Search results for "Optimal transport"
showing 8 items of 8 documents
Optimal maps and exponentiation on finite dimensional spaces with Ricci curvature bounded from below
2013
We prove existence and uniqueness of optimal maps on $RCD^*(K,N)$ spaces under the assumption that the starting measure is absolutely continuous. We also discuss how this result naturally leads to the notion of exponentiation.
Mass transportation on sub-Riemannian structures of rank two in dimension four
2017
International audience; This paper is concerned with the study of the Monge optimal transport problem in sub-Riemannian manifolds where the cost is given by the square of the sub-Riemannian distance. Our aim is to extend previous results on existence and uniqueness of optimal transport maps to cases of sub-Riemannian structures which admit many singular minimizing geodesics. We treat here the case of sub-Riemannian structures of rank two in dimension four.
Slopes of Kantorovich potentials and existence of optimal transport maps in metric measure spaces
2014
We study optimal transportation with the quadratic cost function in geodesic metric spaces satisfying suitable non-branching assumptions. We introduce and study the notions of slope along curves and along geodesics and we apply the latter to prove suitable generalizations of Brenier's theorem of existence of optimal maps.
On one-dimensionality of metric measure spaces
2019
In this paper, we prove that a metric measure space which has at least one open set isometric to an interval, and for which the (possibly non-unique) optimal transport map exists from any absolutely continuous measure to an arbitrary measure, is a one-dimensional manifold (possibly with boundary). As an immediate corollary we obtain that if a metric measure space is a very strict $CD(K,N)$ -space or an essentially non-branching $MCP(K,N)$-space with some open set isometric to an interval, then it is a one-dimensional manifold. We also obtain the same conclusion for a metric measure space which has a point in which the Gromov-Hausdorff tangent is unique and isometric to the real line, and fo…
Equivalent definitions of very strict $CD(K,N)$ -spaces
2023
We show the equivalence of the definitions of very strict $CD(K,N)$ -condition defined, on one hand, using (only) the entropy functionals, and on the other, the full displacement convexity class $\mathcal{DC}_N$. In particular, we show that assuming the convexity inequalities for the critical exponent implies it for all the greater exponents. We also establish the existence of optimal transport maps in very strict $CD(K,N)$ -spaces with finite $N$.
Convexities and optimal transport problems on the Wiener space
2013
The aim of this PhD is to study the optimal transportation theory in some abstract Wiener space. You can find the results in four main parts and they are aboutThe convexity of the relative entropy. We will extend the well known results in finite dimension to the Wiener space, endowed with the uniform norm. To be precise the relative entropy is (at least weakly) geodesically 1-convex in the sense of the optimal transportation in the Wiener space.The measures with logarithmic concave density. The first important result consists in showing that the Harnack inequality holds for the semi-group induced by such a measure in the Wiener space. The second one provides us a finite dimensional and dime…
On deterministic solutions for multi-marginal optimal transport with Coulomb cost
2022
In this paper we study the three-marginal optimal mass transportation problem for the Coulomb cost on the plane $\R^2$. The key question is the optimality of the so-called Seidl map, first disproved by Colombo and Stra. We generalize the partial positive result obtained by Colombo and Stra and give a necessary and sufficient condition for the radial Coulomb cost to coincide with a much simpler cost that corresponds to the situation where all three particles are aligned. Moreover, we produce an infinite class of regular counterexamples to the optimality of this family of maps.
Parallel translations, Newton flows and Q-Wiener processes on the Wasserstein space
2022
- We extend the definition of Lott’s Levi-Civita connection to the Wasserstein space of probability measures having density and divergence. We give an extension of a vector field defined along an absolutely curve onto the whole space so that parallel translations can be introduced as done in differential geometry. In the case of torus, we prove the well-posedness of Lott’s equation for parallel translations.- We prove the well-posedness of the Newton flow equation on the Wasserstein space and show the connections between the relaxed Newton flow equation and the Keller-Segel equation.- We establish an intrinsic formalism for Itô stochastic calculus on the Wasserstein space throughout three k…