Molecules as metric measure spaces with Kato-bounded Ricci curvature

Batu Güneysu; Max von Renesse

doi:10.5802/crmath.76

Probabilités, Physique mathématique

Molecules as metric measure spaces with Kato-bounded Ricci curvature

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

Comptes Rendus. Mathématique, Volume 358 (2020) no. 5, pp. 595-602.

Résumé

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

μ (d x) = e^{- 2 Ψ (x)} d x

has a Bakry-Emery-Ricci tensor which is absolutely bounded by the function $x \mapsto {| x - Σ |}^{- 1}$ , which we show to be an element of the Kato class induced by $ℳ$ . In addition, it is shown that $ℳ$ 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 $\tilde{Ψ}$ 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.

Reçu le : 2019-09-11
Révisé le : 2020-04-30
Accepté le : 2020-05-19
Publié le : 2020-09-14

DOI : 10.5802/crmath.76

Affiliations des auteurs :

Batu Güneysu ¹ ; Max von Renesse ²

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{CRMATH_2020__358_5_595_0,
     author = {Batu G\"uneysu and Max von Renesse},
     title = {Molecules as metric measure spaces with {Kato-bounded} {Ricci} curvature},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {595--602},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {358},
     number = {5},
     year = {2020},
     doi = {10.5802/crmath.76},
     language = {en},
}

TY  - JOUR
AU  - Batu Güneysu
AU  - Max von Renesse
TI  - Molecules as metric measure spaces with Kato-bounded Ricci curvature
JO  - Comptes Rendus. Mathématique
PY  - 2020
SP  - 595
EP  - 602
VL  - 358
IS  - 5
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.76
LA  - en
ID  - CRMATH_2020__358_5_595_0
ER  -

%0 Journal Article
%A Batu Güneysu
%A Max von Renesse
%T Molecules as metric measure spaces with Kato-bounded Ricci curvature
%J Comptes Rendus. Mathématique
%D 2020
%P 595-602
%V 358
%N 5
%I Académie des sciences, Paris
%R 10.5802/crmath.76
%G en
%F CRMATH_2020__358_5_595_0

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/

Bibliographie
Cité par

[1] Michael Aizenman; Barry Simon Brownian motion and Harnack inequality for Schrödinger operators, Commun. Pure Appl. Math. (1982) no. 2, pp. 209-273 | DOI | Zbl

[2] Luigi Ambrosio; Nicola Gigli; Giuseppe Savaré Metric measure spaces with Riemannian Ricci curvature bounded from below, Duke Math. J., Volume 163 (2014) no. 7, pp. 1405-1490 | DOI | MR | Zbl

[3] Luigi Ambrosio; Nicola Gigli; Giuseppe Savaré Bakry-Emery curvature condition and Riemannian Ricci curvature bounds, Ann. Probab., Volume 43 (2015) no. 1, pp. 339-404 | DOI | Zbl

[4] Kathrin Bacher; Karl-Theodor Sturm Localization and tensorization properties of the Curvature-Dimension condition for metric measure spaces, J. Funct. Anal., Volume 259 (2010) no. 1, pp. 28-56 | DOI | MR | Zbl

[5] Dominique Bakry; Michel Émery Diffusions hypercontractives, Séminaire de probabilités XIX, 1983/84 (Lecture Notes in Mathematics), Volume 1123, Springer, 1985, pp. 177-206 | DOI | Numdam | MR | Zbl

[6] Mathias Braun; Batu Güneysu Heat flow regularity, Bismut’s derivative formula, and pathwise Brownian couplings on Riemannian manifolds with Dynkin bounded Ricci curvature (2020) (https://arxiv.org/abs/2001.10297)

[7] Fabio Cavalletti; Emanuel Milman The globalization theorem for the curvature dimension condition (2016) (https://arxiv.org/abs/1612.07623)

[8] Fabio Cavalletti; Karl-Theodor Sturm Local curvature-dimension condition implies measure-contraction property, J. Funct. Anal., Volume 262 (2012) no. 12, pp. 5110-5127 | DOI | MR | Zbl

[9] Dario Cordero-Erausquin; Robert J. McCann; Michael Schmuckenschläger A Riemannian interpolation inequality à la Borell, Brascamb and Lieb, Invent. Math., Volume 146 (2001) no. 2, pp. 219-257 | DOI | Zbl

[10] Matthias Erbar; Kazumasa Kuwada; Karl-Theodor Sturm On the equivalence of the entropic curvature-dimension condition and Bochner’s inequality on metric measure spaces, Invent. Math., Volume 201 (2015) no. 3, pp. 993-1071 | DOI | MR | Zbl

[11] Søren Fournais; Thomas Østergaard Sørensen Pointwise estimates on derivatives of Coulombic wave functions and their electron densities (2018) (https://arxiv.org/abs/1803.03495)

[12] Nicola Gigli On the differential structure of metric measure spaces and applications, Memoirs of the American Mathematical Society, 1113, American Mathematical Society, 2015 | Zbl

[13] Alexander Grigorʼyan Heat kernel and analysis on manifolds, AMS/IP Studies in Advanced Mathematics, 47, American Mathematical Society, 2009 | MR

[14] Batu Güneysu Covariant Schrödinger semigroups on Riemannian manifolds, Operator Theory: Advances and Applications, 264, Springer, 2017 | Zbl

[15] Maria Hoffmann-Ostenhof; Thomas Hoffmann-Ostenhof; Thomas Østergaard Sørensen Electron wavefunctions and densities for atoms, Ann. Henri Poincaré, Volume 2 (2001) no. 1, pp. 27-100 | MR | Zbl

[16] Tosio Kato On the eigenfunctions of many-particle systems in quantum mechanics, Commun. Pure Appl. Math., Volume 10 (1957), pp. 151-177 | DOI | MR | Zbl

[17] Kazumasa Kuwada; Kazuhiro Kuwae Radial processes on ${RCD}^{*} (K, N)$ spaces, J. Math. Pures Appl., Volume 126 (2019), pp. 72-108 | DOI | MR | Zbl

[18] Kazuhiro Kuwae; Masayuki Takahashi Kato class measures of symmetric Markov processes under heat kernel estimates, J. Funct. Anal., Volume 250 (2007) no. 1, pp. 86-113 | DOI | MR | Zbl

[19] John Lott; Cédric Villani Ricci curvature for metric-measure spaces via optimal transport, Ann. Math., Volume 169 (2009) no. 3, pp. 903-991 | DOI | MR | Zbl

[20] Andrea Mondino; Aaron Naber Structure Theory of Metric-Measure Spaces with Lower Ricci Curvature Bounds I, J. Eur. Math. Soc., Volume 21 (2019) no. 6, pp. 1809-1854 | DOI | MR | Zbl

[21] Felix Otto; Cédric Villani Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality, J. Funct. Anal., Volume 173 (2000) no. 2, pp. 361-400 | DOI | MR | Zbl

[22] Max-K. von Renesse; Karl-Theodor Sturm Transport Inequalities, Gradient Estimates, Entropy and Ricci Curvature, Commun. Pure Appl. Math., Volume 58 (2005) no. 7, pp. 923-940 | DOI | MR | Zbl

[23] Barry Simon Schrödinger Semigroups, Bull. Am. Math. Soc., Volume 7 (1982) no. 3, pp. 447-526 | DOI | Zbl

[24] Peter Stollmann; Jürgen Voigt Perturbation of Dirichlet forms by measures, Potential Anal., Volume 5 (1996) no. 2, pp. 109-138 | DOI | MR | Zbl

[25] Karl-Theodor Sturm On the geometry of metric measure spaces. I., Acta Math., Volume 196 (2006) no. 1, pp. 65-177 | DOI | Zbl

[26] Karl-Theodor Sturm On the geometry of metric measure spaces. II, Acta Math., Volume 196 (2006) no. 1, pp. 133-177 | DOI | MR | Zbl

[27] Anton Thalmaier; James Thompson Derivative and divergence formulae for diffusion semigroups, Ann. Probab., Volume 47 (2019) no. 2, pp. 743-773 | DOI | MR | Zbl

Cité par Sources :

Commentaires - Politique

Ces articles pourraient vous intéresser

Ricci curvature of metric spaces

Yann Ollivier

C. R. Math (2007)

Model spaces for sharp isoperimetric inequalities

Emanuel Milman

C. R. Math (2012)