Related papers: Optimal transportation, topology and uniqueness
In the semi-discrete version of Monge's problem one tries to find a transport map $T$ with minimum cost from an absolutely continuous measure $\mu$ on $\mathbb{R}^d$ to a discrete measure $\nu$ that is supported on a finite set in…
The aim of this article is to show that the Monge-Kantorovich problem is the limit of a sequence of entropy minimization problems when a fluctuation parameter tends down to zero. We prove the convergence of the entropic values to the…
We introduce fast algorithms for generalized unnormalized optimal transport. To handle densities with different total mass, we consider a dynamic model, which mixes the $L^p$ optimal transport with $L^p$ distance. For $p=1$, we derive the…
In this paper, we investigate Monge-Kantorovich problems for which the absolute continuity of marginals is relaxed. For $X,Y\subseteq\mathbb{R}^{n+1}$ let $(X,\mathcal{B}_X,\mu)$ and $(Y,\mathcal{B}_Y,\nu)$ be two Borel probability spaces,…
The duality theory of the Monge--Kantorovich transport problem is analyzed in a general setting. The spaces $X, Y$ are assumed to be polish and equipped with Borel probability measures $\mu$ and $\nu$. The transport cost function $c:X\times…
We study a multi-marginal optimal transport problem with surplus $b(x_{1}, \ldots, x_{m})=\sum_{\{i,j\}\in P} x_{i}\cdot x_{j}$, where $P\subseteq Q:=\{\{i,j\}: i, j \in \{1,2,...m\}, i \neq j\}$. We reformulate this problem by associating…
In this note, we present a unified approach to the problem of existence of a potential for the optimal transport problem with respect to non-traditional cost functions, that is, costs that assume infinite values. We establish a new method…
The theory of Monge-Kantorovich Optimal Mass Transport (OMT) has in recent years spurred a fast developing phase of research in stochastic control, control of ensemble systems, thermodynamics, data science, and several other fields in…
This is our first paper on the extension of our recent work on the Lax-Oleinik commutators and its applications to the intrinsic approach of propagation of singularities of the viscosity solutions of Hamilton-Jacobi equations. We…
Given a smooth Riemannian manifold $(M,g)$, compact and without boundary, we analyze the dynamical optimal mass transport problem where the cost is given by the sum of the kinetic energy and the relative entropy with respect to a reference…
The duality theory of the Monge-Kantorovich transport problem is investigated in an abstract measure theoretic framework. Let $(\mathcal{X},\mathcal{F},\mu)$ and $(\mathcal{Y},\mathcal{G},\nu)$ be any probability spaces and…
We give a necessary and sufficient condition on the cost function so that the map solution of Monge's optimal transportation problem is continuous for arbitrary smooth positive data. This condition was first introduced by Ma, Trudinger and…
By disintegration of transport plans it is introduced the notion of transport class. This allows to consider the Monge problem as a particular case of the Kantorovich transport problem, once a transport class is fixed. The transport problem…
Many numerical and learning algorithms rely on the solution of the Monge-Kantorovich problem and Wasserstein distances, which provide appropriate distributional metrics. While the natural approach is to treat the problem as an…
The optimal transportation problem, first suggested by Gaspard Monge in the 18th century and later revived in the 1940s by Leonid Kantorovich, deals with the question of transporting a certain measure to another, using transport maps or…
We consider the optimal mass transportation problem in $\RR^d$ with measurably parameterized marginals, for general cost functions and under conditions ensuring the existence of a unique optimal transport map. We prove a joint measurability…
We analyze several problems of Optimal Transport Theory in the setting of Ergodic Theory. In a certain class of problems we consider questions in Ergodic Transport which are generalizations of the ones in Ergodic Optimization. Another class…
A result of Hohloch links the theory of integer partitions with the Monge formulation of the optimal transport problem, giving the optimal transport map between (Young diagrams of) integer partitions and their corresponding symmetric…
We study the Kantorovich-Rubinstein transhipment problem when the difference between the source and the target is not anymore a balanced measure but belongs to a suitable subspace $X(\Omega)$ of first order distribution. A particular…
We prove uniqueness and Monge solution results for multi-marginal optimal transportation problems with a certain class of surplus functions; this class arises naturally in multi-agent matching problems in economics. This result generalizes…