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The dual problem of optimal transportation in Lorentz-Finsler geometry is studied. It is shown that in general no solution exists even in the presence of an optimal coupling. Under natural assumptions dual solutions are established. It is further shown that the existence of a dual solution implies that the optimal transport is timelike on a set of full measure. In the second part the persistence of absolute continuity along an optimal transportation under obvious assumptions is proven and a solution to the relativistic Monge problem is provided. [ABSTRACT FROM AUTHOR]

Recently, Gigli developed a Sobolev calculus on non-smooth spaces using module theory. In this paper it is shown that his theory fits nicely into the theory of differentiability spaces initiated by Cheeger, Keith and others. A relaxation procedure for Lp-valued subadditive functionals is presented and a relationship between the module generated by a functional and the one generated by its relaxation is given. In the framework of differentiability spaces, which includes so called PI- and RCD(K,N)-spaces, the Lipschitz module is pointwise finite dimensional. A general renorming theorem together with the characterization above shows that the Sobolev spaces of differentiability spaces are reflexive. [ABSTRACT FROM AUTHOR]

Let ( M , g ) be a smooth Riemannian manifold and G a compact Lie group acting on M effectively and by isometries. It is well known that a lower bound of the sectional curvature of ( M , g ) is again a bound for the curvature of the quotient space, which is an Alexandrov space of curvature bounded below. Moreover, the analogous stability property holds for metric foliations and submersions. The goal of the paper is to prove the corresponding stability properties for synthetic Ricci curvature lower bounds. Specifically, we show that such stability holds for quotients of RCD ⁎ ( K , N ) -spaces, under isomorphic compact group actions and more generally under metric-measure foliations and submetries. An RCD ⁎ ( K , N ) -space is a metric measure space with an upper dimension bound N and weighted Ricci curvature bounded below by K in a generalized sense. In particular, this shows that if ( M , g ) has Ricci curvature bounded below by K ∈ R and dimension N , then the quotient space is an RCD ⁎ ( K , N ) -space. Additionally, we tackle the same problem for the CD / CD ⁎ and MCP curvature-dimension conditions. We provide as well geometric applications which include: A generalization of Kobayashi's Classification Theorem of homogeneous manifolds to RCD ⁎ ( K , N ) -spaces with essential minimal dimension n ≤ N ; a structure theorem for RCD ⁎ ( K , N ) -spaces admitting actions by large (compact) groups ; and geometric rigidity results for orbifolds such as Cheng's Maximal Diameter and Maximal Volume Rigidity Theorems. Finally, in two appendices we apply the methods of the paper to study quotients by isometric group actions of discrete spaces and of (super-)Ricci flows. [ABSTRACT FROM AUTHOR]

In this paper we investigate the relationship between a general existence of transport maps of optimal couplings with absolutely continuous first marginal and the property of the background measure called essentially non-branching introduced by Rajala–Sturm (2014) [27] . In particular, it is shown that the qualitative non-degeneracy condition introduced by Cavalletti–Huesmann (2015) [6] implies that any essentially non-branching metric measure space has a unique transport maps whenever the initial measure is absolutely continuous. This generalizes a recently obtained result by Cavalletti–Mondino (2017) [8] on essentially non-branching spaces with the measure contraction condition MCP ( K , N ) . In the end we prove a measure rigidity result showing that any two essentially non-branching, qualitatively non-degenerate measures on a fixed metric spaces must be mutually absolutely continuous. This result was obtained under stronger conditions by Cavalletti–Mondino (2016) [7] . It applies, in particular, to metric measure spaces with generalized finite dimensional Ricci curvature bounded from below. [ABSTRACT FROM AUTHOR]

In this article, a proof of the interpolation inequality along geodesics in p-Wasserstein spaces is given. This interpolation inequality was the main ingredient to prove the Borel-Brascamp-Lieb inequality for general Riemannian and Finsler manifolds and led Lott-Villani and Sturm to define an abstract Ricci curvature condition. Following their ideas, a similar condition can be defined and for positively curved spaces one can prove a Poincaré inequality. Using Gigli's recently developed calculus on metric measure spaces, even a q-Laplacian comparison theorem holds on q-infinitesimal convex spaces. In the appendix, the theory of Orlicz-Wasserstein spaces is developed and necessary adjustments to prove the interpolation inequality along geodesics in those spaces are given. [ABSTRACT FROM AUTHOR]

Based on the idea of a recent paper by Ambrosio–Gigli–Savaré (2014) [5] , we show that the L 2 -gradient flow of the q -Cheeger energy, called q -heat flow, solves a generalized gradient flow problem of the Renyi entropy functional in the p -Wasserstein. For that, a further study of the q -heat flow is presented including a condition for its mass preservation. Under a convexity assumption on the upper gradient, which holds for all q ≥ 2 , one gets uniqueness of the gradient flow and the two flows can be identified. Smooth solutions of the q -heat flow are solutions to the parabolic q -Laplace equation, i.e. ∂ t f t = Δ q f t . [ABSTRACT FROM AUTHOR]

We systematically investigate the problem of representing Markov chains by families of random maps, and which regularity of these maps can be achieved depending on the properties of the probability measures. Our key idea is to use techniques from optimal transport to select optimal such maps. Optimal transport theory also tells us how convexity properties of the supports of the measures translate into regularity properties of the maps via Legendre transforms. Thus, from this scheme, we cannot only deduce the representation by measurable random maps, but we can also obtain conditions for the representation by continuous random maps. Finally, we present conditions for the representation of Markov chain by random diffeomorphisms. [ABSTRACT FROM AUTHOR]

We introduce a new definition of nonpositive curvature in metric spaces and study its relation to the existing notions of nonpositive curvature in comparison geometry. The main feature of our definition is that it applies to all metric spaces and does not rely on geodesics. Moreover, a scaled and a relaxed version of our definition are appropriate in discrete metric spaces, and are believed to be of interest in geometric data analysis. [ABSTRACT FROM AUTHOR]