Given a complete and separable metric space $X$ and a cost function $c\colon X\times X\to \mathbb R$, one defines its Wasserstein space as the collection of sufficiently concentrated Borel probability measures endowed with a metric that is calculated by means of optimal transport plans. The most common example is when the cost function is the $p$th power of the distance: $c(x,y) = d(x,y)^p$, $p'>0$, in which case the Wasserstein space is denoted by the symbol $\mathcal{W}_p(X)$. This notion has strong connections to many flourishing areas of science (including economics, computer science, machine learning, PDEs, etc.), moreover, the $p$-Wasserstein space itself is an interesting object, being a natural measure theoretic analogue of $L^p$ spaces.

Kloeckner and Bertrand wrote a series of papers about the geometric properties of quadratic, $p=2$, Wasserstein spaces. For instance, in 2010 Kloeckner characterized the isometry groups $\mathrm{Isom}\left(\mathcal{W}_2(\mathbb{R}^n)\right)$, $n\in\mathbb{N}$, and in 2016 they described $\mathrm{Isom}\left(\mathcal{W}_2(X)\right)$ for every negatively curved geodesically complete Hadamard space $X$.

In my talk I will show a few possible generalisations/extensions of these results. First, I will present our result on the characterisation of the isometry group $\mathrm{Isom}(\mathcal{W}_p(E))$ for all parameters $p '> 0$ and separable (not necessarily finite dimensional) real Hilbert spaces $E$. Then, I will continue with our most recent result which completely describes the isometry group $\mathrm{Isom}(\mathcal{W}_1(X))$ for all metric spaces $X$ that satisfy a strict triangle inequality (in the sense that $d(x,y) = d(x,z)+d(z,y)$ holds if and only if $z\in\{x,y\}$). As a consequence we easily obtain a characterisation of $\mathrm{Isom}(\mathcal{W}_p(X))$ for all $0'<p'<1$ and arbitrary metric space $X$. If time permits, I will also explain some work in progress on the quantum version of Wasserstein spaces.

Contact Jacob Schach Møller jacob@math.au.dk, for Zoom details.