Isometric study of Wasserstein spaces – the real line

Abstract

Recently Kloeckner described the structure of the isometry group of the quadratic Wasserstein space W2(R n ). 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 Isom (Wp(R)), the isometry group of the Wasserstein space Wp(R) for all p ∈ [1, ∞) \ {2}. We show that W2(R) is also exceptional regarding the parameter p: Wp(R) is isometrically rigid if and only if p = 2. Regarding the underlying space, we prove that the exceptionality of p = 2 disappears if we replace R by the compact interval [0,1]. Surprisingly, in that case, Wp([0, 1]) is isometrically rigid if and only if p = 1. Moreover, W1([0, 1]) admits isometries that split mass, and Isom (W1([0, 1])) cannot be embedded into Isom (W1(R)).

The isometry group of Wasserstein spaces: the Hilbertian case

“…We prove (b) in Subsection 3.2 as Theorem 3.16. In Subsection 3.3, we show that for $$p = 2k$$ with $$k\in \mathbb {N}, k\geqslant 2$$ isometries map measures supported on a line into measures supported on another line, which allows us to utilize our recent result from [11], see Theorem 3.18. Finally, in Section 4 we prove isometric rigidity of 1‐Wasserstein spaces over metric spaces $$(X,\rho )$$ satisfying the strict triangle inequality $$$$\begin{equation*} \rho (x,y) < \rho (x,z) + \rho (z,y) \quad (x,y,z,\in X, z\notin \lbrace x,y\rbrace ), \end{equation*}$$$$see Theorem 4.6.…”

“…We shall see in Section 2 that, contradicting to intuition, such a Polish space exists for all parameters $$p\geqslant 1$$. In [11], we showed that the 1‐Wasserstein space built over the line segment [0,1] possesses isometries that send Dirac measures into measures typically supported on two points — hence split mass. Using this example, in Section 2 we construct another Polish space which illustrates that the answer to the above question is indeed affirmative for all parameters $$p \geqslant 1$$.…”

“…Such isometries are called exotic isometries. Motivated by this latter striking result, in [11] we gave a complete characterization of isometries of $$p$$‐Wasserstein spaces built on the real line $$\mathbb {R}$$ for all parameters $$1\leqslant p<\infty$$. It turned out that the $$p=2$$ case is exceptional in the sense that if $$p\ne 2$$ then all isometries of $$\mathcal {W}_p(\mathbb {R})$$ are trivial.…”

“…Bertrand and Kloeckner dedicated a whole series of papers [2, 3, 17–19] to understand and describe some important geometric properties of 2‐Wasserstein spaces including the structure of their isometries. For more results concerning the structure of isometries with respect to different probability metrics, we refer the reader to the papers [4, 5, 8–12, 16, 23, 27, 31].…”

“…In this section, we answer the aforementioned questions [17, Questions 1‐2] affirmatively by showing that for all $$p > 1$$ there exists a Polish space $$X$$ such that the $$p$$‐Wasserstein space $$\mathcal {W}_p(X)$$ possesses mass splitting isometries. For $$p=1$$, this question was recently answered by the authors in [11]. We recall the details below, as we shall manipulate this example in order to answer the question in the case of strictly convex cost ($$p>1$$).…”