A new $L^\infty$ estimate in optimal mass transport

Let Ω be a bounded Lipschitz regular open subset of R d and let µ, ν be two probablity measures on Ω. It is well known that if µ = f dx is absolutely continuous, then there exists, for every p > 1, a unique transport map T p pushing forward µ on ν and which realizes the Monge-Kantorovich distance W p (µ, ν). In this paper, we establish an L ∞ bound for the displacement map T p x − x which depends only on p, on the shape of Ω and on the essential infimum of the density f .

Theory of Probability & Its Applications, 2006

Given two Borel probability measures µ and ν on R d such that dν/dµ = g, we consider certain mappings of the form T (x) = x + F (x) that transform µ into ν. Our main results give estimates of the form

Calculus of Variations and Partial Differential Equations, 2011

The optimal mass transportation was introduced by Monge some 200 years ago and is, today, the source of large number of results in analysis, geometry and convexity. Here I investigate a new, surprising link between optimal transformations obtained by different Lagrangian actions on Riemannian manifolds. As a special case, for any pair of non-negative measures λ + , λ − of equal mass

Calculus of Variations and Partial Differential Equations, 2009

We introduce a new class of distances between nonnegative Radon measures in R d . They are modeled on the dynamical characterization of the Kantorovich-Rubinstein-Wasserstein distances proposed by Benamou and Brenier (Numer Math 84:375-393, 2000) and provide a wide family interpolating between the Wasserstein and the homogeneous W −1, p γ -Sobolev distances. From the point of view of optimal transport theory, these distances minimize a dynamical cost to move a given initial distribution of mass to a final configuration. An important difference with the classical setting in mass transport theory is that the cost not only depends on the velocity of the moving particles but also on the densities of the intermediate configurations with respect to a given reference measure γ . We study the topological and geometric properties of these new distances, comparing them with the notion of weak convergence of measures and the well established Kantorovich-Rubinstein-Wasserstein theory. An example of possible applications to the geometric theory of gradient flows is also given.

2002

Monge's problem refers to the classical problem of optimally transporting mass: given Borel probability measures µ + = µ -on R n , find the measure preserving map s(x) between them which minimizes the average distance transported. Here distance can be induced by the Euclidean norm, or any other uniformly convex and smooth norm d(x, y) = x -y on R n . Although the solution is never unique, we give a geometrical monotonicity condition singling out a particular optimal map s(x). Furthermore, a local definition is given for the transport cost density associated to each optimal map. All optimal maps are then shown to lead to the same transport density a ∈ L 1 (R n ).

SIAM Journal on Control and Optimization, 2003

We show that the optimal regularity result for the transport density in the classical Monge-Kantorovich optimal mass transport problem, with the measures having summable densities, is a Sobolev differentiability along transport rays.

Journal De Mathematiques Pures Et Appliquees, 2005

Given two absolutely continuous probability measures f± in R2, we consider the classical Monge transport problem, with the Euclidean distance as cost function. We prove the existence of a continuous optimal transport, under the assumptions that (the densities of) f± are continuous and strictly positive in the interior part of their supports, and that such supports are convex, compact, and

2020

The presentation covers prerequisite results from Topology and Measure Theory. This is then followed by an introduction into couplings and basic definitions for optimal transport. The Kantrorovich problem is then introduced and an existence theorem is presented. Following the setup of optimal transport is a brief overview of the Wasserstein distance and a short proof of how it metrizes the space of probability measures on a COMPACT domain. This presentation is a detailed examination of Villani's "Optimal Transport: Old and New" chapters 1-4 and part of 6.

Applied Mathematics Letters, 2009

We address the question of how to represent Kantorovich potentials in the mass transportation (or Monge-Kantorovich) problem as a signed distance function from a closed set. We discuss geometric conditions on the supports of the measure f + and f − in the Monge-Kantorovich problem which ensure such a representation. Finally, we obtain, as a by-product, the continuous differentiability of the potential on the transport set.

Consider a cloud of particles (say, a smoke cloud) drifted in the atmosphere. We can measure the density of the cloud at any instant of time we wish. What can we say about its velocity?