**EXAMPLE 5.10 (One atom on the lower levels).** — Consider $(\mu_t)_{t \in [0,1]}$ such that for every $t$, $\mu_t$ has exactly one atom and this atom is between the quantile levels $A(t) = 0$ and $B(t)$. An example is $\mu_t = B(t)\delta_0 + (1-B(t))\mathcal{E}(1)$, where $\delta_0$ is the Dirac mass in zero and $\mathcal{E}(1)$ the exponential law of parameter 1. No regularity assumption is made on $B$. Similar observations as in Example 5.9 permit us to specify the kernel between time $s$ and $t > s$. 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:

$$ \ell_{s,t}(x, \cdot) = \begin{cases} \alpha_{s,t}^{-1} \lambda_{[0,\alpha_{s,t}]} & \text{if } x \le \alpha_{s,t}, \\ \delta_x & \text{if } x > \alpha_{s,t}. \end{cases} $$

A particle of quantile value $\le \alpha_{s,t}$ 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 than $\alpha_{s,t}$, the particle keeps on with the same level until time $t$ as if it were the quantile process.

## 5.4. TRANSFORMATIONS OF MARKOV-QUANTILE PROCESSES

**EXAMPLE 5.11 (Markov-quantile processes).** — According to $(\mu_t)_t$, a quantile process may be Markov or not. Recalling Remark 1.8 (b), if $\mathfrak{Q}$ is Markov it coincides with the Markov-quantile process. As proved in [23, Prop. 3], the criterion is the following: the process is not Markov if and only if there exists $\alpha \ne \alpha' \in [0, 1]$ and $t_1 < t_2 < t_3$ such that the $\alpha$-quantile and the $\alpha'$-quantile of $\mu_{t_2}$ are equal but that those of $\mu_{t_1}$ and $\mu_{t_3}$ differ. This can be summarized saying that “X’s are forbidden”, where X refers to the shape of the letter, the four ends being $G_{\mu_1}(\alpha)$, $G_{\mu_1}(\alpha')$, $G_{\mu_3}(\alpha)$ and $G_{\mu_3}(\alpha')$, the intersection being $G_{\mu_2}(\alpha) = G_{\mu_2}(\alpha')$. Other letters like O, Y and Z are allowed.

**REMARK 5.12 (Reversal of time and twist of space and time).** — If $(X_t)_{t \in \mathbb{R}}$ is the Markov-quantile process attached to $(\mu_t)_{t \in \mathbb{R}}$ then $X_{-t}$ is the Markov-quantile process in $\text{Marg}((\mu_{-t})_{t \in \mathbb{R}})$. This comes from Theorem A (iv) on the limit of products of quantile couplings and the fact that $^t\mathfrak{Q}(\mu, \nu) = \mathfrak{Q}(\nu, \mu)$. More generally, for every homeomorphism $\varphi$ from $\mathbb{R}$ into $\mathbb{R}$, $X_{\varphi(t)}$ has law the Markov-quantile measure of $\text{Marg}((\mu_{\varphi(t)})_{t \in \mathbb{R}})$. Of course non-injective monotone continuous map $\varphi$ may be used too.

Moreover, if for all $t$, $f_t: \mathbb{R} \to \mathbb{R}$ are strictly monotone functions with the same orientation for all $t$, the process $f_t(X_t)$ is a Markov-quantile process.

**REMARK 5.13.** — The Markov-quantile process being time-reversible, and as a coupling $P \in \text{Marg}(\mu, \nu)$ has increasing kernel if and only if $^tP \in \mathcal{M}^>(\nu, \mu)$ and $P \preceq_{10} Q$ is equivalent to $^tP \preceq_{10} ^tQ$ (recall Remark 3.16 and Definition 3.4), points (ii) and (iii) of in Theorem A can be replaced by

(ii') For every $s < t$, $\mathfrak{M}\mathfrak{Q}^{s,t} \in \mathcal{M}^>(\mu_s, \mu_t)$, and

(iii') for every $s < t$, $\mathfrak{M}\mathfrak{Q}^{s,t}$ is a minimal coupling for $\preceq_{10}$ among the processes satisfying (i) and (ii').
## 5.5. OPEN QUESTIONS

### 5.5.1. Markovinification

(a) We may interpret the resulting process in Theorem 2.26 as the Markov process that has infinitesimally the same transitions as $P$. However, as it depends on the choice of the partitions, this Markov process is not a priori uniquely determined. At which conditions is this Markov process uniquely determined and how can it be characterized? If the initial process is a quantile process, we proved in Theorem A that the answer is yes, without condition, and characterized it using orderings.

(b) In Theorem B we proved that Markovinification of the quantile process occurs for processes $(\mathcal{Q}_{[R_n]})_n$ in place of consistent (see Definition 2.14) transport plans. Does an *existence* statement analogous to Theorem 2.26 happen when we consider sequences $(P_{[R_n]})_n$?

### 5.5.2. The Kellerer theorem in dimension $d \ge 2$. —
Kellerer proved Theorem 1.17 for martingales in $\mathbb{R}$. For measures $(\mu_t)_t$ on $\mathbb{R}^d$, increasing in convex order, it is known that there exists an associated martingale [19, 3] but not whether one of them is Markov. This is a major question. Note that another interpretation of the question in higher dimension is possible when considering martingales indexed by a multidimensional set (see [18, Prob. 7b]). This problem was solved in [23].

### 5.5.3. Markov Kamae-Krengel theorem. —
Kamae and Krengel proved in [26] that if $(\mu_t)_{t \in \mathbb{R}}$ are measures on a partially ordered Polish space $E$ such that $t \mapsto \mu_t$ is increasing for the stochastic order, in the sense that $t \mapsto \int fd\mu_t$ is increasing for any increasing bounded $f: E \mapsto \mathbb{R}$, there exists an increasing process $(X_t)_t$ with law in $\text{Marg}(\mu)$. We proved in Theorem A and C that if $E$ is $\mathbb{R}$, the process can moreover be Markov. A natural problem is whether this is also true for any $E$.

### 5.5.4. Strong Markov property. —
The Markov-quantile process is not always the unique Markov process that minimizes the energy $\mathcal{E}$, as is shown in Example 5.4. However in this example $\mathcal{M}\mathcal{Q}$ is strongly Markovian. Is the Markov-quantile process strongly Markovian for every $(\mu_t)_t$? Does the strong Markov property characterize it for curves of finite energy?

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Manuscript received 3rd February 2020
accepted 20th October 2021

CHARLES BOUBEL, Institut de Recherche Mathématique Avancée, UMR 7501, Université de Strasbourg et CNRS

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NICOLAS JUILLET, Institut de Recherche Mathématique Avancée, UMR 7501, Université de Strasbourg et CNRS

7 rue René Descartes, 67000 Strasbourg, France

E-mail : nicolas.juillet@uha.fr

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