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arXiv · 1505.04335

Harmonic Measures on the Sphere via Curvature-Dimension

Emanuel Milman
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Abstract

We show that the family of probability measures on the $n$-dimensional unit sphere, having density proportional to: \[ S^n \ni y \mapsto \frac{1}{|y - x|^{n+α}}, \] 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.

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Emanuel Milman. 2015-05-16. Harmonic Measures on the Sphere via Curvature-Dimension. https://arxiv.org/abs/1505.04335

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