scholarly journals A remark on the optimal transport between two probability measures sharing the same copula

2014 ◽  
Vol 84 ◽  
pp. 131-134 ◽  
Author(s):  
A. Alfonsi ◽  
B. Jourdain
Keyword(s):  
Optimal Transport ◽  
Probability Measures

Schrödinger dynamics and optimal transport of measures

2021 ◽  
pp. 2150016
Author(s):  
Lorenzo Zanelli
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Transport Theory ◽  
Optimal Transport ◽  
Time Dependent ◽  
Probability Measures ◽  
Flat Torus ◽  
Optimal Transport Theory

In this paper, we recover a class of displacement interpolations of probability measures, in the sense of the Optimal Transport theory, by means of semiclassical measures associated with solutions of Schrödinger equation defined on the flat torus. Moreover, we prove the completing viewpoint by proving that a family of displacement interpolations can always be viewed as a path of time-dependent semiclassical measures.

scholarly journals A GEOMETRIC STUDY OF WASSERSTEIN SPACES: HADAMARD SPACES

2012 ◽  
Vol 04 (04) ◽  
pp. 515-542 ◽  
Author(s):  
JÉRÔME BERTRAND ◽  
BENOÎT KLOECKNER
Keyword(s):  
Large Scale ◽  
Euclidean Plane ◽  
Optimal Transport ◽  
Probability Measures ◽  
Hadamard Space ◽  
Geodesic Boundary ◽  
Wasserstein Space ◽  
Scale Properties ◽  
Positively Curved ◽  
Geometric Study

Optimal transport enables one to construct a metric on the set of (sufficiently small at infinity) probability measures on any (not too wild) metric space X, called its Wasserstein space [Formula: see text]. In this paper we investigate the geometry of [Formula: see text] when X is a Hadamard space, by which we mean that X has globally non-positive sectional curvature and is locally compact. Although it is known that, except in the case of the line, [Formula: see text] is not non-positively curved, our results show that [Formula: see text] have large-scale properties reminiscent of that of X. In particular we define a geodesic boundary for [Formula: see text] that enables us to prove a non-embeddablity result: if X has the visibility property, then the Euclidean plane does not admit any isometric embedding in [Formula: see text].

scholarly journals Formulation and properties of a divergence used to compare probability measures without absolute continuity

2022 ◽  
Author(s):  
Paul Dupuis ◽  
Yixiang Mao
Keyword(s):  
Relative Entropy ◽  
Model Uncertainty ◽  
Absolute Continuity ◽  
Optimal Transport ◽  
Transport Cost ◽  
Probability Measures

This paper develops a new divergence that generalizes relative entropy and can be used to compare probability measures without a requirement of absolute continuity. We establish properties of the divergence, and in particular derive and exploit a representation as an infimum convolution of optimal transport cost and relative entropy.  Also included are examples of computation and approximation of the divergence, and the demonstration of properties that are useful when one quantifies model uncertainty.

scholarly journals An Optimal Transport View of Schrödinger's Equation

2012 ◽  
Vol 55 (4) ◽  
pp. 858-869 ◽  
Author(s):  
Max-K. von Renesse
Keyword(s):  
Symplectic Structure ◽  
Optimal Transport ◽  
Riemannian Metric ◽  
Symplectic Space ◽  
Probability Measures ◽  
Extended System ◽  
Third Law ◽  
Classical Potential ◽  
Wasserstein Space ◽  
Complex Valued

AbstractWe show that the Schrödinger equation is a lift of Newton's third law of motion on the space of probability measures, where derivatives are taken with respect to the Wasserstein Riemannian metric. Here the potential μ → F(μ) is the sum of the total classical potential energy (V, μ) of the extended system and its Fisher information . The precise relation is established via a well-known (Madelung) transform which is shown to be a symplectic submersion of the standard symplectic structure of complex valued functions into the canonical symplectic space over the Wasserstein space. All computations are conducted in the framework of Otto's formal Riemannian calculus for optimal transportation of probability measures.

scholarly journals OPTIMAL PATHS RELATED TO TRANSPORT PROBLEMS

2003 ◽  
Vol 05 (02) ◽  
pp. 251-279 ◽  
Author(s):  
QINGLAN XIA
Keyword(s):  
Probability Measure ◽  
Weak Topology ◽  
Optimal Transport ◽  
Vector Measure ◽  
Probability Measures ◽  
Actual Cost ◽  
Total Cost ◽  
Optimal Paths ◽  
Transport Path ◽  
Transport Problems

In transport problems of Monge's types, the total cost of a transport map is usually an integral of some function of the distance, such as |x - y|p. In many real applications, the actual cost may naturally be determined by a transport path. For shipping two items to one location, a "Y shaped" path may be preferable to a "V shaped" path. Here, we show that any probability measure can be transported to another probability measure through a general optimal transport path, which is given by a vector measure in our setting. Moreover, we define a new distance on the space of probability measures which in fact metrizies the weak * topology of measures. Under this distance, the space of probability measures becomes a length space. Relations as well as related problems about transport paths and transport plans are also discussed in the end.

scholarly journals On discontinuity of planar optimal transport maps

2015 ◽  
Vol 07 (02) ◽  
pp. 239-260 ◽  
Author(s):  
Otis Chodosh ◽  
Vishesh Jain ◽  
Michael Lindsey ◽  
Lyuboslav Panchev ◽  
Yanir A. Rubinstein
Keyword(s):  
Optimal Transport ◽  
Main Idea ◽  
Regularity Theory ◽  
Probability Measures ◽  
Bounded Domains ◽  
Quadratic Cost ◽  
Extrinsic Geometry ◽  
Discontinuous Map ◽  
Transport Maps ◽  
Monge Ampere

Consider two bounded domains Ω and Λ in ℝ2, and two sufficiently regular probability measures μ and ν supported on them. By Brenier's theorem, there exists a unique transportation map T satisfying T#μ = ν and minimizing the quadratic cost ∫ℝn ∣T(x) - x∣2 dμ(x). Furthermore, by Caffarelli's regularity theory for the real Monge–Ampère equation, if Λ is convex, T is continuous. We study the reverse problem, namely, when is T discontinuous if Λ fails to be convex? We prove a result guaranteeing the discontinuity of T in terms of the geometries of Λ and Ω in the two-dimensional case. The main idea is to use tools of convex analysis and the extrinsic geometry of ∂Λ to distinguish between Brenier and Alexandrov weak solutions of the Monge–Ampère equation. We also use this approach to give a new proof of a result due to Wolfson and Urbas. We conclude by revisiting an example of Caffarelli, giving a detailed study of a discontinuous map between two explicit domains, and determining precisely where the discontinuities occur.

scholarly journals Characterization of barycenters in the Wasserstein space by averaging optimal transport maps

2018 ◽  
Vol 22 ◽  
pp. 35-57 ◽  
Author(s):  
Jérémie Bigot ◽  
Thierry Klein
Keyword(s):  
Optimal Transport ◽  
Probability Measures ◽  
Reference Measure ◽  
Precise Characterization ◽  
Fixed Reference ◽  
Random Probability ◽  
Wasserstein Space ◽  
Transport Maps ◽  
Random Probability Measures

This paper is concerned by the study of barycenters for random probability measures in the Wasserstein space. Using a duality argument, we give a precise characterization of the population barycenter for various parametric classes of random probability measures with compact support. In particular, we make a connection between averaging in the Wasserstein space as introduced in Agueh and Carlier [SIAM J. Math. Anal. 43 (2011) 904–924], and taking the expectation of optimal transport maps with respect to a fixed reference measure. We also discuss the usefulness of this approach in statistics for the analysis of deformable models in signal and image processing. In this setting, the problem of estimating a population barycenter from n independent and identically distributed random probability measures is also considered.

scholarly journals SAMPLING OF ONE-DIMENSIONAL PROBABILITY MEASURES IN THE CONVEX ORDER AND COMPUTATION OF ROBUST OPTION PRICE BOUNDS

2019 ◽  
Vol 22 (03) ◽  
pp. 1950002 ◽  
Author(s):  
AURÉLIEN ALFONSI ◽  
JACOPO CORBETTA ◽  
BENJAMIN JOURDAIN
Keyword(s):  
Real Line ◽  
Optimal Transport ◽  
Option Price ◽  
Order Moment ◽  
Probability Measures ◽  
Convex Order ◽  
Empirical Measures ◽  
The Real ◽  
Martingale Optimal Transport ◽  
Transport Problems

For [Formula: see text] and [Formula: see text] two probability measures on the real line such that [Formula: see text] is smaller than [Formula: see text] in the convex order, this property is in general not preserved at the level of the empirical measures [Formula: see text] and [Formula: see text], where [Formula: see text] (resp., [Formula: see text]) are independent and identically distributed according to [Formula: see text] (resp., [Formula: see text]). We investigate modifications of [Formula: see text] (resp., [Formula: see text]) smaller than [Formula: see text] (resp., greater than [Formula: see text]) in the convex order and weakly converging to [Formula: see text] (resp., [Formula: see text]) as [Formula: see text]. According to  Kertz & Rösler(1992) , the set of probability measures on the real line with a finite first order moment is a complete lattice for the increasing and the decreasing convex orders. For [Formula: see text] and [Formula: see text] in this set, this enables us to define a probability measure [Formula: see text] (resp., [Formula: see text]) greater than [Formula: see text] (resp., smaller than [Formula: see text]) in the convex order. We give efficient algorithms permitting to compute [Formula: see text] and [Formula: see text] (and therefore [Formula: see text] and [Formula: see text]) when [Formula: see text] and [Formula: see text] have finite supports. Last, we illustrate by numerical experiments the resulting sampling methods that preserve the convex order and their application to approximate martingale optimal transport problems and in particular to calculate robust option price bounds.

Sign in / Sign up

Export Citation Format

Share Document