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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.

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.

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.

Publié le :
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},
     pages = {437--449},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 26},
     number = {2},
     year = {2017},
     doi = {10.5802/afst.1540},
     language = {en},
     url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1540/}
}
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/

[1] Dominique Bakry L’hypercontractivité et son utilisation en théorie des semigroupes, Lectures on probability theory (Saint-Flour, 1992) (Lect. Notes Math.) Volume 1581, Springer, 1994, pp. 1-114

[2] Dominique Bakry; Michel Émery Diffusions hypercontractives, Sémin. de probabilités XIX, Univ. Strasbourg 1983/84 (Lect. Notes Math.) Volume 1123, Springer, 1985, pp. 177-206

[3] Dominique Bakry; Ivan Gentil; Michel Ledoux Analysis and geometry of Markov diffusion operators, Grundlehren der Mathematischen Wissenschaften, Volume 348, Springer, 2014, xx+552 pages

[4] Dominique Bakry; Zhongmin Qian Volume comparison theorems without Jacobi fields, Current trends in potential theory (Theta Series in Advanced Mathematics) Volume 4, Theta, 2005, pp. 115-122

[5] Franck Barthe; Yutao Ma; Zhengliang Zhang Logarithmic Sobolev inequalities for harmonic measures on spheres, J. Math. Pures Appl., Volume 102 (2014) no. 1, pp. 234-248 | Article

[6] Sergey G. Bobkov Large deviations and isoperimetry over convex probability measures with heavy tails, Electron. J. Probab., Volume 12 (2007), pp. 1072-1100 (electronic only) | Article

[7] Sergey G. Bobkov; Michel Ledoux Weighted Poincaré-type inequalities for Cauchy and other convex measures, Ann. Probab., Volume 37 (2009) no. 2, pp. 403-427 | Article

[8] Christer Borell Convex measures on locally convex spaces, Ark. Mat., Volume 12 (1974), pp. 239-252 | Article

[9] Herm Jan Brascamp; Elliott H. Lieb On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation, J. Funct. Anal., Volume 22 (1976), pp. 366-389 | Article

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

[11] Dario Cordero-Erausquin; Robert J. McCann; Michael Schmuckenschläger Prékopa-Leindler type inequalities on Riemannian manifolds, Jacobi fields, and optimal transport, Ann. Fac. Sci. Toulouse, Volume 15 (2006) no. 4, pp. 613-635 | Article

[12] Sylvestre Gallot; Dominique Hulin; Jacques Lafontaine Riemannian geometry, Universitext, Springer, 2004, xv+322 pages

[13] Mikhael Gromov; Vitali D. Milman A topological application of the isoperimetric inequality, Am. J. Math., Volume 105 (1983) no. 4, pp. 843-854 | Article

[14] Alexander V. Kolesnikov; Emanuel Milman Brascamp–Lieb type inequalities on weighted Riemannian manifolds with boundary (https://arxiv.org/abs/1310.2526, to appear in J. Geom. Anal.)

[15] Alexander V. Kolesnikov; Emanuel Milman Poincaré and Brunn–Minkowski inequalities on the boundary of weighted Riemannian manifolds (https://arxiv.org/abs/1310.2526)

[16] Michel Ledoux The geometry of Markov diffusion generators, Ann. Fac. Sci. Toulouse, Volume 9 (2000) no. 2, pp. 305-366 | Article

[17] Michel Ledoux The concentration of measure phenomenon, Mathematical Surveys and Monographs, Volume 89, American Mathematical Society, 2001, x+181 pages

[18] Andre Lichnerowicz Variétés riemanniennes à tenseur C non négatif, C. R. Acad. Sci., Paris, Sér. A, Volume 271 (1970), pp. 650-653

[19] Andre Lichnerowicz Variétés kählériennes à première classe de Chern non negative et variétés riemanniennes à courbure de Ricci généralisée non negative, J. Differ. Geom., Volume 6 (1971), pp. 47-94 | Article

[20] Joram Lindenstrauss; Lior Tzafriri Classical Banach spaces. II, Ergebnisse der Mathematik und ihrer Grenzgebiete, Volume 97, Springer, 1979, x+243 pages

[21] John Lott Some geometric properties of the Bakry-Émery-Ricci tensor, Comment. Math. Helv., Volume 78 (2003) no. 4, pp. 865-883 | Article

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

[23] Emanuel Milman Beyond traditional Curvature-Dimension I: new model spaces for isoperimetric and concentration inequalities in negative dimension (https://arxiv.org/abs/1409.4109, to appear in Trans. Amer. Math. Soc.)

[24] Emanuel Milman Beyond traditional Curvature-Dimension II: Graded Curvature-Dimension condition and applications (manuscript)

[25] Emanuel Milman Sharp isoperimetric inequalities and model spaces for the curvature-dimension-diameter condition, J. Eur. Math. Soc. (JEMS), Volume 17 (2015) no. 5, pp. 1041-1078 | Article

[26] Frank Morgan Geometric measure theory (a beginner’s guide), Elsevier/Academic Press, 2009, viii+249 pages

[27] Shin-ichi Ohta (K,N)-convexity and the curvature-dimension condition for negative N, J. Geom. Anal., Volume 26 (2016) no. 3, pp. 2067-2096 | Article

[28] Grisha Perelman The entropy formula for the Ricci flow and its geometric applications (2002) (https://arxiv.org/abs/math/0211159)

[29] Zhongmin Qian Estimates for weighted volumes and applications, Q. J. Math., Oxf. Ser., Volume 48 (1997) no. 190, pp. 235-242 | Article

[30] 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 | Article

[31] Gideon Schechtman; Michael Schmuckenschläger A concentration inequality for harmonic measures on the sphere, Geometric aspects of functional analysis (Israel, 1992–1994) (Oper. Theory, Adv. Appl.) Volume 77, Birkhäuser, 1995, pp. 256-273

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

[33] Cédric Villani Optimal transport - old and new, Grundlehren der Mathematischen Wissenschaften, Volume 338, Springer, 2009, xxii+973 pages

[34] Feng-Yu Wang Analysis for diffusion processes on Riemannian manifolds, Advanced Series on Statistical Science and Applied Probability, Volume 18, World Scientific, 2014, xii+379 pages

[35] Guofang Wei; Will Wylie Comparison geometry for the Bakry-Emery Ricci tensor, J. Differ. Geom., Volume 83 (2009) no. 2, pp. 377-405 | Article