Stochastic Monge–Kantorovich problem and its duality

In this article, we prove the existence of a stochastic optimal transference plan for a stochastic Monge–Kantorovich problem by measurable selection theorem. A stochastic version of Kantorovich duality and the characterization of stochastic optimal transference plan are also established. Moreover, Wasserstein distance between two probability kernels is also discussed.     

The obstacle problem for a fractional Monge–Ampère equation

Abstract

We study the obstacle problem for a nonlocal, degenerate elliptic Monge–Ampère equation. We show existence and regularity of a unique classical solution to the problem and regularity of the free boundary.     

Causal transport plans and their Monge–Kantorovich problems

This paper investigates causal optimal transport problems. Within this framework, primal attainments and dual formulations are obtained under standard hypothesis, for the related variational problems. Causal transport plans are intrinsically related to martingales by a preserving property. Specific concretizations yield primal problems equivalent to several classical problems of stochastic control, and of stochastic calculus; trivial filtrations yield usual problems of optimal transport.     

Abstract Cyclical Monotonicity and Monge Solutions for the General Monge–Kantorovich Problem

Abstract cyclical monotonicity is studied for a multivalued operator F : X L, where L R ^X . A criterion for F to be L-cyclically monotone is obtained and connections with the notions of L-convex function and of its L-subdifferentials are established. Applications are given to the general Monge–Kantorovich problem with fixed marginals. In particular, we show that in some cases the optimal measure is unique and generated by a unique (up to the a.e. equivalence) optimal solution (measure preserving map) for the corresponding Monge problem.     

Optimal boundary regularity for a singular Monge–Ampère equation

In this paper we study the optimal global regularity for a singular Monge–Ampère type equation which arises from a few geometric problems. We find that the global regularity does not depend on the smoothness of domain, but it does depend on the convexity of the domain. We introduce $(a,\eta )$ type to describe the convexity. As a result, we show that the more convex is the domain, the better is the regularity of the solution. In particular, the regularity is the best near angular points.     

Analytic solutions for the approximated 1-D Monge–Kantorovich mass transfer problems

This paper mainly investigates the approximation of a global maximizer of the 1-D Monge–Kantorovich mass transfer problem through the approach of nonlinear differential equations with Dirichlet boundary. Using an approximation mechanism, the primal maximization problem can be transformed into a sequence of minimization problems. By applying the canonical duality theory, one is able to derive a sequence of analytic solutions for the minimization problems. In the final analysis, the convergence of the sequence to a global maximizer of the primal Monge–Kantorovich problem will be demonstrated.     

Symmetric Monge–Kantorovich problems and polar decompositions of vector fields

We address the problem of whether a bounded measurable vector field from a bounded domain Ω into \({\mathbb{R}^d}\) is N-cyclically monotone up to a measure preserving N-involution, where N is any integer larger than 2. Our approach involves the solution of a multidimensional symmetric Monge–Kantorovich problem, which we first study in the case of a general cost function on a product domain Ω ^N . The polar decomposition described above corresponds to a special cost function derived from the vector field in question (actually N ? 1 of them). The problem amounts to showing that the supremum in the corresponding Monge–Kantorovich problem when restricted to those probability measures on Ω ^N which are invariant under cyclic permutations and with a given first marginal μ, is attained on a probability measure that is supported on a graph of the form x → (x, Sx, S ² x,..., S ^N-1 x), where S is a μ-measure preserving transformation on Ω such that S ^N  = I a.e. The proof exploits a remarkable duality between such involutions and those Hamiltonians that are N-cyclically antisymmetric.     

Optimality Conditions for Smooth Monge Solutions of the Monge–Kantorovich problem

The Monge–Kantorovich problem (MKP) with given marginals defined on closed domains and a smooth cost function is considered. Conditions are obtained (both necessary ones and sufficient ones) for the optimality of a Monge solution generated by a smooth measure-preserving map . The proofs are based on an optimality criterion for a general MKP in terms of nonemptiness of the sets for special functions on X × X generated by c and f. Also, earlier results by the author are used when considering the above-mentioned nonemptiness conditions for the case of smooth .     

Wave packets and the quadratic Monge–Kantorovich distance in quantum mechanics

In this paper, we extend the upper and lower bounds for the “pseudo-distance” on quantum densities analogous to the quadratic Monge–Kantorovich(–Vasershtein) distance introduced in [F. Golse, C. Mouhot, T. Paul, Commun. Math. Phys. 343 (2016) 165–205] to positive quantizations defined in terms of the family of phase space translates of a density operator, not necessarily of rank 1 as in the case of the Töplitz quantization. As a corollary, we prove that the uniform as $?\to 0$ convergence rate for the mean-field limit of the N-particle Heisenberg equation holds for a much wider class of initial data than in [F. Golse, C. Mouhot, T. Paul, Commun. Math. Phys. 343 (2016) 165–205]. We also discuss the relevance of the pseudo-distance compared to the Schatten norms for the purpose of metrizing the set of quantum density operators in the semiclassical regime.     

Monge–Kantorovich Norms on Spaces of Vector Measures

One considers Hilbert space valued measures on the Borel sets of a compact metric space. A natural numerical valued integral of vector valued continuous functions with respect to vector valued measures is defined. Using this integral, different norms (we called them Monge–Kantorovich norm, modified Monge–Kantorovich norm and Hanin norm) on the space of measures are introduced, generalizing the theory of (weak) convergence for probability measures on metric spaces. These norms introduce new (equivalent) metrics on the initial compact metric space.     

On the Monge–Kantorovich problem with additional linear constraints

The Monge–Kantorovich problem with the following additional constraint is considered: the admissible transportation plan must become zero on a fixed subspace of functions. Different subspaces give rise to different additional conditions on transportation plans. The main results are stated in general form and can be carried over to a number of important special cases. They are also valid for the Monge–Kantorovich problem whose solution is sought for the class of invariant or martingale measures. We formulate and prove a criterion for the existence of an optimal solution, a duality assertion of Kantorovich type, and a necessary geometric condition on the support of the optimal measure similar to the standard condition for c-monotonicity.     

Optical Design of Single Reflector Systems and the Monge–Kantorovich Mass Transfer Problem

We consider the problem of designing a reflector that transforms a spherical wave front with a given intensity into an output front illuminating a prespecified region of the far-sphere with prescribed intensity. In earlier approaches, it was shown that in the geometric optics approximation this problem is reduced to solving a second order nonlinear elliptic partial differential equation of Monge–Ampere type. We show that this problem can be solved as a variational problem within the framework of Monge–Kantorovich mass transfer problem. We develop the techniques used by the authors in their work Optical Design of Two-Reflector Systems, the Monge–Kantorovich Mass Transfer Problem and Fermat's Principle [Preprint, 2003], where the design problem for a system with two reflectors was considered. An important consequence of this approach is that the design problem can be solved numerically by tools of linear programming. A known convergent numerical scheme for this problem was based on the construction of very special approximate solutions to the corresponding Monge–Ampere equation. Bibliography: 14 titles.     

Multi-marginal Monge–Kantorovich transport problems: A characterization of solutions

We shall present a measure theoretical approach that, together with the Kantorovich duality, provides an efficient tool to study the optimal transport problem. Specifically, we study the support of optimal plans where the cost function does not satisfy the classical twist condition in the two marginal problem as well as in the multi-marginal case when twistedness is limited to certain subsets.     

Remarks on the Monge–Kantorovich problem in the discrete setting

In Optimal Transport theory, three quantities play a central role: the minimal cost of transport, originally introduced by Monge, its relaxed version introduced by Kantorovich, and a dual formulation also due to Kantorovich. The goal of this Note is to publicize a very elementary, self-contained argument extracted from [9], which shows that all three quantities coincide in the discrete case.     

Optimal Entropy-Transport problems and a new Hellinger–Kantorovich distance between positive measures

Inventiones mathematicae - We develop a full theory for the new class of Optimal Entropy-Transport problems between nonnegative and finite Radon measures in general topological spaces. These...     

From the Schrödinger problem to the Monge–Kantorovich problem

The aim of this article is to show that the Monge–Kantorovich problem is the limit, when a fluctuation parameter tends down to zero, of a sequence of entropy minimization problems, the so-called Schrödinger problems. We prove the convergence of the entropic optimal values to the optimal transport cost as the fluctuations decrease to zero, and we also show that the cluster points of the entropic minimizers are optimal transport plans. We investigate the dynamic versions of these problems by considering random paths and describe the connections between the dynamic and static problems. The proofs are essentially based on convex and functional analysis. We also need specific properties of Γ-convergence which we didn?t find in the literature; these Γ-convergence results which are interesting in their own right are also proved.