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 bar right arrow T-t is an element of L-p (R-d, R-d; sigma), where Tt is a transport map (optimal or not) from a reference measure s to a measure mu(t) which lies along an absolutely continuous curve t bar right arrow 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.