Event Details

Isometric study of Wasserstein spaces -- an overview

There is a long history and vast literature of surjective distance preserving maps (i.e. isometries) on different kind of metric spaces. The starting point of our work is Molnar's paper [1] in which he characterized Levy isometries of the space of all cumulative distribution functions of one-dimensional random variables. As a generalization, with Gy.P. Geher, we gave a complete description of the structure of Levy-Prokhorov isometries on the space of all Borel probability measures on an arbitrary separable real Banach space [2]. Maybe the most intensively studied metric nowadays (when speaking about spaces of Borel probability measures) is the p-Wasserstein metric. Kloeckner and Bertrand wrote a whole sequence of papers about isometric properties of p-Wasserstein spaces when p=2 (see e.g. [3] and [4] and the references therein).

In this talk I will survey first some of the earlier results in the subject, and then I will present some new results on isometric embeddings (i.e., not necessarily surjective distance preserving maps) of p-Wasserstein spaces.

The talk is based on a joint work with Gy. P. Geher (University of Reading) and Daniel Virosztek (IST Austria).

[1] L. Molnar, Levy isometries of the space of probability distribution functions, Journal of Mathematical Analysis and Applications 380(2) 847-852.

[2] Gy. P. Geher and T. Titkos, A characterisation of isometries with respect to the Levy-Prokhorov metric, Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, (in press) DOI: 10.2422/2036-2145.201702_011

[3] B. Kloeckner, A geometric study of Wasserstein spaces: Euclidean spaces, Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 9 (2010) no. 2, 297-323.

[4] J. Bertrand and B. Kloeckner, A Geometric Study of Wasserstein Spaces: Isometric Rigidity in Negative Curvature. International Mathematics Research Notices 2016:5, 1368-1386.