Harmonic Measures on the Sphere via Curvature-Dimension

Abstract

We show that the family of probability measures on the $n$-dimensional unit sphere, having density proportional to: \[ S^n \ni y \mapsto \frac{1}{|y - x|^{n+\alpha}}, \] satisfies the Curvature-Dimension condition $CD(n-1-\frac{n+\alpha}{4},-\alpha)$, for all $|x| < 1$, $\alpha \geq -n$ and $n\geq 2$. The case $\alpha = 1$ 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.