The Holonomy of Optimal Mass Transport: The Gaussian-Linear Case

Abstract

The theory of Monge-Kantorovich Optimal Mass Transport (OMT) has in recent years spurred a fast developing phase of research in stochastic control, control of ensemble systems, thermodynamics, data science, and several other fields in engineering and science. We herein introduce a new type of transportation problems. The salient feature of these problems is that particles/agents in the ensemble are labeled and their relative position along their journey is of interest. Of particular importance in our program are control laws that steer ensembles along cycles ensuring that individual particles return to their original position. This feature is in contrast with the classical theory of optimal transport where the primary object of study is the path of probability densities, without any concern about particle labels. In the theory that we present, we focus on the case Gaussian distributions and linear dynamics, and explore a hitherto unstudied sub-Riemannian structure of Monge-Kantorovich transport where the relative position of particles along their journey is modeled by the holonomy of the transportation schedule. From this vantage point, we discuss several other problems of independent interest.