Abstract
This paper proves that the separation convergence toward the uniform distribution abruptly occurs at times around ln(n)∕n for the (time-accelerated by 2) Brownian motion on the sphere with a high dimension n. The arguments are based on a new and elementary perturbative approach for estimating hitting times in a small noise context. The quantitative estimates thus obtained are applied to the strong stationary times constructed in (Arnaudon, Coulibaly-Pasquier and Miclo (2020)) to deduce the wanted cut-off phenomenon.
Keywords
hitting times; separation discrepancy; small noise one-dimensional diffusions; spherical Brownian motions; strong stationary times;
Replaces
Marc Arnaudon, Koléhè Coulibaly-Pasquier, and Laurent Miclo, “On the separation cut-off phenomenon for Brownian motions on high dimensional spheres”, TSE Working Paper, n. 24-1510, February 2024.
Reference
Marc Arnaudon, Koléhè Coulibaly-Pasquier, and Laurent Miclo, “On the separation cut-off phenomenon for Brownian motions on high dimensional spheres”, Bernoulli, vol. 30, n. 2, May 2024, revised June 10, 2026, pp. 1007–1028.
Published in
Bernoulli, vol. 30, n. 2, May 2024, revised June 10, 2026, pp. 1007–1028