Isometric study of Wasserstein spaces – the real line

2020/02/03 by György Pál Gehér, Tamás Titkos, Dániel Virosztek · 14 citations
Mathematics · Medicine · Physics and Astronomy · #Advanced Neuroimaging Techniques and Applications #Geometric Analysis and Curvature Flows #Geometry #Geometry and complex manifolds #Isometric exercise #Line (geometry) #Mathematical analysis #Mathematics #Physical therapy #Pure mathematics #Real line #math-ph #math.FA #math.MG #math.MP #math.PR #msc:46E27. #msc:54E40 #msc:60A10 #msc:60B05

paper · pdf · doi:10.1090/tran/8113

published in Transactions of the American Mathematical Society 373(8), 5855-5883 (American Mathematical Society) · 32 pages, 7 figures. Accepted for publication in Trans. Amer. Math. Soc

arxiv created 2020/02/03 · openalex publication_date 2020/02/12 · arxiv updated 2020/07/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

Recently Kloeckner described the structure of the isometry group of the quadratic Wasserstein space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper W 2 left-parenthesis double-struck upper R Superscript n Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">W</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal W₂(\mathbb Rⁿ)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . 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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper I normal s normal o normal m left-parenthesis script upper W Subscript p Baseline left-parenthesis double-struck upper R right-parenthesis right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">I</mml:mi> <mml:mi mathvariant="normal">s</mml:mi> <mml:mi mathvariant="normal">o</mml:mi> <mml:mi mathvariant="normal">m</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">W</mml:mi> </mml:mrow> <mml:mi>p</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathrm Isom(\mathcal W_p(\mathbb R))</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , the isometry group of the Wasserstein space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper W Subscript p Baseline left-parenthesis double-struck upper R right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">W</mml:mi> </mml:mrow> <mml:mi>p</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal W_p(\mathbb R)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for all <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p element-of left-bracket 1 comma normal infinity right-parenthesis minus StartSet 2 EndSet"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo> ∈ </mml:mo> <mml:mo stretchy="false">[</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mi mathvariant="normal"> ∞ </mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo class="MJX-variant"> ∖ </mml:mo> <mml:mo fence="false" stretchy="false"></mml:mo> <mml:mn>2</mml:mn> <mml:mo fence="false" stretchy="false"></mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">p ∈ [1, ∞ )∖ \2\</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We show that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper W 2 left-parenthesis double-struck upper R right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">W</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal W₂(\mathbb R)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is also exceptional regarding the parameter <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> : <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper W Subscript p Baseline left-parenthesis double-struck upper R right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">W</mml:mi> </mml:mrow> <mml:mi>p</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal W_p(\mathbb R)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is isometrically rigid if and only if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p not-equals 2"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo> ≠ </mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotat

Citations

• Optimal Transport
• Topics in Optimal Transportation
• Scalable Bayes via Barycenter in Wasserstein Space
• Ricci curvature for metric-measure spaces via optimal transport
• Calculation of the Wasserstein Distance Between Probability Distributions on the Line
• Asymptotic coupling and a general form of Harris’ theorem with applications to stochastic delay equations

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