text:
\subsection{Framework}
Let $I$ be an interval of $\rr$ so that its interior $I^0=(x_0, y_0)$ where $ -\infty \le x_0 < y_0 \le +\infty$. Consider the diffusion generator on $I$ :
$$
\LL  = a(x) \frac{d^2}{d x^2}+ b(x) \frac{d}{dx}$$
with the Neumann boundary condition at $\partial I =\{x_0,y_0\}\cap \rr$, where the coefficients $a,b:I \to \rr$ satisfy  :

{\it
{\bf (H1)} $ 0 < a \in C^1(I)$ and $b$ is Borel measurable and locally bounded on $I$.}

We can write $\LL$ in Feller's form in terms of the scale function $s$ and the speed function $m$ :
$$
\LL =\frac{d}{dm} \frac{d}{ds}
$$
 where $s,m$ are  determined by their derivatives given by
$$
s^\prime(x)=\exp\left( -\int_c^x \frac{b(u)}{a(u)} du\right) \mbox{ and } m^\prime(x) = \frac{1}{a(x)s^\prime(x)}
$$
where $c$ is some fixed point in $I$.

Let $C_0^\infty (I)$ be the space of infinitely differentiable real functions $f$ on $I$ with compact support and $C_{0,N}^\infty(I)$ be the space of all functions $f$ in $C_0^\infty (I)$ such that $ f^\prime |_{\partial I} =0$ (i.e. satisfying the Neumann boundary condition).
Let $(X_t)$ be the diffusion process on the interval $I$ generated by $\LL$ with initial value $x$.

We will assume :

{\it
{\bf (H2)} The diffusion process is non-explosive or equivalently ( \cite{IM})  :
$$
\begin{aligned}
&\int_c^{y_0} s^\prime(x)\left( \int_c^x m^\prime(y) dy\right)dx= +\infty \quad &\mbox{if } y_0 \notin I;\\
&\int_{x_0}^{c} s^\prime(x) \left( \int_x^c m^\prime(y)\ dy\right)dx= +\infty \quad &\mbox{if } x_0 \notin I.
\end{aligned}
$$

{\bf (H3)}  The speed measure  $dm=m^\prime (x)dx$ is finite, i.e. $m(I):=\int_I m'(x)dx<+\infty$.

{\bf (H4)} The generator $\LL$ defined on $ C_{0,N}^\infty(I)$ is essentially self-adjoint on $L^2(I,dm)$ or equivalently (\cite{DJ,EB}) :
$$
s\notin L^2((x_0,c],dm) \mbox{ if } x_0 \notin I;\quad s\notin L^2([c,y_0),dm) \mbox{ if } y_0 \notin I.
$$
}

Throughout this paper we assume that {\bf (H1)}-{ \bf (H4)} are satisfied. In this case, $\mu(dx):= \frac{m^\prime(x)}{m(I)} dx$ is the unique invariant probability measure of the diffusion $(X_t)$. Let $(P_t)$ be the transition semigroup of $(X_t)$, $\LL_2$ be the generator of $(P_t)$ on $L^2(I,\mu)$ with domain $\dd(\LL_2)$, which is an extension of $(\LL, C_{0,N}^\infty(I))$.

\subsection{Exponential convergence in $L^1$-Wasserstein metric}
Given an absolutely continuous and increasing function $\rho: I \to \rr$, let $d_{\rho}(x,y) =|\rho(x)-\rho(y)|$ be the metric on $I$ associated with $\rho$.
Define the Lipschitzian norm of $f: I \to \rr$ with respect to (w.r.t. in short) $d_{\rho}$ as
$$\|f\|_{Lip(\rho)}:=\sup_{x,y \in I, x \neq y} \frac{|f(x)-f(y)|}{|\rho(x)-\rho(y)|}.$$

For any two probability measures $\mu$, $\nu$ on $I$, the Wasserstein distance between $\mu$ and $\nu$ w.r.t. a given metric $d(x,y)$ on $I$ is defined by
$$
W_{1,d} (\mu,\nu)= \inf_{\pi} \iint_{I^2} d(x,y)\pi(dx,dy)
$$
where $\pi $ runs over all couplings of $\mu$ and $\nu$, i.e. all probability measures $\pi$ on $I^2$ with the first and second marginal distributions $\mu$ and $\nu$, respectively. We say that $(P_t)$ is exponential convergent in $W_{1,d_\rho}$ if there exist some constants $K \ge 1$ and $\delta>0$ such that
\beq\label{exp-C}
W_{1,d_\rho} (P_t(x,\cdot),P_t(y,\cdot) ) \le d_\rho(x,y) K e^{-\delta t}, \ \forall x,y\in I,\ t\ge 0.
\nneq
By Kantorovich duality relation, this is equivalent to
\beq\label{exp-C2}
\| P_t\|_{Lip(\rho)}:=\sup_{f:\|f\|_{Lip(\rho)}\le 1}\|P_t f\|_{Lip(\rho)} \le Ke^{-\delta t}, \ t\ge 0.
\nneq

In this paper, we are interested in the exponential convergence of $(P_t)$ in $W_{1,d_\rho}$.  When $\rho\in L^1(I,\mu)$, consider the Banach space
$$C_{Lip(\rho),0}:=\{f: I \to \rr: \|f\|_{Lip(\rho)} < +\infty \mbox{ and } \mu(f)=0\}$$
equipped with the Lipschitzian norm $\|\cdot\|_{Lip(\rho)}$. We have immediately

\bprop
If $(P_t)$ is exponential convergent in $W_{1,d_\rho}$, i.e.  \eqref{exp-C} holds and $\rho\in L^2(I, \mu)$, then the Poisson operator $(-\LL_2)^{-1} =\int_0^{+\infty} P_t dt : C_{Lip(\rho),0}\to C_{Lip(\rho),0}$ is bounded, and
$$
\|(-\LL_2)^{-1} \|_{Lip(\rho)}\le \int_0^\infty \|P_t\|_{Lip(\rho)}dt \le \frac{K}{\delta}<+\infty.
$$
\nprop

Recall the following result in Djellout and the third named author \cite{DW}.
\bthm
Assume {\bf (H1)}-{\bf (H4)} and let $\rho$ be a function on $I$ such that $\rho \in C^1(I) \cap L^2(I,\mu)$, $\rho^\prime >0$ everywhere,
the Poisson operator
$$
(-\LL_2)^{-1}: C_{Lip(\rho),0}(I) \to C_{Lip(\rho),0}(I)
$$
is well defined and bounded if and only if (iff in short)
\beq\label{cW}
c_W(\rho):= \sup_{x \in I } \frac{s^\prime(x)}{\rho^\prime (x)}\int_x^{y_0} [\rho(y) -\mu(\rho)] m^\prime(y) dy < +\infty.
\nneq
In that case, its norm is given by
$$
\|(-\LL_2)^{-1}\|_{Lip(\rho)} =c_W(\rho).
$$
\nthm

In other words a necessary condition for the exponential convergence of $(P_t)$ in $W_{1,d_\rho}$ is $c_W(\rho)<+\infty$.
The objective of this paper is to show that this necessary condition becomes sufficient in a quite general situation and some sharp estimates of the involved constants $\delta, K$ could be obtained.

\subsection{Some comments on the literature}
For the one-dimensional diffusions, the Poincar\'e inequality (equivalent to the exponential convergence in $L^2(\mu)$) and the log-Sobolev inequality (equivalent to the exponential convergence in entropy) can be characterized by means of the generalized Hardy inequality (\cite{BaRo, Chen05}). For the characterization of  Latala-Oleszkiewicz inequality (\cite{LO}) between Poincar\'e and log-Sobolev, see Barthe and Robertho  \cite{BaRo}. See \cite{CW16} for the characterization of  the Sobolev inequality.

For the transport and isoperimetric inequalities, see Djellout and Wu \cite{DW} for sharp estimates of constants.

On the other hand the exponential convergence in the $L^p$-Wasserstein metric is also a very active subject of study. For the early study on this question, the reader is referred to  \cite{Chen92, CW95, CW97, Chen05}.
Renesse and Sturm \cite{RS05} showed that for $\LL=\Delta -\nabla V\cdot\nabla$ on a Riemannian manifold, if the exact exponential convergence (\ref{exp-C})  in $W_{1,d}$ (with $K=1, d$ being the Riemannian metric) holds, then the Bakry-Emery's curvature  must be bounded from below by $\delta$. If one allowed $K>1$  in
 (\ref{exp-C}), Eberle \cite{EB05} found sharp sufficient conditions for  high dimensional  interacting diffusions, by means of reflected coupling.    The reader is referred to this last paper for an overview of literature.

The general idea of proving the exponential convergence in  $W_{1,d}$  for a high dimenioanl diffusion is to use the reflection coupling $(X_t,Y_t)$ and then to compare $d(X_t,Y_t)$ with some one-dimensional diffusion. But curiously a general study on the exponential convergence in $W_{1,d}$ of one-dimensional diffusion is absent : that is the objective of our study.

\subsection{Organization}
Our paper is organized as follows. In the next section, we state the main result and present several corollaries. In Section 3, we provide several examples to illustrate our theorem. In Section 4, we prove the main result in the compact case. The proof of the general case is given in Section 5.

\section{Main Result}
For a function $f$ on $I$, the $\sup$-norm of $f$ is defined by $\|f\|_\infty =\sup_{x \in I} |f(x)|$. Recall that the conditions {\bf (H1)-(H4)} are always assumed.

\subsection{Main theorem}
Our main result in this paper is :
\bthm \label{mr}
Assume $\rho \in C^1(I) \cap L^2(I,\mu)$ and $ \rho^\prime (x) >0,\ \forall x \in I $. Suppose that $c_W(\rho) <+\infty $.
\benu[(a)]
\item If moreover

{\bf (C)} there exists some positive $C^2$-function $\varphi$ such that :
\benu[(1)]
\item for all $x \in I$, $\rho^\prime (x) \le \varphi (x) \le C \rho^\prime (x)$ for some constant $C \ge 1$ ;
\item for some constant $M \ge 0$,
\beq\label{test f}
(a\varphi^\prime +b \varphi)^\prime \le M \rho^\prime
\nneq
in the distribution sense on $I^0 = (x_0,y_0)$,
\nenu
then for any constant $\alpha \in (0,\frac{1}{M})$,
\beq\label{r1}
W_{1,d_\rho} (P_t(x,\cdot),P_t(y,\cdot) ) \le d_\rho(x,y) K e^{-\delta t},\ \forall x,y\in I, \ t\ge 0,
\nneq
where
\beq
\begin{aligned}
&\delta=\delta(\alpha)=\frac{1-\alpha M}{C \alpha +c_W(\rho)}, \ K=K(\alpha)=\frac{1}{\alpha}\left\| \frac{u+\alpha \varphi }{\rho^\prime}\right\|_\infty, \\
&u(x)=s^\prime(x)\int_x^{y_0} [\rho(y) -\mu(\rho)] m^\prime(y) dy.
\end{aligned}
\nneq
\item
For the metric $d_{\tilde{\rho}}$ where
$
\tilde{\rho}^\prime (x) =u(x)
$ given above,
we have
\beq\label{r2}
W_{1,d_{\tilde{\rho}} }(P_t(x,\cdot),P_t(y,\cdot) ) \le \exp\left(-\frac{t}{c_W(\rho)}\right)d_{\tilde{\rho}}(x,y),\ \forall x,y\in I, \ t\ge 0.
\nneq
\nenu
\nthm

\brmk{\rm As $\LL$ is symmetric on $L^2(I,\mu)$,
by the result in \cite{Wu04}, for any metric $d_\rho$, if \eqref{exp-C} holds, the spectral gap $\lambda_1$ of $\LL$ in $L^2(I,\mu)$ satisfies
$$
\delta \le \lambda_1.
$$
In other words, the best possible exponential convergence rate $\delta$ in any Wasserstein metric $W_{1,d\rho}$ of $(P_t)$ is the spectral gap $\lambda_1$.  Chen's variational formula for the spectral gap $\lambda_1$ says (\cite{Chen99})
$$
\lambda_1=\sup_\rho \frac 1 {c_W(\rho)}.
$$
As $\delta=\delta(\alpha)\to\frac 1 {c_W(\rho)}$ when $\alpha\to 0+$, our estimate on the exponential convergence rate $\delta$ is sharp.

If moreover $\lambda_1>0$ is associated with an eigenfunction $\rho$, which can be chosen to be increasing (\cite{Chen92}), then one verifies easily that
$u(x)=\frac 1{\lambda_1}\rho'(x)$. Thus $c_W(\rho)= \frac 1{\lambda_1}$ and we get from Part (b) above that
$$
W_{1,d_{\rho} }(P_t(x,\cdot),P_t(y,\cdot) ) \le \exp\left(-\lambda_1 t\right)d_{\rho}(x,y),\ \forall x,y\in I, \ t\ge 0.
$$
}
\nrmk

\subsection{Corollaries and examples}

We present now a corollary for illustrating the extra condition {\bf (C)} in Part (a) of Theorem \ref{mr}.

\bcor\label{cor22} Assume that $I=\rr $, $a\equiv 1$ and $\rho(x)=x$, i.e. $d_{\rho} $ is the Euclidean metric on $\rr$.  Suppose that
$$
c_W(\rho)=\sup_{x \in \rr} u(x)< +\infty
$$
where $u(x):=e^{V(x)} \int_x^{+\infty} [y -\mu(y)] e^{-V(y)} dy$ and $V$ is some primitive of $-b(x)$. If
the Bakry-Emery's curvature is bounded from below, i.e.
$V^{\prime \prime}\ge - M$ for some constant $M \ge 0$,  then for any $\alpha\in (0,1/M)$,
\beq\label{rr1}
W_{1,d_\rho} (P_t(x,\cdot),P_t(y,\cdot) ) \le |x-y| K e^{-\delta t},\ \forall x,y\in \rr, \ t\ge 0,
\nneq
where
$$
\begin{aligned}
\delta&=\delta(\alpha)=\frac{1-\alpha M}{ \alpha +c_W(\rho)}, \ K=K(\alpha)=\frac{\left\| u+\alpha \right\|_\infty}{\alpha}. \\
\end{aligned}
$$
\ncor
\bprf
It is obtained by Part (a) of Theorem \ref{mr} with $\varphi(x)=1$ simply.
\nprf

\bcor\label{cor23} Let $I=\rr $ and $\rho(x)=x$ (corresponding to the Euclidean metric). Assume $\int_\rr x^2 d\mu(x)<+\infty$. If
 there exist some constants $C_1 \ge 1$ and $L>|\mu(\rho)|$ large enough such that  $B(x)=\int_0^x\frac{b(y)}{a(y)} dy\to -\infty$ as $x\to\pm\infty$ and
\beq\label{bx1}
b(x)\ne0, \ \frac{x-\mu(\rho)}{-b(x)}\le C_1   ,\quad \mbox{if } |x|>L,
\nneq
then $c_W(\rho)<+\infty$ and $(P_t)$ is exponential convergent in $W_{1,d_{\tilde{\rho}}}$, where
$$ \tilde{\rho}'(x)=u(x)=e^{-B(x)}\int_x^{+\infty} (y-\mu(\rho)) \frac 1{a(y)}e^{B(y)} dy .$$
Furthermore, if either  $b'\le M$  for some non-negative constant $M$, or
\beq\label{bx2}
\frac{ x-\mu(\rho)}{-b(x)} \ge \frac{1}{C_1},\quad \mbox{if } |x|>L,
\nneq
then $(P_t)$ is exponential convergent in $W_{1,d_{\rho}}$ with $\rho(x)= x$.
\ncor

\bprf
For $x>L$, by the mean value theorem of Cauchy, there exists some $ \xi \in (x,+\infty)$ such that
$$
\begin{aligned}
u(x)= \frac{-e^{B(\xi)}(\xi-\mu(\rho))}{e^{ B(\xi)} b(\xi)} = \frac{\xi-\mu(\rho)}{-b(\xi)} \le C_1.
\end{aligned}
$$
For $x <-L$, by the similar proof, we also have $u(x) \le C_1$. Then $c_W(\rho) < +\infty$.  Hence by Part (b) of Theorem \ref{mr}, the exponential convergence \eqref{r2} in the metric $W_{1,d_{\tilde\rho}}$ holds.

Furthermore,  if moreover $b'\le M$ for some non-negative constant $M$, applying Theorem \ref{mr}(a) for
$\varphi=1$, we get the exponential convergence in $W_{1,d_{\rho}}$. In the case where (\ref{bx2}) is satisfied,
we have $u(x) \in [1/C_2,C_2]$ for all $x\in\rr$,  for some constant $C_2\ge C_1$ by the mean value theorem of Cauchy. Then
$$\frac 1{C_2}d_{\tilde\rho}\le d_{\rho}\le C_2 d_{\tilde\rho} .$$ Thus we get the exponential convergence in  $W_{1,d_{\rho}}$-metric :
$$
W_{1,d_{\rho}}(P_t(x,\cdot), P_t(y,\cdot))\le C_2^2 \exp\left(-\frac{t}{c_W(\rho)}\right)|x-y|, \ \forall x,y\in\rr, t\ge0.
$$
\nprf

A curious point in Corollary \ref{cor23} above is that our sufficient condition above for the exponential convergence in  $W_1$-metric associated with the Euclidean distance, depends very few upon the volatility coefficient $a(x)$ (except the conditions {\bf (H1)-(H4)}).

We give an example of Corollary \ref{cor23} :
\bexa{\rm
Let $I=\rr $, $\rho(x)=x$, i.e. $d_{\rho} $ is the Euclidean metric on $\rr$.
Consider the generator
$$
\LL  = \frac{d^2 }{dx^2}+b(x)\frac{d}{dx},
$$
where $ b(x) = -V'(x)= -x(2+\sin x)$.

For this example $\mu(dx) =\frac{1}{C} e^{-V(x)}dx$, where $V(x)=x^2-x\cos x + \sin x$, $C$ is the normalization constant . It is easy to see that the hypotheses {\bf (H1)-(H4)} are all satisfied, and
$ \mu (\rho^2)= \frac{1}{C} \int_\rr x^2 \exp(-x^2+x\cos x - \sin x)dx < +\infty $. Notice that the Bakry-Emery's curvature
$$
V^{\prime\prime}(x)= 2+\sin x + x\cos x
$$
is not bounded from below.

Since
$$
B(x)=-V(x)=-x^2+x\cos x - \sin x \to -\infty \mbox{ as } x \to \pm \infty
$$
and when $L>|\mu (\rho)|$ large enough,
$$
\frac{1}{4} \le \frac{ x-\mu(\rho)}{-b(x)}=\frac{ x-\mu(\rho)}{x(2+\sin x)} \le 4, \mbox{ if }|x| \ge L ,
$$
by Corollary \ref{cor23}, $(P_t)$ generated by $\LL$ is exponential convergent in $W_{1,d_{\rho}}$.

}

\nexa

We present now another example to illustrate the extra condition {\bf (C)}.

\bexa
{\rm
Let $I=\rr, a(x)=1,b(x)=-V'(x)$ where $V(x)$ is an even function such that $V'(x)=x^n(n+1+\sin x) $ for
$x \ge 0$, here $n \ge 2 $.

Since
$$
V^{\prime \prime} (x) = n x^{n -1} (n +1 +\sin x)+ x^n \cos x
$$
is unbounded on $[0,+\infty)$, the Bakry-Emery's curvature is unbounded from below.

For $\rho(x) = x$, we see that,
$$
\lim_{x \to +\infty} u(x)=\lim_{x \to +\infty}e^{V(x)} \int_x^{+\infty} (y-\mu(\rho))e^{-V(y)}dy=\lim_{x \to +\infty}\frac{x -\mu(\rho)}{x^n( n +1 + \sin x)}=0.
$$
Similarly $\lim_{x \to -\infty} u(x)=0$, then
$$
c_W(\rho) =\sup_{x \in \rr} u(x) < +\infty.
$$
We choose the following $\varphi$ :
$$
\varphi(x)=
\begin{cases}
\frac{1}{n+1-\sin x}, \quad &\mbox{if } x \le - L;\\
f(x), \quad &\mbox{if } x \in [-L,L];\\
\frac{1}{n+1+\sin x}, \quad &\mbox{if } x \ge L
\end{cases}
$$
where $L>|\mu(\rho)|$ is a positive constant and $ f(x)$ is $C^2[-L,L]$-function such that $\varphi(x)$ is a $C^2$-function on $\rr$.

For $x \ge L$,
$$
\begin{aligned}
(a\varphi' +b\varphi)'&= \varphi^{\prime \prime} -V'\varphi'-V^{\prime \prime} \varphi\\
&=\frac{(n+1) \sin x + \cos^2 x +1}{ (n +1 +\sin x)^3}-n x^{n -1}\\
&\le M_1
\end{aligned}
$$
where $M_1$ is a positive constant.

For $x<-L$, similarly we have $(a\varphi' +b\varphi)' \le M_2 $ for some constant $M_2>0$.

For $ x \in [-L,L]$, by continuity of $(a\varphi' +b\varphi)'$, $(a\varphi' +b\varphi)' \le M_3$ for some constant $M_3>0$.

Then $(a\varphi' +b\varphi)' \le \max \{M_1,M_2,M_3\}$ on $\rr$.
By Theorem \ref{mr}(a),  the exponential convergence \eqref{r1} w.r.t. $d_\rho$ holds.
}
\nexa

\subsection{Main idea}
We explain now the main idea in Theorem \ref{mr}. The crucial point is that in the actual one-dimensional case, we would have formally
the following commutation relation
\beq\label{a11}
(P_tf)' = P_t^D f'
\nneq
where $P_t^D$ is the semigroup generated by $\LL^D g = (a g' +bg)'$. Then by the Kantorovitch duality relation,
the exponential convergence of $(P_t)$ in $W_{1,d_\rho}$ is equivalent to that of $P_t^D$ to $0$ in the Banach space $b_V\BB$ of all Borel-measurable functions $g$ such that
the norm  $\|g\|_V:=\sup_{x\in I} \frac{|g(x)|}{V(x)}<+\infty$, where $V(x)=\rho'(x)$. An easy sufficient condition
to this last exponential convergence is $\LL^ D U \le -\delta U$ for some positive constant $C$ and some function $U$ such that $\rho'\le U\le C\rho'$.

To see the meaning of the necessary condition $c_W(\rho)<+\infty$, notice that $u$ is a particular solution of $\LL^D u =-\rho'$, $u$ should  be bounded by $C \rho'$ if $P_t^D$ converges exponentially to $0$ in the norm $\|\cdot\|_{\rho'}$. Moreover our extra condition {\bf (C)} says simply that there is some function $\varphi\ge \vep \rho'$ in the domain of $\LL^D$ in $b_V\BB$.

However the formal approach above is very difficult to be realized in the general case. It can be realized rigorously in the compact
 case ($I=[x_0,y_0]$) when $a,b$ are quite regular : see \S 4. The general case can be treated by approximation, as the involved constants $\delta, K$ have explicit expressions.

\section{ Several Examples}
Recall that for $\rho(x)=\int_c^x\frac{1}{\sqrt{a(y)}} dy$ the associated metric $d_{\rho}(x,y)=|\rho(x)-\rho(y)|$ is the intrinsic metric $d_X$ of the diffusion $(X_t)$. In this section, we present  several examples and study  their exponential convergence in the $W_1$-metric associated with  the intrinsic distance $d_X$.

\bexa[Ornstein-Uhlenbeck generator]{\rm
Let $I=\rr$. Consider the Ornstein-Uhlenbeck generator
$$
\LL  = \frac{d^2 }{dx^2}- x \frac{d}{dx},
$$
then $\mu(dx)=\frac{1}{2\pi}e^{-\frac{x^2}{2}}dx$ is the invariant probability measure of $\LL$.
For $\rho(x)=x$, we have $u(x)=1$, thus $c_W(\rho) =1$. Since
$ \tilde{\rho}(x)=x=\rho(x)$, by Part (b) of Theorem \ref{mr},  $ \|P_t\|_{Lip(\rho)} \le e^{-t}$ (well known). We know that the spectral gap $\lambda_1$ of $\LL$ is $1$, which shows that Theorem \ref{mr} is sharp.
\
}
\nexa

\bexa \label{exa22}
{\rm
Let $I=\rr $, $b(x)=-V'(x)$ where $V(x)=C_1|x|^r$,  $C_1$ and $r$ are positive constants, and $a\in C^1(\rr)$ which is bounded i.e. $\frac 1C_2 \le  a(x)\le C_2$ for some constant $C_2\ge1$.
Then $\mu (dx) = \frac 1{Za(x)}e^{B(x)} dx$ ($Z$ being the normalization constant), $B(x)=\int_0^x \frac {b(y)}{a(y)} dy$. For $ \rho (x)=x$, we see that $\frac 1{\sqrt {C_2}}d_\rho\le d_X\le \sqrt{C_2}d_\rho$ and
$$
c_W(\rho)= \sup_{x\in \rr}e^{-B(x)}\int_x^{+\infty} \frac1{a(y)}e^{B(y)} (y-\mu(\rho)) dy.
$$
}
\nexa
For this example it is well known that the spectral gap exists (i.e. $\lambda_1>0$) iff $r\ge 1$. About the exponential convergence in $W_1$ associated to the Euclidean metric,  we have the following result :
\bcor In the above Example \ref{exa22}, for $\rho (x) =x$, $(P_t)$ is exponential convergent in $W_{1,d_\rho}$ iff $r\ge 2$.
\ncor
\bprf At first we can check easily that all assumptions {\bf (H1)-(H4)} hold.
For the necessity, we only need to prove that $c_W(\rho) = +\infty$ when $r <2$. By the L'Hospital criterion,
$$
\begin{aligned}
\lim_{x \to +\infty} e^{-B(x)}\int_x^{+\infty} \frac1{a(y)} e^{B(y)} (y-\mu(\rho)) dy
&=\lim_{x \to +\infty}
 \frac{ -e^{B(x)} (x-\mu(\rho))}{-C_1r |x|^{r-1}e^{B(x)}}\\
&=\lim_{x \to +\infty}\frac{1}{ C_1 r (r-1) |x|^{r-2}}= +\infty.
\end{aligned}
$$
By the definition of $c_W(\rho)$, we have $c_W (\rho)=+\infty$.

For the sufficiency, by the proof above we see that $c_W(\rho) < +\infty$ when $r \ge 2 $. Letting $\varphi=1$, we have
$$
(a\varphi' +b\varphi)'=b^\prime=-C_1 r(r-1)|x|^{r-2}\le 0,
$$
then the condition in Part (a) of Theorem \ref{mr} is satisfied with $C=1$ and $M=0$. Hence by Theorem \ref{mr}(a),  $(P_t)$ is exponential convergent in $W_{1,d_\rho}$ i.e. \eqref{r1} holds with $\delta =\frac{1}{c_W(\rho)+\alpha}$ and $K=\frac{c_W(\rho)+\alpha}{\alpha}$, where $\alpha >0$ is arbitrary. By the equivalence between $d_\rho$ and $d_X$, the exponential convergence \eqref{r1} w.r.t. $d_X$ holds with the same $\delta$ above and $K$ replace by $C_2K$.
\nprf

\brmk
{\rm  What happens if $r\in [1,2)$ ? In fact if one takes an odd and increasing  $C^\infty$ -function $\rho$ such that $\rho(x)=e^{c x^{2-r}}$ for $x\ge 1$ where $c>0$ is arbitrary if $r\in (1,2)$  and $0<c<C_1/2$ if $r=1$.

By Cauchy's mean value theorem,
$$
\frac 1C \le \frac{u(x)}{\rho'(x)} \le C
$$
for some constant $C>1$. Therefore by Theorem \ref{mr}(b), when $r\in [1,2)$, $P_t$ is exponentially convergent in $W_{1,d_\rho}$ for $\rho(x)$ given above.
}
\nrmk

\bexa[Jacobi diffusion]{\rm
Let $I=(0,1)$, $a(x)=x(1-x)$ and $b(x)=-x+\frac{1}{2}$, then $\mu(x) =\frac{1}{ \pi\sqrt{x(1-x)}}$. For
$\rho =\frac{\pi}{2}+ \arcsin(2x-1)$, $d_\rho=d_X$ (the intrinsic metric), we  see that
$$
c_W(\rho) =\sup_{x \in (0,1)} \left( \frac{\pi^2}{8} -\frac{1}{2} \arcsin^2 (2x-1) \right) =\frac{\pi^2}{8},
$$
and by calculus we have
$$
(a \rho^{\prime \prime} +b\rho^\prime)^\prime = 0.
$$
We can choose $\varphi=\rho^\prime$, then
$$
(a \varphi^{ \prime} +b\varphi)^\prime \le 0.
$$
Hence the exponential convergence \eqref{r1} w.r.t. $d_X$ holds with
$$
\begin{aligned}
\delta=\frac{8}{ 8\alpha +\pi^2} \mbox{ and }
K=\frac{\pi^2+8\alpha}{8\alpha}
\end{aligned}
$$
where $\alpha >0$ is arbitrary.
}
\nexa

\bexa[Continuous branching process]
{\rm
Let $I=(0,+\infty)$, $a(x)=2 x$ and $b(x)=-2x+1$, then $\mu(x) =\frac{1}{\sqrt{\pi}} \frac{e^{-x}}{\sqrt{x}}$. This process arises as diffusion limit of discrete space branching process. For $ \rho =\sqrt{2x}$, $d_\rho=d_X$, we see that
$$
c_W(\rho) =\sup_{x \in \rr^+} \left( 1-\frac{e^x}{\sqrt{\pi}} \int_x^\infty \frac{e^{-y}}{\sqrt{y}} dy\right) =1,
$$
and by calculus we have
\beq\label{branching1}
(a \rho^{\prime \prime} +b\rho^\prime)^\prime = -\rho^\prime.
\nneq
We note that
$$
\tilde {\rho}^\prime = \rho^\prime,
$$
by Part (b) of Theorem \ref{mr} the exponential convergence
\eqref{r1} w.r.t. $d_X$ holds with
$$
\begin{aligned}
\delta=1 \mbox{ and }
K=1.
\end{aligned}
$$
Moreover from (\ref{branching1}) we see that the increasing function $\rho-\mu(\rho)$ is an eigenfunction of $-\LL$, and its associated eigenvalue  $1$ must be the spectral gap $\lambda_1$. This example shows again that Theorem \ref{mr} is sharp.
}
\nexa

\bexa[Reflected Bessel diffusion process]{\rm
Let $I=(0,1]$, $a(x)=\frac{1}{2}$ and $b(x)=\frac{\beta-1}{2x}$ where $\beta > 1$ is a constant (the dimension), then $\mu(dx) =\beta x^{\beta-1}dx$. For $ \rho =x$, $\sqrt{2}d_\rho= d_X$, we see that
$$
c_W(\rho) =\sup_{x \in (0,1]} \frac{2}{\beta+1} (x-x^2) =\frac{1}{2(\beta+1)} < \frac{1}{4}.
$$
Choose $\varphi=\rho^\prime=1$, then
$$
(a \varphi^{ \prime} +b\varphi)^\prime =b^\prime =- \frac{\beta-1}{2x^2}  \le 0.
$$
Hence by Theorem \ref{mr}, the exponential convergence \eqref{r1} w.r.t. $d_X$ holds with
$$
\begin{aligned}
\delta=\frac{2(\beta+1)}{ 2\alpha (\beta+1)+1} \mbox{ and }
K=\frac{1}{2}\left(\frac{1}{2\alpha(\beta+1)}+1\right)
\end{aligned}
$$
where $\alpha >0$ is arbitrary.}
\nexa

\section{Compact case}
In this section, we prove the main result Theorem \ref{mr} in the compact case i.e. $I=[x_0,y_0]$ is a bounded closed interval of $\rr$.
\subsection{$C^\infty $-case on compact interval}
Assume $a$, $b \in C^\infty [x_0,y_0]$. For all $f \in C_N^\infty [x_0,y_0]$ where $C_N^\infty [x_0,y_0]= \{f\in C^\infty[x_0,y_0] \mbox{ and } f^\prime |_{\{x_0,y_0\}} =0\}$, $ u(t,x) :=P_t f (x) \in C^\infty (\rr^+ \times [x_0,y_0])$ and it satisfies
$$
\begin{cases}
\partial_t u=\LL u=au^{\prime \prime}+b u^\prime;\\
\partial_x u(t,x_0)=\partial_x u(t,y_0) =0 .
\end{cases}
$$
Let $v(t,x)=\partial_x u(t,x)$, it satisfies
\beq\label{PDE}
\begin{cases}
\partial_t v =(av^\prime+bv)^\prime:=\LL^D v;\\
v(t,x_0)= v(t,y_0) =0,
\end{cases}
\nneq
in other words $v(t,x)$ satisfies the Dirichlet boundary condition. For all $g \in C_D^\infty[x_0,y_0] := \{ g\in C^\infty[x_0,y_0] :  g |_{\{x_0,y_0\}} =0\}$, we define $\LL^D$ as follows :
$$
\LL^D g= (ag^\prime+bg)^\prime.
$$
$(\LL^D, C_D^\infty [x_0,y_0]) $ generates a unique $C_0$-semigroup $(P_t^D)$ on the Banach space $C_D[x_0,y_0]$ of continuous functions on $I=[x_0,y_0]$ satisfying $f(x_0)=f(y_0)=0$. Now we show that :
\blem\label{lem41}
For all $f \in C_D[x_0,y_0]$,
\beq\label{semigp}
P_t^D f(x)=\ee^x 1_{[t < \tau_{\partial I}]} f(X_t)D_t
\nneq
where
\benu
\item
$(X_t)$ is the diffusion generated by $\LL$ with the Neumann boundary condition :
\beq\label{Refdiffu}
dX_t=\sqrt{2 a } (X_t) dB_t+b(X_t) dt +dL_t^{x_0}-dL_t^{y_0}
\nneq
where $(B_t)$ is the Brownian motion, $L_t^{x_0}$ (resp. $ L_t^{y_0}$) is the local time of $(X_t)$ at $x_0$ (resp. $y_0$) ;
\item
$$
\tau_{\partial I} =\inf\{ t\ge 0: X_t \in \{x_0,y_0\} \}
$$
is the first hitting time to the boundary ;
\item
for $t < \tau_{\partial I}$,
$$
 D_t =\exp \left(\int_0^t \frac{a^\prime}{\sqrt{2 a}}(X_s) d B_s + \int_0^t b^\prime (X_s) ds - \frac{1}{4}\int_0^t \frac{{a^\prime}^2}{a}(X_s)ds \right).
$$
\nenu
\nlem
\bprf
For $g\in C_D[x_0,y_0]$, let
$$
\tilde{P}_t^D g(x)=\ee^x 1_{[t < \tau_{\partial I}]} g(X_t)D_t=\ee^x  g(X_t^D)D_t
$$
where $ (X^D_t)$ satisfying $X^D_t=X_t$ for $t<\tau_{\partial I}$ and $X^D_t=X_{\tau_{\partial I} }$ for $t\ge\tau_{\partial I}$, is the killed process at $\partial I$.
First, by the Markov property of $(X_t)$, it is easy to see that $(\tilde{P}_t^D)$ is a $C_0$-semigroup on $C_D[x_0,y_0]$.
Then we prove \eqref{semigp}. When $ 0 \le t <\tau_{\partial I}$, for any $g \in C_D^\infty [x_0,y_0]$,
by It\^{o} formula,
$$
\begin{aligned}
d(D_t g(X_t))&=D_t d g(X_t)+g(X_t) d D_t+ d\langle D, g(X)\rangle_t\\
&=D_t \big(a(X_t) g^{\prime \prime}(X_t)dt + b(X_t) g^\prime (X_t)dt\big)\\
&\quad + D_t\big(g(X_t) b^\prime(X_t)dt+ a^\prime (X_t) g^\prime (X_t)dt\big)+ dM_t\\
&=D_t \LL^D g(X_t) dt + dM_t,
\end{aligned}
$$
where
$$
M_t=\int_0^t D_s\left(\sqrt{2a}(X_s)g^\prime(X_s) +\frac{a^\prime}{\sqrt{2a}}(X_s) g(X_s)\right)dB_s
$$ is a martingale. And for $t\ge \tau_{\partial I}$, $D_t g(X^D_t)=0$. Then for $x \in (x_0,y_0)$,
$$
\begin{aligned}
&\quad \tilde{P}^D_tg(x)-g(x)\\
&=\ee^x(D_t g(X_t^D)) -g(x)\\
&=\ee^x \left(\int_0^t 1_{[s < \tau_{\partial I}]} D_s\LL^D g(X_s)ds + M_{t\wedge \tau_{\partial I}}\right)\\
&=\ee^x \int_0^t 1_{[s < \tau_{\partial I}]} D_s \LL^D g(X_s)ds\\
&=\int_0^t \tilde{P}_s^D \LL^D g(x)ds.
\end{aligned}
$$
Then by the uniqueness, $\tilde{P}^D_tg(x)=P^D_tg(x)$.
\nprf

For every $f\in C_N^\infty[x_0,y_0]$, since $ v(t,x)=\partial_x u(t,x)=(P_t f(x))^\prime$ satisfies the partial differential equation \eqref{PDE} with the initial value condition $ v(0,x)=f^\prime (x)$, it is given by
$$
v(t,x) = P_t^D f^\prime(x).
$$
Hence
\beq\label{PPD}
(P_t f)^\prime = P_t^D f^\prime, \mbox{ for all } t\ge0, \ f\in C_N^\infty[x_0,y_0].
\nneq

Recall that for an everywhere positive function $V$, the $V$-norm of $f$ is defined by $\|f\|_{V}=\sup_{x\in I} \frac{|f(x)|}{V(x)}$.

\blem\label{lem13}
If there exists some positive constant $\delta$ and a $C^2$-function $V: [x_0,y_0] \to \rr^+$ such that $C_1 \le \frac{V}{\rho^\prime} \le C_2$ ($C_1,C_2$ are positive constants) and
\beq\label{LDV}
(a V^\prime +bV)^\prime \le -\delta V \mbox{ on } (x_0,y_0),
\nneq
then
\beq\label{Pd}
\|P_t^D \|_V \le e^{-\delta t}, \forall t\ge 0.
\nneq
Moreover, we have
\beq\label{P}
\|P_t \|_{Lip(\rho)} \le K e^{-\delta t}, \forall t \ge 0
\nneq
where $K=\|\frac{V}{\rho^\prime}\|_{\infty}\| \frac{\rho^\prime}{V} \|_{\infty}$.
\nlem

\bprf The only delicate point is that the test function $V$ does not necessarily belong to the domain of definition of the generator $\LL^D$, in fact $V(x_0), V(y_0)$ may be different of $0$.
At first, we prove \eqref{Pd}. For this purpose it is enough to show $P_t^D V(x) \le e^{-\delta t} V(x)$. Let
$$
Y_t= e^{\delta t} D_t V(X_t) I_{[t<\tau_{\partial I }]},
$$
we only need to prove that $(Y_t)$ is a supermartingale. For $ t<\tau_{\partial I }$, by It\^{o} formula and \eqref{LDV},
$$
d Y_t = \delta Y_t dt+ e^{\delta t}D_t (a V^\prime +bV)^\prime(X_t)dt+dM_t \le dM_t
$$
where $M_t$ is a local martingale up to $\tau_{\partial I}$. Then $(Y_t)$ is a supermartingale by Fatou's lemma. Thus by Lemma \ref{lem41},
$$
e^{\delta t} P_t^D V(x)= \ee^x Y_t \le Y_0=V(x)
$$
where \eqref{Pd} follows.

Now we prove \eqref{P},  which is equivalent to
\beq\label{Pbis}
\|P_tf\|_{Lip(\rho)} \le K e^{-\delta t} \|f\|_{Lip(\rho)},\ \forall f\in C^\infty[x_0,y_0].
\nneq
At first  for $f\in C_N^\infty[x_0,y_0]$,
 we have by \eqref{Pd},

$$
\begin{aligned}
\|P_t f\|_{Lip(\rho)}&=\sup_{x\in I} \left|\frac{(P_t f)^\prime}{\rho^\prime} \right|\\
&\le \| P_t^D f^\prime\|_ V \left\| \frac{V}{\rho^\prime} \right\|_\infty\\
&\le e^{-\delta t} \| f^\prime\|_V \left\|\frac{V}{\rho^\prime} \right\|_\infty\\
&\le \left\|\frac{V}{\rho^\prime}\right\|_\infty \left\|\frac{\rho^\prime }{V}\right\|_\infty e^{-\delta t}\|f\|_{Lip(\rho)}\\
&=:K e^{-\delta t}\|f\|_{Lip(\rho)}.
\end{aligned}
$$
Now for every $f\in C^\infty[x_0,y_0]$ and $n \in \nn^+$, let $f_n=f(x_0)+\int_{x_0}^x \psi_n(y) f'(y)dy$ where $\psi_n(x)=1$ for $x\in[x_0+1/n,y_0-1/n]$ and $\psi_n$ is $C^\infty$-smooth, valued in $[0,1]$, with compact support contained in $(x_0,y_0)$. For each $n \in \nn^+$,  as $f_n$ satisfies the Neumann boundary condition and $\|f_n\|_{Lip(\rho)}\le \|f\|_{Lip(\rho)}$, we have
$$|P_tf_n(x)-P_tf_n(y)|\le K e^{-\delta t} \|f\|_{Lip(\rho)}|\rho(x)-\rho(y)|$$
where \eqref{Pbis} follows by letting $n\to\infty$.
\nprf

Now we turn to :

\bprf[Proof of Theorem \ref{mr} in the compact and $C^\infty$-case]
{\bf Part (a).} Since $\left\| \frac{\rho^\prime }{ u+\alpha \varphi}\right\|_\infty\le \frac{1}{\alpha}$,  it is enough to show \eqref{r1} holds with $\delta=\frac{1-\alpha M}{C \alpha +c_W(\rho)}$, $
K=\left\| \frac{u+\alpha \varphi }{\rho^\prime}\right\|_\infty \left\| \frac{\rho^\prime }{ u+\alpha \varphi}\right\|_\infty$.

By Lemma \ref{lem13}, we only need to find a $C^2$-function $V: [x_0,y_0] \to \rr^+$ such that
$C_1\le\frac{V}{\rho^\prime} \le C_2$ for some positive constants $C_1,C_2$ and
\beq\label{LV}
(aV^\prime +b V)^\prime \le -\delta V \mbox{ on } (x_0,y_0).
\nneq
Consider the following equation
\beq\label{Lu}
\LL^D u= -\rho^\prime \mbox{ on } (x_0,y_0),\ u(x_0)=u(y_0)=0.
\nneq
It is explicitly solvable and the unique solution satisfying the Dirichlet boundary condition is
$$
u(x)=s^\prime(x)\left(\int_x^{y_0}  (\rho(y)-\mu(\rho)) m^\prime(y) dy \right).
$$
It is easy to see $u(x)>0$ for all $x \in (x_0,y_0)$.

Since $ (a \varphi^\prime+b \varphi)^\prime \le M \rho^\prime$, for any constant $\alpha \in (0,\frac{1}{M})$, we can choose
$$
V=\alpha \varphi + u.
$$
First notice that $\sup_{x \in I} \frac{u(x)}{\rho^\prime(x)}=c_W(\rho) < +\infty$, we have
$$
0<\alpha \le \frac{V(x)}{\rho^\prime(x)} \le C \alpha+c_W(\rho).
$$
Moreover
$$
\begin{aligned}
(a V^\prime+b V)^\prime \le \alpha M \rho^\prime -\rho^\prime \le -\frac{1-\alpha M}{C \alpha +c_W (\rho)} V =:-\delta V,
\end{aligned}
$$
then by Lemma \ref{lem13}, we get the desired result.

{\bf Part (b).} It is enough to show \eqref{exp-C2} holds with $\delta=\frac{1}{c_W(\rho)}$, $
K=1$.
Consider the unique solution $u$ of \eqref{Lu} with Dirichlet boundary condition.
Since $ c_W(\rho) <+\infty$, then
$$
(a u^\prime+b u^\prime)^\prime= -\rho^\prime\le - \frac{1}{c_W(\rho)} u.
$$
Since $\tilde{\rho}^\prime =u$, by  Lemma \ref{lem13} with $V=u$, we have
\beq\label{P2}
\|P_t^D\|_u \le e^{-\delta t}.
\nneq
Then
$$
\begin{aligned}
\|P_t \|_{Lip(\tilde{\rho})} =\|P_t^D  \|_u \le e^{-\delta t},
\end{aligned}
$$
which is \eqref{r2}. The proof is finished.
\nprf

\subsection{General case on compact interval}
In this subsection, we prove the main result when $a \in C^1[x_0,y_0]$ and $b$ is Borel measurable and bounded on $[x_0,y_0]$.
\bprf[Proof of Theorem \ref{mr}]
{\bf Part (a).}

{\bf First reduction :} $a,b\in C^1[x_0,y_0]$. Taking $a_\e, b_\e \in C^\infty[x_0,y_0]$ such that $a_\e(x) \to a(x)$, $a_\e^\prime(x) \to a^\prime (x)$, $b_\e(x) \to b(x)$, $b_\e^\prime(x) \to b^\prime (x)$ uniformly over $[x_0,y_0]$ as $\e \to 0$ (i.e. $a_\e\to a$ and $b_\e\to b$ in $C^1[x_0,y_0]$).
Since
$$
u_\e(x)=\exp\left( -\int_c^x \frac{b_\e(y)}{a_\e(y)} dy\right)\int_x^{y_0} \frac{1}{a_\e(y)}[\rho(y) -\mu_\e(\rho)] \exp\left( \int_c^y \frac{b_\e(z)}{a_\e(z)} dz\right) dy
$$
($c$ is a fixed constant in $[x_0,y_0]$),
we see that
$u_\e(x) \to u(x)$ uniformly over $[x_0,y_0]$ as $\e \to 0$, then
$ c_W(\rho, \e)$ defined in \eqref{cW} associated with $(a_\e,b_\e)$ converges to $c_W(\rho)$ associated with $(a,b)$ as $\e \to 0$.

Moreover the condition in \eqref{test f} is satisfied for $(a_\e,b_\e)$ with some constant $M_\e>M$ and $ M_\e \to M$ as $\e \to 0$.
By the result of the $C^\infty$-case in Part (a) of Theorem \ref{mr}, the semigroup $(P_t^\e)$ generated by $\LL^\e =a_\e \frac{d^2}{dx^2}+b_\e \frac{d}{dx}$ satisfies
\beq\label{exp-Ce}
\|P_t^\e \|_{Lip(\rho)} \le K_\e e^{-\delta_\e t}
\nneq
where
$
\delta_\e=\delta_\e(\alpha)=\frac{1-\alpha M_\e}{C \alpha +c_W(\rho,\e)} \mbox{ and }
K_\e=K_\e(\alpha)=\left\| \frac{u_\e+\alpha \varphi }{\rho^\prime}\right\|_\infty \left\| \frac{\rho^\prime }{ u_\e+\alpha \varphi}\right\|_\infty,
$
where $\alpha \in (0,\frac{1}{M_\e})$.
Obviously, $\delta_\e \to \delta$ and $ K_\e \to  K$ as $\e \to 0$.

Now we only need to show for all $f \in  C^\infty[x_0,y_0]$,
\beq\label{Pcov}
P_t^\e f(x) \to P_t f(x), \ \forall x \in [x_0,y_0] \mbox{ as }\e \to 0.
\nneq
But the process $(X_t^\e)$ generated by $\LL^\e$ with the refelection Neumann boundary condition converges in law to $(X_t)$ (well-known in the theory of SDE, \cite{IW}).   Then the convergence above holds.

{\bf Second Reduction :} $a\in C^1[x_0,y_0]$, $b$ is Borel measurable and bounded on $[x_0,y_0]$.
It is enough to show that the semigroup $P_t^{(\delta)}$ generated by $\LL$ on $[x_0+\delta, y_0-\delta]$ with the Neumann boundary conditon at $\{ x_0+\delta, y_0 -\delta\}$ satisfies the conclusion, for all $\delta>0$ small enough, since $P_t^{(\delta)}f\to P_tf$ and all involved constants associated with  $P_t^{(\delta)}$ converge to those related to $P_t$ . Thus working on $P_t^{(\delta)}$ if necessary, we may assume without loss of generality that $a(x)$, $b(x)$ are defined on
$[x_0 -\delta ,y_0+\delta]$ and the condition (\ref{test f}) in Part (a) holds on $(x_0-\delta,y_0+\delta)$.

For any $0<\e<\delta$, taking $ p_\e (x) =\frac{1}{\e } p(\frac{x}{\e})$ where $p$ is a positive  $C^\infty$-function on $\rr$ such that its support is contained in $[-1,1]$ and $\int_\rr p(x)dx=1$. Let
$$
b_\e :=\frac{(b \varphi)*p_\e}{\varphi} \in C^1[x_0-\e,y_0+\e]
$$
where $f*g(x)=\int_\rr f(x-y)g(y)dy$ is the convolution.
By condition \eqref{test f}, we have
$$
( a \varphi^\prime + b \varphi )^\prime * p_\e \le M \rho^\prime *p_\e \le (M+\eta_1(\e)) \rho^\prime\ \text{ on }\ [x_0,y_0]
$$
where $\eta_1(\vep)$ is some positive constant tending to $0$ as $\vep\to 0$.
Since  on $I=[x_0,y_0]$,
$$
\begin{aligned}
&(a \varphi^\prime)^\prime * p_\e \ge (a \varphi^\prime)^\prime -\eta_2(\e) \rho^\prime \mbox{ and }(b \varphi)^\prime *p_\e =(b_\e \varphi)^\prime
\end{aligned}
$$
where $\eta_2(\vep)$ is some positive constant tending to $0$ as $\vep\to 0$.
Then
$$
(a \varphi^\prime)^\prime+ (b_\e \varphi)^\prime\le (a \varphi^\prime)^\prime*p_\e +\eta_2(\e) \rho^\prime + (b \varphi)^\prime *p_\e\le (M+\eta(\e)) \rho^\prime
$$
on $[x_0,y_0]$ where $\eta(\e):=\eta_1(\e)+\eta_2(\e)$.
By the result in the first reduction, for the semigroup $(P_t^\e)$ generated by $\LL^\e f = a f^{\prime\prime}+b_\e f'$ with the Neumann boundary condition,  for any $\alpha\in (0,1/M)$ fixed, we have for any $\e>0$ sufficiently small,
 $\alpha<1/(M+\eta(\e))$,
\beq\label{exp-Ce2}
\|P_t^\e \|_{Lip(\rho)} \le K_\e e^{-\delta_\e t}
\nneq
where
$
\delta_\e(\alpha)=\frac{1-\alpha M_\e}{C \alpha +c_W(\rho,\e)} \mbox{ and }
K_\e(\alpha)=\left\| \frac{u_\e+\alpha \varphi }{\rho^\prime}\right\|_\infty \left\| \frac{\rho^\prime }{ u_\e+\alpha \varphi}\right\|_\infty.
$
$$
u_\e(x)=\exp\left( -\int_c^x \frac{b_\e(y)}{a (y)} dy\right)\int_x^{y_0} \frac{1}{a(y)}[\rho(y) -\mu_\e(\rho)] \exp\left( \int_c^y \frac{b_\e(z)}{a(z)} dz\right) dy
$$
($c$ is a fixed constant in $[x_0,y_0]$).

Since $b_\e(x) \to b(x)$ in measure $dx$ on $[x_0,y_0]$ as $\e \to 0$ and $b_\e\ (\e >0)$, $b$ are uniformly bounded,
we see that
$u_\e(x) \to u(x)$ as $\e \to 0$, then
$ c_W(\rho, \e)$ defined in \eqref{cW} associated with $(a,b_\e)$ converges to $c_W(\rho)$ associated with $(a,b)$ as $\e \to 0$. Moreover,
$ \delta_\e  \to \delta $, $K_\e\to K$ as $\e \to 0$.

Now we only need to show for all $f \in  C_N^\infty[x_0,y_0]$,
$$P_t^\e f(x) \to P_t f(x), \ \forall x \in [x_0,y_0] \mbox{ as }\e \to 0.$$
For all $0<\e<\delta $, consider the diffusion $X_t^\e$ on $[x_0,y_0]$ satisfying
$$
d X_t^\e =\sqrt{2 a}(X_t^\e) d B_t+ b_\e(X_t^\e)dt +d L_t^{x_0,\e}- d L_t^{y_0,\e}
$$
with initial value $x$ and the diffusion $X_t^0$ on $[x_0,y_0]$ satisfying
$$
d X_t^0=\sqrt{2 a}(X_t^0) d B_t+ d L_t^{x_0}- d L_t^{y_0}
$$
with the same initial value $x$.
Then for any $f \in C[x_0,y_0]$, we have by Girsanov's formula,
$$
\begin{aligned}
P_t^\e f(x) &= \ee^x\left[ f(X_t^0) \exp\left(  \int_0^t \frac{b_\e}{ \sqrt{2a}}(X_s^0) d B_s- \frac{1}{4} \int_0^t \frac{b_\e^2}{a} (X_s^0) ds\right)\right]\\
&=\ee^x\left[ f(X_t^0) \exp(  Y_t^\e- \frac 12 \<Y^\e\>_t)\right]
\end{aligned}
$$
where $Y_t^\e :=  \int_0^t \frac{b_\e}{ \sqrt{2a}}(X_s^0) d B_s$. Set
$Y_t=\int_0^t \frac{b}{ \sqrt{2a}}(X_s^0) d B_s$.
Since $\{ \<Y^\e\>_t, \e>0 \}$ is uniformly bounded, the family of  exponential martingales
$\{\exp( Y_t^\e-  \frac 12 \<Y^\e\>_t), \e >0 \}$ is uniformly integrable,  then it is enough to show
$$
\lim_{\e \to 0}\ee^x  \<Y^\e- Y\>_t =  \lim_{\e \to 0}\ee^x \int_0^t \frac{(b_\e-b)^2}{2a}(X_s^0)ds =0.
$$
This follows from the dominated convergence theorem,
since $ \{b_\e, \e >0\}$, $b$ are uniformly bounded, $b_\e \to  b$ in measure $dx$,  and the transition probability
$\pp_x(X^0_s\in dx)$ is absolutely continuous for every $s>0$ by the ellipticity assumption {\bf (H1)}.

Taking $\e \to 0$ in both sides of \eqref{exp-Ce2}, we get the desired result.

{\bf Part (b).} The proof is similar to Part (a), we omit the proof.

\nprf

\section{ General non-compact case}
In this section, we consider the general case : $I=(x_0,y_0)$, $I=[x_0,y_0)$ or $I=(x_0,y_0]$. We only prove Theorem
\ref{mr} in the open interval case.

\bprf[Proof of Theorem \ref{mr}] {\bf Part (a).} For $I=(x_0,y_0)$, $\forall n \in \nn^*$, let $ I_n=[x_n, y_n]$, where $x_n<y_n$, $x_n\downarrow x_0$ and $ y_n \uparrow y_0$ as $ n \to +\infty$ so that the point $c$ in the definition of $s'(x)$ and $m'(x)$ belongs to $(x_1,y_1)$. Let $P_t^{(n)}$ be the semigroup generated by
$\LL_n f(x):=\LL f(x)$ for $x\in I_n$ and $f\in C_N^\infty[x_n,y_n]$ (i.e. satisfying the Neunmann  boundary condition at $x_n,y_n$). Let $X_t^{(n)}$ be the diffusion on the interval $I_n$ generated by $\LL_n $ with the Neunmann boundary condition :
$$
dX_t^{(n)}=\sqrt{2 a } (X_t^{(n)}) dB_t+b(X_t^{(n)}) dt +dL_t^{x_n}-dL_t^{y_n}.
$$

By the result in Section 4 for compact case, we have for all $n>0$ and any arbitrary $\alpha \in (0,\frac{1}{M})$,
\beq\label{exp-C3}
\|P_t^{(n)}\|_{Lip(\rho)} \le K_n e^{-\delta_n t}
\nneq
where
$$\begin{aligned}
\delta_n &= \frac{1-\alpha M}{ C \alpha+c_W(\rho,n)},\ c_W(\rho,n)=\sup_{x\in [x_n,y_n]}\frac{u_n(x)}{\rho'(x)},\\
K_n &=\frac{1}{\alpha}\sup_{x \in I_n}\left| \frac{u_n(x)+\alpha \varphi(x)}{ \rho'(x)} \right|,\\
u_n(x)&=s'(x)\int_x^{y_n} [\rho(y) -\mu_n(\rho)] m'(y) dy, \ x\in I_n
\end{aligned}
$$
with $\mu_n(dx)=\frac{1_{I_n}(x) m'(x)}{m(I_n)} dx$. We see that $u_n(x)\to u(x)$ uniformly as $ n \to +\infty$ over the compact interval $I_N$ for any $N\ge 1$ fixed.

Let $\xi_0\in I$ such that $\rho(\xi_0)=\mu(\rho)$. Fix some $N$ and $\delta>0$ so that $[\xi_0-\delta,\xi_0+\delta]\subset I_N$.  Notice that for $n\ge N$,  if $x\in I_n$ and $x\ge \xi_0+\delta$,
$$\aligned
u_n(x) &\le s'(x)\int_x^{y_0} [\rho(y) -\mu_n(\rho)] m'(y) dy\\
&=u(x) +  [\mu(\rho)-\mu_n(\rho)]s'(x)m([x,y_0]).
\endaligned
$$
For the last term above, if $x\ge \xi_0+\delta$

$$\aligned
s'(x)m([x,y_0))&\le \frac 1{\rho(\xi_0+\delta)-\mu(\rho)} s'(x) \int_{x}^{y_0} [\rho(y)-\mu(\rho)] m'(y)dy\\
&=\frac {u(x)}{\rho(\xi_0+\delta)-\mu(\rho)}.
\endaligned$$

Therefore there is some constant $A_1>0$ such that
$s'(x)m([x,y_0))\le A_1 u(x)$ for all $x\ge \xi_0$.

Now if $x\le \xi_0-\delta$,
$$\aligned
u_n(x) &= s'(x)\int_{x_n}^{x} [-\rho(y) +\mu_n(\rho)] m'(y) dy\\
&\le u(x) +[\mu_n(\rho)-\mu(\rho)] s'(x)m((x_0,x])
\endaligned$$
and similarly we have $s'(x)m((x_0,x])\le A_2 u(x)$ for all $x\le \xi_0$ and some constant $A_2>0$.  Summarizing the previous discussion we get
$$
u_n(x)\le u(x) + |\mu(\rho)-\mu_n(\rho)| Au(x), \ x\in I_n
$$
where $A =\max\{A_1, A_2\}$,
which implies that
$$\limsup_{n\to\infty}c_W(\rho,n)\le c_W(\rho),\ \limsup_{n\to\infty}K_n\le K.$$
(Notice that $\liminf_{n\to\infty}c_W(\rho,n)\ge c_W(\rho),\ \liminf_{n\to\infty}K_n\ge K$ hold always.)

Now for the exponential convergence in Part (a), it remains to show that for any $f \in C_0^\infty (x_0,y_0)$ and $x\in (x_0,y_0)$,
$$
\lim_{n\to \infty } P_t^{(n)}f(x) = P_tf(x).
$$
Denote the first hitting time of $(X_t)$ to the boundary of $I_n$ by
$$
\tau_{\partial I_n}= \inf \{ t\ge 0: X_t \in \{x_n,y_n\} \},
$$
we have
$$
X_t^{(n)}=X_t, \ \forall t \in [0,\tau_{\partial I_n}).
$$
By the non-explosion assumption {\bf (H2)}, we have for any $t\ge0$ and $x\in I$ fixed,
\beq\label{limtau}
\lim_{n \to \infty} \pp_x (\tau_{\partial I_n} >t)=1,
\nneq
then
$$
\begin{aligned}
| P_t^{(n)}f(x)-P_t f(x)|&=|\ee^x f(X_t^{(n)})-\ee^x f(X_t) |\\
&\le 2\|f\|_\infty\pp_x(t \ge \tau_{\partial I_n}) \to 0 \mbox{ as } n \to \infty,
\end{aligned}
$$
by \eqref{limtau}.

\medskip
{\bf Part(b).}  For  any $f \in C_0^\infty (x_0,y_0)$ and $x<y$ in $(x_0,y_0)$, the support of $f$ and $\{x,y\}$ are contained in $I_N$ for some $N\ge1$ large enough. Then if $N\le n\to \infty$, recalling that $\tilde \rho'(x)=u(x)$,
$$\sup_{x\in (x_n,y_n)}\frac{|f'(x)|}{u_n(x)}= \sup_{x\in I_N} \frac{|f'(x)|}{u_n(x)}\to  \sup_{x\in I_N} \frac{|f'(x)|}{u(x)} =\|f\|_{Lip(\tilde \rho)}$$
for $u_n\to u$ uniformly over $I_N$. Since $\displaystyle \lim_{n\to+\infty} c_W(\rho,n)= c_W(\rho)$ as shown in Part (a), we get by Part (b) in the compact case,
$$\aligned
|P_tf(x)-P_tf(y)|&=\lim_{n\to\infty}|P_t^{(n)}f(x)- P_t^{(n)}f(y)|\\
&\le \lim_{n\to\infty} \exp\left(-\frac {t}{c_W(\rho,n)}\right) \int_{x}^y u_n(r) dr  \sup_{z\in (x_n,y_n)}\frac{|f'(z)|}{u_n(z)}\\
&= \exp\left(-\frac {t}{c_W(\rho)}\right)[\tilde\rho(y)-\tilde \rho(x)] \|f\|_{Lip(\tilde \rho)}
\endaligned$$
where the desired result follows by Kantorovitch duality characterization.

\nprf

redpajama_set_name: RedPajamaArXiv


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Home / #IPKat / access to healthcare / Amaka Vanni / Brazil / india / Léon Dijkman / nigeria / Novartis v. Union of India / Patent Games in the Global South / right to health / TRIPs / TWAIL / [Book review] Patent Games in the Global South
[Book review] Patent Games in the Global South
Léon Dijkman Friday, November 20, 2020 - #IPKat, access to healthcare, Amaka Vanni, Brazil, india, Léon Dijkman, nigeria, Novartis v. Union of India, Patent Games in the Global South, right to health, TRIPs, TWAIL
On this Kat's night stand during the long, locked-down nights in Florence was Amaka Vanni's monograph Patent Games in the Global South: Pharmaceutical Patent Law-Making in Brazil, India and Nigeria (Hart 2020). Vanni is not just a legal scholar, but also a documentary filmmaker—and it shows in her writing. The from-the-ground-up account of pharmaceutical patents in Brazil, India and Nigeria she offers is timely and perceptive, and like a good documentary, it manages to be both objective and intimate.
Vanni's work comes at a time when structural racism and inequality are – at long last – being discussed in earnest [see Katpost here]. Feelings of hopelessness and defeat may all too easily take hold of us when considering the gross injustices perpetrated against former colonies and the echoes they carry still today. At the time of writing, atrocities are being committed by the government of Vanni's home country, Nigeria, against peaceful protesters: once again, it seems that the world stands by idly.
Patent Games in the Global South shows that, far from being insulated from these perverse relations of power, intellectual property [IP] laws can be tools for perpetuating colonial dominance at the expense of local populations. The strength of the book is that it tells us there is hope, and Vanni's meticulous research may yet lead the way towards a more equitable system of IP in non-Western jurisdictions.
The book consists of four main parts: a theoretical background and three chapters devoted to Brazil, India and Nigeria. In each of these chapters, Vanni traces the historical roots of the country's patent law, examines how the current-day law impacts the population's access to healthcare and the involvement of civil society and other non-state actors in the pharmaceutical patent system. The overall inquiry is how each country managed to – or missed opportunities to – exploit flexibilities in the TRIPs agreement to safeguard its population's access to heath rights. The book's most interesting insights are based on field work conducted by Vanni in all three countries over the course of two years.
TWAIL: Third-World Approaches to International Law
Vanni emphasizes repeatedly that she identifies with a school of thought known as Third-World Approaches to International Law [TWAIL]. She employs a 'Twailian lens' for "contextualising, historicising, mapping and analysing patent law regime and reforms in the countries under study", in hopes of providing "a starting point from which to begin the process of revolutionary reimagination and reordering" [37].
The book cites a recent chapter by James Thuo Gathii that is helpful in better understanding the TWAIL research project [here]. To be honest, though, I am still not convinced. To be sure, the project's goals are important: exposing implicit biases and hidden power structures is crucial to foster the type of open and inclusive debate that TWAIL scholars strive for. But I am not persuaded that it is a 'methodology' as Vanni would have it, instead of advocacy, legitimate and necessary though it may be.
The distinction matters, because framing TWAIL as a methodology exposes it to the very criticism it levels at international IP norms, i.e. that their ostensible neutrality obfuscates a clear agenda. Why not just come out as a critic of these norms on its own right? Vanni's mastery of the subject is such that no appeals to a broader movement are necessary to make her arguments persuasive.
If anything, applying the same 'Twailian lens' to all case studies exposes the book to a weakness of TWAIL that Vanni herself recognizes, i.e. that "TWAIL overlooks the asymmetries between and within the Third World" [52]. The chapter on Nigeria illustrates the point: Vanni shows that this country's local conditions (history, industry, civil society) differ significantly from those in Brasil and India and that this is reflected in how the Nigerian patent system currently functions. In light of these differences, it seems oddly reductionist to assume that Brasil, India and Nigeria are all allies in a uniform struggle against Western IP norms. Rather, as Vanni's careful case studies show, each country is fighting its own fight, and TWAIL may obscure the specifics of those fights by absorbing them in a single movement.
Pharmaceutical patents in Brasil, India and Nigeria
The heart of the book are three case studies demonstrating the way that TRIPs' flexibilities can be utilized to remove or lower patent-related obstacles to access to healthcare for the local population. All three countries were opposed to inclusion of IP provisions in TRIPs and all share the heritage of colonial patent laws which served not to foster domestic innovation, but to further strengthen the colonial power's grip on the country's economy. However, the patent laws of these countries have since developed differently. The key take-aways are as follows:
Brazil was quick to implement TRIPs-compliant patent rules due to economic pressure [78]. Since then, however, the Brazilian Supreme Federal Tribunal has given teeth to the constitutional right to health in its decision in Dina Rosa Vieira v. Municipality of Porta Alegre (2000). This right is now used as a basis to curtail excessive patent protection, both by courts [86] and in Brazil's compulsory licensing strategy [92].
India's pharmaceutical patent ecosystem is characterized by the strong presence of generic companies. For instance, section 3(d) of the Indian Patents Act raises the bar for follow-on patents on pharmaceuticals in an effort to combat evergreening. Vanni cites the Indian Supreme Court's decision in Novartis v. Union of India (2013) as striking an "intelligent balance" between the interests of its patient population and its nascent originator pharmaceutical industry [137]. For her, India is a prime example of "how civil society groups and generic manufacturers have both used the language of the human right to life and the need for pharmaceutical autonomy in political action to promote public policy change" [154].
Nigeria is described as having been less successful so far in creating "progressive patent legislation" [164]. Vanni recalls that in the 1970's, brand name pharmaceutical companies had set up manufacturing plants in Nigeria to act as a point of distribution for West Africa which, by 1980, employed 10 000 Nigerians [185]. But, after the oil bust of the 1980's, those companies went in rapid decline as Nigerian currency devalued, the country fell into poverty and the operations were no longer sustainable [193]. Crucially, the presence of these pharmaceutical companies had not fostered a transfer of know-how to the local population, so that nothing was left after their departure. Consequently, the pharmaceutical industry in present-day Nigeria is primarily focused on local production of basic drugs, without original research being undertaken. The sadness of the situation is aptly captured by one of Vanni's interviewees, who remarks that "We have nothing to protect, really" [181].
How can these differences be explained? Vanni attributes them primarily to the presence, in the case of Brazil and India, or absence, in the case of Nigeria, of a local industry and concomitant civil society groups that exert influence over local lawmakers to deploy a patent strategy to serve the country's interest. In that respect and several others, "Nigeria has something to learn from these countries" [209].
Perhaps other countries might be a source of inspiration, too. England, in particular, comes to mind. Until 1977, the Patents Act contained a provision that allowed any interested person to apply for a compulsory license to a patent in respect of a substance capable of being used as food or medicine; the license would be granted unless it appeared to the comptroller of patents that there were "good reasons for refusing the application." And recently, English courts have spearheaded limitation of patent exclusivity to safeguard patient interests, at least in cases where patients would immediately be deprived of life-saving drugs [here and here]. Here, too, a truly global exchange could improve patent laws on both sides of the North-South divide.
Patent Games in the Global South is an important book because it provides empirical support to voices questioning the wisdom and equity of imposing Western-style IP regimes in developing countries [the argument is well summarized by Keith Maskus in this chapter, unfortunately not available open acces]. What is more, it analyses what is needed to improve the situation, and gives concrete examples of how TRIPs flexibilities might be operationalized to best serve a country's interests. There is much more I could say about this book, but I hope to have made clear that it is an impressive achievement that no scholar or policy maker in the area of patent law and access to health can afford to overlook.
ISBN: 9781509927395 / 9781509927401 (PDF eBook)
Available from Bloomsbury Professional; list price GBP 65
[Book review] Patent Games in the Global South Reviewed by Léon Dijkman on Friday, November 20, 2020 Rating: 5
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redpajama_set_name: RedPajamaCommonCrawl


Milwaukee Art Center. Personal Selections from the Mrs. Harry Lynde Bradley Collection. New Orleans: Bradley Family Foundation, 1975, color illus. p. 31.
Isaac Delgado Museum of Art, Selections from the Collection of Mrs. Harry Lynde Bradley, New Orleans, Louisiana: Isaac Delgado Museum of Art, 1969, cat. 39.
Milwaukee Art Center. The Collection of Mrs. Harry Lynde Bradley. Milwaukee, Wisconsin: Milwaukee Art Center, 1968, cat. 44, b/w illus. p. 49.
Klee Lui-Meme, Berggruen & Co., Paris, France, 1961.


text:
package net.rkunze.maven.compiler.jsr308javac;

import org.codehaus.plexus.compiler.AbstractCompilerTest;
import org.codehaus.plexus.compiler.CompilerConfiguration;
import org.codehaus.plexus.util.StringUtils;

import java.io.File;
import java.util.ArrayList;
import java.util.Arrays;
import java.util.Collection;
import java.util.Collections;
import java.util.Iterator;
import java.util.LinkedHashMap;
import java.util.List;
import java.util.Map;
import java.util.TreeSet;
import static junit.framework.Assert.assertEquals;
import static org.codehaus.plexus.PlexusTestCase.getBasedir;
import org.codehaus.plexus.compiler.CompilerException;
import org.codehaus.plexus.compiler.CompilerMessage;
import org.codehaus.plexus.util.FileUtils;

/**
 * @author Jason van Zyl
 */
public class JavacJSR308CompilerTest
    extends AbstractCompilerTest
{
    private static final String PS = File.pathSeparator;

    public void setUp()
        throws Exception
    {
        super.setUp();
        setCompilerDebug( true );
        setCompilerDeprecationWarnings( true );

    }

    protected String getRoleHint()
    {
        return "javac+jsr308";
    }

    protected int expectedErrors()
    {

        // javac output changed for misspelled modifiers starting in 1.6...they now generate 2 errors per occurrence, not one.
        if ( "1.5".compareTo( getJavaVersion() ) < 0 )
        {
            return 4;
        }
        else
        {
            return 3;
        }
    }

    protected int expectedWarnings()
    {
        return 2;
    }

    protected Collection<String> expectedOutputFiles()
    {
        return Arrays.asList( new String[]{ "org/codehaus/foo/Deprecation.class", "org/codehaus/foo/ExternalDeps.class",
            "org/codehaus/foo/JSR308Nullness.class", "org/codehaus/foo/Person.class", "org/codehaus/foo/ReservedWord.class" } );
    }

    protected Collection<String> expectedOutputFilesNullnessChecker()
    {
        return Arrays.asList( new String[]{ "org/codehaus/foo/Deprecation.class", "org/codehaus/foo/ExternalDeps.class",
            "org/codehaus/foo/Person.class" } );
    }

    public void internalTest( CompilerConfiguration compilerConfiguration, List<String> expectedArguments ) throws CompilerException
    {
        String[] actualArguments = JavacJSR308Compiler.buildCompilerArguments( compilerConfiguration, new String[0] );

        assertEquals( "The expected and actual argument list sizes differ.", expectedArguments.size(),
                      actualArguments.length );

        for ( int i = 0; i < actualArguments.length; i++ )
        {
            assertEquals( "Unexpected argument", expectedArguments.get( i ), actualArguments[i] );
        }
    }

    public void testBuildCompilerArgs13() throws CompilerException
    {
        List<String> expectedArguments = new ArrayList<String>();

        CompilerConfiguration compilerConfiguration = new CompilerConfiguration();

        compilerConfiguration.setCompilerVersion( "1.3" );

        populateArguments( compilerConfiguration, expectedArguments, true, true );

        internalTest( compilerConfiguration, expectedArguments );
    }

    public void testBuildCompilerArgs14() throws CompilerException
    {
        List<String> expectedArguments = new ArrayList<String>();

        CompilerConfiguration compilerConfiguration = new CompilerConfiguration();

        compilerConfiguration.setCompilerVersion( "1.4" );

        populateArguments( compilerConfiguration, expectedArguments, false, false );

        internalTest( compilerConfiguration, expectedArguments );
    }

    public void testBuildCompilerArgs15() throws CompilerException
    {
        List<String> expectedArguments = new ArrayList<String>();

        CompilerConfiguration compilerConfiguration = new CompilerConfiguration();

        compilerConfiguration.setCompilerVersion( "1.5" );

        populateArguments( compilerConfiguration, expectedArguments, false, false );

        internalTest( compilerConfiguration, expectedArguments );
    }

    public void testBuildCompilerArgsUnspecifiedVersion() throws CompilerException
    {
        List<String> expectedArguments = new ArrayList<String>();

        CompilerConfiguration compilerConfiguration = new CompilerConfiguration();

        populateArguments( compilerConfiguration, expectedArguments, false, false );

        internalTest( compilerConfiguration, expectedArguments );
    }

    public void testBuildCompilerDebugLevel() throws CompilerException
    {
        List<String> expectedArguments = new ArrayList<String>();

        CompilerConfiguration compilerConfiguration = new CompilerConfiguration();

        compilerConfiguration.setDebug( true );

        compilerConfiguration.setDebugLevel( "none" );

        populateArguments( compilerConfiguration, expectedArguments, false, false );

        internalTest( compilerConfiguration, expectedArguments );
    }

    // PLXCOMP-190
    public void testJRuntimeArguments() throws CompilerException
    {
        List<String> expectedArguments = new ArrayList<String>();

        CompilerConfiguration compilerConfiguration = new CompilerConfiguration();

        // outputLocation
        compilerConfiguration.setOutputLocation( "/output" );
        expectedArguments.add( "-d" );
        expectedArguments.add( new File( "/output" ).getAbsolutePath() );

        // classpath (automatically added by JavacJSR308Compiler)
        expectedArguments.add("-Xbootclasspath/p:"
                + ClasspathConfig.getAnnotatedJDK(System.getProperty("java.version")).getAbsolutePath()
                + PS + ClasspathConfig.getCompilerJar().getAbsolutePath() + PS );
        expectedArguments.add( "-classpath" );
        expectedArguments.add( ClasspathConfig.getCheckerJar().getAbsolutePath() + PS );

        // targetVersion
        compilerConfiguration.setTargetVersion( "1.7" );
        expectedArguments.add( "-target" );
        expectedArguments.add( "1.7" );

        // sourceVersion
        compilerConfiguration.setSourceVersion( "1.7" );
        expectedArguments.add( "-source" );
        expectedArguments.add( "1.7" );

        // customCompilerArguments
        Map<String, String> customCompilerArguments = new LinkedHashMap<String, String>();
        customCompilerArguments.put( "-J-Duser.language=en_us", null );
        compilerConfiguration.setCustomCompilerArgumentsAsMap( customCompilerArguments );
        // don't expect this argument!!

        internalTest( compilerConfiguration, expectedArguments );
    }

    /* This test fails on Java 1.4. The multiple parameters of the same source file cause an error, as it is interpreted as a DuplicateClass
     * Setting the size of the array to 3 is fine, but does not exactly test what it is supposed to - disabling the test for now
    public void testCommandLineTooLongWhenForking()
        throws Exception
    {
        JavacJSR308Compiler compiler = (JavacJSR308Compiler) lookup( org.codehaus.plexus.compiler.Compiler.ROLE, getRoleHint() );

        File destDir = new File( "target/test-classes-cmd" );
        destDir.mkdirs();

        // fill the cmd line arguments, 300 is enough to make it break
        String[] args = new String[400];
        args[0] = "-d";
        args[1] = destDir.getAbsolutePath();
        for ( int i = 2; i < args.length; i++ )
        {
            args[i] = "org/codehaus/foo/Person.java";
        }

        CompilerConfiguration config = new CompilerConfiguration();
        config.setWorkingDirectory( new File( getBasedir() + "/src/test-input/src/main" ) );
        config.setFork( true );

        List messages = compiler.compileOutOfProcess( config, "javac", args );

        assertEquals( "There were errors launching the external compiler: " + messages, 0, messages.size() );
    }
    */

    private void populateArguments( CompilerConfiguration compilerConfiguration, List<String> expectedArguments,
                                    boolean suppressSourceVersion, boolean suppressEncoding ) throws CompilerException
    {
        // outputLocation

        compilerConfiguration.setOutputLocation( "/output" );

        expectedArguments.add( "-d" );

        expectedArguments.add( new File( "/output" ).getAbsolutePath() );

        // Bootstrap classpath for the annotated JDK and the JSR308 javac
        expectedArguments.add("-Xbootclasspath/p:"
                + ClasspathConfig.getAnnotatedJDK(System.getProperty("java.version")).getAbsolutePath()
                + PS + ClasspathConfig.getCompilerJar().getAbsolutePath() + PS);

        // classpathEntires

        List<String> classpathEntries = new ArrayList<String>();

        classpathEntries.add( "/myjar1.jar" );

        classpathEntries.add( "/myjar2.jar" );

        compilerConfiguration.setClasspathEntries( classpathEntries );

        expectedArguments.add( "-classpath" );

        expectedArguments.add( ClasspathConfig.getCheckerJar().getAbsolutePath() + PS + "/myjar1.jar" + PS + "/myjar2.jar" + PS );

        // sourceRoots

        List<String> compileSourceRoots = new ArrayList<String>();

        compileSourceRoots.add( "/src/main/one" );

        compileSourceRoots.add( "/src/main/two" );

        compilerConfiguration.setSourceLocations( compileSourceRoots );

        expectedArguments.add( "-sourcepath" );

        expectedArguments.add( "/src/main/one" + PS + "/src/main/two" + PS );

        // debug

        compilerConfiguration.setDebug( true );

        if ( StringUtils.isNotEmpty( compilerConfiguration.getDebugLevel() ) )
        {
            expectedArguments.add( "-g:" + compilerConfiguration.getDebugLevel() );
        }
        else
        {
            expectedArguments.add( "-g" );
        }

        // showDeprecation

        compilerConfiguration.setShowDeprecation( true );

        expectedArguments.add( "-deprecation" );

        // targetVersion

        compilerConfiguration.setTargetVersion( "1.7" );

        expectedArguments.add( "-target" );

        expectedArguments.add( "1.7" );

        // sourceVersion

        compilerConfiguration.setSourceVersion( "1.7" );

        if ( !suppressSourceVersion )
        {
            expectedArguments.add( "-source" );

            expectedArguments.add( "1.7" );
        }

        // sourceEncoding

        compilerConfiguration.setSourceEncoding( "iso-8859-1" );

        if ( !suppressEncoding )
        {
            expectedArguments.add( "-encoding" );

            expectedArguments.add( "iso-8859-1" );
        }

        // customerCompilerArguments

        Map<String, String> customerCompilerArguments = new LinkedHashMap<String, String>();

        customerCompilerArguments.put( "arg1", null );

        customerCompilerArguments.put( "foo", "bar" );

        compilerConfiguration.setCustomCompilerArgumentsAsMap( customerCompilerArguments );

        expectedArguments.add( "arg1" );

        expectedArguments.add( "foo" );

        expectedArguments.add( "bar" );

    }

    @SuppressWarnings( "unchecked" )
    public void testCompilingSourcesWithNullnessChecker() throws Exception
    {
        List<CompilerMessage> messages = new ArrayList<CompilerMessage>();
        Collection<String> files = new TreeSet<String>();

        for ( CompilerConfiguration compilerConfig : getJsr308CompilerConfigurations() )
        {
            File outputDir = new File( compilerConfig.getOutputLocation() );

            org.codehaus.plexus.compiler.Compiler compiler = (org.codehaus.plexus.compiler.Compiler) lookup( org.codehaus.plexus.compiler.Compiler.ROLE, getRoleHint() );

            messages.addAll( compiler.performCompile( compilerConfig ).getCompilerMessages() );

            if ( outputDir.isDirectory() )
            {
                files.addAll( normalizePaths( FileUtils.getFileNames( outputDir, null, null, false ) ) );
            }
        }

        final int expectedCompilerErrors = 6;
        final int expectedCompilerWarnings = 1;

        int numCompilerErrors = compilerErrorCount( messages );

        int numCompilerWarnings = messages.size() - numCompilerErrors;

        if ( expectedCompilerErrors != numCompilerErrors )
        {
            System.out.println( numCompilerErrors + " error(s) found:" );
            for ( CompilerMessage error : messages )
            {
                if ( !error.isError() )
                {
                    continue;
                }

                System.out.println( "----" );
                System.out.println( error.getFile() );
                System.out.println( error.getMessage() );
                System.out.println( "----" );
            }

            assertEquals( "Wrong number of compilation errors.", expectedCompilerErrors, numCompilerErrors );
        }

        if ( expectedCompilerWarnings != numCompilerWarnings )
        {
            System.out.println( numCompilerWarnings + " warning(s) found:" );
            for ( CompilerMessage error : messages )
            {
                if ( error.isError() )
                {
                    continue;
                }

                System.out.println( "----" );
                System.out.println( error.getFile() );
                System.out.println( error.getMessage() );
                System.out.println( "----" );
            }

            assertEquals( "Wrong number of compilation warnings.", expectedCompilerWarnings, numCompilerWarnings );
        }

        assertEquals( new TreeSet<String>( normalizePaths( expectedOutputFilesNullnessChecker() ) ), files );
    }

    private List<CompilerConfiguration> getJsr308CompilerConfigurations()
        throws Exception
    {
        String sourceDir = getBasedir() + "/src/test-input/src/main";

        @SuppressWarnings( "unchecked" ) List<String> filenames =
            FileUtils.getFileNames( new File( sourceDir ), "**/*.java", null, false, true );
        Collections.sort( filenames );

        List<CompilerConfiguration> compilerConfigurations = new ArrayList<CompilerConfiguration>();

        int index = 0;
        for ( Iterator<String> it = filenames.iterator(); it.hasNext(); index++ )
        {
            String filename = it.next();

            CompilerConfiguration compilerConfig = new CompilerConfiguration();

            compilerConfig.setDebug( true );

            compilerConfig.setShowDeprecation( true );

            compilerConfig.setClasspathEntries( getJsr308Classpath() );

            compilerConfig.addSourceLocation( sourceDir );

            compilerConfig.setOutputLocation( getBasedir() + "/target/" + getRoleHint() + "/classes-jsr308-" + index );

            FileUtils.deleteDirectory( compilerConfig.getOutputLocation() );

            compilerConfig.addInclude( filename );

            compilerConfig.setAnnotationProcessors(new String[] {
                "org.checkerframework.checker.nullness.NullnessChecker"
            });

            compilerConfig.setTargetVersion( "1.8" );

            compilerConfig.setSourceVersion( "1.8" );

            compilerConfig.setVerbose(true);

            compilerConfigurations.add( compilerConfig );

        }

        return compilerConfigurations;
    }

    protected List<String> getJsr308Classpath()
        throws Exception
    {
        List<String> cp = getClasspath();

        File file = getLocalArtifactPath( "org.checkerframework", "checker-qual", "${checker-framework.version}", "jar" );

        assertTrue( "test prerequisite: checker-qual library must be available in local repository, expected "
                        + file.getAbsolutePath(), file.canRead() );

        cp.add( file.getAbsolutePath() );

        return cp;
    }

    private List<String> normalizePaths( Collection<String> relativePaths )
    {
        List<String> normalizedPaths = new ArrayList<String>();
        for ( String relativePath : relativePaths )
        {
            normalizedPaths.add( relativePath.replace( File.separatorChar, '/' ) );
        }
        return normalizedPaths;
    }
}

redpajama_set_name: RedPajamaGithub


This is my personal favorite when it comes to lectures, Simply awesome!
First off Rick is hilarious! Second, His work with just a coin/marker is unparalleled! This is the best there is when it comes to the most magic possible with just a little bit of pocket space (or none! You can do this with even a straw and a sugar packet, although thats hardly necessary someone will always have a pen and coin). Its also not that difficult! Believe me this lecture will leave you very satisfied but also hungry for more! I think it pairs great with The Marvelous Pencyclopedia, so take a look there as well!
After watching this Rick became one of my favorite performers (Right under Bill Malone in the #1 spot ;), Im confident you will love him and his HomeSchooled lecture too!
This guy his bloody hilarious! Loved every moment! Haha.


St Catherine's Hill has remnants of an Iron Age hill fort and a 12th-century chapel. In the mid-17th century a maze was created on parts of the Hill, which still exists. The site is now managed as a grassland nature reserve.
St Catherine's Hill dates back to the 12th century BC. The Hill was used as an Iron Age fort at least from the 6th century BC, but was later abandoned. St Catherine's Hill was unoccupied until the 12th century AD, when St Catherine's Chapel was constructed on the hill. The Hill was used as sports grounds for Winchester College for hundreds of years, until 1868. It is now partially owned by the Old Wykehamist Lodge of Freemasons.
St Catherine's Hill is an Iron Age site covering 9.3 hectares and is an oval shape. This site was defended with a line of bank and ditch, which followed the hill's contours. After the 1st century BC the Hill was unoccupied until the 12th century AD. At this time, the Hill was the location for St Catherine's Chapel, which was built in a cruciform shape and did not have any aisles.
Apart from the Hill's interesting history and its recreational role for walking and viewing the surrounding countryside, the Hill is a maze. The maze was designed between 1647 and 1710. It is called the Mizmaze and is a labyrinth cut into the turf, which has Christian and pre-Christian symbolism. The Mizmaze is curious in design due to its uninterrupted path which leads from the entry point to the centre. It is suggested that between 1830 and 1840 the maze was wrongly recut so that its present condition is an exact negative of the original. The Iron Age Hill fort is a scheduled ancient monument. The grazed chalk grassland, with representative flora and insect fauna, is a Hants and Isle of Wight Naturalists Trust Nature Reserve.
St Catherine's Hill is an ancient monument that dates back to the 12th century BC. Excavations in 1927-8 showed that occupation on the hill probably began in the 6th century BC. The Iron Age fort was sacked in the 1st century BC and was not occupied thereafter. The foundations of St Catherine's Chapel remain on the site. For centuries 'Hills' was the playground for Winchester College until 1868, when level playing fields near the College were created. Since 1894 the traditional association between Winchester College and St Catherine's Hill has been maintained by a school-organised gathering twice a year.
In 1930 the Old Wykehamist Lodge of Freemasons purchased St Catherine's Hill from the Ecclesiastical Commissioners. The Old Wykehamist Lodge purchased the Hill in order to ensure access rights for the members of the Lodge. There are also grazing rights let elsewhere on the Hill. These various ownerships of St Catherine's Hill complicate the maintenance of this ancient monument.


text:
Sweet Mandarin: The Courageous True Story of Three Generations of Chinese Women and Their Journey from East to West by Helen Tse - 279 pgs
In Sweet Mandarin, Helen Tse gives us the intelligent multi-generational saga of three enterprising and resourceful Chinese women who faced incredible odds to make their dreams and fortunes come to fruition. The story begins with Lily, Helen's grandmother, in a rural village in China. Facing incredible poverty and with a family to provide for, Lilly's father, Leung, has the initiative to break away from his traditional role as a farmer and strikes out to create his own business, which soon begins to prosper. Moving his family from the destitute village to the more bustling city of Hong Kong, Lily and her family seem to be moving upwards. Then an unspeakable tragedy occurs, leaving the family penniless and at the mercy of inhospitable relatives. Lily realizes the situation she and her family face and searches for employment as a housemaid to the affluent British expatriates in China. Soon Lily immigrates to Britain and restarts her life as a small business owner, the proprietor of a Chinese restaurant. Through the struggles of operating the business and raising her children alone, Tse acquaints us with this remarkably strong woman who must face overwhelming trials in order to give herself and her children a better life. The story continues through the tale of Mabel, Lily's daughter, who is raised mostly in Britain, working long hours from childhood at her mother's restaurant counter. Eventually, Mabel takes up the family business and creates her own Chinese restaurant with her husband Eric. Interspersed with these two women's stories is the story of Helen, Mabel's daughter. Helen begins her career as lawyer but ultimately finds her happiness in opening her own Chinese restaurant, Sweet Mandarin, from which the title of the book is based. In elegant prose, the three women's stories are woven together to create a beautiful tapestry of a bold and valiant family of women who never let their struggles get the best of them.
Of the stories in this book, Lily's was featured most heavily. We see the whole picture of her life, from her humbling situation as a child to her rise as a beloved housemaid, the triumphs and ordeals are painted with compelling energy. I was particularly struck by her forced involvement in the Japanese occupation of China in the 1940's, and her eventual departure from China, where she left her family while she built a new life for them. Though sometimes reserved in her expressions of love for her children, her outward resolve to give them a more hopeful future was inspiring. Sometimes it seemed as though she was a tough nut to crack, but in reality, had she not had the boldness to act as she did, her family might not have survived some of the situations that they faced. Some parts of Lily's story were more difficult to digest, for Lily was not always the admirable woman that some would wish she would be. The situations regarding the loss of her first restaurant were upsetting, but I appreciated the author's candor in addressing the fact that her grandmother was just as human as the rest of us, with flaws that any of us could have had. Much less was revealed about Mabel and Helen, and I choose to see this book as Helen's tribute to the sacrifice and success of her grandmother Lily. The legacy that she built for her family sustained them and drew them closer together as a group.
One of the wonderful things in this book was the description of various foods that were a hallmark to the family's home and restaurants. The intricacies of Lily's Curry recipe, and the depiction of Mabel's Claypot Chicken were indeed mouthwatering. I also liked the way the narrative shifted between the stories of the three women. It made the story less choppy and episodic, while still describing the aspects of all three's lives. The author did a very good job of painting the political and societal aspects of China from the 1920's to today, including the focus on why male children are particularly valued above female children in that part of the world. As I was reading, I really felt I understood the sacrifices and joy of the main characters, which is a true measure of success in any book.
This book was an involving story spanning many years and situations. I very much enjoyed the peek into a story that I think many would enjoy. There are many books about China and it's culture, but this book is unique, not only in the story it tells, but in the spirited strength of it's characters. Great book. I have included a link to a television interview with the author, who talks about the inspiration for this book and gives more information.
Labels: Memoir
a comment for both *Sweet Mandarin* and *Monique and the Mango Rains* ... I'm really enjoying your reviews, we seem to drift toward the same type of books!
Library Cat said...
You have really been reading some fantastic literature. I wanted to snag Monique and the Mango Rains and was so disappointed when I did not! In another life, I think I would be a midwife!
Zibilee said...
sheistoofondofbooks & library cat,
Thanks so much for the comments! I actually got Monique and the Mango Rains by contacting the publisher directly. I still have the contact info, and they seem really happy to send out the books to anyone who will review them, so if you'd like that information e-mail me at: heather@figearo.net, and I can pass it on!
An Infamous Army: A Novel of Wellington, Waterloo,...
The Matchmaker of Périgord: A Novel - by Julia Stu...
Aberrations by Penelope Przekop - 336 pgs
So Long at the Fair by Christina Schwarz - 244 pgs...
The Forbidden Daughter by Shobhan Bantwal - 336 pg...
Sweet Mandarin: The Courageous True Story of Three...
Monique and the Mango Rains: Two Years with a Midw...
Months and Seasons by Christopher Meeks - 170 pgs
One More Year by Sana Krasikov - 196 pgs


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GREY, Henry (1683-1740), of Horton, Northumb. and Billingbear, Berks.
b. 17 Aug. 1683, 2nd s. of Richard Neville, M.P., of Billingbear; yr. bro. of Grey Neville and uncle of Richard Neville Aldworth. m. Elizabeth, 1st da. of James Griffin, M.P., 2nd Baron Griffin of Braybrooke, sis. and coh. of Edward, 3rd Baron, s.p. suc. to the Northumbrian estates of his maternal uncle Ralph Grey, M.P., 4th Baron Grey of Warke, 1706, and to his e. bro.'s estates at Billingbear 1723.
charged his pistols in the morning, taking them with him ... he sent a letter to his wife and another to Sir Robert Walpole, dated from a tavern, I think in Drury Lane: he told his wife that she would see him no more and Sir Robert, that he desired he would dispose of his borough ... since then search has been made for him but to no purpose.
In November 1732 he was among several parliamentary candidates considered by the Berwick corporation 'for a third man, to make some sport as they call it'.2 Successful in 1734 for Reading, he voted with the Government till his death, 9 Sept. 1740.
See Rowland, Hist. Fam. of Neville, table v.
1. HMC Portland, vii. 426; Dr. Geo. Clarke to Edw Nicholas, 19 Feb. and 1 Mar. 1726, Egerton mss 2540, ff. 588-9, 596-7.
2. Geo. Liddell to Sir Robt. Walpole, 24 Nov. 1732, Cholmondeley (Houghton) mss.


Photos of Judy Garland pre-recording "Everybody Sing" in 1937 for the musical film "Broadway Melody of 1938", which was released in 1938. Her singing voice is incredible, an absolutely beautiful and flawless singing voice and you'll know what we mean after you listen to it.
She was only a 15 year old child at the time, but sang like an adult, if not better than any adult singer in those days. There's no denying that Judy Garland is a very talented and gifted singer indeed, and MGM must have been delighted to have a superstar of the calibre of Judy Garland at their disposal.
"Everybody Sing" by 15 year old Judy Garland.


text:
McMullen & Wing secure 10-year refit of award-winning VvS1
New Zealand superyacht builder McMullen & Wing is pleased to announce it has secured the 10-year refit of 34-metre motoryacht VvS1. Designed by Gregory C. Marshall and built in New Zealand in 2007, VvS1 has a busy 2016 summer cruising schedule ahead before being hauled at McMullen & Wing's Auckland shipyard in April 2016 for an extensive programme of works.
VvS1 won the World Superyacht Award in 2008 for Best Motoryacht Under 40 Metres, as well as three more awards at World Yacht Trophies in Cannes. The yacht is owned by New Zealanders Sir Michael and Lady Christine Hill, two of the country's most prominent yacht owners.
McMullen & Wing CEO Michael Eaglen is delighted to have confirmed the refit for the Hills. "Michael and Christine and their family are very active yachties. They have explored many of the world's traditional yachting hotspots, only to return even more passionate about New Zealand's stunning coastline," he says. "They are proud flag bearers for our beautiful country and it is an honour to play our part in their ongoing enjoyment of the unique and stunning yachting that New Zealand and the South Pacific offers and to help maintain their magnificent and award-winning New Zealand-built yacht in pristine condition."
The yard is proud to be applying their world standard expertise and craftsmanship to a quality yacht that reflects its luxury expedition ethos. With many decades of experience in refit work and a longstanding working relationship with Marshall – who penned the yard's acclaimed 45m Big Fish and the inbuild 50m Hull 1016 – McMullen & Wing's reputation for superior quality work is an excellent match for the esteemed VvS1, and is a project that the company is delighted to be working on.
With the refit project being centred on survey work, work undertaken will be largely technical. "The focus of the project will be the yacht's 10-year Lloyds special survey and replacing the generators and shore power converter, but we will also be undertaking some general maintenance and improvements around the yacht," says Richard Low, Projects Manager at McMullen & Wing, himself an accomplished superyacht captain. "We're looking forward to working closely with the captain and crew to deliver a top quality cost-effective programme of refit works and taking advantage of the availability of McMullen & Wing's outstanding central Auckland facilities."


Tinsley Meadows Primary Academy is proud to have three Specialist Leaders of Education on its workforce.
Specialist Leaders of Education (SLEs) focus on developing leadership capacity. This role is about developing other leaders so that they have the skills to lead their own teams and improve practice in their own schools.
Please see below for biographies of our SLEs and for more information on how you can broker support for your school.
CLAIRE NORMAN (Specialist Leader of Education for Initial Teacher Training and Assistant Head at Tinsley Meadows).
Claire's biography is available by clicking on the link below. Her services as an SLE can be brokered through Sheffield Teaching School Alliance. You can also contact the school directly to access these services.
Jenna's biography will appear here soon. Jenna's services as an SLE can be brokered through Learning Unlimited Teaching School Alliance. You can also contact the school directly to broker these services.
Emma's biography will appear here soon. Emma's services as an SLE can be brokered through Learning Unlimited Teaching School Alliance. You can also contact the school directly to broker these services.