*Accepted Paper*

**Inserted:** 17 nov 2008

**Last Updated:** 9 feb 2010

**Journal:** Proceedings of the Edinburgh Mathematical Society

**Year:** 2008

**Abstract:**

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 $p^{-1}$-Hölder continuous.

**Keywords:**
Wasserstein distance, Monge-Ampère equation

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