A New L-infinity estimate in Optimal Transport

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

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.

Calculus of Variations and Partial Differential Equations

For cost functions c(x, y) = h(x − y) with h ∈ C 2 homogeneous of degree p ≥ 2, we show L ∞-estimates of Tx − x on balls, where T is an h-monotone map. Estimates for the interpolating mappings T t = t(T − I) + I are deduced from this.

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

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.

Nonlinear Analysis, 2016

We introduce the setting of extended metric-topological measure spaces as a general "Wiener like" framework for optimal transport problems and nonsmooth metric analysis in infinite dimension. After a brief review of optimal transport tools for general Radon measures, we discuss the notions of the Cheeger energy, of the Radon measures concentrated on absolutely continuous curves, and of the induced "dynamic transport distances". We study their main properties and their links with the theory of Dirichlet forms and the Bakry-Émery curvature condition, in particular concerning the contractivity properties and the EVI formulation of the induced Heat semigroup.

Lecture Notes in Mathematics, 2003

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 ).

2002

The Monge-Kantorovich problem is to move one distribution of mass onto another as efficiently as possible, where Monge's original criterion for efficiency was to minimize the average distance transported. Subsequently studied by many authors, it was not until 1976 that Sudakov showed solutions to be realized in the original sense of Monge, i.e., as mappings from R n to R n . A second proof of this existence result formed the subject of a recent monograph by Evans and Gangbo [7], who avoided Sudakov's measure decomposition results by using a partial differential equations approach. In the present manuscript, we give a third existence proof for optimal mappings, which has some advantages (and disadvantages) relative to existing approaches: it requires no continuity or separation of the mass distributions, yet our explicit construction yields more geometrical control than the abstract method of Sudakov. (Indeed, this control turns out to be essential for addressing a gap which has recently surfaced in Sudakov's approach to the problem in dimensions n ≥ 3; see the remarks at the end of this section.) It is also shorter and more flexible than either, and can be adapted to handle transportation on Riemannian manifolds or around obstacles, as we plan to show in a subsequent work . The problem considered here is the classical one:

Archive for Rational Mechanics and Analysis, 2014

In this article, we generalize the Wasserstein distance to measures with different masses. We study the properties of such distance. In particular, we show that it metrizes weak convergence for tight sequences.