Related papers: Duality for rectified Cost Functions
We consider the modified Monge-Kantorovich problem with additional restriction: admissible transport plans must vanish on some fixed functional subspace. Different choice of the subspace leads to different additional properties optimal…
Suppose that $c(x,y)$ is the cost of transporting a unit of mass from $x\in X$ to $y\in Y$ and suppose that a mass distribution $\mu$ on $X$ is transported optimally (so that the total cost of transportation is minimal) to the mass…
We present a general method, based on conjugate duality, for solving a convex minimization problem without assuming unnecessary topological restrictions on the constraint set. It leads to dual equalities and characterizations of the…
We consider some repulsive multimarginal optimal transportation problems which include, as a particular case, the Coulomb cost. We prove a regularity property of the minimizers (optimal transportation plan) from which we deduce existence…
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 consider optimal transport problems where the cost is optimized over controlled dynamics and the end time is free. Unlike the classical setting, the search for optimal transport plans also requires the identification of optimal "stopping…
In this series of lectures we introduce the Monge-Kantorovich problem of optimally transporting one distribution of mass onto another, where optimality is measured against a cost function c(x,y). Connections to geometry, inequalities, and…
We study solutions to the multi-marginal Monge-Kantorovich problem which are concentrated on several graphs over the first marginal. We first present two general conditions on the cost function which ensure, respectively, that any solution…
We consider an extension of the Monge-Kantorovitch optimal transportation problem. The mass is transported along a continuous semimartingale, and the cost of transportation depends on the drift and the diffusion coefficients of the…
We investigate existence of dual optimizers in one-dimensional martingale optimal transport problems. While [BNT16] established such existence for weak (quasi-sure) duality, [BHP13] showed existence for the natural stronger pointwise…
Given two compact metric spaces $X$ and $Y$, a Lipschitz continuous cost function $c$ on $X \times Y$ and two probabilities $\mu \in\mathcal{P}(X),\,\nu\in\mathcal{P}(Y)$, we propose to study the Monge-Kantorovich problem and its duality…
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…
Capacity constrained optimal transport is a variant of optimal transport, which adds extra constraints on the set of feasible couplings in the original optimal transport problem to limit the mass transported between each pair of source and…
In this paper, we want to establish some general results in the Lorentzian optimal transport theory that have well-known Riemannian counterparts. As a first result, we will provide non-trivial assumptions on the measures to ensure strong…
Multimarginal optimal transport (MOT) has gained increasing attention in recent years, notably due to its relevance in machine learning and statistics, where one seeks to jointly compare and align multiple probability distributions. This…
Many results in probability (most famously, Strassen's theorem on stochastic domination), characterize some relationship between probability distributions in terms of the existence of a particular structured coupling between them. Optimal…
We introduce and study a multi-marginal optimal partial transport problem. Under a natural and sharp condition on the dominating marginals, we establish uniqueness of the optimal plan. Our strategy of proof establishes and exploits a…
It is well-known that the optimal transport problem on the real line for the classical distance cost may not have a unique solution. In this paper we recover uniqueness by considering the transport problems where the costs are a power…
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…
In this paper we extend the duality theory of the multi-marginal optimal transport problem for cost functions depending on a decreasing function of the distance (not necessarily bounded). This class of cost functions appears in the context…