GermanWikipedia - /page/entry.mu

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<title>Optimaler Transport</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading">Optimaler Transport</h1>

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr">In der <a href="Mathematik" title="Mathematik">Mathematik</a> bezeichnet Optimaler Transport eine Theorie, die aus der analytischen Modellierung des <a href="Transportproblem" title="Transportproblem">Transportproblems</a> entstanden ist. <a href="John_Lott_(Mathematiker)" title="John Lott (Mathematiker)">Lott</a> und <a href="C%C3%A9dric_Villani" title="Cédric Villani">Villani</a> sowie <a href="Karl-Theodor_Sturm" title="Karl-Theodor Sturm">Sturm</a> gaben mit Hilfe des optimalen Transports eine synthetische Definition von <a href="Ricci-Kr%C3%BCmmung" class="mw-redirect" title="Ricci-Krümmung">Ricci-Krümmungs</a>-Schranken in allgemeinen metrischen Räumen.<a href="#cite_note-1">[1]</a><a href="#cite_note-2">[2]</a>

Optimaler Transport ist ursprünglich ein (auf <a href="Gaspard_Monge" title="Gaspard Monge">Monge</a> und <a href="Leonid_Kantorovich" class="mw-redirect" title="Leonid Kantorovich">Kantorovich</a> zurückgehendes) klassisches Problem, das ausgehend von einer gegebenen Anfangsverteilung und einer gewünschten Endverteilung nach dem günstigsten Transport sucht, bei dem die Anfangs- in die Endverteilung überführt wird.

Die Anfangs- und Endverteilungen werden durch Dichtefunktionen (<a href="Wahrscheinlichkeitsma%C3%9F" title="Wahrscheinlichkeitsmaß">Wahrscheinlichkeitsmaße</a>) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">

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<annotation encoding="application/x-tex">{\displaystyle \mu }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9fd47b2a39f7a7856952afec1f1db72c67af6161.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }">

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<annotation encoding="application/x-tex">{\displaystyle \nu }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c15bbbb971240cf328aba572178f091684585468.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }" loading="lazy"> auf <a href="Metrischer_Raum" title="Metrischer Raum">metrischen Räumen</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

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<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> modelliert. Die Kostenfunktion ist eine gegebene Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c:X\times Y\rightarrow \mathbb {R} }">

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<annotation encoding="application/x-tex">{\displaystyle c:X\times Y\rightarrow \mathbb {R} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1fead873bd4f0032af993f8a5a745f1659c9bc6e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.83ex; height:2.176ex;" alt="{\displaystyle c:X\times Y\rightarrow \mathbb {R} }" loading="lazy">. Der Wert <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c(x,y)}">

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<annotation encoding="application/x-tex">{\displaystyle c(x,y)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aa1be48bb4ececebce224db625fbcaa632b22b95.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.335ex; height:2.843ex;" alt="{\displaystyle c(x,y)}" loading="lazy"> gibt die Kosten für den Transport von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> nach <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

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<annotation encoding="application/x-tex">{\displaystyle y}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"> an. Ein typisches Beispiel ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c(x,y)=\|x-y\|}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c(x,y)=\|x-y\|}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c1c8a4bb54a71c09ff73c15cf52118c382c5d2d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.084ex; height:2.843ex;" alt="{\displaystyle c(x,y)=\|x-y\|}" loading="lazy">, falls <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> Teilmengen eines <a href="Normierter_Vektorraum" class="mw-redirect" title="Normierter Vektorraum">normierten Vektorraumes</a> sind, oder allgemeiner <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c(x,y)=h(x-y)}">

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<annotation encoding="application/x-tex">{\displaystyle c(x,y)=h(x-y)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a75d18b2545985f41b2d8d03c8e867a99981d00a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.908ex; height:2.843ex;" alt="{\displaystyle c(x,y)=h(x-y)}" loading="lazy"> für eine differenzierbare Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}">

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<annotation encoding="application/x-tex">{\displaystyle h}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b26be3e694314bc90c3215047e4a2010c6ee184a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}" loading="lazy">.

<div class="mw-heading mw-heading2"><h2 id="Monge-Problem">Monge-Problem</h2></div>

Gesucht wird eine <a href="Injektivit%C3%A4t" class="mw-redirect" title="Injektivität">injektive</a> Abbildung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r:X\rightarrow Y}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle r:X\rightarrow Y}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b927b1988913983b77144a2cb9b88555f3e60fe8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.353ex; height:2.176ex;" alt="{\displaystyle r:X\rightarrow Y}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu (r^{-1}(B))=\nu (B)}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu (r^{-1}(B))=\nu (B)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c72910de3f5722c0f2a6b62c3ea52d9f5467f36c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.07ex; height:3.176ex;" alt="{\displaystyle \mu (r^{-1}(B))=\nu (B)}" loading="lazy"> für alle <a href="Messbare_Menge" class="mw-redirect" title="Messbare Menge">messbaren Mengen</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\subset Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B\subset Y}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e7921e1b64009c92cbcb5193f70f3358d9183c10.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.636ex; height:2.176ex;" alt="{\displaystyle B\subset Y}" loading="lazy">, welche das Funktional

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \int _{X}c(x,r(x))d\mu (x)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/602548f55362ad06a175905200ff04e94dae38cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:18.436ex; height:5.676ex;" alt="{\displaystyle \int _{X}c(x,r(x))d\mu (x)}" loading="lazy"></dd></dl>

minimiert.

Es gibt Beispiele, in denen das Monge-Problem keine Lösung besitzt, z. B. falls <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9fd47b2a39f7a7856952afec1f1db72c67af6161.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" loading="lazy"> ein <a href="Diracma%C3%9F" title="Diracmaß">Diracmaß</a> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \nu }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c15bbbb971240cf328aba572178f091684585468.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }" loading="lazy"> die Summe von mindestens zwei Diracmaßen ist.

<div class="mw-heading mw-heading2"><h2 id="Kantorovich-Problem">Kantorovich-Problem</h2></div>

Ein relaxiertes Problem wurde 1942 von Kantorovich betrachtet. Das Kantorovich-Problem sucht nach einem Wahrscheinlichkeitsmaß <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }">

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<annotation encoding="application/x-tex">{\displaystyle \pi }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9be4ba0bb8df3af72e90a0535fabcc17431e540a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }" loading="lazy"> auf dem <a href="Produkttopologie" title="Produkttopologie">Produktraum</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\times Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X\times Y}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1613c1ff4b6fbfb6c80a8da83e90ad28f0ab3483.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.594ex; height:2.176ex;" alt="{\displaystyle X\times Y}" loading="lazy"> mit

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \pi (A\times Y)=\mu (A),\quad \pi (X\times B)=\nu (B)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0fdaaa3d66ddc3f30b5d29711d76261d2c1cc45e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.537ex; height:2.843ex;" alt="{\displaystyle \pi (A\times Y)=\mu (A),\quad \pi (X\times B)=\nu (B)}" loading="lazy"></dd></dl>

für alle <a href="Kompakter_Raum" title="Kompakter Raum">kompakten</a> Mengen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\subset X,B\subset Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A\subset X,B\subset Y}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/09aa0804f5f179487cc98cf2cebc0cc44a8ebc53.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.491ex; height:2.509ex;" alt="{\displaystyle A\subset X,B\subset Y}" loading="lazy">, welches das Funktional

<msub>

</mrow>

</msub>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \int _{X\times Y}c(x,y)d\pi (x,y)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1800c7c1698b8e90d13c0d259b320e2eb329cc43.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:20.056ex; height:5.676ex;" alt="{\displaystyle \int _{X\times Y}c(x,y)d\pi (x,y)}" loading="lazy"></dd></dl>

minimiert.

Kantorovich bewies, dass ein solches Wahrscheinlichkeitsmaß immer existiert.

Falls <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=Y=\mathbb {R} ^{n}}">

<msup>

</mrow>

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</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X=Y=\mathbb {R} ^{n}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e9358b84af8e0ffb132dc61c2ddbd2b8be317e85.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.847ex; height:2.343ex;" alt="{\displaystyle X=Y=\mathbb {R} ^{n}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c(x,y)=h(x-y)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c(x,y)=h(x-y)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a75d18b2545985f41b2d8d03c8e867a99981d00a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.908ex; height:2.843ex;" alt="{\displaystyle c(x,y)=h(x-y)}" loading="lazy"> für eine <a href="Konvexe_Funktion" class="mw-redirect" title="Konvexe Funktion">strikt konvexe Funktion</a> h, dann ist die Lösung des Kantorovich-Problems von der Form

<msub>

</mrow>

</msub>

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<annotation encoding="application/x-tex">{\displaystyle \pi =(\operatorname {id} ,r)_{*}\mu }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d1d3e678685f4284573332fb0ce41c8de22d7073.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.718ex; height:2.843ex;" alt="{\displaystyle \pi =(\operatorname {id} ,r)_{*}\mu }" loading="lazy"></dd></dl>

für eine injektive Abbildung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r:X\rightarrow Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r:X\rightarrow Y}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b927b1988913983b77144a2cb9b88555f3e60fe8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.353ex; height:2.176ex;" alt="{\displaystyle r:X\rightarrow Y}" loading="lazy">. Insbesondere hat in diesem Fall auch das Monge-Problem eine Lösung.<a href="#cite_note-3">[3]</a>

<div class="mw-heading mw-heading2"><h2 id="Wasserstein-Metriken">Wasserstein-Metriken</h2></div>

<div class="hauptartikel" role="navigation">→ Hauptartikel: <a href="Wasserstein-Metrik" title="Wasserstein-Metrik">Wasserstein-Metrik</a></div>

Für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\geq 1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\geq 1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cf93c1353080a21b276e79058d82c19c40310653.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.52ex; height:2.509ex;" alt="{\displaystyle p\geq 1}" loading="lazy"> und Wahrscheinlichkeitsmaße <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu ,\nu }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu ,\nu }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/489a4ac1b5d76663fa5328e6f3c1d382655cb496.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.668ex; height:2.176ex;" alt="{\displaystyle \mu ,\nu }" loading="lazy"> auf einem metrischen Raum X sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma (\mu ,\nu )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Gamma (\mu ,\nu )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4a583718b514bd9154fa1e038fbf32b6f1c1c88e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.93ex; height:2.843ex;" alt="{\displaystyle \Gamma (\mu ,\nu )}" loading="lazy"> die Menge aller Wahrscheinlichkeitsmaße auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\times X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X\times X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0008a48abb9837a4cb0f495dd85d0ffda22ead92.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.8ex; height:2.176ex;" alt="{\displaystyle X\times X}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi (A\times X)=\mu (A),\pi (X\times B)=\nu (B)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi (A\times X)=\mu (A),\pi (X\times B)=\nu (B)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3fdfa8f62ced07a14e8aee8e184355b3415284e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.421ex; height:2.843ex;" alt="{\displaystyle \pi (A\times X)=\mu (A),\pi (X\times B)=\nu (B)}" loading="lazy"> für alle kompakten Mengen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A,B\subset X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A,B\subset X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bda7b1e5455c1de48b5de555a434821b1f1d85c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.62ex; height:2.509ex;" alt="{\displaystyle A,B\subset X}" loading="lazy">. Dann definiert

<msub>

</mrow>

</msub>

<msup>

<mrow>

<mrow>

</mrow>

</munder>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W_{p}(\mu ,\nu ):=\left(\inf _{\gamma \in \Gamma (\mu ,\nu )}\int _{X\times X}d(x,y)^{p}\,\mathrm {d} \gamma (x,y)\right)^{1/p}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d0a2d8cc9f101589f2bdf41dd096e386a8b5972f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:47.466ex; height:6.676ex;" alt="{\displaystyle W_{p}(\mu ,\nu ):=\left(\inf _{\gamma \in \Gamma (\mu ,\nu )}\int _{X\times X}d(x,y)^{p}\,\mathrm {d} \gamma (x,y)\right)^{1/p}}" loading="lazy"></dd></dl>

den p-ten Wasserstein-Abstand zwischen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9fd47b2a39f7a7856952afec1f1db72c67af6161.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \nu }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c15bbbb971240cf328aba572178f091684585468.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }" loading="lazy">.

Der p-te Wasserstein-Abstand <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{p}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W_{p}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8daa318a0c0f1dcb48604b379590d72c7392c615.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.253ex; height:2.843ex;" alt="{\displaystyle W_{p}}" loading="lazy"> definiert eine Metrik auf der Menge aller Wahrscheinlichkeitsmaße auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">, deren p-tes Moment endlich ist.

Wenn <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> eine konvexe Teilmenge des <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c510b63578322050121fe966f2e5770bea43308d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p>1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p>1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0f127e7a5f2449ddf3edb8164c2d2439120641f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.52ex; height:2.509ex;" alt="{\displaystyle p>1}" loading="lazy"> ist, dann sind die <a href="Geod%C3%A4te" title="Geodäte">Geodäten</a> der p-ten Wasserstein-Metrik von der Form

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t\mapsto (F_{t})_{*}\pi }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bf84e5bc86d0ba16a9b39591a2f3476ebd9f48b9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.97ex; height:2.843ex;" alt="{\displaystyle t\mapsto (F_{t})_{*}\pi }" loading="lazy">,</dd></dl>

wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{t}:X\times X\rightarrow X}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{t}:X\times X\rightarrow X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c10a3abec9bd4fc57267e39019d6be0918ea5dfa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.652ex; height:2.509ex;" alt="{\displaystyle F_{t}:X\times X\rightarrow X}" loading="lazy"> die durch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{t}(x,y)=tx+(1-t)y}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{t}(x,y)=tx+(1-t)y}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5a5185189003ace551ba65543e4295f42390ba8c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.564ex; height:2.843ex;" alt="{\displaystyle F_{t}(x,y)=tx+(1-t)y}" loading="lazy"> definierte Abbildung und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9be4ba0bb8df3af72e90a0535fabcc17431e540a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }" loading="lazy"> die Lösung des Kantorovich-Problems zu <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c(x,y)=d(x,y)^{p}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c(x,y)=d(x,y)^{p}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/528a94970b7c1f4eb2ed6acbd0b50222006041f3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.037ex; height:2.843ex;" alt="{\displaystyle c(x,y)=d(x,y)^{p}}" loading="lazy"> ist.

<div class="mw-heading mw-heading2"><h2 id="Ricci-Krümmungs-Schranken">Ricci-Krümmungs-Schranken</h2></div>

Es sei M eine kompakte Riemannsche Mannigfaltigkeit mit dem durch die Volumenform gegebenen Wahrscheinlichkeitsmaß. Dann hat M genau dann nichtnegative Ricci-Krümmung, wenn es zu je zwei Wahrscheinlichkeitsmaßen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu ,\nu }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu ,\nu }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/489a4ac1b5d76663fa5328e6f3c1d382655cb496.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.668ex; height:2.176ex;" alt="{\displaystyle \mu ,\nu }" loading="lazy"> eine verbindende Geodäte (bzgl. der W2-Wasserstein-Metrik) gibt, entlang derer das Entropie-Funktional <a href="Konvexe_und_konkave_Funktionen" title="Konvexe und konkave Funktionen">konvex</a> ist.

In Verallgemeinerung dieser Eigenschaft gaben Lott und Villani sowie Sturm eine synthetische Definition nichtnegativer Ricci-Krümmung in allgemeinen metrischen Räumen.

<div class="mw-heading mw-heading2"><h2 id="Quellen">Quellen</h2></div>

<li id="cite_note-1"><a href="#cite_ref-1">↑</a> <a href="John_Lott_(Mathematiker)" title="John Lott (Mathematiker)">John Lott</a>, <a href="C%C3%A9dric_Villani" title="Cédric Villani">Cédric Villani</a>: <a rel="nofollow" class="external text" href="http://math.berkeley.edu/~lott/LottVillani.pdf">Ricci curvature for metric-measure spaces via optimal transport.</a> In: <a href="Annals_of_Mathematics" title="Annals of Mathematics">Annals of Mathematics</a>. Bd. 169, 2009, S. 903–991, (PDF; 552 kB), <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.4007/annals.2009.169.903">10.4007/annals.2009.169.903</a>.

</li>

<li id="cite_note-2"><a href="#cite_ref-2">↑</a> Karl-Theodor Sturm: <style data-mw-deduplicate="TemplateStyles:r261891140">

/* start https://de.wikipedia.org/ */

.mw-parser-output .webarchiv-memento a{color:inherit}

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</style><a rel="nofollow" class="external text" href="https://web.archive.org/web/20070628202244/http://sfb611.iam.uni-bonn.de/uploads/203-komplett.pdf">On the geometry of metric measure spaces.</a> (<a href="Webarchivierung#Begrifflichkeiten" title="Webarchivierung">Memento</a> vom 28. Juni 2007 im <a href="Internet_Archive" title="Internet Archive">Internet Archive</a>) In: <a href="Acta_Mathematica" title="Acta Mathematica">Acta Mathematica</a>. Bd. 196, Nr. 1, 2006, 65–131, (PDF; 591 kB), <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007/s11511-006-0002-8">10.1007/s11511-006-0002-8</a>.

</li>

<li id="cite_note-3"><a href="#cite_ref-3">↑</a> Wilfrid Gangbo, Robert J. McCann: <a rel="nofollow" class="external text" href="http://www.math.toronto.edu/mccann/papers/geometry.pdf">The geometry of optimal transportation.</a> In: Acta Mathematica. Bd. 177, Nr. 2, 1996, 113–161, (PDF; 2,8 MB), <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007/BF02392620">10.1007/BF02392620</a>.

</li>

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