Markovian representation of the continuity equation

It has been established by Lisini that absolutely continuous curves (of order 2) $\mu:t ↦ \mu_t$ in the Wasserstein space over a metric space $X$ can be represented by an action-minimizing probability measure on the space of absolutely continuous curves. We will show that in the basic case of the real line ($X=\mathbb{R}$), this measure can moreover be asked to be Markovian. This is a special case of a more general result, with other consequences, where no continuity assumptions are made on the family $\mu$. (joint work with Charles Boubel)