text:
Browser: Scott Cairns. "Fireproof." "The Day the Earth Stood Still."
|In Blog, Looking Elsewhere
A new Scott Cairns poem is posted at the Denver Literary Examiner, one of several included in the new issue of Poetry. Cairns also has new poems in the new 20th anniversary issue of Image.
Did the recent movie Fireproof inspire you or change your life?
Have you been waiting for me to publish a review of it?
If so, forgive me. I missed a lot of movies in 2008, and that was one of them.
I was recently informed that, since I write reviews for Christian readers, Christian readers will "justifiably be expecting [me] to make an informed decision" on Fireproof — moreso than, say, Wendy and Lucy — because Fireproof is a movie "squarely" in a Christian moviegoer "demographic."
I replied that I had very little moviegoing time these days (with my long-overdue novel on my desk), and if I did have a couple of hours I would probably make it a priority to see something from the top of the list of the year's most acclaimed films, I was informed that I was "showing signs of elitism and film critic snobbery."
My goodness, I hope not. I apologize if I've come across as snobbish for prioritizing Rachel Getting Married over Fireproof. I haven't taken the time to catch up with that film yet. (Nor have I had time for many of the films that are regularly appearing in the top ten lists of other film critics, Christian or otherwise.)
To help make up part of my debt to those expectant Christian moviegoers, I'm opening the phones here. I have no desire whatsoever to turn up my nose against Fireproof, or against Christian filmmakers, or against anything that might have "a positive impact." I'm happy to provide a platform for people to sing this movie's praises, and to talk about how it has impressed them or changed their life. I haven't seen the film, so I cannot, and should not, make any presumptuous judgments.
So… I pass you the megaphone. If Fireproof is one of 2008's most inspiring films, let the world know here. And forgive me for neglecting it or spreading any kind of "snobbish" or "elitist" vibes…
And while we're at it:
What were the most inspiring and uplifting films you saw this year?
For me, a film's "message" isn't enough to make it inspiring. I need beauty. Excellence. Poetry. Truth. Something real, crafted with integrity. Blatant messages accompanied by sentimental music or close-ups of tear-stained faces don't inspire me. Might as well send me a Hallmark card. No… I'm hungry for something real, something that lets me discover meaning, rather than something that shovels meaning upon me.
(And by the way, have you seen Flight of the Red Balloon? It's as buoyant as the subject of its title. Or Silent Light? If there was any film this year that inspiringly portrayed the beautiful scandal of Divine Intervention, well…)
I'm hope I find a good opportunity to see Fireproof. If and when that happens, believe me: I'll share my thoughts without holding back.
At Image, A.G. Harmon considers The Day the Earth Stood Still
iowaboy277
Most inspiring and uplifting films this year? I'll start the nominations with two movies that feature characters who find themselves reaching out across cultures to help others they might normally try to avoid. I hope Richard Jenkins gets some Oscar love for his excellent first starring role in The Visitor. And I'm still feeling the charm of the Israelis and Egyptians from last night's viewing of The Band's Visit.
And what I wouldn't give to see a show on the "Three Girls and Their Buddy" tour….
The current issue of The Other Journal, which ends this week, also includes a new Scott Cairns poem (with audio) and some other cool stuff, like a two-part interview with Kathleen Norris. Anyway, here's the Cairns poem: http://www.theotherjournal.com/article.php?id=476.
Rick de Gier
One of the most uplifiting films I saw last year was Mike Leighs Happy-go-lucky. The main character comes across as naively optimistic at first, but as the story unfolds, you realize she does understand what the world is like, but chooses not to let it get her down. In a brilliant climatic scene where she's attacked by her obsessive driving instructor, she shows her true colors, not judging him, reaching out, but also standing up for herself. Because she's portrayed as a normal, flawed person we can relate to, she's one of the most inspiring movie characters I've seen. A lot more Christ-like than some characters I've come across in Christian books and films.
One that also touched me was another British movie, And when did you last see your father? The direction is a tad heavy-handed at times, but it features the most true to life father-son relationship I've seen in a movie. There's no big resolution, but there are subtle hints of hope and comfort.
I find often movies can be uplifting and comforting just by showing the world as it really is, with a lot of ugliness, but also unexpected beauty.
Jeremy Landes
I'll bite the bait. I can't stand certain Christian movies where the creators heap cliches atop each other to spoon-feed a message to hungry/thirsty Christian-only mouths. So I'm not sure why I even bought a ticket for Fireproof. I was at a church men's retreat at the time, during a several-hour break, and I just wanted to redeem the few personal hours I'd been given. I considered going to see Righteous Kill instead–my faith was minimal that Fireproof was something worth seeing. Things were tense at home, too, and I wanted to forget marital stress, rather than watching characters in the same boat. The movie grabbed me during the first scene, when we hear a little girl tell her mom, "I want to marry daddy." Tears started forming, as I realized that I wanted my own infant daughter to be loved by a husband who would truly respect and honor her–the way I hadn't been doing lately with my bride. I saw my own selfishness reflected in Kirk Cameron's character. I felt the wife character's pain as she coped and reacted with anger/resentment to her husband's anger, his porn addiction, his choosing work over her, and many other elements. Yet, I didn't ever choose a side in the fighting–a tricky script manuever, I thought.
Basically, God helped me see a lot of sin patterns in my life through watching Fireproof, I wept, and then I went through some catharsis. It was powerful. Maybe it was a one-time event that weekend when God wanted to speak something specific to me, but I was grateful. The comic relief was…very relieving when it needed to be. The movie was suspenseful in places, and I was pleasantly surprised by the slow, careful character development. I think the filmmakers didn't do such a terrific job with the conversion scene in the middle–could have been a lot more artful and less quick. I respected the effort, though, since something needed to happen in the character of Caleb or the movie would have just kept going Groundhog Day style–an endless cycle of failure. It was refreshing to see that everything didn't immediately get better afterward. I'm more tempted now to watch Facing the Giants, despite the description of it by Nicolosi(?) as porn for Christians. In Fireproof, he filmmakers created a two-hour drama that related to my life and, it seems, many other men who have seen it–not sure what it's doing for women. It wasn't a commercial for Christianity–I believed these characters could exist, hurt each other, and change. For that reason, I recommend seeing it.
Greg Wright
Hey, Jeremy. Your experience of Fireproof was not a one-off deal, strictly the result of circumstances. It's had the same effect on an awful lot of people, me included — and I'm about as snarky and skeptical when it comes to "Christian" film as anyone (short of being actually mean-spirited).
I just watched the film again with my wife yesterday morning, and it had her (and I, again) pretty well wracked with emotion at several points. Low budget and flawed, yes, but some really powerful filmmaking at work, too.
Preach It, Andrew Stanton!
Response Time!


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\section{Introduction}

The study of multiparameter processes goes back to the 70' and the theory developed for years covers multiple properties of random fields (we refer to the recent books \cite{Khoshnevisan(2002)} and \cite{Adler.Taylor(2007)} for a modern review). For instance, Cairoli and Walsh \cite{Cairoli.Walsh(1975),Walsh(1978),Walsh(1986)} have deeply investigated the extension of the martingale and stochastic integral theories to the two-parameter framework.
A vast literature also concerns the Markovian aspects of random fields. Similarly to the case of martingales, different interesting Markov properties can be formalized for multiparameter processes. Among these, the most commonly studied ones are \emph{sharp-Markov} \cite{Levy(1945),Dalang.Walsh(1992)}, \emph{germ-Markov} \cite{McKean(1963),Pitt(1971),Kunsch(1979)} and \emph{$\ast$-Markov} \cite{Cairoli(1971),Korezlioglu.Lefort.ea(1981)} properties.
We refer to \cite{Balanca(2012)} for a more complete description of these concepts.
The study multiparameter processes is still a very active area of research, particularly the analysis of sample paths and geometric properties (see e.g. \cite{Ayache.Shieh.ea(2011), Dalang.Nualart.ea(2011), Khoshnevisan.Xiao(2005), Tudor.Xiao(2007), Xiao(2009)a}).

Set-indexed processes constitute a natural generalization of multiparameter stochastic processes and their local regularity have been considered in the Gaussian case since the early work of Dudley \cite{Dudley(1973)} (see also \cite{Adler.Feigin(1984), Alexander(1986), Bass.Pyke(1985)}). Extending the literature on random fields, several different subjects have been recently investigated, including set-indexed martingales \cite{Ivanoff.Merzbach(2000)}, set-indexed Markov \cite{Ivanoff.Merzbach(2000)a, Balan.Ivanoff(2002), Balanca(2012)} and L\'evy processes \cite{Adler.Feigin(1984), Bass.Pyke(1984), Herbin.Merzbach(2011)},  and set-indexed fractional Brownian motion \cite{Herbin.Merzbach(2006), Herbin.Merzbach(2009)}. Although the set-indexed formalism appears to be more abstract, it usually offers a simpler and more condensed way to express technical concepts of multiparameter processes. For instance, the present work intensively uses the \ensuremath{\mathcal C}\xspace-Markov property introduced and developed in \cite{Balanca(2012)}. In the latter, the Chapman-Kolmogorov equation related to transition probabilities turns out to be more easily expressed using the set-indexed formalism than the two-parameter framework.

In this paper, we follow the framework established by Ivanoff and Merzbach in the context of set-indexed martingales \cite{Ivanoff.Merzbach(2000)}.
An {\em indexing collection} $\mathcal{A}$ is constituted of compact subsets of a locally compact metric space $\mathcal{T}$ equipped with a Radon measure on the $\sigma$-field generated by $\mathcal{A}$.
\ensuremath{\mathcal A(u)}\xspace and \ensuremath{\mathcal C}\xspace respectively denote the class of finite unions of sets belonging to \ensuremath{\mathcal A}\xspace and the collection of increments $C = A\setminus B$, where $A\in\ensuremath{\mathcal A}\xspace$ and $B\in\ensuremath{\mathcal A(u)}\xspace$. Finally, \ensuremath{{\emptyset'}}\xspace denotes the set $\cap_{A\in\ensuremath{\mathcal A}\xspace} A$, which usually plays a role equivalent to $0$ in $\ensuremath{\mathbf{R}}\xspace_+^N$.
In the present article, we suppose that the collection $\ensuremath{\mathcal A}\xspace$ and the measure $m$ satisfy the following assumptions:
\begin{enumerate}[\it (i)]
  \item \ensuremath{{\emptyset'}}\xspace is a nonempty set and \ensuremath{\mathcal A}\xspace is closed under arbitrary intersections;
  \item \emph{Shape hypothesis}: for any $A, A_1,\dotsc,A_k\in\ensuremath{\mathcal A}\xspace$ with $A\subseteq\cup_{i=1}^k A_i$, there exists $i\in\brc{1,\dotsc,k}$ such that $A\subseteq A_i$;
	\item $m(\ensuremath{{\emptyset'}}\xspace) = 0$ and $m$ is monotonically continuous on \ensuremath{\mathcal A}\xspace, i.e. for any increasing sequence $(A_n)_{n\in\ensuremath{\mathbf{N}}\xspace}$ in \ensuremath{\mathcal A}\xspace,
	\[
    \lim_{n\rightarrow \infty} m(A_n) = m\pthb{ \overline{\cup_{k\in\ensuremath{\mathbf{N}}\xspace}A_k} }.
  \]
\end{enumerate}
For sake of readability, we restrict properties of \ensuremath{\mathcal A}\xspace to the strictly required ones in the sequel. The particular case of $\mathcal{A}=\{ [0,t];\;t\in\ensuremath{\mathbf{R}}\xspace_+^N \}$ shows that the set-indexed formalism extends the multiparameter setting. Another simple example satisfying \emph{Shape} can be constructed on the $\ensuremath{\mathbf{R}}\xspace^3$-unit sphere: $\ensuremath{\mathcal A}\xspace = \brc{A_{\theta,\varphi}; \theta\in [0,\pi) \text{ and } \varphi\in [0,2\pi)}$ where $A_{\theta,\varphi} = \brc{(1,\widehat\theta,\widehat\varphi) : \widehat\theta\in\ivff{0,\theta} \text{ and } \widehat\varphi\in\ivff{0,\varphi}}$.    We refer to \cite{Ivanoff.Merzbach(2000)} for a more complete definition of an indexing collection used in the general theory of set-indexed martingales.

We investigate the existence and properties of a set-indexed extension of the Ornstein-Uhlenbeck (OU) process, originaly introduced in \cite{Uhlenbeck.Ornstein(1930)} and then widely used in the literature to represent phenomena in physics, biology and finance (e.g. see \cite{Frank.Daffertshofer.Beek(2000), {Lansky.Rospars(1995)}, Masuda(2004)}).
A well-known integral representation of the real-parameter OU process $X=\{X_t;\;t\in\ensuremath{\mathbf{R}}\xspace_+\}$ is given by
\begin{equation}  \label{eq:ou_integral}
\forall t\in\ensuremath{\mathbf{R}}\xspace_+;\quad  X_t = X_0\ e^{-\lambda t} + \int_0^t \sigma\ e^{\lambda(s-t)} \,\dt W_s,
\end{equation}
where $\lambda$ and $\sigma$ are positive parameters and the initial distribution $\nu=\mathcal{L}(X_0)$ is independent of the Brownian motion $W$. Furthermore, $X$ is a Markov process characterized by the following transition densities, for all $t\in\ensuremath{\mathbf{R}}\xspace_+$ and $x,y\in\ensuremath{\mathbf{R}}\xspace$;
\begin{equation} \label{eq:ou_kernel}
  p_t(x;y) = \frac{1}{\sigma_t\sqrt{2\pi}} \exp\bktbb{-\frac{1}{2\sigma^2_t}\pthb{ y - x e^{-\lambda t} }^2 } \qquad\text{where } \sigma^2_t = \frac{\sigma^2}{2\lambda}\pthb{1-e^{-2\lambda t}}.
\end{equation}
Two particular cases of initial distribution will be of specific interest in the sequel:
\begin{enumerate}[ \bf 1.]
	\item If $\nu = \delta_x$, $x\in\ensuremath{\mathbf{R}}\xspace$, $X$ is a Gaussian process with the following mean and covariance, for all $s,t\in\ensuremath{\mathbf{R}}\xspace_+$,
	\begin{equation}  \label{eq:ou_deltax}
	  \esp[_x]{X_t} = x e^{-\lambda t}\qquad\text{and}\qquad\cov_x(X_s,X_t) = \frac{\sigma^2}{2\lambda}\pthb{ e^{-\lambda\abs{t-s}} - e^{-\lambda(t+s)} }.
	\end{equation}
	\item If $\nu \sim \ensuremath{\mathcal N}\xspace\pth{ 0,\frac{\sigma^2}{2\lambda} }$, $X$ is a \emph{stationary Ornstein-Uhlenbeck process}, i.e. a zero-mean Gaussian process such that
	\begin{equation}  \label{eq:ou_stationary}
	  \forall s,t\in\ensuremath{\mathbf{R}}\xspace_+;\quad \esp[_\nu]{X_s X_t} = \frac{\sigma^2}{2\lambda}e^{-\lambda\abs{t-s}}.
	\end{equation}
\end{enumerate}

Since a set-indexed extension of the OU process cannot be directly derived from the integral representation \eqref{eq:ou_integral}, we first focus on the stationary process described in \eqref{eq:ou_stationary}. A natural way to extend this covariance to the set-indexed framework is to substitute the absolute value $\abs{t-s}$ with $d(U,V)$ where $d$ is a distance defined on the elements of $\mathcal{A}$. Similarly to the case of the set-indexed fractional Brownian motion described in \cite{Herbin.Merzbach(2006)}, we consider the choice $d(U,V)=m(U\disy V)$, where $\disy$ denotes the symmetric difference.

In Section \ref{sec:siou}, we first define a \emph{stationary set-indexed Ornstein-Uhlenbeck} process (ssiOU) as a zero-mean Gaussian process $X = \brc{X_U; U\in\ensuremath{\mathcal A}\xspace}$ such that
\begin{equation} \label{eq:ssiou_cov}
  \forall U,V\in\ensuremath{\mathcal A}\xspace;\quad \esp{X_U X_V} = \frac{\sigma^2}{2\lambda} e^{-\lambda m(U\disy V)},
\end{equation}
where $\lambda$ and $\sigma$ are positive parameters. Stationarity and Markov properties of this set-indexed process are studied, and lead to the complete characterization proved in Theorem~\ref{th:ssiou_charac}. Then, using the Markov kernel obtained, we are able to introduce in Definition~\ref{def:siou} a general \emph{set-indexed Ornstein-Uhlenbeck process}, whose law is consistent with the covariance structure \eqref{eq:ou_deltax} in particular case of initial Dirac distributions.

Finally, in Section \ref{sec:mpou}, we prove that in the multiparameter case, the set-indexed Ornstein-Uhlenbeck has a natural integral representation which generalizes expression \eqref{eq:ou_integral}.

\section{A stationary set-indexed Ornstein-Uhlenbeck process} \label{sec:siou}

In this section, we define a set-indexed extension of the stationary Ornstein-Uhlenbeck process, defined by the Gaussian covariance structure (\ref{eq:ou_stationary}).

\subsection{Definition and first properties}

As a preliminary to the definition, we need to prove that the expected covariance function of the process is positive definite in the same way as Lemma 2.9 of \cite{Herbin.Merzbach(2006)}.

\begin{lemma}\label{lem:cov}
If $\mathcal{A}$ is an indexing collection, $m$ a Radon measure on the $\sigma$-field generated by $\mathcal{A}$ and $\lambda, \sigma$ positive constants, the function $\Gamma : \mathcal{A}\times\mathcal{A}\rightarrow\ensuremath{\mathbf{R}}\xspace$ defined by
\begin{align*}
\forall U,V\in\ensuremath{\mathcal A}\xspace;\quad \Gamma(U,V) = \frac{\sigma^2}{2\lambda} e^{-\lambda m(U\disy V)},
\end{align*}
is positive definite.
\end{lemma}

\begin{proof}
 Let $f_1,f_2,\ldots,f_k$ be in $L^2(m)$ and $u_1,u_2,\ldots,u_k$ be in $\ensuremath{\mathbf{R}}\xspace$. Let \ensuremath{\mathcal V}\xspace be the vector space $\ensuremath{\mathcal V}\xspace=\spanv\pth{f_1,\ldots,f_k}$. Since $f\mapsto e^{-\frac{1}{2}\norm{f}_{L^2(m)}^2}$ is positive definite, there exists a Gaussian vector $X$ on the finite-dimensional space $\ensuremath{\mathcal V}\xspace$ such that
  \[
    \forall \lambda>0,\ \forall f\in\ensuremath{\mathcal V}\xspace;\quad \espb{e^{i\sqrt{2\lambda}\scpr{X}{f}}} = e^{-\lambda\norm{f}_{L^2(m)}^2}.
  \]
The non-negative definition of $f\mapsto e^{-\lambda\norm{f}_{L^2(m)}^2}$ can be written
\begin{align*}
\sum_{i=1}^k\sum_{j=1}^k u_i u_j e^{-\lambda\norm{f_i-f_j}^2_{L^2}}
= \sum_{i=1}^k\sum_{j=1}^k u_i u_j\espb{e^{i\sqrt{2\lambda}\scpr{X}{f_i-f_j}}}
= \normbb{\sum_{i=1}^k u_i e^{i\sqrt{2\lambda}\scpr{X}{f_i}}}_{L^2(\Omega)}^2 \geq 0.
\end{align*}
For any $U_1,\ldots,U_k\in\ensuremath{\mathcal A}\xspace$, the previous result is applied to $f_1=\ensuremath{\mathbf{1}}\xspace_{U_1},\ldots,f_k=\ensuremath{\mathbf{1}}\xspace_{U_k}\in L^2(m)$.
As in the proof of Lemma 2.9 in \cite{Herbin.Merzbach(2006)}, we remark that
\begin{align*}
\forall i,j\in\brc{1,\dotsc,k};\quad
m(U_i\disy U_j)
= m\pthb{\abs{ \ensuremath{\mathbf{1}}\xspace_{U_i}-\ensuremath{\mathbf{1}}\xspace_{U_j} }}
= \normb{f_i-f_j}_{L^2(m)}^2,
\end{align*}
and we deduce
\[
\sum_{i=1}^k\sum_{j=1}^k u_i u_j e^{-\lambda m(U_i\disy U_j)} \geq 0
\]
which proves the result.
\end{proof}

\noindent According to Lemma \ref{lem:cov}, we can define

\begin{definition}\label{def:ssiou}
Given the indexing collection $\mathcal{A}$ and positive real numbers $\lambda$ and $\sigma$, any Gaussian process $\{X_U;\;U\in\mathcal{A}\}$ such that for all $U,V\in\mathcal{A}$,
\begin{equation*}
\esp{X_U} = 0\quad\textrm{and}\quad \esp{X_U X_V} = \frac{\sigma^2}{2\lambda} e^{-\lambda m(U\disy V)},
\end{equation*}
is called a {\em stationary set-indexed Ornstein-Uhlenbeck (ssiOU) process}.
\end{definition}

The covariance structure of the Gaussian process coming from Definition~\ref{def:ssiou} directly implies the $L^2$-continuity and stationarity properties.

\begin{proposition} \label{prop:ssiou_continuity}
The stationary set-indexed Ornstein-Uhlenbeck process $X$ of Definition~\ref{def:ssiou} is $L^2$-monotone inner- and outer-continuous, i.e. for any increasing sequence $(U_n)_{n\in\ensuremath{\mathbf{N}}\xspace}$ in \ensuremath{\mathcal A}\xspace, such that $\overline{\cup_{k\in\ensuremath{\mathbf{N}}\xspace}U_k}\in\ensuremath{\mathcal A}\xspace$ and for any decreasing sequence $(V_n)_{n\in\ensuremath{\mathbf{N}}\xspace}$ in \ensuremath{\mathcal A}\xspace,
\[
\lim_{n\rightarrow\infty} \espb{\abs{X_{U_n}-X_{\overline{\cup_{k\in\ensuremath{\mathbf{N}}\xspace}U_k}}}^2} = 0
\quad\text{and}\quad
\lim_{n\rightarrow\infty} \espb{\abs{X_{V_n}-X_{\cap_{k\in\ensuremath{\mathbf{N}}\xspace}V_k}}^2} = 0.
\]
\end{proposition}

\begin{proof}
  Let $(U_n)_{n\in\ensuremath{\mathbf{N}}\xspace}$ be an increasing sequence in \ensuremath{\mathcal A}\xspace such that $\overline{\cup_{k\in\ensuremath{\mathbf{N}}\xspace}U_k}\in\ensuremath{\mathcal A}\xspace$. Then using equation \eqref{eq:ssiou_cov}, we have
\begin{align*}
\forall n\in\ensuremath{\mathbf{N}}\xspace;\quad
\espb{\abs{X_{U_n}-X_{\overline{\cup_{k\in\ensuremath{\mathbf{N}}\xspace}U_k}}}^2}
= \frac{\sigma^2}{2\lambda} \pthb{ 2 - 2e^{-\lambda m\pth{\overline{\cup_{k\in\ensuremath{\mathbf{N}}\xspace}U_k}\setminus U_n}} }.
\end{align*}
  According to Assumption \textit{(iii)} on \ensuremath{\mathcal A}\xspace and $m$, $\lim_{n\rightarrow \infty} m\pth{\overline{\cup_{k\in\ensuremath{\mathbf{N}}\xspace}U_k}\setminus U_n} = 0$. Therefore, the $L^2$-monotone inner-continuity follows, and similarly, the outer-continuity of $X$.
\end{proof}

The stationarity increments property for set-indexed processes has been introduced in \cite{Herbin.Merzbach(2009)} in the context of fractional Brownian motion, and it has constitued the key property to derive deep understanding of the set-indexed L\'evy processes in \cite{Herbin.Merzbach(2011)}. The stationarity property defined below is closely related to these two previous works.

\begin{proposition} \label{prop:ssiou_stationarity}
The stationary set-indexed Ornstein-Uhlenbeck process $X$ of Definition~\ref{def:ssiou} is $m$-stationary, i.e. for any $k\in\ensuremath{\mathbf{N}}\xspace$, $V\in\ensuremath{\mathcal A}\xspace$ and increasing sequences $(U_i)_{1\leq i\leq k}$ and $(A_i)_{1\leq i\leq k}$ in \ensuremath{\mathcal A}\xspace such that $m(U_i\setminus V)=m(A_i)$ for all $i\in\brc{1,\dotsc,k}$, $X$ satisfies
\[
\pth{X_{U_1},\ldots,X_{U_k}} \stackrel{(d)}{=} \pth{X_{A_1},\ldots,X_{A_k}}.
\]
\end{proposition}

\begin{proof}
Let $V$, $(U_i)_{1\leq i\leq k}$ and $(A_i)_{1\leq i\leq k}$ be as in the statement. Without any loss of generality, we suppose that $V\subseteq U_i$. Then, for all $j\geq i$, as $U_i\subseteq U_j$ and $A_i\subseteq A_j$,
  \begin{align*}
    m(U_i\disy U_j)
    = m(U_j) - m(U_i)
    = m(U_j\setminus V)-m(U_i\setminus V)
    = m(A_j) - m(A_i)
    = m(A_j\disy A_i).
  \end{align*}
Therefore, we deduce the expected equality, since $X$ is a centered Gaussian process and for all $i,j\in\brc{1,\dotsc,k}$,
$\esp{X_{U_i}X_{U_j}}
    = \displaystyle\frac{\sigma^2}{2\lambda} e^{-\lambda m(U_i\disy U_j)}
    = \frac{\sigma^2}{2\lambda} e^{-\lambda m(A_j\disy A_i)}
    = \esp{X_{A_i}X_{A_j}}$.
\end{proof}

We observe that the definition of stationarity is given in a strict sense, since it concerns the invariance of finite-dimensional distributions under a form of {\it measure-invariant translation}. In the classic theory of stationary random fields, a weaker property relying on the correlation function is usually defined (see \cite{Yaglom(1987)}): $C(s,t)=\esp{X_sX_t}$ only depends on the difference $t-s$. The weak definition of stationarity for one-parameter processes can be naturally extended to the multiparameter case, but it appears that this straightforward extension is not the most relevant. Indeed, the stationarity of increments defined using Lebesgue measure or their invariance under translation appeared to be more interesting to study multiparameter processes (see e.g. L\'evy and fractional Brownian sheets), and this fact explains the form of the set-indexed extension for the stationarity property.

\subsection{Markov property and characterisation of the stationary set-indexed Ornstein-Uhlenbeck process}

To investigate the Markov property, we first need to recall a few notations used in \cite{Balanca(2012)}. Let $C\in\ensuremath{\mathcal C}\xspace$ such that $C = A\setminus B$, with $B\in\ensuremath{\mathcal A(u)}\xspace$ and $B\subseteq A\in\ensuremath{\mathcal A}\xspace$. Since the assumption \emph{Shape} holds on \ensuremath{\mathcal A}\xspace, Definition 1.4.5 in \cite{Ivanoff.Merzbach(2000)} states that there exists a unique \emph{extremal representation} $\brc{A_i}_{i\leq k}$ of $B$, i.e. such that $B = \cup_{i=1}^k A_i$ and for all $i\neq j$, $A_i\nsubseteq A_j$.

Then, let $\ensuremath{\mathcal A}\xspace_\ell$ be the semilattice $\brc{A_1\cap\dotsb\cap A_k,\dotsc,A_1\cap A_2,A_1\dotsc,A_k}\subset\ensuremath{\mathcal A}\xspace$ and \ensuremath{{ \bm{\mathcal{A}_{C}}} }\xspace be defined as the following subset of $\ensuremath{\mathcal A}\xspace_\ell$,
\begin{equation}\label{eq:def_Cfrontier}
  \ensuremath{{ \bm{\mathcal{A}_{C}}} }\xspace = \brc{U\in\ensuremath{\mathcal A}\xspace_\ell;\, U\nsubseteq B^\circ} \overset{\mathrm{def}}{=} \brc{\Cpi{1},\dotsb,\Cpi{n}}, \qquad\text{where }n=\#(\ensuremath{{ \bm{\mathcal{A}_{C}}} }\xspace).
\end{equation}
The notation \vXCp{C} refers to the random vector $\vXCp{C} = \pthb{ X_{\Cpi{1}},\dotsc,X_{\Cpi{n} } }$, and similarly \vxCp{C} is used for a vector of variables. Thereby, according to \cite{Balanca(2012)}, the extension $\Delta X$ of $X$ on the class \ensuremath{\mathcal C}\xspace satisfies
\begin{align} \label{eq:inc_exc_formula}
\Delta X_C &\overset{\mathrm{def}}{=} X_A - \bktbb{ \sum_{i=1}^k X_{A_i} - \sum_{i<j} X_{A_i \cap A_j} + \dotsb + (-1)^{k+1} X_{A_1\cap\dotsb\cap A_k} } \nonumber\\
&= X_A - \bktbb{ \sum_{i=1}^{n} (-1)^{\eCpi{i}} X_{\Cpi{i}} },
\end{align}
where $(-1)^{\eCpi{i}}$ represents the sign of the term $X_{\Cpi{i}}$ in the inclusion-exclusion formula.
In other words, \eqref{eq:inc_exc_formula} says that every term $X_U$ in the previous inclusion-exclusion formula such that $U\notin\ensuremath{{ \bm{\mathcal{A}_{C}}} }\xspace$ is cancelled by another term in the sum.

Finally $\brc{ \ensuremath{\mathcal F}\xspace_A; A\in\ensuremath{\mathcal A}\xspace }$ denotes the natural filtration generated by $X$, and for all $B\in\ensuremath{\mathcal A(u)}\xspace$ and $C\in\ensuremath{\mathcal C}\xspace$, $\ensuremath{\mathcal F}\xspace_B$ and $\Gc{C}$ respectively correspond to
\begin{equation} \label{eq:def_filtrations}
  \ensuremath{\mathcal F}\xspace_B = \bigvee_{A\in\ensuremath{\mathcal A}\xspace,A\subseteq B}\,\ensuremath{\mathcal F}\xspace_A \qquad\text{and}\qquad \Gc{C} = \bigvee_{B\in\ensuremath{\mathcal A(u)}\xspace,B\cap C=\ensuremath{{\emptyset}}\xspace}\,\ensuremath{\mathcal F}\xspace_B.
\end{equation}
We note that these filtrations are not necessarily outer-continuous.

In the following result, we prove that the ssiOU process satisfies the \ensuremath{\mathcal C}\xspace-Markov property introduced in \cite{Balanca(2012)}.

\begin{proposition} \label{prop:ssiou_c_markov}
The stationary set-indexed Ornstein-Uhlenbeck process $X$ of Definition~\ref{def:ssiou} is a \ensuremath{\mathcal C}\xspace-Markov process with respect to its natural filtration $(\ensuremath{\mathcal F}\xspace_A)_{A\in\ensuremath{\mathcal A}\xspace}$, i.e. for all $C = A\setminus B$ with $A\in\ensuremath{\mathcal A}\xspace$, $B\in\ensuremath{\mathcal A}\xspace(u)$ and all Borel function $f:\ensuremath{\mathbf{R}}\xspace\rightarrow\ensuremath{\mathbf{R}}\xspace_+$, $X$ satisfies
\begin{equation}  \label{eq:def_cmarkov}
  \espc{f(X_A)}{\Gc{C}} = \espc{f(X_A)}{\vXCp{C}} \overset{\mathrm{def}}{=} \Pc{C} f(\vXCp{C}) \quad \Pr\textrm{-a.s}.
\end{equation}
\end{proposition}

\begin{proof}
Let $C=A\setminus B$ be in \ensuremath{\mathcal C}\xspace, $\brc{A_i}_{i\leq k}$ be the extremal representation of $B$ and $U$ be in $\ensuremath{\mathcal A}\xspace$ such that $U\cap C=\ensuremath{{\emptyset}}\xspace$. We first note that $U\cap A = (U\cap C)\cup(U\cap B) = U\cap B\in\ensuremath{\mathcal A}\xspace$. Thus, since \ensuremath{\mathcal A}\xspace satisfies the \emph{Shape} hypothesis, there exists $l\in\brc{1,\dotsc,k}$ such that $U\cap B= U\cap A_l$.
Consider the following quantity $I_U$,
\begin{align*}
I_U &= \espB{ X_U \pthB{ X_A - \sum_{i=1}^k X_{A_i}e^{-\lambda m(A\setminus A_i)} + \sum_{1\leq i<j \leq k} X_{A_i\cap A_j} e^{-\lambda m(A\setminus A_i\cap A_j)} + \\
& \qquad\qquad\qquad\qquad \dotsb + (-1)^{k} X_{A_1\cap \dotsb \cap A_k}e^{-\lambda m(A\setminus A_1\cap \dotsb \cap A_k)} } } \\
&= \frac{\sigma^2}{2\lambda} \pthB{ e^{-\lambda m(A \disy U)} - \sum_{i=1}^k e^{-\lambda (m(A_i \disy U) + m(A\setminus A_i))} + \sum_{1\leq i<j \leq k} e^{-\lambda ( m(A_i\cap A_j \disy U) + m(A\setminus A_i\cap A_j))} + \\
& \qquad\qquad\qquad\qquad \dotsb + (-1)^{k} e^{-\lambda (m(A_1\cap \dotsb \cap A_k \disy U)+m(A\setminus A_1\cap \dotsb \cap A_k))} } \\
&= \frac{\sigma^2}{2\lambda} e^{-\lambda (m(A)+m(U))} \pthB{ e^{-2\lambda m(A\cap U)} - \sum_{i=1}^k e^{-2\lambda m(A_i \cap U)} + \sum_{1\leq i<j \leq k} e^{-2\lambda m(A_i\cap A_j\cap U)} + \\
& \qquad\qquad\qquad\qquad \dotsb + (-1)^{k} e^{-2\lambda m(A_1\cap \dotsb \cap A_k \cap U)} }.
\end{align*}

\noindent Let us introduce the set-indexed function $h:A\mapsto e^{-2\lambda m(A\cap U)}$. Since the assumption \emph{Shape} holds, $h$ admits an extension $\Delta h$ on \ensuremath{\mathcal A(u)}\xspace based on an inclusion-exclusion formula. Thus, we have $I_U = \frac{\sigma^2}{2\lambda} e^{-\lambda (m(A)+m(U))} \pthb{ h(A\cap U) - \Delta h(B\cap U)}$. But since $A\cap U = B\cap U = A_l\cap U\in\ensuremath{\mathcal A}\xspace$ and $h$ coincides with $\Delta h$ on \ensuremath{\mathcal A}\xspace, we obtain $I_U = 0$.

Therefore, $I_U = 0$ for all $U\in\ensuremath{\mathcal A}\xspace$ such that $U\cap C=\ensuremath{{\emptyset}}\xspace$ and as $X$ is a Gaussian process, we can claim that the random variable
\[
X_A - \sum_{i=1}^k X_{A_i}e^{-\lambda m(A\setminus A_i)} + \dotsb + (-1)^{k} X_{A_1\cap \dotsb \cap A_k}e^{-\lambda m(A\setminus A_1\cap \dotsb \cap A_k)}
\]
and $\Gc{C}$ are independent.
Since the previous random variable is expressed as an inclusion-exclusion formula, equation \eqref{eq:inc_exc_formula} shows that it can also be expressed as
\[
X_A - Z_C \quad\textrm{with}\quad
Z_C = e^{-\lambda m(A)}\bktbb{ \sum_{i=1}^{n} (-1)^{\eCpi{i}} X_{\Cpi{i}}\,e^{\lambda m(\Cpi{i})} }.
\]
Notice that $Z_C$ is $\vXCp{C}$-measurable (and then $\Gc{C}$-measurable) and $X_A - Z_C$ is independent of $\Gc{C}$ (and then independent of $\vXCp{C}$). Hence, using a classic property of the conditional expectation, we have
\[
  \espc{f(X_A)}{\Gc{C}} = \esp{f(X_A-Z_C + Z_C) \mid \Gc{C}} = \espc{f(X_A)}{Z_C}.
\]
We similarly obtain the equality $\espc{f(X_A)}{\vXCp{C}} = \espc{f(X_A)}{Z_C}$ which ends the proof.
\end{proof}
Intuitively, the \ensuremath{\mathcal C}\xspace-Markov property can be understood as following: For any increment $C = A\setminus B$, the $\sigma$-field $\Gc{C}$ represents the past, described as \emph{strong} as it contains all the information inside the regions $B$ satisfying $C = A\setminus B$. The vector \vXCp{C} itself gathers the minimum information related to the "border" points of $C$, and finally, $X_A$ represents the future value of the process. Then, Equation \eqref{eq:def_cmarkov} simply states that conditioning the future with respect the full history $\Gc{C}$ or the vector $\vXCp{C}$ are equivalent.

According to Proposition 2.9 in \cite{Balanca(2012)}, we can deduce that the set-indexed Ornstein-Uhlenbeck process also satisfies set-indexed \emph{sharp-Markov} and \emph{Markov} properties whose definitions can be found in \cite{Ivanoff.Merzbach(2000)a}.
In the multiparameter case, it implies that this process is \emph{sharp-Markov} and \emph{germ-Markov} with respect to finite unions of rectangles (see \cite{Ivanoff.Merzbach(2000)a, Balanca(2012)}). The question whether this implication remains true for more complex sets has not been investigated yet (see \cite{Dalang.Walsh(1992),Dalang.Walsh(1992)a} for answers in the particular case of Brownian and L\'evy sheets).

As a consequence of the previous proposition, we can derive the \ensuremath{\mathcal C}\xspace-transition system \ensuremath{\mathcal P}\xspace and the initial law that characterize entirely a ssiOU process.

\begin{corollary} \label{cor:ssiou_markov_dens}
The \ensuremath{\mathcal C}\xspace-transition system $\ensuremath{\mathcal P}\xspace = \brc{\Pc{C}(\vxCp{C};\Gamma);\,C\in\ensuremath{\mathcal C}\xspace, \Gamma\in\mathcal{B}(\ensuremath{\mathbf{R}}\xspace)}$ of the stationary set-indexed Ornstein Uhlenbeck process of Definition~\ref{def:ssiou} is characterized by the following transition densities, for all $C=A\setminus B\in\ensuremath{\mathcal C}\xspace$:
\begin{equation} \label{eq:ssiou_markov_dens}
\pc{C}(\vxCp{C};y) = \frac{1}{\sigma_C\sqrt{2\pi}} \exp\bktBB{ -\frac{1}{2\sigma_C^2} \pthbb{ y - e^{-\lambda m(A)}\bktbb{ \sum_{i=1}^{n} (-1)^{\eCpi{i}}\, x_{\Cpi{i}}\, e^{\lambda m(\Cpi{i})} } }^2 },
\end{equation}
where
\[
\sigma_C^2 = \frac{\sigma^2}{2\lambda} \pthbb{ 1 - e^{-2\lambda m(A)} \bktbb{ \sum_{i=1}^{n} (-1)^{\eCpi{i}} e^{2\lambda m(\Cpi{i})} } }.
\]
Furthermore, the initial law is given by $X_\ensuremath{{\emptyset'}}\xspace\sim\ensuremath{\mathcal N}\xspace(0,\tfrac{\sigma^2}{2\lambda})$.
\end{corollary}

\begin{proof}
Let $C = A\setminus B$ be in \ensuremath{\mathcal C}\xspace and let $Z_C$ and $Y_C$ be the following Gaussian variables
\[
Z_C = e^{-\lambda m(A)}\bktbb{ \sum_{i=1}^{n} (-1)^{\eCpi{i}} X_{\Cpi{i}}\,e^{\lambda m(\Cpi{i})} } \quad\text{and}\quad Y_C = X_A - Z_C.
\]
Since the process $X$ is centered, $\esp{Z_C} = \esp{Y_C} = 0$. We note $\sigma_C^2$ the variance of $Y_C$. Using the independence of $Y_C$ and $\ensuremath{\mathcal G}\xspace^{*}_C$, shown in the proof of Proposition \ref{prop:ssiou_c_markov}, and the fact that $Z_C$ is $\ensuremath{\mathcal G}\xspace^{*}_C$-measurable, we have for any measurable function $f:\ensuremath{\mathbf{R}}\xspace\rightarrow\ensuremath{\mathbf{R}}\xspace_+$,
\begin{align*}
\espc{f(X_A)}{\ensuremath{\mathcal G}\xspace^{*}_C}
= &\espc{f( Z_C + Y_C )}{\ensuremath{\mathcal G}\xspace^{*}_C} \\
= &\frac{1}{\sigma_C\sqrt{2\pi}}  \int_\ensuremath{\mathbf{R}}\xspace f\pth{u + Z_C} \exp\pthbb{-\frac{ u^2}{2\sigma_C^2}} \dt u\\
= &\frac{1}{\sigma_C\sqrt{2\pi}}  \int_\ensuremath{\mathbf{R}}\xspace f\pth{v} \exp\pthbb{-\frac{ (v-Z_C)^2}{2\sigma_C^2}} \dt v
\overset{\mathrm{def}}{=} \int_\ensuremath{\mathbf{R}}\xspace f(v)\, \pc{C}(\vXCp{C};v) \,\dt v.
\end{align*}
Equation \eqref{eq:ssiou_markov_dens} follows from this last equality.
It remains to prove the expression of the variance $\sigma_C^2$. We first note that, as $X_{\Cpi{i}}$ is $\ensuremath{\mathcal G}\xspace^{*}_C$-measurable and $Y_C$ is independent of $\ensuremath{\mathcal G}\xspace^{*}_C$,  $\esp{X_{\Cpi{i}} Y_C} = 0$ for any $i\in\brc{1,\dotsc,n}$. Therefore, we have
\begin{align*}
\sigma_C^2 = \esp{X_A Y_C}
&= \esp{X_A^2} - e^{-\lambda m(A)} \bktbb{ \sum_{i=1}^{n} (-1)^{\eCpi{i}} \,\espb{X_A X_{\Cpi{i}}} \,e^{\lambda m(\Cpi{i})} } \\
&= \frac{\sigma^2}{2\lambda}\pthbb{ 1 - e^{-\lambda m(A)} \bktbb{ \sum_{i=1}^{n} (-1)^{\eCpi{i}} \,e^{-\lambda m(A\disy \Cpi{i})}\,e^{\lambda m(\Cpi{i})} } } \\
&= \frac{\sigma^2}{2\lambda} \pthbb{ 1 - e^{-2\lambda m(A)} \bktbb{ \sum_{i=1}^{n} (-1)^{\eCpi{i}} e^{2\lambda m(\Cpi{i})} } }.
\end{align*}
\end{proof}

The following result shows that properties exhibited in Propositions \ref{prop:ssiou_continuity}, \ref{prop:ssiou_stationarity} and  \ref{prop:ssiou_c_markov} lead to a complete characterization of the stationary set-indexed Ornstein-Uhlenbeck process.

\begin{theorem} \label{th:ssiou_charac}
A set-indexed mean-zero Gaussian process $X=\brc{X_U;\,U\in\ensuremath{\mathcal A}\xspace}$ is a stationary set-indexed Ornstein-Uhlenbeck process if and only if the three following properties hold:
\begin{enumerate}[(i)]
\item $L^2$-monotone inner- and outer-continuity;
\item $m$-stationarity;
\item \ensuremath{\mathcal C}\xspace-Markov property.
\end{enumerate}
\end{theorem}

\begin{proof}
We already know that the stationary set-indexed Ornstein-Uhlenbeck process of Definition~\ref{def:ssiou} satisfies these three properties.
Conversely, let $X$ be a zero-mean Gaussian set-indexed process which is $L^2$-monotone inner- and outer-continuous, $m$-stationary and \ensuremath{\mathcal C}\xspace-Markov. Without any loss of generality, we suppose $\esp{X_\ensuremath{{\emptyset'}}\xspace^2}=1$.

We first consider an increasing and continuous function $f:\ensuremath{\mathbf{R}}\xspace_+\rightarrow\ensuremath{\mathcal A}\xspace$, i.e. an {\em elementary flow} in the terminology of \cite{Ivanoff.Merzbach(2000), Herbin.Merzbach(2009)}, such that $f(0)=\ensuremath{{\emptyset'}}\xspace$. Since $m$ is monotonically continuous on \ensuremath{\mathcal A}\xspace (Condition {\it (iii)} of the indexing collection), the function $\theta:t\mapsto m[f(t)]$ is continuous, $\theta(0)=0$ and the pseudo-inverse $\theta^{-1}(t)=\inf\{u:\theta(u)>t\}$ satisfies $\theta\circ\theta^{-1}(t)=t$.
Then, the projected one-parameter process $X^{m,f} = \{X_{f\circ\theta^{-1}(t)};\;t\in\ensuremath{\mathbf{R}}\xspace_+\}$ is a centered one-parameter Gaussian process which is $L^2$-continuous, stationary (see \cite{Herbin.Merzbach(2009)}) and Markov (see \cite{Balanca(2012)}, Proposition 2.10).
Therefore, $X^{m,f}$ is a one-dimensional Ornstein-Uhlenbeck process (see e.g. \cite{Samorodnitsky.Taqqu(1994)}).  Since $\esp{(X^f_0)^2}=1$, there exists $\lambda_f > 0$ such that for all $s,t\in\ensuremath{\mathbf{R}}\xspace_+$,
\begin{align*}
\esp{X^{m,f}_s X^{m,f}_t} = e^{-\lambda_f \abs{t-s}} = e^{-\lambda_f \abs{m[f\circ\theta^{-1}(t)]-m[f\circ\theta^{-1}(s)]}} = e^{-\lambda_f m[f\circ\theta^{-1}(s)\disy f\circ\theta^{-1}(t)] }.
\end{align*}

Let us prove the constant $\lambda_f$ does not depend on the function $f$. Let $f_1$ and $f_2$ be two different elementary flows which satisfy the previous conditions. Then, as $m(\ensuremath{{\emptyset'}}\xspace)=0$, for any $t>0$, we know that $m(f_1\circ\theta_1^{-1}(t)\setminus\ensuremath{{\emptyset'}}\xspace) = m(f_2\circ\theta_2^{-1}(t))=t$, and therefore, according to the $m$-stationarity of $X$,
$\pthb{X^{m,f_1}_t,X^{m,f_1}_0} \overset{\mathrm{d}}{=} \pth{X^{m,f_2}_t,X^{m,f_2}_0}$,
which implies $e^{-\lambda_{f_1} t} = \esp{X^{m,f_1}_t X^{m,f_1}_0} = \esp{X^{m,f_2}_t X^{m,f_2}_0} = e^{-\lambda_{f_2} t}$, i.e. $\lambda_{f_1} = \lambda_{f_2}\overset{\mathrm{def}}{=}\lambda$.

For all $U,V\in\ensuremath{\mathcal A}\xspace$ such that $U\subseteq V$, there exists $f$ which goes through $U$ and $V$. We obtain
\[
\esp{X_U X_V} = e^{-\lambda \abs{m(V)-m(U)}} = e^{-\lambda m(U\disy V)}.
\]
Finally let $U,V\in\ensuremath{\mathcal A}\xspace$. From the previous equation, we observe that
\[
\esp{(X_V - e^{-\lambda m(V\setminus U)} X_{U\cap V} )X_{U\cap V}} = e^{-\lambda m(V\disy (U\cap V))} - e^{-\lambda m(V\setminus U)} = 0.
\]
Therefore, since $X$ is a Gaussian process, $\espc{X_V}{X_{U\cap V}} = e^{-\lambda m(V\setminus U)} X_{U\cap V}$, and using the \ensuremath{\mathcal C}\xspace-Markov property applied to $C=U\setminus V$ with the fact $\vXCp{U\setminus V}=X_{U\cap V}$, we obtain the expected covariance,
\begin{align*}
\esp{X_U X_V} = \espb{ X_U \,\espc{X_V}{\ensuremath{\mathcal G}\xspace^{*}_{U\setminus V}} }
&= \espb{ X_U \,\espc{X_V}{X_{U\cap V}} } \\
&= e^{-\lambda m(V\setminus U)} \, \espb{ X_U  X_{U\cap V} } \\
&= e^{-\lambda m(V\setminus U)} \cdot e^{-\lambda m(U\setminus V)} = e^{-\lambda m(U\disy V)}.
\end{align*}
\end{proof}

\section{Definition of a general \emph{set-indexed Ornstein-Uhlenbeck process}}
\label{sec:gensiou}

Using the \ensuremath{\mathcal C}\xspace-Markov property obtained in Proposition \ref{prop:ssiou_c_markov} and the \ensuremath{\mathcal C}\xspace-transition system \ensuremath{\mathcal P}\xspace from Corollary \ref{cor:ssiou_markov_dens}, we can finally define a general \emph{set-indexed Ornstein-Uhlenbeck process}.

\begin{definition} \label{def:siou}
A process $X$ is called a \emph{set-indexed Ornstein-Uhlenbeck process} if
\begin{enumerate}[\it (i)]
\item $X_\ensuremath{{\emptyset'}}\xspace\sim\nu$, where $\nu$ is a given initial probability distribution;
\item $X$ is \ensuremath{\mathcal C}\xspace-Markov with a \ensuremath{\mathcal C}\xspace-transition system given by \eqref{eq:ssiou_markov_dens}.
\end{enumerate}
\end{definition}

Theorem 2.2 in \cite{Balanca(2012)} proves the existence of such processes in the canonical probability space $(\ensuremath{\mathbf{R}}\xspace^{\ensuremath{\mathcal A}\xspace}, \Pr_{\nu})$ for any initial probability distribution $\nu$. Then, $\Pr_{\nu}$ is the probability measure on $\ensuremath{\mathbf{R}}\xspace^{\ensuremath{\mathcal A}\xspace}$ under which the canonical process defined by $X_U(\omega)=\omega(U)$ for all $\omega\in\ensuremath{\mathbf{R}}\xspace^{\ensuremath{\mathcal A}\xspace}$ is a set-indexed Ornstein-Uhlenbeck process.
In the particular case of Dirac initial distribution, the complete determination of the laws of $X$ is given by the following result.

\begin{proposition} \label{prop:siou_deltax}
For any $x\in\ensuremath{\mathbf{R}}\xspace$, under the probability $\Pr_x$, the canonical set-indexed Ornstein-Uhlenbeck process $X$ is the Gaussian process defined by the covariance structure
\begin{align} \label{eq:siou_deltax1}
\forall U\in\ensuremath{\mathcal A}\xspace;\quad &\esp[_x]{X_U} = x\ e^{-\lambda m(U)}, \\
\forall U,V\in\ensuremath{\mathcal A}\xspace;\quad &\cov_x(X_U,X_V) = \frac{\sigma^2}{2\lambda}\pthb{ e^{-\lambda m(U\disy V)} - e^{-\lambda( m(U)+m(V) )} }.\label{eq:siou_deltax2}
\end{align}
\end{proposition}

\begin{proof}
We first check that $X$ is a Gaussian process under the probability $\Pr_x$. \\
Let $A_1,\dotsc,A_k\in\ensuremath{\mathcal A}\xspace$ and $\lambda_1,\dotsc,\lambda_k\in\ensuremath{\mathbf{R}}\xspace$. Without any loss of generality, we can suppose that $\ensuremath{\mathcal A}\xspace_\ell = \brc{A_0=\ensuremath{{\emptyset'}}\xspace,A_1,\dotsc,A_k}$ is a semilattice and we denote $C_i = A_i\setminus(\cup_{j=0}^{i-1} A_j)$ for all $i\in\brc{1,\dotsc,k}$. Then, using notations from Corollary \ref{cor:ssiou_markov_dens}, we have
\begin{align*}
\espbb[_x]{\exp\pthbb{ i\sum_{j=1}^k \lambda_j X_{A_j} }}
&= \espbb[_x]{\exp\pthbb{ i\sum_{j=1}^{k-1} \lambda_j X_{A_j} } \,\espcB[_x]{\exp\pthb{i\lambda_k X_{A_k}}}{\ensuremath{\mathcal G}\xspace_{C_k}^*} } \\
&= \espbb[_x]{\exp\pthbb{ i\sum_{j=1}^{k-1} \lambda_j X_{A_j} } \,\exp\pthb{i\lambda_k Z_{C_k}} \,\espb[_x]{\exp\pth{i\lambda_k Y_{C_k}}} } \\
&= \exp\pthbb{-\frac{\lambda_k^2\sigma^2_{C_k}}{2}} \,\espbb[_x]{\exp\pthbb{ i\sum_{j=1}^{k-1} \lambda'_j X_{A_j} } },
\end{align*}
since $Z_{C_k}$ is weighted sum of $X_V$, $V\in\brc{A_0,\dotsc,A_{k-1}}$.
Therefore, by induction on $k$, we get the characteristic function of a Gaussian variable.

In order to obtain the mean and the covariance functions, we consider the case $k=3$, with the semi-lattice $\{\ensuremath{{\emptyset'}}\xspace, A_1=A_2\cap A_3, A_2, A_3\}$.
We compute
\begin{align*}
\espB[_x]{\exp\pthB{ i( \lambda_2 X_{A_2} + \lambda_3 X_{A_3} ) } }
&= \exp\pthB{-\frac{1}{2}\lambda_3^2\sigma^2_{C_3}} \,\espB[_x]{\exp\pthB{ i( \lambda_2 X_{A_2} + \lambda_3 Z_{C_3} ) } } \\
&= \exp\pthB{-\frac{1}{2}\lambda_3^2\sigma^2_{C_3}} \,\espB[_x]{\exp\pthB{ i( \lambda_2 X_{A_2} + \lambda_3 e^{-\lambda m(A_3\setminus A_1)} X_{A_1} ) } }.
\end{align*}
Using the \ensuremath{\mathcal C}\xspace-Markov property applied to $C_2=A_2\setminus A_1$, we get
\begin{align*}
\espB[_x]{\exp\pthB{ i( \lambda_2 X_{A_2} + \lambda_3 X_{A_3} ) } }
&= \exp\pthB{-\frac{1}{2}\lambda_3^2\sigma^2_{C_3} - \frac{1}{2}\lambda_3^2\sigma^2_{C_2}} \\
&\qquad\times\espB[_x]{\exp\pthB{ i\pthb{ \lambda_2 e^{-\lambda m(A_2\setminus A_1)} + \lambda_3 e^{-\lambda m(A_3\setminus A_1)} } X_{A_1} } }.
\end{align*}
Then, the \ensuremath{\mathcal C}\xspace-Markov property applied to $C_1=A_1\setminus\ensuremath{{\emptyset'}}\xspace$ leads to
\begin{align*}
&\espB[_x]{\exp\pthB{ i( \lambda_2 X_{A_2} + \lambda_3 X_{A_3} ) } } \\
&\qquad\qquad = \exp\pthB{-\frac{1}{2}\lambda_3^2\sigma^2_{C_3} - \frac{1}{2}\lambda_3^2\sigma^2_{C_2} - \frac{1}{2}\pthb{ \lambda_2 e^{-\lambda m(A_2\setminus A_1)} + \lambda_3 e^{-\lambda m(A_3\setminus A_1)} }^2 \sigma_{C_1}^2 } \\
&\qquad\qquad\qquad\qquad\times\espB[_x]{\exp\pthB{ i \pthb{ \lambda_2 e^{-\lambda m(A_2\setminus A_1)} + \lambda_3 e^{-\lambda m(A_3\setminus A_1)}}  e^{-\lambda m(A_1)} X_\ensuremath{{\emptyset'}}\xspace } } \\
&\qquad\qquad = \exp\pthB{-\frac{1}{2}\lambda_3^2\sigma^2_{C_3} - \frac{1}{2}\lambda_3^2\sigma^2_{C_2} - \frac{1}{2}\pthb{ \lambda_2 e^{-\lambda m(A_2\setminus A_1)} + \lambda_3 e^{-\lambda m(A_3\setminus A_1)} }^2 \sigma_{C_1}^2 } \\
&\qquad\qquad\qquad\qquad\times\exp\pthB{ i\pthb{ \lambda_2 e^{-\lambda m(A_2)} + \lambda_3 e^{-\lambda m(A_3)} } x }.
\end{align*}
The mean of $X$ comes from the last line. The covariance is obtained from the cross term in front of $\lambda_2\lambda_3$:
  \begin{align*}
    \sigma_{C_1}^2 e^{-\lambda m(A_2\setminus A_1)} e^{-\lambda m(A_3\setminus A_1)}
    &= \frac{\sigma^2}{2\lambda}\pthb{1 - e^{-2\lambda m(A_1)}} e^{-\lambda m(A_2\disy A_3)} \\
    &= \frac{\sigma^2}{2\lambda}\pthb{ e^{-\lambda m(A_2\disy A_3)} - e^{-\lambda( m(A_2)+m(A_3) )} },
  \end{align*}
  since $A_1=A_2\cap A_3$ and $\sigma_{C_1}^2 = \frac{\sigma^2}{2\lambda}\pthb{1 - e^{-2\lambda m(A_1)}}$.
\end{proof}

\section{Multiparameter Ornstein-Uhlenbeck process} \label{sec:mpou}

In the particular case of the indexing collection $\ensuremath{\mathcal A}\xspace=\brc{\ivff{0,t}; t\in\ensuremath{\mathbf{R}}\xspace_+}$ endowed with the Lebesgue measure $m$, the set-indexed Ornstein-Uhlenbeck processes studied in Sections \ref{sec:siou} and \ref{sec:gensiou} reduce to the classical one-dimensional Ornstein-Uhlenbeck process.

In the multiparameter setting, a natural extension of the stationary Ornstein-Uhlenbeck process can be defined by
\begin{equation} \label{eq:mpsou}
  \forall t\in\ensuremath{\mathbf{R}}\xspace_+^N;\quad Y_t = \int_{-\infty}^{t} \sigma\, e^{\scpr{\alpha}{u-t}}\dt W_u.
\end{equation}
where $\sigma > 0$, $\alpha=(\alpha_1,\dots,\alpha_N)\in\ensuremath{\mathbf{R}}\xspace^N$ with $\alpha_i > 0$ and $W$ is the Brownian sheet. The covariance of this process is given by
\[
  \esp{Y_s Y_t} = \prod_{i=1}^N \int_{-\infty}^{s_i\wedge t_i} \sigma^2 e^{\alpha_i(u_i-s_i-t_i)}\,\dt u_i = \frac{\sigma^2}{\prod_{i=1}^N \alpha_i} \exp\brcbb{ -\sum_{i=1}^N \alpha_i (s_i+t_i - s_i\wedge t_i) }.
\]
Hence, $Y$ is a stationary set-indexed Ornstein-Uhlenbeck process on the space $\ensuremath{\mathbf{R}}\xspace_+^N$ endowed with the indexing collection $\ensuremath{\mathcal A}\xspace=\brc{\ivff{0,t}; t\in\ensuremath{\mathbf{R}}\xspace^N_+}$ and the measure $m_\alpha$ defined on the Borel $\sigma$-field by
\begin{equation}\label{eq:m-alpha}
\forall A\in\ensuremath{\mathcal B}\xspace(\ensuremath{\mathbf{R}}\xspace^N); \quad m_\alpha(A) = \sum_{i=1}^N \alpha_i \lambda_1 \pth{A\cap e_i},
\end{equation}
where $\lambda_1$ is the Lebesgue measure on $\ensuremath{\mathbf{R}}\xspace$ and $e_1,\dotsc,e_N$ are the axes of $\ensuremath{\mathbf{R}}\xspace^N$: $e_1=\ensuremath{\mathbf{R}}\xspace\times\brc{0}^{N-1}$, $e_2=\brc{0}\times\ensuremath{\mathbf{R}}\xspace\times\brc{0}^{N-2}$, \ldots

\medskip

The following proposition extends this result to the general set-indexed Ornstein-Uhlenbeck process defined in Section \ref{sec:gensiou}, proving that it also has a natural integral representation in the particular multiparameter case.

\begin{proposition} \label{prop:siou_mpou}
  Let $Y=\{Y_t;\;t\in\ensuremath{\mathbf{R}}\xspace_+^N\}$ be the multiparameter process defined by
  \begin{equation} \label{eq:mpou}
    \forall t\in\ensuremath{\mathbf{R}}\xspace_+^N;\quad Y_t = e^{-\scpr{\alpha}{t}}\bktbb{Y_0 + \sigma\int_{\ivof{-\infty,t}\setminus\ivof{-\infty,0}} e^{\scpr{\alpha}{u}}\dt W_u },
  \end{equation}
  where $\sigma > 0$, $\alpha=(\alpha_1,\dots,\alpha_N)\in\ensuremath{\mathbf{R}}\xspace^N$ with $\alpha_i > 0$ for each $i\in\brc{1,\dotsc,N}$, $W$ is the Brownian sheet and $Y_0$ is a random variable independent of $W$.

  \noindent Then, $Y$ is a set-indexed Ornstein-Uhlenbeck process of Definition~\ref{def:siou} on the space $(\ensuremath{\mathcal T}\xspace,\ensuremath{\mathcal A}\xspace,m_\alpha)$, with $\ensuremath{\mathcal A}\xspace=\{[0,t];\;t\in\ensuremath{\mathbf{R}}\xspace_+^N\}$ and $m_{\alpha}$ defined in \eqref{eq:m-alpha}.
\end{proposition}
\begin{proof}
  First we observe that the measure $m_\alpha$ satisfies, for all $s,t\in\ensuremath{\mathbf{R}}\xspace_+^N$,
  \[
    m_\alpha\pthb{ \ivff{0,s}\cap\ivff{0,t} } = \sum_{i=1}^N \alpha_i (s_i\wedge t_i) = \scpr{\alpha}{s\curlywedge t}\quad \text{where} \quad s\curlywedge t := (s_1\wedge t_1,\dotsc,s_N\wedge t_N).
  \]
  Let $t_1,\dotsc,t_k$ be in $\ensuremath{\mathbf{R}}\xspace_+^N$ and $\lambda_1,\dots,\lambda_k$ in $\ensuremath{\mathbf{R}}\xspace$.
  For any fixed $x_0\in\ensuremath{\mathbf{R}}\xspace$, $Y^{x_0}$ denotes the Gaussian process defined by
  \[
    \forall t\in\ensuremath{\mathbf{R}}\xspace_+^N;\quad Y^{x_0}_t = e^{-\scpr{\alpha}{t}}\bktbb{x_0 + \sigma\int_{A_t}e^{\scpr{\alpha}{u}}\dt W_u },
  \]
  where $A_t = \ivof{-\infty,t}\setminus\ivof{-\infty,0}$.

  Let $Y$ be the $\ensuremath{\mathbf{R}}\xspace_+^N$-indexed process defined by \eqref{eq:mpou} and denote by $\nu$ the law of $Y_0$. Since $Y_0$ and $W$ are independent, we have
  \begin{align*}
    \espB{e^{i \sum_{j=1}^k \lambda_j Y_{t_j} }}
    = \int_\ensuremath{\mathbf{R}}\xspace \espB{e^{i \sum_{j=1}^k \lambda_j Y^{x_0}_{t_j} }} \, \nu(\dt x_0),
  \end{align*}
  Let us determine the mean and covariance of the process $Y^{x_0}$, for any $x_0\in\ensuremath{\mathbf{R}}\xspace$:
  \[
  \forall t\in\ensuremath{\mathbf{R}}\xspace_+^N; \quad \espb{Y^{x_0}_t} = x_0 \, e^{-\scpr{\alpha}{t}} = x_0\, e^{-m_\alpha(\ivff{0,t})}
  \]
  and for all $s,t\in\ensuremath{\mathbf{R}}\xspace_+^N$,
  \begin{align*}
    \cov(Y^{x_0}_s,Y^{x_0}_t)
    &= \sigma^2 e^{ -\scpr{\alpha}{s+t} }\espbb{ \int_{A_s} e^{\scpr{\alpha}{u}} \dt W_u \int_{A_t} e^{\scpr{\alpha}{u}} \dt W_u }
    = \sigma^2 e^{ -\scpr{\alpha}{s+t} } \int_{A_{s\curlywedge t}} e^{2\scpr{\alpha}{u}} \dt u \\
    &= \frac{\sigma^2}{2^n \prod_{j=1}^n \alpha_j }  e^{ -\scpr{\alpha}{s+t} }\pthb{ e^{2\scpr{\alpha}{s\curlywedge t}} - 1 } \\
    &= \frac{\sigma^2}{2^n \prod_{j=1}^n \alpha_j }  e^{ -m_\alpha(\ivff{0,s})-m_\alpha(\ivff{0,t}) }\pthb{ e^{2 m_\alpha(\ivff{0,s}\cap\ivff{0,t}) } - 1 } \\
    &= \frac{\widetilde{\sigma}^2}{2} \pthB{ e^{-m_\alpha(\ivff{0,s}\disy\ivff{0,t})} -  e^{ -\pth{m_\alpha(\ivff{0,s})+m_\alpha(\ivff{0,t})} } }
    = \cov_{x_0}(X_{\ivff{0,s}},X_{\ivff{0,t}}),
  \end{align*}
  where $X$ is the canonical set-indexed Ornstein-Uhlenbeck process with parameters $(\widetilde{\sigma},\lambda=1)$ with notations of Proposition~\ref{prop:siou_deltax}. Therefore, the process $Y^{x_0}$ has the same law as $X_{[0,\bullet]}$ starting from $x_0$ and
  \[
    \espB{e^{i \sum_{j=1}^k \lambda_j Y^{x_0}_{t_j} }} = \espB[_{x_0}]{e^{i \sum_{j=1}^k \lambda_j X_{\ivff{0,t_j}} }}.
  \]
  Consequently
  \[
    \espB{e^{i \sum_{j=1}^k \lambda_i Y_{t_j} }} = \int_\ensuremath{\mathbf{R}}\xspace \espB[_{x_0}]{e^{i \sum_{j=1}^k \lambda_j X_{\ivff{0,t_j}} }} \, \nu(\dt x_0) = \espB[_\nu]{e^{i \sum_{j=1}^k \lambda_j X_{\ivff{0,t_j}} }},
  \]
  which states that $Y$ and $X_{[0,\bullet]}$ have the same law and concludes the proof.
\end{proof}

\begin{remark}
We have exhibited an unusual measure $m_\alpha$ on $\ensuremath{\mathbf{R}}\xspace^N$, which only charges the axes $(e_i)_{i\leq N}$. This measure is also interesting when the set-indexed Brownian motion (siBM) is considered on the space $(\ensuremath{\mathcal T}\xspace,\ensuremath{\mathcal A}\xspace,m_\alpha)$ with $\alpha=(1,\dotsc,1)$, as it corresponds to a classic multiparameter process called the additive Brownian motion (see e.g. \cite{Khoshnevisan(2002)}). Conversely, since we know that the Brownian sheet is a siBM on the space $(\ensuremath{\mathcal T}\xspace,\ensuremath{\mathcal A}\xspace,\lambda)$, where $\lambda$ is the Lebesgue measure, we could also define a different multiparameter Ornstein-Uhlenbeck process using the Lebesgue measure instead of $m_\alpha$.
\end{remark}

\begin{remark}
  A different multiparameter extension of the Ornstein-Uhlenbeck process has already been introduced in the literature (e.g. see \cite{Wang(1988), Wang(1995)} and \cite{Graversen.Pedersen(2011)}). It admits an integral representation given by,
  \begin{equation} \label{eq:mpou2}
    \forall t\in\ensuremath{\mathbf{R}}\xspace_+^N;\quad Y_t = e^{-\scpr{\alpha}{t}}\bktbb{Y_0 + \sigma\int_0^{t}e^{\scpr{\alpha}{u}}\dt W_u }.
  \end{equation}
  If we consider a Markov point of view, the definition given in Proposition \ref{prop:siou_mpou} seems more natural. Indeed, as described in \cite{Balanca(2012)}, the transition probabilities of the process described in Equation \eqref{eq:mpou2} do not strictly correspond to those of the set-indexed Ornstein-Uhlenbeck, and can not be extended to the set-indexed formalism. Furthermore, we observe that the model \eqref{eq:mpou2} does not embrace the natural stationary case described in equation \eqref{eq:mpsou}.
\end{remark}

\

redpajama_set_name: RedPajamaArXiv


On a lightly floured surface roll out half of the puff pastry dough into a 9 inch square. Cut dough into four (four and a half inch) squares; place on baking sheet. Repeat with remaining dough. In a large bowl toss apples sliced with lemon juice. In small bowl, combine 1 tablespoon of the sugar and cinnamon; stir into apples. Lay apple slices on pastry, overlapping slightly. Bake in a 400 ° F oven for 15 minutes; sprinkle the remaining 1 tablespoon of sugar over apples. Return tarts to oven and bake about 10 minutes longer or until pastry is puffed and golden. To prepare the glaze, in a small bowl, combine apricot jam and brandy or orange marmalade and liqueur; brush the glaze over warm tarts. Serve warm or at room temperature. Makes 8 tarts.


text:
Victor Martinez on the Pirates: "I have no respect for no one on that team"
By Craig CalcaterraJul 2, 2015, 8:30 AM EDT
In this age of "you gotta tip your cap to the other team" kinds of quotes it's rare to hear real animosity coming from a major league clubhouse. But you heard it yesterday from Victor Martinez, referring to the Pirates. And not just at a single player. He is not pleased with that entire team. From the Detroit Free Press:
"I have no respect for no one on that team, including Cole and their coaching staff."
The context: Justin Verlander hit Starling Marte with a pitch on Tuesday night. It was not, it appeared, intentional, as it came in a 1-2 count, there had been nothing to provoke Verlander and, to be fair, Verlander's command has not been fantastic lately. Later, however, when Martinez was up, Gerrit Cole hit him in the ribs in what everyone on the Tigers, Brad Ausmus included, thinks was intentional. Martinez:
"Yeah, everyone knows that," Martinez said when asked by WXYT-FM reporter Jeff Riger if he thought Cole's pitch was intentional . . . If they think that Verlander hit Marte with a 1-2 count — he was battling that at-bat — if they really think we did it on purpose, they're playing the wrong sport."
No matter what you think of this situation — and you have to be opposed to guys throwing at other guys intentionally — you have to love Cole's comment when told of Martinez's words: "That's his opinion, man." The dude abides.
For purposes of interleague play, the Tigers and the Pirates are designated as "rivals," mostly because each lacks a traditional or natural rival in the other league. Now, one assumes, the rivalry actually has some teeth.
Tags: Detroit Tigers, Gerrit Cole, Justin Verlander, Pittsburgh Pirates, Starling Marte, Victor Martinez

redpajama_set_name: RedPajamaCommonCrawl


text:
Make an Impact: Donate Bitcoin to Projects on the Ground in Nepal
Posted April 28th, 2015 by Connie Gallippi & filed under Professional.
This post was submitted by Elizabeth Ploshay, Connie Gallippi, Victoria van Eyk, and Sarah Martin
Last week, Nepal was hit with a magnitude 7.8 earthquake – the worst in 80 years. More than 4,000 people died in the wreckage and hundreds more are without food, water, or shelter.
Natural disasters shake not just the core of our communities, but also the core of our humanity. Many of us feel a deep seated need to reach out and help during major catastrophes. But it can sometimes seem overwhelming when the scale is so large, and hard to know how to make a real difference.
Three leaders in the Bitcoin ecosystem – the BitGive Foundation, ChangeTip, and BitPay – have solutions to help:
The BitGive Foundation is running a campaign to raise donations for Medic Mobile, a nonprofit organization that uses mobile technology to improve health service delivery. Here's a message from the Medic Mobile Team:
"The Medic Mobile team is actively exploring how we can help with local, coordinated relief efforts after the earthquake that devastated Kathmandu on Saturday, April 25th. Our third largest office is based there and very thankfully all of our teammates are safe. The Bitcoin community has already been so supportive of our efforts in the past and we'd be grateful for that support again. Every donation will go directly to our Nepal office and help as we deploy the right technology. Thank you from our whole team."
Click here for more information about the campaign and how you can contribute.
ChangeTip is encouraging the Bitcoin community to contribute to the Red Cross via Twitter tips. More than 400 ChangeTip community members contributed to the Red Cross from Sunday to Monday as part of an ongoing, week-long campaign. The campaign currently exceeds more than $3,200 in micropayments.
BitPay also processes donations for Save the Children, an international relief organization mobilizing humanitarian assistance in Nepal. Save the Children began accepting bitcoin donations last fall after a successful 2013 campaign with the BitGive Foundation to raise funds for the Philippines Typhoon Haiyan relief effort. To donate in bitcoin to Save the Children in bitcoin click here.
How can you help support the people, families, and communities in Nepal? Reach out to more charitable organizations and encourage them to accept bitcoin donations. BitPay charges no (0%) transaction fees, and nonprofits receive all (100%) of their contributions in their local, fiat currency.
As we all know, Bitcoin is not just a fast and easy way to send money. It's an incredibly powerful tool to help those most in need. Let's join together and show the world how Bitcoin can make a meaningful difference in real people's lives.
Elizabeth Ploshay is an Account Manager at BitPay and Board Director at The Bitcoin Foundation, Connie Gallippi is the Founder and Executive Director of the BitGive Foundation, Victoria van Eyk is the Vice President of Community Development at ChangeTip, and Sarah Martin is the Head of Special Projects at the Digital Currency Council (DCC).
Make an Impact: Donate Bitcoin to Projects on the Ground in Nepal – Digital Currency Council | AllCryptoCurrencies.com


The Value of Information Exchange in Electric Vehicle Charging.
Event: 35th International Conference on Information Systems (ICIS) , 14.-17. Dez. 2014 , Auckland, New Zealand.
Renewable energy integration is accompanied by highly volatile energy generation, which urges energy suppliers to use costly countermeasures to prevent energy imbalances, grid instabilities and power outages. Therefore, using demand-side approaches to shift flexible demand over time is a promising opportunity. In the case of electric vehicles, research papers mostly discuss vehicle drivers' individual charging strategies based upon pricing information. The objective of this paper is to quantify the aggregate economic benefit of an advanced metering approach wherein electric vehicle drivers simply provide information about the start of the next trip to the energy supplier, who can then optimize the charging strategy for all drivers based on this information. By using a quantitative model and a multi-agent simulation for evaluation, we analyze original data from Germany to conclude that advanced metering can enable significant savings. Finally, we present a pricing scheme that would incentivize the drivers to provide truthful information.


We survived our first week back to school, but it has been crazy around here. It's been a huge adjustment getting back on a schedule. Even though we were late our first day, I made everyone run to the front porch for a quick picture. Instead of cute signs, we keep it simple and use our fingers to say what grade we are in.
Oh, and guess what I got in the mail this weekend?
I'm so excited. I'll have some fun projects to show you and possibly a giveaway.
That fabric cutter looks pretty cool!!


Location: Charlottesville, Va. | Newport News, Va.
The Short Story: The Rams closed out the Virginia Challenge and CNU Captains Classic on Saturday, setting pace for the conference championships taking place in two weeks.
Marcus Finnie took third place overall in long jump after recording a mark of 7.14m.
The 4x400 meter relay team set their best time of the season with a 3:11.25. Bigal Harrison, Chukwuezugo Aguolu, Kahlil Shepard, and Devon Thompson each contributed to the sixth place finish.
Also earning a sixth place finish was Ian Davis in the 110 meter hurdles. The sophomore ran a 14.57 time.
Placing fifth in the javelin was Shawn Scornaienchi. His mark was 58.09m.
Jamik Alexander competed in the 200m, completing the half-lap in 21.76 seconds.
1:56.55 was all Austin Conway needed in the 800m for sixth overall. Coming in one spot behind Conway was sophomore Jordan Fountain at 1:57.05.
Placing fourth overall in the 1500 was sophomore Bryce Catlett. The distance runner recorded a time of 3:58.25.
Junior Bashir Idris took gold in the men's triple jump after clearing a distance of 14.09m.
Coming in at second in the shot put was freshman Ebenezer Belayneh.
The next time the Rams compete will be for the Atlantic 10 conference title. The meet begins Saturday, May 5.


text:
Youth Action Movement Eswatini
Flas Website
YAM on Facebook
Yam on Instagram
YAM on Twitter
Sexual Rights Are Human Rights!
The Family Life Association of Swaziland (FLAS) took a paradigm and strategic shift in 1999 after the IPPF conference to focus on young people and integrate Sexual and Reproductive and Health Rights programing. The focus has brought about a lot of controversy amongst the conservative and religious sector especially when SRHR is used in conjunction with young people. Hence the objective of today's article is to explain what are Sexual and Reproductive Health and Rights as well as the essence of providing comprehensive SRHR services and information to young people in Eswatini.
Sexual and Reproductive Health and Rights (SRHR) encompasses all of the rights and issues surrounding a person's Sexual and Reproductive life. These rights are closely linked with other internationally recognised human rights, such as the right to privacy, the right to education and information, the right to equality and freedom from violence and all forms of discrimination, the right to the highest attainable standard of health etc.
The term SRHR combines four separate but interrelated concepts: Sexual Health (SH), Reproductive Health (RH), Sexual Rights (SR) and Reproductive Rights (RR). Sexual and Reproductive Health cover the right to decide if, when and how often to have children, right to live free from diseases, the right to have access to accurate, comprehensive and confidential information. On the other hand, Sexual and Reproductive Rights cover the sexual pleasure, the right to sexual privacy, the right to have the full range of contraceptives, the right to choose your partner as well.
SRHR therefore encompasses many human rights that are internationally, regionally and nationally recognised and protected. These Rights are still considered extremely controversial, especially when this term is used in conjunction to young people. They are particular contentious as they also recognise and protect sexual and gender diversity, issues which are not yet accepted in many communities.
SRHR and Youth Situation in Eswatini
FLAS primarily focus on young people aged 10-24 years to provide comprehensive and integrated SRHR services and information. The reasons the Association focuses in the area of SRHR are outlined below:
INADEQUATE and LIMITED SRHR POLICY IMPLEMENTATION
The country has signed and ratified a number of SRH international and regional policies, some of which have been operationalised through policies and legislation. However, there still remains a gap in terms of implementation due to the unawareness by the population – at all levels and absence of operational instruments such as guidelines and legislation. This is notable in areas affecting women and girls. In spite of the many efforts to stimulate interest in Sexual and Reproductive Health and Rights matters, knowledge and support remains relatively weak across the board.
Although there has been a notable improvement in the policy framework in the country over the years, there still remain some barriers to making healthy choices. The policy environment with respect to social and programmatic enablers is particularly challenging with concerns ranging from the protection of adolescent girls from sexual violence, discrimination and the enabling environment for meaningful consultation with and engagement of adolescents.
HIGH MATERNAL, NEO NATAL AND CHILD MORTALITY
Maternal and new-born deaths during childbirth are a result of a complex mix of factors such as inadequate services, low levels of community awareness, physical and financial barriers as well as traditional cultural values and gender norms. In Swaziland, all of these factors play a role and are further complicated by the severe HIV epidemic.
Swaziland has one of the highest maternal mortality rate at 589 per 100 000 live births and, the infant mortality also stands at 85 per 1000 live births. The high maternal mortality is as a result of various factors and compounded by the HIV situation where 26% of the reproductive population is living with HIV and HIV prevalence.
LIMITED ACCESS TO SRHR SERVICES ESPECIALLY TO VULNERABLE AND MARGINALISED GROUPS
In a country where the general dominating trends are pervasive poverty and economic inequality, access to quality Sexual and Reproductive Health services remains a key priority area. The social marginalization of vulnerable groups calls for stimulation of a better policy landscape and service delivery mechanisms. Limited access to appropriate information, gender inequality, abuse of cultural norms also add to the barriers that limit service availability and as well as service uptake in settings where those services have been made available
With over 50% of the country's total population below age 20, there is a need to prioritise young people and enhancing their capacities to make informed decisions. One of the most striking manifestations of this is that, 45% of the under-18 populations is orphaned or vulnerable (MICS 2015), with one in every four Swazi children having lost one or both parents. These youths are at increased risk of childhood poverty, abuse, violence and low educational attainment, which in turn further increases their vulnerability to contracting HIV (NERCHA 2014).
It is vital to the future of Swaziland that young people be afforded opportunities for social, personal and professional growth and development. Most urgently, they require access to activities that will equip them with the information, confidence and ability to not only participate fully in society, but also navigate the multitude of life choices and issues that lay before them.
FLAS together with IPPF will strive to work towards a world where women, men and young people everywhere have control over their own bodies, and their destinies. A world where they are free to choose parenthood or not; free to decide how many children they will have and when; free to pursue healthy sexual lives without fear of unwanted pregnancies and sexually transmitted infections, including HIV. A world where gender or sexuality are no longer a source of inequality or stigma. We will not retreat from doing everything we can to safeguard these important choices and rights for current and future generations.
For Enquires/ Comments
Email; maxwell@flas.org.sz
Cell: 7618 5770/ 7918 5770
Comprehensive Sexuality Education (CSE) for Youth People
Comprehensive Sexuality Education (CSE) For Young People.
On 7 December 2013, Ministers of Health and Education from 20 Countries in Eastern and Southern Africa met to discuss the common challenges facing young people in the region. While they recognised that significant progress had been made to address the needs of adolescents and young people with respects to ensuring access to life skills based HIV and Comprehensive Sexuality Education (CSE) and youth friendly sexual and reproductive health (SRH) services: they realised that there was still more work to be done. Notably the Eastern and Southern Africa remains the region mostly affected by HIV, its faces high rates of adolescent maternal mortality and among other social and public health challenges.
The 20 countries then crafted the Ministerial Commitment on CSE and SRH services for Adolescents and Young people in Eastern and Southern Africa – otherwise known as the ESA commitment. In it, they agreed to work collaboratively towards…
Young People, Education and Crime, Gender based violence & Substance Abuse.
On this video we look at Young People, Education and Crime, Gender based violence & Substance Abuse. Meet Sandy a social worker from Phumelela Projects who works with children with the law. -
"Not that we don't have dreams as young people. We do have dreams but we are discouraged by the poor access to youth programme funds. The responses are not welcoming; they willl say we need some capital before we can even be considered for additional funding and we do not have this. So we will go to crime." - Young person, aged 21
Criminal involvement and low educational attainment among young people are two social problems with common roots and overlapping features. Not much is known about how these two problems are interrelated. Criminal involvement may causally influence educational attainment, and vice versa. At the same time, criminal involvement and educational attainment outcomes can both be driven by multiple, potentially overlapping, observed and unobserved factors, such…