Multiplicative Schrodinger problem and the Dirichlet transport

We consider a Monge-Kantorovich optimal transport problem on the unit simplex with a cost function given by the log of the Euclidean inner product. We show that the transport is the large deviation limit of multiplication by the Dirichlet (or, gamma) process and suitable normalization. This is a multiplicative counterpart to the Wasserstein-2 transport that is carried by adding Brownian motion to an initial mass distribution (called the Schrodinger problem by Leonard). The potential function and the Lagrangian of this transport appear to be closely related to the Wasserstein diffusion (Brownian motion on the Wasserstein space) put forward by Sturm and other coauthors, although it is unclear what the exact nature of this relationship is.