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

Source abstract

Abstract We discuss the problem of the regularity-in-time of the map t ↦ T t ∊ L p (ℝ d , ℝ d ; σ), where T t is a transport map (optimal or not) from a reference measure σ to a measure μ t which lies along an absolutely continuous curve t ↦ μ t 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.