apply in many problems of interest. Perhaps more
closely related to the present paper is the work of Kogoj [Kog17], which uses
analytic methods from potential theory to establish a cone-type criterion for
the existence of a generalized solution, in the sense of Perron-Wiener, of the
Dirichlet problem for a hypoelliptic $L$ in parabolic form. More importantly,
[Kog17] assumes the existence of a _well-behaved_ fundamental solution and
that the diagonal of $\sigma(x)\sigma(x)^{T}$ never vanishes. Note that
probabilistically the cone-type condition is natural as it gives particle a
space to exit the domain, and indeed for uniformly elliptic operators, such
condition is sufficient to guarantee boundary regularity (in the potential-
theoretic sense). As described above, additional conditions are required in
hypoelliptic setting, and the supplementary criteria provided in [Kog17]
appear difficult to check in concrete scenarios. We refer the reader to [GS90,
LTU17, NS87] for earlier, related results. We also refer to the recent work of
the authors [CFH21] which provides probabilistic methods for determining
regular points (defined below in Section 3) on the boundary using the theory
of large deviations and laws of the iterated logarithm. Note that [CFH21] was
partly inspired by the work of Lachal [Lac97], where a functional law of the
iterated logarithm was obtained for the iterated Kolmogorov stochastic
differential equation. However, applications to solvability of (1.2) were not
discussed in detail in [CFH21] or [Lac97].

Another goal of the present work is to provide a relatively self-contained
presentation on the topic in this paper, referring primarily to graduate-level
textbooks to obtain needed results. Our hope is that the level of the paper is
similar to the level in Oksendal’s book [Oks13], and that the interested
graduate student who has had an introductory course in stochastic analysis
will find the paper readable. We did this for two reasons. First, prior to
starting this project, at times we found it difficult to locate results for
(1.2) in the hypoelliptic setting and this sentiment was also confirmed in
conversations with colleagues. Second, there were claims in the literature
about (1.2) that seemed both correct and intuitive, but we could not locate a
proof in the existing literature. Thus, we hope this paper will serve as a
resource to which the (stochastic) analysis community can refer as needed,
since either there is a physical proof in the paper or the statement is easy
to locate in the literature using the provided references.

The organization of this paper is as follows. In Section 2, we introduce
further notation, assumptions, terminology and make preliminary remarks about
hypoellipticity. Hörmander’s theorem, in particular, will be discussed in
detail in Section 2. In Section 3, we elaborate on the importance of boundary
behavior for $x_{t}$ as its relates to solvability of (1.2) in the classical
sense. In particular, in Section 3 one can find our definition of _regular_
and _irregular_ points on $\overline{U}$. Section 4 focuses on special
equations of the form (1.2), in particular those involving functionals of
$\tau$, and these special equations are used later to formulate our main
general results in Section 6. Section 6 provide conditions on $L$, $\tau$, and
$u_{\text{stoc}}$ that guarantee $Lu_{\text{stoc}}=-f$ on $U$ in the sense of
distributions, hence in the classical sense if $L$ is hypoelliptic and $f$
smooth. Green’s functions and a generalization of Bony’s form of the Harnack
inequality [Bon69, Theorem 7.1] are discussed in Section 5, while in Section
7, as an application of these results, we re-derive the transience and
recurrence dichotomy for degenerate diffusions.

##  2\. Notation and Preliminary Remarks

In this section, we introduce notation, assumptions and terminology used
throughout the paper. We also make a few preliminary remarks. First, we fix
some standard notation.

###  2.1. Basic notation

For Borel sets $V,V_{1}\subset\mathbf{R}^{k}$, $V_{2}\subset\mathbf{R}^{n}$,
and $W\subset\mathbf{R}^{\ell}$, we use the following notation.

  * –

$C(V;W)$ denotes the set of continuous functions $\phi:V\to W$;

  * –

$C^{j}(V;W)$, $j\geq 1$, denotes the set of $j$-times continuously
differentiable functions $\phi:V\to W$;

  * –

$C^{\infty}(V;W)=\bigcap_{j=1}^{\infty}C^{j}(V;W)$;

  * –

$C^{j_{1},j_{2}}(V_{1}\times V_{2};W)$ denotes the set of functions
$\phi=\phi(x,y):V_{1}\times V_{2}\to W$ which are $j_{1}$-times continuously
differentiable in $x$ and $j_{2}$-times continuously differentiable in $y$;

  * –

$B(V;W)$ denotes the set of bounded, Borel measurable functions $\phi:V\to W$;

  * –

In any of the above functions spaces, a subscript of $0$ indicate that the
function is moreover compactly supported in its domain of definition, e.g.
$C_{0}^{j}$ versus $C^{j}$;

  * –

For $1\leq p\leq\infty$, $L^{p}(V)$ denotes the set of measurable functions
$\phi:V\to\mathbf{R}$ such that
$\|\phi\|_{L^{p}(V)}^{p}:=\int_{V}|\phi|^{p}\,dx<\infty$ if $p\in[1,\infty)$
and $\|\phi\|_{L^{\infty}(V)}:=\text{ess sup}_{x\in V}|\varphi(x)|<\infty$ if
$p=\infty$;

  * –

$\text{Lip}(V;W)$ denotes the set of functions $\phi:V\to W$ which are
Lipschitz continuous on $V$;

  * –

For any $s\in\mathbf{R}$, $H^{s}(\mathbf{R}^{k})$ denotes the usual Sobolev
space $W^{s,2}(\mathbf{R}^{k})$;

  * –

If the target $W$ is clear from context or not important, we write
$C^{j}(V)=C^{j}(V;W)$, $C_{0}^{j}(V)=C_{0}^{j}(V;W)$, $B(V)=B(V;W)$, etc. When
the context is clear, we may also drop $V$ and write $C^{j},C_{0}^{j}$, etc.

  * –

$\mathcal{B}_{V}$ denotes the set of Borel measurable subsets of $V$;

  * –

$\mathcal{B}$ denotes the set of Borel measurable subsets of the open set
$\mathcal{X}$, where $\mathcal{X}$ is as in the introduction.

  * –

For any $B\in\mathcal{B}$ bounded with $\overline{B}\subset\mathcal{X}$, we
define

$\displaystyle\tau_{B}=\inf\\{t>0\,:\,x_{t}\in B^{c}\\},$ (2.1)

where $\inf\emptyset:=\infty$.

  * –

We define a sequence $\mathcal{X}_{n}\subset\mathcal{X}$, $n\geq 1$ of bounded
open sets with $\overline{\mathcal{X}_{n}}\subset\mathcal{X}_{n+1}$ for all
$n$ and $\bigcup_{n=1}^{\infty}\mathcal{X}_{n}=\mathcal{X}$.

  * –

At times, we will also need a sequence $U_{n}\subset U$, $n\geq 1$, of bounded
open sets with $\overline{U_{n}}\subset U_{n+1}$ and
$\bigcup_{n=1}^{\infty}U_{n}=U$.

  * –

We define

$\displaystyle\tau=\tau_{U}:=\lim_{k\to\infty}\tau_{U\cap\mathcal{X}_{k}}.$
(2.2)

and

$\displaystyle\tau_{\mathcal{X}}=\lim_{k\to\infty}\tau_{\mathcal{X}_{k}}.$
(2.3)

###  2.2. Nonexplosivity of $x_{t}$

Depending on the behavior of the coefficients $b$ and $\sigma$, the solution
$x_{t}$ of equation (1.3) evolving on the neighborhood $\mathcal{X}$ of
$\overline{U}$ is only a priori defined locally in time until the process
exits $\mathcal{X}$. That is, by the standard existence and uniqueness theorem
for SDEs [Oks13], for any initial condition $x\in\mathcal{X}$ at time $t=0$,
equation (1.3) has a unique (pathwise) solution $x_{t}$ for all times $0\leq
t<\tau_{\mathcal{X}}$. However, throughout the paper, we will assume that
$\tau_{\mathcal{X}}$ is almost surely infinite as in the following definition.

######  Definition 2.1.

We say that the process $x_{t}$ is _non-explosive_ if

$\displaystyle\mathbf{P}_{x}\\{\tau_{\mathcal{X}}<\infty\\}=0\,\,\,\text{ for
all }x\in\mathcal{X},$ (2.4)

where $\mathbf{P}_{x}$ denotes the probability $\mathbf{P}$ but indicates that
$x_{0}=x$.

Nonexplosivity implies that for all initial conditions $x\in\mathcal{X}$,
equation (1.3) has a unique pathwise solution $x_{t}$ defined on $\mathcal{X}$
for all finite times $t\geq 0$ almost surely. Furthermore, $t\mapsto
x_{t}:[0,\infty)\to\mathcal{X}$ is continuous almost surely.

######  Remark 2.2.

Suppose that $U$ is bounded. Since $b$ and $\sigma$ are smooth and defined on
an open neighborhood of $\overline{U}$, one can extend $b$ and $\sigma$ to
functions $\hat{b}\in
C^{\infty}(\mathbf{R}^{\mathfrak{m}};\mathbf{R}^{\mathfrak{m}})$ and
$\hat{\sigma}\in C^{\infty}(\mathbf{R}^{\mathfrak{m}};M_{\mathfrak{m}\times
r})$, respectively, such that $\hat{b}=b$ on $\overline{U}$ and
$\hat{\sigma}=\sigma$ on $\overline{U}$ and such that $\hat{b},\hat{\sigma}$
both have bounded derivatives of all orders, e.g. extend $b$, $\sigma$ by zero
outside of a larger neighborhood and mollify. Consequently, replacing
$\mathcal{X}$ with $\mathbf{R}^{\mathfrak{m}}$ and $b,\sigma$ in (1.3) with
$\hat{b},\hat{\sigma}$ we find that the resulting solution $\hat{x}_{t}$ is
nonexplosive. Note that the problem (1.2) and its formal stochastic
representation (1.4) remain unchanged regardless of the chosen extension, so
nonexplosivity is implicitly present. On the other hand, if $U$ is unbounded,
then one must verify nonexplosivity separately, typically using Lyapunov-type
methods. See 2.4 below for further details.

By nonexplosivity, $x_{t}$ is a well-defined Markov process on $\mathcal{X}$.
It thus induces a corresponding Markov semigroup $\\{\mathcal{P}_{t}\\}_{t\geq
0}$ which acts on functions $\phi\in B(\mathcal{X};\mathbf{R})$ via

$\displaystyle\mathcal{P}_{t}\phi(x):=\mathbf{E}_{x}\phi(x_{t})$ (2.5)

and dually on $\mathcal{B}$-measures $\nu$ via

$\displaystyle\nu\mathcal{P}_{t}(B)=\int_{\mathcal{X}}\mathcal{P}_{t}\mathbf{1}_{B}(x)\nu(dx),\,\,B\in\mathcal{B}.$
(2.6)

We let

$\displaystyle\mathcal{P}_{t}(x,B):=\mathcal{P}_{t}\mathbf{1}_{B}(x),\,\,\,\,\,\,\,\,B\in\mathcal{B},\,x\in\mathcal{X},$
(2.7)

denote the corresponding Markov transitions.

Note that if $L$ is as in (1.1) and $\phi\in
C^{1,2}_{0}([0,\infty)\times\mathcal{X};\mathbf{R})$, then _Dynkin’s formula_

$\displaystyle\mathbf{E}_{x}\phi(\sigma,x_{\sigma})=\phi(0,x)+\mathbf{E}_{x}\int_{0}^{\sigma}(\partial_{t}+L)\phi(s,x_{s})\,ds$
(2.8)

holds for any bounded stopping time $\sigma$ with respect to the filtration
$\mathcal{F}_{t}$. Dynkin’s formula allows one to study various properties of
$L$ by analyzing path properties of the stochastic process $x_{t}$.

######  Remark 2.3.

By a standard localization procedure applied to the equation (1.3) on
$\mathcal{X}_{k+1}$, the same formula (2.8) holds for any bounded stopping
time $\sigma\leq\tau_{\mathcal{X}_{k}}$ and any $\phi\in
C^{1,2}([0,\infty)\times\mathcal{X}_{k+1})$ regardless if $x_{t}$ is
nonexplosive.

As a simple application of Dynkin’s formula, we briefly recall the following
basic method for checking nonexplosivity. See also [Kha11, MT93, RB06].

######  Lemma 2.4.

Suppose there exists $w\in C^{2}(\mathcal{X};[0,\infty))$ and constants
$C,D>0$ such that

$\displaystyle
w_{k}:=\inf_{x\in\partial\mathcal{X}_{k}}w(x)\to\infty\,\,\text{ as
}\,\,k\to\infty\qquad\text{ and }\qquad Lw\leq Cw+D\,\,\text{ on
}\,\,\mathcal{X}.$

Then $x_{t}$ is nonexplosive as in (2.4).

###### Proof.

2.3, nonnegativity of $w$ and Dynkin’s formula together imply that if
$\tau_{k}=\tau_{\mathcal{X}_{k}}$, then

$\displaystyle w_{k}e^{-Ct}\mathbf{P}_{x}\\{\tau_{k}\leq t\\}$
$\displaystyle\leq\mathbf{E}_{x}w(x_{t\wedge\tau_{k}})e^{-C(t\wedge\tau_{k})}\mathbf{1}_{\\{\tau_{k}<t\\}}\leq\mathbf{E}_{x}w(x_{t\wedge\tau_{k}})e^{-C(t\wedge\tau_{k})}$
$\displaystyle=w(x)+\mathbf{E}_{x}\int_{0}^{t\wedge\tau_{k}}-Cw(x_{s})e^{-Cs}+e^{-Cs}Lw(x_{s})\,ds$
$\displaystyle\leq w(x)+\int_{0}^{t}e^{-Cs}D\,ds\leq w(x)+D.$

Rearranging the above we obtain $\mathbf{P}_{x}\\{\tau_{k}\leq t\\}\leq
e^{Ct}(w(x)+D)/w_{k}$. Passing $k\to\infty$, we find that
$\mathbf{P}_{x}\\{\tau_{\mathcal{X}}\leq t\\}=0$ for any given $t\geq 0$ and
$x\in\mathcal{X}$ and (2.4) follows. ∎

###  2.3. The process $x_{t}$ stopped on $\partial U$

We will often use the process $x_{t}$ _stopped_ on the boundary $\partial U$.
That is, define the stopping time

$\displaystyle\tau_{0}=\inf\\{t\geq 0\,:\,x_{t}\notin U\\}.$ (2.9)

Then the _stopped process_ $\tilde{x}_{t}$ is defined by

$\displaystyle\tilde{x}_{t}:=x_{t\wedge\tau_{0}},\,\,t\geq 0.$ (2.10)

Because $x_{t}$ is assumed to be nonexplosive, the stopped process
$\tilde{x}_{t}$ is a well-defined Markov process distributed on $\overline{U}$
for all times $t\geq 0$. We let $\tilde{\mathcal{P}}_{t}$ and
$\tilde{\mathcal{P}}_{t}(x,\,\cdot\,)$, $x\in\overline{U}$, respectively
denote the Markov semigroup and transitions associated to $\tilde{x}_{t}$.

######  Remark 2.5.

For the process $x_{t}$ with $x_{0}\in\overline{U}$, $\tau_{0}$ is the first
time $x_{t}$ hits the boundary. Note that $\tau_{0}$ is in general different
from the first positive exit time $\tau$ from $U$ when $x_{t}$ initiates on
the boundary $\partial U$, as the process started there may first enter the
region $U$ before exiting.

###  2.4. Hypoellipticity

Let us first define precisely the term _hypoelliptic_.

######  Definition 2.6.

Let $V\subset\mathbf{R}^{k}$ be non-empty, open set and $M$ be a differential
operator with coefficients belonging to $C^{\infty}(V)$. We say that $M$ is
_hypoelliptic_ on $V$ if for any distribution $v$ on $V$ with $Mv\in
C^{\infty}(W)$ for some $W\subset V$ nonempty open, we have $v\in
C^{\infty}(W)$.

Fundamental to our analysis are the smoothing properties of the differential
operator $L$ afforded by hypoellipticity. In the classical paper [Hör67],
Hörmander (see Theorem 2.9 below) provides a sufficient condition for
hypoellipticity on open $V\subset\mathbf{R}^{k}$ for operators $M$ of the form

$\displaystyle M=a+X_{0}+\tfrac{1}{2}\sum_{\ell=1}^{j}X_{\ell}^{2},$ (2.11)

where $a\in C^{\infty}(V;\mathbf{R})$ and $X_{0},X_{1},\ldots,X_{j}$ are
$C^{\infty}$ vector fields on $V$.

After some algebraic manipulations, note that

$\displaystyle L,L^{*},\partial_{t}\pm L,\partial_{t}\pm L^{*},$

where $L$ is as in (1.1) and $L^{*}$ denotes the formal $L^{2}(dx)$-adjoint of
$L$, can all be written in the form (2.11) on the respective open sets
$\mathcal{X},\mathcal{X},(0,\infty)\times\mathcal{X},(0,\infty)\times\mathcal{X}$.
For example, observe that if we let
$X_{i}=\sum_{j=1}^{\mathfrak{m}}\sigma_{ji}(x)\partial_{x_{j}}$,
$i=1,\ldots,r$, and $Y_{0}=\sum_{j=1}^{\mathfrak{m}}b_{j}(x)\partial_{x_{j}}$,
then

$\displaystyle L$
$\displaystyle=Y_{0}-\sum_{\ell=1}^{m}\bigg{[}\sum_{i=1}^{r}\sum_{j=1}^{m}\sigma_{ji}(x)\partial_{x_{j}}(\sigma_{\ell
i}(x))\bigg{]}\frac{\partial}{\partial
x_{\ell}}+\tfrac{1}{2}\sum_{j=1}^{r}X_{j}^{2}$ (2.12)
$\displaystyle=:X_{0}+\tfrac{1}{2}\sum_{j=1}^{r}X_{j}^{2}.$ (2.13)

To introduce Hörmander’s condition which implies hypoellipticity, we first
define the _Lie bracket_ $[X,Y]$ of differentiable vector fields
$X=\sum_{i}X^{i}\partial_{x_{i}}$ and $Y=\sum_{j}Y^{j}\partial_{x_{j}}$ on an
open set in $\mathbf{R}^{k}$ by

$\displaystyle[X,Y]=\sum_{j=1}^{k}\sum_{i=1}^{k}(X^{i}\partial_{x_{i}}(Y^{j})-Y^{i}\partial_{x_{i}}(X^{j}))\partial_{x_{j}}.$

######  Definition 2.7.

Suppose $V\subset\mathbf{R}^{k}$ is an open set and $M$ is an operator of the
form (2.11), where $X_{0},X_{1},\ldots,X_{m}$ are $C^{\infty}$ vector fields
on $V$ and $a\in C^{\infty}(V)$. Define the following $C^{\infty}$ vector
fields on $V$:

$\displaystyle X_{j},$ $\displaystyle j\in\\{0,1,\ldots,m\\}$ (2.14)
$\displaystyle[X_{i},X_{j}],$ $\displaystyle i,j\in\\{0,1,\ldots,m\\}$
$\displaystyle[[X_{i},X_{j}],X_{k}],$ $\displaystyle
i,j,k\in\\{0,1,2,\ldots,m\\}$ $\displaystyle\vdots$ $\displaystyle\vdots$

If the vector fields in (2.14) span the tangent space at all points in $V$, we
say $M$ satisfies the _Hörmander condition_ on $V$.

######  Remark 2.8.

If $M$ of the form (2.11) is elliptic, then $X_{1},X_{2},\ldots,X_{m}$ span
the tangent space at all points in $V$. Thus the brackets in (2.14) can be
viewed as a generalization of ellipticity. Relating this back to $L$ in (1.1)
and the equation (1.3), one can interpret Lie brackets in (2.13) as a
propagation of the randomness implicitly through the equation.

Hörmander’s theorem provides even more refined estimates on the smoothing
effect along every bracket in the list (2.14). Essentially, the more brackets
one takes to reach a certain direction, the smaller the smoothing effect
occurs along that direction. While we do not state this general result
rigorously, we need below the following simpler version to establish a
generalization of Bony’s Harnack inequality in Section 5.

######  Theorem 2.9 (Hörmander 1967 [Hör67]).

If $M$ defined in (2.11) satisfies the Hörmander condition on a nonempty open
set $V\subset\mathbf{R}^{k}$, then $M$ is hypoelliptic on $V$. More
specifically, there exists $\delta>0$ such that for any $\psi_{1},\psi_{2}\in
C^{\infty}_{0}(V)$ with $\psi_{2}=1$ on an open neighborhood of
$\text{\emph{supp}}(\psi_{1})$ we have

$\displaystyle\|\psi_{1}u\|_{H^{s+\delta}(\mathbf{R}^{k})}\leq
C_{s}(\|\psi_{2}Mu\|_{H^{s}(\mathbf{R}^{k})}+\|\psi_{2}u\|_{H^{s}(\mathbf{R}^{k})})$
(2.15)

for any $s\in\mathbf{R}$ and any distribution $u$ on $V$ with
$\psi_{2}u,\psi_{2}Mu\in H^{s}(\mathbf{R}^{k})$. In (2.15), $C_{s}$ is a
constant depending only on $s$, $\psi_{1}$, $\psi_{2}$, and the domain $V$ and
all functions are assumed to be zero outside of their compact supports.

######  Remark 2.10.

Note that hypoellipticity is a consequence of the inequality (2.15) by a
bootstrapping argument. As remarked above, although we mostly use
hypoellipticity somewhat independently of Hörmander’s result, we need the
precise estimate (2.15) for the proof of a generalization of Bony’s Harnack
inequality [Bon69].

Often, instead of listing operators that are hypoelliptic, we make a simper
hypothesis. See the _parabolic Hörmander condition_ below, which ensures that
an entire list of operators is hypoelliptic.

######  Definition 2.11.

Suppose that $V\subset\mathbf{R}^{k}$ is open and $M$ is an operator of the
form (2.11) where the $X_{0},X_{1},\ldots,X_{m}$ are $C^{\infty}$ vector
fields on $V$ and $a\in C^{\infty}(V)$. Let
$\mathcal{V}_{0}=\\{X_{1},X_{2},\ldots,X_{m}\\}$ and for $k\geq 1$ inductively
define

$\displaystyle\mathcal{V}_{k}=\\{[X_{i},X]\,:\,X\in\mathcal{V}_{k-1},i=0,1,2,\ldots,m\\}\qquad\textrm{and}\quad\mathcal{V}=\bigcup_{k=0}^{\infty}\mathcal{V}_{k}\,.$

If $\mathcal{V}$ spans $\mathbf{R}^{\mathfrak{m}}$ at all points in $V$, then
we say that $M$ satisfies the _parabolic Hörmander condition_ on $V$.

######  Remark 2.12.

Observe that for $M$ to satisfy the parabolic Hörmander condition, one cannot
include $X_{0}$ in the list of a spanning set. Rather, $X_{0}$ must be first
commuted with another vector field, for example $[X_{0},X_{1}]$.

######  Example 2.13.

Let

$\displaystyle M=\partial_{x_{2}}^{2}+\partial_{x_{1}}=:X_{1}^{2}+X_{0}.$

Then, $M$ satisfies the Hörmander condition on $\mathbf{R}^{2}$ but $M$ does
not satisfy the parabolic Hörmander condition on any open set in
$\mathbf{R}^{2}$.

Because the parabolic Hörmander condition limits the fields that can be taken
in a spanning set, a routine calculation shows that if $V\subset\mathcal{X}$
is non-empty, open and $L$ as in (1.1) is written in the form (2.13) and
satisfies the parabolic Hörmander condition on $V$, then all operators

$\displaystyle L,\,L^{*},\,L+\beta,\,L^{*}+\beta,\,\partial_{t}\pm
L,\,\partial_{t}\pm L^{*}$ (2.16)

where $\beta\in\mathbf{R}$ are hypoelliptic on the respective sets

$\displaystyle V,\,V,\,V,\,V,\,(0,\infty)\times V,\,(0,\infty)\times V.$
(2.17)

Below, we will see that hypoellipticity of the operators in (2.16) decides the
existence and regularity of densities related to the law of $x_{t}$.

###  2.5. Assumption list

Here we provide an almost complete list of assumptions used in the paper. It
is meant as a reference except for 2.14. Thus, aside from 2.14, the reader
should skip the rest of this section and come back to consult particular
assumptions used later in the paper.

######  Remark 2.14.

Throughout, we assume that $b\in
C^{\infty}(\mathcal{X};\mathbf{R}^{\mathfrak{m}})$ and $\sigma\in
C^{\infty}(\mathcal{X};M_{\mathfrak{m}\times r})$ without explicitly
mentioning it. Also, if $U$ is bounded, one does _not_ need to explicitly
assume that $x_{t}$ is nonexplosive using the extension argument from 2.2.

Depending on the context, we apply the following assumptions as needed.

(U00) $U\subset\mathbf{R}^{\mathfrak{m}}$ is nonempty, open set and $V:=U^{c}$
is nonempty, closed set with $\mathcal{X}\supset\overline{U}$ open.

(U0) $U\subset\mathbf{R}^{\mathfrak{m}}$ is nonempty, open set and
$\mathcal{X}\supset\overline{U}$ is an open set.

(NE) $x_{t}$ is nonexplosive as in 2.1.

(L1) $L$ is hypoelliptic on $U$.

(L2) $\partial_{t}-L^{*}$ is hypoelliptic on $(0,\infty)\times U$.

(L3) $\partial_{t}-L$ is hypoelliptic on $(0,\infty)\times U$.

(L4) $L^{*}$ is hypoelliptic on $U$.

(PH) $L$ satisfies the parabolic Hörmander condition on $U$ as in 2.11.

(UID($g,x_{*}$)) Fix $g:\overline{U}\to\mathbf{R}$ measurable and
$x_{*}\in\partial U$. Then for some $\delta>0$ the family

$\displaystyle\mathcal{G}_{g,\delta}(x_{*}):=\\{g(x_{\tau}(x))\,:\,|x-x_{*}|<\delta,\,x\in
U\\}$ (2.18)

is uniformly integrable. Here, $\tau$ is as in (2.2) and $x$ in $x_{\tau}(x)$
means $x_{0}=x$.

(UID($g$)) Fix $g:\overline{U}\rightarrow\mathbf{R}$ measurable. Then, for
every $x_{*}\in\partial U$, condition (UID($g,x_{*}$)) is satisfied.

(UIP($f,x_{*}$)) Fix $x_{*}\in\partial U$ and $f:\overline{U}\to\mathbf{R}$
measurable. Then for some $\delta>0$,

$\displaystyle\mathcal{G}_{\delta}^{f}(x_{*}):=\\{\textstyle{\int_{0}^{\tau}f(x_{s}(x))\,ds}\,:\,|x-x_{*}|<\delta,\,x\in
U\\}$ (2.19)

is uniformly integrable.

(UIP($f$)) Fix $f:\overline{U}\rightarrow\mathbf{R}$ measurable. Then for
every $x_{*}\in\partial U$, condition (UIP($f,x_{*}$)) is satisfied.

(CE($x_{*}$)) Fix $x_{*}\in\partial U$ and recall $\mathcal{X}_{n}$ introduced
in (2.3). For every $\delta_{1}>0$ there exists $n\in\mathbf{N}$ and
$\delta_{2}>0$ such that $|x-x_{*}|<\delta_{2}$, $x\in U$, implies

$\displaystyle\mathbf{P}_{x}\\{\tau_{\mathcal{X}_{n}}<\tau\\}<\delta_{1}.$
(2.20)

(CE) For every $x_{*}\in\partial U$, condition (CE($x_{*}$)) is satisfied.

In 6.7, we provide sufficient conditions for (CE($x_{*}$)) to hold.

##  3\. Remarks on Boundary Behavior

In order to solve the equation (1.2) in the classical sense, understanding the
behavior of the process $x_{t}$ satisfying the equation (1.3) near the
boundary $\partial U$ is critical. In this section, we explore various
conditions related to boundary behavior used in the literature to ensure well-
posedness of equation (1.2) (in the classical sense) when $L$ fails to be
uniformly elliptic.

###  3.1. Nice points and regular points

######  Definition 3.1.

We call $x_{*}\in\partial U$ _nice_ if there exists an open neighborhood
$U_{x_{*}}\subset\mathcal{X}$ of $x_{*}$ and a function $w\in
C^{2}(U_{x_{*}})$ satisfying the following conditions:

  * (i)

$w>0$ on $U_{x_{*}}\setminus\\{x_{*}\\}$ and $w(x_{*})=0$;

  * (ii)

$Lw<0$ on $U_{x_{*}}$.

######  Remark 3.2.

We show in 3.9 below that if $x_{*}\in\partial U$ is nice and the relevant
hypoellipticity is satisfied, then the process $x_{t}$ exits $U$
instantaneously when started from $x_{*}$. However, proving that $x_{*}$ is
nice; that is, finding a Lyapunov function $w$ in 3.1, can be highly
nontrivial or even impossible. One can ensure $x_{*}\in\partial U$ is nice
provided $\partial U$ has an exterior normal vector to $U$ at $x_{*}$ and
randomness pointing in the direction of the normal vector (see 3.10 below).
Intuitively, the process $x_{t}$ projected onto this normal direction for
small times behaves like a scaled one-dimensional Brownian motion. Then, the
process must exit the domain instantaneously as the one-dimensional Brownian
motion has no preferred direction and dominates the motion in small times. See
[CFH21, Section 7] for further details.

######  Remark 3.3.

If $U$ is furthermore assumed to be a bounded and $L$ is assumed to be
hypoelliptic, then one can show that the Dirichlet problem (1.2) with $f\equiv
0$, has a unique classical solution if all points on the boundary are nice.
Although the main result in [Ram97] establishes this fact, it is not exactly
stated in this way.

In this paper, we find it more convenient to phrase our hypotheses in terms of
stopping times for the process $x_{t}$ solving (1.3). Hence, we define

$\displaystyle\overline{\tau}=\inf\\{t\geq
0\,:\,x_{t}\notin\overline{U}\\}\,,$ (3.1)

and recall the first positive exit time $\tau$ from $U$ defined in equation
(2.2).

######  Definition 3.4.

We call an interior point $x\in U$ _regular_ if
$\mathbf{P}_{x}\\{\tau=\overline{\tau}\\}=1$. A boundary point
$x_{*}\in\partial U$ is called _regular_ if
$\mathbf{P}_{x_{*}}\\{\overline{\tau}=0\\}=1$. Points on $\overline{U}$ are
called _irregular_ otherwise. We call the set $U$ _interior regular_ if all
$x\in U$ are regular. We call $U$ _boundary regular_ if all $x_{*}\in\partial
U$ are regular.

######  Remark 3.5.

Observe that if $x\in U$ is regular, then the process $x_{t}$ started from $x$
exits $U$ and $\overline{U}$ at the same time. In particular, $x_{t}$ cannot
reach $\partial U$ and return to the interior of $U$ with positive
probability. On the other hand, $x_{*}\in\partial U$ being regular means
$x_{t}$ initiated at $x_{*}$ must exit $\overline{U}$ instantaneously.

######  Remark 3.6.

Note that the event $\\{\overline{\tau}=0\\}$ belongs to the _germ sigma
field_ $\bigcap_{t>0}\mathcal{F}_{t}$, hence has probability $0$ or $1$ by
Blumenthal’s $0$-$1$ law. Thus, $x_{*}\in\partial U$ being regular is
equivalent to $\mathbf{P}_{x_{*}}\\{\overline{\tau}=0\\}>0$.

Using the strong Markov property for $x_{t}$, the next result states that
boundary regular implies interior regular.

######  Proposition 3.7.

If $U$ is boundary regular, then $U$ is interior regular.

###### Proof.

Let $x\in U$ and suppose $\tau=\infty$. Then $\overline{\tau}\geq\tau=\infty$,
so $\tau=\overline{\tau}$. If the event $\\{\tau<\infty\\}$ has positive
probability, then the strong Markov property and boundary regularity gives

$\displaystyle\mathbf{P}_{x}\\{\tau=\overline{\tau},\,\tau<\infty\\}=\mathbf{E}_{x}[\mathbf{E}_{x}\mathbf{1}_{\\{\tau=\overline{\tau},\,\tau<\infty\\}}|\mathcal{F}_{\tau}]=\mathbf{E}_{x}\mathbf{1}_{\\{\tau<\infty\\}}\mathbf{P}_{x_{\tau}}\\{\overline{\tau}=0\\}=\mathbf{P}_{x}\\{\tau<\infty\\}.$

This finishes the proof since

$\displaystyle\mathbf{P}_{x}\\{\tau=\overline{\tau}\\}$
$\displaystyle=\mathbf{P}_{x}\\{\tau=\overline{\tau},\,\tau=\infty\\}+\mathbf{P}_{x}\\{\tau=\overline{\tau},\,\tau<\infty\\}$
$\displaystyle=\mathbf{P}_{x}\\{\tau=\infty\\}+\mathbf{P}_{x}\\{\tau<\infty\\}=1.$

∎

As the next example shows, the converse of 3.7 is false even if the parabolic
Hörmander condition is satisfied.

######  Example 3.8.

Let $U\subset\mathbf{R}^{2}$ be the interior of the open square with vertices
$(-1,1)$, $(1,1)$, $(1,-1)$, $(-1,-1)$, and consider the following SDE on $U$

$\displaystyle dx^{1}_{t}$ $\displaystyle=-(x_{t}^{2})^{2}\,dt\,,$
$\displaystyle dx^{2}_{t}$ $\displaystyle=\sqrt{2}\,dB_{t}\,,$

where $B_{t}$ is a standard, one-dimensional Brownian motion. Let $I$ be the
open segment connecting $(1,-1)$ and $(1,1)$. Note that all points on
$I\subset\partial U$ are not regular since $x_{t}^{1}$ is decreasing for all
times. On the other hand, $x_{t}=(x_{t}^{1},x_{t}^{2})$ has generator

$\displaystyle
L=-(x^{2})^{2}\partial_{x^{1}}+\partial_{x^{2}}^{2}=:X_{0}+X_{1}^{2},$

which satisfies the parabolic Hörmander condition on $\mathbf{R}^{2}$ since

$X_{1}=\partial_{x^{2}}\,\,\text{ and
}\,\,[X_{1},[X_{1},X_{0}]]=-2\partial_{x^{1}}$

span the tangent space at all points in $\mathbf{R}^{2}$. Also, observe that
the process initiated at $x\in U$ can only exit $U$ on $\partial U\setminus
I$. Hence, $U$ is interior regular since every point on $\partial U\setminus
I$ is regular.

Next, we clarify a the relationship between boundary regular and nice.

######  Proposition 3.9.

Suppose $x_{*}\in\partial U$ is nice and that $\partial_{t}\pm L$ and
$\partial_{t}\pm L^{*}$ are hypoelliptic on $(0,\infty)\times
B_{\delta}(x_{*})$ for some $\delta>0$ such that
$B_{\delta}(x_{*})\subset\mathcal{X}$. Then, $x_{*}$ is regular.

###### Proof.

Let $x_{*}\in\partial U$ be nice and fix $\delta>0$ such that $\partial_{t}\pm
L^{*}$, $\partial_{t}\pm L^{*}$ are hypoelliptic on $(0,\infty)\times
B_{\delta}(x_{*})$. Choose an open neighborhood $U_{x_{*}}$ of $x_{*}$ such
that there exists a function $w\in C^{2}(U_{x_{*}})$ satisfying properties (i)
and (ii) in 3.1. Without loss of generality, we may assume that
$V:=U_{x_{*}}=B_{\delta}(x_{*})$. It follows that the distribution $\mu_{t}$
of the stopped process $x_{t}^{V}:=x_{t\wedge\tau_{V}}$ (see 4.3) satisfies
Fokker-Planck equation $(\partial_{t}+L^{*})(\mu_{t})=0$ in the sense of
distributions on $(0,\infty)\times V$. Thus, when restricted to subsets of
$(0,\infty)\times V$, $\mu_{t}$ is absolutely continuous with respect to
Lebesgue measure with density $p_{t}$ which is smooth on $(0,\infty)\times V$.
In other words, the law of $x_{t}^{V}$ restricted to subsets of $V$, for fixed
$t$, is absolutely continuous with respect to Lebesgue measure on $V$.

If $x_{*}$ is not regular, then $\mathbf{P}_{x_{*}}\\{\overline{\tau}>0\\}=1$.
Thus, there exists $\epsilon>0$ such that $B_{\epsilon}(x_{*})\subset V$ and
$\mathbf{P}_{x_{*}}\\{\tau_{B_{\epsilon}(x_{*})}<\overline{\tau}\\}>0$, for
otherwise the distribution of $x_{t}^{V}$ started at $x_{*}$ would have non-
zero mass concentrated at $x_{*}$, violating the absolute continuity of
$\mu_{t}$ above. For
$\sigma(t):=t\wedge\tau_{B_{\epsilon}(x_{*})}\wedge\overline{\tau}$, Dynkin’s
formula (see (2.8)) and positivity of $w$ on $\partial B_{\epsilon}(x_{*})$
gives

$\displaystyle
c\mathbf{P}_{x_{*}}\\{\tau_{B_{\epsilon}(x_{*})}<\overline{\tau}\wedge
t\\}\leq\mathbf{E}_{x_{*}}w(x_{\sigma(t)})\leq w(x_{*})=0$

for some constant $c>0$ independent of $t$. Passing $t\rightarrow\infty$ we
obtain $\mathbf{P}_{x_{*}}\\{\tau_{B_{\epsilon}(x_{*})}<\overline{\tau}\\}=0$,
a contradiction. ∎

######  Remark 3.10.

Fix $x_{*}\in\partial U$ and as in 3.9 suppose that $\partial_{t}\pm L$ and
$\partial_{t}\pm L^{*}$ are hypoelliptic on $(0,\infty)\times
B_{\delta}(x_{*})$ for some $\delta>0$ such that
$B_{\delta}(x_{*})\subset\mathcal{X}$. Moreover, suppose that $\partial U$
satisfies the _exterior sphere condition_ at $x_{*}$; that is, there exists
$\lambda>0$ such that $x_{*}\in\overline{B_{\lambda}(x^{\prime})}$ and
$B_{\lambda}(x^{\prime})\subseteq\overline{U}^{c}$, where
$B_{\lambda}(x^{\prime})$ is the open ball centered at
$x^{\prime}=x_{*}+\lambda\nu(x_{*})$ with $\nu(x_{*})$ being a unit exterior
normal vector to $\partial U$ at $x_{*}$. It is standard to see that we can
decrease $\lambda$ in the definition of exterior sphere condition if
necessary.

Then, if the noise has a non-zero component in $\nu(x_{*})$ at $x_{*}$, then
$x_{*}$ is regular, see for example [Bon69]. Such an assumption in fact
implies that $x_{*}$ is nice. Indeed, suppose

$\displaystyle\sum_{i,j=1}^{\mathfrak{m}}(\sigma(x_{*})\sigma^{T}(x_{*}))_{ij}v_{i}(x_{*})v_{j}(x_{*})>0.$
(3.2)

Then, by choosing $\beta>0$ large enough, the function

$\displaystyle
w(x)=e^{-\beta|x^{\prime}-x_{*}|^{2}}-e^{-\beta|x^{\prime}-x|^{2}}$ (3.3)

satisfies 3.1 at $x_{*}$ on $U_{x_{*}}=B(x_{*},|x^{\prime}-x_{*}|/2)\cap U$ if
$x^{\prime}=x_{*}+\lambda\nu(x_{*})$ and $\lambda>0$ is sufficiently small.
Note that (3.2) and large $\beta>0$ allows one to to disregard terms in $Lw$
of order $\beta$.

###  3.2. Modification of the domain

We conclude this section by restating [CFH21, Corollary 7.10] in our context.
Intuitively, it asserts that starting from a _reasonable_ domain $U$, provided
there is noise in a fixed direction for all points on the boundary $\partial
U$ (see 3.10), one can slightly modify $U$ to produce an approximate domain
$V$ which is boundary regular. Thus, by 3.7, $V$ is both interior regular and
boundary regular.

In order to state the result, we say that $U$ has _non-flat_ boundary
$\partial U$ if for each $x_{*}\in\partial U$ and every $r>0$, the set
$\partial U\cap B_{r}(x_{*})$ is not a subset of a hyperplane. Then the
following result was proved in [CFH21, Corollary 7.10].

######  Theorem 3.11.

Suppose that $U\subset\mathbf{R}^{\mathfrak{m}}$, $\mathfrak{m}\geq 2$, is a
bounded, convex, non-flat domain with $C^{1}$ boundary $\partial U$. Suppose,
furthermore, that there exists a fixed unit vector
$v\in\mathbf{R}^{\mathfrak{m}}$ such that $v$ belongs to the column space of
$\sigma(x)$ for all $x\in\partial U$. Then, for every $\epsilon>0$, there
exists a non-empty, open convex domain $V_{\epsilon}\subset U$ with piecewise
linear boundary $\partial V_{\epsilon}$ such that $V_{\epsilon}$ and
$V_{\epsilon}^{c}$ are boundary regular for $x_{t}$ and
$|U|-|V_{\epsilon}|<\epsilon$. Here, $|\cdot|$ denotes Lebesgue measure on
$\mathbf{R}^{\mathfrak{m}}$.

To prove 3.11 one defines $V_{\epsilon}$ to be the interior of the convex hull
of a sufficient number of points
$x_{1},x_{2},\ldots,x_{n(\epsilon)}\in\partial U$ selected independently and
randomly according to Hausdorff measure on $\partial U$. One can then show
that with probability one, the faces generated by these points on $\partial
V_{\epsilon}$ are not parallel to the fixed vector $v$ in 3.11, so that the
process exits $\overline{V_{\epsilon}}$ instantaneously when started there.
The fact that the volumes of $V_{\epsilon}$ and $U$ are close is intuitive
provided $n(\epsilon)$ is large enough, which was rigorously proved in [SW03].

##  4\. Interior regularity of functionals of $\tau$

Our goal in this section is to obtain interior smoothness of various
functionals of the first positive exit time $\tau$ from $U$. Given the
relevant hypoellipticity, the key first step is to show that the law of the
stopped process $\tilde{x}_{t}$, introduced in Section 2.3, when restricted to
Borel subsets of $U$, denoted by $\mathcal{B}_{U}$, has a density
$\tilde{p}_{t}(x,y)$ with respect to Lebesgue measure on $U$. Furthermore, the
mapping $(t,x,y)\mapsto\tilde{p}_{t}(x,y)\in C^{\infty}((0,\infty)\times
U\times U)$ and satisfies, respectively, the forward and backward Kolmogorov
equations in the classical sense:

$\displaystyle\partial_{t}\tilde{p}_{t}(x,y)=L_{y}^{*}\tilde{p}_{t}(x,y)\,\,\,\text{
on }\,\,\,(0,\infty)\times U,\,x\in U\text{ fixed},$ (4.1)
$\displaystyle\partial_{t}\tilde{p}_{t}(x,y)=L_{x}\tilde{p}_{t}(x,y)\,\,\,\text{
on }\,\,\,(0,\infty)\times U,\,y\in U\text{ fixed}.$ (4.2)

This is done in Section 4.2. In Section 4.3, we deduce interior smoothness and
the equations satisfied by

$\displaystyle x\mapsto\mathbf{E}_{x}\varphi(\tau):U\to\mathbf{R}$ (4.3)

for some choices of smooth $\varphi:[0,\infty)\to\mathbf{R}$, for example
$\varphi(x)=x^{k}$, $k\in\mathbf{N}$ or
$\varphi(x)=\mathbf{1}_{(t,\infty)}(x)$ for fixed $t$.

First, however, we need to establish a few auxiliary results.

###  4.1. Auxiliary results

######  Lemma 4.1.

Suppose that conditions (U0) and (NE) are satisfied. Then, the following
assertions hold true.

  * (i)

For any $t>0$, $x\in\mathcal{X}$, and $M>0$ we have

$\lim_{y\to
x}\mathbf{E}\bigg{\\{}\sup_{s\in[0,t]}|x_{s}(x)-x_{s}(y)|^{2}\wedge
M\bigg{\\}}=0\,,$

where $s\mapsto x_{s}(z)$ satisfies (1.3) with $x_{0}(z)=z$.

  * (ii)

If $x_{*}\in\partial U$ is regular, then for any $\epsilon,\delta>0$, there is
$\rho>0$ such that $\mathbf{P}_{x}\\{\tau\geq\delta\\}\leq\epsilon$ for any
$x\in U$ with $|x-x_{*}|<\rho$. That is, as $x\rightarrow x_{*}\in\partial
U,x\in U$, $\tau(x)\rightarrow\tau(x_{*})$ in probability.

###### Proof.

The proof of part (i) follows a standard Grönwall-type comparison argument.
Let $\tau_{k}=\tau_{\mathcal{X}_{k}}$ and we denote the dependence of
$\tau_{k}$ on the initial condition $x$ of $x_{t}$ by writing $\tau_{k}(x)$.
Observe that for $x,y\in\mathcal{X}$ and $s\leq\tau_{k}(x)\wedge\tau_{k}(y)$,
the processes $x_{s}(x)$ and $x_{s}(y)$ respectively agree pathwise with
processes $x_{s,k}(x)$ and $x_{s,k}(y)$ satisfying Itô SDEs with globally
Lipschitz coefficients (cf. 2.2). Thus if $\Delta_{t}=x_{t}(x)-x_{t}(y)$ and
$\Delta_{t,k}=x_{t,k}(x)-x_{t,k}(y)$, then for any $\sigma\leq
t\wedge\tau_{k}(x)\wedge\tau_{k}(y)$ with $t\geq 0$ deterministic, we have the
estimate

$\displaystyle\mathbf{E}\sup_{s\in[0,\sigma]}|\Delta_{s}|^{2}\leq\mathbf{E}\sup_{s\in[0,t]}|\Delta_{s,k}|^{2}.$

Hence, to estimate $\mathbf{E}\sup_{s\in[0,\sigma]}|\Delta_{s}|^{2}$ it
suffices, for a given deterministic time $t\geq 0$ to estimate
$\mathbf{E}\sup_{s\in[0,t]}|\Delta_{s}|^{2}$, where
$\Delta_{t}=x_{t}(x)-x_{t}(y)$ satisfies

$\Delta_{t}=x-y+\int_{0}^{t}[b(x_{s}(x))-b(x_{s}(y))]\,ds+\int_{0}^{t}[\sigma(x_{s}(x))-\sigma(x_{s}(y))]\,d\mathbf{W}_{s}$
(4.4)

with globally Lipschitz $b,\sigma$ on $\mathbf{R}^{\mathfrak{m}}$ and
Lipschitz constants depending on $k$. Hence, there exists a constant $Q_{k}>0$
such that

$\displaystyle|b(x)-b(y)|+|\sigma(x)-\sigma(y)|\leq Q_{k}|x-y|\,\,\,\text{ for
all }\,\,\,x,y\in\mathbf{R}^{\mathfrak{m}}.$

Using Doob’s maximal inequality and Itô isometry, we have

$\displaystyle\mathbf{E}\sup_{s\in[0,t]}|\Delta_{s}|^{2}$ $\displaystyle\leq
9|x-y|^{2}+9Q_{k}^{2}\mathbf{E}\sup_{s\in[0,t]}\left(\int_{0}^{s}|x_{u}(x)-x_{u}(y)|\,du\right)^{2}$
$\displaystyle\qquad+9\mathbf{E}\sup_{s\in[0,t]}\left|\int_{0}^{s}(\sigma(x_{u}(x))-\sigma(x_{u}(y)))d\mathbf{W}_{u}\right|^{2}$
$\displaystyle\leq
9|x-y|^{2}+C_{k}(t)\int_{0}^{t}\mathbf{E}\sup_{s\in[0,u]}|\Delta_{s}|^{2}du$

for some constant $C_{k}(t)$ depending only on $k,t$. Consequently by
Grönwall’s inequality, we have

$\mathbf{E}\sup_{s\in[0,t]}|\Delta_{s}|^{2}\leq
9|x-y|^{2}e^{C_{k}(t)t}=:|x-y|^{2}D_{k}(t).$ (4.5)

Now if $b,\sigma$ are no longer globally Lipschitz and
$\delta_{k}=\text{dist}(\mathcal{X}_{k},\partial\mathcal{X}_{k+1})>0$, we have
by Chebyshev’s inequality and (4.5)

$\displaystyle\mathbf{E}\sup_{s\in[0,t]}|\Delta_{s}|^{2}\wedge M$
$\displaystyle=\mathbf{E}\sup_{s\in[0,t]}|\Delta_{s}|^{2}\wedge
M(\textbf{1}\\{\tau_{k}(x)<t\\}+\textbf{1}\\{\tau_{k}(x)\geq t\\})$
$\displaystyle\leq
M\mathbf{P}_{x}\\{\tau_{k}<t\\}+\mathbf{E}_{x}\sup_{s\in[0,t]}|\Delta_{s,k}|^{2}\mathbf{1}\\{\tau_{k}(x)\geq
t,\tau_{k+1}(y)\geq
t\\}+M\mathbf{P}\\{\sup_{s\in[0,t]}|\Delta_{s,k}|^{2}>\delta_{k}^{2}\\}$
$\displaystyle\leq
M\mathbf{P}_{x}\\{\tau_{k}<t\\}+D_{k}(t)|x-y|^{2}\,+\frac{|x-y|^{2}D_{k}(t)}{\delta_{k}^{2}}M.$

Passing $y\to x$ and then $k\to\infty$ using nonexplosivity of $x_{t}(x)$, we
obtain (i).

To obtain (ii), suppose there exist $\epsilon_{0},\delta_{0}>0$ and a sequence
$\\{x_{n}\\}\subset U$ with $x_{n}\rightarrow x_{*}$ such that

$\displaystyle\mathbf{P}_{x_{n}}\\{\tau\geq\delta_{0}\\}\geq\epsilon_{0}$

for all $n$. For any $\epsilon>0$, let $U^{+}_{\epsilon}$ be given by

$\displaystyle
U_{\epsilon}^{+}=\\{x\in\mathcal{X}\,:\,\text{dist}(x,\overline{U})<\epsilon\\}.$
(4.6)

By path continuity, $\overline{\tau}=\lim_{\epsilon\rightarrow
0}\tau_{U_{\epsilon}^{+}}$ almost surely. Since $x_{*}$ is regular, there is
$\epsilon\in(0,1)$ so that

$\displaystyle\mathbf{P}_{x_{*}}\\{\tau_{U_{\epsilon}^{+}}<\delta_{0}/2\\}\geq
1-\epsilon_{0}/2\,,$

and consequently for any $n$

$\displaystyle\mathbf{P}\\{\tau(x_{n})\geq\delta_{0},\tau_{U_{\epsilon}^{+}}(x_{*})<\delta_{0}/2\\}\geq\epsilon_{0}/2\,,$

where the initial condition is indicated in the stopping times above. By
conclusion (i) with $M=1$, and $\epsilon<1$ for any large enough $n$ one has
$\mathbf{P}\\{\sup_{s\in[0,\delta_{0}/2]}|x_{s}(x_{n})-x_{s}(x_{*})|>\epsilon/2\\}\leq\epsilon_{0}/4$.
But then

$\displaystyle\frac{\epsilon_{0}}{2}$
$\displaystyle\leq\mathbf{P}\left\\{\tau(x_{n})\geq\delta_{0},\tau_{U_{\epsilon}^{+}}(x_{*})<\frac{\delta_{0}}{2},\sup_{s\in[0,\delta_{0}/2]}|x_{s}(x_{n})-x_{s}(x_{*})|\leq\frac{\epsilon}{2}\right\\}$
$\displaystyle\qquad+\mathbf{P}\left\\{\sup_{s\in[0,\delta_{0}/2]}|x_{s}(x_{n})-x_{s}(x_{*})|>\frac{\epsilon}{2}\right\\}\leq
0+\frac{\epsilon_{0}}{4},$

a contradiction. ∎

In addition to 4.1, we need one more auxiliary result. Although this result is
basic, it is used repeatedly throughout the paper.

######  Lemma 4.2.

Let $V\subset\mathbf{R}^{k}$ be open and let $\mathcal{L}$ and
$\mathcal{L}_{n}$, $n\in\mathbf{N}$, be linear second-order differential
operators with $C^{\infty}(V)$ coefficients. Suppose that $\\{v_{n}\\}$ be a
uniformly bounded sequence of measurable functions $v_{n}:V\to\mathbf{R}$ and
that the following conditions are satisfied:

  * -

For every $\phi\in C_{0}^{\infty}(V)$,
$(\mathcal{L}_{n}-\mathcal{L})^{*}\phi\to 0$ as $n\to\infty$ in $L^{1}(V,dx)$
where the $*$ denotes the formal adjoint with respect to $L^{2}(V,dx)$-inner
product;

  * -

For some measurable function $f$ on $V$ which is bounded on compact subsets of
$V$, $\mathcal{L}_{n}v_{n}\to-f$ as $n\rightarrow\infty$ on $V$ in the sense
of distributions.

  * -

$v_{n}$ converges to $v_{\infty}$ in the sense of distributions on $V$ as
$n\rightarrow\infty$.

Then, $\mathcal{L}v_{\infty}=-f$ on $V$ in the sense of distributions.
Furthermore, if $\mathcal{L}$ is hypoelliptic on $V$ and $f\in C^{\infty}(V)$,
then $v_{\infty}\in C^{\infty}(V)$ and $\mathcal{L}v_{\infty}=-f$ on $V$ in
the classical sense.

###### Proof.

For fixed $\varphi\in C_{0}^{\infty}(V)$, conditions of the statement and the
Dominated Convergence Theorem imply

$\displaystyle\int_{V}v_{\infty}\mathcal{L}^{*}\varphi\,dx=\lim_{n\rightarrow\infty}\int_{V}v_{n}\mathcal{L}^{*}\varphi\,dx$
$\displaystyle=\lim_{n\to\infty}\int_{V}v_{n}\mathcal{L}_{n}^{*}\varphi-\lim_{n\to\infty}\int_{U}v_{n}(\mathcal{L}_{n}-\mathcal{L})^{*}\varphi\,dx$
$\displaystyle=\int_{V}-f\varphi\,dx.$

Thus $\mathcal{L}v_{\infty}=-f$ on $V$ in the sense of distributions. The
remaining assertion is an immediate consequence of hypoellipticity. ∎

###  4.2. The forward and backward Kolmogorov equations for the stopped
process

Recall that if $x_{t}$ is nonexplosive, then the stopped process
$\tilde{x}_{t}$, see (2.10), is a well-defined, continuous-time Markov process
distributed on $\overline{U}$. Moreover, the associated semigroup and
transitions are respectively denoted by $\tilde{\mathcal{P}}_{t}$ and
$\tilde{\mathcal{P}}_{t}(x,\,\cdot\,)$.

Our main result in this section is the following.

######  Theorem 4.3.

Suppose that (U0), (NE), and (L2) are satisfied.

  * (i)

For all $x\in U$ and $t>0$, the restriction of the measure
$\tilde{\mathcal{P}}_{t}(x,\,\cdot\,)$ to Borel subsets of $U$ is absolutely
continuous with respect to Lebesgue measure on $U$ with density
$\tilde{p}_{t}(x,y)$, and for fixed $x\in U$ the mapping
$(t,y)\mapsto\tilde{p}_{t}(x,y)\in C^{\infty}((0,\infty)\times U)$.
Furthermore, the forward Kolmogorov equation (4.1) is satisfied in the
classical sense.

  * (ii)

If (L3) is furthermore satisfied, then $(t,x,y)\mapsto\tilde{p}_{t}(x,y)$
belongs to $C^{\infty}((0,\infty)\times U\times U)$ and the backward
Kolmogorov equation (4.2) is satisfied in the classical sense.

######  Remark 4.4.

The above result is natural and understood by experts in the field. However,
we found it difficult to locate a complete proof of part (ii) as it is much
more subtle than part (i). While part (i) follows almost immediately from
Dynkin’s formula, part (ii) requires several nontrivial approximations and
steps.

######  Remark 4.5.

Intuitively the result above holds because, while the measure
$\tilde{\mathcal{P}}_{t}(x,\,\cdot\,)$ has a singular component on the
boundary if the process exits $U$ by time $t$ with positive probability, when
restricted to subsets of $U$ this singularity is not seen.

######  Remark 4.6.

Note that in the statement above we may take $U=\mathcal{X}$, in which case
$\tilde{x}_{t}$ coincides with the original process $x_{t}$ on $\mathcal{X}$,
and thus the statement is about the law of $x_{t}$ in $\mathcal{X}$.

###### Proof of 4.3.

To prove part (i), fix $t>0$, $x\in U$, and take any $\psi\in
C^{\infty}([0,t]\times U)\cap C^{\infty}_{0}((0,t)\times U)$. Thus, in
particular, $\psi$ is compactly supported in $(0,t)\times U$. Then, Dynkin’s
formula yields

$\displaystyle
0=\mathbf{E}_{x}\psi(t,\tilde{x}_{t})=\mathbf{E}_{x}\psi(t,x_{t\wedge\tau_{0}})$
$\displaystyle=\psi(0,x)+\mathbf{E}_{x}\int_{0}^{t\wedge\tau_{0}}(L+\partial_{s})\psi(s,x_{s})\,ds$
$\displaystyle=0+\mathbf{E}_{x}\int_{0}^{t}(L+\partial_{s})\psi(s,\tilde{x}_{s})\,ds\,,$

where $\tau_{0}$ from (2.9) is the first exit time from $U$. Hence,

$\mathbf{E}_{x}\int_{0}^{t}(L+\partial_{s})\psi(s,\tilde{x}_{s})=\int_{0}^{t}\int_{U}(L+\partial_{s})\psi(s,y)\tilde{p}_{s}(x,dy)ds=0$

for any $\psi\in C_{0}^{\infty}((0,t)\times U)$. That is,
$(L^{*}_{y}-\partial_{t})(\tilde{p}_{t}(x,dy))=0$ on $(0,\infty)\times U$ in
the sense of distributions. By hypoellipticity of $\partial_{t}-L^{*}$ on
$(0,\infty)\times U$ as in (L2), part (i) follows.

Part (ii) is more involved as there are several layers of approximations. We
divide the proof into two main steps.

_Step 1_. Let $\psi\in C_{0}^{\infty}(U)$ and consider the mapping
$(t,x)\mapsto u(t,x):=\tilde{\mathcal{P}}_{t}\psi(x)$. We claim that $u\in
C^{\infty}((0,\infty)\times U)$ and $\partial_{t}u=Lu$ in the classical sense
on $(0,\infty)\times U$.

_Step 2_ .We prove part (ii) by taking a sequence of approximations of
$\tilde{p}_{t}(x,y)$ of the form $\tilde{\mathcal{P}}_{t}\psi_{n}(x)$,
$\psi_{n}\in C_{0}^{\infty}(U)$.

_Proof of Step 1_. First observe that since $x_{t}$ is nonexplosive, the
stopped process $\tilde{x}_{t}=x_{t\wedge\tau_{0}}$ with initial condition
$\tilde{x}_{0}=x\in U$ satisfies the Itô SDE

$\displaystyle
d\tilde{x}_{t}=b(\tilde{x}_{t})\mathbf{1}_{U}(\tilde{x}_{t})\,dt+\sigma(\tilde{x}_{t})\mathbf{1}_{U}(\tilde{x}_{t})\,d\mathbf{W}_{t}.$

We first show $u\in C^{\infty}((0,\infty)\times U)$ and $\partial_{t}u=Lu$ in
the classical sense on $(0,\infty)\times U$ in the special case when
$b,\sigma\in C^{\infty}(\mathcal{X})$ are bounded with bounded derivatives of
all orders.

Recall that $U_{n}$ is a sequence of bounded open sets with
$\overline{U}_{n}\subset U$ and $U_{n}\uparrow U$ as $n\to\infty$. Suppose
that $\varphi_{n}\in C^{\infty}(\overline{U};[0,1])$ satisfies
$\varphi_{n}(x)=1$ on $U_{n}$ and $\varphi_{n}(x)=0$ for $x\in U_{n+1}^{c}$.
Consider a sequence of approximating processes $x_{t}^{n}$ with $x_{0}^{n}=x$
and

$\displaystyle dx_{t}^{n}$
$\displaystyle=b(x_{t}^{n})\varphi_{n}(x_{t}^{n})\,dt+\sigma(x_{t}^{n})\varphi_{n}(x_{t}^{n})\,d\mathbf{W}_{t}.$
(4.7)

Claim. If $b,\sigma\in C^{\infty}(\mathcal{X})$ are bounded with bounded
derivatives of all orders, then for any $t>0$ fixed,

$\mathbf{E}_{x}\sup_{r\in[0,t]}|\tilde{x}_{r}-x_{r}^{n}|^{2}\to 0\,\,\text{ as
}\,\,n\to\infty.$ (4.8)

###### Proof of (4.8)..

Observe that

$\displaystyle\tilde{x}_{t}-x_{t}^{n}$
$\displaystyle=\int_{0}^{t}\varphi_{n}(x_{s}^{n})[b(\tilde{x}_{s})-b(x_{s}^{n})]\,ds+\int_{0}^{t}\varphi_{n}(x_{s}^{n})[\sigma(\tilde{x}_{s})-\sigma(x_{s}^{n})]\,d\mathbf{W}_{s}$
$\displaystyle\qquad+\int_{0}^{t}(1-\varphi_{n}(x_{s}^{n}))\mathbf{1}_{U}(\tilde{x}_{s})b(\tilde{x}_{s})\,ds+\int_{0}^{t}(1-\varphi_{n}(x_{s}^{n}))\mathbf{1}_{U}(\tilde{x}_{s})\sigma(\tilde{x}_{s})\,d\mathbf{W}_{s}.$

Hence, squaring both sides, taking the supremum and using Doob’s maximal
inequality we find that for all $t\leq T$

$\displaystyle\mathbf{E}\sup_{r\in[0,t]}|\tilde{x}_{r}-x_{r}^{n}|^{2}$
$\displaystyle\leq
C_{1}\int_{0}^{t}\mathbf{E}\sup_{r\in[0,s]}|\tilde{x}_{r}-x_{r}^{n}|^{2}\,ds+C_{2}\int_{0}^{t}\mathbf{E}[(1-\varphi_{n}(x_{s}^{n}))\mathbf{1}_{U}(\tilde{x}_{s})]^{2}\,ds\,,$

where the constants
$C_{1}=C_{1}(T,\|b\|_{\text{Lip}},\|\sigma\|_{\text{Lip}})>0$ and
$C_{2}=C_{2}(T,\|b\|_{L^{\infty}},\|\sigma\|_{L^{\infty}})>0$ do _not_ depend
on $n$. Grönwall’s inequality then implies that for all $t\leq T$

$\displaystyle\mathbf{E}\sup_{r\in[0,t]}|\tilde{x}_{r}-x_{r}^{n}|^{2}$
$\displaystyle\leq
C_{2}e^{tC_{1}}\int_{0}^{t}\mathbf{E}[(1-\varphi_{n}(x_{s}^{n}))\mathbf{1}_{U}(\tilde{x}_{s})]^{2}\,ds\,.$

Next, observe that if $\tilde{x}_{s}(\omega)\in U$ for some $s\geq 0$, then by
path continuity and the definition of the stopped process,
$\tilde{x}_{u}(\omega)\in U_{n_{0}}$ for all $u\in[0,s]$ for some
$n_{0}=n_{0}(\omega,s)\in\mathbf{N}$. Hence, by definition,
$x_{u}^{n}(\omega)\equiv\tilde{x}_{u}(\omega)$ for all $u\in[0,s]$ and $n\geq
n_{0}$, and consequently by the Bounded Convergence Theorem

$\displaystyle\lim_{n\to\infty}\int_{0}^{t}\mathbf{E}[(1-\varphi_{n}(x_{s}^{n}))\mathbf{1}_{U}(\tilde{x}_{s})]^{2}\,ds=0\,.$

Hence, $\mathbf{E}\sup_{r\in[0,t]}|\tilde{x}_{r}-x_{r}^{n}|^{2}\to 0$ as
$n\to\infty$, establishing (4.8). ∎

Let $u_{n}(t,x):=\mathbf{E}_{x}\psi(x_{t}^{n})$. It follows that for fixed
$t>0$, the mapping $x\mapsto u_{n}(t,x)$ belongs to the space
$C^{2}(\mathbf{R}^{\mathfrak{m}})$ [Fri75, Theorem 5.5]. Define an extended
version $A_{n}$ of the generator of $x_{t}^{n}$ by

$\displaystyle A_{n}g(x):=\lim_{t\rightarrow
0}\frac{\mathbf{E}_{x}g(x_{t}^{n})-g(x)}{t}\,,$ (4.9)

where the domain of $A_{n}$ consists of all measurable functions
$g:\mathbf{R}^{\mathfrak{m}}\to\mathbf{R}$ such that the limit in (4.9) exists
for each $x$. Next, by [Oks13, Theorem 8.1.1], for fixed
$x\in\mathbf{R}^{\mathfrak{m}}$, $t\mapsto u_{n}(t,x)\in
C^{1}(\mathbf{R}^{\mathfrak{m}})$ and by Markov property $u_{n}$ satisfies on
$(0,\infty)\times\mathbf{R}^{\mathfrak{m}}$ the generalized backward equation

$\displaystyle\partial_{t}u_{n}=A_{n}u_{n}.$

Since $x\mapsto u_{n}(t,x)$ is globally $C^{2}$, Dynkin’s formula implies that
$A_{n}u=L_{n}u$ where $L_{n}$ is the _classical_ generator of $x_{t}^{n}$,
which is a second-order differential operator with
$C^{\infty}(\mathbf{R}^{\mathfrak{m}})$ coefficients. Moreover, these
coefficients agree with the coefficients of $L$ on $U_{n}$. In particular, by
(L3), $L-\partial_{t}$ is hypoelliptic on $(0,\infty)\times U_{n}$, and
therefore $u_{n}\in C^{\infty}((0,\infty)\times U_{n})$ and
$\partial_{t}u_{n}=Lu_{n}$ on $(0,\infty)\times U_{n}$ in the classical sense.
To obtain a similar result for
$u(t,x)=\tilde{\mathcal{P}}_{t}\psi(x)=\mathbf{E}_{x}\psi(\tilde{x}_{t})$, we
note that $u_{n}$ is uniformly bounded. Furthermore, $u_{n}\to u$ pointwise as
$n\to\infty$ since by Claim

$\displaystyle|u_{n}(t,x)-u(t,x)|^{2}\leq\|\psi\|_{\text{Lip}}^{2}\mathbf{E}_{x}|\tilde{x_{t}}-x_{t}^{n}|^{2}\to
0$ (4.10)

as $n\to\infty$. By 4.2, $u\in C^{\infty}((0,\infty)\times U)$ and
$\partial_{t}u=Lu$ in the classical sense on $(0,\infty)\times U$. Thus Step 1
is completed in the case when $b,\sigma$ have bounded derivatives of all
orders.

To complete Step 1, assume that $b\in
C^{\infty}(\mathcal{X};\mathbf{R}^{\mathfrak{m}})$ and $\sigma\in
C^{\infty}(\mathcal{X};M_{\mathfrak{m}\times r})$ are not necessarily bounded
and $x_{t}$ is nonexplosive. For the sequence $\\{U_{n}\\}$ above, note that
the restrictions of $b,\sigma$ to $U_{n}$ can be extended to functions on
$\mathbf{R}^{\mathfrak{m}}$ which are bounded with bounded derivatives all
orders. Setting $\tau_{n}=\inf\\{t>0\,:\,\tilde{x}_{t}\notin U_{n}\\}$, we
have shown above that if $\tilde{x}_{t}^{n}:=\tilde{x}_{t\wedge\tau_{n}}$,
then $(t,x)\mapsto\tilde{u}_{n}(t,x):=\mathbf{E}_{x}\psi(\tilde{x}_{t}^{n})$
satisfies $\tilde{u}_{n}\in C^{\infty}((0,\infty)\times U_{n})$ and
$\partial_{t}\tilde{u}_{n}=L\tilde{u}_{n}$ on $(0,\infty)\times U_{n}$ in the
classical sense. However, since $\psi$ is bounded, then $\tilde{u}_{n}$ is
also uniformly bounded in $n$ and by (4.10), $\tilde{u}_{n}\to u$ pointwise on
$(0,\infty)\times U$. Step 1 now follows after applying 4.2.

_Proof of Step 2_. Let $\chi\in
C_{0}^{\infty}(\mathbf{R}^{\mathfrak{m}};[0,\infty))$ with
$\int_{\mathbf{R}^{\mathfrak{m}}}\chi(x)\,dx=1$. Let
$\chi_{\epsilon}(x)=\epsilon^{-\mathfrak{m}}\chi(\epsilon^{-1}x)$ and extend
$\tilde{p}_{t}(x,y)$ to be zero for all $(t,x,y)\notin(0,\infty)\times U\times
U$. For any
$(t,x,y)\in(0,\infty)\times\mathbf{R}^{\mathfrak{m}}\times\mathbf{R}^{\mathfrak{m}}$,
define

$\displaystyle
u_{\epsilon}(t,x,y):=\int_{\mathbf{R}^{\mathfrak{m}}}\tilde{p}_{t}(x,z)\chi_{\epsilon}(y-z)\,dz.$
(4.11)

Consider any bounded open $V\subset U$ with $\overline{V}\subset U$. For each
fixed $y\in\overline{V}$, there is $\epsilon_{0}(y)>0$ small enough such that
$z\mapsto\chi_{\epsilon}(y-z)\in C_{0}^{\infty}$ is compactly supported in $U$
for all $\epsilon\in(0,\epsilon_{0}(y)]$. It follows by compactness of
$\overline{V}$, that we may choose $\epsilon_{0}>0$ such that
$z\mapsto\chi_{\epsilon}(y-z)$ is compactly supported in $U$ for all
$y\in\overline{V}$ and all $\epsilon\in(0,\epsilon_{0}]$. Thus, for any
$\epsilon\in(0,\epsilon_{0}]$ and $(t,x)\in(0,\infty)\times U$, the formula
(4.11) makes sense and $y\mapsto u_{\epsilon}(t,x,y)\in C^{\infty}(V)$. Also,
by Step 1, for each $y\in V$ and $\epsilon\in(0,\epsilon_{0}]$, $(t,x)\mapsto
u_{\epsilon}(t,x,y)\in C^{\infty}((0,\infty)\times U)$ and
$\partial_{t}u_{\epsilon}(t,x,y)=L_{x}u_{\epsilon}(t,x,y)$ on
$(0,\infty)\times U$ in the classical sense. To conclude the result, it
suffices to show for any $\varphi\in C_{0}((0,\infty)\times U\times V)$ with
$\varphi$ supported in $K\subset(0,\infty)\times U\times V$ compact, that

$\displaystyle\lim_{\epsilon\to
0}\int_{K}\varphi(t,x,y)[u_{\epsilon}(t,x,y)-\tilde{p}_{t}(x,y)]dx\,dy\,dt=0\,.$

Indeed, simply replace $\varphi$ in the above formula by
$(\partial_{t}-L_{x})^{*}\varphi$, where $\varphi$ is smooth with compact
support.

First, since $\int_{\mathbf{R}^{\mathfrak{m}}}\chi(z)\,dz=1$, a standard
substitution gives for all $\epsilon\in(0,\epsilon_{0}]$ and $y\in V$

$\displaystyle
u_{\epsilon}(t,x,y)-\tilde{p}_{t}(x,y)=\int_{\mathbf{R}^{\mathfrak{m}}}[\tilde{p}_{t}(x,y-\epsilon
z)-\tilde{p}_{t}(x,y)]\chi(z)\,dz.$

Next, let $\hat{p}_{t}(x,y)=\tilde{p}_{t}(x,y)$ on $K$ and
$\hat{p}_{t}(x,y)=0$ otherwise. By Tonelli’s theorem,
$(t,x,y)\mapsto\hat{p}_{t}(x,y)\in
L^{1}(\mathbf{R}\times(\mathbf{R}^{\mathfrak{m}})^{2})$. Moreover,

$\displaystyle\bigg{|}\int_{K}\varphi(t,x,y)[u_{\epsilon}(t,x,y)-\tilde{p}_{t}(x,y)]\,dx\,dy\,dt\bigg{|}$
$\displaystyle\leq\|\varphi\|_{L^{\infty}}\int_{\mathbf{R}\times(\mathbf{R}^{\mathfrak{m}})^{3}}|\hat{p}_{t}(x,y-\epsilon
z)-\hat{p}_{t}(x,y)|\chi(z)dz\,dx\,dy\,dt.$

Since $\hat{p}_{t}\in L^{1}(\mathbf{R}\times(\mathbf{R}^{\mathfrak{m}})^{2})$,
for any $\delta>0$ there is $\psi\in
C_{0}^{\infty}((\mathbf{R}^{\mathfrak{m}})^{3})$ within $L^{1}$ distance
$\delta$ of $\hat{p}_{t}$. Thus, by translation invariance and Fubini’s
theorem

$\displaystyle\bigg{|}\int_{K}\varphi(t,x,y)[u_{\epsilon}(t,x,y)-\tilde{p}_{t}(x,y)]\,dx\,dy\,dt\bigg{|}$
$\displaystyle\leq
2\delta\|\varphi\|_{L^{\infty}}+\|\varphi\|_{L^{\infty}}\int_{(\mathbf{R}^{\mathfrak{m}})^{4}}|\psi(t,x,y-\epsilon
z)-\psi(t,x,y)|\chi(z)dz\,dx\,dy\,dt.$

Since $\psi$ is compactly supported, passing $\epsilon\to 0$, using Dominated
Convergence theorem, and then $\delta\to 0$ finishes the proof of 4.3(ii). ∎

###  4.3. Interior smoothness of functionals of the first positive exit time

In this section we analyze quantities of the form

$\displaystyle\mathbf{E}_{x}\varphi(\tau),\,\,x\in U,$ (4.12)

for various functions $\varphi$ and $\tau$ as in (2.2). The first two choices

$\varphi(x)=\mathbf{1}\\{|x|<\infty\\}\,\,\,\text{ and }\,\,\,\varphi(x)=x,$

that respectively yield in (4.12)

$\displaystyle\mathbf{P}_{x}\\{\tau<\infty\\}\qquad\text{ and
}\qquad\mathbf{E}_{x}\tau$ (4.13)

are of significant importance for analyzing recurrence and transience in
Section 7 below. Throughout, we assume the parabolic Hörmander condition (PH)
for simplicity, which in particular implies that (L1)–(L4) are satisfied.

######  Proposition 4.7.

Suppose that (U00), (NE), and (PH) are satisfied. For any
$(t,x)\in(0,\infty)\times U$, define

$\displaystyle u_{1}(t,x)=\mathbf{P}_{x}\\{\tau>t\\}\qquad\text{ and }\qquad
u_{2}(x)=\mathbf{P}_{x}\\{\tau=\infty\\}.$ (4.14)

Then:

  * (i)

$u_{1}\in C^{\infty}((0,\infty)\times U)$ and $\partial_{t}u_{1}=Lu_{1}$ on
$(0,\infty)\times U$ in the classical sense;

  * (ii)

$u_{2}\in C^{\infty}(U)$ and $Lu_{2}=0$ on $U$ in the classical sense.

###### Proof.

For (i), observe that for any $x\in U$,
$u_{1}(t,x)=\mathbf{P}_{x}\\{\tau>t\\}=\tilde{\mathcal{P}}_{t}(x,U)$ where
$\tilde{\mathcal{P}}_{t}(x,\,\cdot\,)$ is the transition kernel of the stopped
process $\tilde{x}_{t}=x_{t\wedge\tau_{0}}$. Let $K\subset U$ be compact and
by 4.3 $\tilde{\mathcal{P}}_{t}(x,K)=\int_{K}\tilde{p}_{t}(x,y)\,dy$.
Furthermore, by 4.3 and the Dominated Convergence Theorem, it follows that
$(t,x)\mapsto\tilde{\mathcal{P}}_{t}(x,K)$ is $C^{\infty}((0,\infty)\times U)$
and $\partial_{t}\tilde{\mathcal{P}}_{t}(x,K)=L\tilde{\mathcal{P}}_{t}(x,K)$
on $(0,\infty)\times U$ in the classical sense. To obtain the same result for
$\tilde{\mathcal{P}}_{t}(x,U)$, let $K_{n}\subset U$ be a sequence of compact
sets with $K_{n}\uparrow U$ as $n\rightarrow\infty$. Then, by the monotone
convergence theorem, the sequence $(\tilde{\mathcal{P}}_{t}(x,K_{n}))_{n}$ is
uniformly bounded and converges pointwise to $\tilde{\mathcal{P}}_{t}(x,U)$.
The assertion (i) follows after applying 4.2.

To obtain (ii), first observe that

$u_{2}(x)=\mathbf{P}_{x}\\{\tau=\infty\\}=\lim_{t\rightarrow\infty}\tilde{\mathcal{P}}_{t}(x,U).$

Thus we seek to apply an argument similar to that in 4.2. To this end, note
that

$\displaystyle\bar{u}(t,x):=\int_{t}^{t+1}\tilde{\mathcal{P}}_{s}(x,U)\,ds\rightarrow
u_{2}(x)\,\,\,\text{ pointwise as }\,\,\,t\to\infty$

and that $\bar{u}(t,x)$ is uniformly bounded on $[0,\infty)\times U$. Thus,
for any function $\varphi\in C_{0}^{\infty}(U)$ and any $t>0$, using
integration by parts and the Fubini-Tonelli Theorem we have

$\displaystyle 0$
$\displaystyle=\int_{t}^{t+1}\int_{U}(L-\partial_{s})\tilde{\mathcal{P}}_{s}(x,U)\varphi(x)\,dx\,ds$
$\displaystyle=\int_{U}\bar{u}(t,x)L^{*}\varphi(x)\,dx-\int_{U}[\tilde{\mathcal{P}}_{t+1}(x,U)-\tilde{\mathcal{P}}_{t}(x,U)]\varphi(x)\,dx.$

By passing $t\rightarrow\infty$ and using the Dominated Convergence Theorem,
$u_{2}$ solves $Lu_{2}=0$ in the sense of distributions on $U$. Part (ii)
follows by hypoellipticity of $L$ on $U$. ∎

We next state and prove a regularity result for $\mathbf{E}_{x}\tau$; that is,
for $\varphi(x)=x$ in (4.12).

######  Proposition 4.8.

Suppose that (U00), (NE), and (PH) are satisfied and
$\mathbf{P}_{x}\\{\tau<\infty\\}=1$ for all $x\in U$.

  * (i)

If $U_{0}\subset U$ is bounded open with $\overline{U_{0}}\subset U$, then
$v_{0}(x):=\mathbf{E}_{x}\tau_{U_{0}}$ is bounded on $\overline{U_{0}}$.
Furthermore, $v_{0}\in C^{\infty}(U_{0})$ and $Lv_{0}=-1$ on $U_{0}$ in the
classical sense.

  * (ii)

Let $U_{0}\subset U$ be an open set and suppose that
$v(x):=\mathbf{E}_{x}\tau$ is finite on a dense set $D\subset U_{0}$. Then,
$v\in C^{\infty}(U_{0})$ and $v$ solves $Lv=-1$ on $U_{0}$ in the classical
sense.

######  Remark 4.9.

The technical hypothesis in (ii) that $v$ is finite on a dense set and not
everywhere is needed below in Section 7. We defer the proof of 4.8(ii) until
after we prove a hypoelliptic Harnack inequality (5.5).

###### Proof of 4.8(i).

Let $\sigma=\tau_{U_{0}}$ and note that $\mathbf{P}_{x}\\{\tau<\infty\\}=1$
imply that for all $x\in\overline{U_{0}}$, there exists $t>0,\alpha\in(0,1/2)$
such that $\mathbf{P}_{x}\\{\sigma>t\\}\leq(1-2\alpha)$. By 4.7(i), there
exists $\epsilon=\epsilon(x)>0$ such that
$\mathbf{P}_{y}\\{\sigma>t\\}\leq(1-\alpha)$ for all $y\in B_{\epsilon}(x)$.
Using compactness of $\overline{U_{0}}$ and taking a finite subcover, it
follows that there exists $t_{*}>0,\alpha_{*}\in(0,1)$ such that
$\mathbf{P}_{y}\\{\sigma>t_{*}\\}\leq(1-\alpha_{*})$ for all
$y\in\overline{U_{0}}$. Since $\tau$ is almost surely finite, so is $\sigma$
by $U_{0}\subset U$ and path continuity. Thus, for any $x\in\overline{U_{0}}$
we have

$\displaystyle\mathbf{E}_{x}\sigma=\sum_{m=1}^{\infty}\mathbf{E}_{x}\sigma\mathbf{1}\\{\sigma\in[(m-1)t_{*},mt_{*})\\}\leq
3t_{*}+\sum_{m=3}^{\infty}mt_{*}\mathbf{P}_{x}\\{\sigma>(m-2)t_{*}\\}.$ (4.15)

By the Markov property, for $x\in\overline{U_{0}}$ and $m\in\mathbf{N}$ we
have

$\displaystyle\mathbf{P}_{x}\\{\sigma>mt_{*}\\}=\mathbf{E}_{x}[\mathbf{E}_{x}\mathbf{1}\\{\sigma>mt_{*}\\}|\mathcal{F}_{(m-1)t_{*}}]$
$\displaystyle=\mathbf{E}_{x}\mathbf{1}\\{\sigma>(m-1)t_{*}\\}\mathbf{P}_{x_{(m-1)t_{*}}}\\{\sigma>t_{*}\\}$
$\displaystyle\leq(1-\alpha_{*})\mathbf{P}_{x}\\{\sigma>(m-1)t_{*}\\}\leq(1-\alpha_{*})^{m}$

and (4.15) implies for all $x\in\overline{U_{0}}$

$\displaystyle\mathbf{E}_{x}\sigma\leq
3t_{*}+\sum_{m=3}^{\infty}mt_{*}(1-\alpha_{*})^{m-2},$

which is finite and bounded independently of $x\in\overline{U_{0}}$.

Next, in order to show that $v_{0}\in C^{\infty}(U_{0})$ and satisfies
$Lv_{0}=-1$ on $U_{0}$ in the classical sense, observe that

$\displaystyle
v_{0}(x)=\mathbf{E}_{x}\sigma=\int_{0}^{\infty}\mathbf{P}_{x}\\{\sigma>t\\}\,dt=:\int_{0}^{\infty}u(t,x)\,dt.$

4.7 yields $u\in C^{\infty}((0,\infty)\times U_{0})$ and $Lu=\partial_{t}u$ on
$U_{0}$ in the classical sense. Thus if $v_{n}(x)=\int_{1/n}^{n}u(t,x)\,dt$
for $n\in\mathbf{N}$, it follows that $v_{n}\in C^{\infty}(U_{0})$ and for all
$x\in U_{0}$

$\displaystyle Lv_{n}(x)=\int_{\frac{1}{n}}^{n}Lu(t,x)\,dt$
$\displaystyle=\int_{\frac{1}{n}}^{n}\partial_{t}u(t,x)\,dt$
$\displaystyle=u(n,x)-u(1/n,x)=\mathbf{P}_{x}\\{\sigma>n\\}-\mathbf{P}_{x}\\{\sigma>1/n\\}\to-1$

as $n\to\infty$ in the sense of distributions on $U_{0}$. Since the sequence
$v_{n}$ is uniformly bounded by bounded $v_{0}$ and $v_{n}\rightarrow v_{0}$
pointwise on $U_{0}$, the result follows from 4.2. ∎

We next consider the expression (4.12) for more general
$\varphi:[0,\infty)\rightarrow\mathbf{R}$. First we need an auxiliary lemma,
which is a representation result for the expected value of certain random
variables.

######  Lemma 4.10.

Suppose that (U00) and (NE) are met and let $\varphi:[0,\infty)\to\mathbf{R}$
be $C^{1}$, strictly increasing and satisfy $\varphi(t)\to\infty$ as
$t\to\infty$. Then

$\displaystyle\mathbf{E}_{x}\varphi(\tau)=\varphi(0)+\int_{0}^{\infty}\varphi^{\prime}(t)\mathbf{P}_{x}\\{\tau>t\\}\,dt.$
(4.16)

###### Proof.

Note that by shifting the formula (4.16), it suffices to assume that
$\varphi(0)=0$. Since $\varphi$ is $C^{1}$, strictly increasing with
$\varphi(t)\to\infty$ as $t\to\infty$, $\varphi$ has a $C^{1}$ inverse
$\varphi^{-1}$ mapping $[0,\infty)$ onto $[0,\infty)$. Moreover,
$\varphi^{-1}$ is strictly increasing on $[0,\infty)$. It thus follows that
$\varphi^{-1}(t)\to\infty$ as $t\to\infty$. Then,

$\displaystyle\mathbf{E}_{x}(\varphi(\tau)\wedge
N)=\int_{0}^{N}\mathbf{P}_{x}\\{\varphi(\tau)>t\\}\,dt$
$\displaystyle=\int_{0}^{N}\mathbf{P}_{x}\\{\tau>\varphi^{-1}(t)\\}\,dt$
$\displaystyle=\int_{0}^{\varphi^{-1}(N)}\varphi^{\prime}(t)\mathbf{P}_{x}\\{\tau>t\\}\,dt.$

Passing $N\to\infty$ and using the Monotone Convergence Theorem finishes the
proof. ∎

Next, we state and prove our main result for general $\varphi$.

######  Proposition 4.11.

Suppose that (U00), (NE), and (PH) are satisfied. Let $\varphi\in
C^{2}([0,\infty);\mathbf{R})$ be strictly increasing with
$\varphi(t)\to\infty$ as $t\to\infty$ with derivative $\varphi^{\prime}\in
C^{1}([0,\infty);\mathbf{R})$ which is strictly increasing with
$\varphi^{\prime}(t)\to\infty$ as $t\to\infty$. Suppose furthermore that

$v_{1}(x):=\mathbf{E}_{x}\varphi(\tau)\qquad\text{ and }\qquad
v_{2}(x)=\mathbf{E}_{x}\varphi^{\prime}(\tau)$

are bounded on compact subsets of $U$. Then $Lv_{1}=-v_{2}$ on $U$ in the
sense of distributions.

######  Remark 4.12.

Compare the equation $Lv_{1}=-v_{2}$ with $Lv=-1$ when $\varphi(x)=x$.
Although such a $\varphi$ does not satisfy the hypotheses above, we still have
the analogous conclusion.

###### Proof of 4.11.

By 4.10, we have

$\displaystyle
v_{1}(x)=\varphi(0)+\int_{0}^{\infty}\varphi^{\prime}(t)\mathbf{P}_{x}\\{\tau>t\\}\,dt\qquad\text{
and }\qquad
v_{2}(x)=\varphi^{\prime}(0)+\int_{0}^{\infty}\varphi^{\prime\prime}(t)\mathbf{P}_{x}\\{\tau>t\\}\,dt.$

In the latter formula, simply apply 4.10 to $\varphi^{\prime}$ instead of
$\varphi$. Consider the sequence of functions on $U$ given by

$\displaystyle
v_{1,n}(x):=\varphi(0)+\int_{1/n}^{n}\varphi^{\prime}(t)\mathbf{P}_{x}\\{\tau>t\\}\,dt\,.$

Observe that by 4.7, $v_{1,n}\in C^{\infty}(U)$ and

$\displaystyle
Lv_{1,n}(x)=\int_{1/n}^{n}\varphi^{\prime}(t)L_{x}(\mathbf{P}_{x}\\{\tau>t\\})\,dt$
$\displaystyle=\int_{1/n}^{n}\varphi^{\prime}(t)\partial_{t}(\mathbf{P}_{x}\\{\tau>t\\})\,dt$
$\displaystyle=\varphi^{\prime}(n)\mathbf{P}_{x}\\{\tau>n\\}-\varphi^{\prime}(1/n)\mathbf{P}_{x}\\{\tau>1/n\\}$
$\displaystyle\qquad-\int_{1/n}^{n}\varphi^{\prime\prime}(t)\mathbf{P}_{x}\\{\tau>t\\}\,dt\,.$

Since for any $x\in U$ we have $\mathbf{E}_{x}\varphi^{\prime}(\tau)<\infty$,
it follows that

$\lim_{n\to\infty}\varphi^{\prime}(n)\mathbf{P}_{x}\\{\tau>n\\}\leq\lim_{n\to\infty}\mathbf{E}_{x}\varphi^{\prime}(\tau)\mathbf{1}\\{\tau\geq
n\\}=0$ (4.17)

on $U$. Furthermore, since $\mathbf{P}_{x}\\{\tau=0\\}=0$ for any $x\in U$, it
also follows that on $U$

$\lim_{n\to\infty}\varphi^{\prime}(1/n)\mathbf{P}_{x}\\{\tau>1/n\\}=\varphi^{\prime}(0)\mathbf{P}_{x}\\{\tau>0\\}=\varphi^{\prime}(0).\,$
(4.18)

In addition, $\varphi^{\prime}$ is strictly increasing, and therefore
$\varphi^{\prime\prime}\geq 0$ and by the monotone convergence theorem,
$Lv_{1,n}\to-v_{2}$ pointwise as $n\to\infty$ on $U$. Since $v_{1},v_{2}$ are
bounded on compact subsets of $U$ and $v_{1,n}\to v_{1}$ as $n\to\infty$, 4.2
implies $Lv_{1}=-v_{2}$ on $U$ in the sense of distributions. ∎

Let us note some more specific examples and results that follow as an
immediate consequence of the previous results.

######  Corollary 4.13.

Suppose that (U00), (NE), and (PH) are satisfied. Then:

  * (i)

If for some $k\in\mathbf{N}$, $k\geq 2$,
$v_{1}^{k}(x):=\mathbf{E}_{x}\tau^{k}$ is bounded on compact subsets of $U$,
then $v_{1}^{k}\in C^{\infty}(U)$ and $Lv_{1}^{k}=-k\mathbf{E}_{x}\tau^{k-1}$
on $U$ in the classical sense.

  * (ii)

Suppose that for some $\delta>0$,
$w_{\delta}(x):=\mathbf{E}_{x}e^{\delta\tau}$ is bounded on compact subsets of
$U$. Then $w_{\delta}\in C^{\infty}(U)$ and $Lw_{\delta}=-\delta w_{\delta}$.

###### Proof.

For (i), set $\varphi(t)=t^{k}$ and note that $v_{1}^{k}(x)$ and
$v_{2}^{k}(x):=k\mathbf{E}_{x}\tau^{k-1}$ are both bounded on compact subsets
of $U$. By 4.11, $Lv_{1}^{k}=-v_{2}^{k}$ on $U$ in the sense of distributions.
By induction and 4.8, $v_{2}^{k}\in C^{\infty}(U)$ so that $v_{1}^{k}\in
C^{\infty}(U)$ and $Lv_{1}^{k}=-v_{2}^{k}$ in the classical sense.

For (ii), set $\varphi(t)=e^{\delta t}$. Then, 4.11 implies
$(L+\delta)w_{\delta}=0$ in the sense of distributions. However, $L+\delta$ is
hypoelliptic on $U$, so that $w_{\delta}\in C^{\infty}(U)$ and
$Lw_{\delta}=-\delta w_{\delta}$ in the classical sense. ∎

##  5\. Green’s functions and Bony’s Harnack inequality

In this section, we explore _Green’s functions_ in the hypoelliptic setting
for an open set $U$. As a consequence, we generalize Bony’s form of the
Harnack inequality [Bon69].

Let $\beta>0$ be a constant and suppose $f\in C^{\infty}(U)\cap
B(\overline{U})$. Often (see [Bon69]) one refers to the _Green’s operator_
$G_{\beta}$ as the ‘mapping’ $f\mapsto v$, where $v$ is the ‘unique’ solution
of the Poisson problem

$\left\\{\begin{aligned} (L-\beta)v&=-f\qquad&&\text{ on }U\,,\\\
u&=0\qquad&&\text{ on }\partial U.\end{aligned}\right.$ (5.1)

When $\beta=0$ and $L$ is hypoelliptic on $U$, the uniqueness of solutions of
(5.1) heavily depends on the structure of the diffusion $x_{t}$ driven by $L$
near $\partial U$. This is also the case when (5.1) and $\beta>0$. Here, we
employ stochastic methods to define and deduce properties of $G_{\beta}f$
without the uniqueness.

Formally, our ‘best guess’ of solution of (5.1) would be

$\displaystyle v(x)=\mathbf{E}_{x}\int_{0}^{\tau}f(x_{s})e^{-\beta s}\,ds\,,$
(5.2)

where $\tau=\inf\\{t>0\,:\,x_{t}\notin U\\}$. The expression (5.2) is
motivated by a formal application of Itô’s formula to the function $e^{-\beta
t}v(x_{t})$, where $v$ is a presumed classical solution of (5.1). Even though
it is not clear nor necessarily true that $v$ solves (5.1), $v$ in (5.2) is
well-defined for any function $f\in B(\overline{U})$. Note that $v$ in (5.2)
is well-defined even if $\tau=\infty$, due to the presence of the
exponentially decaying factor $e^{-\beta s}$.

######  Remark 5.1.

Under further conditions on the stopping time $\tau$, one can define $G_{0}$
or even sometimes $G_{-\beta}$ for $\beta>0$ small enough. Here, for
simplicity, we keep $\beta$ positive to avoid any further complexities of
assumptions on $\tau$. Additionally, under similar assumptions, one can
replace $\beta$ by a spatially dependent function $\beta=\beta(x)\in
C(\overline{U})$ to arrive at the Feynman-Kac formula

$\displaystyle\mathbf{E}_{x}\int_{0}^{\tau}f(x_{s})e^{-\int_{0}^{s}\beta(x_{v})\,dv}\,ds,$
(5.3)

provided (5.3) makes sense.

###  5.1. Definition and properties of $G_{\beta}$

Given the preliminary remarks above, we now define $G_{\beta}$.

######  Definition 5.2.

Let (U00) and (NE) be satisfied. For any $\beta>0$ and any
$f:\overline{U}\to[0,\infty)$ measurable, define $G_{\beta}f:U\to[0,+\infty]$
by

$\displaystyle G_{\beta}f(x):=\mathbf{E}_{x}\int_{0}^{\tau}f(x_{s})e^{-\beta
s}\,ds.$ (5.4)

If $f:\overline{U}\to\mathbf{R}$ is measurable and $G_{\beta}|f|(x)<\infty$
for all $x\in U$, we define

$\displaystyle
G_{\beta}f(x):=G_{\beta}f_{+}(x)-G_{\beta}f_{-}(x)=\mathbf{E}_{x}\int_{0}^{\tau}f(x_{s})e^{-\beta
s}\,ds$ (5.5)

where $f_{+}$, $f_{-}$ denote the positive and negative parts of $f$,
respectively. We call $G_{\beta}$ the _Green’s operator of order_ $\beta>0$.
Let

$\mathcal{D}_{\beta}:=\\{f:\overline{U}\to\mathbf{R}\text{
measurable}\,:\,x\mapsto G_{\beta}|f|(x)\text{ is bounded on compacts in
}U\\}$

be the _domain_ of $G_{\beta}$.

######  Remark 5.3.

Observe that the expression (5.5) is better behaved compared with the formal
stochastic representation corresponding to the usual Poisson problem (i.e.
with $\beta=0$) due to the presence of the exponentially decaying factor
$e^{-\beta s}$.

Often it is convenient to express the operator $G_{\beta}f$ as a traditional
integral operator, meaning that for any _reasonable_ function $f$ on
$\overline{U}$ and any $x\in U$:

$\displaystyle G_{\beta}f(x)=\int_{\overline{U}}f(y)g_{\beta}(x,y)\,dy$ (5.6)

for some kernel $g_{\beta}$ to be referred to as the _Green’s function_
associated to $G_{\beta}$. However, we must be careful as there are subtleties
in defining $g_{\beta}(x,y)$ for $y\in\partial U$ and for $x=y$ in $U$.

To obtain a workable expression for $g_{\beta}$, under the assumptions (U00)
and (NE) we recall the stopped process $\tilde{x}_{t}=x_{t\wedge\tau_{0}}$
from (2.10) distributed on $\overline{U}$, where $\tau_{0}=\inf\\{t\geq
0\,:\,x_{t}\notin U\\}$. We also recall its transition kernel
$\tilde{\mathcal{P}}_{t}(x,\,\cdot\,)$ defined on Borel subsets of
$\overline{U}$. Note that, for any $x\in U$ and any $f\in\mathcal{D}_{\beta}$,
the Dominated Convergence theorem implies

$\displaystyle\mathbf{E}_{x}\int_{0}^{\tau_{0}}f(\tilde{x}_{s})\mathbf{1}_{U}(\tilde{x}_{s})e^{-\beta
s}\,ds$ $\displaystyle=\lim_{\delta\downarrow
0}\mathbf{E}_{x}\int_{0}^{(\tau_{0}-\delta)\vee
0}f(\tilde{x}_{s})\mathbf{1}_{U}(\tilde{x}_{s})e^{-\beta s}\,ds$
$\displaystyle=\lim_{\delta\downarrow
0}\mathbf{E}_{x}\int_{0}^{(\tau_{0}-\delta)\vee 0}f(\tilde{x}_{s})e^{-\beta
s}\,ds$
$\displaystyle=\mathbf{E}_{x}\int_{0}^{\tau_{0}}f(\tilde{x}_{s})e^{-\beta
s}\,ds.$

Hence, for any $f\in\mathcal{D}_{\beta}$ and $x\in U$, we have by Fubini’s
theorem

$\displaystyle G_{\beta}f(x)$
$\displaystyle=\mathbf{E}_{x}\int_{0}^{\tau_{0}}f(\tilde{x}_{s})e^{-\beta
s}\,ds=\mathbf{E}_{x}\int_{0}^{\infty}\mathbf{1}_{U}(\tilde{x}_{s})f(\tilde{x}_{s})e^{-\beta
s}\,ds$ (5.7)
$\displaystyle=\int_{0}^{\infty}\int_{U}f(y)\tilde{\mathcal{P}}_{s}(x,dy)e^{-\beta
s}\,ds=\int_{U}f(y)\int_{0}^{\infty}\tilde{\mathcal{P}}_{s}(x,dy)e^{-\beta
s}\,ds$ $\displaystyle=:\int_{U}f(y)g_{\beta,x}(dy)\,,$

where for each $x\in U$, $g_{\beta,x}$ is a finite Borel measure on $U$ given
by

$\displaystyle g_{\beta,x}(A)=\int_{0}^{\infty}\tilde{\mathcal{P}}_{s}(x,A\cap
U)e^{-\beta s}\,ds.$ (5.8)

In particular, the purported Green’s ‘function’ $g_{\beta,x}$, $x\in U$, does
not charge the boundary $\partial U$. In order to show that $g_{\beta,x}$ has
a density with respect to Lebesgue measure on $U$, we prove the following
result:

######  Theorem 5.4.

Let $\beta>0$, and suppose that (U00), (NE), and (PH) are satisfied. Then:

  * (i)

For every $x\in U$, the finite measure $g_{\beta,x}$ on Borel subsets of $U$
given by (5.8) is absolutely continuous with respect to Lebesgue measure on
$U$ with density $g_{\beta}(x,y)$.

  * (ii)

The mapping $(x,y)\mapsto g_{\beta}(x,y)\in C^{\infty}(U\times
U\setminus\text{\emph{Diag}})$, where $\text{\emph{Diag}}=\\{(x,x)\,:\,x\in
U\\}$. Furthermore, for fixed $x\in U$,
$(L_{y}^{*}-\beta)g_{\beta}(x,y)=-\delta_{x}$ in the sense of distributions on
$U$ and, for fixed $y\in U$, $(L_{x}-\beta)g_{\beta}(x,y)=-\delta_{y}$ in the
sense of distributions on $U$.

  * (iii)

For any $f\in\mathcal{D}_{\beta}\cap C(U)$, $(L_{x}-\beta)G_{\beta}f=-f$ on
$U$ in the sense of distributions.

  * (iv)

If $u\in C^{2}(U;[0,\infty))$ satisfies $Lu=0$ on $U$, then $u\geq\beta
G_{\beta}u\text{ on }U.$

###### Proof.

First, we show (i). By 4.3, for $s>0$ and $x\in U$, the measure
$\tilde{\mathcal{P}}_{s}(x,\,\cdot\,\cap U)$ is absolutely continuous with
respect to Lebesgue measure on $U$ with density $\tilde{p}_{t}(x,y)$.
Furthermore, $(t,x,y)\mapsto\tilde{p}_{t}(x,y):(0,\infty)\times U\times
U\mapsto\mathbf{R}$ is $C^{\infty}$ and $\tilde{p}_{t}$ satisfies both the
forward and backward Kolmogorov equations in the classical sense as in 4.3(i),
(ii). Thus, if $|A\cap U|=0$, then $\tilde{\mathcal{P}}_{s}(x,A\cap U)=0$ for
any $s>0$ and by (5.8) one has $g_{\beta,x}(A)=0$. Hence, by the Radon-Nikodym
theorem, for each $x\in U$, $g_{\beta,x}$ has density $y\mapsto
g_{\beta}(x,y)$ which belongs to $L^{1}(U)$. This concludes (i).

Next, we establish parts (ii) and (iii) simultaneously. Note that for any
$f\in C^{\infty}_{0}(U)$, we have for any $x\in U$,

$\displaystyle
G_{\beta}((L-\beta)f)(x)=\int_{U}(L-\beta)f(y)g_{\beta}(x,y)dy.$ (5.9)

On the other hand, because $f$ vanishes outside of a compact set in $U$,
Dynkin’s formula (2.8) applied to
$f(x_{t\wedge\tau_{0}})e^{-\beta(t\wedge\tau_{0})}$ gives

$\displaystyle\mathbf{E}_{x}f(x_{t\wedge\tau_{0}})e^{-\beta(t\wedge\tau_{0})}$
$\displaystyle=f(x)+\mathbf{E}_{x}\int_{0}^{t\wedge\tau_{0}}(L-\beta)f(x_{s})e^{-\beta
s}ds.$ (5.10)

Since $f$ is compactly supported in $U$ and $\beta>0$, after passing
$t\to\infty$ we have

$\mathbf{E}_{x}f(x_{t\wedge\tau_{0}})e^{-\beta(t\wedge\tau_{0})}\to 0.$

By (5.5) and (5.10),

$\displaystyle G_{\beta}(L-\beta)f(x)=-f(x)$ (5.11)

on $U$ for all $f\in C_{0}^{\infty}(U)$. By combining (5.9) and (5.11) we
obtain

$\displaystyle(L^{*}_{y}-\beta)g_{\beta}(x,\cdot)=-\delta_{x}\,\,\,\text{ on
}U$ (5.12)

in the sense of distributions. Hence, by hypoellipticity of $L^{*}-\beta$, for
each $x\in U$, the mapping $y\mapsto g_{\beta}(x,y)\in
C^{\infty}(U\setminus\\{x\\})$.

To obtain regularity in the $x$ variable, for any $f\in C(U)\cap
B(U)\subset\mathcal{D}_{\beta}$, consider the sequence of approximations

$\displaystyle
G_{\beta}^{n}f(x):=\int_{1/n}^{n}\int_{U}f(y)\tilde{p}_{s}(x,y)\,dy\,e^{-\beta
s}\,ds=:\int_{U}f(y)g_{\beta}^{n}(x,y)\,dy$

and let $h\in C^{\infty}_{0}(U)$. Employing 4.3 and the Dominated Convergence
Theorem (in order to interchange the integral and derivatives) it follows that

$\displaystyle\int_{U}h(x)(L-\beta)_{x}G_{\beta}^{n}f(x)\,dx$
$\displaystyle=\int_{U}\int_{U}h(x)f(y)\int_{1/n}^{n}\partial_{t}(\tilde{p}_{t}(x,y)e^{-\beta
t})\,dt\,dy\,dx$ $\displaystyle=\int_{U}h(x)[e^{-\beta
n}\mathbf{E}_{x}f(\tilde{x}_{n})-e^{-\beta/n}\mathbf{E}_{x}f(\tilde{x}_{1/n})]\,dx.$

Passing $n\to\infty$ and using the Dominated Convergence Theorem, we find that
$G_{\beta}$ satisfies

$(L-\beta)_{x}G_{\beta}f(x)=-f(x)$ (5.13)

in the sense of distributions on $U$. More generally, if
$f\in\mathcal{D}_{\beta}\cap C(U)$, then for every $N\in\mathbf{N}$ define
$f_{N}:=f\wedge N\in C(U)\cap B(U)$. Then $(L-\beta)_{x}G_{\beta}f_{N}=-f_{N}$
in the sense of distributions on $U$. Passing $N\to\infty$, it follows by 4.2
and definition of $\mathcal{D}_{\beta}$, that $(L-\beta)_{x}G_{\beta}f=-f$ in
the sense of distributions on $U$. This proves (iii).

To finish the rest of (ii), let $f,h\in C_{0}^{\infty}(U)$ and note that by
(5.13)

$\int_{U\times
U}[(L-\beta)_{x}^{*}h](x)g_{\beta}(x,y)f(y)\,dydx=-\int_{U}h(x)f(x)\,dx.$
(5.14)

Since $f$ and $h$ are arbitrary,

$\displaystyle(L-\beta)_{x}g_{\beta}(x,y)=-\delta_{y}(x)$ (5.15)

on $U$ in the sense of distributions, and therefore the hypoellipticity of
$L-\beta$ implies that $x\mapsto g_{\beta}(x,y)$ is
$C^{\infty}(U\setminus\\{y\\})$, as desired.

Finally, for part (iv), for $n\geq 1$ let $U_{n}$ be a sequence of bounded
open sets with $U_{n}\uparrow U$ and $\overline{U}_{n}\subset U$. Set
$\tau_{n}=\inf\\{t\geq 0\,:\,x_{t}\notin U_{n}\\}$. Then, Dynkin’s formula
(2.8), $Lu=0$, and $u\geq 0$ imply for any $x\in U$,

$\displaystyle 0\leq\mathbf{E}_{x}e^{-\beta
t\wedge\tau_{0}\wedge\tau_{n}}u(x_{t\wedge\tau_{0}\wedge\tau_{n}})=u(x)-\beta\mathbf{E}_{x}\int_{0}^{t\wedge\tau_{0}\wedge\tau_{n}}u(x_{s})e^{-\beta
s}\,ds.$

Rearranging and passing $n\to\infty$ and $t\to\infty$ gives the result by the
Monotone Convergence Theorem. ∎

We now apply 5.4 to obtain a version of Harnack’s inequality originally due to
Bony [Bon69]. Our assumptions are weaker as we do not assume the uniqueness of
the solution of (5.1). Before we proceed, let

$\displaystyle\mathcal{H}_{U}=\\{u\in C^{2}(U;[0,\infty))\,:\,Lu=0\text{ on
}U\\}.$ (5.16)

Of course, if $L$ is hypoelliptic on $U$, then any distribution $u$ with
$Lu=0$ on $U$ must belong to $C^{\infty}(U)$ and satisfy $Lu=0$ on $U$ in the
classical sense.

######  Theorem 5.5 (Harnack inequality).

Suppose that (U00), (NE), and (PH) are satisfied. Consider any compact set
$K\subset U$, any set $D\subset U$ which is dense in $U$ and any multiindex
$\alpha$. Then, there exist points $y_{1},y_{2},\ldots,y_{k}\in D$ and a
constant $c>0$ such that the following inequality

$\displaystyle\sup_{x\in K}|D^{\alpha}u(x)|\leq
c[u(y_{1})+u(y_{2})+\cdots+u(y_{k})]$ (5.17)

is satisfied for all $u\in\mathcal{H}_{U}$.

###### Proof.

From this point in the paper, the argument is a slight modification of the
proof of [Bon69, Lemma 7.1]. Fix any multi-index $\alpha$. Let $K\subset U$ be
compact and $x_{0}\in K$. We show that there exist an open neighborhood
$V\subset U$ of $x_{0}$, an element $y\in D$, and a constant $c>0$ such that

$\displaystyle u(y)\geq c\sup_{x\in V}|D^{\alpha}u(x)|\,\,\text{ for all
}u\in\mathcal{H}_{U}.$ (5.18)

The result then follows from (5.18) using compactness of $K$. By 5.4, fixing
$\beta>0$, we obtain for all $y\in U$ and all $u\in\mathcal{H}_{U}$

$\displaystyle u(y)\geq\beta\int_{U}u(z)g_{\beta}(y,z)\,dz.$

We next claim that there exists $y\in D\setminus\\{x_{0}\\}$ such that
$g_{\beta}(y,x_{0})>0$. If not, by 5.4, $g_{\beta}(\cdot,x_{0})=0$ on
$U\setminus\\{x_{0}\\}$, a contradiction to
$(L-\beta)_{y}g_{\beta}(y,x_{0})=-\delta_{x_{0}}(x)$ on $U$ in the sense of
distributions (see (5.15)). Thus fix $y\in D\setminus\\{x_{0}\\}$ so that
$g_{\beta}(y,x_{0})>0$ and by continuity, choose disjoint neighborhoods $W$ of
$y$ and $X$ of $x_{0}$ and a constant $c>0$ such that

$\displaystyle g_{\beta}(w,x)\geq c\,\,\text{ for all }(w,x)\in W\times X.$

Nonnegativity of $g_{\beta}$ then implies for fixed $y\in D$

$\displaystyle u(y)\geq\beta c\int_{X}u(x)\,dx$

for all $u\in\mathcal{H}_{U}$. In order to bound the integral
$c\int_{X}u(x)\,dx=c\|u\|_{L^{1}(X)}$ from below, we bootstrap (2.15) to
obtain for any $s>0$ and $t<0$ and any open neighborhood $V$ of $x_{0}$ with
$\overline{V}\subseteq X$, the existence of a constant $C_{s,t}$ depending
only on $s,t$ and $V,X$ such that

$\displaystyle\|u\|_{H^{s}(V)}\leq C_{s,t}\|u\|_{H^{t}(X)}\,\,\text{ for all
}u\in\mathcal{H}_{U}.$

For sufficiently negative $t$ we have $L^{1}\hookrightarrow H^{t}$, and there
exists $C>0$ independent of $u$ such that

$\displaystyle\sup_{x\in V}|D^{\alpha}u(x)|\leq
C\int_{X}u(x)\,dx\leq\frac{C}{\beta c}u(y).$

∎

Given the previous result, we return to the proof of 4.8(ii).

###### Proof of 4.8(ii).

Take a sequence of bounded open sets $U_{n}$, $n\geq 1$, with
$\overline{U_{n}}\subset U$ and $U_{n}\uparrow U$ as $n\to\infty$. By 4.8(i),
for any $n\geq 1$, if $\tau_{n}:=\tau_{U_{n}}$, then
$w_{n}(x):=\mathbf{E}_{x}\tau_{n}$ is bounded on $\overline{U_{n}}$ and
$C^{\infty}(U_{n})$ with $Lw_{n}=-1$ on $U_{n}$ in the classical sense. In
addition, since $U_{n}\subset U$, we have $w_{n}\leq v$. Let $x\in U_{0}$ and
fix $\delta>0$ such that $\overline{B_{\delta}(x)}\subset U_{0}$. Fix
$m\in\mathbf{N}$ such that $U_{m}\supset\overline{B_{\delta}(x)}$. Then, for
each $n\geq m$, the function $z_{n}=w_{n}-w_{m}$ satisfies $Lz_{n}=0$ on
$B_{\delta}(x)$ and for any $y\in D$, $0\leq z_{n}(y)\leq v(y)<\infty$. By
5.5, $z_{n}$ has bounded derivatives of all orders independently of $n$ on
$\overline{B_{\delta}(x)}$. Since $m$ is fixed, $w_{n}=z_{n}+w_{m}$ has has
bounded derivatives of all orders independently of $n$ on
$\overline{B_{\delta}(x)}$. Then, $w_{n}$ converges to a function
$\overline{v}$, uniformly on compact subsets of $U_{0}$.

The assertion (ii) follows from 4.2 once we show that $v=\overline{v}$ on
$U_{0}$. Since $U_{n}\uparrow U$, then $\tau_{n}\uparrow\tau$ and
$v=\overline{v}$ follows from the monotone convergence theorem. This finishes
the proof of (ii). ∎

##  6\. General results on the formal stochastic solution

###  6.1. Preliminary remarks

Throughout this section, we assume that $U\subset\mathbf{R}^{\mathfrak{m}}$ is
non-empty, open set with non-empty complement $U^{c}$. We return to one of our
main goals: satisfaction of (1.2) by the formal stochastic representation,
which for the Poisson equation is given by (provided it makes sense)

$\displaystyle
v_{\text{stoc},0}(x)=\mathbf{E}_{x}\int_{0}^{\tau}f(x_{s})\,ds=\int_{0}^{\tau_{0}}f(\tilde{x}_{s})\,ds,\,\,x\in
U,$ (6.1)

where $\tilde{x}_{t}=x_{t\wedge\tau_{0}}$ is the process stopped at the
boundary $\partial U$. For example if $f\equiv 1$,
$v(x):=\mathbf{E}_{x}\tau=\mathbf{E}_{x}\tau_{0}=\mathbf{E}_{x}\int_{0}^{\tau_{0}}1\,ds$,
$x\in U$ and as long as the hypotheses of 4.8(ii) are satisfied, $v\in
C^{\infty}(U)$ and $Lv=-1$ on $U$ in the classical sense. Thus, we expect that
under analogous assumptions, $v_{\text{stoc},0}\in C^{\infty}(U)$ and
$Lv_{\text{stoc},0}=-f$ in the classical sense on $U$.

Additionally, if $g\in C^{\infty}_{0}(\mathcal{X})$ for the formal stochastic
solution of the Dirichlet problem

$\displaystyle u_{\text{stoc},0}(x)=\mathbf{E}_{x}g(x_{\tau}),$ (6.2)

Dynkin’s formula applied to $g$ yields for any $x\in U$, $t\geq 0$

$\displaystyle u_{\text{stoc},0}(x,t):=\mathbf{E}_{x}g(x_{\tau\wedge t})$
$\displaystyle=g(x)+\mathbf{E}_{x}\int_{0}^{\tau\wedge t}Lg(x_{s})\,ds.$ (6.3)

Thus by formally passing $t\to\infty$, rearranging and using (6.1), one has
that $u_{\text{stoc},0}-g$ satisfies $L(u-g)=Lg$, and interior smoothness and
$Lu_{\text{stoc},0}=0$ on $U$ in the classical sense follows. Then, by an
approximation argument and 4.2, the same properties follow for more general
$g$.

###  6.2. Interior smoothness for $v_{\text{stoc},0}$ and $u_{\text{stoc},0}$

We begin with the interior smoothness for $v_{\text{stoc},0}$. Suppose (U00)
and (NE) hold and for any measurable $f:\overline{U}\to[0,\infty)$, define
$G_{0}f:U\to[0,\infty]$ by (cf. (5.4))

$\displaystyle G_{0}f(x)=\mathbf{E}_{x}\int_{0}^{\tau}f(x_{s})\,ds\,,$ (6.4)

where $\tau=\tau_{0}$ is as in (2.9). For any $f:\overline{U}\to\mathbf{R}$
measurable with $G_{0}|f|(x)<\infty$ for all $x\in U$, we let (cf. (5.5))

$\displaystyle
G_{0}f(x):=G_{0}f_{+}(x)-G_{0}f_{-}(x)=\mathbf{E}_{x}\int_{0}^{\tau}f(x_{s})\,ds$
(6.5)

and

$\displaystyle\mathcal{D}_{0}(U)=\\{f:\overline{U}\to\mathbf{R}\text{
measurable}\,:\,x\mapsto G_{0}|f|(x)\text{ is bounded on compact sets in
}U\\}.$ (6.6)

######  Theorem 6.1 (Interior Smoothness for $v_{\text{stoc},0}$).

Let $f\in C(U)\cap\mathcal{D}_{0}(U)$ and assume that (U00), (NE), and (PH)
are satisfied. Suppose, moreover $\mathbf{P}_{x}\\{\tau<\infty\\}=1$ for all
$x\in U$. Then

$\displaystyle Lv_{\text{stoc},0}=-f\,\,\text{ on }\,\,U$ (6.7)

in the sense of distributions. Also, if $f\in
C^{\infty}(U)\cap\mathcal{D}_{0}(U)$, then $v_{\text{stoc},0}\in
C^{\infty}(U)$ and $Lv_{\text{stoc},0}=-f$ on $U$ in the classical sense.

###### Proof.

Suppose $f\in C(U)\cap\mathcal{D}_{0}(U)$. Let $U_{n}$ be bounded, open set
with $\overline{U}_{n}\subset U$ and $U_{n}\uparrow U$ as $n\to\infty$. For
$n,k\in\mathbf{N}$ define

$\displaystyle
v_{n,k}(x):=\int_{1/k}^{k}\int_{U_{n}}\tilde{p}_{t}(x,y)f(y)\,dy\,dt.$

By 4.3, $v_{n,k}\in C^{\infty}(U)$. Fubini’s Theorem then gives

$\displaystyle
Lv_{n,k}(x)=\int_{\frac{1}{k}}^{k}\int_{U_{n}}L_{x}\tilde{p}_{t}(x,y)f(y)\,dy\,dt$
$\displaystyle=\int_{\frac{1}{k}}^{k}\int_{U_{n}}\partial_{t}\tilde{p}_{t}(x,y)f(y)\,dy\,dt$
$\displaystyle=\mathbf{E}_{x}\mathbf{1}_{U_{n}}(\tilde{x}_{k})f(\tilde{x}_{k})-\mathbf{E}_{x}\mathbf{1}_{U_{n}}(\tilde{x}_{1/k})f(\tilde{x}_{1/k})$
$\displaystyle\to-\mathbf{1}_{U_{n}}(x)f(x)$

as $k\to\infty$ in the sense of distributions on $U$, where we used that $f$
is bounded, continuous on $U_{n}$ and for all $x\in U$,
$\mathbf{E}_{x}\mathbf{1}_{U_{n}}(\tilde{x}_{k})\leq
1-\mathbf{P}\\{\tilde{x}_{k}\in\partial U\\}\to 0$ as $k\to\infty$ by
$\mathbf{P}_{x}\\{\tau<\infty\\}=1$.

Since $f$ is bounded on $U_{n}$, by the Dominated Convergence theorem
$v_{n,k}\to v_{n}$ as $k\to\infty$, where

$v_{n}(x):=\int_{0}^{\infty}\int_{U_{n}}\tilde{p}_{t}(x,y)f(y)\,dy\,dt=\mathbf{E}_{x}\int_{0}^{\tau}\mathbf{1}_{U_{n}}(\tilde{x}_{s})f(\tilde{x}_{s})\,ds.$

Then, by 4.2, $Lv_{n}=-f$ on $U_{n}$ in the sense of distributions. Splitting
$f$ into positive and negative parts (using $f\in\mathcal{D}_{0}(U)$) and
using the Monotone Convergence Theorem, we obtain that $v_{n}$, which is
locally uniformly bounded on $U$, converges pointwise to $v_{\text{stoc},0}$
as $n\to\infty$. By employing 4.2 again, we obtain the desired result. ∎

Given the previous result, we next investigate the interior smoothness of
$u_{\text{stoc},0}$.

######  Theorem 6.2 (Interior smoothness for $u_{\text{stoc},0}$).

Assume (U00), (NE), and (PH) are satisfied and
$\mathbf{P}_{x}\\{\tau<\infty\\}=1$ for all $x\in U$. Suppose $g\in
C(\mathcal{X})$ and $u_{\text{stoc},0}(x)=\mathbf{E}_{x}g(x_{\tau})$ is
bounded on compact subsets of $U$. Then, $u_{\text{stoc},0}\in C^{\infty}(U)$
and $Lu_{\text{stoc},0}=0$ on $U$ in the classical sense.

######  Remark 6.3.

Note that by the Tietze Extension Theorem we can replace the assumption $g\in
C(\mathcal{X})$ with $g$ being continuous only on a neighborhood of $\partial
U$.

###### Proof.

First let $U_{0}\subset U$ be bounded, open with $\overline{U_{0}}\subset U$.
By 4.8(i), $v_{0}(x):=\mathbf{E}_{x}\tau_{U_{0}}$ is bounded on $U_{0}$. Thus,
for any open $V_{0}$ with $V_{0}\supset\overline{U}_{0}$ and any $g\in
C^{2}(V_{0})$, the boundedness of $U_{0}$ implies $Lg\in
C(U_{0})\cap\mathcal{D}_{0}(U_{0})$ (with respect to $\tau_{U_{0}}$). By 6.1,
the function

$\displaystyle v_{0}(x)=\mathbf{E}_{x}\int_{0}^{\tau_{U_{0}}}Lg(x_{s})\,ds$
(6.8)

satisfies $Lv_{0}=-Lg$ on $U_{0}$ in the sense of distributions. For
$u_{0}(x):=\mathbf{E}_{x}g(x_{\tau_{U_{0}}})$, Dynkin’s formula, the Dominated
Convergence Theorem, and $\mathbf{E}_{x}\tau_{U_{0}}<\infty$ imply

$\displaystyle u_{0}(x)=\lim_{t\to\infty}\mathbf{E}_{x}g(x_{\tau_{U_{0}}\wedge
t})=g(x)+\lim_{t\to\infty}\mathbf{E}_{x}\int_{0}^{\tau_{U_{0}}\wedge
t}Lg(x_{s})\,ds=g(x)+\mathbf{E}_{x}\int_{0}^{\tau_{U_{0}}}Lg(x_{s})\,ds.$

Hence, by (6.8) and $Lv_{0}=-Lg$,

$Lu_{0}=Lg-Lg=0\,\,\,\text{ on }\,\,U_{0}$

in the sense of distributions. Since $L$ is hypoelliptic, $u_{0}\in
C^{\infty}(U_{0})$ and $Lu_{0}=0$ on $U_{0}$ in the classical sense.

Next assume $g$ is merely continuous and supported on a bounded neighborhood
$W_{0}$ of the boundary $\partial U_{0}$, and without loss of generality
assume $g\equiv 0$ on $W_{0}^{c}$. Fix any $\psi\in
C_{0}^{\infty}(\mathbf{R}^{\mathfrak{m}};[0,\infty))$ with
$\int_{\mathbf{R}^{\mathfrak{m}}}\psi\,dx=1$, denote
$\psi_{\epsilon}(x)=\epsilon^{-\mathfrak{m}}\psi(\epsilon^{-1}x)$ and set

$\displaystyle
g_{\epsilon}(x)=\int_{\mathbf{R}^{\mathfrak{m}}}g(y)\psi_{\epsilon}(x-y)\,dy\in
C_{0}^{\infty}(\mathbf{R}^{\mathfrak{m}})\qquad\text{ and }\qquad
u_{\epsilon}(x)=\mathbf{E}_{x}g_{\epsilon}(x_{\tau_{U_{0}}}).$ (6.9)

Since $g_{\epsilon}$ is smooth, by the first part of the proof,
$u_{\epsilon}\in C^{\infty}(U_{0})$ and $Lu_{\epsilon}=0$ on $U_{0}$ in the
classical sense. Furthermore, $g$ is bounded, and therefore $u_{\epsilon}$ is
uniformly bounded and converges pointwise to
$u_{0}(x)=\mathbf{E}_{x}g(x_{\tau_{U_{0}}})$ on $U_{0}$ as $\epsilon\to 0$. It
follows by 4.2 and hypoellipticity of $L$ that $u_{0}\in C^{\infty}(U_{0})$
and $Lu_{0}=0$ on $U_{0}$ in the classical sense.

Finally, let $U_{n}$, $n\in\mathbf{N}$, be a sequence of bounded open sets
with $\overline{U_{n}}\subset U$ and $U_{n}\uparrow U$. Suppose that $g\in
C(\mathcal{X};[0,\infty))$ is nonnegative and by assumption
$u_{+}(x):=\mathbf{E}_{x}g(x_{\tau})$ is bounded on compact subsets of $U$.
Set $\tau_{n}=\inf\\{t>0\,:\,x_{t}\notin U_{n}\\}$ and note that we already
proved that $u_{n,N}(x):=\mathbf{E}_{x}[g(x_{\tau_{n}})\wedge N]\in
C^{\infty}(U_{n})$ with $Lu_{n,N}=0$ on $U_{n}$ in the classical sense. After
passing $n\to\infty$, 4.2, the Dominated Convergence Theorem, and
$\tau_{n}\to\tau$ imply that
$u_{\infty,N}(x):=\mathbf{E}_{x}[g(x_{\tau})\wedge N]$ satisfies
$u_{\infty,N}\in C^{\infty}(U)$ with $Lu_{\infty,N}=0$ on $U$ in the classical
sense. Passing $N\to\infty$ and again applying 4.2 and Monotone convergence
theorem, we find that $u_{+}\in C^{\infty}(U)$ with $Lu_{+}=0$ on $U$ in the
classical sense. The result follows after decomposing $g$ into positive and
negative parts, $g=g_{+}-g_{-}$. ∎

###  6.3. Boundary behavior

Note that under the hypotheses of 6.1 and 6.2, the formal stochastic solution
$u_{\text{stoc}}$ belongs to (1.4) is $C^{\infty}(U)$ and $u_{\text{stoc}}$ is
also classical solution of $Lu_{\text{stoc}}=-f$ on $U$, provided $f\in
C^{\infty}(U)$. The next natural problem is to determine satisfaction of the
boundary condition in (1.2). We explore it first for the Dirichlet part
$u_{\text{stoc},0}$.

######  Theorem 6.4.

Fix $x_{*}\in\partial U$ and $g\in C(\mathcal{X})$. Assume (U00), (NE), and
$\mathbf{P}_{x}\\{\tau<\infty\\}=1$ for all $x\in U$. If (UID($g,x_{*}$)) and
(CE($x_{*}$)) are satisfied and $x_{*}$ is regular as in 3.4, then
$u_{\text{stoc},0}(x)\to g(x_{*})$ as $x\to x_{*},x\in U$.

######  Remark 6.5.

Note that if $U$ is bounded, by extending $b,\sigma$ to
$\mathbf{R}^{\mathfrak{m}}$ so that $x_{t}$ is nonexplosive as in 2.2,
conditions (UID($g,x_{*}$)) and (CE($x_{*}$)) are satisfied for every
$x_{*}\in\partial U$. Indeed, $g$ is continuous on $\mathbf{R}^{\mathfrak{m}}$
and bounded on $\overline{U}$. Moreover, one can set $\mathcal{X}_{n}=B_{n}$
so that for all $n$ large enough
$\mathbf{P}_{x}\\{\tau_{\mathcal{X}_{n}}<\tau\\}=0$ for all $x\in U$.

######  Remark 6.6.

As discussed above, checking that a given point $x_{*}\in\partial U$ is
regular can be challenging. See [Kog17, CFH21] for some criteria. However, if
it makes sense to slightly modify the set $U$, then the regularity of $x_{*}$
can be easier to verify. See 3.11 for further information.

###### Proof of 6.4.

By (UID($g,x_{*}$)), choose $\delta>0$ so that $\mathcal{G}_{g,\delta}(x_{*})$
as in (2.18) is uniformly integrable. For $|x-x_{*}|<\delta$ and
$n\in\mathbf{N}$, write

$\displaystyle u_{\text{stoc},0}(x)-g(x_{*})$
$\displaystyle=\mathbf{E}_{x}[(g(x_{\tau})-g(x_{*}))\mathbf{1}\\{x_{\tau}\notin\overline{\mathcal{X}_{n}}\\}]+\mathbf{E}_{x}[(g(x_{\tau})-g(x_{*}))\mathbf{1}\\{x_{\tau}\in\overline{\mathcal{X}_{n}}\\}]$
$\displaystyle=:T_{1}(x,n)+T_{2}(x,n).$

Fix $\epsilon>0$ and we claim that there exists $n\in\mathbf{N}$ and
$\delta_{1}\in(0,\delta]$ such that $|T_{1}(x,n)|<\epsilon/2$ for all
$|x-x_{*}|<\delta_{1},x\in U$. Since the family
$\mathcal{G}_{g,\delta}(x_{*})$ is uniformly integrable, there is
$\epsilon^{\prime}>0$ be such that

$\epsilon^{\prime}<\frac{\epsilon}{4(|g(x_{*})|+1)}$

and whenever $\mathbf{P}\\{A\\}<\epsilon^{\prime}$ we have
$\mathbf{E}|X|\mathbf{1}_{A}<\tfrac{\epsilon}{4}$ for all
$X\in\mathcal{G}_{g,\delta}(x_{*})$. However, by (CE($x_{*}$)), there exists
$n\in\mathbf{N},\delta_{1}\in(0,\delta]$ such that $|x-x_{*}|<\delta_{1}$,
$x\in U$ implies

$\displaystyle\mathbf{P}_{x}\\{x_{\tau}\notin\overline{\mathcal{X}_{n}}\\}\leq\mathbf{P}_{x}\\{\tau_{\mathcal{X}_{n}}<\tau\\}<\epsilon^{\prime}.$

Hence, for such $n$ and $\delta_{1}$, uniformly integrability gives
$|T_{1}(x,n)|<\tfrac{\epsilon}{2}$ for all $|x-x_{*}|<\delta_{1},x\in U$,
establishing the claim.

We next claim that for this choice of $n$, there exists
$\delta_{2}\in(0,\delta_{1}]$ such that $|T_{2}(x,n)|<\tfrac{\epsilon}{2}$,
thus finishing the proof of the result. Indeed, since $g$ is continuous on the
compact set $\overline{\mathcal{X}_{n}}$, there is a Lipschitz function $h\in
C(\overline{\mathcal{X}_{n}})$ such that
$\|g-h\|_{L^{\infty}(\overline{\mathcal{X}_{n}})}<\frac{\epsilon}{8}$. Hence,
for any $\epsilon_{*}>0$ we have

$\displaystyle|T_{2}(x,n)|$
$\displaystyle\leq\frac{\epsilon}{4}+\mathbf{E}_{x}[(h(x_{\tau})-h(x_{*}))\mathbf{1}\\{x_{\tau}\in\overline{\mathcal{X}_{n}}\\}]$
$\displaystyle\leq\frac{\epsilon}{4}+C_{1}\mathbf{E}|x_{\tau}(x)-x_{*}|\wedge
C_{2}\mathbf{1}\\{\tau<\epsilon_{*}\\}+2\|h\|_{L^{\infty}}\mathbf{P}_{x}\\{\tau\geq\epsilon_{*}\\}$
$\displaystyle\leq\frac{\epsilon}{4}+C_{1}\mathbf{E}\sup_{s\in[0,\epsilon_{*}]}|x_{s}-x_{*}|\wedge
C_{2}+2\|h\|_{L^{\infty}}\mathbf{P}_{x}\\{\tau\geq\epsilon_{*}\\}=:\frac{\epsilon}{4}+T_{2}^{1}+T_{2}^{2}\,,$

where $C_{1},C_{2}>0$ are constants depending only on $n$ and $h$. To estimate
$T_{2}^{1}$, observe that

$\displaystyle T_{2}^{1}$
$\displaystyle=C_{1}\mathbf{E}\sup_{s\in[0,\epsilon_{*}]}|x_{s}(x)-x_{*}|\wedge
C_{2}$ $\displaystyle\leq
C_{1}\big{(}\mathbf{E}\sup_{s\in[0,\epsilon_{*}]}|x_{s}(x)-x_{s}(x_{*})|^{2}\wedge
C_{2}^{2}\big{)}^{1/2}+C_{1}\big{(}\mathbf{E}\sup_{s\in[0,\epsilon_{*}]}|x_{s}(x_{*})-x_{*}|^{2}\wedge
C_{2}^{2}\big{)}^{1/2}$

where the last inequality follows by triangle and Cauchy-Schwarz inequalities.
Using 4.1(i) and path continuity of $x_{t}$, there exits
$\delta_{2}\in(0,\delta_{1}]$ and $\epsilon_{*}>0$ small enough so that
$T_{2}^{1}\leq\epsilon/8$ for all $x$ with $|x-x_{*}|<\delta_{2}$. For
$T_{2}^{2}$, 4.1(ii) and the regularity of $x_{*}$, ensure that
$T_{2}^{2}<\epsilon/8$ if $\delta_{2}>0$ is sufficiently small. This finishes
the proof of the result. ∎

######  Remark 6.7.

Note that if $U$ us bounded, then (CE($x_{*}$)) is trivially satisfied by
choosing $n$ large such that $U\subset\mathcal{X}_{n}$. Here, we verify
(CE($x_{*}$)) for unbounded $U$ if $x\mapsto\mathbf{E}_{x}\tau$ is bounded on
$B_{\delta_{2}}(x_{*})\cap U$ for some $\delta_{2}>0$ and hypotheses of 2.4
are satisfied.

For any $t>0$, we have

$\displaystyle\mathbf{P}_{x}\\{\tau_{\mathcal{X}_{n}}<\tau\\}\leq\mathbf{P}_{x}\\{\tau\geq
t\\}+\mathbf{P}_{x}\\{\tau_{\mathcal{X}_{n}}\leq t\\}.$

Fix any $\delta_{1}>0$. Then, Chebychev’s inequality gives for all $x\in
B_{\delta}(x_{*})\cap U$ that

$\displaystyle\mathbf{P}_{x}\\{\tau\geq t\\}\leq t^{-1}\mathbf{E}_{x}\tau\leq
C/t\,.$

Fix $t>0$ large enough independent of $x$, such that
$\mathbf{P}_{x}\\{\tau\geq t\\}<\frac{\delta_{1}}{2}$. As in the proof of 2.4,
we have

$\displaystyle w_{n}e^{-Ct}\mathbf{P}_{x}\\{\tau_{\mathcal{X}_{n}}\leq
t\\}\leq w(x)+D\,.$

with $w_{n}\to\infty$ as $n\to\infty$. For already fixed $t>0$, there is
$n\in\mathbf{N}$ large enough so that
$\mathbf{P}_{x}\\{\tau_{\mathcal{X}_{n}}\leq t\\}<\frac{\delta_{1}}{2}$ for
each $x\in B_{\delta_{2}}(x_{*})$ and (2.20) is satisfied.

Next, we derive sufficient conditions on the Poisson part $v_{\text{stoc},0}$
of $u_{\text{stoc}}$ that ensure $v_{\text{stoc},0}(x)\to 0$ as $x\to
x_{*}\in\partial U$, $x\in U$.

######  Theorem 6.8.

Suppose (U00) and (NE) are satisfied and that $x_{*}\in\partial U$ is regular
as in 3.4. Suppose, furthermore, that conditions (UIP($f,x_{*}$)) and
(CE($x_{*}$)) are satisfied for some $f:U\to\mathbf{R}$ which is measurable
and bounded on $\mathcal{X}_{n}\cap U$ for all $n$. If for every $x\in U$,
$\mathbf{P}_{x}\\{\tau<\infty\\}=1$, then $v_{\text{stoc},0}(x)\to 0$ as $x\to
x_{*}$, $x\in U$.

###### Proof.

Let $\epsilon>0$ and by (UIP($f,x_{*}$)) there is $\delta_{1}>0$ such that for
all $x\in U$ with $|x-x_{*}|<\delta_{1}$ the family
$\mathcal{G}_{\delta_{1}}^{f}(x_{*})$ in (2.19) is uniformly integrable. For
simplicity, set $Y_{t}=\textstyle{\int_{0}^{t}f(x_{s})\,ds}$. Then, since $f$
is bounded on $\mathcal{X}_{n}\cap U$, for any $\delta>0$

$\displaystyle|v_{\text{stoc},0}(x)|$
$\displaystyle\leq|\mathbf{E}_{x}Y_{\tau}\mathbf{1}\\{\tau_{\mathcal{X}_{n}}<\tau\\}|+|\mathbf{E}_{x}Y_{\tau}\mathbf{1}\\{\tau_{\mathcal{X}_{n}}\geq\tau\\}|$
$\displaystyle\leq|\mathbf{E}_{x}Y_{\tau}\mathbf{1}\\{\tau_{\mathcal{X}_{n}}<\tau\\}|+|\mathbf{E}_{x}Y_{\tau\wedge\delta}\mathbf{1}\\{\tau_{\mathcal{X}_{n}}\geq\tau,\tau\leq\delta\\}|+|\mathbf{E}_{x}Y_{\tau}\mathbf{1}\\{\tau>\delta\\}|$
$\displaystyle\leq|\mathbf{E}_{x}Y_{\tau}\mathbf{1}\\{\tau_{\mathcal{X}_{n}}<\tau\\}|+\delta\|f\|_{L^{\infty}(\mathcal{X}_{n}\cap
U)}+|\mathbf{E}_{x}Y_{\tau}\mathbf{1}\\{\tau>\delta\\}|.$

By uniform integrability, there exists $\epsilon^{\prime}>0$ such that
$\mathbf{P}\\{A\\}<\epsilon^{\prime}$ implies
$\mathbf{E}_{x}Y\mathbf{1}_{A}<\epsilon/3$ for all $|x-x_{*}|<\delta_{1}$. By
(CE($x_{*}$)), there are $n\in\mathbf{N},\delta_{2}\in(0,\delta_{1}]$ such
that $\mathbf{P}_{x}\\{\tau_{\mathcal{X}_{n}}<\tau\\}<\epsilon^{\prime}$ for
all $|x-x_{*}|<\delta_{2}$, $x\in U$. For this choice of $n$, let

$\delta=\frac{\epsilon}{3(\|f\|_{L^{\infty}(\mathcal{X}_{n}\cap U)}+1)}.$

By making $\delta_{2}>0$ smaller if necessary, by 4.1(ii) we can ensure that
$\mathbf{P}_{x}\\{\tau>\delta\\}<\epsilon^{\prime}$ for all
$|x-x_{*}|<\delta_{2}$, $x\in U$. The result follows since for
$|x-x_{*}|<\delta_{2}$, $x\in U$, we have $|v_{\text{stoc},0}(x)|<\epsilon$
and $\epsilon>0$ is arbitrary. ∎

We can now combine the previous results and relate them back to the original
problem (1.2).

######  Corollary 6.9.

Assume (U00), (NE), and $\mathbf{P}_{x}\\{\tau<\infty\\}=1$ for all $x\in U$.
Let $g\in C(\mathcal{X})$ and $f\in C^{\infty}(U)\cap\mathcal{D}_{0}(U)$ be
such that $u_{\text{stoc},0}$ is bounded on compact subsets of $U$ and $f\in
B(\mathcal{X}_{n}\cap U)$ for all $n$. If $U$ is boundary regular for $x_{t}$
and the conditions (UID($g$)), (UIP($f$)), and (CE) are satisfied, then
$u_{\text{stoc}}$ is a classical solution of (1.2). If $U$ is furthermore
assumed to be bounded, then $u_{\text{stoc}}$ is the unique classical solution
of (1.2).

###### Proof.

The only assertion we have left to prove is the uniqueness, when $U$ is
bounded.

Let $\\{U_{n}\\}$ be a sequence of bounded open subsets of $U$ with
$\overline{U}_{n}\subset U_{n+1}$ and $U=\cup_{n}U_{n}$. If $u$ is a classical
solution of (1.2), then $u$ is smooth on $U_{n}$ for each $n$, and by Dynkin’s
formula (2.8) we have for any $x\in U$ and $n\in\mathbf{N}$ large enough

$\displaystyle\mathbf{E}_{x}u(x_{t\wedge\tau_{U_{n}}})=u(x)+\mathbf{E}_{x}\int_{0}^{t\wedge\tau_{U_{n}}}Lu(x_{s})\,ds=u(x)-\mathbf{E}_{x}\int_{0}^{t\wedge\tau_{U_{n}}}f(x_{s})\,ds,$

and therefore

$\displaystyle
u(x)=\mathbf{E}_{x}u(x_{t\wedge\tau_{U_{n}}})+\mathbf{E}_{x}\int_{0}^{t\wedge\tau_{U_{n}}}f(x_{s})\,ds.$

Since $u$ is continuous on the compact set $\overline{U}$, $u$ is bounded on
$U$. Also, since $g\in C(\mathcal{X})$ and $f\in
C^{\infty}(U)\cap\mathcal{D}_{0}(U)$, passing $t\rightarrow\infty$ and then
$n\rightarrow\infty$, boundary regularity and the Dominated Convergence
Theorem imply that

$\displaystyle u(x)=u_{\text{stoc}}(x).$

∎

##  7\. The Transience and Recurrence Dichotomy for Degenerate Diffusions

The goal of this section is to carefully establish the dichotomy between
transience and recurrence for degenerate diffusions by adapting the classic
cycle constructions of Khasminskii [Kha60, Has60], which was carried out in
the setting of elliptic diffusions on Euclidean space. Note that this has been
done previously using the language of invariant control sets as in [Kli87].
However, we found a gap in the arguments in [Kli87] that we could not fix in
an obvious way (see 7.22 below). Moreover, some regularity claims in [Kli87]
could not be verified without calculations analogous to ones in the previous
sections. Although it is known that there are alternative, probabilistic paths
which circumnavigate these issues (we refer, in particular, to the work of
Harris [Har56], the survey paper of Baxendale [Bax11] and the work of Meyn-
Tweedie [MT12]), here we establish the dichotomy using classical ideas of
Khasminskii and regularity properties established above. Note that this
approach traces back to Maruyama and Tanaka [MT57] and Watanabe [WM58] in the
case of a one-dimensional, elliptic diffusion. We also refer to the works
[AKDR66, GT15] which we found helpful.

###  7.1. Nice diffusions

In this subsection, we briefly introduce the structural assumptions we make on
the diffusion $x_{t}$ satisfying (1.3) in this section. Recall that by
condition (NE), $x_{t}\in\mathcal{X}$ for all $t\geq 0$ and any initial
condition $x_{0}=x\in\mathcal{X}$.

In order to formulate our results, we need a notion of _irreducibility_ of
$x_{t}$ as introduced in the following definition.

######  Definition 7.1.

Suppose that condition (NE) is satisfied. We call $x_{t}$ _irreducible_ if for
any $x,y\in\mathcal{X}$ and $\delta>0$ we have
$\mathbf{P}_{y}\\{\tau_{B_{\delta}(x)^{c}}<\infty\\}>0$.

Note that irreducibility means that, for all $x,y\in\mathcal{X}$, the process
started at $y\in\mathcal{X}$ enters an arbitrarily small neighborhood of
$x\in\mathcal{X}$ with positive probability. Thus, the process can transition
between arbitrarily small neighborhoods of any two points in $\mathcal{X}$.

######  Remark 7.2.

Comparing terminology, if $x_{t}$ is irreducible in the sense of 7.1, then
$\mathcal{X}$ is the unique invariant control set of $x_{t}$ as in [Kli87].
Certainly, the methods used below can be applied in more general settings,
e.g. if there is more than one invariant control set or if the process $x_{t}$
eventually enters an invariant control set from a larger set to not return to
other parts of space. For our purposes, one irreducible set $\mathcal{X}$ is
sufficient.

We are ready to introduce the assumptions we impose on the diffusion $x_{t}$
in this section.

######  Definition 7.3.

We say that the diffusion $x_{t}$ is _nice_ if the following conditions are
met:

  * (i)

Condition (NE) is satisfied;

  * (ii)

The generator $L$ of $x_{t}$ is satisfies the parabolic Hörmander condition on
$\mathcal{X}$ as in 2.11;

  * (iii)

$x_{t}$ is irreducible as in 7.1.

One key property of a nice diffusion employed below is that the process leaves
bounded sets in $\mathcal{X}$ sufficiently fast.

######  Proposition 7.4.

Suppose that $x_{t}$ is a nice diffusion and $U\subset\mathcal{X}$ is
nonempty, open and bounded with $\overline{U}\subset\mathcal{X}$. Then there
exists $\delta>0$ such that

$\displaystyle\sup_{x\in\overline{U}}\mathbf{E}_{x}e^{\delta\tau_{\overline{U}}}<\infty.$
(7.1)

###### Proof.

The proof of this result follows a similar reasoning used in the proof of
4.8(i). Let $z\in\mathcal{X}\setminus\overline{U}$ and fix $\epsilon>0$ such
that $B_{\epsilon}(z)\subset\mathcal{X}\setminus\overline{U}$. By
irreducibility of $x_{t}$, for all $x\in\overline{U}$ there exists $t=t(x)>0$
and $\alpha=\alpha(x)\in(0,1)$ such that

$\displaystyle\mathbf{P}_{x}\\{\tau_{B_{\epsilon}(z)^{c}}>t\\}\leq 1-\alpha.$

Applying 4.7(i) along with the parabolic Hörmander condition on $\mathcal{X}$
and compactness of $\overline{U}$, there exists $t_{*}>0$ and
$\alpha_{*}\in(0,1)$ independent of $x$ so that

$\displaystyle\mathbf{P}_{x}\\{\tau_{\overline{U}}>t_{*}\\}\leq\mathbf{P}_{x}\\{\tau_{B_{\epsilon}(z)^{c}}>t_{*}\\}\leq
1-\alpha_{*}\,\,\,\text{ for all }\,\,\,x\in\overline{U}.$ (7.2)

Following the proof of 4.8(i), the Markov property implies that

$\displaystyle\mathbf{P}_{x}\\{\tau_{\overline{U}}\geq
mt_{*}\\}\leq(1-\alpha_{*})^{m-1}$

for all $m\in\mathbf{N}$ and all $x\in\overline{U}$. Hence
$\tau_{\overline{U}}<\infty$, $\mathbf{P}_{x}$-almost surely. Furthermore,
choosing $\delta=\delta(t_{*},\alpha_{*})>0$ small enough so that $e^{\delta
t_{*}}(1-\alpha_{*})<1$, it follows that for any $x\in\overline{U}$:

$\displaystyle\mathbf{E}_{x}e^{\delta\tau_{\overline{U}}}=\sum_{m=1}^{\infty}\mathbf{E}_{x}e^{\delta\tau_{\overline{U}}}\mathbf{1}\\{(m-1)t_{*}\leq\tau_{\overline{U}}<mt_{*}\\}$
$\displaystyle\leq\sum_{m=1}^{\infty}e^{\delta
mt_{*}}\mathbf{P}_{x}\\{\tau_{\overline{U}}\geq(m-1)t_{*}\\}$
$\displaystyle\leq(1-\alpha_{*})^{-2}\sum_{m=1}^{\infty}(e^{\delta
t_{*}}(1-\alpha_{*}))^{m}<\infty.$

∎

######  Remark 7.5.

Yet another way to rephrase the conclusion of 7.4 is that the process $x_{t}$
exits any bounded domain in $\mathcal{X}$ in _logarithmic time_ or
_exponentially fast_ on average.

###  7.2. Recurrence and transience for nice diffusions

We start with the definition of transience and recurrence.

######  Definition 7.6.

Suppose the diffusion $x_{t}$ is nice. We say that a point $x\in\mathcal{X}$
is _recurrent_ if for any $\delta>0$ and any $y\in\mathcal{X}\setminus
B_{\delta}(x)$

$\displaystyle\mathbf{P}_{y}\\{\tau_{B_{\delta}(x)^{c}}<\infty\\}=1.$ (7.3)

Otherwise, we say $x\in\mathcal{X}$ is _transient_.

Our next goal is to prove that points in $\mathcal{X}$ are either all
recurrent, in which case we call $x_{t}$ _recurrent_ , or all transient, in
which case we call $x_{t}$ _transient_. Thus, the dichotomy between transience
and recurrence is established in the following proposition. Afterwards, we
establish further properties of transience and recurrence.

######  Proposition 7.7.

Assume $x_{t}$ is nice. If $x\in\mathcal{X}$ is recurrent, then all points in
$\mathcal{X}$ are recurrent. Consequently, either all points in $\mathcal{X}$
are recurrent or all points in $\mathcal{X}$ are transient.

###### Proof.

Suppose $x\in\mathcal{X}$ is recurrent and let $y\in\mathcal{X}$ with $x\neq
y$. We show that $y$ is also recurrent. Suppose that $\delta>0$ is any
positive real number such that $x\notin B_{2\delta}(y)$ and
$\overline{B_{2\delta}(y)}\subset\mathcal{X}$. By irreducibility in 7.1 and
path continuity (if $x_{t}\in B_{\delta}(y)$ for some t, then the inclusion
holds for rational $t$), there exists $t_{*}>0$ such that

$\displaystyle 2\alpha:=P_{t_{*}}(x,B_{\delta}(y))>0.$ (7.4)

Since $L$ satisfies the parabolic Hörmander condition on $\mathcal{X}$,
$w\mapsto P_{t_{*}}(w,B_{\delta}(y))$ is continuous at $x$ by 4.6 and 4.3. In
particular, there exists $\epsilon>0$ such that $B_{\epsilon}(x)\cap
B_{2\delta}(y)=\emptyset$, $\overline{B_{\epsilon}(x)}\subset\mathcal{X}$ and
$P_{t_{*}}(w,B_{\delta}(y))\geq\alpha$ for all
$w\in\overline{B_{\epsilon}(x)}$. Define stopping times
$\sigma_{j},j=0,1,\ldots$, inductively as follows:

$\displaystyle\sigma_{1}$ $\displaystyle=\inf\\{t\geq 0\,:\,x_{t}\in
B_{\epsilon}(x)\\},$ $\displaystyle\sigma_{2}$
$\displaystyle=\sigma_{1}+t_{*}$ $\displaystyle\sigma_{3}$
$\displaystyle=\inf\\{t\geq\sigma_{2}\,:\,x_{t}\in B_{\epsilon}(x)\\}$
$\displaystyle\vdots$ $\displaystyle\qquad\vdots$ $\displaystyle\sigma_{2k}$
$\displaystyle=\sigma_{2k-1}+t_{*}$ $\displaystyle\sigma_{2k+1}$
$\displaystyle=\inf\\{t\geq\sigma_{2k}\,:\,x_{t}\in B_{\epsilon}(x)\\}.$

for $k\geq 2$. Then for all $j\geq 1$, the stopping time $\sigma_{j}$ is
almost surely finite since $x_{t}$ is nice and $x$ is recurrent. Next, by the
strong Markov property we have for each $j\geq 0$:

$\displaystyle\mathbf{P}_{w}\\{\tau_{B_{\delta}(y)^{c}}>\sigma_{2j+2}\\}$
$\displaystyle=\mathbf{E}_{w}\mathbf{E}_{w}[\mathbf{1}\\{\tau_{B_{\delta}(y)^{c}}>\sigma_{2j+2}\\}|\mathcal{F}_{\sigma_{2j+1}}]$
$\displaystyle=\mathbf{E}_{w}\mathbf{1}\\{\tau_{B_{\delta}(y)^{c}}>\sigma_{2j+1}\\}\mathbf{E}_{x_{\sigma_{2j+1}}}\mathbf{1}\\{\tau_{B_{\delta}(y)^{c}}>t_{*}\\}$
$\displaystyle\leq(1-\alpha)\mathbf{P}_{w}\\{\tau_{B_{\delta}(y)^{c}}>\sigma_{2j}\\}.$

Thus, by induction,
$\mathbf{P}_{w}\\{\tau_{B_{\delta}(y)^{c}}>\sigma_{2j}\\}\leq(1-\alpha)^{j}$
for all $j\geq 0$. Finally, the Borel-Cantelli Lemma implies
$\mathbf{P}_{w}\\{\tau_{B_{\delta}(y)^{c}}<\infty\\}=1$ for all
$w\in\mathcal{X}$, and therefore $y$ is recurrent. ∎

We also have the following corollary of 7.7.

######  Corollary 7.8.

Suppose $x_{t}$ is nice. If $x_{t}$ is recurrent, then for any
$x,y\in\mathcal{X}$ and any open set $U_{y}\subset\mathcal{X}$ containing $y$:

$\displaystyle\mathbf{P}_{x}\\{\omega\,:\,\exists\,s_{j}(\omega)\in(0,\infty)\uparrow\infty\text{
for which }x_{s_{j}}\in U_{y}\text{ for all }j\\}=1.$

######  Remark 7.9.

7.8 states that if $x_{t}$ is recurrent, then for all $x\in\mathcal{X}$,
almost surely the process started from $x$ visits infinitely often any
neighborhood of any $y\in\mathcal{X}$.

###### Proof of 7.8.

Let $x,y\in\mathcal{X}$ and $U_{y}\subset\mathcal{X}$ be open with $y\in
U_{y}$. Fix $z\in\mathcal{X}$ with $z\neq x$ and $z\neq y$. Choose $\delta>0$
such that $B_{\delta}(y)\subset U_{y}$, $B_{\delta}(z)\cap
B_{\delta}(x)=\emptyset$, $B_{\delta}(z)\cap B_{\delta}(y)=\emptyset$ and
$B_{\delta}(z)\subset\mathcal{X}$. Define stopping times
$\sigma_{i},i=0,1,\ldots$ inductively by

$\displaystyle\sigma_{0}$ $\displaystyle=0;$ $\displaystyle\sigma_{1}$
$\displaystyle=\inf\\{t\geq\sigma_{0}\,:\,x_{t}\in B_{\delta}(y)\\}$
$\displaystyle\sigma_{2}$ $\displaystyle=\inf\\{t\geq\sigma_{1}+1\,:\,x_{t}\in
B_{\delta}(z)\\}$ $\displaystyle\vdots$ $\displaystyle\qquad\vdots$
$\displaystyle\sigma_{2k+1}$
$\displaystyle=\inf\\{t\geq\sigma_{2k}\,:\,x_{t}\in B_{\delta}(y)\\}$
$\displaystyle\sigma_{2k+2}$
$\displaystyle=\inf\\{t\geq\sigma_{2k+1}+1\,:\,x_{t}\in B_{\delta}(z)\\}.$

By recurrence, $\sigma_{k}$, $k\geq 0$ is almost surely finite. The result
follows by setting $s_{j}=\sigma_{2j+1}$. ∎

The next proposition further explores implications of transience for a nice
diffusion $x_{t}$.

######  Proposition 7.10.

Suppose $x_{t}$ is nice and transient. For any $x,y\in\mathcal{X}$ there
exists $\delta>0$ small enough such that
$\overline{B_{\delta}(y)}\subset\mathcal{X}$ and

$\displaystyle\mathbf{P}_{x}\\{\omega\,:\,\exists\,t_{0}(\omega)\in[0,\infty)\text{
such that }x_{t}(\omega)\notin B_{\delta}(y)\,\,\forall t\geq
t_{0}(\omega)\\}=1.$ (7.5)

###### Proof.

Since $x_{t}$ is transient, there exist $\delta_{1}>0$, $y\in\mathcal{X}$, and
$z\in\mathcal{X}\setminus B_{2\delta_{1}}(y)$ such that

$\displaystyle\mathbf{P}_{z}\\{\tau_{B_{\delta_{1}}(y)^{c}}=\infty\\}=:2\alpha>0.$
(7.6)

By 4.7, there exists $\epsilon>0$ such that
$\overline{B_{\epsilon}(z)}\subset\mathcal{X}$ and

$\mathbf{P}_{w}\\{\tau_{B_{\delta_{1}}(y)^{c}}=\infty\\}\geq\alpha$

for all $w\in B_{\epsilon}(z).$ In order to obtain a contradiction, for every
$\delta>0$ define

$E_{\delta}(x):=\\{\omega\,:\,\exists\,s_{j}(\omega)\in(0,\infty)\uparrow\infty\text{
such that }x_{s_{j}}(\omega)\in B_{\delta}(y),x_{0}=x\\}$ (7.7)

and assume there is $x\in\mathcal{X}$ such that for each $\delta>0$

$\displaystyle\mathbf{P}\\{E_{\delta}(x)\\}>0.$ (7.8)

Note that for each $t\geq 0$, one has $E_{\delta}(x)=E_{\delta}(x_{t}(x))$,
where $x_{t}(x)$ is the process with $x_{0}(x)=x$. Next, we claim that, on the
set $E_{\delta}(x)$ for all $\delta>0$ small enough, the process $x_{t}$
almost surely enters $\overline{B_{\epsilon/2}(z)}$.

To prove the claim, first observe that since $\mathcal{X}$ is irreducible,

$\mathbf{P}_{y}\\{\tau_{B_{\epsilon/2}(z)^{c}}<\infty\\}>0.$

Thus, by path continuity (cf. the proof of 7.7) there exists $t_{*}>0$ such
that $P_{t_{*}}(y,B_{\epsilon/2}(z))=:2a>0$. Since $w\mapsto
P_{t_{*}}(w,B_{\epsilon/2}(z))$ is continuous, there exists
$\delta\in(0,\delta_{1})$ such that
$\overline{B_{\delta}(y)}\subset\mathcal{X}$ and
$P_{t_{*}}(w,B_{\epsilon/2}(z))\geq a$ for all $w\in\overline{B_{\delta}(y)}$.
Fix such $\delta\in(0,\delta_{1})$ and inductively define stopping times
$\zeta_{i}$, $i=1,2,\ldots$, by

$\displaystyle\zeta_{1}$ $\displaystyle=\inf\\{t\geq 0\,:\,x_{t}\in
B_{\delta}(y)\\}\,,$ $\displaystyle\zeta_{2}$
$\displaystyle=\zeta_{1}+t_{*}\,,$ $\displaystyle\vdots$
$\displaystyle\qquad\vdots$ $\displaystyle\zeta_{2k+1}$
$\displaystyle=\inf\\{t\geq\zeta_{2k}\,:\,x_{t}\in B_{\delta}(y)\\}\,,$
$\displaystyle\zeta_{2k+2}$ $\displaystyle=\zeta_{2k+1}+t_{*}.$

By the definition of $E_{\delta}(x)$, the stopping times $\zeta_{i}$ are
almost surely finite on $E_{\delta}(x)$. Then, the strong Markov property (cf.
the proof of 7.7) yields

$\displaystyle\mathbf{P}_{x}\\{\tau_{B_{\epsilon/2}(z)^{c}}>\zeta_{2j}\,|\,E_{\delta}(x)\\}\leq(1-a)^{j}$
(7.9)

for any $j$. Thus the Borel-Cantelli lemma implies that
$\mathbf{P}_{x}\\{\tau_{B_{\epsilon/2}(z)^{c}}<\infty\,|\,E_{\delta}(x)\\}=1$,
establishing the claim.

Next, define stopping times $\sigma_{i}^{\prime}$, $i=1,2,\ldots,$ by

$\displaystyle\sigma_{1}^{\prime}$ $\displaystyle=\inf\\{t\geq 0\,:\,x_{t}\in
B_{\delta}(y)\\}\,,$ $\displaystyle\sigma_{2}^{\prime}$
$\displaystyle=\inf\\{t\geq\sigma_{1}^{\prime}\,:\,x_{t}\in
B_{\epsilon/2}(z)\\}\,,$ $\displaystyle\vdots$ $\displaystyle\qquad\vdots$
$\displaystyle\sigma_{2k+1}^{\prime}$
$\displaystyle=\inf\\{t\geq\sigma_{2k}^{\prime}\,:\,x_{t}\in
B_{\delta}(y)\\}\,,$ $\displaystyle\sigma_{2k+2}^{\prime}$
$\displaystyle=\inf\\{t\geq\sigma_{2k+1}^{\prime}\,:\,x_{t}\in
B_{\epsilon/2}(z)\\}.$

By a similar argument to the one used above, it also follows that on the event
$E_{\delta}(x)$, $\sigma_{j}^{\prime}$ is finite almost surely for all $j\geq
1$. Observe that
$\mathbf{P}(\sigma_{j}^{\prime}<\infty)\geq\mathbf{P}(E_{\delta}(x))\geq c$,
where $c$ is independent of $j$. However, by the strong Markov property and
iteration, if $j\geq 2$ we have

$\displaystyle\mathbf{P}_{x}\\{\sigma_{2j}^{\prime}<\infty\\}$
$\displaystyle=\mathbf{E}_{x}\mathbf{E}_{x}[\mathbf{1}\\{\sigma_{2j}^{\prime}<\infty\\}\,|\,\mathcal{F}_{\sigma_{2j-2}^{\prime}}]$
$\displaystyle=\mathbf{E}_{x}\mathbf{1}\\{\sigma_{2j-2}^{\prime}<\infty\\}\mathbf{P}_{x_{\sigma_{2j-2}^{\prime}}}\\{\sigma_{2}^{\prime}<\infty\\}\leq(1-\alpha)\mathbf{P}_{x}\\{\sigma_{2j-2}^{\prime}<\infty\\}\leq(1-\alpha)^{j-1}.$

Thus Borel-Cantelli implies that $\sigma_{2j}^{\prime}<\infty$ for only
finitely many $j$, a contradiction. ∎

As an immediate consequence, we have the following corollary.

######  Corollary 7.11.

If $x_{t}$ is transient, then for any compact set $K\subset\mathcal{X}$ and
$x\in\mathcal{X}$

$\displaystyle\mathbf{P}_{x}\\{\omega\,:\,\exists\,t_{0}^{K}(\omega)\in[0,\infty)\text{
such that }x_{t}(\omega)\notin K\,\,\forall t\geq t_{0}^{K}(\omega)\\}=1$

and

$\displaystyle\lim_{t\to\infty}\mathbf{P}_{x}\\{x_{t}\in K\\}=0\,.$

###### Proof.

Fix $x\in\mathcal{X}$ and for any $y\in K$ fix $\delta_{y}>0$ and
$t_{0}(\omega)=t_{0,y}(\omega)$ such that the conclusion of 7.10 holds true.
From the open cover $\\{B_{\delta(y)}(y)\\}_{y\in K}$ of $K$ choose a finite
subcover and define $t_{0}^{K}$ to be the maximum of $t_{0,y}$ in this finite
subcover. For the second conclusion, we note that

$\mathbf{P}_{x}\\{x_{t}\in
K\\}\leq\mathbf{P}_{x}\\{t_{0}^{K}(\omega)>t\\}\rightarrow 0$

as $t\rightarrow\infty$, where $t_{0}^{K}$ is as in the first assertion. ∎

###  7.3. Invariant measures

A central interest in the theory of stochastic differential equations is the
large-time behavior of the process $x_{t}$. In particular, we are interested
in the relationship between recurrence, transience and the existence of
_invariant measures_. Such measures are the random analogues of equilibrium
points of deterministic ordinary differential equations.

To introduce invariant measures, throughout this section we again assume
$x_{t}$ is a nice diffusion. In particular, $x_{t}$ is non-explosive process
on $\mathcal{X}$, and consequently the process $x_{t}$ is Markov with Markov
semigroup $(\mathcal{P}_{t})_{t\geq 0}$. Recalling that $\mathcal{B}$ denotes