On isometric embeddings of Wasserstein spaces - the discrete case

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:480

Article

Geher, Gyorgy Pal¹  Titkos, Tamas^2,3  Virosztek, Daniel⁴ 

[1] Univ Reading, Dept Math & Stat, POB 220, Reading RG6 6AX, Berks, England

[2] Hungarian Acad Sci, Alfred Renyi Inst Math, Realltanoda U 13-15, H-1053 Budapest, Hungary

[3] BBS Univ Appl Sci, Alkotmany U 9, H-1054 Budapest, Hungary

[4] IST Austria, Campus 1, A-3400 Klosterneuburg, Austria

关键词: Wasserstein space;    Isometric embeddings;    Probability measures;    Discrete metric;   

DOI  :  10.1016/j.jmaa.2019.123435

来源: Elsevier

【 摘 要 】

The aim of this short paper is to offer a complete characterization of all (not necessarily surjective) isometric embeddings of the Wasserstein space W-p(chi), where chi is a countable discrete metric space and 0 < p < infinity is any parameter value. Roughly speaking, we will prove that any isometric embedding can be described by a special kind of chi x (0, 1]-indexed family of nonnegative finite measures. Our result implies that a typical non-surjective isometric embedding of W-p(chi) splits mass and does not preserve the shape of measures. In order to stress that the lack of surjectivity is what makes things challenging, we will prove alternatively that W-p(chi) is isometrically rigid for all 0 < p < infinity. (C) 2019 Elsevier Inc. All rights reserved.

【 授权许可】

Free