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Harmonic Measures on the Sphere via Curvature-Dimension
Emanuel Milman
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 26 (2017) no. 2, p. 437-449

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

Sny1|y-x|n+α,

satisfies the Curvature-Dimension condition CD(n-1-n+α 4,-α), for all |x|<1, α-n and n2. 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.

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

Sny1|y-x|n+α,

satisfait la condition de Courbure-Dimension CD(n-1-n+α 4,-α), pour tout |x|<1, α-n et n2. 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.

Published online : 2017-04-13
DOI : https://doi.org/10.5802/afst.1540
@article{AFST_2017_6_26_2_437_0,
     author = {Emanuel Milman},
     title = {Harmonic Measures on the Sphere via Curvature-Dimension},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 26},
     number = {2},
     year = {2017},
     pages = {437-449},
     doi = {10.5802/afst.1540},
     language = {en},
     url = {https://afst.centre-mersenne.org/item/AFST_2017_6_26_2_437_0}
}
Milman, Emanuel. Harmonic Measures on the Sphere via Curvature-Dimension. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 26 (2017) no. 2, pp. 437-449. doi : 10.5802/afst.1540. afst.centre-mersenne.org/item/AFST_2017_6_26_2_437_0/

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