The optimal mass transport problem for relativistic costs

Abstract

In this paper, we study the optimal mass transportation problem in \({\mathbb{R}^{d}}\) for a class of cost functions that we call relativistic cost functions. Consider as a typical example, the cost function c(x, y) = h(x − y) being the restriction of a strictly convex and differentiable function to a ball and infinite outside this ball. We show the existence and uniqueness of the optimal map given a relativistic cost function and two measures with compact support, one of the two being absolutely continuous with respect to the Lebesgue measure. With an additional assumption on the support of the initial measure and for supercritical speed of propagation, we also prove the existence of a Kantorovich potential and study the regularity of this map. Besides these general results, a particular attention is given to a specific cost because of its connections with a relativistic heat equation as pointed out by Brenier (Extended Monge–Kantorovich Theory. Optimal Transportation and Applications, 2003).