TITLE: Why do the even Bernoulli numbers grow so fast?
QUESTION [1 upvotes]: Question is in the title.
We have:
$$B_{2n} \sim (-1)^{n-1} 4 \sqrt {\pi n} \left( \frac n {\pi e} \right)^{2n}$$

REPLY [6 votes]: Recall the generating function
$$f(z) = \sum_{q\ge 0} B_q \frac{z^q}{q!} = \frac{z}{e^z-1}.$$
We can treat it as if it were rational as the poles are countable with
no accumulation point, so we may use the techniques of extracting coefficients from rational generating functions, with the asymptotics being given by inverse powers of the dominant poles.
The pole at $z=0$ is cancelled by the factor $z$ in the numerator. The
nearest two poles are at $\pm 2\pi i,$ both at the same distance from zero ($2\pi$).
Computing the residues we obtain
$$\mathrm{Res}(f(z); z=\pm 2\pi i) = \pm 2\pi i$$
and from this we get the singular decomposition
$$\frac{2\pi i}{z-2\pi i} - \frac{2\pi i}{z+2\pi i}.$$
Turn this into geometric series to get
$$\frac{1}{z/(2\pi i)-1} - \frac{1}{z/(2\pi i)+1}
= - \frac{1}{1-z/(2\pi i)} - \frac{1}{1+z/(2\pi i)}
\\= -\sum_{q\ge 0} \frac{z^q}{(2\pi i)^q}
-\sum_{q\ge 0} (-1)^q \frac{z^q}{(2\pi i)^q}
= -2 \sum_{q\ge 0} \frac{z^{2q}}{(2\pi i)^{2q}}
\\ = -2 \sum_{q\ge 0} (-1)^q \frac{z^{2q}}{(2\pi)^{2q}}
= 2 \sum_{q\ge 0} (-1)^{q+1} \frac{z^{2q}}{(2\pi)^{2q}}.$$
Extracting coeffcients and since $B_{2n}  = (2n)! [z^{2n}] f(z)$ we find that
$$B_{2n} \sim (2n)! \times 2 \times
\frac{(-1)^{n+1}}{(2\pi)^{2n}}.$$
Using Stirling's formula this becomes
$$\sqrt{4\pi n} \left(\frac{2n}{e}\right)^{2n}
\times 2 \times
\frac{(-1)^{n+1}}{(2\pi)^{2n}}.$$
This is
$$4 \times (-1)^{n+1} \times \sqrt{\pi n}
\left(\frac{n}{\pi e}\right)^{2n}.$$
Addendum. To  justify rigorously the above asymptotics  we need to
verify that  there isn't an additional entire  component not accounted
for in the complete singular decomposition.
To do this introduce the sum
$$g(w) =
\sum_{q\ge 1}\left(\frac{1}{w/q-1}-\frac{1}{w/q+1}\right)
= 2 \sum_{q\ge 1} \frac{1}{w^2/q^2-1}.$$
This  can  be  evaluated  using  a  standard  technique  from  complex
variables and setting
$$G(z) = 1/(w^2/z^2-1)\times\pi\cot(\pi z)$$
we have
$$g(w) =
-
\left(
\mathrm{Res}(G(z); z=0)
+\mathrm{Res}(G(z); z=w)
+\mathrm{Res}(G(z); z=-w)
\right).$$
Computing the residues we get
$$g(w) = \pi w \cot(\pi w).$$
Now we claim that
$$f(z) = -\frac{1}{2}z + \sum_{q\ge 1}
\left(\frac{2\pi i q}{z-2\pi i q}-\frac{2\pi i q}{z+2\pi i q}\right).$$
The sum is $g(z/(2\pi i))$ so we get
$$-\frac{1}{2}z + \frac{z}{2i}  \cot\left(\frac{z}{2i}\right)
= -\frac{1}{2}z + \frac{z}{2i}  \times i \times
\frac{e^{z/2}+e^{-z/2}}{e^{z/2}-e^{-z/2}}
\\= \frac{1}{2} z
\left(-1 + \frac{e^z+1}{e^z-1}\right)
= \frac{1}{2}z \frac{2}{e^z-1} = \frac{z}{e^z-1}.$$
This shows  rigorously that the  entire part is  $$-\frac{1}{2}z$$ and
hence makes no  contribution to $B_{2n}$ where $n\ge  1$ and the above
singular decomposition is justified.


TITLE: What does this function calculate?
QUESTION [1 upvotes]: Can anyone confirm to me what this function calculates?
Function m(t);
p=9t;
q=p+32;
r = q/5;
Return r;
End function

Ok so inputting numbers from $0-9$ (in numerical order) the output is as follows:
$$6.4,
8.2,
10,
11.8,
13.6,
15.4,
17.2,
19,
20.8,
22.6
$$
Ok so clearly each value is being incremented by $1.8$. But would it be correct to say that the function adds increments by $1.8$?

REPLY [1 votes]: r = q/5 $\rightarrow$ r = (p + 32)/5 $\rightarrow$ r = (9t + 32)/5
Hence r = 9/5t + 32/5
equation of line is y = mx + b, so as we can see this is just equation of line of slope 9/5 and y-intercept of 32/5.


text:
\begin{document}
\title{A Terracini Lemma for osculating spaces \\
with applications to Veronese surfaces}
\author{Edoardo Ballico and Claudio Fontanari}
\date{}
\maketitle

\begin{small}
\begin{center}
\textbf{Abstract}
\end{center}
Here we present a partial generalization to higher order osculating spaces
of the classical Lemma of Terracini on ordinary tangent spaces.
As an application, we investigate the secant varieties to the osculating
varieties to the Veronese embeddings of the projective plane.
\vspace{0.5cm}

\noindent AMS Subject Classification: 14N05.

\noindent Keywords: osculating space, secant variety, osculating
variety, Veronese surface.
\end{small}

\vspace{0.5cm}

\section{Introduction}
The geometry of defective varieties, whose secant varieties have dimension
less than the expected one, is a subtle and intriguing subject which has
been investigated by several authors, both classical and modern (see for
instance the introduction and the references in \cite{BF3}). The main
tool for understanding defective varieties is provided by the celebrated
Lemma of Terracini (see \cite{Terracini:11} for the original statement and
\cite{Dale:84}, \cite{Adl:87} for modern versions):

\begin{Lemma}\label{basicone} Let $X \subset \PP^r$ be an integral
non-degenerate projective variety of dimension $n$ and let $h \le r$ be
a positive integer. Take $h+1$ general points $p_0, \ldots, p_h$ of $X$
and let $P \in < p_0, \ldots, p_h >$ be a general point in the
$h$-secant variety $\Sec_h(X)$ of $X$. Then the tangent space
$T_P(\Sec_h(X))$ to $\Sec_h(X)$ at $P$ is given by $T_P(\Sec_h(X))
= < \bigcup_{i=0}^h T_{p_i}(X) >$.
\end{Lemma}

The generalization of the above result from tangent spaces to higher
order osculating spaces (respectively, from double points to points of
higher multeplicity) is a very delicate problem, which has recently
attracted a great deal of interest (for instance, we are aware of
work in progress by Ciliberto on these topics). Here we present our
attempts in this direction, which have been inspired by the less known
paper \cite{Terracini:22} of Terracini: indeed, our Lemma~\ref{Terracini}
should be regarded as a rough extension of Lemma~\ref{basicone}.

As an application, we are going to investigate the secant varieties to the
osculating varieties $T(m,V_{2,d})$ of order $m$ to the
$d$-Veronese surface $V_{2,d}$.
It is known that the tangential variety to $V_{2,d}$ is not $h$-defective
unless $d=3$ and $h=1$ (see \cite{CGG} and \cite{B}) and that the
$2$-osculating variety of $V_{2,d}$ is not $h$-defective unless $d=4$ and
$h=1$ (see \cite{BF1}). Other related results are collected in the very
recent pre-print \cite{BCGI}. Here instead we prove the following:

\begin{Proposition}\label{special}
Fix integers $d \ge 1$, $h \ge 1$, $m \ge 1$.
If
$$
\frac{(d+2)(d+1)}{2}  \le 2(h+1) + (h+1)\frac{(m+2)(m+1)}{2}
$$
then $T(m,V_{2,d})$ is $h$-defective for:

(a) $h=1$, $m+2 \le d \le 2m+2$;

(b) $h=2$, $3(m+2)/2 \le d \le 2m+2$;

(c) $h=4$, $2m+4 \le d \le (5m+8)/2$;

(d) $h=5$, $12(m+2)/5 \le d \le (5m+8)/2$;

(e) $h=6$, $21(m+2)/8 \le d \le (8m+14)/3$;

(f) $h=7$, $48(m+2)/17 \le d \le (17m+32)/6$.

\noindent If instead
$$
\frac{(d+2)(d+1)}{2}  \ge 2(h+1) + (h+1)\frac{(m+2)(m+1)}{2}
$$
then $T(m,V_{2,d})$ is $h$-defective for:

(a) $h=1$, $m+1 \le d \le 2m$;

(b) $h=2$, $3(m+1)/2 \le d \le 2m$;

(c) $h=4$, $2m+2 \le d \le (5m+3)/2$;

(d) $h=5$, $12(m+1)/5 \le d \le (5m+3)/2$;

(e) $h=6$, $21(m+1)/8 \le d \le (8m+6)/3$;

(f) $h=7$, $48(m+1)/17 \le d \le (17m+15)/6$.
\end{Proposition}

\begin{Theorem}\label{nonspecial}
Fix integers $d \ge 1$, $h \ge 1$, $m \le 18$.
Then
$$
\dim (\Sec_h(T(m,V_{2,d}))) = \expdim (\Sec_h(T(m,V_{2,d}))) =
$$
$$
= \min \left\{2(h+1) + (h+1)\frac{(m+2)(m+1)}{2}
- 1, \frac{(d+2)(d+1)}{2} - 1\right\}
$$
for

(a) $h=1$ and either $d < m+1$ or $d > 2m+2$;

(b) $h=2$ and either $d < 3(m+1)/2$ or $d > 2m+2$;

(c) $h=4$ and either $d < 2m+2$ or $d > (5m+8)/2$;

(d) $h=5$ and either $d < 12(m+1)/5$ or $d > (5m+8)/2$;

(e) $h=6$ and either $d < 21(m+1)/8$ or $d > (8m+14)/3$;

(f) $h=7$ and either $d < 48(m+1)/17$ or $d > (17m+32)/6$;

(g) $h=3$ or $h \ge 8$.

\end{Theorem}

We deduce the above statements from the classification of $(-1)$-special
linear systems on the projective plane due to Ciliberto and Miranda
(\cite{CilMir:00}). Indeed, our proof shows that Theorem~\ref{nonspecial}
holds for every integer $m$ such that any special linear system of plane
curves with base points of equal multiplicity $m+2$ is $(-1)$-special.

We are sincerely grateful to the anonymous referee for her/his crucial
remarks on an earlier version of the present paper.

This research is part of the T.A.S.C.A. project of I.N.d.A.M., supported by
P.A.T. (Trento) and M.I.U.R. (Italy).

\section{The results}
Let $X \subset \PP^r$ be a non-degenerate integral projective variety of
dimension $n$ defined over the complex field $\C$.
If $p \in X$ and $m$ is a non-negative integer, let
$T^m_p(X)$ denote the $m$-osculating space to $X$ at $p$ (in particular,
we have $T^0_p(X)= \{p\}$ and $T^1_p(X)=T_p(X)$, the usual tangent space).
Fixed non-negative integers $m_0, \ldots, m_h$, define the higher order
join
$$
J(m_0, \ldots, m_h, X) := \overline{\bigcup_{p_0, \ldots, p_h}
< T^{m_0}_{p_0}(X), \ldots, T^{m_h}_{p_h}(X) >},
$$
where $p_0, \ldots, p_h$ are general points on $X$. In particular,
if $m_0 = \ldots = m_h = m$, we have
$$
J(m, \ldots, m, X) = \Sec_h(T(m,X)),
$$
where $T(m,X)$ is the $m$-osculating variety of $X$ (see for instance
\cite{BF2}, Definition~1) and $\Sec_h$ denotes the $h$-secant variety
(see for instance \cite{DioFon:01}, Definition~1.1).

\begin{Lemma}\label{Terracini} Notation as above.

(i) The expected dimension of $J(m_0, \ldots, m_h, X)$ is
$$
\expdim(J(m_0, \ldots, m_h, X)) = \min \left\{
(h+1)n + \sum_{i=0}^h {{m_i + n} \choose n} - 1, r \right\}.
$$

(ii) There is a natural inclusion
$$
T_P(J(m_0, \ldots, m_h, X)) \subseteq
<T^{m_0+1}_{p_0}(X), \ldots, T^{m_h+1}_{p_h}(X) >,
$$
where $p_0, \ldots, p_h$ are general points on $X$ and $P$ is general in
$< T^{m_0}_{p_0}(X), \ldots,$ $T^{m_h}_{p_h}(X) >$.

(iii) If $r \ge (h+1)n + \sum_{i=0}^h {{m_i + n} \choose n} - 1$ and
$\dim < T^{m_0}_{p_0}(X), \ldots, T^{m_h}_{p_h}(X) > =
\sum_{i=0}^h {{m_i + n} \choose n} - 1 - \delta$, then
$$
\dim J(m_0, \ldots, m_h, X)) \le (h+1)n + \sum_{i=0}^h {{m_i + n}
\choose n} - 1 - \delta.
$$

(iv) If $r \ge \sum_{i=0}^h {{m_i + n+1} \choose n} - 1$ and
$\dim < T^{m_0+1}_{p_0}(X), \ldots, T^{m_h+1}_{p_h}(X) > =
\sum_{i=0}^h {{m_i + n+1} \choose n} - 1$, then
$$
\dim(J(m_0, \ldots, m_h, X)) = \expdim(J(m_0, \ldots, m_h, X)).
$$

(v) If $n = 2$, $\dim <T^{m}_{p_0}(X), \ldots, T^{m}_{p_h}(X) > =
\min \left\{(h+1) \frac{(m+2)(m+1)}{2} - 1, r \right\}$ and
$\dim \Sec_h(T(m,X)) < \min \left\{2(h+1) + (h+1)\frac{(m+2)(m+1)}{2} - 1, r
\right\}$, then
$$
T_P(\Sec_h(T(m,X))) = <T^{m+1}_{p_0}(X), \ldots, T^{m+1}_{p_h}(X) >,
$$
where $p_0, \ldots, p_h$ are general points on $X$ and $P$ is general in
$< T^{m}_{p_0}(X), \ldots,$ $T^{m}_{p_h}(X) >$.
\end{Lemma}

\proof For $0 \le i \le h$ let
\begin{eqnarray*}
p_i: U_i \subseteq \C^n &\longrightarrow& X \\
t_{i1}, \ldots, t_{in} &\longmapsto& p_i(t_{i1}, \ldots, t_{in})
\end{eqnarray*}
be a local parametrization of $X$ (in the euclidean topology)
centered in a general point of $X$.
A general point of $J(m_0, \ldots, m_h, X)$ is of the form:
$$
P = p_0(t_{01}, \ldots, t_{0n}) +
\sum_{i=1}^h \lambda_i p_i(t_{i1}, \ldots, t_{in}) +
\sum_{ {i = 0} \atop{ {1 \le \vert I
\vert \le m_i}} }^h \lambda_i^I p_i^I(t_{i1}, \ldots, t_{in})
$$
where $p^I$ denotes the derivative of $p$ corresponding to the
multi-index $I = (a_1, a_2, \ldots)$ with every $a_j \in \{1, \ldots, n \}$.
Notice that $P$ depends on $(h+1)n + \sum_{i=0}^h {{m_i + n} \choose n} - 1$
parameters, hence (i) follows.

The tangent space $T_P(J(m_0, \ldots, m_h))$ is the linear span
$$
< \{ P \} \cup \left\{ \frac{\partial P}{\partial t_{ij}}
\right\}_{ {i = 0, \ldots, h} \atop {j = 1, \ldots, n}}
\cup \left\{ \frac{\partial P}{\partial \lambda_i}
\right\}_{i = 0, \ldots, h}
\cup \left\{ \frac{\partial P}{\partial \lambda^I_i}
\right\}_{ {i = 0, \ldots, h}
\atop { 1 \le \vert I \vert \le m_i }} > =
$$
$$
= < \{ p_i \}_{i = 0, \ldots, h} \cup \{ p_i^I \}_{ {i = 0, \ldots, h}
\atop { 1 \le \vert I \vert \le m_i }} \cup
\left\{ \sum_{ {\vert I \vert = m_i} }
\lambda_i^I p_i^{I \cup \{j\}} \right\}_{ {i = 0, \ldots, h} \atop
{j = 1, \ldots, n}} >,
$$
hence (ii), (iii), and (iv) follow.

Finally we have to check (v). Since
$<T^{m}_{p_0}(X), \ldots, T^{m}_{p_h}(X) >$ is of the expected dimension
but $\Sec_h(T(m,X))$ is not, there is at least one $i \in \{0, \ldots, h \}$
and $j(i) \in \{1,2\}$ such that $\sum_I \lambda_i^I p_i^{I \cup \{j(i)\}}$ is
a linear combination of the other points spanning $T_P(\Sec_h(T(m,X))$.
Since the coefficients $\lambda_i^I$ are general, by specializing all but
one coefficients $\lambda_i^I$ to $0$ and the remaining one to $1$ in all
possible ways, we obtain:
\begin{eqnarray*}
T_P(\Sec_h(T(m,X)) &=&
< \{ p_k \}_{k = 0, \ldots, h} \cup \{ p_k^I \}_{ {k = 0, \ldots, h}
\atop { 1 \le \vert I \vert \le m }} \cup \\
& & \cup \{ p_i^I \}_{\vert I \vert = m+1} \cup
\left\{ \sum_{ {\vert I \vert = m} }
\lambda_k^I p_k^{I \cup \{j\}} \right\}_{ {k \ne i} \atop
{j = 1, \ldots, n}} >.
\end{eqnarray*}
Moreover, since $m_0 = \ldots = m_h = m$, the same is true for every
$i \in \{0, \ldots, h \}$, hence the claim follows.

\qed

Now we turn to the promised applications.
We point out that our arguments rely on the main classification result
of \cite{CilMir:00}, from which we borrow also notation and terminology.

\emph{Proof of Proposition~\ref{special}.}
In the former case, we have
$$
\expdim (\Sec_h(T(m,V_{2,d}))) = \frac{(d+1)(d+2)}{2} - 1
$$
for every $m \ge 1$, hence the claim follows directly from
Lemma~\ref{Terracini}~(ii) and the speciality of the
linear system $\mathcal{L}_d((m+2)^{h+1})$
(see \cite{CCMO:03}, Theorem~2.4).
In the latter case, the claim follows from Lemma~\ref{Terracini}~(iii)
and the speciality of the linear system $\mathcal{L}_d((m+1)^{h+1})$
(see again \cite{CCMO:03}, Theorem~2.4).

\qed

\emph{Proof of Theorem~\ref{nonspecial}.}
Notice first of all that, since $m+2 \le 20$, any special
$\mathcal{L}_d((m+2)^{h+1})$ is $(-1)$-special by \cite{CCMO:03}.
Next, if $\frac{(d+1)(d+2)}{2} \ge (h+1)\frac{(m+3)(m+2)}{2}$,
just apply Lemma~\ref{Terracini}~(iv).
Finally, if $\frac{(d+1)(d+2)}{2} \le (h+1)\frac{(m+3)(m+2)}{2}$,
argue by contradiction and conclude by Lemma~\ref{Terracini}~(v).

\qed

config: arxiv


TITLE: Smallest packaging of rectangles
QUESTION [0 upvotes]: I'm trying to stack $n$ rectangles with size $a \times b$ to minimize the distance from the origin to the furthest point ($L$).
I attach a couple of images to clarify.

Thanks for you help!
Jorge

REPLY [0 votes]: A greedy algorithm will work.
It suffices to show that if we have two boxes stack on top of each other, the higher one will have a greater L. Hence we just maintain a priority queue of all the possible locations to put a box, then choose the one with the smallest L, place it and insert the new possible locations into the priority queue.


TITLE: Cardinality function proof
QUESTION [0 upvotes]: I need help solving the following exercise:

Let $X$ be a set with $|X| = n\in \mathbb N_0$, $f:\mathcal P(X) \to \{k\in \mathbb N_0: 0 \leq k \leq n\}$ satisfying
  a) $\forall Y \in \mathcal P(X), Y \neq \emptyset: f(Y) > 0$
  b) $\forall Y, Z \in \mathcal P(X): f(Y \cup Z) = f(Y \cap Z) + f((Y \cup Z)\backslash (Y \cap Z))$
  Show that $\forall Y \in \mathcal P(X): f(Y) = |Y|$

I already proved that $f(\emptyset) = 0$ by choosing $Y = X$ and $Z = \emptyset$ using the second rule, but I don't know how to continue from here, since I know nothing about the cardinality but that it's $>0$ for all $Y \neq \emptyset$. Would appreciate any hint.

REPLY [1 votes]: HINT: Show first that if $A,B\subseteq X$ are disjoint, then $f(A\cup B)=f(A)+f(B)$. Then show by induction on $k$ that if $A_1,\ldots,A_k$ are pairwise disjoint, $f\left(\bigcup_{i=1}^kA_k\right)=\sum_{i=1}^kf(A_i)$. Finally, let $X=\{x_k:k=1,\ldots,n\}$, and consider $f(X)=f\left(\bigcup_{k=1}^n\{x_k\}\right)$, bearing in mind that the codomain of $f$ is $\{0,1,\ldots,n\}$.


text:
\begin{document}

\section{Introduction}

There are many ways of specifying Markov processes,
the most popular being as solutions of stochastic
equations, as solutions of martingale problems, or in
terms of solutions of the Kolmogorov forward equation
(the Fokker-Planck equation or the master equation
depending on context).  The solution of a stochastic
equation explicitly gives a process while a solution
of a martingale problem gives the distribution of a
process and a solution of a forward equation gives the
one dimensional distributions of a process.
Typically, these approaches are
equivalent (assuming that there is a stochastic equation
formulation) in the sense that existence of a solution
specified by one method implies existence of
corresponding solutions to
the other two (weak existence for the stochastic
equation) and hence uniqueness for one method implies
uniqueness for the other two (distributional uniqueness
for the stochastic equation).

One approach to proving uniqueness for a forward
equation and hence for the corresponding martingale
problem is to verify a condition on the generator similar
to the range condition of the Hille-Yosida theorem. (See
Corollary \ref{rngunq}.)
We show that the original generator $A$ of our martingale
problem (or a restriction of the original generator $A$,
in the case of martingale
problems with boundary conditions) can be extended
to a generator $\hat {A}$ such that every solution of
the martingale problem for $A$ is a solution for $\hat {A}$ and $
\hat {A}$
satisfies the range condition of Corollary \ref{rngunq}.
Our extension is constructed by including viscosity solutions of
the resolvent equation
\begin{equation}\lambda u-Au=h.\label{ireso}\end{equation}

Viscosity solutions have been used to study value
functions in stochastic optimal
control and optimal stopping theory since the very
beginning (see the classical references \cite{CIL92}, \cite{Ph98},
as well as \cite{FS06}). It may be interesting to note
that, in the context of
Hamilton-Jacobi equations, the idea of studying a parabolic
equation by solving a resolvent
equation in the viscosity sense appears already in \cite{CL83}, Section VI.3,
where it is applied to a model problem.
The methodology is also important for related
problems in finance (for example \cite{ST00},
\cite{BKR01}, \cite{KK04}, \cite{BCM10}, \cite{CPD12} and
many others).

Viscosity solutions have also been used to study the
partial differential equations associated with
forward-backward stochastic differential equations
(\cite{MZ11},\cite{EKTZ14}) and in the theory of large deviations (\cite{FK06}).

The basic data for our work is an operator
$A\subset C_b(E)\times C_b(E)$ on a complete, separable metric space
$E$.  We offer an abstract definition of viscosity
sub/supersolution for (\ref{ireso}) (which for
integro-differential operators in ${\bf R}^d$ is equivalent to the
usual one) and prove, under very general conditions, that
the martingale problem for $A$ has a unique solution if
the comparison principle for $(\ref{ireso})$ holds.

We believe the interest of this result is twofold: on one
hand it clarifies the general connection between
viscosity solutions and martingale problems; on the
other, there are still many martingale problems, for
instance in infinite dimension, for which
uniqueness is an open question.

We also discuss two alternative abstract definitions
of viscosity sub/supersolution that might be especially
useful in infinite dimensional spaces.
All our proofs work under these alternative definitions
as well.

The first alternative definition is a modification of
a definition suggested to us by Nizar Touzi and used in
\cite{EKTZ14}. Being a stronger definition (it allows
for more test functions), it should be easier to prove
comparison results under this definition.

The second alternative definition appears in \cite{FK06}
and is a stronger definition too.
Under this definition, a sort of converse of our main
result holds, namely if $h$ belongs to $\overline {{\cal R}(\lambda
-A)}$ (under
uniform convergence), then the comparison principle for
semisolutions of (\ref{ireso}) holds (Theorem
\ref{converse}).  When $E$ is compact,
this definition is equivalent to our main definition,
hence the comparison principle holds for semisolutions
in that sense as well.

Next we consider stochastic processes that must satisfy
some boundary conditions, for example, reflecting
diffusions.  Boundary conditions are expressed in terms
of an operator $B$ which enters into the formulation of a
{\em constrained martingale problem\/} (see \cite{Kur90}).  We
restrict our attention to models in which the boundary
term in the constrained martingale problem is expressed
as an integral against a local time.  Then it still holds
that uniqueness of the solution of the constrained
martingale problem follows from the comparison
principle between viscosity sub and supersolutions of
(\ref{ireso}) with the appropriate boundary conditions.
Notice that, as for the standard martingale problem,
uniqueness for the constrained martingale problem
implies that the solution is Markovian (see \cite{Kur90},
Proposition 2.6).

In the presence of boundary conditions, even for
${\bf R}^d$-valued diffusions, there are examples for which
uniqueness of the martingale problem is not known.
Processes in domains with boundaries that are only
piecewise smooth or with boundary operators that are
second order or with directions of reflection that are
tangential on some part of the boundary continue to be a
challenge.  In this last case, as an example of an application
of our results, we use the comparison principle proved
in \cite{PK05} to obtain uniqueness.

The strategy of our proofs has been initially inspired by the proof
of Krylov's selection theorem for martingale problems
that appears in \cite{EK86}
and originally appeared in unpublished work of
\cite{GG77}. In that proof the generator is
recursively extended in such a way that
there are always solutions of the martingale problem
for the extended generator, but eventually only one.
If uniqueness fails for the original martingale problem,
there is more than one way
to do the extension. Conversely if, at each stage of the
recursion, there is only one
way to do the extension and all solutions of the
martingale problem for the original generator remain
solutions for the extended generator, then
uniqueness must hold for the original generator.

Analogously, assuming the comparison principle
for (\ref{ireso}) (or (\ref{ireso}) with the appropriate
boundary conditions) holds for a large enough class of
functions $h$, we construct an
extension $\hat {A}$ of the original operator $A$
(of a restriction of the original operator $A$,
in the case of constrained martingale problems)
such that all solutions of the martingale problem (the
constrained martingale problem) for $A$
are solutions of the martingale problem for $\hat {A}$, and such
that uniqueness holds for $\hat {A}$. Actually, in the case of
ordinary martingale problems the extension, although
possible, is not needed, because
the comparison principle for $\mbox{\rm (\ref{ireso})}$ directly yields a condition
($(\ref{lapid})$) that, if valid for a large enough class of
functions $h$, ensures uniqueness of the one-dimensional distributions
of solutions to the martingale problem, and hence
uniqueness of the solution. The extension is needed,
instead, for constrained martingale problems.

A few works on viscosity solutions of partial
differential equations
and weak solutions of stochastic differential equations
have appeared in recent years.
For diffusions in ${\bf R}^d$, \cite{BS12}, assuming a comparison
principle exists,
show that the backward equation has a
unique viscosity solution, and it follows that the
corresponding stochastic differential equation has a
unique weak solution. For Markovian forward-backward
stochastic differential equations, \cite{MZ11} also derive
uniqueness of the weak solution from existence of a
comparison principle for the corresponding partial
differential equation. In the non-Markovian case, the
associated partial differential equation becomes path
dependent. \cite{EKTZ14} propose the notion of viscosity
solution of (semilinear) path dependent partial
differential equations on the space of
continuous paths already mentioned above and prove a comparison principle.

The rest of this paper is organized as follows: Section
\ref{backgrnd} contains some background material on
martingale problems and on viscosity solutions; Section
\ref{mgpunq} deals with martingale problems;
the alternative definitions of viscosity solution are
discussed in Section \ref{altdef};
Section \ref{cmgpunq} deals with martingale
problems with boundary conditions; finally in Section
\ref{examples}, we present two examples, including the
application to diffusions with degenerate direction of reflection.

\section{Background}\label{backgrnd}

\subsection{Martingale problems}

Throughout this paper we will assume that $(E,r)$ is a
complete separable metric space, $D_E[0,\infty )$ is the space of
cadlag, $E$-valued functions endowed with the Skorohod topology,
${\cal B}(D_E[0,\infty )))$ is the $\sigma$-algebra of Borel sets of $
D_E[0,\infty )$,
$C_b(E)$ denotes the space of bounded, continuous
functions on $(E,r)$,  $B(E)$ denotes the space of
bounded, measurable functions on $(E,r)$,
 and ${\cal P}(E)$ denotes the space of
probability measures on $(E,r)$. $||\cdot ||$ will denote the
supremum norm on $C_b(E)$ or $B(E)$.

\begin{definition}\label{mgp}
A measurable
stochastic process $X$ defined on a probability
space $(\Omega ,{\cal F},{\mathbb P})$ is a solution of
the {\em martingale problem\/} for
\[A\subset B(E)\times B(E),\]
provided there exists a filtration $\{{\cal F}_t\}$ such that $X$ and
$\int_0^{\dot{}}g(X(s))ds$ are $\{{\cal F}_t\}$-adapted, for every $
g\in B(E)$, and
\begin{equation}M_f(t)=f(X(t))-f(X(0))-\int_0^tg(X(s))ds\label{mg}\end{equation}
is a $\{{\cal F}_t\}$-martingale for each $(f,g)\in A$.  If $X(0)$ has distribution $
\mu$,
we say $X$ is a solution of the martingale problem for $(A,\mu )$.
\end{definition}\

\begin{remark}
Because linear combinations of martingales are martingales,
without loss of generality, we can, but need not,
assume that $A$ is linear and that $(1,0)\in A$.

We do not, and cannot for our purposes, require $A$ to be
single valued.  In particular, the operator $\hat {A}$ defined in
the proof of Theorem \ref{cmgpb} will typically not be
single valued.
\end{remark}\medskip

In the next sections
we will restrict our attention to
processes $X$ with sample paths in $D_E[0,\infty )$.

\begin{definition}\label{Dmgp}
A stochastic process $X$, defined on a probability
space $(\Omega ,{\cal F},{\mathbb P})$, with sample paths in $D_E[0,
\infty )$ is a solution of
the {\em martingale problem\/} for
\[A\subset B(E)\times B(E),\]
{\em in} $D_E[0,\infty )$ provided there exists a filtration $\{{\cal F}_
t\}$ such
that $X$ is $\{{\cal F}_t\}$-adapted and
\begin{equation}M_f(t)=f(X(t))-f(X(0))-\int_0^tg(X(s))ds\label{Dmg}\end{equation}
is a $\{{\cal F}_t\}$-martingale for each $(f,g)\in A$.  If $X(0)$ has distribution $
\mu$,
we say $X$ is a solution of the martingale problem for
$(A,\mu )$ in $D_E[0,\infty )$.
\end{definition}\

\begin{remark}
Since $X$ has sample paths in $D_E[0,\infty )$, the fact that
it is $\{{\cal F}_t\}$-adapted implies that
$\int_0^{\dot{}}g(X(s))ds$ is $\{{\cal F}_t\}$-adapted, for every $
g\in B(E)$.
\end{remark}\medskip

\begin{remark}\label{cadlag}
The requirement that $X$ have sample paths in $D_E[0,\infty )$ is
usually fulfilled provided the state space $E$ is selected
appropriately.
Moreover, if $A\subset C_b(E)\times C_b(E)$, ${\cal D}(A)$ is dense in $
C_b(E)$ in the topology of
uniform convergence on compact sets and
for each compact $K\subset E$, $\varepsilon >0$, and $T>0$, there
exists a compact $K'\subset E$ such that
\[P\{X(t)\in K',t\leq T,X(0)\in K\}\geq (1-\varepsilon )P\{X(0)\in
K\},\]
for every progressive process such that (\ref{mg}) is a martingale,
then every such process
has a modification with sample paths in $D_E[0,\infty )$ (See
Theorem 4.3.6 of \cite{EK86}.)
\end{remark}\medskip

\begin{remark}\label{fdd}
The martingale property is equivalent to the
requirement that
\[E[(f(X(t+r))-f(X(t))-\int_t^{t+r}g(X(s))ds)\prod_ih_i(X(t_i))]=
0\]
for all choices of $(f,g)\in A$, $h_i\in B(E)$, $t,r\geq 0$, and
$0\leq t_i\leq t$.  Consequently, the property of being a solution
of a martingale problem is a property of the finite-
dimensional distributions of $X$.

In particular, for the martingale problem in $D_E[0,\infty )$,
the property of being a solution is a property of the
distribution of $X$ on $D_E[0,\infty )$.
Much of what follows in the next sections will be
formulated in terms of the collection $\Pi\subset {\cal P}(D_E[0,
\infty ))$ of
distributions of solutions of the martingale problem. For
some purposes, it will be convenient to assume that $X$ is
the canonical process defined on
$(\Omega ,{\cal F},\mathbb{P})=(D_E[0,\infty ),{\cal B}(D_E[0,\infty
))),P)$, for some
$P\in\Pi$.
\end{remark}\medskip

In view of Remark \ref{fdd} it is clear that
uniqueness of the solution to the martingale problem for
an operator $A$ is to be meant as uniqueness of the
finite-dimensional distributions.

\begin{definition}\label{uniq}
We say that {\em uniqueness holds for the martingale }
{\em problem for} $A$  if, for every $
\mu$, any two solutions of the martingale
problem for $(A,\mu )$  have the same finite-dimensional
distributions. If we restrict our attention to solutions in $D_E[0,\infty)$, then uniqueness holds if any two solutions for $(A,\mu)$ have the same distribution on $D_E[0,\infty)$
\end{definition}

One of the most important consequences of the
martingale approach to Markov processes is that
uniqueness of one-dimensional distributions implies
uniqueness of finite-dimensional distributions.

\begin{theorem}\label{1dfid}
Suppose that for each $\mu\in {\cal P}(E)$,
any two solutions of the martingale
problem for $(A,\mu )$  have the same one-dimensional
distributions.  Then any two solutions have the same
finite-dimensional distributions. If any two solutions of the martingale
problem for $(A,\mu )$  in $D_E[0,\infty)$ have the same one-dimensional
distributions, then they have the same
distribution on $D_E[0,\infty)$.
\end{theorem}

\begin{proof}
This is a classical result. See for instance Theorem 4.4.2
and Corollary 4.4.3 of \cite{EK86}.
\end{proof}

For $\mu\in {\cal P}(E)$ and $f\in B(E)$ we will use the notation
\begin{equation}\mu f=\int_Ef(x)\mu (dx).\label{notat-i}\end{equation}

\begin{lemma}\label{resid}Let $X$ be a $\{{\cal F}_t\}$-adapted stochastic process
with sample paths in $D_E[0,\infty )$, with initial distribution
$\mu$, $f,g\in B(E)$ and $\lambda >0$. Then $(\mbox{\rm \ref{Dmg}}
)$ is
a $\{{\cal F}_t\}$-martingale if and only if
\begin{equation}M_f^{\lambda}(t)=e^{-\lambda t}f(X(t))-f(X(0))+\int_0^te^{-\lambda
s}(\lambda f(X(s))-g(X(s))ds\label{resmg}\end{equation}
is a $\{{\cal F}_t\}$-martingale. In particular, if $(\mbox{\rm \ref{Dmg}}
)$ is a $\{{\cal F}_t\}$-martingale
\begin{equation}\mu f=E[\int_0^{\infty}e^{-\lambda s}(\lambda f(X
(s))-g(X(s)))ds].\label{inrg}\end{equation}
\end{lemma}

\begin{proof}\ The general statement is a special case of Lemma
4.3.2 of \cite{EK86}. If $f$ is
continuous, as will typically be the case in the next sections, $M_f$ will be cadlag and
we can apply It\^o's formula to obtain
\[e^{-\lambda t}f(X(t))-f(X(0))=\int_0^t(-f(X(s))\lambda e^{-\lambda
s}+e^{-\lambda s}g(X(s)))ds+\int_0^te^{-\lambda s}dM_f(s),\]
where the last term on right is a
$\{{\cal F}_t\}$-martingale. (Note that, since all the processes involved are cadlag,
we do not need to require the filtration $\{{\cal F}_
t\}$
to satisfy the `usual conditions'.)
Conversely, if $(\mbox{\rm \ref{resmg}})$ is a
$\{{\cal F}_t\}$-martingale, the assertion follows by applying It\^o's
formula to $f(X(t))=e^{\lambda t}\Big(e^{-\lambda t}f(X(t))\Big)$.

In particular, if $(\mbox{\rm \ref{Dmg}})$ is a $\{{\cal F}_t\}$-martingale
\[E[f(X(0)]-E[e^{-\lambda t}f(X(t))]=E[\int_0^te^{-\lambda s}(\lambda
f(X(s))-g(X(s))ds],\]
and the second statement follows by taking $t\rightarrow\infty$.
\end{proof}

\begin{lemma}\label{rnlem}
Let $X$ be a solution of the martingale problem for $A\subset C_b(E)\times C_b(E)$ in
$D_E[0,\infty )$ with respect to a filtration $\{{\cal F}_t\}$.
Let $\tau\geq 0$ be a finite $\{{\cal F}_t\}$-stopping time and
$H\geq 0$ be a ${\cal F}_{\tau}$-measurable random variable such that
$0<E[H]<\infty$. Define $P^{\tau ,H}$ by
\begin{equation}P^{\tau ,H}(C)=\frac {E[H{\bf 1}_C(X(\tau +\cdot
))]}{E[H]},\quad C\in {\cal B}(D_E[0,\infty )).\label{phdist}\end{equation}
Then $P^{\tau ,H}$ is the distribution of a solution of the martingale
problem for $A$ in $D_E[0,\infty )$.
\end{lemma}

\begin{proof}
Let $(\Omega ,{\cal F},\mathbb{P})$ be the probability space
on which $X$ is defined, and define $\mathbb{P}^H$ on $(\Omega ,{\cal F}
)$ by
\[\mathbb{P}^H(C)=\frac {E^{\mathbb{P}}[H{\bf 1}_C]}{E^{\mathbb{P}}
[H]},\quad C\in {\cal F}.\]
Define  $X^{\tau}$ by $X^{\tau}(t)=X(\tau +t)$.
$X^{\tau}$ is adapted to the filtration $\{{\cal F}_{\tau +t}\}$ and
for  $0\leq t_1<\cdots <t_n<t_{n+1}$ and  $f_1,\cdots ,f_n\in B(E
)$,
\begin{eqnarray*}
&&E^{\mathbb{P}^H}\Big[\Big\{f(X^{\tau}(t_{n+1}))-f(X^{\tau}(t_n)
)-\int_{t_n}^{t_{n+1}}Af(X^{\tau}(s))ds\Big\}\,\Pi_{i=1}^nf_i(X^{
\tau}(t_i))\Big]\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\\
&&\quad =\frac 1{E^{\mathbb{P}}[H]}E^{\mathbb{P}}\Big[H\Big\{f(X(
\tau +t_{n+1}))-f(X(\tau +t_n))\\
&&\qquad\qquad\qquad\qquad -\int_{\tau +t_n}^{\tau +t_{n+1}}Af(X(
s))ds\Big\}\,\Pi_{i=1}^nf_i(X(\tau +t_i))\Big]\qquad\qquad\qquad\qquad
\qquad\qquad\\
&&\quad =0\end{eqnarray*}
by the optional sampling theorem.
Therefore, under $\mathbb{P}^H$, $X^{\tau}$ is a solution of the
martingale problem. $P^{\tau ,H}$, given by (\ref{phdist}),
is the distribution of $X^{\tau}$ on $D_E[0,\infty )$.
\end{proof}

\begin{lemma}\label{ext} Let $\lambda >0$. Suppose $u,h\in B(E)$ satisfy
\begin{equation}\mu u=E[\int_0^{\infty}e^{-\lambda t}h(X(t))dt],\label{mgcond}\end{equation}
for every solution of the martingale problem for $A$  in $D_E[0,\infty )$ with
initial distribution $\mu$ and for every $\mu\in {\cal P}(E)$.
Then
\begin{equation}u(X(t))-\int_0^t(\lambda u(X(s))-h(X(s)))ds\label{mgprop}\end{equation}
is a $\left\{{\cal F}_t^X\right\}$-martingale for every solution $
X$ of the martingale
problem for $A$ in $D_E[0,\infty )$.
\end{lemma}

\begin{proof}\ The lemma is Lemma 4.5.18 of \cite{EK86},
but we repeat the proof here for the convenience of the
reader.

Let $X$ be a solution of the martingale problem for $A$ on a
probability space $(\Omega ,{\cal F},\mathbb{P})$. For $t\geq 0$ and
$B\in {\cal F}_t$ with $\mathbb{P}(B)>0$, define $\mathbb{Q}$ on $
(\Omega ,{\cal F})$ by
\[\mathbb{Q}(C)=\frac {E^{\mathbb{P}}[{\bf 1}_B{\bf 1}_C]}{\mathbb{
P}(B)},\quad C\in {\cal F}.\]
Then
\begin{eqnarray*}
&&E^{\mathbb{P}}[{\bf 1}_Be^{\lambda t}\int_t^{\infty}e^{-\lambda
s}h(X(s))ds]=E^{\mathbb{P}}[{\bf 1}_B\int_0^{\infty}e^{-\lambda s}
h(X(t+s))ds]\\
&&=\mathbb{P}(B)E^{\mathbb{Q}}[\int_0^{\infty}e^{-\lambda s}h(X(t
+s))ds].\end{eqnarray*}
By the same arguments as in the proof of Lemma
\ref{rnlem}, $X(t+\cdot )$ is a solution of the martingale
problem for $A$ on the probability space $(\Omega ,{\cal F},\mathbb{
Q})$
with  respect to the filtration $\{{\cal F}_{t+\cdot}\}$, with initial
distribution
$\nu (C)=\mathbb{Q}(X(t)\in C)=\mathbb{P}(X(t)\in C|B)$.
Hence, by the assumptions of the lemma,
\[E^{\mathbb{P}}[{\bf 1}_Be^t\int_t^{\infty}e^{-\lambda s}h(X(s))
ds]=\mathbb{P}(B)\nu u=E^{\mathbb{P}}[{\bf 1}_Bu(X(t))].\]
Therefore
\[E[\int_0^{\infty}e^{-\lambda s}h(X(s))ds|{\cal F}_t]=e^{-\lambda
t}u(X(t))+\int_0^te^{-\lambda s}h(X(s))ds,\]
and the assertion follows from Lemma \ref{resid} with
$f=u$ and $g=\lambda u-h$.
\end{proof}

Our approach to proving uniqueness of the solution of
the martingale problem for $(A,\mu )$ relies on the following
theorem.

\begin{definition}\ A class of functions $M\subset {\cal C}_b(E)$ is
{\em separating\/} if, for $\mu ,\nu\in {\cal P}(E)$, $\mu f=\nu
f$ for all
$f\in M$ implies $\mu =\nu$.
\end{definition}

\begin{theorem}\label{lapunq}
For each $\lambda >0$, suppose that
\begin{equation}E[\int_0^{\infty}e^{-\lambda s}h(X(s))ds]=E[\int_0^{\infty}e^{-
\lambda s}h(Y(s))ds],\label{lapid}\end{equation}
for any two solutions of the martingale problem for
$A$ in $D_E[0,\infty )$ with the same initial distribution and
for all $h$ in a separating class of functions $M_{\lambda}$.
Then  any two solutions of the martingale problem for
$A$ in $D_E[0,\infty )$ with the same initial distribution
have the same distribution on $D_E[0,\infty )$.
\end{theorem}\

\begin{proof}
The proof is actually implicit in the proof of Corollary
4.4.4 of \cite{EK86}, but we give it here for clarity.
For any two solutions of the martingale problem for
$A$ in $D_E[0,\infty )$ with the same initial distribution,
we have, for all $h\in M_{\lambda}$,
\begin{equation}\int_0^{\infty}e^{-\lambda s}E[h(X(s))]ds=\int_0^{
\infty}e^{-\lambda s}E[h(Y(s))]ds.\label{lamid}\end{equation}
Consider the measures on $(E,{\cal B}(E))$ defined by
\[m^X(C)=\int_0^{\infty}e^{-\lambda s}E[1_C(X(s))]ds,\quad m^Y(C)
=\int_0^{\infty}e^{-\lambda s}E[1_C(Y(s))]ds.\]
Then $(\ref{lamid})$ reads
\[m^Xh=m^Yh,\]
and, as $M_{\lambda}$ is separating, (\ref{lamid}) holds for all
$h\in B(E)$.  Since the identity holds for each $\lambda >0$,
by uniqueness of the Laplace transform,
\[E[h(X(s))]=E[h(Y(s))],\qquad\mbox{\rm for almost every }s\geq 0
,\]
and by the right continuity of $X$ and $Y$, for all
$s\geq 0$. Consequently, $X$ and
$Y$ have the same one-dimensional distributions, and hence
by Theorem \ref{1dfid}, the same finite-dimensional
distributions and the same distribution on $D_E[0,\infty )$.
\end{proof}

\begin{corollary}\label{rngunq}
Suppose that for each $\lambda >0$,
${\cal R}(\lambda -A)$ is separating.  Then for each initial distribution
$\mu\in {\cal P}(E)$, any two cadlag solutions of the martingale problem for
$(A,\mu )$ have the same distribution on $D_E[0,\infty )$.
\end{corollary}

\begin{proof}
The assertion follows immediately from Lemma \ref{ext}
and Theorem \ref{lapunq}.
\end{proof}

Martingale problems and dissipative operators are closely
related.

\begin{definition}
A linear operator $A\subset B(E)\times B(E)$ is {\em dissipative\/} provided
\[\Vert\lambda f-g\Vert\geq\lambda\Vert f\Vert ,\]
for each $(f,g)\in A$ and each $\lambda >0$.
\end{definition}

\begin{lemma}\label{diss}
Suppose that for each $x\in E$, there exists a solution of the
martingale problem for $(A,\delta_x)$. Then $A$ is dissipative.
\end{lemma}

\begin{proof}
By Lemma \ref{resid},
\[|f(x)|\leq E[\int_0^{\infty}e^{-\lambda s}|\lambda f(X(s))-g(X(
s))|ds]\leq\frac 1{\lambda}\Vert\lambda f-g\Vert .\]
\end{proof}

\subsection{Viscosity solutions}
\label{viscsol}

Let $A\subset C_b(E)\times C_b(E)$.  Theorem \ref{rngunq} implies that
if for each $\lambda >0$, the equation
\begin{equation}\lambda u-g=h\label{reso}\end{equation}
has a solution $(u,g)\in A$ for every $h$ in a class
of functions $M_{\lambda}\subseteq C_b(E)$ that is separating,
then for each initial distribution $\mu$, the martingale problem for
$(A,\mu )$ has at most one solution.
Unfortunately in many situations it is hard to verify
that $(\ref{reso})$ has a solution in $A$. Thus one is lead to consider
a weaker notion of solution, namely the notion of
{\em viscosity solution}.

\begin{definition}\label{visc}{\bf (viscosity (semi)solution)}

Let $A$ be as above, $\lambda >0$, and $h\in C_b(E)$.

\begin{itemize}

\item[a)]$u\in B(E)$ is a {\em viscosity subsolution\/} of (\ref{reso})
if and only if $u$ is upper semicontinuous  and
if $(f,g)\in A$ and $x_0\in E$ satisfy
\begin{equation}\sup_x(u-f)(x)=(u-f)(x_0),\label{subtest}\end{equation}
then
\begin{equation}\lambda u(x_0)-g(x_0)\leq h(x_0).\label{subeq}\end{equation}

\item[b)]$u\in B(E)$ is a {\em viscosity supersolution\/} of $(\mbox{\rm \ref{reso}}
)$
if and only if $u$ is lower semicontinuous  and
if $(f,g)\in A$ and $x_0\in E$ satisfy
\begin{equation}\inf_x(u-f)(x)=(u-f)(x_0),\label{suptest}\end{equation}
then
\begin{equation}\lambda u(x_0)-g(x_0)\geq h(x_0).\label{supeq}\end{equation}
\end{itemize}

A function $u\in C_b(E)$ is a
{\em viscosity solution\/} of (\ref{reso}) if it is both a
subsolution and a supersolution.
\end{definition}

In the theory of viscosity solutions, usually existence of
a viscosity solution follows by existence of a viscosity
subsolution and a viscosity supersolution, together with
the following {\em comparison principle}.

\begin{definition}\label{compar} The {\em comparison principle }
holds for $(\mbox{\rm \ref{reso}})$ when every subsolution is
pointwise less than or equal to every supersolution.
\end{definition}

\begin{remark}\label{do-in-v}
To better motivate the notion of viscosity solution in the
context of martingale problems, assume that there exists
a solution of the martingale problem for $(A,\delta_x)$ for each
$x\in E$.  Suppose that
there exists $v\in C_b(E)$ such that
\begin{equation}e^{-\lambda t}v(X(t))+\int_0^te^{-\lambda s}h(X(s
))ds\label{lammgp}\end{equation}
is a $\{{\cal F}^X_t\}$-martingale for every solution $X$ of the
martingale problem for $A$.
Let $(f,g)\in A$ and $x_0$ satisfy
\[\sup_x(v-f)(x)=(v-f)(x_0).\]
Let $X$ be a solution of the martingale problem for $(A,\delta_{x_
0})$.
Then
\[e^{-\lambda t}(v(X(t))-f(X(t)))+\int_0^te^{-\lambda s}(h(X(s)-\lambda
f(X(s))+g(X(s)))ds\]
is a $\{{\cal F}_t^X\}$-martingale by Lemma \ref{resid}, and
\begin{eqnarray*}
&&E[\int_0^te^{-\lambda s}(\lambda v(X(s))-g(X(s))-h(X(s)))ds]\\
&&\qquad\qquad =E[\int_0^te^{-\lambda s}\lambda (v(X(s))-f(X(s)))
ds]\\
&&\qquad\qquad\qquad\qquad +E[e^{-\lambda t}(v(X(t)-f(X(t)))-(v(x_
0)-f(x_0))]\\
&&\qquad\qquad\leq 0.\end{eqnarray*}
Dividing by $t$ and letting $t\rightarrow 0$, we see that
\[\lambda v(x_0)-g(x_0)\leq h(x_0),\]
so $v$ is a subsolution for (\ref{reso}).  A similar argument
shows that it is also a supersolution and hence a
viscosity solution. We will give conditions such  that if the comparison
principle holds for some $h$, then a viscosity solution $v$ exists and (\ref{lammgp}) is a
martingale for every solution of the martingale
problem for $A$.
\end{remark}\medskip

In the case of a domain with boundary, in order to
uniquely determine the solution of the martingale problem for $A$ one usually
must specify some boundary conditions, by means of
boundary operators $B_1\ldots ,B_m$.

Let $E_0\subseteq E$ be an open set and let
\[\partial E_0=\cup_{k=1}^mE_k,\]
for disjoint, nonempty Borel sets $E_1,\ldots ,E_m$.

Let $A\subseteq C_b(\bar {E}_0)\times C_b(\bar {E}_0)$, $B_k\subseteq
C_b(\bar {E}_0)\times C(\bar {E}_0)$, $k=1,...,m$,
be linear operators with a common domain ${\cal D}$.
For simplicity we will assume that $ $$E$ is compact (hence
the subscript $b$ will be dropped) and that $A,B_1,\ldots ,B_m$ are
single valued.

\begin{definition}
Let $A,B_1\ldots ,B_m$ be as above, and let $\lambda >0$.  For
$h\in C_b(\bar {E}_0)$, consider the equation
\begin{eqnarray}
\lambda u-Au&=&h,\quad\mbox{\rm on }E_0\label{creso}\\
-B_ku&=&0,\quad\mbox{\rm on }E_k,\quad k=1,\cdots ,m.\nonumber
\end{eqnarray}

\begin{itemize}
\item[a)]$u\in B(\bar {E}_0)$ is a {\em viscosity subsolution\/} of (\ref{creso})
if and only if $u$ is upper semicontinuous  and
if $f\in {\cal D}$ and $x_0\in\bar {E}_0$ satisfy
\begin{equation}\sup_x(u-f)(x)=(u-f)(x_0),\label{csubtest}\end{equation}
then
\begin{eqnarray}
\lambda u(x_0)-Af(x_0)\leq h(x_0),&\qquad&\mbox{\rm if }x_0\in E_
0,\label{interior}\\
(\lambda u(x_0)-Af(x_0)-h(x_0))\wedge\min_{k:x_0\in\bar {E}_k}(-B_
kf(x_0))\leq 0,&\qquad&\mbox{\rm if }x_0\in\partial E_0.\label{bndry}
\end{eqnarray}

\item[b)]$u\in B(\bar {E}_0)$ is a {\em viscosity supersolution\/} of
$(\mbox{\rm \ref{creso}})$
if and only if $u$ is lower semicontinuous  and
if $f\in {\cal D}$ and $x_0\in\bar {E}_0$ satisfy
\begin{equation}\inf_x(u-f)(x)=(u-f)(x_0),\label{csuptest}\end{equation}
then
\begin{eqnarray}
\lambda u(x_0)-Af(x_0)\geq h(x_0),&\qquad&\mbox{\rm if }x_0\in E_
0,\label{csupeq}\\
(\lambda u(x_0)-Af(x_0)-h(x_0))\vee\max_{k:x_0\in\bar {E}_k}(-B_k
f(x_0))\geq 0,&\qquad&\mbox{\rm if }x_0\in\partial E_0.\nonumber\end{eqnarray}
\end{itemize}

A function $u\in C(\bar {E}_0)$ is a
{\em viscosity solution\/} of (\ref{creso}) if it is both a
subsolution and a supersolution.

\end{definition}

\begin{remark}
The above definition, with the `relaxed' requirement that
on the boundary either the interior inequality or the
boundary inequality be satisfied by at least one among
$-B_1f,\cdots ,-B_mf$ is the standard one in the theory of
viscosity solutions
where it is used in particular because it is stable under
limit operations and because it can be localized. As will be
clear in Section \ref{cmgpunq}, it suits perfectly our
approach to martingale problems with boundary conditions.
\end{remark}\medskip

\section{Comparison principle and uniqueness for
martingale problems}\label{mgpunq}

In this section, we restrict our attention to
\[A\subset C_b(E)\times C_b(E)\]
and consider the martingale problem for $A$ in $D_E[0,\infty )$.
Let $\Pi\subset {\cal P}(D_E[0,\infty ))$ denote the collection of distributions
of solutions of
the martingale problem for $A$ in $D_E[0,\infty )$,
and, for $\mu\in {\cal P}(E)$, let
$\Pi_{\mu}\subset\Pi$ denote the subcollection with initial distribution
$\mu$.  If $\mu =\delta_x$, we will write $\Pi_x$ for $\Pi_{\delta_
x}$. In this section
$X$ will be the canonical process on $D_E[0,\infty )$.
Assume the following condition.

\begin{condition}\label{H}\hfill

\begin{itemize}
\item[a)] ${\cal D}(A)$ is dense in $C_b(E)$ in the topology of
uniform convergence on compact sets.

\item[b)] For each $\mu\in {\cal P}(E)$, $\Pi_{\mu}\neq\emptyset$.

\item[c)]
If ${\cal K}\subset {\cal P}(E)$ is compact, then $\cup_{\mu\in {\cal K}}
\Pi_{\mu}$ is compact.  (See
Proposition \ref{comp} below.)
\end{itemize}
\end{condition}

\begin{remark}\label{Jak}
In working with these conditions, it is simplest to take
the usual Skorohod topology on $D_E[0,\infty )$. (See, for
example, Sections 3.5-3.9 of \cite{EK86}.) The results of
this paper also hold if we take the Jakubowski topology
 (\cite{Jak97}).  The $\sigma$-algebra of
Borel sets ${\cal B}(D_E[0,\infty ))$ is
the same for both topologies and, in fact, is simply the
smallest $\sigma$-algebra under which all mappings of the form
$x\in D_E[0,\infty )\rightarrow x(t)$, $t\geq 0$, are measurable.

It is also relevant to note that mappings of the form
\[x\in D_E[0,\infty )\rightarrow\int_0^{\infty}e^{-\lambda t}h(x(
t))dt,\quad h\in C_b(E),\lambda >0,\]
are continuous under both topologies.  The Jakubowski
topology could be particularly useful for extensions of
the results of Section \ref{cmgpunq} to constrained
martingale problems in which the boundary terms are
not local-time integrals.
\end{remark}\medskip

\begin{proposition}\label{comp}In addition to Condition
\ref{H}(a), assume that
for each compact $K\subset E$, $\varepsilon >0$, and $T>0$, there
exists a compact $K'\subset E$ such that
\[P\{X(t)\in K',t\leq T,X(0)\in K\}\geq (1-\varepsilon )P\{X(0)\in
K\},\quad\forall P\in\Pi .\]
Then Condition \ref{H}(c) holds.
\end{proposition}

\begin{proof}
The assertion is part of the thesis of Theorem 4.5.11 (b) of \cite{EK86}.
\end{proof}

Let $\lambda >0$, and for $h\in C_b(E)$, define
\begin{equation}u_{+}(x)=u_{+}(x,h)=\sup_{P\in\Pi_x}E^P[\int_0^{\infty}
e^{-\lambda t}h(X(t))dt],\label{uplus}\end{equation}
\begin{equation}\quad u_{-}(x)=u_{-}(x,h)=\inf_{P\in\Pi_x}E^P[\int_
0^{\infty}e^{-\lambda t}h(X(t))dt].\label{uminus}\end{equation}
\begin{equation}\pi_{+}(\Pi_{\mu},h)=\sup_{P\in\Pi_{\mu}}E^P[\int_
0^{\infty}e^{-\lambda t}h(X(t))dt],\label{piplus}\end{equation}
\begin{equation}\pi_{-}(\Pi_{\mu},h)=\inf_{P\in\Pi_{\mu}}E^P[\int_
0^{\infty}e^{-\lambda t}h(X(t))dt].\label{piminus}\end{equation}

\begin{lemma}\label{intrep}
Under Condition \ref{H}, for $h\in C_b(E)$,
$u_{+}(x,h)$ is upper semicontinuous (hence measurable), and
\begin{equation}\pi_{+}(\Pi_{\mu},h)=\int_Eu_{+}(x,h)\mu (dx)\qquad
\forall\mu\in {\cal P}(E).\label{intrep2}\end{equation}
The analogous result holds for $u_{-}$ and $\pi_{-}$.
\end{lemma}

\begin{proof}
For $\pi_{+}$, the lemma is a combination of Theorem 4.5.11(a), Lemma 4.5.8,
Lemma 4.5.9 and Lemma 4.5.10 of \cite{EK86}, but we recall here
the main steps of the proof for the convenience of the
reader. Throughout the proof, $h$ will be fixed and
will be omitted. In addition we will use the notation $\pi_{+}(\Pi_{
\mu},h)=\pi_{+}(\mu )$.

First of all let us show that, for $\mu_n\rightarrow\mu$,
\[\limsup_{n\rightarrow \infty}\pi_{+}(\mu_n)\leq\pi_{+}(\mu ).\]
In fact, by the compactness of $\Pi_{\mu}$ there is $P\in\Pi_{\mu}$ that
achieves the supremum. Moreover, by
the compactness of $\Pi_{\mu}\cup\cup_n\Pi_{\mu_{n_i}}$, for every convergent
subsequence $\{\pi_{+}(\mu_{n_i})\}=\left\{E^{P_{n_i}}[\int_0^{\infty}
e^{-\lambda t}h(X(t))dt]\right\}$, we can extract a
subsequence $\left\{P_{n_{i_j}}\right\}$ that converges to some $
P\in\Pi_{\mu}$.
Since $\int_0^{\infty}e^{-\lambda t}h(x(t))dt]$ is continuous on $
D_E[0,\infty )$, we then have
\[\lim_{i\rightarrow\infty}\,\pi_{+}(\mu_{n_i})=\lim_{j\rightarrow\infty}\,E^{P_{n_{i_j}}}[\int_0^{\infty}
e^{-\lambda t}h(X(t))dt]=E^P[\int_0^{\infty}e^{-\lambda t}h(X(t))
dt]\leq\pi_{+}(\mu ).\]

This yields, in particular, the upper semicontinuity (and
hence the measurability) of $u_{+}(x)=\pi_{+}(\delta_x)$.

Next, Condition \ref{H} (b) and (c) implies that, for $\mu_1,\mu_
2\in {\cal P}(E)$,
$0\leq\alpha\leq 1$,
\[\pi_{+}(\alpha\mu_1+(1-\alpha )\mu_2)=\alpha\pi_{+}(\mu_1)+(1-\alpha
)\pi_{+}(\mu_2),\]
(Theorem 4.5.11(a), Lemma 4.5.8 and Lemma 4.5.10 of
\cite{EK86}. We will not recall this part of the
proof). This yields, for $\{\mu_i\}\subset {\cal P}(E)$, $\alpha_
i\geq 0$, $\sum_i\alpha_i=1$:
\[\pi_{+}(\sum_i\alpha_i\mu_i)=\sum_1^N\alpha_i\,\pi_{+}(\frac 1{
\sum_1^N\alpha_i}\sum_1^N\alpha_i\mu_i)+\sum_{N+1}^{\infty}\alpha_
i\,\pi_{+}(\frac 1{\sum_{N+1}^{\infty}\alpha_i}\sum_{N+1}^{\infty}
\alpha_i\mu_i),\]
and hence
\[\Big|\pi_{+}(\sum_i\alpha_i\mu_i)-\sum_i\alpha_i\pi_{+}(\mu_i)\Big
|\leq 2\frac {||h||}{\lambda}\sum_{N+1}^{\infty}\alpha_i,\qquad\forall
N.\]

Finally, for each $n$, let $\{E^n_i\}$ be a countable collection of disjoint
subsets of $E$ with diameter less than $\frac 1n$ and such that
$E=\bigcup_iE^n_i$. In addition let $x^n_i\in E^n_i$ satisfy
$ $$u_{+}(x^n_i)\geq\sup_{E^n_i}u_{+}(x)-\frac 1n$. Define $u_n(x
)=\sum_iu_{+}(x^n_i)1_{E^n_i}(x)$
and $\mu_n=\sum_i\mu (E^n_i)\delta_{x^n_i}.$
Then $\{u_n\}$ converges to $u_{+}$ pointwise and boundedly, and
$\{\mu_n\}$ converges to $\mu$. Therefore
\[\int_Eu_{+}(x)\mu (dx)=\lim_n\int_Eu_n(x)\mu (dx)=\lim_n\sum_i\pi_{
+}(\delta_{x^n_i})\mu (E^n_i)=\lim_n\,\pi_{+}(\mu_n)\leq\pi_{+}(\mu
).\]
To prove the opposite inequality, let
$\mu_i^n(B)=\mu (B\cap E^n_i)/\mu (E^n_i)$, for $\mu (E^n_i)>0$, and
$u_n(x)=\sum_i\pi_{+}(\mu^n_i)1_{E^n_i}(x)$. For each $x\in E$, for every $
n$
there exists a (unique) $i(n)$ such that $x\in E_{i(n)}^n$. Then
$u_n(x)=\pi_{+}(\mu^n_{i(n)})$ and $\mu^n_{i(n)}\rightarrow\delta_
x$, hence $\limsup_nu_n(x)\leq u_{+}(x).$
Therefore
\[\pi_{+}(\mu )=\pi_{+}(\sum_i\mu (E^n_i)\,\mu_i^n)=\int_Eu_n(x)\mu
(dx)\leq\int_E\limsup_n\,u_n(x)\,\mu (dx)\leq\int_Eu_{+}(x)\mu (d
x),\]
where the last but one inequality follows from the fact
that the $u_n$ are uniformly bounded.

To prove the assertion
for $\pi_{-}$ use the fact that $\pi_{-}(\Pi_{\mu},h)=-\pi_{+}(\Pi_{
\mu},-h)$.
\end{proof}

\begin{lemma}\label{subsol}Assume
that Condition \ref{H} holds.  Then $u_{+}$ is a viscosity
subsolution of (\ref{reso}) and $u_{-}$ is a viscosity
supersolution of the same equation.
\end{lemma}

\begin{proof}
Since
$u_{-}(x,h)=-u_{+}(x,-h)$ it is enough to consider $u_{+}$.  Let
$(f,g)\in A$.  Suppose $x_0$ is a point such that
$u_{+}(x_0)-f(x_0)=\sup_x(u_{+}(x)-f(x))$.  Since we can always
add a constant to $f$, we can assume $u_{+}(x_0)-f(x_0)=0$.
By compactness (Condition \ref{H}(c)), we have
\[u_{+}(x_0)=E^P\left[\int_0^{\infty}e^{-\lambda t}h(X(t)dt\right
]\]
for some $P\in\Pi_{x_0}$.

For $\epsilon >0$, define
\begin{equation}\tau_{\epsilon}=\epsilon\wedge\inf\{t>0:r(X(t),x_0)\geq\epsilon\mbox{\rm \ or }
r(X(t-),x_0)\geq\epsilon \}\label{tau}\end{equation}
and let $H_{\epsilon}=e^{-\lambda\tau_{\epsilon}}$.
Then, by Lemma \ref{resid},
\begin{eqnarray*}
0&&=u_{+}(x_0)-f(x_0)\\
&&=E^P\left[\int_0^{\infty}e^{-\lambda t}\left(h(X(t))-\lambda f(
X(t))+g(X(t))\right)dt\right]\\
&&=E^P\left[\int_0^{\tau_{\epsilon}}e^{-\lambda t}\left(h(X(t))-\lambda
f(X(t))+g(X(t))\right)dt\right]\\
&&\qquad\qquad\quad +E^P\left[e^{-\lambda\tau_{\epsilon}}\int_0^{
\infty}e^{-\lambda t}\left(h(X(t+\tau_{\epsilon}))-\lambda f(X(t+
\tau_{\epsilon}))+g(X(t+\tau_{\epsilon}))\right)dt\right]\\
&&=E^P\left[\int_0^{\tau_{\epsilon}}e^{-\lambda t}\left(h(X(t))-\lambda
f(X(t))+g(X(t))\right)dt\right]\\
&&\qquad\quad\qquad +E^P[H_{\epsilon}]E^{P^{\tau_{\epsilon},H_{\epsilon}}}\left
[\int_0^{\infty}e^{-\lambda t}\left(h(X(t))-\lambda f(X(t))+g(X(t
))\right)dt\right].\\
\end{eqnarray*}
Setting $\mu_{\epsilon}(\cdot )=P^{\tau_{\epsilon},H_{\epsilon}}(
X(0)\in\cdot )$, by Lemma
\ref{rnlem} and Lemma \ref{resid}, the above chain of
equalities can be continued as (with the notation
$(\mbox{\rm \ref{notat-i}})$)
\[\begin{array}{rcl}
&&\leq E^P\left[\int_0^{\tau_{\epsilon}}e^{-\lambda t}\left(h(X(t
))-\lambda f(X(t))+g(X(t))\right)dt\right]+E^P[H_{\epsilon}](\pi_{
+}(\Pi_{\mu_{\epsilon}},h)-\mu_{\epsilon}f)\end{array}
,\]
and, by Lemma \ref{intrep},
\begin{eqnarray*}
&&=E^P\left[\int_0^{\tau_{\epsilon}}e^{-\lambda t}\left(h(X(t))-\lambda
f(X(t))+g(X(t))\right)dt\right]+E^P[H_{\epsilon}](\mu_{\epsilon}u_{
+}-\mu_{\epsilon}f)\\
&&=E^P\left[\int_0^{\tau_{\epsilon}}e^{-\lambda t}\left(h(X(t))-\lambda
f(X(t))+g(X(t))\right)dt\right]+E^P\left[e^{-\lambda\tau_{\epsilon}}
(u_{+}(X(\tau_{\epsilon}))-f(X(\tau_{\epsilon}))\right]\\
&&\leq E^P\left[\int_0^{\tau_{\epsilon}}e^{-\lambda t}\left(h(X(t
))-\lambda f(X(t))+g(X(t))\right)dt\right],\end{eqnarray*}
where the last inequality uses the fact that
$u_{+}-f\leq 0$. Therefore
\begin{eqnarray*}
0&\leq&\lim_{\epsilon\rightarrow 0}\frac {E^P\left[\int_0^{\tau_{
\epsilon}}e^{-\lambda t}\left(h(X(t))-\lambda f(X(t))+g(X(t))\right
)dt\right]}{E^P[\tau_{\epsilon}]}\\
&=&h(x_0)-\lambda f(x_0)+g(x_0)\\
&=&h(x_0)-\lambda u_{+}(x_0)+g(x_0).\end{eqnarray*}
\end{proof}

\begin{corollary}\label{sol}
Let $h\in C_b(E)$.  If in addition to Condition \ref{H},
the comparison principle holds for
equation (\ref{reso}), then
$u=u_{+}=u_{-}$ is the unique viscosity solution of equation
(\ref{reso}).
\end{corollary}

\begin{theorem}\label{mgpb}
Assume that Condition \ref{H} holds.  For $\lambda >0$, let $M_{\lambda}$
be the set of $h\in C_b(E)$, such that the comparison
principle holds for (\ref{reso}).  If for each $\lambda >0$,
$M_{\lambda}$ is separating, then uniqueness holds for the martingale
problem for $A$ in $D_E[0,\infty )$.
\end{theorem}

\begin{proof}
If the comparison principle for (\ref{reso}) holds for some
$h\in C_b(E)$, then by Lemma \ref{subsol}, $u_{+}=u_{-}$.  Then,
by the definition of $u_{+}$ and $u_{-}$ and Lemma \ref{intrep}, for any two solutions
$P_1,P_2\in\Pi_{\mu}$, we must have
\[E^{P_1}[\int_0^{\infty}e^{-\lambda t}h(X(t))dt]=E^{P_2}[\int_0^{
\infty}e^{-\lambda t}h(X(t))dt].\]
Consequently, if $M_{\lambda}$ is separating, Theorem \ref{lapunq}
implies $P_1=P_2$.
\end{proof}

\begin{remark}\label{extunq}
Another way of viewing the role of the comparison
principle in the issue of uniqueness for the martingale
problem for $A$ is the following.

Suppose the comparison principle holds for some $h$ and
let $u_{+}=u_{-}=u$. By Lemmas \ref{intrep} and \ref{ext},
\[u(X(t))-\int_0^t(\lambda u(X(s))-h(X(s)))ds,\]
is a $\left\{{\cal F}_t^X\right\}$-martingale for every $P\in\Pi$.
Then, even though
$u_{+}$ and $u_{-}$ are defined in nonlinear ways, the linearity of
the martingale property ensures
that $A$ can be extended to the linear span $A^u$ of
$A\cup \{(u,\lambda u-h)\}$ and every solution of the martingale
problem for $A$ will be a solution of the martingale
problem for $A^u$. By applying this procedure to
all functions $h\in M_{\lambda}$, we obtain an extension $\hat {A}$ of $
A$ such
that every solution of the martingale
problem for $A$ will be a solution of the martingale
problem for $\hat {A}$ and such that ${\cal R}(\lambda -\hat {A})
\supset M_{\lambda}$ and hence is
separating. Therefore uniqueness follows from Corollary
\ref{rngunq}.

Notice that, even if the comparison
principle does not hold, under Condition \ref{H}, by
Lemma 4.5.18 of \cite{EK86}, for each $\mu\in {\cal P}(E)$, there
exists $P\in\Pi_{\mu}$ such that under $P$
\[u_{+}(X(t))-\int_0^t(\lambda u_{+}(X(s))-h(X(s)))ds\]
is a $\{{\cal F}_t^X\}$-martingale.
\end{remark}\medskip

\begin{remark}
If, for some $h$, there exists $(u,g)\in\hat {A}$ such that $\lambda
u-g=h$
(essentially $u$ is the analog of a stochastic solution as
defined in \cite{SV72b}), then, by Lemma \ref{resid} and
Remark \ref{do-in-v}, $u$ is a viscosity solution of $(\mbox{\rm \ref{reso}}
)$.
\end{remark}\medskip

\section{Alternative definitions of viscosity solution} \label{altdef}
Different definitions of viscosity solution may be useful,
depending on the setting.  Here we discuss two other
possibilities.  As mentioned in the Introduction, the first,
which is stated in terms of
solutions of the martingale problem, is a modification of
a definition used in
\cite{EKTZ14}, while the second is a stronger version
of definitions in \cite{FK06}.
We show that Lemma \ref{subsol} still holds under these
alternative definitions and hence all the results of Section
\ref{mgpunq} carry over. Both definitions are stronger in
the sense that the inequalities (\ref{subeq}) and
(\ref{supeq}) required by the previous definition are
required by these definitions. Consequently, in both
cases, it should be easier to prove the comparison
principle.

${\cal T}$ will denote the set of $\{{\cal F}_t^X\}$-stopping
times.

\begin{definition}\label{visc2}{\bf (Stopped viscosity (semi)solution)}

Let $A\subset C_b(E)\times C_b(E)$, $\lambda >0$, and $h\in C_b(E
)$.

\begin{itemize}

\item[a)]$u\in B(E)$ is a {\em stopped viscosity subsolution\/} of (\ref{reso})
if and only if $u$ is upper semicontinuous  and
if $(f,g)\in A$, $x_0\in E$, and there exists
a strictly positive $\tau_0\in {\cal T}$ such that
\begin{equation}\sup_{P\in\Pi_{x_0},\,\tau\in {\cal T}}\frac {E^P
[e^{-\lambda\tau\wedge\tau_0}(u-f)(X(\tau\wedge\tau_0))]}{E^P[e^{
-\lambda\tau\wedge\tau_0}]}=(u-f)(x_0),\label{subtest2}\end{equation}
 then
\begin{equation}\lambda u(x_0)-g(x_0)\leq h(x_0).\label{subeq2}\end{equation}

\item[b)]$u\in B(E)$ is a {\em stopped viscosity supersolution\/} of (\ref{reso})
if and only if $u$ is lower semicontinuous  and
if $(f,g)\in A$,  $x_0\in E$, and there exists
a strictly positive $\tau_0\in {\cal T}$ such that
\begin{equation}\inf_{P\in\Pi_{x_0},\,\tau\in {\cal T}}\frac {E^P
[e^{-\lambda\tau\wedge\tau_0}(u-f)(X(\tau\wedge\tau_0))]}{E^P[e^{
-\lambda\tau\wedge\tau_0}]}=(u-f)(x_0),\label{suptest2}\end{equation}
then
\begin{equation}\lambda u(x_0)-g(x_0)\geq h(x_0).\label{supeq2}\end{equation}

\end{itemize}

A function $u\in C_b(E)$ is a
{\em stopped viscosity solution\/} of (\ref{reso}) if it is both a
subsolution and a supersolution.

\end{definition}

\begin{remark}
If $(u-f)(x_0)=\sup_x (u-f)(x)$, then (\ref{subtest2}) is satisfied. Consequently,
every stopped sub/supersolution in the sense of Definition \ref{visc2} is a
sub/supersolution in the sense of Definition \ref{visc}.
\end{remark}\medskip

\begin{remark}
Definition \ref{visc2} requires $(\ref{subeq2})$
($(\ref{supeq2})$) to hold only at points $x_0$ for which
$(\ref{subtest2})$ ($(\ref{suptest2})$) is verified for
some $\tau_0$. Note that, as in Definition \ref{visc}, such an $x_
0$ might not exist.

Definition \ref{visc2} essentially requires a local maximum
principle and is  related to the notion of
characteristic operator as given in \cite{Dyn65}.
\end{remark}\medskip

For Definition \ref{visc2}, we have the following analog
of Lemma \ref{subsol}.

\begin{lemma}\label{subsol2}Assume
that Condition \ref{H} holds.  Then
$u_{+}$ given by (\ref{uplus}) is a stopped viscosity
subsolution of (\ref{reso}) and $u_{-}$ given by (\ref{uminus})
is a stopped viscosity
supersolution of the same equation.
\end{lemma}

\begin{proof}
Let $(f,g)\in A$.  Suppose $x_0$ is a point such that
$(\ref{subtest2})$ holds for $u_{+}$ for some
$\tau_0\in {\cal T}$, $\tau_0>0$.
Since we can always
add a constant to $f$, we can assume $u_{+}(x_0)-f(x_0)=0$.
By the same arguments used in the proof of Lemma
\ref{subsol}, defining $\tau_{\epsilon}$ and $H_{\epsilon}$ in the same way, we
obtain, for some $P\in\Pi_{x_0}$(independent of $\epsilon )$,
\begin{eqnarray*}
0&&=u_{+}(x_0)-f(x_0)\\
&&\leq E^P\left[\int_0^{\tau_{\epsilon}\wedge\tau_0}e^{-\lambda t}\left
(h(X(t))-\lambda f(X(t))+g(X(t))\right)dt\right]\\
&&\qquad\qquad\quad +E^P\left[e^{-\lambda\tau_{\epsilon}\wedge\tau_
0}(u_{+}(X(\tau_{\epsilon}\wedge\tau_0))-f(X(\tau_{\epsilon}\wedge
\tau_0))\right]\\
&&\leq E^P\left[\int_0^{\tau_{\epsilon}\wedge\tau_0}e^{-\lambda t}\left
(h(X(t))-\lambda f(X(t))+g(X(t))\right)dt\right],\end{eqnarray*}
where the last inequality uses $(\ref{subtest2})$
and the fact that
$u_{+}(x_0)-f(x_0)=0$.  Then the result follows as in
Lemma \ref{subsol}.
\end{proof}

The following is essentially Definition 7.1 of \cite{FK06}.

\begin{definition}\label{visc3}{\bf (Sequential viscosity (semi)solution)}

Let $A\subset C_b(E)\times C_b(E)$, $\lambda >0$, and $h\in C_b(E
)$.

\begin{itemize}

\item[a)]$u\in B(E)$ is a {\em sequential viscosity subsolution }
of (\ref{reso})
if and only if $u$ is upper semicontinuous  and
for each $(f,g)\in A$ and each sequence $y_n\in E$ satisfying
\begin{equation}\lim_{n\rightarrow\infty}(u-f)(y_n)=\sup_x(u-f)(x
),\label{subtest3}\end{equation}
we have
\begin{equation}\limsup_{n\rightarrow\infty}(\lambda u(y_n)-g(y_n
)-h(y_n))\leq 0.\label{subeq3}\end{equation}

\item[b)]$u\in B(E)$ is a {\em sequential viscosity supersolution\/} of (\ref{reso})

if and only if $u$ is lower semicontinuous  and
for each $(f,g)\in A$ and each sequence $y_n\in E$ satisfying
\begin{equation}\lim_{n\rightarrow\infty}(u-f)(y_n)=\inf_x(u-f)(x
),\label{suptest3}\end{equation}
we have
\begin{equation}\liminf_{n\rightarrow\infty}(\lambda u(y_n)-g(y_n
)-h(y_n))\geq 0.\label{supeq3}\end{equation}
\end{itemize}

A function $u\in C_b(E)$ is a
{\em sequential viscosity solution\/} of (\ref{reso}) if it is both a
subsolution and a supersolution.
\end{definition}

\begin{remark}\label{3vs1}
For $E$ compact, every viscosity sub/supersolution is a
sequential viscosity sub/supersolution.
\end{remark}\medskip

For sequential viscosity semisolutions, we have the following analog
of Lemma \ref{subsol}.  $C_{b,u}(E)$ denotes the space of
bounded, uniformly continuous functions on $E$.

\begin{lemma}\label{subsol3}
For $\epsilon >0$, define
\[\tau_{\epsilon}=\epsilon\wedge\inf\{t>0:r(X(t),X(0))\geq\epsilon\mbox{\rm \ or }
r(X(t-),X(0))\geq\epsilon \}.\]
Assume $A\subset C_{b,u}(E)\times C_{b,u}(E)$,  for each $\epsilon
>0$,
$\inf_{P\in\Pi}E^P[\tau_{\epsilon}]>0$, and
that Condition \ref{H} holds.  Then, for $h\in C_{b,u}(E)$,
$u_{+}$ given by $(\ref{uplus})$ is a sequential viscosity
subsolution of (\ref{reso}) and $u_{-}$ given by $(\ref{uminus})$
is a sequential viscosity
supersolution of the same equation.
\end{lemma}

\begin{proof}
 Let $(f,g)\in A$.  Suppose $\{y_n\}$ is a sequence such that
$(\ref{subtest3})$ holds for $u_{+}$.
Since we can always
add a constant to $f$, we can assume $\sup_x(u_{+}-f)(x)=0$.
Let $H_{\epsilon}=e^{-\lambda\tau_{\epsilon}}$. Then, by the same arguments as in
Lemma \ref{subsol}, we have, for some
$P_n\in\Pi_{y_n}$(independent of $\epsilon )$,
\begin{eqnarray*}
(u_{+}-f)(y_n)\leq E^{P_n}\left[\int_0^{\tau_{\epsilon}}e^{-\lambda
t}\left(h(X(t))-\lambda f(X(t))+g(X(t))\right)dt\right],\end{eqnarray*}
where we have used the fact that
$\sup_x(u_{+}-f)(x)=0$. Therefore
\[\frac {(u_{+}-f)(y_n)}{E^{P_n}[\tau_{\epsilon}]}\leq\frac {E^{P_
n}\left[\int_0^{\tau_{\epsilon}}e^{-\lambda t}\left(h(X(t))-\lambda
f(X(t))+g(X(t))\right)dt\right]}{E^{P_n}[\tau_{\epsilon}]}.\]
Replacing $\epsilon$ by $\epsilon_n$ going to zero sufficiently slowly so
that the left side converges to zero, the uniform
continuity of $f$, $g$, and $h$ implies the right side is
asymptotic to $h(y_n)-\lambda f(y_n)+g(y_n)$ giving
\begin{eqnarray*}
0&\leq&\liminf_{n\rightarrow\infty}(h(y_n)-\lambda f(y_n)+g(y_n))\\
&=&\liminf(h(y_n)-\lambda u_{+}(y_n)+g(y_n).\end{eqnarray*}
\end{proof}

The following theorem is essentially Lemma 7.4 of
\cite{FK06}.  It gives the intuitively natural result that
if $h\in\overline {{\cal R}(\lambda -A)}$ (where the closure is taken under uniform
convergence),
then the comparison principle
holds for sequential viscosity semisolutions of $\lambda u-Au=h$.

If $E$ is compact, the same results hold for viscosity
semisolutions, by Remark \ref{3vs1}.

\begin{theorem}\label{converse}
Suppose $h\in C_b(E)$ and there
exist $(f_n,g_n)\in A$ satisfying $\sup_x|\lambda f_n(x)-g_n(x)-h
|\rightarrow 0$.
Then the comparison principle holds for sequential
viscosity semisolutions of $(\ref{reso})$.
\end{theorem}

\begin{proof}
Suppose $\underline u$ is a sequential viscosity subsolution.
Set $h_n=\lambda f_n-g_n$.
For $\epsilon_n>0$, $\epsilon_n\rightarrow 0$, there exist $y_n\in
E$
satisfying
$\underline u(y_n)-f_n(y_n)\geq\sup_x(\underline u(x)-f_n(x))-\epsilon_
n/\lambda$ and
$\lambda\underline u(y_n)-g_n(y_n)-h(y_n)\leq\epsilon_n$.
Then
\begin{eqnarray*}
\sup_x(\lambda\underline u(x)-\lambda f_n(x))&\leq&\lambda\underline
u(y_n)-\lambda f_n(y_n)+\epsilon_n\\
&\leq&h(y_n)+g_n(y_n)-\lambda f_n(y_n)+2\epsilon_n\\
&=&h(y_n)-h_n(y_n)+2\epsilon_n\\
&\rightarrow&0.\end{eqnarray*}
Similarly, if $\overline u$ is a supersolution of $\lambda u-Au=h$,
\[\liminf_{n\rightarrow\infty}\inf_x(\overline u(x)-f_n(x))\geq 0
,\]
and it follows that $\underline u\leq\overline u$.
\end{proof}

\section{Martingale problems with boundary
conditions} \label{cmgpunq}

The study of stochastic processes that are constrained
to some set $\bar {E}_0$ and must satisfy some boundary
condition on $\partial E_0$, described by one or more boundary
operators $B_1,\ldots ,B_m$, is typically carried out by
incorporating the boundary condition in the definition of
the domain ${\cal D}(A)$ (see Remark \ref{0bdry} below).
However, this approach restricts the problems that can
be dealt with to fairly regular ones, so we follow the
formulation of a constrained martingale problem given in
\cite{Kur90}.  (See also \cite{Kur91,KS01}).

Let $E_0\subseteq E$ be an open set and let
\[\partial E_0=\cup_{k=1}^mE_k,\]
for disjoint, nonempty Borel sets $E_1,\ldots ,E_m$.  Let
$A\subseteq C_b(\bar {E}_0)\times C_b(\bar {E}_0)$, $B_k\subseteq
C_b(\bar {E}_0)\times C_b(\bar {E}_0)$, $k=1,...,m$, be
linear operators with a common domain ${\cal D}$ such that
$(1,0)\in A$, $(1,0)\in B_k$, $k=1,...,m$.  For simplicity we will
assume that $ $$E$ is compact (hence the subscript $b$ will be
dropped) and that $A,B_1,\ldots ,B_m$ are single-valued.

\begin{definition}\label{cmgp}
A stochastic process $X$ with sample paths in $D_{\bar {E}_0}[0,\infty
)$ is
a solution of the {\em constrained martingale problem\/} for
$(A,E_0;B_1,E_1;\ldots ;B_m,E_m)$ provided there exist a filtration
$\{{\cal F}_t\}$ and continuous, nondecreasing processes $\gamma_
1,\ldots ,\gamma_m$
such that $X$, $\gamma_1,\ldots ,\gamma_m$ are $\{{\cal F}_t\}$-adapted,
\[\gamma_k(t)=\int_0^t{\bf 1}_{\bar {E}_k}(X(s-))d\gamma_k(s),\]
and for each $f\in {\cal D}$,
\begin{equation}M_f(t)=f(X(t))-f(X(0))-\int_0^tAf(X(s))ds-\sum_{k
=1}^m\int_0^tB_kf(X(s-))d\gamma_k(s)\label{cmg}\end{equation}
is a $\{{\cal F}_t\}$-martingale.
\end{definition}

\begin{remark}\label{adap}
$\gamma_1,\ldots ,\gamma_m$ will be called {\em local times\/} since $
\gamma_k$ increases
only when $X$ is in $\bar {E}_k$.  Without loss of
generality, we can assume that the $\gamma_k$ are $\{{\cal F}^X_t
\}$-adapted.
(Replace $\gamma_k$ by its dual, predictable projection on $\{{\cal F}_
t^X\}$.)
Definition \ref{cmgp}\ does not require
that the $\gamma_k$ be uniquely determined
by the distribution of $X$, but if $\gamma_k^1$ and $\gamma_k^2$, $
k=1,\ldots ,m$, are
continuous and satisfy
the martingale requirement with the same filtration,
we must have
\[\sum_{k=1}^m\int_0^tB_kf(X(s-))d\gamma_k^1(s)-\sum_{k=1}^m\int_
0^tB_kf(X(s-))d\gamma_k^2(s)=0,\]
since this expression will be a continuous martingale with
finite variation paths.
\end{remark}\medskip

\begin{remark}
The main example of a constrained martingale
problem in the sense of the above definition is the
constrained martingale problem that describes a reflected
diffusion process.  In this case, $A$ is a second order elliptic
operator and the $B_k$ are first order
differential operators. Although there is a vast
literature on this topic, there are still relevant cases of
reflected diffusions that have not been uniquely characterized
as solutions of martingale problems or stochastic
differential equations. In Section \ref{degref}, the results
of this section are used in one of these cases. More
general constrained diffusions where the $B_k$ are second
order elliptic operators, for instance diffusions with sticky
reflection, also satisfy Definition \ref{cmgp}.

Definition \ref{cmgp}\ is a special case of a more general
definition of constrained martingale problem given in
\cite{KS01}.  This broader definition allows for more
general boundary behavior, such as models considered in
\cite{CK06}.
\end{remark}\medskip

Many results for solutions of martingale problems carry
over to solutions of constrained martingale problems. In
particular Lemma \ref{resid} still holds.
In addition the following lemma holds.

\begin{lemma}\label{cres}
Let $X$ be a stochastic process with sample paths in
$D_{\bar {E}_0}[0,\infty )$, $\gamma_1,\ldots ,\gamma_m$ be continuous, nondecreasing processes
such that $X$, $\gamma_1,\ldots ,\gamma_m$ are $\{{\cal F}_t\}$-adapted. Then for
$f\in {\cal D}$ such that $(\ref{cmg})$ is a $\{{\cal F}_t\}$-martingale and $
\lambda >0$,
\begin{eqnarray*}
M_f^{\lambda}(t)&=&e^{-\lambda t}f(X(t))-f(X(0))+\int_0^te^{-\lambda
s}(\lambda f(X(s))-Af(X(s)))ds\\
&&\qquad\qquad\qquad -\sum_{k=1}^m\int_0^te^{-\lambda s}B_kf(X(s-
))d\gamma_k(s)\end{eqnarray*}
is a $\{{\cal F}_t\}$-martingale.
\end{lemma}

\begin{proof}
Since ${\cal D}\subset C(E)$, $M_f$ is cadlag, we can apply It\^o's formula
to
$e^{-\lambda t}f(X(t))$ and obtain
\begin{eqnarray*}
e^{-\lambda t}f(X(t))-f(X(0))&=&\int_0^t(-f(X(s))\lambda e^{-\lambda
s}+e^{-\lambda s}Af(X(s)))ds\\
&&\qquad\qquad +\sum_{k=1}^m\int_0^te^{-\lambda s}B_kf(X(s-))d\gamma_
k(s)+\int_0^te^{-\lambda s}dM_f(s).\end{eqnarray*}
\end{proof}

Lemma \ref{rnlem} is replaced by Lemma
\ref{ctrans} below.

\begin{lemma}\label{ctrans}

\begin{itemize}

\item[a)]The set of distributions of
solutions of the constrained martingale problem for
$(A,E_0;B_1,E_1;\ldots ;B_m,E_m)$ is convex.

\item[b)]Let $X$, $\gamma_1,\ldots ,\gamma_m$ satisfy Definition
\ref{cmgp}.  Let $\tau\geq 0$ be a bounded $\{{\cal F}_t\}$-stopping time and
$H\geq 0$ be a ${\cal F}_{\tau}$-measurable random variable such that
$0<E[H]<\infty$. Then the measure $P^{\tau ,H}\in {\cal P}(D_E[0,
\infty )$ defined by
\begin{equation}P^{\tau ,H}(C)=\frac {E[H\mbox{\rm \{\bf 1}_
C(X(\tau +\cdot ))]}{E[H]},\quad C\in {\cal B}(D_{\overline {E_0}}
[0,\infty )),\label{cphdist}\end{equation}
is the distribution of
a solution of the constrained martingale problem for
$(A,E_0;B_1,E_1;$ $\ldots ;B_m,E_m)$.

\end{itemize}
\end{lemma}

\begin{proof}
Part (a) is immediate.  For Part (b), let $(\Omega ,{\cal F},\mathbb{
P})$
be the probability space on which $X$, $\gamma_1,\ldots ,\gamma_m$ are
defined, and define $\mathbb{P}^H$ by
\[\mathbb{P}^H(C)=\frac {E^{\mathbb{P}}[H{\bf 1}_C]}{E^{\mathbb{P}}
[H]},\quad C\in {\cal F}.\]
Define  $X^{\tau}$ and $\gamma_k^{\tau}$ by $X^{\tau}(t)=X(\tau +
t)$ and
$\gamma_k^{\tau}(t)=\gamma_k(\tau +t)-\gamma_k(\tau )$.
$X^{\tau}$ and the $\gamma_k^{\tau}$ are adapted to the filtration $
\{{\cal F}_{\tau +t}\}$ and
for  $0\leq t_1<\cdots <t_n<t_{n+1}$ and  $f_1,\cdots ,f_n\in B(E
)$,
\begin{eqnarray*}
&&E^{\mathbb{P}^H}\Big[\Big\{f(X^{\tau}(t_{n+1}))-f(X^{\tau}(t_n)
)-\int_{t_n}^{t_{n+1}}Af(X^{\tau}(s))ds-\sum_{k=1}^m\int_{t_n}^{t_{
n+1}}B_kf(X^{\tau}(s-))d\gamma^{\tau}_k(s)\Big\}\\
&&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\Pi_{i=1}^nf_i(
X^{\tau}(t_i))\Big]\\
&&\quad =\frac 1{E^{\mathbb{P}}[H]}E^{\mathbb{P}}\Big[H\Big\{f(X(
\tau +t_{n+1}))-f(X(\tau +t_n))\\
&&\qquad\qquad\qquad\qquad -\int_{\tau +t_n}^{\tau +t_{n+1}}Af(X(
s))ds-\sum_{k=1}^m\int_{\tau +t_n}^{\tau +t_{n+1}}B_kf(X(s-))d\gamma_
k(s)\Big\}\\
&&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\Pi_{i=1}^nf_i(
X(\tau +t_i))\Big]\\
&&\quad =0\end{eqnarray*}
by the optional sampling theorem.
Therefore, under $\mathbb{P}^H$, $X^{\tau}$ is a solution of the constrained
martingale problem with local times $\gamma^{\tau}_1,\cdots ,\gamma^{
\tau}_m$.
$P^{\tau ,H}$, given by (\ref{cphdist}),
is the distribution of $X^{\tau}$ on $D_{\overline {E_0}}[0,\infty
)$.
\end{proof}

As in Section \ref{mgpunq}, let $\Pi$ denote the set of
distributions of solutions of the constrained martingale
problem and
$\Pi_{\mu}$ denote the set of
distributions of solutions with initial condition $\mu$.
In the rest of this
section $X$ is the canonical process on $D_E[0,\infty )$
and $\gamma_1,\ldots ,\gamma_m$ are a set of $\{{\cal F}^X_t\}$-adapted local times (see
Remark \ref{adap}).
We assume that the following conditions hold.
See Section \ref{suffsect} below for
settings in which these conditions are valid. Recall that
we are assuming $E$ is compact.

\begin{condition}\label{Hc}\hfill

\begin{itemize}

\item[a)] ${\cal D}$ is dense in $C(\bar {E}_0)$ in the topology of
uniform convergence.

\item[b)] For each $\mu\in {\cal P}(\bar {E}_0)$, $\Pi_{\mu}\neq\emptyset$ (see Proposition
\ref{suffHc}).

\item[c)] $\Pi$ is compact
(see Proposition \ref{suffHc}).

\item[d)] For each $P\in\Pi$ and $\lambda >0$, there exist $\gamma_
1,\ldots ,\gamma_m$
satisfying the requirements of Definition \ref{cmgp}
such that
 $E^P\Big[\int_0^{\infty}e^{-\lambda t}d\gamma_k(t)\Big]<\infty$, $
k=1,\cdots ,m$ (see Proposition \ref{suffHc}).
\end{itemize}
\end{condition}

\begin{remark}
For $P\in\Pi$, Condition \ref{Hc}(d) and Lemma \ref{cres} give
\begin{equation}\mu f=E[\int_0^{\infty}e^{-\lambda s}(\lambda f(X
(s))-Af(X(s)))ds-\sum_{k=1}^m\int_0^{\infty}e^{-\lambda s}B_kf(X(
s-))d\gamma_k(s)].\label{cresid}\end{equation}
\end{remark}\medskip

\begin{remark}
We can take the topology on $D_E[0,\infty )$ to be either the
Skorohod topology or the Jakubowski topology (see
Remark \ref{Jak}).
\end{remark}\medskip

The definitions of $u_{+}$, $u_{-}$, $\pi_{+}$ and $\pi_{-}$
are still given by
(\ref{uplus}), (\ref{uminus}), (\ref{piplus}) and
(\ref{piminus}). With Condition \ref{Hc} replacing
Condition \ref{H}, the proof of Lemma \ref{intrep} carries
over (Lemma \ref{ctrans} above guarantees that
Lemmas 4.5.8 and 4.5.10 in \cite{EK86} can be applied).

\begin{lemma}\label{csubsol}Assume
Condition \ref{Hc} holds.  Then $u_{+}$ is a viscosity
subsolution of $(\ref{creso})$ and $u_{-}$ is a viscosity
supersolution of the same equation.
\end{lemma}

\begin{proof}
The proof is similar to the proof of Lemma \ref{subsol},
so we will only sketch the argument.
For $f\in {\cal D}$, let $x_0$ satisfy
$\sup_{x\in\bar {E}_0}(u_{+}-f)(x)=u_{+}(x_0)-f(x_0)$. By adding a constant to
$f$ if necessary, we
can assume that $u_{+}(x_0)-f(x_0)=0$.

With $\tau_{\epsilon}$ as in the proof of
Lemma \ref{subsol}, by Lemmas \ref{cres}, \ref{ctrans}
and \ref{intrep}, and the compactness of $\Pi_{x_0}$ (Condition
\ref{Hc}(c)), for some $P\in\Pi_{x_0}$ (independent of $\epsilon$) we have
\begin{eqnarray*}
0\leq E^P\bigg[\int_0^{\tau_{\epsilon}}e^{-\lambda t}\left(h(X(t)
)-\lambda f(X(t))+Af(X(t))\right)dt\\
+\sum_{k=1}^m\int_0^{\tau_{\epsilon}}e^{-\lambda t}B_kf(X(t-))d\gamma_
k(t)\bigg],\end{eqnarray*}
Dividing the
expectations by $E^P[\lambda^{-1}(1-e^{-\lambda\tau_{\epsilon}})+
\sum_{k=1}^m\int_0^{\tau_{\epsilon}}e^{-\lambda t}d\gamma_k(t)]$ and
letting $\epsilon\rightarrow 0$, we must have
\[0\leq (h(x_0)-\lambda f(x_0)+Af(x_0))\vee\max_{k:x_0\in\bar {E}_
k}B_kf(x_0),\]
which, since $f(x_0)=u_{+}(x_0)$, implies (\ref{bndry}), if
$x_0\in\partial E_0$, and (\ref{interior}), if $x_0\in E_0$.
\end{proof}

\begin{corollary}\label{csol}
Let $h\in C(\bar {E}_0)$.  If, in addition to Condition \ref{Hc},
the comparison principle holds for
equation (\ref{creso}), then
$u=u_{+}=u_{-}$ is the unique viscosity solution of equation
(\ref{creso}).
\end{corollary}

The following theorem is the analog of Theorem \ref{mgpb}.

\begin{theorem}\label{cmgpb}
Assume Condition \ref{Hc} holds.  For $\lambda >0$, let $M_{\lambda}$
be the set of $h\in C(\bar {E}_0)$ such that the comparison
principle holds for (\ref{creso}).  If for every $\lambda >0$,
$M_{\lambda}$ is separating, then the distribution of the
solution $X$ of the
constrained martingale problem for $(A,E_0;B_1,E_1;\ldots ;B_m,E_
m)$
is uniquely determined.
\end{theorem}

\begin{proof}\
The proof of this result is in the spirit of Remark
\ref{extunq}.
Let $\hat {A}$ be the collection of $(f,g)\in B(\bar {E}_0)\times
B(\bar {E}_0)$ such that
$f(X(t))-\int_0^tg(X(s))ds$ is a $\left\{{\cal F}^X_t\right\}$-martingale for all $
P\in\Pi$.
Denote by $\hat{\Pi}$ the set of the distributions of solutions of
the martingale problem for $\hat {A}$, and by $\hat{\Pi}_{\mu}$ the set of
solutions with initial distribution $\mu$. Then, by construction,
for each $\mu\in {\cal P}(\bar {E}_0)$, $\Pi_{\mu}\subseteq\hat{\Pi}_{
\mu}$.  By the comparison principle,
Lemmas \ref{csubsol},
\ref{intrep} and \ref{ext}, for each $h\in M_{\lambda}$ and
$u=u_{+}=u_{-}$ given by (\ref{uplus}) (or equivalently,
(\ref{uminus})),
$(u,\lambda u-h)$ belongs to $\hat {A}$, or equivalently the pair $
(u,h)$
belongs to $\lambda -\hat {A}$.  Consequently ${\cal R}(\lambda -
\hat {A})\supseteq M_{\lambda}$ is separating
and the thesis follows from Corollary \ref{rngunq}.
\end{proof}

\begin{remark}\label{0bdry}
Differently from Remark \ref{extunq}, the operator $\hat {A}$ is not
an extension of $A$
as an operator on the domain ${\cal D}$, but it is an extension of $
A$
restricted to the domain
${\cal D}_0=\{f\in {\cal D}:\,B_kf(x)=0\,\,\,\forall x\in\bar {E}_
k,\,k=1,\cdots ,m\}$.
The distribution of the solution $X$ of the
constrained martingale problem for
$(A,E_0;B_1,E_1;\ldots ;B_m,E_m,\mu )$
is uniquely determined even though the same might not
hold for the solution of the
martingale problem for $(A\Big|_{{\cal D}_0},\mu )$.
\end{remark}\medskip

\subsection{Sufficient conditions for the validity of
Condition \ref{Hc}}\label{suffsect}

In what follows, we assume that $E-E_0=\cup_{k=1}^m\tilde {E}_k$, where
the $\tilde {E}_k$ are disjoint Borel sets satisfying $\tilde {E}_
k\supset E_k$,
$k=1,\ldots ,m$.

\begin{proposition}\label{suffHc}
Assume Condition \ref{Hc}(a)  and that the following
hold:

\begin{itemize}

\item[i)] There exist linear operators
$\tilde {A},\tilde {B}_1,\cdots ,\tilde {B}_m:\tilde {{\cal D}}\subseteq
C(E)\rightarrow C(E)$ with $\tilde {{\cal D}}$ dense in $C(E)$, $
(1,0)\in\tilde {A}$,
$(1,0)\in\tilde {B}_k$, $k=1,...,m$, that are extensions of $A,B_
1,\ldots ,B_m$
in the sense that for every $f\in {\cal D}$ there exists $\tilde {
f}\in\tilde {{\cal D}}$ such
that $f=\left.\tilde f\right|_{\bar {E}_0}$, $Af=\left.\tilde A\tilde
f\right|_{\bar {E}_0}$, and $B_kf=\left.\tilde B_k\tilde f\right|_{
\bar {E}_0}$,
$k=1,...,m$, and such that the martingale problem for
each of $\tilde {A},\tilde {B}_1,\cdots ,\tilde {B}_m$ with initial condition $
\delta_x$ has a
solution for every $x\in E$.

\item[ii)] If $E\neq\bar {E}_0$, there exists $\varphi\in\tilde {{\cal D}}$ such that
$\varphi =0$ on $\bar {E}_0$, $\varphi >0$ on $E-\bar {E}_0$ and $
\tilde {A}\varphi =0$ on $\bar {E}_0$, $\tilde {B}_k\varphi\leq 0$ on
$\overline {\tilde {E}_k}$, $k=1,...,m$.

\item[iii)] There exists $\left\{\varphi_n\right\}$, $\varphi_n\in
{\cal D}$, such that
$\sup_{n,x}|\varphi_n(x)|<\infty$ and $B_k\varphi_n(x)\geq n$ on $
\bar {E}_k$ for all $k=1,...,m$.

\end{itemize}

Then b) c) and d) in Condition \ref{Hc} are verified.

\end{proposition}

\begin{proof}
{\bf Condition \ref{Hc}(b)}.  We will obtain a solution of the
constrained martingale problem for
$(A,E_0;B_1,E_1;\ldots ;B_m,E_m)$ by constructing a solution of the
constrained martingale problem for
$(\tilde {A},E_0;\tilde {B}_1,\tilde {E}_1;\ldots ;\tilde {B}_m,\tilde {
E}_m)$ and showing that any such
solution that starts in $\bar {E}_0$ stays in $\bar {E}_0$ for all times.
Following \cite{Kur90}, we will construct a solution of
the constrained martingale problem for
$(\tilde {A},E_0;\tilde {B}_1,\tilde {E}_1;\ldots ;\tilde {B}_m,\tilde {
E}_m)$ from a solution of the
corresponding patchwork martingale problem.

$\tilde {A},\tilde {B}_1,\cdots ,\tilde {B}_m$ are dissipative operators by i) and Lemma
\ref{diss}.
Then, by Lemma 1.1 in \cite{Kur90}, for each initial distribution
on $E$, there exists a solution of the
patchwork martingale problem for
$(\tilde {A},E_0;\tilde {B}_1,\tilde {E}_1;\ldots ;\tilde {B}_m,\tilde {
E}_m)$. In addition, if $E\neq\bar {E}_0$, by ii) and the same
argument used in the proof of Lemma 1.4 in
\cite{Kur90}, for every solution $Y$ of
the patchwork martingale problem for
$(\tilde {A},E_0;\tilde {B}_1,\tilde {E}_1;\ldots ;\tilde {B}_m,\tilde {
E}_m)$ with initial distribution
concentrated on $\bar {E}_0$,  $Y(t)\in\bar {E}_0$ for all $t\geq
0$.
Therefore $Y$ is also a solution of
the patchwork martingale problem for
$(A,E_0;B_1,E_1;\ldots ;B_m,E_m)$.
By iii) and Lemma 1.8, Lemma 1.9,
Proposition 2.2, and Proposition 2.3 in \cite{Kur90},
from $Y$, a solution $X$ of the constrained
martingale problem for $(A,E_0;B_1,E_1;\ldots ;B_m,E_m)$
can be constructed.

\vspace{.1in}

\noindent {\bf Condition \ref{Hc}(c)}.  If $X$ is a solution of the
constrained martingale problem for
$(A,E_0;$ $B_1,E_1;\ldots ;B_m,E_m)$
and $\gamma_1,\ldots ,\gamma_m$ are associated local
times, then $\eta_0(t)=\inf\{s:\,s+\,\gamma_1(s)+\ldots +\gamma_m
(s)>t\}$ is strictly
increasing and diverging to infinity as $t$ goes to infinity,
with probability one, and $Y=X\circ\eta_0$ is a solution of the
patchwork martingale problem for
$(A,E_0;B_1,E_1;\ldots ;B_m,E_m)$, $\eta_0,\eta_1=\gamma_1\circ\eta_
0,\ldots ,\eta_m=\gamma_m\circ\eta_0$ are
associated increasing processes (see the proof of
Corollary 2.5 of \cite{Kur90}).  Let $\left\{(X^n,\gamma_1^n,\ldots
,\gamma_m^n)\right\}$ be a
sequence of solutions of the constrained martingale
problem for $(A,E_0;B_1,E_1;\ldots ;B_m,E_m)$ with initial
conditions $\left\{\mu^n\right\}$, $\mu^n\in {\cal P}(E)$, with associated local times.
Since ${\cal P}(E)$ is compact, we may assume, without loss of
generality, that $\left\{\mu^n\right\}$ converges to $\mu$.  Let
$\left\{(Y^n,\eta_0^n,\eta_1^n,\ldots ,\eta_m^n)\right\}$ be the sequence of the corresponding
solutions of the patchwork martingale problem and
associated increasing processes.  Then by the density of ${\cal D}$
and Theorems 3.9.1 and 3.9.4 of \cite{EK86},
$\left\{(Y^n,\eta_0^n,\eta_1^n,\ldots ,\eta_m^n)\right\}$ is relatively compact under the
Skorohod topology on
$D_{E\times {\bf R}^{m+1}}[0,\infty )$.

Let $\left\{(Y^{n_k},\eta_0^{n_k},\eta_1^{n_k},\ldots ,\eta_m^{n_
k})\right\}$ be a subsequence converging
to a limit $(Y,\eta_0,\eta_1,\ldots ,\eta_m)$.  Then $Y$ is a solution of the
patchwork martingale problem for $(A,E_0;B_1,E_1;\ldots ;B_m,E_m)$
with initial condition $\mu$ and $\eta_0,\eta_1,\ldots ,\eta_m$ are associated
increasing processes.  By iii) and Lemma 1.8 and Lemma 1.9
in \cite{Kur90}, $\eta_0$ is strictly increasing and diverging to
infinity as $t$ goes to infinity, with probability one.  It
follows that $\left\{(\eta_0^{n_k})^{-1}\right\}$ converges to $(
\eta_0)^{-1}$ and hence
$\left\{(X^{n_k},\gamma_1^{n_k},\ldots ,\gamma_m^{n_k})\right\}=\left
\{(Y^{n_k}\circ (\eta_0^{n_k})^{-1},\eta_1^{n_k}\circ (\eta_0^{n_
k})^{-1},\ldots ,\eta_m^{n_k}\circ (\eta_0^{n_k})^{-1})\right\}$
converges to $(Y\circ (\eta_0)^{-1},\eta_1\circ (\eta_0)^{-1},\ldots
,\eta_m\circ (\eta_0)^{-1})$ and
$Y\circ (\eta_0)^{-1}$ with associated local times
$\eta_1\circ (\eta_0)^{-1},\ldots ,\eta_m\circ (\eta_0)^{-1}$ is a solution of the constrained
martingale problem for $(A,E_0;B_1,E_1;\ldots ;B_m,E_m)$ with
initial condition $\mu$.

\vspace{.1in}

\noindent {\bf Condition \ref{Hc}(d)}.  Let $\varphi_1$ be the function of
iii) for $n=1$.  By Lemma \ref{cres} and iii) we have
\[\begin{array}{l}E\Big[\sum_{k=1}^m\int_0^te^{-\lambda s}d\gamma_k(s)\Big]\\
\qquad\leq e^{-\lambda t}E[\varphi_1(X(t))]-E[\varphi_1(X(0))]+\int_0^te^{-\lambda
s}E[\lambda\varphi_1(X(s))-A\varphi_1(X(s))]ds.
\end{array}\]
\end{proof}

\section{Examples}\label{examples}

Several examples of application of the results of the
previous sections can be given by exploiting comparison
principles proved in the literature. Here we will discuss
in detail two examples.

The first example is a class of diffusion processes
reflecting in a domain $D\subseteq {\bf R}^d$ according to an oblique direction
of reflection which may become tangential.
This case is not covered by the existing literature on reflecting
diffusions, which assumes that the direction of reflection is
uniformly bounded away from the tangent hyperplane.

The second example is a large class of jump diffusion processes
with jump component of
unbounded variation and possibly degenerate diffusion
matrix. In this case uniqueness results are already
available in the literature (see e.g. \cite{IW89},
\cite{Gra92}, \cite{KP96}) but we believe it is
still a good benchmark to show how our method works.

\subsection{Diffusions with degenerate oblique
direction of reflection}\label{degref}

Let $D\subseteq {\bf R}^d$, $d\geq 2$, be a bounded domain with $
C^3$ boundary,
i.e.
\[\begin{array}{cc}
D=\{x\in {\bf R}^d:\,\psi (x)>0\},\quad\partial D=\{x\in {\bf R}^
d:\,\psi (x)=0\},\\
|\nabla\psi (x)|\,>0,\quad\mbox{\rm for }x\in\partial D,\end{array}
\]
for some function $\psi\in C^3({\bf R}^d)$, where $\nabla$ denotes the
gradient, viewed as a row vector. Let
$l:\bar {D}\rightarrow {\bf R}^d$ be a vector field in $C^2(\bar {
D})$ such that
\begin{equation}|l(x)|>0\mbox{\rm \ and }\langle l(x),\nu (x)\rangle
\geq 0,\quad\forall x\in\partial D,\label{drdir1}\end{equation}
$\nu$ being the unit inward normal vector field, and let
\begin{equation}\partial_0D=\{x\in\partial D:\,\langle l(x),\nu (
x)\rangle =0\}.\label{drdir2}\end{equation}
We assume that $\partial_0D$ has dimension $d-2$. More precisely,
for $d\geq 3$, we assume that $\partial_0D$ has a finite number of connected
components, each of the form
\begin{equation}\{x\in\partial D:\,\psi (x)=0,\tilde{\psi }(x)=0\}
,\label{drdir3}\end{equation}
where $\psi$ is the function above and $\tilde{\psi}$ is another function
in $C^2({\bf R}^d)$ such that the level set $\{x\in\partial D:\,\tilde{
\psi }(x)=0\}$
is bounded and $|\nabla\tilde{\psi }(x)|\,>0$ on it.
For $d=2$, we assume that $\partial_0D$ consists of a finite
number of points.
In addition, we assume that $l(x)$ is never tangential to $\partial_
0D$.

Our goal is to prove uniqueness of the reflecting
diffusion process with generator of the form
\begin{equation}Af(x)=\frac 12tr\left(D^2f(x)\sigma (x)\sigma (x)^
T\right)+\nabla f(x)b(x),\label{drop}\end{equation}
where $\sigma$ and $b$ are Lipschitz continuous
functions on $\bar {D}$, and direction of reflection $l$.
We will characterize this reflecting diffusion process as
the unique solution of the constrained martingale
problem for $(A,D;B,\partial D)$, where $A$ is given by
(\ref{drop}),
\begin{equation}Bf(x)=\nabla f(x)l(x),\label{drbop}\end{equation}
and the common domain of $A$ and $B$ is ${\cal D}=C^2(\overline D
)$. Our
tools will be the results of Section \ref{cmgpunq} and the
comparison principle proved  by
\cite{PK05}.

\begin{proposition}
Condition \ref{Hc} is verified.
\end{proposition}

\begin{proof}
Condition \ref{Hc}a) is obviously verified. Therefore we
only need to prove that the assumptions of Proposition
\ref{suffHc} are satisfied.
Let $0<r<1$ be small enough that for $d(x,\partial D)<\frac 43r$
the normal projection of $x$ on $\partial D$, $\pi_{\nu}(x)$, is well
defined and $|\nabla\psi (x)|\,>0$. Set $U(\bar {D})=\left\{x:\,d(x,\bar
D)<r\right\}$.
Let $\chi (c)$ be a nondecreasing function in $C^{\infty}({\bf R}
)$ such that
$0\leq\chi (c)\leq 1$, $\chi (c)=1$ for $c\geq\frac {2r}3$, $\chi
(c)=0$ for $c\leq\frac r3$.
We can extend $l$ to a Lipschitz continuous vector field on $\overline {
U(\bar {D})}$
by setting, for $x\in$$\overline {U(\bar {D})}-\bar {D}$,
\[\begin{array}{cc}
l(x)=(1-\chi (d(x,\partial D)))\,l(\pi_{\nu}(x)).\end{array}
\]
We can also extend $\sigma$ and $b$ to Lipschitz continuous
functions on $\overline {U(\bar {D})}$ by setting, for $x\in$$\overline {
U(\bar {D})}-\bar {D}$,
\[\begin{array}{cc}
\sigma (x)=(1-\chi (d(x,\partial D)))\,\sigma (\pi_{\nu}(x)),\\
b(x)=(1-\chi (d(x,\partial D)))\,b(\pi_{\nu}(x)).\end{array}
\]
Clearly, both the martingale problem for $A$, with domain
$C^2(\overline {U(\bar {D})})$, and the martingale problem for $B$, with the
same domain, have a solution for every initial condition
$\delta_x$, $x\in\overline {U(\bar {D})}$. Since every $f\in C^2(\bar
D)$
can be extended to a function $\tilde {f}\in C^2(\overline {U(\bar {
D})})$ and
\[Af=\left.\left(A\tilde f\right)\right|_{\bar {D}},\quad Bf=\left
.\left(B\tilde f\right)\right|_{\bar {D}},\]
Condition (i) in Proposition \ref{suffHc} is verified.

Next, consider the function $\varphi$ defined as
\[\varphi (x)\left\{\begin{array}{ll}
=0,&\mbox{\rm for }x\in\bar {D},\\
=\exp\{\frac {-1}{d(x,\partial D)}\},&\mbox{\rm for }x\in\overline {
U(\bar {D})}-\bar {D},\end{array}
\right.\]
where $U(\bar {D})$ is as above. Since $\partial D$ is of class $
C^3$,
$\varphi\in C^2(\overline {U(\bar {D})})$. Moreover
\[\nabla\varphi (x)=-\,|\nabla\varphi (x)|\,\nu (\pi_{\nu}(x)),\quad\mbox{\rm for }
x\in\overline {U(\bar {D})}-\bar {D}.\]
Therefore $\varphi$ satisfies Condition (ii) in Proposition \ref{suffHc}.

Finally, in order to verify iii) of Proposition
\ref{suffHc}, we just need to modify slightly the proof of
Lemma 3.1 in \cite{PK05}. Suppose first that $\partial_0D$ is
connected. Let $\tilde{\psi}$ be the function in $(\ref{drdir3})$. Since
$l(x)$ is never tangent to $\partial_0D$, it must hold $\nabla\tilde{
\psi }(x)l(x)\neq 0$
for each $x\in\partial_0D$,
and hence, possibly replacing $\psi$ by $-\psi$, we can assume
that
\begin{equation}\tilde{\psi }(x)=0,\quad\nabla\tilde{\psi }(x)l(x
)>0,\quad\quad\forall\,x\in\partial_0D.\label{skD}\end{equation}
Let $U(\partial_0D)$ be a neighborhood of $\partial_0D$ such that
$\inf_{\overline {U(\partial_0D)}}\nabla\tilde{\psi }(x)l(x)>0$, and for each $
n\in {\bf N}$, set
\[\begin{array}{c}
\partial_0^nD=\Big\{x\in\partial D\cap\overline {U(\partial_0D)}:\,\,
|\tilde{\psi }(x)|<\frac 1{2n}\Big\},\\
\tilde {C}_n=\frac 1{\inf_{\partial_0^nD}\nabla\tilde{\psi }(x)l(x)},\\
C_n=\frac {\tilde {C}_n\sup_{\partial D}|\nabla\tilde{\psi }(x)l(x)|+1}{\inf_{
\partial D-\partial_0^nD}\nabla\psi (x)l(x)}.\end{array}
\]
Let $\chi_n$ be a function in $C^{\infty}({\bf R})$ such that $\chi_
n(c)=nc$
for $|c|\leq\frac 1{2n}$, $\chi_n(c)=-1$ for $c\leq -\frac 1n$,
$\chi_n(c)=1$ for $c\geq\frac 1n$, $0\leq\chi_n'(c)\leq n$ for every $
c\in {\bf R}$, and define
\[\varphi_n(x)=\chi_n(C_n\psi (x))+\tilde {C}_n\chi_n(\tilde{\psi }
(x)).\]
Then $|\varphi_n(x)|$ is bounded by $1+\frac 1{\inf_{\partial_0D}
\nabla\tilde{\psi }(x)l(x)}$ and we
have, for $x\in\partial_0^nD$,
\[\nabla\varphi_n(x)l(x)=n\Big[C_n\nabla\psi (x)l(x)+\tilde {C}_n\nabla\tilde{\psi }
(x)l(x)\Big]\geq n,\]
and for $x\in\partial D-\partial_0^nD$,
\[\nabla\varphi_n(x)l(x)=nC_n\nabla\psi (x)l(x)+\tilde {C}_n\chi_n'(\tilde{
\psi }(x))\nabla\tilde{\psi }(x)l(x)\geq n.\]

If $\partial_0D$ is not connected, there is a function $\tilde{\psi}^
k$
satisfying $(\ref{skD})$ for each connected component
$\partial_0^kD$. Let $U^k(\partial_0^kD)$ be neighborhoods such that
$\inf_{\overline {U^k(\partial_0^kD)}}\nabla\tilde{\psi}^k(x)l(x)>0$. We can assume, without loss of generality, that
$\overline {U^k(\partial_0^kD)}\subseteq V^k(\partial_0^kD)$, where $\overline {
V^k(\partial_0^kD)}$ are
pairwise disjoint and $\tilde{\psi}^k$ vanishes outside $V^k(\partial_
0^kD)$.
Then, defining $\partial_0^{k,n}D$ and $\tilde {C}_n^k$ as above,
\[C_n^k=\frac {\tilde {C}_n^k\sup_{\partial D}|\nabla\tilde{\psi}^k(x)
l(x)|+1}{\inf_{\partial D-\cup_k\partial_0^{k,n}D}\nabla\psi (x)l(x)},\]
and $\varphi_n^k$ as above, $\varphi_n(x)=\sum_k\varphi_n^k(x)$ verifies
iii) of Proposition \ref{suffHc}.
\end{proof}.

Theorem 2.6 of \cite{PK05} gives the comparison
principle for a class of linear and nonlinear equations
that includes, in particular, the partial differential
equation with boundary conditions
\begin{equation}\begin{array}{ll}
\lambda u(x)-Au(x)=h(x),&\mbox{\rm in }D,\\
-Bu(x)=0,&\mbox{\rm on }\partial D,\end{array}
\label{drpde}\end{equation}
where $h$ is a Lipschitz continuous function, and $A$, $B$ are
given by (\ref{drop}), (\ref{drbop}) and verify, in
addition to the the assumptions formulated at the
beginning of this section, the following local condition on
$\partial_0D$.

\begin{condition}\label{PK}\hfill

For every $x_0\in\partial_0D$, let $\phi$ be a $C^2$ diffeomorphism from
the closure of a
suitable neighborhood $V$ of the origin into the closure of a suitable
neighborhood of $x_0$, $U(x_0)$, such that $\phi (0)=x_0$ and the
$d\mbox{\rm th}$ column of $J\phi (z)$, $J_d\phi (z)$, satisfies
\begin{equation}J_d\phi (z)=-l(\phi (z)),\quad\forall z\in\phi^{-
1}\left(\overline {\partial D\cap U(x_0)}\right).\label{drchc}\end{equation}
Let $\tilde {A}$,
\[\tilde {A}f(z)=\frac 12tr\left(D^2f(z)\tilde\sigma (z)\tilde\sigma^
T(z)\right)+\nabla f(z)\tilde {b}(z),\]
be the operator such that
\[\tilde {A}(f\circ\phi )(z)=Af(\phi (z)),\quad\forall z\in\phi^{
-1}\left(\overline {D\cap U(x_0)}\right).\]
Assume
\begin{itemize}

\item[a)] $\tilde {b}^i$, $i=1,...,d-1$, is a function of the first $
d-1$
coordinates $(z^1,...,z^{d-1})$ only, and $\tilde {b}_d$ is a function of $
z_d$
only.

\item[b)] $\tilde{\sigma}^{ij}$, $i=1,...,d-1$, $j=1,...,d$ is a function of the
first $d-1$ coordinates $(z^1,...,z^{d-1})$ only.
\end{itemize}
\end{condition}

\begin{remark}\label{drnorm}
For every $x_0\in\partial_0D$, some coordinate of $l(x_0)$, say the
$d\mbox{\rm th}$ coordinate, must be nonzero. Then in
(\ref{drchc}) we can choose $U(x_0)$ such that in $\overline {
U(x_0)}$
$l^d(x)\neq 0$ and we can replace $l(x)$ by $l(x)/|l^d(x)|$,
since this normalization does not change the boundary
condition of (\ref{drpde}) in $\overline {D\cap U(x_0)}$ (i.e. any
viscosity sub/supersolution of (\ref{drpde}) in $\overline {D\cap
U(x_0)}$
is a viscosity sub/supersolution of (\ref{drpde}) in $\overline {
D\cap U(x_0)}$ with
the normalized vector field and conversely).

Moreover, since (\ref{drchc}) must be verified only in
$\phi^{-1}\left(\overline {\partial D\cap U(x_0)}\right)$, in the construction of $
\phi$ we can use any
$C^2$ vector field $\bar {l}$ that agrees with $l$, or the above
normalization of $l,$ on $\overline {\partial D\cap U(x_0)}$.
\end{remark}\medskip

Therefore, whenever Condition \ref{PK} is satisfied
Theorem \ref{cmgpb} applies and there exists one and only
one diffusion process reflecting in $D$ according to the
degenerate oblique direction of reflection $l$.

The following is a concrete example
where Condition \ref{PK} is satisfied.

\begin{example}
{\rm Let
\[D=B_1(0)\subseteq {\bf R}^2.\]
and suppose the direction of reflection $l$ satisfies
(\ref{drdir1}) with the strict inequality at
every $x\in\partial D$ except at $x_0=(1,0)$, where
\[l(1,0)=\left[\begin{array}{c}
0\\
-1\end{array}
\right].\]
Of course, in a neighborhood of $x_0=(1,0)$ we can always
assume that $l$ depends only on the second coordinate $x_2$.
In addition, by Remark \ref{drnorm}, we can suppose
\[l_2(x)=-1.\]
Consider
\[\sigma (x)=\sigma =\left[\begin{array}{cc}
1&0\\
0&0\end{array}
\right].\]
and a drift $b$ that, in a neighborhood of $x_0=(1,0)$,
depends only on the second coordinate, i.e.
\[b(x)=b(x_2).\]
Assume that, in a neighborhood of $x_0=(1,0)$, the
direction of reflection $l$ is parallel to $b$.
Then we can find a nonlinear change of coordinates
$\phi$ such that Condition \ref{PK} is verified, namely
\[\begin{array}{c}
\phi_2(z)=z_2\\
\phi_1(z)=-\int_0^{z_2}l_1(\zeta_2)d\zeta_2+z_1+1,\end{array}
\]
which yields
\[\tilde{\sigma }(z)=\sigma ,\quad\tilde {b}_1(z)=0,\quad\tilde {
b}_2(z)=b_2(z_2).\]

}
\end{example}

\subsection{Jump diffusions with degenerate diffusion
matrix}\label{jd}.

Consider the operator
\[Af=Lf+Jf\]
\begin{equation}Lf(x)=\frac 12tr\left(a(x)D^2f(x)\right)+\nabla f
(x)b(x)\label{JD-op}\end{equation}
\[Jf(x)=\int_{{\mathbb R}^{d'}-\{0\}}\left[f(x+\eta (x,z))-f(x)-\nabla
f(x)\eta (x,z)I_{|z|<1}\right]m(dz),\]
where $\nabla is$ viewed as a row vector.
\noindent
Assume:

\begin{condition}\label{JD-ex}\hfill

\begin{itemize}

\item[a)] $a=\sigma\sigma^T$, $\sigma$ and $b$ are continuous.

\item[b)] $\eta (\cdot ,z)$ is continuous for every $z$, $\eta (x,
\cdot )$ is
Borel measurable for every $x$,

\noindent
$\sup_{|z|<1}|\eta (x,z)|<+\infty$ for
every $x$ and
\[|\eta (x,z)|I_{|z|<1}\leq\rho (z)(1+|x|)I_{|z|<1},\]
for some positive, measurable function $\rho$ such that
$\lim_{|z|\rightarrow 0}\rho (z)=0$.

\item[c)] $m$ is a Borel measure such that
\[\int_{{\mathbb R}^{d'}-\{0\}}\left[\rho (z)^2I_{|z|<1}+I_{|z|\geq
1}\right]m(dz)<+\infty .\]
\end{itemize}

\end{condition}

\noindent
Then, with ${\cal D}(A)=\{f+c:f\in C^2_c({\mathbb R}^d),c\in {\mathbb R}
\}$,
$A\subset C_b(E)\times C_b(E)$ and $A$ satisfies Condition \ref{H}.

A comparison principle for bounded subsolutions and
supersolutions of the equation $(\ref{reso})$
when $A$ is given by $(\ref{JD-op})$ is proven in
\cite{JK06}, as a special case of a more general result,
under the following assumptions:

\begin{condition}\label{JD-comp}\hfill

\begin{itemize}
\item[a)] $\sigma$ and $b$ are Lipschitz continuous.

\item[b)]
\[|\eta (x,z)-\eta (y,z)|I_{|z|<1}\leq\rho (z)|x-y|I_{|z|<1}.\]
\item[c)] $h$ is uniformly continuous.
\end{itemize}

\end{condition}

Then, under the above assumptions, our result of Theorem
\ref{mgpb} applies and uniqueness of the solution of the
martingale problem for $A$ is granted.