We introduce an optimal transport problem on the unit simplex which can be regarded as a multiplicative analogue of the Euclidean quadratic transport. The transports are given in terms of the gradient maps of exponentially concave functions, and can be interpreted probabilistically using a particle system of Dirichlet processes. The optimal transport map induces a dual geometry - in the sense of information geometry - on the simplex which has constant negative curvature. Moreover, under a novel displacement interpolation the entropy is "semi-convex", leading to interesting questions beyond the usual setting of Wasserstein space. The talk is based on joint work with Soumik Pal.