Mass transportation with LQ cost functions
For mathematicians and researchers in optimal transport, this provides a theoretical extension of classical results to a new class of cost functions, but it is incremental as it builds on existing frameworks.
The paper generalizes Brenier's theorem to optimal transport with LQ cost functions, proving existence and uniqueness of an optimal transport map that is the gradient of a convex function up to a linear change of coordinates in the controllable case, with regularity results and analysis of the non-controllable case.
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.