  Let $\mu$ = ($\mu$t) t$\in$R be a 1-parameter family of probability measures
on R. In [13] we introduced its "Markov-quantile" process: a process X = (Xt)
t$\in$R that resembles at most the quantile process attached to $\mu$, among
the Markov processes attached to $\mu$, i.e. whose family of marginal laws is
$\mu$. In this article we look at the case where $\mu$ is absolutely continuous
in the Wasserstein space P 2 (R). Then, X is solution of a Benamou-Brenier
transport problem with intermediate marginals $\mu$t. It provides a Markov
Lagrangian probabilistic representation of the continuity equation, moreover
the unique Markov process:-obtained as a limit for the finite dimensional
topology of quantile processes where the past is made independent of the future
at finitely many times.-or, alternatively, obtained as a limit of processes
linearly interpolating $\mu$. This raises new questions about ways to obtain
Markov Lagrangian representations of the continuity equation, and to seek
uniqueness properties in this framework.
