相关论文: Isometric study of Wasserstein spaces --- the real…

Motivated by Kloeckner's result on the isometry group of the quadratic Wasserstein space $\mathcal{W}_2\left(\mathbb{R}^n\right)$, we describe the isometry group $\mathrm{Isom}\left(\mathcal{W}_p (E)\right)$ for all parameters $0 < p <…

We study $p-$Wasserstein spaces $ \mathcal{W}_p(\mathbb{R}^n, d_N)$ over $\mathbb{R}^n$ equipped with a norm metric $d_N$. We show that, if the norm is smooth enough, then the Wasserstein space is isometrically rigid whenever $p \neq 2$. We…

We study the structure of isometries of the quadratic Wasserstein space $\mathcal{W}_2\left(\mathbb{S}^n,\|\cdot\|\right)$ over the sphere endowed with the distance inherited from the norm of $\mathbb{R}^{n+1}$. We prove that…

Our purpose in this paper is to study isometries and isometric embeddings of the $p$-Wasserstein space $\mathcal{W}_p(\mathbb{H}^n)$ over the Heisenberg group $\mathbb{H}^n$ for all $p>1$ and for all $n\geq 1$. First, we create a link…

The aim of this short paper is to offer a complete characterization of all (not necessarily surjective) isometric embeddings of the Wasserstein space $\mathcal{W}_p(\mathcal{X})$, where $\mathcal{X}$ is a countable discrete metric space and…

We study $p$-Wasserstein spaces over the branching spaces $\mathbb{R}^2$ and $[-1,1]^2$ equipped with the maximum norm metric. We show that these spaces are isometrically rigid for all $p\geq1,$ meaning that all isometries of these spaces…

The aim of this paper is to prove that the $p$-Wasserstein space $\mathcal{W}_p(X)$ is isometrically rigid for all $p\geq 1$ whenever $X$ is a countable graph metric space. As a consequence, we obtain that for every countable group $H$ and…

In this paper, we study isometries of $p$-Wasserstein spaces. In our first result, for every complete and separable metric space $X$ and for every $p\geq1$, we construct a metric space $Y$ such that $X$ embeds isometrically into $Y$, and…

We study the Wasserstein space (with quadratic cost) of Euclidean spaces as an intrinsic metric space. In particular we compute their isometry groups. Surprisingly, in the case of the line, there exists a (unique) "exotic" isometric flow.…

Let $(X,d,\mathfrak{m})$ be a metric measure space. The study of the Wasserstein space $(\mathbb{P}_p(X),\mathbb{W}_p)$ associated to $X$ has proved useful in describing several geometrical properties of $X.$ In this paper we focus on the…

Given a metric space X, one defines its Wasserstein space W2(X) as a set of sufficiently decaying probability measures on X endowed with a metric defined from optimal transportation. In this article, we continue the geometric study of W2(X)…

In this paper we study the isometric rigidity of certain classes of metric spaces with respect to the $p$-Wasserstein space. We prove that spaces that split a separable Hilbert space are not isometrically rigid with respect to…

This paper aims to study isometries of the $1$-Wasserstein space $\mathcal{W}_1(\mathbf{G})$ over Carnot groups endowed with horizontally strictly convex norms. Well-known examples of horizontally strictly convex norms on Carnot groups are…

We study the geometry of the space of measures of a compact ultrametric space X, endowed with the L^p Wasserstein distance from optimal transportation. We show that the power p of this distance makes this Wasserstein space affinely…

We define new differential structures on the Wasserstein spaces $\mathcal{W}_p(M)$ for $p > 2$ and a general Riemannian manifold $(M,g)$. We consider a very general and possibly degenerate second order partial differential flow equation…

The Wasserstein distance $\mathcal{W}_p$ is an important instance of an optimal transport cost. Its numerous mathematical properties as well as applications to various fields such as mathematical finance and statistics have been well…

For a Polish space $X$, we define the Shape space $\mathcal{S}_p(X)$ to be the Wasserstein space $W_p(X)$ modulo the action of a subgroup $G$ of the isometry group $ISO(X)$ of $X$, where the action is given by the pushforward of measures.…

We study the geometric structure of the space of random measures $\mathcal{P}_p(\mathcal{P}_p(X))$, endowed with the Wasserstein on Wasserstein metric, where $(X, d)$ is a complete separable metric space. In this setting, we prove a metric…

We investigate the stability of the Wasserstein distance, a metric structure on the space of probability measures arising from the theory of optimal transport, under metric ultralimits. We first show that if $(X_{i},d_{i})_{i\in\mathbb{N}}$…

The aim of this article is to write the $p$-Wasserstein metric $W_p$ with the $p$-norm, $p\in [1,\infty)$, on $\R^d$ in terms of copula. In particular for the case of one-dimensional distributions, we get that the copula employed to get the…