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The metric geometry nature of Wasserstein spaces of probability measures: On the point of view of submetry projections
Yenisey, Mehmet
Yenisey, Mehmet
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Abstract
The Wasserstein spaces of probability measures are central objects in optimal transport theory and in probability theory. We formulate a theorem regarding the metric geometry origins of the Wasserstein spaces: Motivated by certain hidden symmetry considerations arising in the study of Hamilton-Jacobi equations in spaces of probability measures that are associated to a mathematical reinterpretation of certain physical models, for $p\in [1,\infty)$ and a separable Hilbert space $E$, we set forth to rigorously represent the Wasserstein space $(\mathcal{P}_p(E),W_p)$ as a so-called metric quotient space induced by metric foliation. Namely, for the following sequence of Lebesgue spaces, $$L_p^{(n)}(E):=L_p([0,1]^n,\mathcal{B}([0,1]^n),\otimes_{i=1}^nm;\ E),\ \quad n\in \N,$$ where $m$ denotes Lebesgue measure on $([0,1],\mathcal{B}([0,1]))$, we show that the Wasserstein space $(\mathcal{P}_p(E),W_p)$ is isometrically isomorphic to a metric quotient space induced by a metric foliation of the direct limit metric space, $\displaystyle\lim_{\longrightarrow}\ L_p^{(n)}(E)=:\mathcal{L}_p(E)$, which is defined as a direct limit in a category of metric spaces.
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Date
2022-08-31
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University of Kansas
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Mathematics, direct limit, metric foliation, metric quotient space, Prokhorov, submetry, Wasserstein