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On Hölder continuity-in-time of the optimal transport map towards measures along a curve

Abstract We discuss the problem of the regularity-in-time of the map t ↦ Tt ∊ Lp(ℝd, ℝd; σ), where Tt is a transport map (optimal or not) from a reference measure σ to a measure μt which lies along an absolutely continuous curve t ↦ μt in the space ($(\mathscr{P}_p(\mathbb{R}^d),W_p)$)). We prove that in most cases such a map is no more than 1/p-Hölder continuous.