Leírás
Recently Kloeckner described the structure of the isometry group of the quadratic Wasserstein space $W_2(\mathbb{R}^n)$ (see https://arxiv.org/abs/0804.3505, MR2731158). It turned out that the case of the real line is exceptional in the sense that there exists an exotic isometry flow. Following this line of investigation, we compute the isometry group of the Wasserstein space $W_p(\mathbb{R})$ for all $p \in [1, \infty)\setminus\{2\}$. We make use of the fact that $W_p(\mathbb{R})$ naturally embeds into $L^p([0,1])$, hence allowing the use of functional analytic tools. Namely, we show that $W_2(\mathbb{R})$ is also exceptional regarding the parameter $p$: $W_p(\mathbb{R})$ is isometrically rigid if and only if $p\neq 2$.
Regarding the underlying space, we prove that the exceptionality of $p=2$ disappears if we replace $\mathbb{R}$ by the compact interval $[0,1]$. Surprisingly, in that case, $W_p([0,1])$ is isometrically rigid if and only if $p\neq1$. Moreover, $W_1([0,1])$ admits isometries that split mass, and which therefore cannot be extended into an isometry of $W_1(\mathbb{R})$. Using this phenomenon we also answer a question of Kloeckner about the existence of isometries in quadratic Wasserstein spaces that split mass.
The talk is based on the paper https://arxiv.org/abs/2002.00859 which is a joint work with Tamás Titkos (Rényi Institute) and Dániel Virosztek (IST Austria).
Zoom link:
https://zoom.us/j/137790396?pwd=dDdLV0RDdlZ4cEpud2tTTTJVVHluUT09