Mass transportation with LQ cost functions

TL;DR

This paper explores optimal transport in Euclidean space with LQ cost functions, establishing existence, uniqueness, and regularity of optimal maps, and extending Brenier's Theorem to this setting.

Contribution

It generalizes Brenier's Theorem for LQ cost functions, proving existence and uniqueness of optimal transport maps and analyzing their regularity.

Findings

Optimal transport maps exist and are unique under certain conditions.

In the controllable case, the optimal map is a gradient of a convex function up to linear transformation.

Regularity results for the optimal transport maps are established.

Abstract

We study the optimal transport problem in the Euclidean space where the cost function is given by the value function associated with a Linear Quadratic minimization problem. Under appropriate assumptions, we generalize Brenier's Theorem proving existence and uniqueness of an optimal transport map. In the controllable case, we show that the optimal transport map has to be the gradient of a convex function up to a linear change of coordinates. We give regularity results and also investigate the non-controllable case.