Harmonic Measures on the Sphere via Curvature-Dimension

Emanuel Milman

doi:10.5802/afst.1540

Emanuel Milman ¹

¹ Department of Mathematics, Technion - Israel Institute of Technology, Haifa 32000, Israel

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 26 (2017) no. 2, pp. 437-449.

Résumé
Abstract

On montre que la famille de mesures de probabilités sur la sphère n-dimensionelle, dont les densités sont proportionnelles a :

S^{n} ∋ y \mapsto \frac{1}{{| y - x |}^{n + α}},

satisfait la condition de Courbure-Dimension $CD (n - 1 - \frac{n + α}{4}, - α)$ , pour tout $| x | < 1$ , $α \geq - n$ et $n \geq 2$ . Le cas $α = 1$ correspond à la distribution de probabilité qu’un mouvement Brownian partant de $x$ atteigne la sphère (aussi appelee la “mesure harmonique” sur la sphère). En guise d’applications, des inegalités isopérimetriques et de trou spectral, ainsi que des estimées de concentration seront presentées. Nous discuterons aussi de possibles extensions de nos resultats.

We show that the family of probability measures on the $n$ -dimensional unit sphere, having density proportional to:

S^{n} ∋ y \mapsto \frac{1}{{| y - x |}^{n + α}},

satisfies the Curvature-Dimension condition $CD (n - 1 - \frac{n + α}{4}, - α)$ , for all $| x | < 1$ , $α \geq - n$ and $n \geq 2$ . The case $α = 1$ corresponds to the hitting distribution of the sphere by Brownian motion started at $x$ (so-called “harmonic measure” on the sphere). Applications involving isoperimetric, spectral-gap and concentration estimates, as well as potential extensions, are discussed.

Publié le : 2017-04-12

DOI : 10.5802/afst.1540

Affiliations des auteurs :

Emanuel Milman ¹

¹ Department of Mathematics, Technion - Israel Institute of Technology, Haifa 32000, Israel

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Harmonic {Measures} on the {Sphere} via {Curvature-Dimension}},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
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Emanuel Milman. Harmonic Measures on the Sphere via Curvature-Dimension. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 26 (2017) no. 2, pp. 437-449. doi : 10.5802/afst.1540. https://afst.centre-mersenne.org/articles/10.5802/afst.1540/

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