Comptes Rendus
Probabilités, Physique mathématique
Molecules as metric measure spaces with Kato-bounded Ricci curvature
Comptes Rendus. Mathématique, Volume 358 (2020) no. 5, pp. 595-602.

Set Ψ:=log(Ψ ˜), with Ψ ˜>0 the ground state of an arbitrary molecule with n electrons in the infinite mass limit (neglecting spin/statistics). Let Σ 3n be the set of singularities of the underlying Coulomb potential. We show that the metric measure space given by 3n with its Euclidean distance and the measure

μ(dx)=e -2Ψ(x) dx

has a Bakry-Emery-Ricci tensor which is absolutely bounded by the function x|x-Σ| -1 , which we show to be an element of the Kato class induced by . In addition, it is shown that is stochastically complete, that is, the Brownian motion which is induced by a molecule is nonexplosive. Our proofs reveal a fundamental connection between the above geometric/probabilistic properties and recently obtained derivative estimates for Ψ ˜ by Fournais/Sørensen, as well as Aizenman/Simon’s Harnack inequality for Schrödinger operators. Moreover, our results suggest to study general metric measure spaces having a Ricci curvature which is synthetically bounded from below/above by a function in the underlying Kato class.

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DOI : 10.5802/crmath.76

Batu Güneysu 1 ; Max von Renesse 2

1 Institut für Mathematik, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099 Berlin, Germany
2 Fakultät für Mathematik und Informatik, Universität Leipzig, Ritterstraße 26, 04109 Leipzig, Germany
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits
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     title = {Molecules as metric measure spaces with {Kato-bounded} {Ricci} curvature},
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Batu Güneysu; Max von Renesse. Molecules as metric measure spaces with Kato-bounded Ricci curvature. Comptes Rendus. Mathématique, Volume 358 (2020) no. 5, pp. 595-602. doi : 10.5802/crmath.76. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.76/

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