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$C_{loc}^{1,1}$ optimal pairs in the dual optimal transport problem for a Lorentzian cost along displacement interpolations
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abstract
We consider the optimal transportation problem on a globally hyperbolic spacetime with a cost function $c$, which corresponds to the optimal transportation problem on a complete Riemannian manifold where the cost function is given by the squared Riemannian distance. Building upon methods of weak KAM theory, we will establish the existence of $C_{loc}^{1,1}$ optimal pairs for the dual optimal transport problem for probability measures along displacement interpolations.
Forward citations
Cited by 2 Pith papers
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Semiconvexity of (weak) Kantorovich potentials in the Lorentzian optimal transport problem
In globally hyperbolic spacetimes with cost −d^p, weak Kantorovich potentials are locally semiconvex on an open set of full measure, yielding a unique optimal transport map T with ∇φ+∇_x c=0.
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On the locus of multiple maximizing geodesics on a globally hyperbolic spacetime
In globally hyperbolic spacetimes, the locus of multiple maximizing geodesics is locally contractible and homotopy equivalent to the causal future minus the Lorentzian Aubry set.
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