Kantorovich potentials and continuity of total cost for relativistic cost functions

Abstract

In this paper we consider the optimal mass transport problem for relativistic cost functions, introduced in [12] as a generalization of the relativistic heat cost. A typical example of such a cost function is ct(x, y) = h(y−x t), h being a strictly convex function when the variable lies on a given ball, and infinite otherwise. It has been already proved that, for every t larger than some critical time T > 0, existence and uniqueness of optimal maps hold; nonetheless, the existence of a Kantorovich potential is known only under quite restrictive assumptions. Moreover, the total cost corresponding to time t has been only proved to be a decreasing rightcontinuous function of t. In this paper, we extend the existence of Kantorovich potentials to a much broader setting, and we show that the total cost is a continuous function. To obtain both results the two main crucial steps are a refined "chain lemma" and the result that, for t > T , the points moving at maximal distance are negligible for the optimal plan.

On the existence of dual solutions for Lorentzian cost functions

“…The conditions are met on a weakly dense subset of pairs of measures by Corollary 2.9. Theorem 2.8 generalizes results in [6,7,17]. The condition of strict timelikeness is related to parts of the definition of q-separated measures in [17].…”

“…The conclusion of Theorem 2.12 under assumption (ii) on the other hand has no counterpart there. The theorem generalizes [6, Corollary 3.6] and [7,Theorem C]. Note that Theorem 2.12 is proven indirectly and relies on a very similar construction as the 1 + 1-dimensional example in Section 3.1.…”

“…Conversely the existence of a dual solution necessitates that only a negligible part of mass is transported along lightlike geodesics, see Theorem 2.12 the second major result of the present paper. A related result in special relativity is [7,Theorem C].…”

Causal evolution of probability measures and continuity equation

“…Lorentzian optimal transport theory is a new fast-developing field of mathematical physics. Starting with the work of Brenier [17], who was the first to study the Monge-Kantorovich problem with relativistic cost functions, several authors successfully adapted the tools of standard, Riemannian optimal transport theory to the setting of Lorentzian (or even Lorentz-Finsler) manifolds [14,15,36,59]. It is readily motivated by physical considerations, including the so-called 'early universe reconstruction problem' [18,[30][31][32], relationships between general relativity and the second law of thermodynamics [41,48], and the notion of causality and causal evolution of nonlocal objects [24,25,43].…”

A study of the dual problem of the one-dimensional L∞-optimal transport problem with applications

“…Similar results have been generalized to a much larger class of cost functions, first for strictly convex costs with respect to the difference [14,15,7] and later on for those satisfying the so-called twist condition, see [13,11]. These arguments cannot be reproduced for the cost functionals for which the existence of solutions (ϕ, ψ) is not known (see for instance [3,18] for recent progresses in this direction in special cases) and even, for our L ∞ problem (1.1), the meaning of "dual formulation" is itself not clear. Yet, although the functional γ → y − x L ∞ γ is not convex, it is still level-convex (in the sense that the level sets of the functional are convex sets and this, sometimes, goes under the name of quasi-convex).…”