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, pp. 1007–1028.
Published in
Bernoulli, vol. 30, n. 2, May 2024, pp. 1007–1028