Leírás
University of Szeged, Bolyai Institute, Analysis seminar
Abstract. I will report on our study of Wasserstein isometries --- a joint work with György Pál Gehér (University of Reading) and Tamás Titkos (Rényi Institute, Budapest). More precisely, I will present the description of non-surjective isometries of Wasserstein spaces over the countable discrete metric space and the unit interval, as well as the structure of surjective isometries of Wasserstein spaces over the real line.
It turned out that non-surjective Wasserstein isometries over the discrete metric space form a large family and can be described by a special kind of $N \times (0,1]$-indexed family of nonnegative finite measures. For the unit interval, we obtain that the a-priori non-surjective isometries are actually surjective, and the isometry group of the Wasserstein space is the Klein group $C_2 \times C_2$ for $p=1$, and the two-element group $C_2$ for $p>1$. For the real line, we show that the p-Wasserstein space is isometrically rigid --- that is, its isometry group coincides with that of the real line --- if and only if $p$ is not equal to 2. A promising approach to characterize non-surjective Wasserstein isometries on the $d$-dimensional torus will also be shown.