\documentclass[JEP,XML,SOM,Unicode,NoEqCountersInSection,NoFloatCountersInSection,published]{cedram} \datereceived{2020-02-03} \dateaccepted{2021-10-20} \dateepreuves{2021-11-03} \usepackage[scr=rsfs,cal=euler]{mathalfa} \multlinegap0pt \renewcommand{\theenumi}{\alph{enumi}} \renewcommand{\theenumii}{\roman{enumii}} \newenvironment{enumeratei} {\bgroup\def\theenumi{\roman{enumi}}\def\theenumii{\arabic{enumii}}\begin{enumerate}} {\end{enumerate}\egroup} \makeatletter \let\p@enumii\relax \makeatother \newcommand{\ldot}{\mathbin{.}} \newcommand{\lcdots}{\ldot\cdots\ldot} \newcommand{\RedefinitSymbole}[1]{% \expandafter\let\csname old\string#1\endcsname=#1 \let#1=\relax \newcommand{#1}{\csname old\string#1\endcsname\,}% } \RedefinitSymbole{\forall} \RedefinitSymbole{\exists} \newcommand\mto{\mathchoice{\longmapsto}{\mapsto}{\mapsto}{\mapsto}} \newcommand{\psfrac}[2]{\sfrac{(#1)}{#2}} \newcommand{\spfrac}[2]{\sfrac{#1}{(#2)}} \usepackage[all,cmtip]{xy}\let\labelstyle\textstyle \usepackage{stmaryrd} \usepackage{bbm} \usepackage{mathtools} \theoremstyle{plain} \newtheorem{themBJ}{Theorem} \renewcommand*{\thethemBJ}{\Alph{themBJ}} \newtheorem{them}{Theorem}[section] \newtheorem{pro}[them]{Proposition} \newtheorem{pronotation}[them]{Proposition/Notation} \newtheorem{prodefinition}[them]{Proposition/Definition} \newtheorem{lemnotation}[them]{Lemma/Notation} \newtheorem{cor}[them]{Corollary} \newtheorem{lem}[them]{Lemma} \newtheorem*{claim}{Claim} \theoremstyle{definition} \newtheorem{vocabulary}[them]{Vocabulary} \newtheorem{defi}[them]{Definition} \newtheorem{convention}[them]{Convention} \newtheorem{notation}[them]{Notation} %? 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Its quantile process $(G_t)_{t\in\mathbb{R}}:\mathopen]0,1\mathclose[\to\mathbb{R}^\mathbb{R}$, given by $G_t(\alpha)=\inf\{x\in\mathbb{R}:\mu_t(\mathopen]-\infty,x\mathclose])\geqslant\alpha\}$, is not Markov in general. We modify it to build the Markov process we call ``Markov-quantile'': there is a \emph{unique} Markov process $X$ with marginals $\mu_t$, being a limit for the finite dimensional topology of quantile processes where the past is made independent of the future at finitely many times (many non-Markovian limits exist in general). Strikingly, no regularity is required for the family~$\mu$. Moreover, if $\mu$ is increasing for the stochastic order, $X$ has increasing trajectories. This is an analogue of a result of Kellerer dealing with the convex order. In a companion paper \cite{BouJui2} it is also proved that if $\mu$ is absolutely continuous in the Wasserstein space $\mathcal{P}_2(\mathbb{R})$, $X$ is solution of a Benamou--Brenier transport problem with marginals $\mu_t$, providing a Markov representation of the continuity equation, unique in the sense above. \end{abstract} \alttitle{Le processus Markov-quantile attaché à une famille de marges} \altkeywords{Processus markovien, processus quantile, transport optimal, équation de continuité, processus croissant, théorème de Kellerer, transport optimal martingale optimal, peacocks, copule} \begin{altabstract} Soit $\mu=(\mu_t)_{t\in \mathbb{R}}$ une famille à un paramètre de mesures de probabilité sur $\mathbb{R}$. Son processus quantile $(G_t)_{t\in\mathbb{R}}:\mathopen]0,1\mathclose[\to\mathbb{R}^\mathbb{R}$, donné par $G_t(\alpha)=\inf\{x\in\mathbb{R}:\mu_t(\mathopen]-\infty,x\mathclose])\geqslant\alpha\}$, n'est en général pas markovien. Nous le modifions pour construire le processus markovien que nous nommons \og Markov-quantile\fg: il existe un \emph{unique} processus markovien $X$ de marges $\mu_t$ qui est limite, pour la convergence fini-dimensionnelle, de processus quantiles dont le passé est rendu indépendant du futur en un nombre fini d'instants (beaucoup de limites non markoviennes existent en général). Il est frappant qu'aucune hypothèse de régularité sur la famille $\mu$ n'est nécessaire. En outre, si $\mu$ est croissante pour l'ordre stochastique, les trajectoires de $X$ sont croissantes. Ceci est un analogue d'un résultat de Kellerer traitant de l'ordre convexe. Dans un article associé [8] on montre aussi que si $\mu$ est absolument continue dans l'espace de Wasserstein $\mathcal{P}_2(\mathbb{R})$, $X$ est solution d'un problème de transport de Benamou--Brenier avec marges $\mu_t$, et fournit donc une représentation markovienne de l'équation de continuité, unique dans le sens donné plus haut. \end{altabstract} \maketitle \vspace*{-2\baselineskip} \tableofcontents \mainmatter \section{Introduction}\label{sec:intro} Rather surprisingly, the basic question of the existence of a \emph{Markov} martingale with prescribed marginals $(\mu_t)_{t\in\R}$ remains open since the partial result \cite{Ke72} of 1972. More generally, only the method of \cite{Ke72}, rediscovered or revisited by several authors, seems to be available to provide Markov processes with prescribed marginals---and satisfying an additional constraint, as being a martingale. In this paper we give two new methods---one of them rests on order arguments---to solve this type of question. We apply them to measures $(\mu_t)_{t\in\R}$ in stochastic order. This detailed introduction explains precisely this context and all our results. \subsection{Formal structure of the article}\label{subsec:formal} We prove three main theorems, stated and labeled as \ref{them:a}, \ref{them:b}, and \ref{them:c} in this introduction and proved in the order~\ref{them:c},~\ref{them:a},~\ref{them:b} in the paper; see also their interdependence in Figure \ref{fig:un}, where appears also the Main Theorem of a companion paper \cite{BouJui2}. Theorem \ref{them:a} answers the Main Problem p.\,\pageref{problem} below, and is a general theoretical result in Probability Theory; it builds a certain stochastic process with given marginals: the \emph{Markov-quantile} process. Theorem \ref{them:b} gives a convergence result to it. Theorems \ref{them:c} presents an application of the Markov-quantile process to another context (Martingales and a theorem of Kellerer \cite{Ke72,Ke73}), giving by the way the Main Problem additional motivations, see Section~\ref{subsec:intro_kellerer} and Figure \ref{fig:un}. We also prove Theorem \ref{them:sous_n}, linked with Theorem \ref{them:c}. Being a bit more technical, it is not stated in this introduction. Another application (to Optimal Transport and the Continuity Equation) in presented in \cite{BouJui2}. In this introduction we first give the very necessary notions to state Theorems \ref{them:a}--\ref{them:b} as quickly as possible in Section~\ref{subsec:resultats}, then state Theorem \ref{them:c} in Section~\ref{subsec:intro_kellerer}. We quickly comment on \cite{BouJui2} in Section~\ref{subsec:intro_transport}. We slow the flow in Section~\ref{subsec:intro_quantilemarkovien} to give a qualitative insight into the Main Problem and its difficulties, that also shows why it is a natural problem in itself. We give in Section~\ref{subsec:organisation} the structure of the article and in Section~\ref{subsec:notation} an index of our notation. This paper treats of Measure, Probability, and Transport Theories. To be understood by a large panel of readers, we give the definitions of the more specific notions of each of these fields, or \emph{Reminders} about them if needed. \subsection{Our results and their motivation}\label{subsec:resultats} Take $(\espace_\tau)_{\tau\in\Tt}$ a (finite or not) family of measurable spaces; $\prod_{\tau\in\Tt}\espace_\tau$ is endowed with its cylindrical $\sigma$-algebra, generated by the preimages of those of the factors by the projections. \begin{reminder}\label{reminder:process} A \emph{process} is a family $X=(X_\tau)_{\tau\in\Tt}$ of measurable maps \[ X_\tau:\Omega\to \espace_\tau, \] called random variables, defined on the same probability space $(\Omega,\P)$. In this article, contrary to what may be considered usual, no measurability condition is required on $\Omega\times \Tt$. For every $\Tt'\subset \Tt$, $(X_\tau)_{\tau\in\Tt'}$ defines a map $F_{\Tt'}$ from~$\Omega$ to $\prod_{\tau\in\Tt'}\espace_\tau$. The \emph{law} of $(X_\tau)_{\tau\in\Tt'}$ is the pushed-forward probability measure $(F_{\Tt'})_\#\P$ on $\prod_{\tau\in\Tt'}\espace_\tau$, which is also called the \emph{marginal law} of the measure $(F_{\Tt})_\#\P$ on $\prod_{\tau\in\Tt'}\espace_\tau$. \end{reminder} Now let $\mu_\tau$ be a probability measure on $E_\tau$, for each $\tau$. \begin{notation}\label{notation:marg} For all measurable space $E$, $\m(E)$\ and $\p(E)$\ are the spaces of measures and probability measures on $E$. If $\Tt'\subset\Tt$, $\pr^{\Tt'}$ is the projection $\prod_{\tau\in\Tt}E_\tau\to\prod_{\tau\in\Tt'}E_\tau$; in case $\Tt=\{\tau_1,\ldots,\tau_m\}$ is finite, $\pr^{\tau_1,\ldots,\tau_m}$ means $\pr^{\{\tau_1,\ldots,\tau_m\}}$. When $P\in\p\bigl(\prod_{\tau\in\Tt}\espace_\tau\bigr)$ and $ss,\quad \law(X_t|\,(X_{u})_{u\leqslant s})=\law(X_t|\,X_s), \end{equation} where $\law(X_t|\,(X_u)_{u\leq s})$ is the law of $X_t$ conditionally to the $\sigma$-algebra generated by the $X_u$ for $u\leq s$. See also Definition \ref{defi:markov2}, where the Markov property is introduced only through notions defined in this article. For point \eqref{q:b}, $\como\in\ma((\mu_t)_{t\in\R})$ is defined as follows. \begin{reminder}[The quantile process]\label{remind:quantile}The quantiles of a measure $\mu\in\p(\R)$ generalize the notion of the median, which is the quantile of level $\sfrac{1}{2}$. The \emph{quantile of~$\mu$ of level $\alpha$} is the smallest real number $x_\mu(\alpha)$ such that $\mu(\op]-\infty,x_\mu(\alpha)])\geq \alpha$ and $\mu([x_\mu(\alpha),+\infty\clo[)\geq 1-\alpha$. The quantile process $X=(X_\tau)_{\tau\in \Tt}$, defined on $\Omega=[0,1]$ with the Lebesgue measure, is given by $X_t(\alpha)=x_{\mu_t}(\alpha)$, and we denote $\law(X)$ by $\como\in \ma((\mu_t)_{t\in \Tt})$. In particular, $\law(X_t)=\mu_t$ for every $t\in \Tt$. See Definition \ref{defi:quantile2} for full details. \end{reminder} Alternatively, $\como$ is also implicitly defined by the fact that it is the unique measure such that for all $x\in\R$ and all $t>s$, if $X$ is a process of law $\como$: \begin{equation}\label{eq:quantile_as_minimum}\law(X_{t}|\,X_{s}\leqslant x)=\min\{\law(Y_{t}|\,Y_{s}\leqslant x): \law(Y)\in\ma((\mu_t)_{t\in\R})\}, \end{equation} where this minimum is with respect to the stochastic order: \begin{reminder}\label{remind:sto} The \emph{stochastic order} on $\p(\R)$ is defined by: $\mu\leqs\nu$ if, for all $x\in\R$, $\mu(\op]-\infty,x])\geqslant\nu(\op]-\infty,x])$. \end{reminder} The idea of the minimality property \eqref{eq:quantile_as_minimum} is the following. Among the processes with marginals $(\mu_t)_t$, the quantile process is the one that perturbs the least possible the order in which the elements of mass are distributed at each time: for any two times $s0$, $\law(X_t|X_0)=\mu_t$. Now $\law(X_t|X_0=0,\,X_{-1}=i)=\delta_i$ for $i\in\{0,1\}$, so that \eqref{eq:def_markovien} is false for $u=0$. As~proved in \cite[Prop.\,3]{Ju_seminaire}, $\como$ is Markov except if a phenomenon of this type happens, see details in Example \ref{ex:quantile}. In particular, $\como$ is Markov when the measures $\mu_t$ have no atoms (see a direct proof in Remark \ref{rem:demo_nonatomique}). \item\label{p2:rem:como_markovien} When $\como\in\ma((\mu_t)_{t\in \R})$ is Markov, $\mq=\como$. Indeed, then, $\como^{s,t}=\como^{s,r_1}\lcdots \como^{r_n,t}$ for every $s\leq r_1<\cdotss$, $\mq^{s,t}=\como^{s,t}$. Since both processes are Markov they coincide in law (see Corollary~\ref{cor:consist}). \end{enumerate} \end{rem} \begin{rem}[Justification of the name ``Markov-quantile'']While Properties \eqref{item:markov} and \eqref{item:transitions} of Theorem \ref{them:a}\eqref{p1:them:a} are satisfied by the product measure (law of the independent process) $\bigotimes_{t\in \R} \mu_t$, the quantile process $\como$ satisfies \eqref{item:transitions} and \eqref{item:minimal} in the sense that it satisfies \eqref{item:transitions} and its couplings $\como^{s,t}$ are minimal among those of the measures satisfying \eqref{item:transitions}. In fact, Theorem \ref{them:a} is constructive and builds $\mq$ as a modification of~$\como$; therefore we call this measure $\mq$ the \emph{``Markov-quantile'' measure} attached to $(\mu_t)_{t\in\R}$. \end{rem} In fact, a deeper convergence statement holds than that resulting from point \eqref{item:limite_de_compose} above. Indeed we introduce the following notion of a measure in $\ma((\mu_t)_{t\in\R})$ ``turned into a Markov law at a finite set of instants'', denoted in a way that is consistent with the notation of Theorem \ref{them:a}\eqref{item:limite_de_compose}. \begin{pronotation}\label{pro:markovinise_fini1} If $P\in\ma((\mu_t)_{t\in\R})$ and $R\subset \R$ is finite, there is a unique measure in $\ma((\mu_t)_{t\in\R})$, denoted by $P_{[R]}$, such that: -- $P_{[R]}$ is the law of a family of variables $(X_t)_{t\in\R}$ that is ``Markov at the instants of\/ $R$'' \ie \eqref{eq:def_markovien} holds with ``$\,\forall s\in R$'' instead of ``$\,\forall s\in\R$'', -- for the closure $I$ of each connected component of $\R\setminus R$, $(\pr^I)_\# P_{[R]}=(\pr^I)_\# P$. \end{pronotation} This proposition follows from the way we define $P_{[R]}$ in Definition \ref{defi:quantile_discretement_markovien}, using the concatenation of transport plans given by Definition \ref{defi:concatenation}. We show: \begin{themBJ}\label{them:b}There is an increasing sequence $(R_n)_{n\in\N}$ of finite subsets of $\R$ such that $\como_{[R_n]}\in\ma((\mu_t)_{t\in\R})$ converges weakly to $\mq$. \end{themBJ} \begin{reminder}\label{reminder:weak_convergence} A sequence $(P_n)_n$ of (probability) measures on some measurable topological space $E$ \emph{converges weakly} to $P$ if, for all bounded continuous function $f$, $\int f\dd P_n\to\int f\dd P$. For $E=\R^\R$ with the weak topology, this convergence amounts to the weak convergence of all finite marginals. \end{reminder} \begin{rem} Of course, not every sequence $(R_n)_{n\in \N}$ is admissible for Theorem \ref{them:b}. In fact, we prove a more precise version of Theorem \ref{them:b}, see Theorem \ref{them:deuxieme_sec}. It introduces the notion of \emph{essential atomic times} of $(\mu_t)_t$ that turn out to be the times contained in $\bigcup_n R_n$ for all sequence $(R_n)_{n\in \N}$ admissible for Theorem \ref{them:b}; see also Remark \ref{rem:choix} in this introduction. \end{rem} \subsubsection*{Our problem: a classical type of question} The problem of defining a measure or a process $P$ with given marginals and additional properties is a general problem that includes the Main Problem and has already been explored several times in pure and applied Probability Theory as well as in Analysis or Dynamics. Without claiming exhaustiveness on this rich topic we review some research streams and provide references. A result related to Theorem \ref{them:a}\eqref{item:croissant} is proved by Kamae and Krengel in \cite{KaKr}. The measures $(\mu_t)_{t\in \R}$ are in $\p(E)$, where $E$ is a partially ordered Polish space. Assuming the measures in stochastic order, in a suitable sense, the authors prove that there exists an increasing process in $\ma(\mu)$. Other orders can be considered together with expected properties on the processes. For $E=\R^d$, Chapter 8 of \cite{SS} proposes plenty of orders. In Stochastic Analysis and Mathematical Finance, the topic of peacocks and their associated martingales is closely related to our problem. ``Peacock'' stands for PCOC: Processus Croissant pour l'Ordre Convexe (French), that is, increasing process for the convex order. One aims at defining a martingale in $\ma(\mu)$, where $\mu=(\mu_t)_{t\in \R}$ are the marginals of some peacock, using various techniques. Most of the time the peacock is part of a specific class, so the purpose is more specific than the work of Kellerer presented in Section~\ref{subsec:intro_kellerer}. In most of this literature (see e.g., \cite{Gy,Dupire,BrSh,MaYo02,Ju_ejp,HTT,HiRo13,Hob16,pages} and the references therein) the martingales may or not be Markov. The papers by Lowther \cite{Low,Low2} on limits of diffusion processes for the finite dimensional convergence permitted some authors to refocus on the Markov setting (see, e.g., \cite{BHS,HiRoYo14,Ju_seminaire}), rediscovering Kellerer's work by the way. Lowther's proof consists in adapting the local volatility coefficient of a SDE without drift---as indicated by Dupire in his very influential note \cite{Dupire} on financial engineering---in~order to match the marginals of $(\mu_t)_{t\in \R}$ mollified in time and space. The solution being Markov-Lipschitz the ``demollification'' happens in such a way that the Markov property is preserved at the limit. At last, an important example in the topic are the fake Brownian motions, that are processes sharing some of the properties characterizing the standard Brownian motion: they are continuous Markov martingales with marginals $\mu_t=\mathcal{N}(0,t)$. See, e.g., \cite{MaYo02,HaKl,Albin,Ol,Hob_fake,HTT} for examples of constructions. \subsection{Relations to Kellerer's Theorem}\label{subsec:intro_kellerer} If you forget about $\mq$ itself, Theorem~\ref{them:a}\eqref{item:croissant} gives the following existence property. \begin{cor}\label{coro:c} If $(\mu_t)_{t\in \R}\in\p(\R)^\R$ is increasing for $\leqs$, there exists a Markov process $X=(X_{t})_{t\in\R}:\Omega\to\R^\R$ such that $\law(X)\in\ma((\mu_t)_t)$ and that the trajectories $t\mto X_t(\omega)$ are increasing. \end{cor} This extends to the case of the stochastic order the famous theorem of Kellerer on martingales and submartingales with given marginals, Theorem \ref{them:kellerer_intro} below. Our Theorem \ref{them:c} recovers, with a different proof, Kellerer's result, as well as (simultaneously) Corollary \ref{coro:c} on increasing processes. The proof of Theorem \ref{them:c} is also completely independent of that of Theorem \ref{them:a}. Moreover, the method used to show it leads to an existence statement for certain Markov processes, Theorem \ref{them:sous_n}, omitted in this introduction. To state Theorem \ref{them:c} we need to recall two definitions. \begin{figure}[ht] \hspace*{\fill}\xymatrix@=.7cm{ &\txt{Theorem \ref{them:c}\\ (and Theorem \ref{them:sous_n})\\ {\em Martingale Theory \& Peacocks}% }\ar[dr]\\ \txt{Lemma \ref{lem:closed}}\ar[ur]\ar[dr]&\txt{Theorem \ref{them:a}\\\hspace*{-1.2cm}{\em Fundamental Probability Theory}\hspace*{-1.2cm}}\ar[d]\ar[r]^-{\txt{\scriptsize immediate}}&\txt{Corollary \ref{coro:c},\\ \ie enhancement\\of Theorem \ref{them:c}\\ in the case of $\leqs$}\\ &\txt{Theorem \ref{them:b}\\ {\em Stochastic Processes}}\ar[d]&\\ &\hspace*{-3cm}\left(\mbox{$\begin{array}{c}\text{Main Theorem of the companion paper \cite{BouJui2}}\\ \text{(and other theorems of it)}\\ \text{\em Optimal Transport \& Continuity Equation} \end{array}$ }\right)\hspace*{-3cm} }\hspace*{\fill} \caption{Main theorems: interdependence and field of application.}\label{fig:un} \end{figure} \begin{defi}\label{defi:leqc}Two measures $\mu$ and $\nu$ on $\R$, with finite first moments, are said to be in convex order $\leqc$, respectively in convex stochastic order $\leqcs$, if for every convex, respectively convex increasing function $\varphi$: \begin{align}\label{eq:order} \int \varphi\, \dd \mu\leq \int \varphi\, \dd \nu. \end{align} \end{defi} Notice that $\mu\leqs\nu$ if and only if \eqref{eq:order} holds for (bounded) increasing functions $\varphi$. Now we define a martingale. We do it in the Markov framework (all we need), where this is a bit simpler. We add in \eqref{item:increasing_coupling} a terminology of our own. \skpt \begin{defi}\label{defi:martingale} \begin{enumerate} \item \label{item:martingale}\ A measure $P$ on $(\R^d)^2$ is a \emph{martingale coupling} if for every non-negative continuous bounded function $f:\R^d\to \R$: \begin{equation}\label{eq:def_martingale}\iint f(x)(y-x) \dd P(x,y)=0. \end{equation} For $\Tt\subset\R$, a Markov measure $P$ on $(\R^d)^{\Tt}$ is a Markov martingale if for every $\{s,t\}\subset\Tt$ with $s0}$---actually here, we determine it completely. A modification makes $X$ Markov. Namely, consider the concatenation $X^{[0]}$ at time $0$ of $t\in \R^-\mto\alpha(-t)$ and $t\in \R^+\mto\beta t$, where $(\alpha,\beta)$ is uniformly picked in $[0,1]^2$, in place of $\alpha=\beta$ picked in $[0,1]$. Then $X^{[0]}$ is the only possible answer to the Main Problem: it is Markov (hence $(X_t)_{t<0}$ and $(X_t)_{t>0}$ are independent) and equal to the quantile process where the latter is Markov. Moreover, it satisfies the properties of Theorem \ref{them:a}, so it is the Markov-quantile process. \end{ex} \begin{defi}\label{defi:atomic}A measure without atom is said to be diffuse. We say that $t$ is an \emph{atomic time} of a family $(\mu_t)_{t\in\R}$ of measures if $\mu_t$ is not diffuse. \end{defi} \begin{ex}\label{ex2}Now take $\mu_0$ any non-diffuse measure and, for $t\neq 0$, $\mu_t$ any diffuse measure, for instance $\mu_0\ast \theta_t$, where $\theta_t$ stands for $\mu_t$ of Example \ref{ex1}, or $\mu_t=\la$. The quantile process is $(X_t)_t=(\min\{x\in \R:\,\mu_t(\op]-\infty,x])\geq \alpha\})_t$; restricted to $\R^-$ or $\R^+$ it is Markov. Pick $\beta$ uniformly on $[0,1]$. On $([0,1]^2,\la\otimes \la)$, we consider the modified process $X^{[0]}$ defined by: \begin{align*} X_t^{[0]}(\alpha,\beta)= \begin{cases} X_t(\alpha)&\text{if }t\leq 0\\ X_t(\alpha^-+\beta(\alpha^+-\alpha^-))&\text{it }t>0, \end{cases} \end{align*} where $\alpha^+=\alpha^-=\alpha$ if $\mu_0(X_t(\alpha))=0$, and otherwise $\op]\alpha^-,\alpha^+\clo[$ is the maximal open interval such that $X_0(\alpha')=X_0(\alpha)$ for all $\alpha'$ in it. Let us introduce $I_\alpha=\{\alpha\}$ in the first case and $I_\alpha=\op]\alpha^-,\alpha^+\clo[$ in the second one. Conditionally on the fact that~$X_0^{[0]}$ equals some atom $x_0=X_0(\tilde\alpha)$ of $\mu_0$, the values of $X^{[0]}_t$ for $t>0$ are made independent of those of $X^{[0]}_t$ for $t<0$. Indeed, the vector $(\alpha,\alpha^- +\beta(\alpha^+-\alpha^-))$ is uniformly distributed on $I_{\tilde\alpha}\times I_{\tilde\alpha}$: this generalizes Example \ref{ex1}; $X^{[0]}$ is Markov and has law\/ $\mq\in \ma(\mu)$. \end{ex} \begin{ex}When the set $R$ of atomic times of $(\mu_t)_{t\in \R}$ is finite, one may repeat at each $r\in R$ the independence operation described above at time $0$, to produce a Markov process. Moreover, one may check that it does not matter in which order, \hbox{because} these operations commute. The resulting process is indeed our Markov-quantile process. With the notation of the paper, its law is $\como_{[R]}$. Similarly it is easy to imagine the Markov-quantile process when the atomic times form a locally finite set, like, e.g.,~$\mathbb{Z}$. \end{ex} \begin{rem} See Section~\ref{sec:exemples} for more examples of Markov-quantile processes. \end{rem} The situation becomes complicated when the set $\A$ of atomic times is not locally finite or even worse, uncountable. Consider the following a priori reasonable approach. Let $(R_n)_{n\in \N}$ be a nested family of finite sets such that $R_\infty=\bigcup_n R_n$ is dense in $\A$. We consider the sequence $(\como_{[R_n]})_n$ \label{page:tentative_limite} and hope for a limit $Q$. Then we encounter three problems: \begin{itemize} \item By compactness of $\ma((\mu_t)_t)$, $(\como_{[R_n]})_n$ has an accumulation point (see similar reasonings in \cite{HiRo13,Ju_ejp,KaKr,Lis,Vi2}), but it has no reason to be unique. \item If $\A$ is uncountable, this limit needs not to satisfy the Markov property \eqref{eq:def_markovien} at times $s\in\A\setminus R_\infty$, at which we did not perform the modification of Examples \ref{ex1}--\ref{ex2}. A continuity assumption on $t\mto \mu_t$ could let hope to yield it (this was used, with other goals, for measures in stochastic or convex order, see, e.g., \cite{BHS,HiRoYo14}) but we do not make such an assumption. Also in the ``space of quantile levels'', the irregularity may be maximal: the set $\{(t,\alpha)\in\R\times[0,1]:\,X_t(\alpha)$ is an atom of $\mu_t\}$ needs not to be measurable. \item Anyway, limits of Markov processes are in general not Markov so here, property~\eqref{eq:def_markovien} is not ensured even at $s\in R_\infty$. To our knowledge, before the present paper this type of problem had principally one solution, based on Lipschitz kernels (see Lemma \ref{lem:lipschtz_implique_cinq}), first discovered by Kellerer \cite{Ke72,Low,HiRoYo14,BeJu16, BHS}. However, see \cite[Lem.\,5.3]{Naga} for a different statement. \end{itemize} Another consequence of this non stability of the Markov property is that it is also not possible to consider the sequence of quantile processes for mollified curves $\mu^{(n)}=(\mu_t\ast \theta_n)_t$, relying on the fact that all the measures $\mu_t^{(n)}$ are diffuse, so that each $\como\in\ma((\mu^{(n)}_t)_{t\in\R})$ is Markov.\enlargethispage{.5\baselineskip}% \skpt \begin{rem}\label{rem:choix} \begin{enumerate} \item In fact, the convergence in our Theorem \ref{them:a}\eqref{item:limite_de_compose}, and hence in its enhancement Theorem \ref{them:b}, rests on the \emph{order} $\leqlc$ introduced in Definition \ref{defi:geqlc}: for all $s$ and $t>s$, the choice of the times $r_i$ follows from that of a sequence in some set of measures, tending to the supremum of this set for $\leqlc$, see Lemma \ref{lem:existenceT}, in particular its point~\eqref{p3:def_L_R}. \item As the examples above suggest, for all $s$ and $t>s$, the composition appearing in point \eqref{item:limite_de_compose} of Theorem \ref{them:a} needs only using couplings $\como_{r_i,r_{i+1}}$ where the times $r_i$ are atomic. Adding non-atomic times has no effect. Similarly, in Theorem \ref{them:b}, that provides nested finite sets $R_n$ such that $\como_{[R_n]}\to\mq$, $R=\bigcup_nR_n$ may avoid all non-atomic times. Now it appears moreover, but only as a consequence of Theorem \ref{them:a} once it is proved, that all the atomic times of $(\mu_t)_t$ do not play the same role: \begin{itemize} \item Some are ``essential'' (see Definition \ref{defi:essentiel}). All of them that lie in $\op]s,t\clo[$ must eventually appear among the $r_i$ in Theorem \ref{them:a}\eqref{item:limite_de_compose}, and all of them must belong to $R$ in Theorem \ref{them:b}. This is possible as they turn out to be at most countable (Proposition \ref{pro:denombrable}). -- One may choose the $r_i$ in Theorem \ref{them:a}, or $R$ in Theorem \ref{them:b}, so that they avoid any fixed finite set of the other atomic times. \end{itemize} Therefore, the intersection of the sets $R$ satisfying the convergence property of Theorem \ref{them:b} is the set of the essential atomic times. The existence, for any sequence $(\mu_t)_t$, of the set of its essential atomic times, at most countable even if the set $\{(t,\alpha)\in\R\times[0,1]:\,X_t(\alpha)$ is an atom of $\mu_t\}$ is not measurable, is in itself a significant result of this article. Perhaps does this notion admit generalizations when the set of parameters or the measurable space, both equal to $\R$ in this work, are more general spaces. \end{enumerate} \end{rem} \begin{rem} It is also very important to notice that as soon as the set \[ \{(t,\alpha)\in\R\times[0,1]:\,X_t(\alpha)\text{ is an atom of }\mu_t\} \] is regular enough (see the examples of Section~\ref{sec:exemples} for clearly stated instances of this), the Markov-Quantile process may be explicitly computed, as it is done in the important Example \ref{ex:integers}. More generally, certain properties of this set and of $\mq$ are linked, see the whole of Section~\ref{sec:exemples}. Through its various examples, that section gives also an intuition of how $\mq$ behaves. \end{rem} \subsection{Organization of the paper}\label{subsec:organisation} In Section~\ref{sec:concatenation+kellerer+boujui} we introduce in Section~\ref{subsec:kernels_markov} kernels and transport plans, their composition and concatenation, and the Markov property expressed in this language. We give in Section~\ref{subsec:proof_kelsto} the structure of Kellerer's work \cite{Ke72,Ke73}, explain why it motivates our reasoning towards Theorem \ref{them:c}, and prove the latter. However, we postpone the introduction of one auxiliary notion, and the proof of Lemma \ref{lem:transitions_croissantes_limite} and of the essential Lemma \ref{lem:closed} to Section~\ref{sec:auxiliaire}. In Section~\ref{subsec:markovinification} we state and prove the ``Markovinification'' Theorem \ref{them:sous_n}. In Section~\ref{sec:auxiliaire} we introduce the auxiliary notions and results leading to the proofs of Lemmas \ref{coro}, \ref{lem:closed} and \ref{lem:transitions_croissantes_limite}, then also used in Section~\ref{sec:construction}, namely: \begin{enumerate} \item the ``lower orthant'' and stochastic orders, related suprema and the notion of increasing kernel in Section~\ref{subsec:stochastic_order}, \item the quantile transport and the notion of minimal coupling in Section~\ref{ss:minimality}, \item two distances, $\rho$ and $\tilde \rho$, inducing the weak topology on spaces of transport plans $\ma(\mu,\nu)$ in Section~\ref{subsec:rho}. \end{enumerate} Lemma \ref{coro} is proved in Section~\ref{subsec:proof_coro} and Lemmas \ref{lem:closed} and \ref{lem:transitions_croissantes_limite} are in Section~\ref{subsec:proof_lemmas}. Along the way, Section~\ref{subsec:kernels_markov}, Section~\ref{subsec:stochastic_order}, and Section~\ref{ss:minimality} also give all the background to understand in detail the three properties \eqref{item:markov}--\eqref{item:minimal} characterizing $\mq$ in Theorem~\ref{them:a}. In Section~\ref{sec:construction} we prove Theorems \ref{them:a} and \ref{them:b}: in Section~\ref{subsec:little_t} we explain how the situation may be pushed forward to the space $[0,1]$ of ``levels of quantiles'', in Section~\ref{subsec:preuve_premier} we~prove Theorem \ref{them:a}, \ie build the Markov-quantile process, and in Section~\ref{subsec:preuve_deuxieme} we state and prove Theorem \ref{them:deuxieme_sec} which is a more precise and complete version of Theorem \ref{them:b}. To do this we introduce the essential atomic times of $(\mu_t)_t$. In Section~\ref{sec:exemples} we exhibit the Markov-quantile process in a series of examples, state three last remarks about Theorem \ref{them:a}, and give open questions. \begin{note} When we introduce various tools, sometimes classical, we do it in a way and with remarks adapted to our context. The reader already knowing them may read quickly, taking notice of our few specific remarks, which are useful in the rest of the article. \end{note} \Subsection{Notation}\label{subsec:notation} \begin{enumerate} \item We gather in Table \ref{table:notation} the places where the notation we use widely is introduced, so that the reader can find them quickly if needed. \begin{table}[htb]\renewcommand{\arraystretch}{1.2} \mb\scriptsize\hspace*{\fill}\begin{tabular}{|p{.52\textwidth-.8ex}|l|}\hline We introduce: &in:\\\hline $\m(E)$, $\p(E)$, $\pr^{\Tt'}$, $P^t$ (similarly $\Gamma^t$), $P^{s,t}$, the set $\ma((\mu_\tau)_{\tau\in\Tt})$ & Notation \ref{notation:marg}\\\hline the quantile process $\como$ & Reminder \ref{remind:quantile}\\\hline $\como$ again, together with the quantile coupling $\como(\mu,\nu)$ & Definitions \ref{defi:quantile1} and \ref{defi:quantile_process}, Notation \ref{not:multi}\\\hline the stochastic order $\leqs$ & Reminder \ref{remind:sto}\\\hline $\mq$ & Definition \ref{pro:comp_conv}\eqref{bbb}\\\hline if \mn$P$ is some process, $R\subset\R$ and $\sharp R<\infty$\mb, $P_{[R]}$ & Notation \ref{pro:markovinise_fini1}, Definition \ref{defi:quantile_discretement_markovien}\\\hline $\leqc$ and $\leqcs$ & Definition \ref{defi:leqc}\\\hline $\la$ & Notation \ref{not:leb}\\\hline the composition $k\ldot k'$ of kernels & Section~\ref{ss:compos}\\\hline the kernels $k_P$ & Notation \ref{notation:k_P}\\\hline $\idk_E$ & Notation \ref{not:ident}\\\hline $\join(\mu,k)$ & Notation \ref{notation_join}\\\hline the transport plans $\id_{2,\mu}$ and $\id_{n,\mu}$ & Notation \ref{notation:transport_id}\\\hline if \mn$P$\mb\ is some process, $\trans P$ & Definition \ref{defi:transpose}\\\hline $P\circ P'$ & Definition \ref{defi:concatenation}\\\hline $\mathcal{N}^\ML_{s,t}$ & Definition \ref{defi:lipschitz}\\\hline $\mathcal{N}^\IT_{s,t}$ & Notation \ref{nota:Nik},\\\hline $x\leq y$ when \mn $(x,y)\in(\R^d)^2$\mb & Notation \ref{notation:intervallesRd}\\\hline $F_\mu$ or $F[\mu]$, if \mn$\mu$\mb\ is some measure & Definition \ref{def:cumul}\\\hline $\mu\leqlc\nu$ & Definition \ref{defi:geqlc}\\\hline $\lcsup_\tau P_\tau$ & Definition \ref{def:lcsup}\\\hline $\m(\mu)$, $\p(\mu)$, $\mdec(\mu)$ and $\pdec(\mu)$ & Notation \ref{notation:mdec}\\\hline $G_\mu$ & Definition \ref{defi:quantile2}\\\hline $\mathclose{\rfloor\hspace*{-1.1ex}\lceil}$ & Notation \ref{notation:intervallesRd}\\\hline the distance $\rho$ & Notation \ref{pro:rho}\\\hline $\tilde \rho$ & Section~\ref{ss:another}\\\hline the kernels $q_r,\, k_r$ and $\trans k_r$ & Notation \ref{notation:kernels}\\\hline $A_{r,x}$, $A_{r}$ and $\ell_r$ & Notation \ref{nota:Asx}\\\hline $L_R$, for \mn$R\subset\R$\mb & Notation \ref{lem:existenceT}\\\hline $\ell_R$ & Notation \ref{nota:kernel_tS}\\\hline $\lev$ & Definition \ref{defi:lev}\\\hline \end{tabular}\hspace*{\fill}\normalsize \mn \vspace*{.5\baselineskip} \caption{\label{table:notation}Places where our notation is introduced} \end{table} \item In this article, $\R^\ast$ or $\N^\ast$ means $\R\smallsetminus\{0\}$ or $\N\smallsetminus\{0\}$. If $E'\subset E$, $\mbox{\mb$\one_{E'}$\mn}:E\to\{0,1\}$ is the indicator function of $E'$ and, if $\mu\in\m(E)$, \mb$\mu\lfloor_{E'}$\mn\ stands for $\one_{E'}\mu$. The Dirac measure at $x\in E$ is denoted by \mb$\delta_x$\mn\ and \mb$\la$\mn\ stands for the Lebesgue measure on $[0,1]$, $\R$ or~$\R^d$. Most of the time we deal with $\la|_{[0,1]}$ so, when there is no ambiguity, we simply write $\la$ in this case. If $f:E\to F$ is measurable and $\mu\in\m(E)$, ${\text{\mb$f_\#\mu$\mn}}\in\m(F)$ is defined by $f_\#\mu(B)=\mu(f^{-1}(B))$. Product measures are denoted by \mb$\mu\otimes \nu$\mn. If $f$ and $g$ are functions, \mb$f\otimes g$\mn\ stands for $(x,y)\mto(f(x),g(y))$. \item Recall also Convention \ref{conv:croissant}: introducing $\{r_1,\ldots,r_m\}$ or $m$-tuples $(r_k)^m_{k=1}$ of real numbers, we mean implicitly that $r_1<\ldots\mu_t$ (for any $sb$ is ($a\leq b$ or ``$a$ and $b$ are not comparable''). To avoid ambiguity, we prefer to call it with a positive term, namely ``increasing''. Similarly, we use ``non increasing'' instead of ``decreasing''. This convention is chosen in many articles dealing with our topic, e.g., \cite{SS,KaKr,Low2,HiRoYo14} \end{enumerate} \end{vocabulary} \subsubsection*{Acknowledgements} The authors wish to thank Ji\v rí \v Cern\'y, Martin Huesmann, Christian Léonard, Emmanuel Opshtein and Xiaolu Tan for bibliographic or editorial suggestions as well as Michel Émery and Erwan Hillion for discussions on examples related to this work. \section{An extension of a theorem of Kellerer}\label{sec:concatenation+kellerer+boujui} \subsection{The Markov property, composition and concatenation of kernels and transport plans}\label{subsec:kernels_markov} Everywhere $E$, $E'$, $E''$ etc.\ are topological spaces (or sometimes Polish spaces) and $\mathcal{B}(E)$, $\mathcal{B}(E')$ and $\mathcal{B}(E'')$ their Borel $\sigma$-algebras. \begin{defi} A probability kernel, or kernel $k$ from $E$ to $E'$ is a map\vspace*{-3pt}\enlargethispage{.5\baselineskip}% \[ k:E\times \mathcal{B}(E')\to [0,1] \] such that $k(x,\cdot)$ is a probability measure on $E'$ for every $x$ in $E$ and $k(\cdot,B)$ is a measurable map for every $B\in {\mathcal B}(E')$. \end{defi} Probability kernels are usually interpreted as transition matrices, see Remark \ref{rem:matrix}: after one step a particle at $x$ in $E$ arrives at a random position in $E'$, distributed with respect to $k(x,\cdot)$. We often have that interpretation in mind. \begin{remnotation}\label{notation:k_P}Every transport plan $P\in\p(E\times E')$ can be disintegrated with respect to its first marginal $P^1\eqdefup(\pr^1)_\#P$ and a kernel that we denote by $k_P$, defined from $E$ to $E'$, so that:\vspace*{-3pt} \[\iint f(x,y)\,\dd P(x,y)=\int\left(\int f(x,y)\,k_P(x,\dd y)\right)\,\dd P^1(x)\] for every bounded continuous function $f$. Observe that $x\mto k(x,\cdot)$ is $P^1$-almost surely uniquely determined. \end{remnotation} \subsubsection{Composition and action of kernels}\label{ss:compos} Kernels $k$ from $E$ to $E'$ and $k'$ from $E'$ to $E''$ can be composed as follows:\vspace*{-3pt} \[(k\ldot k')(x,A)=\int_{E'} k'(y,A) k(x,\dd y).\] Similarly, acting on the right, kernels from $E$ to $E'$ \emph{transport}, or \emph{send} positive measures $\theta$ on $E$ on positive measures on $E'$. Acting on the left, they send (adequately integrable) functions $f:E'\to \R$, on functions $E\to \R$:\vspace*{-3pt} \[(\theta\ldot k)(A)=\int_{E'}k(y,A)\theta(\dd y)\quad \text{and:}\quad (k\ldot f)(x)=\int_{E'} f(y) k(x,\dd y).\] \noindent Associativity holds, e.g., $(k\ldot k')\ldot k''=k\ldot(k'\ldot k'')$, and $\theta\ldot(k\ldot f)=(\theta\ldot k)\ldot f$, where the action of measures on functions is the obvious one. This is consistent with the following remark. \begin{rem}\label{rem:matrix} We recall the usual interpretation of the composition as matrix multiplication. If $E=\{x_i\}_{i=1}^{n}$, $E'=\{y_j\}_{j=1}^{n'}$, $E''=\{z_k\}_{k=1}^{n''}$ are finite, a measure $\theta\in\m(E)$ is a row vector $(\theta(\{x_i\}))_{i=1}^{n}=(\theta_i)_{i=1}^{n}$, a kernel $k$ from $E$ to $E'$ is a matrix $k=((k_{i,j})_{i=1}^{n})_{j=1}^{n'}$, where $(k_{i,j})_{j=1}^{n'}$ is the measure $k(x_i,\,\cdot\,)\in\m(E')$, viewed as a vector, and a function $f$ from $E''$ to $\R$ is a (column) vector $f=(f(z_j))_{j=1}^{n''}$. Then, taking $k'=((k'_{j,k})_{j=1}^{n'})_{k=1}^{n''}$ a kernel from $E'$ to $E''$, $\theta\ldot k$, $k\ldot k'$ and $k'\ldot f$ introduced above have the same sense as products of matrices. \end{rem} \begin{notation}\label{not:ident} We denote by $\idk_E$ the identity kernel (that acts trivially) $(x,B)\mto\delta_x(B)=\one_B(x)$. \end{notation} \begin{notation}\label{notation_join} With $\mu\in\m(E)$ and $k$ a kernel from $E$ to $E'$ is naturally associated the law $\join(\mu,k)\in\m(\espace\times \espace')$, having $\mu$ as first marginal and the family $(k(x_0,\cdot))_{x_0\in E}$ as laws (on $E'$) conditioned by $x_0\in E$:\vspace*{-3pt} \[\forall B,B'\in{\mathcal B}(E)\times{\mathcal B}(E'),\quad \join(\mu,k)(B\times B')=\int_Bk(x,B')\dd\mu(x).\] In particular, $P=\join(P^1,k_P)$. \end{notation} \subsubsection{Composition of transport plans}\label{subsubsec:composition} If $P\in\ma(\mu,\mu')$ and $Q\in\ma(\mu',\mu'')$, we can compose them in a similar way as we compose kernels, getting the product:\vspace*{-3pt} \[P\ldot Q\eqdefup\join(\mu,k_P\ldot k_Q)\in\ma(\mu,\mu''),\quad \text{so that: }k_{P\ldot Q}=k_P\ldot k_Q.\] \begin{notation}\label{notation:transport_id} We denote $((\Id_E)_{i=1}^n)_\#\mu\in\ma((\mu)_{i=1}^n)$ by $\Id_{n,\mu}$ or simply $\id_n$ when there is no ambiguity. With $n=2$, $\Id_{2,\mu}=\join(\mu,\idk_E)$. It is moreover the identity transport: $\id_{2,\mu}{}\ldot P=P=P\ldot \id_{2,\mu'}$. \end{notation} \subsubsection{Action of transport plans on measures and functions}\label{subsubsec:action_des_transports} If $\mu\in\m(E)$ and $\mu'\in\m(E')$, transport plans $P\in\ma(\mu,\mu')$ have an action similar to that of kernels from $E$ to $E'$, on ($\mu'$-almost surely defined) classes of functions $f$ and on measures $\theta\in\m(E;\mu)$ absolutely continuous with respect to $\mu$. For instance, the latter are transported in $\m(E',\mu')$, as follows: \[\text{if $\theta\ll \mu$ has density $g$ and $B'\in{\mathcal B}(E')$,}\quad(\theta\ldot P)(B')=\int_{B'}\int_E g(x)\dd P(x,y).\] If $k$ is a kernel from $E$ to $E'$, $\theta\ldot k=\theta\ldot \join(\mu,k)$. Conversely $\theta\ldot P=\theta\ldot k_P$. \begin{defi}\label{defi:transpose} Take $P\in\ma(\mu_1,\ldots,\mu_k)$, or $P\in\ma((\mu_t)_{t\in\R})$. We define its transpose $\trans P\in\ma(\mu_k,\ldots,\mu_1)$, \resp $\trans P\in\ma((\mu_{-t})_{t\in\R})$ by: $ \trans P(B_1\times \ldots\times B_k)=P(B_k\times\ldots\times B_1)$, \resp $\trans P(\prod_iB_{t_i})=P(\prod_iB_{-t_i})$. \end{defi} The notation $\trans P$ is a reference to a transposed matrix in Remark \ref{rem:matrix}. For $P=\ma(\mu,\nu)$ we will often consider the bilinear map \[B:(\theta,f)\mto (\theta P)f=\theta (Pf)=\iint g(x)f(y)\,P(\dd x,\dd y),\] where $g$ is the density of $\theta$ with respect to $\mu$. The case $\theta=\mu\lfloor_{\op]-\infty,x]}$, $f=\one_{\op]-\infty,y]}$ with $B(\theta,\nu)=P(\op]-\infty,x]\times\op]-\infty,y])$ is of special interest, see Section~ \ref{subsec:stochastic_order}. \subsubsection{Concatenation of transport plans}\label{subsec:caten} (See, e.g., \cite[p.\,111]{Ke72}, \cite[p.\,23]{Vi2}.) \begin{defi}\label{defi:concatenation} If $\mu_i\in\p(E_i)$ for $i\in\{1,2,3\}$, if $P_{1,2}\in\ma(\mu_1,\mu_2)$ and $P_{2,3}\in\ma(\mu_2,\mu_3)$, their concatenation $P_{1,2}\circ P_{2,3}$ is the unique $R\in\p(\R^3)$ such that for every $ (B_1,B_2,B_3)\in\mathcal{B}(E_1)\times\mathcal{B}(E_2)\times\mathcal{B}(E_3)$: \begin{align}\label{eq:concatenation} R(B_1\times B_2\times B_3)&=\int_{x\in B_1}\int_{y\in B_2}\int_{z\in B_3} \dd\mu_1(x)k_{1,2}(x,\dd y)k_{2,3}(y,\dd z). \end{align} In particular, $R\in\ma((\mu_1,\mu_2,\mu_3)$, $(\pr^{1,2})_\#R=P_{1,2}$, and $(\pr^{2,3})_\#R=P_{2,3}$. \end{defi} \begin{rem} Let $k_{i,j}$ be a disintegration kernel for $P_{i,j}$ and $P_{2,1}\eqdefup\trans P_{1,2}$. The right side of \eqref{eq:concatenation} also reads: $\iint_{B_1\times B_2} \int_{B_3}k(y,\dd z)\dd P_{1,2}(x,y)$, hence also: \begin{equation} \int_{y\in B_2} \iint_{x\in B_1,\,z\in B_3}k_{2,1}(y,\dd x)k_{2,3}(y,\dd z)\mu_2(\dd y).\label{eq:concatenation3} \end{equation} Concatenation is ``reversed when time is reversed'' in the sense that $\trans P_{2,3}\circ{}\trans P_{1,2}=\trans(P_{1,2}\circ P_{2,3})$; this is immediate after \eqref{eq:concatenation3}. Formula \eqref{eq:concatenation} gives immediately that $\circ$ is associative, leading to its following generalization: \[P_{1,2}\circ\ldots\circ P_{n-1,n}\Bigl({\prod_{i=1}^n}B_i\Bigr)=\int_{(x_i)_i\in \prod_iB_i} \hspace{-4ex}\dd\mu_1(x_1)k_{1,2}(x_1,\dd x_2)\ldots k_{n-1,n}(x_{n-1},\dd x_n).\] \end{rem} \skpt \begin{rem} \begin{enumerate} \item In Section~\ref{subsubsec:composition}, one can also define $P\ldot Q$ as $\pr^{E,E''}_\#(P\circ Q)$. \item The composition and concatenation of transport plans find an easy interpretation in terms of random variables. If $(X_1,X_2,X_3)$ is a random vector with $\law(X_1,X_2)=P$ and $\law(X_2,X_3)=Q$ such that $(X_i)_{i\in\{1,2,3\}}$ is a Markov process ($X_1$ and $X_3$ are independent conditionally on $X_2$), then $P\ldot Q$ is the law of $(X_1,X_3)$ and $P\circ Q$ the law of $(X_1,X_2,X_3)$, see \eqref{eq:concatenation3}. \end{enumerate} \end{rem} \subsubsection{The Markov Property} We introduce the Markov property here in an alternative, equivalent way to the usual one. \begin{remdefi}\label{defi:markov2}As recalled in \eqref{eq:def_markovien}, a process $(X_t)_{t\in\R}$ is said to be \emph{Markov}~if:\vspace*{-3pt} \[ \forall s\in\R,\forall t>s,\quad \law(X_t|\,(X_{u})_{u\leqslant s})=\law(X_t|\,X_{s}). \] Denoting $\law((X_t)_t)\in\p(\R^\R)$ by $P$, this is equivalent to the fact that for all finite subset $S=\{s_1,\ldots,s_d\}$ of $\R$, $(\pr^S)_\#P$ is the concatenation $P^{s_1,s_2}\circ\cdots\circ P^{s_{d-1},s_d}$, where $P^{s_i,s_{i+1}}$ denotes $(\pr^{\{s_i,s_{i+1}\}})_\#P$. More generally, we say that any measure $P\in\p(\R^\R)$ is Markov if it satisfies this property. We extend these definition in the obvious way to processes indexed on subsets $R\subset \R$ and measures on $\p(\R^R)$. \end{remdefi} We recall the Kolmogorov--Daniell theorem and its usual corollary on Markov measures. \begin{pro}[Kolmogorov--Daniell theorem]\label{pro:inductive_limit} Let $E$ be a Polish space and, for each finite subset $S$ of\/ $\R$, $\mu_S$ be a probability measure on $E^S$. If $(\pr^{S'})_\#\mu_S=\mu_{S'}$ for any $S'\subset S$, there exists a unique $P\in \p(E^\R)$ with $(\pr^S)_\#P=\mu_S$ for all $S$. \end{pro} One of the most usual applications of Proposition \ref{pro:inductive_limit} is for measures $\mu_S$ of type $\mu_{s_1,s_2}\circ\cdots\circ \mu_{s_{d-1},s_d}$, where $S=\{s_1,\ldots,s_d\}$. \begin{cor}\label{cor:consist} Let $(\mu_{s,t})_{ss$, $P^{s,t}$ has increasing kernel, there exists a Markov measure $P'$ in $\ma((\mu_t)_t)$ such that each $P^{\prime s,t}$ is a limit of products $P^{s,r_1}\lcdots P^{r_m,t}$ with $\{r_1,\ldots,r_m\}\subset\op]s,t\clo[$. One may take $P'$ such that for each $(s,t)$, the limit is obtained with a sequence $(\{r^n_1,\ldots,r^n_{m(n)}\})_n$ such that $\max_{k=0}^{m(n)} |r_{k+1}-r_k|\to_{n\to\infty}0$, where $(r_0,r_{m+1})$ stands for $(s,t)$. \end{them} \begin{proof} If $t>s$, setting $P_{[R]}^{s,t}\eqdef P^{s,r_1}\lcdots P^{r_m,t}$ for all $R=\{r_1,\ldots,r_m\}\subset\op]s,t\clo[$, we~introduce \[ \mathcal{N}_{s,t}^{\rm (P)}=\bigcap_{\sigma>0}\overline{\{P_{[R]}^{s,t}:\,\max\nolimits_{k=0}^m |r_{k+1}-r_k|\leqslant\sigma\}}, \] where $(r_0,r_{m+1})$ stands for $(s,t)$. It is included in $\mathcal{N}_{s,t}^\IT$ by Lemma \ref{lem:transitions_croissantes_limite}, and it satisfies Assumptions \eqref{them:existence1}, \eqref{them:existence3} and \eqref{them:existence4} of Theorem \ref{them:existence}. Indeed, for \eqref{them:existence1}, $\mathcal{N}_{s,t}^{\rm (P)}\neq\varnothing$ as an intersection of nested non-empty compact (closed in the compact space $\ma(\mu_s,\mu_t)$) sets; \eqref{them:existence3} is true by definition, and \eqref{them:existence4} by Proposition \ref{pro:continuity}. Thus by Remark \ref{rem:otherNs}\eqref{rem:otherNsa}, $\mathcal{N}_{s,t}^{\rm (P)}$ satisfies all the assumptions of Theorem \ref{them:existence}. We are done. (Notice that the alternative definition $\mathcal{N}_{s,t}^{\rm (P)}=\overline{\{P_{[R]}^{s,t}\}}$ would have given the same result, except its last sentence.) \end{proof} Note that if $P$ is Markov the spaces $\mathcal{N}_{s,t}^{\rm (P)}$ and $\overline{\{P_{[R]}^{s,t}\}}$ are both reduced to $\{P^{s,t}\}$, so that the Markov measure obtained from any of them is $P$ itself. This conservation property also holds locally on intervals $I\subset \R$ if $(P_t)_{t\in I}$ is Markov. Notice also that Theorem \ref{them:sous_n} does not require $(\mu_t)_{t\in\R}$ to be increasing for $\leqs $. Theorem \ref{them:sous_n} links Section~\ref{sec:concatenation+kellerer+boujui} with the Markov-quantile process $\mq$ built in Section~\ref{sec:construction}. Indeed, taking $P=\como$, $\como^{s,t}$ is in $\mathcal{N}_{s,t}^\IT$ for all $s0,\,\exists\eta>0,\,\forall y\geqslant x,\quad \|y-x\|_\infty\leqslant\eta\implies F(x)\leqslant F(y)\leqslant F(x)+\varepsilon. \end{equation} \item \label{item:Flimite} $\lim_{\min_i(x_i)\to-\infty} F(x)=0$ and $\lim_{\min_i(x_i)\to+\infty} F(x)=1$, \item \label{item:Fdifferences_croisees} for every $h=(h_1,\ldots,h_d)\in[0,+\infty\clo[^d$ and $x \in \R^d$, the quantity \[ \sum_{\eps} \sigma(\eps) F(x+\eps h), \] which is the measure of the rectangle $]x,x+h]\subset \R^d$, is non-negative. Here $\eps=(\eps_1,\ldots,\eps_d)$ ranges over $\{0,1\}^d$, $\sigma$ is $1$ if $\sum \eps_i$ is even, $-1$ otherwise, and $\eps h$ means $(\eps_1h_1,\ldots,\eps_dh_d)$. \end{enumerate} \end{reminder} \begin{defi}\label{defi:geqlc} If $d\in\N^\ast$ and $m\in\op]0,+\infty\clo[$, following \cite[\S 6.G]{SS}, we define the \emph{lower orthant order} on $\{\mu\in\m(\R^d):\,\mu(\R^d)=m\}$ by: $\mu\leqlc\nu$ if $F_\mu\geqslant F_\nu$. \end{defi} \begin{defi}\label{def:lcsup}We call lower orthant supremum of a family $(P_\tau)_{\tau\in\Tt}$ of measures of same mass $m$ on $\R^d$, the smallest upper bound of $\{P_\tau\} _\tau$ for $\leqlc$, if it exists, \ie a~measure $P$ of mass $m$ such that: \begin{itemize} \item for every $\tau$, $P_\tau\leqlc P$, \item $P\leqlc Q$ as soon as $P_\tau\leqlc Q$ for every $\tau$. \end{itemize} By definition, if it exists it is unique. We denote it by $\lcsup_\tau P_\tau$. Similarly we define $\lcinf_\tau P_\tau$. \end{defi} \skpt \begin{rem}\label{rem:lcsup} \begin{enumerate} \item \label{p:lcsupa} In Reminder \ref{remind:F}, \eqref{item:Fonctions_partielles} and the first limit of \eqref{item:Flimite} pass to the infimum of functions that are both monotone and upper semi-continuous. If moreover $(P_\tau)_\tau$ has an upper bound $P$, then the second limit of \eqref{item:Flimite} holds. Indeed, for all $\tau$, $F[P_\tau]\geqslant F[P]$, so~that $\inf_\tau F[P_\tau]\geqslant F[P]$, and besides $\lim_{\min_i(x_i)\to+\infty}F[P](x)=1$. Thus, if $(P_\tau)_\tau$ is bounded from above, $\inf_\tau F[P_\tau]$ satisfies \eqref{item:Fonctions_partielles}--\eqref{item:Flimite} of Reminder \ref{remind:F}, so is a cumulative distribution function if and only if it satisfies \eqref{item:Fdifferences_croisees}. \item \label{p:lcsupb} If \eqref{item:Fdifferences_croisees} of Reminder \ref{remind:F} is satisfied by the functions $F[P_\tau]$, then $(P_\tau)_\tau$ has a lower orthant supremum, and $F[\lcsup_\tau P_\tau]=\inf_\tau F[P_\tau]$. \end{enumerate} \end{rem} \begin{remnotation}\label{nota:stosup} If $d=1$ the order $\leqlc$ is usually called stochastic order and denoted by $\leqs$; we will then call ``stochastic supremum'' the lower orthant supremum of Definition \ref{def:lcsup} and denote it by $\stosup$. \end{remnotation} \skpt \begin{lem}[Existence criteria for $\lcsup$] \label{lem:sto_order} \begin{enumerate} \item \label{item:suite_croissante_majoree} If, for $\leqlc$, a sequence $(P_n)_{n\in \N}\!\in\!(\p(\R^d))^\N$ is bounded from above, and increasing, \ie $n\!\leqslant\! m\Rightarrow P_{n}\!\leqlc\! P_{m}$, then $\lcsup_nP_n$ exists and $(P_n)_{n}$ converges weakly~to~it. \item \label{item:treillis} If a family $(P_\tau)_{\tau\in\Tt}$ is bounded from above for $\leqlc$ and if for every $\tau,\tau'\in \Tt$ there exists $\sigma\in \Tt$ such that $P_\sigma\geqlc P_\tau$ and $P_\sigma\geqlc P_{\tau'}$, then $\lcsup_\tau P_\tau$ exists and there is an increasing sequence $(P_{\tau_n})_n$ that converges weakly to it. \end{enumerate} The results extend in an obvious way to measures of mass $m>0$ in $\m(\R)$. \end{lem} \skpt \begin{proof} \eqref{item:suite_croissante_majoree} Set $F\eqdef\inf_nF[P_n]$. After Remark \ref{rem:lcsup}, showing that $F$ satisfies \eqref{item:Fdifferences_croisees} of Reminder \ref{remind:F} ensures the existence of $\lcsup_nP_n$. Consider $M\eqdef\sum_{\eps} \sigma(\eps) F(x+\eps h)$ as in Reminder \ref{remind:F}\eqref{item:Fdifferences_croisees}. Since $F[P_n]$ is decreasing, $F$ is its simple limit, so $M$ is the simple limit of $\sum_{\eps} \sigma(\eps) F[P_n](x+\eps h)$, hence $M\geqslant0$. We are done. The weak convergence is given by the pointwise convergence of the cumulative distribution functions, see Reminder~\ref{remind:billingsley}. \eqref{item:treillis} This is a diagonal construction. Let $\mathcal{C}=\bigcup_{k\in\N}\{x_k\}$ be a countable dense set in $\R^d$. Set $F\eqdef\inf_{\tau\in\Tt}F_\tau$. Then for every $(k,n)\in\N^{\ast2}$ we find $\tau_{k,n}\in \Tt$ such that $F[P_{\tau_{k,n}}](x_k)\leqslant F(x_k)+\sfrac1n$. For each $n$, by a finite induction using the assumption of~\eqref{item:treillis} on the $P_{\tau_{k,n}}$, we find $\sigma_{n}\in \Tt$ such that: $\forall k\leqslant n,\, F[P_{\sigma_{n}}](x_k)\leqslant F(x_k)+\sfrac1n$ and $(P_{\sigma_n})_n$ is increasing. Hence: \begin{equation}\label{eq:convergence_sur_partie_dense} \forall x\in \mathcal{C},\quad F[P_{\sigma_{n}}](x)\to F(x). \end{equation} By \eqref{item:suite_croissante_majoree}, $P=\lcsup_n P_{\sigma_n}$ exists and: \begin{equation}\label{eq:conv_sur_Rd} \forall x\in \R^d,\quad F[P_{\sigma_{n}}](x)\to F[P](x). \end{equation} Let us prove that \eqref{eq:convergence_sur_partie_dense} holds for any $x\in\R^d$, so that $F=F[P]$. Assume by contradiction that, for some $x$, $F(x)<\inf_nF[P_{\sigma_n}](x)$, \ie by definition of $F$, that for some $\tau\in\Tt$, $F(x)1$. Consider, e.g., \[ \mathcal{S}=\{P_1,P_2\}=\bigl\{\tfrac12(\delta_{(1,0)}+\delta_{(0,1)}),\tfrac12(\delta_{(0,0)}+\delta_{(2,2)})\bigr\}\subset\p(\R^2), \] then $\inf\{F[P_1],F[P_2]\}$ does not satisfy \eqref{item:Fdifferences_croisees} of Reminder \ref{remind:F} and $\mathcal{S}$ has no lower orthant supremum: both $P\eqdef\frac12(\delta_{(1,1)}+\delta_{(2,2)})$ and $P'\eqdef\frac12(\delta_{(0,2)}+\delta_{(2,0)})$ are upper bounds for it but no upper bound $P''$ satisfies $P''\leqlc P$ and $P''\leqlc P'$ (observe $F[P]$ and $F[P']$). \end{enumerate} \end{rem} \begin{rem}In the following we use several times the Lebesgue differentiation theorem for Borel measures; a reference is, e.g., \cite[\S2.8--2.9]{Fe}. \end{rem} \begin{prodefinition}\label{prodef:transitions} Take $\mu$ and $\nu$ in $\p(\R)$. We say that a transport plan $P\in\ma(\mu,\nu)$ has \emph{increasing kernel} if one (and then any) of the following statements holds: \begin{enumerate} \item\label{prodef:transitionsa} \label{item:transitions_croissantes_application_croissante} Initial definition: if $\theta,\theta'\ll \mu$ and $\theta$ and $\theta'$ have he same mass, then $\theta\leqs \theta'$ implies $\theta \ldot P\leqs \theta'\ldot P$. \item\label{prodef:transitionsb} \label{pt:condexp} For every increasing $h:\R\to[0,1]$, $P\ldot h$ is $\mu$-almost surely increasing, \ie more exactly, there is an increasing $\tilde{h}:\R\to [0,1]$ such that, for every bounded continuous function $g$: \[\int g(x)h(y) \dd P(x,y)= \int g(x)\tilde{h}(x)\dd \mu(x).\] \item\label{prodef:transitionsc} There exists a kernel $k$ in the $\mu$-equivalence class of $k_P$ such that $x\mto k(x,\cdot)$ is increasing from $(\R,\leq)$ to $(\p(\R),\leqs)$. \item\label{prodef:transitionsd} There exists a random vector $(X,Y)$ with $\law(X,Y)=P$ such that $x\in\R\mto\law(Y|\,X=x)$ is increasing from $(\R,\leq)$ to $(\p(\R),\leqs)$ (in the sense of the $\mu$\nobreakdash-equi\-valence classes of {\small increasing} functions: it is increasing on a set of full measure $F\subset \R$). \end{enumerate} \end{prodefinition} \begin{rem}Be cautious that having increasing kernel is distinct from being an increasing coupling, a notion defined in Definition \ref{defi:martingale}\eqref{item:increasing_coupling}. \end{rem} \begin{proof}[Proof of the equivalence in Proposition \ref{prodef:transitions}] Statements \eqref{prodef:transitionsc} and \eqref{prodef:transitionsd} are essentially a change of notation. (To get \eqref{prodef:transitionsd} $\Rightarrow$ \eqref{prodef:transitionsc}, notice that for $y$ in the $\mu$-null set $\R\setminus E$ of \eqref{prodef:transitionsd}, $k(y,\cdot)$ can be defined as $\stosup_{x\in E,\,x0$. Since each $b_i\in \R$ can be approached from the right by a sequence of non-atomic points for $\mu_i$, there exists $b'\geqslant b$ such that \[ \mu_1(\op]b_1,b'_1])+\cdots+ \mu_d(\op]b_d,b'_d])<\eps\AND \mu_1(b'_1)+\cdots+ \mu_d(b'_d)=0, \] hence in particular $Q(\partial R')=0$ for every $Q\in \ma((\mu_i)_{1\leq i\leq d})$ so that \eqref{pro:rhoi} applies to~$R'$ for $P$ and all the $P_n$. Thus:\vspace*{-3pt} \begin{align*} |P_n(R)- P(R)|&\leq|P_n(R)- P_n(R')|+|P_n(R')-P(R')|+|P(R')- P(R)|\\ &\leq 2\eps+|P_n(R')-P(R')|, \end{align*} and if \eqref{pro:rhoi} holds, $|P_n(R')-P(R')|\leqslant\eps$ for $n$ great enough, hence \eqref{pro:rhoii} follows. \eqref{pro:rhoii} $\Rightarrow$ \eqref{pro:rhoiii}. We apply an adapted version of the prior argument. Suppose \eqref{pro:rhoii} and take $\eps>0$. For every $i$ there exists a finite sequence\vspace*{-3pt} \[ -\infty=b_0^{(i)}(\eps)<\cdots< b^{(i)}_{N(\eps,i)}(\eps)=+\infty \] avoiding the big atoms of $\mu_i$, \ie so that $\mu_i(\op]b_k^{(i)},b_{k+1}^{(i)}\clo[)< \eps$ (this is classical and is proved, e.g., in \cite[\S 12]{Bi}, which deals with the modulus of continuity of càdlàg paths). Every $R=\op]-\infty,b]$ contains some rectangle $R^-_\eps$ and is included in the interior of some rectangle $R^+_\eps$, both bounded by consecutive points $(b^{(i)}_k)_{i,k}$. Using again \eqref{eq:rectangles} for the first and last terms: \begin{align*} |P_n(R)- P(R)|&\leq\underbrace{|P_n(R)- P_n(R_\eps^-)|}_{\leqslant d\eps}+\underbrace{|P_n(R_\eps^-)-P(R_\eps^-)|}_{\underset{n\to\infty}\longrightarrow0\ \text{by \eqref{pro:rhoii}}\quad\text{\bf$(\ast)$}}+\underbrace{|P(R_\eps^-)- P(R)|}_{\leqslant d\eps}, \end{align*} where {\bf$(\ast)$} is uniform as the rectangles $R_\eps^-$ are finitely many. We get \eqref{pro:rhoiii}. \end{proof} \subsubsection{Another expression for $\rho$; an alternative distance $\tilde \rho$}\label{ss:another} Let $\mu_1,\ldots,\mu_d$ be probability measures on $\R$ and $P$, $Q$ stand for elements of $\ma((\mu_i)_{i=1}^d)$. \begin{notation} When $f$, $g$ and $(f_i)_{i=1}^d$ are functions, by a slight abuse in this subsection, $f\ldot g$ denotes the function $(x,y)\mto f(x)g(y)$ and\/ $\prod_{i=1}^df_i$ the function $(x_1,\ldots ,x_d)\mto \prod_{i=1}^df_i(x_i)$ \end{notation} \begin{rem}\label{rem:product}By definition, $\rho(P,Q)=\sup\bigl|\int \prod_{i=1}^df_i \,\dd P-\int \prod_{i=1}^df_i \,\dd Q \bigr|$, where each $f_i$ ranges over $\{\one_{\op]-\infty,x]}:\,x\in\R\}$. But in fact: \end{rem} \skpt \begin{prodefinition}\label{pro:product} \begin{enumerate} \item\label{pro:producta} For any $P$ and $Q$ in $\ma((\mu_i)_{i=1}^d)$:\vspace*{-3pt} \begin{equation}\label{eq:rho_comme_sup} \rho(P,Q)=\sup\biggl|\int\prod_{i=1}^df_i \,\dd P-\int \prod_{i=1}^df_i \,\dd Q \biggr|, \end{equation} where each $f_i$ ranges over $\{f:\R\to[0,1]: f\text{ is decreasing}\}$. \item\label{pro:productb} Proposition \ref{pro:rho} may be stated with a distance $\tilde\rho$ based on $x\in \R^n\mto P([x,+\infty\clo[)$ in place of $F[P]$. Then $\tilde\rho$ satisfies \eqref{eq:rho_comme_sup} with \emph{increasing} functions $f_i$. \end{enumerate} \end{prodefinition} \begin{proof} Once \eqref{pro:producta} is shown, \eqref{pro:productb} is clear. To show \eqref{pro:producta}, by Remark \ref{rem:product}, we have only to prove $\geqslant$. For $f:\R\to [0,1]$ a decreasing function and $I(t)\eqdef f^{-1}([t,+\infty\clo[)$, $f(x)=\int_0^1 \one_{I(t)}(x)\dd t$, thus for $(f_i)_{i=1}^d$ such functions: \[\biggl(\prod_{i=1}^df_i\biggr) (x_1,\ldots, x_d)=\int_{t\in[0,1]^d}\one_{R(t)}(x_1,\ldots,x_d)\dd t,\] where $R(t)=I(t_1)\times\cdots\times I(t_d)$. Therefore: \begin{align*} \biggl|\int \prod_{i=1}^d f_i\,\dd P-\int \prod_{i=1}^d f_i \,\dd Q\biggr|&=\biggl|\int_{t\in[0,1]^d}P(R(t))-Q(R(t))\dd t\biggr|\\ &\leq \int_{t\in[0,1]^d}|P(R(t))-Q(R(t))|\dd t\\ &\leq \rho(P,Q).\qedhere \end{align*} \end{proof} Proposition \ref{pro:product} has a corollary in the case of transport plans ($d=2$). \begin{pro}\label{pro:multiplicationcontractante}Take $\mu$, $\mu'$ in $\p(\R)$ and $P$, $Q$ in $\ma(\mu,\mu')$. Then: \[\rho(P,Q)=\sup\{\rho(\theta\ldot P, \theta \ldot Q):\theta\in\mdec(\mu)\ \text{and $\theta$ has density bounded by $1$}\},\] where the distance $\rho$ on the right of the equality is that on $\p(\R)$. Hence if, for some $\nu\in \p(\R)$, $R\in\ma(\nu,\mu)$ is a transport preserving $\mdec$, \ie $\forall\eta\in\mdec(\nu)$, $\eta\ldot R\in\mdec(\mu)$, then $\rho(R\ldot P,R\ldot Q)\leqslant\rho(P,Q)$. \end{pro} \begin{proof} For the first equality, as $\int f\ldot g\, \dd P=((f\mu)\ldot P)\ldot g$, by Proposition \ref{pro:product}: $\rho(P,Q)=\sup_{f,g} |((f\mu)\ldot P)\ldot g-((f\mu)\ldot Q)\ldot g|$ and $\rho(\theta\ldot P, \theta \ldot Q)=\sup_g |(\theta\ldot P)\ldot g-(\theta\ldot Q)\ldot g|$. Now $\theta$ is as in the proposition if and only if $\theta=f\mu$ with $f:\R\to[0,1]$, decreasing. The result follows. Then if $R\in\ma(\nu,\mu)$ is as claimed, \[ \rho(RP,RQ)=\sup\{\rho(\theta\ldot P, \theta \ldot Q)|\,\theta=\bar\theta\ldot R, \] where $\bar\theta\in\mdec(\mu)$ and $\bar\theta$ has density bounded by 1$\}$. This set is included in that of the proposition, since the action of $R$ on $\bar\theta$ does not increase the maximum of its density. \end{proof} \subsection{Proof of Lemma \ref{coro} and remarks about it}\label{subsec:proof_coro} We prove Lemma \ref{coro} on the existence of a process consisting exclusively of increasing paths. Our proof requires the use of $\stoinf$ and $\stosup$ introduced in Section~\ref{subsec:stochastic_order}. \begin{proof}[Proof of Lemma \ref{coro}] Let $(\mu_t)_{t\in \R}$ be an increasing family of probability measures for $\leqs$ and $P$ be a Markov measure in $\ma((\mu_t)_{t\in \R})$ such that for every $S=\{s_1,\ldots,s_d\}$, the measure $(\pr^S)_\#P$ is concentrated on \[ \{(x_1,\ldots,x_d)\in \R^d:\,x_1\leq\ldots \leq x_d\}. \] Set $\mu_{t^-}\eqdef\stosup_{st}\mu_s$, that are also the left and right limits of $(\mu_t)_t$ for the weak topology, see Remark \ref{rem:stosup_existe}. Since $(\mu_t)_{t\in \R}$ is increasing for $\leqs$, we have $\mu_{t^-}\leqs \mu_t \leqs \mu_{t^+}$ and $t\in\R$ is a discontinuity time of $(\mu_t)_{t\in \R}$ for the weak topology if and only if $\mu_{t^-}\neq\mu_{t^+}$. Such points are at most countably many. Indeed, the regions in $\R^2$ between the graphs of $F[\mu_{t^-}]$ and $F[\mu_{t^+}]$, for the discontinuity times~$t$, are disjoint and of positive Lebesgue measure, hence are at most countably many, by $\sigma$-additivity of the measure. Let $C$ be a countable dense subset of $\R$ containing the discontinuity points. Introduce:\vspace*{-3pt} \[N=\{x\in \R^\R:\,\exists (s,t)\in C^2, sx(t)\} .\] Being a countable union of $P$-null sets, $N$ is $P$-null. Now take $(X_t)_{t\in \R}$ with law $P$, e.g., take the canonical process $\Omega\eqdef\R^\R$ and $X_t=\pr^t:x\in \R^\R\mto x(t)$. We define $(\tilde X_t)_{t}$ as null functions on $N$, and as follows on $\Omega\setminus N$: \begin{itemize} \item for $t\in C$, $\tilde X_t(\omega)=X_t(\omega)$, \item for $t\notin C$, $\tilde X_t(\omega)=\lim_{sX_t(\omega)\}$ is a union of null sets, hence is null, so almost surely, $X_t\geqslant \sup_{s0$ and for every $n\in \N$, $\mu\text{-}\mathrm{esssup}_{[0,x]}\tilde{h}_n=\mu\text{-}\mathrm{essinf}_{[x,1]}\tilde{h}_n=\tilde{h}_n(x)$. In case $\mu=\la$, since $\tilde h_n$ is increasing, hence has at most a countable number of discontinuity points, $\mu(A)$ is $1$. This also holds in the general case and is given by the increasing functions $\tilde{h}_n\circ G_\mu$, we leave the details to the reader. Now we take any $x\in A$ and prove $\tilde h_n(x)\to\tilde h_0(x)$. It is enough to prove $\limsup_{n}\tilde{h}_n(x)\leq \tilde{h}_0(x)$, since $\liminf_{n}\tilde{h}_n(x)\leq \tilde{h}_0(x)$ can be proved symmetrically. Suppose, for contradiction:\vspace*{-3pt} \[\limsup_{n}\tilde{h}_n(x)\geq \tilde{h}_0(x)+\eps,\quad \text{for some $\eps>0$.}\] As $\tilde{h}_{0}$ is increasing and $\tilde{h}_0(x)=\mu\text{-}\mathrm{essinf}_{[x,1]}\tilde{h}_0$ there exists $y>x$ such that $\mu[x,y]>0$ and:\vspace*{-3pt} \[\tilde{h}_0(x)\leq\frac1{\mu[x,y]} \int_{[x,y]}\tilde{h}_0\,\dd\mu\leq \tilde{h}_0(x)+\eps/2.\] This contradicts the facts that $\frac1{\mu[x,y]} \int_{[x,y]}\tilde{h}_n\,\dd\mu\to_{n\to\infty}\frac1{\mu[x,y]} \int_{[x,y]}\tilde{h}_0\,\dd\mu$, obtained with $g=\one_{[x,y]}$ in \eqref{eq:lem:pointwise}, and that $\tilde h_n$ is increasing. \end{proof} \begin{pro}\label{pro:continuity} Let $P_n$ tend to $P_0$ in $\ma(\mu_1,\ldots,\mu_d,\eta)$ and $P_n'$ tend to $P_0'$ in $\ma(\eta,\nu)$. Assume moreover that $P'_n$ has increasing kernel for every $n$. Then $P_n'\circ P_n$ tends to $P_0'\circ P_0$. \end{pro} \begin{proof} By Propositions \ref{pro:product}\eqref{pro:productb} and \ref{pro:rho}, we must show that\vspace*{-3pt} \[ \int f(x)g(y)h(z)\dd (P_n\circ P'_n)(x,y,z)\to \int f(x)g(y)h(z)\dd (P_0\circ P_0')(x,y,z), \] where $f((x_i)_{i=1}^d)=\prod_if_i(x_i)$ and $g$, $h$ and the $f_i$ are any increasing functions from $\R$ to $[0,1]$. For all $n\geqslant0$: \begin{align*} \int f(x)g(y)h(z)\dd (P_n\circ P'_n)(x,y,z)&=\int f(x)g(y)\int h(z) k_{P'_n}(y,\dd z)\dd P_n(x,y)\\ &=\int f(x)(g\tilde h_n)(y)\dd P_n, \end{align*} where $\tilde h_n=P_{n}{}\ldot h$, by Definition \ref{defi:concatenation} and Section~\ref{subsubsec:action_des_transports}. By Lemma \ref{lem:pointwise}, $(g\tilde{h}_n)_n$ pointwise converges to $g\tilde{h}_0$, almost surely. Hence by the dominated convergence theorem: \[\left|\int f(x)(g\tilde h_n)(y)\dd P_0-\int f(x)(g\tilde h_0)(y)\dd P_0\right|\to0.\] Moreover, $\tilde{\rho}(P_n,P_0)$ tends to zero and for every $n$, $g\tilde h_n$ is an increasing functions taking values in $[0,1]$. Hence, again by Propositions \ref{pro:product}\eqref{pro:productb} and \ref{pro:rho}: \begin{equation}\label{eq:triangle} \left|\int f(x)(g \tilde h_n)(y)\dd P_n-\int f(x)(g \tilde h_n)(y)\dd P_0\right|\to0. \end{equation} Therefore with the triangle inequality, \eqref{eq:triangle} holds with $\tilde h_0$ in place of $\tilde h_n$ on the right, which is exactly what we claimed. \end{proof} \skpt \begin{rem}[Counterexamples to Proposition \ref{pro:continuity}] \begin{enumerate} \item Case where the mar\-gin\-als are not fixed. Take $d=1$, $\mu=\nu=\frac{1}{2}(\delta_{-1}+\delta_1)$ and $\eta=\frac{1}{2}(\delta_{-1/n}+\delta_{1/n})$ and its limit $\delta_0$ in the parameter $n$. We consider the quantile couplings $P_n=\frac{1}{2}(\delta_{-1,-1/n}+\delta_{1,1/n})\in\ma(\mu_0,\mu_1)$ and $P^{\prime}_n=\trans\!P_n\in\ma(\mu,\eta)$. They have increasing kernel but not fixed marginals. On the one hand, \[ P_n\circ P^{\prime}_n=\frac{1}{2}(\delta_{-1,-1/n,-1}+\delta_{1,1/n,1})\underset{n\to\infty}{\longrightarrow}\frac{1}{2}(\delta_{-1,0,-1}+\delta_{1,0,1}). \] On the other hand, \[ (\lim_n(P_n)_{n})\circ(\lim_n(P^{\prime}_n)_{n})=\frac{1}{4}(\delta_{1,0,1}+\delta_{1,0,-1}+\delta_{-1,0,1}+\delta_{-1,0,-1}). \] \item Case where the transitions are not increasing. See the example in \cite{Ke72} after the proof of Satz 14. Here the marginals are fixed but the transport plans neither have Lipschitz nor increasing kernel. \end{enumerate} \end{rem} We can now prove Lemma \ref{lem:closed}. It follows directly from Proposition \ref{pro:continuity}. \begin{proof}[Proof of Lemma \ref{lem:closed}]\label{proof:lem:closed} Consider a sequence $P^n_{1,2}\circ\cdots\circ P^n_{d,d+1}$ with limit $P\in\ma(\mu_1,\ldots,\mu_{d+1})$ and denote $\pr^{i,i+1}_{\#}P$ by $P_{i,i+i}$. Since $(\pr^{i,i+1})_\#$ is continuous, $P_{i,i+1}$ is the limit of $(P_{i,i+1}^n)_n$, hence $P_{i,i+1}\in\mathcal{I}_{i,i+1}$. Proposition \ref{pro:continuity} used by induction shows that $P=P_{1,2}\circ\cdots\circ P_{n,n+1}$. \end{proof} \section{Construction and characterization of the Markov-quantile process}\label{sec:construction} \emph{Now $(\mu_t)_{t\in \R}$ is a family of probability measures on $\R$, $F_t$ denotes the cumulative distribution function $F_{\mu_t}$ of $\mu_t$ and $G_t$ its quantile function, see Section~\ref{ss:minimality}.} In this section we build the Markov-quantile measure $\mq$ and prove Theorems A and B. Our proof of Theorems A--\ref{them:b} is based on transport plans $L_{[s,t]}\in \ma(\la,\la)$, defined in Section~\ref{subsec:little_t}, that are the 2-marginals of an important auxiliary process law called the quantile level measure $\lev\in\ma(\la_t)_{t\in\R}$, where each $\lambda_t$ is a copy indexed by $t$ of $\la=\la_{[0,1]}$. Here is the link between $\lev$ and $\mq$: $G_t$ maps $([0,1],\la_t)$ to $(\R\cup\{\pm \infty\},\mu_t)$; set $G=\otimes_{t\in\R}G_t$, so that $G_\#$ maps $\ma((\lambda_t)_{t\in\R})$ to $\ma((\mu_t)_{t\in\R})$. Then, as proved in the proof Theorem \ref{them:deuxieme_sec} p.\,\pageref{lieu_preuve_glev}:\vspace*{-3pt} \begin{equation}\label{eq:Glev} G_\#\lev=\mq. \end{equation} \begin{rem} Readers familiar with Mathematical Statistics may compare $L_{[s,t]}$ with the notion of \emph{copula} and \eqref{eq:Glev} with Sklar's theorem. In particular, the (bivariate) cumulative distribution function of $\lev^{s,t}=L_{[s,t]}$ is a copula associated with $\mq^{s,t}$. It is defined as the pointwise maximum of the family of copulas made of the cumulative distribution functions associated with $(L_{[s,t]\cap R})_{R\subset \R}$, where $R$ ranges over finite sets, see Definitions \ref{defi:geqlc} and \ref{def:lcsup}, Lemma \ref{lem:sto_order}, Proposition \ref{pro:rho} and Notation \ref{lem:existenceT}. Although the link is immediate we will not use further the terminology of Copula Theory. \end{rem} This section is divided in three. In Section~\ref{subsec:little_t} we define the coupling $L_R$ for all $R\subset \R$, using the key monotonicity Lemma \ref{lem:mono2}. In Section~\ref{subsec:preuve_premier} we define $\mq$ and prove Theorem \ref{them:a}, purely via the 2-marginals $(\mq^{s,t})_{s0.\\ \end{cases} \end{align*} \end{notation} \skpt \begin{rem}\label{rem:finesse} \begin{enumerate} \item \label{p1:rem:finesse} Notice that $\trans{q}_s \ldot q_t=\join(\mu_s,\trans{k}_s \ldot k_t)=\como^{s,t}$, where $\como^{s,t}=\como(\mu_s,\mu_t)$, see Definition \ref{defi:quantile_process}. Indeed, $\trans{q}_s \ldot q_t=\trans{q}_s \ldot\id_{\la,2}{}\ldot q_t$, then apply \eqref{p2:rem:finesse} below, which will also be useful farther. \item \label{p2:rem:finesse} If some ordered pair $(U,V)$ of variables has law $T\in \ma(\la,\la)$, then\vspace*{-3pt} \[ \ma(\mu_s,\mu_t)\ni\trans{q}_s \ldot T\ldot q_t=\law(G_s(U),G_t(V))=(G_s\otimes G_t)_\#T. \] Indeed, $\trans{q}_s \ldot T\ldot q_t=\pr^{1,4}_\#(\trans{q}_s \circ T\circ q_t)$ and the 4-times process $(G_{\mu_s}(U),U,V,G_{\mu_t}(V))$ has law $\trans{q}_s \circ T\circ q_t$ and is Markov, since the $\sigma$-fields spanned by $U$ and $\{U,G_{\mu_s}(U)\}$ are the same. \item \label{p3:rem:finesse} From \eqref{p1:rem:finesse} we get $\trans q_r\ldot q_r=\id_{\mu_r,2}$, so that $\trans q_r\ldot q_r\ldot\trans q_r=\trans q_r$ and $q_r\ldot\trans q_r\ldot q_r=q_r$. \emph{However}, $q_r\ldot\trans q_r\neq \id_{\la,2}$. Indeed, $k_r\ldot\trans{k}_r$ maps any quantile level $\alpha\in \op]0,1\clo[$ on itself \emph{except when} $G_r(\alpha)$ is an atom of $\mu_r$. Actually, $\mu_r$-almost surely:\vspace*{-3pt} \begin{align*} k_r\ldot\trans{k}_r(\alpha)= \begin{cases} \delta_\alpha&\text{if }\mu_r(G_{r}(\alpha))=0\\ (\alpha^+-\alpha^-)^{-1}\la|_{]\alpha^-,\alpha^+[}&\text{if }\alpha\in]\alpha^-,\alpha^+[, \end{cases} \end{align*} where $]\alpha^-,\alpha^+[$ denotes any set $A_{r,x}$ as follows. \end{enumerate} \end{rem} \skpt \begin{notation}[Atomic levels, $\ell_r$]\label{nota:Asx} \begin{enumerate} \item \label{p:Asx} We denote by $A_{r,x}$ the interval $\op]F_r(x^-),F_r(x)\clo[$ of quantile levels merged by $G_r$ on some atom $x$ of $\mu_r$, and by $A_r$ the set $\bigcup_{\mu_r(x)>0}A_{r,x}\!\subset\! [0,1]$ of ``atomic levels'' of~$\mu_r$. \item \label{p:l_R_fini} We denote $k_r\ldot\trans{k}_r$ described in Remark \ref{rem:finesse}\eqref{p3:rem:finesse} by $\ell_r$. \end{enumerate} \end{notation} \begin{rem}\label{rem:action_l}Measures $\theta\ll \la$ are transported by $\ell_r$ as follows: $\theta\ldot \ell_r$ coincides with~$\theta$ on $\op]0,1\clo[\setminus A_r$, and on each $A_{r,x}$ it has constant density and mass $\theta(A_{r,x})$, \ie equals $(\alpha^+-\alpha^-)^{-1}\theta(A_{r,x})$. Equivalently, $F[\theta\ldot \ell_r]$ is continuous, equal to $F[\theta]$ on $\op]0,1\clo[\setminus A_{r,x}$, and affine on each connected component $A_{r,x}$ of $A_r$. \end{rem} The product $\como^{s,r_1}{}\ldot\como^{r_1,r_2}\lcdots \como^{r_{m-1},r_m}{}\ldot\como^{r_m,t}$ appearing in Theorem \ref{them:a} is more deeply analyzed in Theorem \ref{them:b}. Its kernel reads: \begin{equation}\label{eq:composee_l} \begin{aligned} (\trans k_s\ldot k_{r_1})\ldot(\trans k_{r_1}{}\ldot k_{r_2})\lcdots (\trans k_{r_{m-1}}&\ldot k_{r_m})\ldot(\trans k_{r_m}{}\ldot k_t)\\ &= \trans k_s\ldot(k_{r_1}{}\ldot\trans k_{r_1})\lcdots (k_{r_m}{}\ldot\trans k_{r_m})\ldot k_t\\ &=\trans k_s\ldot\ell_{r_1}\lcdots \ell_{r_m}{}\ldot k_t. \end{aligned} \end{equation} In Remark \ref{rem:comp_mini}, \eqref{eq:composee_l} is further commented and re-expressed for transports in place of kernels. For now, it leads to introduce the following kernel. \begin{notation}\label{notation:Ts} Take $R=\{r_1,\ldots,r_m\}\!\subset\!\R$. We denote the kernel \hbox{$\ell_{r_1}{}\ldot\ell_{r_2}{}\ldot\cdots\ldot\ell_{r_m}$} from $\op]0,1\clo[$ to itself by $\ell_R$, and $\join(\la;\ell_R)\in\ma(\la,\la)$ by $L_R$. If $R=\varnothing$, $\ell_\varnothing$ is the identity kernel and $L_\varnothing$ the identity transport. \end{notation} Notice that $\ell_{\{r\}}=\ell_r$ and that for any $R$, $\la\ldot\ell_R=\la$. Moreover, $\ell_R$ only depends on $(A_r)_{r\in R}$. The following lemma is particularly simple. \begin{lem}\label{lem:mono1} Let $R\subset \R$ be a finite set and $\mu,\,\nu$ be in $\m(\la)$. \begin{enumerate} \item \label{item:mono1a} If $\mu\leqs \nu$ then $\mu \ldot\ell_R\leqs \nu\ldot\ell_R$, \ie $\join(\la; \ell_R)$ has increasing kernel. \item \label{item:mono1b} If $\mu$ is in $\mdec(\la)$, then so is $\mu \ldot\ell_R$, and $\mu\leqs \mu\ldot\ell_R$. \end{enumerate} \end{lem} \begin{proof} For $r\in \R$, one easily checks that $\ell_r$ is increasing and stabilizes $\mdec$; \eqref{item:mono1a}~and the first point of \eqref{item:mono1b} follow. If $\mu\in \mdec(\la)$, to see that $\mu\leqs \mu\ldot\ell_R$, look at the cumulative distribution functions. They coincide off the components of $A_r$. Now on each of those, $F_{\mu\ldot\ell_R}$ is affine whereas $F_\mu$ is concave since $\mu$ has decreasing density, so necessarily $F_\mu\geqslant F_{\mu\ldot\ell_R}$. \end{proof} \begin{figure} \begin{center} \def\svgwidth{{.7\columnwidth}} \input{figure1ter.pdf_tex} \caption{Composition of kernels $\ell_r$}\label{figure} \end{center} \end{figure} We add a remark, linked with Figures \ref{figure}--\ref{figure3}, on the principle of Theorem \ref{them:a}'s proof. A reader only looking for the formal proof itself may skip it. \begin{rem}\label{rem:qualitative}We will not (directly) obtain the couplings $\mq^{s,t}$ as a limit of products $\como^{s,t}_{[R_n]}$, for some finite sets $R_n$ with dense union, as suggested in Section~\ref{subsec:intro_quantilemarkovien} p.\,\pageref{page:tentative_limite}---to show this does not work. We aim at obtaining $\mq^{s,t}$ as a \emph{supremum}, of~the set $\{\como^{s,t}_{[R]} : R\ \text{finite and } R\subset]s,t[\}$ (see Theorem \ref{them:a}\eqref{item:limite_de_compose} for the notation), and actually we do it on the space of quantile levels, \ie we look for a supremum of $\{\ell_R : R\ \text{finite and } R\subset]s,t[\}$. The question is to find the adequate quantity, or order relation, for which a supremum (and hopefully then a maximum) shall be sought. First, Figure \ref{figure} makes us observe how kernels of the type $\ell_R$ act on measures of $\p(\op]0,1\clo[)$. It displays the action of $\ell_{R}$ with $R=\{r_1,r_2,r_3,r_4\}$ on some Dirac measure~$\delta$. The vertical segment on the left is the space $\op]0,1\clo[$ of quantile levels. We suppose that each $\mu_{r_i}$ has a single atom and draw vertically, at abscissa $r_i$, the interval $A_{r_i}$ (see Notation \ref{nota:Asx}\eqref{p:Asx}). The drawing is in the case where $\delta=\delta_{x}$ with $x\in A_{r_1}$. Then, see Remark \ref{rem:action_l}: $\ell_{\{r_1\}}$ maps $\delta$ on the uniform probability measure on $A_{r_1}$; in turn, $\ell_{\{r_2\}}$ leaves the latter unchanged outside of $A_{r_2}$ and makes it uniform on $A_{r_2}$, etc.\ The first drawing shows a ``possible trajectory of an element of mass at $x$'' transported by the discrete Markov chain with transition kernels $(\ell_{r_i})_{i=1}^4$. Since we take $x \in A_{r_1}$, it is displaced by $\ell_{r_1}$ to $x '$, picked uniformly at random in $A_{r_1}$. In case $x '\not\in A_{r_2}$, as in the figure, it is unchanged by $\ell_{r_2}$; then in case $x '\in A_{r_3}$ (figure), it is displaced by $\ell_{r_3}$ to a random $x ''\in A_{r_3}$, and finally, in case $x ''\in A_{r_4}$, displaced by $\ell_{r_4}$ to a random $x '''\in A_{r_4}$. The second drawing shows the successive measures $\delta_x $, $\delta_x \ldot\ell_{r_1}$, $\delta_x \ldot\ell_{r_1}{}\ldot\ell_{r_2}$ etc., the level of grey being proportional to the value of their density. So each $\ell_{r_i}$ ``spreads'' a little more the mass of $\delta\ldot\ell_{r_1}\lcdots \ell_{r_{i-1}}$, replacing it by its mean (measure of constant density) on each connected component of $A_{r_i}$. If~$\theta\ll\la$, this averaging process lowers the total variation of the density at each step: at most, you get the measure with constant density one, \ie $\la$ itself, on which all the transports~$\ell_{R}$ act trivially. Thus a natural idea is to consider that, if ${R'}\subsetneq R$ are finite and $\theta\ll\la$, the density of $\theta\ldot \ell_{R}$ will be closer to $\one_{\op]0,1\clo[}$, for some adequate distance, than that of $\theta\ldot \ell_{R'}$. Unfortunately, this is the case if $R'=R\cap\op]-\infty,t]$ for some $t$ but not in general. Figure \ref{figure2} shows, from top to bottom, the Dirac measure $\delta_x$ and the graph of the densities of: \begin{itemize} \item $\delta_x \ldot\ell_{r_1}$, $\delta_x \ldot\ell_{r_1}{}\ldot\ell_{r_2}$, $\delta_x \ldot\ell_{r_1}{}\ldot\ell_{r_2}{}\ldot\ell_{r_3}$, and $\delta_x \ldot\ell_{r_1}{}\ldot\ell_{r_2}{}\ldot\ell_{r_3}{}\ldot\ell_{r_4}$ (on the left), \item $\delta_x \ldot\ell_{r_1}$, $\delta_x \ldot\ell_{r_1}{}\ldot\ell_{r_2}$, and $\delta_x \ldot\ell_{r_1}{}\ldot\ell_{r_2}{}\ldot\ell_{r_4}$ (on the right), \end{itemize} in the case $x=\frac23$, $A_{r_1}=\op]\frac13,\frac56\clo[$, $A_{r_2}=\op]\frac23,1\clo[$, $A_{r_3}=\op]\frac12,\frac56\clo[$, $A_{r_4}=\op]0,\frac23\clo[$. Then $\delta\ldot\ell_{\{r_1,r_2,r_4\}}=\la$, \ie has exactly density $\one_{\op]0,1\clo[}$, whereas $\delta\ldot\ell_{R}\neq\la$. A remedy is to look the kernels $\ell_R$ act \emph{on measures of density $\frac1x\one_{\op]0,x\clo[}$}. For the latter, and more generally any element of $\pdec(\op]0,1\clo[)$, which is stable by the action of the couplings $\ell_R$, the idea above works, with the stochastic order. This is provided by Lemmas~\ref{lem:mono2} and~\ref{lem:existenceT} below; see also Remark \ref{rem:difference}. Figure \ref{figure3} gives the example of the kernels~$\ell_{r_i}$ of Figure \ref{figure2} acting on the measure $\nu$ of density $\frac1x\one_{\op]0,x\clo[}$ with $x=\sfrac23$: one gets \hbox{$\nu\ldot\ell_{r_1}{}\ldot\ell_{r_2}{}\ldot\ell_{r_3}{}\ldot\ell_{r_4}\geqs\nu\ldot\ell_{r_1}{}\ldot\ell_{r_2}{}\ldot\ell_{r_4}$}. \end{rem} \begin{figure} \def\svgwidth{.7\textwidth} \hspace*{\fill}\hspace*{\fill}\mbox{\input{figure2bis.pdf_tex}}\hspace*{\fill} \caption{Composition of kernels $\ell_r$, acting on some Dirac measure~$\delta_x$. On each column, the horizontal interval is the space $\op]0,1\clo[$ of quantile levels. From top to bottom, the horizontal bars represent successive atomic intervals $A_{r_i}$ and, below each one, the density of~$\delta_x$ transported by the composition of the successive corresponding kernels $\ell_{r_1}$, \ldots, $\ell_{r_i}$. On the right, $A_{r_3}$ is omitted.}\label{figure2} \end{figure} \begin{figure}[t] \def\svgwidth{.7\textwidth} \hspace*{\fill}\hspace*{\fill}\hspace*{\fill}\mbox{\input{figure3.pdf_tex}}\hspace*{\fill}\hspace*{\fill} \caption{The kernels $\ell_r$ of Figure \ref{figure2}, acting on the measure of density $\frac1x\one_{\op]0,x\clo[}$. The global setup of this figure is the same as that of Figure~\ref{figure2}.}\label{figure3} \end{figure} Though simple, the next lemma is a key of our construction of $\mq$. \begin{lem}\label{lem:mono2} Let $R\subset R'$ be two finite subsets of\/ $\R$ and $\mu\in\mdec(\la)$. Then $\mu \ldot\ell_R\leqs \mu \ldot\ell_{R'}$. \end{lem} \begin{proof} Using an induction on the cardinal difference, it is enough to prove this if $R'$ has one more element than $R$, say $r'$. We order it with the elements $r_i$ of $R$: $r_1<\cdotss$, and $R=\{r_1,\ldots,r_m\}\subset[s,t]$, $\como^{s,t}_{[R]}$ denotes the coupling $\como(\mu_{s},\mu_{r_1}).\como(\mu_{r_2},\mu_{r_3}).\cdots.\como(\mu_{r_{m}},\mu_{t})\in\ma(\mu_s,\mu_t)$, see Theorem \ref{them:a}\eqref{item:limite_de_compose}. \end{reminder} \skpt \begin{prodefinition}\label{pro:comp_conv} \begin{enumerate} \item \label{aaa} The set $\{\como^{s,t}_{[R]}\,|\,R\subset \R\ \text{and}\ \sharp R<\infty\}$ has a lower orthant supremum and there is a nested sequence $(R_n)_{n\in \N}$ of finite sets such that $\como^{s,t}_{[R_n]}$ tends to it. We denote it by $\mu_{s,t}$. \item \label{bbb} The family $(\mu_{s,t})_{ss$, $\mq^{s,t}=(G_s\otimes G_t)_\#L_{]s,t[}=(G_s\otimes G_t)_\#L_{[s,t]}$. \end{enumerate} \end{prodefinition} To show Proposition \ref{pro:comp_conv} we first state the following crucial relation between $\como^{s,t}_{[R]}\in \ma(\mu_s,\mu_t)$ and $L_R\in \ma(\la,\la)$ defined in Section~\ref{subsec:little_t}. \begin{rem}\label{rem:comp_mini} Take any finite subset $R$ of $\R$ and $(s,t)$ with $s0$ we find a finite $R'\subset R$ such that $\rho(L_R,L_{R'})\leqslant\eps$. As $R=\bigcup_\tau R_\tau$ there is a $\tau_0$ such that $R_{\tau_0}\supset R'$. Then: $\tau\geqslant \tau_0\Rightarrow L_{R'}\leqlc L_{R_\tau}\leqlc L_{R}\Rightarrow \rho(L_{R'},L_{R_\tau})\leqslant\eps$ by the remark. \end{proof} \begin{lem}\label{lem:convergence_couples} There is a nested sequence $(R_n)_{n\in\N}$ of finite subsets of $\R$ such that $ss$, $L_{R_n\cap\op]s,t\clo[}\to_{n\to\infty}L_{\op]s,t\clo[}$. Let us prove it. Fix $\eps>0$. By Lemma \ref{lem:croissance_lc} (use a subsequence $(k_n)_n$ such that $(\op]u_{k_n},u'_{k_n}\clo[)_n$ is an exhaustion of $\op]s,t\clo[$), there exists $k$ such that $\rho(L_{\op]s,t\clo[},L_{\op]u_k,u'_k\clo[})<\eps/2$. Then $L_{\op]u_k,u_k'\clo[\cap R_n}\leqlc L' \leqlc L_{\op]s,t\clo[}$, where $L'$ stands for $L_{\op]s,t\clo[\cap R_n}$ or $L_{\op]u_k,u_k'\clo[}$ for any $k$. For $n\geq\max\{k,2/\eps\}$: \[\rho(L_{\op]s,t\clo[},L_{\op]u_k,u'_k\clo[\cap R_n})\leq \rho(L_{\op]s,t\clo[},L_{\op]u_k,u'_k\clo[})+ \rho(L_{\op]u_k,u'_k\clo[},L_{\op]u_k,u'_k\clo[\cap R_n})<\eps.\] Therefore with $L'=L_{\op]s,t\clo[\cap R_n}$ we obtain the desired convergence to $L_{\op]s,t\clo[}$. \end{proof} \begin{defi}\label{defi:essentiel}Let $I$ be an interval. If, for some interval $J\supset I$ such that $J\setminus I$ is disconnected (then for all such smaller $J'$), $L_{J}\neq L_{J\setminus I}$, we call $I$ an \emph{essential atomic interval of $(\mu_t)_{t\in\R}$}. If $I=\{t\}$ is essential, we call $t$ an \emph{essential atomic time of $(\mu_t)_{t\in\R}$}. \end{defi} To check the parenthesis in the definition, suppose that $L_{J'}= L_{J'\setminus I}$ and get $L_{J}= L_{J\setminus I}$ by Proposition \ref{pro:compo}. \begin{pro}\label{pro:denombrable}If a nested sequence $(R_n)_n$ is as in Lemma \ref{lem:convergence_couples} then $\bigcup_nR_n$ contains all the essential atomic times of $(\mu_t)_t$. In particular, these times are at most countably many. \end{pro} \begin{proof}We show a contrapositive result. Suppose that $t$ is an essential atomic time, that $st$ are such that $L_{\op]s,s'\clo[\setminus\{t\}}\neq L_{\op]s,s'\clo[}$, and that $(R_n)_n$ is a nested sequence of finite sets such that $t\not\in\bigcup_nR_n$, that $L_{R_n\cap\op]s,t\clo[}\to L_{\op]s,t\clo[}$ and that $L_{R_n\cap\op]t,s'\clo[}\to L_{\op]t,s'\clo[}$. Then: \begin{align*} L_{R_n\cap\op]s,s'\clo[}&=L_{R_n\cap\op]s,t\clo[}{}\ldot L_{R_n\cap\op]s,t\clo[}\ \ \text{by Proposition \ref{pro:compo}}\\ &\to L_{\op]s,t\clo[}{}\ldot L_{\op]t,s'\clo[}\ \ \text{by assumption and by Proposition \ref{pro:continuity}}\\ &=L_{\op]s,s'\clo[\setminus\{t\}}\neq L_{\op]s,s'\clo[}\ \ \text{by Proposition \ref{pro:compo} and the assumption}. \end{align*} Therefore, $(R_n)_n$ cannot be as in Lemma \ref{lem:convergence_couples}. \end{proof} \begin{rem}\label{rem:intervalles_essentiels} Be careful, the property to be an essential atomic interval is true in general neither for a union of two such (intersecting) intervals, nor for their intersection, nor for an interval containing a such interval or included in it. \end{rem} Remark \ref{rem:essentiel_divers} and Example \ref{ex:intervalle_essentiel} are qualitative comments on the notions introduced in Definition \ref{defi:essentiel}. They are independent of the proof of Theorem \ref{them:deuxieme_sec} and may be skipped in a first reading. \pagebreak[2] \skpt \begin{rem}\label{rem:essentiel_divers} \begin{enumerate} \item Essential atomic times are of course atomic. Indeed, else, $L_{\{t\}}$ is the identity transport so $L_I\ldot L_{\{t\}}{}\ldot L_{I'}=L_I\ldot L_{I'}$ for all intervals with $\sup I\leqslant t \leqslant \inf I'$ and $t$ cannot be essential. \item \label{item:suffisante} Suppose, to simplify, that some $\mu\in\p(\R)$ has exactly one atom $x$ and consider a family $(\mu_t)_{t\in\R}$ such that $\mu=\mu_0$. An obvious sufficient condition for $0$ to be unessential is to choose $\mu_t$ such that for a certain sequence $t_n\to0$, $\mu_{t_n}$ has an atom $x_n$ such that $A_{t_n,x_n}\supset A_{t,x}$ (see Notation \ref{nota:Asx}), or even only \[ \forall \varepsilon>0,\,\exists n_0:\quad n\geqslant n_0\implies A_{t_n,x_n}+\op]-\varepsilon,\varepsilon\clo[\supset A_{t,x}: \] ``atoms merging the same levels of quantile as $x$ merges at $t=0$, accumulate on $0$''. Indeed, taking possibly a subsequence, we may suppose that $(t_n)_n$ tends to zero from the right or the left (say, from the right). For any $s>t$, the function $x\mto k_{L_{[t_n,s\clo[}}(x,\espcdot)$ (when it is defined \ie $t_ns$: \begin{align*} L_{R\cap\op]s,t\clo[}&=\lcsup\{L_{R_n\cap\op]s,t\clo[}\}\ \ \text{by Lemma \ref{lem:croissance_lc}}\\ &\geqlc\lcsup\{L_{R'_n\cap\op]s,t\clo[}\}\ \ \text{as $R'_n\subset R_n$ and by Lemma \ref{lem:existenceT}\eqref{p1:lem:existenceT}}\\ &=\lim_{n\to\infty}L_{R'_n\cap\op]s,t\clo[}=L_{\op]s,t\clo[}\ \ \text{\begin{tabular}[t]{l}by Lemma \ref{lem:croissance_lc} and\\ by property of the sets $R'_n$.\end{tabular}} \end{align*} But by definition, $L_{R\cap\op]s,t\clo[}\leqlc L_{\op]s,t\clo[}$, thus $L_{R\cap\op]s,t\clo[}= L_{\op]s,t\clo[}$. \end{proof} To provide a clear proof of Theorem \ref{them:deuxieme_sec}\eqref{item:them:deuxieme_b} we introduce an auxiliary notion in Definition \ref{defi:fits} below. Suppose that some $P\in \ma(\mu,\mu_1,\ldots,\mu_k)$ is disintegrated as $P=\join(\mu,k_{P})$ and that $g:\R\to \R$ and $h:\R^k\to \R^k$ are measurable maps. May we disintegrate $(g\otimes h)_\#P$? In case $g$ is into one easily checks $(g\otimes \id)_\#P=\join(g_\#\mu,k^g_P)$, where $k^g_P$ is defined by $k^g_P(y,\cdot)= k^g_P(g^{-1}(y),\cdot)$, $g_\#\mu$-almost surely. Otherwise, the next notion and lemma will enable us to obtain a similar disintegration, and associated properties. \begin{defi}\label{defi:fits} We say that $g:\R\to \R$ \emph{fits} $P\in \ma(\mu,\mu_1,\ldots,\mu_k)$ if there exists a kernel $k^g_P$ such that $k^g_P(g(x),\cdot)=k_{P}(x,\cdot)$, $\mu$-almost surely. \end{defi} \skpt \begin{rem}\label{rem:random_from_center}\begin{enumerate} \item \label{item:rem:random_from_center_a} If $g$ fits $P=\join(\mu,k)$ and $h$ is a measurable map from $\R^k$ into itself we can disintegrate $(g\otimes h)_\#P$ as $\join(g_\#\mu,h_\#k^g_P)$, where $h_\#k^g_P(y,\cdot)= k^g_P(y,h^{-1}(\cdot))$. \item \label{item:rallonge} If $g$ fits $P$, it also fits $P\ldot P'$ and $P\circ P'$. \item \label{item:rem:random_from_center_commute_avec_circ} If $g$ fits $P\in \ma(\mu,\nu_1,\ldots,\nu_k)$ or $Q\in\ma(\mu,\nu'_1,\ldots,\nu'_{k'})$, and if $f:\R^k\to\R^k$ and $h:\R^{k'}\to\R^{k'}$ are measurable maps, then: \[(f\otimes g \otimes h)_\#(\trans P\circ Q)=(f\otimes g)_\#\trans P\circ (g \otimes h)_\#Q.\] \end{enumerate} The proofs are direct, using the definitions of a kernel, composition and concatenation in Section~\ref{subsec:caten}. Notice that, in case $\mu=\la\lfloor_{[0,1]}$ and $g$ is a quantile function (which is the only case in which we will use the remark), point \eqref{item:rem:random_from_center_commute_avec_circ} is a particular case of Lemma \ref{lem:distributivite}\eqref{item:lem:distributiviteB} below. In the language of this lemma, ``$g$ fits $\trans P$ or $Q$'' means that $\trans P\ldot L=\trans P$ or $L\ldot Q=Q$, which both imply in particular that $\trans P\ldot L\ldot Q=\trans P\ldot Q$. \end{rem} We will need the following little technical result. \begin{lem}\label{lem:int_fg}Take $a\!0$ such that if $s\!\in\!\op]t-\eps,t\clo[$, the kernel $x\mto \trans\ell_{\op]s,t\clo[}(x,\espcdot)$ (see Notation \ref{nota:Asx}) is not constant on $A_{t,b}$, \item the atom $b$ is said to be right-essential if there is some $\eps\!>\!0$ such that if $s\!\in\!\op]t,t+\eps\clo[$, the kernel $x\mto\ell_{\op]t,s\clo[}(x,\espcdot)$ is not constant on $A_{t,b}$, \end{itemize} Let us call ``left-'' or ``right-essential'' an atomic time $t$ such that $\mu_t$ has at least one left, respectively right essential atom. Left- or right-essential atomic times must not be essential. However, they are also at most countably many. Here is a sketch of proof, written for right-essential times: show that any such time is a discontinuity point of some function $\varphi_s:\op]-\infty,s]\ni u\mto L_{[u,s]}$. Now if some $\varphi_s$ is discontinuous at $t$, all~$\varphi_{s'}$ are, for $s'\in\op]t,s]$ (use Remark \ref{rem:multiplicationcontractante}). Hence, the union of the sets of discontinuity points of the functions $\varphi_s$, for $s\in\R$, is the same as their union for $s\in\Q$. Finally the claim below, left to the reader, implies that for each $\varphi_s$, this set is at most countable, which gives the result. \begin{claim} If $\psi:u\mto M_u\in \ma(\la,\la)$ is increasing or decreasing for $\leqlc$, it has at most countably many (weak) discontinuity points. \end{claim} \end{rem} Now we can prove the end of Theorem \ref{them:deuxieme_sec}, \ie its parts \eqref{item:them:deuxieme_b} and \eqref{item:them:deuxieme_c}. \begin{proof}[Proof of Theorem \ref{them:deuxieme_sec}\eqref{item:them:deuxieme_b}--\eqref{item:them:deuxieme_c}]\label{proof:them:deuxieme} Let us prove point \eqref{item:them:deuxieme_b}. Take $R\subset\R$ and $\lev_R$ and $\lev$ in $\ma((\la_t)_{t\in\R})$ given by Definition \ref{defi:lev}. We need the following claim, that extends Remark \ref{rem:mini_et_Q}, and relies eventually on Remark \ref{rem:finesse}\eqref{p2:rem:finesse}: \begin{claim} If $R$ is finite, $G_\#\lev_{R}=\como_{[R]}$, where $G=(\otimes_{t\in\R} G_t)$. \end{claim} Let us prove it. Take $R\subset\R$ finite and $n$ its cardinal. We must prove that for any finite $S\!=\!\bigcup_{i=1}^N\{s_i\}$, $G_\#\lev_R^{s_1,\ldots,s_N}\!=\!\como_{[R]}^{s_1,\ldots,s_N}$, where, by an abuse of notation we~will often make use of, $G$ stands for $\otimes_{i=1}^N G_{s_i}$. It suffices to prove it in the case $S\supset R$, which we suppose now. We introduce the cardinals $\ell_0,\ldots,\ell_n$ of the subsets of $S\setminus R$ situated between to consecutive elements $r_k$ and $r_{k+1}$ of $R\cup\{\pm\infty\}$. We re-index these subsets as $\{s_1^k,\ldots,s_{\ell_k}^k\}$. Since $\lev_R$ is Markov, $(\pr^S)_\#\lev_R=B_0\circ\cdots\circ B_m$, where: \begin{align*} B_k=L_{r_{k}}\circ\id_{\ell_{k}}\circ L_{r_{k+1}}= \begin{cases} \id_{\ell_0}\circ L_{r_1}&\text{if }k=0\\ L_{r_m}\circ \id_{\ell_m}&\text{if }k=m\\ L_{[r_k,r_{k+1}]}=L_{r_{k}}. L_{r_{k+1}}&\text{if }\ell_k=0\\ L_{[r_k,r_{k+1}]}=L_{r_{k}}\circ L_{r_{k+1}}&\text{if }\ell_k=1\\ L_{r_{k}}\circ\underbrace{\id_2\circ\cdots\circ\id_2}_{\ell_k-1}\circ L_{r_{k+1}}&\begin{nnarray}[t]{l}\text{otherwise,}\\\text{\quad \ie if }\ell_k\geq 2,\end{nnarray} \end{cases} \end{align*} hence we must prove that: $G_\#(B_1\circ\cdots\circ B_m)=\como_{[R]}^{s_1,\ldots,s_N}$. For all $r\in R$, the quantile function $G_r=G_{\mu_r}$ fits $q_r=\join(\mu_r,x\mto \delta_{G_r(x)})$, so we may take $k^{G_r}_{q_r}:x\mto \delta_x$ as given by Definition \ref{defi:fits}. Now, by Remark \ref{rem:random_from_center}\eqref{item:rallonge}, it also fits $L_{\{r\}}=\trans L_{\{r\}}=q_r\ldot\trans{q_r}$. Therefore, by Remark \ref{rem:random_from_center}\eqref{item:rallonge} and \eqref{item:rem:random_from_center_commute_avec_circ}, $G_\#(B_1\circ\cdots\circ B_m)=B'_1\circ\ldots\circ B'_m$, where: \begin{align*} B'_k&=G_\# B_k=\begin{cases} (\otimes_{s\leqslant r_1} G_{s})_\#\id_{\ell_0}\circ L_{r_1}&\text{if }k=0\\ (\otimes_{s\geqslant r_m} G_{s})_\#L_{r_m}\circ \id_{\ell_m}&\text{if }k=m\\ (G_{r_k}\otimes G_{r_{k+1}})_\#(L_{r_{k}}{}\ldot L_{r_{k+1}})&\text{if }\ell_k=0\\ (G_{r_k}\otimes G_{s_1^k}\otimes G_{r_{k+1}})_\#(L_{r_{k}}\circ L_{r_{k+1}})&\text{if }\ell_k=1\\ (\otimes_{r_k\leqslant s\leqslant r_{k+1}}G_s)_\#(L_{r_{k}}\circ\id_{\ell_k}\circ L_{r_{k+1}})&\text{otherwise, \ie }\ell_k\geq 2. \end{cases} \end{align*} \noindent Proving that $B'_k=\como(\mu_{r_k},\mu_{s^k_1},\ldots,\mu_{s^k_{\ell_k-1}},\mu_{r_{k+1}})$ will now prove the claim. For simplicity we assume $k\notin \{0,m\}$ and $\ell_k\geq 1$ but the other cases, which are simpler, can be proved similarly. Observe that: \begin{itemize} \item by definition of $\como$ and $G$, \[ \como(\mu_{r_k},\mu_{s_1^k},\ldots,\mu_{s^k_{\ell_k}},\mu_{r_{k+1}})=\law((G_{r_k}(U),\ldots,G_{r_{k+1}}(U)), \] where $U$ is a variable of law $\la$ on $[0,1]$, \item $(\otimes_{r_k\leqslant s\leqslant r_{k+1}}G_s)_\#(L_{r_{k}}\circ\id_{\ell_k}\circ L_{r_{k+1}})$ is the law of \[ (G_{r_k}(U_1),G_{s^k_1}(U_2),\ldots,G_{s^k_{\ell_k}}(U_2),G_{r_{k+1}}(U_3)), \] where $\law(U_1,U_2,\ldots,U_2,U_3)=B_k$. In particular, $\law(U_i)=\la$ for $i=1,2,3$, $\law(U_1,U_2)=L_{r_k}$, and $\law(U_2,U_3)=L_{r_{k+1}}$. \end{itemize} So only the first and last variables may differ. Now, notice that by Remark \ref{rem:finesse}\eqref{p2:rem:finesse} applied to $T=L_{\{r\}}=q_r\ldot\trans q_r$, if two variables $(U,V)$ satisfy $\law(U,V)=L_{\{r\}}$, the law of $(G_{r}(U),G_r(V))$ is \[ \trans q_r\ldot L_{\{r\}}{}\ldot q_r=\trans q_r\ldot q_r\ldot\trans q_r\ldot q_r=\trans q_r\ldot q_r=\id_{\mu_r,2}, \] therefore $G_{r}(U)=G_r(V)$ almost surely. Using this with $r={r_k}$ and $(U,V)=(U_1,U_2)$, respectively $r=r_{k+1}$ and $(U,V)=(U_2,U_3)$, we get respectively $G_{r_k}(U_1)=G_{r_k}(U_2)$ and $G_{r_{k+1}}(U_2)=G_{r_{k+1}}(U_3)$ almost surely. The claim is proved. Now take $R\subset \R$ satisfying \eqref{eq:convergence_doug} and $(R_n)_n$ a nested exhaustion of it by finite sets. Let $\widetilde\lev\in\ma((\la_t)_{t\in\R})$ be the Markov process having $\widetilde\lev{}^{s,t}= L_{R\cap[s,t]}$ as 2\nobreakdash-mar\-ginals. Recall that $L_{R\cap[s,t]}=L_{R\cap \{s\}}{}\ldot L_{R\cap\op]s,t\clo[}{}\ldot L_{R\cap \{t\}}=L_{R\cap \{s\}}{}\ldot L_{\op]s,t\clo[}{}\ldot L_{R\cap \{t\}}=\lim_{n\to\infty}\lev_{R_n\cap[s,t]}=\lim_{n\to\infty}\lev_{R_n}^{s,t}$. It exists by Corollary \ref{cor:consist}, whose assumption is satisfied by Proposition \ref{pro:compo}. All these transport plans have increasing kernel, hence by Lemma \ref{lem:closed}\label{appel_lemme_closed}, for every $S=\bigcup_{i=1}^k\{s_i\}$: \begin{multline} \lev_{R_n}^{s_1,\ldots,s_k}=\lev_{R_n}^{s_1,s_2}\circ\cdots\circ \lev_{R_n}^{s_{k-1},s_k}\label{eq:lev_egal_concatenation}\\ \to\widetilde \lev^{s_1,s_2}\circ\cdots\circ \widetilde\lev{}^{s_{k-1},s_k}=\widetilde\lev{}^{s_1,\ldots,s_k}. \end{multline} Thus $\lev_{R_n}$ converges weakly to $\widetilde\lev$, and then by Remark \ref{rem:diese_contractant}, $G_\# \lev_{R_n}\to G_\# \widetilde\lev$. We are left with the tasks to prove $G_\# \lev_{R_n}=\como_{[R_n]}$ and $G_\# \widetilde\lev=\mq$. The former is our claim above. Let us prove the latter. \begin{note} \label{lieu_preuve_glev} At the beginning of Section~\ref{sec:construction} we announced \eqref{eq:Glev}, \ie $G_\#\lev=\mq$. In~fact, we prove $G_\# \widetilde\lev=\mq$, which is a bit more difficult. To get \eqref{eq:Glev} the same arguments work, the final reasoning with Lemma \ref{lem:distributivite}\eqref{item:lem:distributiviteB} being replaced by a direct use of Remark \ref{rem:random_from_center}\eqref{item:rem:random_from_center_commute_avec_circ}, as for all $\{r_1,\ldots,r_\ell\}$, $G_{r_\ell}$ fits $\lev^{r_1,\ldots,r_\ell}$. \end{note} We recall that $\mq$ was defined as the unique Markov law with the same marginals of dimension 2 as $G_\# \lev$. But by \eqref{eq:convergence_doug} and Proposition \ref{pro:comp_conv}\eqref{ccc}, the 2-marginals of $G_\#\widetilde\lev$ and $G_\#\lev$ are equal. Hence it is sufficient to prove that $G_\# \widetilde \lev$ is Markov, \ie that for all $(s_1,\ldots,s_k)$, $(G_\#\widetilde\lev)^{s_1,\ldots,s_k}=(G_\#\widetilde\lev)^{s_1,s_2}\circ\ldots\circ(G_\#\widetilde\lev)^{s_{1-1},s_k}$. Since $\widetilde\lev$ is Markov, $(\widetilde\lev)^{s_{2},\ldots,s_{k}}=\widetilde\lev{}^{s_{2},s_{3}}\circ\ldots\circ\widetilde\lev{}^{s_{k-1},s_{k}}$; besides, notice the following fact, that we will prove a bit below: \begin{fact} For all $(s_i)_{i=1}^k\in\R^k$, $\lev^{s_1,\ldots,s_k}$ viewed as a transport plan from $(\R,\la)$ to $(\R^{k-1},\la^{\otimes k-1})$ has increasing kernel (for the order $\leqlc$ instead of $\leqs$). \end{fact} Hence we may conclude by using $k-1$ times Lemma \ref{lem:distributivite}\eqref{item:lem:distributiviteB}. Let us check the first step. Since the measures $\trans(\widetilde\lev{}^{s_1,s_2})$ and $\widetilde\lev{}^{s_2,\ldots,s_k}$ have increasing kernel, we have only to show that $\widetilde\lev{}^{s_1,s_2}.\widetilde\lev{}^{s_{2},s_3}\!=\!\widetilde{\lev}^{s_1,s_{2}}{}\ldot L_{\{s_{2}\}}.\widetilde{\lev}^{s_{2},s_3}$, \ie by definition of~$\widetilde\lev$, that $L_{R\cap[s_1,s_2]}{}\ldot L_{R\cap[s_2,s_3]}=L_{R\cap[s_1,s_2]}{}\ldot L_{s_2}{}\ldot L_{R\cap[s_2,s_3]}$. This amounts to checking that:\vspace*{-3pt} \[ \begin{cases} L_{\semi s_1,s_2\clo[}{}\ldot L_{\op]s_2,s_3\semi}=L_{\semi s_1,s_2\clo[}{}\ldot L_{s_2}{}\ldot L_{\op]s_2,s_3\semi}&\ \text{if}\ s_2\not\in R\\ L_{\semi s_1,s_2]}{}\ldot L_{[s_2,s_3\semi}=L_{\semi s_1,s_2]}{}\ldot L_{s_2}{}\ldot L_{[s_2,s_3\semi}&\ \text{otherwise.}\end{cases} \] The second point is true by Proposition \ref{pro:compo}, and if $s_2\not\in R$, by Proposition \ref{pro:denombrable}, $s_2$ is not an essential atomic time, which is the wanted equality. We finally must prove the fact stated above. Actually it is true for any concatenation $P_1\circ\ldots\circ P_k$ of couplings $P_i$ from $\R$ to $\R$ with increasing kernel —and $\lev^{s_1,\ldots,s_k}$ is of this type, see~\eqref{eq:lev_egal_concatenation}. We check this for $k=2$; the same argument, applied by induction, gives the general case. Take $B_2$ and $B_3$ two intervals of the type $\op]-\infty,a]$ and $x\leqslant x'$; we must show that: $k_{P_1\circ P_2}(x,B_2\times B_3)\geqslant k_{P_1\circ P_2}(x',B_2\times B_3)$. As $P_2$ has increasing kernel, the function $k_{P_2}(\espcdot,B_3)$ is decreasing (thus also $\one_{B_2}k_{P_2}(\espcdot,B_3)$). As $P_1$ has increasing kernel, $k_{P_1}(x,\espcdot)\leqs k_{P_1}(x',\espcdot)$. Thus:\vspace*{-3pt} \[k_{P_1\circ P_2}(x,B_2\times B_3)\!=\!\int k_{P_1}(x,\dd y) \one_{B_2}k_{P_2}(y,B_3)\!\geqslant\! \int k_{P_1}(x',\dd y) \one_{B_2}k_{P_2}(y,B_3),\] the desired result. This proves part \eqref{item:them:deuxieme_b} of the theorem. Now we prove part \eqref{item:them:deuxieme_c}. For point \eqref{item:ordre_facultatif}, each 2-margin $\como_{[R_n]}^{s,t}$ and $\como_{[R'_n]}^{s,t}$ tends respectively to $\lcsup_n\{\como_{[R_n]}^{s,t}\}$ and $\lcsup_n\{\como_{[R'_n]}^{s,t}\}$, which are equal by Lemma \ref{lem:croissance_lc} since $\bigcup_nR_n=\bigcup_nR'_n$. Besides, if $R''\supset R$, taking a nested sequence $(R''_n)_n$ of fini\-te sets such that $\bigcup_n R''_n=R''$ and $R''_n\supset R_n$ for all $n$, we get that, for any~$s$ and~$t>\nobreak s$, $\lim_n\como_{[R''_n]}^{s,t}\geqlc \lim_n\como_{[R_n]}^{s,t}=\mq^{s,t}$, but by the minimality property~of \hbox{Theorem}~\ref{them:a}\eqref{item:minimal}, for all $n$, $\mq^{s,t}\geqlc\como_{[R''_n]}^{s,t}$. Therefore, $\lim_n\como_{[R''_n]}^{s,t}=\mq^{s,t}$ and the result follows. For \eqref{item:inatomique_facultatif}, it is sufficient to show that for any finite subsets~$R$ and~$E$ of $\R$ with~$\mu_t$ diffuse for all $t\in E$, $\como_{[R\cup E]}^{s,t}=\como_{[R]}^{s,t}$. This follows plainly from the definitions. After Definition \ref{defi:quantile_discretement_markovien} it amounts to showing that for all $t\in E$, all $\{s_1,\ldots,s_k\}\subset\op]-\infty,t\clo[$ and all $\{s'_1,\ldots,s'_{k'}\}\subset\op]t,+\infty\clo[$, $\como^{s_1,\ldots,s_k,t}\circ\como^{t,s'_1,\ldots,s'_{k'}}=\como^{s_1,\ldots,s_{k},t,s'_1,\ldots,s'_{k'}}$. This comes from a trivial case of Remark \ref{rem:random_from_center}\eqref{item:rem:random_from_center_commute_avec_circ} applied with all the measures equal to $\la_{[0,1]}$, $P=\trans\Id_{\la,k+1}$ (\ie $P=\Id_{\la,k+1}$), $Q=\Id_{\la,k'+1}$ and $(f,g,h)=(\otimes_{i=1}^k G_{s_i},G_t,\otimes_{i=1}^{k'} G_{s'_i})$. Indeed, as $\mu_t$ is diffuse, $G_t$ is injective, hence trivially fits $P$ and $Q$. Then:\vspace*{-3pt}\enlargethispage{.6\baselineskip}% \begin{align*} \como^{s_1,\ldots,s_k,t}&\circ\como^{t,s'_1,\ldots,s'_{k'}}\\ &\hspace*{-5mm}=(\otimes_{i=1}^k G_{s_i}\otimes G_t)_\#\Id_{\la,k+1}\circ(G_t\otimes (\otimes_{i=1}^{k'} G_{s'_i}))_\#\Id_{\la,k'+1}\ \ \text{by definition of $\como$}\\ &\hspace*{-5mm}=((\otimes_{i=1}^k G_{s_i})\otimes G_t\otimes (\otimes_{i=1}^{k'} G_{s'_i}))_\#\Id_{\la,k+1}\circ\Id_{\la,k'+1}\ \ \text{by the remark}\\ &\hspace*{-5mm}=((\otimes_{i=1}^k G_{s_i})\otimes G_t \otimes (\otimes_{i=1}^{k'} G_{s'_i}))_\#\Id_{\la,k+k'+1}\\ &\hspace*{-5mm}=\como^{s_1,\ldots,s_k,t,s'_1,\ldots,s'_{k'}}. \end{align*} To alleviate the writing, we prove the rest of \eqref{item:inatomique_facultatif}, with $\sharp E'=1$ (the general proof is alike). Take $R$ given by point \eqref{item:them:deuxieme_a} and $t\in\op]s,s'\clo[$ some unessential atomic time of $(\mu_t)_t$, then $L_{R_n\cap \op]s,s'\clo[\setminus\{t\}}=L_{R_n\cap \op]s,t\clo[}{}\ldot L_{R_n\cap \op]t,s'\clo[}\to L_{\op]s,t\clo[}{}\ldot L_{\op]t,s'\clo[}$ by Lemma \ref{lem:closed}. Now $L_{\op]s,t\clo[}{}\ldot L_{\op]t,s'\clo[}=L_{\op]s,s'\clo[}$ since $t$ is unessential. Thus $R'=R\setminus\{t\}$ satisfies \eqref{eq:convergence_doug}, hence~\eqref{eq:cv}, by point \eqref{item:them:deuxieme_b}. Let us prove \eqref{item:essentiel_necessaire}. Suppose that $I$ is some essential atomic interval, \ie there is an interval $J\supset I$ such that $J\setminus I$ is disconnected and $L_{J} \neq L_{J\setminus I}$, and assume that $I\cap R=\varnothing$. Since $L_{J} \geqlc L_{J\setminus I}$, this means that there is some $(a,a')\in\op]0,1\clo[^2$ such that $L_{J}([0,a]\times[0,a'])< L_{J\setminus I}([0,a]\times[0,a'])$. Denote $(\inf J,\sup J)\in\R^2$ by $(s,s')$. If~$a\not\in A_{s}$ and $a'\not\in A_{s'}$ (see Notation \ref{nota:Asx}), \ie if $G([0,a^{(\prime)}])=[0,G(a^{(\prime)})]$, then, pushing the inequality by $G$, and reminding that, since $I\cap R=\varnothing$, $L_{J\setminus I}\geqlc \lim_n L_{R_n\cap J}$, so that $L_{J\setminus I}([0,a]\times[0,a'])\leqslant \lim_n L_{R_n\cap J}([0,a]\times[0,a'])$, we get:\vspace*{-2pt} \begin{equation}\label{eq:mq_neq_como_Rn} \mq^{s,s'}(\op]-\infty,G(a)]\times\op]-\infty,G(a')])< \lim_{n\to\infty}\como_{[R_n]}^{s,s'}(\op]-\infty,G(a)]\times\op]-\infty,G(a')]). \end{equation} Therefore $ \lim_{n\to\infty}\como_{[R_n]}$ cannot be equal to $\mq$ and we are done. If $a\in A_s$, let $A=\op]a_0,a_1\clo[$ be the connected component of $A_s$ containing $a$; notice that $\{a_0,a_1\}\cap A=\varnothing$. We prove that \eqref{eq:mq_neq_como_Rn} holds with $a_0$ or $a_1$ in place of $a$. By construction of~$L_E$ for any set~$E$ having $s$ as minimum, the functions $b\mto k_{L_E}(b,\,.\,)$, the values of which are measures on $[0,1]$, are constant on $A$. Applying this for $E=L_{\{s\}\cup J}$ and $E=L_{\{s\}\cup J\setminus I}$, we get that either \eqref{item:421bci} below is true, or the restrictions of $L_{ J}$ and $L_{ J\setminus I}$ to $[0,a_0]\times[0,a']$ coincide, thus necessarily \eqref{item:421bcii} is true (we let $\{s\}\cup J$ appear instead of~$J$ but this does not matter by Remark \ref{rem:comp_mini}): \begin{enumeratei} \item\label{item:421bci} $L_{\{s\}\cup J}([0,a_0]\times[0,a'])< L_{\{s\}\cup J\setminus I}([0,a_0]\times[0,a'])$, \item\label{item:421bcii} on $A$, $k_{L_{\{s\}\cup J}}(b,[0,a'])< k_{L_{\{s\}\cup J\setminus I}}(b,[0,a'])$, \end{enumeratei} \noindent and if \eqref{item:421bci} is false and \eqref{item:421bcii} is true then $L_{ J}([0,a_1]\times[0,a'])< L_{ J\setminus I}([0,a_1]\times[0,a'])$. Hence anyway \eqref{eq:mq_neq_como_Rn} holds with $a_0$ or $a_1$ in place of $a$. Proceed symmetrically for $a'$ if~$a'\in A_{s'}$. This shows point \eqref{item:essentiel_necessaire}. \end{proof} \begin{rem}\label{rem:demo_nonatomique} The proof of Theorem \ref{them:deuxieme_sec}\eqref{item:them:deuxieme_c}\eqref{item:inatomique_facultatif} shows directly the last sentence of Remark \ref{rem:como_markovien}\eqref{p1:rem:como_markovien}. Indeed it shows that if every $\mu_t$ is diffuse, then for all finite $R\subset\R$, $\como_{[R]}=\como$, which gives an expression of the Markov property, see Definition \ref{defi:markov2}. \end{rem} \section{Examples and open questions}\label{sec:exemples} \Subsection{Example of Markov-quantile processes attached to discrete measures on $\N$} In this section $x_+$ is the positive part $\max\{0,x\}$ of any $x\in\R$. \begin{ex}[Discrete measures]\label{ex:integers} Let $(\mu_t)_{t\in [0,1]}$ be concentrated on $\N$ for every~$t$ and assume that for every $k\in \N$ the map $A_k:t\mto \sum_{i=0}^k \mu_t(i)$ is in $\mathcal{C}^1([0,1])$ and piecewise monotone (e.g., $A_k$ is analytic). Let moreover $A_{-1}$ be the zero constant function. We assume that:\vspace*{-3pt}\enlargethispage{.3\baselineskip}% \[\max\Bigl(\frac{-A'_k(t)}{\mu_t(k)},\frac{A'_{k-1}(t)}{\mu_t(k)}\Bigr)\] is bounded from above for $(t,k)\in[0,1]\times \mathbb{N}$. Then, using the characterization of the Markov-quantile process as a limit of quantile couplings, namely Theorem \ref{them:a}\eqref{item:limite_de_compose}, it can be proved that the Markov-quantile process $(X_t)_{t\in [0,1]}$ is the time continuous Markov chain with jump rate $q_{k,k+1}=\sfrac{(-A'_k(t))_+}{\mu_t(k)}$ from $k$ to $k+1$, and $q_{k,k-1}=\sfrac{(A'_{k-1}(t))_+}{\mu_t(k)}$ from $k$ to $k-1$ and $q_{k,j}=0$ for $|j-k|\neq 1$. Denoting $\P(X_t=k)$ by $p_k$ it means that the so-called forward Kolmogorov--Chapman system is satisfied:\vspace*{-3pt} \begin{align*} \frac{\dd p_k}{\dd t}(t)= \begin{cases} p_{1}q_{1,0}-p_0q_{0,1}&\text{if }k=0,\\ p_{k+1}q_{k+1,k}+p_{k-1}q_{k-1,k}-p_k(q_{k,k-1}+q_{k,k+1})&\text{if }k\in \N^*, \end{cases} \end{align*} where the derivative is a right derivative. Recall that the jump rate is defined for $i\neq j$~by:\vspace*{-3pt} \[ q_{i,j}(t)=\lim_{h\to 0^+}\frac{\P(X_{t+h}=j|\,X_t=i)}{h}. \] The classical theory that can be read in Feller's book \cite[Chap.\,XVII, \S 9]{Fe_t1} and the references therein (see also \cite{Doe}) ensures that our process is solution of the forward Kolmogorov--Chapman system. The uniqueness of the solution for a Markov process is obtained from the uniform bound on the rates $q_{i,j}(t)$. In place of a complete proof let us compute the jump rate in a typical case. Notice before that similar computations can be found in \cite[\S 4]{Ju_ejp}. We are looking for the jump rate $q_{k,k+1}(t)$ in the case of $A:=A_{k-1}$ and $B:=A_k$ locally decreasing on the right of $t$. At every time $t$ the atomic measure $\mu_t$ is completely described by the partition of the interval $[0,1]$ of quantile levels through the sequence $(A_k(t))_{k\in \N}$. Indeed, $]A_{k-1}(t),A_k(t)[\subset [0,1]$ is the interval of the quantile levels of the atom $\mu_t(k)\delta_k$. Recall that both $A$ and $B$ are in $\mathcal{C}^1([0,1])$. We can assume that $h$ is so small that $B(t+h)>A(t)$. For $\{r',r''\}\subset[t,t+h]$ with $r'k$. We aim at proving that the conditional probability to be still in $k$ at time $t+h$ is:\vspace*{-3pt} \begin{align}\label{eq:grando} 1-\frac{B'(t)}{(B-A)(t)}\,h+O(h^2). \end{align} Since the probability to jump more than twice is $O(h^2)$, \eqref{eq:grando} furnishes the announced jump rate $q_{k,k+1}(t)=(-A'_k(t))_+/\mu_t(k)$ in the case of decreasing functions. So let us prove \eqref{eq:grando}. We consider a partition $R=\{r_0,\ldots,r_{m}\}$ of $[t,t+h]$ with $(r_0,r_{m})=(t,t+h)$ and the discrete quantile Markov chain associated with it. As $A$ and $B$ are decreasing, note that no mass can leave the quantile level interval $[A,B]$ and come back on it in the same time interval $[t,t+h]$. On $[r_n,r_{n+1}]$ the probability to stay in the interval is \[ \frac{B(r_{n+1})-A(r_n)}{B(r_n)-A(r_n)}=1-\frac{B(r_n)-B(r_{n+1})}{B(r_n)-A(r_n)} \] as one can easily convince oneself with a picture similar to the left part of Figure \ref{figure}. We let the proof of the following fact to the reader: there exists $\delta=O(h)$ such that for every $r'0$, the process $(X_t)_{t}$ starts with slope $-1$, is flat on $[\psfrac{1+X_0}{2},\psfrac{5-X_0}{2}]$ and continues with slope $1$ after $\psfrac{5-X_0}{2}$. \end{ex} \Subsection{Theoretic Markov-quantile processes} \begin{ex}[One atom with regular level functions]\label{ex:swimming_atom} Take $(\mu_t)_{t\in{[0,1]}}$ such that for every $t$, $\mu_t$ has exactly one atom $x_t\in \R$ and the interval of quantile levels of this atom at time $t$ is $\op]A(t),B(t)\clo[$. Assume moreover that $A$ and $B$ are of class $\mathcal{C}^1$ and piecewise monotone. Then the Markov-quantile process $(X_t)_{t\in [0,1]}$ can be described using two Poisson point processes of jump rates $(A')_+/(B-A)$ and $(B')_-/(B-A)$. Conditionally on $F_{\mu_t}(X_t)\in \op]A(t), B(t)\clo[$, we have $X_t=x_t$ until the next time $t_0\geq t$ in the point process. Then the process $(X_t)_t$ leaves $x_t$ and starts a piece of quantile trajectory constant in the space $[0,1]$ of quantile levels with value $A(t_0)$ or $B(t_0)$. The process may hit again $x_t$ if there exists some $t_1>t_0$ with $A(t_1)=x_{t_0}$, or $B(t_1)=x_{t_0}$ respectively. \end{ex} The next remark is of general interest and particularly significant with respect to Remark \ref{rem:a_bouger}. It presents the Markov-quantile process as one end of the spectrum of processes of law in $\ma(\mu)$ that satisfies \eqref{item:transitions} of Theorem \ref{them:a}, \ie have increasing kernels, the other end being the independent process. \begin{rem}\label{rem:a_bouger} The minimality condition \eqref{item:minimal} of Theorem \ref{them:a} satisfied by the Markov-quantile process $(X_t)_t$ attached to some $(\mu_t)_t$ can also be stated as follows. For every process $(Y_t)_{t\in \R}$ satisfying \eqref{item:markov} and \eqref{item:transitions} of Theorem \ref{them:a}, for every $s0$, we may take $r$ such that $\P(X_r=0)$ is small enough so that $\P(X_r=0|X_s=0)\leqslant\eps$, $\P(X_r=1|X_s=0)\geqslant1-\eps$, and $|\P(X_t=0|X_r=1)-\P(X_t=0)|\leqslant\eps$. Then:\vspace*{-3pt} \begin{align*} |\P(X_t=0|X_s=0)&-\P(X_t=0)|\\ &=\bigl|\P(X_t=0|X_r=0)\P(X_r=0|X_s=0)\\ &\hspace*{1.5cm}+\P(X_t=0|X_r=1)\P(X_r=1|X_s=0)-\P(X_t=0)\bigr|\\ &\hspace*{6.7cm}\text{since $X$ is Markov}\\[-5pt] &\leqslant|\P(X_t=0|X_r=1)-\P(X_t=0)|+2\eps\\ &\leqslant3\eps, \end{align*} which is the wanted independence. Hence, we assume that $m_{s,t}>0$ and that $a$ takes values in $[m_{s,t},1-m_{s,t}]$ in $\mu_t=a(t)\delta_0+b(t)\delta_1$. We are left with the task to prove that independence is equivalent to a infinite total variation of $a$ on $[s,t]$. Let $\theta_0$ be the uniform measure on $[0,a(s)]$. Our goal reduces to establishing $\la=\theta_0 \ell_{[s,t]}\eqdef\stosup_R \theta_0\ell_{r_1}\ell_{r_2}\cdots \ell_{r_m}$, where $R$ ranges among the partitions $\{r_0,\ldots, r_m\}$ with $(r_0,r_m)=(s,t)$. For the measures under consideration, if $a(r_{k-1})\leq a(r_k)\leq a(r_{k+1})$ or $a(r_{k-1})\geq a(r_k)\geq a(r_{k+1})$ it holds $\ell_{r_{k-1}} \ell_k \ell_{k+1}=\ell_{r_{k-1}} \ell_{r_{k+1}}$. Therefore we can assume without loss of generality that the sequence $(a(r_k))_{k=0,\ldots,m}$ has increments with alternating sign, for instance $a(r_{2k+1})\geq a(r_{2k})$ for every $k$. We define $\theta_n=\theta_0 \ell_{r_0}\cdots \ell_{r_n}$. The measure $\theta_n$ can be written in the form:\vspace*{-3pt} \[ \theta_n=d_n\la_{[0,a(r_n)]}+d'_n\la_{[a(r_n),1]}\in \mdec, \] where $d_n=a(r_n)^{-1}\theta_n([0,a(r_n)])$ in fact parametrizes the complete measure. Note, after Remark \ref{rem:a_bouger} that $\theta_n\leqs \la$, which means $d_n\geq 1\geq d'_n$. As $a(r_n)\in[m_{s,t}, 1-m_{s,t}]$, the sequence converges to $\la$ if and only if $d_n\to 1$. Recalling the effect of the kernel $\ell_{r_{n+1}}$, described on Figure \ref{figure2} and defined in Notation \ref{nota:Asx}\eqref{p:l_R_fini}, we find:\vspace*{-5pt} \begin{align*} d_{n+1}-1= \begin{cases} (d_{n}-1)\dfrac{a(r_{n+1})}{a(r_n)}&\text{if }a(r_{n+1})s$. Let $\alpha_{s,t}$ be $\sup_{r\in[s,t]}B(r)$. Then $L_{[s,t]}$ is simply the uniform measure of mass $\alpha_{s,t}$ on $[0,\alpha_{s,t}]^2$ plus the one-dimensional uniform measure of mass $1-\alpha_{s,t}$ on the diagonal between $(\alpha_{s,t},\alpha_{s,t})$ and $(1,1)$. The same for the kernel $\ell_{s,t}$ reads: \[ \ell_{s,t}(x,\cdot)= \begin{cases} \alpha_{s,t}^{-1}\la\lfloor_{[0,\alpha_{s,t}]}&\text{if }x\leq \alpha_{s,t},\\ \delta_x&\text{if }x>\alpha_{s,t}. \end{cases} \] A particle of quantile value $\leq \alpha_{s,t}$ at time $s$ is uniformly mapped at time $t$ on the particles of quantile levels $[0,\alpha_{s,t}]$. If the quantile value at time $s$ is greater that $\alpha_{s,t}$, the particle keeps on with the same level until time $t$ as if it were the quantile process. \end{ex} \Subsection{Transformations of Markov-quantile processes} \begin{ex}[Markov-quantile processes]\label{ex:quantile} According to $(\mu_t)_t$, a quantile process may be Markov or not. Recalling Remark \ref{rem:como_markovien} \eqref{p2:rem:como_markovien}, if $\como$ is Markov it coincides with the Markov-quantile process. As proved in \cite[Prop.\,3]{Ju_seminaire}, the criterion is the following: the process is \emph{not} Markov if and only if there exists $\alpha\neq \alpha'\in [0,1]$ and $t_1