Optimal Transport Theory and Wasserstein Geometry for Gradient Flows in Metric Spaces
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| Formato: | Recurso digital |
| Lenguaje: | inglés |
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2025
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| _version_ | 1866901627623964672 |
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| author | Revista, Zen MFC, 10 |
| author_facet | Revista, Zen MFC, 10 |
| contents | <div>Mathematical Applications of Science Fiction</div> <div> </div> <p>We present a rigorous derivation of the existence and unique-<br>ness of gradient flows for semi-convex functionals on met-<br>ric measure spaces satisfying the Riemannian Curvature-<br>Dimension condition RCD∗(K, N ). By constructing a<br>motivic transport operator within the Wasserstein space<br>W2(X), we extend the Jordan-Kinderlehrer-Otto (JKO)<br>scheme to non-smooth structures exhibiting perfectoid sin-<br>gularities. We establish the Evolution Variational Inequal-<br>ity (EVI) for the generalized entropy functional, proving<br>K-contraction of the flow. Furthermore, we analyze the<br>spectral convergence of the discrete-time approximations<br>using a Hilbert-Schmidt embedding of the transport plans,<br>providing a robust framework for dissipative dynamics in<br>crystalline geometries. The analysis avoids standard co-<br>homological obstructions by utilizing a derived symplectic<br>formalism on the tangent bundle of the probability space.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_18102234 |
| institution | Zenodo |
| language | eng |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Optimal Transport Theory and Wasserstein Geometry for Gradient Flows in Metric Spaces Revista, Zen MFC, 10 <div>Mathematical Applications of Science Fiction</div> <div> </div> <p>We present a rigorous derivation of the existence and unique-<br>ness of gradient flows for semi-convex functionals on met-<br>ric measure spaces satisfying the Riemannian Curvature-<br>Dimension condition RCD∗(K, N ). By constructing a<br>motivic transport operator within the Wasserstein space<br>W2(X), we extend the Jordan-Kinderlehrer-Otto (JKO)<br>scheme to non-smooth structures exhibiting perfectoid sin-<br>gularities. We establish the Evolution Variational Inequal-<br>ity (EVI) for the generalized entropy functional, proving<br>K-contraction of the flow. Furthermore, we analyze the<br>spectral convergence of the discrete-time approximations<br>using a Hilbert-Schmidt embedding of the transport plans,<br>providing a robust framework for dissipative dynamics in<br>crystalline geometries. The analysis avoids standard co-<br>homological obstructions by utilizing a derived symplectic<br>formalism on the tangent bundle of the probability space.</p> |
| title | Optimal Transport Theory and Wasserstein Geometry for Gradient Flows in Metric Spaces |
| url | https://doi.org/10.5281/zenodo.18102234 |