On the Relation Between Optimal Transport and Schrödinger Bridges: A Stochastic Control Viewpoint

Abstract

We take a new look at the relation between the optimal transport problem and the Schr\"{o}dinger bridge problem from the stochastic control perspective. We show that the connections are richer and deeper than described in existing literature. In particular: a) We give an elementary derivation of the Benamou-Brenier fluid dynamics version of the optimal transport problem; b) We provide a new fluid dynamics version of the Schr\"{o}dinger bridge problem; c) We observe that the latter provides an important connection with optimal transport without zero noise limits; d) We propose and solve a fluid dynamic version of optimal transport with prior; e) We can then view optimal transport with prior as the zero noise limit of Schr\"{o}dinger bridges when the prior is any Markovian evolution. In particular, we work out the Gaussian case. A numerical example of the latter convergence involving Brownian particles is also provided.