Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

Part I. Examples and Fundamental Concepts Introduction 1. First examples 2. Equivalence, classification, and invariants 3. Principle classes of asymptotic invariants 4. Statistical behavior of the orbits and introduction to ergodic theory 5. Smooth invariant measures and more examples Part II. Local Analysis and Orbit Growth 6. Local hyperbolic theory and its applications 7. Transversality and genericity 8. Orbit growth arising from topology 9. Variational aspects of dynamics Part III. Low-Dimensional Phenomena 10. Introduction: What is low dimensional dynamics 11. Homeomorphisms of the circle 12. Circle diffeomorphisms 13. Twist maps 14. Flows on surfaces and related dynamical systems 15. Continuous maps of the interval 16. Smooth maps of the interval Part IV. Hyperbolic Dynamical Systems 17. Survey of examples 18. Topological properties of hyperbolic sets 19. Metric structure of hyperbolic sets 20. Equilibrium states and smooth invariant measures Part V. Sopplement and Appendix 21. Dynamical systems with nonuniformly hyperbolic behavior Anatole Katok and Leonardo Mendoza.

Introduction to the Modern Theory of Dynamical Systems: EQUILIBRIUM STATES AND SMOOTH INVARIANT MEASURES

Optimal Transport: A Semi-Discrete Approach

The Wasserstein distance between two probability measures on a metric space
is a measure of closeness with applications in statistics, probability, and
machine learning. In this work, we consider the fundamental question of how
quickly the empirical measure obtained from $n$ independent samples from $\mu$
approaches $\mu$ in the Wasserstein distance of any order. We prove sharp
asymptotic and finite-sample results for this rate of convergence for general
measures on general compact metric spaces. Our finite-sample results show the
existence of multi-scale behavior, where measures can exhibit radically
different rates of convergence as $n$ grows.

Sharp asymptotic and finite-sample rates of convergence of empirical measures in Wasserstein distance

Three-dimensional geometry and topology. Volume 1, by William P. Thurston, (edited by Sylvio Levy). Princeton Mathematical Series, 35. £29.95. 1997. ISBN 0-691-08304-5 (Princeton University Press).

We consider the problem of approximating a probability measure defined on a metric space by a measure supported on a finite number of points. More specifically we seek the asymptotic behavior of the minimal Wasserstein distance to an approximation when the number of points goes to infinity. The main result gives an equivalent when the space is a Riemannian manifold and the approximated measure is absolutely continuous and compactly supported.

/pdf/approximation-by-finitely-supported-measures-55jzt1i9ib.pdf

Approximation by finitely supported measures

We study the Wasserstein space (with quadratic cost) of Euclidean spaces as an intrinsic metric space. In particular we compute their isometry groups. Surprisingly, in the case of the line, there exists a (unique) "exotic" isometric flow. This contrasts with the case of higher-dimensional Euclidean spaces, where all isometries of the Wasserstein space preserve the shape of measures. We also study the curvature and various ranks of these spaces.

/pdf/a-geometric-study-of-wasserstein-spaces-euclidean-spaces-3wfwfl672l.pdf

A geometric study of Wasserstein spaces: Euclidean spaces

We study the Wasserstein space (with quadratic cost) of Euclidean spaces as an intrinsic metric space. In particular we compute their isometry groups. Surprisingly, in the case of the line, there exists a (unique) ``exotic'' isometric flow. This contrasts with the case of higher-dimensional Euclidean spaces, where all isometries of the Wasserstein space preserve the shape of measures. We also study the curvature and various ranks of these spaces.

/pdf/a-geometric-study-of-wasserstein-spaces-euclidean-spaces-5bjwrdim1n.pdf

Given a compactly supported probability measure on a Riemannian manifold, we study the asymptotic speed at which it can be approximated (in Wasserstein distance of any exponent p) by finitely supported measure. This question has been studied under the names of ``quantization of distributions'' and, when p=1, ``location problem''. When p=2, it is linked with Centroidal Voronoi Tessellations.

/pdf/approximation-by-finitely-supported-measures-3mwf464b19.pdf

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 W(X). In this paper we investigate the geometry of W(X) 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- W(X) is not non-positively curved, our results show that W(X) have large-scale properties reminiscent of that of X. In particular we define a geodesic boundary for W(X) 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 W(X).

/pdf/a-geometric-study-of-wasserstein-spaces-hadamard-spaces-4arwp71lu3.pdf

Benoît Kloeckner

Papers

Approximation by finitely supported measures

A geometric study of Wasserstein spaces: Euclidean spaces

A geometric study of Wasserstein spaces: Euclidean spaces

Approximation by finitely supported measures

A geometric study of Wasserstein spaces: Hadamard spaces