Description
ELTE Analízis tanszék szemináruma
Absztrakt:
I will give an overview of our systematic study of Wasserstein isometries.
In the past few years, we considered several natural Polish spaces, such as real Hilbert spaces, finite-dimensional tori, spheres, the unit interval, or a countable space endowed with the discrete metric, and described the isometry groups of $p$-Wasserstein spaces over them. Isometries of the underlying spaces always give rise to isometries of the Wasserstein spaces by the push-forward of measures. These isometries are called trivial.
The question we have to answer reads as follows: Is the Wasserstein space more symmetric than the underlying space?
Are there non-trivial isometries?
If so, how to describe them?
We will show several examples of highly non-trivial Wasserstein isometries. Some of those even split mass. That is, they send Dirac masses to non-Dirac measures. On the other hand, we will show that in many cases, the isometry groups of the Wasserstein space and the underlying space coincide. This phenomenon is called isometric rigidity.
Based on the works [Geher, Titkos, Virosztek: (1) J. Math. Anal. Appl. 480 (2019), 123435. (2) Trans. Amer. Math. Soc. 373 (2020), 5855–5883. (3) J. London Math. Soc. (2022), https://doi.org/10.1112/jlms.12676. (4) Mathematika (2022), to appear, https://arxiv.org/abs/2203.04054.]