Nicola Gigli, Tapio Rajala and Karl-Theodor Sturm Optimal maps and exponentiation on finite dimensional spaces with Ricci curvature bounded from below J. Geom. Anal. 2015 http://arxiv.org/abs/1305.4849
Abstract: We prove existence and uniqueness of optimal maps on RCD∗(K,N) spaces under the assumption that the starting measure is absolutely continuous. We also discuss how this result naturally leads to the notion of exponentiation and to the local-to-global property of RCD∗(K,N) bounds.