Multiplicative Schrödinger problem and the Dirichlet transport

Soumik Pal, Ting-Kam Leonard Wong

Published 2018-07-16Version 1

We consider an optimal transport problem on the unit simplex whose solutions are given by gradients of exponentially concave functions and prove two main results. One, we show that the optimal transport is the large deviation limit of a particle system of Dirichlet processes transporting one probability measure on the unit simplex to another by coordinatewise multiplication and normalizing. The structure of our Lagrangian and the appearance of the Dirichlet process relate our problem closely to the entropic measure on the Wasserstein space as defined by von-Renesse and Sturm in the context of Wasserstein diffusion. The limiting procedure is a triangular limit where we allow simultaneously the number of particles to grow to infinity while the `noise' goes to zero. The method, which generalizes easily to other cost functions, including the Wasserstein cost, provides a novel combination of the Schr\"odinger problem approach due to C. L\'eonard and the related Brownian particle systems by Adams et al. which does not require gamma convergence. Two, we analyze the behavior of entropy along the lines of transport. The base measure on the simplex is taken to be the Dirichlet measure with all zero parameters which relates to the finite-dimensional distributions of the entropic measure. The interpolating curves are not the usual McCann lines. Nevertheless we show that entropy plus the transport cost remains convex, which is reminiscent of the semiconvexity of entropy along lines of McCann interpolations in negative curvature spaces. We also obtain, under suitable conditions, dimension-free bounds of the optimal transport cost in terms of entropy.