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We study the optimal transport problem in the Euclidean space where the cost function is given by the value function associated with a Linear Quadratic minimization problem. Under appropriate assumptions, we generalize Brenier's Theorem…
We consider the Monge-Kantorovich optimal transportation problem between two measures, one of which is a weighted sum of Diracs. This problem is traditionally solved using expensive geometric methods. It can also be reformulated as an…
A simple procedure to map two probability measures in $\mathbb{R}^d$ is the so-called \emph{Knothe-Rosenblatt rearrangement}, which consists in rearranging monotonically the marginal distributions of the last coordinate, and then the…
Unlike convex case, a local equilibrium point of a nonconvex Nash-Cournot oligopolistic equilibrium problem may not be a global one. Finding such a local equilibrium point or even a stationary point of this problem is not an easy task. This…
The purpose of this paper is to show that in a finite dimensional metric space with Alexandrov's curvature bounded below, Monge's transport problem for the quadratic cost admits a unique solution.
We investigate the properties of convex functions in the plane that satisfy a local inequality which generalizes the notion of sub-solution of Monge-Ampere equation for a Monge-Kantorovich problem with quadratic cost between non-absolutely…
We introduce graphs associated to transport problems between discrete marginals, that allow to characterize the set of all optimizers given one primal optimizer. In particular, we establish that connectivity of those graphs is a necessary…
We study the asymptotic behavior of solutions to the second boundary value problem for a parabolic PDE of Monge-Amp\`ere type arising from optimal mass transport. Our main result is an exponential rate of convergence for solutions of this…
A least-squares method for solving the hyperbolic Monge-Amp\`ere equation with transport boundary condition is introduced. The method relies on an iterative procedure for the gradient of the solution, the so-called mapping. By formulating…
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…
We give a new probabilistic construction of solutions to real Monge-Amp\`ere equations in R^n satisfying the second boundary value problem with respect to a given target convex body P) which fits naturally into the theory of optimal…
We study optimal transport-based distributionally robust optimization problems where a fictitious adversary, often envisioned as nature, can choose the distribution of the uncertain problem parameters by reshaping a prescribed reference…
In this note we prove that, if the cost function satisfies some necessary structural conditions and the densities are bounded away from zero and infinity, then strictly $c$-convex potentials arising in optimal transportation belong to…
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…
In this paper, we prove second derivative estimates together with classical solvability for the Dirichlet problem of certain Monge-Ampere type equations under sharp hypotheses. In particular we assume that the matrix function in the…
We consider a large population dynamic game in discrete time where players are characterized by time-evolving types. It is a natural assumption that the players' actions cannot anticipate future values of their types. Such games go under…
The Monge-Kantorovich problem is revisited by means of a variant of the saddle-point method without appealing to $c$-conjugates. A new abstract characterization of the optimal plans is obtained in the case where the cost function takes…
The inverse reflector problem arises in geometrical nonimaging optics: Given a light source and a target, the question is how to design a reflecting free-form surface such that a desired light density distribution is generated on the…
The aim of this paper is to obtain quantitative bounds for solutions to the optimal matching problem in dimension two. These bounds show that up to a logarithmically divergent shift, the optimal transport maps are close to be the identity…
We provide a solution to the problem of optimal transport by Brownian martingales in general dimensions whenever the transport cost satisfies certain subharmonic properties in the target variable, as well as a stochastic version of the…