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Let $\mu$ = ($\mu$t) t$\in$R be a 1-parameter family of probability measureson 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 $\mu$, amongthe 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 continuousin the Wasserstein space P 2 (R). Then, X is solution of a Benamou-Breniertransport problem with intermediate marginals $\mu$t. It provides a MarkovLagrangian probabilistic representation of the continuity equation, moreoverthe unique Markov process:-obtained as a limit for the finite dimensionaltopology of quantile processes where the past is made independent of the futureat finitely many times.-or, alternatively, obtained as a limit of processeslinearly interpolating $\mu$. This raises new questions about ways to obtainMarkov Lagrangian representations of the continuity equation, and to seekuniqueness properties in this framework.

On absolutely continuous curves in the Wasserstein space over R and their representation by an optimal Markov process