On Holder continuity in time of the optimal transport map towards measures along a curve.

We discuss the problem of the regularity-in-time of the map $t\mapsto T_t\in L^p(\R^d,\R^d;\sigma)$, where $T_t$ is a transport map (optimal or not) from a reference measure $\sigma$ to a measure $\mu_t$ which lies along an absolutely continuous curve $t\mapsto\mu_t$ in the space $P_p(\R^d),W_p$. We prove that in most cases such a map is no more than 1/p-Holder continuous.