{"paper":{"title":"Harmonic Measures on the Sphere via Curvature-Dimension","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG","math.FA","math.SP"],"primary_cat":"math.MG","authors_text":"Emanuel Milman","submitted_at":"2015-05-16T22:55:42Z","abstract_excerpt":"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."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.04335","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}